Skip to content Skip to navigation

Connexions

You are here: Home » Content » Perfect Reconstruction FIR Filter Banks

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

Perfect Reconstruction FIR Filter Banks

Module by: Phil Schniter

Summary: This module will drop the restrictive QMF conditions and focus on using FIR filters to achieve perfect reconstruction from filterbanks.

FIR Perfect-Reconstruction Conditions

The QMF design choices prevented the design of a useful (i.e., frequency selective) perfect-reconstruction (PR) FIR filterbank. This motivates us to re-examine PR filterbank design without the overly-restrictive QMF conditions. However, we will still require causal FIR filters with real coefficients.

Recalling that the two-channel filterbank (Figure 1),

Figure 1
Figure 1 (PR_f1.png)
has the input/output relation:
Yz=12XzX-z H 0 z H 1 z H 0 -z H 1 -z G 0 z G 1 z Y z 1 2 X z X -z H 0 z H 1 z H 0 z H 1 z G 0 z G 1 z (1)
we see that the delay-ll perfect reconstruction requires
2z-l0= H 0 z H 1 z H 0 -z H 1 -z G 0 z G 1 z 2 z l 0 H 0 z H 1 z H 0 z H 1 z G 0 z G 1 z (2)
where Hz= H 0 z H 1 z H 0 -z H 1 -z H z H 0 z H 1 z H 0 z H 1 z or, equivalently, that
G 0 z G 1 z =H -1z2z-l0=1detHz H 1 -z- H 1 z- H 0 -z H 0 z2z-l0=2detHzz-l H 1 -z-z-l H 0 -z G 0 z G 1 z H z -1 2 z l 0 1 H z H 1 z H 1 z H 0 z H 0 z 2 z l 0 2 H z z l H 1 z z l H 0 z (3)
where
detHz= H 0 z H 1 -z- H 0 -z H 1 z H z H 0 z H 1 z H 0 z H 1 z (4)
For FIR G 0 z G 0 z and G 1 z G 1 z , we require 1 that
detHz=cz-k H z c z k (5)
for c∈ℝ c and k∈ℤ k . Under this determinant condition, we find that
G 0 z G 1 z =2z-l-kc H 1 -z- H 0 -z G 0 z G 1 z 2 z l k c H 1 z H 0 z (6)
Assuming that H 0 z H 0 z and H 1 z H 1 z are causal with non-zero initial coefficient, we choose k=l k l to keep G 0 z G 0 z and G 1 z G 1 z causal and free of unnecessary delay.

Summary of Two-Channel FIR-PR Conditions

Summarizing the two-channel FIR-PR conditions: H 0 z∧ H 1 z=  causal real-coefficient  FIR H 0 z H 1 z   causal real-coefficient  FIR ∀c,c∈ℝ∧l∈ℤ:detHz=cz-l c c l H z c z l G 0 z=2c H 1 -z G 0 z 2 c H 1 z G 1 z=-2c H 0 -z G 1 z -2 c H 0 z

Footnotes

  1. Since we cannot assume that FIR H 0 z H 0 z and H 1 z H 1 z share a common root.

Comments, questions, feedback, criticisms?

Send feedback