# Analysis and Optimization of A Double-IRS Cooperatively Assisted System with
A Quasi-Static Phase Shift Design

Gengfa Ding, Feng Yang, Lianghui Ding, and Ying Cui The authors are with the
Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai
200240, China. The paper will be presented in part in SPAWC 2022 [1].

###### Abstract

The analysis and optimization of single intelligent reflecting surface
(IRS)-assisted systems have been extensively studied, whereas little is known
regarding multiple-IRS-assisted systems. This paper investigates the analysis
and optimization of a double-IRS cooperatively assisted downlink system
(D-IRS-C), where a multi-antenna base station (BS) serves a single-antenna
user with the help of two multi-element IRSs, connected by an inter-IRS
channel. The channel between any two nodes is modeled with Rician fading. The
BS adopts the instantaneous CSI-adaptive maximum-ratio transmission (MRT)
beamformer, and the two IRSs adopt a cooperative quasi-static phase shift
design. The goal is to maximize the average achievable rate, which can be
reflected by the average channel power of the equivalent channel between the
BS and user at low channel estimation and phase adjustment costs and
computational complexity. First, we obtain tractable expressions of the
average channel power of the equivalent channel in the general (Rician
factor), pure line of sight (LoS), and pure non-line of sight (NLoS) regimes,
respectively. Then, we jointly optimize the phase shifts of the two IRSs to
maximize the average channel power of the equivalent channel in these regimes.
The optimization problems are challenging non-convex problems. We obtain
globally optimal closed-form solutions for some cases and propose
computationally efficient iterative algorithms to obtain stationary points for
the other cases. Next, we compare the computational complexity for optimizing
the phase shifts and the optimal average channel power of D-IRS-C with those
of a counterpart double-IRS non-cooperatively assisted system (D-IRS-NC) and a
counterpart single-IRS-assisted system (S-IRS) at a large number of reflecting
elements in the three regimes. Finally, we numerically demonstrate notable
gains of the proposed solutions over the existing solutions at different
system parameters. To our knowledge, this is the first work that optimizes the
quasi-static phase shift design of D-IRS-C and characterizes its advantages
over the optimal quasi-static phase shift design of the counterpart D-IRS-NC
and S-IRS.

###### Index Terms:

Intelligent reflecting surface (IRS), double IRSs, cooperation, quasi-static
phase shift design, coordinate descent, optimization.

##  I Introduction

Stringent requirements for future wireless networks, such as ultra-high data
rate and energy efficiency, cannot be fully achieved with the existing
wireless communication technologies. Intelligent reflecting surface (IRS),
which consists of nearly passive, low-cost, reflecting elements with
reconfigurable parameters, has been recently recognized as a promising
solution for improving spectrum and energy efficiency [2]. There have been
extensive studies on IRS. In what follows, we restrict our attention to the
optimization of base station (BS) beamforming and IRS phase shifts for an IRS-
assisted single-cell network. In the existing works on optimal designs for
IRS-assisted systems, BS beamforming designs are usually adaptive to
instantaneous channel state information (CSI), whereas the phase shift designs
can be classified into instantaneous CSI-adaptive phase shift designs (which
adapt to instantaneous CSI) [3, 4, 5, 6, 7, 8] and quasi-static phase shift
designs (which adapt to CSI statistics and do not change over time slots
during a certain period) [9, 10, 11, 12, 13, 14, 15]. Notice that a quasi-
static (also termed statistical) phase shift design yields a low phase
adjustment cost at some performance sacrifice compared with an instantaneous
CSI-adaptive phase shift design. Considering the practical implementation
issue, a quasi-static phase shift design may be more valuable [9, 10, 11, 12,
13, 14, 15].

Early research on IRS investigates the optimal design for an IRS-assisted
system with a single IRS. For example, in [3, 4, 5, 6, 10, 11, 9, 12, 13], the
authors optimize the BS beamformer and IRS phase shifts to minimize the
transmit power [3], outage probability [12], and average transmit power [13]
and maximize the weighted sum rate [4], secrecy rate [5], energy efficiency
[6], and ergodic rate [9, 10, 11]. Specifically, [3, 4, 5, 6] adopt
instantaneous CSI-adaptive phase shift designs, whereas [9, 10, 11, 12, 13]
consider quasi-static phase shift designs.

Later study regarding IRS concentrates on the optimal design for an IRS-
assisted system with more than one IRS. For instance, in [7, 8, 14, 15], the
authors optimize the BS beamformer and IRS phase shifts to minimize the outage
probability [14] and maximize the sum secrecy rate [7], received signal power
[8], and ergodic achievable rate [15]. In particular, [7, 8] adopt
instantaneous CSI-adaptive phase shift designs, whereas [14, 15] consider
quasi-static phase shift designs. Notice that the abovementioned works on
multi-IRS-assisted systems [7, 8, 14, 15] ignore channels between any two
IRSs, referred to as inter-IRS channels, and hence cannot capture their
interaction. To achieve the full potential of multi-IRS-assisted systems, [16,
17, 18, 19, 20, 21, 22, 23] explicitly model inter-IRS channels in double-IRS-
assisted systems [16, 17, 18, 19, 20, 21, 22] and multi-IRS-assisted systems
[23] and consider IRS cooperation. Specifically, in [16, 17], the authors
propose cascaded channel estimation methods for double-IRS cooperatively
assisted systems. In [18, 19], the authors show that the power gain with
respect to (w.r.t.) the total number of reflecting elements of a double-IRS-
assisted system is higher in order than that of the counterpart single-IRS-
assisted system. In [20, 21, 22, 23], the authors optimize the BS beamformer
and IRS phase shifts to maximize the signal-to-interference-plus-noise ratio
(SINR) [20], capacity [21], secrecy rate [22], and received signal power [23].

The existing works on double-IRS cooperatively assisted systems [18, 19, 16,
17, 20, 21, 22, 23] have two main limitations. Firstly, [18, 19, 16, 17, 20,
21, 22, 23] simply assume that the direct channel between the BS and user is
entirely blocked for tractability. Secondly, [18, 19, 16, 17, 20, 21, 22, 23]
solely consider instantaneous CSI-adaptive phase shift designs, which incur
higher channel estimation and phase adjustment costs and computational
complexities. Therefore, the analysis and optimization of the double-IRS
cooperatively assisted systems with a more general channel model (e.g.,
capturing the direct channel) and a more cost-effective phase shift design
remain open.

In this paper, we shall shed some light on the above issue. Specifically, we
consider a double-IRS cooperatively assisted system (D-IRS-C), where a multi-
antenna BS serves a single-antenna user with the help of two multi-element
IRSs connected by an inter-IRS channel. The antennas at the BS and the
reflecting elements at the IRSs are arranged in uniform rectangular arrays
(URAs). The channel between any two nodes is modeled with Rician fading. In
particular, the line of sight (LoS) components do not change during the
considered time, and the non-line of sight (NLoS) components vary from time
slot to time slot. The BS adopts the instantaneous CSI-adaptive maximum-ratio
transmission (MRT) beamformer. The two IRSs adopt a cooperative quasi-static
phase shift design to improve the average achievable rate at low channel
estimation and phase adjustment costs and computational complexity. As the
average achievable rate can be approximately reflected by the average channel
power of the equivalent channel between the BS and user with a negligible
approximation error, this paper focuses on the analysis and optimization of
the average channel power111If not specified otherwise, the average channel
power means the average channel power of the equivalent channel between the BS
and user. rather than the average achievable rate. The main contributions of
this paper are summarized as follows.

  * •

Analysis of Average Channel Power: First, we characterize the influences of
the phase shifts of the two IRSs on the average channel power and divide the
channel conditions into four cases accordingly. Then, we obtain the tractable
expressions of the average channel power in the general (Rician factor), pure
LoS, and pure NLoS regimes, respectively.

  * •

Optimization of Average Channel Power: First, we jointly optimize the phase
shifts of the two IRSs to maximize the average channel power in the general
and pure LoS regimes, respectively. The corresponding optimization problems
are challenging non-convex problems. We obtain globally optimal closed-form
solutions for some cases and propose computationally efficient iterative
algorithms to obtain stationary points for the other cases based on the
coordinate descent (CD) and block coordinate descent (BCD) methods. Then, in
each regime, we characterize the optimal average channel power when the total
number of reflecting elements is large.

  * •

Comparison with Counterpart IRS-Assisted Systems: First, we analyze and
optimize the average channel powers of a counterpart double-IRS non-
cooperatively assisted system (D-IRS-NC) and a counterpart single-IRS-assisted
system (S-IRS). Then, we compare the computational complexities for
calculating the quasi-static phase shift designs and optimal average channel
powers of D-IRS-C, D-IRS-NC, and S-IRS. Specifically, we show that D-IRS-C can
achieve a better performance and computational complexity tradeoff than D-IRS-
NC and S-IRS when the total number of reflecting elements is sufficiently
large.

  * •

Numerical Results: We numerically demonstrate notable gains of the proposed
solutions over the existing quasi-static phase shift designs. Furthermore, we
numerically show that for D-IRS-C, the proposed quasi-static phase shift
design is more desirable than the respective instantaneous CSI-adaptive phase
shift design as long as the LoS components are sufficiently large.

Notation: Boldface lower-case letters (e.g., $\mathbf{x}$), boldface upper-
case letters (e.g., $\mathbf{X}$), non-boldface letters (e.g., $x$), and
calligraphic upper-case letters (e.g., $\mathcal{X}$) denote vectors,
matrices, scalars, and sets, respectively. $\mathbf{X}^{H}$, $\mathbf{X}^{T}$,
$\mathrm{tr}(\mathbf{X})$, and $\|\mathbf{X}\|_{F}$ denote the conjugate
transpose, transpose, trace, and Frobenius norm of a matrix, respectively.
$\mathrm{vec}(\mathbf{X})$ denotes the vectorization of a matrix.
$\mathrm{diag}(\mathbf{x})$ denotes a square diagonal matrix with the elements
in $\mathbf{x}$ on its main diagonal. $\mathrm{Diag}(\mathbf{X})$ denotes the
leading diagonal of matrix $\mathbf{X}$. ${\left\|\mathbf{x}\right\|_{2}}$
denotes the Euclidean norm of a vector. $\mathfrak{R}\\{x\\}$ and
$\left|x\right|$ denote the real part and modulus of a complex number,
respectively. $\angle(\mathbf{x})$ denotes the argument of each element of
$\mathbf{x}$. $\mathbb{E}\left[\cdot\right]$ denotes the expectation w.r.t.
all random variables in the brackets. $\otimes$ denotes the kronecker product.
$\mathbb{C}^{a\times b}$ and $\mathbb{R}^{a\times b}$ denote the space of
$a\times b$ complex-valued and real-valued matrices, respectively.
$\mathcal{CN}(\mu,\sigma^{2})$ denotes a circularly symmetric complex Gaussian
random variable with mean $\mu$ and variance $\sigma^{2}$. $\mathbf{1}_{T}$
and $\mathbf{0}_{T}$ denote the $T$ dimentional vectors with all elements
being 1 and 0, respectively. $X\overset{d}{\sim}Y$ represents that random
variables $X$ and $Y$ follow the same distribution.

##  II System Model

As shown in Fig. 1, we consider a double-IRS cooperatively assisted (downlink)
system (D-IRS-C), where a multi-antenna BS, represented by $S$, communicates
to a single-antenna user, represented by $U$, with the help of two multi-
element IRSs, indexed by 1 and 2, which are connected by an inter-IRS
channel.222The proposed framework can be extended to an IRS cooperatively
assisted system with multiple IRSs and users. In this paper, we consider two
IRSs and one user to obtain first-order design insights. We assume that IRS 1
and IRS 2 are placed closer to the BS and user, respectively. We consider four
channels, i.e., the direct channel $SU$, cascaded channels $(S1,1U)$,
$(S2,2U)$, and $(S1,12,2U)$.333As in [20, 21], we ignore the cascaded channel
$(S2,21,1U)$ and the cascaded channels passing each IRS more than once, e.g.,
the cascaded channel $(S1,12,21,1U)$, as they have much larger path loss. Note
that [20, 21] also ignore the direct channel $SU$ for simplicity. The
locations of the BS and IRSs are fixed and known, and the user’s location is
static during a certain period and known.444The user’s location can be
estimated using standard localization methods such as time difference of
arrival (TDOA) techniques [24]. The BS is equipped with a URA of $M_{S}\times
N_{S}$ antennas. Each IRS $l\in\mathcal{L}$ is equipped with a URA of
$M_{l}\times N_{l}$ reflecting elements, where
${\mathcal{L}}\triangleq\\{1,2\\}$ denotes the set of IRS indices.555As in
[18, 17, 20, 21], the optimization of the element allocation between the two
IRSs and the IRSs’ locations, which can be approximately tackled using
exhaustive search or heuristic methods [25], are out of the scope of this
paper. Let $T_{i}\triangleq M_{i}N_{i}$, $i=S,1,2$, denote the numbers of the
BS’s antennas and IRSs’ reflecting elements. Let $T\triangleq T_{1}+T_{2}$
denote the total number of reflecting elements in the system. Let
$\mathcal{M}_{i}\triangleq\\{{1,2,...,M_{i}}\\}$,
$\mathcal{N}_{i}\triangleq\\{{1,2,...,N_{i}}\\}$, and
$\mathcal{T}_{i}\triangleq\\{{1,2,...,T_{i}}\\}$ denote the corresponding sets
of indices. The phase shifts of each IRS’s reflecting elements can be
determined by a smart controller attached to it. The BS communicates to the
two IRS controllers to configure the two IRSs’ phase shifts via separate
reliable wireless links so that both IRSs jointly assist the downlink
transmission from the BS to the user.

Figure 1: Double-IRS cooperatively assisted system (D-IRS-C).

We consider a narrow-band system and adopt the block-fading model for small-
scale fading. Let $\mathbf{H}_{Sl}\in\mathbb{C}^{T_{l}\times T_{S}}$,
$\mathbf{H}_{12}\in\mathbb{C}^{T_{2}\times T_{1}}$,
$\mathbf{h}_{SU}^{H}\in\mathbb{C}^{1\times T_{S}}$, and
$\mathbf{h}_{lU}^{H}\in\mathbb{C}^{1\times T_{l}}$ represent the channel
matrix between the BS and IRS $l\in\mathcal{L}$, the channel matrix between
IRS 1 and IRS 2, the channel vector between the BS and user, and the channel
vector between IRS $l\in\mathcal{L}$ and the user, respectively. Notice that
in contrast with the existing works on double-IRS cooperatively assisted
systems [18, 19, 16, 17, 20, 21, 22], we consider the direct channel between
the BS and user. As the IRSs are usually far above the ground where scattering
is relatively weak, we adopt the Rician fading model666The Rician fading model
is more general than the Rayleigh fading model. This paper considers the
Rician fading model to obtain first-order design insights for quasi-static
phase shift design. Note that quasi-static phase shift design generally takes
effect as long as channel components contain deterministic phases (e.g., in
clustered delay line (CDL) models [26]). for all small-scale fading channels
[10, 11, 9, 12, 13, 14, 15]. Specifically,

$\displaystyle\mathbf{H}_{ab}$
$\displaystyle=\sqrt{\alpha_{ab}}\bigg{(}\sqrt{\frac{K_{ab}}{K_{ab}+1}}\bar{\mathbf{H}}_{ab}+\sqrt{\frac{1}{K_{ab}+1}}\tilde{\mathbf{H}}_{ab}\bigg{)},ab\in\\{S1,S2,12\\},$
(1) $\displaystyle\mathbf{h}_{ab}^{H}$
$\displaystyle=\sqrt{\alpha_{ab}}\bigg{(}\sqrt{\frac{K_{ab}}{K_{ab}+1}}\bar{\mathbf{h}}_{ab}^{H}+\sqrt{\frac{1}{K_{ab}+1}}\tilde{\mathbf{h}}_{ab}^{H}\bigg{)},ab\in\\{1U,2U,SU\\},$
(2)

where $\alpha_{ab}>0$ represents the large-scale fading power; $K_{ab}\geq 0$
represents the Rician factor;
$\tilde{\mathbf{H}}_{ab}\in\mathbb{C}^{T_{b}\times T_{a}}$ and
$\tilde{\mathbf{h}}_{ab}^{H}\in\mathbb{C}^{1\times T_{a}}$ represent the
random normalized NLoS components in a slot with elements independently and
identically distributed (i.i.d.) according to $\mathcal{CN}(0,1)$;
$\bar{\mathbf{H}}_{ab}\in\mathbb{C}^{T_{b}\times T_{a}}$ and
$\bar{\mathbf{h}}_{ab}^{H}\in\mathbb{C}^{1\times T_{a}}$ represent the
deterministic normalized LoS components with unit-modulus elements. Note that
$\bar{\mathbf{H}}_{ab}$ and $\bar{\mathbf{h}}_{ab}^{H}$ do not change during
the considered period, as the locations of the BS, IRSs, and user are assumed
to be invariant [10, 11, 9, 12, 14].

Let $\lambda$ and $d\left(\leq\frac{\lambda}{2}\right)$ denote the wavelength
of transmission signals and the distance between two adjacent reflecting
elements or antennas in each row and each column of the URAs. Define
$f(x^{(h)},x^{(v)},m,n)\triangleq 2\pi\frac{d}{\lambda}\sin x^{(v)}((m-1)\cos
x^{(h)}+(n-1)\sin x^{(h)})$,
$\mathbf{A}(x^{(h)},x^{(v)},M,N)\triangleq\left(e^{jf(x^{(h)},x^{(v)},m,n)}\right)_{m=1,...,M,n=1,...,N}$,
and
$\mathbf{a}(x^{(h)},x^{(v)},M,N)\triangleq\mathrm{vec}\left(\mathbf{A}(x^{(h)},x^{(v)},M,N)\right)\in\mathbb{C}^{MN}$.
Then, $\bar{\mathbf{H}}_{ab}$ and $\bar{\mathbf{h}}_{ab}$ are modeled as [10,
11, 9, 12]:

$\displaystyle\bar{\mathbf{H}}_{ab}$
$\displaystyle=\mathbf{a}_{A,ab}\mathbf{a}^{H}_{D,ab},ab\in\\{S1,S2,12\\},$
(3) $\displaystyle\bar{\mathbf{h}}_{ab}$
$\displaystyle=\mathbf{a}_{D,ab},ab\in\\{1U,2U,SU\\},$ (4)

where
$\mathbf{a}_{A,ab}\triangleq\mathbf{a}(\delta_{ab}^{(h)},\delta_{ab}^{(v)},M_{b},N_{b})$
and
$\mathbf{a}_{D,ab}\triangleq\mathbf{a}(\varphi_{ab}^{(h)},\varphi_{ab}^{(v)},M_{a},N_{a})$.
Here, $f(\theta^{(h)},\theta^{(v)},m,n)$ represents the difference of the
corresponding phase changes over the LoS component;
$\delta_{ab}^{(h)}\left(\delta_{ab}^{(v)}\right)$ represents the azimuth
(elevation) angle of arrival (AoA) of a signal from node $a$ to the URA at
node $b$; $\varphi_{ab}^{(h)}\left(\varphi_{ab}^{(v)}\right)$ represents the
azimuth (elevation) angle of departure (AoD) of a signal from the URA at node
$a$ to node $b$.

To reduce the channel estimation and phase adjustment costs, we consider a
quasi-static phase shift design [10, 11, 9, 12, 13, 14, 15]. To be specific,
the phase shifts of the two IRSs do not change with the NLoS components from
slot to slot and remain constant during the considered period. Let
$\bm{\theta}_{l}\triangleq\left(\theta_{l,m,n}\right)_{m\in\mathcal{M}_{l},n\in\mathcal{N}_{l}}\in\mathbb{R}^{M_{l}\times
N_{l}}$ denote the constant phase shifts of IRS $l$, where the phase shift of
its $(m,n)$-th element satisfies $\theta_{l,m,n}\in\left[0,2\pi\right)$. For
notation convenience, we introduce
$\bm{\phi}_{l}\triangleq\left(\phi_{l,t}\right)_{t\in\mathcal{T}_{l}}=\mathrm{vec}\left(\bm{\theta}_{l}\right)\in\mathbb{R}^{T_{l}}$
to represent the phase shifts of IRS $l$, where its $t$-th element satisfies:

$\phi_{l,t}\in\left[0,2\pi\right),t\in\mathcal{T}_{l},l\in\mathcal{L}.$ (5)

The equivalent channel between the BS and user, denoted by
$\mathbf{h}_{e}^{H}(\bm{\phi}_{1},\bm{\phi}_{2})\in\mathbb{C}^{1\times
T_{S}}$, can be expressed as:

$\mathbf{h}_{e}^{H}(\bm{\phi}_{1},\bm{\phi}_{2})=\mathbf{h}_{SU}^{H}+\sum_{l\in\mathcal{L}}\mathbf{h}_{lU}^{H}\mathrm{diag}{(\mathbf{v}_{l}^{H})}\mathbf{H}_{Sl}+\mathbf{h}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\mathbf{H}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\mathbf{H}_{S1},$
(6)

where
$\mathbf{v}_{l}\triangleq\left(e^{-j\phi_{l,t}}\right)_{t\in\mathcal{T}_{l}}\in\mathbb{C}^{T_{l}}$.
Note that
$\mathbf{h}_{lU}^{H}\mathrm{diag}{(\mathbf{v}_{l}^{H})}\mathbf{H}_{Sl}$
represents the cascaded channel $(Sl,lU)$, and
$\mathbf{h}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\mathbf{H}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\mathbf{H}_{S1}$
represents the cascaded channel $(S1,12,2U)$. We adopt the following
assumptions on CSI: $\alpha_{ab}$, $K_{ab}$, $ab\in\\{S1,S2,SU,1U,2U,12\\}$
are known; $\delta_{ab}^{(h)}\left(\delta_{ab}^{(v)}\right)$,
$\varphi_{ab}^{(h)}\left(\varphi_{ab}^{(v)}\right)$,
$ab\in\\{S1,S2,SU,1U,2U,12\\}$ are known, implying that
$\bar{\mathbf{H}}_{ab}$, $ab\in\\{S1,S2,12\\}$ and $\bar{\mathbf{h}}_{ab}$,
$ab\in\\{1U,2U,SU\\}$ (i.e., CSI statistics) are known;
$\mathbf{h}_{e}(\bm{\phi}_{1},\bm{\phi}_{2})$ (i.e., instantaneous CSI) is
known to the BS and user at each slot for given $\bm{\phi}_{1}$ and
$\bm{\phi}_{2}$.

######  Remark 1 (CSI Estimation).

Note that $\alpha_{ab}$ and $K_{ab}$ can be obtained by standard offline
channel measurement [27]. Given the BS, IRSs, and user’s locations,
$\delta_{ab}^{(h)}\left(\delta_{ab}^{(v)}\right)$ and
$\varphi_{ab}^{(h)}\left(\varphi_{ab}^{(v)}\right)$ can be easily obtained
using angle estimation techniques [28] at the beginning of the considered
communication period. Thus, the cost for estimating CSI statistics for quasi-
static phase shift design is negligible. With time division duplexing (TDD),
at each slot, $\mathbf{h}_{e}(\bm{\phi}_{1},\bm{\phi}_{2})$ can be obtained by
setting the phase shifts to the optimized values, sending one pilot symbol
from the user ($T_{S}$ pilot symbols from the BS sequentially), and estimating
the received signal at the BS (user), using standard channel estimation
methods [29]. Thus, the cost for estimating instantaneous CSI for quasi-static
phase shift design is much lower than that for instantaneous CSI-adaptive
phase shift design [17, 20, 21].777Instantaneous CSI $\mathbf{h}_{SU}$,
$\text{diag}(\mathbf{h}_{1U})\mathbf{H}_{S1}$,
$\text{diag}(\mathbf{h}_{2U})\mathbf{H}_{S2}$, and
$\text{diag}(h_{2U,t}\mathbf{H}_{12,:,t})\mathbf{H}_{S1}$, $t=1,..,T_{1}$,
where $h_{2U,t}$ denotes the $t$-th element of $\mathbf{h}_{2U}$, are required
per slot in [17, 20, 21]. To estimate such instantaneous CSI, totally
$\left\lceil\frac{\left(T_{2}+1\right)T_{1}}{T_{S}}\right\rceil+T_{1}+T_{2}+2$
(if $T_{S}<T_{1}$) or $2T_{2}+T_{1}+3$ (if $T_{S}>T_{1}$) pilot symbols have
to be sent [17].

The BS serves the user via linear beamforming. Let
${\mathbf{w}}\in\mathbb{C}^{T_{S}}$ with $\|{\mathbf{w}}\|_{2}^{2}=1$
represent the normalized linear beamforming vector at the BS. Then, the
received signal at the user can be expressed as:

$\displaystyle
Y=\mathbf{h}_{e}^{H}(\bm{\phi}_{1},\bm{\phi}_{2})\mathbf{w}\sqrt{P_{S}}X_{S}+Z,$
(7)

where $P_{S}$ represents the transmit power of the BS, $X_{S}\in\mathbb{C}$
with $\mathbb{E}\left[|X_{S}|^{2}\right]=1$ represents an information symbol
for the user, and $Z\sim\mathcal{CN}(0,\sigma^{2})$ denotes the additive white
gaussian noise (AWGN).

We consider coding within each slot. To maximize the achievable rate (singal-
to-noise ratio, SNR) at each slot, we adopt the instantaneous CSI-adaptive MRT
beamformer at the BS [10, 11, 9, 12, 13, 15]:888It is obvious that
$\mathbf{w}$ in (8) is optimal for the maximization of the instantaneous SNR
w.r.t. $\mathbf{w}$ under $\|\mathbf{w}\|_{2}^{2}=1$ at each slot and hence is
optimal for the average rate maximization.

$\displaystyle\mathbf{w}=\frac{\mathbf{h}_{e}(\bm{\phi}_{1},\bm{\phi}_{2})}{\|\mathbf{h}_{e}(\bm{\phi}_{1},\bm{\phi}_{2})\|_{2}}.$
(8)

Thus, for given $\bm{\phi}_{1}$ and $\bm{\phi}_{2}$, the average achievable
rate of D-IRS-C over slots, $C(\bm{\phi}_{1},\bm{\phi}_{2})$ (bit/s/Hz), is
given by:

$\displaystyle
C(\bm{\phi}_{1},\bm{\phi}_{2})\triangleq\mathbb{E}\left[\log_{2}\left(1+\frac{P_{S}}{\sigma^{2}}\left\|\mathbf{h}_{e}(\bm{\phi}_{1},\bm{\phi}_{2})\right\|_{2}^{2}\right)\right].$
(9)

Therefore, we would like to analyze and maximize
$C(\bm{\phi}_{1},\bm{\phi}_{2})$. For tractability, we approximate
$C(\bm{\phi}_{1},\bm{\phi}_{2})$ with its upper bound,
$\log_{2}\left(1+\frac{P_{S}}{\sigma^{2}}\gamma(\bm{\phi}_{1},\bm{\phi}_{2})\right)$,
obtained based on Jensen’s inequality as in [9, 10, 11]. Here,
$\gamma\left(\bm{\phi}_{1},\bm{\phi}_{2}\right)\triangleq\mathbb{E}\left[\|\mathbf{h}_{e}(\bm{\phi}_{1},\bm{\phi}_{2})\|_{2}^{2}\right]$
represents the average channel power of D-IRS-C. Later in Section VI, we will
numerically show that the upper bound is a good approximation of
$C(\bm{\phi}_{1},\bm{\phi}_{2})$. In Section III and Section IV, we analyze
and maximize $\gamma\left(\bm{\phi}_{1},\bm{\phi}_{2}\right)$ instead of
$\log_{2}\left(1+\frac{P_{S}}{\sigma^{2}}\gamma(\bm{\phi}_{1},\bm{\phi}_{2})\right)$,
which is an increasing function of $\gamma(\bm{\phi}_{1},\bm{\phi}_{2})$. Note
that in what follows, we let $ab$, $\overline{ab}$, and $\widetilde{ab}$
represent the Rician channel, LoS channel, and NLoS channel between node $a$
and node $b$, respectively. The Rician channel $ab$ can be viewed as the
composition of the LoS channel $\overline{ab}$ and NLoS channel
$\widetilde{ab}$.

##  III Analysis

In this section, we analyze the average channel power of D-IRS-C in the
general and special regimes, respectively.

###  III-A Analysis in General Regime

In this part, we consider the analysis of the average channel power of D-IRS-C
in the general regime. First, we characterize the influences of
$\bm{\phi}_{1}$ and $\bm{\phi}_{2}$ on $\gamma(\bm{\phi}_{1},\bm{\phi}_{2})$.

######  Lemma 1 (Influences of Phase Shifts of D-IRS-C).

(i) If $K_{S1}(K_{1U}+K_{12})=0$ (i.e., $K_{S1}=0$ or $K_{1U}=K_{12}=0$), then
$\gamma(\bm{\phi}_{1},\bm{\phi}_{2})$ does not change with $\bm{\phi}_{1}$.
(ii) If $K_{2U}(K_{12}+K_{S2})=0$ (i.e., $K_{2U}=0$ or $K_{12}=K_{S2}=0$),
then $\gamma(\bm{\phi}_{1},\bm{\phi}_{2})$ does not change with
$\bm{\phi}_{2}$.

###### Proof:

Please refer to Appendix A. ∎

Lemma 1 indicates that IRS $l$ takes effect if and only if there exist two LoS
channels through which signals can arrive at IRS $l$ from a neighbor node and
depart from IRS $l$ to a neighbor node. Based on Lemma 1, we can divide the
channel conditions into four cases according to the values of Rician factors,
as shown in Fig. 2.999The special cases listed in this paper are those which
not only correspond to special channel setups but also have closed-form
solutions or iterative algorithms with closed-form updates in each iteration.

Figure 2: The channel setups of D-IRS-C in the four cases.

  * •

Case 0 ($K_{S1}=0$ or $K_{1U}=K_{12}=0$ and $K_{12}=K_{S2}=0$ or $K_{2U}=0$):
$\gamma(\bm{\phi}_{1},\bm{\phi}_{2})$ does not change with $\bm{\phi}_{1}$ or
$\bm{\phi}_{2}$ and hence is rewritten as $\gamma^{(0)}$.

  * •

Case 1 ($K_{S1}>0$, $K_{1U}+K_{12}>0$, and $K_{12}=K_{S2}=0$ or $K_{2U}=0$):
$\gamma(\bm{\phi}_{1},\bm{\phi}_{2})$ changes only with $\bm{\phi}_{1}$ and
hence is rewritten as $\gamma^{(1)}(\bm{\phi}_{1})$.

  * •

Case 2 ($K_{S1}=0$ or $K_{1U}=K_{12}=0$, $K_{12}+K_{S2}>0$, and $K_{2U}>0$):
$\gamma(\bm{\phi}_{1},\bm{\phi}_{2})$ changes only with $\bm{\phi}_{2}$ and
hence is rewritten as $\gamma^{(2)}(\bm{\phi}_{2})$.

  * •

Case 3 ($K_{S1}>0$, $K_{1U}+K_{12}>0$, $K_{12}+K_{S2}>0$, and $K_{2U}>0$):
$\gamma(\bm{\phi}_{1},\bm{\phi}_{2})$ changes with both $\bm{\phi}_{1}$ and
$\bm{\phi}_{2}$ and is also written as
$\gamma^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$.

Then, we characterize the average channel power of D-IRS-C in the four cases
of channel conditions. For ease of exposition, we introduce several new
notations. Firstly, for $ab\in\\{S1,S2,12,1U,2U,SU\\}$, we define
$L_{\overline{ab}}\triangleq\frac{K_{ab}\alpha_{ab}}{K_{ab}+1}$ and
$L_{\widetilde{ab}}\triangleq\frac{\alpha_{ab}}{K_{ab}+1}$, which can be
interpreted as the large-scale fading powers of the LoS channel
$\overline{ab}$ and NLoS channel $\widetilde{ab}$, respectively. Obviously,
$L_{\overline{ab}}$ and $L_{\widetilde{ab}}$ both increase with $\alpha_{ab}$,
$L_{\overline{ab}}$ increases with $K_{ab}$, and $L_{\widetilde{ab}}$
decreases with $K_{ab}$. Secondly, we define
$L_{a_{1}b_{1},a_{2}b_{2}}\triangleq\prod\limits_{i=1}^{2}L_{a_{i}b_{i}},(a_{1}b_{1},a_{2}b_{2})\in\bigcup\limits_{l\in\mathcal{L}}\left(\\{\overline{Sl},\widetilde{Sl}\\}\times\\{\overline{lU},\widetilde{lU}\\}\right)$
and
$L_{a_{1}b_{1},a_{2}b_{2},a_{3}b_{3}}\triangleq\prod\limits_{i=1}^{3}L_{a_{i}b_{i}},(a_{1}b_{1},a_{2}b_{2},a_{3}b_{3})\in\\{\overline{S1},\widetilde{S1}\\}\times\\{\overline{12},\widetilde{12}\\}\times\\{\overline{2U},\widetilde{2U}\\}$,
which can be interpreted as the large-scale fading powers of the cascaded
channels $(a_{1}b_{1},a_{2}b_{2})$ and $(a_{1}b_{1},a_{2}b_{2},a_{3}b_{3})$,
respectively. Thirdly, for $a\in\\{S,1\\},l\in\mathcal{L},b\in\\{2,U\\}$, we
define:

$\displaystyle\bm{\Delta}_{\overline{al},\overline{lb}}$
$\displaystyle\triangleq\angle\left(\mathrm{diag}(\mathbf{a}_{D,lb}^{H})\mathbf{a}_{A,al}\right)\in\mathbb{R}^{T_{l}},$
(10) $\displaystyle r_{\overline{al},\overline{ab}}$
$\displaystyle\triangleq\mathbf{a}_{D,al}^{H}\mathbf{a}_{D,ab}\in\mathbb{C},$
(11) $\displaystyle r_{\overline{ab},\overline{lb}}$
$\displaystyle\triangleq\mathbf{a}_{A,ab}^{H}\mathbf{a}_{A,lb}\in\mathbb{C}.$
(12)

Note that $\bm{\Delta}_{\overline{al},\overline{lb}}$ represents the sum of
the phase changes over the LoS channel $\overline{al}$ and LoS channel
$\overline{lb}$; $r_{\overline{al},\overline{ab}}$ depends on
$\varphi_{al}^{(h)},\varphi_{al}^{(v)},\varphi_{ab}^{(h)}$, and
$\varphi_{ab}^{(h)}$, which are determined only by the placement of the URA at
node $a$ and the locations of IRS $l$ and node $b$;
$r_{\overline{12},\overline{S2}}$ depends on
$\delta_{12}^{(h)},\delta_{12}^{(v)},\delta_{S2}^{(h)}$, and
$\delta_{S2}^{(v)}$, which are determined only by the placement of the URA at
IRS 2 and the locations of the BS and IRS 1. Finally, we define:

$\displaystyle\mathbf{A}_{11}$
$\displaystyle\triangleq\mathrm{diag}\left(\bar{\mathbf{h}}_{1U}^{H}\right)\bar{\mathbf{H}}_{S1}\bar{\mathbf{H}}_{S1}^{H}\mathrm{diag}\left(\bar{\mathbf{h}}_{1U}\right)\in\mathbb{C}^{T_{1}\times
T_{1}},\mathbf{A}_{12}\triangleq\mathrm{diag}(\mathbf{a}_{D,12}^{H})\bar{\mathbf{H}}_{S1}\bar{\mathbf{H}}_{S1}^{H}\mathrm{diag}(\mathbf{a}_{D,12})\in\mathbb{C}^{T_{1}\times
T_{1}},$ $\displaystyle\mathbf{A}_{21}$
$\displaystyle\triangleq\mathrm{diag}\left(\bar{\mathbf{h}}_{2U}^{H}\right)\bar{\mathbf{H}}_{S2}\bar{\mathbf{H}}_{S2}^{H}\mathrm{diag}\left(\bar{\mathbf{h}}_{2U}\right)\in\mathbb{C}^{T_{2}\times
T_{2}},\mathbf{A}_{22}\triangleq\mathrm{diag}(\bar{\mathbf{h}}_{2U}^{H})\bar{\mathbf{H}}_{12}\bar{\mathbf{H}}_{12}^{H}\mathrm{diag}(\bar{\mathbf{h}}_{2U})\in\mathbb{C}^{T_{2}\times
T_{2}},$ $\displaystyle\mathbf{A}_{3}$
$\displaystyle\triangleq\mathrm{diag}\left(\bar{\mathbf{h}}_{2U}^{H}\right)\bar{\mathbf{H}}_{12}\mathrm{diag}\left(\mathbf{a}_{A,S1}\right)\in\mathbb{C}^{T_{2}\times
T_{1}},$ $\displaystyle\mathbf{b}_{11}$
$\displaystyle\triangleq\mathrm{diag}\left(\bar{\mathbf{h}}_{1U}^{H}\right)\bar{\mathbf{H}}_{S1}\bar{\mathbf{h}}_{SU}\in\mathbb{C}^{T_{1}},\mathbf{b}_{12}\triangleq\mathrm{diag}\left(\mathbf{a}_{D,12}^{H}\right)\bar{\mathbf{H}}_{S1}\bar{\mathbf{H}}_{S2}^{H}\mathbf{a}_{A,12}\in\mathbb{C}^{T_{1}},$
$\displaystyle\mathbf{b}_{21}$
$\displaystyle\triangleq\mathrm{diag}\left(\bar{\mathbf{h}}_{2U}^{H}\right)\bar{\mathbf{H}}_{S2}\bar{\mathbf{h}}_{SU}\in\mathbb{C}^{T_{2}},\mathbf{b}_{22}\triangleq\mathrm{diag}\left(\bar{\mathbf{h}}_{2U}^{H}\right)\bar{\mathbf{H}}_{12}\bar{\mathbf{h}}_{1U}\in\mathbb{C}^{T_{2}},$
$\displaystyle\mathbf{B}_{1}$
$\displaystyle\triangleq\mathrm{diag}\left(\bar{\mathbf{h}}_{2U}^{H}\right)\bar{\mathbf{H}}_{12}\in\mathbb{C}^{T_{2}\times
T_{1}},$ $\displaystyle\mathbf{B}_{2}$
$\displaystyle\triangleq\bar{\mathbf{H}}_{S1}\bar{\mathbf{H}}_{S1}^{H}\mathrm{diag}\left(\bar{\mathbf{h}}_{1U}\right)\in\mathbb{C}^{T_{1}\times
T_{1}},\mathbf{B}_{3}\triangleq\bar{\mathbf{H}}_{S1}\bar{\mathbf{H}}_{S2}^{H}\mathrm{diag}\left(\bar{\mathbf{h}}_{2U}\right)\in\mathbb{C}^{T_{1}\times
T_{2}},$ $\displaystyle\mathbf{B}_{4}$
$\displaystyle\triangleq\mathrm{diag}\left(\bar{\mathbf{h}}_{2U}^{H}\right)\bar{\mathbf{H}}_{12}\mathrm{diag}\left(\bar{\mathbf{H}}_{S1}\bar{\mathbf{h}}_{SU}\right)\in\mathbb{C}^{T_{2}\times
T_{1}},\mathbf{B}_{5}\triangleq\mathrm{diag}\left(\bar{\mathbf{h}}_{2U}^{H}\right)\bar{\mathbf{H}}_{S2}\bar{\mathbf{H}}_{S1}^{H}\mathrm{diag}\left(\bar{\mathbf{h}}_{1U}\right)\in\mathbb{C}^{T_{2}\times
T_{1}}.$

Based on the above notations, we characterize the average channel power of
D-IRS-C in the general regime as follows.

######  Theorem 1 (Average Channel Power of D-IRS-C in General Regime).

$\displaystyle\gamma^{(0)}=$
$\displaystyle\alpha_{SU}T_{S}+\sum_{l=1}^{2}(L_{\widetilde{Sl},\overline{lU}}+L_{\overline{Sl},\widetilde{lU}}+L_{\widetilde{Sl},\widetilde{lU}})T_{S}T_{l}$
$\displaystyle+\big{(}L_{\widetilde{S1},\overline{12},\widetilde{2U}}+L_{\overline{S1},\widetilde{12},\overline{2U}}+L_{\overline{S1},\widetilde{12},\widetilde{2U}}+L_{\widetilde{S1},\widetilde{12},\overline{2U}}+L_{\widetilde{S1},\widetilde{12},\widetilde{2U}}\big{)}T_{S}T_{1}T_{2},$
(13) $\displaystyle\gamma^{(1)}(\bm{\phi}_{1})=$
$\displaystyle\mathbf{v}_{1}^{H}\left(L_{\overline{S1},\overline{1U}}\mathbf{A}_{11}+L_{\overline{S1},\overline{12},\widetilde{2U}}T_{2}\mathbf{A}_{12}\right)\mathbf{v}_{1}$
$\displaystyle+2\mathfrak{R}\left\\{\mathbf{v}_{1}^{H}\left(\sqrt{L_{\overline{SU}}L_{\overline{S1},\overline{1U}}}\mathbf{b}_{11}+\sqrt{L_{\overline{S2},\widetilde{2U}}L_{\overline{S1},\overline{12},\widetilde{2U}}}\mathbf{b}_{12}\right)\right\\}+\gamma^{(0)},$
(14) $\displaystyle\gamma^{(2)}(\bm{\phi}_{2})=$
$\displaystyle\mathbf{v}_{2}^{H}\left(L_{\overline{S2},\overline{2U}}\mathbf{A}_{21}+L_{\widetilde{S1},\overline{12},\overline{2U}}T_{S}\mathbf{A}_{22}\right)\mathbf{v}_{2}$
$\displaystyle+2\mathfrak{R}\left\\{\mathbf{v}_{2}^{H}\left(\sqrt{L_{\overline{SU}}L_{\overline{S2},\overline{2U}}}\mathbf{b}_{21}+\sqrt{L_{\widetilde{S1},\overline{1U}}L_{\widetilde{S1},\overline{12},\overline{2U}}}T_{S}\mathbf{b}_{22}\right)\right\\}+\gamma^{(0)},$
(15) $\displaystyle\gamma^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})=$
$\displaystyle\gamma^{(1)}(\bm{\phi}_{1})+\gamma^{(2)}(\bm{\phi}_{2})-\gamma^{(0)}+L_{\overline{S1},\overline{12},\overline{2U}}T_{S}\mathbf{v}_{2}^{H}\mathbf{A}_{3}\mathbf{v}_{1}^{*}\mathbf{v}_{1}^{T}\mathbf{A}_{3}^{H}\mathbf{v}_{2}$
$\displaystyle+2\mathfrak{R}\Big{\\{}\mathbf{v}_{1}^{H}\mathrm{diag}(\mathbf{v}_{2}^{H}\mathbf{B}_{1})\left(\sqrt{L_{\overline{S1},\overline{1U}}L_{\overline{S1},\overline{12},\overline{2U}}}\mathbf{B}_{2}\mathbf{v}_{1}+\sqrt{L_{\overline{S2},\overline{2U}}L_{\overline{S1},\overline{12},\overline{2U}}}\mathbf{B}_{3}\mathbf{v}_{2}\right)$
$\displaystyle+\mathbf{v}_{2}^{H}\left(\sqrt{L_{\overline{SU}}L_{\overline{S1},\overline{12},\overline{2U}}}\mathbf{B}_{4}\mathbf{v}_{1}^{*}+\sqrt{L_{\overline{S1},\overline{1U}}L_{\overline{S2},\overline{2U}}}\mathbf{B}_{5}\mathbf{v}_{1}\right)\Big{\\}}.$
(16)

###### Proof:

Please refer to Appendix B.∎

Theorem 1 indicates that the average channel power of D-IRS-C in Case 0 of the
general regime does not depend on $\bm{\phi}_{1}$ or $\bm{\phi}_{2}$, and
hence a quasi-static phase shift design is void in Case 0.

###  III-B Analysis in Pure LoS Regime

In this part, we analyze the average channel power when the Rician factors go
to infinity, corresponding to the regime of pure LoS channels.101010Pure LoS
channels usually exist in the indoor scenario or far above the ground where
scattering is relatively weak [30]. Apparently, Cases 0, 1, and 2 are void in
the pure LoS regime, and the pure LoS regime can be regarded as a special case
of Case 3 of the general regime. Based on Theorem 1, we have the following
result.

######  Corollary 1 (Average Channel Power of D-IRS-C in Pure LoS Regime).

As $K_{S1}$, $K_{S2}$, $K_{12}$, $K_{SU}$, $K_{1U}$,
$K_{2U}\rightarrow\infty$,
$\gamma^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})\rightarrow\bar{\gamma}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$,
where

$\displaystyle\bar{\gamma}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$
$\displaystyle\triangleq\alpha_{SU}T_{S}+\alpha_{S1}\alpha_{1U}\mathbf{v}_{1}^{H}\mathbf{A}_{11}\mathbf{v}_{1}+\alpha_{S2}\alpha_{2U}\mathbf{v}_{2}^{H}\mathbf{A}_{21}\mathbf{v}_{2}+\alpha_{S1}\alpha_{12}\alpha_{2U}T_{S}\mathbf{v}_{2}^{H}\mathbf{A}_{3}\mathbf{v}_{1}^{*}\mathbf{v}_{1}^{T}\mathbf{A}_{3}^{H}\mathbf{v}_{2}$
$\displaystyle+2\mathfrak{R}\Big{\\{}\sqrt{\alpha_{SU}\alpha_{S1}\alpha_{1U}}\mathbf{v}_{1}^{H}\mathbf{b}_{11}+\sqrt{\alpha_{SU}\alpha_{S2}\alpha_{2U}}\mathbf{v}_{2}^{H}\mathbf{b}_{21}$
$\displaystyle\quad\quad\quad+\mathbf{v}_{1}^{H}\mathrm{diag}(\mathbf{v}_{2}^{H}\mathbf{B}_{1})\Big{(}\sqrt{\alpha_{S1}\alpha_{1U}\alpha_{S1}\alpha_{12}\alpha_{2U}}\mathbf{B}_{2}\mathbf{v}_{1}+\sqrt{\alpha_{S2}\alpha_{2U}\alpha_{S1}\alpha_{12}\alpha_{2U}}\mathbf{B}_{3}\mathbf{v}_{2}\Big{)}$
$\displaystyle\quad\quad\quad+\mathbf{v}_{2}^{H}\Big{(}\sqrt{\alpha_{SU}\alpha_{S1}\alpha_{12}\alpha_{2U}}\mathbf{B}_{4}\mathbf{v}_{1}^{*}+\sqrt{\alpha_{S1}\alpha_{1U}\alpha_{S2}\alpha_{2U}}\mathbf{B}_{5}\mathbf{v}_{1}\Big{)}\Big{\\}}.$
(17)

As the Rician factors go to infinity, the large-scale fading power of each
channel or cascaded channel consisting of at least one NLoS channel goes zero.
Thus, Theorem 1 readily implies Corollary 1.

###  III-C Analysis in Pure NLoS Regime

In this part, we analyze the average channel power of D-IRS-C when the Rician
factors are zero, corresponding to the regime of pure NLoS channels.
Obviously, Cases 1, 2, and 3 are void in the pure NLoS regime, and the pure
NLoS regime can be regarded as a special case of Case 0 in the general regime.
From Theorem 1, we have the following result.

######  Corollary 2 (Average Channel Power of D-IRS-C in Pure NLoS Regime).

If $K_{S1}=K_{S2}=K_{12}=K_{SU}=K_{1U}=K_{2U}=0$, then
$\gamma^{(0)}=\tilde{\gamma}^{(0)}$, where

$\displaystyle\tilde{\gamma}^{(0)}\triangleq\alpha_{SU}T_{S}+\alpha_{S1}\alpha_{1U}T_{S}T_{1}+\alpha_{S2}\alpha_{2U}T_{S}T_{2}+\alpha_{S1}\alpha_{12}\alpha_{2U}T_{S}T_{1}T_{2}.$
(18)

When the Rician factors become zero, the large-scale fading power of each
channel or cascaded channel consisting of at least one LoS channel becomes
zero. As a result, Theorem 1 readily implies Corollary 2. Analogously, a
quasi-static phase shift design is ineffective for D-IRS-C in the pure NLoS
regime.

##  IV Optimization

In this section, we maximize the average channel power of D-IRS-C w.r.t. the
phase shifts in the general and special regimes, respectively.

###  IV-A Optimization in General Regime

In this part, we maximize the average channel power of D-IRS-C w.r.t. the
phase shifts in the general regime. As shown in Theorem 1, $\gamma^{(0)}$ is
irrelevant to $\bm{\phi}_{1}$ and $\bm{\phi}_{2}$ in the general regime. Thus,
we optimize the phase shifts only for Cases 1, 2, and 3 in the general regime
in the sequel.

####  IV-A1 Optimization for Case $l=1,2$ of General Regime

For $l=1,2$, as $\gamma^{(l)}(\bm{\phi}_{l})$ depends only on $\bm{\phi}_{l}$,
we optimize $\bm{\phi}_{l}$ in Case $l$.

######  Problem 1 (Optimization for D-IRS-C in Case $l=1,2$ of General
Regime).

For Case $l=1,2$,

$\displaystyle\gamma^{(l)\star}\triangleq\max_{\bm{\phi}_{l}}$
$\displaystyle\quad\gamma^{(l)}(\bm{\phi}_{l})$ $\displaystyle s.t.$
$\displaystyle\quad\phi_{l,t}\in[0,2\pi),t\in\mathcal{T}_{l},$ (19)

where $\gamma^{(1)}(\bm{\phi}_{1})$ and $\gamma^{(2)}(\bm{\phi}_{2})$ are
given by (14) and (15), respectively. Let $\bm{\phi}_{l}^{\star}$ denote an
optimal solution for Case $l$, where $l\in\mathcal{L}$.

Problem 1 is a challenging non-convex problem with a very complex objection
function and $T_{l}$ optimization variables. However, by the triangle
inequality and Cauchy-Schwartz inequality, we can obtain globally optimal
closed-form solutions of Problem 1 for Case 1 with $K_{1U}=0$ or $K_{12}=0$
and for Case 2 with $K_{S2}=0$ or $K_{12}=0$, respectively. For ease of
exposition, define $\Lambda(x)=x-2\pi\lfloor\frac{x}{2\pi}\rfloor\in[0,2\pi)$,
$x\in\mathbb{R}$. Note that $\Lambda(\cdot)$ can be used to provide phase
shifts $\bm{\phi}_{l}$, $l\in\mathcal{L}$ satisfying (5) [10].

######  Theorem 2 (Optimal Solution of Problem 1).

(i) The unique optimal solution of Problem 1 for Case 1 with $K_{1U}=0$ or
$K_{12}=0$ is given by:

$\displaystyle\bm{\phi}_{1}^{\star}=\left\\{\begin{aligned}
&\Lambda(-\bm{\Delta}_{\overline{S1},\overline{12}}-\angle(r_{\overline{S1},\overline{S2}}r_{\overline{S2},\overline{12}})\mathbf{1}_{T_{1}}),&K_{1U}=0,\\\
&\Lambda(-\bm{\Delta}_{\overline{S1},\overline{1U}}-\angle(r_{\overline{S1},\overline{SU}})\mathbf{1}_{T_{1}}),&K_{12}=0.\end{aligned}\right.$
(20)

(ii) The unique optimal solution of Problem 1 for Case 2 with $K_{S2}=0$ or
$K_{12}=0$ is given by:

$\displaystyle\bm{\phi}^{\star}_{2}=\left\\{\begin{aligned}
&\Lambda(-\bm{\Delta}_{\overline{12},\overline{2U}}-\angle(r_{\overline{12},\overline{1U}})\mathbf{1}_{T_{2}}),&K_{S2}=0,\\\
&\Lambda(-\bm{\Delta}_{\overline{S2},\overline{2U}}-\angle(r_{\overline{S2},\overline{SU}})\mathbf{1}_{T_{2}}),&K_{12}=0.\end{aligned}\right.$
(21)

###### Proof:

Please refer to Appendix C.∎

From Theorem 2, we can draw the following conclusions for D-IRS-C. Firstly, in
Case 1 with $K_{1U}=0$, the optimal phase changes over the cascaded LoS
channel $(\overline{S1},\overline{12})$ are identical to the phase changes
over the LoS channel $\overline{S2}$. Secondly, in Case 2 with $K_{S2}=0$, the
optimal phase changes over the cascaded LoS channel
$(\overline{12},\overline{2U})$ are identical to the phase changes over the
LoS channel $\overline{1U}$. Finally, in Case 1 and Case 2 both with
$K_{12}=0$, the optimal phase changes over the cascaded LoS channel
$(\overline{Sl},\overline{lU})$ are identical to the phase changes over the
LoS channel $\overline{SU}$.

However, for Case 1 with $K_{12}$, $K_{1U}>0$ and Case 2 with $K_{S2}$,
$K_{12}>0$, we cannot derive a globally optimal solution with the same method.
Instead, we propose a computationally efficient iterative algorithm to obtain
stationary points of Problem 1 for Case 1 with $K_{12}$, $K_{1U}>0$ and Case 2
with $K_{S2}$, $K_{12}>0$, respectively, based on the coordinate descent
method. The idea is that at each step of one iteration,
$\phi_{l,t},t\in\mathcal{T}_{l}$ are sequentially updated by analytically
solving the coordinate optimizations, each with a single variable.
Specifically, in each coordinate optimization, we maximize
$\gamma^{(l)}(\bm{\phi}_{l})$ w.r.t. $\phi_{l,t}$ for some
$t\in\mathcal{T}_{l}$ with the other phase shifts being fixed. For ease of
exposition, we rewrite $\bm{\phi}_{l}$ as $(\phi_{l,t},\bm{\phi}_{l,-t})$,
where $\bm{\phi}_{l,-t}$ represents the elements of $\bm{\phi}_{l}$ except for
$\phi_{l,t}$. Then, given $\bm{\phi}_{l,-t}$ obtained in the previous step,
the coordinate optimizations w.r.t. $\phi_{l,t}$ for Case 1 with $K_{12}$,
$K_{1U}>0$ and Case 2 with $K_{S2}$, $K_{12}>0$ are formulated as follows.

######  Problem 2 (Coordinate Optimization w.r.t. $\phi_{l,t}$ for D-IRS-C in
Case $l=1,2$ of General Regime).

For Case $l=1,2$ and for all $t\in\mathcal{T}_{l}$,

$\displaystyle\phi_{l,t}^{\dagger}\triangleq\operatorname*{argmax}_{\phi_{l,t}\in[0,2\pi)}$
$\displaystyle\quad\gamma^{(l)}(\phi_{l,t},\bm{\phi}_{l,-t}).$

Problem 2 is a single variable optimization problem and hence is much simpler
than Problem 1. Define:

$\displaystyle\mathbf{A}_{1}\triangleq
L_{\overline{S1},\overline{1U}}\mathbf{A}_{11}+L_{\overline{S1},\overline{12},\widetilde{2U}}T_{2}\mathbf{A}_{12},\quad\quad\quad\quad\
\mathbf{A}_{2}\triangleq
L_{\overline{S2},\overline{2U}}\mathbf{A}_{21}+L_{\widetilde{S1},\overline{12},\overline{2U}}T_{S}\mathbf{A}_{22},$
(22)
$\displaystyle\mathbf{b}_{1}\triangleq\sqrt{L_{\overline{SU}}L_{\overline{S1},\overline{1U}}}\mathbf{b}_{11}+\sqrt{L_{\overline{S2},\widetilde{2U}}L_{\overline{S1},\overline{12},\widetilde{2U}}}\mathbf{b}_{12},\mathbf{b}_{2}\triangleq\sqrt{L_{\overline{SU}}L_{\overline{S2},\overline{2U}}}\mathbf{b}_{21}+\sqrt{L_{\widetilde{S1},\overline{1U}}L_{\widetilde{S1},\overline{12},\overline{2U}}}T_{S}\mathbf{b}_{22}.$
(23)

After some basic algebraic manipulations, we can derive a closed-form optimal
solution of Problem 2.

######  Lemma 2 (Optimal Solution of Problem 2).

For Case $l=1,2$ and for all $t\in\mathcal{T}_{l}$, the unique optimal
solution of Problem 2 is given by:

$\displaystyle\phi_{l,t}^{\dagger}=\Lambda\left(-\angle\left(\sum_{k\in\mathcal{T}_{l},k\neq
t}A_{l,t,k}e^{-j\phi_{l,k}}+b_{l,t}\right)\right),$ (24)

where $A_{l,t,k}$ and $b_{l,t}$ represent the $(t,k)$-th element of
$\mathbf{A}_{l}$ given by (22) and the $t$-th element of $\mathbf{b}_{l}$
given by (23), respectively.

###### Proof:

Please refer to Appendix D. ∎

The details of the coordinate descent-based algorithm for Case 1 with
$K_{12}$, $K_{1U}>0$ and Case 2 with $K_{S2}$, $K_{12}>0$ are summarized in
Algorithm 1. Since Problem 2 can be solved analytically, the computation time
of Algorithm 1 is relatively short. Specifically, the computational complexity
of Step 4 of Algorithm 1 is $\mathcal{O}(T_{l})$. Hence the overall
computational complexity of each iteration of Algorithm 1 is
$\mathcal{O}(T_{l}^{2})$. Furthermore, we know that every limit point
generated by Algorithm 1 is a stationary point of Problem 2, as Problem 2 has
a unique optimal solution [31, Proposition 2.7.1].

Algorithm 1 Coordinate Descent Algorithm for Obtaining A Stationary Point of
Problem 1

1: Initialization: Select $\bm{\phi}_{l}$ satisfying the constraints in (5) as
the initial point.

2: repeat

3: for $t=1,...,T_{l}$ do

4: Calculate $\phi_{l,t}^{\dagger}$ according to (24), and set
$\phi_{l,t}=\phi_{l,t}^{\dagger}$.

5: end for

6: until some convergence criterion is met.

####  IV-A2 Optimization for Case 3 of General Regime

As $\gamma^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$ depends on $\phi_{1}$ and
$\phi_{2}$, we jointly optimize $\phi_{1}$ and $\phi_{2}$ in Case 3.

######  Problem 3 (Optimization for D-IRS-C in Case 3 of General Regime).

For Case 3,

$\displaystyle\gamma^{(3)\star}\triangleq\max_{\bm{\phi}_{1},\bm{\phi}_{2}}$
$\displaystyle\quad\gamma^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$ $\displaystyle
s.t.$ $\displaystyle\quad\eqref{eq:unitconstrint},$

where $\gamma^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$ is given by (16). Let
$(\bm{\phi}_{1}^{\star},\bm{\phi}_{2}^{\star})$ denote an optimal solution.

Apparently, Problem 3 is a more challenging non-convex problem than Problem 1,
as it has a more complex objective function and more optimization variables.
However, by the triangle inequality and Cauchy-Schwartz inequality, we can
still obtain a globally optimal closed-form solution of Problem 3 for Case 3
with $K_{S2}=K_{1U}=0$ or $K_{12}=K_{SU}=0$.

######  Theorem 3 (Optimal Solution of Problem 3).

(i) For Case 3 with $K_{S2}=K_{1U}=0$, any
$(\phi^{\star}_{1},\phi^{\star}_{2})$ with
$\bm{\phi}_{1}^{\star}=\Lambda\left(-\bm{\Delta}_{\overline{S1},\overline{12}}+\psi\mathbf{1}_{T_{1}}-\frac{1}{2}\angle\left(r_{\overline{S1},\overline{SU}}\right)\mathbf{1}_{T_{1}}\right)$
and
$\bm{\phi}_{2}^{\star}=\Lambda\left(-\bm{\Delta}_{\overline{12},\overline{2U}}-\psi\mathbf{1}_{T_{2}}-\frac{1}{2}\angle\left(r_{\overline{S1},\overline{SU}}\right)\mathbf{1}_{T_{2}}\right)$,
for all $\psi\in\mathbb{R}$, is an optimal solution of Problem 3. (ii) For
Case 3 with $K_{12}=K_{SU}=0$, any $(\phi^{\star}_{1},\phi^{\star}_{2})$ with
$\phi^{\star}_{1}=\Lambda(-\bm{\Delta}_{\overline{S1},\overline{1U}}+\psi\mathbf{1}_{T_{1}})$,
$\phi^{\star}_{2}=\Lambda(-\bm{\Delta}_{\overline{S2},\overline{2U}}+\psi\mathbf{1}_{T_{2}}+\angle(r_{\overline{S1},\overline{S2}})\mathbf{1}_{T_{2}})$,
for all $\psi\in\mathbb{R}$, is an optimal solution of Problem 3.

###### Proof:

Please refer to Appendix E. ∎

Theorem 3 indicates two facts for D-IRS-C in the general regime. Firstly, in
Case 3 with $K_{S2}=K_{1U}=0$, the optimal phase changes over the cascaded LoS
channel $(\overline{S1},\overline{12},\overline{2U})$ are the same as the
phase changes over the LoS channel $\overline{SU}$. Secondly, in Case 3 with
$K_{12}=K_{SU}=0$, the optimal phase changes over the cascaded LoS channels
$(\overline{S1},\overline{1U})$ and $(\overline{S2},\overline{2U})$ are
identical.

However, for Case 3 with $K_{S2}>0$, $K_{1U}>0$, and $K_{12}+K_{SU}>0$, we
cannot derive a globally optimal solution with the same method. Instead, we
propose a computationally efficient iterative algorithm based on the
coordinate descent method. The idea is that at each iteration,
$\phi_{l,t},t\in\mathcal{T}_{l},l\in\mathcal{L}$ are sequentially updated by
analytically solving the coordinate optimizations. Specifically, in each
coordinate optimization, we maximize $\gamma(\bm{\phi}_{1},\bm{\phi}_{2})$
w.r.t. $\phi_{l,t}$ for some $t\in\mathcal{T}_{l},l\in\mathcal{L}$ with the
other phase shifts being fixed. For ease of exposition, we rewrite
$(\bm{\phi}_{1},\bm{\phi}_{2})$ as
$(\phi_{l,t},\bm{\phi}_{l,-t},\bm{\phi}_{-l})$, where $\bm{\phi}_{l,-t}$
represents the elements of $\bm{\phi}_{l}$ except for $\phi_{l,t}$, and $-l$
represents the element in $\mathcal{L}\backslash\\{l\\}$. Then, given
$\bm{\phi}_{l,-t}$ and $\bm{\phi}_{-l}$ obtained in the previous step, the
coordinate optimization w.r.t. $\phi_{l,t}$ is formulated as follows.

######  Problem 4 (Coordinate Optimization w.r.t. $\phi_{l,t}$ for D-IRS-C in
Case 3 of General Regime).

For Case 3 and for all $t\in\mathcal{T}_{l},l\in\mathcal{L}$,

$\displaystyle\phi_{l,t}^{\dagger}\triangleq\operatorname*{argmax}_{\phi_{l,t}\in[0,2\pi)}$
$\displaystyle\gamma(\phi_{l,t},\bm{\phi}_{l,-t},\bm{\phi}_{-l}).$

Problem 4 is a single variable non-convex optimization problem and hence is
much simpler than Problem 3. Define:

$\displaystyle\mathbf{C}_{1}$
$\displaystyle\triangleq\mathbf{A}_{1}+L_{\overline{S1},\overline{12},\overline{2U}}T_{S}\mathbf{A}_{3}^{T}\mathbf{v}_{2}^{*}\mathbf{v}_{2}^{T}\mathbf{A}_{3}^{*}+2\sqrt{L_{\overline{S1},\overline{1U}}L_{\overline{S1},\overline{12},\overline{2U}}}\mathfrak{R}\\{\mathrm{diag}(\mathbf{v}_{2}^{H}\mathbf{B}_{1})\mathbf{B}_{2}\\}\in\mathbb{C}^{T_{1}\times
T_{1}},$ (25a) $\displaystyle\mathbf{C}_{2}$
$\displaystyle\triangleq\mathbf{A}_{2}+L_{\overline{S1},\overline{12},\overline{2U}}T_{S}\mathbf{A}_{3}\mathbf{v}_{1}^{*}\mathbf{v}_{1}^{T}\mathbf{A}_{3}^{H}+2\sqrt{L_{\overline{S2},\overline{2U}}L_{\overline{S1},\overline{12},\overline{2U}}}\mathfrak{R}\\{\mathbf{B}_{1}\mathrm{diag}(\mathbf{v}_{1}^{H})\mathbf{B}_{3}\\}\in\mathbb{C}^{T_{2}\times
T_{2}},$ (25b)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathbf{d}_{1}\triangleq\mathbf{b}_{1}+\sqrt{L_{\overline{S2},\overline{2U}}L_{\overline{S1},\overline{12},\overline{2U}}}\mathrm{diag}(\mathbf{v}_{2}^{H}\mathbf{B}_{1})\mathbf{B}_{3}\mathbf{v}_{2}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\sqrt{L_{\overline{SU}}L_{\overline{S1},\overline{12},\overline{2U}}}\mathbf{B}_{4}^{T}\mathbf{v}_{2}^{*}+\sqrt{L_{\overline{S1},\overline{1U}}L_{\overline{S2},\overline{2U}}}\mathbf{B}_{5}^{H}\mathbf{v}_{2}\in\mathbb{C}^{T_{1}},$
(26a)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathbf{d}_{2}\triangleq\mathbf{b}_{2}+\sqrt{L_{\overline{S1},\overline{1U}}L_{\overline{S1},\overline{12},\overline{2U}}}\mathbf{B}_{1}\mathrm{diag}(\mathbf{v}_{1}^{H})\mathbf{B}_{2}\mathbf{v}_{1}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\sqrt{L_{\overline{SU}}L_{\overline{S1},\overline{12},\overline{2U}}}\mathbf{B}_{4}\mathbf{v}_{1}^{*}+\sqrt{L_{\overline{S1},\overline{1U}}L_{\overline{S2},\overline{2U}}}\mathbf{B}_{5}\mathbf{v}_{1}\in\mathbb{C}^{T_{2}}.$
(26b)

After some basic algebraic manipulations, we can derive a closed-form optimal
solution of Problem 4.

######  Lemma 3 (Optimal Solution of Problem 4).

For Case 3 and all $t\in\mathcal{T}_{l}$, $l\in\mathcal{L}$, the unique
optimal solution of Problem 4 is given by:

$\displaystyle\phi_{l,t}^{\dagger}=\Lambda\bigg{(}-\angle\bigg{(}\sum_{k\in\mathcal{T}_{l},k\neq
t}C_{l,t,k}e^{-j\phi_{l,k}}+d_{l,t}\bigg{)}\bigg{)},$ (27)

where $C_{l,t,k}$ represents the $(t,k)$-th element of $\mathbf{C}_{l}$ given
by (25), and $d_{l,t}$ represents the $t$-th element of $\mathbf{d}_{l}$ given
by (26).

###### Proof:

Please refer to Appendix F. ∎

The details of the coordinate descent-based algorithm are summarized in
Algorithm 2. The computational complexities of Step 4 and Step 6 of Algorithm
2 are $\mathcal{O}(T_{1}T_{2}T_{l})$ and $\mathcal{O}(T_{l})$,
$l\in\mathcal{L}$. Hence, the computational complexity of each iteration of
Algorithm 2 is $\mathcal{O}(T_{1}^{2}T_{2}+T_{2}^{2}T_{1})$. If $T_{1}=cT$ and
$T_{2}=(1-c)T$ for some $c\in(0,1)$, then the computational complexity of each
iteration of Algorithm 2 becomes $\mathcal{O}(T^{3})$. As Problem 4 can be
solved analytically, the computation efficiency of Algorithm 2 is relatively
high. Furthermore, we know that every limit point generated by Algorithm 2 is
a stationary point of Problem 3, as Problem 4 has a unique optimal solution
[31, Proposition 2.7.1].

Algorithm 2 Coordinate Descent Algorithm for Obtaining A Stationary Point of
Problem 3

1: Initialization: Select $\bm{\phi}_{1}$ and $\bm{\phi}_{2}$ satisfying the
constraints in (5) as the initial point.

2: repeat

3: for $l=1,2$ do

4: Calculate $\mathbf{C}_{l}$ and $\mathbf{d}_{l}$ according to (25) and (26),
respectively.

5: for $t=1,..,T_{l}$ do

6: Calculate $\phi_{l,t}^{\dagger}$ according to (27), and set
$\phi_{l,t}=\phi_{l,t}^{\dagger}$.

7: end for

8: end for

9: until some convergence criterion is met.

####  IV-A3 Optimal Average Channel Power in General Regime at Large Number of
Reflecting Elements

We characterize the average channel power of D-IRS-C in Case 0 and the optimal
average channel power of D-IRS-C in Cases 1, 2, and 3 of the general regime at
Large $T_{1}$, $T_{2}$, and $T$.

######  Theorem 4 (Optimal Average Channel Power of D-IRS-C in General
Regime).

(i) $\gamma^{(0)}\stackrel{{\scriptstyle
T_{1},T_{2}\rightarrow\infty}}{{\sim}}(L_{\widetilde{S1},\overline{12},\widetilde{2U}}+L_{\overline{S1},\widetilde{12},\overline{2U}}+L_{\overline{S1},\widetilde{12},\widetilde{2U}}+L_{\widetilde{S1},\widetilde{12},\overline{2U}}+L_{\widetilde{S1},\widetilde{12},\widetilde{2U}})T_{S}T_{1}T_{2}$,
$\gamma^{(1)\star}\stackrel{{\scriptstyle
T_{1},T_{2}\rightarrow\infty}}{{\sim}}L_{\overline{S1},\overline{12},\widetilde{2U}}T_{S}T_{1}^{2}T_{2}$,
$\gamma^{(2)\star}\stackrel{{\scriptstyle
T_{1},T_{2}\rightarrow\infty}}{{\sim}}L_{\widetilde{S1},\overline{12},\overline{2U}}T_{S}T_{1}T_{2}^{2}$,
and $\gamma^{(3)\star}\stackrel{{\scriptstyle
T_{1},T_{2}\rightarrow\infty}}{{\sim}}L_{\overline{S1},\overline{12},\overline{2U}}T_{S}T_{1}^{2}T_{2}^{2}$.
(ii) If $T_{1}=cT$ and $T_{2}=(1-c)T$ for some $c\in(0,1)$, then
$\gamma^{(0)}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}(L_{\widetilde{S1},\overline{12},\widetilde{2U}}+L_{\overline{S1},\widetilde{12},\overline{2U}}+L_{\overline{S1},\widetilde{12},\widetilde{2U}}+L_{\widetilde{S1},\widetilde{12},\overline{2U}}+L_{\widetilde{S1},\widetilde{12},\widetilde{2U}})c(1-c)T_{S}T^{2}$,
$\gamma^{(1)\star}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}L_{\overline{S1},\overline{12},\widetilde{2U}}c^{2}(1-c)T_{S}T^{3}$,
$\gamma^{(2)\star}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}L_{\widetilde{S1},\overline{12},\overline{2U}}c(1-c)^{2}T_{S}T^{3}$,
and $\gamma^{(3)\star}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}L_{\overline{S1},\overline{12},\overline{2U}}c^{2}(1-c)^{2}T_{S}T^{4}$.111111$f(x,y)\stackrel{{\scriptstyle
x,y\rightarrow\infty}}{{\sim}}g(x,y)$ means
$\lim_{x,y\rightarrow\infty}\frac{f(x,y)}{g(x,y)}=1$, and
$f(x)\stackrel{{\scriptstyle x\rightarrow\infty}}{{\sim}}g(x)$ means
$\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=1$.

###### Proof:

Please refer to Appendix G.∎

By Theorem 4 and [21, Theorem 1], for D-IRS-C in the general regime, the
optimal quasi-static phase shift design for Case 3 achieves the same average
power gain in order w.r.t. the total number of reflecting elements $T$ (i.e.,
$\Theta(T^{4})$) as the optimal instantaneous CSI-adaptive phase shift design
but with lower channel estimation and phase adjustment costs. Besides, for
D-IRS-C in the general regime, the optimal average channel powers in Case 1
and Case 2 (i.e., $\Theta(T^{3})$) are equivalent in order and are higher in
order than the average channel power in Case 0 (i.e., $\Theta(T^{2})$) and
lower in order than the average channel power in Case 3 (i.e.,
$\Theta(T^{4})$).

###  IV-B Optimization in Pure LoS Regime

In this part, we maximize $\bar{\gamma}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$
w.r.t. $\bm{\phi}_{1}$ and $\bm{\phi}_{2}$ in the pure LoS regime.

######  Problem 5 (Optimization for D-IRS-C in Pure LoS Regime).

$\displaystyle\bar{\gamma}^{(3)\star}\triangleq$
$\displaystyle\quad\max_{\bm{\phi}_{1},\bm{\phi}_{2}}\bar{\gamma}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$
$\displaystyle s.t.$ $\displaystyle\quad\eqref{eq:unitconstrint},$

where $\bar{\gamma}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$ is given by (17). Let
$\left(\bar{\bm{\phi}}_{1}^{\star},\bar{\bm{\phi}}_{2}^{\star}\right)$ denote
an optimal solution.

As $\bar{\gamma}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$ has a simpler form than
$\gamma^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$, Problem 5 is more tractable than
Problem 3. We propose a computationally efficient iterative algorithm to
obtain a stationary point of Problem 5 using the block coordinate descent
method. Specifically, we divide the optimization variables into $T_{2}+1$
disjoint blocks, i.e., $\bm{\phi}_{1}$, $\phi_{2,t},t\in\mathcal{T}_{2}$, with
the first block consisting of $T_{1}$ coordinates and each of the remaining
blocks consisting of only one coordinate, and sequentially update each block
by analytically solving the corresponding block coordinate and coordinate
optimization problems. Here, we treat $\bm{\phi}_{1}$ as a single block, as
the optimization problem w.r.t. $\bm{\phi}_{1}$ can be analytically solved,
with a lower computational complexity than the separate optimization problems
w.r.t. $\bm{\phi}_{1,t},t\in\mathcal{T}_{1}$, which will be seen shortly. The
block coordinate optimization w.r.t. $\bm{\phi}_{1}$ and the coordinate
optimizations w.r.t. $\bm{\phi}_{2,t},t\in\mathcal{T}_{2}$ are formulated as
follows.

######  Problem 6 (Block Coordinate Optimization w.r.t. $\bm{\phi}_{1}$ for
D-IRS-C in Pure LoS Regime).

In the pure LoS regime,

$\displaystyle\bar{\bm{\phi}}_{1}^{\dagger}\triangleq$
$\displaystyle\quad\operatorname*{argmax}_{\bm{\phi}_{1}}\bar{\gamma}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$
$\displaystyle s.t.$
$\displaystyle\quad\phi_{1,t}\in[0,2\pi),t\in\mathcal{T}_{1}.$

######  Problem 7 (Coordinate Optimization w.r.t. $\bm{\phi}_{2,t}$ for
D-IRS-C in Pure LoS Regime).

In the pure LoS regime, for all $t\in\mathcal{T}_{2}$,

$\displaystyle\bar{\phi}_{2,t}^{\dagger}\triangleq$
$\displaystyle\quad\operatorname*{argmax}_{\phi_{2,t}\in[0,2\pi)}\bar{\gamma}^{(3)}(\phi_{2,t},\phi_{2,-t},\bm{\phi}_{1}).$

Define:

$\displaystyle\bar{\mathbf{C}}_{2}$
$\displaystyle\triangleq\alpha_{S2}\alpha_{2U}\mathbf{A}_{21}+\alpha_{S1}\alpha_{12}\alpha_{2U}T_{S}\mathbf{A}_{3}\mathbf{v}_{1}^{*}\mathbf{v}_{1}^{T}\mathbf{A}_{3}^{H}$
$\displaystyle\quad+2\sqrt{\alpha_{S2}\alpha_{2U}\alpha_{S1}\alpha_{12}\alpha_{2U}}\mathfrak{R}\\{\mathbf{B}_{1}\mathrm{diag}(\mathbf{v}_{1}^{H})\mathbf{B}_{3}\\}\in\mathbb{C}^{T_{2}\times
T_{2}},$ (28) $\displaystyle\bar{\mathbf{d}}_{2}$
$\displaystyle\triangleq\sqrt{\alpha_{SU}\alpha_{S2}\alpha_{2U}}\mathbf{b}_{21}+\sqrt{\alpha_{S1}\alpha_{1U}\alpha_{S1}\alpha_{12}\alpha_{2U}}\mathbf{B}_{1}\mathrm{diag}(\mathbf{v}_{1}^{H})\mathbf{B}_{2}\mathbf{v}_{1}$
$\displaystyle\quad+\sqrt{\alpha_{SU}\alpha_{S1}\alpha_{1U}}\mathbf{B}_{4}\mathbf{v}_{1}^{*}+\sqrt{\alpha_{S1}\alpha_{1U}\alpha_{S2}\alpha_{2U}}\mathbf{B}_{5}\mathbf{v}_{1}\in\mathbb{C}^{T_{2}}.$
(29)

After some basic algebraic manipulations, we can derive closed-form optimal
solutions of Problem 6 and Problem 7 using the triangle inequality and Cauchy-
Schwartz inequality.

######  Lemma 4 (Optimal Solutions of Problem 6 and Problem 7).

(i) The unique optimal solution of Problem 6 is given by:

$\displaystyle\bar{\bm{\phi}}_{1}^{\dagger}$
$\displaystyle=\Lambda\Big{(}-\angle\left(\mathrm{diag}\left(\sqrt{\alpha_{S1}\alpha_{1U}}\bar{\mathbf{h}}_{1U}^{H}+\sqrt{\alpha_{S1}\alpha_{1U}\alpha_{S1}\alpha_{12}\alpha_{2U}}\mathbf{v}_{2}^{H}\mathbf{B}_{1}\right)\mathbf{a}_{A,S1}\right)$
$\displaystyle\quad\quad\ \
-\angle\big{(}\mathbf{a}^{H}_{D,S1}(\sqrt{\alpha_{SU}}\bar{\mathbf{h}}_{SU}+\sqrt{\alpha_{S2}\alpha_{2U}}\bar{\mathbf{H}}_{S2}^{H}\mathrm{diag}(\mathbf{v}_{2})\bar{\mathbf{h}}_{2U})\big{)}\mathbf{1}_{T_{1}}\Big{)}.$
(30)

(ii) For all $t\in\mathcal{T}_{2}$, the unique optimal solution of Problem 7
is given by:

$\displaystyle\bar{\phi}_{2,t}^{\dagger}=\Lambda\left(-\angle\left(\sum_{k\in\mathcal{T}_{2},k\neq
t}\bar{C}_{2,t,k}e^{-j\phi_{2,k}}+\bar{d}_{2,t}\right)\right),$ (31)

where $\bar{C}_{2,t,k}$ represents the $(t,k)$-th element of
$\bar{\mathbf{C}}_{2}$ given by (28), and $\bar{d}_{2,t}$ represents the
$t$-th element of $\bar{\mathbf{d}}_{2}$ given by (29).

###### Proof:

Please refer to Appendix H.∎

Algorithm 3 Block Coordinate Descent Algorithm for Obtaining A Stationary
Point of Problem 5

1: Initialization: Select $\bm{\phi}_{1}$ and $\bm{\phi}_{2}$ satisfying the
constraints in (5) as the initial point.

2: repeat

3: Calculate $\bar{\bm{\phi}}_{1}^{\dagger}$ according to (4), and set
$\bm{\phi}_{1}=\bar{\bm{\phi}}_{1}^{\dagger}$.

4: Calculate $\bar{\mathbf{C}}_{2}$ and $\bar{\mathbf{d}}_{2}$ according to
(28) and (29), respectively.

5: for $t=1,..,T_{2}$ do

6: Calculate $\bar{\phi}_{2,t}^{\dagger}$ according to (31), and set
$\phi_{2,t}=\bar{\phi}_{2,t}^{\dagger}$.

7: end for

8: until some convergence criterion is met.

The details of the block coordinate descent algorithm are summarized in
Algorithm 3. The computational complexities of Step 3, Step 4, and Step 6 are
$\mathcal{O}(T_{1}T_{2})$, $\mathcal{O}(T_{1}T_{2}^{2})$, and
$\mathcal{O}(T_{2})$, respectively. Hence the computational complexity of each
iteration of Algorithm 3 is $\mathcal{O}(T_{1}T_{2}^{2})$. If $T_{1}=cT$ and
$T_{2}=(1-c)T$ for some $c\in(0,1)$, then the computational complexity of each
iteration of Algorithm 3 becomes $\mathcal{O}(T^{3})$. Since Problem 6 and
Problem 7 have unique optimal solutions, respectively, every limit point
generated by Algorithm 3 is a stationary point of Problem 5 [31, Proposition
2.7.1].

The following lemma characterizes how the optimal average channel power of the
pure LoS regime scales $T_{1}$, $T_{2}$, and $T$, when they are large.

######  Lemma 5 (Optimal Average Channel Power of D-IRS-C in Pure LoS Regime).

(i) $\bar{\gamma}^{(3)\star}\stackrel{{\scriptstyle
T_{1},T_{2}\rightarrow\infty}}{{\sim}}\alpha_{S1}\alpha_{12}\alpha_{2U}T_{S}T_{1}^{2}T_{2}^{2}.$
(ii) If $T_{1}=cT$ and $T_{2}=(1-c)T$ for some $c\in(0,1)$, then
$\bar{\gamma}^{(3)\star}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}\alpha_{S1}\alpha_{12}\alpha_{2U}c^{2}(1-c)^{2}T_{S}T^{4}.$

###### Proof:

Following the proof of Theorem 4, we can show Lemma 5. ∎

###  IV-C Optimization in Pure NLoS Regime

As shown in Corollary 2, the average channel power in the pure NLoS regime,
i.e., $\tilde{\gamma}^{(0)}$, does not change with $\bm{\phi}_{1}$ or
$\bm{\phi}_{2}$. Thus, there is no need to further optimize the phase shifts
in this regime. However, the deployment of reflecting elements still
influences $\tilde{\gamma}^{(0)}$. The following lemma characterizes how
$\tilde{\gamma}^{(0)}$ scales $T_{1}$, $T_{2}$, and $T$, when they are large.

######  Lemma 6 (Optimal Average Channel Power of D-IRS-C in Pure NLoS
Regime).

(i) $\tilde{\gamma}^{(0)}\stackrel{{\scriptstyle
T_{1},T_{2}\rightarrow\infty}}{{\sim}}\alpha_{S1}\alpha_{12}\alpha_{2U}T_{S}T_{1}T_{2}$.
(ii) If $T_{1}=cT$ and $T_{2}=(1-c)T$ for some $c\in(0,1)$, then
$\tilde{\gamma}^{(0)}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}\alpha_{S1}\alpha_{12}\alpha_{2U}c(1-c)T_{S}T^{2}$.

###### Proof:

By showing that
$\lim_{T_{1},T_{2}\rightarrow\infty}\frac{\tilde{\gamma}^{(0)}}{\alpha_{S1}\alpha_{12}\alpha_{2U}T_{S}T_{1}T_{2}}=1$
and
$\lim_{T_{1},T_{2}\rightarrow\infty}\frac{\tilde{\gamma}^{(0)}}{\alpha_{S1}\alpha_{12}\alpha_{2U}c(1-c)T_{S}T^{2}}=1$,
we can show $\tilde{\gamma}^{(0)}\stackrel{{\scriptstyle
T_{1},T_{2}\rightarrow\infty}}{{\sim}}\alpha_{S1}\alpha_{12}\alpha_{2U}T_{S}T_{1}T_{2}$
and $\tilde{\gamma}^{(0)}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}\alpha_{S1}\alpha_{12}\alpha_{2U}c(1-c)T_{S}T^{2}$,
respectively.∎

##  V Comparison

In this section, we consider two counterpart IRS-assisted systems, namely
double-IRS non-cooperatively assisted system (D-IRS-NC) and single-IRS-
assisted system (S-IRS), as shown in Fig. 3. The assumptions adopted for the
two systems are similar to those for D-IRS-C in Section II. First, we analyze
and optimize the average channel powers of their quasi-static phase shift
designs. Then, we compare their optimal quasi-static phase shift designs with
the optimal quasi-static phase shift design of D-IRS-C in computational
complexity and average channel power.

(a) Double-IRS non-cooperatively assisted system (D-IRS-NC).

(b) Single-IRS-assisted system (S-IRS).

Figure 3: Counterpart IRS-assisted systems.

###  V-A Double-IRS Non-cooperatively Assisted System (D-IRS-NC)

In this part, we analyze and optimize the average channel power of the quasi-
static phase shift design of D-IRS-NC, which is almost the same as D-IRS-C but
has no inter-IRS channel. More specifically, there exist three channels, i.e.,
the direct channel $SU$ and cascaded channels $(S1,1U)$, $(S2,2U)$,
represented by $\mathbf{h}_{SU}^{H}$,
$\mathbf{h}_{1U}^{H}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\mathbf{H}_{S1}$, and
$\mathbf{h}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\mathbf{H}_{S2}$,
respectively, as shown in Fig. 3 (a). Thus, for D-IRS-NC, the equivalent
channel between the BS and user, denoted by
$\mathbf{h}_{DNC,e}^{H}(\bm{\phi}_{1},\bm{\phi}_{2})\in\mathbb{C}^{1\times
T_{S}}$, is given by:

$\displaystyle\mathbf{h}_{DNC,e}^{H}(\bm{\phi}_{1},\bm{\phi}_{2})=\mathbf{h}_{SU}^{H}+\sum_{l\in\mathcal{L}}\mathbf{h}_{lU}^{H}\mathrm{diag}{(\mathbf{v}_{l}^{H})}\mathbf{H}_{Sl},$
(32)

where $\mathbf{H}_{Sl}$ and $\mathbf{h}_{SU}^{H}$, $\mathbf{h}_{lU}^{H}$ are
given by (1) and (2), respectively. Note that
$\mathbf{h}_{DNC,e}^{H}(\bm{\phi}_{1},\bm{\phi}_{2})$ in (32) is identical to
$\mathbf{h}_{e}^{H}(\bm{\phi}_{1},\bm{\phi}_{2})$ in (6) with $\alpha_{12}=0$.
Let
$\gamma_{DNC}(\bm{\phi}_{1},\bm{\phi}_{2})\triangleq\mathbb{E}\left[\|\mathbf{h}_{DNC,e}(\bm{\phi}_{1},\bm{\phi}_{2})\|_{2}^{2}\right]$
represent the average channel power of the equivalent channel of D-IRS-NC.

First, we characterize the influences of $\bm{\phi}_{1}$ and $\bm{\phi}_{2}$
on $\gamma_{DNC}(\bm{\phi}_{1},\bm{\phi}_{2})$.

######  Lemma 7 (Influences of Phase Shifts of D-IRS-NC).

For $l\in\mathcal{L}$, if $K_{Sl}K_{lU}=0$ (i.e., $K_{Sl}=0$ or $K_{lU}=0$),
then $\gamma_{DNC}(\bm{\phi}_{1},\bm{\phi}_{2})$ does not change with
$\bm{\phi}_{l}$.

###### Proof:

Following the proof of Lemma 1, we can show Lemma 7. ∎

Based on Lemma 7, we can also divide the channel conditions of D-IRS-NC into
four cases.

  * •

Case 0 ($K_{S1}=0$ or $K_{1U}=0$ and $K_{S2}=0$ or $K_{2U}=0$):
$\gamma_{DNC}(\bm{\phi}_{1},\bm{\phi}_{2})$ does not change with
$\bm{\phi}_{1}$ or $\bm{\phi}_{2}$ and hence is rewritten as
$\gamma_{DNC}^{(0)}$.

  * •

Case 1 ($K_{S1}$, $K_{1U}>0$ and $K_{S2}=0$ or $K_{2U}=0$):
$\gamma_{DNC}(\bm{\phi}_{1},\bm{\phi}_{2})$ changes only with $\bm{\phi}_{1}$
and hence is rewritten as $\gamma_{DNC}^{(1)}(\bm{\phi}_{1})$.

  * •

Case 2 ($K_{S1}=0$ or $K_{1U}=0$ and $K_{S2}$, $K_{2U}>0$):
$\gamma_{DNC}(\bm{\phi}_{1},\bm{\phi}_{2})$ changes only with $\bm{\phi}_{2}$
and hence is rewritten as $\gamma_{DNC}^{(2)}(\bm{\phi}_{2})$.

  * •

Case 3 ($K_{S1}$, $K_{1U}$, $K_{S2}$, $K_{2U}>0$):
$\gamma_{DNC}(\bm{\phi}_{1},\bm{\phi}_{2})$ changes with both $\bm{\phi}_{1}$
and $\bm{\phi}_{2}$ and is also written as
$\gamma_{DNC}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$.

Next, we characterize the average channel power of D-IRS-NC in the general,
pure LoS, and pure NLoS regimes.

######  Theorem 5 (Average Channel Power of D-IRS-NC in General Regime).

$\displaystyle\gamma_{DNC}^{(0)}=$
$\displaystyle\alpha_{SU}T_{S}+\sum_{l=1}^{2}(L_{\widetilde{Sl},\overline{lU}}+L_{\overline{Sl},\widetilde{lU}}+L_{\widetilde{Sl},\widetilde{lU}})T_{S}T_{l}$
(33) $\displaystyle\gamma_{DNC}^{(1)}(\bm{\phi}_{1})=$ $\displaystyle
L_{\overline{S1},\overline{1U}}\mathbf{v}_{1}^{H}\mathbf{A}_{11}\mathbf{v}_{1}+2\mathfrak{R}\left\\{\sqrt{L_{\overline{SU}}L_{\overline{S1},\overline{1U}}}\mathbf{v}_{1}^{H}\mathbf{b}_{11}\right\\}+\gamma_{DNC}^{(0)},$
(34) $\displaystyle\gamma_{DNC}^{(2)}(\bm{\phi}_{2})=$ $\displaystyle
L_{\overline{S2},\overline{2U}}\mathbf{v}_{2}^{H}\mathbf{A}_{21}\mathbf{v}_{2}+2\mathfrak{R}\left\\{\sqrt{L_{\overline{SU}}L_{\overline{S2},\overline{2U}}}\mathbf{v}_{2}^{H}\mathbf{b}_{21}\right\\}+\gamma_{DNC}^{(0)},$
(35) $\displaystyle\gamma_{DNC}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})=$
$\displaystyle\gamma_{DNC}^{(1)}(\bm{\phi}_{1})+\gamma_{DNC}^{(2)}(\bm{\phi}_{2})+\sqrt{L_{\overline{S1},\overline{1U}}L_{\overline{S2},\overline{2U}}}\mathbf{v}_{2}^{H}\mathbf{B}_{5}\mathbf{v}_{1}-\gamma_{DNC}^{(0)}.$
(36)

###### Proof:

Following the proof of Theorem 1, we can show Theorem 5. ∎

Similarly, $\gamma_{DNC}^{(0)}$, $\gamma_{DNC}^{(1)}(\bm{\phi}_{1})$,
$\gamma_{DNC}^{(2)}(\bm{\phi}_{2})$, and
$\gamma_{DNC}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$ in Theorem 5 are identical
to $\gamma^{(0)}$, $\gamma^{(1)}(\bm{\phi}_{1})$,
$\gamma^{(2)}(\bm{\phi}_{2})$, and $\gamma^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$
in Theorem 1 with $\alpha_{12}=0$ (i.e.,
$L_{\overline{12}}=L_{\widetilde{12}}=0$), respectively. According to the
definitions of the four cases, Cases 0, 1, and 2 of D-IRS-NC are void in the
pure LoS regime, and Cases 1, 2, and 3 of D-IRS-NC are void in the pure NLoS
regime. Besides, Theorem 5 readily implies the following.

######  Corollary 3 (Average Channel Power of D-IRS-NC in Pure LoS Regime).

As $K_{S1}$, $K_{S2}$, $K_{SU}$, $K_{1U}$, $K_{2U}\rightarrow\infty$,
$\gamma_{DNC}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})\rightarrow\bar{\gamma}_{DNC}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$,
where

$\displaystyle\bar{\gamma}_{DNC}^{(3)}$
$\displaystyle(\bm{\phi}_{1},\bm{\phi}_{2})\triangleq\alpha_{SU}T_{S}+\alpha_{S1}\alpha_{1U}\mathbf{v}_{1}^{H}\mathbf{A}_{11}\mathbf{v}_{1}+\alpha_{S2}\alpha_{2U}\mathbf{v}_{2}^{H}\mathbf{A}_{21}\mathbf{v}_{2}$
$\displaystyle+2\mathfrak{R}\left\\{\sqrt{\alpha_{SU}\alpha_{S1}\alpha_{1U}}\mathbf{v}_{1}^{H}\mathbf{b}_{11}+\sqrt{\alpha_{SU}\alpha_{S2}\alpha_{2U}}\mathbf{v}_{2}^{H}\mathbf{b}_{21}+\sqrt{\alpha_{S1}\alpha_{1U}\alpha_{S2}\alpha_{2U}}\mathbf{v}_{2}^{H}\mathbf{B}_{5}\mathbf{v}_{1}\right\\}.$
(37)

######  Corollary 4 (Average Channel Power of D-IRS-NC in Pure NLoS Regime).

If $K_{S1}$, $K_{S2}$, $K_{SU}$, $K_{1U}$, $K_{2U}=0$, then
$\gamma_{DNC}^{(0)}=\tilde{\gamma}_{DNC}^{(0)}$, where

$\displaystyle\tilde{\gamma}_{DNC}^{(0)}\triangleq\alpha_{SU}T_{S}+\alpha_{S1}\alpha_{1U}T_{S}T_{1}+\alpha_{S2}\alpha_{2U}T_{S}T_{2}.$
(38)

Analogously, $\bar{\gamma}_{DNC}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$ in
Corollary 3 and $\tilde{\gamma}_{DNC}^{(0)}$ in Corollary 4 are identical to
$\bar{\gamma}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$ in Corollary 1 with
$\alpha_{12}=0$ and $\tilde{\gamma}^{(0)}$ in Corollary 2 with
$\alpha_{12}=0$, respectively. Noting that $\gamma_{DNC}^{(0)}$ and
$\tilde{\gamma}_{DNC}^{(0)}$ do not change with $\bm{\phi}_{1}$ and
$\bm{\phi}_{2}$, for D-IRS-NC, we optimize the phase shifts only in Cases 1,
2, and 3 of the general regime and in Case 3 of the pure LoS regime.
Specifically, for D-IRS-NC, the maximization of
$\gamma_{DNC}^{(l)}(\bm{\phi}_{l})$ w.r.t. $\bm{\phi}_{l}$ in Case $l=1,2$ of
the general regime is formulated as:

$\displaystyle\begin{aligned}
\gamma_{DNC}^{(l)\star}\triangleq\max_{\bm{\phi}_{l}}&\quad\gamma_{DNC}^{(l)}(\bm{\phi}_{l})\\\
s.t.&\quad\phi_{l,t}\in[0,2\pi),t\in\mathcal{T}_{l},\end{aligned}$ (39)

where $\gamma_{DNC}^{(1)}(\bm{\phi}_{1})$ and
$\gamma_{DNC}^{(2)}(\bm{\phi}_{2})$ are given by (34) and (35), respectively.
Let $\bm{\phi}_{DNC,l}^{\star}$ denote an optimal solution for Case $l=1,2$.
The optimal solution of the problem in (39) is given below.

######  Lemma 8 (Optimal Solution of Problem in (39)).

The unique optimal solution of Problem in (39) for Case $l=1,2$ is given by:

$\displaystyle\bm{\phi}_{DNC,l}^{\star}=\Lambda(-\bm{\Delta}_{\overline{Sl},\overline{lU}}-\angle(r_{\overline{Sl},\overline{SU}})\mathbf{1}_{T_{l}}).$
(40)

###### Proof:

Following the proof of Theorem 2, we can show Lemma 8. ∎

Lemma 8 can be viewed as a special case of Theorem 2. Lemma 8 indicates that
for Case $l=1,2$ of D-IRS-NC in the general regime, the optimal phase changes
over the cascaded LoS channel $(\overline{Sl},\overline{lU})$ are identical to
the phase changes over the LoS channel $\overline{SU}$. Furthermore, the
computational complexity for calculating $\bm{\phi}_{DNC,l}^{\star}$ based on
Lemma 8 is $\mathcal{O}(T)$.

For D-IRS-NC, the maximization of
$\gamma_{DNC}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$ w.r.t. $\bm{\phi}_{1}$ and
$\bm{\phi}_{2}$ in Case 3 of the general regime is formulated as:

$\displaystyle\gamma_{DNC}^{(3)\star}\triangleq\max_{\bm{\phi}_{1},\bm{\phi}_{2}}$
$\displaystyle\quad\gamma_{DNC}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$ (41)
$\displaystyle s.t.$ $\displaystyle\quad\eqref{eq:unitconstrint},$

where $\gamma_{DNC}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$ is given by (36). Let
$(\bm{\phi}_{DNC,1}^{\star},\bm{\phi}_{DNC,2}^{\star})$ denote an optimal
solution. Besides, for D-IRS-NC, the maximization of
$\gamma_{DNC}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$ w.r.t. $\bm{\phi}_{1}$ and
$\bm{\phi}_{2}$ in Case 3 of the pure LoS regime is formulated as:

$\displaystyle\bar{\gamma}_{DNC}^{(3)\star}\triangleq\max_{\bm{\phi}_{1},\bm{\phi}_{2}}$
$\displaystyle\quad\bar{\gamma}_{DNC}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$ (42)
$\displaystyle s.t.$ $\displaystyle\quad\eqref{eq:unitconstrint},$

where $\bar{\gamma}_{DNC}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$ is given by (3).
Let $(\bm{\bar{}}{\phi}_{DNC,1}^{\star},\bm{\bar{}}{\phi}_{DNC,2}^{\star})$
denote an optimal solution.

The optimal solutions of the problem in (41) and the problem in (42) are given
below.

######  Lemma 9 (Optimal Solutions of Problem in (41) and Problem in (42)).

The unique optimal solutions of the problem in (41) and the problem in (42)
are given by:

$\displaystyle\bm{\phi}_{DNC,1}^{\star}=\bm{\bar{}}{\bm{\phi}}_{DNC,1}^{\star}$
$\displaystyle=\Lambda(-\bm{\Delta}_{\overline{S1},\overline{1U}}-\angle(r_{\overline{S1},\overline{SU}})\mathbf{1}_{T_{1}}),$
(43)
$\displaystyle\bm{\phi}_{DNC,2}^{\star}=\bm{\bar{}}{\bm{\phi}}_{DNC,2}^{\star}$
$\displaystyle=\Lambda(-\bm{\Delta}_{\overline{S2},\overline{2U}}-(\angle(r_{\overline{S2},\overline{SU}})-\angle(r_{\overline{S1},\overline{S2}}))\mathbf{1}_{T_{2}}).$
(44)

###### Proof:

Following the proof of Theorem 2, we can show Lemma 9. ∎

Lemma 9 indicates that for D-IRS-NC in Case 3 of the general and pure LoS
regimes, the optimal phase changes over the cascaded LoS channels
$(\overline{S1},\overline{1U})$ and $(\overline{S2},\overline{2U})$ and the
LoS channel $\overline{SU}$ are identical. Furthermore, the computational
complexities for calculating $\bm{\phi}_{DNC,1}^{\star}$
($\bm{\bar{}}{\bm{\phi}}_{DNC,1}^{\star}$) and $\bm{\phi}_{DNC,2}^{\star}$
($\bm{\bar{}}{\bm{\phi}}_{DNC,2}^{\star}$) based on Lemma 9 are
$\mathcal{O}(T)$.

Finally, we characterize the optimal average channel power of D-IRS-NC in the
general, pure LoS, and pure NLoS regimes at large $T_{1}$, $T_{2}$, and $T$.

######  Theorem 6 (Optimal Average Channel Power of D-IRS-NC).

(i) $\gamma_{DNC}^{(0)}\stackrel{{\scriptstyle
T_{1},T_{2}\rightarrow\infty}}{{\sim}}\sum_{l\in\mathcal{L}}(L_{\widetilde{Sl},\overline{lU}}+L_{\overline{Sl},\widetilde{lU}}+L_{\widetilde{Sl},\widetilde{lU}})T_{S}T_{l}$,
$\gamma_{DNC}^{(1)\star}\stackrel{{\scriptstyle
T_{1},T_{2}\rightarrow\infty}}{{\sim}}L_{\overline{S1},\overline{1U}}T_{S}T_{1}^{2}$,
$\gamma_{DNC}^{(2)\star}\stackrel{{\scriptstyle
T_{1},T_{2}\rightarrow\infty}}{{\sim}}L_{\widetilde{S2},\overline{2U}}T_{S}T_{2}^{2}$,
and $\gamma_{DNC}^{(3)\star}\stackrel{{\scriptstyle
T_{1},T_{2}\rightarrow\infty}}{{\sim}}T_{S}(\sum_{l\in\mathcal{L}}\sqrt{L_{\overline{Sl},\overline{lU}}}T_{l})^{2}$;
$\tilde{\gamma}_{DNC}^{(0)}\stackrel{{\scriptstyle
T_{1},T_{2}\rightarrow\infty}}{{\sim}}\sum_{l\in\mathcal{L}}\alpha_{Sl}\alpha_{lU}T_{S}T_{l}$,
and
$\bar{\gamma}_{DNC}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})\stackrel{{\scriptstyle
T_{1},T_{2}\rightarrow\infty}}{{\sim}}T_{S}(\sum_{l\in\mathcal{L}}$
$\sqrt{\alpha_{Sl}\alpha_{lU}}T_{l})^{2}.$ (ii) If $T_{1}=cT$ and
$T_{2}=(1-c)T$ for some $c\in(0,1)$, then
$\gamma_{DNC}^{(0)}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}c(L_{\widetilde{S1},\overline{1U}}+L_{\overline{S1},\widetilde{1U}}+L_{\widetilde{S1},\widetilde{1U}})T_{S}T+(1-c)(L_{\widetilde{S2},\overline{2U}}+L_{\overline{S2},\widetilde{2U}}+L_{\widetilde{S2},\widetilde{2U}})T_{S}T$,
$\gamma_{DNC}^{(1)\star}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}cL_{\overline{S1},\overline{1U}}T_{S}T^{2}$,
$\gamma_{DNC}^{(2)\star}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}(1-c)L_{\overline{S2},\overline{2U}}T_{S}T^{2}$,
and $\gamma_{DNC}^{(3)\star}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}(c\sqrt{L_{\overline{S1},\overline{1U}}}+(1-c)\sqrt{L_{\overline{S2},\overline{2U}}})^{2}T_{S}T^{2}$;
$\tilde{\gamma}_{DNC}^{(0)}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}(c\alpha_{S1}\alpha_{1U}+(1-c)\alpha_{S2}\alpha_{2U})T_{S}T$,
and
$\bar{\gamma}_{DNC}^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}(c\sqrt{\alpha_{S1}\alpha_{1U}}+(1-c)\sqrt{{\alpha_{S2}\alpha_{2U}}})^{2}T_{S}T^{2}.$

###### Proof:

Following the proof of Theorem 4, we can show Theorem 6. ∎

By Theorem 6 and [8, Proposition 2], for D-IRS-NC, the optimal quasi-static
phase shift design achieves the same average power gain in order w.r.t. $T$
(i.e., $\Theta(T^{2})$) as the optimal instantaneous CSI-adaptive phase shift
design in [8]. Besides, for D-IRS-NC in the general regime, the optimal
average channel powers in Cases 1, 2, and 3 (i.e., $\Theta(T^{2})$) are
equivalent in order and are higher in order than the average channel power in
Case 0 (i.e., $\Theta(T)$).

######  Remark 2 (Quasi-static Phase Shift Design for D-IRS-NC).

The results in Section V.A extend the analysis and optimization results on
quasi-static phase shift design for a multi-IRS non-cooperatively assisted
system in [14], where the BS is equipped with a single antenna. Moreover, note
that the optimal average channel power of D-IRS-NC with a quasi-static phase
shift design and a large number of reflecting elements has not been
characterized in the existing literature.

###  V-B Single-IRS-Assisted System (S-IRS)

In this part,  we analyze and optimize the average channel power of the quasi-
static phase shift design of S-IRS where the BS serves the user with the help
of one IRS indexed by 0 and equipped with a URA of $M_{0}\times N_{0}$
elements, as shown in Fig. 3 (b). Define
$\mathcal{M}_{0}\triangleq\\{1,..,M_{0}\\}$,
$\mathcal{N}_{0}\triangleq\\{1,..,N_{0}\\}$, and
$\mathcal{T}_{0}\triangleq\\{1,..,M_{0}N_{0}\\}$. Let
$\bm{\phi}_{0}\triangleq\left(\phi_{0,t}\right)_{t\in\mathcal{T}_{0}}\in\mathbb{R}^{T_{0}\times
1}$ denote the constant phase shifts of IRS 0, where its $t$-th element
$\phi_{0,t}$ satisfies:

$\displaystyle\phi_{0,t}\in[0,2\pi),t\in\mathcal{T}_{0}.$ (45)

Accordingly, denote
$\mathbf{v}_{0}\triangleq\left(e^{-j\phi_{0,t}}\right)_{t\in\mathcal{T}_{0}}\in\mathbb{C}^{T}$.
For a fair comparison, we let $M_{0}N_{0}=T$ and place IRS 0 at IRS 1’s
location in D-IRS-C. The other setups of S-IRS remain the same as those of
D-IRS-C. Let $\mathbf{H}_{S0}\in\mathbb{C}^{T\times T_{S}}$ and
$\mathbf{h}_{0U}^{H}\in\mathbb{C}^{1\times T_{S}}$ represent the Rician
channel between the BS and IRS 0 and the Rician channel between IRS 0 and the
user, respectively. Specifically, we have:

$\displaystyle\mathbf{H}_{S0}$
$\displaystyle=\sqrt{\alpha_{S0}}\bigg{(}\sqrt{\frac{K_{S0}}{K_{S0}+1}}\bar{\mathbf{H}}_{S0}+\sqrt{\frac{1}{K_{S0}+1}}\tilde{\mathbf{H}}_{S0}\bigg{)},$
$\displaystyle\mathbf{h}_{0U}^{H}$
$\displaystyle=\sqrt{\alpha_{0U}}\bigg{(}\sqrt{\frac{K_{0U}}{K_{0U}+1}}\bar{\mathbf{h}}_{0U}^{H}+\sqrt{\frac{1}{K_{0U}+1}}\tilde{\mathbf{h}}_{0U}^{H}\bigg{)},$

where $\alpha_{S0}$, $\alpha_{0U}>0$ represent the large-scale fading powers;
$K_{S0}$, $K_{0U}\geq 0$ represent the Rician factors;
$\tilde{\mathbf{H}}_{S0}\in\mathbb{C}^{T\times T_{S}}$ and
$\tilde{\mathbf{h}}_{0U}^{H}\in\mathbb{C}^{1\times T_{S}}$ represent the
random normalized NLoS components in a slot with elements i.i.d. according to
$\mathcal{CN}(0,1)$;
$\bar{\mathbf{H}}_{S0}=\mathbf{a}_{A,S0}\mathbf{a}_{D,S0}^{H}\in\mathbb{C}^{T\times
T_{S}}$ and
$\tilde{\mathbf{h}}_{0U}^{H}=\mathbf{a}^{H}_{D,0U}\in\mathbb{C}^{1\times
T_{S}}$ represent the deterministic normalized LoS components with unit-
modulus elements. Then, for S-IRS, the equivalent channel between the BS and
user, denoted by $\mathbf{h}_{SGL,e}^{H}(\bm{\phi}_{0})\in\mathbb{C}^{1\times
T_{S}}$, is given by:

$\displaystyle\mathbf{h}_{SGL,e}^{H}(\bm{\phi}_{0})=\mathbf{h}_{SU}^{H}+\mathbf{h}_{0U}^{H}\mathrm{diag}(\mathbf{v}_{0}^{H})\mathbf{H}_{S0}.$

Let
$\gamma_{SGL}(\bm{\phi}_{0})\triangleq\mathbb{E}\left[\|\mathbf{h}_{SGL,e}(\bm{\phi}_{0})\|_{2}^{2}\right]$
represent the average channel power of the equivalent channel of S-IRS.

For ease of illustration, denote
$\mathbf{A}_{0}\triangleq\mathrm{diag}(\bar{\mathbf{h}}_{0U}^{H})\bar{\mathbf{H}}_{S0}\bar{\mathbf{H}}_{S0}^{H}\mathrm{diag}(\bar{\mathbf{h}}_{0U})\in\mathbb{C}^{T\times
T}$ and $\mathbf{b}_{0}\triangleq\mathrm{diag}(\bar{\mathbf{h}}_{0U}^{H})$
$\bar{\mathbf{H}}_{S0}\bar{\mathbf{h}}_{SU}\in\mathbb{C}^{T}$. Now, we
characterize the average channel power of S-IRS in the general, pure LoS, and
pure NLoS regimes.

######  Lemma 10 (Average Channel Power of S-IRS).

(i) For any $K_{S0},K_{0U}\geq 0$,
$\gamma_{SGL}(\bm{\phi}_{0})=L_{\overline{S0},\overline{0U}}\mathbf{v}_{0}^{H}\mathbf{A}_{0}\mathbf{v}_{0}+2\mathfrak{R}\left\\{\sqrt{L_{\overline{SU}}L_{\overline{S0},\overline{0U}}}\mathbf{v}_{0}^{H}\mathbf{b}_{0}\right\\}+\alpha_{SU}T_{S}+(L_{\widetilde{S0},\overline{0U}}+L_{\overline{S0},\widetilde{0U}}+L_{\widetilde{S0},\widetilde{0U}})T_{S}T.$
(ii) As $K_{S0},K_{0U}\rightarrow\infty$,
$\gamma_{SGL}(\bm{\phi}_{0})\rightarrow\bar{\gamma}_{SGL}(\bm{\phi}_{0})\triangleq\alpha_{S0}\alpha_{0U}\mathbf{v}_{0}^{H}\mathbf{A}_{0}\mathbf{v}_{0}+\alpha_{SU}T_{S}+2\mathfrak{R}\left\\{\sqrt{\alpha_{SU}\alpha_{S0}\alpha_{0U}}\mathbf{v}_{0}^{H}\mathbf{b}_{0}\right\\}$.
(iii) If $K_{S0}=K_{0U}=0$, then
$\gamma_{SGL}(\bm{\phi}_{0})=\tilde{\gamma}_{SGL}$, where
$\tilde{\gamma}_{SGL}\triangleq\alpha_{SU}T_{S}+\alpha_{S0}\alpha_{0U}T_{S}T$.

###### Proof:

Following the proof of Theorem 1, we can show Lemma 10. ∎

By Corollary 4 and Lemma 10, if
$\alpha_{Sl}\alpha_{lU}=\alpha_{S0}\alpha_{0U},l=1,2$, then
$\tilde{\gamma}_{SGL}=\tilde{\gamma}_{DNC}^{(0)}$. Noting that
$\tilde{\gamma}_{SGL}$ does not change with $\bm{\phi}_{0}$, we optimize the
phase shifts only in the general and pure LoS regimes. Specifically, in the
general regime, the maximization of $\gamma_{SGL}(\bm{\phi}_{0})$ w.r.t.
$\bm{\phi}_{0}$ is formulated as:

$\displaystyle\gamma_{SGL}^{\star}\triangleq\max_{\bm{\phi}_{0}}$
$\displaystyle\quad\gamma_{SGL}(\bm{\phi}_{0})$ (46) $\displaystyle s.t.$
$\displaystyle\quad\eqref{eq:singleConstriant}.$

Let $\bm{\phi}_{0}^{\star}$ denote an optimal solution of the problem in (46).
In the pure LoS regime, the maximization of
$\bar{\gamma}_{SGL}(\bm{\phi}_{0})$ w.r.t. $\bm{\phi}_{0}$ is formulated as:

$\displaystyle\bar{\gamma}_{SGL}^{\star}\triangleq\max_{\bm{\phi}_{0}}$
$\displaystyle\quad\bar{\gamma}_{SGL}(\bm{\phi}_{0})$ (47) $\displaystyle
s.t.$ $\displaystyle\quad\eqref{eq:singleConstriant}.$

Let $\bar{\bm{\phi}}_{0}^{\star}$ denote an optimal solution of the problem in
(47).

Define
$\bm{\Delta}_{\overline{S0},\overline{0U}}\triangleq\angle\left(\mathrm{diag}(\mathbf{a}_{D,0U}^{H})\mathbf{a}_{A,S0}\right)\in\mathbb{R}^{T}$
and
$r_{\overline{S0},\overline{SU}}\triangleq\mathbf{a}_{D,S0}^{H}\mathbf{a}_{D,SU}\in\mathbb{C}$.
The optimal solutions of the problem in (46) and problem in (47) are given
below.

######  Lemma 11 (Optimal Solutions of Problem in (46) and Problem in (47)).

The unique optimal solutions of the problem in (46) and problem in (47) are
given by:

$\displaystyle\bm{\phi}_{0}^{\star}=\bar{\bm{\phi}}_{0}^{\star}=\Lambda\left(-\bm{\Delta}_{\overline{S0},\overline{0U}}-\angle\left(r_{\overline{S0},\overline{SU}}\right)\mathbf{1}_{T}\right).$
(48)

###### Proof:

Following the proof of Theorem 2, we can show Lemma 11. ∎

Lemma 11 indicates that for S-IRS in the general and pure LoS regimes, the
optimal phase changes of over the cascaded LoS channel
$(\overline{S0},\overline{0U})$ are identical to the phase changes of over the
LoS channel $\overline{SU}$. Furthermore, the computational complexities for
calculating $\bm{\phi}_{0}^{\star}$ and $\bar{\bm{\phi}}_{0}^{\star}$ based on
Lemma 11 are $\mathcal{O}(T)$.

Finally, for S-IRS, we characterize the optimal average channel power in the
general and pure LoS regimes and the average channel power in the pure NLoS
regime at large $T$.

######  Lemma 12 (Optimal Average Channel Power of S-IRS).

$\gamma_{SGL}^{\star}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}L_{\overline{S0},\overline{0U}}T_{S}T^{2}$,
$\bar{\gamma}_{SGL}^{\star}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}\alpha_{S0}\alpha_{0U}T_{S}T^{2}$, and
$\tilde{\gamma}_{SGL}\stackrel{{\scriptstyle
T\rightarrow\infty}}{{\sim}}\alpha_{S0}\alpha_{0U}T_{S}T$.

###### Proof:

Following the proof of Theorem 4, we can show Lemma 12. ∎

By Lemma 12 and [3, Proposition 2], for S-IRS, the optimal quasi-static phase
shift design achieves the same average power gain in order w.r.t. $T$ (i.e.,
$\Theta(T^{2})$) as the optimal instantaneous CSI-adaptive phase shift design
in [3]. Besides, for S-IRS, the optimal average channel powers in the general
and pure LoS regimes (i.e., $\Theta(T^{2})$) are equivalent in order and are
higher in order than the average channel power in the pure NLoS regime (i.e.,
$\Theta(T)$).

######  Remark 3 (Quasi-static Phase Shift Design for S-IRS).

The results in Section V.B extend the analysis and optimization results on
quasi-static phase shift design for a less general S-IRS in [9] where the
direct channel is modeled as Rayleigh fading. Moreover, note that the optimal
average channel power of S-IRS with a quasi-static phase shift design and a
large number of reflecting elements has not been characterized in the existing
literature.

###  V-C Comparison

In this part, we compare the optimal quasi-static phase shift design of
D-IRS-C with those of D-IRS-NC and S-IRS in computational complexity and
average channel power. Specifically, based on the optimization and analytical
results in Section IV, Section V.A, and Section V.B, we summarize the
computational complexities and average channel powers of the optimal quasi-
static phase shift designs of D-IRS-C, D-IRS-NC, and S-IRS in the general,
pure LoS, and pure NLoS regimes in Table I. Here, we are interested in the
growth rates of the computational complexity and average channel power w.r.t.
the total number of reflecting elements.121212The computational complexity
results are derived from those in Section IV, Section V.A, and Section V.B by
letting $T_{1}=cT$ and $T_{2}=(1-c)T$ for some $c\in(0,1)$ and
$T\rightarrow\infty$. In some cases, there is no need to optimize the phase
shifts, and hence we use $\mathcal{O}(1)$ for the computational complexity and
consider the average channel power.

  * •

Computational Complexity: In each case where the phase shift optimization is
necessary, the closed-form optimal quasi-static phase shift design of D-IRS-C
(if it exists) has the same computational complexity in order as the closed-
form optimal quasi-static phase shift designs of D-IRS-NC and S-IRS, whereas
the numerical quasi-static phase shift design of D-IRS-C has higher
computational complexity in order than the closed-form optimal quasi-static
phase shift designs of D-IRS-NC and S-IRS.

  * •

Optimal Average Channel Power: (i) In the general regime, the optimal average
channel power of D-IRS-NC in Cases 1, 2, and 3 and that of S-IRS are identical
in order. Besides, the optimal average channel powers of D-IRS-NC and S-IRS
are identical in order in the pure LoS and pure NLoS regimes, respectively.
This is because the underlying difference between D-IRS-NC and S-IRS lies in
path loss which does not influence the order of growth of the optimal average
channel power w.r.t. $T$. (ii) In the general regime, the optimal average
channel power of D-IRS-C in each case is higher in order than the optimal
average channel power of D-IRS-NC in the same case and the optimal average
channel power of S-IRS. In the pure LoS and pure NLoS regimes, respectively,
the optimal average channel power of D-IRS-C is higher in order than the
optimal average channel powers of D-IRS-NC and S-IRS. This is because the
additional inter-IRS channel in D-IRS-C can convey signals from the BS to the
user.

  * •

Tradeoff: D-IRS-C achieves a better performance and computational complexity
tradeoff than D-IRS-NC and S-IRS in the cases with closed-form optimal quasi-
static phase shift designs and the cases that do not require optimizing phase
shifts and achieves a different performance and computational complexity
tradeoff in the other cases. For quasi-static phase shift designs, as phase
shifts remain constant during a certain period, the computational complexity
for optimizing phase shifts is negligible. Therefore, D-IRS-C is more
desirable than D-IRS-NC and S-IRS in practice.

TABLE I: Comparisons. Regime | System | Computational Complexity |  | Optimal Average
---
Channel Power
General | D-IRS-C | Case 0 | $\mathcal{O}(1)$ | $\Theta(T^{2})$
Case 1 |  $\mathcal{O}(T^{2})$ (numerical), $\mathcal{O}(T)$ (analytical) | $\Theta(T^{3})$
Case 2
Case 3 |  |  $\mathcal{O}(T^{3})$ (numerical),
---
$\mathcal{O}(T)$ (analytical)
$\Theta(T^{4})$
D-IRS-NC | Case 0 | $\mathcal{O}(1)$ | $\Theta(T)$
Case 1 | $\mathcal{O}(T)$ (analytical) | $\Theta(T^{2})$
Case 2
Case 3
S-IRS |  $\mathcal{O}(T)$ (analytical) | $\Theta(T^{2})$
Pure LoS | D-IRS-C (Case 3) |  $\mathcal{O}(T^{3})$ (numerical) | $\Theta(T^{4})$
D-IRS-NC (Case 3) |  $\mathcal{O}(T)$ (analytical) | $\Theta(T^{2})$
S-IRS |  $\mathcal{O}(T)$ (analytical) | $\Theta(T^{2})$
Pure NLoS | D-IRS-C (Case 3) | $\mathcal{O}(1)$ | $\Theta(T^{2})$
D-IRS-NC (Case 3) | $\mathcal{O}(1)$ | $\Theta(T)$
S-IRS | $\mathcal{O}(1)$ | $\Theta(T)$

##  VI Numerical Results

In this section, we numerically evaluate the average rates of the phase shift
designs for D-IRS-C, D-IRS-NC, S-IRS, and the counterpart system without any
IRS (W/O-IRS) [32]. As shown in Fig. 4, for all systems, the BS and user are
located at $(0,-25,1.2)$ and $(0,25,1)$ (in m), respectively; for D-IRS-C and
D-IRS-NC, IRS 1 and IRS 2 are located at $(-x,-y,5)$ and $(-x,y,5)$,
respectively; for S-IRS, IRS 0 can be located at the locations of IRS 1, IRS
2, or the midpoint between IRS 1 and IRS 2, and the corresponding systems are
termed S-IRS-Pos-1, S-IRS-Pos-2, S-IRS-Pos-Mid, respectively.

In the simulation, we set $d=\frac{\lambda}{2}$, $x=5$, $y=20$,
$M_{S}=N_{S}=2$, $M_{1}=N_{1}=M_{2}=N_{2}=10$, $P_{S}=5\ \text{dBm}$,
$\sigma^{2}=-104\ \text{dBm}$, $\varphi_{S1}^{(h)}=\varphi_{S1}^{(v)}=\pi/6$,
$\varphi_{S2}^{(h)}=\varphi_{S2}^{(v)}=\pi/4$,
$\varphi_{12}^{(h)}=\varphi_{12}^{(v)}=\pi/5$,
$\varphi_{SU}^{(h)}=\varphi_{SU}^{(v)}=\pi/3$,
$\varphi_{1U}^{(h)}=\varphi_{1U}^{(v)}=\pi/8$,
$\varphi_{2U}^{(h)}=\varphi_{2U}^{(v)}=\pi/9$,
$\delta_{S1}^{(h)}=\delta_{S1}^{(v)}=\pi/6$,
$\delta_{S2}^{(h)}=\delta_{S2}^{(v)}=\pi/5$,
$\delta_{12}^{(h)}=\delta_{12}^{(v)}=\pi/4$,
$K_{S1}=K_{S2}=K_{1U}=K_{2U}=K_{SU}=K_{12}=K=10\ \text{dB}$, if not specified
otherwise. We set $\alpha_{ab}=1/(1000d_{ab}^{\bar{\alpha}_{ab}})$ (i.e.,
$-30+10\bar{\alpha}_{ab}\log_{10}(d_{ab})$ dB), $ab=S1,S2,12,1U,2U,SU$, where
$\bar{\alpha}_{ab}$ represents the corresponding path loss exponent. We choose
$\bar{\alpha}_{12}=2.2$, $\bar{\alpha}_{S1}$, $\bar{\alpha}_{2U}$,
$\bar{\alpha}_{S2}$, $\bar{\alpha}_{1U}=2.3$, and $\bar{\alpha}_{SU}=3.7$ [3,
10, 20].131313IRSs are usually placed far above the ground, and their
locations are carefully selected. Thus, the inter-IRS channel experiences the
fewest obstacles and weakest scattering, and the channels $S1$, $S2$, $1U$,
and $2U$ experience fewer obstacles and weaker scattering. Besides, the BS and
user are usually located on the ground. Thus, the direct channel $SU$
experiences the most obstacles and strongest scattering [3, 10, 20]. We
evaluate seven schemes, all adopting the instantaneous CSI-adaptive MRT
beamformer at the BS. For any system SYS with IRS, the optimal quasi-static
phase shift design in the general, pure LoS, and pure NLoS regimes are
referred to SYS-Gen, SYS-LoS, and SYS-NLoS, respectively. Note that -Gen is
sometimes omitted for notation simplicity. Besides, for D-IRS-C, D-IRS-C-
Random chooses the phase shifts uniformly at random [10], and D-IRS-C-ICSI
adopts the optimal instantaneous CSI-adaptive phase shift design as in [20].

Figure 4: The system topology in Section VI.

Figure 5: Average rate versus $P_{S}$.

Figure 6: Average rate versus $T_{S}$.

Figure 7: Average rate versus $T_{1}$ at $T_{2}=200-T_{1}$.

Figure 8: Average rate versus $x$ and $y$.

(a) General regime.

(b) Pure LoS regime.

(c) Pure NLoS regime.

Figure 9: Average rate versus $T$ at $T_{1}=T_{2}=T/2$.

(a) Results for general regime.

(b) Results for special regimes.

Figure 10: Average rate versus $K$.

Fig. 7, Fig. 7, Fig. 7, Fig. 8, Fig. 10, and Fig. 10 illustrate the average
rate versus the transmit power $P_{S}$, number of antennas at the BS $T_{S}$,
number of IRS 1’s reflecting elements $T_{1}$, IRS locations, total number of
reflecting elements in the system $T$, and Rician factor $K$, respectively. In
these figures, D-IRS-C-MC represents the numerical expectation of the rate of
D-IRS-C, and each other curve represents the analytical upper bound of the
average rate of a scheme based on Jensen’s inequality. From Fig. 7, Fig. 7,
Fig. 7, Fig. 10, and Fig. 10 (a), we can see that D-IRS-C-MC and D-IRS-C are
very close to each other, indicating that the upper bound,
$\log_{2}\left(1+\frac{P_{S}}{\sigma^{2}}\gamma(\bm{\phi}_{1},\bm{\phi}_{2})\right)$,
is a good approximation of $C(\bm{\phi}_{1},\bm{\phi}_{2})$. In Fig. 10,
D-IRS-C-T, D-IRS-NC-T, and S-IRS-Pos-1-T represent the asymptotic results for
D-IRS-C, D-IRS-NC, and S-IRS-Pos-1 at large $T$, respectively. For each of
them, the gap between the general result at any $T$ and asymptotic result at
large $T$ decreases with $T$, which is in accordance with Theorem 4, Lemma 5,
Lemma 6, Theorem 6, and Lemma 12. In Fig. 10 (b), the gap between D-IRS-C-Gen
and D-IRS-C-LoS (D-IRS-C-NLoS) decreases (increases) with $K$, in accordance
with Corollary 1 (Corollary 2); the gap between D-IRS-NC-Gen and D-IRS-NC-LoS
(D-IRS-NC-NLoS) decreases (increases) with $K$, in accordance with Corollary 3
(Corollary 4); and the gap between S-IRS-Pos-1-Gen and S-IRS-Pos-1-LoS (S-IRS-
Pos-1-NLoS) decreases (increases) with $K$, in accordance with Lemma 10.

From Fig. 7 and Fig. 7, we see that the average rate of each scheme increases
with $P_{S}$ and $T_{S}$, respectively. From Fig. 7, we see that the average
rate of the proposed solution is maximized when the two IRSs have the same
number of elements, mainly due to the symmetric channel setup. From Fig. 8, we
see that the average rates of D-IRS-C, D-IRS-NC, and S-IRS-Pos-1 are maximized
when IRS 1 and IRS 2 are placed closest to the BS and user, respectively. From
Fig. 10, we observe that the average rate of each scheme for IRS-assisted
systems increases with $T$, mainly due to the increment of reflecting signal
power. Fig. 10 shows that the average rate of each quasi-static phase shift
design increases with $K$, mainly due to the increment of the channel power of
each LoS component.

From Fig. 7, Fig. 7, Fig. 7, Fig. 10, and Fig. 10, we can see that D-IRS-C
outperforms D-IRS-C-Random, D-IRS-Pos-Mid, and D-IRS-NC at any total number of
reflecting elements and outperforms D-IRS-NC, S-IRS-Pos-1, and S-IRS-Pos-2 as
long as the total number of reflecting elements is large enough. Specifically,
the gain of D-IRS-C over D-IRS-C-Random comes from the optimization of the
quasi-static phase shifts, and the gain of D-IRS-C over D-IRS-NC derives from
the effective utilization of the inter-IRS channel  that can convey signals
from the BS to the user. The performance gains of D-IRS-C over D-IRS-NC,
S-IRS-Pos-1, and S-IRS-Pos-2 depend on the average channel power of the
cascaded channel $(S1,12,2U)$, which increases with $T_{1}$, $T_{2}$ (as shown
in Fig. 10) and $K_{S1}$, $K_{12}$, $K_{2U}$ (as shown in Fig. 10).
Furthermore, D-IRS-C-ICSI outperforms D-IRS-C at the sacrifice of increased
channel estimation and phase adjustment costs and overall computational
complexity. Thus, quasi-static phase shift design has practical sense as long
as the LoS components are sufficiently large.

##  VII Conclusion

This paper investigated the analysis and optimization of the quasi-static
phase shift design for D-IRS-C. Furthermore, this paper compared the optimal
quasi-static phase shift design of D-IRS-C with those of D-IRS-NC and S-IRS.
Both analytical and numerical results demonstrate notable gains of the
proposed solutions over the existing solutions and reveal insights into
designing practical IRS-assisted systems. This work can be extended to IRS-
cooperatively-assisted systems with multiple IRSs and users.

## Appendix A: Proof of Lemma 1

First, we prove Statement (i). As
$\gamma\left(\bm{\phi}_{1},\bm{\phi}_{2}\right)=\mathbb{E}\left[\|\mathbf{h}_{e}(\bm{\phi}_{1},\bm{\phi}_{2})\|_{2}^{2}\right]$,
with $\mathbf{h}_{e}(\bm{\phi}_{1},\bm{\phi}_{2})$ given by (6), it is
sufficient to show
$\mathbf{h}^{H}_{e}(\bm{\phi}_{1},\bm{\phi}_{2})\overset{d}{\sim}\mathbf{h}_{SU}^{H}+\mathbf{h}_{2U}^{H}\mathrm{diag}(\mathbf{v}_{2}^{H})\mathbf{H}_{S2}+(\mathbf{h}_{1U}^{H}+\mathbf{h}_{2U}^{H}\mathrm{diag}(\mathbf{v}_{2}^{H})\mathbf{H}_{12})\mathbf{H}_{S1}$,
which is irrelevant to $\mathbf{v}_{1}$. By (6), it is equivalent to show
$\mathbf{H}_{S1}\overset{d}{\sim}\mathrm{diag}(\mathbf{v}_{1}^{H})\mathbf{H}_{S1}$
or
$\mathbf{H}_{12}\overset{d}{\sim}\mathbf{H}_{12}\mathrm{diag}(\mathbf{v}_{1}^{H})$
and
$\mathbf{h}_{1U}^{H}\overset{d}{\sim}\mathbf{h}_{1U}^{H}\mathrm{diag}(\mathbf{v}_{1}^{H})$.
First, we show that
$\mathbf{H}_{S1}\overset{d}{\sim}\mathrm{diag}(\mathbf{v}_{1}^{H})\mathbf{H}_{S1}$
if $K_{S1}=0$. If $K_{S1}=0$, then
$\mathbf{H}_{S1}=\sqrt{\alpha_{S1}}\tilde{\mathbf{H}}_{S1}$. As all elements
of $\tilde{\mathbf{H}}_{S1}$ are i.i.d. $\mathcal{CN}(0,1)$,
$\mathrm{vec}(\mathbf{H}_{S1})\sim\mathcal{CN}(0,\alpha_{S1}\mathbf{I}_{T_{S}T_{1}})$.
Besides, we have:

$\displaystyle\mathrm{vec}(\mathrm{diag}(\mathbf{v}_{1}^{H})\mathbf{H}_{S1})\stackrel{{\scriptstyle(a)}}{{=}}(\mathbf{I}_{T_{S}}\otimes\mathrm{diag}(\mathbf{v}_{1}^{H}))\mathrm{vec}(\mathbf{H}_{S1})\stackrel{{\scriptstyle(b)}}{{\sim}}\mathcal{CN}(0,\alpha_{S1}\mathbf{I}_{T_{S}T_{1}}).$
(49)

where (a) is due to [33, Theorem 7.16], and (b) is due to [34, (A.26)] and the
fact that $\mathbf{I}_{T_{S}}\otimes\mathrm{diag}(\mathbf{v}_{1}^{H})$ is an
unitary matrix (noting that $\mathrm{diag}(\mathbf{v}_{1}^{H})$ is an unitary
matrix). Thus, we have
$\mathbf{H}_{S1}\overset{d}{\sim}\mathrm{diag}(\mathbf{v}_{1}^{H})\mathbf{H}_{S1}$.
Next, we show
$\mathbf{H}_{12}\overset{d}{\sim}\mathbf{H}_{12}\mathrm{diag}(\mathbf{v}_{1}^{H})$
and
$\mathbf{h}_{1U}^{H}\overset{d}{\sim}\mathbf{h}_{1U}^{H}\mathrm{diag}(\mathbf{v}_{1}^{H})$
if $K_{12}=K_{1U}=0$. If $K_{12}=K_{1U}=0$, then
$\mathbf{H}_{12}=\sqrt{\alpha_{12}}\tilde{\mathbf{H}}_{12}$ and
$\mathbf{h}_{1U}^{H}=\sqrt{\alpha_{1U}}\tilde{\mathbf{h}}_{1U}^{H}$. As all
elements of $\tilde{\mathbf{H}}_{12}$ and $\tilde{\mathbf{h}}_{1U}^{H}$ are
i.i.d. $\mathcal{CN}(0,1)$,
$\mathrm{vec}(\mathbf{H}_{12})\sim\mathcal{CN}(0,\alpha_{12}\mathbf{I}_{T_{1}T_{2}})$
and $\mathbf{h}_{1U}^{H}\sim\mathcal{CN}(0,\alpha_{1U}\mathbf{I}_{T_{1}})$.
Besides, we have:

$\displaystyle\mathrm{vec}(\mathbf{H}_{12}\mathrm{diag}(\mathbf{v}_{1}^{H}))\stackrel{{\scriptstyle(c)}}{{=}}(\mathrm{diag}(\mathbf{v}_{1}^{H})\otimes\mathbf{I}_{T_{2}})\mathrm{vec}(\mathbf{H}_{12})\stackrel{{\scriptstyle(d)}}{{\sim}}\mathcal{CN}(0,\alpha_{12}\mathbf{I}_{T_{1}T_{2}}),$
(50)
$\displaystyle\mathbf{h}_{1U}^{H}\mathrm{diag}(\mathbf{v}_{1}^{H})\stackrel{{\scriptstyle(e)}}{{\sim}}\mathcal{CN}(0,\alpha_{1U}\mathbf{I}_{T_{1}}).$
(51)

where (c) is due to [33, Theorem 7.16], (d) is due to [34, A.26] and the fact
that $\mathrm{diag}(\mathbf{v}_{1}^{H})\otimes\mathbf{I}_{T_{2}}$ is unitary
matrix, and (e) is due to the fact that $\mathrm{diag}(\mathbf{v}_{1}^{H})$ is
an unitary matrix. Thus, we have
$\mathbf{H}_{12}\overset{d}{\sim}\mathbf{H}_{12}\mathrm{diag}(\mathbf{v}_{1}^{H})$
and
$\mathbf{h}_{1U}^{H}\overset{d}{\sim}\mathbf{h}_{1U}^{H}\mathrm{diag}(\mathbf{v}_{1}^{H})$.
Therefore, we complete the proof of Statement (i).

Next, we prove Statement (ii). Similarly, it is sufficient to show
$\mathbf{h}_{2U}\overset{d}{\sim}\mathbf{H}_{2U}\mathrm{diag}(\mathbf{v}_{2}^{H})$
or
$\mathbf{H}_{12}\overset{d}{\sim}\mathrm{diag}(\mathbf{v}_{2}^{H})\mathbf{H}_{12}$
and
$\mathbf{H}_{S2}\overset{d}{\sim}\mathrm{diag}(\mathbf{v}_{2}^{H})\mathbf{H}_{S2}$.
Following the proof for Statement (i), we can show that
$\mathbf{h}_{2U}\overset{d}{\sim}\mathbf{H}_{2U}\mathrm{diag}(\mathbf{v}_{2}^{H})$
if $K_{2U}=0$, and
$\mathbf{H}_{12}\overset{d}{\sim}\mathrm{diag}(\mathbf{v}_{2}^{H})\mathbf{H}_{12}$
and
$\mathbf{H}_{S2}\overset{d}{\sim}\mathrm{diag}(\mathbf{v}_{2}^{H})\mathbf{H}_{S2}$
if $K_{12}=K_{S2}=0$. Therefore, we can show Statement (ii).

## Appendix B: Proof of Theorem 1

In what follows, we derive the expression of
$\gamma(\bm{\phi}_{1},\bm{\phi}_{2})$ for all
$K_{S1},K_{S2},K_{12},K_{1U},K_{2U},K_{SU}\geq 0$, based on which we can
readily obtain $\gamma^{(0)}$, $\gamma^{(1)}(\bm{\phi}_{1})$,
$\gamma^{(2)}(\bm{\phi}_{2})$, and $\gamma^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$
given by (13), (14), (15), and (16), respectively. Note that for all
$K_{S1},K_{S2},K_{12},K_{1U},K_{2U},K_{SU}\geq 0$, we have:

$\displaystyle\gamma(\bm{\phi}_{1},\bm{\phi}_{2})$
$\displaystyle=\mathbb{E}\left[\|\mathbf{h}_{e}(\bm{\phi}_{1},\bm{\phi}_{2})\|_{2}^{2}\right]\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}\left[\|\underbrace{\mathbf{h}_{SU}^{H}}_{\triangleq\mathbf{x}_{0}^{H}}+\sum_{l\in\mathcal{L}}\underbrace{\mathbf{h}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\mathbf{H}_{Sl}}_{\triangleq\mathbf{x}_{l}^{H}}+\underbrace{\mathbf{h}_{2U}^{H}\mathrm{diag}(\mathbf{v}_{2}^{H})\mathbf{H}_{12}\mathrm{diag}(\mathbf{v}_{1}^{H})\mathbf{H}_{S1}}_{\triangleq\mathbf{x}_{3}^{H}}\|_{2}^{2}\right]$
$\displaystyle=\sum_{i=0}^{3}\mathbb{E}\big{[}\|\mathbf{x}_{i}^{H}\|_{2}^{2}\big{]}+2\sum_{i=0}^{2}\sum_{j=i+1}^{3}\mathfrak{R}\left\\{\mathbb{E}\big{[}\mathbf{x}_{j}^{H}\mathbf{x}_{i}\big{]}\right\\},$
(52)

where $(a)$ is due to (6). Thus, it remains to calculate
$\mathbb{E}\big{[}\|\mathbf{x}_{i}^{H}\|_{2}^{2}\big{]},i=0,1,2,3$ and
$\mathbb{E}\big{[}\mathbf{x}_{j}^{H}\mathbf{x}_{i}\big{]},i=0,1,2,j=i+1,..,3$.
As all elements of $\tilde{\mathbf{h}}_{1U}$, $\tilde{\mathbf{h}}_{2U}$,
$\tilde{\mathbf{h}}_{SU}$, $\tilde{\mathbf{H}}_{S1}$,
$\tilde{\mathbf{H}}_{S2}$, and $\tilde{\mathbf{H}}_{12}$ are i.i.d.
$\mathcal{CN}(0,1)$, we can easily show that multiplying any of them with an
independent random matrix or a constant matrix producing a random matrix (or
vector) with zero mean. This fact will be used in the following proof.

(i) We have:

$\displaystyle\mathbb{E}\big{[}\|\mathbf{x}_{0}^{H}\|_{2}^{2}\big{]}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}\left[\|\sqrt{L_{\overline{SU}}}\bar{\mathbf{h}}_{SU}^{H}+\sqrt{L_{\widetilde{SU}}}\tilde{\mathbf{h}}_{SU}^{H}\|^{2}_{2}\right]\stackrel{{\scriptstyle(b)}}{{=}}L_{\overline{Sl}}\|\bar{\mathbf{h}}_{SU}^{H}\|^{2}_{2}+L_{\widetilde{Sl}}\mathbb{E}\left[\|\tilde{\mathbf{h}}_{SU}^{H}\|^{2}_{2}\right]\stackrel{{\scriptstyle(c)}}{{=}}\alpha_{SU}T_{S},$
(53)

where $(a)$ is due to (2), $(b)$ is due to
$\mathbb{E}[\bar{\mathbf{h}}_{SU}^{H}\tilde{\mathbf{h}}_{SU}]=0$, and $(c)$ is
due to $\|\bar{\mathbf{h}}_{SU}^{H}\|^{2}_{2}=T_{S}$ and
$\mathbb{E}\left[\|\tilde{\mathbf{h}}_{SU}^{H}\|^{2}_{2}\right]=T_{S}$.

(ii) For $l\in\mathcal{L}$, we have:

$\displaystyle\mathbb{E}\left[\|\tilde{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\bar{\mathbf{H}}_{Sl}\|_{2}^{2}\right]$
$\displaystyle=\mathbb{E}\left[\mathrm{Tr}\left(\tilde{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\bar{\mathbf{H}}_{Sl}\bar{\mathbf{H}}_{Sl}^{H}\mathrm{diag}(\mathbf{v}_{l})\tilde{\mathbf{h}}_{lU}\right)\right]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}\left[\mathrm{Tr}\left(\bar{\mathbf{H}}_{Sl}\bar{\mathbf{H}}_{Sl}^{H}\tilde{\mathbf{h}}_{lU}\tilde{\mathbf{h}}_{lU}^{H}\right)\right]\stackrel{{\scriptstyle(b)}}{{=}}\mathrm{Tr}\left(\bar{\mathbf{H}}_{Sl}\bar{\mathbf{H}}_{Sl}^{H}\mathbb{E}\left[\tilde{\mathbf{h}}_{lU}\tilde{\mathbf{h}}_{lU}^{H}\right]\right)$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}\mathrm{Tr}\left(\bar{\mathbf{H}}_{Sl}\bar{\mathbf{H}}_{Sl}^{H}\right)=T_{S}T_{l},$
(54)

where $(a)$ is due to
$\tilde{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\overset{d}{\sim}\tilde{\mathbf{h}}_{lU}^{H}$
(which is shown in Appendix A) and
$\mathrm{Tr}(\mathbf{A}\mathbf{B}\mathbf{C})=\mathrm{Tr}(\mathbf{B}\mathbf{C}\mathbf{A})$,
and $(e)$ is due to
$\mathbb{E}\left[\tilde{\mathbf{h}}_{lU}\tilde{\mathbf{h}}_{lU}^{H}\right]=\mathbf{I}_{T_{l}}$.
Similarly, we can show:

$\displaystyle\mathbb{E}\left[\|\bar{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\tilde{\mathbf{H}}_{Sl}\|_{2}^{2}\right]=T_{S}T_{l},\quad\mathbb{E}\left[\|\tilde{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\tilde{\mathbf{H}}_{Sl}\|_{2}^{2}\right]=T_{S}T_{l}.$
(55)

Thus, we have:

$\displaystyle\mathbb{E}\big{[}\|\mathbf{x}_{l}^{H}\|_{2}^{2}\big{]}$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}\mathbb{E}\big{[}\|(\sqrt{L_{\overline{lU}}}\bar{\mathbf{h}}_{lU}^{H}+\sqrt{L_{\widetilde{lU}}}\tilde{\mathbf{h}}_{lU}^{H})\mathrm{diag}(\mathbf{v}_{l}^{H})(\sqrt{L_{\overline{Sl}}}\bar{\mathbf{H}}_{Sl}^{H}+\sqrt{L_{\widetilde{Sl}}}\tilde{\mathbf{H}}_{Sl}^{H})\|_{2}^{2}\big{]}$
$\displaystyle\stackrel{{\scriptstyle(e)}}{{=}}L_{\overline{Sl},\overline{lU}}\|\bar{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\bar{\mathbf{H}}_{Sl}\|_{2}^{2}+L_{\widetilde{Sl},\overline{lU}}\mathbb{E}\left[\|\tilde{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\bar{\mathbf{H}}_{Sl}\|_{2}^{2}\right]$
$\displaystyle\quad+L_{\overline{Sl},\widetilde{lU}}\mathbb{E}\left[\|\bar{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\tilde{\mathbf{H}}_{Sl}\|_{2}^{2}\right]+L_{\widetilde{Sl},\widetilde{lU}}\mathbb{E}\left[\|\tilde{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\tilde{\mathbf{H}}_{Sl}\|_{2}^{2}\right]$
$\displaystyle\stackrel{{\scriptstyle(f)}}{{=}}L_{\overline{Sl},\overline{lU}}\|\bar{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\bar{\mathbf{H}}_{Sl}\|_{2}^{2}+(L_{\widetilde{Sl},\overline{lU}}+L_{\overline{Sl},\widetilde{lU}}+L_{\widetilde{Sl},\widetilde{lU}})T_{S}T_{l}$
$\displaystyle\stackrel{{\scriptstyle(g)}}{{=}}L_{\overline{Sl},\overline{lU}}\mathbf{v}_{l}^{H}\mathrm{diag}{(\bar{\mathbf{h}}_{lU}^{H})}\bar{\mathbf{H}}_{Sl}\bar{\mathbf{H}}_{Sl}^{H}\mathrm{diag}{(\bar{\mathbf{h}}_{lU})}\mathbf{v}_{l}+(L_{\widetilde{Sl},\overline{lU}}+L_{\overline{Sl},\widetilde{lU}}+L_{\widetilde{Sl},\widetilde{lU}})T_{S}T_{l},$
(56)

where $(d)$ is due to (1) and (2), $(e)$ is due to
$\mathbb{E}\left[\tilde{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\bar{\mathbf{H}}_{Sl}\right]=\mathbf{0}_{1\times
T_{S}}$,
$\mathbb{E}\left[\tilde{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\tilde{\mathbf{H}}_{Sl}\right]=\mathbf{0}_{1\times
T_{S}}$, and
$\mathbb{E}\left[\tilde{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\tilde{\mathbf{H}}_{Sl}\right]=\mathbf{0}_{1\times
T_{S}}$, $(f)$ is due to (54) and (55), and $(g)$ is due to
$\bar{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})=\mathbf{v}_{l}^{H}\mathrm{diag}{(\bar{\mathbf{h}}_{lU}^{H})}$.

(iii) Similarly to the derivation of (54), we have:

$\displaystyle\mathbb{E}\left[\|\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\bar{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\bar{\mathbf{H}}_{S1}\|_{2}^{2}\right]=\|\bar{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\bar{\mathbf{H}}_{S1}\|_{F}^{2},$
$\displaystyle\mathbb{E}\left[\|\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\bar{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\tilde{\mathbf{H}}_{S1}\|_{2}^{2}\right]=T_{S}\|\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\bar{\mathbf{H}}_{12}\|_{2}^{2},$
$\displaystyle\mathbb{E}\left[\|\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\bar{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\tilde{\mathbf{H}}_{S1}\|_{2}^{2}\right]=\mathbb{E}\left[\|\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\tilde{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\bar{\mathbf{H}}_{S1}\|_{2}^{2}\right]$
$\displaystyle=\mathbb{E}\left[\|\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\tilde{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\bar{\mathbf{H}}_{S1}\|_{2}^{2}\right]=\mathbb{E}\left[\|\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\tilde{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\tilde{\mathbf{H}}_{S1}\|_{2}^{2}\right]$
$\displaystyle=\mathbb{E}\left[\|\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\tilde{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\tilde{\mathbf{H}}_{S1}\|_{2}^{2}\right]=T_{S}T_{1}T_{2}.$
(57)

Thus, we have:

$\displaystyle\mathbb{E}\big{[}\|\mathbf{x}_{3}^{H}\|_{2}^{2}\big{]}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}\left[\|(\sqrt{L_{\overline{2U}}}\bar{\mathbf{h}}_{2U}^{H}+\sqrt{L_{\widetilde{2U}}}\tilde{\mathbf{h}}_{2U}^{H})\mathrm{diag}{(\mathbf{v}_{2}^{H})}(\sqrt{L_{\overline{12}}}\bar{\mathbf{H}}_{12}+\sqrt{L_{\widetilde{12}}}\tilde{\mathbf{H}}_{12})\mathrm{diag}{(\mathbf{v}_{1}^{H})}(\sqrt{L_{\overline{S1}}}\bar{\mathbf{H}}_{S1}+\sqrt{L_{\widetilde{S1}}}\tilde{\mathbf{H}}_{S1})\|_{2}^{2}\right]$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}L_{\overline{S1},\overline{12},\overline{2U}}\|\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\bar{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\bar{\mathbf{H}}_{S1}\|_{2}^{2}+L_{\overline{S1},\overline{12},\widetilde{2U}}\mathbb{E}\left[\|\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\bar{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\bar{\mathbf{H}}_{S1}\|_{2}^{2}\right]$
$\displaystyle\quad+L_{\widetilde{S1},\overline{12},\overline{2U}}\mathbb{E}\left[\|\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\bar{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\tilde{\mathbf{H}}_{S1}\|_{2}^{2}\right]+L_{\widetilde{S1},\overline{12},\widetilde{2U}}\mathbb{E}\left[\|\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\bar{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\tilde{\mathbf{H}}_{S1}\|_{2}^{2}\right]$
$\displaystyle\quad+L_{\overline{S1},\widetilde{12},\overline{2U}}\mathbb{E}\left[\|\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\tilde{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\bar{\mathbf{H}}_{S1}\|_{2}^{2}\right]+L_{\overline{S1},\widetilde{12},\widetilde{2U}}\mathbb{E}\left[\|\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\tilde{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\bar{\mathbf{H}}_{S1}\|_{2}^{2}\right]$
$\displaystyle\quad+L_{\widetilde{S1},\widetilde{12},\overline{2U}}\mathbb{E}\left[\|\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\tilde{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\tilde{\mathbf{H}}_{S1}\|_{2}^{2}\right]+L_{\widetilde{S1},\widetilde{12},\widetilde{2U}}\mathbb{E}\left[\|\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\tilde{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\tilde{\mathbf{H}}_{S1}\|_{2}^{2}\right]$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}L_{\overline{S1},\overline{12},\overline{2U}}\|\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\bar{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\bar{\mathbf{H}}_{S1}\|_{2}^{2}+L_{\overline{S1},\overline{12},\widetilde{2U}}\|\bar{\mathbf{H}}_{12}\mathrm{diag}(\mathbf{v}_{1}^{H})\bar{\mathbf{H}}_{S1}\|_{F}^{2}$
$\displaystyle\quad+L_{\widetilde{S1},\overline{12},\overline{2U}}T_{S}\|\bar{\mathbf{h}}_{2U}\mathrm{diag}(\mathbf{v}_{2}^{H})\bar{\mathbf{H}}_{12}\|_{2}^{2}$
$\displaystyle\quad+\big{(}L_{\widetilde{S1},\overline{12},\widetilde{2U}}L_{\overline{S1},\widetilde{12},\overline{2U}}+L_{\overline{S1},\widetilde{12},\widetilde{2U}}+L_{\widetilde{S1},\widetilde{12},\overline{2U}}+L_{\widetilde{S1},\widetilde{12},\widetilde{2U}}\big{)}T_{S}T_{1}T_{2}$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}L_{\overline{S1},\overline{12},\overline{2U}}T_{S}\mathbf{v}_{2}^{H}\left(\mathrm{diag}(\bar{\mathbf{h}}_{2U}^{H})\bar{\mathbf{H}}_{12}\mathrm{diag}(\mathbf{a}_{A,S1})\right)\mathbf{v}_{1}^{*}\mathbf{v}_{1}^{T}\left(\mathrm{diag}(\bar{\mathbf{h}}_{2U}^{H})\bar{\mathbf{H}}_{12}\mathrm{diag}(\mathbf{a}_{A,S1})\right)^{H}\mathbf{v}_{2}$
$\displaystyle\quad+L_{\overline{S1},\overline{12},\widetilde{2U}}T_{2}\mathbf{v}_{1}^{H}\mathrm{diag}(\mathbf{a}^{H}_{D,12})\bar{\mathbf{H}}_{S1}\bar{\mathbf{H}}_{S1}^{H}\mathrm{diag}(\mathbf{a}_{D,12})\mathbf{v}_{1}+L_{\widetilde{S1},\overline{12},\overline{2U}}T_{S}\mathbf{v}_{2}^{H}\mathrm{diag}(\bar{\mathbf{h}}_{2U}^{H})\bar{\mathbf{H}}_{12}\bar{\mathbf{H}}_{12}^{H}\mathrm{diag}(\bar{\mathbf{h}}_{2U})\mathbf{v}_{2}$
$\displaystyle\quad+\big{(}L_{\widetilde{S1},\overline{12},\widetilde{2U}}+L_{\overline{S1},\widetilde{12},\overline{2U}}+L_{\overline{S1},\widetilde{12},\widetilde{2U}}+L_{\widetilde{S1},\widetilde{12},\overline{2U}}+L_{\widetilde{S1},\widetilde{12},\widetilde{2U}}\big{)}T_{S}T_{1}T_{2},$
(58)

where $(a)$ is due to (1) and (2), $(b)$ is due to

$\displaystyle\mathbb{E}\left[\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\bar{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\bar{\mathbf{H}}_{S1}\right]=\mathbb{E}\left[\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\bar{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\tilde{\mathbf{H}}_{S1}\right]$
$\displaystyle=\mathbb{E}\left[\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\bar{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\tilde{\mathbf{H}}_{S1}\right]=\mathbb{E}\left[\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\tilde{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\bar{\mathbf{H}}_{S1}\right]$
$\displaystyle=\mathbb{E}\left[\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\tilde{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\bar{\mathbf{H}}_{S1}\right]=\mathbb{E}\left[\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\tilde{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\tilde{\mathbf{H}}_{S1}\right]$
$\displaystyle=\mathbb{E}\left[\tilde{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}\tilde{\mathbf{H}}_{12}\mathrm{diag}{(\mathbf{v}_{1}^{H})}\tilde{\mathbf{H}}_{S1}\right]=\mathbf{0}_{1\times
T_{S}},$

$(c)$ is due to (57), and $(d)$ is due to
$\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}{(\mathbf{v}_{2}^{H})}=\mathbf{v}_{2}^{H}\mathrm{diag}{(\bar{\mathbf{h}}_{2U}^{H})}$
and
$\mathrm{diag}{(\mathbf{v}_{1}^{H})}\mathbf{a}_{A,S1}=\mathrm{diag}{(\mathbf{a}_{A,S1})}\mathbf{v}_{1}^{*}$.

(iv) For $l\in\mathcal{L}$, we have:

$\displaystyle\mathbb{E}\big{[}\mathbf{x}_{l}^{H}\mathbf{x}_{0}\big{]}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{E}\big{[}(\sqrt{L_{\overline{lU}}}\bar{\mathbf{h}}_{lU}^{H}+\sqrt{L_{\widetilde{lU}}}\tilde{\mathbf{h}}_{lU}^{H})\mathrm{diag}(\mathbf{v}_{l}^{H})(\sqrt{L_{\overline{Sl}}}\bar{\mathbf{H}}_{Sl}^{H}+\sqrt{L_{\widetilde{Sl}}}\tilde{\mathbf{H}}_{Sl}^{H})(\sqrt{L_{\overline{SU}}}\bar{\mathbf{h}}_{SU}+\sqrt{L_{\widetilde{SU}}}\tilde{\mathbf{h}}_{SU})\big{]}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\sqrt{L_{\overline{SU}}L_{\overline{Sl},\overline{lU}}}\bar{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\bar{\mathbf{H}}_{Sl}\bar{\mathbf{h}}_{SU}\stackrel{{\scriptstyle(c)}}{{=}}\sqrt{L_{\overline{SU}}L_{\overline{Sl},\overline{lU}}}\mathbf{v}_{l}^{H}\mathrm{diag}(\bar{\mathbf{h}}_{lU}^{H})\bar{\mathbf{H}}_{Sl}\bar{\mathbf{h}}_{SU},$
(59)

where $(a)$ is due to (1) and (2), $(b)$ is due to
$\mathbb{E}\left[\tilde{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\bar{\mathbf{H}}_{Sl}\right]=\mathbb{E}\left[\bar{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\tilde{\mathbf{H}}_{Sl}\right]=\mathbb{E}\left[\tilde{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})\tilde{\mathbf{H}}_{Sl}\right]=\mathbf{0}_{1\times
T_{S}}$ and
$\mathbb{E}\left[\tilde{\mathbf{h}}_{SU}\right]=\mathbf{0}_{T_{S}}$, $(c)$ is
due to
$\bar{\mathbf{h}}_{lU}^{H}\mathrm{diag}(\mathbf{v}_{l}^{H})=\mathbf{v}_{l}^{H}\mathrm{diag}(\bar{\mathbf{h}}_{lU}^{H})$.
Similarly, we can show:

$\displaystyle\mathbb{E}\big{[}\mathbf{x}_{3}^{H}\mathbf{x}_{0}\big{]}=$
$\displaystyle\sqrt{L_{\overline{SU}}L_{\overline{S1},\overline{12},\overline{2U}}}\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}(\mathbf{v}_{2}^{H})\bar{\mathbf{H}}_{12}\mathrm{diag}(\mathbf{v}_{1}^{H})\bar{\mathbf{H}}_{S1}\bar{\mathbf{h}}_{SU}$
$\displaystyle=$
$\displaystyle\sqrt{L_{\overline{SU}}L_{\overline{S1},\overline{12},\overline{2U}}}\mathbf{v}_{2}^{H}\mathrm{diag}(\bar{\mathbf{h}}_{2U}^{H})\bar{\mathbf{H}}_{12}\mathrm{diag}(\bar{\mathbf{H}}_{S1}\bar{\mathbf{h}}_{SU})\mathbf{v}_{1}^{*},$
(60) $\displaystyle\mathbb{E}\big{[}\mathbf{x}_{2}^{H}\mathbf{x}_{1}\big{]}=$
$\displaystyle\sqrt{L_{\overline{S1},\overline{1U}}L_{\overline{S2},\overline{2U}}}\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}(\mathbf{v}_{2}^{H})\bar{\mathbf{H}}_{S2}\bar{\mathbf{H}}_{S1}^{H}\mathrm{diag}(\mathbf{v}_{1})\bar{\mathbf{h}}_{1U}$
$\displaystyle=$
$\displaystyle\sqrt{L_{\overline{S1},\overline{1U}}L_{\overline{S2},\overline{2U}}}\mathbf{v}_{2}^{H}\mathrm{diag}(\bar{\mathbf{h}}_{2U}^{H})\bar{\mathbf{H}}_{S2}\bar{\mathbf{H}}_{S1}^{H}\mathrm{diag}(\bar{\mathbf{h}}_{1U})\mathbf{v}_{1},$
(61) $\displaystyle\mathbb{E}\big{[}\mathbf{x}_{3}^{H}\mathbf{x}_{1}\big{]}=$
$\displaystyle\sqrt{L_{\overline{S1},\overline{1U}}L_{\overline{S1},\overline{12},\overline{2U}}}\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}(\mathbf{v}_{2})\bar{\mathbf{H}}_{12}\mathrm{diag}(\mathbf{v}_{1}^{H})\bar{\mathbf{H}}_{S1}\bar{\mathbf{H}}_{S1}^{H}\mathrm{diag}(\mathbf{v}_{1})\bar{\mathbf{h}}_{1U}$
(62)
$\displaystyle+\sqrt{L_{\widetilde{S1},\overline{1U}}L_{\widetilde{S1},\overline{12},\overline{2U}}}T_{S}\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}(\mathbf{v}_{2}^{H})\bar{\mathbf{H}}_{12}\bar{\mathbf{h}}_{1U}$
$\displaystyle=$
$\displaystyle\mathbf{v}_{1}^{H}\mathrm{diag}(\mathbf{v}_{2}^{H}\mathrm{diag}(\bar{\mathbf{h}}_{2U}^{H})\bar{\mathbf{H}}_{12})\left(\sqrt{L_{\overline{S1},\overline{1U}}L_{\overline{S1},\overline{12},\overline{2U}}}\bar{\mathbf{H}}_{S1}\bar{\mathbf{H}}_{S1}^{H}\mathrm{diag}(\bar{\mathbf{h}}_{1U})\mathbf{v}_{1}\right)$
$\displaystyle+\sqrt{L_{\widetilde{S1},\overline{1U}}L_{\widetilde{S1},\overline{12},\overline{2U}}}T_{S}\mathbf{v}_{2}^{H}\mathrm{diag}(\bar{\mathbf{h}}_{2U}^{H})\bar{\mathbf{H}}_{12}\bar{\mathbf{h}}_{1U},$
(63) $\displaystyle\mathbb{E}\big{[}\mathbf{x}_{3}^{H}\mathbf{x}_{2}\big{]}=$
$\displaystyle\sqrt{L_{\overline{S2},\overline{2U}}L_{\overline{S1},\overline{12},\overline{2U}}}\bar{\mathbf{h}}_{2U}^{H}\mathrm{diag}(\mathbf{v}_{2}^{H})\bar{\mathbf{H}}_{12}\mathrm{diag}(\mathbf{v}_{1}^{H})\bar{\mathbf{H}}_{S1}\bar{\mathbf{H}}_{S2}^{H}\mathrm{diag}(\mathbf{v}_{2})\bar{\mathbf{h}}_{2U}$
$\displaystyle+\sqrt{L_{\overline{S2},\widetilde{2U}}L_{\overline{S1},\overline{12},\widetilde{2U}}}\mathrm{Tr}\left(\bar{\mathbf{H}}_{12}\mathrm{diag}(\mathbf{v}_{1}^{H})\bar{\mathbf{H}}_{S1}\bar{\mathbf{H}}_{S2}^{H}\right)$
$\displaystyle=$
$\displaystyle\mathbf{v}_{1}^{H}\mathrm{diag}(\mathbf{v}_{2}^{H}\mathrm{diag}(\bar{\mathbf{h}}_{2U}^{H})\bar{\mathbf{H}}_{12})\left(\sqrt{L_{\overline{S2},\overline{2U}}L_{\overline{S1},\overline{12},\overline{2U}}}\bar{\mathbf{H}}_{S1}\bar{\mathbf{H}}_{S2}^{H}\mathrm{diag}(\bar{\mathbf{h}}_{2U})\mathbf{v}_{2}\right)$
$\displaystyle+\sqrt{L_{\overline{S2},\widetilde{2U}}L_{\overline{S1},\overline{12},\widetilde{2U}}}\mathbf{v}_{1}^{H}\mathrm{diag}\left(\mathbf{a}^{H}_{D,12}\right)\bar{\mathbf{H}}_{S1}\bar{\mathbf{H}}_{S2}^{H}\mathbf{a}_{A,12},$
(64)

Finally, substituting (53)-(64) into (Appendix B: Proof of Theorem 1), we can
obtain the expression of $\gamma(\bm{\phi}_{1},\bm{\phi}_{2})$ for all values
of the Rician factors. Based on the general expression of
$\gamma(\bm{\phi}_{1},\bm{\phi}_{2})$, we can readily obtain $\gamma^{(0)}$,
$\gamma^{(1)}(\bm{\phi}_{1})$, $\gamma^{(2)}(\bm{\phi}_{2})$, and
$\gamma^{(3)}(\bm{\phi}_{1},\bm{\phi}_{2})$. Therefore, we complete the proof
of Theorem 1.

## Appendix C: Proof of Theorem 2

Before proving Theorem 2, we first show the following lemma.

######  Lemma 13.

Suppose $\mathbf{A}=\mathbf{x}\mathbf{y}^{H}\in\mathbb{C}^{M\times N}$ with
$\mathbf{x}\in\mathbb{C}^{M}$ and $\mathbf{y}\in\mathbb{C}^{N}$,
$\mathbf{v}\triangleq\left(e^{-j\phi_{n}}\right)_{n\in\mathcal{N}}\in\mathbb{C}^{N}$
with $\mathcal{N}\triangleq\\{n=1,...,N\\}$, $\mathbf{b}\in\mathbb{C}^{M}$,
and $c_{1},c_{2}\in\mathbb{R}$. The unique optimal solution of the problem

$\displaystyle\max_{\bm{\phi}}$ $\displaystyle\qquad
c_{1}\mathbf{v}^{H}\mathbf{A}^{H}\mathbf{A}\mathbf{v}+c_{2}\mathfrak{R}\left\\{\mathbf{v}^{H}\mathbf{A}^{H}\mathbf{b}\right\\}$
(65) $\displaystyle\mathrm{s.t.}$
$\displaystyle\qquad\phi_{n}\in[0,2\pi),n\in\mathcal{N},$

is
$\bm{\phi}^{\star}=\Lambda\left(-\angle(\mathbf{y})-\angle(\mathbf{x}^{H}\mathbf{b})\mathbf{1}_{N}\right)$.

###### Proof:

We have:

$\displaystyle\mathfrak{R}\left\\{\mathbf{v}^{H}\mathbf{y}\mathbf{x}^{H}\mathbf{b}\right\\}\overset{(a)}{\leq}|\mathbf{v}^{H}\mathbf{y}\mathbf{x}^{H}\mathbf{b}|=|\mathbf{v}^{H}\mathbf{y}||\mathbf{x}^{H}\mathbf{b}|,$
(66)

where $(a)$ is due to $\mathfrak{R}\\{x\\}\leq|x|,x\in\mathbb{C}$ and holds
with equality if and only if:

$\displaystyle\angle(\mathbf{v}^{H}\mathbf{y})+\angle(\mathbf{x}^{H}\mathbf{b})=0.$
(67)

Besides, we have:

$\displaystyle|\mathbf{v}^{H}\mathbf{y}|=\left|\sum_{n=1}^{N}e^{j\phi_{n}}|y_{n}|e^{j\angle(y_{n})}\right|=\left|\sum_{n=1}^{N}|y_{n}|e^{j(\phi_{n}+\angle(y_{n}))}\right|\stackrel{{\scriptstyle(b)}}{{\leq}}\sum_{n=1}^{N}|y_{n}|,$
(68)

where $(b)$ is due to the triangle inequality and holds with equality if and
only if

$\displaystyle\bm{\phi}=-\angle(\mathbf{y})+\psi\mathbf{1}_{N},\psi\in\mathbb{R}.$
(69)

Thus, we have:

$\displaystyle
c_{1}\mathbf{v}^{H}\mathbf{A}^{H}\mathbf{A}\mathbf{v}+c_{2}\mathfrak{R}\left\\{\mathbf{v}^{H}\mathbf{A}^{H}\mathbf{b}\right\\}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}c_{1}|\mathbf{v}^{H}\mathbf{y}|^{2}\|\mathbf{x}\|_{2}^{2}+c_{2}\mathfrak{R}\left\\{\mathbf{v}^{H}\mathbf{y}\mathbf{x}^{H}\mathbf{b}\right\\}$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{\leq}}c_{1}|\mathbf{v}^{H}\mathbf{y}|^{2}\|\mathbf{x}\|_{2}^{2}+c_{2}|\mathbf{v}^{H}\mathbf{y}||\mathbf{x}^{H}\mathbf{b}|$
$\displaystyle\stackrel{{\scriptstyle(e)}}{{\leq}}c_{1}\left(\sum_{n=1}^{N}|y_{n}|\right)^{2}\|\mathbf{x}\|_{2}^{2}+c_{2}\left(\sum_{n=1}^{N}|y_{n}|\right)|\mathbf{x}^{H}\mathbf{b}|,$

where $(c)$ is due to $\mathbf{A}=\mathbf{x}\mathbf{y}^{H}$, $(d)$ is due to
(66), and $(e)$ is due to (68). $(d)$ and $(e)$ both hold with equality if and
only if $\bm{\phi}$ satisfied (69) and (67). By noting that
$\phi_{n}\in[0,2\pi),n\in\mathcal{N}$, the optimal solution of the problem in
(65) is
$\bm{\phi}^{\star}=\Lambda\left(-\angle(\mathbf{y})-\angle(\mathbf{x}^{H}\mathbf{b})\mathbf{1}_{N}\right)$.∎

Now, we prove Theorem 2. First, we prove Statement (i). Consider $K_{12}=0$.
By (14), we have:

$\gamma^{(1)}(\bm{\phi}_{1})=L_{\overline{S1},\overline{1U}}\mathbf{v}_{1}^{H}\mathbf{G}_{11}^{H}\mathbf{G}_{11}\mathbf{v}_{1}+2\sqrt{L_{\overline{SU}}L_{\overline{S1},\overline{1U}}}\mathfrak{R}\\{\mathbf{v}_{1}^{H}\mathbf{G}_{11}^{H}\mathbf{g}_{11}\\}+\gamma^{(0)},$

where
$\mathbf{G}_{11}\triangleq\bar{\mathbf{H}}_{S1}^{H}\mathrm{diag}\left(\bar{\mathbf{h}}_{1U}\right)$
and $\mathbf{g}_{11}\triangleq\mathbf{h}_{SU}$. By (3), (4), and Lemma 13, we
have
$\bm{\phi}_{1}^{\star}=\Lambda(-\angle(\mathrm{diag}(\mathbf{a}^{H}_{D,1U})\mathbf{a}_{A,S1})$
$-\angle(\mathbf{a}_{D,S1}^{H}\mathbf{a}_{D,SU})\mathbf{1}_{T_{1}})\overset{(e)}{=}\Lambda\left(-\bm{\Delta}_{\overline{S1},\overline{1U}}-\angle(r_{\overline{S1},\overline{SU}})\mathbf{1}_{T_{1}}\right)$,
where $(e)$ is due to (10) and (11). Then, consider $K_{1U}=0$. By (14), we
have:

$\gamma^{(1)}(\bm{\phi}_{1})=L_{\overline{S1},\overline{12},\widetilde{2U}}T_{2}\mathbf{v}_{1}^{H}\mathbf{G}_{12}^{H}\mathbf{G}_{12}\mathbf{v}_{1}+2\sqrt{L_{\overline{S2},\widetilde{2U}}L_{\overline{S1},\overline{12},\widetilde{2U}}}\mathfrak{R}\left\\{\mathbf{v}_{1}^{H}\mathbf{G}_{12}^{H}\mathbf{g}_{12}\right\\}+\gamma^{(0)},$

where
$\mathbf{G}_{12}\triangleq\bar{\mathbf{H}}_{S1}^{H}\mathrm{diag}(\mathbf{a}_{D,12})$
and $\mathbf{g}_{12}\triangleq\bar{\mathbf{H}}_{S2}^{H}\mathbf{a}_{A,12}$. By
(3), (4), and Lemma 13, we have
$\bm{\phi}_{1}^{\star}=\Lambda\big{(}-\angle(\mathrm{diag}(\mathbf{a}^{H}_{D,12})$
$\mathbf{a}_{A,S1})-\angle(\mathbf{a}_{D,S1}^{H}\mathbf{a}_{D,S2}\mathbf{a}^{H}_{A,S2}\mathbf{a}_{A,12})\mathbf{1}_{T_{1}}\big{)}\overset{(f)}{=}\Lambda(-\bm{\Delta}_{\overline{S1},\overline{12}}-\angle(r_{\overline{S1},\overline{S2}}$
$r_{\overline{S2},\overline{12}})\mathbf{1}_{T_{1}})$, where $(f)$ is due to
(10), (11), and (12).

Next, we prove Statement (ii). Consider $K_{12}=0$. By (15), we have:

$\gamma^{(2)}(\bm{\phi}_{2})=L_{\overline{S2},\overline{2U}}\mathbf{v}_{2}^{H}\mathbf{G}_{21}^{H}\mathbf{G}_{21}\mathbf{v}_{2}+2\sqrt{L_{\overline{SU}}L_{\overline{S2},\overline{2U}}}\mathfrak{R}\\{\mathbf{v}_{2}^{H}\mathbf{G}_{21}^{H}\mathbf{g}_{21}\\}+\gamma^{(0)},$

where
$\mathbf{G}_{21}\triangleq\bar{\mathbf{H}}_{S2}^{H}\mathrm{diag}\left(\bar{\mathbf{h}}_{2U}\right)$
and $\mathbf{g}_{21}\triangleq\mathbf{h}_{SU}$. By (3), (4), and Lemma 13, we
have
$\bm{\phi}_{2}^{\star}=\Lambda(-\angle(\mathrm{diag}(\mathbf{a}^{H}_{D,2U})\mathbf{a}_{A,S2})$
$-\angle(\mathbf{a}_{D,S2}^{H}\mathbf{a}_{D,SU})\mathbf{1}_{T_{2}})\overset{(g)}{=}\Lambda\left(-\bm{\Delta}_{\overline{S2},\overline{2U}}-\angle(r_{\overline{S2},\overline{SU}})\mathbf{1}_{T_{2}}\right)$,
where $(g)$ is due to (10) and (11). Then, consider $K_{S2}=0$. By (15), we
have: