Skip to content Skip to navigation

OpenStax_CNX

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

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

Perfect Reconstruction FIR Filter Banks

Module by: Phil Schniter. E-mail the author

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=12( XzX-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
( 2zl 0 )=( 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 -1z( 2zl 0 )=1detHz( H 1 z H 1 z H 0 z H 0 z )( 2zl 0 )=2detHz( zl H 1 (z) (zl 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=czk H z c z k
(5)
for cR c and kZ k . Under this determinant condition, we find that
G 0 z G 1 z=2z(lk)c 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 ,cRlZ:detHz=czl 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.

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

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

Lenses

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