• TABLE OF CONTENTS
HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 The power cepstrum and the complex...
 An explanation of the algorithm...
 Experimental results and inter...
 Summary and conclusions
 Definition of the terminology used...
 Mathematical anaylsis of the power...
 Homomorphic filtering and the complex...
 complex cepstrum with additive...
 Composite signal with distorted...
 Computer plots and listing
 Reference
 Biographical sketch
 Copyright














Title: Signal detection and extraction by cepstrum techniques
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Title: Signal detection and extraction by cepstrum techniques
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Table of Contents
    Title Page
        Page i
    Dedication
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
        Page vi
    List of Tables
        Page vii
    List of Figures
        Page viii
        Page ix
        Page x
        Page xi
        Page xii
    Abstract
        Page xiii
        Page xiv
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
    The power cepstrum and the complex cepstrum
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
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        Page 19
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        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
    An explanation of the algorithm for wavelet extraction
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
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        Page 46
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        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
    Experimental results and interpretation
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
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        Page 76
        Page 77
        Page 78
    Summary and conclusions
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
    Definition of the terminology used in the dissertation
        Page 84
        Page 85
    Mathematical anaylsis of the power cepstrum
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
    Homomorphic filtering and the complex cepstrum
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
    complex cepstrum with additive noise
        Page 101
        Page 102
        Page 103
        Page 104
    Composite signal with distorted echoes
        Page 105
        Page 106
        Page 107
    Computer plots and listing
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
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        Page 200
    Reference
        Page 201
        Page 202
    Biographical sketch
        Page 203
        Page 204
    Copyright
        Copyright
Full Text















Signal Detection and Extraction by Cepstrum Techniques


By

Robert Chester Kemerait















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
1971








































To Jimmy C. Perkins















ACKNOWLEDGEMENTS


The author is indebted to Dr. D. G. Childers, chairman of

the supervisory committee, for his guidance and valuable encourage-

ment in every phase of this study. Sincere thanks are expressed

to the members of the supervisory committee, Drs. L. W. Couch II and

N. W. Perry, Drs. A. H. Nevis and L. E. Jones III.

A special thanks goes to Mrs. Carol Halpeny who drafted the

figures and to Mrs. Jeanne Ojeda who typed this manuscript.

The financial assistance provided by the U.S. Government

(G.I. Bill), the State of Florida (teaching assistantship), and the

NIH National Eye institute EY00581 and the Office of Naval Research

Contract N00014-68-A-0173-0014 (research assistantship) are gratefully

acknowledged.

Finally, a special note of thanks is expressed to the author's

wife, Janet, for her constant encouragement, help, and patience.















TABLE OF CONTENTS


ACKNOWLEDGEMENTS...............................................

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

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

ABSTRACT........................................................

CHAPTERS:


INTRODUCTION.......................................

THE POWER CEPSTRUM AND THE COMPLEX CEPSTRUM..........

The Power Cepstrum..................................

History.........................................

Formulation....................................

Results..........................................

The Complex Cepstrum.................................

History..........................................

Formulation....................................

Phase "Unwrapping" .............................

Linear Filtering................................

Summary ....................................... ......

AN EXPLANATION OF THE ALGORITHM FOR WAVELET
EXTRACTION..........................................

Introduction....................................

The Computation of the Power Cepstrum ...............

The Computation of the Complex Cepstrum...............


Page

iii

vii

viii

xiii



1

9

9

9

11

13

13

13

14

15

21

27


29

29

30

31


I

II


III










TABLE OF CONTENTS (Continued)

Page

Phase Unwrapping................................ 31

Inverse Fast Fourier Transform (IFFT)............ 33

Linear Filtering..................................... 33

Comb Filter..................................... 34

Short-pass Filter............................... 34

Long-pass Filter................................ 34

Wavelet Estimation................................... 35

Inverse Procedure................................ 35

Results of the Two Examples..................... 36

IV EXPERIMENTAL RESULTS AND INTERPRETATION.............. 53

Noiseless Case...................................... 53

The Power Cepstrum .............................. 53

The Complex Cepstrum............................ 60

Minimum-Maximum Phase Impulse Trains............ 61

Linear Filtering................................ 62

Impulse Train Extraction........................ 64

Wavelet Extraction.............................. 65

Noisy Case ........................................... 69

The Power Cepstrum.............................. 69

The Complex Cepstrum............................ 70

Linear Filtering................................ 71

Wavelet Extraction.............................. 72

Distorted Echo Removal ............................... 72

Interpretation of Results....................... 75











TABLE OF CONTENTS (Continued)


Page

V SUMMARY AND CONCLUSIONS............................... 79

Noiseless Case........................................ 79

The Power Cepstrum............................... 79

The Complex Cepstrum and Wavelet Extraction...... 80

Noisy Case........................................ 81

Power Cepstrum................................... 81

Complex Cepstrum and Wavelet Extraction.......... 82

Distorted Echo Removal................................ 82

Future Research....................................... 83

APPENDICES

A DEFINITION OF THE TERMINOLOGY USED IN THE
DISSERTATION................... ...................... 84

B MATHEMATICAL ANALYSIS OF THE POWER CEPSTRUM........... 86

C HOMOMORPHIC FILTERING AND THE COMPLEX CEPSTRUM....... 93

D COMPLEX CEPSTRUM WITH ADDITIVE NOISE.................. 101

E COMPOSITE SIGNAL WITH DISTORTED ECHOES................ 105

F COMPUTER PLOTS AND LISTING............................ 108

REFERENCES...................................................... 201

BIOGRAPHICAL SKETCH............................................ 203














LIST OF TABLES


Table Page

1 Computed Coefficients Using Two Terms of the Series.... 90

2 Computed Coefficients Using Three Terms of the
Series................................................. 90

3 Harmonics Overlapping with tl = 0.93 Seconds........... 91

4 Harmonics Overlapping with t1 = 1.71 Seconds........... 92















LIST OF FIGURES


Figure Page

1 Typical composite signal with one echo.................. 2

2 Canonic representation of a homomorphic system........... 6

3 System characteristics for the N and N-1 operations..... 6

4 Block diagram of cepstrum analysis...................... 10

5 Typical phase curve with a linear phase component
present................................................. 16

6 Complex cepstrum with a linear phase component present.. 19

7 Comb filtering the complex cepstrum..................... 23

8 Short-pass filtering the complex cepstrum............... 24

9 Long-pass filtering the complex cepstrum................ 26

10 Normalized VER signal used as the wavelet............... 37

11 Composite signal composed of the wavelet and a single
echo.................................................... 38

12 Log magnitude.......................................... 39

13 Unwrapped phase curve................................... 40

14 Power cepstrum. ...................................... 41

15 Complex cepstrum data................................... 42

16 Complex cepstrum ........................................ 43

17 Recovered wavelet...................................... 44

18 Recovered wavelet expanded ................................ 45

19 Composite signal composed of the wavelet and a single
echo .................................................. 46

20 Log magnitude.......................................... 47


viii









LIST OF FIGURES (Continued)


Figure Page

21 Unwrapped phase curve.................................. 48

22 Power cepstrum......................................... .. 49

23 Complex cepstrum............ ...... .................... 50

24 Recovered echo......................................... 51

25 Recovered echo expanded ............................... 52

26 Effect of the amplitude and epoch time of a single echo
on the observed magnitude of the power cepstrum peak... 54

27 Magnitudes of the power cepstrum peaks as a function
of the summation of the echo amplitudes for the two
echo case.............................................. 56

28 Magnitude of the power cepstrum peak of the second
echo as a function of the time difference between the
two echoes ...... ....................................... 57

29 Mean square error (MSE) as a function of the echo
magnitude............................................... 66

30 Effects of smoothing on the mean square error (MSE)
of the recovered wavelet for the case of additive
noise......................................... .. ....... 73

31 Two methods used to generate a composite signal
consisting of the wavelet plus a distorted echo......... 74

32 Mean square error (MSE) of the recovered wavelet as
a function of the echo distortion using additive
noise.................................................. 76

33 Effect of echo truncation on the power cepstrum peak
magnitude.......................................... 77

34 Model for the distorted echo case........................ 106

35 Composite signal consisting of a wavelet and four
echoes with identical amplitudes (0.2) and delays
of 56, 71, 91, and 111 samples respectively.............. 111

36 Log magnitude...................... ............ ........ 112

37 Unwrapped phase.......................................... 113








LIST OF FIGURES (Continued)

Figure Page

38 Power cepstrum ................................ ...... 114

39 Complex cepstrum....................................... 115

40 Recovered wavelet..................................... 116

41 Recovered wavelet (expanded scale).................... 117

42 Composite signal consisting of a wavelet and a single
echo delayed 56 samples and having an amplitude of 0.4. 119

43 Log magnitude......................................... 120

44 Unwrapped phase.................................. ...... ... 121

45 Power cepstrum ........................................ 122

46 Complex cepstrum...................................... 123

47 Recovered wavelet..................................... 124

48 Recovered wavelet (expanded scale)...................... 125

49 Composite signal consisting of a wavelet and a single
echo delayed 56 samples and having an amplitude of 0.4. 127

50 Log magnitude.......................................... 128

51 Unwrapped phase ................. ... ................... 129

52 Power cepstrum ........................................ 130

53 Complex cepstrum...................................... 131

54 Recovered wavelet..................................... 132

55 Recovered wavelet (expanded scale)...................... 133

56 Power cepstrum........................................ 135

57 Complex cepstrum...................................... 136

58 Recovered wavelet..................................... 137

59 Recovered wavelet (expanded scale)...................... 138

60 Power cepstrum ........................................ 140









LIST OF FIGURES (Continued)

Figure Page

61 Complex cepstrum...................................... 141

62 Recovered impulse train................................. 142

63 Composite signal (256 data points) consisting of a
wavelet and a single echo delayed 56 samples and
having an amplitude of 0.4............................. 144

64 Log magnitude......................................... 145

65 Unwrapped phase....................................... 146

66 Power cepstr.um........................................... 147

67 Complex cepstrum....................................... 148

68 Recovered wavelet..................................... 149

69 Recovered wavelet (expanded scale)...................... 150

70 Composite signal (512 data points) consisting of a
wavelet.and a single echo delayed 56 samples and
having an amplitude of 0.4.............................. 152

71 Log magnitude.................................. ....... 153

72 Unwrapped phase....................................... 154

73 Power cepstrum........................................ 155

74 Complex cepstrum...................................... 156

75 Recovered wavelet..................................... 157

76 Recovered wavelet (expanded scale)...................... 158

77 Composite signal (512 data points) consisting of a
wavelet and a single echo delayed 56 samples and
having an amplitude of 0.4.............................. 160

78 Log magnitude......................................... 161

79 Unwrapped phase....................................... 162

80 Power cepstrum.. ....................................... 163

81 Complex cepstrum...................................... 164








LIST OF FIGURES (Continued)


Figure Page

82 Recovered wavelet..................................... 165

83 Recovered wavelet (expanded scale)..................... 166

84 Single echo with added noise............................ 168

85 Composite signal consisting of a wavelet and a single
noisy echo delayed 56 samples and having an amplitude
of 0.4 before distorted by the noise................... 169

86 Log magnitude........................... ......... .... 170

87 Unwrapped phase...................................... 171

88 Power cepstrum................................ ......172

89 Complex cepstrum........................................ 173

90 Recovered wavelet..................................... 174

91 Recovered wavelet (expanded scale)...................... 175

92 Composite signal consisting of a wavelet and a single
echo (truncated by 5%) delayed 56 samples and having
an amplitude of 0.4................................... 177

93 Log magnitude...................................... 178

94 Unwrapped phase........................................ 179

95 Power cepstrum......................................... 180

96 Complex cepstrum..................................... 181

97 Recovered wavelet...................................... 182

98 Recovered wavelet (expanded scale)...................... 183

99 Computer program used to perform cepstrum analysis..... 184


xii












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

SIGNAL DETECTION AND EXTRACTION BY CEPSTRUM TECHNIQUES

By

Robert Chester Kemerait

August, 1971


Chairman: Dr. D. G. Childers
Major Department: Electrical Engineering

The extraction of an unknown wavelet from a composite waveform

immersed in additive noise has been the focus of this investigation;

the waveform consisted of the wavelet and its distorted echoes.

The investigations were motivated by research on brain waves,

namely the visual evoked response (VER), which is thought to be a

composite signal camouflaged by background activity. One objective

was to evolve a procedure by which a signal composed of overlapping

multiple VER's masked by noise could be decomposed into its separate

contributors. This, in turn, would lead to better diagnostic proce-

dures and enhance our understanding of how brain functioning is

reflected in brain waves.

The extraction procedure that was successfully developed combined

two methods, namely the cepstrum and the complex cepstrum; the properties

of these methods were investigated in detail. These techniques were then

applied to detect and extract a distorted echo immersed in noise.

It was determined that there is a threshold effect exhibited

when truncation is used to distort the echo. When additive noise was


xiii










present, echo removal was best achieved by introducing a linear phase

component in conjunction with amplitude smoothing.

It was further demonstrated that cepstrum techniques could be

applied satisfactorily to a wide range of composite waveforms composed

of distorted wavelets. This dissertation shows how this nonlinear

analysis procedure can extract echo wavelet information, i.e., determine

the epoch times of the echoes, estimate their relative amplitudes in the

presence of noise, and even extract the waveform of the distorted echo.


xiv















CHAPTER I

INTRODUCTION


Certain classes of physical systems can be analyzed by non-

linear analysis techniques. Of particular interest to the personnel

of the Visual Sciences Laboratory at the University of Florida is

the problem of how to decompose a composite waveform comprised of

multiple wavelets evoked by visual stimulation. Similar composite

signals may occur in a variety of physical situations and a few of

the more interesting, as well as those to be dealt with in this

dissertation, are described in the following paragraphs. A typical

situation is shown in Figure 1.

One example which is believed to give rise to a composite

signal results from stimulating the retina via a light source.

This evokes a change in the electroencephalogram (EEG) known as the

visual evoked response (VER). The individual elicitations can be

synchronized and summed to improve the signal to noise ratio (SNR).

These VER signals are thought to arise from spatially disparate

sources, the potentials of which combine to form the recorded signal.

Thus, the VER might be modeled as a summation of nearly identical

wavelets with different amplitudes and arrival times. Thus, the

composite signal can be considered to be composed of a basic signal

(from the source closest to the electrode) and its echoes (the signals

originating from other sources which are located at a greater distance


















NORMALIZED
AMPLITUDE

1.5





1.0






0.5


0 1.0


1.9 TIME (SECONDS)


Typical composite signal with one echo


Figure 1.










from the electrode). Therefore, it would seem reasonable that the

determination of the epoch times of these "echoes" should yield

significant information about the spatial arrangement of sources

located in the visual cortex. One objective is to decompose the com-

posite signal. Any procedure needed to decompose this signal should

be capable of handling composite signals with echoes either greater

than or less than unity (i.e., relative to some arbitrarily selected

reference), since these are independent active sources.

Composite signals can arise from physical phenomena in other

fields as well, and these signals also demonstrate the need for a

system to decompose composite signal data. Examples would include

echoes found in multipath radar systems and spurious echoes en-

countered in sonar systems. Other examples treated extensively in

the literature are speech signals (1) and seismological data (2).

Speech signals would, at first glance, appear to be markedly different

from those being considered in this investigation, since the speech

signal is usually represented as the convolution cf several components

rather than as a summation of a signal and one or more echoes. But if

it is considered that the composite signals (reference wavelet plus

echoes) can be represented mathematically by the convolution of the

reference wavelet and a train of impulses, it can be seen that the

speech signal falls into the same class of signals as those being

examined in this research.

One long-term purpose of the investigation of composite signals

immersed in additive noise is that it will provide some insight toward

eventually achieving the goal of the separation of unknown additive

stochastic processes.










The process of decomposing the convolved signals referred to

above is called deconvolution. Several procedures for achieving this

have proven successful; the more familiar methods include inverse

filtering (3), Wiener filtering (4), and decision theory (4).

Each of these methods has proven successful under individual

circumstances; for example, in the application of the inverse filtering,

a signal is transformed by a linear time invariant system, the function

of which is the reciprocal of the Fourier transform of the signal com-

ponents to be removed. Even so, this method has a serious limitation;

the signal to be removed must be known, and the SNR of the convolved

signal must be quite large. Investigations attempting to employ a

Wiener filter to improve the estimates of decomposition have proven

to be of little value (4). For high input SNR this filter is unnecessary;

for low input SNR the system acts like a matched filter which is un-

suitable for decomposition unless one wishes to decompose signals where

the time of occurrence between signals is approximately twice the length

of the signal duration. Decision theory can be applied in the noise case

to estimate the echo amplitude and arrival times. This is a straight-

forward approach if the signal waveshape is known.

The particular method of deconvolution of a composite signal that

will be investigated and expanded in this dissertation was proposed in

two seemingly unrelated parallel approaches. One approach, published

in 1963 by Bogert et al. (2), dealt with an echo detection method they

called cepstrum, by paraphrasing the word spectrum; the other work,

published in 1965 by A. V. Oppenheim (5), dealt with homomorphic systems.

R. W. Schafer (6), in 1968, published the results of a specific applica-

tion of Oppenheim's theory; he called this the complex cepstrum.










Oppenheim's proposal deals with homomorphic systems and is a theory

that certain classes of nonlinear systems share basic properties and

can be represented as transformations between vector spaces; they

therefore obey a generalized principal of superposition. These

specific nonlinear systems were called homomorphic to emphasize the

fact that they are represented by algebraically defined linear trans-

formations. The two relations that apply to this specific problem are



4(x-y) = K(x) + K(y) (1-1)


((xc) = c'*(x) (1-2)


The block diagram of such a system is given in Figures 2 and 3.

It is apparent that the logarithm function and the exponential

function satisfy this system, and the intuitive application of the

logarithm function by Bogert et al. also fits this homomorphic system.

The difference between their approach and that introduced by

R. W. Schafer is that they were interested only in epoch detection and

therefore could disregard the phase information; Schafer wanted to recover

the original wavelet information and this necessitated saving and utilizing

this phase information. Bogert et al. utilized the upper block diagram

in Figure 3, while Schafer's method consisted of using both the upper and

the lower block diagrams.

This dissertation extends these two areas of investigation, and

deals with a composite signal which consists of multiple echoes in addi-

tion to the reference wavelet signal. Following detailed exploration,

mathematically and experimentally, of the limitations of the two methods,












] + + + + ][

LINEAR
N FILTER N
L


Figure 2.


Canonic representation of a homomorphic system


N Operation


+ + +

FAST FOURIER EXPONENTIAL
y(n) TRANSFORM EXP (

(FFT)


N-1 Operation
N Operation


*

INVERSE
FAST FOURIER
--- TRANSFORM

(IFFT)


Figure 3. System characteristics for the N and N-1 operations










two other important facets of these techniques are examined: the case

with additive noise and the case where distortion is encountered by

the transmitting medium (e.g., brain tissue in the VER case). Through

these investigations a measure of SNR is empirically determined; results

are obtained through the application of the complex cepstrum; several

modifications are introduced to improve the performance of the wavelet

recovery at various SNR levels. The measure of "goodness" for judging

the performance of these techniques used in this doctoral research is

primarily the mean square error (MSE) criterion; this method appears to

be as satisfactory as any other. To facilitate investigation of the

distorted echo, the echo was distorted by two different methods:

1. Noise was added to only the echo (the MSE was plotted

versus the SNR to measure the results) and

2. The echo was truncated (the MSE was plotted versus the

amount of truncation).

The detailed discussion of the power and complex cepstrum

techniques that are developed and utilized in this research is pre-

sented in Chapter II. A mathematical treatment of the two methods

is given in Appendices B and C. This treatment includes examples of

the method used to "unwrap" the phase curve, three methods of linear

filtering, the investigation of additive Gaussian noise, the removal

of a linear phase component, and the investigation of distorted echoes.

The cepstrum techniques explained in Chapter III are illustrated

by two detailed examples. The examples deal with the noise-free,

single echo case; one situation considers the echo amplitude less than





8




that of the reference wavelet and the other considers the echo

amplitude greater than that of the reference wavelet.

Chapter IV examines in a great deal more detail the results

presented illustratively in Chapter III. Here it is demonstrated

that the developed cepstrum technique can determine the epoch times

of several overlapping echoes, and can extract the reference wavelet

or its distorted echo from the composite signal. And, finally,

Chapter V presents the summary and conclusions.















CHAPTER II

THE POWER CEPSTRUM AND THE COMPLEX CEPSTRUM


We present here a brief history of the origin and development

of the power cepstrum and the complex cepstrum techniques. In addi-

tion, the several mathematical operations that form the basis for

these techniques are'systematically discussed.

By definition, the power cepstrum of a function (usually a

function of time) is the power spectrum of the logarithm of the power

spectrum of that function. Similarly, the complex cepstrum of a

function is defined as the inverse Fourier transform of the logarithm

of the Fourier transform of that function. These techniques are

summarized in Figure 4.


The Power Cepstrum


History

The nonlinear analysis technique called cepstrum, set forth by

Bogert et al. (2) in 1963, originated from an intuitive approach

directed toward finding the epoch times of the echoes of a composite

signal. This procedure was subsequently applied to speech signals by A.

M. Noll (1) in 1964 since speech signals were believed to be the con-

volution of two or more signals in the time domain. Thus, a method for

deconvolution was sought, or, equivalently, since convolution in the

time domain corresponds to multiplication in the frequency domain, a














POWER CEPSTRUM


FAST
x(n) FOURIER


TRANSFORM


INVERSE

FAST

FOURIER

TRANSFORM


COMPOSITE SIGNAL


THREE TYPES OF
LINEAR FILTERS
USED ARE:
1. COMB
2. SHORT-PASS
3. LONG-PASS


FAST

FOURIER

TRANSFORM


arg[Y]


CONVERSION

TO

X + iY


FILTERED
COMPLEX CEPSTRUM


ESTIMATE OF WAVELET


Figure 4. Block diagram of cepstrum analysis


__~~~C










method for demodulation was pursued; this is the same situation that

Bogert et al. were faced with when dealing with the echo case in their

research.


Formulation

Faced with a multiplication, Bogert et al. (2) were intuitively

led to a mathematical manipulation that would produce a sum of terms.

In order to accomplish this task, the two or more functions that are

convolved in the time domain must be composed of frequency components

that are significantly different from one another. If this is the case,

the procedure can be implemented by using a mathematical function that

operates on a product and yields a sum. Such a function is obviously

the logarithm.

To explain the power cepstrum analysis, an example of a simple

additive echo (the mathematical derivation of the power cepstrum for

this example is found in Appendix B) can be examined; the formula for

the total signal can be written as follows:



z(t) = y(t) + a oy(t t ) (2-1)


The power spectrum of z(t) can be represented by the following expansion



(W) = _y(w)[1 + a2 + 2a cos wto] (2-2)


This results in the product of two terms and thus one is led to apply

the logarithmic function to the above expression. Such an application

gives


log z() = log ) logo(g (1 + a2 + 2a cos Wto) (2-3)
zy0 0 0










The second term can now be expanded by an appropriate convergent

infinite series; or, when a is much less than unity, the infinite

series can be approximated by 2a cos wt Thus, the logarithm of
o o
the power spectrum is obtained with a nearly cosinusoidal ripple,

the parameters of which are simply related to the echo parameters a
o

and t For the general case when a is not much less than unity the
0 0

higher order terms in the series expansion must be considered. This

general case and the limitations of the power cepstrum will be examined

further in Chapter IV. Since the log power spectrum is a function of

frequency, the frequency of this ripple in the log power spectrum is

to, the units of which are ripples per cycle per second or seconds.

A ripple of this kind will usually be obscured by the irregu-

larities in the log power spectrum itself. Therefore, it is necessary

to borrow from the available techniques used for detection of periodic

phenomena obscured by noise. The result of the additional application

of the power spectrum is called the cepstrum, or, in the terminology

of this dissertation, the power cepstrum. The power spectrum of the

log power spectrum yields a peak (delta function) in the cepstrum domain

at the quefrency of t seconds; this indicates the desired epoch time.

Quefrency is one of the paraphrased words innovated by Bogert et al. (2)

to impress upon the reader that this analysis technique parallels that

of spectral analysis. Appendix A contains a complete list of these

paraphrased words.

Two comments should be added at this point. First, a peak will

be observed at t in the cepstrum domain if the log power spectrum of

the given function is relatively smooth with respect to the ripple which










will mean that there are no erroneous peaks in the power cepstrum

that could be interpreted as epochs of additional echoes. Second,

the phase information is discarded in calculating the power spectrum;

therefore the wavelet information is no longer recoverable.


Results

The mathematical derivation of the power cepstrum is contained

in Appendix B. These derivations show that it is possible to determine

the epoch of a single echo for a wide range of echo amplitudes (less

than and greater than that of the reference wavelet) and for a range

of echo delay times that extend the length of the reference wavelet.

The procedure is limited in that the cepstrum peak due to the echo is

lost in the background; this limitation is primarily due to inaccuracies

in computation rather than to a limitation in the cepstrum technique

itself. This is true for echoes considerably less than and greater

than the reference wavelet.

Problems occur when the power cepstrum technique is applied to

a composite signal consisting of the wavelet plus multiple echoes. Here

an erroneous echo can be detected if the echoes are not multiples of one

another. Peaks occur in the power cepstrum at the sums and differences

of the epoch times of the echoes. This problem will be explained in

detail in Chapter IV.


The Complex Cepstrum


History

The complex cepstrum technique was first described in a doctoral










dissertation by R. Schafer (6) in 1968, and the determination of the

wavelet information is the major innovation of this method. This is

the principal feature that is missing in the power cepstrum approach.

The complex cepstrum is a specific application of the homomorphic

systems theory set forth by A. Oppenheim (5) in 1965; the parallel work

of Bogert et al. was later shown to be a specific application of the

homomorphic systems theory as well.


Formulation

To explain the complex cepstrum procedure, a simple example of

a signal with one echo will be derived. The expression for a composite

signal consisting of a wavelet convolved with a train of weighted

samples is



x(n) = s(n) p(n) (2-4)


0O
p(n) = ak 6(n-nk) (2-5)
k=0


where s(n) and x(n) are the original and the distorted waveforms,

respectively. At this point, a method of separating the echo train

from the original waveform s(n) is desired; therefore, the complex

logarithm of the z-transform of x(n) is determined and then some form

of linear filtering is introduced to achieve signal separation. For

the single echo case, the impulse train is written as


p(n) = 6(n) + ao 6(n-no) .


(2-6)










The z-transform X(z), assuming that x(n) is z-transformable, evaluated

on the unit circle is


-j wn
X(z) l = X(e j) = S(e j)[l + a e o] (2-7)
z=ejo


where the contribution of the echo is a periodic function of w with a

period of 27/no. It is apparent that the repetition rate increases

as n increases so that greater echo delay times cause more rapid

fluctuations in the spectrum. The log spectrum, i.e., the log of

X(e ), is



log X(e) = log S(e j) + log (1 + a e o) (2-8)



Since the logarithm of the periodic waveform remains periodic with the

same repetition rate, the echo is represented in the log spectrum as an

additive periodic component. The necessary use of the complex logarithm

function results in multivalued phase information in the calculation of

the complex logarithm unless corrected for. This is discussed below.


Phase "Unwrapping"

In general, the value of the phase ARG [x(ea + jW)] will be a

discontinuous function of w since the phase is evaluated as the principal

value, i.e., modulo 27. This causes abrupt changes to appear in the

phase curve which can be corrected by "unwrapping" it in order to satisfy

the requirement that it be continuous and odd. (See Figure 5 for an

example of phase "unwrapping.") The reason the phase must be continuous

and odd, and the log magnitude must be even, is that this insures that








------ ------ ----- -------- -


0 N/2

--- -----a--.------ --------
a.


0 N/2


Figure 5.


Typical phase curve with a linear phase component present.
(a) principal value of the argument, (b) unwrapped phase,
and (c) linear phase component


+7E




-7E


1


I


+67E





+4 i


+3x


1. -~










the complex spectrum, the inverse Fourier transform of the log

magnitude, and the phase components will be a real quality. This

latter condition is imposed since the data considered in this disserta-

tion are real. However, the complex cepstrum will not be an even

function of time as demonstrated in reference 7.

Linear phase removal.--In Figure 5, an example is shown of a

linear phase component added to the phase of the wavelet signal. This

linear portion of the phase curve might be due to the wavelet itself,

or due to the addition of zeros to the sampled composite signal. This

latter situation arises when one is applying a fast Fourier transform

algorithm. Here the appendage of additional data points is usually

necessary since most FFT's require the number of data points to be a

power of two. Obviously most sampled composite waveforms to be de-

convolved will not have a total number of sample points equivalent to

a power of two; therefore, the alternatives are the truncation of the

data, a slower or faster sampling rate, or the addition of a sufficient

number of zeros to satisfy this requirement. The latter alternative

is probably the easiest solution.

The z-transform of a typical input signal with zeros added is
m. m
1 o
I (1 ak z )2 (l -bkz)
X(z) = Azr k=l k=l (2-9)
i Po
-1 0
I ( ck z ) d (1 dkz)
k=l k=l


where zr is the term that contributes the linear part to the phase

curve. This may be completely or partially caused by the addition of










zeros as was discussed previously. In the computation of the complex

cepstrum the logarithm of the above expression must be taken, which

yields


log (zr) = log (er(a + ) = r + jwr



and the inverse Fourier transform will yield a curve such as is shown

in Figure 6. If this linear phase portion is of sufficient magnitude,

it will dominate the complex cepstrum so that important characteristics

such as the peaks of the echoes and the complex cepstrum features of

the signal s(n) will be totally masked. In some cases this is desirable,

but, when it is not, the linear portion of the phase curve can be removed

from the unwrapped phase, making the (N/2-1) and the (N/2+1) points equal

to zero. In cases of this nature, the inverse Fourier transform does not

yield a plot like that shown in Figure 6. In addition to changing the

appearance of the complex cepstrum, the linear phase removal also dis-

places in time the recovered wavelet, depending on the amount of the

linear phase component that was removed.

Minimum-maximum phase situations.--A minimum phase system is one

that has no poles or zeros outside the unit circle in the z-domain, where-

as the maximum phase system has no poles or zeros inside the unit circle.

The advantages of a minimum phase situation are explained in

detail in reference 6. A few of the major points are summarized here,

along with the special applications to this research. If a minimum phase

situation is encountered and recognized, then it is possible to calculate

the complex cepstrum, utilizing only the real part, i.e., the log magnitude

of the data, by applying the Hilbert transform properties. These properties

are





19






0 N/2 N


0 N/2 N


/
/
/
/


/
\-,

\ /

0 N
S/2 ~ N

/


Complex cepstrum with a linear phase
present. (a) principal value of the
(b) unwrapped phase, and (c) complex


-moT


-mo 1


Figure 6.


component
argument,
cepstrum


+T -





-_7


m" -










X(eJ) = x(n) e-jn (2-10)
n=O
Tr
Ev[x(n)] = -j log IX(e j)j ejndw (2-11)
-7r


^(n) = Ev[x(n)] u(n) (2-12)


where u(n) = 2 n > 0

=1 n= 0

=0 n < 0


where the hat refers to the complex cepstrum. Thus, the obvious

advantage of a minimum phase sequence is that the phase information

can be discarded computationally and the wavelet is recoverable.

With this type of sequence, the computation time can be reduced if

time is of prime importance. The main drawback to this procedure

is that, for the general case, it is not usually known whether the

composite waveform data is minimum phase. In order for the data to

be minimum phase, the following three conditions must be satisfied.



x(n) =0 n < 0

x(0) # 0

I Ix(n)j <
n=0


To determine that these conditions have been met, the complex cepstrum

must be computed. When the above conditions have been met, Equations

(2-10)to (2-12) can be used for the calculation of the complex cepstrum.

Quite obviously, if these tests must be performed to determine that the










given waveform is minimum phase, the complex cepstrum has already been

calculated; for this reason it is unnecessary to determine if the com-

posite waveform is minimum phase, for the objective has already been

attained. It should be added, however, that the complex cepstrum can

be calculated by using the Hilbert transform properties even if the

composite waveform is not minimum phase. This use of the Hilbert

transform will cause the complex cepstrum to be aliased and this may or

may not be a tolerable condition. Another case where the minimum phase

concept could prove to be important is one in which deconvolution is

applied to many runs of data (speech data, for example) which can be

shown to have similar phase characteristics. In this case, if the first

computation reveals that the signal waveform is minimum phase, the rest

of the data could be deconvolved by using the log magnitude of each and

neglecting the phase components.


Linear Filtering

The concept of the homomorphic system introduced by A.V.Oppenheim

(5) attempts to separate the contributions of s(n) and p(n) by removing

the variations in the log spectrum. By linear filtering, it is possible

to filter the complex log voltage spectrum with a kernel designed to

remove the periodic components. To achieve this linear filtering, a

convolution in the frequency domain is required; the same result can also

be achieved by multiplying the complex cepstrum (time domain) by an

appropriate fixed weighting. Two filters are used to perform this multi-

plication to remove the echo peaks from the complex cepstrum; the third

filter is used to remove the complex cepstrum of the wavelet. A discussion

of the three filters follows.










"Comb" filter technique.--The complex cepstrum is multiplied

by a "comb" filter which is unity valued at all sample points except

those points which have peaks caused by the echo. At these points,

the filter is zero. For a simple echo that has been shown to have a

periodic variation in the log spectrum, peaks are found in the complex

cepstrum only at values of n which are multiples of the echo time n .

Figure 7 illustrates the complex cepstrum before and after filtering

along with the appropriate "comb" filter. After the comb filter has

been applied, the filtered complex cepstrum is smoothed at these zeroed

points by the averaging of the preceding and subsequent points.

"Short-pass" filter.--Another way to remove the echo from a

composite signal is to use the "short-pass" filter method illustrated

in Figure 8 and originated by R. Schafer (6). The short-pass filter

technique utilizes the fact that the complex cepstrum peak of the echo

with the shortest delay is usually well separated from the origin. If

this is the case, then only modest distortion will be introduced in the

recovered wavelet if the complex cepstrum is forced to take on zero

values (analogous to low-pass filtering) for the time samples greater

than or equal to no. This method will not only remove the peak at no,

but will remove all its harmonics, as well as those peaks caused by other

echoes with epoch times greater than n This idea is predicated on the

assumption that the complex cepstrum of the convolved signals does not

overlap. Obviously, this is not possible in practical situations as

there will always be some overlap; the amount of overlap that is tolerable

is the most important criterion to be determined before the short-pass

filter is applied.








I (n) |


Sno 2no 3no


h (n)


2no


3no


I (n) h(n)



T T V, ,


IA(n)


2no


3no


2no


3no


Comb filtering the complex cepstrum. (a) complex cepstrum,
(b) appropriate comb filter, (c) comb filtered complex
cepstrum, and (d) comb filtered and smoothed complex cepstrum


Figure 7.


I I~


r









x(n) |


h(n)









no

b.







l?(n) l









no

c.

Figure 8. Short-pass filtering the complex cepstrum. (a) complex
cepstrum, (b) appropriate short-pass filter, and (c)
short-pass filtered complex cepstrum










An example of an ideal situation with no overlapping is where

the signal s(n) is maximum phase and the wave-train p(n) is minimum

phase (all echo amplitudes are less than unity). Since the wavelet

signal s(n) is generally an unknown, the usual approach is to insure

that the complex cepstrum of the sequence s(n) dies out rapidly so

that s(n) is nearly zero for n = n and for sample values above n

and below n A knowledge of the epoch time of the echo is required

for either the short-pass or the comb filter. The comb filter's

performance is superior, however, because it distorts the complex

cepstrum of the wavelet less. Also, the distortion caused by the

application of the comb filter can be smoothed more easily than that

of the short-pass filter.

"Long-pass" filtering.--"Long-pass" filtering of the complex

cepstrum will also determine the epochs of the echoes; this can also

be applied to the power cepstrum as well. To use this method, the time

of the epoch of the first echo must be known approximately. From this

point on, the regular inverse procedure is applicable, but, instead

of recovering the wavelet signal, the delta functions at the epoch

times of the echoes are obtained. Echo detection becomes progressively

more satisfactory as the complex cepstrum of the wavelet is more success-

fully removed, or separated, from the "impulse train." This detection

procedure has problems similar to those for the power cepstrum; multiple

echoes can yield erroneous peaks at sum and difference times whose

amplitudes, in turn, depend upon the individual echo amplitudes.




















-no


Ix(n)


h(n)


no o
-no no


Figure 9.


Long-pass filtering the complex cepstrum. (a) complex
cepstrum, (b) appropriate long-pass filter, and (c)
long-pass filtered complex cepstrum


I ( ri) h(n)











Summary


The advantages of utilizing the power cepstrum technique to

determine the echo epoch time are:

1. The formulation of the power cepstrum employs the use

of the real logarithm.

2. The power cepstrum technique yields a better indication

of the echo epoch times, even in the presence of noise.

The disadvantages are:

1. The power cepstrum is a real-valued function and all phase

information is lost; therefore, it is impossible to recover

the reference wavelet.

2. The power cepstrum was found to exhibit an extremely large

SNR threshold for the noisy case.

The advantage of utilizing the complex cepstrum technique informa-

tion can be recovered since the phase data are retained.

The disadvantages of the complex cepstrum technique are:

1. The complex cepstrum formulation uses the complex logarithm;

this leads to multivalued functions unless corrected for.

This occurs because the principal value of the logarithm of

a product of complex signals is not always the sum of the

principal values of the individual complex signals. For

this reason, the phase of a given function cannot be computed

from the real and imaginary parts of the Fourier transform

of a single point, but, rather, the computation of the phase

requires knowledge of all preceding points. This means that

the phase must be unwrapped, or, in other words, a continuous

plot must be made.










2. The echo peaks in the complex cepstrum are very difficult

to find.

The power cepstrum technique has been incorporated into the

complex cepstrum algorithm for the purposes of this investigation;

this makes it possible to retain the good features of each method,

namely, both epoch time identification and wavelet recovery.















CHAPTER III

AN EXPLANATION OF THE ALGORITHM FOR WAVELET EXTRACTION


Introduction

In this chapter the procedures used to compute the power

cepstrum and the complex cepstrum are explained in detail. In order

to facilitate the explanation, the cepstrum techniques are applied

to two specific examples which extract a wavelet from a composite

waveform. In example one the composite waveform consists of the

wavelet and a single echo whose amplitude is less than that of the

wavelet; the second example lets the echo amplitude be greater than

that of the wavelet.

The wavelet used is a VER signal. A composite waveform is

artificially constructed by superimposing a VER wavelet on a delayed

attenuated replica (echo). The delay is 56 samples, and the attenua-

tion is 0.4 for the first example and 1.8 for the second example.

Since the total number of data points needed must be a power of two,

256 points were used; therefore 10 zeros were arbitrarily added to the

beginning of the wavelet and 56 to the end. Thus the echo has 66 zeros

at the beginning and none at the end.

The composite signal is then analyzed by the cepstrum program.

The desired output is the epoch time of the echo and an estimate of

the wavelet waveform. To measure the performance of this technique,

the recovered wavelet and the original wavelet are compared, point by

point, with the individual differences of each squared, summated (256











of them), and then divided by 256. This will be recognized as the mean

square error (MSE).

A block diagram of the cepstrum procedure is given in Figure 4,

and the computer results for the two examples appear at the end of this

chapter.


The Computation of the Power Cepstrum


Recall that the power cepstrum procedure determines the epoch

times of the echoes contained in the composite signal. These epoch

times are then utilized in the linear filtering portion of the complex

cepstrum technique.

The power cepstrum procedure is implemented via the following

steps:

1. Compute the power spectrum of the sampled composite signal

(the composite signal is real in this example). This

computation is accomplished by Fourier transforming the

composite signal using the fast Fourier transform (FFT)

subroutine (the input array has the sampled composite

signal as the real part with the imaginary part being zeros);

then the square root of the sum of the squares of the real

and imaginary parts of the output array of the FFT operation

is obtained. The array of real numbers, obtained from the

square root operation, is the desired power spectrum.

2. Compute the real logarithm of the power spectrum that was

obtained in step 1. The logarithm of the power spectrum is

also an array of real numbers.










3. Compute the power cepstrum by repeating step 1 with one

exception: the input array consists of the logarithm

of the power spectrum as its real part, and the imaginary

part is zeroed. In this case, the output will be the

power cepstrum rather than the power spectrum.


The Computation of the Complex Cepstrum


Since the procedure for determining the complex cepstrum starts

by Fourier transforming the sampled composite signal, as does the

procedure for the power cepstrum, the array obtained from the first

application of the FFT subroutine discussed in step 1 of the preceding

section is used as the starting point of this section. The procedures

for the computation of the two cepstrum techniques theoretically differ

from this point on, as the power cepstrum derivation requires the use

of the real logarithm routines.

To achieve the calculation of the complex logarithm, the real

logarithm and the arctangent routine are employed. As is done for the

power cepstrum formulation, the logarithm is taken of the square root

of the sum of the squares of the real and imaginary parts of the FFT

output. The results of this manipulation are the real components of

the final array. The imaginary components of the final array are

obtained by taking the arctangent of the ratio of even to odd terms of

the output FFT array. This quantity is the phase.


Phase Unwrapping

At this point, the unwrapping procedure is applied. This

procedure is implemented by a computer algorithm which performs two










tests for each of the 256 data points. These tests determine if each

data point has a value less than or greater than some epsilon value

from the preceding data point. If this value is different from the

preceding one by more than or less than this epsilon value, a quantity,

which is a modulo 2T, is added or subtracted from the phase so that the

difference is less than the epsilon value. The value of epsilon selected

depends upon the data, sampling rate, etc., and in this case a value of

T served quite effectively. If the calculated phase is bounded above

and below by the epsilon value of T, then the phase is left unchanged.

As was stated earlier, the original input which is real (a

sampled composite signal) is Fourier transformed; the real part of the

output is an even function while the imaginary part is an odd function.

The phase is calculated by the arctangent subroutine. This calculation

yields an odd function of frequency. However, it can be shown for some

waveforms that the unwrapping will cause the phase curve to lose its

oddness.

"Oddness" of the phase.--Oddness is a required feature in computing

the complex cepstrum. This necessary condition is satisfied by an algorithm

which unwraps the phase for the first (N/2 1) points, sets the point at

N/2 to zero, and then sets the points from (N/2 + 1) to N at the negative,

mirror image of the first (N/2 1) points.

Linear phase component.--After unwrapping the phase curve, it is

possible that the phase curve will consist of two parts, a linear portion

and the actual phase of the signal (shown in Figure 5). When a linear

phase component exists, it will cause the complex cepstrum to have an

appearance similar to that shown in Figure 6. The VER signals, strangely











enough, do have a linear phase component; thus, complex cepstrums

similar to Figure 6 were obtained for all of the computer runs.

The method employed to extract the linear component of the

phase is quite simple; it consists of a rotation of the first

(N/2 1) points of the unwrapped phase curve about the zero point

(zero frequency) until the value of the (N/2 l)th point is zero.

The algorithm explained in the phase unwrapping section is then

implemented without change to determine the remaining (N/2 + 1) data

points. The use of this algorithm to compute the remaining phase

insures that the oddness of that phase curve has not been lost by

removal of the linear phase component.


Inverse Fast Fourier Transform (IFFT)

The complex cepstrum is computed via the inverse fast Fourier

transform. The input array to the IFFT consists of the real part

(log magnitude) and the odd part (phase). Since these parts of the

array are even and odd functions respectively, the IFFT output is

real, i.e., there are no imaginary components. Thus, the complex

cepstrum for the real-valued composite signal used here is also a

real-valued function of time. However, it is mathematically demon-

strated in reference 7 that the complex cepstrum is not an even function

of time.


Linear Filtering


The complex cepstrum, which has been computed from the sampled

composite signal, is filtered by one of the three possible methods

discussed in the following paragraphs.










Comb Filter

Because a desired result of the complex cepstrum analysis is

the extraction of the wavelet, the comb filter is one of two types

of linear filtering that are used in this research. The peaks in the

complex cepstrum, caused by the echo, are removed by applying the

comb filter. The remainder of the complex cepstrum is unaffected.

Since the power cepstrum is calculated for only positive time, it is

not possible to determine from the power cepstrum whether the impulse

train is minimum or maximum phase. Therefore, the data points in the

complex cepstrum for both positive and negative time that correspond

to the power cepstrum peaks are set to zero.

Smoothing.--The points in the complex cepstrum that are zeroed

by the comb filtering are "smoothed" to reduce distortion in the

recovered wavelet. The smoothing is accomplished by replacing the

zeroed points with the average of the preceding and subsequent points.


Short-pass Filter

The other type of filter that is used to extract the wavelet

from the complex cepstrum of the composite signal is the short-pass

filter. This filter sets the complex cepstrum points (no 1) to N/2

and N/2 to (- no + 1) to zero. The remaining procedure is identical

to that shown in Figure 4; in this instance the block labeled linear

filter is the short-pass filter.


Long-pass Filter

Long-pass filtering is another method for the detection of

the echo epoch times. This procedure extracts from the complex










cepstrum the same information obtained from the power cepstrum, i.e.,

the echo epoch time.

This technique sets the sampled points 0 < n < (no 1) and

(- n + 1) < n < 0 of the complex cepstrum to zero and then performs

the inverse procedure of that used for determining the complex cepstrum.

Figure 4 applies to the long-pass filtering case as well as to the

short-pass filtering case.

To reiterate, long-pass filtering does not yield an estimate

of the wavelet waveform but, rather, an estimate of the echo epoch

times in the form of a train of delta functions.


Wavelet Estimation


Inverse Procedure

In order to recover an estimate of the wavelet after filtering,

the inverse procedure to that used to obtain the complex cepstrum of

the composite signal is applied. Thus, the smoothed complex cepstrum

is the real part with the imaginary part of the input FFT array set to

zero. The output of this transformation is the logarithm of the Fourier

transform of the desired wavelet; thus, the exponential of the output

is calculated to yield the Fourier transform of the wavelet. This opera-

tion is accomplished by multiplying the exponential of the real part by

the cosine of the imaginary part and placing this product as the real

part in a new array. Similarly, the imaginary part of this new array

is the product of the exponential of the real part of the old array

multiplied by the sine of the imaginary part of the old array. This new

array is now inverse Fourier transformed to give the extracted wavelet.










Results of the Two Examples

For the two examples under consideration, the epoch times are

properly detected. In each case the single echo was delayed 56 samples.

The results of both examples are presented in Figures 10 through 25.

Amplitudes less than unity.--For the first example the echo

amplitude was 0.4 times that of the wavelet. The mean square error (MSE)
-3
of the extracted wavelet is 0.107 x 10-3. For the purposes of comparison,

when a computer simulation was performed with the composite signal con-

sisting of just the wavelet itself, the MSE was 0.703 x 100. Usually

the recovered wavelet appears nearly identical to the original input

wavelet.

Echo amplitude greater than unity.--The second example is one

in which the composite signal consists of the wavelet plus one echo, the

amplitude of which is 1.8 times that of the wavelet. The procedure for

echo removal is identical to that of the other example, and the complex

cepstrum procedure recovers the echo rather than the wavelet itself.

This was predicted mathematically as is shown in Appendix C. The MSE
-2
is found to be 0.33 x 10-2 for this example. Again, the recovered echo

of the wavelet appears nearly identical to the original wavelet.










0.6oo +----------------- ------------------** -----* ---*---------------- --*----'-----***-""**-****""
I I I I I 1 I
I I I I I I 1 1 1
I I i I I I I I I
I I I I I I I I I I
I I I I I I 1 I
SI !. I I I I I I
1 I I I i I I
SI I I I I I1
1 I I II I I I I
0.669 ----------*--------------------- ----- ----------- ------ ---------- - ---------*
I I I I ;I I I 1 I I
I I I I I I I I I 1
I I I i I I K I I I
I I I I I I K I1
I I I I I I I I
I I I If I 1 I i 1 1
I I I **I 1 I
I I I [ e : I I I
I I I i I I I I

I 1 I K I I I I I
1 1 I I I I I I K
I I I > I ? I KI 1
I 1 I ** I *' ** I I I I 1
SI 4 I 1 I **.I *I I I I
I I I I K 1 ** I I I
I I I I 1* I I I
1 I I 1 I I I I I


I I I 1 I I I I
I I* I I I ** I *** *I K 1
II *1 I I
I I I I I I I* I I
I 4 I I 1- 1 *" I I
I I I I I I 1
I I I I I I I I I a I
: I I I *G f I I
I IK ; 1 I i 1 1
I I I I I I
-0.326e --0-----. r----------ma--'-l d VR ----sigl ud ---s te ---w
Abscissa is the sample number (sampling I I 001 samples per second) *
I I I I I I I ..* I
I I I I I I K I I I
II I I I i I I I 1 I I
I I II I I I I I I I
I I I I 1 1 .1 I i 1 i
I i I* i I I i 1
I *! I ; I I 1 i
I I I I K I I I I I
-07 -------*------*----------- ---------------------

1.000 20.900 4C.00 60.7CO a8O.600 L0).50 12U.OO 140.300 160.200 iO.109IO'J 20'J.,00



Figure 10. Normalized VER signal used as the wavelet
Ordinate is the normalized amplitude (1.6 millivolts peak-to-peak)
Abscissa is the sample number (sampling rate is 100 samples per second)








910.721 **---- .. .....----- ---*---- -.-* -. ...... -...- .....+*.-.--**
i I I o 1 I 1 1 I I I !
*t 1 I I 1 I I2
II I I I i :
II I t t I I I I
SI K K 1I
I I I I I I t 1
I II I K I I I
I I 1 I I 1 I I
SI II 1 I I 1
1I !* I I i II
608.921 ---------+--*-----------.--*------*-- -- .--- -+-- --------*- ------ + --* +* ------.--
1 1 : I I z
I I 1 I i I I I I I
I I* t I I I I
I I I I 0 I I I I I
I I I Ia I I 1 I I
1 I I* I I I I I
I I K I 1 I I I I I
I I I I I K I I
S! I *I ; i 1 I
307.121 +------*------- ---***- ---+ .--* -*-"*-- ---- -'- --4 '.....-,+ -.. -- *
i 1 I I I I I I I
I* I I I 1 I 1
K I I *I I I I I
i I I I I I
I I I I I K 1 1 I I ;
S *i : 1 *** IK I I i


S*I I I i l I I I I I
5.322 ****-----4-----*-+-------- r---*-*- ------ ----*-+*-n-*-i--~-----.-r-** .--,-i-,-r,-..-*--
I I i i I I I
I *I I I 7 I 1 I I *9 I
I I 1 I I *** I I I
I .* I 1 I i *t* I i* **
I 1 I I I i !i I ** ** I
I 1 I I I I I I I
SI c 1 I I i I 1 I I I
I I I I I I *. I K I
I I I K I I ; I I I
-296.478 ----------t------------------ ----- -*---------- -------------* -----,* -------- -------- *----*-.-
I i I I I I I *.*** I 1 I
I I I I i* i 4 i* I 1 I
I I I i I I I I I
I I I I I I I I i I
I I I I I I 1 I I I
I I I I I I I I I I
I I I I I I I I I
I K 1 I I i I I I
I i I I I II I i
-598.278 +---------+----*--- +--------------*-----,---4-*-------*----. ****-- -***---*---** i

1.000 26.500 52.000 77.!00 103.CCO 128. 500 354.000 L79.500 2C5.j)0C 30.500 25'.00S


Figure 11. Composite signal composed of the wavelet and a single echo
Echo delay is 0.56 seconds (56 samples)
Echo amplitude is 0.4









************* *


SI 1 I I I I I I

I I I I I I I I
I I I I I 1 I 1 I
i 0 I I I I 1 I I I
1. I I I 1 I I I 1
I I I 1 i 1 1 .1
I.. II I 1 1 1 1 1 1
I I I I I I I I
9.114 +---------+---------+---*------ ----+------~-"`"""'"~ ,"'--"**->-**-a +---w****
1 1 I I II I I

1 I I I I I I I I
I I I I I I I I I
I I I I I I I I
I I I 1 1 I I
I I I I 1 I I I
1 0 I I I I I t 1
I 1 1 I I I I I
I I I I I

I I i I I I
7.863 +.--****-+--**-----+--.-- ----.-++------- **,--******* -- -********** .

I I I I I I I I I
S1 1 1 I i
I I I I I I 1 1
I I 1 I I I
I 1 1 I I I
I 1 i I I I I I 1
SI I I
S* I I I I I

I I 1 I
1 I i 1 I I I
I I I I I
I I I I 1 1 1 I
I 1 I I I I I
I 1 I I I I I *
I I ** 1 I I ** I
I 1 I ** I I I I I *
I ** 1 It I o *f 0 *


& I
I I
I I
I 1
I I
I *

I 1
I


1 1
I *1
I I




1 1I
I I



1 i

I I
I I
1 *




I I
I I
I I
I I!




i I
I I
S I I




I 1
1 *l I
** I




* I

* I I


5.361 *---------------0-*+-*------* -- **---*----*- --- ---***-4 -- ----4-* ----4------- 4
I I ** ** *E 1 I *1 ***I ** I I
SI *I I 0* >.clfr 1 I I *" I I
S* l I I I I I

I 1 I *** 1 ** I *I |
I I I I 1o o* *I 0 I I
I i I ** I I I1 I 1
1 o I* 1I I I i I
II I 1 1 I I 1 I i
4.111 +--------+--------- ----------------- -- .---------- ----------- I---------+-------*------- "


1.000 26.500


52.000 77.5C0 103.000 178.500 154.000 179.500 205.001 230.500 256.-00


Figure 12. Log magnitude









117.341 *-----4----------*-- f---- *.-------- -. .*.*.- *....-.--**4 4**-<..."-*---..
I K I I ** 1 I I
I I 1 I 1 I : I
I I 1 I iK I I
I I I I I I I K I I
I K I I I I i I K 1
I I I I I I I I I


70.405 ------------------------------------- -----------Do, ---- ----------*--------*--------
I I I I I 1 I
I KI I I : I I I
I I I I I IK I I I K I
70.405 ------ --,----- ------- ---------_----------------------- -- ------ +

I I I I I I I I

I I I I I I I j
Si I I I I
I I 1 I I ** I I I

SI i I I I 1 I I

I ** I I I I I **1 I I
1 I1 I I I I e I I
S** I I I I i I I

-23.468 t ------------- ow --- 4 -------------------------------- --------
I** I I I K 1 I I
I ** I I* I I *
I I I I I I I I o I : o*
I-* I I I I I I K* I ** I
I I I I I I I ** *** I

so-
I KI ** II I I I *** I
SK .I ** K K I K I I
-23.468 *.------------- --------------to-------- --------------------------------
I I I ** I I I I I I. K
I I I .* I I I I I I 1
I I' I I I I I I I
I K I *K I I I I I I I
I i 1 ** I I I I I I I
[ K 1 I. K 1 K K I I
SI I I 1 I I I
SI I I K K I I K I I
I I I I I 1 I I I I
-70.405 -------------*--+---------+----- ------------- I ------+----- -.*+***+--------- -------
I I I 1 ** I I i I I
I I II K '* I I I I I I
I I I I ** I I I I I
I I I I I. I I I I I I
I I I I I I I I I I I
I I I I I ** I I I I 1 I
1 I I i K *e i i I I I I
I 1 I I I I I I I I
I I I I K **l I I I I I
-117.341 +------------------------------------ -----------*----. .-----*.--- ---------

1.000 26.500 52.000 77.500 103.000 178.500 154.000 179.500 205.000 230.500 256.000


Figure 13. Unwrapped phase curve









0.177 +----- ------- -------+--- ---- ----4----* **---*
I I I 1 I I I 1 i I I
I I I I I i I I I
1 I I I I I I I I
I I I I I I I I 1 1
SII I I I I I I
I I I I I I I I
I I I I I I 1 I I I 1
I I I I I I I I I
I I 1 1 I 1 I I I i
0.141 .*------- ----.----+-. .--4---4--- .---...--*-.-.--.-- .--.----.-- .---+--' +*
I I 1 I I I I I I I I
I I I I 1 I I II I I
I I I I I I I I I I
I I I I I I I : I I I
l I I I I I I I I I
I I I I I I I I I I I
I I I I I I I I I I I
I I I I I I ( I I I
I I I I I I I I I I
0.106 +---------+-**--------+----- ----+-----* ----+----******-- ***************** ****
I I I I I I I I I I I
0.106 .... ... .+. -- .. ... ... ....- - +. . .". ..+ ..--.-- - -- - .....

I I i I I I I I I I I
I I I 1 I I I I I I I

I I I I I I I I I I I
I I I I I I I
I I I I. I i I I I



0.071 +--****---+~-------- -*-------o- -t----**-+---------+***---**--*--***---*******---+* -*-
I I 1I I I I: I I I
I I I I i I !
I I I I I 1 I I I 1
0,071 + ..... ..........--------- --.--.-......---- ..--- --.--- .4..-----4-------+ I-I
I 1 1 1 1 1 I
I I I I I I I I I I

I I I I I I I I I
II *1 I 1 I 1 I I
I 1 I. I I i 1 1 1

I I I I I I I I I
1 I I I* 1 1 1 "I i
0.035 4------*-------------+----+---* -+--*.4.- *-a-- --a---.-----------+----- ---r *---.-
I L I I ** i ** I 1 I
I 4 I *I I I[o t If ft I
I l* .** I* 1* I I [ *' i *It I
1 I I I ** ** I *** I 1* I e I
I l 1* *l ** I *t I !* t I 4! 1 I'I I
I t*** I ;I t f0 I I o 0F 3I I
I I I ** < *** I *0 *I II *1 :
I 1* ** f I ** **. olr ** I 0 f *e1 2 I
I I *I I I I !* I I I
-0.000 ******-r *+------- ----* -------------- --a--- 4--+-------- -r--*-'----s -----*u-+---*-----+--*****

1.000 26.500 52.000 77.5C0 103.000 128.500 154.00O 179.500 205.000 230.5'0 256.000


Figure 14. Power cepstrum










5.844 -19.'734 26.544 -8.741 11.3C9 -7.013 6.236 -5. 77 4.362 -4.446
3.743 -3.303 3.387 -2.623 2.926 -2.359 2.378 -2.247 1.966 -2.095
1.748 -1.033 1.662 -1.5*8 1.592 -1.410 1.474 -1.312 1.329 -1.4
1.221 -1.154 1.142 -1. 48 1.0 19 -0.992 1.000 -0.53 0c903 *-.090
0.912 -0.900 0.755 -0.847 0.75) -0.769 0.731 -0.707 0.69' -0.*S-
0.641 -3.668 C.608 -C.2?5 0,.Lt -'..575 0.585 -0.525 0.546 -0.547
0.486 -0.558 0.426 -0.527 0.435 -0.465 0.333 -0.407 0.480 -0G434
0.393 -0.372 0C. 9 -0.351 C.374 -0.352 0.337 -0.34? 0.21; -0.638
0.249 -0.359 0.234 -0.333 0.261 -0.300 0.184 -0.23, 0.210 -0.240
0.229 -0.193 0.?48 --0.224 0.221 -0.710 0.124 -0.253 0.125 -0.265
0.1D7 -0.159 O0.1i -0.127 0.153 -0.109 0.105 -0.1 0.072 -0.149
0.079 -0.130 0.05 -0.127 0.0(0 -0.060 0.0' 3 -0.0/2 0.023 -2.117
0.016 -O.062 -0.027 -0,059 -1).C28 -0.063 -0.028 -0.041 -0.031 c.CC9
-0.o1 0.348 -0.041 0.019 0.001 0.c0O -0.040 0.047 -0.052 -0.018
-0.375 0.041 -0.126 C.060 -0.106 0.092 -0.113 0.126 -0.121 0.13C
-0.106 0.178 -0.136 0.171 -0.112 0.117 -0.161 0.149 -0.176 0.189
-0.200 0.176 -0.208 0.122 -0.?63 0.173 --0.303 0.226 -0.263 0. t61
-0.228 0.315 -0. E2 0.333 -0.290 0.295 -0.340 0.310 -0.346 0.275
-0-.72 0.3.2 -r.'lO 0.360 -(,.1 0C.356 -0.425 0.378 -0.'1? .4. 1
-0.49 .L -0.468 0.4 ,5 -0..43 0.f 9. -0.4 3 0.'50 -0.507 0.514
-0.620 0.545 -0.631 0.576 -0.t 2 0.643 -0.7U00 3..70 -0.717 C.S59
-0.779 C.14j -3. I 2 0.313 -0.634 0.892 -0.818 0.904 -3.958 0.906
-1.046 0.929 -1.L26 1.041 -1.159 1.159 ->.200 1.?79 -1.283 1.324
-1.418 1.439 -1.5S6 1.538 -1,662 1.7d7 -1.818 2,054 -2.011 2.247
-2.393 2.362 -2.926 2.596 *-3..7. 3.260 -3.825 4.431 -4.393 7.762
-6.132 7.208 -10.9:6 9.327 --21.945 21.195


Figure 15. Complex cepstrum data








3.743 .-.---o-* ...--. -----*...*....---..-:--..+.------ -.----* ------- --------.-----..--
I I I I I I
I I I I I I 1 K K K I
S I I I I I I I
SI K II I I I
*I I I I I I i I
I I I I ( I I I I
I I I I K K I I I I
K I I 1 I I I I
I I I I I I I I I I
2.320 +--*--- ------- ------- -- -.-..------- ,----- ----. --------+---------o*--_
I I K I K I K I I I I
I *I I 1 i I I e I
I I I I I I I i I
*I I L I K I II II *
*I I K I I I I [ I* I
I I I I i i I* I
I I* I I K i i I o I
KI : I K K I I I
I I I 1 I i I I I i
0.896 *---+--.---.---*----+---.---.----------- 4-----------------.--_------. ~ --))o------...
KI I **. I I I KI I I
I IK K I I K I **+'' I I
I I I ** t I I I *** I I I
S I I **** I I ** ***** I I ;
I I I I** ***#* ** K *** **** i I 1 I
***.e I I 1 *1 ******I. ****. I I I I ****
I- I I 1 *****.* ****** i I I

I I '.I *.. I I I I
-0.527 ------------------*-.**--- ----------+------+-------*-------------------------------*------------+---,.*.--,---_--..---
I I '*' I I I K* .'* I I
I I **I I I I I I I
I 1 *. I K I II I K 1 I
SI* I I I I I I
I 1* I I I I I I I st I
I I I I I I I I I
I I I I I I I I I
I I i I I I I 1 I
I I I I i I I
-1.950 +-----*-------- +-------- -+------+ +-_--..- -*--- -- ----- ..--......<
I I I I I I I I I
I I I I i i I I
I I I K I I I K I
I I I I I I KI K
I I I K I K K I I


I I I I I I I
1 I I KI I I K .
-1.3T3 .---*----------------*-------------------------------+--- -------------+*----+------.+*, -****-

1.000 26.500 52.000 77.500 103.000 128.500 15'.i00 179.500 205.0')0 230.500 256.000


Figure 16. Complex cepstrum









909.615 +-------------*------------ ** ****"" *
I I I I I I I I I
II I I I I I I
I I I I I I I I I I I
t 1 I 1 I 1 I I I I .
I I I I I I 1 I I I
I I I i 1 I 1 I 1 i i
I I I I I I r I I I
I I I I I I I I I I I
I I 1 I I 4 I I I I I
609.348 +----- ----+------ -------- ------- 4-.. ---- ---4-*. .* -- .- .*-, 4..--- +.
I II I I I I I I I I
I I I I 1 I I I 1I
I .I a I I T I I I I
I I I I I I I I I I
I 1 I I I I I
II I 1* I i I I I I I
I I* I I I I I I I
I I I *I I I I I I
SI I I I I I I
309.080 +---. .--- +--.------. ----- .-.. ----- ------*.. -------- .------ --- -"* ** ** **------. .* .....-
I 1I I *I I I I I II I
I I. I I I I I i i I I
I I I I I 1 I I 1 I
I I I I e **I 1 [ I I I
I I I I I I I I
I 1 I I I** I iI I
I I I I I I I I I
1 I I I* K I I I I
I .e 1 I *I *. I I 1 I i
8.813 -- --* -+---- ----9-- -- 9 ----- 4-- weq9.** 9**O***
I I I I I- I I I I
I I I I I I I I
I I I I I i I i 1 I !
I I I I **I I I I I

I I I i **. 1 I I I
I I II I* I I I
I I I I I I 1 I
I I 1 I 1 *8 I I I
-291.455 .-------.-------- -- ..*---- -- --. .------ -... ----*-* ---* *- ---.---.."" -* --.~*
I I I I I I I .I I I 1
1 I I I I K* I 1 I
I I I I I I IK I I I
II I I I I I ** I I I I
SI I I I I I I I
I I I I I 1 I I I I I
I I I I I I I I I I
I I I I I I I
I I I I I I I i
-591.723 .---------+----*---.--.---------^----- ---------*.--+*...--...-4.

1.000 26.500 52.000 77.500 103.000 128.500 154.000 179.500 205.000 230.5"0 256.000


Figure 17. Recovered wavelet









1.000 +---------.------.--------- **---- -- -+-- -- >* -..*----- -- --.--- --- ----...-------
I I I I I I I I I
i I. I I I I K I t
iI I 1 I I I
II I I* i I I I I I
SI I I* I I 1 I
I I 1 i I I I
K I I I I I I I I
IK K[ I I I I IK





0.339 -------------------------.. --- -.. -.. -o -.....- --- --- -.. ---- -. .- ----- .. .. .- ,
0.670 -----------------------------------------4------------4-----------.---------------*--
II I I I I I 1 I 1
I I I II. I I I I
SI ,' I 1 1I I i i ; t
I I I I I I I I
I 1 I I I I I K I I I
i I *,*1 I I I I
I *.I* I I I I I I
I I I I I I I
I I I 1 i 1 I I
0.339 ----------------*------ --*-~---.------------------ --------- .--- ----------- ------
K I I I I I I I i
I! 2 I I I I I I
I I u I R o I *. I I I I I I
SI I K ** e ** K* I I
I I I I I ** *IK I I K I
I K I I I ** I I I
II I I i **K I I I I

*** I I I l" i -
0.009 *---*---+-*---------------------------- -----*---*-*-.---4.- ***-**---*--->*-----4---- + n
I ** I* I I I K K K !
1 I I I I I *I i I I
I I I I i I I I I
** I I I *I I
I I I K I I l ** *I K !
II i K 1 i I K *.** I I I
I I I ** I I I
I I I I I I K.*. I I K
I I 1 K I K* *. *1
*0.321 .------------------------------.----+-----..-- ---..-------+---****--
I I I I I I I I I I
I I IK I I I I *0* I
II KI I K K I : e I K
I K* I I I I I I 1
I I t I I I I
I I I I I I I I I I i
I I I I I I I
I I I I I 1 1 I
iI I I I I I ?I I
-0.652 +-----------+------*-------------------+-------+-----+-------------+---------- ----+

1.000 20.900 40.800 60.700 80.600 100.500 120.400 140.300 160.200 130.100 C0.000
ERR 0.1069064E-03


Figure 18. Recovered wavelet expanded
ERR is the mean square error





1679.210B --------------- -----*------------ ---'------*-*-- *--- -------- ''-----
I 11 I I1 1 I
II I I I I I I I I
I I I I I I i I
II I o 1 I I g I
I I I I i 1
I 1 I I I I I I
1 I I I I I
I I I I Ii I. I
1157.338 +----- -.-- -- -.. ---. + ---~- + ------ -- --"'-' ""4 --.
I 1 I I I I
I I 1
I 1 I 1 1 I I 1 I
I I I I I I I I I

I I I** I I I I I I I I
I I 1 1i I I I I i

I I I ** I t1 I I I I i


1 1 1 77.I5,0 IO I 1 1 2 2
I I II I I I I i I
635.4594 .---- .- -- *--- ******- .--.-. ** ----+-----.-** **- .- *--- .^.'- .*--.--
I I 1 I* 1 I I I I I
i I I .* I I I I I I !
I I 1* 1 1 1I I
I I I I I I I I I I I
I I *I I I I I I
II 1i 1 I 1 1 I I I I
i I I I I I
I I** I I *I I

113.580 +*-*-************+**--- *+"* 4--"- 4-
I ** *I I I I I I
11[3 80 ...." ..* .... --- .. ... .... .... ..... --- --- .. -- .. --- -...... ---....-- -- .... --

*. *I I 1 I I I 9
I ** *! I I I I I 9 I I I
I **** I I i *I I I 11! I I I
I I I I I I I *JI 1 I I
I I I I I I 1 1 I I I
I I I I I ** i e I I
I I I *I I I 1 a tI *** **.* I I
1 1 I 1 t '* t a 1 1
-409.300 ------- -* --4 .---4 -.---4 ---- *----. *9---- -."-9----- *--4 '-* *.* --**
1 I I 1 I II 1 l I
I I I I I I I **

I I I I 1 I ** I
I I I 1 1 1 I I I *1 i
I I I I I I ** 1
I II I 1 1 I I 1 I I I
I I 1 I i I I I
I I I I 91 I I I 1 I
-930.179 .-------.+-----.-------- .--*--.----..----4----*+*. *-----.*.----

1.003 26.500 52.000 77.5 C 103.000 128.500 154.300 179.500 205.000 230.500 s56.000


Figure 19. Composite signal composed of the wavelet and a single echo
Echo delay is 0.56 seconds
Echo amplitude is 1.8







11.027 .-------- +**. ----- --------+.---. .**..-----....-----... ,- ......, .--*.----
I I I 1 I I I I
I 1 1 I I 7 I I I I I
I I I I I I 1 I I
I I I I 1 I I I I I
I I I I I I I I i I
1 I1 I I I I I I ** I
I I I I I I I I1 *
i 1 I I I 1
I IK I I K I I 1 I I **
9.712 +-----. -*- .--.---. .-- ---------+--..--. **-- ----*--- ---,--------- -**-***--- ------..
I I I I I I I I I
SII I I I I I I I
I I I I I I I I I I
I I I I I I I I I
S* ** I I I I I I I
I ** I I I I I I I I I I
I I I I I I I I I
i I I I 1 I 1 1 I 0 I
*l 1 I I I I I I I
8.397 +--------- .. ------- ----.------4-------------- + ---.--- --- ----.. ..--- "...**-*
I I I I I I I I I 1
I I 1 1 I I I I I
1 1 I I I I I I *

I I 1 i I I I K I I I
I *I I I I I I I
II I I I 1 I I
I 1* I I I I' I
I e I I I 1 I I lK I I
I I I I I 1 1I I
7.082 +---- --* ..*------.-.----- ..---*----. -----* -" "---- .--.-- ..-
I I I I I I I I *K*
I I I I I I I
I I ** I I I II I
I I* I I *I !
I 0 I I ** 0 *I I
I* ** I ** I* I 0 I 00 K I I
I I ** I 1 *I* I *1. 0
I I I *I I ** I* *I I I
I ** I ** I I *I e [** I
5.767 --..---- ---.. -------+ ----------- -.--*. -*--- -*-*-- **---- *.---*-----------***- **-----*
I I I *I o* ** I I 1 I i I
I I I *I ** I i K I* I I
I I I I I I I I I I
I I I* I *< I 1* ** *I * I t ii I I
I I I I 0 I o I I I i I
I I I I I I I I I I I
1 I I *I I Ii I I I
I I I I 1* i *1 I 1 I I
I I 1 I I I I I I
*.452 ,.---..---4-+---* .---. .-- ---- 0-- ----- +------ --- ----4 ...-.*- ...*.----+*---* -----------....

1.000 26.5C00 52.000 77.500 103.000 128.300 154 000 179.5P0 205.000 230.500 256.000


Figure 20. Log magnitude










10.105 +*- .** .*-**.---.--.- ..... --- -------.-~ --- +-*4- **.. .-. **---.--**.*-1*..----~L ******
I *I I I I I I I
I I I I I 1 I I I I
I I I I I I I I I I I
I I I 1 I I I I I I I
I I I I I I I I I
I I I I I I I I I I
I 1.. I 1 1 I I I I I I
I ... ** I I I I I I I
I I *** I. I I I I i
6.063 +--------+ -- *-***-**- ----- --------- ,----- -- ---- -------------------
I I 1 I I I I I I
I I **** *1 ** I I I I I I I
I I I *** I I I I I i
I I I I .* *.* i I I I I I
I I I I I,* *** *I1 I I I I
I I I I .. I I I I i I
I I I I I ** I l I I I
S 1 I I I I II tI I I
2.021 +--------- -+ ------------------ ------.-+- -----* --- -- ---* ---***-
I I I I I I I I I I
I I I I II I I I
I I I I I i 1 I I I I
I* 1 I i I I I I I I I
I I I I I I 1 *
I I I I I I I *
SI I I I I I I I I
I I I I I I I I I 1
I I I I I I I I I I .
-2.02l ---------- -------+------- ---.------+----- +----------*- .-- ---**---+-- ***-+
I I I I I I I I I
I I I i I I.* I; I I
I I I I I I ** I I 1 I I
I I I I I*-. ** ** I I I I
; I I I I I *a *. I I I I
I I I I I .** .. 1 I. 1 i I
I. I I I I I i I ** I I
I I I 7 I I I ** ... ** I I
I 1 1 i 7 I ** **** a 1 1
S-6063 .---------+--- ----+----- --------4--------+--------- --*----- -+---*-* *a. *<( *---
I I I I I I *I r. *1** I
I I I I I I I I I *u* e < ** I
I I I I I I I I I *I *
I 1 I I I 1 I 1 1 I
I I I I I I I I I I
I I I I I I I I I 1
1 I I I I I I I I I
I I I I I I I I I
I I I I I I I I
-10.105 +------- ---+-------+--- ---4------+---.--+--- ..--- .-....-*---.----^---+--r------- -***. -

1.000 26.500 52.000 7T.500 103.000 125.500 154.000 1Th9.00 205.000 230.500 256.000


Figure 21. Unwrapped phase curve







0.252 +--------+.--------+ *.---4 ,**- 4 4 4 4 +..*..
SI I I I I I I I
I I I I I I
I I I I 1 I K I 1
I I I I iK I I I I
I I I I I I I I I
I I I I I I I I I
I I I I I I I I I I I
I I I II I I I
I I I I I I I
0.201 .-----------------.---------4-- ----- -------*- "- ** -... -4".-- -
I I I I I I I I I
I I I I I I I
I I I I I I I I
I I I I I ( I I I I
iI I I I 1 I I I

I I I I I I 1 I
SI I I I I I I I
I I I I I 1 I I
0.151 --------+--------------------- --- -------**.- ---+**----+--- .**
SI I I I I I I I
I I I I I 1 I I

I I I I I I 1 I I I I
I I I It 1 I I I
I I I I I K I I I I
I I I I I I 1 1 I I
II I I I I I
II I ; II I K I
0.101 +****---+-----*-*****---* -----********-----""*************-""" *****
I I I I I I I I
1 I I I I I I I I I I
I K I I I I i I (
II I 1 I I I I I
I I I I 1 I I I 1 I
I I I I I I I I I
II I I II I I I I I
I K I I K K I I I I
K K 1 l K K 1 1 I I

II I I I K I I i I
0.050 +---------*-------- ---------+*--------+------**------* -+------*----- --'.~** *--
I KI I I I K I K I
I I I IK I I i I I I
i I I * I * l *. I I
I I I* .I.* I ic. K I i
I ** I* 1 I ** I. 40 4 i o 1
I I* *. I * on ** 4* i e *o I ft I IK
I 1. 4*** .. K I K 4.1 I I 1* I
S ** ** I ** I *I* I c* I O **I 4elo :
K *!* 1 .* ** **44 I 4* ** [K I ** 04 I
-0.000 ***--***+-- ---- ---.--------+.-------......---+---.-**--.------ -

1.000 26.500 52.000 77.500 103.000 128.500 154.000 179.500 205.000 230.500 236.000


Figure 22. Power cepstrum










0.231 ------------4---- -------------------------c--------------+- ------- ---------------
SI 1 I I I I
I I I I I I I I
i I I I I I I
SI I I I I I I
II I I I I I
K I I I I I I I
I I [* I I K I I I I
1 1 I 1
I I I I i I I t I i L
0.131 --------- ------------ -----------------+------- t----- ------- --------
I i ** I I I I I I I I
I I *I I I I I I I I
I I I I I I I
I K KI I K I I I 1 I
I I I K i I I I I
K I I1 I I I I I
I I* I I I I K I I I
S*I K I I I K e K I
I I I ** I
0.030 --- ------ ------------- .--....-------- .- -,.. --, -., .
0.030 ----------------------------------*------ -------- ----------------------------- -
II ** I I I
I I ** I I I *** ** I KK I
** I ** *K *I I I *** K*
I I I I I *K aI * I I
I I K I ** *I *tI ***. i K *. ( I a I
I i I* **I ** l ** i a :
II I I I l I
I I I * I I ( ** I !. K
I I I IK I I I ** I !o I I
-0.070 +----------*-------- ------- -------+---.---- +*,*---- -_ ---,-------,-----.-oo..e-,,.--4-,.-------
I I ** K K K I I I I I
I i II ** I I I I I
I *I I I I I I I I
I i K I K I iK K
I II I I I I
I I I I I *I I I
I I i K I I I I *IK I I
I I I I II 1 I I I
I i I I KI I t
-0.IT1 +-----.------- --.----- ------------------ ..-----.------------ ------------- ---------.....--,.
1 I II ; K I IK !i
I I I 1 I I I 1
I I I K I I 1 I I
I i I IK I I I I I
I I I I K I i I I
I I I I I :I i
II I I i I I 1 1 I
I I I I I I I I I t
I I I I I I I K ? I
-0.272 +---- *------. ---- --+--.,-- ...... ----+ ..------ *------ ..---.----.....---,-.....

1.000 26.500 52.000 77.5C0 103.000 1,8.500 15*.000 179.500 205.000 230.500 256.000


Figure 23. Complex cepstrum











1649.988 ------+---------*------ --- --+-**------ --+----- -----*--4..-. --.*. --
I I I I I I I I I I
I I I I I I I I I I
I I I I I I I I I I I
I I I I I I I I I I I
I I I I I I I I I
I I I I I I I !
I I 1 I I I I I 1 I
I I I I I I I I I I I
I I I I I I i I I I I
1111.368 .---------+---------.--------4------f--+4.-.-----+--4------.-----.,- -- ----4----,4-.--.
I i I i II I i I I
I I I I I I I I I I I
I I 1 1 1 *1 1 I 1 I 1
I I I I I I I I I
I I I I I I I I I I I
I I I 1 I I I I I I
I I I I I I i I I
I I 1* I I I
I I I I I I 1 I I i
572.748 +--**- *-*----4-4** ***4* -0 --- -**-*-+**-*----********-+** 4--**- *-* -+
I I I I I I I I I I
I 1 I I 1 I 1 I I
I I I I t I I I I
I I I I I 1 I I
I I I I I ** I I I ]
iI I I ** Ir *9 i I I I
I I l I I I I *I I 1 I
I I I 11 I i .I I I I
I I I I I l I 1 1n
34.128 +******0 0 .----- -----*---* ------- t---------+-*** -****- **---****-*-** .i
1** l** I*** I .... 1* I ** I I
I I I I i i 1I i I
I I I i ** .1 I i I
I I I *I I I l I* ** I I
I I I *1 I i ** I I
1 I I *I I I I 1 *I ** 1 I
I I I I I I I I h I I
I I I I I I I [ ** I I I
I I I I *I I I 1 1*** I
~504.492 ------+---------+--------+---------+--------- ------ +-----4 ---- ----- *****.-****
I I I I I I I I I ** 1 I
I I I 1 I I I I i *I *. I
I I I I I I I I ***I I
I I I I I 1 I 1 1 I
I I I I I I I 1 I I
I I I I I I I I I I i
I 1 1 I *1 I I I I
I I I I I i I 1 I I I
I I I I I I I I I I 1
******** +--- -+-------t------+---*--+------------ ----------*----**** ***-** --*--- -4^+

1.000 26.500 52.000 77.500 103.000 18.500 154 000 179.500 205.000 230.500 256.000


Figure 24. Recovered echo










1.000 --------------------***-------------------------- ---*- ***----------* ----*
I I I I I I I I I 1 I
I I I I I I I t
SI I I I 1 I 1
I I I I I 1 I
I I I 1 I I I I I I
I I I I I I I I
I I K I ( I I
I I I I I I I I I
I I I I* I I I I
0.673 ----------------- --------- ------- ------ -- --+--------------.---+----------.**-
I I I I I I I I I I I
I I I I I I I I I I I
I I I I. I 8 I I I I I
I I I I I I I 1 I
1 I I I I II 1 Ii
I I I I I- I I I I
I *I K I 1' 1 I I I I
I I I I I I I I

0.346 --------- +--------+-- 4-----+----..4----
I I I I I I I I I I
I I I I I I I 1 I i I
I i *. I I I I I I I I I
I I I I 1 I I I K I (
I I I I n,. I I I I
I I I I 1 4* ** 1 1 I I
I i I I I I I I I I
*I I 1 I 1 1 I I I
I I I I I I 4 I I I I
0.020 ----------+-- ------------------------------ -*---rr-- *-4-------s-- 4-r---- *-4
1**** 1* 1 I I I I I I|
I 14 1 I I I 1* 1 l I I I
I i I I 1 I 4 I I I
I 4**. K I 1 i I ** *I ** **r I I
I ***I I I I i I 4 *! I I
I I I I I I .1 0 I 0I I
SI I I I I I 1* I I
SI I I 1 I I I I1 I
I I I I I 1 I **+ I
-0.307 ---------------------------4------------------------------ -------
I I I I I I 1 I I e I I
1 I I I i I I I I
I I I I I I I 1
I I I 1 1 I I I I I
I 1 I I I I I I
I I I I i K I I I I
I I I I I I I I I I
I I I I K I I I I I
I i I I t
-0.63-4 ---------+--------**---------.----------- ----------------- .-- --------.----,----*-----.----- +

1.000 20.9C0 40.BOO t0.7CO 80.e0. 1lC.530 120.400 140.930 160.200 163.1OC 20.O000
ERR 0.32q91304E-02


Figure 25. Recovered echo expanded















CHAPTER IV

EXPERIMENTAL RESULTS AND INTERPRETATION


Experiments using the cepstrum techniques were undertaken with

two main objectives in mind. The first objective was to duplicate the

results which appear in the following references: Bogert et al. (2),

Schafer (6), Senmoto (7), and Halpeny (8); the second objective was to

extend these experiments to obtain results for other situations. The

results of these experiments and their interpretation are presented in

this chapter.


Noiseless Case


Here the cepstrum techniques are applied to the noiseless com-

posite signal that is formed from the wavelet and its echoes. The

expected results from each experiment are (1) the detection of the echo

epoch times and (2) the extraction of the wavelet or echo from the

composite signal.


The Power Cepstrum

Single echo.--The computer results for two of the single echo

cases are presented in Chapter III and selected plots for the remainder

of the cases are presented in Appendix F. The results contained in the

individual plots are summarized in Figure 26.

For the single echo case, no difficulty is encountered in detecting

the echo epochs whether the amplitude of the echo is less than or greater










NORMALIZED
POWER CEPSTRUM
PEAK MAGNITUDE
1.00





0.75





0.50





0.25


-t .---- -. to = 0.41 sec.

/ / \N

//

//
// / \.

to = 0.56 sec.- \

/;/
//
//

/
/
/
/


MAXIMUM AMPLITUDE = 0.418


0 25 150 175 '100 25 50
ECHO AMPLITUDE IN % OF THE WAVELET AMPLITUDE

Figure 26. Effect of the amplitude and epoch time of a single echo on the observed
magnitude of the power cepstrum peak










than unity. The corresponding amplitudes of the delta functions in

the power cepstrum domain are within the expected limits of computa-

tional error.

Interpretation of results.--An important point is demonstrated

in Figure 26 which shows the ambiguity encountered if an attempt is

made to estimate the amplitude of the original echo. As an example,

if an echo is detected and the value of the peak in the power cepstrum

is the normalized value 0.7, it is not possible to tell whether the

echo's amplitude is 0.575 or 1.65 times that of the wavelet. These

normalized values are taken from Figure 26 in which the epoch time

equals 0.41 seconds. Thus for the single echo case it is possible to

estimate the echo amplitude with the possible exception of this

ambiguity.

Multiple echoes.--Several computer plots for the multiple echo

case are included in Appendix F, and the main features of these plots

are summarized in Figures 27 and 28 which show that serious complica-

tions arise in the multiple echo case if the echoes are not integer

multiples of one another, i.e., if the echo delays are able to assume

any value. An example of the problem encountered can be seen for the

two-echo composite signal in Figure 27 where it is shown that ambiguous

peaks in the power cepstrum begin to be encountered when the sum of the

amplitudes of the two echoes ranges from approximately 0.8 to 1.0. A

similar condition exists when their amplitudes sum to be values in a

corresponding range above 1.0. The peaks that are first encountered

are located in the power cepstrum at the time difference (tl to) of

the first two echo epochs. A detailed explanation for this phenomenon

is given in Appendix B.












% OF ECHO
MAGNITUDE
RELATIVE
TO A1


50-





40-


D---IFFERENCE ECHO, AD
/ ---1st ECHO, Al
2nd ECHO, A
2


/
/
/
/


SECOND ECHO
MAGNITUDE


/
/
/


DIFFERENCE ECHO
MAGNITUDE


- I'~J


3 50 1 00
% OF ECHO MAGNITUDE RELATIVE TO


150
WAVELET AMPLITUDE


Figure 27. Magnitudes of the power cepstrum peaks as a function of the summation
of the echo amplitudes for the two echo case


307





20-





10-


200


I I









100 -


% MAGNITUDE OF
SECOND ECHO
RELATIVE TO
80 -
FIRST ECHO





60 -





40





20


N
N


'Note this point is
--- -- not zero.
i I i ..i i


80 100
SEPARATION IN % OF LENGTH OF WAVELET


Figure 28.


Magnitude of the power cepstrum peak of the second echo as a function of
the time difference between the two echoes


20


40


b


t


S '60










The other important result for the two-echo case is summarized

in Figure 28; this figure shows that the peak in the power cepstrum

caused by the second echo is affected by the amount of separation

between the two echoes in the composite signal. (In Figure 28 the

cepstrum peak of the second echo is not zero when the time difference

is exactly the width of the wavelet itself. It merely appears so

because of the choice of the scale.) There is no difficulty, mathe-

matically or empirically, in letting this time difference be greater

than the total time duration of the wavelet.

Interpretation of results.--Initially, the two most important

questions concerning multiple echoes that arose from this study were:

1. Which log series expansion could be used to approximate

the portion of the power spectrum containing the echo's

epoch times,

and

2. Would the power cepstrum approach yield valid results for

composite signals consisting of multiple echoes?


As shown in Appendix B, the logarithmic function of interest is


log (1 + a + 2a cos t 0)


From the mathematical derivations given in Appendix B, it can be seen

that the appropriate series expansion to be applied is




2 2 i
For the series to converge, a must satisfy ja0 + 2a cos t0W1 < 1.










For composite signals containing two or more echoes, the

mathematical derivations indicate the possibility of obtaining delta

functions in the power cepstrum at the sum and difference times of

the epochs of the echoes. These power cepstrum peaks may have

amplitudes equal to or greater than those peaks located at the epoch

times due to the echoes themselves. Because the series is infinite,

it is extremely difficult to predict mathematically if, and where,

these erroneous peaks will appear, assuming that the initial assumption

is made that the epochs and the number of echoes that form the composite

signal are unknown quantities.

However, the situation of a wavelet and two echoes was thoroughly

investigated experimentally and a threshold phenomenon was found to occur,

i.e., valid epoch detection became impossible at a. = 0.8 as shown in
i

Figure 27. In Appendix B the first three terms of the series approxi-

mating the logarithm for the two-echo case are calculated. Through the

use of these three terms, it is possible to speculate on the effect that

the sum and difference terms will have.

In conclusion, several comments on this investigation of the

properties of the power cepstrum are listed:

1. The infinite series expansion log (1 + x) = x x /2 + "'

mathematically explains all the various possibilities (delays,

magnitudes, etc.) of single and multiple echoes.

2. In the analysis of composite signals consisting of multiple

echoes, erroneous cepstrum peaks are observed at the sum and

difference times of the epochs of these echoes. If the total

number of echoes is small, an algorithm might be implemented











to determine if an observed peak is the sum or difference

of the others, and, if it is, then this peak could be

omitted. The implementation would consist of having the

computer check all the sums and differences of the detected

peaks to determine if any correspond to one of the other

detected peaks. In the event that a sum or difference

of any two of the detected peaks corresponds to a third

detected peak, this third detected peak could be eliminated

as one which is not caused by a true echo.

3. If the multiple echoes are small enough in amplitude (or,

correspondingly, sufficiently greater than unity) so that

the problem given in item 2 above does not occur, it appears

that there is no limit to the number of echoes that can be

detected. In a particular experiment four were detected

easily; each had an amplitude of 0.2.


The Complex Cepstrum

Single echo.--Several simulations are performed for various echo

amplitudes less than and greater than unity. The results of two of these

experiments were shown in Chapter III. Results from additional experi-

ments with and without a linear phase component are contained in Appendix

F.

Since the VER signal (the wavelet signal) has a linear component

inherent in its phase information, and is nonminimum phase, the complex

cepstrums for the single echo with amplitudes less than and greater than

unity appeared nearly identical. Since the composite signal had a linear

phase component, the peaks in the complex cepstrum of the composite signal










reflecting the echo were hardly visible. The removal of this linear

phase component causes a drastic change in the appearance of the

complex cepstrum; yet the peaks due to the echo are only slightly more

evident than before.

Interpretation of results.--The computer simulation results

demonstrate the need for the use of the power cepstrum technique to

determine the echo epochs. If there is no prior knowledge of the echo

epoch, it is extremely difficult to locate the desired peaks in the

complex cepstrum. These peaks must be smoothed for satisfactory

wavelet extraction. For the data (the VER signal) used, locating the

echo peaks in the complex cepstrum is difficult with or without the

presence of a linear phase component. As explained in the next section,

the VER signal was found to be nonminimum phase, thereby masking the

peaks for echo amplitudes both less than and greater than unity.

Multiple echo.--The appearance of the plotted complex cepstrum

data for the multiple echo case is nearly identical to that of the

single echo case with and without the linear phase component.


Minimum-Maximum Phase Impulse Trains

Minimum and maximum phase time series were computer simulated

by making the echo amplitudes less than and greater than unity, respec-

tively. For multiple echoes with amplitudes less than unity, the zeros

of the appropriate z-transform are located inside the unit circle which

is, by definition, a minimum phase condition. Correspondingly, if the

echo amplitudes are greater than unity the zeros are all located outside

the unit circle in the z-domain which is defined as a maximum phase

condition.











The computer results show that the complex cepstrum of the

wavelet was nonzero for negative and positive time; thus it is

classified as a nonminimum phase waveform.

Interpretation of the results.--When the added echo amplitudes

are less than unity (the minimum phase case), peaks are located in the

complex cepstrum for positive time only. When the added echo magni-

tudes are greater than unity (the maximum phase case), peaks are

located in the complex cepstrum for negative time only. Since the wave-

let (VER signal) was found to be nonminimum phase, the algorithm that

calculates the complex cepstrum by using the log magnitude and phase

data is the only one that can be used to give a complex cepstrum without

aliasing. As stated in Chapter II, the complex cepstrum can be computed

from the log magnitude via the Hilbert transform for nonminimum phase

situations, but aliasing of the complex cepstrum is the penalty paid.


Linear Filtering

Two types of linear filters were used to remove the peaks caused

by the echoes in the complex cepstrum. These filters were the comb and

the short-pass.

Comb filtering.--The peaks due to the single echo are satisfac-

torily removed from the complex cepstrum by means of the comb filter.

For the special multiple echo case in which the echoes are integer

multiples of one another, the comb filter is the same as for the one-echo

case, but for multiple echoes with random epoch times the filter becomes

very complex due to the additional peaks situated at the sum and dif-

ference times of the epochs of the various echoes.











Interpretation of results.--The only difference in the comb

filter between the single and multiple echo cases is the additional

complexity due to the number of peaks that are detected in the power

cepstrum. As explained previously, the number of detected peaks may

be considerably larger than the number of echoes contained in the

composite signal to be decomposed. Epoch detection has been found

to be as nearly dependent on the amplitude of the echoes as it is on

their number.

Short-pass filtering.--The short-pass filter removes the echo

peaks from the complex cepstrum as does the comb filter.

After the epoch time of the echo has been determined by the

power cepstrum, the short-pass filter can be implemented. Results

for several experiments are given in Appendix F. These results show

that the complex cepstrum is distorted more by this filter than by

the comb filter.

The implementation of the short-pass filter for the multiple

echo case is identical to that for the single echo case. For a certain

range of multiple echo amplitudes, discussed in the power cepstrum

section, the first peak to appear is that at the difference of the

epoch times and will accordingly be the first data point to have its

value zeroed. For example, in the two-echo case the time of this peak

is less than the shorter of the two epochs.

Interpretation of results.--The amount of distortion depends

on the epoch time of the echo; the smaller the epoch time, the greater

the distortion. Similarly, the faster the complex cepstrum of the

wavelet tends to zero, the smaller the distortion caused by the short-

pass filter. The amount of distortion is at least as severe for the










multiple echo case as it was for the single echo case. In fact it

is usually impractical to apply this filter for multiple echoes

because of the large amount of distortion introduced by it.

As a summary, the two linear filters that were used in the

removal of the echo peaks from the complex cepstrum are compared.

The advantages of the comb filter are that

1. It introduces the least distortion in the recovery of

the wavelet for the noise-free case and for most noisy

cases.

2. It causes the least distortion for multiple echo cases

as well. This is due to the fact that a difference

peak can occur at a time that is less than any of the

epoch times of the echoes. Thus, the short-pass filter

will make the complex cepstrum take on zero values at

the erroneous epoch time which, in turn, introduces

considerable distortion in the recovered wavelet.

Thus, the comb filter can be applied to all cases. For very small

epoch times, the amount of distortion introduced is theoretically

the same as for the large echo epoch cases.

The disadvantage of the comb filter is that for multiple

echoes the short-pass filter is much simpler to apply since the only

information required is the time of the first echo peak in the complex

cepstrum.


Impulse Train Extraction

Long-pass filtering.--Long-pass filtering was successfully

performed on a composite signal which consisted of the wavelet and a










single echo. The signal obtained from the wavelet extraction proce-

dure was the impulse train rather than the wavelet; the determination

of this train is also the purpose of the power cepstrum technique.

Interpretation of results.--While multiple peaks can be

detected by this method, this does not imply that these peaks are

due to echoes at these times, as was discussed previously. Another

important consideration is that the farther the impulse train is

removed from the origin, the more easily the impulse train can be

detected and recovered. Experiments were conducted using extremely

brief echo delays. Even so, the recovered wavelet train was very easy

to identify (see Appendix F). However, the power cepstrum procedure

was still superior.


Wavelet Extraction
-l
This section deals with the inverse procedure (the N- block

in Figure 2). Here the input is the filtered and smoothed complex

cepstrum and the output is the estimated wavelet. The wavelet extrac-

tion performance is measured in terms of the MSE of the recovered

wavelet.

Single echo.--The results for the single echo case without noise

are excellent for echoes with amplitudes less than and greater than the

amplitudes of the reference wavelet. However, there is a definite

degradation in the recovered wavelet as the amplitude of the echo

approaches that of the wavelet. This degradation in performance is

demonstrated by the MSE values displayed in Figure 29.

Multiple echoes.--The results obtained for wavelet extraction

from a composite signal composed of several echoes were also excellent.








10-1-


MSE








1-2












10-3
-3
10 -


I


/
/

/
I
I
I
I
I

I
I


I

I
I


4


I


I
I
I
I


10 20 140


Figure 29.


I I I I I


60 80 100 120 140 160
% AMPLITUDE OF


180
WAVELET


200


Mean square error (MSE) as-a function of the echo
magnitude (epoch time of single echo is 0.56 seconds)


N.
--4










For example, for the situation of four echoes with amplitudes of 0.2,

the four epoch times were detected by the power cepstrum; the cor-

responding points in the complex cepstrum were smoothed and the
-2
resulting MSE was 0.20 x 10 It should be noted that for this case

the peaks at the difference times were quite small and therefore were

not smoothed. Failure to smooth these additional peaks in the com-

plex cepstrum causes the complex cepstrum that is the input to the

inverse procedure for wavelet extraction to remain slightly distorted.

This slight distortion is reflected in the extracted wavelet. Thus,

the obtained MSE is not the smallest possible error that can be attained

for the four-echo case.

The results of several computer runs when two echoes were

present, both with amplitudes greater than unity, are described next.

In one experiment both echo amplitudes were set to 1.5 and no results

were obtained. This case is mathematically similar to the one-echo

case in which the echo amplitude was the same as that of the wavelet

(see Appendix C).

When a computer simulation was performed for two echoes with

amplitudes of 1.2 and 1.5, the highest peak occurred in the power

cepstrum at the time difference of the echo epochs. Using the MSE

obtained for the wavelet extraction of the two-echo case with both

echo amplitudes equaling 0.2 as a desirable goal, we found that we had

to filter and smooth the difference echo peak in addition to the peaks

at the echo epoch times in order to obtain a similar MSE value. It was

found that the recovered signal was the echo that had the largest

amplitude in the composite signal.










Interpretation of results.--For the single echo case the MSE

increased as the amplitude of the echo approached that of the wavelet

(see Figure 29). The number of series terms required to approximate

the expanded function with the same error increases as the amplitude

a approaches unity. Thus, the computational error is due to trunca-

tion error. Also, it is predicted mathematically that the analysis

program will recover the echo rather than the wavelet for the case

where the magnitude of the echo is greater than that of the wavelet.

This result was confirmed experimentally with a noticeable increase

in the MSE as the amplitude of the echo approached that of the wavelet.

The deconvolution of a composite signal consisting of a wave-

let plus two or more echoes presents considerable difficulty. The

procedure is simplified if the epochs of the additional echoes are

integer multiples of the first echo. For one echo the series expansion

for the logarithm causes peaks at all integer multiples of the epoch

time of the echo (see Appendix C); therefore, these appropriate time

samples have to be smoothed to extract the wavelet with the least

distortion. For the multiple echo case with the epoch times being

integer multiples of each other, the additional echoes will cause peaks

(epoch times, sum and difference of epoch times, etc.) to add to or

subtract from the magnitudes at the time samples having peaks caused

by the single echo. Thus, the comb filter will be identical for the

single and multiple echo cases if the first echo epoch is the same as

that for the single echo case.










Noisy Case


This section presents the experimental results and their

interpretation for the situations where Gaussian noise is added

to the composite signal. The frequency content of the VER signal

(again used as the reference wavelet) is almost exclusively contained

between DC and 2 Hertz. An algorithm generates low-pass filtered

noise in this same frequency range (see Appendix F).


The Power Cepstrum

Single echo.--The signal to noise ratio, SNR, was decreased

until the cepstral peak was not detectable. This occurred at a SNR

of 20 db for the example of a single echo delayed 56 samples and

attenuated by 0.4. Another computer simulation where the echo

amplitude was changed to 1.8 yielded similar results.

Multiple echoes.--The only experiment performed was that for

two echoes. These echoes had amplitudes of 0.2 and delays of 56 and

71 samples. The peaks could not be detected for SNR's below approx-

imately 20 db.

Interpretation of results.--The performance of the power

cepstrum for the additive noise case gave results that differed from

the earlier investigations of 0. Halpeny (8). It was soon discovered

that part of the discrepancy was due to the fact that the noise

spectra differed considerably; for the experiments in this research,

the noise had been low-pass filtered. Thus, these results were better

by a factor of 5 (or a SNR = 20); this is still significantly different,

however. We were not able to find a reason for this further discrepancy.










We had originally expected to find additional peaks in the

complex cepstrum caused by the noise, and, with a decrease in the

SNR, the magnitudes of these peaks were expected to increase

accordingly. These were not the results that were obtained, since

a decrease in the SNR merely caused the magnitude of the echo peaks

to decrease while the background peaks (noise) remained relatively

constant. Thus, when the SNR was 20 db the echo peaks decreased

to the point where they became lost in the background noise.

This observed phenomenon is mathematically explained in

Appendix D. Stated briefly, the additive noise distorts the co-

sinusoidal components in the log power spectrum with decreasing

SNR until finally the peaks in the power cepstrum can no longer be

recognized. Smoothing of the log magnitude was found to improve

the detection performance; this will be explained in the section

on linear filtering.


The Complex Cepstrum

Single echo.--Since the composite signal used in this

research had a linear phase component, the effects of additive

noise were not as obvious in the complex cepstrum as they were in

the log magnitude and in the phase curve. The plot of the log

magnitude becomes more noisy with decreasing SNR. Similarly, the

slope of the phase curve changes with the decreasing SNR.

For the examples simulated, with the linear phase component

removed, the complex cepstrum, the log magnitude, and the phase curves

became more distorted with decreasing SNR.










Multiple echo.--The results (effect on the log magnitude, phase,

and the complex cepstrum) were identical to those for the single echo

when noise was added.


Linear Filtering

The results of the three linear filtering methods (short-pass,

long-pass, and comb filtering) discussed in the noiseless case apply

to the noisy case as well for SNR values above the power cepstrum

peak detection threshold which is 20 db for the wavelet used. Below

this SNR it is not possible to apply any of these filters unless a

method is found to lower the threshold value of the power cepstrum

detection capability. Such a method was found and implemented. The

method of improvement used is the Hanning smoothing of the log magni-

tude and with this smoothing favorable results were obtained down to

a SNR = 2; this was the lowest SNR attempted.

Smoothing.--To improve the wavelet extraction it is necessary

to improve the complex cepstrum of this wavelet that is distorted by

noise. As mentioned previously, the addition of a linear phase com-

ponent by the addition of zeros to the time series causes the complex

cepstrum to be less affected by additive noise. Hanning smoothing was

subsequently applied to the log magnitude, the phase, and the complex

cepstrum. The smoothing of the log magnitude caused an improvement

in the MSE of the extracted wavelet of 3 db while there was no notice-

able improvement obtained from the smoothing of only the phase and/or

the complex cepstrum.










Wavelet Extraction

This part of the analysis procedure is identical for both the

noise-free and noisy cases. The results obtained as a result of

smoothing show that the MSE can be reduced (see Figure 30).

Interpretation of results.--The log magnitude and the phase

curve both deteriorated with the decrease in the SNR. When the record

length of a noisy composite signal is doubled by the addition of zeros,

a considerable improvement is effected in the recovered wavelet as

shown in Figure 30. In Appendix D it is shown that the addition of

zeros multiplies the Fourier transform of the composite signal by an
+Je6
e term, thus adding a large linear component to the phase which

tends to override the effect of the noise on the signal phase and

also lessens the distortion introduced by the noise.

From the results obtained, the most effective means of improving

wavelet extraction in the presence of noise is the addition of a linear

phase component in conjunction with amplitude smoothing of the log

magnitude.


Distorted Echo Removal


As shown in Figure 31, distortion is introduced via two methods.

The first adds Gaussian noise to the echo only (see Figure 31a). This

is followed by a measurement of the SNR; comb filtering is then applied

to the complex cepstrum and the SNR of the recovered wavelet is then

calculated.

The second method is to truncate the added echo. The effect of

this truncation on the peak detection capability of the power cepstrum














10-1



MSE







10-2
10-2


~- N.

'` ,


256 DATA POINTS

512 DATA POINTS

512 DATA POINTS WITH MANNING SMOOTHING


0 10 20 30 (db 40
SNR (db)

Figure 30. Effects of smoothing on the mean square error (MSE) of the recovered wavelet for the
case of additive noise


0\












GAUSSIAN NOISE


Figure 31.


Two methods used to generate a composite signal
consisting of the wavelet plus a distorted echo.
(a) adding noise to the echo, and (b) truncating
the echo










and on the wavelet extraction procedure is then determined. The

mean square error (MSE) of the recovered wavelet versus the SNR is

plotted in Figure 32. For the truncated echo experiments, a threshold

occurs at approximately 20 per cent truncation of the echo as is

demonstrated in Figure 33. Above this threshold the peak in the power

cepstrum shifts point for point with the amount of echo truncation.

When the detected peaks are comb filtered and smoothed in the complex

cepstrum, the recovered wavelet is only slightly distorted. This is

determined by visual comparison and by comparison of the mean square

errors for each simulation.


Interpretation of Results

Two forms of echo distortion were used. In the first Gaussian

noise was added to the echo. Here it was found that it was impossible

to detect the epoch of the echo by the power cepstrum with a SNR below

15 db. The recovered wavelet was a "reasonable" reproduction of the

original wavelet for SNR's down to 10 db. The investigation showed

that the cepstrum analysis techniques can be used to decompose a com-

posite signal for modest signal to noise ratios.

The other method used to distort echoes was truncation. An

interesting and important phenomenon was observed; there is a threshold

phenomenon above which truncation of the echo displaces the peak observed

in the power cepstrum. The immediate implication of this is that if the

echo is truncated (as during recording) the power cepstrum will yield

an erroneous epoch time that is dependent on the amount of truncation.

This displacement of the power cepstral peak does not affect the wavelet















FAILED TO DETECT
CEPSTRUM PEAK
THRESHOLD


Figure 32.


Mean square error (MSE) of the recovered wavelet as
a function of the echo distortion using additive
noise


10-1


MSE


SNR (db)


iN',


10-3



















1.0-
MAXIMUM AMPLITUDE = 0.265 \
I \
NORMALIZED /
POWER
CEPSTRUM f
PEAK
MAGNITUDE A
-/ \ /
/ / '












SI II
0.5- \




BACKGROUND NOISE









0 5 10 15 20 25
% TRUNCATION (% SHIFT OF POWER CEPSTRUM PEAK)

Figure 33. Effect of echo truncation on the power cepstrum
peak magnitude
peak magnitude






78



recovery capability of the complex cepstrum since the comb filter

would be shifted accordingly. Thus, the comb filter effectively

removes the echo from the complex cepstrum as long as the truncation

is below the threshold value.














CHAPTER V

SUMMARY AND CONCLUSIONS


In any investigation and development of an analysis technique,

it is necessary to have some method for measuring the results obtained

from the application of that method to given data; in other words, a

performance index is needed. With the perspective gained through the

use of this performance index improvements may be brought about in the

analysis techniques or insight may be gained to be used to develop new

methods. Unfortunately there are no other analysis techniques, linear

or nonlinear, that will satisfactorily extract an unknown wavelet from

a composite signal composed of this wavelet and an unknown number of

its echoes; for this reason there is no way to compare the results of

this investigation with results from other research. Thus, somewhat

arbitrarily, a performance index (the mean square error) was selected;

the mean square error was calculated between the original wavelet and

the one that was extracted by the application of the analysis technique.

This was done in both the noise-free and noisy cases.

The VER signal was used as the reference wavelet in the examples

because of the interest of the personnel of the Visual Sciences Laboratory

who are attempting to analyze and decompose brain waves.


Noiseless Case


The Power Cepstrum

The power cepstrum technique demonstrated the ability to determine

the epoch times of the echoes for the single and multiple echo cases with










echo amplitudes less and greater than that of the reference wavelet;

however, in one situation this technique is limited since erroneous

peaks are found in the power cepstrum for the multiple echo case when

the summation of the individual echoes falls in the range of from

approximately 0.8 to unity. A corresponding range occurs above unity

as well.

Several additional comments pertinent to the power cepstrum

are that

1. The magnitude of peaks for the multiple echo case are

dependent on the magnitude of the initial epoch time

and the time separation between epochs.

2. The amplitude of the echoes with respect to the amplitude

of the reference wavelet can be estimated, but with less

certainty as the number of echoes increases.


The Complex Cepstrum and Wavelet Extraction

By observing the results obtained from this investigation, the

following conclusions have been drawn

1. The cepstrum techniques will satisfactorily determine the

epoch times of the "impulse train" and will also recover,

with a high degree of accuracy, the wavelet information

from a composite signal without noise; this can be done for

a wide range of echo amplitudes and for a large number of

echoes. The only apparent limitation is that imposed by

computational accuracy.

2. Three known exceptions to item 1 are:

a. For a single echo whose amplitude is the same as that

of the reference wavelet.










b. For the multiple echo case when all of the echo

amplitudes are less than unity except for one or

more with unity amplitudes.

c. For multiple echoes where two or more have amplitudes

larger than the wavelet and are of equal value.

3. The complex cepstrum procedure will extract the wavelet

when the composite signal occupies less than 25 per cent

of the time record to be analyzed. The amount of the

total time record that the composite signal occupies is

not the primary limitation.

4. The mean square error increases for the extracted wavelet

in the single echo case as the amplitude of the echo

approaches that of the reference wavelet.

5. Comb filtering is superior to the short-pass filtering.


Noisy Case


Power Cepstrum

From the experimental results that were obtained, the following

conclusions are drawn.

1. The power cepstrum is more susceptible to noise than the

complex cepstrum.

2. The power cepstrum technique exhibits a threshold effect

for the additive noise situation, and with the VER signal

as the reference wavelet this threshold is SNR = 20 db using

band-limited Gaussian noise.










3. The threshold can be significantly lowered by a Hanning

type of amplitude smoothing applied to the log magnitude.


Complex Cepstrum and Wavelet Extraction

Based on the results obtained from the computer simulations

of the complex cepstrum technique, the following conclusions have

been reached.

1. The complex cepstrum technique does not exhibit a threshold

in the presence of additive noise.

2. The log magnitude data are more susceptible to noise than

the phase data are.

3. The MSE of the extracted wavelet can be significantly lowered

(8 db) by simultaneously adding a linear phase component and

amplitude smoothing.

In conclusion, the cepstrum techniques are severely limited in

deconvolving composite waveforms when the composite waveform is immersed

in noise. The power cepstrum technique exhibits a threshold limitation

for the additive noise case. This threshold was lowered 16 db by a Hanning

smoothing of the log magnitude before the final application of the power

spectrum. The SNR for the wavelet extraction procedure was lowered by

8 db by the combination of adding a linear phase component and the same

log magnitude smoothing used for the power cepstrum.


Distorted Echo Removal


Investigations into the deconvolution of a wavelet with a dis-

torted echo showed that a somewhat distorted echo can be detected and

satisfactorily removed even when it was distorted by additive noise.











The procedure used to distort the echo by truncation gave an unexpected

result: a shift occurred in the power cepstrum peak. The most obvious

conclusion to be drawn from these distorted echo results is that caution

must be exercised when truncated data are analyzed by cepstral methods.

It was also observed that if the cepstrum technique were applied to a

data record in which the echo or echoes were believed to occur and none

were detected, then three possible explanations can be advanced:

1. There were no echoes present,

2. The SNR was too low, or

3. The echo was distorted above the truncation threshold.


Future Research


In summary, it has been shown that this particular nonlinear

technique can be successfully applied to decompose a composite signal

that consists of an unknown wavelet and two or more moderately distorted

echoes in the presence of additive noise. With this as a small step,

perhaps a similar nonlinear method can be derived that will be successful

in separating two completely different unknown stochastic signals. The

possibilities for accomplishing this difficult task seem to lie in a non-

linear method similar to the one investigated here.

Further research should be directed toward the improvement of

smoothing techniques for the case when noise is present and toward a

better understanding of the threshold phenomenon observed in the distorted


echo case.
















APPENDIX A

DEFINITION OF THE TERMINOLOGY USED IN THE DISSERTATION


Power Cepstrum

In order to arrive at meaningful terminology for the newly

invented nonlinear analysis technique, Bogert et al. (2) suggested

the use of certain "paraphrased" words for the various quantities

encountered. They arrived at this idea as they found themselves

operating in the frequency domain in ways customary for the time

domain, and vice versa. Thus, the "paraphrased" words were invented

to avoid future confusion.

The most important words include the following. Their

meanings will be readily understood.


Power Cepstrum Domain

quefrency - -

cepstrum - -

complex dedomulation - -

cross cepstrum - -

liftering - -

lifter - -

long-pass lifter - -

short-pass lifter - -

rahmonic - -

repiod -- -


Frequency Domain

frequency

spectrum

complex demodulation

cross spectrum

filtering

filter

high-pass filter

low-pass filter

harmonic

period










Power Cepstrum Domain Frequency Domain

alanysis - analysis

darius - radius

gamnitude - magnitude

lopar - polar

saphe - phase


Complex Cepstrum

The complex cepstrum technique set forth by R. Schafer (6)

did not have the problem of "inventing" a new parameter for the com-

plex cepstrum domain. This was because the complex cepstrum is

obtained from the inverse Fourier transform of a function in the

frequency domain; thus it is merely a function of time. Since the

filtering performed in the complex cepstrum (time) domain resembles

that usually done in the frequency domain, Schafer used the following

mixed terminology and it has also been adopted in this research.

Complex Cepstrum Domain Frequency Domain

short-pass filter - low-pass filter

long-pass filter - high-pass filter

cepstrum - spectrum















APPENDIX B

MATHEMATICAL ANALYSIS OF THE POWER CEPSTRUM


The following paragraphs provide the mathematical formulation

used and expanded for this dissertation. It includes the formulation

as presented by Bogert et al. (2) and as it has been expanded by this

investigation.

Let y(t) designate the wavelet for the specific physical situa-

tion of interest. An echo is represented as a delayed replica of the

wavelet multiplied by a constant. The sum of the original wavelet and

the echoes then forms the composite waveform. The single echo case is

given as


z(t) = y(t) + ao y(t-to)


(B-l)


Now we can calculate the

or by ensemble averaging

we are able to write


autocorrelation function by time averaging

if we assume ergodicity. With this assumption


(B-2)


R (T,t) = R (T)
Y y


where


R (T) =
Y


E{y(t) y(t + T)}


(B-3)


and


R (T) = E{z(t) z(t + T)} .
z


(B-4)




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