Several consequences of the reflection principle in signal processing

A simple principle from the Theory of Complex Variables rules out intervals of zero power spectrum for causal signals. The principle also explains why the familiar filters shift phase. A large class of stable filters is introduced: the Schwarz-Christoffel filters with polygonal Nyquist plot. The Reflection Principle, of the Theory of Complex Variables, is the observation that a function F(s) analytic on a domain D gives rise to a second function F-(s-) analytic on the domain D-, the reflection of D through the real axis. [1] (Here the over-bar denotes complex conjugation.) It follows that if F(s) is real and continuous on an open subinterval of the real axis contained as part of the boundary of the domain D, then F-(s-) is an analytic continuation of F(s) to D-. With a change of variables this simple principle can yield results usually obtained with much deeper Payley-Weinerlike [2] gap theorems.