Skip to content Skip to navigation

Connexions

You are here: Home » Content » Perfect Reconstruction (PR)

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of 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.

    • External bookmarks
  • E-mail the author

Recently Viewed

This feature requires Javascript to be enabled.

Perfect Reconstruction (PR)

Module by: Nick Kingsbury

Summary: This module discusses perfect reconstruction (PR).

We are now able to generalize our analysis for arbitrary filters H0 H0 , H1 H1 , G0 G0 and G1 G1 . Substituting this equation and this equation in our discussion of 2-band filter bank into this earlier equation and then using this equation and this equation from the same discussion, we get:

X ^ z=12G0zY0z+Y0-z+12G1zY1z+Y1-z=12G0zH0zXz+12G0zH0-zX-z+12G1zH1zXz+12G1zH1-zX-z=12XzG0zH0z+G1zH1z+12X-zG0zH0-z+G1zH1-z X ^ z 1 2 G0 z Y0 z Y0 z 1 2 G1 z Y1 z Y1 z 1 2 G0 z H0 z X z 1 2 G0 z H0 z X z 1 2 G1 z H1 z X z 1 2 G1 z H1 z X z 1 2 X z G0 z H0 z G1 z H1 z 1 2 X z G0 z H0 z G1 z H1 z (1)
If we require X ^ z≡Xz X ^ z X z - the Perfect Reconstruction (PR) condition - then:
G0zH0z+G1zH1z≡2 G0 z H0 z G1 z H1 z 2 (2)
and
G0zH0-z+G1zH1-z≡0 G0 z H0 z G1 z H1 z 0 (3)
Identity Equation 3 is known as the anti-aliasing condition because the term in X-z X z in Equation 1 is the unwanted aliasing term caused by down-sampling y0 y0 and y1 y1 by 2.

It is straightforward to show that the expression for H0 H0 , H1 H1 , G0 G0 and G1 G1 , given in this equation, this equation, this equation and this equation for the filters based on the Haar transform, satisfy Equation 2 and Equation 3. They are the simplest set of filters which do.

Before we look at more complicated PR filters, we examine how the filter structures of this figure may be extended to form a binary filter tree (and the discrete wavelet transform).

Comments, questions, feedback, criticisms?

Send feedback