Partial-Period Correlations of Zadoff–Chu Sequences and Their Relatives

Partial-period correlations of sequences are an important performance measure of communication systems employing them, but are known to be notoriously difficult to analyze. In this paper, we present a systematic approach to partial-period correlations as a generalization of the Paterson-Lothian approach. For a pair of sequences (not necessarily distinct), we introduce the linear phase-shifting sequences of one of them and analyze their full-period correlations, called the induced correlations, with the other one. By exploiting the induced correlations, we obtain the partial-period correlation properties of the given pair. We then investigate Zadoff-Chu sequences, and give some theoretical results on their partial-period autocorrelations, including closed-form expressions, and the first and second moments. We also derive an upper bound on the magnitudes of the partial-period cross-correlations of Zadoff-Chu sequences. Finally, we explore generalized chirp-like discrete Fourier transform sequences and show that their partial-period cross-correlations have the same magnitudes as the partial-period autocorrelations of the employed Zadoff-Chu sequence up to permutation.