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A new adaptive algorithm for the real-time equalization of acoustic fields

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Title:
A new adaptive algorithm for the real-time equalization of acoustic fields
Creator:
Spaulding, Jeffrey James, 1960- ( Dissertant )
Principe, Jose ( Thesis advisor )
Taylor, Dr. ( Reviewer )
Childers, Dr. ( Reviewer )
Green, Dr. ( Reviewer )
Siebein, Dr. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1992
Language:
English
Physical Description:
vii, 220 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Acoustic data ( jstor )
Adaptive filters ( jstor )
Approximation ( jstor )
Correlations ( jstor )
Eigenvalues ( jstor )
Equalization ( jstor )
Error rates ( jstor )
Input data ( jstor )
Microphones ( jstor )
Signals ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis Ph. D
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
This dissertation presents a solution for the problem of acoustic equalization. Acoustic data are collected from a microphone at a listener location, and compared with a source (CD player). An adaptive signal processing algorithm, based on the Least Mean Squares (LMS), warps the CD signal to account for the filtering effects of the listening room. The algorithm to adapt the coefficients of a multirate equalizer is computationally efficient and can be performed in real-time with current microprocessor technology. As music is nonstationary, the LMS algorithm will need to undergo rapid convergence to a new set of optimal filter coefficients each time the input signal statistics change. As the LMS is a gradient descent algorithm, fast convergence implies a step size selection which operates at the algorithm's edge of stability. An analytic expression exists to determine the step size for uncorrelated input data as a function of the maximum eigenvalue of the input data. This dissertation extends this work to cover the case of correlated signals such as music. This result provides verification of heuristic rules that have been proposed i the literature. To date, formulations involving the maximum input signal power have been utilized as an estimate of the maximum eigenvalue. In this dissertation, a new method of determining the maximum eigenvalue is proposed by allowing the LMS algorithm to diverge. For a large and consistent domain of initial condition, iterations, and purposely divergent step sizes, the maximum eigenvalue dominates the rate of algorithm divergence. Simulations were pursued to determine the bounds of these variables for robust operation. This methodology was utilized to analyze the performance of the adaptive equalizer on selected music epochs, and to validate the theory put forward in this dissertation.
Thesis:
Thesis (Ph. D.)--University of Florida, 1992.
Bibliography:
Includes bibliographical references (leaves 218-219)
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Jeffrey James Spaulding.

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University of Florida
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University of Florida
Rights Management:
Copyright Jeffrey James Spaulding. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
027916460 ( ALEPH )
26708651 ( OCLC )
AJH1378 ( NOTIS )

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A NEW ADAPTIVE ALGORITHM FOR THE REAL-TIME EQUALIZATION
OF ACOUSTIC FIELDS













BY
JEFFREY JAMES SPAULDING


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1992






















Dedicated to


my Lord Jesus Christ
through Mary, Queen of Virgins,
in partial reparation of sins.














ACKNOWLEDGMENTS


I would like to express my gratitude to my advisor, Dr. Jose Principe, for his encouragement and support through the course of my studies and research. I would also like to express my gratitude to my supervisory committee members, Professor Taylor, Professor Childers, Professor Green, and Professor Siebein.

My work would not have been possible without the help and support of the students of the Computational Neuroengineering Laboratory. I am deeply appreciative. In addition I would like to express my thanks to Dr. Richard Henry and Mr. Phil Brink of the United States Air Force for their constant support.

I would like to thank my friends and family. It has been difficult to have not had the time to devote to the people I love. I appreciate their understanding. I wish to especially thank my grandmother, Mrs. Ruth Moore, whose bequest made my studies possible.

Finally, I wish to thank my mother and father without whom I would never have had the strength or courage to complete my program.














TABLE OF CONTENTS



ACKNOWLEDGMENTS .i AB STRACT. vi CHAPTERS


I BACKGROUND . 1
Introduction. 1 Basic Physics of Room Acoustics . 4 Psychoacoustics . 23
Consequences of Room Acoustics and Psychoacoustics
in the Solution of the Inverse Problem . 34
2 AN EQUALIZATION STRATEGY FOR REAL-TIME
SELF-ADJUSTMENT. 39
Introduction . 39 State-of-the-Art Equalizers. 41 The Adkins-Principe Equalizer Architecture . 48 Automatic Adjustment Using Adaptive Flters . 57
3 LEAST MEAN SQUARES (LMS) ALGORITHM
CONVERGENCE WITH UNCORRELATED INPUT 63
The Wiener-Hopf Solution. 64 The Gradient Descent Algorithm . 68 The LMS Algorithm with Uncorrelated Input . 84
4 Least Mean Squares (LMS) Algorithm Convergence
WITH CORRELATED INPUT. 91
Convergence of the LM4S Algorithm with Correlated Input . . 91
Conditions on g. for Convergence of . 93 Conditions on g. for High Convergence Speed of . 101 Experimental Results . 102 Conclusions . 108













5 DETERMINATION OF kmax WITH A DIVERGENT ,
LMS (DLMS) Algorithm . 118
Determination of Xmax with a Divergent Gradient
D escent Algorithm . 115
Parameter Selection for the Divergent Gradient Descent
(DGD) Algorithm . 118
Determination of Xmax with a Divergent LMS Algorithm . 120
Parameter Selection for the Divergent LMS Algorithm . 128
Integration of the DIMS Algorithm into the Equalizer Architecture. 139
Summary and Conclusions . 143
6 Validation of Concepts . 146
Test Plan . 146 Test R esults . 158
7 CONCLUSIONS AND RECOMMENDED
EXTENSIONS OF RESEARCH . 199
R eview . 199 Recommended Extensions to this Research . 205

APPENDIX
ECVT Plots of Contiguous Epochs of Audio Data . 208

R EFER EN C E S . 218 BIOGRAPHICAL SKETCH . 220














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy


A NEW ADAPTIVE ALGORITHM FOR THE REAL-TIME EQUALIZATION OF ACOUSTIC FIELDS


By

Jeffrey James Spaulding

May 1992

Chairman: Dr. Jose C. Principe
Major Department: Electrical Engineering


This dissertation presents a solution for the problem of acoustic equalization. Acoustic data are collected from a microphone at a listener location, and compared with a source (CD player). An adaptive signal processing algorithm, based on the Least Mean Squares (LMS), warps the CD signal to account for the filtering effects of the listening room. The algorithm to adapt the coefficients of a multirate equalizer is computationally efficient and can be performed in real-time with current microprocessor technology.

As music is nonstationary, the LMS algorithm will need to undergo rapid convergence to a new set of optimal filter coefficients each time the input signal statistics change. As the LMS is a gradient descent algorithm, fast convergence implies a step size selection which operates at the algorithm's edge of stability. An analytic expression exists to determine the step size for uncorrelated input data as a function of the maximum eigenvalue of the input data. This dissertation extends this work to cover














the case of correlated signals such as music. This result provides verification of heuristic rules that have been proposed in the literature.

To date, formulations involving the maximum input signal power have been utilized as an estimate of the maximum eigenvalue. In this dissertation a new method of determining the maximum eigenvalue is proposed by allowing the LMS algorithm to diverge. For a large and consistent domain of initial condition, iterations, and purposely divergent step sizes, the maximum cigenvalue dominates the rate of algorithm divergence. Simulations were pursued to determine the bounds of these variables for robust operation. This methodology was utilized to analyze the performance of the adaptive, equalizer. on. selected music epochs, and to validate the theory put forward in this dissertation.












CHAPTER 1
BACKGROUND

Introduction

For millennia architects have designed structures with the intent of minimizing distortion of acoustic signals propagating from a speaker or a musical instrument to a listening audience. From the Greek amphitheaters of antiquity to the modem concert halls of today technologies developed over the centuries have resulted in sophisticated techniques which have greatly improved acoustic fidelity. Only in this century has technology been sufficiently advanced to approach the problem of acoustic distortion from the perspective of equalization. Equalization anticipates the filtering introduced by the enclosed space and pre-processes the acoustic signal to take these effects into account. In this way when the acoustic signal arrives at the listener, the effects of the equalizer filtering and the room induced filtering cancel each other.

The revolutionary development of the Compact Disk (CD) player, first introduced to the market in 1982, has completely changed the direction of audio engineering. Research in audio engineering is being directed towards creating all-digital audio systems. )&rth the tremendous improvement in microprocessor speed, audio signals can now be processed digitally in real-time. The research presented in this dissertation will provide a design for an equalizer which will be capable of real-time self-adjustment. The design will be focussed on providing a system to perform equalization in a normal home listening room with standard audio components. It will be based on a new equalizer architecture developed by Atkins and Principe at the University of Florida [1], which uses a multirate filter bank. The Atkins- Principe architecture is enhanced so that filter coefficients in each band of the filter bank will be continuously updated in real-time, with coefficient updates being provided using principles from adaptive signal processing.

In chapter two background information will be provided on the current state-of-the-art of acoustic equalization. Background information on room acoustics, psychoacoustics, and the state-of-theart of equalizers will lead to conclusions regarding the limitations of acoustic equalizers. An improved strategy of adaptive equalization based on the Adkins-Principe equalizer architecture is pro-








posed. Chapter two will discuss in detail the theory and operational characteristics of the Atkins-Principe architecture. It will be shown to be an elegant and extremely efficient implementation strategy. Chapter two will also discuss the relevant considerations in choosing an adaptive algorithm for real-time updating of filter coefficients. On the basis of algorithmic complexity and the speed of the current generation of microprocessors the Least Mean Squares (LMS) algorithm will be justified as an appropriate algorithm for this application.

The LMS algorithm can be described as a stochastic gradient descent method which converges due to its time-averaged behavior. For fast convergence speed and low misadjustment in a nonstationary environment the algorithm must have an adaptive step size which can be precisely controlled. When the statistics of the acoustic signal change dramatically, equalizer settings will be incorrectly set and the adaptation algorithm will need to converge as quickly as possible to the new optimal equalizer settings in the least mean squares sense. LMS step size is the key parameter to adjust in order to maximize convergence speed. The theory guiding the optimal value of step size is well-understood for uncorrelated input data [2-31. Chapter three will present the theory of LMS convergence for the case of uncorrelated input data. The theoretical structure of the Wiener-Hopf formulation, the stochastic gradient descent algorithm, and the LMS algorithm with uncorrelated data will be discussed to provide the necessary framework for the more complicated case of the LMS with correlated input data. Chapter four will extend the theory of the LMS algorithm for the convergence properties of mean square error for the case of correlated input data. This theory is necessary to understand adaptation performance for acoustic data which by nature are correlated signals.

The divergence properties of the stochastic descent algorithm and the LMS algorithm are discussed in chapter five as a simple extension of the convergence properties examined in chapter three. The results of this study indicate that the maximum eigenvalue dominatesa diverging gradient descent algorithm. It will be further shown that the optimal value of the step size, after a change in the statistics of the input data, can be well-approximated as a function of the maximum eigenvalue of the vectors input into the filters performing adap-








tive equalization. The standard technique for estimating Xmax is to assume that it is approximately equal to the power in the input vector times the filter order. From systems theory we know that the power is the sum of all eigenvalues, and thus, is less than ?.max* This estimation is excellent under certain restricted conditions, e.g. low filter orders and high eigenvalue spreads. No such strong statements can be made more generally of this approximation technique., A new method of approximating the maximum eigenvalue of a system by allowing an adaptive gradient descent algorithm to diverge is introduced in chapter five. The method is first developed for the stochastic gradient descent algorithm which is based on the statistical properties of an ensemble of independent, identically distributed systems. The gradient descent algorithm will provide a theoretical framework from which to analyze the more complicated LMS algorithm which is based on the time-averaged properties of a single system.

Chapter six discusses the experiments used to test the equalizer architecture and the algorithms used for real-time self-adjustment. The NeXT computer with a digitizing microphone is used to collect and-process data. The software used to simulate real-time equalization with the proposed algorithms is discussed. Finally the test and evaluation results are presented.

Chapter seven summarizes the results of research presented in the dissertation. Specifically it reviews the success of the new architecture proposed for real-time self-adjustment of an audio equalizer. It reviews the theory of the convergence and divergence properties of the LMS algorithm with statistically correlated input data. It reviews the theory and performance of a new variation of the LMS algorithm which provides an estimate of the maximum eigenvalue of the input vectors into the equalizer filters. Finally chapter seven discusses the many areas of this research that could be extended

Because the physics of room acoustics and the principles of psycho-acoustics motivate many of the key design characteristics of the acoustic equalizer, the remainder of chapter one will provide background information on the pertinent areas of these fields.








Basic Physics of Room Acoustics

Equalization is an inverse problem, i.e. the adaptive equalizer will seek a configuration such that its response will be the inverse of the room-response. This section will develop the most simple features of room acoustics because of the basic understanding it will provide into the nature of the signals and systems on which the algorithms and methods introduced in this dissertation will be applied. As electrical engineers are more familiar with electromagnetic radiation than.acoustic, an effort will be made to develop this introduction to the physics of room acoustics in an analogous manner to the standard method of introducing the theory of electromagnetic waveguides, with the most important similarities pointed out where appropriate. A listening room may be thought of as an acoustic coupler, or communications channel, between an acoustic source and a listener. A further limitation that the room is described as a minimum phase system will be imposed. The description of the acoustic channel is particularly complicated because of the importance of both its temporal and spatial dependency. Because of the wave nature of acoustic fields, the sound energy in an enclosed space may be considered to be the mechanical analogy of electromagnetic radiation in a rectangular waveguide.

A discussion of the basic equations and definitions is first presented. Specifically the pressure equation will be discussed as the principal descriptor of acoustic phenomena. The conservation of momentum equation will be shown to supply boundary conditions for the wave equation. A discussion of the reflection and transmission coefficients are discussed. It is shown that many features from optics have analogous results in acousticse.g. Snell's Law and the Law of Reflection. The prominent factor in determining the reflectivity of a wall is the wall's characteristic impedance. The impedance, together with the room's geometry, define the characteristics of the room. A brief discussion of acoustic impedance is therefore presented. The wave equation for a room is developed in a method analogous to methods used for electromagnetic waveguides. The application of this theory is discussedfor a room which is a rectangular cavity with different types of impedance. Lastly the spatial and temporal response of the room is outlined using methods from ray theory. Much








of the discussion is based on introductory texts of Egan[4], Kinsler et al. [5] and Kuttruff [61.

Basic Equations and Definitions

Sound waves are longitudinal waves of regions of greater and lesser pressure. An
acoustic field is described by a wave equation which expresses the deviation from ambient pressure, p, as a function of spatial location r, and time t. The constant of proportionality between the temporal and spatial derivatives is the speed of propagation of the medium, c.

2 c2vp (1-1)

at2

For a time harmonic plane wave the solution of the pressure equation is of the same form as the expression for a plane wave of electromagnetic radiation. By separation of variables the pressure equation reduces to the following familiar form, where k is the wavevector.


P (r, t) = poe i(t -kr) [Kg (1-2)



The velocity of the vibrating particles, v, is related to the pressure field according to the conservation of momentum. The density of the medium is represented by P.

Dv (1-3)
VP aBy solving the above equations we can relate the pressure field to the velocity field by an acoustic impedance, z = PC.


V (r, t) = Voe i(ot-kr) [M] (1-4)


p (r, t) = pcV(r, t) = zV(r, t)(1)


(1-5)








Sensory equipment, including the human auditory system, responds to the intensity of the field. In the case of acoustic data, this corresponds to the average energy flowing across a unit area normal to the direction of propagation per unit time.



V 2
PI IMI (1-6)


In this discussion pressure waves will be considered to be plane waves in order to simplify the mathematical treatment of the phenomena which win be discussed. Note that for an acoustic point source the phase fronts would be spherical in an analogous manner to electromagnetic point sources. The plane wave approximation is only valid when the radius of curvature of the phase front is very large compared to the wavelength of the radiation. We will assume in this discussion that the dimensions of the rooms being considered are sufficiently large that this approximation is valid for frequencies over 1000 Hz. The room under consideration will be a rectangular cavity. These highly idealized cases are studied so that the important features of room responses are made clear. Before room responses can be meaningfully discussed, the reflection of acoustic waves off a wall must be well-understood.

Reflection of a Plane Wave by a Wall

The electromagnetic equivalent of reflection off a wall is a plane electromagnetic wave reflecting off an infinite flat plate of complex impedance. Like the electromagnetic equivalent the media impedances determine the behavior of the wave for a given geometry. Consider the situation depicted in figure 1. 1. Let pi be the incident pressure field, let Pr be the reflected field, and let pt be the transmitted field, with corresponding wavevectors ki, k, and kt.

Since we are assuming plane waves we can drop the y-dependency from our equations without loss of generalization. Recall that a wavevector K is defined as K = Wc , where c is the velocity of propagation in a given medium, and (0 is the radian frequency. Let co be the speed of propagation in air at standard temperature and pressure (STP) and let


































[x




figure 1. 1 The Law of Reflection and Snell's Law are valid for acoustic
radiation as well as electromagnetic radiation. The figure
above indicates the coordinate system and nomenclature
which will be used throughout the discussion of room
acoustics.



n be the ratio of the speed of propagation of a medium from that of air at STP, i.e. n = co/c . Let ux, uy. and uz be the unit vectors of our coordinate system as depicted in figure 1. 1. Then the incident, reflected, and transmitted wavevectors are expressed by equations 1-7 through 1-9, where I1=nK0 and E) is the angle between the wall normal and the wavevector.

By matching the pressure field at the wall, the reflection and transmission coefficients can be calculated. The pressure waves propagating along the wavevectors given in equa-








Ki = nlK0 [- uxcos i + Uzsinei] (1-7)

Kr = nlK0 [uxcOSOr +u zsinEr] (1-8)

Kt = n2K [- uxcos t + UzsinEt] (1-9)
tions 1-10 through 1-12 are shown below.

Pi Pi, 0e-inlK [- XcosO + ZsinOG] 1k1 (1-10)

-in IKo [XcosOr + ZsinOr]
Pr Pr,0e Ukr (111)

P-t Pt, 0e f2K� [-XcosEGt+ZsinOE] (1-12)

The pressure field must be continuous at all point along the boundary, i.e.
Pi, interface + Pr, interface = Pt, interface . These equations yield the Law of Reflection and Snell's Law precisely in the form as the electromagnetic equivalent.
nlK sin(i = nlK sinEr = n2K0sinEt (1-13)

sinO. = sine Law of Reflection (1-14)
1 r
nl sin5i = n2sinet Snell's Law (1-15)

In addition, the conservation of momentum (equation 1-3) must be satisfied at the wall. This will imply a matching of particle velocities at the interface. Note that because of the plane wave assumption only the derivative with respect to x is non-zero in grad P. The venlK -incKo [-Xcos Oi + ZsinO,] (1-16)
Vi = - cosE).iP. 0 e k


-n 10K -inIKo [Xcos Or + Zsin OrVr = t o s E) UkcSr Pr, 0e kr (1-17)


n2Ko -in2K0 [-Xcos 0 + Zsin EM

Vt P- o tP Oe Ukt (1-18)
t j t 2t,0








locity field propagating along the wavevectors given in equations 1-7 through 1-9 is described by equations 1-16 through 1-18, shown above.

By a proper choice of the coordinate system, i.e. for the wall located at x=0, and by an application of Snell's Law and the Law of Reflection, the phase related terms in V can be ignored, as they are common to Vi, Vr, and Vt. Matching the particle velocities at the interface yields the following expression for the reflection coefficient.

Pi, 0cos9i _ Pr, 0cos(r _ Pt 0cosOt (1-19)
z1 z1 z2


z2Pi, 0Cos i-Z2Pr, ocosEr = Zl (P, 0 + Pr, 0) coset (1-20)


P,0
0 (z1 cose + z2cosEi) = z2cosei-z1 cosEt (1-21)
Pi, 0 t+Z 20(iz


R(,) Pr, 0 Z c E z2csi - z IcsO t-22)
R Ec)= -- (I-2
Pi, 0 z2cose(i +ZIcoset



If we assume medium one is air, i.e. z 1 = poCo, and medium two is a hard wall, i.e. Z2 z1 , the above equation reduces to the form seen in most elementary books on acousz2cosE)i - P0C0
tics, R cosi+ 0C The transmission coefficients are easily derived from the reflection coefficients as follows.


(1-23)
Pt (Pi + Pr) = 1 +R = 2z2cosEi
i Pi Zcos z2cO + Zl czsEt








With the expressions for R and T it is clear that the impedances of the materials are the key acoustic parameter of the material, and it is this parameter which determines a room's acoustic properties.

Acoustic Impedance

Acoustic impedance can be separated into a real and an imaginary part. The real part is the acoustic resistance, the component associated with the dissipation of energy. The imaginary part is the acoustic reactance, the component associated with temporary storage of energy. These components are made explicit in the expression shown below.


z (E) o) = P=R+jX [m Kg (1-24)



The acoustic reactance can be further factored into an inertance and a compliance term. The acoustic inertance term is defined as the effective mass of the element under consideration divided by surface area, M = - with units of Kg/ni2. The acoustic compliance is s2 dx
defined as the volume displacement induced by the application of a unit pressure, C = in units of m4sec2[Kg. A mechanical interpretation of a locally reacting wall is shown in figure 1.2.


z (E,co) =R j oM-ZC (1-25)



Let pressure correspond to voltage, velocity to current, resistance to resistance, compliance to capacitance, and inertance to conductance. Then an acoustic system can be replaced by an equivalent electrical system. It is precisely the behavior of these types of circuits that we wish to equalize.























figure 1.2 The mechanical analogy or room impedance is illustrated by
in the figure by a spring for the reactive term and a damper
for the dissipative term.






Resista uc Resistance AAA


Inductance - Inertnce Voltage-Pressure


Figure 1.2 Condupliance

figure 1.3 A room may be modelled by its electrical circuit equivalent
as shown above.



General Theory of Room Responses


The solution to the wave equation in an enclosed space is a familiar eigenvalue problem. The equation reduces to a Helmholtz equation with boundary equations as shown below. The homogenous part of the Helmholtz equation provides a complete set of orthogonal eigenfunctions and associated eigenvalues, 4n (r) and Xn.


M




B.,
.


I








V2P + K2p = -iopq(r) (1-26)


Pi, interface + Pr, interface = Pt, interface (1-27)

VP dV iOPp
Vp dV - _ (1-28)
As a result our source distribution, q(r), can be expanded as a sum of eigenfunctions.V is the enclosed volume of the room.

qCr) = Cn 1yn(r)
(1-29)
n

Cn = fffq (r) VndV (1-30)


The unknown solutions to the inhomogenous Helmholtz equation can also be expressed as a summation of eigenfunctions.


P. (r) = IDnVn (r) (1-31)
n

Substitute the summations for P, (r) and q(r) into the inhomogenous Helmholtz equation and solve for the unknown { Dn } in terms of the known { Cn) as shown below.


XDn[V24fn (r) + K2Vn (r)1 = -iowpICnfn (r) (1-32)
n n

The equation has a simple solution for a point source located at r0, i.e. q(r) = Q5(r-ro).


P (r) = iQp Vn (r) n (rO)
n J2(K -Kn








Note that Kn it the wavevector associated with X. P. (r) is recognized as the Green's function for the room. The Green's function expresses the acoustic transfer function between two points in the room. Note from the Green's function that acoustic fields satisfy the Reciprocity theorem just as in the electromagnetic analogy. The Green's function can be extended to a source, S, emitting a continuum of frequencies as shown below.
00
S (t) f JQ (,) etd(O (1-34)

-00

P (r) f JP(0)eOtdo (1-35)
-00

Equations 1-33 through 1-35 provide a complete wave theoretical description. A discussion of the implications of this formulation for a rectangular room are now in order. Simple Rectangular Rooms

This section will apply the general wave theory to an empty rectangular cavity characterized by a single impedance. While this example does not represent the complexities of actual rooms, it nevertheless provides insight into the features of room responses with a minimum of mathematical complexity. The connection between the reflection properties of walls and the general wave theory will be made explicit through wall impedances. Three classes of impedances will be considered. Consider the room depicted below.

The wave equation for this geometry is most easily applied in the rectangular coordinate system. The room eigenyalues are found by solving the homogenous Helmholtz equation shown below.

a_ ___ I I 2 T=0(1-36)




The equation is easily solved by the technique of separation of variables. Let
TP (x, y, z) = Vi (x) 'V (y) iy (z) . Substitution of this expression into the wave equation
























figure 1.4 A room with dimensions Lx,Ly, and L, and walls with
uniform impedance will be evaluated using the wave
theoretic formulation.


leads to three ordinary differential equation plus the separation equation as shown below.

d2 2

d2 (x) + K2 = 0 (1-38)
dx

d 2 2 (1-39)
d i (z) + K = 0 (-9
dz 0

K2 K+K K (1-40)


The solution of these equations is trivial.

x (x) =A1 e-ikxx + A2eikx (1-41)


(y)= Ble-ikyY + B2eikyy (1-42)


I (Z) = Ce-ikzz + C2eikzz (1-43)


-y


041-y








The constants A1,A2,B1,B2,C1, and C2 are determined by matching boundary conditions. The solution of three specific examples are outlined. Case . Re[z]--O and Im[z]=very large.

The method of solution of the three ordinary differential equations is identical. Consider the equation in x. As a boundary condition guarantee the conservation of momentum is satisfied at the walls (x=O and'x=Lx).

_pdV MOP iCOp
Vp P=*_9dp+ = 0 (1-44)
V - z dx z

dP
As limIzI - , -+ 0. The boundary conditions are satisfied as follows.

d =>A = A (1-45)
dxy~ 'x= iKXA I- iKxA2= 1 A2

d iKxal (eiKxLx _ e-iKxLx =-2Kxay sinKxLx Kx an

(1-46)

a is any integer. The two boundary conditions have been used to solve for A2 and Kx. Similar solutions are found for B2,C2,Ky, and Kz. Substituting these results in the expression for T (x, y, x) and the separation equation leads to a solution of the room eigenmodes and eigenfrequencies as shown below.


T'(x,y,z) = Dcos _--=--cos cos- (1-47)
Lx LY,


f(a,,Y) =cK - 2 (-) + )+ ( ) (1-48)



The constant D=A1B1C1 is determined from an initial condition. case ii. Re[z] = 0.

Satisfy the new boundary conditions at x=0 and x=Lx. A solution for A2 and Kx can be found by solving the two transcendental equations or by using methods form the theory of










d x x (1-49)
dWtx-VIx= 0 = X + iKx) +A ( ~ ) (-9



d A Kpc.iKxLx iKxLx Kpc -iKxLx -K iKxLx
"x = Ix x 2(--'" x+iKx" ) +A( - x )
(1-50)
conformal mapping. If X < 0, the corresponding eigenfrequencies are lower than for the hard wall case (Im[z]>>). For X > 0 the corresponding eigenfrequencies are higher than the hard wall case. Because the impedance is completely reactive, the room will continue to support standing waves, but with phase terms included in the sinusoidal terms of the eigenmodes.

csii. Arbitrary z.

The form of the boundary conditions are the same as case ii with X replaced by z. Eigenmodes can not support standing waves because of the lossy component of z represented
be(X)
by e- ();

(1-51)
(X t (Yt Z ) COSDR (Z
v(x,y,z) = De- (xrsz) c+x)Cos +y " + )
LX y LY YLz


It is clear from matching boundary conditions in the three cases outlined above that the room impedance is key to understanding the behavior of the room. By using the eigenmode expansion technique and the inhomogenous Helmholtz equation the pressure field can be determined for any location in the room. An initial condition is also required in order to evaluate D. The eigenfrequencies indicate those frequencies at which the room has resonant properties. Note that the number of eigenfrequencies increase as the cube of the upper frequency limit of the source. The wave theoretic technique provides a complete description of the acoustic signal in an enclosed space, and provides a solid theoretical foundation for the field of acoustics.








Temporal ProjMrties of a Listening Room

Each reflection of an acoustic wave must be described by its delay in arrival from the direct acoustic wave at the listener location, the direction from which the reflected wave arrives at the listener location, and the strength of the reflection. In this section we will discuss the time delay and strength of reflected acoustic energy. The rate at which reflected acoustic energy arrives at a listener location is an important factor in evaluating the quality of a listening room. Ray theory provides a simple and relatively accurate method of determining the rooms's temporal properties. This theory is completely analogous to ray theory from geometric optics, and it will therefore be applied with a minimum of discussion. Like geometric optics, geometric acoustics ignores interference and refraction effects. Nevertheless it is accurate for rooms of ordinary size for acoustic frequency content in excess of 1000 Hz.

If a room is constructed of a series of planes, image sources can be found by successively reflecting the original source and each subsequent image source about the planes in ,a lattice constructed by using the room as a unit cell. The lattice and a cross section plane are shown in figure 1.5. A spherical shell is chosen with inner radius at distance ct and outer radius c(t+dt), where dt is much less than t. The number of sources in the spherical shell corresponds to the number of mirror reflections between t and t+dt. The volume of the spherical shell, assuming dt << t, is 47cc 3 i2dt. Note from ray theory that there is only one source per room. The number of image sources can be found by dividing the volume of the shell by the volume of the room. As t increases the number of mirror sources increases as t2.


Number of Reflections = 47cc 3 t 2 dt (1-52)
V


As time increases, the reflections have less and less energy associated with them. The higher order reflections lose energy at each reflecting plane, the amount of which depends on the characteristic impedance of the wall. An approximation of how the energy decreases can be made from geometric acoustics by multiplying the initial energy of the direct ray in-






























-----------X


side the listening room by R2, each time the ray path from an image intersects the room walls or an image room wall.


figure 1.5 Geometric acoustics can be used to explain the temporal
properties of reflected energy in a room. Image sources can be formed by successively reflecting the original source and
each subsequent image source about the planes that are
constructed by using the room as a unit cell.



Figure 1.6 demonstrates the concept of an echogram, a graphic method describing the
temporal response of the room. The height of each line represent the intensity of reflection.









During the discussion of psychoacoustics, the temporal properties of an audio field, which generally indicate a pleasing listening room, will be discussed.








C






degree of shading represents
the number of reflections per unit time

figure 1.6 An echogram indicates that as the time delay from the direct
field increases, the number of reflected waves increase, and
the intensity decreases.



The rate of decay in the intensity field is an important factor in creating a good quality listening room. Recall that the intensity of a wave decreases as a function of the square of the distance travelled, (ct)2 . The intensity of the field will also be decreased due to the attenuating influence of air (or any medium through which the wave traverses). Let a be the e-folding time constant of this attenuation. Each time the acoustic wave reflects off of a wall, the intensity will decrease by the square of the wall reflectance. The decay of the intensity of the acoustic wave is summarized in the equation below, where 10 is the intensity at t=O, and n is the number of reflections of the ray.


0= -actR2nt~ 1 0 - act +ntnRj 2 (1-53)
ct) (Ct)2


It is not possible to follow the path of each ray and perform the above calculation. Instead the calculation is made by computing the average number of reflections of a ray (n),








assuming the acoustic field is diffuse, i.e. the rays are uniformly distributed over all possible directions of propagation. The assumption of a completely diffuse wave is never achieved in an ordinary listening room, and the attenuation of the reverberant field will have large or small deviations from the simple calculations being outlined depending on the amount by which the actual acoustic field differs from our assumption of diffuseness.

First express the number of reflections per second which occur at walls perpendicular to the x-axis for a ray propagating at an angle E with respect to the x-axis, nx (E).

nx (0) = --cose (1-54)

x


Average over all angles E.


(nx)- C (1-55)



Similar results are obtained for (ny) and (nz). The average number of reflections along all three axes is expressed below where S is the surface area of the room and V is the room volume.

C (1 1 1) _ cS
n = (nx)+(ny (z) n L+ y+ z - (1-56)



Substituting the above into the intensity equation yields the following.

10 -act + - tlnIRI2
I -

(Ct) 2 (1-57))


The reverberation time is taken as that time at which I(t)/I0 = IXIO-6, and the expression for this time is known as Sabines' Law.










T- = O.16V 2(-8
4aV- Sln!R12(-8

Spatial Properties of a Listening Room

The spatial distribution of acoustic energy is a key factor in constructing a listening room. A diffuse field can be defined as a field in which the intensity in a spherical shell subtended by a solid angle, dfl, is independent of the direction in which the solid angle is pointed. For the case of a room which is a rectangular cavity with hard walls, no matter how small dfl there exists a time at which the field is diffuse. Of course there are no such walls, and clearly, for a rectangular cavity, the field will not be diffuse. For example, images located in directions along the corners of the room experience more attenuating reflections than the other sources.

Fortunately walls are not perfectly smooth and they partially diffuse the acoustic energy at each reflection. Consider the following simple model. Assume at each reflection the percentage of energy reflected specularly is s. Then I1-s represents the portion of energy which is reradiated by the wall diffusely. This is, of course, an overly simplistic model. All incident energy is reradiated by the wall according to a directional distribution function. Nevertheless this model does demonstrate approximately the diffusion process. In the steady state the power present in the total reflected field and in the field reflected specularly can be related to the reflection coefficient, R, and to s.
00
P CI I R 2n (1-59)
n=

00


nl

The percentage of energy power which has been diffused increases as s decreases as shown below.








00
P-P E (1 _ sn) IRI2n d_ s = n = 1

E IRI2n
n=l


Specular Reflection Constant S


(1-61)


figure 1.7 As the reflection constants decreases and the specular
reflection constant increases, the percentage of diffuse
energy in the room increases.



Even more important than wall roughness are the diffusing effects of room furniture and the deviations of most rooms from a rectangular cavity. Even the presence of people in a room act to diffuse the acoustic field. Furthermore in the discussion of psychoacoustics it will be shown that a pleasant listening environment can be created with far less than a completely diffuse field.


Diffuse Energy = f(RS)








Psychoacoustics

An equalizer will not be capable of exactly reproducing the time varying pressure field recorded in a concert hall or recording studio. The engineering compromises which must be made in designing the equalizer must focus on restoring the most psycho-acoustically significant features of the original signal. The design methodology will be such that importance will be placed on maintaining only those features of the waveform which a listener will perceive in an average home listening room. This brief introduction to psycho-acoustics is intended to motivate the signal processing objectives of the adaptive equalizer. Specifically, the level to which signal coloration can be accepted without a loss of listener enjoyment is discussed. This is largely determined by the frequency response and frequency resolution of our hearing, and of the masking properties of the signal itself. A description of how the room-induced temporal and spatial characteristics of the signal contribute to a pleasing sound is investigated. Finally the dynamic range of human hearing is described. Stereophony is not discussed as this research does not attempt to process signals to alter their stereophonic properties.

Coloration

In the time domain a system neither adds coloration to or removes coloration from a signal if its frequency response can be characterized as shown below.


h (t) = k5 (t -T) (1-62)



The function implies no change in the shape of the waveform. The output is a delayed and amplified (or attenuated) replica of the input. Performing a Fourier transform of the impulse response function gives the frequency domain restrictions for such a system.


H (co) = ke (-3


(1-63)








The frequency domain requirements for an ideal room are the following:

L The system must have a constant magnitude response, IH ((0) 1 k.

ii The phase response must be linearly proportional to frequency, (CO) = -cot.

Violation of L can lead to an unintended emphasis in some frequencies over others in ways not intended by a composer or performer. Violation of ii. gives rise to non-linear phase. Non-linear phase causes certain frequencies to pass through the system faster than others.

To accurately reproduce the sensation produced by the original acoustic waveform. it is essential that the sound reproduction system have a flat magnitude response. Obviously some degradation from this standard will not alter the perception of the listener. Nevertheless, the fundamental difference between middle C and any other note is that the energy of the signal is concentrated at 262 Hz and its harmonics. An attenuation of the frequency response at middle C, if other unmasked information is present in the signal, will negatively impact the fidelity of the signal because of the lessening of emphasis on the C-like quality of the sound. More precisely, we are interested in maintaining the pitch and timbre of the signal. Pitch of a musical signal is primarily deterrruined by frequency, but is also influenced by the shape of the waveforin and the loudness of the signal. The frequency dependency is depicted in figure 1.8, where pitch in mels is plotted against frequency, for a signal with a loudness level of 60 phons. Note that a change in frequency has a greater effect on pitch above 1000 Hz than below.

The loudness dependency of pitch can be quite dramatic for pure sinusoidal tones. For music signals the loudness dependency is much less significant. As an example, to maintain the same pitch when increasing the loudness of a musical signal from 60 to 100 phons requires at most a frequency shift of 2%.

If a signal is not a pure tone, but consists of harmonic tones, due to the nonlinear processing of the ear, sum and difference signals are generated. The difference signals determine the pitch perceived by the listener. For example a signal consisting of the first four harmonics of a 100 Hz tone, all of approximately the same intensity as the fundamental, is








perceived as a signal of approximately 160 mels. If the fundamental is removed, the difference signal is unaltered, and the listener still perceives a 160 mel pitch.These effect s



3500
3 0 0 0 . . . . . . I . .
2 5 0 0 . . . . . . . . . .
2 0 0 0 . . . . . . . . . . . . .
ILS O O . . I . . . . . .
1 0 0 0 . . . . . . . . .
5 0 0 . . . . . .
01
3 4 5 6 7 8 9
In(frequancy)

figure 1.8 Pitch is plotted as a function of frequency for a signal with a
loudness of 60 phons [6].


are not quite so dramatic in music signals as the harmonics are usually considerably less intense than the fundamental.

Assume that a series of non-harmonically related frequencies do not mask each other. If we wish to maintain pitch, the relative loudness of the frequencies must be unaltered by the electronic equipment or the listening room. For a pure sinusoidal it is possible to hear changes in loudness of only 0.5 phons [6]. Note also that the perceived loudness is a function not only of the acoustic intensity of the signal, but also the signal fi-equency. Figure 1.9 illustrates this effect for two equal loudness contours. For frequencies over I OOOHz., as frequency increases, the acoustic intensity must increase to maintain the same loudness level.

Surprisingly, controlling the phase response has no appreciable effect on maintaining pitch. The phase response, except in extreme cases, cannot be perceived by listeners hearing music signals. Thus while a lack of phase distortion is required for the shape of the








wave to be unaltered by a system, the same requirement is not generally necessary for a signal to be psycho-acoustically equivalent. The results of Suzuki, Morita, and Shindo[7] indicate that the effects of nonlinear phase on the quality of transient sound is much smaller compared to the effect one would expect from the degree of change in the signal waveform.



70
.
60 dB contour
6 0 . . . . .
s o . . . . . . . .
4 0 . . . . . . . . . . . . .
3 0 . . . . .
. . . . . . . . . . . . . . . . . . . . . .
10
0- dB contour
-10
0 2000 4000 6000 8000 10000 12000 14000 16 00
frequency [14Z]

figure 1.9 Equal loudness contours are presented as a function of
frequency and acoustic intensity (5].


Hansen and Madsen[8] found similar results. In their experiment a signal had all of its frequency components shifted by 90' in phase, producing the most severe distortion possible in the shape of the waveform while maintaining the magnitude response. Only for fundamental frequencies below 1000 Hz was a distinctive difference perceived by the listeners involved in the test. The relative insensitivity to phase will be seen to be critical to the operation of an adaptive equalizer.

Maintaining signal timbre is also a basic requirement in a good sound system. Those features of the signal which are important to the timbre of sound are poorly understood, and the study of timbre is still an area of active research. The transient behavior of an instru--., ment being excited effects the initial behavior of the pressure field. The transients effect the amount of energy present in each of the overtones. This is an important factor in deter-
















































The masking effects of a 400 Hz signal are demonstrated above. The region of the basilar membrane which responds to the 400 Hz waveform will raise the audibility threshold of higher frequencies which stimulate the same region of the basilar membrane [5].


figure 1.10


mining the unique sound of a particular instrument. Amazingly, our hearing system is able to pick out these cues even when many instruments are orchestrated together.

Perhaps the most problematic factor for which to account is the level of masking caused by certain frequency components of a waveform on others. Masking is the degree in which the threshold of audibility of a signal is raised for non-harmonically related frequency components in the same signal, or in a complex tely separate waveform. Masking occurs when one signal has frequency components which are close to the frequency content in another signal. It has been clearly demonstrated that lower frequency components more effectively mask higher frequency components than vice versa. These effects can be explained by physiological phenomena. Lower frequencies produce harmonics which stimulate the region of the basilar membrane sensitive to higher frequencies, thereby masking the sound of higher frequencies. Figure 1. 10 shows the masking effect of a 400 Hz sinusoidal at 60 dB. The threshold increases most dramatically in the region slightly above 400 Hz.


I


4000 4500 500(


500 1000 1500 2000 2500 3000 3500
asked fre uen Q4ZI








Determining exact design criteria to guarantee a true reproduction of pitch and timbre is made difficult by the complicated dependency on signal frequency content, loudness, masking, and the transient behavior of the energy distribution in the harmonics of instruments. For example, if most of the time signals above a certain frequency are masked by lower frequency signals, it would not be important to control the attenuation of these regions of the spectrum. The simplest analysis assumes the worst case scenario in which masking does not occur, regions of the spectra are of a loudness just at the audibility limit, and frequencies must be controlled at a level at which any error in loudness would be below the 0.5 phon detection limit. To maintain timbre, we wish to be able to control transients in the signal caused by the excitation of instruments. Fortunately the phase of the signal will not have to be carefully preserved because of the relative insensitivity of our hearing to these effects.

Practical implementation of these requirements is possible only if the frequency resolution of hearing is sufficiently large that filters of reasonable order can carry out the necessary inverse filtering. Frequency resolution for two sinusoids is known as the difference limen. It has a relatively constant value of 3 Hz below 500 Hz. Above 500 Hz frequency resolution decreases approximately as a linear function of frequency (see figure 1. 11).



Af = 3 Hz for f < 500 Hz (1-64)
0.0030f for f > 500 Hz



Clearly it will be necessary to control pitch and timbre with a much finer degree of resolution at low frequencies than at high.

The human hearing organ can be described as a parallel filter bank with characteristic bandwidths. Fletcher and Munson [9] found that the detection threshold of a tone in broadband noise is not affected by the noise bandwidth until it falls beneath the "critical bandwidth", which are shown in figure 1. 12.








Psychoacoustic Responses to the Temporal Aspects-of Reflected EneM

Reflections under certain circumstances serve to enhance the quality of sound by adding to the direct wave in a way which is reinforcing. For musical signals the reverberation of the room when within certain bounds gives a pleasing effect. Under other circumstances


8 1 1.2 1.4 1.6 1.8 2
frequency [Hz] X104


figure 1. 11


The difference limen gives the frequency resolution of two sinusoidals. An acoustic equalizer will need to control the frequency response of low frequency bands with a finer resolution than high frequency bands [5].


reflections can cause highly undesirable effects. Although many effects can be explained partially through the physiology of hearing, perhaps the most important factors, at least in the reproduction of music, are the types of environments with which we have been habituated and from which our aesthetic senses have been influenced.

The discussion of the temporal effects of sound fields begins by finishing our discussion of coloration. Strong, evenly-distributed echoes can cause serious degradation in the accurate reproduction of an acoustic wave's spectral distribution. Consider a room with the following simple impulse response.









00

I an5 (t - nTO)
n = 0


(1-65)


1 OFOO 10000
frequency (Hz]
Because our hearing can be described as a parallel filter bank, acoustic equalizers are designed to simulate the critical bandwidths . High quality equalizers are designed to have 1/3 octave bands because of their similarity to critical bandwidths, as seen above.


figure 1. 12


The power spectral density indicates equally spaced resonances.


IH (f) 12 = 1 2
1 - 2acoscoTo +a


(1-66)


Figure 1. 13 illustrates the frequency response of the room for different echo periods, To. As To becomes larger the frequency components become less separated. Because of masking effects, the threshold, at which the spectral distribution of the audio signal is perceived to change, decreases as To increases.

Figure 1. 14 gives results for the threshold level at which an average listener perceivesnoticeable coloration induced by a reflected wave. These measurements are based on lis-


tening tests of six different music selections.





I frequency [1/To Hz]
figure 1.13 As the echo period, To, increases, the frequency components
are more closely separated. Due to masking, coloration
becomes less audible.


In addition to our sensitivity to the frequency domain, our hearing is affected by time domain properties of reflected waves. Reflected energy which is just barely perceived acts to increase the loudness of the signal and to change the apparent size of the source. At a higher loudness and a small delay, the perceived source direction moves. For a large delay the reflected energy is perceived as a highly annoying echo. Figure 1.15 shows the intensity at which a distinct echo is perceived as a function of time delay.

If the delay is sufficiently short, the reflected wave can be 10 dB higher than the direct wave without causing echoing. This phenomenon is know as the Haas effect after the discoverer, and is an important factor in the design of auditoria [10].

Acoustic clarity is a measure of our ability to resolve fine temporal structures in acoustic waves. The faster the tempo and the higher the degree of attack on the instruments, the higher is the necessary degree of room clarity. Many measures have been suggested. All of them rely on determining the amount of acoustic power over a period of time representing "early" reflections divided by a normalization. One index used for determining the clarity of rooms for musical signals is given in equation 1-67. A value of C = 0 dB provides











0

dB
-5



-10


20 40 60
echo delay [msec]


80 100


As the echo delay decreases, the reflectivity of the room surfaces must decrease in order to avoid a perceptible change in the coloration of the acoustic signal. For long delays, relatively intense echoes may exist without causing noticeable coloration [5]. This figure shows the level of l~log(a) at which lesteners perceive coloration.


As the time delay of the echo increases, its intensity must decrease in order to avoid an annoyingly distinct sound. The above figure shows the threshold intensity for hearing distinct echoing [5].


figure 1.14


figure 1.15


- delay of reflected wave [macc]


.








sufficient clarity for fast staccato passages. A value of -3 dB is usually acceptable for symphonic music.

- 80msecs

f h 2 (t) dt
C = 1010g . 0 (1-67)
00

f h 2 (t) dt
- 80msecs


Reverberation time of a field is a measure of how the room damps out an acoustic
source. It is usually determined by measuring the time required for the field intensity to decrease by 60 dB from the moment the acoustic source is turned off. The listening room's reverberant field acts to mask the details of a music performance which listeners find unpleasant, e.g. a minor lack of synchrony in an orchestra. The ideal reverberation time depends on the type of music being performed, and on local styles and current fads. For chamber music a reverberation time consistent with the rooms in which the compositions were originally performed is preferred, which is from 1.4 to 1.6 seconds. For symphonies from the romantic period or pieces utilizing large choirs, a reverberation time of over 2 seconds is preferred. For opera a shorter reverberation time is preferred so that the libretto may be more easily understood. As an example, the La Scala opera house in Milan has a reverberation time of 1.2 seconds [I I].

Psychoacoustic Response to the SRatial Aspects of Reflected Energy

For a room to have a good acoustic ambience the listener must have a sense of spatiousness. Originally researchers believed that spaciousness was caused by hearing spatially incoherent fields. It has been shown, however, that this gives rise only to a phantom source direction, and does not provide a sensation of spaciousness. The following conditions are required.

i. The field must be temporally incoherent.

ii. The intensity of the reflected energy must surpass the audibility threshold.








iii. The reflected energy must have components with time delays of less than 100 msecs.

iv. For early reflections, the field must be spatially incoherent, with components

arriving from the lateral directions being of primary importance.

These requirements are generally met in large rooms, but are increasingly difficult to ensure in smaller and smaller rooms. It has been shown [6] that spaciousness is independent of the style of music being performed.

The dynamic range of human hearing is 140 dB. If our hearing were more sensitive it would not be beneficial as we would start hearing the Brownian motion of air molecules. In processing music digitally, ideally a sufficient number of bits are available to ensure coverage of the entire dynamic range of musical works at a resolution consistent with our physiological capabilities. The standard CD format provides 16 bit encoding of amplitude information (approximately 90 dB). Equalizer technology should aim to provide no further degradation in dynamic range than the limit defined by the CD encoding.

Consequences of Room Acoustics and Psychoacoustics in the Solution of the Inverse Problem

Several important inferences can be drawn about potential solution methods to the room inverse problem from the discussion of room acoustics and psychoacoustics. Two obvious approaches are available as solution methods: 1) a theoretical modelling of the physics of the listening room acoustics, and 2) experimentally measuring the characteristics of the room.

The first approach has been briefly reviewed in the discussion of room acoustics. The key acoustic parameter, complex acoustic impedance, must be carefully measured for each material present in the listening room. Using these impedances and the usually complex geometry of most home listening rooms, the Helmholtz equation must be solved. Note that matching boundary conditions will be impossible to perform analytically, and will require a numerically intensive calculation using a finite elements algorithm. As a result of the computations, a set of room eigenvalues and eigenmodes will be determined, from which the room's Greens function is determined. Even for the case of a simple rectangular room,








the number of room eigenmodes increase as the cube of the upper frequency limit of the source. For a room of average dimensions, i.e. Lx = 5 m., Ly = 4 m., and Lz = 3.1 m., between 0 and 20,000 Hz. there are over 57,000,000 [6] resonant room modes. As an additional complication, except at low frequencies, the modes are extremely dense in the frequency domain, i.e. the half-width of a mode is much greater than the separation between modes. Thus the majority of modes are mutually coupling.

The aforementioned calculations are so computationally burdensome that they can not reasonably be performed. Even if a complete set of modes were available, from which the Greens function inverse could be calculated, the processing problem would not be feasible. For the simple room geometry discussed above, there were 57,000,000 room modes, and an FIR filter length of 100 million tap weights would be required. Realizing that implementation of such a large filter is impossible, a selection of the most important room modes would need to be calculated. Because of the mutually coupling nature of the modes, and the changing spectral distribution of the acoustic source, the "important modes" would be constantly changing.

The rigorous eigenmode expansion of the room response, the subsequent calculations of the room's Greens function, and implementation of the"inverse on hardware, can not be performed. As a simplification in the computational complexity of describing a home listening room, a geometric acoustics approach could be attempted. The previous discussion makes clear, however, that geometric acoustics gives insight into the time average properties of the listening room. Depending on the nature of the acoustic source, the approximations could yield poor results. In addition, the computational complexity of characterizing the complex acoustic impedances for the room materials and the complicated geometry of the listening room, continue to result in a huge computational problem. Figure 1. 16 summarizes the rationale for rejecting approaching the inverse room problem by modelling the physics of the room. -1

The experimental approach must be selected, and this approach is discussed in detail in chapter two. Several implications of this method must be addressed in light of the basic principles of room acoustics and psychoacoustics in order to design an intelligent tech-








nique. The experimental approach can be summarized as follows. The listening room is excited by a broadband noise source. A measurement is made of the acoustic pressure at the listener location. This measurement is compared with the source signal band by band with an octave filter bank. An inverse of the room is approximated by ensuring the energy distribution of the source signal is maintained by the measured signal.


figure 1.16


The figure demonstrates the reasons that modelling the physics of room acoustics is not feasible for solving practical room inverse problems.


The discussion of psychoacoustics makes clear that acoustic high fidelity requires maintaining signal pitch, and the level of spatial and temporal coherency, with sufficient spectral resolution to match the psychoacoustic and physiological characteristics of human hearing. Equalization can not restore true audio fidelity of the acoustic field as perceived at the location of the signal recording. Without a significant increase in the sophistication of the recording process, coherence information will continue to not be encoded on the CD. As a result, properties such as clarity, spatiousness, apparent source size, etc. will not be restored


Reason to Reject Waye TheoTetic Geometric
Approach optics Approach

1. Acoustic Impedances of Materials X X
Required.
2. Complicated Boundary Conditions. X X
3. Computational Complexity. x x
4. Resultant Filter Orders.
5. Discemability of Important X
Room Modes.
6. Limited Accuracy X
7. Time-Avera'ged Properties Only. X








by equalization. The coherence properties of the signal received by the listener in the home listening room will be determined by the listening room itself

By ignoring the signal coherence properties, the inverse problem is radically simplified. In addition we have assumed that the listening rooms are minimum phase. Limiting the inverse problem to minimum phase systems does not significantly reduce the robustness of the technology. From the principles of room acoustics it is known that for a listening room to be non-minimum phase it must have a bad acoustic design, e.g. a rectangular cavity with one axis much longer than the others terminated by walls with very low acoustic resistance. The assumption of minimum phase has the important characteristic that equalization will not require poles to be placed outside the unit circle.

The filter design problem has been reduced in scope to removing gross coloration effects of the stereo electronics and the listening room at a sufficiently high level of spectral resolution that it matches the characteristics of human hearing. The summary of psychoacoustics indicated that the phase response of a system does not affect our perception of pitch. The filter designer is left with the simpler task of calculating the inverse filter which inverts the magnitude response of the listening room at a resolution of approximately 1/3 octave. The following figure demonstrates the simplified set of design criteria.











Design Factor


1. Spatial Coherence

2. Temporal Coherence 3. Magnitude Response
4.Phase Response

5. Frequency Resolution 6.Amplitude Resolution
7. Stability


+


Not encoded on CD Not encoded on CD Not pyschoacoustically important


Must match criti bandwidths


cal


16 bits encoded on CD room responses are usually minimum phase


The figure demonstrates the design criteria for a measurement based equalization strategy.


figure 1.17


Yes/No


Comments


1�!












CHAPTER 2
AN EQUALIZATION STRATEGY FOR REAL-TRVIE SELF-ADJUSTMENT Introduction

Chapter two will introduce the current state-of-the-art equalizer technology. It will be shown that this technology does not make efficient use of an equalizer's limited degrees of freedom. Because of the finer frequency resolution required in the lower frequency bands, high filter orders are required to control the small critical digital frequencies. In the high ft-equency bands however, the spectral resolution will be finer than necessary. The Adkins-Principe architecture significantly improved the traditional strategy by introducing a multi-rate octave filter bank. By downsampling data in the lower bands according to the Nyquist limit, the critical digital frequency requirements are made proportionally less severe, and the required inverse filtering can be performed by filters of significantly lower order. As this is the basic approach of the technology introduced in this dissertation, a complete description of this technique will be reviewed in chapter two.

While this strategy is a significant improvement, it does not make use of the time varying spectral distribution of the source. The Adkins-Principe strategy continues to assign the limited degrees of freedom available in the inverse filters to equalize the entire audio bandpass, even if the source has no power in large spectral regions. It furthermore continues to equalize regions of the bandpass where little or no coloration is being induced by the stereo electronics or the listening room. The approach investigated in this work is to add into the Adkins-Principe architectum adaptive filters. Adaptive filters by their very natum will assign the most processing resources to regions of the audio bandpass in which the most coloration exists, significantly improving the efficiency and performance of equalization. A brief description of the major classes of adaptive algorithms is included. Because of the requirements of real-time processing, as well as the requirement of high fidelity, appropriate classes of adaptive algorithms are limited. Because of the low computational complexity, and satisfactory performance the Least Mean Squares (LMS) algorithm is chosen for further investigation in this research.















LP = Lowpass Filter HP = Highpass Filter DEC = Decimation INT = Internolation


_P1 HDECHHP4H INT H LP-,


The Adkins-Principe equalizer architecture is a parallel multi-rate filter bank.


figure 2.0








State-of-the-Art Acoustic Equalizers

It is clear from the preceding section on the physics of room acoustics that a theoretical modelling of a listening room, even with an unreasonably simplistic geometry and constructed with materials of uniform impedance, is such a computationally burdensome endeavour that it can not be performed for an individual's listening room. The current equalizer technology seeks to excite a-room with broadband noise in order to stimulate a broad range of eigenmodes. This signal is measured at a listener location or locations, and an inverse of the cascade of the home stereo system and the home listening room is estimated. Equalization is not a panacea which will restore a perfect reproduction of the pressure field as propagated by the original rendition of the musical performance. In fact, acoustic equalization can not even reconstruct a pressure field which is psycho-acoustically equivalent. Current technology attempts only to remove gross coloration. The strategy employed is depicted in figure 2. 1. Limitations of this strategy will be examined, with special emphasis on those items which the proposed adaptive equalizer will ameliorate.













figureu2.1 Stt-fteatualize r alcren dege s apralle filte s
banks withfilterofiinstaaraduedbcmpig


thbpwraoled i eachbnyamcopoelctdi



thenlstnnrm with thffcenis which ise caused to citearthe

room.








After room excitation with a broadband source, current techniques break the measured signal into octave or 1/3 octave bands using a set of bandpass filters. The power in each band is compared with the power initially radiated in the same band. The equalizer is adjusted so that its filter coefficients give sufficient gain in the attenuated bands that the received signal has a flat frequency response, or any pre-programmed response which the listener finds acoustically' pleasing. The noise source is turned off, and the CD player is turned on.

Source Excitation

The excitation source utilized in equalizers has been the subject of much debate. Several articles [ 13-14] deal with the varying responses of rooms to different types of sources, i.e. firecrackers, pistol blanks, etc. The most important characteristics of the test signal are the signal's spectral content, the amplitude distribution of the source, and the duration of the signal. State-of-the-art equalizers use pink, Gaussian noise for reasons which are now discussed.

Except for smalL undamped rooms, at low excitation frequencies, the sound pressure amplitude distribution is independent of the volume, shape, or acoustical properties of the room. By examining the room's Green's function it is seen that the pressure field is the superposition of many eigenfrequencies, each with its separate damping coefficient. This superposition is complicated due to the finite half-width, and small separation of the room resonances. Since the eigenmodes are closely spaced and mutually coupling, the field may be considered to be resultant from a single frequency with randomly distributed amplitudes and phases. Applying the Central Urnit Theorem to both real and imaginary terms of the pressure, P, leads to the Raleigh distribution of IPL Figure 2.2 demonstrates the similarity of the distribution function of.pressure to Gaussian excitation.

The spectral content must be sufficiently broad that the eigenmodes excited by a music signal are also excited by Gaussian noise. Pink noise satisfies this requirement and has theadditional advantage that it will simplify the signal processing. Pink noise has the property that equal power is radiated in each octave. In addition, unlike impulsive sources, the


























figure 212 A Gaussian noise source is used with many state-of-the-art
equalizers because of its similarity to the pressure distribution of acoustic data in an enclosed space.



pink noise source is of sufficient duration to "charge" the reactive elements of the listening room, which will allow a better equalization of the room's steady state response. Bandpass Filters

Bandpass filters are designed to have bandwidths that approximately match the bandwidth characteristics of our human physiology. All high quality equalizers employ a digital 1/3 octave parallel filter bank, and from figure 1. 12, it is seen that this resolution will closely match the critical bandwidths as discussed in the psycho-acoustics, review. The center frequencies and upper and lower cut-off frequencies are given in figure 2.3.

To maintain pitch the bandpass filters must be designed with sufficient control of the signal's spectral characteristics, while realizing that too fine a resolution will not improve our subjective sense of pitch and will add processing time and expense to the equalizer. The filter bank is made up of linear FIR lowpass and highpass filters cascaded to form bandpass filters. Performance is limited by the low digital cut-off frequencies associated with the












lower frequency bands, which require high order filters. Control of the band at the level of


the difference limen (fig. 1.11) is not achieved. Except in the most unusual circumstances







1/3 Octave Filter Bank

[and Center Frequencles [Hz] Band Um [Hz)

22
25 28
31.5 35
40 44
so 57

so 22-28 Hz
100 88
125 113
160 141
200 176
250 225
315 283
400 353
500 440
630 565
0 707
1000 880
1130
1250 1414 0
1600 1760
2000 2250
3150 2825
4000 3530 Iz:
5000 4400
6300 5650
8000 7070
10000 8800
1200 11300
16000 14140
20000 17600
22500





figure 2.3 The critical frequencies of an 1/3 octave bandpass filter are

shown above. Because of the similarities with the resolution
characteristics of human hearing, the 1/3 octave filter bank is
most ofterichosen for state-of-the-art equalizers.





the difference limen is an unnecessarily strict standard due to the masking effects of music


signals. The precise requirements are difficult to evaluate do the complex nature of masking.


Signal Measurement


Any system which relies on current microphone technology for precise acoustic measurement is problematic. A situation analogous to the Heisenburg Uncertainty Principle


arises when measuring the field - introduction of the microphone into the field disturbs the








field itself. In minimizing this effect, high quality microphones are made as small as possible while maintaining their sensitivity. In addition measuring the pressure field intensity is not synonymous with measuring signal loudness as was shown in the previous section. The conversion from intensity to loudness can be accounted for in software. A more difficult matter is to completely characterize the microphone directionality, which will differ from the directional properties of our hearing. In practice a precise understanding of the microphone's directional response is unavailable. A high quality microphone must also have a fast response if it is not to smear the details of the signal, affecting the perceived timbre and clarity.

Signal Procusing

Processing is performed by a dedicated chip which calculates the power in each band collected from the CD player. The same procedure is performed on the signal received from the microphone. The two signals are compared, and a calculation of the appropriate filter coefficients is made in order to guarantee a flat frequency response, or any other preset frequency response, across the audio bandpass. Unfortunately the details of the processing are not discussed in the literature, probably because of sensitivity towards proprietary technology on the part of manufacturers. The measurements of current state-of-the-art equalizers indicate they are capable of maintaining a flat frequency response across the audio bandpass to within I dB. At 40 dB between 100 and 1000 Hz for speech and music signals a band ripple of I dB is not perceptible. At the band edges ripple can be as great as 2 dB before listeners can perceive coloration. The number of bits used for filter all processing calculations must be at least 16 in order to avoid degrading the SIN of the CD player. High quality equalizers allow separate equalization to be performed for each channel of the stereo. The period of time required for the equalization can be as high as 15 seconds. The filter coefficients can be stored in one of several memories. A graphic interface is provided to allow the user to manually set his listening preferences, which are taken into account in the cal- culation of filter coefficients. These systems are a large improvement in traditional equalization technology which required time consuming manual adjustment of graphic equalizers, and depended on the skill of the individual making the adjustments.








Limitaftons

In the discussion of room acoustics the temporal characteristics of the acoustic field was introduced using the principals of geometric acoustics. This approach gave a qualitative understanding, but it is of limited accuracy and validity. The spatial characteristics were also presented using a simplistic approach to acoustic reflections. A rigorous theoretical foundation of room acoustics-was made frornthe wave equation. By solving boundary conditions it was shown that the impedances of surface materials, and the room's geometry are the key parameters in developing a Green's function. Because of the oversimplification of geometric acoustics, and the huge computational requirements of a wave theoretical approach, filter coefficients required for an equalizer can not be calculated.

State-of-the-art equalizers determine the room response by exciting a room with a
broadband source. A measurement is made of the acoustic field and the listener location, and an inverse of the room response is calculated for implementation in an octave filter bank. The excitation source, signal measurement, signal processing, and equalizer filters of state-of-the-art equalizers were reviewed. This technology is focussed on removing those effects which are psycho-acoustically most significant. In the presentation of psychoacoustics pitch was discussed as a key feature of the music signal. To preserve pitch the equalizer must remove c . coloration introduced by the stereo electronics and the listening room. Phase response was shown to be relatively unimportant in maintaining pitch. The temporal and spatial characteristics of the acoustic field were discussed as important characteristics in giving a listener the sense of fullness and spaciousness of the sound.

The state-of-the-art equalizer technology described above is able to remove most coloration of the stereo-room combination, and thus maintain pitch. The equalizer will not be able to correct the following effects.

1) The equalizer will not be able to correct coloration due to strong regular reflections. These types of effects are represented by transfer functions that are non-minimum phase, and thus would require the equalizer to have poles outside the unit circle. This should not be a problem unless the room in which the original recording was made, or the home listening room has a terrible acoustic design, e.g. a rectangular cavity with one axis much








longer than the others terminated in a wall with very low acoustic resistance.

2) The perceived timbre will not be affected by the equalizer. Timbre is related transient phenomena, and as the equalizer's filters are time invariant, the equalizer will not be able to restore timbre related features.

3) Equalization can not restore true audio fidelity of the signal because of the loss of information regarding both temporal and spatial coherence properties of the signal. Unfortunately these characteristics are psycho-acoustically important. There are many reasons for the lack of fidelity with respect to field coherence properties. Without a significant increase in the level of sophistication in the recording process, the coherence information will not be included on the CD itself. Microphones used in the recording process, as well as the equalization process, do not encode information regarding the spatial characteristics of the signal. Fast temporal structures may also be lost due to the non-zero time constants in all microphones. In addition the state-of-the-art equalizers utilize only one microphone and cannot account for stereophonic properties of the acoustic field. As a result of the above limitations, properties such as clarity, spaciousness, apparent source size, apparent source location, etc. will not be restored by equalizers. These properties will be maintained by the characteristics of the listening room itself.

4) The equalization of a music reproduction system is valid for only one listener location. If the listener moves locations, the equalization must be performed again. The adaptive equalizer will eliminate this problem.

5) The best source with which to excite the room is the actual acoustic signal of interest
- not pink noise. The equalizer's filters should utilize their limited degrees of freedom to correct frequencies at which distortion is a maximum, and not the entire spectrum. This is a fundamental limitation which affects the degree to which relative loudness, and hence pitch, can be maintained. This limitation will be ameliorated by the adaptive equalizer.








The Adkins-Principe Equalizer Architecture

The Adkins-Principe (AP) architecture [1] uses a multi-rate filter bank to produce an octave band structure by repeatedly filtering and decimating a single input signal. A separate highpass filter equalizes each band, making the processing computationally parallel and greatly reducing the necessary microprocessor speed for real-time operation. The advantage of multi-rate designs lies in their increased efficiency. The elegance of the structure can be seen in the block diagram shown in figure 2.0. The proposed real-time equalization strategy will be based on this architecture. The following section will discuss its characteristics. The advantages of 'this architecture over the current state-of-the-art will be made clear. Its operation will be shown to be consistent with maintaining pitch as described in chapter one. In addition, the features of the architecture which must be carefully controlled to assure proper signal reconstruction will be discussed. Octave Bands Generation and Eq4ualization

The highest frequency band is controlled by passing the original signal through a highpass filter. All lower bands are controlled by a cascade of lowpass, decimation, and highpass filters. This is accomplished as shown in figure 2.4.








64-'


futeKadiite aln aI seodbrnh








The tree structure is an efficient strategy for implementing an octave filter bank. The input to each branching node has been bandlimited and decimated. The lower branch performs equalization with a highpass filter while the upper branch further reduces the CD bandwidth. The characteristics of the upper and lower branches are described below.

Each of the upper branches attempts to perform the following lowpass operation.


HLp = 1, 10) <-7 (2-1)
= 0, otherwise

The upper cutoff of band i is equal to the lower cutoff of band i-i. The sampling rate can be reduced by decimating by M according to Shannon's sampling theorem.

Xde [nfl = x [nMT] (2-2)

The decimation rescales the digital frequencies of the DFT, as shown in figure 2.5.
Clearly the decimation rate must be coupled to the value of the digital cutoff frequency of the lowpass filters in order to avoid aliasing.


Decimation by 2









figure 2.5 Decimation by M in the time domain expands the digital
frequencies by M, and reduces the peak value of the
spectrum by M. This figure demonstrates the effects for M
equals 2.



Decimation reduces the number of operations required for real-time processing by reducing the data rate by M'-' for band i. In addition, because the critical digital frequencies








are increased by M, due to the rescaling of digital frequencies engendered by decimation, the required filter orders for all subsequent filtering operations are considerably smaller, or a finer frequency will be possible with the same order filters. This is an important characteristic due to the finer resolution psycho-acoustically required in the lower frequency bands. Figure 2.6 summarizes the upper branch.

At each branching node of the tree structure the bandlimited signal is also sent to a highpass filter (lower branch) for equalization. If the room were non-filtering the highpass filter would be designed as follows.

HHP = l, I101 < (2-3)

= 0, otherwise

Note that the digital cutoff frequency is jq and not - because of the decimation.
\
Operation of the Upper Branch of the Tree Structure
original signal Lowasl Filter Iowpass signal








U Decimate by M
+ +� + .+ � +.� � . . . .







figure 2.6 This figures demonstrates the processing of a signal
generated as the summation of two sine waves. The low
frequency sine wave is separated with a lowpass filter and
then decimated. The decimated part of the signal is thrown
away.








In the more realistic case of a room which filters its acoustic input, the highpass filters are designed with user supplied gains for each band. The graphic interface used on an equalizer implemented on the NeXT computer workstation [ 15] is shown in figure 2.7. The interface indicates gains for seven bands, and interpolates between the frequencies indicated below.


figure 2.7 The graphic interface for an equalizer using the AP
architecture was implemented on the NeXT computer with a user interface as shown in this figure. The highpass filters are
adjusted according to the selections made by dragging the
slides with a mouse.


The combination of filter length and decimation rate must be such that the requirement of at least 1/3 octave resolution is satisfied. In addition, band ripple, transition bandwidth, and stopband attenuation must be such to guarantee that there is less than ldB band ripple across the entire audio bandpass. The research documented in this dissertation will use the filter design presented by Adkins and Principe [I]. Their extensive simulations resulted in a recommended design of 4 bands, each utilizing symmetric finite impulse response (FIR) filters of order 45, and a constant decimation rate of six. A summary of the frequency res-olution of the design is given in figure 2.8.














Band 2 612.5-3675- 19 0.045 -0.250
Band 3 102-612.5__ 19 j0.045 - 0.250
Band 4 17-102 J 19 J0.045 - 0.250







i U IU 1 37 220510
f,12M4 f1W f/(2 f.12M f
f,/2M3 frequency [Hzj (not to scale)
figure 2.8 Frequency resolution for a four band equalizer with filters of
order 45, and a constant decimation rate of 6, is shown
above.



The highpass FIR filters are designed by taking the inverse FFT of the frequency response requested by the user with the equalizer's graphical interface . The filters are smoothed by truncating symmetrically about time = 0 the resulting time domain representation with a Kaiser window with 0 = 3.5. The symmetric nature of the filters guarantees linear phase. The time domain representation is shifted to guarantee that the filters are causal. A summary of the process is shown in figure 2.9. Signal Reconstruction

After equalization has been performed with highpass ifiters, the data in each band must be summed in order to reconstruct the signal. The summation is complicated because of the different sampling rates at which the individual highpass filters operate. In addition there is an unique delay due to the differing number of convolution operations performed in the various branches of the equalizer. Immediately proceeding equalization, the signal










Grephic Interface Desired Frequency Response


figure 2.9 Operation of the lower branch of the tree structure.
a) User defines desired response; b) Frequency response; c) inverse fourier transform for time damson representation, (d) For filter coefficients multiply time domain representation by
a Kaiser v."'Indow to lessen degradation induced by Gibb's
phenomena.


is interpolated to bring it to the sampling frequency of the. preceding band. Interpolation performs the opposite operation of decimation by inserting M-lI evenly spaced samples of value zero between every pair of data points.


x i~nT x [kT] if k=n/M=an integer (2-4)
0 otherwise

Interpolation acts to recompress the digital frequencies of the equalized time series in manner precisely opposite to the decimation process. These effects are demonstrated in the figure 2. 10.

Images created by interpolation must be removed with an anti-aliasing lowpass filter.
I& 7
Note that the digital cutoff frequency is Tand not -0 because of the interpolation. The lowpass filter should have a gain of M to restore the energy lost in the decimation stage.The output from the anti-imaging filters are summed with the output of the preceding band efficiently by using the inverse tree structure shown in figure 2.11.


Graphic Intedoce


Desired Frequency Response










Interpolation by 3


(a) For decimation by M, M- I evenly spaced samples of zero are placed between each pair of samples, (b) frequency response before interpolation, (c) frequency response after interpolation.


jW
H(e


/\AAAAA,


figure 2. 10


figure 2.11 The inverse tree structure efficiently sums the bands together
after interpolation and anti-imaging, bringing the lower
frequency band up to the sampling rate of the higher
frequency band.



The delay associated with each band is calculated with respect to the highest sampling
rate. Recall that for an FIR filter the delay, Td, is expressed as follows, where Ts, is the








sampling rate of band 1.

(N- 1)
Td 2 TS 1 (2-5)


For band i the total delay as seen at the output of the highpass filter is expressed as follows.


Tdi (N 2 1) 1 Mi] TS, (2-6)
i = 0

The delay at all nodes of the inverse tree structure is dominated by the value of Td in the upper branch. The total delay as seen at the output of the AP structure for a four band system is dominated by Td4 and is given below.


2 T M 3 TS 1 (2-7)
-'Tdtotal d4 2


Processing SDsQd

Efficient use of computing resources can be made because of the differing sampling rates of data in various stages of the architecture. Convolutions are performed at different rates and are staggered in time. Data is sampled such that multiply and accumulate operations are made in non-overlapping time slots. In this way major timing bottlenecks are avoided. The CD player sampling rate determines the value of the time slots (1/44, 100 22.676 microseconds). For the four band design 10 FIR filters are operated. Three filters perform convolutions at the highest sampling rates (44.1 kHz), three perform convolutions at the CD rate decimated by six (7.35 kHz), three perform convolutions at the CD rate decimated by 36 (1225 Hz), and one convolution at the CD rate decimated by 216 (204 Hz). The timing diagram is shown in figure 2.12.

As an example of the computational load of the convolutions indicated in the timing diagram above, consider the TI TMS320C40 25Mflop floating point digital signal processing


























figure 2.12


H , t t
I Ptf tff t t t
II


LPI
HP3
LP2 t I
LPt 216
HP4
I p 216


The timing diagram indicates the efficiency of microprocessor implementation of the AP architecture. Decimation and selective sampling allow a staggering of the convolutions which prevent computational bottlenecks.


chip. This processor performs a multiply and accumulate in one cycle. With one chip dedicated to each band of the equalizer the number of computations which can be performed between CD data samples is limited to the following.


# of computations = 25MFLOPS = 566 44.1KHz


According to the timing diagram above, for FIR filters of order 45, 360 multiply and accumulate operations are required, i.e. 63.6% of the CPU time. Theoretical Performance

The AP architecture is an efficient implementation of a filter bank. The audio signal is broken into octaves using a tree structure. Along the lower branch, a bandlimited and downsampled signal is equalized with a highpass filter. Along the upper branch the signal is further bandlimited and downsampled according to the Nyquist limit. After equalization the octave bands are brought to the original sampling rate by upsampling, and using an in-








verse tree structure. Each band has a distinct delay which must be accounted for when recombining bands. Because of the octave band structure and downsainpling, the critical digital frequencies are raised as much as possible, and lead to a degree of bandwidth resolution and of band ripple, across the entire audio bandpass, that is superior to performance advertized on existing equalizer architectures. From figure 2.5 it is seen that fr-equency resolution is finer than 1/3 octave across the entire bandpass. At worst, resolution i s 1/4 octave, while at best the equalizer has 0.045 octave resolution. Computer simulations of the AP architecture indicate for equalizer settings for a flat response, there is a band ripple of less than 0.2 dB, from 17Hz to 22500 Hz. This value is far superior to the values of 2dB which are measured on equalizers which are commercially available. The spectral resolution and the high degree of control of the highpass filters are precisely the characteristics required to restore the proper pitch of the acoustic signal received at the listener location.

Automatic Adjustment Using Adaptive Filters

Elliot and Nelson [I16]-have proposed substituting the highpass filters in a standard filter bank with adaptive filters designed to minimize the mean square of an error signal which is generated by taking the difference between a pink noise source, and the noise source as collected by a microphone at the listener location. Although this approach has the advantage of a lower computational complexity than the standard frequency domain techniques, the most compelling reason for the use of adaptive filters is overlooked, namely, the use of the music signal itself as a reference signal. This research will substitute adaptive filters for the highpass filters in the AP architecture as shown in figure 2.13.

In each stage the CD reference signal is bandlimited and decimated in precisely the manner discussed above. A microphone signal is filtered and decimated in an analogous manner to the CD signal.' A feedback signal (error signal), which is provided by the difference between the filtered CD and microphone signals, is sent to an adaptation algorithm which updates the adaptive filter weights in such a way that the mean square error is minimized. As an example, the structure for band 2 is shown in figure 2.14.























figure 2.13


The adaptive equalizer architecture is a modification to the AP architecture in which highpass filters are replaced by adaptive filters.


The adaptive filter is outlined in grey. The filtered CD signal is compared with the microphone signal similarly bandlimited an decimated. An error signal is generated and sent to the adaptation algorithm which updates the filter weights in such a way that mean square error is minimized.


figure 2.14


This strategy will improve the state-of-the-art by ameliorating two of the limitations discussed in chapter one. Specifically, if the listener changes location (assuming the micro-








phone is also moved) the filter will automatically correct for the changes in the room's response because of the equalizer's capability for real-time self-adjustment. Depending on the type of adaptation algorithm chosen, the updates will be performed either every sample(44.1 kHz), or after every block of samples(44. I kHz/N, where N is the length of the block). More importantly, the reference signal used for equalization is the CD signal itself instead of pink noise. The adaptive filter, when minimizing mean square error, will automatically utilize its limited degrees of freedom to adjust the frequency response in those portions of the spectrum in which the most energy is present - not the entire audio bandpass.

Several adaptation algorithms have been developed for adaptive signal processing
[17-19]. The choice of algorithms depends on several considerations, namely, the stationarity characteristics of the signal, the data rate of the input signal, the degree of accuracy required in filtering, and. the computational resources available for real-time operation. Acoustic data is highly non-stationary. Because the adaptive filters must find their optimal coefficients in a fraction of the time in which the audio signal is quasi-stationary, the adaptation process must be fast. Secondly, although the adaptation must occur rapidly, the allowable misadjustment of the filter weights must be relatively small for meaningful improvement in the current state-of-the-art. Thirdly, due to the high CD sampling rate, the complexity of the algorithm must not be so great as to preclude carrying out the arithmetic in real-time. This is a serious concern even with the high speed microprocessors currently available. Lastly, any adaptation algorithm must not introduce any noise components which will lower the signal fidelity.

All available adaptation algorithms require trade-offs among the crucial factors outlined above. Extremely rapid convergence can be achieved, but at the expense of increased computational complexity. Computationally efficient algorithms exist, but convergence is slowed, and further serious trade-offs between convergence speed and misadjustment need to be taken into account. Other algorithms may make a good compromise between complexity and convergence speed, but by their nature they contribute undesirable noise into the system. Consider the tap delay line architecture for an FIR filter shown below, where








Yk represents the output of the feedforward filter at time k, dk represents the desired (or reference) signal, and ek represents the difference signal between dk and Yk (error signal).


figure 2.15


An adaptive tapped delay line with desired signal, dk, and error signal, ek.


Assume that the sequence {Xk} is quasi-stationary, i.e. the signal has been chosen over a time window in which it is approximately stationary. The output of the tapped delay line is the inner product of the n+l most current filter weights with the filter input.


Yk =Xk Wk= [xk xk- l Xk2 x -


(2-8)


The filter weights, W*, which minimize the mean square error, E [ (dk - Yk) 2] , are expressed by well-known Wiener formulation: W*=R'P where R is the matrix E[XXT] and P is the cross correlation between d and X, E[dX]. The field of adaptive signal processing is concerned with approximating the solution for W*. There are two standard methods by which the solution is approximated: gradient search algorithms and least squares (LS) al-








gorithms. The gradient search methods can be generalized as follows: Wk+ I = Wk - V , where = E [ (dk -Yk) 2] = mean square error. This common sense approach adjusts the weights constituting vector W in a direction opposite to that of the gradient of the error surface. g. represents a constant which regulates the step size of the weight update. The most common gradient search algorithm is the Least Mean Squares (LMS) algorithm which estimates V = V E [error2] by taking the time average of V [error2] . The approximation is valid over quasi-stationary periods because of the quasi-ergodicity of the signal. The LS algorithms differ by estimating the value of R and P with time averages. LS algorithms can be broken into two categories: algorithms which calculate new estimates over a window of data (block LS algorithms), and algorithms which recursively update the estimate after every sample (RLS algorithms). One example of an LS algorithm is the block covariance method. Let f and P' be the time averaged approximations of R and P, and let co be a windowing function.

lJXXT
Rk = l(dllT

Pk = (2-10)
A1
1


Wk = -Rk Pk (2-11)


Certain classes of algorithm can be ruled out of consideration immediately because of their unsuitability with respect to one or more of the requirements given above. Although the block LS algorithms converge more quickly than gradient search techniques, after every stationary window of data the input data autocorrelation matrix, h, must be inverted. Some of the block algorithms have autocorrelation matrices that are Toeplitz and can use the Durban algorithm for matrix inversion, requiring O(N2) operations, where N is the length of the filter. Many of the LS methods(i.e. the covariance method), however, are not Toeplitz and matrix inversion must be performed using the Cholesky algorithm, requiring O(N3) operations. Furthermore, these algorithms update the weight vector every M samples introducing an undesirable artifact into the spectrum of the acoustic signal.








Algorithms which minimize error in the frequency domain operate on blocks of data. As such, they update filter coefficients at regular intervals, introducing a noise component into the signal. In addition, they require more operations in order to transform the data. Although we are most interested in minimizing the error of the magnitude response of the transform, good time domain equalization will guarantee good frequency domain equalization as well.

The recursive algorithms are more promising. They converge rapidly, and the weight vector is updated every sample. These algorithms unfortunately have a heavy computational load. Figure 2.16 gives the computational requirements of the most familiar versions of the RLS algorithm [18].


REQUIRED COMPUTATIONS PER ALGORTHM ITERATION

ALGORITHM MULTIPLICATIONS ADDITIONS SQUARE
& DIVISIONS ROOTS
Modified Fast
Kalman(CMFK) N15N1
Fast Kalman 8N +5 7N+ 2
Growing Memory
Covariance CGMC) 13N + 6 11N +1
Sliding Window 13N +13 12N+ 7
Covarnance (SWC) ________ _____Normalized GMC 12N +16 7N +2 3
Normalized SWO 23N 11 N 5N
Normalized PLS GIN +17 5N +2 2



figure 2.16 The computational complexity for several adaptive
algorithms are displayed above.



All algorithms, except the Kalman types, required significantly more computations than the LMS algorithm (2N + 1 multiplications and 2N additions). On the basis of computational considerations, the algorithms seriously considered for the equalizer are reduced to these two alone. The choice has been made to utilize a fast LMS method as the adaptation algorithm. Chapters three and four will investigate LMS algorithms in detail.













CHAPTER 3
LEAST MEAN SQUARES (LMS) ALGORITHM CONVERGENCE WITH UNCORRELATED INPUT DATA

The LMS algorithm converges q uickly to an approximately optimum solution (in the mean square error sense) given that the step size has been properly initialized. Because of the low computational complexity of the LJMS algorithm, it has been selected as the adaptive algorithm which will be implemented in the AP architecture. Because the LMS algorithm is a gradient descent technique, there is a trade-off between convergence speed and misadjustment. Misadjustment measures on average how close the LMS solution is to the optimum solution, after algorithm convergence. A smaller step size results in a smaller misadjustment. For fast convergence and low misadjustment an adaptive step size is required. At the beginning of a quasi-stationary segment of data, the value of the step size must be initialized in such a way that convergence speed is maximized. In this stage of the adaptive process the lack of convergence speed provides the major contribution to mean square error, and misadjustment is unimportant. As the filter converges, the step size must be reduced because of the increasing dominance of misadjustment.

Chapter three will investigate the optimal value at which the step size should be set at the beginning of a quasi-stationary segment of acoustic data. The initialization problem has been rigorously solved for uncorrelated input signals. To efficiently utilize the LMS algorithm on music signals understanding must be extended to the case of correlated input data (see figure 4. 1). Chapter three will provide the necessary background information for this extension. Chapter three first presents the Wiener-Hopf formulation of the minimization of mean square error. The characteristics of the error surface are highlighted as part of this discussion. As a result it is clear that a stochastic gradient descent algorithm is a feasible approach to the minimization problem. The properties of gradient descent algorithms are discussed with particular emphasis on the LMS algorithm and it properties. Finally, the convergence speed of the mean square error for the LMS algorithm is minimized as a func-








tion of step size. The background information in this chapter is largely taken from the work of Widrow and Steamns [19], and their notation will be used throughout.

The Weiner-Hoof Formulation

The Weiner-Hopf formulation [20] gives a method in which filter weights of a tapped delay (direct form) filter can be optimized when the input to the filter is a stationary stochastic signal. The results hold for a quasi-stationary signal to a greater or lesser degree depending on the precise statistical nature of the signals. The measure by which the filter weights are optimized is the mean square error (MSE).

Because the Wiener-Hopf equation is the underpinning of all adaptive signal processing algorithms, it will be reviewed in detail. The derivation of the Wiener filter will yield important information regarding the nature of the error surface. As a result it will be obvious that gradient descent techniques will be applicable optimization algorithms. The derivation of the Weiner solution is presented below.

Derivation of the Wiener Solution

Figure 2.15 shows a tapped delay line. The input sequence I xk}1, is a stationary stochastic signal. The output of the filter, yk, is simply the inner product of the n+1 most recent inputs with their respective filter weights, for a filter of order n+1.

W0 (3-1)
W I


=k XTWk- [xkxk xk-2* Xkn] 2



Wn



The error signal is generated by taking the difference between the desired signal, dk, and the filter output.









ek = dk-Yk


dk-XWk


Square the error.


e 2 dk T
ek =(k-XkWk)


(3-3)


d22 T T T
=d-2dkk Wk+XiWkWiXk



Take the expectation of both sides of the above equation.


E [e2] -E [ d 2dkXk Wk + TWkWXk]


(3-4)


Ed2-2E[ T T T
kEd2 -2EtdjCkWk] +E[XkWkWTXkl




By keeping W at a fixed value its subscript can be omitted. Furthermore note that
T = W w
XkWW Xk =*XkXkW


2 2E T T T
E [ek] = E [dk] - 2E [dkk] W +W E [Xkk] W


Define the expectation value of the cross correlation matrix as follows.


kE [dkX] = E[dkxk


dkxk- 1 *


(3-6) dkxk-l]


Define the expectation of the correlation matrix as follows.


(3-5)


(3-2)












T
Rk =E [XkXk] E


XkXk Xk - lXk xk - nXk


XkXk- I Xk- lXk- I


XkXk- 1


XkcXk~n Xk -Xk - n


Xk - nXk - n


(3-7)


The mean square error can now be expressed in a simpler notation. Let = E [e2].


= E [dk] - 2PkM + WTRkW


(3-8)


To minimize , set - to zero and solve for the optimum weights, W*


(3-9)


i- -2P+2RW = 0 dW


(3-10)


W = R-P


The above equation is the expression of the Wiener - Hopf equations. Substituting equation 10 into equation 8 and making use of the symmetry of R results in an expression for the minimum value of , *.


= E [dk] - 2pTR P + (R-1p) TR (R P) = E[d2] 2-pTw


(3-11)


(3-12)








With a stationary signal in which the cross correlation matrix and correlation matrix were perfectly determined, it would be possible to solve the Weiner-Hopf equations for W* and *. Unfortunately in engineering applications of interest neither of these assumptions is generally true. The field of adaptive signal processing is concerned with estimating W* without the necessary a priori information required by the Wiener - Hopf formulation. Error Surfaces

Equations eight and ten provide important information regarding the characteristics of the error surface, which is defined as = f(W). The most critical result is the hyperparabolic nature of the surfaces. This is demonstrated by changing the coordinate system used in the expression for . Let V = W - W* be the value of the displacement of the filter weights from the optimal value. Express MSE in terms of this translated coordinate system, i.e. = f(V) . The transformation is accomplished as follows. Recall equation 3.8.




= E[d] + WRW- 2WTP (3-8)


Equation 3-13 is inspired by assuming the proper solution of f(V) and demonstrating its equivalence to equation 3-8.
(3-13)
= E[d] + wTRW-2WTRR- + TR-1 RR-1 -pTR-1p



Substitute W* for R-1P and* for E [d1 -pjrw*

= * +WTRw-2wTRw�+PTR-1RW (3-14)


Because correlation matrices are Hermitian, (PTR-1) T = R1'P.











- +WTRW2WTR+RR (3-15)


Since WTRWV is a scalar, it must be equal to its own transpose. Again use the Hermitian property of R.


+ ~*WTRW+ WTRW -WTRW -WTRW (3-16)


Note that transposition is a linear operator.


= *+ (W- W)TR(W-W* (3-17)

+' ~ VTR V (3-18)


The error surface is clearly a quadratic function of V. As a result, the following important characteristics are obvious: 1) the error surface is a hyperparaboloid, 2) the error surface must be concave upwards in order to avoid regions of negative MSE (a logical impossibility), 3) the optimal set of filter weights are located at the bottom of the hyperparaboloid, and 4) there is only one minimum (no local minima). These observations are critical in their implication that gradient search techniques are able to find the global minimum. An error surface for both the translated and untranslated coordinates is shown below.

The Gradient Descent Algorithm

The gradient of a surface points in the direction in which the function describing the surface will be most increased. A common sense approach to finding a surface minimum is to move in a direction opposite to that of its gradient. This section will show rigorously the convergence properties for a stationary signal. The trade-off between convergence speed and misadjustment will be demonstrated explicitly. Finally, a description of the difficulty in using gradient descent techniques with quasi-stationary signals is discussed.


































figure 3.1 The minimum of the error surface is at the origin of the
translated coordinate system. The parabolic nature of the
surface makes gradient descent algorithms feasible.



Let g. represent a variable which regulates the step size of the weight adjustment. Gradient descent(GD) methods can be generalized as follows.


Wk+l = Wk-p.Vtk


(3-19)


Figure 3.2 graphically demonstrates how GD methods update themselves on the basis of


past information.


A =f (v) =f (W)



Ld































figure 3.2 Gradient descent methods find the error surface minimum by moving in
a direction opposite to that of the error surface gradient.


The trajectory of the filter weights is the projection of the n+l dimensional error surface onto the n-dimensional weight space. For a hyperparaboloid the approach is intuitive - the greater the distance from the surface minimum the larger the gradient, and hence, the larger the weight updates. At the surface minimum the gradient is zero, and no further adjustment is made. A more rigorous justification follows. Algorithm Convergence

Recall from equation 3-9 that the gradient is expressed as -- 2P+2RW. Substituting this into equation 3-19 yields the following.


Wk+1 = Wk+g(2P-2RWk) (3-20)


= Wk + g.R (2R-1P- 2Wk) (3-21)


=f M


=f(W*)


Gradient Descent
Wk+1 = Wk-.tV~k








Substitute the solution of the Wiener-Hopf equation (equation 3-10) into 3-21.


Wk+l = Wk+2gIR(W* -Wk)


= (I-2iR)Wk +2RW


(3-22)


Switch to the translated coordinate system by subtracting W from both sides of equation 3-22.


Wk+I-W* = (I-2iR)Wk +2IRW*-W*


(3-23)


= (I- 2 gR) (Wk- W*)


Vk +1 = (I-2gR)Vk


(3-24)


Perform a unitary transformation on R. Let A be the diagonal matrix of eigenvalues of R, and let Q be the matrix in which the columns represent the corresponding eigenvectors. This transformation assumes R is not singular.


R = QAQ-1 for A =


X1 0 0 X 2

00
00


and


Q= [qlq2 *


0 * qn]


qi is the eigenvector associated with x.








Substitute the above expression for R into equation 3-24.


-1 (3-25)
Vk+l = (I-29QAQ-1) Vk




Note that the following transformation has the effect of decoupling the weight vectors in the translated coordinate space, i.e. the transformation causes the new coordinates (principal coordinate system) to be colinear with the eigenvectors of the error surface.


V = Q-1V (3-26)



With the above transformation the coordinates are aligned with the principal axes of the hyperparaboloid. This can be shown as follows.

= * + VTRV (3-27)

=*+ T (QAQ-1) V (3-28)


S*+ (QV) TA (Q- V) (3-29)


= * + vT A v (3-30)

The transformation is illustrated in figure 3.3. Returning to the convergence proof, substitute equation 3-26 into equation 3-25.


QVk+I = (I-2gLQAQ-1 )Q k (3-31)



Premultiply by Q-






























figure 3.3


The principal coordinate system has axes which lie along the principal axes of the hyperparabolic error surface.


k + I= Q-1 (I- 2gQAQ-1) Qlk


= (I-2gA)V14 By induction 1 k can be expressed in terms of 4 0.


Vtk = (I - 2gA) ko0


The update equations show their decoupled nature more clearly in matrix notation.


Quadratic Error Surface in the Princpal Coordinate System

-f (Q1V)


: X.
----------------------------------


(3-32)


(3-33)












(1 - 2p.0) k

0 0 0 0


0

(1 - 2p%1)k
0 0 0


0

0
0 0
* 0
. (1 - 2gx n )k


(3-34)
The convergence condition can be seen from equation 3-34. If 0 < 11 < __, then with perfect
max
knowledge of the gradient vector, the weight vector will converge to the optimal solution.


For O -1 -, lim Vf k = 0
rO

(3-35)


Convergence Speed

The speed of convergence is limited by the value of step size, and hence, the largest eigenvalue. Consider convergence along coordinate j. A geometric convergence factor can be defined as follows.


rj= (1 -2g ) (3-36)



Recall the power series for an exponential.


x2 x3
exp(x) = ! 3+. ! +


* 0


(3-37)


Let = I 1 Then T. is approximately the e-folding time of the adaptation process, as can be seen by truncating the first two terms of the exponential power series.


Vtk
Vtk- 1




Vtk - n


i'to






-vt-n-











rj = exp (3-38)
J

The convergence of to 17 is represented by a learning curve which can be approximated as the product of exponential terms.

22 2

learningcurve = (e Te . * * e


Learning is limited to the largest e-folding time, which is directly related to the smallest eigenvalue and the magnitude of the step size. Excess Mean Sauare Error

In the discussion it has been assumed V is known. In fact it is not. Adjustments must be made in the above equations to account for a noisy gradient estimate. Let V be the estimate of V .


Wk+1 = Wk-RV~k (3-40)



Switch to the translated coordinate system.


Vk+1 = Vk-J'tV~k (3-41)



The estimate of the gradient can be expressed as the sum of the gradient plus a noise term.


V~k = V~k+Nk (3-42)--






76


From equations 3-19 and 3-42 we conclude the following.


V~k = 2RVk+Nk


(3-43)


Substitute equation 3-43 into equation 3-41.


Vk+l = Vk-g(2RVk+Nk)

= (I-2tR)Vk-gNk Switch to the principal coordinate system.


QVfk+l = (I-2gR)QVtk-9Nk


Vtk+ 1 = 1 (I- 2tR) QVfk - IQ-1Nk

= (I-2 2A) Vk- 9Q-1Nk


Let Q-'Nk = Ntk be the gradient noise projected onto the principal axes.


Vlk+ 1 = (I-2gA) Vfk - gAk


(3-44) (3-45) (3-46) (3-47)


Note that equation 3-47 now represents a set of uncoupled difference equations. The matrix formulation is given below.











(1 - 2gx 0)
0 0 0 0


Vtk
'tk - 1 vtk-n


o 0
0 0
� 0
0
0 . (1 - 2gtn)


By induction V" k can be expressed in terms of Vt0.


k-I V'k~l = (I-2A)klfO j E (I-2[LA)JNk-J-l j=O


(3-49)


Thus, with gradient noise included, V1k does not approach zero as k -- c. The degree of


k-i
lim Vfk = -9 Y (I- 2gA)JNtk-j_ I k - j=0


suboptimality is determined by the set of eigenvalues, {X ji, the statistical characteristicsof N, and the step size, g.

The difference between the minimum MSE and the average MSE is defined as the excess mean square error. Excess MSE provides a measure of the difference between the ac-


(3-50)


0
(1 - 2tx1)
0 0 0



ntk- I
tk- 2





k-n -


Vtk- 1 Vtk -2




vtk - n -


(3-48)








tual and the optimal performance, averaged over time, and is expressed as follows.

Excess MSE = E[-C*]


=I~ T[fA14


(3-51)


k-I Recall that after adaptation vt k = - ( - 2;tA)JNft k-j- 1. Substitute this into equation 3-51.
j-0


( j=Ok ]


Excess MSE = E


k-I
A - I (i-29A)'iNki--1
i= 0


00 00
:[4 ij_1 i j


(1 - 2tA)j A (i - 2tA) iltk -i - 1] (3-52)


Since (I - 2g.A)J and A are both diagonal, they can be commuted.
00 00
Excess [kj 1A (I-k2-A) i+JNk_- i ]ij
2 T
g2.2XE k-j_ 1A (I-41A+42A2)jhk-JN 1] (3-53)


Assume that N results from independent errors. Then Vi #j there will be no contribution


to excess MSE.









2 E [Ntk T I -2 )2 N k
Excess MSE = ]L E k A (i 2A) Kt



= P2E[ { (t- 2) 2j Ntk


(3-54) (3-55)


Since I - 4pA + 4g2A2 is a diagonal matrix, the convergent infinite series may be simplified as follows.


E (i - 4tA + 42A2)j = (4tA - 4gt2A2) -1
j=0

Substitute this expression into equation 3-55.


(3-56)


Excess MSE =


gE[NtkA (A - gA2) -1N]k]


4E[Adk(l-gA)-lNtk]


In matrix form the decoupled nature of equation 3-57 is evident.


rcess MSE =- - E [tk nk _1
4 iL'1


�. ntk-n]


1
1 - II0


0 0


0 1 0
1
o o
0 0 0

0 0 0


(3-57)


0


0


ntk

ntk-


1


(3-58)








The matrix equation can be further simplified as follows.



Excess MSE �EF nftk* 1j (3-59)
4L10 - 4gX1J




Equations 3-36, 3-38,3-39, and 3-59 provide a clear indication of the trade-off between misadjustment and convergence speed. The smaller is the step size the smaller is the misadjustment. Below a certain upper bound, however, convergence speed is slowed. Figure
3.4 illustrates the implications of equations 3-60 and 3-6 1.


Urn Excess MSE = 0 (3-60)



Ri t- = (3-61)


Adaptation in a Ouasi-Stationarv Environment

The adaptation process for quasi-stationary input data, while conceptually simple to understand, is difficult to describe analytically. The optimal weight vector is now a function of time, i.e. W/ = f (k) , as the error surface changes shape and moves through a n+1 dimensional space for a ifiter of order n. Widrow has an analytical description of the optimal step size for the LMS by examining the case of modelling an unknown system by an LMS tap delay filter [21]. The unknown system is assumed to be a first order Markov process with 17 constrained to be a constant.

The more general problem for step size optimization is unsolved. Since the weight vectors are adapted on the basis of past information, and the error surface is changing shape and moving with time, the weight vector error results from two contributions. The first contribution results from noise in the estimation of the error surface gradient. The second contribution to error results from estimating the gradient of the error surface at iteration k








using the error surface at iteration k-i. Note the formulation for the filter weights for the non-stationary case, in the translated coordinate system.


Vk+l = Wk-Wk = (Wk-E[Wk]) + (E[Wk]-Wk)
(3-62)

Substitution of the above into equation 3-19 yields the following.


k+l + (Wk-E[Wk])TRk(Wk-E[Wk]) +

(E[Wk] - wk*) TRk(E[Wk] -Wk*) + 2E[(Wk-E[Wk])TRk(E[Wk] -Wk*)]

(3-63)

Expanding the third term in equation 3-63 and recognizing Wk is constant over an ensemble, the third term reduces to zero, and k+ 1 reduces to the following.


k+1 = + (error from gradient noise) +
(error from moving error surface) = ;*+ (Wk-E[Wk])TRk(Wk-E[Wk]) + (E[Wk] - Wk*) TRk(E[Wk] - W ) (3-)


Equation 3-64 is as far as one can go without usually unavailable a priori information concerning the stationarity characteristics of the signal. A graphic illustration of the operation of the LMS algorithm with non-stationary input data is shown in figure 3-5.





























MSE


-IExcess MSE


figure 3.4 The larger step size results in faster convergence, but larger
excess MSE. For this reason the equalizer design will utilize
a varying step size.


()




























w * W k+1




W2' W

















figure 3.5 For quasi-stationary adaptation their are two contributions to
error in the gradient: 1) the error induced by the typical form
of gradient noise, and 2) the error induced by error surface
lag, i.e. the error induced by using information from a prior
error surface instead of the current error surface. In the
figure above the highlighted update, AWk+lI is based on
information concerning k instead of k + V*








The LMS Algorithm with Uncorrelated Input Data

The best known and most widely used gradient method for stochastic signals in adaptive signal processing is the LMS algorithm[22]. The method is well-understood for stationary uncorrelated input data. The method can be extended to the quasi-stationary case, although the analytic description loses much of its simplicity and is possible only for the most simple models. This section will provide the theoretical background necessary for a discussion on the LMS algorithm with correlated input data. Stationary Model for Uncorrelated Input

Recall the general form of the gradient descent algorithms, Wk+ I = Wk - LV k. The LMS algorithm makes a crude estimate of V ,which nevertheless is quite effective. Let V represent the estimate of V 4Gradient Descent LMS Approximation


a-- [e2 V = -2ekXk (3-66)



= 2 E ek -, -]


= -2EI ekXk] (3-65)



The LMS approximation replaces the ensemble estimate of V with a time average. As the algorithm operates over several iterations, on average the gradient estimate will be in the correct direction. The LMS descent is illustrated in figure 3.6.



















. .'. . . .











figure 3.6 The LMS algorithm approximates the gradient by replacing
the ensemble average of the gradient descent with a time
average. In the time average the LMS gradient will point in
the direction of the surface gradient.



It is a simple matter to demonstrate that the LMS estimate is unbiased.


E[V ] = E[-2ekXk]

T
-2E[Xk(dk-XkWk)]


(3-67) (3-68) (3-69)


= 2RkWk- 2Pk


= V (3-70)


If the convergence condition expressed in equation 3-35 holds, the LMS algorithm will converge to W* in the mean.










1 For0< p< kmax

Consider next the variance of the estimate. Assume that after k iterations Wk has converged. Then the gradient estimate is just the noise process Nk.
0

V =7'(+Nk (3-72)

= Nk (3-73)


For j > k, Nk = -2ekXk. After convergence ek and Xk are approximately independent.


coy [Vp] = cOv [N] = E[tNtf] (3-74)


= 4E[eXjXJl = 4E[eIE[XXJ] (375)

= 4jRj (3-76)


The above does not imply that the variance of the estimate remains finite, or that the mean square error remains finite. A considerably more stringent requirement on step size is necessary, and is discussed below.

Conditions on L. for Finite Variance for Square Error

Horowitz and Senne[2] have studied the convergence properties of the LMS mean
square error for the case of Gaussian, zero-mean, uncorrelated data. Let X = [Io X" T.n]T and Ck be the diagonal elements of E[Vt vt7.


= E[(dk- Wk)2] = E[(dk-kW*) -X(Wk-W )]

= * -2E[XT(Wk W*) (dk-4xrW )] +


E[(Wk -)W) TX T (3-77)
-J x(xk (w- W)I








Because {xi) is Gaussian and uncorrelated, Xk and Wk are statistically independent. Furthermore, Xk (dk - XTW*) = P - RkW* implies independence. Thus Xk can be factored out of the second term of equation 3-77. Since Xk is zero mean, the term vanishes altogether.


+ E[(WkTX Tt)] (3-78)
k X" +i (Wk - W ' "



Since Xk is uncorrelated, E [Xk4] is a diagonal matrix and can be expressed as follows.



k= + trace E[(Wk-W)E[X j (3-79)
= ;*+ trace V V ] E [X T]} (3-80)


k = + AT(Tk (3-81)



Horowitz and Senne next develop a recursive relationship for (Yk from the definition of the LMS algorithm.


Wk+1 = [I-2gLXkXT]Wk+2tdkXk (3-82)

T *
Subtract W* and perform a unitary transformation. Let ek = dk - X Wk.

V'k+ 1 I 21t(XktkT) ] Vt + 2TeV*TXT (3-83)




Take the expectation of Vt multiplied by its transpose. Because of the independence of ek and Xtk, the expectation of their cross terms is zero.









E[Vt,,k+j = E[ Vtk k] -4g1AE[Vtk k4] +f tV (3-84)

4p.2 (E[XFkXk kltkXtkk]) +4g2i*A Apply the Gaussian factoring theorem to the third term of equation 3-84.

E[Xkx-kV jk]T = 2A2E[Vtk1/k] + (3-85)

trace{ AE[ VtkVk] }A

Substitute the decomposed moment into equation 3-84.
T Vf Vff(3-86)
E[fk+ lif+ 1] = E[Vtk T] -4gAE[ IkVTk] +


8g2A2E[ Vtkvt] + 4g2trace { AE[ VkVtT] } A + 4t2 *A Let Ck = E[ Vt kVt[]. Ck can be decomposed as follows.


Ck+ I [i, i] = (1 -4gi% + 8g2X) Ck [i, i] + (3-87)
n
4g2Xi I X pCk [ i'p] + 4gx2 * Xi pil

A recursive formulation is now possible for (5 . Let F [i, i] = 1 - 4gki + 12 g 2X2 and F[ij] = 4 g.2,.


Gk+1 = FY k + 4g2 *X (3-88)







With equation 3-88 a recursive expression is available for .


Sk+l = k= *+,T[F k+42*,] (3-89)


Horowitz and Senne develop convergence conditions which can be seen most easily from the matrix formulation of equation 3-89. Let "(k = [k0 ',. "',] (3-90)

;k *+ 4;L 2e*;1O 1l1 . L] +
k+l
n


k+l
4g;0 + 1212;L2 40x 4g;L X
'14g 1 1n 4-0Z 1 4gz + 12g2;2 .4gx1 X ,1
0 1 nk,1


4,L0x 1 4gx0 1 . 1 - 4g) + 12g2 Xk,n



1
For every entry of F, 11 F [i,j] 11 < 1 and gi is real if 0: < 3 . This condition on g will guarantee finite mean square error. Conditions on 1L for Maximum Convergence Spmed and FiniteVariance for Sauare Error

Feuer and Weinstein [3] have developed an expression for step size which will optimize convergence speed. The expression is a function of generally unavailable a priori information, i.e. all eigenvalues and the initial misadjustment along each of the principal coordinates of the error surface. Given no information concerning Tk,' it is assumed that all of its elements are equal. Horowitz and Senne note that the convergence rate of is approximately optimized (given no further a priori information) by setting g� as follows.





90



d (1 - 4g)L + 12g2;L2 0
dp max max


(3-92)












CHAPTER 4
LEAST MEAN SQUARES (LMS) ALGORITHM CONVERGENCE WITH CORRELATED INPUT DATA

Music signals are not uncorrelated as can be seen from the autocorrelation function for several musical instruments shown in figure 4.1. To meaningfully discuss step size initialization for an adaptive acoustic equalizer, the understanding of LMS convergence properties must be extended to the case of uncorrelated input data. Even though the adaptive step size will be optimized for fast convergence only when a large change occurs in the error surface, and hence the most recent data sample will be largely uncorrelated with the previous data, the tapped delay line will have an input vector which, for order n+1, still shares n components with the previous vector.

Convergence of LMS with Correlated Input

Several mathematicians and engineers have investigated a completely general convergence proof for the LMS algorithm without success. To date no satisfactory proof has been put forward guaranteeing lim k < M' VA. < Rma. The difficulty in a rigorous proof lies in the assumption of strong correlation, i.e. Vt 2!0, E [XtyX +,] > 0. The theoretical work done in this area is mathematically sophisticated. Fortunately, Macchi and Eweda [23] have provided a discussion of the theoretical contributions to convergence problems which are here summarized.

Lyung [24] has demonstrated convergence almost everywhere for W. under the condition that the step size is a decreasing sequence tending to zero. He creates a non-divergence criterion by erecting a suitable barrier on Vk which will always reflect Vk onto a random compact set. However, he has not shown that lim Wk = ".Daniell [25] has shown that k
can be made arbitrarily small by choosing g. sufficiently small. However he must use the assumption of uniformly asymptotically independent observations. In addition he makesthe restrictive assumption that the conditional moments of observations, given past observations, are uniformly bounded. This condition is not satisfied even for the case of
































figure 4.1 The autocorrelation functions are given above for several
different instruments [6]. a) organ music (Bach);
b)symphony orchestra (Glasunov); c)cembalo music (Frescobaldi); d) singing with piano accompaniment
(Grieg), and e)speech (male voice).



Gaussian input. Farden [26] has found a bound on by making the reasonable assumption of a decreasing autocorrelation function. He further makes use of reflecting barriers for Vk to keep Vk in a compact set. Macchi and Eweda [23] find a bound on by assuming only blocks of M samples are correlated and all moments are bounded. They do not make use of reflecting boundaries.

The proofs referred to above have assumptions which are representative of acoustic signals. Acoustic signals do generally have decreasing autocorrelation functions, and for a sufficient delay, may be assumed to be uncorrelated. In order to determine the step size which will minimize 1 at the beginning of a quasi-stationary segment of data, the work of Horowitz and Senne, and Feuer and Weinstein is extended without the assumption of un-








correlated input. The mathematical expressions which result are simplified, and conditions on g~ for maximum convergence speed for are found. Simulations are subsequently carried out for different degrees of correlation, step size, and filter order.

Conditions on IL for Convergence of

Adding correlation to the input data significantly complicates the expression of for the LMS algorithm. The approach developed below formulates a recursive expression for Vk in terms of V0. The expression is substituted into the equation for (equation 3-79). With the assumption that ek* is small, cross terms including ek* can be ignored. This resuilts in a matrix equation for in terms of the initial conditions of the algorithm. The matrix norm is investigated numerically as a function of gi. These results are then compared with large ensemble averages of computed with the LMS algorithm. To begin, recall the expression for the LMS algorithm (equation 3-86) formulated in the translated coordinate system.


Vk+lI = (I - 2JI9XJ) Vk + 2ge*kXk (4-1)



Recursively formulate Vk in terms of V0 using equation 4-1.

k (termi1)
Vk =11[I -2 gXk _4Tj] VQ+
jj=k

k- Ik-i (term 2)
j (4-2)


2ie * klXk_ (tenn 3)




Full Text
54
figure 2.10 (a) For decimation by M, M-1 evenly spaced samples of zero
are placed between each pair of samples, (b) frequency
response before interpolation, (c) frequency response after
interpolation.
figure 2.11 The inverse tree structure efficiently sums the bands together
after interpolation and anti-imaging, bringing the lower
frequency band up to the sampling rate of the higher
frequency band.
The delay associated with each band is calculated with respect to the highest sampling
rate. Recall that for an FIR filter the delay, Td, is expressed as follows, where Tsl is the


114
H
111
M
Point Demonstrated
Figure
|
9
M
1. Level of correlation of music signals
4.1

2. Validity of the approximation given in equation 4-6
4.4, 4.5
Upl
3 Validity of equation 4-10 for low filter order and
4.6
correlation
4. Validity of equation 4-10 for low filter order and
4.7

high correlation
§¡§
5. Validity of equation 4-10 for low filter order and
4.8
m
a large range of correlation
6. Validity of equation 4-10 for filter order =10 and
4.9
f
a large range of correlation
7. Sets of eigenvalues used for norm experiments
4.10
for filter order = 45
8. Results of norm experiments for filter order = 45
4.11
as a function of step size and iteration for data
§§
with low correlation
9. Results of norm experiments for filter order = 45
4.12
*
as a function of step size and iteration for data
with medium correlation
Hi
10. Results of norm experiments for filter order = 45
4.13
as a function of step size and iteration for data
?.4
with high correlation
, s',,,, > , # wxv >'S% .
lal&SUU lifllllll :
i
figure 4.14 The experimental data catalogued in this chapter is
summarized above.


38
¡MM
Design Factor
Yes/No
Comments
1. Spatial Coherence
No
Not encoded on CD
2. Temporal Coherence
No
Not encoded on CD
3. Magnitude Response
Yes
4.Phase Response
No
Not pyschoacoustically
important
5. Frequency Resolution
Yes
Must match critical
bandwidths
6.Amplitude Resolution
Yes
16 bits encoded on CD
7. Stability
No
room responses are
usually minimum phase
figure 1.17 The figure demonstrates the design criteria for a
measurement based equalization strategy.


164
H;
m
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0Hz 25 51 76 102 Select Plot
frequency
A?
figure 6.12 OThe waterfall plot for CD data in band 1 (dc -102 Hz) is
shown above for a 12 second segment of data.


37
by equalization. The coherence properties of the signal received by the listener in the home
listening room will be determined by the listening room itself.
By ignoring the signal coherence properties, the inverse problem is radically simplified.
In addition we have assumed that the listening rooms are minimum phase. Limiting the in
verse problem to minimum phase systems does not significantly reduce the robustness of
the technology. From the principles of room acoustics it is known that for a listening room
to be non-minimum phase it must have a bad acoustic design, e.g. a rectangular cavity with
one axis much longer than the others terminated by walls with very low acoustic resistance.
The assumption of minimum phase has the important characteristic that equalization will
not require poles to be placed outside the unit circle.
The filter design problem has been reduced in scope to removing gross coloration ef
fects of the stereo electronics and the listening room at a sufficiently high level of spectral
resolution that it matches the characteristics of human hearing. The summary of psycho
acoustics indicated that the phase response of a system does not affect our perception of
pitch. The filter designer is left with the simpler task of calculating the inverse filter which
inverts the magnitude response of the listening room at a resolution of approximately 1/3
octave. The following figure demonstrates the simplified set of design criteria.


BIOGRAPHICAL SKETCH
Jeffrey James Spaulding was bom on March 22nd 1960 in Sanford, Florida. In
May 1982, he was graduated from Cornell University with an A.B. in mathematics, and
commissioned as an officer in the U.S. Air Force. While on active duty in Dayton,
Ohio, he received the M.S. degree in systems engineering at Wright State University.
In August 1987 he enrolled in the Electrical Engineering Department of the University
of Florida to obtain a Ph.D. degree. Since then he has worked with Dr. Jose Principe on
various topics in adaptive signal processing at the Computational Neuroengineering
Laboratory.
220


151
6.5. Every time the error signal changes sign, the value of ji is reduced by a factor. If the
error signal does not change for a number of successive iterations of the algorithm, the step
size is increased by a factor. This algorithm is implemented in the highest band only.
The DLMS algorithm parameters are initialized in the upper right hand quadrant of the
tableau. An On/Ofif switch is provided in order to toggle the highest frequency band be
tween the DLMS and the power normalized LMS algorithms. The number of divergent
time sequences used to approximate the ensemble average of the divergence process is in
put (refer to figure 5.19). The number of iterations each sequence is allowed to diverge, and
the value of y is also input, where p = y/ (N power) and y 1.
A method must be provided to indicate that the input signal statistics have changed suf
ficiently to trigger execution of the DLMS algorithm in order to update the value of X .
Parameters for the DLMS algorithm trigger are input in the lower right quadrant of the tab
leau. If the percent error of the equalized signal exceeds a given threshold value (value of
high error) for a given number of points (# of high points to trigger), and without a mini
mum of improvement in the signals exceeding the threshold error (minimum improve
ment), the DLMS algorithm will be triggered. The algorithm can not be invoked, however,
if it was executed less than a given threshold number of iterations previously (minimum dis
tance to retrigger). A block diagram is shown in figure 6.6.
The allign signals option finds the proper delay for the CD signal at the equalizer. Be
cause there is a propagation delay for the audio signal transmitted through the listening
room, the microphone signal will be delayed with respect to the desired CD signal, which
has a direct electrical path. An hardware implementation would require storing the CD data
in a buffer while waiting to acquire the corresponding microphone data. The delay time for
an ordinary living room is approximately 10 msecs, and at the CD sampling rate of 44.1
kHz, a buffer would be required to hold approximately 500 samples of data. A simple al
gorithm is used to approximately determine this delay time. Background levels are deter
mined for the first 50 samples of data when the music is off. A threshold is given for both
the microphone and CD signals. When a music signal exceeds the threshold times the back
ground level, the first local maximum is located. These local maxima are assumed to be the


214
figure A. 12 The ECVT signals plot is given for band 4 (3675-22050 Hz) for the third
11.33 msec epoch of acoustic data.
figure A.13
The ECVT signals plot is given for band 4 (3675-22050 Hz) for the
fourth 11.33 msec epoch of acoustic data.


69
figure 3.1 The minimum of the error surface is at the origin of the
translated coordinate system. The parabolic nature of the
surface makes gradient descent algorithms feasible.
Let ji represent a variable which regulates the step size of the weight adjustment. Gra
dient descent(GD) methods can be generalized as follows.
wk+i =
(3-19)
Figure 3.2 graphically demonstrates how GD methods update themselves on the basis of
past information.


figure 6.25 Learning curves for a 136 msec epoch of band 3.


216
figure A. 16 The ECVT signals plot is given for band 4 (3675-22050 Hz) for the
seventh 11.33 msec epoch of acoustic data.
figure A. 17
The ECVT signals plot is given for band 4 (3675-22050 Hz) for the
eighth 11.33 msec epoch of acoustic data.


27
mining the unique sound of a particular instrument. Amazingly, our hearing system is able
to pick out these cues even when many instruments are orchestrated together.
Perhaps the most problematic factor for which to account is the level of masking caused
by certain frequency components of a waveform on others. Masking is the degree in which
the threshold of audibility of a signal is raised for non-harmonically related frequency com
ponents in the same signal, or in a completely separate waveform. Masking occurs when
one signal has frequency components which are close to the frequency content in another
signal. It has been clearly demonstrated that lower frequency components more effectively
mask higher frequency components than vice versa. These effects can be explained by
physiological phenomena. Lower frequencies produce harmonics which stimulate the re
gion of the basilar membrane sensitive to higher frequencies, thereby masking the sound of
higher frequencies. Figure 1.10 shows the masking effect of a 400 Hz sinusoidal at 60 dB.
The threshold increases most dramatically in the region slightly above 400 Hz.
to the 400 Hz waveform will raise the audibility threshold of
higher frequencies which stimulate the same region of the
basilar membrane [5].


CHAPTER 2
AN EQUALIZATION STRATEGY FOR REAL-TIME SELF-ADJUSTMENT
Introduction
Chapter two will introduce the current state-of-the-art equalizer technology. It will be shown
that this technology does not make efficient use of an equalizers limited degrees of freedom. Be
cause of the finer frequency resolution required in the lower frequency bands, high filter orders are
required to control the small critical digital frequencies. In the high frequency bands however, the
spectral resolution will be finer than necessary. The Adkins-Principe architecture significantly im
proved the traditional strategy by introducing a multi-rate octave filter bank. By downsampling
data in the lower bands according to the Nyquist limit, the critical digital frequency requirements
are made proportionally less severe, and the required inverse filtering can be performed by filters
of significantly lower order. As this is the basic approach of the technology introduced in this dis
sertation, a complete description of this technique will be reviewed in chapter two.
While this strategy is a significant improvement, it does not make use of the time varying spec
tral distribution of the source. The Adkins-Principe strategy continues to assign the limited degrees
of freedom available in the inverse filters to equalize the entire audio bandpass, even if the source
has no power in large spectral regions. It furthermore continues to equalize regions of the bandpass
where little or no coloration is being induced by the stereo electronics or the listening room. The
approach investigated in this work is to add into the Adkins-Principe architecture adaptive filters.
Adaptive filters by their very nature will assign the most processing resources to regions of the au
dio bandpass in which the most coloration exists, significantly improving the efficiency and perfor
mance of equalization. A brief description of the major classes of adaptive algorithms is included.
Because of the requirements of real-time processing, as well as the requirement of high fidelity,
appropriate classes of adaptive algorithms are limited. Because of the low computational complex
ity, and satisfactory performance the Least Mean Squares (LMS) algorithm is chosen for further
investigation in this research.
39


56
figure 2.12 The timing diagram indicates the efficiency of
microprocessor implementation of the AP architecture.
Decimation and selective sampling allow a staggering of the
convolutions which prevent computational bottlenecks.
chip. This processor performs a multiply and accumulate in one cycle. With one chip ded
icated to each band of the equalizer the number of computations which can be performed
between CD data samples is limited to the following.
# of computations = 44 \k.Hz =
According to the timing diagram above, for FIR filters of order 45,360 multiply and accu
mulate operations are required, i.e. 63.6% of the CPU time.
Theoretical Performance
The AP architecture is an efficient implementation of a filter bank. The audio signal is
broken into octaves using a tree structure. Along the lower branch, a bandlimited and
downsampled signal is equalized with a highpass filter. Along the upper branch the signal
is further bandlimited and downsampled according to the Nyquist limit. After equalization
the octave bands are brought to the original sampling rate by upsampling, and using an in-


CHAPTER 6
VALIDATION OF CONCEPTS
Test Plan
The theory and design of the real-time self-adjusting equalizer has been advanced in chapters
four and five. Chapter six will discuss the validation methodology used with actual acoustic data.
The test plan will be discussed in detail in this section. As a result it will be obvious that a valida
tion tool is required. The software developed for this purpose will be described. Results obtained
by applying the test plan and the validation tool to music signals will be documented.
Architecture
Since the scope of this dissertation does not include hardware implementation of the proposed
equalizer design, a software simulation must be formulated in order to test the major propositions
of the preceding chapters. The data available for the validation consists of the CD signal, and the
unequalized signal collected by a microphone at the listener location. An off-fine equalization
must be performed in such a manner that it is a theoretical equivalent to on-line equalization. The
key to finding an equivalent off-line process is to take advantage of both the linearity of the stereo
electronics and the listening room, and the commutivity of convolution for a cascade of linear sys
tems. This will be pursued to avoid the problems of real-time implementation. Consider the fol
lowing electo-acoustic circuits.
The electro-acoustic circuit in figure 6.1a illustrates the proposed real-time self-adjusting
equalizer. A user-defined frequency response is input according to a particular musical taste, and
the CD signal is filtered accordingly. Note that the adaptation algorithm receives this signal direct
ly. This signal acts as the target (desired) signal for the adaptation process. A further filtered signal
is generated as a cascade of approximately linear systems, namelytered CD signal.the CD player,
equalizer, amplifier, and listening room. Equalizer filter weights are updated according to equa
tions 3-19 and 3-66. Figure 6.1b illustrates the equivalent electro-acoustic circuit used to validate
the proposed algorithms of chapters four and five. Note once again that a CD and error signal are
sent as inputs to the adaptation algorithm. The only modification made to the circuit is the order
146


80
The matrix equation can be further simplified as follows.
n
Excess MSE =
5*
f2
^ n'k-j
1
Ly = 0 J.
(3-59)
Equations 3-36,3-38,3-39, and 3-59 provide a clear indication of the trade-off between
misadjustment and convergence speed. The smaller is the step size the smaller is the mis-
adjustment. Below a certain upper bound, however, convergence speed is slowed. Figure
3.4 illustrates the implications of equations 3-60 and 3-61.
lim Excess MSE = 0 (3-60)
)i > 0
lim T =
pi > 0
(3-61)
Adaptation in a Ouasi-Stationarv Environment
The adaptation process for quasi-stationary input data, while conceptually simple to un
derstand, is difficult to describe analytically. The optimal weight vector is now a function
of time, i.e. W* = f(k), as the error surface changes shape and moves through a n+1 di
mensional space for a filter of order n. Widrow has an analytical description of the optimal
step size for the LMS by examining the case of modelling an unknown system by an LMS
tap delay filter [21]. The unknown system is assumed to be a first order Markov process
with constrained to be a constant.
The more general problem for step size optimization is unsolved. Since the weight vec
tors are adapted on the basis of past information, and the error surface is changing shape
and moving with time, the weight vector error results from two contributions. The first
contribution results from noise in the estimation of the error surface gradient. The second
contribution to error results from estimating the gradient of the error surface at iteration k


201
Chapter two concludes by evaluating the available adaptive signal processing algo
rithms for the audio equalizer application. Widrows LMS algorithm is chosen because of
its excellent convergence speed and low computational complexity.
Chapter Three
Chapter three reviews the theory of the LMS algorithm for uncorrelated input data. The
discussion starts by describing Wiener-Kolmogorov theory, which gives the optimal solu
tion of equalizers operating on stochastic signals. It is shown that the error surfaces gener
ated by varying the filter weights are parabolic, and thus gradient search techniques are
feasible. The convergence speed, and excess mean square error of such algorithms are dis
cussed for both stationary and non-stationary environments. It is shown that increasing the
speed of adaptation results in a higher excess mean square error after algorithm conver
gence.
To alleviate this trade-off, an adaptive step size is proposed. When a significant change
occurs in the error (desired signal minus equalized signal), it is assumed that the error sur
face has moved due to the non-stationarity of music signals. Because the equalizer filters
are no longer operating with proper filter coefficients, and the largest contribution to this
error is caused by lack of convergence speed, the step size should be re-initialized to its
largest value.
Finally chapter three focuses specifically on the LMS algorithm. The work of Horowitz
and Senne [2], which describes the convergence of the mean square error, for uncorrelated
input, is reviewed. Special attention is given to the formulation of the optimum step size.
Chapter Four
Because of the short period of stationarity for audio data, the LMS algorithm must con
verge quickly to an optimum solution, if the audio equalizer is to provide a meaningful im
provement to the state-of-the-art. Unfortunately, acoustic data is not uncorrelated, and the
theory of the convergence of the mean square error must be extended. Instead of assuming
the lack of correlation, the convergence properties were studied by assuming that the adap-


53
figure 2.9 Operation of the lower branch of the tree structure.
a) User defines desired response; b) Frequency response; c)
inverse fourier transform for time damson representation, (d)
For filter coefficients multiply time domain representation by
a Kaiser window to lessen degradation induced by Gibbs
phenomena.
is interpolated to bring it to the sampling frequency of the preceding band. Interpolation
performs the opposite operation of decimation by inserting M-l evenly spaced samples of
value zero between every pair of data points.
= x [kT] if k=n/M=an integer (2-4)
0 otherwise
Interpolation acts to recompress the digital frequencies of the equalized time series in man-
*
ner precisely opposite to the decimation process. These effects are demonstrated in the fig
ure 2.10.
Images created by interpolation must be removed with an anti-aliasing lowpass filter.
K K
Note that the digital cutoff frequency is and not r because of the interpolation. -
M M2
The lowpass filter should have a gain of M to restore the energy lost in the decimation stag-
e.The output from the anti-imaging filters are summed with the output of the preceding
band efficiently by using the inverse tree structure shown in figure 2.11.
nT
M.


34
iii. The reflected energy must have components with time delays of less than 100 msecs.
iv. For early reflections, the field must be spatially incoherent, with components
arriving from the lateral directions being of primary importance.
These requirements are generally met in large rooms, but are increasingly difficult to en
sure in smaller and smaller rooms. It has been shown [6] that spatiousness is independent
of the style of music being performed.
The dynamic range of human hearing is 140 dB. If our hearing were more sensitive it
would not be beneficial as we would start hearing the Brownian motion of air molecules.
In processing music digitally, ideally a sufficient number of bits are available to ensure cov
erage of the entire dynamic range of musical works at a resolution consistent with our phys
iological capabilities. The standard CD format provides 16 bit encoding of amplitude
information (approximately 90 dB). Equalizer technology should aim to provide no further
degradation in dynamic range than the limit defined by the CD encoding.
Consequences of Room Acoustics and Psvchoacoustics
in the Solution of the Inverse Problem
Several important inferences can be drawn about potential solution methods to the room
inverse problem from the discussion of room acoustics and psychoacoustics. Two obvious
approaches are available as solution methods: 1) a theoretical modelling of the physics of
the listening room acoustics, and 2) experimentally measuring the characteristics of the
room.
The first approach has been briefly reviewed in the discussion of room acoustics. The
key acoustic parameter, complex acoustic impedance, must be carefully measured for each
material present in the listening room. Using these impedances and the usually complex
geometry of most home listening rooms, the Helmholtz equation must be solved. Note that
matching boundary conditions will be impossible to perform analytically, and will require
a numerically intensive calculation using a finite elements algorithm. As a result of the
computations, a set of room eigenvalues and eigenmodes will be determined, from which
the rooms Greens function is determined. Even for the case of a simple rectangular room,


210
figure A.4 The ECVT signals plot is given for band 2 (102 612.5 Hz)
for the first 0.4000 second epoch of acoustic data.
figureA.5
The ECVT signals plot is given for band 2 (102 612.5 Hz)
or the second 0.4000 second epoch of acoustic data.


90
4-{ 1-4U.X + 12 ii2X2 ) = 0
d\l max ^ mnyf
(3-91)
max'
* 1
* =6X~
max
(3-92)


p reduction s insufficiently
rapid to avoid stability problems
DIMS algonlhn
1
Delay lime for U DI.MS
determination of p can cause
temporary instability to the IMS
algorithm
learn
DLMS algcnllm cunpietcd
and p updated for I MS algunlhin
figure 6.36 When the CD signal changes sufficiently to justify step size
re-initialization, and the DLMS algorithm triggers, a delay
occurs because of the time averaging being performed by the
DLMS algorithm. The LMS algorithm continues to use an
inappropriate step size.


Pages
184-185
Missing
From
Original


217
figure A. 18 The ECVT signals plot is given for band 3 (102 612.5 Hz)
for the ninth 0.4000 second epoch of acoustic data.


107
to higher filter orders. It will be an important result if a single value of y can be chosen with
which to initialize the step size, given that the maximum eigenvalue is known. This will be
shown to be an essential feature for the robust operation of the Divergent LMS (DLMS) al
gorithms determination of A. and p* (see chapter 5).
Tests were run for a filter order of 10, and the results are summarized in figure 4.9. The
left- hand plots show the eigenvalues of the input data for several different degrees of input
correlation. The degree of eigenvalue spread is directly related to the level of correlation
in the data. For a set of eigenvalues of approximately the same value, the error surface of
the LMS algorithm will be approximately symmetric and data will be uncorrelated. For in
put with a dominant eigenvalue, data will be highly correlated. The right-hand plots show
£ as a function of k and y. Results indicate that y remains relatively constant over a wide
range of input data correlations, and the value remains close to the case for uncorrelated
data, i.e. 7=0.166666
When acoustic data were equalized with y= 0.1 for filter orders of 45, the LMS algo
rithm diverged approximately every 500 iterations. This rate of failure was higher than
could be explained on the basis of a prediction error of (see chapter 5). This is to be
expected because of the implications of the Shi-Kozin theorems. As the filter order increas
es, the domain of p contracts, and y decreases. Norm experiments were performed for or
der 45 systems using a time-averaged equivalent of 100 identically distributed systems, and
the results are presented in figures 4.10 to 4.13. Figure 4.10 plots the eigenvalues of the
input data for several different degrees of input correlation. Figure 4.10a is a three-dimen
sional plot of the logarithm of each eigenvalue for each of eight sets of input data. Figure
4.10b superimposes the logarithm of the eigenvalues for the eight test cases. Figure 4.11
plots the norm of (p) as a function of k and 7 for the three most uncorrelated sets of test
data. Figure 4.12 plots the norm of n| ( p) for the next three most uncorrelated sets of test
data. Figure 4.13 plots the norm of (p) for two highly correlated cases. On the basis
of the simulations, the adaptive equalizer, which operates with order 45 adaptive filters, 7
was set to 0.05 (see chapter six). Algorithm failure rates were reduced to 1/500,000 itera
tions of the LMS algorithm.


171
9.822041
Maxim
7.418776
5.015511
MinTime
0.20898
612 Select Plot-
f
figure 6.18 The waterfall plot for microphone data in band 2 (102-612.5
Hz) is shown above for a ten second segment of data. Note
that the features indicated by boxes 1 and 3 are largely
attenuated, and the feature in box 2 has almost completely
disappeared. However, the spectral features in box 4 are still
clearly defined.


97
Theorem (Shi and Kozin): For p D = (0,2), and =
/ P
Y Y7'*'
AfcA
F*|2J
AAno j A j I < 1 if and only if dim [Xv X2, ...,XnQ] = N, where
[Xj, X2, ...,Xno] is the space spanned by the vectors Xv X2, ...,Xno, and N is length of each
vector.
nK
< 1.
Thus for the LMS algorithm, if D = (0,2/M ) where M = max\\X\\, then
if n0 exists to satisfy the above theorem (obviously n0 > N). If {X} is completely uncor
related, then n0 = N and as the input data becomes more correlated n0 becomes larger.
Two important properties of the behavior of the LMS algorithm can now be inferred.
Conclusion #1: If the Shi-Kozin theorems are satisfied, then as the filter order of the LMS
algorithm increases, the domain D of p becomes increasingly restricted. For if the filter
order increases, then M = max ||X|| increases, and D = (0,2/M2) contracts. This result
is verified in experiments which are described in a following section.
Conclusion #2: The number of vectors, n0, necessary to satisfy the condition
dim [Xv X2, ...,Xno] = N will increase as the correlation between subsequent vectors in
creases. This is intuitive, because correlated vectors will tend to point in approximately the
same direction. If nQ is held fixed, and data correlation is increased, the domain of |i must
necessarily contract. This result is also verified in the experimental results which follow.
Geometric Interpretation
A geometric interpretation of the norm of IITT will validate the intuitive understand
ing of the domain of p for which £ converges, as a function of filter order and input data
correlation. Although the size of the derived domain which follows may present a pessi
mistic upper bound, it will nevertheless provide insight into the general behavior of the
LMS algorithm. As a first step, equation 4-6 can be simplified by expressing the matrix
products as matrix norms. The following step assumes that the input, and linear operators
are bounded. Note that only || OTI| depends on p. To decouple the expectation of the
tk~vE{nrXXTTl}VQ
V^{||nrn||||xxr||}V0
(4-8)


figure 6.9
500 point segments. The learning curve and step size value are also plotted.


172
figure 6.19 The waterfall plot of data equalized in band 2 (102-612.5 Hz)
is a more important indicator of the efficacy of the
equalization than time domain measures. Note that the high
frequency information which was substantially reduced in
the microphone data is largely restored. Spectral features in
boxes 1,2, and 3 are enhanced in order to match their relative
strength as shown in the CD waterfall plot. However, the
features in box 4, especially towards the ends, are not as
sharp as the same features in the CD waterfall, and even the
microphone waterfall plot.


41
State-of-the-Art Acoustic Equalizers
It is clear from the preceding section on the physics of room acoustics that a theoretical
modelling of a listening room, even with an unreasonably simplistic geometry and con
structed with materials of uniform impedance, is such a computationally burdensome en
deavour that it can not be performed for an individuals listening room. The current
equalizer technology seeks to excite a room with broadband noise in order to stimulate a
broad range of eigenmodes. This signal is measured at a listener location or locations, and
an inverse of the cascade of the home stereo system and the home listening room is estimat
ed. Equalization is not a panacea which will restore a perfect reproduction of the pressure
field as propagated by the original rendition of the musical performance. In fact, acoustic
equalization can not even reconstruct a pressure field which is psycho-acoustically equiva
lent. Current technology attempts only to remove gross coloration. The strategy employed
is depicted in figure 2.1. Limitations of this strategy will be examined, with special empha-
1
sis on those items which the proposed adaptive equalizer will ameliorate.
noise
source
bandpa
filters
* equalizer
home
electronic;
¡' ~ Speakers
lstenme
room
filter
coefficients
micr
phor
o-
ie(s)
1
r
bandpass
filters
band by band
comparator
figure 2.1 State-of-the-art equalizers are designed as a parallel filter
banks with filter coefficients that are adjusted by comparing
the power collected in each band by a microphone located in
the listening room, with the noise which is used to excite the
room.


155
figure 6.7 A delay is necessary for the CD signal to be aligned with the
microphone signal, because of the propagation delay of the
acoustic data through the listening room. The algorithm is
described by the block diagram shown above.
with each band. External NeXT .SND files are created for the microphone, CD, and equal
izer signals in each of the four bands, and the reconstituted audio band. Listeners can eval
uate the quality of the audio signals using these files.
The summary option opens a non-editable ASCII file containing information about the
acoustic data collection, i.e. the music selection, the collection site, and collection geome
try. The digital tape recorder interacts with the DSP on the NeXT to collect data via the
digital microphone, and to playback the recording via the NeXT sound drivers. The hide
and quit options push out of view all ECVT panels, and quit the program, respectively.
Evaluation Criteria
A quantitative examination of the equalizer will be performed by examining time do
main and frequency domain characteristics of the CD, microphone, and equalized signals.


3
tive equalization. The standard technique for estimating ^max is to assume that it is approx
imately equal to the power in the input vector times the filter order. From systems theory
we know that the power is the sum of all eigenvalues, and thus, is less than ^max. This es
timation is excellent under certain restricted conditions, e.g. low filter orders and high ei
genvalue spreads. No such strong statements can be made more generally of this
approximation technique: A new method of approximating the maximum eigenvalue of a
system by allowing an adaptive gradient descent algorithm to diverge is introduced in chap
ter five. The method is first developed for the stochastic gradient descent algorithm which
is based on the statistical properties of an ensemble of independent, identically distributed
systems. The gradient descent algorithm will provide a theoretical framework from which
to analyze the more complicated LMS algorithm which is based on the time-averaged prop
erties of a single system.
Chapter six discusses the experiments used to test the equalizer architecture and the al
gorithms used for real-time self-adjustment. The NeXT computer with a digitizing micro
phone is used to collect and process data. The software used to simulate real-time
equalization with the proposed algorithms is discussed. Finally the test and evaluation re
sults are presented.
Chapter seven summarizes the results of research presented in the dissertation. Specif
ically it reviews the success of the new architecture proposed for real-time self-adjustment
of an audio equalizer. It reviews the theory of the convergence and divergence properties
of the LMS algorithm with statistically correlated input data. It reviews the theory and per
formance of a new variation of the LMS algorithm which provides an estimate of the max
imum eigenvalue of the input vectors into the equalizer filters. Finally chapter seven
discusses the many areas of this research that could be extended
Because the physics of room acoustics and the principles of psycho-acoustics motivate
many of the key design characteristics of the acoustic equalizer, the remainder of chapter '
one will provide background information on the pertinent areas of these fields.


215
figure A. 14 The ECVT signals plot is given for band 4 (3675-22050 Hz) for the fifth
11.33 msec epoch of acoustic data.
figure A. 15
The ECVT signals plot is given for band 4 (3675-22050 Hz) for the sixth
11.33 msec epoch of acoustic data.


SLOW CONVERGENCE
divergent iteration
V //
figure 6.31 The ECVT signals plot shows the effects of operating the
LMS algorithm with a conservative initialization of step size,
namely p = 0.02/A.max. The algorithm converges so slowly
that it can not keep up with the nonstationary character of the
acoustic input.


APPENDIX
ECVT PLOTS OF CONTIGUOUS EPOCHS OF AUDIO DATA
This appendix presents several contiguous epochs of audio data presented in ECVT plots for
bands one through four. A better representation of the equalizers effectiveness can be determined
by observing a larger epoch of equalized data. Recall that data is downsampled by different rates
in each band, and thus epochs represent less time in the higher frequency bands than the higher.
Special emphasis is placed on data equalized in band four because of the more sophisticated pro
cessing performed using the major ideas presented in chapter four and five.
Figures A.l through A.3 present two contiguous 0.500 second epochs of data from band 1 (dc
-102 Hz). Figures A.4 through A.6 represent three contiguous 0.400 second epochs of data from
band 2 (102-612.5 Hz). Figures A.7 through A.9 represent three contiguous 68 millisecond epochs
of data from band 3 (612.5 3675 Hz). Figures A.10 through A.18 present seventeen contiguous
11.33 millisecond epochs of data from band 4 (3675-22050 Hz).
figure A. 1 The ECVT signals plot is given for band 1 (dc -102 Hz) for the first
0.5000 second epoch of acoustic data.
208


figure 6.28 The ECVT signals plot is given for an 11.33 msec epoch of
acoustic data from band 4. Step size is being initialized
using |i = 0.10/A^ax. While the LMS filter coefficients are
convergent in the mean, the variance of the square error is
unacceptably large.
00
U)


143
according to equation 5-12, and the adaptive filter is reset to an initial condition.
Summary and Conclusions
This chapter has demonstrated the robust operation of gradient descent algorithms
which estimates The estimates are based on measurements obtained by driving the
algorithms into divergence with a sufficiently large step size. The results of chapter 5 are
summarized below. Figure 5.22 reviews the effects of possible sources of estimation error
for The variable, its effect, and figure reference numbers are provided.
i) The simulations of the divergent gradient descent algorithm indicated good results
over a wide range of step size, and eigenvalue spreads. This was interpreted as an indication
of the likely success of a diverging LMS algorithm.
ii) In the derivation of the DLMS algorithm, two simplifying assumptions were made
in order to express the ratio (3 = Ck^k-1 35 the polynomial (3 = l-4|j.Amax+12^iX2max. It was
demonstrated that regardless of the eigenvalue spread of the input data, a value of p. could
be found guaranteeing that the simplifications induced an arbitrarily small error.
iii) Simulations were performed indicating that for any degree of eigenvalue spread, a
value of p could be found guaranteeing an arbitrarily small approximation error for X^^.
iv) Simulations were performed indicating that for any degree of initial misadjustment
along the principal axes of the error surface, a value of p, could be found guaranteeing an
arbitrarily small approximation error for X^^.
v) Simulations were performed indicating that for any filter order, a value of p. could
be found guaranteeing an arbitrarily small approximation error for X^^.
vi) The derivation of the DLMS algorithm utilized the convergence/divergence equa
tion which is valid only for uncorrelated input data. Because of the relative insensitivity of
the divergence properties of £ to correlation until a>0.9, it is intuitive that DLMS algorithm
would be little effected by correlated input data. Simulations verified this observation.
However, the key parameter of the algorithm, p, which was found to be even more insensi
tive to correlation, guarantees good performance.


Ill
figure 4.11
he left-hand plots give the three dimensional plot of
nk (y) as a function of k and y, where ji = yA^ax, for the
three most uncorrelated sets of test data. The right-hand plot
superimposes all cross sections of the surface plots. For tests
one and two, y was limited to 0.05 to avoid algorithm
divergence.


86
lim
For 0 = W* in the mean (3-71)
max
Consider next the variance of the estimate. Assume that after k iterations Wk has con
verged. Then the gradient estimate is just the noise process Nk.
(3-72)
(3-73)
For j > k, Nk = 2e^Lk. After convergence ek and Xk are approximately independent.
cov [V£] = cov [Nj] =
= 4E [ejXjXj] = 4E [ej] E [.XjXf ]
= 4 ZjRj
(3-74)
(3-75)
(3-76)
The above does not imply that the variance of the estimate remains finite, or that the mean
square error remains finite. A considerably more stringent requirement on step size is nec
essary, and is discussed below.
Conditions on [i for Finite Variance for Square Error
Horowitz and Senne[2] have studied the convergence properties of the LMS mean
square error for the case of Gaussian, zero-mean, uncorrelated data. Let A. = ^ Xl ...
and Gk be the diagonal elements of Zs[ Vt Vt .
= E[(dk-X¡Wk)2] = El(dk-X¡W*) -X¡(Wk-W*)]2
= C-lElxliW^W*) (dk-XTkW*)] +
E [(Wk W*) TXkxl(Wk W*) ]
(3-77)


88
4 Vfjt+1 Vf[+1 ] = e[ vfkvf [] 4nA£[ vtk[] + (3-84)
4u2(£[^[vttVt[xft^]) +4^A
Apply the Gaussian factoring theorem to the third term of equation 3-84.
e[xtkXtfyirfrtiXtl] = 2A2E[vtk\fiTk] + <3-85)
trace {A£[vt*Vt[]}A
Substitute the decomposed moment into equation 3-84.
n
E^t+i^k+l] = E[V¡kV\k]-4\lKE\y¡kV (3-86)
+
8h2A2[ VtjfcVtJ] + 4¡l2trace { Afi[ VfjVtf] } A + 4|I2C*A
can be decomposed as follows.
Ck+1[i,i] = (1 4M-A.,. + 8ii2X2) Ck [, i] +
n
4H% £ XpCjfc [/. /] +4h2£*X.
/7 = 1
(3-87)
A recursive formulation is now possible for CT. Let F [z, /] = 1 4pX(. + 12*l2A,? and
F[i,j] = 4*i %Xr
ert+iTO*+4*12^
(3-88)


165
figure 6.13 The waterfall plot for microphone data in band 1 (dc -102
Hz) is shown above for a 12 second segment of data.


104
LMS iteration
figure 4.5 For a filter order of 2 the approximation given by equation 4-
6 works well for data with a wide range of correlation. The
solid line represents C,k created from an ensemble average of
square error (100 independent identically distributed
systems) generated by running the LMS algorithm. This plot
is overlaid by a dashed curve which gives the predicted mean
square error from the approximation expressed in equation
4-6.
To investigate the performance of equation 4-18 as a method to estimate p*, consider
a first order AR process. For no correlation, we know from the results of Horowitz and
Senne that p* = (1 /6) ^max, and for complete correlation, the LMS algorithm reduces to
adaptation along one axis only, in which case p* = (1/2) ^max- The matrix norms are
shown below for both of these extreme cases for a second order tapped delay line. Since
the input signal is ergodic, a time average replaces the expectation operator in equation 4-
18. The equivalent of 100 independent identically distributed experiments were performed/
The results are shown in figures 4.6 and 4.7.


106
The value of y for correlations from 0.01 to 0.99995 are shown in figure 4.6 for a filter
order of two, where p* = j/Xmax. Notice that the values of p* based on the norm approx
imation are very similar to the results obtained from ensemble averages of LMS experi
ments (100 identical independently distributed systems).
figure 4.8 The results of the determination of p* by equation 4-18
agree with large ensemble averages of the LMS algorithm
for a wide range of input data correlation. Note the relative
insensitivity of y to a.
The key feature to note from figure 4.8 is the relative insensitivity of y to the level of
input data correlation, until the correlation becomes very high (correlation > 0.99). Data
correlations of this magnitude should not occur when the step size is being initialized. Step
size initialization will occur only when the error surface has moved dramatically, which
will by definition decorrelate the input. These results at first seem to contradict the Shi-
Kozin theorems. Increasing data correlation increases the domain of p for which £ con
verges, and hence one would expect y to increase. Note however, that for our autoregres
sive model, X = 1 + eigenvalue spread-1 ^ ^ = t eigenvalue spread -1 Ag d alue
max eigenvalue spread + 1 mm eigenvalue spread + 1
spread increases, Xmax more closely approximates lixil2. This approach is non-linear and
effects the domain of p in the opposite way of increasing correlation. These competing
forces balance to create a relatively constant value for y as a function of input correlation,
until a very high level of data correlation. In the following experiment results are extended


142
figure 5.21 As the adaptive equalizer continues to adapt with a non-optimal step size,
a separate version of both the CD signal and the microphone signal are
sent to the DLMS module. The signals are once again broken into bands
using the adaptive AP architecture.
sired signal is removed. The CD signal is input to an adaptive tapped delay line. The mi
crophone signal acts as the desired signal. The choice of desired signal is immaterial as the
eigenvalues and the divergence characteristics of interest are a function of the input signal
only. The adaptation algorithm is operated with a sufficiently large step size, (4. A series
of n adaptive filters diverge for k iterations. At the kth iteration the k-cycle clock passes
the ratio [mic(k) cd (k)TW (k)] to a module which calculates ¡I
[mic(k- 1) -cd(k-\)TW(k-l)]2


125
Eigenvalue Spread = 1.01
Filter Order = 2
% error
of estimate
Eigenvalue Spread = 1.10
Filter Order = 2
Eigenvalue Spread = 1.10
Filter Order = 2
% error
of estimate
Eigenvalue Spread =1.18
Filter Order = 2
Eigenvalue Spread =1.18
Filter Order = 2
% error
of estimate
figure 5.6 Surface plots and contour plots of estimation error as a
function of step size and divergent iterations for different
levels of eigenvalue spread.


108
CQnglygiQnS
Several interesting conclusions are apparent as a result of this study, including the rule
of thumb used to initialize p for the LMS algorithm. The results have been verified only
for input data that can be modelled as an autoregressive process.
i) A domain for p can be found which guarantees that £ will converge exponentially al
most surely. This bound forces the relaxed projection operators which constitute n to con
tract on average.
ii) The domain size for p, and the value of p for fast convergence are a function of filter
order.
iii) The domain size for p, and the value of p for fast convergence are a function of input
data correlation Because p is normalized with respect to A^^ instead of II xil2, the effects
of better approximating || xil2 and the increasing correlation of input data tend to can
cel, allowing for a relatively constant value of y for low filter orders. These competing forc
es largely obviate the need for a specific understanding of { A,f} in determining y for low
filter orders.
iv) The best value of p for fast convergence of is the value which minimizes linil2.
v) For filter orders of less than ten, norm experiments suggest that for 0.1 convergence will result for a wide range of input data correlation. This result is in good
agreement with the rule of thumb setting of p* = O.l/A^^ commonly used in adaptive
signal processing.
vi) The adaptive filters for the equalizer are of order 45. Based on the simulations of
chapter four, y will be set to 0.05.
vi) Figure 4.14 summarizes the results of experiments presented throughout chapter
four.


196
figure 6.38
The waterfall plot for microphone data in band 4 (3675 -
22050 Hz) is shown above for a 1.8 second segment of data.


124
figure 5.5 Estimation error of £ is plotted above as a function of
divergent iterations, k, and step size, y, where |i =
for a filter of order two and input data with an eigenvalue
spread of 1.18. The contour plot is shown below.


48
The Adkins-Principe Equalizer Architecture
The Adkins-Principe (AP) architecture [1] uses a multi-rate filter bank to produce an
octave band structure by repeatedly filtering and decimating a single input signal. A sepa
rate highpass filter equalizes each band, making the processing computationally parallel
and greatly reducing the necessary microprocessor speed for real-time operation. The ad
vantage of multi-rate designs lies in their increased efficiency. The elegance of the structure
can be seen in the block diagram shown in figure 2.0. The proposed real-time equalization
strategy will be based on this architecture. The following section will discuss its character
istics. The advantages of this architecture over the current state-of-the-art will be made
clear. Its operation will be shown to be consistent with maintaining pitch as described in
chapter one. In addition, the features of the architecture which must be carefully controlled
to assure proper signal reconstruction will be discussed.
Octave Bands Generation and Equalization
The highest frequency band is controlled by passing the original signal through a high-
pass filter. All lower bands are controlled by a cascade of lowpass, decimation, and high-
pass filters. This is accomplished as shown in figure 2.4.
figure 2.4
The tree structure efficiently implements the octave filter bank. At each
node the signal is equalized by a highpass filter along one branch, and
further bandlimited along a second branch.


10
With the expressions for R and T it is clear that the impedances of the materials are the
key acoustic parameter of the material, and it is this parameter which determines a rooms
acoustic properties.
AcQBstic Impedance
Acoustic impedance can be separated into a real and an imaginary part. The real part is
the acoustic resistance, the component associated with the dissipation of energy. The imag
inary part is the acoustic reactance, the component associated with temporary storage of en
ergy. These components are made explicit in the expression shown below.
Z(0,(D) = £ = R+j%
Kg
rrP s
(1-24)
The acoustic reactance can be further factored into an inertance and a compliance term.
The acoustic inertance term is defined as the effective mass of the element under consider-
171 O
ation divided by surface area, M = , with units of Kg/m The acoustic compliance is
5 dx
defined as the volume displacement induced by the application of a unit pressure, C
dp
in units of m4sec2/Kg. A mechanical interpretation of a locally reacting wall is shown in
figure 1.2.
z(0, CO) = R+j((M--^) (i-25)
Let pressure correspond to voltage, velocity to current, resistance to resistance, compli
ance to capacitance, and inertance to conductance. Then an acoustic system can be re
placed by an equivalent electrical system. It is precisely the behavior of these types of cir
cuits that we wish to equalize.


144
vii) Ensemble averages are required to make a true measurement of (3. For ergodic input
data, the ensemble average may be replaced with time averages. For real-time applications,
the time required to perform the time average will be the factor which limits performance.
viii) The DLMS algorithm, when operating in conjunction with the LMS algorithm,
will provide a superior estimate of ^ax, and hence the optimum step size can be set for the
LMS algorithm. It is this concept which is suggested as an architecture for a modification
to the adaptive acoustic equalizer.
Parameter
Effects
Figure
Reference
Limiting
Factor
1. eigenvalue spread
arbitrarily small with proper choice
of p and divergent iterations
5.10, 5.11
no
2. filter initial conditions
arbitrarily small with proper choice
of p and divergent iterations
5.12, 5.13
no
3. filter order
arbitrarily small with proper choice
of p and divergent iterations
5.14, 5.15, 5.16
no
4. data correlation
limiting factor for high data correlation
error < 20% except at very high correlation
5.17, 5.18
yes
5. time average vs.
ensemble average
most serious source of error for real-time
applications, error can be made arbirtrarily
small given sufficient time.
5.19, 5.20
yes
figure 5.22
Possible effects of estimation error are summarized for the
divergent LMS algorithm.


85
figure 3.6 The LMS algorithm approximates the gradient by replacing
the ensemble average of the gradient descent with a time
average. In the time average the LMS gradient will point in
the direction of the surface gradient.
It is a simple matter to demonstrate that the LMS estimate is unbiased.
£[vg =
E [-2 efa]
(3-67)
-2E[Xk(.dk-XTkWk) ]
(3-68)
2RkWk-2Pk
(3-69)
(3-70)
If the convergence condition expressed in equation 3-35 holds, the LMS algorithm will con
verge to W* in the mean.


71
Substitute the solution of the Wiener-Hopf equation (equation 3-10) into 3-21.
Wk+1 = Wk + 2fJt(\f-Wk)
= (I-2\LR)Wk + 2\lRW* (3-22)
Switch to the translated coordinate system by subtracting W* from both sides of equation
3-22.
Wk+l-W* = (I-2\lR) Wk + 2[LRW*-W* (3"23)
= (I-2\lR) (Wk-V?)
Vk+i = V-2[lR)Vk (3-24)
Perform a unitary transformation on R. Let A be the diagonal matrix of eigenvalues of R,
and let Q be the matrix in which the columns represent the corresponding eigenvectors.
This transformation assumes R is not singular.
R = QAQ-1 for A =
x1 0
0
0 0
0 0
0
0

0
0
0
0
X
and
Q =
Q1 ?2
Vn
qi is the eigenvector associated with


52
Band
Frequency
Range [Hz]
# of frequencies
controlled
bandwidth of controlled
frequencies [octaves]
Band 1
3675-22050
19
0.045 0.250
Band 2
612.5-3675
19
0.045 0.250
Band 3
102-612.5
19
0.045 0.250
Band 4
17-102
19
0.045 0.250
*(/ i
u 102
f^M4 /2M3
12 3675
f/2M2 V2M
__fre2uencj^[Hz]__(nottoscale)

f^2
figure 2.8 Frequency resolution for a four band equalizer with filters of
order 45, and a constant decimation rate of 6, is shown
above.
The highpass FIR filters are designed by taking the inverse FFT of the frequency re
sponse requested by the user with the equalizers graphical interface. The filters are
smoothed by truncating symmetrically about time = 0 the resulting time domain represen
tation with a Kaiser window with (3 = 3.5. The symmetric nature of the filters guarantees
linear phase. The time domain representation is shifted to guarantee that the filters are caus
al. A summary of the process is shown in figure 2.9.
Signal Reconstruction
After equalization has been performed with highpass filters, the data in each band must
be summed in order to reconstruct the signal. The summation is complicated because of
the different sampling rates at which the individual highpass filters operate. In addition
there is an unique delay due to the differing number of convolution operations performed
in the various branches of the equalizer. Immediately proceeding equalization, the signal


120
lim
k>
X (1-2 ilk )2*vf?
max
max
= 1
n
i = 0
estimation error of less than 5%. The relationships among p, k, and (Xf) and the error
of the one eigenvalue approximation can be seen most clearly by generating surface plots.
In figures 5.3 through 5.5, the upper plots show the estimation error of ln£ as a function of
y and k, for a filter order of two. The contour plots are shown below. The figures indicate
that even for D ~ 1 a wide range of (p, k) combinations exist at which the one eigenvalue
approximation of C,k is excellent.
The estimation error for XMAX is calculated using equation 5-6, and is shown in figure
5.6 as a function of step size and eigenvalue spread for a second order system. In all cases
the estimation error is less than 0.5%. Figure 5.7 represents stressing test cases for the al
gorithm. A filter order of 100 is considered with ^ = 1.0 and X2 = ^3^100 = For
D = 1.01,2, and 10, Xe^x = tr[R] = 99.0,46.0, and 10.9. For the DSD over this range
of eigenvalue spread, the estimate of ^MAX was significantly better, 0.98 < ^max < 1.01.
Determination of with a Divergent LMS Algorithm
The DLMS algorithm is based on the convergence/divergence equation for uncorrelated
data. Recall from chapter three (equation 3-81) the expression for the mean square error of
the LMS algorithm with uncorrelated input data. Perform the recursion expressed in equa
tion 3-88 in order to express C,k in terms of aQ. Once again assume that C,k+ {
= C + XT[FGk + 4ii2CK
~XTFik+1)GQ + 4\L2ChT
I+ F
I-F
(5-7)


126
Eigenvalue Spread =1.01
Filler Order = 2
% error
of estimate
Eigenvalue Spread = 1.10
Filter Order = 2
% error
of estimate
Eigenvalue Spread = 1.18
Filter Order = 2
Eigenvalue Spread =1.18
Filter Order = 2
% error
of estimati
A
Eigenvalue Spread =1.10
Filter Order = 2
12
.0
o
8

t
as
6
4
2
Y
1.1
1.3
1.5
1.7 1.9 2.1 2.3
figure 5.7 As a stressing test case of the DSD algorithm, the error
surfaces and contour plots for the estimate of X are shown
max
as a function of step size and divergent iterations for three
different eigenvalue spreads with X,j = 1 and


58
figure 2.13 The adaptive equalizer architecture is a modification to the
AP architecture in which highpass filters are replaced by
adaptive filters.
figure 2.14 The adaptive filter is outlined in grey. The filtered CD signal
is compared with the microphone signal similarly
bandlimited an decimated. An error signal is generated and
sent to the adaptation algorithm which updates the filter
weights in such a way that mean square error is minimized.
This strategy will improve the state-of-the-art by ameliorating two of the limitations
discussed in chapter one. Specifically, if the listener changes location (assuming the micro-


102
Thus |i* is the value p which minimizes the maximum of eigenvalue of ITO.
p* = p which minimizes A, (IT^n) on average
= p which minimizes the maximum of
xT(nTn)x
~xrx
on average
(4-17)
Equation 4-17 is not easily characterized analytically. Since IT operates on X, and X is a
stochastic process, p* will be optimal in the average.
In the experiments which follow, X is chosen from the stochastic population of interest,
and p* will be dependent on the statistical properties of the probability space, n. The above
equation is thus further constrained such that Xe Q.
# 2
p pwhich minimizes || nil
= p which minimizes the maximum of E
with Xe Q.
Xe n
(nf(ii) xk)T(nf(n) xk)
xkxk
(4-18)
Experimental Results
In order to understand the effects of correlation, the following autoregressive time series
was used as input into an adaptive tapped delay line, where Zk is a zero mean, Gaussian
random variable, and 0 < a < 1 controls the degree of correlation.
Xk~aXk+l+Zk i4-1^)
As an example of the validity of the approximations made in the derivation of equation
4-6, note the results presented in figure 4.4. Figure 4.4 shows ^ generated form a first or
der autoregressive process with a = 0.9512. The solid line represents C,k created from an
ensemble average of square error (100 independent identically distributed systems) gener-


22
P
d ~
P-P
s
P
X (i-in)ifii2n
n = 1
5>
n = 1
2/i
(1-61)
Diffuse Energy = f(R,S)
figure 1.7 As the reflection constants decreases and the specular
reflection constant increases, the percentage of diffuse
energy in the room increases.
Even more important than wall roughness are the diffusing effects of room furniture and
the deviations of most rooms from a rectangular cavity. Even the presence of people in a
room act to diffuse the acoustic field. Furthermore in the discussion of psychoacoustics it
will be shown that a pleasant listening environment can be created with far less than a com
pletely diffuse field.


109
figure 4.9 The plots on the left indicate the eigenvalues of the input data
for a 10th order LMS filter. The right hand plots represent
the mean square error as a function of k and y, where p. = y/
A.max Note that even for high correlation the value of y
remains relatively near the value for the uncorrelated case,
i.e. y=0.166666


76
From equations 3-19 and 3-42 we conclude the following.
Vi* = 2 RVk + Nk (3-43)
Substitute equation 3-43 into equation 3-41.
V*+l = V*-H(2ifV, + JV,)
= (I-2\lR)Vk-\lNk 0-44)
Switch to the principal coordinate system.
2Vf*+1 = (/-2n)fivVniVt (W5)
vt*+1 = e-1(/-2nff)2Vft-n21A'*
= (/-2(*A) V^k-V.QTlNk (3-46)
Let Q~1Nk = N$ £ be the gradient noise projected onto the principal axes.
Vf*+1=(/-2)lA )Vtk-\lNfk (3-47)
Note that equation 3-47 now represents a set of uncoupled difference equations. The matrix
formulation is given below.


CHAPTER 4
LEAST MEAN SQUARES (LMS) ALGORITHM CONVERGENCE
WITH CORRELATED INPUT DATA
Music signals are not uncorrelated as can be seen from the autocorrelation function for
several musical instruments shown in figure 4.1. To meaningfully discuss step size initial
ization for an adaptive acoustic equalizer, the understanding of LMS convergence proper
ties must be extended to the case of uncorrelated input data. Even though the adaptive step
size will be optimized for fast convergence only when a large change occurs in the error
surface, and hence the most recent data sample will be largely uncorrelated with the previ
ous data, the tapped delay line will have an input vector which, for order n+1, still shares n
components with the previous vector.
Convergence of LMS with Correlated Input
Several mathematicians and engineers have investigated a completely general conver
gence proof for the LMS algorithm without success. To date no satisfactory proof has been
put forward guaranteeing lim < M, Vjx < u The difficulty in a rigorous proof lies
K max
in the assumption of strong correlation, i.e. Vx > 0, £ [XtXt+1] > 0. The theoretical work
done in this area is mathematically sophisticated. Fortunately, Macchi and Eweda [23]
have provided a discussion of the theoretical contributions to convergence problems which
are here summarized.
Lyung [24] has demonstrated convergence almost everywhere for Wk under the condi
tion that the step size is a decreasing sequence tending to zero. He creates a non-divergence
criterion by erecting a suitable barrier on Vk which will always reflect Vk onto a random
compact set. However, he has not shown that lim W, = W*. Daniell [25] has shown that
£ can be made arbitrarily small by choosing ji sufficiently small. However he must use the
assumption of uniformly asymptotically independent observations. In addition he
makesthe restrictive assumption that the conditional moments of observations, given past
observations, are uniformly bounded. This condition is not satisfied even for the case of
91


44
lower frequency bands, which require high order filters. Control of the band at the level of
the difference limen (fig. 1.11) is not achieved. Except in the most unusual circumstances
1/3 Octave Filter Bank
figure 2.3 The critical frequencies of an 1/3 octave bandpass filter are
shown above. Because of the similarities with the resolution
characteristics of human hearing, the 1/3 octave filter bank is
most often chosen for state-of-the-art equalizers.
the difference limen is an unnecessarily strict standard due to the masking effects of music
signals. The precise requirements are difficult to evaluate do the complex nature of mask
ing.
Signal Measurement
Any system which relies on current microphone technology for precise acoustic mea
surement is problematic. A situation analogous to the Heisenburg Uncertainty Principle
arises when measuring the field introduction of the microphone into the field disturbs the


141
averages in the estimation of XTnax=1.3. a) The evolution of
logierror2) for 20 divergent series of five iterations each of
the DLMS algorithm; b) The divergent series superimposed;
c) The approximations of the largest eigenvalue for 50
experiments using the DLMS algorithm and the power
approximation; d) The probability density function of the
DLMS algorithm approximation of
successive estimates of C,k + j/^is adjusted by setting the delays z~l to an appropriate val
ue. Note that there is a trade-off between the quality of the ensemble average of C,k+ X/C,k,
and the time required to make a new estimate of A. Thus the larger the delay 1, the better
will be the ensemble average of e2k+ x/e\, and the longer the delay in updating kmax- The
eigenvalues of the input are the only feature measured in the DLMS algorithm. These
eigenvalues are independent of the desired signal. In addition, given the algorithm is oper
ating in a proper domain for p and k, the evolution of C, depends only on p, ^max, k, and
G0 (equation 5-10). This is so because although ek = dk 2dkyk + yk, yk dk 2dkyk.
Although oQ does depend on the desired signal, since the ratio of succeeding values of £,
namely C,k+ {/^k, is measured, the oQ-dependency, and hence the dependency of the de


81
using the error surface at iteration k-1. Note the formulation for the filter weights for the
non-stationary case, in the translated coordinate system.
Vk+l = Wk~Wk = (Wk-E[Wk]) + (E[Wk] -W*)
(3-62)
Substitution of the above into equation 3-19 yields the following.
C* +, = C+ (Wk-E[Wk])TRk(Wk-E[Wk]) +
(£ [Wk] W*) TRk (E [Wk] W*) +
2E[(Wk-E[Wk\)TRk(E[Wk]-W*)]
(3-63)
Expanding the third term in equation 3-63 and recognizing W¡* is constant over an ensem
ble, the third term reduces to zero, and { reduces to the following.
tSJc+ j = ^* + (error from gradient noise) +
(error from moving error surface)
= (Wk-E[Wk])TRk(Wk-E[Wk]) +
(E lWk] W*) TRk (E [1Vkl W*) (3-64)
Equation 3-64 is as far as one can go without usually unavailable a priori information
concerning the stationarity characteristics of the signal. A graphic illustration of the oper
ation of the LMS algorithm with non-stationary input data is shown in figure 3-5.


137
The Effects of Input Data Correlation
From chapter four (equation 4-17) it is evident that the analytic description of the LMS
algorithm with correlated input data is complex. An important result of chapter four was
the relative insensitivity of y to input data correlation. It is intuitive that a similar result
should apply to a diverging LMS algorithm. The following plot presents the results of a
large ensemble average (100 experiments) of a diverging LMS algorithm with input with
different levels of correlation, a. The input data is modelled as a first order autoregressive
process (equation 4-19).
figure 5.17 The LMS algorithm was operated with a step size
sufficiently large to guarantee algorithm divergence. This
figure gives the logarithm of the mean square error after
taking the ensemble average of one hundred experiments for
each level of input data correlation for a first order
autoregressive model.


205
Recommended Extensions to this Research
Several interesting lines of investigation are suggested by the research presented in this
dissertation. These areas are placed in three general categories: 1) additional theoretical de
velopment, 2) additional testing, and 3) alternative applications.
Additional Theoretical Development
The value of y*. where p* = y*/^max> was found by minimizing IIITIL This minimization
was found numerically. A far more aesthetically pleasing conclusion would be an analytic
expression for y*. The difficulty lies in the complexity of the ri-matrix. An attempt at an
analytic solution was made with the Mathematica software. Even for small order filters,
analytic solutions generated many pages of output. Due to this researchers inability to find
a suitable methodology, research in this area was suspended. Nevertheless, this seems a
most compelling area of study of the LMS algorithm for stationary, correlated input.
Because acoustic data is modelled as an autoregressive sequence, y* was found for data
generated in such a manner. While this approach is adequate for audio applications, there
are situations in which the LMS algorithm is utilized on input data better described by other
models. The results presented for the autoregressive case may yield a completely different
solution to data modelled in other ways. It is recommended that linil be minimized for input
data described by different models, and that these results be compared with those presented
in chapter five.
The value of y* is dependent on filter order. In the absence of an exact analytic descrip
tion a set of curves should be generated describing y* for a wide range of filter order, and
input data correlation.
Feuer and Weinstein [3] found that y* was dependent upon the initial misadjustment
along each of the principal axes of the error surface, for the case of uncorrelated input data.
It seems clear that this will be likewise the case with correlated data. This dependency was
ignored in the research presented in this dissertation, because of the impracticability of
knowing a priori the value of these misadjustments. Nevertheless, these dependencies are


131
using equation 5-12. The approximations of equations 5-8 and 5-9 are reflected in equation
5-12 by approximating P with equation 5-10. The estimate of ^-MAX has an error of less
than 1% over a broad range of step sizes and eigenvalue spreads.
ilStiiiii sl
figure 5.11 The approximation of XMAX has less than a one percent
prediction error over a broad range of step size and
eigenvalue spread.
The Effect of Initial Misadiustment
The analysis of the DLMS algorithm has assumed equal initial misadjustment along
each of the principal axes. For successful operation of the algorithm, it must estimate XMAX


42
After room excitation with a broadband source, current techniques break the measured
signal into octave or 1/3 octave bands using a set of bandpass filters. The power in each
band is compared with the power initially radiated in the same band. The equalizer is ad
justed so that its filter coefficients give sufficient gain in the attenuated bands that the re
ceived signal has a flat frequency response, or any pre-programmed response which the
listener finds acousticallypleasing. The noise source is turned off, and the CD player is
turned on.
Source Excitation
The excitation source utilized in equalizers has been the subject of much debate. Sev
eral articles [13-14] deal with the varying responses of rooms to different types of sources,
i.e. firecrackers, pistol blanks, etc. The most important characteristics of the test signal are
the signals spectral content, the amplitude distribution of the source, and the duration of
the signal. State-of-the-art equalizers use pink, Gaussian noise for reasons which are now
discussed.
Except for small, undamped rooms, at low excitation frequencies, the sound pressure
amplitude distribution is independent of the volume, shape, or acoustical properties of the
room. By examining the rooms Greens function it is seen that the pressure field is the su
perposition of many eigenfrequencies, each with its separate damping coefficient. This su
perposition is complicated due to the finite half-width, and small separation of the room
resonances. Since the eigenmodes are closely spaced and mutually coupling, the field may
be considered to be resultant from a single frequency with randomly distributed amplitudes
and phases. Applying the Central Limit Theorem to both real and imaginary terms of the
pressure, P, leads to the Raleigh distribution of IPI. Figure 2.2 demonstrates the similarity
of the distribution function of pressure to Gaussian excitation.
The spectral content must be sufficiently broad that the eigenmodes excited by a music
signal are also excited by Gaussian noise. Pink noise satisfies this requirement and has
theadditional advantage that it will simplify the signal processing. Pink noise has the prop
erty that equal power is radiated in each octave. In addition, unlike impulsive sources, the


139
The Effect of Replacing Ensemble Averages with Time Averages
To this point we have assumed the availability of an ensemble of systems with which to
calculate the expectation of the error squared. Obviously this situation will not arise in the
case of adaptive equalization, and time averaging will have to replace ensemble averaging.
The time average approximation of the ensemble characteristics of the data is the limiting
factor of the algorithm. An engineering compromise must be made between the accuracy
of the time average estimate of the mean square error, and the speed with which the maxi
mum eigenvalue of the input data is estimated. An excellent estimate of C,k could be made
by averaging 100 consecutive sequences of data diverging over k iterations of the DLMS
algorithm. Unfortunately, after 100k samples there may be little point in determining
x The following plot demonstrates the accuracy of the estimate of x by averaging
twenty divergent sequences of five divergent iterations. The data was generated by using a
first order autoregressive process. Figure 5.19a shows plots log(error2) versus divergent it
erations and the divergent sequence. Figure 5.19b overlays the twenty divergent sequences.
Figure 5.19c shows both the estimate of X^* derived from the DLMS algorithm, and the
power estimation method. Fifty experiments were performed. Note that the estimation
based on the DLMS algorithm is far more accurate, and has a much smaller variance. The
probability density function of the eigenvalue estimation is plotted in figure 5.19d. The ex
periments indicated the DLMS algorithm is an unbiased estimator. The mean of the exper
imental results using the DLMS algorithm matched the true value of X,^ = 1.8. As the
number of diverging sequences is increased the variance of the estimate decreases. Figure
5.20 show a similar experiment for which Xj^ = 1.3. These two values of eigenvalues al
low the algorithm to be verified for a wide range of eigenvalue spread.
Integration of the DLMS Algorithm into the Equalizer Architecture
For a calculation of the initial step size, u = v/X X must be determined. The
architecture described in figure 2-13 is modified to integrate the DLMS strategy into the
adaptive equalizer. When a rapid increase occurs in the mean square error, the indication
is that a significant change in the characteristics of the input signal has occurred. As a re
sult, the eigenvalues of the system have changed, and the step size of the LMS algorithm


47
longer than the others terminated in a wall with very low acoustic resistance.
2) The perceived timbre will not be affected by the equalizer. Timbre is related transient
phenomena, and as the equalizers filters are time invariant, the equalizer will not be able
to restore timbre related features.
3) Equalization can not restore true audio fidelity of the signal because of the loss of
information regarding both temporal and spatial coherence properties of the signal. Unfor
tunately these characteristics are psycho-acoustically important. There are many reasons
for the lack of fidelity with respect to field coherence properties. Without a significant in
crease in the level of sophistication in the recording process, the coherence information will
not be included on the CD itself. Microphones used in the recording process, as well as the
equalization process, do not encode information regarding the spatial characteristics of the
signal. Fast temporal structures may also be lost due to the non-zero time constants in all
microphones. In addition the state-of-the-art equalizers utilize only one microphone and
cannot account for stereophonic properties of the acoustic field. As a result of the above
limitations, properties such as clarity, spatiousness, apparent source size, apparent source
location, etc. will not be restored by equalizers. These properties will be maintained by the
characteristics of the listening room itself.
4) The equalization of a music reproduction system is valid for only one listener loca
tion. If the listener moves locations, the equalization must be performed again. The adap
tive equalizer will eliminate this problem.
5) The best source with which to excite the room is the actual acoustic signal of interest
- not pink noise. The equalizers filters should utilize their limited degrees of freedom to
correct frequencies at which distortion is a maximum, and not the entire spectrum. This is
a fundamental limitation which affects the degree to which relative loudness, and hence
pitch, can be maintained. This limitation will be ameliorated by the adaptive equalizer.


84
The LMS Algorithm with Uncorrelated Input Data
The best known and most widely used gradient method for stochastic signals in adap
tive signal processing is the LMS algorithm[22]. The method is well-understood for sta
tionary uncorrelated input data. The method can be extended to the quasi-stationary case,
although the analytic description loses much of its simplicity and is possible only for the
most simple models. This section will provide the theoretical background necessary for a
discussion on the LMS algorithm with correlated input data.
Stationary Model for Uncorrelated Input
Recall the general form of the gradient descent algorithms, Wk+X = Wk pV The
LMS algorithm makes a crude estimate of V which nevertheless is quite effective. Let
represent the estimate of V
LMg Approximation
Gradient Descent
v£ = -2ekXk (3-66)
= -2E[ekXk (3-65)
The LMS approximation replaces the ensemble estimate of V £ with a time average. As
the algorithm operates over several iterations, on average the gradient estimate will be in
the correct direction. The LMS descent is illustrated in figure 3.6.


73
figure 3.3 The principal coordinate system has axes which lie along the
principal axes of the hyperparabolic error surface.
Vfjc + x = Q"1(I-2\iQAQ~l)QVfk
= (/-2nA)ri* (3-32)
By induction Vt k can be expressed in terms of 0.
Vtk = (/-2nA)*Vt0
(3-33)
The update equations show their decoupled nature more clearly in matrix notation.


5 DETERMINATION OF WITH A DIVERGENT
LMS (DLMS) Algorithm 118
Determination of Xrnxx with a Divergent Gradient
Descent Algorithm 115
Parameter Selection for the Divergent Gradient Descent
(DGD) Algorithm 118
Determination of with a Divergent LMS Algorithm 120
Parameter Selection for the Divergent LMS Algorithm 128
Integration of the DLMS Algorithm into the Equalizer Architecture. 139
Summary and Conclusions 143
6 Validation of Concepts 146
Test Plan 146
Test Results 158
7 CONCLUSIONS AND RECOMMENDED
EXTENSIONS OF RESEARCH 199
Review 199
Recommended Extensions to this Research 205
APPENDIX
ECVT Plots of Contiguous Epochs of Audio Data 208
REFERENCES 218
BIOGRAPHICAL SKETCH 220
v


132
over a robust domain of CQ. The following plots demonstrate the percent error in estimat
ing with the terms associated with the largest eigenvalue, as a function of p and G().
The initial misadjustment along the principal axis not associated with the largest eigenvalue
is made an integer multiple of the misadjustment along the axis associated with the largest
eigenvalue, i.e. aQ k = constan/a0 x The estimate of £3 is calculated using the as-
sumptions of equations 5-8 and 5-9, and the estimate is compared with £3 as calculated by
equation 5-7.
2 1< o <10
-60
Xmin
2 1< Eigenvalue Spread = 5
Eigenvalue Spread = 2
2 1 Eigenvalue Spread = 10
-45
1 Eigenvalue Spread = 3
figure 5.12 Approximation error created by approximating by
^max( 1 "4M^max+12p. X ma* jc^Xmax increases as the
misadjustment-along axes, other than that of the associated
with the largest eigenvalue, becomes substantially larger
than the axis associated with the largest eigenvalue.


TABLE OF CONTENTS
ACKNOWLEDGMENTS in
ABSTRACT. vi
CHAPTERS
1 BACKGROUND 1
Introduction 1
Basic Physics of Room Acoustics 4
Psychoacoustics 23
Consequences of Room Acoustics and Psychoacoustics
in the Solution of the Inverse Problem 34
2 AN EQUALIZATION STRATEGY FOR REAL-TIME
SELF-ADJUSTMENT. 39
Introduction 39
State-of-the-Art Equalizers 41
The Adkins-Principe Equalizer Architecture 48
Automatic Adjustment Using Adaptive Filters 57
3 LEAST MEAN SQUARES (LMS) ALGORITHM
CONVERGENCE WITH UNCORRELATED INPUT.... 63
The Wiener-Hopf Solution 64
The Gradient Descent Algorithm 68
The LMS Algorithm with Uncorrelated Input 84
4 Least Mean Squares (LMS) Algorithm Convergence
WITH CORRELATED INPUT... 91
Convergence of the LMS Algorithm with Correlated Input 91
Conditions on p. for Convergence of £.. 93
Conditions on p. for High Convergence Speed of £ 101
Experimental Results 102
Conclusions 108
IV


203
cuted by performing a parametric analysis of estimation error as a function of filter order,
initial conditions of the filter weights, step size, divergent iterations, and the method of ap
proximating an ensemble average. With proper selection of step size the estimation error
can be made arbitrarily small.
Chapter Six
Because a hardware implementation of the proposed equalizer was beyond the scope of
this research, an equivalent off-line architecture, which was mathematically equivalent, was
developed for testing purposes. Software tools were developed on the NeXT workstation,
to simulate the operation of the equalizer algorithms. A test plan was outlined, along with
evaluation criteria.
Test results indicated mixed results. The adaptive algorithm yielded good results in
lower frequency bands, where signals have long periods of quasi-stationary behavior, and
resonances are a larger fraction of the sampling frequency (figures 6-11 through 6-20). In
higher frequency bands the results were not as promising despite an increased level of al
gorithmic sophistication and computational complexity (figures 6-22 through 6-24 and fig
ures 6-37 through 6-39).
The results from band 4 equalization indicated that step size was being initialized prop
erly, using the principles of chapters four and five (figures 6-28 through 6-31). Features
which were strong and persistent were equalized well using the adaptive methodology.
Several factors, however, degraded performance.
The value of \DLMS has an associated estimation error. When the error is positive,
LMS adaptation can be temporarily underdamped (figure 6-34). As a result, high frequency
noise is introduced into the spectra. When the algorithm is sufficiently underdamped a fre
quency of one half the sampling rate is emphasized (22.050 kHz). If the estimation error is
negative, convergence will be too slow to follow highly transient features in the CD signal.
Despite these difficulties, step size is not based on a A/^er, which usually has a significant
negative estimation error.' For robust performance a step size based on would need
to be conservative, and adaptive equalization with audio signals would have unacceptably


195
figure 6.37 The waterfall plot for equalized data in band 4 (3675 22050
Hz) is shown above for a 1.8 second segment of data.


18
side the listening room by R2, each time the ray path from an image intersects the room
walls or an image room wall.
figure 1.5 Geometric acoustics can be used to explain the temporal
properties of reflected energy in a room. Image sources can
be formed by successively reflecting the original source and
each subsequent image source about the planes that are
constructed by using the room as a unit cell.
Figure 1.6 demonstrates the concept of an echogram, a graphic method describing the
temporal response of the room. The height of each line represent the intensity of reflection.


This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
May 1992
Winfred M. Phillips
Dean, College of Engineering
Madelyn M. Lockhart
Dean, Graduate School


176
figure 6.23 The waterfall plot for microphone data in band 3 (612.5 -
3675 Hz) is shown above for a four second segment of data.
Notice that the features in box 1 and 2 are distinct, although
the feature in box 2 has been greatly attenuated. Note also,
that no spectral content exists in box 3.


21
0.167
4aV-Sln\R\2
(1-58)
Soarial Properties of a Listening Room
The spatial distribution of acoustic energy is a key factor in constructing a listening
room. A diffuse field can be defined as a field in which the intensity in a spherical shell
subtended by a solid angle, di2, is independent of the direction in which the solid angle is
pointed. For the case of a room which is a rectangular cavity with hard walls, no matter
how small dQ, there exists a time at which the field is diffuse. Of course there are no such
walls, and clearly, for a rectangular cavity, the field will not be diffuse. For example, im
ages located in directions along the comers of the room experience more attenuating reflec
tions than the other sources.
Fortunately walls are not perfectly smooth and they partially diffuse the acoustic energy
at each reflection. Consider the following simple model. Assume at each reflection the per
centage of energy reflected specularly is s. Then 1-s represents the portion of energy which
is reradiated by the wall diffusely. This is, of course, an overly simplistic model. All inci
dent energy is reradiated by the wall according to a directional distribution function. Nev
ertheless this model does demonstrate approximately the diffusion process. In the steady
state the power present in the total reflected field and in the field reflected specularly can be
related to the reflection coefficient, R, and to s.
OO
fEW2' (1-59)
n = 1
n = 1
In
(1-60)
The percentage of energy power which has been diffused increases as s decreases as shown
below.


13
Note that Kn it the wavevector associated with %n. Pw (r) is recognized as the Greens
function for the room. The Greens function expresses the acoustic transfer function be
tween two points in the room. Note from the Greens function that acoustic fields satisfy
the Reciprocity theorem just as in the electromagnetic analogy. The Greens function can
be extended to a source, S, emitting a continuum of frequencies as shown below.
OO
5() = J <2 (CD) eimd(Q (1-34)
OO
OO
P (r) = J P(()ei(0tdG) (1-35)
Equations 1-33 through 1-35 provide a complete wave theoretical description. A dis
cussion of the implications of this formulation for a rectangular room are now in order.
Simple Rectangular Rooms
This section will apply the general wave theory to an empty rectangular cavity charac
terized by a single impedance. While this example does not represent the complexities of
actual rooms, it nevertheless provides insight into the features of room responses with a
minimum of mathematical complexity. The connection between the reflection properties
of walls and the general wave theory will be made explicit through wall impedances. Three
classes of impedances will be considered. Consider the room depicted below.
The wave equation for this geometry is most easily applied in the rectangular coordinate
system. The room eigenvalues are found by solving the homogenous Helmholtz equation
shown below.
a2^ aaV
a*2 dy2 dz2
+ K2W = 0
(1-36)
The equation is easily solved by the technique of separation of variables. Let
V (x, y,z) = \|/ (x) \\r (y) \|/ (z) Substitution of this expression into the wave equation


101
figure 4.3 The maximum value of the vector norm of X must be
determined in order to find the domain for which
Tit...T6tT&...T¡ contracts. In order to determine the domain
for which C, converges exponentially, the average value of a
must be determined. This figure shows results for 3
experiments.
This will have a tendency to increase Dy. However, this tendency will to some extent be
balanced with an increasing value of p. In any case, the step size domain will be influ
enced. Secondly, as the filter order increases, the value of will (3 increase, which will de
crease the domain size, Dy. These results will be shown in the experimental work presented
shortly.
Conditions on p. for High Convergence Speed of C
The optimal value of p. will minimize ¡^. Note that only n!^ and depend on ji.
mm^ = mm|vo|2 J (rfh) F0||2p (X) d (X)
v crin)
= ¡v0¡2 J min [Xmax (nrn) ] ||X0||2P (X) d(X)
(XQ1 h


191
maintained at in insufficiently large value, and adaptation is so slow that features in the CD
signal are not restored.
In conclusion waterfall plots and a learning curve for band 4 are presented for exami
nation. Figures 6.39 through 6.41 show the CD, microphone and equalized signals water
fall plots for approximately 1.8 seconds of acoustic data. This epoch was chosen because
the microphone data was collected with sufficient energy across the entire band 4 bandpass
that equalization would not be limited because of the equivalent architecture approach tak
en towards testing (see figures 6.32 and 6.33).
Consider the 17 kHz tone which persists through the entire 1.8 second segment of micro
phone data, marked in box 1. The equalizer cancels noise features which are strong and
persistent. Note that the tone has been completely removed by observing the equalized sig
nal waterfall plot. Unfortunately it appears that for many cases the contrapositive also
holds. As an example consider the regions marked by boxes 2 and 3. In these portions of
the signal both the CD and microphone signal had diffuse and transient features which were
completely missed in the equalized signal. In certain situations, if the microphone features
were sufficiently transient and diffuse, even if the CD signal had distinct features, the equal
izer would not perform well. Consider as examples features marked in boxes 3 and 4.
There are also examples, however, where transient spectral features are not lost by the
equalizer. This may be the result of better step size estimation, and an errorsurface which
permits faster convergence. The arrows marked 5 through 8 indicate transient spectral fea
tures which were retained, while eliminating some of the noise present in the microphone
signals. Note also that the equalizer picked up the features represented by the circles
marked 9 through 11.
The most disturbing result is the inability of the equalizer to restore the sharpness of the
CD features marked in boxes 12 and 13. It appears that the equalized signal has features
which are even more diffuse than the microphone. Perhaps performance could have been
enhanced by more extensive experimentation with the heuristic step size algorithm used for
band 4 equalization. However, the blurring of features will exist unless a reliable method


177
figure 6.24 The waterfall plot of data equalized in band 3 (612.5-
3675Hz) is a more important indicator of the efficacy of the
equalization than time domain measures. Note that the high
frequency information which was substantially reduced in
the microphone data is largely restored. However, the
adaptive filter introduces low frequency (i.e. less than 612
Hz) content outside of band 3. Of even greater concern is the
blurring of the prominent features in boxes 1 and 2.


95
S = e-2E[X¡Vke\] +ElVlcTXkX¡Vk]
-ElVfxplvj
(4-4)
~ V0E
n t'-2*1**-/i-,-] vrn
Ly=i y = i
Let (n) be expressed as follows.
0
rf(n) = n [i-2(lxk_.xj_.]
j = i
(4-5)
In the new notation C, is expressed as follows.
nkr(n)xkx^nk(n)'
0
(4-6)
Let £2 be the event space of the stochastic X. Then equation 4-6 is restated as follows,
? = vj
V0
J
|_(Xe £2)
nk7(n) xkx^nk (h) ]p (X) dx
0
< IIV 112
- 0
J llnf^xk
(Xe £2)
p(X)dX
sllvo||2 J nK(lO
(Xe £2)
Xkl2p(X)dX
= llVo|2 J KaAtfmXkfpWdm
(Xe £2)
(4-7)


127
For a sufficiently large p, ^ is the product of geometrically increasing terms. For cer
tain combinations, of p, {X.} and k, the geometric ratio associated with XMAX will dom
inate the value of 1 and can be used in the approximation of ^k/C,k t Two
properties of equation 5-7 must be demonstrated for the above proposition to hold.
3(p, 3 4p2C*Xr
/ + fk
L I F
XXrF(+1)a
0
(5-8)
3(p, {X.} k) 3\rF('k + 0*X (1 -4\i\ + 12pX2 )
k+l
0, max
(5-9)
The conditions for which equations 5-8 and 5-9 hold will be discussed in the proceeding
section. The derivation of the DLMS algorithm continues assuming they are satisfied. The
ratio of i t0 £k leads to the following expression.
^k+l
X (1 -4pA, + 12p.2^2 )*+1an
maxv ^ max ^ max' 0, max
X (l-4pA + 12|i2A,2 )* maxv ^ max n max' 0, max
= (1 4pA +
v ^ max
12|i2A,2 )
* m/7 v'
max'
(5-10)
Let P = 1Then XMAX is expressed by equation 5-11. Solve this quadratic equa
tion for XMAX Note that for a sufficiently large value of p, J3fi 1.
B = 1 -4llX + 12|12^2
r n max n **
max
X
max
1
6p
0 + J3P-2)
M
6p
(5-11)
(5-12)


65
ek = dk~ y k (3-2)
= dk-xlwk
Square the error.
4 = Wk-Xlwk)2 (3-3)
= 4-2dkxlwk+xIwkwlxk
Take the expectation of both sides of the above equation.
E[e2k] = E[d2k-2dkxlwk + xlwkwlxk] O'4)
= E[d\] -2Eld^W^ + E X¡WkW¡Xk]
By keeping W at a fixed value its subscript can be omitted. Furthermore note that
xTkwwTxk = wTxkxTkw.
E[e\] = E [d2k] 2E [d^f ] W + WTE [X^] W
Define the expectation value of the cross correlation matrix as follows.
Pk = E [dkXTk] = E[dkxk dkxk_l
(3-6)
Define the expectation of the correlation matrix as follows.


179
Test Results for Band 4 G675-22050 Hz')
Band 4 represents the biggest challenge to adaptive equalization, and as such, the ideas
presented in chapters four and five were implemented, along with a step size reduction al
gorithm. The bandwidth of many of the spectral features is a smaller fraction of the sam
pling frequency than any of the previous bands. In addition, many spectral features have a
very short period during which they can be considered quasi-stationary. Thef tests on band
4 resulted in mixed outcomes. Although the ideas of chapters four and five were supported
by the experimental evidence, the blurring of features, even while using a step size reduc
tion algorithm, was severe. This results in a degradation of the sharpness and clarity of
sound. In addition, segmentation of the data into quasi-stationary periods, using the heu
ristic technique discussed earlier, was problematic.
Comparison with Previous LMS Implementations
First I will present evidence which indicates the partial success of band 4 equalization
using the methods of chapters four and five. Data presented in this section will indicate
band 4 equalization with LMS algorithms using traditional step size initialization tech
niques are unsuccessful. Figure 6.26 shows an 11.33 msec epoch of band 4 acoustic data.
Note that the DLMS estimates of step size resulted in rapid convergence, while maintaining
a low error.
The step sizes chosen by the DLMS algorithm were substantially larger than would
have been chosen by the power normalized LMS algorithm. The equalized signal automat
ically restored the proper power to the band. Finer structure of the CD signal was also well
maintained in the equalized signal. Note the similarity of the equalized signal and CD sig
nal envelopes. This result is particularly encouraging when a comparison is made with the
amplified microphone signal, showed as a dashed line in the first trace of the ECVT plot.
Power is also restored into higher frequency components of band 4. Unfortunately, this am
plification leads to undesired effects. Note the exaggerated high frequency feature identi
fied in the ECVT plot. The adaptive equalizer is also able to remove pulses from the
microphone signal. This is demonstrated in figure 6.27. Note the distinct pulse train in the
amplified microphone signal, and compare it with the CD signal and equalized signal.


116
modified from the standard algorithm. The calculation of ^MAX is based on measurements
obtained by driving the algorithm into divergence by choosing a sufficiently large value for
the step size, p. Recall equations 3-30 and 3-33.
£ = C + VfrAVf (3-30)
Vt* = (/-2(lA)tVt0 (3'33)
Substituting equation 3-33 into 3-30 yields the following.
=" C + Vfo (/ - 2)iA) kTA (I 2(lA) *vt0 (5-1)
Since A and (/ 2pA) are diagonal, they may be commuted.
L = C* + Vfoa-2nA)*7(/-2nA)tAVfo
= C + Vfo(/-2nA)AVf0
(5-2)
A diagonal matrix raised to a power is equal to the elements of the matrix raised to the pow
er, and the product of diagonal matrices is the product of the elements of the matrices.
Equation 5-3 expresses this relationship. For a sufficiently large |i, £ is the product of geo
metrically increasing terms. For certain combinations, of (X, {A,.} and k, the exponential
term associated with "kMAX will dominate the value of C,k/^k_v and this term alone can be
used as an approximation of Equations 5-4 and 5-5 state these relationships more
formally.


49
The tree structure is an efficient strategy for implementing an octave filter bank. The
input to each branching node has been bandlimited and decimated. The lower branch per
forms equalization with a highpass filter while the upper branch further reduces the CD
bandwidth. The characteristics of the upper and lower branches are described below.
Each of the upper branches attempts to perform the following lowpass operation.
hip=1 W 0, otherwise
The upper cutoff of band i is equal to the lower cutoff of band i-1. The sampling rate
can be reduced by decimating by M according to Shannons sampling theorem.
xdec (n7l = f"Mrl (2-2)
The decimation rescales the digital frequencies of the DFT, as shown in figure 2.5.
Clearly the decimation rate must be coupled to the value of the digital cutoff frequency of
the lowpass filters in order to avoid aliasing.
figure 2.5 Decimation by M in the time domain expands the digital
frequencies by M, and reduces the peak value of the
spectrum by M. This figure demonstrates the effects for M
equals 2.
Decimation reduces the number of operations required for real-time processing by re
ducing the data rate by M1"1 for band i. In addition, because the critical digital frequencies


8
K- ~ n,K [-t/ cos0. +wsin0.]
i l o x i z r
d-7)
Kr = [M;ccos0r + Mzsin0r]
d-8)
Kf = n2K^[-uxcosQt + uzsinQt]
(1-9)
tions 1-10 through 1-12 are shown below.
-in^Q [-XcosO + Zsin0.]
Pi ~ Pi,0e ukt
(1-10)
_ inxK0 [Xcos0r + Zsin0r]
Pr ~ Pr,0e ukr
(1-11)
... ~in2KQ [-XcosQj + ZsinGJ
Pt = Pt, 0e ukt
(1-12)
The pressure field must be continuous at all point along the boundary, i.e.
^i, interface+ ^r, interface = ^t, interface These equations yield the Law of Reflection and
Snells Law precisely in the form as the electromagnetic equivalent.
/ijJ? sin0. = /t^sinOj. = n2KQsm&t (1-13)
sin0. = sin Law of Reflection (1-14)
i r
/ijSinGj. = n2sin0f Snells Law (1-15)
In addition, the conservation of momentum (equation 1-3) must be satisfied at the wall.
This will imply a matching of particle velocities at the interface. Note that because of the
plane wave assumption only the derivative with respect to x is non-zero in grad R The ve-
(1-16)
V. = ^cos0.P- -'".V-XcosS^Zsinej
l p, l 1,0 Ki
1
v = Jocose p 0fin^xcose.+zsinej
r (Dp. r r, 0 kr
1
d-17)
v, = ^coS0Aorn2:[-Xcos0'+Zsin'I(tt
(1-18)


77
vt Jfc
(l-2px0)
0
0
0
0
1
1
1
yt*-l
0
(12^)
0
0

0
0

0


0
0

0


.
.
0

k-n
0
0
0
0
... (l-2pxrt)
"tjfc-l
"tjfc-2
By induction
t k-n
Vt k can be expressed in terms of Vt0.
(3-48)
V^k+l
k-1
(3-49)
(/-2M0Mo-il]T (/-2^A)^_;_1
7 = 0
Thus, with gradient noise included, Vt k does not approach zero as k - . The degree of
lim
kt
it-1
-nX (/-2(lA)W*-;_i
j = 0
(3-50)
suboptimality is determined by the set of eigenvalues, {X.} the statistical characteristics
of N$, and the step size, p.
The difference between the minimum MSE and the average MSE is defined as the ex
cess mean square error. Excess MSE provides a measure of the difference between the ac-


166
figure 6.14 The waterfall plot for equalized data in band 1 (dc -102 Hz)
is shown above for a 12 second segment of data.


154
figure 6.5 Step size adaptation is performed in the high frequency band
according to the above block diagram.
figure 6.6
The DLMS algorithm is triggered according to block
diagram.


147
of linear systems which generate the fil Specifically, the acoustic equalization is performed
after collection by the microphone instead of immediately preceding the audio amplifica
tion. We know from the commutivity of convolution that the following equivalence holds,
where x y represents the convolution of x and y.
CD equalize amp room = CD <8> amp room equalizer
figure 6.1 Equalizer architectures. A) Schematic of electo-acoustic
circuit proposed for hardware implementation; B)equi-
valent circuit proposed for algorithm validation.


4
Basic Physics of Room Acoustics
Equalization is an inverse problem, i.e. the adaptive equalizer will seek a configuration
such that its response will be the inverse of the room response. This section will develop
the most simple features of room acoustics because of the basic understanding it will pro
vide into the nature of the signals and systems on which the algorithms and methods intro
duced in this dissertation will be applied. As electrical engineers are more familiar with
electromagnetic radiation than acoustic, an effort will be made to develop this introduction
to the physics of room acoustics in an analogous manner to the standard method of intro
ducing the theory of electromagnetic waveguides, with the most important similarities
pointed out where appropriate. A listening room may be thought of as an acoustic coupler,
or communications channel, between an acoustic source and a listener. A further limitation
that the room is described as a minimum phase system will be imposed. The description of
the acoustic channel is particularly complicated because of the importance of both its tem
poral and spatial dependency. Because of the wave nature of acoustic fields, the sound en
ergy in an enclosed space may be considered to be the mechanical analogy of
electromagnetic radiation in a rectangular waveguide.
A discussion of the basic equations and definitions is first presented. Specifically the
pressure equation will be discussed as the principal descriptor of acoustic phenomena. The
conservation of momentum equation will be shown to supply boundary conditions for the
wave equation. A discussion of the reflection and transmission coefficients are discussed.
It is shown that many features from optics have analogous results in acoustics,e.g. Snells
Law and the Law of Reflection. The prominent factor in determining the reflectivity of a
wall is the walls characteristic impedance. The impedance, together with the rooms geom
etry, define the characteristics of the room. A brief discussion of acoustic impedance is
therefore presented. The wave equation for a room is developed in a method analogous to
methods used for electromagnetic waveguides. The application of this theory is discussed-
for a room which is a rectangular cavity with different types of impedance. Lastly the spa
tial and temporal response of the room is outlined using methods from ray theory. Much


202
tive filters are of sufficient order that the minimum mean square error is negligible. A ma
trix equation is developed in which terms in minimum mean square error are eliminated. It
is demonstrated under these conditions that p." results from minimizing || n|| with respect to
y, where p = y/\max.
In order to understand the implication of this formulation, input data modelled as an au
toregressive series, with various degrees of correlation, were examined. Several interesting
conclusions were apparent, including the rule of thumb used to initialize step size for the
LMS algorithm. It was shown that minimizing || n||, for filter orders less than ten, results
in o.i < / < 0.15 for a wide range of input data correlation. Since the adaptive filters used for
acoustic equalization are of order 45, simulations for a 44th order autoregressive series
were modelled. In order to guarantee fast convergence, it was determined that f = 0.05.
These results confirmed that / is dependent on filter order. This result was expected as a
consequence of the theorems of Shi and Kozin.
Chapter Five
Since f was determined in chapter four, only the value of \max remained to be found
in order to properly set the LMS step size. Typically \max is approximated by the trace of
the autocorrelation matrix of the filter input data. This is easily measured because of this
quantitys relation to the power of the input vector. Depending on the eigenvalue spread of
the autocorellation matrix, and the filter order, this estimation can be bad. Consider the sit
uation in which Aq=1.0 and ^2=^3==^50=0-5- The trace of the autocorrelation matrix is
25.5 times A^^.
Chapter five proposes operating an LMS algorithm off-line in such a manner that the
mean square error rapidly diverges (DLMS algorithm). Under these conditions the geomet
ric ratio which dominates the process is a quadratic function of This equation is
solved for A^^, and all the necessary information for updating the step size of the adaptive
filters embedded in the equalizer is available.
Chapter five discusses the proper domains for divergent step size and divergent itera
tions for a good estimate of A^* using the DLMS algorithm. An error analysis was exe-


16
) = O
(1-50)
conformal mapping. If % < 0, the corresponding eigenfrequencies are lower than for the
hard wall case (Im[z]l). For % > 0 the corresponding eigenfrequencies are higher than
the hard wall case. Because the impedance is completely reactive, the room will continue
to support standing waves, but with phase terms included in the sinusoidal terms of the ei-
genmodes.
case iii. Arbitrary z.
The form of the boundary conditions are the same as case ii with % replaced by z. Ei-
genmodes can not support standing waves because of the lossy component of z represented
5 (x)
by e
V (x, y, z) = De 6 ^ y* Z^ COS
anx
x
It is clear from matching boundary conditions in the three cases outlined above that the
room impedance is key to understanding the behavior of the room. By using the eigenmode
expansion technique and the inhomogenous Helmholtz equation the pressure field can be
determined for any location in the room. An initial condition is also required in order to
evaluate D. The eigenfrequencies indicate those frequencies at which the room has reso
nant properties. Note that the number of eigenfrequencies increase as the cube of the upper
frequency limit of the source. The wave theoretic technique provides a complete descrip
tion of the acoustic signal in an enclosed space, and provides a solid theoretical foundation
for the field of acoustics.


206
of theoretical interest, and it is recommended that they be investigated.
A simple thresholding technique was used to determine when the statistics of the audio
signal had changed sufficiently to merit updating step size via the DLMS algorithm. Trig
gering the algorithm too infrequently may cause the LMS algorithms embedded in the
equalizer to temporarily diverge. Tripping the DLMS algorithm too frequently will coun
teract the reduction of the step size, as the algorithm converges, by the step size adaptation
algorithm Investigation of appropriate analytic measures for the segmentation of audio
signals, and algorithms to perform according to these measures need a much fuller investi
gation.
In addition to the DLMS algorithm, a step size reduction algorithm was implemented
in software. This algorithm, broadly related to the work of Harris et.al.[31], is used to re
duce step size as the LMS algorithm converges, in order to reduce excess mean square error.
The technique is highly heuristic, and like all other techniques to date, is in no sense opti
mal. Significant research is required to find an effective and robust algorithm for audio
data. At a minium, other algorithms now available should be compared with the one im
plemented in this research.
Additional Testing
The proposed audio equalizer was tested by finding an architecture that could be simu
lated with software, and that was mathematically equivalent to the proposed real-time sys
tem. While this provides good evidence of the likely success of the proposed equalizer,
verification can not be complete without implementation of the equalizer in hardware fol
lowed by thorough testing. The testing should have analytic measures, e.g. learning curves,
but listening tests would be most important. These tests should be conducted under condi
tions acceptable to both musicians and psychologists, and would be the subject of an inves
tigation in its own right.
The validation of the software implementation of the equalizer used adaptive filters of
order 45. The optimum value of y was validated against LMS experiments for filters of or
der two. Likewise the DLMS algorithm was validated for filter of order two. This valida-


A NEW ADAPTIVE ALGORITHM FOR THE REAL-TIME EQUALIZATION
OF ACOUSTIC FIELDS
BY
JEFFREY JAMES SPAULDING
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1992


68
c = C* +WrflW-2WrflW* + W*rflW*
(3-15)
Since WtRW* is a scalar, it must be equal to its own transpose. Again use the Hermitian
property of R.
5 = 5* + WTRW+ W*TRW* WTRW* W*TRW (3-16)
Note that transposition is a linear operator.
£ = C+ (W-W*)rfl(W-W*)
= C* + vtrv
^ y
(3-17)
(3-18)
The error surface is clearly a quadratic function of V. As a result, the following impor
tant characteristics are obvious: 1) the error surface is a hyperparaboloid, 2) the error sur
face must be concave upwards in order to avoid regions of negative MSE (a logical
impossibility), 3) the optimal set of filter weights are located at the bottom of the hyperpa
raboloid, and 4) there is only one minimum (no local minima). These observations are crit
ical in their implication that gradient search techniques are able to find the global minimum.
An error surface for both the translated and untranslated coordinates is shown below.
The Gradient Descent Algorithm
The gradient of a surface points in the direction in which the function describing the
surface will be most increased. A common sense approach to finding a surface minimum
is to move in a direction opposite to that of its gradient. This section will show rigorously
the convergence properties for a stationary signal. The trade-off between convergence
speed and misadjustment will be demonstrated explicitly. Finally, a description of the dif
ficulty in using gradient descent techniques with quasi-stationary signals is discussed.


7
figure 1.1 The Law of Reflection and Snells Law are valid for acoustic
radiation as well as electromagnetic radiation. The figure
above indicates the coordinate system and nomenclature
which will be used throughout the discussion of room
acoustics.
n be the ratio of the speed of propagation of a medium from that of air at STP, i.e.
n = Cq/c Let ux, Uy, and uz be the unit vectors of our coordinate system as depicted in
figure 1.1. Then the incident, reflected, and transmitted wavevectors are expressed by equa
tions 1-7 through 1-9, where |A] = n^Q and is the angle between the wall normal
and the wavevector.
By matching the pressure field at the wall, the reflection and transmission coefficients
can be calculated. The pressure waves propagating along the wavevectors given in equa-


64
tion of step size. The background information in this chapter is largely taken from the work
of Widrow and Steams [19], and their notation will be used throughout.
The Weiner-Hopf Formulation
The Weiner-Hopf formulation [20] gives a method in which filter weights of a tapped
delay (direct form) filter can be optimized when the input to the filter is a stationary sto
chastic signal. The results hold for a quasi-stationary signal to a greater or lesser degree de
pending on the precise statistical nature of the signals. The measure by which the filter
weights are optimized is the mean square error (MSE).
Because the Wiener-Hopf equation is the underpinning of all adaptive signal processing
algorithms, it will be reviewed in detail. The derivation of the Wiener filter will yield im
portant information regarding the nature of the error surface. As a result it will be obvious
that gradient descent techniques will be applicable optimization algorithms. The derivation
of the Weiner solution is presented below.
Derivation of the Wiener Solution
Figure 2.15 shows a tapped delay line. The input sequence {xjJ, is a stationary stochas
tic signal. The output of the filter, y^, is simply the inner product of the n+1 most recent
inputs with their respective filter weights, for a filter of order n+1.
yt = xiwk=
*4-2* *
k n
0
Wi
vv
(3-1)
The error signal is generated by taking the difference between the desired signal, d^, and
the filter output.


159
tive equalizer tends to overcompensate. The learning curve indicates good adaptation.
Figures 6.12 through 6.14 present spectral waterfall plots for the CD, microphone, and
equalized signals over a twelve second time epoch. These figures indicate that although the
microphone signal was attenuated, the spectral distribution closely resembled the CD dis
tribution. The waterfall plot for the equalized data indicates the adaptive equalizer ampli
fied the microphone signal without damaging the spectral distribution.
Learning curves using the Signal Editor program are shown in figure 6.15. If the adaptive
filter were completely ineffective, the square error between the equalizer output and the CD
input would follow the behavior of the solid line (top panel). The dashed line indicates the
actual square error between the equalized signal and the CD signal. The difference between
these traces is defined as the improvement induced by the adaptive filter, and is plotted in
the bottom panel. All positive values in the bottom panel indicate successful adaptation.
Test Results for Band 2 (102-612.5 Hz)
The representative ECVT plot shown in figure 6.16 indicates that the adaptive equalizer
captures the envelope, and enhances the high frequency content of the CD waveform for
this 400 msec epoch of equalized acoustic data from band 2. Note also that the equalized
signal matches the amplitude of the CD signal automatically. It is clear from the examina
tion figures 6.11 and 6.16 that less amplification is required for band 2 than band 1. The
ECVT plot also indicates an especially prominent high frequency feature which is attenu
ated in the microphone signal, and is restored in the equalized signal. It is typical with the
ECVT settings used for these experiments for the filter to overcorrect these types of prob
lems. Also note that because a step size reduction algorithm was not implemented, the
learning curve, as shown in the ECVT plot, does not remain at a small value, even in long
segments of quasi-stationary data.
Figures 6.17 through 6.19 show spectral waterfall plots for the CD, microphone, and
equalized signals. The equalized data has largely restored the frequency content which had
been attenuated in the microphone signal. These attenuated regions are indicated in boxes
1, 2, and 3 in figures 6.17 through 6.19.


211
figure A.6 The ECVT signals plot is given for band 2 (102 612.5 Hz)
for the third 0.4000 second epoch of acoustic data.
figure A.7
The ECVT signals plot is given for band 3 (612.5 3675 Hz) for the first
68 millisecond epoch of acoustic data.


CHAPTER 7
CONCLUSIONS AND RECOMMENDED EXTENSIONSOF RESEARCH
This dissertation has suggested a new strategy for the equalization of sound systems utilizing
the principles of adaptive signal processing. Chapter seven will briefly review, and make conclu
sions, about the major propositions heretofore presented. Research in acoustic equalization could
easily consume an entire career, and thus a complete validation, and investigation of many serious
questions had necessarily to be suspended at some point. Thus this work will conclude by recom
mending several areas of additional research that could be conducted.
Review
Chapter One
Equalization of an acoustic field is an inverse problem. As the listening room through which
the field propagates is the system which is the chief object of equalization, chapter one has provided
a review of the basic principles of room acoustics. It was demonstrated that a rigorous and com
plete solution to the room inverse problem is available by modelling the behavior of a room by solv
ing the Helmholtz equation with boundary conditions, using the usual methods of eigenvalue
decomposition.
Because this solution technique is so computationally intensive, it is an impractical approach,
except for the simplest and most artificial cases. In order to simplify the problem it is necessary to
understand what aspects are most important to the acoustic sensor. In the situation described in this
dissertation, the sensor is human hearing, and thus a brief review of psychoacoustics was made. It
was shown that the spatial and temporal coherence of the acoustic fields, the magnitude response
of the frequency domain representation of the signal, and the frequency and amplitude resolution
of the signal were important factors in achieving fidelity.
Because much of the information regarding the coherence of the audio field is not recorded, the
goal of acoustic equalization is limited to maintaining the magnitude response of the spectral do
main representation of the audio signals with sufficient spectral resolution (as determined by psy-
199


145


14
figure 1.4 A room with dimensions Lx,Ly, and Lz and walls with
uniform impedance will be evaluated using the wave
theoretic formulation.
leads to three ordinary differential equation plus the separation equation as shown below.
I(x)+K2x
= 0
(1-37)
dx2
dx2 y
= 0
(1-38)
-4k(Z)+K2
= 0
d-39)
dz2
K2 = K2x + K2y
+ Kz
(1-40)
The solution of these equations is trivial.
\|/(jc) = A^e ikxX + (l-41)
K,) =v~'v+Vv <1_42)
y(z) = Cie-kZ + C2ekZ 0-43>


REFERENCES
[1] A. Z. Adkins, III, An All-Digital Audio Equalizer Design, Masters thesis,
University of Florida, University of Florida, Gainesville, FL, 1989.
[2] L.L. Horowitz and K.D. Senne, Performance Advantage of Complex LMS for
Controlling Narrow-Band Adaptive Arrays, IEEE Transactions on Acoustics,
Speech, and Signal Processing, Vol.. 29, pp. 722-735, June, 1981.
[3] A. Feuer and E. Weinstein, Convergence Analysis of LMS Filters with
Uncorrelated Gaussian Data, IEEE Transactions on Acoustics, Speech, and
Signaly Processing, Vol. 33, pp. 222-229, February, 1985.
[4] D.M. Egan, Architectural Acoustics. McGraw-Hill, New York., NY, 1989.
[5] L.E. Kinsler, A.R. Frey, A.B. Coppens, and J.V. Sanders, Fundamentals of
Acoustics. Wiley, New York., NY, 1962.
[6] Heinrich Kuttruff, Room Acoustics. Applied Science Publishers, London, 1979.
[7] H. Suzuki, S. Morita and T. Shindo, On the Perceptio of Phase Distortion, Journal
of the Audio Engineering Society, Vol. 28, pp. 570-574, September, 1980.
[8] V. Hansen and E. Madsen, On Aural Phase Detection, Journal of the Audio
Engineering Society, Vol. 22, pp. 10-14, January/February, 1974.
[9] Harvey Fletcher and W. A.Musnson, Relation Between Loudness and Masking,
Journal of the Acoustic Society of America, Vol. 9, pp. 1-10, July, 1937.
[10] H. Haas, Uber den Einfluss des Einfachechos auf die Horsamkeit von Sprache,
Ph.D. dissertation, University of Gottingen, 1949.
[11] L. Beranek, Music. Acoustics, and Architecture. John Wiley, New York, NY, 1962.
[12] M. Barron, The Effects of Early Reflections on Subjective Acoustical Quality in
Concert Halls, Journal of Sound Vibration, vol. 15, 1974.
[13] A. Berkhout, M. Boone, and C.. Kesselman, Acoustic Impulse Response
Measurement, Journal of the Audio Engineering Society, October, Vol. 32, pp.
740-745, 1984.
[14] R. Cann and R. Lyon, Acoustic Impulse Response of Interior Spaces, Journal of
the Audio Engineering Society, Vol. 27, pp. 960-963, December, 1979.
[15] K. Gugel, J. Principe, and S. Etamadi, Multi-Rate Sampling Digital Audio
Equalizer, IEEE Proceedings ofthe SOUTHEASTCON, Vol. 1, pp. 499-502,1991.
[16] S.J. Elliot and P. A. Nelson, Multiple-Point Equalization in a Room Using
Adaptive Digital Filters, Journal of the Audio Engineering Society, Vol. 37, pp.
899-907, November, 1989.
[17] S. Haykin, Adaptive Filter Theory. Prentice-Hall, Englewood Cliffs, NJ, 1986.
218


89
With equation 3-88 a recursive expression is available for £.
?t + 1 = C + i-TOk+l = C + \rlFOk + 4il2CX] (3-89)
Horowitz and Senne develop convergence conditions which can be seen most easily from
the matrix formulation of equation 3-89. Let .(T = 0 i
', + 1 = f* + 41-V[loll-1.]
0
(3-90)
1 -4,x0+ 12^2
4Vl
^v
k+1
31, ... X~\
1 J
4hxqx1 1
-4^ + 1211^2 _

4V-
4|Vi
4*iX0xi
1 4nx + 12
n n n n
For every entry of F, || F [ij] || < 1 and p is real if 0 < p. < ^. This condition on p.
max
will guarantee finite mean square error.
Conditions on u. for Maximum Convergence Sneed and FiniteVariance for Square Error
Feuer and Weinstein [3] have developed an expression for step size which will optimize
convergence speed. The expression is a function of generally unavailable a priori informa
tion, i.e. all eigenvalues and the initial misadjustment along each of the principal coordi
nates of the error surface. Given no information concerning it is assumed that all of
its elements are equal. Horowitz and Senne note that the convergence rate of £ is approx
imately optimized (given no further a priori information) by setting p. as follows.


130
proximation never exceeds 1%. Note also the relatively large domain of y for which the
error remains negligible.
figure 5.10 The approximation for C,3 is excellent over a broad rang of
eigenvalue spreads and step sizes. Note that the norm
approximation of figure 5.9 gave too high a value for the
approximation error.
The Effect of Step Size on the Estimation of t v
The following figure gives the approximation error of ^MAX using the DLMS method,
as a function of step size and eigenvalue spread, for (3 = C,4/C,3 and equal initial misadjust-
ment along each of the principal axes, for a filter order of two. Calculations were performed


160
are shown above.


113
figure 4.13 The left-
n(Y)
land plots give the three dimensional plot of
as a function of k and y, where ji = ykmax, for the
two most correlated sets of test data. The right-hand plot
superimposes all cross sections of the surface plots.


11
figure 1.2 The mechanical analogy or room impedance is illustrated by
in the figure by a spring for the reactive term and a damper
for the dissipative term.
figure 1.3 A room may be modelled by its electrical circuit equivalent
as shown above.
General Theory of Room Responses
The solution to the wave equation in an enclosed space is a familiar eigenvalue problem.
The equation reduces to a Helmholtz equation with boundary equations as shown below.
The homogenous part of the Helmholtz equation provides a complete set of orthogonal ei
genfunctions and associated eigenvalues, \|rn (r) and Xn.


the relaxation term in equation 4-12 will have a magnitude less than 1, and T will contract,
as shown below by equation 4-13. Note that cross terms do not appear in equation 4-13,
because the projection operators P and I-P are orthogonal. Figure 2.2 shows a graphic rep-
27imi2f
1_ x
K max)
y-Pjiy]
resentation of the operation of T
figure 4.2 Each operator,7} represents a pair of projections, P¡ and I-P¡.
For a sufficiently restricted step size domain, these operators
have a norm less than 1, i.e. the operators contract.
(4-13)


150
Concetps Validation Tool
figure 6.3 The Equalizer Concepts Validation Tool (ECVT) top level popup menu
is given above. Each choice in the menu will be discussed below.
The information option allows the operator to open a non-editable ASCII file which pro
vides a brief description and complete instructions for the Equalizer Concepts Validation
Tool (ECVT). The initialize applications option brings forth a panel containing the ECVT
initialization tableau. The operator interactively indicates values for key parameters for the
LMS and DLMS algorithms. From figure 6.4 it is seen that there are four sets of parameters
that are input.
The power normalized LMS algorithm parameters are displayed in the upper left quad
rant. The number of seconds of audio data, the filter order, and the initial value for step size
are input by the user. Note that for the highest frequency band (3675 22050 Hz), the value
input for step size is y, where p = y/^max- For all other bands the input value is y, where
(i = y/ (N power) where N is the filter order. An expansion slot is left open to pro
vide for future modification to the algorithm.
Step size reduction parameters are input in the lower left quadrant of the initialization
tableau. A simple heuristic scheme based loosely on the Harris LMS algorithm [31] is uti
lized in the ECVT. A block diagram of the step size adaptation strategy is shown in figure


Because a step size reduction algorithm was not implemented, the spectral features of
the equalized signal are not as sharp as the CD signal. The equalizer filter coefficients os
cillate about their optimum values, blurring the sharpness of features. For some spectral
features, the equalized signal causes degradation, compared even with the microphone sig
nal. The spectral features marked in box 4 in the following figures is one such example.
figure 6.15 Time domain improvement is measured by comparing the
difference between the (CD signal microphone signal) and
'y
(CD signal equalized signal) as shown in the upper trace.
Note that positive values for the lower trace indicate
improvement in the time domain signal received at the
listener location after equalization. This figure covers the
first two seconds of data shown in the waterfall plots above.
Os


Dedicated to
my Lord Jesus Christ
through Mary, Queen of Virgins,
in partial reparation of sins.


170
The waterfall plot for CD data in band 2 (102-612.5 Hz) is
shown above for a ten second segment of data. The four
boxes indicate features of the spectra which will be
examined in the microphone and equalized signal waterfall
plots.
figure 6.17


162
figure 6.10 The logarithm of the square error is plotted as a function of
divergent iterations. Note the linear character of the
divergence when plotted in this manner. Only the slope of
the curves is used in determining the maximum eigenvalue,
and hence maximum step size for the LMS algorithm.


figure 3.2 Gradient descent methods find the error surface minimum by moving in
a direction opposite to that of the error surface gradient.
The trajectory of the filter weights is the projection of the n+1 dimensional error surface
onto the n-dimensional weight space. For a hyperparaboloid the approach is intuitive the
greater the distance from the surface minimum the larger the gradient, and hence, the larger
the weight updates. At the surface minimum the gradient is zero, and no further adjustment
is made. A more rigorous justification follows.
Algorithm Convergence
Recall from equation 3-9 that the gradient is expressed as = -2P+2RW. Substituting
this into equation 3-19 yields the following.
wk+1 = Wk + li(2P-2RWk)
(3-20)


213
figure A. 10 The ECVT signals plot is given for band 4 (3675-22050 Hz) for the first
11.33 msec epoch of acoustic data.
figure A. 11
The ECVT signals plot is given for band 4 (3675-22050 Hz) for the
second 11.33 msec epoch of acoustic data.


78
tual and the optimal performance, averaged over time, and is expressed as follows.
Excess MSE = £[£-£*]
= £[vt[AVtJ
(3-51)
k-1
Recall that after adaptation n = (/ 2\i)jw Substitute this into equation 3-51.
7-0
Excess MSE = E
f
k-1 i T
j = 0
k l
a -nX
^ 1 = 0
= M^XX^ (7 2nA)7?A (/ 2pA) lrtk-i i]
i j
Since (/ 2jxA)^ and A are both diagonal, they can be commuted.
oo oo
Excess MSE = jl 2£ [¡vf a (/ 2,*a) 1 +-Wt / -1 ]
i j
oo
= l2^[N\l-j. Ia (/-4pA + 4p2A2)yAf !]
j
(3-52)
(3-53)
Assume that M results from independent errors. Then Vi j there will be no contribution
to excess MSE.


35
the number of room eigenmodes increase as the cube of the upper frequency limit of the
source. For a room of average dimensions, i.e. Lx = 5 m., Ly = 4 m., and Lz = 3.1 m., be
tween 0 and 20,000 Hz. there are over 57,000,000 [6] resonant room modes. As an addi
tional complication, except at low frequencies, the modes are extremely dense in the
frequency domain, i.e. the half-width of a mode is much greater than the separation between
modes. Thus the majority of modes are mutually coupling.
The aforementioned calculations are so computationally burdensome that they can not
reasonably be performed. Even if a complete set of modes were available, from which the
Greens function inverse could be calculated, the processing problem would not be feasible.
For the simple room geometry discussed above, there were 57,000,000 room modes, and
an FIR filter length of 100 million tap weights would be required. Realizing that implemen
tation of such a large filter is impossible, a selection of the most important room modes
would need to be calculated. Because of the mutually coupling nature of the modes, and
the changing spectral distribution of the acoustic source, the important modes would be
constantly changing.
The rigorous eigenmode expansion of the room response, the subsequent calculations
of the rooms Greens function, and implementation of the inverse on hardware, can not be
performed. As a simplification in the computational complexity of describing a home lis
tening room, a geometric acoustics approach could be attempted. The previous discussion
makes clear, however, that geometric acoustics gives insight into the time average proper
ties of the listening room. Depending on the nature of the acoustic source, the approxima
tions could yield poor results. In addition, the computational complexity of characterizing
the complex acoustic impedances for the room materials and the complicated geometry of
the listening room, continue to result in a huge computational problem. Figure 1.16 sum
marizes the rationale for rejecting approaching the inverse room problem by modelling the
physics of the room.
The experimental approach must be selected, and this approach is discussed in detail in
chapter two. Several implications of this method must be addressed in light of the basic
principles of room acoustics and psychoacoustics in order to design an intelligent tech-


92
figure 4.1 The autocorrelation functions are given above for several
different instruments [6]. a) organ music (Bach);
b)symphony orchestra (Glasunov); c)cembalo music
(Frescobaldi); d) singing with piano accompaniment
(Grieg), and e)speech (male voice).
Gaussian input. Farden [26] has found a bound on C, by making the reasonable assumption
of a decreasing autocorrelation function. He further makes use of reflecting barriers for Vk
to keep Vk in a compact set. Macchi and Eweda [23] find a bound on £ by assuming only
blocks of M samples are correlated and all moments are bounded. They do not make use
of reflecting boundaries.
The proofs referred to above have assumptions which are representative of acoustic sig
nals. Acoustic signals do generally have decreasing autocorrelation functions, and for a -
sufficient delay, may be assumed to be uncorrelated. In order to determine the step size
which will minimize ££ at the beginning of a quasi-stationary segment of data, the work
of Horowitz and Senne, and Feuer and Weinstein is extended without the assumption of un-


ikiiiiiidiL. figure 6.27 ECVT signals for Band 4 demonstrating pulse removal and envelope restoration.


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
A NEW ADAPTIVE ALGORITHM FOR THE REAL-TIME
EQUALIZATION OF ACOUSTIC FIELDS
By
Jeffrey James Spaulding
May 1992
Chairman: Dr. Jose C. Principe
Major Department: Electrical Engineering
This dissertation presents a solution for the problem of acoustic equalization.
Acoustic data are collected from a microphone at a listener location, and compared
with a source (CD player). An adaptive signal processing algorithm, based on the Least
Mean Squares (LMS), warps the CD signal to account for the filtering effects of the
listening room. The algorithm to adapt the coefficients of a multirate equalizer is
computationally efficient and can be performed in real-time with current
microprocessor technology.
As music is nonstationary, the LMS algorithm will need to undergo rapid
convergence to a new set of optimal filter coefficients each time the input signal
statistics change. As the LMS is a gradient descent algorithm, fast convergence implies
a step size selection which operates at the algorithms edge of stability. An analytic
expression exists to determine the step size for uncorrelated input data as a function of
the maximum eigenvalue of the input data. This dissertation extends this work to cover
vi


figure 6.26 The ECVT signals plot is given for band 4 (3675-22050 Hz) for an 11.33
msec epoch This frame is representative of successful equalization.


110
figure 4.10 The eigenvalues for an autoregressive process are indicative
of the degree of input data correlation, a) Gives a surface
representation of the test cases used for the norm
experiments for filter orders of 45; b) Superimposes the
logarithm of the eigenvalues for each test case.


figure 6.20 Time domain improvement is measured by comparing the
difference between the (CD signal microphone signal)2 and
# 'j
(CD signal equalized signal) as shown in the upper trace.
Note that positive values for the lower trace indicate
improvement in the time domain signal received at the
listener location after equalization. This figure covers 800
mseconds of data.


figure 6.4 The initialization tableau of the Equalizer Concepts Validation Tool allows parameters to be set for the
LMS and DLMS algorithms.


ACKNOWLEDGMENTS
I would like to express my gratitude to my advisor, Dr. Jose Principe, for his
encouragement and support through the course of my studies and research. I would also
like to express my gratitude to my supervisory committee members, Professor Taylor,
Professor Childers, Professor Green, and Professor Siebein.
My work would not have been possible without the help and support of the students
of the Computational Neuroengineering Laboratory. I am deeply appreciative. In
addition I would like to express my thanks to Dr. Richard Henry and Mr. Phil Brink of
the United States Air Force for their constant support.
I would like to thank my friends and family. It has been difficult to have not had the
time to devote to the people I love. I appreciate their understanding. I wish to especially
thank my grandmother, Mrs. Ruth Moore, whose bequest made my studies possible.
Finally, I wish to thank my mother and father without whom I would never have had
the strength or courage to complete my program.
in


CHAPTER 3
LEAST MEAN SQUARES (LMS) ALGORITHM CONVERGENCE
WITH UNCORRELATED INPUT DATA
The LMS algorithm converges quickly to an approximately optimum solution (in the
mean square error sense) given that the step size has been properly initialized. Because of
the low computational complexity of the LMS algorithm, it has been selected as the adap
tive algorithm which will be implemented in the AP architecture. Because the LMS algo
rithm is a gradient descent technique, there is a trade-off between convergence speed and
misadjustment. Misadjustment measures on average how close the LMS solution is to the
optimum solution, after algorithm convergence. A smaller step size results in a smaller
misadjustment. For fast convergence and low misadjustment an adaptive step size is re
quired. At the beginning of a quasi-stationary segment of data, the value of the step size
must be initialized in such a way that convergence speed is maximized. In this stage of the
adaptive process the lack of convergence speed provides the major contribution to mean
square error, and misadjustment is unimportant. As the filter converges, the step size must
be reduced because of the increasing dominance of misadjustment.
Chapter three will investigate the optimal value at which the step size should be set at
the beginning of a quasi-stationary segment of acoustic data. The initialization problem has
been rigorously solved for uncorrelated input signals. To efficiently utilize the LMS algo
rithm on music signals understanding must be extended to the case of correlated input data
(see figure 4.1). Chapter three will provide the necessary background information for this
extension. Chapter three first presents the Wiener-Hopf formulation of the minimization
of mean square error. The characteristics of the error surface are highlighted as part of this
discussion. As a result it is clear that a stochastic gradient descent algorithm is a feasible
approach to the minimization problem. The properties of gradient descent algorithms are
discussed with particular emphasis on the LMS algorithm and it properties. Finally, the
convergence speed of the mean square error for the LMS algorithm is minimized as a func-
63


xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008222300001datestamp 2009-02-16setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title A new adaptive algorithm for the real-time equalization of acoustic fieldsdc:creator Spaulding, Jeffrey Jamesdc:publisher Jeffrey James Spauldingdc:date 1992dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082223&v=0000126708651 (oclc)001758321 (alephbibnum)dc:source University of Floridadc:language English


\
figure 2.0
The Adkins-Principe equalizer architecture is a parallel multi-rate filter bank.


129
It is reasonable to assume that equation 5-9 will be valid for those value of p at which
II F*|| is approximately equal to (1 4[iXmax + 12pA,^ax)k. Figure 5.9 plots the percent
age error in the estimate of || F3|| by (1 4pX + 12 )3 as a function of step size
and eigenvalue spread. The results indicate the existence of a large domain of p for which
the one eigenvalue approximation of Fk is valid.
figure 5.9 The norm experiment indicates a large domain for p and
eigenvalue spread for which the one eigenvalue
approximation of Fk should be valid.
r
Figure 5.10 compares the one eigenvalue approximation error of F3 in the formulation
(equation 5-9) as a function of eigenvalue spread and step size. Even for a small number
of divergent iterations a value of step size can be found, regardless of the eigenvalue spread,
for which the approximation error is arbitrarily small. Note that the percent error of the ap-


43
figure 2.2 A Gaussian noise source is used with many state-of-the-art
equalizers because of its similarity to the pressure
distribution of acoustic data in an enclosed space.
pink noise source is of sufficient duration to charge the reactive elements of the listening
room, which will allow a better equalization of the rooms steady state response.
Bandpass Filters
Bandpass filters are designed to have bandwidths that approximately match the band
width characteristics of our human physiology. All high quality equalizers employ a digital
1/3 octave parallel filter bank, and from figure 1.12, it is seen that this resolution will closely
match the critical bandwidths as discussed in the psycho-acoustics review. The center fre
quencies and upper and lower cut-off frequencies are given in figure 2.3.
To maintain pitch the bandpass filters must be designed with sufficient control of the sig
nals spectral characteristics, while realizing that too fine a resolution will not improve our
subjective sense of pitch and will add processing time and expense to the equalizer. The
filter bank is made up of linear FIR lowpass and highpass filters cascaded to form bandpass
filters. Performance is limited by the low digital cut-off frequencies associated with the


36
ique. The experimental approach can be summarized as follows. The listening room is
excited by a broadband noise source. A measurement is made of the acoustic pressure at
the listener location. This measurement is compared with the source signal band by band
with an octave filter bank. An inverse of the room is approximated by ensuring the energy
distribution of the source signal is maintained by the measured signal.
| Reason to Reject
Wave Theoretic
Approach
Geometric
Optics Approach
l 1. Acoustic Impedances of Materials
| Required.
X
X
! 2. Complicated Boundary Conditions.
X
X
s 3. Computational Complexity.
X
X
i 4. Resultant Filter Orders.
X
: 5. Discemability of Important
Room Modes.
X
6. Limited Accuracy.
X
i 7. Time-Averaged Properties Only.
X
figure 1.16 The figure demonstrates the reasons that modelling the
physics of room acoustics is not feasible for solving practical
room inverse problems.
The discussion of psychoacoustics makes clear that acoustic high fidelity requires main
taining signal pitch, and the level of spatial and temporal coherency, with sufficient spectral
resolution to match the psychoacoustic and physiological characteristics of human hearing.
Equalization can not restore true audio fidelity of the acoustic field as perceived at the lo
cation of the signal recording. Without a significant increase in the sophistication of the
recording process, coherence information will continue to not be encoded on the CD. As a
result, properties such as clarity, spatiousness, apparent source size, etc. will not be restored


93
correlated input. The mathematical expressions which result are simplified, and conditions
on jj. for maximum convergence speed for C, are found. Simulations are subsequently car
ried out for different degrees of correlation, step size, and filter order.
Conditions on it for Convergence of $
Adding correlation to the input data significantly complicates the expression of £ for
the LMS algorithm. The approach developed below formulates a recursive expression for
Vk in terms of V0. The expression is substituted into the equation for £ (equation 3-79).
With the assumption that e£ is small, cross terms including can be ignored. This re
sults in a matrix equation for £ in terms of the initial conditions of the algorithm. The ma
trix norm is investigated numerically as a function of ji. These results are then compared
with large ensemble averages of £ computed with the LMS algorithm. To begin, recall the
expression for the LMS algorithm (equation 3-86) formulated in the translated coordinate
system.
Vk+1 = (/-2|lXtX')Vi + 2|U*tX*
(4-1)
Recursively formulate Vk in terms of VQ using equation 4-1.
k
(term 1)
v*=n[/-2^-/wivo+
7 = 1
k\ ,k i' (teim2>
/=1V=1
(4-2)
2Ve*k-lXk- 1
(term 3)


figure 6.21
The ECVT signals plot is given for band 3 (612.5-3675 Hz)
for a 68 msec epoch of acoustic data.


133
The approximation error for can become large if the misadjustment along an axis
becomes significantly greater than the misadjustment along the principal axis associated
with the maximum eigenvalue. Fortunately these errors tend to cancel themselves out in
the DLMS algorithm because the critical measurement involves a ratio of mean square er
rors. Figure 5.13 provides the approximation error of XMAX. XMAX *s calculated using
equation 5-12 and compared with the results of equation 5-7. For the following experiment,
P = c/c3-
figure 5.13 Because of cancellation effects in the DLMS algorithm the
one eigenvalue approximation for XMAX is significantly
better than the estimate of £3. The DLMS method provides
a good approximation for a robust domain of ji and aQ.
The Effects of Filter Order
The results of the DLMS simulations are excellent for a filter order of two. For the al
gorithm to be useful for acoustic equalization, the procedure must operate well for signifi-
.f


23
Psvchoacoustics
An equalizer will not be capable of exactly reproducing the time varying pressure field
recorded in a concert hall or recording studio. The engineering compromises which must
be made in designing the equalizer must focus on restoring the most psycho-acoustically
significant features of the original signal. The design methodology will be such that impor
tance will be placed on maintaining only those features of the waveform which a listener
will perceive in an average home listening room. This brief introduction to psycho-acous
tics is intended to motivate the signal processing objectives of the adaptive equalizer. Spe
cifically, the level to which signal coloration can be accepted without a loss of listener
enjoyment is discussed. This is largely determined by the frequency response and frequen
cy resolution of our hearing, and of the masking properties of the signal itself. A descrip
tion of how the room-induced temporal and spatial characteristics of the signal contribute
to a pleasing sound is investigated. Finally the dynamic range of human hearing is de
scribed. Stereophony is not discussed as this research does not attempt to process signals
to alter their stereophonic properties.
Coloration
In the time domain a system neither adds coloration to or removes coloration from a sig
nal if its frequency response can be characterized as shown below.
h(t) = kd(t-T) i1"62)
The function implies no change in the shape of the waveform. The output is a delayed and
amplified (or attenuated) replica of the input. Performing a Fourier transform of the im
pulse response function gives the frequency domain restrictions for such a system.
//(CO) = ke~im
(1-63)


119
figure 5.1 For each level of eigenvalue spread the one eigenvalue approximations
(dashed curves) are plotted with the true value (solid curves) of
In (^3) The percent error is also plotted to the right.
figure 5.2 it is clear that for any level of eigenvalue spread, given a sufficient number of it
erations the one eigenvalue estimation error can be made arbitrarily small. As the eigen
value spread approaches one, the number of iterations for a given error will become large,
but for typical values of eigenvalue spread (D > 1.1), k = 4 results in an one eigenvalue


117
vf 0
T
XQ (l 2fii.0) 24 0 0 0 ... 0
vf 0
V-1

0 Xj (1 2|iX1) 2* 0 ... 0
*
1

0 0 ... 0


0 0 ... 0

.4-.
0
0 0 0 0 ... X (1 2|iX ) 24
-vt-n_
C* = C + X h (1 V 2*vt? (5-3)
i = o
c.
n n
?+'£\(l-2vXl)2t4f X. (1 2m.) 2tvt?
1 = 0
1 = 0
9Jfc-l
c*+ 2>,(i-2i\)2(*_1)vt? x^(1-2i,v2 1=0 1=0
(5-4)
C X (l-2)lX )2tvt,2
3 (n, {X.}, k) such that mx
1 'fe-i
max' max
X (1 2\\X )2^-1)vt
MAX v UAVJ
MAX'
MAX
= (1 -2jxA, )2 (55)
v ^ max'
It is a simple matter to determine XMAX by solving the resultant quadratic equation. Let
P = C,k/t,k_ l be the measured value, and note also that p 1.


135
as the value of the step size becomes larger, the approximation error becomes smaller. For
tunately the accuracy of the algorithm does not rely on the accuracy of the estimate of C,k,
but on the accuracy of C>k/C,k_l- Figure 5-16 provides the percent error of the DLMS ap
proximation of A,MAX as a function of step size, Oq and eigenvalue spread. A value of y
is found for each value of Oq and eigenvalue spread such that for P = £4/£3, the estima
tion error of ^MAX, using equation 5-12, is less than 10%.
figure 5.15 The percentage error of the one eigenvalue approximation of
£3 is shown above for as a function of |i and eigenvalue
spread.


158
However, if the LMS algorithm operates well, then it is possible that either < ^max
and t>0.05, or and y<0.05. The refutation of these possibilities will be left
to others, and a successful value of ji* will be taken as good evidence for the validity of the
theoretical concepts put forward.
The ECVT plots show the results of the diverging LMS algorithm by plotting divergent
iterations vs. error2. From the rate of divergence, is determined and the LMS step
size is properly set. Recall from chapter five that there is a delay in the application of an
updated value of |X after the statistics of the CD signal have changed. This was due to the
need to approximate the ensemble behavior of the diverging LMS algorithm with a time av
erage. In the ECVT plots for band 4, the spikes in the solid line representing ftp0wer indicate
the times at which the updated value of (i, determined by the DLMS algorithm, is actually
applied to the update equation of the LMS algorithm.
Test Resets
Data presented in this chapter is a portion of Paul Harts Concerto for Guitar and Jazz
Orchestra. Data was collected in the Computational Neuroengineering Laboratory at the
University of Florida. The room is approximately 8 X 15 meters. The speaker and micro
phone geometry was such that the most prominent energy arriving at the microphone had
undergone one reflection off a wall. The microphone signal has experienced severe atten
uation of high frequency content. ECVT signals plots, waterfall plots, and learning curves
are presented for four bands. The results indicate that the technology proposed in this dis
sertation represents a successful approach to acoustic equalization.
Test Results for Band 1 (dc-102 Hz)
Figure 6.11 presents a representative ECVT plot for band 1 for a 0.5 second segment of
audio data. The amplified microphone signal indicates a traditional equalizer will perform
well, because good equalization requires simple amplification across the entire band. It is
also seen that the adaptive equalizer automatically provides the proper amplification. Finer
spectrally dependent amplification is not required. The ECVT plot indicates that while the
amplified microphone signal may have some higher frequency content attenuated, the adap-


33
sufficient clarity for fast staccato passages. A value of -3 dB is usually acceptable for sym
phonic music.
c =
r 80msecj
lOlog
J h2(t)dt
0
J h2(t)dt
_80m sec s
d-67)
Reverberation time of a field is a measure of how the room damps out an acoustic
source. It is usually determined by measuring the time required for the field intensity to
decrease by 60 dB from the moment the acoustic source is turned off. The listening rooms
reverberant field acts to mask the details of a music performance which listeners find un
pleasant, e.g. a minor lack of synchrony in an orchestra. The ideal reverberation time de
pends on the type of music being performed, and on local styles and current fads. For
chamber music a reverberation time consistent with the rooms in which the compositions
were originally performed is preferred, which is from 1.4 to 1.6 seconds. For symphonies
from the romantic period or pieces utilizing large choirs, a reverberation time of over 2 sec
onds is preferred. For opera a shorter reverberation time is preferred so that the libretto may
be more easily understood. As an example, the La Scala opera house in Milan has a rever
beration time of 1.2 seconds [11].
Psvchoacoustic Response to the Spatial Aspects of Reflected Energy
For a room to have a good acoustic ambience the listener must have a sense of spatious-
ness. Originally researchers believed that spaciousness was caused by hearing spatially in
coherent fields. It has been shown, however, that this gives rise only to a phantom source
direction, and does not provide a sensation of spaciousness. The following conditions are
required.
i. The field must be temporally incoherent.
ii. The intensity of the reflected energy must surpass the audibility threshold.


figure 6.33 When white noise is added to the microphone signal, the
adaptive filter performance is greatly improved. Figures
6.32 and 6.33 make clear that all important frequencies
present in the CD signal must also be present in the
microphone, even if highly attenuated, in order for good
equalizer performance using the test architecture.
Fortunately the proposed real-time architecture will not
suffer from this problem.


67
With a stationary signal in which the cross correlation matrix and correlation matrix
were perfectly determined, it would be possible to solve the Weiner-Hopf equations for W*
and £*. Unfortunately in engineering applications of interest neither of these assumptions
is generally true. The field of adaptive signal processing is concerned with estimating W*
without the necessary a priori information required by the Wiener Hopf formulation.
Error Surfaces
Equations eight and ten provide important information regarding the characteristics of
the error surface, which is defined as £ = f(W). The most critical result is the hyperpar
abolic nature of the surfaces. This is demonstrated by changing the coordinate system used
in the expression for Let V = W W* be the value of the displacement of the filter
weights from the optimal value. Express MSE in terms of this translated coordinate system,
i.e. £, = f(V) The transformation is accomplished as follows. Recall equation 3.8.
C = E[dll + WTRW-2WTP (3-8)
Equation 3-13 is inspired by assuming the proper solution of f(V) and demonstrating its
equivalence to equation 3-8.
(3-13)
c = E[d\] + WTRW-2WTRR~1P + PTR~1RR~1P-PTR~1P
Substitute W* for R~XP andC for E [d2k] PT\f.
£ = £* + WTRW-2WTRVf +PTRTlRW*
Because correlation matrices are Hermitian, (PTR~l)T = R~lP.
(3-14)


31
figure 1.13 As the echo period, Tq, increases, the frequency components
are more closely separated. Due to masking, coloration
becomes less audible.
In addition to our sensitivity to the frequency domain, our hearing is affected by time
domain properties of reflected waves. Reflected energy which is just barely perceived acts
to increase the loudness of the signal and to change the apparent size of the source. At a
higher loudness and a small delay, the perceived source direction moves. For a large delay
the reflected energy is perceived as a highly annoying echo. Figure 1.15 shows the intensity
at which a distinct echo is perceived as a function of time delay.
If the delay is sufficiently short, the reflected wave can be 10 dB higher than the direct
wave without causing echoing. This phenomenon is know as the Haas effect after the dis
coverer, and is an important factor in the design of auditoria [10].
Acoustic clarity is a measure of our ability to resolve fine temporal structures in acoustic
waves. The faster the tempo and the higher the degree of attack on the instruments, the
higher is the necessary degree of room clarity. Many measures have been suggested. All
of them rely on determining the amount of acoustic power over a period of time represent
ing early reflections divided by a normalization. One index used for determining the clar
ity of rooms for musical signals is given in equation 1-67. A value of C = 0 dB provides


28
Determining exact design criteria to guarantee a true reproduction of pitch and timbre
is made difficult by the complicated dependency on signal frequency content, loudness,
masking, and the transient behavior of the energy distribution in the harmonics of instru
ments. For example, if most of the time signals above a certain frequency are masked by
lower frequency signals, it would not be important to control the attenuation of these re
gions of the spectrum. The simplest analysis assumes the worst case scenario in which
masking does not occur, regions of the spectra are of a loudness just at the audibility limit,
and frequencies must be controlled at a level at which any error in loudness would be below
the 0.5 phon detection limit. To maintain timbre we wish to be able to control transients in
the signal caused by the excitation of instruments. Fortunately the phase of the signal will
not have to be carefully preserved because of the relative insensitivity of our hearing to
these effects.
Practical implementation of these requirements is possible only if the frequency reso
lution of hearing is sufficiently large that filters of reasonable order can carry out the nec
essary inverse filtering. Frequency resolution for two sinusoids is known as the difference
limen. It has a relatively constant value of 3 Hz below 500 Hz. Above 500 Hz frequency
resolution decreases approximately as a linear function of frequency (see figure 1.11).
= 3 Hz for f< 500 Hz
0.0030/ for f > 500 Hz 1'64)
T
Clearly it will be necessary to control pitch and timbre with a much finer degree of resolu
tion at low frequencies than at high.
The human hearing organ can be described as a parallel filter bank with characteristic
bandwidths. Fletcher and Munson [9] found that the detection threshold of a tone in broad
band noise is not affected by the noise bandwidth until it falls beneath the critical band- '
width, which are shown in figure 1.12.


17
Temporal Properties of a Listening Room
Each reflection of an acoustic wave must be described by its delay in arrival from the
direct acoustic wave at the listener location, the direction from which the reflected wave ar
rives at the listener location, and the strength of the reflection. In this section we will dis
cuss the time delay and strength of reflected acoustic energy. The rate at which reflected
acoustic energy arrives at a listener location is an important factor in evaluating the quality
of a listening room. Ray theory provides a simple and relatively accurate method of deter
mining the roomss temporal properties. This theory is completely analogous to ray theory
from geometric optics, and it will therefore be applied with a minimum of discussion. Like
geometric optics, geometric acoustics ignores interference and refraction effects. Never
theless it is accurate for rooms of ordinary size for acoustic frequency content in excess of
1000 Hz.
If a room is constructed of a series of planes, image sources can be found by succes
sively reflecting the original source and each subsequent image source about the planes in
a lattice constructed by using the room as a unit cell. The lattice and a cross section plane
are shown in figure 1.5. A spherical shell is chosen with inner radius at distance ct and outer
radius c(t+dt), where dt is much less than t. The number of sources in the spherical shell
corresponds to the number of mirror reflections between t and t+dt. The volume of the
spherical shell, assuming dtt, is Atzcit1dt. Note from ray theory that there is only one
source per room. The number of image sources can be found by dividing the volume of the
shell by the volume of the room. As t increases the number of mirror sources increases as
t2.
Number of Reflections =
47tc3/2
dt
(1-52)
As time increases, the reflections have less and less energy associated with them. The
higher order reflections lose energy at each reflecting plane, the amount of which depends
on the characteristic impedance of the wall. An approximation of how the energy decreases
can be made from geometric acoustics by multiplying the initial energy of the direct ray in-


157
er for some bands than others. In each band the equalized signal is given the proper gain to
match the in-band power levels of the CD signals.
Two theoretical propositions have been posited in this dissertation: 1) that can be
estimated by operating the LMS algorithm in such a manner that the algorithm diverges,
and 2) p*can be approximated by where y is a function of input data correlation,
and filter size. These propositions are evaluated by noting whether the estimated maximum
eigenvalue of the input vectors are smaller than the value estimated by the power of the in
put vectors. The power estimate of X^^ acts as the highest acceptable value, because it
estimates the largest eigenvalue by measuring the sum of all eigenvalues. The improved est-
mation will be indicated by a larger LMS step size in the ECVT plots. In addition it will
have to be demonstrated that the step size calculated with the DLMS algorithm, and the val
ue of y determined in chapter four, results in a finite variance of the square error of the LMS
algorithm. A step size for which the variance of the mean square error becomes large acts
as an upper bound for an acceptable y. The increase in step size using the above proposi
tions, and the values for the LMS algorithms square error, are shown in the ECVT plots for
characteristic segments of data.
The validation approach documented here provides strong evidence concerning the va
lidity of the propositions of chapters four and five, although it does not refute all possible
objections. Since the theoretical understanding was well validated for a filter order of two,
it can be considered presumptive evidence of the validity of extrapolating these approaches
to arbitrary order. In addition these methods were tested independently of each other. In
the validation approach of chapter six, p* is approximated by coupling the methods togeth
er for a filter of order 45. Failure of the algorithm to meet the properties set forth in the
preceding paragraph would demonstrate reasons for serious concern for the validity of ei
ther or both methods used in approximating p*. Success of the technique would not defin
itively demonstrate the validity of both chapters four and five. It is possible that these
methods, while not valid, nevertheless counter each other in such a manner, that when
working together p* is properly determined. More explicitly, if p* causes poor perfor
mance of the LMS algorithm, then X^'^5 < ~kmax, and/or y<0.05 (for filter order 45).


140
Eigenvalue Estimates for 50 Experiments Distribution of Eigenvalue Estimates
figure 5.19 The effects of replacing ensemble averages with time
averages in the estimation of ^max=1.8. a) The evolution of
logferror2) for 20 divergent series of five iterations each of
the DLMS algorithm; b) The divergent series superimposed;
c) The approximations of the largest eigenvalue for 50
experiments using the DLMS algorithm and the power
approximation; d) The probability density function of the
DLMS algorithm approximation of
must be re-initialized. A number of diverging series are averaged to obtain an estimate of
C*/C*_ p and the value of Xmax, and hence ji, is determined.
As indicated in figure 5.21, a copy of the CD signal and the microphone signal are sent
to the DLMS module. The CD signal is given a sufficient delay in order to align it with the
microphone signal. Both signals are passed to the adaptive filter bank. In figure 5.23 the
adaptive filters in the filter bank are expanded, and the operation of one stage is demonstrat
ed. Note that each adaptive filter in figure 5.21 is actually a bank of filters which estimate
the ensemble characteristics of the DLMS. The degree of correlation between data used for


the case of correlated signals such as music. This result provides verification of
heuristic rules that have been proposed in the literature.
To date, formulations involving the maximum input signal power have been utilized
as an estimate of the maximum eigenvalue. In this dissertation a new method of
determining the maximum eigenvalue is proposed by allowing the LMS algorithm to
diverge. For a large and consistent domain of initial condition, iterations, and purposely
divergent step sizes, the maximum eigenvalue dominates the rate of algorithm
divergence. Simulations were pursued to determine the bounds of these variables for
robust operation. This methodology was utilized to analyze the performance of the
adaptive equalizer on selected music epochs, and to validate the theory put forward in
this dissertation.
Vll


82
figure 3.4 The larger step size results in faster convergence, but larger
excess MSE. For this reason the equalizer design will utilize
a varying step size.


187
frequency features were enhanced, giving the sharpness present in the CD signal. These
results are presented in figure 6.33. In the proposed architecture for room equalization, the
CD signal is used as the input to the adaptive filter, and the lack of frequency information
will not arise. The problems which occurred in testing may have been reduced if a higher
quality microphone had been used for data acquisition.
Frequently the estimated value of will be slightly overestimated. Although the es
timate is not sufficiently poor that the effects shown in figures 6.28 through 6.31 occur, high
frequency noise is injected into the equalized signal. Note that although the step size re
duction algorithm is operating, the damping is not sufficiently large to entirely alleviate the
problem. An example is shown for an 11.33 msec epoch shown in figure 6.34. In order to
correct this difficulty, the damping can be increased, but this approach can also introduce
serious problems. The step size can be reduced so rapidly that the algorithm adaptation rate
can not keep pace with the nonstationary nature of the acoustic signal.
Poor performance can be introduced by a failure to trigger the DLMS algorithm. Fro-
mobserving the learning curve and the value of the power normalized estimate of step size,
in the region marked in the box in figure 6.35, it is clear that the DLMS algorithm should
have been triggered. One or more of the conditions indicated by the block diagram in figure
6.6 failed, and as a result the adaptive filter did not converge quickly enough to restore the
pulse clearly seen in the CD signal. Once the DLMS algorithm is triggered, and a new step
size is initialized, the equalized signal again closely resembles the CD signal. The simple
heuristic algorithm suggested in this chapter is not adequate for breaking up the CD signal
into quasi-stationary segments. The development of segmentation algorithms is an active
area of research.
A related problem is the delay in calculating after the DLMS algorithm has been
triggered. In figure 6.36 the step size is too large, as shown in the marked section of the
epoch, and the DLMS algorithm is triggered. While the DLMS algorithm is calculating the
new value of the LMS algorithm continues adaptation with the wrong value of [i. In
this case, the equalized signal begins oscillating at 22.050 kHz, because of the under-
damped adaptation. Other epochs will experience the reverse problem, i.e. step size is


2
posed. Chapter two will discuss in detail the theory and operational characteristics of the
Atkins-Principe architecture. It will be shown to be an elegant and extremely efficient im
plementation strategy. Chapter two will also discuss the relevant considerations in choos
ing an adaptive algorithm for real-time updating of filter coefficients. On the basis of
algorithmic complexity and the speed of the current generation of microprocessors the
Least Mean Squares (LMS) algorithm will be justified as an appropriate algorithm for this
application.
The LMS algorithm can be described as a stochastic gradient descent method which
converges due to its time-averaged behavior. For fast convergence speed and low misad-
justment in a nonstationary environment the algorithm must have an adaptive step size
which can be precisely controlled. When the statistics of the acoustic signal change dra
matically, equalizer settings will be incorrectly set and the adaptation algorithm will need
to converge as quickly as possible to the new optimal equalizer settings in the least mean
squares sense. LMS step size is the key parameter to adjust in order to maximize conver
gence speed. The theory guiding the optimal value of step size is well-understood for un
correlated input data [2-3]. Chapter three will present the theory of LMS convergence for
the case of uncorrelated input data. The theoretical structure of the Wiener-Hopf formu
lation, the stochastic gradient descent algorithm, and the LMS algorithm with uncorrelated
data will be discussed to provide the necessary framework for the more complicated case
of the LMS with correlated input data. Chapter four will extend the theory of the LMS al
gorithm for the convergence properties of mean square error for the case of correlated input
data. This theory is necessary to understand adaptation performance for acoustic data
which by nature are correlated signals.
The divergence properties of the stochastic descent algorithm and the LMS algorithm
are discussed in chapter five as a simple extension of the convergence properties examined
in chapter three. The results of this study indicate that the maximum eigenvalue dominates^
a diverging gradient descent algorithm. It will be further shown that the optimal value of
the step size, after a change in the statistics of the input data, can be well-approximated as
a function of the maximum eigenvalue of the vectors input into the filters performing adap-


168
Figure 6.20 shows the learning curve for the 800 msec data epoch, which includes the
data from figure 6.16. Behavior of the equalizer results in improvement for most of this
segment of data. When the signal statistics of the CD change, performance of the equalizer
is temporarily poor, but the adaptation rapidly results in a filter configuration which im
proves the signal quality.
Test Results for Band 3 (612.5-3675 Hz)
The test results for band 3 closely resemble those for band 2. Figure 6.21 shows a typical
ECVT plot for a 68 msecond time epoch. Once again the equalized signal restores the CD
signal envelope, and high frequency content. The learning curve in the ECVT plot indicates
the sub-optimal consequences of not reducing the LMS step size after algorithm conver
gence. These effects are most clearly demonstrated by observing the waterfall plots.
Figures 6.22 through 6.24 give the waterfall plots for four seconds of acoustic data in
band 3. Notice that although frequency content is restored in the equalized signal, distinct
spectral features become diffuse. Consider the features marked in boxes 1 and 2 in all three
waterfall plots. Although the equalized signal restores power into the major spectral fea
tures, they become distinctly blurred. In fact the equalizer degrades the sharpness even
when compared with the microphone signals. Note also that energy is bleeding into fre
quencies below the lower cutoff frequency of band 3, as demonstrated in box 3.
The learning curve for a 136 msec epoch of acoustic data from band 3 is shown in figure
6.25. Once again improvement is seen over almost all the epoch. The adaptation algo
rithm is working to the extent that each band is given an appropriate gain and is properly
normalized to the power present in each band of the CD signal. In addition, more gain is
being given to certain parts of the bandpass than others in order to enhance spectral features
that have been particularly attenuated. The learning curve, however, is not indicative of the
degradation of sound quality induced by the blurring effects discussed above. It will be
shown that this difficulty becomes more limiting in band 4.


CHAPTER 5
DETERMINATION OF k WITH A DIVERGENT LMS (DLMS) ALGORITHM
MAX
It was shown in chapter four that p.* is approximately equal to the value of ji for which llnll2 is
minimized. Step size, jx, is approximated by y/Ama;c, and thus p* can also be expressed as the
value of y which minimizes llnll2. Except for highly correlated input data, 0.05 < y* <0.16, for
filter orders of less than 45. When the error surface changes dramatically, the input data vector is
not highly correlated (a< 0.95), as the most current data sample is statistically unrelated to previous
data samples. It is when such a change occurs that step size should be re-initialized. To do so, only
the value of AMAX is left to be determined. Typically ^MAX is approximated as trace [R] This
approximation can in many cases be poor. Consider the situation in which Aj = 1.0 and
A2 = A3 = .... = A50 = 0.5. The trace of R is 25.5 times greater than AMAX. Even in a less ex
treme case, determining AMAX more precisely will allow greater control of the convergence prop
erties of the algorithm. It will be shown in this chapter that the divergence properties of gradient
descent algorithms are dominated by AMAX. By observing how the LMS algorithm diverges, the
maximum eigenvalue of the input data can be accurately estimated.
Determining AMAX with a divergent LMS algorithm will be investigated in detail. It is routine
to examine the behavior for the gradient descent algorithm before applying it to the more compli
cated LMS algorithm. Chapter five will follow this convention. Proper operation of the divergent
gradient descent algorithm depends on the selection of the divergent step size, |i and the number
of divergent iterations, k. These parameters will be discussed at length. The more complex case
of the divergent LMS algorithm will be examined, and a parametric analysis of the estimation error
of ^MAX will be performed. Finally the integration of the DLMS algorithm into the adaptive equal
izer will be discussed.
Determination of Ami^ with a Divergent Gradient Descent Algorithm
Note that the divergent gradient descent algorithm forces the error surface to be ascended. The
word descent is retained in the algorithm name for convenience, since only the domain of jj. is
115


194
can be found for step size reduction. The effect of the blurring of the spectral features is a
loss of clarity in the audio signal. This effect is obvious and annoying to the listener.
A learning curve is shown in figure 6.42. While there is improvement across most of
the epoch, it is not nearly as dramatic as results presented for bands 1 through 3. Recall
also that much more computational effort was put into band 4, namely a more sophisticated
step size initialization, and a heuristic step size reduction algorithm.
In order to evaluate the band by band equalization in more detail over a greater length
of contiguous time epochs, additional data have been included in the appendix. Band 4 is
emphasized in particular, and 17 contiguous time epochs are shown in ECVT plots. In ad
dition, contiguous ECVT plots are shown for bands 1 through 3.


29
Psvchoacoustic Responses to the Temporal Aspects of Reflected Energy
Reflections under certain circumstances serve to enhance the quality of sound by adding
to the direct wave in a way which is reinforcing. For musical signals the reverberation of
the room when within certain bounds gives a pleasing effect. Under other circumstances
figure 1.11 The difference limen gives the frequency resolution of two
sinusoidals. An acoustic equalizer will need to control the
frequency response of low frequency bands with a finer
resolution than high frequency bands [5].
reflections can cause highly undesirable effects. Although many effects can be explained
partially through the physiology of hearing, perhaps the most important factors, at least in
the reproduction of music, are the types of environments with which we have been habitu
ated and from which our aesthetic senses have been influenced.
The discussion of the temporal effects of sound fields begins by finishing our discussion
of coloration. Strong, evenly-distributed echoes can cause serious degradation in the accu
rate reproduction of an acoustic waves spectral distribution. Consider a room with the fol
lowing simple impulse response.


66
** = E [***£] =E
Vk
%-i


Vk-n
xk-\xk
1-H
1
K*
1


xk \xk-n










I
1
a
Vk-I


1
Si
1
K
1
(3-7)
The mean square error can now be expressed in a simpler notation. Let £ = E [ej¡] .
i DrW/,w,r
= £[d|] -2FW'+W'KjtW
(3-8)
To minimize C, set -r=-7 to zero and solve for the optimum weights, W*
dW
dt
dW
-2P + 2RW = 0
(3-9)
W* = R~lP
(3-10)
The above equation is the expression of the Wiener Hopf equations. Substituting equation
10 into equation 8 and making use of the symmetry of R results in an expression for the
minimum value of .
C* = E [d\] 2PTR~lP + (R~lP) TR (R-'p) (3-11)
= E[dl] -PT\f
(3-12)


19
During the discussion of psychoacoustics, the temporal properties of an audio field, which
generally indicate a pleasing listening room, will be discussed.
degree of shading represents
the number of reflections per unit time
figure 1.6 An echogram indicates that as the time delay from the direct
field increases, the number of reflected waves increase, and
the intensity decreases.
The rate of decay in the intensity field is an important factor in creating a good quality
listening room. Recall that the intensity of a wave decreases as a function of the square of
the distance travelled, (ct)2. The intensity of the field will also be decreased due to the at
tenuating influence of air (or any medium through which the wave traverses). Let a be the
e-folding time constant of this attenuation. Each time the acoustic wave reflects off of a
wall, the intensity will decrease by the square of the wall reflectance. The decay of the in
tensity of the acoustic wave is summarized in the equation below, where Iq is the intensity
at t=0, and n is the number of reflections of the ray.
70 -actr,2nt ^0 act + ntln\R\2
~e K
(ct)2 {ct)2
(1-53)
It is not possible to follow the path of each ray and perform the above calculation. In
stead the calculation is made by computing the average number of reflections of a ray (az),


200
choacoustic considerations). A measurement technique, in which the settings for the filters
which make up the equalizer are determined, is capable of providing good performance in
this more limited definition of equalization.
Chapter Two
The current state-of-the-art equalization strategy operates by exciting the room with a
broadband noise source (e.g. pink noise), and measuring its response at various listener lo
cations with microphones. The collected signal is passed through an 1/3-octave filter bank
to achieve the approximately correct frequency resolution. The power in each band is com
pared with the radiated signal, and filter coefficients are calculated to increment or decre
ment the power of the radiated signal, to match the desired characteristics as input by the
listener.
Chapter two reviews each stage of this technology, and outlines their weaknesses and
limitations. The algorithm proposed in this dissertation eliminates the need to re-equalize
a room when a listener moves locations within the room. More importantly however, the
proposed strategy makes a more efficient use of the limited degrees of freedom available to
the equalizer filters.
These improvements are made by modifying the Adkins-Principe equalizer architec
ture, which is an 1/3-octave filter bank which uses the principle of multi-rate sampling. By
downsampling each band to the Nyquist rate, a higher degree of resolution is available for
a given filter order. By adding adaptive filters to increment or decrement the relative
amount of power in each band, the efficiency of the filters is further increased. Adaptive
filters will automatically apply the most amount of processing resources to those areas of
the band where the room is having the largest effects. In contrast, the conventional FIR fil
ters would equally apply processing resources across the band regardless of the degree of
filtering action of the room, or the amount of power present in the audio signal. Because
the adaptive filters update at the sampling rate of the bands, a change in listener location
during playback of the audio signals will not require a new equalization to be started from
scratch.


212
figure A.8 The ECVT signals plot is given for band 3 (612.5 3675 Hz) for the
second 68 millisecond epoch of acoustic data.
figure A.9
The ECVT signals plot is given for band 3 (612.5 3675 Hz) for the third
68 millisecond epoch of acoustic data.


CHAPTER 1
BACKGROUND
Introduction
For millennia architects have designed structures with the intent of minimizing distortion of
acoustic signals propagating from a speaker or a musical instrument to a listening audience. From
the Greek amphitheaters of antiquity to the modem concert halls of today technologies developed
over the centuries have resulted in sophisticated techniques which have greatly improved acoustic
fidelity. Only in this century has technology been sufficiently advanced to approach the problem
of acoustic distortion from the perspective of equalization. Equalization anticipates the filtering
introduced by the enclosed space and pre-processes the acoustic signal to take these effects into
account. In this way when the acoustic signal arrives at the listener, the effects of the equalizer
filtering and the room induced filtering cancel each other.
The revolutionary development of the Compact Disk (CD) player, first introduced to the market
in 1982, has completely changed the direction of audio engineering. Research in audio engineering
is being directed towards creating all-digital audio systems. With the tremendous improvement in
microprocessor speed, audio signals can now be processed digitally in real-time. The research pre
sented in this dissertation will provide a design for an equalizer which will be capable of real-time
self-adjustment. The design will be focussed on providing a system to perform equalization in a
normal home listening room with standard audio components. It will be based on a new equalizer
architecture developed by Atkins and Principe at the University of Florida [1], which uses a multi
rate filter bank. The Atkins- Principe architecture is enhanced so that filter coefficients in each
band of the filter bank will be continuously updated in real-time, with coefficient updates being
provided using principles from adaptive signal processing.
In chapter two background information will be provided on the current state-of-the-art of acous
tic equalization. Background information on room acoustics, psychoacoustics, and the state-of-the-
art of equalizers will lead to conclusions regarding the limitations of acoustic equalizers. An im
proved strategy of adaptive equalization based on the Adkins-Principe equalizer architecture is pro-
1


46
Limitations
In the discussion of room acoustics the temporal characteristics of the acoustic field was
introduced using the principals of geometric acoustics. This approach gave a qualitative
understanding, but it is of limited accuracy and validity. The spatial characteristics were
also presented using a simplistic approach to acoustic reflections. A rigorous theoretical
foundation of room acoustics was made from the wave equation. By solving boundary con
ditions it was shown that the impedances of surface materials, and the rooms geometry are
the key parameters in developing a Greens function. Because of the oversimplification of
geometric acoustics, and the huge computational requirements of a wave theoretical ap
proach, filter coefficients required for an equalizer can not be calculated.
State-of-the-art equalizers determine the room response by exciting a room with a
broadband source. A measurement is made of the acoustic field and the listener location,
and an inverse of the room response is calculated for implementation in an octave filter
bank. The excitation source, signal measurement, signal processing, and equalizer filters
of state-of-the-art equalizers were reviewed. This technology is focussed on removing
those effects which are psycho-acoustically most significant. In the presentation of psycho
acoustics pitch was discussed as a key feature of the music signal. To preserve pitch the
equalizer must remove coloration introduced by the stereo electronics and the listening
room. Phase response was shown to be relatively unimportant in maintaining pitch. The
temporal and spatial characteristics of the acoustic field were discussed as important char
acteristics in giving a listener the sense of fullness and spatiousness of the sound.
The state-of-the-art equalizer technology described above is able to remove most color
ation of the stereo-room combination, and thus maintain pitch. The equalizer will not be
able to correct the following effects.
1) The equalizer will not be able to correct coloration due to strong regular reflections.
These types of effects are represented by transfer functions that are non-minimum phase,
and thus would require the equalizer to have poles outside the unit circle. This should not
be a problem unless the room in which the original recording was made, or the home lis
tening room has a terrible acoustic design, e.g. a rectangular cavity with one axis much


20
assuming the acoustic field is diffuse, i.e. the rays are uniformly distributed over all possible
directions of propagation. The assumption of a completely diffuse wave is never achieved
in an ordinary listening room, and the attenuation of the reverberant field will have large or
small deviations from the simple calculations being outlined depending on the amount by
which the actual acoustic field differs from our assumption of diffuseness.
First express the number of reflections per second which occur at walls perpendicular
to the x-axis for a ray propagating at an angle 0 with respect to the x-axis, nx(Q) .
nx (0) = ~j~ COS0
(1-54)
Average over all angles 0.
Similar results are obtained for (ny) and (nz). The average number of reflections along
all three axes is expressed below where S is the surface area of the room and V is the room
volume.
Substituting the above into the intensity equation yields the following.
d-57))
The reverberation time is taken as that time at which I(t)/Io = 1X10'6, and the expression
for this time is known as Sabines Law.


equalized agnal muse*
the puke ia the CD gnal

l':ll#IIBllliilli; 1
If
TW DI MS algorithm Is not properly triggered
resulting Id i value of p whkb la too auaL This,
In turn*causes the equalizer to mis a restoring a
fealurtt&frc CD data because of
CD agnal pulses
learn
poorleanng
cbarachlehstics
DLMS algonlhm
i* triggered too
late s~^~>
4-
4
V
Y
divergent iteration
iK|
figure 6.35 ECVT signals for an 11.33 msec epoch with delayed triggering.


112
Y-0.025
Y-0.125
tests 4 to 6
)MX025
7*0.075
I7t(7;k)l
figure 4.12
he left-hand plots give the three dimensional plot of
nk (y) as a function of k and y, where p = for the
three next most uncorrelated sets of test data. The right-hand
plot superimposes all cross sections of the surface plots.


60
yk represents the output of the feedforward filter at time k, dk represents the desired (or
reference) signal, and ek represents the difference signal between dk and yk (error signal).
figure 2.15 An adaptive tapped delay line with desired signal, dk, and
error signal, ek.
Assume that the sequence {Xk} is quasi-stationary, i.e. the signal has been chosen over a
time window in which it is approximately stationary. The output of the tapped delay line
is the inner product of the n+1 most current filter weights with the filter input.
y* =
xlwk=
XkXk-\Xk-2*
m
kn
0
(2-8)
The filter weights, W*, which minimize the mean square error, E [ (dk yk)2], are ex
pressed by well-known Wiener formulation: W*=R*1P, where R is the matrix E[XXT] and
P is the cross correlation between d and X, E[dX]. The field of adaptive signal processing
is concerned with approximating the solution for W*. There are two standard methods by
which the solution is approximated: gradient search algorithms and least squares (LS) al-


74
1
1
(l-2nx0)*
0
0
0
0
vto
vtjfc-l
0 (1
-2^)*
0
0
vt_i

=
0
0

0


0
0

0


.

.
0

vt*-
0
0
0
0
... (l-2xn)k
_vt -n
(3-34)
The convergence condition can be seen from equation 3-34. If o < ji < then with perfect
^max
knowledge of the gradient vector, the weight vector will converge to the optimal solution.
For
0<|i<
1
max
lim
k <*>
= 0
(3-35)
Convergence Speed
The speed of convergence is limited by the value of step size, and hence, the largest ei
genvalue. Consider convergence along coordinate j. A geometric convergence factor can
be defined as follows.
r¡= (1 -
(3-36)
Recall the power series for an exponential.
2 3
exp(x) = l+x+Ji + ^T+
(3-37)
Let Xj =
Then x. is approximately the e-folding time of the adaptation process, as
2|1A.J J
can be seen by truncating the first two terms of the exponential power series.


122
figure 5.3 Estimation error of C, is plotted above as a function of
divergent iterations, k, and step size, y, where p. = y/^max,
for a filter of order two and input data with an eigenvalue
spread of 1.01. The contour plot is shown below.


156
Specifically, learning curves for time domain analysis, and power spectral density plots for
spectral domain analysis will be generated for these signals.
Power spectral density waterfall plots are created using the Spectro program, written by
Parry Cook from the Stanford Center for Computer Research in Music and Acoustics. The
spectra are generated in such a manner that there is a 50% overlap of data segments used
in the 1024 point DFT calculations in order to improve temporal resolution. Hamming win
dows are used on all segments of data to reduce truncation effects. Waterfall plots are gen
erated for the CD, microphone, and equalized signals. Similarity between the CD and
equalized waterfall plots is interpreted as successful equalization.
Although the ear is predominantly a frequency domain sensor, the assumption will be
that rapid time convergence of the equalized signal to the CD signal will corresponds to rap
id convergence of the power spectral densities. Time domain convergence is a stricter stan
dard than convergence for power spectral density. Time domain convergence implies con
vergence of both the power spectral density, and the signal phase. Furthermore, the
assumption will be that convergence will occur without generating temporary, intermediate,
and audible noise in the spectra. This assumption is validated by observing the concomitant
waterfall plots. Convergence in the time domain is measured by generating learning curves
which show the values of (CD signal microphone signal)2 and comparing them with the
value of (CD signal equalized signal) These data are presented using the SignalEditor
program written by Armando Barreto of the Computational Neuro-Engineering Laborato
ry. The top panel generated by this software superimposes both curves, with the former rep
resented by a dashed line, and the latter represented by a solid line. The lower panel rep
resents the improvement in the equalized signal by subtracting the former from the latter.
From the learning curve it is seen that the square error of the equalized signals decrease
after a change in the CD signal statistics, which is indicative of convergence of the LMS
algorithm in the time domain. The square error of the microphone signal is typically much
larger than the equalized signal, due to the significantly reduced power of the acoustic sig
nal collected by the microphone vis a vis the direct electrical connection of the CD signal
into the digitizing microphone. However, it is seen that this difference is significantly high-


105
I ni2 = /(n; iteration) | n|2 = /(^;4)
figure 4.6 Norm plots, a) The norm of is calculated as a function of
LMS iteration and y, where according to equation
4-9 for uncorrelated input data. The approximation is
consistent with the results of Horowitz and Senne; b) the
cross section of the 3-D plot given in (a) is given for LMS
iteration=4.
figure 4.7
The norm was generated for the highly correlated case in a
analogous method used for figure 4-4.


55
sampling rate of band 1.
(N-l)
(2-5)
For band i the total delay as seen at the output of the highpass filter is expressed as follows.
(N-l)
2
5>#
i = 0
(2-6)
The delay at all nodes of the inverse tree structure is dominated by the value of Td in the
upper branch. The total delay as seen at the output of the AP structure for a four band sys
tem is dominated by Td4 and is given below.
ltotal
= 27. -
4
(N 1.) -3_
M Tsl
(2-7)
Processing Speed
Efficient use of computing resources can be made because of the differing sampling
rates of data in various stages of the architecture. Convolutions are performed at different
rates and are staggered in time. Data is sampled such that multiply and accumulate opera
tions are made in non-overlapping time slots. In this way major timing bottlenecks are
avoided. The CD player sampling rate determines the value of the time slots (1/44,100 =
22.676 microseconds). For the four band design 10 HR filters are operated. Three filters
perform convolutions at the highest sampling rates (44.1 kHz), three perform convolutions
at the CD rate decimated by six (7.35 kHz), three perform convolutions at the CD rate dec
imated by 36 (1225 Hz), and one convolution at the CD rate decimated by 216 (204 Hz).
The timing diagram is shown in figure 2.12.
As an example of the computational load of the convolutions indicated in the timing di
agram above, consider the TI TMS320C40 25Mflop floating point digital signal processing


98
two norms, apply the Cauchy-Schwartz theorem to equation 4-8.
1/2
c < [||n7n||2] e[\\xxt\\]) v{
o
(4-9)
Normalize the step size by as follows, fi = where y is a constant. IT
can be expressed as a product of relaxed projection operators, n = TkTk_ t r0, where 7} is
expressed by equation 4-10. Step size is represented by |X = (2y||Xf||2) /^max in order to
f21xill2l
i
, X
V max '
P/fl2J
(4-10)
preserve the geometric interpretation of T Consider that Pk = ['- <**#] projects
a vector onto subspace S, which is the space that has vector normal Note that T can be
expressed in terms of P, and an orthogonally complementary projection operator, I-P. It is
(2
rk->-
r
a-pk>
max
= P
k+
2i^ir
max /
a-pk)
(4-11)
projection onto Sk
projection onto
relaxation term
now possible to determine the conditions under which T contracts. Let T operate on a vec
tor of appropriate dimension, y. If y is restricted to the domain D^(J) = P>^OT£KC/||^/||^]
T¡[y] = P [y] +
l -
27IIXII
2s
(y-P¡ fyi)
(4-12)
v
max y


152
same portion of the original signal, and the offset is determined. When the signal alignment
option is selected, a panel is displayed for the user to enter the thresholds. After the delay
is determined, a graphics window displays the two signals with the CD signal appropriately
shifted. A block diagram of the alignment algorithm is shown in figure 6.7. The panel and
graphics windows are shown in figure 6.8.
In the real-time architecture, it will not be possible to artificially allign signals. As a
result the equalizer filter coefficients will be updated on information delayed by the propa
gation time of the acoustic wave from the speakers to the microphone. This causality prob
lem of adaptive control is fundamental to systems with larger delays. Adaptation in a
normally sized living room will result in feedback control approximately 10 msec in the
past. Highly transient features will be non-optimally handled.
The equalize menu allows for the equalization of any of the four bands as described in
chapters two through five. During equalization, panels indicating the key parameters of the
algorithm are displayed. An example of the graphical information provided is shown in fig
ure 6.9. For 500 point segments of data, the microphone signal, CD signal, the equalized
output, the square error between the CD and equalizer output, and the step size are plotted.
A representation of traditional equalization is given by the dashed trace superimposed on
the microphone signal. An amplification has been given to the microphone signal so that it
matches the power of the CD in the band of interest. For the highest frequency band (3675
- 22050 Hz) the power normalized LMS algorithm step size is superimposed on the step
size as determined by the DLMS algorithm. The scale of the plots of the microphone, CD,
and equalized signals is normalized by the largest value of the CD signal in the 500 point
window being displayed. When the DLMS algorithm is operating, a separate panel dis
plays the diverging sequences of data. An example is shown in figure 6.10, where the x axis
represents the divergent iterations, and the y axis represents the logarithm of the square er
ror.
After all four bands have been equalized, the signals are upsampled to the original CD
sampling rate, and added together taking into account the different filter delays associated


219
[18] M. Honig and D. Messerschmitt, Adaptive Filters: Structures. Algorithms, and
Applications. Klower, Boston, 1989.
[19] B. Widrow and M. Hopf, Adaptive Switching Circuits, IEEE WESCON
Convention Record, pp. 96-140, 1960.
[20] Norbert Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time
Ssries.ii£nginggring Applications Wiley, New York, NY, 1949.
[21] B. Widrow, J. McCool, M. Larimore, and C. Johnson, jr., Stationary and
Nonstationary Learning Characteristics of the LMS Adaptive Filter, Proceedings
of the IEEE, Vol. 64, pp. 1151-1162, August, 1976.
[22] B. Widrow and S. Steams, Adaptive Signal Processing. Prentice-Hall, Englewood
Cliffs, NJ, 1989.
[23] O. Macchi and E. Eweda, Second-Order Convergence Analysis of Stochastic
Adaptive Linear Filtering, IEEE Transactions on Automatic Control, Vol. 28, pp.
76-85, January, 1983.
[24] L. Ljung, Analysis of Recursive Stochastic Algorithms, IEEE Transactions on
Automatic Control, Vol. 22, pp. 551-575, August, 1977.
[25] T.P. Daniell, Adaptive Estimation with Mutually Correlated Training Sequences,
IEEE Transactions on Systems and Cybernetics, Vol. 6, pp. 12-19, January, 1970.
[26] D.C. Farden, J.C. Godingjr., and K. Sayood, On the Desired Behavior of Adaptive
Signal Processing Algorithms, Proceedings of the 1979 IEEE International
Conference on Acoustics, Speech, and Signal Processing, 1979.
[27] H. Fursenberg and H. Kesten, Products of Random Matrices, Annals of
Mathematical Statistics, vol-31, pp. 457-469,1960.
[28] R.R. Bitmead and B.D.O. Anderson, Lyapunov Techniques for the Exponential
Stability of Linear Difference Equations with Random Coefficients, IEEE
Transactions on Automatic Control, Vol. 25, pp. 782-787, May, 1980.
[29] D.H. Shi and F. Kozin, On Almost Sure Convergence of Adaptive Algorithms,
IEEE Transactions on Automatic Contol, Vol. 31, pp- 471-474, May, 1986.
[30] Ariel Corporation, Operating Manual for the DM-N Digital Microphone,
Highland Park, NJ, 1989.
[31] R. Harris, D. Chambries, and F. Bishop, A Variable Step Adaptive Filter
Algorithm, IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol.
34, pp. 309-316, April, 1986.


79
Excess MSE = [tff A (/ 2j*A) 2yiVtjfc]
7
(3-54)
= [l2£
atT^a
<- 7
ATf*
(3-55)
Since / 4pA + 4p2 A2 is a diagonal matrix, the convergent infinite series may be simplified
as follows.
y* (/ 4pA + 4p2A2)^ = (4pA-4p2A2) 1
7 = 0
Substitute this expression into equation 3-55.
Excess MSE =
(3-56)
= ^[^(/-nA)-1^]
In matrix form the decoupled nature of equation 3-57 is evident.
(3-57)
¡ess MSE
f
1
0
0
1
o
i-i0
o
1
o
0
0
[4*4*_i 4*_]
u l
0
nXj
0

.


0
0
0
1
1 '0
4*
"t*_i
4*_
(3-58)


51
In the more realistic case of a room which filters its acoustic input, the highpass filters are
designed with user supplied gains for each band. The graphic interface used on an equalizer
implemented on the NeXT computer workstation [15] is shown in figure 2.7. The interface
indicates gains for seven bands, and interpolates between the frequencies indicated below..
Begin Pause
Stop
*I.j 2 9312S8 92 j-1*432836 -j-4 266888 <4 j-423S74 tt|-6S1971 B.jo 7 ¡5 817022
J SOU! ¡"¡25(12 ¡30002 ¡750112 ¡Tti ¡46WT j 1 Ikhz
figure 2.7 The graphic interface for an equalizer using the AP
architecture was implemented on the NeXT computer with a
user interface as shown in this figure. The highpass filters are
adjusted according to the selections made by dragging the
slides with a mouse.
The combination of filter length and decimation rate must be such that the requirement
of at least 1/3 octave resolution is satisfied. In addition, band ripple, transition bandwidth,
and stopband attenuation must be such to guarantee that there is less than ldB band ripple
across the entire audio bandpass. The research documented in this dissertation will use the
filter design presented by Adkins and Principe [1]. Their extensive simulations resulted in
a recommended design of 4 bands, each utilizing symmetric finite impulse response (FIR)
filters of order 45, and a constant decimation rate of six. A summary of the frequency res
olution of the design is given in figure 2.8.


138
The results of the divergence experiments indicate that the divergence characteristics of
the LMS algorithm for correlated input vary substantially from the uncorrelated case. In
the development of the DLMS algorithm it has been assumed that uncorrelated input data
is utilized. Fortunately, the critical measurement of the DLMS algorithm is the ratio of ^
to £ j As the correlation increases, i.e. asa 1.0, P decreases, and generates an error
in the estimate of A. From the plot below it is seen that this error will be small (less than
10%) unless the input is highly correlated (a > 0.9).
7 a=0.005
/ 7
/ 7 '
/ cc=u.ytx
/ 7 .' i
/ 7 /
/ 7 /
/ // /40.98
/ /
/ 7 '/ /
J /
/ 7
/ / /
/ / 4
/ /a=0.99
a' / /
/ / *
' / /
y/7
1
0 ~ 2 4
divergent iterations
0 0.1 02 03 0.4 03 0 0.7 O 0.9
correlation
% error of L a). estii
estimate
iuwmm4HHiiinii|iiimin|in
0 0.1 02 03 0.4 03 0 0.7 0
correlation
figure 5.18 The critical measurement of the DLMS algorithm is the ratio
of C, to L, . As the correlation increases, i.e. asa 1.0,
the measurement decreases, and generates an error in the
estimate of A.
max


figure 6.34 When the DLMS algorithm underestimates Xmax, the step
size reduction algorithm is able in most cases to keep the
LMS algorithm from pathological behavior. Nevertheless,
high frequency noise is often injected into the equalized signal.


207
tion was used as presumptive evidence that the techniques could be extended to arbitrary
orders. Both techniques were used together for the selection of LMS step size, and the con
vergence properties of the resulting mean square error were observed. Results, which are
strongly indicative of the validity of both propositions, do not refute the possibility that both
algorithms, while not operating as expected by the theoretical development previously pre
sented, counter each other in such a manner that they appear to operate properly, e.g. the
DLMS algorithm overestimates and the predicted value of y* is too low. To lessen
the appeal of this objection, the theoretical results which predict y* and ^ should be
compared with actual LMS experiments for a wider range of filter orders.
Fast Kalman filters were briefly discussed as another adaptation algorithm well-suited
to audio equalization. It is recommended that a software implementation be made, and the
results of equalizations be compared with those obtained via the LMS algorithm.
AUcmatiY£-Appligations
A multi-rate filter bank employing adaptive filters would seem an appealing architec
ture for many types of applications. The application that seems most immediate is as a mea
surement device for the characterization of room responses. The architecture used to
validate the theoretical performance of equalizers could be used directly for this purpose.
The techniques would have the advantage of high spectral resolution because of the varying
sampling rates, and a concentration on characteristics of the room where the most filtering
is occurring, because of the embedded adaptive filters.
Noise cancellation, which is closely related to audio equalization, could make use of the
proposed architecture and algorithms. The techniques would be most useful in environ
ments with broadband noise.
The DLMS algorithm may be useful in communication systems, especially those in
which a training sequence is available, or data which does not have input with a large ei
genvalue spread.
It is hoped that this research, while incomplete, has made a contribution to the theory
of adaptive signal processing, and the development of solutions to audio equalization.


197
figure 6.39 The waterfall plot for equalized data in band 4 (3675 22050
Hz) is shown above for a 1.8 second segment of data.


45
field itself. In minimizing this effect, high quality microphones are made as small as pos
sible while maintaining their sensitivity. In addition measuring the pressure field intensity
is not synonymous with measuring signal loudness as was shown in the previous section.
The conversion from intensity to loudness can be accounted for in software. A more diffi
cult matter is to completely characterize the microphone directionality, which will differ
from the directional properties of our hearing. In practice a precise understanding of the
microphones directional response is unavailable. A high quality microphone must also
have a fast response if it is not to smear the details of the signal, affecting the perceived tim
bre and clarity.
Signal Processing
Processing is performed by a dedicated chip which calculates the power in each band
collected from the CD player. The same procedure is performed on the signal received from
the microphone. The two signals are compared, and a calculation of the appropriate filter
coefficients is made in order to guarantee a flat frequency response, or any other preset fre
quency response, across the audio bandpass. Unfortunately the details of the processing are
not discussed in the literature, probably because of sensitivity towards proprietary technol
ogy on the part of manufacturers. The measurements of current state-of-the-art equalizers
indicate they are capable of maintaining a flat frequency response across the audio bandpass
to within ldB. At 40 dB between 100 and 1000 Hz for speech and music signals a band
ripple of ldB is not perceptible. At the band edges ripple can be as great as 2 dB before
listeners can perceive coloration. The number of bits used for filter all processing calcula
tions must be at least 16 in order to avoid degrading the S/N of the CD player. High quality
equalizers allow separate equalization to be performed for each channel of the stereo. The
period of time required for the equalization can be as high as 15 seconds. The filter coeffi
cients can be stored in one of several memories. A graphic interface is provided to allow
the user to manually set his listening preferences, which are taken into account in the cal- -
culation of filter coefficients. These systems are a large improvement in traditional equal
ization technology which required time consuming manual adjustment of graphic
equalizers, and depended on the skill of the individual making the adjustments.


25
perceived as a signal of approximately 160 mels. If the fundamental is removed, the dif
ference signal is unaltered, and the listener still perceives a 160 mel pitch.These effects
are not quite so dramatic in music signals as the harmonics are usually considerably less
intense than the fundamental.
Assume that a series of non-harmonically related frequencies do not mask each other.
If we wish to maintain pitch, the relative loudness of the frequencies must be unaltered by
the electronic equipment or the listening room. For a pure sinusoidal it is possible to hear
changes in loudness of only 0.5 phons [6]. Note also that the perceived loudness is a func
tion not only of the acoustic intensity of the signal, but also the signal frequency. Figure
1.9 illustrates this effect for two equal loudness contours. For frequencies over 1000Hz.,
as frequency increases, the acoustic intensity must increase to maintain the same loudness
level.
Surprisingly, controlling the phase response has no appreciable effect on maintaining
pitch. The phase response, except in extreme cases, cannot be perceived by listeners hear
ing music signals. Thus while a lack of phase distortion is required for the shape of the


128
Parameter Selection for the Divergent LMS Algorithm
In this section a parametric study will be made of those values of p for which the as
sumptions given in equation 5-8 and 5-9 are valid. The proper selection of p depends on
{ X.} The DLMS algorithm performs well for as little as three divergent iterations, as will
be shown in the proceeding plots. As a result, the number of divergent iterations will not
be included in many of the simulations which follows. Figure 5.8 demonstrates the validity
of equation 5-8 for £3 as a function of step size and eigenvalue spread. The approximation
is computed using equation 5-8 and compared with the value calculated by equation 5-7.
The value of has been chosen to be a larger than expected value, namely, = 0.5£Q.
The percent error is less than 1% in all cases.
figure 5.8 The approximation of equation 5-8 is valid for £3 over a
robust range of p and eigenvalue spread, even when is
large.


30
ACO = 2<"5(f-nr0)
n = O
d-65)
figure 1.12 Because our hearing can be described as a parallel filter
bank, acoustic equalizers are designed to simulate the critical
bandwidths High quality equalizers are designed to have
1/3 octave bands because of their similarity to critical
bandwidths, as seen above.
The power spectral density indicates equally spaced resonances.
1
2 (1-66)
1 -2acoscor0 + a
Figure 1.13 illustrates the frequency response of the room for different echo periods, Tq.
As Tq becomes larger the frequency components become less separated. Because of mask
ing effects, the threshold, at which the spectral distribution of the audio signal is perceived
to change, decreases as Tq increases.
Figure 1.14 gives results for the threshold level at which an average listener perceives
noticeable coloration induced by a reflected wave. These measurements are based on lis
tening tests of six different music selections.


209
figure A.2 The ECVT signals plot is given for band 1 (dc 102 Hz) for
the second 0.5000 second epoch of acoustic data.
figure A.3 The ECVT signals plot is given for band 1 (dc -102 Hz) for
the third 0.5000 second epoch of acoustic data.


134
cantly larger filter orders. In the following set of figures the filter order is increased to ten
in order to provide evidence that the DLMS can be extended to an arbitrary order. In figure
5.14 the LMS algorithm is allowed to diverge for three iterations will a variety of step sizes
and eigenvalue spreads. Initial misadjustment is equal along each principal axis. The min
imum mean square error is chosen to be large, namely, £* = 0.5£0. The results of calcu
lations using equations 5-8 and 5-9 are overlaid.
figure 5.14 The approximation given in equation 5-8 and 5-9 are
overlaid on the true value of £3 for a filter of order 10. Initial
misadjustment was equal along each of the principal axes. In
order to rigorously test the simplifying assumptions of the
DLMS the value of Z¡* was set high, namely £* = 0.5 £Q.
The approximation using equation 5-8 for £3 is excellent, especially for a high level of
eigenvalue spread. The approximation using equation 5-9 is considerably less accurate, es
pecially for low eigenvalue spreads. The percentage error of the one eigenvalue approxi
mation of ^3 is demonstrated for levels of eigenvalue spread in the figure 5.15. Note that


62
Algorithms which minimize error in the frequency domain operate on blocks of data.
As such, they update filter coefficients at regular intervals, introducing a noise component
into the signal. In addition, they require more operations in order to transform the data. Al
though we are most interested in minimizing the error of the magnitude response of the
transform, good time domain equalization will guarantee good frequency domain equaliza
tion as well.
The recursive algorithms are more promising. They converge rapidly, and the weight
vector is updated every sample. These algorithms unfortunately have a heavy computation
al load. Figure 2.16 gives the computational requirements of the most familiar versions of
the RLS algorithm [18].
REQUIRED COMPUTATIONS PER ALGORITHM ITERATION
ALGORITHM
MULTIPLICATIONS
& DIVISIONS
ADDITIONS
SQUARE
ROOTS
Modified Fast
Kalman (MFK)
5N + 13
5N + 13
Fast Kalman
8N +5
7N + 2
Growing Memory
Covariance (GMC1
13N + 6
11N + 1
Sliding Window
Covariance (SWC)
13N + 13
12N +7
Normalized GMC
12N + 16
7N +2
3
Normalized SWC
23N
11N
5N
Normalized PLS '
9N + 17
5N + 2
2
figure 2.16 The computational complexity for several adaptive
algorithms are displayed above.
All algorithms, except the Kalman types, required significantly more computations than
the LMS algorithm (2N + 1 multiplications and 2N additions). On the basis of computa-
tional considerations, the algorithms seriously considered for the equalizer are reduced to
these two alone. The choice has been made to utilize a fast LMS method as the adaptation
algorithm. Chapters three and four will investigate LMS algorithms in detail.


72
Substitute the above expression for R into equation 3-24.
yk+l = U~^QAQ-l)Vk
(3-25)
Note that the following transformation has the effect of decoupling the weight vectors in
the translated coordinate space, i.e. the transformation causes the new coordinates (princi
pal coordinate system) to be colinear with the eigenvectors of the error surface.
yt = Q~^V (3-26)
With the above transformation the coordinates are aligned with the principal axes of the hy
perparaboloid. This can be shown as follows.
£ = C + VtRV
(3-27)
= C + VT(QAQ-l)V
(3-28)
= C + (QTV)TA(Q-'v)
(3-29)
= C + VfrAVf
(3-30)
The transformation is illustrated in figure 3.3. Returning to the convergence proof, substi
tute equation 3-26 into equation 3-25.
Qtfk+i = U-2iiQAQ-1)QVtk
(3-31)
Premultiply by Q ^.


148
From figure 5.21 it is clear that to specifically test the DLMS module, no modification
in processing procedure will be necessary. The only signal requirements necessary to exe
cute the algorithm are the CD signal and an unequalized signal collected by the micro
phone.
Test Signal Acquisition
Signals sent to the adaptation algorithm consist of a CD signal generated by a Sony
CDP-110 compact disc player and a signal collected at the listener location by an ARIEL
DM-N digital microphone. The digital microphone houses a two channel A/D converter
with an adjustable sampling rate. A block diagram is shown in figure 6.2.
The two microphone elements consist of cardioid electret condenser cartridges that
convert sound pressure waves into analog voltages. The elements are mounted such that
they are 45 degrees from the centerline and 90 degrees from each other. Twin amplifiers
increase the signal voltage to a level more appropriate for A/D conversion. Two in-line
mini-jacks allow a direct electrical input to the A/D converters. The amplified microphone
signals are ignored when the mini-jacks are utilized. Twin gain controls allow the operator
to balance the magnitude of the input signals. Sixteen bit A/D converters measure signal
amplitudes at intervals inversely proportional to the sampling rate. Output from the con
verters are serial data buffered and transmitted by cable to a 5MHz serial port on the NeXT
workstation. Total harmonic distortion at 1000 Hz is less than 0.005%. The in-band ripple
across the audio bandpass is less than 0.003dB. Out-of-band rejection is greater than 96dB
[30]. The CD player is connected to one of the mini-jacks. Acoustic data is collected by
the other channel.
Equalizer Concepts Validation Tool
In order to form the LMS and DLMS algorithm computations, an objective C code has
been developed for the NeXT workstation. The package of routines provides the ability to
record and playback NeXT sound files using the ARIEL digital microphone, and equalize
audio data using both LMS and DLMS algorithms. Recall that implementing the DLMS


Internet Distribution Consent Agreement
In reference to the following dissertation:
AUTHOR: Spaulding, Jeffrey
TITLE: A new adaptive algorithm for the real-time equalization of acoustic
fields (record number: 1758321)
PUBLICATION DATE: 1992
I, [t'dC as copyright holder for the
aforementioned dissertation, hereby grant specific and limited archive and distribution rights to
the Board of Trustees of the University of Florida and its agents. I authorize the University of
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This is a non-exclusive grant of permissions for specific off-line and on-line uses for an
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This grant of permissions prohibits use of the digitized versions for commercial-use or-profit.
Personal information blurred
Date of Signature
Please print, sign and return to:
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5/28/2008


5
of the discussion is based on introductory texts of Egan[4], Kinsler et al. [5] and Kuttruff
[6].
Basic Equations and Definitions
Sound waves are longitudinal waves of regions of greater and lesser pressure. An
acoustic field is described by a wave equation which expresses the deviation from ambient
pressure, p, as a function of spatial location r, and time t. The constant of proportionality
between the temporal and spatial derivatives is the speed of propagation of the medium, c.
c2Vp
(l-l)
For a time harmonic plane wave the solution of the pressure equation is of the same form
as the expression for a plane wave of electromagnetic radiation. By separation of variables
the pressure equation reduces to the following familiar form, where k is the wavevector.
P ( r, t)
i (cot-kr)
= Poe
d-2)
The velocity of the vibrating particles, v, is related to the pressure field according to the
conservation of momentum. The density of the medium is represented by P.
V7 9v
Vp = -ps
(1-3)
By solving the above equations we can relate the pressure field to the velocity field by
an acoustic impedance, z = pc.
V(r, /) = V0ei((t kr)
m
s
(1-4)
p(r,t) = pcV(r,t) = zV(r, /)
d-5)


87
Because {x¡} is Gaussian and uncorrelated, Xk and Wk are statistically independent. Fur
thermore, Xk{dk- XlW*) = Pk RkW* implies independence. Thus Xk can be factored
out of the second term of equation 3-77. Since Xk is zero mean, the term vanishes altogeth
er.
et = c + E [ (Wk W*) TXkxl(Wk W*) ] (3-78)
Since Xk is uncorrelated, E [X^J] is a diagonal matrix and £ can be expressed as follows.
= c +race {E[(wk-Wt) (Wt-W)r] E } (3-79)
= c* + trace {£[V^[] £ [XkXTk] } (3-80)
ct = r+x.rot i)
Horowitz and Senne next develop a recursive relationship for (7 from the definition of the
LMS algorithm.
Wk+l = iI 2(lXix[] + 2\ldkXk (3-82)
Subtract W* and perform a unitary transformation. Let ek = dk-X\wk .
vtjt+i = [/-2H(x/x/T)]v/+2^*xtt e-)
Take the expectation of Vt multiplied by its transpose. Because of the independence of
ek and Xf the expectation of their cross terms is zero.


103
As an example of the validity of the approximations made in the derivation of equation
4-6, note the results presented in figure 4.4. Figure 4.4 shows £>k generated form a first or
der autoregressive process with a = 0.9512. The solid line represents ^ created from an
ensemble average of square error (100 independent identically distributed systems) gener
ated by running the LMS algorithm. This plot is overlaid by a dashed curve which gives
the predicted mean square error from the approximation expressed in equation 4-6. The
convergence properties are similar. The approximation has been tested for a wide range of
correlations, namely, 0.01 < a < 0.9995. Figure 4.5 shows results for the two extremes.
figure 4.4 Figure 4.4 indicates that the approximation given by
equation 4-6 is valid for a relatively highly correlated
autoregressive input process. The solid line represents C,k
created from an ensemble average of square error (100
independent identically distributed systems) generated by
running the LMS algorithm. This plot is overlaid by a
dashed curve which gives the predicted mean square error
from the approximation expressed in equation 4-6.


83
Ck+1
figure 3.5 For quasi-stationary adaptation their are two contributions to
error in the gradient: 1) the error induced by the typical form
of gradient noise, and 2) the error induced by error surface
lag, i.e. the error induced by using information from a prior
error surface instead of the current error surface. In the
figure above the highlighted update, A^+1 is based on
information concerning C¡k instead of £,k+l-


61
gorithms. The gradient search methods can be generalized as follows:
Wk+1 = Wk- |iV where C, = E[(dk-yk)2] = mean square error. This common
sense approach adjusts the weights constituting vector W in a direction opposite to that of
the gradient of the error surface, ji represents a constant which regulates the step size of
the weight update. The most common gradient search algorithm is the Least Mean Squares
(LMS) algorithm which estimates V £ = V E [error2] by taking the time average of
V [error ]. The approximation is valid over quasi-stationary periods because of the qua-
si-ergodicity of the signal. The LS algorithms differ by estimating the value of R and P
with time averages. LS algorithms can be broken into two categories: algorithms which
calculate new estimates over a window of data (block LS algorithms), and algorithms which
recursively update the estimate after every sample (RLS algorithms). One example of an
LS algorithm is the block covariance method. Let R and P be the time averaged approxi
mations of R and P, and let co be a windowing function.
/
(2-9)
h = XVA
/
(2-10)
A ^ A \ W.
Wk = Rk Pk
(2-11)
Certain classes of algorithm can be ruled out of consideration immediately because of
their unsuitability with respect to one or more of the requirements given above. Although
the block LS algorithms converge more quickly than gradient search techniques, after every
stationary window of data the input data autocorrelation matrix, R, must be inverted. Some
of the block algorithms have autocorrelation matrices that are Toeplitz and can use the Dur
ban algorithm for matrix inversion, requiring 0(N2) operations, where N is the length of
the filter. Many of the LS methods(i.e. the covariance method), however, are not Toeplitz
and matrix inversion must be performed using the Cholesky algorithm, requiring 0(N ) op
erations. Furthermore, these algorithms update the weight vector every M samples intro
ducing an undesirable artifact into the spectrum of the acoustic signal.


26
wave to be unaltered by a system, the same requirement is not generally necessary for a sig
nal to be psycho-acoustically equivalent. The results of Suzuki, Morita, and Shindo[7] in
dicate that the effects of nonlinear phase on the quality of transient sound is much smaller
compared to the effect one would expect from the degree of change in the signal waveform.
figure 1.9 Equal loudness contours are presented as a function of
frequency and acoustic intensity [5].
Hansen and Madsen[8] found similar results. In their experiment a signal had all of its
frequency components shifted by 90 in phase, producing the most severe distortion pos
sible in the shape of the waveform while maintaining the magnitude response. Only for
fundamental frequencies below 1000 Hz was a distinctive difference perceived by the lis
teners involved in the test. The relative insensitivity to phase will be seen to be critical to
the operation of an adaptive equalizer.
Maintaining signal timbre is also a basic requirement in a good sound system. Those
features of the signal which are important to the timbre of sound are poorly understood, and
the study of timbre is still an area of active research. The transient behavior of an instru-,
ment being excited effects the initial behavior of the pressure field. The transients effect
the amount of energy present in each of the overtones. This is an important factor in deter-


59
phone is also moved) the filter will automatically correct for the changes in the rooms re
sponse because of the equalizers capability for real-time self-adjustment. Depending on
the type of adaptation algorithm chosen, the updates will be performed either every sam-
ple(44.1 kHz), or after every block of samples(44.1kHz/N, where N is the length of the
block). More importantly, the reference signal used for equalization is the 03 signal itself
instead of pink noise. The adaptive filter, when minimizing mean square error, will auto
matically utilize its limited degrees of freedom to adjust the frequency response in those
portions of the spectrum in which the most energy is present not the entire audio bandpass.
Several adaptation algorithms have been developed for adaptive signal processing
[17-19]. The choice of algorithms depends on several considerations, namely, the station-
arity characteristics of the signal, the data rate of the input signal, the degree of accuracy
required in filtering, and the computational resources available for real-time operation.
Acoustic data is highly non-stationary. Because the adaptive filters must find their optimal
coefficients in a fraction of the time in which the audio signal is quasi-stationary, the adap
tation process must be fast. Secondly, although the adaptation must occur rapidly, the al
lowable misadjustment of the filter weights must be relatively small for meaningful
improvement in the current state-of-the-art. Thirdly, due to the high CD sampling rate, the
complexity of the algorithm must not be so great as to preclude carrying out the arithmetic
in real-time. This is a serious concern even with the high speed microprocessors currently
available. Lastly, any adaptation algorithm must not introduce any noise components
which will lower the signal fidelity.
All available adaptation algorithms require trade-offs among the crucial factors outlined
above. Extremely rapid convergence can be achieved, but at the expense of increased com
putational complexity. Computationally efficient algorithms exist, but convergence is
slowed, and further serious trade-offs between convergence speed and misadjustment need
to be taken into account. Other algorithms may make a good compromise between com
plexity and convergence speed, but by their nature they contribute undesirable noise into
the system. Consider the tap delay line architecture for an FIR filter shown below, where


15
The constants Aj,A2,Bi,B2Ci, and C2 are determined by matching boundary conditions.
The solution of three specific examples are outlined.
Case i. Re[z]=0 and Im[z]=very large.
The method of solution of the three ordinary differential equations is identical. Consid
er the equation in x. As a boundary condition guarantee the conservation of momentum is
satisfied at the walls (x=0 and x=Lx).
Vp = ~p~r = --^-P=>^-P+-^-P = 0 (1-44)
at z ax z
dP
As lim|z| -4 o ^ 0 The boundary conditions are satisfied as follows.
£v\x = 0 = iKxAl-iKxA2=SAl = A2 0-45)
i,% = Lx = = 1;
(1-46) *
a is any integer. The two boundary conditions have been used to solve for A2 and Kx. Sim
ilar solutions are found for B2,C2,Ky, and Kz. Substituting these results in the expression
for Y (x, y, x) and the separation equation leads to a solution of the room eigenmodes and
eigenfrequencies as shown below.
x ^ ctnx Brcy ynz
x (x, y, z) = Dcos-ycos-zcos-y
Lx Ly LZ
/(a,P,Y) =
cK = c
2n 2
oc 2 B 2 y 2
(r) +(f) +(r)
L>x *->y LZ
(1-47)
(1-48)
The constant D=AiBjCi is determined from an initial condition.
case ii. Re[z] = 0.
Satisfy the new boundary conditions at x=0 and x=Lx. A solution for A2 and Kx can be
found by solving the two transcendental equations or by using methods form the theory of


57
verse tree structure. Each band has a distinct delay which must be accounted for when re
combining bands. Because of the octave band structure and downsampling, the critical
digital frequencies are raised as much as possible, and lead to a degree of bandwidth reso
lution and of band ripple, across the entire audio bandpass, that is superior to performance
advertized on existing equalizer architectures. From figure 2.5 it is seen that frequency res
olution is finer than 1/3 octave across the entire bandpass. At worst, resolution is 1/4 oc
tave, while at best the equalizer has 0.045 octave resolution. Computer simulations of the
AP architecture indicate for equalizer settings for a flat response, there is a band ripple of
less than 0.2 dB from 17Hz to 22500 Hz. This value is far superior to the values of 2dB
which are measured on equalizers which are commercially available. The spectral resolu
tion and the high degree of control of the highpass filters are precisely the characteristics
required to restore the proper pitch of the acoustic signal received at the listener location.
Automatic Adjustment Using Adaptive Filters
Elliot and Nelson [16] have proposed substituting the highpass filters in a standard filter
bank with adaptive filters designed to minimize the mean square of an error signal which is
generated by taking the difference between a pink noise source, and the noise source as col
lected by a microphone at the listener location. Although this approach has the advantage
of a lower computational complexity than the standard frequency domain techniques, the
most compelling reason for the use of adaptive filters is overlooked, namely, the use of the
music signal itself as a reference signal. This research will substitute adaptive filters for the
highpass filters in the AP architecture as shown in figure 2.13.
In each stage the CD reference signal is bandlimited and decimated in precisely the
manner discussed above. A microphone signal is filtered and decimated in an analogous
manner to the CD signal. A feedback signal (error signal), which is provided by the differ
ence between the filtered CD and microphone signals, is sent to an adaptation algorithm
which updates the adaptive filter weights in such a way that the mean square error is mini
mized. As an example, the structure for band 2 is shown in figure 2.14.


136
Eigenvalue Spread = 2
^uni^TMu)
4)
vP
0s
-1.5
2 1 Eigenvalue-Spread = 5
Eigenvalue Spread = 3
2 1 2 1 f EigenvalueSpread = 10
2 1 figure 5.16 The approximation error of xMAX as a function of |i and aQ
is less than 10% for all eigenvalue spreads, using
P = C4/C3-


figure 6.16 The ECVT signals plot shows a representative 400 msec time epoch for
band 2. Note that in addition to automatically providing the appropriate
gain, the equalized signal restores the CD envelope, and high frequency
information. Prominent features attenuated in the microphone are also
restored in the equalized signal. The learning curve behavior is limited
by a lack of step size reduction after the convergence of the equalizer
filter coefficients.


149
Block Diagram of the ARIEL DM-C Digital Microphone
I
KjSS
left
channel
right
channel
X " ,
figure 6.2 The ARIEL digital microphone block diagram[30] is shown above. One
channel collects acoustic data, while the other receives direct electrical
input from the CD player to the mini-jack. Data is digitized at the
standard CD sampling rate of 44.1 kHz.
algorithm in hardware for low frequency bands would require excessively long, and im-
practical delay lines to be constructed. Therefore the performance of the DLMS algorithm
will be evaluated in software only for the highest frequency band (refer to chapter five for
more detail). The validation tool will be explained below. The top level popup menu is
shown in figure 6.3.


94
Vk+, = (/-211**4)^+ 2|lVt
= (I-2llXkx[) [ (I-2¡lXk_lXk_ () j + 2(ie*j._ 1XJ,_ j]
+ 2|l/X
= (/-2|lZtX)(/-2|lZt_14_I)V4_,
+ 2ile\_, (/- 2(iarix[) JTt_, + 2^Vt
term 1
= (/ 2n*fcx[) (/ 2ixxk_ xxTk_ x) ... (/ 2n*0*5) yQ

a
+ 2e*t_0-2i,xtx¡)xk_
+ 2vSt 2 (/ 2iixkxk) (/ 2,^ _2
+ 2We*0(/-2iiJTtJr[) (j-2nxt_ jAr[_ j)... (/-2x1 x[)x
+ 2e*k_lxt_i
term 3
(4-3)
Assume that the filter order has been chosen to have a sufficiently large number of degrees
of freedom such that V& e* k = (dk X[W*) = 0. Substitute equation 4-3 into equation
3-79, and eliminate all cross terms in e* k.


32
0
dB
O
-10
20 40 60 80 100
echo delay [msec]
figure 1.14 As the echo delay decreases, the reflectivity of the room
surfaces must decrease in order to avoid a perceptible change
in the coloration of the acoustic signal. For long delays,
relatively intense echoes may exist without causing
noticeable coloration [5]. This figure shows the level of
101og(a) at which lesteners perceive coloration.
figure 1.15 As the time delay of the echo increases, its intensity must
decrease in order to avoid an annoyingly distinct sound. The
above figure shows the threshold intensity for hearing
distinct echoing [5].


'
' >
Ai
mJc
9
CD
1

/ h'9
. att
A A ilkj
her frequencies
rnuated
i
\ /\ A a A A a
t r< A A A a A / \ /\ (A r\ A A M A A A A A A /*\ t
{! 1 1 V 7 \1
'i \ 7 v y
* if
U.AAn
V vy \/ y 1
v ' v y
A AAaA
/ V
1
A A
V v v V v vJ V \7 Vv V innr y \ v v y
' v V V v v V V' v
aAaaAAaaAAAaAAAaaAAa
/ higher frequencies
i A f exagerated
UlA aA/\AA AaA A A
va/Vw v V V u v V V Vvyv ^ V V v v v/ v \
a L a A Aa aAa r\A A A A A A y\ A A-A A
/vrwwwy vv/y y vvv'vyvvv^
Hearn
characteistics
I
h
figure 6.11 The ECVT signals plot is given for band 1 (dc-102Hz) for
0.5 seconds of audio data. From the first trace it is seen that
M
M
Hi
a large degree of amplification is required to match the
microphone signal to the CD signal. However, the adaptive
equalizer does not signficantly alter the signal in a spectrally
dependent fashion.


24
The frequency domain requirements for an ideal room are the following:
i. The system must have a constant magnitude response, \H( to) | = k.
ii The phase response must be linearly proportional to frequency, (co) = -cor.
Violation of i. can lead to an unintended emphasis in some frequencies over others in ways
not intended by a composer or performer. Violation of ii. gives rise to non-linear phase.
Non-linear phase causes certain frequencies to pass through the system faster than others.
To accurately reproduce the sensation produced by the original acoustic waveform it is
essential that the sound reproduction system have a flat magnitude response. Obviously
some degradation from this standard will not alter the perception of the listener. Neverthe
less, the fundamental difference between middle C and any other note is that the energy of
the signal is concentrated at 262 Hz and its harmonics. An attenuation of the frequency re
sponse at middle C, if other unmasked information is present in the signal, will negatively
impact the fidelity of the signal because of the lessening of emphasis on the C-like quality
of the sound. More precisely, we are interested in maintaining the pitch and timbre of the
signal. Pitch of a musical signal is primarily determined by frequency, but is also influ
enced by the shape of the waveform and the loudness of the signal. The frequency depen
dency is depicted in figure 1.8, where pitch in mels is plotted against frequency, for a signal
with a loudness level of 60 phons. Note that a change in frequency has a greater effect on
pitch above 1000 Hz than below.
The loudness dependency of pitch can be quite dramatic for pure sinusoidal tones. For
music signals the loudness dependency is much less significant. As an example, to main
tain the same pitch when increasing the loudness of a musical signal from 60 to 100 phons
requires at most a frequency shift of 2%.
If a signal is not a pure tone, but consists of harmonic tones, due to the nonlinear pro
cessing of the ear, sum and difference signals are generated. The difference signals deter
mine the pitch perceived by the listener. For example a signal consisting of the first four
harmonics of a 100 Hz tone, all of approximately the same intensity as the fundamental, is


6
Sensory equipment, including the human auditory system, responds to the intensity of
the field. In the case of acoustic data, this corresponds to the average energy flowing across
a unit area normal to the direction of propagation per unit time.
W1
In this discussion pressure waves will be considered to be plane waves in order to sim
plify the mathematical treatment of the phenomena which will be discussed. Note that for
an acoustic point source the phase fronts would be spherical in an analogous manner to
electromagnetic point sources. The plane wave approximation is only valid when the radius
of curvature of the phase front is very large compared to the wavelength of the radiation.
We will assume in this discussion that the dimensions of the rooms being considered are
sufficiently large that this approximation is valid for frequencies over 1000 Hz. The room
under consideration will be a rectangular cavity. These highly idealized cases are studied
so that the important features of room responses are made clear. Before room responses
can be meaningfully discussed, the reflection of acoustic waves off a wall must be well-un
derstood.
Reflection of a Plane Wave bv a Wall
The electromagnetic equivalent of reflection off a wall is a plane electromagnetic wave
reflecting off an infinite flat plate of complex impedance. Like the electromagnetic equiv
alent the media impedances determine the behavior of the wave for a given geometry. Con
sider the situation depicted in figure 1.1. Let p¡ be the incident pressure field, let pr be the
reflected field, and let pt be the transmitted field, with corresponding wavevectors kj, kp and
kt.
Since we are assuming plane waves we can drop the y-dependency from our equations^,
without loss of generalization. Recall that a wavevector K is defined as K = co/c ,
where c is the velocity of propagation in a given medium, and co is the radian frequency.
Let cq be the speed of propagation in air at standard temperature and pressure (STP) and let


180
That step size has been properly set for improved convergence speed is demonstrated
as follows. If the standard step size initialization is used, namely p = 0.10/Xmax, the
LMS algorithm, while converging in the mean, gives rise to a variance in square error which
is unacceptable, within one 11 msec epoch. This is clearly demonstrated in figure 6.28.
Step size was next initialized at a value 20% greater than suggested by the results of chapter
five (p = 0.06/Xmax). Figure 6.29 shows the same epoch as figure 6.28. The equalizer
begins to exhibit the same undesirable effects as before, although with less severity. Figure
6.30 displays the next epoch illustrating that the step size adaptation algorithm provided
sufficient damping to restore good behavior to the algorithm. This type of behavior indi
cates that step size thus initialized results in a square error which is marginally stable. If an
excessively conservative value is used to initialize the LMS step size, the adaptive equalizer
responds too slowly to the non-stationary environment of the acoustic signals to follow im
portant features. Figure 6.31 demonstrates the resulting slow convergence of the equalized
signal when step size is initialized as p = 0.02/Xmax, for the same epoch of data used pre
viously.
A test was also performed on 250 contiguous epochs of acoustic data in band 4 with p
initialized at This value results in a step size 20% less than the value recom
mended in chapter five. A conservative approach gives a safety margin to allow for inac
curacy in the estimate, XDLMS. For these epochs the average ratio of XDLMS/Xpower was
2.4, and LMS convergence speed was enhanced.
Performance Analysis in Band 4
Testing revealed several limitations of the equalization techniques. One problem which
is apparent in the equivalent equalizer architecture used for testing, however, will not ap
pear in the proposed real-time architecture. In the equivalent equalizer architecture the
adaptive equalizer can not restore frequency content present in the CD signal and wholly
absent in the microphone signal. Figure 6.32 shows an 11.33 msec epoch in which the
equalized signal does not restore a pulse train clearly evident in the CD signal. There is a
lack of sharpness of features across most of the epoch. However when white noise was add
ed to the microphone signal, the equalized signal restored the pulses, and several high


75
r- = 1 X. = exp (-) (3-38)
J J Xy
The convergence of C, to £* is represented by a learning curve which can be approximated
as the product of exponential terms.
2 2
learning curve =
(3-39)
Learning is limited to the largest e-folding time, which is directly related to the smallest ei
genvalue and the magnitude of the step size.
Excess Mean Square Error '
In the discussion it has been assumed V£ is known. In fact it is not. Adjustments must
be made in the above equations to account for a noisy gradient estimate. Let V £ be the
estimate of V£.
"Vu = wk-^k
(3-40)
Switch to the translated coordinate system.
vk+l = vk-^k
(3-41)
The estimate of the gradient can be expressed as the sum of the gradient plus a noise term.
K = ^k+Nk
(3-42)


figure 6.32 When frequency information is present in the CD signal, and
wholly absent in the microphone signal, the adaptive
equalizer will perform poorly. After convergence the
adaptive filter can not generate frequencies not present in the input signal.


204
poor performance (figure 6-31).
A heuristic algorithm was used to trigger the DLMS algorithm. This algorithm attempts
to segment the audio signal into quasi-stationary epochs. The algorithm did not always per
form well. When the segmentation algorithm fails, the LMS algorithm continues to operate
with an inappropriate step size (figure 6-35).
The time required for the DLMS algorithm to approximate A.max leads to a similar ef
fect. While the DLMS algorithm is operating, the LMS algorithm will continue to use an
inappropriate step size (figure 6-36).
In band 4 step size reduction is handled by a heuristic algorithm. Because the spectral
features become a smaller fraction of sampling frequency, poor step size reduction results
in a blurring of the features (figures 6-37 through 6-39). This blurring is partially a result
of excessive gradient noise.
The adaptation of the equalizer will be made on the basis of information delayed in time
by the propagation of the acoustic signal in the listening room. The problem of propagation
delay is fundamental to adaptive control. It is particularly difficult for acoustic applications
because the speed of sound is slow.
It has been seen that adaptive control can lead to better results than merely band by band
amplification of the acoustic signal using traditional equalizer technology. However stan
dard LMS implementations will not operate well. The LMS algorithm with a fixed step size
is not stable because of the large dynamic range of audio signals. The power normalized
LMS algorithm suffers from the poor estimation of by signal power, as well as a lack
of understanding of the algorithms convergence properties with correlated input data. The
techniques of chapters four and five result in a significantly improved ability to track a high
ly non-stationarv acoustic signal, from the standard LMS implementations. Nevertheless
several areas of research require more investigation for robust and reliable operation. The
following discussion offers some suggestions.


121
1
17.5
. 15
1,1 1 eigenvalue approc^'9'
|
-5
12.5
yy
-10
yy
I
10
yy
-IS
7.5
yy
|
-20
yy
i
5
li
\
I
-25
2.5
yy
y
I
-30
-35
2 4 6 8 10 12 14
k (Iterations)
1
17.5
1
15
12.5
10
7.5
4
5
/
2.5
i

2 4 6 8 10 12 14
k (iterations)
Eigenvalue Spread = US
p.7.5-
¡ 15-
12.5
j 10
I 7.5
\ 5
\
\ 2.5
In £ 1 eigenvalue apprcK
j
3
-2
§
-4
|
-6
II
\
|
!
-8
-10
2 4 6 8 10 12 14
kteratton) ,
figure 5.2 For each level of eigenvalue spread the one eigenvalue
approximation (dashed curves) are plotted with the true
value (solid curves) of In (C,k). As the number of divergent
iterations increase, the one eigenvalue approximation
becomes increasing accurate for all levels of eigenvalue
spread.


9
locity field propagating along the wavevectors given in equations 1-7 through 1-9 is de
scribed by equations 1-16 through 1-18, shown above.
By a proper choice of the coordinate system, i.e. for the wall located at x=0, and by an
application of Snells Law and the Law of Reflection, the phase related terms in V can be
ignored, as they are common to V¡, Vp and Vt. Matching the particle velocities at the inter
face yields the following expression for the reflection coefficient.
P,,Ocos0i Pr,Ocos0r Pi,Ocos0,
(1-19)
*2P/, OCOS0rZ2Pr, OCOS0r = Z1 (Pl, 0 + Pr, 0> COS0i O'20
r r 0
(zjCosG^ + ^cosGp = Z2csOf.e os^
UO
d-21)
#(0,G)) =
Pr Q Z2 C0S ^/ Z 1 C0S
Pi q Z2cos- + 2icos
(1-22)
If we assume medium one is air, i.e. z1 = p0c0, and medium two is a hard wall, i.e.
z2 zx the above equation reduces to the form seen in most elementary books on acous-
z2cos0.-poco
tics, R = p- The transmission coefficients are easily derived from the re-
z2cos0f+poco
flection coefficients as follows.
p, (p¡+pr)
7-(0, i i
= 1 +R =
2 Zn COS0.
2 i
d-23)
^2^80^+ 2^080^


I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
of Electrical Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Donald G. Childers
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
is
FredT Taylor
Professor of Electrical Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
David M. Green
Graduate Research Professor of Psychology
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Gary Sii
in


(CD signal microphone signal)2
samples
improvement = (CD signal microphone)2 (CD signal equalized signal)2
AV(/\aA^'yA'Vv'^V~/"'^V
nr7~~. ~
- Sflna) 2 Delay -
samples - ^gnal Salactton
irCh:i2 ~lsU ^y^r~~l D"'Maa"F .
I ,
a -
figure 6.40 Learning in band 4 is disappointing, especially when
compared with the results of bands 1 through 3. Although
their is improvement across most of the band, it is far less
dramatic than in other bands.
VO
00



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12
V2P + K2P = icupq (r)
P + P P
i, interface r, interface t, interface
(1-26)
(1-27)
vP = -Pf = -^P
y ydt z
d-28)
As a result our source distribution, q(r), can be expanded as a sum of eigenfunctions. V
is the enclosed volume of the room.
= XWr)
n
Cn=\\\q(r)yndV
(1-29)
(1-30)
The unknown solutions to the inhomogenous Helmholtz equation can also be expressed
as a summation of eigenfunctions.
'w = EW'-)
(1-31)
n
Substitute the summations for Pa (r) and q(r) into the inhomogenous Helmholtz
equation and solve for the unknown {D} in terms of the known {Cn} as shown below.
ED[ V2v (r) + K\ (r) ] = -i n
n
The equation has a simple solution for a point source located at tq, i.e.
Pa(r) = ifip£
VB ^r> VB (r0)
7 J(K2-K2n)
d-33)


123
figure 5.4 Estimation error of C, is plotted above as a function of
divergent iterations, k, and step size, y, where p = y/^majc>
for a filter of order two and input data with an eigenvalue
spread of 1.10. The contour plot is shown below.


100
In order for nrn to contract, we force all the constituent relaxed projection operators to
contract, as described in equation 4-14. This can be guaranteed by forcing y to belong to
the contraction domain of each T operator. Let Dy(i) represent the contraction domain for
T0Tl TkTk rlr0
[0
T^T
lklk
Wol*1
(4-14)
T
7). Then the contraction domain for II II is given below. Fortunately, we are able to relax
ye P| DAm)
m = 0
min
m
o,
max
mil J
(4-15)
the requirement of equation 4-15 by noting that we only require nrn contract on average.
Thus, a determination of the upper limit of Dy requires information on the higher order sta
tistics of {X}. Note also that k must not be so large that the initial assumption of these der
ivations is violated. A good description of £ must not require terms in e*.
The difficulty of finding the upper bound of Dy is illustrated by figure 4.3. Three real
izations of {l!X1ll2,...,IIX6ll2}are shown. The data are scattered about the variance of IIXII2
according to the higher order statistics of {X}. For each realization there exists a value a
such that Dy= [0, kmax/ to guarantee convergence of £(|| tJtJ+ TrkTk Tj+1 rj2), a value for (3 = must be
found. This average value of a will guarantee that on average the matrix equation for will
result in a contraction.
D
y
max
tJ
(4-16)
Two important conclusions about the behavior of the domain of p. for an exponentially
converging mean square error can be drawn from equation 4-16. As input data correlation
increases, for data modelled as an autoregressive series, the value of will increase.


50
are increased by M, due to the rescaling of digital frequencies engendered by decimation,
the required filter orders for all subsequent filtering operations are considerably smaller, or
a finer frequency will be possible with the same order filters. This is an important charac
teristic due to the finer resolution psycho-acoustically required in the lower frequency
bands. Figure 2.6 summarizes the upper branch.
At each branching node of the tree structure the bandlimited signal is also sent to a
highpass filter (lower branch) for equalization. If the room were non-filtering the highpass
filter would be designed as follows.
n
hhp=1 ll km
= 0, otherwise
(2-3)
JC \
Note that the digital cutoff frequency is and not r because of the decimation.
M m2 \
Operation of the Upper Branch of the Tree Structure
\
figure 2.6 This figures demonstrates the processing of a signal
generated as the summation of two sine waves. The low
frequency sine wave is separated with a lowpass filter and
then decimated. The decimated part of the signal is thrown
away.


96
Furs-
where, the maximum eigenvalue of matrix A is represented by ^^(A). In order to prop
erly restrict the domain of |i such that C. converges requires examination of I rvl 2
K I j i
tenberg and Kesten [27] developed the following important theorem on the product of
random matrices.
Furstenberg-Kesten theorem: If {A*} is a stationary, ergodic NxN matrix sequence for
which £ {log+||Aj||} < oo, where log+t = max (logt, 0), then
Hm-togHA.A.,
* = Hni-^{l0g||AA-i
n ~r
Aill>
with probability one.
Bitmead and Anderson [28] define a sequence {zk} to possess the property of almost sure
exponential convergence to zero (A.S.E.C.), if there exists p > 0, independent of time, such
that lim (1 + P) kZu = 0. Shi and Kozin [29] strengthen the Furstenberg-Kesten theorem
k-* oo
to guarantee convergence in the A.S.E.C. sense for a finite product of random matrices.
Shi-Kozin theorem: Suppose the sequence {Ak} satisfies the conditions of the Fursten
berg-Kesten theorem, then { Ak] converges in the A.S.E.C. sense if and only if there exists
a positive integer n0 for which E {log|| AnAn^ x A1||}<0.
The Shi-Kozin theorem can be directly applied to the LMS algorithm. Let
2. . liog|n?|--n).
n"|| = e 2t1 < 1. If conditions can be found which ensure the e
A, = [/-|tXjfr then = A Ao_ t
Aj. Let
where T| > 0. Then
istence of n0, then C, will have almost sure exponential convergence.
Shi and Kozin studied extensively the implications of their theorem on the domain of (i
for which the vector weights of the Normalized LMS algorithm converge. They prove the
following.


175
figure 6.22 The waterfall plot for CD data in band 3 (612.5 3675 Hz) is
shown above for a four second segment of data. Note that
the features in boxes 1 and 2 are distinct. Note also that the
CD signal has been filtered so that no energy is present below
612 Hz.


118
x2 --x +
MAX JX MAX
(1-P)
4\l2
= 0
_ i + VP
wax 2|1
2|i
(5-6)
Parameter Selection for the Divergent Gradient Descent (DGD) Algorithm
The estimation error of XMAX depends on the set of input data eigenvalues, { A.;} The
DGD algorithm is sufficiently robust that p and k can be selected without a priori informa
tion concerning { X;} In this section a parametric study of p and k, and their relation to
the error in the estimation of ^MAX will be conducted. First the one eigenvalue approxima
tion of £ must be verified (equation 5-5). The divergent step size, the number of divergent
iterations, and the eigenvalue spread are varied, and the one eigenvalue estimation error is
calculated. For input data modelled as an autoregressive process, the degree of eigenvalue
spread corresponds to the degree of correlation in the input. Figure 5.1 demonstrates the
validity of the one eigenvalue approximation of ¡3 for a filter order of two as a function of
eigenvalue spread, D.and y, where y = P^majc- The left-hand plots give the one eigenval
ue approximation (dashed lines) of In £3 with the true value superimposed (solid lines).
The right-hand plots give the percent error of the approximation. Note that even for k = 3,
the one eigenvalue approximation error is less than 10% for V^-max >1.2 and D = 1.1. As
the eigenvalue spread increases the approximation error becomes negligible.
In figure 5.2 the relationship between the percentage error and the number of iterations
and is investigated, for \i^max >1-4 and a filter order of two. Once again the left-hand plots
give the one eigenvalue approximation (dashed lines) of In £3 with the true value superim
posed (solid lines). The right-hand plots give the percent error of the approximation. From