Connexions

You are here: Home » Content » More on Perfect Reconstruction
Content Actions
Lenses

What is a lens?

Lenses

A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual Connexions member, a community, or a respected organization.

This content is ...
Affiliated with (?)
This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.

  • This module is included inLens: Rice University OpenCourseWare
    By: OpenCourseWare ConsortiumAs a part of collection:"Signals and Systems"

    Click the "Rice University OCW" link to see all content affiliated with them.

    Rice University OCW
Also in these lenses

  • This module is included inLens: richb's DSP resources
    By: Richard BaraniukAs a part of collection:"Signals and Systems"

    Comments:

    "My introduction to signal processing course at Rice University."

    Click the "richb's DSP" link to see all content selected in this lens.

    richb's DSP
Tags

(?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

More on Perfect Reconstruction

Module by: Roy Ha, Justin Romberg

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

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

Derive Reconstruction Formulas

What is
<sinπt-nπt-n,sinπt-kπt-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
<·,·>=12π∫-ππⅇ-ⅈωnⅇⅈωkdω · · 1 2 ω ω n ω k (3)
fig1a.png
Subfigure 1.1
fig1b.png
Subfigure 1.2
Figure 1
if n=k n k
< sinc n , sinc k >=12π∫-ππⅇ-ⅈωnⅇⅈωkdω=1 sinc n sinc k 1 2 ω ω n ω k 1 (4)
if n≠k n k
< sinc n , sinc k >=12π∫-ππⅇ-ⅈωnⅇⅈωndω=12π∫-ππⅇⅈωk-ndω=12πsinπk-nⅈk-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,
<sinπt-nπt-n,sinπt-kπt-k>=1ifn=k0ifn≠k t n t n t k t k 1 n k 0 n k (6)
Therefore sinπt-nπt-n n∈ℤ 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 nsinπt-nπt-n 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.

Comments, questions, feedback, criticisms?

Send feedback