Circular Convolution Property of Fourier Series 2.3 2002/09/05 2003/07/24 10:39:02 GMT-5 Justin Romberg jrom@rice.edu Mariyah Poonawala mariyah@rice.edu Prashant Singh prash@ece.rice.edu circular circular convolution convolution fourier fourier series This module looks at the basic circular convolution relationship between two sets of Fourier coefficients.

Signal Circular Convolution Given a signal f t with Fourier coefficients c n and a signal g t with Fourier coefficients d n , we can define a new signal, v t , where v t ⊛ f t g t We find that the Fourier Series representation of y t , a n , is such that a n c n d n . ⊛ f t g t is the circular convolution of two periodic signals and is equivalent to the convolution over one interval, i.e. ⊛ f t g t t 0 T τ 0 T f τ g t τ . Circular convolution in the time domain is equivalent to multiplication of the Fourier coefficients. This is proved as follows a n 1 T t 0 T v t ω 0 n t 1 T 2 t 0 T τ 0 T f τ g t τ ω 0 n t 1 T τ 0 T f τ 1 T t 0 T g t τ ω 0 n t ν ν t τ 1 T τ 0 T f τ 1 T ν τ T τ g ν ω 0 ν τ 1 T τ 0 T f τ 1 T ν τ T τ g ν ω 0 n ν ω 0 n τ 1 T τ 0 T f τ d n ω 0 n τ d n 1 T τ 0 T f τ ω 0 n τ c n d n Take a look at a square pulse with a period, T 1 T 4 :

For this signal c n 1 T n 0 1 2 2 n 2 n What signal has Fourier coefficients a n c n 2 1 4 2 n 2 2 n 2 ?

A triangle pulse train with a period of T 4 .