Discrete-time Fourier Transform - Convolution

Convolution

The Convolution theorem for sequences is:



An important special case is the circular convolution of sequences and defined by where is a periodic summation. The discrete-frequency nature of "selects" only discrete values from the continuous function which results in considerable simplification of the inverse transform. As shown at Convolution_theorem#Functions_of_a_discrete_variable..._sequences:

\begin{align} x_N * y\ &=\ \scriptstyle{DTFT}^{-1} \displaystyle \big\\ &=\ \scriptstyle{DFT}^{-1} \displaystyle \big. \end{align}

For and sequences whose non-zero duration is ≤ N, a final simplification is:



The significance of this result is expounded at Circular convolution and Fast convolution algorithms.

Read more about this topic:  Discrete-time Fourier Transform