MIMO Paraconjugate

Definition: The paraconjugate of $\mathbf{H}(z)$ is defined as

$${\tilde{\mathbf{H}}}(z) \isdef \mathbf{H}^\ast(z^{-1})$$

where $\mathbf{H}^\ast(z)$ denotes transpose of $\mathbf{H}(z)$ followed by complex-conjugation of the coefficients within $\mathbf{H}^T(z)$ (and not the powers of $z$ ). For example, if

$$\mathbf{H}(z)=\left[\begin{array}{c} 1+jz^{-1} \\ [2pt] 1+z^{-2} \end{array}\right]$$

then

$${\tilde{\mathbf{H}}}(z)=\left[\begin{array}{cc} 1-jz & 1+z^2 \end{array}\right]$$