Performance of the maximum likelihood constant frequency estimator for frequency tracking

In this paper, the performance of maximum likelihood (ML) estimators for an important frequency estimation problem is considered when the signal model assumptions are not valid. The motivation for this problem is to understand the robustness of the hidden Markov model-maximum likelihood (HMM-ML) tandem frequency estimator [I], where the signal is divided into time blocks, and the frequency in each time block is estimated using the ML approach under the assumption that the signal has a constant frequency in each time block. In order to analyze the sensitivity of ML estimators to the model assumptions, the mean frequency of a discrete complex tone that has a time-varying (ramp) frequency is estimated under the incorrect assumption that it has a constant frequency. In particular, the behavior of the threshold region with respect to different chirp rates is analyzed, and a simple rule is given. The mean squared error above the threshold region is shown to be constant even at very high SNR levels. These results are supported by simulations. E STIMATING the parameters of a sinusoidal signal from a given discrete set of observation data is an important problem arising in different branches of science and engineering. In the time-series analysis literature, recent results for this problem are due to Walker [2], Hannan [3], and Hasan [4]. In those papers, the signal is assumed to be in the form where t[n] is i.i.d. noise. It has been shown that the leastsquares estimate of Ro, which is asymptotically equivalent to a maximum-likelihood estimate when ~ [ n ] is a Gaussian noise, is the maximizer of the periodogram It has also been shown that the asymptotic variance of the frequency estimation is of order N P 3 . However, in [5], it is concluded that the product of the amplitude of the signal and data size must be quite large for the reliability of the asymptotic analysis results. Manuscript received February 10, 1993; revised December 10, 1993. This work was supported by the Cooperative Research Centre for Robust and Adaptive Systems by the Australian Commonwealth Government under the Cooperative Research Centres Program. The associate editor coordinating the review of this paper and approving it for publication was Prof. John Goutsias. The authors are with the Department of Systems Engineering, Research School of Information Sciences and Engineering, Australian National University, Canberra, Australia. IEEE Log Number 9403760. In [6], the signal is assumed to be in the form of a singlefrequency complex tone, and its parameters are estimated from a finite number of noisy discrete-time observations. In [6], Rife and Boorstyn showed that the maximum-likelihood estimate attains the Cramer-Rao bounds at high signal to noise ratio (SNR). In addition, at low SNR, the existence of a range of SNR's is observed where the mean-squared error (MSE) increases very rapidly with reduction in SNR. This threshold effect can be understood in terms of the "outliers" occurring in the maximization of (2) (see [6] for details). In [7], it has been shown that the approximate phase error variance is a good indicator for the threshold. The threshold effects are also discussed in [8] from an information theoretic point of view. In the previous estimation approaches, the signal is assumed to have a constant frequency during the measurement interval. When the signal has a time-varying frequency over the measurement interval, the frequency can, in principle, be tracked by using either extended Kalman filters (EKF's) [9] (or phaselocked loops (PLL) if the amplitude is known) or a hidden Markov model-maximum likelihood (HMM-ML) tandem estimator [I]. In the EKF approach, the parameters of the signal are estimated by using a Kalman filter where the output matrix of the associated signal model is obtained by linearizing the measurement around the one-step prediction of the parameters. Here, there is an assumption on the random variation of the frequency but no assumption on the deterministic variation of the frequency. In contrast, Streit and Barrett [ I ] assumed that the frequency variation is piecewise constant. Accordingly, they divided the signal time function into fixed finite-sized blocks. Then, the frequency in each block was estimated using a ML approach, and frequency transitions between successive blocks were modeled using hidden Markov models (HMM's). Their algorithm works well even at very low SNR's. Since fixed block sizes are used in [ I ] , the assumption that the signal has constant frequency over these blocks may not always be valid. In order to assess the practical utility of the tandem estimator, it is important to understand the degradation in the performance of the HMM-ML tandem estimator when the signal has a time-varying frequency. The crucial question is as follows: What happens if the frequency is estimated using a ML estimator that is based on the assumption that the signal has a constant, unknown single frequency when, in fact, the signal has a slowly time-varying single frequency? In this paper, we will analyze this problem. The aim is to be able to give rules for the choice of the block length, i.e. design parameters. The techniques in Rife and Boorstyn [6] will be the basis for our analysis and identify some of the fundamental 1053-587)3/94$04.00 O 1994 IEEE 2750 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 42, NO. 10, OCTOBER 1994 mechanisms causing departure from predicted performance. observations by using the maximum-likelihood (ML) estimaThe model that is selected for our purposes is a complex signal tion technique. In that paper, the parameters of s ( t ) in (3) that has the simplest form of time-varying frequency, viz. a (bo and w0) are estimated when Aw(t) = 0 from the noisy ramp. Although selection of the model for the time variation of measurement signal z[n] for n = 0, . . . , N 1. Then, the ML the frequency as a ramp is restrictive, it is useful to understand estimate of the frequency Ro = wo/T is given by the nature of the problem, and it clearly gives some insight for the general problem of estimation of signals that have a more fi o . a% m: x {N 1 I Z ( f l ) I} general time-varying frequency. (6) where Z ( R ) is the discrete-time Fourier transform of z[.], A. Assumptions and Signal Model which is defined by Specifically, the signal that we will analyze in this paper is N-1 a linear FM signal that has the form Z ( R ) : = z[n] enp(-jRn). (7) ~ ( t ) = bo exp[j(wo + nw( t ) ) t ] t E [O, T I ] (3) n=O where Aw(t) = wlt and Tl is the length of the signal; the sampled version of the signal s( t ) is ~ [ n ] = bo exp[j(Ro + Rln)n] (4) where T is the sampling period, Ro = woT, and Rl = w1T2. The sampling frequency f , is defined by f , = 1/T and w, = 2 ~ f , . The measurement data z[n] , for n = 0, . . . , N 1, is assumed to be Like most nonlinear estimators, the ML frequency estimator exhibits threshold effects. At high SNR's, the frequency estimate occurs near the true frequency. As the SNR decreases, the probability that the global maximum of I Z ( R ) I / N lies far from the true frequency increases. The occurrence of these outliers (i.e., the frequency estimates that are far from the true frequency) causes a sudden decrease in the performance of the ML estimator as the SNR is reduced. The mean-squared error can be written as z[n] = s[n] + w [n] E { ( ; w ~ ) ~ ) = (1 q)E{ ( ; ~ 0 ) ~ I NO outlier) ( 5 ) + q E ( ( b W O ) ~ I outlier} (8) where w[n] = W R [ ~ ] + j w ~ [ n ] , and both wR[.] and wI[.] are Gaussian noise sequences with mean zero and variance a2. They are statistically independent of each other. The SNR is defined as the the ratio between the average signal power and the average noise power, which is b;/2a2 for the noisy complex linear FM signals. The signal model is selected as a complex sinusoid in order to compare our results with the results of Rife and Boorstyn [6], and in practice, this kind of signal model is used to avoid leakage problems (see [lo]). In addition, note that one can create a complex signal from its real part by using the Hilbert transform. Care must be taken since, in this case, wR[.] and w ~ [ , ] will no longer be independent noises; however, the cross correlation between wR[.] and wI[.] can be removed by down sampling the complex measurement signal. In the next section, we will summarize the outlier analysis of Rife and Boorstyn for the constant frequency case. In the third section, we will derive the statistics of the periodogram of the complex measurement signal z[n]. The fourth section is devoted to generalization of the outlier analysis for the estimation of the linear FM signal under the incorrect assumption that its frequency is constant. In the last section, mean squared error of the frequency estimates analyzed at different SNR levels and a simple rule for the threshold SNR is given in terms of the chirp rate. Simulation results and theoretical results are compared. 11. OUTLIER ANALYSIS OF RIFE AND BOORSTYN FOR CONSTANT FREQUENCY In [6], the parameters of a complex, single-frequency tone are estimated from a finite number of noisy discrete-time where q denotes the outlier probability, when f o = fs /2. If f o # f s /2 , (8) is still valid, but the terms E { ( ; ~ 0 ) ~ I outlier) and E { ( ; ~ 0 ) ~ I No outlier) will differ. Since the frequency estimate is close to the true frequency when there is no outlier, E{(& ~ 0 ) ~ I No outlier} is approximately equal to the Cramer-Rao bound, which is derived in [6]. In addition, when the Dm size is equal to the data size, the DFT of the noise is i.i.d; therefore, when an outlier occurs, the probability density function of the DFT of the noise is approximately uniform; thus, E { ( ; wo)' I outlier} can be calculated easily. Finally, when wa = w,/2, i.e., when there is no bias, the outlier probability q is calculated as where Zk: = {+ I Z ( R ) 1 ) (k = 0 , . . . , N 1 and Ic # N/2) , f2, ( x ) is the Rayleigh probability density function defined by and f*,,, ( x ) is the Rician probability density function defined