**Calculating sideband frequencies **

Sideband frequencies can be calculated with the following formula where

**C _{ƒ}** = carrier frequency,

OR (in words)

**the carrier frequency (C _{ƒ}) plus and minus all the integer multiples of the modulating frequency (M_{ƒ}) beginning with the C_{ƒ} ± 0.**

OR (for those with serious math anxiety)

C_{ƒ}, C_{ƒ} + M_{ƒ}, C_{ƒ} - M_{ƒ}, C_{ƒ} + 2M_{ƒ}, C_{ƒ} - 2M_{ƒ}, C_{ƒ} + 3M_{ƒ}, C_{ƒ} - 3M_{ƒ}, etc. to infinity and beyond.*

C

For example, a **carrier frequency of 400 **and a **modulating frequency of 50** will produce the spectrum listed below for the first three sideband pairs (indices of n=0 to n=2)

integermultiple index |
fm spectrum
| formula to find spectrum |
math to find spectrum |
---|---|---|---|

n=0 | 400 Hz | C_{ƒ} ± (0 * M_{ƒ}) |
400 + (0 * 50) |

n=1 | 450 Hz, 350 Hz | C_{ƒ} ± (1 * M_{ƒ}) |
400 + (1 * 50), 400 - (1 * 50) |

n=2 | 500 Hz, 300 Hz | C_{ƒ} ± (2 * M_{ƒ}) |
400 + (2 * 50), 400 + (2 * 50) |

A graph of these sideband pairs up to n = 4 looks like this:

**Sideband Phase**

Note that the ODD number indexed pairs (n = 0,3,5, etc) have their lower sideband component in reverse phase to their positive upper sideband counterparts, indicated in the graph as the descending lines. This will become important shortly when we discuss sideband strength and reflected sidebands in the next few pages.

**Carrier:Modulator (C:M) Ratio**

Another method for calculating sidebands, useful when the carrier and modulator are maintaining a constant frequency relationship, is through a ratio of carrier to modulating frequency (C:M). For example, a C_{ƒ} of 100 and an M_{ƒ} of 200 would produce a **C:M ratio** of 1:2 (click here to see how to reduce C:M ratios to their *normal form*). We'll see later that integer ratios that have a carrier value of 1 have certain harmonic properties. It is important to note, however, that *the carrier frequency is not necessarily the fundamental of a harmonic spectrum* in FM, since it may not be the lowest sounding pitch produced. In the graphic example above, with C_{ƒ} = 400 Hz and M_{ƒ} = 50 Hz, the lowest souding sideband is 50 Hz, which would be considered the fundamental.