# Connexions

You are here: Home » Content » Symmetry Properties of the Fourier Transform

### Recently Viewed

This feature requires Javascript to be enabled.

# Symmetry Properties of the Fourier Transform

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

Summary: Looks at some of the consequences of conjugate symmetry of the Fourier transform of real signals.

When x(t)x(t) is real, the Fourier transform has conjugate symmetry, X(-jΩ)=X(jΩ)*X(-jΩ)=X(jΩ)*. It is not hard to see this:

X ( j Ω ) * = - x ( t ) e - j Ω t d t * = - x ( t ) e - j Ω t * d t = - x ( t ) e j Ω t d t = X ( - j Ω ) X ( j Ω ) * = - x ( t ) e - j Ω t d t * = - x ( t ) e - j Ω t * d t = - x ( t ) e j Ω t d t = X ( - j Ω )
(1)

where the second equality uses the definition of a Riemann integral as the limiting case of a summation, and the fact that the complex conjugate of a sum is equal to the sum of the complex conjugates. The third equality used the fact that the complex conjugate of a product is equal to the product of complex conjugates.

Letting X(jΩ)=a(jΩ)+jb(jΩ)X(jΩ)=a(jΩ)+jb(jΩ), it follows that

X ( - j Ω ) = a ( - j Ω ) + j b ( - j Ω ) X ( - j Ω ) = a ( - j Ω ) + j b ( - j Ω )
(2)

and

X ( j Ω ) * = a ( j Ω ) - j b ( j Ω ) X ( j Ω ) * = a ( j Ω ) - j b ( j Ω )
(3)

Equating Equation 2 and Equation 3 gives a(jΩ)=a(-jΩ)a(jΩ)=a(-jΩ) and b(-jΩ)=-b(jΩ)b(-jΩ)=-b(jΩ), which implies that the real and imaginary parts of X(jΩ)X(jΩ) have even and odd symmetry, respectively. A consequence of this is that X(jΩ)=X(jΩ)*=X(-jΩ)X(jΩ)=X(jΩ)*=X(-jΩ), that is, the magnitude of the Fourier transform has even symmetry. It can similarly be shown that the phase of the Fourier transform has odd symmetry.

## 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