Skip to content Skip to navigation


You are here: Home » Content » Aliasing-Cancellation Conditions of Filterbanks


Recently Viewed

This feature requires Javascript to be enabled.

Aliasing-Cancellation Conditions of Filterbanks

Module by: Phil Schniter. E-mail the author

Summary: This module will look at methods and examples of aliasing-cancellation conditions.


It is possible to design combinations of analysis and synthesis filters such that the aliasing from downsampling/upsampling is completely cancelled. Below we derive aliasing-cancellation conditions for two-channel filterbanks. Though the results can be extended to M-channel filterbanks in a rather straightforward manner, the two-channel case offers a more lucid explanation of the principle ideas (see Figure 1).

Figure 1
Figure 1 (alias_f1.png)

Aliasing Cancellation Conditions

The aliasing cancellation conditions follow directly from the input/output equations derived below. Let i01 i 0 1 denote the filterbank branch index. Then

U i z=12p=01 H i z12e(i)πpXz12e(i)πp U i z 1 2 p 0 1 H i z 1 2 p X z 1 2 p
Yz=i=01 G i z U i z2=i=01 G i z12p=01 H i ze(i)πpXze(i)πp=12i=01 G i z( H i zXz+ H i zXz)=12( XzXz )( H 0 z H 1 z H 0 z H 1 z )( G 0 z G 1 z ) Y z i 0 1 G i z U i z 2 i 0 1 G i z 1 2 p 0 1 H i z p X z p 1 2 i 0 1 G i z H i z X z H i z X 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
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 . Hz H z is often called the aliasing component matrix. For aliasing cancellation, we need to ensure that Xz X z does not contribute to the output Yz Y z . This requires that ( H 0 z H 1 z )( G 0 z G 1 z )= H 0 z G 0 z+ H 1 z G 1 z=0 H 0 z H 1 z G 0 z G 1 z H 0 z G 0 z H 1 z G 1 z 0 which is guaranteed by
G 0 z G 1 z= H 1 z H 0 z G 0 z G 1 z H 1 z H 0 z
or by the following pair of conditions for any rational Cz Cz G 0 z=Cz H 1 z G 0 z C z H 1 z G 1 z=(Cz) H 0 z G 1 z C z H 0 z Under these aliasing-cancellation conditions, we get the input/output relation
Yz=12( H 0 z H 1 z H 1 z H 0 z)CzXz Y z 1 2 H 0 z H 1 z H 1 z H 0 z C z X z
where Tz=12( H 0 z H 1 z H 1 z H 0 z)Cz T z 1 2 H 0 z H 1 z H 1 z H 0 z C z represents the system transfer function. We say that "perfect reconstruction" results when yn=xnl y n x n l for some lN l , or equivalently when Tz=zl T z z l .


The aliasing-cancellation conditions remove one degree of freedom from our filterbank design; originally, we had the choice of four transfer functions H 0 z H 1 z G 0 z G 1 z H 0 z H 1 z G 0 z G 1 z , whereas now we can choose three: H 0 z H 1 zCz H 0 z H 1 z C z .

Content actions

Download module as:

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


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