Holographic Intelligent Reflecting Surfaces: Channel Modelling and Capacity Analysis

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Introduction
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Holographic Intelligent Reflecting Surfaces
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In holographic massive MIMO, an antenna surface with (approximately) continuous aperture is used for communication. The surface either actively generates beamformed RF signals or control its reflections of RF signals generated at other locations.

*Notation:* We use a bold symbol lowercase letter and bold symbol capital letter to denote a vector and a matrix, respectively. The trace, conjugate, transpose, conjugate transpose, and inverse of a matrix are denoted by $\text{Tr}[\cdot]$, $(\cdot)^*$, $(\cdot)^{\textrm{T}}$, $(\cdot)^{\textrm{H}}$, and $(\cdot)^{-1}$, respectively; $\propto$ represents both sides of the equation are multiplicatively connected to a constant; $|\cdot|$ denotes the modulus of a complex number or the determinant of a square matrix; $\delta (\cdot)$ denotes the Dirac delta function; $||\cdot||$ denotes the $\ell^2$ norm; $\textrm{diag}\left\{ \boldsymbol{A} \right\}$ represents the vector composed of the diagonal elements of those of $\boldsymbol{A}$. ${\textrm{diag}}\left\{\boldsymbol{a}\right\}$ and $\left[ \boldsymbol{A} \right]_{\textrm{diag}}$ represent the diagonal matrices with the diagonal specified by $\boldsymbol{a}$ and the diagonal elements of $\boldsymbol{A}$, respectively; $\mathbf{a} \times \mathbf{b}$ denotes the cross product between vectors $\mathsf{a}$ and $\mathbf{b}$; $\mathbb{E}_p[\boldsymbol{x}]$ denotes the expectation of $\boldsymbol{x}$ with respect to distribution $p$; $\mathcal{I}_N$ denotes the index set $\{-\frac{N-1}{2},\ldots,\frac{N-1}{2}\}$, where $N$ is a positive odd number; $\partial$ denotes the partial derivative operator. The complex Gaussian distribution of $\boldsymbol{x}$ with mean $\bar{\boldsymbol{x}}$, covariance matrix $\boldsymbol{\Sigma}$, and zero relation matrix is denoted by $\mathcal{CN}(\boldsymbol{x};\bar{\boldsymbol{x}},\boldsymbol{\Sigma})$.

Frequently used symbols are listed in Table \[SymbolTable\].

<span>||C<span>.06</span>|C<span>.35</span>||C<span>.06</span>|C<span>.35</span>||</span> Symbol & Definition & Symbol & Definition\
$d_{\mathsf{t}}$, $d_{\mathsf{r}}$ & Antenna spacing of Tx and Rx, respectively & $N_{\mathsf{t}}$, $N_{\mathsf{r}}$ & Numbers of Tx and Rx antennas, respectively\
$L_{\mathsf{t}}$, $L_{\mathsf{r}}$ & Lengths of Tx and Rx, respectively & $t_p$, $r_q$ & The $p$-th antenna of Tx and the $q$-th antenna of Rx, respectively\
$\varphi_{\mathsf{t}}$, $\varphi_{\mathsf{r}}$ & Angle between $z_{\mathsf{t}}$ axis and $z$ axis, angle between $z_{\mathsf{r}}$ axis and $z$ axis, respectively & $\psi_{\mathsf{t}}$, $\gamma_{\mathsf{t}}$ & Elevation and azimuth anlges of Tx in $x$-$y_t$-$z_t$ coordinate system\
$\psi_{\mathsf{r}}$, $\gamma_{\mathsf{r}}$ & Elevation and azimuth anlges of Rx in $x$-$y_r$-$z_r$ coordinate system & $\xi_{\mathsf{t}}$ & Polarization angle of Tx in $x$-$y$-$z$ coordinate system\
$m_{k,l}$ & The $k$-th along $x$ and the $l$-th along $y$ RE & $g(\boldsymbol{\Psi}_{(k,l),p}, \allowbreak \boldsymbol{\Psi}_{q,(k,l)})$ & RE response function of $m_{k,l}$ along reflection direction $m_{k,l}$-$r_q$ for an incident wave from direction $t_p$-$m_{k,l}$\
$L_{\mathsf{x}}^{\mathsf{tot}}$, $L_{\mathsf{y}}^{\mathsf{tot}}$ & Length of the IRS along $x$- and $y$-axes, respectively & $L_{\mathsf{x}}$, $L_{\mathsf{y}}$ & Lengths of each RE along $x$- and $y$-axes, respectively\
$Q_{\mathsf{x}}$, $Q_{\mathsf{y}}$ & Number of REs along $x$- and $y$-axes, respectively & $S_{\mathsf{x}}$, $S_{\mathsf{y}}$ & RE spacing of IRS along $x$- and $y$-axes, respectively\
$\lambda$, $\kappa$ & Wavelength and wavenumber, respectively & $\Gamma$, $\tau$, $\omega$ & Reflection coefficient, its amplitude, and its phase, respectively\
& & &\
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Cascaded LoS Channel Modelling
==============================

In this section, we first introduce the geometric model of the considered IRS-aided MIMO system, and the response function of each reflecting element of the IRS. Based on that, we derive the attenuation and phase shift of the EM wave experienced in each Tx-IRS-Rx link, and establish an explicit channel representation of the considered system.

IRS-Aided MIMO System {#SecSystem}
---------------------

Consider an IRS-aided MIMO system operating in the sub-THz band with $N_{\mathsf{t}}$ transmit antennas and $N_{\mathsf{r}}$ receive antennas. We assume that the direct path between the receiver and the transmitter is blocked by obstacles. An IRS is deployed to assist the communication between the transmitter and the receiver. The propagation paths from the transmitter to the IRS and from the IRS to the receiver are in LoS, and they are only affected by free-space attenuation. We refer to the above model as the cascaded LoS MIMO channel.

\[b\]<span>0.35</span> ![3-D geometrical model for a cascaded LoS MIMO channel with arbitrarily deployed ULAs at both link ends. (a) The LoS channel between the Tx and the Rx is blocked by obstacles. A cascaded LoS link aided by an IRS is established between the two end. (b) $\varphi_{\mathsf{r}}$ and $\omega_{\mathsf{r}}$ are defined as the elevation and the azimuth angles of $r_0$ in the $x$-$y$-$z$ coordinate system, respectively. (c) $\psi_{\mathsf{r}}$ and $\gamma_{\mathsf{r}}$ are defined as the elevation and the azimuth angles of the principal direction of the Rx in the $x_{\mathsf{r}}$-$y_{\mathsf{r}}$-$z_{\mathsf{r}}$ coordinate system, respectively.<span data-label="SysFig"></span>](GeneralizedAllCenterTileObs.pdf "fig:"){width="\textwidth"}

\[b\]<span>0.35</span> ![3-D geometrical model for a cascaded LoS MIMO channel with arbitrarily deployed ULAs at both link ends. (a) The LoS channel between the Tx and the Rx is blocked by obstacles. A cascaded LoS link aided by an IRS is established between the two end. (b) $\varphi_{\mathsf{r}}$ and $\omega_{\mathsf{r}}$ are defined as the elevation and the azimuth angles of $r_0$ in the $x$-$y$-$z$ coordinate system, respectively. (c) $\psi_{\mathsf{r}}$ and $\gamma_{\mathsf{r}}$ are defined as the elevation and the azimuth angles of the principal direction of the Rx in the $x_{\mathsf{r}}$-$y_{\mathsf{r}}$-$z_{\mathsf{r}}$ coordinate system, respectively.<span data-label="SysFig"></span>](phiomege.pdf "fig:"){width="\textwidth"}

\[b\]<span>0.27</span> ![3-D geometrical model for a cascaded LoS MIMO channel with arbitrarily deployed ULAs at both link ends. (a) The LoS channel between the Tx and the Rx is blocked by obstacles. A cascaded LoS link aided by an IRS is established between the two end. (b) $\varphi_{\mathsf{r}}$ and $\omega_{\mathsf{r}}$ are defined as the elevation and the azimuth angles of $r_0$ in the $x$-$y$-$z$ coordinate system, respectively. (c) $\psi_{\mathsf{r}}$ and $\gamma_{\mathsf{r}}$ are defined as the elevation and the azimuth angles of the principal direction of the Rx in the $x_{\mathsf{r}}$-$y_{\mathsf{r}}$-$z_{\mathsf{r}}$ coordinate system, respectively.<span data-label="SysFig"></span>](xryrzr.pdf "fig:"){width="\textwidth"}

The 3-D geometrical model of the system to be analyzed is illustrated in Fig. \[SysFiga\]. The origin $O$ is taken to be at the center of the IRS. The $x$- and the $y$-axes are taken to be parallel to the sides of the IRS, while the $z$-axis is perpendicular to the IRS. Uniform linear arrays (ULAs) are deployed at both the transmitter and the receiver. The antennas of the transmit ULA (Tx) and the receive ULA (Rx) are uniformly spaced with interval $d_{\mathsf{t}}$ and $d_{\mathsf{r}}$, respectively. The lengths of Tx and Rx are given by $L_{\mathsf{t}} = (N_{\mathsf{t}}-1)d_{\mathsf{t}}$ and $L_{\mathsf{r}} = (N_{\mathsf{r}}-1)d_{\mathsf{r}}$, respectively. Denote by $t_p$ the $p$-th antenna of the Tx, and by $r_q$ the $q$-th antenna of the Rx, where $p \in \mathcal{I}_{N_{\mathsf{t}}}$ and $q\in \mathcal{I}_{N_{\mathsf{r}}}$. For notational brevity, we assume that $N_{\mathsf{t}}$ and $N_{\mathsf{r}}$ are both odd numbers, and thus $t_0$ and $r_0$ are at the centers of the Tx and the Rx, respectively. $D_{\mathsf{t}}$ and $D_{\mathsf{r}}$ denote the distance from the origin to $t_0$ and $r_0$, respectively. To describe the positions of the Tx and the Rx relative to the IRS, we introduce auxiliary coordinate axes $x_{\mathsf{t}}$, $y_{\mathsf{t}}$, $z_{\mathsf{t}}$, $x_{\mathsf{r}}$, $y_{\mathsf{r}}$, $z_{\mathsf{r}}$, and angles $\varphi_{\mathsf{t}}$, $\omega_{\mathsf{t}}$, $\psi_{\mathsf{t}}$, $\gamma_{\mathsf{t}}$, $\varphi_{\mathsf{r}}$, $\omega_{\mathsf{r}}$, $\psi_{\mathsf{r}}$, $\gamma_{\mathsf{t}}$, and $\gamma_{\mathsf{r}}$, where the subscripts $\mathsf{t}$ and $\mathsf{r}$ are for distinguishing the transmit and receive two sides in the communication link. We take the IRS-Rx side as an example to explain these auxiliary coordinate axes and angles. As in Fig. \[SysFigb\], $\varphi_{\mathsf{r}}$ and $\omega_{\mathsf{r}}$ are defined as the elevation and the azimuth angles of $r_0$ in the $x$-$y$-$z$ coordinate system, respectively, with $\varphi_{\mathsf{r}}\in [0,\frac{\pi}{2}]$ and $\omega_{\mathsf{r}} \in [0,2\pi)$. With the center of the Rx determined by $D_{\mathsf{r}}$, $\varphi_{\mathsf{r}}$, and $\omega_{\mathsf{r}}$, we further describe the orientation of the Rx. The $z_{\mathsf{r}}$-axis is taken to be in the direction from the origin to $t_0$, and the $y_{\mathsf{r}}$-axis is taken on the $z$-$z_{\mathsf{r}}$ plane and perpendicular to the $z_{\mathsf{r}}$-axis, with the angle between the positive directions of the $z$- and the $z_{\mathsf{r}}$- axes greater than $\varphi_{\mathsf{r}}$. The $x_{\mathsf{r}}$-axis is defined by its unit vector $\mathbf{n}_{x_{\mathsf{r}}} = \mathbf{n}_{y_{\mathsf{r}}}\times \mathbf{n}_{z_{\mathsf{r}}}$, where $\mathbf{n}_{y_{\mathsf{r}}}$ and $\mathbf{n}_{z_{\mathsf{r}}}$ are the unit vectors of the $y_{\mathsf{r}}$ and $z_{\mathsf{r}}$ axes, respectively, and $\times$ is the cross product operator. Define the direction from $r_{-\frac{N_{\mathsf{r}}-1}{2}}$ to $r_{\frac{N_{\mathsf{r}}-1}{2}}$ as the principal direction of the Rx. As illustrated in Fig. \[SysFigc\], $\psi_{\mathsf{r}}$ and $\gamma_{\mathsf{r}}$ are defined as the elevation and the azimuth angles of the principal direction of the Rx in the $x_{\mathsf{r}}$-$y_{\mathsf{r}}$-$z_{\mathsf{r}}$ coordinate system, respectively, with $\psi_{\mathsf{r}}\in [0,\pi]$ and $\gamma_{\mathsf{r}} \in [0,2\pi)$. Such a geometrical model with parameters $\varphi_{\mathsf{r}}$, $\omega_{\mathsf{r}}$, $\psi_{\mathsf{r}}$, $\gamma_{\mathsf{r}}$, and $D_{\mathsf{r}}$ are competent to describe any location and orientation of the Rx relative to the IRS. For the Tx-IRS side, axes $x_{\mathsf{t}}$-, $y_{\mathsf{t}}$-, $z_{\mathsf{t}}$-, and angles $\varphi_{\mathsf{t}}$, $\omega_{\mathsf{t}}$, $\psi_{\mathsf{t}}$, and $\gamma_{\mathsf{t}}$ are defined in a similar manner as the counterparts in the IRS-Rx side, with $\varphi_{\mathsf{t}}\in [0,\frac{\pi}{2}]$, $\omega_{\mathsf{t}} \in [0,2\pi)$, $\psi_{\mathsf{t}}\in [-\frac{\pi}{2},\frac{\pi}{2}]$ and $\gamma_{\mathsf{t}} \in [0,2\pi)$.

The IRS consists of multiple reflecting elements (REs) evenly spaced on a grid, as illustrated in Fig. \[PlaneelementFig\]. Each RE is composed of numerous sub-wavelength unit cells (also known as meta atoms). Each unit cell contains programmable components (such as tunable varactor diodes or switchable positive-intrinsic-negative (PIN) diodes) and is able to change the properties of an impinging EM wave when reflecting it. In this paper, we consider that the unit cells on one RE are arranged densely enough so that the collection of them acts as a continuous programmable surface. *The surface impedance is suitably designed to realize reflection coefficient $\Gamma=\tau e^{j\beta(x,y)}$, where ${\beta(x,y)}$ is the phase shift applied at point $(x,y)$ on the RE, and $\tau$ is the amplitude of the reflection coefficient which is assumed to be constant across the RE.* This definition of RE is similar to that of *continuous tile* in @najafi_physics-based_2020. The REs are spaced by $S_{\mathsf{x}}$ and $S_{\mathsf{y}}$ along the $x$- and $y$-axes, respectively, with each RE of size $L_{\mathsf{x}} \times L_{\mathsf{y}}$. Each column ($x$ direction) and each row ($y$ direction) of the IRS contain $Q_{\mathsf{x}}$ and $Q_{\mathsf{y}}$ REs, respectively. The RE in the $k$-th row and the $l$-th column of the IRS is denoted by $m_{k,l},~ k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}}$, and $m_{0,0}$ is centered at the origin $O$.  The size of the entire IRS is given by $L_{\mathsf{x}}^{\mathsf{tot}} \times L_{\mathsf{x}}^{\mathsf{tot}}$, where $L_{\mathsf{x}}^{\mathsf{tot}} = (Q_{\mathsf{x}}-1)S_{\mathsf{x}} + L_{\mathsf{x}}$ and $L_{\mathsf{y}}^{\mathsf{tot}} = (Q_{\mathsf{y}}-1)S_{\mathsf{y}} + L_{\mathsf{y}}$. We next characterize the cascaded LoS MIMO channel based on the geometric system model described above.

![Illustration of a planar IRS of size $L_{\mathsf{x}}^{\mathsf{tot}} \times L_{\mathsf{y}}^{\mathsf{tot}}$ comprised of REs of size $L_{\mathsf{x}} \times L_{\mathsf{y}}$ with spacing $S_{\mathsf{x}}$ and $S_{\mathsf{y}}$. Each RE works as a continuous surface.<span data-label="PlaneelementFig"></span>](PlaneTileFar.pdf){width="0.5\linewidth"}

RE Response Function
--------------------

In this subsection, we consider the impact of each RE imposed on its impinging EM wave in reflection. To simplify the discussion, we make the following basic assumption:

\[REFarFieldAssump\] Each RE of the IRS is in the **far field** of each single Tx and Rx antenna. For an RE of size $L_{\mathsf{x}}\times L_{\mathsf{y}}$, the boundary of its far field and near field is defined as $B_{\textrm{RE}}\triangleq \frac{2\left(L_{\mathsf{x}}^2+L_{\mathsf{y}}^2 \right)}{\lambda}$ @huang_antennas_nodate [Eq. 4.9]. Beyond this boundary, the maximum phase difference of the received signal on an RE is less than $\frac{\pi}{8}$, and thus the curvature of the wavefront on the RE can be neglected.

Assumption \[REFarFieldAssump\] is made for a small RE size relative to the Tx-IRS and IRS-Rx distances. The size of each RE is generally considered to be up to tens of times of the wavelength. Considering an RE with $L_{\mathsf{x}}=L_{\mathsf{y}} = 10\lambda$ for example, the boundary $B_{\textrm{RE}}$ is $2\textrm{m}$ for $\lambda=0.005\textrm{m}$. In most practical IRS-aided communication scenarios, the Tx-IRS and IRS-Rx distances are well beyond this boundary. Therefore, it is reasonable to assume that each RE is in the far field of a Tx/Rx antenna. Under Assumption \[REFarFieldAssump\], the spherical wave radiated by each Tx/Rx antenna can be approximately regarded as a plane wave at each single RE, and the REs work in a similar manner as continuous tiles in @najafi_physics-based_2020. Thus the continuous tile channel model developed in @najafi_physics-based_2020 can be directly borrowed for our purpose, as detailed below.

For a specific RE $m_{k,l}$, $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}}$, when reflecting the impinging plane wave radiated from the Tx antenna $t_p$, to the Rx antenna $r_q$, the impact imposed on the reflected wave can be characterized by a complex response function $g(\boldsymbol{\Psi}_{(k,l),p},\boldsymbol{\Psi}_{q,(k,l)})$, $p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$, which is referred to the *RE response function* of $m_{k,l}$. Specifically, $\boldsymbol{\Psi}_{(k,l),p} = (\delta_{(k,l),p},\vartheta_{(k,l),p},\varsigma_{(k,l),p})$ models the impinging wave, where $\delta_{(k,l),p}$ and $\vartheta_{(k,l),p}$ denote the elevation and azimuth angles of the direction from $t_p$ to $m_{k,l}$ in the $x$-$y$-$z$ coordinate system, respectively, and $\varsigma_{(k,l),p}$ represents the polarization of the impinging wave; $\boldsymbol{\Psi}_{q,(k,l)} = (\delta_{q,(k,l)},\vartheta_{q,(k,l)})$ models the reflecting direction from $m_{k,l}$ to $r_q$, where $\delta_{q,(k,l)}$ and $\vartheta_{q,(k,l)}$ denote the elevation and azimuth angles of the direction from $m_{k,l}$ to $r_q$, in the $x$-$y$-$z$ coordinate system, respectively. The directional angles described above are illustrated in Fig. \[PlaneelementFig\]. Under Assumption \[REFarFieldAssump\], the RE response function of $m_{k,l}$ for $t_p$ and $r_q$ in the far field can be approximately expressed as @najafi_physics-based_2020 [Eq. 1] $$\label{gLimit}
    g(\boldsymbol{\Psi}_{(k,l),p},\boldsymbol{\Psi}_{q,(k,l)}) = \sqrt{4\pi d_{q,(k,l)}^2} e^{\frac{j2\pi d_{q,(k,l)}}{\lambda}} \frac{E_{\mathsf{r}}(\boldsymbol{\Psi}_{q,(k,l)})}{E_{\mathsf{t}}(\boldsymbol{\Psi}_{(k,l),p})},$$ *where $E_{\mathsf{t}}(\boldsymbol{\Psi}_{(k,l),p})$ is a phasor denoting the complex amplitude of the incident electric field impinging from the direction $\boldsymbol{\Psi}_{(k,l),p}$ on the center of $m_{k,l}$, and $E_{\mathsf{r}}(\boldsymbol{\Psi}_{q,(k,l)})$ is a phasor denoting the complex amplitude of the reflected electric field at $r_q$, and $d_{q,(k,l)}$ is the distance from $r_q$ to the center of $m_{k,l}$.* In , besides depending on the impinging wave $\boldsymbol{\Psi}_{(k,l),p}$, the reflecting direction $\boldsymbol{\Psi}_{q,(k,l)}$, and the $m_{k,l}$-$r_q$ distance $d_{q,(k,l)}$, $E_{\mathsf{r}}(\boldsymbol{\Psi}_{q,(k,l)})$ is also affected by the reflection coefficients configured on $m_{k,l}$. Denote by $\Gamma_{k,l}=\tau_{k,l} e^{j \beta_{k,l}(x,y)}$ the reflection coefficient realized at point $(x,y)$ (with the center of $m_{k,l}$ as the origin, $x \in (-L_{\mathsf{x}}/2, L_{\mathsf{x}}/2)$, $y \in (-L_{\mathsf{y}}/2, L_{\mathsf{y}}/2)$) of $m_{k,l}$, where $\beta_{k,l}(x,y)$ is referred to as the phase shift function of $m_{k,l}$, and $\tau_{k,l}$ is the amplitude which is assumed to be constant across $m_{k,l}$, $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}}$. For simplicity, we further assume that all REs are properly designed to achieve a common reflection coefficient amplitude $\tau$, i.e., $\tau_{k,l}=\tau$, $\forall k \in \mathcal{I}_{Q_{{\mathsf{x}}}}$, $\forall l \in \mathcal{I}_{Q_{{\mathsf{y}}}}$.

From @najafi_physics-based_2020 [Proposition 1], for an RE $m_{k,l}$, given the polarization angle $\varsigma_{(k,l),p}$ of the impinging wave, the maximum amplitude of $g(\boldsymbol{\Psi}_{(k,l),p},\boldsymbol{\Psi}_{q,(k,l)})$ is given by $$\label{gAmpMax}
    \left|g(\boldsymbol{\Psi}_{(k,l),p},\boldsymbol{\Psi}_{q,(k,l)})\right|_{\max} = \frac{\sqrt{4\pi}\tau  L_{\mathsf{x}} L_{\mathsf{y}}}{\lambda} \tilde{g}(\boldsymbol{\Psi}_{(k,l),p},\boldsymbol{\Psi}_{q,(k,l)}),$$ where $$\begin{gathered}
    \tilde{g}(\boldsymbol{\Psi}_{(k,l),p},\boldsymbol{\Psi}_{q,(k,l)}) = c(\boldsymbol{\Psi}_{(k,l),p}) \times \\
    \left\|\left[\begin{array}{c}
    \cos \left(\varsigma_{(k,l),p}\right) \cos \left(\delta_{q,(k,l)}\right) \sin \left(\vartheta_{q,(k,l)}\right)-\sin \left(\varsigma_{(k,l),p}\right) \cos \left(\delta_{q,(k,l)}\right) \cos \left(\vartheta_{q,(k,l)}\right) \\
    \sin \left(\varsigma_{(k,l),p}\right) \sin \left(\vartheta_{q,(k,l)}\right)+\cos \left(\varsigma_{(k,l),p}\right) \cos \left(\vartheta_{q,(k,l)}\right)
    \end{array}\right]\right\|_{2},\end{gathered}$$ $$c(\boldsymbol{\Psi}_{(k,l),p}) = \frac{A_{\mathsf{z}}\left(\boldsymbol{\Psi}_{\mathsf{t}}\right)}{\sqrt{A_{\mathsf{x,y}}^2 \left(\boldsymbol{\Psi}_{\mathsf{t}}\right) + A_{\mathsf{z}}^2 \left(\boldsymbol{\Psi}_{\mathsf{t}}\right)}}$$ with $A_{\mathsf{z}}\left(\boldsymbol{\Psi}_{\mathsf{t}}\right) \triangleq \cos(\delta_{(k,l),p})$ and $A_{\mathsf{x,y}} \left(\boldsymbol{\Psi}_{\mathsf{t}}\right) \triangleq  \cos(\varsigma_{(k,l),p})\sin(\delta_{(k,l),p})\cos(\vartheta_{(k,l),p}) + \sin(\varsigma_{(k,l),p}) \allowbreak \sin(\delta_{(k,l),p})  \allowbreak \sin(\vartheta_{(k,l),p})$, $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}}$, $p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$. The maximum amplitude of $g(\boldsymbol{\Psi}_{(k,l),p},\boldsymbol{\Psi}_{q,(k,l)})$ in is achieved when $m_{k,l}$ applies the linear phase shift function $$\label{BetaklSum}
    \beta_{k,l}(x,y) = \beta_{k,l}(x) + \beta_{k,l}(y) + \beta_{k,l} - \frac{\pi}{2},$$ where

\[PhaseShift\] $$\begin{aligned}
        \beta_{k,l}(x) &= -\frac{2\pi}{\lambda} \left(\sin(\delta_{(k,l),p})\cos(\vartheta_{(k,l),p}) + \sin(\delta_{q,(k,l)})\cos(\vartheta_{q,(k,l)})\right) x, \\
        \beta_{k,l}(y) &= -\frac{2\pi}{\lambda} \left(\sin(\delta_{(k,l),p})\sin(\vartheta_{(k,l),p}) + \sin(\delta_{q,(k,l)})\sin(\vartheta_{q,(k,l)})\right) y,
    \end{aligned}$$

and $\beta_{k,l}$ is referred to as the initial phase shift of $m_{k,l}$. Then, the phase of the response function of $m_{k,l}$ is given by $$\label{REPHASE}
    \angle g(\boldsymbol{\Psi}_{(k,l),p},\boldsymbol{\Psi}_{q,(k,l)}) = \beta_{k,l},$$ $\forall p \in \mathcal{I}_{N_{\mathsf{t}}},~ \forall q \in \mathcal{I}_{N_{\mathsf{r}}}$.

In the remainder of this paper, we assume that all the REs apply the phase shift functions of the form , with

\[PhaseShift0\] $$\begin{aligned}
        \beta_{k,l}(x) &= -\frac{2\pi}{\lambda} \left(\sin(\delta_{(k,l),0})\cos(\vartheta_{(k,l),0}) + \sin(\delta_{0,(k,l)})\cos(\vartheta_{0,(k,l)})\right) x, \\
        \beta_{k,l}(y) &= -\frac{2\pi}{\lambda} \left(\sin(\delta_{(k,l),0})\sin(\vartheta_{(k,l),0}) + \sin(\delta_{0,(k,l)})\sin(\vartheta_{0,(k,l)})\right) y,
    \end{aligned}$$

$k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}}$. This enables all the REs to achieve the maximum response amplitude for the impinging wave from transmit antenna $t_0$ to receive antenna $r_0$ with given polarization angle $\varsigma_{(k,l),0}$, $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}}$. Denote by $g_0$ the response of $m_{0,0}$ to the impinging wave from $t_0$ to $r_0$, i.e., $|g_0| = \frac{\sqrt{4\pi}\tau L_{\mathsf{x}} L_{\mathsf{y}}}{\lambda}\tilde{g}(\boldsymbol{\Psi}_{(0,0),0},\boldsymbol{\Psi}_{0,(0,0)})= \frac{\sqrt{4\pi}\tau L_{\mathsf{x}} L_{\mathsf{y}}}{\lambda}\tilde{g}((\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varsigma_{(0,0),0}),(\varphi_{\mathsf{r}},\psi_{\mathsf{r}}))$, $\angle g_0 = \beta_{0,0}$.

To further simplify the channel model, we make the following assumption:

\[SizeAssump\] The distance between the centers of the Tx and the IRS is much greater than the sizes of the Tx and the IRS, and the distance between the centers of the Rx and the IRS is much greater than the sizes of the Rx and the IRS, i.e., $D_{\mathsf{t}} \gg L_{\mathsf{t}}, L_{\mathsf{x}}^{\mathsf{tot}}, L_{\mathsf{y}}^{\mathsf{tot}} $, and $D_{\mathsf{r}} \gg L_{\mathsf{r}}, L_{\mathsf{x}}^{\mathsf{tot}}, L_{\mathsf{y}}^{\mathsf{tot}}$.

Under Assumption \[SizeAssump\], in all the $t_p$-$m_{k,l}$-$r_q$ links, the amplitudes of the RE response functions are approximately the same as $|g_0|$, i.e., $$\label{CommonREAmp}
    |g(\boldsymbol{\Psi}_{(k,l),p},\boldsymbol{\Psi}_{q,(k,l)})|=|g_0|,$$ $\forall k \in \mathcal{I}_{Q_{\mathsf{x}}},~ \forall l \in \mathcal{I}_{Q_{\mathsf{y}}},~ \forall p \in \mathcal{I}_{N_{\mathsf{t}}},~ \forall q \in \mathcal{I}_{N_{\mathsf{r}}}$. This approximation is based on the fact the amplitude of $g(\boldsymbol{\Psi}_{(k,l),p},\boldsymbol{\Psi}_{q,(k,l)})$ is insensitive to the variations in $\boldsymbol{\Psi}_{(k,l),p}$ and $\boldsymbol{\Psi}_{q,(k,l)}$, i.e., under Assumption \[SizeAssump\], the differences of $\boldsymbol{\Psi}_{(k,l),p}$ are small, and so are the differences of $\boldsymbol{\Psi}_{q,(k,l)}$, $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}}$, $p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$. As a justification, we consider an IRS-aided MIMO system with $D_{\mathsf{t}} = D_{\mathsf{r}} = 10 \rm{m}$, $\varphi_{\mathsf{t}} = \varphi_{\mathsf{r}} = \frac{\pi}{3}$, $\omega_{\mathsf{t}}=\frac{3\pi}{2}$, $\omega_{\mathsf{r}} = \frac{\pi}{2}$. The polarization of $t_0$ is set to $\varsigma_{(0,0),0} = \frac{\pi}{3}$. The IRS is of size $L_{\mathsf{x}}^{\mathsf{tot}} \times L_{\mathsf{y}}^{\mathsf{tot}} = 1 \rm{m}  \times 1 \rm{m} $, $Q_{\mathsf{x}} = Q_{\mathsf{y}} = 11$, $S_{\mathsf{x}} = S_{\mathsf{y}} = 0.095 \rm{m} $, $L_{\mathsf{x}} = L_{\mathsf{y}} = 0.05 \rm{m} $. Fig. \[PhysicsFig\] illustrates the values of $\tilde{g}(\boldsymbol{\Psi}_{(k,l),0},\boldsymbol{\Psi}_{0,(k,l)})$ over all the REs, $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}}$. We observe that the variation of $\tilde{g}(\boldsymbol{\Psi}_{(k,l),0},\boldsymbol{\Psi}_{0,(k,l)})$ over the REs is mild, and the deviation from $\tilde{g}(\boldsymbol{\Psi}_{(0,0),0},\boldsymbol{\Psi}_{0,(0,0)})$ is less than $10\%$. The above numerical example justifies the validity of the approximation in .

Assumption \[SizeAssump\] does not necessarily mean that the IRS is in the far field of the Tx and the Rx. For an IRS of the size $L_{\mathsf{x}}^{\mathsf{tot}} \times L_{\mathsf{y}}^{\mathsf{tot}}$, the boundary of its far field and near field is given by $B_{\textrm{IRS}} = 2 \frac{\left(L_{\mathsf{x}}^{\mathsf{tot}}\right)^2+\left(L_{\mathsf{y}}^{\mathsf{tot}}\right)^2}{\lambda}$ @huang_antennas_nodate [Eq. 4.9]. Taking $L_{\mathsf{x}}^{\mathsf{tot}} = L_{\mathsf{y}}^{\mathsf{tot}} = 1\textrm{m}$ and $\lambda=0.005\rm{m}$ (which means $f=60\rm{GHz}$) for an example, the boundary $B_{\rm{IRS}}$ is $800\rm{m}$. However, from the above example, we see that $D_{\mathsf{t}}=D_{\mathsf{r}}=10\rm{m}$ is sufficiently large to ensure the approximation in .

![Thermogram for the $\tilde{g}(\boldsymbol{\Psi}_{(k,l),0},\boldsymbol{\Psi}_{0,(k,l)})$ over different $t_0$-$m_{k,l}$-$r_0$ links, $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}}$. $D_{\mathsf{t}} = D_{\mathsf{r}} = 10  \rm{m} $, $\varphi_{\mathsf{t}} = \varphi_{\mathsf{r}} = \frac{\pi}{3}$, $\omega_{\mathsf{t}}=\frac{3\pi}{2}$, $\omega_{\mathsf{r}} = \frac{\pi}{2}$. The polarization of $t_0$ is set to $\varsigma_{(0,0),0} = \frac{\pi}{3}$. The IRS is of size $L_{\mathsf{x}}^{\mathsf{tot}} \times L_{\mathsf{y}}^{\mathsf{tot}} = 1 \rm{m}  \times 1 \rm{m} $, $Q_{\mathsf{x}} = Q_{\mathsf{y}} = 11$, $S_{\mathsf{x}} = S_{\mathsf{y}} = 0.095 \rm{m} $, $L_{\mathsf{x}} = L_{\mathsf{y}} = 0.05 \rm{m} $.<span data-label="Justification"></span>](REphase.pdf){width=".6\textwidth"}

Link Distance
-------------

The path loss and path phase shift of the EM wave experienced in each cascaded LoS link are related to the link distance. Thus we need to describe the link distance first. Denote by $d_{(k,l),p}$ the distance from $t_p$ to $m_{k,l}$, and by $d_{q,(k,l)}$ the distance from $m_{k,l}$ to $r_q$, $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}}$, $p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$.

Based on the axes and parameters defined in Section \[SecSystem\], the coordinates of the Tx antenna $t_p$, the Rx antenna $r_q$, and the center of the RE $m_{k,l}$ can be written into explicit expressions. $d_{(k,l),p}$ and $d_{q,(k,l)}$ can be analytically derived by calculating the Euclidean distances between $m_{k,l}$ and $t_p$ and between $m_{k,l}$ and $r_q$, respectively. Define $\mathbf{a}_{{\mathsf{t}},p}$ as the vector from the origin to $t_p$, $\mathbf{a}_{{\mathsf{r}},q}$ as the vector from the origin to $r_q$, and $\mathbf{a}_{k,l}$ as the vector from the origin to the center of $m_{k,l}$. From Fig. \[SysFig\], these vectors are given by

$$\begin{aligned}
        \mathbf{a}_{{\mathsf{t}},p} &= (pd_{\mathsf{t}}\sin \psi_{\mathsf{t}} \cos\gamma_{\mathsf{t}})\mathbf{n}_{x_{\mathsf{t}}} + (pd_{\mathsf{t}}\sin \psi_{\mathsf{t}} \sin\gamma_{\mathsf{t}})\mathbf{n}_{y_{\mathsf{t}}} + (D_{\mathsf{t}}+pd_{\mathsf{t}}\cos \psi_{\mathsf{t}})\mathbf{n}_{z_{\mathsf{t}}},~ p \in \mathcal{I}_{p} \label{ant} \\
        \mathbf{a}_{{\mathsf{r}},q} &= (qd_{\mathsf{r}}\sin \psi_{\mathsf{r}} \cos\gamma_{\mathsf{r}})\mathbf{n}_{x_{\mathsf{r}}} + (qd_{\mathsf{r}}\sin \psi_{\mathsf{r}} \sin\gamma_{\mathsf{r}})\mathbf{n}_{y_{\mathsf{r}}} + (D_{\mathsf{r}}+qd_{\mathsf{r}}\cos \psi_{\mathsf{r}})
        \mathbf{n}_{z_{\mathsf{r}}},~ q \in \mathcal{I}_{q} \label{anr} \\
                        \mathbf{a}_{k,l} & = k S_{\mathsf{x}} \mathbf{n}_x + l S_{\mathsf{y}} \mathbf{n}_y \\
        & = (k S_{\mathsf{x}} \sin \omega_{\mathsf{t}}-lS_{\mathsf{y}}\cos \omega_{\mathsf{t}})\mathbf{n}_{x_{\mathsf{t}}} + (kS_{\mathsf{x}}\cos \varphi_{\mathsf{t}}\cos \omega_{\mathsf{t}} + lS_{\mathsf{y}}\cos \varphi_{\mathsf{t}}\sin \omega_{\mathsf{t}})\mathbf{n}_{y_{\mathsf{t}}} \notag \\
        & \quad +  (kS_{\mathsf{x}}\sin \varphi_{\mathsf{t}} \cos \omega_{\mathsf{t}} + lS_{\mathsf{y}}\sin \varphi_{\mathsf{t}} \sin \omega_{\mathsf{t}})\mathbf{n}_{z_{\mathsf{t}}},~ k\in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}} \label{aijt} \\
        & = (k S_{\mathsf{x}} \sin \omega_{\mathsf{r}}-lS_{\mathsf{y}}\cos \omega_{\mathsf{r}})\mathbf{n}_{x_{\mathsf{r}}} + (kS_{\mathsf{x}}\cos \varphi_{\mathsf{r}}\cos \omega_{\mathsf{r}} + lS_{\mathsf{y}}\cos \varphi_{\mathsf{r}}\sin \omega_{\mathsf{r}})\mathbf{n}_{y_{\mathsf{r}}} \notag \\
        & \quad +  (kS_{\mathsf{x}}\sin \varphi_{\mathsf{r}} \cos \omega_{\mathsf{r}} + lS_{\mathsf{y}}\sin \varphi_{\mathsf{r}} \sin \omega_{\mathsf{r}})\mathbf{n}_{z_{\mathsf{r}}},~ k\in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}} \label{aijr}
    \end{aligned}$$

where $\mathbf{n}_x$, $\mathbf{n}_y$, $\mathbf{n}_z$, $\mathbf{n}_{x_{\mathsf{t}}}$, $\mathbf{n}_{y_{\mathsf{t}}}$, $\mathbf{n}_{z_{\mathsf{t}}}$, $\mathbf{n}_{x_{\mathsf{r}}}$, $\mathbf{n}_{y_{\mathsf{r}}}$, and $\mathbf{n}_{z_{\mathsf{r}}}$ are the unit vectors of the $x$, $y$, $z$, $x_{\mathsf{t}}$, $y_{\mathsf{t}}$, $z_{\mathsf{t}}$, $x_{\mathsf{r}}$, $y_{\mathsf{r}}$, and $z_{\mathsf{r}}$ axes, respectively. Based on and , $d_{(k,l),p}$ is given by

\[dklpORI\] $$\begin{aligned}
        d_{(k,l),p} & = ||\mathbf{a}_{{\mathsf{t}},p} - \mathbf{a}_{k,l}|| \\
                                                        &= \left[(pd_{\mathsf{t}}\sin \psi_{\mathsf{t}} \cos\gamma_{\mathsf{t}}-k S_{\mathsf{x}} \sin \omega_{\mathsf{t}}+lS_{\mathsf{y}}\cos \omega_{\mathsf{t}})^2 \right. \notag \\
        & \quad \left. + (pd_{\mathsf{t}}\sin \psi_{\mathsf{t}} \sin\gamma_{\mathsf{t}}-kS_{\mathsf{x}}\cos \varphi_{\mathsf{t}}\cos \omega_{\mathsf{t}} - lS_{\mathsf{y}}\cos \varphi_{\mathsf{t}}\sin \omega_{\mathsf{t}})^2 \right. \notag \\
        & \quad \left. + (D_{\mathsf{t}} +pd_{\mathsf{t}}\cos\psi_{\mathsf{t}}-kS_{\mathsf{x}}\sin \varphi_{\mathsf{t}} \cos \omega_{\mathsf{t}} - lS_{\mathsf{y}}\sin \varphi_{\mathsf{t}} \sin \omega_{\mathsf{t}})^2\right]^{\frac{1}{2}} \label{Exactklp} \\
        &\approx \frac{(pd_{\mathsf{t}}\sin \psi_{\mathsf{t}} \cos\gamma_{\mathsf{t}}-k S_{\mathsf{x}} \sin \omega_{\mathsf{t}}+lS_{\mathsf{y}}\cos \omega_{\mathsf{t}})^2}{2(D_{\mathsf{t}} +pd_{\mathsf{t}}\cos\psi_{\mathsf{t}}-kS_{\mathsf{x}}\sin \varphi_{\mathsf{t}} \cos \omega_{\mathsf{t}} - lS_{\mathsf{y}}\sin \varphi_{\mathsf{t}} \sin \omega_{\mathsf{t}})} \notag \\
        & \quad + \frac{(pd_{\mathsf{t}}\sin \psi_{\mathsf{t}} \sin\gamma_{\mathsf{t}}-kS_{\mathsf{x}}\cos \varphi_{\mathsf{t}}\cos \omega_{\mathsf{t}} - lS_{\mathsf{y}}\cos \varphi_{\mathsf{t}}\sin \omega_{\mathsf{t}})^2}{2(D_{\mathsf{t}} +pd_{\mathsf{t}}\cos\psi_{\mathsf{t}}-kS_{\mathsf{x}}\sin \varphi_{\mathsf{t}} \cos \omega_{\mathsf{t}} - lS_{\mathsf{y}}\sin \varphi_{\mathsf{t}} \sin \omega_{\mathsf{t}})} \notag \\
        & \quad + D_{\mathsf{t}} +pd_{\mathsf{t}}\cos\psi_{\mathsf{t}}-kS_{\mathsf{x}}\sin \varphi_{\mathsf{t}} \cos \omega_{\mathsf{t}} - lS_{\mathsf{y}}\sin \varphi_{\mathsf{t}} \sin \omega_{\mathsf{t}}, \label{Approxklpmid} \\
        &\approx \frac{(pd_{\mathsf{t}}\sin \psi_{\mathsf{t}} \cos\gamma_{\mathsf{t}}-k S_{\mathsf{x}} \sin \omega_{\mathsf{t}}+lS_{\mathsf{y}}\cos \omega_{\mathsf{t}})^2}{2D_{\mathsf{t}}} \notag \\
        & \quad + \frac{(pd_{\mathsf{t}}\sin \psi_{\mathsf{t}} \sin\gamma_{\mathsf{t}}-kS_{\mathsf{x}}\cos \varphi_{\mathsf{t}}\cos \omega_{\mathsf{t}} - lS_{\mathsf{y}}\cos \varphi_{\mathsf{t}}\sin \omega_{\mathsf{t}})^2}{2D_{\mathsf{t}}} \notag \\
        & \quad + D_{\mathsf{t}} +pd_{\mathsf{t}}\cos\psi_{\mathsf{t}}-kS_{\mathsf{x}}\sin \varphi_{\mathsf{t}} \cos \omega_{\mathsf{t}} - lS_{\mathsf{y}}\sin \varphi_{\mathsf{t}} \sin \omega_{\mathsf{t}}, \label{Approxklp}
    \end{aligned}$$

$k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}},~ p \in \mathcal{I}_{N_{\mathsf{t}}}$, where gives the exact path length, and two approximations are made in and . Specifically, in , a Taylor series expansion is made to ; and in , the denominator is simplified to $2D_{\mathsf{t}}$. Both approximations are valid when $D_{\mathsf{t}} \gg L_{\mathsf{t}}, L_{\mathsf{x}}^{\mathsf{tot}}, L_{\mathsf{y}}^{\mathsf{tot}} $, which is guaranteed by Assumption \[SizeAssump\]. Similarly, based on and , we obtain

\[dqklORI\] $$\begin{aligned}
        d_{q,(k,l)} & = ||\mathbf{a}_{{\mathsf{r}},q} - \mathbf{a}_{k,l}|| \\
                                        &\approx \frac{(qd_{\mathsf{r}}\sin \psi_{\mathsf{r}} \cos\gamma_{\mathsf{r}}-k S_{\mathsf{x}} \sin \omega_{\mathsf{r}}+lS_{\mathsf{y}}\cos \omega_{\mathsf{r}})^2}{2D_{\mathsf{r}}} \notag \\
        & \quad + \frac{(qd_{\mathsf{r}}\sin \psi_{\mathsf{r}} \sin\gamma_{\mathsf{r}}-kS_{\mathsf{x}}\cos \varphi_{\mathsf{r}}\cos \omega_{\mathsf{r}} - lS_{\mathsf{y}}\cos \varphi_{\mathsf{r}}\sin \omega_{\mathsf{r}})^2}{2D_{\mathsf{r}}} \notag \\
        & \quad + D_{\mathsf{r}} +qd_{\mathsf{r}}\cos\psi_{\mathsf{r}}-kS_{\mathsf{x}}\sin \varphi_{\mathsf{r}} \cos \omega_{\mathsf{r}} - lS_{\mathsf{y}}\sin \varphi_{\mathsf{r}} \sin \omega_{\mathsf{r}}, \label{Approxqkl}
    \end{aligned}$$

$k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$.

Path Loss and Phase Shift Model
-------------------------------

With the link distances given in and , we are now ready to describe the path loss and path phase shift model. Combining the discussions in the preceding subsections, the total path loss of the $t_p$-$m_{k,l}$-$r_q$ link is given by $$\label{PL}
    \textrm{PL}_{q,(k,l),p} = \frac{4\pi \left|g(\boldsymbol{\Psi}_{(k,l),p},\boldsymbol{\Psi}_{q,(k,l)})\right|^2}{\lambda^2} \textrm{PL}_{(k,l),p}\textrm{PL}_{q,(k,l)},$$ $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}},~ p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$. In , $\textrm{PL}_{(k,l),p} \triangleq \left(\frac{\lambda}{4\pi d_{(k,l),p}}\right)^2$ and $\textrm{PL}_{q,(k,l)} \triangleq \left(\frac{\lambda}{4\pi d_{q,(k,l)}}\right)^2$ stand for the path losses of the LoS $t_p$-$m_{k,l}$ and $m_{k,l}$-$r_q$ channels, respectively, where $d_{(k,l),p}$ and $d_{q,(k,l)}$ are the distances from $m_{k,l}$ to $t_p$ and $r_q$, respectively. Under Assumption \[SizeAssump\], all the LoS path loss $\textrm{PL}_{(k,l),p}$ are approximately the same as $\textrm{PL}_{(0,0),0}$, and all the LoS path loss $\textrm{PL}_{q,(k,l)}$ are approximately the same as $\textrm{PL}_{0,(0,0)}$, i.e., $$\label{LoSPLApprox}
    \textrm{PL}_{(k,l),p} = \textrm{PL}_{(0,0),0},~ \textrm{PL}_{q,(k,l)} = \textrm{PL}_{0,(0,0)},$$ $\forall k \in \mathcal{I}_{Q_{\mathsf{x}}},~ \forall l \in \mathcal{I}_{Q_{\mathsf{y}}},~ \forall p \in \mathcal{I}_{N_{\mathsf{t}}},~ \forall q \in \mathcal{I}_{N_{\mathsf{r}}}$. Similar approximations for the LoS path loss have been adopted when the size of the antenna array is much smaller relative to the link distance; see, e.g., @bohagen_construction_2005 [@bohagen_design_2007; @bohagen_optimal_2007; @bohagen_spherical_2009; @wang_tens_2014]. From and , the total path losses of the Tx-IRS-Rx links reduce to a constant over the indices $k$, $l$, $p$, $q$, i.e., $$\label{CommonPL}
    \textrm{PL}_{q,(k,l),p} = \eta_0^2,$$ $\forall k \in \mathcal{I}_{Q_{\mathsf{x}}},~ \forall l \in \mathcal{I}_{Q_{\mathsf{y}}},~ \forall p \in \mathcal{I}_{N_{\mathsf{t}}},~ \forall q \in \mathcal{I}_{N_{\mathsf{r}}}$, where $\eta_0 \triangleq \sqrt{\frac{4\pi \left|g_0 \right|^2}{\lambda^2}\textrm{PL}_{(0,0),0}\textrm{PL}_{0,(0,0)}}=\frac{\tau L_{\mathsf{x}} L_{\mathsf{y}}}{4\pi D_{\mathsf{t}}D_{\mathsf{r}}}\allowbreak \tilde{g}((\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varsigma_{(0,0),0}),(\varphi_{\mathsf{r}},\psi_{\mathsf{r}}))$.

The received signal via the $t_p$-$m_{k,l}$-$r_q$ link experiences a total path phase shift of $$\label{totalpps}
    \zeta_{q,(k,l),p} \triangleq \zeta_{(k,l),p} + \zeta_{q,(k,l)} + \angle g(\boldsymbol{\Psi}_{(k,l),p},\boldsymbol{\Psi}_{q,(k,l)}),$$ where $$\begin{aligned}
\label{ZETA}
    \zeta_{(k,l),p} \triangleq \frac{2\pi d_{(k,l),p}}{\lambda} ~ \textrm{and} ~
    \zeta_{q,(k,l)} \triangleq \frac{2\pi d_{q,(k,l)}}{\lambda}\end{aligned}$$ are the LoS path phase shifts of the links $t_p$-$m_{k,l}$ and $m_{k,l}$-$r_q$, respectively, $\angle g(\boldsymbol{\Psi}_{(k,l),p},\boldsymbol{\Psi}_{q,(k,l)}) = \beta_{k,l}$ is the phase of the RE response function $g(\boldsymbol{\Psi}_{(k,l),p},\boldsymbol{\Psi}_{q,(k,l)})$, $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}},~ p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$.

Channel Representation
----------------------

The considered IRS-aided MIMO system can be generally described by $$\label{LinearModel}
    \boldsymbol{y} = \boldsymbol{H}\boldsymbol{x} + \boldsymbol{w},$$ where $\boldsymbol{H}\in \mathbb{C}^{N_{\mathsf{r}} \times N_{\mathsf{t}}}$ denotes the overall channel coefficient matrix; $\boldsymbol{x}\in \mathbb{C}^{N_{\mathsf{t}}}$ denotes the transmit signal vector; $\boldsymbol{y} \in \mathbb{C}^{N_{\mathsf{r}}}$ denotes the receive signal vector; $\boldsymbol{w}\in \mathbb{C}^{N_{\mathsf{r}}}$ denotes the circularly symmetric complex Gaussian noise that follows $\mathcal{CN}(\boldsymbol{0},\sigma_w^2 \boldsymbol{I})$. The element in the $\left(q+\frac{N_{\mathsf{r}}+1}{2}\right)$-th row and the $\left(p+\frac{N_{\mathsf{t}}+1}{2}\right)$-th column of $\boldsymbol{H}$, denoted by $h_{q,p}$, represents the overall channel between the Tx antenna $t_p$ and the Rx antenna $r_q$, $p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$.

We now represent the channel matrix $\boldsymbol{H}$ based on the total path loss and the phase shift model described in the preceding two subsections. Specifically, denote by

\[hklp\] $$\begin{aligned}
        h_{(k,l),p} &\triangleq e^{-j\zeta_{(k,l),p}} \\
        &=
        \begin{multlined}[t]\label{hklpb}
            \exp \left\{ -j \left( \frac{\pi(pd_{\mathsf{t}}\sin \psi_{\mathsf{t}} \cos\gamma_{\mathsf{t}}-k S_{\mathsf{x}} \sin \omega_{\mathsf{t}}+lS_{\mathsf{y}}\cos \omega_{\mathsf{t}})^2}{\lambda D_{\mathsf{t}}} \right. \right. \\
            \left. \left. \quad + \frac{\pi(pd_{\mathsf{t}}\sin \psi_{\mathsf{t}} \sin\gamma_{\mathsf{t}}-kS_{\mathsf{x}}\cos \varphi_{\mathsf{t}}\cos \omega_{\mathsf{t}} - lS_{\mathsf{y}}\cos \varphi_{\mathsf{t}}\sin \omega_{\mathsf{t}})^2}{\lambda D_{\mathsf{t}}} \right. \right. \\
            \left. \left. \quad + \frac{2\pi\left(D_{\mathsf{t}} +pd_{\mathsf{t}}\cos\psi_{\mathsf{t}}-kS_{\mathsf{x}}\sin \varphi_{\mathsf{t}} \cos \omega_{\mathsf{t}} - lS_{\mathsf{y}}\sin \varphi_{\mathsf{t}} \sin \omega_{\mathsf{t}}\right)}{\lambda} \right) \right\}
        \end{multlined}
    \end{aligned}$$

and

\[hqkl\] $$\begin{aligned}
        h_{q,(k,l)} &\triangleq e^{-j\zeta_{q,(k,l)}} \\
        &=
        \begin{multlined}[t]\label{hqklb}
            \exp \left\{ -j \left( \frac{\pi(qd_{\mathsf{r}}\sin \psi_{\mathsf{r}} \cos\gamma_{\mathsf{r}}-k S_{\mathsf{x}} \sin \omega_{\mathsf{r}}+lS_{\mathsf{y}}\cos \omega_{\mathsf{r}})^2}{\lambda D_{\mathsf{r}}} \right. \right. \\
            \left. \left. \quad + \frac{\pi(qd_{\mathsf{r}}\sin \psi_{\mathsf{r}} \sin\gamma_{\mathsf{r}}-kS_{\mathsf{x}}\cos \varphi_{\mathsf{r}}\cos \omega_{\mathsf{r}} - lS_{\mathsf{y}}\cos \varphi_{\mathsf{r}}\sin \omega_{\mathsf{r}})^2}{\lambda D_{\mathsf{r}}} \right. \right. \\
            \left. \left. \quad + \frac{2\pi\left(D_{\mathsf{r}} +qd_{\mathsf{r}}\cos\psi_{\mathsf{r}}-kS_{\mathsf{x}}\sin \varphi_{\mathsf{r}} \cos \omega_{\mathsf{r}} - lS_{\mathsf{y}}\sin \varphi_{\mathsf{r}} \sin \omega_{\mathsf{r}}\right)}{\lambda} \right) \right\}
        \end{multlined}
    \end{aligned}$$

the normalized LoS channels between the Tx antenna $t_p$ and the RE $m_{k,l}$ and between $m_{k,l}$ and the Rx antenna $r_q$, respectively, $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}},~ p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$. and are based on and the link distances in and . Let $$\begin{aligned}
    \boldsymbol{H}_{\mathsf{t}} & \triangleq
    \begin{bmatrix}
    h_{\left(-\frac{Q_{\mathsf{x}}-1}{2},-\frac{Q_{\mathsf{y}}-1}{2}\right),-\frac{N_{\mathsf{t}}-1}{2}} & \cdots & h_{\left(-\frac{Q_{\mathsf{x}}-1}{2},-\frac{Q_{\mathsf{y}}-1}{2}\right),\frac{N_{\mathsf{t}}-1}{2}}  \\
    \vdots & \ddots & \vdots \\
            h_{\left(\frac{Q_{\mathsf{x}}-1}{2},\frac{Q_{\mathsf{y}}-1}{2}\right),-\frac{N_{\mathsf{t}}-1}{2}} & \cdots & h_{\left(\frac{Q_{\mathsf{x}}-1}{2},\frac{Q_{\mathsf{y}}-1}{2}\right),\frac{N_{\mathsf{t}}-1}{2}}
    \end{bmatrix} \in \mathbb{C}^{Q_{\mathsf{x}}Q_{\mathsf{y}}\times N_{\mathsf{t}}} \label{Ht} ~ \textrm{and} \\
    \boldsymbol{H}_{\mathsf{r}} & \triangleq 
    \begin{bmatrix}
    h_{-\frac{N_{\mathsf{r}}-1}{2},\left(-\frac{Q_{\mathsf{x}}-1}{2},-\frac{Q_{\mathsf{y}}-1}{2}\right)} & \cdots & h_{-\frac{N_{\mathsf{r}}-1}{2},\left(\frac{Q_{\mathsf{x}}-1}{2},\frac{Q_{\mathsf{y}}-1}{2}\right)}  \\
    \vdots & \ddots & \vdots \\
    h_{\frac{N_{\mathsf{r}}-1}{2},\left(-\frac{Q_{\mathsf{x}}-1}{2},-\frac{Q_{\mathsf{y}}-1}{2}\right)} & \cdots & h_{\frac{N_{\mathsf{r}}-1}{2},\left(\frac{Q_{\mathsf{x}}-1}{2},\frac{Q_{\mathsf{y}}-1}{2}\right)} 
    \end{bmatrix} \in \mathbb{C}^{N_{\mathsf{r}}\times Q_{\mathsf{x}}Q_{\mathsf{y}}} \label{Hr}\end{aligned}$$ be the normalized LoS Tx-IRS and IRS-Rx channels, respectively, where the element in the $\left(\left(k+\frac{Q_{\mathsf{x}}-1}{2}\right)Q_{\mathsf{y}}+\frac{Q_{\mathsf{y}}+1}{2}+l\right)$-th row and the $\left(p+\frac{N_{\mathsf{t}}+1}{2}\right)$-th column of $\boldsymbol{H}_{\mathsf{t}}$ is given by $h_{(k,l),p}$, and the element in the $\left(q+\frac{N_{\mathsf{r}}+1}{2}\right)$-th row and the $\left(\left(k+\frac{Q_{\mathsf{x}}-1}{2}\right)Q_{\mathsf{y}}+\frac{Q_{\mathsf{y}}+1}{2}+l\right)$-th column of $\boldsymbol{H}_{\mathsf{r}}$ is given by $h_{q,(k,l)}$, $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}},~ p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$. Based on , denote by $$\label{thetakl}
    \theta_{k,l}\triangleq e^{j\beta_{k,l}}$$ the phase shift imposed by the RE $m_{k,l}$, and define a diagonal matrix $$\label{PSI}
    \boldsymbol{\Theta} \triangleq \textrm{diag}\left\{ \left[ \theta_{-\frac{Q_{\mathsf{x}}-1}{2},-\frac{Q_{\mathsf{y}}-1}{2}},\ldots, \theta_{\frac{Q_{\mathsf{x}}-1}{2},\frac{Q_{\mathsf{y}}-1}{2}} \right] \right\}$$ that accounts for the phase shifts imposed by all the REs of the IRS, with $\theta_{k,l}$ in the $\Big(\left(k+\frac{Q_{\mathsf{x}}-1}{2}\right)Q_{\mathsf{y}}+ \frac{Q_{\mathsf{y}}+1}{2} + l \Big)$-th diagonal position, $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}}$. Based on the above expressions, the IRS-aided MIMO channel matrix is given by the following proposition:

\[ChannelProp\] Under Assumptions \[REFarFieldAssump\] and \[SizeAssump\], the overall cascaded LoS MIMO channel $\boldsymbol{H}$ is given by $$\label{H}
        \boldsymbol{H} = \eta_0 \boldsymbol{H}_{\mathsf{r}} \boldsymbol{\Theta} \boldsymbol{H}_{\mathsf{t}}.$$

From , all the Tx-IRS-Rx links share a common total path loss $\eta_0$. From , , , and , the total path phase shift of the received signal experienced in the $t_p$-$m_{k,l}$-$r_q$ link (in exponential) is $h_{(k,l),p} h_{q,(k,l)} \theta_{k,l}$, with $h_{(k,l),p}$ defined in , $h_{q,(k,l)}$ defined in , and $\theta_{k,l}$ defined in , $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}},~ p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$. Therefore, $\boldsymbol{H}$ is given by .

The cascaded LoS MIMO channel model given in has a similar form to those proposed in the existing works, such as in @najafi_physics-based_2020 [@tarable_meta-surface_2020; @ozdogan_intelligent_2020; @zhang_capacity_2020]. The difference is that, based on the link distances derived under the coordinate system, our model characterize the path loss and the path phase shift precisely. From and , we see that the LoS path phase shifts $\zeta_{(k,l),p}$ and $\zeta_{q,(k,l)}$ contain quadratic terms of antenna indices $k$ and $l$. This means that a spherical wavefront on different REs are taken into account in our model. When $D_{\mathsf{t}}$ and $D_{\mathsf{r}}$ tend to infinity, the quadratic terms of $k$ and $l$ in $\zeta_{(k,l),p}$ and $\zeta_{q,(k,l)}$ diminish, and $\zeta_{(k,l),p}$ and $\zeta_{q,(k,l)}$ reduce to linear functions over $l$. This means that in the far field the entire IRS see a plane wavefront originating from a Tx/Rx antenna, and our model in reduces to the plane wave model adopted in @najafi_physics-based_2020 [@tarable_meta-surface_2020; @ozdogan_intelligent_2020; @zhang_capacity_2020; @li_joint_2019; @cai_hierarchical_nodate].

In the literature, designing the phase shifts imposed by the REs in reflection is referred to as *passive beamforming*, which largely affects the overall channel properties. In our model, passive beamforming (PB) corresponds to design the initial phase of each RE phase shift function, i.e., $\beta_{k,l}$ in , $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}}$, or the equivalent $\boldsymbol{\Theta}$ in . In @li_joint_2019, the authors show that when the Tx and the Rx are in the far-field of the IRS, the PB scheme that compensates the LoS path phase shift differences over Tx-IRS-Rx links, leads to the maximum receive signal-to-noise ratio (SNR). Under such a PB design, the phase shift imposed by each RE is a linear function of the RE indices, and the IRS achieves anomalous reflection based on the generalize Snell's law @yu_light_2011. Similar PB schemes that employ linear phase shifts (over the RE indices) are also discussed in @tarable_meta-surface_2020 [@najafi_physics-based_2020; @cai_hierarchical_nodate]. But in the channel model considered here, since the curvature of the wavefront on the IRS is taken into account, PB with linear phase shifts is insufficient to compensate the path phase shifts differences over Tx-IRS-Rx links. This inspires us to come up with a new PB scheme, named *reflective focusing*, as detailed in the following subsection.

Channel Representation with Reflective Focusing
-----------------------------------------------

In this subsection, we describe a special channel representation with reflective focusing. Reflective focusing aims to make the EM wave radiated by a certain Tx antenna, after being reflected by different REs, coherently superimposed at a certain Rx antenna. This can be achieved by properly designing the phase shifts of RE phase shift functions. Thus, reflective focusing is a special PB scheme. Later, we will show that the cascaded LoS MIMO channel with reflective focusing has the potential to support full spatial multiplexing.

To be specific, the reflective focusing for the pair of Tx antenna $t_p$ and Rx antenna $r_q$ can be achieved by letting $$\label{REFOCUSING}
    \beta_{k,l} = \frac{d_{(k,l),p} + d_{q,(k,l)}}{\lambda},$$ where $d_{(k,l),p}$ and $d_{q,(k,l)}$ are the $t_p$-$m_{k,l}$ and $m_{k,l}$-$r_q$ link distances given in and , respectively, $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}}$. With the RE phase shift in , the EM wave emitted from $t_p$, after being reflected by different REs, have the same phase at $r_q$.

Without loss of generality, we henceforth always assume that the Tx-Rx antenna pair concerned in reflective focusing is $(t_0,r_0)$, achieved by letting $$\label{REfocusing0}
    \beta_{k,l} = \bar{\beta}_{k,l} \triangleq \frac{d_{(k,l),0} + d_{0,(k,l)}}{\lambda},$$ $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}}$. Based on channel model , with reflective focusing in , the channel between the Tx antenna $t_p$ and the Rx antenna $r_q$ is given by

\[hqp\] $$\begin{aligned}
        h_{q,p} &\triangleq \eta_0 \sum_{k=-\frac{Q_{\mathsf{x}}-1}{2}}^{\frac{Q_{\mathsf{x}}-1}{2}} \sum_{l=-\frac{Q_{\mathsf{y}}-1}{2}}^{\frac{Q_{\mathsf{y}}-1}{2}}  h_{(k,l),p} \bar{\beta}_{k,l} h_{q,(k,l)} \\
                                &= \eta_0 \sum_{k=-\frac{Q_{\mathsf{x}}-1}{2}}^{\frac{Q_{\mathsf{x}}-1}{2}} \sum_{l=-\frac{Q_{\mathsf{y}}-1}{2}}^{\frac{Q_{\mathsf{y}}-1}{2}} \exp \left\{ -\frac{j2\pi}{\lambda} \left[\frac{(pd_{\mathsf{t}}\sin \psi_{\mathsf{t}} \cos\gamma_{\mathsf{t}})^2}{2D_{\mathsf{t}}} + 
        \frac{(qd_{\mathsf{r}}\sin \psi_{\mathsf{r}} \cos \gamma_{\mathsf{r}})^2}{2D_{\mathsf{r}}} \right. \right. \notag \\
        &\quad \left. \left. -\frac{pd_{\mathsf{t}}\sin \psi_{\mathsf{t}} \cos\gamma_{\mathsf{t}}(k S_{\mathsf{x}} \sin \omega_{\mathsf{t}}-lS_{\mathsf{y}}\cos \omega_{\mathsf{t}})}{D_{\mathsf{t}}} -\frac{qd_{\mathsf{r}}\sin \psi_{\mathsf{r}} \cos\gamma_{\mathsf{r}}(k S_{\mathsf{x}} \sin \omega_{\mathsf{r}}-lS_{\mathsf{y}}\cos \omega_{\mathsf{r}})}{D_{\mathsf{r}}}  \right. \right. \notag \\
        &\quad \left. \left.  -\frac{pd_{\mathsf{t}}\sin \psi_{\mathsf{t}} \sin\gamma_{\mathsf{t}}(kS_{\mathsf{x}}\cos \varphi_{\mathsf{t}}\cos \omega_{\mathsf{t}} + lS_{\mathsf{y}}\cos \varphi_{\mathsf{t}}\sin \omega_{\mathsf{t}})}{D_{\mathsf{t}}} + pd_{\mathsf{t}}\cos\psi_{\mathsf{t}} \right. \right. \notag \\
        &\quad \left. \left.  -\frac{qd_{\mathsf{r}}\sin \psi_{\mathsf{r}} \sin\gamma_{\mathsf{r}}(kS_{\mathsf{x}}\cos \varphi_{\mathsf{r}}\cos \omega_{\mathsf{r}} + lS_{\mathsf{y}}\cos \varphi_{\mathsf{r}}\sin \omega_{\mathsf{r}})}{D_{\mathsf{r}}} + qd_{\mathsf{r}}\cos\psi_{\mathsf{r}} \right] \right\} \notag \\
        &= \eta_0 P_{q,p} Q_{q,p}
    \end{aligned}$$

where $$\begin{aligned}
\label{Pqpfinal}
    P_{q,p} &\triangleq \exp \left\{\frac{-j2\pi}{\lambda} \left( \frac{\left(d_{\mathsf{t}}\sin \psi_{\mathsf{t}} \cos\gamma_{\mathsf{t}} \right)^2}{2D_{\mathsf{t}}} p^2 + \frac{\left(qd_{\mathsf{r}}\sin \psi_{\mathsf{r}} \cos \gamma_{\mathsf{r}} \right)^2}{2D_{\mathsf{r}}} q^2 + pd_{\mathsf{t}}\cos\psi_{\mathsf{t}} + qd_{\mathsf{r}}\cos\psi_{\mathsf{r}}  \right)\right\}\end{aligned}$$ is irrelevant to the indices $k$ and $l$, and

\[Qqpfinal\] $$\begin{aligned}
        Q_{q,p} &\triangleq \sum_{k=-\frac{Q_{\mathsf{x}}-1}{2}}^{\frac{Q_{\mathsf{x}}-1}{2}} \sum_{l=-\frac{Q_{\mathsf{y}}-1}{2}}^{\frac{Q_{\mathsf{y}}-1}{2}} \exp \left\{\frac{j2\pi}{\lambda} \left[ \left( \frac{d_{\mathsf{t}} S_{\mathsf{x}} \sin \psi_{\mathsf{t}} (\cos\gamma_{\mathsf{t}} \sin \omega_{\mathsf{t}}+ \sin\gamma_{\mathsf{t}} \cos \varphi_{\mathsf{t}} \cos\omega_{\mathsf{t}} )}{D_{\mathsf{t}}}p  \right. \right. \right. \notag \\
        & \quad \left. \left. \left. + \frac{d_{\mathsf{r}} S_{\mathsf{x}} \sin \psi_{\mathsf{r}} (\cos\gamma_{\mathsf{r}} \sin \omega_{\mathsf{r}}+ \sin\gamma_{\mathsf{r}} \cos \varphi_{\mathsf{r}} \cos\omega_{\mathsf{r}} )}{D_{\mathsf{r}}}q \right)k \right. \right. \notag \\
        & \quad + \left. \left. \left( \frac{d_{\mathsf{t}}S_{\mathsf{y}}\sin \psi_{\mathsf{t}} (-\cos \gamma_{\mathsf{t}} \cos \omega_{\mathsf{t}} + \sin\gamma_{\mathsf{t}}\cos \varphi_{\mathsf{t}} \sin \omega_{\mathsf{t}})}{D_{\mathsf{t}}}p \right. \right. \right. \notag \\
        & \quad \left. \left. \left. \frac{d_{\mathsf{r}}S_{\mathsf{y}}\sin \psi_{\mathsf{r}} (-\cos \gamma_{\mathsf{r}} \cos \omega_{\mathsf{r}} + \sin\gamma_{\mathsf{r}}\cos \varphi_{\mathsf{r}} \sin \omega_{\mathsf{r}})}{D_{\mathsf{r}}}q \right)l \right] \right\} \label{Qqp3} \\    
        &= \frac{\sin\left(\pi \left( C_{\mathsf{t},\mathsf{x}}p + C_{\mathsf{r},\mathsf{x}}q \right)\right)\sin\left(\pi \left( C_{\mathsf{t},\mathsf{y}}p + C_{\mathsf{r},\mathsf{y}}q \right)\right)}{\sin\left(\frac{\pi}{Q_{\mathsf{x}}} \left( C_{\mathsf{t},\mathsf{x}}p + C_{\mathsf{r},\mathsf{x}}q \right)\right)\sin\left(\frac{\pi}{Q_{\mathsf{y}}} \left( C_{\mathsf{t},\mathsf{y}}p + C_{\mathsf{r},\mathsf{y}}q \right)\right)},\label{Qqpfrac}
    \end{aligned}$$

$p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$, with

\[C\] $$\begin{aligned}
        C_{\mathsf{t},\mathsf{x}} &\triangleq \frac{d_{\mathsf{t}}S_{\mathsf{x}}Q_{\mathsf{x}}A_{\mathsf{t},\mathsf{x}} \sin\psi_{\mathsf{t}}\cos(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{x}}(\omega_{\mathsf{t}},\varphi_{\mathsf{t}}))}{\lambda D_{\mathsf{t}}},\label{Ctx}\\
        C_{\mathsf{t},\mathsf{y}} &\triangleq \frac{d_{\mathsf{t}}S_{\mathsf{y}}Q_{\mathsf{y}}A_{\mathsf{t},\mathsf{y}}\sin\psi_{\mathsf{t}}\cos(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{y}}(\omega_{\mathsf{t}},\varphi_{\mathsf{t}}))}{\lambda D_{\mathsf{t}}}, \label{Cty} \\
        C_{\mathsf{r},\mathsf{x}} &\triangleq \frac{d_{\mathsf{r}}S_{\mathsf{x}}Q_{\mathsf{x}}A_{\mathsf{r},\mathsf{x}}\sin\psi_{\mathsf{r}}\cos(\gamma_{\mathsf{r}}-\bar{\gamma}_{\mathsf{r},\mathsf{x}}(\omega_{\mathsf{r}},\varphi_{\mathsf{r}}))}{\lambda D_{\mathsf{r}}},\\
        C_{\mathsf{r},\mathsf{y}} &\triangleq \frac{d_{\mathsf{r}}S_{\mathsf{y}}Q_{\mathsf{y}}A_{\mathsf{r},\mathsf{y}}\sin\psi_{\mathsf{r}}\cos(\gamma_{\mathsf{r}}-\bar{\gamma}_{\mathsf{r},\mathsf{y}}(\omega_{\mathsf{r}},\varphi_{\mathsf{r}}))}{\lambda D_{\mathsf{r}}}, \label{Cr} \\
        A_{\mathsf{t},\mathsf{x}} &\triangleq \sqrt{\sin^2 \omega_{\mathsf{t}} + \cos^2 \varphi_{\mathsf{t}} \cos^2 \omega_{\mathsf{t}}}, ~\cos \bar{\gamma}_{\mathsf{t},\mathsf{x}}= \frac{\sin \omega_{\mathsf{t}}}{A_{\mathsf{t},\mathsf{x}}}, ~ \sin \bar{\gamma}_{\mathsf{t},\mathsf{x}}= \frac{\cos \varphi_{\mathsf{t}}\cos \omega_{\mathsf{t}}}{A_{\mathsf{t},\mathsf{x}}}, \label{Atx}\\
        A_{\mathsf{t},\mathsf{y}} &\triangleq \sqrt{\cos^2 \omega_{\mathsf{t}} + \cos^2 \varphi_{\mathsf{t}} \sin^2 \omega_{\mathsf{t}}}, ~\cos \bar{\gamma}_{\mathsf{t},\mathsf{y}}= -\frac{\cos \omega_{\mathsf{t}}}{A_{\mathsf{t},\mathsf{y}}}, ~ \sin \bar{\gamma}_{\mathsf{t},\mathsf{y}}= \frac{\cos \varphi_{\mathsf{t}}\sin \omega_{\mathsf{t}}}{A_{\mathsf{t},\mathsf{y}}},\label{Aty}\\
        A_{\mathsf{r},\mathsf{x}} &\triangleq \sqrt{\sin^2 \omega_{\mathsf{r}} + \cos^2 \varphi_{\mathsf{r}} \cos^2 \omega_{\mathsf{r}}}, ~\cos \bar{\gamma}_{\mathsf{r},\mathsf{x}}= \frac{\sin \omega_{\mathsf{r}}}{A_{\mathsf{r},\mathsf{x}}}, ~ \sin \bar{\gamma}_{\mathsf{r},\mathsf{x}}= \frac{\cos \varphi_{\mathsf{r}}\cos \omega_{\mathsf{r}}}{A_{\mathsf{r},\mathsf{x}}},\label{Arx}\\
        A_{\mathsf{r},\mathsf{y}} &\triangleq \sqrt{\cos^2 \omega_{\mathsf{r}} + \cos^2 \varphi_{\mathsf{r}} \sin^2 \omega_{\mathsf{r}}}, ~\cos \bar{\gamma}_{\mathsf{r},\mathsf{y}}= -\frac{\cos \omega_{\mathsf{r}}}{A_{\mathsf{r},\mathsf{y}}}, ~ \sin \bar{\gamma}_{\mathsf{r},\mathsf{y}}= \frac{\cos \varphi_{\mathsf{r}}\sin \omega_{\mathsf{r}}}{A_{\mathsf{r},\mathsf{y}}}.\label{Ary}
                    \end{aligned}$$

The simplification from to employs the geometric sum formula and the trigonometric addition formulas. Denote by $\boldsymbol{P}\in \mathbb{C}^{N_{\mathsf{r}}\times N_{\mathsf{t}}}$ and $\boldsymbol{Q}\in \mathbb{C}^{N_{\mathsf{r}}\times N_{\mathsf{t}}}$ the matrices with $P_{q,p}$ and $Q_{q,p}$ in the $\left(q+\frac{N_{\mathsf{t}}+1}{2}\right)$-th row and the $\left(p+\frac{N_{\mathsf{r}}+1}{2}\right)$-th column, respectively, $p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$. Then the channel with reflective focusing can be expressed as $$\label{HPQ}
    \boldsymbol{H} = \eta_0 \boldsymbol{P} \odot \boldsymbol{Q},$$ where $\odot$ is the Hadamard product operator. As shown in what follows, this expression is useful in the derivation of generalized Rayleigh distances as well as in the channel capacity analysis.

Full Multiplexing Region of the Cascaded LoS MIMO Channel
=========================================================

In this section, we study the full multiplexing region (FMR) of the cascaded LoS MIMO system. We first give the FMRs of the LoS channels in the transmit and receive sides, and then derive an inner bound of the FMR of the overall cascaded LoS MIMO channel under reflective focusing.

FMRs of the Tx-IRS and the IRS-Rx Channels
------------------------------------------

In this subsection, we study the FMRs of the two single-side LoS MIMO channels, i.e., the Tx-IRS and the IRS-Rx channels. Roughly speaking, for a LoS MIMO channel with $M$ antennas at Tx, $N$ antennas at Rx, and fixed antenna spacings, FMR is the union of Tx-Rx distances that allows the MIMO channel to support $\min\{M,N\}$ simultaneous spatial streams with equal channel quality @wang_tens_2014. Within the FMR, the LoS MIMO system can fully exploit the multiplexing gain brought by the use of antenna arrays. In our model, the IRS can be viewed as a $Q_{\mathsf{x}} \times Q_{\mathsf{y}}$ uniform rectangular array (URA). Thus, the Tx-IRS (or IRS-Rx) channel can be viewed as a MIMO channel with a ULA (or URA) and a URA (or ULA) deployed at the transmitter and the receiver. Studying the FMRs of the Tx-IRS and IRS-Rx channels will help understand the properties of the overall cascaded LoS MIMO channel.

In our model, the Tx-IRS and IRS-Rx channels are given by $\boldsymbol{H}_{\mathsf{t}} \in \mathbb{C}^{Q_{\mathsf{x}}Q_{\mathsf{y}}\times N_{\mathsf{t}}}$ in and $\boldsymbol{H}_{\mathsf{r}} \in \mathbb{C}^{N_{\mathsf{r}}\times Q_{\mathsf{x}}Q_{\mathsf{y}}}$ in , respectively. Elements of $\boldsymbol{H}_{\mathsf{t}}$ and $\boldsymbol{H}_{\mathsf{r}}$ are complex exponentials with the phases determined by the link distances. We assume that the number of REs on the IRS is no less than the number of antennas in both Tx and Rx, i.e., $Q_{\mathsf{x}}Q_{\mathsf{y}} \geq N_{\mathsf{t}},N_{\mathsf{r}}$. For a Tx-IRS channel to support spatial multiplexing of $N_{\mathsf{t}}$ data streams, the $N_{\mathsf{t}}$ columns of $\boldsymbol{H}_{\mathsf{t}}$ are required to satisfy $$\begin{aligned}
\label{TOrthCrtt}
    \langle \boldsymbol{h}_{\mathsf{t};p_1}, \boldsymbol{h}_{\mathsf{t};p_2} \rangle = 
    \left\{\begin{array}{ll}
        Q_{\mathsf{x}}Q_{\mathsf{y}}, & p_1 = p_2 \\
        0, & p_1 \neq p_2   
    \end{array}\right.,
        ~ p_1 \in \mathcal{I}_{N_{\mathsf{t}}},~ p_2 \in \mathcal{I}_{N_{\mathsf{t}}},\end{aligned}$$ where $\langle \cdot,\cdot \rangle$ denotes the inner-product operator, and $\boldsymbol{h}_{\mathsf{t};p}$ denotes the $\left(p+\frac{N_{\mathsf{t}}+1}{2}\right)$-th column of $\boldsymbol{H}_{\mathsf{t}}$. In , the columns of $\boldsymbol{H}_{\mathsf{t}}$ are orthogonal, and the $N_{\mathsf{t}}$ eigenchannels of $\boldsymbol{H}_{\mathsf{t}}$ share a common channel gain of $Q_{\mathsf{x}}Q_{\mathsf{y}}$. The corresponding Tx-IRS channel is said to be able to support *full multiplexing* of $N_{\mathsf{t}}$ spatial streams. Similarly, for the IRS-Rx channel to support full multiplexing of $N_{\mathsf{r}}$ spatial streams, the rows of $\boldsymbol{H}_{\mathsf{r}}$ satisfy the orthogonality and equal-gain requirement as $$\begin{aligned}
\label{TOrthCrtr}
    \langle \boldsymbol{h}_{{\mathsf{r}},q_1}, \boldsymbol{h}_{{\mathsf{r}},q_2} \rangle = 
    \left\{\begin{array}{ll}
        Q_{\mathsf{x}}Q_{\mathsf{y}}, & q_1 = q_2 \\
        0, & q_1 \neq q_2   
    \end{array}\right.,
        ~ q_1 \in \mathcal{I}_{N_{\mathsf{r}}},~ q_2 \in \mathcal{I}_{N_{\mathsf{r}}},\end{aligned}$$ where $\boldsymbol{h}_{{\mathsf{r}},q}$ denotes the $\left(q+\frac{N_{\mathsf{r}}+1}{2}\right)$-th column of $\boldsymbol{H}_{\mathsf{r}}$. When the Tx-IRS (or IRS-Rx) distance $D_{\mathsf{t}}$ (or $D_{\mathsf{r}}$) is within the corresponding FMR, the orthogonality and equal-gain requirement on $\boldsymbol{H}_{\mathsf{t}}$ (or $\boldsymbol{H}_{\mathsf{r}}$) can be met by appropriately adjusting the Tx (Rx) orientation. We now present the formal definitions of the FMRs of the Tx-IRS and IRS-Rx channels.

For given IRS-Tx direction $(\varphi_{\mathsf{t}},\omega_{\mathsf{t}})$, the **Tx-IRS FMR $\mathcal{D}_{\mathsf{t}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}})$** is a set of $D_{\mathsf{t}}$ that enables the columns of $\boldsymbol{H}_{\mathsf{t}}$ to satisfy by tuning $\psi_{\mathsf{t}}$ and $\gamma_{\mathsf{t}}$. For given IRS-Rx direction $(\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$, the **IRS-Rx FMR $\mathcal{D}_{\mathsf{r}}(\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$** are defined in a similar way as $\mathcal{D}_{\mathsf{t}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}})$ by replacing the subscript $\mathsf{t}$ by $\mathsf{r}$ and by .

We have the following results on $D_{\mathsf{t}}^{\mathsf{R}}$ and $D_{\mathsf{r}}^{\mathsf{R}}$.

\[propSingleSideDray\] For the Tx-IRS channel, if $Q_{\mathsf{x}} \geq N_{\mathsf{t}}$, $D_{\mathsf{t}}^{\mathsf{R}} \geq D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}} \triangleq \frac{d_{\mathsf{t}} S_{\mathsf{x}}Q_{\mathsf{x}}A_{\mathsf{t},\mathsf{x}}}{\lambda}$ with $A_{\mathsf{t},\mathsf{x}}$ in ; if $Q_{\mathsf{y}} \geq N_{\mathsf{t}}$, $D_{\mathsf{t}}^{\mathsf{R}} \geq D_{\mathsf{t},\mathsf{y}} \triangleq \frac{d_{\mathsf{t}} S_{\mathsf{y}}Q_{\mathsf{y}}A_{\mathsf{t},\mathsf{y}}}{\lambda }$ with $A_{\mathsf{t},\mathsf{y}}$ in ; if $Q_{\mathsf{x}} \geq N_{\mathsf{t}}$ and $Q_{\mathsf{y}} \geq N_{\mathsf{t}}$, $D_{\mathsf{t}}^{\mathsf{R}}$ is given by $$\label{DT}
        D_{\mathsf{t}}^{\mathsf{R}} = \max\left\{ D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}, D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}} \right\}.$$ When $D_{\mathsf{t}} \leq D_{\mathsf{t}}^{\mathsf{R}}=D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}$, the orthogonality and equal-gain requirement can be satisfied by letting $$\begin{aligned}
\label{txangleCrt}
        |\cos(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{x}})| = 1 ~\textup{and} ~ |\sin\psi_{\mathsf{t}}| = \frac{D_{\mathsf{t}}}{D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}}, ~ \textup{or}~ |\cos(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{x}})| = \frac{D_{\mathsf{t}}}{D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}}  ~\textup{and} ~ |\sin\psi_{\mathsf{t}}| = 1.
    \end{aligned}$$ When $D_{\mathsf{t}} \leq D_{\mathsf{t}}^{\mathsf{R}}=D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}}$, the orthogonality and equal-gain requirement can be satisfied by letting $$\begin{aligned}
\label{tyangleCrt}
        |\cos(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{y}})| = 1 ~\textup{and} ~ |\sin\psi_{\mathsf{t}}| = \frac{D_{\mathsf{t}}}{D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}}}, ~ \textup{or}~ |\cos(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{y}})| = \frac{D_{\mathsf{t}}}{D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}}}  ~\textup{and} ~ |\sin\psi_{\mathsf{t}}| = 1.
    \end{aligned}$$ The above results literally hold for the IRS-Rx channel by replacing the subscript $\mathsf{t}$ by $\mathsf{r}$, by , by , and by .

The proof of Proposition \[propSingleSideDray\] can be found in Appendix \[proofSingleSideDray\]. Proposition \[propSingleSideDray\] gives the boundaries of the FMRs on the Tx-IRS side and the IRS-Rx side, and the solution of the orientation angles of the Tx and the Rx to achieve full multiplexing between Tx/Rx and the IRS. Fig. \[Eigsa\] and Fig. \[Eigsb\] illustrate the eigenvalues of $\frac{1}{Q_{\mathsf{x}}Q_{\mathsf{y}}} \boldsymbol{H}_{\mathsf{t}}^{\text{H}} \boldsymbol{H}_{\mathsf{t}}$ against $D_{\mathsf{t}}$, with $N_{\mathsf{t}}=N_{\mathsf{r}}=5$, $Q_{\mathsf{x}} = Q_{\mathsf{y}} =15$, $\varphi_{\mathsf{t}}=\frac{\pi}{6}$, $\omega_{\mathsf{t}} = \frac{7\pi}{6}$, $d_{\mathsf{t}}=S_{\mathsf{x}}=S_{\mathsf{y}}=0.1$m, and $\lambda=0.005$m. Based on these parameters we have $D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}} = \frac{d_{\mathsf{t}} S_{\mathsf{x}}Q_{\mathsf{x}}A_{\mathsf{t},\mathsf{x}}}{\lambda}=27.0416$m, and $D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}} = \frac{d_{\mathsf{t}} S_{\mathsf{y}}Q_{\mathsf{y}}A_{\mathsf{t},\mathsf{y}}}{\lambda}=29.0474$m. Since $D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}} > D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}$, the boundary of the FMR is given by $D_{\mathsf{t}}^{\mathsf{R}} = D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}}$. In Fig. \[Eigsa\] and Fig. \[Eigsb\], the orientation of Tx (i.e., $\psi_{\mathsf{t}}$ and $\gamma_{\mathsf{t}}$) are set to respectively meet and by letting $\frac{D_{\mathsf{t}}}{D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}}=1$ and $\frac{D_{\mathsf{t}}}{D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}}}=1$. $D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}$ (or $D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}}$) is the largest $D_{\mathsf{t}}$ that enables $\boldsymbol{H}_{\mathsf{t}}$ to satisfy the orthogonality and equal-gain requirement in under the orientation setting in (or ).

\[b\]<span>0.48</span> ![Eigenvalues of $\frac{1}{Q_{\mathsf{x}}Q_{\mathsf{y}}} \boldsymbol{H}_{\mathsf{t}}^{\text{H}} \boldsymbol{H}_{\mathsf{t}}$ against $D_{\mathsf{t}}$. $N_{\mathsf{t}}=N_{\mathsf{r}}=5$, $Q_{\mathsf{x}} = Q_{\mathsf{y}} =15$, $\varphi_{\mathsf{t}}=\frac{\pi}{6}$, $\omega_{\mathsf{t}} = \frac{7\pi}{6}$, $d_{\mathsf{t}}=S_{\mathsf{x}}=S_{\mathsf{y}}=0.1$m, $\lambda=0.005$m. (a) $\psi_{\mathsf{t}}=\frac{\pi}{2}$, $\gamma_{\mathsf{t}} = \bar{\gamma}_{\mathsf{t},\mathsf{x}}$, $D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}} = 27.0416$m. (b) $\psi_{\mathsf{t}}=\frac{\pi}{2}$, $\gamma_{\mathsf{t}} = \bar{\gamma}_{\mathsf{t},\mathsf{y}}$, $D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}} = 29.0474$m.<span data-label="EigsFig"></span>](Dtx.pdf "fig:"){width="\textwidth"}

\[b\]<span>0.48</span> ![Eigenvalues of $\frac{1}{Q_{\mathsf{x}}Q_{\mathsf{y}}} \boldsymbol{H}_{\mathsf{t}}^{\text{H}} \boldsymbol{H}_{\mathsf{t}}$ against $D_{\mathsf{t}}$. $N_{\mathsf{t}}=N_{\mathsf{r}}=5$, $Q_{\mathsf{x}} = Q_{\mathsf{y}} =15$, $\varphi_{\mathsf{t}}=\frac{\pi}{6}$, $\omega_{\mathsf{t}} = \frac{7\pi}{6}$, $d_{\mathsf{t}}=S_{\mathsf{x}}=S_{\mathsf{y}}=0.1$m, $\lambda=0.005$m. (a) $\psi_{\mathsf{t}}=\frac{\pi}{2}$, $\gamma_{\mathsf{t}} = \bar{\gamma}_{\mathsf{t},\mathsf{x}}$, $D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}} = 27.0416$m. (b) $\psi_{\mathsf{t}}=\frac{\pi}{2}$, $\gamma_{\mathsf{t}} = \bar{\gamma}_{\mathsf{t},\mathsf{y}}$, $D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}} = 29.0474$m.<span data-label="EigsFig"></span>](Dty.pdf "fig:"){width="\textwidth"}

From , the overall cascaded LoS MIMO channel $\boldsymbol{H}$ is jointly determined by the Tx-IRS channel $\boldsymbol{H}_{\mathsf{t}}$, the IRS-Rx channel $\boldsymbol{H}_{\mathsf{r}}$, the passive beamforming matrix $\boldsymbol{\Theta}$, and the path loss $\eta_0$. That means even if both the Tx-IRS channel and the IRS-Rx channel can support full multiplexing, the full multiplexing capability of the overall cascaded LoS MIMO channel cannot be necessarily guaranteed. As such, we need to consider the FMR of the overall cascaded LoS MIMO channel, as detailed in the next subsection.

FMR of the cascaded LoS MIMO Channel
------------------------------------

We first give a definition to the *full multiplexing* of the cascaded LoS MIMO channel. Without loss of generality, we henceforth always assume $N_{\mathsf{t}}\geq N_{\mathsf{r}}$. A cascaded LoS MIMO channel (with $N_{\mathsf{t}}\geq N_{\mathsf{r}}$) is qualified to support full multiplexing if the columns of $\boldsymbol{H}$ satisfy

\[NtgeqNr\] $$\label{NtgeqNra}
        \langle \boldsymbol{h}_{p_1}, \boldsymbol{h}_{p_2} \rangle = \eta_0^2 Q_{\mathsf{x}}^2 Q_{\mathsf{y}}^2,~ \textrm{for}~ p_1 = p_2,$$ and $$\label{NtgeqNrb}
        \langle \boldsymbol{h}_{p_1}, \boldsymbol{h}_{p_2} \rangle =0,~ \textrm{for}~ p_1 \neq p_2,$$

where $p_1 \in \mathcal{I}_{N_{\mathsf{t}}},~ p_2 \in \mathcal{I}_{N_{\mathsf{t}}}$, and $\boldsymbol{h}_{p}$ denotes the $\left(p+\frac{N_{\mathsf{t}}+1}{2}\right)$-th column of $\boldsymbol{H}$. In , $\eta_0^2 Q_{\mathsf{x}}^2 Q_{\mathsf{y}}^2$ is the overall channel gain by taking into account the power gain of an eigenchannel of $\boldsymbol{H}_{\mathsf{t}}$ in , that of $\boldsymbol{H}_{\mathsf{r}}$ in , and the total path loss $\eta_0^2$. With the above notion of full multiplexing, we now present the formal definition of the FMR of the cascaded LoS MIMO channel.

\[DefnGDRay\] For given IRS-Tx direction $(\varphi_{\mathsf{t}},\omega_{\mathsf{t}})$ and IRS-Rx direction $(\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$, the full multiplexing region of the cascaded LoS MIMO channel, referred to as the **Tx-IRS-Rx FMR $\mathcal{D}_{\mathsf{c}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$**, are the union of all $(D_{\mathsf{t}},D_{\mathsf{r}})$ pairs that enable $\boldsymbol{H}$ to satisfy by tuning $\psi_{\mathsf{t}}$, $\gamma_{\mathsf{t}}$, $\psi_{\mathsf{r}}$ and $\gamma_{\mathsf{r}}$.

We have the following results on the Tx-IRS-Rx FMR $\mathcal{D}_{\mathsf{c}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$.

\[propGray\] For a cascaded LoS MIMO channel with $N_{\mathsf{t}}+N_{\mathsf{r}}-2<2Q_{\mathsf{x}}$, and $N_{\mathsf{t}}+N_{\mathsf{r}}-2<2Q_{\mathsf{y}}$, the Tx-IRS-Rx FMR $\mathcal{D}_{\mathsf{c}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$ at least covers the union of two regions $\mathcal{D}_{\mathsf{c},\mathsf{x}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$ and $\mathcal{D}_{\mathsf{c},\mathsf{y}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$. Specifically, the boundary of $\mathcal{D}_{\mathsf{c},\mathsf{x}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$ is given by $$\begin{aligned}
        D_{\mathsf{r}} &= D_{\mathsf{r},\mathsf{x}}^{\mathsf{R}}, ~ \textup{for}~ D_{\mathsf{t}} \leq D_{\mathsf{t},\mathsf{x}}^*, ~ \textup{and}~  \label{DcxStart}\\
        D_{\mathsf{r}} &= D_{\mathsf{r},\mathsf{x}}^{\mathsf{R}} \max\left\{|\cos(\gamma_{\mathsf{x}}^+(D_{\mathsf{t}})-\bar{\gamma}_{\mathsf{r},\mathsf{x}})|, |\cos(\gamma_{\mathsf{x}}^+(D_{\mathsf{t}})-\bar{\gamma}_{\mathsf{r},\mathsf{x}})|\right\},  ~ \textup{for}~ D_{\mathsf{t},\mathsf{x}}^* \leq D_{\mathsf{t}} \leq D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}, ~ \textup{and}~ \label{Dcx2} \\
        D_{\mathsf{t}} &= D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}, \label{Dcx3}
    \end{aligned}$$ where $D_{\mathsf{t},\mathsf{x}}^*$ is defined by $$\begin{aligned}
        D_{\mathsf{t},\mathsf{x}}^* &\triangleq D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}|\cos(\gamma_{\mathsf{x}}^*-\bar{\gamma}_{\mathsf{t},\mathsf{x}})|~ \textup{with} \\
        \tan(\gamma_{\mathsf{x}}^*) &= \frac{\cos \bar{\gamma}_{\mathsf{t},\mathsf{x}}-G_{\mathsf{x}}^* \cos \bar{\gamma}_{\mathsf{t},\mathsf{y}}}{G_{\mathsf{x}}^* \sin \bar{\gamma}_{\mathsf{t},\mathsf{y}}-\sin \bar{\gamma}_{\mathsf{t},\mathsf{x}}} ~ \textup{and}~ \\
        G_{\mathsf{x}}^* &\triangleq \frac{A_{\mathsf{t},\mathsf{x}}A_{\mathsf{r},\mathsf{y}} \cos(\bar{\gamma}_{\mathsf{r},\mathsf{x}}-\bar{\gamma}_{\mathsf{r},\mathsf{y}})}{A_{\mathsf{t},\mathsf{y}}A_{\mathsf{r},\mathsf{x}}},
    \end{aligned}$$ and $\gamma_{\mathsf{x}}^+(D_{\mathsf{t}})$ and $\gamma_{\mathsf{x}}^-(D_{\mathsf{t}})$ are defined by $$\begin{aligned}
\label{gammaxPstv}
        \tan(\gamma_{\mathsf{x}}^+(D_{\mathsf{t}})) = \frac{\cos \bar{\gamma}_{\mathsf{r},\mathsf{x}}-G_{\mathsf{x}}^+(D_{\mathsf{t}}) \cos \bar{\gamma}_{\mathsf{r},\mathsf{y}}}{G_{\mathsf{x}}^+(D_{\mathsf{t}}) \sin \bar{\gamma}_{\mathsf{r},\mathsf{y}}-\sin \bar{\gamma}_{\mathsf{r},\mathsf{x}}} ~ \textup{and}~ \tan(\gamma_{\mathsf{x}}^-(D_{\mathsf{t}})) = \frac{\cos \bar{\gamma}_{\mathsf{r},\mathsf{x}}-G_{\mathsf{x}}^-(D_{\mathsf{t}}) \cos \bar{\gamma}_{\mathsf{r},\mathsf{y}}}{G_{\mathsf{x}}^-(D_{\mathsf{t}}) \sin \bar{\gamma}_{\mathsf{r},\mathsf{y}}+\sin \bar{\gamma}_{\mathsf{r},\mathsf{x}}},
    \end{aligned}$$ respectively, with $$\begin{aligned}
        G_{\mathsf{x}}^+(D_{\mathsf{t}}) &\triangleq \frac{A_{\mathsf{t},\mathsf{y}}A_{\mathsf{r},\mathsf{x}}\left( \cos(\bar{\gamma}_{\mathsf{t},\mathsf{x}}-\bar{\gamma}_{\mathsf{t},\mathsf{y}})- \sin(\bar{\gamma}_{\mathsf{t},\mathsf{x}}-\bar{\gamma}_{\mathsf{t},\mathsf{y}}) \sqrt{\left(\frac{D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}}{D_{\mathsf{t}}}\right)^2-1} \right)}{A_{\mathsf{t},\mathsf{x}}A_{\mathsf{r},\mathsf{y}}} ~ \textup{and} ~ \label{GxPstv}\\
        G_{\mathsf{x}}^-(D_{\mathsf{t}}) &\triangleq \frac{A_{\mathsf{t},\mathsf{y}}A_{\mathsf{r},\mathsf{x}}\left( \cos(\bar{\gamma}_{\mathsf{t},\mathsf{x}}-\bar{\gamma}_{\mathsf{t},\mathsf{y}})+ \sin(\bar{\gamma}_{\mathsf{t},\mathsf{x}}-\bar{\gamma}_{\mathsf{t},\mathsf{y}}) \sqrt{\left(\frac{D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}}{D_{\mathsf{t}}}\right)^2-1} \right)}{A_{\mathsf{t},\mathsf{x}}A_{\mathsf{r},\mathsf{y}}}. \label{DcxEnd}
    \end{aligned}$$ The boundary of $\mathcal{D}_{\mathsf{c},\mathsf{y}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$ is defined by literally replacing the subscripts $\mathsf{x}$ by $\mathsf{y}$ and $\mathsf{y}$ by $\mathsf{x}$ in -.

Proposition \[propGray\] gives at least an inner bound of $\mathcal{D}_{\mathsf{c}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$. It is derived under reflective focusing. The proof of Proposition \[propGray\] can be found in Appendix \[proofGray\]. We have the following remarks on Proposition \[propGray\].

When both the centers of the Tx and the Rx are located on the $y$-$z$ or $x$-$z$ planes, i.e., $\omega_{\mathsf{t}} \in \left\{0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\right\}$ and $\omega_{\mathsf{r}} \in \left\{0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\right\}$, $\mathcal{D}_{\mathsf{c},\mathsf{x}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$ and $\mathcal{D}_{\mathsf{c},\mathsf{y}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$ are two rectangular areas in the $D_{\mathsf{t}}$-$D_{\mathsf{r}}$ plane, where $\mathcal{D}_{\mathsf{c},\mathsf{x}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$ are bounded by $D_{\mathsf{t}} = 0$, $D_{\mathsf{t}} = D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}$, $D_{\mathsf{r}} =0$, and $D_{\mathsf{r}} = D_{\mathsf{r},\mathsf{x}}^{\mathsf{R}}$, and $\mathcal{D}_{\mathsf{c},\mathsf{y}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$ are bounded by $D_{\mathsf{t}} = 0$, $D_{\mathsf{t}} = D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}}$, $D_{\mathsf{r}} =0$, and $D_{\mathsf{r}} = D_{\mathsf{r},\mathsf{y}}^{\mathsf{R}}$.

Mutual Information Maximization: Analytical Results
===================================================

In this section, we study the mutual information (MI) maximization problem of the considered IRS-aided MIMO system. The system input-output model is given by , and the overall channel model is given by . For simplicity, we assume that the transmit signal employs CSCG signaling, and the transmit power is uniformly allocated on the $N_{\mathsf{t}}$ transmit antennas, i.e., $\boldsymbol{x} \sim \mathcal{CN}(\boldsymbol{0},\mathcal{P}\boldsymbol{I})$, where $\mathcal{P}$ is the power allocated to each transmit antenna. Then, the MI between the transmit signal $\boldsymbol{x}$ and the received signal $\boldsymbol{y}$ is given by $$\label{Iori}
    \mathcal{I}(\boldsymbol{x};\boldsymbol{y}) = \log \det\left( \frac{\mathcal{P}}{\sigma_w^2} \boldsymbol{H}^{\textrm{H}} \boldsymbol{H} + \boldsymbol{I} \right),$$ where $\boldsymbol{H}$ is the channel matrix in , and $\sigma_w^2$ is the noise power. We aim to maximize $\mathcal{I}(\boldsymbol{x};\boldsymbol{y})$ over the IRS phases $\boldsymbol{\Theta}$ and the orientation angles of Tx and Rx $\mathcal{A} \triangleq \left\{\gamma_{\mathsf{t}},\psi_{\mathsf{t}},\gamma_{\mathsf{r}},\psi_{\mathsf{r}}\right\}$, while the other parameters (such as the wavelength, the Tx-IRS and IRS-Rx distances, Tx and Rx directions, and the spacing and the size parameters of Tx and Rx) are fixed. This problem is formulated as

\[MIori\] $$\begin{aligned}
        \max_{\mathbf{\Theta},\mathcal{A}} &~ \mathcal{I}(\boldsymbol{x};\boldsymbol{y}) \\
        \text { s.t. } & \left|\theta_{k,l}\right|=1,~ \forall k \in \mathcal{I}_{Q_{\mathsf{x}}},~ \forall l \in \mathcal{I}_{Q_{\mathsf{y}}}, \\
        & \psi_{\mathsf{t}} \in [0,\pi], ~ \gamma_{\mathsf{t}} \in [-\pi/2,\pi/2], \label{tconstraint} \\
        & \psi_{\mathsf{t}} \in [0,\pi], ~ \gamma_{\mathsf{t}} \in [-\pi/2,\pi/2]. \label{rconstraint}
    \end{aligned}$$

Problem generally does not have a closed-form solution, and its numerical solution will be discussed in Section \[Algo\]. Here, we provide an upper bound of $\mathcal{I}(\boldsymbol{x};\boldsymbol{y})$, and derive the analytical solution of in the high and low SNR regimes. To this end, by recalling $N_{\mathsf{t}} \geq N_{\mathsf{r}}$, we denote by $\mu_{\mathsf{H};1},\ldots, \mu_{\mathsf{H};N_{\mathsf{r}}}$ the singular values of $\boldsymbol{H}$. Then by denoting $\rho \triangleq \frac{\mathcal{P}}{\sigma_w^2}$ as the transmit SNR, $\mathcal{I}(\boldsymbol{x};\boldsymbol{y})$ can be recast into $$\label{MIsvd}
    \mathcal{I}(\boldsymbol{x};\boldsymbol{y}) = \sum_{i=1}^{N_{\mathsf{r}}} \log\left( 1 + \rho \mu_{\mathsf{H};i}^2 \right).$$ Further, denote by $\mu_{\mathsf{t};1},\ldots, \mu_{\mathsf{t};N_{\mathsf{t}}}$ and $\mu_{\mathsf{r};1},\ldots, \mu_{\mathsf{r};N_{\mathsf{r}}}$ the singular values of $\boldsymbol{H}_{\mathsf{t}}$ and $\boldsymbol{H}_{\mathsf{r}}$ in descending order, respectively. The following lemma provides an upper bound of $\mathcal{I}(\boldsymbol{x};\boldsymbol{y})$.

\[leqLemma\] $$\label{MIinequ}
        \mathcal{I}(\boldsymbol{x};\boldsymbol{y}) \leq \sum_{i=1}^{N_{\mathsf{r}}} \log\left( 1+\rho \eta_0^2 \mu_{\mathsf{r};i}^2 \mu_{\mathsf{t};i}^2 \right).$$ $\mathcal{I}(\boldsymbol{x};\boldsymbol{y})$ reaches the right-hand side of if and only if $$\label{SVDequal}
        \mu_{\mathsf{H};i}=\eta_0 \mu_{\mathsf{r};i} \mu_{\mathsf{t};i},~ i=1,\ldots,N_{\mathsf{r}}.$$

Recall that $\boldsymbol{H} = \eta_0 \boldsymbol{H}_{\mathsf{r}} \boldsymbol{\Theta} \boldsymbol{H}_{\mathsf{t}}$ in . Let $\boldsymbol{A} \triangleq \eta_0 \boldsymbol{H}_{\mathsf{r}} \boldsymbol{\Theta}$, then $\boldsymbol{H}$ can be rewritten as $\boldsymbol{H} = \boldsymbol{A}\boldsymbol{H}_{\mathsf{t}}$. Since $\boldsymbol{\Theta}$ is a diagonal matrix with normalized diagonal elements, $\boldsymbol{H}_{\mathsf{r}} \boldsymbol{\Theta}$ share the same singular values with $\boldsymbol{H}_{\mathsf{r}}$, and the singular values of $\boldsymbol{A}$ are given by $\left\{\mu_{\mathsf{A};1},\ldots, \mu_{\mathsf{A};N_{\mathsf{r}}}\right\} = \left\{\eta_0 \mu_{\mathsf{r};1},\ldots, \eta_0 \mu_{\mathsf{r};N_{\mathsf{r}}}\right\}$. Define two real-valued function $f(x) = \log(1+\rho x^2)$ and $\chi(t) \equiv f(e^t) = \log(1+\rho e^{2t})$. Based on , @horn1986topics [Theorem 3.3.14(c)], and the fact that $\chi(t)$ is increasing and convex over $t$ in $[0,+\infty)$, we have

$$\begin{aligned}
            \mathcal{I}(\boldsymbol{x};\boldsymbol{y}) &\leq \sum_{i=1}^{N_{\mathsf{r}}} \log\left( 1+\rho \mu_{\mathsf{A};i}^2 \mu_{\mathsf{t};i}^2 \right) \label{lemmaleq1} \\
            &= \sum_{i=1}^{N_{\mathsf{r}}} \log\left( 1+\rho \eta_0^2 \mu_{\mathsf{r};i}^2 \mu_{\mathsf{t};i}^2 \right),
        \end{aligned}$$

where the equality in holds if and only if $\mu_{\mathsf{H};i}=\mu_{\mathsf{A};i} \mu_{\mathsf{t};i}= \eta_0 \mu_{\mathsf{r};i} \mu_{\mathsf{t};i},~ i=1,\ldots,N_{\mathsf{r}}$, as given in . This concludes the proof.

To proceed, we present a relaxed version of problem as

\[OptSVD\] $$\begin{aligned}
\label{MIRHS}
        \max_{\mathcal{U}_{\mathsf{t}},\mathcal{U}_{\mathsf{r}}} &~  \sum_{i=1}^{N_{\mathsf{r}}} \log\left( 1+\rho \eta_0^2 \mu_{\mathsf{r};i}^2 \mu_{\mathsf{t};i}^2 \right) \\
        \text { s.t. } & \sum_j^{N_{\mathsf{t}}} \mu_{\mathsf{t};j}^2 = N_{\mathsf{t}}Q_{\mathsf{x}}Q_{\mathsf{y}}, \label{Htconstraint} \\
        & \sum_i^{N_{\mathsf{r}}} \mu_{\mathsf{r};i}^2 = N_{\mathsf{r}}Q_{\mathsf{x}}Q_{\mathsf{y}}, \label{Hrconstraint}
    \end{aligned}$$

where $\mathcal{U}_{\mathsf{t}} \triangleq \left\{\mu_{\mathsf{t};1},\ldots, \mu_{\mathsf{t};N_{\mathsf{t}}}\right\}$ and $\mathcal{U}_{\mathsf{r}} \triangleq \left\{\mu_{\mathsf{r};1},\ldots, \mu_{\mathsf{r};N_{\mathsf{r}}}\right\}$. Eqs. and account for the total channel gain constraints of $\boldsymbol{H}_{\mathsf{t}}$ and $\boldsymbol{H}_{\mathsf{r}}$, respectively.

The maximum MI given by is an upper bound of the maximum MI given by .

From Lemma \[leqLemma\], we see that the objective function of problem is always lower than that of problem . Further, since elements of $\boldsymbol{H}_{\mathsf{t}}$ and $\boldsymbol{H}_{\mathsf{r}}$ are normalized complex exponentials, the total power constraints on $\boldsymbol{H}_{\mathsf{t}}$ and $\boldsymbol{H}_{\mathsf{r}}$ in and always hold, regardless of the values of $\gamma_{\mathsf{t}}$, $\psi_{\mathsf{t}}$, $\gamma_{\mathsf{r}}$, $\psi_{\mathsf{r}}$. Thus and are looser constraints than and . Therefore, with a greater objective function and relaxed constraints, the maximum MI given by is greater than that given by .

The following propositions provide the asymptotic solutions to as $\rho \to \infty$ and $\rho \to 0$.

\[propHighSNR\] With $\rho \to \infty$, the maximum MI of problem is achieved when the power gains of $\boldsymbol{H}_{\mathsf{t}}$ and $\boldsymbol{H}_{\mathsf{r}}$ are evenly allocated to their respective first $N_{\mathsf{r}}$ eigenmodes, i.e.,

\[rhoinfsvd\] $$\begin{aligned}
            \mu_{\mathsf{t};1}^2 &= \cdots = \mu_{\mathsf{t};N_{\mathsf{r}}}^2 = \frac{N_{\mathsf{t}}}{N_{\mathsf{r}}} Q_{\mathsf{x}} Q_{\mathsf{y}}, \\
            \mu_{\mathsf{r};1}^2 &= \cdots = \mu_{\mathsf{r};N_{\mathsf{r}}}^2 = Q_{\mathsf{x}}Q_{\mathsf{y}}.
        \end{aligned}$$

\[propLowSNR\] With $\rho \to 0$, is maximized when the power gains of $\boldsymbol{H}_{\mathsf{t}}$ and $\boldsymbol{H}_{\mathsf{r}}$ are concentrated on their respective first eigenmode, i.e.,

\[rho0svd\] $$\begin{aligned}
            \mu_{\mathsf{t};1}^2 &= N_{\mathsf{t}}Q_{\mathsf{x}}Q_{\mathsf{y}},~ \mu_{\mathsf{t};j}^2 = 0, ~ j = 2,\ldots,N_{\mathsf{t}}, \label{rho0svda} \\
            \mu_{\mathsf{r};1}^2 &= N_{\mathsf{r}}Q_{\mathsf{x}}Q_{\mathsf{y}},~ \mu_{\mathsf{r};i}^2 = 0, ~ i = 2,\ldots, N_{\mathsf{r}}. \label{rho0svdb}
        \end{aligned}$$

The proofs of Propositions \[propHighSNR\] and \[propLowSNR\] can be found in Appendix \[proofHighLowSNR\].

Mutual Information Maximization: Iterative Algorithm Design {#Algo}
===========================================================

In this section, we present the proposed algorithm to solve the MI optimization problem in . The key idea is to alternately update $\mathcal{A}$ and $\boldsymbol{\Theta}$. Specifically, Subsection \[SecTheta\] provides the solution to $\boldsymbol{\Theta}$ with fixed $\mathcal{A}$, and Subsection \[SecA\] provides the solution to $\mathcal{A}$ with fixed $\boldsymbol{\Theta}$. Finally, the overall algorithm is summarized in Subsection \[SecOverAlgo\].

Solve $\boldsymbol{\Theta}$ for Fixed $\mathcal{A}$ {#SecTheta}
---------------------------------------------------

We first consider the MI maximization problem over the passive beamforming $\boldsymbol{\Theta}$. With fixed Tx and Rx orientations $\mathcal{A}$, $\boldsymbol{H}_{\mathsf{t}}$ and $\boldsymbol{H}_{\mathsf{r}}$ can be formulate by following and , respectively, and the MI optimization problem is equivalent to $$\label{OriginalProblem}
    \begin{array}{ll}
        \begin{aligned}
        \min_{\boldsymbol{\Theta}} &~ -\mathcal{I}(\boldsymbol{x};\boldsymbol{y})=\mathbb{E}_{\boldsymbol{x}, \boldsymbol{y}} \log \frac{p(\boldsymbol{x}|\boldsymbol{y})}{p(\boldsymbol{x})} \\
        \text { s.t. } & \left|\beta_{k,l}\right|=1,~ \forall k \in \mathcal{I}_{Q_{\mathsf{x}}},~ \forall l \in \mathcal{I}_{Q_{\mathsf{y}}}
        \end{aligned}
    \end{array}$$ where $p(\boldsymbol{x}|\boldsymbol{y})$ is the distribution of $\boldsymbol{x}$ conditioned on $\boldsymbol{y}$. This problem can be recast as: $$\label{program}
    \begin{array}{ll}
        \begin{aligned}
        \min_{\mathbf{\Theta}} &~ \mathcal{Q}(\boldsymbol{\Theta}) \\
        \text { s.t. } & \left|\beta_{k,l}\right|=1,~ \forall k \in \mathcal{I}_{Q_{\mathsf{x}}},~ \forall l \in \mathcal{I}_{Q_{\mathsf{y}}} \\
        & \mathcal{Q}(\boldsymbol{\Theta}) \triangleq \min_{\boldsymbol{\Phi}, \boldsymbol{\Sigma} \succeq 0} \mathcal{J}(\boldsymbol{\Phi},\boldsymbol{\Sigma})
        \end{aligned}
    \end{array}$$ where $\mathcal{J}(\boldsymbol{\Phi},\boldsymbol{\Sigma})\triangleq \mathbb{E}_{\boldsymbol{x},\boldsymbol{y}} \left(\log \operatorname{det}(\boldsymbol{\Sigma})+\left\|\boldsymbol{\Sigma}^{-\frac{1}{2}}(\boldsymbol{x}-\boldsymbol{\Phi} \boldsymbol{y})\right\|_{2}^{2}\right)$; $\boldsymbol{\Phi} \in \mathbb{C}^{N_{\mathsf{t}}\times N_{\mathsf{r}}}$ and $\boldsymbol{\Sigma} \in \mathbb{C}^{N_{\mathsf{t}}\times N_{\mathsf{t}}}$ are auxiliary parameter matrices. Note that $\mathcal{Q}(\boldsymbol{\Theta})$ is a variational representation of $\mathbb{E}_{\boldsymbol{x},\boldsymbol{y}} \log \frac{p(\boldsymbol{x}|\boldsymbol{y})}{p(\boldsymbol{x})}$. The equivalence between and can be proved by using the Blahut-Arimoto algorithm @cover_elements_nodate [Sec. 10.8]. The optimization problem in is non-convex due to the unit modulus constraint on $\beta_{k,l},~ \forall k \in \mathcal{I}_{Q_{\mathsf{x}}},~ \forall l \in \mathcal{I}_{Q_{\mathsf{y}}}$. To solve , we borrow a two-step alternating optimization method from @yan_passive_2020, that is, first to solve $\mathcal{Q}(\boldsymbol{\Theta})$ for given $\boldsymbol{\Theta}$, and then to optimize $\boldsymbol{\Theta}$ for given $\mathcal{Q}(\boldsymbol{\Theta})$. These two steps are executed iteratively and finally give an updated $\boldsymbol{\Theta}$.

### Solve $\mathcal{Q}(\boldsymbol{\Theta})$ for Given $\boldsymbol{\Theta}$

We first solve $\mathcal{Q}(\boldsymbol{\Theta})$ for given $\boldsymbol{\Theta}$. For given $\boldsymbol{\Theta}$, the closed-form solution to $\mathcal{Q}(\boldsymbol{\Theta})=\min_{\boldsymbol{\Phi}, \boldsymbol{\Sigma} \succeq 0} \allowbreak \mathcal{J}(\boldsymbol{\Phi},\boldsymbol{\Sigma})$ is given by $$\begin{aligned}
    \boldsymbol{\Phi} &= \boldsymbol{C}_{\boldsymbol{x}\boldsymbol{y}}\boldsymbol{C}_{\boldsymbol{y}\boldsymbol{y}}^{-1}, \label{PHI} \\
    \boldsymbol{\Sigma} &= \mathcal{P} \boldsymbol{I} - \boldsymbol{C}_{\boldsymbol{x}\boldsymbol{y}} \boldsymbol{C}_{\boldsymbol{y}\boldsymbol{y}}^{-1} \boldsymbol{C}_{\boldsymbol{x}\boldsymbol{y}}^{\textrm{H}}, \label{SIGMA}\end{aligned}$$ where $\boldsymbol{C}_{\boldsymbol{x}\boldsymbol{y}}$ is the covariance matrix of $\boldsymbol{x}$ and $\boldsymbol{y}$ given by $$\boldsymbol{C}_{\boldsymbol{x}\boldsymbol{y}} = \mathcal{P} \eta_0 \left( \boldsymbol{H}_{\mathsf{r}} \boldsymbol{\Theta} \boldsymbol{H}_{\mathsf{t}} \right)^{\rm{H}}$$ and $\boldsymbol{C}_{\boldsymbol{y}\boldsymbol{y}}$ is the covariance matrix of $\boldsymbol{y}$ given by $$\boldsymbol{C}_{\boldsymbol{y}\boldsymbol{y}} = \mathcal{P} \eta_0^2 \boldsymbol{H}_{\mathsf{r}} \boldsymbol{\Theta} \boldsymbol{H}_{\mathsf{t}} \boldsymbol{H}_{\mathsf{t}}^{\rm{H}} \boldsymbol{\Theta}^{\rm{H}} \boldsymbol{H}_{\mathsf{r}}^{\rm{H}} + \sigma_w^2 \boldsymbol{I}.$$ The solutions provided in and are obtained by setting the first-order derivatives of $\mathcal{J}(\boldsymbol{\Phi},\boldsymbol{\Sigma})$ over $\boldsymbol{\Phi}$ and $\boldsymbol{\Sigma}$ to zero.

### Optimization of $\boldsymbol{\Theta}$ for Given $\mathcal{Q}(\boldsymbol{\Theta})$

Next we minimize $\mathcal{Q}(\boldsymbol{\Theta})$ over $\boldsymbol{\Theta}$. For given $\mathcal{Q}(\boldsymbol{\Theta})$, the optimization problem in reduces to $$\label{OptForGivenTheta}
    \begin{array}{ll}
        \begin{aligned}
            \min_{\boldsymbol{\Theta}} &~ \mathbb{E}_{\boldsymbol{x},\boldsymbol{y}} \left\|\boldsymbol{\Sigma}^{-\frac{1}{2}}(\boldsymbol{x}-\boldsymbol{\Phi} \boldsymbol{y})\right\|_{2}^{2} \\
            \text { s.t. } & \left|\beta_{k,l}\right|=1,~ \forall k \in \mathcal{I}_{Q_{\mathsf{x}}},~ \forall l \in \mathcal{I}_{Q_{\mathsf{y}}}.
        \end{aligned}
    \end{array}$$ By omitting the terms irrelevant to $\boldsymbol{\Theta}$ and defining $$\begin{aligned}
    \boldsymbol{\Lambda} &\triangleq \mathcal{P} \eta_0^2 \left[ \sum_{j=0}^{N_{\mathsf{t}}} \textrm{diag}\left\{ \boldsymbol{h}_{\mathsf{t};j}^{\textrm{H}} \right\} \boldsymbol{H}_{\mathsf{r}}^{\textrm{H}} \boldsymbol{\Phi}^{\textrm{H}} \boldsymbol{\Sigma}^{-1} \boldsymbol{\Phi} \boldsymbol{H}_{\mathsf{r}} \textrm{diag} \left\{ \boldsymbol{h}_{\mathsf{t};j} \right\}, \right] \label{LAMBDA} \\
    \boldsymbol{\alpha} &\triangleq -\mathcal{P} \eta_0 \textrm{diag}\left\{ \boldsymbol{H}_{\mathsf{r}}^{\textrm{H}} \boldsymbol{\Phi}^{\textrm{H}} \boldsymbol{\Sigma}^{-1} \boldsymbol{H}_{\mathsf{t}}^{\textrm{H}} \right\}, \label{ALPHA}\end{aligned}$$ where $\boldsymbol{h}_{\mathsf{t};j}$ denotes the $j$-th column of $\boldsymbol{H}_{\mathsf{t}}$, $j=1,\ldots,N_{\mathsf{t}}$, the optimization problem in is converted to $$\label{QCQP}
    \begin{array}{ll}
        \begin{aligned}
            \min_{\boldsymbol{\Theta}} &~ \boldsymbol{\beta}^{\textrm{H}} \boldsymbol{\Lambda} \boldsymbol{\beta} + \boldsymbol{\alpha}^{\textrm{H}} \boldsymbol{\beta} + \boldsymbol{\beta}^{\textrm{H}} \boldsymbol{\alpha} \\
            \text { s.t. } & \left|\beta_{k,l}\right|=1,~ \forall k \in \mathcal{I}_{Q_{\mathsf{x}}},~ \forall l \in \mathcal{I}_{Q_{\mathsf{y}}}
        \end{aligned}
    \end{array}$$ where $\boldsymbol{\beta}$ The optimization problem in is a quadratically constrained quadratic program (QCQP) and can be solved by the majorization-minimization (MM) algorithm proposed in @pan_multicell_2020.

\[t\] Initialization: Initialize $\boldsymbol{\beta}$ randomly

The iterative algorithm for MI maximization over $\boldsymbol{\Theta}$ is presented in Algorithm \[PBAlgo\]. The algorithm first adopt a randomly initialized $\boldsymbol{\beta}$, and then iteratively solve $\mathcal{Q}(\boldsymbol{\Theta})$ and optimize $\boldsymbol{\Theta}$. The MM algorithm is employed in Lines 5 to 7, where $\lambda_{\max}$ in Line 6 is the largest eigenvalue of $\boldsymbol{\Lambda}$, and $\arg(\cdot)$ in Line 7 returns the argument(s) of the input in an elementwise manner. The stopping criterion 1 in Line 2 is generally set as $\mathcal{Q}(\boldsymbol{\Theta})$ being reduced by less than a predetermined threshold $\epsilon_{\boldsymbol{\Theta}}>0$ in one iteration, or the maximum number of iteration being reached. The stopping criterion 2 in Line 5 is generally set as the objective function in being reduced by less than a predetermined threshold $\epsilon_{\textrm{MM}}>0$ in one iteration, or the maximum number of iteration being reached. Algorithm \[PBAlgo\] is guaranteed to converge to at least a local minimum of the original problem , and finally output the updated $\boldsymbol{\Theta}$.

Solve $\mathcal{A}$ for Fixed $\boldsymbol{\Theta}$ {#SecA}
---------------------------------------------------

Now we consider the MI maximization problem over the Tx and Rx orientations. With fixed passive beamforming matrix $\boldsymbol{\Theta}$, the MI optimization problem is equivalent to $$\label{MIA}
    \begin{array}{ll}
        \begin{aligned}
        \min_{\mathcal{A}} &~ \mathcal{M}(\boldsymbol{x};\boldsymbol{y}) \triangleq -\mathcal{I}(\boldsymbol{x};\boldsymbol{y}) \\
        \text { s.t. } & \psi_{\mathsf{t}} \in [0,\pi], ~ \gamma_{\mathsf{t}} \in [-\pi/2,\pi/2], \\
        & \psi_{\mathsf{t}} \in [0,\pi], ~ \gamma_{\mathsf{t}} \in [-\pi/2,\pi/2].
        \end{aligned}
    \end{array}$$

For the problem in , we find a stationary point of $\mathcal{M}(\boldsymbol{x};\boldsymbol{y})$ with respect to $\mathcal{A}$ via gradient descent (GD). Specifically, let $\boldsymbol{m}\triangleq \left[\gamma_{\mathsf{t}},\psi_{\mathsf{t}},\gamma_{\mathsf{r}},\psi_{\mathsf{r}} \right]$, and denote the $\boldsymbol{m}$ obtained in the previous iteration as $\boldsymbol{m}^{(\mathsf{old})}$. Denote by $\nabla \mathcal{M}(\boldsymbol{x};\boldsymbol{y}) \triangleq \left[\frac{\partial \mathcal{M}(\boldsymbol{x};\boldsymbol{y})}{\partial\gamma_{\mathsf{t}}},\frac{\partial \mathcal{M}(\boldsymbol{x};\boldsymbol{y})}{\partial\psi{\mathsf{t}}},\frac{\partial \mathcal{M}(\boldsymbol{x};\boldsymbol{y})}{\partial\gamma_{\mathsf{r}}},\frac{\partial \mathcal{M}(\boldsymbol{x};\boldsymbol{y})}{\partial\psi{\mathsf{r}}} \right]$ the gradient of $\mathcal{M}(\boldsymbol{x};\boldsymbol{y})$ with respect to $\boldsymbol{m}$. The update of $\boldsymbol{m}$ is given by $$\boldsymbol{m}^{(\mathsf{new})} = \boldsymbol{m}^{(\mathsf{old})} - \epsilon \nabla \mathcal{M}(\boldsymbol{x};\boldsymbol{y})$$ where $\epsilon>0$ is an appropriate step size that can be selected from the backtracking line search to satisfy $$\label{linesearch}
    \mathcal{M}(\boldsymbol{x};\boldsymbol{y})\Big|_{\boldsymbol{m}=\boldsymbol{m}^{(\mathsf{new})}} \leq \mathcal{M}(\boldsymbol{x};\boldsymbol{y})\Big|_{\boldsymbol{m}=\boldsymbol{m}^{(\mathsf{old})}}.$$ The derivation of gradient $\nabla \mathcal{M}(\boldsymbol{x};\boldsymbol{y})$ is given in Appendix \[gradient\]. The GD-based algorithm for MI maximization over $\mathcal{A}$ is presented in Algorithm \[GDS\]. The stopping criterion 3 in Line 2 is generally set as $\mathcal{M}(\boldsymbol{x};\boldsymbol{y})$ being reduced by less than a predetermined threshold $\epsilon_{\mathcal{A}}>0$ in one iteration, or the maximum number of iteration being reached.

\[t\] Initialization: Initialize $\mathcal{A}$ randomly

Overall Algorithm {#SecOverAlgo}
-----------------

\[t\] Initialization: Initialize $\mathcal{A}$ and $\boldsymbol{\Theta}$ randomly

The overall proposed algorithm for MI maximization is summarized in Algorithm \[OAAlgo\]. The input argument $\sigma_w^2$ is the CSCG noise variance; $\mathcal{P}$ is the average transmit power on each Tx antenna; $\eta_0$ is the total path loss; $d_{\mathsf{t}}$ and $d_{\mathsf{r}}$ are the antenna spacings of Tx and Rx, respectively; $N_{\mathsf{t}}$ and $N_{\mathsf{r}}$ are the numbers of antennas of Tx and Rx, respectively; $S_{\mathsf{x}}$ and $S_{\mathsf{y}}$ are the RE spacings in the $x$- and the $y$- directions of the IRS, respectively; $Q_{\mathsf{x}}$ and $Q_{\mathsf{y}}$ are the number of REs in the $x$- and the $y$- directions of the IRS, respectively; $\lambda$ is the wavelength; $D_{\mathsf{t}}$ and $D_{\mathsf{r}}$ are the distances between the centers of Tx and the IRS and between the centers of Rx and the IRS, respectively. The algorithm iteratively calls Algorithms \[PBAlgo\] and \[GDS\] to update $\boldsymbol{\Theta}$ and $\mathcal{A}$, respectively, so as to maximize the MI $\mathcal{I}(\boldsymbol{x};\boldsymbol{y})$. The convergence of Algorithm \[OAAlgo\] is guaranteed since $\mathcal{I}(\boldsymbol{x};\boldsymbol{y})$ increases monotonically in each step. The stopping criterion 4 in Line 2 is generally set as $\mathcal{M}(\boldsymbol{x};\boldsymbol{y})$ being reduced by less than a predetermined threshold $\epsilon_{\textrm{OA}}>0$ in one iteration, or the maximum number of iteration being reached.

Numerical Results
=================

Conclusion
==========

The conclusion goes here.

Proof of Proposition \[propSingleSideDray\] {#proofSingleSideDray}
===========================================

It suffices to prove the FMR on the Tx-IRS side. The FMR on the IRS-Rx side can be proved in a similar manner. By recalling $h_{(k,l),p} = e^{-j\zeta_{(k,l),p}}$ and $\zeta_{(k,l),p} = \frac{2\pi d_{(k,l),p}}{\lambda}$, $\langle \boldsymbol{h}_{\mathsf{t};p_1}, \boldsymbol{h}_{\mathsf{t};p_2} \rangle$ in can be rewritten as

$$\begin{aligned}
        \langle \boldsymbol{h}_{\mathsf{t};p_1}, \boldsymbol{h}_{\mathsf{t};p_2} \rangle &= \sum_{k = -\frac{Q_{\mathsf{x}}-1}{2}}^{\frac{Q_{\mathsf{x}}-1}{2}} \sum_{l = -\frac{Q_{\mathsf{y}}-1}{2}}^{\frac{Q_{\mathsf{y}}-1}{2}} h_{(k,l),p_1} h_{(k,l),p_2}^* \\
        &= \sum_{k = -\frac{Q_{\mathsf{x}}-1}{2}}^{\frac{Q_{\mathsf{x}}-1}{2}} \sum_{l = -\frac{Q_{\mathsf{y}}-1}{2}}^{\frac{Q_{\mathsf{y}}-1}{2}} e^{\frac{j2\pi}{\lambda}\left( d_{(k,l),p_2} - d_{(k,l),p_1} \right)}, \label{IPhtp1htp2}
    \end{aligned}$$

$p_1 \in \mathcal{I}_{N_{\mathsf{t}}},~ p_2 \in \mathcal{I}_{N_{\mathsf{t}}}$. By plugging and in , and defining $$\begin{aligned}
    C_{p_1,p_2} &\triangleq \exp\left\{ \frac{j2\pi}{\lambda}\left( (p_2-p_1)d_t\cos\psi_{\mathsf{t}}+ \frac{(p_2^2-p_1^2)d_t^2\sin^2 \psi_{\mathsf{t}}}{2D_t} \right)\right\}\end{aligned}$$ we have

\[ip\] $$\begin{aligned}
        \langle \boldsymbol{h}_{p_1},\boldsymbol{h}_{p_2} \rangle &= C_{p_1,p_2} \sum_{k = -\frac{Q_{\mathsf{x}}-1}{2}}^{\frac{Q_{\mathsf{x}}-1}{2}} e^{j2\pi (p_1-p_2) C_{\mathsf{t},\mathsf{x}} k} \sum_{l = -\frac{Q_{\mathsf{y}}-1}{2}}^{\frac{Q_{\mathsf{y}}-1}{2}} e^{j2\pi (p_1-p_2) C_{\mathsf{t},\mathsf{y}} l} \label{ip1} \\
        &= C_{p_1,p_2} \frac{\sin(\pi (p_1-p_2) C_{\mathsf{t},\mathsf{x}}) \sin(\pi (p_1-p_2) C_{\mathsf{t},\mathsf{y}})}{\sin (\frac{\pi}{Q_{\mathsf{x}}} (p_1-p_2) C_{\mathsf{t},\mathsf{x}}) \sin (\frac{\pi}{Q_{\mathsf{y}}} (p_1-p_2) C_{\mathsf{t},\mathsf{y}})}, \label{ip2} \\
        &= C_{p_1,p_2} A_{p_1,p_2}^{\mathsf{x}} A_{p_1,p_2}^{\mathsf{y}} \label{ip3},
    \end{aligned}$$

$p_1 \in \mathcal{I}_{N_{\mathsf{t}}},~ p_2 \in \mathcal{I}_{N_{\mathsf{t}}}$, where $A_{p_1,p_2}^{\mathsf{x}} \triangleq \frac{\sin(\pi (p_1-p_2) C_{\mathsf{t},\mathsf{x}})}{\sin (\frac{\pi}{Q_{\mathsf{x}}} (p_1-p_2) C_{\mathsf{t},\mathsf{x}})}$ and $A_{p_1,p_2}^{\mathsf{y}} \triangleq \frac{\sin(\pi (p_1-p_2) C_{\mathsf{t},\mathsf{y}})}{ \sin (\frac{\pi}{Q_{\mathsf{y}}} (p_1-p_2) C_{\mathsf{t},\mathsf{y}})}$, with $C_{\mathsf{t},\mathsf{x}}$ and $C_{\mathsf{r},\mathsf{y}}$ defined in and , respectively. The simplification from to employs the expression of a geometric sum.

From , for $\boldsymbol{H}_{\mathsf{t}}$ to meet the orthogonality requirements in , a necessary and sufficient condition is

\[AxAycond\] $$\begin{aligned}
        A_{p_1,p_2}^{\mathsf{x}} &= Q_{\mathsf{x}} ~ \textup{and}~ A_{p_1,p_2}^{\mathsf{y}}=Q_{\mathsf{y}},~  \forall p_1 = p_2, \label{AxAyconda}\\
        A_{p_1,p_2}^{\mathsf{x}} &= 0 ~ \textrm{or} ~ A_{p_1,p_2}^{\mathsf{y}}=0,~  \forall p_1 \neq p_2 \label{AxAycondb}
    \end{aligned}$$

with $p_1 \in \mathcal{I}_{N_{\mathsf{t}}},~ p_2 \in \mathcal{I}_{N_{\mathsf{t}}}$. When $p_1=p_2$, is naturally satisfied. To meet , a necessary and sufficient condition is $$\label{CtxLim}
    |C_{\mathsf{t},\mathsf{x}}| \in \mathbb{Z}^+ ~ \textup{and} ~ \frac{C_{\mathsf{t},\mathsf{x}}(p_1-p_2)}{Q_{\mathsf{x}}} \notin \mathbb{Z},~ \forall p_1 \neq p_2,~ p_1,p_2 \in \mathcal{I}_{N_{\mathsf{t}}}$$ or $$\label{CtyLim}
    |C_{\mathsf{t},\mathsf{y}}| \in \mathbb{Z}^+ ~ \textup{and} ~ \frac{C_{\mathsf{t},\mathsf{y}}(p_1-p_2)}{Q_{\mathsf{y}}} \notin \mathbb{Z},~ \forall p_1 \neq p_2,~ p_1,p_2 \in \mathcal{I}_{N_{\mathsf{t}}}$$ where $\mathbb{Z}^+$ and $\mathbb{Z}$ are the sets of positive integers and all integers, respectively. We first consider how to meet . By recalling that $C_{\mathsf{t},\mathsf{x}}=\frac{d_{\mathsf{t}}S_{\mathsf{x}}Q_{\mathsf{x}}A_{\mathsf{t},\mathsf{x}}\sin\psi_{\mathsf{t}}\cos(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{x}})}{\lambda D_{\mathsf{t}}}$ is inversely proportional to $D_{\mathsf{t}}$ in , we note that to ensure $|C_{\mathsf{t},\mathsf{x}}| \in \mathbb{Z}^+$, $D_{\mathsf{t}}$ is upper-bounded by $D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}\triangleq \frac{d_{\mathsf{t}}S_{\mathsf{x}}Q_{\mathsf{x}}A_{\mathsf{t},\mathsf{x}}}{\lambda}$. When $D_{\mathsf{t}} \leq D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}$, we can always ensure $|C_{\mathsf{t},\mathsf{x}}|=1$ by letting $\psi_{\mathsf{t}}$ and $\gamma_{\mathsf{t}}$ satisfy . Otherwise, when $D_{\mathsf{t}}> D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}$, $|C_{\mathsf{t},\mathsf{x}}|$ is always less than $1$. Futhermore, since $(p_1-p_2) \in \left\{-N_{\mathsf{t}}+1,\ldots,N_{\mathsf{t}}-1\right\}$, $p_1,p_2 \in \mathcal{I}_{N_{\mathsf{t}}}$, $$|C_{\mathsf{t},\mathsf{x}}|=1 ~\textup{and}~ Q_{\mathsf{x}} \geq N_{\mathsf{t}}$$ is a sufficient condition for $\frac{C_{\mathsf{t},\mathsf{x}}(p_1-p_2)}{Q_{\mathsf{x}}} \notin \mathbb{Z}$. Thus when $Q_{\mathsf{x}} \geq N_{\mathsf{t}}$ and $D_{\mathsf{t}} \leq D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}$, is a sufficient condition for , as well as for the orthogonality and equal-gain requirement in . Similarly, to meet , $D_{\mathsf{t}}$ is upper-bounded by $D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}} \triangleq \frac{d_{\mathsf{t}}S_{\mathsf{y}}Q_{\mathsf{y}}A_{\mathsf{t},\mathsf{y}}}{\lambda}$. When $Q_{\mathsf{y}} \geq N_{\mathsf{t}}$ and $D_{\mathsf{t}} \leq D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}}$, is a sufficient condition for , and also for the orthogonality and equal-gain requirement in .

Based on the above, when $Q_{\mathsf{x}} \geq N_{\mathsf{t}}$ and $Q_{\mathsf{y}} \geq N_{\mathsf{t}}$, the boundary of the FMR of the Tx-IRS channel is given by $\max\{D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}},D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}}\}$. Within this boundary, the orthogonality and equal-gain requirement in can be met by letting $\psi_{\mathsf{t}}$ and $\gamma_{\mathsf{t}}$ satisfy or .

Proof of Proposition \[propGray\] {#proofGray}
=================================

We prove Proposition \[propGray\] based on reflective focusing. Recall that, with reflective focusing, the overall channel matrix $\boldsymbol{H}$ can be expressed by the Hadamard product of two matrices $\boldsymbol{P}$ and $\boldsymbol{Q}$ in , with the elements of $\boldsymbol{P}$ and $\boldsymbol{Q}$, denoted by $P_{q,p}$ and $Q_{q,p}$, given in and , respectively, $p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$. Let $Q_{q,p}^{\mathsf{x}}\triangleq\frac{\sin\left(\pi \left( C_{\mathsf{t},\mathsf{x}}p + C_{\mathsf{r},\mathsf{x}}q \right)\right)}{\sin\left(\frac{\pi}{Q_{\mathsf{x}}} \left( C_{\mathsf{t},\mathsf{x}}p + C_{\mathsf{r},\mathsf{x}}q \right)\right)}$ and $Q_{q,p}^{\mathsf{y}}\triangleq\frac{\sin\left(\pi \left( C_{\mathsf{t},\mathsf{y}}p + C_{\mathsf{r},\mathsf{y}}q \right)\right)}{\sin\left(\frac{\pi}{Q_{\mathsf{y}}} \left( C_{\mathsf{t},\mathsf{y}}p + C_{\mathsf{r},\mathsf{y}}q \right)\right)}$, then we have $Q_{q,p}=Q_{q,p}^{\mathsf{x}}Q_{q,p}^{\mathsf{y}}$, $p \in \mathcal{I}_{N_{\mathsf{t}}}$, $q \in \mathcal{I}_{N_{\mathsf{r}}}$. From and , when $$\begin{aligned}
    \textup{sign}\left(C_{\mathsf{t},\mathsf{x}}C_{\mathsf{r},\mathsf{x}}\right) &= \textup{sign}\left(C_{\mathsf{t},\mathsf{y}}C_{\mathsf{r},\mathsf{y}}\right), \label{CSign} \\
    |C_{\mathsf{t},\mathsf{x}}|&=|C_{\mathsf{r},\mathsf{x}}| = 1, \label{CEquala} \\
    |C_{\mathsf{t},\mathsf{y}}|&=|C_{\mathsf{r},\mathsf{y}}|=C_{\mathsf{y}} \label{CEqualb}\end{aligned}$$ where $C_{\mathsf{y}}$ is an arbitrary constant, we obtain $$\label{pplusq1}
    Q_{q,p}^{\mathsf{x}} = \frac{\sin\left(\pi \left( p + q \right)\right)}{\sin\left(\frac{\pi}{Q_{\mathsf{x}}} \left( p + q \right)\right)}~\textup{and}~Q_{q,p}^{\mathsf{y}} = \frac{\sin\left(\pi C_{\mathsf{y}} \left( p + q \right)\right)}{\sin\left(\frac{\pi C_{\mathsf{y}}}{Q_{\mathsf{y}}} \left(p + q \right)\right)},  ~p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}},$$ or $$\label{pplusq2}
    Q_{q,p}^{\mathsf{x}} = \frac{\sin\left(\pi \left( p - q \right)\right)}{\sin\left(\frac{\pi}{Q_{\mathsf{x}}} \left( p - q \right)\right)}~\textup{and}~Q_{q,p}^{\mathsf{y}} = \frac{\sin\left(\pi C_{\mathsf{y}} \left( p - q \right)\right)}{\sin\left(\frac{\pi C_{\mathsf{y}}}{Q_{\mathsf{y}}} \left(p - q \right)\right)},  ~p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}.$$ Particularly, $Q_{q,p}^{\mathsf{x}}=Q_{\mathsf{x}}$ and $Q_{q,p}^{\mathsf{y}}=Q_{\mathsf{y}}$ for $q=-p$ in , and for $q=p$ in , $p \in \mathcal{I}_{N_{\mathsf{t}}}$, $q \in \mathcal{I}_{N_{\mathsf{r}}}$. Since $p+q$ and $p-q$ are integers, the numerators of $Q_{q,p}^{\mathsf{x}}$ in and are always zeros. Furthermore, when $$\label{NtNrQxQy}
    N_{\mathsf{t}}+N_{\mathsf{r}}-2 < 2Q_{\mathsf{x}},$$ we have $\frac{\left|p+q\right|}{Q_{\mathsf{x}}} < 1$ and $\frac{\left|p+q\right|}{Q_{\mathsf{y}}} < 1$. Then, the denominators of $C_{q,p}^{\mathsf{x}}$ in and are nonzero except for the cases of $p-q=0$ and $p+q=0$, respectively. Based on the above, when , , , and are met, the elements of $\boldsymbol{Q}$ are given by $$\label{Qqppp}
    Q_{q,p}=\left\{\begin{array}{ll}
        Q_{\mathsf{x}}Q_{\mathsf{y}}, & q = -p ~\textup{or}~ q=p \\
        0, & \rm{otherwise}
    \end{array}\right.
    ,~ p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}.$$ Correspondingly, the elements of $\boldsymbol{H}$ in are given by $$\label{tildeHqp}
    h_{q,p}=\left\{\begin{array}{ll}
        \eta_0 Q_{\mathsf{x}} Q_{\mathsf{y}} P_{q,p}, & q = -p ~\textup{or}~ q=p  \\
        0, & \rm{otherwise}
    \end{array}\right.
    ,~ p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}.$$ By noting $|P_{q,p}|=1$, we see that , , , and give a sufficient condition of the orthogonality requirements in . By replacing the subscripts $\mathsf{x}$ by $\mathsf{y}$ and $\mathsf{y}$ by $\mathsf{x}$ in the above results, we see that with $$\begin{aligned}
    &|C_{\mathsf{t},\mathsf{y}}|=|C_{\mathsf{r},\mathsf{y}}| = 1,\\
    &|C_{\mathsf{t},\mathsf{x}}|=|C_{\mathsf{r},\mathsf{x}}|=C_{\mathsf{x}},~\textup{and} \\
    &N_{\mathsf{t}}+N_{\mathsf{r}}-2 < 2Q_{\mathsf{y}}\end{aligned}$$ is also a sufficient condition of the orthogonality requirements in . Thus, to prove Proposition \[propGray\], it suffices to show that for a cascaded LoS system with $N_{\mathsf{t}}+N_{\mathsf{r}}-2 < 2Q_{\mathsf{x}}$, within the region $\mathcal{D}_{\mathsf{c},\mathsf{x}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$ bounded by , , and , , , and can be satisfied by adjusting $\psi_{\mathsf{t}}$, $\gamma_{\mathsf{t}}$, $\psi_{\mathsf{r}}$, and $\gamma_{\mathsf{r}}$.

To this end, we solve the equations given by , , and . For notational simplicity we abbreviate $\bar{\gamma}_{\mathsf{t},\mathsf{x}}(\omega_{\mathsf{t}},\varphi_{\mathsf{t}}))$ by $\bar{\gamma}_{\mathsf{t},\mathsf{x}}$, $\bar{\gamma}_{\mathsf{t},\mathsf{y}}(\omega_{\mathsf{t}},\varphi_{\mathsf{t}}))$ by $\bar{\gamma}_{\mathsf{t},\mathsf{y}}$, $\bar{\gamma}_{\mathsf{r},\mathsf{x}}(\omega_{\mathsf{r}},\varphi_{\mathsf{r}}))$ by $\bar{\gamma}_{\mathsf{r},\mathsf{x}}$, and $\bar{\gamma}_{\mathsf{r},\mathsf{y}}(\omega_{\mathsf{r}},\varphi_{\mathsf{r}}))$ by $\bar{\gamma}_{\mathsf{r},\mathsf{y}}$. we first rewrite as $$\begin{aligned}
    |C_{\mathsf{t},\mathsf{x}}|&=\frac{d_{\mathsf{t}}S_{\mathsf{x}}Q_{\mathsf{x}}A_{\mathsf{t},\mathsf{x}} \sin\psi_{\mathsf{t}}|\cos(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{x}})|}{\lambda D_{\mathsf{t}}}=1 ~\textup{and}~ \label{CtxrxEqual1a} \\
    |C_{\mathsf{r},\mathsf{x}}|&=\frac{d_{\mathsf{r}}S_{\mathsf{x}}Q_{\mathsf{x}}A_{\mathsf{r},\mathsf{x}}\sin\psi_{\mathsf{r}}|\cos(\gamma_{\mathsf{r}}-\bar{\gamma}_{\mathsf{r},\mathsf{x}})|}{\lambda D_{\mathsf{r}}}=1, \label{CtxrxEqual1b}\end{aligned}$$ where $\mathcal{B}_{\mathsf{t}}\triangleq \left\{ \omega_{\mathsf{t}},\varphi_{\mathsf{t}} \right\}$, $\mathcal{B}_{\mathsf{r}}\triangleq \left\{ \omega_{\mathsf{r}},\varphi_{\mathsf{r}} \right\}$. To meet and , $D_{\mathsf{t}}$ and $D_{\mathsf{r}}$ are naturally bounded by $D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}} \triangleq \frac{d_{\mathsf{t}} S_{\mathsf{x}}Q_{\mathsf{x}}A_{\mathsf{t},\mathsf{x}}}{\lambda}$ and $D_{\mathsf{t},\mathsf{y}}^{\mathsf{R}} \triangleq \frac{d_{\mathsf{t}} S_{\mathsf{y}}Q_{\mathsf{y}}A_{\mathsf{t},\mathsf{y}}}{\lambda}$, respectively. Next, can be rewritten as $$\label{signtransform}
    \textup{sign}\left(\frac{C_{\mathsf{t},\mathsf{x}}}{C_{\mathsf{t},\mathsf{y}}}\right) = \textup{sign}\left(\frac{C_{\mathsf{r},\mathsf{x}}}{C_{\mathsf{r},\mathsf{y}}}\right).$$ By combining , , and , we obtain $$\label{CequalNosign}
    \frac{C_{\mathsf{t},\mathsf{x}}}{C_{\mathsf{t},\mathsf{y}}} = \frac{C_{\mathsf{r},\mathsf{x}}}{C_{\mathsf{r},\mathsf{y}}}.$$ By plugging and into , to meet , , and , a necessary condition is $\gamma_{\mathsf{t}}$ and $\gamma_{\mathsf{r}}$ to satisfy

\[CosRatio\] $$\begin{aligned}
                                        \frac{ \cos(\gamma_{\mathsf{r}}-\bar{\gamma}_{\mathsf{r},\mathsf{y}})}{\cos(\gamma_{\mathsf{r}}-\bar{\gamma}_{\mathsf{r},\mathsf{x}})} &= \frac{A_{\mathsf{t},\mathsf{y}}A_{\mathsf{r},\mathsf{x}}\cos(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{y}})}{A_{\mathsf{t},\mathsf{x}}A_{\mathsf{r},\mathsf{y}} \cos(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{x}})} \\
        & = \frac{A_{\mathsf{t},\mathsf{y}}A_{\mathsf{r},\mathsf{x}}\left( \cos(\bar{\gamma}_{\mathsf{t},\mathsf{x}}-\bar{\gamma}_{\mathsf{t},\mathsf{y}}) - \sin(\bar{\gamma}_{\mathsf{t},\mathsf{x}}-\bar{\gamma}_{\mathsf{t},\mathsf{y}}) \tan(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{x}}) \right)}{A_{\mathsf{t},\mathsf{x}}A_{\mathsf{r},\mathsf{y}} }.
    \end{aligned}$$

Let $$\label{Gx}
    G_{\mathsf{x}}(\gamma_{\mathsf{t}}) \triangleq  \frac{A_{\mathsf{t},\mathsf{y}}A_{\mathsf{r},\mathsf{x}}\left( \cos(\bar{\gamma}_{\mathsf{t},\mathsf{x}}-\bar{\gamma}_{\mathsf{t},\mathsf{y}}) - \sin(\bar{\gamma}_{\mathsf{t},\mathsf{x}}-\bar{\gamma}_{\mathsf{t},\mathsf{y}}) \tan(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{x}}) \right)}{A_{\mathsf{t},\mathsf{x}}A_{\mathsf{r},\mathsf{y}} },$$ then can be rewritten as $$\label{tanG}
    \tan \gamma_{\mathsf{r}} = \frac{\cos \bar{\gamma}_{\mathsf{r},\mathsf{y}}-G_{\mathsf{x}}(\gamma_{\mathsf{t}})\cos \bar{\gamma}_{\mathsf{r},\mathsf{x}}}{G_{\mathsf{x}}(\gamma_{\mathsf{t}})\sin \bar{\gamma}_{\mathsf{r},\mathsf{x}}-\sin \bar{\gamma}_{\mathsf{r},\mathsf{y}}}.$$ Rewrite , , , and as

$$\begin{aligned}
        &|C_{\mathsf{t},\mathsf{x}}|=\frac{d_{\mathsf{t}}S_{\mathsf{x}}Q_{\mathsf{x}}A_{\mathsf{t},\mathsf{x}} \sin\psi_{\mathsf{t}}|\cos(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{x}})|}{\lambda D_{\mathsf{t}}}=1, \label{ProofOritx}\\
        &|C_{\mathsf{r},\mathsf{x}}|=\frac{d_{\mathsf{r}}S_{\mathsf{x}}Q_{\mathsf{x}}A_{\mathsf{r},\mathsf{x}}\sin\psi_{\mathsf{r}}|\cos(\gamma_{\mathsf{r}}-\bar{\gamma}_{\mathsf{r},\mathsf{x}})|}{\lambda D_{\mathsf{r}}}=1, \label{ProofOrity} \\
        &\frac{ \cos(\gamma_{\mathsf{r}}-\bar{\gamma}_{\mathsf{r},\mathsf{y}})}{\cos(\gamma_{\mathsf{r}}-\bar{\gamma}_{\mathsf{r},\mathsf{x}})} = \frac{A_{\mathsf{t},\mathsf{y}}A_{\mathsf{r},\mathsf{x}}\cos(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{y}})}{A_{\mathsf{t},\mathsf{x}}A_{\mathsf{r},\mathsf{y}} \cos(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{x}})} \label{ProofOrigamma}
    \end{aligned}$$

Note that the results derived in is irrelevant to $\psi_{\mathsf{t}}$ and $\psi_{\mathsf{r}}$, thus from and , at the boundary of $\mathcal{D}_{\mathsf{c},\mathsf{x}}(\varphi_{\mathsf{t}},\omega_{\mathsf{t}},\varphi_{\mathsf{r}},\omega_{\mathsf{r}})$, there must be $$\label{psiboundary}
    \sin \psi_{\mathsf{t}}=1 ~ \textup{and} ~ \sin \psi_{\mathsf{r}}=1,$$ i.e., $\psi_{\mathsf{t}}=\psi_{\mathsf{r}}=\frac{\pi}{2}$. By plugging into , we have $$|\cos(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{x}})| = \frac{D_{\mathsf{t}}}{D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}},$$ and thus $$\label{TanSign}
    \tan(\gamma_{\mathsf{t}}-\bar{\gamma}_{\mathsf{t},\mathsf{x}}) = \sqrt{\left(\frac{D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}}{D_{\mathsf{t}}}\right)^2-1}~ \textup{or} ~ -\sqrt{\left(\frac{D_{\mathsf{t},\mathsf{x}}^{\mathsf{R}}}{D_{\mathsf{t}}}\right)^2-1}.$$ By plugging into , we obtain $G_{\mathsf{x}}^+$ and $G_{\mathsf{x}}^-$ defined in and , respectively. Then by plugging $G_{\mathsf{x}}=G_{\mathsf{x}}^+$ and $G_{\mathsf{x}}=G_{\mathsf{x}}^-$ into , we obtain the restriction for $\gamma_{\mathsf{r}}$ shown in .

Proof of Proposition \[propHighSNR\] and \[propLowSNR\] {#proofHighLowSNR}
=======================================================

Maximizing the RHS of is equivalent to $$\label{opt}
\begin{aligned}
\max_{\{\mu_{\mathsf{t};i}\},\{\mu_{\mathsf{r};i}\}} \quad & \sum_{i=1}^{N_{\mathsf{r}}} \log(1+\rho \mu_{\mathsf{r};i}^2 \mu_{\mathsf{t};i}^2),\\
\textrm{s.t.} \quad & \sum_{i=1}^{N_{\mathsf{r}}} \mu_{\mathsf{r};i}^2 = N_{\mathsf{r}}Q_{\mathsf{x}}Q_{\mathsf{y}},\\
  &\sum_{i=1}^{N_{\mathsf{r}}} \mu_{\mathsf{t};i}^2 = N_{\mathsf{t}} Q_{\mathsf{x}} Q_{\mathsf{y}}. \\
\end{aligned}$$ Under different SNR levels, the solution to has different forms.

### High SNR

When $\rho \to \infty$, we have $$\begin{aligned}
    \arg \max_{\{\mu_{\mathsf{t};i}\},\{\mu_{\mathsf{r};i}\}} \quad \sum_{i=1}^{N_{\mathsf{r}}} \log(1+ \rho \mu_{\mathsf{r};i}^2 \mu_{\mathsf{t};i}^2) & = \arg \max_{\{\mu_{\mathsf{t};i}\},\{\mu_{\mathsf{r};i}\}} \quad \log\left(\prod_{i=1}^{N_{\mathsf{r}}} \mu_{\mathsf{r};i}^2 \mu_{\mathsf{t};i}^2\right) \\
    &= \arg \max_{\{\mu_{\mathsf{t};i}\},\{\mu_{\mathsf{r};i}\}} \quad \prod_{i=1}^{N_{\mathsf{r}}} \mu_{\mathsf{r};i}^2 \prod_{i=1}^{N_{\mathsf{r}}} \mu_{\mathsf{t};i}^2 \\
    &= \left\{\arg \max_{\{\mu_{\mathsf{r};i}\}} \quad \prod_{i=1}^{N_{\mathsf{r}}} \mu_{\mathsf{r};i}^2,~ \arg \max_{\{\mu_{\mathsf{t};i}\}} \quad \prod_{i=1}^{N_{\mathsf{r}}} \mu_{\mathsf{t};i}^2 \right\}.\end{aligned}$$ The optimal solution of is $$\begin{aligned}
    \mu_{\mathsf{t};1}^2 &= \cdots = \mu_{\mathsf{t};N_{\mathsf{r}}}^2 = \frac{N_{\mathsf{t}}}{N_{\mathsf{r}}} Q_{\mathsf{x}}Q_{\mathsf{y}}, \\
    \mu_{\mathsf{r};1}^2 &= \cdots = \mu_{\mathsf{r};N_{\mathsf{r}}}^2 = Q_{\mathsf{x}}Q_{\mathsf{y}}.\end{aligned}$$ At this time, columns of $\boldsymbol{H}_{\mathsf{t}}$ are orthogonal, and rows of $\boldsymbol{H}_{\mathsf{r}}$ are orthogonal.

### Low SNR

When $\rho \to 0$, we have $$\begin{aligned}
   \sum_{i=1}^N \log(1+\rho \mu_{\mathsf{r};j}^2 \mu_{\mathsf{t};i}^2) & = \rho \sum_{i=1}^N \mu_{\mathsf{r};j}^2 \mu_{\mathsf{t};i}^2 \\
   & \leq \rho \sum_{i=1}^N \mu_{\mathsf{r};j}^2 \sum_{i=1}^N \mu_{\mathsf{t};i}^2 \label{sigmaleq} \\
   & = \rho N_{\mathsf{t}} N_{\mathsf{r}} Q_{\mathsf{x}}^2 Q_{\mathsf{y}}^2\end{aligned}$$ The equality in holds if and only if $\mu_{\mathsf{t};1}^2 = N_{\mathsf{t}}Q_{\mathsf{x}}Q_{\mathsf{t}}$, and $\mu_{\mathsf{r};1}^2 = N_{\mathsf{r}}Q_{\mathsf{x}}Q_{\mathsf{y}}$. At this time, $\boldsymbol{H}_{\mathsf{t}}$ and $\boldsymbol{H}_{\mathsf{r}}$ are rank one.

Gradient Calculation {#gradient}
====================

Based on and , the objective function $\mathcal{M}(\boldsymbol{x};\boldsymbol{y})$ in can be rewritten as

$$\begin{aligned}
        \mathcal{M}(\boldsymbol{x};\boldsymbol{y}) &= - \mathcal{I}(\boldsymbol{x};\boldsymbol{y})  \\
        &= -\log \det\left( \frac{\mathcal{P}}{\sigma_w^2} \boldsymbol{H}^{\textrm{H}} \boldsymbol{H} + \boldsymbol{I} \right) \\
        & = -\log \det\left( \frac{\mathcal{P}}{\sigma_w^2} \boldsymbol{H}_{\mathsf{t}}^{\mathsf{H}} \boldsymbol{\Theta}^{\mathsf{H}} \boldsymbol{H}_{\mathsf{r}}^{\mathsf{H}} \boldsymbol{H}_{\mathsf{r}} \boldsymbol{\Theta} \boldsymbol{H}_{\mathsf{t}} + \boldsymbol{I} \right) \\
        & = -\log \det(\Delta)
    \end{aligned}$$

where $\Delta \triangleq \frac{\mathcal{P}}{\sigma_w^2} \boldsymbol{H}_{\mathsf{t}}^{\mathsf{H}} \boldsymbol{\Theta}^{\mathsf{H}} \boldsymbol{H}_{\mathsf{r}}^{\mathsf{H}} \boldsymbol{H}_{\mathsf{r}} \boldsymbol{\Theta} \boldsymbol{H}_{\mathsf{t}} + \boldsymbol{I}$. Based on the chain rule of derivative and the Jacobi's formula, for $\varrho \in \mathcal{A}$, the partial derivative of $\mathcal{M}(\boldsymbol{x};\boldsymbol{y})$ with respect to $\varrho$ is given by $$\label{Mvarrho}
    \frac{\partial \mathcal{M}(\boldsymbol{x};\boldsymbol{y})}{\partial \varrho} = -\textrm{Tr}\left[ \Delta^{-1} \frac{\partial \Delta}{\partial \varrho} \right].$$ The partial derivatives of $\Delta$ with respect to $\gamma_{\mathsf{t}}$, $\psi_{\mathsf{t}}$, $\gamma_{\mathsf{r}}$, and $\psi_{\mathsf{r}}$ are given by $$\begin{aligned}
    \frac{\partial \Delta}{\partial \gamma_{\mathsf{t}}} &= - \frac{\mathcal{P}}{\sigma_w^2} \frac{\partial \boldsymbol{H}_{\mathsf{t}}^{\mathsf{H}}}{\gamma_{\mathsf{t}}} \boldsymbol{\Theta}^{\mathsf{H}} \boldsymbol{H}_{\mathsf{r}}^{\mathsf{H}} \boldsymbol{H}_{\mathsf{r}} \boldsymbol{\Theta} \boldsymbol{H}_{\mathsf{t}} - \frac{\mathcal{P}}{\sigma_w^2} \boldsymbol{H}_{\mathsf{t}}^{\mathsf{H}} \boldsymbol{\Theta}^{\mathsf{H}} \boldsymbol{H}_{\mathsf{r}}^{\mathsf{H}} \boldsymbol{H}_{\mathsf{r}} \boldsymbol{\Theta} \frac{\partial \boldsymbol{H}_{\mathsf{t}}}{\partial \gamma_{\mathsf{t}}},  \\
    \frac{\partial \Delta}{\partial \psi_{\mathsf{t}}} &= - \frac{\mathcal{P}}{\sigma_w^2} \frac{\partial \boldsymbol{H}_{\mathsf{t}}^{\mathsf{H}}}{\psi_{\mathsf{t}}} \boldsymbol{\Theta}^{\mathsf{H}} \boldsymbol{H}_{\mathsf{r}}^{\mathsf{H}} \boldsymbol{H}_{\mathsf{r}} \boldsymbol{\Theta} \boldsymbol{H}_{\mathsf{t}} - \frac{\mathcal{P}}{\sigma_w^2} \boldsymbol{H}_{\mathsf{t}}^{\mathsf{H}} \boldsymbol{\Theta}^{\mathsf{H}} \boldsymbol{H}_{\mathsf{r}}^{\mathsf{H}} \boldsymbol{H}_{\mathsf{r}} \boldsymbol{\Theta} \frac{\partial \boldsymbol{H}_{\mathsf{t}}}{\partial \psi_{\mathsf{t}}},  \\
    \frac{\partial \Delta}{\partial \gamma_{\mathsf{r}}} &= - \frac{\mathcal{P}}{\sigma_w^2} \boldsymbol{H}_{\mathsf{t}} \boldsymbol{\Theta}^{\mathsf{H}} \frac{\partial \boldsymbol{H}_{\mathsf{r}}^{\mathsf{H}}}{\partial \gamma_{\mathsf{r}}} \boldsymbol{H}_{\mathsf{r}} \boldsymbol{\Theta} \boldsymbol{H}_{\mathsf{t}} - \frac{\mathcal{P}}{\sigma_w^2} \boldsymbol{H}_{\mathsf{t}}^{\mathsf{H}} \boldsymbol{\Theta}^{\mathsf{H}} \boldsymbol{H}_{\mathsf{r}}^{\mathsf{H}} \frac{\partial\boldsymbol{H}_{\mathsf{r}}}{\partial \gamma_{\mathsf{r}}} \boldsymbol{\Theta}  \boldsymbol{H}_{\mathsf{t}},  \\
    \frac{\partial \Delta}{\partial \psi_{\mathsf{r}}} &= - \frac{\mathcal{P}}{\sigma_w^2} \boldsymbol{H}_{\mathsf{t}} \boldsymbol{\Theta}^{\mathsf{H}} \frac{\partial \boldsymbol{H}_{\mathsf{r}}^{\mathsf{H}}}{\partial \psi_{\mathsf{r}}} \boldsymbol{H}_{\mathsf{r}} \boldsymbol{\Theta} \boldsymbol{H}_{\mathsf{t}} - \frac{\mathcal{P}}{\sigma_w^2} \boldsymbol{H}_{\mathsf{t}}^{\mathsf{H}} \boldsymbol{\Theta}^{\mathsf{H}} \boldsymbol{H}_{\mathsf{r}}^{\mathsf{H}} \frac{\partial\boldsymbol{H}_{\mathsf{r}}}{\partial \psi_{\mathsf{r}}} \boldsymbol{\Theta}  \boldsymbol{H}_{\mathsf{t}},\end{aligned}$$ respectively. Based on and , elements of $\frac{\partial \boldsymbol{H}_{\mathsf{t}}}{\partial \gamma_{\mathsf{t}}}$, $\frac{\partial \boldsymbol{H}_{\mathsf{t}}}{\partial \psi_{\mathsf{t}}}$, $\frac{\partial \boldsymbol{H}_{\mathsf{r}}}{\partial \gamma_{\mathsf{r}}}$, and $\frac{\partial \boldsymbol{H}_{\mathsf{r}}}{\partial \psi_{\mathsf{r}}}$ are given by

= -j { -j ( + + ) } ( + ),

= -j { -j ( + + ) } ( + - ),

= -j { -j ( + + ) } ( + ),

\[phqklppsir\] = -j { -j ( + + ) } ( + - ),

respectively, $k \in \mathcal{I}_{Q_{\mathsf{x}}},~ l \in \mathcal{I}_{Q_{\mathsf{y}}},~ p \in \mathcal{I}_{N_{\mathsf{t}}},~ q \in \mathcal{I}_{N_{\mathsf{r}}}$. Based on -, we calculate the partial derivative of $\mathcal{M}(\boldsymbol{x};\boldsymbol{y})$ with respect to $\gamma_{\mathsf{t}}$, $\psi_{\mathsf{t}}$, $\gamma_{\mathsf{r}}$, and $\psi_{\mathsf{r}}$, and construct the gradient of $\mathcal{M}(\boldsymbol{x};\boldsymbol{y})$ with respect to $\boldsymbol{m}$ as $$\label{finalgrad}
    \nabla \mathcal{M}(\boldsymbol{x};\boldsymbol{y}) = \left[\frac{\partial \mathcal{M}(\boldsymbol{x};\boldsymbol{y})}{\partial\gamma_{\mathsf{t}}},\frac{\partial \mathcal{M}(\boldsymbol{x};\boldsymbol{y})}{\partial\psi{\mathsf{t}}},\frac{\partial \mathcal{M}(\boldsymbol{x};\boldsymbol{y})}{\partial\gamma_{\mathsf{r}}},\frac{\partial \mathcal{M}(\boldsymbol{x};\boldsymbol{y})}{\partial\psi{\mathsf{r}}} \right].$$

<span>Michael Shell</span> Biography text here.

<span>John Doe</span> Biography text here.

<span>Jane Doe</span> Biography text here.

[^1]: M. Shell was with the Department of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, 30332 USA e-mail: (see http://www.michaelshell.org/contact.html).

[^2]: J. Doe and J. Doe are with Anonymous University.

[^3]: Manuscript received April 19, 2005; revised August 26, 2015.

{'source': 'arXiv_out/2109'}