Typical Peak Sidelobe Level of Binary Sequences
For a binary sequence S-n = {s(i) : i = 1, 2, ..., n} is an element of {+/- 1}(n), n > 1, the peak sidelobe level (PSL) is defined as M(S-n) = max(k=1,2,...,n-1)vertical bar Sigma(n-k)(i=1)s(i)s(i+k)vertical bar. It is shown that the distribution of M(S-n) is strongly concentrated, and asymptotically almost surely gamma(S-n) = M(S-n)/root n ln n is an element of [1 -o(1), root 2]. Explicit bounds for the number of sequences outside this range are provided. This improves on the best earlier known result due to Moon and Moser that the typical gamma(S-n) is an element of [o(1/root ln n), 2], and settles to the affirmative the conjecture of Dmitriev and Jedwab on the growth rate of the typical peak sidelobe. Finally, it is shown that modulo some natural conjecture, the typical gamma(S-n) equals root 2.
peak sidelobe level (psl) concentration aperiodic autocorrelation random binary sequences autocorrelation second moment method