10.6 Deconvolution via Impulse-Response Optimisation

Another approach to deconvolution is represented by the method of optimising the shape of a specially defined impulse response. It is based on the notion of a cascade connection of the given deteriorating system and the linear restoration filter to be designed. Such a cascade should ideally have a unit transfer, i.e. its overall impulse response should be (in continuous version) c( t) = ?( t). This cannot be practically achieved, as was shown in Section 10.2. However, it is possible to introduce a suitable criterion evaluating the shape of the impulse response and to then use this criterion in the optimisation process constrained by a condition limiting excessive increase in noise in consequence of restoration.

The method is formulated assuming continuous-signal degradation by the convolutional effect of a continuously working linear system and by additive noise; the discretisation (and usually A/D conversion) is included only at the input of the discrete restoration system. We suppose a known impulse response h( t) of the distorting system; the optimal impulse response { m _n} of the FIR restoration filter of length M is to be found. The output signal of the complete chain can be represented quasicontinuously; for the overall impulse response expressed as convolution, we thus obtain

(10.62)

A suitable measure of restoration quality is the slenderness of the resulting impulse response c( t), characterised by the criterion

(10.63)

where w( t)...