Bat Chirps With Good Properties: Zak Space Construction of Perfect Polyphase Sequences

Previously, a discretization of the linear FM chirp of length N = KL2, L and KL is an element of Z, was given and the conditions for its minimal Zak space support were derived. Chirps satisfying these conditions are known as finite chirps. In this work, subsets of finite chirps of length N = L-2, L a prime, are examined. The investigation leads to a new, Zak space construction of general polyphase sequence sets of size L - 1 with optimal auto and cross-correlation properties, known as perfect sequence sets. It is shown that perfect sequence sets are closely related to sets of finite chirps and, in particular, include the sets of Zadoff-Chu sequences (which are identical with subsets of finite chirps) and the sets of generalized Frank sequences (which are identical with sets of modulations of finite chirps), as special cases. The entire collection of perfect sequence sets is then given by a partition of the set of perfect auto correlation sequences, obtained by right coset decomposition of the group of all permutations with respect to a certain cyclic group. The construction suggests several further generalizations that can be obtained by operating exclusively on subgroups of the permutation group.