- #1
H_man
- 145
- 0
Homework Statement
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...