##
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 , where denotes the identity matrix, and denotes the

*Hermitian transpose*(complex-conjugate transpose) of :

### 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

*coefficients*within (and not the powers of ). For example, if

#### 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.*,

*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*paraunitary filter bank*must therefore obey

- A
*synthesis filter bank*corresponding to analysis filter bank is defined as that filter bank which inverts the analysis filter bank,*i.e.*, satisfies*perfect reconstruction filter bank*. When a filter bank transfer function 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
:
- 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

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Allpass Problems

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Paraunitary FiltersC.4