Skip to content Skip to navigation

Connexions

You are here: Home » Content » More on Perfect Reconstruction

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions 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 Connexions 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 ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • OrangeGrove display tagshide tags

    This module is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange GroveAs a part of collection:"Signals and Systems"

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • richb's DSP display tagshide tags

    This module is included inLens: richb's DSP resources
    By: Richard BaraniukAs a part of collection:"Signals and Systems"

    Comments:

    "My introduction to signal processing course at Rice University."

    Click the "richb's DSP" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

More on Perfect Reconstruction

Module by: Roy Ha, Justin Romberg. E-mail the authors

User rating (How does the rating system work?)
Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

:
(0 ratings)

Summary: This module examines the idea and formula behind perfect reconstruction in more depth.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Introduction

In the previous module on reconstruction, we gave an introduction into how reconstruction works and briefly derived an equation used to perform perfect reconstruction. Let us now take a closer look at the perfect reconstruction formula:

ft=∑n=-∞∞ f s sinπt−nπt−n f t n f s t n t n (1)
We are writing ft f t in terms of shifted and scaled sinc functions. sinπt−nπt−n n∈ℤ t n t n n is a basis for the space of -ππ bandlimited signals. But  wait . . . .

Derive Reconstruction Formulas

What is

<sinπt−nπt−n,sinπt−kπt−k>=? t n t n t k t k ? (2)
This inner product can be hard to calculate in the time domain, so let's use Plancharel Theorem
<·,·>=12π∫-ππⅇ-ⅈωnⅇⅈωkdω · · 1 2 ω ω n ω k (3)

Figure 1
(a)
Figure 1(a) (fig1a.png)
(b)
Figure 1(b) (fig1b.png)

if n=k n k

< sinc n , sinc k >=12π∫-ππⅇ-ⅈωnⅇⅈωkdω=1 sinc n sinc k 1 2 ω ω n ω k 1 (4)
if n≠k n k
< sinc n , sinc k >=12π∫-ππⅇ-ⅈωnⅇⅈωndω=12π∫-ππⅇⅈωk−ndω=12πsinπk−nⅈk−n=0 sinc n sinc k 1 2 ω ω n ω n 1 2 ω ω k n 1 2 k n k n 0 (5)

Note:

In Equation 5 we used the fact that the integral of sinusoid over a complete interval is 0 to simplify our equation.
So,
<sinπt−nπt−n,sinπt−kπt−k>=1ifn=k0ifn≠k t n t n t k t k 1 n k 0 n k (6)
Therefore sinπt−nπt−n n∈ℤ t n t n n is an orthonormal basis (ONB) for the space of -ππ bandlimited functions.

Sampling:

Sampling is the same as calculating ONB coefficients, which is inner products with sincs

Summary

One last time for ft f t -ππ bandlimited

Synthesis

ft=∑n=-∞∞ f s nsinπt−nπt−n f t n f s n t n t n (7)

Analysis

f s n=ft| t=n f s n t n f t (8)
In order to understand a little more about how we can reconstruct a signal exactly, it will be useful to examine the relationships between the fourier transforms (CTFT and DTFT) in more depth.

Content actions

Give Feedback:

E-mail the module authors | Rate module ( How does the rating system work?)

Rating system

Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

(0 ratings)

Download:

Module PDF

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.

| A lens (?)

Definition of a lens

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions 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 Connexions 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