Skip to content Skip to navigation

Connexions

You are here: Home » Content » Periodic Extension to DTFS

Navigation

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? 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.

This content is ...

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.
 

Periodic Extension to DTFS

Module by: Roy Ha. E-mail the author

Summary: This module looks at the periodic extensions of the DTFS coefficients and how the coefficients play a critical role in manipulating signals.

Introduction

Now that we have an understanding of the discrete-time Fourier series (DTFS), we can consider the periodic extension of c⁢k c k (the Discrete-time Fourier coefficients). Figure 1 shows a simple illustration of how we can represent a sequence as a periodic signal mapped over an infinite number of intervals.

Figure 1
(a) vectors
Figure 1(a) (fig1a.png)
(b) periodic sequences
Figure 1(b) (fig1b.png)

Exercise 1

Why does a periodic extension to the DTFS coefficients c⁢k c k make sense?

Solution

Aliasing: b k =ei⁢2⁢πN⁢k⁢n b k 2 N k n

b k + N =ei⁢2⁢πN⁢(k+N)⁢n=ei⁢2⁢πN⁢k⁢n⁢ei⁢2⁢π⁢n=ei⁢2⁢πN⁢n= b k b k + N 2 N k N n 2 N k n 2 n 2 N n b k
(1)
→ DTFS coefficients are also periodic with period NN.

Examples

Example 1: Discrete time square wave

Figure 2
Figure 2 (fig2.png)

Calculate the DTFS c⁢k c k using:

c⁢k=1N⁢∑n=0N−1f⁢n⁢e−(i⁢2⁢πN⁢k⁢n) c k 1 N n 0 N 1 f n 2 N k n
(2)
Just like continuous time Fourier series, we can take the summation over any interval, so we have
c k =1N⁢∑n=− N 1 N 1 e−(i⁢2⁢πN⁢k⁢n) c k 1 N n N 1 N 1 2 N k n
(3)
Let m=n+ N 1 m n N 1 (so we can get a geometric series starting at 0)
c k =1N⁢∑m=02⁢ N 1 e−(i⁢2⁢πN⁢(m− N 1 )⁢k)=1N⁢ei⁢2⁢πN⁢k⁢∑m=02⁢ N 1 e−(i⁢2⁢πN⁢m⁢k) c k 1 N m 0 2 N 1 2 N m N 1 k 1 N 2 N k m 0 2 N 1 2 N m k
(4)
Now, using the "partial summation formula"
∑n=0Man=1−aM+11−a n 0 M a n 1 a M 1 1 a
(5)
c k =1N⁢ei⁢2⁢πN⁢ N 1 ⁢k⁢∑m=02⁢ N 1 e−(i⁢2⁢πN⁢k)m=1N⁢ei⁢2⁢πN⁢ N 1 ⁢k⁢1−e−(i⁢2⁢πN⁢(2⁢ N 1 +1))1−e−(i⁢k⁢2⁢πN) c k 1 N 2 N N 1 k m 0 2 N 1 2 N k m 1 N 2 N N 1 k 1 2 N 2 N 1 1 1 k 2 N
(6)
Manipulate to make this look like a sinc function (distribute):
c k =1N⁢e−(i⁢k⁢2⁢π2⁢N)⁢(ei⁢k⁢2⁢πN⁢( N 1 +12)−e−(i⁢k⁢2⁢πN⁢( N 1 +12)))e−(i⁢k⁢2⁢π2⁢N)⁢(ei⁢k⁢2⁢πN⁢12−e−(i⁢k⁢2⁢πN⁢12))=1N⁢sin2⁢π⁢k⁢( N 1 +12)Nsinπ⁢kN= digital sinc c k 1 N k 2 2 N k 2 N N 1 1 2 k 2 N N 1 1 2 k 2 2 N k 2 N 1 2 k 2 N 1 2 1 N 2 k N 1 1 2 N k N digital sinc
(7)

Note:

It's periodic! Figure 3, Figure 4, and Figure 5show our above function and coefficients for various values of N 1 N 1 .

Figure 3: N 1 =1 N 1 1
(a) Plot of f⁢n f n . (b) Plot of c⁢k c k .
Figure 3(a) (dtfs1.png)Figure 3(b) (dtfs1a.png)
Figure 4: N 1 =3 N 1 3
(a) Plot of f⁢n f n . (b) Plot of c⁢k c k .
Figure 4(a) (dtfs2.png)Figure 4(b) (dtfs2a.png)
Figure 5: N 1 =7 N 1 7
(a) Plot of f⁢n f n . (b) Plot of c⁢k c k .
Figure 5(a) (dtfs3.png)Figure 5(b) (dtfs3a.png)

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