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

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.

DTFT

$$X\left(\omega \right)=\sum _{n=-\infty}^{\infty}x\left(n\right){e}^{-(j\omega n)}$$

1

Inverse DTFT

$$x\left(n\right)=\frac{1}{2\pi}{\int}_{0}^{2\pi}X\left(\omega \right){e}^{j\omega n}d\omega $$

2

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

$$y\left[n\right]=\sum _{k=-\infty}^{\infty}x\left[k\right]h\left[n-k\right]$$

3

$$y\left[n\right]=x\left[n\right]*h\left[n\right]$$

4

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

$$F(f*g)=F\left(f\right)\xb7F\left(g\right)$$

5

$$F(f\xb7g)=F\left(f\right)*F\left(g\right)$$

6

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

$$f*g={F}^{-1}(F\left(f\right)\xb7F\left(g\right))$$

7

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).