Frequency modulation using GLSL?

Question

1. Context

I use GLSL to plot the amplitude of the waveform at a given frequency as follows:

Displaying simple waveforms like the ones above is a trivial task, it's just a matter of using the right equations (the available GLSL code is here ). Now I am trying to show the results of frequency modulation between two waveforms.

---

2. Research

After some research, I found two possible ways to do this:

• Using a dsp-like approach , as far as I know, using phase batteries (see the first answer here ) in combination with search tables is the proposed approach when working with signal processing. As a GLSL beginner, I realized that this is not possible with GLSL shaders because they cannot store and increment variables in several GPU loops.

• Using a pure mathematical approach , this includes equations such as Chowning a simple formula fm . Formula FM http://img577.imageshack.us/img577/4820/fmformula.png
  This formula works great in some situations (a cosine wave modulating another cosine or cosine modulating a sawtooth saw ), but I could not find a general equation that would work in each case (when a sawtooth wave modulates the cosine, I expect the carrier frequency to be modulated by a sawtooth amplitude but all I get is a carrier wave, apparently unmodulated).

---

3. Questions

• What would be the best approach to solve this problem?
• Is it possible to use a DSP-like approach in GLSL?
• If not, is there any general FM equation flexible enough to do all the work?

Given my lack of skills in all the disciplines involved here (audio dsp, computer science, gpu programming, math), I would not be surprised if I miss something really simple here. Be patient.

Answer 1

You are right that the conventional dsp-like phase battery approach is not well suited for parallel computing on a GPU; therefore, the “pure mathematical” method is probably the best option.

A generalization of the simple FM Chowning formula to more general frequency modulation functions is provided on the Wikipedia page for frequency modulation (the very first equation on this page). The key point is that the argument to the cos function is the phase, which, as the equation in Wikipedia shows, is an integral over time of the frequency. In FM, the frequency is usually the carrier plus modulation: for example, with the Chowning formula for simple sinusoidal FM, the frequency as a function of time t is

 f(t) = f_c - M * f_m * sin(f_m * t) 

where f_c is the carrier frequency, M is the amount of modulation, and f_m is the modulation frequency. It integrates with phase

 p(t) = f_c * t + M * cos(f_m * t) 

which corresponds to the phase in the equation in your question.

To modulate the cosine with a sawtooth saw, f(t) will be a sawtooth wave (plus the carrier frequency), and therefore, to find p(t) , you will need to find the time integral of the sawtooth wave. It's relatively simple (it should be a piecewise-quadratic function), but the people from math.stackexchange should be able to help if you have difficulties.

(Note. I have formulated everything here in terms of time t , but it could also just be a space x .)