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:

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Properties of Paraunitary Systems
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MIMO Paraconjugate