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Aliasing-Cancellation Conditions of Filterbanks

Module by: Phil Schniter

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

Introduction

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).
alias_f1.png
Figure 1

Aliasing Cancellation Conditions

The aliasing cancellation conditions follow directly from the input/output equations derived below. Let i∈ 01 i 0 1 denote the filterbank branch index. Then
U i z=12∑p=01 H i z12ⅇ-ⅈπpXz12ⅇ-ⅈπp U i z 1 2 p 0 1 H i z 1 2 p X z 1 2 p (1)
Yz=∑i=01 G i z U i z2=∑i=01 G i z12∑p=01 H i zⅇ-ⅈπpXzⅇ-ⅈπp=12∑i=01 G i z H i zXz+ H i -zX-z=12XzX-z 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 (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 . Hz H z is often called the aliasing component matrix. For aliasing cancellation, we need to ensure that X-z 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 (3)
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 -zCzXz Y z 1 2 H 0 z H 1 z H 1 z H 0 z C z X z (4)
where Tz=12 H 0 z H 1 -z- H 1 z H 0 -zCz 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=xn-l y n x n l for some l∈ℕ l , or equivalently when Tz=z-l T z z l .
note: 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 .

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