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USE OF FREQUENCY TRACKERS IN LASER DOPPLER VELOCIMETRY FOR SOUND FIELD MEASUREMENT: COMPARATIVE STUDY OF TWO ESTIMATORS
Mechanical Systems and Signal Processing (2003) 17(2), 329d344 doi:10.1006/mssp.2001.1390, available online at http://www.idealibrary.com on
USE OF FREQUENCY TRACKERS IN LASER DOPPLER VELOCIMETRY FOR SOUND FIELD MEASUREMENT: COMPARATIVE STUDY OF TWO ESTIMATORS C. MELLET Laboratoire d+acoustique de l+UniversiteH du Maine, CNRS-UMR 6613, Avenue Olivier Messiaen, 72085 Le Mans cedex 9, France. E-mail: [email protected]
J.-C. VALIERE Laboratoire d+ED tudes AeH roacoustiques CNRS-UMR 6609, 40, Av. du Recteur Pineau 86022 Poitiers cedex, France. E-mail: [email protected] AND
V. VALEAU Pole Sciences et Technologie, LEPTAB, UniversiteH de La Rochelle Avenue Michel CreH peau, 17042 La Rochelle Cedex 1, France. E-mail: [email protected] (Received 20 April 2000, accepted 17 January 2001) This paper deals with the processing of signals resulting from the measurement of sound "elds using the laser Doppler velocimetry technique. For such an application, the problem lies in the demodulation of signals whose instantaneous frequencies have small and fast variations. Two frequency trackers are systematically compared using Monte-Carlo simulations. The "rst tracker is parametric and is an adaptation of Kalman "ltering, while the second is non-parametric and based on the cross Wigner}Ville distribution. As opposed to time}frequency techniques, Kalman "ltering does not su!er from a limiting short-time assumption; its adaptability makes it able to accurately track any instantaneous frequency variation in a noiseless environment, even when very low modulation indices are involved. For low signal-to-noise ratios, the Wigner}Ville-based estimator performs better; for moderate signal-to-noise ratios (higher than 12 dB), Kalman "ltering represents the best compromise in terms of bias and variance. 2003 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
The characterisation of complex sound "elds is an important concern in acoustics for applications concerning noise control or source identi"cation. Scalar pressure measurement is easily achievable using microphones, but vectorial particle velocity is seldom measured as few techniques are available for this measurement. Furthermore, the simultaneous measurement of both quantities allows the determination of acoustic impedance and energy #ux [1], leading to straightforward characterisations of some acoustic systems. Hot-wire anemometry and a technique based on velocity gradient estimation using microphones are sometimes used to estimate the acoustic particle velocity, but they have many drawbacks. Firstly, these methods are intrusive, which disturbs the acoustic "eld, and secondly they raise calibration problems if an accurate estimation is required. These problems are overcome 0888}3270/03/#$30.00/0
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with the use of the laser Doppler velocimetry (LDV), an optical method widely used in #uid mechanics. This technique is based on the measurement of the light frequency shift (or Doppler frequency) scattered by a seeding particle crossing an interference fringe pattern generated by the intersection of two coherent laser beams. A great advantage of this device is that it is a non-intrusive technique and as such can be used in near-"eld measurement, where acoustic phenomena are quite sensitive to gauge presence; moreover, the size of the interference pattern allows a very good spatial discretisation of the measurement [2]. The determination of particle velocity with LDV requires the Doppler frequency of the recorded data to be estimated. The nature of data processing depends on the application. In #uid mechanics, the determination of the #ow velocity at a given instant usually lies in the estimation of the frequency of an amplitude modulated sine wave (usually called burst), corrupted by additive and possibly multiplicative noise; moreover, the use of medium seeding usually leads to a random sampling of the velocity variation. In this context, correlation-based methods, fast Fourier transforms and maximum likelihood techniques [2] have been successfully used. Some authors have also explored the capability of the LDV technique to measure the acoustic velocity superimposed upon a #ow. The techniques of signal processing used the principle of photo-correlation [3] or tools dedicated to non-uniformly sampled data [4]. Earlier measurements have been reported by Taylor [5], using an apparatus allowing the measurement of the particle acoustic velocity with no #ow. The estimation of the velocity amplitude was then based on the analysis of the averaged density spectrum of the signal [5], but this approach led to high variance estimates and appeared to be strongly dependent on the seeding conditions. Recently, a "rst experiment has been carried out in the laboratory in order to test this technique over a large range of acoustic levels and frequencies, with promising results [6], and current work aims to carefully assess and improve the accuracy of the technique for acoustic applications [7]. It was pointed out that as velocity variations due to an acoustic "eld lead to temporal variations of the frequency of the Doppler signals, the instantaneous frequency (IF) introduced by Ville [8] and Gabor [9] is a very relevant quantity to consider, as it is simply proportional to velocity variations. The mathematical de"nition of the IF is 1 d(t) f (t)" G 2 dt
(1)
where (t) represents the phase of the analytic signal associated with the real signal recorded by the photo-multiplier. Previous studies have shown the abilities and limitations of time}frequency analysis for Doppler signal IF estimation in the context of LDV measurement [6, 10]. A model of the signal has been established which makes possible the use of parametric methods to estimate the IF law; in a recent study [11], such a method, the Kalman "ltering, was shown to give promising results. This paper deals with the use of an adaptation of Kalman "ltering to estimate the IF variation in the context of LDV application to acoustics. First, the LDV device and the signal model are presented (Section 2). Section 3 introduces the Kalman "ltering principle and its adaptation to IF estimation of laser Doppler signals when a sound is measured. The last part (Section 4) consists of a comparison of the results obtained by this estimation with a &reference' estimator based on the cross Wigner}Ville distribution, which is known to be an e$cient IF estimator, even with low signal-to-noise ratios (SNR) [12].
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Figure 1. Principle of the measurement set-up of LDV experiment.
2. MEASUREMENT PRINCIPLE
2.1. LDV APPARATUS DESCRIPTION LDV is an optical technique initially used to measure #uid velocities in #ows. The set-up of this experiment, displayed in Fig. 1, is composed of optical and signal processing materials. It can be distinguished into two parts: the optical part which generates and collects the signal, and the data processing part which estimates the velocity. The light source is an argon-ion laser divided into two coherent laser beams by a splitter. These two beams are focused on a small area of the space. Their superposition generates an interference fringe pattern called the probe volume. The size of this volume and the interfringe depend on the angle, , formed by the two laser beams, and on the laser wavelength . Moreover, an opto-acoustic modulator (Bragg cell) is introduced on the path of one beam to produce a controlled shift of the light frequency (F "40 MHz). This @ modulation induces a displacement of the fringe pattern, and this makes the determination of the velocity sign possible. When a particle in motion crosses the probe volume, it scatters some light whose intensity is collected by a photo-multiplier. The medium has to be seeded "rst with a particle in order to obtain signi"cant scattered light. Accordingly, a fog produced by the chemical reaction between an aerosol and air is used. Some preprocessing and signal acquisition are then carried out. The collected signal, whose spectrum is centred around F "40 MHz, is frequency shifted around a frequency @ which allows its digital acquisition, and is low-pass "ltered by proper Bessel "ltering. This frequency shifting is made by using a function of a Burst Spectrum Analyser (BSA, DANTEC). It is worth mentioning that under certain conditions (high velocity or low frequency), this device makes the estimation of the particle acoustic velocity possible. The signal is recorded on a 12-bits card included in a Concurrent Computer Workstation (MAXION) and custom signal processings are achieved by means of time}frequency or parametric techniques. These processings are the main subject of this paper.
332 2.2.
C. MELLET E¹ A¸. SIGNAL MODEL
2.2.1. Steady yow case When a particle crosses the probe volume with a constant V, it scatters some light which is frequency shifted because of a Doppler e!ect. The intensity of the scattered light is collected by the photo-multiplier, and its variation is a sine whose frequency f can be T written as v sin (/2) f" " v. T i /2
(2)
v is the projection of the particle velocity V onto an axis x perpendicular to the interference fringes. So f is proportional to v, and to the fringe spacing, i, which depends on the optical B parameters and . Now with a Bragg cell e!ect, the frequency to be measured is the sum of the Bragg frequency F "40 MHz and of the frequency f . Then, the so-called Doppler @ T frequency ( f ) can be written as B sin (/2) v. (3) f "F # B @ /2 2.2.2. Application to acoustics Let us assume that a sound "eld is generated by a sine-wave source. The subsequent acoustic velocity induces an oscillation, which is usually relatively small, of the medium particles. The projection of the corresponding particle velocity, v, onto the axis x is written as v(t)"< cos (2 f t#) (4) ?A where < and are the amplitude and phase of the velocity, f the acoustic frequency ?A imposed by the source. It can be shown [3] that the Doppler frequency is a function of time and its variation f (t) can be written as B v(t) < f (t)"F # "F # cos (2 f t#). (5) B @ @ ?A i i Then the recorded signal is a sinusoidally frequency-modulated signal as (Fig. 2)
< (6) x(t)"A cos 2F t# sin (2 f t#)# @ ?A if ?A with phase depending on the instant when a seeding particle happens to be in the probe volume. As the spectrum of x(t) is narrowband, the signal can be frequency shifted around a lower frequency F , the so-called carrier frequency: A < x(t)"A cos 2F t# sin (2 f t#)# . (7) A ?A if ?A A comprehensive treatment of a complete model may be found in [3]. 2.2.3. Analytical signal The unknown parameters are < and . They are estimated from the IF law of the analytic signal z(t) associated to x(t): z(t)"x(t)#i¹H [x(t)]
(8)
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Figure 2. Example of a real LDV signal.
where ¹H represents the Hilbert transform and is de"ned as 1 ¹H[x(t)]" vp
!t d. x()
(9)
vp is interpreted as &Cauchy principal values', which under some conditions on the signal produces the quadrature signal of x(t). Then, the analytical signal z(t) is written as
< z(t)"A exp i 2F t# sin (2 f t#)# A ?A if ?A
.
(10)
The quantity
(R f (t)" L R G F #4 cos (2 f t#). A G ?A
(11)
The IF has a sine wave variation, whose amplitude is proportional to the acoustic velocity and phase is the phase of the acoustic velocity. For example, for a 110 dB SPL sound wave, the velocity amplitude < is about 15 mm/s, and for our optical set-up, the IF variation amplitude is 15 kHz. For a 65 dB SPL sound wave, < is only 100 m/s, and the IF shift is only 100 Hz. Parameters < and are obtained from the estimation of f (t), by using G a lock-in procedure. This requires the acoustic frequency f to be known, which is the case ?A in many applications.
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3. METHODS OF SIGNAL PROCESSING
3.1. INTRODUCTION In former studies [10, 13], time}frequency distributions were used to estimate the parameters < and . It was shown that the estimation error exceeds 20% for low modulation indices (lower than 0.2) in a noisy environment. In order to improve these results, some parametric methods are used, as they have the advantage to use a priori informations to model the system which produced the signal. So, an adaptation of Kalman "ltering has been carried out to follow the IF variations [14]. Indeed, Kalman "ltering have been proved to be e!ective for processing non-stationary processes [15]. 3.2. KALMAN FILTER For many years, Kalman "ltering has been used in the development of automatic control, especially in the tracking of object trajectories [15]. Indeed, Kalman "ltering, by estimating the position, velocity and acceleration, allows to "t the trajectory of a #ying object and hence permits a control of its direction. A modi"cation of this method has been carried out to track the IF [14]. Kalman "lter is a recursive way to estimate the value of the state vector X with respect to I the minimisation of the mean-square error estimation. Kalman "lter is based on the description as a dynamic system having the form X "FX #< (12) I I\ I where < represents a white-noise sequence, X is the state vector composed of characterI I istic components of the studied system and F is the matrix of transition, independent of time, which predicts the evolution of the system. In the application of the measurement of the velocity by LDV, the state vector X is I composed of the phase of the discrete analytic signal z[k] [equation (10)] and its two successive derivatives which are, respectively, proportional to the IF f [k] and the modulaG tion ratio [k]. The parameters of the velocity (< and ) are so deduced from the evolution of the IF. The evolution of the system has been constructed under the assumption of a linear frequency variation between two consecutive instants. The transition matrix, F, is then de"ned as
1 ¹ 2C C F" 0 1 ¹ C 0 0 1
(13)
in which ¹ is the sampling period of the signal. The term < [equation (12)] has been C I introduced in the model to account for the deviation between the real signal frequency modulation and the linear approximation. This term allows the application of Kalman "ltering to any frequency-modulated signals. Then, the measurement is described by [k]"HX #w (14) I I where [k] is the phase of the observed signal previously transformed in analytic signal, H is a matrix relating the measurement data to the state de"ned by H"(1 0 0)
(15)
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and w represents a white noise sequence, mutually independent and independent of X , I I de"ned by its mean and variance, respectively, Ew "0 and R "Ew .R An important I I I assumption in the Kalman "ltering scheme is that the noises w and < have to be I I non-correlated. An estimate X) of the state X is computed from the data [0], [1], 2 , I> I> I> [k#1] so as to minimise the mean-square error in the estimate. The estimate is computed as a function only of the measurement [k#1] and the previous best estimate XK . The estimate is so given as the linear combination of the predicted estimate I I as XK "FXK (16) I> I I I and the predicted error estimation [k#1]!HXK : I> I XK "XK #KK ([k#1]!HXK ). (17) I> I> I> I I> I> I The gain vector K) can be considered as being chosen to minimise the mean-square error I> and is given by PK H2 I> I (18) KK " I> HPK H2#R I> I I> depending on the covariance of the error in the predicted estimate PK is de"ned as I> I PK "E(X !XK ) (X !XK )2 . (19) I> I I I> I I I> I This matrix can also be calculated recursively by PK "FPK F2#Q . (20) I> I I I I The P is the covariance of the error in the estimate XK : I> I> I> I> PK "E(X !XK )(X !XK )2 I> I> I> I> I> I> I> I "F[PK !K HPK ] F2. (21) I> I I> I> I The use of this technique requires the estimation of the variances R and Q to be included I I in the correction carried out on the estimation at each time, which represents a di$culty due to unknown behaviour of these noises. So the "rst variance, R , is estimated by using I a Robbins}Monro algorithm [16]. The second variance, Q , is substituted by the inverse of I a constant coe$cient, [14]: FPK F2 PK " I I . I> I
(22)
This value, which lies between 0 and 1, must be chosen to be large for slow-frequency variations and smaller for quick variations. Consequently, the main advantage of this method is its adaptability, because any frequency modulation can be extracted by an adapted choice of the term . In order to assess the performance of this estimator, a comparison is made with an e$cient IF estimator based on a joint time}frequency distribution, the cross Wigner}Ville distribution, whose principle is presented in the next section.
RE ) denoting the mathematical expectation and ¹ the symbol of transposition.
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3.3. CROSS WIGNER}VILLE-BASED ESTIMATOR The basic idea of time}frequency distributions is to obtain a representation of the signal energy in the time}frequency plane. Among these distributions, the Wigner}Ville distribution (WVD) is widely used:
\ zV t#2 z*V t!2 exp (!2 f) d
=
>
(23)
where z is the analytic signal associated to a real signal x(t). The symbol * represents the V complex conjugate. Rao and Taylor have suggested [17] that the peak detection of the WVD is an e$cient IF estimator of signal with linear frequency modulation and high SNR. Conversely, it appears to be less adapted for arbitrary frequency-modulated signals. Moreover, a degradation of the estimation occurs at low SNR [12]. For low SNR signals, an adaptation of the cross Wigner}Ville distribution (XWVD) has been proposed as an IF estimator, leading to a more robust estimation [18, 19]. The principle of this method is, for a given time t, to build a linearly frequency-modulated reference signal z (t). The XWVD between the signals z (t) and z (t) is de"ned as P V P X=
\ hF () zV t#2 z*P t!2 exp (!2 f ) d.
X=
(25)
To estimate the IF at the discrete instant n, the XWVD-based algorithm performs recursively. The modulation law of the reference signal z (t) is linear, and computed at each P iteration by using the following relation: f !f f #f L\ L L\ K # L F 2
(26)
are, where K denotes a discrete time vector of duration , and the terms f and f F L L\ respectively, the IF estimation obtained at instant n at the previous iteration and the IF estimation at instant (n!1) (for the "rst iteration, f is simply determined from the L WVD peak). The windowed XWVD between z (t) and z (t) is then calculated, and the V P search of its peak leads to a new IF estimation f . This procedure is repeated until L convergence is obtained. Convergence has been shown to occur for signals whose IF variation is close to linear [19], and this estimator shows better performance than WVD for noisy signals [12]. Nevertheless, as the XWVD scheme is adapted to linearly frequency-modulated signals, the time window must have a short duration when the IF variation is fast, which induces IF estimations with bad resolutions. The consequences are illustrated in the following section, where various analyses of sinusoidally frequency-modulated signals, such as the LDV signals presented in Section 2.2, are provided.
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4. RESULTS
In order to assess the performances of Kalman "ltering for estimating the acoustic velocity from LDV signals, several tests are performed on simulated signals synthesized from the signal model established in Section 2.2:
< z [n]"A exp j 2F n¹ # sin (2 f n¹ #)# #e[n¹ ]. (27) V A C if ?A C C ?A In this expression, z [n] represents the discrete analytical signal associated to x [equation V (7)], ¹ is the sampling period and e is a white Gaussian cyclic centred noise with variance C 2 . The sampling frequency F "1/¹ is set to 375 kHz. The estimated parameters are the C C acoustic velocity amplitude < and phase . All simulations are made with the following set of parameters: F "47 kHz, f "5 kHz, "/4, "0, A"1, d"2 ms, where d is the C ?A duration of the analysed signal. Simulations consist of a comparison between the two estimators presented previously. Results are presented "rst in a noiseless environment in Section 4.1, and for di!erent values of the SNR (10 log(A/2 )) in Section 4.2. For each case, results are shown as a function of the signal modulation index "
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Figure 3. Relative error of the velocity magnitude estimation in a noiseless environment as a function of : *;*, XWVD; *䉫*, Kalman "ltering.
Figure 4. Velocity-phase estimation in a noiseless environment as a function of . The simulated value of the phase is !453: *;*, XWVD; *䉫*, Kalman "ltering.
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variation. Simultaneously, this involves a loss in frequency resolution; then small amplitudes of IF variations (i.e. small velocity <) become more di$cult to track. This is why a Wigner}Ville-type method is carried out for small values of the modulation index. Conversely, no FFT is involved in the Kalman "lter scheme, so that no frequency resolution problems are encountered. Moreover, the assumption made is not constraining, as only linearity of IF variation between two signal samples is required. The signal is sampled in order to respect the Shannon criterion, and, following the Carson rule, the maximum frequencies contained in the signal are of the order of F #(#1) f . So, the carrier ! ?A frequency F being practically chosen to be around 10 * f , the sampling period is always ! ?A less than that of a 20th of the modulation period. Then for any modulation, the assumption of linearity of IF variation between two signal samples, used in Kalman "ltering, is always su$cient. In any case, the term < in state equation (12) can account for any deviation from I this model. As a consequence, Kalman "ltering is able to perform much better than the XWVD estimator in a noiseless environment, which is observed in the simulations. Nevertheless, this conclusion has to be tempered when processing is performed in a noisy environment, as observed in the next section. 4.2. NOISY ENVIRONMENT In presence of a random component e[n¹ ], the estimators characteristics need to be C analysed from a statistical point of view. For a given value of the SNR and of the modulation index , the results displayed in this section are obtained with Monte-Carlo simulations: the estimated values of the parameters are obtained from 100 realisations of the signal described by equation (27). In order to accurately assess the performances of the estimators, the bias and variance of the parameters estimations are considered separately in Sections 4.2.1. and 4.2.2. In the latter section, the estimation variances are compared with the Cramer}Rao bounds of the estimation. 4.2.1. Convergence value In this subsection, an investigation of the limits of the estimation is made for small modulation indices. Figures 5 and 6 display the relative estimation error (
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C. MELLET E¹ A¸.
Figure 5. Relative error of the velocity magnitude estimation as a function of for a 30 dB SNR: *;*, XWVD; *䉫*, Kalman "ltering.
Figure 6. Relative error of the velocity magnitude estimation as a function of for a 15 dB SNR: *;*, XWVD; *䉫*, Kalman "ltering.
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Figure 7. Velocity-phase estimation as a function of for a 15 dB SNR. The simulated value of the phase is !453: *;*, XWVD; *䉫*, Kalman "ltering.
beyond the threshold, the estimate of the phase tends towards zero, whatever the simulated value of this phase be. This is a result of a phase estimation with a high variance and carried out modulo ; the estimate converges towards 0. 4.2.2. Variance and Cramer}Rao Bound (CRB) In order to study the statistical e$ciency of the estimators, the inverse of the estimator variance in dB is compared with the theoretically derived Cramer}Rao bounds [13]. The estimation variance is approximated with the variance obtained from the 100 Monte-Carlo Vsimulations. The comparison of the two estimators shows that estimation variance tends towards the CRB from some threshold value of the SNR (Figs 8 and 9). The Kalman "ltering asymptotically follows the CRB without never reaching it, while the XWVD estimator variance reaches the CRB and becomes e$cient. The threshold under which the variance rises very rapidly when the SNR decreases is around 6 and 12 dB, respectively, for the XWVD and Kalman estimators. The XWVD gives the best performance in terms of variance and proves its utility to analyse signals embedded in noise with a low SNR. Kalman "ltering has a greater sensibility to noise: the fact that the assumption of non-correlation of the sequences < and w [equations (12, 14)] becomes invalid when the SNR decreases is the I I more likely explanation. Figure 9 displays the variance inverse for the phase estimation. Results are di!erent: Kalman "lter variance is around the CRB, while the XWVD estimation does not reach the CRB. The XWVD estimation has a higher variance for the phase than for the magnitude, and this results in the Kalman "lter having a higher bias. These observations show the
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C. MELLET E¹ A¸.
Figure 8. Inverse in dB of the estimators variance as a function of the SNR. Comparison with the Cramer}Rao bounds. Estimated parameter of the velocity magnitude: **, CRB; *;*, XWVD; *䉫*, Kalman "ltering.
Figure 9. Inverse in dB of the estimators variance as a function of the SNR. Comparison with the Cramer}Rao bounds. Estimated parameter of the velocity phase: **, CRB; *;*, XWVD; *䉫*, Kalman "ltering.
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di$culties to obtain an estimator with a low bias and e$ciency, a compromise must be found to obtain an estimator with a good precision to permit the analysis of real data. Nevertheless, the Kalman "lter will be preferred to estimate the parameter of signal with low modulation index but with a moderate SNR.
5. CONCLUSION
The problem of the estimation of the acoustic particle velocity by means of the laser Doppler velocimetry (LDV) technique has been studied. This application involves the estimation of the IF of a signal with a low modulation index. Two types of analyses have been used to perform the analysis: the "rst is a parametric technique, the Kalman "ltering and the second is the Cross Wigner}Ville-based (XWVD) estimator and is non-parametric. The IF estimation is one original application of Kalman "ltering and does not imply the usual trade-o! of time}frequency analysis between time and frequency resolution. As a consequence, it has been shown that Kalman "ltering is able to perform the IF tracking for any modulation index in a noiseless environment. Indeed, the adaptivity of the system model, introduced by the term < in the algorithm [equation (12)], allows IF variations to I be followed for high acoustic frequency and low intensity level. As a result, this term limits the noise immunity of this method. While for moderate SNR (higher than 12 dB), Kalman "ltering represents the best compromise in terms of bias and variance, the XWVD estimator, although its nature prevents the processing of signal with very low modulation indices, has a better noise robustness. Now, the performance of Kalman "ltering for real data has to be assessed; as LDV signals in acoustics are characterised by low modulation indices and a good quality (moderate SNR), the use of Kalman "ltering in the LDV experiment appears as an exciting challenge.
REFERENCES 1. H. BAILLIET, P. LOTTON, M. BRUNEAU, V. GUSSEV, J.-C. VALIEE RE and B. GAZENGEL 2000 Applied Acoustics 60, 1}11. Acoustic power #ow in a thermoacoustic resonator by means of laser doppler anemometry (LDA) and microphonic measurement. 2. O. BESSON and F. GALTIER 1996 IEEE ¹ransactions on Signal Processing 44, 3056}3068. Estimating particles velocity from laser measurements: maximum likelihood and Cramer}Rao bounds. 3. D. HANN and C. A. GREATED 1993 Measurement Science and ¹echnology 4, 157}164. Acoustic measurements in #ows using photon correlation spectroscopy. 4. T. LOIZEAU and Y. GERVAIS 1997 Acta Acustica, 83, 945}954. Measurement of the acoustic velocity by laser Doppler anemometry. 5. K. J. TAYLOR 1976 Journal of Acoustical Society of America 59, 691}694. Absolute measurement of acoustic particle velocity. 6. J.-C. VALIERE, P. HERZOG, V. VALEAU and G. TOURNOIS 2000 Journal of Sound and
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12. B. BOASHASH and P. O'SHEA 1990 Proceedings of the SPIE Conference on Advanced Acoustics, Speech and Signal Processing, San Diego, 1348, July. Algorithms for instantaneous frequency estimation: a comparative study. 13. V. VALEAU 1999 ¹hse de doctorat, ;niversiteH du Maine. Mesure de la vitesse acoustique particulaire par aneH momeH trie laser Doppler: estimation de freH quence instantaneH e a` variation sinusodale, validation de la mesure. 14. I. VINCENT 1995 ¹hse de doctorat, Ecole Doctorale Sciences pour l1IngeH nieur de Nantes. Classi"cations de signaux non stationnaires. 15. H. W. SORENSON 1980 Kalman Filtering: ¹heory and Application. New York: IEEE Press. 16. H. ROBBINS and S. MONRO 1951 Annals of Mathematical Statistics 22, 400}407. A stochastic approximation method. 17. Rao and F. Taylor 1990 Electronic ¸etters 26, 246}248. Estimation of the instantaneous frequency using the discrete Wigner distribution. 18. B. BOASHASH, G. JONES and P. O'SHEA 1989 Proceedings of the SPIE Advanced Algorithms and Architectures for Signal Processing I< V1152, 382}400. Instantaneous frequency of signals: concepts, estimation techniques and applications. 19. B. BOASHASH and P. O'SHEA 1993 IEEE ¹ransactions on Signal Processing 41, 1439}1445. Use of the cross Wigner Ville distribution for the estimation of instantaneous frequency.
USE OF FREQUENCY TRACKERS IN LASER DOPPLER VELOCIMETRY FOR SOUND FIELD MEASUREMENT: COMPARATIVE STUDY OF TWO ESTIMATORS C. MELLET Laboratoire d+acoustique de l+UniversiteH du Maine, CNRS-UMR 6613, Avenue Olivier Messiaen, 72085 Le Mans cedex 9, France. E-mail: [email protected]
J.-C. VALIERE Laboratoire d+ED tudes AeH roacoustiques CNRS-UMR 6609, 40, Av. du Recteur Pineau 86022 Poitiers cedex, France. E-mail: [email protected] AND
V. VALEAU Pole Sciences et Technologie, LEPTAB, UniversiteH de La Rochelle Avenue Michel CreH peau, 17042 La Rochelle Cedex 1, France. E-mail: [email protected] (Received 20 April 2000, accepted 17 January 2001) This paper deals with the processing of signals resulting from the measurement of sound "elds using the laser Doppler velocimetry technique. For such an application, the problem lies in the demodulation of signals whose instantaneous frequencies have small and fast variations. Two frequency trackers are systematically compared using Monte-Carlo simulations. The "rst tracker is parametric and is an adaptation of Kalman "ltering, while the second is non-parametric and based on the cross Wigner}Ville distribution. As opposed to time}frequency techniques, Kalman "ltering does not su!er from a limiting short-time assumption; its adaptability makes it able to accurately track any instantaneous frequency variation in a noiseless environment, even when very low modulation indices are involved. For low signal-to-noise ratios, the Wigner}Ville-based estimator performs better; for moderate signal-to-noise ratios (higher than 12 dB), Kalman "ltering represents the best compromise in terms of bias and variance. 2003 Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
The characterisation of complex sound "elds is an important concern in acoustics for applications concerning noise control or source identi"cation. Scalar pressure measurement is easily achievable using microphones, but vectorial particle velocity is seldom measured as few techniques are available for this measurement. Furthermore, the simultaneous measurement of both quantities allows the determination of acoustic impedance and energy #ux [1], leading to straightforward characterisations of some acoustic systems. Hot-wire anemometry and a technique based on velocity gradient estimation using microphones are sometimes used to estimate the acoustic particle velocity, but they have many drawbacks. Firstly, these methods are intrusive, which disturbs the acoustic "eld, and secondly they raise calibration problems if an accurate estimation is required. These problems are overcome 0888}3270/03/#$30.00/0
2003 Elsevier Science Ltd. All rights reserved.
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with the use of the laser Doppler velocimetry (LDV), an optical method widely used in #uid mechanics. This technique is based on the measurement of the light frequency shift (or Doppler frequency) scattered by a seeding particle crossing an interference fringe pattern generated by the intersection of two coherent laser beams. A great advantage of this device is that it is a non-intrusive technique and as such can be used in near-"eld measurement, where acoustic phenomena are quite sensitive to gauge presence; moreover, the size of the interference pattern allows a very good spatial discretisation of the measurement [2]. The determination of particle velocity with LDV requires the Doppler frequency of the recorded data to be estimated. The nature of data processing depends on the application. In #uid mechanics, the determination of the #ow velocity at a given instant usually lies in the estimation of the frequency of an amplitude modulated sine wave (usually called burst), corrupted by additive and possibly multiplicative noise; moreover, the use of medium seeding usually leads to a random sampling of the velocity variation. In this context, correlation-based methods, fast Fourier transforms and maximum likelihood techniques [2] have been successfully used. Some authors have also explored the capability of the LDV technique to measure the acoustic velocity superimposed upon a #ow. The techniques of signal processing used the principle of photo-correlation [3] or tools dedicated to non-uniformly sampled data [4]. Earlier measurements have been reported by Taylor [5], using an apparatus allowing the measurement of the particle acoustic velocity with no #ow. The estimation of the velocity amplitude was then based on the analysis of the averaged density spectrum of the signal [5], but this approach led to high variance estimates and appeared to be strongly dependent on the seeding conditions. Recently, a "rst experiment has been carried out in the laboratory in order to test this technique over a large range of acoustic levels and frequencies, with promising results [6], and current work aims to carefully assess and improve the accuracy of the technique for acoustic applications [7]. It was pointed out that as velocity variations due to an acoustic "eld lead to temporal variations of the frequency of the Doppler signals, the instantaneous frequency (IF) introduced by Ville [8] and Gabor [9] is a very relevant quantity to consider, as it is simply proportional to velocity variations. The mathematical de"nition of the IF is 1 d(t) f (t)" G 2 dt
(1)
where (t) represents the phase of the analytic signal associated with the real signal recorded by the photo-multiplier. Previous studies have shown the abilities and limitations of time}frequency analysis for Doppler signal IF estimation in the context of LDV measurement [6, 10]. A model of the signal has been established which makes possible the use of parametric methods to estimate the IF law; in a recent study [11], such a method, the Kalman "ltering, was shown to give promising results. This paper deals with the use of an adaptation of Kalman "ltering to estimate the IF variation in the context of LDV application to acoustics. First, the LDV device and the signal model are presented (Section 2). Section 3 introduces the Kalman "ltering principle and its adaptation to IF estimation of laser Doppler signals when a sound is measured. The last part (Section 4) consists of a comparison of the results obtained by this estimation with a &reference' estimator based on the cross Wigner}Ville distribution, which is known to be an e$cient IF estimator, even with low signal-to-noise ratios (SNR) [12].
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Figure 1. Principle of the measurement set-up of LDV experiment.
2. MEASUREMENT PRINCIPLE
2.1. LDV APPARATUS DESCRIPTION LDV is an optical technique initially used to measure #uid velocities in #ows. The set-up of this experiment, displayed in Fig. 1, is composed of optical and signal processing materials. It can be distinguished into two parts: the optical part which generates and collects the signal, and the data processing part which estimates the velocity. The light source is an argon-ion laser divided into two coherent laser beams by a splitter. These two beams are focused on a small area of the space. Their superposition generates an interference fringe pattern called the probe volume. The size of this volume and the interfringe depend on the angle, , formed by the two laser beams, and on the laser wavelength . Moreover, an opto-acoustic modulator (Bragg cell) is introduced on the path of one beam to produce a controlled shift of the light frequency (F "40 MHz). This @ modulation induces a displacement of the fringe pattern, and this makes the determination of the velocity sign possible. When a particle in motion crosses the probe volume, it scatters some light whose intensity is collected by a photo-multiplier. The medium has to be seeded "rst with a particle in order to obtain signi"cant scattered light. Accordingly, a fog produced by the chemical reaction between an aerosol and air is used. Some preprocessing and signal acquisition are then carried out. The collected signal, whose spectrum is centred around F "40 MHz, is frequency shifted around a frequency @ which allows its digital acquisition, and is low-pass "ltered by proper Bessel "ltering. This frequency shifting is made by using a function of a Burst Spectrum Analyser (BSA, DANTEC). It is worth mentioning that under certain conditions (high velocity or low frequency), this device makes the estimation of the particle acoustic velocity possible. The signal is recorded on a 12-bits card included in a Concurrent Computer Workstation (MAXION) and custom signal processings are achieved by means of time}frequency or parametric techniques. These processings are the main subject of this paper.
332 2.2.
C. MELLET E¹ A¸. SIGNAL MODEL
2.2.1. Steady yow case When a particle crosses the probe volume with a constant V, it scatters some light which is frequency shifted because of a Doppler e!ect. The intensity of the scattered light is collected by the photo-multiplier, and its variation is a sine whose frequency f can be T written as v sin (/2) f" " v. T i /2
(2)
v is the projection of the particle velocity V onto an axis x perpendicular to the interference fringes. So f is proportional to v, and to the fringe spacing, i, which depends on the optical B parameters and . Now with a Bragg cell e!ect, the frequency to be measured is the sum of the Bragg frequency F "40 MHz and of the frequency f . Then, the so-called Doppler @ T frequency ( f ) can be written as B sin (/2) v. (3) f "F # B @ /2 2.2.2. Application to acoustics Let us assume that a sound "eld is generated by a sine-wave source. The subsequent acoustic velocity induces an oscillation, which is usually relatively small, of the medium particles. The projection of the corresponding particle velocity, v, onto the axis x is written as v(t)"< cos (2 f t#) (4) ?A where < and are the amplitude and phase of the velocity, f the acoustic frequency ?A imposed by the source. It can be shown [3] that the Doppler frequency is a function of time and its variation f (t) can be written as B v(t) < f (t)"F # "F # cos (2 f t#). (5) B @ @ ?A i i Then the recorded signal is a sinusoidally frequency-modulated signal as (Fig. 2)
< (6) x(t)"A cos 2F t# sin (2 f t#)# @ ?A if ?A with phase depending on the instant when a seeding particle happens to be in the probe volume. As the spectrum of x(t) is narrowband, the signal can be frequency shifted around a lower frequency F , the so-called carrier frequency: A < x(t)"A cos 2F t# sin (2 f t#)# . (7) A ?A if ?A A comprehensive treatment of a complete model may be found in [3]. 2.2.3. Analytical signal The unknown parameters are < and . They are estimated from the IF law of the analytic signal z(t) associated to x(t): z(t)"x(t)#i¹H [x(t)]
(8)
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Figure 2. Example of a real LDV signal.
where ¹H represents the Hilbert transform and is de"ned as 1 ¹H[x(t)]" vp
!t d. x()
(9)
vp is interpreted as &Cauchy principal values', which under some conditions on the signal produces the quadrature signal of x(t). Then, the analytical signal z(t) is written as
< z(t)"A exp i 2F t# sin (2 f t#)# A ?A if ?A
.
(10)
The quantity
(R f (t)" L R G F #4 cos (2 f t#). A G ?A
(11)
The IF has a sine wave variation, whose amplitude is proportional to the acoustic velocity and phase is the phase of the acoustic velocity. For example, for a 110 dB SPL sound wave, the velocity amplitude < is about 15 mm/s, and for our optical set-up, the IF variation amplitude is 15 kHz. For a 65 dB SPL sound wave, < is only 100 m/s, and the IF shift is only 100 Hz. Parameters < and are obtained from the estimation of f (t), by using G a lock-in procedure. This requires the acoustic frequency f to be known, which is the case ?A in many applications.
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3. METHODS OF SIGNAL PROCESSING
3.1. INTRODUCTION In former studies [10, 13], time}frequency distributions were used to estimate the parameters < and . It was shown that the estimation error exceeds 20% for low modulation indices (lower than 0.2) in a noisy environment. In order to improve these results, some parametric methods are used, as they have the advantage to use a priori informations to model the system which produced the signal. So, an adaptation of Kalman "ltering has been carried out to follow the IF variations [14]. Indeed, Kalman "ltering have been proved to be e!ective for processing non-stationary processes [15]. 3.2. KALMAN FILTER For many years, Kalman "ltering has been used in the development of automatic control, especially in the tracking of object trajectories [15]. Indeed, Kalman "ltering, by estimating the position, velocity and acceleration, allows to "t the trajectory of a #ying object and hence permits a control of its direction. A modi"cation of this method has been carried out to track the IF [14]. Kalman "lter is a recursive way to estimate the value of the state vector X with respect to I the minimisation of the mean-square error estimation. Kalman "lter is based on the description as a dynamic system having the form X "FX #< (12) I I\ I where < represents a white-noise sequence, X is the state vector composed of characterI I istic components of the studied system and F is the matrix of transition, independent of time, which predicts the evolution of the system. In the application of the measurement of the velocity by LDV, the state vector X is I composed of the phase of the discrete analytic signal z[k] [equation (10)] and its two successive derivatives which are, respectively, proportional to the IF f [k] and the modulaG tion ratio [k]. The parameters of the velocity (< and ) are so deduced from the evolution of the IF. The evolution of the system has been constructed under the assumption of a linear frequency variation between two consecutive instants. The transition matrix, F, is then de"ned as
1 ¹ 2C C F" 0 1 ¹ C 0 0 1
(13)
in which ¹ is the sampling period of the signal. The term < [equation (12)] has been C I introduced in the model to account for the deviation between the real signal frequency modulation and the linear approximation. This term allows the application of Kalman "ltering to any frequency-modulated signals. Then, the measurement is described by [k]"HX #w (14) I I where [k] is the phase of the observed signal previously transformed in analytic signal, H is a matrix relating the measurement data to the state de"ned by H"(1 0 0)
(15)
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and w represents a white noise sequence, mutually independent and independent of X , I I de"ned by its mean and variance, respectively, Ew "0 and R "Ew .R An important I I I assumption in the Kalman "ltering scheme is that the noises w and < have to be I I non-correlated. An estimate X) of the state X is computed from the data [0], [1], 2 , I> I> I> [k#1] so as to minimise the mean-square error in the estimate. The estimate is computed as a function only of the measurement [k#1] and the previous best estimate XK . The estimate is so given as the linear combination of the predicted estimate I I as XK "FXK (16) I> I I I and the predicted error estimation [k#1]!HXK : I> I XK "XK #KK ([k#1]!HXK ). (17) I> I> I> I I> I> I The gain vector K) can be considered as being chosen to minimise the mean-square error I> and is given by PK H2 I> I (18) KK " I> HPK H2#R I> I I> depending on the covariance of the error in the predicted estimate PK is de"ned as I> I PK "E(X !XK ) (X !XK )2 . (19) I> I I I> I I I> I This matrix can also be calculated recursively by PK "FPK F2#Q . (20) I> I I I I The P is the covariance of the error in the estimate XK : I> I> I> I> PK "E(X !XK )(X !XK )2 I> I> I> I> I> I> I> I "F[PK !K HPK ] F2. (21) I> I I> I> I The use of this technique requires the estimation of the variances R and Q to be included I I in the correction carried out on the estimation at each time, which represents a di$culty due to unknown behaviour of these noises. So the "rst variance, R , is estimated by using I a Robbins}Monro algorithm [16]. The second variance, Q , is substituted by the inverse of I a constant coe$cient, [14]: FPK F2 PK " I I . I> I
(22)
This value, which lies between 0 and 1, must be chosen to be large for slow-frequency variations and smaller for quick variations. Consequently, the main advantage of this method is its adaptability, because any frequency modulation can be extracted by an adapted choice of the term . In order to assess the performance of this estimator, a comparison is made with an e$cient IF estimator based on a joint time}frequency distribution, the cross Wigner}Ville distribution, whose principle is presented in the next section.
RE ) denoting the mathematical expectation and ¹ the symbol of transposition.
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3.3. CROSS WIGNER}VILLE-BASED ESTIMATOR The basic idea of time}frequency distributions is to obtain a representation of the signal energy in the time}frequency plane. Among these distributions, the Wigner}Ville distribution (WVD) is widely used:
\ zV t#2 z*V t!2 exp (!2 f) d
=
>
(23)
where z is the analytic signal associated to a real signal x(t). The symbol * represents the V complex conjugate. Rao and Taylor have suggested [17] that the peak detection of the WVD is an e$cient IF estimator of signal with linear frequency modulation and high SNR. Conversely, it appears to be less adapted for arbitrary frequency-modulated signals. Moreover, a degradation of the estimation occurs at low SNR [12]. For low SNR signals, an adaptation of the cross Wigner}Ville distribution (XWVD) has been proposed as an IF estimator, leading to a more robust estimation [18, 19]. The principle of this method is, for a given time t, to build a linearly frequency-modulated reference signal z (t). The XWVD between the signals z (t) and z (t) is de"ned as P V P X=
\ hF () zV t#2 z*P t!2 exp (!2 f ) d.
X=
(25)
To estimate the IF at the discrete instant n, the XWVD-based algorithm performs recursively. The modulation law of the reference signal z (t) is linear, and computed at each P iteration by using the following relation: f !f f #f L\ L L\ K # L F 2
(26)
are, where K denotes a discrete time vector of duration , and the terms f and f F L L\ respectively, the IF estimation obtained at instant n at the previous iteration and the IF estimation at instant (n!1) (for the "rst iteration, f is simply determined from the L WVD peak). The windowed XWVD between z (t) and z (t) is then calculated, and the V P search of its peak leads to a new IF estimation f . This procedure is repeated until L convergence is obtained. Convergence has been shown to occur for signals whose IF variation is close to linear [19], and this estimator shows better performance than WVD for noisy signals [12]. Nevertheless, as the XWVD scheme is adapted to linearly frequency-modulated signals, the time window must have a short duration when the IF variation is fast, which induces IF estimations with bad resolutions. The consequences are illustrated in the following section, where various analyses of sinusoidally frequency-modulated signals, such as the LDV signals presented in Section 2.2, are provided.
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4. RESULTS
In order to assess the performances of Kalman "ltering for estimating the acoustic velocity from LDV signals, several tests are performed on simulated signals synthesized from the signal model established in Section 2.2:
< z [n]"A exp j 2F n¹ # sin (2 f n¹ #)# #e[n¹ ]. (27) V A C if ?A C C ?A In this expression, z [n] represents the discrete analytical signal associated to x [equation V (7)], ¹ is the sampling period and e is a white Gaussian cyclic centred noise with variance C 2 . The sampling frequency F "1/¹ is set to 375 kHz. The estimated parameters are the C C acoustic velocity amplitude < and phase . All simulations are made with the following set of parameters: F "47 kHz, f "5 kHz, "/4, "0, A"1, d"2 ms, where d is the C ?A duration of the analysed signal. Simulations consist of a comparison between the two estimators presented previously. Results are presented "rst in a noiseless environment in Section 4.1, and for di!erent values of the SNR (10 log(A/2 )) in Section 4.2. For each case, results are shown as a function of the signal modulation index "
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Figure 3. Relative error of the velocity magnitude estimation in a noiseless environment as a function of : *;*, XWVD; *䉫*, Kalman "ltering.
Figure 4. Velocity-phase estimation in a noiseless environment as a function of . The simulated value of the phase is !453: *;*, XWVD; *䉫*, Kalman "ltering.
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variation. Simultaneously, this involves a loss in frequency resolution; then small amplitudes of IF variations (i.e. small velocity <) become more di$cult to track. This is why a Wigner}Ville-type method is carried out for small values of the modulation index. Conversely, no FFT is involved in the Kalman "lter scheme, so that no frequency resolution problems are encountered. Moreover, the assumption made is not constraining, as only linearity of IF variation between two signal samples is required. The signal is sampled in order to respect the Shannon criterion, and, following the Carson rule, the maximum frequencies contained in the signal are of the order of F #(#1) f . So, the carrier ! ?A frequency F being practically chosen to be around 10 * f , the sampling period is always ! ?A less than that of a 20th of the modulation period. Then for any modulation, the assumption of linearity of IF variation between two signal samples, used in Kalman "ltering, is always su$cient. In any case, the term < in state equation (12) can account for any deviation from I this model. As a consequence, Kalman "ltering is able to perform much better than the XWVD estimator in a noiseless environment, which is observed in the simulations. Nevertheless, this conclusion has to be tempered when processing is performed in a noisy environment, as observed in the next section. 4.2. NOISY ENVIRONMENT In presence of a random component e[n¹ ], the estimators characteristics need to be C analysed from a statistical point of view. For a given value of the SNR and of the modulation index , the results displayed in this section are obtained with Monte-Carlo simulations: the estimated values of the parameters are obtained from 100 realisations of the signal described by equation (27). In order to accurately assess the performances of the estimators, the bias and variance of the parameters estimations are considered separately in Sections 4.2.1. and 4.2.2. In the latter section, the estimation variances are compared with the Cramer}Rao bounds of the estimation. 4.2.1. Convergence value In this subsection, an investigation of the limits of the estimation is made for small modulation indices. Figures 5 and 6 display the relative estimation error (
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Figure 5. Relative error of the velocity magnitude estimation as a function of for a 30 dB SNR: *;*, XWVD; *䉫*, Kalman "ltering.
Figure 6. Relative error of the velocity magnitude estimation as a function of for a 15 dB SNR: *;*, XWVD; *䉫*, Kalman "ltering.
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Figure 7. Velocity-phase estimation as a function of for a 15 dB SNR. The simulated value of the phase is !453: *;*, XWVD; *䉫*, Kalman "ltering.
beyond the threshold, the estimate of the phase tends towards zero, whatever the simulated value of this phase be. This is a result of a phase estimation with a high variance and carried out modulo ; the estimate converges towards 0. 4.2.2. Variance and Cramer}Rao Bound (CRB) In order to study the statistical e$ciency of the estimators, the inverse of the estimator variance in dB is compared with the theoretically derived Cramer}Rao bounds [13]. The estimation variance is approximated with the variance obtained from the 100 Monte-Carlo Vsimulations. The comparison of the two estimators shows that estimation variance tends towards the CRB from some threshold value of the SNR (Figs 8 and 9). The Kalman "ltering asymptotically follows the CRB without never reaching it, while the XWVD estimator variance reaches the CRB and becomes e$cient. The threshold under which the variance rises very rapidly when the SNR decreases is around 6 and 12 dB, respectively, for the XWVD and Kalman estimators. The XWVD gives the best performance in terms of variance and proves its utility to analyse signals embedded in noise with a low SNR. Kalman "ltering has a greater sensibility to noise: the fact that the assumption of non-correlation of the sequences < and w [equations (12, 14)] becomes invalid when the SNR decreases is the I I more likely explanation. Figure 9 displays the variance inverse for the phase estimation. Results are di!erent: Kalman "lter variance is around the CRB, while the XWVD estimation does not reach the CRB. The XWVD estimation has a higher variance for the phase than for the magnitude, and this results in the Kalman "lter having a higher bias. These observations show the
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Figure 8. Inverse in dB of the estimators variance as a function of the SNR. Comparison with the Cramer}Rao bounds. Estimated parameter of the velocity magnitude: **, CRB; *;*, XWVD; *䉫*, Kalman "ltering.
Figure 9. Inverse in dB of the estimators variance as a function of the SNR. Comparison with the Cramer}Rao bounds. Estimated parameter of the velocity phase: **, CRB; *;*, XWVD; *䉫*, Kalman "ltering.
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di$culties to obtain an estimator with a low bias and e$ciency, a compromise must be found to obtain an estimator with a good precision to permit the analysis of real data. Nevertheless, the Kalman "lter will be preferred to estimate the parameter of signal with low modulation index but with a moderate SNR.
5. CONCLUSION
The problem of the estimation of the acoustic particle velocity by means of the laser Doppler velocimetry (LDV) technique has been studied. This application involves the estimation of the IF of a signal with a low modulation index. Two types of analyses have been used to perform the analysis: the "rst is a parametric technique, the Kalman "ltering and the second is the Cross Wigner}Ville-based (XWVD) estimator and is non-parametric. The IF estimation is one original application of Kalman "ltering and does not imply the usual trade-o! of time}frequency analysis between time and frequency resolution. As a consequence, it has been shown that Kalman "ltering is able to perform the IF tracking for any modulation index in a noiseless environment. Indeed, the adaptivity of the system model, introduced by the term < in the algorithm [equation (12)], allows IF variations to I be followed for high acoustic frequency and low intensity level. As a result, this term limits the noise immunity of this method. While for moderate SNR (higher than 12 dB), Kalman "ltering represents the best compromise in terms of bias and variance, the XWVD estimator, although its nature prevents the processing of signal with very low modulation indices, has a better noise robustness. Now, the performance of Kalman "ltering for real data has to be assessed; as LDV signals in acoustics are characterised by low modulation indices and a good quality (moderate SNR), the use of Kalman "ltering in the LDV experiment appears as an exciting challenge.
REFERENCES 1. H. BAILLIET, P. LOTTON, M. BRUNEAU, V. GUSSEV, J.-C. VALIEE RE and B. GAZENGEL 2000 Applied Acoustics 60, 1}11. Acoustic power #ow in a thermoacoustic resonator by means of laser doppler anemometry (LDA) and microphonic measurement. 2. O. BESSON and F. GALTIER 1996 IEEE ¹ransactions on Signal Processing 44, 3056}3068. Estimating particles velocity from laser measurements: maximum likelihood and Cramer}Rao bounds. 3. D. HANN and C. A. GREATED 1993 Measurement Science and ¹echnology 4, 157}164. Acoustic measurements in #ows using photon correlation spectroscopy. 4. T. LOIZEAU and Y. GERVAIS 1997 Acta Acustica, 83, 945}954. Measurement of the acoustic velocity by laser Doppler anemometry. 5. K. J. TAYLOR 1976 Journal of Acoustical Society of America 59, 691}694. Absolute measurement of acoustic particle velocity. 6. J.-C. VALIERE, P. HERZOG, V. VALEAU and G. TOURNOIS 2000 Journal of Sound and
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12. B. BOASHASH and P. O'SHEA 1990 Proceedings of the SPIE Conference on Advanced Acoustics, Speech and Signal Processing, San Diego, 1348, July. Algorithms for instantaneous frequency estimation: a comparative study. 13. V. VALEAU 1999 ¹hse de doctorat, ;niversiteH du Maine. Mesure de la vitesse acoustique particulaire par aneH momeH trie laser Doppler: estimation de freH quence instantaneH e a` variation sinusodale, validation de la mesure. 14. I. VINCENT 1995 ¹hse de doctorat, Ecole Doctorale Sciences pour l1IngeH nieur de Nantes. Classi"cations de signaux non stationnaires. 15. H. W. SORENSON 1980 Kalman Filtering: ¹heory and Application. New York: IEEE Press. 16. H. ROBBINS and S. MONRO 1951 Annals of Mathematical Statistics 22, 400}407. A stochastic approximation method. 17. Rao and F. Taylor 1990 Electronic ¸etters 26, 246}248. Estimation of the instantaneous frequency using the discrete Wigner distribution. 18. B. BOASHASH, G. JONES and P. O'SHEA 1989 Proceedings of the SPIE Advanced Algorithms and Architectures for Signal Processing I< V1152, 382}400. Instantaneous frequency of signals: concepts, estimation techniques and applications. 19. B. BOASHASH and P. O'SHEA 1993 IEEE ¹ransactions on Signal Processing 41, 1439}1445. Use of the cross Wigner Ville distribution for the estimation of instantaneous frequency.