Two-Channel Critically Sampled Filter Banks
Figure 11.15 shows a simple two-channel band-splitting filter bank,
followed by the corresponding synthesis filter bank which
reconstructs the original signal (we hope) from the two channels. The
analysis filter
is a half-band lowpass filter, and
is a complementary half-band highpass filter. The synthesis filters
and
are to be derived. Intuitively, we expect
to be a lowpass that rejects the upper half-band due to the
upsampler by 2, and
should do the same but then also
reposition its output band as the upper half-band, which can be
accomplished by selecting the upper of the two spectral images in the
upsampler output.
The outputs of the two analysis filters in Fig.11.15 are
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(12.16) |
Using the results of §11.1, the signals become, after downsampling,
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(12.17) |
After upsampling, the signals become
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|
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After substitutions and rearranging, we find that the output

For perfect reconstruction, we require the aliasing term to be zero. For ideal half-band filters cutting off at





In this case, synthesis filter










Referring again to (11.18), we see that we also need the
non-aliased term to be of the form
where

![]() |
(12.21) |
That is, for perfect reconstruction, we need, in addition to aliasing cancellation, that the non-aliasing term reduce to a constant gain


Let
denote
. Then both constraints can be expressed in
matrix form as follows:
![]() |
(12.22) |
Substituting the aliasing-canceling choices for
and
from
(11.19) into the filtering-cancellation constraint (11.20), we
obtain
The filtering-cancellation constraint is almost satisfied by ideal zero-phase half-band filters cutting off at




Next Section:
Amplitude-Complementary 2-Channel Filter Bank
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Multirate Noble Identities