10.7 Generalised Discrete Mean-Square-Error Minimisation

The method that we shall introduce in this section is, in a sense a generalisation of Wiener filtering. The concept assumes that both the original signal and noise are realisations of stochastic, not necessarily stationary, processes and it also uses the optimisation criterion of mean-square error (10.8). The model of deterioration is again eqn (10.47). The restoration is a generalisation of filtering via frequency domain; we suppose that the vectors have already been prolonged to the length P and the matrices to size P P by zero padding, as in the second part of Section 10.5.

A simplifying facet of this method is that the transform need not necessarily be of a DFT type, because the convolution property is not used. On the other hand, the restoration-filter frequency transfer is not expressed by a vector but rather by a matrix and the output in the transform domain is thus given by a regular matrix product of the frequency-response matrix and the input-signal spectrum. This leads to removing the rule, valid for linear convolution systems, that any output harmonic component can only originate based on an input component of the same frequency. The described filter therefore is generally not convolutional.

The estimate of the original is obviously

(10.69)

where A is the core of the transform used and M is the matrix transfer of the filter in the transform domain. According to the principle of orthogonality, the differences...