# IRS-assisted Multi-cell Multi-band Systems: Practical Reflection Model and
Joint Beamforming Design ††thanks: W. Cai, R. Liu, M. Li, and Y. Liu are with
the School of Information and Communication Engineering, Dalian University of
Technology, Dalian 116024, China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>yangliu_613@dlut.edu.cn). ††thanks:
Q. Wu is with the State Key Laboratory of Internet of Things for Smart City,
University of Macau, Macau 999078, China (email: qingqingwu@um.edu.mo).
††thanks: Q. Liu is with the School of Computer Science and Technology, Dalian
University of Technology, Dalian 116024, China (e-mail: qianliu@dlut.edu.cn).

Wenhao Cai, Rang Liu, Ming Li,
Yang Liu, Qingqing Wu, and Qian Liu

###### Abstract

Intelligent reflecting surface (IRS) has been regarded as a promising and
revolutionary technology for future wireless communication systems owing to
its capability of tailoring signal propagation environment in an
energy/spectrum/hardware-efficient manner. However, most existing studies on
IRS optimizations are based on a simple and ideal reflection model that is
impractical in hardware implementation, which thus leads to severe performance
loss in realistic wideband/multi-band systems. To deal with this problem, in
this paper we first propose a more practical and more tractable IRS reflection
model that describes the difference of reflection responses for signals at
different frequencies. Then, we investigate the joint transmit beamforming and
IRS reflection beamforming design for an IRS-assisted multi-cell multi-band
system. Both power minimization and sum-rate maximization problems are solved
by exploiting popular second-order cone programming (SOCP), Riemannian
manifold, minimization-majorization (MM), weighted minimum mean square error
(WMMSE), and block coordinate descent (BCD) methods. Simulation results
illustrate the significant performance improvement of our proposed joint
transmit beamforming and reflection design algorithms based on the practical
reflection model in terms of power saving and rate enhancement.

###### Index Terms:

Intelligent reflecting surface, practical reflection model, multi-cell multi-
band systems, beamforming optimization.

##  I Introduction

With the rapid development of wireless communication networks and the
popularizing of various intelligent devices, the demands for high transmission
rate and low latency have also been exponentially growing in the last decades
[1]. To accommodate these constantly increasing demands, several key
technologies, such as heterogeneous dense networks, massive multiple-input
multiple-output (MIMO), and millimeter-wave (mmWave) communications, have been
proposed to improve system performance for the fifth-generation (5G) and
beyond communication networks [2, 3]. However, it seems that we are
approaching the limit of the theories obtained from such technologies.
Furthermore, the required high hardware complexity and consequently high
energy consumption in practical implementations remain the bottlenecks for
large-scale deployment [4]. Therefore, new technologies are desired to provide
fundamental advances for future wireless networks in a more
energy/spectrum/hardware-efficient fashion.

The innovative concept of intelligent reflecting surface (IRS) has recently
emerged as such a promising technology [4]-[10]. An IRS is a two-dimensional
(2D) planar array consisting of numerous reflecting elements, which are
implemented by reconfigurable electromagnetic (EM) internals with very low
power consumption. Each reflecting element independently adjusts the phase-
shift and amplitude of incident EM waves in a programmable manner, which
collaboratively achieves reflection beamforming and reshapes propagation
environments for wireless communications. Deploying an IRS in existing
communication networks brings the capability of creating a favorable
propagation environment and provides more degrees of freedom (DoFs) for
network optimizations. Furthermore, these lightweight, hardware-efficient, and
cost-effective reflecting elements provide the IRS with portability and
mobility for various practical applications and enable large-scale IRSs to
produce higher passive beamforming gains to significantly improve the
transmission quality of service (QoS). Therefore, IRS has been envisioned as a
revolutionary technology owing to its capability of creating a smart and
reconfigurable wireless propagation environment in a hardware-efficient way
and has drawn significant attentions within both industry and academic
communities [4].

In order to take advantage of the IRS mentioned above, extensive researches on
deploying IRS in various wireless communication systems have been conducted to
improve the performance under different metrics. By judiciously adjusting IRS
elements, the reflected signals are intelligently elaborated to achieve
performance improvement in terms of spectral efficiency [11], energy
efficiency [12], transmit power [13, 14], sum-rate [15]-[17], etc., for
single-user/multi-user MIMO/multi-input single-output (MISO) [11]-[16],
wideband orthogonal frequency division multiplexing (OFDM) [17], or multi-cell
systems [18]. In addition, researchers have explored the designs of IRS with
low-resolution phase-shift [19] and phase error [20] or under imperfect
channel state information (CSI) [21]. Moreover, IRS has been applied in
various novel applications, such as physical layer security [22, 23], index
modulation [24], passive information transmission [25], etc.

In the above applications, it is assumed that each reflecting element has an
ideal IRS reflection model, which induces the same constant amplitude yet
variable phase-shift response to the incident signals. Based on this ideal IRS
reflection model, a lot of existing algorithms, e.g., majorization
minimization (MM), Riemannian manifold optimization, semidefinite relaxation
(SDR), etc., have been readily employed in IRS reflection optimizations.
Unfortunately, such an ideal IRS is difficult to be realized by current
hardware circuit techniques. Therefore, existing designs with the ideal
reflection model inevitably suffer from severe performance loss in realistic
systems since the responses of practical hardware circuits are quite different
from the ideal one [26]-[30].

In order to unlock the full potential IRS in realistic systems, many different
practical IRS reflection models are derived to illustrate the mechanism that
affects the reflection coefficient [26]-[29], [31]. The seminal work [26]
presented a two-dimensional amplitude-phase reflection model, which shows a
fundamental relationship between the reflection amplitude and phase-shift for
narrowband systems. However, it has been verified that the reflection
amplitude and phase-shift vary with the frequency of incident signals [27].
This two-dimensional reflection model, which does not consider the effect of
signal frequency, cannot be readily applied in wideband/multi-band systems. To
tackle this issue, in our previous work [28], we analyzed the responses to
signals at different frequencies and established a three-dimensional
amplitude-frequency-phase reflection model. Since this sophisticated model
brings huge difficulties for joint beamforming and reflection designs, a
simplified version was developed in [29] for typical wideband OFDM systems.
Besides, the authors in [31] derived a more general reflection model using
rigorous scattering parameter network analysis, which mainly focuses on simple
IRS-aided communication systems and thus leads to a complicated model for a
certain complicated communication system, e.g., the multi-cell multi-band
system considered in this paper. Therefore, the existing IRS reflection model
is too complicated to be used for the joint beamforming and IRS reflection
designs. Deriving a more straightforward and more tractable practical IRS
reflection model for the multi-cell multi-band systems remains an open
problem.

Motivated by the above, in this paper we propose a simplified practical IRS
reflection model for IRS-assisted multi-cell multi-band systems and then
investigate associated joint transmit beamforming and IRS reflection
beamforming designs. Specifically, multiple base stations (BSs), which belong
to the same service provider but operate at different frequency bands,
simultaneously serve the users in their cells with the aid of one IRS. To the
best of our knowledge, this problem has not been investigated in the
literature yet. Our main contributions are summarized as follows.

  * •

We first analyze the IRS phase-shift response to the incident signals at
different frequencies and provide a lean practical frequency-dependent IRS
reflection model. Compared with our previous work, this newly proposed model
significantly reduces the required computational complexity and is more
suitable for realistic multi-cell multi-band systems.

  * •

Then, we investigate the power minimization problem, which aims to minimize
the total transmit power subject to the signal-to-interference-plus-noise
ratio (SINR) constraints of all users and the practical IRS reflection model.
A three-step algorithm is proposed to solve for the transmit beamforming and
the practical IRS reflection beamforming by utilizing the second-order cone
programming (SOCP), the Riemannian manifold optimization, and the
minimization-majorization (MM) method after some sophisticated
transformations.

  * •

The sum-rate maximization problem is also investigated to maximize the sum-
rate subject to the total transmit power constraint and the practical IRS
reflection model. In order to effectively handle this non-convex problem, we
exploit the weighted minimum mean square error (WMMSE) approach to convert the
original problem into a solvable multi-variable optimization, which is then
handled by the block coordinate descent (BCD) method. Based on the proposed
lean practical frequency-phase model, the closed-form solutions of IRS phase-
shift and service selection of each element can be simultaneously obtained for
the sum-rate maximization problem, which can significantly improve the
efficiency of the algorithm.

  * •

Finally, extensive simulation results are illustrated to show the
effectiveness of our proposed algorithms and validate the significant
performance improvement achieved by considering the proposed practical IRS
reflection model in the designs for multi-cell multi-band communication
networks.

Figure 1: An IRS-assisted multi-cell multi-band system.

Notation: Boldface lower-case and upper-case letters indicate column vectors
and matrices, respectively. $\mathbb{C}$ and $\mathbb{R}^{+}$ denotes the set
of complex and positive real numbers, respectively. $(\cdot)^{*}$,
$(\cdot)^{T}$, $(\cdot)^{H}$, and $(\cdot)^{-1}$ denote the conjugate,
transpose, conjugate-transpose operations, and the inversion of a matrix,
respectively. $\mathbb{E}\\{\cdot\\}$ and $\mathfrak{R}\\{\cdot\\}$ denote
statistical expectation and the real part of a complex number, respectively.
$\mathbf{I}_{L}$ indicates an $L\times L$ identity matrix. $||\mathbf{a}||$
and $||\mathbf{a}||_{0}$ denote the $\ell_{2}$ norm and $\ell_{0}$ norm of a
vector $\mathbf{a}$, respectively. $\odot$ denotes the Hadamard product. In
addition, $\angle{\theta}$ denotes the angle of a complex number $\theta$.

##  II System Model and Practical Reflection Model

###  II-A System Model

We consider an IRS-assisted multi-cell multi-band wireless communication
system as shown in Fig. 1, where an IRS composed of $M$ reflecting elements is
deployed to simultaneously assist the downlink communications in $S$ cells.
Specifically, in the $s$-th cell, $s=1,2,\ldots,S$, the BS equipped with
$N_{\text{t}}$ transmit antennas serves $K_{s}$ single-antenna users at the
frequency $f_{s}$. Let $\mathcal{S}\triangleq\\{1,\ldots,S\\}$ denote the set
of BSs, $\mathcal{K}_{s}\triangleq\\{1,\ldots,K_{s}\\}$ denote the set of the
users served by the $s$-th BS, and $\mathcal{M}\triangleq\\{1,\ldots,M\\}$
denote the set of IRS reflecting elements. Without loss of generality, we
assume $f_{1}>f_{2}>\cdots>f_{S}$ in the considered multi-cell multi-band
system. In addition, the IRS is controlled by an IRS controller through a
dedicated control link and only the first-order reflection is considered due
to significant path loss.

Denote
$\mathbf{z}_{s}\triangleq[z_{s,1},\ldots,z_{s,K_{s}}]^{T}\in\mathbb{C}^{K_{s}}$
as the transmitted symbols for the users served by the $s$-th BS,
$\mathbb{E}\\{\mathbf{z}_{s}\mathbf{z}_{s}^{H}\\}=\mathbf{I}_{K_{s}}$, and
$\mathbf{W}_{s}\triangleq[\mathbf{w}_{s,1},\ldots,\mathbf{w}_{s,K_{s}}]\in\mathbb{C}^{N_{\text{t}}\times
K_{s}}$ as the precoder matrix of the $s$-th BS, $\forall
s\in\mathcal{S},k_{s}\in\mathcal{K}_{s}$. Since the BSs operate at different
frequencies, the inter-cell interference can be easily eliminated through
receiving filters at the users. Thus, the received baseband signal at the
$k_{s}$-th user served by the $s$-th BS can be expressed as

$y_{s,k_{s}}=\left(\mathbf{h}^{H}_{\text{r},s,k_{s}}\bm{\Theta}_{s}\mathbf{G}_{s}+\mathbf{h}^{H}_{\text{d},s,k_{s}}\right)\sum_{k_{s}\in\mathcal{K}_{s}}\mathbf{w}_{s,k_{s}}z_{s,k_{s}}+n_{s,k_{s}},$
(1)

where $\mathbf{h}_{\text{r},s,k_{s}}\in\mathbb{C}^{M}$,
$\mathbf{G}_{s}\in\mathbb{C}^{M\times N_{\text{t}}}$, and
$\mathbf{h}_{\text{d},s,k_{s}}\in\mathbb{C}^{N_{\text{t}}}$ represent the
baseband equivalent channels from the IRS to the $k_{s}$-th user, from the
$s$-th BS to the IRS, and from the $s$-th BS to the $k_{s}$-th user,
respectively. It is noted that the quasi-static flat-fading Rayleigh channel
model111The proposed practical IRS reflection model and associated design
algorithms are also suitable for other channel fading models, e.g., Rician
fading. is adopted for all channels and we assume that all the CSIs are
perfectly known at the BSs given existing efficient channel estimation
approaches [32]-[39]. For example, the CSIs can be acquired at different BSs
based on the received pilot signals from their users with acceptable training
overhead. $n_{s,k_{s}}\sim\mathcal{C}\mathcal{N}(0,\sigma^{2})$ denotes the
additive white Gaussian noise (AWGN) at the $k_{s}$-th user.
$\bm{\Theta}_{s}\triangleq\text{diag}\\{\bm{\theta}_{s}\\}$, and
$\bm{\theta}_{s}\triangleq[\theta_{s,1},\ldots,\theta_{s,M}]^{T}$ is the IRS
reflection coefficient vector for the signals transmitted by the $s$-th BS,
i.e., the signals at the frequency $f_{s}$. We emphasize that with the same
IRS settings, the IRS reflection vectors are different for the incident
signals at different frequencies due to practical hardware characteristics,
and they are inherently correlated as described in the next subsection.

With the received signal in (1), the SINR of the $k_{s}$-th user served by the
$s$-th BS is given by

$\gamma_{s,k_{s}}=\frac{\big{|}\big{(}{\mathbf{h}^{H}_{\text{r},s,k_{s}}}\bm{\Theta}_{s}\mathbf{G}_{s}+\mathbf{h}_{\text{d},s,k_{s}}^{H}\big{)}\mathbf{w}_{s,k_{s}}\big{|}^{2}}{\sum^{j\in\mathcal{K}_{s}}_{j\neq
k_{s}}\big{|}\big{(}{\mathbf{h}^{H}_{\text{r},s,k_{s}}}\bm{\Theta}_{s}\mathbf{G}_{s}+\mathbf{h}_{\text{d},s,k_{s}}^{H}\big{)}\mathbf{w}_{s,j}\big{|}^{2}+\sigma^{2}}.$
(2)

Figure 2: The equivalent circuit of a reflecting element.

###  II-B Practical IRS Reflection Model

An IRS is typically implemented by a printed circuit board (PCB), in which
semiconductor devices [40], e.g., positive-intrinsic-negative (PIN) diodes,
are embedded to tune the reflection response by varying the impedance of each
reflecting element. As discussed in [26]-[29], the reflection response of a
reflecting element can be described by an equivalent parallel resonant circuit
as shown in Fig. 2, where $L_{1}$, $L_{2}$, $C$, and $R$ denote the equivalent
inductances, variable capacitance, and the loss resistance, respectively.
Thus, the impedance of a reflecting element can be written as

$Z(C,f)=\frac{j2{\pi}fL_{1}(j2{\pi}fL_{2}+\frac{1}{j2{\pi}fC}+R)}{j2{\pi}fL_{1}+(j2{\pi}fL_{2}+\frac{1}{j2{\pi}fC}+R)}.$
(3)

Then, the reflection coefficient $\theta$ that describes the effect of a
reflecting element on the incident EM waves is given by

$\theta(C,f)=\frac{Z(C,f)-Z_{0}}{Z(C,f)+Z_{0}},$ (4)

where $Z_{0}=377\Omega$ denotes the free space impedance. From the microwave
theory (4), we see that the reflection coefficient $\theta(C,f)$ is a function
of the capacitance $C$ and the frequency $f$ of incident signals. In other
words, the reflection coefficient of each reflecting element is controlled by
the variable capacitance and the same reflecting element exhibits different
amplitude and phase-shift responses to the signals at different frequencies.
The creditability and correctness of the microwave theory can also be verified
by the experimental and simulation results in [27], [30]. We have analyzed
this phenomenon and established a sophisticated three-dimensional amplitude-
frequency-phase model in our initial work [28], and further provided a
simplified version for IRS optimization in wideband systems [29]. However,
this practical IRS reflection model is composed of complex arc-tangent and
Witch of Agnesi functions, which are too complicated and inconvenient for the
joint beamforming and IRS reflection designs in the considered multi-cell
multi-band systems. This motivates us to derive a simpler approximated
reflection model.

To establish a simplified and tractable IRS reflection model, we first plot
the phase-shift response $\angle\theta$ versus the capacitance in Fig. 3
according to (3) and (4). We take the scenario that $S=3$, $f_{1}=2.605$GHz,
$f_{2}=2.345$GHz, and $f_{3}=1.885$GHz as an example222The proposed IRS
reflection model can be easily extended to other frequency combinations and
similar conclusions can be obtained., which adopts three available frequencies
of China Mobile 4G network. In addition, we choose a practical surface-mount
diode SMV1231-079 [41] with parameters $L_{1}=2.5$nH, $L_{2}=0.7$nH, and
$R=1\Omega$ to implement the IRS. It can be noticed that the phase-shift
response varies with the frequency of incident signal as analyzed above. In
addition, we observe that for a given frequency $f$, the phase-shift can be
accurately tuned in a specific capacitance range but maintains almost the same
in other ranges. Furthermore, the tunable capacitance ranges of different
frequencies are nearly not overlapped when the difference of frequencies is
relatively large. In other words, when we adjust the capacitance to provide a
tunable phase-shift for a certain frequency, the phase-shift response for
other frequencies is almost unchanged. In this way, the capacitance values in
Fig. 3 can be divided into four ranges (the green, orange, blue, and gray
areas), which exhibit four different reflection responses for the signals of
three different frequencies.

Figure 3: The phase-shift response versus capacitance for different
frequencies ($S=3$, $f_{1}=2.605$GHz, $f_{2}=2.345$GHz, $f_{3}=1.885$GHz).

Take the orange area as an example. When the capacitance varies from 1.3pF to
2pF, the phase-shift response for signals at 2.345GHz changes by
$260^{\circ}$, while 1.885GHz changes by $40^{\circ}$ and 2.605GHz changes by
$45^{\circ}$. Therefore, we can approximately consider this capacitance range
as that the reflecting element serves the BS operating at 2.345GHz by
providing an ideal $(0,2\pi]$ phase-shift and meanwhile exhibits a fixed
$0/2\pi$ phase-shift for other BSs. Since the IRS consists of lots of
reflecting elements, the performance loss due to the approximation for each
independent reflecting element is negligible. Moreover, considering the
hardware complexity and cost in practical implementations, the tunable phase-
shift values are usually discrete (e.g., 1-bit or 2-bit resolution), which
also supports our approximation. Correspondingly, the green and blue areas
indicate that the BSs operating at 2.605GHz and 1.885GHz are served by this
reflecting element, respectively. Meanwhile, in the gray areas, IRS provides
fixed and uncontrollable phase-shifts to all the BSs. Therefore, IRS cannot
generate effective beamforming for any BSs.

TABLE I: A simplified representation of Fig. 3. | $C_{1}$ | $C_{2}$ | $C_{3}$ | $C_{\text{no}}$
---|---|---|---|---
2.605 GHz | $(0,2\pi]$ | $0$ | $0$ | $0/2\pi$
2.345 GHz | $2\pi$ | $(0,2\pi]$ | $0$ | $0/2\pi$
1.885 GHz | $2\pi$ | $2\pi$ | $(0,2\pi]$ | $0/2\pi$

Based on the above descriptions, we simplify the phase-shift response for
different operating frequencies in Table I, where $C_{1}$, $C_{2}$, $C_{3}$,
and $C_{\text{no}}$ correspond to the capacitance ranges of the green, orange,
blue, and gray areas in Fig. 3, respectively. We can clearly see that by
controlling the capacitance of a reflecting element, a certain BS (i.e., the
signal with a certain frequency) is selected to be served by this reflecting
element with an ideal (i.e., $0-2\pi$ tunable) phase-shift meanwhile the other
BSs have a fixed $0/2\pi$ phase-shift, or none of the BSs are selected. This
frequency-selective characteristic of each reflecting element motivates us to
describe a practical phase-shift response using the product of an ideal phase-
shift and a binary indicator. This simplified IRS phase-shift response
transforms the extremely complicated nonlinear functional relationship between
phase-shift and frequency into a binary functional relationship, which can
further facilitate the joint transmit beamforming and IRS phase-shift design.
Thus, for the $m$-th reflecting element, we define its service selection
vector as $\mathbf{a}_{m}\triangleq[a_{1,m},\ldots,a_{S,m}]^{T}$,
$a_{s,m}\in\\{0,1\\}$. Specifically, $a_{s,m}=1$ indicates that the $s$-th BS
is selected to be served with a fully tunable phase-shift response provided by
the $m$-th reflecting element, and $a_{s,m}=0$ represents that the $m$-th
reflecting element exhibits a fixed $0/2\pi$ phase-shift for the $s$-th BS.
For example, the four status in Table I can be represented by
$\mathbf{a}_{m}=[1,0,0]^{T}$, $\mathbf{a}_{m}=[0,1,0]^{T}$,
$\mathbf{a}_{m}=[0,0,1]^{T}$, and $\mathbf{a}_{m}=[0,0,0]^{T}$, respectively.
In addition, according to the above analysis, each reflecting element can
serve at most one BS. Thus, the constraint for the service selection vector of
the $m$-th reflecting element can be expressed as

$\left\|\mathbf{a}_{m}\right\|_{0}\leq 1,\;\;a_{s,m}\in\\{0,1\\},\;\;\forall
s,m.$ (5)

We further define an $S\times M$ service selection matrix for the IRS as
$\mathbf{A}\triangleq[\mathbf{a}_{1},\ldots,\mathbf{a}_{M}]$. It is noted that
the $s$-th row of $\mathbf{A}$ is also the service selection vector for the
$s$-th BS and indicates which reflecting elements serve this BS. For brevity,
we denote the service selection vector for the $s$-th BS as
$\widetilde{\mathbf{a}}_{s}\triangleq[a_{s,1},\ldots,a_{s,M}]^{T}\in\\{0,1\\}^{M}$,
which leads to

$\mathbf{A}=[\mathbf{a}_{1},\ldots,\mathbf{a}_{M}]=[\widetilde{\mathbf{a}}_{1},\ldots,\widetilde{\mathbf{a}}_{S}]^{T}.$
(6)

Denote the ideal phase-shift vector $\bm{\phi}_{s}$ for the $s$-th BS as
$\bm{\phi}_{s}\triangleq[\phi_{s,1},\ldots,\phi_{s,M}]^{T},\;\phi_{s,m}\in(0,2\pi]$.
The practical phase-shift $\angle\bm{\theta}_{s}$ is thus given by

$\angle\bm{\theta}_{s}=\bm{\phi}_{s}\odot\widetilde{\mathbf{a}}_{s}.$ (7)

Besides, as shown in [26]-[29], the reflection amplitude also varies with the
frequency of signals. However, for the signals at a certain frequency, i.e.,
transmitted by a certain BS, each reflecting element has the same reflection
amplitude degradation, which can be compensated at the transmitter side using
some advanced hardware techniques [42]. Therefore, neglecting the practical
reflection amplitude response will not affect the optimizations for the
considered multi-cell multi-band systems. In this way, the practical
reflection vector for the $s$-th BS can be written as
$\bm{\theta}_{s}=\exp\left(\jmath\angle\bm{\theta}_{s}\right)$. In summary,
the practical IRS reflection model is formulated as

$\displaystyle\bm{\theta}_{s}=\exp\left(\jmath\bm{\phi}_{s}\odot\widetilde{\mathbf{a}}_{s}\right),\;\;\forall
s,$ (8a) $\displaystyle\phi_{s,m}\in(0,2\pi],\;\;\forall s,\;m,$ (8b)
$\displaystyle\left\|\mathbf{a}_{m}\right\|_{0}\leq
1,\;\;a_{s,m}\in\\{0,1\\},\;\;\forall s,\;m.$ (8c)

Given the above system model and practical IRS reflection model, the joint
transmit beamforming and IRS reflection designs for the power minimization and
sum-rate maximization problems will be investigated in Sec. III and Sec. IV,
respectively.

##  III Algorithm for Power Minimization Problem

In this section, we aim to jointly optimize the transmit beamformers
$\mathbf{W}\triangleq[\mathbf{W}_{1},\ldots,\mathbf{W}_{S}]$ for all BSs, the
ideal IRS phase-shifts
$\bm{\Phi}\triangleq[\bm{\phi}_{1},\ldots,\bm{\phi}_{S}]$ for all BSs, and the
service selection matrix $\mathbf{A}$ to minimize the total transmit power,
subject to users’ SINR requirements and the practical reflection model.
Therefore, the power minimization problem is formulated as

$\displaystyle\min\limits_{\mathbf{W},\bm{\Phi},\mathbf{A}}~{}~{}$
$\displaystyle\sum_{s\in\mathcal{S}}\sum_{k_{s}\in\mathcal{K}_{s}}\left\|\mathbf{w}_{s,k_{s}}\right\|^{2}$
(9a) s.t. $\displaystyle\gamma_{s,k_{s}}\geq\Gamma_{s,k_{s}},\;\;\forall
s,\;k_{s},$ (9b)
$\displaystyle\bm{\theta}_{s}=\exp\left(\jmath\bm{\phi}_{s}\odot\widetilde{\mathbf{a}}_{s}\right),\;\;\forall
s,$ (9c) $\displaystyle\phi_{s,m}\in(0,2\pi],\;\;\forall s,\;m,$ (9d)
$\displaystyle\left\|\mathbf{a}_{m}\right\|_{0}\leq
1,\;\;a_{s,m}\in\\{0,1\\},\;\;\forall s,\;m,$ (9e)

where $\Gamma_{s,k_{s}}>0$ denotes the minimum SINR requirement of the
$k_{s}$-th user served by the $s$-th BS. The non-convex NP-hard problem (9) is
very difficult to solve due to the following three reasons. First, the coupled
variables brought by the constraints (9b) and (9c) make this multi-variable
problem hard to solve. Second, the frequency-selective model introduced by the
constraint (9c) causes significant difficulty to the IRS phase-shift design.
Third, the practical reflection responses for different BSs are different and
related by a binary service selection matrix in constraints (9c)-(9e). In
order to tackle these difficulties, we utilize the characteristics of the
proposed practical reflection model to decompose the original problem (9) into
two sub-problems and develop a three-step algorithm to solve the original
problem as presented below.

As described in Sec. II-B, by tuning the capacitance, each reflecting element
independently selects at most one BS to provide an ideal/tunable phase-shift.
This fact motivates us to decompose the original problem into service
selection design and the joint transmit beamforming, ideal phase-shift design
sub-problems. In particular, in order to better evaluate the ability of each
BS in exploiting IRS and determine the service selection, we first assume that
the IRS is ideal/tunable for all BSs (i.e.,
$\widetilde{a}_{s}=\mathbf{1},\forall s$) and jointly design the transmit
beamforming and ideal phase-shifts for each BS. Then, the service selection
matrix $\mathbf{A}$ is optimized to select appropriate reflecting elements for
serving each BS. Finally, with the obtained $\mathbf{A}$ the ideal phase-
shifts of selected reflecting elements and associated transmit beamforming are
jointly designed for each BS. It is noted that the joint transmit beamforming
and ideal phase-shift design with ideal IRS reflection model (i.e.,
$\mathbf{A}=1$) or the practical IRS reflection model follows the same route,
which is presented in Sec. III-A in details. In addition, the procedure of the
service selection design is described in Sec. III-B.

###  III-A Joint Transmit Beamforming and Ideal Phase-Shift Design

Based on the above analysis, with fixed $\mathbf{A}$ (i.e.,
$\widetilde{\mathbf{a}}_{s}$, $s=1,\ldots,S$), the joint transmit beamforming
and ideal phase-shift optimization problem for the $s$-th BS can be expressed
as

$\displaystyle\min\limits_{{\mathbf{W}}_{s},{\bm{\phi}}_{s}}~{}~{}$
$\displaystyle\sum_{k_{s}\in\mathcal{K}_{s}}\left\|{\mathbf{w}}_{s,k_{s}}\right\|^{2}$
(10a) s.t. $\displaystyle\gamma_{s,k_{s}}\geq\Gamma_{s,k_{s}},\;\;\forall
k_{s},$ (10b)
$\displaystyle{\bm{\theta}}_{s}=\exp\left(\jmath\bm{\phi}_{s}\odot\widetilde{\mathbf{a}}_{s}\right),$
(10c) $\displaystyle\phi_{s,m}\in(0,2\pi],\;\;\forall m.$ (10d)

Before solving this problem, we present a sufficient condition for the
feasibility of problem (10). If the equivalent channels
$\mathbf{G}_{s}^{H}\mathbf{H}_{\text{r},s}+\mathbf{H}_{\text{d},s},\forall s$
are full rank, where
$\mathbf{H}_{\mathrm{d},s}\triangleq\left[\mathbf{h}_{\mathrm{d},s,1},\cdots,\mathbf{h}_{\mathrm{d},s,K_{s}}\right]\in\mathbb{C}^{N_{\text{t}}\times
K_{s}}$ and
$\mathbf{H}_{\mathrm{r},s}\triangleq\left[\mathbf{h}_{\mathrm{r},s,1},\cdots,\mathbf{h}_{\mathrm{r},s,K_{s}}\right]\in\mathbb{C}^{M\times
K_{s}}$, then problem (10) is feasible for any finite QoS requirement
$\Gamma_{k_{s}}$. The specific proof is shown in Proposition 1 in [13]. Note
that this full rank assumption has been widely adopted in the literature [17],
[19], [21]. In fact, the channel coefficient matrix always satisfies full
rank, when it follows some continuous independently identical distributions
(i.i.d.), e.g. Rayleigh or Ricing. Therefore, in this paper, we also assume
that the equivalent channels are full rank and consequently problem (10) is
feasible. Moreover, some special cases that any equivalent channel is low
rank, the problem can also be handled by finding a suitable initial point or
the method in [43].

Notice that problem (10) is a non-convex problem due to the coupled variables
in constraint (10b) and the non-convex reflection constraint (10c). Therefore,
we utilize the BCD method to iteratively solve for $\mathbf{W}_{s}$ and
$\bm{\phi}_{s}$, which is described in the rest of this subsection.

$\displaystyle\mathbf{D}_{s}$
$\displaystyle\triangleq\sum_{k_{s}\in\mathcal{K}_{s}}\Big{(}\widehat{\mathbf{d}}_{s,k_{s},k_{s}}\widehat{\mathbf{d}}_{s,k_{s},k_{s}}^{H}-\Gamma_{s,k_{s},k_{s}}\sum_{j\in\mathcal{K}_{s}\atop
j\neq
k_{s}}\widehat{\mathbf{d}}_{s,k_{s},j}\widehat{\mathbf{d}}_{s,k_{s},j}^{H}\Big{)},$
(12a) $\displaystyle\mathbf{b}_{s}$
$\displaystyle\triangleq\sum_{k_{s}\in\mathcal{K}_{s}}\Big{[}\big{(}b_{s,k_{s},k_{s}}^{*}+\mathbf{1}^{H}\overline{\mathbf{d}}_{s,k_{s},k_{s}}^{*}\big{)}\mathbf{d}_{s,k_{s},k_{s}}-\Gamma_{s,k_{s}}\sum_{j\in\mathcal{K}_{s}\atop
j\neq
k_{s}}\big{(}b_{s,k_{s},j}^{*}+\mathbf{1}^{H}\overline{\mathbf{d}}_{s,k_{s},j}^{*}\big{)}\mathbf{d}_{s,k_{s},j}\Big{]}.$
(12b)

1) Update $\mathbf{W}_{s}$: Given the service selection matrix $\mathbf{A}$
and the ideal phase-shift vector $\bm{\phi}_{s}$, the practical IRS reflection
vector $\bm{\theta}_{s}$ is fixed. Thus, the combined effective channel from
the $s$-th BS to the $k_{s}$-th user is determined as
$\mathbf{h}_{s,k_{s}}^{H}\triangleq{\mathbf{h}^{H}_{\text{r},s,k_{s}}}{\bm{\Theta}}_{s}\mathbf{G}_{s}+\mathbf{h}_{\text{d},s,k_{s}}^{H}$.
Then, the sub-problem for optimizing $\mathbf{W}_{s}$ is formulated as

$\displaystyle\min\limits_{{\mathbf{W}}_{s}}~{}~{}$
$\displaystyle\sum_{k_{s}\in\mathcal{K}_{s}}||{\mathbf{w}}_{s,k_{s}}||^{2}$
(11a) s.t.
$\displaystyle\frac{|\mathbf{h}_{s,k_{s}}^{H}{\mathbf{w}}_{s,k_{s}}|^{2}}{\sum^{j\in\mathcal{K}_{s}}_{j\neq
k_{s}}|\mathbf{h}_{s,k_{s}}^{H}{\mathbf{w}}_{s,j}|^{2}+\sigma^{2}}\geq\Gamma_{s,k_{s}},\;\;\forall
k_{s},$ (11b)

which is a standard SOCP problem and can be easily solved using the popular
convex optimization toolbox such as CVX [44].

2) Update $\bm{\phi}_{s}$: After obtaining the transmit beamforming
$\mathbf{W}_{s}$, the objective of the power minimization problem (10) is
determined, which makes the IRS phase-shift design as a feasibility-check
problem with lots of possible solutions, which may not guarantee the converge
of the iteration algorithm. Therefore, another proper objective function with
respect to $\bm{\phi}_{s}$ is required to accelerate the convergence. A widely
used method is forcing the QoS constraint (10b) to be more stricter to provide
additional DoFs for the power minimization in the next iteration.

For brevity, we first define

$\displaystyle\mathbf{d}_{s,k_{s},j}$
$\displaystyle\triangleq\text{diag}(\mathbf{h}^{H}_{\text{r},s,k_{s}})\mathbf{G}_{s}{\mathbf{w}}_{s,j},$
(13a) $\displaystyle b_{s,k_{s},j}$
$\displaystyle\triangleq\mathbf{h}^{H}_{\text{d},s,k_{s}}{\mathbf{w}}_{s,j},$
(13b)

and reformulate the QoS constraint (10b) as

$\displaystyle\big{|}\bm{\theta}_{s}^{T}$
$\displaystyle\mathbf{d}_{s,k_{s},k_{s}}+b_{s,k_{s},k_{s}}\big{|}^{2}$ (14)
$\displaystyle-\Gamma_{s,k_{s}}\big{(}\sum_{j\in\mathcal{K}_{s}\atop j\neq
k_{s}}\big{|}\bm{\theta}_{s}^{T}\mathbf{d}_{s,k_{s},j}+b_{s,k_{s},j}\big{|}^{2}+\sigma^{2}\big{)}\geq
0.$

Then, the optimization problem for $\bm{\phi}_{s}$ is formulated to maximize
the sum of the left-hand side of (14) as

$\displaystyle\max\limits_{{\bm{\phi}}_{s}}$
$\displaystyle\sum_{k_{s}\in\mathcal{K}_{s}}\Big{(}\big{|}\bm{\theta}_{s}^{T}\mathbf{d}_{s,k_{s},k_{s}}+b_{s,k_{s},k_{s}}\big{|}^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\Gamma_{s,k_{s}}\sum_{j\in\mathcal{K}_{s}\atop
j\neq
k_{s}}\big{|}\bm{\theta}_{s}^{T}\mathbf{d}_{s,k_{s},j}+b_{s,k_{s},j}\big{|}^{2}\Big{)}$
(15a) s.t.
$\displaystyle~{}~{}{\bm{\theta}}_{s}=\exp\left(\jmath\bm{\phi}_{s}\odot\widetilde{\mathbf{a}}_{s}\right),$
(15b) $\displaystyle~{}~{}\phi_{s,m}\in(0,2\pi],\;\;\forall m.$ (15c)

Different from the IRS reflection designs with the ideal reflection model, we
notice that the practical reflection coefficient $\theta_{s,m}$ is determined
by both the ideal phase-shift $\phi_{s,m}$ and the service selection indicator
$a_{s,m}$ in constraint (15b). Particularly, when $a_{s,m}=0$, $\theta_{s,m}$
is fixed as 1 regardless of the value $\phi_{s,m}$, i.e., optimizing
$\phi_{s,m}$ cannot influence the objective value (15). Therefore, we only
need to optimize the phase-shift vector of the selected reflecting elements.
Following this analysis, we divide the set of reflecting elements
$\mathcal{M}$ into the set of selected reflecting elements
$\mathcal{I}_{s}\triangleq\\{m|a_{s,m}=1\\}$ and the set of others
$\mathcal{E}_{s}\triangleq\\{m|a_{s,m}=0\\}$, and then correspondingly
partition the phase-shift vector $\bm{\phi}_{s}$ into
$\widehat{\bm{\phi}}_{s}$ which contains $\phi_{s,m},\forall
m\in\mathcal{I}_{s}$ and $\overline{\bm{\phi}}_{s}$ which consists of
$\phi_{s,m},\forall m\in\mathcal{E}_{s}$, the practical reflection
$\bm{\theta}_{s}$ into $\widehat{\bm{\theta}}_{s}$ and
$\overline{\bm{\theta}}_{s}$, and $\mathbf{d}_{s,k_{s},j}$ into
$\widehat{\mathbf{d}}_{s,k_{s},j}$ and $\overline{\mathbf{d}}_{s,k_{s},j}$. It
is obvious that $\widehat{\bm{\phi}}_{s}$ is the required optimization
variable, $\overline{\bm{\theta}}_{s}=\mathbf{1}$ and constraint (15b) is
converted to $\widehat{\bm{\theta}}_{s}=\exp(\jmath\widehat{\bm{\phi}}_{s})$.
Thus, $\widehat{\bm{\phi}}_{s}$ can be obtained by taking the angle of
$\widehat{\bm{\theta}}_{s}$, and problem (15) with respect to
$\widehat{\bm{\theta}}_{s}$ can be reformulated as

$\displaystyle\min\limits_{\widehat{\bm{\theta}}_{s}}~{}~{}$
$\displaystyle-\widehat{\bm{\theta}}_{s}^{T}\mathbf{D}_{s}\widehat{\bm{\theta}}^{*}_{s}-\widehat{\bm{\theta}}_{s}^{T}\mathbf{b}_{s}-\mathbf{b}_{s}^{H}\widehat{\bm{\theta}}_{s}$
(16a) s.t.
$\displaystyle|\widehat{\theta}_{s,m}|=1,\;\;m=1,\ldots,\left|\mathcal{I}_{s}\right|,$
(16b)

where $\mathbf{D}_{s}$ and $\mathbf{b}_{s}$ are defined in (12) as shown at
the top of this page.

It can be observed that the main difficulty to tackle problem (16) is the non-
convex unit modulus constraint (16b). In the literature of IRS optimization,
non-convex relaxation, e.g., SDP/SDR, MM, and alternating minimization methods
are usually exploited to handle constraint (16b) on the Euclidean space.
However, a fast convergence and satisfactory performance cannot be guaranteed
since either the objective function or the constraint is relaxed using these
methods. Therefore, we propose to solve problem (16) on the Riemannian space.
Specifically, the unit modulus constraint (16b) forms a smooth Riemannian
manifold, on which problem (16) becomes an unconstrained problem. Since the
Riemannian manifold resembles a Euclidean space at each point, lots of classic
algorithms developed on the Euclidean space, e.g., steepest descent, conjugate
gradient, quasi-Newton methods, have found their counterparts on the
Riemannian manifold [45], [46]. In addition, since the objective function
(16a) is a typical quadratic function with respect to
$\widehat{\bm{\theta}}_{s}$, the required first-order and second-order
derivatives in the above methods can be easily calculated as
$-2\mathbf{D}_{s}\widehat{\bm{\theta}}^{*}_{s}-2\mathbf{b}_{s}^{H}$ and
$-2\mathbf{D}_{s}$, respectively. Therefore, problem (16) can be readily
solved using the advanced Riemannian manifold based algorithms. Due to space
limitations, the details are omitted in this paper.

After obtaining the solution $\widehat{\bm{\theta}}_{s}^{\star}$ of problem
(16), the ideal phase-shift $\phi_{s,m}^{\star},\forall m$, is updated by

$\phi_{s,m}^{\star}=\left\\{\begin{array}[]{lr}\angle\widehat{\theta}_{s,m}^{\star},&a_{s,m}=1,\\\
\phi_{s,m},&a_{s,m}=0.\end{array}\right.$ (17)

###  III-B Service Selection Design

With obtained transmit beamforming $\mathbf{W}_{s}$ and ideal phase-shift
vector $\bm{\phi}_{s}$ for each BS, the original problem (9) is transformed
into a service selection design problem. Since the service selection matrix
$\mathbf{A}$ cannot directly affect (9a), we should formulate another
optimization problem to achieve better QoS for all users excepting to provide
a larger feasible area, which can assist the joint transmit beamforming and
practical phase-shift design in the next stage. Thus, similar to problem (15),
the optimization problem for updating $\mathbf{A}$ is formulated as

$\displaystyle\max\limits_{\mathbf{A}}$
$\displaystyle~{}\sum_{s\in\mathcal{S}}\sum_{k_{s}\in\mathcal{K}_{s}}\Big{(}\big{|}\bm{\theta}_{s}^{T}\mathbf{d}_{s,k_{s},k_{s}}+b_{s,k_{s},k_{s}}\big{|}^{2}$
$\displaystyle\hskip 51.21504pt-\Gamma_{s,k_{s}}\sum_{j\in\mathcal{K}_{s}\atop
j\neq
k_{s}}\big{|}\bm{\theta}_{s}^{T}\mathbf{d}_{s,k_{s},j}+b_{s,k_{s},j}\big{|}^{2}\Big{)}$
(18a) s.t.
$\displaystyle~{}~{}{\bm{\theta}}_{s}=\exp\left(\jmath\bm{\phi}_{s}\odot\widetilde{\mathbf{a}}_{s}\right),\;\;\forall
s,$ (18b) $\displaystyle~{}\left\|\mathbf{a}_{m}\right\|_{0}\leq 1,\;\;\forall
m,$ (18c) $\displaystyle~{}~{}a_{s,m}\in\\{0,1\\},\;\;\forall s,\;m.$ (18d)

Since the objective function is implicit with the variable $\mathbf{A}$, we
first re-write it by substituting the equality constraint (18b) into (18).
Thus, the term $\bm{\theta}_{s}^{T}\mathbf{d}_{s,k_{s},j}+b_{s,k_{s},j}$ in
(18a) can be equivalently re-written as

$\displaystyle\bm{\theta}_{s}^{T}\mathbf{d}_{s,k_{s},j}+b_{s,k_{s},j}$
$\displaystyle=\big{(}\widetilde{\mathbf{a}}_{s}^{T}\odot
e^{\jmath\bm{\phi}_{s}^{T}}\big{)}\mathbf{d}_{s,k_{s},j}+(\bm{1}-{\widetilde{\mathbf{a}}_{s}})^{T}\mathbf{d}_{s,k_{s},j}+b_{s,k_{s},j}$
(19a)
$\displaystyle=\widetilde{\mathbf{a}}_{s}^{T}\text{diag}\\{e^{\jmath\bm{\phi}_{s}}-\bm{1}\\}\mathbf{d}_{s,k_{s},j}+\bm{1}^{T}\mathbf{d}_{s,k_{s},j}+b_{s,k_{s},j}$
(19b)
$\displaystyle=\widetilde{\mathbf{a}}_{s}^{T}\widetilde{\mathbf{d}}_{s,k_{s},j}+\widetilde{b}_{s,k_{s},j},$
(19c)

where for simplicity we define

$\displaystyle\widetilde{\mathbf{d}}_{s,k_{s},j}$
$\displaystyle\triangleq\mathrm{diag}(e^{\jmath\bm{\phi}_{s}}-\bm{1})\mathbf{d}_{s,k_{s},j},\;\;\forall
s,\;k_{s},\;j,$ (20a) $\displaystyle\widetilde{b}_{s,k_{s},j}$
$\displaystyle\triangleq\bm{1}^{T}\mathbf{d}_{s,k_{s},j}+{b}_{s,k_{s},j},\;\;\forall
s,\;k_{s},\;j.$ (20b)

Then, by plugging (19c) into the objective (18), expanding the quadratic
terms, and ignoring the constant terms, the original problem (18) can be
equivalently re-formulated as

$\displaystyle\max\limits_{\mathbf{A}}$
$\displaystyle~{}\sum_{s\in\mathcal{S}}\widetilde{\mathbf{a}}_{s}^{T}\widetilde{\mathbf{E}}_{s}\widetilde{\mathbf{a}}_{s}-\sum_{s\in\mathcal{S}}\widetilde{\mathbf{a}}_{s}^{T}\widetilde{\mathbf{D}}_{s}\widetilde{\mathbf{a}}_{s}+\sum_{s\in\mathcal{S}}\mathfrak{R}\\{\widetilde{\mathbf{a}}_{s}^{T}\widetilde{\bm{\beta}}_{s}\\}$
(21a) s.t. $\displaystyle~{}\left\|\mathbf{a}_{m}\right\|_{0}\leq
1,\;\;\forall m,$ (21b) $\displaystyle~{}~{}a_{s,m}\in\\{0,1\\},\;\;\forall
s,\;m,$ (21c)

where we define

$\displaystyle\widetilde{\mathbf{E}}_{s}\triangleq$
$\displaystyle~{}\sum_{k_{s}\in\mathcal{K}_{s}}(1+\Gamma_{s,k_{s}})\mathbf{d}_{s,k_{s},j=k_{s}}\mathbf{d}^{H}_{s,k_{s},j=k_{s}},\;\;\forall
s,$ (22a) $\displaystyle\widetilde{\mathbf{D}}_{s}\triangleq$
$\displaystyle~{}\sum_{k_{s}\in\mathcal{K}_{s}}\Gamma_{s,k_{s}}\sum_{j\in\mathcal{K}_{s}}\mathbf{d}_{s,k_{s},j}\mathbf{d}^{H}_{s,k_{s},j},\;\;\forall
s,$ (22b) $\displaystyle\widetilde{\bm{\beta}}_{s}\triangleq$
$\displaystyle~{}2\sum_{k_{s}\in\mathcal{K}_{s}}\big{(}\mathbf{d}_{s,k_{s},j=k_{s}}b^{*}_{s,k_{s},j=k_{s}}$
$\displaystyle\hskip 51.21504pt-\Gamma_{s,k_{s}}\sum_{j\neq
k_{s}}\mathbf{d}_{s,k_{s},j}b^{*}_{s,k_{s},j}\big{)},\;\;\forall s.$ (22c)

It can be observed that problem (21) cannot be directly solved due to the non-
convex objective function (21a) and the non-smooth constraints (21b) and
(21c). Therefore, we propose to transform both the constraints and the
objective function into more tractable forms in the followings.

Algorithm 1 Joint Transmit Beamforming and Practical IRS Reflection Design for
Power Minimization Problem

0: $\mathbf{h}_{\text{r},s,k_{s}}^{H}$, $\mathbf{G}_{s}$,
$\mathbf{h}_{\text{d},s,k_{s}}^{H}$, $\Gamma_{s,k_{s}}$, $\forall
s\in\mathcal{S}$, $\forall k_{s}\in\mathcal{K}_{s}$, $\sigma^{2}$.

0: $\mathbf{W}_{s}^{\star}$, $\bm{\phi}_{s}^{\star}$, $\forall
s\in\mathcal{S}$, $\mathbf{A}^{\star}$.

1: Initialize $\mathbf{A}=\mathbf{1}$, $\bm{\Phi}$.

2: for $s=1$ to $S$ do

3: while no convergence for (10a) do

4: Calculate $\mathbf{W}_{s}^{\star}$ by solving (11) with CVX.

5: Obtain $\widehat{\bm{\theta}}_{s}^{\star}$ by solving (16) with Riemannian
manifold optimizations.

6: Construct ideal phase-shift $\bm{\phi}_{s}^{\star}$ by (17).

7: end while

8: end for

9: while no convergence for $\mathbf{A}$ do

10: Obtain $\mathbf{A}$ by solving (27).

11: end while

12: Repeat steps 2-8.

13: Return $\mathbf{W}_{s}^{\star}$, $\bm{\phi}_{s}^{\star}$, $\forall
s\in\mathcal{S}$,
$\mathbf{A}^{\star}=[\mathbf{a}_{1}^{\star},\ldots,\mathbf{a}_{M}^{\star}]$.

Considering that each element of $\mathbf{A}$ is binary as shown in constraint
(21c), the 0-norm constraint (21b) can be equivalently transformed into a
1-norm constraint and re-formulated as:

$||\mathbf{a}_{m}||_{0}=||\mathbf{a}_{m}||_{1}=\mathbf{1}^{T}\mathbf{a}_{m}\leq
1,\;\;\forall m.$ (23)

Then, by utilizing the difference-of-convex function method in [47], the
binary constraint (21c) is be transformed into:

$\displaystyle 0\leq a_{s,m}\leq 1,\;\;\forall s,\;m,$ (24a)
$\displaystyle\sum_{s\in\mathcal{S}}\big{(}\mathbf{1}^{T}\widetilde{\mathbf{a}}_{s}-\widetilde{\mathbf{a}}^{T}_{s}\widetilde{\mathbf{a}}_{s}\big{)}\leq
0.$ (24b)

Since the constraint (24b) is non-convex, we further apply the penalty method
to move it to the objective and re-formulate the optimization problem as

$\displaystyle\max\limits_{\mathbf{A}}$
$\displaystyle~{}\sum_{s\in\mathcal{S}}\widetilde{\mathbf{a}}_{s}^{T}\widetilde{\mathbf{E}}_{s}\widetilde{\mathbf{a}}_{s}-\sum_{s\in\mathcal{S}}\widetilde{\mathbf{a}}_{s}^{T}\widetilde{\mathbf{D}}_{s}\widetilde{\mathbf{a}}_{s}+\sum_{s\in\mathcal{S}}\mathfrak{R}\\{\widetilde{\mathbf{a}}_{s}^{T}\widetilde{\bm{\beta}}_{s}\\}$
$\displaystyle~{}~{}-\tau\sum_{s\in\mathcal{S}}\big{(}\mathbf{1}^{T}\widetilde{\mathbf{a}}_{s}-\widetilde{\mathbf{a}}^{T}_{s}\widetilde{\mathbf{a}}_{s}\big{)}$
(25a) s.t. $\displaystyle~{}~{}\mathbf{1}^{T}\mathbf{a}_{m}\leq 1,\;\;\forall
m,$ (25b) $\displaystyle~{}~{}0\leq a_{s,m}\leq 1,\;\;\forall s,\;m,$ (25c)

where $\tau>0$ is a penalty factor. It is proved in [48] that the transform is
equivalent when $\tau$ has a moderately high value. After obtaining the
preferable linear constraints (25b) and (25c), the non-convex objective
function (25a) is the major obstacle. In order to efficiently solve this
problem, we employ the MM method and seek for a linear surrogate function of
(25a) by utilizing the first-order Taylor expansion. Specifically, a lower-
bounded surrogate function of the first term and last term in (25a) in the
$t$-th iteration is derived as

$\displaystyle\sum_{s\in\mathcal{S}}\widetilde{\mathbf{a}}_{s}^{T}\widetilde{\mathbf{E}}_{s}\widetilde{\mathbf{a}}_{s}-\tau\sum_{s\in\mathcal{S}}\big{(}\mathbf{1}^{T}\widetilde{\mathbf{a}}_{s}-\widetilde{\mathbf{a}}^{T}_{s}\widetilde{\mathbf{a}}_{s}\big{)}$
(26a)
$\displaystyle\geq\sum_{s\in\mathcal{S}}\big{[}({\widetilde{\mathbf{a}}_{s}^{t})}^{T}\widetilde{\mathbf{E}}_{s}\widetilde{\mathbf{a}}_{s}^{t}+2\mathfrak{R}\big{\\{}(\widetilde{\mathbf{a}}_{s}-\widetilde{\mathbf{a}}_{s}^{t})^{T}\widetilde{\mathbf{E}}_{s}\widetilde{\mathbf{a}}_{s}^{t}\big{\\}}\big{]}$
$\displaystyle~{}~{}~{}~{}-\tau\sum_{s\in\mathcal{S}}\big{[}\mathbf{1}^{T}\widetilde{\mathbf{a}}_{s}-{(\widetilde{\mathbf{a}}^{t}_{s})}^{T}\widetilde{\mathbf{a}}^{t}_{s}-2{(\widetilde{\mathbf{a}}^{t}_{s})}^{T}(\widetilde{\mathbf{a}}_{s}-\widetilde{\mathbf{a}}^{t}_{s})\big{]},$
(26b)

where the vector $\widetilde{\mathbf{a}}^{t}_{s}$ denotes the solution of the
last iteration.

Based on the above derivations, by substituting the surrogate function (26)
into (25a), the optimization problem in the $t$-th iteration can be formulated
as

$\displaystyle\max\limits_{\mathbf{A}}$
$\displaystyle~{}~{}-\sum_{s\in\mathcal{S}}\widetilde{\mathbf{a}}_{s}^{T}\widetilde{\mathbf{D}}_{s}\widetilde{\mathbf{a}}_{s}+\sum_{s\in\mathcal{S}}\mathfrak{R}\\{(\widetilde{\mathbf{a}}_{s}^{T}\bm{\beta}_{s}\\}+c$
(27a) s.t. $\displaystyle~{}~{}\mathbf{1}^{T}\mathbf{a}_{m}\leq 1,\;\;\forall
m,$ (27b) $\displaystyle~{}~{}0\leq a_{s,m}\leq 1,\;\;\forall s,\;m.$ (27c)

where

$\displaystyle\bm{\beta}_{s}\triangleq$
$\displaystyle~{}2\widetilde{\mathbf{E}}_{s}\widetilde{\mathbf{a}}^{t}_{s}+\tau\widetilde{\mathbf{a}}^{t}_{s}+\widetilde{\bm{\beta}}_{s},\;\;\forall
s,$ (28a) $\displaystyle c\triangleq$
$\displaystyle~{}\sum_{s\in\mathcal{S}}\big{[}({\widetilde{\mathbf{a}}_{s}^{t})}^{T}\widetilde{\mathbf{E}}_{s}\widetilde{\mathbf{a}}_{s}^{t}-2\mathfrak{R}\\{(\widetilde{\mathbf{a}}_{s}^{t})^{T}\widetilde{\mathbf{E}}_{s}\widetilde{\mathbf{a}}_{s}^{t}\\}$
$\displaystyle~{}~{}~{}~{}~{}-(\widetilde{\mathbf{a}}_{s}^{t})^{T}\lambda_{s}\widetilde{\mathbf{a}}_{s}^{t}-\tau{(\widetilde{\mathbf{a}}^{t}_{s})}^{T}\widetilde{\mathbf{a}}^{t}_{s}\big{]}.$
(28b)

It is obvious that problem (27) is a convex problem that can be efficiently
solved by various existing methods.

###  III-C Summary and Complexity Analysis

Summary: Based on the above derivations, the joint transmit beamforming and
IRS reflection design for the power minimization problem (9) is
straightforward and summarized in Algorithm 1. Given initialization
$\mathbf{A}=\mathbf{1}$, the transmit beamforming and ideal phase-shifts for
each BS are iteratively updated. Note that the reformulated objective function
(13) is utilized in the iteration, theoretical convergence analysis cannot be
easily obtained. But the simulation results shown in Sec. V illustrate that
the proposed algorithm will converge fast to a local optimum. Then, the
service selection matrix $\mathbf{A}^{\star}$ is optimized based on the
obtained transmit beamforming and ideal phase-shift results. Finally,
$\mathbf{W}_{s}^{\star}$ and $\bm{\phi}_{s}^{\star}$ are solved one more time
with the obtained service selection matrix $\mathbf{A}^{\star}$.

Complexity analysis: In each iteration, updating the transmit beamforming
$\mathbf{W}_{s}$ for each BS by solving a SOCP problem has a complexity of
$\mathcal{O}(N_{\mathrm{t}}^{4.5}K_{s}^{3.5})$; updating the ideal phase-shift
vector $\widehat{\bm{\theta}}_{s}$ using Riemannian manifold optimizations has
a complexity of $\mathcal{O}(M^{1.5})$ at most. The complexity for updating
the service selection matrix $\mathbf{A}$ is $\mathcal{O}((SM)^{3})$.
Therefore, the total complexity of the proposed algorithm is of order
$\mathcal{O}\\{\sum_{s=1}^{S}(2N_{\mathrm{t}}^{4.5}K_{s}^{3.5}+2M^{1.5})+(SM)^{3}\\}$.

##  IV Algorithms for Sum-rate Maximization Problem

In this section, we investigate the sum-rate maximization problem in the
considered multi-cell multi-band systems. It is worth noting that the power
minimization problem and the sum-rate maximization problem cannot be directly
converted from one to another. In order to effectively handle this non-convex
NP-hard optimization problem, the WMMSE approach and BCD method are employed
to iteratively solve for each variable with a closed-form solution.

Specifically, our goal is to jointly optimize the transmit beamformers
$\mathbf{W}$ for the users of all BSs, the ideal IRS phase-shifts $\bm{\Phi}$,
and the service selection matrix $\mathbf{A}$ to maximize the sum-rate,
subject to the transmit power budget of each BS and the practical IRS
reflection model. Therefore, the optimization problem is formulated as

$\displaystyle\max\limits_{\mathbf{W},\bm{\Phi},\mathbf{A}}~{}~{}$
$\displaystyle\sum_{s\in\mathcal{S}}\sum_{k_{s}\in\mathcal{K}_{s}}\log_{2}(1+\gamma_{s,k_{s}})$
(29a) s.t.
$\displaystyle\sum_{k_{s}\in\mathcal{K}_{s}}\left\|\mathbf{w}_{s,k_{s}}\right\|^{2}\leq
P_{s},\;\;\forall s,$ (29b)
$\displaystyle\bm{\theta}_{s}=\exp\left(\jmath\bm{\phi}_{s}\odot\widetilde{\mathbf{a}}_{s}\right),\;\;\forall
s,$ (29c) $\displaystyle\phi_{s,m}\in(0,2\pi],\;\;\forall s,\;m,$ (29d)
$\displaystyle\left\|\mathbf{a}_{m}\right\|_{0}\leq
1,\;\;a_{s,m}\in\\{0,1\\},\;\;\forall s,\;m,$ (29e)

where $P_{s}>0$ denotes the transmit power budget of the $s$-th BS. Seeking
for the solution to this non-convex NP-hard problem is very difficult, not
only due to the complicated objective function (29a) that contains the
fractional terms in $\log({\cdot})$, but also because of the coupled variables
in the objective function (29a) and constraints (29c). Therefore, in the
followings we first employ the WMMSE approach to convert the original
optimization problem (29) into a more tractable multi-variate problem and then
use the BCD method to iteratively solve for each variable.

Although the WMMSE approach has been widely used for solving the sum-rate
maximization problem, we would like to emphasize that the IRS beamforming
design with the practical reflection model will be quite different and
difficult due to the complicated constraint of each phase-shift element.
Therefore, we need to develop an efficient algorithm to solve this sum-rate
maximization problem (29), which for the first time considers the frequency-
selective characteristic of the IRS in multi-cell networks.

###  IV-A Problem Reformulation by WMMSE

Following the derivations in [49], a scalar $\nu_{s,k_{s}}\in\mathbb{C}$ is
applied for the $k_{s}$-th user to estimate the transmitted signal
$z_{s,k_{s}}$. Then, the MSE of the $k_{s}$-th user can be calculated as

$\displaystyle\text{MSE}_{s,k_{s}}=$
$\displaystyle\mathbb{E}\big{\\{}(\nu_{s,k_{s}}^{*}y_{s,k_{s}}-z_{s,k_{s}})(\nu_{s,k_{s}}^{*}y_{s,k_{s}}-z_{s,k_{s}})^{*}\big{\\}}$
(30) $\displaystyle=$
$\displaystyle\sum_{j\in\mathcal{K}_{s}}\big{|}\nu_{s,k_{s}}^{*}({\mathbf{h}^{H}_{\text{r},s,k_{s}}}\bm{\Theta}_{s}\mathbf{G}_{s}+\mathbf{h}_{\text{d},s,k_{s}}^{H})\mathbf{w}_{s,j}\big{|}^{2}$
$\displaystyle-2\mathfrak{R}\big{\\{}\nu_{s,k_{s}}^{*}({\mathbf{h}^{H}_{\text{r},s,k_{s}}}\bm{\Theta}_{s}\mathbf{G}_{s}+\mathbf{h}_{\text{d},s,k_{s}}^{H})\mathbf{w}_{s,k_{s}}\big{\\}}$
$\displaystyle+|\nu_{s,k_{s}}|^{2}\sigma^{2}+1,\;\;\forall s,\;k_{s}.$

Introducing the MSE weights $\mu_{s,k_{s}}\in\mathbb{R}^{+},\forall s,k_{s}$,
the sum-rate maximization problem (29) is equivalently reformulated as:

$\displaystyle\min\limits_{\mathbf{W},\bm{\Phi},\mathbf{A},\bm{\nu},\bm{\mu}}~{}~{}$
$\displaystyle\sum_{s\in\mathcal{S}}\sum_{k_{s}\in\mathcal{K}_{s}}\left(\mu_{s,k_{s}}\text{MSE}_{s,k_{s}}-\log_{2}\mu_{s,k_{s}}\right)$
(31a) s.t. $\displaystyle(\text{\ref{eq:sum rate problem
b}})-(\text{\ref{eq:sum rate problem e}}),$ (31b)

where $\bm{\mu}$ and $\bm{\nu}$ denote the vectors that contain the auxiliary
variables $\mu_{s,k_{s}}$ and $\nu_{s,k_{s}},\forall s,k_{s}$, respectively.
Since the complicated $\log_{2}(1+\gamma_{s,k_{s}})$ term in the objective
function is transformed into a polynomial term and a very simple
$\log_{2}(\cdot)$ term, the reformulated optimization problem (31) is much
more tractable. Specifically, the obtained multi-variate optimization problem
(31) can be solved using the typical BCD method. The details for updating each
variable are presented in the next subsection.

$\displaystyle\mathbf{B}_{s}$
$\displaystyle\triangleq\sum_{k_{s}\in\mathcal{K}_{s}}\mu_{s,k_{s}}|\nu_{s,k_{s}}|^{2}\text{diag}\\{\mathbf{h}_{\text{r},s,k_{s}}^{H}\\}\mathbf{G}_{s}\sum_{j\in\mathcal{K}_{s}}\mathbf{w}_{s,j}\mathbf{w}_{s,j}^{H}\mathbf{G}_{s}^{H}\text{diag}\\{\mathbf{h}_{\text{r},s,k_{s}}\\},\forall
s,$ (34a) $\displaystyle\mathbf{c}_{s}$
$\displaystyle\triangleq\sum_{k_{s}\in\mathcal{K}_{s}}\mu_{s,k_{s}}\nu_{s,k_{s}}^{*}\text{diag}\\{\mathbf{h}_{\text{r},s,k_{s}}^{H}\\}\mathbf{G}_{s}\mathbf{w}_{s,k_{s}}\Big{(}1-\nu_{s,k_{s}}\sum_{j\in\mathcal{K}_{s}}\mathbf{w}_{s,j}\mathbf{w}_{s,j}^{H}\mathbf{h}_{\text{d},s,k_{s}}\Big{)},\forall
s.$ (34b)

###  IV-B Block Update by BCD

####  IV-B.1 Update $\bm{\nu}$

With given the transmit beamformer $\mathbf{W}$, the ideal IRS phase-shift
$\bm{\Phi}$, the service selection matrix $\mathbf{A}$, and the MSE weight
vector $\bm{\mu}$, the design of $\bm{\nu}$ for each user is independent. The
optimization problem of solving for $\nu_{s,k_{s}}$ can be expressed as

$\displaystyle\min_{\nu_{s,k_{s}}}{}$
$\displaystyle|\nu_{s,k_{s}}|^{2}\Big{(}\sum_{j\in\mathcal{K}_{s}}\big{|}({\mathbf{h}^{H}_{\text{r},s,k_{s}}}\bm{\Theta}_{s}\mathbf{G}_{s}+\mathbf{h}_{\text{d},s,k_{s}}^{H})\mathbf{w}_{s,j}\big{|}^{2}+\sigma^{2}\Big{)}$
(32)
$\displaystyle-2\mathfrak{R}\big{\\{}\nu_{s,k_{s}}^{*}({\mathbf{h}^{H}_{\text{r},s,k_{s}}}\bm{\Theta}_{s}\mathbf{G}_{s}+\mathbf{h}_{\text{d},s,k_{s}}^{H})\mathbf{w}_{s,k_{s}}\big{\\}},$

which is an unconstrained quadratic convex problem. Thus, setting the first-
order derivative with respect to $\nu_{s,k_{s}}$ to zero, the optimal solution
$\nu_{s,k_{s}}^{\star}$ can be easily calculated as

$\nu_{s,k_{s}}^{\star}=\frac{({\mathbf{h}^{H}_{\text{r},s,k_{s}}}\bm{\Theta}_{s}\mathbf{G}_{s}+\mathbf{h}_{\text{d},s,k_{s}}^{H})\mathbf{w}_{s,k_{s}}}{\sum_{j\in\mathcal{K}_{s}}\big{|}({\mathbf{h}^{H}_{\text{r},s,k_{s}}}\bm{\Theta}_{s}\mathbf{G}_{s}+\mathbf{h}_{\text{d},s,k_{s}}^{H})\mathbf{w}_{s,j}\big{|}^{2}+\sigma^{2}}.$
(33)

####  IV-B.2 Update $\bm{\mu}$

Fixing the transmit beamformer $\mathbf{W}$, the ideal IRS phase-shift
$\bm{\Phi}$, the service selection matrix $\mathbf{A}$, and the variable
$\bm{\nu}$, the optimization problem with respect to each independent MSE
weight $\mu_{s,k_{s}}$ can be formulated as

$\min_{\mu_{s,k_{s}}}~{}~{}\mu_{s,k_{s}}\text{MSE}_{s,k_{s}}-\log_{2}\mu_{s,k_{s}}.$
(35)

Similarly, the optimal solution $\mu_{s,k_{s}}^{\star}$ can be obtained by
applying the typical first-order optimality condition as

$\mu_{s,k_{s}}^{\star}=\text{MSE}_{s,k_{s}}^{-1}\overset{(a)}{=}1+\gamma_{s,k_{s}},$
(36)

where $(a)$ is derived by substituting the optimal $\nu_{s,k_{s}}^{\star}$ in
(33) into $\text{MSE}_{s,k_{s}}$ in (30).

####  IV-B.3 Update $\mathbf{W}$

After obtaining the variable $\bm{\nu}$, the MSE weight $\bm{\mu}$, the ideal
IRS phase-shift $\bm{\Phi}$, and the service selection matrix $\mathbf{A}$,
the optimization problem for designing the transmit beamformer
$\mathbf{W}_{s}$ for the $s$-th BS can be formulated as

$\displaystyle\min\limits_{\mathbf{W}_{s}}$
$\displaystyle\sum_{k_{s}\in\mathcal{K}_{s}}\mu_{s,k_{s}}\Big{(}\sum_{j\in\mathcal{K}_{s}}|\overline{\mathbf{h}}^{H}_{s,k_{s}}\mathbf{w}_{s,j}|^{2}-2\mathfrak{R}\\{\overline{\mathbf{h}}^{H}_{s,k_{s}}\mathbf{w}_{s,k_{s}}\\}\Big{)}$
(37a) s.t.
$\displaystyle\sum_{k_{s}\in\mathcal{K}_{s}}||\mathbf{w}_{s,k_{s}}||^{2}\leq
P_{s},$ (37b)

where we define
$\overline{\mathbf{h}}_{s,k_{s}}^{H}\triangleq\nu_{s,k_{s}}^{*}({\mathbf{h}^{H}_{\text{r},s,k_{s}}}\bm{\Theta}_{s}\mathbf{G}_{s}+\mathbf{h}_{\text{d},s,k_{s}}^{H})$
for brevity. Note that (37) is a convex optimization problem, which can be
readily solved using standard convex optimization algorithms, e.g., the
interior-point algorithm. In order to reduce the execution time, we employ the
typical Lagrange multiplier method to obtain a closed-form solution. Attaching
a Lagrange multiplier $\lambda_{s}\geq 0$ to the power constraint (37b), the
Lagrange function of problem (37) can be formulated as

$\displaystyle\mathcal{L}(\mathbf{W}_{s},\lambda_{s})$
$\displaystyle\triangleq\sum_{k_{s}\in\mathcal{K}_{s}}\mu_{s,k_{s}}\Big{(}\sum_{j\in\mathcal{K}_{s}}\big{|}\overline{\mathbf{h}}^{H}_{s,k_{s}}\mathbf{w}_{s,j}\big{|}^{2}-2\mathfrak{R}\big{\\{}\overline{\mathbf{h}}^{H}_{s,k_{s}}\mathbf{w}_{s,k_{s}}\big{\\}}\Big{)}$
$\displaystyle~{}~{}~{}~{}+\lambda_{s}\Big{(}\sum_{k_{s}\in\mathcal{K}_{s}}||\mathbf{w}_{s,k_{s}}||^{2}-P_{s}\Big{)}$
(38a)
$\displaystyle=\sum_{k_{s}\in\mathcal{K}_{s}}\Big{(}\mathbf{w}^{H}_{s,k_{s}}\sum_{j\in\mathcal{K}_{s}}\mu_{s,j}\overline{\mathbf{h}}_{s,j}\overline{\mathbf{h}}^{H}_{s,j}\mathbf{w}_{s,k_{s}}-2\mu_{s,k_{s}}\mathfrak{R}\big{\\{}\overline{\mathbf{h}}_{s,k_{s}}\mathbf{w}_{s,k_{s}}\big{\\}}$
$\displaystyle~{}~{}~{}~{}+\lambda_{s}\mathbf{w}^{H}_{s,k_{s}}\mathbf{w}_{s,k_{s}}\Big{)}-\lambda_{s}P_{s}.$
(38b)

Then, setting
$\frac{\partial\mathcal{L}(\mathbf{W}_{s},\lambda_{s})}{\mathbf{w}_{s,k_{s}}}=\mathbf{0}$,
the optimal beamforming vector $\mathbf{w}_{s,k_{s}}^{\star}$ is given by

$\mathbf{w}_{s,k_{s}}^{\star}=\mu_{s,k_{s}}\Big{(}\sum_{j\in\mathcal{K}_{s}}\mu_{s,j}\overline{\mathbf{h}}_{s,j}\overline{\mathbf{h}}^{H}_{s,j}+\lambda_{s}\mathbf{I}_{N_{\text{t}}}\Big{)}^{-1}\overline{\mathbf{h}}_{s,k_{s}},$
(39)

where the Lagrange multiplier $\lambda_{s}\geq 0$ should be guaranteed to
satisfy the complementarity slackness condition of the power constraint (37b).
Plugging the obtained transmit beamformers
$\mathbf{w}_{s,k_{s}}^{\star},\forall k_{s},$ into the power constraint (37b),
the optimal solution of $\lambda_{s}$ can be easily obtained by one
dimensional search methods, e.g., the bisection search method.

####  IV-B.4 Update $\bm{\Phi}$ and $\mathbf{A}$

Since the ideal IRS phase-shift $\bm{\Phi}$ is closely associated with the
service selection $\mathbf{A}$ in designing the practical IRS reflection
$\bm{\Theta}$ for all BSs, a joint design for $\bm{\Phi}$ and $\mathbf{A}$ is
necessary to achieve a satisfactory performance, which has not been considered
in the existing literature. Thus, with the fixed variable $\bm{\nu}$, MSE
weight $\bm{\mu}$, and transmit beamformer $\mathbf{W}$, the optimization
problem for the joint ideal IRS phase-shift $\bm{\Phi}$ and service selection
matrix $\mathbf{A}$ design is written by

$\displaystyle\min\limits_{\bm{\Phi},\mathbf{A}}$
$\displaystyle\sum_{s\in\mathcal{S}}\sum_{k_{s}\in\mathcal{K}_{s}}\mu_{s,k_{s}}\Big{[}\sum_{j\in\mathcal{K}_{s}}\big{|}\nu_{s,k_{s}}^{*}({\mathbf{h}^{H}_{\text{r},s,k_{s}}}\bm{\Theta}_{s}\mathbf{G}_{s}+\mathbf{h}_{\text{d},s,k_{s}}^{H})\mathbf{w}_{s,j}\big{|}^{2}$
$\displaystyle~{}~{}~{}~{}~{}-2\mathfrak{R}\big{\\{}\nu_{s,k_{s}}^{*}({\mathbf{h}^{H}_{\text{r},s,k_{s}}}\bm{\Theta}_{s}\mathbf{G}_{s}+\mathbf{h}_{\text{d},s,k_{s}}^{H})\mathbf{w}_{s,k_{s}}\big{\\}}\Big{]}$
(40a) s.t. $\displaystyle~{}~{}(\text{\ref{eq:sum rate problem
c}})-(\text{\ref{eq:sum rate problem e}}).$ (40b)

For brevity, we define $\mathbf{B}_{s}$ and $\mathbf{c}_{s}$, which are
presented in (34) on the top of this page, and rearrange problem (40) as

$\displaystyle\min\limits_{\bm{\Phi},\mathbf{A}}~{}~{}$
$\displaystyle\sum_{s=1}^{S}\Big{(}\bm{\theta_{s}}^{H}\mathbf{B}_{s}\bm{\theta_{s}}-2\mathfrak{R}\big{\\{}\bm{\theta_{s}}^{H}\mathbf{c}_{s}\big{\\}}\Big{)}$
(41a) s.t.
$\displaystyle\bm{\theta}_{s}=\exp\left(\jmath\bm{\phi}_{s}\odot\widetilde{\mathbf{a}}_{s}\right),\;\;\forall
s,$ (41b) $\displaystyle\phi_{s,m}\in(0,2\pi],\;\;\forall s,\;m,$ (41c)
$\displaystyle\left\|\mathbf{a}_{m}\right\|_{0}\leq
1,\;\;a_{s,m}\in\\{0,1\\},\;\;\forall s,\;m.$ (41d)

Due to the joint optimization of the $M$ reflecting elements and the non-
convex constraint (41d) of each reflecting element, it is very difficult to
directly seek for a optimal solution to problem (41). Therefore, we attempt to
decompose problem (41) into $M$ sub-problems, each of which jointly designs
the ideal phase-shifts and service selection of a certain reflecting element
with the other reflecting elements fixed. To this end, we first split the
objective function (41a) as

$\displaystyle\sum_{s=1}^{S}\Big{(}\bm{\theta_{s}}^{H}\mathbf{B}_{s}\bm{\theta_{s}}-2\mathfrak{R}\big{\\{}\bm{\theta_{s}}^{H}\mathbf{c}_{s}\big{\\}}\Big{)}$
(42a)
$\displaystyle\overset{(b)}{=}\sum_{s=1}^{S}\sum_{m=1}^{M}\Big{[}\sum_{n=1}^{M}\mathbf{B}_{s}(m,n)\theta^{*}_{s,m}\theta_{s,n}-2\mathfrak{R}\big{\\{}\theta_{s,m}^{*}\mathbf{c}_{s}(m)\big{\\}}\Big{]}$
(42b) $\displaystyle\overset{(c)}{=}\sum_{s=1}^{S}\Big{[}\sum_{n\neq
m}\big{(}\mathbf{B}_{s}(m,n)\theta_{s,m}^{*}\theta_{s,n}+\mathbf{B}_{s}(n,m)\theta_{s,n}^{*}\theta_{s,m}\big{)}$
$\displaystyle~{}~{}~{}~{}+\mathbf{B}_{s}(m,m)|\theta_{s,m}|^{2}-2\mathfrak{R}\big{\\{}\theta_{s,m}^{*}\mathbf{c}_{s}(m)\big{\\}}\Big{]},$
(42c)
$\displaystyle\overset{(d)}{=}\sum_{s=1}^{S}\Big{(}2\mathfrak{R}\big{\\{}\zeta_{s,m}\theta_{s,m}^{*}\big{\\}}+\mathbf{B}_{s}(m,m)\Big{)},$
(42d)

where $\zeta_{s,m}\triangleq\sum_{m\neq
n}\mathbf{B}_{s}(m,n)\theta_{s,n}-\mathbf{c}_{s}(m)$ for brevity. In
particular, step $(b)$ unfolds the quadratic term of $\bm{\theta}_{s}$, step
$(c)$ separates the reflection coefficients of the $m$-th reflecting elements
and the others, and step $(d)$ is derived based on
$\mathbf{B}_{s}=\mathbf{B}_{s}^{H}$ and $|\theta_{s,m}|=1$.

Algorithm 2 Joint Transmit Beamforming and Practical IRS Reflection Design for
Sum-Rate Maximization Problem

0: $\mathbf{h}_{\text{r},s,k_{s}}^{H}$, $\mathbf{G}_{s}$,
$\mathbf{h}_{\text{d},s,k_{s}}^{H}$, $P_{s}$, $\forall s\in\mathcal{S}$,
$\forall k_{s}\in\mathcal{K}_{s}$, $\sigma^{2}$.

0: $\mathbf{w}_{s,k_{s}}^{\star},\forall s\in\mathcal{S},\forall
k_{s}\in\mathcal{K}_{s}$, $\bm{\phi}_{m}^{\star},\forall m\in\mathcal{M}$,
$\mathbf{A}^{\star}$.

1: Initialize $\mathbf{A}$, $\bm{\Phi}$, $\mathbf{W}$.

2: while no convergence of the objective (31a) do

3: Update $\nu_{s,k_{s}},\forall s\in\mathcal{S},\forall
k_{s}\in\mathcal{K}_{s}$ by (33).

4: Update $\mu_{s,k_{s}},\forall s\in\mathcal{S},\forall
k_{s}\in\mathcal{K}_{s}$ by (36).

5: Update $\mathbf{w}_{s,k_{s}},\forall s\in\mathcal{S},\forall
k_{s}\in\mathcal{K}_{s}$ by (39).

6: while no convergence of the objective (41a) do

7: for $m=1:M$ do

8: Obtain $s^{\star}$ by solving (45).

9: Update $\phi_{s^{\star},m}$ by (44).

10: Update $\mathbf{a}_{m}$ by (46).

11: end for

12: end while

13: end while

14: Return $\mathbf{w}_{s,k_{s}}^{\star},\forall s\in\mathcal{S},\forall
k_{s}\in\mathcal{K}_{s}$, $\bm{\phi}_{m}^{\star},\forall
m\in\mathcal{M}$,$\mathbf{A}^{\star}=[\mathbf{a}_{1}^{\star},\ldots,\mathbf{a}_{M}^{\star}]$.

Based on the derivations in (42), when we only consider the $m$-th reflecting
element, its corresponding ideal phase-shifts vector
$\widetilde{\bm{\phi}}_{m}\triangleq[\phi_{1,m},\ldots,\phi_{S,m}]^{T}$ of all
BSs and service selection vector $\mathbf{a}_{m}$ are jointly designed. Thus,
after discarding some constant and irrelevant terms, the $m$-th sub-problem is
formulated as

$\displaystyle\min\limits_{\widetilde{\bm{\phi}}_{m},\mathbf{a}_{m}}~{}~{}$
$\displaystyle\sum_{s=1}^{S}\left|\zeta_{s,m}\right|\cos\left(\angle\zeta_{s,m}-\phi_{s,m}a_{s,m}\right)$
(43a) s.t. $\displaystyle\phi_{s,m}\in(0,2\pi],\;\;\forall s,$ (43b)
$\displaystyle\left\|\mathbf{a}_{m}\right\|_{0}\leq
1,\;\;a_{s,m}\in\\{0,1\\},\;\;\forall s.$ (43c)

We observe that the binary vector $\mathbf{a}_{m}$ contains at most one non-
zero entry according to constraint (43c). If $a_{s,m}=0$, there is no need to
optimize $\phi_{s,m}$ since it cannot change the objective value. Meanwhile,
if $a_{s,m}=1$, there must exist a $\phi_{s,m}=\pi+\angle\zeta_{s,m}$ that
achieves the minimum value $-|\zeta_{s,m}|$ of the $s$-th term in (43a).
Therefore, the optimal solution $\mathbf{a}_{m}^{\star}$ to problem (43) is a
non-zero vector. We assume the $s^{\star}$-th entry of
$\mathbf{a}_{m}^{\star}$ is 1 and thus only the $s^{\star}$-th entry of
$\widetilde{\bm{\phi}}_{m}$ is updated by

$\phi_{s^{\star},m}=\pi+\angle\zeta_{s^{\star},m}.$ (44)

Then, the optimization problem (43) is converted into

$\displaystyle\min\limits_{s^{\star}}~{}~{}$
$\displaystyle\sum_{\begin{subarray}{c}s=1\\\ s\neq
s^{\star}\end{subarray}}^{S}\left|\zeta_{s,m}\right|\cos\angle\zeta_{s,m}-\left|\zeta_{s^{\star},m}\right|$
(45a) s.t. $\displaystyle s^{\star}=1,2,\ldots,S,$ (45b)

which can be easily solved by exhaustively searching its feasible set (45b).
After finding $s^{\star}$, the optimal ideal phase-shift $\phi_{s^{\star},m}$
is updated by (44) and the optimal service selection vector
$\mathbf{a}_{m}^{\star}$ is determined by

$a_{s^{\star},m}=1,\;\;a_{s,m}=0,\;\forall s\in\mathcal{S},s\neq s^{\star}.$
(46)

Figure 4: An illustration of the relative position among the BSs, IRS, and
users.

###  IV-C Summary and Complexity Analysis

Summary: Based on the above derivations, the joint transmit beamforming and
practical IRS reflection design for sum-rate maximization problem is
straightforward and summarized in Algorithm 2. With an appropriate
initialization, the variable $\bm{\nu}$, the MSE weight $\bm{\mu}$, the
transmit beamformer $\mathbf{W}$, and the practical IRS reflection
$\bm{\Theta}=\exp(\bm{\Phi}\odot\mathbf{A}),$ are iteratively updated until
the convergence is met. Since the objective function value at each step is
nondecreasing, we can guarantee that the BCD method can strictly converge to a
local optimum point. In addition, in the initialization, the service selection
matrix $\mathbf{A}$ and the ideal phase-shift $\bm{\Phi}$ are randomly
generated from their corresponding feasible sets, and the transmit beamformer
of each BS adopts the typical MMSE beamformer.

Complexity analysis: In each iteration, updating $\bm{\nu}$ has a complexity
of $\mathcal{O}\\{\sum_{s\in\mathcal{S}}K_{s}(K_{s}+1)N_{\text{t}}M^{2}\\}$;
updating $\bm{\mu}$ has a complexity of
$\mathcal{O}\\{\sum_{s\in\mathcal{S}}K_{s}^{2}N_{\text{t}}M^{2}\\}$; updating
transmit beamformer $\mathbf{W}$ requires about
$\mathcal{O}\\{\sum_{s\in\mathcal{S}}N_{\text{t}}K_{s}(3M^{2}+N^{2}_{\text{t}})\\}$
operations. Finally, the order of complexity for updating ideal IRS phase-
shifts $\mathbf{\Phi}$ and service selection matrix $\mathbf{A}$ is about
$\mathcal{O}\\{\sum_{s\in\mathcal{S}}[(M-1)(K_{s}^{3}M^{2}N_{\mathrm{t}}^{2}+K_{s}^{3}MN_{\mathrm{t}}^{2})]\\}$,
Therefore, the total complexity of the proposed algorithm is
$\mathcal{O}\big{\\{}\sum_{s\in\mathcal{S}}K_{s}\big{[}(2K_{s}+1)N_{\text{t}}M^{2}+K_{s}(3M^{2}+N_{\text{t}}^{2})+K_{s}^{2}(MN_{\text{t}}^{2}(M-1)(M+N_{\text{t}}))\big{]}\big{\\}}$.

##  V Simulation Results

In this section, extensive simulation results are presented to demonstrate the
significance of using the proposed practical IRS reflection model in the IRS-
assisted multi-cell multi-band system and the effectiveness of our proposed
algorithms. All simulations are implemented on a computer with Intel(R)
Core(TM) i7-9700 CPU, NVIDIA GeForce GT-710 GPU, Matlab Version 2020b, CVX
toolbox Version 2.2, and Manopt toolbox Version 7.0. We assume that the multi-
cell multi-band system consists of $S=3$ or $S=4$ cells/BSs. In each cell, the
BS equipped with $N_{\text{t}}=16$ antennas serves $K=K_{s}=3,\forall s,$
single-antenna users. An IRS composed of $M=64$ reflecting elements is
deployed to assist the downlink communications for the considered system. The
noise power is set as $\sigma^{2}=-70$dBm. The QoS requirement of each user
and the power budget of each BS are the same, i.e.,
$\Gamma=\Gamma_{s,k_{s}}=5\textrm{dB},\forall s,\forall k_{s}$, and
$P=P_{s}=-5\textrm{dB},\forall s$. In addition, the distance-dependent channel
path loss333This assumption is widely used in the existing works [7], [14],
[15], [20], [22]-[24], [26]-[30]. Moreover, proposed practical IRS reflection
model and associated algorithms are also suitable for other realistic
propagation environments in [50]. is modeled as
$\eta(d)=C_{0}(\frac{d}{d_{0}})^{-\alpha}$, where $C_{0}=-30$dB denotes the
signal attenuation at the reference distance $d_{0}=1$m, and $\alpha$ denotes
the path loss exponent. We set the path loss exponents for the BS-IRS, IRS-
user, and BS-user channels as 2.5, 2.8, and 3.5, respectively.

Figure 5: Transmit power versus the number of iterations ($N_{\text{t}}=16$,
$S=3$, $K=3$, $\Gamma=5$dB).

(a) $S=3$ (b) $S=4$

Figure 6: Total transmit power versus the QoS requirement $\Gamma$ ($M=64$,
$N_{\text{t}}=16$, $K=3$).

(a) $S=3$ (b) $S=4$

Figure 7: Total transmit power versus the number of IRS reflecting elements
$M$ ($\Gamma=5$dB, $N_{\text{t}}=16$, $K=3$).

(a) IRS-user distance $D$ (b) IRS-BS distance $L$

Figure 8: Total transmit power versus the distance between BSs, IRS, and users
($M=64$, $\Gamma=5$dB, $N_{\text{t}}=16$, $S=3$, $K=3$).

A two-dimensional coordinate system is shown in Fig. 4 to demonstrate the
position relationship of different devices in the considered systems from a
top-down view. For both cases of $S=3$ BSs and $S=4$ BSs, the IRS is located
at $(0,0)$, and the BSs are randomly distributed at a distance of $L=52$m away
from the IRS. Since the IRS is deployed to enhance the transmission quality
for the edge users, we assume that the users in each cell are randomly
distributed $D=2$m away from the IRS.

Figure 9: Sum-rate versus the number of iterations ($P=-5$dB,
$N_{\text{t}}=16$, $S=3$, $K=3$).

(a) $S=3$ (b) $S=4$

Figure 10: Sum-rate versus the power budget $P$ ($M=64$, $N_{\text{t}}=16$,
$K=3$).

(a) $S=3$ (b) $S=4$

Figure 11: Sum-rate versus the number of IRS reflecting elements $M$
($P=-5$dB, $N_{\text{t}}=16$, $K=3$).

(a) IRS-user distance $D$ (b) IRS-BS distance $L$

Figure 12: Sum-rate versus the distance between BSs, IRS, and users ($M=64$,
$P=-5$dB, $N_{\text{t}}=16$, $S=3$, $K=3$).

###  V-A Power Minimization Problem

In this subsection, we show the simulation results for the power minimization
problem in Figs. 5-8. The transmit power versus the number of iterations is
first presented in Fig. 5 to show the convergence of the proposed algorithm.
It can be observed that the convergence can be met within 12 iterations under
different settings and the scheme with lesser IRS reflecting elements has
faster convergence due to the reduced dimension of variables. These
convergence results support the low-complexity implementation.

Fig. 6 shows the total transmit power versus the QoS requirement $\Gamma$ for
the cases that $S=3$ and $S=4$. Our algorithm based on the proposed practical
IRS reflection model in this paper is denoted as “w/ IRS, proposed”. The
practical IRS reflection model for the considered multi-cell multi-band
systems exhibits that each reflecting element is independently selected to
serve a certain BS. Therefore, for comparison we also include i) the scenario
that all reflecting elements are selected to serve the same BS, i.e., the IRS
provides tunable phase-shifts for a certain BS while exhibits fixed
$0^{\circ}$ phase-shifts for other BSs, which is denoted as “w/ IRS, w/o
selection”; ii) the scheme that each reflecting element is randomly selected
to serve a certain BS, which is denoted as “w/ IRS, random selection”; iii)
the case without deploying the IRS, which is denoted as “w/o IRS”. We can
easily observe from Fig. 6 that the proposed algorithm requires less transmit
power than the “w/ IRS, random selection” and “w/o IRS” schemes for all
transmit power ranges, which validates the advantages of deploying IRS in
wireless communication systems. Moreover, the proposed algorithm outperforms
the “w/ IRS, w/o selection” scheme, which verifies the importance of joint
designing the transmit beamforming, ideal phase-shift design, and IRS service
selection.

Then, the transmit power versus the number of IRS reflecting elements $M$ is
plotted in Fig. 7. A similar conclusion can be drawn as that from Fig. 6. We
also observe that the transmit power of all schemes decreases with the
increasing number of IRS reflecting elements, and the power reduction of our
proposed algorithm is more remarkable, which validates the advancement of the
proposed practical model and design algorithm in playing the role of the IRS.

To show the impact of IRS location, Fig. 8 illustrates the total transmit
power as functions of the IRS-user distance $D$ and IRS-BS distance $L$. It
can be observed in Fig. 8 (a) that for the schemes with IRS, when the users
move away from the IRS towards the BSs, the transmit power increases at first,
because the reflected signals from the IRS becomes weaker, and then decreases
thanks to stronger signals from BSs. It is worth noting that our proposed
algorithm always has better performance and service coverage. For the “w/o
IRS” scheme, the transmit power keeps decreasing since the users move closer
to their corresponding BSs. When users are randomly distributed at a distance
of $D=2$m away from the IRS, the total transmit power versus the IRS-BS
distance $L$ is shown in Fig. 8 (b). The longer distance between BS and IRS,
the more significant performance improvement of deploying IRS. Moreover, our
proposed algorithm always outperforms the others under different IRS-user and
IRS-BS distance settings.

(a) Sum-rate maximization (b) Power minimization

Figure 13: Performance comparison between the practical response and the
proposed model ($M=64$, $N_{\text{t}}=16$, $S=3$, $K=3$).

###  V-B Sum-rate Maximization Problem

In this subsection, the simulation results for the sum-rate maximization
problem are demonstrated in Figs. 9-12. A similar fast convergence performance
as that for the power minimization problem is also observed in Fig. 9.

Fig. 10 shows the sum-rate versus the power budget $P$ for the scenarios that
$S=3$ and $S=4$. It can be seen that more transmit power provides a larger
sum-rate for all schemes and the proposed algorithm always significantly
outperforms the others under different settings. Specifically, our proposed
algorithm can provide about 3dB gain compared with the scheme without BS
selection, and more than 4dB gain compared with the scheme with random BS
selection. The sum-rate versus the number of IRS reflecting elements $M$ is
illustrated in Fig. 11. Similar conclusion can be drawn as that from Fig. 7.
Note that our proposed algorithm has a larger performance improvement with the
increase of $M$. This is because more reflecting elements can provide larger
passive beamforming gain, while the practical model and proposed design
algorithm can utilize all IRS elements more efficiently. The sum-rate versus
the IRS-user distance $D$ and IRS-BS distance $L$ are presented in Fig. 12 (a)
and (b), respectively. The impact of the IRS locations on the system
performance is consistent with that in Fig. 8, which further demonstrates the
importance of IRS deployment with the proposed design algorithm in practical
systems.

###  V-C Analysis of Model Errors

In order to illustrate the impact of the phase approximation of the proposed
IRS model in Table. I on the degradation of system performance, Fig. 13
compares the performances using the microwave theory based actual response
expressed in (4), and the proposed simplified model (8). Specifically, when
the microwave theory (4) is used, the phase-shift response of other
frequencies is not approximated as unchanged, but also varied accordingly.
However, the microwave theory (4) is extremely complicated and no existing
algorithm can be directly adopted. Thus, we iteratively design the IRS phase-
shift for each element by one-dimensional exhaustive search with fixed other
elements. As seen from Fig. 13, the microwave theory based actual response (4)
can provide slightly better performance than our proposed simplified model.
However, the exhaustive search algorithm that enables the design under complex
nonlinear equation (4) causes significantly higher computational complexity.
For example, for a 64-element IRS-assisted system, the beamforming design with
the microwave theory (4) takes an average of 46.217 seconds using the
exhaustive search method, while the proposed simplified model (8) only
requires an average of 1.072 seconds using our proposed beamforming design
algorithm. Therefore, we can conclude that our proposed model can facilitate
the beamforming design with negligible performance loss.

##  VI Conclusions

In this paper, we first derived a practical yet tractable IRS reflection model
for an IRS-assisted multi-cell multi-band system. Based on the proposed model,
we investigated the joint transmit beamforming and IRS reflection designs for
both power minimization and sum-rate maximization problems. Efficient
algorithms were proposed to solve them by exploiting SOCP, Riemannian
manifold, WMMSE, BCD, and the proposed efficient search algorithms Simulation
results demonstrated significant performance improvement of the proposed
algorithms, which confirmed the importance of using the proposed practical
reflection model and the joint beamforming and IRS reflection designs in IRS-
assisted multi-cell multi-band systems. Moreover, there are many issues of
IRS-assisted systems with the practical reflection model worth being
investigated in future works, including more sophisticated beamforming design
algorithms, fast channel estimation, influence of CSI errors, as well as
learning-based methods, etc.

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