Sequences with identical autocorrelation functions

TL;DR

This paper investigates when two sequences, especially binary ones, share the same autocorrelation function, revealing that nontrivial cases are rare and identifying specific sequence lengths where this occurs.

Contribution

It provides necessary and sufficient conditions for equicorrelationality considering sequence alphabets and catalogs lengths with nontrivial cases up to 44.

Findings

Nontrivial equicorrelationality among binary sequences is rare.

Identified sequence lengths up to 44 where nontrivial equicorrelationality occurs.

Defined and analyzed the concept of equivocal and unequivocal lengths.

Abstract

Aperiodic autocorrelation is an important indicator of performance of sequences used in communications, remote sensing, and scientific instrumentation. Knowing a sequence's autocorrelation function, which reports the autocorrelation at every possible translation, is equivalent to knowing the magnitude of the sequence's Fourier transform. The phase problem is the difficulty in resolving this lack of phase information. We say that two sequences are equicorrelational to mean that they have the same aperiodic autocorrelation function. Sequences used in technological applications often have restrictions on their terms: they are not arbitrary complex numbers, but come from a more restricted alphabet. For example, binary sequences involve terms equal to only $+1$ and $-1$. We investigate the necessary and sufficient conditions for two sequences to be equicorrelational, where we take their alphabet into consideration. There are trivial forms of equicorrelationality arising from modifications that predictably preserve the autocorrelation, for example, negating a binary sequence or reversing the order of its terms. By a search of binary sequences up to length $44$, we find that nontrivial equicorrelationality among binary sequences does occur, but is rare. An integer $n$ is said to be equivocal when there are binary sequences of length $n$ that are nontrivially equicorrelational; otherwise $n$ is unequivocal. For $n \leq 44$, we found that the unequivocal lengths are $1$--$8$, $10$, $11$, $13$, $14$, $19$, $22$, $23$, $26$, $29$, $37$, and $38$. We pose open questions about the finitude of unequivocal numbers and the probability of nontrivial equicorrelationality occurring among binary sequences.