Analyzing an FM signal

Posted on by John

radio dial

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

\cos( 2\pi f_c t + \beta \sin(2\pi f_m t) )

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.

\cos( 2\pi f_c t + \beta \sin(2\pi f_m t) ) = \sum_{k=-\infty}^\infty J_n(\beta) \cos(2\pi(f_c + nf_m)t)

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

J_n(\beta) \approx \frac{\beta^n}{2^n \, n!}

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

\cos(A + B) &=& \cos A \cos B - \sin A \sin B \\ 2\cos A \cos B &=& \cos(A-B) + \cos(A+B) \\ 2\sin A \sin B &=& \cos(A-B) - \cos(A+B)

and three Bessel function identities

\cos( z \sin \theta) &=& J_0(z) + 2\sum_{k=1}^\infty J_{2k}(z) \cos(2k\theta) \\ \sin( z \sin \theta) &=& 2\sum_{k=0}^\infty J_{2k+1}(z) \sin((2k+1)\theta) \\ J_{-n}(z) &=& (-1)^n J_n(z)

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

\cos( 2\pi f_c t + \beta \sin(2\pi f_m t) )

and apply the sum identity for cosines to get

\cos(2\pi f_c t) \cos(\beta \sin(2\pi f_m t)) - \sin(2\pi f_c t) \sin(\beta \sin(2\pi f_m t))

Now let’s take the first term

\cos(2\pi f_c t) \cos(\beta \sin(2\pi f_m t))

and apply one of our Bessel identities to expand it to

J_0(\beta) \cos(2\pi f_c t) + \sum_{k=1}^\infty J_{2k}(\beta) \left\{ \cos(2\pi (f_c - 2k f_m)t) + \cos(2\pi(f_c + 2k f_m)t) \right\}

which can be simplified to

\sum_{n \,\, \mathrm{even}} J_n(\beta) \cos(2\pi(f_c + nf_m)t)

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

\sin(2\pi f_c t) \sin(\beta \sin(2\pi f_m t))

to

\sum_{k=1}^\infty J_{2k+1}(\beta) \left\{ \cos(2\pi (f_c - (2k+1) f_m)t) - \cos(2\pi(f_c + (2k+1) f_m)t) \right\}

which simplifies to

\sum_{k=1}^\infty J_{2k+1}(\beta) \left\{ \cos(2\pi (f_c - (2k+1) f_m)t) - \cos(2\pi(f_c + (2k+1) f_m)t) \right\}

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

\cos( 2\pi f_c t + \beta \sin(2\pi f_m t) ) = \sum_{k=-\infty}^\infty J_n(\beta) \cos(2\pi(f_c + nf_m)t)

***

More on mathematics and radio.

5 thoughts on “Analyzing an FM signal”

  1. tcmJOE

    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.

  3. Javier Glz.

    What would change if I want to take in acount the imaginary part of the frecuency modulation signal?

  5. Hank Largey

    Really great explanation, I think another way to examine this would be to use the Taylor series expansion of the sin in the argument, this may be a bit tedious but would better explain where the Bessel function multiplier comes from.

Comments are closed.