# OpenStax_CNX

You are here: Home » Content » Digital Signal Processing (Ohio State EE700) » Bi-Orthogonal Perfect Reconstruction FIR Filterbanks

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

Course by: Phil Schniter. E-mail the author

# Bi-Orthogonal Perfect Reconstruction FIR Filterbanks

Module by: Phil Schniter. E-mail the author

Summary: This module looks at Bi-Orthogonal PR-FIR filterbanks and shows how they are similar to orthogonal designs yet provide linear-phase filters.

## Bi-Orthogonal Filter Banks

Due to the minimum-phase spectral factorization, orthogonal PR-FIR filterbanks will not have linear-phase analysis and synthesis filters. Non-linear phase may be undesirable for certain applications. "Bi-orthogonal" designs are closely related to orthogonal designs, yet give linear-phase filters. The analysis-filter design rules for the bi-orthogonal case are

• Fz F z : zero-phase real-coefficient halfband such that Fz= n =(N1)N1fnzn F z n N 1 N 1 f n z n , where NN is even.
• z(N1)Fz= H 0 z H 1 z z N 1 F z H 0 z H 1 z
It is straightforward to verify that these design choices satisfy the FIR perfect reconstruction condition detHz=czl H z c z l with c=1 c 1 and l=N1 l N 1 :
detHz= H 0 z H 1 z H 0 z H 1 z=z(N1)Fz1(N1)z(N1)Fz=z(N1)(Fz+Fz)=z(N1) H z H 0 z H 1 z H 0 z H 1 z z N 1 F z 1 N 1 z N 1 F z z N 1 F z F z z N 1
(1)
Furthermore, note that z(N1)Fz z N 1 F z is causal with real coefficients, so that both H 0 z H 0 z and H 1 z H 1 z can be made causal with real coefficients. (This was another PR-FIR requirement.) The choice c=1 c 1 implies that the synthesis filters should obey G 0 z=2 H 1 z G 0 z 2 H 1 z G 1 z=-2 H 0 z G 1 z -2 H 0 z From the design choices above, we can see that bi-orthogonal analysis filter design reduces to the factorization of a causal halfband filter z(N1)Fz z N 1 F z into H 0 z H 0 z and H 1 z H 1 z that have both real coefficients and linear-phase. Earlier we saw that linear-phase corresponds to root symmetry across the unit circle in the complex plane, and that real-coefficients correspond to complex-conjugate root symmetry. Simultaneous satisfaction of these two properties can be accomplished by quadruples of roots. However, there are special cases in which a root pair, or even a single root, can simultaneously satisfy these properties. Examples are illustrated in Figure 1:

The design procedure for the analysis filters of a bi-orthogonal perfect-reconstruction FIR filterbank is summarized below:

1. Design a zero-phase real-coefficient filter Fz= n =(N1)N1fnzn F z n N 1 N 1 f n z n where N is a positive even integer (via, e.g., window designs, LS, or equiripple).
2. Compute the roots of Fz F z and partition into a set of root groups G 0 G 1 G 2 G 0 G 1 G 2 that have both complex-conjugate and unit-circle symmetries. Thus a root group may have one of the following forms: G i = a i a i *1 a i 1 a i * G i a i a i 1 a i 1 a i G i = a i a i *  ,   | a i |=1    a i a i 1 G i a i a i G i = a i 1 a i   ,   a i R    a i a i G i a i 1 a i G i = a i   ,   a i =±1    a i a i ± 1 G i a i Choose 1 a subset of root groups and construct H ^ 0 z H ^ 0 z from those roots. Then construct H ^ 1 z H ^ 1 z from the roots in the remaining root groups. Finally, construct H ^ 1 z H ^ 1 z from H ^ 1 z H ^ 1 z by reversing the signs of odd-indexed coefficients.
3. H ^ 0 z H ^ 0 z and H ^ 1 z H ^ 1 z are the desired analysis filters up to a scaling. To take care of the scaling, first create H ~ 0 z=a H ^ 0 z H ~ 0 z a H ^ 0 z and H ~ 1 z=b H ^ 1 z H ~ 1 z b H ^ 1 z where aa and bb are selected so that n h ~ 0 n=1=n h ~ 1 n n h ~ 0 n 1 n h ~ 1 n . Then create H 0 z=c H ~ 0 z H 0 z c H ~ 0 z and H 1 z=c H ~ 1 z H 1 z c H ~ 1 z where cc is selected so that the property z(N1)Fz= H 0 z H 1 z z N 1 F z H 0 z H 1 z is satisfied at DC (i.e., z=ej0=1 z 0 1 ). In other words, find cc so that n h 0 nm h 1 n1m=1 n h 0 n m h 1 n 1 m 1 .

## Footnotes

1. Note that H ^ 0 z H ^ 0 z and H ^ 1 z H ^ 1 z will be real-coefficient linear-phase regardless of which groups are allocated to which filter. Their frequency selectivity, however, will be strongly influenced by group allocation. Thus, you many need to experiment with different allocations to find the best highpass/lowpass combination. Note also that the length of H 0 z H 0 z may differ from the length of H 0 z H 0 z .

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

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

#### Collection 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?

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

#### 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?

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