(adsbygoogle = window.adsbygoogle || []).push({}); 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 = W_{0}+/- W_{1}+/- W_{2}

The equation for this FM wave is y(t) = Acos(W_{0}t+W_{1}COS(W_{2}t)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(W_{0}t+W_{1}COS(W_{2}t)t ) =

= A{cos(W_{0}t)cos(W_{1}COS(W_{2}t)t ) -

sin(W_{0}t)sin(W_{1}sin(W_{2}t)t )}

we are given that W_{1}t<<1

and W_{1}^{2}t -> 0

so if we assume cos(W_{1}COS(W_{2}t)t )~1

Then we have:

= A{cos(W_{0}t)-sin(W_{0}t)W_{1}cos(W_{2}t)t )}

= A{cos(W_{0}t) - sin(W_{0}t+W_{2}t)W_{1}+

W_{1}cos(W_{0}t)sin(W_{2}t)

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

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# Homework Help: Frequency components of FM wave

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