Circular Convolution Property of Fourier Series

Discussion in 'Homework Help' started by blah2222, Dec 3, 2011.

  1. blah2222

    Thread Starter Well-Known Member

    May 3, 2010
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    Hey all, I know that I've been asking a ton of questions but I have yet again hit another stumbling block along the way. I am becoming very comfortable with Fourier Series and have not moved onto Transforms yet, so this question is not regarding those.

    I am just wondering what this means in terms of Fourier Series. By definition, in order to perform the circular convolution of two signals, do they both have to have the same period, T, or can they be any two periodic signals? Also, I understand that the main result of this is that if you have two functions f(t) and g(t) and perform the circulation convolution between them to result in h(t), the complex exponential Fourier Series coefficients of v(t), Vn, will be the product of the coefficients of f(t) and g(t).

    Here are my notes from class and I am wondering why Vn is the product of both coefficients times the period T. But in the example on this site, T is no where to be found?

    Here are my notes: Vn = (T)(Fn)(Gn)

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    Here is the site that says: Vn = (Fn)(Gn)

    [​IMG]

    Thanks!
     
  2. blah2222

    Thread Starter Well-Known Member

    May 3, 2010
    554
    33
    Maybe I should clarify my questions if they aren't clear.

    1) Is circular convolution restricted to signals of the same period?

    2) Is the circular convolution result that the coefficient of the resultant signal is the product of the coefficients of the input signals or the latter multiplied by the period of the signals, (assuming it is the same for both)?

    Thanks
     
  3. blah2222

    Thread Starter Well-Known Member

    May 3, 2010
    554
    33
    Upon re-reading this over and over again I came to the key difference between the two. It is how they differ in their Circular Convolution (CC) definitions by a scaling factor (1/T).

    In my notes CC is defined as:

    g(t) = f1(t)*f2(t) = \int ^T_0 f1(tau)f2(t - tau)dtau ---> Gn = T(F1n)(F2n)

    Where as the website is defining it as:

    g(t) = f1(t)*f2(t) = \frac{1}{T}\int ^T_0 f1(tau)f2(t - tau)dtau --> Gn = (F1n)(F2n)

    Which is correct?
     
  4. blah2222

    Thread Starter Well-Known Member

    May 3, 2010
    554
    33
    EDIT:

    They are both correct as they both account for the (1/T) scaling factor, but one is implemented in the CC definition and the other is in the coefficient result.

    Found this out on my own.

    Thanks be to me...
     
  5. thatoneguy

    AAC Fanatic!

    Feb 19, 2009
    6,357
    718
    Nice work! And, you are welcome. We are great listeners sometimes. :D
     
    blah2222 likes this.
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