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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
|
(12.16) |
Using the results of §11.1, the signals become, after
downsampling,
|
(12.17) |
After upsampling, the signals become
After substitutions and rearranging, we find that the output
is a filtered replica of the input signal plus an aliasing term:
For perfect reconstruction, we require the aliasing term to be
zero. For ideal half-band filters cutting off at
, we
can choose
and
and the aliasing term is zero
because there is no spectral overlap between the channels, i.e.,
, and
.
However, more generally (and more practically), we can force the
aliasing to zero by choosing synthesis filters
In this case, synthesis filter
is still a lowpass, but the
particular one obtained by
-rotating the highpass analysis
filter around the unit circle in the
plane. Similarly, synthesis
filter
is the
-rotation (and negation) of the analysis
lowpass filter
on the unit circle.
For this choice of synthesis filters
and
, aliasing is
completely canceled for any choice of analysis filters
and
.
Referring again to (11.18), we see that we also need the
non-aliased term to be of the form
where
is of the form
|
(12.21) |
That is, for perfect reconstruction, we need, in addition to aliasing
cancellation, that the non-aliasing term reduce to a constant gain
and/or delay
. We will call this the filtering cancellation
constraint on the channel filters. Thus perfect reconstruction
requires both aliasing cancellation and filtering cancellation.
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
, since in that
case we have
and
. However, the
minus sign in (11.23) means there is a discontinuous sign flip as
frequency crosses
, which is not equivalent to a linear
phase term. Therefore the filtering cancellation constraint fails for
the ideal half-band filter bank! Recall from above, however, that
ideal half-band filters did work using a different choice of
synthesis filters, relying instead on their lack of spectral overlap.
The presently studied case from (11.19) arose from so-called
Quadrature Mirror Filters (QMF), which are discussed further
below. First, however, we'll look at some simple special cases.
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