**Bessel Functions and Sideband Strength**

As * I* increases, each sideband pair follows its own path of increasing and decreasing strength called a

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 =

The video below visually and audibly demonstrates the first 6 orders (0-5) of sidebands as * I* increases from 0 (i.e. no modulation) to 5. The carrier frequency is 700 Hz and the modulating frequency is 85 Hz. See if you can aurally track individual frequencies, such as the carrier (red) as they increase, decrease or disappear as

**Two Examples**

Here are two examples of the spectra produced for fixed values of * I,* computed by simply looking at the vertical example lines above. The first value of

The second example shows a higher value of * I* at a mod index of 4, which also includes some negative strengths where the Bessel function has dipped below 0. Note, for example, that the n=0 carrier frequency is now in inverted phase.

In general, as * I* increases, we can infer that more and more frequencies become audible. This can be a problem for digital synthesis, where the upper sidebands may reach the Nyquist frequency (see the digital audio chapter) and alias. FM is not band-limited. For this reason, most digital synthesis will have a limit on the maximum value of

Here are two audio examples. The first has a C:M ratio of 1:2 creating a harmonic spectrum, the second a C:M ratio of 1:1.31 creating an inharmonic spectrum. In both examples, the modulation index is increased slowly over 10 seconds from 0 (no modulation) to 15. Play these several times and focus on different frequencies as they move through their Bessel functions. Start with the carrier frequency (the first frequency you will hear). Listen as it immediately begins to lose strength, completely disappears and then reappears. Listen to the effects of phase cancellation in the first harmonic example, which will have far fewer discreet frequencies than the inharmonic one.

Example 1 C:M ratio = 1:2 | Example 2 C:M ratio = 1:1.31 |
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