# Latency Minimization for Intelligent Reflecting Surface Aided Mobile Edge
Computing

Tong Bai, , Cunhua Pan, , Yansha Deng, ,
Maged Elkashlan, , Arumugam Nallanathan, and Lajos Hanzo T. Bai, C. Pan, M.
Elkashlan, and A. Nallanathan are with the School of Electronic Engineering
and Computer Science, Queen Mary University of London, London, E1 4NS, U.K.
(e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>a.nallanathan@qmul.ac.uk).Y. Deng is with the Department of Engineering,
King’s College London, London, WC2R 2LS, U.K. (e-mail:
yansha.deng@kcl.ac.uk).L. Hanzo is with the School of Electronics and Computer
Science, University of Southampton, Southampton, SO17 1BJ, U.K. (e-mail:
lh@ecs.soton.ac.uk).(Corresponding author: Cunhua Pan and Tong Bai)

###### Abstract

Computation off-loading in mobile edge computing (MEC) systems constitutes an
efficient paradigm of supporting resource-intensive applications on mobile
devices. However, the benefit of MEC cannot be fully exploited, when the
communications link used for off-loading computational tasks is hostile.
Fortunately, the propagation-induced impairments may be mitigated by
intelligent reflecting surfaces (IRS), which are capable of enhancing both the
spectral- and energy-efficiency. Specifically, an IRS comprises an IRS
controller and a large number of passive reflecting elements, each of which
may impose a phase shift on the incident signal, thus collaboratively
improving the propagation environment. In this paper, the beneficial role of
IRSs is investigated in MEC systems, where single-antenna devices may opt for
off-loading a fraction of their computational tasks to the edge computing node
via a multi-antenna access point with the aid of an IRS. Pertinent latency-
minimization problems are formulated for both single-device and multi-device
scenarios, subject to practical constraints imposed on both the edge computing
capability and the IRS phase shift design. To solve this problem, the block
coordinate descent (BCD) technique is invoked to decouple the original problem
into two subproblems, and then the computing and communications settings are
alternatively optimized using low-complexity iterative algorithms. It is
demonstrated that our IRS-aided MEC system is capable of significantly
outperforming the conventional MEC system operating without IRSs.
Quantitatively, about $20\leavevmode\nobreak\ \%$ computational latency
reduction is achieved over the conventional MEC system in a single cell of a
$300\leavevmode\nobreak\ \rm{m}$ radius and $5$ active devices, relying on a
$5$-antenna access point.

###### Index Terms:

Intelligent reflecting surface, mobile edge computing, latency minimization.

##  I Introduction

###  I-A Motivation and Scope

In the Internet-of-Things (IoT) era, myriads of machines and sensors are
envisioned to be connected [1]. However, since these devices typically have
limited computing capabilities, resource-intensive applications cannot be
readily supported by these devices due to their resultant excessive
computational latency. Aimed at tackling this issue, powerful computing nodes
can be deployed at the edge of the network (typically co-located with the
access points (APs)) [2]. As a benefit, the computational latency of these
resource-intensive applications can be reduced, by employing both local
computing on the devices and edge computing for processing these computational
tasks, provided that these tasks can be successfully off-loaded. This paradigm
is referred to as mobile edge computing (MEC) [3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14]. At the time of writing, the potential of this MEC paradigm has
not been fully exploited, predominantly because the computation off-loading
link is far from perfect. For example, the devices located at the cell edge
typically suffer from a low off-loading success rate, and/or their computation
off-loading may impose higher latency than computing their tasks locally.
Hence these devices have to rely on their own computing resources, which is
however often incapable of supporting resource-intensive applications.
Therefore, it is imperative to improve the performance of MEC systems from a
communications perspective.

The recent advances in programmable meta-materials [15] facilitate the
construction of intelligent reflecting surfaces (IRSs) [16] for enhancing both
the spectral- and energy-efficiency of wireless communications. Specifically,
an IRS is comprised of an IRS controller and a large number of passive
reflecting elements. Under the instructions of the IRS controller, each IRS
reflecting element is capable of adjusting both the amplitude and the phase of
the signals reflected, thus collaboratively modifying the signal propagation
environment. The gain attained by IRSs is based on the combination of both the
virtual array gain and the reflection-aided beamforming gain. To elaborate,
the virtual array gain can be achieved by combining both the direct and IRS-
reflected signals, while the reflection-aided beamforming gain is realized by
proactively controlling the phase shift induced by the IRS elements. By
beneficially combining these two types of gains, the IRS becomes capable of
boosting the devices’ off-loading success rate, hence improving the potential
of MEC systems. In this treatise, our attention is focused on investigating
the role of IRSs in MEC systems.

###  I-B Related Works

####  I-B1 Design of Mobile Edge Computing Systems

At the current state-of-the-art, MEC systems can be categorized into [5]:
single-user [6, 7, 8, 9] and multi-user systems [10, 11, 12, 13, 14]. Among
the design metrics of single-user MEC systems, the computation off-loading
strategy plays a crucial role. More explicitly, the binary off-loading
strategy of [6] was proposed to decide whether the task is executed locally at
the mobile device or remotely at the edge-cloud node. By contrast, Wang _et
al._ [7] conceived a partial off-loading scheme for data-partitioning oriented
applications, where a fraction of the data can be processed at the mobile
device, while the rest at the edge. However, in realistic multi-user systems,
inter-user interference is imposed both on the radio communications link and
on the computing node at the edge, which may erode the overall performance of
the MEC system. In order to cope with this hindrance, Sardellitti _et al._
[10] jointly optimized the transmit precoding matrices and the computational
resources allocated to each user in a multi-cell multi-user scenario, while
Sheng _et al._ [17] proposed an energy-efficient algorithm to optimize the
resource allocation of terminals, radio access networks, and edge servers in a
multi-carrier scenario. For the system where the devices have to make their
off-loading decisions locally, Chen _et al._ [11] provided a distributed joint
computation off-loading and channel selection policy relying on classic game
theory. Recently, a specific user association scheme was also developed for
multi-user systems served by multiple edge computing nodes [13], while a
mobility-aware dynamic service scheduling algorithm was proposed for MEC
systems [14]. Furthermore, Yang _et al._ [18] conceived an over-the-air
computation aided federated learning algorithm for reducing both the latency
and the power consumption, and for preserving the users’ privacy in MEC
systems. At the time of writing, the computation offloading issue of the
devices in the face of hostile communications environments has not been well
addressed. Against this background, in this paper the performance is improved
by invoking IRSs. Let us now continue by reviewing the relevant research
contributions on IRSs as follows.

####  I-B2 Intelligent Reflecting Surface Aided Wireless Networks

In order to explore the benefits of IRSs in wireless communications, extensive
research efforts have been invested into their ergodic capacity analysis [19],
channel estimation [20], and practical reflection phase shift modeling [21],
as well as into the associated phase shift design [22, 23, 24, 25, 26, 27, 28,
29]. Specifically, a joint design of the IRS phase shift and of the precoding
at the AP was proposed for minimizing the transmit power, while maintaining
the target receive signal-to-interference-plus-noise ratio (SINR) [22],
relying on the sophisticated techniques of the semidefinite relaxation and of
alternating optimization. These investigations were then extended to the more
practical discrete phase shift setting [23]. However, the excessive
computational complexity of the algorithm developed in [22] prohibits its
application in large-scale IRSs. In order to reduce the complexity, Guo _et
al._ [24] proposed three low-complexity algorithms, while Pan _et al._ [26]
provided a pair of majorization-minimization (MM) algorithms and complex
circle manifold methods for multi-cell scenarios. Furthermore, in order to
reduce the overhead during the IRS channel estimation, Yang _et al._ [29]
grouped the IRS elements, where each group shares the same phase shift
coefficient, and optimized the power allocation and phase shift in orthogonal
frequency division multiplexing (OFDM)-based wireless systems. Apart from the
conventional communications scenarios, the role of IRSs was also investigated
both in terms of improving physical-layer security [30, 31, 32, 33], and
simultaneous wireless information and power transfer (SWIPT) [34, 35], where
substantial gains were achieved. These impressive research contributions
inspired us to exploit the beneficial role of IRSs in MEC systems.

###  I-C Contributions and Organizations

Our main contributions are the employment of IRSs in MEC systems, and the
joint design of computing and communications for minimizing the computational
latency of IRS-aided MEC systems, detailed as follows.

  * •

_New IRS-aided MEC system design and latency minimization problem
formulation:_ In order to further exploit the potential of MEC systems, we
first propose IRS-aided MEC systems, for assisting the computational task off-
loading of mobile devices. A latency-minimization problem is formulated for
multi-device scenarios, which optimizes the computation off-loading volume,
the edge computing resource allocation, the multi-user detection (MUD) matrix,
and the IRS phase shift, subject to both the total edge computing capability
and to the IRS phase shift constraints. Owing to the coupling effect of
multiple optimization variables, the latency-minimization problem cannot be
solved directly. Hence, relying on the block coordinate descent (BCD)
technique, the original problem is decoupled into two subproblems for
alternatively optimizing computing and communications settings.

  * •

_Computing design:_ Given a fixed communications setting, we decouple the
computation off-loading volume and the edge computing resource allocation,
again using the BCD technique. Our analysis reveals that given a fixed edge
computing resource allocation, the optimal computation off-loading volume can
be determined by assuming the equivalence of the latency induced by local
computing and by edge computing. Given a fixed computation off-loading volume,
the subproblem is proved to be a convex problem, and the optimal edge
computing resource can be found by relying on the KKT conditions and on the
classic bisection search method.

  * •

_Communications design:_ Given a computing setting, the objective function
(OF) becomes available in a non-convex sum-of-ratios form, which cannot be
solved using the algorithms developed in [22, 24, 26, 35]. To tackle this
challenge, this problem is transformed to an equivalent parameterized form by
introducing auxiliary variables. Our analysis reveals that this equivalent
form can be decomposed into a series of tractable subproblems. Then, an
iterative algorithm is developed to find the solution. In each iteration, the
auxiliary variables are updated using the modified Newton’s method, while upon
reformulating this series of tractable subproblems by exploiting the
equivalence between the weighted sum-rate maximization problem and the
weighted mean square error (MSE) minimization problem, closed-form expressions
are provided for the MUD matrix and for the IRS phase shift, using the
weighted minimum MSE method and MM algorithm, respectively. Our analysis
reveals that the proposed algorithm exhibits a low complexity.

  * •

_Study of the single-device scenario:_ In order to complete the
investigations, the single-device scenario is also studied, where neither the
edge computing resource allocation nor the multi-user interference has to be
considered. A low-complexity iterative algorithm is proposed by simplifying
the algorithm developed in the aforementioned multi-device scenario.

  * •

_Numerical validations and evaluations:_ The numerical results verify the
convergence of the proposed algorithms, and quantify the performance of our
IRS-aided MEC system in terms of its latency in diverse simulation
environments.

The rest of the paper is organized as follows. In Section II, we establish the
system model and formulate the latency minimization problem. The solution of
this latency minimization problem is provided in Section III. In Section IV,
we investigate the solution of the special case, where a single device is
served by the MEC system. Our numerical results are discussed in Section V.
Finally, our conclusions are offered in Section VI.

_Notation:_ In this paper, scalars are denoted by italic letters. Boldface
lower- and upper-case letters denote vectors and matrices, respectively;
$\mathbb{C}^{M\times N}$ represents the space of $M\times N$ complex matrices;
$\boldsymbol{I}_{N}$ denotes an $N\times N$ identity matrix; $j$ denotes the
imaginary unit, i.e. $j^{2}=-1$. The maths operations used throughout the
paper are summarized in Table I.

Table I: Math operations Notation | Operation
---|---
${\boldsymbol{x}}^{T}$ |  the transpose of $\boldsymbol{x}$
${\boldsymbol{X}}^{T}$ |  the transpose of $\boldsymbol{X}$
${\boldsymbol{x}}^{*}$ |  the complex conjugate of $\boldsymbol{x}$
${\boldsymbol{X}}^{*}$ |  the complex conjugate of $\boldsymbol{X}$
${\boldsymbol{x}}^{H}$ |  the Hermitian transpose of $\boldsymbol{x}$
${\boldsymbol{X}}^{H}$ |  the Hermitian transpose of $\boldsymbol{X}$
${\boldsymbol{X}}^{-1}$ |  the inverse of $\boldsymbol{X}$
$\bigodot$ | the Hadamard product
$\Re\\{\cdot\\}$ | the real part of a complex number
$\text{arg}\\{\cdot\\}$ | the argument of a complex number
$|\cdot|$ | the absolute value of a scalar
$\|\cdot\|$ | the 2-norm of a vector
$\lfloor\cdot\rfloor$ | the floor of a scalar
$\lceil\cdot\rceil$ | the ceiling of a scalar
$\text{diag}(\boldsymbol{x})$ | the diagonal matrix where
the diagonal elements are $\boldsymbol{x}$
$\text{diag}(\boldsymbol{X})$ | the vector whose elements are
the diagonal elements of $\boldsymbol{X}$
$\mathcal{CN}(0,\sigma^{2})$ | Circularly Symmetric Complex Gaussian
associated with zero-mean and variance $\sigma^{2}$

##  II System Model and Problem Formulation

In this section, our system model is elaborated on, from both communications
and computing perspectives. Following this, a latency-minimization problem is
formulated for our IRS-aided MEC system, detailed as follows.

###  II-A Communications Model

As shown in Fig. 1, we consider an MEC system operating in a single-cell
scenario, where $K$ single-antenna devices may opt for off-loading a certain
fraction of or all of their computational tasks to an edge computing node via
an $M$-antenna AP through the wireless transmission link. The edge computing
node and the AP are assumed to be co-located and connected using high-
throughput low-latency optical fiber. Then, the latency imposed by the data
communication between the AP and the edge computing node is deemed to be
negligible. An IRS comprised of $N$ reflecting elements is placed in the cell
for assisting the devices’ computation off-loading. We assume that both the
antenna spacing at the AP and the element spacing of the IRS are high enough
so that the small-scale fading associated both with two different antennas and
with two different reflecting elements is independent, respectively.

Figure 1: Illustration of the system model, where a $N$-element intelligent
reflecting surface (IRS) assists the computation off-loading of $K$ single-
antenna devices to the edge computing node via the $M$-antenna access point.

The equivalent baseband channels spanning from the $k$-th device to the AP,
and from the $k$-th device to the IRS, as well as from the IRS to the AP are
denoted by $\boldsymbol{h}_{d,k}\in\mathbb{C}^{M\times 1}$,
$\boldsymbol{h}_{r,k}\in\mathbb{C}^{N\times 1}$, and
$\boldsymbol{G}\in\mathbb{C}^{M\times N}$, respectively. These channels are
assumed to be perfectly estimated111Naturally, this assumption is idealistic.
Hence the algorithm developed in this paper can be deemed to represent the
best-case bound for the latency performance of realistic scenarios. and quasi-
static, hence remaining near-constant when devices are scheduled for off-
loading their computational tasks. As for the IRS, we simply set the amplitude
reflection coefficient to $1$ for all reflection elements and denote the phase
shift coefficient vector by
$\boldsymbol{\theta}=[\theta_{1},\theta_{2},\ldots,\theta_{N}]^{T}$, where
$\theta_{n}\in[0,2\pi)$ for all $n\in\\{1,2,\ldots,N\\}$.222Due to the
associated hardware limitations, only a limited number of discrete phase
shifts can be provided for each IRS element in practice [23]. Our proposed
algorithms provide the best-case bound for the latency of realistic scenarios.
The phase-quantization effects are evaluated in Section V-A3. Then, we have
the reflection-coefficient matrix of the IRS
$\boldsymbol{\Theta}=\text{diag}\big{\\{}e^{j\theta_{1}},e^{j\theta_{2}},\ldots,e^{j\theta_{N}}\big{\\}}$,
where $j$ represents the imaginary unit. It is assumed that the IRS phase
shift setting is calculated at the AP in accordance with both the channel and
computing dynamics, which is then sent to the IRS controller along the
dedicated channel. The composite device-IRS-AP channel is modeled as a
concatenation of the device-IRS link, the IRS reflection characterized by its
phase shift, and the IRS-AP link.

Here, it is assumed that the computation off-loading of the $K$ devices takes
place over a given frequency band $B$ within the same time resource. Upon
denoting the off-loading power, and off-loading signal of the $K$ devices, as
well as the noise vector by $p_{t}$,
$\boldsymbol{s}=[s_{1},s_{2},\ldots,s_{K}]^{T}$, and
$\boldsymbol{n}=[n_{1},n_{2},\ldots,n_{M}]^{T}$, respectively, the signal
$\boldsymbol{y}\in\mathbb{C}^{M\times 1}$ received at the AP is readily
formulated as

$\displaystyle\boldsymbol{y}=\sqrt{p_{t}}\boldsymbol{H}\boldsymbol{s}+\boldsymbol{n}=\sqrt{p_{t}}\sum^{K}_{k=1}\big{(}\boldsymbol{h}_{d,k}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,k}\big{)}s_{k}+\boldsymbol{n},$
(1)

where we assume $n_{m}\sim\mathcal{CN}(0,\sigma^{2})$ for $m=1,2,\ldots,M$.
Furthermore, we define
$\boldsymbol{h}_{k}\triangleq\boldsymbol{h}_{d,k}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,k}$
and
$\boldsymbol{H}\triangleq\big{[}\boldsymbol{h}_{1},\boldsymbol{h}_{2},\ldots,\boldsymbol{h}_{K}\big{]}$.
As a computational complexity compromise at the AP, a linear MUD technique is
invoked. Upon denoting the MUD matrix by $\boldsymbol{W}\in\mathbb{C}^{M\times
K}$, the signal recovered at the AP is obtained as

$\displaystyle\hat{\boldsymbol{s}}=\boldsymbol{W}^{H}\boldsymbol{y}=\boldsymbol{W}^{H}(\sqrt{p_{t}}\boldsymbol{H}\boldsymbol{s}+\boldsymbol{n}).$
(2)

As for the $k$-th device, its recovered signal is formulated as

$\displaystyle\hat{s}_{k}=\boldsymbol{w}_{k}^{H}\Bigg{[}\sqrt{p_{t}}\sum^{K}_{j=1}\big{(}\boldsymbol{h}_{d,j}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,j}\big{)}s_{j}+\boldsymbol{n}\Bigg{]},$
(3)

where $\boldsymbol{w}_{k}$ is the $k$-th column of the matrix
$\boldsymbol{W}$. Then, the SINR of the $k$-th device’s signal recovered is
given by

$\gamma_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta})=\frac{p_{t}\big{|}\boldsymbol{w}_{k}^{H}\big{(}\boldsymbol{h}_{d,k}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,k}\big{)}\big{|}^{2}}{p_{t}\sum^{K}_{j=1,j\neq
k}\big{|}\boldsymbol{w}_{k}^{H}\big{(}\boldsymbol{h}_{d,j}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,j}\big{)}\big{|}^{2}+\sigma^{2}|\boldsymbol{w}_{k}^{H}|^{2}}.$
(4)

Accordingly, upon assuming a perfect capacity-achieving transmission scheme is
invoked, we arrive at the maximum achievable computation off-loading rate of
the $k$-th device, formulated as

$R_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta})=B\log_{2}\big{[}1+\gamma_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta})\big{]}.$
(5)

###  II-B Computing Model

We consider the data-partitioning based application of [7], where a fraction
of the data can be processed locally, while the other part can be off-loaded
to the edge node. The computing model is detailed for the local and edge
computing as follows.

  * •

_Local computing:_ For the $k$-th device, $L_{k}$, $\ell_{k}$, and $c_{k}$ are
used to represent its total number of bits to be processed, its computation
off-loading volume in terms of the number of bits, and the number of CPU
cycles required to process a single bit, respectively. As for the local
computing, upon denoting the computational capability at the $k$-th device in
terms of the number of CPU cycles per second by $f^{l}_{k}$, the time required
for carrying out the local computation is formulated as
$D^{l}_{k}(\ell_{k})=(L_{k}-\ell_{k})c_{k}/f^{l}_{k}$.

  * •

_Edge computing:_ we denote the maximum number of executable CPU cycles at the
edge and the computational capability allocated to the $k$-th device by
$f^{e}_{\text{total}}$ and $f^{e}_{k}$, respectively, which obey
$\sum^{K}_{k=1}f^{e}_{k}\leq f^{e}_{\text{total}}$. Here, it is assumed that
the edge computing for the $k$-th device only begins its operation, when all
its $\ell_{k}$ bits are completely off-loaded. In this case, the total latency
of edge computing is jointly constituted by the computation off-loading, and
by the edge computing, as well as by the end-to-end delay of sending the
computational result back. Given that the computation result is typically of a
small size [5], the feedback latency can be negligible, upon using the
technique of ultra-reliable low-latency communications [36]. Then, the total
latency imposed by the computation off-loading and the edge computing is given
by
$D^{e}_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta},\ell_{k},f^{e}_{k})=\ell_{k}/R_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta})+\ell_{k}c_{k}/f^{e}_{k}$.

To this end, the latency of the $k$-th device can be readily calculated by
selecting the maximum value between those imposed by the local and by the edge
computing, formulated as

$\displaystyle
D_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta},\ell_{k},f^{e}_{k})$
$\displaystyle=$
$\displaystyle\max\big{\\{}D^{l}_{k}(\ell_{k}),D^{e}_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta},\ell_{k},f^{e}_{k})\big{\\}}$
$\displaystyle=$
$\displaystyle\max\bigg{\\{}\frac{(L_{k}-\ell_{k})c_{k}}{f^{l}_{k}},\frac{\ell_{k}}{R_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta})}+\frac{\ell_{k}c_{k}}{f^{e}_{k}}\bigg{\\}}.$

###  II-C Problem Formulation

In this paper, we aim for minimizing the weighted computational latency of all
the devices, by jointly optimizing the computation off-loading volume
$\boldsymbol{\ell}=[\ell_{1},\ell_{2},\ldots,\ell_{K}]^{T}$, the edge
computing resources
$\boldsymbol{f}^{e}=[f^{e}_{1},f^{e}_{2},\ldots,f^{e}_{K}]^{T}$ allocated to
each device, the MUD matrix $\boldsymbol{W}$, and the IRS phase shift
$\boldsymbol{\theta}$. Specifically, the weighted delay minimization problem
is formulated as

$\displaystyle\mathcal{P}0\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\min}\limits_{\boldsymbol{W},\boldsymbol{\theta},\boldsymbol{\ell},\boldsymbol{f}^{e}}\sum^{K}_{k=1}\varpi_{k}D_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta},\ell_{k},f^{e}_{k})$
$\displaystyle\text{s.t.}\quad 0\leq\theta_{n}<2\pi,\quad n=1,2,\ldots,N,$
(7a) $\displaystyle\quad\quad\ell_{k}\in\\{0,1,\ldots,L_{k}\\},\quad
k=1,2,\ldots,K,$ (7b) $\displaystyle\quad\quad\sum^{K}_{k=1}f^{e}_{k}\leq
f^{e}_{\text{total}},$ (7c) $\displaystyle\quad\quad f^{e}_{k}\geq 0,\quad
k=1,2,\ldots,K.$ (7d)

where $\varpi_{k}$ represents the weight of the $k$-th device. (7a) specifies
the range of the phase shift of the IRS elements; (7b) indicates that the
computation off-loading volume should be an integer between $0$ and $L_{k}$
for the $k$-th device; Finally, (7c) and (7d) restrict the range of the edge
computing resources allocated to each device.

######  Remark 1.

In Problem $\mathcal{P}0$, we have a total of four optimization variables,
namely, the off-loading volume, edge computing resource allocation, MUD
matrix, and IRS phase shift. The optimization of the former two variables is
related to the computing setting, while the optimization of the other two
specifies the communications design. The difficulties of solving Problem
$\mathcal{P}0$ are owing to three aspects. The first one is the segmented form
of the OF. The second one is the coupling effect between the MUD matrix
$\boldsymbol{W}$ and the IRS phase shift vector $\boldsymbol{\theta}$. The
final one is that the OF is non-convex regarding the phase shift
$\boldsymbol{\theta}$. Hence, it is an open challenge to obtain a globally
optimal solution directly. In this paper, a locally optimal solution is
provided. Specifically, upon using the popular BCD technique for decoupling
the communications and computing designs, the segmented form of the OF can be
easily transformed to a more tractable form. Similarly, optimal solutions can
be provided for the MUD matrix $\boldsymbol{W}$ and for the IRS phase shift
vector $\boldsymbol{\theta}$, after they are decoupled using the BCD technique
in the communications design. To tackle the non-convexity regarding
$\boldsymbol{\theta}$, the Majorization-Minimization (MM) algorithm is
invoked, which is capable of iteratively approaching a locally optimal
solution at low complexity.

##  III Joint Optimization of Computing and Communications Setting

The joint optimization of computing and computations settings is realized
relying on the BCD technique. The pivotal idea of the BCD technique is to
optimize one of the variables while fixing the other variables in an
alternating manner, until the convergence of the OF is achieved. In the rest
of this section, the joint optimization of the off-loading volume and of the
edge computing resource allocation is presented while fixing the
communications setting, followed by the joint optimization of the MUD matrix
and of the IRS phase shift while fixing the computing setting. Our goal is the
joint optimization both of the communications and of the computing design.

###  III-A Joint Optimization of the Off-loading Volume and the Edge Computing
Resource Allocation While Fixing the Communications Settings

Given an MUD matrix $\boldsymbol{W}$ and an IRS phase shift vector
$\boldsymbol{\theta}$, Problem $\mathcal{P}0$ can be simplified to

$\displaystyle\mathcal{P}1\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\min}\limits_{\boldsymbol{\ell},\boldsymbol{f}^{e}}\sum^{K}_{k=1}\varpi_{k}D_{k}(\ell_{k},f^{e}_{k})$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P0_constraint2},\eqref{eqn:P0_constraint3},\eqref{eqn:P0_constraint4}.$
(8a)

The optimization of $\boldsymbol{\ell}$ and $\boldsymbol{f}^{e}$ can be
decoupled, relying on the aforementioned BCD technique, detailed as follows.

####  III-A1 Optimization of $\boldsymbol{\ell}$

The value of $\boldsymbol{\ell}$ can be optimized, with the aid of the
proposition below.

######  Proposition 1.

Given an MUD matrix $\boldsymbol{W}$, and an IRS phase shift coefficient
vector $\boldsymbol{\theta}$. as well as an edge computing resource allocation
vector $\boldsymbol{f}^{e}$, the optimal number of off-loaded bits is given by

$\displaystyle\ell^{*}_{k}=\mathop{\arg\min}\limits_{\hat{\ell}_{k}\in\big{\\{}\lfloor\hat{\ell}^{*}_{k}\rfloor,\lceil\hat{\ell}^{*}_{k}\rceil\big{\\}}}D_{k}(\hat{\ell}_{k}),$
(9)

where $\lfloor\cdot\rfloor$ and $\lceil\cdot\rceil$ represent the floor and
ceiling operations, respectively, and $\hat{\ell}^{*}_{k}$ is selected for
ensuring that the value of $D^{l}_{k}(\hat{\ell}_{k})$ becomes equivalent to
that of $D^{e}_{k}(\hat{\ell}_{k})$, i.e.

$\displaystyle\hat{\ell}^{*}_{k}=\frac{L_{k}c_{k}R_{k}f^{e}_{k}}{f^{e}_{k}f^{l}_{k}+c_{k}R_{k}\big{(}f^{e}_{k}+f^{l}_{k}\big{)}}.$
(10)

Proof:  See Appendix A. $\blacksquare$

Algorithm 1 Joint optimization of $\boldsymbol{\ell}$ and
$\boldsymbol{f}^{e}$, given $\boldsymbol{W}$ and $\boldsymbol{\theta}$

0: $\boldsymbol{h}_{r,k}$, $\boldsymbol{h}_{d,k}$, $\boldsymbol{G}$, $B$,
$p_{t}$, $\sigma^{2}$, $K$, $\varpi_{k}$, $L_{k}$, $c_{k}$, $f^{l}_{k}$,
$f^{e}_{\text{total}}$, $t^{\text{max}}_{1}$, $\epsilon$, $\boldsymbol{W}$,
$\boldsymbol{\theta}$, and ${\hat{\boldsymbol{f}}^{e}}$ satisfying (7c) and
(7d)

0: Optimal ${\boldsymbol{\ell}}^{*}$ and ${\boldsymbol{f}^{e}}^{*}$, given
$\boldsymbol{W}$ and $\boldsymbol{\theta}$ 1\. Initialization

initialize $t_{1}=0$, $\epsilon_{1}^{(0)}=1$,
${\boldsymbol{f}^{e}}^{(0)}\leftarrow\hat{\boldsymbol{f}}^{e}$

calculate $\boldsymbol{R}$ using (5) 2\. Joint optimization of
$\boldsymbol{\ell}$ and $\boldsymbol{f}^{e}$

while $\epsilon_{1}^{(t_{1})}>\epsilon$ $\&\&$ $t_{1}<t^{\text{max}}_{1}$ do

$\bullet$ calculate $\boldsymbol{\ell}^{(t_{1}+1)}$ using (10)

$\bullet$ calculate ${\boldsymbol{f}^{e}}^{(t_{1}+1)}$ and $\mu$ by using (16)
and the bisection search method, respectively

$\bullet$
$\epsilon_{1}^{(t_{1}+1)}=\frac{\big{|}\text{obj}\big{(}\boldsymbol{\ell}^{(t_{1}+1)},{\boldsymbol{f}^{e}}^{(t_{1}+1)}\big{)}-\text{obj}\big{(}\boldsymbol{\ell}^{(t_{1})},{\boldsymbol{f}^{e}}^{(t_{1})}\big{)}\big{|}}{\text{obj}\big{(}\boldsymbol{\ell}^{(t_{1}+1)},{\boldsymbol{f}^{e}}^{(t_{1}+1)}\big{)}}$

$\bullet$ $t_{1}\leftarrow t_{1}+1$

end while3. Output optimal ${\boldsymbol{\ell}}^{*}$ and
${\boldsymbol{f}^{e}}^{*}$

${\boldsymbol{\ell}}^{*}\leftarrow\boldsymbol{\ell}^{(t_{1})}$ and
${\boldsymbol{f}^{e}}^{*}\leftarrow{\boldsymbol{f}^{e}}^{(t_{1})}$

####  III-A2 Optimization of $\boldsymbol{f}^{e}$

Here, the edge computing resource allocation $\boldsymbol{f}^{e}$ is
optimized, while fixing the MUD matrix $\boldsymbol{W}$, the IRS phase shift
coefficient vector $\boldsymbol{\theta}$, and the off-loading volume
$\boldsymbol{\ell}$. Specifically, upon substituting (10) into the OF of
Problem $\mathcal{P}1$, the problem can be reformulated as:

$\displaystyle\mathcal{P}1\text{-}E\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\min}\limits_{\boldsymbol{f}^{e}}\sum^{K}_{k=1}\frac{\varpi_{k}(L_{k}c^{2}_{k}R_{k}+L_{k}c_{k}f^{e}_{k})}{f^{e}_{k}f^{l}_{k}+c_{k}R_{k}(f^{e}_{k}+f^{l}_{k})}$
$\displaystyle\text{s.t.}\quad\eqref{eqn:P0_constraint3},\eqref{eqn:P0_constraint4}.$
(11a)

Problem $\mathcal{P}1\text{-}E$ can be proved to be a convex optimization
problem following the proposition below.

######  Proposition 2.

Problem $\mathcal{P}1\text{-}E$ is a convex optimization problem.

Proof:  See Appendix B. $\blacksquare$

Since Problem $\mathcal{P}1\text{-}E$ is convex and the Slater’s condition
[37] is satisfied 333In words, the Slater’s condition for convex programming
states that strong duality holds if all constraints are satisfied and the
nonlinear constraints are satisfied with strict inequalities., the
Karush–Kuhn–Tucker (KKT) may be imposed on the problem for finding its optimal
solution. Specifically, the Lagrangian function associated with Problem
$\mathcal{P}1\text{-}E$ is given by

$\displaystyle\mathcal{L}(\boldsymbol{f}^{e},\mu,\boldsymbol{\nu})$
$\displaystyle=$
$\displaystyle\sum^{K}_{k=1}\frac{\varpi_{k}(L_{k}c^{2}_{k}R_{k}+L_{k}c_{k}f^{e}_{k})}{c_{k}R_{k}f^{l}_{k}+(f^{l}_{k}+c_{k}R_{k})f^{e}_{k}}$
(12)
$\displaystyle+\mu\bigg{(}\sum^{K}_{k=1}f^{e}_{k}-f^{e}_{\text{total}}\bigg{)},$

where the variable $\mu$ is the non-negative Lagrange multiplier, while the
optimal edge computing resource allocation vector ${\boldsymbol{f}^{e}}^{*}$
and the optimal Lagrange multiplier $\mu^{*}$ should satisfy the following KKT
conditions, for $k=1,2,\ldots,K$:

$\displaystyle\frac{\partial\mathcal{L}}{\partial
f^{e}_{k}}=\frac{-\varpi_{k}L_{k}c^{3}_{k}R^{2}_{k}}{\big{[}c_{k}R_{k}f^{l}_{k}+(f^{l}_{k}+c_{k}R_{k}){f^{e}_{k}}^{*}\big{]}^{2}}+\mu^{*}=0,$
(13)
$\displaystyle\mu^{*}\bigg{(}\sum^{K}_{k=1}{f^{e}_{k}}^{*}-f^{e}_{\text{total}}\bigg{)}=0,$
(14) $\displaystyle{f^{e}_{k}}^{*}\geq 0.$ (15)

The value of $f^{e}_{k}$ can be directly derived from (13) for a given $\mu$,
which is written as

$\displaystyle
f^{e}_{k}=\frac{\sqrt{\frac{\varpi_{k}L_{k}c^{3}_{k}R^{2}_{k}}{\mu}}-c_{k}R_{k}f^{l}_{k}}{f^{l}_{k}+c_{k}R_{k}},\quad
k=1,\ldots,K.$ (16)

In order to ensure $f^{e}_{k}\geq 0$ in (16), we have
$\sqrt{\frac{\varpi_{k}L_{k}c^{3}_{k}R^{2}_{k}}{\mu}}-c_{k}R_{k}f^{l}_{k}\geq
0$, which is reformulated as
$\mu\leq\frac{\varpi_{k}L_{k}c_{k}}{{f^{l}_{k}}^{2}}$. Given that $\mu\neq 0$
in (16), the optimal $\mu^{*}$ can be found in the range of
$(\mu_{l},\mu_{u}]=\Big{(}0,\mathop{\min}\limits_{k}\big{(}\frac{\varpi_{k}L_{k}c_{k}}{{f^{l}_{k}}^{2}}\big{)}\Big{]}$
to ensure (14), using the well-known bisection search method associated with
the termination coefficient of $\epsilon$, because $\sum^{K}_{k=1}{f^{e}_{k}}$
can be proved to be monotonically decreasing with respect to $\mu$. The
procedure of solving Problem $\mathcal{P}1$ is summarized in Algorithm 1. The
complexity of Algorithm 1 is dominated by calculating
${\boldsymbol{f}^{e}}^{(t_{1}+1)}$ using (16) and by calculating $\mu$ using
the bisection search method. Its complexity is on the order of
$\mathcal{O}\big{(}\log_{2}(\frac{\mu_{u}-\mu_{l}}{\epsilon})K\big{)}$. Thus
the total complexity of Algorithm 1 is
$\mathcal{O}\big{(}t^{\text{max}}_{1}\log_{2}(\frac{\mu_{u}-\mu_{l}}{\epsilon})K\big{)}$.

###  III-B Joint Optimization of the MUD Matrix and the IRS Phase Shift
Coefficient While Fixing the Computing Settings

Given an off-loading volume vector $\boldsymbol{\ell}$ and an edge computing
resource allocation vector $\boldsymbol{f}^{e}$, Problem $\mathcal{P}0$ is
reformulated as

$\displaystyle\mathcal{P}2\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\min}\limits_{\boldsymbol{W},\boldsymbol{\theta}}\sum^{K}_{k=1}\varpi_{k}D_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta})$
$\displaystyle\text{s.t.}\quad 0\leq\theta_{n}<2\pi,\quad n=1,2,\ldots,N.$
(17a)

######  Remark 2.

The challenges of solving Problem $\mathcal{P}2$ are due to two aspects. The
first one is the segmented form of
$D_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta})$ that is caused by the
operation $\max$ as detailed in (II-B), while the second issue is that the OF
is the summation of fractional functions, with respect to $\boldsymbol{W}$ and
$\boldsymbol{\theta}$ as shown in the OF of Problem $\mathcal{P}2\text{-}E1$
below, which makes the problem a non-convex sum-of-ratios optimization. In
order to tackle these two issues, we transform the problem as follows.

####  III-B1 Problem Transformation

As detailed in Proposition 1, the optimal solution of Problem $\mathcal{P}0$
results in $D_{k}=D^{l}_{k}=D^{e}_{k}$. Hence upon replacing $D_{k}$ by
$D^{e}_{k}$ and removing the constant terms, Problem $\mathcal{P}2$ is
reformulated as:

$\displaystyle\mathcal{P}2\text{-}E1\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\min}\limits_{\boldsymbol{W},\boldsymbol{\theta}}\sum^{K}_{k=1}\frac{\varpi_{k}\ell_{k}}{R_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta})}$
$\displaystyle\text{s.t.}\quad 0\leq\theta_{n}<2\pi,\quad n=1,2,\ldots,N.$
(18a)

It is then rewritten as the following equivalent form:

$\displaystyle\mathcal{P}2\text{-}E2\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\min}\limits_{\boldsymbol{W},\boldsymbol{\theta},\boldsymbol{\beta}}\sum^{K}_{k=1}\beta_{k}$
$\displaystyle\text{s.t.}\quad\frac{\varpi_{k}\ell_{k}}{R_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta})}\leq\beta_{k},\quad
k=1,2,\ldots,K,$ (19a) $\displaystyle\quad\quad 0\leq\theta_{n}<2\pi,\quad
n=1,2,\ldots,N.$ (19b)

The following proposition may assist us in solving Problem
$\mathcal{P}2\text{-}E2$.

Algorithm 2 Joint optimization of $\boldsymbol{W}$ and $\boldsymbol{\theta}$,
given $\boldsymbol{\ell}$ and $\boldsymbol{f}^{e}$

0: $\boldsymbol{h}_{r,k}$, $\boldsymbol{h}_{d,k}$, $\boldsymbol{G}$, $B$,
$p_{t}$, $\sigma^{2}$, $\varpi_{k}$, $\boldsymbol{\ell}$, and
$\boldsymbol{f}^{e}$

0: Optimal $\boldsymbol{W}^{*}$ and $\boldsymbol{\theta}^{*}$, given
$\boldsymbol{\ell}$ and $\boldsymbol{f}^{e}$ 1\. Initialization

initialize $t_{2}=0$, $\zeta\in(0,1)$, $\epsilon\in(0,1)$, and
$\boldsymbol{\theta}^{(0)}$ satisfying (7a)

calculate $\boldsymbol{W}^{(0)}$ and $\boldsymbol{R}^{(0)}$ using (31) and
(34), respectively

calculate $\boldsymbol{\lambda}^{(0)}$ and $\boldsymbol{\beta}^{(0)}$ using
(26) 2\. Joint optimization of $\boldsymbol{W}$, $\boldsymbol{\theta}$,
$\boldsymbol{\lambda}$ and $\boldsymbol{\beta}$

repeat

$\bullet$ update $\boldsymbol{W}^{(t_{2}+1)}$ and
$\boldsymbol{\theta}^{(t_{2}+1)}$ using Algorithm 3

$\bullet$ update $\boldsymbol{\lambda}^{(t_{2}+1)}$ and
$\boldsymbol{\beta}^{(t_{2}+1)}$ as follows
$\lambda_{k}^{(t_{2}+1)}=\lambda_{k}^{(t_{2})}-\frac{\zeta^{i^{(t_{2}+1)}}\chi_{k}\big{(}\lambda_{k}^{(t_{2})}\big{)}}{R_{k}\big{(}\boldsymbol{w}_{k}^{(t_{2}+1)},\boldsymbol{\theta}^{(t_{2}+1)}\big{)}},$
(20) and
$\beta^{(t_{2}+1)}=\beta^{(t_{2})}-\frac{\zeta^{i^{(t_{2}+1)}}\kappa_{k}\big{(}\beta_{k}^{(t_{2})}\big{)}}{R_{k}\big{(}\boldsymbol{w}_{k}^{(t_{2}+1)},\boldsymbol{\theta}^{(t_{2}+1)}\big{)}},$
(21) where $i^{(t_{2}+1)}$ is the smallest integer among
$i\in\\{1,2,3,\ldots\\}$ satisfying
$\displaystyle\sum^{K}_{k=1}\bigg{|}\chi_{k}\bigg{(}\lambda_{k}^{(t_{2})}-\frac{\zeta^{i}\chi_{k}(\lambda_{k}^{(t_{2})})}{R_{k}(\boldsymbol{w}_{k}^{(t_{2}+1)},\boldsymbol{\theta}^{(t_{2}+1)})}\bigg{)}\bigg{|}^{2}$
$\displaystyle\quad+\sum^{K}_{k=1}\bigg{|}\kappa_{k}\bigg{(}\beta^{(t_{2})}-\frac{\zeta^{i}\kappa_{k}(\beta^{(t_{2})})}{R_{k}(\boldsymbol{w}_{k}^{(t_{2}+1)},\boldsymbol{\theta}^{(t_{2}+1)})}\bigg{)}\bigg{|}^{2}\quad\quad$
$\displaystyle\quad\leq(1-\epsilon_{3}\zeta^{i})^{2}\sum^{K}_{k=1}\Big{[}\big{|}\chi_{k}\big{(}\lambda_{k}^{(t_{2})}\big{)}\big{|}^{2}+\big{|}\kappa_{k}\big{(}\beta^{(t_{2})}\big{)}\big{|}^{2}\Big{]}.$
(22)

$\bullet$ $t_{2}\leftarrow t_{2}+1$

until the following conditions are achieved
$\displaystyle\lambda_{k}^{(t_{2})}R_{k}(\boldsymbol{w}_{k}^{(t_{2})},\boldsymbol{\theta}^{(t_{2})})-1=0,$
(23)
$\displaystyle\beta_{k}R_{k}(\boldsymbol{w}_{k}^{(t_{2})},\boldsymbol{\theta}^{(t_{2})})-\varpi_{k}\ell_{k}=0$
(24) 3\. Output optimal ${\boldsymbol{W}}^{*}$ and ${\boldsymbol{\theta}}^{*}$

${\boldsymbol{W}}^{*}\leftarrow\boldsymbol{W}^{(t_{2})}$ and
${\boldsymbol{\theta}}^{*}\leftarrow{\boldsymbol{\theta}}^{(t_{2})}$

######  Proposition 3.

If $(\boldsymbol{W}^{*},\boldsymbol{\theta}^{*},\boldsymbol{\beta}^{*})$ is
the solution of Problem $\mathcal{P}2\text{-}E2$, a
$\boldsymbol{\lambda}^{*}=[\lambda_{1},\lambda_{2},\ldots,\lambda_{K}]$ exists
that $(\boldsymbol{W}^{*},\boldsymbol{\theta}^{*})$ satisfies the KKT
conditions of the following problem, when we set
$\boldsymbol{\beta}=\boldsymbol{\beta}^{*}$ and
$\boldsymbol{\lambda}=\boldsymbol{\lambda}^{*}$

$\displaystyle\mathcal{P}2\text{-}E3\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\min}\limits_{\boldsymbol{W},\boldsymbol{\theta}}\sum^{K}_{k=1}\lambda_{k}\big{[}\varpi_{k}\ell_{k}-\beta_{k}R_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta})\big{]}$
$\displaystyle\text{s.t.}\quad 0\leq\theta_{n}<2\pi,\quad n=1,2,\ldots,N.$
(25a)

Furthermore, $(\boldsymbol{W}^{*},\boldsymbol{\theta}^{*})$ also satisfies the
following equations, when we set $\boldsymbol{\beta}=\boldsymbol{\beta}^{*}$
and $\boldsymbol{\lambda}=\boldsymbol{\lambda}^{*}$

$\begin{cases}\lambda_{k}=\frac{1}{R_{k}(\boldsymbol{w}_{k}^{*},\boldsymbol{\theta}^{*})},\quad
k=1,2,\ldots,K,\\\
\beta_{k}=\frac{\varpi_{k}\ell_{k}}{R_{k}(\boldsymbol{w}_{k}^{*},\boldsymbol{\theta}^{*})},\quad
k=1,2,\ldots,K.\end{cases}$ (26)

Correspondingly, if $(\boldsymbol{W}^{*},\boldsymbol{\theta}^{*})$ is a
solution to Problem $\mathcal{P}2\text{-}E3$ and satisfies (26) when we set
$\boldsymbol{\beta}=\boldsymbol{\beta}^{*}$ and
$\boldsymbol{\lambda}=\boldsymbol{\lambda}^{*}$,
$(\boldsymbol{W}^{*},\boldsymbol{\theta}^{*},\boldsymbol{\beta}^{*})$ is the
solution of Problem $\mathcal{P}2\text{-}E2$ associated with the Lagrange
multiplier $\boldsymbol{\lambda}=\boldsymbol{\lambda}^{*}$.

Proof:  See Appendix C. $\blacksquare$

To this end, the sum-of-ratios form in Problem $\mathcal{P}2\text{-}E1$ has
been transformed to a parameterized subtractable form in Problem
$\mathcal{P}2\text{-}E3$, which can be solved in two steps [38, 39, 40]: the
first step is to obtain $\boldsymbol{W}^{*}$ and $\boldsymbol{\theta}^{*}$ by
solving Problem $\mathcal{P}2\text{-}E3$, given $\boldsymbol{\beta}$ and
$\boldsymbol{\lambda}$; the second step is to update $\boldsymbol{\beta}$ and
$\boldsymbol{\lambda}$ using the modified Newton’s method until the
convergence is achieved. The procedure is summarized in Algorithm 2, where we
have

$\displaystyle\chi_{k}(\lambda_{k})=\lambda_{k}R_{k}(\boldsymbol{w}_{k}^{*},\boldsymbol{\theta}^{*})-1,\quad
k=1,2,\ldots,K,$ (27)
$\displaystyle\kappa_{k}(\beta_{k})=\beta_{k}R_{k}(\boldsymbol{w}_{k}^{*},\boldsymbol{\theta}^{*})-\varpi_{k}\ell_{k},\quad
k=1,2,\ldots,K.$ (28)

The complexity of Algorithm 2 is analyzed at the end of Section III-B.

Let us now focus our attention on the first step of solving Problem
$\mathcal{P}2\text{-}E3$, i.e. optimizing $\boldsymbol{W}^{*}$ and
$\boldsymbol{\theta}^{*}$, given a set of $\boldsymbol{\beta}$ and
$\boldsymbol{\lambda}$ as well as an off-loading volume vector
$\boldsymbol{\ell}$. In this case, Problem $\mathcal{P}2\text{-}E3$ can be
simplified to
$\mathop{\max}\limits_{\boldsymbol{W},\boldsymbol{\theta}}\sum^{K}_{k=1}\lambda_{k}\beta_{k}R_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta})$
subject to (25a), which constitutes a weighted sum-rate maximization problem.
As revealed in [41], maximizing the weighted sum-rate can be accomplished via
weighted MSE minimization. The latter problem is easier to handle, because it
is convex regarding each optimization variable, while fixing others. As such,
we focus our attention on constructing the corresponding weighted MSE
minimization problem. Specifically, following Theorem I in [41], we introduce
an auxiliary weight variable $\Upsilon_{k}$ for the $k$-th device and
formulate the corresponding weighted MSE minimization problem as:

$\displaystyle\mathcal{P}2\text{-}E4\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\min}\limits_{\boldsymbol{W},\boldsymbol{\theta}}\sum^{K}_{k=1}\big{[}\Upsilon_{k}e_{k}(\boldsymbol{W},\boldsymbol{\theta})$
$\displaystyle\qquad-\lambda_{k}\beta_{k}\log_{2}(\lambda_{k}^{-1}\beta_{k}^{-1}\Upsilon_{k})-\lambda_{k}\beta_{k}\big{]}$
$\displaystyle\text{s.t.}\quad 0\leq\theta_{n}<2\pi,\quad\forall
n\in\\{1,2,\ldots,N\\},$ (29a)

where the mathematical expression of $\\{\Upsilon_{k}\\}$ is given in Section
III-B3 and $e_{k}$ represents the MSE of the $k$-th user, which is given by

$\displaystyle e_{k}(\boldsymbol{W},\boldsymbol{\theta})$
$\displaystyle\triangleq$
$\displaystyle\mathbb{E}\big{[}(\hat{s}_{k}-s_{k})(\hat{s}_{k}-s_{k})^{H}\big{]}$
(30) $\displaystyle=$
$\displaystyle\big{[}\sqrt{p_{t}}\boldsymbol{w}^{H}_{k}(\boldsymbol{h}_{d,k}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,k})-1\big{]}\times$
$\displaystyle\big{[}\sqrt{p_{t}}\boldsymbol{w}^{H}_{k}(\boldsymbol{h}_{d,k}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,k})-1\big{]}^{H}$
$\displaystyle+p_{t}\sum^{K}_{j\neq
k}\boldsymbol{w}_{k}^{H}(\boldsymbol{h}_{d,j}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,j})(\boldsymbol{h}_{d,j}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,j})^{H}\boldsymbol{w}_{k}$
$\displaystyle+\sigma^{2}\boldsymbol{w}^{H}_{k}\boldsymbol{w}_{k}.$

As such, compared to Problem $\mathcal{P}2\text{-}E3$, Problem
$\mathcal{P}2\text{-}E4$ becomes more tractable, because given an IRS phase
shift coefficient vector, the OF of Problem $\mathcal{P}2\text{-}E4$ is convex
regarding an optimization variable, while fixing the other one. Again, the BCD
technique is invoked for solving this problem as follows.

####  III-B2 MUD Matrix Design

In Problem $\mathcal{P}2\text{-}E4$, fixing the phase shift coefficient vector
$\boldsymbol{\theta}$ and the auxiliary variable $\Upsilon_{k}$, the MUD
vector can be obtained by forcing the first-order derivative of the OF with
respect to $\boldsymbol{w}_{k}$ as $0$. After several steps of mathematical
manipulations, it is readily observed that the above minimization is
equivalent to minimizing the weighted MSE. Then, the MUD vector is given by
[42, Sec. 6.2.3]

$\displaystyle\boldsymbol{w}_{k}=\sqrt{p_{t}}\boldsymbol{J}^{-1}(\boldsymbol{h}_{d,k}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,k}),$
(31)

where
$\boldsymbol{J}=p_{t}\sum^{K}_{j=1}(\boldsymbol{h}_{d,j}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,j})(\boldsymbol{h}_{d,j}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,j})^{H}+\sigma^{2}\boldsymbol{I}_{M}$.

####  III-B3 Auxiliary Variable Design

Fixing $\boldsymbol{\theta}$ and $\boldsymbol{w}_{k}$, the optimal auxiliary
variable can be obtained by minimizing the OF of Problem
$\mathcal{P}2\text{-}E4$ with respect to $\Upsilon_{k}$, given by

$\displaystyle\Upsilon_{k}=\lambda_{k}\beta_{k}(e_{k})^{-1}.$ (32)

Furthermore, substituting (31) into (30), the MSE becomes

$\displaystyle
e^{\text{MMSE}}_{k}=1-p_{t}(\boldsymbol{h}_{d,k}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,k})^{H}\boldsymbol{J}^{-1}(\boldsymbol{h}_{d,k}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,k}).$

Bearing in mind that the relationship between the SINR and the MSE of the
system equipped with the minimum mean square error (MMSE) MUD is given by
$\gamma_{k}=(e^{\text{MMSE}}_{k})^{-1}-1$ [42, Sec. 6.2.3], (5) may be
reformulated as

$\displaystyle R_{k}=-B\log_{2}\big{(}e^{\text{MMSE}}_{k}\big{)}.$ (34)

####  III-B4 IRS Phase Shift Coefficient Design

In this subsection, we focus our attention on optimizing the reflection phase
shift coefficients $\boldsymbol{\theta}$, while fixing the auxiliary variable
$\Upsilon_{k}$ and the MUD matrix $\boldsymbol{W}$. Specifically, by
substituting (30) into the OF of Problem $\mathcal{P}2\text{-}E4$ and removing
the terms that are independent of the phase shift coefficient vector
$\boldsymbol{\theta}$, Problem $\mathcal{P}2\text{-}E4$ is reformulated as:

$\displaystyle\mathcal{P}2\text{-}E5\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\min}\limits_{\boldsymbol{\theta}}\sum^{K}_{k=1}\sum^{K}_{j=1}\Upsilon_{k}p_{t}\boldsymbol{w}^{H}_{k}\boldsymbol{h}_{j}\boldsymbol{h}_{j}^{H}\boldsymbol{w}_{k}$
$\displaystyle\quad-\sum^{K}_{k=1}\Upsilon_{k}\sqrt{p_{t}}\boldsymbol{h}_{k}^{H}\boldsymbol{w}_{k}-\sum^{K}_{k=1}\Upsilon_{k}\sqrt{p_{t}}\boldsymbol{w}_{k}^{H}\boldsymbol{h}_{k}$
$\displaystyle\text{s.t.}\quad 0\leq\theta_{n}\leq 2\pi,\quad\forall
n\in\\{1,2,\ldots,N\\},$ (35)

where the first and the second terms in the OF can be respectively formulated
in expansion forms as

$\displaystyle\quad\sum^{K}_{k=1}\sum^{K}_{j=1}\Upsilon_{k}p_{t}\boldsymbol{w}^{H}_{k}\boldsymbol{h}_{j}\boldsymbol{h}_{j}^{H}\boldsymbol{w}_{k}$
$\displaystyle=\sum^{K}_{k=1}\sum^{K}_{j=1}\big{(}\Upsilon_{k}p_{t}\boldsymbol{w}_{k}^{H}\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,j}\boldsymbol{h}_{r,j}^{H}\boldsymbol{\Theta}^{H}\boldsymbol{G}^{H}\boldsymbol{w}_{k}$
$\displaystyle\quad+\Upsilon_{k}p_{t}\boldsymbol{w}_{k}^{H}\boldsymbol{h}_{d,j}\boldsymbol{h}_{r,j}^{H}\boldsymbol{\Theta}^{H}\boldsymbol{G}^{H}\boldsymbol{w}_{k}$
$\displaystyle\quad+\Upsilon_{k}p_{t}\boldsymbol{w}_{k}^{H}\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,j}\boldsymbol{h}_{d,j}^{H}\boldsymbol{w}_{k}+\Upsilon_{k}p_{t}\boldsymbol{w}_{k}^{H}\boldsymbol{h}_{d,j}\boldsymbol{h}_{d,j}^{H}\boldsymbol{w}_{k}\big{)},$
(36)

and

$\displaystyle\quad\sum^{K}_{k=1}\Upsilon_{k}\sqrt{p_{t}}\boldsymbol{h}_{k}^{H}\boldsymbol{w}_{k}$
$\displaystyle=\sum^{K}_{k=1}\big{(}\Upsilon_{k}\sqrt{p_{t}}\boldsymbol{h}_{d,k}^{H}\boldsymbol{w}_{k}+\Upsilon_{k}\sqrt{p_{t}}\boldsymbol{h}_{r,k}^{H}\boldsymbol{\Theta}^{H}\boldsymbol{G}^{H}\boldsymbol{w}_{k}\big{)}.$
(37)

Upon defining
$\boldsymbol{A}\triangleq\sum^{K}_{k=1}\Upsilon_{k}p_{t}\boldsymbol{G}^{H}\boldsymbol{w}_{k}\boldsymbol{w}_{k}^{H}\boldsymbol{G}$,
$\boldsymbol{B}\triangleq\sum^{K}_{j=1}\boldsymbol{h}_{r,j}\boldsymbol{h}_{r,j}^{H}$,
$\boldsymbol{C}\triangleq\sum^{K}_{k=1}\sum^{K}_{j=1}\Upsilon_{k}p_{t}\boldsymbol{h}_{r,j}\boldsymbol{h}_{d,j}^{H}\boldsymbol{w}_{k}\boldsymbol{w}_{k}^{H}\boldsymbol{G}$,
and
$\boldsymbol{D}\triangleq\sum^{K}_{k=1}\Upsilon_{k}\sqrt{p_{t}}\boldsymbol{h}_{r,k}\boldsymbol{w}_{k}^{H}\boldsymbol{G}$,
Problem $\mathcal{P}2\text{-}E5$ may be rewritten as:

$\displaystyle\mathcal{P}2\text{-}E6\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\min}\limits_{\boldsymbol{\theta}}\text{tr}(\boldsymbol{\Theta}^{H}\boldsymbol{A}\boldsymbol{\Theta}\boldsymbol{B})+\text{tr}\big{[}\boldsymbol{\Theta}^{H}(\boldsymbol{C}-\boldsymbol{D})^{H}\big{]}$
$\displaystyle\quad+\text{tr}\big{[}\boldsymbol{\Theta}(\boldsymbol{C}-\boldsymbol{D})\big{]}$
$\displaystyle\text{s.t.}\quad 0\leq\theta_{n}\leq 2\pi,\quad\forall
n\in\\{1,2,\ldots,N\\}.$ (38)

Defining $\boldsymbol{\phi}\triangleq[\phi_{1},\ldots,\phi_{N}]^{T}$ where
$\phi_{n}=e^{j\theta_{n}}$, and
$\boldsymbol{v}=\big{[}[\boldsymbol{C}-\boldsymbol{D}]_{1,1},\ldots,[\boldsymbol{C}-\boldsymbol{D}]_{N,N}\big{]}^{T}$,
we have

$\displaystyle\text{tr}(\boldsymbol{\Theta}^{H}\boldsymbol{A}\boldsymbol{\Theta}\boldsymbol{B})=\boldsymbol{\phi}^{H}(\boldsymbol{A}\odot\boldsymbol{B})\boldsymbol{\phi},$
(39)

where $\odot$ represents the Hadamard product, and

$\displaystyle\text{tr}\big{[}\boldsymbol{\Theta}^{H}(\boldsymbol{C}-\boldsymbol{D})^{H}\big{]}=\boldsymbol{v}^{H}\boldsymbol{\phi}^{*},\text{
}\text{tr}\big{[}\boldsymbol{\Theta}(\boldsymbol{C}-\boldsymbol{D})\big{]}=\boldsymbol{\phi}^{T}\boldsymbol{v}.$
(40)

Further defining
$\boldsymbol{\Psi}\triangleq\boldsymbol{A}\odot\boldsymbol{B}$, we may
equivalently rewrite Problem $\mathcal{P}2\text{-}E6$ as:

$\displaystyle\mathcal{P}2\text{-}E7\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\min}\limits_{\boldsymbol{\phi}}f(\boldsymbol{\phi})=\boldsymbol{\phi}^{H}\boldsymbol{\Psi}\boldsymbol{\phi}+2\Re\big{\\{}\boldsymbol{\phi}^{H}\boldsymbol{v}^{*}\big{\\}}$
$\displaystyle\text{s.t.}\quad|\phi_{n}|=1,\quad\forall
n\in\\{1,2,\ldots,N\\}.$ (41)

Problem $\mathcal{P}2\text{-}E7$ is a non-convex one because of the unit
modulus constraint on $\phi_{n}$. In the following, the MM algorithm [43] is
invoked for solving this problem, which has two steps. In the majorization
step, we construct a continuous surrogate function
$g(\boldsymbol{\phi}|\boldsymbol{\phi}^{t})$, which represents the upperbound
of $f(\boldsymbol{\phi})$. Then in the minimization step, $\boldsymbol{\phi}$
is updated by
$\boldsymbol{\phi}^{t+1}\in\mathop{\arg\min}\limits_{\boldsymbol{\phi}}g(\boldsymbol{\phi}|\boldsymbol{\phi}^{t})$.
As such, we may initialize $\boldsymbol{\phi}^{0}$ that satisfies the
constraint (41), and then use the MM algorithm to generate a sequence of
feasible vectors $\\{\boldsymbol{\phi}^{t}\\}$, where $t$ refers to the
iteration index. Now the surrogate function is constructed with the aid of the
proposition below.

Algorithm 3 Joint optimization of $\boldsymbol{W}$ and $\boldsymbol{\theta}$,
given $\boldsymbol{\lambda}$ and $\boldsymbol{\beta}$

0: $\boldsymbol{h}_{r,k}$, $\boldsymbol{h}_{d,k}$, $\boldsymbol{G}$, $B$,
$p_{t}$, $\sigma^{2}$, $\varpi_{k}$, $t^{\text{max}}_{2}$, $\epsilon$,
$\boldsymbol{\lambda}$ and $\boldsymbol{\beta}$

0: Optimal $\boldsymbol{W}^{*}$ and $\boldsymbol{\theta}^{*}$, given
$\boldsymbol{\lambda}$ and $\boldsymbol{\beta}$ 1\. Initialization

initialize $t_{3}=0$, $\epsilon_{3}^{(0)}=1$, $\boldsymbol{\theta}^{(0)}$
satisfying (7a) 2\. Joint optimization of $\boldsymbol{W}$ and
$\boldsymbol{\theta}$

while $\epsilon_{3}^{(t_{2})}>\epsilon$ $\&\&$ $t_{3}<t^{\text{max}}_{3}$ do

$\bullet$ calculate $\boldsymbol{W}^{(t_{3}+1)}$ using (31)

$\bullet$ calculate $\boldsymbol{\Upsilon}^{(t_{3}+1)}$ using (32)

$\bullet$ calculate ${\boldsymbol{\theta}}^{(t_{3}+1)}$ by solving Problem
$\mathcal{P}2\text{-}E6$ with the aid of the MM algorithm

$\bullet$
$\epsilon_{3}^{(t_{3}+1)}=\frac{\big{|}\text{obj}\big{(}\boldsymbol{W}^{(t_{3}+1)},{\boldsymbol{\theta}}^{(t_{3}+1)}\big{)}-\text{obj}\big{(}\boldsymbol{W}^{(t_{3})},{\boldsymbol{\theta}}^{(t_{3})}\big{)}\big{|}}{\text{obj}\big{(}\boldsymbol{W}^{(t_{3}+1)},{\boldsymbol{\theta}}^{(t_{3}+1)}\big{)}}$

$\bullet$ $t_{3}\leftarrow t_{3}+1$

end while3. Output optimal ${\boldsymbol{W}}^{*}$ and
${\boldsymbol{\theta}}^{*}$, given $\boldsymbol{\lambda}$ and
$\boldsymbol{\beta}$

${\boldsymbol{W}}^{*}\leftarrow\boldsymbol{W}^{(t_{3})}$ and
${\boldsymbol{\theta}}^{*}\leftarrow{\boldsymbol{\theta}}^{(t_{3})}$

######  Proposition 4.

Denoting the maximum eigenvalue of $\boldsymbol{\Psi}$ by
$\hat{\lambda}_{max}$ and given a solution $\boldsymbol{\phi}^{t}$ at the
$t$-th iteration, we have the inequality below

$\displaystyle f(\boldsymbol{\phi})$ $\displaystyle\leq$
$\displaystyle\boldsymbol{\phi}^{H}\hat{\lambda}_{max}\boldsymbol{I}_{N}\boldsymbol{\phi}-2\Re\big{\\{}\boldsymbol{\phi}^{H}(\hat{\lambda}_{max}\boldsymbol{I}_{N}-\boldsymbol{\Psi})\boldsymbol{\phi}^{t}\big{\\}}$
(42)
$\displaystyle+(\boldsymbol{\phi}^{t})^{H}(\hat{\lambda}_{max}\boldsymbol{I}_{N}-\boldsymbol{\Psi})\boldsymbol{\phi}^{t}+2\Re\big{\\{}\boldsymbol{\phi}^{H}\boldsymbol{v}^{*}\big{\\}}.$

Proof:  See [26, 44]. $\blacksquare$

Algorithm 4 Joint Optimization of $\boldsymbol{\ell}$, $\boldsymbol{f}^{e}$,
$\boldsymbol{W}$ and $\boldsymbol{\theta}$

0: $\boldsymbol{h}_{r,k}$, $\boldsymbol{h}_{d,k}$, $\boldsymbol{G}$, $B$,
$p_{t}$, $\sigma^{2}$, $\varpi_{k}$, $L_{k}$, $c_{k}$, $f^{e}_{\text{total}}$,
and $\epsilon$

0: Optimal $\boldsymbol{\ell}$, $\boldsymbol{f}^{e}$, $\boldsymbol{W}$ and
$\boldsymbol{\theta}$ 1\. Initialization

initialize $t_{4}=0$, $\epsilon_{4}^{(0)}=1$

initialize $\boldsymbol{\theta}^{(0)}$ satisfying (7a) and
${\boldsymbol{f}^{e}}^{(0)}$ satisfying (7c) and (7d)

calculate $\boldsymbol{W}^{(0)}$ using (31) 2\. Joint optimization of
$\boldsymbol{\ell}$ and $\boldsymbol{f}^{e}$, given $\boldsymbol{W}^{(t_{4})}$
and $\boldsymbol{\theta}^{(t_{4})}$

calculate $\boldsymbol{\ell}^{(t_{4}+1)}$ and
${\boldsymbol{f}^{e}}^{(t_{4}+1)}$ using Algorithm 1 3\. Joint optimization of
$\boldsymbol{W}$ and $\boldsymbol{\theta}$, given
$\boldsymbol{\ell}^{(t_{4}+1)}$ and ${\boldsymbol{f}^{e}}^{(t_{4}+1)}$

calculate $\boldsymbol{W}^{(t_{4}+1)}$ and ${\boldsymbol{\theta}}^{(t_{4}+1)}$
using Algorithm 2 4\. Convergence checking

$\epsilon_{4}^{(t_{4})}=\frac{\big{|}\text{obj}\big{(}\boldsymbol{\ell}^{(t_{4}+1)},{\boldsymbol{f}^{e}}^{(t_{4}+1)},\boldsymbol{W}^{(t_{4}+1)},{\boldsymbol{\theta}}^{(t_{4}+1)}\big{)}-\text{obj}\big{(}\boldsymbol{\ell}^{(t_{4})},{\boldsymbol{f}^{e}}^{(t_{4})},\boldsymbol{W}^{(t_{4})},{\boldsymbol{\theta}}^{(t_{4})}\big{)}\big{|}}{\text{obj}\big{(}\boldsymbol{\ell}^{(t_{4}+1)},{\boldsymbol{f}^{e}}^{(t_{4}+1)},\boldsymbol{W}^{(t_{4}+1)},{\boldsymbol{\theta}}^{(t_{4}+1)}\big{)}}$

if $\epsilon_{4}^{(t_{4})}>\epsilon$ $\&\&$ $t_{4}<t^{\text{max}}_{4}$ holds
then

$t_{4}=t_{4}+1$

Go to Step 2

else

integerize $\ell^{(t_{4}+1)}$ by (9)

Output the optimal $\boldsymbol{\ell}^{*}$, ${\boldsymbol{f}^{e}}^{*}$,
$\boldsymbol{W}^{*}$ and $\boldsymbol{\theta}^{*}$

end if

Here, the terms on the right side of (42) is defined by our surrogate function
$g(\boldsymbol{\phi}|\boldsymbol{\phi}^{t})$. Then, Problem
$\mathcal{P}2\text{-}E7$ at the $t$-th iteration is reformulated as

$\displaystyle\mathcal{P}2\text{-}E8\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\min}\limits_{\boldsymbol{\phi}}g(\boldsymbol{\phi}|\boldsymbol{\phi}^{t})$
$\displaystyle\text{s.t.}\quad|\phi_{n}|=1,\quad\forall
n\in\\{1,2,\ldots,N\\}.$ (43)

Since
$(\boldsymbol{\phi}^{t})^{H}(\hat{\lambda}_{max}\boldsymbol{I}_{N}-\boldsymbol{\Psi})\boldsymbol{\phi}^{t}$
is a constant for a given $\boldsymbol{\phi}^{t}$ and we have
$\boldsymbol{\phi}^{H}\hat{\lambda}_{max}\boldsymbol{I}_{N}\boldsymbol{\phi}=M\hat{\lambda}_{max}$,
Problem $\mathcal{P}2\text{-}E8$ can be equivalently written as

$\displaystyle\mathcal{P}2\text{-}E9\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\max}\limits_{\boldsymbol{\phi}}\Re\Big{\\{}\boldsymbol{\phi}^{H}\big{[}(\hat{\lambda}_{max}\boldsymbol{I}_{N}-\boldsymbol{\Psi})\boldsymbol{\phi}^{t}-\boldsymbol{v}^{*}\big{]}\Big{\\}}$
$\displaystyle\text{s.t.}\quad|\phi_{n}|=1,\quad\forall
n\in\\{1,2,\ldots,N\\}.$ (44)

Then, the optimal solution of Problem $\mathcal{P}2\text{-}E9$ is readily
given by

$\displaystyle\boldsymbol{\phi}^{t+1}=e^{j\arg\\{(\hat{\lambda}_{max}\boldsymbol{I}_{N}-\boldsymbol{\Psi})\boldsymbol{\phi}^{t}-\boldsymbol{v}^{*}\\}}.$
(45)

Accordingly, the optimal solution to Problem $\mathcal{P}2\text{-}E6$ can be
obtained as

$\displaystyle\boldsymbol{\theta}^{t+1}=\arg\\{(\hat{\lambda}_{max}\boldsymbol{I}_{N}-\boldsymbol{\Psi})\boldsymbol{\phi}^{t}-\boldsymbol{v}^{*}\\}.$
(46)

The termination condition of the MM algorithm is given by
$\big{|}f(\boldsymbol{\phi}^{t+1})-f(\boldsymbol{\phi}^{t})\big{|}/f(\boldsymbol{\phi}^{t+1})\leq\epsilon$
or $t\geq t^{\text{max}}_{\text{MM}}$. The procedure of solving Problem
$\mathcal{P}2\text{-}E3$ is summarized in Algorithm 3.

The complexity of Algorithm 3 is dominated by its Step 2. Specifically, the
complexity of calculating $\boldsymbol{W}^{(t_{3}+1)}$ by (31) is on the order
of $\mathcal{O}\big{(}\max\\{KM^{3},KMN^{2}\\}\big{)}$; the complexity of
calculating $\boldsymbol{\Upsilon}^{(t_{3}+1)}$ by (32) is on the order of
$\mathcal{O}(K)$. With regard to the calculation of
${\boldsymbol{\theta}}^{(t_{3}+1)}$ using the MM algorithm, the complexity of
calculating the eigenvalue $\lambda_{max}$ of $\boldsymbol{\Psi}$ is on the
order of $\mathcal{O}(N^{3})$, while for each iteration of the MM algorithm,
the main complexity lies in the calculation of $\boldsymbol{\phi}^{t+1}$ in
(45), whose complexity is on the order of $\mathcal{O}(N^{2})$. Hence the
complexity of the MM algorithm is
$\mathcal{O}(N^{3}+t^{\text{max}}_{\text{MM}}N^{2})$. Summing these three
terms together, we obtain the total complexity of Algorithm 3 as
$\mathcal{O}\big{(}\max\\{N^{3}+t^{\text{max}}_{\text{MM}}N^{2},KM^{3},KMN^{2}\\}\big{)}$.
Finally, the complexity of Algorithm 2 is mainly dependent on updating
$\boldsymbol{W}^{(t_{2}+1)}$ and $\boldsymbol{\theta}^{(t_{2}+1)}$ using
Algorithm 3, because all other steps are given by explicit mathematical
expressions.

###  III-C Overall Algorithm to Solve Problem $\mathcal{P}0$

Based on the above discussions, we provide the detailed description of the BCD
algorithm used for solving Problem $\mathcal{P}0$ in Algorithm 4. Note that a
decreasing OF value of Problem $\mathcal{P}0$ is guaranteed in Step 2 and Step
3. Furthermore, the OF value has a lower bound due to the constraint on the
total edge computing resources. Hence, Algorithm 4 is guaranteed to converge.

The computational complexity of Algorithm 4 is mainly dependent on its Step 2
and Step 3, whose complexities have been analyzed in the above subsections.
Furthermore, the simulation results in Section V show that Algorithm 4
converges rapidly, which demonstrates the low complexity of our algorithms.

##  IV Specific Case Study: The Single-Device Scenario

In order to fully characterize the IRS-aided MEC system, a special case is
investigated in this section, where a single device is served by the MEC
system. The optimization problem of the single-device scenario becomes much
simpler for the following reasons. Firstly, the edge computing resource
allocation no longer has to be considered, because all the edge computing
resources can be assigned to this single device. Secondly, the sum-of-ratios
form in Problem $\mathcal{P}2\text{-}E1$ becomes a single-ratio form, which
implies that the optimization problem is more tractable. Thirdly, the multi-
user interference does not have to be considered, when the detection vector
and the IRS phase shift coefficient vector are optimized. The joint
optimization is detailed as follows.

Problem $\mathcal{P}0$ can be simplified for the single-device scenario as

$\displaystyle\mathcal{P}3\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\min}\limits_{\boldsymbol{w},\boldsymbol{\theta},\ell}D(\boldsymbol{w},\boldsymbol{\theta},\ell)$
$\displaystyle\text{s.t.}\quad 0\leq\theta_{n}<2\pi,\quad n=1,2,\ldots,N,$
(47a) $\displaystyle\quad\quad\ell\in\\{0,1,\ldots,L\\},$ (47b)

where the OF $D(\boldsymbol{w},\boldsymbol{\theta},\ell)$ becomes

$\displaystyle D(\boldsymbol{w},\boldsymbol{\theta},\ell)$ $\displaystyle=$
$\displaystyle\max\bigg{\\{}\frac{(L-\ell)c}{f^{l}},\frac{\ell}{R(\boldsymbol{w},\boldsymbol{\theta})}+\frac{\ell
c}{f^{e}_{\text{total}}}\bigg{\\}}.$ (48)

As illustrated in Proposition 1, for a given set of $\boldsymbol{w}$ and
$\boldsymbol{\theta}$, $D(\boldsymbol{w},\boldsymbol{\theta},\ell)$ achieves
its minimum value when $\ell$ is selected to ensure
$\frac{(L-\ell)c}{f^{l}}=\frac{\ell}{R(\boldsymbol{w},\boldsymbol{\theta})}+\frac{\ell
c}{f^{e}_{\text{total}}}$. Therefore, the optimal value of the relaxation of
$\ell$ is given by

$\displaystyle\hat{\ell}^{*}=\frac{LcRf^{e}_{\text{total}}}{f^{e}_{\text{total}}f^{l}+cR\big{(}f^{e}_{\text{total}}+f^{l}\big{)}}.$
(49)

Then, Problem $\mathcal{P}3$ is reformulated as

$\displaystyle\mathcal{P}3\text{-}E1\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\min}\limits_{\boldsymbol{w},\boldsymbol{\theta}}\frac{\ell}{R(\boldsymbol{w},\boldsymbol{\theta})}+\frac{\ell
c}{f^{e}_{\text{total}}}$ $\displaystyle\text{s.t.}\quad
0\leq\theta_{n}<2\pi,\quad n=1,2,\ldots,N,$ (50a)

which is equivalent to

$\displaystyle\mathcal{P}3\text{-}E2\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\max}\limits_{\boldsymbol{w},\boldsymbol{\theta}}R(\boldsymbol{w},\boldsymbol{\theta})$
$\displaystyle\text{s.t.}\quad 0\leq\theta_{n}<2\pi,\quad n=1,2,\ldots,N.$
(51a)

Substituting (4) and (5) into the OF of Problem $\mathcal{P}3\text{-}E2$, and
taking several steps of mathematical manipulation, Problem
$\mathcal{P}3\text{-}E2$ may be equivalently transformed into

$\displaystyle\mathcal{P}3\text{-}E3\mathrel{\mathop{\mathchar 58\relax}}$
$\displaystyle\mathop{\max}\limits_{\boldsymbol{w},\boldsymbol{\theta}}\frac{p_{t}\big{|}\boldsymbol{w}^{H}\big{(}\boldsymbol{h}_{d}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r}\big{)}\big{|}^{2}}{\sigma^{2}|\boldsymbol{w}^{H}|^{2}}$
$\displaystyle\text{s.t.}\quad 0\leq\theta_{n}<2\pi,\quad n=1,2,\ldots,N.$
(52a)

Again, the BCD technique is invoked for optimizing $\boldsymbol{w}$ and
$\boldsymbol{\theta}$ in Problem $\mathcal{P}3\text{-}E3$. Specifically, given
a $\boldsymbol{\theta}$, $\boldsymbol{w}$ can be optimized following the well-
known maximum ratio combining (MRC) criterion [45], which is given by

$\displaystyle\boldsymbol{w}=\sqrt{p_{t}}(\boldsymbol{h}_{d}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r})/\sigma,$
(53)

while for a given $\boldsymbol{w}$, we have the following inequality for the
OF of Problem $\mathcal{P}3\text{-}E3$,

$\displaystyle\frac{p_{t}\big{|}\boldsymbol{w}^{H}\big{(}\boldsymbol{h}_{d}+\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r}\big{)}\big{|}^{2}}{\sigma^{2}|\boldsymbol{w}^{H}|^{2}}\leq\frac{p_{t}\big{|}\boldsymbol{w}^{H}\boldsymbol{h}_{d}\big{|}^{2}}{\sigma^{2}|\boldsymbol{w}^{H}|^{2}}+\frac{p_{t}\big{|}\boldsymbol{w}^{H}\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r}\big{|}^{2}}{\sigma^{2}|\boldsymbol{w}^{H}|^{2}}.$

The equality in (IV) holds only when the IRS phase shift coefficient obeys
$\arg\\{\boldsymbol{w}^{H}\boldsymbol{h}_{d}\\}=\arg\\{\boldsymbol{w}^{H}\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r}\\}$.
Accordingly, the reflection phase shift vector $\boldsymbol{\theta}$ may be
readily obtained as

$\displaystyle\boldsymbol{\theta}=\arg\\{\boldsymbol{w}^{H}\boldsymbol{h}_{d}\\}-\arg\\{\text{diag}\\{\boldsymbol{w}^{H}\boldsymbol{G}\\}\boldsymbol{h}_{r}\\}.$
(55)

In Algorithm 5, we provide the overall algorithm that is used for solving our
optimization problem for the single-device scenario.

The complexity of Algorithm 5 is dominated by calculating
${\boldsymbol{\theta}}^{(t_{5}+1)}$ and $\boldsymbol{w}^{(t_{5}+1)}$ using
(55) and (53), whose complexities are on the order of
$\mathcal{O}\big{(}\max\\{MN,N^{2}\\}\big{)}$ and of $\mathcal{O}(MN^{2})$,
respectively. Hence the complexity of Algorithm 5 is on the order of
$\mathcal{O}(MN^{2})$.

Algorithm 5 Joint Optimization of $\ell$, $\boldsymbol{w}$ and
$\boldsymbol{\theta}$ proposed for the single-user scenario

0: $\boldsymbol{h}_{r}$, $\boldsymbol{h}_{d}$, $\boldsymbol{G}$, $B$, $p_{t}$,
$\sigma^{2}$, $L$, $c$, $f^{e}_{\text{total}}$, and $\epsilon$

0: Optimal $\ell$, $\boldsymbol{w}$ and $\boldsymbol{\theta}$ 1\.
Initialization

initialize $t_{5}=0$, $\epsilon_{5}^{(0)}=1$

initialize $\boldsymbol{\theta}^{(0)}$ satisfying (7a)

calculate $\boldsymbol{w}^{(0)}$ using (53) 2\. Joint optimization of
$\boldsymbol{w}$ and $\boldsymbol{\theta}$

repeat

$\bullet$ calculate ${\boldsymbol{\theta}}^{(t_{5}+1)}$ and
$\boldsymbol{w}^{(t_{5}+1)}$ using (55) and (53), respectively

$\bullet$
$\epsilon_{5}^{(t_{5})}=\frac{\big{|}\text{obj}\big{(}\boldsymbol{w}^{(t_{5}+1)},{\boldsymbol{\theta}}^{(t_{5}+1)}\big{)}-\text{obj}\big{(}\boldsymbol{w}^{(t_{5})},{\boldsymbol{\theta}}^{(t_{5})}\big{)}\big{|}}{\text{obj}\big{(}\boldsymbol{w}^{(t_{5}+1)},{\boldsymbol{\theta}}^{(t_{5}+1)}\big{)}}$,
where the obj refers to the OF of Problem $\mathcal{P}3\text{-}E3$

$\bullet$ $t_{5}=t_{5}+1$

until $\epsilon_{5}^{(t_{5})}\leq\epsilon$ $||$ $t_{5}>t^{\text{max}}_{5}$ 3\.
Optimization of $\boldsymbol{\ell}$

calculate $\hat{\ell}^{(t_{5}+1)}$ using (49)

integerize $\ell^{(t_{5}+1)}$ by (9)

##  V Numerical Results

In this section, the benefits of deploying the IRS in a MEC system are
evaluated, relying on our algorithms developed in Section III and IV. We
consider a single-cell MEC system for both the single-device and two-device as
well as multi-device scenarios. As shown in Fig. 3, the AP’s coverage radius
is $R=300\leavevmode\nobreak\ \rm{m}$ and the IRS is deployed at the cell
edge. The location of the device is specified both by $d$ and by
$\overline{d}$ in the single-device scenario, while in the two-device
scenario, the devices’ locations are specified by $(d_{1},\overline{d}_{1})$
and $(d_{2},\overline{d}_{2})$, respectively. Furthermore, in the multiple-
device scenario, it is assumed that the devices are uniformly distributed
within a circle, whose size and location are prescribed by its radius $r$, as
well as $d$ and $\overline{d}$, respectively. The default value of these
parameters are set in the “Location model” block of Table II. As for the
communications channel, we consider both the small scale fading and the large
scale path loss. Specifically, the small scale fading is i.i.d. and obeys the
complex Gaussian distribution associated with zero mean and unit variance,
while the path loss in $\rm{dB}$ is given by

$\text{PL}=\text{PL}_{0}-10\alpha\log_{10}\Big{(}\frac{d}{d_{0}}\Big{)},$ (56)

where $\text{PL}_{0}$ is the path loss at the reference distance $d_{0}$; $d$
and $\alpha$ represent the distance of the communications link and its path
loss exponent, respectively. Here we use $\alpha_{ua}$, $\alpha_{ui}$ and
$\alpha_{ia}$ to denote the path loss exponent of the link between the device
and the AP, that of the link between the device and the IRS, as well as that
of the link between the IRS and the AP, respectively. The zero-mean additive
white Gaussian noise associated with the variable of $\sigma^{2}$ is imposed
on the off-loaded signal. The default settings of these parameters are
specified in the “Communications model” block of Table II. The variables
$L_{k}$, $c_{k}$ and $f^{l}_{k}$ obey the uniform distribution, whose ranges
are given in the “Computing model” block of Table II.

Table II: Default simulation parameter setting Description | Parameter and Value
---|---
Location model | $R=300\leavevmode\nobreak\ \rm{m}$
$\overline{d}=\overline{d}_{1}=\overline{d}_{2}=10\leavevmode\nobreak\ \rm{m}$
Communication model | $\text{Bandwidth}=1\leavevmode\nobreak\ \rm{MHz}$
$\text{PL}_{0}=30\leavevmode\nobreak\ \rm{dB}$, $d_{0}=1\leavevmode\nobreak\
\rm{m}$
$\alpha_{ua}=3.5$, $\alpha_{ui}=2.2$, $\alpha_{ia}=2.2$
$M=5$
$p_{t}=1\leavevmode\nobreak\ \rm{mW}$
$\sigma^{2}=3.98\times 10^{-12}\leavevmode\nobreak\ \rm{mW}$
Computing model | $L_{k}=[250,350]\leavevmode\nobreak\ \rm{Kb}$
$c_{k}=[700,800]\leavevmode\nobreak\ \rm{cycle/bit}$
$f^{l}_{k}=[4,6]\times 10^{8}\leavevmode\nobreak\ \rm{cycle/s}$
Weight | $\varpi_{k}=1/K$
Convergence criterion | $\epsilon=0.001$

(a)

(b)

Figure 2: Convergence of the algorithms for (a) the single-device scenario
using Algorithm 5 and (b) the two-device scenario using Algorithm 4. The
parameters are set as follows: $f^{e}_{\text{total}}=50\times
10^{9}\leavevmode\nobreak\ \rm{cycle/s}$; $c_{k}=750\leavevmode\nobreak\
\text{cycle}$, $L_{k}=300\leavevmode\nobreak\ \rm{Kb}$, and
$f^{l}_{k}=0.5\times 10^{9}\leavevmode\nobreak\ \rm{cycle/s}$ for all devices.
(a): $d=280\leavevmode\nobreak\ \rm{m}$; (b):
$d_{1}=d_{2}=280\leavevmode\nobreak\ \rm{m}$.

(a)

(b)

(c)

Figure 3: Top-view setting of (a) the single-device scenario, (b) the two-
device scenario, and (c) the multiple-device scenario.

The following subsections detail our simulation results, in terms of the
properties of our proposed algorithm and of the latency performance in both
the single-device, and two-device, as well as multi-device scenarios in
various simulation environments. The following three schemes are considered:

  * •

_With IRS:_ The off-loading volume, edge computing resource allocation, MUD
matrix, and IRS phase shift are optimized relying on Algorithm 4 and Algorithm
5 in the multi-device and single-device scenarios, respectively.

  * •

_RandPhase:_ The off-loading volume, edge computing resource allocation, as
well as MUD matrix are optimized using Algorithm 4 and Algorithm 5 in multi-
device and single-device scenarios, respectively, while skipping the step of
designing the IRS phase shift, which is randomly set obeying the uniform
distribution in the range of $[0,2\pi)$.

  * •

_Without IRS:_ The composite channel
$\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{h}_{r,k}$ taking into account
the IRS is set to $0$. The off-loading volume, edge computing resource
allocation, and the MUD matrix are designed following Algorithm 4 and
Algorithm 5 in the multi-device and single-device scenarios, respectively.

###  V-A Properties of the Proposed Algorithms

In this subsection, the properties of Algorithm 4 and 5 are investigated, with
the aid of numerical results.

####  V-A1 Convergence

Fig. 2 shows the device-average latency versus the number of iterations under
various settings of the IRS phase shift number, i.e. $N=10$, $20$, and $40$,
for both the single-device and multi-device scenarios. We have the following
two observations. Firstly, a larger number of phase shifts leads to a slightly
slower convergence, especially for the multi-device scenario. This is because
more optimizing variables are involved. Secondly, the proposed algorithms are
capable of achieving a convergence within $5$ iterations, which validates its
practical implementation.

####  V-A2 Impact of the Initialization Settings

(a)

(b)

Figure 4: Simulation results of the the maximum and minimum latency versus the
realization index obtained under $100$ random initialization settings for (a)
the single-device scenario using Algorithm 5 and (b) the two-device scenario
using Algorithm 4. “Max” and “Min” refer to the maximum and minimum value,
respectively. The parameters are set as follows: $N=40$. (a):
$d=280\leavevmode\nobreak\ \rm{m}$; (b): $d_{1}=d_{2}=280\leavevmode\nobreak\
\rm{m}$.

As elaborated on in Remark 1, locally optimal results are provided by our
proposed algorithms. Hence, the results obtained are directly dependent on the
initialization settings of Algorithm 4 and 5. In order to clarify its impact,
Fig. 4 presents the latency performance under different initialization
settings both for single- and multi-device scenarios. Specifically, for each
realization of the wireless channels and computing tasks to be processed,
$100$ locally optimal results are obtained using our proposed algorithms,
where each of the initializations is randomly set. Among these locally optimal
results, the maximum latency value that can be deemed to be the worst-case
result using our proposed algorithms is labeled as “Max” in Fig. 4, while the
minimum latency value is labeled by “Min” in Fig. 4 which may resemble the
globally optimal result. It is shown that these two values are almost
identical for the single-device scenario, while their gap ranges from $2\%$ to
$17\%$ in the multi-device scenario, which implies that our proposed
algorithms are capable of approaching the optimal performance.

####  V-A3 Impact of the Phase Quantization

(a)

(b)

Figure 5: Simulation results of the latency versus the realization index under
different assumptions of IRS phase shifts for (a) the single-device scenario
and (b) the two-device scenario. “Cont.”, “1-bit” and “2-bit” refer to the
assumptions of continuous, 1-bit, and 2-bit phase shifts, respectively. The
parameters are set as follows: $N=40$. (a): $d=280\leavevmode\nobreak\
\rm{m}$; (b): $d_{1}=d_{2}=280\leavevmode\nobreak\ \rm{m}$.

Due to the associated hardware limitation, only a limited number of discrete
IRS phase shifts can be provided in practice [23], which prohibits the direct
implementation of our proposed algorithms. An intuitive practical solution to
this issue is to round the continuous phase shift obtained to its nearest
discrete phase shift. Naturally, a performance loss is imposed, owing to the
associated quantization effect. Fig. 5 evaluates the impact of phase
quantization on the latency, where three practical assumptions are considered.
Specifically, under the assumption of continuous phase shifts, the phase shift
of each IRS element can be set as an arbitrary value in the interval of
$[0,2\pi]$; Determined by a 1-bit control signal, the phase shift of each IRS
element has to be either $0$ or $\pi$ under the assumption of 1-bit phase
shift; for a 2-bit control signal, the phase shift of each IRS element has to
be one of the values in the set of
$\left\\{0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\right\\}$. Particular to the
schemes under discrete phase shift assumptions, the values of
$\boldsymbol{W}$, $\boldsymbol{\ell}$, and $\boldsymbol{f}^{e}$ are updated
based on the quantized phase shifts and then the latency is calculated
accordingly. We have the following observations. Firstly, as expected, the
latency decreases upon increasing the number of discrete phase shifts.
Secondly, the performance gap between the schemes under the assumptions of
continuous phase shifts and $2$-bit phase shifts ranges from $1\%$ to $5\%$,
which implies that the quantization loss becomes negligible for as few as four
phase shifts in practice.

###  V-B Single-Device Scenario

Fig. 6-8 present the latency versus various parameter settings in the single-
device scenario, discussed as follows.

Figure 6: Simulation results of the latency versus the number of the IRS
elements in the single-device scenario, where we set
$d=280\leavevmode\nobreak\ \rm{m}$ and $f^{e}_{\text{total}}=50\times
10^{9}\leavevmode\nobreak\ \rm{cycle/s}$.

####  V-B1 Impact of the Number of Reflecting Elements

Fig. 6 presents the latency versus the number of the reflecting elements, for
the various phase shift design schemes. Our observations are as follows.
Firstly, the performance gap between the schemes “Without IRS” and “RandPhase”
becomes higher upon increasing the number of reflecting elements, which
implies that the IRS is capable of assisting the computation off-loading even
without carefully designing the phase shift. This is because the received SINR
can be improved by deploying an IRS for computation off-loading. The gain was
termed as the virtual array gain in Section I. Secondly, the performance gain
of the scheme “With IRS” over the scheme “RandPhase” is around
$11\leavevmode\nobreak\ \rm{ms}$ when we set $N=10$, while it becomes
$46\leavevmode\nobreak\ \rm{ms}$ when we have $N=100$. This implies that a
sophisticated design of the IRS phase shift response provides a beamforming
gain, and that increasing the number of IRS elements leads to a higher
reflection-based beamforming gain. Combining these two types of gains
together, IRSs are capable of efficiently reducing the latency in MEC systems.

Figure 7: Simulation results of the latency versus the edge computing
capability in the single-device scenario, where we set
$d=280\leavevmode\nobreak\ \rm{m}$ and $N=40$.

####  V-B2 Impact of the Edge Computing Capability

Fig. 7 shows the latency versus the edge computing capability, for various IRS
phase shift schemes. Our observations are as follows. For all these three
schemes, the increase of $f^{e}_{\text{total}}$ drastically reduces the
latency when $f^{e}_{\text{total}}$ is of a small value, while the reduction
of the latency becomes smaller when $f^{e}_{\text{total}}$ reaches a certain
threshold value, say $30\times 10^{9}\leavevmode\nobreak\ \rm{cycle/s}$. This
is because the latency imposed by the edge computing dominates when
$f^{e}_{\text{total}}$ is of a small value, whereas the latency imposed by
computation off-loading plays a dominant role when $f^{e}_{\text{total}}$
reaches a high value. Therefore, it is not necessary to equip the edge
computing node with an extremely powerful computing capability for latency
minimization.

Figure 8: Simulation results of the latency versus the device location in the
single-device scenario, where we set $N=40$ and $f^{e}_{\text{total}}=50\times
10^{9}\leavevmode\nobreak\ \rm{cycle/s}$.

####  V-B3 Impact of the Device Location

Fig. 8 depicts the latency versus the device location, equipped with various
IRS phase shift schemes. Our observations are as follows. In the case where no
IRS is employed, the latency increases upon increasing the distance between
the AP and the device. In the case where the IRS’s phase shift is randomly
set, the advantage of using the IRS becomes visible when the distance between
the device and the IRS is less than $20\leavevmode\nobreak\ \rm{m}$. By
contrast, the benefit of the IRS becomes notable for a much larger coverage of
$100\leavevmode\nobreak\ \rm{m}$ for the “With IRS” scheme. This observation
implies that a sophisticated design of the IRS phase shift response is capable
of extending the coverage of the IRS. Furthermore, the latency reaches its
maximum value at $d=260\leavevmode\nobreak\ \rm{m}$ and thereafter becomes
smaller for the “With IRS” scheme. This is because the direct device-AP link
dominates the computation off-loading when the device’s location obeys $d\leq
260\leavevmode\nobreak\ \rm{m}$, while the composite device-IRS-AP link plays
a dominant role, when we have $d\geq 260\leavevmode\nobreak\ \rm{m}$. This
observation further consolidates that a higher gain can be achieved in the
near-IRS area, where the composite device-IRS-AP link dominates the
computation off-loading.

###  V-C Multi-Device Scenario

Fig. 9-12 present the latency in the two-device scenario, and Fig. 13-14 show
the latency in the multiple-device scenario, which are discussed as follows.

Figure 9: Simulation results of the latency versus the number of the IRS
elements in the two-device scenario, where we set
$d_{1}=260\leavevmode\nobreak\ \rm{m}$, $d_{2}=280\leavevmode\nobreak\
\rm{m}$, and $f^{e}_{\text{total}}=50\times 10^{9}\leavevmode\nobreak\
\rm{cycle/s}$.

####  V-C1 Impact of the Number of IRS Elements

Fig. 9 depicts the latency versus the number of the IRS elements in the two-
device scenario, equipped with various phase shift schemes. Apart from the
insights obtained in the single-device scenario, we also have the following
observations. Firstly, Device 2 outperforms Device 1 for the “With IRS”
scheme, whilst Device 1 has a lower latency both for the “Without IRS” and
“RandPhase” schemes compared to Device 2. This is in accordance with the
comparative relationship between the devices located at
$d=260\leavevmode\nobreak\ \rm{m}$ and at $d=280\leavevmode\nobreak\ \rm{m}$
in terms of the latency using those three phase shift schemes, as shown in
Fig. 8. This also implies that IRSs may change the latency ranking of the
devices in MEC systems. Secondly, upon increasing the number of IRS elements,
Device 2 obtains a higher gain than Device 1. This is because Device 2 is
located closer to the IRS, where the composite device-IRS-AP channel dominates
the computation off-loading. Again, this implies that given a specific path
loss exponent, a higher array and passive beamforming gain may be achieved if
the device is located closer to the IRS.

Figure 10: Simulation results of the latency versus the edge computing
resource in the two-device scenario, where we set
$d_{1}=260\leavevmode\nobreak\ \rm{m}$, $d_{2}=280\leavevmode\nobreak\
\rm{m}$, and $N=40$.

####  V-C2 Impact of the Edge Computing Capability

Fig. 10 presents the latency versus the edge computing capability in the
multi-device scenario, equipped with various phase shift schemes. This
scenario follows similar trends to the single-device case illustrated in Fig.
7.

Figure 11: Simulation results of the latency versus the user location in the
two-device scenario, where we set $d_{2}=280\leavevmode\nobreak\ \rm{m}$,
$N=40$, and $f^{e}_{\text{total}}=50\times 10^{9}\leavevmode\nobreak\
\rm{cycle/s}$.

####  V-C3 Impact of the Device Location

Fig. 11 plots the latency versus the location of Device 1, while fixing the
location of the AP, the IRS, and Device 2. As for the “With IRS” scheme, the
curve of Device 1’s latency intercepts that of Device 2 at
$d_{1}=220\leavevmode\nobreak\ \rm{m}$ and $d_{1}=280\leavevmode\nobreak\
\rm{m}$, which implies that the devices at these two locations have the same
channel gain. In other words, the IRS is capable of assisting the device at
$d=280\leavevmode\nobreak\ \rm{m}$ to achieve the same latency as the device
at $d=220\leavevmode\nobreak\ \rm{m}$. Note that the specific values of these
two equivalent-latency locations are dependent on the specific values of the
path loss exponents of the device-IRS, IRS-AP, and device-AP channels, as
presented below.

Figure 12: Simulation results of the latency versus the path loss exponent in
the two-device scenario, where we set
$\alpha_{ui}=\alpha_{ia}=\alpha_{\text{IRS}}$. The parameters are set as
follows: $d_{1}=220\leavevmode\nobreak\ \rm{m}$,
$d_{2}=280\leavevmode\nobreak\ \rm{m}$, $N=40$, and
$f^{e}_{\text{total}}=50\times 10^{9}\leavevmode\nobreak\ \rm{cycle/s}$.

####  V-C4 Impact of the Path Loss Exponent

Fig. 12 illustrates the latency versus the path loss exponent value associated
with the IRS. It can be observed that the intercept point disappears, when
$\alpha_{\text{IRS}}$ is changed for the “With IRS” scheme. Furthermore, the
latency of devices increases upon increasing $\alpha_{\text{IRS}}$. This is
because higher $\alpha_{\text{IRS}}$ leads to a lower array and beamforming
gain by the IRS. This provides important insights for engineering design: the
location of the IRS should be carefully selected to avoid obstacles, for
achieving a lower $\alpha_{ui}$ and $\alpha_{ia}$.

Figure 13: Simulation results of the device-average latency versus the number
of devices $K$. The parameters are set as follows: $d=280\leavevmode\nobreak\
\rm{m}$, $\overline{d}=10\leavevmode\nobreak\ \rm{m}$,
$r=10\leavevmode\nobreak\ \rm{m}$, $N=40$, and $f^{e}_{\text{total}}=50\times
10^{9}\leavevmode\nobreak\ \rm{cycle/s}$.

####  V-C5 Impact of the Number of Devices

Fig. 13 shows the latency versus the number of devices in the cycle in multi-
device scenario. It can be readily observed that the device-average latency
increases upon increasing the number of devices in the IRS-aided MEC system.
This is partially because of the reduced edge computational resources
allocated to each device and partially due to the reduced beamforming gain
achieved at each device. The former issue may be overcome by equipping the
edge node with more powerful computing capability, while the latter problem
can be solved by deploying more IRSs in the MEC system for forming stronger
beams. Nonetheless, compared to the “Without IRS” scheme, our “With IRS”
scheme is capable of reducing the device-average latency from 177 ms to 139
ms, when we have $5$ devices in the MEC system. This again validates the
benefits of our proposed system.

Figure 14: Simulation results of the device-average latency versus the ICI-to-
noise ratio. The parameters are set as follows: $K=3$,
$d=280\leavevmode\nobreak\ \rm{m}$, $\overline{d}=10\leavevmode\nobreak\
\rm{m}$, $r=10\leavevmode\nobreak\ \rm{m}$, $N=40$, and
$f^{e}_{\text{total}}=50\times 10^{9}\leavevmode\nobreak\ \rm{cycle/s}$.

####  V-C6 Impact of Inter-Cell Interference

In realistic scenarios, inter-cell interference (ICI) also degrades
computation off-loading. To quantify the impact of ICI, Fig. 14 presents the
latency versus the ICI-to-noise power ratio, where the BS is assumed to know
the power of the received interference but not the specific signal transmitted
from other cells. Observe that the benefit of employing IRSs in MEC systems
decreases upon increasing the ICI-to-noise power ratio. To elaborate, when
computation off-loading is used in the face of strong ICI, the fraction of
tasks that can be off-loaded becomes marginal. In this case, the wireless
devices have to rely on their own computing capabilities. In other words, the
potential of IRSs may not be fully exploited. This observation suggests an
important insight for engineering design: the spectrum allocation of adjacent
cells has to be carefully managed for minimizing the ICI in IRS-aided MEC
systems.

##  VI Conclusions

In order to reduce the computational latency, an IRS was proposed for
employment in MEC systems. Based on this model, a latency-minimization problem
was formulated, subject to practical constraints on the total edge computing
capability and IRS phase shifts. Sophisticated algorithms were developed for
optimizing both the computing and communications settings. The benefits of
using IRSs in the MEC system were evaluated under various simulation
environments. Quantitatively, the device-average computational latency was
reduced from $177\leavevmode\nobreak\ \rm{ms}$ to $139\leavevmode\nobreak\
\rm{ms}$, compared to the conventional MEC system operating without IRSs in a
single cell associated with the cell radius of $300\leavevmode\nobreak\
\rm{m}$, a $5$-antenna access point and $5$ active devices. Furthermore, the
rapid convergence of our proposed algorithm was confirmed numerically, which
validates their benefits. As our future work, an energy-minimization based
design will be conceived for IRS-aided MEC systems.

##  Appendix A The proof of Proposition 1

$\hat{\ell}_{k}\in[0,L_{k}]$ is used to represent the relaxation [46] of the
integer value $\ell_{k}\in\\{0,1,\ldots,L_{k}\\}$. Furthermore, given the
values of $\boldsymbol{W},\boldsymbol{\theta}$ and $\boldsymbol{f}^{e}$, we
define the delay associated with $\hat{\ell}_{k}$ to be
$\hat{D}_{k}(\hat{\ell}_{k})\triangleq\max\big{\\{}D^{l}_{k}(\hat{\ell}_{k}),D^{e}_{k}(\hat{\ell}_{k})\big{\\}}$,
which can be reformulated from (II-B) as a segmented form below

$\hat{D}_{k}(\hat{\ell}_{k})=\begin{cases}\frac{(L_{k}-\hat{\ell}_{k})c_{k}}{f^{l}_{k}},&0\leq\hat{\ell}_{k}\leq\frac{L_{k}c_{k}R_{k}f^{e}_{k}}{f^{e}_{k}f^{l}_{k}+c_{k}R_{k}(f^{l}_{k}+f^{e}_{k})},\\\
\frac{\hat{\ell}_{k}}{R_{k}}+\frac{\hat{\ell}_{k}c_{k}}{f^{e}_{k}},&\frac{L_{k}c_{k}R_{k}f^{e}_{k}}{f^{e}_{k}f^{l}_{k}+c_{k}R_{k}(f^{l}_{k}+f^{e}_{k})}<\hat{\ell}_{k}\leq
L_{k}.\end{cases}$ (57)

A glance at (57) reveals that $\hat{D}_{k}(\hat{\ell}_{k})$ decreases upon
increasing $\hat{\ell}_{k}$ in the range of
$\hat{\ell}_{k}\in\bigg{[}0,\frac{L_{k}c_{k}R_{k}f^{e}_{k}}{f^{e}_{k}f^{l}_{k}+c_{k}R_{k}\big{(}f^{e}_{k}+f^{l}_{k}\big{)}}\bigg{]}$,
while $\hat{D}_{k}(\hat{\ell}_{k})$ increases upon increasing $\hat{\ell}_{k}$
in the range of
$\hat{\ell}_{k}\in\bigg{[}\frac{L_{k}c_{k}R_{k}f^{e}_{k}}{f^{e}_{k}f^{l}_{k}+c_{k}R_{k}\big{(}f^{e}_{k}+f^{l}_{k}\big{)}},L_{k}\bigg{]}$.
Therefore, it is readily inferred that $\hat{D}_{k}(\hat{\ell}_{k})$ achieves
its minimum value when we set
$\hat{\ell}_{k}=\frac{L_{k}c_{k}R_{k}f^{e}_{k}}{f^{e}_{k}f^{l}_{k}+c_{k}R_{k}\big{(}f^{e}_{k}+f^{l}_{k}\big{)}}$,
which is denoted by $\hat{\ell}_{k}^{*}$. Bearing in mind that the optimal
value of $\ell_{k}$ has to be an integer, it may be obtained by carrying out
the operation
$\ell^{*}_{k}=\mathop{\arg\min}\limits_{\hat{\ell}\in\big{\\{}\lfloor\hat{\ell}^{*}_{k}\rfloor,\lceil\hat{\ell}^{*}_{k}\rceil\big{\\}}}D_{k}(\hat{\ell}_{k})$.
This completes the proof.

##  Appendix B The proof of Proposition 2

Let us denote the second derivative of the OF of Problem
$\mathcal{P}1\text{-}E$ with respect to $f^{e}_{k}$ by $\Phi_{1\text{-}E}$,
which is calculated as

$\displaystyle\Phi_{1\text{-}E}=\frac{2\varpi_{k}L_{k}c_{k}^{3}R_{k}^{2}(f^{l}_{k}+c_{k}R_{k})}{\big{[}f^{e}_{k}f^{l}_{k}+c_{k}R_{k}\big{(}f^{e}_{k}+f^{l}_{k}\big{)}\big{]}^{3}}.$
(58)

Since the values of $\varpi_{k}$, $c_{k}$, $R_{k}$, $f^{l}_{k}$ are all
positive and we have $L_{k}\geq 0$, $f^{e}_{k}\geq 0$, it may be readily
demonstrated that $\Phi_{1\text{-}E}\geq 0$. Hence the OF is a convex function
with respect to $f^{e}_{k}$. Furthermore, the constraint functions (7c) and
(7d) are all of linear forms. Hence, Problem $\mathcal{P}1\text{-}E$ is shown
to be a strictly convex problem.

##  Appendix C The proof of Proposition 3

The Lagrangian of Problem $\mathcal{P}2\text{-}E2$ is given by

$\displaystyle\mathcal{L}(\boldsymbol{W},\boldsymbol{\theta},\boldsymbol{\beta},\boldsymbol{\lambda})=\sum^{K}_{k=1}\beta_{k}+\sum^{K}_{k=1}\lambda_{k}\big{[}\varpi_{k}\ell_{k}-\beta_{k}R_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta})\big{]}$
(59)

where $\\{\lambda_{k}\\}$ is the non-negative Lagrange multiplier. If
$(\boldsymbol{W}^{*},\boldsymbol{\theta}^{*},\boldsymbol{\beta}^{*})$ is the
solution of Problem $\mathcal{P}2\text{-}E2$, there exists
$\boldsymbol{\lambda}^{*}$ satisfying the following KKT conditions

$\displaystyle\frac{\partial\mathcal{L}}{\partial\theta_{k}}=-\lambda_{k}^{*}\beta_{k}^{*}\triangledown
R_{k}(\boldsymbol{w}_{k}^{*},\boldsymbol{\theta}^{*})=0,\quad k=1,2,\ldots,K,$
(60)
$\displaystyle\frac{\partial\mathcal{L}}{\partial\boldsymbol{w}_{k}}=-\lambda_{k}^{*}\beta_{k}^{*}\triangledown
R_{k}(\boldsymbol{w}_{k}^{*},\boldsymbol{\theta}^{*})=0,\quad k=1,2,\ldots,K,$
(61)
$\displaystyle\frac{\partial\mathcal{L}}{\partial\beta_{k}}=1-\lambda_{k}^{*}R_{k}(\boldsymbol{w}_{k}^{*},\boldsymbol{\theta}^{*})=0,\quad
k=1,2,\ldots,K,$ (62)
$\displaystyle\lambda_{k}^{*}\big{[}\varpi_{k}\ell_{k}-\beta_{k}^{*}R_{k}(\boldsymbol{w}_{k}^{*},\boldsymbol{\theta}^{*})\big{]}=0,\quad
k=1,2,\ldots,K,$ (63) $\displaystyle\lambda_{k}^{*}\geq 0,\quad
k=1,2,\ldots,K,$ (64)
$\displaystyle\varpi_{k}\ell_{k}-\beta_{k}^{*}R_{k}(\boldsymbol{w}_{k}^{*},\boldsymbol{\theta}^{*})\leq
0,\quad k=1,2,\ldots,K,$ (65) $\displaystyle 0\leq\theta_{k}^{*}\leq
2\pi,\quad k=1,2,\ldots,K.$ (66)

Since we have $R_{k}(\boldsymbol{w}_{k},\boldsymbol{\theta})>0$, (62) is
equivalent to

$\displaystyle\lambda_{k}^{*}=\frac{1}{R_{k}(\boldsymbol{w}_{k}^{*},\boldsymbol{\theta}^{*})},\quad\forall
k\in\\{1,2,\ldots,K\\},$ (67)

and then (63) is equivalently written as

$\displaystyle\beta_{k}^{*}=\frac{\varpi_{k}\ell_{k}}{R_{k}(\boldsymbol{w}_{k}^{*},\boldsymbol{\theta}^{*})},\quad\forall
k\in\\{1,2,\ldots,K\\}.$ (68)

Furthermore, Eq. (60), (61) and (66) are exactly the KKT conditions of Problem
$\mathcal{P}2\text{-}E3$, when we set
$\boldsymbol{\lambda}=\boldsymbol{\lambda}^{*}$ and
$\boldsymbol{\beta}=\boldsymbol{\beta}^{*}$. This proves the first conclusion
of Proposition 3. Following the same procedure, the second conclusion of
Proposition 3 may also be readily shown.

## References

  * [1] M. R. Palattella, M. Dohler, A. Grieco, G. Rizzo, J. Torsner, T. Engel, and L. Ladid, “Internet of things in the 5G era: Enablers, architecture, and business models,” IEEE J. Sel. Areas Commun., vol. 34, pp. 510–527, Mar. 2016.
  * [2] FP7 European Project (2012), “Distributed computing, storage and radio resource allocation over cooperative femtocells (TROPIC).” [Online]. http://www.ict-tropic.eu.
  * [3] S. Barbarossa, S. Sardellitti, and P. Di Lorenzo, “Communicating while computing: Distributed mobile cloud computing over 5G heterogeneous networks,” IEEE Signal Process. Mag., vol. 31, pp. 45–55, Nov. 2014.
  * [4] W. Shi, J. Cao, Q. Zhang, Y. Li, and L. Xu, “Edge computing: Vision and challenges,” IEEE Internet Things J., vol. 3, pp. 637–646, March 2016\.
  * [5] Y. Mao, C. You, J. Zhang, K. Huang, and K. B. Letaief, “A survey on mobile edge computing: The communication perspective,” IEEE Commun. Surv. Tutor., vol. 19, pp. 2322–2358, April 2017.
  * [6] W. Zhang, Y. Wen, K. Guan, D. Kilper, H. Luo, and D. O. Wu, “Energy-optimal mobile cloud computing under stochastic wireless channel,” IEEE Trans. Wireless Commun., vol. 12, pp. 4569–4581, Sep. 2013.
  * [7] Y. Wang, M. Sheng, X. Wang, L. Wang, and J. Li, “Mobile-edge computing: Partial computation offloading using dynamic voltage scaling,” IEEE Trans. Commun., vol. 64, pp. 4268–4282, Oct. 2016.
  * [8] J. Ren, G. Yu, Y. Cai, and Y. He, “Latency optimization for resource allocation in mobile-edge computation offloading,” IEEE Trans. Wireless Commun., vol. 17, pp. 5506–5519, Aug. 2018.
  * [9] T. Bai, J. Wang, Y. Ren, and L. Hanzo, “Energy-efficient computation offloading for secure UAV-edge-computing systems,” IEEE Trans. Veh. Technol., vol. 68, pp. 6074–6087, June 2019.
  * [10] S. Sardellitti, G. Scutari, and S. Barbarossa, “Joint optimization of radio and computational resources for multicell mobile-edge computing,” IEEE Trans. Signal Inf. Process. Netw., vol. 1, pp. 89–103, June 2015.
  * [11] X. Chen, L. Jiao, W. Li, and X. Fu, “Efficient multi-user computation offloading for mobile-edge cloud computing,” IEEE/ACM Trans. Netw., vol. 24, pp. 2795–2808, Oct. 2015.
  * [12] X. Lyu, W. Ni, H. Tian, R. P. Liu, X. Wang, G. B. Giannakis, and A. Paulraj, “Optimal schedule of mobile edge computing for Internet of Things using partial information,” IEEE J. Sel. Areas Commun., vol. 35, pp. 2606–2615, Nov. 2017.
  * [13] Y. Dai, D. Xu, S. Maharjan, and Y. Zhang, “Joint computation offloading and user association in multi-task mobile edge computing,” IEEE Trans. Veh. Technol., vol. 67, pp. 12313–12325, Dec. 2018.
  * [14] T. Ouyang, Z. Zhou, and X. Chen, “Follow me at the edge: Mobility-aware dynamic service placement for mobile edge computing,” IEEE J. Sel. Areas Commun., vol. 36, pp. 2333–2345, Oct 2018.
  * [15] T. J. Cui, M. Q. Qi, X. Wan, J. Zhao, and Q. Cheng, “Coding metamaterials, digital metamaterials and programmable metamaterials,” Light: Science & Applications, vol. 3, no. 10, p. e218, 2014.
  * [16] M. Di Renzo, M. Debbah, D.-T. Phan-Huy, et al., “Smart radio environments empowered by reconfigurable AI meta-surfaces: an idea whose time has come,” EURASIP J. Wireless Commun. Netw., vol. 2019, pp. 1–20, May 2019.
  * [17] M. Sheng, Y. Wang, X. Wang, and J. Li, “Energy-efficient multiuser partial computation offloading with collaboration of terminals, radio access network, and edge server,” IEEE Trans. Commun., vol. 68, pp. 1524–1537, March 2020.
  * [18] K. Yang, T. Jiang, Y. Shi, and Z. Ding, “Federated learning via over-the-air computation,” IEEE Trans. Wireless Commun., vol. 19, pp. 2022–2035, March 2020.
  * [19] Y. Han, W. Tang, S. Jin, C.-K. Wen, and X. Ma, “Large intelligent surface-assisted wireless communication exploiting statistical CSI,” IEEE Trans. Veh. Technol., vol. 68, pp. 8238–8242, Aug. 2019.
  * [20] B. Zheng and R. Zhang, “Intelligent reflecting surface-enhanced OFDM: Channel estimation and reflection optimization.” [Online]. Available: https://arxiv.org/abs/1909.03272.
  * [21] S. Abeywickrama, R. Zhang, and C. Yuen, “Intelligent reflecting surface: Practical phase shift model and beamforming optimization.” [Online]. Available: https://arxiv.org/abs/1907.06002.
  * [22] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,” IEEE Trans. Wireless Commun., pp. 1–1, 2019.
  * [23] Q. Wu and R. Zhang, “Beamforming optimization for intelligent reflecting surface with discrete phase shifts.” [Online]. Available: https://arxiv.org/abs/1810.03961.
  * [24] H. Guo, Y.-C. Liang, J. Chen, and E. G. Larsson, “Weighted sum-rate optimization for intelligent reflecting surface enhanced wireless networks.” [Online]. Available: https://arxiv.org/abs/1905.07920.
  * [25] J. Ye, S. Guo, and M.-S. Alouini, “Joint reflecting and precoding designs for SER minimization in reconfigurable intelligent surfaces assisted MIMO systems.” [Online]. Available: https://arxiv.org/abs/1906.11466.
  * [26] C. Pan, H. Ren, K. Wang, W. Xu, M. Elkashlan, A. Nallanathan, and L. Hanzo, “Intelligent reflecting surface for multicell MIMO communications.” [Online]. Available: https://arxiv.org/abs/1907.10864.
  * [27] G. Zhou, C. Pan, H. Ren, K. Wang, W. Xu, and A. Nallanathan, “Intelligent reflecting surface aided multigroup multicast MISO communication systems.” [Online]. Available: https://arxiv.org/abs/1909.04606.
  * [28] C. Huang, G. C. Alexandropoulos, C. Yuen, and M. Debbah, “Indoor signal focusing with deep learning designed reconfigurable intelligent surfaces.” [Online]. Available: https://arxiv.org/abs/1905.07726.
  * [29] Y. Yang, B. Zheng, S. Zhang, and R. Zhang, “Intelligent reflecting surface meets OFDM: Protocol design and rate maximization.” [Online]. Available: https://arxiv.org/abs/1906.09956.
  * [30] M. Cui, G. Zhang, and R. Zhang, “Secure wireless communication via intelligent reflecting surface,” IEEE Wireless Commun. Lett., pp. 1–1, 2019.
  * [31] H. Shen, W. Xu, S. Gong, Z. He, and C. Zhao, “Secrecy rate maximization for intelligent reflecting surface assisted multi-antenna communications,” IEEE Commun. Lett., vol. 23, pp. 1488–1492, Sep. 2019\.
  * [32] D. Xu, X. Yu, Y. Sun, D. W. K. Ng, and R. Schober, “Resource allocation for secure IRS-assisted multiuser MISO systems.” [Online]. Available: https://arxiv.org/abs/1907.03085.
  * [33] J. Chen, Y.-C. Liang, Y. Pei, and H. Guo, “Intelligent reflecting surface: A programmable wireless environment for physical layer security.” [Online]. Available: https://arxiv.org/abs/1905.03689.
  * [34] Q. Wu and R. Zhang, “Weighted sum power maximization for intelligent reflecting surface aided SWIPT.” [Online]. Available: https://arxiv.org/abs/1907.05558.
  * [35] C. Pan, H. Ren, K. Wang, M. Elkashlan, A. Nallanathan, J. Wang, and L. Hanzo, “Intelligent reflecting surface enhanced MIMO broadcasting for simultaneous wireless information and power transfer.” [Online]. Available: https://arxiv.org/abs/1908.04863.
  * [36] M. Bennis, M. Debbah, and H. V. Poor, “Ultrareliable and low-latency wireless communication: Tail, risk, and scale,” Proc. IEEE, vol. 106, pp. 1834–1853, Oct. 2018.
  * [37] S. Boyd and L. Vandenberghe, Convex optimization.  Cambridge University Press, 2004.
  * [38] Y. Jong, “An efficient global optimization algorithm for nonlinear sum-of-ratios problem.” [Online]. Available: http://www.optimization-online.org/DB_FILE/2012/08/3586.pdf, 2012.
  * [39] S. He, Y. Huang, L. Yang, and B. Ottersten, “Coordinated multicell multiuser precoding for maximizing weighted sum energy efficiency,” IEEE Trans. Signal Process., vol. 62, pp. 741–751, March 2013.
  * [40] Y. Pan, C. Pan, Z. Yang, M. Chen, and J. Wang, “A caching strategy towards maximal D2D assisted offloading gain,” IEEE Trans. Mobile Comput., pp. 1–1, 2019.
  * [41] Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, “An iteratively weighted MMSE approach to distributed sum-utility maximization for a MIMO interfering broadcast channel,” IEEE Trans. Signal Process., vol. 59, pp. 4331–4340, Sep. 2011.
  * [42] L.-L. Yang, Multicarrier communications.  John Wiley & Sons, 2009.
  * [43] Y. Sun, P. Babu, and D. P. Palomar, “Majorization-minimization algorithms in signal processing, communications, and machine learning,” IEEE Trans. Signal Process., vol. 65, pp. 794–816, Feb. 2016.
  * [44] J. Song, P. Babu, and D. P. Palomar, “Sequence design to minimize the weighted integrated and peak sidelobe levels,” IEEE Trans. Signal Process., vol. 64, pp. 2051–2064, Apri 2015.
  * [45] A. Goldsmith, Wireless Communications.  Cambridge University Press, 2005.
  * [46] M. L. Fisher, “The Lagrangian relaxation method for solving integer programming problems,” Management Science, vol. 27, pp. 1–18, Jan. 1981\.