# Artificial-Noise-Aided Secure MIMO Wireless Communications via Intelligent
Reflecting Surface

Sheng Hong, Cunhua Pan, Hong Ren, Kezhi Wang, and Arumugam Nallanathan This
work was supported by the National Natural Science Foundation of China
(61661032), the Young Natural Science Foundation of Jiangxi Province
(20181BAB202002), the China Postdoctoral Science Foundation (2017M622102), the
Foundation from China Scholarship Council (201906825071).S. Hong is with
Information Engineering School of Nanchang University, Nanchang 330031, China.
(email: shenghong@ncu.edu.cn). C. Pan, H. Ren, and A. Nallanathan are with the
School of Electronic Engineering and Computer Science at Queen Mary University
of London, London E1 4NS, U.K. (e-mail:{c.pan, h.ren,
a.nallanathan}@qmul.ac.uk). K. Wang is with Department of Computer and
Information Sciences, Northumbria University, UK. (email:
kezhi.wang@northumbria.ac.uk).

###### Abstract

This paper considers a MIMO secure wireless communication system aided by the
physical layer security technique of sending artificial noise (AN). To further
enhance the system security performance, the advanced intelligent reflecting
surface (IRS) is invoked in the AN-aided communication system, where the base
station (BS), legitimate information receiver (IR) and eavesdropper (Eve) are
equipped with multiple antennas. With the aim for maximizing the secrecy rate
(SR), the transmit precoding (TPC) matrix at the BS, covariance matrix of AN
and phase shifts at the IRS are jointly optimized subject to constrains of
transmit power limit and unit modulus of IRS phase shifts. Then, the secrecy
rate maximization (SRM) problem is formulated, which is a non-convex problem
with multiple coupled variables. To tackle it, we propose to utilize the block
coordinate descent (BCD) algorithm to alternately update the TPC matrix, AN
covariance matrix, and phase shifts while keeping SR non-decreasing.
Specifically, the optimal TPC matrix and AN covariance matrix are derived by
Lagrangian multiplier method, and the optimal phase shifts are obtained by
Majorization-Minimization (MM) algorithm. Since all variables can be
calculated in closed form, the proposed algorithm is very efficient. We also
extend the SRM problem to the more general multiple-IRs scenario and propose a
BCD algorithm to solve it. Finally, simulation results validate the
effectiveness of system security enhancement via an IRS.

###### Index Terms:

Intelligent Reflecting Surface (IRS), Reconfigurable Intelligent Surfaces,
Secure Communication, Physical Layer Security, Artificial Noise (AN), MIMO.

##  I Introduction

The next-generation (i.e, 6G) communication is expected to be a sustainable
green, cost-effective and secure communication system [1]. In particular,
secure communication is crucially important in 6G communication networks since
communication environment becomes increasingly complicated and the security of
private information is imperative [2]. The information security using
crytographic encryption (in the network layer) is a conventional secure
communication technique, which suffers from the vulnerabilities, such as
secret key distribution, protection and management [3]. Unlike this network
layer security approach, the physical layer security can guarantee good
security performance bypassing the relevant manipulations on the secret key,
thus is more attractive for the academia and industry [4]. There are various
physical-layer secrecy scenarios. The first one is the classical physical-
layer secrecy setting where there is one legitimate information receiver (IR)
and one eavesdropper (Eve) operating over a single-input-single-output (SISO)
channel (i.e., the so-called three-terminal SISO Gaussian wiretap channel) [5,
6]. The second one considers the physical-layer secrecy with an IR and Eve
operating over a multiple-input-single-output (MISO) channel, which is called
as three-terminal MISO Gaussian wiretap channel. The third one is a renewed
and timely scenario with one IR and one Eve operating over a multiple-input-
multiple-output (MIMO) channel, which is named as three-terminal MIMO Gaussian
wiretap channel[7, 8] and is the focus of this paper. For MIMO systems, a
novel idea in physical-layer security is to transmit artificial noise (AN)
from the base station (BS) to contaminate the Eve’s received signal [9, 10,
11]. For these AN-aided methods, a portion of transmit power is assigned to
the artificially generated noise to interfere the Eve, which should be
carefully designed. For AN-aided secrecy systems, while most of the existing
AN-aided design papers focused on the MISO wiretap channel and null-space AN
[7, 12], designing the transmit precoding (TPC) matrix together with AN
covariance matrix for the MIMO wiretap channel is more challenging [13].

In general, the secrecy rate (SR) achieved by the mutual information
difference between the legitimate IR and the Eve is limited by the channel
difference between the BS-IR link and the BS-Eve link. The AN-aided method can
further improve the SR, but it consumes the transmit power destined for the
legitimate IR. When the transmit power is confined, the performance bottleneck
always exists for the AN-aided secure communication. To conquer the dilemma,
the recently proposed intelligent reflecting surface (IRS) technique can be
exploited. Since higher SR can be achieved by enhancing the channel quality in
the BS-IR link and degrading the channel condition in the BS-Eve link, the IRS
can serve as a powerful complement to AN-aided secure communication due to its
capability of reconfiguring the wireless propagation environment.

The IRS technique has been regarded as a revolutionary technique to control
and reconfigure the wireless environment [14, 15],[16]. An IRS comprises an
array of reflecting elements, which can reflect the incident electromagnetic
(EM) wave passively, and the complex reflection coefficient contains the phase
shift and amplitude. In practical applications, the phase shifts of the
reflection coefficients are discrete due to the manufacturing cost[17].
However, many works on IRS aided wireless communications are based on the
assumption of continuous phase shifts [18],[19]. To investigate the potential
effect of IRS on the secure communication, we also assume continuous phase
shifts to simplify the problem. We evaluate its impact on the system
performance in the simulation section. Theoretically, the reflection amplitude
of each IRS element can be adjusted for different purpose [20]. However,
considering the hardware cost, the reflection amplitude is usually assumed to
be 1 for simplicity. Hence, by smartly tuning the phase shifts with a
preprogrammed controller, the direct signals from the BS and the reflected
signals from the IRS can be combined constructively or destructively according
to different requirements. In comparison to the existing related techniques
which the IRS resembles, such as active intelligent surface [21], traditional
reflecting surfaces[22], backscatter communication [23] and amplify-and-
forward (AF) relay [24], the IRSs have the advantages of flexible
reconfiguration on the phase shifts in real time, minor additional power
consumption, easy installation with many reflecting elements, etc.
Furthermore, due to the light weight and compact size, the IRS can be
integrated into the traditional communication systems with minor modifications
[25]. Because of these appealing virtues, IRS has introduced into various
wireless communication systems, including the single-user case [26, 27], the
downlink multiuser case [28, 18, 29, 30, 31], mobile edge computing [32],
wireless information and power transfer design [33], and the physical layer
security design [34, 35, 36, 37].

IRS is promising to strengthen the system security of wireless communication.
In [34, 36, 38], the authors investigated the problem of maximizing the
achievable SR in a secure MISO communication system aided by IRS, where both
the legitimate user and eavesdropper are equipped with a single antenna. The
TPC matrix at the BS and the phase shifts at the IRS were optimized by an
alternate optimization (AO) strategy. To handle the nonconvex unit modulus
constraint, the semidefinite relaxation (SDR) [39], majorization-minimization
(MM) [18, 40], complex circle manifold (CCM) [41] techniques were proposed to
optimize phase shifts. An IRS-assisted MISO secure communication with a single
IR and single Eve was also considered in [35], but it was limited to a special
scenario, where the Eve has a stronger channel than the IR, and the two
channels from BS to Eve and IR are highly correlated. Under this assumption,
the transmit beamforming and the IRS reflection beamforming are jointly
optimized to improve the SR. Similarly, a secure IRS-assisted downlink MISO
broadcast system was considered in [37], and it assumes that multiple
legitimate IRs and multiple Eves are in the same directions to the BS, which
implies that the IR channels are highly correlated with the Eve channels. [42]
considered the transmission design for an IRS-aided secure MISO communication
with a single IR and single Eve, in which the system energy consumption is
minimized under two assumptions that the channels of access point (AP)-IRS
links are rank-one and full-rank. An IRS-assisted MISO network with
cooperative jamming was investigated in [2]. The physical layer security in a
simultaneous wireless information and power transfer (SWIPT) system was
considered with the aid of IRS [43]. However, there are a paucity of papers
considering the IRS-assisted secure communication with AN. A secure MISO
communication system aided by the transmit jamming and AN was considered in
[44], where a large number of Eves exist, and the AN beamforming vector and
jamming vector were optimized to reap the additional degrees of freedom (DoF)
brought by the IRS. [45] investigated the resource allocation problem in an
IRS-assisted MISO communication by jointly optimizing the beamforming vectors,
the phase shifts of the IRS, and AN covariance matrix for secrecy rate
maximization (SRM), but the direct BS-IRs links and direct BS-Eves link are
assumed to be blocked.

Although a few papers have studied security enhancement for an AN-aided system
through the IRS, the existing papers related to this topic either only studied
the MISO scenario or assumed special settings to the channels. The
investigation on the MIMO scenario with general channel settings is absent in
the existing literature. Hence, we investigate this problem in this paper by
employing an IRS in an AN-aided MIMO communication system for the physical
layer security enhancement. Specifically, by carefully designing the phase
shifts of the IRS, the reflected signals are combined with the direct signals
constructively for enhancing the data rate at the IR and destructively for
decreasing the rate at the Eve. As a result, the TPC matrix and AN covariance
matrix at the BS can be designed flexibly with a higher DoF than the case
without IRS. In this work, the TPC matrix, AN covariance matrix and the phase
shift matrix are jointly optimized. Since these optimization variables are
highly coupled, an efficient algorithm based on the block coordinate descent
(BCD) and MM techniques for solving the problem is proposed.

We summarize our main contributions as follows:

  1. 1.

This is the first research on exploiting an IRS to enhance security in AN-
aided MIMO communication systems. Specifically, an SRM problem is formulated
by jointly optimizing the TPC matrix and AN covariance matrix at the BS,
together with the phase shifts of the IRS subject to maximum transmit power
limit and the unit modulus constraint of the phase shifters. The objective
function (OF) of this problem is the difference of two Shannon capacity
expressions, thus is not jointly concave over the three highly-coupled
variables. To handle it, the popular minimum mean-square error (MMSE)
algorithm is used to reformulate the SRM problem.

  2. 2.

The BCD algorithm is exploited to optimize the variables alternately. Firstly,
given the phase shifts of IRS, the optimal TPC matrix and AN covariance matrix
are obtained in closed form by utilizing the Lagrangian multiplier method.
Then, given the TPC matrix and AN covariance matrix, the optimization problem
for IRS phase shifts is transformed by sophisticated matrix manipulations into
a quadratically constrained quadratic program (QCQP) problem subject to unit
modulus constraints. To solve it, the MM algorithm is utilized, where the
phase shifts are derived in closed form iteratively. Based on the BCD-MM
algorithm, the original formulated SRM problem can be solved efficiently.

  3. 3.

The SRM problem is also extended to the more general scenario of multiple
legitimate IRs. A new BCD algorithm is proposed to solve it, where the optimal
TPC matrix and AN covariance matrix are obtained by solving a QCQP problem,
and the unit modulus constraint is handled by the penalty convex-concave
procedure (CCP) method.

  4. 4.

The simulation results confirm that on the one hand, the IRS can greatly
enhance the security of an AN-aided MIMO communication system; on the other
hand, the phase shifts of IRS should be properly optimized. Simulation results
also show that larger IRS element number and more transmit power is beneficial
to the security. Moreover, properly-selected IRS location and good channel
states of the IRS-related links are important to realize the full potential of
IRS.

This paper is organized as follows. Section II provides the signal model of an
AN-aided MIMO communication system assisted by an IRS, and the SRM problem
formulation. The SRM problem is reformulated in Section III, where the BCD-MM
algorithm is proposed to optimize the TPC matrix, AN covariance matrix and
phase shifts of IRS. Section IV extends the SRM problem to a more general
scenario of multiple IRs. In Section V, numerical simulations are given to
validate the algorithm efficiency and security enhancement. Section VI
concludes this paper.

_Notations_ : Throughout this paper, boldface lower case, boldface upper case
and regular letters are used to denote vectors, matrices, and scalars
respectively. ${\bf{X}}\odot{\bf{Y}}$ is the Hadamard product of $\bf X$ and
$\bf Y$. ${\rm{Tr}}\left({\bf{X}}\right)$ and $\left|{\bf{X}}\right|$ denote
the trace and determinant of ${\bf{X}}$ respectively. ${{\mathbb{C}}^{M\times
N}}$ denotes the space of $M\times N$ complex matrices. ${\rm{Re}}\\{\cdot\\}$
and $\arg\\{\cdot\\}$ denote the real part of a complex value and the
extraction of phase information respectively. ${\rm{diag}}\\{\cdot\\}$ is the
operator for diagonalization. ${\cal C}{\cal N}({\bm{\mu}},{\bf{Z}})$
represents a circularly symmetric complex gaussian (CSCG) random vector with
mean ${\bm{\mu}}$ and covariance matrix ${\bf{Z}}$.
${\left(\cdot\right)^{\rm{T}}}$, ${\left(\cdot\right)^{\rm{H}}}$ and
${\left(\cdot\right)^{\rm{\ast}}}$ denote the transpose, Hermitian and
conjugate operators respectively. $(\cdot)^{\star}$ stands for the optimal
value, and $(\cdot)^{{\dagger}}$ means the pseudo-inverse. $[\cdot]^{+}$ is
the projection onto the non-negative number, i.e, if $y=[x]^{+}$, then
$y=\rm{max}\\{0,x\\}$.

##  II Signal Model and Problem Formulation

###  II-A Signal Model

We consider an IRS-aided communication network shown in Fig. 1 that consists
of a BS, a legitimate IR and an Eve, all of which are equipped with multiple
antennas. The number of transmit antennas at the BS is ${{N}_{T}}\geq 1$, and
the numbers of receive antennas at the legitimate IR and Eve are
${{N}_{I}}\geq 1$ and ${{N}_{E}}\geq 1$ respectively. To ensure secure
transmission from the BS to the IR, the AN is sent from the BS to interfere
the eavesdropper to achieve strong secrecy.

Figure 1: An AN-aided MIMO secure communication system with IRS.

With above assumptions, the BS employed the TPC matrix to transmit data
streams with AN. The transmitted signal can be modeled as

$\displaystyle{\bf{x}}={\bf{Vs}}+{\bf{n}},$ (1)

where ${\bf{V}}\in{{\mathbb{C}}^{{{N}_{T}}\times d}}$ is the TPC matrix; the
number of data streams is $d\leq\min({{N}_{T}},{{N}_{I}})$; the transmitted
data towards the IR is
$\mathbf{s}\sim\mathcal{C}\mathcal{N}(0,{\mathbf{I}_{d}})$; and
$\mathbf{n}\in{\cal C}{\cal N}({\bm{0}},{\bf{Z}})$ represents the AN random
vector with zero mean and covariance matrix $\mathbf{Z}$.

Assuming that the wireless signals are propagated in a non-dispersive and
narrow-band way, we model the equivalent channels of the BS-IRS link, the BS-
IR link, the BS-Eve link, the IRS-IR link, the IRS-Eve link by the matrices
$\mathbf{G}\in{{\mathbb{C}}^{M\times{{N}_{T}}}}$,
${{\bf{H}}_{b,I}}\in{{\mathbb{C}}^{{{N}_{I}}\times{{N}_{T}}}}$,
${{\bf{H}}_{b,E}}\in{{\mathbb{C}}^{{{N}_{E}}\times{{N}_{T}}}}$,
${{\bf{H}}_{R,I}}\in{{\mathbb{C}}^{{{N}_{I}}\times
M}}$,${{\bf{H}}_{R,E}}\in{{\mathbb{C}}^{{{N}_{E}}\times M}}$, respectively.
The phase shift coefficients of IRS are collected in a diagonal matrix defined
by ${\bf{\Phi}}=\text{diag }\\!\\!\\{\\!\\!\text{
}{{\phi}_{1}},\cdots,{{\phi}_{m}},\cdots,{{\phi}_{M}}\text{
}\\!\\!\\}\\!\\!\text{ }$ and ${{\phi}_{m}}={{e}^{j{{\theta}_{m}}}}$, where
${{\theta}_{m}}\in[0,2\pi]$ denotes the phase shift of the $m$-th reflection
element. The multi-path signals that have been reflected by multiple times are
considered to be absorbed and diffracted, then the signal received at the
legitimate IR is given by

${\bf{y}}_{I}=({\bf{H}}_{b,I}+{\bf{H}}_{R,I}{\bf{\Phi}}{\bf{G}}){\bf{x}}+{\bf{n}}_{I},$
(2)

where ${{\bf{n}}_{I}}$ is the random noise vector at IR obeying the
distribution
${{\bf{n}}_{I}}\sim\mathcal{C}\mathcal{N}({\bf{0}},\sigma_{I}^{2}{\bf{I}}_{{{N}_{I}}})$.
The signal received at the Eve is

$\displaystyle{\bf{y}}_{E}=({\bf{H}}_{b,E}+{\bf{H}}_{R,E}{\bf{\Phi}}{\bf{G}}){\bf{x}}+{{\bf{n}}_{E}},$
(3)

where ${\bf{n}}_{E}$ is the Eve’s noise vector following the distribution
${\bf{n}}_{E}\sim\mathcal{C}\mathcal{N}({\bf{0}},\sigma_{E}^{2}{\bf{I}}_{{{N}_{E}}})$.

Assume that the BS has acquired the prior information of all the channel state
information (CSI). Then the BS is responsible for optimizing the IRS phase
shifts and sending them back to the IRS controller through a separate low-rate
communication link such as wireless links [20],[17] or wired lines [46]. The
assumption of perfect CSI knowledge is idealistic, since the CSI estimation
for IRS networks is challenging. However, the algorithms developed allow us to
derive the relevant performance upper bounds for realistic scenarios in the
presence of realistic CSI errors. Recently, we have investigated the design of
robust and secure transmission in IRS-aided MISO wireless communication
systems in [47] by considering the statistical CSI error model associated with
the cascaded channels for the eavesdropper. Its extension to the MIMO scenario
will be studied in our future work.

Upon substituting $\bf{x}$ into (2), ${\bf{y}}_{I}$ can be rewirtten as

$\displaystyle{\bf{y}}_{I}={\hat{\bf{H}}_{I}}({\bf{V}}{\bf{s}}+{\bf{n}})+{{\bf{n}}_{I}}{\rm{=}}{\hat{\bf{H}}_{I}}{\bf{V}}{\bf{s}}+{\hat{\bf{H}}_{I}}{\bf{n}}+{{\bf{n}}_{I}},$
(4)

where
${{\hat{\bf{H}}}_{I}}\overset{\triangle}{=}{{\bf{H}}_{b,I}}+{{\bf{H}}_{R,I}}{\bf{\Phi}}{\bf{G}}$
is defined as the equivalent channel spanning from the BS to the legitimate
IR. Then, the data rate (bit/s/Hz) achieved by the legitimate IR is given by

$\displaystyle{R_{I}}({\bf{V}},{\bf{\Phi}},{\bf{Z}})={\rm{log}}\left|{{\bf{I}}+{{\hat{\bf{H}}}_{I}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{I}^{H}{\bf{J}}_{I}^{-1}}\right|,$
(5)

where ${{\bf{J}}_{I}}$ is the interference-plus-noise covariance matrix given
by
${{\bf{J}}_{I}}\overset{\triangle}{=}{{\hat{\bf{H}}}_{I}}{\bf{Z}}{{\hat{\bf{H}}}_{I}}^{H}+\sigma_{I}^{2}{{\bf{I}}_{{{N}_{I}}}}$.

Upon substituting $\bf{x}$ into (3), ${{\bf{y}}_{E}}$ can be rewritten as

$\displaystyle{\bf{y}}_{E}={\hat{\bf{H}}_{E}}({\bf{Vs}}+{\bf{n}})+{\bf{n}}_{E}={\hat{\bf{H}}_{E}}{\bf{Vs}}+{\hat{\bf{H}}_{E}}{\bf{n}}+{\bf{n}}_{E},$
(6)

where
${{\hat{\bf{H}}}_{E}}\overset{\triangle}{=}{{\bf{H}}_{b,E}}+{{\bf{H}}_{R,E}}{\bf{\Phi}}{\bf{G}}$
is defined as the equivalent channel spanning from the BS to the Eve. Then,
the data rate (bit/s/Hz) achieved by the Eve is given by

$\displaystyle{R_{E}}({\bf{V}},{\bf{\Phi}},{\bf{Z}})={\rm{log}}\left|{{\bf{I}}+{{\hat{\bf{H}}}_{E}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{E}^{H}{\bf{J}}_{E}^{-1}}\right|,$
(7)

where ${{\bf{J}}_{E}}$ is the interference-plus-noise covariance matrix given
by
${{\bf{J}}_{E}}\overset{\triangle}{=}{{\hat{\bf{H}}}_{E}}{\bf{Z}}{{\hat{\bf{H}}}_{E}}^{H}+\sigma_{E}^{2}{{\bf{I}}_{{{N}_{E}}}}$.
The achievable secrecy rate is given by

$\displaystyle{{\rm{C}}_{AN}}{\rm{(}}{\bf{V}},{\bf{\Phi}},{\bf{Z}})$
$\displaystyle{\rm{=}}[{R_{I}}({\bf{V}},{\bf{\Phi}},{\bf{Z}})-{R_{E}}({\bf{V}},{\bf{\Phi}},{\bf{Z}}){]^{+}}$
$\displaystyle={\rm{log}}\left|{{\bf{I}}+{{\hat{\bf{H}}}_{I}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{I}^{H}{\bf{J}}_{I}^{-1}}\right|-{\rm{log}}\left|{{\bf{I}}+{{\hat{\bf{H}}}_{E}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{E}^{H}{\bf{J}}_{E}^{-1}}\right|$
$\displaystyle={\rm{log}}\left|{{\bf{I}}+{{\hat{\bf{H}}}_{I}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{I}^{H}{{({{\hat{\bf{H}}}_{I}}{\bf{Z}}{{\hat{\bf{H}}}_{I}}^{H}+\sigma_{I}^{2}{{\bf{I}}_{{N_{I}}}})}^{-1}}}\right|$
$\displaystyle\quad-{\rm{log}}\left|{{\bf{I}}+{{\hat{\bf{H}}}_{E}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{E}^{H}{{({{\hat{\bf{H}}}_{E}}{\bf{Z}}{{\hat{\bf{H}}}_{E}}^{H}+\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}})}^{-1}}}\right|.$
(8)

###  II-B Problem Formulation

With the aim for maximizing SR, the TPC matrix ${\bf{V}}$ at the BS, the AN
covariance matrix ${\bf{Z}}$ at the BS, and the phase shift matrix
${\bf{\Phi}}$ at the IRS should be optimized jointly subject to the
constraints of the maximum transmit power and unit modulus of phase shifts.
Hence, we formulate the SRM problem as

$\displaystyle\
\underset{{\bf{V}},{\bf{\Phi}},{\bf{Z}}}{\mathop{\text{missing}}{max}}\ \
{{\rm{C}}_{AN}}{\rm{(}}{\bf{V}},{\bf{\Phi}},{\bf{Z}}{\rm{)}}$ (9a)
$\displaystyle\ \ \text{s.t.}\quad\
{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{\bf{Z}}{\rm{)}}\leq{P_{T}},$ (9b)
$\displaystyle\quad\quad\quad{\bf{Z}}\succeq 0,$ (9c)
$\displaystyle\quad\quad\quad\\!\left|{{{{\phi}}_{m}}}\right|=1,m=1,\cdots,M,$
(9d)

where ${P_{T}}$ is the maximum transmit power limit. The optimal value of SR
in (9) is always non-negative, which can be proved by using contradiction.
Assume that the optimal value of SR is negative, then we can simply set the
TPC matrix ${\bf{V}}$ to zero matrix, and the resulted SR will be equal to
zero, which is larger than a negative SR.

By variable substitution ${\bf{Z}}={{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}$, where
${{\bf{V}}_{E}}\in{{\mathbb{C}}^{{{N}_{T}}\times{{N}_{T}}}}$, Problem (9) is
equivalent to

$\displaystyle\
\underset{{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}}{\mathop{\text{missing}}{max}}\
\ {{\rm{C}}_{AN}}{\rm{(}}{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}{\rm{)}}$ (10a)
$\displaystyle\ \ \text{s.t.}\quad\
{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}{\rm{)}}\leq{P_{T}},$
(10b)
$\displaystyle\quad\quad\quad\\!\left|{{{{\phi}}_{m}}}\right|=1,m=1,\cdots,M,$
(10c)

where the OF of (10a) is obtained by substituting
${\bf{Z}}={{\bf{V}}_{E}}{{\bf{V}}_{E}}^{H}$ into (II-A). In (10a), the
expression of OF is difficult to tackle, and the variables of ${\bf{V}}$,
${\bf{V}}_{E}$ and ${\bf{\Phi}}$ are coupled with each other, which make
Problem (10) difficult to solve. In addition, the unit modulus constraint
imposed on the phase shifts in (10c) aggravates the difficulty. In the
following, we provide a low-complexity algorithm to solve this problem.

##  III A Low-Complexity Algorithm of BCD-MM

Firstly, the OF of Problem (10) is reformulated into a more tractable
expression equivalently. Then, the BCD-MM method is proposed for optimizing
the TPC matrix ${\bf{V}}$, ${\bf{V}}_{E}$, and the phase shift matrix
${\bf{\Phi}}$ alternately.

###  III-A Reformulation of the Original Problem

Firstly, the achievable SR
${{\rm{C}}_{AN}}{\rm{(}}{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}{\rm{)}}$ in (II-A)
can be further simplified as

$\displaystyle{{\rm{C}}_{AN}}\rm{(}{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}\rm{)}$
$\displaystyle{\rm{=log}}\left|{{\bf{I}}_{{N_{I}}}+{{\hat{\bf{H}}}_{I}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{I}^{H}{{({{\hat{\bf{H}}}_{I}}{\bf{Z}}{{\hat{\bf{H}}}_{I}}^{H}+\sigma_{I}^{2}{{\bf{I}}_{{N_{I}}}})}^{-1}}}\right|{\rm{+log}}\left|{{{\hat{\bf{H}}}_{E}}{\bf{Z}}{{\hat{\bf{H}}}_{E}}^{H}+\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}}}\right|$
$\displaystyle\quad-{\rm{log}}\left|{{{\hat{\bf{H}}}_{E}}{\bf{Z}}{{\hat{\bf{H}}}_{E}}^{H}+\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}}+{{\hat{\bf{H}}}_{E}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{E}^{H}}\right|$
$\displaystyle=\underbrace{{\rm{log}}\left|{{\bf{I}}_{{N_{I}}}+{{\hat{\bf{H}}}_{I}}{\bf{V}}{{\bf{V}}^{H}}\hat{\bf{H}}_{I}^{H}{{({{\hat{\bf{H}}}_{I}}{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}{{\hat{\bf{H}}}_{I}}^{H}+\sigma_{I}^{2}{{\bf{I}}_{{N_{I}}}})}^{-1}}}\right|}_{{f_{1}}}$
$\displaystyle\quad{\rm{+}}\underbrace{{\rm{log}}\left|{{{\bf{I}}_{{N_{E}}}}+{{\hat{\bf{H}}}_{E}}{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}{{\hat{\bf{H}}}_{E}}^{H}(\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}})^{-1}}\right|}_{{f_{2}}}$
$\displaystyle\quad\underbrace{-{\rm{log}}\left|{{{\bf{I}}_{{N_{E}}}}+\sigma_{E}^{-2}{{\hat{\bf{H}}}_{E}}({\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}})\hat{\bf{H}}_{E}^{H}}\right|}_{{f_{3}}}.$
(11)

The expression in $f_{1}$ represents the data rate of the legitimate IR, which
can be reformulated by exploiting the relationship between the data rate and
the mean-square error (MSE) for the optimal decoding matrix. Specifically, the
linear decoding matrix ${{\bf{U}}_{I}}\in{{\mathbb{C}}^{{{N}_{T}}\times{d}}}$
is applied to estimate the signal vector $\hat{{\bf{s}}}$ for the legitimate
IR, and the MSE matrix of the legitimate IR is given by

$\displaystyle{{\bf{E}}_{I}}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}})$
$\displaystyle\buildrel\Delta\over{=}{{\mathbb{E}}_{{\bf{s}},{\bf{n}},{{\bf{n}}_{I}}}}\left[{(\hat{\bf{s}}-{\bf{s}}){{(\hat{\bf{s}}-{\bf{s}})}^{H}}}\right]$
$\displaystyle{\rm{=}}({{\bf{U}}_{I}}^{H}{{\hat{\bf{H}}}_{I}}{\bf{V}}-{\bf{I}}_{d}){({{\bf{U}}_{I}}^{H}{{\hat{\bf{H}}}_{I}}{\bf{V}}-{\bf{I}}_{d})^{H}}+{{\bf{U}}_{I}}^{H}({{\hat{\bf{H}}}_{I}}{{\bf{V}}_{E}}{{\bf{V}}_{E}}^{H}{{\hat{\bf{H}}}_{I}}^{H}{\rm{+}}\sigma_{I}^{2}{{\bf{I}}_{{N_{I}}}}){{\bf{U}}_{I}}.$
(12)

By introducing an auxiliary matrix ${{\bf{W}}_{I}}\succeq 0$,
${{\bf{W}}_{I}}\in{{\mathbb{C}}^{{d}\times{d}}}$ and exploiting the fact 3) of
Lemma 4.1 in [48], we have

$\displaystyle f_{1}$
$\displaystyle{\rm{=}}\mathop{\text{missing}}{max}\limits_{{{\bf{U}}_{I}},{{\bf{W}}_{I}}\succeq
0}h_{1}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I}})$
$\displaystyle\buildrel\Delta\over{=}\mathop{\text{missing}}{max}\limits_{{{\bf{U}}_{I}},{{\bf{W}}_{I}}\succeq
0}\log\left|{{{\bf{W}}_{I}}}\right|-{\rm{Tr}}({{\bf{W}}_{I}}{{\bf{E}}_{I}}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}}))+d.$
(13)

$h_{1}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I}})$ is concave with
respect to (w.r.t.) each matrix of the matrices
${{\bf{U}}_{I}}$,${\bf{V}}$,${{\bf{V}}_{E}}$,${{\bf{W}}_{I}}$ by fixing the
other three matrices. According to the facts 1) and 2) of Lemma 4.1 in [48],
the optimal ${{\bf{U}}^{\star}_{I}}$, ${{\bf{W}}^{\star}_{I}}$ to achieve the
maximum value of
$h_{1}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I}})$ is given by

$\displaystyle{{\bf{U}}^{\star}_{I}}{\rm{=}}\text{arg}\mathop{\text{missing}}{max}\limits_{{{\bf{U}}_{I}}}h_{1}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I}}){\rm{=}}({\hat{\bf{H}}_{I}}{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}{\hat{\bf{H}}_{I}}^{H}{\rm{+}}\sigma_{I}^{2}{{\bf{I}}_{{N_{I}}}}{\rm{+}}{\hat{\bf{H}}_{I}}{\bf{V}}{{\bf{V}}^{H}}{\hat{\bf{H}}_{I}}^{H})^{-1}{\hat{\bf{H}}_{I}}{\bf{V}},$
(14)
$\displaystyle{{\bf{W}}^{\star}_{I}}{\rm{=}}\text{arg}\mathop{\text{missing}}{max}\limits_{{{\bf{W}}_{I}}\succeq
0}h_{1}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I}}){\rm{=[}}{{\bf{E}}^{\star}_{I}}({{\bf{U}}^{\star}_{I}},{\bf{V}},{{\bf{V}}_{E}}){]^{-1}},$
(15)

where ${{\bf{E}}^{\star}_{I}}$ is obtained by plugging the expression of
${{\bf{U}}^{\star}_{I}}$ into
${{\bf{E}}_{I}}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}})$ as

$\displaystyle{{\bf{E}}^{\star}_{I}}({{\bf{U}}^{\star}_{I}},{\bf{V}},{{\bf{V}}_{E}}){\rm{=}}({{\bf{U}}^{{\star}H}_{I}}{\hat{\bf{H}}_{I}}{\bf{V}}-{\bf{I}}_{d}){({{\bf{U}}^{{\star}H}_{I}}{\hat{\bf{H}}_{I}}{\bf{V}}-{\bf{I}}_{d})^{H}}+{{\bf{U}}^{{\star}H}_{I}}({\hat{\bf{H}}_{I}}{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}{\hat{\bf{H}}_{I}}^{H}{\rm{+}}\sigma_{I}^{2}{{\bf{I}}_{{N_{I}}}}){{\bf{U}}^{{\star}}_{I}}{{\rm{}}}.$
(16)

Similarly, by introducing the auxiliary variables ${{\bf{W}}_{E}}\succeq 0$,
${{\bf{W}}_{E}}\in{{\mathbb{C}}^{{{N}_{T}}\times{{N}_{T}}}}$,
${{\bf{U}}_{E}}\in{{\mathbb{C}}^{{{N}_{E}}\times{{N}_{T}}}}$, and exploiting
the fact 3) of Lemma 4.1 in [48], we have

$\displaystyle f_{2}$
$\displaystyle{\rm{=}}\mathop{\text{missing}}{max}\limits_{{{\bf{U}}_{E}},{{\bf{W}}_{E}}\succeq
0}h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})$
$\displaystyle\buildrel\Delta\over{=}\mathop{\text{missing}}{max}\limits_{{{\bf{U}}_{E}},{{\bf{W}}_{E}}\succeq
0}\log\left|{{{\bf{W}}_{E}}}\right|-{\rm{Tr}}({{\bf{W}}_{E}}{{\bf{E}}_{E}}({{\bf{U}}_{E}},{{\bf{V}}_{E}}))+{N}_{t},$
(17)

$h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})$ is concave w.r.t each
matrix of the matrices ${{\bf{U}}_{E}}$,${{\bf{V}}_{E}}$,${{\bf{W}}_{E}}$ when
the other two matrices are given. According to the facts 1) and 2) of Lemma
4.1 in [48], the optimal ${{\bf{U}}^{\star}_{E}}$, ${{\bf{W}}^{\star}_{E}}$ to
achieve the maximum value of
$h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})$ is given by

$\displaystyle{{\bf{U}}^{\star}_{E}}{\rm{=}}\text{arg}\mathop{\text{missing}}{max}\limits_{{{\bf{U}}_{E}}}h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}}){\rm{=}}(\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}}{\rm{+}}{\hat{\bf{H}}_{E}}{{\bf{V}}_{E}}{{{\bf{V}}^{H}_{E}}}{\hat{\bf{H}}_{E}}^{H})^{-1}{\hat{\bf{H}}_{E}}{{\bf{V}}_{E}},$
(18)
$\displaystyle{{\bf{W}}^{\star}_{E}}{\rm{=}}\text{arg}\mathop{\text{missing}}{max}\limits_{{{\bf{W}}_{E}}\succeq
0}h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}}){\rm{=[}}{{\bf{E}}^{\star}_{E}}({{\bf{U}}^{\star}_{E}},{{\bf{V}}_{E}}){]^{-1}},$
(19)

where ${{\bf{E}}^{\star}_{E}}$ is obtained by plugging the expression of
${{\bf{U}}^{\star}_{E}}$ into ${{\bf{E}}_{E}}({{\bf{U}}_{E}},{{\bf{V}}_{E}})$
as

$\displaystyle\begin{array}[]{l}{{\bf{E}}^{\star}_{E}}({{\bf{U}}^{\star}_{E}},{{\bf{V}}_{E}})=({{\bf{U}}_{E}}^{{\star}H}{{\hat{\bf{H}}}_{E}}{\bf{V}}_{E}-{\bf{I}}_{N_{T}}){({{\bf{U}}^{{\star}H}_{E}}{{\hat{\bf{H}}}_{E}}{\bf{V}}_{E}-{\bf{I}}_{N_{T}})^{H}}+{{\bf{U}}^{{\star}H}_{E}}(\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}}){{\bf{U}}^{\star}_{E}}.\end{array}$
(21)

By using the Lemma 1 in [13], we have

$\displaystyle f_{3}$
$\displaystyle{\rm{=}}\mathop{\text{missing}}{max}\limits_{{{\bf{W}}_{X}}\succeq
0}h_{3}({{\bf{V}}},{{\bf{V}}_{E}},{{\bf{W}}_{X}})$
$\displaystyle{\rm{=}}\mathop{\text{missing}}{max}\limits_{{{\bf{W}}_{X}}\succeq
0}\log\left|{{{\bf{W}}_{X}}}\right|-{\rm{Tr}}({{\bf{W}}_{X}}{{\bf{E}}_{X}}({{\bf{V}}},{{\bf{V}}_{E}}))+{N}_{E},$
(22)

where ${{\bf{W}}_{X}}\succeq 0$,
${{\bf{W}}_{X}}\in{{\mathbb{C}}^{{{N}_{E}}\times{{N}_{E}}}}$ are the
introduced auxiliary variable, and

$\displaystyle\begin{array}[]{l}{{\bf{E}}_{X}}({{\bf{V}}},{{\bf{V}}_{E}})\buildrel\Delta\over{=}{{{\bf{I}}_{{N_{E}}}}+\sigma_{E}^{-2}{{\hat{\bf{H}}}_{E}}({\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}})\hat{\bf{H}}_{E}^{H}}.\end{array}$
(24)

$h_{3}({{\bf{V}}},{{\bf{V}}_{E}},{{\bf{W}}_{X}})$ is concave w.r.t each matrix
of the matrices ${{\bf{V}}},{{\bf{V}}_{E}},{{\bf{W}}_{X}}$ when the other two
matrices are given. The optimal ${{\bf{W}}^{\star}_{X}}$ to achieve the
maximum value of $h_{3}({{\bf{V}}},{{\bf{V}}_{E}},{{\bf{W}}_{X}})$ is

$\displaystyle{{\bf{W}}^{\star}_{X}}{\rm{=}}\text{arg}\mathop{\text{missing}}{max}\limits_{{{\bf{W}}_{X}}\succeq
0}h_{3}({{\bf{V}}},{{\bf{V}}_{E}},{{\bf{W}}_{X}}){\rm{=[}}{{\bf{E}}_{X}}({{\bf{V}}},{{\bf{V}}_{E}}){]^{-1}}.$
(25)

By substituting (III-A), (III-A), (III-A) into (III-A), we have

$\displaystyle{{\rm{C}}_{AN}}{\rm{(}}{\bf{V}},{{\bf{V}}_{E}}{\rm{)}}$
$\displaystyle=\mathop{\text{missing}}{argmax}\limits_{{{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}}}{{\rm{C}}^{l}_{AN}}({{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}},{\bf{V}},{{\bf{V}}_{E}}),$
(26)

where

$\displaystyle{{\rm{C}}^{l}_{AN}}({{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}},{\bf{V}},{{\bf{V}}_{E}})\buildrel\Delta\over{=}$
$\displaystyle
h_{1}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I}})+h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})$
$\displaystyle+h_{3}({{\bf{V}}},{{\bf{V}}_{E}},{{\bf{W}}_{X}}).$ (27)

Obviously,
${{\rm{C}}^{l}_{AN}}({{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}},{\bf{V}},{{\bf{V}}_{E}})$
is a concave function for each of the matrices
${{\bf{U}}_{I}}$,${{\bf{W}}_{I}}$,${{\bf{U}}_{E}}$,${{\bf{W}}_{E}}$,${{\bf{W}}_{X}}$,${\bf{V}}$,${{\bf{V}}_{E}}$
when the other six matrices are given. By substituting (26) into Problem (10),
we have the following equivalent problem:

$\displaystyle\ \underset{{{\bf{U}}_{I}},{{\bf{W}}_{I}}\succeq
0,{{\bf{U}}_{E}},{{\bf{W}}_{E}}\succeq 0,{{\bf{W}}_{X}}\succeq
0,{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}}{\mathop{\text{missing}}{max}}\ \
{{\rm{C}}^{l}_{AN}}({{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}},{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}})$
(28a) $\displaystyle\quad\quad\quad\quad\quad\quad\text{s.t.}\quad\
{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}_{E}}^{H}{\rm{)}}\leq{P_{T}},$
(28b) $\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\ \
\left|{{{{\phi}}_{m}}}\right|=1,m=1,\cdots,M.$ (28c)

To solve Problem (28), we apply the BCD method, each iteration of which
consists the following two sub-iterations. Firstly, with given
${\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}$, update
${{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}}$
by using (14), (15), (18), (19), (25) respectively. Secondly, with given
${{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}}$,
update ${\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}$ by solving the following
subproblem:

$\displaystyle\
\underset{{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}}{\mathop{\text{missing}}{min}}\
\
-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}})-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{U}}^{H}_{I}}{{{\mathbf{\hat{H}}}}_{I}}\mathbf{V})+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V}}\mathbf{V})$
$\displaystyle\quad\quad\quad\quad-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}^{H}_{E}}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}})-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}^{H}_{E}}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+\text{Tr}({{\mathbf{V}}^{H}_{E}}{{\mathbf{H}}_{VE}}{{\mathbf{V}}_{E}})$
(29a) $\displaystyle\ \
\text{s.t.}\quad{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}{\rm{)}}\leq
P_{T},$ (29b)
$\displaystyle\quad\quad\quad\\!\left|{{{{\phi}}_{m}}}\right|=1,m=1,\cdots,M,$
(29c)

where

$\displaystyle{{\mathbf{H}}_{V}}={{\mathbf{\hat{H}}}_{I}}^{H}{{\mathbf{U}}_{I}}{{\mathbf{W}}_{I}}{{\mathbf{U}}^{H}_{I}}{{\mathbf{\hat{H}}}_{I}}+\sigma_{E}^{-2}\mathbf{\hat{H}}_{E}^{H}{{\mathbf{W}}_{X}}{{\mathbf{\hat{H}}}_{E}},$
(30)
$\displaystyle{{\mathbf{H}}_{VE}}={{\mathbf{\hat{H}}}_{I}}^{H}{{\mathbf{U}}_{I}}{{\mathbf{W}}_{I}}{{\mathbf{U}}^{H}_{I}}{{\mathbf{\hat{H}}}_{I}}+{{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{U}}_{E}}{{\mathbf{W}}_{E}}{{\mathbf{U}}^{H}_{E}}{{\mathbf{\hat{H}}}_{E}}+\sigma_{E}^{-2}{{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{W}}_{X}}{{\mathbf{\hat{H}}}_{E}}.$
(31)

Problem (29) is obtained from Problem (28) by taking the
${{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}}$
as constant values, and the specific derivations are given in Appendix A.

It is obvious that Problem (29) is much easier to tackle than Problem (10) due
to the convex quadratic OF in (29a). Now, we devote to solve Problem (29)
equivalently instead of Problem (10), and the matrices ${\bf{V}}$,
${\bf{V}}_{E}$, and phase shift matrix $\mathbf{\Phi}$ will be optimized.

###  III-B Optimizing the Matrices ${\bf{V}}$ and ${\bf{V}}_{E}$

In this subsection, the TPC matrix ${\bf{V}}$ and matrix ${\bf{V}}_{E}$ are
optimized by fixing $\mathbf{\Phi}$. Specifically, the unit modulus constraint
on the phase shifts $\mathbf{\Phi}$ is removed, and the updated optimization
problem reduced from Problem (29) is given by

$\displaystyle\ \
\underset{{\bf{V}},{{\bf{V}}_{E}}}{\mathop{\text{missing}}{min}}\ \
-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}})-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{U}}^{H}_{I}}{{{\mathbf{\hat{H}}}}_{I}}\mathbf{V})+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V}}\mathbf{V})$
$\displaystyle\quad\quad\quad\quad-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}^{H}_{E}}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}})-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}^{H}_{E}}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+\text{Tr}({{\mathbf{V}}^{H}_{E}}{{\mathbf{H}}_{VE}}{{\mathbf{V}}_{E}})$
(32a) $\displaystyle\ \
\text{s.t.}\quad{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}^{H}_{E}}{\rm{)}}\leq
P_{T}.$ (32b)

The above problem is a convex QCQP problem, and the standard optimization
packages, such as CVX [49] can be exploited to solve it. However, the
calculation burden is heavy. To reduce the complexity, the near-optimal closed
form expressions of the TPC matrix and AN covariance matrix are provided by
applying the Lagrangian multiplier method.

Since Problem (32) is a convex problem, the Slater’s condition is satisfied,
where the duality gap between Problem (32) and its dual problem is zero. Thus,
Problem (32) can be solved by addressing its dual problem if the dual problem
is easier. For this purpose, by introducing Lagrange multiplier $\lambda$ to
combine the the constraint and OF of Problem (32), the Lagrangian function of
Problem (32) is obtained as

$\displaystyle\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right)$
$\displaystyle\\!\buildrel\Delta\over{=}\\!-{\rm{Tr}}\left({{\bf{W}}_{I}}{{\bf{V}}^{H}}{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}\right)\\!-\\!{\rm{Tr}}\left({{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{{\bf{\hat{H}}}}_{I}}{\bf{V}}}\right)\\!+\\!{\rm{Tr}}\left({{{\bf{V}}^{H}}{{\bf{H}}_{V}}{\bf{V}}}\right)\\!-\\!{\rm{Tr}}\left({{{\bf{W}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}}\right)$
$\displaystyle\quad-{\rm{Tr}}\left({{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}}\right)+{\rm{Tr}}\left({{\bf{V}}_{E}^{H}{{\bf{H}}_{VE}}{{\bf{V}}_{E}}}\right)+\lambda[{{\rm{Tr}}\left({{\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}}\right)}-P_{T}]$
$\displaystyle=-{\rm{Tr}}\left({{{\bf{W}}_{I}}{{\bf{V}}^{H}}{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}}\right)-{\rm{Tr}}\left({{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{{\bf{\hat{H}}}}_{I}}{\bf{V}}}\right)+{\rm{Tr}}\left[{{{\bf{V}}^{H}}\left({{{\bf{H}}_{V}}+\lambda{\bf{I}}}\right){\bf{V}}}\right]$
$\displaystyle\quad-{\rm{Tr}}\left({{{\bf{W}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}}\right)\\!-\\!{\rm{Tr}}\left({{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}}\right)\\!+\\!{\rm{Tr}}\left[{{\bf{V}}_{E}^{H}\left({{{\bf{H}}_{VE}}\\!+\\!\lambda{\bf{I}}}\right){{\bf{V}}_{E}}}\right]\\!-\\!\lambda{P_{T}}.$
(33)

Then the dual problem of Problem (32) is

$\displaystyle\mathop{\max}\limits_{\lambda}\quad\quad{\rm{}}h\left(\lambda\right)$
(34a) $\displaystyle\text{s.t.}\quad\quad{\rm{}}\lambda\geq 0,$ (34b)

where $h\left(\lambda\right)$ is the dual function given by

$\displaystyle
h\left(\lambda\right)\buildrel\Delta\over{=}\mathop{\min}\limits_{{\bf{V}},{{\bf{V}}_{E}}}{\rm{}}\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right).$
(35)

Note that Problem (35) is a convex quadratic optimization problem with no
constraint, which can be solved in closed form. The optimal solution
${\bf{V}^{\star}},{{\bf{V}^{{\star}}}_{E}}$ for Problem (35) is

$\displaystyle[{\bf{V}^{\star}},{{\bf{V}^{{\star}}}_{E}}]=\text{arg}\mathop{\min}\limits_{{\bf{V}},{{\bf{V}}_{E}}}{\rm{}}\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right).$
(36)

By setting the first-order derivative of
$\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right)$ w.r.t.
${{{\bf{V}}}}$ to zero matrix, we can obtain the optimal solution of
${\bf{V}}$ as follows:

$\displaystyle\frac{\partial{\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right)}}{{\partial{\bf{V}}}}=\bf{0},$
(37a)
$\displaystyle\frac{\partial{\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right)}}{{\partial{{\bf{V}}_{E}}}}=\bf{0}.$
(37b)

The left hand side of Equation (37a) can be expanded as

$\displaystyle\frac{\partial{\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right)}}{{\partial{\bf{V}}}}$
$\displaystyle=\frac{{\partial{\rm{Tr}}\left[{{{\bf{V}}^{H}}\left({{{\bf{H}}_{V}}+\lambda{\bf{I}}}\right){\bf{V}}}\right]}}{{\partial{\bf{V}}}}-\left({{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{{\bf{\hat{H}}}}_{I}}}\right)^{H}-\left({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}}\right)$
$\displaystyle=2\left({{{\bf{H}}_{V}}+\lambda{\bf{I}}}\right){\bf{V}}-2\left({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}}\right).$
(38)

The equation (37a) becomes

$\displaystyle\left({{{\bf{H}}_{V}}+\lambda{\bf{I}}}\right){\bf{V}}=\left({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}}\right).$
(39)

Then the optimal solution ${\bf{V}^{\star}}$ for Problem (36) is

$\displaystyle{{\bf{V}}^{\star}}$
$\displaystyle=\left({{{\bf{H}}_{V}}+\lambda{\bf{I}}}\right)^{{\dagger}}\left({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}}\right)$
$\displaystyle\buildrel\Delta\over{=}{{\bf{\Theta}}_{V}}\left(\lambda\right)\left({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}}\right).$
(40)

Similarly, we solve Problem (36) by setting the first-order derivative of
$\mathcal{L}\left({{\bf{V}},{{\bf{V}}_{E}},\lambda}\right)$ w.r.t.
${{{\bf{V}}_{E}}}$ to zero matrix, which becomes

$\displaystyle
2\left({{{\bf{H}}_{VE}}+\lambda{\bf{I}}}\right){{\bf{V}}_{E}}-2{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}=\bf{0}.$
(41)

Then the optimal solution ${\bf{V}}_{E}^{\star}$ for Problem (36) is

$\displaystyle{\bf{V}}_{E}^{\star}$
$\displaystyle=\left({{{\bf{H}}_{VE}}+\lambda{\bf{I}}}\right)^{{\dagger}}{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}$
$\displaystyle\buildrel\Delta\over{=}{{\bf{\Theta}}_{VE}}\left(\lambda\right){\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}.$
(42)

Once the optimal solution ${\lambda}^{\star}$ for Problem (34) is found, the
final optimal ${\bf{V}^{\star}},{\bf{V}}_{E}^{\star}$ can be obtained. The
value of ${\lambda}^{\star}$ should be chosen in order to guarantee the
complementary slackness condition as

$\displaystyle\lambda[{\rm{Tr(}}{\bf{V}^{\star}}{{\bf{V}}^{{\star}H}}{\rm{+}}{{\bf{V}}_{E}^{{\star}}}{{\bf{V}}^{{\star}H}_{E}}{\rm{)}}-P_{T}]=0.$
(43)

We define

$\displaystyle P(\lambda)$
$\displaystyle\buildrel\Delta\over{=}{\rm{Tr(}}{\bf{V}^{\star}}{{\bf{V}}^{{\star}H}}{\rm{+}}{{\bf{V}}_{E}^{{\star}}}{{\bf{V}}^{{\star}H}_{E}}{\rm{)}}={\rm{Tr(}}{\bf{V}^{\star}}{{\bf{V}}^{{\star}H}}{\rm{)}}+{\rm{Tr(}}{{\bf{V}}_{E}^{{\star}}}{{\bf{V}}^{{\star}H}_{E}}{\rm{)}},$
(44)

where

$\displaystyle{\rm{Tr}}\left({{\bf{V}}^{\star}{{\bf{V}}^{{\star}H}}}\right)$
$\displaystyle={\rm{Tr}}\left({{{\bf{\Theta}}_{V}}\left(\lambda\right)({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})}({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})^{H}{{\bf{\Theta}}^{H}_{V}}\left(\lambda\right)\right)$
$\displaystyle={\rm{Tr}}\left({{{\bf{\Theta}}^{H}_{V}}\left(\lambda\right){{\bf{\Theta}}_{V}}\left(\lambda\right)({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})}({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})^{H}\right),$
(45)
$\displaystyle{\rm{Tr}}\left({{{\bf{V}}_{E}^{{\star}H}}{\bf{V}}_{E}^{\star}}\right)$
$\displaystyle={\rm{Tr}}\left({{{\bf{\Theta}}_{VE}}\left(\lambda\right)({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})}({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})^{H}{{\bf{\Theta}}^{H}_{VE}}\left(\lambda\right)\right)$
$\displaystyle={\rm{Tr}}\left({{{\bf{\Theta}}^{H}_{VE}}\left(\lambda\right){{\bf{\Theta}}_{VE}}\left(\lambda\right)({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})}({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})^{H}\right).$
(46)

Then $P(\lambda)$ becomes

$\displaystyle
P(\lambda)={\rm{Tr}}\left({{{\bf{\Theta}}^{n}_{V}}({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})^{H}}\right)+{\rm{Tr}}\left({{{\bf{\Theta}}^{n}_{VE}}({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})^{H}}\right),$
(47)

where

$\displaystyle{{\bf{\Theta}}^{n}_{V}}$
$\displaystyle={{\bf{\Theta}}^{H}_{V}}\left(\lambda\right){{\bf{\Theta}}_{V}}\left(\lambda\right)=\left({{{\bf{H}}_{V}}+\lambda{\bf{I}}}\right)^{{\dagger}H}\left({{{\bf{H}}_{V}}+\lambda{\bf{I}}}\right)^{{\dagger}},$
(48) $\displaystyle{{\bf{\Theta}}^{n}_{VE}}$
$\displaystyle={{\bf{\Theta}}^{H}_{VE}}\left(\lambda\right){{\bf{\Theta}}_{VE}}\left(\lambda\right)=\left({{{\bf{H}}_{VE}}+\lambda{\bf{I}}}\right)^{{\dagger}H}\left({{{\bf{H}}_{VE}}+\lambda{\bf{I}}}\right)^{{\dagger}}.$
(49)

To find the optimal ${\lambda^{\star}}\geq 0$, we first check whether
$\lambda=0$ is the optimal solution or not. If

$\displaystyle
P(0)={\rm{Tr}}\left({{{\bf{V}}^{{\star}H}}(0){\bf{V}}^{\star}(0)}\right)+{\rm{Tr}}\left({{\bf{V}}_{E}^{{\star}H}(0){{\bf{V}}_{E}}^{\star}(0)}\right)\leq{P_{T}},$
(50)

then the optimal solutions are given by ${{\bf{V}}^{\star}}={{\bf{V}}}(0)$ and
${{\bf{V}}_{E}^{\star}}={{\bf{V}}_{E}}(0)$. Otherwise, the optimal
$\lambda^{\star}>0$ is the solution of the equation $P(\lambda)=0$.

It is ready to verify that ${{\bf{H}}_{V}}$ and ${{\bf{H}}_{VE}}$ is a
positive semidefinite matrix. Let us define the rank of ${{\bf{H}}_{V}}$ and
${{\bf{H}}_{VE}}$ as $r_{V}={\rm{rank}}({\bf{H}}_{V})\leq N_{T}$ and
$r_{VE}={\rm{rank}}({\bf{H}}_{VE})\leq N_{T}$ respectively. By decomposing
${{\bf{H}}_{V}}$ and ${{\bf{H}}_{VE}}$ by using the singular value
decomposition (SVD), we have

${{\bf{H}}_{V}}=\left[{{{\bf{P}}_{V,1}},{{\bf{P}}_{V,2}}}\right]{{\bf{\Sigma}}_{V}}{\left[{{{\bf{P}}_{V,1}},{{\bf{P}}_{V,2}}}\right]^{\rm{H}}},{{\bf{H}}_{VE}}=\left[{{{\bf{P}}_{{VE},1}},{{\bf{P}}_{{VE},2}}}\right]{{\bf{\Sigma}}_{VE}}{\left[{{{\bf{P}}_{{VE},1}},{{\bf{P}}_{{VE},2}}}\right]^{\rm{H}}},$
(51)

where ${\bf{P}}_{V,1}$ comprises the first $r_{V}$ singular vectors associated
with the $r_{V}$ positive eigenvalues of ${{\bf{H}}_{V}}$, and
${\bf{P}}_{V,2}$ includes the last $N_{T}-r_{V}$ singular vectors associated
with the $N_{T}-r_{V}$ zero-valued eigenvalues of ${{\bf{H}}_{V}}$,
${{\bm{\Sigma}}_{V}}={\rm{diag}}\left\\{{{{\bm{\Sigma}}_{V,1}},{{\bf{0}}_{\left({{N_{T}}-{r_{V}}}\right)\times\left({{N_{T}}-{r_{V}}}\right)}}}\right\\}$
with ${\bm{\Sigma}}_{V,1}$ representing the diagonal submatrix collecting the
first $r_{V}$ positive eigenvalues. Similarly, the first $r_{VE}$ singular
vectors corresponding to the $r_{VE}$ positive eigenvalues of
${{\bf{H}}_{VE}}$ are contained in ${\bf{P}}_{VE,1}$, while the last
$N_{T}-r_{VE}$ singular vectors corresponding to the $N_{T}-r_{VE}$ zero-
valued eigenvalues of ${{\bf{H}}_{VE}}$ are held in ${\bf{P}}_{VE,2}$.
${{\bm{\Sigma}}_{VE}}={\rm{diag}}\left\\{{{{\bm{\Sigma}}_{{VE},1}},{{\bf{0}}_{\left({{N_{T}}-{r_{VE}}}\right)\times\left({{N_{T}}-{r_{VE}}}\right)}}}\right\\}$
is a diagonal matrix with ${\bm{\Sigma}}_{{VE},1}$ representing the diagonal
submatrix gathering the first $r_{VE}$ positive eigenvalues. By defining
${{\bf{P}}_{V}}\buildrel\Delta\over{=}\left[{{{\bf{P}}_{V,1}},{{\bf{P}}_{V,2}}}\right]$
and
${{\bf{P}}_{VE}}\buildrel\Delta\over{=}\left[{{{\bf{P}}_{{VE},1}},{{\bf{P}}_{{VE},2}}}\right]$,
and substituting (51) into (48) and (49), $P(\lambda)$ becomes

$\displaystyle
P(\lambda)={\rm{Tr}}\left({[{\left({{{\bf{P}}_{V}}{{\bf{\Sigma}}_{V}}{\bf{P}}_{V}^{H}+\lambda{{\bf{P}}_{V}}{\bf{P}}_{V}^{H}}\right)^{-1}}{\left({{{\bf{P}}_{V}}{{\bf{\Sigma}}_{V}}{\bf{P}}_{V}^{H}+\lambda{{\bf{P}}_{V}}{\bf{P}}_{V}^{H}}\right)^{-1}}]({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})^{H}}\right)$
$\displaystyle\\!\\!+\\!\\!{\rm{Tr}}\\!\left(\\!{[\\!{\left(\\!{{{\bf{P}}_{VE}}{{\bf{\Sigma}}_{VE}}{\bf{P}}_{VE}^{H}\\!+\\!\lambda{{\bf{P}}_{VE}}{\bf{P}}_{VE}^{H}}\\!\right)^{-1}}\\!\\!{\left({{{\bf{P}}_{VE}}{{\bf{\Sigma}}_{VE}}{\bf{P}}_{VE}^{H}\\!+\\!\lambda{{\bf{P}}_{VE}}{\bf{P}}_{VE}^{H}}\right)^{-1}}\\!]\\!(\\!{{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}}\\!)\\!({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})^{H}}\\!\right)$
$\displaystyle={\rm{Tr}}\left({[{\left({{{\bf{\Sigma}}_{V}}+\lambda{\bf{I}}}\right)^{-2}}]{\bf{Z}}_{V}}\right)+{\rm{Tr}}\left({[{\left({{{\bf{\Sigma}}_{VE}}+\lambda{\bf{I}}}\right)^{-2}}]{\bf{Z}}_{VE}}\right)$
$\displaystyle{=}\sum\limits_{i=1}^{r_{V}}\left[{\frac{{{\left[{{\bf{Z}}_{V}}\right]}_{i,i}}}{{{\left({{{\left[{{\Sigma}_{V}}\right]}_{i,i}}\\!+\\!\lambda}\right)}^{2}}}}\right]+\sum\limits_{i=1}^{r_{VE}}\left[{\frac{{{\left[{{\bf{Z}}_{VE}}\right]}_{i,i}}}{{{\left({{{\left[{{\Sigma}_{VE}}\right]}_{i,i}}\\!+\\!\lambda}\right)}^{2}}}}\right]+\sum\limits_{i={r_{V}}+1}^{{N_{T}}}{\left[{\frac{{{\left[{{\bf{Z}}_{V}}\right]}_{i,i}}}{{{\left({\lambda}\right)}^{2}}}}\right]}\\!+\\!\sum\limits_{i={r_{VE}}+1}^{{N_{T}}}{\left[{\frac{{{\left[{{\bf{Z}}_{VE}}\right]}_{i,i}}}{{{\left({\lambda}\right)}^{2}}}}\right]},$
(52)

where
${{\bf{Z}}_{V}}={\bf{P}}_{V}^{H}({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}})^{H}{{\bf{P}}_{V}}$
and
${{\bf{Z}}_{VE}}={\bf{P}}_{VE}^{H}({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})({{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}})^{H}{{\bf{P}}_{VE}}$.
${\left[{{{\bf{Z}}_{V}}}\right]}_{i,i}$,
${\left[{{{\bf{Z}}_{VE}}}\right]}_{i,i}$,
${\left[{{{\Sigma}}_{V}}\right]}_{i,i}$, and
${\left[{{{\Sigma}}_{VE}}\right]}_{i,i}$ represent the $i$th diagonal element
of matrices ${{{\bf{Z}}_{V}}}$, ${{\bf{Z}}_{VE}}$, ${{{\Sigma}}_{V}}$, and
${{{\Sigma}}_{VE}}$, respectively. The first line of (III-B) is obtained by
substituting (51) into the expression of $P({\lambda})$ in (47). It can be
verified from the last line of (III-B) that $P({\lambda})$ is a monotonically
decreasing function.

Then, the optimal $\lambda^{\star}$ can be obtained by solving the following
equation,

$\displaystyle\sum\limits_{i=1}^{r_{V}}\\!\left[{\frac{{{\left[{{{\bf{Z}}_{V}}}\right]}_{i,i}}}{{{{\left({{{\left[{{{\Sigma}}_{V}}\right]}_{i,i}}+\lambda}\right)}^{2}}}}}\right]\\!+\\!\\!\sum\limits_{i=1}^{r_{VE}}\\!\left[{\frac{{{\left[{{{\bf{Z}}_{VE}}}\right]}_{i,i}}}{{{{\left({{{\left[{{{\Sigma}}_{VE}}\right]}_{i,i}}+\lambda}\right)}^{2}}}}}\right]\\!+\\!\\!\sum\limits_{i={r_{V}}+1}^{{N_{T}}}\\!{\left[{\frac{{{\left[{{{\bf{Z}}_{V}}}\right]}_{i,i}}}{{{{\left({\lambda}\right)}^{2}}}}}\right]}\\!+\\!\\!\sum\limits_{i={r_{VE}}+1}^{{N_{T}}}\\!{\left[{\frac{{{\left[{{{\bf{Z}}_{VE}}}\right]}_{i,i}}}{{{{\left({\lambda}\right)}^{2}}}}}\right]}=P_{T}.$
(53)

To solve it, the bisection search method is utilized. Since $P(\infty)=0$, the
solution to Equation (53) must exist. The lower bound of $\lambda^{\star}$ is
a positive value approaching zero, while the upper bound of $\lambda^{\star}$
is given by

${\lambda^{\star}}<\sqrt{\frac{{\sum\limits_{i=1}^{{N_{T}}}{{{\left[{{{\bf{Z}}_{V}}}\right]}_{i,i}}}}+{\sum\limits_{i=1}^{{N_{T}}}{{{\left[{{{\bf{Z}}_{VE}}}\right]}_{i,i}}}}}{{{P_{T}}}}}\buildrel\Delta\over{=}\lambda^{{\rm{ub}}}.$
(54)

which can be proved as

$\displaystyle{P}(\lambda^{{\rm{ub}}})$
$\displaystyle=\sum\limits_{i=1}^{r_{V}}{\frac{{{{\left[{{{\bf{Z}}_{V}}}\right]}_{i,i}}}}{{{{\left({{{\left[{{{\Sigma}}_{V}}\right]}_{i,i}}+{\lambda^{{\rm{ub}}}}}\right)}^{2}}}}}+\sum\limits_{i=1}^{r_{VE}}{\frac{{{{\left[{{{\bf{Z}}_{VE}}}\right]}_{i,i}}}}{{{{\left({{{\left[{{{\Sigma}}_{VE}}\right]}_{i,i}}+{\lambda^{{\rm{ub}}}}}\right)}^{2}}}}}+\sum\limits_{i={r_{V}}+1}^{{N_{T}}}{\left[{\frac{{{\left[{{\bf{Z}}_{V}}\right]}_{i,i}}}{{{\left({\lambda^{{\rm{ub}}}}\right)}^{2}}}}\right]}\\!+\\!\sum\limits_{i={r_{VE}}+1}^{{N_{T}}}{\left[{\frac{{{\left[{{\bf{Z}}_{VE}}\right]}_{i,i}}}{{{\left({\lambda^{{\rm{ub}}}}\right)}^{2}}}}\right]}$
$\displaystyle<\sum\limits_{i=1}^{{N_{T}}}{\frac{{{{\left[{{{\bf{Z}}_{V}}}\right]}_{i,i}}}}{{{{\left({\lambda^{{\rm{ub}}}}\right)}^{2}}}}}+\sum\limits_{i=1}^{{N_{T}}}{\frac{{{{\left[{{{\bf{Z}}_{VE}}}\right]}_{i,i}}}}{{{{\left({\lambda^{{\rm{ub}}}}\right)}^{2}}}}}={P_{T}}.$
(55)

When the optimal $\lambda^{{\star}}$ is found, the optimal matrices
${{\bf{V}}^{{\star}}}$ and ${{\bf{V}}_{E}^{{\star}}}$ can be obtained by
substituting $\lambda^{\star}$ into (III-B) and (III-B).

###  III-C Optimizing the Phase Shifts $\mathbf{\Phi}$

In this subsection, the phase shift matrix $\mathbf{\Phi}$ is optimized by
fixing ${{\bf{V}}}$ and ${{\bf{V}}_{E}}$. The transmit power constraint in
Problem (29) is only related with ${{\bf{V}}}$ and ${{\bf{V}}_{E}}$, thus is
removed. Then, the optimization problem for $\mathbf{\Phi}$ reduced from
Problem (29) is formulated as

$\displaystyle\ \ \underset{{\bf{\Phi}}}{\mathop{\text{missing}}{min}}\ \
{g_{0}}(\mathbf{\Phi})\buildrel\Delta\over{=}-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}})-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{{\mathbf{\hat{H}}}}_{I}}\mathbf{V})+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V}}\mathbf{V})$
$\displaystyle\quad\quad\quad\quad\quad\quad\
-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}})-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+\text{Tr}({{\mathbf{V}}_{E}}^{H}{{\mathbf{H}}_{VE}}{{\mathbf{V}}_{E}})$
(56a) $\displaystyle\ \
\text{s.t.}\quad\left|{{{{\phi}}_{m}}}\right|=1,m=1,\cdots,M.$ (56b)

By the aid of complex mathematical manipulations, which are given in details
in Appendix B, Problem (56) can be transformed into a form that can facilitate
the MM algorithm. Based on the derivations in Appendix B, the OF
${g_{0}}(\mathbf{\Phi})$ can be equivalently transformed into

$\displaystyle{g_{0}}(\mathbf{\Phi})={\rm{Tr}}\left({{{\bf{\Phi}}^{H}{\bf{D}}^{H}}}\right)+{\rm{Tr}}\left({{\bf{\Phi
D}}}\right)+{\rm{Tr}}\left[{{{\bf{\Phi}}^{H}}{{\bf{B}}_{VE}}{\bf{\Phi}}{{\bf{C}}_{VE}}}\right]+{\rm{Tr}}\left({{{\bf{\Phi}}^{H}}{{\bf{B}}_{V}}{\bf{\Phi}}{{\bf{C}}_{V}}}\right)+C_{t},$
(57)

where $C_{t}$, ${\bf{D}}$, ${{\bf{C}}_{VE}}$, ${{\bf{C}}_{V}}$,
${{\bf{B}}_{VE}}$ and ${{\bf{B}}_{V}}$ are constants for ${\bf{\Phi}}$, and
are given in Appendix B.

By exploiting the matrix properties in [50, Eq. (1.10.6)], the trace operators
can be removed, and the third and fourth terms in (57) become as

$\displaystyle{\rm{Tr}}\left({{{\bm{\Phi}}^{\rm{H}}}{\bf{B}}_{VE}{\bm{\Phi}}{\bf{C}}_{VE}}\right)={{\bm{\phi}}^{\rm{H}}}\left({{\bf{B}}_{VE}\odot{{\bf{C}}_{VE}^{\rm{T}}}}\right){\bm{\phi}},$
(58a)
$\displaystyle{\rm{Tr}}\left({{{\bm{\Phi}}^{\rm{H}}}{\bf{B}}_{V}{\bm{\Phi}}{\bf{C}}_{V}}\right)={{\bm{\phi}}^{\rm{H}}}\left({{\bf{B}}_{V}\odot{{\bf{C}}_{V}^{\rm{T}}}}\right){\bm{\phi}},$
(58b)

where
${\bm{\phi}}\buildrel\Delta\over{=}{\left[{{e^{j{\theta_{1}}}},\cdots,{e^{j{\theta_{m}}}},\cdots,{e^{j{\theta_{M}}}}}\right]^{\rm{T}}}$
is a vector holding the diagonal elements of ${\bm{\Phi}}$.

Similarly, the trace operators can be removed for the first and second terms
in (57) as

${\rm{Tr}}\left({{{\bm{\Phi}}^{\rm{H}}}{{\bf{D}}^{\rm{H}}}}\right)={{\bf{d}}^{\rm{H}}}({{\bm{\phi}}}^{*}),{\rm{Tr}}\left({{\bm{\Phi}}{\bf{D}}}\right)={\bm{\phi}}^{\rm{T}}{\bf{d}},$
(59)

where
${\bf{d}}={\left[{{{\left[{\bf{D}}\right]}_{1,1}},\cdots,{{\left[{\bf{D}}\right]}_{M,M}}}\right]^{\rm{T}}}$
is a vector gathering the diagonal elements of matrix ${\bf{D}}$.

Hence, Problem (56) can be rewritten as

$\displaystyle{\mathop{\min}\limits_{{\bm{\phi}}}\quad{{\bm{\phi}}^{\rm{H}}}{\bm{\Xi}}{\bm{\phi}}+{\bm{\phi}}^{\rm{T}}{\bf{d}}+{{\bf{d}}^{\rm{H}}}({{\bm{\phi}}}^{*})}$
(60a)
$\displaystyle\textrm{s.t.}\quad\left|{{\phi_{m}}}\right|=1,m=1,\cdots,M,$
(60b)

where
$\bm{\Xi}={\bf{B}}_{VE}\odot{{\bf{C}}_{VE}^{\rm{T}}}+{\bf{B}}_{V}\odot{{\bf{C}}_{V}^{\rm{T}}}$.
$\bm{\Xi}$ is a semidefinite matrix, because it is a sum of two semidefinite
matrices, both of which are Hadamard products of two semidefinite matrices. It
is observed that ${\bf{B}}_{VE}$, ${{\bf{C}}_{VE}^{\rm{T}}}$, ${\bf{B}}_{V}$
and ${{\bf{C}}_{V}^{\rm{T}}}$ are semidefinite matrices. Then, the Hadamard
products of ${{\bf{B}}_{VE}\odot{{\bf{C}}_{VE}^{\rm{T}}}}$ and
${{\bf{B}}_{V}\odot{{\bf{C}}_{V}^{\rm{T}}}}$ are semidefinite according to the
Property (9) on Page 104 of [50]. Problem (60) can be further simplified as

$\displaystyle{\mathop{\min}\limits_{\bm{\phi}}\quad
f({\bm{\phi}})\buildrel\Delta\over{=}{{\bm{\phi}}^{\rm{H}}}{\bm{\Xi}}{\bm{\phi}}+2{\rm{Re}}\left\\{{{{\bm{\phi}}^{\rm{H}}}({{\bf{d}}}^{*})}\right\\}}$
(61a)
$\displaystyle\textrm{s.t.}\quad\left|{{\phi_{m}}}\right|=1,m=1,\cdots,M.$
(61b)

The Problem (61) can be solved by the SDR technique [28] by transforming the
unimodulus constraint into a rank-one constraint, however, the rank-one
solution cannot always be obtained and the computation complexity is heavy for
the SDR method. Thus, we propose to solve Problem (61) efficiently by the MM
algorithm as [25], where the closed-form solution can be obtained in each
iteration. Details are omitted for simplicity.

###  III-D Overall Algorithm to Solve Problem (10)

To sum up, the detailed execution of the overall BCD-MM algorithm proposed for
solving Problem (10) is provided in Algorithm 1. The MM algorithm is exploited
for solving the optimal phase shifts ${\bf{\Phi}}^{(n+1)}$ of Problem (61) in
Step 5. The iteration process in MM algorithm ensures that the OF value of
Problem (61) decreases monotonically. Moreover, the BCD algorithm also
guarantees that the OF value of Problem (29) monotonically decreases in each
step and each iteration of Algorithm 1. Since the OF value in (29a) has a
lower bound with the power limit, the convergence of Algorithm 1 is
guaranteed.

Algorithm 1 BCD-MM Algorithm

1: Parameter Setting. Set the maximum number of iterations $n_{\rm{max}}$ and
the first iterative number $n=1$; Give the error tolerance $\varepsilon$.

2: Variables Initialization. Initialize the variables ${\bf{V}}^{(1)}$,
${\bf{V}}_{E}^{(1)}$ and ${\bf{\Phi}}^{(1)}$ in the feasible region; Compute
the OF value of Problem (10) as
${\rm{OF(}}{{\bf{V}}^{(1)}},{{\bf{V}}^{(1)}_{E}},{\bf{\Phi}}^{(1)}{\rm{)}}$;

3: Auxiliary Variables Calculation. Given
${\bf{V}}^{(n)},{{\bf{V}}^{(n)}_{E}}$, ${\bf{\Phi}}^{(n)}$, compute the
optimal matrices
${{\bf{U}}^{(n)}_{I}},{{\bf{W}}^{(n)}_{I}},{{\bf{U}}^{(n)}_{E}},{{\bf{W}}^{(n)}_{E}},{{\bf{W}}^{(n)}_{X}}$
according to (14), (15), (18), (19), (25) respectively;

4: Matrices Optimization. Given
${{\bf{U}}^{(n)}_{I}},{{\bf{W}}^{(n)}_{I}},{{\bf{U}}^{(n)}_{E}},{{\bf{W}}^{(n)}_{E}},{{\bf{W}}^{(n)}_{X}}$,
solve the optimal TPC matrix ${\bf{V}}^{(n+1)}$ and equivalent AN covariance
matrix ${{\bf{V}}^{(n+1)}_{E}}$ of Problem (36) with the Lagrangian multiplier
method;

5: Phase Shifts Optimization. Given
${{\bf{U}}^{(n)}_{I}},{{\bf{W}}^{(n)}_{I}},{{\bf{U}}^{(n)}_{E}},{{\bf{W}}^{(n)}_{E}},{{\bf{W}}^{(n)}_{X}}$
and ${\bf{V}}^{(n+1)},{{\bf{V}}^{(n+1)}_{E}}$, solve the optimal phase shifts
${\bf{\Phi}}^{(n+1)}$ of Problem (61) with the MM algorithm;

6: Termination Check. If
${{\left|\\!{{\rm{OF}}\\!(\\!{\bf{V}}^{(n+1)}\\!,\\!{{\bf{V}}^{(n+1)}_{E}}\\!,\\!{\bf{\Phi}}^{(n+1)}\\!)\\!\\!-\\!\\!{\rm{OF}}\\!(\\!{\bf{V}}^{(n)}\\!,\\!{{\bf{V}}^{(n)}_{E}}\\!,\\!{\bf{\Phi}}^{(n)}\\!)}\\!\right|}/\\!{{\rm{OF}}\\!(\\!{\bf{V}}^{(n+1)}\\!,\\!{{\bf{V}}^{(n+1)}_{E}}\\!,\\!{\bf{\Phi}}^{(n+1)}\\!)}}\\!<\varepsilon$
or $n\geq n_{\rm{max}}$, terminate. Otherwise, update $n\leftarrow n+1$ and
jump to step 2.

Based on the algorithm description, the complexity analysis of the proposed
BCD-MM algorithm is performed. In Step 3, computing the decoding matrices
${{\bf{U}}^{(n)}_{I}}$ and ${{\bf{U}}^{(n)}_{E}}$ costs the complexity of
${\cal O}(N_{I}^{3})+{\cal O}(N_{E}^{3})$, while calculating the auxiliary
matrices ${{\bf{W}}^{(n)}_{I}}$, ${{\bf{W}}^{(n)}_{E}}$, and
${{\bf{W}}^{(n)}_{X}}$ consumes the complexity of ${\cal O}(d^{3})+{\cal
O}(N_{T}^{3})+{\cal O}(N_{E}^{3})$. The complexity of calculating the TPC
matrix ${\bf{V}}^{(n+1)}$ and AN covariance matrix ${{\bf{V}}^{(n+1)}_{E}}$ in
Step 4 can be analyzed according to the specific process of Lagrangian
multiplier method based on the fact that the complexity of computing product
${\bf{XY}}$ of complex matrices ${\bf{X}}\in{{\mathbb{C}}^{m\times n}}$ and
${\bf{Y}}\in{{\mathbb{C}}^{n\times p}}$ is ${\cal O}\left({mnp}\right)$. By
assuming that $N_{T}>N_{I}({\rm{or\ }}N_{E})>d$, the complexity of computing
the matrices $\\{{{\mathbf{H}}_{V}},{{\mathbf{H}}_{VE}}\\}$ in (30) and (31)
is ${\cal O}(N_{T}^{3})+{\cal O}(2N_{T}^{2}d)+{\cal O}(2N_{T}^{2}N_{E})$;
while the complexity of calculating ${\bf{V}}^{*}$, ${\bf{V}}_{E}^{*}$ in
(III-B) and (III-B) is ${\cal O}(2N_{T}^{3})$. The SVD decomposition of
$\\{{{\mathbf{H}}_{V}},{{\mathbf{H}}_{VE}}\\}$ requires the computation
complexity of ${\cal O}(2N_{T}^{3})$, while calculating $\\{{\bf{Z}}_{V}\\}$
and $\\{{\bf{Z}}_{VE}\\}$ requires the complexity of ${\cal
O}(N_{T}^{2}N_{I})+{\cal O}(2N_{T}^{3})$. The complexity of finding the
Lagrangian multipliers $\\{\lambda\\}$ is negligible. Thus, the overall
complexity for ${\bf{V}}^{(n+1)}$, ${\bf{V}}_{E}^{(n+1)}$ is about ${\cal
O}({\rm{max}}\\{2N_{T}^{3},2N_{T}^{2}N_{E}\\})$. In step 5, obtaining optimal
${\bf{\Phi}}^{(n+1)}$ by the MM algorithm need a complexity of $C_{MM}={\cal
O}(M^{3}+T_{MM}M^{2})$, where $T_{MM}$ is the iteration number for
convergence. Based on the complexity required in Step 3, 4 and 5, the overall
complexity $C_{\rm{BCD-MM}}$ of the BCD-MM algorithm can be evaluated by

$C_{\rm{BCD-MM}}={\cal O}({\rm{max}}\\{2N_{T}^{3},2N_{T}^{2}N_{E},C_{MM}\\}).$
(62)

##  IV Extension to the Multiple-IRs Scenario

###  IV-A Problem Formulation

Consider a multicast extension where there are $L\geq 2$ legitimate IRs, and
they all intend to receive the same message. The signal model for the MIMO
multi-IRs wiretap channel scenario is

$\displaystyle{\bf{y}}_{I,l}={\hat{{\bf{H}}}_{I,l}}({\bf{V}}{\bf{s}}+{\bf{n}})+{{\bf{n}}_{I,l}},l=1,\cdots,L,$
(63)

where
${{\hat{\bf{H}}}_{I,l}}\overset{\triangle}{=}{{\bf{H}}_{b,I,l}}+{{\bf{H}}_{R,I,l}}{\bf{\Phi}}{\bf{G}}$.
The subscript $l$ indicates the $l$th IR, and the other notations are the same
as (4) and (6). Under these settings, the achievable SR is given by [51]

$\displaystyle R_{s}{\rm{(}}{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}{\rm{)}}$
$\displaystyle=\underset{l=1,\cdots,L}{\min}\\{{R_{I,l}}({\bf{V}},{\bf{\Phi}},{\bf{Z}})-{R_{E}}({\bf{V}},{\bf{\Phi}},{\bf{Z}})\\},$
(64)

where
${R_{I,l}}({\bf{V}},{\bf{\Phi}},{\bf{Z}})={\rm{log}}\left|{{\bf{I}}+{{\hat{{\bf{H}}}}_{I,l}}{\bf{V}}{{\bf{V}}^{H}}\hat{{\bf{H}}}_{I,l}^{H}{\bf{J}}_{I,l}^{-1}}\right|$
and
${{\bf{J}}_{I,l}}\overset{\triangle}{=}{{\hat{{\bf{H}}}}_{I,l}}{\bf{Z}}{{\hat{{\bf{H}}}}_{I,l}}^{H}+\sigma_{I,l}^{2}{{\bf{I}}_{{{N}_{I}}}}$.

Then the multicast counterpart of the AN-aided SRM problem (10) is formulated
as

$\displaystyle\
\underset{{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}}{\mathop{\text{max}}}\ \
{R_{s}}{\rm{(}}{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}{\rm{)}}$ (65a)
$\displaystyle\ \ \text{s.t.}\quad\
{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}_{E}^{H}}{\rm{)}}\leq{P_{T}},$
(65b)
$\displaystyle\quad\quad\quad\\!\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M.$
(65c)

The objective function of Problem (65) can be rewritten as

$\displaystyle R_{s}{\rm{(}}{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}{\rm{)}}$
$\displaystyle=\underset{l=1,\cdots,L}{\min}\\{\underbrace{{\rm{log}}\left|{{\bf{I}}_{{N_{I}}}+{{\hat{{\bf{H}}}}_{I,l}}{\bf{V}}{{\bf{V}}^{H}}\hat{{\bf{H}}}_{I,l}^{H}{{({{\hat{{\bf{H}}}}_{I,l}}{{\bf{V}}_{E}}{{\bf{V}}_{E}^{H}}{{\hat{{\bf{H}}}}_{I,l}}^{H}+\sigma_{I,l}^{2}{{\bf{I}}_{{N_{I}}}})}^{-1}}}\right|}_{{f_{1,l}}}\\}$
$\displaystyle\quad{\rm{+}}\underbrace{{\rm{log}}\left|{{{\bf{I}}_{{N_{E}}}}+{{\hat{{\bf{H}}}}_{E}}{{\bf{V}}_{E}}{{\bf{V}}_{E}^{H}}{{\hat{{\bf{H}}}}_{E}}^{H}(\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}})^{-1}}\right|}_{{f_{2}}}$
$\displaystyle\quad\underbrace{-{\rm{log}}\left|{{{\bf{I}}_{{N_{E}}}}+\sigma_{E}^{-2}{{\hat{{\bf{H}}}}_{E}}({\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{{\bf{V}}_{E}^{H}})\hat{{\bf{H}}}_{E}^{H}}\right|}_{{f_{3}}},$
(66a)
$\displaystyle=\underset{l=1,\cdots,L}{\min}\\{\mathop{\text{max}}\limits_{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\succeq
0}h_{1,l}({{\bf{U}}_{I,l}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I,l}})\\}+\mathop{\text{max}}\limits_{{{\bf{U}}_{E}},{{\bf{W}}_{E}}\succeq
0}h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})$
$\displaystyle\quad+\mathop{\text{max}}\limits_{{{\bf{W}}_{X}}\succeq
0}h_{3}({\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{X}}).$ (66b)

The lower bound to the first term of (66b) can be found as

$\displaystyle\underset{l=1,\cdots,L}{\min}\\{\mathop{\text{max}}\limits_{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\succeq
0}h_{1,l}({{\bf{U}}_{I,l}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I,l}})\\}$ (67a)
$\displaystyle\geq\mathop{\text{max}}\limits_{\\{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\succeq
0\\}_{l=1}^{L}}\\{\underset{l=1,\cdots,L}{\min}h_{1,l}({{\bf{U}}_{I,l}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I,l}})\\},$
(67b)

where (67b) holds due to the fact that $\underset{x}{\min}\
\underset{y}{\max}f(x,y)\geq\underset{y}{\max}\ \underset{x}{\min}f(x,y)$ for
any function $f(x,y)$. Here by exchanging the positions of
$\mathop{\text{max}}\limits_{\\{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\succeq
0\\}_{l=1}^{L}}$ and $\underset{l=1,\cdots,L}{\min}$ in (67a), we can find a
lower bound to $R_{s}{\rm{(}}{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}{\rm{)}}$ as

$\displaystyle
f_{ms}({\bf{V}},{{\bf{V}}_{E}},\\{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\\}_{l=1}^{L},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}})$
$\displaystyle\triangleq\mathop{\text{max}}\limits_{{\bf{V}},{{\bf{V}}_{E}},\\{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\succeq
0\\}_{l=1}^{L},{{\bf{U}}_{E}},{{\bf{W}}_{E}}\succeq 0,{{\bf{W}}_{X}}\succeq
0}\\{\underset{l=1,\cdots,L}{\min}h_{1,l}({{\bf{U}}_{I,l}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I,l}})$
$\displaystyle\quad+h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})+h_{3}({\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{X}})\\}.$
(68)

We simplify Problem (65) by maximizing a lower bound to its original objective
as follows,

$\displaystyle\
\underset{{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}},\\{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\succeq
0\\}_{l=1}^{L},{{\bf{U}}_{E}},{{\bf{W}}_{E}}\succeq 0,{{\bf{W}}_{X}}\succeq
0}{\mathop{\text{max}}}f_{ms}({\bf{V}},{{\bf{V}}_{E}},\\{{{\bf{U}}_{I,l}},{{\bf{W}}_{I,l}}\\}_{l=1}^{L},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}})$
(69a) $\displaystyle\ \ \text{s.t.}\quad\
{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}_{E}^{H}}{\rm{)}}\leq{P_{T}},$
(69b)
$\displaystyle\quad\quad\quad\\!\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M.$
(69c)

To solve the multicast AN-aided SRM problem in (69), a BCD-QCQP-CCP algorithm
is proposed.

###  IV-B BCD Iterations for Problem (69)

The equivalent SRM problem in (69) provides a desirable formulation for BCD
algorithm. In particular, one can show that problem (69) is convex w.r.t.
either ${\bf{V}},{{\bf{V}}_{E}}$ or ${\bf{\Phi}}$ or $\\{{{\bf{U}}_{I,l}}$,
${{\bf{W}}_{I,l}}\\}_{l=1}^{L}$, ${{\bf{U}}_{E}}$, ${{\bf{W}}_{E}}$,
${{\bf{W}}_{X}}$. By fixing ${\bf{\Phi}}$, the iteration process for problem
(69) is as follows. Let ${\bf{V}}^{n}$, ${{\bf{V}}_{E}^{n}}$,
$\\{{{\bf{U}}_{I,l}^{n}},{{\bf{W}}_{I,l}^{n}}\\}_{l=1}^{L},{{\bf{U}}_{E}^{n}},{{\bf{W}}_{E}^{n}},{{\bf{W}}_{X}^{n}}$
denote the BCD iterate at the $n$th iteration. The BCD iterates are generated
via

$\displaystyle{{\bf{U}}_{I,l}^{n}}=\text{arg}\mathop{\text{max}}\limits_{{{\bf{U}}_{I,l}}}h_{1,l}({{\bf{U}}_{I,l}},{\bf{V}}^{n},{{\bf{V}}_{E}^{n}},{{\bf{W}}_{I,l}^{n}})$
$\displaystyle\qquad\>\>{\rm{=}}({\hat{{\bf{H}}}_{I,l}}{{\bf{V}}_{E}^{n}}{{\bf{V}}_{E}^{nH}}{\hat{{\bf{H}}}_{I,l}}^{H}{\rm{+}}\sigma_{I,l}^{2}{{\bf{I}}_{{N_{I}}}}{\rm{+}}{\hat{{\bf{H}}}_{I,l}}{\bf{V}}^{n}{{\bf{V}}^{nH}}{\hat{{\bf{H}}}_{I,l}}^{H})^{-1}{\hat{{\bf{H}}}_{I,l}}{\bf{V}}^{n},$
(70a)
$\displaystyle{{\bf{W}}_{I,l}^{n}}=\text{arg}\mathop{\text{max}}\limits_{{{\bf{W}}_{I,l}}\succeq
0}h_{1,l}({{\bf{U}}_{I,l}^{n}},{\bf{V}}^{n},{{\bf{V}}_{E}^{n}},{{\bf{W}}_{I,l}}){\rm{=[}}{{\bf{E}}_{I,l}}({{\bf{U}}_{I,l}^{n}},{\bf{V}}^{n},{{\bf{V}}_{E}^{n}}){]^{-1}}$
$\displaystyle{\rm{=}}\text{[}({{\bf{U}}_{I,l}^{{n}H}}{\hat{{\bf{H}}}_{I,l}}{\bf{V}}^{n}-{\bf{I}}_{d}){({{\bf{U}}_{I,l}^{{n}H}}{\hat{{\bf{H}}}_{I,l}}{\bf{V}}^{n}-{\bf{I}}_{d})^{H}}+{{\bf{U}}_{I,l}^{{n}H}}({\hat{{\bf{H}}}_{I,l}}{{\bf{V}}_{E}^{n}}{{\bf{V}}_{E}^{nH}}{\hat{{\bf{H}}}_{I,l}}^{H}{\rm{+}}\sigma_{I,l}^{2}{{\bf{I}}_{{N_{I}}}}){{\bf{U}}_{I,l}^{{n}}}]^{-1},$
(70b)
$\displaystyle{{\bf{U}}_{E}^{n}}{\rm{=}}\text{arg}\mathop{\text{max}}\limits_{{{\bf{U}}_{E}}}h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}^{n}},{{\bf{W}}_{E}^{n}})$
$\displaystyle\qquad\>\>{\rm{=}}(\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}}{\rm{+}}{\hat{{\bf{H}}}_{E}}{{\bf{V}}_{E}^{n}}{{\bf{V}}_{E}^{nH}}{\hat{{\bf{H}}}_{E}}^{H})^{-1}{\hat{{\bf{H}}}_{E}}{{\bf{V}}_{E}^{n}},$
(70c)
$\displaystyle{{\bf{W}}_{E}^{n}}{=}\text{arg}\mathop{\text{max}}\limits_{{{\bf{W}}_{E}}\succeq
0}h_{2}({{\bf{U}}_{E}^{n}},{{\bf{V}}_{E}^{n}},{{\bf{W}}_{E}}){\rm{=[}}{{\bf{E}}_{E}}({{\bf{U}}_{E}^{n}},{{\bf{V}}_{E}^{n}}){]^{-1}}$
$\displaystyle\qquad\>\>{\rm{=}}[({{\bf{U}}_{E}}^{{n}H}{{\hat{{\bf{H}}}}_{E}}{\bf{V}}_{E}^{n}-{\bf{I}}_{N_{T}}){({{\bf{U}}_{E}^{{n}H}}{{\hat{{\bf{H}}}}_{E}}{\bf{V}}_{E}^{n}-{\bf{I}}_{N_{T}})^{H}}+{{\bf{U}}_{E}^{{n}H}}(\sigma_{E}^{2}{{\bf{I}}_{{N_{E}}}}){{\bf{U}}_{E}^{n}}]^{-1},$
(70d)
$\displaystyle{{\bf{W}}_{X}^{n}}{\rm{=}}\text{arg}\mathop{\text{max}}\limits_{{{\bf{W}}_{X}}\succeq
0}h_{3}({\bf{V}}^{n},{{\bf{V}}_{E}^{n}},{{\bf{W}}_{X}}){\rm{=[}}{{\bf{E}}_{X}}({\bf{V}}^{n},{{\bf{V}}_{E}^{n}}){]^{-1}}$
$\displaystyle\qquad\>\>{\rm{=}}[{{{\bf{I}}_{{N_{E}}}}+\sigma_{E}^{-2}{{\hat{{\bf{H}}}}_{E}}({\bf{V}}^{n}{{\bf{V}}^{nH}}+{{\bf{V}}_{E}^{n}}{{\bf{V}}_{E}^{nH}})\hat{{\bf{H}}}_{E}^{H}}]^{-1}.$
(70e)

The parameters ${\bf{V}}^{n}$, ${{\bf{V}}_{E}^{n}}$, ${\bf{\Phi}}^{n}$ are
obtained by solving the following problem

$\displaystyle\
\underset{{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}}{\mathop{\text{max}}}\ \
\underset{l=1,\cdots,L}{\min}\\{h_{1,l}({{\bf{U}}_{I,l}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I,l}})\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad+h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})+h_{3}({\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{X}})$
(71a) $\displaystyle\ \ \text{s.t.}\quad\
{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}_{E}^{H}}{\rm{)}}\leq{P_{T}},$
(71b)
$\displaystyle\quad\quad\quad\\!\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M.$
(71c)

###  IV-C Optimizing the Matrices ${\bf{V}}$ and ${{\bf{V}}_{E}}$

By fixing ${\bf{\Phi}}$, Problem (71) can be written more compactly as

$\displaystyle\ \underset{{\bf{V}},{{\bf{V}}_{E}}}{\mathop{\text{min}}}\ \
\underset{l=1,\cdots,L}{\max}\\{-\text{Tr}({{\bf{W}}_{I,l}}{{\bf{V}}^{H}}{{\hat{{\bf{H}}}}_{I,l}}^{H}{{\bf{U}}_{I,l}})-\text{Tr}({{\bf{W}}_{I,l}}{{\bf{U}}_{I,l}}^{H}{{\hat{{\bf{H}}}}_{I,l}}{\bf{V}})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V,l}^{i}}\mathbf{V})+\text{Tr}({{\mathbf{V}}_{E}^{H}}{{\mathbf{H}}_{VE,l}^{i}}{{\mathbf{V}}_{E}})-{C}_{l}\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}_{E}^{H}}{{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{U}}_{E}})-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}^{H}}{{\mathbf{\hat{H}}}_{E}}{{\mathbf{V}}_{E}})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V}^{e}}\mathbf{V})+\text{Tr}({{\mathbf{V}}_{E}^{H}}{{\mathbf{H}}_{VE}^{e}}{{\mathbf{V}}_{E}})$
(72a) $\displaystyle\ \ \ \ \ \text{s.t.}\quad\
{\rm{Tr(}}{\bf{V}}{{\bf{V}}^{H}}{\rm{+}}{{\bf{V}}_{E}}{{\bf{V}}_{E}^{H}}{\rm{)}}\leq{P_{T}},$
(72b)

where

$\displaystyle{{\mathbf{H}}_{V}^{e}}\text{(}\mathbf{\Phi}\text{)}=\sigma_{E}^{-2}\mathbf{\hat{H}}_{E}^{H}{{\mathbf{W}}_{X}}{{\mathbf{\hat{H}}}_{E}},$
(73a)
$\displaystyle{{\mathbf{H}}_{VE}^{e}}\text{(}\mathbf{\Phi}\text{)}={{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{U}}_{E}}{{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}^{H}}{{\mathbf{\hat{H}}}_{E}}+\sigma_{E}^{-2}{{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{W}}_{X}}{{\mathbf{\hat{H}}}_{E}},$
(73b)
$\displaystyle{{\mathbf{H}}_{V,l}^{i}}\text{(}\mathbf{\Phi}\text{)}={{\mathbf{\hat{H}}}_{I,l}}^{H}{{\mathbf{U}}_{I,l}}{{\mathbf{W}}_{I,l}}{{\mathbf{U}}_{I,l}^{H}}{{\mathbf{\hat{H}}}_{I,l}},$
(73c)
$\displaystyle{{\mathbf{H}}_{VE,l}^{i}}\text{(}\mathbf{\Phi}\text{)}={{\mathbf{\hat{H}}}_{I,l}}^{H}{{\mathbf{U}}_{I,l}}{{\mathbf{W}}_{I,l}}{{\mathbf{U}}_{I,l}^{H}}{{\mathbf{\hat{H}}}_{I,l}},$
(73d)
$\displaystyle{C}_{l}=\log\left|{{\bf{W}}_{I,l}}\right|+d-\text{Tr}({{\bf{W}}_{I,l}}+\sigma_{I,l}^{2}{{\bf{W}}_{I,l}}{{\bf{U}}_{I,l}}^{H}{{\bf{U}}_{I,l}}).$
(73e)

The Problem (72) is a convex QCQP problem, we can obtain its optimal solution
using a general-purpose convex optimization solver.

###  IV-D Optimizing the Phase Shifts $\mathbf{\Phi}$

By fixing ${\bf{V}},{{\bf{V}}_{E}}$, the optimization problem for the phase
shift matrix $\mathbf{\Phi}$ reduced from Problem (72) is formulated as

$\displaystyle\ \underset{\mathbf{\Phi}}{\mathop{\text{min}}}\ \
{g_{0}}(\mathbf{\Phi})\triangleq\underset{l=1,\cdots,L}{\max}\\{-\text{Tr}({{\bf{W}}_{I,l}}{{\bf{V}}^{H}}{{\hat{{\bf{H}}}}_{I,l}}^{H}{{\bf{U}}_{I,l}})-\text{Tr}({{\bf{W}}_{I,l}}{{\bf{U}}_{I,l}}^{H}{{\hat{{\bf{H}}}}_{I,l}}{\bf{V}})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V,l}^{i}}\mathbf{V})+\text{Tr}({{\mathbf{V}}_{E}^{H}}{{\mathbf{H}}_{VE,l}^{i}}{{\mathbf{V}}_{E}})-{C}_{l}\\}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}_{E}^{H}}{{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{U}}_{E}})-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}^{H}}{{\mathbf{\hat{H}}}_{E}}{{\mathbf{V}}_{E}})$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V}^{e}}\mathbf{V})+\text{Tr}({{\mathbf{V}}_{E}^{H}}{{\mathbf{H}}_{VE}^{e}}{{\mathbf{V}}_{E}})$
(74a) $\displaystyle\ \ \ \ \ \text{s.t.}\quad\
\\!\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M.$ (74b)

By complex mathematical manipulations, the OF ${g_{0}}(\mathbf{\Phi})$ can be
equivalently transformed into

$\displaystyle{g_{0}}(\mathbf{\Phi})$
$\displaystyle\triangleq\underset{l=1,\cdots,L}{\max}\\{{g_{0,l}^{i}}(\mathbf{\Phi})\\}+{g_{0}^{e}}(\mathbf{\Phi}),$
(75)

where

$\displaystyle{g_{0,l}^{i}}(\mathbf{\Phi})$
$\displaystyle={\rm{Tr}}\left({{\bf{\Phi}}^{H}{\bf{D}}_{l}^{iH}}\right)+{\rm{Tr}}\left({\bf{\Phi}}{\bf{D}}_{l}^{i}\right)+{\rm{Tr}}\left[{{\bf{\Phi}}{{\bf{C}}_{VE}}{{\bf{\Phi}}^{H}}{{\bf{B}}_{VE,l}^{i}}}\right]+{\rm{Tr}}\left({{\bf{\Phi}}{{\bf{C}}_{V}}{{\bf{\Phi}}^{H}}{{\bf{B}}_{V,l}^{i}}}\right)+C_{l}^{i},$
(76a) $\displaystyle{g_{0}^{e}}(\mathbf{\Phi})$
$\displaystyle={\rm{Tr}}\left({{\bf{\Phi}}^{H}{\bf{D}}^{eH}}\right)+{\rm{Tr}}\left({\bf{\Phi}}{\bf{D}}^{e}\right)+{\rm{Tr}}\left[{{\bf{\Phi}}{{\bf{C}}_{VE}}{{\bf{\Phi}}^{H}}{{\bf{B}}_{VE}^{e}}}\right]+{\rm{Tr}}\left({{\bf{\Phi}}{{\bf{C}}_{V}}{{\bf{\Phi}}^{H}}{{\bf{B}}_{V}^{e}}}\right)+C^{e},$
(76b)

and

$\displaystyle{\bf{D}}_{l}^{i}$
$\displaystyle={\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,I,l}^{H}{{\bf{M}}_{I,l}}{{\bf{H}}_{R,I,l}}-{\bf{GV}}{{\bf{W}}_{I,l}}{\bf{U}}_{I,l}^{H}{{\bf{H}}_{R,I,l}},$
(77a) $\displaystyle{{\bf{C}}_{VE}}$
$\displaystyle={\bf{G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}},$ (77b)
$\displaystyle{{\bf{C}}_{V}}$
$\displaystyle={\bf{G}}{\bf{V}}{\bf{V}}^{H}{{\bf{G}}^{H}},$ (77c)
$\displaystyle{{\bf{B}}_{VE,l}^{i}}$
$\displaystyle=\left({{\bf{H}}_{R,I,l}^{H}{{\bf{U}}_{I,l}}{{\bf{W}}_{I,l}}{\bf{U}}_{I,l}^{H}{{\bf{H}}_{R,I,l}}}\right),$
(77d) $\displaystyle{{\bf{B}}_{V,l}^{i}}$
$\displaystyle=\left({{\bf{H}}_{R,I,l}^{H}{{\bf{U}}_{I,l}}{{\bf{W}}_{I,l}}{\bf{U}}_{I,l}^{H}{{\bf{H}}_{R,I,l}}}\right),$
(77e) $\displaystyle C_{l}^{i}$
$\displaystyle={\rm{Tr}}\left[{{\bf{H}}_{b,I,l}}{{\bf{V}}_{X}}{\bf{H}}_{b,I,l}^{H}{{\bf{M}}_{I,l}}\right]\\!+\\!{\rm{Tr}}\left[{{{\bf{U}}_{I,l}}{\bf{W}}_{I,l}^{H}{{\bf{V}}^{H}}{\bf{H}}_{b,I,l}^{H}}\right]\\!+\\!{\rm{Tr}}\left[{{{\bf{H}}_{b,I,l}}{\bf{V}}{{\bf{W}}_{I,l}}{\bf{U}}_{I,l}^{H}}\right]-{C}_{l},$
(77f) $\displaystyle{\bf{M}}_{I,l}$
$\displaystyle={{\bf{U}}_{I,l}}{{\bf{W}}_{I,l}}{\bf{U}}_{I,l}^{H},$ (77g)
$\displaystyle{\bf{D}}^{e}$
$\displaystyle=\sigma_{E}^{-2}{\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}+{\bf{G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{R,E}}-{\bf{G}}{{\bf{V}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{\bf{H}}_{R,E}},$
(77h) $\displaystyle{{\bf{B}}_{VE}^{e}}$
$\displaystyle=\left({\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}+{\bf{H}}_{R,E}^{H}{{\bf{U}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{\bf{H}}_{R,E}}}\right),$
(77i) $\displaystyle{{\bf{B}}_{V}^{e}}$
$\displaystyle=\left({\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}}\right),$
(77j) $\displaystyle C^{e}$
$\displaystyle=\sigma_{E}^{-2}{\rm{Tr}}\left[{{\bf{H}}_{b,E}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}\right]+{\rm{Tr}}\left[{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}\right]$
$\displaystyle+{\rm{Tr}}\left[{{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}}\right]+{\rm{Tr}}\left[{{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}}\right].$
(77k)

Similarly, Problem (74) can be further simplified as

$\displaystyle\underset{{\bm{\phi}}}{\mathop{\text{min}}}$
$\displaystyle\underset{l=1,\cdots,L}{\max}\\{{{\bm{\phi}}^{{\rm{H}}}}{\bm{\Xi}_{l}^{i}}{\bm{\phi}}+2\textrm{Re}[{\bm{\phi}}^{{\rm{H}}}{\bf{d}}_{l}^{i*}]+C_{l}^{i}\\}+{{\bm{\phi}}^{{\rm{H}}}}{\bm{\Xi}^{e}}{\bm{\phi}}+2\textrm{Re}[{\bm{\phi}}^{{\rm{H}}}{\bf{d}}^{e*}]+C^{e}$
(78a) s.t. $\displaystyle\quad\quad\
\\!\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M,$ (78b)

where

$\displaystyle{\bm{\Xi}_{l}^{i}}$
$\displaystyle={\bf{B}}_{VE,l}^{i}\odot{{\bf{C}}_{VE}^{{\rm{T}}}}+{\bf{B}}_{V,l}^{i}\odot{{\bf{C}}_{V}^{{\rm{T}}}},$
(79a) $\displaystyle{\bm{\Xi}^{e}}$
$\displaystyle={\bf{B}}_{VE}^{e}\odot{{\bf{C}}_{VE}^{{\rm{T}}}}+{\bf{B}}_{V}^{e}\odot{{\bf{C}}_{V}^{{\rm{T}}}},$
(79b) $\displaystyle{\bf{d}}_{l}^{i}$
$\displaystyle={\left[{{{\left[{\bf{D}}_{l}^{i}\right]}_{1,1}},\cdots,{{\left[{\bf{D}}_{l}^{i}\right]}_{M,M}}}\right]^{{\rm{T}}}},$
(79c) $\displaystyle{\bf{d}}^{e}$
$\displaystyle={\left[{{{\left[{\bf{D}}^{e}\right]}_{1,1}},\cdots,{{\left[{\bf{D}}^{e}\right]}_{M,M}}}\right]^{{\rm{T}}}}.$
(79d)

By using the Lemma 1 in [25], Problem (78) will be recast as

$\displaystyle\underset{{\bm{\phi}}}{\mathop{\max}}$
$\displaystyle\underset{l=1,\cdots,L}{\min}\\{2\textrm{Re}[{{\bm{\phi}}^{{\rm{H}}}}{\bf{q}}_{l}^{i,t}]-C_{q,l}^{i,t}\\}+2\textrm{Re}[{{\bm{\phi}}^{{\rm{H}}}}{\bf{q}}^{e,t}]-C_{q}^{e,t}$
(80a) s.t. $\displaystyle\quad\quad\
\\!\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M,$ (80b)

where

$\displaystyle{\bf{q}}_{l}^{i,t}$
$\displaystyle=\left({{\lambda_{l,{\rm{\max}}}^{i}}{{\bf{I}}_{M}}-{\bm{\Xi}_{l}^{i}}}\right){{\bm{\phi}}^{t}}-{\bf{d}}_{l}^{i*},$
(81a) $\displaystyle C_{q,l}^{i,t}$
$\displaystyle=2M{\lambda_{l,{\rm{\max}}}^{i}}-{\left({{\bm{\phi}}^{t}}\right)^{{\rm{H}}}}\left({\bm{\Xi}_{l}^{i}}\right){{\bm{\phi}}^{t}}+C_{l}^{i},$
(81b) $\displaystyle{\bf{q}}^{e,t}$
$\displaystyle=\left({{\lambda_{{\rm{\max}}}^{e}}{{\bf{I}}_{M}}-{\bm{\Xi}^{e}}}\right){{\bm{\phi}}^{t}}-{\bf{d}}^{e*},$
(81c) $\displaystyle C_{q}^{e,t}$
$\displaystyle=2M{\lambda_{{\rm{\max}}}^{e}}-{\left({{\bm{\phi}}^{t}}\right)^{{\rm{H}}}}\left({\bm{\Xi}^{e}}\right){{\bm{\phi}}^{t}}+C^{e},$
(81d)

and ${\lambda_{l,{\rm{\max}}}^{i}}$ is the the maximum eigenvalue of
${\bm{\Xi}_{l}^{i}}$, and ${\lambda_{{\rm{\max}}}^{e}}$ is the maximum
eigenvalue of ${\bm{\Xi}^{e}}$. By defining
${\bf{q}}_{l}^{ie,t}\triangleq{\bf{q}}_{l}^{i,t}+{\bf{q}}^{e,t}$ and
$C_{q,l}^{ie,t}\triangleq C_{q,l}^{i,t}+C_{q}^{e,t}$, Problem (80) can be
rewritten as

$\displaystyle\underset{{\bm{\phi}}}{\mathop{\max}}$
$\displaystyle\underset{l=1,\cdots,L}{\min}\\{2\textrm{Re}[{{\bm{\phi}}^{{\rm{H}}}}{\bf{q}}_{l}^{ie,t}]-C_{q,l}^{ie,t}\\}$
(82a) s.t. $\displaystyle\quad\quad\
\\!\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M,$ (82b)

which is equivalent to the following problem

$\displaystyle\underset{{\bm{\phi}},z}{\mathop{\max}}$ $\displaystyle\quad z$
(83a) s.t. $\displaystyle
2\textrm{Re}[{{\bm{\phi}}^{{\rm{H}}}}{\bf{q}}_{l}^{ie,t}]-C_{q,l}^{ie,t}\geq
z,l=1,\cdots,L,$ (83b)
$\displaystyle\left|{{\phi}_{m}}\right|=1,m=1,\cdots,M.$ (83c)

We note that the above problem is still non-convex due to the unit-modulus
constraints. To deal with the non-convex constraints, the penalty CCP method
is applied. Following the penalty CCP framework, the constraint (83c) are
firstly equivalently rewritten as $-1\leq\left|{{\phi}_{m}}\right|^{2}\leq
1,m=1,\cdots,M$. The non-convex parts of the resulting constraints are then
linearized by
$\left|{{\phi}_{m}^{\text{(n)}}}\right|^{2}-2\textrm{Re}({{\phi}_{m}^{\text{(*)}}}{{\phi}_{m}^{\text{(n)}}})\leq-1,m=1,\cdots,M$.
We finally have the following convex subproblem of ${\bm{\phi}}$ as

$\displaystyle\underset{{\bm{\phi}},z}{\mathop{\max}}$ $\displaystyle\quad
z-\lambda^{(t)}\left\|\mathbf{b}\right\|_{1}$ (84a) s.t. $\displaystyle\
2\textrm{Re}[{{\bm{\phi}}^{{\rm{H}}}}{\bf{q}}_{l}^{ie,t}]-C_{q,l}^{ie,t}\geq
z,$ (84b)
$\displaystyle\left|{{\phi}_{m}^{\text{(n)}}}\right|^{2}-2\textrm{Re}({{\phi}_{m}^{\text{(*)}}}{{\phi}_{m}^{\text{(n)}}})\leq
b_{m}-1,m=1,\cdots,M,$ (84c) $\displaystyle\left|{{\phi}_{m}}\right|^{2}\leq
1+b_{M+m},m=1,\cdots,M.$ (84d)

where $\mathbf{b}=[b_{1},\cdots,b_{2M}]^{T}$ are slack variables imposed over
the equivalent linear constraints of the unit-modulus constraints, and
$\left\|\mathbf{b}\right\|_{1}$ is the penalty term in the OF.
$\left\|\mathbf{b}\right\|_{1}$ is scaled by the regularization factor
$\lambda^{(t)}$ to control the feasibility of the constraints. The specific
steps of the penalty CCP method can be referred in [52].

##  V Simulation Results

In this section, numerical simulations are carried out to evaluate the
assistance of the IRS on the AN-aided MIMO secure communication system. We
focus on the scenario of the standard three-terminal MIMO Guassian wiretap
channel shown in Fig. 2, where there are one BS, one legitimate IR and one
Eve, all with multiple antennas. The distance from the BS to the IRS is
$d_{BR}=50$ m. We assume that the line connecting the IR and Eve is parallel
to the line connecting the BS and the IRS, and that the vertical distance
between them is $d_{v}=2$ m.

Figure 2: The three-terminal MIMO communication scenario in simulation.

The large-scale path loss is modeled as
${\rm{PL=}}{{\rm{P}}{{\rm{L}}_{0}}-10\alpha{{\log}_{10}}\left({\frac{d}{{{d_{0}}}}}\right)}$,
where ${\rm{P}}{{\rm{L}}_{0}}$ is the path loss at the reference distance
$d_{0}=1$ m, $\alpha$ is the path loss exponent, $d$ is the link distance. In
our simulations, we set ${\rm{P}}{{\rm{L}}_{0}}=-30$ dB. The path loss
exponents of the links from BS to Eve, from BS to IR, from IRS to Eve and from
IRS to IR are ${\alpha_{{\rm{BE}}}}=3.5$, ${\alpha_{{\rm{BI}}}}=3.5$,
${\alpha_{{\rm{RE}}}}=2.5$ and ${\alpha_{{\rm{RI}}}}=2.5$ respectively. The
path-loss exponents of the link from BS to IRS is set to be
${\alpha_{{\rm{BR}}}}=2.2$, which means that the IRS is well-located, and the
path loss is negligible in this link.

For the direct channels from the BSs to the Eve and IR, the small-scale fading
is assumed to be Rayleigh fading due to extensive scatters. However, for the
IRS-related channels, the small-scale fading is assumed to be Rician fading.
Specifically, the small-scale channel can be modeled as

$\displaystyle\tilde{\mathbf{H}}$
$\displaystyle=\left(\sqrt{\frac{\beta}{1+\beta}}\tilde{\mathbf{H}}^{LOS}+\sqrt{\frac{1}{1+\beta}}\tilde{\mathbf{H}}^{NLOS}\right),$
(85)

where $\beta$ is the Rican factor, $\tilde{\mathbf{H}}^{LOS}$ denotes the
deterministic line of sight (LoS) component of the IRS-related channel, and
$\tilde{\mathbf{H}}^{NLOS}$ denotes the non-LoS (NLoS) component of the IRS-
related channel, which is modeled as Rayleigh fading. By assuming the antennas
at the BS, IRS, Eve and IR are arranged in a uniform linear array (ULA), the
$\tilde{\mathbf{H}}^{LOS}$ can be modeled as
$\tilde{\mathbf{H}}^{LOS}=\mathbf{a}_{r}\mathbf{a}_{t}^{H}$, where
$\mathbf{a}_{t}$ and $\mathbf{a}_{r}$ are the steering vectors of the
transmitting and receiving arrays respectively. The $\mathbf{a}_{t}$ and
$\mathbf{a}_{r}$ are defined as,

$\displaystyle\mathbf{a}_{t}$
$\displaystyle=\left[\begin{array}[]{cccc}1,&\exp(j2\pi\frac{d_{t}}{\lambda}\sin\varphi_{t}),&\cdots,&\exp(j2\pi\frac{d_{t}}{\lambda}(N_{t}-1)\sin\varphi_{t})\end{array}\right]^{T},$
(86b) $\displaystyle\mathbf{a}_{r}$
$\displaystyle=\left[\begin{array}[]{cccc}1,&\exp(j2\pi\frac{d_{r}}{\lambda}\sin\varphi_{r}),&\cdots,&\exp(j2\pi\frac{d_{r}}{\lambda}(N_{r}-1)\sin\varphi_{r})\end{array}\right]^{T}.$
(86d)

In (86), $\lambda$ is the wavelength; $d_{t}$ and $d_{r}$ are the element
intervals of the transmitting and receiving array; $\varphi_{t}$ and
$\varphi_{r}$ are the angle of departure and the angle of arrival; $N_{t}$ and
$N_{r}$ are the number of antennas/elements at the transmitter and receiver,
respectively. We set $\frac{d_{t}}{\lambda}=\frac{d_{r}}{\lambda}=0.5$, and
$\varphi_{t}=\tan^{-1}(\frac{y_{r}-y_{t}}{x_{r}-x_{t}})$,
$\varphi_{r}=\pi-\varphi_{t}$, where $(x_{t},y_{t})$ is the location of the
transmitter, and $(x_{r},y_{r})$ is the location of the receiver.

If not specified, the simulation parameters are set as follows. The IR’s noise
power and the Eve’s noise power are $\sigma_{I}^{2}=-75$ dBm and
$\sigma_{E}^{2}=-75$ dBm. The numbers of BS antennas, IR antennas, and Eve
antennas are $N_{T}=4$, $N_{I}=2$, and $N_{E}=2$ respectively. There are $d=2$
data streams and $M=50$ IRS reflection elements. The transmit power limit is
$P_{T}=15$ dBm, and the error tolerance is $\varepsilon=10^{-6}$. The
horizontal distance between the BS and the Eve is $d_{BE}=44$ m. The
horizontal distance between the BS and the IR is selected from $d_{BI}=[10\
\text{m},70\ \text{m}]$. The channels are realized 200 times independently to
average the simulation results.

###  V-A Convergence Analysis

The convergence performance of the proposed BCD-MM algorithm is investigated.
The iterations of the BCD algorithm are termed as outer-layer iterations,
while the iterations of the MM algorithm are termed as the inner-layer
iterations. Fig. 4 shows three examples of convergence behaviour for $M=$10,
20 and 40 phase shifts of IRS. In Fig. 4, the SR increases versus the
iteration number, and finally reaches a stable value. It is shown that the
algorithm converges quickly, almost with 20 iterations, which demonstrates the
efficiency of the proposed algorithm. Moreover, a larger converged SR value is
reached with a larger $M$, which means that better security can be obtained by
using more IRS elements. However, more IRS elements bring a heavier
computation, which is demonstrated in Fig. 4 in the form of a slower
convergence speed with more phase shifters.

Figure 3: Convergence behaviour of the BCD algorithm.

Figure 4: Convergence behaviour of the MM algorithm.

Specifically, we evaluate the convergence performance of the MM algorithm used
for solving the optimal IRS phase shifts. The inner-layer iterative process of
the MM algorithm in the first iteration of the BCD algorithm is shown in Fig.
4. The SR value increases as the iteration number increases, and finally
converges to a stable value. According with the convergency performance in the
out-layer iteration, similar conclusions can be drawn for the inner-layer
iteration, which is that a higher converged SR value can be obtained with more
phase shifts but at the cost of lower convergence speed. The reason for the
lower convergence speed with larger $M$ value is that more optimization
variables are introduced, which require more computation complexity.

###  V-B Performance Evaluation

In this subsection, our proposed algorithm is evaluated by comparing the
simulation results to three schemes of

  1. 1.

RandPhase: The phase shifts of the IRS are randomly selected from $[0,2\pi]$.
In this scheme, the MM algorithm is skipped, and only the TPC matrix and AN
covariance matrix are optimized.

  2. 2.

No-IRS: Without the IRS, the channel matrices of IRS related links become zero
matrices, which is ${\bf{H}}_{R,I}={\bf{0}}$, ${\bf{H}}_{R,E}={\bf{0}}$ and
${{\bf{G}}}={\bf{0}}$. This scheme results a conventional AN-aided
communication system, and only the TPC matrix and AN covariance matrix need to
be optimized.

  3. 3.

BCD-QCQP-SDR: The BCD algorithm is utilized. However, the TPC matrix and the
AN covariance matrix is optimized by tackling Problem (32) as a QCQP problem,
which is solved by the general CVX solvers, e.g. Sedumi or Mosek. The phase
shifts of IRS are optimized by solving Problem (61) with the SDR technique.

####  V-B1 Impact of Transmit Power

Figure 5: Achievable SR versus the transmit power limit.

Figure 6: Achievable SR versus the number of phase shifts $M$.

To evaluate the impact of the transmit power limit $P_{T}$, the average SRs
versus the transmit power limit for various schemes are given in Fig. 6, which
demonstrates that the achieved SRs of three schemes increase as the power
limit $P_{T}$ increases. It is observed that the BCD-MM algorithm
significantly outperforms the other three benchmark schemes over the entire
range of transmit power limits. By comparing the RandPhase scheme to the No-
IRS scheme, we find that the RandPhase scheme is better than the No-IRS scheme
for obtaining higher SR, and that the SR gap increases with the power limit
$P_{T}$. The reason is that, for the RandPhase scheme, the IR is closer to the
IRS than the Eve is, and more signal power from the IRS can be acquired by the
IR than that by the Eve, while for the No-IRS scheme, the IR is further from
the BS than the Eve is, and less signal power from the BS can be acquired by
the IR than that by the Eve. This comparison result signifies that even the
phase shifts of IRS are random, the IRS can enhance the system security. In
comparison to the no-IRS scheme, the SR gain achieved by the proposed
algorithm is very obvious, and increases greatly with the power limit $P_{T}$,
which confirms the effectiveness and benefits of employing the IRS. By
comparing the proposed scheme and the RandPhase scheme, we find that the
security gain obtained for the proposed scheme is much greater than that for
the RandPhase scheme. That’s because the phase shifts of IRS are properly
designed to enhance the signal received at the IR more constructively, and
weaken the signal received at the Eve more destructively. This comparison
result emphasizes that optimizing the phase shifts of IRS is important and
necessary. By comparing the proposed BCD-MM algorithm and the BCD-QCQP-SDR
algorithm, we observe that in terms of the SR performance, the proposed BCD-MM
algorithm is better BCD-QCQP-SDR, and the performance gain increases with the
$P_{T}$. Moreover, the proposed BCD-MM algorithm is much more efficient than
the BCD-QCQP-SDR algorithm. The superiority of the proposed algorithm is
further validated.

####  V-B2 Impact of the Phase Shifts Number

The averaged SR performance of four schemes with various phase shifts number
$M$ is shown in Fig. 6, which demonstrates that the proposed BCD-MM algorithm
is significantly superior to the other three schemes. We observe that the SR
achieved by the BCD-MM scheme obviously increases with $M$, while the
RandPhase scheme only shows a slight improvement as $M$ increases, and the No-
IRS scheme has very low SRs irrelative with $M$. Larger the element number $M$
of IRS is, more significant the performance gain obtained by the proposed
algorithm is. For example, when $M$ is small as $M=10$, the SR gain of the
BCD-MM relative to the No-IRS is only 1.3 bit/s/Hz, while this SR gain becomes
9.5 bit/s/Hz when $M$ increases to $M=100$. The performance gain for the
proposed algorithm originates from two perspectives. On the one hand, a higher
array gain can be obtained by increasing $M$, since more signal power can be
received at the IRS with larger $M$. On the other hand, a higher reflecting
beamforming gain can be obtained by increasing $M$, which means that the sum
of coherently adding the reflected signals at the IRS elements increases with
$M$ by appropriately designing the phase shifts. However, only the array gain
can be exploited by the RandPhase scheme, thus the SRs for it increase very
slowly, and remain at much lower values than those for the proposed algorithm.
These results further confirm that more security improvements can be archived
by using a large IRS with more reflect elements and optimizing the phase
shifts properly, however there may bring the computation complexity problem.
In comparison to the BCD-QCQP-SDR algorithm, the proposed BCD-MM algorithm can
achieve the higher SR, and the SR performance gap increases with $M$.

####  V-B3 Impact of the relative location of IRS

Figure 7: Achievable SR versus the location of the IR $d_{\rm{BI}}$.

Figure 8: Achievable SR versus the path loss exponent of IRS-related links.

Fig. 8 illustrates the achieved SRs for four schemes with various BS-IR
horizontal distance $d_{BI}$, where the BS-Eve distance is fixed to be
$d_{BE}=44$ m. It is observed that the proposed BCD-MM algorithm is the best
among the four schemes for obtaining the highest SR value. When the IR moves
far away from the BS, the SRs decrease for the four schemes, however, the SRs
achieved for the RandPhase, the proposed BCD-MM algorithm and the BCD-QCQP-SDR
algorithm increase greatly when the IR approaches the IRS. The achieved SRs at
different BS-IR distances of the RandPhase scheme and the no-IRS scheme are
almost the same, except for $d_{BI}\in[40\text{m},50\text{m}]$, in which case
the IRS brings a prominent security enhancement when IR becomes close to it
even with random IRS phase shifts. Similarly, the proposed BCD-MM algorithm
and the BCD-QCQP-SDR algorithm can achieve almost the same SRs, except for
$d_{BI}\in[40\text{m},50\text{m}]$, in which case the IR is close to the IRS,
and the proposed BCD-MM algorithm is superior to the BCD-QCQP-SDR algorithm.
For other BS-IR distances where the IR is far from the IRS, the SRs of
RandPhase scheme are similar with those of the No-IRS scheme due to the not
fully explored potential of IRS. By optimizing the phase shifts of IRS, the
SRs are enhanced at different BS-IRS distances. And the SR gain of the
proposed BCD-MM algorithm over the RandPhase scheme increases when the IR
moves close to the IRS ($d_{BI}\in[40\text{m},50\text{m}]$). This signifies
that as long as the IRS is deployed close to the IR, significant security
enhancement can be achieved by the IRS in an AN-aided MIMO communication
system. Moreover, it is highly recommended that the IRS phase shifts should be
optimized to prevent the system security degrading into the level of No-IRS
scheme.

####  V-B4 Impact of the Path Loss Exponent of IRS-related Links

In the above simulations, the path loss exponents of the IRS-related links
(including the BS-IRS link, IRS-IR link and IRS-Eve link) are set to be low by
assuming that the IRS is properly located to obtain clean channels without
heavy blockage. Practically, such kind of settings may not always be sensible
due to the real-field environment. Thus, it is necessary to investigate the
security gain brought by the IRS and our proposed algorithm with higher values
of IRS-related path loss exponents. For the sake of analysis, we assume the
path-loss exponents of the links from BS to IRS, from IRS to IR and from IRS
to Eve are the same as
${\alpha_{{\rm{BR}}}}={\alpha_{{\rm{RI}}}}={\alpha_{{\rm{RE}}}}\buildrel\Delta\over{=}{\alpha_{{\rm{IRS}}}}$.
Then, the achieved SRs versus the path-loss exponent ${\alpha_{{\rm{IRS}}}}$
of IRS-related links are shown in Fig. 8, which demonstrates that the SR
obtained by the BCD-MM algorithm decreases as ${\alpha_{{\rm{IRS}}}}$
increases, and finally drops to the same SR value which is achieved by the
RandPhase, BCD-QCQP-SDR and No-IRS schemes. The reason is that larger
${\alpha_{{\rm{IRS}}}}$ means more severe signal attenuation in the IRS-
related links, and more weakened signal received and reflected at the IRS. In
comparison to the BCD-QCQP-SDR algorithm, the proposed BCD-MM algorithm can
achieve the higher SR when the channel state of the IRS related channels is
good, i.e., ${\alpha_{{\rm{IRS}}}}$ is low, and achieve almost the same SR
when ${\alpha_{{\rm{IRS}}}}$ is large. Similarly, the performance gains
brought by our proposed algorithm over the RandPhase and No-IRS schemes is
significant with a small ${\alpha_{{\rm{IRS}}}}$. Specifically, for
${\alpha_{{\rm{IRS}}}}=2$ (almost ideal channels), the security gain is up to
9.6 bit/s/Hz over the No-IRS scheme, and 6.8 bit/s/Hz over the RandPhase
scheme. Therefore, the security gain of IRS-assisted systems depends on the
channel conditions of the IRS-related links. This suggests that it is much
preferred to deploy the IRS with fewer obstacles, in which case, the
performance gain brought by the IRS can be explored thoroughly. Fig. 8 also
shows that when ${\alpha_{{\rm{IRS}}}}$ is small, the RandPhase scheme can
obtain security gain over the No-IRS scheme, but this security gain decreases
to zero when ${\alpha_{{\rm{IRS}}}}$ becomes large. However, the SR gain of
the RandPhase scheme over the No-IRS scheme is almost negligible in comparison
to the SR gain of the proposed scheme over the No-IRS scheme, which
demonstrates that the necessity of jointly optimizing the TPC matrix, AN
covariance matrix and the phase shifts at the IRS.

####  V-B5 Impact of the Number of Data Streams

Compared with the MISO scenario, a significant advantage of the MIMO scenario
is that multiple data streams can be transmitted to the users. To evaluate the
impact of the number of data streams on the SR, the average SRs versus the
transmit power limit for various numbers of data streams are given in Fig. 10.

Figure 9: Achievable SR versus the transmit power limit for various numbers of
data streams.

Figure 10: Achievable SR versus the reflection amplitude $\eta$.

The number of transmit antennas is $N_{T}=4$. The Rician fading channels are
utilized. The path loss exponents are ${\alpha_{{\rm{BR}}}}=2.2$,
${\alpha_{{\rm{BE}}}}=3.5$, ${\alpha_{{\rm{BI}}}}=2.5$,
${\alpha_{{\rm{RE}}}}=3.5$ and ${\alpha_{{\rm{RI}}}}=2.5$ respectively. The
Rician factor is $\beta=3$. The number of phase shifts is $M=50$. As shown in
Fig. 10, the SR increases with the transmit power limit and larger number of
data streams result the higher SR. When the transmit power limit is low,
marginal performance gains are achieved by increasing the number $d$ of data
streams. When the transmit power limit is high, significant performance gains
can be achieved by increasing the number $d$ of data streams. This means that
a greater number of data streams ensure the higher SR, and the performance
gain expands with the transmit power limit. For the case of $d=1$, the SR
performance of $N_{I}=N_{E}=4$ and $N_{I}=N_{E}=1$ is further compared. It is
revealed that the SR obtained by four receiving antennas is higher than the SR
obtained by one single receiving antenna when the transmit power limit is
relatively low. With the increase of transmit power limit, the SR performance
gain brought by multiple receiving antennas decreases. When the transmit power
limit is high enough, the SR performance is saturated, and the SR performance
of the multiple receiving antennas and single receiving antenna becomes the
same.

####  V-B6 Impact of the Reflection Amplitude

Due to the manufactural and hardware reasons, the signals reflected by the IRS
may be attenuated. Then, in Fig. 10, we study the impact of the reflection
amplitude on the security performance. The transmit power limit is 10dBm. We
assume that the reflection amplitudes of all the IRS elements are same as
$\eta$, and that the phase shift matrix of the IRS is rewritten as
${\bf{\Phi}}=\eta\text{diag }\\!\\!\\{\\!\\!\text{
}{{\phi}_{1}},\cdots,{{\phi}_{m}},\cdots,{{\phi}_{M}}\text{
}\\!\\!\\}\\!\\!\text{ }$. As expected, the SR achieved by the IRS-aided
scheme increases with $\eta$ due to less power loss. As $\eta$ increases, the
superiority of the proposed BCD-MM algorithm over the other algorithms becomes
more obvious. The reflection amplitude has a great impact on the security
performance. Specifically, when $\eta$ increases from 0.2 to 1, the SR
increases over 3.6 bit/s/Hz for the proposed BCD-MM algorithm.

####  V-B7 Impact of the Discrete Phase Shifts

In practice, it is difficult to realize continuous phase shifts at the
reflecting elements of the IRS due to the high manufacturing cost. It is more
cost-effective to implement only discrete phase shifts with a small number of
control bits for each element, e.g., 1-bit for two-level (0 or $\pi$) phase
shifts. Thus, the impact of the controlling bits $b$ of the discrete phase
shifts on the security performance is investigated in Fig. 12. The transmit
power limit is 10dBm. It is shown that the SR with continuous phase shifts of
the IRS is higher than those with discrete phase shifts. The limited discrete
phase shifts inevitably cause SR performance degradation. The SR of the IRS
with discrete phase shifts increases with the number of controlling bits $b$,
and becomes saturated when $b\geq 4$, which means that the SR loss is
inevitable even when the number of controlling bits $b$ is high. For the
proposed BCD-MM algorithm, the maximum SR gap between the continuous phase
shifts and the discrete phase shifts is 1.4 bit/s/Hz.

Figure 11: Achievable SR versus the discrete phase bits $b$.

Figure 12: Achievable SR versus the transmit power limit for multiple IRs.

####  V-B8 Multiple IRs Scenario

Finally, we consider the multiple IRs scenario to investigate the security
enhancement brought by the IRS on the AN-aided MIMO communication systems. The
horizontal distances between the BS and the two IRs are selected as
$d_{BI,1}=47$m and $d_{BI,2}=49$m. Considering the heavy computational load,
the element number of the IRS is assumed to be $M=20$. The proposed BCD-QCQP-
CCP algorithm is utilized to perform the joint optimization of the TPC matrix,
AN covariance matrix and the phase shifts of the IRS. The achieved SRs for the
proposed algorithm, the random IRS scheme and the No-IRS scheme are shown in
Fig. 12. By comparing with the Random IRS scheme and the No-IRS scheme, the
proposed BCD-QCQP-CCP algorithm can optimize the phase shifts of the IRS, thus
achieve the higher SR. The SR gain increases with the power limit $P_{T}$.
However, the performance gain is not as much as that in Fig. 6. On the one
hand, the element number of the IRS is set to be lower than that in Fig. 6 due
to the heavy computational load. On the other hand, it is more difficult for
the IRS to adjust the phase shifts to guarantee higher SRs for more legitimate
IRs.

##  VI Conclusions

In this paper, we propose to enhance the security of AN-aided MIMO secure
communication systems by exploiting an IRS. To exploit the IRS efficiently, we
formulate a SRM problem by jointly optimizing the TPC matrix at the BS, the
covariance matrix of AN and phase shifts at the IRS with the constraints of
transmit power limit and unit-modulus of phase shifts. To solve this non-
convex problem, we propose to use the BCD algorithm to decouple the
optimization variables, and optimize them iteratively. The optimal TPC matrix
and AN covariance matrix were obtained in semi-closed form by the Lagrange
multiplier method, and the phase shifts at the IRS were obtained in closed
form by an efficient MM algorithm. Various simulations validated that
significant security gains can be achieved by the proposed algorithm with IRS.
Furthermore, useful suggestions for choosing and deploying the IRS are
provided.

##  Appendix A Derivation of the Problem (29)

By substituting $h_{1}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}},{{\bf{W}}_{I}})$
of (III-A), $h_{2}({{\bf{U}}_{E}},{{\bf{V}}_{E}},{{\bf{W}}_{E}})$ of (III-A)
and $h_{3}({{\bf{V}}},{{\bf{V}}_{E}},{{\bf{W}}_{X}})$ of (III-A) into (III-A),
we have

$\displaystyle{{\rm{C}}^{l}_{AN}}({{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}},{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}})=\log\left|{{{\bf{W}}_{I}}}\right|-{\rm{Tr}}({{\bf{W}}_{I}}{{\bf{E}}_{I}}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}}))+\log\left|{{{\bf{W}}_{E}}}\right|$
$\displaystyle\qquad\qquad\qquad-{\rm{Tr}}({{\bf{W}}_{E}}{{\bf{E}}_{E}}({{\bf{U}}_{E}},{{\bf{V}}_{E}}))+\log\left|{{{\bf{W}}_{X}}}\right|-{\rm{Tr}}({{\bf{W}}_{X}}{{\bf{E}}_{X}}({{\bf{V}}},{{\bf{V}}_{E}}))+d+N_{t}+N_{E}$
$\displaystyle\qquad=C_{g_{0}}-{\underbrace{{\rm{Tr}}({{\bf{W}}_{I}}{{\bf{E}}_{I}}({{\bf{U}}_{I}},{\bf{V}},{{\bf{V}}_{E}}))}_{g_{1}}}-{\underbrace{{\rm{Tr}}({{\bf{W}}_{E}}{{\bf{E}}_{E}}({{\bf{U}}_{E}},{{\bf{V}}_{E}}))}_{g_{2}}}-{\underbrace{{\rm{Tr}}({{\bf{W}}_{X}}{{\bf{E}}_{X}}({{\bf{V}}},{{\bf{V}}_{E}}))}_{g_{3}}},$
(87)

where
$C_{g_{0}}\buildrel\Delta\over{=}\log\left|{{{\bf{W}}_{I}}}\right|+\log\left|{{{\bf{W}}_{E}}}\right|+\log\left|{{{\bf{W}}_{X}}}\right|+d+N_{t}+N_{E}$.

$C_{g_{0}}$ contains the constant terms irrelated with
${\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}}$. By putting matrix functions
${{\bf{E}}_{I}}$, ${{\bf{E}}_{E}}$ and ${{\bf{E}}_{X}}$ into (A), we expand
${g_{1}}$, ${g_{2}}$, and ${g_{3}}$ respectively as follows.

(1) ${{g}_{1}}$ can be reformulated as

$\displaystyle{{g}_{1}}$
$\displaystyle=\text{Tr}({{\bf{W}}_{I}}[({\bf{I}}-{{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{\bf{V}}){{({\bf{I}}-{{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{\bf{V}})}^{H}}+{{\bf{U}}_{I}}^{H}({{{\hat{\bf{H}}}}_{I}}{{\bf{V}}_{E}}{{\bf{V}}_{E}}^{H}{{{\hat{\bf{H}}}}_{I}}^{H}+\sigma_{I}^{2}{{\bf{I}}_{{{N}_{I}}}}){{\bf{U}}_{I}}])$
$\displaystyle=\text{Tr}({{\bf{W}}_{I}}[({\bf{I}}-{{\bf{V}}^{H}}{{{\hat{\bf{H}}}}_{I}}^{H}{{\bf{U}}_{I}}-{{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{\bf{V}}+{{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{\bf{V}}{{\bf{V}}^{H}}{{{\hat{\bf{H}}}}_{I}}^{H}{{\bf{U}}_{I}})$
$\displaystyle\quad+({{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{{\bf{V}}_{E}}{{\bf{V}}_{E}}^{H}{{{\hat{\bf{H}}}}_{I}}^{H}{{\bf{U}}_{I}}\text{+}{{\bf{U}}_{I}}^{H}\sigma_{I}^{2}{{\bf{I}}_{{{N}_{I}}}}{{\bf{U}}_{I}})]).$
(88)

By gathering the constant terms related with ${{\bf{W}}_{I}},{{\bf{U}}_{I}}$
in ${{C}}_{g_{1}}$, ${{g}_{1}}$ can be simplified as

$\displaystyle{{g}_{1}}$
$\displaystyle=-\text{Tr}({{\bf{W}}_{I}}{{\bf{V}}^{H}}{{{\hat{\bf{H}}}}_{I}}^{H}{{\bf{U}}_{I}})-\text{Tr}({{\bf{W}}_{I}}{{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{\bf{V}})+\text{Tr}({{\bf{V}}^{H}}{{{\hat{\bf{H}}}}_{I}}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{\bf{V}})$
$\displaystyle\quad+\text{Tr}({{\bf{V}}_{E}}^{H}{{{\hat{\bf{H}}}}_{I}}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{{\bf{U}}_{I}}^{H}{{{\hat{\bf{H}}}}_{I}}{{\bf{V}}_{E}})+{{C}}_{g_{1}},$
(89)

where
${{C}}_{g_{1}}\buildrel\Delta\over{=}\text{Tr}({{\bf{W}}_{I}}+\sigma_{I}^{2}{{\bf{W}}_{I}}{{\bf{U}}_{I}}^{H}{{\bf{U}}_{I}})$.

(2) ${{g}_{2}}$ can be reformulated as

$\displaystyle{{g}_{2}}$
$\displaystyle=\text{Tr}({{\mathbf{W}}_{E}}[(\mathbf{I}-{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}}){{(\mathbf{I}-{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})}^{H}}+\sigma_{E}^{2}{{\mathbf{U}}_{E}}^{H}{{\mathbf{U}}_{E}}])$
$\displaystyle=\text{Tr}({{\mathbf{W}}_{E}}[(\mathbf{I}-{{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}}-{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}}+{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}}{{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}}$
$\displaystyle\quad+\sigma_{E}^{2}{{\mathbf{U}}_{E}}^{H}{{\mathbf{U}}_{E}}]).$
(90)

By gathering the constant terms related with
${{\mathbf{W}}_{E}},{{\mathbf{U}}_{E}}$ in ${{C}}_{g_{2}}$, ${{g}_{2}}$ can be
simplified as

$\displaystyle{{g}_{2}}=-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}})-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+\text{Tr}({{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}}{{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+{{C}}_{g_{2}},$
(91)

where
${{C}}_{g_{2}}\buildrel\Delta\over{=}\text{Tr}({{\bf{W}}_{E}}+\sigma_{E}^{2}{{\bf{W}}_{E}}{{\bf{U}}_{E}}^{H}{{\bf{U}}_{E}})$.

(3) ${{g}_{3}}$ can be reformulated as

$\displaystyle{{g}_{3}}=\text{Tr}({{\mathbf{W}}_{X}}({{{\bf{I}}_{{N_{E}}}}+\sigma_{E}^{-2}{{\hat{\bf{H}}}_{E}}({\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{{\bf{V}}_{E}}^{H})\hat{\bf{H}}_{E}^{H}})).$
(92)

By gathering the constant terms related with ${{\mathbf{W}}_{X}}$ in
${{C}}_{g_{3}}$, ${{g}_{3}}$ can be simplified as

$\displaystyle{{g}_{3}}=\sigma_{E}^{-2}\text{Tr}({{\mathbf{V}}^{H}}\mathbf{\hat{H}}_{E}^{H}{{\mathbf{W}}_{X}}{{{\mathbf{\hat{H}}}}_{E}}\mathbf{V})+\sigma_{E}^{-2}\text{Tr}({{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{W}}_{X}}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+{{C}}_{g_{3}},$
(93)

where ${{C}}_{g_{3}}\buildrel\Delta\over{=}\text{Tr}({{\bf{W}}_{X}})$.

By substituting (89), (91) and (93) into (A), we have

$\displaystyle{{\rm{C}}^{l}_{AN}}({{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}},{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}})=\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}})+\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{{\mathbf{\hat{H}}}}_{I}}\mathbf{V})$
$\displaystyle-\text{Tr}({{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}}{{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{{\mathbf{\hat{H}}}}_{I}}\mathbf{V})-\text{Tr}({{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}}{{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{{\mathbf{\hat{H}}}}_{I}}{{\mathbf{V}}_{E}})+\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}})$
$\displaystyle+\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})-\text{Tr}({{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}}{{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})-\sigma_{E}^{-2}\text{Tr}({{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}}_{E}^{H}{{\mathbf{W}}_{X}}{{{\mathbf{\hat{H}}}}_{E}}\mathbf{V})$
$\displaystyle-\sigma_{E}^{-2}\text{Tr}({{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{W}}_{X}}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+C_{g},$
(94)

where $C_{g}\buildrel\Delta\over{=}C_{g_{0}}-C_{g_{1}}-C_{g_{2}}-C_{g_{3}}$.

Equation (A) can be rewritten more compactly as

$\displaystyle{{\rm{C}}^{l}_{AN}}({{\bf{U}}_{I}},{{\bf{W}}_{I}},{{\bf{U}}_{E}},{{\bf{W}}_{E}},{{\bf{W}}_{X}},{\bf{V}},{{\bf{V}}_{E}},{\bf{\Phi}})=C_{g}+\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}})+\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{{\mathbf{\hat{H}}}}_{I}}\mathbf{V})$
$\displaystyle\quad-\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V}}\mathbf{V})+\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}})+\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})-\text{Tr}({{\mathbf{V}}_{E}}^{H}{{\mathbf{H}}_{VE}}{{\mathbf{V}}_{E}}).$
(95)

where

$\displaystyle{{\mathbf{H}}_{V}}={{\mathbf{\hat{H}}}_{I}}^{H}{{\mathbf{U}}_{I}}{{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{\mathbf{\hat{H}}}_{I}}+\sigma_{E}^{-2}\mathbf{\hat{H}}_{E}^{H}{{\mathbf{W}}_{X}}{{\mathbf{\hat{H}}}_{E}}.$
(96)
$\displaystyle{{\mathbf{H}}_{VE}}={{\mathbf{\hat{H}}}_{I}}^{H}{{\mathbf{U}}_{I}}{{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{\mathbf{\hat{H}}}_{I}}+{{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{U}}_{E}}{{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{\mathbf{\hat{H}}}_{E}}+\sigma_{E}^{-2}{{\mathbf{\hat{H}}}_{E}}^{H}{{\mathbf{W}}_{X}}{{\mathbf{\hat{H}}}_{E}}.$
(97)

By substituting (95) into Problem (28), and removing the constant term
$C_{g}$, we arrive at the Problem (29).

##  Appendix B Derivation of the new OF form in (57)

The objective function of Problem (56) is

$\displaystyle{g_{0}}(\mathbf{V},{{\mathbf{V}}_{E}},\mathbf{\Phi})=$
$\displaystyle-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{V}}^{H}}{{{\mathbf{\hat{H}}}}_{I}}^{H}{{\mathbf{U}}_{I}})-\text{Tr}({{\mathbf{W}}_{I}}{{\mathbf{U}}_{I}}^{H}{{{\mathbf{\hat{H}}}}_{I}}\mathbf{V})+\text{Tr}({{\mathbf{V}}^{H}}{{\mathbf{H}}_{V}}\mathbf{V})$
$\displaystyle-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{V}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}^{H}{{\mathbf{U}}_{E}})-\text{Tr}({{\mathbf{W}}_{E}}{{\mathbf{U}}_{E}}^{H}{{{\mathbf{\hat{H}}}}_{E}}{{\mathbf{V}}_{E}})+\text{Tr}({{\mathbf{V}}_{E}}^{H}{{\mathbf{H}}_{VE}}{{\mathbf{V}}_{E}}).$
(98)

The third term of (B) is

$\displaystyle{\rm{Tr}}\left({{{\bf{V}}^{H}}{{\bf{H}}_{V}}{\bf{V}}}\right)$
$\displaystyle={\rm{Tr}}\left[{{{\bf{V}}^{H}}\left({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{{\bf{\hat{H}}}}_{I}}+\sigma_{E}^{-2}{\bf{\hat{H}}}_{E}^{H}{{\bf{W}}_{X}}{{{\bf{\hat{H}}}}_{E}}}\right){\bf{V}}}\right]$
$\displaystyle={\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{I}}{\bf{V}}{{\bf{V}}^{H}}{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}}\right]+\sigma_{E}^{-2}{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}{\bf{V}}{{\bf{V}}^{H}}{\bf{\hat{H}}}_{E}^{H}{{\bf{W}}_{X}}}\right].$
(99)

The sixth term of (B) is

$\displaystyle{\rm{Tr}}\left({{\bf{V}}_{E}^{H}{{\bf{H}}_{VE}}{{\bf{V}}_{E}}}\right)=$
$\displaystyle{\rm{Tr}}\left[{{\bf{V}}_{E}^{H}\left({{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{{\bf{\hat{H}}}}_{I}}+{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{{\bf{\hat{H}}}}_{E}}+\sigma_{E}^{-2}{\bf{\hat{H}}}_{E}^{H}{{\bf{W}}_{X}}{{{\bf{\hat{H}}}}_{E}}}\right){{\bf{V}}_{E}}}\right]$
$\displaystyle=$
$\displaystyle{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{I}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}}\right]+{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}}\right]$
$\displaystyle+\sigma_{E}^{-2}{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}{{\bf{W}}_{X}}}\right].$
(100)

The summation of Equation (B) and Equation (B) is

$\displaystyle{\rm{Tr}}\left({{{\bf{V}}^{H}}{{\bf{H}}_{V}}{\bf{V}}}\right)+{\rm{Tr}}\left({{\bf{V}}_{E}^{H}{{\bf{H}}_{VE}}{{\bf{V}}_{E}}}\right)=$
$\displaystyle{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{I}}\left({{\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}}\right){\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}}\right]$
$\displaystyle+\sigma_{E}^{-2}{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}\left({{\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}}\right){\bf{\hat{H}}}_{E}^{H}{{\bf{W}}_{X}}}\right]$
$\displaystyle+{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}}\right].$
(101)

By defining
${{\bf{V}}_{X}}=\left({{\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}}\right)$
and ${{\bf{M}}_{I}}={{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}$, the first
part of (B) can be derived as

$\displaystyle{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{I}}\left({{\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}}\right){\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}}\right]$
$\displaystyle={\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{I}}{{\bf{V}}_{X}}{\bf{\hat{H}}}_{I}^{H}{{\bf{M}}_{I}}}\right]$
$\displaystyle={\rm{Tr}}\left[{\left({{{\bf{H}}_{b,I}}+{{\bf{H}}_{R,I}}{\bf{\Phi
G}}}\right){{\bf{V}}_{X}}\left({{\bf{H}}_{b,I}^{H}+{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,I}^{H}}\right){{\bf{M}}_{I}}}\right]$
$\displaystyle={\rm{Tr}}\left[{\left({{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}+{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,I}^{H}+{{\bf{H}}_{R,I}}{\bf{\Phi
G}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}+{{\bf{H}}_{R,I}}{\bf{\Phi
G}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,I}^{H}}\right){{\bf{M}}_{I}}}\right]$
$\displaystyle={\rm{Tr}}[{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}{{\bf{M}}_{I}}+{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}+{{\bf{H}}_{R,I}}{\bf{\Phi
G}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}{{\bf{M}}_{I}}$
$\displaystyle\quad+{{\bf{H}}_{R,I}}{\bf{\Phi
G}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}].$
(102)

The derivation in (B) can be used for the second and third parts of (B).

Based on the derivation in (B), it is obvious that the second part of (B) can
be derived as

$\displaystyle\sigma_{E}^{-2}{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}\left({{\bf{V}}{{\bf{V}}^{H}}+{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}}\right){\bf{\hat{H}}}_{E}^{H}{{\bf{W}}_{X}}}\right]$
$\displaystyle=\sigma_{E}^{-2}{\rm{Tr}}[{{\bf{H}}_{b,E}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}+{{\bf{H}}_{b,E}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}+{{\bf{H}}_{R,E}}{\bf{\Phi
G}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}$
$\displaystyle\quad+{{\bf{H}}_{R,E}}{\bf{\Phi
G}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}].$
(103)

Based on the derivation in (B) and by defining
${{\bf{M}}_{E}}={{\bf{U}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}$, it is obvious
that the third part of (B) can be derived as

$\displaystyle{\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}}\right]$
$\displaystyle={\rm{Tr}}\left[{{{{\bf{\hat{H}}}}_{E}}\left({{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}}\right){\bf{\hat{H}}}_{E}^{H}{{\bf{M}}_{E}}}\right]$
$\displaystyle={\rm{Tr}}[{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}+{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,E}^{H}{{\bf{M}}_{E}}+{{\bf{H}}_{R,E}}{\bf{\Phi
G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}$
$\displaystyle\quad+{{\bf{H}}_{R,E}}{\bf{\Phi
G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,E}^{H}{{\bf{M}}_{E}}].$
(104)

By adding (B), (B) and (B), and gathering constant terms irreverent with
${\bf{\Phi}}$, Equation (B) becomes

$\displaystyle{\rm{Tr}}\left({{{\bf{V}}^{H}}{{\bf{H}}_{V}}{\bf{V}}}\right)+{\rm{Tr}}\left({{\bf{V}}_{E}^{H}{{\bf{H}}_{VE}}{{\bf{V}}_{E}}}\right)$
$\displaystyle={\rm{Tr}}\left[{{{\bf{\Phi}}^{H}}\left({{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{b,E}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}+{\bf{H}}_{R,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}}\right)}\right]$
$\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi}}\left({{\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}+{\bf{G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{R,E}}}\right)}\right]$
$\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi
G}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}\left({{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}}\right)}\right]$
$\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi
G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}{\bf{H}}_{R,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{R,E}}}\right]+C_{{t}_{1}},$
(105)

where

$\displaystyle
C_{{t}_{1}}={\rm{Tr}}\left[{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}{{\bf{M}}_{I}}\right]+\sigma_{E}^{-2}{\rm{Tr}}\left[{{\bf{H}}_{b,E}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}\right]+{\rm{Tr}}\left[{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}\right].$
(106)

The first term of ${g_{0}}(\mathbf{V},{{\mathbf{V}}_{E}},\mathbf{\Phi})$ is
derived as

$\displaystyle{\rm{Tr}}\left({{{\bf{W}}_{I}}{{\bf{V}}^{H}}{\bf{\hat{H}}}_{I}^{H}{{\bf{U}}_{I}}}\right)\\!=\\!{\rm{Tr}}\left({{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}{{\bf{V}}^{H}}{\bf{\hat{H}}}_{I}^{H}}\right)\\!=\\!\underbrace{{\rm{Tr}}\left[{{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}{{\bf{V}}^{H}}{\bf{H}}_{b,I}^{H}}\right]}_{C_{{t}_{2}}(\text{constant
for
}\mathbf{\Phi})}\\!+\\!{\rm{Tr}}\left[{{\bf{H}}_{R,I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}{{\bf{V}}^{H}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}}\right].$
(107)

The second term of ${g_{0}}(\mathbf{V},{{\mathbf{V}}_{E}},\mathbf{\Phi})$ is
derived as

$\displaystyle{\rm{Tr}}\left({{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{{\bf{\hat{H}}}}_{I}}{\bf{V}}}\right)={\rm{Tr}}\left({{{{\bf{\hat{H}}}}_{I}}{\bf{V}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}}\right)={\rm{Tr}}\left[{\left({{{\bf{H}}_{b,I}}+{{\bf{H}}_{R,I}}{\bf{\Phi
G}}}\right){\bf{V}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}}\right]$
$\displaystyle=\underbrace{{\rm{Tr}}\left[{{{\bf{H}}_{b,I}}{\bf{V}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}}\right]}_{C_{{t}_{3}}(\text{constant
for }\mathbf{\Phi})}+{\rm{Tr}}\left[{{\bf{\Phi
GV}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{\bf{H}}_{R,I}}}\right].$ (108)

The fourth term of ${g_{0}}(\mathbf{V},{{\mathbf{V}}_{E}},\mathbf{\Phi})$ is
derived as

$\displaystyle{\rm{Tr}}\left({{{\bf{W}}_{E}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}{{\bf{U}}_{E}}}\right)={\rm{Tr}}\left({{{\bf{U}}_{E}}{{\bf{W}}_{E}^{H}}{\bf{V}}_{E}^{H}{\bf{\hat{H}}}_{E}^{H}}\right)$
$\displaystyle=\underbrace{{\rm{Tr}}\left[{{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}}\right]}_{C_{{t}_{4}}(\text{constant
for
}\mathbf{\Phi})}+{\rm{Tr}}\left[{{\bf{H}}_{R,E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}}\right].$
(109)

The fifth term of ${g_{0}}(\mathbf{V},{{\mathbf{V}}_{E}},\mathbf{\Phi})$ is
derived as

$\displaystyle{\rm{Tr}}\left({{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}}\right)={\rm{Tr}}\left({{{{\bf{\hat{H}}}}_{E}}{{\bf{V}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}}\right)$
$\displaystyle=\underbrace{{\rm{Tr}}\left[{{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}}\right]}_{C_{{t}_{5}}(\text{constant
for }\mathbf{\Phi})}+{\rm{Tr}}\left[{{\bf{\Phi
G}}{{\bf{V}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{\bf{H}}_{R,E}}}\right].$
(110)

By including the first term in (107), the second term in (B), the fourth term
in (B), the fifth term in (B), and the sum of the third and six terms in (B)
of ${g_{0}}(\mathbf{V},{{\mathbf{V}}_{E}},\mathbf{\Phi})$ and gathering
constant terms irreverent with ${\bf{\Phi}}$, we have

$\displaystyle{g_{0}}(\mathbf{\Phi})=-\rm{Equation\
}\eqref{firstoffirstconst}-\rm{Equation\
}\eqref{secondoffirstconst}-\rm{Equation\
}\eqref{fourthoffirstconst}-\rm{Equation\
}\eqref{fifthoffirstconst}+\rm{Equation\ }\eqref{third_sixoffirstconst_form2}$
$\displaystyle={\rm{Tr}}\left[{{{\bf{\Phi}}^{H}}\left(\begin{array}[]{l}{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{b,E}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}+{\bf{H}}_{R,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}\\\
-{\bf{H}}_{R,I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}{{\bf{V}}^{H}}{{\bf{G}}^{H}}-{\bf{H}}_{R,E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}\end{array}\right)}\right]$
(112)
$\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi}}\left(\begin{array}[]{l}{\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}+{\bf{G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{R,E}}\\\
-{\bf{GV}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{\bf{H}}_{R,I}}-{\bf{G}}{{\bf{V}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{\bf{H}}_{R,E}}\end{array}\right)}\right]$
(114) $\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi
G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}\left({{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}+{\bf{H}}_{R,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{R,E}}}\right)}\right]$
$\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi
GV}}{{\bf{V}}^{H}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}\left({{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}}\right)}\right]+C_{t}$
$\displaystyle={\rm{Tr}}\left[{{{\bf{\Phi}}^{H}}\left(\begin{array}[]{l}{\bf{H}}_{R,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{b,I}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{b,E}}{{\bf{V}}_{X}}{{\bf{G}}^{H}}+{\bf{H}}_{R,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{b,E}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}\\\
-{\bf{H}}_{R,I}^{H}{{\bf{U}}_{I}}{\bf{W}}_{I}^{H}{{\bf{V}}^{H}}{{\bf{G}}^{H}}-{\bf{H}}_{R,E}^{H}{{\bf{U}}_{E}}{\bf{W}}_{E}^{H}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}\end{array}\right)}\right]$
(116)
$\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi}}\left(\begin{array}[]{l}{\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,I}^{H}{{\bf{M}}_{I}}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{G}}{{\bf{V}}_{X}}{\bf{H}}_{b,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}+{\bf{G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{\bf{H}}_{b,E}^{H}{{\bf{M}}_{E}}{{\bf{H}}_{R,E}}\\\
-{\bf{GV}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{\bf{H}}_{R,I}}-{\bf{G}}{{\bf{V}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{\bf{H}}_{R,E}}\end{array}\right)}\right]$
(118) $\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi
G}}{{\bf{V}}_{E}}{\bf{V}}_{E}^{H}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}\left({{\bf{H}}_{R,I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}+{\bf{H}}_{R,E}^{H}{{\bf{U}}_{E}}{{\bf{W}}_{E}}{\bf{U}}_{E}^{H}{{\bf{H}}_{R,E}}}\right)}\right]$
$\displaystyle\quad+{\rm{Tr}}\left[{{\bf{\Phi
GV}}{{\bf{V}}^{H}}{{\bf{G}}^{H}}{{\bf{\Phi}}^{H}}\left({{\bf{H}}_{R,I}^{H}{{\bf{U}}_{I}}{{\bf{W}}_{I}}{\bf{U}}_{I}^{H}{{\bf{H}}_{R,I}}+\sigma_{E}^{-2}{\bf{H}}_{R,E}^{H}{{\bf{W}}_{X}}{{\bf{H}}_{R,E}}}\right)}\right]+C_{t},$
(119)

where

$\displaystyle
C_{t}=C_{{t}_{1}}+C_{{t}_{2}}+C_{{t}_{3}}+C_{{t}_{4}}+C_{{t}_{5}}.$ (120)

Then ${g_{0}}(\mathbf{\Phi})$ becomes

$\displaystyle{g_{0}}(\mathbf{\Phi})={\rm{Tr}}\left({{{\bf{\Phi}}^{H}{\bf{D}}^{H}}}\right)+{\rm{Tr}}\left({{\bf{\Phi
D}}}\right)+{\rm{Tr}}\left[{{\bf{\Phi}}{{\bf{C}}_{VE}}{{\bf{\Phi}}^{H}}{{\bf{B}}_{VE}}}\right]+{\rm{Tr}}\left({{\bf{\Phi}}{{\bf{C}}_{V}}{{\bf{\Phi}}^{H}}{{\bf{B}}_{V}}}\right)+C_{t}$
$\displaystyle={\rm{Tr}}\left({{{\bf{\Phi}}^{H}{\bf{D}}^{H}}}\right)+{\rm{Tr}}\left({{\bf{\Phi
D}}}\right)+{\rm{Tr}}\left[{{{\bf{\Phi}}^{H}}{{\bf{B}}_{VE}}{\bf{\Phi}}{{\bf{C}}_{VE}}}\right]+{\rm{Tr}}\left({{{\bf{\Phi}}^{H}}{{\bf{B}}_{V}}{\bf{\Phi}}{{\bf{C}}_{V}}}\right)+C_{t},$
(121)

where