Conditional mean and maximum likelihood approaches to multiharmonic frequency estimation
نویسندگان
چکیده
The performance of an extended Kalman iilter (EKE') applied to the problem of estimating the (assumed constant) parameten (fundamental frequency, harmonic phases, and amplitudes) of a complex multiharmonic signal measured in noise is shown to he asymptotically (i.e., as the number of measurements tends to infinity) efliuent. The Cramer-Rao (CR) bounds associated with the estimation problem are derived for the case where the measurements commence at an arbitrarg time distinct from zero. T HIS paper is concerned with the discrete-time estimation of the parameters of a signal comprising a sum of harmohically related sinusoidal tones contaminated by additive noise, given noisy observations of the tones over a finite interval. Such a harmonic series is paramemzed by its fundamental frequency and its harmonic phases (and amplitudes). The amplitudes, frequency, and phases are assumed not to vary over the interval of observation The work of [I] and [2] analyzed the maximum likelihood approach applied to a closely related parameter estimation problem, with consideration in [l] of the single-tone case and in [Z] the general multiple-tone case. The latter did not consider the special situation where tbe tones are harmonically related, but rather one where no special relationship between the tones exists. The work of [3] contained, in part, a derivation of the Cramer-Rao (CR) bounds for the case of a real muitihannonic signal. measurements of which are assumed to commence at time t = 0. (The CR bounds are lower bounds on the estimation error variance for any unbiased estimator.) One of the results of this paper is the generalization of the derivation of [3] to the case of a complex multiharmonic signal, with noisy measurements assumed to start at an arbinary time distinct from zero. Reasons for preferring a complex signal formulation m discussed in Section 11. The details of our slightly generalized derivation are summarized in Theorem 4.1. (We remark that this theorem is essentially no different Manuscript received March 13. 1991; revised April 14. 1993. This work was supported by the Ddence Science and Technology Organizalion. the Australian Telecommunicsdons and Uecmnics Research Board he ANU Centre for Information Science Research. and the Cooperative Research Crnm for Robust and Adsprive Systems. The assoeiale editor cmrdinating the nview of chis paper 2nd appmving it for publication was Dr. David Rossi. B. James is with the Industrial Systems Omup. Depanment or Electrical md Electronic Engineering. Imperial College of Science, Technology and Medicine. London. SW7 ?BT. England. B. D. 0 . .inJmon and R. C. Williamson are with the Department of Systems Enpimering. Resemh School of Physical Sciences and Engineering. Austnlim Nsuonal University. Canberra. A.C.T.. Auslrdlia. IEEE Log Number 940338.5. from that which could be condensed from the treatment of [31 in which no explicit theorem statement is made.) The statement of Theorem 4.1 sets the scene for the major result of this paper, which is primarily concerned with a particular approximate conditional mean estimator, the extended Kalman filter (EKF). The estimation problem is cast in statespace form, and two parameters are defined which reflect the knowledge of the frequency and phases prior to estimation. The so-called information formulation of the EKF equations is applied to the resulting state-space signal model, and, after derivation of approximate expressions governing the performance of the EKF, it is argued (through appeal to Theorem 4.1) that the EKF is asymptotically efficient for sufficiently high signal-to-noise-ratio (SNR). Finally, conclusions are drawn and directions of future research are discussed. The estimation problem with which we are concerned is to determine the "best" estimate (in some sense) of a constant, but unknown, parameter vector, given a finite set of noisy observations of some function of the parameter vector. For the muitiharmonic (MH) estimation problem with m harmonics, the parameters of interest are the amplitudes bl, . . . b, of each of the harmonics, their relative phases 81, . . , O m , and the fundamental frequency wo. The parameter vector is then defined to be and an arbitrary estimate & of a0 is defined by The underlying real s i sa l comprising rn harmonics is a nonlinear function of the parameter vector a0 and is defined by along with its in-quadranrre counterpart (perhaps obtained via a Hilben transform: see Appendix A and the remark below):
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ورودعنوان ژورنال:
- IEEE Trans. Signal Processing
دوره 42 شماره
صفحات -
تاریخ انتشار 1994