Multi-Input, Multi-Output (MIMO)
for all , where denotes the identity matrix, and denotes the Hermitian transpose (complex-conjugate transpose) of :
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
Definition: The paraconjugate of is defined as
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 ).
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 :
- In the square case (), the matrix determinant,
, is an allpass filter.
- Therefore, if a square
contains FIR elements, its
determinant is a simple delay:
An -channel filter bank can be viewed as an MIMO filter
A paraunitary filter bank must therefore obey
We can note the following properties of paraunitary filter banks:
- A synthesis filter bank
to analysis filter bank
is defined as that filter bank
which inverts the analysis filter bank, i.e., satisfies
- The channel filters in a paraunitary filter bank
are power complementary:
is FIR, the corresponding synthesis filter
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.
is FIR, each synthesis filter,
, is simply the of its corresponding
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.
The Haar filter bank is defined as
For more about paraunitary filter banks, see Chapter 6 of .