# Connexions

You are here: Home » Content » Circular Shifts

### Lenses

What is a lens?

#### 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.

#### In these lenses

• Lens for Engineering

This module is included inLens: Lens for Engineering
By: Sidney Burrus

Click the "Lens for Engineering" link to see all content selected in this lens.

### Recently Viewed

This feature requires Javascript to be enabled.

# Circular Shifts

Module by: Justin Romberg. E-mail the author

Summary: The module looks at circular shifting and how it is can be used as a tool to represent the shifting of a periodic sequence.

The many properties of the DFT become really straightforward (very similar to the Fourier Series) once we have once concept down: Circular Shifts.

## Circular shifts

We can picture periodic sequences as having discrete points on a circle as the domain

Shifting by mm, fn+m f n m , corresponds to rotating the cylinder mm notches ACW (counter clockwise). For m=-2 m -2 , we get a shift equal to that in the following illustration:

To cyclic shift we follow these steps:

1) Write fn f n on a cylinder, ACW

2) To cyclic shift by mm, spin cylinder m spots ACW fnf (( n + m )) N f n f (( n + m )) N

### Example 1

If fn=01234567 f n 0 1 2 3 4 5 6 7 , then f (( n - 3 )) N =34567012 f (( n - 3 )) N 3 4 5 6 7 0 1 2

It's called circular shifting, since we're moving around the circle. The usual shifting is called "linear shifting" (shifting along a line).

### Notes on circular shifting

f (( n + N )) N =fn f (( n + N )) N f n Spinning NN spots is the same as spinning all the way around, or not spinning at all.

f (( n + N )) N =f (( n - ( N - m ) )) N f (( n + N )) N f (( n - ( N - m ) )) N Shifting ACW mm is equivalent to shifting CW Nm N m

f (( - n )) N f (( - n )) N The above expression, simply writes the values of fn f n clockwise.

## Circular shifts and the DFT

### Theorem 1: Circular Shifts and DFT

If fn DFT Fk f n DFT F k then f (( n - m )) N DFT e(i2πNkm)Fk f (( n - m )) N DFT 2 N k m F k (i.e. circular shift in time domain = phase shift in DFT)

#### Proof

fn=1Nk=0N1Fkei2πNkn f n 1 N k 0 N 1 F k 2 N k n
(1)
so phase shifting the DFT
fn=1Nk=0N1Fke(i2πNkn)ei2πNkn=1Nk=0N1Fkei2πNk(nm)=f (( n - m )) N f n 1 N k 0 N 1 F k 2 N k n 2 N k n 1 N k 0 N 1 F k 2 N k n m f (( n - m )) N
(2)

## Conclusion

With the concept of the circular shifts well-established, we can now better understand the convenience afforded by the circular convolution as opposed to the redundant linear convolution, and better understand the convolution as applied to the DFT.

## 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