Typical Peak Sidelobe Level of Binary Sequences

Abstract

For a binary sequence Sn = {si: i=1,2,...,n} E [epsilon] {±1}n [superscript n] , n > 1, the peak sidelobe level (PSL) is defined as M(Sn [subscript n])= max [subscript k=1,2,...,n-1| [divided by] E [epsilon superscript n-k subscript i=1 s [subscript 1] S [subscript 1 = k]. It is shown that the distribution of M(Sn) is strongly concentrated, and asymptotically almost surely y [gamma] {S [subscript n])=M(Sn [subscript n] [divided by] [square root of] n 1n n E [epsilon] [1-o(1), [square root of] 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 Y [gamma] (Sn {subscript n]) E [epsilon] [o(1 [divided by] [square root of] 1n 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 Y [gamma](Sn [subscript n]) equals [square root of] 2 .