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Reconstruction

Module by: Justin Romberg

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

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)
recon_blk.png
Figure 1: Reconstruction block diagram with lowpass filter (LPF).
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)
recon_blk2.png
Figure 2:

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)
receg_f1.pngreceg_f2.png
Subfigure 3.1
Subfigure 3.2
Figure 3: Zero Order Hold
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.
recf_f1.png
Subfigure 4.1: First Order Hold
recf_f2.png
Subfigure 4.2: Second Order Hold
recf_f3.png
Subfigure 4.3: ∞ Order Hold
Figure 4: Nth Order Hold Examples (nth order hold is equal to an nth order B-spline)

Ultimate Reconstruction Filter

question: What is the ultimate reconstruction filter?
Recall that (see Figure 5)
recon_blk3.png
Figure 5: Our current reconstruction block diagram. Note that each of these signals has its own corresponding CTFT or DTFT.
If Gⅈω G ω has the following shape (Figure 6):
square_wv.png
Figure 6: Ideal lowpass filter
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 ?

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