A Constructive Representation for the Fourier Dual of the Zadoff–Chu Sequences

In this paper, a complex matrix C consisting of a set of perfect sequences is studied. The matrix C is constructed by taking the inverse discrete Fourier transform (IDFT) of a diagonal matrix, in which the diagonal elements comprise an arbitrary periodically perfect sequence gamma. Properties of the matrix C are presented. In addition, the Fourier dual E of the matrix C is investigated. When gamma is a Zadoff-Chu sequence for the case of N even, M=1, and g=0, an explicit representation for the matrix E is derived.