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Continuous-Time Fourier Transform (CTFT)

Module by: Richard Baraniuk, Melissa Selik. E-mail the authors

Summary: Details the Continuous-Time Fourier Transform.

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Introduction

Due to the large number of continuous-time signals that are present, the Fourier series provided us the first glimpse of how me we may represent some of these signals in a general manner: as a superposition of a number of sinusoids. Now, we can look at a way to represent continuous-time nonperiodic signals using the same idea of superposition. Below we will present the Continuous-Time Fourier Transform (CTFT), also referred to as just the Fourier Transform (FT). Because the CTFT now deals with nonperiodic signals, we must now find a way to include all frequencies in the general equations.

Equations

Continuous-Time Fourier Transform

ℱ⁢Ω=∫−∞∞f⁢t⁢e−(i⁢Ω⁢t)d t ℱ Ω t f t Ω t
(1)

Inverse CTFT

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

Warning:

Do not be confused by notation - it is not uncommon to see the above formula written slightly different. One of the most common differences among many professors is the way that the exponential is written. Above we used the radial frequency variable Ω Ω in the exponential, where Ω=2⁢π⁢f Ω 2 f , but one will often see professors include the more explicit expression, i⁢2⁢π⁢f⁢t 2 f t , in the exponential. Click here for an overview of the notation used in Connexion's DSP modules.

The above equations for the CTFT and its inverse come directly from the Fourier series and our understanding of its coefficients. For the CTFT we simply utilize integration rather than summation to be able to express the aperiodic signals. This should make sense since for the CTFT we are simply extending the ideas of the Fourier series to include nonperiodic signals, and thus the entire frequency spectrum. Look at the Derivation of the Fourier Transform for a more in depth look at this.

Relevant Spaces

The Continuous-Time Fourier Transform maps infinite-length, continuous-time signals in L2L2 to infinite-length, continuous-frequency signals in L2L2. Review the Fourier Analysis for an overview of all the spaces used in Fourier analysis.

Figure 1: Mapping L 2 ⁢R L 2 in the time domain to L2⁢R L2 in the frequency domain.
Figure 1 (CTFTspacee.png)

Example Problems

Exercise 1

Find the Fourier Transform (CTFT) of the function

f⁢t={e−(α⁢t)  if  t≥00  otherwise   f t α t t 0 0
(3)

Solution

In order to calculate the Fourier transform, all we need to use is Equation 1, complex exponentials, and basic calculus.

ℱ⁢Ω=∫−∞∞f⁢t⁢e−(i⁢Ω⁢t)d t =∫0∞e−(α⁢t)⁢e−(i⁢Ω⁢t)d t =∫0∞e(−t)⁢(α+i⁢Ω)d t =0−-1α+i⁢Ω ℱ Ω t f t Ω t t 0 α t Ω t t 0 t α Ω 0 -1 α Ω
(4)
ℱ⁢Ω=1α+i⁢Ω ℱ Ω 1 α Ω
(5)

Exercise 2

Find the inverse Fourier transform of the square wave defined as

X⁢Ω={1  if  |Ω|≤M0  otherwise   X Ω 1 Ω M 0
(6)

Solution

Here we will use Equation 2 to find the inverse FT given that t≠0 t 0 .

x⁢t=12⁢π⁢∫−MMei⁢Ωtd Ω =12⁢π⁢ei⁢Ωt| Ω , Ω = ei⁢w =1π⁢t⁢sinM⁢t x t 1 2 Ω M M Ω t Ω w 1 2 Ω t 1 t M t
(7)
x⁢t=Mπ⁢(sinc⁢M⁢tπ) x t M sinc M t
(8)

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