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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 fromdownsampling/upsampling is completely cancelled. Below we derive aliasing-cancellation conditions for two-channelfilterbanks. Though the results can be extended to M-channel filterbanks in a rather straightforward manner, thetwo-channel case offers a more lucid explanation of the principle ideas (see [link] ).

Aliasing cancellation conditions

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

U i z 1 2 p 0 1 H i z 1 2 p X z 1 2 p
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 H z H 0 z H 1 z H 0 z H 1 z . H z is often called the aliasing component matrix . For aliasing cancellation, we need to ensure that X z does not contribute to the output 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 which is guaranteed by
G 0 z G 1 z H 1 z H 0 z
or by the following pair of conditions for any rational C z G 0 z C z H 1 z G 1 z C z H 0 z Under these aliasing-cancellation conditions, we get theinput/output relation
Y z 1 2 H 0 z H 1 z H 1 z H 0 z C z X z
where T z 1 2 H 0 z H 1 z H 1 z H 0 z C z represents the system transfer function. We saythat "perfect reconstruction" results when y n x n l for some l , or equivalently when T z z l .
The aliasing-cancellation conditions remove one degree of freedom from our filterbank design; originally, we had thechoice of four transfer functions H 0 z H 1 z G 0 z G 1 z , whereas now we can choose three: H 0 z H 1 z C z .

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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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