New results in signal design for the AWGN channel
Abstract
The design of M average power constrained signals in white Gaussian noise (AWGN) is addressed. The longstanding strong simplex conjecture which postulates that the optimal signals, under an average power constraint, are the vertices of a regular simplex, is disproven. A signal set is shown that performs better than the regular simplex at low signaltonoise ratios for an M greater than or equal to 7. This leads to a theorem stating that for any fixed M greater than or equal to 7 no signal set exists that is optimal at all signaltonoise ratios. Furthermore it is found that the optimal solution at low signaltonoise ratios is not an equal energy solution for any M greater than or equal to 7. The regular simplex is shown to be the unique polytope that maximizes the minimum distance between signals. This result leads to the corollary that a signal set which maximizes the minimum distance is not necessarily optimum. However, the regular simplex is shown to be globally optimum in the sense of uniquely maximizing the union bound at all signaltonoise ratios.
 Publication:

NASA STI/Recon Technical Report N
 Pub Date:
 January 1993
 Bibcode:
 1993STIN...9326451S
 Keywords:

 Random Noise;
 Signal Generators;
 Signal Processing;
 Simplex Method;
 White Noise;
 Monotone Functions;
 Signal To Noise Ratios;
 Transmission Loss;
 Communications and Radar