Receiver performance evaluation using photocountingcumulants with application to atmospheric turbulence
Paul R. Prucnal
A new technique for evaluating optical communication system performance directly from the photocountingcumulants is presented. This technique is most useful in situations where exact, explicit, closed form solu-tions exist for the photocounting cumulants, but do not exist for the photocounting distribution, such as inthe generalized case of communication through lognormal atmospheric turbulence. Using this technique,theoretical probability of error curves are presented for communication through lognormal atmospheric tur-bulence, for a superposed coherent-in-chaotic signal, embedded in additive independent Poisson noise, with
arbitrary ratio of sampling time to source coherence time, arbitrary ratio of coherent to chaotic component,arbitrary mean frequencies of the coherent and chaotic components, and where the chaotic component need
not be stationary and may have arbitrary spectral distribution. Since no solution exists for the photocount-ing distribution itself in this generalized case, the corresponding performance calculation has not previouslybeen possible. The case described applies to the detection of radiation originating from a multimode laser
or scattered from a rough target, and passing through atmospheric turbulence. A special case of these re-
sults is shown to be in excellent agreement with previous calculations [Rosenberg and Teich, Applied Optics
12, 2625 (1973)] for a lognormally modulated coherent signal.
1. Introduction
Light transmitted through the turbulent atmosphereundergoes intensity modulation that results in alteredphotocounting statistics. Indeed, a substantial amountof turbulence will tend to dominate the overall fluctu-ations in photocounts, regardless of the original sourcestatistics. The effect of turbulence, which is typical ofarbitrary intensity modulation,1 is an accentuation ofphoton-bunching, resulting in an increased magnitudeof the photocounting cumulants, an increased coeffi-cient of variation, and an overall broadening of thephotocounting distribution.2
The common assumption that turbulence induceslognormal intensity modulation 3 has lead to difficultiesin obtaining analytic solutions for the photocountingstatistics, 2 and has placed a variety of restrictions on thesolutions that have been obtained. Photocountingstatistics have been obtained with the restriction thatthe detector counting interval T is much shorter thanthe characteristic fluctuation time Ta of the turbulence.3
The author is with Columbia University, Columbia RadiationLaboratory, Department of Electrical Engineering, New York, NewYork 10027.
Typically, Ta 1 msec,4 so that the assumption T << Ta
is quite reasonable. Approximate implicit solutionshave been obtained for the N-fold joint photocountingdistribution for a coherent source,5 and the one-foldphotocounting distribution for an interfering mixtureof coherent and chaotic radiation,6 with the restrictionthat the ratio : of T to twice the coherence time of thesource is much smaller than unity. Under the addi-tional restriction that the ratio y of the coherent com-ponent to the chaotic component is much larger thanunity, an exact explicit recursive solution for the pho-tocounting distribution has been obtained.7
Some aspects of the related communications problemhave been investigated, but are limited by the abovedescribed restrictions and approximations on theunderlying photocounting distributions, as well as ad-ditional approximations used in evaluating the systemperformance. Binary pulse code modulation (BPCM)has been investigated for a lognormally modulated co-herent signal in interfering chaotic noise of arbitrarybandwidth, based upon an exact recursive solution ofthe photocounting distribution, for 3 << 1, and y >> 1.7BPCM has also been investigated for a lognormallymodulated interfering mixture of coherent and chaoticsignal in the presence of additive independent Poissonnoise, based upon an approximate implicit solution ofthe photocounting distribution, for : << 1 and arbitraryy.6 Pulse position modulation (PPM) and BPCM arrayreceivers have been investigated for a lognormally
modulated coherent signal in additive independentnoninterfering Poisson noise, based upon an approxi-mate implicit solution for the photocounting distribu-tion, with an additional approximation in evaluating theperformance, for: >> 1.2,8,9
In contrast to the many restrictions and approxima-tions on the photocounting distributions, the N-foldphotocounting cumulants have been obtained2 exactlyunder a much more general set of conditions: for alognormally modulated mixture of coherent and chaoticradiation, with arbitrary : and y, where the chaoticcomponent need not be stationary and may have arbi-trary spectral distribution, and the mean frequencieswc and WT of the coherent and chaotic components,respectively, need not coincide. The case of arbitrary/ and y applies particularly to the scattering of radia-tion from a rough target, and the output of a nonlockedmultimode laser, where beating between the modesoccurs. 0 -1 2
Based upon the preceding discussion, it is apparentthat a technique for evaluating communications systemperformance directly from the photocounting cumu-lants would be useful. Indeed, such a technique wouldbe useful in any situation where exact explicit closedform solutions exist for the photocounting cumulants,but do not exist for the photocounting distributions.The purpose of this paper is therefore twofold: to de-scribe a new technique for evaluating system perfor-mance directly from the photocounting cumulants; andto apply this technique to communication through theturbulent atmosphere, for which the photocountingcumulants are known in far more generality than thephotocounting distributions themselves.
II. Theory
The performance of a communication system inwhich a sequence of bits i E 0,11 are transmitted witha priori probabilities Pi can be characterized by theBayes risk
where the constant Cij is the cost of deciding that i wastransmitted, given that j was transmitted. For thespecial case of zero cost for a correct decision and unitycost for an incorrect decision, A? reduces to the proba-bility of error:
Pr(E = POPF + P1(1 - PD).
It has previously been shown,16 using normalizingtransformations, that (3) can be expressed as
PD(F) = 2 erfctD(F), (4)2
where
tD() g(t) - (gi(r)) (5)[2 vargi (r)] 1/
2
The known function gi (r) transforms the pdf p(r I i) intothe Gaussian (normal) pdf N[(gi(r)), vargi(r)] wherei E 0,11. The exact form of gi is known in the case ofnormal, chi-squared, noncentral chi-squared, and log-normal pdfs,3 and approximate expressions for gi havebeen developed for a variety of pdfs such as the Poisson,binomial, and negative-binomial.17 Furthermore,Kendall and Stuart' 8 have derived the expression
_ Di ((r)) (6)
where k is a constant, and the integral is evaluated at(r) = r for pdfs for which the variance can be expressedas a function D of the mean
D((r)) = (varr)1/2 . (7)
It is apparent that (4) provides an alternative solutionfor system performance in cases where gi (r) is known,and is most useful where the pdf p (r I i) in (3) cannot becalculated easily.
The probabilities of detection and false alarm can beobtained from (4) on a case-by-case basis, by finding thegi corresponding to the particular pdf, and solving forits mean and variance. However, a general expressionfor PD(F), in terms of the ordinary and central momentsof an arbitrary pdf, can be obtained using (6), as follows.Expanding (gi (r)) in a Taylor series about r = (r), andusing (6), yields
k f Di 2 n=3 n! (DBJ 1 (8)
where the exponent (n) denotes the nth derivative withrespect to (r), nU denotes the nth ordinary moment ofr, and where Di is understood to be a function of (r).Expanding (gi(r)2) in a Taylor series about r = (r), andsubtracting the square of the expansion in (8), yields1 - fun/]( -i
k2 n=3 n! 1m=Cn- (2D.) (D, ]
[ n _( 1(m-1] (9)(2)
Provided single-threshold processing at the receiver isoptimum,13-15 the probabilities of detection and falsealarm are
PD(F) = , p(ri)dr, (3)
where D(F) corresponds to i = 1(0), t is the threshold,and p (r I i) is the conditional probability density func-tion (pdf) of the observation r given that i was trans-mitted.
Here [n/2] denotes the integer part of n/2,
Cn = 2, Cn = Cn-1, Cn = bCn:I + Cn-1. (10)
The constant b = 2 for k = 1, and b = for k > 1. Theprobabilities of detection and false alarm are then ex-pressed directly in terms of the ordinary and centralmoments of the pdf, by substituting (5)-(10) in (4).This expression can be computed with the desired de-gree of accuracy by truncating the expansions (8) and
(9) appropriately. As an illustration, we note that whenall terms of degree greater than one are truncated, (4)reduces to the simple form
PD(F) - erfc II d ifd It X D dlr)]2 2 V L D t . DiI~~
ru 00 .5
a: lu5
0cr
-310 -
-4 01FE 10M0
105
-610
-7
100 4 8 12 16 20
SIGNAL-TO-NOISE RATIO
Fig. 1. Probability of error Pr(e) vs SNR y for single-detectormaximum likelihood photocounting receiver. A lognormally mod-ulated coherent signal (y >> 1 or i >> 1) is embedded in-additive in-dependent noninterfering Poisson noise with (n) = 1. Solid curvesrepresent results of Rosenberg and Teich (Ref. 8). Solid points
represent results of present paper.
100
(11)
The accuracy of the approximation in (11) will be dis-cussed for a particular example in Sec. III.
We note that PD(F) as given in (4) and (5) is exact, andcan be computed for any pdf p (r i) for which the nor-malizing transformation is known. When gi is notknown, but the variance and ordinary moments of p (r I i)are known, PD(F) can be obtained approximately using(8)-(10). Finally, knowing only the variance of p(r li),PD(F) is obtained with less accuracy from (11). Forsimplicity, (11) will be used in the remainder of thepaper.
Ill. Lognormal Modulation
Using the technique described in the previous section,PD(F) will be computed for communication through theturbulent atmosphere. For the case of a lognormallymodulated mixture of coherent and chaotic radiation,with arbitrary /, y, c, and COT, and where the chaoticcomponent need not be stationary and may have arbi-trary spectral distribution, the photocounting cumu-lants have been computed by Rosenberg and Teich,2
and may be used in (8) and (9). In particular, thevariance is given by
(12)
where s is the number of signal photocounts, and or is thelog intensity standard deviation. If we assume the areaof the detector is much less than the coherence area ofthe detected process, then the integrated intensity isgiven by
t+TW=a f I(t)dt, (13)
where the intensity I is integrated over the countinginterval, and a is the detector quantum efficiency. Wodenotes the integrated intensity in the absence of tur-bulence.
As an example, for a chaotic component of Lorentzianspectrum, (12) reduces to
vars = (s) + b(s) 2 , (14)
where
[ 2(y + +2)l ]1 (15)(y + 1)2
and B1 and B2 are given in Ref. 2. For simplicity, we letCRc = CO)T, for which B, and B2 reduce to
BP = 2(e-PO + p3 - 1)/(p 3)2 p E 1,21. (16)
The specific case of additive independent Poisson noisewill be considered here, for which PF can be calculatedusing conventional techniques. The variance of thesignal plus noise is then D2 = (n) + vars. Using (11)and (14) the probability of detection is
is the SNR. Note that (17) applies for c > 0, and thatthe case c = 0 corresponds to a Poisson signal plus noise,which can occur for y = 0, for or = 0 and = a, or for o-= 0 and y = a. Note also that the more general case ofarbitrary noise that is additive and independent can betreated as easily by setting D2 = varn and D 2= vars +varn, and then using (11) and (14) to calculate PD(F).
IV. Discussion
Theoretical probability of error curves correspondingto the results of the previous section [see (17) and (2)]will now be discussed. In particular, we will begin bycomparing a special case of (17) to the results ofRosenberg and Teich.5
The probability of error Pr(E) vs the SNR -y is plottedin Fig. 1 for a lognormally modulated coherent signal(y >> 1 or 3 >> 1), embedded in additive independent
Fig. 2. Probability of error Pr(e) vs SNR -y for single-detectorNeyman-Pearson photocounting receiver. A lognormally modulatedsuperposed coherent-in-chaotic signal (of Lorentzian spectrum withW = oT) is embedded in additive independent noninterfering Poissonnoise. Curves labelled correspond to >> 1, curves labelled 0,4correspond to << 1 and y = 4, and curves labelled 0,0 correspond to
0 << 1 andy = 0. Here n) = 1, t = 6, and PF = 5.94 X 10-4
.
noninterfering Poisson noise of mean (n) = 1, for thecases = 0 (quiescent atmosphere), = 0.5 (light tur-bulence), a = 1.0 (moderate turbulence), and o = 1.5(saturated turbulence). The solid curves represent theprevious results of Rosenberg and Teich [see Fig. 2(a),ref. 8] for the optimum single-detector maximum like-lihood photocounting receiver, based upon the optimumthreshold t (see Fig. 1, ref. 8). The solid points repre-sent the corresponding results of the present paper forthe special case y >> 1 or >> 1, based upon the sameoptimum threshold t. Figure 1 shows excellent agree-ment between the previous results of Rosenberg andTeich and the corresponding results of the presentpaper, over 4 orders of magnitude. Thus the approxi-mations leading to (11) (i.e., truncation of terms withdegree greater than one) provide sufficient accuracy inthe present case.
The Pr(E) vs y is plotted in Figs. 2-5 for a lognormallymodulated mixture of interfering coherent and chaoticradiation, where the chaotic component has Lorentzianspectrum with c, = T, embedded in additive inde-pendent noninterfering Poisson noise. The optimumsingle-detector Neyman-Pearson photocounting re-ceiver is presented. Figures 2 and 4 correspond to (n)= 1, t = 6, and P = 5.94 X 10-4, whereas Figs. 3 and 5correspond to n) = 4, t = 19 and P = 5.16 X 10-8.
In Figs. 2 and 3, the dashed curves, solid curves, anddotted curves represent the cases cr = 0, a. = 0.5, and f= 1.5, respectively. The curves labelled correspondto >> 1, the curves labelled 0,4 correspond to << 1 andy = 4, and the curves labelled 0,0 correspond to << 1and y = 0.
In Figs. 4 and 5, af = 0.5 in all cases. The dottedcurves and solid curves represent the cases y = 4 and y<< 1, respectively. Each curve is labelled correspondingto its value of , for the cases << 1, /3 = 2,/ = 20 and/>>1.
In examining Figs. 2-5, it is clear that the Pr(E) isreduced in each case by increasing the SNR y, as ex-pected. It is also apparent that the curves intersect atyt = 5 in Figs. 2 and 4, and at yt = 3.75 in Figs. 3 and 5.For y > -Yt, the Pr(E) is reduced by increasing either or y, or decreasing a, while holding the other two pa-rameters constant. The best performance is obtainedfor the coherent quiescent case ( = , a = 0), the worstperformance for the chaotic saturated case ( << 1, y <<1, o = 1.5), and intermediate performance is obtainedfor the superposed case (y = 4, ur fixed). For y < -Yt,precisely the opposite behavior occurs. These effectsare explained below.
The parameters , y, and af determine the broadnessof the signal photocounting distribution through theparameter b [see (15)]. The broadness parameter bincreases as decreases, y decreases, or a increases. Inthe coherent limit (y > 1 or >> 1) the broadening ef-fect of the lognormal atmosphere is expressed by
b = ee2 1
100
(20)
-110
W_0
o -2, 10
a.
0
10
1041 4 8 12 16
SIGNAL-TO-NOISE RATIO20
Fig. 3. Same as Fig. 2, with (n) = 4, t = 19 and P = 5.16 X 10-8.
Fig. 4. Probability of error Pr(c) vs SNR y for a single-detectorNeyman-Pearson photocounting receiver. A lognormally modulatedsuperposed coherent-in-chaotic signal (of Lorentzian spectrum withX = UT) is embedded in additive independent noninterfering Poissonnoise. In all cases a = 0.5. Curves labelled corresponding to 3 << 1,1 = 2, = 20 and = . Here n) = 1, t = 6, and PF = 5.94 X 10-
4.
whereas in the quiescent limit (a = 0), the broadeningeffect of interfering coherent and chaotic source fluc-tuations is expressed by
b = B 2 + 2B,(y + 1)2 1
(Y+ -+ 2
We note that the parameters /3, y, and enter the Pr(E)solely through the broadness parameter b [see (17)].
Now consider the case where the mean signal plusnoise count is less than the threshold [i.e., ,y < (t/ (n ) )- 1]. Increasing b leads to a higher probability that thesignal plus noise count is less than t and therefore ahigher Pr(E). In contrast, for the case where the meansignal plus noise count is greater than the threshold [i.e.,- > (t/(n)) - 1], increasing b leads to a higher proba-bility that the signal plus noise count is greater than t,and therefore a lower Pr(E). The boundary of these twoeffects occurs at t = (t/ ( n)) - 1 as is seen in Figs. 2-5.Indeed it may seem rather surprising at first that for y< yt, better system performance is obtained for in-creased fluctuations in photocounts. Actually, more
fluctuations in the source enhances the probability ofi higher photocounts, and increased fluctuations in the
atmosphere together with a fixed SNR requires an in-creased average signal at the source.
Several other effects are apparent as well in Figs. 2and 3. The Pr(E) is less sensitive to y for large o. Forexample, for the case 3>> 1 in Fig. 2, the ratio of Pr(E)
0 at = to the Pr(E) at y = 20 is 1580 for = 0, com-2 pared to 2.3 for a = 1.5. Also, the Pr(E) is relatively
-- 2 insensitive to y and /3 for large a, where the atmospheric20 fluctuations tend to dominate the overall fluctuations20 in photocounts. For example, for y = 20 in Fig. 2, theOD ratio of Pr(E) for >> 1 to the Pr(E) for << 1 and y <<
1, is 165.7 for o = 0, 7.7 for = 0.5, and 1.2 for = 1.5.Finally, the Pr(E) is least sensitive to a in the chaoticcase ( << 1, y << 1), which has the largest pre-existingfluctuations.
An alternate way to illustrate the same results is tocompare the SNR necessary to yield a given Pr(E). InFig. 2, Pr(E) = 0.0773 is attained for y = 7.8, / >> 1, and_r = 0, for y = 10, >> 1, and cr = 0.5, for y = 10.5, <<1,y = 4, and o= 0.5, fory = 13.5,<< 1,y =4, and=0.5 and for y = 20,3 << 1, y = 0, and = 0. It is ap-parent that with more fluctuations in the source, largerincreases in are necessary to maintain the specified
16 20 level of performance.
100
a:0
a:a:
L-J
0a:a.
104 i I I I I I 1 4 8 12 16 20
SIGNAL-TO-NOISE RATIO
Fig. 5. Same as Fig. 4, with (n) = 4, t = 19 and PF = 5.16 X 10-8.
A new technique has been presented for evaluatingPD(F) exactly when normalizing transformations for thenoise and signal-plus-noise distributions are known [see(4) and (5)]. If the normalizing transformations are notknown, PD(F) can be evaluated approximately from thevariance and ordinary photocounting moments of thenoise and signal-plus-noise distributions [see (8)-(10)].Finally, knowing only the photocounting variance, asimple but approximate expression for PD(F) has beenpresented [see (11)]. Using this technique, PD(F) hasbeen computed for communication through lognormalatmospheric turbulence, for a superposed coherent-in-chaotic signal, embedded in additive independentPoisson noise, with arbitrary , y, w and coT, and wherethe chaotic component need not be stationary and mayhave arbitrary spectral distribution. Theoretical Pr(E)curves are presented, and in the special cases of: >> 1or y >> 1, excellent agreement is found with previousresults 8 for a purely coherent signal. It is shown thatthe Pr(E) depends solely on a broadness parameter b,which itself is a function of /3, y, and . For y > ytimproved performance is obtained with reduced fluc-tuations in photocounts, whereas for y < t improvedperformance is obtained with increased fluctuations inphotocounts.
This work was supported by the Joint ServicesElectronics Program (U.S. Army, U.S. Navy, and U.S.Air Force) under contract DAAG29-79-C-0079.
References1. P. R. Prucnal and M. C. Teich, J. Opt. Soc. Am. 69, 539 (1979).2. S. Rosenberg and M. C. Teich, J. Appl. Phys. 43, 1256 (1972).3. B. Saleh, Photoelectron Statistics (Springer, New York, 1978).4. M. C. Teich and S. Rosenberg, Appl. Opt. 12, 2616 (1973).5. M. C. Teich and S. Rosenberg, Opto-electron. 3, 63 (1971).6. P. Diament and M. C. Teich, Appl. Opt. 10, 1664 (1971).7. G. Lachs and S. R. Laxpati, J. Appl. Phys. 44, 3322 (1973).8. S. Rosenberg and M. C. Teich, Appl. Opt. 12, 2625 (1973).9. S. Rosenberg and M. C. Teich, IEEE Trans. Inf. Theory IT-19,
807 (1973).10. M. C. Teich and R. Y. Yen, IEEE Trans. Aerosp. Electron. Syst.
AES-8, 13 (1972).11. J. Pehina, Acta Phys. Pol. A52, 559 (1977).12. J. K. Jao and M. Elbaum, Proc. IEEE 66, 781 (1978).13. M. C. Teich, P. R. Prucnal and G. Vannucci, Opt. Lett. 1, 208
(1977).14. P. R. Prucnal and M. C. Teich, Appl. Opt. 17, 3576 (1978).15. P. R. Prucnal and M. C. Teich, IEEE Trans. Inf. Theory IT-25,
213 (1979).16. P. R. Prucnal, Appl. Opt. 19, 3606 (1980); J. Math. Psychol. 21,
168 (1980).17. F. J. Anscombe, Biometrika 35, 246 (1948).18. M. G. Kendall and A. Stuart, The Advanced Theory of Statistics
(Hafner, New York, 1966), p. 88.
Books continued from page 3596nological problems. The author's style is clear and readable. Thebook is well bound, the print is legible, and misprints are few. If thereis a criticism it is the uneven choice of topics, which includes solar cellsand photographic film. The inclusion of these has not diminishedthe book's value, however, because these chapters are both informa-tive and interesting. The preface states that the book is suitable forundergraduates; while this is true, it will probably find more use ina first-level graduate course specializing in optoelectronics.
In summary, OPTOELECTRONICS DEVICES AND OPTICALIMAGING TECHNIQUES is recommended for scientists and en-gineers who desire a working knowledge of optoelectronics.
H. C. SCHAU
Interferometers: Principles of Engineering Theory andApplications. YU. V. KOLOMIITSOV. Mashinostroenie, Len-ingrad, 1976. 296 pp. 1R. 13 kop.
This book is a treatment of specific types of interferometers andtheir use in accomplishing a range of measurements in physics andengineering. The principles of operation of the different instrumentsare discussed, but there is also a large amount of technical detail aimedat facilitating the use of such instruments in practice.
Chapter 1 discusses basic principles of the interference of light andcovers topics such as fringe formation, coherence, interference of twocoherent light sources, and the division of a light wave. Chapter 2deals with interference in plates since many interferometers are basedon this principle. Interference in plane parallel and wedge shapedplates, as well as in plates bounded by spherical surfaces, is discussed.Multibeam interference and interference in two plates are also dealtwith.
Chapter 3 is concerned with the special features of interference inwhite light and with compensators. Topics such as the nonuniformityof thickness of glass in the ray paths and the compensation of thisnonuniformity are discussed. The layout and characteristics of thedifferent basic types of interferometer are treated in Chap. 4. Thesecan be summarized as single-plate interferometers such as Fizeau,Michelson, and Fabry-Perot, two-plate interferometers, Fresnel-typeinterferometers, and holographic interferometers.
Chapter 5 deals with the causes leading to a reduction in contrastof the interference fringe pattern. These include the fact that thelight source may not be monochromatic, unevenness in the intensitiesof the interfering light beams, and the presence of scattered light.Other causes such as different states of polarization of the interferingbeams, the finite dimensions of the light source, and the deformationof the wave surfaces are also considered. Chapter 6 treats the re-quirements on the optical and mechanical components of interfer-ometers. The quality of preparation of the basic optical componentsis discussed, as well as the accuracy with which they must be fittedinto the interferometer. There is a fairly detailed treatment of theaccuracy required in the mechanical components of interferometers,and a thorough discussion of how to adjust the instruments for op-eration.
Chapter 7 is devoted to interferometers for measuring lengths.Instruments treated include those for making absolute length mea-surements, interferometers for measuring objects internally and ex-ternally, as well as those for measuring small displacements and laserinstruments for measuring large displacements. Specific Soviet andWestern commercial interferometers are discussed. Chapter 8 dealswith quality control of the shape and microgeometry of surfaces.Topics covered include quality control of flat surfaces, sphericalsurfaces, and surfaces of complex form. Chapter 9 discusses inter-ferometers for investigating optical systems, for measuring refractiveindices, and for investigating inhomogeneities in transparent sub-
