# Joint Precoding for Active Intelligent Transmitting Surface Empowered
Outdoor-to-Indoor Communication in mmWave Cellular Networks

Xie Xie, Chen He, Feifei Gao, Zhu Han,
and Z. Jane Wang Xie Xie and Chen He are with the School of Information
Science and Technology, Northwest University, Xi’an, 710069, China.
Corresponding author: Chen He (email: chenhe@nwu.edu.cn). Feifei Gao is with
the Department of Automation, Tsinghua University, Beijing, 100084, China. Zhu
Han is with the Department of Electrical and Computer Engineering, University
of Houston, TX, USA. Z. Jane Wang is with the Department of Electrical and
Computer Engineering, The University of British Columbia, Vancouver, BC
V6T1Z4, Canada.

###### Abstract

Outdoor-to-indoor communications in millimeter-wave (mmWave) cellular networks
have been one challenging research problem due to the severe attenuation and
the high penetration loss caused by the propagation characteristics of mmWave
signals. We propose a viable solution to implement the outdoor-to-indoor
mmWave communication system with the aid of an active intelligent transmitting
surface (active-ITS), where the active-ITS allows the incoming signal from an
outdoor base station (BS) to pass through the surface and be received by the
indoor user-equipments (UEs) after shifting its phase and magnifying its
amplitude. Then, the problem of joint precoding of the BS and active-ITS is
investigated to maximize the weighted sum-rate (WSR) of the communication
system. An efficient block coordinate descent (BCD) based algorithm is
developed to solve it with the suboptimal solutions in nearly closed-forms. In
addition, to reduce the size and hardware cost of an active-ITS, we provide a
block-amplifying architecture to partially remove the circuit components for
power-amplifying, where multiple transmissive-type elements (TEs) in each
block share a same power amplifier. Simulations indicate that active-ITS has
the potential of achieving a given performance with much fewer TEs compared to
the passive-ITS under the same total system power consumption, which makes it
suitable for application to the size-limited and aesthetic-needed scenario,
and the inevitable performance degradation caused by the block-amplifying
architecture is acceptable.

###### Index Terms:

Active intelligent transmitting surfaces, millimeter-wave, power
amplification.

##  I Introduction

Millimeter-wave (mmWave) cellular networks are capable of supplying the ever-
increasing demand of rates for advanced fifth-generation communications thanks
to their abundant available bandwidth. However, a fundamental challenge for
mmWave communications is that the mmWave signal experience a severe
attenuation and a high penetration loss compared with the lower frequency
bands, which makes mmWave signals highly vulnerable to obstacles [1, 2, 3].
The emerging technique of reconfigurable intelligent surfaces (RISs) have been
proposed as a promising candidate for alleviating the unfavorable properties
of mmWave signals. RIS is an ultra-thin metasurface comprising multiple
programmable elements, which enables to achieve a high beamforming gain by
smartly manipulating the incident signal for proactively customizing the radio
propagation environment [4, 5, 6]. More importantly, RIS can significantly
reduce the outage caused by the presence of random blockages through
establishing virtual line-of-sight (LoS) links between base stations (BSs) and
user equipments (UEs), which can considerably enhance the reliability of
mmWave communications, especially in harsh urban propagation environments [7,
8, 9]. These benefits have inspired a lot of work to investigate RIS-assisted
mmWave communication networks and verify that RIS in favor of enhancing the
signal strength, extending the service range, and improving the spectral- and
energy-efficiency [10, 11, 12, 13].

On the other hand, outdoor-to-indoor communication in mmWave cellular networks
is a common communication scenario as the most mobile data traffic is consumed
indoors, however, which is challenging since BSs and UEs are located on the
opposite side of building structures [14], e.g., walls and windows, while
penetrating them leads a severe attenuation (measurements have shown that
around 40 and 28 dB for tinted-glass and brick [15], respectively), which
limits the feasibility for an outdoor BS communicates with indoor UEs inside
the buildings. Relay enabled system is a potential solution for outdoor-to-
indoor mmWave communications but has some drawbacks, e.g., expensive hardware
components and high signal processing complexity [16], while these
shortcomings of relays can be overcome by utilizing the superiority of the
RISs [17, 18]. However, the widely studied RISs-assisted systems focused on
traditional reflective-type RISs, also referred to intelligent reflecting
surfaces (IRSs) [19], as objects hanging on walls or facades of buildings,
which face a placement restriction: both BSs and UEs have to be located on the
front side of an IRS, and hence IRSs can only achieve a half-space coverage
and are not up to the outdoor-to-indoor communication. As a remedy, some works
[20, 21] innovative deployed multiple IRSs in a cooperative multi-hop manner
to bypass the obstacle. Nevertheless, this does not apply to outdoor-to-indoor
communications due to no matter where IRSs are placed, the signal after
reflecting cannot bypass the wall to UEs behind the IRSs [17, 19].

To challenge this restriction and facilitate more flexible deployment of RIS,
the novel concept of transmissive(refractive)-type RIS, also named intelligent
transmitting surfaces (ITS) has been proposed [22, 23]. ITS is a meta-material
reflection-less surface structure [24], which is equipped with a large number
of transmissive-type elements (TEs), provides beam shaping, steering, and
focus capabilities [25]. A key feature of ITS is that the incoming signal can
pass through each TE, and then the refracted signal can be received by UEs
behind the ITS. More importantly, ITSs can be flexibly deployed by coexisting
with existing infrastructures in the middle of a communication environment,
such as embedded in walls or windows between two different environments
(outdoor-to-indoor, room-to-room, etc.) [19, 17]. ITSs can fully unleash the
potential of RISs on breaking the half-space (reflection space) limitation of
signal propagation manipulated by IRS, and so that covers the back side space
(transmission space) 111It is worth noted that by integrating IRS and ITS
together, a highly flexible full-space manipulation of signal propagation can
be achieved [22, 26, 27]. In other works, the authors proposed a pair of novel
resemble prototypes, simultaneously transmitting and reflecting RISs (STAR-
RISs) [28] and intelligent Omni-surface (IOSs) [29], where the surfaces can
simultaneously transmit and reflect the incident signals. ITS or IRS can be a
special full transmission or reflection mode of these concepts [26]..

However, identically with the reflective cascaded channel in the IRS-assisted
communication, the transmissive cascaded channel in the ITS-assisted
communication will still suffer a multiplicative fading effect [30], which
potentially causes the ITS achieve a poor performance gain. A common approach
to improve the performance gain of RISs is increasing the number of
reflective-type elements (REs) or TEs [31]. Intuitively speaking, this
approach is feasible for the IRS but might not be practical for the ITS due to
that the number of TEs is limited by the physical size of windows and the
aesthetic appearance effect, which might be the price to be paid for that the
ITS is embedded with existing building structures [19]. Additionally, the
substantial electromagnetic penetration loss when the mmWave signal impinges
upon and penetrates the ITS cannot be negligible [17]. These concerns imply
that the performance gain reaped by the ITS might be worthy weak.

To this end, the deployment of an active-ITS will bring a notable leap forward
for improving performance in the outdoor-to-indoor mmWave communication. The
noticeable difference between the active-ITS and the passive-ITS is that the
former can refract the incoming signals after amplifying amplitudes and
shifting phases simultaneously, rather than only refracting them after
changing phases as done by the latter, which offer an extra flexibility to
reconfigure the incident signal. Identical to the principle of active REs [32,
33, 34], an active TE equips with an additional power amplifier for magnifying
the signal amplitude [35]. In this way, active-ITS brings an extra degree of
freedom (DOF) in the beamforming, and the number of TEs of active-ITS can be
beneficially reduced compared with the passive-ITS with a same performance
[36, 37]. The active-ITS seems to act as an active concave lens [13] that
directly refracts the incident signal with power amplification at the
electromagnetic level. Nevertheless, it still takes the advantages of the
passive-ITS that does not have expensive and power-consuming ratio-frequency
(RF) chains, and hence has no capability for signal processing [38, 39, 40],
which is fundamentally different from conventional amplify-and-forward (AF)
relays [30].

Motivated by the above background, we aim to utilize the superiority of an
active-ITS to alleviate the negative effects of the mmWave signal so that to
implement the challenging but important outdoor-to-indoor communication system
in the mmWave cellular network with a high performance. Particularly, the main
contributions of this paper are summarized as follows:

  * •

To our best knowledge, this is the first work to implement outdoor-to-indoor
mmWave communication with the aid of an active-ITS, where the incoming signal
can be refracted to the back side of the active-ITS after magnifying its
amplitude and manipulating its phase.

  * •

To jointly optimize the precoding matrix of both the BS and active-ITS in the
outdoor-to-indoor mmWave communication system, we formulate a weighted sum-
rate (WSR) maximization problem, which is non-convex and challenging to obtain
its corresponding globally optimal solution. Consequently, we propose a
compromising approach to solve it, where the original problem is transformed
into an equivalent form, and then a block coordinate descent (BCD) based
algorithm is employed to obtain the suboptimal solutions in nearly closed-
forms of the linear precoding matrix of the BS, and the power amplification
factor matrix and the transmissive phase-shifting matrix of the active-ITS.

  * •

In order to reduce the size and hardware cost of the active-ITS, we provide a
block-amplifying architecture to partially remove the circuit components for
power-amplifying, where multiple TEs in each block share a same power
amplifier. Then, we extend the proposed algorithm under the element-amplifying
architecture to jointly optimize the precoding matrix under the block-
amplifying architecture.

  * •

Simulation results demonstrate that the active-ITS can significantly improve
the WSR performance of the outdoor-to-indoor mmWave communication system, and
the inevitable performance loss caused by the block-amplifying ITS
architecture is acceptable.

The organization of the rest of this work is organized as follows. In Section
II, we describe the active-ITS empowered outdoor-to-indoor communications in
the mmWave cellular network model, and formulate the WSR maximization problem.
In Section III, we propose an efficient joint design algorithm to solve it.
Then, in Section IV, the proposed algorithm is extended to solve the problem
with the block-amplifying ITS architecture. Simulation results are presented
in Section V, and finally the paper is concluded in Section VI.

Notations: Scalars, vectors, and matrices are presented by lower-case, bold-
face lower-case, and bold-face upper-case letters, respectively.
$\mathcal{C}\mathcal{N}\left(\mathbf{0},\mathbf{I}\right)$ denotes the
circularly symmetric complex Gaussian (CSCG) distribution with zero mean and
covariance matrix $\mathbf{I}_{N}$, where $\mathbf{I}_{N}$ denotes an $N\times
N$ identity matrix. $\operatorname{Vecd}\left\\{\cdot\right\\}$ forms a vector
out of the diagonal of its matrix argument. $\circ$ denote the Hadamard
products. $\mathbf{A}^{\operatorname{H}}$ and
$\operatorname{Tr}\left(\mathbf{A}\right)$ denote the Hermitian and trace
operators of matrix $\mathbf{A}$, respectively.
$\operatorname{Re}\left\\{a\right\\}$ is the real part of a scalar $a$.

##  II SYSTEM MODEL AND PROBLEM FORMULATION

Fig. 1: A system model schematic of the active-ITS empowered outdoor-to-indoor
mmWave communication.

As shown in Fig. 1, we consider the downlink of an outdoor-to-indoor mmWave
communication scenario, where the direct-link channels from the BS to UEs are
assumed severely blocked by the concrete-wall and the tinted-glass, and hence,
fall into almost complete outage [10]. Thanks to the transmissive
characteristic of the ITS, the incoming mmWave signal can pass through the
surface when impinge upon it, and hence, the signal can be refracted from the
outdoor BS to the indoor UEs through the ITS. To alleviate the severe
attenuation of mmWave signals and greatly reduce the number of TEs of the ITS
for application to the size-limited window, we assume each active TE of the
ITS can not only shift the phase of the incident signal, but also amplify the
amplitude of the incident signal with the aid of the power supply [35]. In
addition, a smart controller is attached to the active-ITS and responsible to
operate the active-ITS with the transmissive coefficients, which are
coordinated by the BS [22, 41, 42, 43].

###  II-A Channel Model

Let $M_{t}$, $M_{r}$, and $N$ denote the number of the transmitting antennas,
the receiving antennas, and the TEs equipped by the BS, UEs, and active-ITS,
respectively. The complicated uniform planar arrays (UPA) antenna
configuration is employed at the BS, UEs, and active-ITS, which is more
practical than uniform linear array (ULA) for RIS-assisted systems[11]. In
addition, due to the low diffraction from objects of mmWave signals, mmWave
channels are usually characterized according to the widely used Saleh-
Valenzuela model [10, 11, 12, 13]. Hence, the channel matrices from the BS to
the active-ITS and from the active-ITS to the $k$-th UE are denoted by
$\mathbf{G}\in\mathbb{C}^{N\times M_{t}}$ and
$\mathbf{H}_{k}\in\mathbb{C}^{N\times M_{r}},\forall k\in K$, respectively,
which can be mathematically determined as follows

$\displaystyle\mathbf{G}$
$\displaystyle=\sqrt{\frac{M_{t}N}{L_{\text{BI}}}}\sum\nolimits_{l=1}^{L_{\text{BI}}}{\alpha_{l}\mathbf{a}_{\text{ITS}}\left(\nu_{l}^{\text{AOA}},\vartheta_{l}^{\text{AOA}}\right)\mathbf{a}_{\text{BS}}^{\operatorname{H}}\left(\nu_{l}^{\text{AOD}},\vartheta_{l}^{\text{AOD}}\right)},$
(1a) $\displaystyle\mathbf{H}_{k}$
$\displaystyle=\sqrt{\frac{M_{r}N}{P_{\text{IU}}}}\sum\nolimits_{p=1}^{P_{\text{IU}}}{\beta_{p}\mathbf{a}_{\text{UE}}\left(\nu_{p}^{\text{AOA}},\vartheta_{p}^{\text{AOA}}\right)\mathbf{a}_{\text{ITS}}^{\operatorname{H}}\left(\nu_{p}^{\text{AOD}},\vartheta_{p}^{\text{AOD}}\right)},\forall
k\in K,$ (1b)

where $L_{\text{BI}}$ and $P_{\text{IU}}$ are the number of propagation paths
between the BS and the active-ITS, and between the active-ITS and the $k$-th
UE, and where variables $\alpha_{l}$ and $\beta_{p}$ denote the complex gains
for links, with $l=1$ and $p=1$ stand for the LoS path and the remaining are
NLoS paths.
$\mathbf{a}_{\text{BS}}\left(\nu_{l}^{\text{AOD}},\vartheta_{l}^{\text{AOD}}\right)$
and
$\mathbf{a}_{\text{ITS}}\left(\nu_{p}^{\text{AOD}},\vartheta_{p}^{\text{AOD}}\right)$
are transmit array response vectors of the BS and the active-ITS,
$\mathbf{a}_{\text{UE}}\left(\nu_{l}^{\text{AOA}},\vartheta_{l}^{\text{AOA}}\right)$
and
$\mathbf{a}_{\text{ITS}}\left(\nu_{p}^{\text{AOA}},\vartheta_{p}^{\text{AOA}}\right)$
are receive array response vectors of the UEs and the active-ITS,
respectively, where $\nu^{\text{AOA}}$ ($\nu^{\text{AOD}}$) and
$\vartheta^{\text{AOA}}$ ($\vartheta^{\text{AOD}}$) denote the azimuth and
elevation angles of arrival (departure), respectively. The typical array
response vector for the UPA model can be mathematically expressed by

$\displaystyle\mathbf{a}\left(\nu,\vartheta\right)=$
$\displaystyle\frac{1}{\sqrt{HV}}\left[1,\cdots,e^{j\frac{2\pi
d}{\lambda}\left(h\sin\left(\vartheta\right)\sin\left(\nu\right)+v\cos\left(\vartheta\right)\right)},\right.$
$\displaystyle\qquad\qquad\qquad\qquad\cdots,\left.e^{j\frac{2\pi
d}{\lambda}\left(\left(H-1\right)\sin\left(\vartheta\right)\sin\left(\nu\right)+\left(V-1\right)\cos\left(\vartheta\right)\right)}\right]^{\operatorname{T}},$
(2)

where $H$ and $V$ denote the number of TEs (antennas) in the horizontal and
vertical directions of the active-ITS (BS/UEs), and the element spacing is set
to be $d=\lambda/2$. To unveil the theoretical upper bound of performance gain
achieved by the active-ITS, we follow the assumption [29, 18, 41, 42, 43] that
the full instantaneous channel state information (CSI) knowledge can be
perfectly estimated at the BS and all the calculations are performed at the
BS.

###  II-B System Model

Let
$\mathbf{A}=\operatorname{diag}\left(a_{1},a_{2},\cdots,a_{N}\right)\in\mathbb{R}^{N\times
N}$ and
$\bm{\Theta}=\operatorname{diag}\left(e^{j\varphi_{1}},e^{j\varphi_{2}},\cdots,e^{j\varphi_{N}}\right)\in\mathbb{C}^{N\times
N}$ represent the power amplification factor matrix and the transmissive
phase-shifting matrix at the active-ITS with $a_{n}\geq 1$ and
$\varphi_{n}\in\left[0,2\pi\right)$, $\forall n\in N$.

The signal from the BS is given by
$\mathbf{x}=\sum_{k=1}^{K}{\mathbf{W}_{k}\mathbf{s}_{k}},\forall k\in K$,
where $\mathbf{s}_{k}\in\mathbb{C}^{s\times 1}$ represents $s$ desired streams
for the $k$-th UE and satisfies
$\mathbf{s}_{k}\sim\mathcal{C}\mathcal{N}\left(\bm{0}_{s},\mathbf{I}_{s}\right)$,
and $\mathbf{W}_{k}\in\mathbb{C}^{M_{t}\times s}$ denotes the linear precoding
matrix. As a result, the signal received at the active-ITS is given by

$\displaystyle\mathbf{y}_{\operatorname{ITS}}^{r}=\mathbf{G}\mathbf{x}+\bm{\upsilon},$
(3)

where $\bm{\upsilon}\in\mathbb{C}^{N\times 1}$ is additive white Gaussian
noise (AWGN) and satisfying
$\bm{\upsilon}\sim\mathcal{C}\mathcal{N}\left(\bm{0}_{N},\delta^{2}\mathbf{I}_{N}\right)$.

Then, by employing an active-ITS, the refracted signal can be denoted by

$\displaystyle\mathbf{y}_{\operatorname{ITS}}^{t}=\kappa\mathbf{A}\bm{\Theta}\left(\mathbf{G}\mathbf{x}+\bm{\upsilon}\right),$
(4)

where $\kappa$ characterizes the extent of loss corresponding to the
absorption and the reflection of the signal power when the mmWave signal
impinges upon and penetrates the ITS. The noise $\bm{\upsilon}$ after power
amplifying cannot be negligible, which is significantly different from the
conventional passive-ITS [33]. For simplicity of presentation, let
$\bm{\Phi}\triangleq\mathbf{A}\circ\bm{\Theta}$ denotes the transmissive
coefficient matrix, where each diagonal element is denoted by
$\Phi_{n,n}=a_{n}e^{j\varphi_{n}},\forall n\in N$.

The signal received by the $k$-th UE can be mathematically expressed as
follows

$\displaystyle\mathbf{y}_{k}=$
$\displaystyle\kappa\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k}\mathbf{s}_{k}+\sum\nolimits_{i=1,i\neq
k}^{K}{\kappa\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{i}\mathbf{s}_{i}}+\kappa\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\bm{\upsilon}+\mathbf{n}_{k},\forall
k\in K,$ (5)

where $\mathbf{n}_{k}\in\mathbb{C}^{M_{r}\times 1}$ represents the thermal
noise at the $k$-th UE and satisfying
$\mathbf{n}_{k}\sim\mathcal{C}\left(\bm{0}_{M_{r}},\sigma_{k}^{2}\mathbf{I}_{M_{r}}\right)$.

Therefore, the signal-to-interference-plus-noise-ratio (SINR) at the $k$-th UE
is given by

$\displaystyle\bm{\Gamma}_{k}=\kappa^{2}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k}\mathbf{W}_{k}^{\operatorname{H}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{V}_{k}^{\operatorname{-1}},\forall
k\in K,$ (6)

where $\mathbf{V}_{k}$ denotes the interference-plus-noise covariance matrix
and can be expressed by

$\displaystyle\mathbf{V}_{k}$ $\displaystyle=\sum\nolimits_{i=1,i\neq
k}^{K}{\kappa^{2}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{i}\mathbf{W}_{i}^{\operatorname{H}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}}+\kappa^{2}\delta^{2}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}+\sigma_{k}^{2}\mathbf{I}_{M_{r}},\forall
k\in K.$ (7)

Accordingly, the achievable data rate of the $k$-th UE is given by

$\displaystyle\mathcal{R}_{k}=\log\left|\mathbf{I}+\bm{\Gamma_{k}}\right|,\forall
k\in K.$ (8)

###  II-C Problem Formulation

In this paper, we aim to maximize the WSR of the outdoor-to-indoor mmWave
communication system by joint optimizing the linear precoding matrix at the
BS, i.e., $\mathbf{W}$, and the transmissive coefficient matrix at the active-
ITS, i.e., $\bm{\Phi}$, while satisfying the constraints of the maximum power
budgets at the BS and active-ITS, and the transmissive phase-shifting at each
TE. Based on the above discussions, the WSR maximization problem can be
formulated as follows

$\displaystyle\mathcal{P}_{1}:\mathop{\max}_{\mathbf{W},\mathbf{\Phi}}\quad$
$\displaystyle\mathcal{R}\left(\mathbf{W},\mathbf{\Phi}\right)={\sum\nolimits_{k=1}^{K}{\alpha_{k}\log\left|\mathbf{I}+\bm{\Gamma_{k}}\right|}},$
(9a) $\displaystyle\operatorname{s.t.}\quad$
$\displaystyle\sum\nolimits_{k=1}^{K}{\|\mathbf{W}_{k}\|_{\operatorname{F}}^{2}}\leq
P_{\operatorname{BS}}^{\max},$ (9b)
$\displaystyle\sum\nolimits_{k=1}^{K}{\kappa^{2}\|\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k}\|_{\operatorname{F}}^{2}}+\delta^{2}\kappa^{2}\|\mathbf{\Phi}\|_{\operatorname{F}}^{2}\leq
P_{\operatorname{ITS}}^{\max},$ (9c)
$\displaystyle\angle\bm{\Phi}_{n,n}\in\left[0,2\pi\right),\forall n\in N,$
(9d)

where $\bm{\alpha}=\left[\alpha_{1},\alpha_{2},\cdots,\alpha_{K}\right]$
denotes the weighting factor vector with each element $\alpha_{k}$ is a
predetermined value depending on the fairness and the required quality of
service for applications, and where $P_{\operatorname{BS}}^{\max}$ and
$P_{\operatorname{ITS}}^{\max}$ are the budgets of the transmission power and
the amplification power at the BS and the active-ITS, respectively.

Remark 1: If we remove the amplifying power budget constraint in (9c) and set
$\mathbf{A}=\mathbf{I}$, Problem $\mathcal{P}_{1}$ reduce to the WSR
maximization problem for conventional passive-RIS assisted communication
systems [41, 42, 43]. However, the additional constraint and the optimization
of the power amplification factor matrix [44] make Problem $\mathcal{P}_{1}$
even more challenging to solve.

##  III The Proposed Suboptimal Joint Precoding Algorithm

The formulated problem $\mathcal{P}_{1}$ is non-convex and arduous to tackle
optimally, in this section, we propose an efficient joint precoding algorithm
to handle it with suboptimal solutions. Particularly, to facilitate the
solution development, we first reformulate the original problem as an
equivalent but tractable form, and then a BCD-based method is provided to
solve the transformed problem.

###  III-A Problem Reformulation and Decomposition

First, to deal with the complexity of the objective function in the formulated
WSR maximization problem $\mathcal{P}_{1}$, we transform it into an equivalent
weighted minimum mean-square error (WMMSE) minimization problem [45, 41, 42].

Let us assume that the MMSE estimator
$\mathbf{U}=\left\\{\mathbf{U}_{k}\in\mathbb{C}^{M_{r}\times s},\forall k\in
K\right\\}$ are applied to UEs, and accordingly, the estimated signal vector
of the $k$-th UE is given by

$\displaystyle\mathbf{\hat{s}}_{k}=\mathbf{U}_{k}^{\operatorname{H}}\mathbf{y}_{k},\forall
k\in K.$ (10)

Therefore, the MMSE matrix of the $k$-th UE is given by

$\displaystyle\mathbf{E}_{k}=$
$\displaystyle\mathbb{E}_{\mathbf{x},\bm{\upsilon},\mathbf{n}}\left[\left(\mathbf{\hat{s}}_{k}-\mathbf{s}_{k}\right)\left(\mathbf{\hat{s}}_{k}-\mathbf{s}_{k}\right)^{\operatorname{H}}\right]$
$\displaystyle=$
$\displaystyle\sum\nolimits_{i=1}^{K}{\kappa^{2}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{i}\mathbf{W}_{i}^{\operatorname{H}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k}}+\kappa^{2}\delta^{2}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k}$
$\displaystyle+\sigma_{k}^{2}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{U}_{k}+\mathbf{I}_{s}-\kappa\mathbf{U}_{k}^{\operatorname{H}}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k}-\kappa\mathbf{W}_{k}^{\operatorname{H}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k},\forall
k\in K.$ (11)

By introducing an auxiliary variable $\mathbf{F}_{k}\in\mathbb{C}^{s\times s}$
for the $k$-th UE and defining $\mathbf{F}=\left\\{\mathbf{F}_{k},\forall k\in
K\right\\}$, problem $\mathcal{P}_{1}$ can be equivalently transformed into

$\displaystyle\mathcal{P}_{2}:\mathop{\max}_{\mathbf{F},\mathbf{U},\mathbf{W},\mathbf{\Phi}}\;$
$\displaystyle{\sum\nolimits_{k=1}^{K}{\alpha_{k}h_{k}\left(\mathbf{F},\mathbf{U},\mathbf{W},\mathbf{\Phi}\right)}}$
(12a) $\displaystyle\operatorname{s.t.}\;$
$\displaystyle\eqref{p11}-\eqref{p13},$ (12b)

where
$h_{k}\left(\mathbf{F},\mathbf{U},\mathbf{W},\mathbf{\Phi}\right)=\log\left|\mathbf{F}_{k}\right|-\operatorname{Tr}\left(\mathbf{F}_{k}\mathbf{E}_{k}\right)+s,\forall
k\in K$.

Although the above problem has been significantly simplified compared with
problem $\mathcal{P}_{1}$, it is still challenging to tackle. Fortunately,
based on the fact that the objective function in (12a) is concave with respect
to any one of the four variables (i.e., $\mathbf{U}$, $\mathbf{F}$,
$\mathbf{W}$, and $\bm{\Phi}$) when the other three variables being fixed,
which makes the problem more amenable. In the following, we provide an
efficient BCD-based algorithm for solving problem $\mathcal{P}_{2}$ in an
iterative manner.

Note that the MMSE decoding matrix $\mathbf{U}$ and the auxiliary matrix
$\mathbf{F}$ do not exist in any constraints in problem $\mathcal{P}_{2}$,
which implies that with fixed $\mathbf{W}$ and $\bm{\Theta}$, the calculation
of $\mathbf{F}^{\star}$ and $\mathbf{U}^{\star}$ constitute a pair of
unconstrained optimization problems. Therefore, the optimal solutions can be
obtained by setting the first-order partial derivatives of the objective
function of problem $\mathcal{P}_{2}$ with respect to $\mathbf{U}_{k},\forall
k\in K$ and $\mathbf{F}_{k},\forall k\in K$ to be zeros, respectively. After
some matrix manipulations, the optimal closed-form solutions can be
respectively determined by

$\displaystyle\mathbf{U}_{k}^{\star}=\left(\mathbf{\bar{V}}_{k}\right)^{\operatorname{-1}}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k},\forall
k\in K,$ (13)

where
$\mathbf{\bar{V}}_{k}=\mathbf{V}_{k}+{\kappa^{2}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k}\mathbf{W}_{k}^{\operatorname{H}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}}$,
and

$\displaystyle\mathbf{F}_{k}^{\star}=\left(\mathbf{E}_{k}^{\star}\right)^{\operatorname{-1}},\forall
k\in K,$ (14)

where $\mathbf{E}_{k}^{\star}$ is determined by

$\displaystyle\mathbf{E}_{k}^{\star}=\mathbf{I}_{s}-\mathbf{W}_{k}^{\operatorname{H}}\mathbf{G}^{\operatorname{H}}\bm{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{\bar{V}}_{k}^{\operatorname{-1}}\mathbf{H}_{k}^{\operatorname{H}}\bm{\Phi}\mathbf{G}\mathbf{W}_{k},\forall
k\in K.$ (15)

Once both $\mathbf{U}^{\star}$ and $\mathbf{F}^{\star}$ are determined, the
remaining work is to optimize the linear precoding matrix $\mathbf{W}$ and the
transmissive coefficient matrix $\mathbf{\Phi}$. Note that $\mathbf{W}$ and
$\mathbf{\Phi}$ are intricately coupled in $\mathbf{E}_{k},\forall k\in K$, we
consider decomposing problem $\mathcal{P}_{2}$ into two subproblem with
respect to these two variables. In particular, the two subproblems can be
expressed by

$\displaystyle\mathcal{P}_{3}:\mathbf{W}^{\star}=\arg\mathop{\min}_{\mathbf{W}}$
$\displaystyle{\sum_{k=1}^{K}{\alpha_{k}\operatorname{Tr}\left(\mathbf{F}_{k}\mathbf{E}_{k}\right)}},\operatorname{s.t.}\;\eqref{p11}\;\text{and}\;\eqref{p12},$
(16a)
$\displaystyle\mathcal{P}_{4}:\mathbf{\Phi}^{\star}=\arg\mathop{\min}_{\mathbf{\Phi}}$
$\displaystyle{\sum_{k=1}^{K}{\alpha_{k}\operatorname{Tr}\left(\mathbf{F}_{k}\mathbf{E}_{k}\right)}},\operatorname{s.t.}\;\eqref{p12}\;\text{and}\;\eqref{p13}.$
(16b)

In the rest of this section, we propose a pair of methods to solve the above
two subproblems.

###  III-B Linear Precoding Matrix Optimization

In this subsection, we focus on the subproblem state in (16a) for optimizing
linear precoding matrix $\mathbf{W}^{\star}$ at the BS with the fixed
transmissive coefficient matrix $\bm{\Phi}$. Since the part terms of the
objective function in problem (16a), i.e.,
$\operatorname{Tr}\left({\alpha_{k}\kappa^{2}\delta^{2}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k}}\right)$,
and
$\operatorname{Tr}\left({\sigma_{k}^{2}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{U}_{k}}\right)+\mathbf{I}_{s}$
are irrelevant constants with respect to $\mathbf{W}$, and hence, by omitting
them, the optimal solution of $\mathbf{W}$ can be determined by solving the
following subproblem

$\displaystyle\mathcal{P}_{3\mbox{-}1}:\mathop{\min}_{\mathbf{W}}\quad$
$\displaystyle\sum_{k=1}^{K}{\operatorname{Tr}\left({\mathbf{W}_{k}^{\operatorname{H}}\mathbf{Q}\mathbf{W}_{k}}\right)}-\sum_{k=1}^{K}{\operatorname{Tr}\left({\alpha_{k}\kappa\mathbf{F}_{k}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k}}\right)}$
$\displaystyle-\sum_{k=1}^{K}{\operatorname{Tr}\left({\alpha_{k}\kappa\mathbf{F}_{k}\mathbf{W}_{k}^{\operatorname{H}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k}}\right)},$
(17a) $\displaystyle\operatorname{s.t.}\quad$
$\displaystyle\sum_{k=1}^{K}{\operatorname{Tr}\left({\mathbf{W}_{k}^{\operatorname{H}}\mathbf{W}_{k}}\right)}\leq
P_{\operatorname{BS}}^{\max},$ (17b)
$\displaystyle\sum_{k=1}^{K}{\operatorname{Tr}\left({\kappa^{2}\mathbf{W}_{k}^{\operatorname{H}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k}}\right)}\leq\hat{P}_{\operatorname{ITS}}^{\max},$
(17c)

where $\hat{P}_{\operatorname{ITS}}^{\max}\triangleq
P_{\operatorname{ITS}}^{\max}-\operatorname{Tr}\left(\delta^{2}\kappa^{2}\bm{\Phi}^{\operatorname{H}}\bm{\Phi}\right)$,
and
$\mathbf{Q}=\sum_{k=1}^{K}{\alpha_{k}\kappa^{2}\mathbf{G}^{\operatorname{H}}\bm{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k}\mathbf{F}_{k}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{H}_{k}^{\operatorname{H}}\bm{\Phi}}\mathbf{G}$.

It can be proved that $\mathbf{Q}$ is a positive semi-definite matrix,
therefore the objective function in (17a) is a convex function. In addition,
the constraints in (17b) and (17c) are convex with respect to $\mathbf{W}$,
and so problem $\mathcal{P}_{3\mbox{-}1}$ constitutes a convex optimization
problem, which can be effectively solved by employing convex solver toolbox,
e.g., CVX [46]. Instead of relying on the generic solver with high
computational complexity, we provide the following duality method to further
proceed with problem $\mathcal{P}_{3\mbox{-}1}$ efficiently.

By introducing a pair of auxiliary variables $\lambda\geq 0$ and $\mu\geq 0$,
the constraints in (17b) and (17c) can be combined as

$\displaystyle\lambda\sum_{k=1}^{K}{\operatorname{Tr}\left({\mathbf{W}_{k}^{\operatorname{H}}\mathbf{W}_{k}}\right)}+\mu\sum_{k=1}^{K}{\operatorname{Tr}\left({\kappa^{2}\mathbf{W}_{k}^{\operatorname{H}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k}}\right)}\leq
P_{\operatorname{sum}}^{\max},$ (18)

where the constant $P_{\operatorname{sum}}^{\max}$ is defined by
$P_{\operatorname{sum}}^{\max}\triangleq\lambda
P_{\operatorname{BS}}^{\max}+\mu\hat{P}_{\operatorname{ITS}}^{\max}$.
Consequently, problem $\mathcal{P}_{3\mbox{-}1}$ can be recast by

$\displaystyle\mathcal{P}_{3\text{-}2}:g\left(\lambda,\mu\right)\triangleq\left\\{\mathop{\min}_{\mathbf{W}}\,\eqref{p21o},\;\operatorname{s.t.}\,\eqref{p22wa}\right\\}.$
(19)

It is evident that a feasible solution of problem $\mathcal{P}_{3\mbox{-}1}$
is also feasible for problem $\mathcal{P}_{3\mbox{-}2}$. The relation of the
optimal values between the two problems is shown in the following result.

######  Lemma 1.

The optimal value of problem $\mathcal{P}_{3\text{-}2}$ for any given pair of
$\left\\{\lambda,\mu\right\\}$, $\lambda\geq 0$, $\mu\geq 0$, is an lower
bound on the optimal value of problem $\mathcal{P}_{3\text{-}1}$. Furthermore,
the optimal value of the dual problem, i.e.,
$\mathop{\max}_{\lambda,\mu}\,g\left(\lambda,\mu\right)$, is equal to that of
problem $\mathcal{P}_{3\text{-}1}$.

###### Proof.

This lemma follows Propositions 4 and 5 in [47], and hence, is omitted here
for brevity. ∎

With the fixed variables $\left\\{\lambda,\mu\right\\}$, by introducing a
Lagrangian multiplier $\varepsilon\geq 0$ associated with the combined
constraint in (18), the corresponding Lagrangian function can be represented
by

$\displaystyle\mathcal{L}\left(\mathbf{W},\varepsilon\right)=$
$\displaystyle\sum_{k=1}^{K}{\operatorname{Tr}\left({\mathbf{W}_{k}^{\operatorname{H}}\mathbf{\hat{Q}}\left(\varepsilon\right)\mathbf{W}_{k}}\right)}-\sum_{k=1}^{K}{\operatorname{Tr}\left({\alpha_{k}\kappa\mathbf{F}_{k}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k}}\right)}$
$\displaystyle-\sum_{k=1}^{K}{\operatorname{Tr}\left({\alpha_{k}\kappa\mathbf{F}_{k}\mathbf{W}_{k}^{\operatorname{H}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k}}\right)}-\varepsilon
P_{\operatorname{sum}}^{\max},$ (20)

where
$\mathbf{\hat{Q}}\left(\varepsilon\right)\triangleq\mathbf{Q}+\varepsilon\left(\lambda\mathbf{I}_{M_{t}}+\mu\kappa^{2}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\right)$.

Accordingly, the Karush–Kuhn–Tucker (KKT) conditions of problem
$\mathcal{P}_{3\text{-}2}$ are given by

$\displaystyle\nabla_{\mathbf{W}_{k}}{\mathcal{L}\left(\mathbf{\tilde{W}},\varepsilon\right)}=2\frac{\partial\mathcal{L}\left(\mathbf{\tilde{W}},\varepsilon\right)}{\partial{\mathbf{W}_{k}^{\ast}}}$
$\displaystyle=\bm{0},\forall k,$ (21a) $\displaystyle\varepsilon
f\left(\mathbf{W}_{k}\left(\varepsilon\right)\right)$ $\displaystyle=0,\forall
k,$ (21b)

where
$f\left(\mathbf{W}_{k}\left(\varepsilon\right)\right)=\lambda\sum_{k=1}^{K}{\operatorname{Tr}\left({\mathbf{W}_{k}^{\operatorname{H}}\mathbf{W}_{k}}\right)}+\mu\sum_{k=1}^{K}{\operatorname{Tr}\left({\kappa^{2}\mathbf{W}_{k}^{\operatorname{H}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k}}\right)}-P_{\operatorname{sum}}^{\max}$.

From the first KKT condition in (21a), with a fixed $\varepsilon$, the optimal
solution of $\mathbf{W}_{k},\forall k$ is calculated by

$\displaystyle\mathbf{W}_{k}\left(\varepsilon\right)=\alpha_{k}\kappa\mathbf{\hat{Q}}\left(\varepsilon\right)^{\operatorname{-1}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k}\mathbf{F}_{k},\forall
k\in K.$ (22)

Then, the dual variable $\varepsilon$ should be determined for satisfying the
second KKT condition in (21b). We search $\varepsilon$ for two cases, i.e.,
$\varepsilon=0$ and $\varepsilon>0$. For the former case, if the inverse of
$\mathbf{Q}$ exists and $f\left(\mathbf{W}_{k}\left(0\right)\right)\leq 0$
holds, the optimal solution for the $k$-th UE can be obtained by

$\displaystyle\mathbf{W}_{k}=\mathbf{W}_{k}\left(0\right)=\alpha_{k}\kappa\mathbf{Q}^{\operatorname{-1}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k}\mathbf{F}_{k},\forall
k\in K.$ (23)

Otherwise, we have to find $\varepsilon$ for ensuring
$f\left(\mathbf{W}_{k}\left(\varepsilon\right)\right)=0$ satisfied. Note that
$f\left(\mathbf{W}_{k}\left(\varepsilon\right)\right)$ is a monotonically
decreasing function of $\varepsilon$ that enables a bisection search method to
find $\varepsilon$ [41]. Then, the remaining work is to solve the following
dual problem

$\displaystyle\mathop{\max}_{\lambda,\mu}\,g\left(\lambda,\mu\right).$ (24)

We employ the iterative subgradient method [42] to find the solutions. First,
the subgradient directions of the above problem are determined by

$\displaystyle
f_{P_{\text{BS}}}=\sum_{k=1}^{K}{\operatorname{Tr}\left({\mathbf{W}_{k}^{\operatorname{H}}\mathbf{W}_{k}}\right)}-P_{\text{BS}}^{\max},$
(25a) $\displaystyle
f_{P_{\text{ITS}}}=\sum_{k=1}^{K}{\operatorname{Tr}\left({\kappa^{2}\mathbf{W}_{k}^{\operatorname{H}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k}}\right)}-\hat{P}_{\text{ITS}}^{\max}.$
(25b)

Then, the values of $\lambda$ and $\mu$ can be updated in an iterative manner,
the details are summarized in Algorithm 1, where $\xi$ denotes the step size
of the subgradient algorithm, $\epsilon$ represents the tolerance, and the
superscript stands for the number of iteration. The overall dual-based method
for optimizing the linear precoding matrix at the BS is summarized in
Algorithm 2, and the optimal solution $\mathbf{W}^{\star}$ can be obtained
when the algorithm converged.

Algorithm 1 Iterative Subgradient algorithm

1: Initialize: $\lambda^{\operatorname{0}}$, $\mu^{\operatorname{0}}$, and
iteration index $p=0$.

2: repeat

3: $\lambda^{{p+1}}$$\leftarrow\lambda^{{p}}+\xi f_{P_{\text{BS}}}$,

4: $\mu^{{p+1}}$ $\leftarrow\mu^{{p}}+\xi f_{P_{\text{ITS}}}$.

5: until $\left|\lambda^{{p+1}}-\lambda^{{p}}\right|\leq\epsilon$ and
$\left|\mu^{{p+1}}-\mu^{{p}}\right|\leq\epsilon$.

Algorithm 2 Dual-based Algorithm for Calculating $\mathbf{W}$

1: Initialize: $\lambda^{\operatorname{0}}$, $\mu^{\operatorname{0}}$,
$\mathbf{W}^{\operatorname{0}}$, $\varepsilon^{\operatorname{0}}$, and
iteration index $r=0$.

2: Calculate the value of the function in (17a) as
$v\left(\mathbf{W}^{\operatorname{0}}\right)$.

3: repeat

4: Calculate $\mathbf{W}^{\operatorname{r+1}}$ and search
$\varepsilon^{\operatorname{r+1}}$,Update $\lambda^{\operatorname{r+1}}$ and
$\mu^{\operatorname{r+1}}$ by employing Algorithm 1,

5: until
$|v\left(\mathbf{W}^{\operatorname{r+1}}\right)-v\left(\mathbf{W}^{\operatorname{r}}\right)|/v\left(\mathbf{W}^{\operatorname{r}}\right)\leq\varepsilon$.

6: Output: $\mathbf{W}^{\star}=\mathbf{W}^{\operatorname{r}}$.

###  III-C Transmissive Coefficient Matrix Optimization

Now, we consider to solve the subproblem of optimizing the transmissive
coefficient matrix at the active-ITS. With the fixed $\mathbf{W}^{\star}$ by
employing Algorithm 2 and define
$\mathbf{\tilde{W}}=\sum_{i=1}^{K}{\mathbf{W}_{i}\mathbf{W}_{i}^{\operatorname{H}}}$,
after dropping the constant terms, problem $\mathcal{P}_{4}$ can be simplified
expressed by

$\displaystyle\mathcal{P}_{4}^{\prime}:\mathop{\min}_{\mathbf{\Phi}}\;$
$\displaystyle\sum_{k=1}^{K}{\alpha_{k}\kappa^{2}\operatorname{Tr}\left(\mathbf{F}_{k}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{\tilde{W}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k}\right)}$
$\displaystyle+\sum_{k=1}^{K}{\alpha_{k}\kappa^{2}\delta^{2}\operatorname{Tr}\left(\mathbf{F}_{k}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k}\right)}$
$\displaystyle-\sum_{k=1}^{K}{\alpha_{k}\kappa\operatorname{Tr}\left(\mathbf{F}_{k}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k}\right)}$
$\displaystyle-\sum_{k=1}^{K}{\alpha_{k}\kappa\operatorname{Tr}\left(\mathbf{F}_{k}\mathbf{W}_{k}^{\operatorname{H}}\mathbf{G}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k}\right)}$
(26) $\displaystyle\operatorname{s.t.}\;$
$\displaystyle\eqref{p12}\;\operatorname{and}\;\eqref{p13}.$

Note that the above problem is still difficult to tackle. In the following, we
transform problem $\mathcal{P}_{4}^{\prime}$ into an equivalent form by
employing some further algebraic manipulations. First, by defining
$\mathbf{D}_{k}=\alpha_{k}\mathbf{H}_{k}\mathbf{U}_{k}\mathbf{F}_{k}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{H}_{k}^{\operatorname{H}}$,
$\mathbf{D}=\sum_{k=1}^{K}{\mathbf{D}_{k}}$, and
$\mathbf{T}=\kappa^{2}\mathbf{G}\mathbf{\tilde{W}}\mathbf{G}^{\operatorname{H}}$,
we have

$\displaystyle\sum_{k=1}^{K}{\alpha_{k}\kappa^{2}\operatorname{Tr}\left(\mathbf{F}_{k}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{\tilde{W}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k}\right)}$
$\displaystyle=\operatorname{Tr}\left(\mathbf{\Phi}^{\operatorname{H}}\mathbf{D}\mathbf{\Phi}\mathbf{T}\right),$
(27a)
$\displaystyle\sum_{k=1}^{K}{\alpha_{k}\operatorname{Tr}\left(\mathbf{F}_{k}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k}\right)}$
$\displaystyle=\operatorname{Tr}\left(\mathbf{\Phi}^{\operatorname{H}}\mathbf{D}\mathbf{\Phi}\right),$
(27b)
$\displaystyle\sum_{k=1}^{K}{\kappa^{2}\operatorname{Tr}\left({\mathbf{W}_{k}^{\operatorname{H}}\mathbf{G}^{\operatorname{H}}\mathbf{\Phi}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k}}\right)}$
$\displaystyle=\operatorname{Tr}\left(\mathbf{\Phi}^{\operatorname{H}}\mathbf{\Phi}\mathbf{T}\right).$
(27c)

Then, by defining
$\mathbf{C}_{k}=\alpha_{k}\kappa\mathbf{H}_{k}\mathbf{U}_{k}\mathbf{F}_{k}\mathbf{W}_{k}^{\operatorname{H}}\mathbf{G}$
and $\mathbf{C}=\sum_{k=1}^{K}{\mathbf{C}_{k}}$, we have

$\displaystyle\sum_{k=1}^{K}{\alpha_{k}\kappa\operatorname{Tr}\left(\mathbf{F}_{k}\mathbf{U}_{k}^{\operatorname{H}}\mathbf{H}_{k}^{\operatorname{H}}\mathbf{\Phi}\mathbf{G}\mathbf{W}_{k}\right)}$
$\displaystyle=\operatorname{Tr}\left(\mathbf{C}^{\operatorname{H}}\mathbf{\Phi}\right),$
(28a)
$\displaystyle\sum_{k=1}^{K}{\alpha_{k}\kappa\operatorname{Tr}\left(\mathbf{F}_{k}\mathbf{W}_{k}^{\operatorname{H}}\mathbf{G}\mathbf{\Phi}^{\operatorname{H}}\mathbf{H}_{k}\mathbf{U}_{k}\right)}$
$\displaystyle=\operatorname{Tr}\left(\mathbf{\Phi}^{\operatorname{H}}\mathbf{C}\right).$
(28b)

Further, we define a sequence of equalities as follows

$\displaystyle\operatorname{Tr}\left(\mathbf{\Phi}^{\operatorname{H}}\mathbf{D}\mathbf{\Phi}\mathbf{T}\right)$
$\displaystyle\triangleq\bm{\phi}^{\operatorname{H}}\mathbf{\Omega}\bm{\phi},$
(29a)
$\displaystyle\operatorname{Tr}\left(\mathbf{\Phi}^{\operatorname{H}}\mathbf{C}\right)$
$\displaystyle\triangleq\bm{\phi}^{\operatorname{H}}\mathbf{c},$ (29b)
$\displaystyle\operatorname{Tr}\left(\mathbf{\Phi}\mathbf{C}^{\operatorname{H}}\right)$
$\displaystyle\triangleq\mathbf{c}^{\operatorname{H}}\bm{\phi},$ (29c)

where $\bm{\Omega}=\mathbf{D}\circ\mathbf{T}^{\operatorname{T}}$,
$\bm{\phi}=\operatorname{Vecd}\left(\bm{\Phi}\right)$, and
$\mathbf{c}=\operatorname{Vecd}\left(\mathbf{C}\right)$, with
$\operatorname{Vecd}\left(\mathbf{X}\right)$ forms a vector out of the
diagonal of its matrix argument. The above equalities given above follow from
the properties in [48, Theorem 1.11].

Here, our object is to obtain the transmissive coefficient vector of the
active-ITS, $\bm{\phi}$, which consists of power amplification factor and
phase-shifting, i.e., $\mathbf{a}$ and $\bm{\theta}$, where
$\mathbf{a}=\operatorname{Vecd}\left(\mathbf{A}\right)$ and
$\bm{\theta}=\operatorname{Vecd}\left(\bm{\Theta}\right)$, respectively.
Consequently, we have $\bm{\phi}=\mathbf{a}\circ\bm{\theta}$.

According to the fact that
$\operatorname{Tr}\left(\bm{\Phi}^{\operatorname{H}}\bm{\Phi}\right)=\mathbf{a}^{\operatorname{T}}\mathbf{a}$,
we have

$\displaystyle\operatorname{Tr}\left(\bm{\Phi}^{\operatorname{H}}\bm{\Phi}\right)$
$\displaystyle=\sum_{n=1}^{N}{a_{n}^{2}},$ (30a)
$\displaystyle\operatorname{Tr}\left(\mathbf{\Phi}^{\operatorname{H}}\mathbf{D}\mathbf{\Phi}\right)$
$\displaystyle=\sum_{n=1}^{N}{a_{n}^{2}D_{\operatorname{n,n}}},$ (30b)
$\displaystyle\operatorname{Tr}\left(\mathbf{\Phi}^{\operatorname{H}}\mathbf{\Phi}\mathbf{T}\right)$
$\displaystyle=\sum_{n=1}^{N}{a_{n}^{2}T_{\operatorname{n,n}}}.$ (30c)

In addition, the constraint on the amplification power budget in (9d) can be
recast by

$\displaystyle\sum_{n=1}^{N}{a_{n}^{2}T_{\operatorname{n,n}}}+\delta^{2}\kappa^{2}\sum_{n=1}^{N}{a_{n}^{2}}\leq
P_{\operatorname{ITS}}^{\max}.$ (31)

Based on the above discussion, problem $\mathcal{P}_{4}^{\prime}$ can be
equivalently transformed as follows

$\displaystyle\mathcal{P}_{4\mbox{-}1}:\mathop{\min}_{\mathbf{a},\bm{\theta}}\;$
$\displaystyle
f\left(\mathbf{a},\bm{\theta}\right)=\left(\mathbf{a}^{\operatorname{T}}\circ\bm{\theta}^{\operatorname{H}}\right)\mathbf{\Omega}\left(\mathbf{a}\circ\bm{\theta}\right)+\delta^{2}\kappa^{2}\sum_{n=1}^{N}{a_{n}^{2}D_{\operatorname{n,n}}}$
$\displaystyle-\left(\mathbf{a}^{\operatorname{T}}\circ\bm{\theta}^{\operatorname{H}}\right)\mathbf{c}-\mathbf{c}^{\operatorname{H}}\left(\mathbf{a}\circ\bm{\theta}\right),$
(32a) $\displaystyle\operatorname{s.t.}\;$
$\displaystyle\sum_{n=1}^{N}{a_{n}^{2}\left(T_{\operatorname{n,n}}+\delta^{2}\kappa^{2}\right)}\leq
P_{\operatorname{ITS}}^{\max},$ (32b)
$\displaystyle\angle\theta_{n}=\varphi_{n}\in\left[0,2\pi\right),\forall n\in
N.$ (32c)

Although problem $\mathcal{P}_{4-1}$ has been significantly simplified
compared with problem $\mathcal{P}_{4}^{\prime}$, it is still non-convex. As a
result, the dual-based algorithm cannot be employed here due to the dual gap
is not zero. Therefore, we consider transform the above challenging problem
into an equivalent form by employing a price mechanism [49]:

$\displaystyle\mathcal{P}_{4\mbox{-}1}^{\prime}:\mathop{\min}_{\mathbf{a},\bm{\theta}}\;$
$\displaystyle
h\left(\mathbf{a},\bm{\theta}\right)=f\left(\mathbf{a},\bm{\theta}\right)+\eta
g\left(\mathbf{a}\right)$ (33a) $\displaystyle\operatorname{s.t.}\;$
$\displaystyle\eqref{p312}.$ (33b)

where $\eta\geq 0$ is the introduced price of the function
$g\left(\mathbf{a}\right)=\sum_{n=1}^{N}{a_{n}^{2}\left(T_{\operatorname{n,n}}+\delta^{2}\kappa^{2}\right)}$,
which is the left hand of constraint in (32b). The value of $\eta$ can be
obtained by employing the bisection search method similar to search
$\varepsilon$ [49].

Then, for a given $\eta$, we propose an efficient element-wise alternate
sequential optimization (ASO) algorithm [18, 50] to obtain the high-quality
suboptimal solutions of the power amplification factor vector $\mathbf{a}$ and
the transmissive phase-shifting vector $\bm{\theta}$ [44].

More specifically, we further express the terms of the objective function
$h\left(\mathbf{a},\bm{\theta}\right)$ as follows

$\displaystyle\left(\mathbf{a}^{\operatorname{T}}\circ\bm{\theta}^{\operatorname{H}}\right)\mathbf{\Omega}\left(\mathbf{a}\circ\bm{\theta}\right)$
$\displaystyle=$ $\displaystyle\sum_{i=1,i\neq
n}^{N}{\left(\mathbf{a}^{\operatorname{T}}\circ\bm{\theta}^{\operatorname{H}}\right)\bm{\Omega}_{:,i}a_{i}\theta_{i}}+\left(\mathbf{a}^{\operatorname{T}}\circ\bm{\theta}^{\operatorname{H}}\right)\bm{\Omega}_{:,n}a_{n}\theta_{n}$
$\displaystyle=$ $\displaystyle\sum_{i=1,i\neq
n}^{N}{a_{n}\theta_{n}^{*}\Omega_{n,i}a_{i}\theta_{i}}+\sum_{j=1,j\neq
n}^{N}{\sum_{i=1,i\neq
n}^{N}{a_{j}\theta_{j}^{*}\Omega_{j,i}a_{i}\theta_{i}}}+a_{n}\theta_{n}^{*}\Omega_{n,n}a_{n}\theta_{n}+\sum_{i=1,i\neq
n}^{N}{a_{i}\theta_{i}^{*}\Omega_{i,n}a_{n}\theta_{n}},$ (34)

and

$\displaystyle\left(\mathbf{a}^{\operatorname{T}}\circ\bm{\theta}^{\operatorname{H}}\right)\mathbf{c}=a_{n}\theta_{n}^{*}c_{n}+\sum_{i=1,i\neq
n}^{N}{a_{i}\theta_{i}^{*}c_{i}}.$ (35)

It can be readily verified that
$\mathbf{\Omega}=\mathbf{D}\circ\mathbf{T}^{\operatorname{T}}$ is a positive
semi-definite matrix due to $\mathbf{D}$ and $\mathbf{T}^{\operatorname{T}}$
are semi-definite matrices, and hence, we have
$\Omega_{n,i}=\Omega_{i,n}^{*}$. Similarly, the constraint on the
amplification power budget in (31) can be expressed by
$a_{n}^{2}\left(T_{\operatorname{n,n}}+\delta^{2}\kappa^{2}\right)\leq{\bar{P}}_{\operatorname{ITS}}^{\max}$
where
${\bar{P}}_{\operatorname{ITS}}^{\max}=P_{\operatorname{ITS}}^{\max}-\sum_{i=1,i\neq
n}^{N}{a_{i}^{2}\left(T_{{i,i}}+\delta^{2}\kappa^{2}\right)}$. Clearly, the
objective function $h\left(\mathbf{a},\bm{\theta}\right)$ can be recast as an
equivalent function with respect to the transmissive coefficients of $n$-th
active TE, i.e., $a_{n}$ and $\theta_{n}$, which is given by

$\displaystyle f\left(a_{n},\theta_{n}\right)=$ $\displaystyle
2\operatorname{Re}\left\\{a_{n}\theta_{n}^{*}\left(\sum_{i=1,i\neq
n}^{N}{\Omega_{n,i}a_{i}\theta_{i}}-c_{n}\right)\right\\}+a_{n}^{2}\Omega_{n,n}+\delta^{2}\kappa^{2}{a_{n}^{2}D_{\operatorname{n,n}}}+\eta
a_{n}^{2}\left(T_{\operatorname{n,n}}+\delta^{2}\kappa^{2}\right)+\chi_{n},$
(36)

where $\chi_{n}$ is a constant value with respect to the $n$-th pair of
optimization variables $\left\\{a_{n},\theta_{n}\right\\}$, which is given by

$\displaystyle\chi_{n}=\sum_{j=1,j\neq n}^{N}{\sum_{i=1,i\neq
n}^{N}{a_{j}\theta_{j}^{*}\Omega_{j,i}a_{i}\theta_{i}}}+\delta^{2}\kappa^{2}\sum_{i=1,i\neq
n}^{N}{a_{i}^{2}D_{\operatorname{i,i}}}+2\operatorname{Re}\left\\{\sum_{i=1,i\neq
n}^{N}{a_{i}\theta_{i}^{*}c_{i}}\right\\}+\eta\sum_{i=1,i\neq
n}^{N}{a_{i}^{2}\left(T_{{i,i}}+\delta^{2}\kappa^{2}\right)}.$ (37)

Consequently, we can only investigate the following problem for sequentially
optimizing a pair of values $\left\\{a_{n},\theta_{n}\right\\}$ while fixing
the remaining $N-1$ pairs. By omitting the constant term $\chi_{n}$ in
$f\left(a_{n},\theta_{n}\right)$, which has no impact on optimizing
$\left\\{a_{n},\theta_{n}\right\\}$, and defining $e_{n}=\sum_{i=1,i\neq
n}^{N}{\Omega_{n,i}a_{i}\theta_{i}}-c_{n}$, we have the following surrogate
problem with respect to $\left\\{a_{n},\theta_{n}\right\\}$, and takes the
forms

$\displaystyle\mathcal{P}_{4\text{-}2}:\mathop{\min}_{a_{n},\theta_{n}}\;$
$\displaystyle
2\operatorname{Re}\left\\{a_{n}\theta_{n}^{*}e_{n}\right\\}+a_{n}^{2}\Omega_{n,n}+\delta^{2}\kappa^{2}{a_{n}^{2}D_{\operatorname{n,n}}}+\eta
a_{n}^{2}\left(T_{\operatorname{n,n}}+\delta^{2}\kappa^{2}\right),$ (38a)
$\displaystyle\operatorname{s.t.}\;$ $\displaystyle\eqref{p312}.$ (38b)

Note that $\theta_{n}$ only appear in the constraints of (32c), we decompose
problem $\mathcal{P}_{4\text{-}2}$ into two subproblems as follows

$\displaystyle\mathcal{P}_{4\text{-}3\text{-}1}$
$\displaystyle:\mathop{\min}_{\theta_{n}}\;\operatorname{Re}\left\\{a_{n}\theta_{n}^{*}e_{n}\right\\},\,\operatorname{s.t.}\eqref{p312},$
(39a) $\displaystyle\mathcal{P}_{4\text{-}3\text{-}2}$
$\displaystyle:\mathop{\min}_{a_{n}}\;\eqref{p32},$ (39b)

In particular, an equivalent expression for problem
$\mathcal{P}_{4\text{-}3\text{-}1}$ is given by

$\displaystyle\mathop{\min}_{\varphi_{n}}\quad$
$\displaystyle\cos\left(-\varphi_{n}+\angle e_{n}\right),$ (40a)
$\displaystyle\operatorname{s.t.}\quad$
$\displaystyle\varphi_{n}\in\left[0,2\pi\right).$ (40b)

The optimal solution of the above problem is

$\displaystyle\varphi_{n}=\angle e_{n}-\pi.$ (41)

Note that Problem $\mathcal{P}_{4\text{-}3\text{-}2}$ is a unconstrained
optimization problem, and with the obtained solution of $\varphi_{n}$ in (41),
the value of (40a) can be calculated as $-1$, and hence, the term of
$2\operatorname{Re}\left\\{a_{n}\theta_{n}^{*}e_{n}\right\\}$ in (38a) reduce
to $-2a_{n}$. Consequently, the solution to
$\mathcal{P}_{4\text{-}3\text{-}2}$ is given in a semi-closed-form expression
as follows

$\displaystyle
a_{n}=\frac{1}{\delta^{2}\kappa^{2}D_{n,n}+\Omega_{n,n}+\eta\left(T_{n,n}+\delta^{2}\kappa^{2}\right)}.$
(42)

Based on the above discussions, a pair of
$\left\\{a_{n},\theta_{n}\right\\},n=1,\cdots,N$ can be conducted by
successively optimizing with the other $N-1$ pairs being fixed, and then
repeat until the convergence is attained. The details of the element-wise ASO
algorithm are summarized in Algorithm 3. It can be readily verified Algorithm
3 is guaranteed to converge and obtain a high quality suboptimal solution
[51].

Algorithm 3 Element-wise ASO Algorithm for Calculating $\bm{\Phi}$

1: Initialize $\mathbf{a}^{0}$, $\bm{\theta}^{0}$, and iteration index $l=0$.

2: Let the value of the function in (32a) as
$\rho^{0}\left(\mathbf{a}^{0},\bm{\theta}^{0}\right)$.

3: repeat

4: Sequentially Optimizing: $\mathbf{a}_{n}^{l+1}$ and
$\bm{\varphi}_{n}^{l+1}$, $\forall n\in N$, by using (42) and (41),
respectively;

5: Calculate $\rho^{l+1}\left(\mathbf{a}^{l+1},\bm{\theta}^{l+1}\right)$;

6: until $\left|\rho^{l+1}-\rho^{l}\right|\leq\varepsilon$, output
$\mathbf{a}^{\star}\triangleq\mathbf{a}^{l}$,
$\bm{\varphi}^{\star}\triangleq\bm{\varphi}^{l}$.

###  III-D Convergence and Complexity Analysis

The detailed description of the proposed BCD-based joint precoding algorithm
is summarized in Algorithm 4, we update one of the variables while the others
being fixed. Algorithm 4 guarantees to converge which can be proved as
follows. For each updating step, with the fixed other variables, it can be
readily proved that the value of the objective function
$\mathcal{R}\left(\mathbf{W},\bm{\Phi}\right)$ is monotonically increasing
after each updating step. Additionally, since the power budget constraints at
the BS and active-ITS, $\mathcal{R}$ is upper bounded by a finite value, thus
Algorithm 4 guarantees to converge.

Then, let us now briefly analyze the total computational complexity of
Algorithm 4. The complexities for updating $\mathbf{U}$ and $\mathbf{F}$ in
steps 4 and 5 are $\mathcal{O}\left(KM_{r}^{3}\right)$ and
$\mathcal{O}\left(Ks^{3}\right)$, respectively. For step 6 that update the
linear precoding matrix of the BS by employing Algorithm 2, the main
complexity contribution of Algorithm 2 is calculating the inverse operation,
which is on the order of $\mathcal{O}\left(KM_{t}^{3}\right)$, and the number
of iterations required for employing Algorithm 2 is
$\mathcal{I}_{\operatorname{Dual}}$. The complexity of Algorithm 3 is
$\mathcal{O}\left(N^{2}\right)$ for each iteration, and the number of
iterations is $\mathcal{I}_{\operatorname{ASO}}$. Therefore, the total
complexity of Algorithm 4 is
$\mathcal{O}\left(\mathcal{I}_{\operatorname{BCD}}\left(KM_{r}^{3}+Ks^{3}+\mathcal{I}_{\operatorname{Dual}}KM_{t}^{3}+\mathcal{I}_{\operatorname{ASO}}N^{2}\right)\right)$,
where $\mathcal{I}_{\operatorname{BCD}}$ is the total number of iterations of
Algorithm 4.

Algorithm 4 BCD-based Joint Precoding Algorithm for Solving Problem
$\mathcal{P}_{1}$

1: Initialize $\mathbf{W}^{\left(\operatorname{0}\right)}$,
$\mathbf{A}^{\left(\operatorname{0}\right)}$,
$\bm{\Theta}^{\left(\operatorname{0}\right)}$,and iteration index $t=0$.

2: repeat

3: Calculate
$\bm{\Phi}^{\left(\operatorname{t}\right)}=\mathbf{A}^{\left(\operatorname{t}\right)}\circ\bm{\Theta}^{\left(\operatorname{t}\right)}$;

4: Updating $\mathbf{U}^{\left(\operatorname{t+1}\right)}$ by using (13);

5: Updating $\mathbf{F}^{\left(\operatorname{t+1}\right)}$, by using (14);

6: Updating $\mathbf{W}^{\left(\operatorname{t+1}\right)}$, by employing
Algorithm 2;

7: Updating $\bm{\Theta}^{\left(\operatorname{t+1}\right)}$ and
$\mathbf{A}^{\left(\operatorname{t+1}\right)}$, by employing Algorithm 3;

8: until
${{\left|{{\mathcal{R}^{\left(\operatorname{t+1}\right)}}-{\mathcal{R}^{\left(\operatorname{t}\right)}}}\right|}}<\varepsilon$,
output
$\mathbf{W}^{\star}\triangleq\mathbf{W}^{\left(\operatorname{t}\right)}$,
$\mathbf{A}^{\star}\triangleq\mathbf{A}^{\left(\operatorname{t}\right)}$, and
$\bm{\Theta}^{\star}\triangleq\bm{\Theta}^{\left(\operatorname{t}\right)}$.

##  IV The Proposed Block-Amplifying architecture of active-ITS

In this section, we provide a block-amplifying architecture of active-ITS to
partially remove the circuit components for power-amplifying, which is
beneficial for reducing the surface size and hardware cost. More specifically,
as illustrated in Fig. 2, the total active TEs are divided into several
blocks, and the TEs in each block are assumed to share a same power amplifier.
Compared with element-amplifying architecture, the block-amplifying
architecture can reduce the scale of the power-amplifying circuit, which makes
it further suitable for application to the size-limited scenario. In addition,
by partially removing the circuit components for power amplifying, the total
power consumption of the block-amplifying architecture reduce to
$NP_{\operatorname{SW}}+RP_{\operatorname{DC}}$, where $R$ ($R\leq N$) denotes
the number of blocks, $P_{\operatorname{SW}}$ represent the power consumption
for switching phase-shifting of each TE, and $P_{\operatorname{DC}}$ is the
direct current biasing power for power-amplifying of each block [36]. Clearly,
the power consumption of the block-amplifying architecture is less than that
of the element-amplifying architecture, i.e.,
$N\left(P_{\operatorname{SW}}+P_{\operatorname{DC}}\right)$, which means that
the more power will be used to perform the amplitude-magnifying, while it can
compensate for some inevitable performance loss caused by such a architecture.
For example, we consider a block-amplifying ITS with $N=60$ ,$R=10$,
$P_{ITS}^{\max}=15$ dBm (31.6 mW), and follow the setting in [34, 36] that
$P_{\operatorname{SW}}=-10$ dBm (0.1 mW) and $P_{\operatorname{DC}}=-5$ dBm
(0.316 mW), and hence the total power consumption of the active-ITS for the
element-amplifying architecture and block-amplifying architecture are 24.96 mW
and 9.16 mW, and corresponding the power used for performing amplitude-
magnifying are $6.64$ mW and $22.44$ mW, respectively.

Fig. 2: Block-amplifying architecture of active-ITS.

Note that the variables $\mathbf{U}$, $\mathbf{F}$, $\mathbf{W}$, and
$\mathbf{\Theta}$ can be directly obtained as that under element-amplifying
architecture case, and hence, the remaining work is to optimize the power
amplifying factor matrix, i.e., $\mathbf{A}$, under the block-amplifying
architecture case. In the following, we extend the proposed ASO algorithm from
the element-wise form into the block-wise form, which can optimize
$\mathbf{A}$ in a block-by-block manner.

Remark 2: From a mathematical point of view, since the element-amplifying
architecture can be interpreted as a special case ($R=N$) of the block-
amplifying architecture, the proposed ASO Algorithm under the block-wise form
is a superset of the element-wise form.

Particularly, we partition the power amplifying factor $\mathbf{a}$ into $R$
blocks, i.e.,
$\mathbf{a}=\left[\mathbf{a}_{1}^{\operatorname{T}},\mathbf{a}_{2}^{\operatorname{T}},\cdots,\mathbf{a}_{R}^{\operatorname{T}}\right]\in\mathbb{R}^{N\times
1}$, where $\mathbf{a}_{r}\in\mathbb{R}^{{\hat{N}}_{r}\times 1}$ is the power
amplifying factor vector corresponding to ${\hat{N}}_{r}$ TEs in the $r$-th
block, which is given by

$\displaystyle\mathbf{a}_{r}=\left[{{\mathbf{a}}_{r}\left(1\right),{\mathbf{a}}_{r}\left(2\right),\cdots,{\mathbf{a}}_{r}\left({\hat{N}}_{r}\right)}\right]^{\operatorname{T}}={a}_{r}\bm{1}_{{\hat{N}}_{r}}^{\operatorname{T}},$
(43)

where $\mathbf{a}_{r}\left({m}\right)$ is the
$\left(\sum_{l=1}^{r-1}{{\hat{N}}_{l}}+m\right)$-th element of $\mathbf{a}$
with $r=2,3,\cdots,R$ and $m=1,2,\cdots,{\hat{N}}_{r}$.

Here, let us revisit problem $\mathcal{P}_{4\text{-}1}$ stated in (32). First,
$\left(\mathbf{a}^{\operatorname{T}}\circ\bm{\theta}^{\operatorname{H}}\right)\mathbf{\Omega}\left(\mathbf{a}\circ\bm{\theta}\right)$
can be expanded as

$\displaystyle\left(\mathbf{a}^{\operatorname{T}}\circ\bm{\theta}^{\operatorname{H}}\right)\mathbf{\Omega}\left(\mathbf{a}\circ\bm{\theta}\right)$
$\displaystyle=$ $\displaystyle\sum_{i=1,i\neq
r}^{R}{\left(\mathbf{a}^{\operatorname{T}}\circ\bm{\theta}^{\operatorname{H}}\right)\bm{\Omega}_{i}\left(\mathbf{a}_{i}\circ\bm{\theta}_{i}\right)}+\left(\mathbf{a}^{\operatorname{T}}\circ\bm{\theta}^{\operatorname{H}}\right)\bm{\Omega}_{r}\left(\mathbf{a}_{r}\circ\bm{\theta}_{r}\right)$
$\displaystyle=$ $\displaystyle\sum_{i=1,i\neq
r}^{R}{\left(\mathbf{a}_{i}^{\operatorname{T}}\circ\bm{\theta}_{i}^{\operatorname{H}}\right)\bm{\Omega}_{i,r}\left(\mathbf{a}_{r}\circ\bm{\theta}_{r}\right)}+\sum_{i=1,i\neq
r}^{R}{\sum_{j=1}^{R}{\left(\mathbf{a}_{j}^{\operatorname{T}}\circ\bm{\theta}_{j}^{\operatorname{H}}\right)\bm{\Omega}_{j,i}\left(\mathbf{a}_{i}\circ\bm{\theta}_{i}\right)}}+\left(\mathbf{a}_{r}^{\operatorname{T}}\circ\bm{\theta}_{r}^{\operatorname{H}}\right)\bm{\Omega}_{r,r}\left(\mathbf{a}_{r}\circ\bm{\theta}_{r}\right),$
$\displaystyle=$ $\displaystyle\sum_{i=1,i\neq
r}^{R}{\left(\left(\mathbf{a}_{i}^{\operatorname{T}}\circ\bm{\theta}_{i}^{\operatorname{H}}\right)\bm{\Omega}_{i,r}\left(\mathbf{a}_{r}\circ\bm{\theta}_{r}\right)+{\left(\mathbf{a}_{i}^{\operatorname{T}}\circ\bm{\theta}_{i}^{\operatorname{H}}\right)\bm{\Omega}_{i,r}\left(\mathbf{a}_{r}\circ\bm{\theta}_{r}\right)}\right)}$
$\displaystyle+\sum_{i=1,i\neq
r}^{R}{\sum_{j=1}^{R}{\left(\mathbf{a}_{j}^{\operatorname{T}}\circ\bm{\theta}_{j}^{\operatorname{H}}\right)\bm{\Omega}_{j,i}\left(\mathbf{a}_{i}\circ\bm{\theta}_{i}\right)}}+\left(\mathbf{a}_{r}^{\operatorname{T}}\circ\bm{\theta}_{r}^{\operatorname{H}}\right)\bm{\Omega}_{r,r}\left(\mathbf{a}_{r}\circ\bm{\theta}_{r}\right),$
(44)

where
$\bm{\Omega}=\left[\bm{\Omega}_{1},\bm{\Omega}_{2},\cdots,\bm{\Omega}_{R}\right]$
and
$\bm{\theta}=\left[\bm{\theta}_{1}^{\operatorname{T}},\bm{\theta}_{2}^{\operatorname{T}},\cdots,\bm{\theta}_{R}^{\operatorname{T}}\right]^{\operatorname{T}}$,
with

$\displaystyle\bm{\Omega}_{r}$
$\displaystyle=\left[\bm{\Omega}_{1,r},\bm{\Omega}_{2,r},\cdots,\bm{\Omega}_{R,r}\right]^{\operatorname{T}}\in\mathbb{C}^{N\times{\hat{N}}_{r}},$
(45a) $\displaystyle\bm{\theta}_{r}$
$\displaystyle=\left[\bm{\theta}_{r}\left(1\right),\bm{\theta}_{r}\left(2\right),\cdots,\bm{\theta}_{r\left({{\hat{N}}_{r}}\right)}\right]^{\operatorname{T}}\in\mathbb{C}^{{\hat{N}}_{r}\times
1},$ (45b)

where $\bm{\theta}_{r}\left({m}\right)$ denotes the
$\left(\sum_{l=1}^{r-1}{{\hat{N}}_{l}}+m\right)$-th element of $\bm{\theta}$
with $r=2,3,\cdots,R$ and $m=1,2,\cdots,{\hat{N}}_{r}$.

By employing the property
$\bm{\Omega}_{i,r}=\bm{\Omega}_{i,r}^{\operatorname{H}}$ , (IV) can be
rewritten as

$\displaystyle\left(\mathbf{a}^{\operatorname{T}}\circ\bm{\theta}^{\operatorname{H}}\right)\mathbf{\Omega}\left(\mathbf{a}\circ\bm{\theta}\right)$
$\displaystyle=$ $\displaystyle 2\operatorname{Re}\left\\{\sum_{i=1,i\neq
r}^{R}{\left(\mathbf{a}_{i}^{\operatorname{T}}\circ\bm{\theta}_{i}^{\operatorname{H}}\right)\bm{\Omega}_{i,r}\left(\mathbf{a}_{r}\circ\bm{\theta}_{r}\right)}\right\\}+\sum_{i=1,i\neq
r}^{R}{\sum_{j=1}^{R}{\left(\mathbf{a}_{j}^{\operatorname{T}}\circ\bm{\theta}_{j}^{\operatorname{H}}\right)\bm{\Omega}_{j,i}\left(\mathbf{a}_{i}\circ\bm{\theta}_{i}\right)}}$
$\displaystyle+\left(\mathbf{a}_{r}^{\operatorname{T}}\circ\bm{\theta}_{r}^{\operatorname{H}}\right)\bm{\Omega}_{r,r}\left(\mathbf{a}_{r}\circ\bm{\theta}_{r}\right).$
(46)

Then, note that the term $\sum_{i=1,i\neq
r}^{R}{\left(\mathbf{a}_{i}^{\operatorname{T}}\circ\bm{\theta}_{i}^{\operatorname{H}}\right)\bm{\Omega}_{i,r}\left(\mathbf{a}_{r}\circ\bm{\theta}_{r}\right)}$
in (IV) can be expanded by

$\displaystyle\sum_{i=1,i\neq
r}^{R}{\left(\mathbf{a}_{i}^{\operatorname{T}}\circ\bm{\theta}_{i}^{\operatorname{H}}\right)\bm{\Omega}_{i,r}\left(\mathbf{a}_{r}\circ\bm{\theta}_{r}\right)}=$
$\displaystyle\sum_{i=1,i\neq
r}^{R}{a_{r}a_{i}\left(\bm{1}_{{\hat{N}}_{r}}^{\operatorname{T}}\circ\bm{\theta}_{r}^{\operatorname{H}}\right)\bm{\Omega}_{r,i}\left(\bm{1}_{{\hat{N}}_{i}}\circ\bm{\theta}_{i}\right)}$
$\displaystyle=$ $\displaystyle\sum_{i=1,i\neq
r}^{R}{a_{r}a_{i}\bm{\theta}_{r}^{\operatorname{H}}\bm{\Omega}_{r,i}\bm{\theta}_{i}}.$
(47)

Similarly, we have

$\displaystyle\left(\mathbf{a}_{r}^{\operatorname{T}}\circ\bm{\theta}_{r}^{\operatorname{H}}\right)\bm{\Omega}_{r,r}\left(\mathbf{a}_{r}\circ\bm{\theta}_{r}\right)$
$\displaystyle=a_{r}^{2}\bm{\theta}_{r}^{\operatorname{H}}\bm{\Omega}_{r,r}\bm{\theta}_{r},$
(48a)
$\displaystyle\left(\mathbf{a}^{\operatorname{T}}\circ\bm{\theta}^{\operatorname{H}}\right)\mathbf{c}$
$\displaystyle=a_{r}\bm{\theta}_{r}^{\operatorname{H}}\mathbf{c}_{r}+\sum\nolimits_{i=1,i\neq
r}^{R}{a_{i}\bm{\theta}_{i}^{\operatorname{H}}\mathbf{c}_{i}}.$ (48b)

By defining $\mathbf{d}=\operatorname{Vecd}\left(\mathbf{D}\right)$ and
$\mathbf{t}=\operatorname{Vecd}\left(\mathbf{T}\right)$, and then partition
the two vector into $R$ blocks, i.e.,
$\mathbf{d}=\left[\mathbf{d}_{1}^{\operatorname{T}},\mathbf{d}_{2}^{\operatorname{T}},\cdots,\mathbf{d}_{R}^{\operatorname{T}}\right]\in\mathbb{C}^{N\times
1}$ and
$\mathbf{t}=\left[\mathbf{t}_{1}^{\operatorname{T}},\mathbf{t}_{2}^{\operatorname{T}},\cdots,\mathbf{t}_{R}^{\operatorname{T}}\right]\in\mathbb{C}^{N\times
1}$, we have

$\displaystyle\operatorname{Tr}\left(\bm{\Phi}^{\operatorname{H}}\mathbf{D}\bm{\Phi}\right)$
$\displaystyle=a_{r}^{2}\sum_{p=1}^{{\hat{N}}_{r}}{\mathbf{d}_{r}\left(p\right)}+\sum_{i=1,i\neq
r}^{R}{a_{i}^{2}\sum_{q=1}^{{\hat{N}}_{i}}{\mathbf{d}_{i}\left(q\right)}},$
(49a)
$\displaystyle\operatorname{Tr}\left(\bm{\Phi}^{\operatorname{H}}\bm{\Phi}\mathbf{T}\right)$
$\displaystyle=a_{r}^{2}\sum_{p=1}^{{\hat{N}}_{r}}{\mathbf{t}_{r}\left(p\right)}+\sum_{i=1,i\neq
r}^{R}{a_{i}^{2}\sum_{q=1}^{{\hat{N}}_{i}}{\mathbf{t}_{i}\left(q\right)}}.$
(49b)

Similarly, we have

$\displaystyle\operatorname{Tr}\left(\bm{\Phi}^{\operatorname{H}}\bm{\Phi}\right)$
$\displaystyle=a_{r}^{2}{\hat{N}}_{r}+\sum\nolimits_{i=1,i\neq
r}^{R}{a_{i}^{2}{{\hat{N}}_{i}}}.$ (50)

Consequently, problem $\mathcal{P}_{4\text{-}1}$ can be reformulated by

$\displaystyle\mathcal{P}_{5}:\mathop{\min}_{a_{r}}\;$ $\displaystyle
f\left(a_{r}\right)=a_{r}^{2}m_{r}+2\operatorname{Re}\left\\{a_{r}z_{r}\right\\}+\chi_{r},$
(51a) $\displaystyle\operatorname{s.t.}\;$ $\displaystyle
a_{r}^{2}g_{r}\leq{\hat{P}}_{\operatorname{ITS}}^{\max},$ (51b)

where

$\displaystyle m_{r}$
$\displaystyle=\bm{\theta}_{r}^{\operatorname{H}}\bm{\Omega}_{r,r}\bm{\theta}_{r}+\delta^{2}\kappa^{2}\sum_{p=1}^{{\hat{N}}_{r}}{\mathbf{d}_{r}\left(p\right)},$
(52a) $\displaystyle z_{r}$ $\displaystyle=\sum_{i=1,i\neq
r}^{R}{a_{i}\bm{\theta}_{r}^{\operatorname{H}}\bm{\Omega}_{r,i}\bm{\theta}_{i}}-\bm{\theta}_{r}^{\operatorname{H}}\mathbf{c}_{r},$
(52b) $\displaystyle g_{r}$
$\displaystyle=\sum_{p=1}^{{\hat{N}}_{r}}{\mathbf{t}_{r}\left(p\right)}+\kappa^{2}\delta^{2}{\hat{N}}_{r},$
(52c) $\displaystyle{\hat{P}}_{\operatorname{ITS}}^{\max}$
$\displaystyle=P_{\operatorname{ITS}}^{\max}-\sum_{i=1,i\neq
r}^{R}{a_{i}^{2}\sum_{q=1}^{{\hat{N}}_{i}}{\mathbf{t}_{i}\left(q\right)}}-\kappa^{2}\delta^{2}\sum_{i=1,i\neq
r}^{R}{a_{i}^{2}{{\hat{N}}_{i}}},$ (52d)

and where $\chi_{r}$ denotes a constant value with respect to $a_{r}$ and does
not affect its solution, and is given by

$\displaystyle\chi_{r}=\sum_{i=1,i\neq
r}^{R}{\sum_{j=1}^{R}{a_{j}a_{i}\bm{\theta}_{j}^{\operatorname{H}}\bm{\Omega}_{j,i}\bm{\theta}_{i}}}+2\operatorname{Re}\left\\{\sum_{i=1,i\neq
r}^{R}{a_{i}\bm{\theta}_{i}^{\operatorname{H}}\mathbf{c}_{i}}\right\\}+\kappa^{2}\delta^{2}\sum_{i=1,i\neq
r}^{R}{a_{i}^{2}\sum_{q=1}^{{\hat{N}}_{i}}{\mathbf{d}_{i}\left(q\right)}}.$
(53)

The above problem can be solved similarly to (42) and hence it is omitted for
brevity.

Based on the above discussions, $\mathbf{a}_{r},\forall r$ can be conducted by
successively optimizing with the other $R-1$ blocks being fixed, and then
repeat until the convergence is attained [51].

##  V Simulation Result

Fig. 3: The simulated active-ITS empowered outdoor-to-indoor mmWave
communication.

In this section, numerical results are provided to evaluate the WSR
performance achieved by the proposed BCD-based joint precoding algorithm under
the two architectures of active-ITSs.

###  V-A Simulation Setup

We consider a three-dimensional setup as illustrated in Fig. 3, where the BS
at the outdoor communicates with UEs at the indoor with the aid of an active-
ITS. The BS, active-ITS, and UEs are assumed in a straight line, and heights
of them are 3m, 6m, and 1.5m, respectively, and the total horizontal distances
$d_{\text{BU}}=d_{\text{BI}}+d_{\text{IU}}=50$ m. The $K=4$ UEs are uniformly
and randomly distributed in a circle centered at
$(0\operatorname{m},50\operatorname{m})$ with a radius of $5$ m.

For array response vector, the azimuth angles of arrival
$\upsilon^{\operatorname{AOA}}$ and departure $\upsilon^{\operatorname{AOD}}$
are uniformly distributed in the interval $\left[-\pi,\pi\right)$. The
elevation angles of arrival $\vartheta^{\operatorname{AOA}}$ and departure
$\vartheta^{\operatorname{AOD}}$ are uniformly distributed in the interval
$\left[-\pi/2,\pi/2\right)$. The complex gains of LOS paths
$\alpha_{l},\forall l$ and $\beta_{p},\forall p$ are independently distributed
with $\mathcal{C}\mathcal{N}\left(0,10^{-0.1{\operatorname{PL}}}\right)$,
where the distance-dependent path loss $\operatorname{PL}$ for both the LoS
and the NLoS paths can be modeled by

$\displaystyle\operatorname{PL}=\operatorname{PL_{0}}+10b\log_{10}\left(D\right)+\zeta\,\text{dB},$
(54)

where $D$ denotes the individual link distance and $\zeta$ denotes lognormal
shadow fading following
$\zeta\sim\mathcal{N}\left(0,\sigma_{\zeta}^{2}\right)$. As the real-world
channel measurements for the carrier frequency of 28 GHz suggested in [3,
Table I], the values of the parameters in (54) are set as
$\operatorname{PL_{0}}=61.4$, $b=2$, and $\sigma_{\zeta}=5.8$ dB for the LoS
paths, and $\operatorname{PL_{0}}=72$, $b=2.92$, and $\sigma_{\zeta}=8.7$ dB
for NLoS paths. The bandwidth $B=251$ MHz, the noise power is
$\sigma^{2}=-174+10\log_{10}B=-90$ dBm. Unless otherwise stated, the other
adopted simulation parameters are set as follows: the distance between the BS
and ITS of $d_{\operatorname{BI}}=45$ m, the circuit power consumption for
phase-shifting and power-amplifying are $P_{\operatorname{SW}}=-10$ dBm and
$P_{\operatorname{DC}}=-5$ dBm [34], the power budgets of the BS and active-
ITS are $P_{\operatorname{BS}}^{\max}=30$ dBm and
$P_{\operatorname{ITS}}^{\max}=30$ dBm, the number of TEs of $N=10\times 4$,
the number of antennas of $M_{t}=4\times 2$ and $M_{r}=2\times 2$, the
weighting factor of $\alpha_{k}=1,\forall k$, the penetration efficiency of
$\kappa=0.8$, and the target accuracy of $\varepsilon=10^{-3}$.

Without loss of generality, for the block-amplifying ITS architecture, we
adopt an evenly-block strategy, where the number of TEs in each block is equal
and set as 5 and 10, i.e., “Block-amplifying ITS, $N_{B}=5$” and “Block-
amplifying ITS, $N_{B}=10$”. We also introduce a baseline scheme “Block-
amplifying ITS, $N_{B}=N$”, where all TEs of ITS share the same power
amplifier, so all the active TEs have identical amplitudes, i.e.,
$a_{n}=a,\forall n\in N$ [36]. In addition, due to that the active-ITS
requires an additional power, i.e., $P_{\operatorname{ITS}}^{\max}$, and
hence, for fair comparisons, we assume that the power budget at the BS for the
passive-ITS scheme (referred as “Without-amplifying ITS”) is allocated an
additional amount of $P_{\operatorname{ITS}}^{\max}$, i.e.,
$P_{\operatorname{BS,passive}}^{\max}=P_{\operatorname{ITS}}^{\max}+P_{\operatorname{BS,active}}^{\max}$,
so that all the compared schemes have the same total power consumption.

Fig. 4: WSR against the number of iterations, $\mathcal{I}_{BCD}$.

###  V-B Convergence Behavior

In this subsection, we investigate the convergence behavior of the proposed
BCD-based algorithm for the different architectures of active-ITSs with
different numbers of TEs. As shown in Fig. 4, we plot the curves of the WSR
performance achieved by the proposed algorithm for the active-ITS schemes
(include Element-amplifying ITS and Block-amplifying ITS) against the number
of iterations. In particular, it can be observed that the curve with $N=60$
TEs converges slower than that of $N=40$ TEs, and the curves of both the
element-amplifying architecture and the block-amplifying architecture converge
to corresponding stationary points after a few iterations. The observation is
consistent with our expectations. It can be concluded that the proposed
algorithm has good convergence behavior and the convergence speed is sensitive
to the number of TEs of an active-ITS, and the impact of $N_{B}$ on the
convergence speed is slight.

###  V-C The impact of key parameters on WSR performances

In this subsection, we study the WSR performance achieved by the proposed
algorithms for both the active-ITS scheme and the passive-ITS scheme (i.e.,
Without-amplifying ITS) versus the key parameters of the considered system
setting. All the simulation results as follows are averaged over 1,000
independent channel realizations.

Fig. 5: WSR against the position of the active-ITS, $d_{\operatorname{BI}}$.

In Fig. 5, we investigate the WSR performance for different locations of the
ITS by varying the distance between the BS and ITS, i.e.,
$d_{\operatorname{BI}}$. One important observation is that with increasing
$d_{\operatorname{BI}}$, the WSR performance achieved by the active-ITS scheme
monotonically increase, which is different from the passive-ITS scheme, which
achieve the highest and the worst WSR performances when passive-ITS are
deployed close to the BS(UEs) and at the middle location due to the
multiplicative fading effect. This interesting observation indicates that the
active-ITS should to be deployed closer to UEs for fully unleashing its
potential on achieving a better WSR performance. Intuitively speaking, the
deployment strategy meets the requirements of a practical outdoor-to-indoor
communication scenario where the active-ITS is usually very close to the UEs,
i.e., $d_{\operatorname{IU}}\ll d_{\operatorname{BI}}$. In addition, it can be
observed that the active-ITS schemes perform constantly better than passive-
ITS scheme in all locations, which implies that the active-ITS brings a
significant performance gain compared with the passive-ITS due to the
amplified signal is only attenuated once. More importantly, it can be seen
that the WSR of “Element-amplifying ITS” scheme outperforms that of the
“Block-amplifying ITS” schemes, which can be attributed to that the block-
amplifying architecture of active-ITS inevitably lost some DoF for proactively
configuring the wireless channels. This observation indicates the importance
of carefully optimizing the power amplification factor matrix at the active-
ITS for enhancing the WSR performance. Nevertheless, considering that such a
block-amplifying architecture of active-ITS can reduce the scale of the power-
amplifying circuit for application to space-limited scenarios, the resulting
slight performance gap is acceptable.

The WSR performances versus the number of TEs with
$P_{\operatorname{ITS}}^{\max}=30$ dBm and $P_{\operatorname{ITS}}^{\max}=20$
dBm are shown in Fig. 6. It illustrates that with the increasing number of
TEs, both curves of the active-ITS and the passive-ITS schemes monotonically
increases, since more TEs introduce more DoFs to proactively configure the
wireless channel which yields a higher WSR performance. Meanwhile, it also
illustrates the superiority of the active-ITS that brings a large benefit to
WSR performance compared with the passive-ITS. Particularly, the usage of
active ITSs can greatly reduce the number of TEs compared with passive ITS
case for reaping a given performance level, and hence greatly decrease both
the size and the complexity of ITS, which is beneficial for embedding in
space-limited and aesthetic-needed building structures. It can be explained by
that when equipped a small number of active TEs in the active-ITS, according
to the constraint in (9c), a larger power amplification factor will be
allocated at each active TE. In addition, it can be observed that with a small
number of TEs, the performance degradation between element-amplifying ITS and
block-amplifying ITS is slight with $P_{\operatorname{ITS}}^{\max}=20$ dBm,
while the performance gap in the large $N$ regime also is acceptable. This
observation is because that for a large number of TEs at the active-ITS, the
block-amplifying architecture has more power to amplify the signal, which can
compensate for some of the loss of performance. This observation highlights
the effectiveness of the proposed block-amplifying architecture.

Fig. 6: WSR against the number of TEs.

We plot the WSR performance against the amplifying power budget at ITS in Fig.
7. As expected, it can be observed that all the WSR performances of all
schemes increase as $P_{\operatorname{ITS}}^{\max}$ increase. More
specifically, the WSR performances achieved by the active-ITS schemes
outperform that of passive-ITS, which indicates that the active-ITS can
exploit a part of the power for fighting with the multiplicative path loss
effect influence and achieving a higher WSR performance than the passive-ITS
scheme for the same power consumption. In addition, it can be concluded from
Fig. 7 that the active-ITS scheme can significantly reduce the power
consumption compared to the passive-ITS scheme when achieving a given
performance level. More importantly, it can be observed that the performance
degradation caused by the block-amplifying architecture of the active-ITS is
slight when $P_{ITS}^{\max}=15$ dBm, which is because that the more power is
used for the circuit static power consumption in the small $P_{ITS}^{\max}$
regime.

Fig. 7: WSR against the amplifying power budget
$P_{\operatorname{ITS}}^{\max}$ at the active-ITS. Fig. 8: WSR against the
refracting efficiency $\kappa$.

Fig. 8 illustrates the WSR performance against the refracting efficiency of
ITSs. It can be observed that the refracting efficiency of ITS has a
substantial impact on the WSR performance, whereas expected, with the
increasing $\kappa$, the WSR achieved by the active-ITS schemes increased
significantly. The refracting efficiency of an ITS is determined by the
hardware components, which is used to characterize the power loss caused by
signal absorption and signal reflection when the mmWave signal impinges upon
and penetrate the ITS. This figure demonstrates the benefits of active-ITS
that can exploit the additional power amplifiers to compensate for the power
loss caused by the practical hardware component.

##  VI Conclusion

In this paper, an active-ITS empowered outdoor-to-indoor mmWave communication
system was investigated, where the active-ITS benefited on greatly reducing
the number of TEs while maintaining a given performance compared with the
passive-ITS. To jointly optimize the linear precoding matrix of the BS and
transmissive coefficients of the active-ITS, we formulated a WSR maximization
problem, and then a BCD-based joint precoding algorithm was proposed for
solving it. In order to further reduce the size and hardware cost of active-
ITS, a block-amplifying architecture was proposed to partially remove the
hardware components for power amplifying. And then we extended the proposed
BCD-based joint precoding algorithm into the block-wise form for optimizing
the transmissive coefficients of the active-ITS under the block-amplifying
architecture. Simulation results demonstrated that the active-ITS could
significantly enhance the system performance, and the inevitable performance
degradation caused by the block-amplifying ITS architecture was acceptable.

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