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Detection techniques forvalidating Doppler estimates in heterodyne lidar

Barry J. Rye and R. Michael Hardesty

We investigate the ability of detection techniques based on the likelihood ratio to discriminate betweenheterodyne lidar Doppler estimates at low signal levels using examples generated by simulation. Thedistinction between estimates that are regarded as acceptable and as spurious is based on the Cramer–Rao lower bound. The conditional false alarm probability, which ordinarily describes recording detec-tion of a signal when none is present, is then found to be an approximate upper bound on the probabilityof selection of a spurious estimate. The method is superior theoretically to similar techniques based ondetection functions other than the likelihood ratio. The likelihood ratio also provides a basis for repro-cessing rejected data in the light of contextual information provided by those estimates that are accepted.© 1997 Optical Society of America

1. Introduction

Unbiased Doppler frequency shift estimates can beobtained from pulsed heterodyne lidar systems withperformance limited essentially by a Cramer–Raolower bound ~CRLB!. But this is achieved only if athreshold, which is characterized mainly by the totalenergy in the signal, is surpassed. At low signallevels, it may be necessary to augment the data sam-ple from a single-range gate by accumulation of spec-tral or correlation function estimates.1,2 Theseestimates are obtained by independent measure-ments of returns from the range gate within a shorttime compared with the period characteristic of grossfluctuations in the wind speed being measured. Weaddress two remaining questions here. First, howdo we know, in practice, when the accumulation hasbeen sufficient to ensure optimal performance?Second, what can be done to evaluate an estimatewhen sufficient accumulation has not or cannot beused?Somewhat similar problems have been addressed

B. J. Rye is with the Environmental Technology Laboratory,Cooperative Institute for Research in Environmental Science ~Uni-versity of Colorado and National Oceanic and Atmospheric Admin-istration!, 325 Broadway, Boulder, Colorado 80303. R. M.Hardesty is with the National Oceanic and Atmospheric Adminis-tration Environmental Technology Laboratory, 325 Broadway,Boulder, Colorado 80303.Received 3 June 1996; revised manuscript received 23 Septem-

ber 1996.0003-6935y97y091940-12$10.00y0© 1997 Optical Society of America

1940 APPLIED OPTICS y Vol. 36, No. 9 y 20 March 1997

in Doppler radar by the introduction of the consen-sus3 and statistical averaging method.4 In eachcase, the time series data are divided into segments,and an estimate is obtained from each segment. Theconsensus technique searches for clustering withinthe set of estimates obtained; if insufficient clusteringis obtained, the data set is rejected. We have arguedearlier that, if the reason for the spurious estimatesthat degrade estimator performance lies in normalreceiver noise as is usually the case in lidar, consen-sus appears to offer no advantage over accumulation,1and that detection ~or discrimination! techniques ofthe type we consider in this paper improve estimateselectivity compared with consensus.5 The differ-ence is that, in radar, spurious estimates are oftenoutliers. That is, they are not consequences of theaerosol backscatter and system statistics but arisefrom clutter, real objects that are not wind borne.For example, the statistical averaging method wasintroduced recently to remove outliers caused bybackscatter from birds, which give rise to signals thatcan be distinguished from atmospheric returns be-cause their statistics are non-Gaussian. In this pa-per we continue to assume that Doppler lidars are notsusceptible to outliers and consider the problem inthe ideal limit that spurious estimates are noise re-lated and can be described by the use of the ~known!statistics of the receiver noise. Two other tech-niques used in Doppler radar remain of interest.The first is context analysis in which an estimate iscompared with others from neighboring range gatesand observation times. Different forms of this tech-nique, some using pattern recognition, are used in

radar to distinguish outliers, but we comment hereinthat detection leads naturally to a context analysismethod that might be implemented in lidar. Thesecond technique is the coherence field, or normalizedcoherent power diagnostic, in which a function of thefirst lag term in the time series autocorrelation func-tion ~ACF! ~indicating presence of a coherent signal!is recorded, and data are rejected if this function doesnot lie above a threshold value. This approach isimplemented in Doppler processors in both radarsand lidars, and although apparently undocumented,has some similarity with the method described here.Our purpose, then, is to describe and illustrate a

straightforward method of selecting estimates at lowsignal levels that uses standard Doppler processingand is based closely on standard detection techniquesand to suggest routes whereby it might be extended.Some details of the assumptions and notation that weuse are summarized in Appendix A. We have de-scribed this method earlier in conference5 and work-shop papers.6

2. Estimate Statistics

A. Estimate Scatter: Histogram-Based Techniques

Maximum likelihood ~ML! estimators of the Dopplerlidar frequency shift7–9 have the properties ofmatched filters in the spectral domain.7,8,10 As-sume, for the present, that the frequency shift F1 isthe only unknown. From the measured data set x,the algorithms generate a log likelihood that is afunction of frequency F, ln@p~x, F!#. The term p~x,F! is the probability of obtaining the data set x if F isthe Doppler shift. According to Bayes theorem, itcan also be interpreted as the probability of F beingequal to the shift, given only the data set ~i.e., withoutany a priori information concerning the value of theshift!. Thus the ML frequency estimate F1* of theDoppler frequency shift F1 is that at which the max-imum in the log likelihood occurs, so that ln@p~x,F!#max 5 ln@p~x, F1*!#.Figure 1 contains a scatterplot characteristic of fre-

quency estimates F1* and the peak log-likelihood val-ues ln@p~x, F1*!# from which they are derived in a lowsignal regime. It is clear that such estimates, dis-tributed across the entire receiver search band, mustbe evaluated to assess their reliability. The ten-dency of estimates to cluster around the true value isdescribed by the histogram of frequency estimates,which is the expected value ~EV! of the linear densityof the abscissa in Fig. 1. An example of a histogramobtained by using simulations is given in Fig. 2~a!,which is taken from an earlier paper5:

g~F1*! 5 ~1 2 a! 1a

Î2psg

expF2~F1* 2 F1!

2

2sg2 G , (1)

where the first term represents a uniform back-ground distribution, the Gaussian function repre-sents the clustering of estimates around F1, and aand sg are the fitted parameters. Figure 2~b! showsthat the width of this Gaussian peak, sg is no greater

than sCR, indeed it is rather lower, and that the ratiosgysCR diminishes as the signal level decreases. Ex-tensive studies of the estimate histogram for variousML estimators have been presented by Frehlich andYadlowsky9 since amplified by Frehlich.11

B. Detection-Based Techniques

The detection technique is based not on the histogrambut on the probability density function ~PDF! of the loglikelihoods, which is the EV of the linear density of theordinates in Fig. 1. It is apparent that many of theclustered estimates in Fig. 1 are characterized by log-likelihood values in excess of those of the others andcould, in principle, be distinguished from them by theuse of a threshold. In ordinary detection theory, thelog likelihood is used in this way to determine thepresence of a signal in a linear communication chan-nel. Determination of the position ~in time or, forradar, in range! of the return signal from a target is,however, a nonlinear problem12 because the measure-ment equation that relates the measurements to theestimate is nonlinear. This is also true in Dopplerlidar for determination of the frequency shift, which isthe position ~in frequency! of the signal. For thesenonlinear problems, a single test may not be availableto assign a detection probability for F1*. Use of theword detection in this context is perhaps undesirable,especially as in heterodyne lidar it is often alreadyused ambiguously to refer to both the photodetector~really the photomixer! and any subsequent signal rec-tifier in the receiver. We have continued with it be-cause of its links in processing theory.We begin by discussing detection proper, or the de-

cision on the presence of the signal, that necessitatesour using the complete likelihood function ~generally afunction of signal strength and width as well as of

Fig. 1. Scatterplot of 180 pairs of values of the maximum loglikelihood ln@p~xuF!#max and the frequency estimate F1*, the latterplotted relative to the receiver bandwidth f1* 5 F1*yFS ~see Ap-pendix A!. The mean spectrum is described by Eqs. ~2! and ~16!,and data samples are processed to obtain F1* by correlation withthe modified Levin filter function @Eq. ~7!#. The sample size isM 5 64, the relative frequency of the signal shift and bandwidthare, respectively, f1 5 0.2 and f2 5 F2yFS 5 0.02, and the widebandsignal-to-noise ratio d 5 210 dB ~a1 ' 1.2, Npc ' 10!.

20 March 1997 y Vol. 36, No. 9 y APPLIED OPTICS 1941

frequency shift!. The likelihood ratio, on which suchdecisions are based, is first derived in an approxima-tion ~a derivation without the approximation is givenin Appendix B!, and its application is illustrated andcompared with some simpler but less powerful alter-natives including the coherence field. Then we takeadvantage of the assumption ~see Section 1! that spu-rious estimates are noise related and that the PDF oftheir likelihoods is characterized mainly by a subset ofthe PDF for estimates obtained when the signal isabsent. When this is valid, a decision about the pres-ence of the signal becomes nearly equivalent to a de-cision as to whether the frequency estimate is one ofthose clustered about the true value.

3. Log of the Likelihood Ratio or Log-LikelihoodDifference

In the frequency domain, the signal and noise aredescribed by spectral power densities. The EV of thetotal spectral power in the ith discrete channel is the

Fig. 2. ~a! Histogram ~squares! of estimates obtained from simu-lations for parametersM 5 64, f1 5 0.0, f2 5 0.02, d 5 210 dB, anda fit ~solid curve! to these values obtained from Eq. ~1!. Thedashed curve is a Gaussian curve with a standard deviation equalto the Cramer–Rao lower bound sCR ~Appendix A!. ~b! Parame-ters of curves fitted to histograms of the type exemplified in ~a!.The parameters are the same as for ~a! except that the effect ofincrease in the sample size is illustrated by an accumulation ofreturns from n pulses at constant M. The squares give the ratiosgysCR and the curve is the fitted value of a @Eq. ~1!#.


sum of the EV’s of the signal and the noise powers:

fi 5 fi~S! 1 fi

~N!. (2)

Binary decision theory12 is based on the selection orrejection of a hypothesis H0. Here the hypothesis isthat only noise is present, and we write H0:f

~S! 5 0.Rejection of H0 is regarded as acceptance of its alter-native, which is written H1:f

~S! . 0. Decisions onwhether to accept H0 or H1 for a given data sampleare based on the value of the likelihood ratio:

L~x! 5@p1~xuH1!#max@p0~xuH0!#max

, (3)

where p1 is the likelihood function obtained by ourprocessing the data set x and assuming that H1 andp0 are the likelihood obtained assuming H0. Un-known parameters within p0 and p1 are assumed tobe set to the values that maximize p0 and p1, i.e., theyare ML estimates. In the frequency domain thedata set x is the sample spectral density.Under H0, the spectrum is that of noise. If the

noise is white, fi~N! 5 f~N! and fluctuations in the

spectral components xi are uncorrelated. If we as-sume an accumulation of n spectra and normalize sothat EV~xi! 5 f~N!, the statistics of the spectral com-ponents are chi-square1:

p0~xuH0! 5 )i50

M21 S nf ~N!Dn xin21 exp@2nxiyf ~N!#

G~n!, (4)

where G~n! 5 ~n 2 1!! An M point power spectrum,derived from a data time series containingM complexdata points, is assumed throughout ~Appendix A!.The log likelihood can be written as

ln@p0~xuH0!# 5 q~x! 1 r@x; f ~N!#,

q~x! 5 M@n ln~n! 2 lnG~n!#

1 ~n 2 1! (i50

M21

ln~xi!,

r~x; f ~N!! 5 2n (i50

M21

$ln@f ~N!# 1 @xiyf ~N!#%. (5)

The ML estimate of f~N! could be obtained empiri-cally through iteration by finding the value of f~N!

that maximizes r~x; f~N!!. However, we can easilyshow by differentiating r~x; f~N!! with respect to f~N!

that the ML estimate is simply the average of the xi,as would be expected.For the calculation under H1 with signal present,

we use Levin’s approximation13 that the spectralcomponents remain uncorrelated. Use of theBrovko–Zrnic likelihood function7–9 derived by ourassuming realistic statistics for the time domain datais better at high signal levels,9 and a derivation of thelikelihood ratio with thismodel, which is analogous towhat follows here, is given in Appendix B. Levin’sapproximation is simpler and leads to an analyticexpression for the CRLB ~Appendix A! and to satis-factory estimates in the weak signal and white noise

regime ~where the spectrally uncorrelated noise dom-inates the signal!, which is the regime of prime in-terest. The expression for the likelihood function isidentical with that in Eq. ~4! when f~N! is replaced byfi. It can be written as

ln@p1~xuH1!# 5 q~x! 2 n (i50

M21

@ln~fi! 1 ~xiyfi!#,

5 q~x! 1 r@x; f ~N!# 1 ln$L@x, f ~N!, f ~S!#%,(6)

where the log of the likelihood ratio @Eq. ~3!#, or thelog-likelihood difference ~LLD!, which is the functionwe seek, is

ln$L@x, f ~N!, f ~S!#% 5 n (i50

M21 F xif ~N!

fi~S!

fi2 lnS fi

f ~N!DG . (7)

ML estimates of the parameters of fi can be foundeither from Eq. ~6! by maximizing the expression r@x,f# 5 2n •i ln@fi# 2 n •i xiyfi, in which the last termis the correlation of the data xi with Levin’s matchedfilter function 21yfi, or from Eq. ~7!, in which thefirst term on the right-hand side is the correlation ofthe data ~normalized to fN! with the modified Levinfilter function 1fi

~S!yfi. Eq. ~7! provides the LLDdirectly and was used to construct Fig. 1.Although the spectral density of the noise f~N! can

usually be assumed to be known, this is not the casefor fi

~S!. The unknown spectrum is usually replacedwith its ML estimate fi

~S!* and the LLD with thegeneralized log-likelihood difference ~gLLD! based onthese estimates:

ln$Lg@x, f ~N!, f ~S!*#% 5 n (i50

M21 H xif ~N!

fi~S!*

fi*2 lnFfi*

f ~N!GJ .(8)

In the examples that follow, we assume the signalbandwidth F2 to be known when we determine fi

~S!*

and the gLLD on the basis that the signal bandwidthcan be dominated in lidar by the pulse and not theatmospheric return. This leaves two unknowns, theshift F1 and a signal strength parameter, which can bethe wideband signal-to-noise ratio ~SNR! d that mea-sures signal power, the signal energy expressed as aneffective signal photocount Npc,9 or the degeneracy pa-rameter a1 ~see Appendix A!. In some cases we con-sider the signal strength to be known,whereas in othercases we estimate it using a one-dimensional numeri-cal iteration as the value that maximizes the gLLD.The estimate F1* is always obtained as the frequencycoordinate of the peak in the spectral correlation termand is an output of this iteration. In the limit of lowa1, the modified Levin function approaches the classi-cal matched filter value fi

~S!, which is Lee’s filter func-tion2 and enables estimates F1* to be formed that areindependent of the knowledge of signal strength.

4. Detection of the Presence of Signals

A. Conditional False Alarm and Detection Probabilities

Given the gLLD ~ln Lg!* available after a measure-ment, signal detection is portrayed ordinarily as acomparison of ~ln Lg!* with a threshold ln Lg9. Inparticular, the total false alarm probability is de-fined12 as the probability that a data sample pro-duces an estimate greater than ln Lg9 when thesignal might be, but in fact is not, present. Theconditional false alarm probability PFA~ln Lg9! is theprobability, as a function of the threshold, of thisconclusion being obtained with data samples fromwhich the signal is excluded; that is, PFA is condi-tioned on H0. It is calculated from the PDF,PDF0$ln@Lg~x!#%, of gLLD values obtained fromnoise-only data samples:

PFA~ln Lg9! 5 CDF0~ln Lg9!

5 1 2 *2`

ln~Lg9!

PDF0$ln@Lg~x!#% d$ln@L~gx!#%.

(9)

Here the conventions are that the PDF is normalizedso that its indefinite integral is unity, and the com-plementary distribution function ~CDF! is 1 minusthe distribution function, as usually defined. Exam-ples of PDF0 and CDF0 are given in Fig. 3. Theutility of PFA arises because PDF0$ln@Lg~x!#% can becalculated with Eq. ~7! with data samples synthe-sized or observed in the absence of a signal. Figure4 is a graph of a type from which PFA~ln Lg9! can beinferred, given the known or estimated values of thedegeneracy parameter a1 or a1*.The conditional detection probability ~CDP! ~condi-

tioned on H1! at a given threshold ln Lg9 is defined

Fig. 3. Example of a probability density function PDF0 ~dottedcurve! and a complementary distribution function CDF0 ~solidcurve! of the log-likelihood difference ln~Lg9! for noise-only datawith M 5 64 obtained from simulations. The signal parametersthat are needed in the processing algorithm @Eqs. ~7! and ~16!# aref2 5 0.05 and a1 5 2.5 ~d ' 25 dB, Npc ' 20!.


similarly by

PD~ln Lg9! 5 CDF1~ln Lg9!

5 1 2 *2`

ln~Lg9!

PDF1$ln@Lg~x!#% d$ln@Lg~x!#%,

(10)

where PDF1$ln@Lg~x!#% is the PDF of gLLD valuesobtained with a signal present. The total detectionprobability pD was obtained by multiplying PD withthe a priori probability of a signal being present, andthe total false alarm probability pFA was obtained bymultiplying PFA with the a priori probability of itsbeing absent.12 The a priori probabilities are inde-pendent of the data set andmay ormay not be known.

B. Alternatives to the Use of the Log-LikelihoodDifference for Detection

According to the Neymann–Pearson theorem, the useof the likelihood ratio, or in this paper the gLLD,leads to the most powerful hypothesis test, i.e., thehighest probability for detecting the presence of thesignal for a given false alarm probability.12 At thispoint it is useful to compare results from this testwith those of similar but simpler tests also used inDoppler radar and lidar that are based on the use ofthe zeroth and first lags of the sample covariance.By writing the heterodyne output as the complexdiscrete time series of in-phase and quadrature com-ponents zi 5 zi

~p! 1 jzi~q!, a contribution to the kth lag

of the covariance is rk~i! 5 zi~cc!zi1k, where the super-

script ~cc! indicates complex conjugate. For our sub-optimal detection algorithms, we used the followingfunctions:

~i! r0~i! 5 ~zi~p!!2 1 ~zi

~q!!2, relating to the power of zi,~ii! Re@r1~i!# 5 zi

~p!zi11~p! 1 zi

~q!zi11~q! , ~Re is the real

part!, being the sum of the first lag covariance terms

Fig. 4. Log-likelihood difference ~gLLD! values taken from CDFcurves ~as in Fig. 3! for noise-only data at different PFA @Eq. ~9!#.The range gate and relative signal bandwidth are constant ~M 564, f2 5 0.05!, and the gLLD is shown as a function of degeneracy.The PFA are 30% for the lowest ~heavy solid! curve and decrease forhigher curves through 3%, 0.3%, 0.03%, and 0.003%.


in each of the in-phase and quadrature componentsseparately, and

~iii! ur1~i!u2 5 ~zi

~p! 1 zi~q!!~zi11

~p! 1 zi11~q! !, which, in

addition to the self-correlation terms in Re@r1~i!#, alsocontains the cross-correlation terms zi

~p!zi11~q! 1

zi~q!zi11

~p! .

As sample estimates of these parameters over arange gate containing M values of zi, we used r*0, r*1,r*, m*1, and m* where

r*0 5 ~1yM! (i51

M

r0~i!,

r*1 5 @1y~M 2 1!# (i51

M21

Re@r1~i!#, r* 5 r*1yr*0,

m*0 5 @1y~M 2 1!# (i51

M21

r02~i!

m*1 5 H@1y~M 2 1!# (i51

M21

ur1~i!u2J1y2

, m* 5 m*1ym*0.

(11)

The estimates of rk and mk are thus unbiased esti-mators of the covariance of the original ~ungated!data. Their use in detection depends on the assump-tion that the noise PDF is known. The single sam-ple estimates r* and m* attempt to normalize first lagto power estimates within the sample.Results from a simulation are shown in Fig. 5.

Values of PFA~ln Lg9! for the gLLD detector werefound from noise-only data as for Fig. 4. For thesuboptimal ACF-based detection functions of Eq.~11!, conditional false alarm probabilities were ob-tained in a similar way—in effect, from Eq. ~9! withthe sample estimates ~r*0, etc.! substituted for ln@Lg#.Samples containing a signal were then processed todetermine PD~ln Lg9! and the corresponding CDP’sfor the covariance-based functions. Figure 5 is aplot of the CDP’s obtained with the thresholds ln@Lg9#in Eqs. ~9! and ~10!. They were chosen to give aconditional false alarm probability of 1% for each ofthe detection functions. Only one plot is shown be-cause the results are typical and seem physicallyplausible. The gLLD detector derived as in Section3 with Levin’s algorithm gives the highest PD at allsignal levels, despite the fact that these data weregenerated by a simulated time series, which leads tocorrelated spectral components and Levin’s approachbeing suboptimal. The best of the suboptimal detec-tion functions is r*0. The difference between r*0 andthe gLLD is that the gLLD estimates the signalstrength assuming the shape and bandwidth F2 ofthe signal spectrum ~Section 3!, whereas r*0 is anunfiltered estimate of signal strength over the entirereceiver bandwidth FSmade without any informationabout the signal. Estimation of signal strength fromr*0 is suboptimal and can lead to inferior estimates atlow signal levels.14 Of the functions that attempt toexploit the partial coherence of the signal with thefirst lag,m*1, performs rather better than r*1, presum-ably because it includes the cross-correlation terms.

The single-sample ratios r* and m* are degradedheavily by noise. The reason the first lag functionsdo not perform as well as r*0 is presumably also re-lated to noise, because the first lag terms are moredifficult to extract from noise than the zeroth lag termat low signal levels.

5. Validation of the Doppler Estimate

A. Known Signal Strength

The main problem in Doppler lidar is to decide, notwhether a signal is present, but whether a ML esti-mate of frequency shift F1* is acceptable, i.e.,whether it lies close to the true shift. To quantifywhat is regarded as acceptable, we divide the Dopplersearch bandwidth FS into an acceptance band aroundthe true shift, occupying frequencies ~ f1 6 D!FS,where f1 5 F1yFS and 2DFS is the acceptance band-width and the remainder. For example, with thedata of Fig. 1, we might define D 5 0.1 so that theacceptance band covers relative frequencies between0.1 and 0.3. We prefer to link the acceptance band-width to a theoretical limit on the precision of thefrequency estimate such as the CRLB ~see AppendixA! andwriteD 5 rCRsCR. Definition of the factor rCRis somewhat arbitrary but, if we wish to use the con-ventional analysis outlined in Subsection 4.A leadingto the false alarm probability, it is advantageous to

Fig. 5. Conditional detection probability PD ~for with-signal data!at thresholds set to correspond to a false alarm probability ~withnoise-only data! of PFA 5 1%. The parameters for the simulationwere M 5 64, f2 5 0.05. The PD, shown as a function of signalstrength ~Npc!, compare the use of the gLLD with the differentdetection functions defined in Eq. ~11!. These functions are thegLLD ~solid curve, open circles!, r*0 ~dashed curve, open squares!,m*1 ~solid curve!, r*1 ~dashed curve, solid squares!, r* ~dashed curve,open triangles!, and m* ~dashed curve, crosses!.

regard the entire region outside the acceptance bandas containing only noise-related estimates. The val-ues of rCR should, therefore, be sufficiently large toencompass all signal-related estimates. Followingthe discussion of Subsection 2.A, where sg # sCR andthe estimate histogram appears to be approximatelyGaussian, a choice of rCR ' 3.3 would be conservative@if rCR 5 3.3 and sg 5 sCR, 99.99% of the estimatesdescribed by the Gaussian term in Eq. ~1! are in-cluded within the acceptance band#. Spurious esti-mates are now defined as those outside theacceptance band.The question is whether the conditional false alarm

probability, which is defined in Eq. ~9! from noise-only data and intended to quantify a decision on thehypothesisH0 that only noise is present, can be help-ful in deciding whether a with-signal Doppler esti-mate lies inside the acceptance band. The link isthat estimates outside the acceptance band are noise-related and, when considered in isolation, shouldhave a PDF closely related to that obtained fromnoise-only data. In fact, these two PDF’s are notidentical, one reason being that smaller noise-relatedpeaks in the likelihood function, which would contrib-ute to the PDF for noise-only data, tend to be domi-nated by signal-related peaks when a signal ispresent. On these grounds, we should expect thatthe conditional false alarm probability will set anupper bound on the total probability of spurious es-timates.This argument can be quantified as follows. In or-

dinary detection theory, the PDF’s and CDF’s that areconditional on H1 ~signal present! are subdivided intodetections ~hits!with conditional probabilityPD~lnLg9!@Eq. ~10!# or misses with PM~ln Lg9! 5 1 2 PD~ln Lg9!.Here, for the lidar returns, we assume a signal is al-ways present at some level. The estimates are stillcharacterized by CDF1~ln Lg!, which we now regard asthe total complementary distribution function. Thehypothesis is that the estimate is spurious. The sub-script 10 is used for spurious estimates and 11 foracceptable estimates. To avoid confusion with thehit, miss, and false alarm terminology of linear detec-tion theory, we define the conditional probabilities ofthe system presenting us with a winner ~acceptableand above threshold! as PW, a sell ~acceptable but be-low threshold! as PS, or a bluff ~spurious but abovethreshold! as PB. ~The origins of these terms arefound in the Dictionary and Glossary pages currentlyreferred to at the website http:yywww.yahoo.comyRec-reationyGamesyCard_GamesyPokery.! For a giventhreshold level ln Lg9,

PW~ln Lg9! 5 CDF11~ln Lg9!,

PS~ln Lg9! 5 1 2 PW~ln Lg9!,

PB~ln Lg9! 5 CDF10~ln Lg9!. (12)

For a low Lg9 5 l9, Ptotal~ln l9! 5 PW~ln l9! 1 PB~ln l9!3 1. We call the EV of the proportion of acceptableestimates the a priori probability ratio R1. It couldbe determined from an estimate histogram9,11 ~Fig. 2!


independently of the likelihood ratio estimation.The proportion of spurious estimates is 12R1; if theyare a subset of estimates obtained with noise-onlydata, then

pB~ln Lg9! 5 ~1 2 R1!PB~ln Lg9! # PFA~ln Lg9! (13)

as expected. In a similar way we can argue that

pW~ln Lg9! 5 R1PW~ln Lg9! # PD~ln Lg9!. (14)

The total probability that the system offers a sell isthen

pS~ln Lg9! 5 R1PS~ln Lg9! 5 R1 2 pW~ln Lg9!. (15)

Figures 6~a! and 6~b! are graphs that generallysupport this argument but also show its limitations.The curves of Fig. 6~a! show the fraction of wins anderrors for given PFA 5 1% as the width of the accep-tance band rCR is varied. Clearly, a choice of rCR '3 separatesmost of them, but there is some indicationof a further increase in the standard deviation ratherthan saturation for all rCR . 3. The standard devi-ation of the wins is somewhat less than sCR, as ex-pected from the discussion of Fig. 1 in Subsection 2.A.Figure 6~b! shows examples of the probability distri-butions with rCR 5 3.3. The relationship given inEq. ~13!, pB # PFA, is satisfied for high PFA; indeed, itis overly conservative for PFA $ 10%. On the otherhand, PFA values less than approximately 0.1–1%may not bound pB at the highest signal levels whenthe distinction between win and bluff is sensitive tothe profile of the estimate histogram at the edge ofthe acceptance band. This is consistent with therecent observation that the wings of the estimatehistogram may not be describable by a single Gauss-ian.11More general plots of the probabilities as a function

of Npc and a for a given signal bandwidth are shownin Fig. 7. From Fig. 7, plotted for PFA 5 1%, the winrate can be improved by increasing a1 ~reducing sig-nal bandwidth! if a1 is below approximately 1, butthis ceases once a1 . 1, and the statistics are domi-nated by those of the correlated signal. The win rateis then characterized mainly by Npc. It becomes es-sentially 100% when Npc . 60, which is not surpris-ing. Figure 7 generalizes previous resultsindicating that accumulation becomes sufficient forachievement of the CRLB independent of bandwidthfor signals of about this level.1,9 The win rate be-comes essentially zero when Npc , 2, or when thesignal energy is less than approximately twice thenoise energy. Thus the detection technique can ex-tend the useful range of signal strengths over approx-imately 1 order of magnitude only. However theimprovement is striking. If a1 . 1 and Npc is re-duced by a factor of 6, from 60 to 10, pW is reducedonly from 100% to approximately 40%.Figures 8~a! and 8~b! display the probabilities of

bluffs and sells for various constant values of a1.Figure 8~a! shows that the problem of having pB notbounded by PFA is most serious for a1 . 1 and athigher signal levels ~Npc . 20!. Figure 8~b! shows


the probability of one not recognizing acceptable es-timates pS~ln Lg9!, which is of interest mainly be-cause these might be recoverable in a second stage ofprocessing by the use of context methods ~see Section1!; it appears to be greatest for low a1, a1 , 1. Ingeneral, the bluff rate in Fig. 8~a! is independent ofdegeneracy a1 for a1 , 1, whereas the results of Fig.8~b! @which, like those of Fig. 7, are for acceptable

Fig. 6. ~a! The fraction of frequency shift estimates F1* fallingwithin 6DFS of the true value F1 as D is increased relative to aconstant value of sCR. Only estimates with gLLD ln Lg9 greaterthan a threshold value, set to make PFA 5 1%, are considered.Wins ~estimates within 6DFS of F1! are shown as a fine solid curveand sells as a heavy curve. The standard deviation of the wins isshown as a dashed curve ~right-hand ordinate!. Note that thisstandard deviation is generally less than sCR, which is possiblebecause it characterizes only selected estimates. The parametersused in the simulations are M 5 64, f1 5 0.0, f2 5 0.02, d 5 210dB ~a1 ' 0.5, Npc ' 10!. ~b! Example of complementary distribu-tion functions CDF0 ~ln Lg9! for noise-only data ~heavy solid curve!and CDF1 ~ln Lg9! for with-signal data ~fine solid curve! obtainedfrom simulations withM 5 64, f2 5 0.05, and degeneracy a1 5 2.5~d ' 25 dB,Npc ' 20!. The dotted curves are the total winner andbluff probabilities pW and pB obtained with signal by our multiply-ing CDF1~ln Lg9! with the a priori ratio R1 and ~1 2 R1!, respec-tively @Eqs. ~13! and ~15!#. In general, at a given value of thelog-likelihood difference, ln Lg9, PFA @from CDF0~ln Lg9!, Eq. ~9!# isgreater than pB, in agreement with the inequality in Eq. ~13!, andPD is greater than pW. However, closer examination of the PFAand pB curves at low false alarm probability values as shown on theright-hand ordinate, indicates that this inequality no longer ap-plies when PFA , 0.5%.

estimates# are independent of a1 for a1 . 1. Theresults shown in these graphs are not affectedstrongly by the choice of signal bandwidth F2.

Fig. 7. ~a!Contour plot of the total win probability pW as a functionof effective photocountNpc and photocount degeneracy a1 for a signalof bandwidth f2 5 0.05 and PFA 5 1%. The dashed curves corre-spond to constant values of the sample size M 5 Npcy@=~2p! f2a1#equal to 10 ~upper left!, 100, and 1000 ~lower right!. ~b! Graph oftotal probability of a bluff pB as a function of Npc for various a1,corresponding to different sections through ~a! ~i.e., 10 , M , 1000!.The line markers indicate a1 5 0.1 ~solid squares!, 0.3 ~plus signs!,1.0 ~open squares!, 3.0 ~crosses!, and 10.0 ~solid triangles!. ~c!Graph of the probability pS of unrecognized acceptable estimates asa function of Npc for the same values of a1 as in ~b!.

The probability of winners shown in Fig. 7 is notthe same as the fraction of good estimates that hasbeen defined in different ways, each based on theestimate histogram, by Frehlich and Yadlowsky,9who use the value of the parameter a in Eq. ~1!, andby Anderson,15 whose definition is equivalent to the apriori ratio R1 when DFS is chosen to be equivalent toa constant Doppler shift of 1 mys rather than basingit, as we have done, on sCR. Basing the acceptance

Fig. 8. ~a!Comparison of the probabilities of wins, bluffs, and sellsat a false alarm probability PFA of 1% obtained with ~curves! andwithout ~points! prior knowledge of signal strength. The param-eters are f2 5 0.05 andM 5 64. Wins are shown as solid squaresand a heavy curve, bluffs as open squares and a thin curve, andsells as crosses and a dashed curve. The signal strength is pa-rameterized by Npc and is the true value. In general, there islittle difference between results obtained with the two methods,i.e., little is gained from prior knowledge. The main exception tothis is at low signal levels where pB . PFA. ~b! Standard deviation~referred to receiver bandwidth! of the winners for various falsealarm probabilities for the same parameters as ~a!. The pointsrefer to values obtained without prior knowledge of signalstrength, open squares for PFA 5 0.1%, solid squares for PFA 51.0%, and crosses for PFA 5 10%. Somewhat surprisingly, theseresults all have a standard deviation that approximates the valueexpected from the CRLB corresponding to the true signal param-eters ~heavy curve!. This is not the case for the results obtainedwith prior knowledge ~solid curve for PFA 5 0.1% and dashed curvefor PFA 5 10%!, where the standard deviation is much less than theCRLB for low signal levels ~the technique selects only the peak ofthe scatter plot of Fig. 1! and greater than the CRLB at high signallevels, especially for the higher PFA ~5 10%!when estimates withinthe wings on the estimate distribution are included among thewins.


bandwidth DFS on sCR does, however, entail a para-dox: The win probability can increase as the signallevel is decreased. In fact, when the signal leveldecreases, the acceptance bandwidth 2DFS 52rCRsCR eventually becomes greater than the re-ceiver bandwidth, all estimates become acceptable,and acceptability becomes meaningless. Thegreater the signal bandwidth, the larger the sCR andthe higher the minimum usable signal strength atwhich this occurs. It is also the reason that bluffsare so low and sells so high at low signal levels inFigs. 8~a! and 8~b!. A further minor problem is thatwhen the data are discretely sampled, the frequencyspectrum is circular or periodic, implying that if uF1u. ~0.5 2 D!FS, then the acceptance band wrapsaround the receiver bandwidth and covers regions ateach end of the normalized frequency range 20.5 to0.5; this complication is ignored here.

B. Estimated Signal Strength

Our simulations have been extended to compare theresults of using the generalized likelihood ratio with-out knowing the signal level. An estimate of thesignal level ~either Npc or a for a givenM and f2! wasobtained, maximizing the gLLD function of Eq. ~8! bynumerical reiteration. Each corresponding Dopplerestimate was cataloged as acceptable if it lay within3.3 times the true CRLB of the true frequency shiftand as spurious otherwise. The coordinates of theestimate, namely, gLLD ln~Lg! and its degeneracy,were then, in effect, compared with the curve for aparticular value of the false alarm probability PFA,like one of those shown in Fig. 4. If the estimate wasacceptable, then the estimate was recorded as a win-ner if the point described by the coordinates lay below~i.e., it corresponds to higher PFA than! the curve andas a sell otherwise. For the spurious estimates, abluff was recorded if the point lay above the curve.The results, which are not greatly dependent on

operating parameters, are exemplified in Fig. 9.Clearly, prior knowledge of the signal strengthmakesonly aminor difference, the probability of a bluff beingmore sensitive to it than the probability of a win.

6. Comments and Conclusions

We have examined the use of techniques for deter-mining the presence of meaningful signals in lidarreturns and the nonlinear problem of validating aDoppler-shift estimate. The former technique ischaracterized by a ~conditional! false alarm probabil-ity PFA that can be determined from noise-only dataand, therefore, is known; the latter technique is char-acterized by a ~total! probability of a bluff pB, which,as we have shown, can be approximately bounded byPFA. It is usual to portray detection techniques interms of comparing the estimated gLLD with athreshold value set at a particular PFA ~see Subsec-tion 4.A!. In other applications, where extremelylow error rates are critical, data are likely to be dis-carded if the gLLD is below the threshold. Thisshould not be the case in Doppler lidar, where advan-tage can be taken of reprocessing data by using con-


textual analysis based on the estimates classified aswins.Therefore thresholding in lidar is recommended

when one is deciding whether to accept an estimateimmediately or to submit the data for reprocessing.One application in which thresholding might be use-ful is in the reduction of data storage requirements bythe elimination of on-line data. For this to be prac-tical, the data products that are needed to make adecision should be rapidly computable, and it is herethat the alternative techniques discussed in Subsec-tion 4.B would be useful. Apparently, a simple ~sub-optimal! return energy estimate formed by onesquaring the raw data is the best substitute for thefull gLLD, and, in principle, little advantage can begained by one using other than the zeroth lag of thecomplex covariance estimate. Thresholding withthese suboptimal methods should be used conserva-tively to avoid rejecting data that might be useful inlater processing. These techniques are robust in thesense that they do not depend on the validity of thein-built assumptions of more sophisticated process-ing algorithms.The limitations of our approach exemplified in

Figs. 7 and 8 are ~1! accumulation is sufficient toensure acceptable estimates when the effective pho-tocount is greater than approximately 60, and essen-tially no estimates can be validated if it is belowapproximately 2, so that the range of signal strengthsover which detection techniques are useful seemslimited to approximately 1 order of magnitude; and~2! the upper bound on pB cannot be set by PFA whenPFA , 1% and is unduly conservative when PFA .10% ~see Subsection 5.A!. The first limitation isoverstated because the technique is useful at all sig-nal levels in providing an answer to our first questionin Section 1: How do we know when accumulationhas been sufficient to ensure optimal performance?The answer is that it is unnecessary to determinewhether accumulation in any given case has beensufficient; it is only necessary to determine whetherPFA is sufficiently low for the estimate to be accept-able. In principle, data could continue to be takenfrom a repetitively pulsed lidar until and only untilthis condition wasmet by analogy with the sequentialobserver of detection theory.11 The second limita-tion is partly a consequence of our having to set asharp boundary between acceptable and unaccept-able estimates on the basis of the CRLB, with manyof the bluffs presumably coming from just outsidethat boundary ~see Fig. 6!; it is probably immaterialif the detection is supplemented by contextual anal-ysis and not used alone for rejecting lidar data.Two further developments that lie beyond the

scope of the present paper suggest themselves.First, similar methods can be applied to other Dopp-ler lidars that use spectral resolution with direct de-tection rather than heterodyning; an example of acontour plot analogous to Fig. 7 for a photon-countinginterferometer instrument is given elsewhere.6Second, relative to the goal of recovering in a secondstage of processing some of the estimates classified as

sells, the estimates accepted following the initialthreshold could form the contextual basis for repro-cessing data from the remaining range gates. Onemechanism for incorporating these might be based onthe use of the Kalman filter and smoother techniquesthat we have discussed previously.16 At that time,we assumed the estimates to be given and the origi-nal data unavailable, and we used the filter simply toautomate the rejection of spurious estimates and out-liers. Although the Kalman algorithm is usually de-rived in terms of minimizing mean-square error, itcan be derived from ML considerations. This en-ables the contextual or a priori information fromneighboring range gates to be presented in the formof a ~spectral! likelihood function that can be applieddirectly, with Bayes theorem, to modify the likelihoodfunctions used here, thus replacing the ML estimatesconsidered in Subsection 2.A with maximum a poste-riori estimates.

Appendix A: Formulas, Notation, and AssumptionsUnderlying the Simulated Data

In the simulations, the atmospheric parameterswithin a range gate are assumed constant. The re-turn signal prior to range-gating is modeled with aGaussian spectral density fS~i!:

fi~S!

f ~N! 5 a1 expF2~Fi 2 F1!

2

2F22 G , (A1)

where Fi is the frequency of the ith spectral compo-nent ~0 # i , M!, F1 the Doppler frequency shift, andF2 the bandwidth of the received signal. Because ofthe use of the discrete Fourier transform ~DFT! it isalso useful to use the relative frequencies f1 5 FiyFS5 iyM, where FS is the receiver bandwidth, the Dopp-ler search bandwidth, and the sampling frequency foreach of the in-phase and quadrature data time series~negative Doppler shifts are by convention aliasedinto the range My2 # i , M!. In addition, f1 5F1yFS and f2 5 F2yFS. The parameter a1 as definedin Eq. ~A1! is the maximum possible SNR in thespectrum a1 5 dy@=~2p! f2#, where d 5 SyN 5 •ifi

~S!DFy•i f~N!DF 5 •i fi~S!y~Mf~N!! is the wideband

SNR. a1, which is constant over pulse accumula-tion, serves to distinguish the regime ~a1 , 1! wherethe noise from wideband background ~ideally localoscillator shot noise and characterized by f~N!! iscomparable to fluctuations in the signal itself ~speck-le or fading!. Although lidar processing theory isinherited from Doppler radar and is entirely classi-cal, and photons are not counted in heterodyne lidarsystems, semiclassical measures of signal strengthare useful in Doppler lidar because the backgroundnoise can be written fN 5 ~QF!hn, where QF is anoise figure and hn is the quantum energy.17 Inparticular, the ratio of signal energy E~S! 5 •i fi

~S! tobackground noise spectral density, which is often use-ful12 in the processing of theory to characterize esti-mator performance, can be written as an effective

photocount9 Npc 5 E~S!yf~N!. Hence9

Npc 5(i50

M21

fi~S!

~QF!hn5 dM. (A2)

With this signal spectrum and notation, the CRLB,sCR, calculated from the Levin formula with contin-uous rather than discrete variables, is13,18,19

sCR 5F2

ÎNpc

1

Îg1~a1!,

g1~a1! 5a1

Î2p *2`

` x2 exp~2x2!dx@1 1 a1 exp~2x2y2!#2

. (A3)

Asymptotically, g1~a1! } a1p, where p 5 1 for a1 ,, 1

and p 5 21 for a1 .. 1. It is maximized at g1~3.27!5 0.203.The semiclassical interpretation of a1 is the photo-

count degeneracy or effective photocount per speck-le20 a1 5 Npcym1. To reconcile this definition withthe preceding one, the speckle count is defined hereby

m1 5M

(k50

2M21

mk~S!

5M

2 ReS(k50

M21

mk~S!D 2 1

, (A4)

with the ~normalized! autocorrelation function ~ACF!of the original ~ungated! signal mk

~S! 5 rk~S!yr0

~S! and thezero lag covariance r0

~S! 5 S. The summation overthe two-sided ACF ~0 # k , 2M, which includesnegative lags! can be reduced in Eq. ~A4! to anM-point one-sided ACF ~0 # k , M! because mk

~S! isHermitian @mk

~S! 5 m2k~S!~cc!, where ~cc! indicates com-

plex conjugate#. With the 2M-point DFT relations,written as rk

~S! 5 @FSy~2M!# •i Fi~S! exp@2piky~2M!#

and Fi~S! 5 ~1yFS! •k rk

~S! exp@22piky~2M!#, then forF1 5 0,

a1 5 d (k50

2M21

mk~S! 5 d

(k50

2M21

rk~S!

r0~S! 5 2Npc

f0~S!

(i50

2M21

fi~S!

5f0

~S!

f ~N! .

(A5)

If F1 Þ 0, the spectral peak is simply shifted so thata1 is, in general, the maximum SNR as before.Our simulations are intended to illustrate the ar-

guments in the text and are derived usually fromsample spectra constructed with components thathave independent fluctuations, which is Levin’s ap-proximation ~the exception is in Section 4, where timeseries were used!. We have generally and conserva-tively assumed long range gates, e.g., M 5 64, thatcorrespond to a range gate of length of approximately200 m for a typical 2-mm lidar, although in practiceLevin’s algorithm works quite well at the low signallevels of interest for lower M. Our main results, inFig. 7, use values of M as low as 10.


Appendix B: Derivation of the Log-LikelihoodDifference with the Brovko–Zrnic Likelihood Function

The likelihood function of a data sample drawn froma complex time series with Gaussian statistics andcharacterized by its correlation function rather thanits spectrum has been used by a number of authors7–9following the original derivation attributed toBrovko.8 Here we largely follow the notation ofChornoboy,7 amended to be consistent with that usedin the main text. The log-likelihood function is

ln@p~Z, R!# 5 2lnuRu 2 Z1R21Z, (B1)

where the covariance matrix R 5 ND~Gd 1 I!D~cc! 5N@DGdD~cc! 1 I#, uRu 5 det R, the complex data areZ 5 ~z0, z1, . . . , zM21!

T, the superscript T indicatestranspose, the superscript1 indicates complex trans-pose, the identity matrix I 5 diag @1 1 . . . 1#, thesignal component G 5 G~ f2! is a symmetric M 3 MToeplitz matrix with terms rik 5 exp@22puk 2iu f2!

2y2# @Gaussian tomatch the spectrum of Eq. ~16!#,D 5 diag @1 exp~2jv! exp~22jv! . . . exp~2j~m21!v!#, v5 2pf1, and N 5 f~N!FS and d are as before ~seeAppendix A!.If one processes the data assuming no signal, then

d 5 0, R 5 NI, and ln@p0~ZuH0!# 5 2ln~NMuIu! 2Z1N21I21Z 5 2M ln~N! 2 ~N!21 •i uziu

2, from whichthe ML estimate of N is •i uziu

2yM as expected. Ifone assumes a signal, then

ln@p1~ZuH1!# 5 2M ln~N! 2 lnuD@Gd 1 I#D~cc!u

2 N21Z1@D~Gd 1 I!D~cc!#21Z.(B2)

On the right-hand side, the second term lnuD~Gd 1I!D~cc!u 5 lnuGd 1 Iu because uABCu 5 uAiBiCu anduDu~cc! 5 uD~cc!u. In the last term, by the use of~ABC!21 5 C21B21A21, D 5 ~D~cc!!21, and D21 5D~cc!,

Z1@D~Gd 1 I!D~cc!#21Z 5 Z1DGD~cc!Z

5 2ReF(m50

M21 S (i50

M2m21

zi~cc!gi,i1mzi1mD

3 exp~2jvm!G 2 (i50

M21

uzi u2gi,i,

(B3)

where g are the elements of the centrosymmetric ma-trix G 5 ~Gd 1 I!21. The final expression is equiv-alent to a 2M-point DFT7,21 @see Eq. ~A4!#. The LLDln@L~Z!# 5 ln@p1~ZuH1!yp0~ZuH0!# becomes

ln~L! 5 N21Z1DCD~cc!Z 2 lnuGd 1 Iu

5 N21 S2Re (m50

M21

(i50

M2m21

zi~cc!Ci,i1mzi1mexp~2jvm!

2 (i50

M21

uzi u2Ci,iD 2 ln uGd 1 Iu, (B4)

where c are the elements of C 5 I 2 ~Gd 1 I!21.Within Eqs. ~B3! and ~B4!, the unweighted summa-


tion •i zi~cc!zi1m can be regarded as ~M times! either a

biased estimate of rm~S! or an unbiased estimate of

rm~S!mm

~W!. Here mm~W! 5 1 2 umuy~M 2 1! is the ACF of

the rectangular window function used to select ~or inlidar, to range gate! the data sample, and r~S!m~W! isthe covariance of the gated signal plus noise.With the relation @I 2 ~Gd 1 I!21# 5 Gd~Gd 1 I!21,

the LLD can also be written as

ln~L! 5 ~1yN!Z1DGdGD~cc!Z 2 lnuGd 1 Iu. (B5)

A comparison of Eqs. ~B2! and ~B3! with ~B5!, showsthat the principal difference is that in Eq. ~B5! thecovariance terms are weighted by the elements of thematrix GdG 5 1Gd~Gd 1 I!21 rather than those of2~Gd 1 I!21. The analogy with Eqs. ~5! and ~7!,where in Eq. ~7! the spectral domain data are corre-lated with the modified Levin function f~S!yf ratherthan @in Eq. ~5!# with 21yf, is obvious. The formGdG indicates that, in the limit of low d ~G ' I!, theweighting of the covariance terms will be given by theri,i1m, which are independent of i as in Levin’s ap-proximation. To compute ln~L!, C 5 I 2 G is sim-pler to implement, and the number of points withinthe DFT of Eq. ~B4! ~which is not divided byM! can beextended to an integer that is a power of 2 by zeropadding without affecting the value of ln~L! obtained.

We are grateful to R. G. Frehlich and R. E. Lataitisfor comments in internal review and toMarilyn Steegfor editing the original manuscript.

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