10.5 Constrained Deconvolution

Another concept of restoration is represented by constrained deconvolution. This method does not impose the requirement of minimising estimation errors but only requests that the original noise power in the restored signal be preserved. It comes from the general tendency of deconvolution methods to lead to a substantial degradation in signal-to-noise ratio and therefore it seeks an approximate approach which would correct the convolution distortion without increasing the noise level.

The method uses the notion of a discrete model of deterioration consisting in distortion caused by a discrete linear system with a finite impulse response { h _n}, n = 0, 1, , J - 1, and additive noise { ? _n},

(10.47)

For a finite section { x _n} of the input signal, M samples long, obviously n ÃŽ ?0, M + J - 2 ?, i ÃŽ ?0, M - 1 ?. The model equation can then be rewritten into vector form

(10.48)

where the nonzero elements of the matrix are H _n,i = h _{n - i}, ( n - i) ÃŽ ?0, J - 1 ? so that it is a circulant matrix. The noise energy of the observed signal, that is before restoration, is

(10.49)

If the noise is centred, its energy is given by the variance,

We shall define the residuum r as being the difference vector dependent...