Chapter Four: Synthesis

### 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

*increases on the horizontal axis.*

**I**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 =

*+ 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.*

**I**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).