HIGH-RESOLUTION INSTANTANEOUS FREQUENCY ESTIMATION BASED ON TIME-VARYING AR MODELING

Peijun Shan and A. A. (Zouis) Beex

Systems Group -DSP Research Laboratory The Bradley Department of Electrical and Computer Engineering

VIRGlNIA TECH Blacksburg, Virginia 2406 1-0 1 1 1, USA

(e-mail: pshun@vf.edu and beex@vt.edu)

ABSTRACT Time-varying AR (TVAR) modeling based instantaneous fieqmcy (IF) estimation has been considered a poor estimator since k e q proposed in 1984. As a result, not much further investigation of this melthod was reported in the literature. We present our recent worlk on this method, which leads us to conclude that, contrary to the prevailing opinion, the TVAR based IF estimation performs well, and is especially suitable for those practical casa whae only a short data record is available and linear IF law can not be assumed. WAR model based IF estimation is thus worthy of further research. We compare TVAR based IF estimation with the Wigner-Ville Distribution (WVD) peak based method, which reveals performance ceilings due to end-effects, fiequency quantization error, and bias associated with the WVD based approach.

1. "JXODUCTION For an observed signal consistkg of smgle or multiple time- varying mow-band fkquency components, such as frequency modulated components in white noise, it is of primary interest to estimate the: instantaneous frequency (IF) of each component. This problem arises, for example, in the fields of radar, wireless communications, and underwater acoustics.

Time-varying autoregressive (TVAR) model based IF estimation, fist proposed by Shmnan and Friedlander in 1984 [8], has been considered a poor estimator since being proposed [l, 81. As a result, not much Mer investigation of this method has been reported in the literature over the past decade. This paper presents our recent work on this method, which leads us to conclude that the TVAR based IF estimator is a fairly good one, and especially advantageous for those practical cases where only a short data record is available and linear IF law can not be assumed. Further research may thus be worthwhile.

Boashash [I] reviewed and compared various IF estimation algorithms in terms of statistical performance as well as computational complexity. The estimators based on the peaks of the Wigner-Ville Distributions (WVD) and Cross WVD's (XWVD) were reported to provide superior statistical performance with reasonable complexity. In addition, WVD peak based IF estimation was shown to be optimal for linear FM signals with high to moderate SNR [7]. Therefore, in this paper,

in order to evaluate the performance of the. WAR based IF estimator, we choose the WVD peak based IF estimator to compare against and choose a linear FM signal with high to moderate SNR as one of the scenarios.

Our work reveals that the performance of the WVD based method deteriorates for three reasons: the end-effects, the fiequency quantization error, and the bias for general nonlinear FM. As in the stationary case, we expect that a paramdc model, such as an autoregressive model, may perform significantly better by exploiting a priori knowledge about the signal. We c o b this by comparing the statistical performance of the WAR based and the WVD based algorithms through simulations.

The rest of the paper is organized as follows. In Section 2 the TVAR model is reviewed briefly for easy reference. In Section 3, we describe the TVAR based IF estimation method. In Section 4, we compare the statistical performance of the TVAR based and WVD peak based IF estimators.

2. TVAR MODEL REVIEW We briefly review the time-varying AR model. For details of the TVAR model, please refer to Grenier [2] and Hall, Oppenheim and Willsky [3].

A discrete-time time-varying autoregressive (TVAR) process x( t ) of order p is expressed as

x(t> = -2 a, (t>x(t - i> + e(t ) (1) i d

where e(t) is a stationary white noise process with zero mean and variance G * , and the TVAR coeffcients {a,@),

i = 1,2;..,p } are modeled by linear combinations of a set of basis time functions { u,(f), k = OJ, . - . ,q }:

where { u,( t ) ,k=O, l , . . . ,q } can be any appropriate set of basis functions. If { u,(t) } are chosen as powers of time, then { ai(t) } are polynomial functions of time t . If U, ( r ) are trigonometric functions, then (2) is a finite order Fourier series expansion. In any case, the TVAR model is described completely by the set of parametem { a, i = 13,. -,p; k = OJ,. - , u2 1.

0-7803-5073-1/98/$10.00 01998 IEEE 109

The estimation of { ult } aims at minimizing the total squared prediction error in predicting the sequence x(t) :

(3)

Ewe defme the generalized covariance function as

thenthesolution { u I k , i=l,2,--p, k=1,2;-q} thatminimizes (3) can be solved for from the generalized covariance equations:

15 j<p ,OSZ<q ( 5 ) This is a system of p(q + 1) linear equations.

3. W A R BASED IF ESTIMATION In the stationary case? it is well known that tones in white noise can be modeled as an AR process and high-resolution frequency estimation can be achieved through modeling [4]. We can extend this idea to the time-varying case to achieve high-resolution IF estimation through WAR modeling.

For a signal consisting of A4 FM components in white noise with high to moderate SNR, we model the signal with a TVAR model, with order p=M for complex exponential FM components and p=2M for real signals. For low SNR, a higher order may be useful to let the extra poles model the noise [5]. The time-varying transfer function [6] corresponding to the TVAR model can be expressed as

(6) 1 H(z , t) =

l+$,(t)z-I 1-1

By rooting the denominator polynomial formed by the TVAR cwfilcient estimates at each time instant t , we can obtain the p poles as fimctions of time: p, (t), i = 1,2,...,p . At each instant, the poles associated with the FM components appear on or close to the unit circle while the poles associated with the noise, if anyy appear away h m and inside the unit circle.

The instantaneous angles of the poles associated with the FM components can be used as an estimate of the instantaneous frequencies f , ( t ) :

$1 ( t ) = arg PI (0 for (PI (t>l* 1 (7)

Our procedure for IF estimation based on the WAR model consists of the following steps: 1. Choose the basis fimctions ut(t) ,k =1,2,..,,q , and the

orders q andp. 2. Calculate the generalized covariance fimction according to

(41 solve for U, from equation (5), and construct the TVAR coefficients U, (t) by (2).

3. Root the time-varying poles pI(t),i=1,2,...p at each instant t .

4. Distinguish the poles on or close to the unit circle, and find their time-varying angles as the IF estimates of the FM components.

In addition, if a model for the frequency variation f , ( t ) is available, such as a polynomial function, a more accurate estimate of f , (t) can be achieved by fitting f , (n) to that known model.

The basis function set and orders p and q should be selected using U priori knowledge of the signal. For a signal consisting of continuous FM components such as, for example, an echo from moving targets, it is appropriate to use powers of time as the basis function set Other alternatives include trigonometric functions [3] and wavelets [9].

4. SIMULATION RESULTS To observe the performance of the TVAR based IF estimator, we compare it with the WVD peak based IF estimator. We start with a linear FM signal and next consider observation of a nonlinear FM signal, each for high to moderate SNR.

4.1 Linear FM Signal Simulations We use a linear FM signal with 32 samples at unit sampling rate, which clups from frequency 0.1 to 0.41 Hz according to f ( t ) = O . l + O . O l t , t = O J , ~ ~ ~ , 3 1 , and is corrupted by additive white Gaussian noise. The signal is generated using the IF with unit amplitude and a random initial phase. The reason for choosing a linear FM signal is that the WVD peak based IF estimator is optimal for this type of signals with high to moderate SNR [1,7l. The analytical signal in complex noise is used. The model involved is a frst-order autoregressive model (p=l) with the time-varying coefficient represented as a third-order polynomial function (q=3), denoted as WAR(1,3). The generalized covariance method [3] is used to identify the TVAR parameters.

In Figure 1, the reciprocal of the MSE is plotted against SNR, for both WAR based and WVD peak based IF estimation. We ran 100 simulations for each SNR value and the MSE was calculated from the 100 IF estimates at samples t from 1 to 31. In this case, the TVAR based method o u t p e ~ o m the WVD based method and the end-effects of the WVD calculation produce a distinct performance ceiling for the WVD based IF estimates. The end- effects are due to the fact that, with finite-length data records, the number of data samples involved in calculating the discrete WVD is proportional to the distance from the closer end of the data record.

Next, we compare the two methods without the influence of the end-effects. The MSE of the IF estimates is evaluated for the time indices t=8 through 23, i.e. during the central half of the 32- sample data record. The results are shown in Figure 2. While the TVAR based estimation did not change appreciably, the WVD based estimation performs significantly better away from the ends of the data record. However? another performance ceiling materializes for WVD based IF estimation at SNR above 6 dB. This ceding is due to the frequency quantization error of the discrete WVD arising from the finite length of the data records. For 32-sample sequences, the time-frequency bins for the discrete

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WVD tm ofthe size of 1 sample period in time by 0.5/32=1/64 Hz in fkquency, ifno zempadding is applied.

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30

20

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A

!i s

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, I 0 10 20 30

SNR (dB)

F’igure 1.1IMSE vs. SNR for IF estimation on the full time duration: WAX (solid) and WVD peak (dots).

L A 10 20 30

SNR (dB)

Figure 2.lIMSE vs. SNR for IF estimation near the time center of the data record WAR (solid) and WVD peak (dots).

When using WVD based IF estimation, for short records with moderate to high SNR, the associated fresuency quantization error dominates the IF estimation mor at the center of the data record, while the end-effects of the WVD calculation produce estimation errors that dominate the overall IF estimation perfinmame. In comparison, TVAR based IF estimation is free of the frequency quantization error and its endeffects are slight. Here we chose the WVD peak based IF estimator to compare against, but out conclusion may be generalized to other time- frequency distribution (TFD) peak or moment based IF estimators

To reduce the frequency quantization error, inhpolation on the discrete WVD or zero-padding on the WVD kernel [7] or data is necessary. In this way the WVDbasedIF estimator may approach the optimal performance, i.e. the Cramer-Rao Bound [l], for linear FM at the center of the data record. Figure 3 shows the correspondmg simulation result for IF estimates at the center half

lo 0

P I .

of the data records (samples t=&-23) when zaos are padded so that a 4096-pint FFT is used in calculating the WVD, rather than the earlier 32-pint FFT. At moderate to high SNR, the MSE of the TVAR based estimator is about lodB higher than the Cramer- Rao Bound, which is tolerable in most cases.

SNR (dB)

Figure 3. l/MSE vs. SNR for IF estimation near the time center of the data record TVAR (solid) , WVD peak (dots) with sufficient zero-padding, and the Cmner-Rao Bound (dashed line).

4.2 Non-Linear FM Signal Simulations Now, we compare TVAR and WVD based IF estimators in the case of a nonlinear FM signal. This simulation is the same as in the linear FM case except that here the desired IF law is f(t)=0.05+0.004t2, t=0,1,...,31 . Considenng the higher order of the IF law, we increased the AR coefficients’ order to q=5.2ero-padding to 40% pints is applied in the WVD method so that the effect of frequency quantization emor is virtually removed. The 1 M E results are plotted versus SNR in Figure 4, which represents IF estimation at the center half of the time record.

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0 10 20 30 SNR (d6)

F’igure 4.1MSE vs. SNR for IF estimation near the time center of the data record for a nonlinear FM Signal: WAR (solid) with q=5, and WVD peak (dots) With zero-padding to 4096 points.

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Comparing Figure 4 and Figure 3, the TVAR based method performs about the same as in the linear FM case while the WVD based method degrades noticeably. This means that the TVAR based method is significantly less sensitive to the nonlinearity of the IF law. The new performance ceiling of the WVD method in Figure 4 is due to the bias of the WVD based IF estimation. That is, when the IF law is not ideally linear, the peak of the WVD shifts from the position of the actual ins&ntaneous fieqmcy.

Selecting a higher order q for WAR based IF estimation, such as q=5 in Figure 4, reduces the AR coefficient fitting error which dominates the performance at high SNR, but also accommodates more noise into the model which dominates the p e a f o m c e at low SNR. Figure 5 gives the performance of the WAR method when selecting order p 3 . Here the l/MSE of the TVAR method incr- slowly with SNR due to the AR coefficient fitting error, but is about 4 dB higher at low SNR than when using q=5.

lo L 0

SNR (dB)

Figure 5 . l M E vs. SNR for IF estimation near the time center of the data record for a nonlinear FM signal: TVAR (solid) with q=3, and WVD peak (dots).

Figure 6 helps to observe and explain the performance degradation of the WVD based method for the nonlinear IF law. In Figure 6, we show the actual IF and 100 estimates based on the WVD (with enough zero-padding) at each instant, for SNR=20 dB. Besides the end-effects, which produce significant IF estimation variance near the ends of the data record, we also see that the WVD based IF estimation is severely biased toward the inner side of the IF w e .

Other advantages of TVAR model based IF estimation, though not explored in this paper, include its capability to estimate- multi- component signals without first having to separate the components [SI, its ability to resolve closely spaced components due to its high resolution, and its moderate complexity.

5. CONCLUSION The WAR based IF estimator is a fairly good one, and especially advantageous for those practical cases where only a short data record is available and linear IF laws can not be assumed a priori. The optimality of the WVD based method requires the following simultanmus conditions: 1) a linear FM signal, 2) the time instances of the estimated IF are far from the data ends, 3) generous zero-paddmg or frequency interpolation, and 4) high

SNR. Violation of any of the conditions can make the WVD based method perform worse than the TVAR based method. The TVAR based method, though not optimal in any case, is more robust and performs satisfactorily in various situations. In addition, the benefit of the TVAR based method being high- resolution is highly desired for signals consisting of multiple closely spaced FM components.

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Figure 6. Actual IF and 100 estimates at each instant, for SNR=20 dB, using the WVD peak based IF estimator with zero-padding to length 4096.

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M. G. Hall, A. V. Oppenheim and A. S. Willsky, “Time- varying parametric modeling of speech,” Signal Processing, vol. 5, no. 3, pp. 267-285, 1983. S. M. Kay, Modern spectml estimation: theov and application, Prentice-Hall, Englewood Cliffs, NJ, 1988. R. Kumaresan, D. W. Tufts, and L. L. Scharf, “A F’rony method for noisy data: choosing the signal components and selecting the order in exponential signal models,” Proceedings o$ L!?EE, vol. 72, n0.2, pp. 230-233, Feb. 1984. L. Ljung, System identiication: theov for the user, Chapter 5, PrenticeHall, Upper Saddle River, NJ, 1987. P. Rao and F. J. Taylor, “Estimation of the instantaneous frequency using the discrete Wiper distribution,” ElectmniesLetters, vol. 26, no. 4, pp. 246-248, Feb. 1990. K. C. Sharman and B. Friedlandex, “Time-varying autoregressive modeling of a class of nonstationary signals,” Proc. Z W S P ’84, vol. 2, pp. 22.2.1-22.2.4, San Diego, CA, March 1984 M. K. TsatsaniS and G. B. Giatmakis, “Time-varying system identification and model validation using wavelets”, IEEE Trans. on Signal Processing vol. 41, no. 12, pp. 3512-3523, Dec 1993.

pp. 899-911.A~. 1983.

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