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

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

or

Integrating both sides of this equation with respect to yields that the total energy in equals the total energy out, as required by the definition of losslessness.

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
- MIMO Paraconjugate
- MIMO Paraunitary Condition
- Properties of Paraunitary Systems
- Properties of Paraunitary Filter Banks

- Paraunitary Filter Examples

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University