With the above definition for paraconjugation of a MIMO transfer-function matrix, we may generalize the MIMO allpass condition Eq. (D.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: