Skip to content Skip to navigation

Connexions

You are here: Home » Content » Orthogonal Perfect Reconstruction FIR Filterbank

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.

      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
  • E-mail the author
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    		(0 ratings)

Recently Viewed

This feature requires Javascript to be enabled.

Orthogonal Perfect Reconstruction FIR Filterbank

Module by: Phil Schniter

Summary: This module introduces the ideas behind and design issues of Orthogonal Perfect Reconstruction of FIR filterbanks.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Orthogonal PR Filterbanks

The FIR perfect-reconstruction (PR) conditions leave some freedom in the choice of H 0 z H 0 z and H 1 z H 1 z . Orthogonal PR filterbanks are defined by causal real-coefficient even-length-NN analysis filters that satisfy the following two equations:

1= H 0 z H 0 z-1+ H 0 -z H 0 -z-1 1 H 0 z H 0 z H 0 -z H 0 z (1)
H 1 z=±z-(N−1) H 0 -z-1 H 1 z ± z N 1 H 0 z (2)
To verify that these design choices satisfy the FIR-PR requirements for H 0 z H 0 z and H 1 z H 1 z , we evaluate detHz H z under the second condition above. This yields
detHz=± H 0 z H 1 z-1− H 0 -z H 1 z=-z-(N−1) H 0 z H 0 z-1+ H 0 -z H 0 -z-1=z-(N−1) H z ± H 0 z H 1 z H 0 -z H 1 z z N 1 H 0 z H 0 z H 0 -z H 0 z z N 1 (3)
which corresponds to c=-1 c -1 and l=N−1 l N 1 in the FIR-PR determinant condition detHz=cz-l H z c z l . The remaining FIR-PR conditions then imply that the synthesis filters are given by
G 0 z=-2 H 1 z-1=2z-(N−1) H 0 z-1 G 0 z -2 H 1 z 2 z N 1 H 0 z (4)
G 1 z=2 H 0 -z=2z-(N−1) H 1 z-1 G 1 z 2 H 0 z 2 z N 1 H 1 z (5)
The orthogonal PR design rules imply that H 0 ⅇⅈω H 0 ω is "power symmetric" and that H 0 ⅇⅈω H 1 ⅇⅈω H 0 ω H 1 ω form a "power complementary" pair. To see the power symmetry, we rewrite the first design rule using z=ⅇⅈω z ω and -1=ⅇ±ⅈπ -1 ± π , which gives
1= H 0 ⅇⅈω H 0 ⅇ-ⅈω+ H 0 ⅇⅈω−π H 0 ⅇ-ⅈω−π=| H 0 ⅇⅈω|2+| H 0 ⅇⅈω−π|2=| H 0 ⅇⅈω|2+| H 0 ⅇⅈπ−ω|2 1 H 0 ω H 0 ω H 0 ω H 0 ω H 0 ω 2 H 0 ω 2 H 0 ω 2 H 0 ω 2 (6)
The last two steps leveraged the fact that the DTFT of a real-coefficient filter is conjugate-symmetric. The power-symmetry property is illustrated in Figure 1:

Figure 1
Figure 1 (ortho_f1.png)

Power complementarity follows from the second orthogonal PR design rule, which implies | H 1 ⅇⅈω|=| H 0 ⅇⅈπ−ω| H 1 ω H 0 ω . Plugging this into the previous equation, we find

1=| H 0 ⅇⅈω|2+| H 1 ⅇⅈω|2 1 H 0 ω 2 H 1 ω 2 (7)
The power-complimentary property is illustrated in Figure 2:

Figure 2
Figure 2 (ortho_f2.png)

Comments, questions, feedback, criticisms?

Send feedback