Paraunitary Filter Banks
Paraunitary filter banks form an interesting subset of perfect
reconstruction (PR) filter banks. We saw above that we get a PR filter
bank whenever the
synthesis polyphase matrix
times the
analysis polyphase matrix
is the identity matrix
, i.e., when
![]() |
(12.70) |
In particular, if


Paraconjugation is the generalization of the complex conjugate
transpose operation from the unit circle to the entire
plane. A
paraunitary filter bank is therefore a generalization of an
orthogonal filter bank. Recall that an orthogonal filter bank
is one in which
is an orthogonal (or unitary) matrix, to
within a constant scale factor, and
is its transpose (or
Hermitian transpose).
Lossless Filters
To motivate the idea of paraunitary filters, let's first review some properties of lossless filters, progressing from the simplest cases up to paraunitary filter banks:
A linear, time-invariant filter
is said to be
lossless (or
allpass) if it preserves signal
energy. That is, if the input signal is
, and the output
signal is
, then we have
![]() |
(12.71) |
In terms of the


![]() |
(12.72) |
Notice that only stable filters can be lossless since, otherwise,

It is straightforward to show that losslessness implies
![]() |
(12.73) |
That is, the frequency response must have magnitude 1 everywhere on the unit circle in the

![]() |
(12.74) |
and this form generalizes to


The paraconjugate of a transfer function may be defined as the
analytic continuation of the complex conjugate from the unit circle to
the whole
plane:
![]() |
(12.75) |
where





![]() |
(12.76) |
in which the conjugation of

We refrain from conjugating
in the definition of the paraconjugate
because
is not analytic in the complex-variables sense.
Instead, we invert
, which is analytic, and which
reduces to complex conjugation on the unit circle.
The paraconjugate may be used to characterize allpass filters as follows:
A causal, stable, filter
is allpass if and only if
![]() |
(12.77) |
Note that this is equivalent to the previous result on the unit circle since
![]() |
(12.78) |
To generalize lossless filters to the multi-input, multi-output (MIMO) case, we must generalize conjugation to MIMO transfer function matrices.
A
transfer function matrix
is
said to be lossless
if it is stable and its frequency-response matrix
is
unitary. That is,
![]() |
(12.79) |
for all





![]() |
(12.80) |
Note that
is a
matrix
product of a
times a
matrix. If
, then
the rank must be deficient. Therefore, we must have
.
(There must be at least as many outputs as there are inputs, but it's
ok to have extra outputs.)
A lossless
transfer function matrix
is paraunitary,
i.e.,
![]() |
(12.81) |
Thus, every paraunitary matrix transfer function is unitary on the unit circle for all



Lossless Filter Examples
The simplest lossless filter is a unit-modulus gain
![]() |
(12.82) |
where




A lossless FIR filter can only consist of a single nonzero tap:
for some fixed integer




Every finite-order, single-input, single-output (SISO), lossless IIR filter (recursive allpass filter) can be written as
![]() |
(12.84) |
where








The normalized DFT matrix is an
order zero paraunitary
transformation. This is because the normalized DFT matrix,
,
, where
, is a unitary matrix:
![]() |
(12.85) |
Properties of Paraunitary Filter Banks
Paraunitary systems are essentially multi-input, multi-output (MIMO)
allpass filters. Let
denote the
matrix transfer
function of a paraunitary system. In the square case (
), the
matrix determinant,
, is an allpass filter.
Therefore, if a square
contains FIR elements, its determinant
is a simple delay:
for some integer
.
An
-channel analysis filter bank can be viewed as an
MIMO filter:
![]() |
(12.86) |
A paraunitary filter bank must therefore satisfy
![]() |
(12.87) |
More generally, we allow paraunitary filter banks to scale and/or delay the input signal:
![]() |
(12.88) |
where


We can note the following properties of paraunitary filter banks:
- The synthesis filter bank is simply the paraconjugate of the
analysis filter bank:
(12.89)
That is, since the paraconjugate is the inverse of a paraunitary filter matrix, it is exactly what we need for perfect reconstruction. - The channel filters
are power complementary:
(12.90)
This follows immediately from looking at the paraunitary property on the unit circle. - When
is FIR, the corresponding synthesis filter matrix
is also FIR.
- When
is FIR, each synthesis filter,
, is simply the
of its corresponding analysis filter
:
(12.91)
whereis the filter length. (When the filter coefficients are complex,
includes a complex conjugation as well.) This follows from the fact that paraconjugating an FIR filter amounts to simply flipping (and conjugating) its coefficients. As we observed in (11.83) above (§11.5.2), only trivial FIR filters of the form
can be paraunitary in the single-input, single-output (SISO) case. In the MIMO case, on the other hand, paraunitary systems can be composed of FIR filters of any order.
- FIR analysis and synthesis filters in paraunitary filter banks
have the same amplitude response. This follows from the fact
that
, i.e., flipping an FIR filter impulse response
conjugates the frequency response, which does not affect its amplitude response
.
- The polyphase matrix
for any FIR paraunitary perfect reconstruction filter bank can be written as the product of a paraunitary and a unimodular matrix, where a unimodular polynomial matrix
is any square polynomial matrix having a constant nonzero determinant. For example,
Examples
Consider the Haar filter bank discussed in §11.3.3, for which
![]() |
(12.92) |
The paraconjugate of

![]() |
(12.93) |
so that
![]() |
(12.94) |
Thus, the Haar filter bank is paraunitary. This is true for any power-complementary filter bank, since when


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Filter Banks Equivalent to STFTs
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Perfect Reconstruction Filter Banks