NUSC Technical Report 6339

EVEC

Performance of the Optimum- - and Several Suboptimum

Receivers for ThresholdDetection of Known Signals in

Additive, White, Non- Gaussian Noise

Raymond F. IngramRobert Houle

Submarine Electromagnetk Systems Department

24 November 1980

N1USCNaval Underwater Systems Center

i * .Newport, Rhode Island * New London, Connecticut

. vdmd for pmoi rdma iribume wunimaiM

LA.

8i i 29 c'

This research was conducted under NUSC's IR/IED Project No. A51200,"Nonlinear Noise Processing," Principal Investigator, R. F. Ingram (Code 341),Program Element, 61152N, Navy Subproject and Task No. ZR 000 0101, ProgramManager, I. H. Probus (NAVMATO8TI).

The Technical Reviewer for this report was A. H. Nuttali (Code 3302:).

The authors would like to thank R. J. Aiksnoras (Code 341) for his assistance inmeasuring ELF noise distributions a.nd the fitting of them to Middleton's Class BNoise Model.

ReviewedtadApproved: 24 Novemaber 1983

W.A. Von WiukiAsnistaUt Technical Director for Tedkulop

The authors of this report are located at theNew London Laboratory. Naval Underwater Systems Center

New London, Connecticut 06320.

REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORMBE i2. GOVT ACCESSION NO~I 3. RECIPIENT'S CATALOG -4U&'ER

N -T.R-63:39 1 ~_ _ _ _ _ _ _

- - ________S. TYPE O F REPORT &PERIOO-COVfeWE0--

/ <BERFORNANCE OF THE_0PT1IMUN AND 5EVERAL SUBOPTINMU1 / , '.

-'AECEIVERS FOR JHRESHOLD DETECTION OF KNOUW•N __________

SIGNALS IN .ADD'ITIVE, YIrTE, NON-CAUSSIkN NOISE. 6. PERFORMING ORG. REPORT -um8ER

Rayond F. !IngranRobert/Houle

9 PERFORMING ORGANIZ-A.ICH 'dAME AM.O AODRESS 1.PROGRXU TLML. -C - rASICARE.A & WORK UNI

T NUMBERS

Naval Underwater Systems Center / Al0S! A51200New London Laboratory ZROO00101New London, CT 06320

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ELF Noise Optimum Threshold ReceiverELF Receiver Propagation Validation SystemMiddleton's Class B Noise Model Suboptimnu ReceiverNon-Gaussian Noise

Nonlinear Receiver21 £.3STRACT (C-crajue ow ,e-" side if nec~essryn d itenzify -r bloc* mtber

"\The optimum and several suboptimum receivers for the detection of kno-nsignals in additive, white, non-Gaussian noise are identified. The performanceof these nonlinear receivers is parametrically compared with the optimumGaussian receiver using Middleton's Class B Noise Model. The performance of theextremely low frequency (ELF) receiver for the Propagation Validation System isshown to be wi-_hin 0.5 dB of the optimum for high-level ELF noise.---._

DD, c.c.- 1473 S=1.o,, OF IN ov SS IS OSSOLET-SS) l 6

.X

TII 6339

TABLE OF CONTENTS S4

Page

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

INiTRODUCTION.....................................................

OPTIMUJM RECEIVER STRUCTURE....... ... ... ..... ... ... ..... ... ........ 1

PERFORMANCE MEASURE........ ... ... ..... ... ........ ... ..... ..... 3

CANONICAL NOISE MODEL...... ... ... ..... ... ... ..... ... ... ..... ..... 5

COMPARISON' OF NONLINEAR DEVICES IN CLASS B NOISE ENVIRONMENT . .. 7

EL! RECEIVER PERFORMANCE.... ..... ... ........ ... ........ ... ....... 8

CONCLUSIONS....... ... ... ..... ... ........ ... ........ ... ..... ..... 9

REFERENCES........ ... ........ ... ..... ... ... ..... ... ... ..... .... 25

APPENDI -DRVTO OF MME IMPROVEMENT FACTORSFOR SEVERAL NONLINEARITIES...................................A-1

APPENDIX B--NUMERICAL. COMPUTATION OF THE HYPERGEOE!ETRICAND GANMMA FUNCTIONS.........................................B-i

bAccess

, L'Avail-- .C2

DisfI

i/HRvreBlank

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LIST OF ILLUSTRATIONS

Figure Page

1 Block Diagram of Optimum Receiver ..... 102 Typical Noise Densities and Resulting Transfer Functions 103 Simple Suboptimal Nonlinearities ........... 114 Middleton's Class B Noise Probability Density Function,

Where A. = 1.0 and Various a . ................. 115 Middleton's Class B Noise Probability Density Function,

Where a = 1.0 and Various A,, ... ..... ...... 126 Varia.:ion of Iopt and I p/o- with Dynamic Range 137 Improvement Factor for O~ptimum Nonlinearity,

Where A. = 0.01 .............. ....... 148 Improvement Factor for Optimum Nonlinearity,

Where A. = 0.1 .1. .......................... is9 Improvement Factor for Optimum Nonlinearity,

Where A, = 1.0 .................... .. .. 1610 Improvement Factor for Optimum Nonlinearity,

'here = 1.s ......................... . 1711 Clipper Performance Relative to Optimum Performance,

2here Am = 0.01 .......................... is-12 Clipper Performance Relative to Optimum Performance,

Where a = 1.0 . ............... ........... 1913 Clipper Performance Relative to Optimum Performance,

Where Am = 1.5 ........ ....... ........ 20I4 Hole Puncher Performance Relative to ODtimum Receiver 2115 APD of High Level ELF Noise, Where Aa = 1.5 and a = 1.4 . 2216 APD of High Level ELF Noise, here A. = 1.0 and a = 1.2 . . 2317 APD of High Level ELF Noise, Where Aa = 1.0 and

A-1 Optim Receiver Structure for Threshold Detection . 4. A-1A B-I Graphical Representation of the Quadratic Function(n + W} zn+l

(a), a + .-. b(t - . . . -.... B-3

B-2 Flow Diagram for Evaluating the Hypergeometric Function . B-SB-3 Flow Diagram for Evaluating the Gama Function . . ..... B-6

__ •ii i/ivSReverse Blank •

TR 6

PERFORMANCE OF THE OPTIMUM AND SEVERAL SUBOPTIMUMRECEIVERS FOR THRESHOLD DETECTION OF KNOWN SIGNALS

IN ADDITIVE, %HITE, NON-GAUSSIAN NOISE

INTRODUCTION

The additive noise encountered at a receiver input is often non-Gaussian.If the non-Gaussian nature of the noise is Tot taken into account in thedesign of the receiver, significant performance degradation of the receivercan be expected. The purpose of this technical report is to describe the per-formance that can be expected from the optimum and several suboptimum receiversused for detecting known threshold signals in an additive, white (i.e.,statistically independent noise samples), non-Gaussian noise environment.

The optimum receiver structure for detecting known threshold signals

in additive, white, non-Gaussian noise is the same as that which should beused if the noise were Gaussian, except that a zero memory nonlinearity isplaced between the receiver input and the Gaussian detector. The input-output characteristic of the nonlinearity is given by -d/dx (In pn(x)), wherepn(x) is the density function of the noise alone. The measure of performanceimprovement obtained by including the nonlinear device is given by the ratio

of the signal to noise ratio (SNR) at the receiver output with the nonlinear-ity in the circuit to the SNR at the receiver output without the nonline-arity. The magnitude of this improvement is evaluated here using Middieton'sClass B Noise Model. The performance of several more easily implementedsuboptium nonlinearities, i.e., clipper, hole runcher, and hard limiter,are also evaluated using Middleton's model; and typical performancesrelative to the optimum are presented. Finally, the amplitude probabilitydistributions (APD's) of high level extremely low frequency (ELF) noise aregiven and compared with the APD's resulting from Middleton's model. Fromthese APD's and the parametric results presented, the expected performanceof the optimum and suboptiium nonlinearities is derived.

S..OPTIMUM RECEIVER STRUCTURE

The optimum receiver structure for the threshold detection of binarycoherent signals in an additive, white-noise environment is well known. •The derivation of this structure is summarized here.

Consider a discrete time, memoryless commmication channel in which oneof two known signals, = [SIS12, "' SN]' or_ = [S 2 1 , S22, - S2N]',is transmitted every T seconds by successively transmitting the signal compo-

nents, Sji, (i = 1, 2, ---, N), every At seconds. Each digit, zi, in thereceived sequence of digits z, is the sum of one of two known signals and awhite-noise component, hi. The receiver, then, mast decide between the twohypotheses, Hl: z = S 1 + n and H2 : z = _2 + n, in such a manner that some

I criterion is optimized. For example, one may wish to minimize the expected

A-

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risk (Bayes Test) if costs have been assigned to all joint outcomes ofcorrectly or incorrectly choosing a particular hypothesis. Alternatively,lacking such a cost structure, one may minimize the probability of an in-correct choice (Ideal Observer Scheme).

Because the noise is white and the signal components are known con-pletely, the conditional probability density function of the received vector,z, can be expressed in terms of the density functions of the individual noisecomponents, Pn (ai); i.e.,

Np(z/H) = pz(z ,_ z .. z/d) A l p ( - Si) Ci = 1,2)

The receiver decides between Hl and H2 by comparing pz(z/H1) withpz(z/H2) relative to some constant threshold, A, determinea b--y the a prioripiobabilities of Hl and H2 and the optimality criterion. This is to say thatthe receiver forms the following likelihood ratio as its test statistic:

-; NS pn(Zi - Sli) H,

A pz(z/H1) i=l > i A

P 21 NPn(Zi - S2i) H2

Since the logarithm function is ronoton-ic, an equivalent test is

N N HIn A = Ir. p r:._Si • I p(i-Si In XinA -WS.i d IniLl P'n~li -'2i)~ ' A.i=l 1" z "1

Considering the case of threshold detectioa, lit the signal components,S -i(j = 1,2, i = 1,2 ... , N), be sufficiently s€-a3l so that each term in theat-ove expression can be expanded in a first order Taylor series about z.The log likelihood ratio test then becomes1

S~~~In A n(i Sli In pn(zi) •_i~l •i

dz1

n [ in - In Pn(Z> nX-inPn(zi) S 2i~ d lpzzi 1- <

i3l 1H4

~--------- -- ~- 2

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orN N

lnA "= -n pn lnp( i Pn(zi) "InA (1)1=1 Aidz. n iJ 21i7 dz. in<S• H2

Equation (1) implies the block diagram of figure 1 for the optir"alireceiver structure. Note that it is just a correlation receiver in whicha zero memory function, - d/dzi (in pn(zi)), operates on the received signalprior to the correlation. Note also that the receiver is canonical; i.e.,its general form and decision logic is not based on any particular model ofthe noise environment. However, the receiver must "knoW" (or estimate) thenoise density function in order to implement the - d/dzi (ln pn(zi) operationin the block diagram. If the noise environment is changing slowly in time, Athe optimal receiver must be adaptive.

When the noise is Gaussian, the - d/dzi (In pn(Zi)) function is linear inzi and, as such, may be incorporated as a gain factor (weight) within thesignal component, Si,, so that the optimum receiver structure is exactlythat of a correlator:

d d z1 /2 2 -Gaussian noise d In pn(zi) = -

~d .i z . . n 2iia

For a non-Gaussian noise density, however, the derivative of the log densityfunction is not a linear function of the received waveforms. Thus, the"optimum receiver structure is a correlator preceded by a zero memory circuitwhose input-output characteristics depend cnly on the noise probabilityder.sity function.

As an example, it has been proposed that naturally occurring electro-mag-ictic noise below 100 MHz is Gaussian distributed with occasional largeamplitude noise bursts, which occur often enough to affect the tails of thenoise densil ". Several authors"- 6 have suggested that the density functiondescribing t -ese tails folloiws an e-aIzil (0 < k < 1) law. The smaller thek, the more impulsive the noise (see figure 2a). For this noise density,the input-autput characteristic of the optimal nonlinearity is shown in 7figure 2b. As the noise becomes more impulsive, the nonlinear devicesuppresses the larger noise excursions to a greater degree.pI

t PERFORMANCE MEASURE

After the optimal receiver structure has been derived, a natural questionto ask is how much performance improvement can be expected using this receivercompared with one in which the nonlinearity prior to the correlator is sub-optimal or absent? Usually the performance of a receiver is measurcd by theprobability of error for a given signal energy. This requires that theprobability density function of the test statistic be known or, alternatively,

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that the probability of error can be bounded in some manner. Under theassumption of threshold-type signals, the correlator must sum many receivedsymbols to increase the SNR for satisfactory performance. As was seenpreviously, the nonlinear device tends to suppress large amplitude vari-

ations; thus, each received digit is bounded. (From a practical point ofview, the received symbols are always bounded owing to the finite responsesof amplifiers, filters, etc., even when the receiver is linear.) The teststatistic, therefore, is the sum of independent, finite variance randomvariables and, by the Central Limit The~rei, is asymptotically Gaussiandistributed. Since the density function of a normal random variable iscompletely specified by its mean and variance, the SNR at the output of thecorrelator may be used to express the receiver performance. P!ence, definethe improvement factor, I, as the ratio of the SNR's at the output of thereceivers with and without a nonlinear noise processor:

SNRNL SNR at output of receiver with nonlinearity presentNIR = S -§,W -at output of receiver with nonlinearity removed

Assuming threshold binary equal-energy signals, the improvement factorfor the optimal receiver is (see appendix A)

[ad 1

CF~J [ d l ]2 2~ d - ( pn(- Iopt - In p pn) dz =o p(z) dz , (2a)

where a- is the variance of a received symbol (i.e,, the power of the noisecomponent). When the noise is Gaussian, the improvement factor equals one.

Although the optimal receiver structure is canonical in form, it may bedifficult to implement. This leads one to investigate simpler, suboptimalnonlinear processors using the improvement factor to compare the performanceof various processors. Appendix A derives the improvement factor for threecommon nonlinear devices: the hard limiter, the clipper, and the holepuncher. The results are listed here for convenience:

1HL 4 2 2 (0) (2b)

nI'= 2 pn(z) dz +i c2(1 - p)(2c)

IH= l2 p - 2c Pn'(c) 2 J' z pn(z) dz , (2d)

4

S... . .. . ..-.- , r- T i

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where

p Pn(z) dz = Probability of being in linear range of nonlinearity

-C

c = Linear range of nonlinearity (see figure 3).

The input-output characteristics of the above devices are shown infigure 3. Note that the hard limiter is the optimum nonlinearity when thenoise has a density function of the form pn(z) = Ae-lz[K Comparing figure 2bwith figure 3, one sees that the clipper is optimal and that the hole punJceris near optimal when the noise is Gaussian with tails distributed as e-!zlfor k = 1 and 0, respectively. It should also be pointed out that althoughthe noise power, a 2 , appears in each of the above expressions, it does notaffect the comparison of one nonlinear device with another, though it cer-tainly is important when comparing a nonlinear receiver with a linear one.

CANONICAL NOISE MODEL

As is evident in the above discussion, the optimal receiver structureis highly dependent on the probability density function of the noise.Although various noise models have been proposed for particular environments,the only general model available to date is that suggested by Middleton.- 9

His model, the Class B Noise Model, which is analytically tractable and is inexcellent agreement with measured data for a variety of cases, is canonicalin nature. That is to say the noise density function, which is characterizedby various parameters, does not change form.

Based on the bandwidth of the interference relative to the receiverbandwidth, Middleton defines two major classes of noise. Of interest hereis the Class B Noise Model in which the interference is highly impulsive,resulting in a noise bandwidth much greater than the receiver's front endbandwidth. For this case, the model for the amplitude noise probabilitydensity is expressable as two distinct functions: one valid for izi : zoI•Band the other for JzJ > ZoB . The second function approaches zero muchmore rapidly than the first as z ÷ . It was already pointed out that anonlinear device operat-ng on threshold signals in this type of noise envi-ronment will suppress large voltage excursions in the received waveform.Thus, the noise power at the output of the nonlinear device is pA-imarilydetermined by the noise density function, pn(z), for Izi less than zc ,i.e., the value at which significant suppression occurs. The more impul-4• sive the noise environment, the more likely zc is much smaller than zoBFor this reason, only the first function in Middleton's model is used here.Thus,

CI A-( p(z) t- ( E -l Fl (!+ 1 1Pnm. " 2 2z P M'S~M=O

I sz . B (3)

S.. . .

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where

0<ca<2., A >0

r(-) = Garma Function

F (a;b;x) = Confluent Hypergeometric Function.

Three values, A. , , and ZoB , define the noise model. One cannormalize the probability density function defined by these three values tounit variance, for I z < zo8 , by introducing a fourth parameter, 1 . Thisis done by changing the variable of integration from z to z as follows:

a2(z z 2po(z) dzoB n

oBiit pn(1 d- 4=> 1 o= .pC2 d•_ * 4)1

OZB

where

z 4b ZB ro h

-Z oB

R 1/0 (z ____ ___ ___

m m

1 (-I) A (ml+l:F(nt +1 !; 2J 5pMz a! -2 Jl 2 2

:=m 0

Z4Typical density functions normalized for unit power with zoB 10 are

plotted on a log-log scale in figures 4 and S. Note that for constant A.(figure 4), the more impulsive densities, i.e., those whose tails approachzero more slowly, are characterized by smaller a values. In fact, one cansee in figure 5 that the slope of the tails is determined by the a parameter.

* Although the role of A. is not as readily apperent as that of a, it isrelated to the slope of the density function during the transition from theGaussian portion of the distribution to the non-Gaussian tails and to therange over which the density function is Gaussian-like. The "bendover"1 point,ZoB, is that point beyond which equation (3) no longer adequately describesthe true density.

4-6= .A

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COMPARISON OF NONLINEAR DEVICES IN CLASS B NOISE ENVIRONQENT

The improvement factor of a nonlinear receiver is a function of the totalnoise power, the shape of the noise density function that yields that power,and the input-output characteristics of the nonlinear device. Equation (4)indicates that the noise power depends on ZoB, the bendover point inMiddleton's Class B Noise Model. Intuitively, the more ZoB exceeds thesuppression range of a nonlinear device, the greater the improvement factorfor that device. An example of this can be seen in figure 6. Note that theincrease in the improvement factor is entirely due to the increase in thevariance once zoB exceeds a particular value.

Figures-7, 8, 9, and 10 show the improvement factor Zor the optimal non-linear receiver versus ioB for various noise statistics. (These results

were obtained using equations (2a) and (4) and normalizing as in equation (5).)For these calculations pn(z) for z > ZoB was assumed to be zero. As Aaapproaches zero (figure 7), the density function is Gaussian over a greaterrange of values and, therefore, the improvement factor depends strongly onthe shape of the tails. If the tails are decaying rapidly (larger a), onlymodest improvements are to be expected. On the other hand, if the tails IA,

Sapproach zero very slowly (small a), significant processing gains may beachieved using the optimum nonlinear receiver over a linear one. As A.increases (figures 8,9, and 10), the noise is Gaussian over a smalier rangeof values; and improvements will occur at smaller values of ZoB. Here, also,the shape of the tails greatly affect the processing gain. In general, fora given zoB, the improvement factors increase as Aa increases and as adecreases.

One of the primary goals of this report is to rate the performance ofsimpler suboptimal nonlinear receivers. The improvement factor for the hardlimiter, the clipper, and the hole puncher were evaluated under a variety ofnoise conditions using Middleton's model and equations (2b)-(2d). Typicalresults are shown in figures 11-14. The performance of the clipper and the

D: hole puncher are functions of the "clip level," i.e., the percentage of timethe input is in the nonlinear portion of the device's operating characteris-

tics. The hard limiter is a special case of the clipper in which the cliplevel is set at 100 percent.

Figures 11, 12, and 13 present the ratio of the clipper improvementfactor to optimum improvement factor versus clip percentage for various Aand a. While the optimum clip level, i.e., the clip level that results inthe best performance, depends on the noise parameters defining the densityfunction, the performance of the clipper relative to the optimum nonlinearity4 •is not a strong function of the percentage of time that the input is in thenonlinear portion of the transfer function. In fact, by setting the cliplevel to obtain 90 percent clipping, the clipper performance will remainwithin about 4.5 dB of the optimum over a large range of noise parameters.When figures 7-10 are used as an absolute measure, the optimal improvementfactor is large ior just those values of a for which the clippers' perform-ance decreases. As the optimal performance drops in value due to an in-

creasing a, the clipper's performance increases so that, relatively speak-ing, the clipper is always close to the optimal.

-"- , ,- 7

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Even simpler than the clipper is the hard limiter, which approaches theperformance of the clipper without the requirement cf adaptive control of I"the clip level. The more impulsive the noise, the closer the hard liiiter's

performance approaches that of the optimum clipper.

The relative performance of the hole puncher (figure 14) is very sensi- Ttive to both the clip level and the parameters of the noise density; but,when adjusted properly, it essentially can achieve optimal performance. Thus,while the transfer characteristic is easily implemented, logic is requiredto estimate the noise parameters, and adaptive control is needed to maintainthe proper clip level.

The four nonlinear devices discussed in the report form a hierarchy ofreceivers based on ct.mplexity and performance. Although each applicationdetermines the exact tradeoffs, there are many cases in which the simplersuboptimal devices are more cost effective than the optimal.

ELF RECEIVER PERFORMANCE

In this section the general results obtained in the previous sectionsare applied to the ELF range of electromagnetic comimunications. Several digi-tal tape recordings of t1-pical high level ELF noise from the Saipan area wereanalyzed to yield amplitude probability distributions. These measured distri-butions are compared with distributions computed from Middleton's Class BNoise Model , equation (5), and the parameters A., a, and ZOB are determined.From these parameter values and figures 7, 8, and 9 the expected performance"for the optimum and suboptimum nonlinearities can be determined.

Figures 15-17 are plots of the amplitude probability distributions oftypical high level ELF noise from Saipan. The noise was bandpass filteredto 20-150 Hz and each distribution was measured over 400 s. The ordinateof these figures is the percentage of tine that the magnitude of the noiseexceeds the abscissa level; the reference level of the abscissa is the rmslevel of the noise. The individual squares on the plots are points from

- ��Middleton's model. From these plots it is clear that the amplitude proba-bility distributions and density functions of actual ELF noise can bedescribed accurately by the amplitude distributions and densities resultingfrom Middleton's model. For these high levels of ELF noise, the parametervalues range over A. = 1.0 - 1.5, a = 1.2 - 1.4, and ZoB approaching 40 dB.An optimally designed nonlinear receiver could be expected to perform I0-20 dBbetter than a linear receiver for these parameter ranges; however, the c--tualperformance will depend on the particular combination of parameter values.If a clipper were utilized and set to clip between 20-80 percent of the time,

A its performance would be within about 1 dB of the optimum nonlinearity.

The current Propagation Validation System (PVS) ELF receiver utilizesa clipper that adjusts itself so that the received signal and noise is clipped40 percent of the time. This rather simple device, which does not necessitateany complex noise parameter estimation, provides performnance that is

8

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comparable to that provided with the optimm nonlinearity. A detailedanalysis of the nonlinear noise processing, including the aspects leadingto the design and measured performance utilized in the Navy's ELFreceivers, is contained in references 1 and 10.

"CONCLUSIONS

The optimm receiver for detecting known threshold signals in additive,white, non-Gaussian noise has been descriled and its performance is definedas a function of the noise amplitude probability density function. Theoptimum receiver's performance has been calculated and plotted versus variousranges of the parameters defining Middleton's Class B Noise Model. Promthese plots it is apparent that the optimum nonlinear receiver can yield verysignificant performance improvements relative to the receiver that is optimin Gaussian noise. Implementation of the optima nonlinearity, however, canbe rather complex, requiring real-time estimation of the noise parameters.

"The performances of several suboptimum nonlinearities were calculatedand their performance relative to the optimum was plotted. A properlyadjusted hole puncher yields performance within 1 dS of the optimau; however,its' relative performance is very sensitive to the percentage of time theinput is suppressed. Proper adjustment requires estimating the noise param-eters, as is also required for the optimum nonlinearity. The performances ofthe clipper depend on the noise parameter values and the percentage of clip,as does the hole puncher; however, it is less sensitive to these values thanis the hole puncher. A simple clipper adjusted to clip 90 percent of the timeyields performance within 4.5 dB of the optimum over a wide range of noiseparameters; the relative performance improves as the noise becomes lessimpulsive. The hard limiter, requiring no adjustment, yields performance within2 dB of the optimum clipper; the performance relative to the optimu clipperiaproves as the noise becomes more impulsive.

The instantaneous amplitude probability density function resulting fromMiddleton's model fits measured high level ELF noise quite closely withA• = 1.0 - l.S, a =.2 - 1.4, and ;oB 2 40 dB. For this range of parameters,a clipper adjusted to clip 40 percent of the time, as does the PVS ELFreceiver, provides performance within 0.5 dB of the optimm.

Reco3mendations for future work include

=i• I. Determine optimua receiver's performance sensitivity to inaccuraciesin parameter estimation.

2. Extend results to determine performance of these thresholdreceivers for nonthreshold signals.

3. Compare predicted performance of the optimum receiver, as definedhere, at very low frequency (VLF) with measured performance of existing VLFreceivers.

4 9"-------------------------e- :t m'..- -r W..A-- -- •4V

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Figure 1. Block Diagram of Optimum~ Receiver

100.

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OUT

fC

1*' -C C - C 1

1ý1 I lt'Z LOIERI IbW C1??MX ICI tMi '.1CM

Figure 3. Simple Suboptimal Nonlinearities

1.0

. . . .1.: 10.SI • *.-" f_ .. z. :*10

S* l.0 :.. * 3.1 10 1

- f - 0

-42

Figure 4. Middleton~s Class 8 Noise Probability Density Function,M~ere A =1.0 and Various a

_11•"-

1->

TR 63391

gL

AS

I [z

.30 .30 .~ *z1 to ao so D O- 5t .0 -=6 -• 0 3c •o .0 zo+ .

Figure S. Middleton's Class B Noise Probability Density Function,Mhere a = 1.0 and Various A

-- C

i12

- -- I l

'" "TR 6339

30OW! +

Ion

10 ,03

--- • '• . 40"-

.60

-10 10 -O 30 ZO 70 so

4 2SFigure 6. Variation of I and Iop With Dynamic Rangeopt o

13

- ----- -77--7 77- -- + .

TR 6339

rI

A • 0A01

Fu 7 -nfz

l • .0

4T

14

•"I1•Figure 7. Thprovemsent Factor for Optiurm Nonlineaity, WhereA = 0.01

m~ii

14 :

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to __l

5

I 0.-I

'P i8I m

ii I

SO30:0.. s

TR 6339

4I .0 ;

I-!I III.

.,°• Figure 9. Inprovement Factor for OptiJma Nonlinearity, Where Aa 1.016.

'---0

rmI0

0•IIIKu2 3 0 03

.0lo

Fiue9 mrvetFatrfrOtauNninaiy hr .

"F ci1

' , ,Ii°! 1 l i i I / I i

TR 6339

iV

1 171

1_

jlilI

Figure 10. Improveaent Factor for 0ptiwm Nonlinearity, khere A = 1.5

=17

1m'S"--.... . . . ."-I'' • , • . . .

Ii

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I-I

iz-4C

II • , I I I I I

o 10 " 30 4* so bo 70 so I(O

Figure 11. Clipper Performance Relative to Optimum Performance,Where A =0.01

Ia

1-ii

Z..2fl*4 iiC ..- t•+c

=TR 6339

mA

Il

'h e r e A 1 .0

SIt II I "

t2 -shreA = .

"-1:

11

ill I

I I: l -"

ll It :

!.! •-•Fig 12.CliperPer~riace elatve o Oti~lrerozan19

TR 6339

.0 . .....

-Iz

a 19. 20 so Wa ~ ~ 70 so 90 Ice

-~~--lXLTM~ CL~r

Figure 13. Clipper Performance Relative to Optimum Performance,NWhere A O= 1.5

20I

0 .5 0.-4

\| I' A -.10

¢-•" aI IS• =1 I 't

7/I I

•' I :

= ~IS- I 1

*, \I I

-J II I5

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n t

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-I" / n -

To R 6339

g.1

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so ~A 1.. 1.4

70

°-- 30so

P 40

I-.0$•o.oea

0.0000I F I-'0 -. o -50 .-4 -30 -_10 -10 a 10 30 zo so 0 O

Figure 1S. APD of High Level ELF Noise, Where a = 1.5 and a = 1.4

22

TR 6339

SIAPVC - TAM NO. XG~-031. RECM 1-1005Sa a o am- S- M 3015-s MWEi

70 2

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,i il -.-

TR 6339

91

u s .• . S ~1?A . -A R A N T A . 3 k ) . A .,- 0 3 1 . a 7 0 0 -8 0 0*00U0 AMO FXK PUEV0WIUS NOS 1S3 a

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$ 0ý

00.010

0~0o . €c ~i ~I I ! ! ! ! ! I 1 I-70 - , -SO -£0 -30o .2' -10 0 to 20 30 £0 so 60 70

Figure 17. APD of High Level ELF Noise, Where A = 1.0 anda= 1.2 -1.4 a

24<I I 1l-4

TR 6339

REFERENCES

1. J. E. Evans and A. S. Griffiths, "Design of a Sanguine Noise ProcessorEased on World-Wide Extremely Low Frequency (ELF) Recordings," IEEETransactions on Comunications, vol. C0M-22, April 1974, pp. 528-539.

2. S. S. Rappaport and L. Kurz, "An Optimal Nonlinear Detector for DigitalData Transnission Through Non-Gaussian Channels," IEEE Transactions inComamnications Technology, vol. COM-14, June 1966, pp. 266-274.

3. 0. Antonov, "Optimal Detection of Signals in Non-Gaussian Noise," RadioEngineering and Electronic Physics (USSR), vol. 12, 1967, pp. 541-548.

4. A. D. Watt and E. L. Maxwell, "Measured Statistical Characteristics ofVLF Atmospheric Radio Noise," Proceedings of the Institute of RadioEngineers, Inc.. vol. 45, no. 1, January 1957, pp. 55-62.

5. W. Q. Crichlow, C. J. Roubique, A. D. Spaulding, and W. M. Berry."Determination of the Amplitude Probability Distribution of AtmosphericRadio Noise from Statistical Moments," Journal of Research of theNational Bureau of Standards, vol. 64 D, no. 1, January 1960, pp. 49-56.

6. H. Yuhada, T. Ishida, and M. Higashimara, "Measurement of the .AmplitudeProbability Distribution, of Atmospheric Noise," Journal of the Radio atResearch Laboratory, Japan, vol. 3, no. 11, January 1956, pp. 101-109.

7. D. Middleton, Statistical-Physical Models of Man-Made Radio-Noise,Part I: First-Order Probability Models of the Instantaneous Amplitude,Office of Telecommunications, Technical Report 0T-74-36, U.S. GovernmentPrinting Office, Washington, DC 20402, April 1974.

8. D. Middleton, Statistical Physical Models of Man-Made and Natural RadioNoise, Part II: First-Order Probability Models of the Envelope andPhase, Office of Telecunmications Technical Report OT-76-86, U.S.

* -_ Government Printing Office, Washington, DC 20402, April 1976.

E. D. Middleton, Statistical-Physical Models of Man-Made and Natural RadioNoise, Part III: First-Order Probability Models of the InstantaneousAmplitude of Class B Interference, Office of Telecounications TechnicalReport NTIA-CR-78-1, U.S. Government Printing Office, Washington, DC

* 20402, June 1978.

10. A. Griffiths, ELF Noise Processing, Lincoln laboratory Technical Report490, Massachusetts Institute of Technology, Lexington, January 1972(DOC AD-739907).

25/26-- Reverse Blank

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Appendix A

DERIVATION OF THE IMPROVEMMT FACTORSFOR SEVERAL NONLINEARITIES

In this appendix the improvement factors for the optimum nonlinearity,the hard limiter, the clipper, and the hole puncher are derived. The i2prove-ument factor is defined as the signal to noise ratio (SIM) at the correlatoroutputwith the appropriate nonlinearity in the signal pathdiviaed by theSNR with the nonlinearity removed. The improvement factor is an indicationof the performance improvement or degradation that is expected owing toinclusion of the nonlinear device in the signal path. The test statistic atthe correlator output is the sum of independent, finite variance randomvariables; and by the Central Limit Theorem it is asymptotically Gaussiandistributed- Hence, the output SMR is sufficient for describing receiverperformance for large N.

The receiver structure of interest is shown in figure A-1. Each inputsample, xi, is the sum of a completely known signal sample, Si, and an inde-pendent noise sample, ni. The signal samples are considered to have comefrom one of two binary equal-energy signals; the noise samples are identicallydistributed zero mean with variance 04 and have a symetric first orderamplitude probability density function. The yi are the outputs of a zeromemory nonlinearity, f(xj). The test statistic, 6, is the sum over i of Nproducts of the form Sliy The SNR at the correlator output is defined asthe ratio of the square of the expected value of 6 and the variance of 6.Remembering that the Si are completely known, the SNR at the correlator outputwith the nonlinearity in the circuit is easily found to be given by(N2

•i• S•:L= S2Saiy 1

i=1 lAl

SNR%'

S Vat(y i)

* IFigure Aol. Optimum Receiver Structure for Threshold Detection

With the nonlinearity removed•, the SNR at the correlator output is given by

Si

NL= i= .. (A-2)

X

0n

S..

EAl

TR 6339

resulting in an improvement factor, I, of J

SNL n(iil1\i___ Ll ~(A-3)

In order to evaluate the improvement factor for the various nonlinear-ities of interest, the mean and variarce of the samples at the nonlinear out-put are nbeded.

For threshold signals in independent, identically distributed noise, the* optium nonlinearity takes the form

yi = f(xi) = - In PhC(x) ( A-4)

where pn (xi) is the amplitude probability density function of the noise-onlyi samples. The expected value of yi is given by

r= i In Pn (xi) Pn(xi - dx (A-S)

It is possible to expand pn(xi -Si) in a Taylor series about xi, and theterms in Si of order two and greater can be dropped for threshold signals.Then, using the fact that pn(xi) is symmetrical, yi is found to be

S~Iyi SAL (A-6)

Using a similar approach, we find the variance of the nonline-Ar output to be

Var(yi) = L(l - S(A-7)

The improvement factor for the optimum nonlinearity is, then, given by

2n]) n 1

A-2

TR 6339

2with equality as S. L approaches zero. a

1

The hard limiter is defined by a transfer function such that y-. +x- > 0 and Y-. -1 for x. < 0. The expected value of y- is given b

2 pn(x- S,) dxi] 1 (A-9)

where the expression within the brackets is sinply the probability of y= +1.Since we are interested in comparing the various nonlinearities under similarsignal conditions, i.e., threshold signals, Si can be assumed small. Forsmall S- and a symmetrical noise density function,

y.=2S. pO (A-10)Yi i Pn(0)

and

2 2Var1- ( Pn(0) . (A-11)

Substituting equations (A-10) and (A-Il) into (A-3) yields the improvementfor the hard limiter,

4L (0) (A-12)

2 2with equality as 4S, p (0) approaches zero.

The clipper is described by a transfer function such that

II

-c x- <- c, (A-13)] i

where c is a constant pre-ater than zero. The expected value of yi is foundfrom

•|•iil Yi=I xi Pn(xi _Si)dx +' ni 4-S~x f[- (- n pxi -$,d

(A-34)

A-

TR 6339

"Again, letting S. beccue small,1

y= Sip--,

where c

p = 21 pn(x) dxi (A-15)0

In a similar manner, the variance of the clipper output can be found:

Var(yi)= 2 xi Pn(xi) dxi c2(I - p) (A-16)•ii

0

Substituting the above expressions into (A-3) yields the improvement factorfor the clipper:

2

* = 0 (A-17)

2 .. i Pn(Xi) dx1 + l - p)

The hole puncher is sinilar to the clipper except that the output is setto zero if the input exceeds the threshold c. The mean and variance at theoutput of the hole puncher are found by following the procedure utilized withthe clipper. The mean is given by

• -= p[ - 2c pn(c)] (A-18)

where p was defined in equation (A-15) and pn(c) is the noise density functionP \ evaluated at c. The variance is given by

j2

Var(yi2) = p (x-) dxi (A-19)f ni

0

and the improvement factor by

nI -c] (A-20)

2 x pn(x.i) dx

0

AA-4

TR 6339

It is interesting to note that while the SNR's at the correlator outputdepend on the signal structure,the improvement factors for the four nonlin-earities of interest are independent o-7 the values of the signal samples.This is a result of the small SNR assumption.

I-

A-S/A-6Reverse Blank

- -"-. -- - -- -AZ°n ~~n. - -- •

TR 6339

!N -- •

Appendix B

NUMERICAL COPUTA..TION OF THEHYPERGEOMETRIC AND GAMMA FRNCTIONS I1

Inherent within the results of this report is the need to evaluate thedensity function , pn(z), for Middleton's Class B Noise Model and its deriv-ative, P. (z). These functions are expressed in equations (B-l) and (B-2);p' (z) is derived from pn(z) with the aid of the following identities:

dadz iFl(a;b; z) = iFl(a + 1; b 1; z)ýff1 1] 1b1-

ar(a) r(a + 1)

ii•, (_l)m Am

a__i .= = - r + A a +l 1 2) Iz < Z

p'(z) dIz M4 (1z 1 ~ 2 ~ ;jj<B

.... r- 2 /1 1F l(_ 2 Y •- Il< oBS-0

-n -1) le~a (3M +X _ma2) I'Pn = nz d9 E4 R.C-)! 1 1 )ll 2 ;2 ; ow"

3=0

(B-2)where

"0 o< a<2 A >0

r(a) =Gamma Function

1F, (a; b; c)= Confluent Hypergeometric Function.

Thus, both p (z) and its derivative require computating the HypergeometricFunction, lF1(a; b; -z), and the Gamma Function, rCa). For all values of aand the index of summation, a, indicated in equations (B-l) and (B-2), theparameters a, b, and z satisfy a > b > 0. and z > 0. This appendix outlinesthe algorithms used in this report to compute the Hypergeometric and GammaFunctions for these ranges of arguments.

The Confluent Hypergeometric Function, 1 F (a; b; -z), is defined inequaticn (B-3). This equation, however, is suitable for the numerical corn-putation of iFl(a; b; -z) only for small arguments of a and z. Thus, equa-tions (B-4) through (B-6) are needed to evaluate the Hypergeometric Functionfor a broader range of arguments, in particular large z;

B-1

TR 6339

°iii

( Ca)( zF (a; b; -z) =• (B-3)

i• i•-1=0 (b) i!

1 (a; b; -z) e- Cb- a; b; z) (B-4)

(b - a) 1 F (a - 1b; -z) + (2a - b - z) 1Fl(a;b; -z) - a 1 Fl(a + 1: b; -z) = 0

(B-5)

1F 1 (a;b; -z) 1 - a r(b) Nia (.Z > 0), (B-6)r b-a i=0 i! z

where

(a). = a(a + 1) (a + 2)-.-(a + i - 1) = , (a + j); (a) = 11J=o

N = index of the smallestterm in equation (B-6).

Whereas equations (B-3) through (B-5) are identities, equation (B-6)is an asymptotic series approximation to the Hypergeometric Function validonly for large z with respect to a and b. Note, however, that this as)mptoticseries is divergent for any given z since

((a)n (a + 1 - b)n

n.- n n(li n n li (n 1 + 1)! z /

n-- -, it n-' ,(a (n n ) n )n

I lin (a +n) (a + I -b + n) _O

n-- (n + 1) z

- •. Hence, the point of truncation of the series, N, is important when usingequation (B-6) to approximate the Hypergeometric Function.

To understand the behavior of the terms of this asymptotic expansion,* consider the following:

(a)n+ (a + I b)n÷ (a)n (a + I -b)n:~ tn+l tn n +I. n+l -n' n

•i(a)n (a + I b)n

(nn! ((a + n) (a + 1 - b + n) -(n . l)z]

B-2

TR 6339

n~lSince (a)n(a + 1 b) /(n + W z is positive, the sign of tn+ - t isi*. determined by the quaZIratic equation

[(a + n) (a + 1- b + n) - (n + 1) z]

2 2=.[n + (2a + 1- b -z)n + (a + a -ab +z)],

which has roots at

nln 2 = (z + b - 2a - 1 4 + 2z(1 + b 2a) + (1 b )/2

Figure (B-i) is a graphical representation of this quadratic function.SI.

Figure B-1. Graphical Representation of the Quadraticn+1

Function (n + ) t)(a) (a + I - b) n+l n

If nI and n2 are both less than zero, equation (B-6) diverges immediatelyand should not be used to approximate the Hypergeometric Function for thegiven values of a, b, and z. If, however, nI and n 2 are of the opposite sign,the terms of the series initially decrease in value until n = P2 and, then,grow indefinitely. When both n, and n 2 are positive, the terms initiallyincrease in value, then decrease, and finally grow indefinitely again. Inthese latter two cases, one can find the smallest term of the series (i.e., thestopping point) by comparing the (n + 1)th term with th- nth term only afterthe inde- exceeds nI. Thus, if n > nl, tn+l > tn and tn < to = 1 are allsatisfied, the stopping index N equals n. If this last condition, tn < t isthe only one not satisfied, the first term of the expansion, to, must be •hesmallest and N is set equal to 0.

The computation of pn(z) or pA(z) for a particular z requires evaluatingthe Hypergeometric Function via a series in which the "a" argument [ (mra/2)"+ b)] is a linear function of the index of summation and is, therefore,increasing as more terms are included in the series. For large z, one hopesthat the asymptotic expression (equation (B-6)) can be used, and that thetfr(-)/m! factor in equations (B-1) and (B-2) decreases rapidly enough sothat only a few terms in the summation need be evaluated to satisfactorilyapproximate pn(z) or p n(z). By choosing to use equation (B-6) only when z > 10,one can guarantee that the asymptotic expression will be valid to approximate

0Fi(a;b; -z), for m = 0,1,2, a = ima/2) + b, b = 1/2 or 312, and 0 < a < 2.Ifhowever, the AI r(*)i!= factor is not decreasing rapidly enough to limitthe number of terms required in equation (B-I) or (B-2), so that a = [(m2.12) +b]becomes too large to use equation (B-6) to approximate the HypergeometricFunction for the given z, then equation (9-6) may be used on a - I or a - 2 --- ,

- and then equation (B-S) can be utilized to iterate on "a" back to its originalvalue.

B-3...............-..-

TR 6339

This iteration technique was also used to evaluate the Gamma Function,

r(z), which appears with positive arguments in equations (B-1) and (B-2) andnegative arguments in equation (B-6). A polynomial approximation to r(z)(equation (B-7)) was used on the fractional part of the argument, therebyrequiring one to use the recursion relationship in equation (B-8) to step theargument up or down to its desired value:

8 *1

r (z + 1) 'T 1 ÷+ ci zi 0 <_ z < I (B-7)

zr(z) = r(z + 1) (B-8)

i |i ~-Z (1/12z_)r(z) = e (z-½5 ee 30 < z < 55 ( CB-9)

where

cl = -0.577191652 c 5 -0.756704078

c= 0.988205891 c6 = 0.48219 9-"

c= -0.897056937 c 7 -0.19352 7818

c = 0.918206857 C 0.03586 8343

Figures B-2 and B-3 are flow diagrams of the algorithms used in thisreport to evaluate the Hypergeometric and the Gala Functions, respectively.

-B-4

TR 6339

b4

inu aTI

% D

I3 -SI

Is-E M IXi-Fiur B-.ETDigaAorEUAatIgte lpreotti Fnto

CUW1T

US q AI.;9- -

TR 6339

USE EQUAriON 8-7 ON

., z . INT(-z) - 1.

ITERATE ON z USING?z INTEGER? NO EQUATION B-8 I.[.,

r i z ) z I I i I .1 I

I1'I EUTONB8NO r(z} - MAX VAILUE T-.r - -IT( -

|lISI I!QIIAT|ON B-aI:IZ) - r( z Y

VNI) EQuATION B1-7 0 < a I? RTR

ITO (:ONIBUIrICýi

USNE IiQLAI ION 8-7 ON' = - - I 1I )! ) - I,

I TIRAl II USING; EQUATIONIl-- , I.>,

8

12I)i z(r(z- I

• I USE EQUATION B-930 z R55EYSU-R2-/2

It •r(z) C V/ (±! 2W Ie/12z

RETURN

Figure B-3. Flow Diagram for Evaluating the Gamma Function

B-6

'R 6339

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