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# Reconstruction

Module by: Justin Romberg. E-mail the author

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

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## 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)

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=gt fimp t=gtn= fs nδtn= f ~ t=n= fs n(gtδtn)=n= fs ngtn 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)

### 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: gt={1  if  0<t<τ0  otherwise   g t 1 0 t τ 0

fs n=n= fs ngtn fs n n fs n g t n
(2)

##### 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.

## Ultimate Reconstruction Filter

### question:

What is the ultimate reconstruction filter?

Recall that (see Figure 5)

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

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= fs ngtn=n= fs nsinπ(tn)π(tn)=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 π π , it can be reconstructed perfectly from its samples on the integers fs n=ft|t=n fs n t n f t

ft=n= fs nsinπ(tn)π(tn) 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(tn)π(tn) t n t n equal at integers other than n?
• What is the support of sinπ(tn)π(tn) t n t n ?

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