Fourier Series/Transformations and Convolution

  • Thread starter Yosty22
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  • #1
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Homework Statement



(f*g)(x) = integral from -pi to pi of (f(y)g(x-y))dy
f(x) = ∑cneinx
g(x) = ∑dneinx

en is defined as the Fourier Coefficients for (f*g) {the convolution} an is denoted by:

en = 1/(2pi) integral from -pi to pi of (f*g)e-inx dx

Evaluate en in terms of cn and dn

Hint: somewhere in the integral, the substitution z = x - y might be helpful.

Homework Equations



For convolution, if C(x) = the integral from -infinity to infinity of (f(y)g(x-y))dx, the Fourier transform of C(x) is equal to the Fourier transform of f times the Fourier transform of g.

The Attempt at a Solution



I'm really not too sure where to begin with this other than understanding the definition in the Relevant Equations section. Before this problem, all we were told was this definition. I do not see any way this is related to the Fourier series mentioned in the problem, let alone how to incorporate just the coefficients of the series into the integral.

Any help to point me in the right direction would be greatly appreciated.
 

Answers and Replies

  • #2
vela
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Do you understand how that result, the theorem you mention in the relevant equations section, was obtained? If not, start there. You might try something similar in solving the problem.
 
  • #3
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I am pretty confused. The only thing jumping out to me on what to do is to substitute substitute (f*g)(x) = integral from -pi to pi of (f(y)g(x-y))dy into the integral for en and work it like a double integral. I am not exactly too sure if this would work though because I cannot think of where cn and dn would come in though (since he wants the answer in terms of those coefficients).
 

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