A contribution to the signal-design problem for incoherent-phase communication systems

In a previous study, a general expression for the detection probability was given for the transmission of an arbitrary -ary signaling alphabet through the incoherent-phase channel which adds white Gaussian noise. Necessary and sufficient conditions were obtained demonstrating that with no dimensionality or bandwidth restrictions, the incoherent orthogonal signal structure locally minimizes the error probability for all signal-to-noise ratios. The previous expression is improved herein, from which it is verified that, when the signal dimensionality and bandwidth are restricted to be less than that necessary for the previous optimum, necessary conditions to locally extremize the probability of error are satisfied by the signal-waveform inner product matrix whose dimensionality is as large as possible. The form of the solution obtained is a block diagonal matrix with each block the regular simplex inner-product matrix.