# OpenStax-CNX

You are here: Home » Content » m11 - Properties of the DFT

### Recently Viewed

This feature requires Javascript to be enabled.

# m11 - Properties of the DFT

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

Summary: The DFT has many powerful properties, many of which are a result of the orthogonal basis function. If calculated using the fast Fourier transform (FFT), it is a powerful computational tool.

## Properties of the DFT

The properties of the DFT are extremely important in applying it to signal analysis and to interpreting it. The main properties are given here using the notation that the DFT of a length- N N complex sequence x ( n ) x ( n ) is { x ( n ) } = C ( k ) { x ( n ) } = C ( k ) .

1. Linear Operator: { x ( n ) + y ( n ) } = { x ( n ) } + { y ( n ) } { x ( n ) + y ( n ) } = { x ( n ) } + { y ( n ) }
2. Unitary Operator: F 1 = 1 N F T F 1 = 1 N F T
3. Periodic Spectrum: C ( k ) = C ( k + N ) C ( k ) = C ( k + N )
4. Periodic Extensions of x ( n ) x ( n ) : x ( n ) = x ( n + N ) x ( n ) = x ( n + N )
5. Properties of Even and Odd Parts: x ( n ) = u ( n ) + j v ( n ) x ( n ) = u ( n ) + j v ( n ) and C ( k ) = A ( k ) + j B ( k ) C ( k ) = A ( k ) + j B ( k ) u u v v A A B B | C | | C | θ θ even 0 even 0 even 0 odd 0 0 odd even π / 2 π / 2 0 even 0 even even π / 2 π / 2 0 odd odd 0 even 0
6. Cyclic Convolution: { h ( n ) x ( n ) } = { h ( n ) } { x ( n ) } { h ( n ) x ( n ) } = { h ( n ) } { x ( n ) }
7. Multiplication: { h ( n ) x ( n ) } = { h ( n ) } { x ( n ) } { h ( n ) x ( n ) } = { h ( n ) } { x ( n ) }
8. Parseval: n = 0 N 1 | x ( n ) | 2 = 1 N k = 0 N 1 | C ( k ) | 2 n = 0 N 1 | x ( n ) | 2 = 1 N k = 0 N 1 | C ( k ) | 2
9. Shift: { x ( n M ) } = C ( k ) e j 2 π M k / N { x ( n M ) } = C ( k ) e j 2 π M k / N
10. Modulate: { x ( n ) e j 2 π K n / N } = C ( k K ) { x ( n ) e j 2 π K n / N } = C ( k K )
11. Down Sample or Decimate: { x ( K n ) } = 1 K m = 0 K 1 C ( k + L m ) { x ( K n ) } = 1 K m = 0 K 1 C ( k + L m ) where N = L K N = L K
12. Up Sample or Stretch: If x s ( 2 n ) = x ( n ) x s ( 2 n ) = x ( n ) for integer n n and zero otherwise,then { x s ( n ) } = C ( k ) { x s ( n ) } = C ( k ) , for k = 0 , 1 , 2 , , 2 N 1 k = 0 , 1 , 2 , , 2 N 1
13. N Roots of Unity: ( W N k ) N = 1 ( W N k ) N = 1 for k = 0 , 1 , 2 , , N 1 k = 0 , 1 , 2 , , N 1
14. Orthogonality:
k = 0 N 1 e j 2 π m k / N e j 2 π n k / N = { N if  n = m 0 if  n m . k = 0 N 1 e j 2 π m k / N e j 2 π n k / N = { N if  n = m 0 if  n m .
(1)
15. Diagonalization of Convolution: If cyclic convolution is expressed as a matrix operation by y = H x y = H x with H H given by ((Reference)), the DFT operator diagonalizes the convolution operator H H , or F T H F = H d F T H F = H d where H d H d is a diagonal matrix with the N N values of the DFT of h ( n ) h ( n ) on the diagonal. This is a matrix statement of Property 6. Note the columns of F F are the N N eigenvectors of H H , independent of the values of h ( n ) h ( n ) .

## Content actions

### Give feedback:

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