On the derivatives of the homomorphic transform operator and their application to some practical signal processing problems

The derivatives of the homomorphic transform operator and its inverse are obtained with Banach algebra techniques. The derivatives can be applied to two important practical problems. The first problem involves a standard method for designing recursive multidimensional digital filters. The method uses spectral factorization or Hilbert transform methods to stabilize a given weighting sequence without changing its magnitude response in the frequency domain. The derivatives of the stabilization process are required by the design algorithm. The derivatives of the homomorphic transform, together with the chain rule for Frechet derivatives, can be used to calculate the derivatives of stabilization analytically. The second problem is the analysis of homomorphic deconvolution techniques in the presence of additive disturbances. The fact that the homomorphic transform maps convolution to addition makes it very useful in certain blind deconvolution problems. However, the highly nonlinear nature of the homomorphic transform complicates analysis of the effects of additive disturbances. In most practical applications, the presence of additive measurement noise is to be expected. The derivatives obtained in this paper are shown to be very useful in obtaining approximations of the effects of additive disturbances