# Frequency components of FM wave

#### H_man

1. The problem statement, all variables and given/known data

I wanted to show that a particular wave is composed of the 4 frequencies

W = W0 +/- W1 +/- W2

The equation for this FM wave is y(t) = Acos(W0t+W1COS(W2t)t )

I tried showing the different frequency components by performing a fourier transform on the equation y(t). However I got a bit stuck with integrating the cosine term as part of the exponential function. I thought about expanding it as a Taylor series but that doesn't really help.

If this was a practical problem I would just use Matlab and a DFT and hey presto. But trying to do this analytically is posing a problem.

Expanding in terms of trigonometric identities to show the point also doesn't seem to get me closer to a solution.

y(t) = Acos(W0t+W1COS(W2t)t ) =

= A{cos(W0t)cos(W1COS(W2t)t ) -
sin(W0t)sin(W1sin(W2t)t )}

we are given that W1t<<1
and W12t -> 0
so if we assume cos(W1COS(W2t)t )~1

Then we have:

= A{cos(W0t)-sin(W0t)W1cos(W2t)t )}

= A{cos(W0t) - sin(W0t+W2t)W1 +
W1 cos(W0t)sin(W2t)

But I don't see a solution from here....
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

#### H_man

Its okay. I just found the solution. It involves Bessel Functions....

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