On complementary sequences

Abstract

A set of complex-valued sequences {S _j} ^k_j=1 , where S_j =(s _j1, s_j2, ..., s_j N) is called complementary if the sum R(·) of their auto-correlation functions {Rs _j (·)} ^k_kj=1 satisfies

$$R\left( \tau \right) = \sum\limits_{j = 1}^k {\sum\limits_{i = 1}^{N - \tau } {s_{ji} s_{ji}^* + \tau = 0} } ,\forall \tau \ne 0.$$

In this paper, we introduce a new family of complementary pairs of sequences over the alphabet α₃={+1,−1,0}. The inclusion of zero in the alphabet, which may correspond to a pause in transmission, leads both to a better understanding of the conventional binary case, where the alphabet is α ₂={+1,−1}, and to new nontrivial constructions over the ternary alphabet α ₃. For every length N, we derive restrictions on the location of the zero elements and on the form of the member sequences of the pair. We also derive a bound on the minimum number of zeros, necessary for the existence of a complementary pair of length N over α ₃. The bound is tight, as it is met by some of the proposed constructions, for infinitely many lengths.