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 $z$ plane as follows:

Theorem: Every lossless $p\times q$ transfer function matrix $\mathbf{H}(z)$ is paraunitary, i.e.,

$${\tilde{\mathbf{H}}}(z) \mathbf{H}(z) = \mathbf{I}_q$$

By construction, every paraunitary matrix transfer function is unitary on the unit circle for all $\omega$ . Away from the unit circle, the paraconjugate ${\tilde{\mathbf{H}}}(z)$ is the unique analytic continuation of $\overline{\mathbf{H}^T(\ejo )}$ (the Hermitian transpose of $\mathbf{H}(\ejo )$ ).

Example: The normalized DFT matrix is an $N\times N$ order zero paraunitary transformation. This is because the normalized DFT matrix, $\mathbf{W}=[W_N^{nk}]/\sqrt{N},\,n,k=0,\ldots,N-1$ , where $W_N\isdef e^{-j2\pi/N}$ , is a unitary matrix:

$$\frac{\mathbf{W}^\ast}{\sqrt{N}} \frac{\mathbf{W}}{\sqrt{N}} = \mathbf{I}_N$$