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MIMO Paraconjugate


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

$\displaystyle {\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

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

then

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

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