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Discrete Time Convolution and the DTFT

Module by: Stephen Kruzick, Dan Calderon. E-mail the authors

Summary: 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.

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

DTFT

X⁢ω=∑ n =−∞∞x⁢n⁢e−(j⁢ω⁢n) X ω n x n j ω n
(1)

Inverse DTFT

x⁢n=12⁢π⁢∫02⁢πX⁢ω⁢ej⁢ω⁢nd ω x n 1 2 ω 0 2 X ω j ω n
(2)

Demonstration

Figure 1: Interact (when online) with a Mathematica CDF demonstrating the Discrete Convolution. To Download, right-click and save as .cdf.
DiscreteConvolutionDemo

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

y⁢n=∑ k =−∞∞x⁢k⁢h⁢n−k y n k x k h n k
(3)
As with continuous-time, convolution is represented by the symbol *, and can be written as
y⁢n=x⁢n*h⁢n y n x n h n
(4)
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 )
(5)
F ( f · g ) = F ( f ) * F ( g ) F ( f · g ) = F ( f ) * F ( g )
(6)

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 ) )
(7)

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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