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

DTFT
$X⁢ω=∑ n =−∞∞x⁢n⁢e−(j⁢ω⁢n) X ω n x n j ω n$
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Inverse DTFT
$x⁢n=12⁢π⁢∫02⁢πX⁢ω⁢ej⁢ω⁢nd ω x n 1 2 ω 0 2 X ω j ω n$
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# 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

$y⁢n=∑ k =−∞∞x⁢k⁢h⁢n−k y n k x k h n k$
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As with continuous-time, convolution is represented by the symbol *, and can be written as
$y⁢n=x⁢n*h⁢n y n x n h n$
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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).