Multiple pulse detection in atmospheric turbulence

Russell E. Warren

Expressions are derived for the probability of detecting at least m threshold crossings in n transmitted laserpulses in the presence of atmospheric turbulence assuming uncorrelated Gaussian receiver noise. The limit-ing cases of no pulse-to-pulse correlation and complete correlation are evaluated explicitly. In general, wefind the presence of correlation can significantly modify the detection statistics and should be included insystems calculations.

1. Introduction

The effect of atmospheric turbulence in modifyingthe detection statistics of a laser receiver is of concernto systems designers required to estimate the perfor-mance of laser ranging or communications systemsoperating in turbulence. While the problem of singlepulse detection has been treated by Fried andSchmeltzer' and more recently by McMillan andBarnes,2 apparently no attention has been given to thegeneralization to multiple pulse detection. This gen-eralization is necessary to describe more sophisticateddetection schemes, which depend on the detection of mthreshold crossings given n transmitter pulses forbackground suppression, for example. This is a non-trivial problem given the solution to the one pulse casedue to the possibility of pulse-to-pulse correlation,which must be considered for high laser prf's. It will beseen that this correlation has a strong effect on the de-tection probability. We perform the analysis assuminguncorrelated Gaussian receiver noise and a multivariatelog-normal distribution for the turbulence fluctuations.Two limiting cases of complete and no pulse-to-pulsecorrelation are treated in detail.

11. Analysis

Following Ref. 2 the single pulse log-normal proba-bility of a pulse having energy between E and E + dEis given by

Pl(E)dE = 2 exp [ (2n 2] d (lnE), (1)

where lnE* is the mean log-signal in the presence ofturbulence, and aE is the log-energy rms variance cor-rected for saturation effects and receiver averaging. Weassume the receiver detection to be characterized byGaussian noise of rms c-N with threshold T so the singlepulse detection probability for a given energy E is

p(E,T) = (2ir)12ow ' drT p [e - 2a2] ] (2)

For the single pulse case the detection probability isthen

P(1,1) = dEPi(E)p(E,T). (3)

We generalize Eq. (3) to the multiple pulse case by re-placing Eq. (1) by the joint log-normal density P,(E 1 ,tl,... EEtn) for observing pulses 1, ... ,n at times t1,... tn having energies E1,...E. E. The probability ofobserving exactly k threshold crossings in n pulses isthen (n/k)B(nk), where

B(n,k) = dE... dEnPn(E1,t1 . .*Entn)

* p(E1,T) . .. p(Ek,T)q(Ek+1,T) . .. q (En, T), (4)

and q(E,T) 1 - p(E,T) is the probability a pulse ofenergy E will be below threshold T. This model as-sumes the receiver noise is uncorrelated but allows forcorrelation in the turbulence induced fluctuations. Theprobability of at least m threshold crossings in n pulsesP(n,m) is easily seen to be

P (n)B ,k),z= k

The author is with Pacific-Sierra Research Corporation, SantaMonica, California 90404.

Received 17 April 1978.0003-6935/78/0901-2721$0.50/0.© 1978 Optical Society of America.

(5)

where (n/k) is the binomial coefficient. Equations (2),(4), and (5) will give the desired result once Pn is spec-ified.

We evaluate the joint log-normal density Pn usingtheorem 2.4 of Ref. 3 to the effect that the same trans-formation that maps the log-normal variate x to thenormal variate y, i.e., y = lnx, maps the multivariate

.1 September 1978 / Vol. 17, No. 17 / APPLIED OPTICS 2721

1.0

ZZ 0.9

ID0n

o

o

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

M

Fig. 1. Probability of detecting at least m pulses out of 10 for athreshold-to-noise ratio 7.02 and SNR 7.02 assuming no pulse-to-

pulse correlation and H = 1.

2

u2:

t;

ID,

:

1.0a 0.4 \ \ 0

. 0.9 \

-J 08

.70 - 0. 0.02

2 0.5 \

0.4-

0>. 0.3

0.2

00.1

2 3 4 5 6 7 8 9 10

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Fig. 3. Probability of detecting at least m pulses out of 10 for athreshold-to-noise ratio 7.02 and SNR 10 assuming no pulse-to-pulse

correlation and H = 1.

2

E

I-

Lo

o

10

M

Fig. 2. Probability of detecting at least m pulses out of 10 for athreshold-to-noise ratio 7.02 and SNR 7.02 assuming complete

pulse-to-pulse correlation and H = 1.

log-normal distribution to a multivariate normal dis-tribution. Thus, we may write

P.(Elstl*. Et,)dEl .. dE0

1r/ A 12exp [- a; (In E') (A-1)jj (In Ej)]

X d (lnEj) ... d (lnEj), (6)

where A is the covariance matrix having elements

aij= ((lnEj/E!)(lnEj/E)) (7)

and I A I is the determinate of A. A is understood to bea function of t1, ... ,t, and represents the temporalcorrelation of the pulses in the presence of turbulence.We assume it is reasonable to write aij as aE

where C is the normalized temporal correlation between

Fig. 4. Probability of detecting at least m pulses out of 10 for athreshold-to-noise ratio 7.02 and SNR 10 assuming complete

pulse-to-pulse correlation and H = 1.

pulses at times ti and tj and further assume temporalstationarity so aij = EC (I t -tj D ).

The correlation function C can be computed as theFourier transform of the power spectral density of thelog-amplitude fluctuations if the Taylor frozen-in hy-pothesis is used.4 In the general case, Eq. (4) requiresthe evaluation of an n-fold multiple integral using theestimated correlation matrix. In this paper we limitourselves to evaluating the two limiting cases of nopulse-to-pulse correlation and complete correlation.

For pulse separations sufficiently long, we expectthere to be no correlation between successive realiza-tions of the random refractive index profile so aij a' bij. This causes Eq. (6) to factor into a product ofsingle pulse distributions as

P.(Etj,. . .E.,t.) -P1 (Elt) .. . P1(E03t0 ). (8)

2722 APPLIED OPTICS / Vol. 17, No. 17 / 1 September 1978

In this limit B(n,k) becomesno [C- 1k

B(n,k) [ dEP,(E,t)p(E,T)]correlation Jo

X SOdEP1(E,t)q(E,T) *n- (9)

At the other extreme, for pulse separations so shortthat each pulse samples effectively the same turbulenceprofile, aij u, and the multivariate distribution be-comes

Pn(Elti. Ent.) P(El,t)S(E 2 - E) ... (En En-1).(10)

Then B(n,k) becomescomplete O

B(n,k) copltef dEPl(E,t)[p(E,T)]k[q(E,T)]n-k. (11)correlation °

Ill. Examples

Equations (9) and (11) can be evaluated numericallyby substituting Eq. (1) with the change of variable E/E*= exp(V/2uEw). It can be shown2 that E* = (E)exp(-o-E/2), where (E) is the mean received signal withturbulence. Introducing the vacuum signal-to-noiseratio SNR - EO/1N for E0 the vacuum signal, wehave

E- = H. SNR exp(-ao/2) exp(v\12aEw), (12)UJN

where H (E) E0 is the reduction of the mean signaldue to turbulence; it is a function of the laser beamshape and receiver area as well as the turbulencestrength. We also introduce the threshold-to-noiseratio TNR given in terms of the pulse width r and theaverage false alarm rate FAR as

TNR = [2 ln ( ) )] 1/2 (13)2-\/3,rFARIWith these substitutions Eqs. (9) and (11) become

noB(n,k) Pk(l -p)n-k, (14)

correlation

where

P = dw exp(-w 2 )p(w) (157r - (1and

complete 1B(n,k) r - dw eXp(_W2)[p(W)jk[j - p(w)]n-k.

correlation x/lrs-x

(16:Here

p(w) = 1 [1 + erf (H SNR exp('/2) exp(//2aEW)-TNR ]

Equations (14) and (16) are now easily evaluated nu-merically using Gauss-Hermite quadrature. 5 For thecases evaluated here a quadrature order of thirty wasused.

We apply these results to the case of a laser with pulsewidth T = 20 nsec and a FAR of 1/h. This gives TNR= 7.02. The two cases considered are SNR = TNR andSRN = 10. These correspond to 50% and 99.9% prob-ability of detection for one pulse in vacuum. We fur-ther assume H can be set at 1.0 by increasing thetransmitter energy appropriately. Figures 1 and 2 (3and 4) plot the probability of at least m thresholdcrossings for n = 10 for the uncorrelated and completelycorrelated cases for SNR = 7.02 (SNR = 10). The dif-ferent curves in each figure represent scintillationstrengths of E = 0. to 0.7 in steps of 0.2. The upperlimit represents the largest observed scintillation.

IV. Conclusions

There are several conclusions to be drawn from theseresults: (1) Correlation has a strong effect on P(n,m);for high laser prf's (>500 Hz depending on the temporalpower spectrum of the scintillation) there is a bunchingof pulses and flattening of the probability curves withincreasing o-E. This means, in the limit of completecorrelation, that the receiver can see most of the pulsesif it can see one. (2) P(n, 1) decreases with increasingcorrelation, which implied that the probability of nothreshold crossings in n pulses [=1 - P(n, 1)] increaseswith increasing correlation. (3) Although increasingturbulence always lowers P(n,m) for given n,m in theuncorrelated case, correlation can result in higher de-tection probabilities for m n in the presence of tur-bulence compared to the vacuum (assuming H can bemaintained at 1.).

The author expresses appreciation to A. R. Shapiroof Pacific-Sierra Research Corporation for suggestingthe problem and G. J. Hall, also of PSR, for helpfultechnical discussions. This research was supported bythe Naval Research Laboratory, Washington, D.C.,under contract N00173-78-M-E298.

References1. D. L. Fried and R. A. Schmeltzer, Appl. Opt. 6,1729 (1967).2. R. W. McMillan and N. P. Barnes, Appl. Opt. 15, 2501 (1976).3. J. Aitchison and J. A. C. Brown, The Lognormal Distribution

(Cambridge U.P., Cambridge, 1957).4. V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave

Propagation, translated from the Russian, originally publishedin 1967 (National Technical Information Service, Springfield, Va.,1971).

5. A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas(Prentice-Hall, Englewood Cliffs, N.J., 1966).

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