Waveform Design using Multi-Tone Feedback Frequency Modulation

This paper introduces a waveform design method using Multi-Tone Feedback Frequency Modulation (MT-FFM), a generalization of the single oscillator feedback FM method developed by Tomisawa in The MT-FFM utilizes a collection of K harmonically related oscillators each governed by a design parameter z(k). These are in turn utilized as a discrete set of design parameters that may be modified to generate a richer set of modulation functions than in the single oscillator case. The waveforms resulting from these modulation functions can be designed to possess user defined properites by modifying the design parameters z(k). The MT-FFM's modulation function is represented in exact closed-form using a Kapteyn series composed of Generalized Bessel Functions (GBF). This paper describes the structure of the MT-FFM waveform, derives the Kapteyn series representation of the waveform's modulation function, and demonstrates the design method with a waveform design example.