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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.

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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 sinπ(tn)π(tn) f t n f s t n t n
(1)
We are writing ft f t in terms of shifted and scaled sinc functions. sinπ(tn)π(tn) nZ t n t n n is a basis for the space of π π bandlimited signals. But  wait . . . .

### Derive Reconstruction Formulas

What is

sinπ(tn)π(tn),sinπ(tk)π(tk)=? 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
·,·=12πππe(iωn)eiωkdω · · 1 2 ω ω n ω k
(3)

if n=k n k

sinc n , sinc k =12πππe(iωn)eiωkdω=1 sinc n sinc k 1 2 ω ω n ω k 1
(4)
if nk n k
sinc n , sinc k =12πππe(iωn)eiωndω=12πππeiω(kn)dω=12πsinπ(kn)i(kn)=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(tn)π(tn),sinπ(tk)π(tk)={1  if  n=k0  if  nk t n t n t k t k 1 n k 0 n k
(6)
Therefore sinπ(tn)π(tn) nZ 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 ft f t π π bandlimited

#### Synthesis

ft=n= f s nsinπ(tn)π(tn) 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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