Skip to content Skip to navigation

Connexions

You are here: Home » Content » m14 - Evaluation of the DTFT by the DFT

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

m14 - Evaluation of the DTFT by the DFT

Module by: C. Sidney Burrus. E-mail the author

Summary: Samples of the DTFT can be calculated using the DFT

Evaluation of the DTFT by the DFT

If the DTFT of a finite sequence is taken, the result is a continuous function of ω ω . If the DFT of the same sequence is taken, the results are N N evenly spaced samples of the DTFT. In other words, the DTFT of a finite signal can be evaluated at N N points with the DFT.

X ( ω ) = D T F T { x ( n ) } = ∑ n = − ∞ ∞ x ( n ) e − j ω n X ( ω ) = D T F T { x ( n ) } = ∑ n = − ∞ ∞ x ( n ) e − j ω n
(1)
and because of the finite length
X ( ω ) = ∑ n = 0 N − 1 x ( n ) e − j ω n . X ( ω ) = ∑ n = 0 N − 1 x ( n ) e − j ω n .
(2)
If we evaluate ω ω at N N equally space points, this becomes
X ( 2 π N k ) = ∑ n = 0 N − 1 x ( n ) e − j 2 π N k n X ( 2 π N k ) = ∑ n = 0 N − 1 x ( n ) e − j 2 π N k n
(3)
which is the DFT of x ( n ) x ( n ) . By adding zeros to the end of x ( n ) x ( n ) and taking a longer DFT, any density of points can be evaluated. This is useful in interpolation and in plotting the spectrum of a finite length signal. This is discussed further in Chapter (Reference).

There is an interesting variation of the Parseval's theorem for the DTFT of a finite length- N N signal. If x ( n ) ≠ 0 x ( n ) ≠ 0 for 0 ≥ n ≥ N − 1 0 ≥ n ≥ N − 1 , and if L ≥ N L ≥ N , then

∑ n = 0 N − 1 | x ( n ) | 2 = 1 L ∑ k = 0 L − 1 | X ( 2 π k / L ) | 2 = 1 π ∫ 0 π | X ( ω ) | 2 ⅆ ω . ∑ n = 0 N − 1 | x ( n ) | 2 = 1 L ∑ k = 0 L − 1 | X ( 2 π k / L ) | 2 = 1 π ∫ 0 π | X ( ω ) | 2 ⅆ ω .
(4)
The second term in (Equation 4) says the Riemann sum is equal to its limit in this case.

Content actions

Download module 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 ...

Add 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