# Secure Hybrid Digital and Analog Precoder for mmWave Systems with low-
resolution DACs and finite-quantized phase shifters

Ling Xu, Feng Shu, , Guiyang Xia,
Yijin Zhang, Zhihong Zhuang, and Jiangzhou Wang This work was supported in
part by the National Natural Science Foundation of China (Nos. 61771244,
61472190, 61702258, 61501238, and 61602245)(Corresponding authors: Feng Shu,
Ling Xu, and Zhihong Zhuang). Ling Xu, Feng Shu, Guiyang Xia, Yijin Zhang and
Zhihong Zhuang are with School of Electronic and Optical Engineering, Nanjing
University of Science and Technology, 210094, CHINA. E-mail:{shufeng,
<EMAIL_ADDRESS>Wang is with the School of Engineering and
Digital Arts, University of Kent, Canterbury CT2 7NT, U.K. Email:
<EMAIL_ADDRESS>
###### Abstract

Millimeter wave (mmWave) communication has been regarded as one of the most
promising technologies for the future generation wireless networks because of
its advantages of providing a ultra-wide new spectrum and ultra-high data
transmission rate. To reduce the power consumption and circuit cost for mmWave
systems, hybrid digital and analog (HDA) architecture is preferred in such a
scenario. In this paper, an artificial-noise (AN) aided secure HDA beamforming
scheme is proposed for mmWave MISO system with low resolution digital-to-
analog converters (DACs) and finite-quantized phase shifters on RF. The
additive quantization noise model for AN aided HDA system is established to
make an analysis of the secrecy performance of such systems. With the partial
channel knowledge of eavesdropper available, an approximate expression of
secrecy rate (SR) is derived. Then using this approximation formula, we
propose a two-layer alternately iterative structure (TLAIS) for optimizing
digital precoder (DP) of confidential message (CM), digital AN projection
matrix (DANPM) and analog precoder (AP). The inner-layer iteration loop is to
design the DP of CMs and DANPM alternatively given a fixed matrix of AP. The
outer-layer iteration loop is in between digital baseband part and analog
part, where the former refers to DP and DANPM, and the latter is AP. Then for
a given digital part, we propose a gradient ascent algorithm to find the
vector of AP vector. Given a matrix of AP, we make use of general power
iteration (GPI) method to compute DP and DANPM. This process is repeated until
the terminal condition is reached. Simulation results show that the proposed
TLAIS can achieve a better SR performance compared to existing methods,
especially in the high signal-to-noise ratio region.

###### Index Terms:

Hybrid digital and analog, mmWave, security, artificial Noise, low-resolution
digital-to-analog converter(DAC)

##  I Introduction

With the rapid development of wireless communication technologies as well as
explosive access of mobile terminals, the demand of wireless data traffic
grows exponentially. However, the spectrum of current exiting wireless systems
is highly congested, which brings a bottleneck for increasing wireless access
rate ulteriorly. Millimeter wave (mmWave) communication, whose frequency band
ranges from 30GHz to 300GHz, emerges as a promising candidate way of
addressing the problem of spectrum congestion [1, 2, 3, 4].

Nevertheless, although the mmWave communication becomes more and more popular
due to its abundant available and under-utilized spectrum, its realistic
applications are under many constraints such as the severe free space path
loss and rain attenuation [5]. To overcome these challenges, multiple-input-
multiple-output (MIMO) is usually adopted to compensate for these shortages.
Meanwhile, hybrid digital and analog (HDA) architecture has been employed to
further reduce the energy consumption of mmWave MIMO systems [6, 7, 8, 9, 10,
11, 12, 13]. The design of hybrid precoder was formulated as a problem of
sparse signal reconstruction by exploiting the inherent spatial sparse
structure of mmWave channel in [6, 7], where orthogonal matching pursuit (OMP)
algorithm was used to achieve a comparatively good performance. In [8], the
authors proposed three methods of innovative alternative minimization for both
fully-connected and partially-connected hybrid combiner in mmWave MIMO
systems, which confirms the feasibility and effectiveness of hybrid precoder
in mmWave MIMO architecture. In [9], an energy-efficient successive-
interference-cancelation-based algorithm with low complexity was proposed for
mmWave MIMO systems. Authors in [10] and [11] extended conventional flat-
fading mmWave channels to broadband frequency-selective mmWave channels, where
the joint analog beamforming for the entire band and respective baseband
precoder for each sub-band were required to be designed carefully.
Specifically, in [10] the heuristic algorithms in OFDM systems were developed
for two scenarios: single-user MIMO and multiple-user multiple-input-single-
output scenarios. In [11], the authors developed a novel method by dynamically
establishing the structure of sub-arrays. In [12] , the hybrid analog and
digital (HAD) architecture was used at receiver to make a measurement of
direction of arrival (DOA). Here, three low-complexity DOA estimation methods
were proposed and the corresponding HAD Cramer-Rao lower bound was also
derived. In [13], a robust beamforming method using HAD receive structure was
proposed to achieve interference compression.

However, the aforementioned research works mainly focus on HDA mmWave systems
without taking security into consideration. Wyner first proposed a discrete,
memoryless wiretap channel model to investigate the secrecy of wireless
communication [14]. Based on the framework of Wyner’s wiretap channel,
multiple-antenna technique [15] is applied to enhance security and beamforming
has been proven to be secrecy-capacity-achieving under the circumstances that
the desired receiver has single antenna and full channel knowledge are
available for all terminals[16, 17]. If the transmitter can not obtain the
simultaneous channel state information (CSI) of eavesdropper or only know the
partial CSI of eavesdropper, artificial noise (AN) was shown to be an
effective aiding method to strengthen security [18, 19, 20, 21]. Recently,
some researchers paid a close attention to security in mmWave MIMO channel.
Authors in [22] made a systematical investigation of the security performance
in HDA mmWave systems under two different CSIs: full and partial. Here,
maximum ratio transmission (MRT) method was extended to such scenarios and
minimum secrecy outage probability was adopted to design the corresponding
hybrid precoder. Considering the sparse features of millimeter wave channels
in [23], a discrete angular domain channel model was proposed to meticulously
derive the secure performance for the proposed transmission schemes in slow
fading multipath channels. In [24], an AN-aided hybrid precoder is proposed to
maximize the lower bound of average secrecy rate (SR).

In practical applications, if low-resolution digital-to-analog converters
(DACs) or analog-to-digital converters (ADCs) are adopted, the average power
consumption of hybrid transceivers will be greatly reduced. Actually, a high
-resolution DACs/ADCs can be power-hungry [25]. In [26], how to strike a
trade-off between energy efficiency and spectrum efficiency was investigated
for analog, hybrid, and digital receivers with low resolution ADCs,
respectively. To further reduce energy consumption, hybrid beamforming and
digital beamforming with low resolution ADCs are studied in [27]. Here, the
authors find that digital beamforming equipped with finite resolution ADCs can
achieve a higher achievable rate and can be more energy efficient than hybrid
beamforming in the low signal-to-noise ratio (SNR) region. In [28], taking
both low resolution DACs and RF losses in hybrid into account in mmWave MIMO
system, a quantized hybrid transmitter with additive quantization noise model
(AQNM) has been constructed for both fully-connected and partially-connected
hybrid architecture, and a lower bound of achievable rate is presented.

However, to the best of our knowledge, there are few literature of making an
investigation on how to achieve a security in hybrid mmWave systems with low
resolution DACs. Therefore, in our paper, we propose an AN-aided hybrid mmWave
transmitter with low resolution DACs and finite-quantized phase shifters. The
transmitted signals with the help of AN are first precoded by digital precoder
(DP) in baseband, then passing through low-resolution DACs and RF chains
before analog precoder (AP). Here, the AQN model is adopted to approximate the
quantized signal as a linear output, which simplifies the analysis of further
secrecy rate. Since the optimization problem is non-convex and intractable to
tackle, an alternate iteration algorithm is resorted to maximize the
approximate expression of secrecy rate. Our main contributions are summarized
as follows:

  1. 1.

An AN-aided secure hybrid precoding system model with low-resolution DACs and
finite-quantized phase shifters on RF is established. By taking two kinds of
quantization errors (QEs) into consideration, the proposed model is completely
distinct from conventional non-secure hybrid precoding system without
considering QEs and the AQN quantized model is adopted in our system model to
approximate the distortion from QE of low-resolution DACs as a linear output.
With partial eavesdropping channel knowledge available, an approximate
expression for secrecy rate (SR) is derived. This expression converts the
original intractable problem into a more easy-to-handle one. Thus, the
approximate SR (ASR) expression will significantly simplify the optimization
and design of digital precoding (DP) vector of confidential messages, digital
AN projection matrix (ANPM) and analog precoding (AP) vector in what follows.

  2. 2.

A two-layer alternatively iterative structure (TLAIS) is proposed to maximize
the ASR by taking quantization errors from both low-resolution DAC and phase
shifters into consideration. Given AP vector, the DP vector of confidential
messages and ANPM are alternatively computed within an interior iterative loop
by making use of general power iterative (GPI) method with the aim to maximize
the approximate SR. In particular, by complex Kronecker product manipulation,
the problem of maximizing ASR with respect to the optimization matrix ANPM is
converted into one with respect to an optimization vector being the
vectorization of the corresponding ANPM. Given the DP vector and digital ANPM,
a steepest ascent algorithm is used to attain the updated AP vector. The above
process is repeated until the terminal condition is reached. Finally, the
phases of AP vector is directly taken out as the inputs of the finite
quantized phase shifters on RF. More importantly, to reduce the computation
complexity, we abstract the non-zero elements in analog part by taking
advantage of the sparsity of analog precoder.

The remainder of this paper is organized as follows. Section II presents the
system model of AN-aided HDA mmWave multiple input single output (MISO) system
with low resolution DACs and finite-quantized phase shifters, and the
approximate expression of SR is given in this section. Based on the
approximate formula of SR, a TLAIS among DP of confidential message, digital
AN projection matrix (DANPM) and AP is proposed in Section III. Performance
analysis and simulation evaluations are presented in Section IV. Finally, we
draw our conclusions in Section V.

Notation: throughout the paper, matrices, vectors, and scalars are denoted by
letters of bold upper case, bold lower case, and lower case, respectively.
$(\cdot)^{T}$, $(\cdot)^{*}$ and $(\cdot)^{H}$ denote transpose, conjugate,
and conjugate transpose, respectively. $\parallel\cdot\parallel_{2}$ and
$\parallel\cdot\parallel_{F}$ denote the $l_{2}$ norm of a vector and
Frobenius norm of a matrix, respectively. $\text{Tr}(\cdot)$ and
$\mathrm{vec}(\cdot)$ are matrix trace and matrix vectorization; $\otimes$ and
$\odot$ indicate the Kronecker products and Hadamard products between two
matrices, respectively. $\mathrm{diag}(\mathbf{A})$ returns a diagonal matrix
consisting of the corresponding diagonal elements of matrix $\mathbf{A}$.
$\mathrm{Diagblk}(\mathbf{a}_{1},\mathbf{a}_{2},\cdots,\mathbf{a}_{N})$
returns the block diagonal matrix concatenated of
$\mathbf{a}_{1},\mathbf{a}_{2},\cdots,\mathbf{a}_{N}$. And
$\mathbb{N}[\mathbf{a}]$ returns a vector consisting of the non-zero elements
in $\mathbf{a}$. $\mathbf{A}(m,n)$ denotes the element in $m^{th}$ row and
$n^{th}$ column, $\mathbf{A}(m_{1}:m_{2},n_{1}:n_{2})$ returns a matrix
consisting of $m_{1}^{th}$ to $m_{2}^{th}$ row and $n_{1}^{th}$ to
$n_{2}^{th}$ column in $\mathbf{A}$.

$\displaystyle R_{b}=\log_{2}\left(1+\frac{\beta
P(1-\eta)^{2}\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}}{(1-\beta)P(1-\eta)^{2}\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{T}_{BB}\mathbf{T}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}+\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{R}_{\mathbf{n}_{q}\mathbf{n}_{q}}\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}+\sigma^{2}}\right)$
(10) $\displaystyle R_{e}=\mathbb{E}\left[\log_{2}\left(1+\frac{\beta
P(1-\eta)^{2}\mathbf{h}_{e}\mathbf{F}_{RF}\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{h}_{e}^{H}}{(1-\beta)P(1-\eta)^{2}\mathbf{h}_{e}\mathbf{F}_{RF}\mathbf{T}_{BB}\mathbf{T}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{h}_{e}^{H}+\mathbf{h}_{e}\mathbf{F}_{RF}\mathbf{R}_{\mathbf{n}_{q}\mathbf{n}_{q}}\mathbf{F}_{RF}^{H}\mathbf{h}_{e}^{H}+\sigma^{2}}\right)\right]$
(11)

##  II Channel and System Models

Figure 1: Hybrid architecture transmitter with low resolution DACs

###  II-A System model

In this section, we consider a system model with HDA transmitter of low
resolution DACs as shown in Fig. 1. There are three network nodes: Alice, Bob,
and Eve. Working as a transmitter, Alice uses a partially connected hybrid
architecture in this paper, where each RF chain is connected to a subset of
antennas. Assume the transmit antennas at transmitter is $N_{T}$, and the
number of RF chains is $K$, then it is clear that $\mathrm{N_{T}=KT}$ with
each sub-array having $T$ antennas.

The transmit signal can be expressed as follows:

$\displaystyle\mathbf{x}=\mathbf{F}_{RF}Q_{b}(\sqrt{\beta
P}\mathbf{f}_{BB}s+\sqrt{(1-\beta)P}\mathbf{T}_{BB}\mathbf{z})$ (1)

where $P$ is the effective transmitted power, $\beta$ denotes the power
allocation (PA) factor of confidential message, and $1-\beta$ indicates the PA
factor of AN. $\mathbf{f}_{BB}\in\mathbb{C}^{K\times 1}$ and
$\mathbf{T}_{BB}\in\mathbb{C}^{K\times K}$ represent the digital beamforming
vector of confidential message and AN projection matrix, respectively. To
satisfy the transmit power constraint, we have
$\mathbb{E}\\{\mathbf{x}\mathbf{x}^{H}\\}=P_{T}$.
$\mathbf{F}_{RF}\in\mathbb{C}^{N\times K}$ is the analog precoding matrix with
the following structure:

$\displaystyle\mathbf{F}_{RF}$
$\displaystyle=\mathrm{Diagblk}(\mathbf{f}_{RF1},\cdots,\mathbf{f}_{RFK})$
$\displaystyle=\left[{\begin{array}[]{*{20}{c}}{\mathbf{f}_{RF1}}&{\mathbf{0}}&{\cdots}&{\mathbf{0}}\\\
{\mathbf{0}}&{\mathbf{f}_{RF2}}&{\cdots}&{\mathbf{0}}\\\
{\vdots}&{\vdots}&{\ddots}&{\vdots}\\\
{\mathbf{0}}&{\mathbf{0}}&{\cdots}&{\mathbf{f}_{RFK}}\end{array}}\right]$ (6)

where $\mathbf{f}_{RFk}$ denotes a vector defined as

$\displaystyle\mathbf{f}_{RFk}=[\exp(j\varphi_{k,1}),\exp(j\varphi_{k,2}),\cdots,\exp(j\varphi_{k,M})]^{T}.$
(7)

where

$\displaystyle\varphi_{k,i}\in\mathcal{F}_{RF}=\left\\{0,\frac{2\pi}{2^{b_{ps}}},\cdots,\frac{2\pi
2^{b_{ps}-1}}{2^{b_{ps}}}\right\\}$ (8)

where $\mathcal{F}_{RF}$ represents the set of $b_{ps}$-bit-quantized phases
with $b_{ps}$ being the number of quantization bits of phase shifters at RF.
In this paper, we apply the additive quantization noise (AQN) model in [26,
28] to approximate the DACs quantization as a linear output for the simplicity
of analysis, which can be formulated as

$\displaystyle Q_{b}(\mathbf{u})\approx(1-\eta)\mathbf{u}+\mathbf{n}_{q}$ (9)

where $\eta$ is defined as the reciprocal of signal-to-quantization-noise
ratio. $\mathbf{n}_{q}$ is the additive quantization noise vector and
uncorrelated with the input $\mathbf{u}$, that is,
$\mathbb{E}[\mathbf{u}\mathbf{n}_{q}^{H}]=\mathbb{E}[\mathbf{n}_{q}\mathbf{u}^{H}]=\mathbf{0}$.
Therefore, the transmitted signal in (1) can be rewritten as

$\displaystyle\mathbf{x}\approx$ $\displaystyle\sqrt{\beta
P}(1-\eta)\mathbf{F}_{RF}\mathbf{f}_{BB}s$
$\displaystyle+\sqrt{(1-\beta)P}(1-\eta)\mathbf{F}_{RF}\mathbf{T}_{BB}\mathbf{z}+\mathbf{F}_{RF}\mathbf{n}_{q}$
(10)

According to [26], the covariance of the quantization noise can be given by

$\displaystyle\mathbf{R}_{\mathbf{n}_{q}\mathbf{n}_{q}}=\eta(1-\eta)\mathrm{diag}\\{\mathbf{R}_{uu}\\}$
(11)

where $\mathbf{R}_{uu}=\beta
P\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}+(1-\beta)P\mathbf{T}_{BB}\mathbf{T}_{BB}^{H}$.
Therefore, the received signal at Bob can be represented as

$\displaystyle y_{b}=$ $\displaystyle\sqrt{\beta
P}(1-\eta)\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{f}_{BB}s$
$\displaystyle+\sqrt{(1-\beta)P}(1-\eta)\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{T}_{BB}\mathbf{z}+\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{n}_{q}+n_{b}$
(12)

Similarly, the receive signal at eavesdropper can be given as

$\displaystyle y_{e}$ $\displaystyle=\sqrt{\beta
P}(1-\eta)\mathbf{h}_{e}\mathbf{F}_{RF}\mathbf{f}_{BB}s$
$\displaystyle+\sqrt{(1-\beta)P}(1-\eta)\mathbf{h}_{e}\mathbf{F}_{RF}\mathbf{T}_{BB}\mathbf{z}+\mathbf{h}_{e}\mathbf{F}_{RF}\mathbf{n}_{q}+n_{e}$
(13)

where $n_{b}\sim\mathcal{C}\mathcal{N}(0,\sigma_{b}^{2})$ and
$n_{e}\sim\mathcal{C}\mathcal{N}(0,\sigma_{e}^{2})$ are additive white
Gaussian noise (AWGN) at Bob and Eve, respectively. For the convenience of
analysis, we set $\sigma_{b}^{2}=\sigma_{e}^{2}$.

Therefore, the achievable rate at Bob and Eve can be formulated in (10) and
(11), respectively. And we can express the optimization problem of maximizing
the SR as

$\displaystyle\max\limits_{\mathbf{F}_{RF},\mathbf{f}_{BB},\mathbf{T}_{BB},\beta}{\kern
1.0pt}~{}~{}~{}~{}~{}~{}R_{s}=R_{b}-R_{e}$
$\displaystyle\text{subject~{}to}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathbf{F}_{RF}\in\mathcal{F}_{RF},\mathbb{E}\\{\mathbf{x}\mathbf{x}^{H}\\}=P_{T},0\\!\leq\beta\leq\\!1$
(12)

Since it is assumed that only the partial knowledge of wiretap channel at Eve
is available, we can only obtain the average achievable rate of Eve.
Meanwhile, because of partially-connected structure, we have
$\mathbf{F}_{RF}^{H}\mathbf{F}_{RF}=\mathbf{I}_{K}$. To obey the power
constraint, we should have
$\|\mathbf{f}_{BB}\|^{2}=\|\mathbf{T}_{BB}\|_{F}^{2}=1$, and
$P=\frac{P_{T}}{(1-\eta)^{2}+\eta(1-\eta)}$. Therefore, the objective function
in optimization problem (II-A) can be lower bounded as:

$\displaystyle R_{s}$
$\displaystyle=R_{b}-\mathbb{E}[\log_{2}\left\\{1+\mathrm{SINR}_{e}\right\\}]$
$\displaystyle\overset{\text{(a)}}{\geq}R_{b}-\log_{2}\left\\{1+\mathbb{E}[\mathrm{SINR}_{e}]\right\\}$
$\displaystyle=\tilde{R}_{s}$ (13)

where $(a)$ holds due to

$\displaystyle\mathbb{E}[\log_{2}(x)]\leq\log_{2}(1+\mathbb{E}[x]).$ (14)

Therefore, replacing the objective function $R_{s}$ in (II-A) by
$\tilde{R}_{s}$ forms the following simplified optimization problem

$\displaystyle\max\limits_{\beta,\mathbf{f}_{BB},\mathbf{T}_{BB},\mathbf{F}_{RF}}{\kern
1.0pt}~{}~{}~{}~{}~{}~{}\tilde{R}_{s}=R_{b}-\log_{2}\left\\{1+\mathbb{E}[\mathrm{SINR}_{e}]\right\\}$
$\displaystyle\text{subject~{}to}~{}~{}~{}\mathbf{F}_{RF}\in\mathcal{F}_{RF},\|\mathbf{f}_{BB}\|^{2}=1,\|\mathbf{T}_{BB}\|_{F}^{2}=1,0\\!\leq\beta\leq\\!1$
(15)

###  II-B Channel model

In what follows, a narrow-band clustered mmWave channel model is used with $L$
propagation paths, which can be described as follows:

$\displaystyle\mathbf{h}=\sqrt{\frac{N_{t}}{L}}\sum_{l=1}^{L}g_{l}\mathbf{a}_{t,l}^{H}(\phi_{l})$
(16)

where $\sqrt{\frac{N_{t}}{L}}$ is the normalized factor, $l$ stands for the
index of paths, and $L$ is the number of channel paths. The path gains
$g_{l}\sim\mathcal{C}\mathcal{N}(0,1)$ depicts the complex gain of the
$l^{th}$ path. $\mathbf{a}_{t}(\phi_{l})$ is the corresponding response vector
of transmit antenna array, with $\phi_{l}$ denotes the azimuth angles,
respectively. For uniform linear array with $N$ elements, the array response
vector can be represented as

$\displaystyle\mathbf{a}(\phi)=\frac{1}{\sqrt{N}}[1,e^{-j\frac{2\pi}{\lambda}dsin(\phi)},\cdots,e^{-j\frac{2\pi}{\lambda}(N-1)dsin(\phi)}]^{T}$
(17)

where $\lambda$ and $d$ are the wavelength of the signal and distance spacing
between the antenna elements.

Since we consider the mmWave channel with partial channel knowledge of Eve,
i.e., the full knowledge of angles of departure (AoD) and the distribution of
the gains of mmWave paths $\alpha_{l}$. As we can see from (16), the channel
can be written more compactly as

$\displaystyle\mathbf{h}=\sqrt{\frac{N_{t}}{L}}\mathbf{g}\mathbf{A}_{t}$ (18)

with $\mathbf{g}=[g_{1},g_{2},\cdots,g_{L}]$ and
$\mathbf{A}_{t}=[\mathbf{a}_{t,1}^{H},\mathbf{a}_{t,2}^{H},\cdots,\mathbf{a}_{t,L}^{H}]^{T}$.
Since the each element in $\mathbf{g}$ follows the complex Gaussian
distribution with zero mean and unit variance, $\mathbf{g}$ is the i.i.d.
complex Gaussian vector with probability density function as

$\displaystyle f(\mathbf{g})=\frac{1}{\pi^{L}}e^{-\mathbf{g}^{H}\mathbf{g}}.$
(19)

Therefore, the expectation $\mathbb{E}\\{\mathrm{SINR}_{e}\\}$ can be
approximated as (20). The approximation in $(b)$ is similarly adopted in [29,
30, 31].

##  III Design and Optimization of DP AP and ANPM

In this section, we propose a TLAIS of maximizing the ASR by taking
quantization errors from both low-resolution DAC and phase shifters into
account as shown in Fig. 2. For the outer loop, given the initial DP vector of
confidential messages and ANPM, a steepest ascent algorithm is used to attain
the updated AP vector by maximizing the approximate SR (ASR). After completing
this, the optimization process turns to the inner loop. Given the AP matrix,
the DP vector of confidential messages and ANPM are alternatively attained
within an interior iterative loop by using GPI method with the aim to maximize
the ASR. The two loops are repeated until their individual terminal conditions
are satisfied.

Figure 2: Schematic diagram of TLAIS
$\displaystyle\mathbb{E}\\{\mathrm{SINR}_{e}\\}=\mathbb{E}\left\\{\frac{\beta
P(1-\eta)^{2}\frac{N_{t}}{L}\mathbf{g}_{e}\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}\mathbf{g}_{e}^{H}}{(1-\beta)P(1-\eta)^{2}\frac{N_{t}}{L}\mathbf{g}_{e}\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{T}_{BB}\mathbf{T}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}\mathbf{g}_{e}^{H}+\frac{N_{t}}{L}\mathbf{g}_{e}\mathbf{A}_{te,m}\mathbf{F}_{RF}\mathbf{R}_{\mathbf{n}_{q}\mathbf{n}_{q}}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}\mathbf{g}_{e}^{H}+\sigma^{2}}\right\\}$
$\displaystyle\overset{\text{(b)}}{\approx}\frac{\mathbb{E}\left\\{\beta
P(1-\eta)^{2}\frac{N_{t}}{L}\mathbf{g}_{e}\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}\mathbf{g}_{e}^{H}\right\\}}{\mathbb{E}\left\\{(1-\beta)P(1-\eta)^{2}\frac{N_{t}}{L}\mathbf{g}_{e}\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{T}_{BB}\mathbf{T}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}\mathbf{g}_{e}^{H}+\frac{N_{t}}{L}\mathbf{g}_{e}\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{R}_{\mathbf{n}_{q}\mathbf{n}_{q}}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}\mathbf{g}_{e}^{H}+\sigma^{2}\right\\}}$
$\displaystyle=\frac{\beta
P(1-\eta)^{2}\frac{N_{t}}{L}\mathbb{E}\left\\{\mathrm{tr}[\mathbf{g}_{e}^{H}\mathbf{g}_{e,m}\mathbf{A}_{te,m}\mathbf{F}_{RF}\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}]\right\\}}{(1-\beta)P(1-\eta)^{2}\frac{N_{t}}{L}\mathbb{E}\left\\{\mathrm{Tr}[\mathbf{g}_{e}^{H}\mathbf{g}_{e}\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{T}_{BB}\mathbf{T}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}]+\frac{N_{t}}{L}\mathrm{Tr}[\mathbf{g}_{e}^{H}\mathbf{g}_{e}\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{R}_{\mathbf{n}_{q}\mathbf{n}_{q}}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}]+\sigma^{2}\right\\}}$
$\displaystyle=\frac{\beta
P(1-\eta)^{2}\frac{N_{t}}{L}\mathrm{tr}[\mathbb{E}\left\\{\mathbf{g}_{e}^{H}\mathbf{g}_{e,m}\right\\}\mathbf{A}_{te,m}\mathbf{F}_{RF}\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}]}{(1-\beta)P(1-\eta)^{2}\frac{N_{t}}{L}\mathrm{tr}[\mathbb{E}\left\\{\mathbf{g}_{e}^{H}\mathbf{g}_{e}\right\\}\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{T}_{BB}\mathbf{T}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}]+\frac{N_{t}}{L}\mathrm{Tr}[\mathbb{E}\left\\{\mathbf{g}_{e}^{H}\mathbf{g}_{e}\right\\}\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{R}_{\mathbf{n}_{q}\mathbf{n}_{q}}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}]+\sigma^{2}}$
$\displaystyle=\frac{\beta
P(1-\eta)^{2}\frac{N_{t}}{L}\mathrm{tr}[\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te,m}^{H}]}{(1-\beta)P(1-\eta)^{2}\frac{N_{t}}{L}\mathrm{Tr}[\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{T}_{BB}\mathbf{T}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}]+\frac{N_{t}}{L}\mathrm{Tr}[\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{R}_{\mathbf{n}_{q}\mathbf{n}_{q}}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}]+\sigma^{2}}$
(20) $\displaystyle\mathrm{SINR}_{b}$ $\displaystyle=\frac{\beta
P(1-\eta)^{2}\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}}{(1-\beta)P(1-\eta)^{2}\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{T}_{BB}\mathbf{T}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}+\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{R}_{\mathbf{n}_{q}\mathbf{n}_{q}}\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}+\sigma^{2}}$
$\displaystyle=\frac{\beta
P(1-\eta)^{2}\mathrm{tr}[\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}]}{(1-\beta)P(1-\eta)^{2}\mathrm{tr}[\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{T}_{BB}\mathbf{T}_{BB}^{H}]+\mathrm{tr}[\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{R}_{\mathbf{n}_{q}\mathbf{n}_{q}}]+\sigma^{2}}$
$\displaystyle=\frac{\beta
P(1-\eta)^{2}\mathrm{vec}(\mathbf{F}_{RF})^{H}[\mathbf{F}_{BB}\otimes\mathbf{H}_{b}]\mathrm{vec}(\mathbf{F}_{RF})}{(1-\beta)P(1-\eta)^{2}\mathrm{vec}(\mathbf{F}_{RF})^{H}[(\mathbf{T}_{BB}\mathbf{T}_{BB}^{H})^{T}\otimes\mathbf{H}_{b}]\mathrm{vec}(\mathbf{F}_{RF})+\mathrm{vec}(\mathbf{F}_{RF})^{H}[\mathbf{R}_{nqnq}^{T}\otimes\mathbf{H}_{b}]\mathrm{vec}(\mathbf{F}_{RF})+\sigma^{2}}$
$\displaystyle=\frac{\mathbf{d}^{H}\mathbf{\Gamma}_{b1}\mathbf{d}}{\mathbf{d}^{H}[\mathbf{\Gamma}_{b2}+\mathbf{\Gamma}_{b3}+\frac{\sigma^{2}}{K}\mathbf{I}_{N}t]\mathbf{d}}$
(30)

###  III-A Design of AP matrix

Observing the structure of the numerator in (20), we can simplify it to the
following expression

$\displaystyle~{}~{}~{}\mathrm{Tr}[\mathbf{A}_{te,m}\mathbf{F}_{RF}\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te,m}^{H}]$
$\displaystyle=\mathrm{Tr}[\mathbf{F}_{RF}^{H}\mathbf{A}_{te,m}^{H}\mathbf{A}_{te,m}\mathbf{F}_{RF}\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}]$
$\displaystyle\overset{\text{(c)}}{=}\mathrm{vec}(\mathbf{F}_{RF})^{H}[(\mathbf{f}_{BB}\mathbf{f}_{BB}^{H})^{T}\otimes(\mathbf{A}_{te,m}^{H}\mathbf{A}_{te,m})]\mathrm{vec}(\mathbf{F}_{RF})$
$\displaystyle=\mathrm{vec}(\mathbf{F}_{RF})^{H}[\mathbf{F}_{BB}\otimes\mathbf{A}]\mathrm{vec}(\mathbf{F}_{RF})$
(21)

where
$\mathbf{F}_{BB}=(\mathbf{f}_{BB}\mathbf{f}_{BB}^{H})^{T}\in\mathbb{C}^{K\times
K}$ and
$\mathbf{A}=\mathbf{A}_{te,m}^{H}\mathbf{A}_{te,m}\in\mathbb{C}^{N_{T}\times
N_{T}}$. In the above equation, $(c)$ holds due to the fact that

$\displaystyle\mathrm{Tr}\left(\mathbf{P}^{H}\mathbf{Z}\mathbf{P}\mathbf{W}\right)=\mathrm{vec}(\mathbf{P})^{H}\left(\mathbf{W}^{T}\otimes\mathbf{Z}\right)\mathrm{vec}(\mathbf{P}).$
(22)

Since the analog beamforming matrix $\mathbf{F}_{RF}$ (III-A) is block
diagonal and most of its elements are zeros, i.e., the dimension of
$\mathbf{F}_{RF}$ is $N_{T}\times K$. In other words, there are $N_{T}$ non-
zero elements and $(K-1)N_{T}$ zero elements for matrix $\mathbf{F}_{RF}$, so
the matrix $\mathbf{F}_{RF}$ can be viewed as a sparse matrix. In order to
reduce the computational complexity and dimension in the following, only non-
zero elements of $\mathbf{F}_{RF}$ are required to be extracted. Because there
are $N_{T}$ non-zero elements in $\mathbf{F}_{RF}$, let us define

$\displaystyle\mathbf{d}=\mathbb{N}[\mathrm{vec}(\mathbf{F}_{RF})]=[\mathbf{f}_{RF1},\mathbf{f}_{RF2},\cdots,\mathbf{f}_{RFK}]^{T}\in\mathbb{C}^{N\times
1}$ (23)

represent the mathematic operation of extracting all non-zero elements from
$\mathbf{F}_{RF}$. To match the above operation, now, let us extract extract
the corresponding $N\times N$ elements from $NK\times NK$ matrix
$\left(\mathbf{F}_{BB}\otimes\mathbf{A}\right)$ in (III-A). The rule of
extracting method is as follows: matrix $\mathbf{F}_{BB}\otimes\mathbf{A}$ can
be partitioned into $K\times K$ or $K^{2}$ block matrices with each being
$N\times N$ submatrix,

each block matrix can be expressed as $\mathbf{F}_{BB}(m,n)\otimes\mathbf{A}$
with $m=1,\cdots,K,n=1,\cdots,K$. Only $M\times M$ elements per block matrix
are extracted with the following rule:

$\displaystyle\mathbf{\Gamma}\left((m-1)M+1:mM,(n-1)M+1:nM\right)$
$\displaystyle=$
$\displaystyle[\mathbf{F}_{BB}(m,n)\otimes\mathbf{A}]\left(\\!(m-1)\\!M+1\\!:\\!mM,\\!(n-1)\\!M+1\\!:\\!nM\right)$
(24)

Via the above manipulation, we have a simple form of (III-A) as follows

$\displaystyle\mathrm{vec}(\mathbf{F}_{RF})^{H}[\mathbf{F}_{BB}\otimes\mathbf{A}]\mathrm{vec}(\mathbf{F}_{RF})=\mathbf{d}^{H}\mathbf{\Gamma}\mathbf{d}$
(25)

In the same fashion, (20) can be rewritten in a more concise way as follows:

$\displaystyle\mathbb{E}\\{\mathrm{SINR}_{e}\\}$
$\displaystyle\approx\frac{\mathbf{d}^{H}\mathbf{\Gamma}_{e1}\mathbf{d}}{\mathbf{d}^{H}\mathbf{\Gamma}_{e2}\mathbf{d}+\mathbf{d}^{H}\mathbf{\Gamma}_{e3}\mathbf{d}+\sigma^{2}}$
$\displaystyle=\frac{\mathbf{d}^{H}\mathbf{\Gamma}_{e1}\mathbf{d}}{\mathbf{d}^{H}[\mathbf{\Gamma}_{e2}+\mathbf{\Gamma}_{e3}+\frac{\sigma^{2}}{K}\mathbf{I}_{Nt}]\mathbf{d}}$
(26)

Similarly, the achievable rate of Eve is also rewritten as

$\displaystyle\log_{2}\left(1+\mathbb{E}[\mathrm{SINR}_{e}]\right)\approx\log_{2}\left(\frac{\mathbf{d}^{H}\mathbf{A}_{e,1}\mathbf{d}}{\mathbf{d}^{H}\mathbf{A}_{e,2}\mathbf{d}}\right)$
(27)

with

$\displaystyle\mathbf{A}_{e,1}=\mathbf{\Gamma}_{e1}+\mathbf{\Gamma}_{e2}+\mathbf{\Gamma}_{e3}+\frac{\sigma^{2}}{K}\mathbf{I}_{Nt},$
(28)

and

$\displaystyle\mathbf{A}_{e,2}=\mathbf{\Gamma}_{e2}+\mathbf{\Gamma}_{e3}+\frac{\sigma^{2}}{K}\mathbf{I}_{Nt}.$
(29)

Similarly, the $\mathrm{SINR}_{b}$ can be also expressed as (III). Therefore,
we can further write the achievable rate of Bob as

$\displaystyle
R_{b}=\log_{2}\left(\frac{\mathbf{d}^{H}\mathbf{A}_{b,1}\mathbf{d}}{\mathbf{d}^{H}\mathbf{A}_{b,2}\mathbf{d}}\right)$
(31)

with

$\displaystyle\mathbf{A}_{b,1}=\mathbf{\Gamma}_{b1}+\mathbf{\Gamma}_{b2}+\mathbf{\Gamma}_{b3}+\frac{\sigma^{2}}{K}\mathbf{I}_{Nt}$

and

$\displaystyle\mathbf{A}_{b,2}=\mathbf{\Gamma}_{b2}+\mathbf{\Gamma}_{b3}+\frac{\sigma^{2}}{K}\mathbf{I}_{Nt}.$
(33)

Finally, the optimization problem of maximizing $\tilde{R}_{s}$ (II-A) can be
further recasted as

$\displaystyle\tilde{R}_{s}$
$\displaystyle=R_{b}-\log_{2}\left(1+\mathbb{E}[\mathrm{SINR}]\right)$
$\displaystyle\approx\log_{2}\left(\frac{\mathbf{d}^{H}\mathbf{A}_{b,1}\mathbf{d}}{\mathbf{d}^{H}\mathbf{A}_{b,2}\mathbf{d}}\cdot\frac{\mathbf{d}^{H}\mathbf{A}_{e,2}\mathbf{d}}{\mathbf{d}^{H}\mathbf{A}_{e,1}\mathbf{d}}\right)$
(34)

Since the $\tilde{R}_{s}$ is the non-convex function of $\mathbf{d}$. In
particular, all elements in $\mathbf{d}$ have the unit modulus, which
satisfies $\mathbf{d}\in\mathcal{F}_{RF}$. Therefore, a gradient ascent (GA)
method is used to compute the AP matrix. Let us define

$\displaystyle
f(\mathbf{d})=\frac{\mathbf{d}^{H}\mathbf{A}_{b,1}\mathbf{d}}{\mathbf{d}^{H}\mathbf{A}_{b,2}\mathbf{d}},g(\mathbf{d})=\frac{\mathbf{d}^{H}\mathbf{A}_{e,2}\mathbf{d}}{\mathbf{d}^{H}\mathbf{A}_{e,1}\mathbf{d}}.$
(35)

The gradient of $\tilde{R}_{s}$ with respect to $\mathbf{d}$ in can be given
by

$\displaystyle\nabla_{\mathbf{d}}=\frac{(f^{{}^{\prime}}(\mathbf{d})g(\mathbf{d})+f(\mathbf{d})g^{{}^{\prime}}(\mathbf{d}))}{f(\mathbf{d})g(\mathbf{d})\mathrm{ln}2}$
(36)

where

$\displaystyle
f^{{}^{\prime}}(\mathbf{d})=\frac{\mathbf{A}_{b,1}^{H}\mathbf{d}(\mathbf{d}^{H}\mathbf{A}_{b,2}\mathbf{d})-(\mathbf{d}^{H}\mathbf{A}_{b,1}\mathbf{d})\mathbf{A}_{b,2}^{H}\mathbf{d}}{(\mathbf{d}^{H}\mathbf{A}_{b,2}\mathbf{d})^{2}},$
(37a) $\displaystyle
g^{{}^{\prime}}(\mathbf{d})=\frac{\mathbf{A}_{e,2}^{H}\mathbf{d}(\mathbf{d}^{H}\mathbf{A}_{e,1}\mathbf{d})-(\mathbf{d}^{H}\mathbf{A}_{e,2}\mathbf{d})\mathbf{A}_{e,1}^{H}\mathbf{d}}{(\mathbf{d}^{H}\mathbf{A}_{e,1}\mathbf{d})^{2}}.$
(37b)

After obtaining the $\nabla_{\mathbf{d}}$, we will renew the value
$\mathbf{d}^{(t)}$ of $\mathbf{d}$ by
$\mathbf{d}^{(t-1)}+\alpha\nabla_{\mathbf{d}}$ with $\alpha$ being the
searching step. The detailed process of GA algorithm proposed by us is listed
in Algorithm 1.

Algorithm 1 Gradient ascent algorithm for analog precoder

Input:
(1) Initialize $\mathbf{F}_{RF}^{(0)}$ according to (79) and extract non-zero
elements in $\mathbf{F}_{RF}^{(0)}$ to initialize $\mathbf{d}^{(0)}$;
(2) Fixed $\mathbf{f}_{BB}$ and $\mathbf{T}_{BB}$, compute $R_{s}^{(0)}$ and
set $t=1,\alpha$, threshold value $\alpha_{min},\epsilon$;
for $\alpha>\alpha_{min}$

1: Compute $\nabla_{\mathbf{d}}^{(t-1)}$ according to (36) and
$\mathbf{d}^{(t)}=\mathbf{d}^{(t-1)}+\alpha\nabla_{\mathbf{d}}^{(t-1)}$,
reform
$\mathbf{F}_{RF}^{(t)}=\frac{1}{\sqrt{M}}\exp\\{j\angle(\mathbf{d}^{(t)})\\}$;

2: Compute $R_{s}^{(t)}$ using $\mathbf{T}_{BB}$, $\mathbf{f}_{BB}$ and
$\mathbf{F}_{RF}^{(t)}$, search the optimal $\beta^{t}$ by 1-D search;

3: If $R_{s}^{(t)}-R_{s}^{(t-1)}>\epsilon$
$\mathbf{d}^{(t)}=\mathbf{d}^{(t-1)}+\alpha\nabla_{\mathbf{d}}^{(t-1)}$;else
$\mathbf{d}^{(t)}=\mathbf{d}^{(t-1)}$; $\alpha=\frac{\alpha}{2}$;

4: $t=t+1$;

end for
Output: $\mathbf{F}_{RF}^{(t)}$.

###  III-B Design of DP vector of confidential message

In this section, we will design the DP vector $\mathbf{f}_{BB}$ assuming the
other two precoders $\mathbf{F}_{RF}$ and $\mathbf{T}_{BB}$ are known in
advance. To optimize $\mathbf{f}_{BB}$, the achievable rate of Bob is
represented as a function of $\mathbf{f}_{BB}$ as follows

$\displaystyle R_{b}$
$\displaystyle=\log_{2}\left(1+\frac{\xi_{1}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{f}_{BB}}{\xi_{2}\mathbf{h}_{b}\mathbf{F}_{RF}\mathrm{diag}[\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}]\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}+\gamma_{b}}\right)$
$\displaystyle\overset{\text{(d)}}{=}\log_{2}\left(1+\frac{\xi_{1}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{f}_{BB}}{\xi_{2}\mathbf{h}_{b}\mathbf{F}_{RF}[(\mathbf{f}_{BB}\mathbf{f}_{BB}^{H})\odot\mathbf{I}_{K}]\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}+\gamma_{b}}\right)$
$\displaystyle\overset{\text{(e)}}{=}\log_{2}\left(1+\frac{\xi_{1}\mathbf{f}_{BB}^{H}\tilde{\mathbf{h}}_{b}^{H}\tilde{\mathbf{h}}_{b}\mathbf{f}_{BB}}{\xi_{2}\mathbf{f}_{BB}^{H}[\mathbf{I}_{K}\odot(\tilde{\mathbf{h}}_{b}^{H}\tilde{\mathbf{h}}_{b})]\mathbf{f}_{BB}+\gamma_{b}}\right)$
$\displaystyle=\log_{2}\left(\frac{\mathbf{f}_{BB}^{H}\mathbf{Q}_{b}\mathbf{f}_{BB}}{\mathbf{f}_{BB}^{H}\mathbf{P}_{b}\mathbf{f}_{BB}}\right)$
(38)

where

$\displaystyle\xi_{1}=\beta P(1-\eta)^{2}$ (39)
$\displaystyle\xi_{2}=\eta(1-\eta)\beta P,$ (40) $\displaystyle\gamma_{b}$
$\displaystyle=(1-\beta)P(1-\eta)^{2}\|\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{T}_{BB}\|^{2}$
$\displaystyle+\eta(1-\eta)(1-\beta)P$ (41)
$\displaystyle\tilde{\mathbf{h}}_{b}=\mathbf{h}_{b}\mathbf{F}_{RF},$ (42)
$\displaystyle\mathbf{Q}_{b}=\xi_{2}[\mathbf{I}_{K}\odot(\tilde{\mathbf{h}}_{b}^{H}\tilde{\mathbf{h}}_{b})]+\gamma_{b}\mathbf{I}_{K}+\xi_{1}\tilde{\mathbf{h}}_{b}^{H}\tilde{\mathbf{h}}_{b},$
(43)

and

$\displaystyle\mathbf{P}_{b}=\xi_{2}[\mathbf{I}_{K}\odot(\tilde{\mathbf{h}}_{b}^{H}\tilde{\mathbf{h}}_{b})]+\gamma_{b}\mathbf{I}_{K}.$
(44)

Equality $(d)$ in (III-B) is achieved because
$\mathrm{diag}[\mathbf{A}]=\mathbf{A}\odot\mathbf{I}$ and $(e)$ holds due to
the fact that
$\mathrm{Tr}[(\mathbf{A}\odot\mathbf{I})\mathbf{B}]=\mathrm{Tr}[\mathbf{A}(\mathbf{I}\odot\mathbf{B})]$.

Now, we can rewrite $\mathbb{E}\\{\mathrm{SINR}_{e}\\}$ as a function of
$\mathbf{f}_{BB}$ from (20) as follows:

$\displaystyle\mathop{\mathbb{E}[\mathrm{SINR}_{e}]}\limits_{\mathbf{f}_{BB}}\approx\frac{\lambda_{1}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{f}_{BB}}{\lambda_{2}\mathrm{tr}[\mathbf{A}_{te}\mathbf{F}_{RF}\mathrm{diag}\\{\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}\\}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}]+\gamma_{e}}$
$\displaystyle=\frac{\lambda_{1}\mathbf{f}_{BB}^{H}\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te}\mathbf{f}_{BB}}{\lambda_{2}\mathrm{tr}[\tilde{\mathbf{A}}_{te}(\mathbf{f}_{BB}\mathbf{f}_{BB}^{H})\odot\mathbf{I}_{K}\tilde{\mathbf{A}}_{te}^{H}]+\gamma_{e}}$
$\displaystyle=\frac{\lambda_{1}\mathbf{f}_{BB}^{H}\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te}\mathbf{f}_{BB}}{\lambda_{2}\mathrm{tr}[(\mathbf{f}_{BB}\mathbf{f}_{BB}^{H})\\{\mathbf{I}_{K}\odot(\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te})\\}]+\gamma_{e}}$
$\displaystyle=\frac{\mathbf{f}_{BB}^{H}(\lambda_{1}\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te})\mathbf{f}_{BB}}{\mathbf{f}_{BB}^{H}(\lambda_{2}[\mathbf{I}_{K}\odot\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te}]+\gamma_{e}\mathbf{I}_{K})\mathbf{f}_{BB}}$
(45)

where

$\displaystyle\lambda_{1}=\beta P(1-\eta)^{2}N_{t}/L,$ (46)
$\displaystyle\lambda_{2}=\eta(1-\eta)\beta PN_{t}/L,$ (47)
$\displaystyle\gamma_{e}=(1-\beta)P(1-\eta)^{2}\frac{N_{t}}{L}\mathrm{tr}[\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{T}_{BB}\mathbf{T}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}]\\!+\\!$
$\displaystyle\\!\frac{N_{t}}{L}\\!\mathrm{tr}[\mathbf{A}_{te}\mathbf{F}_{RF}\eta(1\\!-\\!\eta)\mathrm{diag}\\!\\{\\!(1\\!-\\!\beta)P\mathbf{T}_{BB}\mathbf{T}_{BB}^{H}\\!\\}\\!\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}]\\!+\\!\sigma^{2},$
(48)

and $\tilde{\mathbf{A}}_{te}=\mathbf{A}_{te}\mathbf{F}_{RF}$. Using the above
expression of $\mathbb{E}\\{\mathrm{SINR}_{e}\\}$ , we directly have the
average achievable rate at Eve as follows:

$\displaystyle\log_{2}\left(1+\mathbb{E}[\mathrm{SINR}_{e}]\right)\approx\log_{2}\left(\frac{\mathbf{f}_{BB}^{H}\mathbf{Q}_{e}\mathbf{f}_{BB}}{\mathbf{f}_{BB}^{H}\mathbf{P}_{e}\mathbf{f}_{BB}}\right)$
(49)

where

$\displaystyle\mathbf{Q}_{e}=\lambda_{2}[\mathbf{I}_{K}\odot\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te}]+\gamma_{e}\mathbf{I}_{K}+\lambda_{1}\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te}$
(50)

and

$\displaystyle\mathbf{P}_{e}=\lambda_{2}[\mathbf{I}_{K}\odot\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te}]+\gamma_{e}\mathbf{I}_{K}.$
(51)

Combining (III-B)and (49) yields the expression of $\tilde{R}_{s}$ as

$\displaystyle\tilde{R}_{s}=R_{b}-\log_{2}\left(1+\mathbb{E}[\mathrm{SINR}_{e}]\right)$
$\displaystyle\approx\log_{2}\left(\frac{\mathbf{f}_{BB}^{H}\mathbf{Q}_{b}\mathbf{f}_{BB}}{\mathbf{f}_{BB}^{H}\mathbf{P}_{b}\mathbf{f}_{BB}}\cdot\frac{\mathbf{f}_{BB}^{H}\mathbf{P}_{e}\mathbf{f}_{BB}}{\mathbf{f}_{BB}^{H}\mathbf{Q}_{e}\mathbf{f}_{BB}}\right)$
(52)

Observing the form of (III-B) is the product of fractional quadratic
functions, one efficient way to address this problem is GPI algorithm [32].
Therefore, we can obtain the solution to $\mathbf{f}_{BB}$ by resorting to GPI
algorithm.

###  III-C Design of ANPM

In this subsection, we will optimize the digital ANPM by fixing
$\mathbf{F}_{RF}$ and $\mathbf{f}_{BB}$. Under this condition, we can rewrite
the $R_{b}$ as a function of $\mathbf{T}_{BB}$ as follows:

$\displaystyle R_{b}(\mathbf{T}_{BB})$
$\displaystyle\\!=\\!\mathrm{log}_{2}\left(1\\!+\\!\frac{\kappa_{b}}{\alpha_{1}\|\tilde{\mathbf{h}}_{b}\mathbf{T}_{BB}\|^{2}\\!+\\!\alpha_{2}\tilde{\mathbf{h}}_{b}\mathrm{diag}\\{\mathbf{T}_{BB}\mathbf{T}_{BB}^{H}\\}\tilde{\mathbf{h}}_{b}^{H}\\!+\\!\omega_{b}}\right)$
$\displaystyle\\!=\\!\mathrm{log}_{2}\\!\left(\\!1\\!+\\!\frac{\kappa_{b}}{\alpha_{1}\|\tilde{\mathbf{h}}_{b}\\!\mathbf{T}_{BB}\\!\|^{2}\\!+\\!\alpha_{2}\mathrm{tr}\\!\\{\\!\mathbf{T}_{BB}^{H}[\mathbf{I}\\!\odot\\!(\tilde{\mathbf{h}}_{b}^{H}\tilde{\mathbf{h}}_{b})]\mathbf{T}_{BB}\\!\\}\\!\\!+\\!\omega_{b}}\\!\right)\\!$
$\displaystyle\\!=\\!\mathrm{log}_{2}\left(1+\frac{\kappa_{b}}{\alpha_{1}\|\tilde{\mathbf{h}}_{b}\mathbf{T}_{BB}\|^{2}+\alpha_{2}\mathrm{tr}\\{\mathbf{T}_{BB}^{H}\mathbf{C}_{b}\mathbf{T}_{BB}\\}+\omega_{b}}\right)$
$\displaystyle\\!=\\!\mathrm{log}_{2}\left(\frac{\mathrm{Tr}\\{\mathbf{T}_{BB}^{H}\mathbf{E}_{b}\mathbf{T}_{BB}\\}}{\mathrm{Tr}\\{\mathbf{T}_{BB}^{H}\mathbf{F}_{b}\mathbf{T}_{BB}\\}}\right)$
$\displaystyle\\!=\\!\mathrm{log}_{2}\left(\frac{\mathbf{w}^{H}\left(\mathbf{I}\otimes\mathbf{E}_{b}\right)\mathbf{w}}{\mathbf{w}^{H}\left(\mathbf{I}\otimes\mathbf{F}_{b}\right)\mathbf{w}}\right)$
(53)

where

$\displaystyle\mathbf{w}=\mathrm{vec}\left(\mathbf{T}_{BB}\right),$ (54)
$\displaystyle\alpha_{1}=(1-\beta)P(1-\eta)^{2},$ (55)
$\displaystyle\alpha_{2}=\eta(1-\eta)(1-\beta)P,$ (56)

$\displaystyle\kappa_{b}=\beta
P(1-\eta)^{2}\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H},$
(57) $\displaystyle\omega_{b}=\eta(1-\eta)\beta
P\tilde{\mathbf{h}}_{b}\mathrm{diag}[\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}]\tilde{\mathbf{h}}_{b}^{H}+\sigma^{2},$
(58)

$\displaystyle\mathbf{C}_{b}=\mathbf{I}\odot(\tilde{\mathbf{h}}_{b}^{H}\tilde{\mathbf{h}}_{b}),$
(59)
$\displaystyle\mathbf{E}_{b}=\alpha_{1}\tilde{\mathbf{h}}_{b}^{H}\tilde{\mathbf{h}}_{b}+\alpha_{2}\mathbf{C}_{b}+\omega_{b}\mathbf{I}_{K}+\kappa_{b}\mathbf{I}_{K},$
(60)

and

$\displaystyle\mathbf{F}_{b}=\alpha_{1}\tilde{\mathbf{h}}_{b}^{H}\tilde{\mathbf{h}}_{b}+\alpha_{2}\mathbf{C}_{b}+\omega_{b}\mathbf{I}_{K}.$
(61)

To obtain the $\tilde{R}_{s}$, $\mathbb{E}\\{\mathrm{SINR}_{e}\\}$ is
rewritten as a function of $\mathbf{T}_{BB}$ as follows:

$\displaystyle\mathop{\mathbb{E}[\mathrm{SINR}_{e}]}\limits_{\mathbf{T}_{BB}}$
$\displaystyle\\!\approx\\!\frac{\kappa_{e}}{\alpha_{3}\mathrm{tr}[\\!\mathbf{T}_{BB}^{H}\\!\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te}\\!\mathbf{T}_{BB}\\!]\\!+\\!\\!\alpha_{4}\\!\mathrm{tr}[\mathbf{T}_{BB}^{H}[\mathbf{I}_{K}\\!\odot\\!(\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te})]\mathbf{T}_{BB}]\\!+\\!\omega_{e}}$
$\displaystyle=\frac{\kappa_{e}}{\mathrm{tr}[\mathbf{T}_{BB}^{H}\\{\alpha_{3}\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te}+\alpha_{4}\mathbf{C}_{e}+\omega_{e}\mathbf{I}_{K}\\}\mathbf{T}_{BB}]}$
$\displaystyle=\frac{\kappa_{e}}{\mathrm{tr}[\mathbf{T}_{BB}^{H}\mathbf{F}_{e}\mathbf{T}_{BB}]}$
(62)

where

$\displaystyle\alpha_{3}=(1-\beta)P(1-\eta)^{2}\frac{N_{t}}{L},\alpha_{4}=\eta(1-\eta)(1-\beta)P\frac{N_{t}}{L},$
(63) $\displaystyle\kappa_{e}=\beta
P(1-\eta)^{2}\frac{N_{t}}{L}\mathrm{tr}[\tilde{\mathbf{A}}_{te}\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}\tilde{\mathbf{A}}_{te}],$
(64)

and

$\displaystyle\omega_{e}=\eta(1-\eta)\beta
P\frac{N_{t}}{L}\mathrm{tr}[\tilde{\mathbf{A}}_{te}\mathrm{diag}(\mathbf{f}_{BB}\mathbf{f}_{BB}^{H})\tilde{\mathbf{A}}_{te}^{H}]+\sigma^{2}.$
(65)

Therefore, we can obtain

$\displaystyle\log_{2}\left(1+\mathbb{E}[\mathrm{SINR}]\right)\approx\log_{2}\left(\frac{\mathrm{tr}[\mathbf{T}_{BB}^{H}(\mathbf{F}_{e}+\kappa_{e}\mathbf{I}_{K})\mathbf{T}_{BB}]}{\mathrm{tr}[\mathbf{T}_{BB}^{H}\mathbf{F}_{e}\mathbf{T}_{BB}]}\right)$
$\displaystyle=\log_{2}\left(\frac{\mathrm{tr}[\mathbf{T}_{BB}^{H}\mathbf{E}_{e}\mathbf{T}_{BB}]}{\mathrm{tr}[\mathbf{T}_{BB}^{H}\mathbf{F}_{e}\mathbf{T}_{BB}]}\right)$
$\displaystyle=\log_{2}\left(\frac{\mathbf{w}^{H}\left(\mathbf{I}\otimes\mathbf{E}_{e}\right)\mathbf{w}}{\mathbf{w}^{H}\left(\mathbf{I}\otimes\mathbf{F}_{e}\right)\mathbf{w}}\right)$
(66)

with

$\displaystyle\mathbf{C}_{e}={\mathbf{I}_{K}\odot(\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te})},$
(67)
$\displaystyle\mathbf{E}_{e}=\alpha_{3}\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te}+\alpha_{4}{\mathbf{I}_{K}\odot(\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te})}+\omega_{e}\mathbf{I}_{K}+\kappa_{e}\mathbf{I}_{K},$
(68)

and

$\displaystyle\mathbf{F}_{e}=\alpha_{3}\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te}+\alpha_{4}{\mathbf{I}_{K}\odot(\tilde{\mathbf{A}}_{te}^{H}\tilde{\mathbf{A}}_{te})}+\omega_{e}\mathbf{I}_{K}.$
(69)

Making use of (III-C) and (III-C), the $\tilde{R}_{s}$ can be reformulated as
a function of $\mathbf{w}$ with $\mathbf{F}_{RF}$ and $\mathbf{f}_{BB}$ fixed
as follows:

$\displaystyle\tilde{R}_{s}=R_{b}-\log_{2}\left(1+\mathbb{E}[\mathrm{SINR}]\right)$
$\displaystyle\approx\log_{2}\left(\frac{\mathbf{w}^{H}\left(\mathbf{I}\otimes\mathbf{E}_{b}\right)\mathbf{w}}{\mathbf{w}^{H}\left(\mathbf{I}\otimes\mathbf{F}_{b}\right)\mathbf{w}}\cdot\frac{\mathbf{w}^{H}\left(\mathbf{I}\otimes\mathbf{F}_{e}\right)\mathbf{w}}{\mathbf{w}^{H}\left(\mathbf{I}\otimes\mathbf{E}_{e}\right)\mathbf{w}}\right)$
(70)

where $\mathbf{w}=\mathrm{vec}\left(\mathbf{T}_{BB}\right)$.

Algorithm 2 Iterative method for digital precoders

Input:
(1) Fixed $\mathbf{F}_{RF}$, initialize $\mathbf{f}_{BB}^{(0)}$ and
$\mathbf{T}_{BB}^{(0)}$ and set $k=0$, threshold value $\epsilon$;
(2) Compute $R_{s}^{(0)}$ using $\mathbf{f}_{BB}^{(0)}$,
$\mathbf{T}_{BB}^{(0)}$ and $\mathbf{F}_{RF}$, search the optimal $\beta^{0}$
by 1-D search;
Repeat

1: Solve $\mathbf{T}_{BB}^{(k+1)}$ using $\mathbf{f}_{BB}^{(k)}$ and
$\mathbf{F}_{RF}$ through (III-C);

2: Solve $\mathbf{f}_{BB}^{(k+1)}$ using $\mathbf{T}_{BB}^{(k)}$ and
$\mathbf{F}_{RF}$ through (III-B);

3: Compute $R_{s}^{(k+1)}$ using $\mathbf{T}_{BB}^{(k+1)}$,
$\mathbf{f}_{BB}^{(k+1)}$ and $\mathbf{F}_{RF}$, search the optimal
$\beta^{k+1}$ by 1-D search;

4: $k=k+1$;

Until $R_{s}^{(k+1)}-R_{s}^{(k)}\leq\epsilon$;
Output: $\mathbf{f}_{BB}^{(k)}$ and $\mathbf{T}_{BB}^{(k)}$;

Algorithm 3 Proposed TLAIS algorithm

Input:
(1) Initialize $\mathbf{F}_{RF}^{(0)}$, $\mathbf{f}_{BB}^{(0)}$ and
$\mathbf{T}_{BB}^{(0)}$ according to (79), (III-D) and (87), respectively and
set $p=0$, threshold value $\epsilon$;
(2) For $\mathbf{F}_{RF}^{(0)}$, update $\mathbf{f}_{BB}^{(0)}$ and
$\mathbf{T}_{BB}^{(0)}$ with Algorithm 2;
Repeat

1: For $\mathbf{f}_{BB}^{(p)}$ and $\mathbf{T}_{BB}^{(p)}$, using Algorithm 1
to obtain $\mathbf{F}_{RF}^{(p+1)}$;

2: For $\mathbf{F}_{RF}^{(p+1)}$, using Algorithm 2 to obtain updated
$\mathbf{f}_{BB}^{(p+1)}$ and $\mathbf{T}_{BB}^{(p+1)}$;

3: Compute $R_{s}^{(p+1)}$ using $\mathbf{T}_{BB}^{(p+1)}$,
$\mathbf{f}_{BB}^{(p+1)}$ and $\mathbf{F}_{RF}^{(p+1)}$;

4: $p=p+1$;

Until $R_{s}^{(p+1)}-R_{s}^{(p)}\leq\epsilon$;
Output: $R_{s}^{(p)}$.

Similarly, $\mathbf{w}$ can be solved with GPI, and the corresponding ANPM can
be obtained by reverse operation of vector-to-matrix operator.

Until now, we have completed the design of secure hybrid precoders. Our
iterative idea can be described as follows: for given AP matrix
$\mathbf{F}_{RF}$, the near-optimal DP vector $\mathbf{f}_{BB}$ and ANPM
$\mathbf{T}_{BB}$ are iteratively computed; Then given the DP vector
$\mathbf{f}_{BB}$ and ANPM $\mathbf{T}_{BB}$, we renew the AP matrix
$\mathbf{F}_{RF}$ by GA method. The alternative iterations among
$\mathbf{F}_{RF}$, $\mathbf{f}_{BB}$, and $\mathbf{T}_{BB}$ is repeated until
a stop criterion satisfies. Here, stop criterion is chosen as follows:
$R_{s}^{(p+1)}-R_{s}^{(p)}\leq\epsilon$ with $p$ being the iteration index.
The proposed method is summarized in Algorithm 3.

###  III-D Initialization of TLAIS

In the following, we will show how to set the initial values of the proposed
TLAIS. In other words, we need to make an initialization for three matrices
AP, DP, and ANPM, respectively. Since the optimization problem in (II-A) is
non-convex and intractable to be directly handled. Thus, to simplify it, we
first consider the scenario without AN, i.e. $\beta=1$, as in [24]. In this
case, the achievable rate of Bob can be expressed as

$\displaystyle R_{b}$
$\displaystyle\\!=\\!\log_{2}\\!\left(\\!1\\!+\\!\frac{P(1-\eta)^{2}\|\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{f}_{BB}\|^{2}}{\eta(1-\eta)P\mathbf{h}_{b}\mathbf{F}_{RF}\mathrm{diag}\\{\mathbf{f}_{BB}\mathbf{f}_{BB}^{H}\\}\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}+\sigma_{b}^{2}}\\!\right)\\!$
$\displaystyle\\!=\\!\log_{2}\\!\left(\\!1+\frac{\mu_{1}|\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{f}_{BB}\|^{2}}{\mu_{2}\mathbf{h}_{b}\mathbf{F}_{RF}[(\mathbf{f}_{BB}\mathbf{f}_{BB}^{H})\odot\mathbf{I}_{K}]\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}+\sigma_{b}^{2}}\right)$
$\displaystyle\\!=\\!\log_{2}\\!\left(\\!1+\frac{\mu_{1}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}\mathbf{h}_{b}\mathbf{F}_{RF}\mathbf{f}_{BB}}{\mu_{2}\mathbf{f}_{BB}^{H}[\mathbf{I}_{K}\odot\left(\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}\mathbf{h}_{b}\mathbf{F}_{RF}\right)]\mathbf{f}_{BB}+\sigma_{b}^{2}}\right)$
(71)

where $\mu_{1}=P(1-\eta)^{2}$ and $\mu_{2}=\eta(1-\eta)P$.

If a fully digital architecture is adopted, i.e.,
$\mathbf{F}_{RF}=\mathbf{I}_{Nt}$, then (III-D) can be simplified as

$\displaystyle R_{b}$
$\displaystyle=\log_{2}\\!\left(\\!1+\frac{\mu_{1}\mathbf{f}_{FD}^{H}\mathbf{h}_{b}^{H}\mathbf{h}_{b}\mathbf{f}_{FD}}{\mu_{2}\mathbf{f}_{FD}^{H}[\mathbf{I}_{N_{t}}\odot\left(\mathbf{h}_{b}^{H}\mathbf{h}_{b}\right)]\mathbf{f}_{FD}+\sigma_{b}^{2}}\right)$
$\displaystyle=\log_{2}\left(\frac{\mathbf{f}_{FD}^{H}\mathbf{X}_{b}\mathbf{f}_{FD}}{\mathbf{f}_{FD}^{H}\mathbf{Y}_{b}\mathbf{f}_{FD}}\right)$
(72)

where

$\displaystyle\mathbf{X}_{b}=\mu_{2}[\mathbf{I}_{Nt}\odot(\mathbf{h}_{b}^{H}\mathbf{h}_{b})]+\sigma^{2}\mathbf{I}_{N_{t}}+\mu_{1}\mathbf{h}_{b}^{H}\mathbf{h}_{b},$
(73)

and

$\displaystyle\mathbf{Y}_{b}=\mu_{2}[\mathbf{I}_{Nt}\odot(\mathbf{h}_{b}^{H}\mathbf{h}_{b})]+\sigma^{2}\mathbf{I}_{N_{t}}.$
(74)

Similarly, the $\log_{2}\left(1+\mathbb{E}[\mathrm{SINR_{e}}]\right)$ at Eve
can be rewritten as

$\displaystyle\log_{2}\left(1+\mathbb{E}[\mathrm{SINR}_{e}]\right)$
$\displaystyle\overset{(f)}{\approx}\log_{2}\left(1+\frac{\mu_{3}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{f}_{BB}}{\mu_{4}\mathbf{f}_{BB}^{H}[\mathbf{I}_{K}\odot(\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}\mathbf{A}_{te}\mathbf{F}_{RF})]\mathbf{f}_{BB}+\sigma^{2}}\right)$
$\displaystyle\overset{(g)}{=}\log_{2}\left(1+\frac{\mu_{3}\mathbf{f}_{FD}^{H}\mathbf{A}_{te}^{H}\mathbf{A}_{te}\mathbf{f}_{FD}}{\mu_{4}\mathbf{F}_{FD}^{H}[\mathbf{I}_{K}\odot(\mathbf{A}_{te}^{H}\mathbf{A}_{te})]\mathbf{f}_{FD}+\sigma^{2}}\right)$
$\displaystyle=\log_{2}\left(\frac{\mathbf{f}_{FD}\mathbf{X}_{e}\mathbf{f}_{FD}}{\mathbf{f}_{FD}^{H}\mathbf{Y}_{e}\mathbf{f}_{FD}}\right)$
(75)

where $\mu_{3}=P(1-\eta)^{2}\frac{N_{t}}{L}$,
$\mu_{4}=\eta(1-\eta)P\frac{N_{t}}{L}$. The approximation $(f)$ in (III-D)
holds because $\beta=1$ and $(g)$ is true due to
$\mathbf{F}_{RF}=\mathbf{I}_{N_{t}}$. What’s more,

$\displaystyle\mathbf{X}_{e}=\mu_{4}[\mathbf{I}_{N_{t}}\odot(\mathbf{A}_{te}^{H}\mathbf{A}_{te})]+\sigma^{2}\mathbf{I}_{N_{t}}+\mu_{3}\mathbf{A}_{te}^{H}\mathbf{A}_{te},$
(76)

and

$\displaystyle\mathbf{Y}_{e}=\mu_{4}[\mathbf{I}_{N_{t}}\odot(\mathbf{A}_{te}^{H}\mathbf{A}_{te})]+\sigma^{2}\mathbf{I}_{N_{t}}$
(77)

Therefore, the approximate expression $\tilde{R}_{s}$ of SR can be expressed
as

$\displaystyle\tilde{R}_{s}=R_{b}-\log_{2}\left(1+\mathbb{E}[\mathrm{SINR}]\right)$
$\displaystyle\approx\log_{2}\left(\frac{\mathbf{f}_{FD}^{H}\mathbf{X}_{b}\mathbf{f}_{FD}}{\mathbf{f}_{FD}^{H}\mathbf{Y}_{b}\mathbf{f}_{FD}}\cdot\frac{\mathbf{f}_{FD}^{H}\mathbf{Y}_{b}\mathbf{f}_{FD}}{\mathbf{f}_{FD}^{H}\mathbf{X}_{b}\mathbf{f}_{FD}}\right).$
(78)

Therefore, $\mathbf{f}_{FD}$ can be solved with GPI algorithm. After obtaining
$\mathbf{f}_{FD}$, we can obtain analog beamforming $\mathbf{F}_{RF}$ as

$\displaystyle\mathbf{f}_{RF,l}=\frac{1}{\sqrt{T}}\mathrm{exp}\left\\{jQ_{bps}[\mathbf{f}_{FD,(l-1)M:lM}]\right\\},l=1,2,\cdots,K.$
(79)

where $Q_{bps}$ denotes the operation of quantizing each phase of the
$\mathbf{f}_{FD}$ to its nearest value in discrete phase set
$\mathcal{F}_{RF}$.

Observing (III-D), we can assign the initial value of $\mathbf{f}_{BB}$ by
rewriting (III-D) as

$\displaystyle R_{b}$
$\displaystyle=\log_{2}\left(1+\frac{\mathbf{f}_{BB}^{H}[\mu_{1}\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}\mathbf{h}_{b}\mathbf{F}_{RF}]\mathbf{f}_{BB}}{\mathbf{f}_{BB}^{H}\\{\mu_{2}[\mathbf{I}_{K}\odot(\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}\mathbf{h}_{b}\mathbf{F}_{RF})]+\sigma^{2}\\}\mathbf{f}_{BB}}\right)$
$\displaystyle=\log_{2}\left(\frac{\mathbf{f}_{BB}^{H}\tilde{\mathbf{X}}_{b}\mathbf{f}_{BB}}{\mathbf{f}_{BB}^{H}\tilde{\mathbf{Y}}_{b}\mathbf{f}_{BB}}\right).$
(80)

Similarly, we have the approximate rate at Eve as

$\displaystyle\log_{2}\left(1+\mathbb{E}[\mathrm{SINR}_{e}]\right)$
$\displaystyle\approx\log_{2}\left(1+\frac{\mu_{3}\mathbf{f}_{BB}^{H}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}\mathbf{A}_{te}\mathbf{F}_{RF}\mathbf{f}_{BB}}{\mu_{4}\mathbf{f}_{BB}^{H}[\mathbf{I}_{K}\odot(\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}\mathbf{A}_{te}\mathbf{F}_{RF})]\mathbf{f}_{BB}+\sigma^{2}}\right)$
$\displaystyle=\log_{2}\left(\frac{\mathbf{f}_{BB}^{H}\tilde{\mathbf{X}}_{e}\mathbf{f}_{BB}^{H}}{\mathbf{f}_{BB}^{H}\tilde{\mathbf{Y}}_{e}\mathbf{f}_{BB}}\right)$
(81)

with

$\displaystyle\tilde{\mathbf{X}}_{b}\\!=\\!\mu_{2}[\mathbf{I}_{K}\\!\odot\\!(\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}\mathbf{h}_{b}\mathbf{F}_{RF})]\\!+\\!\sigma^{2}\mathbf{I}_{K}\\!+\\!\mu_{1}\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}\mathbf{h}_{b}\mathbf{F}_{RF},$
(82)
$\displaystyle\tilde{\mathbf{Y}}_{b}=\mu_{2}[\mathbf{I}_{K}\odot(\mathbf{F}_{RF}^{H}\mathbf{h}_{b}^{H}\mathbf{h}_{b}\mathbf{F}_{RF})]+\sigma^{2}\mathbf{I}_{K},$
(83)
$\displaystyle\tilde{\\!\mathbf{X}}_{e}\\!\\!=\\!\mu_{4}[\mathbf{I}_{K}\\!\odot\\!(\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}\mathbf{A}_{te}\\!\mathbf{F}_{RF}\\!)]\\!+\\!\sigma^{2}\mathbf{I}_{K}\\!+\\!\mu_{3}\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}\mathbf{A}_{te}\\!\mathbf{F}_{RF}\\!,$
(84)

and

$\displaystyle\tilde{\mathbf{X}}_{e}=\mu_{4}[\mathbf{I}_{K}\odot(\mathbf{F}_{RF}^{H}\mathbf{A}_{te}^{H}\mathbf{A}_{te}\mathbf{F}_{RF})]+\sigma^{2}\mathbf{I}_{K}.$
(85)

Then the $\tilde{R}_{s}$ can be rewritten as

$\displaystyle\tilde{R}_{s}=R_{b}-\log_{2}\left(1+\mathbb{E}[\mathrm{SINR}]\right)$
$\displaystyle\approx\log_{2}\left(\frac{\mathbf{f}_{BB}^{H}\tilde{\mathbf{X}}_{b}\mathbf{f}_{BB}}{\mathbf{f}_{BB}^{H}\tilde{\mathbf{Y}}_{b}\mathbf{f}_{BB}}\cdot\frac{\mathbf{f}_{BB}^{H}\tilde{\mathbf{Y}}_{e}\mathbf{f}_{BB}}{\mathbf{f}_{BB}^{H}\tilde{\mathbf{X}}_{e}\mathbf{f}_{BB}}\right)$
(86)

Therefore, the initial value of $\mathbf{f}_{BB}$ can be obtained by
maximizing (III-D) via GPI scheme.

To set a good initial value of ANPM, it’s reasonable to assign the initial
value of $\mathbf{T}_{BB}$ by the concept of null-space projection, which
forces the $\mathbf{T}_{BB}$ in the null space of channel of desired user,
thus eliminating the interference of AN to the intended user. The expression
of the initial value of $\mathbf{T}_{BB}$ can be given as follows:

$\displaystyle\mathbf{T}_{BB}=\frac{\mathbf{I}_{K}-\tilde{\mathbf{h}}_{b}^{H}\left(\tilde{\mathbf{h}}_{b}\tilde{\mathbf{h}}_{b}^{H}\right)^{-1}\tilde{\mathbf{h}}_{b}}{\|\mathbf{I}_{K}-\tilde{\mathbf{h}}_{b}^{H}\left(\tilde{\mathbf{h}}_{b}\tilde{\mathbf{h}}_{b}^{H}\right)^{-1}\tilde{\mathbf{h}}_{b}\|_{F}}$
(87)

where $\tilde{\mathbf{h}}_{b}=\mathbf{h}_{b}\mathbf{F}_{RF}$ stands for the
equivalent channel of intended user.

Since the beamforming matrices $\mathbf{f}_{BB}$ and $\mathbf{T}_{BB}$ are
closely related to the power allocation factor $\beta$ in accordance with
(III-B) and (III-C), the reasonable way to find the optimal PA is achieved by
the conventional 1-D linear search.

##  IV Numerical Results

In this section, we will present the numerical simulations to evaluate the SR
performance of our proposed method and compare it with existing methods like
AN-aided method in [24]. Here it is particularly pointed out that since it’s
infeasible to directly apply the algorithm without taking low-resolution DACs
into account in [24] to our paper. Then, the idea in [24] is modified to suit
our model and at the same time choose it as our initial value of our proposed
precoders. The second algorithm is the maximum ratio transmission (MRT)
algorithm. To guarantee the fair of performance comparison, the same AN power
is introduced to MRT. What’s more, to reveal the importance of AN, we
simultaneously make a comparison of our proposed method without AN-aided
together with MRT algorithm. The parameters are chosen as follows. In our
simulation experiment, the transmitter at Alice is equipped with 32 transmit
antennas with 4 RF chains, and mmWave channel model is also adopted in our
paper with $L=12$ and azimuth angles being a uniform distribution over
$[0,2\pi]$.

Figure 3: SR versus SNR with $b_{DAC}=8$ bits.

Fig. 3 demonstrates the SR performance versus SNR with number $b_{DAC}$ of
quantization bits of ADC being 8. As can be seen from this figure, the
proposed TLAIS method performs better than AN-aided method in [24] and MRT
plus AN method in terms of SR performance. From this figure, we find an
important fact that AN has an important impact on SR performance. Taking the
proposed TLAIS as an example, the proposed TLAIS performs much better than the
corresponding one without AN. More important, with the help of AN, the SR
performance of all three methods including our proposed method gradually grows
with increase in SNR. Conversely, without the aid of AN, all methods exist the
effect of SR performance ceil as SNR increases. The main reason that our
proposed method is better than MRT can be explained as follows: unlike our
method, the major goal of MRT algorithm is to maximize the receive SNR at Bob
and don’t concern the receive SNR at Eve. However, our method is to maximize
SR. Thus, our proposed method can achieve a better SR performance.

Figure 4: Optimal PA factor versus SNR.

Fig. 4 shows the optimal PA factor $\beta$ versus SNR with $b_{DAC}=8$, where
$\beta$ is to maximize SR by adjusting PA between confidential message and AN.
From Fig. 4, it is seen that, in the low SNR region, the optimal PA factor
always remains 1, which implies that in this scenario all power should be
allocated to transmit confidential message to guarantee the communication
quality, and AN has a trivial impact on SR performance. Furthermore, as SNR
increases, the optimal PA factor declines gradually. This reveals a fact that
more power should be allocated to AN to interfere with eavesdropper, and
improves the security of communication. Fig. 4 verifies the effectiveness of
AN-aided method especially in the medium and high SNR regions.

Figure 5: SR versus number of quantization bits of DAC with SNR=15dB.

Fig. 5 illustrates the SR versus number $b_{DAC}$ of quantization bits of DAC
with $\mathrm{SNR=15dB}$. For the small value of $b_{DAC}$, the SR is small
and its performance loss is significant because the distortion of low-
resolution DACs is extremely severe. As $b_{DAC}$ increases, the SR
performance improves dramatically, especially when $b_{DAC}$ ranges from $1$
to $6$ bits. It’s noted that the SR performance of five methods all tend to
stable when $b_{DAC}$ reach up to 6. In such a situation, the performance loss
is negligible. What’s more, given a fixed value of $b_{DAC}$, our proposed
method achieves a higher SR performance than other algorithms.

Figure 6: SR versus number of quantization bits of phase shift with SNR=15dB.

Fig. 6 plots the SR versus number $b_{PS}$ of quantization bits of phase
shifters with $\mathrm{SNR=15dB}$ and $b_{DAC}=8$. The proposed TLAIS performs
much better than MRT and MRT with AN in terms of SR for all cases.
Additionally, we also find the fact that their SR tend to a flat rate ceil
with increase in number of quantization bits of phase shifts. From Fig. 6, it
follows that the SR loss due to the effect of quantization error from phase
shifters becomes trivial when $b_{PS}\geq 4$ .

Figure 7: SR versus number of quantization bits of phase shifter with SNR=15dB
and $b_{DAC}=4$.

Fig. 7 plots the SR versus number $b_{PS}$ of quantization bits of phase
shifters with $\mathrm{SNR=15dB}$ and $b_{DAC}=4$. It is noted that Fig. 7
shows the similar performance trend with Fig. 6. In other words, it follows
that our proposed no-AN aided method is still better than MRT plus AN method
in medium and high values of $b_{PS}$. This result is consistent with that
Fig. 5 at $b_{DAC}=4$.

##  V Conclusion

In this paper, a TLAIS was proposed for AN-aided hybrid precoding by taking
the low-resolution DACs and finite-quantized phase shifters into account.
First, we developed a quantized DAC model for AN aided hybrid precoding based
on AQN model including the effect of quantization noise. Then, an approximate
expression of SR with partial channel knowledge of eavesdropper was also
derived. Based on the derived approximate expression,a TLAIS was presented to
optimize the design of DP vector, ANPM, and AP matrix by resorting to the
principle of GPI. Considering the unit modulus constraints of AP, we adopted a
GA algorithm to compute the AP matrix. Furthermore, to reduce the computation
complexity, only the non-zero elements in AP matrix were abstracted by taking
advantage of the sparsity features of partially-connected hybrid architecture
to form an optimization vector. Simulation results indicate that our proposed
method can achieve a better SR performance than existing methods such as MRT
and Max-SR+NSP.

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