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
denote the filterbank branch index. Then
where
.
is often called the
aliasing component matrix .
For aliasing cancellation, we need to ensure that
does not contribute to the output
.
This requires that
which is guaranteed by
or by the following pair of conditions for any rational
Under these aliasing-cancellation conditions, we get theinput/output relation
where
represents the system transfer function. We saythat "perfect reconstruction" results when
for some
, or equivalently when
.
The aliasing-cancellation conditions remove one degree of
freedom from our filterbank design; originally, we had thechoice of four transfer functions
, whereas now we can choose three:
.
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