Skip to content Skip to navigation

Connexions

You are here: Home » Content » Reconstruction

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is '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 (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.

    • External bookmarks
  • E-mail the author
  • Rate this 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)

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 ...

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.

Reconstruction

Module by: Justin Romberg

Summary: This module describes reconstruction (a.k.a. interpolation).

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

The reconstruction process begins by taking a sampled signal, which will be in discrete time, and performing a few operations in order to convert them into continuous-time and, with any luck, into an exact copy of the original signal. A basic method used to reconstruct a -ππ bandlimited signal from its samples on the integer is to do the following steps:

  • turn the sample sequence fsn fs n into an impulse train fimpt fimp t
  • lowpass filter fimpt fimp t to get the reconstruction f ~ t f ~ t (cutoff freq. = π)

Figure 1: Reconstruction block diagram with lowpass filter (LPF).
Figure 1 (recon_blk.png)

The lowpass filter's impulse response is gt g t . The following equations allow us to reconstruct our signal (Figure 2), f ~ t f ~ t .

f ~ t=gtfimpt=gt∑n=-∞∞fsnδt−n= f ~ t=∑n=-∞∞fsngtδt−n=∑n=-∞∞fsngt−n f ~ t g t fimp t g t n fs n δ t n f ~ t n fs n g t δ t n n fs n g t n (1)

Figure 2:
Figure 2 (recon_blk2.png)

Examples of Filters g

Example 1: Zero Order Hold

This type "filter" is one of the most basic types of reconstruction filters. It simply holds the value that is in fsn fs n for ττ seconds. This creates a block or step like function where each value of the pulse in fsn fs n is simply dragged over to the next pulse. The equations and illustrations below depict how this reconstruction filter works with the following gg: gt= 1if0<t<τ0otherwise g t 1 0 t τ 0

fsn=∑n=-∞∞fsngt−n fs n n fs n g t n (2)

Figure 3: Zero Order Hold
(a) (b)
Figure 3(a) (receg_f1.png)Figure 3(b) (receg_f2.png)

question:
How does f ~ t f ~ t reconstructed with a zero order hold compare to the original ft f t in the frequency domain?

Example 2: Nth Order Hold

Here we will look at a few quick examples of variances to the Zero Order Hold filter discussed in the previous example.

Figure 4: Nth Order Hold Examples (nth order hold is equal to an nth order B-spline)
(a) First Order Hold
Figure 4(a) (recf_f1.png)
(b) Second Order Hold
Figure 4(b) (recf_f2.png)
(c) ∞ Order Hold
Figure 4(c) (recf_f3.png)

Ultimate Reconstruction Filter

question:

What is the ultimate reconstruction filter?

Recall that (see Figure 5)

Figure 5: Our current reconstruction block diagram. Note that each of these signals has its own corresponding CTFT or DTFT.
Figure 5 (recon_blk3.png)

If Gⅈω G ω has the following shape (Figure 6):

Figure 6: Ideal lowpass filter
Figure 6 (square_wv.png)

then f ~ t=ft f ~ t f t Therefore, an ideal lowpass filter will give us perfect reconstruction!

In the time domain, impulse response

gt=sinπtπt g t t t (3)
f ~ t=∑n=-∞∞fsngt−n=∑n=-∞∞fsnsinπt−nπt−n=ft f ~ t n fs n g t n n fs n t n t n f t (4)

Amazing Conclusions

If ft f t is bandlimited to -ππ , it can be reconstructed perfectly from its samples on the integers fsn=ft|t=n fs n t n f t

ft=∑n=-∞∞fsnsinπt−nπt−n f t n fs n t n t n (5)

The above equation for perfect reconstruction deserves a closer look, which you should continue to read in the following section to get a better understanding of reconstruction. Here are a few things to think about for now:

  • What does sinπt−nπt−n t n t n equal at integers other than n?
  • What is the support of sinπt−nπt−n t n t n ?

Comments, questions, feedback, criticisms?

Send feedback