In this work, we consider the modeling of signals that are almost, but not quite, harmonic, i.e., composed of sinusoids whose frequencies are close to being integer multiples of a common frequency. Typically, in applications, such signals are treated as perfectly harmonic, allowing for the estimation of their fundamental frequency, despite the signals not actually being periodic. Herein, we provide three different definitions of a concept of fundamental frequency for such inharmonic signals and study the implications of the different choices for modeling and estimation. We show that one of the definitions corresponds to a misspecified modeling scenario, and provides a theoretical benchmark for analyzing the behavior of estimators derived under a perfectly harmonic assumption. The second definition stems from optimal mass transport theory and yields a robust and easily interpretable concept of fundamental frequency based on the signals’ spectral properties. The third definition interprets the inharmonic signal as an observation of a randomly perturbed harmonic signal. This allows for computing a hybrid information theoretical bound on estimation performance, as well as for finding an estimator attaining the bound. The theoretical findings are illustrated using numerical examples.

中文翻译：

---

为几乎谐波信号定义基频

在这项工作中，我们考虑对几乎（但不完全）谐波的信号进行建模，即由频率接近公共频率整数倍的正弦波组成。通常，在应用中，这些信号被视为完美的谐波，允许估计它们的基频，尽管这些信号实际上不是周期性的。在此，我们为此类非谐波信号的基频概念提供了三种不同的定义，并研究了建模和估计的不同选择的含义。我们表明其中一个定义对应于错误指定的建模场景，并为分析在完美调和假设下得出的估计量的行为提供了理论基准。第二个定义源于最优质量传输理论，并基于信号的频谱特性产生了一个稳健且易于解释的基频概念。第三个定义将非谐波信号解释为对随机扰动的谐波信号的观察。这允许计算关于估计性能的混合信息理论界限，以及寻找达到界限的估计量。使用数值例子说明了理论发现。以及寻找达到界限的估计量。使用数值例子说明了理论发现。以及寻找达到界限的估计量。使用数值例子说明了理论发现。