Conjugate Quadrature Filters (CQF)
A class of causal, FIR, two-channel, critically sampled, exact perfect-reconstruction filter-banks is the set of so-called Conjugate Quadrature Filters (CQF). In the z-domain, the CQF relationships are
![]() |
(12.39) |
In the time domain, the analysis and synthesis filters are given by
![\begin{eqnarray*}
h_1(n) &=& -(-1)^n h_0(L-1-n) \\ [5pt]
f_0(n) &=& h_0(L-1-n) \\ [5pt]
f_1(n) &=& -(-1)^n h_0(n) \eqsp - h_1(L-1-n).
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2071.png)
That is,
for the lowpass channel, and each highpass
channel filter is a modulation of its lowpass counterpart by
.
Again, all four analysis and synthesis filters are determined by the
lowpass analysis filter
. It can be shown that this is an
orthogonal filter bank. The analysis filters
and
are power complementary, i.e.,
![]() |
(12.40) |
or
![]() |
(12.41) |
where






![]() |
(12.42) |
The power symmetric case was introduced by Smith and Barnwell in 1984 [272]. With the CQF constraints, (11.18) reduces to
Let
, such that
is a spectral factor of
the half-band filter
(i.e.,
is a nonnegative power
response which is lowpass, cutting off near
). Then,
(11.43) reduces to
The problem of PR filter design has thus been reduced to designing one half-band filter



A simple design of an FIR half-band filter would be to window a sinc function:
![]() |
(12.45) |
where

Note that as a result of (11.43), the CQF filters are power complementary. That is, they satisfy
![]() |
(12.46) |
Also note that the filters


- FIR
- orthogonal
- linear phase
By relaxing ``orthogonality'' to ``biorthogonality'', it becomes possible to obtain FIR linear phase filters in a critically sampled, perfect reconstruction filter bank. (See §11.9.)
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Orthogonal Two-Channel Filter Banks
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Linear Phase Quadrature Mirror Filter Banks