Frequency modulation combines a signal with a carrier wave by changing (modulating) the carrier wave’s frequency.
Starting with a cosine carrier wave with frequency fc Hz and adding a signal with amplitude β and frequency fm Hz results in the combination
The factor β is known as the modulation index.
We’d like to understand this signal in terms of cosines without any frequency modulation. It turns out the result is a set of cosines weighted by Bessel functions of β.
We will prove the equation above, but first we’ll discuss what it means for the amplitudes of the cosine components.
For small values of β, Bessel functions decay quickly, which means the first cosine component will be dominant. For larger values of β, the Bessel function values increase to a maximum then decay like one over the square root of the index. To see this we compare the coefficients for modulation index β = 0.5 and β = 5.0.
First, β = 0.5:
and now for β = 5.0:
For fixed β and large n we have
and so the sideband amplitudes eventually decay very quickly.
Update: See this post for what the equation above says about energy moving from the carrier to sidebands.
To prove the equation above, we need three basic trig identities
and three Bessel function identities
The Bessel function identities above can be found in Abramowitz and Stegun as equations 9.1.42, 9.1.43, and 9.1.5.
And now the proof. We start with
and apply the sum identity for cosines to get
Now let’s take the first term
and apply one of our Bessel identities to expand it to
which can be simplified to
where the sum runs over all even integers, positive and negative.
Now we do the same with the second half of the cosine sum. We expand
which simplifies to
where again the sum is over all (odd this time) integers. Combining the two halves gives our result