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Discrete Time Convolution and the DTFT

By Stephen Kruzick, Dan Calderon

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 2π 2 -periodic) continuous signal in the frequency domain.

DTFT
Xω= n =xne(jωn) X ω n x n j ω n
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Inverse DTFT
xn=12π02πXωejωnd ω x n 1 2 ω 0 2 X ω j ω n
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Demonstration

Figure 1: Interact (when online) with a Mathematica CDF demonstrating the Discrete Convolution.

Convolution Sum

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

yn= k =xkhnk y n k x k h n k
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As with continuous-time, convolution is represented by the symbol *, and can be written as
yn=xn*hn y n x n h n
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Convolution is commutative. For more information on the characteristics of convolution, read about the Properties of Convolution.

Convolution Theorem

Let ff and gg be two functions with convolution f*gf*g.. Let FF be the Fourier transform operator. Then

F ( f * g ) = F ( f ) · F ( g ) F ( f * g ) = F ( f ) · F ( g )
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F ( f · g ) = F ( f ) * F ( g ) F ( f · g ) = F ( f ) * F ( g )
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By applying the inverse Fourier transform F-1F-1, we can write:

f * g = F - 1 ( F ( f ) · F ( g ) ) f * g = F - 1 ( F ( f ) · F ( g ) )
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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).