Frequency modulation (FM) and phase modulation (PM) are well-known synthesis methods, which have been deployed widely in musical instruments. More recently, some synthesisers have implemented direct forms of FM (as opposed to PM), allowing, at least as part of their design, for higher-order modulation topologies. However, such implementations are affected by well-known difficulties that arise in the modulation of frequency, which are normally solved by the use of PM. In this article, we analyse these problems and using a direct comparison with PM, we put forward a solution for the direct application of FM in higher-order modulation arrangements. We begin by reviewing the theory of first-order FM, contrasting it to PM. We then proceed to develop a formulation of second-order FM which is equivalent to the issue-free PM synthesis, and present a closed-form expression for the evaluation of the second-order FM spectrum. We then extend the principle to higher-order topologies, by advancing the concept of an FM operator, analogous to the one used in PM instrument designs. From this we demonstrate that feedback FM is also a practical possibility. Finally, we complement the paper by giving a reference implementation in C++.

翻译：频率调制（FM）与相位调制（PM）是众所周知的合成方法，已广泛应用于乐器领域。近年来，部分合成器实现了直接形式的FM（区别于PM），至少在设计中允许采用高阶调制拓扑结构。然而，此类实现受到频率调制中常见问题的困扰，通常需借助PM解决。本文通过直接对比PM分析这些问题，提出直接应用FM于高阶调制方案的解决方案。我们首先回顾一阶FM理论，并将其与PM进行对比；继而推导与无问题PM合成等价的高阶FM公式，并给出二阶FM频谱计算的闭式表达式。随后通过提出类似于PM乐器设计中的FM算子概念，将原理推广至高阶拓扑结构，进而论证反馈FM的实践可行性。最后，我们提供C++参考实现作为论文补充。