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NUSC Technical Report 81331 October 1987 UPIC FILE COP.)

Detection Performance ofNormalizer for Multiple SignalsSubject to Partially CorrelatedFading With Chi-Squared Statistics

00 Albert H. Nuttall

00 Surface ASW Directorate DTICI" ELECTE f

I DEC 15 1987

Naval Underwater Systems CenterNewport, Rhode Island / New London, Connecticut

Approved for public release; distribution is unlimited.

"l 12 9 024

TR 8133

Preface

This research was conducted under NUSC Project No. A75205, SubprojectNo. ZROO00101, "Applications of Statistical Communication Theory to AcousticSignal Processing," Principal Investigator, Dr. Albert H. Nuttall (Code 304).This technical report was prepared with funds provided by the NUSC In-HouseIndependent Research and Independent Exploratory Development Program,sponsored by the Chief of Naval Research.

The Technical Reviewer for this report was Dr. G. C. Carter (Code 3314).

Reviewed and Approved:

/ W. A.Von WinkleAssociate Technical Director for Research and Technology

UNCL ASSI FI EDSECURITY C.ASSIFCTION OF THIS PA'4E

REPORT DOCUMENTATION PAGEl. REPORT SECURITY CLASSIFICATION lb. RESTRICTIVE MARKINGS

UNCLASSIFIED2a. SECURITY CLASSIFICATION AUTHORITY 3 DISTRIBUTION/AVAILA8ILITY OF REPORT

2 Approved for public release;b. DECLASSIFIC.ATION/}WNGROIN SCEDUl,,E distribution is unlimited.

4. PERFORMING ORGANIZATION REPORT NUMBER(S) S. MONITORING ORGANIZATION REPORT NUMBER(S)

TR 8133

64. NAME OF PERFORMING ORGANIZATION 6b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATIONNaval Underwater Ofjplcabe)Systems Center Code 304

6r ADDRESS (Cty, State. .W ZIPC¢). 7b. ADRESS (Code, St)r, d ZP odo)New London LaboratoryNew London, CT 06320

Sa. NAME OF FUNDING I SPONSORING b. OFFICI SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBERORGANIZATION O f 41110k ,k

Sc ADDRESS (City, State, and ZIP Code) 10. SOURCE OF FUNDING NUMBERS

PROGRAM PROJECT TASK IWORK UNITELEMENT NO. NO. NO. jACCESSION NO.

11, TITLE (Include Secur"t Cawflcation)DETECTION PERFORMANCE OF NORMALIZER FOR MULTIPLE SIGNALS SUBJECT TO PARTIALLY CORRELATEDFADING WITH CHI-SQUARED STATISTICS

12. PERSONAL. AUTHOR(S)Albert H. Nuttall

13a. TYPE OF REPORT 13b. TiME COVERED 14. DATE OF REPORT (Year, Mornt, Oay) S PAGE COUNTFROM _____TO ____1987 October 1

!6 SUPPLEMENTARY NOTATION

17 COSATI COOES 18. SUBJECT TERMS (Continue on reverse if necen/ay and identify by block number)FIELD GROUP SUB-GROUP Normalizer Correlated Fading

False Alarm Probability Chi-squared FadingDetection Probability Multiple Pulses

19 ABSTRACT (Continue on reverse if nece eay and identify by block number)

The falke 31arm and detection probabilities for a multi-pulse signal subject topartially correlated fading, in the presence of Gaussian noise of unknown level, arederived in closed form. The number K of signal pulses, as well as the number L ofnoise-only pulses used to estimate the noise background power level, are arbitrary.The power-fading is characterized by a chi-squared distrji _ on with 2m degrees offreedom and a normalized set of covariance coefficient , all of which can beselected arbitrarily, in order to match an experimental realization or an actualmeasured situation. The performance capability of this processor depends additionallyon the received signal-to-noise rat-o.

This study is an extension of NUSC Technical Report 7707, to cover the case ofa nonconstant threshold. Comparisons of this normalizer with the earlier results(for L =-) enable a quantitative evaluation of the losses incurred by lack of -- 6

20 OiSTRiBUTiON/AVAILABILiTY OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATIONK3UNCLASSIF-EDl4NLIMITED E SAME AS RPT . OTIC USERS IUNCLASSI FI ED

22a NAME OF RESPONS;ILE ;NDIVIOUAL 22b. TELEPHONE(incI Area Coo*) 22c. OFFiCE SYMBOLAlbert H. 'Nuttall (203) 440-4618 Code 304

00 FORM 1473,84 MAR 83 APR edition may be uSed until exhausted SECURITY CLASSIFICATION OF 11IS PAGEAll other editions are oibsolete

UNCLASSI FI ED

l.CL ASS I FI EDsEcumirv CKA"AS*iCAose Or rws PAGE

18. SUBJECT TERMS (Cont'd.)

Unknown Noise LevelConstant False Alarm RateOperatin g Characteristics

19. ABSTRACT (Cont'd.)

knowledge of the noise level. The important capability of constant falsealarm rate is achieved by this normalizer. ,

Plots of the detection probability vs false alarm probability arefurnished for a variety of typical choices of the various parameters;however, the multitude of parameters and cases precludes a comprehensiveall-encompassing compilation of numerical results. Accordingly, a generalprogram in BASIC is listed, whereby additional results of interest to aparticular user can be easily obtaii~ed, once numerical values are assignedto all the parameters.

'7!

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

• .. ..... ..... .. .. .... ......."- ::

A i

UNCLASSIFIEDSECUITY CL.A&SIVICAtOw Or rui PAGE

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TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS .. ....... ............. ..... ii

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

LIST OF SYMBOLS................... .. ........... .. .. . . ....

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

PROBLEM DEFINITION. ....... ............. ....... 2

NORMALIZER PROBABILITIES .. ............. ........... 4

Definitions of Parameters. .. ............. ........ 4

Probabilities for Known Noise Level .. ......... ........ 5

Normalizer Ratio. ........ ............. ..... 6

Normalizer Distributions ... ............. ........ 8

Comparison With Earlier Results. .. ............. .... 11

Special Cases. .. ............. .............. 12

Recursion for Cumulative Distribution Function. ............ 13

Detection and False Alarm Probabilities. .. ............. 15

GRAPHICAL RESULTS. .. ............ ............... 16

SUMMARY. .. ............ ............. ...... 20

APPENDIX A - PROGRAM LISTINGS. .. ............. ....... A-1

REEERFNCES. ....... ............. ........... R-1

* TR 8133

LIST OF ILLUSTRATIONS

Figure Page

1 Time-Frequency Occupancy Diagram ..................... 2

2 ROC for K = 1, m = 1, L = *. .... ................ ..... 21

3 ROC for K = 1, m = 1 , L = 32 .... ................ .... 22

4 ROC for K = 1, m = 1 , L = 16 ... ................ ..... 23

5 ROC for K = 1, m = .5, L =00. ........ . .. . . . ............ 24

6 ROC for K = 1, m = .5, L = 32 ... ............... .... 25

7 ROC for K = 1, m = .5, L = 16 ... ............... .... 26

8 ROC for K = 2, m = 1, p = O, L = 0.. ............. ..... 27

9 ROC for K = 2, m = 1, p = 0, L = 32 .. ............ .. 28

10 ROC for K = 2, m = 1, p = 0, L = 16 .. ............ .. 29

11 ROC for K = 2, m = 1, p = .5, L= *. ......... . .......... 30

12 ROC for K = 2, m = 1, p = .5, L = 32 .. ............ .... 31

13 ROC for K = 2, m = 1, p = .5, L = 16 .. ............ .... 32

14 ROC for K = 2, m = .5, p = 0, 1 = 0. .............. .. 33

15 ROC for K = 2, m = .5, p = 0, L = 32 . ............ 34

16 ROC for K = 2, m = .5, p = 0, L = 16 .. ............ .... 35

17 ROC for K = 2, m = .5, p = .5, L =a. .............. ..... 36

18 ROC for K = 2, m = .5, p = .5, L = 32 ................ 37

19 ROC for K = 2, m = .5, p = .5, L = 16 ................ 38

20 ROC for K = 4, m = 1, p = 0, L =a. ................. 39

21 ROC for K = 4, m = 1, p = 0, L = 32 .. ............ .. 40

22 ROC for K = 4, m = 1, p = 0, L = 16 .. ............ .. 41

23 ROC for K = 4, m = 1, p = .5, L = oo.................. 42

24 ROC for K = 4, m = 1, p = .5, L = 32 .. ............ .... 43

25 ROC for K = 4, m = 1, p = .5, L = 16 .. ............ .... 44

26 ROC for K = 4, m = .5 p = 0, L = . .... ............ 45

27 ROC for K = 4, m = .5, p = 0, L = 32 .. ............ .... 46

28 ROC for K = 4, m = .5, p = 0, L = 16 .. ............ .... 47

29 ROC for K = 4, m = .5, p = .5, L =. .... . . ......... 48

30 ROC for K = 4, m = .5, p = .5, L = 32 ................ 49

31 ROC for K 4, m = .5, p = .5, L = 16 ................ 50

3? SNR for P = .9, K = 1, m = 1 ... ............... ..... 51

33 SNR for PD .9, K = 2, m = 1, p = .5 ...... . . . ....... 51

N0_ii

TR 8133

LIST OF TABLES

Table Page

1 Fundamental Parameters ......... .................. 4

2 Auxiliary Parameters .......... ................... 5

3 Identification of Variables ... ............... .I..11

LIST OF SYMBOLS

K number of signal pulses added, figure 1

L number of noise-only pulses, figure 1

m fading parameter, table 1

#kj normalized covariance coefficient, table 1

ROC Receiver Operating Characteristic

SNR Signal-to-Noise Ratio

FF1 Fast Fourier Transform

Fl average received signal energy per pulse

No single-sided noise spectral density level (watts/Hz)

Ke equivalent number of samples, table 2

N summary parameter, table 2

y sum of K signal squared-envelope samples

2 noise powern

4 a,b auxiliary constants, (2)

R scaled signal-to-noise ratio, (2)

Qy exceedance distribution function of y

A scaled threshold, (3)

Fn(x) exceedance function, (5)

iii

TR 8133

LIST OF SYMBOLS (Cont'd)

en(x) partial exponential, (6)

'yo sum of L noise-only squared-envelope samples

normalizer ratio output, (9)

alternative normalizer ratio, (10)

u threshold, (11)

PU cumulative distribution function of v

f(T) characteristic function, (12)

p(u) probability density function, (13)

H1k(x) hypergeometric function, (25)

PD detection probability, (30)

PF false alarm probability, (31)

P exponential correlation coefficient, (32)

iv

TR 8133

DETECTION PERFORMANCE OF NORMALIZER FOR MULTIPLE SIGNALS SUBJECT

TO PARTIALLY CORRELATED FADING WITH CHI-SQUARED STATISTICS

INTRODUCTION

In a recent study [1], the detection performance capability of a

multiple-pulse system subject to correlated fading was quantitatively

delineated. It was assumed there that the noise level was known, so that a

threshold could be set for an arbitrarily specified false alarm

probability. Then the detection probability was evaluated as a function of

the threshold level, the received signal-to-noise ratio, the number K of

signal pulses, and the fading statistics.

Here we will extend these earlier results to cover the case where,

additionally, the noise level is unknown and must be estimated on the basis

of a finite number L of noise-only samples. The same approximation

technique that was presented in [1] is used to determine the detection

probability of this normalizer system. The reader is referred to [1] for

additional background, motivation, interpretations, and related references.

For the sake of brevity, we will employ the same notation and presume that

the reader has complete familiarity with the earlier material and

development.

TR 8133

PROBLEM DEFINITION

We will couch the problem in a particular setting, one with obvious

appeal and application; however, it should be obvious how to extend this

setting to a more general one, particularly in light of the arbitrary fading

covariance coefficients that are allowed in the analysis.

Suppose a sequence of K tone bursts at a common center frequency are

transmitted, as depicted in figure 1. Each rectangular slot symbolizes

_slots

_ _

Figure 1. Time-Frequency Occupancy Diagram

a tone of duration T1 seconds and approximate frequency bandwidth l/T1

*j Hz. These bursts may be abutting in time or may be arbitrarily separated in

time by several multiples of T At the receiver, K narrowband filters of

bandwidth I/T Hz are sampled at the times of peak signal output (if

present) and their squared envelopes are summed. Depending on the time

separation between pulses, the signal strength may fade considerably; the

2

TR 8133

exact amount and frequency of the fading depends on the distribution of the

fading and the covariance of the fading amplitude of adjacent (as well as

separated) pulses.

It is presumed that, during a single tone of duration Tl seconds, the

fading is essentially constant, resulting in a constant amplitude scaling

and phase shift applied to the pulse. The time separations between pulses

in figure 1 are arbitrary, thereby allowing for an arbitrary degree of

correlation between the fading factors applied to each pulse.

To establish a reference against which this sum of K matched filter

outputs can be compared, for purposes of deciding on the presence or absence

of signal, a group of L nonoverlapping noise-only slots, located arbitrarily

in the time-frequency plane, are also energy-detected and summed. For very

large I, this noise reference is very stable, and performance approaches

that predicted by [1]. However, for moderate values of L and for small

false alarm probabilities of interest, it is important to know how much

degradation in performance is incurred by being forced to use this noisy

* reference.

An obvious implementation of the processing implied by figure 1 is to

employ fast Fourier transforms. The L reference bins can then be an

arbitrary collection of time and/or frequency bins. However, L cannot be so

large that nonstationary and/or nonwhite noises cause their own kind of

errors in noise power estimation. The tradeoff between these conflicting

requirements will be assessed quantitatively in this investigation.

I3

TR 8133

NORMALIZER PROBABILITIES

DEFINITIONS OF PARAMETERS

Very heavy reliance will be made here on the basis that was set up in

[1]. Thus we have the following fundamental parameters of the detection

procedure (the immediate references in tables 1 and 2 are to [1]):

K, number of potential-signal pulses added, (figure 1 and A-ll);

m, signal fading parameter (power-scaling is chi-squared with 2m

p' degrees of freedom), (13);

{Pkjl' normalized covariance coefficients of signal power-scalings

I (15);

average received signal energy per pulse , (9);N, single-sided received noise spectral density level

L, number of noise-only pulses added.

Table 1. Fundamental Parameters

'44

4 TR 8133

In addition, there are two very useful auxiliary parameters that found

frequent use in [1]:

Ke , K/ kj = equivalent number of independent signal pulses, (10);/kj~l

N, m Ke = a summary parameter describing the distribution of

*the sum of power scalings,(A-24) and (A-29).

Table 2. Auxiliary Parameters

None of the parameters, m, Kel N, need be integer. Also, N can be larger

or smaller than K, the number of signal pulses.

PROBABIITILES FOR KNOWN NOISE LEVEL

The probability density function of the sum y [1; (A-ll)] of the K

signal envelope-squared samples is given by [1; (B-4)]

p (u) = exp(-u/a) u K - ; K; for u> (aKN bN(K) 1 1 (a b))

where [1; (A-32),(B-3),(B-7)]

a = 2y 2, b =2a 2 (1 + R) ,l R (2)n n N N(20

The exceedance distribution function Q (u) of output sum y is given byY

several alternative forms in [1; (B-9),(B-11),(B-13)]. For a fixed

threshold (known noise level), the detection probability is

0 ' IINI ' _ ' . . .".. . . ... + . . . .

TR 8133

P D (-A,R,N,K) =

=1- 1 0 (N)n( R )n -u (3)

(I + R) N n " +R K I. 2 2n

and the false alarm probability is [1; (B-10)]

P F = EK- (.A) ,(4)

where we define the exceedance distribution function

E (x) = exp(-x) e (x) (5)n n '

and

n

e n(x) 2 ~Xi/j! (6)

j=0

is the partial exponential [2; 6.5.11]. The results in (3) and (4) should

be used for L =ob, that is, for known noise level.

NORMAl IZER RATIO

. From this point on, L is presumed finite. Suppose a noise level

estimate, y, is obtained, based upon L independent measurements of

noise-only bins. It is assumed that the average noise level in these L bins

is the same as in the K potential-signal bins, but that this noise level is

unknown. If we let

6

IR 8133

y = y(K,E 1 N) (7)

denote the sum of K signal bin outputs with average signal-to-noise ratio

E 1/N then

Yo = y(L,O) (8)

is the corresponding sum of L noise-only bins. Now define ratio

I- y(KE 1 0)(9)YO y(L,O)

for sets of K and L pulses, respectively. The noise contributions to the

total of K + L outputs are presumed independent of each other; however, the

signal fading factors amongst the K signal outputs are correlated to an

arbitrary degree. We are interested in the distribution of this normalizer

ratio, v.

When signal is absent, the ratio v in (9) is independent of the absolute

level of the received noise; therefore, we can expect the normalizer to

achieve the important capability of constant false alarm rate. That means

that a specified false alarm probability can be achieved without knowledge

of the average noise power level.

The quantities y and y0 are the sums of K and L squared-envelope

samples, respectively, and are not the averages of these sampled

quantities. In terms of the sample-average quantities, we could define a

slightly different normalizer ratio

6

TR 8133

._ y/K Ly0/L K (10)

It then readily follows that the cumulative distribution function of random

variable ; is

NKP_.(u) = Prob(u < u) = Prob < L u (11)

in terms of the cumulative distribution function of ratio u in (9). Thus, a

simple scale factor change allows for consideration of the alternative ratio

When we plot the detection probability versus the false alarm

probability, that is, eliminate the threshold, the same performance

characteristics result for random variable v as for '. Accordingly, we will

not use or refer to ' or P_(u) any further, but concentrate solely onU

normalizer ratio v, given by (9).

NORMALIZER -DISTRIBUTIONS

The characteristic function of noise-only random variable y can be

found directly from [I; (A-13)] by setting A to zero and replacing K by L:

f if a) , a 2 2n (12)YO n

8o I -i a -

0' ~ ' - .. r I I

TR 8133

The corresponding probability density function of yo is

u L-1p (u) _ u exp(-u/a) for u > 0 (13)

r(L) aL

The exceedance distribution function is

Qyo(u) = Prob(y > u) = EL-l(u/a) for u > 0 , (14)

in terms of the functions defined in (5) and (6).

The cumulative distribution function of ratio v in (9) is given by

(since y > 0)

P (u) = Prob( < u) = Prob(YI < u)= Prob(y < UYo

0

= f dy p y(y) dx p YO(x) = dy py(y) Qyo (Y/U) =

0 y/u 0

dy exp(-y/a) v K-dN bN (N; K; y E (15)0 a - b NF(K)

, for threshold u > 0, where we used (1) and (14). We now expand EL-1

* according to (5) and (6) and integrate term-by-term to obtain [3; 7.621 4]

(U) = (b +__(_+ub)(K_ F(N K '-;K; -u) (16)'P,

But from (2),

I -9

TR 8133

a 1 Kb - 1 + R ' R N N (17)

0

where the parameters involved are described in tables 1 and 2. Making these

substitutions in (16), thexc follows for the cumulative distribution function

of random variable u,

P (u) 1K. (K K ;u . (18)( + R ) N 6 + ( 1 + u 1 + R 1+ UR 0=0 (l u

An alternative more useful form is obtained when we use [2; 15.3.3]:

u K( 1 + u N (K) 1 + R ,KN ;1+R 1 uP()=( )1 R N -6R FA K N; K;

P(u) + u u + R 1

(19)

for u > 0. This result is very attractive since the negative integer

argument, -,, in the hypergeometric function causes termination of the

series at A terms. Thus, (19) is a closed form (albeit tedious) for the

cumulative distribution function of v, involving a finite number of

elementary functions.

It should be noticed that the absolute noise level a2 does noti na appear in (18) or (19). (The cumulative distribution function for

alternative normalizer ratio " given by (10) can now easily be found by use

a, of (11).)

,10

W% P!

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COMPARISON WITH EARLIER RESULTS

The result (19) for the cumulative distribution function of normalizer

ratio v, operating in a partially correlated fading environment, is an

approximation, having been based upon a characteristic function fitting

procedure explained in [1; (A-24)-(A-28)]. Nevertheless, (19) is identical

with the exact fading result for a related normalizer problem; namely,

agreement with [4; (25)] is achieved under the following identifications:

TR 4783 Here Interpretation

c u threshold

K number of signal pulses

N L number of noise-only pulses

+ + 1 N m Ke, table 2

I KN N

0

Table 3. Identification of Variables

lhe identity of + 1 with N is made by comparing [4; (24A)] with

[1; 'A-29)]. The final identity of ) with R utilizes [4; (24B)] and

[1; (9)]:

SR ET/N E K/NT T - 1 R (20)'u+1 N N

where the arrow indicates transferrance from [4] to [I].

@411

TR 8133

The approach in [4] proceeded as follows: the detection probability

for nonfading signals in all the bins depended only on the total received

signal-to-noise ratio RT. When RT was assigned the fading probability

density function [4; (24A)], the average detection probability in [4; (25)]

resulted. For the special case of fading parameter u = M - 1 there,

'numerous graphical results were given in [4; figures 1-36].

The current results here are more general, in that they allow for

partially correlated fading (through parameter K ) and a more generale

power-fading model (with 2m degrees of freedom). This means that N = m K

-. here is not restricted to be equal to the number of signal pulses, K, but is

arbitrary. Thus the current numerical results will significantly augment

and extend those in [4]. If N = K here, then R = EI/N ° = signal-to-noise

ratio per pulse, and (19) reduces to [4; (15B)], for which many numerical

results were given in [4; figures 1-36].

SPECIAL CASES

For m = 1, which corresponds to Rayleigh amplitude fading, and for

Pkj 'which corresponds to uncorrelated fading, then K e = K, N = K, and

we get from (19),

P (u) = u .R) , (21),=0

in agreement with [4; (15B)].

On the other hand, if R 0, then (18) and (19) both reduce to

a,

TR 8133

L-1

~(o) (u )K ~ (K~t ,(2

S)! (1 +(22)

which is equal to 1 - PF' where PF is the false alarm probability.2.

Since noise level is not involved in (22), threshold u can ben

selected to realize a given PF' once K and L have been specified. This is

a quantitative verification of the expected constant false alarm rate

property of the normalizer.

Finally, in the special case of one signal pulse, K = 1, and Rayleigh

amplitude fading, In = 1, then Ke 1, N = 1, R = E1/N° , and (19) yields

P (u) - 1 +- - (- 1. +; R). (23)Pu ) -I+ u + R + u + R T+ u + R

That is,

I - PU(u) + R (24)

which agrees with [5; (6)] when we make the identifications (from there to

here) of N - L, T/N 4 u, y - R.

RECURSION FOR CUMULATIVE DISTRIBUTION FUNCTION

let the hypergeometric function appearing in (19) be represented as

follows:

(K)

1(xF(-, K - N; K; x) (25)

13

a. '.,'e Y 1 ' ' * ' ..

TR 8133

Then

Ho(x) (26)

while (25) has the recursion [2; 15.2.10]

,XH(x) = [K + 2A1- 2 +(N - K + 1 -. f)x] H _(x) - (K +A - 2)(1 - x)H_ 2 (x)

for 2> 1 , (27)

where we-define H-1 (x) = 0. In terms of (25), the cumulative distribution

function of v in (19) becomes

u 1K N L-1I + RRI

P (u) + u)K ( + u +R_ l +R R lR Ru) (28)0 + u + + u + R) + R I + u<' ,Q=O

This form, in conjunction with recursion (27), was used for all the

numerical results here, for L finite. The parameters appearing in (28) have

all been explained in tables 1 and 2. The explicit dependence on the

fundamental parameters is indicated below:

Ke Ke(K, fPkj),

N : N(m, K, TPkj),

R = R(E 1 /No, m, K, ?Pkj}) . (29)

In addition, the cumulative distribution function in (28) is a function of L

and threshold u.

14

TR 8133

DETECTION AND FALSE ALARM PROBABILITIES

The detection probability is given by

= Prob(u > ujR > 0) = 1 - PvtA), (30)

where P,(u) is available in (28). The false alarm probability is

PF = Prob(v > u R = 0) = 1 - P(°)(u), (31)

where P(o)(u) is available in (22). By allowing threshold u to vary

over a wide range, PD and P values can be obtained and plotted against

each other, resulting in the standard receiver operating characteristics;

the threshold is thereby eliminated from the plotted outputs. Programs for

plotting PD vs PF' both for L finite as well as infinite, are listed in

appendix A.

15

TR 8133

GRAPHICAL RESULTS

Due to the multitude of parameters appearing in this investigation (see

tables 1 and 2), it is impossible to give a comprehensive compilation of

encompassing numerical results. Considering just the covariance

coefficients [Pk 1 for the moment, complete specification requires

assignment of K(K - 1)/2 values to these quantities; to circumvent this

difficulty, we consider numerically, here, only the very special case of

exponential correlation, for which

1k-Pkj = P for 1 < k, j < K (32)

and look at a couple of particular values for p. Our approach here, of

necessity, is to give some representative sample receiver operating

characteristics and a general computer program in BASIC, whereby additional

results can easily by obtained once the user has specified all the

particular values of interest in his application. This program allows for

arbitrary covariance coefficients, [Pkjl' and is not limited to the

specific example (32).

The particular cases we will investigate are as follows:

K 1, 2, 4,

L = 16, 32, ao,

m = .5, 1,

p = 0, .5 (33)

16

TR 8133

All possible combinations of these four fundamental variables lead to 30

plots, which appear below in figures 2-31. (There are only 6 plots for

K = 1, not 12, because the value of p is irrelevant for K = 1). The curves

are indexed by the. per-pulse signal-to-noise ratio, E1/No, in dB. The

false alarm and detection probability pairs range from (poor quality) pair

5,.01) up to (high quality) pairs near (E-l0,.999).

The number of signal pulses, K, is limited to the low values 1, 2, 4,

because these seem to-be the cases of most immediate practical use. The

number of noise-only samples, L, is not evaluated for L = 64 because of the

proximity of the results to those for L =cc; conversely, results are not

presented for L = 8, because a severe degradation in performance occurs, that

probably cannot be tolerated. The fading parameter value m = 1 corresponds

to Rayleigh amplitude fading (exponential power fading), while m = .5

corresponds to a deeper more-damaging form of fadiny. The correlation

coefficient p = 0 corresponds to uncorrelated (independent) fading, while

p = .5 allows for adjacent (equispaced) pulses in figure 1 to have some

dpgree of dependent fading.

An explanation of the initial result in figure 2 follows: for K = 1,

m = 1, L =0 (known noise level), the detection probability is plotted versus

Pl the false alarm probability for values of the latter between IE-10 and .1.

Ihe value of the per-pulse signal-to-noise ratio, F/N in dB, varies over1 o

the range 6, 8, 10, .... 42, giving detection probability values covering

* All the figures are collected together after the Summary section.

17

TR 8133

the range .01 to .999. rhe only difference in the accompanying pair, figures

3 and 4, is that L is reduced to 32 and 16, respectively.

The results in figures 5 through 7 correspond to the worst casesconsidered here. Namely, there is just one (fading) signal pulse, and m is

.5, which means a very deep fading medium; see [1; figure 2]. The values of

signal-to-noise ratio required for L = 16 in figure 7 are so large as to be

physically unrealistic, except for the poorer quality region.

vi

On the other hand, for K = 4 signal pulses, Rayleigh amplitude fading

(m = 1), and uncorrelated fading (p = 0), the results in figures 20 through

22 are very encouraging, being physically reasonable over the whole range of

plotted values. But when m is decreased to .5, and p is increased to .5,

the results in figures 29 through 31, still for K = 4 pulses, indicate

substantially increased signal-to-noise ratio requirements at the higher

quality end of the performance region.

An alternative method of presenting the graphical results, which

accounts for the losses incurred by not knowing the noise level, is to plot

the required value of E I/N vs I., for various values of the remaining

parameters and for specified performance quality in terms of PF and PD'

0A Two such cases are illustrated in figures 32 and 33. They show that theKI..

cost of not knowing the noise level is not severe for the high false alarm

probabilities, but is quite significant for the lower more-desirable false

alarm probabilities. For example, in figure 33 for K = 2 signal pulses, the

signal-to-noise ratio must be about 1.5 dB larger at L 10 noise pulses than

1804

TR 8133

at L = 100, when P = 01. However, if we want to operate at PF = IE-10, the

increased signal-to-noise ratio requirement is about 6 dB per pulse. The

44. numbers are comparable for the K = 1 results in figure 32.

The asymptotes for large L in figures 32 and 33 can be found in some

cases from earlier results in (1]. For example, reference to [1; figure 8]

for K = 2, p = .5 gives EI/N o 16.8 dB, while PF = IE-6, PD = .9, m = 1.

Comparison with figure 33 here reveals that the performance requirement is

virtually at this level by the time that L = 100.

i1

TR 8133

SUMMARY

Although figures 32 and 33 are very informative, allowing for a ready

assessment of the losses incurred by using a finite small value for L, the

number of noise-only pulses, they also illustrate the voluminous compilation

that would be needed for a thorough numerical investigation. For example,

if: detection probabilities PD were of interest for values .5, .9, .99,

.999; number of signal pulses K for values 1, 2 , 10; fading parameter

m for values .5, 1, 2; and fading correlation coefficient p for 0, .5, 1;

,<. this would require a total of 4*10*3*3 = 360 figures. The approach here is

instead to present some representative receiver operating characteristics, in

figures 2 through 31, from which information similar to that in figures 32

and 33 can be extracted, and to list a general program for the generation of

additional receiver operating characteristics for whatever cases may be of

interest to the user.

Some related work on the performance of a log-normalizer subject to

. Weibull or log-normal inputs has been published by the author in [6];

however, no fading was allowed, and the number of signal pulses was limited

to K = 1. In a different vein, the performance of an or-ing device

operating on the output of an incoherent combiner of multiple pulses was

analy7ed in [7]. These works augment and complement the analysis conducted

. •here.

S20

04. w "

,, ,''";':*.- '-,.''., ''!.,w :. - = '

TR 8133

.999 :

.99

.995

c .99

... 95

0

443

-0 .0L .

.2 _

.01 __ I__ ____ _

E-IOE-9 E-8 E-7 E-6 £-5 E-4 E-3 .01.02 .05 .1

Probability of False Alarm

Figure 2. POC for K=I, rn=i, Lo

4 21

TR 8133

4.3

I.95

02

4Q)

.70.

.4)

~0

.02

'I0

E-IOE-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .1

Probability of False Alarm

Figure 3. ROC for K=I, m=1, L=32

42

TR 8133

.998

.98

- .5~0

.6

.5

0

.23

TR 8133

.998

.995 .....

.99

.98

.95

0

.4

p..2

.02

.01

44

TR 8133

.3

.995

.99

.95

0

4~24

.7 .0.............

.6

.4~

a-

00

.35

4.0

E-,0E9 E8 E7 -- 6 E5 E4 E3.1.2 .5 .

Fp I.2 . P:, F r

025

TR 8133

.999

.998

.995up

.995

.95

.9503

.7

0

'a"- -. 5

CL .

2.2

___ ~B)

".02

E-1OE-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .1

Probabi I ty of False AlIarm

F gu re 7 . IRCC t^c r K ,m= .5, L=1

TR 8133

.99

V.9

4P,

U

Q-)

.1.5

V.2

.05

.02

E- 1C0E-9 E-8 E-7 E-6 £-5 E-4 E-3 .01.C2 .05 1

Prcb 1 ity cf False P!armn

F- 9 ure B. M-CC for K=2, m=:,

27

TR 8133

.998

a;-...995

p.99

a.9

0

V 0

.1.,

4..5

(U .

0-6 /

1%1

p..05

U .02

E- 10E-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .

Probability of False Alarm

Figure 9. ROC for K=2, m=I, PO=O, L=32

TR 8133

.95

0

W4~

CU

.5

.4-

C- .35

.00

.0

E-IOE-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .1

Probabil ty of False P.1a rmr

Figure 10". RCC ;or K=2, m=1, p~C, L=16

02

TR 8133

.998

.1~4 .995

.. 99

.99

.98

.95

4-1

01~

4-4

.05

.02

.0L- O - - - - - - - 0 . 2 . 5 .

Probability (dB of Fas -Ra

F~~~~~~~~~ i I 1 O frK2 = =0

TR 8133

998

.995

.99

.98

.95

c .

4 J

4..4

4-)

0a- .

.2

.02

.01E- IC E-9 E-8 E-7 E-6 E-5 E-4 £-3 .101.02 .05 .1

Prcbability corF False R' rm

F ig9u re 12 . rROC F'c r K=2 T,m--, Fv.5, L=32

31

TR 8133

. 999

- .995

.99

.95

0

4J

5l- 10.7l, X C

-

No(dB)

.05

.02 _ _ _ _ _ _ _ _ _ _

E-1OE-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .1

Probability of False Alarm

Figure 13. ROC for K=2, m=1, P=.5, L=16

G.0111111

TR 8133

.99S

03

.69

.9

I; .7__ _ I _ __ 0 ___ _

a. .3 ur T~ C o'K2,r.,~,L

(d 33* S W' ~ *4 No

TR 8133

.999

.998

-V32

.995

V..9

0

4J

.8Q.)

J~.7

0X

.5)

-'.4

.0

IL

a. .2

* .05

0.0

E-IOE-9 E-0 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .1

*Probability of False Alarm

Figure 15. ROC for K=2, m=..5, p=O, L=32

11 pr d

TR 8133

.998

.995

C .9

0

.4j

.8)

.7

~0

1,00

0L

0..2

E-I10E-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .1

Probability of False Alarm

Figure 16. ROC for K=2, m=.5, p=O, L=16

@4 35

TR 8133

995

.99

.98

.95

c .

4J)

4- .7

0

.3

.2N

.05

.01

E-IOE-9 E-8 E-7 E-6 E-5 E-4 E-3 .01 .0? .05

Probability of False Alarm

Figure 17. ROC for K=2, m=.5, P=.5, L=04

-1 .'.

'A TR 8133

A-..999

.995

.99

A,... ...........

4.4J

I'..

A%.5

.. 4

*C .3

(d3B

.05

00

A.0.EAO."- - E6 E5 E4 E3 0 .2 .5 .

Prba ilt of Fl4 lr

Fiu e 1 .R C f r K= ,m . ,p . ,L 3

L3

ZIIII!.I

TR 8133

.998

.95

-~c .902

~20

.40

CL .

U.

X N

E-IOE-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .1

Probability of False Alarm

Figure 19. ROC for K=2, m=.5, p=.5, L=16

TR 8133

.. 99

.998

.91,

.05

C0..

Ad B

.. 02

E- 10E-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .

Probability of False Alarm

Figure 20. ROC for K=4, m1I, =O, L=0

39

TR 8133

.998

.99

c .9000

44 0

0 /0

0

.02

.05,

E-IOE-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .1

Probability of False Riarm

Figure 21. R~OC for K=4, m=1, p=O, L=32

MVW"a "ov" w--N....W.W .-. -- WTTWIWO.R

TR 8133

.999

4.99

.. 99

ZF/' 4l

.99

.95

.4-,,

MI0 .4-0

.4.2

.1.05

050

E-1CE-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .1

Probability of False Riarm

K,-ure 22. POC for K=4, m1I, P=O, L= 16

I4 v'

TR 8133

.999

.998

V.99

;I.95

c .9)

4- .5

0

-No

.0

.02

.01 l-L

E- 10E-9 E-8 E-? E-6 E-5 E-4 E-3 .01.0? .05 .1

Probability o; False R 1a rr

Figure 23. POC for K=4, m=nA, F=.5, L=co

V.'.

* TR 8133

.998 '

.995

.. 98

c .9

0

030

p..8

0 Z

>1 .2A4--

.05

.. 4

'A) __ __ ___ ___0

a-OE- .3 - - £5 E4 E3 .0.2 .5 .

Xrbbit - as ir.2 r 4 O 'r 4 1 .,L3

I( B3

TR 8133

.999

.9. .995

.99

.98

5,95

c .90

r

4-)

.8 lo8

-. 0

.5 AA IZ /7.r

rd-0 .4

0

.. 1 Z

a- .35

.2 .

S.0

555* .02

.01

E-10E-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .1

Probab~ 1 tx' of False P! arm

-~ Figure 25. ROC ;or K=4, m=!, p=. 5 , L=16

N

- - ~~ U- f W W U

TR 8133

* .999

.998

995

.98

c .9_ _

0

"5 .8

4- .

.. 5

-~-0

-5.3

N-o

.02

0.0

E1E9E8 E-7 E6 -5 -4 E-3 .01.02 .05 .1

Prob abIi ty o{ Fa Ise PC a r-

F-.gure 26. PC' Por K= , m=.5, p=O, L=00

TR 8133

.999

.998

.. 9

C-,-

.98

01L)

.6

.05

-0'.

.. 2

E- 10E-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.C2 .05 .1

Probability of False Alarm

Figure 27. POC ;or K=4, m=.5, f=O L=32

TR 8133

.995

.. 9

.95

0

Q3.8~

0

-. 5

~0

ot .3_

.2__

.05

.02

.01 _ _ _

E-IOE-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .1

7.,Probability o-f False Alarm

Figure 28. POC f'or K=4, m=.5, O =.

TR 8133

.999

.9958--

.99

.95

15*, 0

Uz

Q) 1

.5-.5

ra .4

.2.

.0

E- 10E-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .1

ProbabilIi ty a F a se Ha rm

Fiu re 2S. ROC 'or K=4, m=.5, p=. 5 , =

IA d.

v TR 8133

.99

.998

.99

C2

4........ ..

.95

c-. .9

5.5

5..4

0L

5-~a .3 _ _

05

.02 ___

E-I10E-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .1

Probability of False Alarm

Figure 30). ROG 'or K=4, m=.5, p=.5, L=32

* 49

TR 8133

.995__

.9

.9

U2

.4)

- .3

.2

.02 7Z 7Z_E-IOE-9 E-8 E-7 E-6 E-5 E-4 E-3 .01.02 .05 .1

Probability of False Alarm

Figure 31. ROC for K=4, m=.5, p=.5, L=16

Tq u~ r-w wr W W W W Y w v w w v w L V v

TR 8133

~35

30

E-6\

25E-4z

Li 2 0

10 10

L, Number of Noise Pulses

Figure 32. SNR for P,=.9, K1I, m=1

35- \E 8 E-10

30

0 25

0z

N 20

15

L, Number of Noise Pulses

F .igure 33. SNR for PD=.g, K=2, m=I, r=.551/1

Reverse !')lank

TR 8133

APPENDIX A

PROGRAM LISTINGS

There are two programs listed in this appendix, the first for L finite,

the second for L infinite, where L is the number of noise-only pulses used

to establish a reference. The fundamental parameters K,m,L are input in

lines 20, 30, 40, while p is input in line 1400. The particular values of

E1/N (in dB) that are of interest are input in lines 340 and 350.

Provision is made for 20 PD vs PF curves in lines 60-90; this can easily

be changed to accommodate other cases.

The false alarm and detection probabilities are available in lines 1000

and 1130, respectively. The detection probability utilizes R and N as input

variahles; see table 2. The particular covariance programmed in lines

i.390-1430 is exponential, but this, too, can easily be generalized.

To save space, the complete program for L infinite is not listed.

Rather, just the essential false alarm and detection probability routines

are listed at the end of the appendix; these are obviously not functions of

L. The changes required to accommodate this case of infinite L should be

obv i ous.

A -1

TR 8133

10 GENERATE PD-YS-PF NUMBERS FOR FINITE L20 k=4 I NUMBER OF SIiGNAL PULSE'- ADDED, i:

S s=.5 I FAD I NG PFRAMETER, r '.. "ri D0F -40 L=16 I NUMBER OF NOISE PULSES AIDED L

50 DIM U, 100>60 CON, Pd, 1 100),Pd2k10U>,Pd' 100 ,P Fd4, 100, Pd 5, 100:- , C d 1l 00 ', ' 100 ,P '100 :,P d 100 :F 10 I , 1 '- 10 0

:-:A " M R',- 12: 1 C,0 C, F',A :I: 'I A0 C, Pd, 1 4 10 0 : P, '-J 15, 100..- FA FI I Cl C0:, Pd' 1 , - 1,07

CO0N Pd i1 100I, Rd 19 100 P Pd 20 100)100 DIUBLE K,L,I,3 . INTEGERS

110 S=0.S120 FOR I=1 TO V:

130 FOR J=l TO K140 +=S+Ft ,,. I , J i ORM AIL I EEL Ci -V Iv i t I IHC E C CEFF I C I EtIT.

I o NEXT J160 N E XT I170 e=k *K .S EOU I-'ALENT NiUrBl:ER OF I nDEPENDIET FiDEC-.

,. 10 N=Mil=*Ke N m Ke

190 U:O.:00 UU+. 01210 Pf=FNPf(U,K,L)

IF Pf.::.l THEN 200U" I (.1MAi V.>'I ( UJ -. 01, . 1 )

24L U=U+.015 C' P"=FNPi( U, K, L

2 60 IF Ptf.1E-10 THEN 240. 70 U2:U

Del u:(U2-U ,.,- 100.20 FOR 1=O TO 100

1100 U=UL+DelA*IIJ I)U I THRESHOLD VALUESPt I )FNt. (,K, L I PR OBABILITY OF FALSE ALRFM

-. t IHE:.::T I4 FOR 3=1 TO 20

tEl Jdb=2 +2 I Gt L-TC ICI E A I TiO FER PULSE, E1 , ,dE:

I.0 E ic 1r,, i.. 1 ' E 1 rc,,db7 0 P= E I Ko*: N

FOR I=0 TO 100_N.0 U =U ' I .

. ,=FlPd U, P , N, , L FPROAB,,ILIT' CF DETECTIO410 IF 3=1 THEN Pdl' I .'Pd420 IF J=2 THEH Pd2, I.- =Pd

*~ 4 10440 I

590 IF i=19 THEN Pd15,I =Pd,-00 IF J320 THEN Pd' I =F',-,=.10 HE:T I, t20 N EX T J, FOR 1=0 TO 100

0' ,40 Ff', I =FHI'',hi ', P i P I '

_ P..1 I '=FNIr vph ' ,.Pd' I.."%"" ,':-gO F',Rd I ,=FNIml-,ph ,F'2 I '

d- 680

.0 Pd' I, 1, FI r,,.h ' PdI9' I

40 Pd2O I iFtIr,,ph , P,2'0, I

NEXT IC ALL A

0 END

A-2

.4 TR 8133

I 3:90 DEF FN I r- pihi, i 55, 2m.s.23900 IF ::=. 5 THEN RETUn 0910 P=MIN, X, 1.-:kX ',. -20 T=-LOG,:P,

9 0 T=SQR ( T+ T'.>" 94C' P=1 .+T*, 1.4-32733+T- .13-9--'9tT.0:3C1-:&::

, . ')950 R=T-t.2.515517+T ,.S0L.53T.01cl.: , F

960 IF .5 THEN P=-P970 RETURN P980 FNEND990

100 D EF FN t I DOuBLE L FALSE HLHFT F E I l I L IT

1010 IF U.=O. THEN RETUR' 1.10U20 DOUBLE Ls IHTEGER

1030 U1=U+1.1040 K1=K-11050 S=T=EXP(KLOG(UUI))

' 1060 FOR Ls:l TO L-1

1070 T=T*(KI+Li)"/(Ls:*U1)1"" 1080 S=S+T

1090 NEXT Ls

1100 RETURN I.-i1 110 FNEND

', i 120

S1130 DEF FNlP,' , R, N, DOIBLE , DETEC TII FEABILIT','1140 IF U::=0. THEN RETURN 1.

- 1150 DOUBLE Ls I INTEGER

1160 UI=U+l.1170 R1=R+I-.

1 1,0 U2=U,- US1190 Ru:Ia 1+U

. ! -12:00 K 2=h:-2

1210 Nk =N-: + 11220 Y=R 1 Ru

1230 :1=U2-R, R11 40 1 =X,,I,1 50 T = E:.:'F' K CLOG U 1_1 7 +t ILO ,:. U I R u)1 -0 Ho=O.12.'7 0 H= 1 .

12:,: FOR L-==1 TO L-11 :'9u T=T-Y

*1 uu0 J -*I1 I = J +Ls--+ -L ' -. H+ J . :1 Ho Ls

1 20 Ho=H

1330 H=A1 40 S=S + T *-H1350 t4E:i:T L-

a 360 RETUR i -1 .1 370 FNENDS3',30

1390 DEF FNCc, DOUBLE I. J.:-1400 Ph . 5 NORMAL I ::ED C O IA1 C E C CEFF IC I ENT1410 C.' Rh,, BSFIE', I-J , I EP::ONENTIAL BEHA'v'IOF'i420 TETu ,

14 30 FHEND1440 I

A-3

TR 8133

145 U SUB A PLOT Pt' VS PF ON NORMAL PROBABILITY PAPER4 1460 C0NPI Pf , Pd I' Pd2t( Pd. *.j, Pu4' , P,5*

1470 CON P ,6*.,d'Pdt*), Pd * , Pd'.*', Pd10 . ,,Pdl 1' '14:-,0 CM Pd 2 ',Pd1 3.*.'FPd14f.,* Pi15,.± d,15 Pd .16 ,Pd17.

* 1490 COM Pd1S( *. , Pd9(* ), Pd20( -1500 DIl AS1:301,B$ 30)1510 DlNI s:label$, :.30,,Yl ,bel$tl: 3'015 20 D 'I Xc,:ord ' : 30),Ycoord 1:30)

f 150 DIN r-1 " i d , I : 30:' , Ygr i d 3 1 31I 40 DOUBLE N,LA,LV, Nx,1NVM, I !rWTErERS

.. 15 50 1

1560 A5="Probabi l it':- of' Fal se Al ar-ri,"1 570 B$="Prbabili , f DctEct Ior"158U

Lx= 1260; PE I .abel 1 : L.: , > o r ,-, I :L1610 DATA E-1O,E-9,E-8,E-7,E-6,E-5,E-4,E-3,.01 .O2, .05,. 11620 READ Xlabel$(*)

1630 DATA IE-10,IE-9,IE-8,1E-7,1E-6,1E-5,IE-4,.001,.01,.02,.05,.11640 READ Xcoord(*)

11660 L-:,= 18:,70 PEDI Y lat-, $(1:L -' r ,, I : L,'

r 1630 DATA .01, .02, .05, .1,.2, .3, .4, .5, .6. ,.,. 9690 D ATA .95, .98, .99, .995, .99, E.999

1700 READ Ylabel$(*)

1710 DATA .01,.02,.05,.1,.2,.3,.4,5,.6,.,"8, .917 2 0 DATA .95,.98,.99,.995,.993,.9991730 READ Ycoord(*

1740 1

1750 N.= 147 1760 REDIM Xgrid( 1:1.,

1770 DATA 1E-10,1E-9, 1E-8,1E-T,1E-6,IE-5,1E-4173::0 DATA .001, ., 2, 02 .005 .01, . .05,. 11 790 READ :::gr- d *13500131 I0 N,>,lS

1 0 D A TA 01 r, 0 05, 1i , " : 4 5, 6 -.7,' ,.9

" I ::4U DATA .95, ?S, ... ..1-]A5 READ ','gr~d'*I ;.50 R

*..7 U FOR 1=1 TO L,..1 :o rd I Ff=FrH I -':', ,r dt I''1 . 0 1E:T I

N I -600 FOR 1=1 TO Lv

1 LU '1'aor' ,, I .. =FtI ,ri,,ph i Y-: oor d IC, tU HE.T I

I' v FOR I=1 TO tN:I ?40 :,:gri d' I ' =FN I r,vph I ,'gr I' I

11950 lE'<T I1960 FOR I=1 TO Ny1970 Y r'i ':i,: I r',p. it ,'gr g 'l I1 ?:0 NEXT I1? 90 'tl=:.',, t 1 '

g000 .2="gri d k

I Y lE' 2-'1 ' :2-::

A 4

TR 8133

4 G t INIT 200. 20 . VERITICAL PAPER050 PLOTTER 1'- 505, "HPGL"

Sl:1U PRINTER 1*:, 5050 7. C P R I NT " S'34- IU LIMIT PLOTTER 505 ,i. 200. , .,2,g. 1 GDU -

',- Ll a I YIEWPFeT 22 . * . 122.1 l 'i ,'IEWPORT 22. ,85. , . , 122. TOP OF PARFE11 '. I ENPORT 25. , :5 , I B. , . F OTTOM OF PAPEPSU WINDOW XI, -'Y ,Y2

213I FOR 1=l TO N

-'140 rMIOVE Xgridt I I'

1 i DRAW :gri' I d"16Fa HExT I

170 FOR I=1 TO N,.)2180 MOVE Xl, Ygrid(I

2190 DRAW X2,Ygrid(1)220 NEXT I

2210 CSIZE 2.3,.5

2220 LORG 52230 YY I -(Y2-Y ) 022240 FOR I= TO Lx

2250 rIO',VE :c -Door.,-d I Y- - A LABEL NI abel t I

0 HEXT I-,so CSIZE 3.,.5

M -OVE .5*C1+>;:,Y1-.06*'"-Y12300 LABEL AS

1 0 MOVE .5*1 X1 I2l -. 1 * 2 -YI)- LABEL "Figure 31. PO, fr. K=4, r,=.5' =.5, L=16"

2330 CSIZE 2.3,.52340 LORG 82350. .. .1- :K2-X '. 0 1

E.u0 FOR I=1 TO L,,2370 MOV E rdcoc, rje I3:D',so LABEL Yl bel$ I

,29 0 tI:E:T I240 0 LDIR P1 2.2410 C'1IE 3. ,.5

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2440 LABEL B$I

2450 PENUP

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2470 PEHUP

2480 PLOT P-' * ,Pd2'.249 PENUP2500

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40 PLOT Pf *),P,:20'"50 PENUPr u PAUSE

23 0J PRINTER 1-i, C:RTPLOTTER 505 I-, TEPr I1 ATED:, B E ti D

A-50

TR 8133

10 DEF Fr4Pi ,Tir , DOUBLE F FHL':.E HiL i RM F- ROBE;RE I L I T0 DOUBLE J ' INTEGER

&0=T=E:.P -Thr40 FOR I= TO -I

T T *T rir J

HE NEXT JEo RETUJP!S

RNEr4r'

1 10c DE F F nFj j' Thr R' P ,, DOU11BLE T TR 7 7907 HFF.C-I l E r ro r=1. E - 10

130 DOUBLE KidKs 'INTEGERS

140 Et=EXP(Thr)*150 K 1 =K -1.1160 N1 =N-1IIt170 R1I=1 . +RIt .13-0 C0=R Pi

190~c E =T E=1I.:0 0 FOR K==1 TO K1

10 TE=TE*Thrxi-0 E=E*Te

230 N EX T Ks4 £c=B=NFG .Et -E,0.

Jh T~i63 FOR Ksl TO 100070 Te =Te*Thr/K+S

.1 1d7 B;=riaXtB-TE.,0.

0 C Pr TtB10 SS+ Pr

I IF t BC Pr- -Er r or *FIE:',--, THEN 350rIE:-T KS£FRC P I tNT "1000 TERMS FIT: " ;K;N; Thr ;RP; Fr S-

;U P d= 1. -E:+P -T hr -Nt +LO0G' P 10 Pr RE TU1R N Pd

970 FNEr4D

A -

TR 8133

REFERENCES

1. A. H. Nuttall and E. S. Eby, Signal-to-Noise Ratio Requirements for

Detection of Multiple Pulses Subject to Partially Correlated Fading

With Chi-Squared Statistics of Various Degrees of Freedom, NUSC

Technical Report 7707, Naval Underwater Systems Center, New London, CT,

2 June 1986.

2. Handbook of Mathematical Functions, U. S. Department of Commerce,

National Bureau of Standards, Applied Mathematics Series No. 55, U. S.

Government Printing Office, Washington, D.C., June 1964.

3. 1. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and

Products, Academic Press, Inc., New York, NY, 1980.

4. A. H. Nuttall and P. G. Cable, Operating Characteristics for Maximum

Likelihood Detection of Signals in Gaussian Noise of Unknown Level:

I1. Random Signals of Unknown Level, NUSC Technical Report 4783,

Naval Underwater Systems Center, New London, CT, 31 July 1974.

5. D. R. Morgan, "Two-Dimensional Normalization Techniques," IEEE Journal

of Oceanic Fnqineering, vol. OF-I?, no. 1, pp. 130-142, January 1981.

6. A. H. Nuttall, Operating Characteristics of Log-Normalizer for Weibull

and log-Normal Inputs, NUSC Technical Report 8075, Naval Underwater

Systems Center, New L.ondon, CT, 17 August 1987.

1. A. H. Nuttall, Operating Characteristics for Indicator Or-ing of

Incoherently Combined Matched-Filter Outputs, NUSC lechnical Report

8121, Naval Underwater Systems Center, New London, Cl, 21 September

1987.

R - /R -2

Reverse Blank

Ar

Wfl r , - L" ' . . r J, - , r " -. r, '9, -v '-v' -w ' ' V' W , t V v ru'

TR 8133

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