Skip to content Skip to navigation

Connexions

You are here: Home » Content » Reconstruction

Navigation

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. E-mail the author

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

Note: You are viewing an old version of this document. The latest version is available here.

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 −π π bandlimited signal from its samples on the integer is to do the following steps:

  • turn the sample sequence fs ⁢n fs n into an impulse train fimp ⁢t fimp t
  • lowpass filter fimp ⁢t 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 g⁢t g t . The following equations allow us to reconstruct our signal (Figure 2), f ~ ⁢t f ~ t .

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 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 fs ⁢n fs n for ττ seconds. This creates a block or step like function where each value of the pulse in fs ⁢n 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: g⁢t={1  if  0<t<τ0  otherwise   g t 1 0 t τ 0

fs ⁢n=∑n=−∞∞ fs ⁢n⁢g⁢t−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 f⁢t 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 G⁢i⁢ω G ω has the following shape (Figure 6):

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

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

In the time domain, impulse response

g⁢t=sinπ⁢tπ⁢t g t t t
(3)
f ~ ⁢t=∑n=−∞∞ fs ⁢n⁢g⁢t−n=∑n=−∞∞ fs ⁢n⁢sinπ⁢(t−n)π⁢(t−n)=f⁢t f ~ t n fs n g t n n fs n t n t n f t
(4)

Amazing Conclusions

If f⁢t f t is bandlimited to −π π , it can be reconstructed perfectly from its samples on the integers fs ⁢n=f⁢t|t=n fs n t n f t

f⁢t=∑n=−∞∞ fs ⁢n⁢sinπ⁢(t−n)π⁢(t−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 sinπ⁢(t−n)π⁢(t−n) t n t n equal at integers other than n?
  • What is the support of sinπ⁢(t−n)π⁢(t−n) t n t n ?

Content actions

Download module as:

PDF | 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