## Appendix B Derivations
### Transmission-Based Representations
\begin{pmatrix}\mathbf{O}&\mathbf{O}& \{\hat{\mathbf{T}}_{UU}^{-}(-p)\}^{\star}&\mathbf{O}\\ \mathbf{O}&\mathbf{O}&\mathbf{O}\\ \mathbf{O}&\hat{\mathbf{T}}_{UU}^{+}(p)&\mathbf{O}&\mathbf{O}\end{pmatrix} \begin{pmatrix}\hat{\mathbf{F}}_{U}^{-}(p)\\ \hat{\mathbf{F}}_{U}^{+}(p)\\ \hat{\mathbf{F}}_{U}^{-}(p)\end{pmatrix} , \tag{15}\]  
where we have dropped the subscript \(I\) once again for notational convenience. After taking the inverse Fourier transform (as defined in Equation (A3) and applying operator \(\mathcal{Z}\) to the first and third row, we obtain Equation (13), presented in the main text. Appendix C Expression for \(\mathbf{F}\) as a Function of \(\boldsymbol{\tau}\), \(\boldsymbol{\alpha}\) and \(\boldsymbol{\beta}\)  
In this Appendix, we write \(\mathbf{F}\) explicitly as a function of \(\boldsymbol{\tau}\), \(\boldsymbol{\alpha}\) and \(\boldsymbol{\beta}\). We start with the substitution of Equations (20)-(23) into the definition of \(\mathbf{B}_{Mar}\), which is given in the left-hand side of Equation (12). We write the result as  
\[\mathbf{B}_{Mar}=\underbrace{ \begin{pmatrix}\mathbf{G}_{U}^{[p]}\mathcal{R}_{U }\mathcal{Z}\mathbf{E}_{UU}^{S}\\ \mathbf{E}_{UU}^{-}\\ \mathbf{O}\\ \mathbf{O}\end{pmatrix} }_{ \begin{subarray}{c}\boldsymbol{\alpha}&\boldsymbol{ \beta}\end{subarray} } \begin{pmatrix}\boldsymbol{\alpha}&\boldsymbol{0}\\ \boldsymbol{\beta}\end{pmatrix} }_{ \begin{subarray}{c}\boldsymbol{\alpha}&\boldsymbol{ \beta}\end{subarray} }+\underbrace{ \begin{pmatrix}\mathbf{O}\\ \mathbf{O}\\ \mathbf{G}_{U}^{[p]}\mathcal{R}_{U}\mathcal{H}_{A}^{S}\mathbf{E}_{UU}^{S}\\ \mathcal{T}_{A}^{P}\mathcal{Z}\mathbf{E}_{UU}^{P}\end{pmatrix} }_{ \begin{subarray}{c}\boldsymbol{\alpha}&\boldsymbol{0}\\ \boldsymbol{\beta}\end{subarray} } \begin{pmatrix}\boldsymbol{\alpha}&\boldsymbol{0} \\ \boldsymbol{\beta}\end{pmatrix} ^{-\frac{1}{2}}, \tag{16}\]where \(\mathbf{M}_{M_{M_{M_{M}}}}^{l}\) and \(\mathbf{M}_{M_{M_{M}}}^{l}\) have been used to identify the constructed matrices. In a similar way, we can substitute Equations (20)-(23) into the definition of \(\mathbf{B}_{M_{M}}\), which is given in the left-hand side of Equation (15). This leads to  
\[\mathbf{B}_{M_{M}}=\underbrace{ \begin{pmatrix}\mathbf{O}\\ \mathbf{E}_{UL}^{p}\\ \mathbf{O}\\ -\mathbf{G}_{L}^{|p|}\mathcal{T}_{UL}\mathcal{Z}\mathbf{E}_{UL}^{s}\\ \mathbf{M}_{M_{M}}^{l}\end{pmatrix} }_{\mathbf{M}_{M_{M}}^{l}} \begin{pmatrix} \alpha&0\\ 0&\beta\end{pmatrix} ^{\frac{1}{2}}+\underbrace{ \begin{pmatrix}-\Theta_{U}^{|p|} \mathcal{T}_{UL}\mathcal{H}_{S}^{s}\mathbf{E}_{UL}^{s}\\ \mathbf{O}\\ \mathcal{T}_{d}^{P}\mathcal{Z}\mathbf{E}_{UL}^{p}\\ \mathbf{M}_{M_{M}}^{l}\end{pmatrix} }_{\mathbf{M}_{M}^{l}} \begin{pmatrix}\alpha& 0\\ 0&\beta\end{pmatrix} ^{-\frac{1}{2}}, \tag{20}\]  
where \(\mathbf{M}_{M}^{l}\) and \(\mathbf{M}_{M}^{l}\) have been used to identify the constructed matrices. Finally, we may substitute Equations (22) and (23) into the definition of \(\mathbf{B}_{Cau}\), which is given in the left-hand side of Equation (17). This yields  
\[\mathbf{B}_{Cau}=\underbrace{\frac{1}{2} \begin{pmatrix}\mathcal{R}_{UL} \mathcal{Z}\mathbf{E}_{UL}^{s}\\ -\mathbf{E}_{UL}\\ \mathcal{T}_{UL}\mathcal{Z}\mathbf{E}_{UL}^{s}\\ \mathbf{M}_{Cau}^{l}\end{pmatrix} }_{\mathbf{M}_{Cau}^{l}} \begin{pmatrix}\alpha &0\\ 0&\beta\end{pmatrix} ^{\frac{1}{2}}+\underbrace{\frac{1}{2} \begin{pmatrix} \mathbf{O}\\ \mathcal{T}_{UL}\mathcal{H}_{S}^{s}\mathbf{E}_{UL}^{s}\\ \mathcal{R}_{L}\mathcal{H}_{S}^{s}\mathbf{E}_{UL}^{s}\\ -\mathcal{Z}\mathcal{H}_{S}^{s}\mathbf{E}_{UL}^{s}\\ \mathbf{M}_{Cau}^{l}\end{pmatrix} }_{\mathbf{M}_{Cau}^{l}} \begin{pmatrix}\alpha &0\\ 0&\beta\end{pmatrix} ^{-\frac{1}{2}}, \tag{21}\]  
where \(\mathbf{M}_{M}^{l}\) and \(\mathbf{M}_{M}^{l}\) have been used to identify the constructed matrices. Next, we substitute Equations (20)-(21) into (18) and apply the pseudo-inverse of \(\mathbf{A}\) to both sides of the result. This eventually leads to  
\[\mathbf{F}_{m}=\mathbf{A}^{\ddagger}\underbrace{