3270 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 12, DECEMBER 2003 Fast Algorithms for Identification of Periodically Varying Systems Maciej Niedz ´wiecki and Tomasz Klaput Abstract—The problem of identification/tracking of periodi- cally varying systems is considered. When system coefficients vary rapidly with time, the most frequently used weighted least squares (WLS) and least mean squares (LMS) algorithms are not capable of tracking the changes satisfactorily. To obtain good estimation results, one has to use more specialized adaptive filters, such as the basis function (BF) algorithms, which are based on explicit models of parameter changes. Unfortunately, estimators of this kind are numerically very demanding. The paper describes new recursive algorithms that combine low computational requirements, which are typical of WLS and LMS filters, with very good tracking capabilities, which are typical of BF filters. Index Terms—Basis function algorithms, nonstationary pro- cesses, periodically varying systems, system identification. I. INTRODUCTION C ONSIDER the problem of identification/tracking of coef- ficients of a periodically varying system governed by (1) where denotes the normalized discrete time, denotes the system output, is the regression vector made up of the past input samples, is an additive (white) noise, and denotes the vector of periodically varying system coefficients modeled as weighted sums of complex exponentials (2) We will assume that the input sequence is wide-sense stationary ergodic with known covariance matrix and that the frequencies are constant and known a priori. The weighting coefficients in (2) will be regarded as unknown and possibly (slowly) time-varying quantities, even though no explicit dependence of on is shown in (2). One of the challenging potential applications, which under certain conditions admits such formulation of the problem, is adaptive equalization of rapidly fading communication chan- nels. In modern wireless systems, distortion introduced into the transmitted signals is caused mostly by the multipath effect: The Manuscript received February 2, 2002; revised April 10, 2003. This work was supported by KBN under Grant 8 T11A 19919. The associate editor co- ordinating the review of this paper and approving it for publication was Dr. Chong-Yung Chi. The authors are with the Faculty of Electronics, Telecommunications, and Computer Science, Department of Automatic Control, Technical University of Gdan ´sk, Gdan ´sk, Poland (e-mail: maciekn@pg.gda.pl). Digital Object Identifier 10.1109/TSP.2003.819007 signal reaches the receiver along different paths, i.e., with dif- ferent time delays. When the multipath effects are dominated by few strong reflectors (scatterers) and when the transmitter and/or receiver moves with a constant speed, the impulse re- sponse of the channel (along with the transmitter and receiver filters) is periodic and can be modeled in the form (2). In this particular case, denotes the sampled baseband signal re- ceived by the mobile radio system, denotes the sequence of transmitted (complex) symbols, including the known training sequence, stands for the number of different signal paths, and are the corresponding Doppler shifts. For mobile radio channels, the sinusoidal model described above has a long history, which goes back to Aiken [1]. Quite re- cently, a number of papers explored the possibility of using it for equalization purposes (see, e.g., Lindbom et al. [2], Tsatsanis and Giannakis [3], and Giannakis and Tepedelenlioglu [4]). The same model can be used for equalization of aeronautical radio channels [13] and underwater acoustic channels [3], [13]. When the system coefficients vary rapidly with time, the most frequently used weighted least squares (WLS) and least mean squares (LMS) identification algorithms are not capable of tracking the changes satisfactorily. To obtain good estimation results, one has to use more specialized adaptive filters, such as the basis function (BF) algorithms, which make use of explicit models of parameter changes, such as (2). Unfortunately, the BF trackers are computationally very demanding. The paper focuses on the computational and parameter tracking aspects of the complex exponential basis function approach. We show that after suitable modifications, the com- putational complexity of BF algorithms can be significantly reduced. The proposed recursive algorithms combine low computational requirements, which are typical of WLS and LMS filters, with very good tracking capabilities, which are typical of BF filters. Based on some general results available for BF filters, the tracking characteristics of the proposed algorithms, such as its equivalent estimation memory and the associated impulse response, are analyzed. From the qualitative viewpoint, our work resembles the work done by Lindbom [5] and Ahlén et al. [6] on “simplification” of Kalman filter- and Wiener filter-based trackers. II. EXPONENTIALLY WEIGHTED BASIS FUNCTION ESTIMATORS The method of basis functions is based on an explicit model of parameter variation. It is assumed that the parameter trajec- tory can be approximated by a linear combination of known functions of time called the basis functions. Since, in practice, the coefficients of such functional series expansions cannot be 1053-587X/03$17.00 © 2003 IEEE