time-resolved speckle effects on the estimation of laser-pulse arrival times

Vol. 2, No. 5/May 1985/J. Opt. Soc. Am. A 649

Time-resolved speckle effects on the estimation oflaser-pulse arrival times

Bin-Ming Tsai and Chester S. Gardner

Department of Electrical and Computer Engineering, University of Illinois, Urbana, Illinois 61801

Received May 21, 1984; accepted December 20, 1984

When laser pulses are reflected by targets that have range spreads larger than the transmitted pulse width, thewidth of the received pulses will be longer than the correlation length of the speckle-induced fluctuations. As aconsequence, speckle will cause random small-scale fluctuations within the received pulse that will distort itsshape. This phenomenon is called time-resolved speckle. In laser ranging and altimetry, the random pulse distor-tion caused by time-resolved speckle can seriously degrade the timing accuracy of the receivers. In this paper, westudy the statistical properties of time-resolved speckle and the problem of estimating the arrival times of laserpulses in its presence. The maximum-likelihood (ML) estimator of the pulse arrival time is derived, and its perfor-mance is evaluated for pulse reflections from flat diffuse targets. The performance of the ML estimator is com-pared with the performance of several suboptimal estimators. When the signal level is high, speckle noise placesa fundamental limit on the accuracy of the suboptimal estimators. It is shown that the ML estimator performsconsiderably better than the suboptimal estimators and that its accuracy improves as the width of the receiver ob-servation interval increases.

1. INTRODUCTION

The statistical properties of laser speckle have been studiedextensively during the past two decades.1' 2 However, thesestudies have been either for cw laser illumination'- 3 or for apulsed laser in which the width of the received pulse is com-parable to the correlation length of the speckle-inducedfluctuations.4 The latter case applies when the reflectingsurface, or target, is rough on the optical scale but has a rangespread that is much smaller than the laser pulse width. Inthese cases, speckle causes random fluctuations of the totalreceived energy and is an important noise phenomenon intarget detection.

Mode-locking and Q-switching techniques are now usedroutinely to generate laser pulses of a few picoseconds in du-ration. These short pulses provide higher accuracies in ap-plications such as remote sensing and ranging. In practice,many targets, such as the ocean surface, the ground, andman-made objects, have range spreads that far exceed thewidths of the picosecond laser pulses. The reflected pulsesare broadened to about twice the range spread of the target,while the pulse shape is related to the target geometry.5 6 Ifthe range spread of the target is larger than the transmittedpulse width, the width of the received pulse will be longer thanthe correlation length of the speckle-induced fluctuations. Asa consequence, speckle will cause random small-scale fluctu-ations within the received pulse that will distort its shape.This phenomenon is called time-resolved speckle. Similarphenomena can also occur for extended flat diffuse targets,in which the broadening of the received pulse is due to thewave-front curvature of the laser beam, and for continuouslydistributed targets in the atmosphere.

For applications such as target identification and remotesensing of sea states, 6 the waveform of the received pulse isused to characterize the target. Therefore knowledge of thestatistics of the time-resolved speckle is important. In ap-plications such as laser distance ranging, estimation of the

arrival time of the target-reflected pulse in the presence oftime-resolved speckle is the problem.

Estimation of the arrival times of laser pulses was firststudied by Bar-David,7 8 who derived the maximum-likelihood(ML) and minimum-mean-square-error estimators under theassumption that the detected signal follows Poisson statistics. 8

In the presence of speckle, the detected signal will no longerfollow Poisson statistics. Elbaum and Diament9 considereda similar problem of estimating the centroid of a laser radarimages contaminated by speckle. In this paper, we derive theML estimator for arrival times of laser pulses taking into ac-count both shot noise and speckle. The timing results ob-tained have useful applications in laser ranging and altim-etry.

2. STATISTICS OF THE DETECTED SIGNAL

In the typical laser-ranging-system geometry, the laser pulsesare reflected by a diffuse target and detected by a receivingtelescope and photodetector. The photodetector is followedby an amplifier and a low-pass filter. The filter output issampled periodically at an interval T equal to the inverse ofthe filter bandwidth. Thus the sampled values are propor-tional to the number of photons detected in consecutive timebins of width T. Alternatively, the detector could be followedby a photon counter that is sampled every T seconds. In ei-ther case, the receiver observations can be modeled as a pho-tocount vector k:

k = (k,, k2,. .., kJ, (1)

where ki is the photocount within the ith time bin and thetime-bin width is T.

The statistics of ki are related to the signal energy Wi re-ceived during the ith time bin. For reflection from diffusetargets, it has been shown that Wi is approximately gammadistributed,",3 obeying the probability density function

0740-3232/85/050649-08$02.00 © 1985 Optical Society of America

B.-M. Tsai and C. S. Gardner

650 J. Opt. Soc. Am. A/Vol. 2, No. 5/May 1985

G(,., t) = (; )-I exp(-t2 /2ay2 ).p(wi) = [r(mi}wi]-i(Mi~i1i)Mi exp(-MiWi),(2)

where Wi is the mean value of the signal energy in the ith bin.Mi is the signal-to-noise ratio of speckle, which is defined asthe square of the mean divided by the variance

Mi = W 2I/var(Wi). (3)

The mean and the variance of Wi have been derived inprevious papers for reflections from rough diffuse targets.5' 6

They are given by

= Q Jil)Tdt d2pb2 (p,z)lf(t - 0)12 (4)

and

var(Wi) = Q2K X-' dtf :1 dt2 f d 2 pb4 (P, Z)

x If(t1 - t')121f(t2 - 0)12, (5)

where

bj(p, z) = Ia(p, Z)j #r/2

(p) S d2 pla (p, Z)j f3r"/ 2(p), (6)

K = AR [ d 2 p a (p, Z)I 2r (p)] 2 /X 2Z 2

X S d 2 pla(p, Z)14 1r 2 (p), (7)

and

= 2z/c + p2 /cz - 2(p)/c. (8)

Here

Q is the expected received signal energy per pulse,p = (x, y) is the horizontal coordinate vector on the target

surface measured from the center of the laser footprint,a (p, z) is the complex amplitude cross section of the laser

footprint,4r(P) is the power reflectivity of the target,

t(p) is the surface profile of the target,AR is the receiver aperture area,z is the target distance,c is the velocity of light,If(t)l2 is the transmitted laser-pulse intensity, andh(t) is the impulse response of the receiver electronics.

In Eq. (8), J/ is the delay of the reflected pulse. The first termin 1' is the nominal round-trip propagation time from the laserto the center of the target. The second term is the additionaldelay that is due to the curvature of the laser wave front, andthe last term is due to the range spread of the target. In Eq.(7), K is usually referred to as the number of speckle corre-lation cells within the receiver aperture.1

For a Gaussian-shaped transmitted pulse intensity with rmswidth af, Eqs. (4) and (5) can be simplified so that

Wi QF1(iT)T (9)

and

var(Wi) Q2 K 1-1F2 (iT)T,

where

Fi(t) = f d2pb2(p, z)G(qf, t - ),

F2(t) = d2 pb4 (p, z)G(of1 iJ/, t - ),

(1 0)

(13)

Note that G (u, t) is a normalized Gaussian pulse shape withrms width equal to 0.. F(t) is the received pulse shape andF2 (t) is the waveform of the speckle variance of the receivedsignal. Both are functions of the target geometry. In arrivingat expressions (9) and (10), we have made the assumption thatthe width of the time bin T is long compared with the widthof the transmitted pulse but short compared with the widthsof F1 and F2.

By using expressions (9) and (10), Mi can be expressed asMi = KF 1

2(iT)T/F 2 (iT). (14)

Mi, the speckle signal-to-noise ratio, may also be thoughtof as the number of speckle correlation cells within the receiveraperture during the ith time bin. For targets in which therange spread is large compared with the transmitted pulsewidth, Mi will change with time as different regions of thetarget are illuminated by the propagating pulse. At the re-ceiver, the speckle correlation area is inversely related to thearea of illumination on the target. If only one point is illu-minated on the target, the speckle correlation area is infinite,and Mi is one. This is the situation when the leading edge ofthe pulse first reaches the target and when the trailing edgeilluminates the most distant part of the target. When a largenumber of points are simultaneously illuminated, the specklewill be fully developed, and the speckle correlation area willbe inversely related to the area of illumination. If the specklecorrelation area is less than the receiver-aperture area, thenthe number of speckle correlation cells within the aperture willbe larger than one. If the target is completely illuminated bythe laser footprint, Mi will be equal to KS Typically, Mi willbe approximately one near the leading and trailing edges ofthe received pulse and larger than one near the center of thepulse.

If the laser footprint is uniform, or if the target is smallcompared with the laser footprint and the width of thetransmitted laser pulse is small compared with the rangespread of the target, then F1 (t) and F2 (t) given by Eqs. (11)and (12) are approximately equal, and we have

Mi KFl(iT)T. (15)

As expected, since F(iT)T 1, Mi is smaller than or equalto K. For the case in which the receiver cannot resolve thereturn waveform and only the total energy of the receivedpulse is observed, Mi is equal to K. 4 For targets with largerange spread, typically only a part of the target area is illu-minated within one time bin, and Mi will be smaller thanKS

Since W is gamma distributed with parameter Mi, thedistribution of k can be shown to have a probability densityfunction given by4

p(ki) r(k+ + +M) (i--i (1 + M )where

ki = n'Wi/hP = (N)Fl(iT)T.

(16)

(17)

I- 71 is the quantum efficiency of the photodetector, hv is theenergy of one photon, and (N) is the expected number of

(11) photocounts per received pulse. When M is an integer, Eq.(16) is the negative binomial density function. In Eq. (16),

(12) the variance of k is



var(ki) = i + ( 2)/(Mi). (18)

For most laser receivers, the target-reflected laser pulsesare usually much broader than the transmitted pulse.Therefore the width of the time bin, T, is larger than thetransmitted pulse width. In this case, the photocounts fromdifferent time bins are statistically independent. The jointdensity function of ki is then

np(k) = II P(ki).i=1

(19)

Equation (19) forms the basis for the analysis on receivertiming in Section 3.

3. ESTIMATION OF ARRIVAL TIME

In laser ranging and altimetry applications, the arrival timeof the reflected pulse, denoted by r, is of interest. In this case,Eq. (19) can be written explicitly as

p(kl) = II 1 i~i\r k +(1)r/

X 1+ i ()M&0-)

(20)

The ML estimate of the arrival time is the value of T thatmaximizes the likelihood function p(klr) or n[p(k -)].1OThat is,

fSPEC = argmax [lnp(k r)], (21)

where arg max. [F(x)] denotes the argument x that maximizesF(x). By substituting Eq. (20) into Eq. (21) and making theassumption that the received pulse always stays within theobservation interval for all values of r, so that there are no endeffects, the estimator given by Eq. (21) reduces to

In these equations, the expectations are over the observationvector k. Equations (25) and (26) cannot be applied to pulseswhose derivatives are not mean-square integrable. Thisconstraint excludes rectangular pulses from the analysis.

For the ML estimator, it is not difficult to show thatE[H(ro)] is zero. Therefore the ML estimator is unbiased.The variance of H(ro) is given by

var[Ht(ro)] = i {k.2 k (i + (27)

and E[ft(ro)] is given by

E[*(r0 )] = - {=- /- + k (28)

Finally, by substituting these results into Eq. (26), we havefor the variance of the ML estimator

var(-spEc) i_ ki/kL ( + (29)

By using the expressions for ki and Mi given by Eqs. (14) and(17), we can also write relation (29) as

var(TspEc) [ F1 T/( ) (30)

Since H(-) is the likelihood function, the variances givenin relations (29) and (30) for the ML estimator are identicalto the Cram6r-Rao bound. The actual timing error is ex-pected to be higher. However, when the signal-to-noise ratiois high, the actual timing error will approach the Cram6r-Raobound.

As a comparison, the ML estimator derived by Bar-Davidfor the case of negligible speckle noise (i.e., shot-noise-onlycase) is 7,8

TSHOT = arg max [. kE i In ki(T)]- (31)

TSPEC = arg max[H(-j],

nH(O) = 5; [ki + Mi-) - 1/2]ln[ki + Mi(T)]

i=1

-E ki ln 1 + M(T).i= i k -

(22)

(23)

It will be interesting to see how this estimator performs in thepresence of speckle. By employing the same procedure usedabove, we find it to be unbiased, and the variance of its errorin the presence of speckle is given by

(.2-1 n k.2 n .2\

var(fSHOT) = (E - ) + E /l - )i~l hi i=1~ A ~ h

In arriving at Eqs. (22) and (23), we have made use of thefollowing relation:

In rwx _ (x -1/2)ln x - x + (1/2) In 2r, x >> 1.

(24)

The performance of TSPEC will be analyzed based on aprocedure similar to that used by Helstrom10 and Bar-David. 7

We shall make the assumption that the shot noise and speckleare not severe so thatE[(r0)]2 >> var[f(tOo)], whereTo is theactual arrival time of the pulse and H is the second derivativeof H. In this case, the bias and the variance of the ML esti-mator are given approximately by

E(AMSPEC) = E(fSPEC - TO) = -E[(TO-o)]/E[H(To)](25)

and

(N) (i1 FT)1+ K () F2T/

(E F12)2 (32)

The first term in Eq. (32) is the timing error resulting fromshot -noise, and the second term is the additional timing errorresulting from speckle.

When the speckle noise is small compared with shot noise(i.e., when K » (N) ), the timing error of the ML estimatorgiven by relation (29) is approximately given by

var(fPSPEC) hii2 1 ...~j(n ki2)-1 2/( nk 212

(26) Relation (33) is the same as Eq. (32), which is reasonable be-

where


var(fSPEC = var[A(T0)]1E[ft(,r0)]2.


Table 1. Summary of Laser-Pulse Arrival-Time Estimators and Their Variances

Estimator Timing Variances

TSPEC= arg max [ki + Mi(r) - 1/2]ln[ki + MM(r)] { k ki2/k (1 +

kiln [ 'ir)]

TSHOT = arg max E_ ki n ki(r) E -J + E E

7 i=1 i~ as =1M's/\=i~n 1= - (- + -

T~CENT = iTk, ki ' i )2T

2 k +

cause TSHOT is the ML estimator for the case in which speckleis negligible.

When F1 and F2 are approximately equal (i.e., when thelaser footprint is uniform, or when the target is small com-pared with the laser footprint and the width of the transmittedlaser pulse is small compared with the range spread of thetarget), both relation (30) and Eq. (32) reduce to

var(iSpEc) = var(QsHoT) ( + ) I l 4A2J

(1 11(N) Ki B (34)

whereA(t) = [k(t)]1/2 (35)

and

S dtA2(t)B2 =

| dtA2(t)fJ S

. dww2jA(W)I2

J dol A ()1 2

A(t) is the received pulse amplitude, A(w) is its Fouriertransform, and B is the rms bandwidth of the received pulseamplitude. From Eq. (34) we see that the timing error is in-versely proportional to the bandwidth of the received pulse.The term involving 11(N) is the shot-noise contribution tothe timing error. The presence of time-resolved speckle in-troduces additional error that is proportional to 1/K,.

To implement both the ML estimator TSPEC and the sub-

optimal estimator fSHOT, it is necessary to know the meanreceived pulse shape. However, in many cases, the geometryof the target is not known, and prior knowledge of the meanreceived pulse shape is not available. In this case, suboptimalestimators that do not require knowledge of the mean pulseshapes are required. An estimator that does not requireknowledge of the pulse shape calculates the centroid of thereceived pulse. This can be written mathematically as5'6

n nTCENT = E iTki/ ki. (37)

The bias of the centroid estimator is simply

EIAWCENTI = ElICENT - ToiT-T 0 (38)

where i is given by

n _ ii = E i /Eki-i=1 i=1

(39)

The centroid estimator can have a nonzero bias. The varianceof the centroid estimator is given by

var(kcENT) = - E (i-I 2 T2 i +(N) 2 ?=1 m -

= (i - i)2T3 + - (40)

Equation (40) indicates that the variance of the centroidestimator is proportional to the mean-square time spread ofthe received pulse. Within a group of waveforms with thesame time spread, the ML estimator will perform better as theslope or bandwidth of the waveform increases, whereas thecentroid estimator does not take advantage of the increase insignal bandwidth, and its performance will stay the same.

For a Gaussian-shaped received pulse, the centroid esti-mator is identical to fSHOT, so that the performance of thecentroid estimator is close to that of the ML estimator whenspeckle is not severe. However, for non-Gaussian waveforms,the centroid estimator was typically not so good as the MLestimator. The estimators considered above and their vari-ances are summarized in Table 1.

The results obtained above illustrate the effects of time-resolved speckle on the receiver timing performance. Whenshot noise is dominant, the mean-square timing error is pro-portional to 11(N). When speckle is dominant, the timingerror is proportional to 1/K. If KS is much larger than (N),such as in the weak-signal case, the effects of speckle on timingare negligible compared with shot noise. But if (N) is com-parable with or even greater than KS, speckle effects willdominate timing performance.

4. TIMING PERFORMANCE FOR DIFFUSEFLAT TARGETS

To illustrate the speckle effects quantitatively, we will lookat some specific target configurations. The first example isan infinite flat diffuse target. For simplicity, we assume thatthe target has uniform reflectivity and that the laser beam isnormally incident upon the target. This is a good model forreflections from the ground, sands, or walls of large buildings.In this case, the broadening of the received pulse is due towave-front curvature of the laser beam.

If the laser-beam divergence angle is 0 at exp(-1/2) point,



the rms footprint radius at the target plane is z tan 0. In thiscase, we have

K8 = 4'7rAR tan 2 O/X2. (41)

For normal incidence, the mean and the variance of Wi canbe calculated to give

QcT [ C2af 2 c(iT - 2z/C)14z tan 2 0 P8Z2 tan 4 0 2z tan 2 0

X erfc Can2 0 - iT- 2z/) (42)

and

Q2cT [c 2oyf2 c(iT - 2z/c)1var (Wi) expI

2K z tan2 0 r [4z2 tan4 0 z tan 2 0

X erfc (ca - ) (43)

0.75

0.2

0

IBEAM DIVERGENCE = 0.5 mRAD

250 At0

()

-20 0 20 60 100 140 180 0

LU

TIME (t-2Z /C)

0-f

Fig. 1. Mean received waveform for normal incidence upon an in-finite flat diffuse target (z = 500 km, c = 0.5 cm, AR = 100 cm2 , X

= 1 um).

0.75

0.50

0.25

0

BEAM DIVERGENCE = 0.5 mRAD

-600.- 550

LU 500- 450&3 400n 350OL 30000 250i 200

(Y 150-Z 100co 50

0-20

BEAM DIVERGENCE 0.5-0.9mRAD

0 20 60 100 140 180

TIME (t-2Z/C)

c-fFig. 3. Signal-to-noise (S/N) ratio of speckle for normal incidenceon an infinite flat diffuse target (z = 500 km, c uf = 0.5 cm, AR = 100cm2 , X = 1 tm).

lo r

100

I0

= 0.9

102 IO4 106 . 108 1010

NUMBER OF PHOTOELECTRONS

Fig. 4. Rms ranging error using the ML estimator TSPEC for normalincidence upon an infinite flat diffuse target (z = 500 km, c af = 0.5cm, AR = 100 cm2, X = 1 /Im).

Equations (42) and (43) are plotted in Figs. 1 and 2 for threedifferent laser-beam divergence angles. The value of Mi isgiven by

8 =z tan20 f 2 (c f/ af

erfc (2 cUf iT - 2z/C,ercL tan2 0~ cf ),

-20 0 20 60 100 140 180

(44)

which is plotted in Fig. 3. The figure shows that Mi startsincreasing when the leading edge of the laser pulse illuminatesthe target and remains constant after the trailing edge of thepulse arrives at the target. This can be seen from the formulaof Mi shown in Eq. (44). When the entire pulse illuminatesthe target, i is large, and Mi is approximately given by

TIME (t-2Z/C)(-f

Fig. 2. Speckle variance for normal incidence on an infinite flatdiffuse target (z = 500 km, c af = 0.5 cm, AR = 100 cm2 , X = 1 tim).

Mi_ wrARcT/X2z, (45)

which is constant with time (i). Relation (45) does not dependon 0, so the curves of Mi in Fig. 3 for three different beam di-vergence angles coincide with one another.

nU)zLLJ

z

LL

I-

uJa:

bJ

z

-IJ

LUU)


O.E


NCE = 0.9 mRAD

pulse shape. By neglecting the curvature effects, we find thatthe mean and the variance of the received signal energy aregiven by

Wi = QG(UT, iT - Z/C)T (46)

and

var(Wi) = Q2 KlG(1TA/2, iT - 2z/c)T, (47)

where

= (Of2 + 4Z2 tan 2 0 tan 2 011/2°T =2 ofi + 2 (48)

10-2

1 i , I i . J

100 102 104 106 108 1010


Fig. 5. Rms ranging error using the suboptimal estimator TSHOT fornormal incidence upon an infinite flat diffuse target (z = 500 km, cf= 0.5 cm, AR = 100 cm2 , = 1 m).

l01

02

zU)z

rl

10-

BEAM DIVERGENCE =0.9 mRAD

Ks = 90,600

0-2 1 , , , I , , ,10° 102 104 106 108 0 10

NUMBER OF PHOTOELECTRONSFig. 6. Comparison of the rms ranging errors using fSPEC and TSHOTfor normal incidence upon an infinite flat diffuse target (z = 500 km,cu = 0.5 cm, AR = 100 cm

2, X = 1 tm).

The rms timing error of the ML estimator TSPEC was cal-culated using relation (29), and the results are plotted in Fig.4. The rms timing error of Bar-David's estimator TSHOT canbe calculated by using Eq. (32) and the results are plotted inFig. 5. In Fig. 6, we show a comparison of the performancesof the ML estimator TSPEc and the suboptimal TSHOT esti-mator. We see that, at small-signal levels, the timing erroris limited by shot noise and that the performances of the twoestimators are identical. At the high signal limit, the timingaccuracy is limited by speckle, and the ML estimator performsbetter. The implementation of the ML estimator is morecomplicated and requires knowing both the mean receivedpulse shape ki and Mi. In situations in which shot noise isexpected to be the limiting factor, the suboptimal estimatorcan be used without sacrificing accuracy. However, if TSHOT

is to be used in the high-signal-level situations, one should beaware of the limit of ranging accuracy imposed by time-re-solved speckle.

For nonnormal incidence on an infinite flat target, thebroadening resulting from the tilt of the target dominates the

p is the incident angle of the laser beam with respect to thetarget surface normal. The value of Mi is given by

Mi = KsT/2VrUT. (49)

Mi is constant in time (i). Equations (46) and (47) are plottedin Figs. 7 and 8 for three different incident angles. For this

>_ INUI N\ IJL

L 0.75 U)z

-5000 -2000 0 2000 5000

TIME (t-2Z/C)

(fFig. 7. Mean received waveform for nonnormal incidence upon aninfinite flat diffuse target (z = 500 km, cr = 5.0 cm, AR = 100 C2,

= 10m).

LUC-)z

LUC-

LU

U)

0.75

0.50

0.25

0

INCIDENCE ANGLE= 50

-5000 -2000 0 2000 5000

TIME (t-2Z/C)

-fFig. 8. Speckle variance for nonnormal incidence upon an infiniteflat diffuse target (z = 500 km, c = 0.5 cm, AR = 100 cm2 , X = 1Am).

l0

i0LU

0z

:zn

U)

100


10°


BEAM DIVERGENCE= 0.1 mRADvar(fspE) 6\ nTJ 2K8 (N) >>K, (53)

-) 3 \\ KS= ,118 where N is the number of receiver time bins.O 0 \\\From relation (53), we see that the speckle-limited timing

error of the ML estimator depends on the width of the receiverUj \\\observation interval nT. The ML estimator emphasizes0 102 INCIDENCE signal counts in the tails of the received pulse, where theZ ANGLE =20 speckle is less severe [i2IMj <k, in Eq. (18)]. Therefore,

0 in the large-signal limit, the accuracy of the ML estimator: 10I improves as the width of the observation interval increases.

u) In this paper, we have not considered background noise. If2 TSPEC background noise is present, its effects will be dominant in the

10° ' C ' ' 'tails of the received pulse, where the signal counts are low.0° 102 104 106 108 1010 Therefore the performance of the ML estimator is expected

to be most sensitive to background noise when (N) and nTNUMBER OF PHOTOELECTRONS are large.

Fig. 9. Rms ranging error using the ML estimator spEc for non- For nonnormal incidence on flat targets, the mean receivednormal incidence upon an infinite flat diffuse target (z = 500 km, c af pulse is Gaussian shaped, and SHOT is identical to rCENT-= 0.5 cm, AR = 100 cm2 , A 1 Aim). The timing variances of SHOT and T

CENT are

104

103

102

10

100

BEAM DIVERGENCE = 0.1 mRAD

- Ks= 1,118

INC

A

SHOT

10° 102 IC

NUMBER OF

Fig. 10. Rms ranging error usi:upon an infinite flat diffuse targ100 cm2 , X = 1 ,um).

var('rSHoT) = var(fcENT) ITT2 T 2-(N) 2K8,

(54)

Equation (54) is plotted in Fig. 10 versus received signal en-ergy. The shot-noise-limited timing variance is

CIDENCEGLE = 200

_ - ~~10°050 )

0LU

06 108 1010 21

PHOTOELECTRONS Z

ng TSHOT for nonnormal incidenceet (z =SO0km, cof= 0.S cm, AR V

5;

:y

target configuration, the variance of the ML estimator is gapproximately by

r( AP) CT T2 1varf~s~c) (N ) g((N ))

where

g((N)) n~l~aT)dy

,/2j7r-KnT/2e1T)y2 exp(-y2/2)

1 + KS -1(N) exp(-y 2 /2)

Relation (50) is plotted in Fig. 9 versus received signal en-ergy.

In the case in which speckle is negligible, i.e., (N) << K, theshot-noise-limited timing error of the ML estimator is

UT 2var(ifSPEC) ~ N (N) << K8

On the other hand, when (N) >> K, the speckle-limitedtiming error of the ML estimator is

104

103

102

10I

10°iven

BEAM DIVERGENCE = 0.1 mRADKs= 1,118

_ INCIDENCE ANGLE = 5°

~~~~TSHOT_ ,~~~~~~~~~~~~IL , I I I I

10° 1o2 104 106 108 I0 1


Fig. 11. Comparisonofthermsrangingerrorsusing sPEcand TSHOT

(50) for nonnormal incidence upon an infinite flat diffuse target (z = 500km, cf = 0.5 cm, AR = 100 cm2, X = 1 im).

Table 2. Summary of the Timing Variance for theFlat Tilted Target

Speckle-LimitedEstimator Timing Variance Timing Variance

aT2 1 6 120T 3 aT 2

'TSPEC (N) g((N)) 6 nT J 2K

SOT2 aT2 T2(N) 2K 2K8

CNT2 ST2aT2

(N) 2K 2K,

(52)

104

0It0

z52U)

a

Jl


�i

I

(51)


T 2

var(~SHOT) _-(N)' (N) << K,

which is the same as that of the ML estimator TSPEC.speckle-limited timing variance of T

SHOT is

UT 2

var(sHoT) 2K I (N) >> K.

The timing variances of TSPEC and TSHOT are compared inFig.11. We see that the two estimators perform the same inthe shot-noise-limited regime, but, in the speckle-limitedregime, the ML estimator performs considerably better. Theabove results for nonnormal incidence on flat target aresummarized in Table 2.

5. CONCLUSIONS

In laser ranging and altimetry, time-resolved speckle causesrandom small-scale fluctuations within the received pulse thatdistort its shape and limit the timing accuracy of the receivers.In this paper, the ML estimator of the pulse arrival time wasderived under the assumption that the received pulse wasdistorted by shot noise and time-resolved speckle. The per-formance of the estimator was evaluated for pulse reflectionsfrom flat diffuse targets and compared with the performanceof the suboptimal centroid estimator and the suboptimalBar-David ML estimator, which was derived under the as-sumption of no speckle. When the signal level is high, specklenoise dominates the timing error and places a fundamentallimit on the accuracy of the suboptimal estimators. In thespeckle-limited regime, the ML estimator performs consid-erably better than the suboptimal estimators. In the large-signal limit, the accuracy of the ML estimator improves as thewidth of the receiver observation interval (i.e., number of timebins) increases. However, background noise was not con-sidered in this analysis. If background noise is present, itseffects will be most important in the tails of the received pulse,

where the signal counts are low. Since the ML estimatoremphasizes signal counts in the pulse tails, where speckle isless severe, the timing performance of the ML estimator isexpected to be most sensitive to background noise when thereceived pulse energy is high and the receiver observationinterval is large. The results show that our ML estimator issuperior in the speckle-limited regime.

ACKNOWLEDGMENT

The authors would like to thank James B. Abshire of theGoddard Space Flight Center for his many helpful commentsand suggestions. This research was supported in part by theNational Aeronautics and Space Administration under grantNSG-5049.

REFERENCES

1. J. W. Goodman, "Statistical properties of laser speckle pattern,"in Laser Speckle and Related Phenomena, J. C. Dainty, ed.(Springer-Verlag, New York, 1975), pp. 9-65.

2. J. Opt. Soc. Am. 66(11) (1976).3. J. W. Goodman, in Remote Techniques for Capillary Wave

Measurement, K. S. Krishnan and N. A. Peppers, eds. (StanfordResearch Institute Report, Stanford, Calif., 1973).

4. J. W. Goodman, "Some effects of target-induced scintillation onoptical radar performance," Proc. IEEE 53, 1688 (1965).

5. C. S. Gardner, "Target signatures for laser altimeters: an anal-ysis," Appl. Opt. 21, 3932 (1982).

6. B. M. Tsai and C. S. Gardner, "Remote sensing of sea state usinglaser altimeters," Appl. Opt. 21, 3932 (1982).

7. I. Bar-David, "Communication under the Poisson regime," IEEETrans. Inform. Theory IT-15, 31 (1969).

8. I. Bar-David, "Minimum-mean-square-error estimation of photonpulse delay," IEEE Trans. Inf. Theory IT-21, 326 (1975).

9. M. Elbaum and P. Diament, "Estimation of image centroid, sizeand orientation with laser radar," Appl. Opt. 16, 2433 (1977).

10. C. W. Helmstrom, Statistical Theory of Signal Detection (Per-gamon, London, 1960).


time-resolved speckle effects on the estimation of laser-pulse arrival times

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