Skip to content Skip to navigation


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


Recently Viewed

This feature requires Javascript to be enabled.


(What is a tag?)

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

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.


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
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 ?
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

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

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
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


In Equation 5 we used the fact that the integral of sinusoid over a complete interval is 0 to simplify our equation.
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
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 is the same as calculating ONB coefficients, which is inner products with sincs


One last time for ft f t π π bandlimited


ft=n= f s nsinπ(tn)π(tn) f t n f s n t n t n


f s n=ft| t=n f s n t n f t
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.

Content actions

Download module as:

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens


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

What is in a lens?

Lens makers point to 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 member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks