Introduction to Computer Music: Volume One

6. Principles of Audio-rate Frequency Modulation | page 3

Calculating sideband frequencies

Sideband frequencies can be calculated with the following formula where Cƒ=carrier frequency, Mƒ=modulator frequency, n=all positive integers including 0:

Cƒ ± nMƒ   [n=0,1,2,3...]

OR (for those with math anxiety)

the carrier frequency (Cƒ) plus and minus all the integer multiples of the modulating frequency (Mƒ)

OR (for those with really serious math anxiety)

Cƒ, Cƒ + Mƒ, Cƒ - Mƒ, Cƒ + 2Mƒ, Cƒ - 2Mƒ, Cƒ + 3Mƒ, Cƒ - 3Mƒ, etc. to infinity and beyond

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 (n=0 to n=2)

integer
multiple
fm spectrum math to find spectrum
n=0 400 Hz 400 + (0 * 50)
n=1 450 Hz, 350 Hz 400 + (1 * 50), 400 - (1 * 50)
n=2 500 Hz, 300 Hz 400 + (2 * 50), 400 + (2 * 50)

A graph of this example's sideband pairs, related by color, appears below:

Another way of 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 Cƒ of 100 and an Mƒ 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 below that integer ratios that have a carrier value of 1 have certain properties. We could calculate the upper sidebands for this ratio in relation to the carrier frequency as C+M, C+2M, C+3M, C+4M, etc. For our 1:2 example, the first upper sideband would be 1+2=3, the second would be 1 + (2 * 2)=5. If you worked this out in Hz, you would quickly come to the conclusion that these are the odd numbered partials of the carrier frequency. We calculate the lower sidebands similarly as C-M, C-2M, C-3M, C-4M or in our 1:2 example, -1, -3, -5, etc.

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