# Joint Beamforming and Reflecting Design in Reconfigurable Intelligent
Surface-Aided Multi-User Communication Systems

Xiaoyan Ma, Shuaishuai Guo, Haixia Zhang, Yuguang Fang, and Dongfeng Yuan The
work presented in this paper was supported in part by the Project of
International Cooperation and Exchanges NSFC under Grant No. 61860206005 and
in part by National Natural Science Foundation of China under Grant No.
61671278 and 61801266. (_Corresponding author: Haixia Zhang_.) X. Ma and D.
Yuan are with Shandong Provincial Key Laboratory of Wireless Communication
Technologies and School of Information Science and Engineering, Shandong
University, Qingdao 266237, China (email<EMAIL_ADDRESS>dfyuan@sdu.edu.cn).S. Guo, and H. Zhang are with Shandong Provincial Key
Laboratory of Wireless Communication Technologies and School of Control
Science and Engineering, Shandong University, Jinan 250061, China (email:
<EMAIL_ADDRESS>haixia.zhang@sdu.edu.cn).Y. Fang is with the
Computer, Electrical and Mathematical Science and Engineering Division,
Department of Electrical and Computer Engineering University of Florida,
Gainesville, FL, USA (email: fang@ece.ufl.edu).

###### Abstract

Reconfigurable intelligent surface (RIS) provides a promising way to build the
programmable wireless transmission environments in the future. Owing to the
large number of reflecting elements used at the RIS, joint optimization for
the active beamforming at the transmitter and the passive reflector at the RIS
is usually complicated and time-consuming. To address this problem, this paper
proposes a low-complexity joint beamforming and reflecting algorithm based on
fractional programing (FP). Specifically, we first consider a RIS-aided multi-
user communication system with perfect channel state information (CSI) and
formulate an optimization problem to maximize the sum rate of all users. Since
the problem is nonconvex, we decompose the original problem into three
disjoint subproblems. By introducing favorable auxiliary variables, we derive
the closed-form expressions of the beamforming vectors and reflecting matrix
in each subproblem, leading to a joint beamforming and reflecting algorithm
with low complexity. We then extend our approach to handle the case when
transmitter-RIS and RIS-receiver channels are not perfect and develop
corresponding low-complexity joint beamforming and reflecting algorithm with
practical channel estimation. Simulation results have verified the
effectiveness of the proposed algorithms as compared to various benchmark
schemes.

###### Index Terms:

Reconfigurable intelligent surface (RIS), joint beamforming and reflecting
design, cascaded channel estimation, fractional programming

##  I Introduction

Reconfigurable intelligent surface (RIS) has emerged as a promising research
topic in wireless communications [1, 2, 3]. Specifically, RIS is a planar
surface comprising a large number of low-cost passive reflecting elements,
each of which can induce a phase shift to the incident signals [4, 5, 6, 7].
By smartly tuning the phase shifts of all reflecting elements according to the
dynamic wireless channel, the reflected signals can be constructively added to
those from other links to boost the desired signal power, thus significantly
enhancing the communication performance without the need of deploying
additional active base stations (BSs) or relays. Besides, from the
implementation perspective, RISs possess appealing features such as low
profile and lightweight, thus can be easily mounted on walls or ceilings[8].
Since RISs are complementary devices in wireless networks, deploying them in
existing wireless systems does not require to change their standards and
hardware, while only necessary modification of the communication protocols
suffices. Furthermore, RISs are usually much cheaper than active BSs/relays,
which can be easily deployed rapidly [9, 10]. Owing to these advantages, RISs
have attracted intensive attention and many research works have been done,
including high-layer protocol design [11, 12, 13], capacity analysis [14, 15,
16], energy/spectral efficiency maximization [17, 18, 19], physical layer
security design [20, 21, 22], and modulation and coding [23, 24].
Comprehensive surveys of these works on RISs can be found in [25, 26].

New challenges also arise in the design and implementation of RIS-aided
wireless systems. How to jointly optimize the RIS reflection coefficients and
the transmitter beamforming matrix to maximally reap the RIS performance gains
is one of the most crucial problems[27, 28]. Besides, as discussed in [29],
the number of RIS reflecting elements can be large, which indicates that the
joint optimization of the active beamforming at the transmitter and the
passive reflecting at the RIS can be highly complicated and time-consuming. To
address this problem, this paper targets at a low-complexity joint beamforming
and reflecting design for RIS-aided multi-user communication systems.
Moreover, considering that RIS channel estimation (CE) are usually
complicated, therefore, perfect channel state information (CSI) is difficult
to obtain, low-complexity joint beamforming and reflecting design with
practical channel information is further investigated.

###  I-A Related Works

The joint beamforming and reflecting design in RIS-aided communication systems
has aroused much research interest recently. Alternatively optimizing the
beamforming and the reflecting matrices is a common method to solve this
multivariate problem. To clearly review the state-of-the-art, we classify the
related works into two categories according to the application scenarios as
follows.

_1) RIS-Aided Point-to-Point MIMO Systems._ Point-to-point multiple-input
multiple-output (MIMO) system is one of the classical models that have been
well investigated. It is important to explore how much performance gains the
communication systems can obtain from the deployment of RIS. Ye, Guo and
Alouni attempted to answer this question from the system reliability
perspective. It is shown in [30] that RIS can greatly reduce the symbol error
rate (SER) by carefully designing the beamforming and reflecting matrices. In
[30], the joint design of the beamforming and reflecting matrices is optimized
by a gradient descent algorithm with much higher computational complexity.
Different from [30], Zhang and Zhang attempted to explore the performance
improvement brought by RIS from the spectral efficiency perspective. In [31],
they alternatively optimized the beamforming and reflecting matrices for
spectral efficiency maximization. The closed-form beamforming and reflecting
matrices in each optimization step are derived, which can greatly reduce the
computational complexity. Results on RIS-aided point-to-point communication
systems can well demonstrate the impacts of RIS on the communication links,
but cannot be extended to the situations with the coexistence of other users,
i.e., the interference links. As a result, the joint design of the beamforming
and reflecting cannot be directly used in RIS-aided multi-user communication
systems.

_2) RIS-Aided Multi-User Communication Systems._ For RIS-aided multi-user
systems, the joint optimization method based on semidefinite relaxation (SDR)
is one of the most commonly used methods. In [32, 33], the transmit power of a
BS is minimized by decomposing the joint optimization problem into two
subproblems: one is the conventional power-minimization problem in MIMO
systems, while the other is the RIS phase vector optimization problem. Then
the phase optimization problem is solved by SDR with high computational
complexity. Chen et al. maximized the minimum-secrecy-rate in a downlink
multi-input single-output (MISO) broadcast system by jointly optimizing the
beamformers at the transmitter and reflecting coefficients at the RIS, and
proposed an algorithm based on the alternating optimization and the path-
following algorithm to solve the problem [34]. In [35], Guo et al. combined
the fractional programming and alternating direction method of multipliers
(ADMM) to maximize the weighted sum rate of all users at downlink
transmissions in the RIS-aided multi-user communication systems. Although the
alternating optimization algorithms mentioned above achieve quite good
performance, the main shortcoming lies in that the complexity of those
algorithms is extremely high for large-sized RISs. In [36] and [37], the
energy efficiency is maximized in RIS-aided communication systems by employing
zero-forcing (ZF) precoding at the transmitter. Since ZF precoding completely
cancels the inter-user interference, the power allocation at the BS and the
phase optimization at the RIS can be well decoupled. However, the ZF precoding
may amplify the background noise as well, and the performance could be
severely compromised when the channel is ill-conditioned. Moreover, the
derivations are not applicable directly for other precoding schemes. Di et al.
in [38] proposed a hybrid beamforming scheme by assuming that there is a
limited number of discrete phase shifts at RIS. Huang et al. proposed a novel
method leveraging the deep reinforcement learning (DRL) to jointly design the
beamforming and reflecting matrix simultaneously [39].

###  I-B Contributions

In this paper, we propose a low-complexity joint beamforming and reflecting
design method for RIS-aided multi-user communication systems. The main
contributions are summarized as follows.

  * •

We first investigate a RIS-aided multi-user communication system with perfect
separate CSIs and formulate an optimization problem to maximize the sum rate
of all users, which is multivariate and non-convex. To solve this problem, we
decompose the original problem into three disjoint subproblems by utilizing
the novel FP transformations. By introducing favorable auxiliary variables, we
derive the closed-form optimal expressions of the beamforming vectors and
reflecting matrix.

  * •

Based on the closed-form optimal expressions, we design a low-complexity joint
beamforming and reflecting algorithm to maximize the sum rate of all users.
Due to the closed-form optimal expressions used in each iteration, the
proposed algorithm enjoys extreme low complexity compared with traditional
optimization methods.

  * •

Considering perfect CSIs are not always available in practical systems, we
futher modify the proposed algorithm to cope with estimated CSI, the effect of
channel estimation error on the system performance is also investigated.
Numerical results show that the proposed algorithm can be easily combined with
existing CE methods and achieve better sum rate performance than other
benchmark schemes.

The remainder of this paper is organized as follows. The system model and
problem formulation are introduced in Section II. The detailed derivations of
the proposed low-complexity alternating algorithm are presented in Section
III. In Section IV, the proposed low-complexity algorithm is further modified
by taking the practical channel estimation into consideration. Numerical
simulation results are presented in Section V to verify the effectiveness of
the proposed algorithms. Finally, we conclude this paper in Section VI.

###  I-C Notations

The notations used in this paper are listed as follows.
$\mathbb{E}\\{\cdot\\}$ denotes statistical expectation.
$\mathcal{CN}(\mu,\sigma^{2})$ denotes the circularly symmetric complex
Gaussian (CSCG) distribution with mean $\mu$ and variance $\sigma^{2}$.
$||\mathbf{w}||$ denotes the Euclidean norm. $||\mathbf{W}||_{F}$ denotes the
Frobenius norm. $\mathbf{I}_{M}$ represents the $M\times M$ identity matrix.
$\mathbf{G}^{T}$ and $\mathbf{G}^{H}$ denote the transpose and conjugate
transpose of the matrix $\mathbf{G}$, respectively. $\mathbf{G}^{-1}$
represents the inverse of $\mathbf{G}$ while $\mathbf{G}^{+}$ represents the
pseudo-inverse of $\mathbf{G}$. $\text{Tr}[\mathbf{G}]$ is the trace of the
matrix. $\mathop{\mathrm{diag}}(\bm{\phi})$ denotes the diagonal matrix of
vector $\bm{\phi}$. $u_{i,q}$ represents the element at row $i$ and column $q$
in matrix $\mathbf{U}$, $v_{n}$ represents the $n$-th elements of vector
$\mathbf{V}$. For any real number, $\sqrt{a}$ denotes the square root of $a$.
For any complex variable $x$, $|x|$ denotes the absolute value, $\angle x$
represents its phase angle, $x^{*}$ denotes its conjugate, $\Re\\{x\\}$ and
$\Im\\{x\\}$ represent its real part and imaginary part, respectively.

##  II System Model and Problem Formulation

###  II-A System and Channel Model

This paper investigates a RIS-aided multi-user communication system as shown
in Fig. 1, which consists of one BS equipped with $M$ antennas and $K$ signal
antenna users. A RIS with $N$ reflecting elements is deployed between the BS
and users to provide additional high-quality transmission links. In this
paper, the direct links are weak due to the obstacles, thus BS chooses to use
the RIS-aided links to improve the signal quality received at users. The
equivalent channel from user $k$ to RIS and from RIS to BS are denoted by
$\mathbf{h}_{r,k}\in$$\mathbb{C}^{N\times 1}$ and
$\mathbf{G}\in\mathbb{C}^{M\times N}$, respectively, where $k=1,...,K$
represents the number of users.

Figure 1: A RIS-aided multi-user communication system.

The RIS-aided link (i.e., user-RIS-BS link) is modeled as a concatenation of
three components, i.e., the user-RIS link $\mathbf{h}_{r,k}$, RIS phase shift
matrix (i.e., passive beamforming) $\mathbf{\Phi}\in\mathbb{C}^{N\times N}$,
and RIS-BS link $\mathbf{G}$.
$\mathbf{\Phi}=\mathop{\mathrm{diag}}(\bm{\phi})$ is the diagonal matrix of
the phase shift vector $\bm{\phi}$, where
$\bm{\phi}=[\phi_{1},\phi_{2},...,\phi_{N}]^{T}\in\mathbb{C}^{N\times 1}$,
$\phi_{n}=\beta_{n}e^{j\theta_{n}}$ with $n=1,2,...,N$ denotes the phase and
amplitude changes of each reflecting element introduced on the incident
signals. $\theta_{n}\in[0,2\pi)$ and $\beta_{n}\in[0,1]$ denote the phase
shift and the ON/OFF state of the $n$-th reflecting element at the RIS,
respectively. $\bm{\theta}=[\theta_{1},\theta_{2},...,\theta_{N}]^{T}$
represents the phase angle vector of all reflecting elements. In the
subsequent development, we set $\beta_{n}=1$, which represents that all
elements of the RIS are switched on to maximize the signal reflection.

It is assumed that the channels from BS to RIS and from RIS to users suffer
from Rician fading. Similar to [40, 41], we further assume that the antenna
elements form a half-wavelength uniform linear array (ULA) configuration at
the AP and RIS, thus the channel $\mathbf{G}^{H}$ and $\mathbf{h}_{r,k}^{H}$
can be modeled by

$\mathbf{G}^{H}=d_{\mathbf{G}}\left(\sqrt{\frac{\mathcal{F}_{\mathbf{G}}}{\mathcal{F}_{\mathbf{G}}+1}}\bar{\mathbf{G}}^{H}+\sqrt{\frac{1}{\mathcal{F}_{\mathbf{G}}+1}}\tilde{\mathbf{G}}^{H}\right),$
(1)

and

$\mathbf{h}_{r,k}^{H}=d_{r,k}\left(\sqrt{\frac{\mathcal{F}_{r,k}}{\mathcal{F}_{r,k}+1}}\bar{\mathbf{h}}_{r,k}^{H}+\sqrt{\frac{1}{\mathcal{F}_{r,k}+1}}\tilde{\mathbf{h}}_{r,k}^{H}\right),$
(2)

where $d_{\mathbf{G}}$ and $d_{r,k}$ denote the corresponding path loss at
distance $d$; $\mathcal{F}_{\mathbf{G}}$ and $\mathcal{F}_{r,k}$ are the
Rician factors of $\mathbf{G}^{H}$ and $\mathbf{h}_{r,k}^{H}$, respectively;
$\bar{\mathbf{G}}^{H}$ and $\bar{\mathbf{h}}_{r,k}^{H}$ are the Line-of-Sight
(LoS) components and can be expressed by the responses of the ULA. The array
response of a $M$-element ULA is

$\displaystyle\mathbf{a}_{M}(\varrho)=[1,e^{j2\pi\frac{d}{\mathcal{\lambda}}\sin\varrho},...,e^{j2\pi\frac{d}{\mathcal{\lambda}}(M-1)\sin\varrho}],$
(3)

where $\varrho$ is the angular parameters. Under this condition,
$\bar{\mathbf{G}}^{H}$ and $\bar{\mathbf{h}}_{r,k}^{H}$ are expressed as

$\displaystyle\bar{\mathbf{G}}^{H}=\mathbf{a}_{N}^{H}(\eta)\mathbf{a}_{M}(\varrho),$
(4)

and

$\displaystyle\bar{\mathbf{h}}_{r,k}^{H}=\mathbf{a}_{N}(\iota_{k}),$ (5)

where $\eta$, $\varrho$ and $\iota_{k}$ are the angular parameters,
$\tilde{\mathbf{G}}^{H}$ and $\tilde{\mathbf{h}}_{r,k}^{H}$ are the non-LoS
(NLoS) components, whose elements are chosen from $\mathcal{CN}(0,1)$.

###  II-B Problem Formulation

Let $x_{k}$ denote the transmitted signal from the BS to user $k$ with
$\mathbb{E}\\{|x_{k}|^{2}\\}=1,k=1,2,\cdots,K$. Then, the transmitted signals
for all users can be expressed as

$\displaystyle\begin{aligned}
\mathbf{x}=\sum_{k=1}^{K}\bm{w}_{k}x_{k},\end{aligned}$ (6)

where $\bm{w}_{k}\in\mathbb{C}^{M\times 1}$ is the corresponding transmit
beamforming vector for user $k$. The signal received at user $k$ can be
expressed as

$\displaystyle\begin{aligned}
y_{k}=\mathbf{h}_{r,k}^{H}\mathbf{\Phi}\mathbf{G}^{H}\mathbf{x}+n_{k},\end{aligned}$
(7)

where $n_{k}\sim\mathcal{CN}(0,\sigma_{0}^{2})$ denotes the complex additive
white Gaussian noise (AWGN) at $k$-th user. The $k$-th user treats all the
signals for other users (i.e., $x_{1},...,x_{k-1},x_{k+1},...,x_{K}$) as
interference. Thus, the corresponding signal to interference plus noise ratio
(SINR) at user $k$ can be expressed as

$\displaystyle\begin{aligned}
\gamma_{k}=\frac{|\mathbf{h}_{r,k}^{H}\mathbf{\Phi}\mathbf{G}^{H}\bm{w}_{k}|^{2}}{\sum_{i=1,i\neq
k}^{K}|\mathbf{h}_{r,k}^{H}\mathbf{\Phi}\mathbf{G}^{H}\bm{w}_{i}|^{2}+\sigma_{0}^{2}},\end{aligned}$
(8)

with

$\displaystyle\begin{aligned} \sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq
P_{T},\end{aligned}$ (9)

being the power constraint at BS, where $P_{T}$ represents the maximum
transmit power. When BS uses all the power to transmit information for user
$k$, i.e., $||\bm{w}_{k}||^{2}=P_{T}$, $\gamma_{k}$ can achieve the maximum
value with
$\gamma_{k}^{max}=\frac{||\mathbf{h}_{r,k}^{H}\mathbf{\Phi}\mathbf{G}^{H}||^{2}P_{T}}{\sigma_{0}^{2}}$.
Thus, the range of $\gamma_{k}$ is $0\leq\gamma_{k}\leq\gamma_{k}^{max}$.

Let $\mathbf{W}=[\bm{w}_{1},\bm{w}_{2},...,\bm{w}_{K}]\in\mathbb{C}^{M\times
K}$ denote the overall beamforming matrix at the BS. In this paper, we aim to
maximize the sum rate of all users at the downlink transmission phase by
jointly designing the active transmit beamforming matrix $\mathbf{W}$ at the
BS and the passive reflecting matrix $\mathbf{\Phi}$ at the RIS, subject to
the RIS phase shift constraint and the BS transmit power constraint.
Mathematically, the optimization problem is formulated as

$\displaystyle(P_{1})\quad$
$\displaystyle\max\limits_{\mathbf{W},\mathbf{\Phi}}\quad
f_{1}(\mathbf{W},\mathbf{\Phi})=\sum_{k=1}^{K}\log(1+\gamma_{k}),$ (10)
$\displaystyle s.t.\quad\quad|\phi_{n}|=1,\quad n=1,2,...,N,$ (11)
$\displaystyle\quad\quad\quad\sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq P_{T}.$ (12)

In ($P_{1}$), both the objective function $f_{1}(\mathbf{W},\mathbf{\Phi})$
and the phase shift constraints (11) are non-convex. Besides, the joint
optimization of $\mathbf{W}$ and $\mathbf{\Phi}$ is deeply coupled. All these
bring great difficulties to find the global optimal solution. In this paper,
we attempt to find a low-complexity solution to ($P_{1}$) by applying the FP
technique. Details are given in the next section.

##  III Joint Beamforming and Reflecting Design With Perfect CSI

Problem ($P_{1}$) is to maximize a sum-of-log-of-ratio objective, i.e.,
$\sum_{k=1}^{K}\log(1+\gamma_{k})$. It is a typical FP problem owing to the
fractional expression of $\gamma_{k}$ in (8). In order to solve this non-
convex multi-variable problem, we first introduce a series of auxiliary
variables, and then transform ($P_{1}$) into a much low-complex problem by
applying the FP techniques [42, 43]. It is written as

$\displaystyle\begin{aligned}
(P_{1a})\quad&\max\limits_{\bm{\alpha},\mathbf{W},\mathbf{\Phi}}\quad
f_{1a}(\bm{\alpha},\mathbf{W},\mathbf{\Phi}),\\\ &\quad
s.t.\quad\quad|\phi_{n}|=1,\quad n=1,2,...,N,\\\
&\quad\quad\quad\quad\sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq P_{T},\\\
&\quad\quad\quad\quad\alpha_{k}\geq 0,\quad k=1,2,...,K.\end{aligned}$ (13)

where

$\displaystyle\begin{aligned}
f_{1a}(\bm{\alpha},\mathbf{W},\mathbf{\Phi})=&\sum_{k=1}^{K}\log(1+\alpha_{k})-\sum_{k=1}^{K}\alpha_{k}\\\
&+\sum_{k=1}^{K}\frac{(1+\alpha_{k})\gamma_{k}}{1+\gamma_{k}},\\\
\end{aligned}$ (14)

and the parameters $\alpha_{1},\alpha_{2},...,\alpha_{K}$ are the introduced
real auxiliary optimization variables for each user with
$\bm{\alpha}=[\alpha_{1},\alpha_{2},...,\alpha_{K}]^{T}$ being the auxiliary
variable vector. The original problem is transformed into an optimization
problem with decision variables $\bm{\alpha},\mathbf{W}$, and $\mathbf{\Phi}$.
To efficiently solve the problem, we decouple $(P_{1a})$ into three disjoint
subproblems and target to find the closed-form optimal solutions of each
subproblem. Detailed information about how to alternatively optimize
$\bm{\alpha}$, $\mathbf{W}$ and $\mathbf{\Phi}$ are given as follows.

###  III-A Closed-Form Alternating Optimization for Joint Beamforming and
Reflecting Design

_1) Closed-Form Optimal Solution for $\bm{\alpha}$._ First, we aim to find the
optimal closed-form expression for $\bm{\alpha}$ with given $\mathbf{W}$ and
$\mathbf{\Phi}$. Specifically, with fixed $\mathbf{W}$ and $\mathbf{\Phi}$,
the objective function in $(P_{1a})$ can be seen as a non-constrained
optimization problem only with respect to $\bm{\alpha}$.

######  Proposition 1.

When $\mathbf{W}$ and $\mathbf{\Phi}$ are fixed, the optimal $\alpha_{k}$ can
be expressed as

$\displaystyle\begin{aligned} \alpha_{k}^{op}=\gamma_{k}.\end{aligned}$ (15)

###### Proof:

Taking the partial derivative of $f_{1}(\bm{\alpha},\mathbf{W},\mathbf{\Phi})$
with respect to $\alpha_{k}$ yields

$\displaystyle\begin{aligned} \frac{\partial
f_{1}(\bm{\alpha},\mathbf{W},\mathbf{\Phi})}{\partial\alpha_{k}}&=\frac{\gamma_{k}-\alpha_{k}}{(1+\gamma_{k})(1+\alpha_{k})}.\end{aligned}$
(16)

It is easy to verify that $\alpha_{k}^{op}=\gamma_{k}$ maximizes
$f_{1}(\bm{\alpha},\mathbf{W},\mathbf{\Phi})$, since $\frac{\partial
f_{1}(\bm{\alpha},\mathbf{W},\mathbf{\Phi})}{\partial\alpha_{k}}>0$ when
$\alpha_{k}<\gamma_{k}$ and $\frac{\partial
f_{1}(\bm{\alpha},\mathbf{W},\mathbf{\Phi})}{\partial\alpha_{k}}<0$ when
$\alpha_{k}>\gamma_{k}$. Besides, substituting $\alpha_{k}^{op}=\gamma_{k}$
into $(P_{1a})$ yields

$\displaystyle\begin{aligned}
f_{1a}(\bm{\alpha}^{op},\mathbf{W},\mathbf{\Phi})=\sum_{k=1}^{K}\log(1+\gamma_{k})=f_{1}(\mathbf{W},\mathbf{\Phi}),\end{aligned}$

where
$\bm{\alpha}^{op}=[\alpha_{1}^{op},\alpha_{2}^{op},\cdots,\alpha_{K}^{op}]^{T}$.
This guarantees the equivalence between $(P_{1})$ and $(P_{1a})$. ∎

_2) Closed-Form Optimal Solution for Active Beamforming $\mathbf{W}$._ When
$\bm{\alpha}$ is fixed, the variable left for optimization in $(P_{1a})$ is
$\gamma_{k}$ only. Thus the corresponding optimization problem can be
rewritten as

$\displaystyle\begin{aligned}
(P_{2})\quad\max_{\mathbf{W},\mathbf{\Phi}}\quad&\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}\gamma_{k}}{1+\gamma_{k}},\\\
s.t.\quad&|\phi_{n}|=1,\quad n=1,2,...,N,\\\
\quad&\sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq P_{T},\end{aligned}$ (17)

where $\tilde{\alpha}_{k}=1+\alpha_{k}$ for $k=1,2,...,K$. For easy
presentation, we adopt
$\mathbf{h}_{k}^{H}=\mathbf{h}_{r,k}^{H}\mathbf{\Phi}\mathbf{G}^{H}$ to
present the equivalent channel. Thus, the SINR at user $k$ in (8) can be
rewritten as

$\displaystyle\begin{aligned}
\gamma_{k}=\frac{|\mathbf{h}_{k}^{H}\bm{w}_{k}|^{2}}{\sum_{i=1,i\neq
k}^{K}|\mathbf{h}_{k}^{H}\bm{w}_{i}|^{2}+\sigma_{0}^{2}}.\end{aligned}$ (18)

Substituting (18) into (17), we can reformulate the objective function in
$(P_{2})$ to be

$\displaystyle\begin{aligned}
\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}\gamma_{k}}{1+\gamma_{k}}=\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|\mathbf{h}_{k}^{H}\bm{w}_{k}|^{2}}{\sum_{i=1}^{K}|\mathbf{h}_{k}^{H}\bm{w}_{i}|^{2}+\sigma_{0}^{2}}.\end{aligned}$
(19)

Thus, given $\bm{\alpha}$ and $\mathbf{\Phi}$, optimizing $\mathbf{W}$ becomes

$\displaystyle\begin{aligned} (P_{2a})\quad&\max_{\mathbf{W}}\quad
f_{2a}(\mathbf{W})\\!=\\!\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|\mathbf{h}_{k}^{H}\bm{w}_{k}|^{2}}{\sum_{i=1}^{K}|\mathbf{h}_{k}^{H}\bm{w}_{i}|^{2}\\!+\\!\sigma_{0}^{2}},\\\
&s.t.\quad\quad\sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq P_{T}.\end{aligned}$ (20)

It is obvious that $(P_{2a})$ is a multiple-ratio fractional programming
problem. Using the quadratic transform proposed in [42], $(P_{2a})$ can be
reformulated as a biconvex optimization problem with a new objective function
as

$\displaystyle\begin{aligned}
f_{2b}(\mathbf{W},\bm{\beta})=&\sum_{k=1}^{K}2\sqrt{\tilde{\alpha}_{k}}\Re\left\\{{\beta_{k}^{*}\mathbf{h}_{k}^{H}\bm{w}_{k}}\right\\}\\\
&-\sum_{k=1}^{K}|\beta_{k}|^{2}\left(\sum_{i=1}^{K}|\mathbf{h}_{k}^{H}\bm{w}_{i}|^{2}+\sigma_{0}^{2}\right),\end{aligned}$
(21)

where $\beta_{1},\beta_{2},...,\beta_{K}$ are the newly introduced complex
auxiliary variables and $\bm{\beta}=[\beta_{1},\beta_{2},...,\beta_{K}]^{T}$
is the auxiliary variable vector. Then, as proved in [42] and [43], solving
$(P_{2a})$ with respect to $\mathbf{W}$ is equivalent to solving the following
biconvex problem with respect to $\mathbf{W}$ and $\bm{\beta}$,

$\displaystyle\begin{aligned} (P_{2b})\quad&\max_{\mathbf{W},\bm{\beta}}\quad
f_{2b}(\mathbf{W},\bm{\beta}),\\\
&s.t.\quad\sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq P_{T}.\end{aligned}$ (22)

By analyzing $(P_{2b})$, we have the following proposition regarding to its
solution.

######  Proposition 2.

The optimal beamforming vector $\bm{w}_{k}$ can be updated by

$\displaystyle\begin{aligned}
\bm{w}_{k}^{op}=\sqrt{\tilde{\alpha}_{k}}\beta_{k}^{op}\left(\sum_{i=1}^{K}|\beta_{i}^{op}|^{2}\mathbf{h}_{i}\mathbf{h}_{i}^{H}+\lambda^{op}\mathbf{I}_{M}\right)^{-1}\mathbf{h}_{k},\end{aligned}$
(23)

where

$\displaystyle\begin{aligned}
\beta_{k}^{op}=\frac{\sqrt{\tilde{\alpha}_{k}}\mathbf{h}_{k}^{H}\bm{w}_{k}}{\sum_{i=1}^{K}|\mathbf{h}_{k}^{H}\bm{w}_{i}|^{2}+\sigma_{0}^{2}}.\end{aligned}$
(24)

In (23), $\lambda^{op}$ is the dual variable introduced for the power
constraint $\sum_{k=1}^{K}||\bm{w}_{k}^{op}||^{2}=P_{T}$, which can be
expressed as

$\displaystyle\begin{aligned} \lambda^{op}=\min\left\\{\lambda^{op}\geq
0:\sum_{k=1}^{K}||\bm{w}_{k}^{op}||^{2}=P_{T}\right\\}.\end{aligned}$ (25)

###### Proof:

First, when $\mathbf{W}$ is fixed, the partial gradient of
$f_{2b}(\mathbf{W},\bm{\beta})$ with respect to $\beta_{k}$ can be expressed
as

$\displaystyle\begin{aligned} \frac{\partial
f_{2b}(\mathbf{W},\bm{\beta})}{\partial\beta_{k}}\\!=\\!2\sqrt{\tilde{\alpha}_{k}}\mathbf{h}_{k}^{H}\mathbf{w}_{k}\\!-\\!2\beta_{k}\sum_{i=1}^{K}|\mathbf{h}_{k}^{H}\bm{w}_{i}|^{2}\\!-\\!2\beta_{k}\sigma_{0}^{2}.\end{aligned}$
(26)

By solving $\partial f_{2b}(\mathbf{W},\bm{\beta})/\partial\beta_{k}=0$, the
optimal FP auxiliary variable $\beta_{k}^{op}$ for fixed $\mathbf{W}$ can be
derived as that in (24). Next, with fixed $\beta_{k}^{op}$,
$f_{2b}(\mathbf{W},\bm{\beta})$ can be rewritten as

$\displaystyle\begin{aligned}
f_{2b}(\mathbf{W},\bm{\beta}^{op})=&\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\left(\beta_{k}^{op^{*}}\mathbf{h}_{k}^{H}\bm{w}_{k}+\bm{w}_{k}^{H}\mathbf{h}_{k}\beta_{k}^{op}\right)\\\
&-\sum_{i=1}^{K}|\beta_{i}^{op}|^{2}\sum_{k=1}^{K}\bm{w}_{k}^{H}\mathbf{h}_{i}\mathbf{h}_{i}^{H}\bm{w}_{k}.\end{aligned}$
(27)

Considering the power constraint $\sum_{k=1}^{K}||\bm{w}_{k}||^{2}=P_{T}$, we
write the Lagrangian function of $f_{2b}(\mathbf{W},\bm{\beta}^{op})$ as

$\displaystyle\begin{aligned}
&f_{2b}^{(L)}(\lambda,\mathbf{W},\bm{\beta}^{op})\\!=\\!\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\left(\beta_{k}^{op^{*}}\mathbf{h}_{k}^{H}\bm{w}_{k}\\!+\\!\bm{w}_{k}^{H}\mathbf{h}_{k}\beta_{k}^{op}\right)\\\
&\\!-\\!\sum_{i=1}^{K}|\beta_{i}^{op}|^{2}\sum_{k=1}^{K}\bm{w}_{k}^{H}\mathbf{h}_{i}\mathbf{h}_{i}^{H}\bm{w}_{k}\\!-\\!\lambda\left(\sum_{k=1}^{K}\bm{w}_{k}^{H}\bm{w}_{k}\\!-\\!P_{T}\right),\end{aligned}$
(28)

where $\lambda$ is the dual variable introduced for the power constraint.
Taking the partial derivative of
$f_{2b}^{(L)}(\lambda,\mathbf{W},\bm{\beta}^{op})$ with respect to
$\bm{w}_{k}$, we have

$\displaystyle\begin{aligned} &\frac{\partial
f_{2b}^{(L)}(\lambda,\mathbf{W},\bm{\beta}^{op})}{\partial\bm{w}_{k}}=\\\
&2\sqrt{\tilde{\alpha}_{k}}\mathbf{h}_{k}\beta_{k}^{op}\\!-\\!2\sum_{i=1}^{K}|\beta_{i}^{op}|^{2}\sum_{k=1}^{K}\mathbf{h}_{i}\mathbf{h}_{i}^{H}\bm{w}_{k}\\!-\\!2\lambda\mathbf{I}_{M}\bm{w}_{k}.\end{aligned}$
(29)

Then, by solving $\partial
f_{2b}^{(L)}(\lambda,\mathbf{W},\bm{\beta}^{op})/\partial\bm{w}_{k}=0$, we can
get the optimal expression of $\bm{w}_{k}^{op}$ in (23). For $\lambda^{op}$,
taking the partial derivative of
$f_{2b}^{(L)}(\lambda,\mathbf{W},\bm{\beta}^{op})$ with respect to $\lambda$,
we have

$\displaystyle\begin{aligned} &\frac{\partial
f_{2b}^{(L)}(\lambda,\mathbf{W},\bm{\beta}^{op})}{\partial\lambda}=-\sum_{k=1}^{K}\bm{w}_{k}^{H}\bm{w}_{k}\\!+\\!P_{T}.\end{aligned}$
(30)

By solving the equation ${\partial
f_{2b}^{(L)}(\lambda,\mathbf{W},\bm{\beta}^{op})}/{\partial\lambda}=0$, i.e.,
$\sum_{k=1}^{K}||\bm{w}_{k}^{op}||^{2}=P_{T}$, we can obtain all feasible
solutions for $\lambda^{op}$. Among all solutions, the smallest number is
chosen since $\lambda\geq 0$,
$\sum_{k=1}^{K}\bm{w}_{k}^{H}\bm{w}_{k}-P_{T}\leq 0$, and the dual function of
$f_{2b}^{(L)}(\lambda,\mathbf{W},\bm{\beta}^{op})$, i.e.,
$\min\limits_{\lambda}\mathcal{G}(\lambda)=\min\limits_{\lambda}\max\limits_{\mathbf{W},\bm{\beta}^{op}}f_{2b}^{(L)}(\lambda,\mathbf{W},\bm{\beta}^{op})$
is an increasing function with respect to $\lambda$, thus we need to select
the smallest one, since minimizing $\mathcal{G}(\lambda)$ is equals to
maximizing $f_{2b}^{(L)}(\lambda,\mathbf{W},\bm{\beta}^{op})$. Finally, we can
get $\lambda^{op}$ in Eq. (25). ∎

_3) Closed-Form Optimal Solution for Passive Reflecting $\mathbf{\Phi}$._ For
fixed $\bm{\alpha}$, the objective function in $(P_{1a})$ is reduced to
$(P_{2})$ in (17) as discussed above. At the reflecting optimization phase, we
further rewrite the objective function of $(P_{2})$ as an equation of
$\mathbf{\Phi}$, i.e.,

$\displaystyle\begin{aligned}
f_{3}(\mathbf{\Phi})&=\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}\gamma_{k}}{1+\gamma_{k}}\\\
&=\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|\mathbf{h}_{r,k}^{H}\mathbf{\Phi}\mathbf{G}^{H}\bm{w}_{k}|^{2}}{\sum_{i=1}^{K}|\mathbf{h}_{r,k}^{H}\mathbf{\Phi}\mathbf{G}^{H}\bm{w}_{i}+\sigma_{0}^{2}|}.\end{aligned}$
(31)

Recall that $\mathbf{\Phi}=\mathop{\mathrm{diag}}(\bm{\phi})$ with
$\bm{\phi}=[\phi_{1},\phi_{2},...,\phi_{N}]^{T}$, $f_{3}(\mathbf{\Phi})$ can
be further expressed as a function of $\bm{\phi}$, i.e.,

$\displaystyle\begin{aligned}
f_{3}(\bm{\phi})&=\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|\bm{\phi}^{H}\mathop{\mathrm{diag}}(\mathbf{h}_{r,k}^{H})\mathbf{G}^{H}\bm{w}_{k}|^{2}}{\sum_{i=1}^{K}|\bm{\phi}^{H}\mathop{\mathrm{diag}}({\mathbf{h}_{r,k}^{H}})\mathbf{G}^{H}\bm{w}_{i}|^{2}+\sigma_{0}^{2}}\\\
&=\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|\bm{\phi}^{H}\bm{b}_{k,k}|^{2}}{\sum_{i=1}^{K}|\bm{\phi}^{H}\bm{b}_{i,k}|^{2}+\sigma_{0}^{2}},\end{aligned}$
(32)

where
$\bm{b}_{i,k}=\mathop{\mathrm{diag}}(\mathbf{h}_{r,k}^{H})\mathbf{G}^{H}\bm{w}_{i}$
can be treated as the equivalent channel. The corresponding optimization
problem can be transformed to be

$\displaystyle\begin{aligned} (P_{3})\quad&\max_{\bm{\phi}}\quad
f_{3}(\bm{\phi})=\sum_{k=1}^{K}\frac{\tilde{\alpha}_{k}|\bm{\phi}^{H}\bm{b}_{k,k}|^{2}}{\sum_{i=1}^{K}|\bm{\phi}^{H}\bm{b}_{i,k}|^{2}+\sigma_{0}^{2}},\\\
&s.t.\quad\quad|\phi_{n}|=1,\quad n=1,2,...,N.\end{aligned}$ (33)

By solving ($P_{3}$), we obtain the following results.

######  Proposition 3.

The optimal $\phi_{n},n=1,2,...,N$ in the reflecting vector $\bm{\phi}$ is
updated by

$\displaystyle\begin{aligned}
\phi_{n}^{op}=e^{j\theta_{n}^{op}},\end{aligned}$ (34)

with

$\displaystyle\begin{aligned} \theta_{n}^{op}&=\angle B_{2},\\\ \end{aligned}$
(35)

where $B_{2}=v_{n}-\sum_{q=1,q\neq n}^{N}u_{n,q}\phi_{q}$, $v_{n}$ is the
$n$-th element in vector
$\mathbf{V}=\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\varepsilon_{k}^{op{*}}\bm{b}_{k,k}$,
$u_{n,q}$ represents the element at row $i$ and column $q$ in matrix
$\mathbf{U}\\!=\\!\sum_{k=1}^{K}|\varepsilon_{k}^{op}|^{2}\sum_{i=1}^{K}\bm{b}_{i,k}\bm{b}_{i,k}^{H}$,
$\phi_{q}$ denotes the $q$-th element in $\bm{\phi}$ and
$\varepsilon_{k}^{op}$ is proved to be

$\displaystyle\begin{aligned}
\varepsilon_{k}^{op}=\frac{\sqrt{\tilde{\alpha}_{k}}\bm{\phi}^{H}\bm{b}_{i,k}}{\sum_{i=1}^{K}|\bm{\phi}^{H}\bm{b}_{i,k}|^{2}+\sigma_{0}^{2}}.\end{aligned}$
(36)

###### Proof:

See Appendix A. ∎

###  III-B Low-complexity Alternating Algorithm for Joint Beamforming and
Reflecting Design

Based on the above analyses, we propose a low-complexity alternating algorithm
to solve the joint beamforming and reflecting problem. The detailed steps of
the proposed algorithm are listed in Algorithm 1. In Algorithm 1,
$\bm{\alpha}$, $\mathbf{W}$ and $\mathbf{\Phi}$ are alternatively optimized
with the closed-form expressions. Besides, we have the following conclusion.

######  Proposition 4.

The proposed low-complexity alternating algorithm is guaranteed to converge.

###### Proof:

See Appendix B. ∎

The computational complexity of Algorithm 1 mainly comes from the steps for
the beamforming optimization and the phase shift update. In the beamforming
optimization step, the complexity for calculating $\mathbf{W}$ is
$\mathcal{O}(MK+K)$ and the complexity for updating $\bm{\alpha}$ is
$\mathcal{O}(K)$. In the phase shift update step, for each user $k$, the
optimal $\varepsilon_{k}^{op}$ is updated by (36), whose computational
complexity can be estimated as $\mathcal{O}(K)$, the phase shift optimization
of each reflecting element is designed according to (34) at a computational
complexity level of $\mathcal{O}(N)$ operations. Summarizing all above, the
aggregated complexity of Algorithm 1 can be expressed as
$\mathcal{O}\left((N+MK+3K)I_{1}\right)$, with $I_{1}$ denoting the required
number of outer iterations. From the analysis, it is found that the
computational complexity linearly increases with the number of reflecting
elements, i.e., $N$. For more comparison, we list the complexity and running
time of different optimization algorithms for joint beamforming and reflecting
design at Table II in Section V. Observing the comparison, we find that the
computational complexities of SDR based algorithm and ADMM based algorithm are
much more sensitive to $N$ and not suitable for the applications with large
$N$. We will give detailed performance comparisons about these algorithms in
Section V.

##  IV Proposed Joint Beamforming and Reflecting Design With Channel
Estimation

In this section, we will discuss the situation where perfect CSIs are not
available, and we should seek the help from channel estimation (CE) to get
CSI. We consider two types of CEs, separate CE and cascaded CE. For separate
CE [44, 45, 46], the separate BS-RIS and RIS-user channels can be obtained.
Under this circumstances, we can directly use the estimated BS-RIS and RIS-
user channels to replace the perfect CSIs in the proposed algorithm. For
cascaded CE, where only the cascaded BS-RIS-user channel can be obtained, we
take the classical least-square (LS) CE for an example to show how the
proposed algorithm works with the cascaded CSIs. We assume that the uplink and
downlink channels have channel reciprocity. In order to acquire CSIs for the
downlink transmission, uplink cascaded channel estimation is performed at the
beginning of each time interval. However, since the RIS does not have any
signal processing capabilities, only the cascaded user-RIS-BS channel can be
estimated at the BS. During the CE phase, each user sends orthogonal pilot
signals through the cascaded channel. Then, the estimated cascaded CSIs can be
used for downlink beamforming and reflecting design.

Algorithm 1 Low-complexity optimization algorithm for joint beamforming and
reflecting design with perfect CSIs

1: Initialization: Initialize $\mathbf{\Phi}^{0}$ and $\mathbf{W}^{0}$ that
satisfy (11) and (12), respectively. Set the threshold value $\tau$ and the
maximum number of iterations $I_{m}$. Iteration time $i=1$.

2: Step 1: Update $\bm{\alpha}^{i}$ according to (15).

3: Step 2: Update $\bm{\beta}^{i}$ according to (24).

4: Step 3: Update $\mathbf{W}^{i}$ according to (23) and (25).

5: Step 4: Update $\bm{\varepsilon}^{i}$ according to (36).

6: Step 5: Obtain $\bm{\phi}^{*}$ according to (34) and (35) .

7: if $f_{3}(\bm{\phi}^{*})\geq f_{3}(\bm{\phi}^{i-1})$ then

8: $\bm{\phi}^{i}=\bm{\phi}^{*}$

9: else

10: $\bm{\phi}^{i}=\bm{\phi}^{i-1}$

11: end if

12: Set $\mathbf{\Phi}^{i}=\mathop{\mathrm{diag}}(\bm{\phi}^{i})$.

13:
$i=i+1$.UNTIL$f_{1a}(\bm{\alpha}^{i},\mathbf{W}^{i},\mathbf{\Phi}^{i})-f_{1a}(\bm{\alpha}^{i-1},\mathbf{W}^{i-1},\mathbf{\Phi}^{i-1})\leq\tau$
or the iteration time $i=I_{m}$.

###  IV-A Cascaded Channel Estimation

Taking user $k$ as an example and assuming that the transmit power at user $k$
is $P_{k}$, we can write the received pilot signals at the BS as

$\displaystyle\begin{aligned}
y_{k}&=P_{k}\mathbf{G}\mathbf{\Phi}_{l}\mathbf{h}_{r,k}s_{l}+n_{k}\\\
&=P_{k}\mathbf{G}\mathop{\mathrm{diag}}(\mathbf{h}_{r,k})\bm{\phi}_{l}s_{l}+n_{k}\\\
&=P_{k}\mathbf{H}_{k}\bm{\phi}_{l}s_{l}+n_{k},\end{aligned}$ (37)

where $s_{l}$ represents the $l$-th pilot symbol satisfying
$\mathbb{E}\\{|s_{l}|^{2}\\}=1$, $\mathbf{\Phi}_{l}$ is the corresponding RIS
phase shift matrix when transmitting the pilot signal $s_{l}$, $n_{k}$ denotes
the AWGN noise in the uplink channel estimation phase with zero mean and
$\sigma_{k}^{2}$ variance, and
$\mathbf{H}_{k}=\mathbf{G}\mathop{\mathrm{diag}}(\mathbf{h}_{r,k})$ is the
cascaded channel that needs to be estimated. Let $L$ denote the total pilot
signal length used for channel estimation and define
$\mathbf{S}=P_{k}[s_{1},s_{2},...,s_{L}]^{T}\in\mathbb{C}^{L\times 1}$, we
express the received signal during the channel estimation as

$\displaystyle\begin{aligned}
\mathbf{Y}_{k}&=\mathbf{H}_{k}[\bm{\phi}_{1},\bm{\phi}_{2},....,\bm{\phi}_{L}]\mathop{\mathrm{diag}}(\mathbf{S})+\mathbf{N}\\\
&=\mathbf{H}_{k}\mathbf{X}+\mathbf{N},\end{aligned}$ (38)

where
$\mathbf{\Psi}=[\bm{\phi}_{1},\bm{\phi}_{2},....,\bm{\phi}_{L}]\in\mathbb{C}^{N\times
L}$ represents the equivalent RIS beamforming matrix during the channel
estimation phase, $\mathbf{X}=\mathbf{\Psi}\mathop{\mathrm{diag}}(\mathbf{S})$
can be seen as the equivalent training signals due to the fact that
$\mathbf{\Psi}$ and $\mathop{\mathrm{diag}}(\mathbf{S})$ are known during the
channel estimation, and $\mathbf{N}$ denotes the AWGN matrix. In this paper,
we adopt a discrete Fourier transform (DFT) matrix as equivalent RIS
beamforming matrix, i.e.,

$\mathbf{\Psi}=\left[\begin{matrix}1&1&\cdots&1\\\
1&\psi_{n}&\cdots&\psi_{n}^{L-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
1&\psi_{n}^{L-1}&\cdots&\psi_{n}^{(N-1)(L-1)}\end{matrix}\right],$

with $\psi_{n}=e^{2\pi j/L}$. This is because a DFT matrix has the full rank
and exhibits good performance for channel estimation [47].

Let $\mathbf{X}^{+}=\mathbf{X}^{H}(\mathbf{X}\mathbf{X}^{H})^{-1}$ represent
the pseudo-inverse of $\mathbf{X}$ and a LS estimation for the cascaded user-
RIS-BS channel can be expressed as
$\hat{\mathbf{H}}_{k}=\mathbf{Y}_{k}\mathbf{X}^{+}$. The corresponding channel
estimation error can be calculated by

$\displaystyle\begin{aligned}
\mathbf{H}_{k}-\hat{\mathbf{H}}_{k}&=\mathbf{H}_{k}-\mathbf{Y}_{k}\mathbf{X}^{+}=-\mathbf{N}\mathbf{X}^{+},\end{aligned}$
(39)

the expectation of which can be computed as

$\displaystyle\begin{aligned}
\mathbb{E}\left\\{||\mathbf{H}_{k}-\hat{\mathbf{H}}_{k}||_{F}^{2}\right\\}&=\mathbb{E}\left\\{||\mathbf{N}\mathbf{X}^{+}||_{F}^{2}\right\\}\\\
&=M\sigma_{n}^{2}\text{Tr}\left[(P_{k}\mathbf{\Psi}\mathbf{\Psi}^{H})^{-1}\right]\\\
&=\frac{P_{k}M\sigma_{n}^{2}N}{L}.\end{aligned}$ (40)

###  IV-B Proposed Low-complexity Alternating Optimization with Cascaded
Channel Estimation

According to the channel reciprocity, the cascaded BS-RIS-user channel
estimation for downlink data transmission can be expressed by
$\hat{\mathbf{H}}_{k}^{H}$. With $\hat{\mathbf{H}}_{k}^{H}$, the joint
beamforming and reflecting design based on perfect separate CSIs in Section
III has to be revisited. Specifically, with the real cascaded downlink channel
$\hat{\mathbf{H}}_{k}^{H}$ for each user $k$, the SINR in (8) can be modified
to be

$\displaystyle\begin{aligned}
\gamma_{k}^{im}=\frac{|\bm{\phi}^{im^{H}}\hat{\mathbf{H}}_{k}^{H}\bm{w}_{k}^{im}|^{2}}{\sum_{i=1,i\neq
k}^{K}|\bm{\phi}^{im^{H}}\hat{\mathbf{H}}_{k}^{H}\bm{w}_{i}^{im}|^{2}+\sigma_{0}^{2}},\end{aligned}$
(41)

where
$\hat{\mathbf{H}}_{k}^{H}=\mathop{\mathrm{diag}}(\mathbf{h}_{r,k}^{H})\mathbf{G}^{H}$
represents the downlink cascaded channel for user $k$, $\bm{\phi}^{im}$ and
$\bm{w}_{k}^{im}$ denote the RIS reflecting vector and BS beamforming vector
designed based on the cascaded channel estimation $\hat{\mathbf{H}}_{k}^{H}$,
respectively.

Then, the sum rate maximization problem with cascaded channel estimation can
be reformulated as

$\displaystyle(P_{1a}^{im})\max\limits_{\bm{\alpha}^{im},\mathbf{W}^{im},\mathbf{\Phi}^{im}}\quad$
$\displaystyle f_{1}(\bm{\alpha}^{im},\mathbf{W}^{im},\mathbf{\Phi}^{im}),$
(42) $\displaystyle s.t.\quad\quad\quad$ $\displaystyle|\phi_{n}^{im}|=1,\quad
n=1,2,...,N,$ (43) $\displaystyle\sum_{k=1}^{K}||\bm{w}_{k}^{im}||^{2}\leq
P_{T},$ (44) $\displaystyle 0\leq\alpha_{k}^{im},\quad k=1,2,...,K.$ (45)

with

$\displaystyle\begin{aligned}
f_{1}(\bm{\alpha}^{im},\mathbf{W}^{im},\mathbf{\Phi}^{im})=&\sum_{k=1}^{K}\log(1+\alpha_{k}^{im})-\sum_{k=1}^{K}\alpha_{k}^{im}\\\
&+\sum_{k=1}^{K}\frac{(1+\alpha_{k}^{im})\gamma_{k}^{im}}{1+\gamma_{k}^{im}},\end{aligned}$
(46)

and the parameters $\alpha_{1}^{im},\alpha_{2}^{im},...,\alpha_{k}^{im}$ are
the introduced auxiliary optimization variables with
$\bm{\alpha}^{im}=[\alpha_{1}^{im},\alpha_{2}^{im},...,\alpha_{K}^{im}]^{T}$
being the corresponding auxiliary variable vector, $\mathbf{W}^{im}$ and
$\mathbf{\Phi}^{im}$ represent the BS beamforming matrix and RIS reflecting
matrix, respectively, $\phi_{n}^{im}$ is the phase shift of each reflecting
element in $\bm{\phi}^{im}$ with $n=1,2,...,N$.

Different from the system design in Section III with perfect separate CSIs,
the formulated three-variable joint beamforming and reflecting optimization
problem in $(P_{1a}^{im})$ should be solved based on the estimated cascaded
channel $\hat{\mathbf{H}}_{k}^{H}$. That is, the closed-form expressions of
$\bm{\alpha}^{im}$, $\mathbf{W}^{im}$ and $\mathbf{\Phi}^{im}$ need to be
found with respect to $\hat{\mathbf{H}}_{k}^{H}$. Through analyses, we have
the following results, which are similar to Propositions 1-3.

######  Proposition 5.

The optimal expression of $\alpha_{k}^{im}$ with cascaded channel estimation
can be updated by

$\displaystyle\begin{aligned}
\alpha_{k}^{im^{op}}=\gamma_{k}^{im}.\end{aligned}$ (47)

###### Proof:

$\alpha_{k}^{im^{op}}$ can be obtained by solving $\partial
f_{1}(\bm{\alpha}^{im},\mathbf{W}^{im},\mathbf{\Phi}^{im})/\partial\alpha_{k}^{im}=0$.

∎

Given $\bm{\alpha}^{im}$ and $\mathbf{\Phi}^{im}$, the optimization problem
for finding the optimal $\mathbf{W}^{im}$ can be transformed by adopting the
FP technique [42] to be

$\displaystyle\begin{aligned}
(P_{2a}^{im})\quad\max_{\mathbf{W}^{im},\bm{\beta}^{im}}\quad&f_{2a}(\mathbf{W}^{im},\bm{\beta}^{im}),\\\
s.t.\quad\quad&\sum_{k=1}^{K}||\bm{w}_{k}^{im}||^{2}\leq P_{T},\end{aligned}$
(48)

where $f_{2a}(\mathbf{W}^{im},\bm{\beta}^{im})$ can be expressed as

$\displaystyle\begin{aligned}
f_{2a}(\mathbf{W}^{im},\bm{\beta}^{im})\\!=\\!&\sum_{k=1}^{K}2\sqrt{\tilde{\alpha}_{k}^{im}}\Re\left\\{{\beta_{k}^{im^{*}}\mathbf{h}_{k}^{im^{H}}\bm{w}_{k}^{im}}\right\\}\\\
&\\!-\\!\sum_{k=1}^{K}|\beta_{k}^{im}|^{2}\left(\sum_{i=1}^{K}|\mathbf{h}_{k}^{im^{H}}\bm{w}_{i}^{im}|^{2}\\!+\\!\sigma_{0}^{2}\right),\end{aligned}$
(49)

and parameters $\beta_{1}^{im},\beta_{2}^{im},...,\beta_{K}^{im}$ are the
introduced auxiliary optimization variables with
$\bm{\beta}^{im}=[\beta_{1}^{im},\beta_{2}^{im},...,\beta_{K}^{im}]^{T}$ being
the auxiliary variable vector, $\tilde{\alpha}_{k}^{im}=1+\alpha_{k}^{im}$,
and $\mathbf{h}_{k}^{im}=\bm{\phi}^{im^{H}}\hat{\mathbf{H}}_{k}^{H}$ is the
equivalent channel represented by the cascaded channel estimation result
$\hat{\mathbf{H}}_{k}^{H}$.

######  Proposition 6.

The optimal beamforming vector $\bm{w}_{k}^{im^{op}}$ is updated by

$\displaystyle\begin{aligned} &\bm{w}_{k}^{im^{op}}\\!=\\!\\\
&\sqrt{\tilde{\alpha}_{k}^{im}}\beta_{k}^{im^{op}}\left(\sum_{i=1}^{K}|\beta_{i}^{im^{op}}|^{2}\mathbf{h}_{i}^{im}\mathbf{h}_{i}^{im^{H}}\\!+\\!\lambda_{im}^{op}\mathbf{I}_{M}\right)^{-1}\mathbf{h}_{k}^{im},\end{aligned}$
(50)

where

$\displaystyle\begin{aligned}
\beta_{k}^{im^{op}}=\frac{\sqrt{\tilde{\alpha}_{k}^{im}}\mathbf{h}_{k}^{im^{H}}\bm{w}_{k}^{im}}{\sum_{i=1}^{K}|\mathbf{h}_{k}^{im^{H}}\bm{w}_{i}^{im}|^{2}+\sigma_{0}^{2}},\end{aligned}$
(51)

$\lambda_{im}^{op}$ is the dual variable introduced for the power constraint,
which satisfies $\sum_{k=1}^{K}||\bm{w}_{k}^{im^{op}}||^{2}=P_{T}$. That is,
$\lambda_{im}^{op}$ can be expressed as

$\displaystyle\begin{aligned}
\lambda_{im}^{op}=\min\left\\{\lambda_{im}^{op}\geq
0:\sum_{k=1}^{K}||\bm{w}_{k}^{im^{op}}||^{2}=P_{T}\right\\}.\end{aligned}$
(52)

###### Proof:

Similar to the proof of Proposition 2, the optimal expression of
$\beta_{k}^{im}$, i.e., $\beta_{k}^{im^{op}}$, can be obtained by setting
$\partial f_{2a}(\mathbf{W}^{im},\bm{\beta}^{im})/\partial\beta_{k}^{im}=0$.
After $\beta_{k}^{im^{op}}$ is fixed, the optimal expression of
$\bm{w}_{k}^{im^{op}}$ can also be obtained by setting $\partial
f_{2a}(\mathbf{W}^{im},\bm{\beta}^{im})/\partial\bm{w}_{k}^{im}=0$. ∎

At the RIS reflecting matrix optimization phase, since
$\mathbf{\Phi}^{im}=\mathop{\mathrm{diag}}(\bm{\phi}^{im})$ with
$\bm{\phi}^{im}=[\phi_{1}^{im},\phi_{2}^{im},...,\phi_{N}^{im}]^{T}$, the
optimization problem can be rewritten as

$\displaystyle\begin{aligned}
(P_{3a}^{im})\quad\quad\max_{\bm{\phi}^{im},\bm{\varepsilon}^{im}}\quad\quad&f_{3a}(\bm{\phi}^{im},\bm{\varepsilon}^{im}),\\\
s.t.\quad\quad\quad&|\phi_{n}^{im}|=1,\quad n=1,2,...,N,\quad\end{aligned}$
(53)

where

$\displaystyle\begin{aligned}
f_{3a}(\bm{\phi}^{im},\bm{\varepsilon}^{im})&\sum_{k=1}^{K}2\sqrt{\tilde{\alpha}_{k}^{im}}\Re\left\\{\varepsilon_{k}^{im^{*}}\bm{\phi}^{im^{H}}\bm{b}_{k,k}^{im}\right\\}\\\
&\\!-\\!\sum_{k=1}^{K}|\varepsilon_{k}^{im}|^{2}\left(\sum_{i=1}^{K}|\bm{\phi}^{im^{H}}\bm{b}_{i,k}^{im}|\\!+\\!\sigma_{0}^{2}\right),\\\
\end{aligned}$ (54)

$\varepsilon_{1}^{im},\varepsilon_{2}^{im},...,\varepsilon_{N}^{im}$ are the
introduced auxiliary optimization variables with
$\bm{\varepsilon}^{im}=[\varepsilon_{1}^{im},\varepsilon_{2}^{im},...,\varepsilon_{N}^{im}]^{T}$
being the auxiliary variable vector, and
$\bm{b}_{i,k}^{im}=\hat{\mathbf{H}}_{k}^{H}\bm{w}_{i}^{im}$ denotes the
equivalent channel represented by the cascaded channel estimation result
$\hat{\mathbf{H}}_{k}^{H}$.

######  Proposition 7.

The optimal $\phi_{n}^{im}$ ($n=1,2,...,N$) in the reflecting vector
$\bm{\phi}^{im}$ can be updated by

$\displaystyle\begin{aligned}
\phi_{n}^{im^{op}}=e^{j\theta_{n}^{im^{op}}},\end{aligned}$ (55)

with

$\displaystyle\begin{aligned} \theta_{n}^{im^{op}}&=\angle
B_{2}^{im},\end{aligned}$ (56)

where
$B_{2}^{im}\\!=\\!v_{n}^{im}\\!-\\!\sum_{q\\!=\\!1,q\\!\neq\\!n}^{N}u_{n,q}^{im}\phi_{q}^{im}$,
$v_{n}^{im}$ is the $n$-th element in vector
$\mathbf{V}^{im}\\!=\\!\sum_{k\\!=\\!1}^{K}\sqrt{\tilde{\alpha}_{k}^{im}}\varepsilon_{k}^{im^{op{*}}}\bm{b}_{k,k}^{im}$,
$u_{n,q}^{im}$ represents the element at row $i$ and column $q$ in matrix
$\mathbf{U}^{im}\\!=\\!\sum_{k=1}^{K}|\varepsilon_{k}^{im^{op}}|^{2}\sum_{i=1}^{K}\bm{b}_{i,k}^{im}\bm{b}_{i,k}^{im^{H}}$,
$\phi_{q}^{im}$ denotes the $q$-th element in $\bm{\phi}^{im}$, and
$\varepsilon_{k}^{im^{op}}$ is proved to be

$\displaystyle\begin{aligned}
\varepsilon_{k}^{im^{op}}=\frac{\sqrt{\tilde{\alpha}_{k}^{im}}\bm{\phi}^{im^{H}}\bm{b}_{i,k}^{im}}{\sum_{i=1}^{K}|\bm{\phi}^{im^{H}}\bm{b}_{i,k}^{im}|^{2}+\sigma_{0}^{2}}.\end{aligned}$
(57)

###### Proof:

See Appendix C. ∎

###  IV-C Alternating Algorithm for Joint Beamforming and Reflecting Design
With Cascaded CSIs

Based on the above analysis, the proposed low-complexity joint beamforming and
reflecting algorithm can be modified according to the cascaded CE results
$\hat{\mathbf{H}}_{k}$. Similar to Algorithm 1, in Algorithm 2, three
subproblems are alternatively updated with the closed-form optimal expressions
of $\bm{\alpha}^{im^{op}}$, $\bm{w}_{k}^{im^{op}}$ and $\phi_{n}^{im^{op}}$.
Specifically, $\bm{\alpha}^{im^{i}}$ is updated according to (47),
$\bm{\beta}^{im^{i}}$ is updated according to (51), $\mathbf{W}^{im^{i}}$ is
updated according to (50) and (52), $\bm{\phi}^{im^{*}}$ and
$\bm{\varepsilon}^{im^{op}}$ are updated according to (55), (56) and (57),
respectively. Besides, we have the following conclusion.

######  Proposition 8.

Algorithm 2 is guaranteed to converge.

###### Proof:

Proposition 8 can be proved similarly to proposition 4, since the main
difference between Algorithm 1 and Algorithm 2 is that the optimal
$\bm{\alpha}^{im^{op}}$, $\mathbf{w}_{k}^{im^{op}}$ and $\phi_{n}^{im^{op}}$
in Algorithm 2 are expressed by the cascaded channel estimation result
$\hat{\mathbf{H}}_{k}^{H}$ instead of the perfect separate CSIs
$\mathbf{G}^{H}$ and $\mathbf{h}_{r,k}^{H}$. ∎

Figure 2: The simulated RIS-aided $K$-user communication system comprising of one $M$-antenna BS and one $N$-element RIS. TABLE I: Simulation setups. Parameters |  Values
---|---
BS location |  (0 m, 0 m)
RIS central location $D_{1}$ |  (200 m, 0 m)
Transmit power $P_{T}$ at the BS |  10 dBW
Large scale fading for $\mathbf{G}^{H}$ and $\mathbf{h}_{r,k}^{H}$ |  35.6 + 22.0 $\log_{10}d$ dB
Rician factor for $\mathbf{G}^{H}$ and $\mathbf{h}_{r,k}^{H}$ |  10 dB
TABLE II: Complexity comparison of different optimization algorithms for joint
beamforming and reflecting design.

Algorithm |  Complexity |  Running time for each iteration when $M=3,N=6,K=2$ (Unit: second)
---|---|---
Proposed algorithm |  $\mathcal{O}\left((N+MK+3K)I_{1}\right)$ |  0.0695
ADMM based algorithm |  $\mathcal{O}\left((N^{3}+MK+2K)I_{2}\right)$ |  2.3618
MMSE + SDR based algorithm |  $\mathcal{O}\left((N^{6}+MK)I_{3}\right)$ |  438.216
ZF + SDR based algorithm |  $\mathcal{O}\left((N^{6}+MK)I_{4}\right)$ |  435.558

  * *

$I_{n}$ with $n=1,2,3,4$ is the number of outer iterations for different
algorithms.

##  V Performance Evaluation

This section provides numerical results to show the effectiveness of the
proposed low-complexity algorithm for joint beamforming and reflecting in
$(M,N,K)$ RIS-aided multi-user communication systems. This section is further
divided into two subsections. Subsection V-A investigates the performance of
the proposed design with perfect separate CSIs and Subsection V-B studies the
performance of the proposed design with practical channel estimation. As
discussed above, we investigate the performance of the RIS-aided $K$-user
communication system as illustrated in Fig. 2. The $K$ single-antenna users
are assumed to be uniformly and randomly distributed within a circle centered
at ($200$ m, $50$ m) with a radius $30$ m. For clarity, the system parameters
are summarized in Table I. The pathloss is set according to the 3GPP
propagation environment described in [48]. For comparison, we also simulate
the performance of the following schemes:

  * •

ADMM based scheme [49]. It obtains the precoding vectors and phase shift
matrices by solving non-convex problems in the ADMM method.

  * •

MMSE/ZF + SDR based scheme [32]. Classical MMSE or ZF precoding schemes are
directly adopted at the BS side, and the RIS phase shift matrix is obtained
through the SDR method.

###  V-A Performance of the Proposed Design With Perfect CSIs

Figure 3: Convergence behavior of the proposed algorithm in $(4,25,3)$,
$(4,25,2)$ and $(4,36,3)$ RIS-aided multi-user communication systems.

First, we assume that BS knows all the CSIs of the involved channels, i.e.,
all separate CSIs, and investigate the convergence behavior and the
performance in terms of sum rate of the proposed algorithm. Fig. 3 shows the
convergence behavior of the proposed low-complexity algorithm. It is observed
that the proposed algorithm converges as stated in proposition 4 and converges
fast in the depicted $(4,25,3)$, $(4,25,2)$ and $(4,36,3)$ RIS-aided multi-
user communication systems.

Figure 4:  Achievable sum rate of the proposed algorithm, ADMM based algorithm
and MMSE/ZF+SDR algorithm in $(4,36,3)$ RIS-aided multi-user communication
systems.

To demonstrate the superiority of the proposed algorithm, we compare it with
the ADMM based algorithm and traditional MMSE/ZF + SDR based algorithm in
terms of the complexity. We list the complexity and running time of these
algorithms in Table II. All simulations are performed on the same computer
with a 3.2GHz Intel(R) Core(TM) i7-8700 CPU and 16GB RAM. In Fig. 4, we
further show the convergence speed and the achievable sum rate of these
algorithms. As it is shown, the proposed algorithm has lower complexity than
the other algorithms in a $(4,36,3)$ RIS-aided multi-user system. Besides, the
convergence speed of the proposed algorithm is much faster than those of the
traditional MMSE/ZF + SDR based algorithm. The reason is that in each
iteration, the algorithm is updated with the closed-form optimal solutions,
which not only improves the sum rate performance of the algorithm, but also
reduces the complexity and running time.

For more comparison, we show the performance of the proposed algorithm, ADMM
based algorithm and MMSE/ZF + SDR based algorithm in terms of achievable sum
rate. As illustrated in Fig. 5, when $M=4$, $K=2$ are fixed, with the increase
of RIS reflecting elements, the proposed algorithm always provides better
system performance.

Figure 5: Achievable sum rate of the proposed algorithm, ADMM based algorithm
and MMSE/ZF + SDR based algorithm when the number of RIS reflecting elements
increases, where $M=4$ and $K=2$ are fixed.
Figure 6: Achievable sum rate of the proposed algorithm, ADMM based algorithm,
MMSE/ZF + SDR based algorithm when Rician factor changes
($\mathcal{F}_{\mathbf{G}}$ = $\mathcal{F}_{r,k}$), where $M=4$ and $K=2$.
Figure 7: Achievable sum rate of the proposed algorithm when the number of
users $K$ changes, $N=10$ is fixed.

Fig. 6 shows how the sum rate is affected by the channel Rician factor. We
assume that the channel from BS to RIS and from RIS to users follow Rician
fading with Rician factors $\mathcal{F}_{\mathbf{G}}$ and $\mathcal{F}_{r,k}$.
One can observe that with the increasing of Rician factors, the sum rate
increases. This is because when the LoS component is getting stronger and
stronger, the channel state is getting better, thus the sum rate increases
accordingly.

Fig. 7 shows the relationship between the sum rate and the number of user $K$
for $N=10$. It is observed that the obtained sum rate decreases when $K$
increases. It implies that a larger number of users results in a lower average
user rate due to the lower transmit power of each user and severer inter-user
interference. Moreover, it is also found that a higher transmit power or
larger number of BS antennas leads to a higher sum rate.

###  V-B Joint Beamforming and Reflecting Design With Channel Estimation

We investigate the performance of the proposed algorithm with separate and
cascaded CE. The separate CE method in [45] and the cascaded LS CE method in
Section IV-A are adopted. We assume that the channel coherent time is much
larger than the pilot signal length. Simulation results are included in Fig.
8, which show that the achievable sum rate increases as $L$ increases, because
larger $L$ leads to accurate CSI. Also, the proposed algorithm combining with
more advanced channel estimation methods, i.e., separate CE, achieves better
performance. For comparison, we also include the performance of the ADMM based
algorithm and MMSE/ZF + SDR based algorithm with perfect CSIs. It is shown
that with sufficient long pilot signal length, the proposed algorithm with
imperfect CSIs can outperform the ADMM based algorithm and MMSE/ZF + SDR based
algorithm with perfect CSIs, which further verifies the effectiveness of the
proposed algorithms. Besides, we can see that the convergence speed of the
proposed algorithm for $N=81$ in Fig. 8 increases slightly compared with that
for $N=36$ in Fig. 4, but they both can converge within around $10$
iterations.

Figure 8: Achievable sum rate of the proposed algorithm, ADMM based algorithm,
MMSE/ZF + SDR based algorithm under different CSIs, where $M=4$, $N=81$ and
$K=2$.
Figure 9: Channel estimation error versus pilot length $L$.

However, it is worth mentioned that the coherent time duration is limited in
practical scenarios. Given a fixed coherent time, which is comparable to the
pilot signal length, increasing the number of pilot signals no longer always
increases the sum rate. This is because even though increasing the number of
pilot signal length reduces the channel uncertainty, it also reduces the
information transmission time. To demonstrate the impact of pilot signal
length on the sum rate, we simulated the channel estimation accuracy and the
results are shown in Fig. 9. It can be seen that the channel estimation error
decreases when $L$ increases as analyzed in Eq. (40). We then simulate the
performance of the proposed algorithm with different pilot signal length $L$
and time slot length $\Upsilon$ in Fig. 10. We define the effective sum rate
$\tilde{R}=\frac{\Upsilon-L}{\Upsilon}\sum_{k=1}^{K}\log(1+\gamma_{k}^{im})$,
where $L$ represents the time used for channel estimation in one time slot,
and $\Upsilon$ represents the total length of one time slot, thus $\Upsilon-L$
indicates the effective time for information transmission. Lines 1-2 first
increases as $L$ increases since the channel estimation errors decreases as
$L$ increases. Then, when $L$ keeps increasing, the effective sum rate will
decrease since the time slots for information transmission $\Upsilon-L$
reduces and the gain brought by the improved channel accuracy cannot
compensate for the loss caused by the reduction of $\Upsilon-L$.

Figure 10: Effective sum rate of the proposed algorithm with different pilot
signal length $L$ and time slot length $\Upsilon$, where $M=4$ and $K=2$.
Figure 11: The sum rate of the proposed algorithm, ADMM based algorithm and
MMSE/ZF + SDR based algorithm when the horizontal distance from BS to RIS
changes, where $M=4$, $N=64$ and $K=3$.

Fig. 11 illustrates the effective sum rate of all users when moving the RIS
from (180 m, 0 m) to (215 m, 0 m). It can be seen that, when $D_{1}$ increases
from 200 m to 215 m, the sum rate decreases. This is because both of the pass-
loss of channel $\mathbf{G}^{H}$ and $\mathbf{h}_{r,k}^{H}$ increase. However,
when decreasing $D_{1}$ from 200 m to 180 m, the sum rate first increases, and
then decreases, and we get that the optimal location is $D_{1}$=190 m. This is
because the path loss of the RIS-aided link is the product of the path losses
of $\mathbf{G}^{H}$ and $\mathbf{h}_{r,k}^{H}$. Therefore, although the
summation of the transmission distance decreases, the propagation condition
might not become better.

##  VI Conclusion

In this paper, we have investigated a RIS-aided multi-user communication
system where the direct links between the BS and the users are blocked and a
RIS is deployed to provide additional high quality links to improve the system
performance. The downlink sum rate optimization problem is formulated, which
is a computationally intensive task due to the large number of RIS reflecting
elements. To address this non-convex multivariable problem, we have decoupled
the original problem into three disjoint subproblems by utilizing fractional
programming. By introducing auxiliary variables in each subproblem, we are
able to obtain the closed-form expressions of the optimizing decision
variables, and then, based on these, develop a low-complexity alternating
optimization algorithm. We have also shown that, the proposed algorithm enjoys
extremely low complexity $\mathcal{O}\left((N+MK+3K)I_{1}\right)$ with $N$
being the number of RIS reflecting elements. Through extensive numerical
analysis, our results have verified that the proposed algorithm performs
significantly better compared to various benchmark schemes.

##  Appendix A Proof of Proposition 3

The optimization problem $(P_{3})$ is a multiple-ratio problem, which can be
transformed into a biconvex optimization problem according to the quadratic
transform proposed in [42] and [43]. By introducing the new auxiliary variable
vector $\bm{\varepsilon}$, $f_{3}(\bm{\phi})$ can be rewritten as

$\displaystyle\begin{aligned}
f_{3a}(\bm{\phi},\bm{\varepsilon})=&\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\left(\varepsilon_{k}^{*}\bm{\phi}^{H}\bm{b}_{i,k}+\bm{b}_{i,k}^{H}\bm{\phi}\varepsilon_{k}\right)\\\
&-\sum_{k=1}^{K}|\varepsilon_{k}|^{2}\left(\sum_{i=1}^{K}|\bm{\phi}^{H}\bm{b}_{i,k}|^{2}+\sigma_{0}^{2}\right),\end{aligned}$
(58)

where $\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{N}$ are the introduced
complex auxiliary variables and
$\bm{\varepsilon}=[\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{N}]^{T}$
is the auxiliary variable vector. First, the optimal auxiliary variable
$\varepsilon_{k}$, $k=1,2,...,N$ can be solved by setting $\partial
f_{3a}(\bm{\phi},\bm{\varepsilon})/\partial\varepsilon_{k}=0$. Taking partial
derivative of $f_{3a}(\bm{\phi},\bm{\varepsilon})$ with respect to
$\varepsilon_{k}$ yields

$\displaystyle\begin{aligned} \frac{\partial
f_{3a}(\bm{\phi},\bm{\varepsilon})}{\partial\varepsilon_{k}}\\!=\\!2\sqrt{\tilde{\alpha}_{k}}\bm{\phi}^{H}\bm{b}_{i,k}-2\varepsilon_{k}\sum_{i=1}^{K}|\bm{\phi}^{H}\bm{b}_{i,k}|^{2}-2\varepsilon_{k}\sigma_{0}^{2}.\end{aligned}$

By setting $\partial
f_{3a}(\bm{\phi},\bm{\varepsilon})/\partial\varepsilon_{k}=0$, we can get the
optimal $\varepsilon_{k}^{op}$ to be

$\displaystyle\begin{aligned}
\varepsilon_{k}^{op}=\frac{\sqrt{\tilde{\alpha}_{k}}\bm{\phi}^{H}\bm{b}_{i,k}}{\sum_{i=1}^{K}|\bm{\phi}^{H}\bm{b}_{i,k}|^{2}+\sigma_{0}^{2}}.\end{aligned}$
(59)

Substituting $\varepsilon_{k}^{op}$ into $f_{3a}(\bm{\phi},\bm{\varepsilon})$
yields

$\displaystyle\begin{aligned}
f_{3b}(\bm{\phi},\bm{\varepsilon}^{op})=&\sum_{k=1}^{K}2\sqrt{\tilde{\alpha}_{k}}\Re\left\\{\varepsilon_{k}^{op^{*}}\bm{\phi}^{H}\bm{b}_{k,k}\right\\}\\\
&\\!-\\!\sum_{k=1}^{K}|\varepsilon_{k}^{op}|^{2}\sum_{i=1}^{K}\bm{\phi}^{H}\bm{b}_{i,k}\bm{b}_{i,k}^{H}\bm{\phi}\\!-\\!\sum_{k=1}^{K}|\varepsilon_{k}^{op}|^{2}\sigma_{0}^{2}\\\
=&-\bm{\phi}^{H}\mathbf{U}\bm{\phi}+2\Re\left\\{\bm{\phi}^{H}\mathbf{V}\right\\}-\mathbf{C},\end{aligned}$
(60)

where

$\displaystyle\begin{aligned}
&\mathbf{U}\\!=\\!\sum_{k=1}^{K}|\varepsilon_{k}^{op}|^{2}\sum_{i=1}^{K}\bm{b}_{i,k}\bm{b}_{i,k}^{H},\quad\mathbf{V}\\!=\\!\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\varepsilon_{k}^{op{*}}\bm{b}_{k,k},\\\
&\mathbf{C}\\!=\\!\sum_{k=1}^{K}|\varepsilon_{k}^{op}|^{2}\sigma_{0}^{2}.\end{aligned}$
(61)

In order to obtain the closed-form optimal expression of $\bm{\phi}$, we
define $u_{i,q}$ as the element at row $i$ and column $q$ in
$\mathbf{U}\in\mathbb{C}^{N\times N}$, $v_{n}$ represents the $n$-th elements
of vector $\mathbf{V}\in\mathbb{C}^{N\times 1}$. Recall that
$\bm{\phi}=[\phi_{1},\phi_{2},...,\phi_{N}]^{T}$, in (60), the term
$\bm{\phi}^{H}\mathbf{V}$ can be rewritten as

$\begin{split}\bm{\phi}^{H}\mathbf{V}&=[\phi_{1}^{*},\phi_{2}^{*},...,\phi_{N}^{*}]\left[\begin{matrix}v_{1}\\\
v_{2}\\\ \vdots\\\ v_{N}\end{matrix}\right]=\sum_{i=1}^{N}\phi_{i}^{*}v_{i}\\\
&=\phi_{n}^{*}v_{n}+\sum_{i=1,i\neq n}^{N}\phi_{i}^{*}v_{i}.\end{split}$ (62)

The term $\bm{\phi}^{H}\mathbf{U}\bm{\phi}$ in (60) can be rewritten as

$\begin{split}\bm{\phi}^{H}\mathbf{U}\bm{\phi}\\!=\\!&[\phi_{1}^{*},\phi_{2}^{*},...,\phi_{N}^{*}]\left[\begin{matrix}u_{1,1}&u_{1,2}&\cdots&u_{1,N}\\\
u_{2,1}&u_{2,2}&\cdots&u_{2,N}\\\ \vdots&\vdots&\ddots&\vdots\\\
u_{N,1}&u_{N,2}&\cdots&u_{N,N}\end{matrix}\right]\left[\begin{matrix}\phi_{1}\\\
\phi_{2}\\\ \vdots\\\ \phi_{N}\end{matrix}\right]\\\
=&\sum_{i=1}^{n}\sum_{q=1}^{N}\phi_{i}^{*}u_{i,q}\phi_{j}\\\
=&\phi_{n}^{*}u_{n,n}\phi_{n}+2\Re\left\\{\sum_{q=1,q\neq
n}^{N}\phi_{n}^{*}u_{n,q}\phi_{q}\right\\}\\\ &+\sum_{i=1,i\neq
n}^{n}\sum_{q=1,q\neq n}^{n}\phi_{i}^{*}u_{i,q}\phi_{q}.\end{split}$ (63)

Substituting (62) and (63) into (60),
$f_{3b}(\bm{\phi},\bm{\varepsilon}^{op})$ can be rewritten as

$\displaystyle\begin{aligned}
f_{3c}(\phi_{n},\bm{\varepsilon}^{op})\\!\\!=\\!&-\phi_{n}^{*}u_{n,n}\phi_{n}\\!\\!+\\!\\!2\Re\left\\{\phi_{n}^{*}v_{n}\\!-\\!\\!\\!\\!\sum_{q=1,q\neq
n}^{N}\phi_{n}^{*}u_{n,q}\phi_{q}\right\\}\\\ &+2\Re\left\\{\sum_{i=1,i\neq
n}^{N}\phi_{i}^{*}v_{i}\\!-\\!\\!\\!\\!\\!\\!\sum_{i=1,i\neq
n}^{N}\sum_{q=1,q\neq n}^{N}\phi_{i}^{*}u_{i,q}\phi_{q}\right\\}\\\
=&-|\phi_{n}|^{2}B_{1}+2\Re\left\\{\phi_{n}^{*}B_{2}\right\\}+B_{3},\end{aligned}$

where

$\displaystyle\begin{aligned} &B_{1}=u_{n,n},\quad\quad
B_{2}=v_{n}-\sum_{q=1,q\neq n}^{N}u_{n,q}\phi_{q},\\\
&B_{3}=2\Re\left\\{\sum_{i=1,i\neq n}^{N}\phi_{i}^{*}v_{i}-\sum_{i=1,i\neq
n}^{N}\sum_{q=1,q\neq
n}^{N}\phi_{i}^{*}u_{i,q}\phi_{q}\right\\}.\end{aligned}$ (64)

Since $\phi_{n}=e^{j\theta_{n}}$, $|\phi_{n}|^{2}=1$. Finally, the
optimization problem to find optimal $\mathbf{\Phi}$ given $\bm{\alpha}$ and
$\mathbf{W}$ can be expressed as

$\displaystyle\begin{aligned} (P_{3c})\quad&\max_{\phi_{n}}\quad\quad
f_{3c}(\phi_{n},\bm{\varepsilon}^{op})\\!=\\!-B_{1}\\!+\\!2\Re\left\\{\phi_{n}^{*}B_{2}\right\\}+B_{3},\\\
&s.t.\quad\quad\quad|\phi_{n}|=1\quad n=1,2,...,N.\end{aligned}$ (65)

From (65), we can see that maximizing $f_{3c}(\phi_{n},\bm{\varepsilon}^{op})$
is equivalent to maximizing the term $2\Re\\{\phi_{n}^{*}B_{2}\\}$. Thus, the
optimal solution of $\theta_{n}$ in (35) can be obtained. $\blacksquare$

##  Appendix B Convergence analysis of Algorithm 1

In Algorithm 1, Proposition 1 ensures that the update of $\bm{\alpha}$ always
gives an improving value, that is,
$f_{1a}(\bm{\alpha}^{i},\mathbf{W}^{i-1},\mathbf{\Phi}^{i-1})\geq
f_{1a}(\bm{\alpha}^{i-1},\mathbf{W}^{i-1},\mathbf{\Phi}^{i-1})$ always
satisfies in the $i$-th iteration.

For the BS beamforming optimization, when $\bm{\alpha}$ and $\mathbf{\Phi}$
are fixed, (21) is a biconvex problem related to $\bm{w}_{k}$ and $\beta_{k}$.
For fixed $\mathbf{W}$, the optimal $\beta_{k}^{op}$ can be obtained by
setting $\partial f_{2b}(\mathbf{W},\bm{\beta})/\partial\beta_{k}=0$. For
fixed $\beta_{k}$, the optimization problem to find $\mathbf{W}$ can be
expressed as

$\displaystyle\begin{aligned} \max_{\mathbf{W}}\quad
f_{2b}(\mathbf{W})=&\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\left(\beta_{k}^{*}\mathbf{h}_{k}^{H}\bm{w}_{k}+\bm{w}_{k}^{H}\mathbf{h}_{k}\beta_{k}\right)\\\
&-\sum_{i=1}^{K}|\beta_{i}|^{2}\sum_{k=1}^{K}\bm{w}_{k}^{H}\mathbf{h}_{i}\mathbf{h}_{i}^{H}\bm{w}_{k},\\\
s.t.\quad\quad\quad\quad\quad\quad&\sum_{k=1}^{K}||\bm{w}_{k}||^{2}\leq
P_{T}.\end{aligned}$ (66)

Using the Lagrangian function, we can transform the above optimization problem
into

$\displaystyle\begin{aligned}
f_{2b}^{(L)}(\mathbf{W})=&-\sum_{i=1}^{K}|\beta_{i}|^{2}\sum_{k=1}^{K}\bm{w}_{k}^{H}\mathbf{h}\mathbf{h}_{i}^{H}\bm{w}_{k}\\\
&+\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}}\left(\beta_{k}^{*}\mathbf{h}_{k}^{H}\bm{w}_{k}+\bm{w}_{k}^{H}\mathbf{h}_{k}\beta_{k}\right)\\\
&-\lambda\left(\sum_{k=1}^{K}\bm{w}_{k}^{H}\bm{w}_{k}-P_{T}\right).\end{aligned}$
(67)

By setting $\partial f_{2b}^{(L)}(\mathbf{W})/\partial\bm{w}_{k}=0$, we can
get the optimal expression of $\bm{w}_{k}$ in (23). By doing so,
$f_{1a}(\bm{\alpha}^{i},\mathbf{W}^{i},\mathbf{\Phi}^{i-1})\geq
f_{1a}(\bm{\alpha}^{i-1},\mathbf{W}^{i-1},\mathbf{\Phi}^{i-1})$ is guaranteed.

Finally, for fixed $\bm{\alpha}$ and $\mathbf{W}$,
$f_{3a}(\bm{\phi},\bm{\varepsilon})$ in (58) is also a biconvex problem, we
can get the optimal $\varepsilon_{k}^{op}$ by setting $\partial
f_{3a}(\bm{\phi},\bm{\varepsilon})/\partial\varepsilon_{k}=0$. Besides, we
optimize the phase shift $\phi_{n}$ one by one according to (34). Based on all
above analysis, we have $f_{3}(\bm{\phi}^{i})\geq f_{3}(\bm{\phi}^{i-1})$,
which further guarantees that the objective function is non-decreasing. That
is $f_{1a}(\bm{\alpha}^{i},\mathbf{W}^{i},\mathbf{\Phi}^{i})\geq
f_{1a}(\bm{\alpha}^{i-1},\mathbf{W}^{i-1},\mathbf{\Phi}^{i-1})$ after the
$i$-th iteration. Thus, the algorithm generates a non-decreasing sequence.

To prove the convergence of Algorithm 1, we still need to show that the
sequence generated from Algorithm 1 is upper bounded. As shown in (16), for
$0\leq\alpha_{k}\leq\gamma_{k}$ and
$\bm{\alpha}=[\alpha_{1},\alpha_{2},...,\alpha_{K}]^{T}$,
$f_{1a}(\bm{\alpha},\mathbf{W},\mathbf{\Phi})$ is an increasing function with
respect to $\bm{\alpha}$. The maximum value of
$f_{1a}(\bm{\alpha},\mathbf{W},\mathbf{\Phi})$ with respect to $\bm{\alpha}$
is achieved by setting $\alpha_{k}^{op}=\gamma_{k}$, leading to
$f_{1a}(\bm{\alpha},\mathbf{W},\mathbf{\Phi})=f_{1}(\mathbf{W},\mathbf{\Phi})$.
This shows that $f_{1a}(\bm{\alpha},\mathbf{W},\mathbf{\Phi})$ is upper-
bounded by the original objective function $f_{1}(\mathbf{W},\mathbf{\Phi})$.
Besides, since $\gamma_{k}$ ranges from $0$ to $\gamma_{k}^{max}$,
$f_{1}(\mathbf{W},\mathbf{\Phi})$ must have a finite upper bound. Thus,
Algorithm 1 generates a bounded non-decreasing sequence and hence converges to
at least a local optimum.

$\blacksquare$

##  Appendix C Proof of Proposition 7

In (54), the optimal $\varepsilon_{k}^{im}$, i.e., $\varepsilon_{k}^{im^{op}}$
can be obtained by setting $\partial
f_{3a}(\bm{\phi}^{im},\bm{\varepsilon}^{im})/\partial\varepsilon_{k}^{im}=0$
for $k=1,2,...,K$ as shown in (57). After $\bm{\varepsilon}^{im^{op}}$ is
obtained, we further rewrite $f_{3a}(\bm{\phi}^{im},\bm{\varepsilon}^{im})$ as
the following expression

$\displaystyle\begin{aligned}
&f_{3b}(\bm{\phi}^{im},\bm{\varepsilon}^{im^{op}})\\\
\\!=\\!&\sum_{k=1}^{K}2\sqrt{\tilde{\alpha}_{k}^{im}}\Re\left\\{\varepsilon_{k}^{im^{op^{*}}}\bm{\phi}^{im^{H}}\bm{b}_{k,k}^{im}\right\\}\\\
&\\!-\\!\sum_{k=1}^{K}|\varepsilon_{k}^{im^{op}}|^{2}\sum_{i=1}^{K}\bm{\phi}^{im^{H}}\bm{b}_{i,k}^{im}\bm{b}_{i,k}^{im^{H}}\bm{\phi}^{im}\\!-\\!\sum_{k=1}^{K}|\varepsilon_{k}^{im^{op}}|^{2}\sigma_{0}^{2}\\\
&\\!=\\!-\bm{\phi}^{im^{H}}\mathbf{U}^{im}\bm{\phi}^{im}\\!+\\!2\Re\left\\{\bm{\phi}^{im^{H}}\mathbf{V}^{im}\right\\}-\mathbf{C}^{im},\end{aligned}$
(68)

where

$\displaystyle\begin{aligned}
&\mathbf{U}^{im}\\!=\\!\sum_{k=1}^{K}|\varepsilon_{k}^{im^{op}}|^{2}\sum_{i=1}^{K}\bm{b}_{i,k}^{im}\bm{b}_{i,k}^{im^{H}},\mathbf{V}^{im}\\!=\\!\sum_{k=1}^{K}\sqrt{\tilde{\alpha}_{k}^{im}}\varepsilon_{k}^{im^{op^{*}}}\bm{b}_{k,k}^{im},\\\
&\mathbf{C}^{im}=\sum_{k=1}^{K}|\varepsilon_{k}^{im^{op}}|^{2}\sigma_{0}^{2}.\end{aligned}$

The term $\bm{\phi}^{im^{H}}\mathbf{V}^{im}$ in (LABEL:65) can be rewritten as

$\begin{split}\bm{\phi}^{im^{H}}\mathbf{V}^{im}&=\sum_{i=1}^{N}\phi_{i}^{im^{*}}v_{i}^{im}=\phi_{n}^{im^{*}}v_{n}^{im}+\sum_{i=1,i\neq
n}^{N}\phi_{i}^{im^{*}}v_{i}^{im}.\end{split}$ (69)

The term $\bm{\phi}^{im^{H}}\mathbf{U}^{im}\bm{\phi}^{im}$ in (LABEL:65) can
be rewritten as

$\begin{split}\bm{\phi}^{im^{H}}\mathbf{U}^{im}\bm{\phi}^{im}=&\sum_{i=1}^{n}\sum_{j=1}^{N}\phi_{i}^{im^{*}}u_{i,j}^{im}\phi_{j}^{im}\\\
=&\phi_{n}^{im^{*}}u_{n,n}^{im}\phi_{n}^{im}\\!+\\!2\Re\left\\{\sum_{j=1,j\neq
n}^{N}\phi_{n}^{im^{*}}u_{n,j}^{im}\phi_{j}^{im}\right\\}\\\ &+\sum_{i=1,i\neq
n}^{n}\sum_{j=1,j\neq
n}^{n}\phi_{i}^{im^{*}}u_{i,j}^{im}\phi_{j}^{im}.\end{split}$ (70)

Using the simplified expressions in (69) and (70), (LABEL:65) can be rewritten
as

$\displaystyle\begin{aligned}
&f_{3c}(\phi_{n}^{im},\bm{\varepsilon}^{im^{op}})\\!=\\!-|\phi_{n}^{im}|^{2}B_{1}^{im}\\!+\\!2\Re\left\\{\phi_{n}^{im^{*}}B_{2}^{im}\right\\}\\!+\\!B_{3}^{im},\end{aligned}$

where

$\displaystyle\begin{aligned} &B_{1}^{im}=u_{n,n}^{im},\quad\quad
B_{2}^{im}=v_{n}^{im}-\sum_{j=1,j\neq n}^{N}u_{n,j}^{im}\phi_{j}^{im},\\\
&B_{3}^{im}\\!=\\!2\Re\left\\{\sum_{i=1,i\neq
n}^{N}\phi_{i}^{im^{*}}v_{i}^{im}\\!-\\!\\!\\!\\!\sum_{i=1,i\neq
n}^{N}\sum_{j=1,j\neq
n}^{N}\phi_{i}^{im^{*}}u_{i,j}^{im}\phi_{j}^{im}\right\\}.\end{aligned}$

Since $\phi_{n}^{im}=e^{j\theta_{n}^{im}}$, $|\phi_{n}^{im}|^{2}=1$. Finally,
the optimization problem of $\mathbf{\Phi}^{im}$ when $\bm{\alpha}^{im}$ and
$\mathbf{W}^{im}$ are fixed can be expressed as

$\displaystyle\begin{aligned}
\max_{\bm{\phi}^{im}}\quad&f_{3c}(\phi_{n}^{im},\bm{\varepsilon}^{im^{op}})\\!=-B_{1}^{im}\\!+\\!2\Re\left\\{\phi_{n}^{im^{*}}B_{2}^{im}\right\\}\\!+\\!B_{3}^{im},\\\
s.t.\quad&\theta_{n}^{im}\in[0,2\pi)\quad n=1,2,...,N.\end{aligned}$ (71)

It can be seen from (71) that maximize
$f_{3c}(\phi_{n}^{im},\bm{\varepsilon}^{im^{op}})$ is equivalent to maximize
$2\Re\\{\phi_{n}^{im^{*}}B_{2}^{im}\\}$. Thus, the optimal solution of
$\theta_{n}^{im}$ can be obtained by setting $\theta_{n}^{im^{op}}=\angle
B_{2}^{im}$ as shown in (56). $\blacksquare$

## References

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# Charm Leptonic and semi-Leptonic decays at BESIII

(On behalf of the BESIII Collaboration)
Institute of High Energy Physics, CAS, Beijing, China, 100049
E-mail Work supported in part by the National Natural Science Foundation of
China (NSFC) under Contract No. 11675200

###### Abstract:

We present recent BESIII results about the study of mesons which contain at
least one charm quark. We determined the $D_{(s)}^{+}$ decay constants, the
form factors of $D$ semi-leptonic decays, the CKM matrix elements
$|V_{cs(d)}|$.

##  1 Introduction

The study of mesons which contain at least one charm quark is referred to as
open charm physics. It offers the possibility to study up-type quark
transitions. Since the $c$ quark cannot be treated in any mass limit,
theoretical predictions are difficult and experimental input is crucial. The
at-threshold decay topology offers special opportunities to study open charm
decays.

The BESIII is a magnetic spectrometer working at a double-ring $e^{+}e^{-}$
collider operating at center-of-mass energy between 2.0 GeV and 4.6 GeV,
located at the Institute of High Energy Physics (IHEP) in Beijing. The maximum
luminosity of the BEPCII at 3.773 GeV is $1\times 10^{33}\rm cm^{-2}\rm
s^{-1}$ [1]. The samples of interest for the analysis described in the
following were taken at the $D\bar{D}$ threshold (3.773 GeV) and
$D^{*}_{s}D_{s}$ threshold (4.178 GeV) with integrated luminosities of 2.92
fb-1 and 3.19 fb-1, respectively. Throughout in the following, charge
conjugate is implied.

##  2 Pure leptonic $D_{(s)}$ decays

The pure leptonic decay of charged $D_{(s)}^{+}$ mesons proceeds via the
annihilation of the $c$ quark and the anti-$d$ (anti-$s$) quark to a virtual
$W^{\pm}$ boson and its decay to $\ell^{+}\nu_{\ell}$. The decay rate can be
parameterized as:

$\Gamma(D_{(s)}^{+}\to\ell^{+}\nu_{\ell})=\frac{G_{F}^{2}}{8\pi}f^{2}_{D_{(s)}}m^{2}_{\ell}m_{D_{(s)}}(1-\frac{m^{2}_{\ell}}{m^{2}_{D_{(s)}}})^{2}|V_{cd(s)}|^{2},$
(1)

where $G_{F}$ is the Fermi coupling constant, $m_{\ell}$ and $m_{D_{(s)}}$ are
the masses of the lepton and the $D_{(s)}$ meson in the final state,
respectively.

###  2.1 $D^{+}\to\mu^{+}\nu_{\mu}$

By analyzing the data sample accumulated at 3.773 GeV, we first selected the
$D^{-}$ meson sample from its hadronic decay modes, which is called singly
tagged (ST) $D^{-}$. The $D^{+}\to\mu^{+}\nu_{\mu}$ decays are selected in the
recoil side of the ST $D^{-}$. To select the leptonic $D^{+}$ decay with a
missing neutrino, we calculated $U_{\rm miss}=E_{\rm miss}-p_{\rm miss}$ or
$M_{\rm miss}^{2}=E_{\rm miss}^{2}-p_{\rm miss}^{2}$, where $E_{\rm miss}$ and
$p_{\rm miss}$ are the missing energy and missing momentum of the event.
Figure 2 shows the distribution of $M_{\rm miss}^{2}$. After subtracting the
backgrounds, $409.0\pm 21.2$ signal events are retained and the branching
fraction (BF) is measured to be $BF(D^{+}\to\mu^{+}\nu_{\mu})=(3.71\pm
0.19_{stat}\pm 0.06_{syst})\times 10^{-4}$, which is the most precise
measurement to date. Combining this BF measurement and the Particle Data Group
(PDG) [2] values of $D^{+}$ lifetime, $m_{D^{+}}$, $m_{\mu}$ and magnitude of
$|V_{cd}|$ determined from the global Standard Model (SM) fit, the decay
constant is determined to be $f_{D}=(203.2\pm 5.3_{stat}\pm 1.8_{syst})$ MeV.
Alternatively, the magnitude of CKM matrix element —Vcd— is extracted to be
—Vcd— = $0.2210\pm 0.0058_{stat}\pm 0.0047_{syst}$ [3].

Figure 1:  The $M_{\rm miss}^{2}$ distributions of the
$D^{+}\to\mu^{+}\nu_{\mu}$ candidates, where two arrows denote the signal
region.

Figure 2:  Fit of the accepted the $D_{s}^{+}\to\mu^{+}\nu_{\mu}$ candidates
events (BESIII preliminary result).

###  2.2 $D_{s}^{+}\to\mu^{+}\nu_{\mu}$

By analyzing the data taken at 4.178 GeV, we also studied the leptonic
$D^{+}_{s}$ decays. From 14 $D^{-}_{s}$ hadronic decay modes, preliminarily,
$388660\pm 2592$ ST $D_{s}^{-}$ mesons were accumulated. In the system
recoiling against the ST $D_{s}^{-}$, the signal events of
$D_{s}^{+}\to\mu^{+}\nu_{\mu}$ decay were selected. The distribution of
$M_{\rm miss}^{2}$ is shown in Fig. 2. We obtained $1135.9\pm 33.1$
$D_{s}^{+}\to\mu^{+}\nu_{\mu}$, and the preliminary result of absolute BF is
determined to be $BF(D_{s}^{+}\to\mu^{+}\nu_{\mu})=(0.550\pm 0.016_{stat}\pm
0.015_{syst})$%, the decay constant $f_{D_{s}}=245.9\pm 3.6_{stat}\pm
3.5_{syst}$ MeV and $|V_{cs}|=0.987\pm 0.014_{stat}\pm 0.015_{syst}$. The
additional systematic uncertainties according to the input parameters are
negligible for —Vcs— and 0.3 for $f_{D_{s}}$.

##  3 Semi-leptonic $D_{(s)}$ decays

###  3.1 $D^{0(+)}\to Pe^{+}\nu_{e}$ ($P=K^{-},\pi^{-},K^{0},\pi^{0}$)

In the SM, neglecting the lepton mass, the differential decay rate for
$D^{0(+)}\to Pe^{+}\nu_{e}$ ($P=K^{-},\pi^{-},K^{0}$ or $\pi^{0}$) is given by

$\frac{d\Gamma}{dq^{2}}=X\frac{G_{F}^{2}}{24\pi^{3}}|V_{\rm
cd(s)}|^{2}p^{3}|f^{P}_{+}(q^{2})|^{2},$ (2)

where $X$ is a multiplicative factor due to isospin, which equals to 1 for
modes $D^{0}\to K^{-}e^{+}\nu_{e}$, $D^{0}\to\pi^{-}e^{+}\nu_{e}$, $D^{+}\to
K^{0}e^{+}\nu_{e}$, and 1/2 for mode $D^{+}\to\pi^{0}e^{+}\nu_{e}$, $p$ is the
momentum of the pseudo-scalar meson $P$ in the rest frame of the $D$ meson,
$q^{2}$ is the squared four momentum transfer, i.e., the invariant mass of the
electron and neutrino system, $f^{P}_{+}(q^{2})$ is the form factor which
describes the strong interaction between the final state quarks and is usually
parameterized in data analysis.

Based on the data taken at 3.773 GeV, BESIII studies the dynamics of the
$D^{0}\to K^{-}e^{+}\nu_{e}$ and $D^{0}\to\pi^{-}e^{+}\nu_{e}$ decays. The BFs
are measured to be $BF(D^{0}\to K^{-}e^{+}\nu_{e})=(3.505\pm 0.014_{stat}\pm
0.033_{syst})$% and $BF(D^{0}\to\pi^{-}e^{+}\nu_{e})=(0.295\pm 0.004_{stat}\pm
0.003_{syst})$% [5]. Similarly, the BFs are measured to be
$BF(D^{+}\to\bar{K}^{0}e^{+}\nu_{e})=(8.604\pm 0.056_{stat}\pm 0.151_{syst})$%
and $BF(D^{+}\to\pi^{0}e^{+}\nu_{e})=(0.363\pm 0.008_{stat}\pm 0.005_{syst})$%
[6].

We also studied the differential decay rates of these two processes. Figures 3
and 4 show the fit results. We extract the product $f_{+}(0)|V_{cs(d)}|$ and
other form factor parameters. Using the values for
$f_{+}^{K(\pi)}(0)|V_{cs(d)}|$ from the two-parameter $z$-series expansion
fits and with $f^{K}_{+}(0)=0.747\pm 0.011\pm 0.015$ [7] and
$f^{\pi}_{+}(0)=0.666\pm 0.020\pm 0.021$ [8] calculated in LQCD, $|V_{cs}|$ is
obtained to be $0.9601\pm 0.0033\pm 0.0047\pm 0.0239$ ($|V_{cd}|=0.2155\pm
0.0027\pm 0.0014\pm 0.0094$), where the first uncertainties are statistical,
the second ones systematic, and the third ones are due to the theoretical
uncertainties in the form factor calculations. The BESIII results are in good
agreement with the previous measurements, and with the best precision to date.

Figure 3:  Fits to partial decay widths of $D^{0}\to K^{-}e^{+}\nu_{e}$ (left)
and $D^{0}\to\pi^{-}e^{+}\nu_{e}$ (right).  Figure 4:  Fits to partial decay
widths of $D^{+}\to\bar{K}^{0}e^{+}\nu_{e}$ (left) and
$D^{+}\to\pi^{0}e^{+}\nu_{e}$ (right).

###  3.2 $D^{0(+)}\to P\mu^{+}\nu_{\mu}$ ($P=K^{-},\pi^{-},K^{0},\pi^{0}$)

Based on the data sample collected at $\sqrt{s}=3.773$ GeV, BESIII although
studied $D^{0(+)}\to P\mu^{+}\nu_{\mu}$ ($P=K^{-},\pi^{-},K^{0},\pi^{0}$). The
BFs and the lepton universality ratios $R_{LU}=BF(D^{0(+)}\to
P\mu^{+}\nu_{\mu})/BF(D^{0(+)}\to Pe^{+}\nu_{e})$ are measured to be
$BF(D^{0}\to K^{-}\mu^{+}\nu_{\mu})=(3.429\pm 0.019_{stat}\pm 0.035_{syst})$%
($R_{LU}=0.978\pm 0.007_{stat}\pm 0.012_{syst}$) [9],
$BF(D^{+}\to\bar{K}^{0}\mu^{+}\nu_{\mu})=(8.72\pm 0.07_{stat}\pm
0.18_{syst})$% ($R_{LU}=0.988\pm 0.033$) [10],
$BF(D^{0}\to\pi^{-}\mu^{+}\nu_{\mu})=(0.267\pm 0.007_{stat}\pm 0.007_{syst})$%
($R_{LU}=0.905\pm 0.027_{stat}\pm 0.023_{syst}$) and
$BF(D^{+}\to\pi^{0}\mu^{+}\nu_{\mu})=(0.342\pm 0.011_{stat}\pm 0.010_{syst})$%
($R_{LU}=0.942\pm 0.037_{stat}\pm 0.027_{syst}$) [11].

###  3.3 $D_{s}^{+}\to K^{(*)0}e^{+}\nu_{e}$

Using the data sample collected at $\sqrt{s}=4.178$ GeV, BESIII measured
$D_{s}^{+}\to K^{0}e^{+}\nu_{e}$ and $D_{s}^{+}\to K^{*0}e^{+}\nu_{e}$. The
BFs are obtained to be $BF(D_{s}^{+}\to K^{0}e^{+}\nu_{e})=(3.25\pm
0.38_{stat}\pm 0.16_{syst})$% and $BF(D_{s}^{+}\to
K^{*0}e^{+}\nu_{e})=(2.37\pm 0.26_{stat}\pm 0.20_{syst})$% [12]. The first
measurements of the hadronic form-factor parameters are obtained. The result
for $D_{s}^{+}\to K^{0}e^{+}\nu_{e}$ is $f_{+}(0)=0.720\pm 0.084(stat)\pm
0.013(syst)$, and for $D_{s}^{+}\to K^{*0}e^{+}\nu_{e}$ decay, the form-factor
ratios are $r_{V}=V(0)/A_{1}(0)=1.67\pm 0.34(stat)\pm 0.016(syst)$ and
$r_{2}=A_{2}(0)/A_{1}(0)=0.77\pm 0.28(stat)\pm 0.07(syst)$.

###  3.4 $D^{+}_{s}\to\eta(\eta^{\prime})e^{+}\nu_{e}$

By analyzing the data taken at 4.178 GeV, BESIII measured the absolute BFs for
semi-leptonic $D^{+}_{s}\to\eta(\eta^{\prime})e^{+}\nu_{e}$ decays. The
preliminary results are $BF(D^{+}_{s}\to\eta e^{+}\nu_{e})=(2.32\pm
0.06_{stat}\pm 0.06_{syst})$% and
$BF(D^{+}_{s}\to\eta^{\prime}e^{+}\nu_{e})=(0.82\pm 0.07_{stat}\pm
0.03_{syst})$%, and combining the BFs of
$BF_{D^{+}\to\eta(\eta^{\prime})e^{+}\nu_{e}}$, the $\eta-\eta^{\prime}$
mixing angle is determined to be $\phi_{P}=(40.2\pm 1.4_{stat}\pm
0.5_{syst})^{\circ}$. From the first measurements of the dynamics of
$D_{s}^{+}\to\eta(^{\prime})e^{+}\nu_{e}$ decays, the products of the hadronic
form factors $f_{+}^{\eta(^{\prime})}$(0) and the CKM matrix element —Vcs— are
extracted with different form factor parameterizations. For the two parameter
series expansion, the preliminary results are $f_{\eta}$(0)—Vcs— = $0.446\pm
0.005_{stat}\pm 0.004_{syst}$ and $f_{+}^{\eta^{\prime}}$(0)—Vcs—=$0.477\pm
0.049_{stat}\pm 0.011_{syst}$. Taking —Vcs— from the CKMfitter as input, we
determine preliminary $f_{+}^{\eta}(0)=0.458\pm 0.005_{stat}\pm 0.004_{syst}$
and $f_{+}^{\eta^{\prime}}(0)=0.490\pm 0.050_{stat}\pm 0.011_{syst}$.
Alternatively, using the $f_{+}^{\eta(^{\prime})}$(0) calculated by light-cone
sum rules leads to —Vcs—=$1.032\pm 0.012_{stat}\pm 0.009_{syst}\pm
0.079_{theo}$ and $0.917\pm 0.094_{stat}\pm 0.021_{syst}\pm 0.155_{theo}$,
respectively.

##  4  Summary

We present a selection of recent BESIII charm results based on the data sets
collected by BESIII detector near the $D\bar{D}$ threshold (3.773 GeV),
$D^{*}_{s}D_{s}$ threshold (4.178 GeV) with integrated luminosities of 2.93
fb-1 and 3.19 fb-1, respectively. From the leptonic $D_{(s)}$ pure leptonic
and semi-leptonic decays we determined the most precise values for the decay
constant $f_{D^{+}_{(s)}}$, the hadronic form factors $f^{K(\pi)}_{+}(0)$, and
the form factor shape $f^{K(\pi)}_{+}(q^{2})$ which provide important test to
LQCD calculations, and CKM matrix unitary.

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