Title: Real-time composite signal decomposition
CITATION PDF VIEWER THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00098354/00001
 Material Information
Title: Real-time composite signal decomposition
Physical Description: xiii, 187 leaves. : illus. ; 28 cm.
Language: English
Creator: Skinner, David Preston, 1947-
Publication Date: 1974
Copyright Date: 1974
 Subjects
Subject: Signal theory (Telecommunication)   ( lcsh )
Time-series analysis   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 185-186.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00098354
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000585194
oclc - 14198779
notis - ADB3826

Downloads

This item has the following downloads:

realtimecomposit00skin ( PDF )


Full Text











REAL-TIME COMPOSITE SIGNAL DECOMPOSITION


By


DAVID PRESTON SKINNER




















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




UNIVERSITY OF FLORIDA








































Dedicated to Betty became I tove heA.


__















ACKNOWLEDGEMENTS


The author wishes to thank Dr. D.G. Childers, chairman of his

supervisory committee, for his guidance during the preparation of

this dissertation. The example set by him, both intellectually and

personally, has been a highlight of this author's experience at the

University of Florida. The author also wishes to express his gratitude

to the other members of his supervisory committee for their assistance.

Finally, the author would like to express his appreciation to his

parents, Mr. and Mrs. J.P. Skinner, for their encouragement and

love throughout the years, and to his wife for her unflinching support.


____
















TABLE OF CONTENTS


PAGE

ACKNOWLEDGEMENTS ......................................... iii

LIST OF TABLES ....................................... .......... vii

LIST OF FIGURES ................................... ..... .. viii

ABSTRACT .............................................. .... xii

CHAPTER I INTRODUCTION ................................... 1

CHAPTER II THE CEPSTRA .................................... 5

THE POWER CEPSTRUM ........................... 5

History ....................................... 5

Introduction ............................... 6

THE.COMPLEX CEPSTRUM AND WAVELET RECOVERY .... 10

History .................................... 10

Introduction ............................... 11

The Relation Between the Complex and Power
Cepstrum ................................... 18

The Phase Cepstrum ......................... 20

SUMMARY ..................................... 21

CHAPTER III THE PITFALLS OF CEPSTRUM COMPUTATION ........... 23

LINEAR PHASE TERMS ........................... 23

PHASE UNWRAPPING ERRORS ...................... 43

ALIASING ..................................... 47

OVERSAMPLING ................................. 49

SUMMARY ...................................... 49



(iv)








PAGE


CHAPTER IV THE EFFECTS OF SOME DATA PROCESSING TECHNIQUES
ON THE CEPSTRA ................................. 51

WINDOWING THE COMPOSITE SIGNAL ................ 51

The Exponential Window ..................... 53

Interpretation of Results .................. 55

The Common Window .......................... 58

Interpretation of Results .................. 61

Conclusions ................................ 76

The exponential window ............ ....... 76

The common windows ... ..................... 77

WINDOWING THE LOG-SPECTRUM ................... 77

Interpretation of the Results .............. 80

Conclusions ................................ 81

HANNING THE LOG SPECTRUM ..................... 81

Experimental Results ....................... 84

Conclusions ............................... 86

THE ADDITION OF ZEROES ....................... 87

Experimental Results ....................... 91

Conclusions ................................ 110

CHAPTER V ALGORITHMS FOR A REAL-TIME WAVELET RECOVERY
SYSTEM ........................................ 112

THE DFT ALGORITHMS ........................ ... 113

The Cooley-Tukey (decimation in time) and
Sande-Tukey (decimation in frequency)
Algorithms..... ............................ 113

Modifying the Basic FFT Algorithms ......... 114

The Bergland Algorithm ..................... 118





(v)








PAGE


The Hartwell Modification .................. 118

Which Algorithm for Complex Cepstrum
Computation? ................................ 122

COMPUTATION OF NONLINEAR FUNCTIONS ............. 126

Computation of Nonlinear Functions .......... 126

PHASE UNWRAPPING AND LINEAR PHASE REMOVAL ..... 131

LINEAR FILTERING ... .......................... 132

:AN OVERALL LOOK AT SYSTEM PERFORMANCE ......... 133

An.Estimate of System Performance ........... 136

REAL-TIME COMPUTATION OF THE POWER CEPSTRUM ... 136

SUMMARY ...... ................................. 137

CHAPTER VI SUMMARY AND CONCLUSIONS ......................... 139

COMPUTATIONAL PROBLEMS ........................ 139

WINDOWING ..................................... 140

EXTENDING THE DATA RECORD WITH ZEROES ......... 142

ALGORITHMS FOR REAL-TIME WAVELET RECOVERY ..... 142

SUGGESTIONS FOR FUTURE RESEARCH ............... 143

APPENDIX A A COMPARISON OF THE ECHO DETECTION CAPABILITY
OF THE PHASE, POWER AND COMPLEX CEPSTRA ......... 145

APPENDIX B THE EFFECTS OF ADDITIVE NOISE ON THE CEPSTRA
AT HIGH SNR ................................. .... 155

APPENDIX C THE INVERSE TRANSFORM OF THE TRANSFORM OF A
REAL VALUED SERIES UTILIZING THE FORWARD
ALGORITHM .................................. ... 173

APPENDIX D PROGRAM LISTING ................................. 174

BIBLIOGRAPHY ................................................. 185

BIOGRAPHICAL SKETCH ............................. ........... 187


(vi)














LIST OF TABLES


TABLE PAGE

1 Algorithms for Cepstrum Computation ................ 123

2 FFT Processor Performance ............................ 125

3 Throughput Comparison of Stages in the Wavelet
Recovery System ................................... 135


(vii)















LIST OF FIGURES


FIGURE PAGE

1 Computation of the Power and Complex Cepstrum ....... 8

2 Phase Unwrapping .................................. 14

3 Effect of a Delay in the Composite Signal on the
MSE of the Recovered Wavelet ...................... 25

4 Composite Signal ................................... 28

5 Unwrapped Phase Curve .............................. 29

6 Log Magnitude ...................................... 30

7 Complex Cepstrum ................ ......... ........ 31

8 Phase Cepstrum ..................................... 32

9 Power Cepstrum ................... ................. 33

10 Recovered Wavelet .................................. 34

11 Composite Signal .................................... 36

12 Unwrapped Phase Curve ... ..................... ... .37

13 Log Magnitude ....................................... 38

14 Complex Cepstrum ............................... 39

15 Phase Cepstrum ....................................... 40

16 Power Cepstrum ....................................... 41

17 Recovered Wavelet .................................... 42

18 Phase of X(ejWT) .................................... 45

19 Phase Unwrapping Errors ........................... 45

20 MSE of Recovered Wavelet when the Input Data Record
is Exponentially Windowed ......................... 55


(viii)









FIGURE PAGE

21 MSE of the Recovered Wavelet when the Input Data
AreHamming Windowed ................................. 60

22 Composite Signal ................................... 65

23 Unwrapped Phase Curve .............................. 66

24 Log Magnitude .................................. .... 67

25 Complex Cepstrum ................................... 68

26 Phase Cepstrum ..................................... 69

27 Power Cepstrum .................................... 70

28 Recovered Wavelet ................................... 71

29 MSE of the Recovered Wavelet when the Log Spectrum
Is Hamming Windowed ................................ 79

30 MSE of Recovered Wavelet when the Log Spectrum Is
Hanning Smoothed ................................... 85

31 Composite Signal ................................. . 93

32 Unwrapped Phase Curve ............................... 94

33 Log Magnitude ....................................... 95

34 Complex Cepstrum .................................... 96

35 Phase Cepstrum ..................................... 97

36 Power Cepstrum ...................................... 98

37 Recovered Wavelet ................................... 99

38 Composite Signal (Zeroes Added, 512 Points) ......... 101

39 Unwrapped Phase Curve ............................... 102

40 Log Magnitude ....................................... 103

41 Complex Cepstrum .................................... 104

42 Phase Cepstrum ...................................... 105

43 Power Cepstrum ...................................... 106

44 Recovered Wavelet .................................... 107



(ix)









FIGURE PAGE

45 MSE of the Recovered Wavelet when Data Record Is
Extended with Zeroes ................................ 109

46 Cooley-Tukey FFT Algorithm ......................... 115

47 The Sande-Tukey FFT Algorithm (with bit input
reversed) ......................................... 116

48 Bergland Real Valued Input Algorithm ................ 119

49 Bergland Inverse Algorithm ............... ......... 120

50 Computation of log(x) .............................. 128

51 Computation of exp(x) ............................. 129

52 Overall Wavelet Recovery Algorithm .................. 138

53 Composite Signal .................................. 148

54 Unwrapped Phase Curve .......................... 149

55 Log Magnitude ..................................... 150

56 Complex Cepstrum .................................. 151

57 Phase Cepstrum ..................................... 152

58 Power Cepstrum ............. ........................ 153

59 Recovered Wavelet .................................... 154

60 Composite Signal ................................. 158

61 Unwrapped Phase Curve .............................. 159

62 Log Magnitude ..................................... .. 160

63 Complex Cepstrum ................................... 161

64 Phase Cepstrum ..................................... 162

65 Power Cepstrum ......................... ........... 163

66 Recovered Wavelet .................................... 164

.67 Composite Signal (Zeroes Added, 1024 Points) ........ 166

68 Unwrapped Phase Curve .............................. 167




(x)









FIGURE PAGE

69 Log Magnitude ....................................... 168

70 Complex Cepstrum .................................... 169

71 Phase Cepstrum ..................................... 170

72 Power Cepstrum ..................................... 171

73 Recovered Wavelet ................................. 172


(xi)















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



REAL-TIME COMPOSITE SIGNAL DECOMPOSITION

By

David Preston Skinner

December, 1974

Chairman: Donald G. Childers
Major Department: Electrical Engineering


The purpose of this research is to investigate the recovery in

real-time of an unknown wavelet from a composite signal in the presence

of additive noise. Specifically, the composite signal consists of

a wavelet and its echoes.

Two techniques, the power and complex cepstra, are used in the

wavelet recovery procedure. Historically, these techniques have been

computed separately. An alternate definition of the power cepstrum

is presented which closely relates the two techniques,-and makes

computation of the power cepstrum from the complex cepstrum trivial.

This unification led to the discovery of a third cepstrum technique,

the phase cepstrum, which like the power cepstrum is useful in the

determination of echo epoch times.

Several problems inherent in cepstrum computation which must be

overcome to achieve satisfactory wavelet recovery are examined. These


-i.) --









problems include linear phase terms, phase unwrapping errors, aliasing,

and oversampling.

The effects of windowing (as used to reduce leakage in spectral

analysis) at various stages in the wavelet recovery system are examined.

In general, windowing of the input data or log spectrum is detrimental

to wavelet recovery. A windowing of the complex cepstrum may result

in some improvement in wavelet recovery.

The addition of zeroes to the input data record is explored.

This is observed to reduce the phase unwrapping and aliasing problems.

Finally, algorithms suitable for use in a real-time wavelet

recovery system are examined. A system based on the selected algorithms

should be able to achieve sampling rates of around 10 samples/sec.






























(;;iii)














CHAPTER I

INTRODUCTION

Composite signals consisting of the convolution of two or more

simple signals arise in a number of physical situations. The

separation of the contributions of the components of such signals

often yields important information about the underlying physical

processes. Several techniques have been devised for this purpose,

these include inverse filtering [1], Wiener filtering [1], decision

theory [2] and the power and complex cepstrum techniques to be

discussed herein. Each of these techniques is applicable under its

own circumstances.

In particular the power and complex cepstrum have been shown

to be invaluable when the composite signal consists of a basic

wavelet convolved with a train of impulses [3]. One example of this

is a composite signal consisting of a basic wavelet plus echoes

as might arise in sonar or radar applications. The power cepstrum,

defined as the power spectrum of the logarithm of the power spectrum

of the given signal, has been shown to be successful in the deter-

mination of echo arrival times when the composite signal consists of

a basic wavelet and its echoes [3]. Of course, since the power

cepstrum is derived from the power spectrum there is no hope of

reconstructing the original wavelet from it. The complex cepstrum

technique, defined as the Fourier transform of the complex logarithm








of the Fourier transform of the given composite signal, overcomes

this limitation since the phase information is preserved. The success

of both techniques stems from the fact that convolution in the

time domain is equivalent to multiplication in the frequency domain.

Thus by taking the logarithm of the frequency domain function the

contributions from the basic wavelet and impulse train may be

separated into a sum. In the case of the complex cepstrum appropriate

filtering techniques may then achieve separation of the components,

and by performing the inverse operation (to the complex cepstrum)

we may extract the basic wavelet. There is, in fact, no other

technique either linear or nonlinear, which allows the extraction

of an unknown wavelet from a composite signal consisting of the

wavelet and its echoes.

This investigation will primarily be concerned with the cepstrum

techniques as applied to echo detection and extraction. However,

the results are in most cases equally applicable to the other areas

in which cepstrum techniques have found applicability. Among these

areas are seismology [4,5], and speech [6,7].

In the past the cepstrum techniques have been limited to non-

real-time applications. It seems clear that the performance of

these techniques in real-time would aid the research effort in the

above-mentioned fields, and perhaps open new areas of interest.

This investigation is the first into the area of real-time compu-

tation of the power and complex cepstrum, and wavelet recovery. This

is a formidable problem since these techniques involve the.computa-

tion of several Fourier transforms together with nonlinear operations.








The purpose of this research was:

(1) to identify and investigate the problems inherent in

computation of the cepstra, and in wavelet recovery,

(2) to investigate the use of ordinary data handling techniques

(windowing and the addition of zeroes) at various points in

the wavelet recovery algorithm to improve the performance of

these techniques both in noise free, and additive noise

environments, and

(3) to select algorithms suitable for the real-time computation

of the cepstra, and for real-time wavelet recovery, and finally,

to estimate the possible data rate of a system based on the

selected algorithms.

Chapter II reviews the power and. complex cepstrum techniques.

Historically these two techniques have been treated separately. An

alternate definition of the power cepstrum is presented which closely

relates the two techniques, and makes computation of the power

cepstrum from the complex cepstrum trivial. This unification of the

two techniques led to the discovery of a third cepstrum technique,

the phase cepstrum, which like the power cepstrum is useful for the

determination of echo epoch times. This technique is introduced and

discussed. A comparison of this new technique with the complex and

power cepstra for the estimation of the echo amplitudes, and epochs

in the presence of noise is reported in Appendix A.

Chapter III presents some problems of cepstrum computation which

must be overcome to achieve satisfactory wavelet recovery. Some of

these have been reported by other authors ; others were previously









unmentioned. Problems of linear phase terms, phase unwrapping errors,

aliasing, and oversampling are discussed, and methods for their

alleviation are indicated.

Chapter IV analyzes (both theoretically and experimentally)

the effects of windowing at various stages in the wavelet recovery

process, and the effects of extending the input data record with

zeroes. Considerable insight is gained into the performance of

cepstrum techniques in additive noise.

Chapter V discusses the algorithms suitable for use in a

real-time wavelet recovery system, and roughly estimates the per-

formance of such a system. The computation of the power cepstrum

(only) in real-time is also discussed. And, finally, Chapter VI

collects and summarizes the results of the previous chapters.














CHAPTER II

THE CEPSTRA


This chapter presents a brief history of, and introduction to,

the power and complex cepstrum techniques. In addition, a new

cepstrum technique, the phase cepstrum, is introduced along with

an alternate definition of the power cepstrum which serves to unify

the cepstra, and is of computational interest.


THE POWER CEPSTRUM


History

The power cepstrum was first presented by Bogert et al..[4] in

1963 as a heuristic technique for finding the echo epoch times of

a composite signal. Subsequently, A.M. Noll [6] in 1964 successfully

applied the power cepstrum to speech data to determine the pitch

period and for voiced-unvoiced detection. Noll also simulated a

technique for short time power cepstrum computation utilizing the

direct computation of the discrete Fourier transform (DFT). In 1966,

Bogert and Ossanna[8] examined the statistical properties of the

power cepstrum of a Gaussian signal in Gaussian noise. Halpeny [9]

examined the properties of the cepstrum of a composite signal

consisting of a basic wavelet plus multiple echoes in the presence

of additive noise. Kenerait [3] then extended Halpeny's results

and examined the effects of echo truncation on the power cepstrum








(in conjunction with his work on the complex cepstrum). Kemerait's

dissertation shows the power cepstrum to be invaluable in the

detection and estimation of echo amplitudes and epochs.


Introduction

Basically, the power cepstrum, usually referred to in the

literature as simply the cepstrum, is just a clever way of separating

the contributions of two simple signals to the power spectrum of

a composite signal (which is the convolution of the two simple

signals). Of course to effectively separate any two signals there

must be some difference on which we may base the separation; for

the power cepstrum (and analogously for the complex cepstrum) this

difference is that the power spectra (more properly the logarithms

of the power spectra) of the two simple signals must vary (as a

function of w) at different rates; i.e., in Tukey's terminology [4]

they must occupy different ranges of quefrency. Historically the

power cepstrum has been defined as the power spectrum of the logarithm

of the power spectrum of the given signal. In actuality, the power

spectrum of the logarithm of the power spectrum does not exist for

most signals. The power cepstrum is only meaningful when defined

in a sampled data sense (as is the complex cepstrum). Thus the

following definition is proposed: The power cepstrum is the square

of the inverse z-transfonn of the logarithm of the magnitude squared

of the z-transform of the given sequence. Evaluated on the unit

circle this definition (except for the normalization factors associated

with the power spectrum) is precisely the procedure historically used

for estimating the power cepstrum. Thus we may write









Xpc(nT) = (Z-(log IX(z) 2)2 (2-1)


where x (nT) is the power cepstrum.of the sequence x(nT) and X(z)

is the z-transform of x(nT). Consider the signal given below

y(nT) = x(nT) e(nT) (2-2)

where denotes convolution. The magnitude'squared of the z-transform
of the above equation is then

IY(z) 12 = X(z)12 IE(z)12 (2-3)

Taking the logarithm of equation (2-3), we obtain

log lY(z) 2 = log IX(z)12 + log lE(z)(2 (2-4)

We may now return to the time domain by taking the square of the
inverse z-transform of equation (2-4) with the contributions due to
the two signals additively combined (provided they occupy different
ranges of quefrency).


pc(nT) = xpc(nT) + e (nT) (2-5)



Figure 1 shows the computation of the power cepstrum as outlined
above. It is of interest to note that the second squaring
operation will again mix the contributions from the two signals
if they do not occupy distinctly different quefrency ranges (there
will usually be some overlap in practice).


















UQJ
-0
0J 0-
X0 I-

4 >


E

4-



0,



CL
E
0


"LI
*0







'4-





0
41
0


0
O-











5-


*r-


.C
40


0 (-
41







0 t.
C)

C 4- -

40 r

0



=nO

0E
Ir0

C I
SON


C0 E

=) 0

O O







C r- LI..

Z S.- U-









To further clarify this technique consider a composite signal

consisting of a basic wavelet and a single echo


y(nT) = x(nT) + ax(nT-noT)


(2-6)

(2-7)


= x(nT) (6(nT) + a6(nT-n T)).


Taking the magnitude squared of the z-transform of equation (2-7),
we have


jY(z) 2 = |X(z)j2 |(1+az |-no
IY(z)!2 = IX(z)I2 I(l+az )I


(2-8)


Evaluating the above equations on the unit circle (z = ejwT) and
taking the logarithm, we obtain


log lY(e jT)I2 =.log IX(eT) 2 + log(l+a2+2a

= log X(eJiT) 2 + log(l+a2) +




We may now expand the third term on the right
a power series (except for the point values a
to obtain


cos(wn T)) (2-9)

log(l+ 22a cos(wnoT))
1+a
(2-10)

hand side of (2-10) in
= +1 and cos wn T = l)


log (1+ 2a2 cos(wn T)) = (-1)m+l (-2a cos wn T)m/m
l+a m=1 l+a


(2-11)


Thus, the logarithm of the magnitude squared of the z-transform of
the composite signal will contain cosinusoidal ripples whose amplitude
and quefrency (that is, the "frequency" of the ripples) are related
to the echo amplitude and delay respectively. If we take the
inverse z-transform of (21TO) and square, we obtain peaks at quefrencies of




10



n T seconds (note, the units of quefrency are seconds, since (2-10) is a
0
function of frequency the inverse z-transform brings us back into the

time domain) and multiples thereof, These peaks should be detectable

provided loglX(eWT) 2 is approximately quefrency limited to less than

n T, so as not to obscure the peaks. Thus ripples in logjX(ej T) 2

should have a ripple length greater than I/n T. It is now apparent that

the power cepstrum is extremely useful in both echo epoch detection, and

amplitude estimation. It is not obvious from the expansion of (2-11)

what the relationship is between the echo amplitude a and the heights of

the peaks in the power cepstrum, but it will be seen in the discussion

of the relation between the power and complex cepstra that the

relationship is simple. This will be further discussed in Appendix A.

It should be noted that if we can remove the ripple peaks from the power

cepstrum, it is possible to invert the procedure described to obtain an

estimate of the power spectrum of the basic wavelet, but the basic wavelet

itself can not be recovered since the phase information is discarded

(the power cepstrum is totally independent of the signal phase).

THE COMPLEX CEPSTRUM AND WAVELET RECOVERY


History

The complex cepstrum technique originated as an outgrowth of

the homomorphic system theory set forth by A. Oppenheim.[10] in 1965.

The earlier work of Bogert et al. [8] on the power cepstrum has also

been shown to be a specific application of homomorphic system theory.

The complex cepstrum technique was first presented in R.W. Shafer's

doctoral dissertation [7] in which he derived the technique from

homomorphic system theory., analyzed many aspects of the complex









cepstrum in detail, and used the technique in echo detection, wavelet

recovery and speech analysis. In 1971, Kemerait [3] studied echo

detection and wavelet recovery utilizing the complex cepstrum in

the presence of additive noise, and in the case of an echo distorted

by truncation.


Introduction

The primary difference between the power and complex cepstrum

techniques is that the complex cepstrum technique retains the phase

information of the composite signal. Thus the complex cepstrum

technique may be utilized not only for echo detection, but also in

wavelet recovery.

Formally we define the complex cepstrum as the inverse z-transform

of the (complex) logarithm of the z-transform of a given input

sequence

-n
X(z) = Hx(nT)z-n (2-12)
n=-

X(z) = log X(z) (2-13)


x(nT) 2 c log(X(z))zn" dz (2-14)


where c lies within an annular region in which log X(z) has been

defined as single valued and analytic.

Let us now consider application of the complex cepstrum to a

composite signal formed by convolving two simpler signals


y(nT) = x(nT) e(nT) .


(2-15)








Z-transforming (2-15), we obtain the familiar result

Y(z) = X(z)E(z) (2-16)

Now taking the logarithm of both sides of (2-16), we have

Y(z) = log Y(z) = log X(z) + log E(z). (2-17)

Observe that the contributions from the two signals are now

additively combined (just as for the power cepstrum), and that the

phase information of the original signal is retained. We can now

use familiar time domain techniques to achieve separation of the

signals [3]. To complete the calculation of the complex cepstrum

we inverse z-transform equation (2-17) and obtain


9(nT) = i(nT) + e(nT) (2-18)

It should be noted that computation of the complex cepstrum is an

invertible operation; thus if the log E(z) contribution can be

"filtered" from Y(z), so that the complex cepstrum becomes


yR(nT) = x(nT) (2-19)

then we may z-transform (2-19), exponentiate the result and inverse

z-transform to obtain the signal x(nT). Figure 1 shows the overall

system for wavelet recovery. Note that the encircled operation may

be replaced by a digital filtering operation on the log spectrum treating

it as if it were a time domain function. In fact if this "filtering"

operation can be accomplished by multiplying the complex cepstrum

by some function of (nT) (and z-transforming the result), we may call

it frequency invariant (by analogy with time-invariant systems),








since the output of the filter is just the convolution of its input

with the "impulse" response.

It is of interest to note that filtering the original signal

y(nT) is accomplished in the quefrency domain by adding the complex

cepstrum of the impulse response of the filter to the complex cepstrum

of y(nT). Also note that filters which are ordinarily unrealizable

in the time domain that is, the impulse response is nonzero for

negative time) are implementable in this manner.

One problem present in the computation of the complex cepstrum

arises from the fact that the complex logarithm is a multivalued

function. If we compute the imaginary part of the logarithm modulo

2n, that is, evaluated at its principal value then discontinuities

appear in the phase curve. This is clearly not allowable since the

log (X(z)) is the z-transform of x(nT) and thus must be .analytic in

some annular region of the z-plane. This problem may be rectified

by making the following observations:

(1) The imaginary part of log X(z) must be continuous, and

periodic (evaluated on the unit circle) as a function of w with period

2- since it is the z-transform of x(nT).

(2) Since it is required that the complex cepstrum of a real

function be real, it is required that the imaginary part of

log (X(z)) be an odd function of w.

Subject to the above conditions (and provided the phase curve

is sampled at a sufficient rate) we may compute an unwrapped phase

curve having the correct properties with the following algorithm.

Consider the phase curve (modulo 2r) shown in Figure 2(a). If we
















* *


* *


*. a


Figure 2 Phase Unwrapping. (a) phase modulo 2w,
(b) c(k), the correction sequence, and
(c) unwrapped phase curve.


_ _


* 0









are sampling the phase at a rate sufficient to assure that it never

changes by more than a between samples then the.phase may be unwrapped

by adding the correction sequence C(k) to the phase modulo 2T

sequence P(k) where C(k) is determined as follows


C(0) = 0

C(k) = C(k-l) 2T If P(k)-P(k-l) > T

C(k) = C(k-l) + 27T If P(k-l)-P(k) > T


Figure 2(b) shows the correction sequence, and Figure 2(c) shows the

unwrapped phase curve. Alternatively the phase may be unwrapped by

computing the relative phase between adjacent samples of the spectrum.

These phases may then be added to achieve a cumulative (unwrapped)

phase for each point. Both methods have the drawback that the

computation must be done sequentially, i.e., the phase at each point

must be computed before the phase at the next point can be computed.

It should also be noted that if the phase never changes by more

than TT/2 between samples, the phase module T could be computed and

unwrapped with algorithms similar to the above. This is interesting

since it is slightly easier to calculate the phase modulo i than the

phase modulo 27 (the arctangent algorithm is simpler), and many

signals of interest have this property (though noise does not).

There is one class of signals for which phase unwrapping is

unnecessary, namely minimum phase signals. A minimum phase sequence

is a sequence whose z-transform has no poles or zeroes outside the

unit circle. Thus from equation (2-14)


x(nT) = 0 n < 0 .


(2-20)









Let e(nT) denote the even part of R(nT). Then


xe(nT) = (x(nT) + x(-nT)) (2-21)

x(nT) = 2u(nT) Re(nT) (2-22)

where u(nT) = 1 n > 0

= n =

=0 n < 0

but

x (nT) = Z- (log IX(z)) .

and thus x(nT) can be computed from a transform involving only the

real part of log X(z). The complex cepstrum determined in this

manner is equivalent to reconstructing the phase associated with the

magnitude of the spectrum using the Hilbert transform. We also see

that for n 0 the complex cepstrum is identical to the power cepstrum

in this case (except for a factor of 2 and the squaring operation).

From equation (2-20), we see that the complex cepstrum of a minimum

phase sequence is zero at negative quefrencies. Analogously a

maximum phase sequence may be defined (the z-transform has no poles

or zeroes inside the unit circle) which is zero at positive quefrencies.

As will be seen shortly the impulse trains generated in the complex

cepstrum by the presence of a single additive echo are nonzero only

on one side of the origin, and thus will often be referred to as

minimum (or maximum) phase impulse trains. To conclude this section

an example of the application of the complex cepstrum to the single

additive echo case is presented.








In equation (2-15), let e(nT) = 6(nT)+a6(nT-noT).
That is

y(nT) = x(nT) e(nT) = x(nT)+ax(nT-n T) (2-23)

Taking the z-transform and evaluating it on the unit circle, we have

Y(ejT) = X(ejWT) (l+ae-jno) .T (2-24)

Taking the logarithm of both sides of (2-24), we obtain

?(ejT) = log (Y(ejWT))= log (X(eJ T)) + log (l+ae-jn oT). (2-25)

If a < 1 we may expand the right most term in (2-25) in a power series;
thus
2 3
Y(ejWT) = log X(e WT) + ae-jnoT e-2jno + 3 e-3jn .

(2-26)

Inverse z-transforming (2-26) we find the complex cepstrum

2
y(nT) = x(nT) + a6(nT-n T) - 6(nT-2nT)...... (2-27)

Thus the complex cepstrum of the composite signal consists of the

complex cepstrum of the basic wavelet plus a train of 6 functions
located at positive quefrencies at the echo delay (and its multiples)
whose heights are simply related to the echo amplitude. Shafer [7]
has shown that these 6 functions can be effectively removed by
multiplying the.complex cepstrum by a sequence.which is unity.every-
where except at multiples of n T, and zero at these points. After.
this comb filter has been applied the complex cepstrum is smoothed
at the zeroed points by averaging adjacent points. The filtered








complex cepstrum is then used to recover the basic wavelet by inverting

the operations used to compute the.complex cepstrum. Had the echo

amplitude (a) been greater than 1, (2-25) could have been rewritten


y(ejWT) = log (ae-jnomTX(ejT)) + log (1+ 1 eJnoT) (2-28)
a

which may be expanded
nT T + T eJnolT j2wn0T
V(ejT) = log (ae-jno X(ejT 1 T) ej2 (2-29)
2a
Thus the complex cepstrum will again have peaks at the echo delay

(and multiples),but these peaks occur at negative rather than

positive quefrencies and their amplitudes will be related to 1 rather
a
than a. If these peaks are filtered out and the wavelet recovery

procedure is followed we note that the echo rather than the basic

wavelet is recovered.

From the above discussions, we note that the peaks of the impulse

train in the complex cepstrum may never have an amplitude of greater

than unity regardless of the value of (a). It is also interesting

to note that multiplying the original signal by a scale factor only

changes the n=O term of the complex cepstrum, since the scale factor

appears as a shift in the mean of the.log spectrum. Thus the complex

cepstrum is virtually independent of the signal power.

The Relation Between the Complex and Power Cepstrum

It is obvious that the complex and power cepstrum are closely

related; however, the exact nature of this relationship has never

been fully exploited. This is primarily due to the differing

definitions for the two techniques. From equation (2-1) the








relationship between the two cepstra becomes apparent,

x p(nT) = (Z-(log X(z) X*(z))2 (2-30)

= (Z-(log X(z) + log X*(z)))2 (2-31)


Assuming x(nT) is real and evaluating its. z-transform on the unit

circle, we find X (z) = X(z ), thus we may write
Xpc(nT) =(2- log X(z) dz 2
X (nT) ( Jlog X(z) zn- 1dz + 1 -flog X(z-1) zn-l dz)2 (2-32)

Letting z' = z-, we obtain

x (nT) = ( flog X(z) zn-ldz + 2Lflog X(z') -n-dz2 (2-33)
pc 21j zj' ) (2-33)

but the complex cepstrum is by definition

R(nT) =27j log X(z) zn-dz

Thus we may write (2-33) as

x p(nT) = ((nT))+ x(-nT)2 .(2-34)


The power cepstrum is just 4 times the square of the even part of

the complex cepstrum. This also follows from the fact that the power

cepstrum is the square of the inverse transform of twice .the real

part of the log spectrum. Equation (2-34) is of interest since as

pointed out in [3] the power cepstrum is often superior to the

complex cepstrum for epoch detection. Thus in a wavelet recovery

system we may wish to compute both the power and complex cepstrum.

Rather than using the straightforward method outlined in the section


_ __ __ __








on the power cepstrum, we can compute the complex cepstrum, and then

use the simple relation given by (2-34). This allows the use of one

less FFT in the computation process.


The Phase Cepstrum

The inverse transform of the log phase yields peaks at multiples of

the echo epoch in much the same way that the inverse transform of

the log magnitude does. To see this, let us once again examine the

log spectrum for the single additive echo case given in equation (2-25)

A T T ~-jwn T
Y(eJT) = log (X(e3JT)) + log (l+ae ) (2-25)


= logJX(eJT) + j Phase[X(ejwT)] (2-35)

asin n T
+ 2 log (l+a +2a cos wnoT) + jatan-- +acos0nT .
0

The 4th term in equation (2-35) produces ripples in the log phase,

just as the third term produces ripples in the log magnitude. Since

Y(ejWT) is the transform of a real sequence, its magnitude (ReY(ejwT))

is an even function of w, and its phase (ImY(ej T)) is an odd function

of w. Thus the inverse transform of Re(Y(ejaT)) will yield the even

portion of the complex cepstrum and the inverse transform of

jIm(Y(e Ja)) will produce the odd portion of the complex cepstrum.

Since the inverse transform of the term log (1+ae j ) produces

peaks on one side of the origin only, the peaks produced by its

real and imaginary parts must be equal in magnitude and must cancel

on one side of the origin while reinforcing on the other (according

to whether (a) is greater or less than one).


II








From the above argument it is obvious that a quantity called the

phase cepstrum may be defined which should be of use in the estimation

of echo epoch times, and amplitudes. Formally the phase cepstrum

may be defined as the magnitude squared of the inverse z-transform of

twice the phase of the z-transform (or equivalently the imaginary part

of the logarithm of the z-transform) of a given input sequence, which

may be written


Xphc(nT) = jZ- (2 log X(z)-2 log IX(z)1)12' (2-36)

where the factor of 2 has been introduced to eliminate any normalization

factors in the relation between the phase and complex cepstrum. It

is easy to show from (2-36) that the phase cepstrum is related to

the complex cepstrum as below


x p(nT) = (x(nT)-x(-nT))2 (2-37)


Thus the phase cepstrum is to the log phase just what the power

cepstrum is to the log magnitude. Empirically it has been determined

that the phase cepstrum is less useful than the power cepstrum for

the determination of echo epoch times because of its greater sen-

sitivity to noise, but it has proven to be a valuable guide in the

determination of the effects of noise on the signal phase, and may

find application in problems of signal identification. Computation

of the phase cepstrum alone is almost as difficult as that of the

complex cepstrum, since both require the phase unwrapping procedure.

SUMMARY

A new cepstrum technique, the phase cepstrum, has been introduced









which should be useful in the detection of echo epoch times. The

power and phase cepstra have been shown to be (except for a constant)

the square of the even and odd parts of the complex cepstrum.

Because of this close relationship between these techniques, only

the effects of the data handling procedures on the complex cepstrum

are discussed in Chapter IV. The extension of the results to the

power and phase cepstra is obvious. Since the effect of noise on

the phase cepstrum has not been examined previously, this technique

is compared with the power and complex cepstra (in the presence of

additive noise) for the detection of echo epoch times and the esti-

mation of echo amplitudes in Appendix A. Some insight is thereby

gained into the results of previous authors.


___















CHAPTER III
THE PITFALLS OF CEPSTRUM COMPUTATION


Since the computation of the complex cepstrum and subsequent

wavelet recovery involve a number of discrete Fourier transforms

together with nonlinear operations, it is not surprising to find

problems associated with these computations not present in ordinary

spectral analysis. This chapter is devoted to the study of these

problems,some of which have been noted by other authors; others

were previously unknown.


LINEAR PHASE TERMS

The phase of the z-transform of a composite signal will often

contain a linear trend. This may be a property of the signal under

analysis, or may be due to the signal being delayed (that is, not

starting at the beginning of the record). In such cases the z-transform

of a typical input signal can be rewritten


X(z) = z-rX'(z) (3-1)

where z-r is the term which contributes the linear part of the phase

curve. Then


X(z) = log X(z) = log z-r + log X'(z) (3-2)









Consider the contribution of the term log z to the complex cepstrum.


XLp(nT) = 2j flog z-rzn-d (3-3)

Evaluating the above expression on the unit circle z = e T, we

obtain


Lp(nT) = (jwTr)ejnWTjd(wT) (3-4)

Integrating, we find that the contribution of the linear phase term

to the complex cepstrum is


xLP(nT) = 0 n=0 (3-5)

S cos Tn ntO
n

The linear phase component thus causes a rapid oscillation in the

complex cepstrum on which the wavelet and echo information rides.

If the linear phase component is sufficiently large, it will dominate

the complex (and phase) cepstrum,masking important characteristics.

Figure 3 shows the mean square error (MSE) of the recovered

wavelet as a function of the delay in a composite signal. This delay

only affects the phase of the composite signal (introducing a linear

component). Note the rapid increase in MSE as a function of the

delay. For comparison the MSE of the recovered wavelet is shown

when the linear phase term is removed prior to computing the complex

cepstrum. No explanation is available for the very low MSE observed

at the odd delay times. For the composite signal considered (echo

amplitude = .5) the echo peak becomes undetectable in both the phase

and complex cepstra at a delay of 30 sample times (when no linear





















* Complete Phase Removal
* No Phase Removal


SI I I I I
15 30 45 60 75

Signal Delay

Figure 3 Effect of a Delay in the Composite Signal on
the MSE of the Recovered Wavelet.









phase removal is performed). Clearly the removal of linear phase

terms is required for an effective wavelet recovery system.

Linear phase removal results in a dramatic change in the

appearance of the complex cepstrum and also displaces the recovered

wavelet in time (unless corrected for). The linear phase component

is removed by calculating the unwrapped phase of the N/2-1 point,

using this to compute the slope of the linear phase term, and then

subtracting the linear phase term from the signal phase. For

the purpose of linear phase removal the unwrapped phase of the

N/2-1 point can be truncated to an integer multiple of r [3].

This allows the correction of the recovered wavelet (for linear

phase removal) by shifting it an integer number of sample times.

This generally still leaves a small linear phase term present in

the log spectrum. Figure 7 shows the complex cepstrum for this

type of linear phase removal. Note the effects of the remaining

linear phase term are clearly visible.

Figure 14 shows the complex cepstrum of the same signal as

Figure 7, but here the linear phase term was removed entirely prior

to computation of the complex cepstrum. Note the radically different

appearance of the complex cepstrum. In this case, the linear phase

term must be reinserted in the log spectrum of the recovered wavelet

during the inverse complex cepstrum operations.

It should be noted that if the linear phase term is not entirely

removed from the log spectrum of the composite signal then the

echo peaks cannot be smoothed (without introducing considerable

error into the recovered wavelet) by averaging the points adjacent

to them since these points have contributions from the linear phase









Example

Incomplete Linear Phase Removal

Figures 4 through 10


This group of figures illustrates the computation of the cepstra,

and wavelet recovery when the linear phase term removed from the

unwrapped phase curve is based on the unwrapped phase of the N/2-1

point truncated to an integer multiple of r.

The composite signal (256 points) is


y(nT) = x(nT) + .5x(nT-30T)


where x(nT) = nT enT 0 5 n < 64



Figure Number Figure Title


4 Composite Signal
SNR = 40 dB

5 Unwrapped Phase Curve

6 Log Magnitude

7 Complex Cepstrum

8 Phase Cepstrum

9 Power Cepstrum

10 Recovered Wavelet
MSE = 7.97 x 10-7





28










-------------I -- -- ------





I I
M



I









rr--rll~l~lrr~-r~---- ------------;F
o
I 3
Si i a


:-------------- ------
*I 4. i
I I I I I *




C)











-. - I ao i












a l - - /



--------------------------------
3 Ca o 34 a'

























,rrrr--:~clr-*---:--- -r~
I * *

* a I


* S. S S
S. 5 5 r S I*1I II* J.u*1


* S
* 5 a S i
S S
* SI S
*I a


S
-"
ft *
S a























I
: *( xt r n n ^|LII i ---- ~r *.1in _ ^__ r r - __. ^___..,___
* ma : :" i
* S



























i
* S a S













ft
a S S




S











fC__ II"I.rLeDC- *_

*
*~ ~ a, S






S
















tf

__ .. : .
s. S a a K g


S. *
S S
5 S. 5
*---' a------ --: S
* a S S
* a
*l S
* S



*> -......- .. .-- __________


~ ~ ~ a *S
*~~i *5 5
* S C 5 5 5 5-;-- ~ aU~







*r S
*


~~r~rrr~




C

,,~.
I




I
























a a





a == z a: i.
a a
S a




* a




* a
* a
I a
* a
I S
4 a
* a


* ..a-- ~. .. .,


*r a a
* a a a
a a p 3
a a a a a
*~r a .
* a a a. a a
~ ~ l -__ CI -
o) I a
a a
* .
* ~ ~ a *. a
* ? *c -.e---.
* a a a
* a g a a
* a a
C a.a
* a
ac a
* a. a
*~~ ~ a

*~c--Z L I a ~ -- a----- a a a
* a
* aaa
*i a. a

a a



a a


" 01


I ?
0

0 0
o



0 .
.?
L



-0 0


o


U C--- --------- -- ------- ^ ^^---. s
* a a a
a


a


S~~ I a

* ** a- a.^^ m^ c ill~B *, -
* a S

*^ M -a a ^ a a -

,.* a b* S
* a S

*. aSa
I a a aa






a a.

a a a S.
a





a0 a
I
~ i
i '
*5a a
fta as a
a a a





* *^_ a s
Sea a a
S. a p
a. ~ a
a a a a
a at a*rrrr~rrrrorr
a a a
a. a 4
a a
4 . .r-e.-..e
* a j a m


4-,
























S S


S S
S S


')I-P-~--.........-) ...........
: i;

.. I .-
* 9

i 5
S O5
0 3 5
0 0 -


* 0 0*--*0*55O**0*C S ~-00~c0*0r-0*~~ 4 r4 ..r..-.e
* S 5 5 00

S 5 3 0 5

S S 5 0 0
*-"LLCrC-~rrC))IC)- 3 5 8 0 5
~ ~ 3 5 0
S 5 0
~ ~ S S 5 0
5 3 5 0 0
*rr~u c-rr ~c-~r~r S 5 5 0 0
* . .... o o.aS.- oe. I *4 f .5 4 0
SS 5 0 0
S S 5 0
5 3 5 0 5
S 5 5 0 0
*'~~t~~~ Sar~r-r 5 5 0 58 r
* S S 0 O
* S 0 0
S 0 0
~ ~ ~ 3 0
*I S 5
S 5 5 SO
S 5 0 0
* SICIC r~~+,, S V 0 0
~ ~ ~ 5 5 0
* S S
* S0
5 5 S SO
~ ~ 5 S -
+5 0 n 4 5 0 a.. S r...I. 0 05I ~ .~
* S 5 5 S 05--P~r CCF~U; ~C
* SS 0
05
* S 5S 0
S S 5 00
5 3 5 0





5 .4.0?r I 00... S









0 0

-.. o *a..0---- 00 50


* 50









Oo"
0s

SC

S.






5-.
















*I.


5 3 0 *40 *
S SS 0 S
~ ~ ~ S S
~ ~ ~ S *
*~~ S S o S
~ ~ ~ S S 5
S 0 55 0
*r 5 0 5 05
S S 0 5



o a a a
0


SE
S-





x

u
-I-'












3


SL


n



o





















. .. S S .. S .




I





j
U.














:i

__ ^ __ 1 "
U U
I S: U U


I U I S I4

















:a
IU
.
I S I U U ft






i S U
: I U





















ii
I U U S 4


I U U a






U



I U
I S U S
I S
U II U

*ur c~~r S~ C~ u-~ l S~ IIFW I U CL~C II U 4~CI


*~ I t

U U I U
SI U


I I U 4



I I 4


U I S 3 4
I U U I S


I.I I I 5: 4
*^ U S U U
I U I








hS
II 4



* I. I S

*IUII 44
I SII4S4
I I S4
I S US
I I U S4

I S II US
* I| I 4


*. *< I* 5' 5






















! t




i i j i _i
i '
* ,





i
i i
* *










e ii" l "" ~
I i





, --^rCI ^*nr^r,' ^.^. ^I ^ 'Y IO.
1 t1
















: -
_*j _ a_ a a a
*
a aa

a
a a a a a a ^ .


Q C
D C
O O


0
O
O O
O 5
























4f ~ fl S

S S S
S SS S
S S S S
~ ~ S
S S S
* S S
* S S5 5
S S S
S S S S
*~ S
S SS S
* S S
S SS S
* S S
S S S S 5 0L--u~~)~)+~L
S~ S S S
* S S S S
S SS S
S SS S
* 55 S I 4.O.B4ruu -~ 0 B.B45i
* S S S
S SS S
S B5 5

*r~-,~r c- c~~c~r-r S S S S Bu~-
* S S

* S: S S S
* S S S
*~uo S c--~~rrr~lr S S -r-~rur~~ S
* S S S S
*~ S S S
*
*~~ S S
*~ S S
*~ S S
S S S
*---- --~;-. S S S
* S
* S S

* S SS S
~ ~ S: S
*rsoLp~r---o~ICI~- S
* S S S S
* S S S
* S B S
S 5 5B,
S S S S
*~~ S S
* S S SS
* S S SS
~~~ S SS
* S S S S



~~rr-- ~ cr- - ,.----..n- s., S S
* S S SY S
*N S SS
* S
S
S SS S


41-








1-
a r



>I
o X
0

* x

00


S-
:3
tm LI.J
*r









Example

Complete Linear Phase Removal

Figures 11 through 17


This group of figures illustrates the computation of the cepstra,

and wavelet recovery when the linear phase term removed from the

unwrapped phase curve is based on the unwrapped phase of the N/2-1 point

(no truncation).

The composite signal (256 points) is


y(nT) = x(nT) + .5x(nT-30T)


where x(nT) = nT enT 0 5 n < 64



Figure Number Figure Title

11 Composite Signal
SNR = 40 dB

12 Unwrapped Phase Curve

13 Log Magnitude

14 Complex Cepstrum

15 Phase Cepstrum

16 Power Cepstrum

17 Recovered Wavelet
MSE = 8.01 x 10-7
















0


I I I I j
Itt
I i

I I



J i !I
I I IP



a ac
i i "
--------* -













I I s

I I






4.t-M b *--- --k44JJ-l4-- J------------* 4* -UHJ --kMlr-----
Sa i i
Sft^ - ^ t- - ----.. ^ .... .. - ft .
I I I a
a a a
a I I I -
a an
a 0)
I i I *
aIr-lr,,rr,+rrrr~-~r- S-
a~~ a aa
a a a a *. o aIo
a a a
I a aa a .a Vi ILL/
a a aa a tft
I t ft 4* f



a a a a a
a~~~ a t
a a a







C- C 0
0 0 0ad 0
Pa (4 d 0
0r 0 0 0 0 0






















I ;
I I I. I I


IIi I I
I I I j a
-. -I
i I I I IC



I I I I
a a a as

I i i i
I I I I



i I I I I
I I a
I I0.





SI I I
cii
S* I I 1






" i i io
------. ------ ----- -
SI i

Ci





c
I I

-




a- C I I I.
*s,
1 1 I I -*I

(* I I I I1
a a. I I I
I I C1I I C
I I a i a.


r- - c----- - c -- --
m o o












0


6 j ( -
I 1 0
SI ":I I *
! i r i -
I I o
~ ~ t : .

' ^ -.^ i......... .... -I. .. ^ ^

I 1 I o I
I I I i 6
I 6 6 6 i6 s
^--._.-^ -- -- -- - --- -^-0.0






SI !.
i J I 6 * o



*I * 60
*i








I I i I o
I I I I
-~


Ioi






i i i E
0 6
I i e 1 i



i *' I lo
I I *I


I... 1 1




i 6 I i 6 6 i .
1 i I I~ I I o

6 6I 6* 6
6 I 11 0





1* 0 I!
I 6 I I- l
I 6






I I | 1 1
i~~ *i i ri
6 6 f 0. 0 60
*0.










: ~ I
I 60
6 0,
It **i I



CI*
O P1

6 6




39










. --b--t--M--------JU---f --^ >-M-lM*h4M-- k---- - .M- -- h-- -----
I :" Io
4..
SI I H


I iI I I

O *
^ --------- --------





- -- -.-- --.-- --. -- -.. ....... .


4-M*1l-1.11--M 1,------ --- 1---- --f----- --- -- --- ----
I C
I I I IC








j


I E
1 I
I. 1o ,

io D





I I 0
* i U-




*n o o-- ---- 4" -o-- l ----- - - -
- - I -o -
I I iJ S






i j i I 4 .


0 '- C N N




























5|


j!
* 4


jaa




6 4,






.S|


0
0


a a


40
0





ftj


0
I a






0
a0
S a
a a
aC aU(
a c

I a
a *
I a 61


a"rQ CCI aC~Y I a IC((CC-CL~ ~ l aC a
a a a a a
a a aa a

* a a a a
a a a a




* 5 .65
* *" C~-+r~-I *fl~r C ~ IC-a. a..*.-a-)--.,--*.-

a a a S .6


*~ ~~ 6
*an




I ~ ~ a
a a a
a
*I a a


I~ 4
aCC~~"~+rCr~ 6 a a a aIC ~ L~C~)(~I


0^





i i




Sa
I I
I~ a











i
a a
a a aU, Ur ~ CCI~IUCUC~lC
a a a
a a a
a a a
a a



I
a~L~'l' a-CCC I a aCCtUj C"-~CCI




CI C C C


P
R



C,
F

O
N



O
O

Fl



O
O


P




O
D
N







Y1

I







~






41















o




I s
* S__ a_ S S a-- -


_____ ___ __ __ .,

I




\ i
r ___CI~ U I CI~ ____ __ __ __^ <
* a0








; :
I 0.

SS I a a
* 0\ t .0 0 S 0000 S



sI
S


a a*






SI '





I ;: __ ^



_~ 1- ._ ____s
* aa
* S a











I : M ^
a
a a a




I S S S*
* a a a a a 0
* a a a
* a a a a 5
* s
* a SS
~~~ a
S S a a
* S :
*~ a a aa
* a a a .a a a
*--,,r ar~rre au ,w~~r~~ a a a at'
a a a
* a a a a -
SL S S a
*~ a a a


















---- --- i .----
SU :


I -


I I t
4.1-ll-lt-l-*i-l-l---- '--- ------ -- ------








------------ --t11 U kr*f~fjJ Ml44jJ-h~' f~cMr~*- -<<-. *-- --*----
I I 1 1 *



------------ a-- ---- --*
i"






i. o
ii o







I i I ,I













I---------- -
I--------------- ------- ------------- ------------------a. ..

a a' a <* f-
i,,,,-------------------------------------








a a a a a









component which are opposite in sign to the contribution of the

point being smoothed. To smooth the nth point properly when a

linear phase term is present the contributions of the n+2 and n-2

points must be averaged. This form of smoothing results in a

somewhat smaller MSE (than averaging the points adjacent to the

echo peaks) even when the linear phase term is completely removed.

Subject to the above smoothing considerations it has been

determined that the MSE of the recovered wavelet is not significantly

different for the two forms of linear phase removal, though as

mentioned previously the appearance of the complex (and phase)

cepstrum is quite different. This undoubtedly will affect the

detection of small amplitude echo peaks and echo peaks occurring

at low quefrencies. Thus the second form of linear phase removal

is preferred, and is employed in the computer experiments discussed

in this dissertation.


PHASE UNWRAPPING ERRORS

Previous authors have tacitly assumed that the phase of the DFT

of a given sequence may be unwrapped unambiguously. Only recently

has it been pointed out that errors may occur in the phase unwrapping

process [5]. Since the arctangent routine used in the phase

unwrapping algorithm computes the phase modulo 2n, it is evident

that phase unwrapping (explained in Chapter II) is performed correctly

only if the change in phase between samples of the z-transform of the

given sequence is less than Tr. The DFT is just the z-transform

(evaluated on the unit circle) of the sequence sampled at values

w = na =2r/TR where TR = NT is the record length under consideration.









The sampling theorem (in the frequency domain) indicates that Aw

must be less than or equal to 27r/TR in order to reconstruct X(z)

exactly from its samples. Thus the DFT samples the z-transform

at the minimum rate necessary for reconstruction. Sampling at this

rate does not, however, assure us that the phase of the z-transform

(evaluated on the unit circle z=ejWT) does not change by more than

a between samples as can be seen by the following simple examples.

Example 1

Let x(nT) = 6((n-N/2)T) + 6((n-N)T) (3-6)
NT
X(z)= X(e T) = -Je + e-jNT (3-7)
z=ejWT

Consider the phase diagram of X(ej T) shown in Figure 18. It is

evident from Figure 18 that even if the z-transform is sampled at

twice the minimum rate (A = < ) the change in phase can be
R R
greater than w. Thus the phase unwrapping algorithm will yield

incorrect results. To see the effects of this on the computed

phase curve consider Figure 19. Obviously the computed phase curve

differs considerably from the true phase curve.

Example 2

Let x(nT) = y(nT-noT) [O,N) (3-8)
= 0 otherwise

-jwn T c.T
X(ejwT)= e o yj(eJ ) (3-9)

As expected the phase of X(e j) is the sum of a linear phase component

and the phase of Y(ejT ). If w is sampled at the minimum rate, that
n2s
is, w = nAwn = n then



























(b)

Figure 18 Phase of X(eJWT).


(a) w=O, and (b) w=T/TR.


-671


Figure 19 Phase Unwrapping Errors.
(b) phase modulo 2i, and


I


(a) true phase,
(c) unwrapped phase.









jn2r
Xs(eJ) = X(e ) = ej2n(no/N)y (eJ) (3-10)

Thus if no > N/2 the linear component of the phase will change by

more than 7 between samples and unless the phase of Y(ejWT)

counteracts this change, the phase unwrapping algorithm will yield

erroneous results. This was observed in computer experiments when

the composite signal is delayed by more than half the record

length. As expected this not only reduces the echo detectability

in both the phase and complex cepstra, but also severely distorts

the recovered wavelet.

Fortunately,provided the composite signal starts relatively

early in the record, many signals of interest can be unwrapped

correctly since their phase changes slowly. Furthermore many signals

only occupy a portion of the total record, and thus the DFT yields

samples of the z-transform spaced more closely than dictated by the

minimum sampling considerations. Even in the absence of the above

conditions, this problem can be alleviated by extending the data

record with zeroes as this increases the sampling rate of the

z-transform (this will be shown in the next chapter).

These errors often occur when unwrapping the phase of a composite

signal in additive noise at low signal-to-noise ratios (SNR), since

the noise may dominate portions of the spectrum and the phase of

the noise spectrum will change rapidly from sample to sample. This

error in phase unwrapping seems to be actually beneficial in this

case in that it always limits the jumps in phase (to less than 1)

when in actuality the jumps may be somewhat larger. This topic










will be discussed in more detail in the section on the addition of

zeroes in the following chapter.


ALIASING

Even though the DFT yields adequately spaced samples of the

z-transform (X(z)), it is not surprising that aliasing may be a

problem in computation of the cepstra, since the nonlinear complex

logarithm operation will introduce harmonics into X(z).

Consider, for example, the complex cepstrum. Because of the

harmonics introduced into X(z), the complex cepstrum will be infinite

in extent. The DFT will yield a finite N point sequence, thus the

NT NT
contributions from quefrencies nT > and nT < will alias

into the interval (- N n).

The aliasing of the cepstra of the basic wavelets considered

did not prove to be a problem. This might have been expected since

their time duration was never more than 25% of the total record

length.

One manifestation of aliasing is an ambiguity in the determination

of the echo epoch time. A peak at n T in the complex cepstrum may

be caused either by an echo delayed by (N-no)T or by an echo delayed

by n T. This ambiguity cannot be resolved unless the relative echo

amplitude is known to be either greater or less than unity (unless

the original data record is extended by the addition of zeroes).

Aliasing of the echo impulse train can also be a problem. The

aliasing of the higher order terms in the log expansion given in

equation (2-25) is often observed. This is especially troublesome









if (a) is close to one since the amplitudes of the peaks in the

impulse train do not fall rapidly in this case.

Aliasing is also evident when additive noise is introduced into

the observed record. Since noise is present throughout the observed

record it is not surprising that it produces contributions at high

quefrencies in the complex cepstrum and thus may cause aliasing.

In the noise analysis of Appendix B, it is shown that aliasing

is the primary mechanism through which noise is introduced into

the complex cepstrum.at negative quefrencies for high SNR.

As will be seen in the next chapter,adding zeroes to the input

data sequence reduces the aliasing of the complex cepstrum. In

fact, if the number of zeroes added is greater than or equal to

the original record length the ambiguity discussed above can never

occur since the echo delay will always be less than one half the

augmented record length.

Aliasing can also be reduced (for a given composite signal) by

choosing the record length NT as large as possible subject to the

constraints on the number of points analyzable and on the minimum

sampling rate. That this is the case becomes evident if we recall
I 1
that the.samples of the log spectrum are spaced Af = T =T apart;

thus increasing NT increases the "sampling rate" of the log spectrum,

and should minimize aliasing in its inverse DFT. This gain is offset

somewhat by the fact that increasing the duration in time of the

observed data increases the "long time" (high quefrency) contributions

to the cepstra from noise (assuming the composite signal remains

constant). Furthermore the shorter the interval of the data record









that the composite signal occupies,the lower the SNR for a given

noise environment.


OVERSAMPLING

One problem caused by additive noise occurs when the composite

signal is oversampled. Outside the signal band (although.these

components may have been greatly attenuated prior to sampling) noise

often dominates the spectrum. This presents no problem in ordinary

spectral analysis as these components contain little power, but

they may have a marked effect on the cepstra (since the complex

logarithm of the spectrum is taken). Because of this nonlinearity

the regions of low power may contribute as much or more to the

.cepstra as the signal band, and therefore will degrade both echo

detectability and eventual wavelet recovery. This effect is especially

evident in the phase cepstrum since the phase of the spectrum outside

the signal band may change rapidly from point to point. It is difficult

to assess the effect of oversampling on the recovered wavelet, but.

the appearances of the complex, phase and power cepstra are observably

degraded by oversampling.

Oversampling also aggravates the phase unwrapping and aliasing

problems previously noted, since it shortens the data record (if the

number of points is fixed) which in turn implies that the samples of

the log spectrum are spaced farther apart.


SUMMARY

Four problems connected with cepstral analysis have been examined.

These problems are linear phase terms, phase unwrapping errors, aliasing

of the cepstra, and oversampling.


__





50



The presence of a linear phase.term distorts the appearance of

the complex and phase cepstra even when partially removed as in

reference [3]. Such a term degrades not only echo epoch detectability

in the phase and complex cepstra, but also wavelet recovery. The

complete removal of linear phase contributions to the cepstra is a

necessity for satisfactory wavelet recovery.

Phase unwrapping errors are observed to occur if the log phase

changes by more than. between samples. This can be quite detri-

mental to echo epoch detectability in the complex and phase cepstra,

and to wavelet recovery.

Some aliasing of the cepstra is always present since the cepstra

are derived from samples of the log spectrum. This results in an

ambiguity in the echo epoch determination. Both the aliasing and

phase unwrapping problems can be alleviated by extending the original

data record with zeroes.

Finally, oversampling the input data record is observed to degrade

wavelet recovery when additive noise is present. Oversampling also

aggravates the phase unwrapping and aliasing problems, and thus

should be avoided.














CHAPTER IV

THE EFFECTS OF SOME DATA PROCESSING TECHNIQUES
ON THE CEPSTRA


Care must be exercised when computing the complex, phase, and

power cepstra. The repeated use of the DFT may raise the problems

of aliasing, leakage, and the picket fence effect. Since the complex,

phase, and power cepstra are nonlinear techniques, it is not clear

that ordinary data processing techniques, normally used to improve

spectral analysis, are applicable. Two such techniques, windowing

(normally used to reduce leakage) and the addition of zeroes (normally

used to reduce the picket fence effect),are examined to determine

the effects of their application at various stages in the echo

detection and wavelet recovery process. Primarily the effects of

these techniques on the complex cepstrum are discussed, since the

extension of the results to the phase and power cepstra is usually

obvious.


WINDOWING THE COMPOSITE SIGNAL

When the input data record is windowed the applicability of

cepstrum techniques is no longer evident. Consider for a single

additive echo


y(nT) = [x(nT-k T) + ax(nT-k T-n T)]w(nT) (4-1)









where w(nT) = 0 n > L-1 or n < 0

x(nT) = 0 n < 0

w(nT) is the windowing sequence. The basic wavelet x(nT) has the

argument (n-k )T, since we wish it to begin at some unspecified

point (k T) because it is not known where (if at all) in the data

record the composite signal begins. Equation (4-1) is rewritten


y(nT) = [x(nT)*(6(nT-koT)+aS(nT-koT-noT))]w(nT) (4-2)

Z-transforming (4-2), we obtain
-k -n
Y(z) = [z o(X(z)(l+az O))]*W(z) (4-3)

Thus, the contributions of the basic wavelet x(nT) and the echo delay

cannot be separated by taking the logarithm since the term in brackets

is convolved with W(z). Fortunately many practical situations permit

the extraction of the basic wavelet (though there is some error)

and detection of the echo arrival time.

Experiments have been undertaken to determine the effects of

windowing the composite signal on the wavelet recoverability (as

measured by the MSE between the recovered wavelet and the true

wavelet) and on the echo epoch detectability. These effects are

ascertained in both the noise free and the additive noise case.

Unless otherwise noted the composite signal examined is of the form

y(nT) = x(nT-k T) + ax(nT-k T-noT) 0 n [ L-1 (4-4)

and x(nT) =nTebnT 0 n < 64 (4-5)
x(nT)= nTe"bnT 0 n < 64 (4-5)





with L = 256

a =.5

n = 30

k= 0

b = .1, .5, 1.0

T = .333 .


A typical experiment is conducted as follows: a composite signal as

given above is generated. Bandlimited random noise is added to this

sequence and it is windowed. The complex, phase, and power cepstra

are computed. Peaks due to the echo train are smoothed in the complex

cepstrum, which is then inverse transformed to obtain an estimate of

the basic wavelet. This estimate is corrected for the windowing by

multiplication by (w(nT))-l where w(nT) is the windowing sequence.

The mean square error (MSE) is computed between the recovered and the

initial basic wavelet over the entire record length.


The Exponential Window

The exponential window


w(nT) = an 0 S n < L


a < I


(4-6)


= 0 otherwise


has been proposed by Shafer (7) to reduce any error associated with

truncating the echo. To some extent this window also reduces

leakage, but since its properties are quite different from those of

the commonly used windows, it will be studied separately.

Experiments were undertaken to determine the effects of this

window on wavelet recoverability and to determine its effects on









echo detectability both in the noise free and the additive noise

cases.

The following observations are.made on the effects of this

window:

(1) No difficulty is encountered detecting the echo epoch,

though the peaks in the power, phase and complex cepstrum are

reduced in height. The echo detectability threshold in the case of

additive noise is about the same as observed for the rectangular

window (about a signal to noise ratio (SNR) of 8dB).

(2) The MSE of the recovered wavelet is somewhat better than that

obtained with the rectangular window at low SNR (14 dB and below),

but not nearly as good at higher SNR. A comparison of the MSE is

given in Figure 20. At high SNR the MSE in the recovered wavelet

is comparable to that for the rectangularly windowed case if it is

computed only over the known duration of the basic wavelet. Thus

the MSE results shown in Figure 20 reflect the distortions introduced

into the recovered record outside the signal duration by correcting

for the exponential window.


Interpretation of Results

The following derivation is presented in order to clarify the

effects of the exponential window. Consider again the z-transform

of equation (4-1), it is:

L-1
Y(z) = [w(nT)x(nT-k T)+aw(nT)x(nT-k T-n T)]z-n (4-7)
n=0
-n L-n -1
-n '0(o-)
Y(z) = X (z)+az o x(nT-k T)w(nT+n T)z-n (4-8)
n=-n























* Rectangular Window
* Exponential Window


40 .- -
40


Figure 20 MSE of Recovered Wavelet when the Input Data
Record is Exponentially Windowed


10-2+


10-3


20

SNR(dB)







L-1
where X (z) = > w(nT)x(nT-k T)zn .
n=O

-n L-n -1
Y(z) = X (z) + az o x(nT-k T)w(nT+n T)z-n+E (4-9)
n=0 o
-n -1
where El = az o > x(nT-k T)w(nT+n T)z-
n=-n
0

Now letting w(nT) = an 0 S n < L

= 0 otherwise

-n n L-no-1
Y(z) = X (z) + az o a o x(nT-k T) anz-n+E1 (4-10)

-n n L-1 no-LL no- n-n
= X (z)+az O[a 0 o x(nT-k T)anz--zn0 a x(nT+LT-n T-k T)aa nz-
n=O n=O

+ E (4-11)


The term denoted El is an error term. A close examination reveals

that the error E arises because it has not been assumed that the

basic wavelet begins within the windowed record. If this assumption

is made (that is, k must be positive and less than L-1), then the

error term El vanishes. Henceforth this assumption is made unless

otherwise noted. The second term in brackets also is an error term.

It arises because the echo has been truncated. It is the error due

to this term that this window minimizes (due to the presence of the

factor aL). At present it is assumed that the duration of x(nT) is

less than (L-no- o)T so that the truncation error vanishes. With

the above assumption equation (4-11) becomes

n -n n -n
Y(z) = Xw(z) + oaz o Xw(z) = X w(z)(1ha Oaz 0) (4-12)








n -n
Y(z) = log Y(z) = log Xw(z) + log (l+a'oaz o) (4-13)

n
Clearly the presence of the factor a 0 alters the heights of the

peaks which are observed in the complex cepstrum. If a < 1, the

peaks in the complex cepstrum are reduced by the factor (a o)m

where m is the order of the peak (as compared to the rectangularly

windowed case). This is precisely what is observed in the experi-

mental section. If a > 1, and a na > 1 then the peaks in the

no-m
complex cepstrum are enhanced by the factor (ao ) If a > 1, and

a na < 1, then the peaks may be either reduced or enhanced, and

they occur at positive rather than negative quefrencies. The basic

wavelet rather than the echo is then recovered.

It can be seen from the above discussion that aliasing of the

echo peaks can be reduced by choosing a properly. Another possible

advantage to using this window occurs in the case of multiple echos.

Kemerait [3] has indicated that when the sum of the echo amplitudes

is near one,ambiguous peaks occur in the cepstrum. Again by choosing

a properly this situation can be avoided. Assuming that the contri-
n -n
bution to the log spectrum from the term log(l+a z ) can be

filtered out (by transforming and smoothing the corresponding peaks

in the complex cepstrum), the inverse wavelet recovery procedure

should, after correcting for the windowing, yield the basic wavelet.

Experimentally this is found to be true over the duration of the basic

wavelet, but the windowing correction appears to introduce some error

into the recovered record outside this interval and thus the MSE

results shown in Figure 20 are somewhat greater than observed for the

rectangular window at high-SNR. As stated previously,at low SNR









the MSE is substantially lower than that obtained using the rectangular

window. It is also noted that the echo detectability threshold is

about the same as observed for the rectangular window, even though

the peaks in the complex cepstrum are reduced in amplitude. Both

of the above results are undoubtedly due to the fact that the signal

occurred near the beginning of the record and thus is not reduced

as much by the windowing as is the noise which is spread throughout

the record. Had the signal occurred much later in the record, it

is expected that the MSE would increase considerably, and that the

echo detectability would suffer. This has subsequently been verified

experimentally. Essentially, the'cost of reducing the error due

to truncation with the exponential window is that the portion of the

signal occurring late in the:data record is also discriminated against.


The Common Window

Three windows (Hamming, Hanning, and Tapering) commonly used to

reduce leakage in spectrum analysis are examined to determine the

effects of their application. Since these windows exhibit similar

properties, they are discussed as a group in this section with the

Hamming window being discussed first, and the other windows then

compared with it.

.The following observations are made on the effects of the

Hamming window.

(1) Several signals of the form given in equations (4-4) and

(4-5) with (b) ranging from .1 to 1.0 were generated without additive

noise. In all cases the echo epoch is detectable, though the height

of the peaks found in the complex, phase, and power cepstrum are









quite different than are found when the rectangular window.is used.

Substantial peaks at the echo epoch time are often found at both

positive and negative quefrencies in the complex .cepstrum with the

negative quefrency peaks being larger (the rectangularly windowed

sequence has peaks at positive quefrencies only). The peaks are

somewhat smeared. For the echo delay used (n =30), satisfactory

wavelet extraction is not possible except when b=1.0; in this case

extraction of the echo rather than the basic wavelet produces the

least error (since the negative quefrency peaks dominate the complex

cepstrum) though considerable distortion is still evident in the

recovered wavelet.

(2) Since wavelet recovery is only possible when b=1.0, a

number of experiments were conducted with b=1.0 to determine the effects

of the window when noise is added. Figure 21 shows the MSE of the

recovered wavelet as a function of the SNR when the input data

record is Hamming windowed. For reference the MSE obtained with the

rectangular window is also shown. It is noted that the MSE is

sometimes reduced for the Hamming windowed composite signal by

smoothing not only the peaks, but several adjacent points as well.

For this Hamming windowed composite signal the peaks in the

.complex cepstrum occur at negative rather than positive quefrencies.

In the no noise case the negative quefrency peaks are larger than for

the corresponding rectangularly windowed signal. Even with this

increased peak height in the no noise case the SNR at which the echo

epoch becomes undetectable is about 12 dB greater than found for the

rectangularly windowed case.


__





















* Rectangular Window
* Hamming Window


10io


- ----
40


SNR(dB)
Figure 21 MSE of the Recovered Wavelet when
Are Hamming Windowed.


the Input Data


10--









Only a few computer computations were made using the Hanning

and Tapering windows, but general trends have been ascertained from

the results. The Hanning window produces considerably worse results

than the Hamming. Likewise, the Tapering window gives poor results

if a major portion of the composite signal lies in the region of

tapering; however, if the composite signal lies in the untapered region

the results are almost identical to those for the rectangular window.

One additional drawback of these two windows is that they are zero

at the endpoints and thus it is impossible to correct the recovered

wavelet at these points. It is noted that correction for the window

when w(nT)<
introduce some distortion.


Interpretation of results

The following theoretical analysis is intended to clarify the

conditions under which wavelet extraction is possible after using

one of the foregoing windows. Consider (4-7) which is rewritten as

L-1 L-l
Y(z) = w(k T)~2 x(nT-k T)z-n+aw(n T+k T)7 x(nT-k T-n T)z-n
n=O n=O

L-1 L-l
+ 2 x(nT-k T)[w(nT)w(k T)z+a x(nT-kT-n T)[w(nT)-w(n T+k T)]z-n
n=O n=O 0 0 0 0

(4-14)

The third and fourth terms of equation (4-14) can be considered error

terms. It will be seen below that extraction of the wavelet is

possible if these terms are small. This requires the window to be

approximately constant over the duration of x(nT). For the windows

under consideration this requires the duration of x(nT) to be much









less than the record length. The error terms vanish completely when

the window is rectangular. The sum of the third and fourth terms

will be designated as error term E2. Equation (4-14) is now rewritten
as

-n
Y(z) = [w(koT) + az o w(noT+kT)] X(z) + E2 (4-15)

L-1
where X(z) = : x(nT-k T)z-n
n=O

aw(n T+k T)
If = -W-oTTo- >1, then

-n E2
Y(z) = log Y(z) = log[az w(noT+k T)X(z) + E2 ] + log(l+z0o/) .
00 l+z n/

(4-16)
The term log (l+zno/B) may now be expanded as we have done previously
to obtain

-n E2 "o 2no
Y(z) = log[az- w(nT+k T)X(z) + l--+z + 7 . . (4-17)
0 1+z /e 2 0

Thus the peaks in the complex cepstrum occur at negative quefrencies.

If these peaks are removed, then the recovered wavelet will be

-n E
yRW(nT) = Z-l(az w(n T+k T)X(z) + -) (4-18)
S0 0 l+z /2

1
Expanding n in a series, and noting the origin of the error
l+z /8


term E2, we may rewrite equation (4-18) as







yRW(nT) = aw(n T+k T)x(nT-k T-noT)+[x(nT-k T)(w(nT)-w(k T))

+ ax(nT-k T-noT)(w(nT)-W(n T+k T))]*[6(nT)-- 6(nT+noT)

+ -26(nT+2nrT)...] = w(nT)ax(nT-k T-n T)+[x(nT-k T)(w(nT)

w(k T))]*[6(nT)-B-16(nT+n T)+-26(nT+2n T)...]

+ [x(nT-kT-noT)(w(nT)-w(n T+kT))]*[-B (nT+n T)

+ -2(nT+2n T)... (4-19)

After correction for windowing, and a few simple manipulations,
equation (4-19) becomes
w(koT)
YR(nT) = ax(nT-k T-n T)+x(nT-koT)(1 0-

-1 w(nT+noT) -1 w(noT+koT)
w(nT) + W(nT)
w(nT+noT) w(koT)
x(nT+n T-k T)( w(nT) w( nT

-1 w(nT+2noT) -1 w(noT+koT)
w w(nT) w(nT) )+ "' (4-20)

Thus the recovered wavelet consists of the echo plus a number of
distorted and shifted replicas of the basic wavelet. The above
analysis agrees well with the experimental results. For the case
considered (no=30, a=.5 with the Hamming window), =1.25. Thus peaks
are expected at negative quefrencies and since = .8>a, these peaks
should be larger than for the corresponding rectangularly windowed
case. The echo can be recovered by filtering these peaks from the
complex cepstrum. The distortions suggested by equation (4-20) are
evident in the recovered wavelet shown in Figure 28. The recovered










Example

Hamming Windowed Data Record

Figures 22 through 28


This group of figures illustrates the computation of the cepstra,

and wavelet recovery when the input data record is Hamming windowed.

Note the distortions present in the recovered wavelet.

The composite signal (256 points) is


y(nT) = x(nT) + .5x(nT-30T)


where x(nT) = nT e-nT 0 n < 64


Figure Number Figure Title

22 Composite Signal
SNR = 40 dB

23 Unwrapped Phase Curve

24 Log Magnitude

25 Complex Cepstrum

26 Phase Cepstrum

27 Power Cepstrum

28 Recovered Wavelet
MSE = 4.24 x 10-3






65











o
I I 3 I 3 03 '

I I 3 I 1 6
I

:I 33

i' el



.II
:Ii
I lI I *I '




D: !
I 3 !



i i i *




1 1 11i
II I 1 .



II
J *J* *





! ,i 1 i
. 3 I a


SI i I I t I


* 3I I I 43
I I 3 I II I 4


' I 33 3 8

i 3 3 4 1 0,


! ! o Sn
I.~~ a a.


'3>! - ^ 3 ^L ^ .s 430







3 3 3 1 .4 4 M
* i ia .* ;? Z

* 3 .1 I ' *



SI* I I I I C-
I1 I 4 I Ci
:::L









!i 1 .
I I I I I |
I I I I 3 *3 |

I l i 3 a
3 3 I 3 I3
3 ~ ~ ~ I I. 3 '





I I I I
o o 3, *4, *3 K,

0 0
I I I
.J 4;r


3 3 3 3 4 .. I






66












o
a 1 a a'-
e I I Ii I
e I l e !
I ! -
.i!
a a 1 a

a a ai
I I 4 I q

a a
a a a a
II tt i
I a i
i D



I I I o


a a a





!
i I i
1

%

. I I *
* .. C






*! I 1
I I a a a

a a a a




* a I a a WI
* a I I V



SII I
I a:
a a I a a Co






Soa
i i 1 -
a I I a go
a* a a a i -
a a a a a a
-a,



a a' a a. a a,
a I : a
I

I | I 0


'! * < I 1 C
i* D

1 a a t

<^ ~ ~~ r ?" t
I II
4.-. Ia, I 4-------44 ta,

a a I I I 1*4 '


a~ I t
II atI :

I~ ~ a a aa



I a. I
a a I a ~




67






















* 5 6 S I


:~ :
. I ,. 4 6 4









-
S* 4













I I I I t
i I 4 S S S
i S 44 6 S
















I Q-U+ CCU~ 41UIII CIO~~ CI eLI
6 8 -4 6 80
















i .i I









I' i i

elI I
i . I I






SI 5 4' I 8

i i i I e
e i ii
I i i I


8 I 6 5
3 5 I I 4 5t




I 4. S 5

I I I I
SI S 8 I


I I I S I
S 4 1 5
5 8 4 4 5

* 3 4 I S












; *
Ie 6 8 I I
S 5 5 6 IS




I 8 46
I 5 4 6 I 5 C
*I 5 5 4



S4 S S S o
I 3 I 4 4- I 50
I 6 I 5 43


















CI I,
6 6I












* 64 I SC
* I 81 50
I 5 5 4 I I .


* 5; S 4
* I *' 3 5I



0 I* 5 4 8^ 8
* I 5 4 6 6
* 8 485
* 4 S .


I






















* aI a a I
! :
a a.

* .a *
* -*b-**-.b~-*a-* t*ba S *
a- a_ a a^ a. a a
a a a a a. a. a, a ^-.r-^ ^ m nii ^,,, l'L
a a
! a .
a a I I a. a
a a a a I
L


a a a a a.. aa
a a I I a a
a a a t


a aa
a a a a
a * a .

a..





a


a a a .. I
a a~ ac a a a aa

a~ a a a. a


a a a .
*,



a I a a .
a a




a a a a a
a a
a a a




:~ *.I
r*



: .




*



I
a 'a a' a a
a aI I a a
a a a a a. a




S* aaa
4 a- a1 a. C aa
a- aT a a a. a
a a a a 0 a0

a aI


I






69

















: ".:
II I I .
* I I I
* S I I a0
* a I a__ c_ ^



* c
* S S I a c
* a oS




* S I S ,




















c l $l r wr .usrrr
. S a a,









: i ie
aI I S0


















a a a a a c
a a* aa cc c
* S I c


















!
:
a a SN
S I S Sc














a a c,
* a a a










S :*
I I I Sc
* S I S 0 S











I Ia I






a. a I i
at.^-ti-ti-^-^.--_^ ^ ai.--^t,-n- a^"-**-- ** cap
a aI I 5 A-

a. a a a



aI I c 0I


a a a a *O a0
<* a at a* a o














a a a a a c
a







70












a -.-.-----.-- a~ u~~rr~ Il~-~rrr-
So
I
I S SI



S S
I I S a 0-








;- -'"ro
I I S a a t n



















;
1 :g


S ft











i














!-
*_ S_ J S I_ I
II .


S I I 5








































S1 o
* S S S I ft

0I S 0 S I
* I S t 5
* I S1 I S t
* I I 5 ft
* a I S f
* I I 5 f
SI I S a0

I SI I 5 ft C
* S I I S ft
*I I afp

* aS I f
* I S 5f
I S I 5
* I 55f
SI I I0
* I I 5 5 a
* I I 00
~ ~~ I 1
*~ S S If
* I S I 'I 0
* ~ ~ I I0
I S II 0: C

*I S S f
,c~c~ch ec,,,, r,,* I 5 I 5 ftc~e~w
* a I If
I I I~ I 0
* I I 5 f
I I I I-





















. o.

'oc.:




j i . i i'-
a aI I **
a I SI
I S *e
I t<
8 S *! S

*--,*-,-rt-.>--,-,l-i |-, IIn*-i Tw . u M i-i
I S *
II SI *
_* a S I **

*I I *





P" ^r i -
*I I S **







a I


Ii S
*I a I C




i
~ ~ S I S
a S I
S S S a *lc~~~l-- r ~ccl c~~
S
a


*II S a *C
I a a I IC I
I I 8 C
I
S I .
aaC I C C 5 I I
I *C5a

a a
a Ia
I *


a aa
I
a a C a


C^ T* C "
0 0 0 D 0 0


,L~IUc~c~~,


cc~c~,U-.

..
..
.










wavelet of Figure 28 is a result of smoothing only the peaks in the

complex cepstrum. When not only the peaks but several adjacent points

were smoothed the distortions clearly diminished. It is to be noted

from equation (4-18) that if E2 vanishes then the echo may be

recovered exactly;thus E2 is indeed an error term.

If B is less than one, a parallel analysis to that given above

predicts peaks at positive quefrencies, and reveals that a distorted

version of the basic wavelet rather than the echo will be recovered.

The theoretical analysis presented above gives considerable

insight into the wavelet recovery process when the window is

approximately constant over the duration of the basic wavelet.

A second analysis is presented below which gives further insight

into the effects of windowing on epoch detection and wavelet recovery

under different conditions.

Consider equation (4-9), it may be rewritten as

-n L-l
Y(z) = X (z)+az o0 x(nT-k T)w(nT)z-n+E+E3 (4-21)
n=O

-n L-1
where E3 = az 0 x(nT-k T)(w(nT+n T)-w(nT))z-n (4-22)
n=O

-n
thus Y(z) = X (z)(1+az 0)+E+E3 (4-23)


Both El and E3 are considered error terms in that if they are not

present, then one could obtain the complex cepstrum, remove the

contributions due to the echo impulse train, perform the inverse

complex cepstrum operation, correct for the windowing by multiplying

the recovered sequence by (w(nT))-1 and obtain the basic wavelet.









The error term El has already been discussed in the section on

the exponential window, and will again be assumed to vanish. A

close examination of the error E3 reveals it arises from two distinct

sources. E3 can be written as

-no L-no-1 -n L-l
E3 = az no x(nT-k T)[w(nT+n T)-w(nT)]z- +az o0 x(nT-k T)w(nT)zn.
Sn= n=L-n


The first term in the equation above represents an error introduced

by the window because the window weights the basic wavelet and echo

differently, the second term is again the.truncation error due to

the fact that the echo may extend beyond the record under consideration,

and as in the section on the exponential window it is neglected.

This leaves only the.windowing error to be considered; obviously E3

vanishes when w(nT)=w(nT+n T). This again suggests the rectangular

window for wavelet extraction. Other common windows for should be

acceptable, provided x(nT-k T)(w(nT+n T)-w(nT)) is small. For

most commonly used windows (Hamming, Hanning, etc.) this requires

n to be small (i.e., the delay between the basic wavelet and echo

must be short). Carrying the analysis further we may rewrite

equation (4-23) as

E -n
Y(z) = (X(z) + 3 )(1+az ) (4-24)
1+az

thus

E -n
Y(z) = log Y(z)= log( (z)+ -- ) + log (l+az ) (4-25)
From equation (4-25) we see that the side of the complex cepstrum
From equation (4-25) we see that the side of the complex cepstrum


I









on which the echo peaks occur will be determined by the echo amplitude

(a) just as in the rectangularly windowed case.
Assuming that the term log (l+az-n ) can be filtered out, we

can extract the wavelet by (assuming a E
YRW(nT) = 1Z (X (z) + -- )
l+az
= Z1(Xw(z) + E3(1-azno+a2z 2n...)) (4-26)

YRW(nT) = w(nT)x(nT-k T)+[ax(nT-k T-n T)(w(nT+n T)-w(nT))]
RW 0 0 0 0
*[6(nT)-a6(nT-n T)+a26(nT-2n T)...] (4-27)

Multiplying (4-27) by (w(nT))-l, we obtain
w(nT+n T) 2
YR(nT) =x(nT-kT) + [ax(nT+k T-n T)( w(nT) -)]-a x(nT+k T-2n T)
R o o o w(nT) o o
w(nT-n T)
(1- w + .. (4-28)

Thus the basic wavelet is recovered though it is distorted by the
windowing error terms. A similar derivation applies if a>l, yielding
recovery of the echo rather than the basic wavelet, but with similar
distortion terms.
From the two analyses considered it is evident that there are

two cases when we may expect satisfactory wavelet recovery after

windowing with one of the windows commonly used to reduce leakage:
(1) When w(nT) is relatively constant over the duration of
the basic wavelet. This requires the signal duration to be short
compared to the total record length. The experimental results
presented for the signal of the form given in equations (4-4) and (4-5)










(with b=1.0) are an example of this case. Even though the actual

signal duration is a considerable portion of the total record length,

90% of the signal energy lies in a region only 3 or 4% of the total

record length. Thus the effective duration is quite short. The

first theoretical analysis presented accurately predicts the results

of these experiments.

(2) When the window w(nT) is approximately constant over the
w(nT+noT)
echo delay n that is w(nT 1. This requires n to be short

compared to the total record length. Additional computer computations

were conducted to verify the findings of the second analysis.

Utilizing the signal given in equations (4-4) and (4-5) with b=.1,

it is found that echo epoch detection and successful wavelet recovery

(in the no noise case) are possible for echo delays as short as 5

sample times when the rectangular window is used. When the same

signal is Hamming windowed,echo detection and satisfactory wavelet

recovery arepossible for small echo delays (n <10) though even for

n equal 5 the MSE is considerably greater for this window than for

the rectangular window.

It should be noted that no assumptions (other than regarding

the magnitudes of a.and B) were made in the analyses presented until

the filtering of the peaks in the complex cepstrum was considered.

Essentially both analyses are correct up to this point though the

nature and magnitude of the error terms differ. This leads to the

observation that when a1, and neither approximation is valid,

peaks should be present at both positive and negative quefrencies.









One set is predicted by the first theoretical analysis and the other

set is predicted by the second analysis. This is precisely what

was observed in many of the experimental outputs.


Conclusions


The exponential window

Windowing the input data with the exponential window was expected

from theoretical considerations to perform as well as the rectangular

window in the no noise case (and somewhat better if echo truncation

is a problem). Over the duration of the signal this is verified

by the experimental results at high SNR; however, correction for

the exponential window introduces some distortion into the recovered

record outside the signal duration. At low SNR the exponential window

performed better than the rectangular window. This is undoubtedly

due .to the fact that the composite signalexamined occurred at the

beginning of the data record and thus the exponential window tends

to reduce the noise content of the total record much more than the

signal content. Correspondingly when the composite signal occurs

near the end of the data record, the exponential window degrades the

recovered wavelet. It is similarly noted that while the echo epoch

peak in the complex cepstrum is diminished,.the detection threshold

is unchanged when the composite signal occurs near the beginning of

the record but increases when the composite signal occurs later in

the record. Finally it was noted that the exponential window may be

used to reduce aliasing of the echo impulse train, and to prevent

ambiguous peaks in the multiple echo case. Unless echo truncation









or one of the problems cited above is present, an exponential windowing

of the input data sequence appears to offer no advantages over the

rectangular window.


The common windows

Application of the Hamming, Hanning, and Tapering windows is

very detrimental to wavelet recovery. Satisfactory recovery is

not possible unless:

(1) the echo delay is small compared with the total record, or

(2) the duration of the basic wavelet is short compared with

the record length.

Even in these cases wavelet recovery is degraded. It is interesting

to note that in spite of the distortion caused by windowing, echo

epoch detection is always possible (at least in the no noise case)

and that while peaks maybe introduced on both sides of the complex

cepstrum, no shifting of the peaks from the proper echo epoch is

present. Windowing does however, increase the SNR at which echo

epoch determination is possible. This indicates that cepstral

techniques may be useful in resolving signals of similar, but not

identical, shapes. The theoretical analyses presented predict the

errors produced in the recovered wavelet by windowing with the

Hamming window quite well. A similar analysis may give some insight

into the distortion introduced in the recovered wavelet when the

echo is truncated.


WINDOWING THE LOG SPECTRUM

Since the complex cepstrum contains well defined peaks, it

seems reasonable that windowing the log spectrum be considered to









reduce the leakage due to these peaks. Only the effect of the Hamming

window on the log spectrum is studied, as this window seems repre-

sentative of the windows used to reduce leakage and it is nonzero

at its endpoints making correction less difficult. After windowing

the log spectrum, the complex cepstrum is computed and the peaks

present due to the echo are removed ;this sequence is then inverse

transformed to obtain the recovered log spectrum which is (unless

otherwise noted) corrected by multiplying the recovered log spectrum

by the inverse of the windowing sequence.

The following observations are made on the effects of windowing

the log spectrum.

(1) A spreading of the peaks in the complex cepstrum is noted;

e.g., a single positive peak is reduced in amplitude and has two

adjacent negative peaks.

(2) Empirically, it is found that the "filtering" of the complex

cepstrum is critical. Smoothing the complex cepstrum at single peak

points as used previously proves to be totally inadequate, as the

recovered wavelet is virtually unrecognizable even when no noise is

present. When both the peak and the two points adjacent to it are

smoothed, recovery is possible. Figure 29 compares the MSE of the

recovered wavelet (when adjacent points are smoothed) to that

obtained when only rectangular windowing of the log spectrum is used.

As can be seen for the no noise case, windowing the log spectrum

results in about the same MSE as for the rectangularly windowed case,

but performance rapidly deteriorates when noise is added. The wavelet

becomes unrecoverable at a SNR of 14 dB.. The echo detectability

threshold (20 dB) is also increased by 12 dB over the rectangularly

windowed case.






















* Rectangular Window
* Hamming Windowed Log Spectrum


MSE








10-6-
10









-7,
1 -4
-10-7 I I I I -- --4-
10 20 30 40

SNR(dB)


Figure 29 MSE of the Recovered Wavelet when the Log
Is Hamming Windowed.


Spectrum









(3) Several runs have been made with no correction of the

recovered log spectrum, and it was found that wavelet recovery is

impossible if correction is not made.


Interpretation of the Results

Let us consider the effects of windowing the log spectrum in

the single echo case.

-n
Y(z) = log Y(z) = log X(z)+ log (l+az ) (4-29.)

Windowing ?(z), we obtain

-n
W(z)Y(z) = W(z) log (I+az ) + W(z) log X(z)

where W(z) is the window.

Assuming a
power series, and inverse z-transforming equation (4-29), we obtain

the complex cepstrum.

2 3
(nT) = w(nT)*(nT)nT)*(nT) T)*(a6(n-no)- 6(n-2n )+ 6(n-3n )...) .(4-30)


Thus, it is observed that windowing the log spectrum convolves the

complex cepstrum with the transform of the window. In the case of the

Hamming window this means the complex cepstrum is convolved with the

sequence

w(nT) = -.226(nT+T)+.566(nT)-.226(nT-T)

This accounts for the spreading of the peaks noted experimentally.

If the peaks due to the echo can be "filtered" from the complex cepstrum,

then we may obtain the basic wavelet by correcting the recovered










log spectrum and completing the remainder of the wavelet recovery

algorithm. It is observed that the contributions in the complex

cepstrum due to the echo are no longer isolated peaks but in fact

extend to the points immediately adjacent to the main peaks thus

we expect that "filtering" over a single peak will be inadequate

for wavelet recovery as is observed.

The rapid deterioration of the MSE with increasing SNR is prob-

ably also connected with the filtering problem but this is not

well understood. The increase in the echo detectability threshold

is undoubtedly due to the fact that the peaks are reduced in height

by convolution with the sequence given above.


Conclusions

While windowing the log spectrum with the Hamming window reduces

leakage, it is apparent that other factors (principally the problem

with "filtering" the complex cepstrum) tend to degrade wavelet

recovery especially at low SNR. Thus windowing the log spectrum

with the Hamming window cannot be recommended as part of the general

wavelet recovery algorithm. The performance of other similar windows

is expected to be the same.


HANNING THE LOG SPECTRUM

It has been reported that the Hanning smoothing of the log

magnitude (that is, the real part of the log spectrum) results in a

decrease in both the MSE and the echo epoch detectability threshold

in the presence of additive noise [3]. Since this is closely related

to windowing the complex cepstrum, it is appropriate to discuss this

topic at this time. Smoothing consists of convolving the sequence to










be smoothed with a smoothing sequence (e.g., the Hanning smoothing

sequence is .25, .50, and .25). Smoothing is ordinarily used after

an FFT to reduce leakage, as this is equivalent to windowing the

input prior to the FFT with a Hanning window convolutionn in the

frequency domain is equivalent to multiplication in the time domain).

In the present context it serves a quite different function, smoothing

the log spectrum is essentially a frequency invariant low (short)

pass filtering operation. Consider the single additive echo case.

-n
YS(z) = [log X(z) + log (1+az )]*W(2) (4-31)

where W(z) is the smoothing function or equivalently it may be

regarded as the impulse response of a filter. Inverse z-transforming

the above equation, we find


ys(nT) = ((nT)+e(nT))w(nT) (4-32)

where x(nT) is the complex cepstrum of the basic wavelet, e(nT) is

the train of 6 functions due to the presence of an echo, and w(nT)

is the inverse z-transform of the smoothing function.

For the Hanning weights given above, w(nT) = (l+cos -).

Thus we see that Hanning smoothing tends to reduce the high quefrency

NT
(near -) contributions to the complex cepstrum, and is equivalent

to windowing the complex cepstrum.

If only the log magnitude is smoothed, then
-n -n
Ys () = Re(log X(z)+ log(1+az O))*W(z)+jlm(log X(z)+ log(l+az o)

-n -n
= (loglX(z)l+ logll+az O)*W(z)+j(Phase X(z)+Phase (log(l+az 0))).


(4-33)








Evaluating the inverse z-transform on the unit circle z=ejmT and
noting that

ReY (z) = loglX(eJmT) + log(l+a2+2a cos amnT) (4-34)

is an even function of w, and

WT I a sin(n wT)
jImY(z) =jPhase X(ejT) +jatan-l(- cos(n T) (4-35)

is an odd function of w, we see that the transform of equation (4-34)
is the even part of the complex cepstrum y(nT) (i.e., it is

y (nT) = -(y(nT) + y(-nT)) and the transform of equation (4-35) is
the odd part of the complex cepstrum y(nT) (i.e., it is y (nT) =

I (9(nT)-9(-nT))). Thus the smoothed complex cepstrum is

9s(nT) = 9e(nT)w(nT) + 0o(nT) (4-36)

but

Se(nT) = n(9(nT)+y(-nT))= (x(nT)+x(-nT)+^(nT)+e(-nT)) (4-37)

and
yo(nT) = ^((nT)-9(-nT)) =(x(nT)-x(-nT)+e(nT)-e(-nT)) (4-38)

where e(nT) is the train of impulses in the complex cepstrum due to
the echo. From the above analysis it is clear that the log magnitude
terms produce 6 functions on both sides of the origin, and the log
phase terms produce 5 functions which reinforce on one side of the
origin and cancel on the other to produce the one-sided e(nT). When








the log magnitude is smoothed the 6 functions produced by it are

reduced by the window w(nT); thus they no longer cancel on one side

with the 6 functions due to the phase portion, and peaks may be

expected on both sides of the origin (though the peaks on one side

should be substantially greater). Thus we observe that a minimum

phase sequence (normally zero on the negative side of the complex

cepstrum) may produce contributions on both sides of the complex.

cepstrum if the log magnitude alone is smoothed. If both the log

magnitude and phase are smoothed their contributions will be reduced

equally and thus they will still cancel completely on the negative

side of the complex cepstrum.


Experimental Results

A Hanning smoothing performed on the log magnitude and phase

produces a windowing of the complex cepstrum as expected. For the

case considered (n =30) the echo epoch detection threshold appears

to be slightly lower than for the unsmoothed case. This is undoubtedly

due to the fact that the primary echo epoch peak is only slightly

reduced by the window while the peaks associated with noise located

at higher quefrencies are considerably reduced. Had the echo epoch

been at a higher quefrency (e.g., no=100) the detection threshold

would have increased greatly. As can be seen from Figure 30, the MSE

of the recovered wavelet is considerably reduced by the smoothing

process. Only in the no noise case did the unsmoothed process prove

to be superior.

Smoothing the log magnitude (only) produced similar results as seen

in Figure 30, but this is clearly inferior to smoothing both the log

magnitude and phase when additive noise is present. Since the power





85













Rectangular Window
Rectangularly Windowed Complex Cepstrum
10- a Hamming Smoothing of Log Magnitude Only
o Hanning Smoothing of Log Spectrum




10-3





MSE 10-4
0
MSE 10- 0




10-5







10
10-7
10-7 -- I I I

10 20 30 40

SNR(dB)

Figure 30 MSE of Recovered Wavelet when .the Log Spectrum
Is Hanning Smoothed.









cepstrum is independent of phase information, the power cepstrum

in this case is identical to the power cepstrum produced above.

Kemerait [3] has reported observing echo peaks at both positive and

negative quefrencies only when a>l. Our results clearly contained

echo peaks at positive and negative quefrencies whether a>l or a
The positive peaks are dominant when a
dominant when a>l. This clearly confirms the theoretical analysis

presented. Since echo peaks are present at both positive and negative

quefrencies the complex cepstrum must be smoothed on both sides.

If only the dominant peaks are filtered the MSE of the recovered

wavelet is degraded slightly.

An attempt was made to improve the MSE performance by not

smoothing the log spectrum at low frequencies (where most of the

signal power is present) while continuing to smooth the higher frequency

components. This is again essentially a filtering process but in

this case the filtering is no longer frequency invariant. The results

from this type of smoothing process were found to be considerably

worse than those found above, and its use was discontinued.


Conclusions

The Hanning smoothing of the log spectrum is essentially a

frequency invariant low (short) pass filtering operation on the log

spectrum, or equivalently it may be regarded as a windowing of the

complex cepstrum. In this light the effect of smoothing on echo

epoch detection is easily understood, as it does in fact simply reduce

echo peaks occurring at high quefrencies more than those at low

quefrencies.




University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - - mvs