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



Next Section:
Lossless Filter Examples
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Example: Polyphase Analysis of the Weighted Overlap Add Case: 50% Overlap, Zero-Padding, and a Non-Rectangular Window