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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.

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).

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
(1)
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
(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 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
(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 z)CzXz 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 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 .

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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