Sequences with identical autocorrelation spectra

Aperiodic autocorrelation measures the similarity between a finite-length sequence of complex numbers and translates of itself. Autocorrelation is important in communications, remote sensing, and scientific instrumentation. The autocorrelation function reports the aperiodic autocorrelation at every possible translation. Knowing the autocorrelation function of a sequence is equivalent to knowing the magnitude of its Fourier transform. Resolving the lack of phase information is called the phase problem. We say that two sequences are isospectral 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 an alphabet that may reside in a proper subring of the complex field or may come from a finite set of values. For example, binary sequences involve terms equal to only $+1$ and $-1$. In this paper, we investigate the necessary and sufficient conditions for two sequences to be isospectral, where we take their alphabet into consideration. There are trivial forms of isospectrality arising from modifications that predictably preserve the autocorrelation, for example, negating sequences or both conjugating their terms and writing them in reverse order. By an exhaustive search of binary sequences up to length $34$, we find that nontrivial isospectrality among binary sequences does occur, but is rare. We say that a positive integer $n$ is barren to mean that there are no nontrivially isospectral binary sequences of length $n$. For integers $n \leq 34$, we found that the barren ones are $1$--$8$, $10$, $11$, $13$, $14$, $19$, $22$, $23$, $26$, and $29$. We prove that any multiple of a non-barren number is also not barren, and pose an open question as to whether there are finitely or infinitely many barren numbers.