# 9.4 Discrete time convolution and the dtft

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This module describes the relationship between discrete convolution and the DTFT.

## Introduction

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

## 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.

## Dtft

$X(\omega )=\sum_{n=()}$ x n j ω n

## Inverse dtft

$x(n)=\frac{1}{2\pi }\int_{0}^{2\pi } X(\omega )e^{j\omega n}\,d \omega$

## 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

$y(n)=\sum_{k=()}$ x k h n k
As with continuous-time, convolution is represented by thesymbol *, and can be written as
$y(n)=(x(n), h(n))$
Convolution is commutative. For more information on the characteristics of convolution,read about the Properties of Convolution .

## Convolution theorem

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

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

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

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

## 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).

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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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