IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, zyxwvutsrq VOL. ASSP-34, NO. zyxwvu 5, OCTOBER zyxwv 1986 1097 Tracking Properties and Steady-State Performance of RLS Adaptive Filter Algorithms EVANGELOS ELEFTHERIOU AND DAVID D. FALCONER, FELLOW, IEEE Abstract-Adaptive signal processing algorithms derived from LS (least squares) cost functions are known to converge extremely fast and have excellent capabilities to “track” an unknown parameter vector. This paper treats analytically and experimentally the steady-state op- eration of RLS (recursive least squares) adaptive filters with exponen- tial windows for stationary and nonstationary inputs. A new forqula for the “estimation-noise” has been derived involving second- and fourth-order statistics zyxwvutsrqp of the filter input as well as the exponential win- dowing factor and filter length. Furthermore, it is shokn that the ad- aptation process associated with “lag effects’’ depends solely on the exponential weighting parameter zyxwvutsr k. In addition, the calculation of the excess mean square error due to the lag for an assumed Markov channel provides the necessary information about tradeoffs between speed of adaptation and steady-state error, It is also the basis for com- parison to the simple LMS algorithm. In a simple case of channel iden- tification, it is shown that the LMS and RLS adaptive filters have the same tracking behavior. Finally, in the last part, we present new RLS restart procedures applied to transversal structures for mitigating the disastrous results of the third source of noise, namely, finite precision arithmetic. I. INTRODUCTION A N area which is of strong current practical impor- tance and research interest is the adaptive channel es- timation and equalization of rapidly time-varying chan- nels. Adjustment algorithms for adaptive filtering derived from LS (least squares) cost functions are known to con- verge extremely fast and have excellent capabilities work- ing in a time-varying environment. Although various sim- ulation results confirming these facts can be found in the literature, there ’is little theoretical work published de- scribing the steady-state performance characteristics of the RLS (recursive ‘feast squares) adaptive filter. It is known that all adaptive filters capable of adapting atreal-timeratesexperiencelossesinperformancebe- cause their adjustments are based on statistical averages taken with limited sample sizes [l]. For exponentially windowed RLS algorithms, these losses expressed through the excess MSE (mean square error) are a result of two Manuscript recejved December 18, 1985; revised March 18, 1986. This work was supported in part by the Natural Sciences and Engineering Re- search Council under Grant A5828 and in part by the Department of zyxwvuts Corn- munications under Contract 03SU82-00134. E. Eleftheriou wgs with the Department of Systems and Computer En- gineering, Carleton University, Ottawa, Ont., Canada K1S 5B6. He is now with IBM Zurich Research Laboratory, Ruschlikon, Switzerland. D. D. Falconer-is with the Department of Systems and Computer En- gineering, Carlettin University, Ottawa, Ont., Canada K1S 5B6. IEEE LogNumbet‘8609287. main sources of error. The first source of error is attrib- uted to the exponential weightingof the squared error se- quence and therefore to the exponential nature of the es- timators used to estimate the correlation matrix and the cross-correlation vector [3], i.e., finite window effect. This error, which we shall call “estimation-noise,” re- sults in a misadjustment of the coefficient vector of the adaptive filter from its optimal setting. An analogy to the previous situation is encountered with the LMS (least mean square) adaptive filter because of the gradient noise [5]. The second sourceof error is associated with filtering nonstationary signals. This error, which has been called “lag error,” is caused by the attempt of the adaptive sys- tem to track variations of the input signal. At this point, we should emphasize that finite-precision arithmetic also contributes to the excess MSE, and in that sense could be considered as a third source of noise. More important, however, is the fact that roundoff errors trigger numerical instabilities posing potential problems in implementing the RLS adaptive algorithms. The analysis of the stationary and nonstationary char- acteristics of the LMS algorithm can be found in the pi- oneering work of Widrow zyxw et al. in [5]. In [4], the tracking ability of a wide class of adaptive signal processing al- gorithms has been studied. That work develops an upper bound on the squared error between the parameter vector being tracked and the value obtained by the algorithm. Also, in [6] and zyxw [7], a more or less qualitative analysis of a preliminary experimental examination of the response of an adaptive lattice predictor and a Kalman estimator to nonstationary inputs has been presented, respectively. Very recently, the analysis of the behavior of an orthog- onalized LMS adaptive filter in a time-varying environ- ment appeared in [8]. A discussion of the steady-state op- eration limitations of RLS adaptive algorithms, because of the finite precision arithmetic and possible solutions, can be found in the work of Cioffi and Kailath [lo] and Lin [9]. Finally, work more closely related to the subject of the present paper, and especially to the part dealing with the’calculation of the “lag error” time constant, has been presented by Ling and Proakis in [ 111. In this paper we present an attempt at quantitive under- standing of the steady-state performance characteristics of adaptive filters driven by RLS algorithms. A new, more accurate expression for the “estimation-noise” than the one given in [3] and [ll] has been derived. Furthermore, the general tracking ability and the explicit calculation of 0096-3518/86/1000-1097$01.00 zyxwv O 1986 IEEE