How FM Killed the Additive Star, Part I

Additive synthesis, as we have explored earlier in the blog, is an excellent way to create very complex and interesting periodic waves. Also, since additive synthesis is modeled after the Fourier series, it is also quite simple, conceptually and mathematically, to understand. However, an issue which arises with additive synthesis is that it becomes quite expensive to compute waves with many harmonics compounded upon it. An oscillator must be provided for each wave that is intended to be part of the summation; calculating all samples creates a fair bit of overhead. Additionally, while the waveforms are intricate in their own right, studies have shown that a large portion of our ability to recognize instrument timbre is due to the attack and decay patterns of each instrument. Additive synthesis does a poor job of emulating accurately these patterns.

Fortunately, Frequency Modulation Synthesis (FM synthesis) arose to fulfill such requisites. Developed by John Chowning at Stanford during the 1970's, FM synthesis is an alternative means by which complex audio signals may be synthesized. However, the process is much cheaper compared to additive synthesis and provides ample flexibility to model to a very granular level of accuracy the intricate attacks and decays of actual instruments. While FM synthesis is somewhat "recent" in its conception, frequency modulation has seen its fair share of applications in years prior, most notably, in FM radio.

Mathematics
Take a look at the FM synthesis equation below:

Amplitude, which may vary by time, simply controls the height of the wave at a given time t. The carrier frequency is the audio frequency about which all the FM sidebands are clustered. Modulation index indicates the amount by which the modulating frequency will affect the carrier frequency over time. The modulating frequency is an additional audio-ranged frequency which is used to alter the carrier frequency.

A bit confusing? Understandably so. To perhaps elucidate the consequences of the equation, we can simplify its appearance a bit:

where phi is our angular frequency for our carrier frequency and beta is our angular frequency for our modulating frequency. We can safely ignore amplitude and modulation index for now. The resulting equation has a trig identity which is the infinite sum of sinusoidal waves of varying phases, multiplied by a Bessel function (with which I have NO experience... mathematics is black magic!). These infinite sinusoids actually represent the sidebands previously mentioned. They can be thought of as our harmonics when adding periodic waves together in additive synthesis. Cool stuff, no?

For a visual and audio walkthrough on this stuff, check out this link here.

In the next post, I'll provide a code and audio example of FM synthesis so that we can get a more tangible idea of the technique.

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