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.

翻译：非周期自相关衡量有限长复数序列与其自身平移版本之间的相似度。自相关在通信、遥感和科学仪器领域具有重要应用。自相关函数报告了所有可能平移下的非周期自相关值。已知序列的自相关函数等价于已知其傅里叶变换的幅度。消除相位信息缺失的问题被称为相位问题。我们称两个序列具有同谱性，是指它们拥有相同的非周期自相关函数。技术应用中使用的序列通常对其取值有约束：它们并非任意复数，而是源自可能位于复数域真子环或有限值集合的字母表。例如，二元序列仅包含 $+1$ 和 $-1$ 的项。本文在考虑字母表约束的前提下，研究两个序列具有同谱性的充要条件。存在一些通过可预测地保持自相关的修改产生的平凡同谱形式，例如序列取反、共轭项并逆序排列等。通过对长度不超过 $34$ 的二元序列进行穷举搜索，我们发现二元序列中确实存在非平凡同谱性，但出现频率极低。我们将正整数 $n$ 定义为贫瘠数，是指不存在长度为 $n$ 的非平凡同谱二元序列。对于 $n \leq 34$ 的整数，我们发现贫瘠数为 $1$--$8$、$10$、$11$、$13$、$14$、$19$、$22$、$23$、$26$ 和 $29$。我们证明任何非贫瘠数的倍数仍然是非贫瘠数，并提出了关于贫瘠数个数是有限还是无限的开放性问题。