Theory and practice of higher-order frequency modulation synthesis

Frequency modulation (FM) and phase modulation (PM) are well-known synthesis methods, which have been deployed widely in musical instruments. In this paper, we analyse the design of stacked FM synthesis and using a direct comparison with PM, put forward a method we call higher-order frequency modulation (hoFM). We begin by reviewing the theory of first-order FM, contrasting it to PM. We then discuss the problems of extending first-order FM by simply applying the modulation to the frequency, which may result in carrier drift caused by the presence of DC in the modulating signal. We proceed to develop a formulation of second-order FM which is equivalent to the issue-free PM synthesis, and present an expression for the evaluation of the second-order FM spectrum. By virtue of the application of amplitude modulation concurrently with frequency modulation, we are able to eliminate the DC component and thus any carrier drift caused by it. These principles are then extended to higher-order topologies, where we note that in the general case, the modulation signal at each level is amplitude modulated by its own FM input. From this, we are able to advance the concept of an FM operator, analogous to the one used in PM instrument design, to realise hoFM. From this we demonstrate that feedback FM is also a practical possibility. Finally, moving from continuous to discrete time, we develop a reference C++ implementation for computer music applications, and discuss issues relating to digital implementations.