Polynomial annihilating systems and their applications

In this paper, we present a fundamental result of polynomial signal processing. We show that a system annihilates any polynomial input signals up to a given degree if and only if the derivatives of the frequency response of the system vanish. A direct consequence of the result is that the maximally flat filters are ideal for polynomial signals. Our result interprets maximally flat filters in the time domain. Besides, our result suggests that numerical methods derived from the polynomial interpolation are equivalent to the maximally flat filter, and suggests that any maximally flat filters can be analyzed in terms of polynomial interpolation in the time domain.