Skip to content Skip to navigation

Connexions

You are here: Home » Content » Perfect Reconstruction QMF

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 QMF

Module by: Phil Schniter

Perfect Reconstruction QMF

The system transfer function for a QMF bank is

Tz= H 0 2z- H 1 2z=4z-1 P 0 z2 P 1 z2 T z H 0 z 2 H 1 z 2 4 z -1 P 0 z 2 P 1 z 2 (1)
For perfect reconstruction, we need Tz=z-l T z z l for some l∈ℕ l , which implies the equivalent conditions 4z-1 P 0 z2 P 1 z2=z-l 4 z -1 P 0 z 2 P 1 z 2 z l P 0 z2 P 1 z2=14z-l-1 P 0 z 2 P 1 z 2 1 4 z l 1 P 0 z P 1 z=14z-l-12 P 0 z P 1 z 1 4 z l 1 2 For FIR polyphase filters, this can only be satisfied by P 0 z= β 0 z- n 0 P 0 z β 0 z n 0 P 1 z= β 1 z- n 1 P 1 z β 1 z n 1 where we have n 0 + n 1 =l-12 n 0 n 1 l 1 2 and β 0 β 1 =14 β 0 β 1 1 4 .

In other words, the polyphase filters are trivial, so that the prototype filter H 0 z H 0 z has a two-tap response. With only two taps, H 0 z H 0 z cannot be a very good lowpass filter, meaning that the sub-band signals will not be spectrally well-separated. From this we conclude that two-channel 1 perfect reconstruction QMF banks exist but are not very useful.

Footnotes

  1. It turns out that MM-channel perfect reconstruction QMF banks have more useful responses for larger values of MM.

Comments, questions, feedback, criticisms?

Send feedback