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Representation of locally stationary signals using lowpass moments

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Title:
Representation of locally stationary signals using lowpass moments
Creator:
Celebi, Samel, 1966- ( Dissertant )
Principe, Jose ( Thesis advisor )
Harris, John G. ( Reviewer )
Childers, Donald G. ( Reviewer )
Taylor, Fred J. ( Reviewer )
Yerelan, Sencer ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
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Copyright Date:
1995
Language:
English
Physical Description:
vii, 154 leaves : ill. ; 29 cm.

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Approximation ( jstor )
Bandwidth ( jstor )
Estimation methods ( jstor )
Interpolation ( jstor )
Mathematical vectors ( jstor )
Polynomials ( jstor )
Propagation delay ( jstor )
Signals ( jstor )
Spectral methods ( jstor )
Taylor series ( jstor )
Dissertations, Academic -- Electrical and Computer Engineering -- UF ( lcsh )
Electric filters -- Mathematical models ( lcsh )
Electrical and Computer Engineering thesis, Ph. D ( lcsh )
Signal processing -- Mathematical models ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
The gamma filter is a generalized feedforward structure which can implemented as a cascade of identical first order lowpass filters. Although functionally it is an infinite impulse response (IIR) filter, topologically it mimics a finite impulse response (FIR) filter. As a result of this blend, it inherits the computational power of the former and the easy stability analysis of the latter. Being a single input multiple output structure, the gamma filter is characterized by its order and the time scale which linearly warps the time axis. Furthermore, when orthogonalized, the gamma filter transforms into the well-known Laguerre filter. The gamma kernels correspond to the impulse responses of the gamma filter outputs. This family of decaying exponentials form a complete set in the Hilbert space. The first problem attacked in this study is the approximation of a given signal as a weighted sum of gamma kernels such that the mean square error is minimized. conditions for the optimum weights and time scale are derived and they are decoupled from each other. A vector space interpretation of the role of the time scale is also provided. According to this interpretation the time scale parameter has the effect of changing the relative angle between the desired signal and the gamma projection space without altering the structure of the gamma space. Two numerical methods to estimate the time scale are included. The gamma filter produces at its tap, the Poisson moments of the input signal. Poisson moments correspond to the Taylor's series coefficients of the recent magnitude spectrum of the input signal at zero frequency. Localized in time and frequency, Poisson moments offer a computationally inexpensive, alternate time-frequency representation for locally stationary signals, such as speech. In the analog domain the moments can be physically measured and fed into a nonlinear mapper, such as an artificial neural network for tasks such as classification, predilection and identification. For a faithful representation of the spectrum, it is sufficient to observe the moments at the informative rate rather than the usually higher Nyquist rate. This convenience removes a big deal of burden from the shoulders of the processor following the gamma filter. Very much like the Laplace transform, Poisson moments have the virtue of transforming linear differential equations into algebraic ones. Based on this property, one can linearly solve for the unknown parameters of an autoregressive moving average (ARMA) system simply by observing the Poisson moments of the system input and the output at finite number of time instants. Although the method doesn't put any restrictions on the choice of the filter time scale and the observation times, in practice, these parameters play a significant role in determining the accuracy of the results. If the parameters are not selected properly, the result tend to be ill-conditioned. This study provides some insight into the choice of the time scale.
Thesis:
Thesis (Ph. D.)--University of Florida, 1995.
Bibliography:
Includes bibliographical references (leaves 149-153).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Samel Celebi.

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Copyright Samel Celebi. 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.
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REPRESENTATION OF LOCALLY STATIONARY SIGNALS
USING LOWPASS MOMENTS







By


SAMEL CELEBI


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

UNIVERSITY OF FLORIDA


1995














ACKNOWLEDGEMENTS,


Simply looking at this book, it is hard to guess how much of an effort goes into a dissertation study. This book is a record of only a small portion that has been skimmed out of days and nights of hard work often accompanied by financial, mental and emotional distress. In order to overcome these difficulties, one needs continuous support of his/her environment. I feel very fortunate in that sense.

First and foremost, I would like to thank my chairman Professor Jose C. Principe for the encouragement he provided during the course of my studies. His availability to his students and the abundance of the ideas he offered are irreplaceable. Special thanks are due to Dr. John G. Harris whose constructive comments and careful proofreading helped to improve the quality of the thesis. I also thank my committee members Dr. Donald G. Childers, Dr. Fred J. Taylor and Dr. Sencer Yerelan for the guidance they granted me.

During my first semester in the graduate program at the University of Florida, I was sponsored by the Turkish Ministry of Education. For this I am deeply grateful. I would like to take this opportunity to thank my family for their patience and the morale support they gave me.

It has been a pleasure to be a part of the Computational Neuroengineering Laboratory. I benefitted from it both socially and academically. Id like to thank all my labmates for the friendship they extended me.

Finally, I thank Paula very much for being with me during all this. It would be very hard, if not impossible, without her support and love.















TABLE OF CONTENTS


vage

ACKNOW LEDGEM ENTS . ii

ABSTRACT . vi

CHAPTERS

1 INTRODUCTTON . 1

2 GAM M A KERNELS . 4

Introduction . 4
Gamma Kernels . 4
Gamma Filter . 6
Gamma Filter as a M emory M odel . 8
Laguerre Kernels . 9
Laguerre Polynomials . 11
Laguerre Polynomials and the Confluent Hypergeometric Equation 12
Parametric Representation of Signals Using Gamma Kernels . 13
Statement of the Problem . 16
Structure of the Gamma Space . 17
Optimum Value of the W eights . 19
Optimum Value of the T"me Scalek . 22
Previous W ork . 22
Optimum x . 22
Estimation of optimum X using moment matching . 25
Example . 29
Estimation of optimurnk using a matched filter . 32
Online Adaptation of the Weights and the Time Scale . 34
Conclusions . 37





W









3 TIME-FREQUENCY DISTRIBUTIONS . 39

Introduction . 39
Uncertainty Principle . 39 Timie-Frequency Representations. 42
Cohen's Class of Bivariate Distributions .45 Spectrogramn and Its Relation to Wvigner Distribution. 49
Conclusions. 50

4 INTERPOLATION. 52

Introduction . 52 Polynomial Interpolation. 52 General Problem of Finite Interpolation . 53
Example . 54 Systems Possessing the Interpolation Property . 55
Hermite Interpolation . 56
Hermite Interpolation of ARMA Signals . 59 Issues in Hermite Interpolation. 60
Region of convergence. 60 Bandwidth of interpolation . 62 Maximum error versus the bandwidth . 65
Conclusions. 66

5 REPRESENTATION OF LOCALLY STATIONARY SIGNALS USING
POISSON MOMENTS . 69

Introduction . 69 Gamma Filter and the Poisson Moments: Analysis Formula . 70
Reconstruction of a Signal from Its Poisson Moments: Synthesis
Formula . 71
Poisson Moments and the Magnitude Spectrum . 73 Spectral Hermite Interpolation . 75 Spectrogram, with Exponential Decaying Winmdow . 76 Variance and Bias of the Specrogramn Estimator .78 Choice of the Tine Scale. 83 Examples. 87 Asymptotic Properties of the Poisson Moments . 97
Impulse Distribution Expansion Versus Taylor'sSeries Expansion . 97









Choice of Tune Scale and Measurement Time . 98
Poisson Moment of a Signal and Its Tune Delayed Versions . 105
Signal Classification Using the Poisson Moments . 107
Estimation of the Time Delay . 109
Example . 110
Conclusions . 112

6 SYSTEM IDENTIFICATION USING POISSON MOMENTS . 115

Introduction . 115
Poisson Moments and Their Relation to Function Derivatives . 116
Method I . 118
M ethod Il . 119 Exam ple . 119
Effect of ) . . 120
PMF in the Identification of Tune-Varying Parameter Systems . 129
Least Squares and Recursive Least Squares Estimates . 130
Conclusions . 130

7 CONCLUSIONS . 133

APPENDICES

A SUMMARY OF THE PROPERTIES OF THE GAMMA AND THE
LAGUERRE KERNELS . 136

B SUN94ARY OF PARKS'AND MASNADI'S WORKS . 139

C TRvIE-FREQUENCY SPREAD OF THE GAMMA KERNEL . 142

D A PROOF ON THE SCHUR COMPONENT . 144

E COMPUTATION OF THE SCHUR COMPONENT FOR
ARMA SYSTEMS . 147

REFERENCES . 149

BIOGRAPHICAL SKETCH . 154















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


REPRESENTATION OF LOCALLY STATIONARY SIGNALS USING LOWPASS MOMENTS

By

SAMEL CELEBI

December 1995


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

The gamma filter is a generalized feedforward structure which can be implemented as a cascade of identical first order lowpass filters. Although functionally it is an infinite impulse response (11R) filter, topologically it mimics a finite impulse response (FIR) filter. As a result of this blend, it inherits the computational power of the former and the easy stability analysis of the latter. Being a single input multiple output structure, the gamma filter is characterized by its order and the time scale which linearly warps the time axis. Furthermore, when orthogonalized, the gamma filter transforms into the well-known Laguerre filter.

The gamma kernels correspond to the impulse responses of the gamma filter outputs. This family of decaying exponentials form a complete set in the Hilbert space. The first problem attacked in this study is the approximation of a given signal as a weighted sum of gamma kernels such that the mean square error is minimized. Conditions for the optimum weights and time scale are derived and they are decoupled from each other. A vector space









interpretation of the role of the time scale is also provided. According to this interpretation the time. scale parameter has the effect of changing the relative angle between the desired signal and the gamma projection space without altering the structure of the gamma space. Two numerical methods to estimate the time scale are included.
The gamma, filter produces at its taps, the Poisson moments of the input signal. Poisson moments correspond to the Tayloes series coefficients of the recent magnitude spectrum of the input signal at zero frequency. Localized in time and frequency, Poisson moments offer a computationally inexpensive, alternate time-Erequency representation for locally stationary signals, such as speech. In the analog domain the moments can be physically measured and fed into a nonlinear mapper, such as an artificial neural network for tasks such as classification, prediction and identification. For a faithful representation of the spectrum, it is sufficient to observe the moments at the information rate rather than the usually higher Nyquist rate. This convenience removes a big deal of burden from the shoulders of the processor following the gamma. filter.
Very much like the Laplace transform, Poisson moments have the virtue of transforming linear differential equations into algebraic ones. Based on this property, one can Hnearly solve for the unknown parameters of an autoregressive moving average (ARMA) system simply by observing the Poisson moments of the system input and the output at finite number of time instants. Although the method doesn't put any restrictions on the choice of the filter time scale and the observation times, in practice, these parameters play a significant role in determining the accuracy of the results. If the parameters are not selected properly, the results tend to be ill-conditioned. This study provides some insight into the choice of the time scale.















CHAPTER 1


NRODUCTON

The purpose of this study is to find alternate signal representation schemes for locally stationary temporal signals, such as speech. These alternate representations are expected to express the signal in a low dimensional space with sufficient fidelity and little computational cost. Analog implementation and resistance to noise are also among the desirable features of the scheme.

Having restricted the scope of our search to linear structures, we realized that the

desired structure had to incorporate short term memory in order to be able to process nonstationary signals. In that respect, Finite Impulse Response (FIR) filters are not a good choice due to the coupling between the filter size and the depth of the past information they can provide. The other alternative, the Infinite Impulse Response (11R) filters, on the other hand, suffered from instability problems.

Given these constraints, we concentrated our studies on the gamma filter and the famnily of exponential kernels that correspond to its impulse response: the gamma kernels. Developed by Principe and deVries [1].[3], the gamma filter is a generalized feedforward filter. While functionally being an llR, topologically it mimidcs an FIR structure. As an outcome of this blend, gamma filter inherits the computational power of the former and mathematical tractability and the stability of the latter. Furthermore, compared to a FIR filter of the same order, it has only one more degree of freedom.

In the early stages of our research we observed that the gamma filter was more efficient' than the conventional tapped delay line when applied to classical signal processing


1. In terms of the number of parameters required for a given mean square error.









tasks such as prediction and system identification. Encouraged by these results, we have decided to explore this structure in more detail from a representation point of view.

Another finding that motivated us to analyze the gamma filter is the fact that structures similar to the gamma filter are pervasive in various environments. Depending on the medium, the gamma filter may take the form of a cascade of RC circuits, or a leaky transmission line or even a network of synaptic connections [4]. There is also some evidence that some of the preprocessing of the time-varying biological signals are done by structures similar to the gamma filter [5].
The core of this study is composed of analyzing the gamma filter from three perspectives: parametric and nonpararnetric signal representation and system identification. Because of the distinct character of these individual topics, rather than organizing the thesis in the conventional way, where there is a separate results section, we prefer to present the results within the body of each chapter followed by some concluding remarks.
The fiow of the thesis can be summarized as follows: In Chapter 2 we introduce the gamma kernels and attack the problem of representing signals as a weighted sum of these kernels which are shown to be complete in the Hilbert space. The solution of this parametric representation problem turns out be a generalization of the Vvriener-Hopf normal equation. This chapter also provides a vector space framework interpretation of the problem clearly demonstrating the advantage of using the gamma structure over that of the FIR filter. In addition to the derivation of the optimality conditions, we link the gamma kernels to the Kummerfunctions which encompasses several well-known functions such as Bessel, Gaussian, Hermite, Laguerre and exponential.

In Chapter 5, we pick a nonparametric representation point of view and suggest to use the gamma filter as an alternate time-frequency representation scheme. The gamma filter generates at its taps the time moments of the input signal. These are called the Poisson moments and they can be regarded as signatures of the input. In this chapter we show how the Poisson moments carry information about the spectral content of the signal. This









information is localized both in time and frequency and it is readily available at the outputs of the gamma filter as a measurable quantity. Hence, the moments offer a comnputationally inexpensive time-frequency representation.

Chapter 3 is tailored to provide background information on the time-frequency distributions, a topic that is frequently visited in Chapter 5. In doing so, we pay particular attention to the spectrogram. method. By the same token, Chapter 4 provides background information on the interpolation theory with special attention to the Hermite interpolation. This is done in an attempt to prepare the reader for the nonparametric representation problem studied in Chapter 5.

Chapter 6 ponders the problem of identifying autoregressive moving average

(ARMA) systems using the gamma filter. ARMA systems are described by linear differential equations. Very much like the Laplace transform, the Poisson moments transform the linear differential equations into algebraic ones. Using this, we will show how to algebraically determine the unknown parameters of a linear system by observing a finite number of Poisson moments of the system input and the output.

Finally, Chapter 7 underscores the highlights of the study and cites possible future research issues.














CHAPTER 2


GAMMA KERNELS

Introduction

The core of this study is composed of the utilization of the gamma kernels and their orthogonalized counterpart, the Laguerre kernels, in parametric and nonparametric representation of signals and in the identification of systems. This chapter begins with introducing these two family of kernels and elaborates on their properties. Although the gamma and the Laguerre kernels span the same space, individually they help to explain different aspects of the problem. Therefore, we will introduce both families and link them to each other when necessary.

Gamma Kernels

In the course of this study, the term kernel refers to the implementation of an operator. The gamma kernel, in particular, refers to the implementation of a generalized delay operator. The gamma kernel was devised to substitute for unit delay operator which performed poorly in the processing of signals with redundancies. We define the gamma kernels as


g (A.,t 0 n- )! e n = 1 2 3




This family of kernels constitutes a subset of eigenfunctions of the autoregressive moving average (ARMA) systems. Therefore, it is a good candidate for modeling signals that accept the linear model. The kernels are complete in the Hilbert space (02 space) which










spans all the finite energy signals. This implies that any finite energy signal can be approximated arbitraily closely using a finite number of gamma kernels. The parameter X controls the time scale (or the memory depth) of the kernels by changing the width of the kernels. Increasing X has the effect of linearly compressing the kernels, while decreasing X has the opposite effect. Figure 1 depicts the kernels in the time domain for ~=.For reasons that will become clearer in later sections that deal with the memory structures, the area under each kernel is normalized to unity iLe,



fgn(k, t) dt= 1 (2)
0


The Laplace transform of the kernels is given as


Xn
Gn (S) ( ) = (S)


where G (s) mL Ig1,)t)} I



Gammia Filter

Equation (3) suggests a practical implementation for the kernels, in the form of a cascade of identical first order lowpass filters. This structure shown in Figure 2 is called the gamma Filter [1] .The impulse response of the filter from the input to the kth tap is given by gkQ.,t). The K0 order gamma filter transfer function is characterized by a pole at s=-,% with multiplicity K. The time scale X is responsible for setting the position of this pole. The position of the zeros however, are determined by the weights wl,w2,.,wK and the time scale L.









g1(Xt)


gl(O
0.8

0.6 92(%1t)

0.4" 93010
/ g4(1 t)


0.2

r 1t


Figure I Continuous ine gamma kernels for ,-l


Gamma Filter


Kh stage


1 st stage 2nd stage


Figure 2 Gamma filter.









Gamma kernels satisfy the following differential equation d(4



If the time derivative is replaced by the backward time differences, the filter can also be realized in the discrete time domain. For this case, the kernels satisfy the following difference equation


g-k+1 (n) = '%9k(n-l1) + (1- 9k+ I (n -l (5)


The discrete time kernels and their corresponding z-transform are given by



g(n) n- (fi~ Xn-k n :k
(6)





Notice that for the special case of X=1I, the discrete time gamma filter becomes a finite impulse response (FIR) filter. One can think of the gamma filter as a generalized feedforward structure where the unit delay operator z1 is replaced with a generalized delay structure G(z) that accommodates a local feedback. Due to the inclusion of this feedback, the gamma filter becomes a recursive structure. Recursive structures are computationally more efficient than their feedforward counterparts; however, they may suffer from instability problems. That is why HIR filters are almost exclusively used in adaptive signal processing. The instability problem can be taken care of by restricting the feedback connections to be local. In the gamma filter, stability is guaranteed as long as one of the identical stages is stable, which prescribes X>Q for the continuous time and O








The Gamma Filter as a Memory Model

We start this section by quoting two definitions, one for the memory depth and the

other for the memory resolution, proposed by Principe [l1] [2] [6] in an attempt to quantify and compare different memory structures.

Definition: Memory depth D is the temporal mean value of the impulse response g(t) of the filter used for storing temporal information. Considering the impulse response as a probability density, this corresponds to the first moment of the impulse response given by


D t ()dt =dG (s) continuous time
jt~~t-~dsI
0 S =0 (7)


D= ng (n) = -z d z) discrete time
n=O

Definition: Memory resolution R is the number of degrees of freedom (i.e. memory state variables) per unit time.

The conventional digital signal processing structures are built around the tapped delay line (G(z)-z 1) which is an FIR filter. The memory depth of an FIR filter is equal to the number of filter parameters, that is the number of its taps. This strict coupling of the memory depth to the number of filter parameters leads to poor modeling of lowpass frequency signals which usually have a long sustenance in time, but few degrees of freedom. Furthermore, the high memory resolution of the FIR filters makes these filters susceptible to noise on lowpass signals. Contrary to FIRs, memory depth of an IIR filter can be made arbitrarily large at the expense of resolution. These filters, however, suffer from instability problems. The gamma filter in the meantime, incorporates a cascade of generalized delay units in the form of an overall feedforward structure. Restricting the feedbacks to be local solves the instability problem while preserving the decoupling of the memory depth from the number of system parameters. For the case of a Kth order gamma filter, the memory, depth equals









Dgam z~ dGk (Z) I K (8)



The resolution of the gamma filter is given by


Rgam - K (9)



Note that the gamma structure obeys the relation order =depth x resolution (10)
K =DxR


This expression tells us that the gamma filter is able to trade the memory depth for the resolution by varying its time scale parameter X. The memory depth and resolution for the FIR structures, on the other hand, are fixed at DFI=K and RFI=l.

Lazuerre Kernels

The gamma kernels lead to the gamma filter which can be thought of as a link between FIR and HR structures. They are also useful tools in understanding memory models. Although they have other appealing properties such as being produced recursively from each other using convolution, they still lack one important feature: orthogonality.' These kernels are linearly independent of each other, but not orthogonal. They can, however, be orthogonalized using the Gram-Schmidt procedure. When the Gram-Schmidt procedure is applied to the gamma kernels, one obtains the Laguerre kernels that have the form




1. Another advantage of the galmma kernels over the Laguerre kernels is that we can expand them to
the continuous index case where k is a real number


Skt 0 -) k2:1












L,(t) = F,/2 ( (-2---) ti
v=O


n = 0, 1,2, .


(11)


These kernels are complete in the L2 space and they form an orthonormal set such that


Ln(t)Lm(t)dt = Bnm


8 = ( 1 nm 0


n=m n ;m


(12)


The Laplace domain representation of the Laguerre kernels is



Ln (s) = ,/X (s-X)n
(S + X)n+ (13)


Similar to that of the gamma filter, (13) suggests a filter structure in the form of a first order lowpass filter followed by a cascade of identical first order allpass filters. Again, the overall structure is characterized with a single multiple pole at s=-%. Laguerre kernels may be linked to the gamma kernels through the following relation


(14)


Ln(t) = G- E 1 (-2)1)g(X't)
i= 1


One can also realize the Laguerre kernels in the discrete domain. The corresponding zdomain representation will then be Ln () I- 2 1-X ',~
- =-(_XZ n =0,1,2,. (15)


t, x>O:









Before proceeding to the problem of signal representation using the Laguerre kernels, we would like to introduce an orthogonal family of polynomials that shed some light into the origins of the Laguerre kernels. Laguerre Polynomials

A family of polynomials that are closely related to the Laguerre kernels is the

Laguerre polynomials. Laguerre polynomials have been studied extensively in the literature [7]. These polynomials are orthogonal on the interval (0,oo) with respect to the weight function et. In other words,



fLn(t) Lm (t) e-tdt = nm (16)
0


where L (t) is a polynomial. Laguerre polynomials satisfy the equation1


et dn
t)- etd(xle-x) n = 0,1, 2,. (17)
nde


which can be expanded as


n(t) = i i (18)
i=O

Like any other orthogonal polynomial system, Laguerre polynomials can be computed using the triple recursion relation. The recursive relation (also called the Christoffel-Darboux formula) [7] is

1. A more general definition for the Laguerre polynomials also exists where the polynomials
(t) are orthonormal with respect to the weight ae- and they satisfy the following equation [71 ( Ca d-t ", a+,,n( dt )










(2n + I - t) n fn t)(9
f-n +I W = - l (t) n (19)
n+1 n+1


Comparing (11) and (18), Laguerre polynomials can be related to the Laguerre kernels through the expression



Ln (%, t) = 2-Ln (2,%t) e-X (20)



Laguerre Polynomials and the Confluent Hvpergeometric Equation

The second-order ordinary differential equation



t-x(t) + (7-t) dx(t) _-ax (t) = 0 (21)



is called the confluent hypergeometric equation [8][9][10]. The parameters a and y can assume any real or complex value except for 'j0,-1,-2,. The variable t may be complex. One of the solutions of (21) is called the Kummerfunction (or the confluent hypergeometric function) and it is denoted by 1F,(ay ;t). This function can be computed by evaluating the infinite series


axt ac(a+ 1)?t
F, (cc, 7; t) = I+-- + + (22)
y1! 'y(^f +I1) 2!


Many well known transcendental functions, including the Laguerre polynomials, happen to be the special cases of Kummer's function with different selections of the parameters a andy. Table 1 fists some of these functions. For an extensive list see [9]. Referring to Table 1 it can be seen that the Laguerre polynomials satisfy the differential equation










dLn(t)
t-Ln (t) + (1 -t) dt + nL (t) = 0 n = 0, 1, 2,. (23)



whose solution is



Ln (t) = IF1 (-n, 1 ;t) (24)




Parametric Representation of Signals Using the Gamma Kernels

Decomposition of signals in terms of the linear sums of basis functions is a powerful tool that allows the representation of a signal as a discrete table of numbers, in lieu of its original form. The new representation is essentially the projection of the original signal onto the hyperplane spanned by the basis functions Oi(t).



Y(t) = Xwii(t) (25)
i=l1


Choice of these basis functions may vary depending on the application. The basis
functions can be fixed as well as adaptive. Examples of popular basis functions are complex exponentials, which lead to Fourier series expansion , Sinc functions which are used in representing bandlimited signals in terms of their time samples [11] and Gabor functions which have the best time-frequency resolution properties and consequently allow the maximum information transfer rate [12]. Among these, complex exponentials secure themselves a special place by being the eigenfunctions of linear systems. Even though complex exponentials prove to be an invaluable tool in analyzing linear systems, they do not seem to be very efficient (in terms of numbers of bases required) in representing tran-














Table 1 Some well-known transcendental functions derived from Kummer's function Function t a Relation to IF,(a,-y ;t)
-v

Bessel 2jt v+I/2 2v+I F(I +v) et(2) Jv (t)
2it

ModifiedBessel 2t v+1/2 2v+1 r(I+V)e?(P IV )


Coulomb Wave 2jt L+1-jTj 2L+2 etFL (i, t) i--1CL C )


Incomplete Gamma -t a a+l at-aGamma (a, 0


Exponential t a a et


Trigonometric -2jt 1 2 L. sin (t)
t


Hyperbolic 2t 1 2 e sinh (t)
t

n! 1 -n
Hermite t2/2 -n 1/2 (2n)! H2n (t)


Error Function 4t2 1/2 3/2 -yerf(t)


Exp. Integral -t -eXEi (t)


Laguerre t -n 1 (-I) n! L. (t)









sient signals and signals with low degrees of freedom and long sustenance in time. This is

many due to the fixed projection manifold these basis functions define.

A long-neglected family of linear system eigenfunctions is the family of exponentials given by


()= (e 7)U (t) 0:5r


Closely related to the complex exponentials, this family proves to be an efficient tool in representing signals with redundancies [ 13]. In this study, we suggest to use the gamma kernels in linearly expanding a given signal. The gamma kernels can be considered to be a subset of the exponential family where all the time scales are limited to be the same, i.e. ki=X. Laguerre kernels would be an equally good candidate, since these two kernel families span the same space. Both kernels happen to be eigenfunctions of autoregressive moving average (ARMA) systems. Therefore, they are inherently able to capture the dynamics of signals that accept the linear model.

When the number of bases used in the decomposition is finite, the representation problem transforms into an approximation problem. In order to assess the goodness of the approximation (or equivalently the closeness of representation) one needs a measure that is called the norm. Two of the popular norms are L23 and L:* which are the special cases (p=2,-o) of LP norm that is defined as


I/p

LP UW)I = [ifP(t) dt] (27)




The L2 norm which will be used throughout this study finds application in system identification and prediction, while L" has been shown to be of interest for control synthesis









problems.The 12 norm can be interpreted as the Euclidean distance. In optimization it has the advantage of yielding a set of linear equations in terms of the basis weights. The approximations obtained using 13 norm are called the least squares or mean square error approximations.

Statement of the Problem

We would like to approximate a signal d(t) as a weighted sum of gamma (Laguerre) kernels

K
d (t) = y (t) = Ew1gn (X, t) = WTG
n=1 (28)

W = [w, W2, ., W] T weight vector

G= [gI(),t), ., basis vector


such that the mean square error (MSE) J

00
J= = fe2 (t) dt (29)
0
e (t) = d (t) -y (t)


is minimized. In other words, we are trying to approximate the signal d(t) by its projection on the K-dimensional subspace of gamma (Laguerre) kernels. This way we will obtain a parametric representation of d(t) in terms of the weight vector W and the time scale X.
The gamma and Laguerre kernels each constitute a complete set in L2. That means by selecting the approximation order K sufficiently high, the mean square error J can be made arbitrarily small. There are two questions that need to be answered in this optimization problem:

i) What should be the optimum value of the weights?









ii) What should be the optimum value of the time scale . such that the MSE is minimized?


The answer to the first question has been given by Wiener, Kolmogorov and Krein [14][15] early this century. The solution is embraced in the Wiener-Hopf equations (normal equations). Here we will be attacking a generalization of this equation due to the extra time scale parameter X involved in the gamma bases. As a result of this parameter involved, the problem is called a parametric least squares approximation. A thorough discussion on parametric least squares problem can be found in [16]. Before proceeding further with the calculation of the optimum weight and the time scale, we would like to inspect the structure of the space spanned by the set of the gamma bases I gn(,,t) "n=l,2, . }, to get insight into the roles of the weights and the time scale in the approximation. Structure of the Gamma Space

Here we will inspect the gamma bases in a vector space framework. Notice that

despite the fact that the gamma bases are linearly independent, they are not orthogonal. If we regard each basis as a vector in L2 space, then the cosine of the angle between two vectors will be equal to their correlation coefficients. The correlation between the bases gn(t) and gi(t) is given by [6]

0
<9n(X, t) gm (", t) > = Jgn (X, t) gm (X, t) dt
o (30)
0
=f n+mp+m e-2Xtdt
J(n - 1)! (m - 1) !
0

making the substitution X=1P/2 and n+m-2=k-l we can rewrite this as


(k- i)! X _ _ _ _t- (31)
= (n- 1)! (m- 1)! 2n+m-0 (k- 1)!d(









The integrand is recognized as gk(3,t) the area under which is unity due to normalization given in (2). Therefore,



- 2n+- +m-12 (32)



With that in mind, the cross correlation coefficient is given as


(n + m - 2) (33)
2 2 2 (n-)!(m- 1)!
g<9>< gm


hence, we obtain the angle between gn(t) and gi(t) as



Lgngm = Cos[ (n+m-2)!] (34)
=c'-L2VJ(n--M (m - 1)



Equation (34) tells us that the angles among the gamma basis vectors do not change as a function of X. This is an important property of the space constructed using the gamma bases. X conformally maps the basis vectors onto another manifold. Conformal mapping is the class of mapping that preserves the angle relations. Figure 3 demonstrates the angular relation among the first three bases. In spite of the fact that, the space itself is parametric in X, the infrastructure of the space does not change, because X leaves the relation among the bases intact.

We can view X as an extra degree of freedom which has the effect of changing the relative angle of the manifold with respect to the desired signal d(t). Lengths of the bases actually change as a function of . However, in a decomposition problem, scaling of the basis functions is insignificant since a proper choice of the weights will counterbalance it.




















Figure 3 Angular relation among the bases


Figure 4 The effect of changing X on the gamma space


Optimum Value of the Weights

For a fixed X, the best choice of weights can be found from the condition D)J/ W = 0 which yields the normal equations [14]


R.() WPtI, = P(3


012-450 023=300 013-65.90


dtt7A C\cy1t)


(35)








the P vector in (35) holds the crossorrelation values between the bases and d(t). It has the form



P'r. = [.] T (36)


In (35) R(k) is the Gram matrix of the bases. Its (n,m)th element is which was already derived in (32). The result is duplicated here for convenience.



X 2m- dn- 1 (37)


Using (32), the Gram matrix R(X) can be written as R (X) = XRC (38)


where Re is a constant matrix whose (n,m)th element is

1 (n+m-m'-I (39)
2n + m- I n-1 )


For convenience, we will write gn(X,t) as gn(t) from this point on unless there is a possibility of confusion. With that in mind, the optimum weights are given by WPt = R1 P (X) for gamma kernels (40)



If the approximation problem defined in (28) is carried out using the Laguerre kernels instead of the gamma kernels the autocorrelation matrix in (35) becomes an identity matrix and in return the optimum weights have a much simpler form










W'pV = p (k) for Laguerre kernels
(41)
where P() = [.]T



For a given %, the mean squared error J is quadratic in the weights. Therefore WoPt has a single solution and that solution depends on X. That's why this is called a parametric least squares problem.

Let's evaluate the mean squared error when the weights are chosen to be optimum. This is called the Schur complement.



Jsch () = J () W= WP, = T T T T
Jsch W)= =Ed-2WR

R Wopt = P


J.,ch () = Ed - WTOPtP


In (42) Ed denotes the energy of the desired signal d(t). One gets a simpler form for the Schur complement if Laguerre kernels are used as the basis. Combining (41) and (42)

K-i 2
Jsch(W) Ed - WoptWopt - E- _ : (W,= for Laguerre kernels (43)
n=O


The value of the Schur complement Jsch(X) is the same for the gamma kernels and the Laguerre kernels, however, the weights of the decomposition may be different. In the following section we will attempt to minimize Jsch(X) and decouple the optimum value of the time scale X from that of the optimum weights.









Optimm Value of Time Scale X

Previous work

In this section we will address the problem of finding the optimum time scale for signal approximation using the gamma, and the Laguerre kernels and derive the optimality condition that is independent from the optimum weight condition. Determination of the optimum time scale is not a new issue. The literature contains numerous publications on the approximation of signals using the Laguerre kernels [17]. [24], all in the quest of optimum time scale. Although, no exact solution has been found yet, the works of Parks [22] and Masnadi [24] who managed to solve the problem for special cases are worth mentioning here. In his work, Parks gives an estimate of optimum X for a special class of continuous time signals which are defined by two moments. Later, Fu [23] extended this study to the discrete time signals. Meanwhile, Masnadi showed that when the order of approximation was chosen larger than a lower bound and when dealing with signals with no repetitive poles, optimum X was the only real root of a polynomial. He gave the coefficients of this polynomial in terms of the poles of the signal that is to be approximated. Both Parks' and Masnadi's works are summarized in Appendix B. In this study we will derive the optimality conditions for the time scale and devise a moment based method to estimate the optimum value satisfying that condition.

Optimum X

Let's look at the value of the mean squared error for a given X when the optimum weights are chosen. As explained before, this is called the Schur complement and it occurs when e(t), the error of approximation is orthogonal to all the bases used (i.e. normal equation is satisfied) . Before proceeding to the calculation of optimum X's from the condition aj h~ax = o let's take a look at the dependence of Jrch on X. Figure 5 depicts how the dependence of Jsc on X looks like when d(t) is chosen as the impulse response of an









arbitrary third order elliptical filter. This figure is obtained by evaluating the optimum weights in (40) for different values of . and approximation order K and calculating the corresponding mean square error numerically (see Appendix E).The structure of this figure is typical for both continuous time and discrete time cases. As stated before, Jsch curves are identical for the choice of gamma and Laguerre kernels since these two kernel families span the same space.


Figure 5 Typical structure of Jb curves

As can be seen from Figure 5, Jsch curves coincide with each other for Kt order and K+Ist order approximations at the points where aJl,Ch/aX = 0 is satisfied. That is, two consecutive curves touch each other at the local minima or local maxima. A proof of this property can be found in Appendix D. For a discrete time version of this proof see Silva [25]. Since Jsch has the same value for orders K and K+l at the local minima, this implies that the K+Ist kernel does not contribute to the approximation of d(t) at these points . In other words the nonzero component of the K+lst kernel that is orthogonal to the first K


log(Jsch)









kernels is also orthogonal to d(t). For the Laguerre kernels this corresponds to the K+Ist Laguerre kernel LK(X,t) itself, since this is orthogonal to the LO(,,t).LK.I(X,t) by definition. We can write the condition for the optimum , as


ajsch0-) 0 = wK = 0 (44)



Unlike the Laguerre kernel, the K+lst gamma kernel is not orthogonal to its predecessors. Let's denote the component of gK+l(t) that is orthogonal to gl(Xt),., gK(X,,t) with gK+ I (X, t). It can be obtained by subtracting projections of the first K kernels from the K+Ist kernel. Actually, this new kernel turns out be a scaled version of LK(t). For the gamma kernels, the general form of the ith orthogonal component is given by

i
g (%, t) aig, ( t) = A.G = 1, 2,.
v=1
i-I 1(45) where av1= (V- 1)~
T
Ai = [ali. .aii]



The optimality condition of (44) can be rewritten for the gamma kernels as


ajsch 0-) T
-= 0 = AK+ 1P 0%) = 0 (46)


The crosscorrelation vector P was already defined in (36), this time its dimension is K+l. There are no weight terms in (44) and (46) which means that the optimum value of X can be computed separately from that of the weights. Given d(t) one can first determine the best orientation of the manifold (i.e. X) of order K to represent d(t) and then on this









manifold can compute the weights of the bases using the normal equation (40) or (41) depending on the kernel used.

Challenges in computing the optimum time scale

Equations (44) and (46) state the necessary conditions for optimum time scale. However, these are only the necessary conditions, not the sufficient conditions. Due to the nonlinear dependence on X, Jh function has a nonconvex shape. For an order K approximation Jsch has K minima, only one being the global minimum. The optimality conditions specified in (44) and (46) hold for all the minima and therefore does not distinguish the global minimum from the local ones. In a way this is expected since this condition was obtained by setting the gradient Of Jsch to zero. Gradient techniques do not suffice to find the global extremum by themselves in nonconvex functions. They bear only local information about the function characteristics and therefore they need to be backed up by global information about the function itself.

Another challenge in computing the optimum time scale is the involvement of the

crosscorrelation operation. One possibility of overcoming this is to substitute the expectation operator by a time operator, and under the assumption of ergodicity, compute the optimal weight. Unfortunately for the case of (46), there are two problems with that approach: first, the gamma bases are infinite in time, which means that the correlation computed with a finite window may produce severe truncation. The second problem is related to the fact that X is an implicit variable in the correlation. Therefore the conventional procedure to estimate the normal equation does not hold for our case. In the next section we derive a numerical estimation method which overcomes these difficulties by factoring X out of the correlation operation and in return transforming the condition given in (46) into a polynomial equation in terms of X.









Estimation of optimum X using moment matching

In this section we will address the problem of estimating the value of X that minimizes the mean square error J. As stated before, the optimum X has to satisfy the normal equation in (46) (or (44) for Laguerre kernels) while the optimum weights have to satisfy the normal equation in (40) (or (41)). We propose to use the moment matching method which basically attempts to estimate X by equating a finite number of time moments of y(t) to those of d(t), i.e. we want to make


mi{Y(t)} = mijd(t)} I = 0o. K (47)



Using (47) we propose to create K+1 linearly independent equations which will enable us to solve for K weights and k. The operator mi{ } reads as the ith time moment of. Let's first derive the ith time moment of the nth gamma basis, which is defined as



mi [g. 06, ) 1 f Ign ( , t) dt (48)



Matching the moments in time is the dual of Taylor's series approximation around f=0 in the frequency domain. In the time domain if the first K moments are matched, this will correspond to the Kth order Taylor's series approximation performed in the frequency domain.

Inserting the definition of gn(Xt) into (48) yields nni-_ = (n+i- 1)! 1
mif{gn(Xt)} =f-(-. !e -(n- 1) (49)
0









From this we can calculate the ith time moment of the estimate y(t)


mi{y(t)} =mi{GTI.+ ,WK+i,,I}
1- W K+ ,.,


(50)


The weight vector W and basis vector G were already defined in (28). Now their dimension is K+I. Here qj is a row vector given as


(51)


Inserting (50) into (47) one gets


qW
=i m5 {d(t)}


= = O.K


(52)


These K+1 sets of equations can be put into a matrix form.


).QW = AMd


mo {d (t) }1 Md= I
LMr~d(t)J 1+ IXK+ I


where


(53)


FO: l

Uq Jr+xK I


(54)


qi= (i l) (n i (~- 1)! .
. (n- I)! --. (K-1)./ , +1









Now let's force equation (53) to obey the normal equations for the weights, i.e. let's substitute W with 1R;1P yielding


QRCIP = AMd


(55)


We can merge Q and Re"1 together to form a matrix H


(56)


HP = AMd


the unknown crosscorrelation vector P can be solved as


P )H -A Md
K+lxK+l K++l K+ I z


(57)


There is one other normal equation that we have to satisfy.


(58)


AK+IP(%) = 0


Premultiplying (57) with ATK+i yields the polynomial fAX)


T
f()=Ax+iP () = ATH"IAMd = 0


(59)


of order K+1 with constant coefficients. Since we have enforced the normal equations (40) and (45) during the derivation of (59), roots of the polynomial f(,%) turn out to be the estimates of the local minima of Jsch(X). f(%) is a polynomial with constant coefficients and its roots can be computed using numerical techniques. To demonstrate this method an example is included.









Example

In this example, the signal to be decomposed is chosen as


d(t) = te-0.4t-te-'t


t>0


(60)


Figure 6 shows the normalized mean squared error Jsch as a function of X for approximation orders K=3 and K=4. J is computed by determining the optimum weights from (40) for various values of X. Normalization is done with respect to , the energy of the signal to be approximated.


Figure 6 Normalized J as a function ofX


For order K=3, optimum value of X, is around )-Opt--0.2. At X= Xopt, Jnormalized--0.095. Using the moment matching technique, we will get an estimate of Xopt that is close to the theoretical one.


Jacmlized


order K=3









For K=3, Md vector that holds the time moments of d(t) can be computed as




Md = " (61)





Matrices Rc, Q and H can easily be formed as



R 4 6 = 2 88 (62)
L3 2466 612
245 246012





[0 8 -16 161
H - QRcl - -8 56 -12 112 (63)
--_ 432 -992
_528 3504 -7872 6


Also from (45)

T
A4= [1 -6 12 -8] (64)


Substituting these into (59) one gets


f(X) = ATAH-1M (65)

= (0.39+ 2.93X- 131.8X2 + 476X3), = 0


Roots of f() are X=0, X2=-0.04, X3=0.08, X4=0.236. When Xopt is chosen as 0.236 this yields Jnormaijzed=.1 1 which is slightly worse than the theoretical optimum. Figure 7 illustrates the approximation of d(t) obtained with K=3 and X=0.236.



































Figure 7 d(t) and its 3ra order approximation


Although we have matched the first K+1 time moments of y(t) to that of d(t), one still gets consistent estimates of %opt when different moments are used. Table 2 shows the estimates obtained by matching different moments which corroborate our claim.


Table 2 :Time scale estimates


Moments used estimate of

0,1,2,3 0.238
0,1,2,4 0.237
0,1,3,4 0.238
0,2,3.4 0.238
1,2,3,4 0.239
2,3,4,5 0.233
1.2,3,5 0.238


As mentioned before, matching the moments in the time domain is equivalent to matching the Taylor's series coefficients in the frequency domain around the zero frequency. In that sense the method has some limitations. Due to the locality of the informa-









tion provided by the moments, this method would give satisfactory results for signals whose energy is concentrated around the dc frequency. For signals with higher frequency components, the method will tend to approximate the envelope of the signal. In representing bandpass signals it is advisable to use complex time scale. The real part of the time scale can be estimated using the moment method, this time the desired signal being the bandpass signal frequency shifted to the baseband.

The formulation of the estimation method was carried out such that the number of

moments equaled the number of K+1 unknowns (K weights and the time scale). If desired more moments can be used which would transform into an overdetermined problem that can be solved using least square techniques. Estimation of the optimum X using a matched filter

The left hand side of (44) represents the crosscorrelation between the desired signal and the Kth Laguerre kernel. Kuo and Celebi [26] suggested to implement this equation for different values of ), in the form of a parallel organization of matched filters (Figure 8). This arrangement is very popular in communication systems because it implements the optimal receiver for a signal constellation in white noise. However, here we are interested in the zero crossings of the matched filter (correlator) outputs instead of the more familiar peak value. Let us state how the system can be utilized. The desired signal is applied to the block of Figure 8 , and , is varied from 0 to X. in increments of m/m to produce m time signals sj(t) constructed by correlating the input signal with the Kth order Laguerre kernel. A change of the sign from mth correlator output to the m+lst will indicate an extremum point on the MSE curve somewhere in between these two time scale values. The position of the zero-crossing can either be estimated or the time scale interval can be further tightened. The individual values have to be further tested find the value that minimizes the mean square error of the approximation. A rule of thumb for picking up X,. is selecting Xma approximately 2.5 times the bandwidth of d(t). This upper bound stems








from the fact that when X exceeds it the last basis does not have a significant contribution to the approximation.

Figure 9a depicts the dependence of Jsch(X) on X for approximation order K=3 and choice of the desired signal whose Laplace transform is given by


s-3
D (s) = 2
(s + 1) (s + s + 4.25)


Figure 9b illustrates the zero-crossings of the correlator outputs which coincide with the minima of the Jsch curve.


LK( max/mt)


x d,

LK(2 max/m,t)





LK(Xm,t)



_ dmZ


Sign changes between di, d+l


optimum X estimates


Figure 8 Estimation of the optimum time scale using a matched filter implementation


(66)


d(t)


























Figure 9 a) MSE curve for order K=3. b) Weighted correlator outputs


Online Adantation of the Weigzhts and the Time Scale

It is possible to solve the optimization problem of (28) using online adaptation, in other words, by gradually adapting the optimization parameters in the opposite direction of the mean square error gradient, i.e.

dW= ijW W (%,W

(67)

AX = aT (IW
%W)

Since this approach is based on a gradient technique, online adaptation of (67) will only find the stationary points on the Jsch not the global minimum.

Consider the system identification problem depicted in Figure 10. Our task is to adapt the weights and the time scale of the gamma filter such that the mean square error is minimized. This is equivalent to the problem of (28) in the sense that we are approximating the impulse response of the unknown plant as a weighted sum of gamma kernels. The weight update term is obtained by evaluating










dW _ a D 2
-d - -7w W(W W)=- wa-

= -21W


Figure 10 Adaptive system identification problem


where the vector X holds the signals at the taps, i.e.


xl x(t) x W x'*g (t) LXK(oj 1x(t)*g (t)


Evaluating the partial derivative of (68) one gets


VW~t) dW(t)
V W(t) = =dt 21lw


(69)


step size: 1w > 0 (70)


(68)









Notice that the online adaptation is basically an integration operation


W(t) = JVW(u)du


(71)


which inherently implements time averaging. Hence, if the input signal x(t) is ergodic and the step size is chosen sufficiently small, the ensemble average operator of (70) can be conveniently removed yielding the least mean squares (LMS) adaptation rule.
The update term for k can be derived similarly by partial differentiating the mean square error with respect to X, i.e.



AJ De (t) W TaX
dt = -TI)- =-2iu. = 27L
(72)
=2TIL< e (t) W *t aGgxl->




The vector GIx1 holds the gamma kernels and its form was already given in (28). As derived in Appendix A, the derivative of gamma kernels with respect to X can be written as the difference of the kernels. Taking this into account, we can write


A 2TIL ()R.k
T-t =--< t rt2 1


(73)


where


T 1 -


+ [= K+ X









Again, if the input signal x(t) is ergodic and the step size is small enough, the expectation operator in (73) can be conveniently ignored.

Conclusions

This chapter introduced the gamma kernels, a special class of exponentials, and the filter structure associated with it. Gamma kernels were inspired as generalized delay operators with the purpose of substituting the tapped delay lines which performed poorly in the processing signals with redundancies. Gamma kernels are characterized by a single time scale parameter that linearly controls the amount of the delay the kernels induce.

Having linked the gamma kernels to their orthogonal counterpart, the Laguerre kernels, we identified the two as a special case of the Kumnmer'sfunction which entails numerous well-known transcendental functions ( Hermite, Bessel, Gaussian, etc).

The main problem that was attacked in this chapter was that of representing a given signal in the linear projection space of the gamma kernels such that the eudlidian distance between the signal and its projection is minimized. We formulated this problem in a vector space framework and interpreted the role of the time scale as of changing the relative angle between the projection manifold and the signal of interest. The solution of the problem turned out to be an extension of the Wiener-Hopf equation. We decoupled the solution of the optimum time scale from that of the optimum weights. The solution of the optimum time scale problem is unsolved. We developed two techniques, one based on moment matching and the other on matched filtering, to numerically estimate the optimum time scale.

There are several open questions that still must be addressed. Allowing the time scale to be complex would make the processing of the bandpass signals possible. Some related work has been done in the area of Kautz filters [27], but there are many unanswered questions.









In addition to the time moment method presented in this study, other global information measures can be sought to solve the problem of optimum time scale. This global information can help the gradient descent techniques avoid local minima during adaptation.

The highly organized structure of the Jsch curves must be investigated further. There may be some potential information stored in these curve structures that can be extracted to help us position the global time scale. Furthermore, we have observed very similar JSCh curve structures when the gamma kernels are replaced with the Hermite kernels [28]. This indicates the possibility that other members of the Kummer family may exhibit similar phenomena.














CHAPTER 3


TMIE-FREQUENCY REPRESENTATIONS Introduction

Standard spectral estimation methods are based on the implicit assumption that the signals being analyzed are characterized by a stationary spectrum, in other words a spectrum that does not vary in time. This assumption is notjustified for large number of signal processing applications. Spectral characteristics of signals arising in nature often evolve in time due to the changing dynamics of their underlying processes. Prime examples are human speech, seismic signals and biological signals. It is often desirable to view the spectral evolution of these signals. This can be achieved by expressing the time varying signal as a joint time-frequency distribution which depicts the signal energy (or the intensity) per unit time, per unit frequency. There are several time-frequency distribution methods available, each with its own motivation -and properties. The most prominent methods are Wigner-Ville distribution, Page distribution, Rihaczek distribution and the spectrogram. In this chapter we will briefly go over these techniques and show where the proposed method of spectral estimation using poison moments falls in the global picture of time-frequency distributions. We start by introducing the uncertainty principle which is an important concept in understanding time-frequency distributions.

Uncertainty Principle

The Fourier transform is the most commonly used spectral analysis tool. It decomposes a signal into its frequency components. In doing so, however, it fails to reflect when these frequency components occur or how they evolve, at least not directly. Hence, while









possessing an infinite frequency resolution, Fourier transform representation lacks time resolution. In the case of time varying signals where spectral characteristics vary over time, this poses a problem, since the spectral variations can not be seen explicitly in this domain. Instead, the Fourier transform displays an average of the stationary segments of the overall signal. One way of attaining time resolution is by placing a time window over the time segment of interest and evaluating the so called short term Fourier transfiorm (STFI7) of the signal. This method makes sure that only the portion of the signal that is of interest is taken into account. There is an inherent side effect of time windowing, though, which shows itself in the form of spectral smoothing and thereby loss of frequency resolution. Spectral smoothing stems from the fact that the windowing corresponds to the convolution of the original spectrum with that of the window in the frequency domain. Decreasing the window width in an attempt to increase the time resolution win cause the window spectrum to widen which in return further degrades the frequency resolution. Hence, the time resolution is acquired at the cost of the frequency resolution. This stems from the fact that, both the window function and its spectrum cannot be made arbitrarily narrow. For techniques that try to acquire locality in time using windows, there is an inherent trade-off between the time and the frequency resolutions. In his attempt to quantify this relation, Gabor [12] defined the effective time duration' (At) and the effective bandwidth (AD of an arbitrary waveform and showed that their product is lower-bounded by a constant as follows.


A tAf (74)


This is called the uncertainty principle. At a formal level, it resembles to the uncertainty principle in quantum mechanics . The inequality of (74) states that a narrow pulse yields


1. Gabor defines the effective (r.nLs.) duration of a signal f(t) as the variance of its analytic part
eft), where t (t) = f(t) +g(t) . Here (t) symbolizes the Hilbert transform of f(t). A parallel definition holds for the effective bandwidth.









a wide spectrum. Due to the coupling of the time duration to the bandwidth there is always an ambiguity attached to the time fr-equency measurements that employ windowing. This ambiguity stems from our attempt to express a function of one independent variable (time), in terms of two variables, namely time and frequency. The frequency parameter (f) corresponds to the operator -j at in the time domain, and hence it is not an independent variable1 [29]. To summarize, as far as the signal measurements go one should talk about the energy within a band during a specified time interval rather than the instantaneous fr-equency which cannot be measured.

Gaussian shaped signals (with a possible modulation) satisfy (74) with equality. Those are the signals with the smallest time-bandwidth product and they have the form2



V()=ep 2o-& J2q~ +iOJ (75)



Gabor [ 12] coined the phrase elementary signal for these signals. He further suggested expanding a signal into a double sum of delayed and fr-equency shifted versions of these elementary pulses. Later, Hellstrom. [30] made Gabor's expansion exact by using an integral expansion. Gabor believed that in the time-fr-equency plane, each of these pulses represented a quantum of information which he called a logon and depicted these information cells with nonoverlapping rectangles of constant area. It should be remarked that, due to the fact that the delayed versions of Gaussian signals overlap with each other and the fact that Gabor's definition of time-bandwidth product was arbitrary, the approach of representing signals via logons is only an approximate one. Further critique of Gabor's pioneering work can be found in Lerner [3 1]




1. Through out this studyj represents the imaginary unit .�_i
2. Triangular waveforms have slightly worse (10% higher) time.-bandwidth products than the Gaussian waveforms.









Tune-Frequency Representations

The -basic goal of the time-frequency representations is to devise a joint function of time and frequency, that is a distribution, which represents the energy or the intensity of a signal simultaneously in time and frequency. It is required that these distributions possess some desirable properties. Of those properties, nonnegativity, invariance with respect to time delays and frequency shifts in the signal are important ones [32]. Another, desirable feature is the satisfaction of the marginal conditions. This implies that the integral of the distribution over the entire frequency axis at a given time should yield the signal power at that moment.

Gabor [12] and Ville [33] were the first ones to address the problem of time-frequency distributions. Inspired by similar developments in quantum mechanics, their original motivation was to improve on the spectrogramn which is basically a magnitude squared short time Fourier transform (STFT) commonly used for speech signal analysis. The spectral estimation technique that will be introduced in Chapter 5 can be formulated as a spectrogram technique. For that reason we will start with this relatively simple technique and relate it to other time-frequency representations.

We can write the spectrogramn as the magnitude squared Fourier transform of a windowed signal. That is
_2 _0 2

P~tJ f fs(u) h* (u-t =j'd ,fS (V)HW (V _f) 2Nvtdv (76)


Here s(t) is the signal of interest, while h(t) is a window that tapers to zero. As explained in the previous section, like any other window based technique, spectrogram has a timefrequency resolution trade-off associated with it. Equation (76) can be alternatively written as

00 40
P~f f fJ Js(x) s* (y) h* (x -t) h(y -t) e f('x Y) dxdy (77)








This bivariate energy density has the units of Joules per Hertz per sec. Substituting


X =U+


y =u-


we can rewrite it in the form


P(t,f) = f s(u+ ) (u- )h(u-t-j) (u-t+l)e- dudr


In expanding (76) to this equivalent form, we have mapped the one dimensional window h(t) to the two-dimensional kernel h*(t+,r/2)h(t-v/2) which is a function of time and lag 't. If h(t) is equal to rect(t/T), that is a rectangular window of width T, the two dimensional kernel would manifest a diamond shape as depicted in Figure 11.


Figure 11 Two-dimensional spectrogram kernel


The width T of the kernel along the t-axis determines the time resolution, while the reciprocal of the width 2T along the t-axis determines the frequency resolution [32]. Thus, when using the spectrogram method, it is impossible to improve the frequency resolution without inducing a loss of time resolution.


(78)


(79)


h(t-tr/2)




T2 ~t


h t+]2

. . .









The form of the distribution given in (79) can be considered to be the Fourier transform of the time-indexed autocorrelation function R(tj), i.e.


P (t,f) = e-J2XfR (t, r) dt (80)


where



R(t,r) = 4(u-t, )s(u+ )s (u-I)du
-- (81)


*(t,t) h* (t+)h(tAs can be seen in this expression, the kernel function 4(tt) plays an important role in determining the characteristics of the time-indexed autocorrelation which is to be estimated utilizing time averages. Then, a question arises as to which properties the kernel function should have for the estimation of the autocorrelation function. Since a time average would smooth out any time-varying property of nonstationary signals, the kernel function needs to assign a large weight to s(u+t/2)s*(u--d2) when u is close to t and a small value when u is far away from L On the other hand, to a give a reasonable accuracy to the estimate R(tt), the range of the time average should be increased for large value of t compared to the range of the time average for a small value of r.

An alternative selection of the time window is the decaying exponential window, i.e. h(t)=e'Xu(t). This window emphasizes the recent past of the signal1. Exponential window is closely related to the gamma filter structure since it is embedded in the gamma filter in the form of a leaky integrator. In Chapter 5, we will propose to use gamma filter as an alternate time-frequency representation scheme. For now, it should suffice to mention that the proposed method falls under the category of spectrogram with its two-dimen1. Tune-frequency spread of the decaying exponential window is studied in Appendix C.








sional kernel as depicted in Figure 12. As seen in this figure, the kernel assigns larger values to the points close to the origin as desired. In the meantime, it provides a bigger range of time average for large values of , which in return allows a better autocorrelation estimate.


Figure 12 Exponential decaying window kernel a) top view b) 3 dimensional view



Cohen's Class of Bivariate Distributions

In an attempt to circumvent spectrogram's time-frequency resolution trade-off, Ville [33] tried to derive an exact signal energy distribution, based on one represented by Wigner [34] to calculate two dimensional probability distribution in time and frequency of a quantum particle, defining what is known as Wigner-Ville distribution (WD).


1~~i f t=st+)-~~
(t, o) = '-f s* (t - s�(


(82)









Its characteristic function, introduced in modified form to radar theory by Woodward [35], is well-known as the ambiguity function. WD is one of most commonly used time-frequency representations. Following a similar approach that led us to (79), it can be shown that the two-dimensional kernel of WD is 8(r) (Figure 13). That is, while owing infinite time resolution, its frequency resolution is proportional to the data length used. WD can discern nonstationarities. One drawback of WD is the generation of cross-terms. Crossterms occur at the arithmetic mean between any two frequencies (positive or negative) present in the original signal. This interference obscures the output of the WD. Working with the analytic part of the signal will suppress cross-terms due to positive-negative frequency pairs.




T







-T


Figure 13 Two-dimensional kernel associated with the Wigner-Ville distribution


A year after Wigner, Kirkwood [36] came up with another distribution and argued that for certain problems it had a simple form. Following this, Rihaczek [37] rederived this distribution based upon complex energy density functions as


e(t, 0) = I s(t)S* (w) ej (83)
42-7c









In 1952, Page [38] developed the concept of running spectrum and defined an instantaneous power spectrum as the rate of change of energy density spectrum of the signal segment from -00 to t as t is increased.

a 2

P (t, 0) f s (,) e-Jcoc (84)



Levin [39] used the same definition for the segment (t, -o) and defined a new function as the average of both types of instantaneous power spectra.


P (t, CO) = P(t,0) + P(tC) (85)
2

where P+(t,w) is the same as P(tw) defined in (84) except for the fact that the integration limits are (t,o).

It was later realized by Cohen [40] [41] that these time-frequency representations have more in common than is readily apparent, and he proposed what has now become known as the Cohen's class of bilinear time frequency representations, a unifying framework for several of the time-frequency distributions that have been proposed in the literature. The general form of Cohen's bilinear distribution is as follows:


P (t, 0) = 1 fffeJetJ+Jeu (,) s* (t- 2) s (t +IT) dudtdO (86)



The function 4(tj) is the kernel1. By choosing different kernels, different distributions are obtained. For instance, Wigner and Page distributions are obtained by taking (tj)=l and etllV2, respectively (Table 3).


1. In general the kernels may depend explicitly on time and frequency. However, the time and frequency shift invariant distributions are those for which the kernel is independent of time and frequency.









A relatively new class of kernels is due to Choi and Wirlliams [42]. This kernel, named after its inventors is given by expression


(t, ") = e > 0 : scaling factor (87)


This kernel is effective in diminishing the effects of cross terms while retaining most of the properties which are useful for a time-frequency distribution.

All of these distributions have advantages and disadvantages of their own (for a
detailed discussion see Cohen [41]). Of the distributions cited here, WD possesses a large number of desirable properties and provides the most concentration in the time-frequency plane for a signal. This makes WD an attractive tool for the analysis of time-varying spectra. Given the aforementioned connection between the spectrogram and the representation using the gamma filter, we would like to relate the spectrogram to WD, which we take as a reference.

Spectro2ram and Its Relation to Wigner Distribution

As shown in (76), the spectrogram of a signal s(t) is obtained by taking the square
magnitude of the short-time Fourier transform of the signal and its window h(t). The basic properties and effectiveness of the spectrogram for a particular signal depend on the functional form of the window. As seen in Table 3, spectrogram is a member of the Cohen's class of bilinear time-frequency representations. The relationship between the spectrogram and the Wigner distribution of a signal was given by Claasen [43] as a convolution integral.



Sf(t, 0o) = Wf(t, 0) Wh (r-t, wo-0)ddO (88)










Table 3 Some distributions and their kernels

Reference Kernel Distribution

1 IiS (t- )s(t+!)dc
Wigner-Ville J 2


1
eIS -(t)S* (0))ei0j
Kirkwood-Rihaczek



ei~~l' 12 ~ S (.Z) eJ-"d'
Page ;t 2x._.



S e t o r m h* (u h(u + e- jans (z) h (,c - t) & 1
Spectrogram 1 2 2ud 52zs@hrt~~
e--O du


Z 1*
2o (i Re {- s (t) S (w) e"'}
Margenau-Hill




The spectrogram is a time-frequency smoothed version of WD, with the smoothing

function given by the WD of the window h(t). A direct consequence of this is the fact that for a given signal waveform, the effective area in the time-frequency plane of the spectrogram is slightly greater than that of the WD [44]. The time-frequency extent of the spectrogram is minimized when the window is matched to the signal. The nature of the spectrogram, by which it is relatively free of interference terms, has prompted research on determining windows that would maximize concentration of signals in the time-frequency









plane. Since WD provides maximally concentrated distribution and the spectrogram is related to it through (88) it seems obvious that a window function whose WD is most concentrated in time-frequency plane would achieve the minimal spread. The Gaussian window is such a window with its WTD being a bivariate Gaussian function in the timefrequency plane.

In order to achieve optimal concentration, the parameters of the window (namely its time-frequency width and orientation in the time-frequency plane) must be matched to those of the signal components. Such an approach is embodied in a data-adaptive transform [45] where the window parameters are adapted to minimize a concentration measure at each location in time-frequency domain.

Conclusions

This chapter provides background for Chapter 5 where an alternate time-frequency representation scheme that utilizes the gamma filter is proposed. This new representation scheme can be identified as a spectrogramn method that uses a decaying exponential window to discern nonstationarities. In an attempt to familiarize the reader with the general concept of time-frequency distributions, this chapter gave a general outline of the available techniques, paying particular attention to the spectrogram.

Starting out with the uncertainty principle, it was shown that any time-frequency distribution that incorporates windowing, including the spectrogram, trades the time resolution with the frequency resolution. The extent of the trade-off is governed by the timefrequency spread of the window used. One rough way of quantifying the spread is by computing the time-bandwidth product of the window. Time-bandwidth product of the exponential decaying window is derived in Appendix C and compared with that of the gaussian window, which assumes the lowerbound.

Not all time-fr-equency distribution methods are bound by the limitations of the uncertainty principle, such as the Wigner-Ville distribution (WD), Page distribution and Choi-






51


Williams distribution. However, these methods usually distort the distribution in some sense, e.g. by yielding negative results, cross terms or failing to satisfy the marginality conditions.

Among the distributions cited here, WD has the maximum time-frequency concentration. Hence, it can be used as a benchmark. We link the spectrogramn method to the WD using a convolution equation. This equation formulates the spectrogramn as a smoothened version of the WD. It is shown that the amount of smoothing is dictated by the WD of the window used.














CHAPTER 4


HiTERPOLATION

Introduction

Suppose that we have been given certain information about a function. Perhaps, we

know its values, or its derivatives at certain points or maybe its moments. How can we use this finite amount of information to construct other functions that will approximate the original? This is the question we will address in this section. The process of approximation using a finite amount of functional data is called interpolation. Interpolation can be divided into; i) the analysis section where the information on the function is gathered, and ii) the synthesis section, where the approximation is done by fitting prototype functions to the available data. The logic behind the interpolation is that if an approximation agrees with the original function at certain points, then it should also come close to the original function at the intermediate points. It should be noted that the choice of the prototype functions, the way we gather the functional data, as well as the process of function fitting are problem dependent and they rely mostly on experience. In the next section we will introduce the most common form of interpolation; the polynomial interpolation and then proceed to the general problem of finite interpolation.


Polyflon-dal Intervolation

Polynomial interpolation is an approximation technique where a polynomial p,(s) of degree n is equated to the function f(s) at n+1 distinct points. These points are called the interpolation knots. The appropriateness of using the polynomial interpolation for function approximation is justified by the following two theorems.









Theorem: Existence and Uniqueness

Given n+1 distinct (real or complex) points so, s1, .,sn and n+l values f0, fl, . fn, there exists a unique polynomial PN(S) E Pn (set of all nth order polynomials) for which Pn(si)=fi i=0,1,.,n
For proof refer to Davis [46].

Theorem (Weierstrass)

Let f(s) ( C [ab] where C is the set of complex numbers. Given C>O we can find a polynomial PN(S) of sufficiently high degree for which


Vf(s) -p, (s)l <, a:s5b (89)


Together, these two theorems state that one can construct a unique polynomial that fits the function f(s) at given points and the difference between the function and the interpolating polynomial can be made arbitrarily small by selecting the number of points sufficiently large. A natural question that comes to one's mind is whether we can do a similar approximation based on the basis of n+l arbitrary pieces of linear information, rather than the values of the function itself. Further, can we use prototype functions other than polynomials for the approximation? These questions lead to the following general problem.


General Problem of Finite Interpolation

Let X be a linear space of dimension n and let L1,L2,.,Ln be given functionals defined on X. For a given set of values fl, f2,.,fn can we find an element of X, say x such that


Li (x) = fi i = 1, 2, ., n (90)


The answer is yes, if the L, are independent in X* 1.









Lemma: Let X have dimension n. If X1, x2,.xn are independent in X and L1,L2,.,Ln are independent in X* then


IL(xj)I *0


(91)


The converse is also true. See Davis [46] for a proof. Theorem

Let a linear space X have a dimension n and let L1,L2,.,Ln be elements of X*. The interpolation problem of (90) possesses a solution for arbitrary values w1, w2,.,wn if and only if 1, are independent in X*. The solution will be unique [46][47]. Example


Choose four linearly independent functionals as


L1 [f] = Sin [f(0.4)] L3 [] = f(0)


1
L2 I = ff(x) dx
0
df(x)
L4 Uf df Ix=fI


(92)


Given the functional values 1[f(x)] (i=1,.,4) interpolate to f(x)=e'3XSin(x) using the interpolant function h(x)


h(x) = aivi (x)
i=l1


(93)


v1 (x) = Sin (x)

v3 (x) = x


1. X* refers to the space that encloses the functionals L, J-2,.,L.


where


v2 (x) = Cos (x) V4(x) = x2






55


The weights ai of the interpolating function h(x) should be chosen such that Li [f(x) ] = Li [h (x) ] i = 1, ., 4


(94)


This can be put in a matrix form as


L, [vj] . L,[V4]][a, Llt[l [ iii F(95)
L4[v,1] . L4V4] = a4 L4f[J


Weights of the interpolating function h(x) can be found by solving this set of linear equations. Figure 14 illustrates f(x) and its approximation by h(x).


Figure 14 Interpolation of f(x) by h(x)


Systems Possessing the Interpolation Property

Here is a list of commonly used interpolation functional and interpolant function pairs. Example 1: interpolation at discrete points


x=Pn Lj(t)=f(Zj) i=l,2,.


0.12 0.1 0.08 0.06

0.04 0.02


X=Pn


i=1,2,.









Interpolation is done at distinct points using only the function values. Example 2: Taylor interpolation

X=Pn 1i40=f i)(Zo)i=,.

Interpolation is done at only one point using the derivatives of the function. Example 3: Hermite (osculatory) interpolation

X=P2n.1 Li(f)=f)(Zk) i=1,2 k=l,2,.i j=1,2,.,k

Hermite interpolation may be thought of as a combination of Example 1 and Example 2. Interpolation is done at several points using the derivatives of the function in addition to the values.

Example 4: Birkoff (lacunary) interpolation

X=P2nI Li(f)=fOj)(zk) This is similar to the Hermite interpolation, except for the fact that some of the derivatives or the function values might not be available. Example 5: Trigonometric interpolation

X= Linear combination of l,cos(x),Cos(nx),., sin(x),sin(2x),.sin(nx). This is called a trigonometric polynomial of degree n L(f)--ith Fourier series coefficient


Hermite Interpolation

In this study, we will restrict our attention to Hermite interpolation. This is the kind of interpolation where we are trying to match both the values and the derivatives of the interpolant polynomial to that of the function. Given the function values and the derivatives










f(so), /1) (so), . o) (so)
(96)
f(sn), f(1) (Sn)" . jm-,) (Sn)



with the interpolation knots si being distinct and possibly complex, it can be shown that the Hermite polynomial [48]



Hm(s) = n Mi-I Mi-ij- P) (s,) w) dsk (S) i (97)
i=O j=O k=O


where

n n
w(s) = 1 (s-si,)M m = Xmi-1 (98)
i=O i=O

interpolates to the values and the derivatives of f(s) given in (96), i.e.


Hm) (s1) =J(s) O 0:5j < mi (9



It should be noted that when n--O, Hermite interpolation becomes the Taylor's series expansion. For analytic functions the following expression offers a more compact form for the Hermite polynomial [46]

1 f w (t) - w(S)
Hm(S) - j w(t) (t-s)f) (100)
C


The contour C should be selected such that it encloses all the interpolation knots si and f(s) remains analytical inside as depicted in Figure 15. Recalling Cauchy's residue theorem











f(s) = W-tdt (101)

C

One can write the remainder of interpolation (or the error of approximation) as 2jw (t) (t) dt (102)
e (s) af (s) - Hm (s) = yt 12
C

This equation leads to the Cauchy remainder given as


e(s) W (S) m+(M ) (103)
(m + 1)!V


where 4 is an unknown number that lies within the ring min{ Isol,.,Isnl } Although, it is not easy to compute the error of approximation using the Cauchy
remainder, it has an instructive form. As seen in (103), the magnitude of the error depends on the product of the derivative of f(s), which we have no control on, and w(s) already defined in (98). This raises the following question: how can we select the interpolation knots s0,sl,.,sn in the interval [ab] such that the maximum of Iw(s)I is as small as possible, i.e.

Wmin =min max Iw(s)I Isol <- sj 5 (Isn4
Si s (104)


For the case when multiplicity of all the interpolation knots is the same, i.e. m0--ml=.=mn, the answer to that optimization problem is given by the Tschebyscheff polynomials. It turns out that if the interpolation knots si are chosen as the roots of the Tschebyscheff polynomial, Iw(s)I is minimized in the interval [ab]. Further, if the interpolation knots si are all real that dictates [46]










Si = Cos I (2in ) (b-a) +a


Figure 15 Contour of integration for evaluating the Hennite polynomial


Hermite Interpolation of Signals Produced by ARMA Models


Assume that f(s) fits into an ARMA model. i.e.


K
a.
f(s) = I -i (106)
i=l

For this class of functions, the error of approximation given in (102) reduces to the form


K K
es ai = w(s) bi - w(s)y(s)
e (s) =w (s) W (Pi) (S - Pi) s-pi


(107)


Notice that when the function assumes an ARMA model, the approximation error can be written as a product of w(s) and y(s). y(s) has the same poles as f(s) does, however its zeros may be different. The peaks of y(s) should occur at s=Pi. Zeros of w(s) should be selected to coincide with these peaks so as to cancel their effects which implies si=pi.


Poles of f(s)


interpolation points si


i = 0, 1, .,n
mo =MI = . =mn>O:


(105)









Issues in Hermite Interpolation

There are four questions that need to be answered regarding the Hermite interpolation problem:

1) What is the region of convergence of the approximation? i.e. for what values of s does


limr V(s) -Hm(s) = 0 seROC (108)


hold?

2) If two interpolation knots are used, for a given error bound and polynomial order what can be the maximum separation between these two knots? Or assuming that this is a spectral approximation, what is the bandwidth of the approximation? 3) For a given bandwidth and error bound what order of polynomial is required? 4) How does the inclusion of derivative information in addition to the function values help to improve the approximation? i.e. How does the Hermite interpolation compare with the case where multiplicity mi of each knot is unity. Region of convergence

Here, we will give an answer to the first question. Let's consider the simple two point Hermite interpolation case and assume that the complex function fits into an ARMA model. Our purpose is to approximate the spectrum off(s) in between the points so and s1 that lie on the imaginary axis (Figure 16). Select the interpolation knots on the imaginary axis as s0--0 and sl=jou Applying (107) we get


K I
le~~s ~ a = _ (s-s0,) (s-sl) (19
i=1 1-i(is)(is


























Figure 16 Two point Hermite interpolation along the imaginary axis


The condition (108) holds if


g--(S-So) (s-sI)I

for all the poles Pi.
The contours of constant g are called leniiscates. Inside the lemniscate is the region of convergence (ROC). Outside the lenniscate approximation diverges even for large polynomial orders. ROC always encloses the interpolation knots, however for small g (which occurs when a pole of f(s) is close to one of the interpolation knots) the ROC may be composed of more than one disconnected region. Figure 17 illustrates the evolution of the ROC for different g values. In this figure the 'x' mark represents the location of the dominant pole Pd. This is the pole for which 1(Pd-So) (Pd-Sl)l is minimum determines the region of convergence. Intuitively, this is the pole that is closest to either of the interpolation knots.


Im{s)--n




s1=jco


interpolation knots > Refs)









Bandwidth of interpolation


As seen in Figure 17, when the poles of f(s) get close to the imaginary axis, g gets smaller and in return ROC splits into two disconnected regions. As a result, the approximation does not converge in the entireness of the interval s=jQ where fle [O,ow]. The natural question that comes to one's mind is that what is the maximum bandwidth wnax that the approximation would converge? Or in other words, how far apart can so and sI be?


Tm(s) Im(s)
2,. 2

g-1.7 g--0.4

1) 1' Si



0 Re(s) o) Refs)



-2 -1 0 1 -2 -1 0 1
Tm(s) limis)
2 2

g=0.3 g--0.2
1S1 I S1



0 Refs) 0 Re(s)



-2 -1 0 1 -2 -1 0 1


Figure 17 Regions of convergence for various dominant pole locations



Let's concentrate on (110). Substitute s0=0, sl=jo and s=ji. The frequencies inside the ROC have to satisfy









I U) ('2 - 0) < g


Equivalently,


--g < C2 _ CC

This yields two equations


j22-_ jjCD+ g > 0 !i2-- flO _-g > 0


and


(113)


The real Q2 values that satisfy these two equations are given by


)- CO2 + 4g < 0- Q _ 4g
2 2


and


(114)


2 2



This represents two regions. These two regions intersect when


OWm) 2 4g Omax- max g
2


COmax + ,qtmax g
2


(115)


The maximum allowable bandwidth is given by , max = 2,Fg = 2 IpiIPd-jComaI (116)


This nonlinear equation can be solved by considering that the variables satisfy a triangular


relation as illustrated in Figure 18.


(111)


g>O


(112)














x= IPdI


y=lpd-jaIaxl


Figure 18 Triangular relation between the maximum bandwidth and the dominant pole


x and 0 represent the magnitude and the argument of the complex pole Pd, respectively. With this new notation one can rewrite (116) as W max = 2 (117)


The law of cosines dictates that y2 = X2 +2 -2xc,aCosO or
(118)

y2 = x2 + 4xy - 4x yCosO


This quadratic equation in y yields one positive solution given by


y = 2(x-xa CosO) + J4(_x-2CosO)2 +x2 (119)


Substituting this into (117) yields the maximum allowable bandwidth in terms of the magnitude and the argument of the dominant pole.


Omax











Omax =2A2X (X- X3"2Cos0) +" 4 (x- X,,2COS0)2+ X2 (120)

X = IPA o = ZPd



Maximum error versus the bandwidth

In the previous section we derived the bandwidth of convergence when the polynomial order m->o. Here, we will try to find an answer to a more practical question: what is the maximum bandwidth one can achieve for a given error bound 4 and finite polynomial order m. i.e. we want to make

e(jn)l = f(jQ) - Hm (jQ)[_: QE[0, o] 4>0 (121)

m is f'mite

That is

K (jn)(jf-.)fl
e eai n< ! (122)
i =P7 (Pi-- j,)n (,a _-Pi)



where m=2n. Moving the absolute value operator into the summation one can get a pessimistic bound


Kj I . Ia) i'i' j ) _. (123)
I e(j )lj " I TI @-�)n, (,n 1-(1)3)
i=0O P71(i-')


The maximum value that the term I (in) n (j5a -jw)"j can attain in the interval
fle [0, w] is (w/2) 2n. The minimum value that the terms I (j7 - Pi) I and I (p, -jw) I can have is pip where pir is the real part of the pole Pi. With these in mind we can write









2n K I~ 14





An upperbound for the bandwidth is


1 K m
co(n) 52[--)]I y(n) = 1 5+ (125)



For reasonably small error values where 4

Conclusions

This chapter gave an outline of the general theory of interpolation paying special attention to the Hermite interpolation. It was shown that given a set of linearly independent functional information about a function, a prototype function can be made to fit to this information, thereby approximating the function in some sense. A special case of this is when the functional information is presented in the form of function values and derivatives at various points in the domain. For this case, the Hermite interpolation technique can be used to tailor a polynomial whose values and derivatives fit to that of the function at the given points. This is the same kind of interpolation scheme that will be adopted in














maximum percent error


0 0.5 1 1.5 2 2.5 3
w (rad/sec)

Figure 19 Maximum percent error of approximation versus bandwidth w




maximum bandwidth versus order


1 15 2 2.5 3 3.5 4 4.5 5
order n

Figure 20 Maximum bandwidth for a given error bound versus interpolation order n


~1.5









Chapter 5 in approximating the magnitude spectrum of a signal from its frequency derivatives. In. that sense, this chapter can be considered as a prelude aimed at providing the background theory that we will frequently refer in Chapter 5.

Due to the intractability of a general purpose derivation, we analyzed the approximation error mostly in the context of the Hermite interpolation and ARMA signals. It was shown that the approximation error can be decreased by incorporating higher order derivatives into the interpolation. The contribution of the higher order terms is roughly logarithn-fic. That is, the contribution becomes marginal for very high orders. In the case of relatively high approximation errors, addition of high order derivatives may even degrade the approximation.

The separation between the interpolation points plays an important role in determining the approximation error. The maximum separation (that is the bandwidth, in the case of spectral approximation) is dictated by the dominant pole of the function of interest in the vicinity of the interpolation points. The closer the pole is to either of the points, the smaller has to be the bandwidth. Throughout the chapter, we linked the parameters of the interpolation (number of derivatives, bandwidth, dominant pole) to the region of convergence and approximation error. Furthermore, we demonstrated the advantage of using higher order derivatives with an example.

It should be remarked that Hermite interpolation is not the only tool that allows one to approximate a function in terms of its derivatives at distinct points. Hermite interpolation creates a polynomial to do the approximation. Although, not very tractable, one can replace the polynomial with a rational function and fit it to the derivatives of the function of interest. This technique is closely related to the Pade' approximation method [49] [50]. Another prototype function that can be used is the family of Sinco's as suggested by Linden and Abramson [51][52].














CHAPTER 5


REPRESENTATION OF LOCALLY STATIONARY SIGNALS USING LOWPASS MOMENTS

Introduction

In the previous- chapter, we have seen how the incorporation of the derivative information decreases the interpolation error and broadens the region of support. In this chapter, we will demonstrate how to obtain the spectral derivatives of the time-varying signals by using the gamma filter. The gamma, filter generates the so called Poisson moments of the input signal at its taps, which will be shown to correspond to the Taylor's series coefficients, i.e. derivatives of the recent input signal spectrum around the zero frequency. The appeal of the proposed method stems from the fact that in the analog domain, the Poisson moments can readily be measured at the tap outputs of the gamma filter, rather than being computed offline by a computer. Being the Taylor's series coefficients, the moments constitute a nonparametric representation that depicts the temporal characteristics of the input signal. Hence, they are computational correlates of the signal waveform, which can be directly fed into a nonlinear mapper, such as an artificial neural network (ANN) performing classification, identification or prediction. Or alternatively, they can be used to approximate the signal spectrum based on the Hermite interpolation technique that was introduced in the previous chapter.

The gamma filter accommodates an inherent decaying-exponential time-window that makes it possible to localize signals in time and in return represent nonstationary signals. The representation takes place in the form of a spectrogram approximation which was previously introduced under the general topic of time-frequency distributions. In that respect,









the proposed method is an alternate time-frequency distribution that is readily available at the outputs of the gamma filter. Furthermore, if the gamma filters are implemented in analog hardware, it suffices to sample these outputs only at the information rate', rather than the Nyquist rate of the incoming signal. With this convenience, the speed of the processor following the gamma filter becomes independent of the highest frequency of the input signal, being only constrained by the stationarity period of the signal.

We start this chapter by first introducing Poisson moments. This is followed by the

review of the synthesis formula and the analysis of the relationship between the moments and the signal magnitude spectrum.


Gamma Filter and the Poisson Moments: Analysis Formula

Fairman and Shen [53] proposed that a distribution f(t) can be expanded in terms of the derivatives of Dirac's delta function as follows2


f(t) = Efi (to) e-X(t-to) 8() (t- to) (126)
i=O

This infinite summation is identified as the impulse series expansion. The term li(to) is called the ith Poisson Moment of f(t) at t=t0 which is given by [55] gi Wt
f4(to) -f(t) 0Pi(t)[ p(t) - IWi = 0, 1,2,. (127)
t= to xi



'0 ' stands for the convolution operator, while pi(t) is recognized as a scaled version of the continuous-time gamma kernel gi(t) already defined in (1). By definition, f~l(t) is set equal to f(t). Assuming f(t) is causal, the convolution of (127) can be expanded as


1. By information rate, we mean the rate at which a new information packet arrives, i.e. statistical
characteristics of the signal change significantly.
2. For more information on the distribution theory, see Schwartz [54].










toi
A(to) = ff (t- t) e;Lt t ,dt (128)
0

Obviously, the Poisson moment fi(t0) can be physically measured as the i+l't tap value of a scaled gamma filter' with input f(t), rather than evaluating the convolution integral of (128). It is this computational convenience of the Poisson moments that essentially motivates their use (Figure 21).

Equation (127) can be considered to be an analysis formula for the Poisson moments. Before introducing the problem of spectral approximation using the Poisson moments, we would like to introduce the synthesis formula which is the counterpart of the analysis formula in the sense that it dictates how to restore a signal given its Poisson moment samples.



k. time scale


Isttage k+ 1 t alage









Figure 21 Poisson moments are generated by the gamma filter


Reconstruction of a Sip-nal from Its Poisson Moment Sampiles: Synthesis Formula

Having seen how easily the Poisson moments can be obtained using a gamma filter, let's try to develop a synthesis formula that will dictate the reconstruction of the signal from its moment samples. As is well known from Shannon's sampling theory, one can


1. An alternate name for the scaled gamma fiter is Poisson Filter Chain (PFC) [55].









reconstruct a signal f(t) of bandwidth B from its time samples as long as the sampling frequency is equal to or higher than the Nyquist rate 2B [56]. If this condition is met, a bandlimited continuous time signal can be written in terms of its samples as follows


f(t) = f(')sinc (2Bt-n) (129)
n =


This basic reconstruction formula for the case of uniform sampling was later extended by Papoulis [57] to incorporate the samples of multiple linear filter outputs. Papoulis showed that it would be possible to restore a signal given data sampled at 1/Nth the Nyquist rate from the output of N arbitrary filters into which the signal has been fed. Restoration of the input signal from the samples of tap outputs (i.e. Poisson moments) of an Nth order gamma filter would be a special case of this general reconstruction theory. For the case of the gamma filter, the original signal f(t) can be written in terms of its filtered samples (poisson moments) as follows N-I
f(t) = _ f(nTN) kT(t-nTN) TN = N (130)

i = On =-o


The interpolation kernels ki(t) are given by

B
ki (t) = J 'i (u;t) 2xutdu i = 0, 1, .N- 1 (131)
B - 2BIN

Note that ki(t) and 'Fi(u;t) are not Fourier transform pairs. We state without proof that, if they exist, the kernels 'i(u;t) should satisfy the simultaneous set of equations N-I j2zmt
2BIi (u;t) pi (u _ = e TN (132)
i=O









over the parameter set O_ n -1, B-B/N

Poisson Moments and the Magnitude Spectrum

Having seen how the original signal can be restored from its Poisson moments, let's examine the relationship between the moments and the input signal spectrum. Assume a causal signal f(t) and rewrite (128) as


f(to) - jf(to-t)u(to-t)u (t)e dt (133)



where u(t) is the unit step function. Next, introduce fw(to,t), the decaying exponential and rectangular windowed version of the delayed and inverted f(t), such that [ t- t-�2 1eX (14
fw(tot) = f(to-t)rect[ to2 e (134)



where


rect(t) = 1 Itl1/2 (135)
0 else


Figure 22 depicts how the decaying exponential window placed over the signal f(t) emphasizes the recent past of the signal. Substituting (134) into (133) one obtains

"F
A (to) = f w(to, t) dt fw(to, t) - Fw(to, f) (136)
f i ( to









The delay and the time inversion have no effect on the magnitude spectrum which is generally the main concern in speech recognition. The decaying exponential windowing however, helps to emphasize the recent past of the signal, thereby achieving time locality. Its side effect shows itself in the form of frequency domain smoothing. The amount of smoothing is associated with the frequency spread of the decaying exponential window which is discussed in Appendix C.

In addition to the decaying exponential window, there is also a rectangular window, but for to > 4/X, its effect can be safely ignored. Therefore, IFw(toL2)1 can be considered as an approximation to the magnitude spectrum of the recent past of the original signal f(t0).


I f(t)1


Figure 22 Thie exponential window emphasizes the past of the signal prior to t--t0.


Having established the relationship between IFW(tO,Q2)I and the magnitude spectrum of the original signal, let's go back to (136). It can be shown that the ith Poisson moment of f

(to) is related to Fw(to,12) as follows











f1W0 (to, (137)
f' (o) T!_dQ Qo


Analyzing .(137), the moments are recognized as the Taylor series expansion coefficients of FwtQ) around !2=O. That is we can write


FW (to, Q) = D (t)f (138)
i=O

Using a finite number of Poisson moments, a Taylor's series approximation can be

obtained for Fw(to,Q2). In many applications, one does not even need to compute the polynomial of (138). Being the computational correlates of the recent magnitude spectrum, moments themselves suffice to represent the spectrum. Hence, instead of computing the polynomial that approximates the spectrum, the moment vector can be directly fed into a nonlinear mapper, such as an ANN for tasks such as prediction, identification and classification.


Snectral Hermite Interpolation

The Poisson moments carry valuable information about the smoothed spectrum of the input signal in the form of Taylor's series coefficients at zero frequency. One shortcoming of the Taylor's series expansion is however, its locality, i.e. approximation diverges at points away from the origin. Hence, it is not possible to characterize a wide bandwidth of frequencies using a finite number of Poisson moments. As a cure for this shortcoming, one can partition the fr-equency range of interest into several small subbands and carry out the approximation separately for each band. In practice, bandpass filters tuned to different center fr-equencies followed by mixers can be used to shift each band to the origin. The baseband signals at the outputs of these devices can be fed into the gamma filter to obtain









the corresponding moment vectors. Consequently, these moment vectors can be concatenated together to give the overall spectral picture. Since the moment vectors store the spectral derivatives at the edges of each band, Hermite interpolation technique can be utilized to obtain an approximation of the magnitude spectrum .

Tracey and Principe [59] successfully implemented a similar scheme in their smal vocabulary word recognition task using artificial neural networks. A group of constant Q bandpass filters were used to model the cochlea [60]. The authors used a cascade of a square device and an envelope detector instead of the mixer, thereby roughly approximating the power spectrum rather than the magnitude spectrum. As described above, the baseband signals were fed into Gamma filters, outputs of which are the Poisson moments. These moments are further fed into an ANN for word classification (Figure 23).


Spectrouarn with Exponential Decaying Window

Despite the fact that the time windowing results in the degradation of the frequency resolution, due to the smoothing effect that comes along with it, windowing proves to be beneficial in reducing the variance of the spectrogram. As studied in Chapter 3, the spectrograrn technique estimates the power spectral density of a signal in terms of the squared magnitude spectrum of the windowed (filtered) signal (Table 3). Although, in the case of infinite amount of data, the spectrogram estimator is unbiased, it remains to be an inconsistent estimator irrespective of the data size used . The variance of the estimation error can be shown to be proportional to the true power density at a particular frequency [61][62].

There are several methods for reducing the error variance. The ideal way is to ensemble average the spectrogram estimates of all the realizations of the underlying process. Due to the unavailability of independent realizations in practice, this approach might not

1. Although, they might not be mathematically as tractable as the Hermite interpolation is, there
exist other interpolation techniques. See [58] for instance, which utilizes the rational polynomials and [51], [52] that utilize the Sinc functions for that purpose.


































Figure 23 Piecewise spectral approximation via Poisson Moments


be feasible in most applications. For ergodic signals, an alternate way would be averaging the spectrograms; of different segments of the data [63]. This approach trades in the frequency resolution for low estimator variance. Another variance reduction method is due to Daniell [64] who suggested to smooth the spectrogram using a lowpass filter in the frequency domain. Similar to Daniell's method the the method proposed in this study smooths the spectrogramn using a lowpass filtering approach and in return reduce the estimator variance at the cost of frequency resolution.

As illustrated in Figure 24, the spectrogram method can be decomposed into three

operators: time windowing, Fourier transform and the absolute value squared. The Poisson moment (PM) method, on the other hand can be implemented using the following cascade of modules: time windowing, Fourier transformation, absolute value, spectral Taylor's













































Variance and Bias of the St)ectrograrn Estimator


series coefficients, Hermite interpolator and a square device. The first four of these modules are embedded in the Gamma filter. If we ignore the square device, PM method can be thought of as an approximation of the spectrogram, which itself is an estimate of the true power spectrum. For that reason, it becomes a challenge to quantify the performance of the PM method, since there is no reference available to compare it to. One way of quantifying the goodness of an estimator is by determining its bias and variance. In view of this next section is dedicated to the derivation of the bias and the variance of the spectrogram. estimator.


Figure 24 Ganuna filter offers an alternative spectrogram representation at its taps


In this section we will derive the variance and the bias of a spectrogram estimator that utilizes an exponential decaying window. Consider the windowed signal


xW (t) = x (t) e At u (t)


(139)









Using the Wiener-Khintchine theorem, the power spectrum of xw(t) can be written in terms of the autocorrelation function. That is

00
PXW() = f RxW (,) eJftd (140)



where the autocorrelation is

00
Rx (T) = Xw (t) xw (t + -z) dt (141)



The expected value of the spectrogram turns out to be

a*
E{Pxw(O} = fPx(4)W(f-4)d4 (142)



The expectation is carried out over all possible realizations of the signal. Here, Px(f)=E{ IX(f)121 represents the true power spectrum of the signal x(t), while W(f) is given by


W(f) = F{e Atu(t) 0e Atu(t)} = F{te-Xtu(t)} 2 (143)
(,% +j2 t)


Noting that W(f) has lowpass characteristics, the smoothing effect of the window on the spectrum is easily seen from (142). The smoothened spectrogram is biased for finite X, but as X->O0, it becomes unbiased, i.e.


lim E { Px, (14 Px 4)
X--,o


(144)









With these in mind, the covariance of the spectrogram can be written as


cov{Px,(fl)Pxw(f2)} = E{Px (fl)P: (f2)} -EfPxw(fl)}E{P,(f2)} (145)


The first right hand side term can be expanded in terms of the autocorrelation function as



E{Px (fl) P (f2) } = E{ J R r) R (U) e-J2 (flr-f2u)dxdu} (146)



Substituting the definition of the autocorrelation function, one gets

00
E {Pxw(fl)Pw 2)} I =J x (x() x (v) x (u)}e( (147)
0
Se -j2 Vt (T - t) +f2 (u - v) ] dtdzrdudv


This expression involves fourth-order moments, which are difficult to evaluate in the general case. Therefore, we will assume that the signal is white noise. This an oversimplification, but will serve to illustrate the point. So, for the white noise input Nv No
E{x(t)x(T) } = --8(t-T) = -SIT (148)


holds. Then, the expectation term in (147) conveniently expands as N2
E{x(t)x (r)x(v)x(u)} = -[88Vu + tv8,u +8 8v] (149)


This yields


2 2+
E{fP,,w(fl) P.,(f2)}1 + 1 (150)
16 % 2 + n2 ,+f,) 2 X+ R2(f1-f2)2j









Following similar steps, it can be shown that

N
E0Px.(f) 4 (151)


Hence,



coV{exfVl) PxWV2)} 16 -[ 2 12 +.+ .7C 2 (152)



In general, any nonwhite process x(t) can be generated by passing white noise of variance N02/2 through a filter of magnitude-squared response IH(f)12" Because the true power spectrum Px(f) is related by

Noo 2
Px() = IH(f)1 (153)


then, the following is approximately true

1+ 1 154
cov PxW O)PxW2)I- PxOl) Px 2) [222 +2 12 ] (154)



Analyzing this expression, it can be seen that the variance of the estimator decreases with increasing ,.
Having calculated the estimator variance at the output of the spectrogram, the next task is finding out the variance of the spectral derivatives (i.e. Poisson moments). Again, due to the intractability of a general solution, we will work with white noise signal for which (152) holds. Assume that we are interested in the variance of the ith derivative of the spectrum at frequency f=fl


c= [Px(fl) +N(fl)] (155)
df,









N(f) represents the estimation error. The variance of ci can be written as varici} = E (d.N f)21(156)



If we treat the differentiation as a linear filtering operation with magnitude response IH(f)=1fli, the variance can be written in terms of the power spectrum PN(fl,f) of the error term as


var{ci} = f'?iPN(flf)df (157)



The power spectrum of N(f1) is nothing but the Fourier transform of the covariance given in (152), i.e.



PN V(f1 ,) = coy {P. (fl) Px, (f2) } e-2;tfdf2 (158)


Note that


P (f1J) < PN (Of) (159)


holds. We define


- 2 ej(Of N o' = (160)
PN t)PN (0'Jn) 8JfA?+ R2?1


Inserting this into (157), one gets a tight upperbound for the Taylor's series variance as N eN = N(2i)!
var{ci} f ie- 4 2i+1 (161)
--0*









Two important observations that can be deduced from this are, first: selecting large

values of the time scale helps to decrease the variance of the moment variance. Second, as expected, variance of the derivatives increase as a function of the derivative order. These two results can be attributed to the lowpass nature of the time scale and the highpass nature of the derivatives.


Choice of The Time-Scale

The choice of the time scale I has a crucial role in determining the size and the accuracy of the representation matrix whose elements are Poisson moments associated with different bands and measurement times. X by itself, controls several parameters ranging from the effective decaying window length to the maximum allowable bandwidth. This section is devoted to the discussion of the overall effect of the time scale on the representation scheme.

Recall that the Poisson moments correspond to the derivatives of the recent magnitude spectrum around zero frequency. The recency in time is achieved as a result of the decaying exponential window that is embedded in the gamma filter. This exponential window helps to emphasize the recent past of the input signal, thereby localizing it in time. The effective length of the window is controlled by the time scale of the filter. A large time scale implies a small window length. The moments represent the average spectral content of the signal under the window. Hence, when dealing with nonstationary signals, one should avoid averaging segments with different stati4ical characteristics, by keeping the window length smaller than the stationarity period. The stationarity period can be thought of as the rate at which the packets of information arrives. The effective length of the decaying exponential window in the root mean square (mis) sense is derived in Appendix C as



X, (P) = -~--- (162)









where p is the percent energy of the window enclosed under the length xt(I)). According to this, 90% of the window energy is kept within the width xt(0.9)=l.l5/X. Thus, the lowerbound for the time scale can be set as



[n Fi l _ (163)




where TS is the stationarity period of the input signal. For speech signals this period varies in the range 10 ms to 100 ms [65].

Notice that the time scale is constrained by the information rate, not the Nyquist rate which usually assumes a higher value. The information rate also corresponds to the speed at which the Poisson moments need to be sampled irrespective of the maximum frequency content of the input signal. This relatively low sampling rate removes considerable amount of burden over the shoulders of the processor following the gamma filter. Although, the maximum frequency content of the signal has no direct effect on the moment sampling rate, it dictates the number of bands required to cover the entire frequency axis. Based on this argument, one can conclude that the low sampling rate is obtained by parallel processing the frequency domain information using several bandpass filters.

While, the period of stationarity imposes a lowerbound on the time scale, frequency resolution is the determining factor on the other end. As already explained in Chapter 3, the side effect of the time windowing shows itself in the form of frequency domain blurring. That is, the resolution in time domain is attained at the cost of resolution in the frequency domain. Any two frequency components that are closer to each other more than the frequency spread of the decaying window will be identified as one, rather than two separate components. Thus, the time windowing acts as a spectral smoother. Consequently, the choice of the time scale cannot be larger than the minimum frequency resolu-









tion desired in the representation. As derived in Appendix C, the fr-equency spread of the time window, or in other words the frequency resolution of the representation is given by


tan (-!p)
Xf~f 2 X(164)



such that p percent of the total signal power lies within the band fe[-xf(p),xO~)]. Merging (163) and (164) together, one can write the limits of the time scale as


In -I~p<~ 15
Iri P] X 2n Af(15
TS ~ tan ( lp)


where Af is the desired frequency resolution.

Having constrained the time scale from above in terms of the desired frequency resolution, the next question addresses the choice of frequency resolution, itself. Consider the two point Hermite interpolation problem discussed in Chapter 4 where one tries to approximate the recent magnitude spectrum of a signal f(t) within the band of frequencies [O,wo] using the derivatives of the spectrum at the edges of the band. Let pdbe the dominant pole of F(s) in that band'. Since it is based on a first order lowpass filtering, the decaying exponential window can be thought to be inserting an extra pole into the transfer function F(s) at s=-X (Figure 25). We know from Chapter 4 that the size of the region of convergence and the maximum width of the band are inversely proportional to the minimum distance between the dominant pole and the interpolation knots at s0= and sl=jow. For that reason, one should avoid choosing the time scale such that the minimum distance between the pole at s=-X and either of the interpolation knots is smaller than that of the dominant pole.


1. The Dominant pole is defined as the pole with significant weight that is closest to the band of
interest. If there are more than one prospective dominant poles, the center of gravity of these
poles is taken as the dominant pole.



































IM(s)=Q


Figure 25 Thne scale should be comparable with the real part of the dominant pole


Otherwise, the pole at s=-k would become the dominant pole and in return, further narrow the band of approximation. In other words, one should select the time scale such that it maximizes min 1%, IPA I 1Pd - S1 11. This condition is guaranteed to hold if


X > 19te fPd} 1 (166)

Another interpretation of (166) is that in terms of the frequency resolution, there is no benefit in choosing X smaller than the real part of the dominant pole since the dominant pole determines the smoothness of the true spectrum. In other words, there is no reason to try to achieve frequency resolution higher than the sharpness of the true spectrum which is essentially determined by the dominant pole in that band. Hence, provided that the information rate condition (163) is satisfied, a suitable choice for the time-scale that ensures both sufficient frequency and time resolution would be


X - 9te f Pd1 (167)


Refs)









Examples

In this section we would like to present examples of spectral estimates obtained by the Poisson moment method. As a test signal, we picked the speech utterance 'greasy' archived in the TIMIT database [66]. The original data was sampled at 8 KHz and stored in 16 bit data format.
In the case of finite amount of data, the true spectrum is not known. In other words, a reference that the estimates can be compared to is not available. This poses the biggest challenge in quantifying the goodness of the spectral estimates. One can resort to statistical measures such as estimate bias, variance, etc, like we did in this chapter, but these are average measures and do not help to assess the quality of the individual estimates. One problem dependent solution would to be to test several estimation methods in the same application and compare their performances. In our case for instance, one can compare the classification error rate of a speech recognition system that uses the proposed PM method as a preprocessor and other conventional frontends (tapped delay line, FF17, cepstral coefficients, etc) in front of an ANN. This approach is not within the scope of this study, however. Interested reader can refer to the work of Tracey and Principe [59] who compared the PM and tapped delay line preprocesors in an isolated speech recognition problem. .
Figure 26 shows the logarithm of the spectral estimate of a segment of the speech signal. This is obtained by first applying a rectangular window of width 20 ms over the speech signal and then computing the magnitude square of its Fast Fourier Transform . Notice the spiky nature of the spectral estimate.
Figure 27 shows the estimate when the rectangular window is replaced with an exponential window. The time scale of the exponential window was chosen to be 150 rad/sec. This choice is mainly determined by the stationarity duration of the speech sound, which was 20ms for our case. As a result of the window type used, this estimate is smoother than the previous one. Although the original method is proposed to be used in the continuous time domain, we were obliged to carry out the simulation in the discrete time domain





Figure 26 Spectral estimate using a rectangular window length of 20 ms


using highly interpolated versions of the sampled signal.' In this approach, the signal is first exponentially windowed and later its spectrum is computed using the discrete time Fourier transform. The derivatives of the spectrum are computed numerically to give an estimate of the Poisson moments. In Figure 27, the dashed line depicts the estimate obtained by Poisson moments. We used only I moment per band. There were 16 logarithmically distributed bands that cover the frequency axis. This estimate may be thought of as a 16 point Fourier transform of an exponentially windowed signal, which is far from being satisfactory. Figure 28 and Figure 29 display the estimate results for the cases of 3 moments/band and 4 moments/band, respectively. The improvement in the estimates obtained by the utilization of the moments is clearly seen. Finally, in Figure 30 we plot the maximum absolute error between the spectral estimate and its estimate obtained using Poisson moment technique, as a function of the number of moments used per band. This figure demonstrates the decrease in the error achieved by using higher order moments. The


2000 frea (Hz)


3500


1. The original signal was sampled at Wk. We upsampled it by a factor of 40.









reason we chose to use the maximum absolute error stems from the fact that this is a more pessimistic measure compared to the MSE in the sense that it penalizes impulse like errors more and in the case of speech signals, let's us identify errors that occur around the format frequencies where the main features of the signal are kept.

1.2



Exponential windowed
0.8- --------- Estimate

0.6
16 bands
1 moment/band
0.4- ,-I 50

0.2

01
0 Soo 1000 1500 2000 2500 3000 3500
HZ
Figure 27 Spectral estimate using an exponential window (solid line) and its estimate
obtained by Poisson moment technique (dashed line) (I moment/band).

In the rest of this section, we will analyze the dependence of the proposed method on the time scale and the number of bands that are used. For this purpose, we define two error criteria which complement each other. The first one is the power of the spectral estimation error normalized with respect to the power of the desired spectrum (i.e. magnitude spectrum of the exponentially windowed signal). This gives an idea of the average error along the entire spectrum. This criterion however, does not penalize narrow, impulse like errors substantially. For that reason, we pick the maximum absolute estimation error normalized with respect to the peak of the desired spectrum as the second criterion to evaluate the goodness of the estimates. It should be noted that compared to the error power, this is not only a pessimistic criterion, but due to its nonlinear nature, it is also sensitive to the parameters of the method.




































0 So0 1000 1500 2000 2500 3000 3500
frog (Hz)

Figure 28 Spectral estimate using an exponential window (solid line) and its estimate
obtained by Poisson moment technique (dashed line) (3 moments/band).


0 500 1000 1500 2000 2500 3000 3500
freq (Hz)

Figure 29 Spectral estimate using an exponential window (solid line) and its estimate
obtained by Poisson moment technique (dashed line) (4 moments/band).









Maximum Absolute Error

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1 2 3 4
number of moments per band
Figure 30 Maximum normalized absolute estimation error in the entire frequency domain



We estimated the spectrum of segments of 10 'greasy' words that were uttered by the same male speaker. In Figure 3 1, we plot the variation of the normalized error power as a function of the time scale for one of the utterances. The improvement in the estimates as a function of the the time scale is clearly seen. This is somewhat expected because, for large values of the time scale the time window gets smaller decreasing the number of degrees of freedom of the spectral content of the signal.

In Figure 32, we test the PM method using the same signal segment against the second error criterion: normalized maximum absolute error. This plot too, demonstrates the improvement in the estimate as a function of the time scale and the moment order.

As mentioned before, the results displayed in Figure 31 and Figure 32 were obtained using a single word. Figure 33 and Figure 34 demonstrate the results of the same experiment averaged over the set of 10 utterances. In both of these figures, there exists a time scale threshold below which the utilization of high order moments can be actually disad-












vantegous. This is mainly due to the fact that the spectral estimates get spikier for small values of X causing noisier derivative estimates.


Figure 31 Variation of the normalized error power as a function of the time scale for different number of moments M used



Logarithm of the Normalized Maximum Absolute Error


350 400 450 Soo


200 250 1% 300


M=1


-0.4


-0.6



-0.8 r


. M=2
- - - - - - M=3
M--4


100 ISO 200 250 ). 300


350 400 450


Figure 32 Dependence of the normalized maximum absolute error power as a function of the time scale for different number of moments M.















1.5


1


0.5


0


-.0.5


-1


-1.5


-2


-2.5


_______ M=l . M=2 M--4


7-S


Figure 33 Dependence of the average error power on the time scale. (M: number of moments).


100 150


200 250 1%300


350 400 450


Figure 34 Dependence of the average maximumn absolute error on the time scale. (M: number of moments).




An alternate way of displaying the information in Figure 33 and Figure 34 is as shown in Figure 35 and Figure 36. According to these graphs the fourth moment helps to decrease the error for time scale values larger than 200.


Logarithm of average normalized error power


Logarithm of average normalized maximum absolute error


3.4 3.2

3

2.8 2.6 2.4 2.2

2

1.8 1.6 1.4


S. M=l

. M=2

-------- M=3


. .

.


0




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