Optimal minimum variance estimation for non-linear discrete-time multichannel systems

A non-linear operator approach to estimation in discrete-time multivariable systems is described. It involves inferential estimation of a signal which enters a communication channel that contains non-linearities and transport delays. The measurements are assumed to be corrupted by a coloured noise signal correlated with the signal to be estimated. The solution of the non-linear estimation problem is obtained using non-linear operators. The signal and noise channels may be grossly non-linear and are represented in a very general non-linear operator form. The resulting so-called Wiener non-linear minimum variance estimation algorithm is relatively simple to implement. The optimal non-linear estimator is derived in terms of the non-linear operators and can be implemented as a recursive algorithm using a discrete-time non-linear difference equation. In the limiting case of a linear system, the estimator has the form of a Wiener filter in discrete-time polynomial matrix system form. A non-linear channel equalisation problem is considered for the design example.