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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 ?

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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?

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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.

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