Allpass Filters
This appendix addresses the general problem of characterizing all digital allpass filters, including multi-input, multi-output (MIMO) allpass filters. As a result of including the MIMO case, the mathematical level is a little higher than usual for this book. The reader in need of more background is referred to [84,37,98].Our first task is to show that losslessness implies allpass.
Definition:
A linear, time-invariant filter is said to be
lossless if it preserves signal
energy for every input signal. That is, if the input signal is
, and the output signal is
, then we have




Notice that only stable filters can be lossless, since otherwise
can be infinite while
is finite. We further
assume all filters are causalC.1 for
simplicity. It is straightforward to show the following:
Theorem: A stable, linear, time-invariant (LTI) filter transfer function
is lossless if and only if


Proof: We allow the signals and filter impulse response
to be complex. By Parseval's theorem
[84] for the DTFT, we have,C.2 for any signal
,






Since this must hold for all





![$ \omega\in[-\pi,\pi]$](http://www.dsprelated.com/josimages_new/filters/img899.png)

We have shown that every lossless filter is allpass. Conversely, every unity-gain allpass filter is lossless.
Allpass Examples
- The simplest allpass filter is a unit-modulus gain
can be any phase value. In the real case
can only be 0 or
, in which case
.
- A lossless FIR filter can consist only of a single nonzero tap:
, where
is again some constant phase, constrained to be 0 or
in the real-filter case. Since we are considering only causal filters here,
. As a special case of this example, a unit delay
is a simple FIR allpass filter.
- The transfer function of every finite-order, causal,
lossless IIR digital filter (recursive allpass filter) can be written
as
,
We may think of
as the flip of
. For example, if
, we have
. Thus,
is obtained from
by simply reversing the order of the coefficients and conjugating them when they are complex.
- For analog filters, the general finite-order allpass
transfer function is
,
. The polynomial
can be obtained by negating every other coefficient in
, and multiplying by
. In analog, a pure delay of
seconds corresponds to the transfer function
(root of
at
), the polynomial
has a root at
. Thus, the poles and zeros can be paired off as a cascade of terms such as
Paraunitary FiltersC.4
Another way to express the allpass condition
is to
write





Definition: 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:








Examples:
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:
Theorem: A causal, stable, filter is allpass if and only if


Multi-Input, Multi-Output (MIMO)
Allpass Filters
To generalize lossless filters to the multi-input, multi-output (MIMO) case, we must generalize conjugation to MIMO transfer function matrices:
Theorem: A transfer function matrix
is
lossless if and only if
its frequency-response matrix
is unitary, i.e.,
for all






Let
denote the length
output vector at time
, and
let
denote the input
-vector at time
. Then in the
frequency domain we have
, which
implies



We have thus shown that in the MIMO case, losslessness is equivalent to having a unitary frequency-response matrix. A MIMO allpass filter is therefore any filter with a unitary frequency-response matrix.
Note that
is a
matrix product
of a
times a
matrix. If
, then the rank
must be deficient. Therefore,
. (There must be at least as
many outputs as there are inputs, but it's ok to have extra outputs.)
Paraunitary MIMO Filters
In §C.2, we generalized the allpass property
to the entire complex plane as



MIMO Paraconjugate
Definition:
The paraconjugate of
is defined as





![$\displaystyle \mathbf{H}(z)=\left[\begin{array}{c} 1+jz^{-1} \\ [2pt] 1+z^{-2} \end{array}\right]
$](http://www.dsprelated.com/josimages_new/filters/img1628.png)
![$\displaystyle {\tilde{\mathbf{H}}}(z)=\left[\begin{array}{cc} 1-jz & 1+z^2 \end{array}\right]
$](http://www.dsprelated.com/josimages_new/filters/img1629.png)
MIMO Paraunitary Condition
With the above definition for paraconjugation of a MIMO transfer-function
matrix, we may generalize the MIMO allpass condition Eq.(C.2) to the
entire
plane as follows:
Theorem:
Every lossless transfer function matrix
is paraunitary,
i.e.,

By construction, every paraunitary matrix transfer function is
unitary on the unit circle for all . Away from the
unit circle, the paraconjugate
is the unique analytic
continuation of
(the Hermitian transpose of
).
Example:
The normalized DFT matrix is an order zero
paraunitary transformation. This is because the normalized DFT
matrix,
, where
, is a
unitary matrix:

Properties of Paraunitary Systems
Paraunitary systems are essentially multi-input, multi-output (MIMO)
allpass filters. Let
denote the
matrix transfer
function of a paraunitary system. Some of its properties include the
following [98]:
- 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
.
Properties of Paraunitary Filter Banks
An -channel filter bank can be viewed as an
MIMO filter
![$\displaystyle \mathbf{H}(z) = \left[\begin{array}{c} H_1(z) \\ [2pt] H_2(z) \\ [2pt] \vdots \\ [2pt] H_N(z)\end{array}\right]
$](http://www.dsprelated.com/josimages_new/filters/img1640.png)
A paraunitary filter bank must therefore obey




We can note the following properties of paraunitary filter banks:
- A synthesis filter bank
corresponding to analysis filter bank
is defined as that filter bank which inverts the analysis filter bank, i.e., satisfies
is paraunitary, its corresponding synthesis filter bank is simply the paraconjugate filter bank
, or
- The channel filters
in a paraunitary filter bank are power complementary:
- When
is FIR, the corresponding synthesis filter matrix
is also FIR. Note that this implies an FIR filter-matrix can be inverted by another FIR filter-matrix. This is in stark contrast to the case of single-input, single-output FIR filters, which must be inverted by IIR filters, in general.
- When
is FIR, each synthesis filter,
, is simply the
of its corresponding analysis filter
:
is 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.
Note that only trivial FIR filters 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
.
Paraunitary Filter Examples
The Haar filter bank is defined as
![$\displaystyle \mathbf{H}(z) = \frac{1}{\sqrt{2}}\left[\begin{array}{c} 1+z^{-1} \\ [2pt] 1-z^{-1} \end{array}\right]
$](http://www.dsprelated.com/josimages_new/filters/img1655.png)

![$\displaystyle {\tilde{\mathbf{H}}}(z) = \left[\begin{array}{cc} 1+z & 1 - z \end{array}\right] / \sqrt{2}
$](http://www.dsprelated.com/josimages_new/filters/img1656.png)
![$\displaystyle {\tilde{\mathbf{H}}}(z) \mathbf{H}(z) = \left[\begin{array}{cc} 1...
...ight] \left[\begin{array}{c} 1+z^{-1} \\ [2pt] 1-z^{-1} \end{array}\right]
= 1
$](http://www.dsprelated.com/josimages_new/filters/img1657.png)


For more about paraunitary filter banks, see Chapter 6 of [98].
Allpass Problems
- The BiQuad Allpass Section
- Show that every second-order filter having transfer function
, for all
and
. (Typically,
and
are chosen such that the filter is stable, but this is not necessary for the result to hold.)
- Find the zeros of the filter as a function of the poles.
In other words, given two poles, what is the rule for placing the zeros
in order to obtain an allpass filter?
- Find the phase response of the zeros in terms of the phase response of the poles.
- Show that every second-order filter having transfer function
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Elementary Audio Digital Filters