Signal design for phase-incoherent communications

The problem of determining optimal signal waveforms for transmission through a phase-incoherent channel is examined under the following conditions: 1) the channel adds white Gaussian noise to the signal, 2) the receiver is synchronized, 3) the receiver is designed to minimize probability of error, 4) all signals have equal energy, 5) all messages are, a priori, equiprobable, and 6) there is no bandwidth restriction on the signal set. This signal design problem is expressed as a variational problem involving a search for the optimal matrix in a specific class of non-negative-definite matrices, or, equivalently, involving a search for the optimal signal correlation set in a collection of admissible signal correlation sets. For all signal-to-noise ratios, the probability of error is shown to have a local extremum in the class of all admissible signal sets at the orthogonal signal set. This extremum is then classified as a local minimum by proving that the second-order variations are, indeed, always positive. The probability of error is then evaluated numerically for the orthogonal signal structure.