Introduction to Computer Music: Volume One

6. Principles of Audio-rate Frequency Modulation | page 8

As I increases, each sideband pair follows its own path of increasing and decreasing strength called a Bessel function. The Bessel function curves followed are different for each of the n-order sidebands--one of the things that makes frequency modulation so interesting. (To trig students, these are called Bessel functions of the first kind of order n; to non-trig students, it's more like Close Encounters of the Third Kind.) Below is a graph of the first seven orders (beginning with 0) of sideband pairs, showing their relative strength on the vertical axis as I increases on the horizontal axis.

Note that at I = 0 (i.e. no modulation), the carrier (red, n=0) is at full strength. As I increases, several things happen. Firstly, the carrier loses strength, and secondly, each additional order of sideband pairs begins to be heard one by one. A good rule of thumb for predicting how many sideband pairs (n) will be audible for a given value of I is: n = I + 1 with I being rounded off to the nearest integer. Each pair, after its initial peak, will decrease in strength and, for a given value of I, be inaudible as it crosses zero. On the negative side of zero, it will be in reverse phase to any similar frequency on the positive side of zero.

To make matters slightly more complicated, each odd-numbered lower sideband (n=1,3,5,...) follows the inverse of the Bessel function associated with it's pair (equivalent to multiplying its strength by -1).

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