# Circular Convolution Property of Fourier Series

#### blah2222

Joined May 3, 2010
582
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)  Here is the site that says: Vn = (Fn)(Gn) Thanks!

#### blah2222

Joined May 3, 2010
582
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

#### blah2222

Joined May 3, 2010
582
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?

#### blah2222

Joined May 3, 2010
582
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...

#### thatoneguy

Joined Feb 19, 2009
6,359
Nice work! And, you are welcome. We are great listeners sometimes. • blah2222