AL) AIVI 099 NAVAL UNDERWATER SYSTEMS CENTER NEW LONDON CT NEW LO--ETC fff/o 177ERROR PROBA ILIT CHAR ACTERISTICS FOR MUJLTIPLE ALTERNATIVE CMMETIlMAY :1 A H NUTTALL

UPLA', IE1( NUSCTR-b473 N

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NUSC Technical Report 64734 May 1961

Error Probability Characteristics forMultiple Alternative CommunicationWith Diversity, but Without Fading

Albert H. NuttallSurface Ship Sonar Depariment

~-AA

I

SNaval Underwater Systems CenterZ.) Newport, Rhode Island / New London, ConnecticutLUJ

9 Approved for public roles"; distribution unlimited.

Preface

This research was conducted under NUSC IR/IED Project No. A75205, Sub-project No. ZROOOOIO, Applications of Statistical Communication Theory toAcoustic Signal Processing, Principal Investigator Dr. Albert H. Nuttall (Code3302), Program Manager CAPT David F. Parrish, Naval Material Command(MAT 08L); and under NUSC Project No. C61605, JACS Systems Engineering,Principal Investigator Ronald Malone (Code 3222), Sponsoring Activity Naval SeaSystems Command, Program Manager Gary Stringer (SEA 63Y33).

The technical reviewer for this report was Paul Radics, Jr. (Code 3222).

Reviewed and Approved: 4 May 1981

W. A. Von WinkleAssociate Technical Director

- for Technology

The author of this report is located at theNew London Laboratory, Naval Underwater Systems Cemer,

New London, Connecticut 06320.

.... -- - :I T . _

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( oherelin Proccs ,ing Envelope-Squared (Combinat ionDiversits Fading1-ncrgy-Density Ratio M -ary Communicat ion

[:l ror Probability Signal-to-Noise Rat io

.10.,SSTRACT (Continue on rev'erse side, If necessary and Identify by block number)

The M-ary' character error probability, P,, for a communications system employingD-told diversity combination is derived and evaluated for the case of no fading, over therange of parameters M = 2, 3, 4, 8, 16, 32, 64, 128, 256, 512, 1024 andD = 1, 2, 3, 4, 6, 8. 12, 16, 24, 32, 48, 64 for P. in the ran~ge 10-1 to 10-5. The culrves

presented allow for ready determination of the additional signal-to-noise raIo reqiriied for

emlployinlg diversity when it is not necessary, or when uncertainty forces it tlIPon the receiver

prlcessinlg. Extensions to a thresholding receiver arid to a vecry genleral clss ot signal laIdinlg

are also mnade, hut no nuimerical restilis are given tor these later cases.

DD QI FANM1 1473 EDITION OF I NOV GS IS OBSOLETES/N 0102-014-6601 .

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Table of Contents

Page

List of Illustrations .............................................. .List of Sym bols ................................................ iiiIntroduction ................................................... IDefinitions and Technical Results ................................... 2G raphical Results ............................................... 3Correct Decision with a Threshold .................................. 15Extension to Fading Signal ....................................... 15Discussion and Summary ........................................ 17Appendix A. Derivation of M-ary Character Error Probability ........... A-IAppendix B. Binary Error Probability .............................. B-1Appendix C. Program for M-ary Character Error Probability ............ C-IAppendix D. Average Error Probability for Fading Signal .............. D-1Appendix E. Binary Error Probability via Characteristic Functions ........ E-1Appendix F. Bounds on Error Probability ........................... F-IR eferences ................................................... R -1

List of Illustrations

Figure Page

I M-ary Character Error Probability for M = 2 ................. 42 M-ary Character Error Probability for M = 3 ................. 53 W-ary Character Error Probability for M = 4 ................. 64 M-ary Character Error Probability for M = 8 ................. 75 M-ary Character Error Probability for M = 16 ................ 86 M-ary Character Error Probability for M = 32 ................ 97 M-ary Character Error Probability for M = 64 ................ . 0.I8 MWary Character Error Probability for M = 128..................19 W-ary Character Error Probability for M = 256 ............... . 1210 M-ary Character Error Probability for M 512 ............... 13I0 M-ary Character Error Probability for M 1024 ............... 14

Re\ crc I1httikt Rci

II I

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List of Symbols

M Number of multiple alternatives

D Order of diversity

P, M-ary character error probability

P, Probability of correct character decision

E T Total received signal energy

N, Received noise power density level (single-sided)

d r Deflection statistic, (2 ET/N,)'

Bjx) Auxiliary function (equation 3)

e.(x) Partial exponential series (equation 4)

P," Binary error probability fr M = 2

PCD Probability of correct decision

PFA Probability of false alarm

RT Energy-density ratio, ET/N, = di/2

El Average total received signal energy for fading

channel

R Average energy-density ratio, (E, /N,)

v Measure of degree of signal fading (equation 9)

1H Signal fading parameter (equation 10)

Average M-ary character error prohabilii%

f() Characteristic function

iii iiRc' cr,,c laiik

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Error Probability Characteristics for Multiple AlternativeCommunication With Diversity, but Without Fading

Introduction

The M-ary character error probability, PC, for orthogonal multiple alternativecommunication with D-fold diversity, was evaluated in reference I for a wide rangeof parameter values. It was presumed there that the slow signal fading on the I)diversily channels was independent and that the received signal energy "iasdistributed exponentially (Rayleigh fading of received signal voltage).

I, is of interest here to reconsider the system performance for the case where,although the receiver was designed for D-fold diversity, there is in fact no signalfading. The fractionalization of the received signal into the diversity channel, thenrestls in poorer performance than had the signal been confined to one channel andprocessed coherently. This situation can arise naturally in practice, as for example,as a result of nultipath arrivals, or it can conic about intentionally in the systemdesign. It can also be of interest in testing a multiple alternative hardware designunder controlled laboratory conditions, where the time or cost of simulating actualfading conditions is excessive.

The basic framework and background for the communications technique con-idered here have already been presented in reference I and will not be repeated, forie sake of brevity. The reader is presumed to be familiar with the time-bandwidthdt,ration and separation constraints listed in the above reference, particularly pages2-4 and appendix B.

A

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Definitions and Technical Results

The source selects one of M equi-probable symbols and transmits a signal on 1)diversity channels (time and/or frequency) to a receiver. The D channel, devoted toeach of the M alternatives are disjoint (in time and/or frequency) with each olhcrand vith the channels for the other alternatives (see reference I). The receiveremploys matched filtering on each of the totality of MD cells of interest. For eachsignal alternative, the corresponding 1) envelope-squared matched filter outputs aresampled appropriately in time and summed. The largest of these N1 deciionvariables is then declared to be the transmitted signal alternative.

The additive noise at the receiver is assumed to be white Gaussian over the totalband of the possible received signals, with a (single-sided) power density level of Nwatts/H,. It is shown in appendix A that the M-ary character error probability ofthis communications technique is given by

=1 - exp(-d2/2) -°0 D e-y 2/2 1 (dT[B (Y/2)]e D-1 df D-1 dTY)[BDI(y22 -

T (I)

Here "deflection" statistic li is given by2

d 2 ET T total received signal energy2 N noise power density level (2)

0

where ET is the total received signal energy on the D channels; ln(x) is the modified

Bessel function of order n and argument x; auxiliary function Bn(x) is defined asf(x tn ex -)-x ()

Bn (x) dt ex(-t) = 1 - e e W;3

and (recietence 2, equation 6.5.1 I)

n 1 ken) W E l xn k=0 k (4)

is the cponential power series through the term x".

It is interesting and worthwhile to observe from (I) and (2) that tlie exac frac-iionali/ation of the total received signal energy E r amongst tle D diversity branchesis immatcrial in so far as system performance is concerned. The available signalenergy F could be divided equally in the D branches, or it could be concenrted in

iust one branch; in either case, error probability P is identical (assuming M and I)are L ncha ngcd).

For M = 2, expression (I) can be evaluated in closed lorm. Scral alternaic [representations for this binary error probability are given in appendix B, along \'ith an efficient program for ihis special case. The program for evalaiation of (I) illgeneral is given in appendix C.

2

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Graphical Results

For a particular value of M, the character error probability P, in (I) is a functionof the number of diversity channels, D, and the total energy-density ratio (signal-lo-noise ratio) E,/N,. We have plolted P, on a normal probability ordinate versusEr/N, on a dB abscissa (i.e., 10 log E 1 /No), with D as a paramcter, in Ihe range 10 'to 10-1 for PC. Figures I-1I correspond, respectively, to M = 2, 3, 4, 8, 16, 32, 64,128, 256, 512, 1024. The curve for D = I in each figure corresponds to the NoFADING results in reference I, figures 1-7.

We ha\ discovered no simple rule-of-thumb for the additional sigal circry'required ito naintain i fixed P, as D is increased, which covers tlie \whole range of I)and M. Nor did we find a simple expression for P, when it is small (- 10 1), despiteconsiderable effort. The best we can do is to observe that

Pe ;r (M - 1) Pe2 for small Pe (5)

and then use result (B-I0) for the binary error probability, P,,. I) is arbitrary iu (5t.[urther developlnenl of his approach appears in appendix B.

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51 6 \ 7 8 \8\91611,2 3 1\4 15 16 7 1

10-

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10-1 __

D1 2\3\4\681216)24\ \324864 __

1 0-2

10-3

156 7 8 9 10 11 12 13 14 15 16 17 18

ET/No in dB

Figure 2. M-ary Character Error Probability for M = 3

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10-1

ND= 1 \ 4 68 8 \ \ 2\ 32'48 64

10-2

10Q-3

ET/NO in dB

Figure 3. M-ary Character Error Probability for M 4

6

II k 047

]R 647.'

10-1 \N \

-8 2 3 6 8 12 1 2 3 1 148164

10-2"\ " \ \1 \ \ \ \ \

Pe

-\\ It I \ \ \ \ \ \ \

10.18 9 0 1\2 1 4 1 6 1 8 1

ET/NO in dB

Figure 4. M-ary Character Error Probability for M 8

7

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10-1

10-

10-3

1 8 ~9 10 11 12 13 14 15 16 17 18 19

ET/N 0 in dB

Figure 5. M-ary Character Error Probability for M =16

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10O-1

1\ 6)8Q2-1\2453 \ .

E1/ 0-i2dFigure~~~ 6. I-r Chrce Ero xrbbliyfrM3

Pe9

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10-1

D=1 2 3 6 8\12 16 24_ \32 \48 64

10-2

1 0-3

10-

ET/NO in dBFigure 7. MWary Character Error Probability for M =64

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1 0-1

1 2\ 34\ 68\ 12\ 16 24\ \3 248 \64

10-2 \k_ ___ __

10-3 _ _ _ _

10Q-510 11 12 13 14 15 16 17 18 19

ET/No in dB

Figure 8. M-ary Character Error Probability for M =128

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10-1

D1 2\ 3' 4 6\ 12\ 16 24\ \32 48 \64 __

10-2__ _

10-s

1 0-~

10 11 12 13 14 15 16 17 18 19Er/N 0 in dB

Figure 9. M-ary Character Error Probability for M =256

12

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10-1XA XX xx II

10-2

10A-

10-3

10-4

ET/N 0 in dB

Figure 10. MWary Character Error Probability for M =512

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10-1

2\ 234 68\1216 24\32\ 4864,

0-4

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Correct Decision with a Threshold

The processing system above presumed that one of the M signal alternatives wasalways transmnitted. A more general situation occurs when the additional case of nosignal present at all is allowed. Then the largest of' the MI decision variables, iscompared with a threshold. There are three types of probabilities of interest forsignal present (reference 3, pages 2-5); they are the probability of a missed decision,the probability of an incorrect decision, and the probabili\ of' a correct decision.The las( is giv en by a slight generalizat ion of (1) to (reference 3, equation 5)

P CD d f- dy yD eYI'2 I TY)[B D (2/2)J - (6)

shere V is thle thlreshold value.

For signal absewt, we then have false alarm probability (reference 3, eqnat ton 2)

P FA I Prob ntax(y1... YM) < V for d T=0

rv 2 x D- 2 Mf - dy y -G -1!

=1 B- B0 .2 (7)

Here we used the limit of thie integrand of (6) as d1 -0 and (3). Result (7) agrees,withi r.-ereiice 3, equation 25, as it must, since both recetvers are proccssineideti cal noise-only channel outputs. No numerical results oti this more generalsituation, (6) and (7), are presetnted here.

Extension to Fading Signal

All thle earlier restilIts in thIiis report have pert ained to cotitant am phtidc sig nalcom1potetIs on the D diversity branches. Now we presumne that t here is ,lo%% siilialfadin1g, meanintg t hat thle t (ial received signal etnergy Fi is a ratndom variable. Inparticular, we let energy-density ratio (signal-to-noise ratio)

d 2 ERT T T (8)

have probability detsit v In tctiou

RTexp(-R 1 ./i)

p( & W + F(v + 1) f r R > 0(9)

15

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with

•V > -1 (10)+

Here RT is the average received total signal energy on the D branches, and v is aconstant, not necessarily an integer.

This very general model of fading was presented and utilized in reference 4, pagesI 1 -12, in the investigation of a maximum likelihood detector. For example, ihe foursignal fading cases considered by Swerling (reference 5) are subsumed by (9) forparticular choices of v; see table I.

Table I. Particular Cases of General Fading

V 0 D-1 1 2D1)-S%,,cr-lin (ase ] 1 I 2 3 4

Cases I and 3 correspond to common signal fading on all D brancic' , whereascases 2 and 4 correspond io independent and identically disiributed signal fading oneach of the D branches. In particular, case 2, v = D - 1, corresponds to ex-ponential probability density funciion

Pl(R) =L exp for R > 0 R I

for the received energy-density ratio, E,/N,,, on one branch; this is Rayleigh fadingof' the received signal voltage amplitude on each branch. Equation (9) is a D-foldconvolution of (11) when v = D - 1. Case 4, v = 2D - 1, corresponds ioprobability density function

P(R) I 'D (12)

for the received energy-density ratio, E,/N0 , on one branch. Equation (9) is a D-fold convolution of(02) when v = 2D - I.

The average character error probability is evaluated in appendix D; it is given by

i. ( 1) v+I] -

P'- = I-o - 1) ! (Ij + 1

Wfdt t D1 e-~ 0 t M 1 F1(v + 1; ; t---~ (3

This single integral can be evaluated numerically for any v of interest; of course, M,D, and E,/N,, need to be specified also for this numerical procedure.

As a special instance of fading (9), consider case 2 in more detail:

16

I R 0471

D - 1,

RT E T/N0ii=-7 -- average energy-density ratio per branch. ht4

Then, if we use reference 2, equation 13.6.12, and (3) above, (13) speciali/es ito

1 = -f dt q,(t) [ftdx qo(x Ml

%%here

q1(t) t ex fort > 0(D - 1)! (~j+ 1)0

qW xD1ex(-)for x> 0 (16)

arc the prohabihli v density fuinct ions of the decision variables Formed by the sum ot'1) etixelope-sqitared fill er outputs for signal-present and noise-onlix rspctively-.Result (15) is identical to refereite. 1, equation C-1. It is what We would hla\ C got tenhad \% e ax eraged lte signal probability density funl~ction inl (1) - thle funllctionlitiiliplvinge the bracket - \% ibh respect to the signal strength, before e\pressiilg I',inl ittegral form . Iii tact, a more general integral than (15) has already been en-countered and cx alnat ed inl reference 3, equation 26, where a t hresholding operation\k as included inl tlie receiver processor for the fading signal.

F-or M vs 2, general fading result (13) can be evaluated inl closed form; thiesspecial cases are presented iii appendix D, especially (D-5) and (13-8).

Discussion ahd Summary

Irhis report ha' addressed the problem of numerically assessing the additionalsignial-to-noise ratio (i.e., enlergy- density ratio) required when a niontadin signal isbrok cit io D (uLnequlal ) coinponnt s and conmbi ned inctoherent ly. Some relatledsi ujies into the cost of' imperfect ons of' receivers, or the cost of lack of knox% ldgeofithe received signal structure, are presented and evaluated inl refereinces 6 anid 7.

Since tile performance of the processor considered here for t lie noitfading. sipnaldepends only (oin tile total received signal energy, regardless of hoxv t'raetioitali/ed itmtay be an tong the divgersit y channels, these results also a pply to a sit nat ion xM herun tcert aintiy ill th lititle of arrixal or the doppler shtift (if tile receixed signal ca nieslie rceiver to search over several channels, even thlough thle particular reeei\ ed(

signal alternative occupies only one channel. The D- fold stimnitolt tor each NI -atalt ernat ive leads to a loss in performance; stated alteriat ixely, lie cost of' lie un-certainty is addit ioital received signal energy required inl order ito nmainttaini the saille,Lqualiiy of perforitance, p'.

F~or at large signal-to-noise ratio, the N4-ary charaeter errot prohai lii t is pkocii

17

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approximately by (5) in terms of' the binary error probability, P,,. The usual ap-proach is then to use (A-3) with M = 2, or to use (B-4), to cvaluate P,,. However,the situation frequently arises where the characteristic functions of the decisionvariables can be evaluated fairly easily, whereas the corresponding probabilitydensity functions and/or cumulative distribution functions cannot. In that case, analternative representation of* P,, directly in terms of the pertinent characteristicfunctions would be uselu. This problem is addressed in appcndix E, with (liefollowing result,, for the binary error probability:

Pe2 - 2"rTr C_ 0o 1l(

= T 4 o- fl(g

1 +

= . " + " Im{fo( )fl(- )} 17

2 Tr 1 (17)

Here fo(Q) and f1tS) are the characteristic functions of the noise-only and signal-plus-noise decision variables, respectively. C_ is a contour in the complex I-plane

along the real axis, with a small downward identation at I = 0; C + is a similar

contour indented upward at 4 = 0. The last form in (17) can be particularly useful

for numerical evaluation of Pe2 for some characteristic functions.

Another alternative to evaluation of M-ary error probabilities is to use bounds\%hich are simpler to compute than the exact result. This is attractive if' the boundsare tight, at least for the range of signal-to-noise ratios and error probabilities ofinterest. Some general bounds on error probability (which were not used here) are

preentcd in appendix F lot a situation that includes interference as well as noise.

18

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Appendix A

Derivation of M-ary Character Error Probability

To avoid duplication ol etlort, we will rely heavily on reference 8, paves, 2, 3, 8, 9and appendises A and B, especially (B-7)-(B-12). We obtain directly the probabilitydensityv futnct ion of the signal decision variable (sum ol D filter oultim ) &s*

1 /X\~ exp 1 T) cTx/2) for x>O0

atnd that tor any one of the M-1I noise-only decision var-iables a,

1Y (y/2)oy~ (A-1

2(D-_-1)'- exp fo A2

A correct decision is yielded only if x is greater Own all Mi- I independent n~oisevai iahles, thle probability of this evenit is

P C = odx p, (x)[Ifo dy p. Y (A-1)

Butt thle integral on y in (A-3) and (A-2) is given by (3) as B,) (x/2). Then [iv use ot(A-1I) and thie suibstitution x y- 2, (A-3) becomies

=exp (d. 2 /2) 2 ~ dM-1eY/

PC =~ I-T f.d y121_(dTY[ 2l(/2)T (A-4)

I lie NI-ary character error probability is P, = I - P,

Several cheeks on (A-4) are available. First, for the trivial ease M = 1, (A-4)vields, 13 1 , w hichi agrees with lthe fact that there are then nio noise-only ehlta tcII \%0IIrv about at all. Second, a, d -O (A-4) reduces, to P, *I/I i ges~ tIlie ohservat ion that we have a Lomplctely random select ion wit i /ero sigmial-t o-

ItoiseC r'atio. Thlird, for D -~ 1. (A-4) agrees with reference 1 , equat ion 9A O\ lhen t lielatteri is corrected for t lie t ypographical omission of t lie fact or e~p( - F I / N,)).

A programn for thle evaluwi 1i ot of!" P I - P. via (A-4) is, given itn appendi\ C.

"A "defkecion'' interpretation of dj is available from reference 8, equations (13-10), (18-7).(B-4), (A-4), and (B-5).

A-1 A-2Re ctsw Bhkttt

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Appendix B

Binary Error Probability

This appendix xN ill deal \cxlusivelv with the special case o[ N1 2, a hinastdecision. Then (I) becomes, ws itl ihe ident ificalion in (3) and (4),

exp-4/o2 f y 2P 2 d D- T)f yyDeY I D 1(d TY) eD Y22

T

exp d2/2 D-1 D+2k -y-1 ) E dy y e - (d

dD k=O k!2k ID1TY)T

exp(- d2/4) D-1 (D)k F - -d/4)-- D2 k=O k!2 1 -k; 11

To evaluate the integral, we used reference 9, equation 6.631 I: the final tran-formation to (B-I) employed reference 2, equation 13.1.27. An ahrlteriai\c ,'\-prcssion to (B-I) is available via reference 2, equalion 13.6.9:

Pe2 = e D k=0 k- (-2T) (13-2)

If wc c\pand the polynonial in d-/4 that occurs in (B-I) and (B-2), and inerchalL'ctllts, we oblain

= xp( T4 D-1 (dT/-d2 /4) 2Dn-1-n (D + n).e2 4 n

2xn(- 2n. j=O 2J j!

All threc expressions above for P,, are finite sums11', of positke quantities and arereasonable for computer evaluation, ecnvi for fairly large I).

Another expression for P., = I - P2 is available by recalling (A-I) and (.\-21and chaitging (A-3) io

Pc2 = dy po(y ) f dx p,(x)

But Iroum (A-I), upon making thil subsittio C and wsing Iciecte 1(1,e(uationt I, "c find for the inner ititcgial in (11-4),

II.I

TR 6473

~~0 D - 1 2 + d T

f dx p f dt t t exp ( 1 (dt)

Y Y

= Q (dT, Y ) (5)

Fhen (B-4) yields

2 = dy YD-1 exp(-y/2) QD(dTA Y/2)c2 2DD - 1)!

[2-1 (D - 1)] f d2D-1 exp (-u2/2) QD(dT' u)

AIt his point, we will use (he niew integral result

f dx x 2 N1 exp p 2x2/2) QM(a, bx)

-2 N1(N - 1)! b -A I-(i!N )()e (2N (; i eN-1

(B-7)

ss here2 2a

2 2 (B-8)

i his integral and several other new integrals of QM functions, along with theirdersations, will be presented fairly soon by this author in another NUSC technicali epor.

I he uc of (B-7) and (13-8) in (B-6) yields

e p - (-2 (4) D -n)e,_ (d2/4) (B-9)Pe2 = 2D=I n=0

I \pandim the partial c\poncmial expansion in powers of d/4 according to (4),and intlerchinginlg mtn111alions, \\e obtain for the binarv error probability

exp =d 2/4 D- I (d /4) D-l-k (2D-1)

e2 22D1 k! n (-Il)2k=O n=O

I his result is very similar to (B-3p, and of colrse yields ideillical numerical \aIlles;,ho\\evcr, a direct verification of the transf'ormalion that would take the inncr ,umil (B)-101) to tlie inner uum in (13-3), or vice versa, has not beenf discovered.

11-2

TR 6473

(B-10) is a very efficient form for numerical evaluation, because the ,ulnunand otthe inner sum (on n) does not depend on the index (k) of the outer sumt; tlis ad-vantageouts property is not true of (B-3). Thus for a given 1), we can compute thepartial sums on n in (B-10) by starting at k = 1) - I and merely adding one ad-ditional term to the inner sum for each downward unit step in k; these D partialsums are all stored and then used for the outer sum starting at k = 0. Thus, x hatappears to be a double sum in (B-10) actually can be computed by means of' t%,osingle sums of D positive terms. Furthermore, tile inner sum oil i in (13-I0) cain hedone once for a value of D of interest, and then used repeatedly for different valuc,,of d. A program incorporating these observations is listed at the end ofI [ilk ap-pendix. All of the results thus far apply to arbitrary values of encrg.-ln,sity ratioE I / N,.

One reason for dwelling on the evaluation of the binary error probabili\ is thaifor large signal-to-noise ratio, i.e., large energy-densitv ratio E iN. the Ni -ar\character error probability is given approsimately by

Pe A (M - 1) Pe2 for small P (B-Il)

This result may be seen to be valid when it is reali/ed that tile probabilitx of mit\particular noise-alone decision variable exceeding tle signal decision %ariwle ik arare event Mien PC is small. Thus the probability of t sko or more noi,,c ariablc,c\ceediiig tle signal variablc is very rare indeed, and we need onll' account to thesingle noise variable case. Since these events are disjoint and there are Ni-I ,uch

variables, (B-I I) follows immediately.

For large d- /2 = E r/N0, an asymptotic form for P., can be d'c.loped from(11-10); express

D-1exp( d2 4) (dT/4)

e2 22D-1 (D - 1)

[ 1 + 2D(D - 1) + (D - 1) (D - 2) (2D2 D + 1) +d2/ (d T/4)(-2

Therefore, asymptotically,

exp d 2,/4) (d2 /)D-1

P (- as dPe2 " 22D-1 (D - 1) ! T (1-13)

Iut reference Io (B-I 2) reveals that in order for (13-13) to be a good approiaiat innIo P,., rather than just aii asymptotic result, tihe requirement d, >> I) must besatisfied. This requirement, however, is typically too large a signal-to-noise ralio

requirement to be practically useful. We have tound it necessary to resort to (1I-1(1for tile binary error probahility for the range of values of inicresl here.

11-3

TR 6473

A program for the binary error probability P,.2 in BASIC' for the Hcwlctl-Packard 9845 is presented below. It utilizes (B-JO) cl seq., as discussed above. The%ariables are self-explanatory and consistent with the earlier text.

F ir-, r, Lk e f or 1-fo:Id dvvers s , tut ri', ft.,-i n. TP 64,'7?i 1 1=12 I 10 NUMBER OF DIvERSITY CHAFrI~EL&£-u Et nO=70 Et nO>0 TOTAL SIGNAL ENEtw,"GY S";INGLE-' .IDEL, NOISE DENSIT,"30 til=D-1

40 D2=tD*2Zt A=EtnO'2CA DIM S(100..

* U REDIt SI DI .l:0 '. 1 I ) = T = 190 FOR J=1 TO D1

100 T=T*(t2-J*) J110 .- Ii1-J) = ' D- J ) T12 r 4'IE'T II -. 0 S=S, 0)

140 T=Ii-'0 FOP 1= TO DI1 m, T-T*A K170 c 1 'Tc;i0 ttE ','.J i

1'90 FPe=E2.F,:-A"'..2 ':fl'-I :

Ps R IIT D =';D,"Er.0 =";Er.OPE ;e10 END

1_ Et nO = 20 Pe = 1. 451::1'939'3.57E-03

11-4

[R 6473

Appendix C

Program for M-ary Character Error Probability

The expression for P, is given by (1). Since we cannot integrate to -, %%e terminatethe integral at y = L. The error incurred in doing this is given by

exp -dT/ 2) D e,'", 12[Bj(Y2/2)]M_ 1

D-1 dy D- /2 1(dTY)T

ex f,./2) D 22[ , ,d L- 40 dy y e-y2121D ID(T Y)T

DD expe[ 1 - d)d 1L (27- d ')1/

CZ/L _1 dy exp[- 1(y d'\VUT) (2 7611 L L T\

D-exp[- .(L-dT) 2]

We used: the fact that B,(x) is upper-boutnded by I, as may be seen froni (3); theasynptotic behavior of l,,(x) for large x as given by reference 2, equation 9.7. 1; andan integration by parts to yield the final form. For given values of D and d (M ha,,dropped out of tlhi error bound), we sten ! tip by units of I from the value d1 - Iuntil ((-I) is sufficiently small.

We chose error 10-12 in the enclosed program. The reason that the integral in (ItilMust be evaluated very accurately is that we must subtract the integral rc.,ilt, from Iin order to gel the error probability. If we want to evaluate P, accurately in therange of 10-5, then we need the integral to 8 or 9 decimal places to counteract theeffect of subtraction from I required by form (I).

(-I

I R 6473

We numerically integrate (1) for y in the interval (0, L) via the Trapezoidal rule,% ith automatic halving of the interval, until a stable result occurs. It is shown inreference I I, pages 55 and 58, that the Trapezoidal rule is excellent for inlegrands%% hose odd derivatives at the end points of integration are equal. For the applicationhere, the integrand of (1) may be shown to behave as y2i)M as y-0+ . Thus a highnumber of odd derivatives are zero at y = 0 + and virtually zero at y = [ for largeI)M, meaning that the correction terms to the Trapezoidal rule vanish. In fact, (I)%%as also integrated via Simpson's rule and via Simpson's rule with end correction(reference 12, pages 414-418); neither performed as well as the Trapezoidal rule,which is incorporated in the program for P, below. Because of the very small valuesfor the cxponentials and the very large values for the Bessel function, it wa, foundnecessary to first evaluate the logarithm of the integrand of (1) and thcn cx-poientiate it prior to its use in the Trapezoidal rule.

S ' r'1-.-r., FC i:.r" D- ild t ,- '. r-sI t, -,A t r K. Ti 4,-:10 t=8- r >:1 NUMBER OF SIGNRL RLTEFNPT2 ./E:'S

0 = 12 F 0 NUMBER OF DIVERSIT'Y CHANNELS.-0 Etr.O= I E, r.O:: TOTAL SIGNAL ENERG'" ./ SINGLE-;IIDED HOISE DENSITY40 it =,:0'f, -*Et. riO I TOTAL "DEFLECTIOr." STATISTIC50 PRINT "M =";M, "D =";D,"EtrO =";EtrO, "Dt =";Dt.; ':ori , rlDlM1,Dt , Et rO50 MI=11-180 DI =D- I

90 S 1 =0I1C00 '-.2=Dtr1 1 C R=D--...=

120 B=.4 Dt A

140 I . Ft1-0 E =B* 2" ,*E:.P ,-. 5*C.C )/Cltu IF E IE-12 THEN 130I1 70 S. 3.:= ):!3 :;I. *. 51 :0 S 4 =.2

1 ,- Ft ti S .+FtS(S2) )*. 5

10 FOR S7=1 TO S4-1 STEP 22C1 S6=:,6+FNS(SI+S3*S7:,

1 E:T S,7F- R I N T U, I..- "ri . 12DE,?1"; D -E 75+ .:, *5 , 6 - .4

- o =S*.5-".0 '_4 =S:4*220 ':5=: 5+s6

: GOTO 200

(-2

TR 6473

3(00 tEF FNS' V INTEGRRFND310 C' (11D,DIqM,Dt,EtriGO

3310 B=1- E: P'T FNt4E f.T , DI340 IF P-0= THEN RETURN 03 50C S= D*LOG' Y D' ',-T+FNInixlog(Dt.*Y, DI >+M1*LC.G.B '.-Et r-iO'it 0 RETURN EXP(S .*Dt-.-o FNEND

30 DEF F4E. 2.N' SUM OF X-' K,'! FROMl K = 1 TI:,

411la FOR. K=1 TO N420 T=T tXV

4 40C NEXT K450I PETURH4613O FNEND47034:313 DEF FN Iri, lcg X, N LOG OF In"'.490 IF *0THEN 521351"10 IF N=0 THEN RETURN 15113 IF N13.- THEN RETURN 135 21 AlF=.5*X5310 F=15413 FOR I=2 TO N5513 F=F*I5 6.13 NEXT I5 710 F=N*LOG..A.-LOGt'F:50 A=Ai*A5913 S=T=1E-8?6013 FOR hI TO 50016113 T=T*W'4I*(N+I)620 S =S +T

60 IF Tt2E11-.3 THEN 660640 NE2TJ I650 OUTPUT 011;500 TERMS AT ";:N6.E.1 Iri..1cg=FLOG'.S?+20CK324903396'713e RETURN lIn, 1 Iog

mi ' 8 = 12 E t 0 Z.0Dt = 6. 32 ,4 5 532 0:

444766E+01li11r0t1j2,41 9 30Et01 4

4 2 661 0 0 E - 0,2 84f -;5109313E-11 16E

1.4,L 16 '02 50 1lE - 01 3'21 ::4 L16eIj, 05 -00 E- 1 6-'4.184.1b'l1100E -0 1 128

Reverse Blank

TR 6473

Appendix 1)

Average Error Probability for Fading Signal

The energy-density ratio variable RT was defined in (8), and its probabilitydensity was given in (9) for a fading signal. From (I ), the error probability expressedin terms of Rr is given by

Pe(R)- exp(_R) f0- dy -D e-Y2 1 . _( / Y) [B Y2/2)]

(1)-I)

The average error probability is

P = 0 dT p(%) Pe,() (D-2)

Substitution of'(9) and (D- I) in (D-2) and interchange of' itegrals lead to

= 1 - dy yD e-Y/2 [aMPe f~ YD - 1 ( y 21)

2RT I /p)J0 d% D-1 ID1(4[2 Y) VT~ r1- +, 1

(211T) 2 r(D-3)

In the inner integral, let RT = 12/2 and use reference 9, equation 6.631 I. Sim-plification leads to

F 1 - 2 D-1(D - 1)! (P + 1)V+1 0 dy y2D1 e -y2

* [ )D. y 1M Fi + 1; D; Y 2) (D-4)

Use of the variable i= y2/2 in (D-4) yields the result already quoted in (13). Resuls(D-4) and (13) hold for general v and M. When v is chosen as the special value D)-I,the restmlling average error probability is given by (15) and (16).

Specialization to M = 2

Now we let parameter v in (9) be general, but we set M = 2. Thus we \%ill developthe average binary error probability for a gen,.,ral signal fading model. Upon use 01'

)-I

TR 6473

(3) and (4), (13) yields for M - 2,

Pe2 [(D - 1)! (j + 1 )V+ l

D-1 1 tD-l+k 2t*I:. J dt t e 1 F 1 (v + 1; D; + 1 t)k=0 " JO

( + 12) k=O D-1 D---k + 1\ ( D - 1 - B, D + k; D; i 2

(D-5)

This finite suLmI of (jaussian hypergeometric functions is not 100 useful in general.ho%% ever, for Ihe special case v = D - I considered in (14)-(16), (D-5) iiniedialclyyields

= D-I +2 )D + 1 k for V = D - 1;P 2D F )(D-6)

this last result can also be derived directly from (15) and (16) for M - 2, byreference to (3) and (4).

A much more appealing result than (D-5) for average binary error probability P_,for general v is attained if we start with the binary no-fading resull in (B-9) and useidentilication (8):

exp (-i.,2) D-1 D-1) D-1-n 1 (R T k2D- 1 E 2 (D-7)

2 n=O ' k=O

Then the average binary error probability is, by use of (D-7) and (9),

e2 foo p(RT) e2

= 2 2D-1 ,v+l r(v + 1)] n'D ) D-1-n 1

L n=O ~ W= k! 2

*fdR2+k ex( Ti~)

2 2D-1 (I V+ ] -l (D ) D-1-n (v + 1) k p/2 )k-- ~~1 22- kD-1 2h k0 k 1 +,t/21

D-2

TR 6473

[,2D-1 \V+1] D-1 (,v + 1)~ /2 k D-1-k (2D-1)

(D-8)

This closed form result is a finite sum of positive quantities and holds for generalvalues of v(>-1). The summations are very similar to those encountered in non-fading binary error probability (B-10). A program for the evaluation of ()-8),\ ,hich takes advantage of the observations made under (B-10), is given below. Vol-v = !) - 1, (D-8) must reduce to (D-6); we have not discovered the transformationthai accomplishes this corroboration, but we have confirmed the equalitnumerically for v = D - 1, as well as for the alternative, more general resul (i)-5).No numerical results for (D-8) have been presced here.

1 1 F,. r Lge bi .r.- Fe ,re D-fold di,.,ersit-, k.ith f'adirg. TP 6473I0 D=12 D.0 NUMBER OF DIVERSITY CHI-4hEL-.

Nu -,.. I N,.,-1 MEASURE OF FADING, EQS. 9-!2":D '1,.=:3.7 I Mu.>=0 MEASURE OF SIG AL-TO-NOIS.E RATIC40 PRINT "I = ; D"Nu N;Hu "u =";Mu5C D1=t-160 I'rI*270 A=Hu. K2+Hu)80 DIi S(100.)90 REDIM SkDl)100 (.DIl)=T=l110 FOR J=1 TO DI120 T=T*,:.D2- J.) ./J13-':0 S D 1- J )=$(D -5) + T140 NEXT 3

150 S=S':(O)1E0U T=1170 FOR K=1 1O DI180 T=T* (Nu+K )"..."A4190 S=S+T*S(K)200 NEXT K210 PeS ,2""D2-1)*(I+.5*Mu)^(Nu+I')20 PRINT "Pc =";Pe2--0 END

= 1.2 Nu 6.1 M,., = -.

F-E = ,. 10,1551686 3; E-0:3

)-3/1)-4

Re crse Ilanik

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

IR 6473

Appendix E

Binary Error Probability Via Characteristic Functions

Let p,(y) and p,(x) be the general probability density funetions of a n)ois1-olvdecision variable and a signal-plus-noise decision variable, respectively. The binaryprobability of error is then

Pe2 = Prob(x < y) = dx pl(x) J dy p0 (y) ([-I)

No\ by reference 13, equation 6, the exceedance probability in (l-I) ik gi'ell interms ot fhe characeristic finiction ,( ) according to

f dy p0 (y) -1 - P (x)0

= 1 -' fc +00 fo(C) exp(-i~x)

f -df f(E ) exp(-i~x) -2)

The integral with a slash on it is a principal value integral; the contour C_ is acontour on the real axis of the complex I-plane except for a small downwardidentation at 4 0. Employment of (E-2) in (E-l) yields

Pe + .11 dx p, (x) 1 f LEf()ex(ix

= f-L _ f o (- E) f( - )(

where C , is indented upward at 4 = 0. Both contours C, %tar at 4 land go

to4 - +00.

Expression (E-3) gives binary error probability P. 2 directly in terms of charac-teristic functions f0(4) and f (), by means of a contour integral. Since characteristicfunctions satisfy the relation

f(-) = f*(f) for real , , ([-4)

E-1

. ..

TR 6473

an alternative real integral form for (E-3) is

p ~ ' 4 ~Im{fMf(E5

Pe2 f + T" 0 0m£()f(-) ES

E-2

o U

. I II - . -,_

TR 6473

Appendix F

Bounds on Error Probabilily

We consider the situation where there are three different variables upon ,. hich adecision as to the transmitted multiple signal alternative must be made. Ilhe\ ac,respectively, the signal, noise, and interference variables. InI particular, let ratdolmdecision variable S denote the output of the signal channel; S ma it sell be t 1imof' several diversity branch outputs. Let N be the Otl ttl of li0' tle ma.ViIt/1t 0) lh1noise Channels out puts, of which there may be many; each noise channel Ot pmtmay itself be Ote sum of several diversity branch output,. And let I be the ouptt (tthe ma.'im,,um,, of tile interference-plus-noise channel output,, of \s hiCh t.here nia\ bemanv; each interferelnce channel output may it ell' be tile ,ll of several dier,,ilbranch outptts.

For ildlCpendett signal, noise, and interference channel outputs, the l obabilit,of ia co'leel decision is

= Prob(I < S, N < S) = fdx Ps(X) Pi(x) P nx) (I--I)

where p.,(x) is the probability density function of random variable a, and I",(\) i,, i,,corresponding cumulative distribution function. We also express

Pi(x) = Prob(I < x) = 1 - Q.(x) ,

P n(x) = Prob(N < x) = 1 - Qn(X) , (:-2)

where Q,(x) is the exceedance probability of randoni variable a.

The probability of character error is

P e= 1-Pc = fdx P(X)[1 - {1 - Qi(x)}{1 - Qn(x)}1

= fdx Ps(X) [Qi(x) + Qn(X) - Qi(x) Qn()]

= Prob(I > S) + Prob(N > S) - Prob(I > S, N > S)(1-3)

lhe last quantily in (F-3) is generally quite siall for practical ecase of intelet. B111in any ccnt, we have the tipper bound

P e Prob(I > S) + Prob(N > S) (and P e 1) (1-4)e e

Itn addition, since

U --I

TR 6473

fdx PS (x) Qi(x) Qn(x) .fdx p S(x) min{Qi(x)' Qn(x)}

-min(Prob(I > S), Prob(N > S)) 15

s e have t hie low er houind

P > max{Prob(I > S), Prob(N > S)}(1)e

Anal na itive to ( F-4) is obtained as f'ollows:

Prob(I > S, N > S) =Prob(I > S) Prob(N > S11 > S)

>Prob(I > S) Prob(N > S) (F-7)

I lieretoic (1-11) yields tipper bou~nd

P : Prob(I > S) + Prob(N > S) -Prob(I > S) Prob(N > S) (F:.8)

t his is at Iighter tipper hound than (F-4), and is also probably a good appro-Oimationin ma;;v practical eases. The bouind has a rnaximun valuec of' I %%heni cillherProh( I>S) I or Prob( N>S) = 1

I I %C leta = min(Prob(I > S) , Prob(N > S)1

b = max{ProbCI > S), Prob(N > S)} (F-9)

he;;t the rattio o't upper and lower bouinds in (F-8) and (F-6) is

upper bound =(a +b) - ab + a a IOlower boundf b b (-O

Ihis is largest %%hen b a << 1, in which ease

upper bound =- vlower bound 2-a~2;(-i

itiiis thie ratio ofbuisis never g reater than 2.

F-2

'IR 6473

References

1.A. H. N itttall, "'Error Probability Characteristics for On liogoital Mnlt ipicAlternative Communication with D-told lDiersit v.' NUSC Tecchnical Report 4769.19 Juine 1974.

2. lIln/'UO o'lMathemalical k-ulftu)/is, U.S. Dept. ol Coinercc, Nat. ir. otStandards. Applied Mathematics Series No. 55. U.S. (lovt . Plrint ing Office,Washington, DC, J title 1964.

3. A. If. Nut tall, "Operat lug Characteristics for Detect ion of' a I-adinye Signial iii%I A cita Ii e I ocat jols \%it h )- fold Divcrsit v.' N USC Tchnical Rcpoi 4793 2(0Ain-nst 1974.

4. A\. HI. Nuitall and fl. (I. C'able, "Opcraiing Characteristics tor NlasinniI kel ihood le ci ion of Signals in G aussian Noise of' Unkno\% I es ci: 1arm III,

Random Sienals of IUnkntow I- evel,'' NUS(CTechnical Report 4781, 31 .1 nlv 1974.

5. P. Swerling, "Probability of Detection for Fluctuating Targets," IRE Trans-actions on Information Theory, vol. IT-6, no. 2, pp. 269-308, April 1960.

6. A. H. Nttall, ''Opciating Characteristics tot IDetection ot it l-ading Signalwith IK lc~iipcnn Fades and D-fold Diversity in M Alter-native local ions",' NI S(lechnical Report 5739, 25 October 3977.

7. A. HI. Nnt tall. "Detection Probabilities for a Fading, Signal in N1 AlternatwseI ocations w\ith D-lold D~iversity and lncoherent Combination of 13 Bins pct

M iesisBanch. ' N USCTechnical M emorandtim 781096, 11 NIa.\ 1978.

8. P). (J. ('able and A. H1. Nuttall, ''Operating (Characterist ics tor Maiinimilcl1 hood D~et ect ion of' Signals ini Gauissian Noise of Uuk nowk iiLevel: P~art 11,

Phlase- I icolueneilt Signals of, Unk nown I evel,'' NUS( Technical Report 4683. 22April 1974.

9. I. S. (iadshtevn and 1. NI. Ry/hik, Table of Iwilegials, Series, awd lPro(/iI(I%Academic Press. NY. 1965.

10. A. H. Niittall. "Somne Integrals Involving the QN1lFunction. ' NIS T5~echnicalRepot 4755, 15 NMay 1974.

11. P'. I1. D~avis and 1-. Rabinuositi, Numerical Inl',iratii, Blaisdell lInilkuinu(o.. Waltham, NJA. 1967.

)2. ( . I anec/os. Applied A itavsis, Prentice H all. Inci.. I ngless ood (Ills, N .1,1964.

13. ,\. If. Ntall, ''Nnnterical Esaltiation of, ('tinwliative lProbahllit\ lisi ibtli otI min oui Diirctls from Characteristic Functions,'' NI SI Repoti No. 1032., 11

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