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More on Perfect Reconstruction

Module by: Roy Ha, Justin Romberg. E-mail the authors

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Summary: This module examines the idea and formula behind perfect reconstruction in more depth.

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Introduction

In the previous module on reconstruction, we gave an introduction into how reconstruction works and briefly derived an equation used to perform perfect reconstruction. Let us now take a closer look at the perfect reconstruction formula:

ft=n=- f s sinπtnπtn f t n f s t n t n (1)
We are writing ft f t in terms of shifted and scaled sinc functions. sinπtnπtn n t n t n n is a basis for the space of -ππ bandlimited signals. But  wait . . . .

Derive Reconstruction Formulas

What is

<sinπtnπtn,sinπtkπtk>=? t n t n t k t k ? (2)
This inner product can be hard to calculate in the time domain, so let's use Plancharel Theorem
<·,·>=12π-ππ-ωnωkdω · · 1 2 ω ω n ω k (3)

Figure 1
(a)
Figure 1(a) (fig1a.png)
(b)
Figure 1(b) (fig1b.png)

if n=k n k

< sinc n , sinc k >=12π-ππ-ωnωkdω=1 sinc n sinc k 1 2 ω ω n ω k 1 (4)
if nk n k
< sinc n , sinc k >=12π-ππ-ωnωndω=12π-ππωkndω=12πsinπknkn=0 sinc n sinc k 1 2 ω ω n ω n 1 2 ω ω k n 1 2 k n k n 0 (5)

Note:

In Equation 5 we used the fact that the integral of sinusoid over a complete interval is 0 to simplify our equation.
So,
<sinπtnπtn,sinπtkπtk>=1ifn=k0ifnk t n t n t k t k 1 n k 0 n k (6)
Therefore sinπtnπtn n t n t n n is an orthonormal basis (ONB) for the space of -ππ bandlimited functions.

Sampling:

Sampling is the same as calculating ONB coefficients, which is inner products with sincs

Summary

One last time for ft f t -ππ bandlimited

Synthesis

ft=n=- f s nsinπtnπtn f t n f s n t n t n (7)

Analysis

f s n=ft| t=n f s n t n f t (8)
In order to understand a little more about how we can reconstruct a signal exactly, it will be useful to examine the relationships between the fourier transforms (CTFT and DTFT) in more depth.

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