photocounting array receivers for optical communication...

Photocounting Array Receivers for OpticalCommunication Through the Lognormal AtmosphericChannel. 2: Optimum and Suboptimum ReceiverPerformance for Binary Signaling

S. Rosenberg and M. C. Teich

Performance characteristics are obtained for various optimum and suboptimum photocounting array re-ceiver structures obtained in the preceding paper (Part 1). Probability of error curves are presented forthe case of lognormal fading and for a variety of receiver configurations including the BPCM single-de-tector and array-detector processors and the BPOLM and BPIM single-detector processor. In addition tothese approximate optimum processors, the suboptimum aperture integration and MAP receivers areevaluated in order to examine degradation of performance with increasing suboptimality. An optimumamount of diversity is shown to exist for the aperture integration receiver, with a fixed signal energy con-straint. In a related paper (Part 3), we examine a bound to the error probability for M-ary signaling.

1. Introduction

In Part 1 of this set of papers,1 the optimum pho-tocounting array receiver structure was obtained fordependent and independent lognormally faded fieldsamples, with several signaling formats. In general,it was shown that the optimum receiver performsweighted counting. For the special case of equal-energy equally likely orthogonal signals and one de-tector, the optimum receiver was shown to performsimple unweighted photoelectron counting both inthe absence and in the presence of lognormal fading.

In this paper, we evaluate the binary error proba-bilities for the receivers discussed in Part 1. Webegin first by considering the binary error probabili-ty for a single detector containing one coherence areaof the field, with and without fading, assuming anonorthogonal signaling format such as pulse-codemodulation (PCM). We then obtain the error prob-ability for a two-detector array, when the fading atthe two detectors is arbitrarily correlated. The deci-sion regions for this case are presented graphicallyand compared with the zero fading case. We thenpresent similar probability of error curves for-the

S. Rosenberg is with Bell Laboratories, Whippany, New Jersey07981; M. C. Teich is with the Department of Electrical Engi-neering & Computer Science, Columbia University, New York,New York 10027.

Received 26 January 1973.

suboptimum aperture integration and MAP receiversdiscussed in Part 1. The results indicate that thetwo-diversity path aperture integration receiver doesalmost as well as the two-detector optimum array re-ceiver for moderate turbulence levels and low back-ground radiation levels. The MAP receiver alsoshows some regions of operation where the perfor-mance is quite close to optimum, particularly for lowSNR. However, for both the MAP and aperture in-tegration receivers with the log-irradiance standarddeviation near the saturation value, the optimumarray receiver outperforms the suboptimum struc-tures. We include a discussion of the effect of diver-sity and optimum diversity on receiver performance.Finally, we evaluate the error probabilities for binaryorthogonal equal-energy signals such as binary polar-ization and binary pulse-interval modulation(BPOLM and BPIM). In the last paper of the series(Part 3)2 we theoretically investigate the probabilityof error for array detection of M-ary equal-energy,equiprobable orthogonal signals in the presence offlat independent fading.

11. Receiver Performance for Nonorthogonal PCM

The performance of the receiver structures dis-cussed in Part 1 may be measured by the total prob-ability of error. For the binary signaling case thereare two types of errors: choosing the hypothesis HIwhen Ho is true and choosing Ho when H1 is true.The probability of error P(E) is therefore given by

November 1973 / Vol. 12, No. 11 / APPLIED OPTICS 2625

P(e) = wop(HfIHo) + wrp(HoIHi), (1)

where wo, represents the a priori probability of a0(1) being transmitted. Assuming for simplicitythat ro = 7r = 1/2, this becomes

P(W = 1p(L > 01Ho) + Ip(L < 01H),

ground noise counts, respectively. (Dark currentcounts are included in NB.) Equivalently, a decisionis made based upon the criterion,

H.

n > kT,Ho

(2)

which may be evaluated by inserting the appropriatedensity for the likelihood function L under both hy-potheses. We therefore obtain the expression

P(e) = fJ p(LIHo)dL + 2jp(LIH,)dL (3)

for the total probability of error. Since the quantityp(L) is often not available in analytical form, theerror probability may equivalently be obtained bysolving for the region where the likelihood ratio A(n)= 1 and then integrating the density of counts nunder each hypothesis over the appropriate region.This latter method will be used in evaluating theerror probabilities for various receiver structures andsignal formats.

For M equiprobable hypotheses, the probability ofa correct decision P(C), conditioned on a transmit-ted signal S1, is

P(C S1) = p(L > L2,L3,...,LM)

= f p(LI)dLI J f p(L2,L3,

Lm)dL2 ... dLm. (4)

Since the signals are assumed to be equally likely,however, the same expression results for all M wave-forms. The total error probability is simply

P(E) = 1 - P(C)

= - f p(Ll)dL 1 ' *-- f p(L 2,L3,*..,

Lm)d L2 . .. d,,,. (5)

This expression is often analytically intractable forM > 2, and the usual approach is to obtain boundsto this quantity. [In Part 3, we derive an upperbound to the error probability for M-ary equal-ener-gy orthogonal signals such as pulse-interval modula-tion (PIM) and pulse-position modulation (PPM).]

A. Binary Pulse-Code Modulation: Single Detector

We begin with an evaluation of the likelihoodfunction for binary pulse-code modulation (BPCM)with a = 0, i.e., for the quiescent atmosphere. Herea represents the logarithmic-irradiance standard de-viation (symbol definitions are the same as in Part1). For one detector, the likelihood function reducesto the easily obtained result3

n ln(N + 1)-N <0, (6Ho

where Ns and NB are the mean signal and back-

(6b)

with kT -Ns/lln[(NsNB) + 1, truncated to thenext highest integer since n can take on only integralvalues. The optimum receiver in this case performsa simple threshold test based upon the quantity kT.3

The error probability is therefore given by

P(e) = p(n > kTIHo) + Ip(n < kTIHl)

1 , { NB. exp(-NB)

2 nT n!kT-l(Ns + NB)n exp[-(Ns + NB)]

n=o n! 1 - kT-l NBn exp(-NB)

=~ - n=- n!+ kTl(Ns + NB)n exp[-(Ns + NB) (7)

Probability of error curves corresponding to Eq. (7)have been presented elsewhere3 for various parametricvariations of Ns, NB, and NS/NB. It should benoted that the threshold depends on the magnitudeof the signal levels as well as on the ratio of signal-to-noise, y = NS/NB, in contradistinction to theclassical Gaussian communication problem whereonly the SNR enters the probability of error.

.X

o

I-0-J0nzXi

15

14

13

12

1 1

10

9

8

6 8 10 12 14 16 1SIGNAL-TO-NOISE RATIO

Fig. 1. Threshold count kT vs SNR y for optimum and MAPsingle-detector photocounting receivers. The parameter a repre-

sents the log-irradiance standard deviation.

2626 APPLIED OPTICS / Vol. 12, No. 11 / November 1973

(6a)

In the presence of fading, we may obtain the deci-sion threshold from Eq. (23) of Part 1, with D = 1.The solution to this nonlinear implicit equation pro-vides the optimum threshold for a particular set ofvalues for , N, and y = Ns/NB. An equivalentmethod is to find the intersection of pL(n) with po(n)and then to choose the smallest integer n for whichp1 (n) > po(n) as the threshold kT. The latter scheme,as well as all other evaluations of error probabilities,was implemented on the Columbia University IBM-OS360 computer using double-precision arithmeticthroughout in order to minimize roundoff error.

The optimum threshold count kT vs the SNR isindicated by the solid lines in Fig. 1. Parametricvariation for various values of a and NB is presented.Also indicated is the optimum threshold for the zeroturbulence case a = 0. The broken lines representthe suboptimum threshold based on the MAP receiv-er discussed later. As the severity of the turbulenceincreases, the variation of threshold with the SNR yis seen to decrease markedly. This can be under-stood from the rapid broadening of the signal-plus-noise counting distribution as a increases; thus theintersection of pl(n) and po(n) does not shift appre-ciably even though the over-all signal mean countincreases with y.

The error probabilities for the single-detectorthreshold receiver are presented in Fig. 2 for two dif-ferent levels of mean noise count NB. The lowestcurve is that for no turbulence (r = 0); the highercurves correspond.to increasing turbulence as shown.The odd-numbered curves represent the optimumdetector, whereas the even-numbered curves arethose for the suboptimum MAP detector to be dis-cussed subsequently. The rapid increase in errorprobability with is apparent. While the falloff fora = 0 is faster than exponential, curves for o id 0 fallat a much slower rate. The typical improvement inperformance with increasing background and signal(but with fixed y) is evident by examining the curvesfor NB = 1 and 4. This improvement can be shownto be a consequence of conditionally Poisson statis-tics4 and the nature of the discrete threshold.

The equivalent power loss due to the presence offading can be obtained by evaluating the number of

0DC

I-

a. I(

L

0

< Itm0

I

a:0

X

0a.a:

dB of additional power required to maintain a givenerror rate, over that in the absence of fading. Thesolid curves in Fig. 3 indicate the power loss for a va-riety of error rates, noise levels, and turbulence levelsand were obtained from the error curves in Fig. 2.For moderate to high levels of a, the necessary in-crease in power can readily exceed 10 dB at moder-ate error rates. Loss curves analogous to those ofFig. 3 have been presented for binary orthogonalequal-energy signals detected at a thermal-noiselimited point detector.5 While the detrimental effectof the lognormal channel is clear, the losses calculatedin Ref. 5 are approximately 5 dB less than thoseshown by the solid curves in Fig. 3. This difference isreadily attributable to the nonorthogonal binarysignal sets considered here. A similar calculation fordirect detected binary orthogonal signals [binarypulse-interval modulation (BPIM) and binary polar-ization modulation (BPOLM)] shows losses of ap-proximately the same order as for the heterodyne caseand are indicated by the dashed curves in Fig. 3.Thus, in addition to the performance gain of BPOLMover BPCM in the absence of turbulence, the curvesshow that the losses for BPOLM are about 3 dB lessthan for BPCM, for moderate to severe turbulencelevels, for a given error rate.

A discussion of the performance gain of BPOLMover BPCM both in the absence and in the presenceof turbulence will be deferred to a later section,where the performance of binary orthogonal equal-energy signal sets are considered.

B. Binary Pulse-Code Modulation: Array Detector

For an array of D detectors, the approximate opti-mum receiver structure was presented in Eq. (22) ofPart 1. Once again, it is difficult to find the deci-sion boundary because of the complexity of this non-linear implicit equation for n. As in the single de-tector case, we determine the threshold boundary byfinding the intersection of the two probability densi-ty surfaces pl(n) and po(n), in n space. If we definethe optimum boundary threshold kT as the D-di-mensional equivalent of the optimum threshold T,

the error probability is given by

654

Fig. 2. Probability of error P(E) vs SNR y for sin-gle-detector photocounting receiver. Odd-numberedcurves correspond to the optimum receiver whileeven-numbered curves correspond to the subopti-mum MAP receiver. (a) NB = 1, number of coher-ent areas = 1; (b) NB = 4, number of coherent areas

= 1.

0 4 8 12 16 20MEAN COUNT SIGNAL-TO-NOISE RATIO

(b)



(a)

10

'U

10

dB OWR OS DEBOP URULNC

lec fo igedeetruigBPM an PMfrts

II

lene or snge dtetorusnBPOLM anBPMfrts

Open circles and squares indicate hand-calculated theoreticalpoints obtained from curves such as those in Fig. 2.

P~) Ip,(n < k) + Ipo(n > kT), (8)

2 -2LI

where by n > kT we mean unit > kTil for all i = 1,... D. The subscript 0(1) to the probability indi-cates that it is conditioned on Ho(H1). In principlethe above rule can be implemented on a computer.For D > 3, however, the solution becomes exces-sively complex as well as costly. In order to evalu-ate the most conservative advantage of array pro-cessing, we were able to obtain performance curvesfor D = 2, with arbitrary fading correlation. Com-puter algorithms were written to find the solution tothe equation

p1 (nj, n 2) = p(ni, n2 ),

ditional error would be introduced in the perfor-mance if the R = 0 boundary were used for the caseR = 1 and vice versa. The actual error probabili-ties, nevertheless, are radically different because thecounting distributions for the two cases (R = 1 andR = 0) differ considerably. For higher backgroundlevels (NB = 8) and thus higher signal levels as well,the variation of the boundary with R is more signifi-cant, even for moderate a. This is readily explainedby the increased broadening of the signal-plus-noisecounting distribution with increasing Ns, as is evi-dent from the variation of the twofold cumulant whichincreases as NS2 , for a given value of -y.6 Thus theeffect of R becomes more prominent at higher back-ground levels. It should be noted, however, that it isthe relative variation of the two surfaces p1(n) and

N

a:

aI

zz0)

COUNT NUMBER n,

(a)

40

35

30

(9)

as well as to obtain the corresponding error probabil-ity. A typical set of decision boundaries for this caseis shown in Fig. 4 for several values of a, y, NB, and R.

By examining the curves for the optimum thresh-old of the two-detector array, one observes, in analo-gy with the single-detector case, that the relativemagnitude of this threshold does not change drasti-cally with y, even for moderate turbulence levels (a= 0.5). For high turbulence levels [Fig. 4 (b)], thereis even less change. In the two-detector case, how-ever, there is the additional parameter of the fadingcorrelation coefficient to consider.

The variation of the optimum threshold boundaryshape with the correlation coefficient R is seen to bemore drastic for larger a and NB, for fixed y. Forlow background levels (NB = 2), the effect of R onthe shape of the boundary is minimal, while for highbackground levels (NB = 8), the effect is pro-nounced. Thus for low background levels, little ad-

NC

a:wLIM

zzDIL)

COUNT NUMBER n1

(b)

Fig. 4. Optimum threshold boundary kT(nl,n2) for the two-de-

tector array receiver with parametric variation of Nn, y, and R.

(a) a = 0.5; (b) a = 1.5.


10 06

2

Z 6a. 2

I o-20 6

M 2

LU 10_5

t. 60 2

t 6

< IO-5.M 6 0 2 a. 10-6

, ~ I I I..\ ,1

0 4 8 12 16 20

.AN COUNT SIGNAL-TO-NOISE RATIO

(a)

l I I1, \, i 10 4 8 i2 16 20

MEAN COUNT SIGNAL-TO-NOISE RATiO

(b)

10 6~ 2

lo 0I- 1.56 I

10a. 21-2 \\0 6

W 10-3U_ 6

0 2,- 10-4

6

-2

60 2

a. 10.66 o-O.0


( C )

Fig. 5. Probability of error P(e) vs SNR y for two-detector PCM photocounting array receiver. The lower curve of each pair representsthe optimum receiver while the upper curve represents the suboptimum MAP receiver. In all cases NB = 2. (a) R = 0; (b) R = 0.5; (c)

R 1.0.

po(n), and not just the broadening of pi(n), whichgenerates the variation in, the threshold boundaryand thus in the error probability. Therefore onlywhen NB is large (and thus for large Ns as well withy fixed) do the variations in the other parameters inp1 (n) become significant.7

The error probabilities are presented in Fig. 5 forNB = 2 and R = 0, 0.5, and 1.0, as a function ofSNR y, with a as a parameter. The smooth curvescorrespond to the optimum processor. In (a) and(b), the MAP receiver performance is indicated bythe upper curve of each pair associated with a givenvalue of a. As observed in the single-detector re-ceiver behavior, severe losses are induced by the tur-bulence. The effect of the fading correlation coeffi-cient R on the probability of error is apparent incomparing (a), (b), and (c) for a given value of a.Furthermore, the curves for R = 1.0 in Fig. 5(c) areidentical to those for the single-detector receiverwith twice the corresponding mean noise count.There is somewhat more than an order-of-magni-tude increase in error probability as R goes from 0 to1, for moderate turbulence levels, and somewhat lessthan an order-of-magnitude at the saturation value a= 1.5. This latter effect is readily explained by thesevere broadening of the probability density surfacefor a - 1.5, so that the area under the surface be-tween the origin (n 1 = n2 = 0) and the optimumthreshold boundary kT(nl,n2) does not change verymuch as R varies from 0 to 1, even though theboundary itself does. By examining some of thevalues of p(ni,n2 ) along the boundary for NB = 8 and-y = 20, we find that for R = 0 the relative magni-tude of points along the boundary, starting at thepoint n1 = n2 , decreases about 4 orders of magni-tude, whereas for R = 1.0, the decrease is over 7 or-ders of magnitude. Thus the points included in theboundary for R = 1 and left ut for R = 0 are sosmall in magnitude that this significant difference inthe boundaries results in a not so significant differ-ence in error probability.

The conclusion reached on studying the paramet-ric variation of these various curves is that for lowbackground noise levels, and for most turbulencelevels, one can process under the assumption that R- 0. For high background levels, moderate turbu-lence (a 0.5), and low -y, one can process with R =0 as well, while for large -y, the variation of R mustbe taken into account. Finally for large backgroundand severe turbulence, and over a wide range of y,processing with R = 0 should not significantly de-grade performance.

Furthermore, these curves implicitly indicate theeffect of diversity on the performance when the totaldetected signal energy is held constant; the figuresfor R = 0 correspond to D = 2, while those for R = correspond to D = 1, where D is the number of inde-pendent diversity paths. Figure 6 indicates that thepower gain due to diversity for a two-detector arraycan exceed 5 dB at moderate error rates and in-

o 1 x

-0-J

m ~~~~~~~~~~~~~~1.03D o~~~=0.5

a.

0 I 2 3 4 5

dB POWER GAIN DUE TO DIVERSITYFOR TWO-DETECTOR ARRAY

Fig. 6. Probability of error P(E) vs dB power gain due to diversi-ty (D = 2) for two-detector photocounting array receiver. Cir-cles, squares, and triangles represent hand-calculated theoreticalpoints. The dependence on background and turbulence is pre-

sented parametrically.

Novermber 1973 / Vol. 12, No. 1 / APPLIED OPTICS 2629

100°6-2

- 10-16

a. 2. 10-20 6

iL 10_3LI 60 2

a:10-4

m 6Li

0 2a.106'2

ME

z

CEit0itir

U-

0

I--j

0cta.

100 6

210I4

6

210-2

6

2

lo-7

6

6

21O-5

6

210-6

6

210O7

ml0 4 8 12 16 20

-AN COUNT SIGNAL-TO-NOISE RATIO

Fig. 7. Probability of error P(E) vs SNR for the aperture integra-tion receiver. The quantity DNB = 8 is kept constant. Theupper curve of each pair corresponds to D = 1, NB = 8 while the

lower curve corresponds to D = 8, NB = 1.

creases with the severity of turbulence. The ques-tion of optimum diversity will be discussed in moredetail in Sec. II. D.

C. Aperture Integration ReceiverThe performance of the aperture integration re-

ceiver is readily obtained by implementing the re-ceiver structure presented in Part 1. For D equal-strength diversity paths with equal energy, the opti-mum threshold is found by the same method used inSec. II. A. The performance of an aperture integra-tion receiver with constant total noise energy DNB ispresented in Fig. 7 for two values of D. The under-lying parametric change as D is varied is containedin CA through Eq. (26b) of Part 1. The curves thusindicate the effect of aperture averaging of the scintil-lation.

While the structure of the aperture integration re-ceiver has been discussed by others in some detail, 8

the receiver performance presented here has not. Asis evident from the curves, the reduction in errorprobability with increasing D, for the same totalmean count D(Ns + NB), is due to the reduction inCA as well as to the over-all increase in signal power.The odd-numbered curves (upper curve of each pair)correspond to D = 1, NB = 8 while the even-num-bered curves (lower curve of each pair) correspond toD = 8,NB = 1.

D. Optimum Diversity for the Aperture IntegrationReceiver

A more realistic situation occurs when the totaldetected signal energy remains constant (Ns = -yNB)and independent of D, while the total noise energyincreases with D as DNB. The effect of diversity isthen to divide the signal energy among D indepen-dent paths. Thus for each path the mean count due

to signal is Ns/D, while that due to noise is NB. Inthis way, we are comparing the case where all thesignal energy is received at a single small detectorwith the case where the signal energy is dividedamong several diversity paths and received at amuch larger detector. This results in an effective re-duction of a concurrent with an increase of noise.The conditional rate parameter for the aperture in-tegration receiver is then given by W = ZNs + DNB.Under this condition, depending upon the magni-tudes of NB and NS/NB, there will be some optimumvalue of D. This can be explained as follows.

With a signal energy constraint independent of D,the error probability decreases with increasing D, thenumber of independent diversity paths. However,the effect of noise rapidly overtakes the gain due tothe independent scintillations on the D paths.Then, at some point, there is an optimum value forD for a given total value of Ns and NS/DNB. This isentirely analogous to the optimum diversity found inclassical fading channels (e.g., Rayleigh fading 9 ).However, the optimum diversity in the classical de-tection problem does not depend on the magnitudesof the signal and noise energies separately, but onlyon their ratio. This follows from the basic differencethat distinguishes classical Gaussian detection fromconditionally Poisson detection problems, as alreadydiscussed.

The probability of error P(e) is plotted as a func-tion of Ns for several values of D, and for fixedvalues of NB and a, in Fig. 8. As Ns increases, ex-amination of a wide range of such curves indicatesthat the lowest probability of error is obtained forcontinually increasing D. For example, we considerFig. 8(a) where NB = 1 per diversity path and a =0.5. For low Ns the lowest error probability is thatfor D = 1, since at these levels of Ns and a, the gaindue to the aperture averaging of a with increasing Dis overpowered by the increase in noise. However atNs ' 12, we have a sufficiently strong signal so thatincreasing D to 2 (concurrent with an increase in de-tected noise) maintains the lowest error probability.If we now examine Fig. 8(b), where a = 1.0, we ob-serve that the same situation prevails as Ns is in-creased from 0, except that in this case the gain dueto aperture averaging begins to show up at lower sig-nal levels. (The first crossover of the curves occursatNs 3.)

We may translate the data from curves similar tothose presented in Fig. 8 to the form of Figs. 9 and10 where we plot the probability of error vs the num-ber of diversity paths with NB, Ns, and a as parame-ters. The optimum diversity for a given value of Nsis clearly represented as the minimum in the proba-bility of error although the location of this optimumvalue is rather broad for a > 1. The minimum isquite pronounced for moderate values of turbulence,however. For severe turbulence (a = 1.5) we ob-serve that above a certain mean signal count, themore diversity the better. This can be explained bythe nature of the photoelectron counting distribution,


Fig. 8. Probability of error P(E) as a function oftotal mean signal count Ns and diversity D for theaperture integration receiver. NB = 1. (a) a = 0.5;

(b) a = 1.0.

TOTAL MEAN SIGNAL COUNT TOTAL MEAN SIGNAL COUNT

(a) ( b)

which broadens more rapidly for larger mean signalcounts and in effect requires more averaging of scin-tillation.

Thus, depending on the severity of turbulence andthe available signal power, it is clear that control ofdiversity in the aperture integration receiver may bedesirable. In fact for light to moderate turbulence(a < 1.0) too little diversity is generally preferable totoo much. An analogous situation was shown toprevail by Kennedy and Hoversten for heterodynedetection 0 ; however, that case is much simplified bythe basically Gaussian nature of the statistics.

E. Comparison of Two-Detector Array with ApertureIntegration Receiver

From the system designer's point of view, a quan-tity of interest is the relative gain in performance ofthe optimum (or approximate optimum) array pro-cessor over that of the suboptimum aperture integra-tion receiver. Thus one can evaluate the loss in per-formance incurred by use of the (much simpler) sin-gle-detector aperture integration receiver. In thissection, the quantities Ns and NB refer to the meancounts per coherence area, or per detector for thetwo-detector array case.

To examine this quantity, the equivalent increasein power (dB) required by the suboptimum receiverto maintain a given error rate of the optimum arrayreceiver (assuming R = 0) was obtained from thevarious probability of error curves. These calcula-tions were effected by setting D = 2 and choosingseveral values of a and NB; The approximate gainof the uncorrelated two-detector array over theequivalent aperture integration receiver (D = 2) ispresented in Table I. At least for D = 2, the valuescalculated confirm the conjecture made by Hover-sten, et al. 8 that the array detector gain will only besignificant for severe turbulence and high back-ground noise. From the table, the aperture integra-tion receiver with D = 2 is seen to perform quite wellat low to moderate turbulence levels and is only afew dB worse than the uncorrelated two-detectorarray at the saturation value of a, for moderate errorrates.

There are a number of possible sources of error inthe entries in Table I, however. First, calculationswere performed by hand from the various error prob-ability curves, and there is therefore some inherenterror in the technique (which is limited by the reso-lution and geometrical accuracy of the curves gener-ated by computer graphics). Furthermore, it has

cc

0ccE

U-0

I-_

-J3D

'co0CEaL

4

2

101

-2102

2 4 6 8 10 12 14 16NUMBER OF DIVERSITY PATHS D

' (a)

- 4a-

X, 20,s, 10-

Li0 5

-J_ 2

o 102

C55

1 2 4 6 8 10 12 1NUMBER OF DIVERSITY PATHS D

14 16

(b)

Fig. 9. Probability of error P(E) vs number of diversity paths Dfor the aperture integration receiver. The dependence on Ns is

shown parametrically. NB = 1. (a) a = 0.5; (b) a = 1.0.


10

a.

0I.

,0

a.

Ns =5

N s =10

-~~~~~~~_ -Ns =5

NB=Io 0.5

. . . . . I . . . I

Ns= I

Ns=5

Ns= _

Ns=15

- -~~~~~~~~~~~~~~ s 20

NB=l_~~~~~~~~~~~~~~ = 1.0 7

. . . . . . . . . . . . .

5--

1 2 4 6 8 10 12 14

NUMBER OF DIVERSITY PATHS D

(a)

4

2

10'

^; 5

aL

CE 202ICi

U- 50>- 2

_J13

coa, 50

a- 2

Io4

5

16 1 2 4 6 8 10 12 14


(b)

4

2

10'

-. 5'a.a-

20 -2C 10u

m 50

>2I.-; Ir3a,

E0CE

164

16 1 2 4 6 8 10 12 14


(c)

Fig. 10. Probability of error P(E) vs number of diversity paths D for the aperture integration receiver. The dependence on N 5 is shownparametrically. NB = 4. (a) a = 0.5; (b) a = 1.0; (c) a = 1.5.

Table . Gain of Uncorrelated Two-Detector Array Receiver(in dB) over Aperture Integration Detector with D = 2

P(f) N11 o = 0.5 af = 1.0 af = 1.5

10-1 2 0 0.30 1.184 0 0.36 1.328 0 0.41 1.36

10-2 2 0 0.56 > 1.364 0 0.56 >1.568 0 0.60 2.10

10-3 2 04 0 >0.708 0.2 0.78 2.75

10-4 2 04 0.18 0.2 1.10

been assumed that the fading random variable Z inthe aperture integration receiver is well approximat-ed by a lognormal density function for two indepen-dent coherence areas in the aperture (D = 2).Based on recent work, this appears to be a reason-able assumption."",12 Finally, since the receiver ap-erture will rarely contain completely independent co-herence areas and since the aperture integration andarray receivers perform identically for R = 1 (D =2), the relative gain of the two-detector array receiv-er over the aperture integration receiver may be fur-ther reduced. A final comment is in order: Sincewe have evaluated only the two-dimensional case,little can be conjectured about the relative gains oflarger arrays.

F. MAP Receiver

The other suboptimum receiver considered pre-viously' is the MAP receiver in which a maximum aposteriori estimate of the fading based on the ob-

served counts n is obtained and used as the true fad-ing. The performance of this receiver for the singledetector and for the two-detector array has been pre-sented in Figs. 2 and 5 (see also Fig. 1). The uppercurve of each pair corresponding to a given value of ais discontinuous in nature and represents the perfor-mance of the MAP receiver. The discontinuitiesarise because of the discrete threshold involved.The curves periodically approach those for the opti-mum receiver; thus, over small ranges of Ns/NB, andwith low to moderate o- and low Ns and NB, theMAP receiver can perform satisfactorily. Thesuboptimum performance results from the fact thatthe fading estimate Z is used whether or not a signalis present.

Ill. Receiver Performance for Orthogonal BPOLMand BPIM

Because depolarization due to the atmospheredoes not appear to be an important effect at opticalfrequencies,'3 one of the orthogonal signal formatsattractive for optical communication is binary polar-ization modulation (BPOLM) where two orthogonalpolarizations of the optical field represent the binarystates of the signal. It has been shown previously'that the optimum receiver for this equal-energy bi-nary orthogonal signal set is just photoelectroncounting. The receiver consists of a single-detectortwin-channel receiver (see Fig. 11) where each of thechannel inputs corresponds to one of the states ofpolarization separated by a device such as a Wollas-ton prism. The performance of such a system in theabsence of fading has been discussed by other au-thors.3 "14 Peters and Arguello'5 have considered theperformance in the presence of lognormal fading, butthey neglect background noise and dark current. Inthis section we evaluate the performance of such a


4

2

AL5

a.ccE0 2

m 103wLi 50

2

04

.J 103co0Cra-

2

5

Ns=204

.-I Nt4 -

N='60-

N'4 -

t w0~~O5 -

N2 80 -

r\~~ - . I .

Ns4 -

c_ ~~~N S 20-

Nsz40 -

NSz80 -

NB'4 -

= 1.0

N2= 4

N, 20

Ns=40Ns =60-

Ns=80

NB=4

,- 1.5

. . .. I ; I I I . * 16

l

..............

I

LEFT CHANNEL "I"P() = 1 - tpi(k)[ po(n),k=1 I T I

L CHOOSE

)-LARGER

nR

Fig. 11. Block diagram of optimum single-detector twin-channelreceiver for BPOLM with lognormal fading.

system in the presence of both fading and back-ground noise.

Another binary orthogonal signal format of inter-est is binary pulse-interval modulation (BPIM) inwhich a single pulse is sent in one of two disjointtime slots,16 corresponding to one bit of information.Assuming that there is no pulse synchronization un-certainty in the processor, there are two possible op-timum detection schemes for this format: one inwhich there are separate counters for each time slot,analogous to the two possible states of polarizationin BPOLM, and the other where there is a single de-tector whose output is stored and then compared atthe end of the signaling interval. Both of theseschemes can be analyzed by using the twin-channelreceiver and the same decision rules. These rules canbe arrived at from three different detection criteria:The first uses the threshold detector discussed in Sec.II.A, but here the counts from each channel are com-pared with the optimum threshold, and the detectorfor which the threshold is exceeded corresponds tothe signal for that channel.' 4 The second decisionrule chooses the signal as that corresponding to thechannel with the larger count.' 4 The third rule issimilar to the second rule with the addition that arandom choice is made if the counts are equal. 3 Weconsider here only the second and third rules; rule 3turns out to be optimum and is slightly better in per-formance than rule 2, as we now show.

For rule 2 the error probability is given by

(a)

X

a.0

LJ

0

I-

:3

a:0

with p,(n) and po(n) representing the counting dis-tributions of signal-plus-noise and noise, respective-ly, in the presence of fading. The error probabilityfor rule 3 is given by

P() {1- ZP1(k)[Zpo(n)]} + 2 Ep'(n) p(n).k=O n=0 2n=0

(11)

The quantity in braces represents the probabilitythat (no > n1) when a 1 is present and the secondterm is just p(no = ni). Further examination of therelationship between the error probabilities for rules2 and 3 leads to the result

P(e) = P(e) - 1p(ni = no).rule 3 rule 2

(12)

For NB large, the second term on the right-hand sideof Eq. (12) becomes negligible, and the two rulesperform identically. Figure 12 indicates the perfor-mance of a single-detector twin-channel counting re-ceiver for rules 2 and 3 both in the absence and inthe presence of fading. The odd-numbered curvescorrespond to rule 3 and the even-numbered curvesto rule 2. The scallops that appear in the perfor-mance curves for the optimum threshold detector forBPCM (compare Fig. 2) are absent here.

Using probability of error curves for BPOLM andBPIM such as those presented in Fig. 12, we havecalculated the equivalent power loss due to turbu-lence. These curves appeared in Fig. 3 along withsimilar curves for BPCM. In addition, we have cal-culated the equivalent power gain in dB of BPOLM(BPIM) over BPCM both in the absence and in thepresence of fading. The results of these calculationsare in Fig. 13 for NB = 1,4 and a = 0, 0.5, 1.0, and1.5. In the absence of atmosphere, the gain is about2 dB at moderate error rates. As the level of turbu-lence increases, indicated by increasing a, the gain ofBPOLM over BPCM readily exceeds 3 dB at moder-ate levels of turbulence and 5 dB at or near = 1.5.Although the gain of BPOLM over BPCM is more or

Fig. 12. Probability of error P(E) vs SNR y for sin-gle-detector twin-channel receiver using BPIM orBPOLM. The upper of each pair of curves corre-sponds to decision rule 2 while the lower correspondsto decision rule 3. The effect of fading is shown

parametrically. (a) NB = 1; (b) NB = 4.


. (b)


(10)

CV

-2-;

1 10

Fig 1..Promnegi of OL ove PC in .Oa

0, the repctv eqiaen oe lse f h w

0

bie wihte anof -Q5ovrBCMi h

0a.

1 B4 1 _ I

I 2 3 4 5dB GAIN OF BPOLM OVER BPCM

Fig. 13. Performance gain of BPOLM over BPCM (in dB) as afunction of error rate, turbulence, and background for a single-detector counting receiver. Open circles and squares indicate

hand-calculated theoretical points.

less constant over a wide range of error rates for au0, the respective equivalent power losses of the twosignaling formats in the presence of turbulence is dif-ferent. This gain in the presence of fading, com-bined with the gain of BPOLM over BPCM in theabsence of fading, results in the over-all gain curvesin Fig. 13. Thus, where the available power fortransmission is constrained, the binary orthogonalformats appear to be the best choice. Furthermore,for a single-detector receiver, such as the apertureintegration receiver, the optimum processor is just acounting receiver for each binary signal mode,whereas for BPCM, the receiver structure is consid-erably more complex due to the nonlinear weightingof the counts.

If we consider an array of detectors rather than asingle detector, the optimum array processor forBPOLM would be constructed for the two channelscorresponding to the two possible orthogonal signals,and then one would choose the largest output, thatis, L, or Lo, corresponding to a 1 or a 0. However,in this case the processing is no longer simple photo-electron counting; the receiver structure was pre-sented in Fig. 2 of Part 1.' The error probabilitiesare obtained from Eq. (5) with M 2 for equiproba-ble hypotheses.

IV. Summary

In this paper we evaluated receiver performance interms of orthogonal and nonorthogonal binary errorprobabilities for a number of optimum and suibopti-mum photocounting receiver structures presented in

Part 1.1 Both single-detector receivers and array-detector receivers were considered. A lognormalfading channel was assumed, and background radia-tion as well as dark current were taken into account.The aperture integration receiver was found to per-form quite well in comparison with the optimumtwo-detector array receiver. Only for extreme tur-bulence, large background levels, uncorrelated fading,and low error rates did the array processor showappreciable equivalent power gain over the apertureintegration receiver. For BPCM, and a single detec-tor, the effects of turbulence were found to producemore than 10 dB of equivalent power loss at moderateto severe levels of fading. This loss was shown to bereducible by the use of diversity, however. Addingjust one diversity path provided as much as 5 dB ofequivalent power gain at moderate turbulence levels.The existence of an optimum amount of diversity,with a fixed signal energy constraint, was demon-strated graphically for the aperture integrationreceiver. Unlike the analogous heterodyne case, how-ever, the optimum value for D was found to dependstrongly on the background noise level as well as onthe mean count SNR-y.

The performance gain of the binary orthogonal sig-nal formats over the nonorthogonal signal formatswas found to increase with the severity of turbu-lence. Thus, as in the absence of turbulence, we canconjecture that orthogonal signal formats are opti-mum for direct detection systems.17 This is particu-larly attractive for the aperture integration receiversince the optimum processor just compares photo-electron counts from the various orthogonal modes,and the receiver structure is very simple.

This work was supported in part by the NationalScience Foundation and is based on portions of adissertation' 8 "19 submitted by S. Rosenberg to theDepartment of Electrical Engineering and ComputerScience at Columbia University in partial fulfillmentof the requirements for the degree of Doctor of Engi-neering Science.

References1. M. C. Teich and S. Rosenberg (Part 1), Appl. Opt. 12, 2616

(1973) [preceding paper].2. S. Rosenberg and M. C. Teich (Part 3), IEEE Trans. Inform.

Theory IT-19, 807 (1973).3. W. K. Pratt, Laser Communication Systems (Wiley, New

York, 1969).4. D. Middleton and C. J. Gundersdorf, "On Optimum Detec-

tion in Quantum Optics, Part 2: Error Probabilities and Ex-pected Performance," Technical Report AFAL-TR-68-158, AirForce Avionics Laboratory, Wright-Patterson Air Force Base,(1970) pp. 8-13.

5. D. L. Fried and R. A. Schmeltzer, Appl. Opt. 6, 1729 (1967).6. S. Rosenberg and M. C. Teich, J. Appl. Phys. 43, 1256 (1972).7. M. C. Teich and S. Rosenberg, J. Opto-electron. 3, 63 (1971).

Note that Eq. (28) of this article should read B = Q(2) - A-'and IB11/2 should be replaced by -B11/2 throughout. Allfigures, results, conclusions, and other equations remain un-changed.

8. E. V. Hoversten, R. 0. Harger, and S. J. Halme, Proc. IEEE

58, 1626 (1970).


9. J. M. Wozencraft and I. M. Jacobs, Principles of Communica-tion Engineering (Wiley, New York, 1965), pp. 527-550.

10. R. S. Kennedy and E. V. Hoversten, IEEE Trans. Inform.Theory IT-14, 716 (1968).

11. R. L. Mitchell, J. Opt. Soc. Am. 58, 1267 (1968).

12. V. Blumel, L. M. Narducci, and R. A. Tuft, J. Opt. Soc. Am.62, 1309 (1972).

13. A. A. M. Saleh, IEEE J. Quantum Electron. QE-3, 540(1967).

14. T. Kinsel, Proc. IEEE 58, 1666 (1970).

15. W. N. Peters and R. J. Arguello, IEEE J. Quantum Electron.QE-3, 532 (1967).

16. M. Ross, IEEE Trans. Aero. Electron. Syst. (Supplement)AES-3, 324 (1967).

17. R. M. Gagliardi and S. Karp, IEEE Trans. Comm. Tech.COM-17, 208 (1969).

18. For abstract of dissertation, see S. Rosenberg, IEEE Trans.Inform. Theory IT-18, 544 (1972).

19. A talk based on portions of this material was presented at the1972 Annual Meeting of the Optical Society of America; for ab-stract, see S. Rosenberg, J. Opt. Soc. Am. 62, 353A (1972).

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