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SEGNAL EIEE'ECTEDN USING PER'I'URBF‘TION GPERATORS

fliesis $0; the. Same of Ph. D.
WCHIGAN STATE UNIVERSWY
MERE-S C. SALAZAR
196?

THE“

This is to certify that the

thesis entitled
STABILITY OF THE LIKELIHOOO
RATIO IN SIGNAL DETECTION USING

PERTURBATION OPERATORS
presented by

ANDRES CLARENCE SALAZAR

has been accepted towards fulfillment
of the requirements for

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ABSTRACT

STABILITY OF THE LIKELIHOOD RATIO
IN SIGNAL DETECTION USING PER TURBATION OPERATORS

by Andres C. Salazar

Attention has been focused in recent years on the continuity
of a signal detection scheme with respect to a small change in the
noise power. This continuity may correspond to the stability or
robustness of the stochastic decision-making hypothesis test.

The case for stability investigated here is that of detecting
a sure signal sent through zero-mean Gaussian noise with continuous
autocorrelation R(s, t) on the finite interval 0 _<_ t E T with a
maximum likelihood test. Once the entire detection scheme is set
on the LZI: O, T] mathematical platform, the operator RO
corresponding to the original noise autocorrelation is perturbed
with 6 R where e is a small real parameter and R1 is a

1

positive semidefinite operator of norm less than R The likeli-

0 .

hood ratio, an integral part of the detection plan, has a form

 

where ak and w are Fourier coefficients of the sent signal and

k

received signal respectively relative to the eigenfunctions (Pk

of the noise autocorrelation and where )\k are the eigenvalues of

the autocorrelation kernel. With R0 in force the ratio has a

certain variance and with chosen threshold completes the detection

ANDR ES C. SALAZAR

scheme. Now that RO has been perturbed what changes are
instituted in y ?
A brief summary of the work of William L. Root in this

area is followed by an explanation of the difference between that

work and the one contained in this thesis. Static perturbation

 

is a term applied to Root's work since it deals With changes in
the likelihood ratio stochastic properties keeping the parameters
ak and Xk the same before and after perturbation.

Simple examples of dynamic perturbation, a term used to

 

denote consideration of all changes of. the likelihood ratio after
perturbation, are followed by a lengthy discuSSion of changes both
stochastic and function-wise of the ratio when an arbitrarily small
norm perturbing operator is used. The method for finding the
perturbed ratio coefficients is similar to the one used in perturbing
the Hamiltonian operator in quantum mechanics. That. is,

+64) +

R + 6 R1 is the perturbed noise operator while <i>k 1' <1) kl

0

k0
€24) +. and X 2X +€X +€ZX + arethe
k2 .. - k k0 k1 kg .0. . ' ,_. i

th . . . .
perturbed k eigenfunction and eigenvalue. The parameter (T 15
small enough so that first order terms are assumed thig- significant
ones; hence, the solutions to the first Set of recursive equations

(formed for each power of 6)

(R -i

o ko‘chkl 0‘ I'RW

k1 1
(<10k1 lcbko) = o

I :(d3

kl (1)

ko IR1 ko)

ANDRES C . SA LAZAR

are sufficient to determine y 2 Yo + (-3 Y1 and Var v 't Var yo + . . .
+ 6 2 Cov (YO, Y1) the new liklihood ratio form and variance where
v1 and Gov (YO, yl) are rather involved expressions. Upon
determining these new quantities we can account for the way the
detection test is kept Optimum while an 6 ”distance" away from
the original one (yo, Var Y0 and threshold tVO). In this way a

continuity or stability can be shown for a finite variance detection

scheme.

STABILITY OF THE LIKELIHOOD RATIO

IN SIGNAL DETECTION USING PER TURBATION OPERATORS

BY
,-<’/
:1“)

Andre s C\.“\Sa1aza r

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering

1967

/
Li

A“? I L/ 9x
7/27/‘fl

ACKNOWLEDGEMENT

. ./ r.
por lg; corina, el (1121 se pasa pero la manana se acuerda.

 

The topic for this thesis was an outcome of the author's
all too brief acquaintance with William L. Root, a dedicated
scholar and teacher.

My gratitude for a warm and fulfilled educational experience
goes to Professor Richard Dubes, thesis advisor, teacher and good
friend, as well as to the secondary members of the doctoral
committee: J. J. Masterson (Mathematics), H. Salehi (Statistics),
W. Kilmer (Electrical Engineering), J. Strelzoff (Electrical

Engineering).

ii

TAB I. E OF C ONTENTS

Chapter Page
ABSTRACT....................

ACKNOWLEDGEMENT . . . . . . . . . . . . . ii

I INTRODUCTION................. 1
I. Detection of a Known Signal in Noise . . . . 1
II. Autocorrelation Functions . . . . . . . . . . 3

III. L Z(J) Kernels and the Karhunen—Loéve
Expansion.......... 5
IV. Hypothesis Testing for Signal Detection . . . 8
V. Object of Thesis ...... . . . . . . . . . 12

II STATIC STABILITY ANALYSIS . . . . . . . . . 14

I. PincherIe-Goursat Perturbation. . . . . . . . 14
II. Perturbed Likelihood Ratio. . . ....... 16
III. Singularity and Nonsingularity . ...... . 20
IV. Reformulation of Procedure in Special Case. 23

III DYNAMIC PERTURBATION . . . . . . . . . . . 25

I. Comments on Static Perturbation ...... 26
II. Dynamic Perturbation--Simple Examples . . 28
III. An Introduction to First Order Perturbation. 31
IV. Perturbation of a Discrete Spectrum . . . . 41
V. Statistical Properties of the Likelihood
Expansion................... 46

VI. Results Relative to the Threshold . . . . . . 57
IV AN EXAMPLE AND THESIS CONCLUSIONS . . 64

I.AnExamp]e.................. 64
II. COI‘ICLUSIOIIS o o o o o o o o c o o o o o o o o o 68

APPENDIX A. HILBERT SPACE . . . . . . . . 70

I. PrehilbertSp-ace............... 70
II. Hilbe rt SpatC e O O O O O O O O O O O O O O O O O 71

7 .
APPENDIX B. I“, L2 AND OPERATORS . . . 73
IO L2 andfiz O O O O O O O O O O O O O O O O O O O 73
II. Operators................... 74

iii

APPENDIX C. L. KERNELS ............. 7b

2
APPENDIX D. KARHUNEN—LOEVE EXPANSIONS. . 8O

BIBLIOGRAPHY . . . . . . . . . ........... 82

O
1 V"

CHAPTER I
INTRODUCTION

The detection of a finite energy sure signal in additive
Gaussian noise is our major concern in the first stages of our
problem. We shall deal briefly in this chapter with the back-
ground material necessary to set up a plausible detection scheme

for such a signal on a finite interval.

1. DETECTION OF A KNOWN SIGNAL IN NOISE

The known signal will henceforth be denoted a(t) , an
integrable square function on the interval [ O, T] . The total noise
process present in the communication channel as well as in the
receiver is additive and will be described by a zero mean real
second order stochastic process, x(t), continuous in the mean
square sence, i. e. , with continuous autocorrelation function,

R(s,t).
R(s,t) = Ex(t)x(s) 0< s,t< T (1.1)

The symbol E is the expectation operation relative to the
probability measure imposed upon the model. From this measure

two hypotheses can be distinguished.

H : signal is not present in the received waveform.

O

H : signal is present in the received waveform.

l

The received waveform can then be accordingly:

w(t)

x(t) for H0

a(t) + x(t) for H1.

w(t)

Autocorrelation functions, R(x, y), form kernels in L2
space if they are L1 integrable. If they are also wide sense
stationary they have a Fourier transform. This type of co-
variance function has a relative maximum at the origin which
is unsurpassed at any other point but can be equalled and is an
even positive semidefinite function. Second order stationarity
permits by means of the Fourier transform convenient spectral
analysis of the detection problem. Studies of noise through
Fourierspectra, emphasized by N. Wiener, help in determining
in what frequency range the noise power is somewhat attenuated
so that the desired signal power strength can be concentrated
there for a. better overall signal to noise ratio.

The ergodic hypothesis to which the importance of second
order stationarity owes its importance implies stationarity which
claims that the noise energy level and structure will not change
relative to time. The concept of ergodicity and to a lesser degree
the weaker condition of wide sense stationarity is based on
homogeneity and an isotropic media. The situation is random
but the prime assumption is that the "natural course of fluctuation"
is allowed at all times. This assumption may be unreasonable for
a macrosc0pic experiment where several factors causing significant
change in the environment may prevail for different sections in the

communication channel causing truly an unstable media. It may

 

lNorbert Wiener, Time Series, Chapter III (Cambridge, Mass.:
MIT Press, 1949).

 

therefore be presumptuous in some cases to try and measure time
varying noise energy levels and fit them to an ergodic or weaker
wide sense stationary formula for use in signal detection schemes.
No claim should be made concerning the niagni'tude of variation of
the noise energy structure. It would be just as presumptuous to
assume the noise is never ergodic. On the contrary, the tendency
toward ergodicity should always be recognized.

Indeed, the approach in this thesis is to rec0gnize the small
but significant variations about, the ergodic value of the noise energy.
This variation, it is maintained, may be the cause of either detection
test instability and/ or type I and type II errors whose magnitudes

may not conform to estimates given by the wide sense stationary

model.

II. AUTOCORR ELATION FUNCTIONS

Before we set up the detection. scheme we are going to use
it is best to describe the properties of the autocorrelation function.
First, when we speak of autosorrei'atious we do not irnplicity mean

"time avera e" or the wide. sense stationar statistical avera e
g Y g
E :~;(t) x(t+7‘} - R(’r’) .
Specifically, a second order random function x(t) on J,

an index set,with properly defined probability space (52,6, p)

is a family of second order random variables such that

 

E lxit) < 00 for every t (-7 J .

We will always assume the random functions have zero mean.

The second order moments are called variances and the function
defined on the set J x J is termed the autocorrelation function

or covariance function.
R(t,s) 2 Ex(t)x(s) (1.1)

It can be shown by the CBS inequality that the function exists and
is finite almost everywhere (Loéve p. 465). The reverse is also

true; R(s,t) finite implies
2
Elxmy = R(t,t) < co teJ.

Hence, covariances or autocorrelation are a manifest nature of
the second order random function.

The properties of the autocorrelation can be summarized

with a few notes (Lo‘eve pp. 466-8).

1. A function R(s,t) on J x J is an autocorrelation
if and only if it is nonnegative definite (sometimes
termed positive semidefinite).

2. The class of autocorrelations is closed under
additions, multiplications and passages to the
limit.

3. Since every nonnegative number is an autocor relation
it follows then that polynomials of covariances with
positive coefficients and their limits are also
autocorrelations.

4. Given a continuous autocorrelation. on a closed interval
J the stochastic process x(t) is normal or Gaussian

if and only if the random variables

, : 1' .) , .'
kn f} XU’.) ¢n(t' dt (1 Z)

are normal or Gaussian‘, {¢n(t)}:_1 here are
the eigenfunctions corresponding to the discrete
spectra of the autocorrela tion function via the
Karhunen-Lole‘ve expansion (section I. 3).

If the autocorrelation is not positive definite, its eigen-
function sequence is not necessarily a generating set for all
L2(J) where J is the closed interval of the continuously indexed
random process. Later we will see that this point implies that
the detection procedure maybe singular, i. e. there exists a

signal in L2(J) which can be detected without error.

III. LZ(J) KERNELS AND THE KARIIUNEN-LOIIVE EXPANSION

A spectral theoreri’i for stochastic processes known as the
Karhunen-Lo‘eve Expansion plays a. central role in the signal
detection scheme of interest here. It permits the noise process

x(t) to be expressed as

x(t) = 2: x <;‘> (x; O<t<T (:J) (1.3)

where the equal sign really represents convergence in a stochastic
CD
as well as an L2(J) mean square norm. Here the {¢k}l\ 1 are
the eigenfunctions of the autocorrelation R(x, y) , an L2(J) kernel,
and per se have no statistical properties. The Fourier coefficients
Xn assume then the entire stochastic characteristic of x(t) . We
\ - . . . .
have the Karhunen-Loeve Expansmn then splitting the noise process

into two orthogonal sequences (Appendix D), the functions {(Pk}k-l

(I)
and the stochastic set {xk}k 1 . Each set can now be dealt with

more clearly.

We now see the importance of dealing with LZ(J) kernels
and indeed the whole L2(J) space structure. From Appendix C
we see that an LZ(J) kernel is a bivariate function K (x, y) for

which both functions
T ,2, '. 1/2 , T 1/2
A(X) = [ID k \X: y) dyl BKY) = [f K (29y) dX] (1.4)
O

exist almost everywhere in J and belong to class L2(J) . It can
be shown that every symmetric and nonnull LZ(J) kernel has at
least one eigenvalue. Further, any nonnull symmetric L2(J) kernel
has an infinite number of eigenvalues or is a PG kernel, i. e. , a

Pincherle-Goursat kernel whose expression is generally put as

N
2. - . < < <
K we rig in 451(k) «inm N 00, o_ x,y__ T

N _ N , .
where {(1) l and {A } are the eigenvectors and eigenvalues
n 11:1 Tl 11:1
respectively. A Hilbert—Sc hmidt kernel on the other hand is one for
which the following is true:

‘ Z
tiff i K(X, Y) dxdy < m
o

0

This indicates a PG kernel is Hilbert'Schmidt but not necessarily
vice versa.

In our case the autocorrelation function, in view of its
properties, forms an LZ(J) kernel and hence submits to eigen-
function expansions. The Karhunem-Loeve Expansion is such an

expansion. Hence, the second order characteristics of the random

‘1

process will form the properties of the covariance function.

If the L J) kernel, R(x, y), is a positive definite

2(
function then the stochastic process is of the proper type; but

if the kernel is only positive semidefinite then it is improper.
These kernels will correspond to positive L2(J) and nonnegative
LZ(J) operators. From Hilbert space terms we understand
positive operators will yield eigenfunction sets which generate
all of L2(J) space while nonnegative operators will have a null
space which does not merely contain the zero vector.

The common assumption, due in part to the fact that
natural disturbances are commonly of Gaussian distributions, is
that the noise process is of Gaussian character as is the kt‘h
Fourier coefficient of the noise relative to the kth eigenfunction
of R(x, y) . This is true for k r l, 2, 3, . .

The following will partially suniiriarize the background
needed for the detection scheme to be discussed in the next
section.

Let x(t), O: t _<_ T _: J, be a real second order
Gaussian random process continuous in mean square with mean
zero. R(s, t) z E x(t) x(s) is then a symmetric, nonnegative
definite continuous function in [ O, T] x [ O, T] and an integral

operator's kernel in L (J).

2
[Rf] (t) = IT Runs) as) ds = gm 6 Law) (1.5)
O

R is self adjoint, nonnegative definite and Hilbert-Schmidt. If

R is positive definite its eigenvectors will form a total sequence

in LZ(J) .

(D
Denote the eigenvectors and eigenvalues of R by {Cbn} and

n=1
:1:
{kn} 1 respectively. By the spectral theorem for stochastic
n:
processes:

X(t) = nEI Xn¢n(t) (1.5)

using the stochastic mean square as the metric where

T 6
xr1 = f0 x(t) ¢n(t) dt . (1. )
Let
T
ar1 = [O a(t) ¢n(t) dt (1.7

where a(t) is a known waveform in LZ(J) . If w(t) = a(t) + x(t)
then wn at an + xn are uncorrelated random variables for all n .
Since the stochastic process is Gaussian wn , n : 1, 2, . . . , are

also independent.

IV. HYPOTHESIS TESTING FOR SIGNAL DETECTION

For distinguishing between two hypotheses which in our
case is between signal and no signal present in the received
waveform the maximum likelihood test is an Optimum Bayes
procedure. The integral part of this test is the likelihood ratio

of the two distributions assumed for two hypotheses.

Probability density (observation taken/Hl true) f1

Probability density (observation taken/HO true) f0 (1' 8)

 

Use of the ratio in various detection procedures varies only in

regard to selection of the threshold t for deciding the

boundary between the two hypothesis regions. More knowledge

or differing defection penalties of error would influence the

selection of a threshold.

For each experiment A > t implies III is true while

A < t is the result of HO predominant. The ratio A is a
random variable with two distributions defined on the sample

space of possible observations. it then has possibly two means

and variance corresponding to its two distributions.

> :: ‘ "g .
P1 (A. t) IA d Fl(_) (1 9a)
1
A1 = region of sample space for which A > t
< : -
PO(A t) fAT d F0(x) (1.9b)

Two kinds of errors may evolve from this simple hypothesis
test and they are called respectively errors of the first (type 1) kind

and second (type. ll) kind.

Error type I: Prob (l-l true) -2

0 "ii

I
C ho s (311/ H

l (1.10a)

Error type II? Prob (I-I chosen/fl1 true) :— e (1.101))

0 ll
In our case we consider the likelihood ratio for Gaussian
noise studied under second order variations. Several auto-
correlations with different operator spectra and corresponding
eigenfunctions may have Gaussian statistical natures. Second
order properties of a stochastic process have to do with the rules
joining the familial members, this fact corresponding to "energy"

for an engineer and first order properties corresponding to the first

order distribution of each random variable. x(t) . The situation

10

we want to investigate is then the energy or power variations of the
noise in a communication channel and how these variations affect
the general signal detection procedure as exemplified by a general
likelihood ratio. We insist on first order stationarity at all times
so that the general assumption, namely that of a Gaussian
distribution for each random variable in the noise stochastic
process, is valid.

Continuing with the Gaussian example we form a detection
test which is a form of likelihood ratio test when a sure signal
is sent. Denote with f the probability (conditional) density

0

function of no signal and f , signal (Root, 1964).

1
2

 

eXpClz’ £311 in l
1 n:
N/Zir iliz...>.N n

(1.11a)

f0 {w1,w2,w3,...,le O} =

 

2
N (wn-an)

1 1
I eXPEanl x )

\FZTrX X ...)x

 

f1{w1,w2,...,lel} =

 

 

 

12 N n
(1.11b)
f Nw a N a2
_ _ _ n n l n
A — f — exp[23 k - Z Z ——)\ ] (1.12)
O n n
2
Wnan 1 an
n=lnAz>3 h -2-Z—-X— (1.13)
n n

The latter term is simply a constant Ll; so Y can be defined as

1
n+¢=n+-Z->3 k (1.14)

.<
H

11
The decision rule is then to compare Y with a threshold t

>
WW) = Z? nxn < tY (1.15)

 

We compute the means and variances according to the different

hypotheses.
2
a
_ _ _fi _ Z
EOY — O Ely — 23 Xk _ [3 (1.16a)
Z
ak 2
Var y = Var y : E — = B (1.16b)
o l Xk

If L32 is finite no mistake free decision can be made. An example

of this will be shown in Chapter II. We call 52 < 00 the non-

singular case and 52 = 00 the singular case. Actually singularity

as such is usually defined to be the mistake free case if realizable.
Grenander (1950) has shown that the problem can be singular

in two ways. First, the integral Operator kernel R(s, t) may have

a nonzero null space whereas a(t) has a nonzero projection onto this

null space. Hence, there exists an element, p ,

p e L2(J) 3 (plen) : o n=1,2,... (1.17)
(I)

where {4)n} 1 is the set of eigenfunctions for R but
11:

(pla(t)) #0 (1.18)

The operation (p I w(t)) where w(t) = a(t) + x(t) will distinguish
between the two hypotheses with probability one. Second, the
series

2
ii if N=°°

W
gmz
w

W

may diverge so from certain Grenander theorems there exists a test
for distinguishing both hypotheses With probability one.

The question arises as to whether a singular test ever
exists in a linear realizable receiver. One claim is that the
case is inherently prevented in the equipment if not in the
incongruence of the mathematical model to the real noise

process.

V. OBJECT OF THESIS

This then is the problem. Fixing the threshold tY for
the assumed stochastic structure of the noise x(t) and its
autocorrelation function we wish to know how badly the effect
of either not knowing the precise autocorrelation function or of
varying it slightly from the presumed value can affect the
likelihood ratio of the detection test. We wish to know how the
likelihood ratio will be affected first, in its functional form,
second, in its statistical properties and third, in the detection
test apparatus. The problem is then one of p(.Ittlll)dtlu)l’l.
Distrubing the kernel R(x, y) of an L2H) operator does what
to the eigenvalues, eigenfunctions which are an integral part of
the ratio? The perturbation is restricted to an additive and

small operator, 6 R1 .

R(e) = R +€R (1.19)

 

2

U. Grenander, "Stochastic Processes and Statistical
Inference,"Ark. 13431.1:195-277, 1950.

13

The parameter 6 here is a small positive number and R1 is
bounded operator with positive semidefinite symmetric kernel.
Property 3 of the autocorrelation (Section II) permits the kernel
of R(e) to be an autocorrelation also.

In Chapter II William L. Root's work in the analysis of
stability of the likelihood ratio will be reviewed.

The method for finding the perturbed likelihood ratio

coefficients in Chapter III is similar to the one used in perturbing

+€R

the Hamiltonian Operator in quantum mechanics. That is, R0 1

is the perturbed noise operator while

and

t . . .
are the perturbed k h eigenfunction and eigenvalue. The parameter
E is small enough so that first order terms are assumed the
significant ones; hence, the solutions to the first set of recursive

equations (formed for each power of e )

(R0 " Kk0 1’ ‘ka

(¢k1l¢ko) —- o
x

; (x I-R

k1 - (cbko IR1 Cbko)
are sufficient to determine y 2 Yo + eyl and Var y : Var Yo +
. . . + 6 2 Cov(y0, Y1) the new likelihood ratio form and variance

where Y1 and Gov (yo, y are rather involved expressions. Upon

1)
determining these new quantities we can account for the way the
detection test is kept optimum while an 6 “distance" away from the
original one (YO, Var yo, and threshold t ). In this way a continuity

Y 0
or stability can be shown for a finite variance detection scheme.

CHAPTER II
STATIC STABILITY ANALYSIS

In this chapter we wish to review one method used for investi-
gating stability of the likelihood ratio in signal detection (Root 196+)
which sheds light into how the problem may be approached and
studied. The summary of Root's work presented here will help
establish an attitude towards the problem which might not come
about easily since the references may not be readily accessible.

The method consists of perturbing the central or assumed
autocorrelation function with a symmetric PG kernel. It then
will be shown that the likelihood ratio will be subject to a larger
variance than in the unperturbed case. This fact will force larger
type I and type II errors for a fixed threshold. Finally, a summary
of the operations used in such a stability examination will be put
into a Hilbert space trio of equations for the special case in which

ak 2
E (f— < 00 .
k

I. PINCHER LE-COUR SAT PER TUR BATION

We wish to know what effect a perturbation of the auto-
correlation function with a PG kernel will have on the likelihood
ratio. We are still considering a sure signal a(t) on [0, T]
corrupted by an additive Gaussian, zero mean noise process with

continuous positive semidefinite autocorrelation function R t, s)

0(
whose operator has a discrete L2[ 0, T] spectrum. The

corresponding RO normalized eigenfunction sequence denoted

l4

15

by {¢k(t)}::l may or may not be a generative basis for L2[ 0, T] .
The family or random variables representing the noise indexed
by the interval [ O, T] is Gaussian or normal with zero mean before
and after perturbation.

Let R0(t, s) be perturbed by R1(t, s) , the kernel of

operator Rl .

(R1¢k(t)l¢j(t)) ckc. k,j=1,2,...N (2.1)

Otherwise, R1 is the zero Operator for all other k, j . The ¢k's
are eigenfunctions of R0(t, s) . The autocorrelation of the second

order perturbed noise process is

R(t,s)=R(t,s)+R(t,s) O_<_t,s,_<_T, (2.2)

O 1

Defining R in this manner implies it is Hilbert Schmidt, self-

1
adjoint, real, positive semidefinite and continuous since the set
members {¢>k(t) ¢j(s)} are continuous. For the sake of
perturbation we insist Rl's operator norm is equal to 6 , a

small real c onstant.

HZ

NN 2
HR1 = ZZI(R1¢kI¢j)l 2 22¢

(2.3)

Since Rl(t, s) is a positive semidefinite function it is a covariance
function. The sum of two covariance functions is a covariance
function.

The signal' 3 expansion

a(t) =k§1ak «pkm (2.4a) ak2f:a(t)¢k(t) dt (2.4b)

16

makes sense if a(t) belongs to the space [{Cbkl] . For a real

noise process with R0(t, s) as its autocorrelation,

xn = f: x(t)¢n(t) dt (2. 5a) E x x = xn an

n k (2.5b)

k I

II. PERTURBED LIKELIHOOD RATIO

Of interest is the change in variance of the likelihood ratio
when the autocorrelation has been perturbed. From equation 1. 15

the ratio is

 

wk k r
Y(w) : Z (2. 6a) w = f w(t) <1) (t) dt (2.6b)
with decision rule:
accept H1 (signal present) if y(s) > tY

reject Hl (signal absent) if y(w)< tY

where tY is the threshold.
In what follows the subscript indicates hypothesis 0 or 1
while the prime mark means the expectation is with respect to

the measure of the perturbed model.

 

 

00 a:
13' Y(W) = 0 12' \/(W) = z .—
0 1 11:1 )\n
(2.7)
Var ww) = E' Ive/HZ
O 0
Expanding the latter equation,
2 anwn akwk anak
1 .. 1 ________ _. 1
Eoly(w)l _ E0 2 x z X -232 x x Eownwk
n n n k

(2.8)

17

However,
12' w w = E' fTw(t) 4: (t) dt rTw(s)¢ (5) ds
0 n k 0 o n '0 k

T

= [TI ¢k(s)¢ (t) E'w(t)w(s) dtds (2.9)
o n

O

claiming for the moment the measures commute in the latter

step. For hypothesis 0 we have w(t) = x(t) so
E; w(t) w(s) = R(t, s)
where

R(t,s) = R t,s)+Rl(t,s).

0(

Expanding equation 2. 9:

l):'.~"‘

ngn Wk : 1‘ng {R0(t, s) + R1(t, 3)} ¢n(t)¢(s) dtds

0

n

i ank + f:~r:R1(t’s)¢n(t)¢k(s) dt da

ik ank+ (Rlcpnlcpk) (2.10)

where fink is the kronecker delta

 

1 n=k
5 =
“k 0 n7§k .
We conclude that
1 a'nalk
VaroY(W) :22 X x {xnénkHRlcbnlcpkn (2.11a)
n k
a2
92.51 +AZ 952+A2 (2.11b)

:3

 

18

 

 

 

 

where
2 anak
A _ :2 ‘x "k (Rl¢k|¢n)
n (2.12)
2 _ ncn 2 _ Nan n
A-lzkl IAI—Iz, I
n n
Now
2
an
Var'1 WW) = EilNW) -Ex—|
n
2 2
= E) |v(w)l - IE(Y(w))| (2.13)
where
2- anak
I _ I
E1|Y(W)i - E1 22X A wnwk
n k
a'nak
_ __ l
_ 232 X X E1 (xn+an)(xk+ak) (2.14)
n k
2 a'na'k
I _. I
E1 ly(w)| _ >323 X X E1(xnxk+anak)
n k
anak
2 22 x i ”nankHRii’k ¢n)+anak)
n k
a2 a2 a2 a a
k n k n k
z 3:— +>:2————)\ x +22————>\ X (Rlcpnlqak)
k n k n k
2
an ncn 2 an 2
=2—+|2 +(23-—-) (2.15)
R A X
k n k
so
a2 a2 a c 2 a2 2
n 2 n n k
Vari(y(w)) = 21.11 +(Z r) +kz x i -(E r)
n n n k
2
a anon 2
= )I (2.16)

 

19

Finally,

2 Varb y(w) . (2.17)

Vari v(w) = (32+ A

For the unperturbed model, the error probabilities are:

 

Co 2
eI = l S e-u/'Z du (2.18a)
tY

2
6” /Z du . (2. 18b)

The error probabilities for the perturbed model are:

°° 2
ei = J— 5 e"u /Z du (2.19a)
NIZTr t
_l____
BZ+AZ
t -B2 2
eh : _l__ W e-u /Z du . (2.19b)
'\/ZTr B

For a fixed threshold note that ei > eI and eh > eH. Since
the integrals are continuous the error probabilities are

2
continuous relative to the perturbation parameter A . That
is, as [A I goes to zero, ei approaches eI and eh
reduces to eII .

If t = k (33/2 is chosen we have

2
e.u /2 du

_ 1
el _ «f2? «Skpl/Z
(2.20)

—- e du

1/2
6 l 3143 *3 -u2/2
H m 00

20

As (3 aproaches 0° ,eI and eII approach 0. Thus, singularity
is implied. Later, we will {see that, if tY is not} taken in this manner

this case is unstable in that A2 could be finite or infinite for

comparably small 6 .

III. SINGULARITY AND NONSINGULARITY

A few remarks about both the singular and nonsingular cases
of detection in regard to the gross magnitude Of [32 are in order
here. Examples Of both cases will be given.

It may be that R0 as well as R1 are not positive Operators
so that they have a nontrivial nullspace within L2[ 0, T] . In this
case the singular case Of detection exists. That is, there exists
a signal a(t) in L2[ 0, T] which is independent of the RO eigen-
function sequence. It may be shown then that this signal is mistake-
free detectable (Grenander (1950) or Root (1963)). However, this
case of singularity is discounted (Davenport and Root) by the
argument that a linear system is always used to receive the wave-
form w(t) = a(t) + x(t) . Hence, w(t) represents the signal
corrupted by the transmission media noise and any introduced by
the receiver. The important idea here is that the detection does
not occur until _a_f_t_e_1;the receiver whether it introduces noise or
not. Be it as it may, we shall consider this singular case and its
ramifications only briefly.

We will now proceed with specific cases for [32 but first

must consider a most important mathematical lemma.

21

(I)
Landau's Lemma: Given a series 21 qn = 00 then for arbitrary
n:
L: 00

a)
6 > 0 there exists a series 23 z = 6 such that Z q z
n 11:1 11 n
This lemma will be used extensively in the following

. . 2 . .
discussmn Of B examples in detection.

1. singular case

 

a 2
2 _ n
H - 7-7 (X— - 0°
n
(a) N finite
N a c
z nxn : A < (I)
n=l n

(b) For N = 00 and any 6 we have a possibility
g anc
n21 xr1

 

by Landau's Lemma. SO regardless Of the smallness in

perturbation described by 6 we may still have

A2 = 0° or A2 < 00. For fixed tY this may mean

unbounded errors as seen in equations 2.19a, 2.19b.

2. nonsingular fig < 00

 

N
(a) For finiteNand 231 c2 = 6 or for
n:
a2
m
z; 1.11 = (32 < oo
11:]. n
we have
ancn
<
Z X 00

22

 

 

 

 

 

 

 

co 2 anco
(b) For N=°° and Z c =6 it is possible {—} is
11:1 n An n=l
at ancn
unbounded while 2 3:— is convergent so 2 X = 00
k n
is not forbidden by Landau’s Lemma.
3. nonsingular
a2
Z —}-2- : “,2 < <1)
k
n
(a) N finite
a'n Cn
Z < co
n=l kn
°° 2
(b) N infinite Z c =6 then by CBS inequality
)nzl n )
an C:n an 7 P 2 I
< — <
X _ Z (X ) Z cn 00
n n
. 2
4. Singular (3 :00
(a) N finite
a c
Z n)\ n _ A < 00
n

 

 

(b) N infinite Z) c: :6. Then

I.
1.

2

a
EYE <2:
max n

P’ m
sNIsN

so the A = 00 possibility exists.

The preceding cases show that a great part of the stability

°° ak 2 2
question depends on whether 23 (-)\—) = p is finite or not.
= k

23

Smaller variance indicates stability (Root 1963) in that A2 can

not have value range from a finite magnitude to an infinite one.

IV. REFORMULATION OF PROCEDURE IN SPECIAL CASE

A formulation Of the approach presented in this chapter

a 2
k <

to detection stability for the special case of E (f— 00

can be put in terms of the unbounded Operator R0 when R0

is CC.
°° ak 2 -1
If E (r) < 00 then a(t) is amember Of the RO
k=1 k

domain. That is,

 

T
f R0(t, s) g(s) ds = a(t) O i t E T (2.21)
O
has a solution g(t) in L2[ 0, T]
-1
R0 a = g
Acutally, g(t) = 133:1 )‘k since
T 00 an T i so
Rt,s tzz ——- Rt,s¢(tdt=Ea¢(s)=a(s) (2.22)
'ro( )g()k=1)‘nfo( )n) n=1nn
O 5 s E T
RO g = a
Recalling
m a W
k k
Y(W) = 73 x
:1 k

we see readily

(g(t) lw(t)) = v(w)

Z4

 

 

because
a a w
_ n n k k _ n k
(31W) “2 i '2‘?” - 22—f— 6nk
n k n
so a w
k k
(gm = 131 )1. = w) (2.23)

Further, we claim A = (ng/g). This can be seen when we put

A2 from equation 2. 12 in the following form:

2 8'n 3'k
a'kcbk Z) 8'n ¢n )
"k 5:
n

(R1g|g) (2.24)

A

 

(R12)

We then have a trio of equations in compact form describing

the entire detection and perturbation procedures.

WW) = (g(t) )W(t)) "’ Y = (g'lw)
[Rs] (t) = a(t) ~ Rg’ = a (2. 25)
A2 = (Rg(t) | g(t)) ~ A2 = (R l g)

We have thus formulated the detection procedure into a series
of integral equations in L2[ 0, T] . This likelihood ratio's
variance is augmented with A2 which approaches zero in mean

square as Rl approaches the zero operator in Operator norm.

CHAPTER III
DYNAMIC PER TUR BA TION

Dynamic perturbation considers the changes which will occur
to the parameters in the likelihood ratio expression as well as the
changes in the stochastic properties such as variance and mean.
Error probabilities should then point up a "dynamic“:‘change‘in
magnitude since we are using the Optimum hypothesis test in both
the unperturbed and perturbed models as differentiated from the
models in the static case. The continuity property can be studied
under the condition that a true likelihood ratio is being used before
and after perturbation.

The key to this dynamic approach is Operator perturbation
theory. If some type of "small" Operator is added to the original
what changes in the eigenvalues and vectors are instituted? Some
theorems have been developed to answer this question but are
entrenched in a mathematical mire of notation and at times in
quantum theoretic arguments and so must be modified for our
pruposes here.

Besides considering simple examples of dynamic perturb-
ation, this chapter outlines the assumptions needed to establish
for the likelihood ratio a perturbation procedure similar to the
one used to perturb the Hamiltonian Operator in quantum mechanics.
Work by Franz Rellich (1953) will help in formulating the procedure

needed to obtain the first order equation set from the groups of

25

Z6

recursive equations set up by this type of Operator perturbation.
Once the changes in the likelihood ratio parameters are found the
stochastic aspect Of the ratio is investigated after perturbation.
It will be shown that both stochastic and functional facets of the
ratio will change proportionately to a small perturbation (6)
parameter. Continuity of the detection scheme relative to a
small change in noise energy structure will then be implied
through stability Of the finite-variance likelihood ratio hypothesis

test.

I. COMMENTS ON STATIC PERTURBATION

In the previous chapter the changes in detection error
probabilities were examined under the hypothesis that the
perturbation kernel R'(s, 1:) would affect only the variance of the
likelihood ratio. This is to say, the same eigenvalues and eigen-
functions were used in the likelihood ratio while its variance
changed and duly affected the detection error probability integrals.
This approach to the perturbation problem might be called the
“static” one and its basis warrants further discussion and closer
examination before more interpretation is devoted to it. In the
following brief review of the Chapter II approach it is hoped that
the need for a more comprehensive perturbation approach will
become apparent. This second approach is termed the ”dynamic"
one for reasons which will become apparent in due time.

In the perturbation example found in the preceding chapter

the autocorrelation function, R(s, t), is presumed known at first.

27

If Gaussian noise is assumed its eigenvalues and eigenfunctions
establish probability density functions as well as a likelihood ratio
and threshold for the hypothesis test. The variance and means of
the likelihood test statistic are determined. Now if a second order
perturbation is assumed on the autocorrelation function, the change
in the original likelihood ratio comes in second order form, namely
in the variance of the ratio under either hypothesis. The error
probabilities are made greater by the increase in variance. If

the proper threshold is chosen it can be shown, however, that as
the perturbation approaches zero in magnitude the errors will
approach their unperturbed magnitudes. The error probabilities
are hence ”continuous " at a chosen threshold point with respect

to the second order perturbation parameter.

Now the question raised here is why the original ratio was
used with parameters, ak and Xk, made Obsolete by the perturbation
itself. Root (1964) maintains that this procedure yields an indication
of “stability" Of the hypothesis test relative to the second order
noise statistics. An objection tO this reasoning is that it really
is unfair to use Obsolete parameters to show the continuity of the
test despite the fact that this procedure simulates the change in
error probabilities actually experienced by a receiver-processor.
Although a nonadaptive receiver processor would indeed use
obsolete parameters when the noise structure is perturbed, the
question raised here is not that the decision making apparatus
used obsolete parameters but that a true measure of changes in

error probabilities is not Obtained. That is, it is unfair to claim

28

that the change in error probabilities is shown entirely by comparing
the variances of the likelihood test function in the original and
perturbed states and by the effect these variances have on the

error integrals. Consequently, it may be claimed improper
continuity and untrue error changes are the result Of the Chapter

II procedure.

II. DYNAMIC PERTURBATION--SIMPLE EXAMPLES

Following are some rather simple examples Of dynamic
perturbation of a likelihood ratio. We will forego the statistical
analysis Of the perturbation and will only demonstrate models of
continuity in the "function" aspect Of the likelihood ratio. In
essence we will additively perturb the autocorrelation R(s, t)
will kernels which range in simplicity from a diagonal to a general
commuting one and then see what happens to the likelihood ratio

parameters.

Example 1. Perturbation relative to a diagonal PG Operator

N
2
: < <
Let R1(t, s) kEI ck ¢k(s) ¢k(t) O _ t, s _ T
be the perturbing kernel. We have that 2 Ci = 62 and that
R(t, s) + A(t, s) have eigenvalues X + c2 and the same eigen-

N k k
vectors {<1>k(t)}k=1 up to the index N while R(t,s) continues

m
with {<l>k(t)}kzl ; ak will not change in the likelihood detection

 

ratio y.
N a W a) a W
A k k k k
y(w) = E -——-—Z + E k (3.1)
k2]. Xk-l-ck kZNTi‘l k

where " denotes the perturbed models.

29

2 2
A N ak co ak
Var y(w) = Z ———-2- + Z) r— < Var y(w)
k=1 X +c k=N+l k
k k
Var y(w) > Var/9(w) and we certainly have

A
lim Var y(w) = Var y(w)

52-»0
Example 2. Perturbation relative tO a CC diagonal Operator

This example is really an extension of the second in case

CD
'23 c: = 62 < 00 and
k=l
a a2
co W <1)
9(w) : k2 '——'k——1'(-Z- and Var IY\(W) = kzl k 2
:1 Z
Xk + Ck kk + ck

Example 3. Perturbation relative to a general commuting Operator

In the previous examples the pervading quality of the perturbing
Operators was that they commuted with R . These foregoing cases
can really be included in the general case Of operators which commute
with R . It is this type Of Operator which permits the eigenvalue set
to be perturbed while the eigenfunction set remains the same.

Let us assume we have reindexed the eigenfunctions
{¢k}:=l so that R0¢k = kk¢k and R1 43k = )‘kcbk k =1, 2, .. .

with equal multiplicity and same order. If
(RO +6R1)<1>k = >\k(1+6)<1>k
then

AA, A A A i
ch _xqak xk_>.k(1+e)

30

A
with say 6 a small parameter. Since (bk = (Pk, ak remains the

same in the likelihood ratio Y .

m a \N
’Y‘ —_. 2 4i
k=1 Xk(l+6)
312
A2 _ A _ 'k
(3 — Vary _ Z? ——xk(l+€) (3.2)

Case 1. If 6 > 0 then the variance (32 < 00 if (32 < 00;
(32 < (32 < 0° . The other possible cases have already been
discussed.

As noted above, the eigenfunctions do not change in this
type of perturbation. Hence the parameters ak remain the same
and the only "perturbed parameters" are the eigenvalues in the
Operator spectrum. As recalled, this was not the case in the
Chapter II example. When the Operator was perturbed no
guarantee was made that the other parameters which include
eigenfunctions and eigenvalues changed as well as the variance.

The next step would be to broaden the class Of perturbing
operators tO either a general CC class or a bounded class.

SO far we have considered the continuity of the likelihood
ratio relative to changes in the eigenvalue set only. We
accomplished this by examining perturbing operators which
commuted with R . The eigenvector set and thus the ak sequence,

a)
signal Fourier coefficient relative to {¢k(t)} . the R

eigenvector set, remained untouched by the fluctuation. This

type of perturbation can intuitively be seen as an inward or

31

introverted disturbance since the Operator's own structure, namely
the eigenvalue or eigenvector sets, is being used for describing
the varying behavior. This is the simplest type Of perturbation
which can occur. Nevertheless, it does seem plausible that the
noise structure as described by the autocorrelation kernel and
corresponding Operator will in fact vary about a center functional
pattern with additive variations which are themselves describable
by the central function pattern.

The time has come however to consider the more general
case which includes both perturbed eigenvectors and eigenvalues.
Hence, we must consider now the change in ak (wk) , the Fourier
coefficient of the sent (received) signal, as well in the likelihood
ratio which leads to the hypothesis test. We will need to delve a
little into operator perturbation theory in order to extract the
needed mathematical results and formulations. However, these
more general cases will give the broader view into kernel
perturbations needed to deal with the effect second order noise
fluctuation has on a signal detection scheme With likelihood ratio

hypothe sis test.

III. AN INTRODUCTION TO FIRST ORDER PERTURBATION

Before we proceed to investigate the signal detection
problem it would perhaps be appropriate to pause here to
consider what uses perturbation theory has had in other scientific
areas and point out how this use has prompted the study manifested

in this paper. There is indeed more than a passing analog between

32

its use elsewhere (especially in quantum mechanics) and its purpose
here.

The following analysis of the signal detection perturbation
has been instigated by work in quantum mechanics and systems whose
behavior is described by Operators, eigenvectors and eigenvalues.
Corson (1950), and Rellich (1953) and Powell-Craseman (1961)
references all deal with Q-M (quantum mechanics) problems of
perturbing Hamiltonian operators or the differential Operators
appearing in the Schroedinger equation. These references, for the
most part, deal with differential Operators while the Operator of
interest in signal detection has integral kernel representations.
The analogy between differential and integral Operators theories
will be exploited and the theorems develOped for differential
operators will be adopted to our purposes.

In the foregoing chapters it has been pointed out that the
signal detection problem is being split into two subproblems. In
one, the noise autocorrelation function is known or is unchanged in
form. The second investigates the effect Of an unknown additive
perturbation to the autocorrelation or a sudden change in the auto-
correlation. This perturbation incurs change in eigenvalues and
eigenvectors of the unperturbed problem structure. In Q-M the
Operators perturbed are Hamiltonians K and H, i.e., system
energy representations. These Operators are analogous to the

noise energy representation in the time domain, the autocorrelation

33

function. The wave function or eigenfunction Of the H and K

Operator depict the state behavior Of the system, whether atomic

or grossly mechanical, while the eigenfunction of R portrays

the function basis or distinct ”function states” Of the noise, i. e.
co

Mklkzl .

In Q-M, disturbance Of eigenvectors and eigenvalues
resulting from operator perturbation will reduce in a continuous
way to the original unperturbed Operator and corresponding eigen-
structure as the perturbing operator reduces to zero norm. This
continuity hypothesis is hard to understand if the spectra of the
unperturbed and perturbed models are discrete (continuous) and
continuous (discrete) respectively since a discontinuous change
occurs in the vanishing perturbing Operator nature. As will be
stated later in a fuller context, singularities might appear in the
sense discrete spectra may become continuous.

Consider a brief lOOk into first order perturbation for a
system using the Hamiltonian operator H; kn is the nth energy

level while on is the nth state function.

HO+6H1

H(6)

dink) 44110 + 6 ¢n1

xn(6) )‘nO + 6 xnl

where the second subscripts correspond to the order Of perturbation.

Suppose

H(€) link) = Xn(€) Link) .

34
Then substituting, we have

2
Ho LI"no + 6 (H1 q’no + H0 q’ni) : )‘no Ll’no + “x111 L”no + )‘no q’ni) + 0‘6 )

Equating equal powers of 6 :

HO ilan 2 XnO 1.on zeroeth order
(3. 3)

HO ipnl + H1 4an xnl 1.)an + XnO tilnl first order

Taking the Hilbert space product on both sides with qlno we have,

since H0 is self adjoint

(inOlHownl) = (Hoinolwnl)
then,

(‘i’no iHi‘i’no) : )‘ni (inolll‘Pno)

x : (vnOlfilwnO)

“1 W110 W110) (3' 4)

 

The increase of energy level Of the nth state is then the expectation
Of the perturbing Operator relative to the nth original state vector.

The foregoing brief use Of perturbation techniques in Q-M
should demonstrate the analogy with our problem at hand. Although
not done in this thesis, it is hoped that this analogy may be carried
further in due time so that extraction of results on one side may be
adapted to the other.

Finally, we turn to a more general formulation of Operator
perturbation so that we can consider the case of signal detection.

The notation followed is that used in the Appendices A-D . For

35

engineers and especially those in finite state systems the z trans-
form or generating function concept is well known and is pointed out
where applicable. The procedure followed is aligned well with that
found in Rellich, Powell-Crasemann and similar references but is
specifically altered and augmented where necessary for signal
detection perturbation.

Consider the following Operator and vector equation.

(RO+€R1)¢k:>‘k¢k (3.5)

A I\ .
where (bk and k are the new eigenvector and eigenvalue relative

k

to the new and perturbed Operator R + 6 R1 .

0

A A
Assuming 6 is small we suppose X and (pk can be put

k
into the following power series of 6 , a form Of the z transform

or generating function Of 6 parameter.

2

2 Z
(RO+6R1)(¢kO+6¢k1+6 ¢k2+..) — (x +6Xk1+6 xk2+..)(¢ko+6¢k1+6 ¢k2+°°

ko

Of course,

(R0+€R1)§(€) = Mask)

where §(6) and M6) are generating functions with 6 parameter
and M6) a scalar model and §(6) actually a function model, i. e. ,

assuming convergence

§(6) = ¢k0(t)+6¢k1(t)+ (3.6)

If the convergence Of §(6) and M6) were not possible the above

36

formulation is presumptuous. Suppose, however, the convergence

of §(6) and M6) are sufficiently "well behaved" so that term by
term multiplication is permitted.

, 2 2
(RO+6R1)(¢kO+6¢k1+..) — (Rooko+6RO¢kl+6 R0¢k2+..)+6R1¢k0+6 R1¢k1+...

2
R0¢ko + €(R0¢kl+Rl¢kO)+€ (RO¢kZ+R1¢kl) +

(1. 0+6). +62). +..)(¢ko+e¢kl+...)

k kl k2

4- I
kO ’6‘ku ‘i’

2
4’ ”kod’ki)“ ()‘k2¢ko+"k1¢k1

xko kO

+xk0¢k2) +

By equating equal powers Of 6 we can Obtain the following recursive

equations for any k integer:

R0¢ko : >‘ko‘i’im

R0¢kl +R1¢ko : Xkl¢ko+xko¢kl (3'7)

X

R0¢kn ‘ xkocpkn 2 x1mi’1<o+‘\kn.1‘i’kiJ""'I k1¢kn-1 'R1¢kn-1

(3.8)

for arbitrary n, a positive integer. TO find exact values for the
unknowns indexed by n 2% O we revert to Hilbert space methods.

The first equation

37
is really

R0¢k ' )‘k‘i’k. : O

which is the unperturbed equation and yields no new information.
The general equation, however, is Of the form
RO (bkn - xko¢kn : akn a vector also. (3. 9)

(R0¢kn - >\kocpkn lcpko) : (akn lcbko)

Recall our inner product is

T
f a(t) b(t) dt = (al b)
0

Since R0 and R1 are symmetric we have

(R0 <i>

l
7
.9.

knlcbko) (¢kn'RO¢kO)

(akn ii’ko) : (R o

¢kni¢ko) " )‘kom
: )‘ko‘cbkni ¢ko) ' )‘kom
Thus akn is perpendicular to ¢ko for n arbitraryaé O

(akn l¢ko) : )‘knw’kol¢ko)+)‘kn-1(¢kol¢k1)+”°”kimkol‘pkn-i

' (‘i’ko IR1 ¢kn-1)

For n=l

Remain.) -<¢kolR1¢>k0) = (Calm) = o

>\kl “ka l(pko) : mko IRl ¢ko)

38

Wko |R1¢k0)

(<ka |¢k01

>’
I

 

kl (3.10)

If xko had its corresponding eigenvector ¢ko normalized then

this would be further simplified to
)‘k1 = (Cpko IR1 ¢ko)
Referring back to our recursive equation (3. 7) we have the right

side of

-)x

R0¢kl kocbkl Z >‘ki‘i’ko -Rl¢ko

known. Now to put a further restriction on the eigenfunction series.

Let us continue the normalization of eigenfunctions as follows

arbitrary k: ll¢k(e) H‘Z =1 = (¢ko+e¢k1+.. l¢ko+€¢kl'+"°)

Again, using the generating function assumption which permits term

by term multiplication and addition;
1 = ( <1» k0 like)

0 = (¢k0|¢k1) + (¢k1|¢ko)
(3.11a)

Since R0 and R1 are symmetric we must have real perturbing

eigenfunctions so
wkjlcbkp = (¢k,l¢kj)

for all 1, k integers > 0 .

39

We conclude that for n = l
2(¢kol¢k1) = 0 or (¢ko)¢k1) . 0

We have then sufficient conditions to determine uniquely ¢kl

k = l, 2, . . . from the second order equation
(R0 ‘ )‘koll‘l’ki : (M11 ' Rl)¢ko
and the first order equation

(¢kol¢k1) = 0 (3.11b)

Similar equations are set for other powers Of 6:

x R Cb

Pb¢kn -)\ko¢kn : xkn¢ko+xkn-1¢kl+°°°+ k1¢kn-l ’ 1 kn-l

0 = (¢kol¢kn)+(¢k1]¢kn_l)+... +(¢kn|¢k0)

It is readily seen from the preceding argument that the higher orders
of approximating eigenvalues and eigenfunctions are determined in a

sequence which begins with X

k0 and éko and proceeds to as high

an index n as is desired.

Although the disturbing Operator R was in the beginning

I
assumed to be such that the generating functions ¢(6) , M6) were
convergent it need not be necessarily the case especially if no
other restrictions are placed on Rl besides symmetry. It can

happen, for example, that R is an unbounded Operator and this

1

may cause havoc in the a priori generating function convergence.

Such an example is given in Rellich's notes which perhaps was a

40

motivating factor in his investigation and eventual theorem claiming
a "boundedness'condition" is necessary for the perturbing operator.
An unbounded Operator's perturbation may cause an eigenvalue to
move from a finite position on the real line to an infinite one
regardless Of how small an 6 is used. Consequently, the like-
lihood ratio y which is our main concern in this paper in relation
to the distrubance Of its parameters becomes unwieldy. in the sense
of having infinite components (infinite eigenvalue in the denominator
of a series term).

The use Of an unbounded Operator as a perturbation may be
dismissed easily by some workers in signal detection by arguing
the noise autocorrelation function is at all times bounded and hence
yields a bounded Operator as well. Also, the Karhunen-Lobve
expansion leads one to believe pure point spectra are the rule for
autocorrelation functions. The perturbing unbounded Operators
beside causing unbounded spectra at times can cause the disappearance
Of point spectra and replacement with continuous spectra. Hence,
we cannot state that a small perturbation parameter indicates a small
perturbation in the Operator's eigenvector structure. Neither may
only the first order perturbation parameter equations be the
significant ones. Also, the sign of the parameter 6 may have a
large effect on how the perturbed operator behaves. There is no
question that careless generalizations such as "a perturbation of
a bounded Operator does not boundlessly change the eigenvector
structure" is unwarranted. For our case of likelihood ratios there

should exist a doubt of whether perturbations of bounded auto-

41

correlations and associated Operators can indeed cause not small
but very significant changes in the Operator's structure and the
likelihood ratio which is heavily dependent upon it.

Another facet Of the perturbation procedure Of the first
order here concerns the eigenvector sequence of the perturbed
operator. There is no guarantee (Rellich, p. 153) that the new
eigenvector sequence is a basis for L2[ 0, T] . Hence, the

perturbed Operator is to be assumed a positive one at all times.

IV. PERTURBATION OF A DISCRETE SPECTRUM

In the previous discussion we have written the form

2
¢k(6)—¢ko(t)+6¢kl(t)+6 ¢k2(t)+... (3.12)
w(s)—x +1 +2). + (313)
k—koek16k2'” '

k=1,2,3,... 051;: T.(Jinterval)

for; small 6 real in the sense Of a generating function. We
neglected to specify a possible region Of convergence relative
to 6 and more seriously omitted mentioning the type Of norm we
are to consider for the convergence. Since 3.12 is a function
series while Xk(6) is a scalar series we need tO specify
appropriate convergence criteria for both.

From 3.13 we may use the normal mathematical rule for
convergence of a power series since it is a scalar series. Since
we are speaking of Of; operators, i. e. , bounded Operators, we

have

k(

for some scalar M < 00 . We have then that the least upper bound

(sup) Of 6 such that 3.13 converges is any 6 for which
Is! < 1

is true for any k. Note the norm for scalars used here is the
absolute value. Analogously, we use the LZ(J) pseudonorm
instead of the absolute value and apply the Cauchy-Hadamard
theorem (p. 382 Fulks) to 3.12. The radius Of convergence for

3.12 is

1

 

(3.14)

pk Ill/n

Erin sup H (bk

CO
<
so that ¢k(6) converges for ]€] pk . Then for {¢k}k:l the
radius Of convergence is p = ifif {pk} .

An added point we can make here is the existence Of

2
R(6) — RO+€R1+€ R2+"'

and its convergence region. Define an element f(6) in L2H) for
'6] < p as regular if a power series in 6 exists in terms Of a
sequence {f0, f1, . . .} all in LZ(J) . Define an operator R as
regular if there exists a pO ,£ 0 and p0 > 0 real such that for
arbitrary f 6 L2(J) , R(6)f is a generating function sequence
or equivalently is a regular element in L2[ 0, T] for I 6] < pa.
Accordingly, if

)1an 5. .11“an

43

then p = i— if such a d exists. This criterion is in terms Of a

majorant series. Another rule (Riesz—'Nagy p. 373) is

 

M
HRan : 9*1

(llfll + HROfH) n=1,2,...

with M: sup {HRnH} and r real positive providing r exists.
n

Our concern is autocorrelation functions and L2(J) kernels so
2
R(x,y,6) = R0(x, y) +6R1(x, y) +6 R2(x, y) f...

is convergent for ]6] < p and uniform in 0 E x, y, E T (= J

interval) with Rn(x, y) continuous in J . If

I Rn(X. y)l E dnH

for 6 < I? with such a d existent R(x,y,6) is a generating
function or power series in 6 .

This background has then set up the proper attitude for
understanding the theorems (Riesz Nagy p. 373-9, Rellich p. 76,
99-, 153 -) which in a following brief summary can give the
conditions necessary for 3.12, 2.13 and the following equation

for a finite multiplicity prOper value (indexed by k) and

corresponding eigenfunction power series in 6 to exist.
: <
R(d ¢ke1 kalékk) lel r (i15)
k = l, 2, . . .

Note first that we have neglected to state how the domain 4(6)
is reacting relative to 6 . In the beginning, we assumed R was

self adjoint or hypermaximal (closed and symmetric and defined

44

over all LZ(J)). A Hermitian Operator is symmetric but defined

only on a dense subspace Of L J) . For different 6's the domain

2<
40;“) might enlarge or reduce relative to L2(J) . For our
consideration it is desirable and indeed necessary that the domain
’01:“) remain stable with respect to the parameter 6 . There is
no reason to consider an unstable domain in the perturbation Of an
autocorrelation function since we are discussing them in the light
of being defined over all LZ(J) .

Let us then speak Of R(6) 6 Xc defined on L2(J) ,
convergent in a nonnull 6 neighborhood and symmetric. If Xk

is the kth isolated eigenvalue Of R(0) with m multiplicity then

k
there exists generating functions or power series KS)(€),XL2)(€), . . .
1
thmht) and corresponding (13:116), ¢]{2)(6), . . . , ¢E<m)(6) all

convergent in their respective norms in the same 6 neighborhood.

. . _ < < .
Let Xk be isolated in kk d k Xk + (12 for nonzero d1, d2.

The following then holds.

(a) R(e) 125%) = 19(6) ¢fj)(e) (3.16)
(i) _
(b) 1k (0) _ wk
(1) 111%. _
(c) wk (e)|<1>k (-1) - 611

(d) in 11k - di < k < )xk + dé the spectrum Of R(6) has

been split into x(klhe), . . . , kim)(€) for some existent
1 1 - 1 < 1 <
(11, (12 With (11 d1, d2 d‘2 .

 

Convergence Of each of the three types Of power series is
with respect to the metric Of each one respectively (scalar, L2(J),
or Operator norm).

45

The important hypothesis here is that R (6) Gage and that it has a

 

domain independent Of 6 and is convergent in Operator norm in a

 

 

 

g neighborhood. Our previous example Of perturbation R0 + 6 R1

 

treated in section 2 Of this chapter was certainly such a R (6) .

This condition or hypothesis may be weakened tO R(6) hermitian

 

and regular with domain 4(6) valid for the eigenvector set

{¢(i) (6)} m (Rellich Theorem 3 100) Note the weakened
k 1:1 ' ' P° -

 

condition has a varying Oar/{(6) . We treat next a criterion for
CD

the completeness Of {<1)k(6)}kzl .
If the Operator R(O) is positive definite and has a
discrete spectrum we wish rules to guarantee the operator may be
perturbed by R(6) and still remain positive with discrete spectrum.
Nonsingularity in the detection problem, consequently, would be
preserved. This doubt is not to be taken lightly for if nonsingularity
leads to singularity through perturbation (or vice versa) then the
detection or hypothesis test perturbation problem is not well posed.
Equivalently, how strong are the conditions which must be met in
order to preserve positivity for the Operator or completeness Of

(11
the eigenvector {¢k(6)} k:1 sequence?

a)
Criterion: (Rellich p. 153-162) Consider {Rk}k-O with Rk all

hermitian in L2 [ 0, T] with R CC and sequence bounded. Let

0

H Ran 5 k“ H ROfH (3.17)

for some positive constant k and every n index, f 6 L2[ 0, T] .

2
R(6) -— Roi-6R1 +6 R21"...

46

is hermitian bounded, regular for '6] < Ek— with eigenvalues

CD 0)
{)\k(6)}k___1 and eigenfunctions {<]>k(6)}k:1 all convergent power

series in l6] < L}; and
(a) R(6) ¢n(€) = Xn(€) ¢n(€)
(b) {<])k(6)}ookZl complete in L210, T] (3.18)
(c) lkn(6)]-*0 as n-00.

As long as the domain 4(6) is independent Of 6 and R(6) is
regular for some 6 neighborhood and R(0) is self adjoint with
discrete spectrum then R(6) has a regular discrete spectrum.

For an integral Operator 3.17 may be put as (Rellich p. 155)

Huanu): 5 knhulROuH ,

T T 2 .
folfo R(x,y,e)u<y)dyl dx: MZIIIIT R(x,y,01u(y)dy|2 (3.19)
O
or

T T __.__ T T _____
]f f R(x, y, 6) u(x) u(y) dx dy] E Mf f R(x, y, 0) u(x) u(y) dx dy
O O O O

V. STATISTICAL PROPER TIES OF THE LIKELIHOOD EXPANSION

In the preceding section we have explained the conditions
for the existence Of the forms R(6) , ¢k(6) and Xk(6) and some
Of their important properties such as completeness of {¢k(€)}k-l .

Now we wish to consider changes in the statistical instead Of the

function aspects Of the likelihood ratio with perturbation.

 

47

From 2. 6 the likelihood ratio is

where ak is the coefficient Of the sure signal relative tO the kth
eigenfunction Of R(x, y), the autocorrelation function of the noise
process; wk is the coefficient Of the received waveform relative

to the same function, ¢k(t), while k is the statistical variance

k
of nk the Fourier coefficient Of the noise process relative to
¢k(t) . The following expression has meaning when taken in light
Of what we have considered previously in this chapter. The

prime will signify, for the moment, the perturbed likelihood

ratio.

 

(3.20)

We want to know how this perturbed ratio compares with
the unperturbed one either in function parameters or statistically.
Since we have dealt with the first order perturbation of the Operator
kernel it is only consistent that we consider first order approximations
to the eigenvalues and eigenvectors in view of the fact that conver-
gences for the three perturbed parameters may be comparable. It
is indeed impractical to assume one series will converge much
faster than another when dealing with our assumed small 6 parameter.

Expanding from 3.13, 3.12:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I) (1) fl
2. Z‘ :
CD J:0 6 akJ .620 L Wk”
y’ : Z + (3.21)
k:] OJ m
23 \ 6
m:0 km
. ako Wko . V akowkl + Wko akl 2
221-161 *6“ \ +e\ ”3(6)
k0 k1 'ko ’ k1
akOWkO / l akowkl + Wkoakl
:: E \ \ ‘F6 2 'x +-ex
'kO ’ k1 kO k1
1 -(~6 \ )
1‘0 (3.22)
a w on k, n a_ w’ +w a
kc; kO Z (_6 {53‘} + E 2 kc; lir Exko k1 + C(62)
k=1 kO n‘ ko 1m k1
1, w a
é 23 kc; k_c_)__ (l 6 xkl) 1. Z 13:) kip Xko kl + O(EZ)
kO ko k k1
a w , w )c a )1
=2: 1‘3“” +e{245§—~‘11-1-ekk1)12 1‘] k0(1- —-—)\kl)+
'ko 'ko ko kO kO
a _ w , )c
. + _ E k; k) (Xkl) + C(62)
‘kO kO
"W, a w a W,
.2 YO + 6 Z _iSiXLILL + L “1&1 ko _ 23 k)? ko xkl +
kc) ko ko k0
2
+ O(6 ) (3.23a)
. (I) 3. JW
y' 5: y + 6V + C(62) where Y *— E M (3.23c)
O .1 1 k=l )xko

(3.23b)

. 2 ..
Dropping the reference to the terms of order 6 or greater we

have

49
I _’__
Y — YO+€Y1

Let us consider Y1 alone for the moment:

 

00 W 00 a W 00 a W
Y1 = Z kp k1 + 2 k1). ko _ 23 kc; ko xkl ' (3.24)
k:1 k0 k=1 ko k=1 ko kO

The minus sign and corresponding term will not make y' negative

)c
if Eris-l < 1 even when the first two terms Of Y1 vanish. The
ko

question remains as to how important this assumption is. Since

we have assumed convergence Of the )x 6) series we must have

R(

6 xkl < xko for all k especially when the perturbation of the

spectrum is assumed small corresponding to small 6 R But we

1 0
shall discuss this later in more detail. 5 Terms of the first and

zero order are present in Y1 and the first order perturbation

terms are in the numerator. From 3. 7, 3.10, 3.11b:
(R0 ‘ >‘koll‘i’ki : ()‘kil " R1)¢ko
(¢k1|¢ko) = o k=1,2,3,...
)‘kl : Wko iRl (bko) (3° 2'5)

Solving for ¢kl for k=1,2,3,... will yield a for k: l,2,3,...

k1
needed for Y1 through

T
an -- .10 a(t) ¢k1<t1dt
k: l,2,3,...

T _
wkl = 10 w(t)¢kl(t)dt .

 

See discussion after equation 3. 38.

50

Hence, we can find the entries for Y1 from the above 3. 25
equations. Note here that the first order perturbation operator
is 6 R1

R' R +6R

C ompute now7

(3.26)

Var (YO+6Y1) Var Yo + 6 Z COV(Y O, Y

1)

Consider for the moment Cov(YO, Y 1) using 3. 23c, 3. 24

 

)‘ko >‘ko xko )‘ko xko

a w a w a w a w )c
COVWO1 Y1) : Cov 2; M, ()3 M4. 2 M _ 2; k0 k0 k1)
(3. 27)
For the zero hypothesis or no sure signal in the received waveform
w(t) we have wkl = nk1 and wk0 = nk0 for k = 1, 2, . . . . where

nkl and nkO have zero means. Equation 3. 27 then becomes

 

a n a n a n a n )x
k k k l kl k k k l
Covwo. v1) = 21/011 = E(z ——‘;\——9)(2 —):’—k— + z ————)\ 0 - >3 0). 0 1k )
ko kO ko ko kO
(3.28)

When we discussed the perturbation technique on a function and kernel

basis we neglected to affirm that its statistical property via the

 

v 7 v17

6

The effect of this second order perturbation on the first
order statistical nature of n(t) , the noise process, is not one of
an additive stochastic process y(t) necessarily, i. e., n(t)— ' n(t)+ y(t)
is not necessarily true.

7 Cov (Y., Y )j, k > 1 can be shown to be less or Of equal
magnitude to thdt O Var YO, COV (YO, Y ) so that the 62 or higher
orders of 6 coefficients Will diminish their significance.

51

Karhunenioéve expansion has not changed. That is, for the

perturbed process, regardless Of the parameter 6:

1 _.
E nk(€)nj(€) - 6kj>xk(€) . (3.29)
Expanding,
CO (I)
E'n (6)n.(6) = E' Z 11 £62 23 n. 6m
R J [:1 k ITIZO Jm
, .
but E nk(6) nj(6) is equal to )ik(6) ij
E'n(€)n(€)=)\ +€)\ +62)\ +63k +
k j kO k1 k2 k3

For proper convergence Of the 6 generating functions or power

series we may equate like powers Of 6 so:

1 _.
E nkO njO _ xko 6kj

1 1 ._
E nk1 njO + E nk0 njl - xkl ij
(3.30)
1 1 1 _
E nk1 nJ1 I E nk2 njo + E nk0 nJ2 )‘k2 ij
1 1 l 1 _
E nk1 nJZ + E nk2 nJ1+ E nk nj3 + E nk3 njo — k3 kj

simplifying, this bec omes

Varn : k

k0 kO
2 Cov nk1 nk0 : )‘kl
2 Cov nkZ nk0 + Var nk1 : 1k2
2 Cov nk1 nk2 + 2 Cov nk0 nk3 = )ka

 

8We will drop the prime superscript after this point.

52

ANNé

Neg in >00 N +
+ 2a 2: >00 N +
+ 9? 85 >00 N +

_l + mxs Hm>

l

 

1 n

 

mmoumxo 050 0.5 0m

0+

A00? .23 >00 N +

+ $.anva >00 N m0

0
N.

 

r

Ao
+

.. 0%
~30.“ 320000 N a

1L
Nam 20 >00 N +

2030 >00 N +
+ oxn mxc >00 N1

as .Nch >00 N +
+ Can uo>

HA0

N

.mvfldn fiINAVH—H >00

w+

 

m
N +

20 0+

0 O +

n 2025 H0>

0L

a a

20 in >00 N +
Neg men >00 N+

Cad porn >00 N +

 

h OVAC” PXHH >00 N

IL

A005 .005 >00 N +

 

0 m0 unofioflwooo 0:»

w+

 

L e

w+000

 

r + or Er

m N

L i.

ons .Hst >00 N w + Ours new? .1.

Nan 0 + 200 + 8:: .S>

"mafia/020m 0:» 03.03 03 .oosoquou Mom

53

Thus, cov (nkj' nkfl) may not equal zero, although cov (nkj' nml) :

for k}4m; j,l =1,Z,...

We are still considering first order perturbation here so

Var(wk + 6w )=Varn +62cov(n
o 0

k1 k kl’ “1(0)

= Var 11kO +62Enk1nk0 = Xko+€xkl

under hypothesis 0:

Ewk0 = Enko = 0.

C onclude

l
E nk1 r1k0 — —2— Xkl — cov (nko’ nkl) (3.35)

Returning to the original calculation Of cov (Y0, Y1) Of 3.28 we have

 

x I

akonko a.on.1 a.1n.o a.0n.o X.1
COv(Y,Y)=EYY=E EZ( \(JJ+—l—l—-—l—J—-J—)
O O l O l - X. )x. X. X.

J k k0 jO jO 30 30

akOa. ak a.1 ak a. X'l
: 22 “—41—?— En n. +-—£—J——En n. --—-—-9—-12-J—En n.
j k k0 jl Xkoxjo kO 30 k X k. k0 JO

 

 

 

 

,n ) .
J k k0 JO k k 31 kkoxjo kO jk xkoxjo J0 kO jk
a2 a a a2 )\
: Z) kq_ c0v(n n )+ k0 k1 kO ( k1
k xkoxko k0 k1 )ck )ck )ck
But from 3. 35 we have then
a2 a a a2 >\
Cov(Y ,y) = z ———-k0 (L). )+ —————k° 1‘1 - k0 1‘1 (3.30)
O 1 k k2 2 k1 )xk )xk )ck
k0 O O 0

But simplifying 3. 36 reduces to

54

 

2
a a a )c
k kl 1 k k1
Cove/0.11) = 2 79—— - 2— 1 ° 16—) (3.37)
k k0 k0 k0

We can now write 3. 26 in its true form using 3. 23c and 3. 37
Z . Z

 

 

 

a a a a )1
k k k1 1 k k1
Var(YO+6Y1)=Z()\ O)+622 -—)\O———-f {43(7)
k k0 k k0 k0 ko
2
a )c 2a a
:2 Kko (1_€>\k1)+€( 1:1 k0)
k ko k0 k0
aZ )\ 00 2a a
CD
Var(YO+6Y1) = '2: K1” (1 -exkl)“ 2: +119- (3.38)
k=1 k0 k0 k=1 k0

Note that as 6 -* 0 Var(YO+6 Y1) -* Var Yo independent of H Rl I]

(whose influence is manifested in )c and a Further,

k1 k1)°
Var(YO + 6Y1) "‘ Var YO independent of 6 if H R1 H -’ 0 since

then xkl -' 0 as well as ak1 "’ 0 . Therefore, this result 3.38

is consistent with the zeroeth order variance since it collapses

 

 

 

into it if either [I R1 H or e —- o.
6kk1
Since 6 is small x < I certainly in ]6] < 1 since
)‘k'+1 1” >‘1<'+1
We must have + < 1 for convergence if Tl— —’ l . SO
xkl kj kj
l -6 h— > 0 for infinitely many k if '6] < 1. Yet this is
k0
not enough to guarantee
2
a )c
2 Xko (1-155(1) > 0
k0 k0

but it does say the greatest number Of series terms are positive.

The series will be positive if 6km < xko however for k : l, 2, . . .

This requirement is manifested by RO > 6 R or that R0 is a

1

larger operator than 6 R1 which is Obviously the case since the

perturbation is indeed assumed small, much smaller than the

55

 

original Operator. Note the order Of eigenvalues here 6 )‘k1 < xko
and that we are _r_1_ot implying xkl are eigenvalues Of R1 .
2
a R 2a a
Var(Y +EY) = 2 k0 (1-6—5—1-)+6Z k1 k0 (3.40)
O l )1 )1 )1
k0 k0 k0

Could the conclusion be drawn that the new variance is smaller or

larger than the original one? That is, can the following hold true?

 

 

 

 

2
- ako in . zaklako ‘
L 1 ‘1 " Z 1 Z)
k0 k0 k0
2
a )1 a '2
. 1 -
z X10 (Xk1 -2 35—) > o (3.41)
kO k0 ko

For Var(YO + 6 Y1) to be smaller than Var YO we need for k )4 0

xkl akO

————-- 2

Xko ak1

> 0

 

such that the whole sum 3. 41 is positive. It can be seen that if

)1
JEL k = 1, 2, . . . are very small for all k this may be impossible.

X

k0
Otherwise, the restrictions we have placed on our perturbation
technique could permit

< -
Var(y0 + 6 Y1) Var YO

Let us try tO compute bounds for Var (Y0 + 6 Y1) in terms

of Var Y0 and a few parameters we may derive now. Let a(t)

be a signal integrable square on [0, T] = .I . Let perturbation

,andk >)\ k=l,2,... Let

>
be such that R0 R1 k0 k1

R1

1
0 §_ 0»: inf {X31 } < 1
k k0

56

 

 

 

 

)1
0< fi’ = sup {kid} 5 H‘< 00, Haconstant. (3.43)
k k0
= . > 2
m 031p {13..ak1_ Kj ako k 1,2,...} (3.44)
m finite since ak's are finite
: 0 ° < : -
M 1311 {13. Iak1]_ K]. lakO] k 1,2,...} (3.45)
M finite since ak's are finite
then
2 2
a a )1 2a a
2 k0_€z)\ko 1k1+ Z 1:11“):
k0 k0 k0 k0
_<_ Var YO - 6 (Var YO)-ts-+ 2 6 (Var Yo) M (3.46)
2 2
ako ako xk1
VarY -6fi'VarY +26mVarY < 2 -6E -— +
O O 0— )1 )c )\
k0 k0 k0
2a a
+6 2 kxl kO
k0
so

max (0, Var yo (1 - 6(fi-2m)) E Var(YO+€Y1): Var YO(1+6(2M --u1- ))

(3.47)
GEVar(YO+6Y1): (9

For 6 small enough we need not worry about limiting the left hand

side of the inequality so we have

-6(fi'-2m) E A Var _<_6(2M--w-) (3.48)

57
Let G = max (fi-Zm,2M-w-) then
- 6 G E A Var 5 6G

lAVar] : e0 (3.49)

VI. RESULTS RELATIVE TO THE THRESHOLD

In previous sections of this chapter we have compared what
we have called the nuclear likelihood ratio YO with a perturbed
model Y(6) . It perhaps should have been stressed that this
comparison is seemingly unjust if a fixed threshold is to be
involved. As we recall from 1.16 the threshold was chosen for

Y simply because 1') in 1.15 was different from the expression

by a c onstant.

 

 

2
fl Wna'n 1 an
nzlni—zz A --2—E->-\— (1.15)
O n n
2
1 an
n
wnan >
y = z 1 < t (1.16b)
n Y

The constant, Of course, supposedly is L]; . Yet if perturbation Of
the second order properties Of the noise is considered this quantity

is not a constant relative to the disturbance parameter 6 .

 

2
a.
(1,0 = é— 2: “0 = %- p: (3.40)
no
02(0)
w(s) = fi—2 X3375 = é— 03(5) (3.47)

58

Thus, in order to consider the changes in the detection structure
relative to a threshold, we will have to explicate the changes
occurring to 1]) as well. To do this, it is important to use the
expression for the likelihood ratio in 1) form. We must keep in
mind, however, that Y0 and no as well as Y(6) and 17(6)

have the same stochastic variance and properties except mean.

 

wn(e) ans) I aim
n(e) : Z - — Z (3.48)
)1n(6) 2 )1n(6)
= Y(6) “12(6)

From 3. 23c,

2
+6 an2+...)

)1 + 6)1 + 62X
no n1 n

E (ano + 6 ar11

 

L11(6)

N|'—‘

Z+...

Approximating to the first few orders of 6 we have

 

 

 

 

 

é—Eaz +6(2a a )+e‘2(ai2 +20 2a )+...
111(6) 5 no n0 n1 n1 n no
>\nl 2 an
11n0(l +€‘>:-“- +6 )1 )
no no
a2 +(2a a )+O(62)
,_ _l_ 7.3 no no n1
— 2 )‘nl
)1n0(l - (-6 )1——))
no
)1
2 7 2 n1 2
1 {am +e(-anoan1) + 0(6)} (1 -€ 1 + O(6)
é _ 2 no
2 )1
no
2 2
s l_ E ano + {2 anoa'nl _l_ 2 8'no (xnl )} + O( 2)
‘ 2 1 €— 1 " 2 1 1 6

no I10 no no

59

 

 

111(6) : 1|) + 6 1111 (3.50)
l ano nl
1111 - ~2- Z X— (2 anl -ano ) (3.51)
no nO
>\nl
If r— is small for n = 1, 2, . . . 1111 may be positive if it is
no
convergent. However, there is no apparent reason to rule out
1111 < 0.
)1 1
Nevertheless, L]J(€) is still positive since 6 {£— << 1
no
2
1 8'110 )‘n1 1 Z anoa'nl
$k)=§ Z (l-ef—4+e§-Z-———-— (153
no no

so L|J(€) > 0.

There is a strong resemblance between 3. 38 and 3. 52 .

nb

This

means that the unperturbed relations 3. 46 is kept after perturbation

in the since that

we)

= %Var1(t) = 13%)

Now we can compare both expression for 1') before and after

perturbation:

But to the first

nk)

 

O 1
-Y0-Lpo~>:n):)n -22
no
.— an(6) wn(6) 1— 331(6)
‘ 1h“) ’ 2 1h“)
= YK)-¢k)
order:

(YO+€ Y1) -(¢O+€¢l)

‘(Yo'Y0)+dY1'¢N

 

)1 (3. 53a)

(3. 53b)

(3.54a)

60

27(6) = no + enl

 

 

 

(3.54c)
From 3.24 and 3. 51,
a w a w a w X
n1={z———k§’\k1+z—————kik° -2 k)? k"(K1:1)}+... (3.55)
ko ko no ko
x
1 ako k1
- — Z) (2 a - a )
2 xko k1 ko )‘ko
We can see from 1.15 that Var n : : Var y .
Var no 2 Var Yo (3. 56a)
Var 20(6) 2 Var y(6) (3. 56b)
Hence,
AVarn : IVarn(6)-Var no): lVary(6)-Varyol

A Varn : A Vary

The change in variance then is nil

relative to consideration of the
L): expression.

Say the threshold for n is tn and for y is tY
>

< t t

Y Y 77 < T?

are the respective likelihood tests. The differences between
thresholds are

 

2
1 no
At = t -t = — 2
Yo U0 2 xno
2
1 an(€)
At(6)

 

 

 

1 ago
Ato = ~2— EX : 410 (3.55)
no
At(6) 2 41(6) 2 ¢0+6¢1 (3.56)
change in At = At(6) -AtO = 6411 (3.57)

Equation 3. 57 is noteworthy. It gives the magnitude of the change
in threshold needed to preserve a true maximum likelihood test
(or modified) quaranteed by the Neyman Pearson lemma to be the
most powerful test for determining H0 or H1 .

Thus, in order to consider energy changes manifested by 6 R1
and their effect on the detection test used, namely the likelihood
ratio, the first order approximation to the threshold dispersion
between 7L and L required to maintain it is less than the 3. 57
m. To calculate the actual threshold change required to
maintain the likelihood ratio test we need the rule for deciding
what the new threshold will be. Suppose for example the threshold

tn chosen is midway between the means of n under hypothesis 1

and O. From 3. 53a

 

2
1 ano
I“:0 no 2 -2- )‘no
2 (3.58)
_ 1 no
E31% ‘ 2 Z x
no
then t = 0
no

From 3. 54c and 3. 55

E0 77(6) : Eo 770+ 6 E0 771

 

 

 

 

 

 

 

2
a a h
_ L no L ko k1
Eon“) '2 Z i +262). (koi '2 k1)
no ko ko
a2 a a a a a2 X
Eln(6)=:—ano+6{22 §1k°-z klxko -zi39XEL+.
no ko ko ko ko
2
a X
. + "1&2 >\kO Xkl}
ko ko
a2 a a a2 k
En(6)=-l—E n°+€{z_1£l_1_<2_1_2._§2(k1)}
l 2 x x 2 k X
no ko ko ko
t = -1—{E n(6) +E 10(6)}
17(6) 2 o 1
2 k 2
a a a a a a
:%{:—e{2kko(xkl)-ZZ k>okl}+ {2 1:1 ko_:_xko Xkl}}
ko ko ko ko ko ko
t, : O 3.59
n(6) ( )
t t -t : O 3.60
A 77 77(6) no ( )
For the Y test:
E0Y0 : Eono+Eo¢o : Eono+4jo
ElYozElno+E1¢ozElno+qjo
(3.61)
EOY(€) = E0 n(6) +12O M6) = E0 n(6) +446)
E1 Y(6) = E1n(6)+E1LIJ(6) = E1n(6)+ l11(6)
t : -1— {4) +12 n +E n} = t +-1— Ll)
yo 2 o o o l o no 2 o
t -l- LlJ(€)+t
Y(6) 2 ()

 

63

1

t : t _ t : t + — - 3.62

A Y Y(6) YO A n 2 (Me) 410) < >
For example, we have from 3. 51 and 3. 59,

At =t +1—e4) (3.63)

)1 M6) 2 l

a a a2 )x
: _:_ 6 {Z} k: kl _ :_ 2 no xnl }
ko no no

Aty = A 1:Y =é—ELPI (3.64)

The 3. 64 result is an example of the change in threshold required
to maintain the modified maximum likelihood test in a detection

problem with second order perturbation of the form 6 R1 .

 

What we have said about convergence in the past still holds
throughout this section. If convergence occurs then a singular case
is in play. As mentioned in chapter one, we are acutely interested

in the nonsingular case for perturbation. It is the nonsingular case
2

 

a
in which 23 {Ii-9- < 00 is generally found. We see that in this case
ko
)x
3.64 is convergent if k1 E l for k = l, 2, 3, . . . or at least for
ko

infinitely many k . It is significant to note the direct dependence
on 6 of 3.64. If the series is convergent, for small parameter 6

the quantity of changes in threshold necessary is small also.

CHAPTER IV
AN EXAMPLE AND THESIS CONCLUSIONS

The ideas and theory expounded in Chapter III will be put
into more tangible form in this chapter with an example taken in
this case of perturbing a wide sense stationary autocorrelation
with a non wide sense stationary one. The signal chosen will be
a sine wave. After a few comments on this example a general
summary of the thesis will follow in which its results will be

coordinated and explicitly outlined.

I. AN EXAMPLE

In order to avoid tedious mathematical detail the example
to be taken here will be one for which the unperturbed and
perturbed kernels are in simplest terms. The procedure for
extracting the eigenfunctions of the unperturbed kernel is outlined
in Davenport and Root (pp. 99-101 and Appendix p. 371) so that
only the final results of that extraction will be shown here.

The autocorrelation function of the original or nuclear

noise process is

emit.-SI , -T< s,t< T,

Roms) = Rout-s!) =

whose eigenfunctions and eigenvalues can be shown to be solutions
to the differential equation

x 4>”(t)+ 7‘ ix ¢(t) = o (4.1)

 

with 0 < R < 2 only.

64

65

The eigenfunctions and eigenvalues of the unperturbed kernel

are known to be in a split form:

 

 

 

 

 

 

A A . A
¢k(t) = Ck COS bkt (4. 2a) ¢k(t) = ck Sln bkt (4. 2b)
T A 8 'I‘ 1 4 3

bk tan tk — 1 (4.3a) bk cot k — ( . b)

2 2 I
)‘k : b2 +1 (4.4a) Xk 2 [B2 +1 (4. 4b) .

k k

l l
Ck = i fl (4.5a) ck = a, (4. 5b)
sian T sin 2 T
k k
T + 2b T - A
k 2bk

 

 

a non wide sense

BY ChOOSing R1(t,s):e('|tl ”'S')

stationary covariance function the three important equations of

fir st orde r pe rturbation,

(RO - xkol) 41d = (xkll - R1) 41m (3.25a)
Wkl l¢k0) = 0 (3.25b)
)‘k1 : (¢kolR1 4’ko) (3'25”

k: 1,2,...

can now be put into integral forms. Equations 3. 25c, 4. 2 are used

tofind
2
4c
k 2 2
X = : c X (4.6)
kl (1 +1313) k ko
“—1 E Ck: —1 k=l.2.o-- (4.7)
VZT' VT

66

The dual nature of the eigenstructure of R0 incurs a
A

similar calculation for xkl .

A
ikl = o 1<=1,2,

To find ¢k1 equations 3. 25a, 3. 26c are used along with the
new found knowledge of 4. 6, 4. 2 and 4. 4. The resulting calculations

are lengthy but involve only algebraic manipulations.

¢k1_ Bktsinb t+A cosbkt

k k
2 2
Bk ‘ Ck xko
C3 (4.9)
_ k 1 2
Ak — -----2- (2 -Tck cos ka)
b
k
A - o
¢k1 ‘

A few characteristics of this particular example are examined

before Chapter III equations involving the change in variance are

broughtin.

 

 

 

First, we note that kkl < )‘ko for infinitely many k since
)‘ko -’ O as bk —’ 0° and eventually c: xko < 1. From 4. 6
)‘k1 = Ci xko where Ck is bounded above and below
1 < Ck : 1 < ——l——
«I2T' \/ sin 2 ka «11"
T + "2 bk
Second, ak1 is related to ak0 in the following manner from 4. 9:

_ 2 Z . L 2 2
ak1 — ck kko (t 8111 bktl a(t)) + (Z - Tck cos ka) ak0 (4.13)

U‘O
WNWN

 

67

A A
Recalling "k1 = o and 41d = 0, we have 91d = o . (4.14)

The nuclear autocorrelation function R0(t, s) has the split
nature of eigenfunction sets but only one of the sets will be perturbed
by the kernel R1(s,t) = e-' tl - l SI .

If the signal a(t) = A sinwmt, - T j t E T the following

c oefficients result:

 

 

 

 

A _
ak0 — O
ako -_- Ack “risk ' ‘06)“): - 8:101; “gm” (4.15)
k ‘ “m k ”m
2 . . ,
_ _C__k_ {-1- Tczcoszb T} A51n(bk-wm)T Sln(bk+h)m)T

akl‘zz‘k kck b-b.) "b+w

bk k m k m

(4.16)

There exists (Om such that

l
=O=a— e.g. wm=§bEfor somek>0.

ako k1 '

Finally, the variance of the perturbed likelihood ratio can be computed

from 3. 38 for this example.

 

2 2
)‘k1 _ Ckkko _ C2 1. < >‘ko (417)
X _ X — k ko -- T °
ko ko
a2 X 2a a
Var(y +6y)= 2—53(1-e k1)+ 2 1‘1 k0 (3.38)
o l X X X
ko ko ko
For large bk’
2 ~ 1
CR T (4.18)

68

so that the series term of 3.38 approaches

 

1
2 2 T 1 2
(1 —e T)+6(2akO-—-—-2— (2 -cos ka):)
k0 Zbk

(4.19)

 

(4.20)

II. CONCLUSIONS

An attempt has been made in this thesis to underscore the
importance of the stability of a signal detection scheme. An example
of a scheme, namely one involving the likelihood ratio for a sure
signal in zero-mean Gaussian noise with continuous autocorrelation
on a finite interval J , has been perturbed by a slight change in the
second order noise statistics. After reviewing previous work and
setting up simple examples, a more general case of perturbation is
formulated in additively distunbing the autocorrelation function with
an LZ(J) positive semidefinite kernel. The change in form and
variance of the perturbed likelihood ratio is found to be of the
order of 6 , a small positive parameter. Using the parameter
strategically, continuity of the signal detection scheme is shown
through stability of a finite -variance likelihood ratio. The concept
kept in mind throughout Chapters III and IV (Section I) which encompass
the new results is that the detection test was to be kept the Optimum
Bayes procedure both before and after perturbation.

Changes in the form of the likelihood ratio is illustrated by

equation 3. 23a while change in variance is shown by 3. 38. Threshold

69

dispersion between the true Gaussian likelihood ratio and the one
used predominantly in the thesis is demonstrated by 3. 57. In
selecting the point midway between the means of the ratio under
either hypothesis (before and after perturbation), the change in
threshold necessary for an optimum signal detection scheme is
shown to vary directly also with 6 parameter (equation 3. 64).
In Chapter IV, Section I, kernels corresponding to the
R and R operators in equation 3. 25 are chosen, the former

O 1

a wide sense stationary one and the latter a non-wide sense

 

stationary one, so that the forms of the series in the results of
Chapter III involving the changes in form and variance of the
likelihood ratio can be seen more easily. Choosing the sine
wave as the sure signal, the perturbed forms are seen to vary
directly and simply with parameter 6 , most clearly shown for
large index as demonstrated by 4. 20.

The relevance of showing continuity of a detection scheme
is made clear by the fact that there is no guarantee the hypothesis
test for detecting the presence of a sure signal in additive noise is
robust or stable. If not stable relative to small changes in the
noise energy, the error probabilities would be liable to drastic
change and the detection model would then be little more than
useless. What this thesis has £1.23 done is shown how good the additive
Gaussian noise with continuous autocorrelation function model is
in relation to noise experienced in actual signal detection equipment.
We have not tried to defend the model as a good or true one, but have
tried to show its stable character. In this way perhaps, an indirect

defense for its "goodness" may be implied.

APPENDICES

A. HILBER T SPACE

Many concepts in signal detection theory can be put into
simpler mathematical forms when the notion of a hilbert space is
introduced. In this appendix section are outlined some important

hilbert space concepts which are used throughout the text.

 

r
I. PREHILBERT SPACE

A prehilbert space (PHS) is a complex vector space P with I

1

a scalar product of any two vectors, x, y 6 P, denoted by (X! y), E

9

defined with the following properties: L

l. (x, y) 2W (where bar denotes complex conjugate)
2. (x+yl 2) = (xl 2) +(yl 2)

3. (Xx) y) = X(x| y), 1. a scalar.

4. (x) y) > 0 when xfi 0, the zero vector.

The above four statements imply the following elementary theorems:
Theorem:
1. (xly+ 2) z (x! y) + (xl 2)
2. (X) Mr) = MXI Y)
3. (6 ly) = (y [6) = 0
4.1x -yl 2) = (xl 2) -(yl z); (xly - z) = (xly) -<xl 2)
5. If (x) z) : (y( z) for all 2 then x = y.
Every PHS is a metric space with the distance function

defined through the inner product thus:

70

71

2 2
d (x.Y) = (X-YI X-Y) = H X-YH -
Hence, the NORM of a vector in a PHS is given by

Hxll = «f(xlx)

Properties of the norm are expressed through the following:
Theorem:
1. IIXxH = N llxll
2. ”x“ > o if x15 9, the zero vector.
3. u x + y” 2 + ”any” 2 a 211x112 + 2 u y” 2 parallelogram law
4- |(xl y)! =1/4{Hx+yllZ - ll x-yll 2 + ill x+iyll 2 -i ”x-iYH 2}
Polarization identity

5. l (xl y)l E H x” H y” Cauchy-Schwartz-Boniakovsky Inequality
(CBS Inequality)

6- Hx+yll _<_ ”X” + “Y” \
7- Hx-Y” > 0: ”X'Y” =0" x=y (

8. ”x-y” = ”y-x" . , Metric character

 

9. Hx-zll _<_ llx-vll + lly-zll

II. HILB ER T SPACE

Once the norm is defined, it is possible to speak of the
convergence of a sequence in a prehilbert space and only then will

the concept of a hilbert space naturally follow.

 

Definitions. Let xm be a sequence of vectors in a PHS.

1. The sequence of vectors converges to a limit vector x

 

if ”xn-x"-’0 asn-‘CD, i.e., give 6> 0,
EN 3n> N implies ”xn -n” < 6 and x is unique

in the PHS.

72

xmll-w

2. The sequence of vectors is Cauchy if H xn -
as m,n-’00,i.e.,given6> 0 3Nal'Xm-anl <6
if m,n > N.

3. A PHS is Complete if every Cauchy sequence converges.
A PHS which is complete is called a hilbert space.

To find a set of vectors which will generate the entire hilbert

space H is a problem demanding a few more specialized definitions.
(I)

By the word ”generate" we mean that a sequence {xk}k 1 generates
co _
H if for any vector x in H, scalars {ak}k exist such that
CO
x — 1:1 Gk xk .

Definitions .

 

1. For x,yin a PHS, x is orthogonal to y l?

 

(x(y) = 0, that fact denoted by xi y. If Xin ,
co

j=l,2,3,... then X‘L{Xj}‘1 and X‘L[{XJ}]
J:

2. An orthogonal sequence {xj} has xki Xj’ jfi k.

 

3. An orthonormal se_quence {xj} has xk _L Xj’ jf- k
kall :1 for k = l,2,3,...

4. A set S of vectors in a PHS P is total if the only
vector 2 of P orthogonal to every vector of S is

the zero vector 2 = 9 . A total sequence is defined

 

accordingly.
5. An orthonormal basis for a hilbert space H is a set
{xa}

6. A hilbert space is separable if it possesses a total set

 

CD
1 which is both total and orthonormal.
n-

in V, the vector space it is defined over.

B. 12, L2 AND OPERATORS

Two prime examples of hilbert spaces are those denoted by
L2 and 12 . Besides being the best known and most utilized spaces
they provide exceptional insight into general hilbert space theory.
In the following we will try to give an informal definition of these
two hilbert spaces after which we will proceed to give a short

. . . . . 2 2
introduction 1nto Operators and their character 1n L and 1 .

I. L‘2 AND 12

. 2 .

The space 1 is the set of all square summable complex
sequences. Each sequence is a countably infinite dimensional
vector. The inner product associated with 12 is

(alb)=.
J

JLMB
91
0‘

where a=(a1,a2,...) and b=(bl,b2,...)

and both belong to 12 .

The space L2 is the set of all square Lebesque-summable
complex functions over the open infinite plane. An analogous space,
L2[ a, b] is the set of all square Lebesque-summable complex
functions over the finite interval [a, b]. The inner product used

in L2[ a, b] is demonstrated by the following:
b __
fa f(t) g(t) dt = (fl s)

where f(t) and g(t) both belong to L2[a,b].

73

74

II. OPERATOR S

An operator or linear transformation defined on a hilbert

space H is a continuous linear mapping T such that

T(o.x + By) 2 OTx + BTy

is a bona fide vector in H for any x, y in H and any scalars F“.
G. , (3 in the scalar field defined with the vector space.
A bounded operator is one for which there exists a constant

M greater than zero for which the norm of any vector in its range

 

is less than M times the vector's norm. That is, T is bounded
if and only if for all x in H there exists a M > 0 such that
(I TX“: M H x

preceding holds is called the operator norm, denoted by H T”

 

 

. The smallest such constant M such that the

 

”T" = inf {Maconstant> o: ”TX” 5 Mllxll a11xinH.}
M

The collection of all bounded operators is denoted byot c .
It is easily seen that a bounded Operator will carry bounded sets
into bounded sets. If it also carries any bounded set into a
convergent set than it is termed CC, Completely Continuous or
Compact. A CC operator is sometimes referred to as a "small"
Operator since it somewhat "compresses” a vector set which is
bounded into a convergent form.

An Operator's adjoint Operator is defined by the following

equation:

>1:
(Txl y) = (xl Ty) for all x, y in H, a hilbert space.

75

We can now define more Operator types using this "adjoint" notion.

Definitions. Let T be an operator defined on H, a hilbert space.

 

l. T is isometric if T*T = I where I is the identity

 

operator, i.e., Ix = x for all x in H.
2. T is unitary if T*T= TT* = I.

3. T is self adLoint if T* = T .

 

4. T is normal if T*T = TT* .

An Operator T has as a proper value or eigenvalue and x

 

as its corresponding eigenvector if Tx = x. It will be seen in

 

appendix C that self adjoint operators are easily characterized
by their eigenvalues and eigenvectors.

Operators in L‘2 are bivariate kernels of either two complex
variables if L2 is complex function space or two real variables if
L2 is restricted to be a real function space. The space 12 has
infinite dimensional matrices as its operators. Both types of

operators are characterized by the following:

 

 

T
l. (Tx)(s) = f T(t,s)x(t)dt=y(s) DisiT
O .
(Tx 2 y in L2[0,T])_
Z. T : a11 a12 .. x— x1
3.21 3.22 ... X2
L..-
r- '1
Tx = E a .x. = .
11 J y
Ea .x
_21 J
1.. ° _

 

 

(Tx=y in £2 ).

C. LZ KER NELS

The space of functions which provides a platform for the
integral equations we are going to consider is L2[ 0, T] . Integral
operators on this space, denoted by capital roman letters R, K, . . .,
have corresponding kernels R(x, y), K(x, y) some of whose
properties are explained in the following discussion.

If a(t) and (3(t) belong to L2]: 0, T] the general integral
equation of interest can be written

T
f R(t,s) a(t) dt = (3(3) 0: s _<_ T (C.1)
O

Abstractly, Ra = [3 . If for every O6 L2[ 0, T], RC1 belongs to

L2[ 0, T] also then we say R is an L2[ 0, T] kernel and is "defined"

over all L2[ 0, T] . Otherwise R is simply an operator and

R(t,s) is its kernel.

For any element, f(t) 6 L2[ 0, T], its norm, I] f” , is

defined by the following:

 

T 2
limb/10 lie)! dt . (c.21

Convergence in the space L2[ 0, T] is based on this norm.
T 2

lim f [g(t) - Za_f(t)] dt=0 (c.3)
k k

N-“lo o

N
H g(t) = lim 2 a f (t)
N—vw -1 k k

76

77

A kernel K(s, t) is self adj oint or hypermaximal if it is

 

 

defined over all L2]: 0, T] and

K(s,t) = K(t,s) .

If the latter equation holds but K(s, t) is defined only on a dense
subspace of L2[ 0, T] then it is hermitian. Symmetric kernels
are usually thought of as real L2[ 0, T] kernels which have the

following prOperty:
R(t, s) = R(s, t). (C. 4)

Autocorrelation functions form symmetric kernels if they
are a product of a second order real stochastic process. Also,
autocorrelations are either nonnegative or positive kernels (the

latter implies the former):

T
f K(s,t) g(s) g(t) ds dt _>_ 0 nonnegative definite
O

(C. 5)

T
_r K(s,t) g(s) g(t) ds dt > 0 positive definite
o

where K(s, t) is the kernel in question and g(t) is any member of
00k], the domain of operator K which has K(s, t) as its kernel.
Symmetric kernels have at least one eigenvalue X and

corresponding eigenvector Cl) . That is,
T
f K(X.Y)¢(X)dx=>\ <l>(Y) 0: Y: T (C-6)
0

holds for some X and <l>(y) where K(s,t) is a symmetric kernel.

An extension of this is given by the following theorem.

78

Theorem. Every nonzero symmetric L2[ 0, T] kernel
either has an infinite number of eigenvalues or is a PG kernel.
(Tricomi, p. 105).

A Pincherle-Goursat kernel (PG) or a kernel of finite rank

has a finite number of eigenvalues and can be represented by

K(s.y) = .5 x. 4.1x) My) "(0.7)

N N
where {Xj}, 1 and {ij}. 1 are its eigenvalues and eigenvectors,
J: J:
respectively.
m _—
For any symmetric kernel, if .23 XJ. ¢j(s) <l>j(t) converges

J=1
uniformly then

 

CD
K(s,t) = >3 x. ¢.(s) <l>.(t) (C.8)
m (I)
where {Xj} and {ij} are K(s,t)‘s eigenvalues and eigen-
jzl .1: co

functions as above. Whether {¢k}k-l is complete or not the

following is true for any symmetric kernel; K(x, y):

N

K(x, y) = lim .2 x. ¢.(x) oly) (c.9)

Mercer's theorem claims uniform as well as mean square convergence
for symmetric, continuous, positive definite kernels. That is, both
.C. 8 and C. 9 hold for such kernels.

Orthogonality of a function in L2[ 0, T] relative to the kernel
K(s,t) is expressed by

T
f K(s,t) y(s)ds = 0 . (C.10)
O

79

If C. 10 holds for a symmetric kernel than y(s) is orthogonal to
K‘s eigenfunctions (Tricomi p. 109).
Picard's theorem gives a criterion for the expansion of

a function in terms of the kernel's eigenfunction set:

Picard Theorem. The equation

 

for symmetric kernel R(t, s),

T
f R(t, s) x(t) dt = y(s) for 0 5 s _<_ T,
o

has a solution in L2[ 0, T] if and only if

(a) y(t) = l.i.m. 151 on“)

T
(b) Yn = lo y(t) <l>n(t) dt

 

 

(C) E —P- < 00 where {X } are R(t, s) eigenfunctions.
n=l Kn n n21
N Yn<l>n(t)
(d) x(t) = l.i.m. E ' —-3\———- 0< t<T
n=1 n — '—

Q)
The expansion expressed in (d) is unique if {¢n(t)} 1 is complete.

D. KARHUNEN LOEVE EXPANSIONS

Of singular importance in the study of vector spaces or
stochastic processes is a spectral theorem which breaks down
into simpler components either operators defined on the vector
space or the stochastic process. We will give a version of the
spectral theoremin hilbert space (CC self adjoint operators) and
will follow that with the analogous spectral theorem in stochastic

processes known as the Karhunen-Loéve Expansion theorem.

Spectral theorem for self adjoint CC operators.

A nonnull CC self adjoint operator T in a hilbert space

H has:
co
1. an eigenvector set {cbn} 1 corresponding to the
n:
(I)
existent eigenvectors {X } of T.
n n=1
2. Xj -* 0 as j —’ 00
G)
3, {Ch} generate all H if T has no null space except 9 .
— CO
4. Tx 2 Z X. (¢.lx)<l>. forany x in H.

Karhunen-Loéve Expansion theorem.
A stochastic process x(t) with zero mean on [0, T] and
continuous autocorrelation function R(t, s) = E x(t) x(s) has:
1

m
l. a series expansion for x(t) relative to {¢j}j_ the

orthonormal set of R(t, s) eigenfunctions.

00 T
x(t) = n21 xn ¢n(t) where xn = f0 x(t) ¢n(t) dt

80

81

(2:). .

2. Exnxk=0 for n75k and Elxn

' N
3. E]x(t)- Z x (l) (t)]2 —*0 as N-’00 forall t in
n=l n n
[0.T]-

4. the above expansion of x(t) unique and

T
f0 R(t, s) ¢n(s) ds = Xn¢n(t) 0: t: T

 

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Davenport, W. 8., Root, W. L., Introduction to Random Signals
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Fulks, W., Advanced Calculus. New York: Wiley, 1961. 521pp. 1

 

Grenander, U. , Stochastic Processes and Statistical Inference. ..
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Masterson, J. J. , Notes for Hilbert Space Theory Course, j
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Pitcher, T. S. , “Likelihood Ratios for Gaussian Processes, "
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Riesz, F., Sz.-Nagy, B., Functional Analysis. New York:
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Root, W. L., Kelly, E. J., "A representation of vector valued
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Root, W. L., "Singular Gaussian Measures in Detection Theory,"
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Root, W. L. , "Stability in Signal Detection Problems, " Proc.
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Root, W. L. , Detection TheorLCourse Notes, Michigan State
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Slepian, D. , "Some Comments on Detection of Gaussian Signals
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83

Tricomi, F. G., Int_e_gral Equations. New York: Interscience,
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382pp.

Wiener, N., Time Series. Cambridge, Mass.: MIT Press. 1949.

 

Wiener, N. , Fourier Integral and Certain of Its Applications.
New York: Dover. 1933. F“

 

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