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Design of stable allpass filters with prescribed phase characteristics

This paper addresses the problem of obtaining all stable allpass filters that match the given phase characteristics in an optimal manner. In this context, it is shown that the class of all stable allpass systems, that match the associated partial impulse response sequence can be parameterized using the theory of bounded functions. The above parameterization is utilized to design stable allpass filters with prescribed phase responses. The given phase characteristics is first utilized to obtain its partial impulse response sequence and a parametric form of all stable allpass functions, and a least mean square procedure is then employed on its phase characteristics for further optimality. As a part of our analysis, it is also shown that there are no stable allpass solutions in the sense of classical Pade' approximation.