NUSC Technical Report 4233

Operating Characteristics for ContinuousSquare-Law Detection in

Gaussian Noise

ALBERT H. NUTTALLDAVID W. HYDE

OtOtffice of the Director of Scienceand Technology

D DDC

3 April 1972

B

NAVAL UNDERWATER SYSTEMS CENTER

NATIONAL TECHNICALINFORMATION SERVICE

SpdngheL.d Vai. r21S1

Approved for public release; distr;bution unlimited.

ADMINISTRATIVE INFORMATION

This report was prepared wider-NUSC Project No. A-752-05, -statistical Communication withApplication to Sonar Signal Processing" (U), Principal Investigator, Dr. A. H. Nutrali (Code TC),Navy Subproiect and Task No. ZF XX 112 001, Program Mvaaiger, Dr. j. H. Huth(DLP/MAT 031,4).

The Technical Reviewer for this report was Dr. P. G. Cdble (Code TC).

REVIEWED AND APPROVED: 3 Apri1 1972

WHITE SEit1U

_______DiretorW. A. Von Winkle

_Director of Science and Technology

Inquiries concerning thisreportmay be addressed to the authors,New London Laboratory, Naval Underwater Systems Center, New

London, Connecticut 06320

S- -

UNCLASSIFIEDSecturity Classification

S... ... ~~DOCUMENT'CONTROL DATA - R & D . .. ..

('Security classr1.1ijion of title, body of abstract end Indexing annotation must be wimered when the ovetall report Is clatrilled .,] ~ ~~ ORIGiNA TING ACa:TIVITY Une~tr(Corporao yte m author) 2Ate •. RE'PORT S;ECURI TY U CLASFD .ASSIFICA;ION

Newpprt, Rhode Island 02840 b. GROUP

3. REPORT TITLE

OPERATING CHLARACTERISTICS FOR CONTINUOUS SQUARE-LAW DET.ECTIONIN GAUSSIAN NOISE

4. OPSCRIPTIVE NOTES (Type of report and inClusive dates)

Research Report5. AUTHORISI (Fitrt name, .nidda. Initial, last name)

Albert H. NuttallDavid W. Hyde

6. REPON, OATE 70. TOTAL NO. OFPAGES 7b. NO. OF REFS

3 April 1972 114 219a, CONTRACT OR GRANT NO. 98. ORIGINATORVS REPORT NUMBER(Z)

.PROJECT NO. A-752-05 TR 4233

C. ZF XX 112 001 9b. OTHER REPORT NOIS, (Any othr numbers that may be A oesinadd. this report)

10. OISTRInUTION STATEMENT

Approved for public release; distribution unlimited.

It- SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Department of the Navy

The exact detection probabilities and false-alarm probabilities for continuoussquare-law detection of a signal in Gaussian noise are calculated, with no restric-tive assumptions about the size of the observation-time-bandwidth products. Twocases are considered: they are called the low-frequency (LF) and narrowband (NB)cases. In both cases, the signal hps coherent, ana inconerent -components.

Results for the detection probability versus signal-te-oF riitio• (SIN), withfalse-alarm probability as a parameter, are given for both tbe LF and NB casesfor observation-time-bandwidth products equal to g,'(n = 0 (1) 7) and for the frac-tion of coherent signal power equal to 0, .2, 1. -For a speciftec detection proba-bility of . 5 and false-alarm probabilities of O-qn (n = 1 (1)8), the required S/N isalso plotted for the same range of time-bandwi~dth products and fractions of co-herent signal power as above.

47, .U .,1473 UNCLASSIFIEDS/N 0102.014-6600 Security Classification

A"

UNCLASSIFIEDSecurity Classification . A

.KEY WORDS -rifROLE Wr T ROLE WT 'OU-T

Arbitrary Time-Bandwidth Product 1Characteristic. Function ICoherent and Incoherent Signals IContinuous Square-Law Detection I LDetection and False-Alarm Probabilities

Exponential Correlation

Fredholm Determinant

Gaussian Noise

Ii

D DD 4 ° 1473 (BACK)

"(PAGE 2) Security Classification

TR 4233 59LlI

ABSTRACT

The exact detection probabilities and false-alarm pgobabilities for square-law detection of asignal in Gaussian noise are calculated, with no restrictive assumptions about the size of tileobservation-time-bandwidth products. Two cases ate considered: In the first, called the low-frequency (LF) case, the signoi is a combination of a dc component and a Gaussian process com-ponent with exponential correlation. Here the noise is a Gaussian process with identical exponentialcorrelation. In the second case, calIled the narrowband (ND) case, the signalI is a combination of asine wave component and a Gaussian process component with an exponentially damped cosinu-

soidal correlation at the same center frequency as the sine wave. Here the noise is a Gaussianprocess with identical exponentially damped cosinusoidalcorrelation. The two components of thesignal are called the coherent and incoherent components, respectively, in both the LF and NBcases.

The output of the square-law detector is integrated continuously over the observation timeand compared with a threshold for statements about signal presence. It is not assumed that thesquare-law detector output is sampled infrequently enough to yield indeperpdent samples.

Results for the detection plobability versus signa l-to-noise ratio(S/N), with false-alarm proba-bility as a' parameter, are given for both the LF and NB cases for observation-time-bandwidthproducts equal to 2n (n = 0(1)7) and f - the fraction of coherent signal power equal to 0, .2, 1.For a specified detection probability of .5 and false-alarm prbbabilities of 10"n (n = 1(!)8), therequired S/N is also plotted for the same range of time-bandwidth products and fractibns of co-

herent signal power as above. Compari.ons with a Gaussian approximation for large time-band-width products are made; it is shown that the Gaussian assumption is optimistic in predictingperformance of the scuare-law detecror, the exact amount depending on the time-bandwidthproduct.

• REVERSE BLANK

TR 4233

TABLE OF CONTENTS

Page

ABSTRACT... . . . . . .............. . i

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

LIST OF ILLUSTRATIONS ...... . ............ v

LIST OF SYMBC;LS ................ . . . . vii

ABBREVIATIONS ......................... ,.n.*. viii

Section

1 INTRODUCTION................ . I

2 PROBLEM DEFINITION ... ......... 52. 1 Low-Frequency Case ..... ......... . 52.2 Narrowband Case.. .......... 62,3 Performance Comparison between Low-Frequency

and Narrowband Cases ........ ......... 7

3 RESULTS ............ ................ 9

3.1 Low-Frequency Case . ........... 93.2 Narrowband Case ..... ........... 15

i} DISCUSSION. .................. 17

APPENDIX A - COMPLEX ENVELOPE RELATIONS ..... 83

APPENDIX B - DERIVATION OF CHARACTERISTIC FUNCTIONFOR LOW-FPZEQUENCY AND NARROWBAND CASES. . ..... 87

APPENDIX C - DETECTION PROBABILITIFS FOR ONE SAMPL_.E 97

APPENDIX D - ASYMPTOTIC BEHAVIOR OF CHARACTERISTICFUNCTION AND ERROR BOUND ..... .......... 101

APPENDIX E - DI, 'TE CTION PROBABILITY UNDER GAUSSIANASSUMPTION ........................... ... 103

LIST OF REFERENCES.. ................. i5

iii/ivREVERSE BLANK

,.y

-H TR 4233

LIST OF TABLES

• Table Page

1 Thresholds for LF Processor . . . . . . . . . 10

2 Thresholds forNB Processor ..... . . . . 15

LIST OF ILLUSTRATIONS

Figure Page

1 Low-Frequency Processor ................. 19

2 Narrowband Processor. . . ..... . ... . 19

3 Detection Probability; LF, OzO0 . ........ 204 Detection Probability; LF, .6 1 ... ......... 23

5 Detection Probability; LF, j3=2 ......... . 266 Detection Probability; LF, #.4 .. .. 29'7 Detection Probability; LF, f3=8 ........ 32

8 Detection Probability; LF, 3 = 16 .......... 59 Detection Probability; LF, 0 = 32 .... ......... .38

10 Detection Probability; LF, f = 64 .. ......... 41

11 Detection Probability; LF, 0 = 128 .. ........ 4412 Detection Probability; LF, 1 = 128, Gaussian Assumption 4713 Required Signal-io-Noise Ratio; LF ........... 5014 Detection Probability; NB, =0. .... ....... .. 5315 Detection Probability; NB, = 1 .. ......... . 5616 Detection Probability; NB,'= 2 .... ... ........ 5917 Detec'don Probability; NB, 4 .. ......... 6218 Detection Probability; NB, := 8 ......... .. 65

19 Detection Probability; NB, 16. . . . . . . . . 6820 Detection Probability; NB, •= 32 . . . . . . . . . 77121 Detection Probability; NB, •= 64 ........... 7422 Detection Probability; NB, '= 128 .......... 7723 Required Signal-to-Noise Ratio; NB ... ....... . 80

Appendix A

A-1 Single-Sided Filter .. ........... 83

V/vi

REVERSE BLANKA

TR 4233

LIST OF SYMBOLS*

t time

y(t waveform at time t

t, decision variable

c(t coherent cotaponent of received signal

s(t) incoherent component of received signal

n(t) received noiser delay variable

Ras, (r), R(r) correlation function

a 2 ao2 variance

W, 7e effective bandwidth

f frequency; fraction of received coherent signal powerGs~n(f) spectrum

T observation interval

fo center frequency

W frequency band

t time instant

P cumulative distribution function

f66), f (6) characteristic function

X, Th thresholdA normalized threshold

Sf2, time-bandwidth produtzr

PF false-alarm probobility

-PD detection probability

P received coherent signal powerC

RCRSR,d 2 signal-to-noise ratio (power)(D Gaussian cumulative distribution function

x, n auxiliary vrriable

Wk weight

N number cf branches

U unit step function

i(-W/2

*In order of appearance.

vii

TR 4233

S -

LIST OF SYMBOLS (Cont'd)x+(t) positive frequency coipoxient Qf x(t)

xH(t) Hilbert transform of -(t)

St) complex envelope of x(t)

xc(t), xs(t) in-phase and quadrature components of x(t)

H(f) transfer func:ion

Re, Im real, imaginary

r(t), x(t) normalized processes

P' PC, Ps auxiliary random variablescharacteristic function A.rgumenr

3. eigenvalue

'6. eigenfunction

,r. expansion coefficienits

I if j=k: 0 if j~k

F, F• resolvent kernelA

ID, D Fredhora determinant

c 1(t) normalized waveform !

S(u) auxiliary function

y, (a auxiliary variables

E, • envelope and phase •modulations

Prob(x) probability of x IAx narrowband analogue of x

Overbar ensemble average

Underbar complex envelope

ABBREVIATIONS

CDF cumulative distribution function

CF characteristic function

FFT fast Fourier transform

LF low frequency

NB narrowband

PDF probability density function

ROC receiver operating characteristics

RV random variable

S/N signal-to-noise ratio

viii

TR 4233

|. I

I

a' OPERATING CHARACTERISTICS FOR CONTINIUOUS

SQUARE-LAW DETECTION INGAUSSIAN NOISE

over bad1. INTRODUCTION

Often, signals in n6ise are detected by filtering a received process to thefrequency band of interest, squaring the filtered output, continuously integrat-ing over an observation .interval. where the signal is anticipated, and comparingthis decision variable with a threshold 6r with the past history of this variable."In fact, under cert.An conditions, the optimum detector for stochastic signalstakes this form (for' example, Ref. 1, Chapter XM). This form of processorcoild be employed whether the received signal is deterministic (known orunknown), stochastic, or a combination. of both, although it' is not necessarily

optimum in all these cases. This processor is, in fact, frequently adopted forpassive detection of targets of unknown statistics.

Two important parameters that characterize the performance of this proc-essor are the bandwidth of the stdchastic process at the input to the squarerand the pbservation (or integration) time. Practical cases occur where thistime-bandwidth product ranges from the order of uuity to several thousand. Inthe latter case of very large time-bandwidth products, the decision variablecan often be accurately approximated by a Gaussian random variable (RV).However, when the time-bandwidth product is of the order of unity, it can notbe approximated as Gaussian, especially when very low false-alarm probabil-ities or high-detection probabilities are of interebt. The question then arisesas to the exact performance of the processor. Calculation of the performanceis complicated by the difficulty in analyzing the continuous integration of a -on-Gaussian process since a "sum" of dependent random variables is formed. Also,app•-oximation of the performance by using a sum of independent variables istenuous, bdcause the exact value of the P'effective" number of independentsamples touseis not easily a !,rtained. Thus, there is a need for (1) the exactevaluation of the receiver operating characteristics (ROC) of the square-lawdetector followed by an integrator, both as a benchmark against which otherprocessors a-d approximations can be compared, and (2) the determination ofwhen the Gaussian assumption can be validly employed, especially for mediumtime-bandwidth products and very small false-alarm probabilities.

TR 4233

Two cases of signal and noise spectra at the input to the squarer areconsidered: In the low-frequency (LF) case, the signal is a combinationof a dc component and a stationary, zero-mean, Gaussian process componentwith exponential correlation, and the noise is a Gaussian process (of differentpower level) with identical exponential correlation. in the narrowband (NB) case,the signal is a combination of a sine-wave component and a Gaussian processcomponent with an exponentially damped cosinusoidal correlation, and the noiseis a Gaussian process with identical correlation, except for power level. Thetwo components of the signel are called the coherent and incoherent componer'S,respectively, in both the LF and NB cases, for ease of discussion.

Actually, the general derivations of processor performance capability pre-sented here are not restricted to the cases above. They actually allow lor anarbitrary deterministic coherent signal component (which may be known or un-known) and an arbitrary (but identical) correlation function for the incoherentsignal component and the noise. ýt4wever, numerical evaluation of performancecapability requires evaluati:.' ibh resolvent (or reciprocal) kernel and theFredholm determinant of th- c:rrelation function. Hence, numerical resultshave been confined here to the cases given in the paragraph above.

Several practical applications fall under the framework above. For example,if the incoherent component of the received signal in the NB case is absent, wehave detection of a sine wave; this is encountered in active or passive detectionof a target or in binary on-off communication with another party. Or, if thecoherent component is absent in the LF or NB case, we have detection of arandom-signal process in noise; this situation can hold true in long-range trans-mission through a fluctuating medium or in reflection of an acoustic signal offthe time-varying ocean surface. It is often employed in passive detection of anoisy source. Finally, if both signal components are present, relevant examplesin the NB case are reflections of an asoustic signal from a surface with lowroughness, or communication via a medium supporting both a direct (nonfluc-tuating) path and fluctuating paths. Extensions to multitone applications are alsopossible and are discussed in Section 4.

A great deal of work on the statistics of the timc-average power of a GaussianSprocess has been presented in the past. Rice' gave the first four semi-invariants

(or moments) fbr an arbitrary correlation function of the Gaussian process,and approximated the probability density function (PDF) for very small and largeobservation times. Emerson" gave an infinite product for the characteristicfunction (CF) of a filtered squared-Gaussian process and, via the first three

2

TR 4233

semi-invariants, approximated the PDF through Gram-Charlier and Laguerreexpansions. Siegert 4 gave the CF in terms of the solution of an integral equa- jtion obtained by Kac and Siegert. 5 In the former, the CF was In terms of aninfinite product, although it was shown Vhat the resolvent kernel offered an al-ternative approach. No examples were presented. Darling and- Siegert6 andSiegert 7 showed that the CF of a more general functional could be obtainedfrom the solution of two integral equations, aud they obtained the CF in closedform for a finite-time exponential averagerfor a Gaussianprocese with an expo-nential correlation. Slepian8 obtained a series expansion for the PDF and cumu-lative distribution function (CDF), where each ternt was given by aul integral.No extreme probabilities were evaluated and the accuracy was about 1 percent.Jacobs I considered a communications problem with rondom signals and madea finite approximation to the CF by increasing the number of terms until theCDF essentially stabilized, Schwartz'" obtained a closed form CF for correla-tion functions whose corresponding homogeneous Fredholm integral equationreduces to an equivalent linear differential equation. Four examples were given,

and both LF and NB cases were considered. Steenson and Stirling " conducteda simulation and gave results on the PDF and false-alarm rate and showed thatthe gamma distribution accurately fitted the data. Some related results on thedistribution of quadratic forms are given in References 12, 13, and 14.

None of the previous results are of sufficient accuracy to allow calculatingthe cumulative distributions very near zero or unity - the regions of interestfor very small false-alarm and high-detection prcbabilities. The complicatedresults for the CF, even when in closed form, indicate the need for quick andaccurate CDF calculations that do not require moment evaluations or seriesexpansions. Here we make no attempt to find analytic expressions for the PDFor CDF, but proceed directly numerically from the CF of the decision variableto the CDF via a Fast Fourier Transform (FFT). 1, 6 Prhe assumption thatthe incoherent component of the received signal has the same correlation (and,hence, spectrum) as the noise in both the LF and NB cases, except for powerlevels, enables evaluating the CF in closed form,

If the integrator following the squarer were replaced by a sampler and sum-mer, calculation of the CF in closed form can be accomplished fairly simplyonce the eigenvalues of the matrix of sampled correlations are evaluated. Thistechnique allows very general 9nd different signal and noise spectra and will bethe subject of a future report. For izow, however, we restrict ourselves to con-tinuous integration without sampling the squarer output.

3/4REVERSE BLANK

Th4233

2. LPROBLEM DEFINITION

Two different cases- will be considered in tbls report, name~ly,, theLL andNB processors. Block diagrams of each proceasoir will be g'iven-wid the perti-.nent characteristics of the signal and rnoise described.. Then the- expected per-formance of the two cases will be briefly compared for the case of la~rge time-bandwidth products.

2.1 LOW-FREQUENCY CASE

A block diagram of the LF processor is given in Fig, I,. where y(t) is a

LF waveform composed of three components:II~ y(t) = c (t) + S (t) + n(t)(1

In (1),a. ci(t) is a LF deterministic waveform (kýnown or unknown) and is called

the coherent component of the received signal.

b. s(t) is a stationary, zero-mean, LF Gaussian process with correlation

RS(ir)= a82 exp(-2W,, 1,j (2)

and is called the incoherent component of the received signal.

;~ IThe noise' n(t) has the same statistical description as s (t), but the two processesare independent. Its correlation is

R (r) =cr 2 exp(-2W, ITI) (3)

The (double-sided) spectra of s(t) and n(t) are given by

G 0a 2 Wei 2 (4)

where W, can be interpreted as the effective (or statistical) baiidwidth,17 of the

positive-frequency components of the LF random processes:

pdf G Sf)] H2 '0o)()

f00df G', 2f0 dTR' (r).00

-TR 4233

rThe identical spectrai shape of the incoherent component s(t) and the noise

n(t) Is realized in practice by filtering fairly broadband received signal andnoise processes through a low-pass RC filter. It can be interpreted physicallyas overresolving the received signal-with a very-lovw--pass filter.

The decision variable .2 in Fig4 I is given by

fdt[c() + (E) 2 (6)T T 2

and Is to be compared with a threshold for detecting signal presence. (No as-sumption about the size of the time-bandwidth product TW, is made.) Thisis broadband-energy detection and contains three subcases of importance:

a. c(t) = 0. Signal presence indicated by an increased power level in thereceived random process.

b. s(t) = 0. Signal presence indicated by a nonzero, time-varying mean inthe received random process.

c. Both c(t) and s(t) nonzero. Signal presence indicated by adding a non-

zero, time-varying, mean-ranclom process to the received random process.

2.2 NARROWBAND CASE

A block diagram of the ND• processor is given in Fig. 2. In the figure, y(t)is an NB waveform composed of three components, as in (1). The signal c(t) isan NB waveform centered at frequency fo, with a deterndnistic complex enve-lope (known or unknown), and is called the coherent component of the receivedsignal. Both s(t) and n(t) are independent, stationary, zero-mean, NB Gaussianprocesses with correlations

s,,n ( ,n=: a2 eXP(-WIrp)cos(2rrfrX .0 (7)

(It is assumed that foT >> 1, where T is the integration time In Fig. 2.) Bydefinition, a(t) is called the incoherent component of the received signal.

The (double-sided) spectra of s(t) and n(t) are given by

Ac 2 e 1 (8)

6; s,n 4,7 2 ^ 2i'-(f_f )2 Jo (f+f f2 +

-. t • • - . - •- -. _ • • • • ' - - . _ .-" -- * - ÷ •z •

TPR 4233

- where -cari be interpreted as the effective bandwidth of the p6sitlve-frequen,3ycomponents of the NB random processes:

2

A dft%.,.J _ (0o)VV S (9)

(df G!s.() 2 2 (r)

The identical spectral-'shape of the incoherent component s(t) and the noisen(t) are realized in practice by filtering fairly broadband received signal andnoise processes through an RLC filterthat is NB. It can be interpreted physicallyas overresolving the received signal with a very NB filter.

The decision variable . is given by

I=1-- J" & Ic_() t- s(t) + n_(t)12 (10)2T T

where x(t) denotes the complex envelope of x(t) (see Appendix A). The decisionvariable -9 is compared with a threshold in order to detect sipnal presence.(No assumption about the size of the time-bandwidth product TW, Is made.)This is NB energy detection and contains three subcases:

a. c(t) = 0. Signal presence indicated by an increased power level in thereceived random process.

b. s(t) = 0. Signal presence indicated by an NB waveform in the receivedrandom process.

$ c. Both c(t) and s(t) nonzero. Signal presence indicated by adding an NBwaveform and random process to the received random process.

2.3 PERFORMANCE COMPARISON BETWEEN LOW-FREQUENCY AND NARROWBAND CASES

The effective bandwidths of the LF and NB processes, which have been de-fined in (5) and (9), respectively, are measures of the spectral width of thepositive-frequency components of t.he respective processes. A bandlimited low-pass process confined to frequencies (-W, W) must be sampled at least every(2W)-I seconds in order not to lose any information. Thus, in a long time intervalT, there would be (2W)T samples upon which to base a decision as to signal

4 presence or absence. On the other hand, a bandlimlted NB process confined tofrequencies (f, - W/2, f, + W/2) and the corresponding negativefrequencyband

7

11 rz• •• • -m c.•/•9 • •- • • _- •-•...........• o•... _'•- -•-

TR 4233

must have it8 ia-phase and quadrature components sampled at least everyseconds-in order not~to lose any -information, Then in a time T, there Would be-

2(W)T samples upon which to base a decision as tosignal presence. Therefore,both bandlimited processes, LF and NB, yield the same number of informationsamples when W is interpreted as the spectral wIdtih of the positive-frequencycomponents. As a result, in our investigation of-nonbandlimited processes, weexpect that when the effective positive-frequency bandwidths are equal, that is,

, = W,, the performance of the LF and NB processors should be comparablefor large TWI. This will bei borne out by the numerical examples in the nextsection.

For small time-bandwidth products, however, the performance ofthe LF and NB procesaors differ. For example, as T--0 in (6) and (10),

,-.2-*y• (t0), -NB-.-"/2 [Yc2 (t.) + y2 (to)] (see Appendix A). Thus, in thelimit of small time-bandwidth products, LF processing contains one degree offreedom, whereas NB processig contains two degrees of freedom. The per-formance of the NB processor will be better than that of the LF processor forsmall time-bandwidth products.

8

TR 4233

3. RESULTS

The CFs for the LF and NB cases are derlved in Appendix B, the general-f~esdlts being (3-22) and (B-46). Thiese results are specialized to the cases ofiiiterest here, resulting in (B-31) and (B-51), respectively. For the specialcase'of T -- 0, (6) and (10) reduce to a single sample of the input processes,and derivation of the detection probabilities and false-alarm probabilities ispossible in closed form; these derivations are presented in Appendix C.

The CDF P of a nonnegative RV from its CF f can be evaluated numericallyvia use of any of the equivalent forms' 6 :

M()= - d6 o mI e exp (-i e:X) I

=•'o2 d6 ,f(6), sin (e:X) (..02 -d

~ 1=1-fe Y~.•f() cos(%eX), X> 0

SIn (11), f, and fi are the real. and imaginary parts, respectively, of CF f. Theerror in terminating the integrals in (11) at finite limits for computer evalua-tion is considered in Appendix D.

3.1 LOW-FREQUENCY CASE

Suppose 0 in (6) is compared with a threshold X for purposes of signaldetection. We now define a normalized threshold,

xA =_ (12)n a 2

and the time-bandwidth product,

=TWe (13)

Values of the required normalized threshold A, for different values of 0 andspecified false-alarm probability PF are given in Table 1.

Before presenting results for the detection probability, it is convenient todefine several power signal-to-noise ratios (S/Ns). We define the receiled co-herent signal power

PC--1 f dt C2(t)T T (14)

9

TR 42-33

Table 1

THRESHOLDS FOR LF PROCESSOR

01 2 4 8 16 32 j64 128

10- 2.706 2.065 1.841 1.634 1.461 1.329 1. 233 1. 164 1. 1i5

10-2 6.635 4.302, 3.362 2.600 2.065 1.710 1.478 1.324 1.223

10-3 10.828 6. 706 4. 971 3. 564 2. 630 2. 047 1.684 1.455 1. 307

1.0-' 15. 137 9. 180 6. 631 4. 540 3. 182 2. 364 1. 872 1.571 1. 381i 10-3 19. 511 11. 693 8. 322 5. 526 3, 728 2.670 2. 050 1.678 1.448

10-' 23.928 14. 2-0 10.031 6.523 4.272 2. 970 2. 221 1.779 1.511

10-7 28. 374 16.785 11.753 7.527 4.815 3.265 2.387 1.877 1.570

10-8 32. 841 19. 352 13.483 8. 537 5. 389 3. 558 2. 549 1. 970 1.627

The coherent and incoherent S/Ns are defined, respectively, as

•, c ,,•.(15)

and0,a2

(16)

The total received S/N Ws defined as

R C---7- a R2 " RC R (17)n

We also define the fraction f of total received signal power that is coherent as

f P Re (18)p . =.=-

PC +10 R

•o10

' TR 4233

Susing (15), (16), and (1 ). Then w e can express 1-

Rs U-0 R, (19)R5 -- (1-f) R ,

j and, thereby, utilize the t.tal received S/N R and the fraction f of coherent Lisignal power as the fundamental parameters characterizing the performauce

of the processor.

I •In the series of Figs. 3 through 11, to follow, plots of the detection prob-j ability PD versus R1/2 are given, with PF as a parameter. Each figure

corresponds to a different value of j9, specifically 13 0, 1, 2, 4, 8, 16, 32, 64,Sand 128. Each figure is composed of three parts: Part A is for f = 1, part B is

* for f =.2, and part C is for f = 0.

The eqaation

dR = 10 log,, R = 20 log,, R1/ 2 (20)

canbe employedto convert the abscissatodecibels inFigs. 3 through 11. (R isa power S/N, whereas R is a measure of. a voltage S/N. ) Equation (20) is thetotal input S/N expressed in decibels.

The difference in performance for different fractions f of coherent signalpower is very marked for small R. Thus, Fig. 3, for fl= 0, illustrates a sig-nificant drop in P. for large values of R1/ 2 when f is decreased from unity. Infact, this degradatioD in performance occurs even when f is decreased from 1to . 9 (this curve is not shown). As 8 increases, the difference in performance islessened until, for large 13, the oppositebehavior occurs for large RI/2 (forexample, PD ý• .9999). That is, the required value of R1 2 for a specified PD

near unity is less for f = 0 than for f = 1 for very large 13; see, for example,Fig. 11.

On the other hand, for small values of PD, the smaller values of f yieldthe better performance for small fl, whereas the larger values of f give bet-ter performance for large 0. There is, however, rather slight differences inperformance, that is, required S/N R, in the case of small PD.

The reason for this cross-over of the curves is explained by the fact thatwe are considering a fixed receiver and varying the transw'Itted signal composi-tion. The fixed receiver is more nearly optimum for some transmitted signalsthan others, and it is irmpossible to order the performance of the various com-binations of coherent and incoherent signal power.

11

TR 4,10033

The situation is made more striking when the performance of the optimumreceiver for completely coherent reception, f= 1, is considered. For a dcsignal waveform, LF processing, and exponential noise correlation, we canuse the first equation on p. 114 of Reference 1 to evaluate Helstrom's powerS/N d2 for the optimum receiver as

*d2 = Re(1 -TW) R U +TWe)=R( +P)

The false-alarm aud detection probabilities for this case are given by [Ref. 1,p. 106, Eq. (3. 32)]

PF = C-0 , (22)

where* x is selected for prescribed PF" (For a specified performance (PF,PD) of this optimum processor for f = 1, required values of d can be readfrom Ref. 1, p. 88, Fig. IV. 1.) When (22) is superposed on part A of Figs. 3through 11, it lies everywhere above the square-law detection curves, as isexpected. Furthermore, the slope of the straight lines ((221, is greater thanthe slopes of the square-law detection curves. However, when (22) is super-posed (m parts B and C of Figs. 3 through 11, the equation lies below the square-law detection curves for small R and above it for the larger values of R. Thesquare-law detector can give better performance in some regions of R (forf < 1) than the optimum coherent receiver (for f = 1) for the flat transmittedsignal, because the transmitted signal (the flat waveform) is not the optimumone to transmit in this noise spectrum. Rather, a deterministic signal thatoscillates arbitrarily fast in the interval T and, thereby, occupies a region oflow-noise-density does arbitrarily good; there is no optimum signal unle s weimpose some bandwidth constraint. This case has not been considered here.

Two analytical results along a related line are possible for (3 = 0. From(C-7), it is found that the detection probability for fixed P. and fixed R (> 0)always decreases as f increases from zero (for A, > 0) regardless of thevalues of the false-alarm probability and S/N. (A, < 0 and R < 0 are uninterestingsituations.) Of course, for large An, that is, low P•, the rate of decreaseis not very large and may not be noticeable from the plots. However, for largeAn and R, for example, R = An -1, (C-7) gives

*(D is the CDF for a zero-mean unit variance Gaussian RV; see (C-3).

12

TR 4232-

O --(2.- exp(-1/2) ( (23)

which is significant.

For f = 1, we find the same result for 0 = 0; see (C-8). The slope can be-come very negative for large A .; in fact,

p D A s A.af-T f = 1 2 (2-te)I/2 a n

2 =i /2-(24)

R1/ 2 - 1/2 -n

Thus, for 6 = 0, performance of the square-law detector is degraded as f in-creases toward 1 for certain S/N R and normalized thresholds A0 .

For large time-bandwidth products ft, it is anticipated that the decisionvariable S in (6) could be approximated by a Gaussian RV. In Appendix E arepresented tie equations for the detection and false-alarm probabilities underthis Gaussian assumption; see (E-3) and (E-4). In Fig. 12, these equations areplotted for B = 128, where f = 1, . 2, and 0, respectively. These approximatecurves are generally above the exact curves given in Fig. 11 for the same con-ditions; that is, the Gaussian assumption gives optimistic performance pre-dictions compared to the true value. However, the deviation is not great at13=128.

For a specified performance (Pr, PD), the required input S/N R decreasesto zero as T - . In fact, from (E-4), we find

P F"' i(\/7'(1 -A^)) /3-, c , (25)

where normalized threshold A, is selected to keep the argument fixed; that is,

Fv)(A-1) = Th (PF) (26)

is kept fixed at a threshold value that depends on the prescribed PF. Then (E-3)

yields

-PDq(\/R - Th (PF)) as 3 -oo (27)

Thus, the required S/N R decays as T-1/2 as T - ., rather than accordingto T-'; this is attributable to the square-law dAtector characteristic. Noticethat this rate of decay of required R holds regarcdess of the value of f; that

13

TR4233

is, it is independent of the fraction of each type of signal comzponent. AlthoughSth. rule was deduced from the Gaussian assumption, it holds in the generalcase because the decision variable I approaches Gaussian as 0 - or.

Figures4throughllmayheused to obtainthe exact values cf the input S/N Rthat are requiredto attain-a specified performance. In particidar, for P. = 10",n = 1(1)8, and PD =. 5, the required value of R (in decibels) is plotted versus0 in Fig. 13. The three parts of the figure correspond tV f = 1, .2, and 0,respectivaly. The rate of decay of R with increasing 0 is approximately /1-ip,as discussed above. In decibels, the decay is -5 log ,,

The Gaussian assumption, Fig. 12, gives values of S/N R that are optimisticby amounts that depend on the quali;-y of performance desired. For example, theGaussian approximation for j = 128, P. = 10-, and P, .5 indicates arequired S/N of -3. 05 dB ior all f, * whereas the actual required S/N is -1. 98dB, a difference of 1.07 dB. In fact, the -3. 05-dB S/N would yield a Prgreaeter than 10-6 at 6= 128. (For 0= 64, the difference in elecibels increasesto almost 1. 5 dB. ) However, for PF = 1, instead of 10-i, the decibel dif-ference between the exact answer and the Gaussian approximation is only. 28 dBat B = 128. These comparisons have been made here only at the P, = .5 level,and conclusions at other levels can be very different. Figures 3 through 11 fur-nish information for required S/N at other performance levels of interest.

A note of caution is in order at this point: From (C-1), (C-5), and (C-6),for 0 = 0, if f = 1, the decision variable I is given by

k = "(p ' 2 n) (28)

and

PD = (D(R 1 1 2 - A'/ 2). +b(DR-R1/2_AV/2)

P P -A21/( A(92)

*This independence of f for required S/N is true only forPD = .5; see (E-3).

14

_•--_ •-•-• -'•:e•'' "••J• • -_, -,. -. -o -2••i -,= I: - ,••..

TR4233

1/

Now for a fixed P., A,, is fixed, and PDv versus R plots as a straightline on normal-probability paper (for example, see Figs. 3 thrugh 11) for all

values of A'. However, the RV -2 in (28) is definitely not Gaussian. Thus,-a straight-line plot on norml-probability paper can not be interpreted ascrresponding to a Gaussian RV.

3.2 '-ARROINBAND CASE

Most of the comments in the previous subsection are applicable here anawill not be repeated, for the sake of breyity. Alsthe ndoation is similar, exceptthat

! IA A:

3 = W ,T .(30) ,

Values of the required normalized threshold A. for different values of , andspecified PF are given in Table 2. As anticipated in Section 2.A3, the thresh-olds for laige time-bandwidth product f are virtually identical to those for.the.LF case (TCable 1). However, the thresholds for small • are quite different.

Table 2

THRESHOLDS FOR NB PROCESSOR

0 1 2 4 8 16 .32 641 '128

10-1 2.303 1.979 1.799 1.615 1.454 1.326 1.232 1. 16 1.115

10-2 4. 605 3.680 3.123 2.528 2.044 1. 703 1.475 1.324 1. 2ý2

I0-3 6.908 5.381 4.447 3.423 2.593 2.036 1.681 1.454 1.307

10-1 9.210 7. 083 5. 770 4.316 3. 129 2. 349 1.868 1.569 1.381

10-5 11.513 8.784 7.093 5.209 3.658 '2. 652 -.2. 045 1.676 1.448

10-6 .13.816 10.485 8.416 6.101 4. 184 2.948 2.215 1.778 1,510

10-7 16.118 12.186 9. 739 6. 994 !4. 709 3.240 2.380 1. 874 1.569

10- 18.421 13. 887 11.063 7.886 5. 233 3. 528 2.541, 1.968 1.626

15

;R is - tkrug --22, -, -pot ofI&viu- t/:ae e fr h

NP prcsopf - ,qAprpqbý,Fi 1.:ý18~equrp,4eVru~

idnia to ths o,-eL ai; 4 teGqsij .. rjnaincudbJUAI----- Validity______ ___________CaO.____________b~ate

-pje41e Wit a

pltdiun Fg. 14 throuheN22, plotshoe rekuidSIN is lress pre thed fori the L

caidenia tor-sa1 those for--i thetE are; thus, ti al. ia ioteL As iojrinarge old

Requiredctpatuds ofuR for otherformancei frorbhedeiitiens ore ava~ciable fro dthý

pa troiulag he fo2ifee2.au. ff

161

TR 4233

4. DISCUSSION

In the problem considered here, no assumptions about the size o" the time-bandwidth product are necessary, and the decision variable need not be ap-proximately Gaussian. Small false-alarm probabilities can be easily and veryaccurately investigated numerically. The complexity of the closed form CF,even for the relatively simple case considered here, precludes any analyticsolution for the CDF; nevertheless, very accurate answers for the CDF can beobtained. This numerical approach is recommended for other problems of thistype.

For each doubling of the time-bandwidth product, it has been shown that

the required input S/N R decreases by 1.5 dB (for large products) in order tomaintain a prescribed performance (Ps, PD). This result is similar to that cited

for an envelope detector [Ref. J.8, p. 262, and Exercises 8. 8 and 8. 9] and isa result of the squaring operation in the receiver. The improved performancefor f = 0 versus f = 1 at low S/N has also been observed for other processorsoperating in fading media [Ref. 18, pp. 208-9 and pp. 214-5J.

Use of a Gaussian approximation on the decision variable for large time-bandwidth products is in error by approximately 1 dB when the time-bandwidthproduct is 100. For larger products, the Gaussian approximation yields pro-gressively better estimates, at least for detection probabilities that are notextreme and for false-alarm probabilities that exceed 10-8. This applies toboth the LF and NB processors under the conditions investigated here.

Some extensions of the problem studied here are possible. They centeraround the ability to obtain the CF of the decisionvariable either in a closedformor in a reasonable amount of computer time. For example, suppose the decisionvariable -2 is given as a weighted sum,

NW k k k (31)

.: k=lI

where the kth branch output, .O:, has signal sk (t) and noise nk(t) independentof all other branches. (This occurs, for example, if a bank of filters is excitedby broadband processes, and the filter in the kth branch is disjoint with that in

* the jth branch for j 0 k.) The statistical situation obtaining in the kth branchIf can be any one of the six cases listed under (6) and (10) and can be dissimilarin different branches. Furthermore, the time of observation, the effective band-width, center frequency, and S/N can be different in each branch. The CFcorresponding to the RV . in (3"1) is given by

17

TR 4233

N N

k=1 k

where f k ) is the CF of the RVIk inuthe kth branch.

Examples that fit into the framework of (31) include the following:

a. Multitone active detection or communication in a time. varying randommedium that gives rise to the scatter component of the received signal. (Themultiple tones could be transmitted to combat fading or jamming or both.)

b. Multitone passive detection, where the scatter component of the receivedsignal may be due to either medium perturbations or source instabilities.

c. Single-tone passive detection by comparison with a threshold establishedfrom neighboring filter outputs.

A compilation of performance-capability characteristics has been presentedfor continuous processing of both LF and NB processes. However, they arelimited to an identical exponential correlation of the random received signaland noise processes; nonidentical spectra, for example, of the signal and noiseprocesses are disallowed. The only other continuous processor that yieldsclosed-form expressions m.r die CF of the decision variable is an exponentialintegrator [Ref. 7, p. 41]1, rather than the finite-time average consideredhere. However, it, too, requires an exponential correlation function. The onlyreasonable way to generalize to arbitrary (unequal) signal and noise spectca isto consider sampled processors. Often this is the practical situation anyway.The decision variable for the square-law detector can thenbe transformed intoa sum of squares of independent Gaussian variables, and the CF evaluated as afinite product. If the size of this product is not excessive, direct numericalcalculation of the CF and, hence, the CDF by the methods in Refs. 15 and 16is possible. This approach will be the subject of future investigation on moregeneral processors and spectra.

In passing, it is noticed that the detection problem posed here is theoreticallya singular one for the purely incoherent signal; that is, detection canbe accom-plished with zeroprobability of error (see, for example, Ref. 1, pp. 304-306).However, the optimum method of processing is impractical - it would require 1precise knowledge and use of all the eigenvalues of the correlation kernel;equivalently, second-derivatives of the observed process are required. Thephysical impossibility of accomplishing this in a practical case forces us toadopt a suboptimum procedure similar to that considered here.

iJi

TEAM-

tY

Fig. 1. Low-Frequency Processor

IITE

4JJSQUARE-LAW ENV.ELOPE DETECTORj

Fig. 2. Narrowband Processor

li

19

MIT

.9999 /

.999"

.998

n=)- 2 345678.99

.98.

.95 ".

•t: .9

p .8

S~.7"

l PD .5 /

.4

.3

S.1 /.05...02 1/ /.012

.005-/

.002.001/

'00051/

,0001 --V- --- - ,----- - -- • •.-- . --. P -

0 2 4 6 8 10

i~iFig. OVA. f I-

Fig. 3. DetectiLon Probability; LF, • = 0

20

TR 4233

.9999

V998

.99-

.98-

.95.

.9.

.8.

.6 =pD .5-

.4-

.3.4

.2-

.05

.02

.01

.005-

.002-

.001

000 2 466 8 1

Fig. 3. Detectioji Probability; LF, 6 0

21

tR 413S

.98--

.998 L

.98

.9

.44

.278

.005

.001'0005

.00010 2 4 6 8 10

Fig. 3C. f 0

Fig. 3. Detection Probability; LF, 0 = 0

22

- --. R 423 5

.9999

.999

PF=0 n=1 2 3 4567F.99

.98

.95

.9

.8~.7

11 .6pD .5

.4

".3

.2

.1

~05

.02

.005

.002U .001

.0005

.00010 2 4 6 8 10

Fig. 4A. f IFig. 4. Detection Probability; LF, I = 1

I-• 23

TRA4233

.9999 "1"

.999

.998

.99•i<.98. PF "lO-n

.,95 1 nc o.-

.6

.44P D .5. /

.3

.2

.1

.05

.02'

.01

.005

.002

.001.0005

.000140 26 8 10

F 1g. 4B. f .2Fig. 4. D.nocttoai Probabilityl LF, = I

24

TRh 4233

.9999

.999 P :1 n0

.998 n :

.99 2

.983

.95 4

.9.6

.8..' ~.7 "

.6

D .5.4

.3

.2

.1

.05

.02

.01

.005

.002.001

.0005

.00010 2 4 6 8 10

Fig. 4C. f=0Fig. 4. Detection Probability; LF, • = 1

25

TR 4233

S ~~~PF-o"

n=1 23456 8.999

.998

.99

.98%L .95

ipV .9

.7

.6

.00.5.4

.3

.2/

.1

.05

.02

.01

.005

.002

.001.0005

.0001 I- -0 2 4 6 8 10

4 TFig. 5A. f= 1

Fig. 5. Detection Probability; LF, • = 2

26

Li

TR 4233

.0999 ,

i,998 // /

6..99 " S8.98

.95

.97

.6

PD .5

.4

.3

.2

.1

.05

.02

.01

.005

.002

.001

.0005

.0001 •-•...-----. I.I_.•0 4 6 8 10

Fig. 5B. f=.2Fig. 5. Detection Probability; LF, 0= 2

27

•. .9999 i

.999-- 4

-.-"- . - i-

.991 n4 5 6- 0

-- -.9 9 - '

10'

.905

.00

.004.

.63

.05

.42

.01

.005

.0001 5

2 4 6 8 10

Fig. 5C. f=0Fig. 5. Detection Probability; LF, 6= 2

28

-~~'R 4233-~ - -

i I

.9999 =0-,

: I

n- 14-3

.999

.998"

,991

.98

.95

.7

P D .5 "

.1

.05 /

.02

.01

.005

.002

.001.0005 '

.0001 . I ,0 12 3. 4" 5

Fig. 6A. f=:LFig. 6. Detection Probability; LF, P= 4

29'

a --4

.981

* S ! o

i .8

I "7' I !

P D .5 -

, .4

.3Vi

S~.05

.02

.01 -

.005.002 -

• : .001.0005

- .0001

0 :2 6 8 10

* I .9

VR

Fig. 6B . f= .2SFig: 6. Detection Probability., LF, 4

30

.5 I * I

A .. A OP

.99998 1- n.

.999

.998

.99 1

.98-

.95.

Iit.

.8.

.7.

pD .5

) .4 .

.2 /

.1

.05.

.02

.01

.005

.002

.001.0005

.0001 - I__ I0 2 4 6 8 10

Fig. 60. f=0Fig. 6. Detection Probability; LF, f= 4

31

.9999,

.999

.998 .4

.99

.8

.7

.6

D .5.4

.3

.2

.05

.02

.01

.005

.002

.001.0005

.0001 I I I -0 ~ 24

Fig. 7A. f1Fig. 7. Detection Probability; LF, 0=8

32

7TR'443:~

.9999

.999-, P =0 2 56 8.998 1,

.99.98

.95.

.96

4 /

.2

.05..

.02.

7 .01'005-

.002 -~001

.0005

.00010 1 2 3 4

Fig. 7. Detection Probability; LF, 8i.7. =.

33

`4233, ,

.9999~f

Siz 2- 3 45.998''-

'.--~C

__ _ _ _ __ _ _ _ _ __ _ _ _ __ _ _ _ _ __ _ _ _' -- ' '"~^9 _ _ __ _ _ _ _

~. , , ' 4. 9 5

CC '~.9

A~-C ~ ~ -

.09

.00

.005

.4000.23

341

~~<' ~.9999'/ 'I

S1I - .996 J n-.j: -/I06 7 8 "

.98t

.95 , .

.7 /0.6

pC.51'4".32.2

.05

.02

.01

.002.001

.0005

.0001 •[-1---0 .5 1.0 1.5 2.0 2.5

Fig. 8A. f=1

Fig, 8. Detection Probability; LF, fi = 16

F' 35

.9999 - - '.1

.9m8

n=1 2 1 4 5'6 8

).95j-.79.98

.95

.9/"

• ~.7 /

P D .5

.4

.3.2

.05 /

.02

.00

.002

.001.0005

.0001 -I---- ---- I-- ---

0 5 1.0 1.5 2.0 2.5

Fig. SB. f=.2Fig. 8. Detection Probability; LF, = 16

36

.9999•

.9998

.998 / o,.99.

.98--

.997.

.999

p .6.

.5

.3 /.05

.01 /.005/

•D : / /

.01

.002

• o '0001 - j -+----,- 1 -- +----

0.5 1.0 1.5 2,o 2.5

Fig. SC. = 0

Fig. 8, Detection Probobility; LF, 0 = 16

S { 87

. ----

TR 4233 f"

.9999:

""".999 ' p- . 10 -n-

.n998" n 2 3=45 8

;99-r

.98

.95

p'PD .5

.3.7

.05.

.02

.01

.005

.002

.001.0005 .5

.0001 . I I I I . ,

0 .5 1.0 1.5 2.0 2.5

Fig. 9A. f= 1Fig. 9. Detection Probability; LF, . 32

A 38

A --

.9999. /

C ~~999-- U.998

6=-2 34 56 8

.98

.95..

.9

D .5

/.3-

.05 ,,

.02/

~01.,

.005./

.002

.001.0005

S.0,I.. , - -4------; . ,. 3

0 .5 1.0 1.5 2,0 2.5

Ftg. 9B. f=.2

Fig. 9. Detection Probability; LF, / = 32

39

ITRI4233

.9999 1

.999+ PF"O' /

.998-"

nz1 2 3 4567 8".99j•. .98

•, .95 .

.9

.6

PD .5

.2

.05

.02' 1

.01

.005

.002•" ~.00i

.0005

.0001 -Ž:.- -$---

0 .5 1.0 1.5 2.0 2.5

•,MFg. 9C. 1=0- Fig. 9. Detection Probability; LF, (= 32

40

'9~984 PF =10~1` 2- 3 4-56 8

.99

.95.

"17.6+

.2,

.05-

.02-

101./

.005-

.002

.001.0005-

.0001 I I I I -0.3 .6 .9 1.2i.

Fig. IOA. f = 1F Ig. 10. Detection Probability; LF, (=64

41

-¢: .9999

.999'- p Cn

-C

Sn=l 2 345.999.98

.995* :.98

.9

.8

.7 ..6

P D .5

.4

.3

.1

.05

02.01

.005

.002

. 001.0005

• ., . 0 0 0 1'2 0 .3 .6 .9 1.2 1.5

I.g. 10B. f= .2

Fig. 10. Detection , robability; LF, •= 64

42

I.! rtRh,4233:II

.9999"

.999

.998+ IF=10-n : .

.99

.98

.95..

..9

•.7 ;

D .5.

.4 a

.3€

.02

J a0

1 .001.00052

10.001

.9 1.2 1.5

~~~Fig. 0.DtcIOnCrbbliy , f 64oF=Fig. 10. Detection Probability; ,F, 14= 64

i 43'

- -- . -- -, -...

TR4233

,999--

•'t" °.998' -. "F -1O' -7

.,98 !.3 1'

:.8

"" .99 .P:0

3

.05.,/

.02 X

S.01: ,~~005 /. .

.002

.001 -I .0005

[ . 7O l•, :- . • - , - :

50 25 .50 .75 1.00 1.25

SFig, 11A. f I

' Fig. 11. Detection Probabilityý LF, 'f=12b

.44

* I 0

.9999 - I

.999

.998 pF 0n=. PF~1O'.99

.98--

.95-

.9

.7 ,

•.6

D .5.4

.3

.2

- .1

.05

.02.

.01

.005

.002

.001.0005

.0001. 0 .25 .50 .75 1.00 1.25

Fig. lIB. f =.2

Fig. 11. Detection Probability; LF, 3 = 128

45

T4233-

.9999

.998 "P =10-n

n: 23 456 8.99-

.98.

.95.

.84

p .6D .54.4

.05

.02-

.01

.005

.002

.001

.0005

.0021

0 .25 .50 .75 1.00 1.25

Fig. 11C. f =OFig. 11. Detection Probability; LF, =128

46

.999

.993

n 1 2 34. 8

.6/

.21j /01

4.25.017.10 12

/ T

Fi2 !?/./

Fi.1,DtcinPobblt;/,f 2, 1simAsmto

1 -~ /7

tk -ý

-- a-

- -~- a

.98.

p .6

D .5

.01~

P.00

.00+

12B f/Fig D teti n ro abliy;T, , 2-, au sin ss mpio

_ .- -

•. • .,9907T,

.999.980P :1

II

.95 L

"a/// '

.9SN 1

.98

P .

• •: . ~~.3.2. • /

.254

01 / // I

.05..I"

, .0025.001

.0005

0 ,25 .50 .75 L.OO 1.25

Fig. I2C. f=0Fig. 12. Detection Probability; LF, / 128, aussian Assumption

14

S. i2 T1-4ý233

14--

12

9

S6

3.T

O- 3 4

0- 3

2

p F = lO'

i• 3 "= :

<-6

•(-9 ______-H- I

1 2 4 8 16 32 64 128r We

Fig. 13A. f =150Fig. 13. Required Signal-to-Noise Ratio; LF

50

14 - -. -

12-'CH

9-

6-

R(dB) 8

3-- 7

K54 4

0-

•o -3-2

PF,-n

1. 4 8 16 32 64 128

;: TWe

;•Fig. 13B. f =.2

.. Fig. 13. Required Signal-to-Noise Ratio; LF

{ 151

w

-. R (-d - 8

3-6

0723n

P =1

-9t

124 16 32 64 128

52

-... . .. ... .. ..

.9999 1

.999-•

.998 P/ : 0"n

.99.

.98-

.95-

.9.'

.8

.7

.6

P D .5

.4

.3

.2

91.0.01 /

S.005 -

,002..001 /

.0005 //0 2 4 6 10

Fit. 14A. f = IFig. 14. Detection Probability; NB , = 0

53

,IJD ..X -••z•+-a. •::, , . •,• ."••'" , ..•"• •.. - " "?-- 7

.9999 1.999.998

.995

.95

.3.

.31

.32

.005

. 0005

02 46 I

fPig. 14.]Mig. 14. Detecti-m, Probability: NB, 0

54

$ (~90999

.999-* .998

.99-

.98-

.95.

J..4

D .1

.05.

.02.

.01.

.005-

.002-

.001* .0005

10001__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

02 4 FR 6 8 10

Fig. 14C.f=0Fig. .14. Detection Probability; NB, =0

ý'TR.4233

.9999 /-

•'•.999 -

J;99 7

.98

.95

.8

' PD .5.6

.4

.1

• • .005

' .002 -

.002

.001

.0005

.0001 " -- - I I I I

0 2 4 6 8 10

Fig. 15A. f=Fig. 15. Detection Probability; NB, I 1

56

V.

.9999 -

.999I..M8 -

.99, P-l =10 -

.98.

.95..

.7 7

.65-

0.4

00..

.002-

.002-.00051

.0005 .

0 2 4 6 . 8 '10.

Fig. 15B. f=0 2Fig. 15. Detection Probability; NB, 30 1

571

TR-4233 a

.999911 .-- a

.999--

.998 T "

-n1•• ,.99- n="

.98 2.

.95 . 3

1 4.9

.8 8'77

-.6

D .5

.05.. 2

* .05

.02• • • 01

: .005

.001* :.0005

, ,00010 2 4' 68 10

Fig. 15b. f= 0* •Fij. 15. 'Detection Probability; NB, 0 = 1

58•~//

.999 F

", 2345 -

.99 -6

.958

.95

05--

.09

.000

.001.05

2 4 6 8 10

AgoFig. 16A. f 1Fig. 16. Detection Probability; NB, 2

59

4'14233'

.09999

998

.99{

.98

.95

.05.

.02.

.01

.005-

.002-

.001-.0005

.0001-0 2 4 6 8 10

Fig. 16B. f=.2Fig. 16. Detection Probability; NB, 0=2

60

TR 4233

.99991 / *'

-n

F99- 1 '2 3 4.5 6 78

.998 /

.99..

.98

•: .95 ..i' /

I /

P D .5 //

,3.80/

.02

.01

.005

.002

.001.0005

.0001

024 8 I0

F0 i g. 16C. f 0

S~F19. 1 6. Detection Probability; NB, '=2

.061

/9 --- I.9 9 9 9

2

.998

.98

.95.

.9.

.85.

.00

.401

.005

.0001

Fig. 17. Detection Probability; NB, ý= 4

TR42ý3

.9999t/St

.9991 n=l 2 3/4 56

- .991 .98-'95

/ /

,9.• !

.8.i /i

p .1

S.05

.02

.01

.005

.002ý001.0005

.0001 .. .+-- - -

0 2 4 6 8 10

Fig. 17B. f=,2

Fig. 17. Detection Probability; NB, F=i4

S63

Z63

TR 4233

.998-

.98

.95.

.9.

.8

.7-

.6D .5.4

.3

.05.

.02.

.01

.005

.002

.002.0005

.00G;0 2 4 6 8 10

Fig. 17C. f =0Fig. 17. Detection Probability; NB, 4

64

TA 4233

.9999 IIPF 10

-n 1/2-345.998/

.99./

.95 I.

.9.

.8./.7.

.6D .5 /

.3

.02.

.01 /

.005

.002

.001.0005

* .00010 12 3 45

Fig. 18A. f=I1Fig. 18. Detection Probability; NB, =8

65

.9999--

FF4

.999--: /3 5 7.998--

.99

.98.

.95-

.8 D

.7.

.6

.05-

.02

.01

.002-

.002-.0005-

.0005.

0 1 2 3 4

Fig. 18B. f=.2Fig. 18. Detection Probability; NB),~ 8

66

1:,ww

"ip_ = lO"n

I r.9,99-i n = 1

S/, F.99 "

.9t•# ' r

.93.? / / / :

i

PD .5 /

.4,• .• i/!t .2 ,

t .05 ,

i .01

S,005

• .002S.001S.0005 .0001

•, 0

SFig. 18, Detection Probability; NB, 8

÷67

7--~-~

/.6...

.01..005 -

.000 /4----40.5101. . .

.ig Fig /9A f =

190eetonPoaiit;N,51

68

"ThW4233

.9999-1 1//,,

".999".

n99= 10 2 3/ 4 56 78

.99 -f

.98 /,95 //1//

.9 //,//.8 1

.7 //

.. 6 //7

D .5 /

.05

I...~ /

4 .1 /11

.02..

.01

.005

.002--.001.

.0005

.00010 .5 1.0 1.5 2.0 2.5

Fig. 19B. f= .2Fig. 19. Detection Probability; NB, 16

69

.9999 -1

.999-/

.998F. ,: n1 2 3 4 564

-• !.99./

.98 .

.95-

.8

.05

.6

.3i

}" .2

.0505-

.002.001

.0005

.0001 /0 .5 ).0 I.5 2.0 2,5

Fig. 19C. f= 0Fig. 19. Detection Probability; NB,_ 16

70

Th 4233,

I..

.99991

p 1

.9989

n=1

.99 -

.98.

.8 I5.9

.98

apt .6

D .5

.4

.I.05

.002

.001,,

.0005

.o .0001 *----.- --I--- .- I I

0 .5 1.0 ' 1.5 2.0 2.5-

Fig. 20A. f= 1Fig. 20. Detection Probability; NB, = 32

*,

ii '71

.. . . . .

TR 4233-

i• .9999

* . "

n, n1 '2 3 456

I ~ I.995

.95

.7

I*1' .6

-~~ .4/.3

.2-

.05

.005

.002

.001.0005

.00010 .5 1.0 1.5 2.0 2.5

Fig. 20B. f=.2Fig. 20., Detection Probability; NB, j3 32

72

I -. '

'T-

.9999

.998F =10-n~l1 * 3 456 8

.99

.98

.95

.9

.8

.7

.6

d .4

.3

.2

.05

.02

.01

.005

.002.001 /

.00010 .5 1.0 1.5 2.0 2.5

Fig. 20C. f = 0Fig. 20. Detection Probability; NB, 3=2

7S.

I TR 4233

.9999

.999 PF , =0n

.998 F

n:1 2 3 456

.99

.98

.95

.9

.8

.6pD .5

.4

.3

.2

.05,

.01

.05•

.002

.001

.0005

.00010 .3 .6 .9 1.2 1.5

Fig. 21A. f 1Fig. 21. Detection Probability; NB, $= 64

74

.998 .P l&

Fnzl 2 3 45 67

.992 .98-

.9

D .8.7-

.3

.2

.1

.05

.01

.005

.002

.001

.0005/

.00010 .3 .6 .9 1.2 1.5

Fig. 21B3. f= 2

3 Fig. 21. .Detection Probability; NB, 0=64

'754

TB 4233

.9999 /1 /'

.998

.98-

.95.

'I .9./

.8-

.7

.6

.051

.024

.02.

.005-/.002-.001I

.0005

.00010 .3 .6 .9 1.2 1.5

Fig. 21C. f= 0Fig. 21. Detection Probability; NB, 3 = 64

76

.9999.

.998 1 P F '10 " n

n=1 2345 8

.99

.98

.95

.9

.7/

D .5

.4

.j5

.02

.01

.005-

.002

.001.0005-

.00014 Z-0 .25 .50 •75 1.00 1.25

Fig. 22A. f= 1Fig. 22. Detection Probability; NB, •= 128

77

TR.4233

.9999/

.999-

n:l 2.998 1FF.98

.95

.9

.8

.7

.6Pt) .5

.4

.3

.2

.1

.05

.002

.005

.0005.001 f

.0005

0 .25 .50 .75 1.00 1.25

4FRFig. 22B. f= .2

Fig. 22. Detection Probability; NB, = 128

78

TR 4233

.9999.0

.999--F O'

'F n~1 2 34 5 7 8.995

.98 I.95.

.84

D .5! .4

I

.2,

.05

.01

.0051

.002-

.001..0005

0 .25 .50 .75 1.00 1.25

Fig. 22C. f= 0Fig, 22. Detection Probability; NB, •= 128

79

-R4 3

- - : 7 . .

k. • - " -'"

I,'

14 --

.5:,,a :

12 ?D- I -L

-6-

R (dB) . •,[

3--6

I""" N 4 5

4-'

K 2 f

2 4WA 16 32 64 128T We

Fig. 23A. f = 1Fig. 23. Required Signal-to-Noise Ratio; NB

80

'UR.4•233

14-

PF :=-5

12

6-

R (dB) 8

0 27

0-6

S1 2 4 8 )16 32 64 128T We

Fig. 23B. f=.2Fig. 23. Required Signal-to-Noise Ratio; NB

81

S - --- pD-- . .-- ___________________________

-N

§ Ii

MV

i• 1R (d6! 8

2 3

-6

-9t

1 4, 8 16 32 64 128. We

Fig. 23C. f = 0"Filig. 23. ReqW.red Signal-to-Isoise P',atio; NB

82

¢-I

tit- 4233

Appendix A

COMPLEX ENVELOPE RELATIONS

Jr. this appendix, wd collect te OomileX envelope reiatios thait ar re-quired in analyzing the narrowband (NB) processor. Consider a stationary zero-mean process x(t) passed thrdugh a time-invariant linear filter, as depictedin Fig. -1. Here U(f) is the unit step function:

, f <01 (A-i)

Fig. A-i. Single-Sided Filter

Thus, the output x,(t) contains only positive-frequency compor.ents. If thespectrum of x(t) is G, (1), then, from Fig. A-i, the spectrum of x+(t) is

-G' G(0 = 4 U(f) G,,(f (A-2)

Also, since

U(f) = 1 + sgn( = ! + i [-i sgn()] (A-3)

we can expressx,(t) = x(t) + ixii(t) , (A-4)

where x,,(t) is the Hilbert transform of x(t):

1 x(r)XH(t) f dr- (A-5)Ir/o t-r

Let the "center" frequency of G, () be fo and let the complex envelope

x(t) of x(t) be defined as

x(t) = x+(t) exp(-i2i'f 0 t) (A-6)

Then the spectrum of the complex envelope is

83

TR 4233

G, (o=G (f+f+) = 4U(f+fo) G(f+fo)G,(, ... ('+ (A-7)

and the mean magnitude-squared value of-the complex envelope is

l(t)12 dfcf= fdfGG a--2ooo 0 (A-8)

where a2 is the power of process x(t).

Let the complex envelope, which is low-frequency (LF), be represented interms of its real and imaginary components:

x(t) = xC(t) + i X (t) (A-9)

There followb, using (A-4), (A-6), and (A-9),

x(t) = x (t) cos (2ff f 0t) - xs(t) sin (2rrfot) (A-10)

Thus, x,(t) and x5 (t) are the in-phase and quadrature components of x(t).

Now if the process x(t) is filtered by a linear filter with transfer functionH(f), yielding (complex) output y(t), it can be shown that

y(t)y(t-r)= f df exp(i2fffr)Gx(f) H( (-f) (A-li)-o A

Applying this result to Fig. A-i, we obtain

x+(t) x+(t-r) =0 for all r (A-12)

since U(f) and U(-f) are disjoint. Hence, from (A-6),

x(t)_x(t -r) = 0 for all 7 (A-13)

Therefore, using (A-9) and (A-13),

xC(t) x8(t-r) ='Re lx(t)l Im ix(tr-r)

= k(t) + x* (t)] I7 [x (t - ) * (t - r)A

2 i (A-14)

= R(z)_ ý R* (r)0 = - im IR,(r) Ii4 - -2

2= lira df exp(i2nfr) G -' =- f df sin(27ffr) Gx(O

84

TA 4233

where we have defined

R (r) = x(t) !* (r -. r) I (A-15)

and utilized the fact that G. (f) is real (see (A-7)). Now from (A-14), it may beseen that if and only if the LF spectrum G.(f) is even about f = 0, Oten theprocesses x,(t) and x,(t) are uncorrelated for allrelative time Aelays. In termsof the original spectrum G,(f), processes xr(t) and x,(t) are uncorrelated forall time delays if and only if G1 (f) is even about fo (in (0, 2f.)). in the specialcase where x(t) is a Gaussian process (not assumed up to now), x,(t) andx, (t) are independent processes [Ref. 19, p. 41], since x,(t) and x,(t) areGaussian processes (being linear combinations of x(t)).

Lastly (eliminating the Gaussian assumption again), by ,a procedure anal.-ogous to (A-14),

xW(t) x (t-r) = Xs(t) xs(t-r) =I Re IR(r) =(:,,I df cos(27rfr) G(f) .. (A-16)

2 - 2-oo

As a special case, the powers are, using (A--8), :

2-00

x 2(t) =x2s(t) f dfGM 2(A77

85/86

REVERSE BLANK

. =

Appendix.BI • DERIVATION OF CHARACTERISTIC FUNCTION

FOR LOW-FREQUENCY AND:NARROWBAND CASES

This appendix cdnsists of 'wo parts: The first part derives the characteristicftmction (dF) of the decision variable I fr the low-frequeny.c(LF) case, asgiven~~~ue by (6)*;) "a•ee asoc arde- •

given by (6)*; t~h'e seco'nd part does the same for the narrbwban.d (N'B) decisionvariable .2, as given by (10). * An 6xample for eadh case is presented.

LOW-FREQUENCY CASE

From (6),

I-' f dr[c(t)'+ s(t) +n(t)]2. (B-1)ST T

The powers in the ze2 o-meah Gaussian processes s(t) and n(t) are g7 anda 2, respectively. Let us now define

c(t)S• rtt) = ,

s(t) + n(t)

. (t 2 2) 1/2S~(o2 + 2)

Then

5 n(B-3

where

P=T f dtfr(t) + x(t)12 (B-4)TT

Let the correlation of x(t) be R(r), which is general here, except that R(0) = 1,as seen fiom (B-2); s(t) and ný(t) are assumed independent of each other. Atthis point, the deterministic waveform c(t) is also general.

*This equation is in the main text of this report.

87

TR 4233

The CF f# of the random variable (EV)a is related to the CF f. of the RV Ptaccording toi! i

tf)()!= .exp u•)"•. = ep (ia(2 +a o)p) f C((a2 + U,. ý) , (B-5)

using (B-3).

Now we define for the kernel (correlation) R, the efgenvalues j X, } , andeigenfunctions jo} :

'Xi f du R(t-u) 4j(u) = j(t), t tT (B-6)T

Next, we expand x(t) in the set of eigenfunctions according to

x(t) = W x1qSi(t), tcT (B-7)

where

4 x -f dt x(t) i0,(t) , (B-8)T

since the eigenfunctions are orthonormal [Ref. 1, pp. 98-103]. The RVs I xjhave the properties

S=0 ,(B-9)

Xxk = ff dt du R(t-u) p1 (t) k(U = -or "3 jk

using the fact that x(t) has zero-mean and employing (B-8) and (B-6). Sincex, in (B-8) is a linear operation on the Gaussian process x(t), then {xj, ateindependent Gaussian variables.

When we employ (B-7) in (B-4), we obtain

p f dtr2 (t)± , 4---r.x. .L x 2 (B-10)T T Ti' T '

where

r, fT dt r (t) 5,(t) (B-li)

88

"TR 4233

Then the CF of RV p is

f p ( 0) = e x p ( i Q p ) = d t r 2C t ))ex p i TLe "r' x + ( Bx

using (B-10) and the independence of the { The jth average in (B-12) is,letting a = 1/X and using (B-9),

dx X2\/2eexp -- exp -r. X + ix

/" 2a24. 1/2 / 2r2 e 2 /T 2 i (B-13)

exp 1\' 4TT1

where the square root is principal value. Equation (B-12) canthenbe expressedas

r(t, I _f~e( d) r 2 !) } x exp

However, employing (B-11), we get

______ .A.o(t) q .2u)SJ -Ji 2ur/T Jr)Z j - i 2Y/T

(B-15)

& = dtdu r(t) r(u) r (t, u; i 2 /T)V

where r is the resolvent (or reciprocal) kernel of R [Ref. 20, p. 141, Eq. 64and p. 134, Eq. 52]. Substituting (B-15ý into (B-14) and employing (B-2) yields

-1/2 ex[

(B-W6)

2a 2 +0 T it du c(t) c(u) I (t, u; 2 6/TC2 2 T7

s II

{f-:.89

TR 4233

Now, if the coherent signal component is absent, that is, c(t) = 0, this CFmust reduce to

[D 03-17)

"where D is the Fredholm determinant [see Ref. 10, p. 18] (notice that theeigenvalues JX I here are equal to j1/X 1 in Ref. '10). The square root in(B3-17) is not principal value, but must be determined by tracing D(i 2 k /T)from k = 0, where D(0) = 1, in a continuous fashion. This result follows byinspecting the product of principal value square roots in (B-16). Utilizing (B-17)in (B-16), we obtain the CF of p as

f [D(i2 ]el / exp fdt c 2(t)

- 03-18)

2 + 2 T2 dt du c(t) c(u) r"(t, u;i2 / el

This CF depends on the Fredholm determinant, on the resolvent kernel of cor-relation R, and on the waveform c(t). If the Fredholm determinant and thedouble integral on the resolvent kernel can be expressed in closed form, theCF of p is available in closed form.

It is convenient to normalize waveform c(t) as follows: Denote the averagepower in received waveform c(t) over the interval T as

P =1 dt c 2 (t) (B-19)

and define the normalized waveform

c1(t) - P 11 2 c(t) (B-20)

The average power in c I(t) over T is unity. Also we define two S/Ns:

p ca2

C Ii- ~RC c• , RS =_-(J•,1a_ .2 a.2

These are the ratios of coherent signal power to noise power, and incoherentsignal power to noise power, respectively, at the input to the LF processorin Fig. 1. Equation (3-18) can then be expressed as

90

)

TR 4233

fp(t) --[D(i2 • T] I2 ep i -I+Rs

2Rc L 2 1 dt dit c (t) c (u) r"(t, u;2 U/

ýI+R T

Example: Exponential Correlation; Constant Waveform.

We assume that s(t) and n(t) are independent and have the same exponentialcorrelation, as given in (2). * Then, using (B-2),

R(r) = exp(_2We I r) (B-23)

The coherent waveform is taken to be dc:

cII) = 1, t t T (B.-24)

Then, from Ref. 10, Eq. (37),t we have-1/2

fp(4) = exp (j3) sinh (2j8 9 (6/)S)) + cosh (2j60( /ji), (B-25)P(4/13)

' ~c(t)=0

where

/3 W , (B-26)1/2! O(u) -- (I -i2 u)t

The square root for 0 is principal value; that for fp is not, but must be tracedcontinuously from • = 0, where fP(0) = 1.

The resolvent kernel is [Ref. 10, p. 21, Eqs. (29) and (30)]

20 2F (t, u; i2 ý/T, = - [ cos(Ojt) cos(W u) - y2 sin(a)t) sin (ou)&oTy (B-27)

- ysin(coit-uI)] , t,u, uT,

where:•( oT = i2j6(/f)- i28 0 (B-28)

* This equation is in thd main text of this report."I The factor 4 in Ref. 10, Eq. (37), fhould be a 2.

91 91

TR 4233

and

1+ cosh(2/0 q5(ý//)) + 95(6/1) sinh(2A3(//3))

sinh(2,3(//3)) + 9(W/3) [1 + cosh(2 3j0( /f3))]

cosh(P3tb) + 0 sinh(f3i )--i (B-29)

sinh(P3 ) + 0 cosh(,36)

I + 0 tanh(P3 )

S+ tanh (P3 )

There follows, after some manipulation,

.• 2 fdt du r (t, u; i2 e/T) 1T3(I - i2 /3)

sinh (2 j0 ) (B-30)

I [ 1 + cosh (2,) + 0sinh(23q)]

Finally, the CF, (B-22), becomes

SR Cfp(O =exp( 3 ) sinh(2Ij6A)+cosh(2 00)] exp iI....

(B-31)

2 sinh(2/30)1-i2•/3 1 32 1+cosh(2,36 ) + 0sinh(2 3e9)

where 0 is given by (B-28) and (B-26) as

,€= (1 - i2 •/g) 1/2 (B-32)

(As -.- 0, there follows from (B-31)1._ /2 c (B-33)

9 2(I - U e) exp

92.

TR 423•

which may be readily derived as the CF of p in (B-4) for one sample Afr(t) + x(t)] 2 . The CDF is derived in Appendix C.)

The CDF can be found from (B-31) by employing the methods in R U. .15and 16. The square root in (B-31) is not principal value, but must be •Cacedcontinuously from • = 0, where fp(0) = 1.

NARROWBAND CASE

From (10), *

=_ fdt IJ_(t)+,t0+n(t) 2 (B-34)2T JT

The powers in zero-mean Gaussian processes s(t) and n(t) are o2 and ,2u

respectively. We now definec(t)(r)t =-_ _ _ _W (2 + a 2)1/2

+ n" (B-35)

.Lt t) + =(t

Co.s + no),

As a result,(=C2 + a2) p , (B-36)

where

P = 't It(t) +x_(t) (22T JTSA A

The correlation of x(t) is 2R(r), where R(T) is general, except that R (0) = 1(see Appendix A).

The charmcteristic function (CF) of the random variable (RV)O is

PcO f((a2 + U2) (B-38)

* This equation is in -he main text of this report.

iz%3

TR 4233

In order to evaluate the CF f p, we expand r(t) and x(t) in terms of their realand imaginary parts:

r(t) =r(t) + ir3 (t) , 3E(0 = xC(t) + i X(. .

ThenP=-(P, + p) , (3-40)

2C

where4.SPC dt [rc(t) + xc(t)

(3-41)ps, f dt [rs (t) + xs (t) 2

A A

Now if andonly if the spectrum G(f) corresponding to R(r) is even about f = 0(that is, G, (f) even about fo in (0, 2foJ), then xc(t) and x,(t) are independ-A

ent random processes [Ref. 19, p. 41], each with correlation R(r). Then,using (13-40), the low-frequency (LF) results of (B-16), and the symmetry prop-erty (t,u;X) r(u,t; X) of the resolvent kernel of R that is symmetric,we obtain the CF of p as

.- ) exp [idtbI2 (t)1J (B-42)

~~~~ 2T aad ugt *u "(~~ T2

1

- dt du £ (t) s* Mu Pr(t,u; i 1a2 + 01 7r ff1 /Tj

The average power in c(t) over T is, for f. T >> 1,

PC f dt c2(t) -= -T r dt I (t)12 . (B-43)

We now define a normalized waveform,

C1 (t) = P C), (t) , (3-44) -

94

f TR 4233

I' for then

4J7f dtL 1Ct) 2=1. (B-45)

The CF in (B-42) can now be expressed as

fp() [e (xT). p R

= [1ep + R (B-46)I+s

• idtduc(t c(U) •(tu; i 6/T)/1 R TT

following the reasoning developed in (B-16), (BE-7), and (B-18), and using (B-21,.Notice that this CF depends on the complex envelope "c (t), and not just on thephysical envelope Ic1 (t)I of the coherent signal component of the narrowband(NB) deterministic waveform. However, any constant unknown phase in thisNB deterministic waveform is irrelevant because it cancels out of the expres-sion c ,(t) c* (u). That is, if complex envelope cl(t) is expressed as

s,= E(t) exp[i 0(t) +iO] , (B-47)where E(t) is the envelope modulation, 4 (t) is the phase modulation, and 0

is an unknown phase independent o- ime t, then

cl(t) 1(n) ( E(t) E(u) exp [i 0(t) - i O(u)] (B-48)

is independent of 0.

Example: Exponential Correlation; Constant Complex Envelope.

Here we assume the correlation R (r) of complex envelope x(t) to beexponential:

I(r) exp(-O l); 0 e T (B-49)

The correlation of the corresponding NB waveform is given by a form similarto (7). * We also take.for the complex envelope of the coherent signal component

c 1 (t) = 1 , (B-50)

*This equation is in the main text of this report.

Ia 95

TR 4233

which corresponds to a pure sine wave for c(t). When an approach analagousto that given above for the LF case is employed, the CF (B3-46) becomes*

fvC•) - [f (e- e/2, 3 -- A12)]

_j I+ s

A A

~~4 sinh (P35)A 4• A A A 11

1 -i2 4 /f .too)i +0cs (fle÷+€sinh( PO)

where

A 1/2 -52)i • - ~(I - i2 V1/6)1/ ,(B-9

and we have utilized (B-31). (As 4 -* 0, (B-51) yields

Sfj ( -i ' exp ,c (B-53)•+ 1 Rs i e -i

which m'ny be readily derived as the CF of the RV p in (B-40) for one samplb.The cumulative distribution function (CDF) is derived in Appendix C.)

The CDF can be foumd from (13-51) by employing the methods in Refs. 15and 16. There is only one square root involved, that for $ in (B-52), and it isa principal value.

*The leading factors in (13-51) agree with Ref. 10, Eq. (44), when we note that his 13 isA AW WET/2 P/2 here.

96

00

TR 4233

jtAppendix C

"DETECTION PROBABILITIES FOR ONE SAMPLE II As the observation time T in (6)* and (10)* approaches zero, the decision

variables -9 take special forms for which the distributions can be evaluatea inclosed form. Also, the signal and noise spectra are not restricted in any way.

Here we utilize some of the notation developed in Appendix B.

LOW-FREQUENCY PROCESSING

From (6)* and (B-19), for T -• 0,

-(P14 +s) (P +i D)2 (C-i)

Then

"Prob(k>X)=Prob(IP% +x.>ŽX)

00 X% d P!12

-f dx p(x) + f dx p(x)

X C p 4 -00CC -(D + ((aco2 + 0.2)% (a2 + a20

=, IR ý*s((1-CRY (A-)s) + +IR,_ R

since x is Gaussian with variance a2 + a2 Here 4) is the cumulative dis-

tribution function (CDF) for a Gaussian random variable (RV),

( l(x) dt (2ff)"½ exp(-t'/2), (C-3)

and the normalized threshold A, is defined in (12). *

We define the fraction f of the total received signal power that is coherentas

*This equation is in the main text of this report.

97

- -L

I. I

-, ,

TB 4233 *

3 .*

C CP' R• , Rc

f .. E--- , ' (C-4)S + a2 R +R

where R is the total S/N at the input to fig. 1. Then (C-2) yields

P (R)" 2 =' 2\ n + / ,(,D (C-5)

for the detection probability. The false-alarm probability follows by settingBR 0:

-2cD(-Al/ 2) (C-6)

Notice that the signal and noise spectra do not appear in: (C-5) or (C-6).

From (C-5) there follows

3f =-(2n)-I/2 exp 2 nl+ (1+R 3 2 (C- )

af= 2(1fo LIR)

Also,

-(2P0 'I C 1/W/2-1) 1/

af n 12 ('n'A 1) I + (-8)-22 A)!2 - ( -

n Imp [-.i. (2 A 1/2 -2] A"1/2 (A

The meaning and interpretation of thesb equations are given in Section 3 of themain text.

98

5 TR 4233

NARROWBAND PROCESSING

Fropn (B-3'7), (B-40), and (B-41), for T- - 0,

22 ~ 2 S

Them from '(B-36) and (Cz-3),

=ProbR [(r+iic)+(sx) j(C)

Prob~i>X> XPr) ( P24 o2 J

2A \1/2 2A

1+R 5 R

I 2A

1/2 d (k pE)0 Pr+ol 2I+ y 25

2A2 \R/ \1-i 12

00 t(2 r 2 1 (.Qj t2 L) /2A * \1/2 s 1- (+R 11)

where R is the region outside a circlb of radius (2 A, /(l.÷R~ ), fmad Qistabulated in Ref. 21. The 6~rabability of detection is

99

TR 4233

The probability of false alarm is obtained by setting R =0:

P -- =e- A) AC-13)

Again, the signal and noise spectra do not enter (C-12) and (C-13).

41

100

TR4 233

Appendix D

ASYMPTOTIC BEHAVIOR OF CHARACTERISTIC FUNCTIONAND ERROR BOUND

The cumulative distribution function (CDF) canbe found fromthe character-istic function (CF) by use of Ref. 16, Eq. (3)

P P(X) f1 I d_j_ m {f(6)exp(-.-6X (D-1)

(Alternatively, Ref. 16, Eq. (14) or (15), could be used for the problems en-countered here.) Since numerical evaluation of this integral must be limited toa finite interval, it is necessary to evaluate the error incurred by terminatingthe range of the integral. To evaluate this error, the behavior of the CF1f(t) forlarge t mast be known.

• I For the low-frequency (LF) case, the pertinent CF is given in (B-31). Itmay be shown that for large t the second exponential is upper-bounded by unity,

* +and the leading terms behave as()1/4 [_, l+i -.1/1 exp 1-u1/2 1_' -1/2)]

U2 j L (D-2)

exp i~ (.' U1/~2 )las\ 4

where u = A/g. As an example of the accuracy of this asymptotic form, forS= 1 and A = 250, the exact value of the CF is 1. 489,62 •1-7 exp (-i2. 692384),

whereas the asymptotic expansion in (D-2) gives 1.490021 • 10-7 exp (-i2. 693924).An upper bound on the CF for large } follows from (D-2) as

The error in terminating the integral in (D-1) can thenbe related to the integral

I - du 1/4 1 /2)]f- exp [I3(1 Q Ul~

77 U L U

7/ / x [8Q- / (D-4)27/ exp(p) dt t-/22

S- 2~'~ ~ f dt t"33"2 exp(-/3t) <2/-- exp[/(1 - (D-21

Y 101

r°

tR 42ý33

Values of uL used to keep' the error below. specified tolerances..can be-foundeasily from (D-4). These values are, depenfient on't'4o exact value. of #;however,they areindependent of R and Rs and are, there' re, conservative.

10.2

102

TR 4233

Appendix E

DETECTION PROBABILITY UNDER GAUSSIAN ASSUMPTION

In this appendix, we assume the time-bandwidth product 1 large enough sothat the random variable p in (B-4) can be assumed Gaussian. In that avent,we need only compute the mean and variance of p. We have

WP I+RPc +1= R (E-()

using (B-2), (B-19), (B-21), and (19). * The variance of p is given by

Ia2(p) =-ff dt du [4r(t)r(-) R(t-u) + 2R 2(t-u)]

T2 T

2tR 2P-1 + exp (-2p6) 4P-1 + exp(-4,) (E-2)

a +R(1-0 A 2 4P2

using (B-23) and (B-24) for the case of interest here. Then from (B-3), (12), *and (C-3),

Prob(t>X) -Protp>

(P A +an

=(•P) -(1 +S R- f))-(p)/

1+R -A

[I +R1( f)]1/2 +R+Rf---(I+R +7Rf) +2 exp (-2p)+ + -ei4

(E-3)

The probability of false alarm is obtained by sietting R = 0:

__ PF =4( • 1- exp (-4)'I /S(E-4)

* This equation is in the main text of this report.

103/104

-W j

TR 4233VLIST OF REFERENCES

1. C. W. Helstrom, Statistical Theory ofSignal Detection, Pergamon Press,New York, 1960.

1 2. S. 0. Rice, "Mathematical Analysis of Random Noise" (Parts I and 11), BellI System: Technical Journal, vol. 23, no. 1, 1944, pp. 283-332, and "Mathe-

matical Analysis of Random Noise" (Part IIl), Bell SystemTecancal Journal,vol. 24, no. 1, 1945, pp. 46-156.

3. R. C. Emerson, "First Probability Densities for Receivers with SquareLaw Detectors," Journal of Applied Physics, vol. 24, no. 9, September1953, pp. 1168-1176.

4. A. J. F. Siegert, "Passage of Stationary Processes through Linear andNon-Linear Devices," Transabtions of the IRE, PGIT-3, March 1954,pp. 4-25,

5. M. Kac and A. J. F. Siegert, "On the Theory of Noise in Radio Receivers

with Square Law Detectors," Journal of Applied Physics, vol. 18, no. 4,April 1947, pp. 383-397.

6. D. A. Darling and, A. J. F. Siegert, "A Systematic Approach to a Class ofProblems in the Theory of Noise and Other Random Phenomena - Part I,"IRE Transactions on Information Theory, vol. IT-3, no. 1, March 1957.pp. 32-37.

7. A. J. F. Siegert, "A Systematic Approach to a Class of Problems in theSTheory of Noise and Other Random Phenomena - Part II, Examples,"

IRE Transactions on Information Theor, vol. IT-3, no. 1, March 1957,pp. 38-43.

8. D. Slepian, "Fluctuations of Random Noise Power," Bei! System TechnicalJournal, vol. 37, no. 1, January 1958, pp. 163-184.

9. . Jacobs, "Energy Detection of Gaussian Communication Signals," IEEE 10thNationa: .ommunications Symposium, October 1964, pp. 440-448.

10. M. I. Schwartz, "Distribution of the Time-Average Power of a GaussianProcess," IEEE Transactions on Information Theory, vol. IT-16, no. 1,January 1970.

105

TR 4233

11. B. 0. Steenson and N. C. Stirling, "The Amplitude Distribution and FalseAlarm Rate of Filtered Noise," Proceedings of the IEEE, vol. 53, no. 1,January 1965, pp. 42-55.

12. H. E. Robbins, "The Distribution of a Definite Quadratic Form," Annalsof Mathematical Statistics, vol. 19, 1948, pp. 226-270.

13. J. Gurland, "Distributions of Definite and Indefinite Quadratic Forms,"Annals of Mathematical Statistics, vol. 26, no. 1, March 1955, pp. 122-127.

14. U. Grenander, H. 0. Pollak, and D. Slepian, "Distribution of QuadraticForms in Normal Variates," Journal of the Society for Industrial andApplied Mathematics, vol. 7, December 1959, pp. 374-401.

15. A. H. Nuttall, Numerical Evaluation of Cumulative Probability DistributionFunctions Directly from Characteristic Function., NUSC Report No. 1032,11 August 1.969. Also 'Proceedings of the IEEE, vol. 57, no. 11, November1969, pp. 2071-2072.

16, A. H. Nuttall, Alternate Forms and Computational Considerations forNumerical Evaluation of Cumula-ive Probability Distributions Directlyfrom Characteristic Functions, NUSC Report No. NL-3012, 12 August1970. Also Proceedings of the IEEE, vol. 58, no. 11, November 1970,pp. 1872-1873.

17. J. S. Bendat and A. G. Piersol, Measurement and Aan sis of RandomData, John Wiley and Sons, Inc., New York, 1958, p. 265.

18. A. D. Whalen, Detection of Si nls in Noise, Academic Press, New York,1971.

19. W. B. Davenport and W. L. Root, An Introduction to the Theory of RandomSignals and Noise, McGraw-Hill Book Co., Inc., New York, 1958.

20. R. Courant and D. Hilbert. Methods of Mathematical Physics, vol. I,Interscience Publishers, New York, 1953.

21. J. I. Marcum, "Table of Q Functions,'" Rand Corporation, Santa Monica,California, Research Memorandum No. RM-339, 1 January 1950.

106