estimate optimization parameters for incoherent backscatter heterodyne lidar including unknown...

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Estimate optimization parameters for incoherentbackscatter heterodyne lidar including unknownreturn signal bandwidth

Barry J. Rye

The conditions for optimizing the precision of heterodyne atmospheric lidar measurements using ex-tended ~deep! targets are investigated. The minimum standard deviation of each unknown ~returnpower, Doppler shift, and signal bandwidth! is approximately twice the optical limit at best and is onlyweakly dependent on knowledge of the other parameters at optimal power levels. Somewhat strongersignal power levels are needed for bandwidth estimation. Results are displayed as a function of atime–bandwidth product to clarify the trade-off between estimate precision and range weighting. Re-alization under ideal conditions is confirmed by use of simulations. © 2000 Optical Society of America

OCIS codes: 010.3640, 040.2840, 280.3340

1. Introduction

The conditions under which a heterodyne lidar sys-tem using atmospheric backscatter is optimized, inthe sense that its estimate precision is maximized ata given signal level, can be determined by the theoryfor the Cramer–Rao lower bound ~CRLB! on the es-imate variance. In a previous article1 we examined

these conditions for the estimation of the widebandsignal-to-noise ratio ~SNR! d and the Doppler-shiftfrequency f1. The signal bandwidth f2 was consid-ered to be known a priori. Here the discussion isextended to include the case when f2 is also esti-mated.

Previously, with only two unknowns, there were inprinciple four different situations to be considered~estimation of f1 with d either known or unknown andice versa!. However, the nature of the model con-entionally used for the return signal, in which theerm containing f1 is separated from that containing

d and f2, entailed that the CRLB on either d or f1 isindependent of knowledge of the other, so that only

The author is with the Environmental Technology Laboratory,RyEyET2, National Oceanic and Atmospheric Administration, Co-operative Institute for Research in Environmental Sciences, Uni-versity of Colorado, 325 Broadway, Boulder, Colorado 80303. Hise-mail addresss is [email protected].

Received 23 February 2000; revised manuscript received 20 July2000.

0003-6935y00y336086-11$15.00y0© 2000 Optical Society of America

6086 APPLIED OPTICS y Vol. 39, No. 33 y 20 November 2000

two values of the CRLB are independent. Consider-ation of all three unknowns increases the number ofCRLB values in principle to 12, but given the simpli-fied signal model, only five values differ and only fourexpressions are needed to describe them. Neverthe-less, the problem is somewhat more complicated thanbefore.

It is customary in lidar to segment the time seriesinto range gates prior to processing and to assumethat, within each range gate, the return is stationary.The bandwidth described by f2 is that of the signalprior to the spectral broadening that arises from thetime-series windowing implicit in range gating. Thespectrum of this ungated signal is given by the con-volution of the spectrum of the transmitted lidarpulse with that of the atmosphere. If the latter iscomparatively narrow, then f2 is the bandwidth of thepulse, which is known a priori. Otherwise estimatesof f2 might be used to infer the atmospheric contri-bution arising from the turbulence, shear, or varia-tions in the atmosphere over the duration of ameasurement. Although in these cases the returnmay also become nonstationary, they are probablythe main area of application for estimation of f2. Wecontinue to assume stationarity and that the profileof the unwindowed return signal spectrum is Gauss-ian:

ps~ f ! 5d

Î2p f2

expF2~ f 2 f1!

2

2f22 G . (1)

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cf

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c1

The conditions for optimization of bandwidth esti-mates are discussed in Section 3, after a further sum-mary of what is meant here by optimization inSection 2. In Section 4, numerical simulations arealso used to confirm that, at least under the idealconditions that these simulations model, the CRLBon the variance of estimates of d, f1, and f2 can berealized, except in the limiting case when the signalis sufficiently small that estimates are biased bynoise. Use of detection techniques for thresholdingestimates in this limit2 is not considered here. Somesimple numerical examples of optimization condi-tions are described in Section 5. The main resultshave been previously included in a conference paper.3

2. Optimization of Heterodyne Systems

It is a common engineering practice to characterizethe performance of a system by comparison with anideal. When constructing a model for ideal mea-surements of optical spectra arising from incoherentsources, one might assume that ~i! speckle and signalading contribute negligibly to the noise, ~ii! that theptical signal can be converted noiselessly into a rec-ified electrical signal obtainable using direct detec-ion with photon counting, and ~iii! the availability of

lossless system of optical filters to separate therequency components of the signal into differenthannels. Then the noise in each channel is limitedo the Poisson statistics of photodetections, and de-ected signal fluctuations within different channelsre uncorrelated. Under these assumptions, Ladingnd Jensen4 obtained general expressions for the

likelihood function of an optical signal and the Fisherinformation matrix used to determine the CRLB’s onestimate variance. In scattering studies of flowinggases, Seasholtz assumed such an ideal measure-ment system to derive expressions for the CRLB onthe estimate variance of velocity,5 density,6 and tem-perature5 ~which, for the molecular scattering in theollisionless limit that was considered, is obtainedrom signal bandwidth!. It was supposed in each

case that the other parameters were known and thatthe signal spectrum was Gaussian. Under the sameassumptions, Rye and Hardesty7 also derived the ex-pression for the CRLB on the variance of Doppler-shift estimates using essentially the same approachand describe the maximum-likelihood estimator thatwould ordinarily be expected to achieve the bound.The variances for the three unknowns of interest hereare as follows:

sopt2~du f1, f2! 5 d2yN,

sopt2~ f1ud, f2! 5 f2

2yN,

sopt2~ f2ud, f1! 5 f2

2y~2N!, (2)

where the notation s2~xuy! means the variance of xgiven a priori knowledge of the variables in y, and Nis the expected value of the photocount. Some ofthese formulas have been derived earlier by meansother than likelihood analysis. For example, thevariance s2~d! 5 d2yN is a simple extension of the

2

standard formula for the variance of Poisson countingstatistics @s2~N! 5 N#; and the expression s2~ f1! 5f2

2yN was given by Gagne et al.8 in an early analysisof the capabilities of a high-resolution Fabry–Perotinterferometer based on the least-squares method.It is emphasized here that the results acquire greatersignificance when derived by use of the likelihoodmethod, because this can be shown to provide fairlygeneral lower bounds on all estimates of the variablegiven as an assumed statistical model, irrespective ofthe estimation algorithm.9

Heterodyne techniques fall short of the ideal be-cause the photodetector is used as a mixer and not asa rectifier. At the photodetector and before rectifi-cation, any different spatial modes ~speckles! in theignal field are combined coherently, and measure-ent noise ~usually dominated by that of the local

scillator! is added to the signal. At the subsequentetection ~rectification! stage, parameter estimatesre degraded by ~i! signal fading, which survives co-erent combination,10 and ~ii! measurement noise en-ancement that is due to the nonlinearity of theetection process. For purposes of comparison withirect detection systems, it is convenient to charac-erize the signal energy or, strictly, the maximumatio of the energy in the signal to the spectral den-ity of the measurement noise ~the noise power perycle of bandwidth11! by an effective photocount that

we also call N. Then domination of the measure-ment noise by fading leads to estimate variances thatare independent of N at high signal levels, and noiseenhancement leads to variances inversely propor-tional to N2 at low signal levels. Examples areshown in Fig. 1. The consequences for the design ofheterodyne systems were described in earlier pa-pers.1,12

The optimal operating point for measurement ofeach unknown is that at which its variance is closestto the ideal value of Eqs. ~2!, which is also where thevariance as a function of N has slope 1yN, and definesthe conditions that a particular value of the varianceis achievable with the minimal photocount. These

Fig. 1. Plots of the following normalized standard deviations cal-culated for heterodyne systems by use of the Brovko–Zrnic formu-las ~see Appendix A! versus the effective photocount N: thin curve,sBZ~du f1, f2!yd; F, sBZ~ f1ud, f2!yf1; h, =2sBZ~ f2ud, f1!yf2; and boldurve, the ideal optical value for these ratios that, in each case, isy=N @Eqs. ~2!#.

0 November 2000 y Vol. 39, No. 33 y APPLIED OPTICS 6087

btEaaa

mt

d

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Table 1. Optimal Operating Parameters in the Levin Limita

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6

conditions have been characterized previously by twoparameters.1 The first is the average ratio, withinthe ungated data time series, of the photocount to thefade count. This tuning, or degeneracy, parameterwas represented by the ratio of the peak spectraldensity in the signal spectrum of the spectral densityof the measurement noise, the latter being assumedindependent of frequency. For the Gaussian signalspectrum of Eq. ~1!, a 5 dy@=~2p! f2#. The secondparameter is the factor FE ~the estimate noise figure!y which the variance of the heterodyne estimate athe optimal operating point exceeds the ideal value.xamples of plots of FE versus a are given in Ref. 1nd Fig. 2. The minimum value of FE and that of at the optimal operating point are written ~FE!min andmin, respectively.CRLB’s applicable to heterodyne receivers were

calculated by use of likelihood theory based in thespectral ~frequency! domain, as for the ideal optical

odel considered as the start of this section, and inhe time domain. In Levin’s spectral model13 it is

assumed that spectral components within a datasample are uncorrelated, which is true only in thelimit of range gates with duration T 5 MyFs, muchlonger than the correlation time of the return signal1yF2 5 1y~ f2Fs!,14 where M is the number of complex

ata points in the range gate and Fs is the samplingfrequency of the complex, demodulated time series.Apart from this limit, the algorithm is suboptimal foruse with gated signals except when the signal level islow and dominated by uncorrelated noise. Becausewe assume a Gaussian signal spectrum prior to rangegating, we can use expressions for the CRLB’s basedon this model15,16 as a good approximation in the limitf long range gates. These expressions can be sum-


arized, if we write the three estimated parameterss u 5 ~u0, u1, u2! 5 ~d, f1, f2!, by

sLevin2~ui uujÞi! 5 sopt

2~ui uujÞi!

3 F a

Î2p *2`

` f ~i!~x!

g2~x, a!dxG21

, (3)

where the sopt2~uiuujÞi! come from Eqs. ~2! and g~x, a!

5 a 1 exp~x2y2!. The f ~i!~x! and the values of a andFE 5 @sLevin2~uiuujÞi!ysopt

2~uiuujÞi!#min at the optimaloperating points, given a priori knowledge of all pa-rameters other than the one being estimated, arelisted in Table 1.

Derivation of the CRLB’s based on a stationarytime-domain likelihood model is discussed by Zrnic16

and is attributed by him to Brovko.17 The time-series data @or, strictly, individual autocorrelationfunction ~ACF! estimates# are weighted in a way thatimplicitly corrects for the correlation of spectral com-ponents that is due to windowing of the time series.Therefore the processing algorithm and the equa-tions for the CRLB differ from Levin’s in that theydisplay the dependence on the relative magnitudes ofthe correlation time of the signal and the duration ofthe range gate, that is, on the time–bandwith productF2T 5 f2FsT 5 f2M. On the other hand, stationar-ity would appear to be a critical assumption herebecause the ordering in time of the ACF estimatesdetermines their weighting. The Brovko and Zrnic~BZ! theory is used here to define the general CRLB’sfor stationary systems and is discussed in AppendixA.

Note here that, in my notation, use of the lowercase f indicates a frequency normalized to the recip-rocal of the sampling period, which is conventional inanalysis of discrete time series; use of the upper caseF indicates an unnormalized frequency. The excep-tions to this rule are the function f ~i! in Eq. ~3! and

able 1, and the factor FE introduced above.

3. Optimization Parameters for Bandwidth Estimation

In the BZ theory, the covariance model contains thethree estimated parameters d, f1, and f2 and can be

Parameter, uiuuj i f ~i! ~x! ~a!min ~FE!min =~FE!min

Wideband SNR, duf1, f2 0 1 1.46 4.31 2.08oppler shift, f1ud, f2 1 x2 3.27 4.93 2.22andwidth, f2ud, f1 2 1⁄2~x2 2 1!2 9.58 6.25 2.50

aThe values were obtained from CRLB’s calculated for Levin’sspectral model with a Gaussian signal spectrum, given a prioriknowledge of the parameters not estimated. Shown are values ofthe function f ~i!~x! @for use in Eq. ~3!#, the tuning parameter a 5dy@=~2p! f2#, and the estimator noise figure FE, and =FE @the lastbeing the factor by which the standard deviation of the estimateexceeds the ideal value of Eqs. ~2!# at the optimal operating pointsor heterodyne systems, according to this model. The values for d

and f1 were given previously,1 except that the incorrect value pre-iously given for ~FE!min~ f1! is corrected here. The formula for

f ~2!~x! is that of Zrnic16 and also corrects a minor error in one givenpreviously.15
Fig. 2. Dependence of estimator noise figure FE on the tuning
arameter a for estimates of frequency shift f1, the wideband SNRd, and bandwidth f2, with different parameters known a priori.

he minima in these curves are the optimal operating points.he CRLB’s were computed with the Brovko–Zrnic formulas ~seeppendix A! with M 5 10, f2 5 0.05, and different values of N.urves are as follows: 3, FE~du f2!; v, FE~d!; ■, FE~ f1ud, f2!; h,

FE~ f1! ~no a priori knowledge!; 1, FE~ f2ud!;µ, FE~ f2!. The reduc-tion factor of Eq. ~A14! is shown as a dotted curve with open circles.

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f3

s

os

separated into functions dependent ~i! on the fre-quency shift f1 and ~ii! on d and f2 @Eqs. ~A1! and ~A2!#.It follows that the ~symmetric! Fisher informationmatrix from which the CRLB’s are calculated is of theorder of 3 3 3 but contains only four independenterms, the only off-diagonal terms being those de-cribing the correlation of d and f2 estimates. As

well as the three CRLB’s on the variance of estimatesof each of d, f1, and f2 given that the others are known,which can be compared directly with the optical lim-its given in Eqs. ~2!, there are only two other CRLB’sto be found, those relating to the estimation of d if f2is also unknown and the estimation of f2 if d is un-known @Eqs. ~A13!#. Furthermore, the reductionfactor by which knowledge of f2 decreases the CRLBon d and the reduction factor by which knowledge ofd decreases the CRLB on f2 are identical @see Eq.~A14!#. It suffices to compute only the first threeCRLB’s and a single reduction factor.

To facilitate the comparisons made in this section,only the optical limits given in Eqs. ~2! are used tocalculate FE. For example, both sBZ

2~du f1, f2! 5sBZ

2~du f2! and sBZ2~du f1! are divided by sopt

2~du f1, f2!to form FE; sopt

2~d! and sopt2~ f2! are not given in Eqs.

~2! and are not used. Graphs of the noise figure FEcalculated in this way for the five CRLB’s assumingparticular values of the signal time-series parame-ters are shown in Fig. 2. The optimal operatingpoint for estimation of each unknown occurs at theminimum of the noise figure curve. It is immedi-ately apparent from Fig. 2 that, in the vicinity of theoptimal operating points, knowledge of f2 has a neg-igible effect on the noise figure for estimation of d andice versa. The reduction factor is plotted and is

Fig. 3. ~a! Optimal operation ~where the noise figure FE is minimizas a function of f2M for different M. Values of a are calculated wthan the one that is estimated! are known a priori. Curve markef2!; ■, amin~ f2ud, f1!; ~ii! M 5 64: thin curve, amin~du f1, f2!; E, amin~identical to those shown in Ref. 1. The sloping straight dashed~rightmost line!. That for N 5 1 is the asymptotic limit for powecurve with M 5 N for large f2M.1 Horizontal straight dashed linGaussian spectra in the order amin~du f1, f2! ~lowest!, amin~ f1ud, f2!These lines would be asymptotic limits for large f2M if the fade coequal to M.1 ~b! The CRLB’s used for d and f2 are calculated assuf other parameters makes no difference, see Appendix A!. Howehown in ~a!, i.e., they are calculated with other parameters know

2

found to be unity ~meaning no reduction! at a value ofd only slightly larger than that for optimal estimationof d and only slightly smaller than that for optimalstimation of f2. These observations exemplify a

general result that FE at the optimal operating pointis affected little by prior knowledge of the other pa-rameters, but that the value of the tuning parametera at the optimal operating point is altered more sig-nificantly.

This result is examined in greater detail in theplots, shown in Figs. 3 and 4, of amin and ~FE!min asfunction of the time–bandwidth product f2M. These

iagrams extend the results in Figs. 4, 5, 7, and 8 ofef. 1 to include estimation of signal bandwidth asell as frequency shift and SNR. For the case inhich the other parameters are known, the curves forandwidth estimation are similar to those forrequency-shift estimation. Over a wide region with2M ; 1 ~M . 1!, both ~a!min and ~FE!min are broadly

comparable to the Levin limits given in Table 1.However, the value of a is much larger in this regionor estimation of f2 than for f1 or d @Figs. 3~a! and~b!#. If d or f2 is estimated in the absence of knowl-

edge of the other, comparison of Figs. 3~a! and 3~b!hows that a in the usual operating regime is altered,

being reduced below the Levin value for bandwidthestimation and increased for estimation of SNR.This has the effect of bringing closer together thevalues of ~a!min and therefore the values of signalpower, given f2M, for optimal estimation of all threeunknowns. Examination of Figs. 4~a! and 4~b!shows that ~Fe!min is essentially unaffected. We re-call that the Levin limits shown are based on priorknowledge of the parameters that are not estimated.

s characterized by values of the tuning parameter a 5 dy@=~2p! f1#e BZ formula for the CRLB assuming that the parameters ~othere as follows: ~i! M 5 10: solid curve, amin~du f1, f2!; F, amin~ f1ud,f2!; h, amin~ f2ud, f1!. Curves for amin~du f1, f2! and amin~ f1ud, f2! ares indicate values of N 5 1 ~leftmost line!, N 5 10, and N 5 64estimates at low f2M, and each line is an asymptotic limit to theow the Levin values of a for optimal operation with untruncatedamin~ f2ud, f1! ~highest! ~numerical values are given in Table 1!.as not limited, for discretely sampled data, to a maximum valuethat all other parameters are unknown a priori ~for f1 knowledge

the horizontal lines for the Levin estimator are the same as thoseas to facilitate comparison.

ed! iith thrs arf1ud,line

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4. Simulations

The simulations reported here are intended to beillustrative and not exhaustive. The numerical val-ues covered are only M 5 16 with f2 5 0.02 ~not

ntypical for a pulse-broadened spectrum! and f2 50.1. Attention is concentrated on the latter valuebecause it is more characteristic of an atmospheri-cally broadened return signal, which is likely to be ofinterest if f2 is to be estimated. d and the effectivephotocount were varied. The outputs from approx-imately 4000 simulated range gates were averaged togenerate the results. For each range gate, dataproducts from 40 simulated pulse returns were accu-mulated, and d was varied to change the photocount.

imulated data were constructed with the superpo-ition method.18 Most of the results are expressed in

the figures as the standard deviation and bias of es-timates divided by the appropriate CRLB calculatedwith formulas derived by use of the BZ algorithm ~seeAppendix A! as a function of effective photocount N.

The two maximum-likelihood ~ML! estimatorswere used in the following ways. For Levin’s esti-mator, we maximize the log of a likelihood ratio givenby2

ln L 5 (i51

M ( Xi

f ~N! H1 2 F fi~S!

f ~N! 1 1G21J 2 lnF fi

f ~N!G)5 (

i51

M HXifi~S!

f ~N!fi2 lnF fi

f ~N!GJ , (4)

where the ith frequency component of the spectrum isthe sum of the signal and noise spectral components,fi 5 fi

~S! 1 f~N!, the notation f indicates a spectrumcalculated by the estimated value~s! of the un-known~s!, and Xi is the ith frequency component ofthe periodogram of the data. In practice, the spectraused in Eq. ~4! are calculated from ACF estimates.Account was taken of spectral broadening and distor-tion that was due to range-gate truncation by use of

Fig. 4. Minimum noise figures FE for various estimators. ~a! Thare known. The curve markers are the same as in Fig. 3, with FE

aussian spectra. The curves for ~FE!min~du f1, f2! and ~FE!min~ f1u,dre calculated with the BZ formulas for the CRLB with all other p

~a! and ~b! indicate that prior knowledge has little effect on the mon a shown in the curves of Fig. 3.


the so-called biased estimate of the ACF, that is, bywindowing the sample ACF with a triangular func-tion ~this ACF estimate also appears in the BZ algo-ithm!.16,19,20 The signal spectrum f~S! in Eq. ~4!

was formed from a Gaussian ACF modified in a sim-ilar way. Hence the signal spectrum used in theprocessing was not Gaussian, and the estimation al-gorithm differs in this way from that used to calculatethe CRLB in Section 2 and Table 1. There, an un-modified Gaussian spectrum was used so as to beconsistent with the neglect of range-gate windowing.As indicated in Section 2, Levin estimators are ex-pected to be suboptimal at high signal levels.

The BZ estimator maximizes the log of the likeli-hood ratio given by2

ln L 5 ~1yPN!Z1D~I 2 G!D*Z 2 ln~det Q!, (5)

here Z is the complex data vector, Z1 is its complextranspose, D 5 diag$1, exp~2j2pf !, exp~2j4pf !, . . . ,exp@2j~M21!2pf%, D* is its complex conjugate, andG 5 Q21. PN and Q are defined in Appendix A. Foroth estimators, these equations are used iteratively.nitial values of the SNR d and signal bandwidth f2,

either of which may be known or unknown, are usedto form ln L as a function of the frequency shift f1.ln L generally contains a number of maxima, and theestimate of f1 that corresponds to the selected valuesof d and f2 is simply the frequency at which the high-est of these maxima occurs. d andyor f2, if un-known, are then adjusted and the process repeateduntil the largest value of ln L is obtained.

The numerical method needed for the reiterativeoptimization is therefore at most two dimensional. Iused the Numerical Recipes21 routine “brent” whenthere was only one unknown other than f1 and theirsimplex routine “powell” when both d and f2 were toe estimated. In the iteration process, initial valuesor c were obtained with estimates by use of thequarer algorithm, in which the square moduli of

formula is used for the CRLB assuming that all other parametersacing a. Horizontal lines show the Levin values for untruncatedare identical to those shown in Ref. 1. ~b! Minimum noise figureseters unknown a priori. Levin lines are the same as in Fig. 4~a!.um noise figures. This contrasts with the more significant effect

e BZrepl, f2!

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a

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Table 2. Optimal Parameters and Precision for Pulse-Broadened Return from Lidar with Short Time–Bandwidth Producta

toa

complex time-series data are averaged over the rangegate.1 In practice, a minimum initial value for f2 isgiven naturally by the transmitted pulse bandwidth~here I used f2 5 0.02, which usually corresponds toa Doppler shift of approximately 1 mys!, and the up-per value was chosen to be 0.2. It was found thatmore reliable estimates were obtained by use of log~d!and log ~ f2! in the iteration rather than d and f2themselves. This procedure also ruled out negativeestimates of these unknowns.

In addition to estimates formed from the Levin andBZ algorithms, the squarer ~or zeroth-lag covariance!algorithm was used for estimates of d and first-lagcovariance algorithms22 for estimating f1 and f2.The first-lag algorithm for f1 is the well-known pulsepair algorithm. The first-lag algorithm for f2 can bederived most directly from the equation for the firstlag of the ACF @see Eq. ~A1! in Appendix A!, um~1!u 5Ps exp@2~2pf2!2y2# ' Ps@1 2 ~2pf2!2y2# which leads ton estimate20

f2 5@1 2 um~1!uyPS#1y2

Î2 p, (6)

here Ps 5 * pS~d!df 5 PNd. When d is assumed toe unknown, its estimate formed from the squarerlgorithm is substituted.Discussion of the results from some of these simu-

ations is given in the captions to Figs. 5 and 6. Ineneral terms, the BZ estimators lie close to the BZRLB as would be expected. There is little differ-nce between results obtained with the Levin and BZstimators. Of the single-lag estimators, the esti-ator of spectral width given in Eq. ~6! is less satis-

actory than the better known squarer and pulse pairstimators, so that further research, perhaps on theag selection used in this algorithm, or construction ofsimple alternative is desirable. ML estimate bias

oes not appear to be statistically significant exceptor estimates of f2 ~especially, curiously, with d known

rather than estimated! when the value is low ~0.02!and the magnitude of the bias is less than approxi-mately 3%. Bias of estimates of d in all cases ~in-luding those not shown! was less than 0.5%.

5. Numerical Examples

The practical implications of these results can be un-derstood by use of spreadsheet calculations. For a

ui amin SNR ~dB! = ~FE

duf1, f2 2.5 29.3 2.1d 3.4 28.0 2.2f1, f1ud, f2 8.2 24.2 3.3f2 17.8 20.8 5.3f2ud, f1 23.9 0.5 5.1

aOptimal operating parameters for estimation of the unknown gipulse duration TFWHM 5 1 ms ~200 ns!, range-gate duration T 5 1 mo 625 mys!, f2 5 0.019, f2M 5 0.19. Also given are the wideband Sn average 100 effective photocounts. The value of the CRLB is qnd as the standard deviation of the equivalent velocity estimates

2

lidar, the minimum bandwidth of the return signal isthat of the pulse and is related, for a transform-limited pulse of Gaussian profile, to the full width athalf maximum of the pulse length, TFWHM, by F2 .1y~1.7pTFWHM!. With particular choices of receiverbandwidth FS and range gate duration T, we can thencalculate the normalized frequency used here, f2 5

2yFS, the number of ~complex! time-series dataoints, M 5 FST, and the time–bandwidth product,

F2T 5 f2M. Data like those of Figs. 3 and 4 can thenbe used to find, respectively, the values of amin and~FE!min. From these, we calculate, respectively, thewideband SNR for optimal operation, d 5=~2p! f2amin and, with Eqs. ~2!, the corresponding

RLB, sBZ 5 =~FE!mins0.Most existing lidars using heterodyne receivers op-

erate at wavelengths of 2 or 10 mm. Table 2 showsresults of calculations for a 10-mm lidar operating

ith T 5 TFWHM 5 1 mm and FS 5 10 MHz ~corre-ponding to Doppler shifts of 650 mys!, yielding

f2M 5 0.187. Because the bandwidths all scale in-ersely as wavelength, the results apply also to a 2m lidar with T 5 TFWHM 5 200 ns, FS 5 50 MHz

650 mys!, and f2M 5 0.187. These operating con-itions are aimed at minimizing range resolution forhat might be regarded as typical pulse durationsnd correspond to those most often used in the threeeterodyne lidars in our laboratory. Results arehown when all other parameters are known andhen they are unknown, which makes no difference

or the estimation of f1.Table 2 indicates that a low SNR is required. This

follows directly from the low values of aopt shown inFig. 3 and implies that the optimal average photo-count per range gate, for a single pulse, is also small,varying from 1.2 for SNR estimates to approximately11 for bandwidth estimates. Precise optimization isimpossible to achieve for all ranges in a working lidaras it depends on a and the return SNR. As com-mented previously,1 the tuning curves are quitebroad ~Fig. 2!, so that a system approximately opti-

ized for Doppler-shift measurements at a certainange could also be expected to have performanceomparable with the optimal for return power andandwidth estimation over a significant distance.Whereas the condition for optimal estimation de-

ends on a and on the power level of the signal, the

n sBZ~ui!yui ~%! sBZ~ui! ~mys!

85 21 —64 22 —27 — 0.3112 37 0.359 36 0.34

s ui, by use of heterodyne lidar with wavelength 10 mm ~or 2 mm!,0 ns!, receiver bandwidth FS 5 10 MHz ~50 MHz! ~both equivalentobtained from amin! and the number of pulses n needed to generateas the relative error for estimates of SNR ~d! and bandwidth ~ f2!eters per second for Doppler shift ~ f1! and bandwidth.

!min

ven as ~20NR ~uotedin m


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mean energy of the return, or the effective photo-count, determines the precision ~Eq. 2! and ~largely!he detection probability.2 In Table 2, the precision

is calculated assuming a sufficient number of pulsesto acquire 100 photocounts, this being sufficient toensure detection with a negligible number of biasednoise-related estimates @see also Figs. 5~a!–5~c!#. Al-though the precision available with this number ofphotocounts might be satisfactory for Doppler-shiftestimates, that for power and bandwidth estimationwould probably need improving by use of morepulses.

The Levin limits shown in Table 1 can be regardedas giving the operating parameters for a lidar opti-mized for estimate precision by use of a long rangegate whereas use of TFWHM 5 T, as in Table 2, im-

roves range resolution at the expense of precision.n a comparison of the results, the Levin values of

Fig. 5. ~a!–~c! Simulated data performance of BZ and Levin estimthe CRLB’s calculated with the BZ formula based on the same assurelative standard deviations and relative biases found by dividingCRLB sBZ~d! on the standard deviation calculated with the formu

for M 5 16 and f2 5 0.1. ~b! and ~c! are similar with f1 and f2, rF, s~BZ!ysBZ; h, B~Lev!ysBZ; E, s~Lev!ysBZ. These ML estimatorsdeviation is approximately the same as, and any bias is much lessof the BZ and Levin estimators in these cases. The data are reoptimization procedure became unstable. This is not unduly limiestimates obtained with spectral peak picking are expected to becto be a useful guide to performance unless estimates are thresholdtriangles show performance of the estimators based on a single laupward-pointing triangle showing sysBZ!. The estimators are the6!. The last is clearly unreliable but no effort has been made to

matters. ~d! Identical to ~c! except that the estimators of f2 andxample, the filled squares indicate B~BZ!~ f2ud!ysBZ~ f2ud! rather th

results, except that s~Lev!~ f2ud! becomes rather larger than s~BZ!~ f2

uboptimal nature of the Levin algorithm in this limit.


min, and therefore of SNR ~given f2!, are lower thanhose in Table 2 by a significant margin, but theifference in =~FE!min is less marked except for the

estimation of bandwidth. Note that the conditionsfor optimal operation of a system, which depends onCRLB values, should be distinguished from the per-formance of the corresponding estimator. Thus, theoperating conditions for the short range gate ~Table2! are defined by use of the CRLB obtained from theBZ expression. Although these CRLB’s differ fromthose obtained with Levin theory ~Table 1!, the per-ormance of the Levin estimation algorithm for thehort gate may be quite satisfactory at the low valuesf SNR that are implied ~Fig. 5!.Spectral broadening of the return arising from the

tmosphere breaks the link between TFWHM and f2.Broadening that results from velocity turbulence hasbeen considered recently by Frehlich and Cornman,23

given no prior knowledge of other parameters are compared withon and shown as a function of the effective photocount N. ~a! The~d! and s~Lev!~d! and B~BZ!~d! and B~Lev!~d! for estimates of d by thef Appendix A, are shown as a function of the effective photocounttively, replacing d. Curve markers are as follows: ■, B~BZ!ysBZ;ar to work as well as might be expected, that is, their standardn, the CRLB. There is little difference between the performanceted to N . 70 because at lower signal levels our two-parameterbecause, at slightly lower photocounts, both the BZ and the Leviniased by noise-related peaks. The CRLB’s used here then ceasea suitable level of the log likelihood ratio @Eqs. ~4! and ~5!#. Thehe ACF ~the downward-pointing triangle showing BysBZ and thearer for d, the pulse pair for f1, and the first lag f2 estimator of Eq.rmine whether choice of a lag other than the first would improveRLB’s used are based on the assumption that d is known. For

~BZ!~ f2!ysBZ~ f2!, etc. There appears to be little difference in thesBZ~ f2ud! at high signal levels, which is not unexpected given the

atorsmptis~BZ!

las oespecappetha

strictingome bed atg in t

squdetethe Can Bud! '

ua~

f

5

pgd@v

i

a

msasaf

plN

Table 3. Optimal Parameters and Precision for Broadband Return from

who predict bandwidth increase by factors of at mostapproximately 2 in the cases cited. Some artificialsources of turbulence may increase broadening byeven larger factors. To demonstrate the effectsclearly, an example is shown in Table 3 in which thebandwidth is increased by a factor of 5. In additionto the reduction in precision that is expected of thefrequency estimates, the effect is to increase the SNRneeded. As would be expected for a signal of broaderbandwidth, optimal operating parameters are closerto the Levin limit of Table 1.

6. Conclusions

Optical estimates of signal bandwidth are character-ized by a lower CRLB than those of Doppler shift@Eqs. ~2!#, and optimally tuned heterodyne systems

sing incoherent backscatter appear to be capable ofpproaching these ideal values as closely @e.g., 2 ,FE!min , 3 for f2M ; 1, TFWHM , T, Fig. 4# as they

are for estimating shift and return signal power, theconditions for which were found in our earlier paper.Optimal tuning requires more signal for bandwidthestimation ~approximately ten effective photocountsper fade! than for return power @1 , ~a!min , 2# orrequency shift @3 , ~a!min , 4# @fig. 3~a!#. There is

therefore no single optimal tuning condition for esti-mating all three unknowns simultaneously, althoughnearly optimal tuning for all three could be realized.Figures 3 and 4 quantify the trade-off between theoptimal precision of a heterodyne measurement andthe length of the range gate ~hence the range weight-ing! for any given return signal bandwidth ~Section!.As a result of the signal model used ~Section 3! the

CRLB of Doppler-shift estimates is independent of ariori knowledge of the other two parameters, but ineneral the CRLB of return power estimates mayepend on knowledge of bandwidth and vice versaAppendix A, Eqs. ~A13!#. What is perhaps less ob-ious is the result found empirically ~Fig. 2! that the

CRLB on estimates of each of these three parametersis independent of a priori knowledge of the othersclose to the conditions required for optimal tuning.In general, the CRLB’s both of bandwidth estimateswith unknown return power and of power estimateswith unknown bandwidth are degraded by the samefactor @Appendix A, Eq. ~A14!#. The tuning condi-tion for estimating return power estimates is movedsomewhat closer to that for frequency shift estima-tion if the bandwidth is known, as is the tuning con-

2

dition for estimating bandwidth if the return power isknown @Fig. 3~b!#.

Simulations indicate that efficient estimation ~thats, achievement of the CRLB on estimate precision! is

possible without significant bias for all estimates us-ing the ML estimators except at low signal levels,when the occurrence of noise-related estimates hasbeen well documented. Results obtained usingLevin’s algorithm, if allowance is made for modifica-tion of the expected signal spectrum by range-gatewindowing, are similar to those using the BZ algo-rithm at low and intermediate signal levels, but aresomewhat degraded at high signal levels, as ex-pected, because of signal spectral correlation ~Figs. 5nd 6!.

Appendix A

I outline the derivation of formulas, based on a time-domain model, that relate to the CRLB on the vari-ance of estimates of unknowns commonly determinedusing incoherent backscatter heterodyne lidar sys-tems. These formulas are assumed in Sections 1–5.

1. Signal Model

Data obtained from a lidar return are assumed to besampled discretely and complex demodulated. TheACF at lag k of the resulting time series is modeled as

m~k! 5 PN$dexp@2~2pf2 k!2y2#exp~ j2pf1 k! 1 1%, (A1)

where PN is the mean noise power ~assumed to beknown a priori!; and the parameters that may beknown or unknown, and therefore to be estimated,are the wideband SNR d, the frequency shift f1, andthe signal bandwidth f2. The noise statistics are as-sumed to be Gaussian. The assumption of a ~sym-

etrical! Gaussian envelope for the ACF ~and for theignal spectrum! has become conventional for muchnalysis in Doppler radar and lidar but it is not es-ential. The essential features of the model are thessumption of stationarity and the representation of

1 as a frequency shift. ~ f1 is often referred to as themean frequency or the spectral first moment, pre-sumably because the frequency mean is regarded asthe object of measurement.! In the notation usedhere, both f1 and f2 are normalized to the reciprocal ofthe sampling period, which is conventional in time-series analysis.

Details of the resulting covariance model for deriv-ing ML estimators and the CRLB’s on estimate vari-ance were discussed by Zrnic,16 who attributes themethod to Brovko,17 and are also to be found in Chor-noboy19 and Frehlich.20 It is assumed that the timeseries has been passed through a frequency filterwith a top hat or rectangular passband of bandwidthFS, which is related to the sampling period by theNyquist relation. The data sample considered in theanalysis is composed of blocks containing M ~com-

lex! data points drawn from the time series. Foridar, these blocks correspond to range-gated data.ote that M 5 FST, where T is the duration of the

data sample, and f1 5 F1yFS, f2 5 F2yFS, where the

Lidara

ui amin SNR ~dB! =~FE!min n sBZ~ui!yui ~%! sBZ~ui! ~mys!

d 1.9 23.4 2.2 22 22 —f1 3.5 20.9 2.4 13 — 1.11f2 6.4 21.8 2.9 7 20 0.95

aAs Table 2, except that return bandwidth is five times greaterthan the pulse ~ f2 5 0.094, f2M 5 0.94!.


p

v

Bi

f

6

uppercase F’s refer to unnormalized frequencies.Functions appearing in the theory include the covari-ance matrix

R~d, f1, f2! 5 $rkl~d, f1, f2!%

5 $qkl~d, f2!exp@ j2pf1~k 2 1!#% (A2)

@in this notation A and $aij% are used alternatively torepresent the matrix A that is made up of the ele-ments aij#, and

Q 5 dG 1 I, (A3)

where I is the unit ~identity! matrix and G is the realcentrosymmetric Toeplitz matrix

G 5 $gkl% 5 $exp@ 2 ~2p~k 2 l ! f2!2y2#%. (A4)

2. Terms in the Information Matrix

The CRLB’s are found in terms of the components ofa Fisher information matrix

H 5 $hmn% 5 HtrFR21 ]R]um

R21 ]R]un

JD, (A5)

Fig. 6. Statistical significance of the simulation results like thosearious estimates compared with the standard deviation of their m

that were run. @Because s~BZ! ' sBZ ~Fig. 5!, ssdm ' sBZy√~4000!are shown because those for Levin are similar. Curve markers ar

~BZ!~ f2ud!; ƒ, B~BZ!~d!; ‚, B~BZ!L~,104uf2!. It would be expected thatn ~a!, plotted for f1 5 0.1, M 5 16 @as in Fig. ~5!#, this is the case, e

estimates occur. However, it is not the case if the bandwidth isbandwidth itself, and in some of the estimates of d with f2 unknow2 in these cases is less than 0.0006 ~approximately 3% of f2!, wh

optimization algorithm used or rounding errors, rather than anytestimates of d, given no prior knowledge for f2 5 0.1 and M 5 16.Levin’s, and the open triangles to the squarer. Except in the smalwith a recent result of Frehlich,25 who reports bias of up to 10% f


where u 5 ~u0, u1, u2! 5 ~d, f1, f2! represents theotential unknowns and the notation tr@B# indicates

the trace of the matrix B. Frehlich20 pointed outthat a derivation of this formula is given by Porat andFriedlander.24 Writing A~m! 5 $akl

~m!% 5 R21~]Ry]um!, we obtain

hmn 5 tr@A~m!A~n!# 5 (k51

M

(l51

M

akl~m!alk

~n! . (A6)

Performing the differentiations, we obtain

akl~0! 5 ~GG!kl 5 (

m51

M

gkmgml ,

akl~1! 5 j2p(

m51

M

gkm~m 2 l !qml,

akl~2! 5 2 ~2p!2f2d (

m51

M

gkm~m 2 l !2gml , (A7)

where G 5 Q21.Equations ~A6! and ~A7! are adequate for compu-

tation of all the terms in H, but some simplifications

n in Fig. 5 is evaluated. ~a! and ~b! Absolute value of the bias of, ssdm, calculated over the approximately 4000 independent trialsthis test is rather stringent#. Only results for the BZ estimator

follows: ▫, B~BZ!~ f1!; n, B~BZ!~ f1u f2!; h, B~BZ!~ f1ud!; 1, B~BZ!~ f2!; 3,absolute value of almost all bias values should lie below 3ssdm, and

for low signal levels where bias is expected because noise-relatedow, f1 5 0.02, M 5 16 @Fig. 6~b!#, when the bias in estimates ofsomewhat above 3ssdm. The magnitude of the maximum bias inccurs when N ; 100, and is likely to be the consequence of themore fundamental. ~c! The magnitude of the fractional bias ine the filled squares relate to the BZ estimator, the open circles toal regime, the bias is of the order of 0.1%. This appears to conflicte BZ estimator.

showean

, ande asthe

xceptnarr

n, lieich ohingHer

l signor th

a

w

oh

fC

2 2 2
are possible if advantage is taken of the symmetryproperties of the matrices. For example, by noting
$~m 2 l !qml% 5 QL 2 LQ, (A8)

where L 5 diag ~1, 2, . . . , M!, the expression for A~1!

becomes A~1! 5 j2pG~QL 2 LQ!. Using G~QL 2LQ!G 5 LG 2 GL yields

h11 5 2 4p2 tr@~LG 2 GL!~QL 2 LQ!#

5 24p2 tr@$~l 2 k!glk~k 2 l !qkl%#

5 24p2 (k51

M

(l51

M

~k 2 l !2qklglk (A9)

s reported by Frehlich.20 Itis also straightforwardto show that the components h01 and h21 are zero, aspreviously used by Zrnic within the ML analysisbased on Levin’s spectral model. Making use of theidentity tr@BC# 5 tr@CB# and Eq. ~A8!, we obtain

h01 5 j2p tr@G~QL 2 LQ!GG#

5 j~2pyd!tr@GL 2 LG# 5 0, (A10)

hereas

h12 5 2j~2p!3f2d tr@G~QL 2 LQ!G~~GL 2 LG!L

2 L~GL 2 LG!!#

5 2j3~2p!3f2 tr@LGLLQ 2 QLLGL#.(A11)

From symmetry, LGL 5 ~LGL!T, QL 5 ~LQ!T andtherefore QL~LGL! 5 ~LQ!T~LGL!T 5 ~LGLLQ!T.Because tr@BT# 5 tr@B#, it follows that h12 5 0. Thenly nonzero terms in the Fisher matrix are therefore00, h11, h22, and h02 5 h20.16

3. Cramer–Rao Lower Bounds

The CRLB on the variance of each parameter is givenby the corresonding diagonal term in the inverse ofthe information matrix constructed in terms of all theunknown parameters. If all three parameters con-sidered here are unknown a priori, the inverse isgiven by

H21 5 Fh00 0 h02

0 h11 0h20 0 h22

G21

51

h11~h00h22 2 h022!

3 Fh11h22 0 h11h02

0 h00h22 2 h022 0

h11h02 0 h00h11

G . (A12)

In this case, the CRLB on the variance of an estimateof d obtained from the BZ model is sBZ

2~d! 5 1y@h00 2~h02

2yh22!#. Whereas if f1 and f2 are known, the in-ormation matrix becomes the scalar h00; we write theRLB as sBZ

2~du f1, f2! 5 1yh00. Continuing thesecalculations for the different permutations of knownand unknown variables, we find

sBZ2~d! 5 sBZ

2~du f1! 5 1y@h00 2 ~h022yh22!#,

sBZ2~du f2! 5 sBZ

2~du f1, f2! 5 1yh00,

2

sBZ ~ f1! 5 sBZ ~ f1ud! 5 sBZ ~ f1u f2!

5 sBZ2~ f1ud, f2! 5 1yh11,

sBZ2~ f2! 5 sBZ

2~ f2u f1! 5 1y@h22 2 ~h022yh00!#,

sBZ2~ f2ud! 5 sBZ

2~ f2ud, f1! 5 1yh22. (A13)

It can be seen that the CRLB on the variance of thefrequency shift f1 does not depend on prior knowledgeof d or f2. Likewise, knowledge of f1 does not affectthe CRLB of d or f2, although knowledge of f2 doesreduce the CRLB on the variance of d and vice versa.The reduction factor is the same in each case and isgiven by

sBZ2~ f2ud!

sBZ2~ f2!

5sBZ

2~du f2!

sBZ2~d!

5 1 2h02

2

h00h22. (A14)

As illustrated in Fig. 2, the reduction factor is unity~and h02 5 h20 5 0! when a is close to its values at theoptimal operating points. According to Eqs. ~A13!,the information matrix becomes diagonal under thiscondition. The CRLB on the variance of each of theestimated parameters is then independent of knowl-edge of the others.

I am grateful to an anonymous referee for the sug-gestion to include the material of Section 5.

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1

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6

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