Low autocorrelated multi-phase sequences

Abstract

The interplay between the ground state energy of the generalized Bernasconi model to multi-phase, and the minimal value of the maximal autocorrelation function, $C_{max}={\max }_K|C_K|$, $K=1,..N-1$, is examined analytically and the main results are: (a) The minimal value of ${\min }_N{C}_{max}$ is $0.435\sqrt{N}$ significantly smaller than the typical value for random sequences $O(\sqrt{\log N}\sqrt{N})$. (b) ${\min }_N{C}_{max}$ over all sequences of length N is obtained in an energy which is about 30% above the ground-state energy of the generalized Bernasconi model, independent of the number of phases m. (c) The maximal merit factor $F_{max}$ grows linearly with m. (d) For a given N, ${\min }_N{C}_{max}\sim \sqrt{N/m}$ indicating that for m=N, ${\min }_N{C}_{max}=1$, i.e. a Barker code exits. The analytical results are confirmed by simulations.

4 pages, 4 figures

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