Adaptation laws with constant gains, that adjust parameters of linear regression models, are investigated. The class of algorithms includes LMS as its simplest member, while other algorithms such as Wiener LMS may improve performance by including linear filters. Expressions in closed form for the tracking MSE are obtained for rapidly varying parameters of FIR systems with white inputs. This situation may occur in e.g. the tracking of fading communication channels. Stability and convergence in MSE are ascertained by the stability of a transfer function, without assuming independent regressor vectors. A key technique for obtaining these results is a transformation of these adaptation algorithms into linear time-invariant filters, called learning filters, that operate in open loop for slow parameter variations.

For pt.I see ibid., vol.49, p.2207-17 (2001). Low-complexity Wiener LMS (WLMS) adaptation algorithms, of use for channel estimation, have been derived in Lindbom et al. (2001). They are here evaluated on the fast fading radio channels encountered in IS-136 TDMA systems, with the aim of clarifying several issues: How much can channel estimation performance be improved with these tools, as compared to LMS adaptation? When can an improved tracking MSE be expected to result in a meaningful reduction of the bit error rate? Will optimal prediction of future channel estimates significantly improve the equalization? Can one single tracker with fixed gain be used for all encountered Doppler frequencies and SNRs, or must a more elaborate scheme be adopted? These questions are here investigated both analytically and by simulation. An exact analytical expression for the tracking MSE on two-tap FIR channels is presented and utilized. With this tool, the MSE performance and robustness of WLMS algorithms based on different statistical models can be investigated. A simulation study then compares the uncoded bit error rate of detectors, where channel trackers are used in decision directed mode in conjunction with Viterbi algorithms. A Viterbi detector combined with WLMS, based on second order autoregressive fading models possibly combined with integration, provides good performance and robustness at a reasonable complexity.

We present a method for optimizing adaptation laws that are generalizations of the LMS algorithm. Time-varying parameters of linear regression models are estimated in situations where the regressors are stationary or have slowly time-varying properties. The parameter variations are modeled as ARIMA processes and the aim is to use such prior information to provide high-performance filtering, prediction or fixed lag smoothing estimates for arbitrary lags. The method is based on a novel signal transformation that recasts the algorithm design problem into a Wiener design.

Adaptation laws with constant gains, that adjust parameters of linear regression models, are investigated. The class of algorithms includes LMS as its simplest member. Closed-form expressions for the tracking MSE are obtained for parameters described by ARIMA processes. A key element of the analysis is that adaptation algorithms are expressed as linear time-invariant filters, here called learning filters, that work in open loop for slow parameter variations. Performance analysis can then easily be performed for slow variations, and stability is assured by stability of these learning filters.

Adaptation algorithms with constant gains are designed for tracking smoothly time-varying parameters of linear regression models, in particular channel models occurring in mobile radio communications. In a companion paper, an application to channel tracking in the IS-136 TDMA system is discussed. The proposed algorithms are based on two key concepts. First, the design is transformed into a Wiener filtering problem. Second, the parameters are modeled as correlated ARIMA processes with known dynamics. This leads to a new framework for systematic and optimal design of simple adaptation laws based on prior information. The algorithms can be realized as Wiener filters, called learning filters, or as "LMS/Newton" updates complemented by filters that provide predictions or smoothing estimates. The simplest algorithm, named the Wiener LMS, is presented. All parameters are here assumed governed by the same dynamics and the covariance matrix of the regressors is assumed known. The computational complexity is of the same order of magnitude as that of LMS for regressors which are either white or have autoregressive statistics. The tracking performance is, however, substantially improved.