Photocounting Array Receivers for Optical Communication - people

Photocounting Array Receivers for OpticalCommunication Through the Lognormal AtmosphericChannel. 1: Optimum and Suboptimum ReceiverStructures

M. C. Teich and S. Rosenberg

The structure of the optimum direct detection array receiver is obtained for a system consisting of anamplitude-stabilized optical source, a lognormal channel, and a bank of photocounting detectors. Addi-tive independent background radiation and detector dark current are taken into account. Both orthogo-nal and nonorthogonal M-ary signaling formats are considered. Attention is given to detection intervalssmall in comparison with the correlation time of the atmospherically induced fluctuations. A saddle-point integration provides an excellent approximation to the optimum processor, resulting in a considera-bly simplified structure. Suboptimum aperture integration and maximum a posteriori (MAP) receiversare also considered. The performancein related papers (Parts 2 and 3).

1. Introduction

In a previous paper the statistical description ofthe output signal from an array of photoelectroncounters was developed.1 The incident radiationwas considered to pass through the lognormal atmo-spheric channel and to contain additive backgroundradiation as well as coherent signal. Probability dis-tributions for the photoelectron counts, both in thepresence of noise plus faded signal, and in the pres-ence of noise alone, were obtained. Exact expres-sions for the first-, second-, and third-order photo-counting cumulants for lognormally modulatedmixtures of coherent and chaotic radiation were alsocalculated.2 In this paper, these results are used toexamine the problem of optimum detection. Onlyclear-air turbulence 3' 4 is considered; atmosphericscattering and absorption are taken into accountonly insofar as they may uniformly reduce the irra-diance at the receiver. The performance of widefield-of-view receivers employing optical scatteringlinks has been considered elsewhere5 and will not bedealt with here.

The detected process is assumed to be the outputsignal from an array of photodetectors, which directdetect the incident radiation. For detectors with

M. C. Teich is with the Department of Electrical Engineering& Computer Science, Columbia University, New York, New York10027; S. Rosenberg is with Bell Laboratories, Whippany, NewJersey 07981.

Received 26 January 1973.

of these receiver structures and their relative merits are presented

positive gain (e.g., photomultipliers) the thermalnoise introduced at the detector can generally be ne-glected in comparison with the quantum, or shotnoise, of the detected signal and background radia-tion.6

The radiation arriving at the receiver is consideredto have been modified by an effective multiplicativerandom process that characterizes the effects of theturbulent atmosphere.7 Furthermore, for signalbandwidths of interest, the fading may be assumedconstant over the detection interval T. Thus T <Ta, with -i-s the characteristic fluctuation time for theatmospherically induced fading. This is justified bythe relatively long typical coherence time for the at-mosphere (-1 msec). The fading statistics for theirradiance are taken to be lognormal in light of mostexperimental evidence to date.8-1 1 Added to thefaded signal radiation is (signal-independent) back-ground noise modeled by a white zero-mean complexGaussian process that is stationary in time andspace. 7 For most practical receivers, the effect ofthe background radiation on the counting process isequivalent to the addition of independent noise pho-tocounts at each detector in the array.' Thesecounts are Poisson distributed with constant mean,proportional to the background noise power density.The background noise takes into account variousthermal radiation sources such as scattered solar ra-diation, direct solar radiation, moonlight, blackbodyradiation, etc. A block diagram of this communica-tion system is indicated in Fig. 1.

The direct detection photocounting distributionsfor lognormally faded laser radiation have been eval-

2616 APPLIED OPTICS / Vol. 12, No. 11 / November 1973

Fig. 1. Block diagram for the direct detection array photocount-ing communications system. The channel under consideration isthe clear-air turbulent atmosphere. A bar over a given quantity

symbolizes that it is a vector.

uated by Diament and Teich12 "13 and by others' 4-' 6

for a single detector and by Teich and Rosenberg' 2

for an array of detectors. The effects of turbulenceon optical radar and on the binary communicationchannel for heterodyne detection were examined byFried and Schmeltzer,17 and by Heidbreder andMitchell.'8 Kennedy and Hoversten7 presented thestructures and error bounds for M-ary orthogonalsignaling and heterodyne detection for fading on Dindependent paths. Halme' 9 extended Kennedy andHoversten's results to arbitrary correlation and in-vestigated various representations of the detectedfield in the aperture. One of Halme's conclusions isthat only the sampling representation of the field inthe aperture is amenable to analysis; this is the rep-resentation to be used here as well.

For direct detection optical communications in thepresence of lognormal fading, few detailed results areavailable. Peters and Arguello20 have obtained theerror probabilities for a polarization modulation sys-tem, using an ideal amplitude-stabilized source witha single detector per channel, in the presence of log-normal fading. These authors do not include back-ground radiation and dark current, however. Soli-meno et al. 2 ' presented binary error probabilities fordirect detection in the presence of lognormal fadingat a single detector. The case they consider, how-ever, is one in which the fluctuations of the back-ground radiation are discernible. That is, thecounting time T was taken to be significantly small-er than the background radiation coherence time -c,,and thus the counting distribution in the absence ofsignal is Bose-Einstein, instead of Poisson, as consid-ered here. In view of the fractional bandwidth ofavailable optical interference filters, the coherencetime of background radiation is generally 10-12sec, and the assumption that the background statis-tics are resolved is unrealistic. Furthermore, themethod given to obtain the error probabilities saysnothing of the processing to be performed on the ob-served counts. Recently, Hoversten et al.2 2 present-ed some results relating to the structure of optimumdirect detection receivers for ideal laser sources, butthe results they present assume independent fadingat each detector in the array. Except for an errorbound to a single-detector counting receiver, whichis optimum for binary orthogonal equal-energy sig-

nals,23 no quantitative performance results were pre-sented. Explicit evaluation of error probabilitiesand information rates for a single-detector binarypulse-code modulation (BPCM) link, assuming ageneralized laser source and lognormal fading, wererecently presented by Yen et al, 24 however.

In this paper (Part 1) we obtain the optimum re-ceiver structures, based on a minimum probability oferror criterion, for lognormally faded amplitude-sta-bilized radiation with arbitrary fading correlation.This radiation is array detected in the presence ofadditive independent Poisson noise photocounts.Both orthogonal and nonorthogonal M-ary signalingformats are considered. In addition, we examineseveral suboptimal receiver structures.

In Part 2,25 we obtain quantitative results for theperformance of the structures presented in Part 1, asmeasured by the total probability of error per bit, forseveral binary signaling formats. In Part 3,25 we de-rive a theoretical upper bound to the error probabili-ty for M-ary equal-energy, equiprobable orthogonalsignals over D diversity paths, assuming flat inde-pendent fading.

II. Channel Model

Clear-air turbulence produces random fluctuationsin the amplitude and phase of a transmitted opticalwave, caused by the random variation of the opticalindex of refraction, in time and space, along a propa-gation path.26 The resulting effects significantly de-grade the transmitted wave, as measured by space-time fading, loss of coherence, and spreading of thebeam. In this work, we do not specifically considerthe effects of haze, smog, clouds, or other atmo-spheric conditions producing scattering and absorp-tion effects. One result of these effects is a net de-crease in the strength of the optical field at the de-tector. Thus by clear-air turbulence we mean onlythose effects produced by index of refraction varia-tions. A thorough review of the current theory ofatmospheric turbulence has been given by Lawrenceand Strohbehn.4

In addition to the effects of the turbulent atmo-sphere, we include in our channel model the effectsof background radiation as produced by variousthermal sources such as the sun, sky, ambient earth-light, etc.27 When the receiver field of view does notinclude the sun, most of the natural optical radiationfor wavelengths below 3 m is due to reflected orscattered solar radiation. For wavelengths above 3gm the dominant source of background radiation isthe thermal radiation of the earth whose spectralshape is approximately that of a blackbody at 280 K.

The background radiation is modeled as a whitezero-mean complex Gaussian process whose compo-nents in the receiver aperture are independent andstationary in space and time. This latter assump-tion is valid for apertures larger than a few wave-lengths and most signal bandwidths of interest. Thebackground radiation is characterized by its spectralradiance N. which has the units W/m2 sra gm of op-tical bandwidth. In our channel model, the effect of

November 1973 / Vol. 12, No. 11 / APPLIED OPTICS 2617

this background radiation is introduced as an addi-tive, signal-independent noise process that producesphotoelectrons at the mean rate NB.28 Furthermore,we need not consider the cross-mixing of signal andnoise components, since for direct detection in ashot-noise limited regime, these terms are negli-gible.2 8,2 9

Based on the previous sections, we formulate thechannel model as follows. A temporally modulatedlinearly polarized wave with complex envelope givenby X(t,r) = S(t)e(t,r) is transmitted. The real fieldcorresponding to this complex envelope is Re[X(t,r)exp( -j27rvt)]. Here e(t,r) represents the complexanalytic field representation for the source, and S(t)represents the temporal modulation, which in latersections is assumed to be of digital format. Aftertraversing the clear-air turbulent atmosphere, andneglecting the propagation time delay, the field is ofthe form Y(t,r) = exp[iP(t,r)]X(t,r) + eB(t,r), where4(t,r) is a complex Gaussian process completelyspecified by its mean and covariance,4 and whereeB(t,r) is the complex envelope of the relevant polar-ization component of the background noise radiationdiscussed in the previous section. (The complexnoise field generally consists of two orthogonal inde-pendent polarization components of equal meanpower.) After appropriate spatial and temporalpreprocessing to limit the noise (which is assumednot to change the field statistics), the field is sam-pled at an array of point detectors, each of whichproduces photoelectron counts as the observable.The channel model is thus the same as that formu-lated by Kennedy and Hoversten7 for heterodyne de-tection, except that in our direct detection schemewe allow for partially correlated fading at the array.It should be mentioned at least in passing, however,that some deviations from this model can occur,especially at severe turbulence levels.15"16

Ill. Photoelectron Counting DistributionsBased on a first-order quantum-mechanical per-

turbation interaction, it can be shown that the prob-ability of observing a photoelectron emitted from aphotocathode is described by the well-known condi-tionally inhomogeneous Poisson process.30 Thus,the probability of emitting n photoelectrons in atime interval (t, t + T) is given by

We further assume that the field maintains com-plete first-order spatial coherence over each detectorsurface, and thus the spatial integral merely pro-duces a constant, representative of the detector area.With Ad the detector area and a = nAd/hv, the jointphotoelectron counting distribution for an array ofdetectors may be written as" 3'

p(n, t T) = HWiniexp(-Wi))i 1 ni ) wil

(3)

witht+T

Wi = f, ailVi(P ,l~dt'. (4)

The angular brackets indicate an ensemble averageover the statistics of Wil. Equation (3) is often re-ferred to as Mandel's formula.3 ' For simplicity, wefurther assume that the photodetector impulse re-sponse is ideal. That is, we assume that individualphotoelectrons can be resolved.

Considering a radiation source that produces aPoisson counting process conditioned only on thefading, the integrated intensity for the ith detector,Wi, is given by

W = ZiNsi + NB (5)

Here Nsi is the mean count due to the signal energyat the ith detector, Z is the normalized fading inten-sity, and NB includes the contribution of backgroundradiation as well as detector dark current, which canalso be represented by an independent Poisson pro-cess.32 The results derived here apply to an ampli-tude-stabilized laser operated well above thresholdor to a source of arbitrary statistics provided thatT/Cr >> 1. Most thermal and laser sources used inoptical communication systems are likely to fit inthis category.

The conditional counting distribution for an arrayof D detectors is therefore given by

D (ZiNsi + N)ni exp[-(ZiNs + NB)]pAnIZ) = [I . (6)

Averaging over the joint density for the normalizedfading random variables {Zi}, the counting distribu-tion becomes'

p(n; Ns; A) = f p(n; NsIZ)p(Z)dZ,

p(n, t, TIW) = [Wn exp(-W)] / n!.

Here the integrated intensity or rate parameter defined by

W= htT fIV(t, r)JPdt'dA,

where the detector quantum efficiency i7 is assunto be constant over the bandwidth of the detecradiation. The quantity hv is the photon enerand V(t',r) represents the analytic signal. Howeiif the radiation is of a stochastic nature, an additial average over the statistic of V(t',r) is requiredorder to obtain the photoelectron counting distriltion.

(1)

(7a)

where

p(Z) = [(27)DI2 IA1/2ZAZs... ZD]-1 exptl-XtA 1XI. (7b)

(2) The vector X has components given byIl Xi = nZ + ( 2/2), i = 1,2,...,D. (8a)

Here the log-irradiance covariance matrix A containselements given by

Aij = Cln(ri, r), ij = 1,2,...,D, (8b)

where

Ai C1 nI(ri, ri) i2


(8c)

is the log-irradiance variance. The vector r speci-fies the position of the ith detector.

We now apply the method of steepest descent,1,12

but here with the relevant quantities defined as fol-lows:

Xi = nZi0 + (2/2), (9a)

Q/1 )(ni; ZioNsi) = ZN + NB - ZioNsi, (9b)

Qij'2'(n,; Z1oNsi) = - ZioNsi] , (9c)

and.

Q2 (2)

B =

The subscriptThe countingform

..Q1D 2

].A-.QDD(2)

0 represents the stationary podistribution then takes the exp.

t(n) _ p1(n) >A ) Po(n) <

Ho

1 -r7WI

(13)

where pl(n) is the density of n under H1, and po(n)is the density of n under Ho. Here 7r, is the a prioriprobability that a signal is present, while 7ro = 1 -7r is the a priori probability that it is not.

Equivalently, since the logarithm is a monotonicfunction, the logarithmic likelihood ratio L(n) isgiven by

H.

L (n) InA(n) > ln( gHO

(14)

Since we are concerned primarily with digital com-(9d) munication, we assume for simplicity that 7r = Y2.

That is, the hypotheses present and not present aretaken to be equally likely.

int. The likelihood ratio then reduces to the simplelicit form

p(n; Ns; A) =

(ZiNsi + NB)'i exp[-(ZioNsi + NB)])irl nil exp {-_2XotA-Xo}

I (10)

where the stationary point Z is obtained from theequation

Q"')(n;ZNs) - A-X = 0. (11)

The counting distribution given by Eq. (10) was pre-viously evaluated, and graphically presented for thecase D = 2 as a function of the various parameters ofinterest.'

The noise counting distribution is given byp(n; NB) = D NB exp(-NB)

i=1 ni!

H,

A(n)> ,HO

(15)

and the likelihood function becomes

Hi

L(n) > 0.HO

(12)

where it is assumed that the mean noise count ateach detector in the array is NB. In the followingsection we use these results to obtain the optimumreceiver structures based on a minimum probabilityof error constraint.

IV. Optimum Receiver Structures

First we consider the binary signaling problem; wemust decide by some appropriate scheme whetherthe detected photoelectron counts are a result of asignal-plus-noise being present or a result of noisealone. This is referred to as the binary hypothesistesting problem. Let H1 be the hypothesis that asignal is present and Ho the hypothesis that it is not.It can then be shown that based on a Bayes criteri-on, the quantity that minimizes the average risk,and in this case the total probability of error, is ob-tained from the likelihood ratio test.33 This testspecifies that either H1 or Ho be chosen dependingon the result of the inequality for the likelihood ratioA(n):

(16)

The log-likelihood ratio or likelihood function is usu-ally the quantity that reveals the receiver structure.That is, it tells us how to process the observed datan in order to decide whether to choose H1 or Ho.

Similarly it can be shown that for M equally likelyhypotheses, the test that corresponds to minimizingthe total probability of error is the maximum likeli-hood test, where one chooses the kth hypothesis if Lk

L for j = 1, . . ,M. That is, we choose the likeli-hood function that is largest as corresponding to thecorrect hypothesis. If likelihood draws occur, anyrandom choice can be made without affecting thetotal probability of error.33

A. Optimum ProcessorWe begin this section by considering the simple

binary detection problem where there either is, or isnot, a signal present.

From Eqs. (6), (7a), (12), and (13), we obtain


IAI 1/21 -B *11/2

.

A(n) =f ... ( l(ZNs + NB)0 exp[-(ZiNs + NB)])P(Z1, Z2 , ... , ZD)dZ ... dZ,

D[I NB0 i exp(-NB)i=1

where p(Z) is given by Eq. (7b). The likelihoodfunction is then

L(n) = n[ f [ ( N + 1)

X exp(-ZiNsi)Ip(Z)dZl. (18)

The density of the lognormal variates ZL is in gen-eral that of correlated variables, and the receiverstructure is rather complex due to the highly nonlin-ear nature of the functions involved. The saddle-point method, as used in Ref. 1, will allow us to ob-tain a tractable receiver structure that provides anexcellent approximation to the performance of theoptimum structure specified by Eq. (18).

The structure of Eq. (18) can be further simplifiedif we allow Nsi = Ns independent of i, implying thatthe same mean signal energy is present at each de-tector. It has already been assumed that NBi = NB.Furthermore, if the fades at all the detectors are sta-tistically independent and of equal strength ( =a), the optimum receiver structure is given by

L = In ti[(ZJB + I) exp(-ZNs)(Z)dZ]

(19)

where p(Z) is the one-dimensional lognormal distri-bution.' 2 " 3 This structure corresponds to a nonlin-ear weighting of the counts from each detector, be-fore combining.

For M equally likely signals, the optimum receiverforms the structure of Eq. (18) with Nsi replaced byNSk, where again i refers to the ith detector and re-fers to the kth waveform, for each of the possible Mwaveforms. For equal-energy orthogonal signals, Lkis given by Eq. (18) with ni replaced by ni1 and Ns8by Ns.

If we allow D = 1, the test corresponding to Lk >Lj for equal-energy orthogonal signals is

Jo ( -+ 1) exp(-ZNs)p(Z)dZ

> f ZNs + 1 exp(-ZNs)p(Z)dZ, (20)

where nk is the observed count assuming that theenergy detected is associated with the kth signal ofthe M possible orthogonal signals. This is equiva-lent to testing whether nk > n, since the functionalsare monotonically increasing with nk. Thus forM-ary equal-energy orthogonal waveforms and onedetector, in the presence of turbulence, the optimumreceiver is that of unweighted photoelectron count-ing, just as in the absence of turbulence. As will beshown in the next section, the approximate optimumreceiver, based on the likelihood function saddle-point solution for this case, does not reduce to the

, (17)

counting receiver except for D = 1. It should bepointed out that the use of the instantaneous fadelevel Z in calculating the estimator can result in afixed threshold that appears to be independent of Z.

B. Approximate Optimum ProcessorIn order to determine the processing implied by

the likelihood functions given in the previous section,we resort to a saddle-point solution, as was done inobtaining the counting distribution. Applying thismethod' and assuming uniform average irradiance atthe detector array, we obtain the following likelihoodratio:

A(n) = [ (ZioNs + ) exp(-ZioNs)]

xexp - !XoA-'Xol

IA11/21 -B *11/2. (21)

The receiver structure is then given by

L = [Dni In( N + 1) - ZioNs] -!XoA XO

- 21nAI - lnl-B*I, (22)

for the binary case.

Fig. 2. Approximate optimum array receiver for partially corre-lated fading and M-ary signaling. 2, is the solution to the MAPestimator equation for the kth signal waveform, and Xo1 repre-sents Xoh. Uniform average illumination of the array is assumed

so that Ns, represents Nsk rather than Ns8 .


The receiver thus performs weighted counting,where the weights now depend on the solution of thestationary equation, Eq. 11, and thus on the ob-served counts. Bias terms that depend on the co-variance matrix of the fading, and on the matrix B*,must be subtracted. It is in these bias terms thatthe processor weights the counts optimally, deem-phasizing those with strong fades about the meansignal count. If in addition a signal is present (inthe binary case), the modal points ZO} form maxi-mum a posteriori (MAP) estimates, ZI, of the fadingat each detector in the array, as will be shown. Wewill later examine a suboptimum receiver that at-tempts to measure the fading on each signaling in-terval, and using that noisy estimate, processes thecounts as if the fading were known.

For M-ary signals, the approximate optimum pro-cessor forms Eq. (22) for each of the possible Mwaveforms. This receiver is shown in Fig. 2, whereZo is indicated by Z for simplicity of notation.

A receiver structure similar to that obtained here,but for independent flat fading, has been given byHoversten et al.22 The receiver structure solutionsgiven there are based on the Bar-David formulationof the Poisson process, in terms of the time occurren-ces of the individual photoelectron pulses, ratherthan on the total number of pulses observed duringthe detection interval (0,T).3 4 The Bar-David for-mulation is more useful in radar and waveform esti-mation, where time occurrences of events are impor-tant. The solution based on the Mandel formulalends itself more readily to the evaluation of receiverperformance, however, and has been used for thatreason. It should be noted that in contrast to previ-ous results, the strucuture specified here is moregeneral in that it accounts for the possibility of cor-related fading.

C. Independent Fading Samples

The structure of Eq. (22) can be further simplifiedif the fading at each detector is independent of thefading at every other. In that case the covariancematrix A and the matrix B* are diagonal, and thereceiver structure reduces to

L = [niln( N + )

ZioNs - [ln(Zio) + ( 2/2)]2

- Z~~0N8 ~ 2ui2

- 21ln{i2[(Z jN + N )2 + ZioNs] + 1}], (23)

where the Zzo} are now obtained from an uncoupledset of stationary equations given by

niZioNsZ0ON + NB

[ln(Z10) + (2 / 2)]- ¢2 = 0 for i = 1,2,...,D. (24)

Equations (23) and (24) are similar to those given byJ. N. Bucknam and first published by Hoversten etal.

2 2 (The expressions given there do not appear to

be correct, however.) For M-ary signaling, we formM such functionals, where now Ns - Nsk, andchoose the largest. The structure for this receiver isgiven in Fig. 3.

The approximate optimum processors discussed to.this point provide an excellent approximation to theexact optimum processors; their performance will beevaluated and presented in Part' 2 for binarl pulse-code modulation (BPCM), binary polarization mod-ulation (BPOLM), and binary pulse-interval modu-lation (BPIM).25 In Part 3, we consider M-aryequal-energy equiprobable orthogonal signaling with'flat independent fading.2 5 We now turn to somesuboptimum receiver structures that are often con-siderably easier to implement.

V. Suboptimum Receiver Structures

It is of interest to investigate some suboptimumreceiver structures in order to evaluate the tradeoffbetween complexity in processing and degradation ofperformance. In particular, we investigate struc-tures for the aperture integration and MAP receiv-ers.

A. Aperture Integration ReceiverThe aperture integration receiver consists of a sin-

gle large detector encompassing the area covered bythe array of D detectors considered in previous sec-tions. The detector area is assumed to encompass Dindependent coherence areas of the faded signal, plusindependent additive background noise radiation.The integrated intensity W is therefore given by

COUNT FORT SECONDS

L I . I _I COMPUTE lREPEAT FOR MIL TERMSOTHER WAVEFORMS

Fig. 3. Approximate optimum array receiver for independentfading and M-ary signaling. In this case the 2ik are obtainedfrom an uncoupled set of stationary equations. The quantity Ns,represents Nsk, as in Fig. 2, indicating that uniform average illu-

mination is assumed at the detector aperture.


Fig. 4. Approximate optimum aperture integration receiver forBPCM with lognormal fading.

W = ZDN + DNs,

where D Ad/Ac, the number of independent co-herence areas in the detector aperture. The randomvariable Z is now given by

Z A Z(r)dr. (26a)

Based on studies of the statistics of this quanti-ty,3 5'3 6 it is clear that Z is well approximated by alognormal random variable for large D. Further-more, it can be shown that X = lnZ is Gaussian withmean - IA

2 /2 and variance

9A2 = ln[ eD 1+ , (26b)

from which the variance of Z is [exp(er2 )-1]/D.This expression shows the effect of aperture averag-ing of the scintillations.3 7 It should be noted thatthe expression for the variance is exact; the approxi-mation rests on the assumption that Z is lognormal.

Experimental evidence, however, indicates thatthe maximum aperture averaging observed is thatwhich reduces the variance of Z to a minimum valueof about 10% of the unaveraged value.38 Thus thereappears to be a limit on the performance of an aper-ture integration receiver, and further improvementcan only be obtained by resorting to detector arrays.

Based on the foregoing assumptions and analysis,the receiver structure is given by Eq. (23) with thesummation on i dropped, U1

2 replaced with erA2 , and

NS and NB replaced by DNs and DNB, respectively.The receiver then decides that a signal is present if L> 0 and that no signal is present if L < 0. As canbe seen from the equation, the receiver weighs the.counts in a nonlinear fashion before combining, aspreviously, but the over-all receiver structure is con-

siderably simpler than for the array, as illustrated inFig. 4 (compare with Fig. 3).

For M-ary signaling, the optimum aperture inte-gration receiver forms the quantities Lk and choosesthe signal corresponding to the largest. For equal-energy orthogonal signals, however, the modal pointsZok are dependent on nk. The weights are thus datadependent and are different for different values ofnk. Thus, the processing does not appear to reduceto unweighted photoelectron counting. Neverthe-less, since Zo can be shown to be a monotonically in-creasing function of n, with all other parameters con-stant, then for equal-energy orthogonal signals theoperation performed by the approximate optimumreceiver is equivalent to comparing nk with n, thatis, to unweighted counting (see Fig. 5) as shown ear-lier.

B. MAP Receiver

Another possible scheme for reducing the complex-ity of the receiver is one in which a noisy estimate ismade of the fading, under the assumption that a sig-nal is present, and then used in the maximum likeli-hood receiver as if the fading were exactly known.The noisy estimate is obtained from the maximuma posteriori estimate of the fading Z. This is foundfrom the MAP equation 3 3

(a/Z)p(ZIn) = (/aZ)Jp(nIZ)p(Z)/p(n) = 0, (27)

where p(Z n) is the a posteriori density of the fading,given that the photoelectron counts n have been de-tected. The quantity p(njZ) is the conditional den-sity of n given Z. However, since p(n) is indepen-dent of Z, we must evaluate (/aZ)Jp(nIZ)p(Z) orequivalently (/OZ) lnp(nIZ) + np(Z) = 0. Since

D (ZiNs + NB)ni exp[-(ZiNs + NB)]pAnJZ) = [InH (28)

i=1ni

and p(Z) is given in Eq. (7b), the MAP equation be-comes

Q"1)(n, Z) - AX = 0, (29)

which is just the stationary equation, Eq. (11). Thesolution is now Z, and the likelihood ratio and likeli-hood function are thus given by

A(n) =[l( N +A () =riNB 1)'iexp(-ZiNs)(30)

SAMPLE EVERYT/M SECONDS

COUNTER N

ANDINTEGRATOR

C H OOSELARGEST

Fig. 5. Approximate optimum single-detector receiver for M-aryequal-energy orthogonal signals (PPM is shown). The receiver

performs simple unweighted counting.


and

L(n) = EniIn( N + 1) - iNs, (31)i=1

which take the form of weighted counts. As pre-viously, for L ' 0 we decide a signal is present. Thereceiver structure is thus considerably less complexthan the approximate optimum array receiver, butat the cost of some performance. The precise perfor-mance of such a receiver is evaluated in Part 2.25

The results indicate that over several ranges of a,NS/NB, and NB, the MAP receiver performs almostas well as does the optimum receiver based on thesaddle-point solution. It should be noted, however,that this receiver always estimates Z whether a sig-nal is present or not. Thus when a signal is absent,2 is not a valid estimate and the performance issuboptimum, as is indicated by the error probabilitycurves. However, if a signaling scheme is used suchthat there always is some signal present, analogousto a transmitted reference or pilot tone, the esti-mates will be valid, and the problem remains as towhich one to choose or perhaps whether to long termaverage the estimates.

In all of the above receiver structures there is theimplicit assumption that all other parameters, suchas Ns, NB, er and A, are known exactly. Realistical-ly these quantities must be obtained either beforeprocessing takes place or as part of the processor it-self. However, the detected counts inherently con-tain all the information about the state of the chan-nel, and thus by building the appropriate parameterestimators, analogous to the MAP estimator for thefading Z, these quantities can be measured. 2 2 Re-ceivers that perform such channel measurement havebeen suggested in the past, but their performance fordirect detection remains to be evaluated.

VI. Summary

We have obtained the optimum array receiverstructures for lognormally faded amplitude-stabi-lized laser radiation, direct-detected in the presenceof independent additive background radiation for ar-bitrarily correlated fading at the detector array. Re-ceiver structures for both orthogonal and nonortho-gonal M-ary signal formats were presented. The na-ture of the fading statistics resulted in complex re-ceiver structures that were approximated by use ofthe saddle-point technique. Fortunately, the ap-proximation mode used in obtaining these structureswas found to be excellent, in the sense that the per-formance (bit error probability) is very close to thatobtained using the exact receiver structures. In ad-dition to the approximate optimum structures, sev-eral suboptimum structures were also investigated.The performance of these receiver structures andtheir relative merits are presented in accompanyingpapers (Parts 2 and 3).25

This work was supported in part by the NationalScience Foundation and is based on portions of adissertation39' 40 submitted by S. Rosenberg to the

Department 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, J. Opto-electron. 3, 63 (1971):

Note that Eq. (28) of this article should read B = Q(2) - A-1

and I1/ 2 should be replaced by -B 11/2 throughout. Allfigures, results, conclusions, and other equations remain un-changed.

2. S. Rosenberg and M. C. Teich, J. Appl. Phys. 43, 1256 (1972).3. J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).4. R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523

(1970).5. R. S. Kennedy, Proc. IEEE 58, 1651 (1970).6. W. K. Pratt, Laser Communication Systems (Wiley, New

York, 1969), Chap. 9.7. R. S. Kennedy and E. V. Hoversten, IEEE Trans. Inform.

Theory IT-14, 716 (1968).8. G. R. Ochs and R. S. Lawrence, J. Opt. Soc. Am. 59, 226

(1969).9. G. R. Ochs, R. R. Bergman, and J. R. Snyder, J. Opt. Soc.

Am. 59, 231 (1969).10. P. H. Deitz and N. J. Wright, J. Opt. Soc. Am. 59, 527

(1969).11. G. E. Mevers, M. P. Keister, Jr., and D. L. Fried, J. Opt.

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(1970).13. P. Diament and M. C. Teich, Appl. Opt. 10, 1664 (1971).14. J. Peiina and V. PerinovA, Czech. J. Phys. B22, 1085 (1972).15. J. Pehina, V. Perinovd, and R. Hordk, Czech. J. Phys. B23,

to be published (1973); V. I. Tatarski, Zh. Eksp. Teor. Fiz. 61,1822 (1971) [Sov. Phys.-JETP 34, 969 (1972)].

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Proc. IEEE 58, 1691 (1970).39. For abstract of dissertation, see S. Rosenberg, IEEE Trans.

Inform. Theory IT-18, 544 (1972).40. A talk based on portions of this material was presented at the

1972 Annual Meeting of the Optical Society of America; forabstract, see S. Rosenberg, J. Opt. Soc. Am. 62, 353A (1972).

2ND JOINT CONFERENCE ON THE SENSING

OF ENVIRONMENTAL POLLUTANTS

December 10 - 12, 1973 at the

Sheraton Park Hotel in Washington, D.C.

Chairman:M. E. RingenbachNQAA/National

Ocean SurveyRockville, Md. 20852

Technical sessions for the 1973 Conference will feature an information

exchange on the basic phenomenology related to sensors and sensing techniques

for measuring land, water and air environmental quality parameters.

The Conference is directed to representatives of all technical disciplines

concerned with the development, evaluation and use of pollution measuring

techniques.

1. Recent advances in remote sensors (active/passive), in situsensors (chemical/biological-physical), and the extensionof laboratory measurement techniques for field use.

2. Extending the capability of existing sensors -- range ofmeasurement, accuracy, reliability, survivability, etc.

3. Applied research into relatively new techniques of sensingand measuring for field and laboratory applications --bio-accumulator, photographic, chemical, acoustic, semiconductor,electromagnetic, etc.

4. Instrument and data quality standardization, such as establishingdata and instrument standards to provide for maximum interchangeabilityand use of data among the various user groups.

5. The impact of meteorological and oceanic dynamics onpollution analysis and possible global scale pollutionmonitoring -- site selection, modeling, laboratory/simulationsof processes, special instrumentation, common data bank, etc.

6. Sensors and techniques for acoustic, electromagnetic andradiological pollution monitoring.

INSTRUMENT SOCIETY OF AMERICA 400 Stanwix Street, Pittsburgh, Pa. 15222 Telephone (412) 2813171


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