New results in signal design for the AWGN channel

The design of M average power constrained signals in white Gaussian noise (AWGN) is addressed. The long-standing 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 signal-to-noise 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 signal-to-noise ratios. Furthermore it is found that the optimal solution at low signal-to-noise 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 signal-to-noise ratios.