Signal Design for Totally Phase-Incoherent Communications

A set of M equally-likely equal-energy transmittable signals is considered, each of which consists of a linear combination of tones from D free running oscillators (D \leq M) . The oscillator tones are assumed sufficiently disjoint to be orthogonal. The design problem consists of finding the optimal receiver and signal set for various M and D . For the additive white Gaussian noise channel, the optimal receiver first forms the sufficient statistic which consists of noncoherently detecting the energy in each of the D tones. Unlike previous designs of digital transmitters based on minimization of the probability of error, when noncoherent oscillations are employed, the optimal receiver and signal set are dependent on the signal-to-noise ratio. The imposed constraints restrict the signal vectors to the all-positive subspace of the surface of a D -dimen sional sphere. The optimal receiver, signal set, and resulting prob ability of error and channel capacity are determined for M \geq D = 2 for low and high signal-to-noise ratios. Severe performance constraints imposed by using a suboptimal square-law receiver are discussed. Preliminary results have been obtained for the general case M \geq D > 2 .