nun...
’Hl’Ifi‘flupfln

{Niki-5
.3 flmwmf
.7 wJW/wfltkm

T... 811 r

g:
fin,

 

... aw... AM”!
«M2,,»

 

 

 

BY:
_ 3‘6"“

 

 

 
 
 

.ltr. do? a.
.Ier! ii... 9.. .3123”
.4141!!!

fr

 

FOR__ -

. R ZERO

 

 

\ _ , , . x
7 . Du; 4/,
. _ , . . ‘ .. 110.145.”.1...»
A . . . , 4 ; , , . r 4 v , . ~11 l.’l.4 ~II .Il
, ,. . . . .... . tr: Fir-51:4;

 

4.! I. LVSI
>.1..I.I.ct./..

 
 

       

 

 

; We;

,./../r,«..:....,v. , I! t . . v.

     

 

 
   
  

LIBRA. LY
Michigan rate

Univcrs y

This is to certify that the

thesis entitled

A ZERO-MEMORY NONLINEAR
FILTER FOR SIGNAL DETECTION

presented by

PHILIP WILLIAM ALLEN

has been accepted towards fulfillment
of the requirements for

 

 

 

Major professor

Datwdé‘i' 391/7j/

\

0-7639

 

 

 

 

 

 

ABSTRACT

A ZERO-MEMORY NONLINEAR
FILTER FOR SIGNAL DETECTION

By

Philip William Allen

This thesis investigates tfmapossibility of increasing the
probability of detecting a constant signal in non-Gaussian noise
by nonlinear processing of the receiver waveform. The approach
taken uses the first-order statistics of the random observables as
a starting point for the design of any optimal detection system,
in contrast with other applications of nonlinear elements which
are motivated by efforts to improve the signal-to—noise ratio in
the receiver. The particular noise model admitted in this thesis
is characterized by an amplitude probability density function
having the form f(x) = a exp(-b|X|e), 9 6 [1,2]. The noise
combines additively with one of two transmitter symbols {0,A},
where A > 0, during each decision interval [0,T]. The detection
problem is solved inéuihypothesis testing framework.

Receiver design is constrained by a restriction on the
maximum False Alarm Rate (FAR) which must be satisfied for every
9 E [1,2]. This constraint, plus the mathematical intractibility
of the noise family under consideration, suggests the use of non-
parametric detectors. The sign detector is very well suited

because of its ease of implementation and its distribution-free

 

 

Philip William Allen

property. It is also the optimal device for weak signals in

Laplacian noise (8 = 1). Another detector considered is based
on a sample mean. This detector is uniformly most powerful for
a given FAR when 6 = 2. It is suboptimal for other values of
9 but is often used when the underlying PDF's "appear" normal.

This thesis proposes a third detection method, one based
on the inclusion of a nonlinear element. The nonlinear transfer
characteristic is derived from a probability integral transforma—
tion which changes the specified input distribution to a standard
normal distribution. This allows determination of a threshold
value which guarantees satisfaction of the FAR restriction while
avoiding any further randomization at the receiver. This is not
possible, in practice, with a sign-detector because its test
statistic has a discrete distribution.

The transfer characteristic chosen results from a study
of the importance of Monotone Likelihood Ratios (MLR) in signal
detection. A method for determining whether a single-parameter
family of PDF's has MLR is given and applied to the noise model
of interest. The MLR is shown to be invariant under a monotone
nonlinear transformation, and this transformation is the basis
for the nonlinear filter detector design. This detector is
compared to the other detection systems under weak and strong

signal conditions, and is shown to be a viable alternative to both.

 

 

A ZERO4MEMORY NONLINEAR
FILTER FOR SIGNAL DETECTION

By

Philip William Allen

 

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering
and

Systems Science

1972

 

 

TO MARSHA

ii

 

ACKNOWLEDGEMENTS

I wish to thank the General Dynamics Corporation,
Electrodynamics Division for the encouragement and financial
support that made this thesis possible. Professor Richard Dubes,
MSU Department of Computer Science, my teacher and thesis advisor,
was a constant source of help, as was Professor Gerald Park,

MSU Department of Electrical Engineering and Systems Science.
Several helpful suggestions were made by Professors Dennis
Gilliland, V. Mandrekar, and Robert Staudte, all of the MSU
Department of Statistics and Probability.

Perhaps the greatest contribution was made by my wife
whose patience, understanding, and encouragement contributed
greatly to the success of this research. Therefore this thesis

is dedicated to her.

iii

 

TABLE OF CONTENTS

Page
List of Figures vi
Chapter
I INTRODUCTION ..................................... l
1.1 Requirements and Assumptions ... ............. 3
1.2 Parametric and Nonparametric Procedures ..... 4
1.3 Objections to Nonparametric Procedures ...... 5
1.4 Literature Review .................... ....... 6
1.5 Thesis Noise Model ........ .......... . ....... 9
1.6 Central Limit Theorem Errors .... ....... ..... 12
1.7 Thesis Objective ....... ..................... 15
1.8 Organization of the Thesis .................. 15
II STATISTICAL BACKGROUND ........................... 16
2.1 Simple Hypothesis Testing ................... 17
2.2 Composite Hypothesis Testing ................ 19
2.3 Signal-to4Noise Ratios (SNR) ................ 23
2.4 Optimal Detector of Constant Signal in
White, Gaussian, Noise .. ..... . .............. 24
2.5 Measure Transformations and MLR ............. 25
2.6 Summary ........ ......... .................... 29
III ZERO-MEMORY NONLINEAR FILTER . .................... 30
3.1 Clippers, Limiters and Integrators .......... 31
3.2 Measure Transformations ...... . .............. 32
3.3 concaVity 0...... ............................ 37
3.4 NLF Output PDF under H .................... 38
. . 1
3.5 Moment Approxrmatlons ....................... 41
3.6 Summary ........ ............................. 45
IV COMPARISON OF DETECTORS .......................... 46
4.1 Optimal Detector of Constant Signal in
Laplacian Noise ............. . ............... 47
4.2 Weak Signal Detector Evaluation ............. 49
4.3 Strong Signal Detector Evaluation ........... 57
4.4 Simulation ............ ........... . ..... 61
.4,5 Summary ..... ................................ 70

iv

Chapter Page

V GENERAL CONCLUSIONS AND EXTENSIONS . .............. 72
5.1 Conclusions ......... .. ...... ................ 72
5.2 Extensions .................................. 73
BIBLIOGRAPHY .............. ............ . ....... ... 75

APPENDIX I Derivation of J(e:2) vs. 9 ...... 78

APPENDIX II Realization of the Nonlinear
Filter ............................. 82

APPENDIX III Discontinuity of g"(x) at the
Origin ............... .... ..... ..... 85

APP NDIX IV Simulation Data .................... 87

LIST OF FIGURES

Figure Page
1.1 Multishot Communication System ................... 2
1.2 ARE of Sign Test Relative to X. Test ............ 11

“'4
1.3 -Z' vs. 9 ................................ ....... 14
o
2.1 Invariance of MLR to Linear Transformation .. ..... 27
2.2 Invariance of MLR to Nonlinear Transformation .... 28
3.1 "Gaussianizing" Transformation ...... ........ ..... 33
3.2 Laplacian Transfer Characteristic ...... .. ........ 36
3.3 PDF of NLF-Output Under H1 ...................... 42
4.1 Probability Mass Function for SN ................ 50
4.2 Nonlinear Filter Detector .... .................... 54
4.3 Simulation Block Diagram ............ ............. 62
4.4 PD vs. SNR, Laplacian Noise, Sample Size = 4 ... 64
4.5 PD vs. SNR, Gaussian Noise, Sample Size = 4 .... 65
4.6 PD vs. SNR, Laplacian Noise, Sample Size = 12 .. 66
4.7 PD vs. SNR, Gaussian Noise, Sample Size = 12 ... 67
4.8 PD vs. SNR, Laplacian Noise, Sample Size = 20 .. 68
4.9 PD vs. SNR, Gaussian Noise, Sample Size = 20 ... 69
A.1 Divergence vs. 9 ............................... 81
A.2 Piece-Wise Approximation .... ............ ......... 83
A.3 Piece-Wise Approximation Flow Chart . ..... ........ 84

vi

CHAPTER I

INTRODUCTION

This thesis deals with a decision-theoretic approach to a
problem commonly encountered in the context of radio communications,
namely, the detection of a constant signal in additive, stationary,
white noise. When the noise is assumed Gaussian the design and
performance of optimal detectors have been deve10ped by Helstrom
[H1], Van Trees [V1], Hancock and Wintz [H2] and Root [R3], among
others. When the Gaussian noise approximation cannot be made,
application of nonparametric testing methods has been prOposed
by Helstrom [H1], Hancock and Lainiotis [H3], Daly and Rushforth
[D1], and many others.

This thesis will point out two criticisms of nonparametric
approaches and propose a viable alternative based on a probability
integral transformation.

The detection problem of interest is shown in Figure 1.1.
The requirements and assumptions which determine the detector design
are listed in Section 1.1. Section 1.2 reviews parametric and
nonparametric detectors as applied to the detection of constant
signals in noise while Section 1.3 points out some criticisms of
nonparametric detectors as alternatives to parametric methods.
Section 1.4 presents some approaches advanced by others for the

problem, while Section 1.5 discusses the applicability of various

639mm coflumoecsefiso uosmfluaoz HQ ouswfim

 

on. o coo o .9 EN EN .H. 0 .HM HN .H. 0
SI. . . . ...... ul}\<¢<$ 1;; u # —‘|

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

/ K K k r x
m u 5 mafia EH5» mH mmHoz
m
- WIRES "wages mafia
._ .-., . II 8.58
2:33.? FEB $.sz ...Nxbex ......mLmem sag
. gamma: 1 5,.sz
mmHoz

mgHmomm Amzzgo mMHHHZm 24mm.

parametric and nonparametric approaches to the problem. Section
1.6 investigates central limit theorem errors resulting from
tests based on a sample sum. Section 1.7 contains the thesis

objective and Section 1.8 explains the organization of the thesis.

1.1 Requirements and Assumptions

All detectors for the problem considered in this thesis
are to be designed in accordance with the following assumptions.
A1 The N independent, identically distributed samples

taken at the receiver are to be classified as coming from

either the density f(X) or, alternatively, the density

f(X - A), A > 0, where the form of the probability density

function (PDF) is in the family a exp (-b\X‘e).

A2 The design criterion is Neyman-Pearson; i.e., the
probability of signal detection (PD) is to be
maximized subject to a constraint on the False Alarm
Rate (FAR)1.

A3 The data received in each signaling interval are
statistically independent of the data received in all
other intervals.

A4 The source symbol selected for transmission during each
signaling interval is statistically independent of the
symbol selected during any other signaling interval.

A5 The transmitter and receiver are synchronized. A

receiver interval of T seconds [or N samples]

 

The statistical literature refers to power and size rather than
detection probability and False Alarm Rate, respectively.

corresponds to exactly one transmitter symbol.

A6 The wideband linear filter serves to bound the noise
power at the A/D converter (ideal sampler) input without
producing sample-to-sample dependence.

Assumptions A3 and A4 guarantee that the analysis of a
one-shot receiver, wherein the decision about the data received
in the interval (O,T) does not depend on data received in any
other interval, is applicable to the multishot system [H2] of
Figure 1.1.

The noise model was introduced by Kanefsky and Thomas [K2]
as a reasonable representation of impulse noise, one of the most
common and troublesome non-Gaussian noise processes. If an
impulse process is modeled as a train of independently occurring
pulses with small overlapping, then several simple pulse shapes

result in amplitude densities having the form specified in A1 [G1].

1.2 Parametric and Nonparametric Procedures

 

An extensive collection of results has been obtained and
published for problems of detecting signals in additive noise.
The vast majority of these results are based on the assumption
that the underlying noise density is Gaussian and white; thus it
can be completely characterized by two parameters. In many cases
the Gaussian noise assumption is dictated by physical considera-
tions; thermal noise, for example. Often one of the Central Limit
Theorems is invoked to justify consideration of the asymptotic
convergence in distribution of a suitably normalized sum of random

variables to a standard normal distribution. The mathematical

tractability of the Gaussian noise model has provided an abundance
of theoretical results. "There is undoubtedly a temptation to
regard distributions as normal, unless otherwise proven, and to
use the standard normal theory wherever possible." [Kl].

A great amount of interest has centered on the develop—
ment of a different approach to this problem. This nonparametric
approach, formulated in the late 1930's made few assumptions about
the probabilistic structure of the noise and used simple and
unsophisticated methods for detector design. Because these
detectors were based on very general assumptions concerning the
noise density, they displayed relative insensitivity to departures
from the assumptions, a feature called "robustness."

It has been noted [Cl] that whereas parametric procedures
provide exact solutions to problems stated approximately, their
nonparametric counterparts provide approximate solutions to problems
stated exactly. It is in this sense that a nonparametric test
may prove to be superior to an "optimum" test designed under
invalid assumptions.

The consideration of alternatives to optimal detectors
may be prompted by the complexity, both analytic and practical,
of the optimal design. Nonparametric detectors are intended to be
simple to implement at a cost of some deterioration of performance

relative to the optimal detector [T1].

1.3 Objections to Nonparametric Procedures

Asymptotic Relative Efficiency (ARE) provides a convenient

measure of relative performance of two detectors which are

 

 

designed for the same problem. It allows comparison between a
nonparametric detector and an "Optimal" detector and measures
the effect of departures from the assumptions on which the
"optimum" detector is designed. Because ARE is essentially a
large sample, small signal performance measure, its validity has
been criticized for practical receivers [T1], [N1].

Another criticism of small sample nonparametric detectors
is that the distribution of the test statistic is discrete which
may necessitate an additional randomization to achieve the require-
ment of A2. This complication is avoided by the method proposed

in this thesis and introduced in Section 1.7.

1.4 Literature Review

 

Rappaport and Kurz [R1] developed an optimal nonlinear
detector for digital data transmission through non Gaussian channels.
They noted the practical difficulty of evaluating the PDF of
their test statistic and suggested Monte Carlo simulation for any
specified noise PDF and sample size. To avoid the very large
amounts of computer time involved in any such simulation, the
authors investigated the asymptotic performance of the nonlinear
detector which was shown to depend on a single SNR parameter.
Several signaling waveforms were considered in a Cauchy noise
environment and asymptotic system performance was shown to be
relatively independent of the signaling waveform.

A suboptimal nonparametric alternative for constant-signal
detection was described by Hancock and Lainiotis [H3]. Their

detector, based on the Median, or Sign, Test was robust, a

desirable property. It used a nonlinear element (ideal clipper)
to reduce the input samples to binary random variables. The
distribution of the test statistic under the hypothesis HO was
binomial. By evaluating the detector's performance on an
asymptotic basis through the Central Limit Theorem, the randomiza-
tion required for small sample problems was not encountered.

An application of a nonlinear filter (NLF) for the improve-
ment of detection reliability for signals in non-Gaussian noise
was also investigated by Richard and Gore [R2]. The NLF transfer
characteristic was determined from the first-order statistics of
the interfering noise and the expected signal amplitude. A narrow
bandwidth, linear, integrating, low pass filter (LPF) followed
the NLF. The long integration time of the LPF and the Central
Limit Theorem was used to justify the assumption that the Gaussian
model "approximately" described the LPF output. This permitted
the calculation of detection probability and false alarm rate as
functions of the NLF output signal-to-noise ratio (SNR). For a
very special class of non-Gaussian noises, resulting from passing
Gaussian noise through a piece-wise linear nonlinearity, the
computed and experimental SNR improvement factors at the NLF out-
put indicated that a considerable improvement in detection
reliability could be obtained.

Hancock and Wade [H7] investigated the problem of attain-
ing near-optimum reception of known binary signals over wide-
band channels in the presence of narrow-band interference. They
noted that the purpose of the receiver was to divide the space

of received waveforns into two disjoint regions — the ”mark” and

 

 

"Space" regions. The linear receiver which they described reduced
the dimensionality of the received waveform to a univariate statistic
which was compared to a decision threshold. The decision surface
generated by the linear receiver was determined to be a plane in
the received signal space while it was noted that the Bayes
optimum decision surface had the form pn(X - M) = pn(X + M),
with pn(-) the interference PDF, X the received waveform vector,
and M the Mark waveform vector. Optimum reception with a linear
receiver could only be possible under certain conditions, Gaussian
interference being one of them. For more general interference,
the authors described a method by which a nonlinear coordinate
transformation could transform the decision surface into a hyper-
plane. Subsequent processing of the transformed coordinates with
a linear receiver was shown to result in optimum reception.

This thesis departs from these approaches in the following
way. The signal detection problem is viewed at the outset as
one to which the existing literature of statistical decision theory
can be applied. The prOperty of Monotone Likelihood Ratio is
investigated fully and its importance in the detection problem
of this thesis is discussed. Then a particular coordinate trans-
formation is chosen within the framework of the decision-theoretic
approach, rather than from a heuristic notion of signal-to-noise
ratio improvement.

Based on decision-theoretic arguments, this thesis pre-
sents a practical alternative to nonparametric and "approximately”

optimum parametric detectors.

1.5 Thesis Noise Model

Attention is here focused on the noise model to be con-
sidered in this thesis. The noise will be characterized by its

first-order statistics, specifically, its PDF
fX(x) = a exp (-b\x‘e), where e 6 [1,2], (l)

and a, b and 9 are related through:

]‘ fx(x)dx = l (2)
°° 2
[ xzfx(x)dx = OX . (3)

An Optimal detector of a constant signal in additive stationary
noise having the PDF (1) can be determined, at least in theory.
However, the complexity of implementationleads to the considera-
tion of other (perhaps) suboptimal procedures. Furthermore, the
detector designed to be Optimal for one value of 9 may prove
unsatisfactory for some other value of 9. The fact that e is
an unknown parameter having a range [1,2] must be included in
the design of any suboptimal procedure.

Previous work for this detection problem has concentrated
on certain nonparametric detectors. Their application is motivated
by simplicity and robustness. Their utility is often justified
by the criterion of .ARE , a small-signal, large sample performance
measure. Comparison between robust detectors and those based
on an assumed Gaussian noise model show that "optimal" detection

can be misleading when the underlying assumptions are not valid.

 

10

What is very interesting about the noise model admitted
is that the optimal tests for the extreme values, 9 = 1 and
9 = 2, are well known. When 9 = l, the test based on signs
(SN) is locally most powerful (LMP) while the test based on
the sample mean (X) is uniformly most powerful (UMP) for e = 2.
Because of the optimal prOperty of the sign detector, other non-
parametric tests (e.g., the Wilcoxson or van der Waerden tests)
will not be considered in this thesis. Both the sign and sample
mean tests are easily implemented although the former has the
disadvantage of a discrete PDF for its detector statistic.

The ARE of the sign test, SN’ relative to that based on

le<3>
— = hich
3 . w
SN’X 1“ (~33)

is plotted in Figure 1.2. If 1 s e s 1.4, the Sign Test would

the sample mean, X; can be shown to be ARE

be preferred to the X. test if ARE were the sole selection

criterion. The range of 9 for which the sign test is preferred

must be reduced if the following are also considered:

a) ARE may not be a valid comparison measure in small sample
problems.

b) The discrete PDF of the test statistic may require an
additional randomization at the receiver.

c) The optimality property of the sign test is a local pro—
perty, i.e., it is based on a vanishingly small SNR.

For these reasons the test based on X seems preferable to the

Sign test for a large range of 9, but there is a major dis-

advantage to its use. For the small sample sizes considered

here central limit theorem errors must be taken into account

11

ARE

 

 

FIGURE 1.2 ARE of Sign Test Relative to X Test

during any receiver design. Quite obviously,the threshold value
determined by consideration of the design specification on FAR
and assuming that X is Gaussian will be optimal only for e = 2.

It is also obvious that no single detector structure can
be optimal for every member of the family of noise distributions
and all possible Signal to Noise Ratios (SNR), defined as the

A

ratio -— , i.e., the rms signal power divided by the rms noise
Ox
power. Because of this, the design criterion A2 must be

modified (adaptive receivers are of no concern in this thesis

 

 

 

 

12

although they do offer an alternative approach) to state
specifically the terms under which optimization is to be
accomplished.

For the purposes of this thesis, the EAR will be assumed
as the primary criterion of design; i.e., any detector design
must guarantee a FAR which is less than, or equal, to the design
goal, for all e 6 [1,2].

As previously mentioned, the sign test is applicable to
this type problem. Because of its distribution-free properties,
a specified value of FAR can be achieved and maintained for all
possible 9, although it may require an additional randomization
in practical problems.

There has been no consideration of alternatives to the
nonparametric detector because of the mathematical intractability
of the family of noise densities. However it can be, and often
is, argued that the Central Limit Theorem can be applied to this
type of problem directly because the assumptions made ensure the
convergence in distribution of any prOperly normalized statistic
based on the sample sum . For practical sample sizes there will
be some error in this approximation, and this situation is now

investigated.

1.6 Central Limit Theorem Errors

 

The central limit theorem errors are more severe for those
density functions whose shape reduces the speed of convergence.
This section deals with the problem of determining which member

of a family "differs" the most from a standard normal, when the

 

 

13

family is characterized by the PDF of equation (1).
Let Fn(x) be the distribution function of the standardized
sample sum. Then the error using a standard normal approximation

may be written [W1, p. 265]

oz
(— 3—,3)O(3)

Fn(x) - ®(x) -Tq1 (X )

3|

+1[%—!(a ~3>i“‘)<x>+ —?a gimm]

9‘

 

 

 

l J;_ (5 ) 35 (7)
- n3/2 [5! (0/5 - 10013” (x) +"" 7! 013(0'4- -3)€I> (X )
28 03 1
+ 9! 0’3 ,(9)(x)] +0[ 3L2], (4)
n
R a
where aj = O , ”j the j-th central moment, and o the standard
j
deviation of the Xi. By the symmetry of (1), this becomes
4
(ozA- 3)i>( )<x> 1
Fn(x) - @(x) = 4ln + o n3/2 . (5)

The central limit theorem error will be largest for that

member 9 6 [1,2] having the largest fourth central moment.

00 9 0° 9
W, =] a x4e'b‘x‘ dx = 2a 2Exl‘e'bX dx . (6)

Making a change of variables and integrating, yields

23
P(5/9) . (7)
e bS/e

 

w,

Using (A4) and (A7) [Appendix 1] in (7), results in

= fro/exam)
r2(3/e)

 

14

u
The graph of 'f% is shown in Figure 1.3. Thus, when
0
e = l, indexing the Laplacian family, the central limit theorem
errors are largest. This same result is found in Appendix I using

the J-divergence as a measure of difference between members of the

family (1).

“'4

'7.-

0’
6
57-
4.
3r

 

The well-known fact that the Laplacian density has "fatter"
tails than a normal density of the same mean and variance [H6,
p. 43] ensures that the FAR determined by a Gaussian assumption
will be optimistic; i.e., it will underestimate the true FAR
and is therefore not a viable alternative.

Previous efforts [R1], [R2] with nonlinear filters have
demonstrated their capability of improving SNR and detection reli-
ability. As noted by Richard and Gore, SNR is not in itself a signif-
icant performance parameter [R2, p. 440]. Of greater importance is
the exact density of the test statistic from which the FAR and detec-
tion probability can be calculated. This thesis will investigate

one particular way of obtaining this important information.

 

15

1.7 Thesis Objective

This thesis will examine the effect of zero-memory non-
linear filters on the detection problem stated in the introduction.
The transfer characteristic of the NLF will be determined by a
probability integral transformation. Generally speaking,the trans—
formation will insure that the test statistic has a continuous
PDF . The type of noise considered at the receiver input and
the monotonicity of the transfer characteristic of the NLF
guarantee the optimality of a threshold test. Taken together,
they enable satisfaction of any specified false alarm rate without
randomization through an easily implemented threshold detector.
The continuity of the test statistic PDF will not depend on

"large" samples.

1.8 Organization of the Thesis

Chapter II presents a relevant background of statistical
decision theory for this communications problem. Chapter III
concentrates on the development of the applicable nonlinear
filter (NLF) and examines its properties. Chapter IV compares
the NLF detector with other alternative approaches, while
Chapter V summarizes the thesis and suggests some directions for

further research.

CHAPTER II

STATISTICAL BACKGROUND

This chapter summarizes the relevant background in
statistical decision theory. The definitions and theorems are
in large part taken from the books oflehmann[L1], Fraser [F1],
and Ferguson [F2], but are adapted to directly apply to the
detection problems encountered in the context of radio communica-
tions. The primary purpose of this chapter is to present a set
of conditions which guarantee that the optimal test for location,
where optimality is in the sense of UMP, is a threshold test.
To that end Section 2.1 develops the Best Critical Region (BCR)
for Binary Hypothesis Testing problems. The notion of Monotone
Likelihood Ratio (MLR) is used to define the BCR in terms of a

decision threshold. Section 2.2 extends threshold detection to

UMP tests, and presents a sufficient condition by which a location
parameter family can be tested for MLR. Section 2.3 defines some
measures of SNR and ARE from parametric and nonparametric theory,
respectively, and shows how they are related. Section 2.4 develops
the Optimal detector of a constant signal in white, Gaussian noise,
and Section 2.5 investigates the effect of zero memory filters,
linear and nonlinear, on the MLR property. Section 2.6 summarizes

the main results of the chapter.

16

17

2.1 Simple Hypothesis Testing

The following definitions are intended to make precise
the idea of location shifts of a real valued random variable X,
whose cumulative distribution function (CDF) depends on a single
real parameter A. Let F(x‘A) denote the distribution function
of the random variable X when A is the true parameter value;
F is assumed to be absolutely continuous with probability

density function (PDF) f(xlA).

Definition. The parameter A is a location parameter for the
distribution of X if, and only if, f(x‘A) = f(x — A) for some

density f(x).

Definition. A single—parameter family of distributions is said

to have monotone likelihood ratio if whenever A1 < A2 the

f(XlA)

2
likelihood ratio -——T———- is a non—decreasing [or non—increasing]

f(X A)

1

function of x.
A non-randomized decision rule for the binary hypothesis

testing problem
H0: A = A0
H t A = A1 > A0

may be defined by a measurable Subset, no, of the sample space
I, no C I, with the understanding that if the observable variable

X falls in no, announce H is true; otherwise say H is

O 1

true. The set 01 Q 1 - 00 is called the critical region. The

false alarm rate (FAR) of the test is given by g p(X\HO)dX
l

 

18

while the probability of detection (PD) of the test is given

by A p(X‘H1)dX.
1

An optimum test, in the Neyman-Pearson sense, is defined

by a critical region Q satisfying I p(X\HO)dX g PF and
“1
g p(X\H1)dX = PD = maximum. This region will be called a Best
1
Critical Region (BCR) and the test defined by a BCR is called

1

a most powerful (MP) test at FAR PF.

 

Definition. A region Ol<: I is said to be best at FAR P

F
for testing HO against H1 if I p(X\HO)dX = PF and if for
01
a .l _ ' __
every other region “1 for which ['p(X|HO)dX — PF’
01
. > I
£p<XlH1>dX a g'NXlHQdXO 1.0. PDml) a PD(“1)'
l 1
In other words, a region “1 is best at FAR PF if,

out of all regions having the same FAR, its measure under the
alternative (PD) is largest. A general method for finding the
best test of a simple hypothesis against a simple alternative can

be found in the Neyman-Pearson lemma.

Lemma: In the test of a simple hypothesis HO against a simple
alternative H1, the region “1 Q [X: A(X) 2 k} where
A f(X‘Hl) r ‘
A(X) = and k satisfies ; p(X H )dX = P] where
f(X‘HO) r11 0 T

O < PF < 1, iS the BCR at the given FAR.

The proof appears in Hogg and Craig [H4] p. 274 and is

not repeated here.

 

19

The preceeding lemma and the definition of MLR allow

statement and proof of the next theorem.

Theorem: The most powerful test of a simple hypothesis H0
against a simple alternative H1 when the likelihood ratio is
monotone (increasing) is a threshold test defined by the critical

(1)

region 01 = [X: X > R}. If R satisfies i p(X\HO)dX ==PF

then 01 is a BCR at FAR PF.

Proof: By the Neyman-Pearson lemma the region “1 Q [X: A(X) > k}
is best at its FAR where k is determined by £p(A‘HO)dA = PF.

If [\(X) is monotone in X then V X 2 K, [\(X) a: [\(K). Choose

+oo +oo
K a p(X\H )dX = P = p(A\H )dA = p(A\H )dA. Then
[XngK] O F {(XO) 0 k 0

A(K ) = k and the regions [X: X a R], [X: A(X) ? k] are equal.
Because the latter region is MP at its FAR, then the threshold

test is also MP.

2.2 Composite Hypothesis Testing

 

The larger detection problem concerns composite hypotheses;
i.e., when the distributions under HO and H1 depend on a para-
meter which is assumed to take on values in some (real) space R.
Let mi: R be the subset of parameter values such that

{PAAA G R} denotes the family of distributions for the random

variable X when H is true. Then R — w corresponds to the

0
possible parameter values when H1 is true. The hypothesis
testing problem can be restated as H : A 6 w

0

H1: A e R - w

 

20

As before, the problem is to dichotomize the observation Space

I so as to satisfy some criterion of goodness". For this

situation the notion of best test at FAR PF must be redefined.

Definition. In the test of a composite hypothesis HO against

an alternative (simple or composite) the region 01 is said to

be at FAR P if sup ]‘ p(X\A)dX = P
F

When the alternative H is also composite the notion

1

of best test must also be generalized.

F.

Definition. A region Q is said to be Uniformly Most Powerful

l

(UMP) at FAR P for testing H I A G w against H : A G R - w

0 1

and if, for any other region Hi at FAR PF’

it is true that g p(X[ A)dX ; g p(X\ A)dX for each ,A E R - m.
I
1 1

F

if 01 has FAR PF

The Neyman-Pearson lemma guarantees that, in the class

of regions having FAR PF’ there exists one whose PD is

maximum at any fixed element A1 6 R - w. There is no reason
why this region should also maximize the PD for some other
element A2 6 R - m. It is, therefore, not surprising that UMP
tests exist only in Special circumstances. Fortunately, one of

these circumstances is widely found in radio communications prob-

lems and will be investigated in this thesis.

Theorem: If the distribution of X has non-decreasing monotone
likelihood ratio, then the region 01 = [X: X 2 K} where K is

chosen to satisfy I p(X‘HO)dX = P O < P < l, is UMP of FAR

F’ F

PF for the hypothesis testing problem HO: A s A against

0

 

 

21

Proof: See Ferguson [F2], page 210.

This theorem and its counterpart for binary hypothesis

testing illustrate the effect of the MLR property. Tests for

location simplify to threshold tests when the underlying
f(x‘A) = f(x - A) have MLR. Because of the importance of the
MLR prOperty in this thesis, the sufficient conditions under which

a density may have a MLR with respect to a location parameter

are now investigated.

 

Definition. A density function f(X) is called strongly unimodal

if - Ln f(X) is a convex function of X.

Lehmann [L1, p. 330] shows that strong unimodality is a

sufficient condition for a density to have MLR.

Theorem: Let f(x) be a density on R. A sufficient condition

that f(x‘A) = f(x - A) have MLR is that f(x) be strongly unimodal.

Proof: Let f(X) be strongly unimodal and suppose X < X',

. . . f(X -A') f(X' -A')
'. Th d t s

A < A e con 1 ion of MLR in X, namely, f(X _ A) _ f(X' _ A) ,

is equivalent to

log f(X' - A) + log f(X - A') g log f(X - A) + 10g f(X' ‘ A')

The four differences can be arranged, from smallest to largest,
as (X - A'), (X - A), (X' - A'), (X' - A) or (X - A'). (X' - A').
(X - A), (X' - A). In either case by defining

t 9 (X' - X)/(X' - X + A' - A) it is true that

X - A t(x - A') + (1 - t)(X' - a)

x' -A' = (l -t)(X -A')+t(X' -A)

 

22

Define h(X) Q —log f(X); by hypothesis, h(X) is convex. Then

for every t1,t2 and O s l S l

hilt, + (1 - l)t2] s lh(t1) + (1 - x)h(t2)

log flit, + (1 i>t21 2 i log f(t,) + (1 - x)log f(tz)

Letting )\=t, t1=X -A'. t2 =X' -A,
log f(X - a) 2 t log f(X - A') + (l - t) log f(X' - A)

Letting A = 1 - t,

log f(X' - A')

RN

(1 — t) log f(X - A') + t log f(X' - A)
Adding these two equations yields
log f(X - A) + log f(X' - A') 2 log f(X - A') + log f(X' - A),

which is the condition for f(X) to have MLR. Q.E.D.'

This theorem provides a method of ascertaining whether

the family of densities
f(x‘A) = a exp(-b\x — A‘e), 9 fixed,

has a MLR. This is seen to be the case since -Ln f(x) =

-a + b\x\e is convex in x for each fixed 9 6 [1,2].

 

 

23

2.3 Signal-toeNoise Ratios (SNR)

In Section 1.5, the measure of signal-to-noise ratio (SNR)
applicable to this thesis was given as the rms signal power divided
by the rms noise power. Because the detector of this thesis is
to be compared to others in Chapter IV, this section introduces
performance measures which have previously been defined for
parametric and nonparametric detection. The general Gaussian prob—
lem is the hypothesis testing problem in which the densities under
both classes are Gaussian, i = 0,1. When the covariance matrices
under the hypotheses are equal but the means are different, the

quantity

 

2 g [6(L‘H1) - MMHOHZ

d [Var (LlHln

with L the test statistic, has been used as a SNR [V1, p. 99].
This same quantity was specified by Hancock and Lainiotis [H3]
for the problem of detecting an unknown d.c. signal having
magnitude 9 > O in zero-mean noise with a symmetric, but other-
wise largely unspecified, PDF.

The physical justification for this concept of SNR rests
on the idea that the two densities of the received random variable,
conditional on signal present and signal absent, become more
distinguishable as either the distance between their central
locations (means) becomes greater or the concentration of their
values around the central location becomes greater (smaller

variances).

 

24

A theorem of Capon in 1961 [C2] introduced the concept
of efficacy in determining the asymptotic relative efficiency
(ARE) of a test Un relative to the test U:, where both tests
satisfied certain regularity conditions. The ARE of the Un-

*
detector with respect to the Un-detector was given by

A e (Un)
Eu u* = ——e;-— , where e(Un)’
’ e(Un)

the efficacy of the Un-detector, was defined as

lim
00 (Un)

2
A 1 LE (U)
e(U ) = ‘— -a§——Q—JQ— .
n n
n—Roo 9:0
The sample size of the Un—detector is n, and E9(Un)
is the expected value of the detector statistic under the
alternative hypothesis H1, when the signal strength is 9.
The following shows that efficacy depends directly on the pre-
vious measure of SNR and the relationship is derived using the

regularity conditions of Capon.

Given that §9IE9(Un) exists for all 9 G (O,a), and

is continuous at 9 = O, a Taylor Series expansion of E9(Un)
. . . U _ E = a__E
about the origin ylelds E9( n) 0(Un) 9n AG e(Un)‘e=O

2
+ 0(en) as 9n H 0 [H5, p. 54]. Then

E (U ) - E (U )
___°_ . 9 n O n
e(Un) 11m \fn 9n 00(Un)

 

2
1 . The relationship between

n—m

efficacy and SNR is obvious.

2.4 Optimal Detector of Constant Signal in White, Gaussian, Noise

This section develops the optimal detector of a constant

signal in white, Gaussian, noise. According to well-known results

 

 

 

25

on sufficiency, the optimal test can be based on the sufficient

2

statistic Y = Yi' The distribution of Ti under H0 is
2 11 2
fl(0,g_) and under H1, T. is fl(A,§r). The normal family has

er—R
Eta

increasing MLR with reSpect to location (variance fixed), and

therefore the best test having a False Alarm Rate of P is

F
H
l
— >
Y < K
Ho
where K is defined to satisfy
= IEQ. '2 0
PF i ( N ) e dy
_ o -1
Thus K — 7E'erfc* PF’ where the complementary error function is
2
_ E_
e 2

 

du. Hence, the BCR is defined as

[TE T >.Q_ erfc; PF}. The probability of detection for

this critical region is defined as

+002 2_, .1%(-"—‘—‘=1)2
P=£(——g—§)Ze G

D

dy.

P =erfc*[erfc;lP -M].

F o

This equation relates the probability of detection directly to
False Alarm Rate, and is the basis of the receiver operating

characteristic (ROC) [V1, p. 38].

2.5 Measure Transformations and MLR

The purpose of this section is to investigate the effect

that a zero memory filter has on its input with reSpect to the

 

26

desirable property of MLR. In Section 2.5.1 measure transforma-
tions are defined while in Section 2.5.2 the MLR property is

shown to be preserved under monotone measure transformation.

2.5.1 Measure Transformations

Let fX(X) be the density function for a continuous
random variable X, and suppose a real-valued point function
g(~) is specified. The random variable Y = g(X) is defined

and its density function fY(y) is to be determined. Assuming

 

that g(-) is monotone increasing wherever fX(X) ¢ 0 the

density of Y is given by
_ -1 _d_ -l
fY(v> — fx(g <v>>ldV g on . (1)
Proof: See Dubes [D2], p. 243.

2.5.2 Monotone Measure Transformations

 

If X has PDF fx(x[A) which has increasing MLR, and
Y = g(X) where g(-) is monotone increasing and satisfies the
regularity conditions which guarantee (1), then the family of PDF's

for Y,
Q -l .9_ -l
£Y(y\A) — fx(g (y)lA)\dy s (y)\,
has increasing MLR. This is immediate since if A1 > A2,

fY(ylA1) fx(g‘1(y)\A1>

 

fY(y‘A2) fx(g’1(y)lA2)

fx(x‘Al) -l
is monotone increasing, and g (-) is monotone
fx(xIA2)

 

27

increasing. The result is now illustrated with examples of linear

and non-linear monotone transformations on location parameter

families.

Example 1. . Let X under H0 be uniform on [-l,l]. Under

H1, X will be uniform on [-1 + a, l + a], a > O, and suppose

the linear filter has the transfer characteristic aX + b, where
both a and b are known positive constants [for the sake of
simplicity]. Thus f(Y‘Hl) = %; for a(a-l) + b g y a a(a+l) + b,
f(Y‘HO) = %;- for b-a g y g b+a. These densities are shown in

Figure 2.1 and it is easily verified that Y has the MLR property.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f(XlHO) f(YlHO)
a
+x : +Y
-1 +1 b-a b b+a
f(xlnl) f(Y‘Hl)
e
+ J +
-l+a +l+o, X b+a‘o, Y

Figure 2.1 Invariance of MLR to Linear
Transformation

Example 2. Let the input random variable X be uniform on (-%,%)

under HO and uniform (A,A+l) under H1 with -% < A S 0. Let

g(X) be defined as g(X) = sin X for '1; s X s + g, +1 for

E n .
X > 2, -1 for X < - 2 . The Situation is illustrated in Figure 2.2.

 

 

28

 

sin (x)

f (x‘H )
x‘nl i

7 X

a

 

NIZl R
D

X
0 1+A

 

 

 

 

 

 

 

 

sin(l+A)
Figure 2.2 Invariance of MLR to Nonlinear
Transformation
2.5.3 A Transformation to Achieve a Specified Distribution
Let X be a random variable having CDF F.

certain regularity conditions [H6, p. 33], the transformation

Subject to

G‘1<F(x>>

will yield a random variable

Y, having CDF G.

 

29

2.6 Summary

This chapter presented some concepts from statistical
decision theory which are applicable to the communications prob-
lem introduced in Chapter I. The property of Monotone Likelihood
Ratio was shown to be a sufficient condition for a test of loca-
tion (known direction) to be implemented by a threshold test.

The equivalence of the Efficacy of Capon and the SNR of Van Trees
was shown and Monotone Likelihood Ratio was shown to be an invariant
prOperty when a family having it was passed through a nonlinear
filter having an odd-symmetric, monotone increasing transfer
characteristic. A sufficient condition for MLR was derived and
applied to show that the family of noise densities introduced in

Chapter I has the MLR property.

 

CHAPTER III

ZERO-MEMORY NONLINEAR FILTER

This chapter discusses the use of a nonlinear filter
between the sampler and detector of Figure 1.1. Section 3.1,
which is concerned with integrators and clippers, provides
motivation for investigating the measure transformation of
Section 3.2. This section outlines the procedure which
”Gaussianizes” random variables; that is, the transfer char-
acteristic of a zero-memory nonlinear filter (NLF) is determined
through a probability integral transformation, which produces
independent Gaussian random variables at the NLF output, when
the filter is driven by a finite sequence of independent random
variables having a Laplacian probability density function.
Section 3.3 then discusses a Special property of the NLF trans-
fer characteristic while Section 3.4 derives the exact dis-
tribution of the NLF output when the input is noise plus signal.
Section 3.5 develops first and second moment approximations for
the NLF output under H , while Section 3.6 summarizes the main

1

results of the chapter.

30

 

31

3.1 Clippers, Limiters and Integrators

In many practical communications receivers, the input
process is first filtered to remove any noise spectral components
which may lay outside of the signal bandwidth. Cascaded linear,
tuned networks, whose overall bandwidth is wide enough to pass
with considerable amplification the significant Signal components
while rejecting those frequencies containing noise alone,are
widely used. If the input to a filter is a quasi-monochromatic
signal, its output at any time is a weighted time-average of its
past input. When the filter bandwidth is much narrower than the
noise bandwidth at its input, the central limit theorem can be
invoked to justify assuming that the amplitude distribution of
the filtered output is Gaussian.

In other receiver types the input process is first sampled,
thereby reducing the continuous-time input to a finite-length time
series. Many applications of nonparametric statistical methods
have been prompted by the fact that the sampled data process often
has a large variance and consequently may not satisfy the con-
ditions of a central limit theorem [R1]. Therefore, so-called
"optimum" tests based on the Gaussian assumption may be highly
unreliable. The inapplicability of Gaussian methods is also
prevelant in underwater sound receivers which pick up Sporadic
biological noise and telecommunications networks which are subject
to randomly occurring impulses due to lightning, sparks in ignition

systems and faulty connections or switches [Hl, p. 317].

 

Narrow Band

32

A large variance for the input random variable may pre-
clude application of a central limit theorem without some form
of noise suppression. One method is the clipping, or limiting,
of the received signal, the usual justification being that this
increases the SNR. Although the introduction of a clipper before
processing the received signal seems intuitively a good idea, it
is not at all clear that a simple clipping level is in some sense
the best system design that can be achieved. Moreover, when
severe models of impulsive noise amplitudes are considered,
commonly defined SNR's may not be useful criteria of system per-
formance. The structure of the nonlinear device for suppression
should be determined by whatever knowledge is available about
the noise PDF so that a minimal amount of received information

pertinent to the signal decision is destroyed [R1], [R2].

3.2 Measure Transformations

 

The essential problem is sketched in Figure 3.1 and can
be stated as follows: Let X be a random variable having a PDF
which is known completely and which is absolutely continuous with
respect to the Gaussian PDF. Determine the transfer characteristic
of a device such that with the r.v. X applied at the input,
the output random variable, Y, has a Gaussian PDF.

This problem is different from the more common one involved
in functions of random variables. There, the transfer characteristic
of a device is specified and the input is a r.v. having a known
PDF. The problem is to then determine the distribution of the

output. This was discussed in Section 2.5.1. The results of

 

 

cowumEuommcmnH :wcflmflcwflmwsmw: H.m mpswwm

wouawm muoEoEIOMmN umocHHcoz

 

33

Amxxuvaxo M

Euow msu mo

if

WT

 

 

 

 

 

 

 

» .>.u was new age mesueaeea

x

 

.>.p was new mom meeunaeee

 

 

 

 

34

Section 1.6 indicate that the PDF of interest in this thesis is

Laplacian.

Problem Statement: Find a transfer characteristic g(x) which
transforms the random variable X having the Laplacian PDF
fX(x) = a exp (-b‘x\), -m < x < +m, With a and b defined by

AZ-
-0-

X RB to the Gaussian random

E;fx(x)dx = l, mezfx(x)dx

variable Y, which has 0 mean and variance 0%.

Problem Solution: Constrain g(x) to be a monotone, strictly
increasing function on the real line having odd symmetry and

require Y = g(X) a.e. Then
P[Y s y] = P[X g x], with x = g- (y).

Case #1: x < O

yloo -% u2 x
j g_____ du = I +bwd
,/ 211'
-oo -00
y _ a bx
or, T - _'e X < O
c b
O
y < 0

Case #2: x > O

Y/GO _% UZ X
f e du = 1'+ a e-bw dw
[72W 2 A
~00
1. = .1. a _ a -bx

Using the first condition on X, [ fx(x)dx = l, the
-m

. a . .
ratio b. is determined.

(1)

(2)

(3)

(4)

(5)

[ae-b‘x\dx l
-cn
4130
g a e-bxdx = 1
a -bx+°°-l=,a=_1_
be 0 ‘2 b 2
Combining (3), (5) and (8) yields
b
9 X“ = % e X X < O
00
=1-%e'bX x>O

Solving for y in (9) and (10) gives the form of g(x)

bx
9-1 2“ x O
.0 [2] <

> 0

s4
ll

 

'~<
I
Q
Q
rel
I
...I
...4
l
(D
N I
U
><
g
X

Figure 3.2 shows the function g(x) of (12).

Using the second condition on the PDF fx(x)

the value of the coefficient b.

(D m
2 A 2 2 -bx 2
[4x fX(x)dX - ox = 25 a x e dx — ox

Equation (13) is easily put in the form

m b302

2 -
g u e udu = X

 

23

(6)

(7)

(8)

(9)

(10)

required:

(11)

(12)

determines

(13)

where the left-hand-side is immediately recognized as the Gamma

 

 

36

xp

am
oaumapouomumso Hommcmue cmfiowa A

N.m ouswwm

 

 

 

 

37

integral; therefore

 

3 2
b OX
= = i:
23 F(3) 2. 2. (l4)
3 2
. b0x b 22 . . .
Finally, -E;-= 1 = (;)b OX = 4. USlng (8) in (14) yields
2 2
b — 2 , or
0x
b = 2 . (15)
0x

The given PDF can more easily be written in the form
b
fx(x) = E'exp (-b‘x\), -m < x < +m . (16)

3.3 Concavity

This section will prove that the transfer characteristic

determined in the previous section is convex on the region x > O.

 

 

Given
-1 e-bt
g(X) :00 Q (1 " 2 )a ~X>O (17)
write it as
2
_ H_
e-bX +m e 2
2 =i 77:“ - (18>
g/oO

Take derivatives on both sides with respect to x twice

1 2
- g(gfl
-b2e-bx l 2 g_ e 00
._______ =.__ H _ I
2 0,0 g (8 ) 2 T; (19)
°o
' 2

g" = A351 [88' _ boo] . (20)

 

38

Thus to show that g is concave on X > 0 it is sufficient

to Show that gg' - be: < O, for x > O.

2 2
ML) 11..
00 +00 2
g' = 00b e j e du (21)
g/oO
Integrating (21) by parts yields
1, 8— 2
g' = -? - 00b 8 J: 2 du (22)
(g/OO) u
Hence
2
16(57) +00 -15 U2
2
g'g - 00b = -oOb g e O f S—---§——-du (23)
(g/oO) U

All terms on the right hand side are positive for x > 0,

therefore gg' - 03b < O and g”(x) < O for x > 0.

3.4 NLF Output PDF under H

 

1

This section assumes the NLF has the transfer characteristic
Specified in (11) and (12) and that the NLF input has the Laplacian
amplitude PDF of (16) translated by the amount A > O. The PDF
of the NLF output will be derived using the methods of Section
2.5.1. This will show the dependence of the form of the output
PDF on the input SNR, and will be used as justification for certain
approximations made in Chapter IV.

Write

-l
_ ‘1 53.411
fY<y> — fx<g <y>>l dy l (24)

 

39

 

where
_1 e-bx
y = 00 i [1 - 2 ] X > O
_ 6P-1[ebx O (25)
GO -—2 x <
and
fX(x) = g-e'le‘Al, .. < x.<-+.. (26)

It will be shown that the case for A = 0, i.e., x is
distributed according to H0, can be considered as a special
case of this computation, thus demonstrating the ”Gaussianizing"

property of the NLF.

Case 1: y < 0 = x < O

 

 

 

-1 ebx
y = g(X) = 00 61> (T) . (27)
Then g'1<y> = §2n(2i<1->> (28)
Co
-1 g— R'<y/oo>
M = l 0 (29)
dy b 29 (y/oo)
f ( ) = Q b[%'&n2i(y/oo) - A] 1 2¢'(y/00)
Y y 2 e boo 22(y/00)
2
4:3 /
e-bA ' 43A e (y 00)
fY(y) = —;(-)— i: (y/oo) = e ff; 00 y < O (30)

Case 2: y > 0 = x > O

-1 -l
b -b ( - A d

 

 

 

-1 "bx
Y = g(x) = 00 9 [1 ' 2 3
e-bx
l - 2 =¢(y—)
00
x = - B’Ln2[1 - 9(y/OO)]
_ I
dy boo 1 - e(y/oo)
% e-b(X-A) when x > A > 0
fx(x) ’-
% eb(X-A) when A > x > O

Combining (31), (35), and (36) yields

-b[-%‘£n2[1 - @(y/00)l @‘(y/oo)

boo l -9 (Y/OO)

 

I
NICT‘
(D

(D

fY(y) -

-2<y/oo>2

—fl———TT_-C_I__ valid for X > A ,
0

II
(D

-bA

which, by (32) is equivalent to y 2 goo-1[l - 9§——] 9 g(A)-

Combining (31), (35), and (37) yields

..b[—bl Ln2[1-@ (y/00)3 '
3.8 e -_1_ Q (y/oo)

f (y) __
Y b00 l-My/oo)

2
-%(y/o )
e e O

Ev!" °0 [l-O’2(y/oo)]2 .

 

Equation (41) is valid for O < x < A, which is equivalent to
-l e.bA
O<y<00¢ [1--2—-]=g(A)-

(32)

(33)

(34)

(35)

(3 6)

(37)

(38)

(3 9)

(40)

(41)

   

 

41

Combining the results (30), (39), and (41) gives

r -%(Y/oo)2

-bA e
e "772:”36' y < 0

-%(y/oo)2

fY(Y) = 4 ebA° Ema—— y > g(A) (42)

e ~i————- O<y<mm
KOti-w/oonz “2" °b

 

 

When A = O, g(A) = O and (42) reduces to a zero-mean

Gaussian PDF with variance The graph of fy(y) is shown

2
GO-
in Figures 3.3 (a), (b), (c), and (d), for increasing values of

SNR. The assumption of weak signal normality of Y appears well

justified when bA 3 IO’ approximately -29 db.

3.5 Moment Approximations

 

Under the alternative (H1) the PDF for the random
variable X, the NLF input, is specified exactly. It is a replica

of the PDF of X under H translated by A > O. Assumptions

0’
Al and A61 guarantee that under H1, the random variable X has
expectation A > 0, and a known variance, 0:. The problem of

this section is to find an approximation for the mean and variance
of Y, the NLF output random variable, when X contains signal
plus noise.

Although the exact form of the PDF of Y is given in

the previous section, the complexity of determining the moments

 

Chapter I

42

1 Y
3— f (—)
0 c’0

  

SNR = -29db

 

 

SNR = -23db

(b)

 

Y
(—‘)
C0

 

SNR = -15db

(C)

 

 

.4 d SNR = ‘9db

(d)

 

Figure 3.3 PDF of NLF-Output Under H

43

encourages the introduction of simplifying approximations. The
expected value and variance of Y, the NLF output,are derived
below.

Let Y], i = 0,1 indicate the expected value of the

NLF output random variable when hypothesis H1 is true.
00 co
= f = . 3
A. y Y<y|H1>dy [mg(x>f(x\H1>dx (4 >

Expanding g(x) in a Taylor series about A yields (subject to

a few regularity conditions), for some g E (O,A),

r-l _ j
g(x) = g(A) + 2 9‘3ng ”(m +5—AL (”on . <44)

j=l r!

Substituting (44) into (43) and interchanging summation and

integration operations,

§1=Ii8m>+ 2 +8 (”(1) + (X f‘———): (10%)], “lede (45>
j= -1 °

gum: + (iv) (:1

= g(A) + "T— 4! “'x,4

+ (46)

Assuming g(°) is sufficiently smooth to allow consideration

of only the first two terms, gives
2
_ . g"(A)oX
Y1 = g(A) + —"2—— (47)

The second central moment is:

= j‘ g2(x)f(x‘H1)dX

Let h(x) = g2(x). Then, using (47),

44

—2 , Wm 2

wm>=aewmfi+gmmwm1 9%
‘2 2

Y1 = g (A) + [(g'amz + g(A)g"(A)]o: (50)

N t' 11 2 - _2 '_ 2 '
0 ing t at 01 - Y1 - (Y1) , (47) and (56) give the desired

result, namely,

2 . '
01 = [g'(a>]ze§ (51>

 

It must be noted that (47) and (51) are of questionable
validity in the weak signal case. This is because the function
g(x) has a discontinuous second-derivative at the origin.
Although the function is sufficiently smooth away from the origin,
and therefore allows (47) and (51) to be useful in the large SNR
case, other methods of approximation must be sought for the weak
signal situation.

Referring directly to the PDF, Sectionli, for very small

A, (42) becomes

2
-%(1-)
C’0

e-bA e

TTO'O

(5 2)

2

..(LV
0
bA e .
fY (Y) e m;- lf y > 0

Then

 

See Appendix III.

2 2
~1§<Y—> ac?)
O 0 m 0
Ed) e'bAo f y dy + ebAo j y e dy <53)
0 2 0 2
-m n o 0 Jzu o
O O
:V/ETO' sinh(bA)
n 0
:/'2:cr M + 0(A2)
n 0
. 2 0
= — (in; (54)
Ti 0'

X
The variance of Y under H1 and weak signal conditions
will be assumed identical to the variance of Y under the null
hypothesis. By assuming that the variances under the two
hypotheses are equal, an optimum decision rule can be defined.
However, no claim will be made for optimality; the important

feature will be the improvement in performance which may be

attained by using a nonlinear element in the detector.

3.6 Summary

This chapter discussed reasons for using nonlinear elements
in the detection process. Because it is not clear that a simple
clipper is in any sense the best nonlinear element that can be

used, the "Gaussianizing" nonlinear filter was introduced. The

 

nonlinear filter (NLF) that transforms Laplacian variables to Gaussian

random variables was derived. The concavity of the transfer
characteristic on the positive real axis was shown and the output
PDF under H1 was found. Approximations for the mean and
variance of the NLF output were derived and their regions of

validity were discussed.

CHAPTER IV

COMPARISON OF DETECTORS

This chapter evaluates the performance of a detector
which includes the NLF derived in Chapter III. It investigates
the possibility of improving the detection reliability. To do so,

the probability of detection (P ) as a function Of false-alarm

D
rate (FAR) is found. For the weak signal detection problem the
NLF-detector performance is compared to the Optimal detector (the
nonparametric sign test) and also to a suboptimal detector which
is based on the assumption of a Gaussian noise model. The com-
parison is repeated for the case of strong signal.

The measure Of performance of one detector relative to

another is based on the following definition [C1, p. 89].

Definition: Let T1 and T2 represent two tests that test the

 

same HO against the same H1, with the critical regions size a,

and with the same values of 9. The relative efficiency of T1

to T (or "efficiency Of T

2 relative to T2”) is the ratio

1

nZ/nl, where n and n2 are the sample sizes of the tests T

l l

and T2 respectively.
This is a reasonable and useful measure Of the relative
efficiency Of T1 with reSpect to T2. An asymptotic relative

efficiency (ARE) of T with respect to T is Obtained by

l 2

passing to the weak-signal limit (when it exists) as H1 approaches

H0 in a properly restricted sense [C5, p. 296].

46

 

 

47

Section 4.1 develops the Optimal detector for location
alternatives for a sample of size N from the Laplacian density
f(X) = ‘3' exp(‘- b‘X-A‘). Section 4.2 focuses on the weak-signal
detection properties of the sign test, the ”Optimal" test based
on an assumed Gaussian noise model, and the NLF-detector. Sec—
tion 4.2 concludes with a comparison between the Sign detector and
the NLF detector. Section 4.3 investigates the large-signal de-
tection properties of the three detectors. In Section 4.4 a Monte
Carlo simulation is performed to investigate the performance Of the
three detection systems for small sample sizes in both Laplacian
and Gaussian noise. Section 4.5 summarizes the main results.

4.1 Optimal Detector Of Constant Signal in Laplacian Noise

 

The Optimal detector, as was discussed in Chapter II, is
f(x x ---x [H )
based on the likelihood ratio, where A(xp) Q 1 2 n 1 ,

f(X1X2”°Xn\Ho)

 

and f(x1'°-xn‘Hi) is the likelihood function of the sample
, A
xn = (x1x2°'°xh), when hypothesis Hi is true. Let the signal-

plus-noise PDF be of the form f(u) = E'exp(-b‘u-A‘), for

-m < u < +m, with b determined from the relationship
m
2 2
f x f(x)dx = OX (1)
-oo

(See Chapter III, Section 3.2 for this development.)

n n -b\x,-A‘ n -b\x,|
A(X)=(Ilae 1 )/(nae 1) (2)
i=1 i=1
I1
= exp ( -b 2130 Xi-M - WED) (3)
-an(xn)=;|X-A\ -\XI (4)
b 1( ti i.l'

TO proceed further with this analysis the real line is

broken up into 3 disjoint intervals, as follows:

48

Let 11‘! [x‘x 2 A > 0}, 12 = [x‘xi< O] and I3 = [x‘A > x 2 0].
Now for each Xi which falls in 11’ the right-hand-side of (4)
reduces to ‘Xi - A] - \Xi\ = -A. For each X1 in 12, the right-
hand-side of (4) becomes ‘Xi - A‘ - ‘Xi‘ = A; and for each X1

in 13, the right-hand-side Of (4) becomes A - 2Xi. If among the

N observations, Ni are assumed to fall in interval I,, the
i

right-hand-side of (4) may be rewritten as:

A05 -N1)*'Z(A -2X)- (5)
N3 1

 

This can also be written as

MN, - N1) + N31: - z: (A - 2x1). (6)

N3

3 N - (N1 + N2), (6) may be written

Using the fact that N

MN, - N1) + [N - (N1 +N2)]A — 2: Xi (7)
N3
= A[N - 2N1] - 2: xj. (8)
N3

The optimum processor consists of three-level threshold
circuitry followed by a counter and summing unit. If the threshold
circuitry detects a sample in 11’ A units are subtracted from
the counter while a positive A units are added to the counter
when a detected sample falls in 12. Each sample occurring in
I3 is summed and stored. The storage unit and counter indication
are summed after the N samples are processed and this test
statistic is compared to the decision threshold. The exact dis-

tribution of this test statistic can, theoretically, be obtained

although actual analytic evaluation would be cumbersome.

 

 

 

49

4.2 Weak Signal Detector Evaluation

If interest is focused on the vanishingly small SNR case
(A a O), a result from classical hypothesis testing literature
becomes immediately applicable. If X1,X2,...,XN is a sample
from the Laplacian distribution with density %-exp(-‘X - A‘),
with A a location parameter, then the Locally Most Powerful
(LMP) test Of A g 0 against A > O is the sign test [L1, p.
344]. The concept of LMP tests is applicable to communication
problems because of the desirability of achieving Optimum detec-
tion for weak signals. Strong signals will be detected even if
the detector is well below Optimum, as was noted by Capon
[C2, p. 67].

As A a 0, the test statistic of (8) becomes A(N - 2N1),
where N1 is the number of Xi greater than A.N Thus an equi-
valent test can be stated in the form SN(XN) = Z u(Xi) where

i=1
1 if z > 0
11(2) =
0 otherwise

The test based on SN has been investigated by Capon
[C3], Kanefsky and Thomas [K2], and others [C4], [H6], [C5].

It is a well known result [F2] that the statistic SN
is sufficient for p, the probability that X > 0. The dis-
tribution Of SN is binomial with parameters N and % under
H0, and therefore the threshold value for any specified FAR can

be determined. The problem Of randomization can best be explained

by an example.

 

 

 

50

Example 1: Determine the threshold value that will guarantee a

 

FAR of 10% when the test statistic is SN and N = 8. The

probability mass function for SN is shown in Figure 4.1. The

randomization problem becomes obvious unless the FAR is specified

as one of the values (0.0, 0.0039, 0.0351, 0.1445,...).

 

 

 

 

 

 

 

0.2734 .
0.2188
0.1094
0.0312
0.0039
1 _ - 4* x
0 l 2 3 4 5 6 7 8

Figure 4.1 Probability Mass Function for 8

TO achieve the specified FAR Of 10%, randomization Of the

threshold value of X = 6 is required.

0.10

II

(.0039) + (0.0312) + B<0.1094)

_ .0649 é
B ‘ .1094 '594

 

The test is based on SN rejects the hypothesis H0 if

X = 7 or 8 and with probability .594 rejects HO if X = 6.

51

This section continues with the derivation of the asymptotic
distributions of the Sign detector under the hypothesis and

alternative and its probability of detection as a function of FAR.

Write
n N . N-‘
P[SN s n] = z (.)p‘.] (1 - p.) 3, n = 0,1,2,...,N (9)
,_ j i i
j—O
where
A
pi = P[X 2 O‘Hi] . (10)

With p0 = % (by Assumption A2, Chapter I), define K 3

(3) = 2 P . (11)

'In 2

This expression assumes that randomization is unnecessary. BecauSe
asymptotic results (N a w) are Of interest in this chapter, this

will be Of no consequence. On the basis of (11), write

N
N ' N—'
P=z<.>pJ<1-p>J (12)
D sz j l l

with p1 given by (10). For large sample sizes, the DeMOivre-
Laplace limit theorem can be invoked to simplify computation of
the probabilities (11) and (12), [F4].

Defining

R ASN'Npi

. (13)
N’1 J Npi(1-pi)

the theorem states that

 

S

7':
P[z1 s SN,i S zz] N fl(zz) - fl(z1) . (l4)

Then (11) may be written

 

 

 

 

2
u
+°° 1 "2—
= -- 1
PF ] N me du ( 5)
K " 2
ll
4
2
_.2_
+m 1 2
PD =]‘ me du (16)
K - NP1
(Nplfl-pl)
From (15)
_ H_ -1 El
K ’4 erfc* PF + 2 . (17)
Using (17) in (16) yields the receiver Operating characteristic
Of the Sign detector, namely,
8 erfc;1 PF - fN (2p1 - 1)
PD = erfc* (18)
1 -
flRpl( 13,)
where the s superscript indexes the Sign detector ROC.
When X has the Laplacian PDF
b
f(X) =§exp (-b\x\), —..<x<+.o (19)
then
e-bA
p1 = 1 ' 2
-bA (20)

 

1"’1" 2

Substituting these expressions into (18) yields

   

53

 

 

erfc;1 P -'/N (1 ‘ e-bA)
PS = erfc F , . (21>
* - _
D {(2 _ e bA)e bA] 2
Substituting b =‘12' into (21) yields
ox A
— f2 3—
s erfc;1 PF -/N(l - e X)
PD = erfc* A_ L (22)
- f2 A— - /2 2
[( OX) Ox
2 - e e

Equation (22) can be used to plot PD as a function of PF with
N and A—' as parameters. The ROC Of (22) can be compared to
that derived in Chapter II, Section 2.5 for the "optimal" detector

based on an assumed Gaussian noise model, namely,

U0

P = erfc7,,[erfc7;1 PF - [N :1 . (23)

For the case Of small SNR, i.e., A ~ 0, (22) becomes

U

PS = erfc*[erfc;1 P - ,/2N L] . (24)
F OX

Suppose that the numbersof samples N in (23) and (24)

are not necessarily equal, i.e., in (23) let N = NO while in

(24), let N = NS.
If the two tests have the same FAR and PD, then equating

(23) and (24) yields

-1 -1
erfcic pF - fNO g— = erfc* PF - ms %— (25)
X X
NO
or simply fi-'= 2 which is precisely the Pitman ARE for the
s

nonparametric sign detector relative to the ”Optimal" Gaussian

detector.

 

 

54

The weak signal performance Of the NLF detector is now pre-

sented. The results Of Chapter III are summarized in Figure 4.2.

 

 

 

 

Gaussian N
XN , Y Gaussian DECISION
Transforming --—--—
l
Filter Detector H0 or H1

 

 

 

 

 

 

 

 

 

 

 

’ISequence of N iidr) ’Sequence of N iidrv havfhg
having Laplacian PDF Gaussian PDF under H . Under
under HO and H1 H and small A, the r.v. are

assumed Gaussian.

Figure 4.2 Nonlinear filter detector

Well known results concerning the sufficiency of the sample
mean. can be invoked for the problem described in Figure 4.2. The de-

tector based on the sample (YN) mean is easily formulated. Under

00 - 2 O0 O0
—— h' d H ~ '—— '—— —— . Th
N ) w lle un er 1, YN fl(/fi (OX)A, N ) e

expression for P as a function of FAR is then given by

D
C
- £3- 9- 491)» (26)

0 TT OX

HO: YN N 7((09

N _ -1
PD — erfc), [erfc* PF

the validity Of which is based on A N O. The N-superscript

indexes the NLF detector ROC.

 

55

Suppose that the numbers of samples N in (24) and (26)
are not necessarily equal, i.e., in (24) let N = NS and in (26),
let N = NN. If the two tests have the same FAR and PD, then
equating (24) and (26) yields

0'
erfc; -S/'2N- A— : erfc:1PF - LN— é‘ ("Q) A
0x 00 “ CIx
N 2
—S=—é0.636. (27)
NN n

Using (25) and (27) the ARE of the NLF-detector relative

 

to the "Optimal" Gaussian detector is found to be

2

‘19
NN

_Q .
N

21:»

=1. 27

Z
2 I'm

On the basis of ARE, the NLF-detector is better than the
"optimal" Gaussian detector, but not as good as the Sign detector.1
This result could have been anticipated under the assumptions on
which it was based. The following theorem makes this situation
clear.

Tpgpppm; Let X and Y be univariate random variables related
by Y = g(X), g(-) a monotone increasing function wherever fX(X)

is nonzero. Then JX(1:2) = JY(1:2).

0° (X)
Proof: J (1:2) Q j (fX 1(X) - fX 2(X)){,n —§l——— dX . Make the
— x 13,200
change of variable X =1(y), which is monotone increasing and

defined uniquely for each y. Then,

 

Note that the NLF does improve SNR without a corresponding
improvement in detection probability.

 

56

-1
fx1(g (y>> d

EX2<g'1(y)>

 

Jx<1:2> = [m<fx1(g'1(y>> - fx2<g'1<y)>>(§;-g‘1<y>>tn y

m fY1(y)

= [ (fY1(Y) ' fY2(Y))Ln E_—(§)- dy
'm Y2

= JY(l:2)

This formalizes the notion that a nonlinearity whose char-

acteristic is monotonic does not affect PD and FAR [R2, p. 438].

 

This is because the probability that the output exceeds a given
threshold is exactly the same as the probability that the input
exceeds a corresponding threshold, there being a one-to-one
relationship between input and output amplitudes.

The Optimal test for location was shown in Chapter II tO

H
l

N >
be Of the form A(X ) < K, where A(XN) was the likelihood ratio

H0

N
and K was picked so that P[A(X ) > K‘HO] = P By the mono-

F.
tonicity Of the logarithm, an equivalent optimal test is of the

H
l

‘ N > * 7'c
form Ln A(X ) H K , with K satisfying
0

l
"U

P[Ln A(XN) > K*\HO] —

H
"U

7'c
Then P[Ln A(XN) > K [H1]

If Ln A(XN) has a symmetric PDF under H and H and if the

0 1’
variance of An A(XN) is the same under both H0 and H1, then
PD depends on the separation of means of Ln A(XN), the larger

the separation the larger the detection probability will be. Then
E[Ln A(XN)[H1] - E[Ln A(XN)‘HO] is a useful measure of the

distance between distributions. When normalized by the common

57

variance, the result corresponds to the SNR measure of Van Trees

[Section 2.3]. But

f1(X)
fO(X)

 

Run A(XN)\H1] =j‘ gloom dx ,

and

f (x)

°° 1
E[{,n A(XN)‘HO] = ‘5:me (x).(,n “W dX

which yields

Bun A(XN)\H1] - Bun A(XN)\H0]

f (X)

35-6—(5 dX = J(l:0)

= l [f1<x> - fO(X)]tn

which is the J-divergence distance measure introduced in Appendix I.
This shows that the Optimal test for small signals at the nonlinear
filter output is ppp based on the sample mean. However, as pointed
out in Section 3.5, no claim for Optimality was intended; the NLF-

detector is being investigated as a suboptimal detector which pro-

vides a continuous test statistic of known form under the null

hypothesis.

4.3 Strong Signal Detector Evaluation

Letting the sample sizes of the Sign detector and Optimal
Gaussian detector be NS and NO, respectively, and equating (22)

and (23) yields

-1 -8
erfc* PF - /NS (1 - e )

Z

 

= ‘ - -J1 2
[(2 - e-B)e-Bl% erfc*1PF 2 B ( 8)

where 9 g,/2 A/ok. Define h(9) Q l - [(2-e-B)e-B]-§, and write

58

N erfc-lP _ 2
r9- = film—"Ll W + ‘1 ‘ 8 BA (29’
s B S

In the region 0 < B < .7, ‘h(9)[ < .15, and for sufficiently

large NS, (29) becomes

-6 2
—9- -—°- 2 (1;) > 1 . (30)
NS 3

The weak-signal limit (9 a 0) is easily found by apply-

2

ing L'Hospital's rule twice, resulting in

N
.12 = 2 (31)
N

s

in agreement with the conclusion Of Section 4.2. Thus, the large—
sample efficiency improvement of the Sign detector relative to the
Optimal Guassian detector in Laplacian noise is seen to be more

than a local property; i.e., valid at vanishingly small SNR's.

This result, although developed here for the first time,
is not entirely unexpected. Recall that the variance of the Sign
detector under H1 was given by Np1(l - p1), with
p1 é P[X > OlHl]. Note that as p1 increases from % to 1, the
variance decreases thus concentrating more Of the probability
mass about its average. With a fixed threshold, the probability
of detection will dramatically increase.

The large signal mean and variance of the NLF detector
were found in Chapter III, Section 3.5. Invoking the central
limit theorem to evaluate P and FAR, and using those moment

D

approximations with a sample size of NN yields the ROC

 

59
l

N = 00 erfc; PF g(A) + % g"(A)c)2C
PD ““4 ox g'm) ' fNN ox g'm) ' (32)

Equating this to (29) yields

 

 

1
o

O erfc*

12F - /NN (803) + 1? smog)
_ l "1 A
- ox g (A)[erfc* PF -‘/NO (3;?] - (33)

Assuming NN and NO are sufficiently large, (33) becomes

0

 

2 2
N _ [g(A) + 15 g"(A)oX] (34)

N A g"(A)

2

By the mean-value theorem (34) may be written

0

 

° A g'(A) for some g E (O,A).

Z

2
N (gm) + A g'(§) + 1? g"(A)c)2()

N

2
'
2(§7£(-§)l) 2 l by the convexity of g(°)

This requires that \% g"(A)o:\ << \A g'(§)\ or,

‘5"(A). << /2 (f) g

g (é) X ox
g'(§) é g'(A)
of Section 3.3) as

Now which can be written (using equation (20)

n bog - g(A)g'(A)

g'(A) = 2 Sb:
C’0

 

><QN

But for sufficiently large signal-to—noise ratio,

 

6O

Tamil lam =Q A. [2
g'<§) S g'm) Sb OX<</2 (GXHX

Thus, the NLF detector has greater efficiency than the
"Optimal" Gaussian detector. This result could also be

anticipated because the variance of the test statistic, given by

2
[g'(A)] oi, asymptotically approaches zero as the signal amplitude

increases.

Lemma: The variance of the NLF output under H1 becomes vanishingly

small as the input signal amplitude increases. That is

lim Var (Y‘Hl) = O.

A-ooo

Proof: Recall that Var (Y‘Hl) é [g'(A)]20:. Because 0: is a

fixed, known constant it is Sufficient to show that lim g'(A) = O.
A—m

In Section 3.3 it was shown that ho: - gg' 2 0. Then

because g > O, Vx > O, and b02 is finite,

0

bg2
__O._g'>0

g _

Therefore
bgz

. 0

g s——-.
g

But by the definition of g as a monotone increasing function,

the limit as A approaches infinity for the right side is zero.

 

Because g' 2 0 when x > 0, this holds also for the left side. (Q.E.D.

 

 

61

4.4 Simulation

 

This section provides a Monte Carlo evaluation of the three
detection systems for small sample sizes in both Laplacian (e = l)
and Gaussian (e = 2) noise. It is expected that the performance
of the detectors for values of e 6 (1,2) will fall somewhere
between their performance at the extremes. Figure 4.3 is a block
diagram of the overall simulation procedure. Threshold values

yielding a False Alarm Rate of 10% were read into the program,

 

along with the randomization strategy for the Sign detector.
Signal-to-Noise ratios were made dependent on a single parameter,
A, by choosing 00 = ox = l. The NLF was synthesized as discussed
in Appendix II and the number of Monte Carlo experiments performed
during each pass through the program was set at 250. This yields
the interval (.07, .14) as a 95% confidence interval for the FAR,
[01, p. 99] a suitable range for the purposes of this simulation.
The average FAR for each sample size and each detector is shown

in Table 4.1.

 

Laplacian Noise

 

4 8 12 16 20
SIGN .089 .104 .115 .101 .087
XBAR .089 .103 .095 .094 .095
NLF .100 .103 .114 .095 .103

 

Gaussian Noise

 

 

 

4 8 12 16 70
SIGN .098 .088 .120 .108 .086
XBAR .103 .108 .112 .093 .109
NLF .129 .128 .105 .104 .111

 

Table 4.1 False Alarm Rates for Detectors

62

 

CTAI 3

Set thresholds for
specified FAR set nois
type: 1 = Laplace, 2 =
Gaussian set min. and
max. SNR and No. of
Monte Carlo Experimenm

. l

 

 

Determine sample size'
NSAMP = 4*k, k = 1.5
I CHOOSE SNR J<?-"

CHOOSE DETECTOR
l = sign det.

 

 

 

N
ll

XBAR det.
3 = NLF det.

CHOOSE HY OTHESIS 2]
a|1=Ho "Hl -

Perform
Monte Carlo Detection

 

Experiments

l PRINT RESULTS __J

 

 

 

 

 

 

 

 

 

(STOP 3

Figure 4.3 Simulation Block Diagram

 

63

The data are presented in Figures 4.4 through 4.9 for sample
sizes 4, 12 and 20. Appendix IV contains a complete tabulation of
simulation data.

Before the data can be adequately evaluated and conclusions
drawn as to the relative performance of each of the detector systems,
it is important that the simulation procedure be examined so that
any biases appearing in the data be clearly defined as to source
and overall effect on the simulation.

The nonlinear filter implementation is discussed in Appendix
II. As mentioned there, a piece-wise linear approximation to the
continuous transfer characteristic of Section 3.2 was made in ten
uniformly spaced intervals extending from x equals zero, to x
equals three. A linear approximation to g(x) for x greater
than three was based on the slope of the chordbetween x equals
3 and x equals 4. Because the transfer characteristic is con-
cave, this approximation tends to amplify outliers and results in
an increased FAR and PD. However, the effect was not substantial
as evidenced by Table 4.1. It was enough to cause the detection
probability of the NLF-detector to be greater than that of the
optimal detector in Gaussian noise, a theoretical impossibility.
This is particularly evident at small sample sizes. This was the
only noticeable bias in the simulation procedure and therefore
the following conclusions can be made.

1. In Laplacian noise the Sign detector is inferior to both
the XBAR detector and the NLF-detector over the greater part of
the range of experimental SNR's for small samples [See Figures

4.4, 4.6 and 4.8]. As the sample size increases the performance

 

64

\D
O
l

 

H.mu

muwm mHmEmm .omwoz amflomaamg .MZm .m> m

v.0:

Q

¢.NHu

q.¢ mpswwm

o¢n

 

 

......4-.- .- -. -2 .-....

r-||1|4.Tl|-

,. ._ -_--4 .....-w..._... ...—.4

«it .1 rfl -..I'I.ll...

 

 

 

 

 

 

 

 

 

 

 

 

_
_
..l 1 ‘3 1 n l I
_ .. ,
a .
w , v A
i _ M
M I II “a M
_. .
_
u . .
d. V , I .lv 0“ III ‘ IIIIII' II II I ‘EI ‘IIIII lip-1|" ‘11. ‘l .1] I'll
H m M , .‘xl
” . . A
_ .- ------ 0 6H“ . .
q . q.
t . . a
n _ . a. F
. a . .
_
_ _
Y
.
.
r
_
7
:1 I l r i it»! oilliiilllnLllzi yllrtlt
.
w r
_ , , . i
. .. . a M .
i . _ .
,. . h _ _ N
i A V , u ,
‘. . -5 '71 1 J, ‘l 1 l 1 i 11.1w ITI Y
i , v .
v N _ , . ,. r i z _ _
_ , m . _ H , _ A
- I _ i a . i i . i .. _ I
A . . _ . i . _
. .. . . .
. . . h
. . . n
W , . A .
l t IIIT! trajllv rlntllli 11-1le!! I11. 1. y 11
t . .. . _
,,,,, i - l . i i _ _ r n
:-r --ITI- . r _ . fl .
y c ,i . . W M _ _
M m m _ l . fl «
t _ i h . w p

 

 

 

 

 

 

 

 

 

 

 

q u ouwm mamamm .omwoz swwmmsmu .mzm .m> m 0.0 ouawflm

0.0: H.Mu 0.0a «.maa oqi

 

 

a M m _y~. i - w i o
i . w . i
H _ m _ _ u x H
__ s + ,

 

65

 

 

 

..-..lllllid val ....1' IIc‘I I III I)- I. i’ll‘llf (livllrzlll ,l.§. . . I ..#
.

 

 

 

 

 

 

 

 

 

 

 

M w i .
s i..- :w
. . ,, t . . a
m _; M c N m w w
_ A m _ _ .
c w w m r . H m , W
n S a w {1+I. q t L L: 1
. ~.;-1 ~ ~ . _ . _ . a w m
f? -. -;-l . , _ n m _ M __ . . m
....._ ...1 _ _ * M m . _ .u W _
_ . w l. __ _ l _ _ L i , :

 

 

66

Q

NH u oufim magamm .omwoz dwwomaqu .MZm .m> m 0.0 ouswwm

H.m: 0.0: ¢.NH: oq:

 

 

 

00".-lll: . :I‘II. I :[lill‘kllrblvf.l: II 1 11.11:: Ill? :11. . . .u . . .ul lolly! \: . Ilul; |-!I.! 1 lulllllllll; 1.: l:

I

. w

 

 

 

 

-w w in.-- max

DI immm114lziws.-.

.0 ES

 

 

 

_ -..... .._.._.._.J,__

 

-47.--“ .

 

 

 

 

 

Ill .1.

 

 

1 .~

ll‘

 

 

 

I‘dl

 

 

 

 

 

 

 

 

67

ouwm oHaEmm .omfioz awwmwsmu .MZm .m>

NH

mm

5.0 muswwm

o0:

 

 

 

 

 

 

 

 

 

_ _ 7 7 7 ._ 7 .
. . 7 . 7 _ V _
w m .7 w . m . r b .
7 7 7 7 . 7 7 .
7 7 . .
11:7 . . 7 ..1 :7.11 11 1-1..111.11:11,:11 1 : 11:
7 7 7. . 7 _
. 7 7 . _. 7
7 7 7
7 7 . _
1: 1+9 1111.1. . 1 11 1 1 1 .1 .1 H 1 1.11
7 7 7 . .
. 7 7 7
7 7 7 . 7 7
1 7 7
.J][ 7 a MHZ M V .
11.11: : 11-1.3111 1.7, 1 . .7 ,
a [I -ll 4 fig“ 7

 

 

 

-1 :1: _O haw

: 7 11" ——-—~
1
1

 

7
.

77.7

    

111111 : 1111111111 1.1.11 1:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

, 7 7
7 7 ,
. P'I-‘ .7 _
7 . 7
, . _
7 7
_ a
1 1 1 :1: 11 1:4: 11111111: 1111*: 11
7 7
_ 7 7
, -M 7 _
7 . 7 7..
7 I 7 .
7 . .
7 7 . 7
. u— m u
11 _ , 1.: 1

 

 

68

0.0:

ON

 

muwm magnum .mfloz

Him:

0.0:

cmnumfiamg .MZm

Q

.m?

«NH:

w.. ...».m

as-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 71 7 7 7 7 .. 7 . 7 . 1
7 7 7 7 , _, 7. 7 . .7
7 7 7 7 7 7 7 7 .7
. 1. , 1 . 7 I: 111 11111111: - 1r: 1111 _ 7
7 _ 7 _ 7
__ 7 7 7 _
7 7 7
, 7 7 7
7 7. 7 7 7
11:71-
7 , ., 7 7
, 7 7 7 7
.. 7 7 7
7 7 _ 7
7 . 7 .7
7 1 .u 7mHz 7
7, 7 7
7 7 e _
:--n-7 .1. 1 .~.7a.mx:..- 1
7 7 7
10:: O fiHm 7
7 -1 117.
7 7
7
1 1, 11.71111 1: 1
7 7 7 7 .
7 _ 7
. 7 . 1.
_ 7 7 . 7
7 . q
. 7- . . , 1»
_. 7. . 7 ,_ , 1N ,
7 7 7 7 .7 ,
7 7 7 7 7 t. 7 7 7 7
_. 7 7 7 7 7 7
1 7 .r 7 7 7 ,7 7
_, 4- . 7 7 7 ....... 7 1.
. 1-- . . 1 7 7
7 7 7 7 7
71 1 1 - 7 1 7 7 7
7 _ 7 _ 7 7
1 -1- . 7 . 7 7 7 7 , 7 7 ,
7 7 7 _ .7111 7 .1 1.
7 _ _
7 7 1 7 _ 7 7 ,
1: 1 :7 , ,5 [Klwfm 7 7 7 7 7
7- 1- 1 .7 7 ,‘ . 1 7. 7 7 7 7 _
7 7 7 1 7 7 ,.

 

 

 

 

 

 

 

 

69

 

om

 

muwm mHaEwm .omwoz cmflmmsmu .MZm

.m>

Q

m

 

 

 

o

q musmwm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

70

of the sign detector improves to the point of being equivalent to
the other two, in agreement with the result of Section 4.3

[Figure 4.8].

2. In Laplacian noise, the NLF detector is superior to the
XBAR detector for small sample sizes but as sample size increases
the performance of the XBAR detector improves to the point of being
equivalent to that of the NLF—detector. This lends support to the
conclusions of Sections 4.2 and 4.3 [See Figures 4.5, 4.7, 4.9].

3. In Gaussian noise, the XBAR detector provides a substantial
improvement in detection probability compared to the sign detector
for all sample sizes and SNR's [See Figures 4.5, 4.7, 4.9]. How-
ever, the NLF—detector and the XBAR detector appear equivalent for
all SNR's and sample sizes.

4. The PDF of the NLF output can theoretically be obtained for
all values of e 6 [1,2], although this thesis concentrated on the
special case 9 = 1. For the case 6 = 2, the method of Section
2.5.1 was invoked; the resultant PDF was mathematically intractable.
For this reason, the NLF-detector FAR cannot be computed for values
0f 9 6 (1,2]. It therefore suffers the same disadvantage as the

XBAR detector for e 6 [1,2). [See Section 1.5.]

4.5 Summary

The optimal detector of constant signals in Laplacian
noise was derived and the difficulty of evaluating its performance
was discussed. It was shown that for vanishingly small signals a
locally most powerful detector could be defined and easily
implemented. Its performance could be easily evaluated for all

sample sizes. Because the test statistic had a discrete PDF, it

 

71

was pointed out that randomization of threshold values might be

necessary to achieve a specified FAR and this was illustrated by
an example. The weak-signal detection property of the ”Optimal"
detector, i.e., the detector designed on the basis of an assumed

Gaussian noise model, was compared to that of the LMP detector on

the basis of ARE and shown to be markedly inferior. The weak-
signal detection property of the NLF detector was compared to
the LMP detector, and while still inferior, was an improvement

over the "optimal" detector. That this result follows from the

 

type of nonlinear transformation assumed in this thesis was shown
in a theorem. The strong signal detection properties of all three
detectors were compared and again the NLF detector was shown to

be a viable alternative to the others.

A Monte Carlo simulation of the detection problem was
implemented and the three detector systems were evaluated in both
Laplacian and Gaussian noise for a large range of SNR and small
sample sizes. It was demonstrated that the sign detector was
inefficient relative to the XBAR detector in Laplacian noise and
small sample sizes unless the signal level was vanishingly small.
The NLF detector was shown to be equivalent to the XBAR detector
and a definite improvement over the sign detector for a wide range

of SNR's and sample sizes in both Laplacian and Gaussian noise.

CHAPTER V

GENERAL CONCLUSIONS AND EXTENSIONS

The major conclusions of the thesis are reviewed in this i

chapter and possibilities for future research are discussed.

5.1 Conclusions

 

 

This thesis has been concerned with the design of a zero—
memory, nonlinear filter, and its application to the problem of
detecting a constant signal in non Gaussian noise. It was shown
in Chapter III that a nonlinear filter could be designed which
transformed a sequence of independent, identically distributed,
Laplacian random variables to a sequence of independent, identically
distributed, Gaussian random variables. This allowed satisfac-
tion of a specified false alarm rate without randomization at the
receiver, a disadvantage when the intractibility of the input
distribution suggests application of non-parametric procedures.

The transfer characteristic of the nonlinear filter (NLF)
was restricted to be monotonic increasing. In Chapter II, the
property of monotone likelihood ratio (MLR) was shown to be invariant
under this transformation. Thus, Optimal tests for location con-
sisted of threshold tests.

In Chapter IV, the NLF-detector was compared to the Optimal

detector and another based on an assumed Gaussian noise model, for

72

 

73

vanishingly small signals. On the basis of ARE, its performance
was an improvement over the detector based on the Gaussian noise
model, but it was not as good as the locally most powerful sign
detector. A theorem showed that the NLF transfer characteristic
did not affect the discriminability between the hypotheses. It
was then concluded that the NLF detector was not the optimal
detector for location alternatives. For strong signals, both the
Sign detector and the NLF detector were shown to be superior to
that based on an assumed Gaussian noise model. It was shown that
the variance of the test statistic under the alternative for the
sign detector and the NLF detector asymptotically decreased with

increasing signal-to-noise ratio.

5.2 Extensions

 

”Little has ever been done in the matter of finding to
what extent this form of transformation might be utilized in
deriving distribution laws for empirical data, but it has been
used now and then to throw unusual types of distribution curves
into forms that conformed better to established types, especially
to the normal curve." [F3, pp. 273-274]. It was this statement
that provided the direction of this thesis. Although a first
step has been taken in investigating the utility of transforming
distributions, there remains a great deal more to be done. The
choice of a transformation yielding a normal curve is not
necessarily best; perhaps another PDF could be more easily realized
than the method of Appendix II indicates is required for

"Gaussianizing". Further, transforming to another PDF could

 

74

possibly improve the efficiency Of a NLF-detector relative to the
Optimal sign detector.
Another direction which should prove fruitful is the in-
vestigation Of the improvement afforded by a nonlinear filter
when the Optimality criterion is minimum average probability Of
error. This is most Often the Optimality criterion in a i

communications network.

 

 

 

 

BIBLIOGRAPHY

 

[c1]

[C2]

[C3]

[C4]

[C5]

[D1]

[D2]

[F 1]

[F2]

[F3]

[F4]

[01]

[H1]

BIBLIOGRAPHY

Conover, W.J., Practical Nonparametric Statistics,
Wiley, New York, 1971.

 

Capon, J., "On the Asymptotic Efficiency Of Locally
Optimum Detectors," IRE Trans. on Information Theory,
Vol. IT-7, NO. 2.

Capon, J., "OptinmnxCOincidence Procedures for
Detecting Weak Signals in Noise," IRE International
Convention Record, Pt. 4, 1960.

Chadwick, H.D. and Kurz, L., "Two Sequential Nonpara-

metric Detection Procedures," NYU Technical Report
400-155, March 1967.

Carlyle, J.W., "Nonparametric Methods in Detection
Theory," Chapter 8 Of Communication Theory,
A.V. Balakrishinan, Ed., McGraw-Hill, New York, 1968.

 

Daly, R.F. and Rushforth, C.K., "Nonparametric Detection
of a Signal Of Known Form in Additive Noise," IEEE Trans.
on Information Theory, VOl. IT-ll, NO. 1.

Dubes, R.C., The Theory Of Applied Probability, Prentice-
Hall, Inc., Englewood Cliffs, N.J., 1968.

 

Fraser, D.A.S., Nonparametric Methods in Statistics,
Wiley, New York, 1957.

 

Ferguson, T.S., Mathematical Statistics: A Decision
Theoretic Approach, Academic Press, New York, 1967.

 

 

 

Fry, T.C., Probability and Its Engineering Uses,
Van Nostrand CO., Princeton, N.J., 1965.

 

Feller, W., An Introduction to Probability Theory and
Its Applications, 3rd Ed., Wiley, New York, 1957.

 

 

Gilbert, E.N. and Pollack, H.O., ”Amplitude Distribution
Of Shot Noise," Bell System Technical Journal, Vol. 39,
NO. 2, pp. 333-350.

Helstrom, C.W., Statistical Theory Of Signal Detection,
Pergamon Press, Oxford, 1968.

 

75

 

[H2]

[H3]

[H4]
[H5]
[H6]

[H7]

[K1]

[K2]

[K3]
[L1]
[N13
[P1]

LR1]

[R2]

[R3]

 

76

Hancock, J.C. and Wintz, P.A., Signal Detection Theory,
McGraw-Hill, New York, 1966.

 

Hancock, J.C. and Lainiotis, D.G., "Distribution-Free
Detection Procedures," Purdue University Technical
Report TR-EE-64-8, 1964 .

Hogg, R.V. and Craig, A.T., Introduction to Mathematical
Statistics, 3rd Ed., Macmillan, London, 1970.

 

Heathcote, C.R., Probability: Elements of the Mathematical i
Theory, Wiley, New York, 1971.

Héjek, J., Nonparametric Statistics, Holden-Day,
San Francisco, 1969.

 

Hancock, J.C. and Wade, W.D., ”A Digital Wide-Band Non-
linear Receiver Capable Of Near Optimum Reception in the
Presence of Narrow-Band Interference," IEEE Trans. on
Communication Systems, Vol. CS-ll, No. 3.

 

Kendel, M.G. and Stuart, A., The Advanced Theory Of
Statistics,fiVOl. 2, Hafner Publishing CO., New York, 1947.

 

Kanefsky, M. and Thomas, J.B., "On Polarity Detection
Schemes with Non-Gaussian Inputs," Journal of the
Franklin Institute, Vol. 280, NO. 2.

Kullback, 8., Information Theory and Statistics, Dover,
New York, 1968.

Lehmann, E.L., Testing Statistical Hypotheses, Wiley,
New York, 1959.

Noether, G.E., Elements Of Nonparametric Statistics,
Wiley, New York, 1967.

 

Parzen, E., Modern Probability Theory and Its Applications,
Wiley, New York, 1960.

 

Rappaport, 8.8. and Kurz, L., ”An Optimal Nonlinear
Detector for Digital Data Transmission Through Non-
Gaussian Channels," IEEE Trans. on Communications
Technology, Vol. COM-l4, NO. 3.

Richard, R.H. and Gore, W3C., '% Nonlinear Filter for
Non-Gaussian Interference," IEEE Transactions on
Communication Systems, Vol. CS-ll, No. 4.

Root, W.L., "The Detection of Signals in Gaussian Noise,”
NASA, NSG-2-59, September 1965.

[S 11

[T1]

[v1]

[W1]

 

77

Siegel, S., Nonparametric Statistics, McGraw-Hill,
New York, 1956.

 

Thomas, J.B., "Nonparametric Detection," Proceedings
of the IEEE, Vol. 58, No. 5.

Van Trees, H.L., Detection, Estimation, and Modulation
Theory, Part I, Wiley, New York, 1968.

 

 

Wilks, S.S., Mathematical Statistics, Wiley, New York,
1962.

 

 

 

APPENDICES

 

 

 

...

 

APPENDIX I

DERIVATION OF J(e:2) VS. 9

The noise amplitude model can be written in the form

9
fY(y) = a e-b‘y‘ .

-m < y < +m, where the relationship between
the parameters a, b and 9 can be derived from the following

equations:

JV fY(y)dy = 1 and (A1)
°° 2 2
j y fY(y)dy = o . (A2)

This form of noise was chosen because it realistically
represents the amplitude distribution of a noise source consisting
of an additive combination of Gaussian and impulse noise.

By the symmetry Of fY(y), equations (Al) and (A2) can

be written

00 — l- '
g fY(y)dy — 2 and (A 1)
co 2 22‘
= l
g y fY(y)dy 2 (A2)
In equations (Al)' and (A2)' make the substitution u = bye,
1
co 213-1 _u be
gu e du = L23 . (A3)

The left side is recognized as F(%). Therefore,

78

 

79

.l
9
b
1‘03“) = ga— . (A4)
Equation (A2)' results in
3
3 _
co —-1 9 2
g u9 e udu = P_§§QL. . (A5)
Therefore,
2
9 2
3 b
PCS") = 9__2_a_g_ . (A6)

Eliminating the variable a from equations (A4) and (A6) yields

3 Q
I‘(—) 2
b= —9— . (A7)
2Hi)
° 9
The J-divergence can be used in Chapter I as a measure of
discriminability between densities in the family a exp(- b‘x\e),
with e 6 [1,2) and a standard normal density (9 = 2) [K3]. Let
fe(x) represent the PDF Of any member Of the family, and f2(x)

represent the PDF of a standard normal density. By definition

f (X)

J(e:2) = Loam) - f2<x>>log gig; dx (A8)

2
J(e:2) jfe(x)log a /2F exp(§—-- b‘x‘e)dx — ff2(x)log a(/ZH

2
exp (%—'- b‘x‘9)dx

2
log a f2? [femax - log a f2? jf2(x)dx + “22:. - b\x\9)fe(x)dx

2
- f8;— - le\e>f2(x>dx

by ‘x‘e(f2(x) - fe(x))dx + % [J‘xzfemmx - jx2f2(x)dx] .

 

80

The latter term is zero because of the assumed equality

of the second moment (unity) for all members 9 E [1,2].

J(e:2) = by \x\9(£2(x) - fe(x))dx . (A9)

By symmetry Of the PDF's, this is written
2
2b °° ”)2;- ° b 9
J(e:2) =fi-g x9e dx - 2ab£ x9e X dx . '

By changing variables, this may be reduced to

 

‘ a, 9;; a, i
J(e:2) =}—%;£(2y> 2 e'ydy - ga- (fie e'ydy
a1 1
= 2 2 9:}. “2.9. 19 1
/fi b 2 H 2) e (b) me + 1) mm

b —[F—£3—/9)-] 9/2 and using

Substituting from equation (A7) for — F(l/e)

 

_2% = —li—- from equation (A4) yields
6. P(69
9b
9
3 " .1.
_1 2mg) 2 fl “9“ 1)
J(9:2) -Jfi? '—-—I- F( 2 ) " 1
“5) Hg)
2,
3 2
2N“)
1 8 9+1 1
=.__ -——— _-— All
fr? Hg) H2) 9 ( >

The value Of e that maximizes (All) indexes the member

of the family (1), Chapter I, which most differs from a standard

normal, and which can be expected to produce the largest central

limit theorem error.

81

The graph of J(1:2) vs. 9 is shown in Figure Al, and
the maximum divergence occurs when 9 = l, which indexes the

Laplacian distribution.

Jx(e:2)

.14 -b

 

 

 

 

FIGURE A.1 Divergence vs. 9

APPENDIX II

REALIZATION OF THE NLF

The NLF of Section 3.2 was implemented on a digital
computer using well-known techniques of piece-wise linear
approximation. The Odd symmetry property of the transfer char-
acteristic considerably simplified the implementation. In 10
uniformly spaced intervals [See Figure A2] extending from x = O
to x = 3, the equation Of the chord connecting adjacent ordinate
values was determined by the computer. This required the reading
in of ordinate values at the 10 abscissa points. An asymptotic
expression for values Of x > 3 was based on a linear approxima-
tion using the slope Of the chord between x = 3 and x = 4.
Because Of the concavity Of the transfer characteristic, g(x),
this approximation tended to bias the NLF-detector as discussed
in Section 4.4. A flow-chart is shown in Figure A2.

Let Fn(X) be the piece-wise linear approximation to the
continuous Odd symmetric curve g(x) derived in Section 3.2.
Define
NTC: Number of piece-wise linear segments (10)

TC(I): I-th positive ordinate Of the curve g(x), I = 1,2,...,NTC.

RANGE: Largest abscissa value (3)

APSL Slope and intercept of approximating straight line for
APIN abscissa values greater than RANGE.
AINC: Abscissa increment (.3)

82

 

 

TC (NTC) bow-oco.oooo 0.00000. concoooon.

TC (NE ‘1.) a... Outta----...ooo-uc

TC(3)
APIN

O
O
O
TC(2) 2
5 2
. O
, o
' :
TC(I) : :
o z 0
o , o
O . o
: : :
: . : All
1 2 3 4‘{ RANGE
Figure A.2 Piece-Wise Approximation

The numbers in parenthesis are the values used during the simulation

83

slope = APSL

FN(X)

 

A,

b 00.3.09...

 

boa-.--.-.ooooooo -.-.-.O---..-...OO

 

..........-.....--.. ...-.....Q......

L------.----.------..-------..-... .. .....

 

 

  

(See Section 4.4).

A flow chart is shown in Figure A3.

  
     
  
    
  

 

84

 

 

 

 

 

 

 

_ I'
IC=0 z<=\X\ ]
I

 

 

 

[ C = C +-AINC

(A.

\XZC/ ' 1

[AA = APSL*X+APIN I

 

 

 

, I A
[LOC=JJ

OC-l ]

AA = TC(LOC-l) +

 

 

 

(x -C+A INC)
AINC [TC(LOC) —
TC (LOC - 1) ]

 

 

 

 

 

AA =
x*TC(l)/AINC

L

 

 

 

 

L FN=AA3kSI J

Figure A.3 Piece-Wise Approximation Flow Chart

APPENDIX III

DISCONTINUITY OF g"(x) AT THE ORIGIN

The function g(x) is shown in Chapter III, Figure 3.2.
That the second derivative Of g(x), evaluated at the origin is

discontinuous is shown here.

Lemma: Let g(x) be an absolutely continuous function satisfying

g(x) = -g(-x). If g(x) is convex for X > 0, then g(-x) is

 

concave on x < 0. That is, if g"(x) s O for x > 0, then

g"(-X) 2 O for x < 0.
Proof: g(-X) = '8(X)
g'(-X) = + g'(X)

g"(-X) = -g"(X)

Then g”(-x) = -(g"(x)) and g"(x) is s O by hypothesis.
Then g”(-x) 2 O. Q.E.D.
Corollary: All even derivatives, defined and continuous at x = 0,

must vanish at x = 0.

Equation (20) of Chapter III gives an explicit expression

for g"(x), namely

— 2 (g(x>g'<x> - 03b), x > 0.

Thus,

g”(0+) = -b g'(0)-

85

86

The term g'(0) is easily found from (18) to be

g'(0) =

NH3
0‘

Therefore

2

g"(0+) = -b # O

N]:

However, by the above lemma

g"(0-) = b2 121 ’

The second derivative approaches different limits at
x = 0 from the left and right. Therefore, the second derivative

is discontinuous at the origin.

 

87

 

ooo.H moo. mmmm. mo. mom. NH. 0mm. mua. Nmm. «we. mom. woa. m.a+

000. 00H. 0000. 000. 0500. 0H. 000. s0H. ~00. 00H. mam. 0HH. 0.0-
qwm. 00H. 0500. 050H. 05. 0H. «00. 000. 500. 000. «00. NHH. H.0-
000. NHH. «0. 0H. mem. 0500. 055. 00H. 000. 05H. «05. qu. 0.0-
0qm. 500. 0500. 0H. 000. H. 0H0. 05H. 500. 000. 05s. 00H. 0.5H-
00H. 000. 05HH. H. 0H. qu. 05H. 00H. 500. 00H. 0HH. soH. 0.0q-
050. qu. 0550. HH. 00. 050. 000. 500. 000. «00. 000. 000. m.H+
5H0. 00H. 0550. mme. 005. 000. 0N0. NmH. N00. 00H. 505. 000. 0.0-
«05. 000. 05. HH. mNHm. 0000. 500. 05H. 005. NHH. «00. 000. H.0-
000. 00H. 05¢. 0HH. 0500. H. 0N0. 000. 500. 05H. N50. 05H. 0.0-
qom. «NH. 0505. 00. 00H. 050. 050. qu. 0cm. «00. ~05. 00H. «.mH-
qu. 05H. 0500. H. m50H. 000. 050. 000. qu. s0H. 000. 50H. .0q-
000. 05H. 00. 000. 0550. 050. 500. 0HH. s00. NHH. 0q5. 00H. m.H+
0H5. qu. 0505. 0500. 0500. HH. 005. 000. 500. 000. 500. N00. 0.0-
000. 00H. 0500. NH. mm. 00H. «00. 0NH. 0N0. 000. 000. N00. H.0-
0Ns. 0sH. 00. NH. «0. H. 50¢. N50. 000. 00H. 0mm. 00H. 0.0-
005. qu. 055. 0500. m55H. 00. 0mm. qwo. 505. 000. «05. 000. «.5H-
00H. qu. mNHH. 0500. H. 050H. 000. 05H. 000. 000. 50H. ~00. .0q-

00 m<m om m<m 0m m<m 00 m<0 0m 00m 0m m<m mZm

HE ”255 on0 RE max 203
cmHmwsmo cmfiomamma

Nome cofiumHDEHm

>H NHQmem<

88

000.H mNH.
000.H OOH.
0mm. NHH.
on. on.
mos. ©MH.
ooH. owo.
000.H ¢0H.
wwm. wwo.
qqm. NHH.
Nmn. NHH.
oqq. Nmo.
ONH. 0HH.

o.H
o.H

mom.
qow.
qwm.
omH.

o.H

mnwm.
«a.
du.
mmm.
mOH.

 

wOH.
wwo.
wwo.
qu.
¢OH.
NH.
H.
mmo.
mOH.
oo.
mmOH.
mNHH.

0mm.
mm.
qu.
moo.
mom.
qwo.
mmwa.
mnmo.
mom.
0.
mm.
mmao.

NHH.
one.
Nmo.
.H.
wmo.
woo.
mmHH.
mmoH.
mmNH.
mNHH.
mmwo.
mo.

000.H one.
000.H Nmo.
who. qNH.
omw. qNH.
q:m. moo.
NMH. moH.
000.H moo.
Nam. one.
0mm. 0HH.
Nun. omo.
moq. omo.
qNH. moo.

000.H N50.
000. NHH.
000. qu.
0H0. 00H.
H00. 000.
00H. 000.
000.H 000.
000. H00.
500. «00.
005. 500.
N50. 0HH.
00H. s00.

000.H mOH.
000.H one.
Nmm. 0HH.
qow. owo.
son. one.
wNH. «we.
000.H moo.
qwa. mwo.
on. «NH.
NHN. mOH.
Nos. ¢OH.
who. wwo.

m.H+
0.0:
H.m-
0.0:
«.mH-
o.oqu
m.H+
0.0:
H.mn
c.0-
d.NHn
o.o¢n

om

oH

 

 

  

 

,.-:

 

 

 

  

i. .... .... :55...

 

t ...ie 5:... , ..
- ...-.....anuE-i...

, ,. . . , g _ $353.1»...

 

 

 
 

 

\
- :35 ..
EHJSL .5”

:_ s
as?