The geometric significance of spectral nulls

Abstract

The paper considers the effect of channel spectral nulls on the performance in communication systems. It is first demonstrated that there exists a natural coordinate system in which one can study the performance of a broad class of communication systems operating over finite impulse response channels. Moreover, in this coordinate system, there exists a sensible measure of performance which is convex. The key result of this paper is that channels with spectral nulls are geometrically significant; they form the bases of the convex set of all possible channels in the natural coordinate system. The convex geometry immediately implies that the worst performance of a communication system is achieved over some channel having the most spectral nulls.