Analytic Phase Derivatives, All-Pass Filters and Signals of Minimum Phase

It is accepted knowledge that inner functions and outer functions in complex analysis correspond, respectively, to all-pass filters and signals of minimum phase. The knowledge, however, has not been justified for general inner and outer functions. In digital signal processing the correspondence and related results are based on studies of rational functions. In this paper, based on the recent result on positivity of phase derivatives of inner functions, we establish the theoretical foundation for all-pass filters and signals of minimum phase. We, in particular, deal with infinite Blaschke products and general singular inner functions induced by singular measures. A number of results known for rational functions are generalized to general inner functions. Both the discrete and continuous signals cases are rigorously treated.