Adaptive Parametric Algorithms for Processing

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2678 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 37, NO. 6, NOVEMBER 1999

Adaptive Parametric Algorithms for ProcessingCoherent Doppler-Lidar Signal

Jean-Luc Zarader, Alain Dabas, Pierre H. Flamant, Bruno Gas, and Olivier Adam

Abstract In this paper, we study the autoregressive andmoving average (ARMA) lter for lidar signal processing. Aftera short presentation of the atmospheric laser Doppler instrumentproject (ALADIN), we introduce the objective of this paper,which is to extract the Doppler frequency and to retrieve thespectral width of a noised lidar signal. A general presentationof ARMA lters and parametric adaptive algorithms (PAAs) isprovided. Then we present results about the choice of the model,the Doppler frequency estimate, and the spectral width estimate.Finally, we study the possible estimate of SNR, which is biasedby the rst estimates (Doppler frequency and spectral width).

Index Terms Adaptive signal processing, laser radar.

I. INTRODUCTION

C OHERENT Doppler lidars have been developed for morethan 15 years now [25] and have proved very powerfuland useful for remotely measuring winds in the atmosphere.The measurement consists of transmitting a laser pulse into theatmosphere. Along its propagation path, the laser radiation isscattered by the aerosol particles drifting with the wind. Part of the energy is scattered back toward the instrument, where it iscaptured by a telescope and analyzed. For a Doppler system,the analysis aims at a range-resolved estimate of the signalfrequency. According to Dopplers effect, it is equal to thefrequency of the transmitted laser pulse, which is monitoredin real time, plus a shift proportional to the wind velocity alongthe line-of-sight. At the present time, various systems, eitherground-based or airborne [8], [9], [23], [36], [41], have beendeveloped and used operationally in meteorology to documentparticular features of atmospheric dynamics [1][6], [11], [13],[21], [36]. The global (i.e., at the scale of the Earth) andcontinuous survey of atmospheric wind eld, though verypromising for weather prediction, still is not available. Itrequires spaceborne instruments that still are not in existencebecause they are at the limit of what the current technologycan provide. Nevertheless, development programs such as theSpace Readiness Coherent Lidar Experiment (SPARCLE) in

the United States (see http://wwwghcc.msfc.nasa.gov/sparcle/)

Manuscript received August 7, 1997; revised November, 30, 1998. Thiswork was supported by a grant from the European Space Agency (ESA).

J.-L. Zarader and B. Gas are with the Laboratoire des Instruments etSystemes, Universit e Paris VI, Paris, France.

A. Dabas is with the Centre National de Recherche M eteorologique, M eteo-France, Toulouse, France.

P. Flamant is with the Laboratoire de M eteorologie Dynamique, Palaiseau,France.

O. Adam is with the Laboratoire d Etude et de Recherche en Informatique,Signaux et Syst emes, Universit e Paris XII, Paris, France.

Publisher Item Identier S 0196-2892(99)06267-1.

and the atmospheric laser Doppler instrument (ALADIN) inEurope [42].

The processing of signals in the digital domain is a difcultpoint in the design of an instrument. The main reason is that thesignals contain a great deal of detection noise (the SNR is oftenbelow 0 dB) due to the limited powers delivered by existinglasers, the low backscatter coefcient of the atmosphere, and,for spaceborne applications in particular, the long distancefrom the targets. Additionally, the useful part of the signal, theatmospheric echo, is subject to random phase and amplitudeuctuations known as speckles. These occasionally lead to

local fadeouts.Hoping that better processing would alleviate the demanding

specications for high-energy lasers and large-area telescopes,many studies have been devoted to lidar signal processing inthe past. First, the precise nature of the signals has been clar-ied [37], resulting in the provision of accurate and realisticsignal simulation models [40]. Second, the best frequency esti-mators for Doppler lidar have been considered: Pulse-Pair [31],Poly-Pulse-Pair [29], parametric or nonparametric adaptivelters [27], [33], [45], and the so-called maximum likelihoodestimator [16]. Their performances for the range-resolvedretrieval of the signal frequency have been determined, bothon simulated [16], [18], [19] and real [15], [18] signals.

Though most of the studies cited above have focused on theretrieval of the mean frequency (that is, the rst order momentof the signal-power spectrum), the signal power and the widthof its spectrum (zeroth and second order moments of thepower spectrum, respectively) are also valuable parameters.The former measures the light intensity backscattered by theatmosphere, and thus provides information on the structureof the atmosphere (for instance, the presence and altitudeof subvisible clouds). Also, it gives the SNR, which is agood piece of information for the assessment of the qualityof Doppler measurements. Regarding the spectrum width, itcontains information on the level of turbulence and windshear,

since both are responsible for a broadening of the returnedspectrum.The present paper investigates the possible application of

adaptive autoregressive and moving average (ARMA) ltersto coherent lidar for the joint estimate of the signal power,mean frequency, and spectrum width. To our knowledge, suchan investigation has never been conducted before. Indeed,parametric spectral analysis already has been conducted onDoppler radars and lidars, which both deliver similar signals,but Keeler and Lee [26] and Mahapatra and Zrnic [29] dwellon the maximum entropy estimator, which is the adoption of

01962892/99$10.00 1999 IEEE


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ZARADER et al. : ADAPTIVE PARAMETRIC ALGORITHMS 2679

an autoregressive (AR) lter [10]. Filters with MA parts alsohave been used, but the objective is then the determinationof the signal spectrum width. At last, the application of notchlters [33] to lidar signals is investigated by Zarader et al. [45].However, the purpose is then the estimation of the signal meanfrequency only.

This paper will be divided into six sections. In the secondsection, the signal model retained for the whole study ispresented with a discussion of its ability to account for actualDoppler-lidar systems. Also, the principles of adaptive ARMAspectral analysis are reviewed with a particular emphasis onthe numerical methods used for adapting the lter to thesignal. The following sections (III, IV, and V) present theperformances reached by adaptive ARMA for the estimate of the Doppler frequency, spectrum width, and signal power. Thelast section concludes the study.

II. BACKGROUND

A. Doppler-Lidar Signals

Basically, two types of coherent Doppler lidars exist, de-pending on the laser technology. Though both deliver pulseswith different characteristics (see below), regarding in partic-ular the wavelength (around 2 m for solid-state and 10 mfor CO lasers), the nature of the signals generated by theinstrument is similar in both cases. After demodulation, it canbe considered as a complex Gaussian process polluted by astatistically independent noise. The noise is generally assumedto be white, which in reality is assured by a proper matchingof the anti-aliasing analog band-pass lter to the samplingfrequency of the analog to digital (A/D) converter. The atmo-spheric return is never perfectly stationary, since it contains

at least a range dependant power decrease. However, at longdistances, and provided the optical and dynamic propertiesof the atmosphere can be considered spatially homogeneousalong the line-of-sight (which is often the case for clear-airmeasurements), the stationarity is nearly met, at least to thesecond order. Therefore, stationarity generally is considereda basic assumption of lidar signal processing studies. Then,the statistical properties of the signal samples are entirelycharacterized by the autocorrelation function .

At a given time, the lidar signal results from the addition of a great number of laser pulses reected by the backscatteringtargets with a time delay (equal to where is the distanceinstrument target) and a frequency Doppler shift [40]. Since thedistance from the target varies over many half-wavelengths, itfollows that the reected pulses add randomly at detectionlevel. They sometimes add constructively, sometimes destruc-tively. It results in random phase and amplitude uctuationsknown as speckles [14]. At a given time, the probability den-sity of the phase is uniform over , the amplitude followsa Rayleigh distribution, and the power follows a negativeexponential distribution [12], [22]. Then, speckle uctuationsevolve in time due to the renewal of the targets inside theprobed volume (caused by the propagation of the laser pulse)and the relative movements of the targets remaining inside.The result is that the autocorrelation function is basically

equal to the autocorrelation of the laser pulse plus a decorre-lating impact from all the wind velocity uctuations inside theilluminated volume [37]. Correspondingly, the signal spectralpower density (the Fourier transform of the autocorrelation)is basically equal to the power spectrum of the laser pulse(that is, the square magnitude of its Fourier transform) plussome broadening by intrapulse velocity uctuations. In manystudies, the autocorrelation function is given a Gaussianshape

(1)

where is the power of the atmospheric echo, and sets thecorrelation time. Since the spectral power density is theFourier transform of

(2)

the parameter is the signal spectral width. In reality,Gaussian autocorrelations are met with solid-state lidars, be-cause delivered pulses have a Gaussian power prole andno signicant chirp [15], [17]. The autocorrelation functionabove is a good model in this case. With CO lidars however,the pulse power prole combines a short spike and long atail. Furthermore, it contains a signicant frequency chirp[44]. Then, the autocorrelation cannot be Gaussian. However,should the pulse energy within the spike be limited as well asthe frequency chirp, the autocorrelation should not be too farfrom Gaussian. Therefore, (1) and (2) will be systematicallyconsidered in the following. As a consequence, the syntheticsignals we shall use in the next section to test the performancesof the proposed estimators will be generated with Zrnicssimulator [46].

In real applications, the spectrum width is related pri-marily to the laser pulse [17]. Solid-state lasers deliver pulseswith no chirp and a duration varying from s to

s [19]. The spectrum width thus ranges fromkHz to MHz. For 10 m systems, the duration is longer(1.5 s to 3 s), but the large frequency chirps neverthelessbroaden the spectra from 250 to 500 kHz. For both types of system, the duration of the processing windows may varyfrom s to s or more, depending on the requiredrange resolution ( m and m, respectively). So thenondimensional product , which is proportional tothe number of independent realizations of speckle uctuationswithin one processing window, might vary from to

or more (that is, about two decades).

B. Parametric Estimation

Parametric models generated in signal processing are usedfor process identication, prediction of signals, or even (for thepresent subject matter) spectral analysis [28]. The interestingaspect of these models rests in their high-frequency resolution.

We can distinguish two types of lters. First of all, AR (orall pole lters) adapted to the analysis of signals with one orseveral peaks in their spectrum. Therefore, they are interestingfor the processing of lidar signals. Yet, the performancesof such lters depend on the number of samples and on


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noise level. To reduce the effect of disturbances, one tendsto increase the lter order (that is, the number of coefcientsor parameters). This technique amounts to using an ARMAmodel.

An ARMA lter can be dened by the followingrelationship:

(1)

where represents white noise and represents the lidarsignal.

Spectral density can be determined from the transfer func-tion of the lter. It can be expressed as follows by a transform[34]:

(2)

Assuming that , we obtain the spectral density of the signal obtained by ltering input with variance . We

can now write

(3)

Contrary to notch lters (ANFs), ARMA lters with freecoefcients enable one to estimate Doppler frequency andspectral width, and to obtain some information concerning theSNR.

The rst problem met upon carrying an ARMA modeling, isthe choice of the lter order. There exist different tests, calledperformance-complexity criteria, that allow one to obtain amodel with a reduced number of parameters, satisfying signalprocessing. For instance, we can mention the Aka ke criterion(Final Prediction Error) [28], dened by

(4)

where and are, respectively, the number of samplesand the number of parameters of the model. is the a posteriori prediction error. Hereafter, we use this criterionin order to determine the coefcients of the lter. One usesparametric adaptive algorithms (PAAs) [24], [30], [43]. Thecoefcients are calculated recursively. Several methods havebeen developed so that parameters can converge toward theiroptimum values. Nonetheless, the principle remains the same:

minimizing a cost function (or criterion) that depends onthe error made between the lidar signal value and the valuepredicted by the lter. This error is called the prediction error.

In part II-C, we shall detail algorithms based on the exactleast square criterion, such as:

1) recursive least squares (RLS);2) recursive maximum likelihood (RML);3) output error (OE).

We shall then compare the results obtained in the estimate of the Doppler frequency and the spectral width of lidar signalsin Sections III and IV. Finally, in Section V, we shall presentthe results obtained for the evaluation of the SNR.

C. Algorithms

The cost function of these algorithms corresponds tothe average power of the a posteriori prediction error. At time

, it can be expressed as

(5)

with

(6)

where is the a posteriori prediction, and are theinput (or observation) vector and the vector of the estimatedparameters, respectively

(7)

and

(8)The criterion can be written as follows for a block of data:

(9)

This global criterion, calculated from initial time, ensures auniform progression of coefcients. The following methodsgive an estimate of the coefcients of the model, minimizingthis criterion.

1) Algorithm of Recursive Least Squares (RLS): The RLSalgorithm is used to update the lter parameters

(10)is the gain matrix. It varies with the power of the input

signal. Its initialization is important and results in a more orless fast convergence of coefcients toward their optimumvalue.

The problem with this algorithm is that minimization of thecriterion is made on all past times (from initial time 1 to time

). This therefore assumes that the set of coefcients doesnot change with time, or from a physical point of view, thatthe signal is stationary (constant Doppler frequency during theshot). To take into account nonstationary aspects of the lidarsignals, several modications can be introduced. They concern

either the calculation of the errorcost function, or the contentof the input vector.The criterion dened previously takes all data into account

with equal weight. To study the nonstationary process, it isnecessary to introduce a weight favoring recent evolutions of the signal to compare it to past behavior. The new errorcostfunction shows a forgetting factor

(11)

where is within zero and one. The forgetting factor alsocan be found in the evolution of the gain matrix .


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Several proposals were made concerning the choice of theforgetting factor.

a) : We obtain the recursive least squares algorithmagain.

b) with within zero and one: In practice,it is chosen within the interval [0.8; 1]. Previous dataare forgotten at an exponentially growing speed. The

weight is at its maximum for the last error calculated.As a matter of fact, the parameter tunes the timeresolution. When for instance, the estimateis performed on the whole signal from the beginningto step . On the contrary, when , theweight given to component is less than

, and is thus negligible. Through ,the time window corresponding to an estimate is xed.When the weighting function is a negative exponential,there exists no straightforward denition for the timeresolution. For instance, if we set the time resolution of an estimate as the time delay , when the weight

becomes lower than and

, the time resolution is equal to ten timesamples . When and , it is equalto 20 and 50 time samples, respectively.

c) with and within zero and one, chosen close to one: The forgetting factor varies.This forgetting is important during the rst iterations(with convergence), and then it decreases through pro-cessing. The forgetting factor asymptotically tends to oneand thus prevents the adaptation gain from decreasingtoo quickly.

One also can introduce a term that enables oneto maintain a constant trace on gain matrix . In fact,

if the trace of this matrix decreases too quickly, updatingdoes not occur. One avoids decreasing the gain so as tobe able to follow possible variations of parameters withtime. For the study of nonstationary processes through theirparameters, this technique provides for permanent adjustmentof the coefcients of the models.

It is possible to combine these possibilities using a priori ,known signal characteristics.

In our study, a forgetting factor enables minimization of theconvergence time by weighting the rst estimates, which werenot realistic. However, since real signals are not very stationaryin terms of frequency, it is necessary to use either a variableforgetting factor, or a factor providing for constant trace of

the gain matrix. We started analyzing the lidar signals using avariation forgetting factor so as to minimize convergence time,and then maintained a constant gain matrix trace to providefor follow-up of possible evolutions of the characteristics of the signal. For all applications, the forgetting factor remainssmaller than 0.98, which corresponds [(for an exponentialfunction at (see previous section)], to a range resolutionof 180 m for a sampling frequency MHz.

2) Algorithm of Recursive Maximum Likelihood (RML): Inestimation theory, a likelihood variable allows us to knowwhether the associated magnitude is close to the result ex-pected. This algorithm repeats this idea in ltering the obser-

vation vector with an estimate of the model associated to lterinputs. This model is a representation of the most likely biasesintroduced through the various inputs. Thus, the ltering stageis intended to remove the component generated by noises thatare present in inputs.

Therefore, vector is ltered by model , whereis the estimator of the coefcients of part MA at time .

Consequently, the a priori prediction of is

(12)

with

(13)

and are the output and the a posteriori prediction errors,respectively, ltered by .

The parameters are as follows:

(14)

This ltering tends to accelerate decorrelation between the vec-tor of observations and the prediction error. Results obtainedare likely to be more interesting than those drawn from theRLS algorithm. Yet, in practice, they are contingent upon theprecision of the estimate of .

Furthermore, upon initializing, as a correct estimate of thelter coefcients is not available, processing has to start withthe RLS algorithm to then return to this algorithm.

Lidar signals are slightly or highly noised, depending onatmospheric conditions. The use of this method can be usefulfor signals with a low SNR. Extraction of the signal conveyingthe signicant information will be facilitated by the prelteringphase.

3) Algorithm of Output Error (OE): This algorithm differsfrom the RLS algorithm by the choice of the vector of observations. Signal samples are replaced by their a posterioriestimates

(15)

This amounts to considering that signal samples are biasedand that, consequently, the information contained in predictionis more accurate. Thus, estimate of the signal depends indi-rectly on the disturbance through the adaptation algorithm. Inthe case of a good modelization, the a posteriori error tendsasymptotically toward a white noise, guaranteeing an unbiasedestimate of parameters.

In this method, the criteria take into account the informationcontained in previous prediction errors. Performances obtainedusing the exact least squares method can be improved fornoisy signals with a large spectral width. There again, it isappropriate to begin the processing using the RLS algorithmand then to continue with the OE algorithm.

To conclude, it can be noticed that a better matchingbetween the lter model inferring the estimator and the signalleads to a better accuracy. In particular, the CramerRao lower


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bound (CRLB) for frequency estimation is asymptoticallyreached when the signal ts the lter model [28]. So it wouldbe necessary to search for the more appropriate ARMA modelto improve the performance and get closer to the CRLB (seeIII-A). It turns out that for a limited range gate duration, asconsidered in the present study, the CRLB cannot be reachedbut only approached within a certain factor (about three to ten)depending on the model to be used.

III. MEAN FREQUENCY ESTIMATE

These algorithms were developed and applied to lidar sig-nals. Our aim is to estimate the wind velocity on a single-shootbasis (without accumulation on successive realizations) inorder to decrease the speckle noise effect [7]. To comparethe results obtained by different algorithms, we used thesimulation model proposed by Zrnic [46]. These signals, fora constant velocity, allow one to evaluate the bias and thestandard deviation of the estimator. Then, in order to replacethis study in an experimental case, we have used a signalmodel with a time-varying frequency, which corresponds tovarying velocity. As previously indicated (see II-C.1), for thesetwo kinds of signals, the forgetting factor remains smallerthan 0.98. Below, we are presenting the bias and the standarddeviation of the estimator calculated on 200 lidar signals of 2,000 real-valued samples. As often is the case in adaptivesignal processing (and for reasons of complexity), we haveprocessed real signals. One of the advantages is that it is notnecessary to create lines and as for complex signals.

For the th shot, the bias (on points) can be written asfollows:

(16)

with and being real and estimated Doppler frequenciesfor the th shot.

For shots, the bias average is

(17)

Variance on the estimate (for the th shot) is

(18)

For shots, the variance average is

(19)

Results obtained for the estimate of the Doppler frequencydepend on the SNR and on the spectral width.

The quality of the various measurements (wind velocity,return power, and spectral width) will be characterized bybiases and standard deviations. Those characteristics are notenough to fully describe their whole probability distributions.For wind velocity for instance, such distribution is the additionof a Gaussian density of good estimates plus a uniform

density of bad estimates, or outliers. Three independentparameters are therefore necessary to characterize the wholedistribution. However, a comprehensive characterization of the probability density functions of the measurements is farbeyond the scope of the present article. It would requiredetermining the type of distribution for the return power andspectral width, and to our knowledge, these have never beenstudied yet. This would call for a long work that would have avery limited scope, since the distributions are likely to dependon the actual laser pulse used for the measurements.

A. Choice of the Model Order

As previously indicated (see end of II-C.3), performancesof the ARMA models depend on the order of parts MAand AR. The number of inputs retained corresponds to arepresentation that is in conformity with lidar signals. Thus,we tend to increase the order when signals are very noisy.Table I shows results obtained (using the RLS algorithm andafter convergence) for the bias and the standard deviation of the Doppler frequency estimate in relation to thecharacteristics of lidar signals (noise, spectral width). The rstline shows the bias and the second line gives the standarddeviation for the model considered. To convert these resultsto velocity, it is necessary to know the frequency/velocityrelation. For example, for a 10 m laser, 1 m/s is equivalentto 200 kHz. Then, for a sampling frequency MHz,a spectral width of corresponds to 400 kHz (m/s). Table I can be read: If MHz, the spectralwidth , the is , and thedB, the standard deviation is , which corresponds to

kHz, inferior to 1 m/s (Note: standsfor throughout the text).

The best results obtained using the FPE criterion are shownin heavy print. One notes that for a large spectral width(0.05), the model gives the best performance. Forsmaller widths, results can be improved slightly by increasingthe order. For instance, one can use an or a model.

One can explain this result by the fact that, for a largespectral width, the spectrum displays less extreme values. Onetherefore can be satised with a model with a smaller numberof poles and zeros.

To conclude, the model we have chosen is thelter, which offers the advantage of a small complexity(reduced calculation time). Furthermore, one will note that it ispossible to extract the Doppler frequency from the calculation

of the poles of the denominator (order 2) of the transferfunction. Thus, one avoids the long spectrum calculation.

B. Convergence Time

Convergence time corresponds to the time necessary toreach an estimate error in the Doppler frequency less than5%. Thus, it is possible to evaluate the number of iterationsnecessary to obtain a rst reliable estimate of the Dopplerfrequency.

Fig. 1 shows the inuence of initialization of the gain matrixused in parametric algorithms. Each gure corresponds

to the overlay of ten estimates of the Doppler frequency on


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TABLE IBIAS AND STANDARD DEVIATION OF DOPPLER FREQUENCY ESTIMATE FOR DIFFERENT ARMA M ODELS . THE DOPPLER FREQUENCY IS . THE FIRST

COLUMN REPRESENTS THE ARMA M ODEL , LINE CORRESPONDS TO BIAS, AND LINE CORRESPONDS TO STANDARD DEVIATION . BIAS AND STANDARDDEVIATION A RE CALCULATED FOR EVERY SNR (5 dB, 0 dB, AND 0 dB) AND EVERY SPECTRAL W IDTH ( , , AND ).

NOTE: 0 STANDS FOR 0 , AS IN THE BODY OF THE TEXT

the rst 500 points of ten different signals. These signalsare simulated using Zrnics model (spectral width ,SNR dB, and real Doppler frequency (solid line).

A large initial value of the gain matrix enables one toincrease the power of the rst errors, and the consequence isto reduce convergence time (to the detriment of the estimate

variance). In fact, the adaptation gain also intervenes in theupdating of coefcients. When this gain is large, variationsalso will be large, from one estimate to the next. Therefore,one therefore must nd a convergence-variance middle term.

Since this convergence time corresponds to a blind zone,we chose a large initial gain ( within 100 and 1000).

It must be noted that one surely can improve these results indifferent ways. For instance, we can initialize lter coefcientsat a value close to the Doppler frequency, with the latter beingin its turn estimated by another algorithm. Also, we can maketwo estimates: the rst one being made in the direction of increasing time and the second in the direction of decreasingtime.

C. Comparison of Algorithms

In Fig. 2(a), we show the bias in the frequency estimateobtained after convergence of the RLS algorithm. On Fig. 2(b),we show the standard deviation. Calculations were madewith an model for three spectral widths: ,

, and . The normalized Doppler frequency is. For each SNR, bias and standard deviation are cal-

culated on 200 signals of 2000 samples.One notes that, regardless of the spectral width, bias and

standard deviation plots are practically identical. The choiceof the model prevails over the choice of the algorithm. In fact,

contrary to ANF, the model has a larger numberof degrees of freedom and therefore, it is able to make a betterapproximation of the lidar signal spectrum.

On these plots, we note that the bias is belowfor signals for which the SNR is above dB. For slightlynoised signals, estimates are biased slightly. Below dB,

the bias increases from to at dB.Within the same range, the relative error of the Dopplerfrequency increases from 5% to 16%.

Standard deviations are of the same magnitude for all threespectral widths. They are small for signals with an SNR above

dB. They exceed in the opposite case, reachingat dB.

Fig. 3 shows the results obtained with algorithm OE. Theconditions of experimentation are the same with the RLSalgorithm.

There again, we note the similarity of the bias with thestandard deviation. Bias remains relatively constant and below

up to dB. For an SNR exceeding dB,results obtained by the RLS or OE algorithms are comparable.Conversely, the bias reaches for an SNR of dB.It is twice as big as the one obtained with the RLS algorithm.

On the other hand, the standard deviation of the OE esti-mate is reduced compared to the RLS estimate

for SNRs below dB. Above , dBstandard deviations are practically identical.

Fig. 4 shows the bias and the standard deviation obtainedwith the RML algorithm. The conditions of experimentationare the same with the RLS and OE algorithms.

In this case, plots differ slightly for SNRs below dB.This is due to the fact that for the RML algorithm, data are


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(a)

(b)

(c)

Fig. 1. Doppler frequency estimate as a function of time (number of samples)using an model for different initial gain value. The Dopplerfrequency is . (a) Initial gain value . (b) Initial gainvalue (c) Initial gain value .

ltered with an estimate of the MA model, and this estimate isall the more biased due to the fact that the noise is important.

For the bias, the RML algorithm enables one to obtain re-sults that are comparable to those given by the RLS algorithm,with the exception of signals in which spectral width is equalto 0.05 and performances decay as soon as the SNR is lowerthan dB.

The standard deviation is of the same magnitude with allthree algorithms. It is almost constant for SNRsexceeding dB. It reaches at dB.

Bias is minimum with the RLS algorithm, and standarddeviations are of the same magnitude with all three algorithms(except for highly noised signals, for which the OE algorithmenables us to obtain a smaller standard deviation).

(a)

(b)

Fig. 2. Bias and standard deviation of the Doppler frequency estimate as afunction of the SNR for . The signals are processed with an RLSalgorithm and a model . The three curves correspond to threedifferent normalized spectral widths: , , and 3 . (a)Normalized bias. (b) Normalized standard deviation.

In light of its simplicity, we retained the RLS algorithm toestimate the Doppler frequency.

D. Tracking of the Doppler Frequency

Contrary to estimators of the FFT or PPP type, adaptivealgorithms function on sliding windows of variable size.Therefore, it is possible to track, with a fair amount of accuracy, the evolution of the Doppler frequency with time. Toassess this tracking capacity, we simulated a noisy signal witha variable frequency [Fig. 5(a)]. Frequency variations (from

to ) are shown in Fig. 5(b). These variationsare drawn from a real-wind distribution with more or lesssignicant slopes.

This study enabled us to x values, allowing for goodtracking such as the forgetting factor (0.9 to 0.98) and thethreshold of the gain matrix trace (0.01).

In Fig. 6 (plot ), we show the estimate of the Dopplerfrequency with the RLS algorithm for an model.The SNR equals dB.


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(a)

(b)

Fig. 3. Bias and standard deviation of the Doppler frequency estimate as afunction of the SNR for . The signals are processed with an OEalgorithm and a model . The three curves correspond to threedifferent normalized spectral widths: , , and 3 . (a)Normalized bias. (b) Normalized standard deviation.

When one compares the estimate for the Doppler frequency(plot ), one notes that for slow variations of the frequency,the tracking is good. It fails only when the frequency gets closeto the Shannon frequency (samples 2100 to 2500), probablybecause of spectral aliasing. This problem can be solved easilyby limiting the Doppler frequency to a value betweenand .

IV. S PECTRUM WIDTH

We studied the estimate of the spectral width for tworeasons.

1) It is a characteristic of weather disturbances.2) Even if it is a rough value, a value of the spectral width

can be used by another estimator of the spectral width(for instance by Levin [27]).

A. Estimate

To make an estimate of the spectral width, one proceedsin three steps. First, one calculates the spectrum from the

(a)

(b)

Fig. 4. Bias and standard deviation of the Doppler frequency estimate as afunction of the SNR for . The signals are processed with an RMLalgorithm and a model . The three curves correspond to threedifferent normalized spectral widths: , , and 3 . (a)Normalized bias. (b) Normalized standard deviation.

coefcients of the lter. Next, one ts a Gaussian shape andcomputes its standard deviation. Finally, the standard deviationis compared to the real spectrum width.

Fig. 7 sums up the various steps. Fig. 7(a) shows two plots,representing the spectrum calculated from the coefcientsof the ARMA model (solid line) and the tted Gaussianshape (dashed line). In Fig. 7(b), we draw the tted Gaussiancurve and the initial spectrum (solid line). These gures were

obtained in processing a real-valued lidar signal with a spectralwidth and an SNR of ratio dB.As in the Doppler frequency study, we dened a bias and a

variance. For the th shot, the deviation between the estimatedwidth and the real width can be written as follows:

(20)

and the average for shots is

(21)


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TABLE IIBIAS AND STANDARD DEVIATION OF SPECTRAL W IDTH ESTIMATE (I.E., NORMALIZED-TO -SAMPLING FREQUENCY ) FOR DIFFERENT ARMA M ODELS . THE

DOPPLER FREQUENCY IS . THE FIRST COLUMN REPRESENTS THE ARMA M ODEL , LINE CORRESPONDS TO BIAS, AND LINE CORRESPONDS TO STANDARD DEVIATION . BIAS AND STANDARD DEVIATION A RE CALCULATED FOR EVERY SNR (5 dB, 0 dB, AND

0 dB) AND EVERY SPECTRAL W IDTH ( , , AND ). NOTE: 0 STANDS FOR 0 , AS IN THE BODY OF THE TEXT

The estimate variance for the th shot is

(22)

For -shots, the variance average is

(23)

B. Results

As we have done for the estimate of the Doppler frequency,in Table II, we show the bias and standard deviation obtainedon the estimate of the spectral width. We varied the order of themodel. There again, we note that the results obtained are ratherclose when the AR orders range from two to eight. However,we note that we obtain slightly better performances when usingmodels with a higher order , , or . For theestimate of the spectral width, we retained themodel.

Fig. 8 shows the bias and the standard deviation obtainedwith the RLS algorithm for the spectral width estimate andfor signals with spectral widths of 0.01, 0.03, and 0.05. TheDoppler frequency is of . For each SNR, the bias andthe standard deviation are calculated on 200 Lidar signals of 2000 samples.

The bias for the estimate of the spectral width is below0.063. As in the case of the Doppler frequency, bias isnoticeably smaller for signals for which the SNR exceeds

dB. Also, it is below 0.02 for signals with three spectralwidths. For a highly noised signal, the spectrum, inferred from

the parameters of the model, is attened, which causes anoverestimation of the width (0.07). At dB, the relativeerror is of 40% (130% and 500%, respectively) for a spectralwidth 0.05 (0.03 and 0.01, respectively).

Standard deviation is below 0.005 for signals with a smallspectral width and for which the SNR exceeds dB. For

the noise levels considered, the standard deviation remainsbelow 0.02 regardless of the width.

For the other algorithms, the results are nearly the same.For example, the bias and the standard deviation with theRML algorithm and an dB are and

, respectively. The estimate of the spectral width isfair for SNRs greater than dB. It is strongly biased atlower SNRs.

V. S IGNAL POWER

In this section, we will present the performances of para-metric estimators for the evaluation of the SNR. If is thenoised discrete lidar signal

(24)

where and are the useful signal and the noise,respectively, at time .

Evaluating the SNR enables us to do the following.1) Use it as a validation criterion for the estimates of spec-

tral widths. The smaller the SNR, the lesser condencewe can have in the estimate of the Doppler frequency.

2) Optimize the use of the Doppler frequency estimator(Levin for example).


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(a)

(b)

Fig. 5. Frequency unstationary signal as a function of time (number of samples). (a) Noisy signal. (b) True Doppler frequency.

Fig. 6. Tracking of the signal Doppler frequency as a function of time(number of samples). (a) True Doppler frequency. (b) Estimated Dopplerfrequency.

A. Estimate

The SNR can be obtained from the estimated spectrum of li-dar signals. Fig. 9 shows the spectral estimate obtained with an

model on real-valued Zrnic signals with SNRsranging from 5 dB to dB. The corresponding spectral widthis , and the normalized Doppler frequency is .

In Fig. 9(a) and (b), we note that the average estimatednoise increases from 0.2 ( dB) to 0.5 (

(a)

(b)

Fig. 7. Estimate of signal spectral width (for ). (a) Solid line:Estimated Spectrum. Dashed Line: Gaussian approximation. (b) Solid line:True Gaussian Spectrum. Dashed Line: Gaussian approximation.

dB) when SNR decreases. We also note that the width of the Gaussian curve is overestimated when SNR is small (seeSection IV-B).

To estimate the SNR (Fig. 10), we considered normalized

powers associated with1) the noised spectrum (solid line) calculated from the

parameters of the model, referred to as ;2) the Gaussian curve (dashed line) calculated from the

previous spectrum, referred to as (this Gaussiancurve can be inferred from the fact that the estimateof the spectral width depends on the precision obtainedwith ARMA models);

3) the average level (constant value) of the noise, referredto as (this level is calculated from the noise leveldifferentiated from the Gaussian curve and is all themore signicant when the SNR is small).


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TABLE III

, , AND POWERS ESTIMATES WITH AN MODEL AS A FUNCTION OF SNR. THE SIGNALSARE PROCESSED WITH THE RLS ALGORITHM . IS THE NOISE POWER LEVEL , IS THE POWER UNDER THE GAUSSIANCURVE, AND IS THE POWER OF THE ESTIMATED SPECTRUM FOR THREE-SIGNAL W IDTH , , AND

TABLE IV

, , AND POWERS ESTIMATES WITH AN MODEL AS A FUNCTION OF SNR. THE SIGNALS

ARE PROCESSED WITH THE RLS ALGORITHM .

IS THE NOISE POWER LEVEL ,

IS THE POWER UNDER THE GAUSSIANCURVE, AND IS THE POWER OF THE ESTIMATED SPECTRUM FOR THREE SIGNAL W IDTH , , AND

At rst, the spectrum of the noisy signal is calculated.Then, we extract the gaussian shape as indicated in IV-A.Then, the constant noise level is estimated.

B. Results

We have carried out this study for the two models previously

retained, i.e., and .Table III shows the results obtained for magnitudes and

with an model, with signals of 0.01,0.03, and 0.05 spectral widths, and with an SNR exceeding

dB. The parametric algorithm used is the RLS algorithm.gives an information on the normalized SNR. In the-

ory, this ratio should decrease when SNR decreases.gives identical information, but no estimate of the spectralwidth is required. Estimate is biased when SNR is small.

Reading this Table III, we can make three comments.1) increases when noise increases, regardless of the

real spectral width.

2) Ratio decreases when the SNR varies from 5dB to dB. Then, contrary to the expected result,this ratio increases. This behavior can be explained bythe fact that below dB, the bias on the estimateof the width becomes important. Power is thereforeoverestimated, and the same applies to ratio .

3) In the end, decreases when SNR decreases.However, below dB, values of this ratio are veryclose to one. This shows that the signal is drownedby the noise. Therefore, it seems difcult to infer thecorresponding SNR from it.

Table IV shows that the results obtained with themodel are better than those obtained with

the model. This can be explained by a moreappropriate modelization of the signal.

Here again, we note that increases when noise increases.Conversely, evolution of the ratio depends on thespectral width. decreases to dB ( dB and


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(a)

(b)

Fig. 8. Bias and standard deviation of spectral width estimate with an RLS algorithm and a model as a function of SNR.The three curves correspond to three different spectral widths: ,

, and 3 . (a) Normalized Bias. (b) Normalized StandardDeviation.

dB, respectively) for a 0.01 width (0.03 and 0.05, respectively).This behavior is linked to the estimate of the spectral width.In fact, we showed previously (Fig. 8) that the relative bias,over the spectral width, is less important than the size of thereal width. Finally, the behavior of the ratio remainsglobally unchanged compared to the behavior of the ratio notedwith the model.

To conclude, an accurate estimate of the SNR using anARMA model remains difcult and poorly reliable at lowSNRs (below dB). This is due primarily to the fact thatthis evaluation is done after the estimate of the spectral width.

VI. CONCLUSION

Parametric methods, and especially the ARMA model, arewell suited for the processing of lidar signals. Coefcientsof the AR part enable one to locate the typical peak inthe spectrum, and the MA part helps improve the parameterestimates.

(a)

(b)

Fig. 9. Spectral estimation with a model for two differentSNRs, using the RLS algorithm. The Doppler frequency is . (a)

dB. (b) 0 dB.

These algorithms, developed for adaptive signal processing,provide for the updating of the parameters of lters upon everypresentation of a new sample of the lidar signal. Moreover,

primarily in the case of frequency nonstationary signals, itis possible to optimize convergence using various methods(forgetting factor and constant trace).

The use of parametric ARMA models allows us to1) know the spectrum estimated upon each sample of the

lidar echo;2) estimate the Doppler frequency (small bias and standard

deviation even for noised signals);3) estimate the spectral width (evaluation of atmospheric

disturbances, interesting information to optimize otherestimators, LEVIN for example);

4) estimate the SNR (needs the spectral width estimate).


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Fig. 10. Power estimate. Solid line: estimated spectrum; dashed line: Gauss-ian approximation; constant line: estimated noise-power level.

However, as we have shown, the last estimate, SNR, isstrongly biased.

At last, the prospectives of the present work are both avalidation of performances on real signals and a comparison of these performances with other possible predictive estimatorsbased on neural networks.

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Jean-Luc Zarader received the Ph.D. degree inapplied physics in 1989 from the Universit e Pierreet Marie Curie, Paris VI, France.

He is currently Ma tre de Conf erences at theLaboratoire des Instruments et Syst emes on thePerception et Commande team, Universit e ParisVI. He is currently participating in the Europeanairborne Doppler lidar Wind Infrared Doppler lidar(WIND) project and the European Space Agencyproject, ALADIN. He also is working on noisyspeech coding from dynamics and predictive neural

networks. His research interests are in adaptive signal processing and neuralnetworks processing. The applications are in lidar signal processing and inspeech recognition and coding.

Dr. Zarader is a member of FRANcophone de lIng enieurie de la Langue(FRANCIL).

Alain Dabas received the engineering degree fromEcole Polytechnique, Palaiseau, France, in 1988,

the engineering degree from the Ecole Nationalede Meteorologie Toulouse, France, in 1990, andthe Ph.D. degree in atmospheric physics from theUniversit e Pierre et Marie Curie, Paris VI, France,in 1993.

He is currently working at the Centre Nationalde Recherches M eteorologique, Toulouse, France,the research center for the French weather ser-vice Meteo-France. He also is participating in the

ground-based portable Doppler lidar project (LVT), and the European airborneDoppler lidar project (WIND). His current research interests are in lidars(light detection and ranging) for the remote sensing of atmospheric wind.He has specialized in lidar signal processing (spectral analysis of detectedRF signals) and data analysis (retrieval of atmospheric wind eld from lidarmeasurements).

Pierre H. Flamant received the Doctorat-es-science degree in physics fromthe Universit e de Paris VI, France, in 1979.

He is currently Directeur de Recherche at the Centre National de laRecherche Scientique (CNRS). He also is leading a team of ten scientistsat the Laboratoire de M eteorologie Dynamique, Palaiseau, France. His mainresearch interests are in meteorology and climate processes, atmospheric ow,planetary boundary layer dynamics, radiative budget linked to semitransparentclouds and aerosols, and lidar physics and relevant disciplines like signalprocessing and data inversion techniques.

Dr. Flamant is the chair of the International Coordination Group on Laser

Atmospheric Studies (ICLAS) and a member of the International RadiationCommission. He also is a member of several working groups on space-basedlidars at the European Space Agency, Paris, France, and Co-Investigator of thePICASSO-CENA program, recently approved by NASA and the French SpaceAgency CNES, to develop a space-based backscatter lidar to be launched in2003. He has more than 60 published papers in peer-reviewed journals.

Bruno Gas received the Ph.D. degree in electronicsfrom the Universit e dOrsay, Paris XI, France, in1994.

He is currently Maitre de Conferences at theUniversite de Paris VI, France, and is working onthe noisy speech coding problem and lidar spectrumanalysis from neural networks. His main researchinterest is the study of neural predictive models fortemporal data coding.

Dr. Gas is member of the international groupFRANcophone de lIngenieurie de la Langue

(FRANCIL).

Olivier Adam received the engineering degree from the Ecole Sup erieuredIngenieur en Electrotechnique et Electronique de Paris, France, in 1991. Hereceived the Ph.D. in signal processing from the University Pierre et MarieCurie, Paris VI, France, in 1995.

He is currently a Research Teacher with the University of Cr eteil, France,in the Laboratoire dEtudes et de Recherches en Instrumentation Signaux etSystemes (LERISS), Creteil, France. His research interests include biomedicaldomain and expert systems. He also is working with the detection of humanauditory pathology and the classication with neural network approach.

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