# Connexions

You are here: Home » Content » m28 - Upsampling, Signal Stretching and Interpolation

### Recently Viewed

This feature requires Javascript to be enabled.

# m28 - Upsampling, Signal Stretching and Interpolation

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

Summary: Properties of the Fourier transform can be used to describ the effects of sampling, stretching, and interpolating.

## Up--Sampling, Signal Stretching, and Interpolation

In several situations we would like to increase the data rate of a signal or, to increase its length if it has finite length. This may be part of a multi rate system or part of an interpolation process. Consider the process of inserting M 1 M 1 zeros between each sample of a discrete time signal.

y ( n ) = { x ( n / M ) < n > M = 0  (or n = kM) 0 otherwise y ( n ) = { x ( n / M ) < n > M = 0  (or n = kM) 0 otherwise
(1)
For the finite length sequence case we calculate the DFT of the stretched or up--sampled sequence by
C s ( k ) = n = 0 M N 1 y ( n ) W M N n k C s ( k ) = n = 0 M N 1 y ( n ) W M N n k
(2)
C s ( k ) = n = 0 M N 1 x ( n / M ) ⨿ M ( n ) W M N n k C s ( k ) = n = 0 M N 1 x ( n / M ) ⨿ M ( n ) W M N n k
(3)
where the length is now N M N M and k = 0 , 1 , , N M 1 k = 0 , 1 , , N M 1 . Changing the index variable n = M m n = M m gives:
C s ( k ) = m = 0 N 1 x ( m ) W N m k = C ( k ) . C s ( k ) = m = 0 N 1 x ( m ) W N m k = C ( k ) .
(4)
which says the DFT of the stretched sequence is exactly the same as the DFT of the original sequence but over M M periods, each of length N N .

For up--sampling an infinitely long sequence, we calculate the DTFT of the modified sequence in ((Reference)) as C s ( ω ) = n = x ( n / M ) ⨿ M ( n ) e j ω n = m x ( m ) e j ω M m C s ( ω ) = n = x ( n / M ) ⨿ M ( n ) e j ω n = m x ( m ) e j ω M m

= C ( M ω ) = C ( M ω )
(5)
where C ( ω ) C ( ω ) is the DTFT of x ( n ) x ( n ) . Here again the transforms of the up--sampled signal is the same as the original signal except over M M periods. This shows up here as C s ( ω ) C s ( ω ) being a compressed version of M M periods of C ( ω ) C ( ω ) .

The z-transform of an up--sampled sequence is simply derived by:

Y ( z ) = n = y ( n ) z n = n x ( n / M ) ⨿ M ( n ) z n = m x ( m ) z M m Y ( z ) = n = y ( n ) z n = n x ( n / M ) ⨿ M ( n ) z n = m x ( m ) z M m
(6)
= X ( z M ) = X ( z M )
(7)
which is consistent with a complex version of the DTFT in (Equation 5).

Notice that in all of these cases, there is no loss of information or invertibility. In other words, there is no aliasing.

## 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

### Reuse / Edit:

Reuse or edit module (?)

#### Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

#### Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.