IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 6, JUNE 1999 1493 Maximum Likelihood Estimation, Analysis, and Applications of Exponential Polynomial Signals Stuart Golden and Benjamin Friedlander, Fellow, IEEE Abstract— In this paper, we model complex signals by ap- proximating the phase and the logarithm of the time-varying amplitude of the signal as a finite-order polynomial. We refer to a signal that has this form as an exponential polynomial signal (EPS). We derive an iterative maximum-likelihood (ML) estimation algorithm to estimate the unknown parameters of the EPS model. The initialization of the ML algorithm can be performed by using the result of a related paper. A statistical analysis of the ML algorithm is performed using a finite-order Taylor expansion of the mean squared error (MSE) of the estimate about the variance of the additive noise. This perturbation analysis gives a method of predicting the MSE of the estimate for any choice of the signal parameters. The MSE from the perturbation analysis is compared with the MSE from a Monte Carlo simulation and the Cram´ er–Rao Bound (CRB). The CRB for this model is also derived in this paper. Index Terms—Chirp, parameter estimation, time-varying fre- quency. I. INTRODUCTION T HE PROBLEM addressed in this paper is the estimation of complex signals with time-varying amplitude and phase functions. We consider observing a complex signal in identically distributed circular white Gaussian noise . That is, suppose we observe (1) where . We assume that the logarithm of the complex signal is exactly representable by a finite-order polynomial (2) where the coefficients of the polynomial are unknown com- plex parameters. Note that the real parts of the polynomial coefficients specify the envelope of the signal, whereas the imaginary parts of the polynomial coefficients specify the phase of the signal. For the amplitude and phase of the com- plex signal to have a unique correspondence to the amplitude and phase of a real signal, the signal should be taken to be Manuscript received June 12, 1996; revised October 30, 1998. This work was supported by the Office of Naval Research under Contracts N00014-91- J-1602 and N00014-95-1-0912 and by the National Science Foundation under Grant NSF MIP-90-17221. The associate editor coordinating the review of this paper and approving it for publication was Prof. P. C. Ching. S. Golden is with Orincon Corporation, San Diego, CA 92121 USA (e- mail: golden@alum.mit.edu). B. Friedlander is with the University of California, Davis, CA 95616 USA (e-mail: friedlan@ece.ucdavis.edu). Publisher Item Identifier S 1053-587X(99)03642-9. analytic. The signal in (2) is analytic if the Fourier transform of (2) is zero for negative frequencies [11]. We refer to a signal that is in the form of (2) as an exponential polynomial signal (EPS). In the noise-free case, the fitted coefficients would be equal to the coefficients that are obtained from a Taylor expansion of the logarithm of the signal. Thus, in theory, this approach can be used to approximate any signal by using a model order that is large enough to capture the desired behavior. In practice, we would initialize the maximum-likelihood estimation algorithm by using an initial estimate such as the one given in [10]. Since the performance of the initial estimate degrades as the order of the polynomial increases, we would typically use this model when the polynomial order is small. The performance of the initial estimate degrades as the order of the polynomial is increased because a larger nonlinearity of the observation will be required to determine the initial estimate. In this paper, the order of the polynomial is assumed to be known. If the order of the polynomial is not known, then it can be estimated by an information theory criterion such as [9]. Signals that can be modeled as an EPS arise in a number of different applications, including seismic signal processing, speech processing, oceanography, radar, and animal sounds. In [7], the authors suggest using an EPS model for the harmonic coding of speech. In [4], figures are presented that show sensor measurements of recorded underwater explosive charges that closely resemble EPS’s. In [8], Peleg shows how a constant- amplitude polynomial phase model can be used for identifying different types of radar. These and other applications have been suggested for closely related models as well. In [1] and [2], Boashash gives additional references where the instantaneous frequency of a signal has been used to model sonar, biomedical applications, speech, underwater acoustics, and oceanography. In this paper, we consider optimal estimation of the un- known parameters of an EPS observed in additive white Gaussian noise. We derive the ML estimates of the param- eters and determine the MSE of the estimates based upon a perturbation analysis. This perturbation analysis allows us to examine the threshold effect of the ML estimates. Further, using a numerical simulation, we compare the results from the perturbation analysis, a Monte Carlo simulation, and the derived Cram´ er–Rao bound (CRB) of the estimates. II. MAXIMUM LIKELIHOOD ESTIMATION To derive the ML estimate of the signal parameters, we find it convenient to use the following alternative representation of 1053–587X/99$10.00  1999 IEEE