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



for all






















Paraunitary MIMO Filters
In §C.2, we generalized the allpass property



MIMO Paraconjugate
Definition: The paraconjugate of






![$\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.

Theorem: Every lossless








![$ \mathbf{W}=[W_N^{nk}]/\sqrt{N},\,n,k=0,\ldots,N-1$](http://www.dsprelated.com/josimages_new/filters/img1633.png)


Properties of Paraunitary Systems
Paraunitary systems are essentially multi-input, multi-output (MIMO) allpass filters. Let

- 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

![$\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 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)


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Allpass Problems
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Paraunitary FiltersC.4