# OpenStax_CNX

You are here: Home » Content » Derivation of the Fourier Transform

### Recently Viewed

This feature requires Javascript to be enabled.

# Derivation of the Fourier Transform

Module by: Carlos E. Davila. E-mail the author

Summary: The Fourier transform is derived from the Fourier series.

Let's begin by writing down the formula for the complex form of the Fourier Series:

x ( t ) = n = - c n e j n Ω 0 t x ( t ) = n = - c n e j n Ω 0 t
(1)

as well as the corresponding Fourier Series coefficients:

c n = 1 T t 0 t 0 + T x ( t ) e - j n Ω 0 t d t c n = 1 T t 0 t 0 + T x ( t ) e - j n Ω 0 t d t
(2)

As was mentioned in Chapter 2, as the period TT gets large, the Fourier Series coefficients represent more closely spaced frequencies. Lets take the limit as the period TT goes to infinity. We first note that the fundamental frequency approaches a differential

Ω 0 = 2 π T d Ω Ω 0 = 2 π T d Ω
(3)

consequently

1 T = Ω 0 2 π d Ω 2 π 1 T = Ω 0 2 π d Ω 2 π
(4)

The nnth harmonic, nΩ0nΩ0, in the limit approaches the frequency variable ΩΩ

n Ω 0 Ω n Ω 0 Ω
(5)

From equation Equation 2, we have

c n T - x ( t ) e - j Ω t d t c n T - x ( t ) e - j Ω t d t
(6)

The right hand side of Equation 6 is called the Fourier Transform of x(t)x(t):

X ( j Ω ) - x ( t ) e - j Ω t d t X ( j Ω ) - x ( t ) e - j Ω t d t
(7)

Now, using Equation 6, Equation 4, and Equation 5 in equation Equation 1 gives

x ( t ) = 1 2 π - X ( j Ω ) e j Ω t d Ω x ( t ) = 1 2 π - X ( j Ω ) e j Ω t d Ω
(8)

which corresponds to the inverse Fourier Transform. Equations Equation 7 and Equation 8 represent what is known as a transform pair. The following notation is used to denote a Fourier Transform pair

x ( t ) X ( j Ω ) x ( t ) X ( j Ω )
(9)

We say that x(t)x(t) is a time domain signal while X(jΩ)X(jΩ) is a frequency domain signal. Some additional notation which is sometimes used is

X ( j Ω ) = F x ( t ) X ( j Ω ) = F x ( t )
(10)

and

x ( t ) = F - 1 X ( j Ω ) x ( t ) = F - 1 X ( j Ω )
(11)

References

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks