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# 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
$ℱ⁢Ω=∫−∞∞f⁢t⁢e−(i⁢Ω⁢t)d t ℱ Ω t f t Ω t$
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Inverse CTFT
$f⁢t=12⁢π⁢∫−∞∞ℱ⁢Ω⁢ei⁢Ω⁢td Ω f t 1 2 Ω ℱ Ω Ω t$
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# Convolution Integral

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

$y⁢t=∫−∞∞x⁢τ⁢h⁢t−τd τ y t τ x τ h t τ$
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Convolution is such an important tool that it is represented by the symbol $*$, and can be written as
$y⁢t=x⁢t*h⁢t y t x t h t$
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# 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).