Considerations for a Computational Estimation of the Complex FM Spectrum

Abstract

Complex frequency modulation (FM) is a useful method for sound synthesis as it allows the creation of rich timbres using only a small number of waveforms. However, the mathematical expression for the frequency domain representation of a frequency modulated (FM) wave with a non-sinusoidal modulator is a complex equation with multiple summation terms. Its implementation in software requires nested loops, with the number of loops depending on the number of sinusoidal components required to describe the complex modulator wave. Although the spectral representation can be easily computed from the FFT of the time domain waveform, this is at the expense of the insight that can be gained by direct evaluation of the analytic expression for the frequency domain representation. This paper looks at creating a software implementation of the analytic expression that overcomes the need for many nested loops and can be generalized. It will also suggest how it could be applied in a distributed processing environment