Wavefront detection by frequency-wavenumber analysis of three-dimensional array data, by Lewis J. Pinson

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Wavefront detection by frequency-wavenumber analysis of three-dimensional array data, by Lewis J. Pinson
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Thesis--University of Florida.
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Bibliography: leaves 165-169.
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Full Text










Wavefront Detection By Frequency-Wavenumber Analysis
of Three-Dimensional Array Data














By

LEWIS J. PINSON


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
1972





































Dedicated to Iris because she is dedicated to me















ACKNOWLEDGMENTS


The author wishes to express sincere appreciation to

Dr. D. G. Childers, chairman of his supervisory committee, for his

guidance in selecting an interesting area of research and for his

promotion of professionalism. He also wishes to thank Dr. R. L. Isaacson

and Dr. L. W. Couch, the other members of his supervisory committee, and

Dr. L. Jones and Dr. J. Cross for helpful suggestions. A special note

of thanks goes to Dr. R. L. Isaacson who provided the financial support

which made it possible for the author to return to the University of

Florida.

Finally, the author would like to express appreciation to the

many people who offered encouragement to pursue a doctoral degree. A

particular expression of gratitude is extended to his mother and

father-in-law, Mr. and Mrs. C. B. Mann, because their encouragement

was the first.
















TABLE OF CONTENTS


ACKNOWLEDGMENTS.............................................................

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

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

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

CHAPTERS:

I INTRODUCTION. .............................. ...

II ARRAY PROCESSING.....................................

A. Adaptive Processors for Signal-to-Noise Ratio
Improvement.................................... ..

B. Feature Extraction by Digital Filtering...........

C. Wavefront Detection..............................

III FREQUENCY-WAVENUMBER SPECTRUM--THEORY.................

A. Generalized Plane Waves..........................

B. Frequency-Wavenumber Representation of Spatial
Fields..................................

C. Estimate of the Frequency-Wavenumber Spectrum.....

D. High Resolution Adaptive Window...................

E. Three-Dimensional Frequency-Wavenumber Spectrum...

F. Finite Bandpass Signals...........................

IV FREQUENCY-WAVENUMBER SPECTRUM--PARAMETRIC ANALYSIS....

A. Comparison of High Resolution and Conventional
Estimates ........... ...... ...... .. ..... .......

B. Array Geometry ..................................


Page

iii

vii

viii

xi



1

4


4

6

7

8

9


12

13

18

22

25

29


30

32











CHAPTERS:

C. Temporal Window Functions........................

D. Additive Incoherent Noise.......................

E. Statistical Reliability........................

F. Spherical Wavefronts............................

G. Spatial Pre-filtering............................

H. Cost and Computation Requirements................

V FREQUENCY-WAVENUMBER SPECTRUM--SIMULATION ...........

A. Array and Signal Example Parameters..............

B. Comparison of High Resolution and Conventional
Estimates........... ................... ... .

C. Array Geometry ..................................

D. Temporal Window Functions......................

E. Additive Incoherent Noise........................

F. Bandpass Signal Spectral Computation.............

G. Three-Dimensional Simulation.....................

VI INVESTIGATION OF ELECTROPHYSIOLOGICAL DATA BY
FREQUENCY-WAVENUMBER TECHNIQUE......................

A. Electrophysiological Signal Characteristics......

B. Analysis of Human Visual-Evoked Response.........

C. Penicillin-Induced Focal Epilepsy in Rats.......

VII DISCUSSION..... ........... ... .................


APPENDICES:

A

B

C


CROSS POWER SPECTRAL DENSITY......................... 107

SPECTRAL REPRESENTATION OF SPATIO-TEMPORAL FIELDS.... 111

FREQUENCY-WAVENUMBER SPECTRUM IN SPHERICAL
COORDINATES ......................................... 116


34

36

39

42

43

45

48

49


51

53

57

63

68

76


81

81

83

90

103














D EXPERIMENTAL METHOD.................................... 118

E PROGRAM LISTINGS........................... .......... 124

F SINGULARITY OF THE CROSS POWER SPECTRAL DENSITY
MATRIX. .............. ... .... .... ................. 163

BIBLIOGRAPHY................. ............ .... ...... ................ 165

BIOGRAPHICAL SKETCH............................................... 170














LIST OF TABLES

Table Page

1 Cross Power Spectral Density Estimate...................... 12

2 Computation Time and Cost................................. 46















LIST OF FIGURES


Figure Page

1 GENERAL TWO-DIMENSIONAL PLANE WAVE ..................... 11

2 CONVENTIONAL FREQUENCY-WAVENUMBER SPECTRAL ESTIMATE...... 17

3 EFFECT OF FINITE DATA RECORD LENGTH ON RESOLUTION....... 19

4 HIGH RESOLUTION FREQUENCY-WAVENUMBER SPECTRAL ESTIMATE.. 21

5 THREE-DIMENSIONAL PLANE WAVE............................. 23

6 BANDPASS SPECTRAL ESTIMATION........................... 28

7 SAMPLING THEOREM RELATIONSHIPS.......................... 33

8 TEMPORAL WINDOW FUNCTIONS AND CHARACTERISTICS.......... 35

9 TWO-DIMENSIONAL ARRAY GEOMETRY AND PARAMETERS ........... 50

10 CONVENTIONAL AND HIGH RESOLUTION PERSPECTIVE PLOTS...... 52

11 CONVENTIONAL ESTIMATE OF THE FREQUENCY-WAVENUMBER
SPECTRUM .............. ............ ....... .............. 54

12 HIGH RESOLUTION ESTIMATE OF THE FREQUENCY-WAVENUMBER
SPECTRUM ................................................. 55

13 CROSS-SECTION OF CONVENTIONAL AND HIGH RESOLUTION
SPECTRAL ESTIMATES ...................................... 56

14 RELATIVE EFFECT OF ARRAY APERTURE ON RESOLUTION......... 56

15 HIGH RESOLUTION ATTENUATION PLOT FOR 4-ELECTRODE ARRAY.. 58

16 EXAMPLE TEMPORAL WINDOW FUNCTIONS....................... 59

17 EFFECT OF RECTANGULAR WINDOW FUNCTION FOR DISCRETE
FREQUENCY COMPUTATION AND R = 0.1....................... 60

18 EFFECT OF COSINE TAPERED WINDOW FUNCTION FOR DISCRETE
FREQUENCY COMPUTATION AND R = 0.1.............. ........ 61


viii











Figure Page

19 EFFECT OF RECTANGULAR WINDOW FUNCTION FOR DISCRETE
FREQUENCY COMPUTATION AND R = 0.0001.................... 62

20 EFFECT OF COSINE TAPERED WINDOW FUNCTION FOR DISCRETE
FREQUENCY COMPUTATION AND R 0.0001 ..................... 64

21 EFFECT OF INCOHERENT NOISE ON SINGLE WAVEFRONT........... 65

22 EFFECT OF INCOHERENT NOISE ON MULTIPLE WAVEFRONTS........ 65

23 SINGLE-WAVEFRONT SPECTRUM FOR R = 0.001.................. 67

24 TWO-WAVEFRONT SPECTRUM FOR R = 0.01 (COSINE WINDOW)...... 69

25 TWO-WAVEFRONT SPECTRUM FOR R = 0.001 (COSINE WINDOW)..... 70

26 BANDPASS SPECTRUM (R = 0.01, RECTANGULAR WINDOW,
PASSBAND = 6-20 Hz)................ ................. 71

27 BANDPASS SPECTRUM (R = 0.01, COSINE WINDOW, PASSBAND =
6-20 Hz)........ ................ ................... 72

28 NARROW BANDPASS SPECTRUM (R = 0.01, RECTANGULAR WINDOW,
PASSBAND = 11.5-16 Hz)................................ 74

29 NARROW BANDPASS SPECTRUM (R = 0.01, COSINE WINDOW,
PASSBAND = 11.5-16 Hz).................................. 75

30 THREE-DIMENSIONAL ARRAY GEOMETRY AND SIGNAL PARAMETERS... 77

31 THREE-DIMENSIONAL FREQUENCY-WAVENUMBER SPECTRUM.......... 79

32 FREQUENCY-WAVENUMBER SPECTRUM OF VER DATA (0.5 sec)...... 85

33 FREQUENCY-WAVENUMBER SPECTRUM OF FIRST HALF VER DATA
(0-0.25 sec)..... ... .................... .. ....... ... 86

34 FREQUENCY-WAVENUMBER SPECTRUM OF LAST HALF VER DATA
(0.25-0.50 sec).......................................... 87

35 FREQUENCY-WAVENUMBER SPECTRUM WITH MAXIMUM VALUE BANDS... 89

36 ERRAT1 SPECTRUM (f1 = 2.7; f2 = 4.7 Hz; PHASE COM-
PENSATED) ... ........ ............................ ..... 93

37 ERRAT2 SPECTRUM (2 mm grid; fl = 3; f2 = 4; f3 = 5 Hz)... 95











Figure Page

38 ERRAT2 SPECTRUM (1 mm grid; fl = 3; f2 = 4; f3 = 5 Hz)... 96

39 ERRAT3 SPECTRUM FOR PRIMARY HEMISPHERE (f = 2 Hz)........ 98

40 ERRAT3 SPECTRUM FOR SECONDARY HEMISPHERE (f = 2 Hz)...... 99

41 ERRAT3 SPECTRUM FOR PRIMARY HEMISPHERE (fl = 7.8;
f2 = 13.7 Hz)............ ....... ...................... 100

42 ERRAT3 SPECTRUM FOR SECONDARY HEMISPHERE (fl = 9.8;
f2 = 21.5 Hz)................. ........................... 101

D-1 SCHEMATIC DIAGRAM FOR MONITORING AND RECORDING
ELECTRICAL ACTIVITY OF RAT NEOCORTEX ..................... 119

D-2 PREPARATION OF RAT DATA FOR FREQUENCY-WAVENUMBER
ANALYSIS.......................... ...................*.... 122











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

WAVEFRONT DETECTION BY FREQUENCY-WAVENUMBER ANALYSIS
OF THREE-DIMENSIONAL ARRAY DATA

By

Lewis J. Pinson

March, 1972

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

A high resolution frequency-wavenumber spectral estimation

technique is employed to analyze spatio-temporal signals from three-

dimensional sensor arrays, to estimate the spectrum for finite

temporal frequency bands using FFT techniques, and to estimate the

spectrum for discrete temporal frequencies.

Direction and true vector velocity of a plane wavefront propa-

gating across a three-dimensional arrayare given by the frequency-

wavenumber spectrum for discrete temporal frequencies. For finite

temporal frequency bands the wavefront direction and bounds on the

vector velocity are given.

A parametric analysis is performed in which the properties of

the frequency-wavenumber spectral estimate are determined for known

signals and parametric values. Array aperture and the ratio of

incoherent noise power to total signal plus noise power are found to

determine the resolution in the wavenumber domain. Other parameters,

including number of temporal samples per sensor, temporal window function,

and number of sensors, are found to affect statistical reliability and

computation cost.











The frequency-wavenumber spectral estimation technique is

applied to the analysis of visual-evoked response (VER) data and to

the analysis of penicillin-induced epileptiform electrical activity

recorded from the rat neocortex. Results indicate that application

of the frequency-wavenumber spectral analysis method to electro-

physiological data is possible and that it provides information for

increased understanding of brain function.

Application of the expanded frequency-wavenumber spectral

estimation method to spatio-temporal data from the areas of seismology,

oceanography, speech, meteorology, radar and sonar would require in

most cases only a change in parametric inputs.


xii














CHAPTER I

INTRODUCTION

Over the past few years Biomedical Engineering has evolved

from isolated applications of electronic instruments as tools for

medicine to sophisticated and ever-expanding applications of advanced

analytical techniques to biological systems. Other areas of applica-

tion, which include seismology, oceanography, speech, meteorology

and sonar, have need for analytical techniques similar to those

applicable to the analysis of biological data. These various

disciplines have certain common characteristics which makes it

advantageous to develop analytical techniques that have general

applicability. One such characteristic is that each of the areas is

concerned with measuring and interpreting spatio-temporal signals.

The purpose of this research was to investigate a technique for

analyzing spatio-temporal signals and develop useful extensions of

the technique to provide additional information about the signals.

Parametric bounds and example computations were chosen to emphasize

application of the technique to the study of electrophysiological

data. However, application to other areas is implied and in most

cases would require only a change in parametric inputs.

Chapter II gives a brief review of array-processing techniques

which have been applied to other fields such as radar, seismology,

acoustics, and image processing. Some of the properties of these










techniques are discussed and compared with the expected properties of

the frequency-wavenumber technique. Digital filtering array processors

are indicated as having general applicability as wavenumber limiting

pre-filters.

In Chapter III the theoretical development is given for the

frequency-wavenumber spectral estimate. It is shown how the frequency-

wavenumber spectrum provides information for determining direction and

velocity for multiple plane wavefronts. An extension of the frequency-

wavenumber technique to three dimensions is developed and shown to be

valid theoretically. Inclusion of frequency bandpass signals as well

as discrete frequency signals has been accomplished using fast Fourier

transform (FFT) techniques and a spectral integration scheme. Details

of the bandpass feature are given in Chapter III.

Many parameters affect the resolution, cost and reliability

of the frequency-wavenumber spectral estimate. Additive incoherent

noise was found to have a major effect on resolution; however, the

array aperture and the temporal window were found to interact with

the incoherent noise level and thus affected the achievable resolution.

Parametric variation, statistical reliability and computation require-

ments are discussed in Chapter IV.

For spherically symmetric wavefronts the frequency-wavenumber

method gives a scalar output which is insufficient for determining

wavefront direction. Chapter IV discusses the case for spherical

wavefronts as well as the need for spatial pre-filtering.

In Chapter V examples are shown for simulation runs which

verify the theoretical development and parametric analyses.











The frequency-wavenumber technique was applied to electro-

physiological data of two types: 1. human visual-evoked response

data recorded from the scalp over the occipital region of the brain,

and 2. penicillin-induced focal epileptic discharges recorded from

the rat neocortex. Details and results of these experiments are

outlined in Chapter VI.

Finally in Chapter VII, a discussion of the frequency-wavenumber

spectral estimation technique is given in relation to the goals accom-

plished by this research effort. Significance of the experimental

results is discussed along with recommended areas for further investiga-

tions.















CHAPTER II

ARRAY PROCESSING

Included under the category of array processing are several

specific techniques which have been investigated by various researchers.

The characteristics of some of these techniques are discussed in rela-

tion to the array-processing technique termed frequency-wavenumber

spectral estimation.

In general, array-processing techniques are used in applications

where their advantages outweigh the disadvantage of increased processing

complexity. The advantages offered by array-processing techniques

include improved signal-to-noise ratio for facilitating the detection of

weak signals, enhancement or emphasis of specific features or parameters

and the determination of spatial characteristics such as wavefront direc-

tion. Major differences between the various array-processing techniques

usually result from: 1. noise assumptions (stationarity, directionality);

2. signal assumptions (known, unknown, uniform or distorted across the

array); and 3. method employed (spatial filtering, stochastic approxima-

tion, steepest descent, conjugate gradients, noise estimation, etc.).

Various methods of array-processing are discussed below.


A. Adaptive Processors for Signal-to-Noise Ratio Improvement

Although many techniques have been developed for improving

signal-to-noise ratio using array data, most of the techniques are











similar in approach and make the assumption that wavefront direction is

known "a priori."

Lacoss [1] has presented an adaptive linear processor with

variable coefficients in which the coefficients are adjusted by a rule

similar to that used by the projection gradient method of a quadratic

form subject to a linear constraint. The purpose of this processor is

to improve signal detection and assumes that the wavefront direction is

known with 'sufficient' accuracy. Thus the signals from the sensors

can be combined after appropriate time delay adjustments with gains

adjusted to provide optimum detection of an unknown signal. An optimum

processor is one which converges to a minimum variance, unbiased estimate

of the signal. Lacoss derived several iterative methods for reducing

sensitivity of the processor to data anomalies such as noise bursts and

for reducing required computation time and storage requirements. Other

adaptive processing techniques for improving signal-to-noise ratio which

are similar to those of Lacoss have been derived in the literature.

Shor [2] has proposed a gradient adaptive method for narrowband hydrophone

arrays which requires knowledge of the signals' autocorrelation function.

Adams [3] has done work with adaptive on-line array processors for

receiving deep-space probe signals.

Kobayashi [4] has proposed two iterative methods for processing

array data: the method of steepest descent and the method of conjugate

gradients with projection. These methods are very similar to those of

Lacoss and rely on essentially the same mathematical development. No

intermediate computation of covariance matrix functions is required

for the methods used by Kobayashi. As do most methods, however, this











method assumes the only difference between the signals at various sensors

is a time delay which can be compensated.

Burg [5] used a multi-dimensional Wiener filtering approach to

array processing. In this approach the signal and noise are represented

as stationary random processes with known cross-correlation functions.

The output of the processor is a minimum mean-squared-error estimate of

the signal. Again, knowledge of time delays between sensors is required.

Another approach to array processing is that proposed by Claerbout

[6]. The cross-correlation matrix is computed by observing the sensor

outputs in a "fitting interval." This information is then used to provide

a minimum mean-squared-error estimate of the noise for a short, future,

projected interval. The predicted noise is then subtracted from the

signal. This method reduces the noise level but introduces distortion

into the signal.

Capon et al. [7] have discussed the advantages and disadvantages

of time-domain versus frequency-domain array processors. The signal-to-

noise improvement is best with time-domain methods; however, time-domain

methods are more sensitive to non-stationary noise and signal anomalies

than the frequency-domain methods.


B. Feature Extraction by Digital Filtering

The fields of feature extraction and digital filtering are by

far too extensive to be given a comprehensive survey in this document;

however, feature extraction from two-dimensional fields by digital

filtering techniques is a form of array processing and is included for

that reason. In addition to feature extraction, digital filters may










be used in array processors to limit wavenumber content and reduce

aliasing effects for a given spatial sampling rate.

Wavenumber limited spatial filters are discussed further in

Chapter IV.


C. Wavefront Detection

To determine a wavefront propagation direction it is necessary

to use methods which give vector wavenumbers as outputs. Standard

beamforming techniques use adjustable phase differences between

elements of a sensor array to determine wavefront directions. These

methods, however, provide no information regarding wave velocities and

are restricted in resolution to the natural beam pattern of the array.

A method which overcomes these deficiencies and computes the high

resolution vector velocity and direction of propagating wavefronts

has been proposed by Capon [8] for processing two-dimensional data

from the large-aperture seismic array (LASA) in Montana. A comparison

of this high resolution estimate of the frequency-wavenumber spectrum

with conventional estimates showed marked improvement for the high

resolution method. The theoretical development of the frequency-

wavenumber spectrum is given in Chapter III. Extension of Capon's

method to include data from three-dimensional arrays is shown to be

valid theoretically. Methods for including finite frequency bands

are discussed and one method is developed in detail.













CHAPTER III

FREQUENCY-WAVENUMBER SPECTRUM--THEORY

This chapter gives a detailed development of the theoretical

background material supporting the validity of the frequency-wavenumber

technique as a descriptor of array sensed signals. First the properties

of plane waves and their mathematical representations are discussed

and it is shown how the frequency-wavenumber spectrum provides informa-

tion describing plane waves.

The concept of estimation as related to the frequency-wavenumber

spectrum is introduced and related to theoretical concepts. Window

functions are discussed in relation to their effect on accuracy of the

estimate. A high resolution adaptive window is introduced which provides

a major improvement in resolution over that obtainable with conventional

fixed window functions.

The use of a two-dimensional array for measuring three-dimensional

signals is inadequate if it is desired to know true vector velocities of

incoming wavefronts. Determination of wavenumbers in two dimensions

identifies azimuth angle but not elevation angle. This means that for a

given azimuth angle, any number of signals with different elevation angles

may be present. With the two-dimensional analysis these various signals

would be indistinguishable and only their phase velocities could be deter-

mined. A three-dimensional analysis identifies wavenumbers in three

dimensions and uniquely determines wavefront directions. This also makes

it possible to determine true vector velocity.










Theory and methods are discussed for extending the frequency-

wavenumber technique to accept data from a three-dimensional array of

sensors and to accept finite bandwidth data in addition to discrete

frequencies. Capon's technique is valid for only single discrete

frequencies. This could be a major disadvantage where the exact spectral

content of the signal is unknown. Also in many applications it may be

desirable to include a band of frequencies even if spectral content is

known. Rather than compute spectral estimates individually for all

frequencies of interest a method was developed for accepting multiple

frequencies as well as finite continuous frequency bands and efficiently

computing the frequency-wavenumber spectrum.


A. Generalized Plane Waves

A general plane wave in two dimensions can be represented

mathematically by [9],[10]


S(t,x,y) = exp [2iri(ft- i *f)] (1)


where: S(t,x,y) = the spatio-temporal signal amplitude

f = temporal frequency (Hz)

k = vector wavenumber (cycles/m)
g
? = spatial coordinate vector (m)

For non-isotropic media, the vector wavenumber (k) is spatially

dependent and is a function of x and y. For an attenuating medium k

has both real and imaginary components. Thus the general vector wave-

number is given by


k = Re[k (x,y)] + Im[k (x,y)] (2)
g g g










where: Re['] denotes the "real part of"

Im[*] denotes the "imaginary part of"

If the signal is represented in an isotropic, non-attenuating

medium then k is a real, vector constant and is denoted by k. The
g
spatio-temporal signal representation for a two-dimensional, isotropic,

non-attenuating medium then becomes


S(t,x,y) = exp [2Ti(ft kir)] (3)


Figure 1 shows the above relationships and how the vector wave-

number specifies the propagation direction for the plane wavefront.

For simplicity and clarity in demonstrating the frequency-wavenumber

technique an isotropic, non-attenuating medium will be assumed so that

for the following analysis equation (3) is representative of a plane

wave. Some of the properties of a plane wave or any general spatio-

temporal signal which is to be measured by an array of sensors are

often more easily studied by measuring the cross power spectrum between

any two sensors. The cross power spectral density is a K x K matrix

for an array of K sensors and is related to the coherency matrix which

provides a measure of the linear dependence among the sensors. The

cross power spectral density matrix may be obtained in either of two

ways: the correlation method or the direct method. Details of these

two methods are given in Appendix A.

Table 1 provides a comparison of pertinent criteria for the

correlation method versus the direct method for estimating the cross

power spectral density. Because of reduced computation time and less

susceptibility to signal variations the direct method was chosen for

the estimate. Statistical reliability is improved by segmenting the

data as described by Welch [11]. In some cases however, increased














/
1/
/ /


Ik12 = 2 + 2
E = vector wavenumber (gives wavefront direction)


GENERAL:

S(t,x,y)

k


= exp [2ii(ft k *9)]

= Re[k (x,y)] + Im[kg(x,y)]
2 2
= x +y


ISOTROPIC, NON-ATTENUATING MEDIA:

S(t,x,y) = exp [2wi(ft KZ-)]

= exp [2ri(ft Ox yy)]

WHERE: k = Re[k (x,y)] = constant
S= component of in x direction

y = component of k in y direction
y = component of k in y direction


Figure 1. GENERAL TWO-DIMENSIONAL PLANE WAVE










statistical reliability is achieved at the cost of reduced resolving

capability.


Table 1: CROSS POWER SPECTRAL DENSITY ESTIMATE


CORRELATION METHOD


DIRECT METHOD


S 2 p(k) eikX 1
k=-L/2 (k) ) m=1 F F()

L = NO. OF DATA POINTS Fjm(X) IS FOURIER TRANSFORM
DEFINITION j (k) IS COVARIANCE OF DATA IN mth SEGMENT OF
MATRIX jth SENSOR.
M SEGMENTS OF N DATA POINTS
EACH FOR EACH SENSOR

COMPUTATIONALESS
MORE LESS
TIME REQUIRED

STATISTICAL GOOD FOR LARGE L GOOD FOR LARGE M & N
RELIABILITY

SUSCEPTIBILITY
TO SIGNAL VARIA- MORE LESS
TION


B. Frequency-Wavenumber Representation of Spatial Fields

In Appendix A it is shown how the cross power spectral density

function for K sensors is a K x K matrix. The cross power spectral

density matrix is a temporal frequency transform of the cross covariance

matrix for an array and its measured signals. The cross power spectral

density matrix contains information concerning the frequency content

of the signal power. In a similar manner it is possible to obtain

information about the wavenumber content of the signal by performing a

spatial frequency transform on the cross power spectral density function.










The result is a frequency-wavenumber spectrum and is defined as



P(X,k) = J f(A,x) e2i d5 (4)
--00
Vector


where f(X,k) is the cross power spectral density function and the

integral is a vector Fourier transform with respect to the spatial

vector, 1. It follows that the cross power spectral density is then

the Fourier inverse of the frequency-wavenumber spectrum.

00

f(X,x) P(X,k) e- dk (5)
-00
Vector


This development is based on a treatment by Yaglom [12] and details

of the development are given in Appendix B.

It is also shown in Appendix B that if the signal waveform is

a discrete frequency plane wave then the frequency-wavenumber spectrum

is a multi-dimensional delta function about the given frequency and

vector wavenumber, i.e.,



P(X,k) = (X o0) 6( ko) (6)


Thus it is shown that wavefront detection is possible through the use

of the frequency-wavenumber spectrum.


C. Estimate of the Frequency-Wavenumber Spectrum

An exact spectral representation for spatio-temporal signals

would require infinite summations over time and space. Since this is










not practical a spectral representation using a finite number of sensors

(spatial representation) and a finite number of time samples (temporal

representation) is computed and distinguished from the exact representa-

tion by being called an estimate of the exact spectrum. Criteria for

measuring goodness of an estimate have been developed and are discussed

in Chapter IV, Section E on statistical reliability. As was shown

previously there are two major methods for estimating power spectral

densities, the correlation method and the direct method. Because of

its desirable statistical characteristics and computational efficiency

the direct segment method or block averaging technique was used for

estimating the cross power spectral density. The resulting matrix is

denoted by Oj (X)] to distinguish it as an estimate of the exact spectrum

f j(X). This estimate of the cross power spectral density is then used

to obtain an estimate of the frequency-wavenumber spectrum.

To illustrate the technique assume that discrete data are avail-

able from K sensors. Data from each of the sensors are divided into M

non-overlapping segments of N data points each to give L = MN data points

per sensor. The cross power spectral density matrix is computed using

the direct segment method. The Fourier transform of the data in the mth

segment of the jth sensor is



F () a S (nT) ein (7)
N n=ln

j = 1,---,K
m = 1,--,M











where: S M(nT)

X = -2ffT

a
n


= the data in the mth segment from the jth sensor

= normalized frequency

= the weighting coefficients describing the

temporal window used in estimating the

frequency spectrum.


For simplicity assume a = 1; n = 1,---,N. In practice, however, side-

lobe leakage can be reduced and the estimate improved in some cases

by using windows different from the rectangular window described here.

From (7) an estimate for the cross power spectral density is

a K x K matrix whose elements are given by


M
1 F(X) Fm (N)
m=1


=


In order to remove the effect of differences in gain and other causes

of improper sensor equalization, a normalization is performed on f i(t)


A
f 1/
f j& ) = --- 1/2-
IA f^ i ) M fzt )]1


The cross power spectral density matrix is closely related to

coherency, the major difference being that phase information is retained

in the normalized cross power spectrum and lost in the coherency function.

The relationship is given by


Y 2(X) = I2


(10)


where y2 (A) is the coherency function. The coherency function varies
it










from zero to one and is a measure of linear dependence between sensor

elements.

Using the normalized cross power spectral density estimate and

the coordinates of the sensors, we can obtain the conventional estimate

for the frequency-wavenumber spectrum:


K K ik2*(5t 5t)
,) =- IKl wjw* e (11)
K j=l A=l1


where the wj are the weights describing the shape of the spatial window

used to estimate the wavenumber spectrum. Again for simplicity we assume

w = 1; j = 1,---,K so that wj represents the aperture of the sensor

array. As was indicated for the temporal window, the spatial window

can be designed to provide an improved estimate of the spectrum. One

such window, proposed by Capon [8], is a function of the particular

wavenumber being considered. It is different for each wavenumber k
o

and passes undistorted any monochromatic plane wave traveling with a

velocity corresponding to the wavenumber k It also suppresses in

an optimum least squares sense the power of waves traveling at veloc-

ities corresponding to wavenumbers other than k

Figure 2 shows a flow chart describing the estimation procedure

for obtaining the frequency-wavenumber spectrum. It provides a summary

of the procedure discussed up to this point and the resultant output

is called the conventional frequency-wavenumber spectrum of a homogeneous

random field.









SENSOR ARRAY
K SENSORS


SPATIAL WINDOW FUNCTION


SPATIAL TRANSFORM P (A,k)


Figure 2. CONVENTIONAL FREQUENCY-WAVENUMBER SPECTRAL ESTIMATE


MULTI-CHANNEL TEMPORAL SAMPLING FUNCTION

t + f f t- -f


MULTI-CHANNEL SEGMENTING


CROSS POWER SPECTRAL DENSITY
1 F F*
M jm tAm
M(A) fm=l
fji() ) =,, 1/2










D. High Resolution Adaptive Window

A finite data record of discrete values may be considered as

the product of an infinite representation of the data with a rectangular

sampling window as shown in Figure 3a. From Fourier transform theory

we know that multiplication in the time domain implies convolution in

the frequency domain. The result of the convolution is shown in

Figure 3b for the time functions in 3a. An ideal situation would be

to have a window function whose transform is an impulse. This of

course implies a window function of infinite width which implies an

infinite data record length. Thus it is desirable to find a window

function which more closely approximates the ideal but with a finite

data record. More detail on window functions and their characteristics

which have been used in this research is given under Section C of

Chapter IV on the parametric effects of temporal window design. The

choice of which window to use depends on the particular application

and criteria selected to determine a desirable output. There is no

window function which is generally accepted as optimum.

Capon introduces a high resolution estimate for the frequency-

wavenumber spectrum which utilizes a window function that is dependent

on the frequency and wavenumber being considered. The high resolution

frequency-wavenumber spectral estimate using this window is given by


K K i27TT-(5- )
P'(,kI) = A(X,k) A (A,k) fJC(A) ek (12)
j=l a=l1

where: AJ(A,;) is the high resolution window function.

Comparison of equations (11) and (12) illustrates the fact that the

















j)


I'


- I 10


0
c
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i-yi r~
O z
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o
z
I Ii
o
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W B
. W
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ca t
0c
00


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high resolution estimate is the direct result of choosing an adaptive
window function. The window function is given by
K

Aj (,k) =K K (13)
I I q j (X,l)
j=l =l

where: [qj (Xk)] = [[exp {27rit;(i 5)}'[fj( l]]-1 (14)


that is, [q j] is the inverse of the matrix resulting from the product
of the exponential matrix [exp (2lrik*(~j i))] with the cross power
spectral density matrix [f j].
Further, if [q ] is defined as the inverse of [fj]


[ j] = [ j]-1 (15)


then it can be shown that an equivalent expression for the high resolu-
tion frequency-wavenumber spectral estimate is given by

K- K
P'(A,k) = ,q(X) exp [27rik-(< (6)] (16)
j=1 i=1


Thus P' can be computed by simply taking the Hermitian inverse of the
cross power spectral density matrix and then computing the spatial
transform. Equivalency of the two expressions, (12) and (16), is
difficult to show in general; however, computations with low order
matrices have verified their validity.
The estimate P' is the power output of a maximum likelihood
array processor whose characteristics vary with wavenumber in an



































MATRIX INVERSION
[qj = [fj]l-


HIGH RESOLUTION ESTIMATE
P'(A,k) = 1/S.T.


Figure 4. HIGH RESOLUTION FREQUENCY-WAVENUMBER SPECTRAL ESTIMATE










optimum least squares sense. That is, P' provides a minimum variance,

unbiased estimate of the power. For each wavenumber, k it passes

undistorted any monochromatic plane wave with a velocity corresponding

to the wavenumber k and suppresses in an optimum least squares sense

all waves traveling at velocities other than ko

,Figure 4 gives a flow chart outlining the procedure for obtaining

the high resolution frequency-wavenumber spectral estimate. The major

difference between it and the conventional technique is the inclusion

of the matrix inversion step. Since [?it] is in general complex,

additional care must be exercised in implementing the technique.


E. Three-Dimensional Frequency-Wavenumber Spectrum

The theoretical development of the frequency-wavenumber spectrum

is not limited to a specific number of dimensions. Therefore the equations

developed for the two-dimensional frequency-wavenumber spectrum are

directly applicable to detection of three-dimensional wavefronts if the

vectors k and x are extended to include components in three dimensions.

The detection of wavefronts in three dimensions would be particularly

useful for the measurement of seismological events via earth body waves

and the measurement of EEG signals which can certainly be considered to

possess three-dimensional characteristics. Although the theoretical

development is straightforward, difficulties arise in the implementation

of a three-dimensional technique. A pictorial representation of three-

dimensional wavefront characteristics is given below. Then some of the

implementation difficulties are discussed further.

Figure 5 shows how a plane wave might be represented in three

dimensions. Determination of the three wavenumber components kx, ky ,










z



k

k /


-A
a "


-y


mA Azimuth Angle
x aE Elevation Angle

Vector Wavenumber (Direction
of Propagation)

ISOTROPIC, PLANE WAVE IN NON-ATTENUATING THREE-DIMENSIONAL MEDIA:

S(t, ) = exp [2rri(ft *.)]

S(t,x,y,z) = exp [2ri(ft k x k y k z)]
x y z

FREQUENCY-WAVENUMBER SPECTRUM:


P'(A,k) = I qij(A) exp [2wik*(k, .)]
Lj=l X=1 ]


WHERE: [q.Z(A)] = inverse of the cross power spectral density
matrix


Figure 5. THREE-DIMENSIONAL PLANE WAVE











and k specifies not only the azimuth and phase velocity of the signal
z
but provides the added advantage of specifying the elevation angle

and group velocity of the signal. Determination of the wavenumber com-

ponents is accomplished through the estimation of the three-dimensional

frequency-wavenumber spectrum.

The major problems involved in the extension of the frequency-

wavenumber spectrum to three dimensions are complexity of implementation,

increased cost, how to display the data in a meaningful way and how to

design sensor arrays to best sense data in specific applications such

as biological preparations. An order of magnitude increase in computa-

tion time and thus cost results from computation of the spatial transform

in three dimensions over that required for computation in two dimensions.

Fast Fourier transform (FFT) techniques are not readily applicable to

the spatial transform operation. Since most FFT algorithms require that

the number of data points be a power of two and since all FFT techniques

are based on data points at equal sampling intervals, the use of these

techniques places stringent requirements on the sensor arrays to be used.

Equal spacing of electrodes for the measurement of biological signals

may in some cases be impossible or even undesirable. If this is so then

a more general technique for determining the frequency-wavenumber spectrum

is required, which precludes use of the fast Fourier transform.

Extension of the frequency-wavenumber technique to three dimensions

was accomplished and test cases were run using both known input signals

and data recorded from the rat neocortex using a three-dimensional electrode

array. Details of the results of the three-dimensional frequency-wavenumber

analysis are discussed in Chapter V for the known input signal.










Meaningful display methods for the three-dimensional spectrum

depend to some extent on the wavefront complexity. A method which

provides two-dimensional spectra for sequential values of the third

dimension was used for all three-dimensional data runs because of its

ease of comparison with the two-dimensional data runs. This method

worked quite well for simple, known input signals; however, for more

complicated signals it proved to be less clear for demonstrating

wavefront directions.

The particular array geometry to be used for the three-

dimensional spectral estimate is strongly dependent on the specific

application. However, to demonstrate many of the principles and

parametric effects, a simple cubic array geometry with known input

signals was used. For practical application of the technique to

remotely sensed spatio-temporal signals such as the EEG, certain

assumptions about the characteristics of the signal were made.

These assumptions are discussed in Chapter VI along with experimental

results.


F. Finite Bandpass Signals

Recall from the equations defining the conventional and high

resolution frequency-wavenumber spectral estimates that both are func-

tions of X the normalized frequency. This says that for each frequency

contained in the signal a different wavenumber spectrum can be computed.

Clearly this procedure is not practical for the case where exact

frequency content is not known or where a large range of frequency

components is known to compose the signal. The question then arises

as to how the technique may be modified to accept a narrow finite band

of frequencies.










Consider the case of a composite signal given by



s = i s (t,xy,z) (17)
J


where: s (t,x,y,z) = exp [2?i(f t k x k y kz z)] (18)


In order for the technique to identify the components of the signal,

it must compute the cross power spectral density estimate which includes

all the frequencies, fj. For a small number of frequencies this may

be accomplished by computing each component and then adding to get the

total. This particular technique was tested for the case of two

frequency components and appeared to introduce very little distortion

into the frequency-wavenumber spectral estimate. Up to five discrete

frequencies have been included in the spectral estimation technique

with only a minor increase in computation time. For large numbers

of discrete components or where interest in finite bandpass signals

exists an integration technique on the output from an FFT algorithm

can be used. The inclusion of multiple frequencies introduces some

distortion into the estimation of the frequency-wavenumber spectrum,

as does the finite bandpass technique. The extent of this distortion

is determined by a complex interaction of parameters including incoherent

noise ratio, data length, and temporal window function.

A method for including finite bandpass data was developed which

uses the average integrated spectral output over the selected spectral

band. The IBM subroutine, HARM, was used in the fast Fourier transform

spectral computations. If the output of HARM is as illustrated in










Figure 6 where the spectrum shown is for the jth sensor, mth data

segment, then an estimate for the bandpass spectral content is


HIGH
Fjm(A) = Ai*Af (19)
i=ILOW


where: Ai = spectral value at ith frequency

ILOW =FLOW
Af

FHIGH
HIGH = F H+ 1

FLOW = low frequency cuton

FHIGH = high frequency cutoff

1
Af = frequency resolution =
record length (sec)

The Fjm(A) are then used in the direct segment method for computing

cross power spectral density.

It is important to note that the inclusion of multiple frequencies

either as discrete components or finite bands introduces ambiguity into

determination of phase velocities or group velocities. However, if

wavefront directions are of prime importance then this method of accepting

multiple frequencies will provide more information more efficiently.












































H 4-1


H-I


























H4-4

II II
H 44


0
CO
Cx















CHAPTER IV

FREQUENCY-WAVENUMBER SPECTRUM--PARAMETRIC ANALYSIS

In general any spectral analysis technique should produce an

estimate that is statistically reliable and which has high resolution.

It is of interest therefore to identify parameters affecting the

estimate and to quantify their effects on resolution, statistical

reliability, and other measures of goodness of the estimate.

The spatial and temporal sample intervals, Ax and AT, are

determined primarily by the expected bandwidths in their respective

transform domains. These sample intervals are chosen to reduce

aliasing in the transform domain of the periodically repeated

spectrum. Temporal resolution is shown to be inversely proportional

to record length and spatial resolution is inversely proportional to

array aperture. Therefore for fixed sample intervals, Ax and AT,

resolution is improved by increasing the number of sample values.

The effect of an increased number of sample values on cost is partic-

ularly important when discrete signals in time and three spatial

coordinates are considered such as those for the three-dimensional

frequency-wavenumber spectral estimate.

Another parameter which results from choosing the direct segment

method for estimating the cross power spectral density matrix is the

number of segments M into which each data record is divided. The

value of M has an effect on the statistical reliability of the frequency-










wavenumber spectral estimate and should be large, which implies more

data points and greater cost so a trade-off is indicated.

Temporal window design can be utilized to reduce sidelobe

leakage and in some cases improve resolution. Various window func-

tions and their characteristics are discussed.

Included under the parametric analysis are discussions of

spherical wavefronts in relation to the objective of determining

wavefront direction and spatial pre-filters in relation to expected

maximum wavenumbers.


A. Comparison of. High Resolution and Conventional Estimates

It was stated that Capon's high resolution spatial window

function improves resolution over that obtainable with a conventional

window function. It is therefore of interest to give a quantitative

comparison of the conventional and high resolution estimates in rela-

tion to achievable resolution. The following analysis is derived

largely from Capon's paper [8].

Consider a single plane wave (wavenumber = k ) propagating

across an array of sensors plus a component of noise in each sensor

which is incoherent between any pair of sensors. Assume that M, the

number of data segments per sensor, and N, the number of data points

per segment, are large so that an unbiased, consistent estimate of

the frequency-wavenumber spectrum is obtained. Then the cross power

spectral density matrix elements are estimated as


f ,(Xo) = 6j(R) exp [-i2'Tok(x xi)] j,A = 1,---,K (20)










where: A = 2wf T
o o
f = temporal frequency of incident wave (Hz)

lo = vector wavenumber of incident wave (cycles/m)

x = vector position coordinates of jth sensor

and

6 z(R) = 1 ; J =
(21)
= l-R; j A


R incoherent noise power (22)
total power of sensor output


Using the above expression for the input signal and the equations for

conventional and high resolution estimates of the frequency-wavenumber

spectrum it can be shown that


P( =,o) = (1-R)IB(Ako)2 + R/K (23)

and

p( ) R 1-R + (R/K) (24)
S1-R + 2(R/K) Po(c)


where: Ar = k (25)
o o

and
K
B(k) = exp [27ri*rc ] (26)
S=1

and K equals the number of sensors. For K =
o


P(ZE) = P' (o,Eo) = l-R + R/K


(27)










In the neighborhood of k consider those values of k such

that the conventional estimate gives a value close to its peak value

of P(Xk).

Since R is small (0 : R 1) and K is usually large (about 16)

the neighborhood for the conventional estimate can be quite large.

For these values of k, the high resolution estimate P'(A ,k) becomes


p .(o R 1-R + (R/K)
P'(A k) = (28)
=o K I-R + 2(R/K) (l-R)

or

1
P'(Ao,k) =- (1-R + R/K) (29)


which is already 3 db down from its peak value. Thus P' has a much

steeper falloff and gives a higher resolution estimate for the

frequency-wavenumber spectrum.


B. Array Geometry

The data to be represented by their frequency-wavenumber spectrum

consist of finite groups of spatio-temporal samples. That is, spatial

sampling is a result of a finite number of discrete sensors located

within a finite aperture. Temporal sampling is the result of discrete

sampling of a finite time record from each sensor. The relationship

between these parameters and the frequency-wavenumber spectrum is given

by an application of the sampling theorem, modified to include multi-

dimensional signals. For example the transform pair represented in

Figure 7 demonstrates properties which hold for any bandlimited,

truncated, discrete signal which includes temporal and spatial signals.



























IT 2B
^1/T -j<


-1/AT

REQUIRE: AT S 1/2B

RESOLUTION IN $ PLANE = 1/T

Figure 7. SAMPLING THEOREM RELATIONSHIPS


In general, the spectral resolution varies inversely with the record

length and the required sample interval varies inversely with the

bandwidth of the signal.

Sampling theorem concepts can be applied to determine the

effects of array geometry on the resolution of the frequency-wavenumber

spectrum. Application of these concepts is, however, complicated by

the additional property of dimensionality for spatial fields. For

example, equal sample intervals for a spatial field has meaning only

if referenced to some direction in the field. For a three-dimensional

field, equal sampling intervals can be achieved in directions represented

by three perpendicular coordinate axes by constructing a cubic sensor

array. However, given no physical limitations on the array size or










shape, Petersen and Middleton [13] have shown that the optimum array

(i.e., minimum number of sample points per unit hypervolume) is not

in general rectangular. The optimum array is represented by the

centers of hyperspheres in their closest packed arrangement. For

two dimensions the optimum array is rhombic and for three dimensions

the tightest packing of spheres is achieved with a hexagonal-close-

packed configuration. These arrangements not only provide maximum

sampling rates for a given number of electrodes but also provide

equal sample intervals between any two adjacent sample points. For

an equal number of elements and a given spatial sampling rate, arrays

other than these have a reduced array aperture and resolution is not

as good.

For any particular array design the sampling theorem relation-

ship still holds that resolution in a given wavenumber component is

inversely proportional to the aperture of the array in that dimension.

In most applications the array design is determined by physical

limitations of the recording area. This is particularly true in the

case of human EEG recordings in which the array geometry is determined

by scalp contour. The rationale for array geometries used in the

experimental data examples is given in Section G and in Chapter VI.


C. Temporal Window Functions

It was shown in Chapter III that the spectrum of a truncated

signal is given by the convolution of the signal transform with the

transform of the window function. An ideal temporal window function

would require an infinite length of time; therefore, it is of interest

to investigate various finite temporal window functions relative to












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o
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their sidelobe levels, bandwidths, sharpness of cutoff, and statistical

reliability. The simplest window function is a rectangle which has a
sin x
transform of the form In general no window function is optimum.

Rather, the best window design for a particular application depends on

the requirements for resolution, statistical reliability and sidelobe

leakage.

Several window functions and their characteristics are shown

in Figure 8. For the frequency-wavenumber analyses two window functions

were used. The rectangular window function was chosen for its simplicity

and as a basis for comparison. The cosine tapered window function was

also used. Because of its reduced sidelobe level the cosine window is

expected to provide a more reliable estimate than the rectangular window.

For the frequency-wavenumber spectral estimate, the particular temporal

window used had little effect on spectral resolution in comparison with

other parameters such as additive incoherent noise level and array

aperture. Effects of the various parameters on resolution are demon-

strated in Chapter V.


D. Additive Incoherent Noise

Since the high resolution estimate requires a matrix inversion

of the cross power spectral density matrix [f], it is of importance

to determine under what conditions this inverse exists.

From matrix theory it is known that an Hermitian matrix has

an inverse if the matrix is positive definite [15]. Capon et al. [7]

have shown that the cross power spectral density matrix [f] when

computed by the block averaging method is non-negative definite.

However, this is not sufficient to guarantee that an inverse exists.










In fact, if the number of data blocks M is less than the number of

sensors K then the matrix is singular. This comes from the fact

that [f] is the sum of M matrices of rank unity.


M
[2] = [q ] (30)
m=l


where: [q ] is a 1 x K row matrix whose jth element is given by


N inX
mJ n=~ jn+N(m-1 ) e (31)


Since the rank of [f] cannot exceed the sum of the ranks of its

constituent parts, then [f] is a K order matrix with rank M. Clearly

if M < K the matrix is singular. Thus a necessary but still not

sufficient condition for [2] to have an inverse is that M a K.

To insure that an inverse exists the matrix [r] is modified

to give the matrix f' which can be shown to be positive definite.



[f'] = (1 R)[f] + R[I] (32)



where: R = ratio of incoherent noise power to total power at a

sensor

and: [I] is the identity matrix

Thus [f'] is a K x K matrix with elements given by


f' = (1-R) ? +R; m n
mn mm
m n mm (33)
= (l-R) fm ; m n









Now [f'] is positive definite if for any function aj, the quadratic

form Q given by


K K
Q = a Bafja (34)
j=1 =l

is positive. Using the definition for [f] and [f'], we have



1-R M K N inX K
-L im Il J ajSjn+N(m.-) e + R I jajl
j=1

7r S X 5 7T (35)


The first term above is always non-negative for it represents

a positive constant (l-R) multiplied by the quadratic form for [2]

which has previously been demonstrated to be non-negative definite.

Thus the only way for Q to be zero is if


K 2 =
Slaj12 0 (36)
j=l

But this implies that aj = 0; j = 1,---,K which is a contradiction.

Therefore [f'] is positive definite and its inverse exists.

The quantitative dependence of the high resolution frequency-

wavenumber estimate on the magnitude of additive incoherent noise is

derived from expressions given in Section A and is given by


pA ) R 1-R + (R/K)
p o' K------ +K 2Wi(-) (37)
1K -R + (R/K) (1-R) I e o j










Recall that the peak value for P'(o ,k=k o) is


P'(o,(0) = 1-R + (R/K) (38)


so that



P'(Ao' ) R/K
_____ 0 _R/K (39)
P'(A ,k ~ *K 2Ti(k-o 2
(R/K) + (1-R) 1 1 e e
j=l


Effects on resolution of the additive incoherent noise are

demonstrated in Chapter V by artificially injecting noise into a

known signal. ( See also Appendix F.)


E. Statistical Reliability

Two desirable properties of any estimator are that it be un-

biased and consistent. For the random process P it is then required

of the estimate P that


E[PN] = P (unbiased)
(40)

lim E[(P FN)2] = 0 (consistent)
N-Kx

That is, the mean value of the estimate should equal the mean value of

the random process and as the number of observations goes to infinity,

the mean square error between the estimate and the random process should

go to zero.

Capon and Goodman [16] show that the conventional and high resolu-

tion frequency-wavenumber spectral estimates are asymptotically unbiased,










that is, in the limit as N (number of data points per segment) gets

large the bias approaches zero. The expected value for the conven-

tional estimate is given by



E[P(X )] = f f P(x,) B(i -k 12 (-A)2 dk (41)
-7 --o0
Vector

K i27trk
where: B(k) =i e (42)
j=1


is the beamforming response pattern of the array and,

2
WN(x)I2 1 sin (N/2)x (43)
sin x

is the Bartlett window resulting from the rectangular time window chosen

for the temporal transform.

Thus P will be an unbiased estimator for P if in the limit the

frequency-wavenumber window approaches a delta function.



f J IWN(x-X)22 B(k-ko I E dk- 6(x-Xo) 6(-k) (44)

Vector


Following Goodman [17] and Capon and Goodman [16], it can be

shown using the Complex Wishart distribution that the high resolution

estimate is unbiased if it is multiplied by M/(M-K+1). That is the

expected value of P' is










= M-K+ P(x,) IB,(X,r,,zo)I 2- IWN(x-_X)12 dx d9

-7r
Vector (45)


K
where: B'(,k,k ) = 1 A.(A,iK ) exp [27ri(K- )* ] (46)
j=l

is the high resolution beamforming pattern of the array and is a func-

tion of Z Since in many cases all spectral values are normalized,

the question of bias is not so important. Also for M >> K the high

resolution estimate gives the same expected value as the conventional

estimate.

Assuming that the noise received by the array is a multi-

dimensional Gaussian process, it can be shown for large N that the

variance of the conventional estimate is given by



VAR [P(A,k)] = {E[P(A, )]}2 ; k = o
(47)
E[)1 ]}2


Since the variance goes to zero as M goes to infinity the estimate is

consistent. Again using the Complex Wishart distribution it can be

shown that the high resolution estimate is a multiple of a chi-square

variable with 2(M-K+1) degrees of freedom and variance given by


VAR [P'(A,k)] = M- K EP'+(X,)]2 (48)


Thus both the conventional and high resolution estimates for the

frequency-wavenumber spectrum are consistent and can be made to be











unbiased. Details of the Complex Wishart distribution and derivations

of the means and variances for the estimates are given in [16], [17].


F. Spherical Wavefronts

The development of the frequency-wavenumber spectrum has been

based on the assumption that incoming signals may be represented by

sums of plane waves. Since plane waves are represented by vector

wavenumbers the frequency-wavenumber spectrum determines wavefront

direction and, for a given temporal frequency, the velocity of the

wavefront. In many applications such as measurement of seismic

events from great distances through the earth's crust, the assump-

tion of plane wavefronts is a good one indeed. For electrophysiological

data, where smaller dimensions are found, the assumption of plane wave-

fronts is probably less valid; however, such an assumption may not

invalidate the usefulness of the computed frequency-wavenumber spectrum.

An analysis of the frequency-wavenumber method applied to spher-

ical wavefronts indicated that velocity and wavenumber magnitude are

determined but wavefront direction has little meaning. This comes from

the fact that the wavenumber for a spherically symmetric wavefront is

a scalar quantity. Thus if the spherical wavefront is given by

2ri(k r-f t)
S(r) = e r (49)



where: k = scalar wavenumber (cycles/cm)

f = temporal frequency (Hz)

o = 27rf
0 0











then it is shown in Appendix C that the frequency-wavenumber technique

gives a delta function about k = k and X = .



P(X,k) = 6(A-Xo) 6(k-ko) (50)


where: P(A,k) = frequency-wavenumber spectrum

From the results given, it is clear that the frequency-wavenumber

technique does not determine wavefront direction for the spherical wave-

front. For the case of spherical wavefronts, the objective should be

to determine coordinates for wave sources relative to the array of

sensors. Accomplishment of this objective requires the use of methods

other than the frequency-wavenumber method.

To apply the frequency-wavenumber method to the analysis of

electrophysiological data it is assumed that the signals may be

represented by approximate plane wavefronts resulting from sources at

distances large compared to the array size or by a summation of plane

wavefronts.


G. Spatial Pre-filtering

In order to reduce aliasing effects it is important to know the

signal bandwidths (spatial and temporal) to establish maximum sample

intervals. It has been established that most EEG temporal frequencies

are below 50 Hz with most of the energy at 2 Hz or below for averaged

visual evoked responses [18]. No limits have been established for

spatial frequencies; therefore, some exploratory flexibility is

maintained for initial experiments with spatial signals. Spatial










pre-filtering to limit wavenumber content and reduce errors due to

aliasing in the spatial frequency domain was considered for applica-

tion to the experimental data. The maximum wavenumber content

allowable without spectral overlap was determined by electrode

spacing for the various experimental data runs. Two types of

experimental data were used: 1. human visual-evoked response data

recorded previously by the University of Florida Visual Sciences

Laboratory, and 2. data recorded from the neocortex of rats.

Minimum electrode spacing for the visual-evoked response (VER)

data was two centimeters and had been determined previously by

crosstalk considerations for scalp measurements [19]. Minimum

spacing for the rat data was determined by available precision for

array construction to be one millimeter.

Digital spatial filters for pre-filtering the experimental

data were not used for two reasons: 1. restrictions on array

geometry (uniform sampling rates) imposed by available multi-

dimensional filtering techniques would make application of the filter

to the VER data impossible and 2. the current state-of-the-art for

implementation of digital spatial pre-filters is such that their

characteristics are not well understood.

Rather than introduce ambiguity and possible distortion into

the experimental data, an alternate method for checking wavenumber

content was used. The method involves a comparison of the wavenumber

spectra for a given data set which has been computed for two different

spatial sampling rates. Such a comparison was made for data from a

rat in which three by three matrices of two and one millimeter grids










respectively were used for recording the data. This method allows

detection of aliasing errors which would indicate that a higher spatial

sampling rate was required.

It is expected that incorporation of spatial pre-filtering into

the analysis technique would eventually be desirable. For further

information on digital filtering and spatial filtering the reader is

referred to [20] and [21].


H. Cost and Computation Requirements

Many factors were involved in the determination of computation

time and cost for individual computer runs; however, some factors

appeared to dominate over others. Computation time and therefore cost

was proportional to the number of electrodes, the total number of

wavenumber points in the output spectrum, and the total number of

sample points of the spatio-temporal field (K x M x N). For the band-

pass computation method, increases in N should have less effect because

of the use of the FFT algorithm in computing power spectral estimates

of the N temporal data samples. This was verified by experimental

results. The savings in computation time by use of the bandpass method

was low but is expected to increase as the number of time samples is

increased.

Shown in Table 2 are some of the time and cost figures for

various experimental runs. These values could be reduced by the use

of a binary program deck which would eliminate compilation time.

This was not done for the experimental investigation because of frequent

program changes. Also, the requirement that several arrays in the pro-

gram must be dimensioned exactly make the use of an object deck impractical



















00





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47


unless a fixed electrode geometry were to be used in analyzing numerous

experimental data trials.

The first two lines in Table 2 are representative of the two-

dimensional simulation experiments. The VER data program included an

interpolation routine which added to the required computation time.















CHAPTER V

FREQUENCY-WAVENUMBER SPECTRUM--SIMULATION

The validity of the concepts of the frequency-wavenumber

analysis technique was tested via a series of simulation experiments.

Characteristics of the frequency-wavenumber spectral estimate were

observed for known signal and noise conditions. The effects of

certain parametric variations were demonstrated by these experiments

and theoretical concepts were verified by the results.

Because of its relative simplicity and lower cost to implement,

a two-dimensional analysis was used for those experiments in which the

principle to be demonstrated did not depend on the use of three-

dimensional computations. Relative cost versus performance was dis-

cussed in Chapter IV. The simulation experiments consisted of software

implementation of the frequency-wavenumber analysis technique. The

computer language used was FORTRAN IV and runs were made on an IBM 360/65

computer. Listings of the various programs used are given in Appendix E.

Storage requirements for the program were large (120,000 locations for

a two-dimensional analysis of a 16-electrode array and 256,000 locations

for a three-dimensional analysis of a 16-electrode array); therefore,

implementation on a small computer would not be feasible without suitable

program segmentation. For fewer electrodes this might be possible and

could provide useful results. However, as indicated by the results of

simulation experiments, the technique worked best for large numbers of

electrodes.










A. Array and Signal Example Parameters

The array used in the two-dimensional simulation experiments

is shown in Figure 9. It is a two-dimensional rectangular array of

sensors with equal spacing in both dimensions. The 4 x 4 (16-electrode)

array was used for most of the experiments; however, the 2 x 2 (circled

electrodes) array was used for some of the experiments to test the

resolving power of the array for limited spatial information.

Shown also in Figure 9 are the signal parameters used in the

two-dimensional simulation experiments. Note that the spatial and

temporal sampling rates were conservatively high at eight times the

signal frequency and wavenumber instead of twice the maximum frequency

as required by the sampling theorem. For analysis of practical and

non-artificial data, sampling rates slightly higher than twice the

Nyquist rate were used to reduce aliasing effects caused by non-ideal

bandlimited and wavenumber limited signals. Application of the

frequency-wavenumber technique to electrophysiological data is dis-

cussed in Chapter VI. It is emphasized that the achievable resolution

for a given number of sensors and a given number of data points is

improved for sample rates nearer to twice the Nyquist rate than that

achieved with a sample rate of eight times the maximum signal frequencies.

Computer results consisting of two-dimensional plots of the

frequency-wavenumber spectral estimates for the various simulated condi-

tions are included in this chapter along with perspective plots and

cross-sectional views. The directions shown in Figure 9 for the vector

wavenumbers were selected so that a vector drawn from the peak value in

the frequency-wavenumber spectral plot to the wavenumber origin represents









y

X_- --X- -X- -X
I I I
I I I
I I I


[-ye = 2 cm.


-I--X- -X- --X


--- --4- --


-.4


-- xS = 2 cm.


- r


a) ARRAY GEOMETRY


S1 = cos [2lr(flt -

S2 = cos [2r(f2t -

53 = cos [27T(f3t -


Bix yly)]

2x Y2y)]

03x y3y)]


COMPOSITE SIGNALS

S = Sl + S2

S = S1 + S3


TS = 0.01 sec = sample time

fl = 1/8 Ts 12.5 Hz

f2,f3 = 12.5 40 Hz variable


81 = -.0625 cycle/cm
Y, = +.0625 cycle/cm
Sl


02 = +.0625 cycle/cm 83 = +.0625 cycle/cm
Y2 = +.0625 cycle/cm y3 = -.0625 cycle/cm
S2 S3


b) SIGNAL PARAMETERS


Figure 9. TWO-DIMENSIONAL ARRAY GEOMETRY AND PARAMETERS










the wavefront direction. The particular convention selected is not

important except that interpretation of results must be consistent.

Array geometry and signal parameters used in the three-dimensional

simulation experiment are discussed in Section G. Results of the three-

dimensional simulation are given in the form of a series of two-dimensional

wavenumber plots for x and y wavenumber components with the z wavenumber

component being constant for a given plot.


B. Comparison of High Resolution and Conventional Estimates

For the first experiment a comparison was made of the obtainable

resolution by the high resolution estimation procedure relative to that

obtainable with the conventional estimation procedure.

A perspective plot on a two-dimensional wavenumber grid is shown

in Figure 10 for results obtained by the two methods. The peak for both

methods occurs at kx = -.0625 and k = +.0625 as it should. It is clear
x y

that the high resolution method provides a sharper peak and thus improves

the resolution over the conventional method. One important point to be

emphasized is the role of noise in the high resolution estimate. As was

discussed in Chapter IV, a small amount of incoherent noise is added to

the cross power spectral density matrix to insure that its inverse exists.

The level of the injected noise has an effect on the resolution obtained

in the high resolution estimate. Thus the results of Figure 10 are

noise-free for the conventional estimate and noisy (10% in this case)

for the high resolution estimate. It is shown later in the experimental

results that the high resolution method can be improved even more when

the noise level is reduced.






















k
-0625 x

y /

I /


CONVENTIONAL FREQUENCY-WAVENUMBER SPECTRAL ESTIMATE



k





.0625


Sx
-.0625





HIGH RESOLUTION FREQUENCY-WAVENUMBER SPECTRAL ESTIMATE

Figure 10. CONVENTIONAL AND HIGH RESOLUTION PERSPECTIVE PLOTS










Shown in Figure 11 is a two-dimensional plot of values for the

conventional frequency-wavenumber spectral estimate. The values shown

are in db attenuation so that the peak occurs at a value of zero.

Only integer values are printed and the actual spectral values are

rounded to their lowest integer values. Figure 12 shows a plot of the

high resolution estimate under identical signal conditions. For clarity

values of attenuation larger than 20 db are not printed.

Figure 13 shows a section view of the results obtained by the

two methods. This is a plot of spectral values along the k = .0625

line.


C. Array Geometry

From the sampling theorem it can be shown that resolution is

inversely proportional to the data length. For temporal signals the

data length is taken to be the record length in seconds and the

resultant effect is on frequency resolution. For spatial transforms

the data length is equivalent to array aperture in meters and the

resultant effect is on wavenumber resolution.

An experimental run was made to test the effect of array

aperture on wavenumber resolution. In one case a 16-electrode array

was used with an aperture of 3xs by 3ys (see Figure 9). In the other

case a 4-electrode array with an aperture of xs by ys, that is, one-

third as large in x and y directions as the 16-electrode array was

used. A section view of the results obtained from the high resolution

method for these two arrays is shown in Figure 14. The three-db band-

widths are indicated and it is observed that for equal noise levels

an aperture which is three times larger yields a resolution bandwidth




































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16 electrodes



S-- Conventional


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Figure 13.


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1.0 k = +.0625 cycles/cm

0.707 high resolution only

0.7074 electrodes

^---4 electrodes


-.0625 0 +.0625


RELATIVE EFFECT OF ARRAY APERTURE ON RESOLUTION


Figure 14.










which is one-third as large. Thus experimental results agree with

expected results as implied by the sampling theorem.

Additional indication of the resolution dependence on aperture

is demonstrated by comparing Figure 12 with Figure 15 which gives the

high resolution attenuation plot for the 4-electrode example.


D. Temporal Window Functions

Two temporal window functions were tested to determine their

relative effects on the frequency-wavenumber spectral estimate. The

rectangular and cosine tapered window functions used in the simulation

runs are shown in Figure 16. The temporal data segments were multiplied

by one of these two window functions before Fourier transformation both

for the discrete frequency computations and fast Fourier transform (FFT)

computations.

For the discrete frequency computation method and a relatively

high incoherent noise ratio the resolution of the frequency-wavenumber

spectrum was the same for both rectangular and cosine tapered temporal

window functions. For a ratio of incoherent noise power to total

signal plus noise power of R = 0.1 and discrete frequency computation,

the frequency-wavenumber spectrum of two plane wavefronts was identical

for both the rectangular and cosine tapered temporal windows. The

spectrum for the rectangular window is shown in Figure 17 and the

spectrum for the cosine tapered window is shown in Figure 18.

As the incoherent noise ratio is reduced, some smearing is

introduced into the two-wavefront spectrum for both rectangular and

cosine tapered temporal windows. Figure 19 shows the spectrum for
a rectangular window with an incoherent noise ratio of R = 10-4
a rectangular window with an incoherent noise ratio of R = 10



































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Figure 20 shows the spectrum under identical conditions except that

a cosine tapered window function was used. The figures show that

smearing is only slightly higher for the cosine tapered window.

The above results apply only to the simulation runs where

discrete frequency computations were used to compute the cross power

spectral density. For the bandpass method, the results are different

and less predictable. It is shown in Section F that for the bandpass

spectral computation, the cosine tapered window function causes less

error due to sidelobe interaction and produces a frequency-wavenumber

spectrum with more clearly distinguishable wavefront directions.

Further quantitative effects of incoherent noise ratio on resolution

are discussed in Section E.


E. Additive Incoherent Noise

The relative effect of additive incoherent noise on the

resolution obtained with the high resolution frequency-wavenumber

technique was tested by adding known amounts of noise to the cross

power spectral density matrix before inversion. For the example

signals used, this proved to be a necessary step to insure non-

singularity of the matrix. Values of the ratio of incoherent noise

power to total signal plus noise power of R = 0.0001, 0.001, 0.01 and

0.1 were used to determine the effect of incoherent noise power ratio

on resolution. A cross-sectional view of spectra obtained for a

single plane wavefront and for various values of R is shown in

Figure 21. Shon in Figure 22 are the spectral profiles for a signal

composed of two plane wavefronts.






64
















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Figure 21. EFFECT OF INCOHERENT NOISE ON SINGLE WAVEFRONT


1.0


-.0625 +.0625


Figure 22. EFFECT OF INCOHERENT NOISE ON MULTIPLE WAVEFRONTS










Ideally it is desirable to have zero noise level; however, the

matrix inversion requires in many cases a finite noise level to insure

non-singularity ( See Appendix F ). This was true of the simple plane

wavefront example signals used in the simulation. For the example

signals and the matrix inversion subroutine used, a minimum value of

R = 6.0 x 10-5 was found to be necessary for the matrix to be non-

singular. However, because of the spectral smearing at low noise levels

a value of R = 0.001 to 0.01 was found to be best. For data recorded

from practical sensor arrays it is expected that any superficial noise

addition would not be necessary because of the inherent noise in the

data and its general complexity. This was verified by results obtained

from application of the frequency-wavenumber technique to the analysis

of electrophysiological data. As shown in Figures 21 and 22 resolution

is improved for both single and multiple component signals as the noise

is reduced. However, there is a non-linear interaction present which

causes the resolution to improve less if multiple wavefronts compose

the signal. Figure 23 shows the high resolution db attenuation plot

for one wavefront incident on the 16-electrode array. The incoherent

noise ratio is R = 0.001. The improvement in resolution for reduced

noise level can be seen by comparing this figure with Figure 12 where

the noise ratio was R = 0.1. Figure 23 shows that even the closest

wavenumber points to the actual signal values had spectral magnitudes

that were greater than 20 db down from the peak value. Further reductions

in noise are possible; however, noise ratios less than R = 0.001 were

found to introduce smearing into the spectral estimate in the form of

sidelobe enhancement when more than one wavefront comprised the signal.






67



















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Spectral plots for two-wavefront signals are shown in Figures 24

and 25 for incoherent noise ratios of R = 0.01 and R = 0.001 respectively.

Cosine tapered window functions were applied to the temporal data before

transformation. As was mentioned in Section D the spectral smearing seen

in Figures 24 and 25 was only slightly greater (within 1 db) than that

observed with a rectangular temporal window.


F. Bandpass Signal Spectral Computation

The bandpass spectral computation method described in Chapter IV

was simulated by applying the method to signals representing single

plane wavefronts.

For a signal represented by S1 in Figure 9 the frequency-wavenumber

spectrum was computed using the bandpass method for a passband from six

to twenty Hz. The spectrum was computed for both a rectangular temporal

window and a cosine tapered temporal window with respective spectral

plots as shown in Figures 26 and 27. Spectral sharpness is best for the

rectangular temporal window as it was for the method using discrete

computation of the cross power spectral density. However, for signals

containing multiple wavefronts and for low incoherent noise levels the

bandpass spectrum using a cosine tapered window is more reliable than

the bandpass spectrum using a rectangular temporal window. This

conclusion is based on simulation experiments for the two window func-

tions in which the two wavefronts were more distinguishable for the

cosine tapered temporal window. This might be expected since cosine

tapering is a form of smoothing and should produce a spectrum with less

error due to sidelobe effects. An unexplained dependence on specific

wavenumber content of the computed spectrum of multiple-wavefront







69

















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signals was also observed. For example a signal composed of wavefronts

S1 and S3 (see Figure 9) was successfully resolved by the frequency-

wavenumber method; whereas, a signal composed of wavefronts Sl and S2

produced an ambiguous spectrum for the same frequency passband. It was

found, however, that judicious selection of the passband effectively

reduced ambiguity in the frequency-wavenunber spectrum. For example,

the composite signal of S1 (12.5 Hz) and S2 (15 Hz) was poorly resolved

by the frequency-wavenumber spectrum for a passband of six to twenty Hz

regardless of the value of incoherent noise injected into the signal or

the choice of temporal window. The same example was run with a different

passband (11.5 to 16 Hz) which produced an improved estimate of the

frequency-wavenumber spectrum over that obtained with the larger pass-

band. Figure 28 shows the narrow bandpass spectrum for a rectangular

window and Figure 29 shows the narrow bandpass spectrum for a cosine

tapered window. A sharper spectrum is achieved with the rectangular

window; however, a more accurate spectrum is obtained by using the cosine

tapered window. The results shown in Figures 28 and 29 are consistent

with the expected result of cosine tapering, which is a form of smoothing.

In general, smoothing produces a broader spectral estimate with higher

statistical reliability and less error due to sidelobes.

Thus the results of experiments using the bandpass computation

method indicate that the selected pEssband should be narrow and carefully

selected to improve the frequency-wavenumber spectral estimate; this

may call for spectrum analysis prior to data processing. Also for

improved reliability of the spectral estimate, a cosine tapered window

was better than a rectangular window when using the bandpass method.
























































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spectrum, both rectangular and cosine tapered windows produced similar

spectra,with the cosine window producing slightly (1 db) more smearing

for a low incoherent noise ratio.


G. Three-Dimensional Simulation

Because of the complexity and projected cost of three-dimensional

frequency-wavenumber computations and since most parametric interactions

were sufficiently demonstrated by a two-dimensional analysis only one

simulation experiment was designed to test the three-dimensional technique.

The signal consisted of a single three-dimensional plane wave-

front as shown in Figure 30. The 3ight-electrode cubic array shown in

Figure 30 was used in the analysis.

Since the signal was a single plane wavefront and by reducing

the incoherent noise ratio to R = 10-4 for improved resolution, signif-

icant savings in computation time and cost were possible by selecting

an eleven-point matrix of wavenumber values in three dimensions instead

of the twenty-one-point matrices used in the two-dimensional analyses.

Results of the three-dimensional frequency-wavenumber simulation

experiment are shown in Figure 31. Two-dimensional attenuation plots

were generated for an x and y wavenumber grid for specific values of

the z-direction wavenumber. In Figure 31a is shown schematically how

the spectral plots were generated. Figures 31b, c and d give the

attenuation plots for z-direction wavenumbers corresponding to the

signal wavenumber (center plot) and one wavenumber point each side of

the signal wavenumber. Because of the high resolution obtained for the

three-dimensional simulation all spectral values were greater than






77







Electrode spacing = 2 cm.


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a) ARRAY GEOMETRY


S = cos [2w(ft k x ky k z)]


f = 12.5 Hz

k = -.075 cycle/cm

k = +.075 cycle/cm

k = -.075 cycle/cm


b) SIGNAL PARAMETERS


Figure 30. THREE-DIMENSIONAL ARRAY GEOMETRY AND SIGNAL PARAMETERS




























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twenty db down from the peak value as indicated in Figure 31. For this

reason, only the three attenuation plots shown are included.

Thus the three-dimensional frequency-wavenumber spectral computa-

tion is demonstrated. For the application of this technique to multiple-

wavefront signals and bandpass computation the parametric interactions

discussed earlier should have analogous effects. Increases in array

aperture, number of data segments, and number of data points which are

required for improved resolution and statistical reliability produce a

greater increase in computation requirements and cost for the three-

dimensional analysis method.















CHAPTER VI

INVESTIGATION OF ELECTROPHYSIOLOGICAL DATA
BY FREQUENCY-WAVENUMBER TECHNIQUE

The frequency-wavenumber technique was applied to two types

of electrophysiological data to determine if the technique might

provide useful information about the data. Human visual-evoked

response (VER) EEG data and penicillin-induced focal epileptic

activity in the rat neocortex were analyzed via the frequency-

wavenumber method to demonstrate the application of the technique

to these data. Succint interpretation of the results appears

premature at this time; however, the results indicated that certain

characteristics of the data might be categorized by application of

the frequency-wavenumber spectral technique.


A. Electrophysiological Signal Characteristics

For the data analyzed, a power spectral analysis was first

performed to determine the temporal frequency content. Recall from

Chapter V that spectral smearing and ambiguity in the wavenumber

domain were less for those examples in which the selected temporal

frequency band most closely approximated the true frequency content

of the signal. The expected wavenumber content of EEG signals has

not yet been determined; however, physical constraints of electrode

size and array spacing determined the maximum wavenumbers which pro-

vided non-aliased spectra. Ideally the data should be subjected to











a wavenumber-limited spatial pre-filtering process prior to application

of the frequency-wavenumber spectrum. However, a more expedient method

for checking the wavenumber content was designed and tested using the

rat data. The method consists of computing the frequency-wavenumber

spectrum for electrode arrays of different spatial sampling rates and

then comparing the resultant spectra for differences due to aliasing.

The VER data had been previously recorded by the University

of Florida Visual Sciences Laboratory with an array geometry designed

for determining characteristics other than the expected wavenumber

content. For a description of the experimental method for recording

the VER and related discussions of its implications see [18], [22]

and [23]. Section B gives the frequency-wavenumber spectral plot for

VER data obtained from a subject with intermittent exotropy (subject

DLH).

For data recorded from the rat neocortex the spatial sampling

rate was restricted by the techniques available for the electrode

array manufacture and by the electrode diameter. The stainless steel

wire electrodes had a diameter of approximately 0.4 mm. Thus an array

of these electrodes in their closest spaced arrangement could efficiently

sample signals with wavenumbers no higher than 12.5 cycles/cm. To keep

tolerance errors low a three-electrode array with 2,0 mm spacing was

fabricated. Sampling intervals of 1.0 mm were obtained by micro-

positioning of the array. For the 1.0 mm sampling interval signals

with wavenumbers as high as 5.0 cycles/cm could be sampled without

aliasing errors. Spectra produced by the 2.0 mm and 1.0 mm sampling

grids were compared to determine relative errors due to aliasing.











Section C describes the results of the application of the frequency-

wavenumber technique to the analysis of focal epileptic discharges

from rat neocortex. Details of the experimental method employed for

collection and analysis of these data are presented in Appendix D.


B. Analysis of Human Visual-Evoked Response

Visual evoked response data recorded from an array of sixteen

electrodes were analyzed via the frequency-wavenumber method. Individual

signals represented averaged EEG responses to visual stimulation of the

subject and were recorded from scalp electrodes positioned over the

occipital region of the brain. The electrode array consisted of sixteen

electrodes located on two concentric circles of radii 2.0 cm and 4.0 cm

respectively. Eight electrodes were uniformly spaced on each circle.

An electrode at the center of the circles was used as a reference.

Electrical response activity was monitored for a period of 500 msec

after stimulus presentation and the average of fifty such responses

was used as the VER data for a given electrode position. All VER

responses for the various electrode positions were referenced to the

center electrode prior to analysis using the frequency-wavenumber

method.

Analysis of the VER data indicated a need for higher temporal

sampling rates than required by the sampling theorem for the expected

maximum temporal frequency content of the signals. This conclusion

was based on the requirement that each time record be divided into a

number of segments greater than or equal to the number of electrodes

(16 for the VER data) and on the observation that short time lapse











transient events comprised the visual-evoked response. Thus a large

number of data points for short observation times required an increased

sampling rate. Additional temporal sample points for the existing VER

data were obtained by a linear interpolation between the true sample

values.

An initial analysis of the VER data using the frequency-

wavenumber method was performed for temporal frequencies of 2.0, 5.0

and 10.0 Hz. The maximum wavenumber included in the analysis was

given by the inverse of four times the electrode spacing or 0.125

cycles/cm. The resultant frequency-wavenumber spectral estimate

for the 500 msec visual-evoked response is shown in Figure 32. The

fact that all spectral values were at either zero or greater than

twenty db attenuation levels is partially explained by subsequent

analyses in which the 500 msec response time is divided into two

250 msec time periods. The frequency-wavenumber spectrum for the

first 250 msec is shown in Figure 33 and the spectrum for the last

250 msec is shown in Figure 34. These computations were for temporal

frequencies of 8.0, 9.0, and 10.0 Hz to include observed power spectral

peaks near these frequencies. Inclusion of the 2.0 Hz temporal fre-

quency in the spectral computation for the 500 msec record length

probably contributed to the ambiguity observed in that spectrum.

Spectra for the subdivided response indicate that spatial signal

sources changed with time for the 500 msec observation period so that

the frequency-wavenumber spectrum for the total observation time

exhibited a smearing effect of different transient spatial sources.

This observation in conjunction with the required large number of








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data points for input to the frequency-wavenumber spectral estimation

technique formed the basis for the conclusion that higher temporal

sampling rates are required.

As indicated by Figure 33 the frequency-wavenumber spectrum for

the first 250 msec time period following stimulus presentation gave a

peak value at k = k = 0. This may be the result of a plane wave
x y
propagating in the km direction or may be the result of a synchronous

active discharge exhibited by neuronal populations under each electrode

and triggered by a common source. The latter explanation is considered

most likely since similar results were obtained with signals recorded

from the rat neocortex. Details of the result of application of the

frequency-wavenumber spectral estimation technique to analysis of

artificially induced focal epilepsy are discussed in Section C.

The frequency-wavenumber spectrum for the last 250 msec of

the VER exhibits bands of uniform maximum and minimum values. Again

this may be partially explained by attributing it to a temporal

smearing of numerous disparate spatial sources. If the problem is

analyzed in reverse by taking the inverse spatial Fourier transform

of a flat spectral band, a different possible explanation is obtained.

Given a frequency-wavenumber spectral plot as shown in Figure

35, the inverse spatial Fourier transform is defined by

00

f(xX) = f P(k,X) e-2ITi-x dk


W W (51)


= e 2i(kxx+k y dydx
oW 2 W
o 2