9.4 Discrete time convolution and the dtft

Introduction

This module discusses convolution of discrete signals in the time and frequency domains.

The discrete-time convolution

Discrete time fourier transform

The DTFT transforms an infinite-length discrete signal in the time domain into an finite-length (or $2\pi$ -periodic) continuous signal in the frequency domain.

Demonstration

Convolution sum

As mentioned above, the convolution sum provides a concise, mathematical way to express the output of an LTI system basedon an arbitrary discrete-time input signal and the system's impulse response. The convolution sum is expressed as

Convolution theorem

Let $f$ and $g$ be two functions with convolution $f*g$ .. Let $F$ be the Fourier transform operator. Then

By applying the inverse Fourier transform $F^{-1}$ , we can write:

Conclusion

The Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain (e.g., time domain) corresponds to point-wise multiplication in the other domain (e.g., frequency domain).