#### 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:

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Properties of Paraunitary Systems

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MIMO Paraconjugate