The z Transform of a Realizable Time Function Relation of the Real and Imaginary Parts and Applications to Deconvolution

The problem of factoring a spectral density into its minimum phase part can arise in deconvolution problems even when one is solving for the unrealizable filter. Section II of this paper shows explicitly how the need for this relation arises in finding the optimum filter for the dereverberation problem. Although the minimum phase relation for Fourier transforms of time-continuous functions is widely known, the corresponding relation for z transforms of time-discrete sequences is less widely known. Section III gives a brief expository treatment of the z transform. The exposition emphasizes the real part-imaginary part relation and how this may be used to factor a z-transform function into two parts, one corresponding to a minimum phase realizable time function. Section IV concludes with a discussion of the computational aspects of this relation and some examples which show the increased accuracy obtainable by use of the z transform (as opposed to the Fourier transform) for time-discrete sequences.