  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.
