Stochastic iterative algorithms for signal set design for Gaussian channels and optimality of the L2 signal set

We propose a stochastic iteration approach to signal set design. Four practical stochastic iterative algorithms are proposed with respect to equal and average energy constraints and sequential and batch modes. By simulation, a new optimal signal set, the L2 signal set (consisting of a regular simplex set of three signals and some zero signals), is found under the strong simplex conjecture (SSC) condition (equal a priori probability and average energy constraint) at low signal-to-noise ratios (SNR). The optimality of the L1 signal set, the confirmation of the weak simplex conjecture, and two of Dunbridge´s (1967) theorems are some of the results obtained by simulations. The influence of SNR and a priori probabilities on the signal sets is also investigated via simulation. As an application to practical communication system design, the signal sets of eight two-dimensional (2-D) signals are studied by simulation under the SSC condition. Two signal sets better than 8-PSK are found. Optimal properties of the L2 signal set are analyzed under the SSC condition at low SNRs. The L2 signal set is proved to be uniquely optimal in 2-D space. The class of signal sets S(M, K) (consisting of a regular simplex set of K signals and M-K zero signals) is analyzed. It is shown that any of the signal sets S(M, K) for 3⩽K⩽M-1 disproves the strong simplex conjecture for M⩾4, and if M⩾7, S(M, 2) (the L1 signal set) also disproves the strong simplex conjecture. It is proved that the L2 signal set is the unique optimal signal set in the class of signal sets S(M, K) for all M⩾4. Several results obtained by Steiner (see ibid., vol.40, no.5, p.721-31, 1994) for all M⩾7 are extended to all M⩾4. Finally, we show that for M⩾7, there exists an integer K´<M such that any of the signal sets consisting of K signals equally spaced on a circle and M-K zero signals, for 4⩽K⩽K´, also disprove the strong simplex conjecture