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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.