Optimality for Coherent Systems When the Dimensionality Is Restricted to D ≤ M − K, Where K ≤ M/2

Abstract

We have obtained solutions to the signal-design problem for the following cases: (1) where the allowed bandwidth, and therefore the allowed dimensionality, is unrestricted, in which case the solution is the regular simplex signal structure (requiring M − 1 dimensions); (2) where the dimensionality is restricted to be such that it is less than that required for the regular simplex, D ≤ M − 2, in which case the solution consists of two lower-dimensional regular simplices placed orthogonal to one another, one simplex containing two signals and the other M − 2; and (3) where the dimensionality is D = 2. In this chapter we look for optimal signal structures in the class of admissible signal sets whose dimensionality is restricted to D ≤ M − K, where K is any given positive integer which is at most M/2. We shall prove that the concept of placing lower-dimensional regular simplices mutually orthogonal to one another can be extended and is indeed the solution for these cases. That is, for a given K the solution to the signal-design problem is shown to consist of K lower-dimensional regular simplices placed mutually orthogonal to one another, where the number of signals in each regular simplex is at least 2. More precisely, in the class of all admissible signal structures whose dimensionality is D ≤ M − K, such signal structures are proved to be local extrema for all signal-to-noise ratios.