**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 *f*_{c} Hz and adding a signal with amplitude β and frequency *f*_{m} 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.

## Component amplitudes

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.

## Proof

To prove the equation above, we need three basic trig identities

and a 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

to

which simplifies to

where again the sum is over all (odd this time) integers. Combining the two halves gives our result

**Related post**: Visualizing Bessel functions

Your final formula sums over values of k but has n in the body. This also happens when you state the formula right before the proof.

Thanks. I just fixed that.

You could also take a Hilbert transform of the signal compute the instantaneous frequency (d phase(y_analytic(t))/dt) and get right back at the 2*pi*f_c + beta sin(2 pi f_m t) 😉

Which seems a little specious, but Hilbert analysis (and the Hilbert-Huang Transform, https://cds.cern.ch/record/1115835?ln=no) is great for working with a non-stationary FM signal.