A set ofMequally-likely equal-energy transmittable signals is considered, each of which consists of a linear combination of tones fromDfree 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 variousMandD. 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 theDtones. 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 aD-dimen sional sphere. The optimal receiver, signal set, and resulting prob ability of error and channel capacity are determined forM geq D = 2for 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 caseM geq D > 2.