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
.