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More on Perfect Reconstruction

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

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

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

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:

f⁢t=∑n=−∞∞ f s ⁢sinπ⁢(t−n)π⁢(t−n) f t n f s t n t n
(1)
We are writing f⁢t f t in terms of shifted and scaled sinc functions. sinπ⁢(t−n)π⁢(t−n) n∈Z t n t n n is a basis for the space of −π π bandlimited signals. But  wait . . . .

Derive Reconstruction Formulas

What is

〈sinπ⁢(t−n)π⁢(t−n),sinπ⁢(t−k)π⁢(t−k)〉=? 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
〈·,·〉=12⁢π⁢∫−ππe−(i⁢ω⁢n)⁢ei⁢ω⁢kdω · · 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 〉=12⁢π⁢∫−ππe−(i⁢ω⁢n)⁢ei⁢ω⁢kdω=1 sinc n sinc k 1 2 ω ω n ω k 1
(4)
if n≠k n k
〈 sinc n , sinc k 〉=12⁢π⁢∫−ππe−(i⁢ω⁢n)⁢ei⁢ω⁢ndω=12⁢π⁢∫−ππei⁢ω⁢(k−n)dω=12⁢π⁢sinπ⁢(k−n)i⁢(k−n)=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,
〈sinπ⁢(t−n)π⁢(t−n),sinπ⁢(t−k)π⁢(t−k)〉={1  if  n=k0  if  n≠k t n t n t k t k 1 n k 0 n k
(6)
Therefore sinπ⁢(t−n)π⁢(t−n) n∈Z t n t n n is an orthonormal basis (ONB) for the space of −π π bandlimited functions.

sampling:

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

Summary

One last time for f⁢t f t −π π bandlimited

Synthesis

f⁢t=∑n=−∞∞ f s ⁢n⁢sinπ⁢(t−n)π⁢(t−n) f t n f s n t n t n
(7)

Analysis

f s ⁢n=f⁢t| 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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