of Linear Regression Models
(the Wiener LMS Algorithm)
PREDICTION OF MOBILE RADIO CHANNELS,
|hh( . )||is the parameter estimate (column vector)|
|fi(t)||is the transposed regressor matrix|
|eps(t)||is the prediction error, or output error|
|F||is obtained directly from the hypermodel, while|
|G||is in general a matrix of transfer functions.|
The polynomial matrix F will be given directly by the hypermodel. The adjustment of G which provides a minimal sum of squared parameter errors can be obtained by solving a linear (Wiener) design problem. This problem is illustrated below, where R is the covariance matrix of the regressors and the leftmost block represents the hypermodel. The rational matrix G in the adaptation law (a) is uniquely determined by the resulting optimal stable rational matrix Lk below.
The Wiener problem can in general be solved via a spectral factorization and a bilateral Diophantine equation. In simplified but powerful variants, the design equations become trivial, and no equations need to be solved. The design can also be made robust against uncertainties in the hypermodels.
Adaptation laws for parameter prediction (k>0), filtering (k=0) and fixed lag smoothing (k<0) can be derived. The use of smoothing improves the attainable performance, while multistep prediction of channel coefficients is of interest e.g. in adaptive Viterbi receivers for TDMA mobile radio systems, or in power control algorithms in CDMA.
In digital communications, the resulting tracking algorithms can be applied to the multi-input multi-output FIR channels appearing in multiuser detection, in CDMA systems as well as in OFDM systems. In other applications, the method can be applied to time-varying output error models and various functional series models. A restriction is that analysis and design has not yet been developed for models with lagged output data as regressors, such as ARMA models.
This work has also resulted in an analysis leading to exact expressions for parameter error covariances for fast time-variations, in situations of relevance for mobile communications.