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Continuous Time Convolution and the CTFT

Module by: Stephen Kruzick. E-mail the author

Summary: This module describes the relationship between continuous convolution and the fourier transform.

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−(j⁢Ω⁢t)d t ℱ Ω t f t Ω t
(1)

Inverse CTFT

f⁢t=12⁢π⁢∫−∞∞ℱ⁢Ω⁢ej⁢Ω⁢td Ω f t 1 2 Ω ℱ Ω Ω t
(2)

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
(3)
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
(4)
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. To Download, right-click and save target as .cdf.
CTFTdenoiseDemo

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 ) = 12⁢π F ( f ) * F ( g ) F ( f · g ) = 1 2 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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