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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