Skip to content Skip to navigation Skip to collection information

OpenStax-CNX

You are here: Home » Content » Signals, Systems, and Society » Symmetry Properties of the Fourier Transform

Navigation

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

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.

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

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

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | More downloads ...

Add:

Collection to:

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? tag icon

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

Module to:

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? tag icon

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