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Homework Help: Frequency components of FM wave

  1. Dec 12, 2009 #1
    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
     
  2. jcsd
  3. Dec 12, 2009 #2
    Its okay. I just found the solution. It involves Bessel Functions....
     
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