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Continuous Time Convolution and the CTFT

By Stephen Kruzick

Introduction

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

Continuous Time Fourier Transform

The CTFT transforms a infinite-length continuous signal in the time domain into an infinite-length continuous signal in the frequency domain.

CTFT
Ω=fte(iΩt)d t Ω t f t Ω t
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Inverse CTFT
ft=12πΩeiΩtd Ω f t 1 2 Ω Ω Ω t
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Convolution Integral

The convolution integral expresses the output of an LTI system based on an input signal, xt x t , and the system's impulse response, ht h t . The convolution integral is expressed as

yt=xτhtτd τ y t τ x τ h t τ
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Convolution is such an important tool that it is represented by the symbol * , and can be written as
yt=xt*ht y t x t h t
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Convolution is commutative. For more information on the characteristics of the convolution integral, read about the Properties of Convolution.

Demonstration

Figure 1: Interact (when online) with a Mathematica CDF demonstrating Use of the CTFT in signal denoising.

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).