# The Optimal Pilot Power Allocation Strategy for Multi-IRS Assisted
Communication Systems

Jiancheng An, Chao Xu, Lu Gan,
Chau Yuen, and Lajos Hanzo L. Hanzo would like to acknowledge the financial
support of the Engineering and Physical Sciences Research Council projects
EP/N004558/1, EP/P034284/1, EP/P034284/1, EP/P003990/1 (COALESCE), of the
Royal Society’s Global Challenges Research Fund Grant as well as of the
European Research Council’s Advanced Fellow Grant QuantCom.J. An and L. Gan
are with the School of Information and Communication Engineering, University
of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China.
(e-mail<EMAIL_ADDRESS>ganlu@uestc.edu.cn).C. Xu and L. Hanzo are
with the School of Electronics and Computer Science, University of
Southampton, Southampton, SO17 1BJ, U.K. (e-mail<EMAIL_ADDRESS>lh@ecs.soton.ac.uk).C. Yuen is with the Engineering Product Development (EPD)
Pillar, Singapore University of Technology and Design, Singapore 487372
(e-mail: yuenchau@sutd.edu.sg).

###### Abstract

Intelligent reflecting surface (IRS) is a promising technology that enables
the precise control of the electromagnetic environment in future wireless
communication networks. To leverage the IRS effectively, the acquisition of
channel state information (CSI) is crucial in IRS-assisted communication
systems, which, however, is challenging. In this paper, we propose the optimal
pilot power allocation strategy for the channel estimation of IRS-assisted
communication systems, which is capable of further improving the achievable
rate performance with imperfect CSI. More specifically, _first of all_ , we
introduce a multi-IRS assisted communication system in the face of practical
channel estimation errors. Furthermore, the ergodic capacity with imperfect
CSI is derived in an explicit closed-form expression under the single-input
single-output (SISO) consideration, which is dependent on the specific pilot
power for estimating different reflection channels. _Secondly_ , we formulate
the optimization problem of maximizing the ergodic capacity with imperfect
CSI, subject to the constraint of the average uplink pilot power. _Thirdly_ ,
the method of Lagrange multipliers is invoked to solve the ergodic rate
maximizing problem and thus to obtain the optimal pilot power allocation
strategy. The resultant pilot power allocation solution suggests allocating
more amount of power to the pilots for estimating the weak reflection
channels, which is counter-intuition but effectively improve the ergodic
capacity with imperfect CSI. Besides, we also elaborate on the expense of the
proposed pilot power allocation strategy upon analyzing the peak-to-average-
power-ratio (PAPR) increase quantitatively. _Finally_ , the extensive
simulation results verify our analysis and reveal some interesting results.
For example, for the user in the vicinity of a large IRS, it is suggested to
switch off other IRSs and only switch on the IRS nearest the user. On the
contrary, for the user near a small IRS, it is better to switch on all IRSs
and perform the optimal pilot power allocation for enhancing the achievable
rate performance.

###### Index Terms:

Intelligent reflecting surface (IRS), reconfigurable intelligent surface
(RIS), multiple IRSs, channel estimation, power allocation, ergodic channel
capacity, imperfect channel state information (CSI).

##  I Introduction

Recently, intelligent reflecting surface (IRS) [1] and its variants, e.g.,
reconfigurable intelligent surface (RIS) [2, 3], large intelligent
surface/antenna (LISA) [4, 5], have arisen as a promising technology to build
the future spectral- and energy-efficient wireless networks in a cost-
efficient manner [6, 7]. Specifically, IRS is a programmable metasurface
consisted of a huge number of passive reflecting elements, each of which can
alter the incident signals by imposing an independent magnitude and/or phase
shift [1, 8]. As a result, the propagation environment between transmitters
and receivers can be customized according to the specific quality-of-service
(QoS) requirements, which brings new degree-of-freedom for enhancing the
wireless channel capacity and mitigating the inter-channel interference [9].
By contrast to the conventional cooperative communications with active relays,
the IRS elements passively reflect the incident signals without requiring any
radio-frequency (RF) chains [10, 11]. The recollection of the diffused energy
in the air makes IRS an excellent option for green communications [12, 13].
Besides, IRS is capable of operating in the full-duplex (FD) mode without
incurring the self-interference as in the traditional FD relay [1, 14].
Moreover, the low-cost material and patch structure render the IRS can be
easily attached to the environment objects, e.g., the walls, ceilings,
building surfaces [9, 15]. Thanks to these potential benefits, it is expected
to deploy massive IRSs in wireless networks to facilitate the future 6G
ecosystem [16, 7, 17].

TABLE I:  Comparison of the proposed scheme to existing research results on IRS-assisted systems. | $\star$ |  [18]-2020 |  [19]-2020 |  [20]-2020 |  [21]-2020 |  [22]-2019 |  [23]-2020 |  [24]-2019 |  [25]-2020
---|---|---|---|---|---|---|---|---|---
Multiple IRSs | $\checkmark$ | — | $\checkmark$ | — | — | — | — | $\checkmark$ | —
Ergodic capacity | $\checkmark$ | $\checkmark$ | — | $\checkmark$ | $\checkmark$ | $\checkmark$ | — | — | $\checkmark$
With imperfect CSI | $\checkmark$ | — | — | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | — | $\checkmark$
Pilot power allocation | $\checkmark$ | — | — | — | — | — | — | — | $\checkmark$
IRS selection | $\checkmark$ | — | — | — | — | — | — | — | —
System setup | SISO | SISO | MU-MISO | SISO | SISO | MIMO | MU-MISO | SISO | SISO
Phase-shift model | Continuous | Continuous | Continuous | Continuous | Discrete | Continuous | Continuous | Continuous | Continuous

Nevertheless, the integration of IRS into existing wireless networks faces new
challenges, one of which is the optimization of the IRS reflection
coefficients for facilitating the coherent superposition of dispersed intended
signals and/or interference cancellation. Specifically, the optimization of
the IRS reflection coefficients is generally coupled with the transmit
beamforming design, which is different from the conventional systems without
IRS [26]. In [10], _Wu et al._ considered the IRS-assisted multi-input single-
output (MISO) and multi-user MISO systems by formulating the transmit power
minimizing problem, subject to the individual signal-to-interference-plus-
noise-ratio (SINR) requirements at each user. The formulated problem is then
solved by semidefinite relaxation (SDR) and alternating optimization (AO)
algorithms to obtain the high-quality approximate solution of the transmit
beamforming vectors for multiple users and reflection coefficients at the IRS.
Besides, [10] demonstrated that the IRS behaves quadratic power law with the
number of reflecting elements, which is promising for energy-efficient
deployment [12]. Soon afterwards, [27] extended the research results of [10]
upon considering the practical IRS elements with discrete phase shifters.
Moreover, [27] showed that, for the IRS equipped with a large number of
reflecting elements, the power loss at the receiver relative to that equipped
continuous phase shifters is about $-3.9$ dB for $1$-bit quantized phase
shifter and only $-0.9$ dB for $2$-bit quantized phase shifter. Furthermore,
[28] investigated the capacity of IRS-assisted point-to-point multi-input
multi-output (MIMO) systems, where the IRS reflection coefficients and
transmit precoding are optimized alternatively by assuming that the other is
given. Following this, the IRS assisted multi-user and multi-cell MIMO systems
are reported in [29] and [30], respectively, where the rate maximizing problem
is solved by AO algorithm and fractional programming technique. The energy-
efficiency maximization for IRS assisted multi-user MISO system is
investigated in [31], where the AO algorithm assisted by the gradient descent
and majorization-minimization (MM) algorithms are invoked to solve the
formulated problem. Recently, the optimization of reflection coefficients at
IRS has been performed under the joint consideration with other existing
techniques, e.g., millimeter-wave (mmWave) communications [32], terahertz
(THz) communications [33, 34], visible light communication (VLC) [35],
orthogonal frequency division multiplexing (OFDM) [36], deep learning (DL)
[37, 38], unmanned aerial vehicle (UAV) [39, 40], index modulation (IM) [41],
non-orthogonal multiple access (NOMA) [42, 43], physical layer security (PLS)
[44, 45], cell-free network [46], mobile edge computing (MEC) [47], and
simultaneous wireless information and power transfer (SWIPT) [48, 49].

Although significant performance gains offered by IRS have been reported in
the open literature, the prior of perfect CSI for all channels involved are
generally assumed [10, 27, 28, 29, 30, 31, 50]. In practice, the acquisition
of accurate CSI, especially the reflection channel coefficients, is a
difficult task due to the passive component at IRS as well as its large number
of reflecting elements [9]. One of the considerations is the semi-passive
channel estimation by installing only a small number of sensing elements to
estimate the channels of the BS-IRS link and the user-IRS link, respectively
[9, 51, 38]. During the channel estimation phase, the IRS works at the sensing
mode by switching off all reflecting elements. The activated sensors receive
the pilots from the BS/user and then estimate the channels associated with the
sensors. Furthermore, the actual channels of the BS-IRS link and the user-IRS
link are obtained by sophisticated signal processing techniques, e.g.,
compressed sensing, and machine learning [38]. It is noted that the efficiency
of the IRS adopting the semi-passive channel estimation approach is heavily
dependent on the number of sensors, the quantization level of the analog-to-
digital converter (ADC) at the sensors as well as the sensing duration [9].

For the extreme implementation without any sensing elements mounted on the
IRS, the IRS becomes fully passive [8, 15]. Due to the deficiency of baseband
signal processing capability, the IRS can no longer estimate the BS-IRS link
and the user-IRS link separately. Instead, some recent research endeavour to
estimate the cascaded reflection channels, which is the Khatri-Rao product of
the BS-IRS link and the user-IRS link [9, 52]. Along this line of thought, the
most straightforward method is to estimate the direct and cascaded reflection
channels in sequence upon switching on only a single element at each time
slot, and thus the direct channel and reflection channels can be estimated
separately without mutual interference [53]. However, it is obvious to note
that the ON/OFF method suffers from severe power loss due to the frequent
switching off the IRS. To solve this problem, [54] proposed a novel channel
estimation method by designing orthogonal IRS reflection coefficient vector
during the channel estimation phase, thus the cascaded reflection channels can
be obtained without the power loss as well as the mutual interference. In
addition, [55] noted that the reflection channels associated with different
users share the common BS-IRS link, and then proposed a novel three-phase
channel estimation method, which estimates a set of low-dimensional scaling
factors instead of directly estimating reflection channel coefficients.
Furthermore, the authors of [56] reduced the pilot overhead by arranging the
IRS elements into groups and estimating only a single channel coefficient for
each group, which suffers from the performance penalty. Recently, several new
research results have been reported by reducing the dependency of the IRS
reflection optimization on accurate CSI. In [57], the statistic CSI is
exploited to perform the phase shift optimization at the expense of system
performance. Our prior work has demonstrated that multiple independent random
phase shifts can gradually approach the optimal performance, where only the
effective sum channel is estimated instead of separately estimating the direct
channel and reflection channels [58]. The interested reader may refer to [9,
59, 60] and references therein to find the most state-of-the-art channel
estimation methods for IRS-assisted wireless systems.

We note that the aforementioned channel estimation methods mainly focus on the
pilot design, reflection coefficients optimization and specific estimation
algorithms, which are crucial and challenging tasks in IRS-assisted wireless
systems [9, 55, 56, 50]. Nevertheless, there are still some points that can be
exploited to further improve the efficiency of the IRS. For example, existing
channel estimation methods generally adopt the constant-envelope pilot signals
for each user/antenna without considering the pilot power allocation during
the channel estimation phase. In [25], we have discussed the pilot power
allocation in the single-IRS assisted communication systems and performed the
power allocation between the pilots used for estimating the direct link and
reflection links. Nonetheless, there exist some limits to the rate performance
improvement [25]. On one hand, the user gains the performance benefits only in
the vicinity of the IRS due to the high path loss, which is the area where the
path loss of the reflection channels is significantly severe than that of the
direct channel. On the other hand, the IRS is generally equipped with a large
number of reflecting elements to compensate for the high path loss. Due to the
asymmetry between the direct link and reflection links, the displayed
performance improvement in [25] is limited, e.g., only effective for a
small/moderate number of reflecting elements at the IRS. Furthermore, we note
that much of the prior works focus on the communication systems assisted by a
single IRS, few of them focus on the design of multiple-IRS assisted
communication systems [19, 24].

Against the above background, in this paper, we propose the optimal pilot
power allocation strategy for multi-IRS assisted wireless systems. For the
sake of exposition, the features of the proposed pilot power allocation scheme
are compared to existing research results on IRS-assisted systems in Table I,
where $\star$ denotes the proposed pilot power allocation scheme. More
specifically, the contributions of this paper are summarized as follows:

  * •

First of all, we introduce a multi-IRS assisted single-input single-output
(SISO) communication system considering the practical cases with channel
estimation errors, where the most straightforward ON/OFF method is adopted for
the sake of brevity. Furthermore, we derive the ergodic achievable rate with
imperfect CSI in an explicit closed-form expression, which is dependent on the
specific pilot power used for estimating different reflection channels.

  * •

Secondly, we formulate the capacity maximizing problem subject to the
constraint of the average pilot power to obtain the specific pilot power
solution at each time slot. We then solve the problem by the method of
Lagrange multipliers based on some effective approximation. The resultant
pilot power allocation strategy suggests allocating the same amount of power
to the pilots used for estimating the reflection channels via the same IRS
while allocating more amount of power to the pilots used for estimating the
weak reflection channels. Furthermore, we elaborate on the PAPR increase of
pilot signals in the proposed pilot power allocation strategy.

  * •

Finally, extensive simulation results verify the derived ergodic capacity and
the optimal pilot power allocation strategy. Besides, some interesting results
are also summarized. For example, for the user in the vicinity of a large IRS,
it is suggested to switch off other IRSs and only switch on the IRS nearest
the user; while for the user near a small IRS, it should switch on all IRSs
and perform the optimal pilot power allocation. Besides, our simulation also
implicitly verify that the IRS should be deployed near the user and
distributed IRS deployment can bring extra performance improvements.

The rest of this paper is structured as follows. Section II introduces the
system model, while Section III derives the ergodic capacity with imperfect
CSI. Furthermore, Section IV provides the problem formulation, and Section V
solves the formulated problem and gives a general discussion. In Section VI,
extensive simulations are provided to verify our analysis. Finally, Section
VII concludes the paper.

_Notations:_ Scalars are denoted by italic letters, vectors/matrices are
denoted by bold-face lower/upper-case letters. ${\left(\cdot\right)^{T}}$,
${\left(\cdot\right)^{H}}$, and ${\left(\cdot\right)^{*}}$ represent the
transpose, Hermitian transpose and conjugate, respectively.
${\mathbb{C}^{x\times y}}$ denotes the space of $x\times y$ complex-valued
matrices. For a complex-valued vector ${\bf{x}}$,
$\text{diag}\left({\bf{x}}\right)$ denotes a diagonal matrix with each
diagonal element being the corresponding element in ${\bf{x}}$.
$\log\left(\cdot\right)$ denotes the logarithmic function, while
$\mathbb{E}\left\\{\cdot\right\\}$ stands for the expected value. The
distribution of a circularly symmetric complex Gaussian (CSCG) random variable
with mean $\mu$ and variance ${\sigma^{2}}$ is denoted by
$\mathcal{CN}\left({\mu,{\sigma^{2}}}\right)$.
$\mathcal{R}\left\\{z\right\\}$, $\mathcal{I}\left\\{z\right\\}$, and
$\left|z\right|$ denote the real part, the imaginary part, and the magnitude
of a complex number $z$, respectively.

Figure 1: Schematic of a multi-IRS assisted communication system.

##  II System Model

Let us consider an IRS-assisted downlink communication system, as shown in
Fig. 1, where multiple IRSs are deployed at the cell edge to enhance the
communication link from the BS to the user. For the sake of brevity, we
consider both the BS and user are equipped with a single antenna, and the
direct link between the BS and the user is blocked by a local building near
the user. We denote the number of IRSs as $K$. The set of the IRSs is denoted
by $\mathcal{K}=\left\\{{1,\cdots,K}\right\\}$. Furthermore, we denote the
number of reflecting elements at the $k$th IRS as ${M_{k}},\ \forall
k\in\mathcal{K}$, while the corresponding set of reflecting elements is
denoted by ${\mathcal{M}_{k}}=\left\\{{1,\cdots,{M_{k}}}\right\\},\ \forall
k\in\mathcal{K}$. All the IRSs are connected with an IRS controller by the
wired/wireless backhaul for feeding back the reflection coefficients. Due to
the high path loss, the reflection signals via multiple IRSs is assumed to be
below the receiver sensitivity and thus can be ignored.

Furthermore, we assume the quasi-static block-fading model is adopted for all
channels involved in Fig. 1. Specifically, the baseband equivalent channels
from the BS to the $k$th IRS, and from the $k$th IRS to the user are denoted
by
${{\bf{u}}_{k}}={\left[{{u_{k,1}},\cdots,{u_{k,{M_{k}}}}}\right]^{T}}\in{\mathbb{C}^{{M_{k}}\times
1}}$, and
${\bf{v}}_{k}^{H}={\left[{{v_{k,1}},\cdots,{v_{k,{M_{k}}}}}\right]^{*}}\in{\mathbb{C}^{1\times{M_{k}}}}$,
respectively, where ${u_{k,{m_{k}}}}$ and ${v_{k,{m_{k}}}}$ denote the
channels from the BS and the user, respectively, to the $m_{k}$th element at
the $k$th IRS. Let
${{\bf{\Phi}}_{k}}=\text{diag}\left({{\varphi_{k,1}},{\varphi_{k,2}},\cdots,{\varphi_{k,{M_{k}}}}}\right)\in{\mathbb{C}^{{M_{k}}\times{M_{k}}}}$
as the reflection coefficients matrix at the $k$th IRS, where
${{\varphi_{k,{m_{k}}}}}$ denotes the reflection coefficient of the
${m_{k}}$th element at the $k$th IRS. In this paper, we focus on the phase
shift design by assuming the magnitude at each IRS element to be
$\left|{{\varphi_{k,{m_{k}}}}}\right|=1$ for the maximum reflection power
collection111In fact, the signal reflected by the IRS may experience a slight
power loss for the IRS configuration. Besides, the magnitude of reflection
coefficients may independently adjusted and/or coupled with the phase shift
[9]. The power allocation considering more practical IRS element model will be
left for our future research topic.. In addition, we assume that the phase
shift can be continuously adjusted in $\left[{0,2\pi}\right)$ to gain
essential insight. As a result, the effective channel from the BS to the user
is modeled as the superposition of $K$-cluster reflection channels, i.e.,
$h=\sum\limits_{k=1}^{K}{{\bf{v}}_{k}^{H}{{\bf{\Phi}}_{k}}{{\bf{u}}_{k}}}$.

In this paper, we focus on the narrowband communications. Denote the signal
transmitted from the BS over the given band as $s$, which has the normalized
power, i.e., ${\left|s\right|^{2}}=1$. Therefore, the baseband signal received
at the user can be expressed as

$\displaystyle r=$
$\displaystyle\left({\sum\limits_{k=1}^{K}{{\bf{v}}_{k}^{H}{{\bf{\Phi}}_{k}}{{\bf{u}}_{k}}}}\right)\sqrt{q}s+n$
$\displaystyle=$
$\displaystyle\left({\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{{u_{k,{m_{k}}}}{\varphi_{k,{m_{k}}}}v_{k,{m_{k}}}^{*}}}}\right)\sqrt{q}s+n,$
(1)

where $q$ is the downlink average transmit power at the BS,
$n\sim\mathcal{CN}\left({0,\sigma_{n}^{2}}\right)$ is the additive white
Gaussian noise (AWGN) with $\sigma_{n}^{2}$ denoting the average noise power
at the user.

For the link via the ${m_{k}}$th element at the $k$th IRS, the reflection
channel consists of three parts, namely, the BS-IRS link ${u_{k,{m_{k}}}}$,
the IRS phase shift ${\varphi_{k,{m_{k}}}}$ and the IRS-user link
$v_{k,{m_{k}}}^{*}$, where ${\varphi_{k,{m_{k}}}}$ can be configured by the
IRS controller to facilitate the coherent superposition of different
reflection links. Nonetheless, the configuration of ${\varphi_{k,{m_{k}}}}$
needs the prior knowledge of the instantaneous CSI, i.e., ${u_{k,{m_{k}}}}$,
$v_{k,{m_{k}}}^{*}$, which, as stated earlier, is challenging to be obtained
separately due to the deficiency of baseband signal processing capability at
the IRS. To solve this problem, some novel channel estimation methods have
been proposed to estimate the cascaded reflection links instead [53, 54, 55,
56]. Therefore, (II) can be rewritten as

$\displaystyle
r=\left({\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{{\varphi_{k,{m_{k}}}}{h_{k,{m_{k}}}}}}}\right)\sqrt{q}s+n,$
(2)

where we have ${h_{k,{m_{k}}}}={u_{k,{m_{k}}}}v_{k,{m_{k}}}^{*}$ denoting the
cascaded reflection channels via the ${m_{k}}$th element at the $k$th IRS.

Next, we consider the practical channel estimation in the uplink phase, which
is crucial for fascinating the coherent superposition of the reflection
signals via different IRS elements. We assume the time division duplex (TDD)
mode for uplink and downlink transmissions and thus the CSI acquisition in the
downlink can be obtained based on the channel reciprocity. Moreover, we adopt
the ON/OFF channel method, where the cascaded reflection channels are
estimated by only switching on a single reflecting element at each time slot
and thus the pilot overhead required is $\sum\nolimits_{k=1}^{K}{{M_{k}}}$.
Denote the uplink pilot symbol at the
$\left({\sum\nolimits_{i=1}^{k-1}{{M_{i}}}+{m_{k}}}\right)$th time slot as
${x_{k,{m_{k}}}}$, which is used for estimating the reflection channel via the
$m_{k}$th element at the $k$th IRS upon switching on only the corresponding
reflecting element. Denote the normalized pilot symbols at the
$\left({\sum\nolimits_{i=1}^{k-1}{{M_{i}}}+{m_{k}}}\right)$th time slot as
${x_{k,{m_{k}}}}$, i.e., ${\left|{{x_{k,{m_{k}}}}}\right|^{2}}=1$. Therefore,
the uplink channel estimation model can be formulated by

$\displaystyle{y_{k,{m_{k}}}}={\left({{\varphi_{k,{m_{k}}}}{h_{k,{m_{k}}}}}\right)^{*}}\sqrt{{p_{k,{m_{k}}}}}{x_{k,{m_{k}}}}+{z_{k,{m_{k}}}},$
(3)

where ${p_{k,{m_{k}}}}$ is the uplink pilot power at the
$\left({\sum\nolimits_{i=1}^{k-1}{{M_{i}}}+{m_{k}}}\right)$th time slot,
${z_{k,{m_{k}}}}\sim\mathcal{CN}\left({0,\sigma_{z}^{2}}\right)$ is the AWGN
at the BS, ${\varphi_{k,{m_{k}}}}$ is the IRS reflection coefficient at the
$\left({\sum\nolimits_{i=1}^{k-1}{{M_{i}}}+{m_{k}}}\right)$th time slot, which
can be set to be an arbitrary complex value satisfying
$\left|{{\varphi_{k,{m_{k}}}}}\right|=1$.

_Remark 1:_ As stated earlier, we note that the ON/OFF method suffers from the
power loss due to the fact of frequently switching off the IRS elements.
Although recent novel channel estimation methods endeavour to improve the
channel estimation accuracy upon optimizing the reflection coefficients at the
IRS, they can not be readily applied for the practical implementation with
discrete phase shifters at each IRS element. For example, considering the
1-bit quantized phase shifters at each IRS element, it is infeasible to design
orthogonal reflection coefficients vectors for the arbitrary number of
reflecting elements. Instead, such so-called improved methods suffer from
severe mutual interference. In view of this, in this paper, we adopt the
ON/OFF method to help us gain insight into the pilot power allocation.

According to (3), the least squares (LS) estimate of the cascaded reflection
channel via the $m_{k}$th element at the $k$th IRS can be obtained by

$\displaystyle{{\hat{h}}_{k,{m_{k}}}}=\frac{1}{{\sqrt{{p_{k,{m_{k}}}}}}}y_{k,{m_{k}}}^{*}{x_{k,{m_{k}}}}\varphi_{k,{m_{k}}}^{*}={h_{k,{m_{k}}}}+{\varepsilon_{k,{m_{k}}}},$
(4)

where
${\varepsilon_{k,{m_{k}}}}=\frac{1}{{\sqrt{{p_{k,{m_{k}}}}}}}z_{k,{m_{k}}}^{*}{x_{k,{m_{k}}}}\varphi_{k,{m_{k}}}^{*}$
denotes the estimation error with respect to ${h_{k,{m_{k}}}}$, which is of
the distribution
${\varepsilon_{k,{m_{k}}}}\sim\mathcal{CN}\left({0,\delta_{k,{m_{k}}}^{2}}\right)$
with
$\delta_{k,{m_{k}}}^{2}={{{\sigma_{z}^{2}}\mathord{\left/{\vphantom{{\sigma_{z}^{2}}{{p_{k,{m_{k}}}}}}}\right.\kern-1.2pt}{{p_{k,{m_{k}}}}}}}$
denoting the mean square error (MSE) of the LS channel estimates.

Based on the estimated ${\hat{h}_{k,{m_{k}}}}$, the IRS controller will adjust
the reflection coefficient of the $m_{k}$th element at the $k$th IRS.
Specifically, the reflection coefficient of the $m_{k}$th element at the $k$th
IRS can be obtained by

$\displaystyle{\hat{\varphi}_{k,{m_{k}}}}=\frac{{\hat{h}_{k,{m_{k}}}^{*}}}{{\left|{{{\hat{h}}_{k,{m_{k}}}}}\right|}}=\frac{{h_{k,{m_{k}}}^{*}+\varepsilon_{k,{m_{k}}}^{*}}}{{\left|{{h_{k,{m_{k}}}}+{\varepsilon_{k,{m_{k}}}}}\right|}}.$
(5)

After optimizing the reflection coefficients and feeding them back to the
corresponding IRS, the actual channel between the BS and the user is thus
formulated by
$h=\sum\nolimits_{k=1}^{K}{\sum\nolimits_{{m_{k}}=1}^{{M_{k}}}{{{\hat{\varphi}}_{k,{m_{k}}}}{h_{k,{m_{k}}}}}}$.
As a result, the achievable rate of the IRS-assisted SISO system under
imperfect CSI can be obtained by

$\displaystyle
R={\log_{2}}\left({1+\frac{q}{{\sigma_{n}^{2}}}{{\left|{\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{{{\hat{\varphi}}_{k,{m_{k}}}}{h_{k,{m_{k}}}}}}}\right|}^{2}}}\right).$
(6)

##  III Ergodic Capacity with Imperfect CSI

In this section, we will derive the explicit closed-form expression of the
ergodic capacity with imperfect CSI. First of all, the ergodic achievable rate
can be expressed by [26]

$\displaystyle\mathbb{E}\left\\{R\right\\}=\mathbb{E}\left\\{{{{\log}_{2}}\left({1+\text{SNR}}\right)}\right\\}\leq{\log_{2}}\left({1+\mathbb{E}\left\\{{\text{SNR}}\right\\}}\right),$
(7)

where the average signal-to-noise-ratio (SNR) is defined by

$\displaystyle\mathbb{E}\left\\{{\text{SNR}}\right\\}=\frac{q}{{\sigma_{n}^{2}}}\mathbb{E}\left\\{{{{\left|{\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{{{\hat{\varphi}}_{k,{m_{k}}}}{h_{k,{m_{k}}}}}}}\right|}^{2}}}\right\\}.$
(8)

We note that the right hand side (RHS) of (7) provides a upper bound of the
ergodic capacity. In fact, under the high SNR and low SNR, the ergodic
achievable rate has the approximate expression of
$\mathbb{E}\left\\{R\right\\}={\log_{2}}\left({\mathbb{E}\left\\{{\text{SNR}}\right\\}}\right)$
and
$\mathbb{E}\left\\{R\right\\}={{\mathbb{E}\left\\{{\text{SNR}}\right\\}}\mathord{\left/{\vphantom{{\mathbb{E}\left\\{{\text{SNR}}\right\\}}{\ln
2}}}\right.\kern-1.2pt}{\ln 2}}$, respectively, both of which are heavily
dependent on the average SNR [61].

Hence, for the sake of brevity, in the following we will focus on the average
channel gain by omitting the transmit SNR
${q\mathord{\left/{\vphantom{q{\sigma_{n}^{2}}}}\right.\kern-1.2pt}{\sigma_{n}^{2}}}$
from (8). Specifically, upon substituting (5) into (8), the average channel
gain can be expressed by

$\displaystyle\mathbb{E}\left\\{{{{\left|{\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{{{\hat{\varphi}}_{k,{m_{k}}}}{h_{k,{m_{k}}}}}}}\right|}^{2}}}\right\\}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left\\{{{{\left|{\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{\frac{{\left({h_{k,{m_{k}}}^{*}+\varepsilon_{k,{m_{k}}}^{*}}\right){h_{k,{m_{k}}}}}}{{\left|{{h_{k,{m_{k}}}}+{\varepsilon_{k,{m_{k}}}}}\right|}}}}}\right|}^{2}}}\right\\}$
$\displaystyle\overset{\left(a\right)}{=}$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{\mathbb{E}\left\\{{{{\left|{{h_{k,{m_{k}}}}}\right|}^{2}}}\right\\}}}$
$\displaystyle+$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{\mathbb{E}\left\\{{\mathcal{R}\left\\{{\frac{{\left({h_{k,{m_{k}}}^{*}+\varepsilon_{k,{m_{k}}}^{*}}\right){h_{k,{m_{k}}}}}}{{\left|{{h_{k,{m_{k}}}}+{\varepsilon_{k,{m_{k}}}}}\right|}}}\right\\}}\right\\}}}$
$\displaystyle\quad\quad\times\sum\limits_{{{\tilde{m}}_{k}}=1,{{\tilde{m}}_{k}}\neq{m_{k}}}^{{M_{k}}}{\mathbb{E}\left\\{{\mathcal{R}\left\\{{\frac{{h_{k,{{\tilde{m}}_{k}}}^{*}+\varepsilon_{k,{{\tilde{m}}_{k}}}^{*}{h_{k,{{\tilde{m}}_{k}}}}}}{{\left|{{h_{k,{{\tilde{m}}_{k}}}}+{\varepsilon_{k,{{\tilde{m}}_{k}}}}}\right|}}}\right\\}}\right\\}}$
$\displaystyle+$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{\mathbb{E}\left\\{{\mathcal{I}\left\\{{\frac{{h_{k,{m_{k}}}^{*}+\varepsilon_{k,{m_{k}}}^{*}{h_{k,{m_{k}}}}}}{{\left|{{h_{k,{m_{k}}}}+{\varepsilon_{k,{m_{k}}}}}\right|}}}\right\\}}\right\\}}}$
$\displaystyle\quad\quad\times\sum\limits_{{{\tilde{m}}_{k}}=1,{{\tilde{m}}_{k}}\neq{m_{k}}}^{{M_{k}}}{\mathbb{E}\left\\{{\mathcal{I}\left\\{{\frac{{h_{k,{{\tilde{m}}_{k}}}^{*}+\varepsilon_{k,{{\tilde{m}}_{k}}}^{*}{h_{k,{{\tilde{m}}_{k}}}}}}{{\left|{{h_{k,{{\tilde{m}}_{k}}}}+{\varepsilon_{k,{{\tilde{m}}_{k}}}}}\right|}}}\right\\}}\right\\}}$
$\displaystyle+$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{\mathbb{E}\left\\{{\mathcal{R}\left\\{{\frac{{\left({h_{k,{m_{k}}}^{*}+\varepsilon_{k,{m_{k}}}^{*}}\right){h_{k,{m_{k}}}}}}{{\left|{{h_{k,{m_{k}}}}+{\varepsilon_{k,{m_{k}}}}}\right|}}}\right\\}}\right\\}}}$
$\displaystyle\times\sum\limits_{\tilde{k}=1,\tilde{k}\neq
k}^{K}{\sum\limits_{{m_{\tilde{k}}}=1}^{{M_{\tilde{k}}}}{\mathbb{E}\left\\{{\mathcal{R}\left\\{{\frac{{\left({h_{\tilde{k},{m_{\tilde{k}}}}^{*}+\varepsilon_{\tilde{k},{m_{\tilde{k}}}}^{*}}\right){h_{\tilde{k},{m_{\tilde{k}}}}}}}{{\left|{{h_{\tilde{k},{m_{\tilde{k}}}}}+{\varepsilon_{\tilde{k},{m_{\tilde{k}}}}}}\right|}}}\right\\}}\right\\}}}$
$\displaystyle+$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{\mathbb{E}\left\\{{\mathcal{I}\left\\{{\frac{{\left({h_{k,{m_{k}}}^{*}+\varepsilon_{k,{m_{k}}}^{*}}\right){h_{k,{m_{k}}}}}}{{\left|{{h_{k,{m_{k}}}}+{\varepsilon_{k,{m_{k}}}}}\right|}}}\right\\}}\right\\}}}$
$\displaystyle\times\sum\limits_{\tilde{k}=1,\tilde{k}\neq
k}^{K}{\sum\limits_{{m_{\tilde{k}}}=1}^{{M_{\tilde{k}}}}{\mathbb{E}\left\\{{\mathcal{I}\left\\{{\frac{{\left({h_{\tilde{k},{m_{\tilde{k}}}}^{*}+\varepsilon_{\tilde{k},{m_{\tilde{k}}}}^{*}}\right){h_{\tilde{k},{m_{\tilde{k}}}}}}}{{\left|{{h_{\tilde{k},{m_{\tilde{k}}}}}+{\varepsilon_{\tilde{k},{m_{\tilde{k}}}}}}\right|}}}\right\\}}\right\\}}},$
(9)

where $\left(a\right)$ holds as the reflection channel ${h_{k,{m_{k}}}}$ and
estimation error ${\varepsilon_{k,{m_{k}}}}$ associated with different
reflecting elements are independent of each other. Furthermore, the second and
third terms at the RHS of (III) reflect the effects of the channel estimation
errors from the same IRS, while the fourth and fifth terms reflect the effects
of the channel estimation errors from other IRSs.

Furthermore, in the practical implementation, the BS and IRS are generally
installed at the high-altitude sites with few local scatters, which means that
the BS-IRS link is more likely to form the line-of-sight (LoS) propagation
link with high Rician factor and long channel coherence time. Therefore, it is
reasonable to assume that the small-scale fading of the cascaded reflection
channels is mainly determined by the IRS-user link, i,e.,
${h_{k,{m_{k}}}}\sim\mathcal{CN}\left({0,\beta_{k}^{2}}\right)$, where
$\beta_{k}^{2}$ denotes the path loss of the cascaded reflection channels via
the $k$th IRS. Note that all the ${M_{k}}$ elements in the $k$th IRS
experience the same degree of path loss. As a result, for the $m_{k}$th
reflecting element at the $k$th IRS, we have
$\mathbb{E}\left\\{{{{\left|{{h_{k,{m_{k}}}}}\right|}^{2}}}\right\\}=\beta_{k}^{2}$.
In addition, according to the conclusions of [58], it is not difficult to
verify that, assuming
${h_{k,{m_{k}}}}\sim\mathcal{CN}\left({0,\beta_{k}^{2}}\right)$ and
${\varepsilon_{k,{m_{k}}}}\sim\mathcal{CN}\left({0,\delta_{k,{m_{k}}}^{2}}\right)$
for $\forall k\in\mathcal{K},\ \forall{m_{k}}\in{\mathcal{M}_{k}}$, we have

$\displaystyle\mathbb{E}\left\\{{\mathcal{R}\left\\{{\frac{{\left({h_{k,{m_{k}}}^{*}+\varepsilon_{k,{m_{k}}}^{*}}\right){h_{k,{m_{k}}}}}}{{\left|{{h_{k,{m_{k}}}}+{\varepsilon_{k,{m_{k}}}}}\right|}}}\right\\}}\right\\}=$
$\displaystyle\frac{{\sqrt{\pi}\beta_{k}^{2}}}{{2\sqrt{\beta_{k}^{2}+\delta_{k,{m_{k}}}^{2}}}},$
(10)
$\displaystyle\mathbb{E}\left\\{{\mathcal{I}\left\\{{\frac{{\left({h_{k,{m_{k}}}^{*}+\varepsilon_{k,{m_{k}}}^{*}}\right){h_{k,{m_{k}}}}}}{{\left|{{h_{k,{m_{k}}}}+{\varepsilon_{k,{m_{k}}}}}\right|}}}\right\\}}\right\\}=$
$\displaystyle 0.$ (11)

Upon substituting (10) and (11) into (III), the average channel gain can be
further simplified as

$\displaystyle\mathbb{E}\left\\{{{{\left|{\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{{{\hat{\varphi}}_{k,{m_{k}}}}{h_{k,{m_{k}}}}}}}\right|}^{2}}}\right\\}$
$\displaystyle=$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{\beta_{k}^{2}}}$
$\displaystyle+$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{\frac{{\sqrt{\pi}\beta_{k}^{2}}}{{2\sqrt{\beta_{k}^{2}+\delta_{k,{m_{k}}}^{2}}}}\sum\limits_{{{\tilde{m}}_{k}}=1,{{\tilde{m}}_{k}}\neq{m_{k}}}^{{M_{k}}}{\frac{{\sqrt{\pi}\beta_{k}^{2}}}{{2\sqrt{\beta_{k}^{2}+\delta_{k,{{\tilde{m}}_{k}}}^{2}}}}}}}$
$\displaystyle+$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{\frac{{\sqrt{\pi}\beta_{k}^{2}}}{{2\sqrt{\beta_{k}^{2}+\delta_{k,{m_{k}}}^{2}}}}\sum\limits_{\tilde{k}=1,\tilde{k}\neq
k}^{K}{\sum\limits_{{m_{\tilde{k}}}=1}^{{M_{\tilde{k}}}}{\frac{{\sqrt{\pi}\beta_{\tilde{k}}^{2}}}{{2\sqrt{\beta_{\tilde{k}}^{2}+\delta_{\tilde{k},{m_{\tilde{k}}}}^{2}}}}}}}}$
$\displaystyle=$ $\displaystyle\sum\limits_{k=1}^{K}{{M_{k}}\beta_{k}^{2}}$
$\displaystyle+$
$\displaystyle\frac{\pi}{4}\sum\limits_{k=1}^{K}{\beta_{k}^{4}\sum\limits_{{m_{k}}=1}^{{M_{k}}}{\frac{1}{{\sqrt{\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{m_{k}}}}}}}}}\sum\limits_{{{\tilde{m}}_{k}}=1,{{\tilde{m}}_{k}}\neq{m_{k}}}^{{M_{k}}}{\frac{1}{{\sqrt{\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{{\tilde{m}}_{k}}}}}}}}}}}}$
$\displaystyle+$
$\displaystyle\frac{\pi}{4}\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{\frac{{\beta_{k}^{2}}}{{\sqrt{\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{m_{k}}}}}}}}}\sum\limits_{\tilde{k}=1,\tilde{k}\neq
k}^{K}{\sum\limits_{{m_{\tilde{k}}}=1}^{{M_{\tilde{k}}}}{\frac{{\beta_{\tilde{k}}^{2}}}{{\sqrt{\beta_{\tilde{k}}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{\tilde{k},{m_{\tilde{k}}}}}}}}}}}}}},$
(12)

which will be further reduced to

$\displaystyle\mathbb{E}\left\\{{{{\left|h\right|}^{2}}}\right\\}=$
$\displaystyle\sum\limits_{k=1}^{K}{{M_{k}}\beta_{k}^{2}}+\frac{\pi}{4}\sum\limits_{k=1}^{K}{\beta_{k}^{2}{M_{k}}\left({{M_{k}}-1}\right)}$
$\displaystyle+$
$\displaystyle\frac{\pi}{4}\sum\limits_{k=1}^{K}{{M_{k}}{\beta_{k}}\sum\limits_{\tilde{k}=1,\tilde{k}\neq
k}^{K}{{M_{\tilde{k}}}{\beta_{\tilde{k}}}}},$ (13)

if there are no channel estimation errors. It is not difficult to verify that
(III) reveals the quadratic power law with the number of reflecting elements,
which is consistent with the conclusions of [10, 60].

Upon substituting (12) into (7), the approximate ergodic channel capacity can
be obtained, which is omitted here for the sake of brevity. We note that in
(12), the average channel gain is dependent on the power of pilots used for
estimating different reflection channels. Therefore, we desire to find the
optimal pilot power strategy that maximizes the ergodic achievable rate with
imperfect CSI, which will be addressed in the next section.

##  IV Problem Formulation

In this section, we will provide the problem formulation. More explicitly, we
endeavour to optimize the pilot power for maximizing the ergodic channel
capacity with imperfect CSI. Based on this consideration, the optimization
problem can be formulated by

$\displaystyle\begin{array}[]{*{20}{l}}{\left({P1}\right)}&{\mathop{\max}\limits_{\mathcal{P}}}&{\mathbb{E}\left\\{R\right\\}}\\\
{}&{s.t.}&{\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{{p_{k,{m_{k}}}}}}=\sum\limits_{k=1}^{K}{{M_{k}}}p},\end{array}$
(16)

where $p$ is the average uplink pilot power at the user; while

$\displaystyle\mathcal{P}=\left\\{{{p_{1,1}},\cdots,{p_{1,{M_{1}}}},\cdots,{p_{K,1}},\cdots,{p_{K,{M_{K}}}}}\right\\}$
(17)

is the set of the specific pilot power for entire
$\sum\nolimits_{k=1}^{K}{{M_{k}}}$ time slots during the channel estimation
phase.

Upon substituting (12) into (16) and removing the constant terms and positive
coefficient, the problem $\left({P1}\right)$ in (16) can be reduced to

$\displaystyle\begin{array}[]{*{20}{l}}{\left({P2}\right)}&{\mathop{\max}\limits_{\mathcal{P}}}&{\varphi\left(\mathcal{P}\right)}\\\
{}&{s.t.}&{\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{{p_{k,{m_{k}}}}}}=\sum\limits_{k=1}^{K}{{M_{k}}}p},\end{array}$
(20)

where the equivalent objective function $\varphi\left(\mathcal{P}\right)$ is
defined by

$\displaystyle\varphi\left(\mathcal{P}\right)=$
$\displaystyle\sum\limits_{k=1}^{K}{\beta_{k}^{4}\sum\limits_{{m_{k}}=1}^{{M_{k}}}{\frac{1}{{\sqrt{\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{m_{k}}}}}}}}}}\sum\limits_{{{\tilde{m}}_{k}}=1,{{\tilde{m}}_{k}}\neq{m_{k}}}^{{M_{k}}}{\frac{1}{{\sqrt{\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{{\tilde{m}}_{k}}}}}}}}}}}$
$\displaystyle+$
$\displaystyle\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{\frac{{\beta_{k}^{2}}}{{\sqrt{\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{m_{k}}}}}}}}}}}\sum\limits_{\tilde{k}=1,\tilde{k}\neq
k}^{K}{\sum\limits_{{m_{\tilde{k}}}=1}^{{M_{\tilde{k}}}}{\frac{{\beta_{\tilde{k}}^{2}}}{{\sqrt{\beta_{\tilde{k}}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{\tilde{k},{m_{\tilde{k}}}}}}}}}}}}.$
(21)

##  V The Proposed Optimal Pilot Power Allocation Strategy

In this section, we will use the method of Lagrange multipliers to obtain the
solution of the problem $\left({P2}\right)$. First of all, the Lagrangian
function $f\left({\mathcal{P},\lambda}\right)$ is defined as follows

$\displaystyle
f\left({\mathcal{P},\lambda}\right)=\varphi\left(\mathcal{P}\right)-\lambda\left({\sum\limits_{k=1}^{K}{\sum\limits_{{m_{k}}=1}^{{M_{k}}}{{p_{k,{m_{k}}}}}}-\sum\limits_{k=1}^{K}{{M_{k}}}p}\right),$
(22)

where $\lambda$ is the Lagrange multiplier.

Next, we proceed to find the stationary point of the Lagrange function upon
yielding the first-order partial derivatives of
$f\left({\mathcal{P},\lambda}\right)$ to be zero. More specifically, for the
$m_{k}$th element at the $k$th IRS, we have

$\displaystyle\frac{{\partial f}}{{\partial{p_{k,{m_{k}}}}}}=$
$\displaystyle\frac{{\beta_{k}^{4}\sigma_{z}^{2}}}{{p_{k,{m_{k}}}^{2}\sqrt{{{\left({\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{m_{k}}}}}}}\right)}^{3}}}}}\sum\limits_{{{\tilde{m}}_{k}}=1,{{\tilde{m}}_{k}}\neq{m_{k}}}^{{M_{k}}}{\frac{1}{{\sqrt{\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{{\tilde{m}}_{k}}}}}}}}}}$
$\displaystyle+$
$\displaystyle\frac{{\beta_{k}^{2}\sigma_{z}^{2}}}{{p_{k,{m_{k}}}^{2}\sqrt{{{\left({\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{m_{k}}}}}}}\right)}^{3}}}}}\sum\limits_{\tilde{k}=1,\tilde{k}\neq
k}^{K}{\sum\limits_{{m_{\tilde{k}}}=1}^{{M_{\tilde{k}}}}{\frac{{\beta_{\tilde{k}}^{2}}}{{\sqrt{\beta_{\tilde{k}}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{\tilde{k},{m_{\tilde{k}}}}}}}}}}}}$
$\displaystyle-$ $\displaystyle\lambda=0.$ (23)

Furthermore, let us first focus on the set of the first-order partial
derivatives associated with the $k$th IRS, i.e., $\left\\{{{{\partial
f}\mathord{\left/{\vphantom{{\partial
f}{\partial{p_{k,1}}}}}\right.\kern-1.2pt}{\partial{p_{k,1}}}},\cdots,{{\partial
f}\mathord{\left/{\vphantom{{\partial
f}{\partial{p_{k,{M_{k}}}}}}}\right.\kern-1.2pt}{\partial{p_{k,{M_{k}}}}}}}\right\\}$.
Upon comparing arbitrary pairs of $\left({{m_{k}}\in{\mathcal{M}_{k}},\
{{m^{\prime}}_{k}}\in{\mathcal{M}_{k}}}\right)$, we have

$\displaystyle\frac{{\beta_{k}^{2}}}{{p_{k,{m_{k}}}^{2}\sqrt{{{\left({\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{m_{k}}}}}}}\right)}^{3}}}}}\sum\limits_{{{\tilde{m}}_{k}}=1,{{\tilde{m}}_{k}}\neq{m_{k}}}^{{M_{k}}}{\frac{1}{{\sqrt{\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{{\tilde{m}}_{k}}}}}}}}}}$
$\displaystyle+$
$\displaystyle\frac{1}{{p_{k,{m_{k}}}^{2}\sqrt{{{\left({\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{m_{k}}}}}}}\right)}^{3}}}}}\sum\limits_{\tilde{k}=1,\tilde{k}\neq
k}^{K}{\sum\limits_{{m_{\tilde{k}}}=1}^{{M_{\tilde{k}}}}{\frac{{\beta_{\tilde{k}}^{2}}}{{\sqrt{\beta_{\tilde{k}}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{\tilde{k},{m_{\tilde{k}}}}}}}}}}}}$
$\displaystyle=$
$\displaystyle\frac{{\beta_{k}^{2}}}{{p_{k,{{m^{\prime}}_{k}}}^{2}\sqrt{{{\left({\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{{m^{\prime}}_{k}}}}}}}\right)}^{3}}}}}\sum\limits_{{{\tilde{m}}_{k}}=1,{{\tilde{m}}_{k}}\neq{{m^{\prime}}_{k}}}^{{M_{k}}}{\frac{1}{{\sqrt{\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{{\tilde{m}}_{k}}}}}}}}}}$
$\displaystyle+$
$\displaystyle\frac{1}{{p_{k,{{m^{\prime}}_{k}}}^{2}\sqrt{{{\left({\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{{m^{\prime}}_{k}}}}}}}\right)}^{3}}}}}\sum\limits_{\tilde{k}=1,\tilde{k}\neq
k}^{K}{\sum\limits_{{m_{\tilde{k}}}=1}^{{M_{\tilde{k}}}}{\frac{{\beta_{\tilde{k}}^{2}}}{{\sqrt{\beta_{\tilde{k}}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{\tilde{k},{m_{\tilde{k}}}}}}}}}}}}.$
(24)

We note that it is hard to obtain the solution of (V) since ${M_{k}}$
variables, i.e., $\left\\{{{p_{k,1}},\cdots,{p_{k,{M_{k}}}}}\right\\}$ in (V)
are coupled with each other. Upon constructing the same terms on both sides of
(V), we have

$\displaystyle\frac{{\beta_{k}^{2}\left({{A_{k}}-{1\mathord{\left/{\vphantom{1{\sqrt{\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{m_{k}}}}}}}}}}\right.\kern-1.2pt}{\sqrt{\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{m_{k}}}}}}}}}}\right)}}{{p_{k,{m_{k}}}^{2}\sqrt{{{\left({\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{m_{k}}}}}}}\right)}^{3}}}}}+\frac{{{B_{k}}}}{{p_{k,{m_{k}}}^{2}\sqrt{{{\left({\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{m_{k}}}}}}}\right)}^{3}}}}}$
$\displaystyle=$
$\displaystyle\frac{{\beta_{k}^{2}\left({{A_{k}}-{1\mathord{\left/{\vphantom{1{\sqrt{\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{{m^{\prime}}_{k}}}}}}}}}}\right.\kern-1.2pt}{\sqrt{\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{{m^{\prime}}_{k}}}}}}}}}}\right)}}{{p_{k,{{m^{\prime}}_{k}}}^{2}\sqrt{{{\left({\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{{m^{\prime}}_{k}}}}}}}\right)}^{3}}}}}+\frac{{{B_{k}}}}{{p_{k,{{m^{\prime}}_{k}}}^{2}\sqrt{{{\left({\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{{m^{\prime}}_{k}}}}}}}\right)}^{3}}}}},$
(25)

where ${A_{k}}$ and ${B_{k}}$ are defined by

$\displaystyle{A_{k}}=\sum\limits_{{m_{k}}=1}^{{M_{k}}}{\frac{1}{{\sqrt{\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k,{m_{k}}}}}}}}}},\
{B_{k}}=\sum\limits_{\tilde{k}=1,\tilde{k}\neq
k}^{K}{\sum\limits_{{m_{\tilde{k}}}=1}^{{M_{\tilde{k}}}}{\frac{{\beta_{\tilde{k}}^{2}}}{{\sqrt{\beta_{\tilde{k}}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{\tilde{k},{m_{\tilde{k}}}}}}}}}}}},$
(26)

which are independent of ${m_{k}}$ and ${m^{\prime}_{k}}$.

According to (V), we can easily obtain that
${p_{k,{m_{k}}}}={p_{k,{{m^{\prime}}_{k}}}}$ for
$\forall{m_{k}}\in{\mathcal{M}_{k}},\
\forall{m^{\prime}_{k}}\in{\mathcal{M}_{k}}$. As a result, the pilot power for
estimating the cascaded reflection channels via the same IRS can be obtained
by

$\displaystyle{p_{k,1}}=\cdots={p_{k,{M_{k}}}}={p_{k}},\quad\forall
k\in\mathcal{K},$ (27)

where ${p_{k}}$ is the average pilot power allocated for the $k$th IRS. (27)
means that one should allocate identical amount of power to the pilots for
estimating the channel coefficients of the reflection link via the same IRS,
which is consistent with our intuition, since they generally experience the
same large-scale fading. In the next, we will focus on the pilot power
allocation among different IRSs having different path loss.

More explicitly, upon substituting (27) into (V) and simplifying, we have

$\displaystyle\frac{{\partial f}}{{\partial{p_{k}}}}=$
$\displaystyle\frac{{\beta_{k}^{2}\sigma_{z}^{2}}}{{p_{k}^{2}\sqrt{{{\left({\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k}}}}}\right)}^{3}}}}}\sum\limits_{\tilde{k}=1}^{K}{\frac{{\beta_{\tilde{k}}^{2}{M_{\tilde{k}}}}}{{\sqrt{\left({\beta_{\tilde{k}}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{\tilde{k}}}}}}\right)}}}}$
$\displaystyle-$
$\displaystyle\frac{{\beta_{k}^{4}\sigma_{z}^{2}}}{{p_{k}^{2}{{\left({\beta_{k}^{2}+\frac{{\sigma_{z}^{2}}}{{{p_{k}}}}}\right)}^{2}}}}-\lambda=0,\quad
k\in\mathcal{K}.$ (28)

Nevertheless, it is not trivial to solve the equation set with $K$ equations
in (V), since (V) is actually an equation of higher degree with respect to
$\left\\{{{p_{1}},\cdots,{p_{K}}}\right\\}$, which are coupled with each
other. Therefore, in the next, we will find a high-quality approximate
solution instead. Considering that for the practical implementation, one may
expect that the communication systems operate at a moderate SNR region for
meeting the specific QoS requirement, i.e.,
${{{p_{k}}\beta_{k}^{2}}\mathord{\left/{\vphantom{{{p_{k}}\beta_{k}^{2}}{\sigma_{z}^{2}}}}\right.\kern-1.2pt}{\sigma_{z}^{2}}}\geq\gamma$,
where $\gamma$ is the minimum SNR requirement. Therefore, it is a reasonable
assumption that
$\beta_{k}^{2}+{{\sigma_{z}^{2}}\mathord{\left/{\vphantom{{\sigma_{z}^{2}}{{p_{k}}}}}\right.\kern-1.2pt}{{p_{k}}}}\approx\beta_{k}^{2},\
\forall k\in\mathcal{K}$. Based on such an approximation, (V) is reduced to

$\displaystyle\frac{{\sigma_{z}^{2}}}{{p_{k}^{2}{\beta_{k}}}}\sum\limits_{\tilde{k}=1}^{K}{{\beta_{\tilde{k}}}{M_{\tilde{k}}}}-\frac{{\sigma_{z}^{2}}}{{p_{k}^{2}}}-\lambda=0.$
(29)

Furthermore, recalling that the average pilot power constraint of
$\sum\limits_{k=1}^{K}{{p_{k}}{M_{k}}}=\sum\limits_{k=1}^{K}{{M_{k}}}p$, the
optimal pilot power allocation strategy can be obtained by

$\displaystyle{p_{k}}=\frac{{\sqrt{\left({\frac{1}{{{\beta_{k}}}}\sum\limits_{\hat{k}=1}^{K}{{\beta_{\hat{k}}}{M_{\hat{k}}}}-1}\right)}}}{{\sum\limits_{\tilde{k}=1}^{K}{{M_{\tilde{k}}}\sqrt{\left({\frac{1}{{{\beta_{\tilde{k}}}}}\sum\limits_{\hat{k}=1}^{K}{{\beta_{\hat{k}}}{M_{\hat{k}}}}-1}\right)}}}}\sum\limits_{\tilde{k}=1}^{K}{{M_{\tilde{k}}}}p.$
(30)

Figure 2: Comparison of two pilot power allocation schemes, where we have
$K=2$, ${M_{1}}={M_{2}}=1$, and $\beta_{1}^{2}=16\beta_{2}^{2}$. It is noted
that the proposed optimal pilot power allocation strategy is capable of
improving the accuracy of the phase shift in the statistical sense.

In the practical deployment, each IRS is generally equipped a large number of
reflecting elements to compensate for the high path loss. Therefore, we have
$\frac{1}{{{\beta_{k}}}}\sum\limits_{\hat{k}=1}^{K}{{\beta_{\hat{k}}}{M_{\hat{k}}}}\gg
1$ for $\forall k\in\mathcal{K}$. Accordingly, the power allocation strategy
in (30) are further simplified to

$\displaystyle{p_{k}}=\frac{{\sum\limits_{\tilde{k}=1}^{K}{{M_{\tilde{k}}}}p}}{{\sqrt{{\beta_{k}}}\sum\limits_{\tilde{k}=1}^{K}{\frac{{{M_{\tilde{k}}}}}{{\sqrt{{\beta_{\tilde{k}}}}}}}}},$
(31)

which is the important conclusion in this paper. The conclusion of (31)
suggests that one should allocate more amount of power to obtain a more
accurate estimate of the weak reflection channels, which is a little counter-
intuition. Next, we will consider two special cases to help us understand this
conclusion more deeply.

###  V-A ${M_{1}}=\cdots={M_{K}}$

Let us first consider the case that all IRSs are equipped with the same number
of reflecting elements, i.e., ${M_{1}}=\cdots={M_{K}}$. Hence, the optimal
pilot power allocation strategy in (31) is reduced to

$\displaystyle{p_{k}}=\frac{{Kp}}{{\sqrt{{\beta_{k}}}\sum\limits_{\tilde{k}=1}^{K}{\frac{1}{{\sqrt{{\beta_{\tilde{k}}}}}}}}},$
(32)

which characterizes the inverse square-root pilot power scalling law with
respect to the path loss of the corresponding reflection channel, i.e.,
${p_{k}}\propto{1\mathord{\left/{\vphantom{1{\sqrt{{\beta_{k}}}}}}\right.\kern-1.2pt}{\sqrt{{\beta_{k}}}}}$.

More explicitly, we give a simple example to explain the conclusions in (32).
For the sake of elaboration, we consider a communication system assisted by
two IRSs, each of which is equipped with only a single reflecting element. The
specific channel coefficients are shown in Fig. 2, where the statistical
channel gain of such two links are set to be
${\beta_{1}^{2}}=16{\beta_{2}^{2}}$. It should be noted that the phase
diffusion of the channel estimate of the weak channel is more severe than that
for the strong channel. Therefore, estimating a more accurate estimate of the
weak channel at the expense of a more wore estimate of the strong channel is a
compromise option. As can be seen from Fig. 2, the proposed optimal pilot
power allocation scheme is capable of improving the accuracy of the IRS phase
shift in the statistical sense.

###  V-B ${\beta_{1}}=\cdots={\beta_{K}}$

Next we consider the case that all IRSs experience the same degree of path
loss, i.e., ${\beta_{1}}=\cdots={\beta_{K}}$. Hence, the optimal pilot power
allocation strategy in (31) is reduced to ${p_{1}}=\cdots={p_{K}}=p$, which
means the proposed pilot power allocation strategy is reduced to the identical
pilot power allocation strategy. In fact, when multiple IRSs experience the
same degree of path loss, it is noted that distributed IRS deployment is
equivalent to the localized IRS deployment with
$\sum\nolimits_{k=1}^{K}{{M_{k}}}$ reflecting elements. In summary, the
proposed pilot power allocation strategy operates effectively in the
distributed deployment of IRSs with different path loss.

_Remark 2:_ The optimal pilot power allocation scheme improves the ergodic
achievable rate with imperfect CSI. Nevertheless, the performance improvement
is achieved at the expense of the PAPR increase. More specifically, the PAPR
of the pilot symbols in the optimal pilot power allocation strategy is defined
by [61]

$\displaystyle{\text{PAPR}_{o}}=\frac{{\max\left\\{{{p_{k}}}\right\\}}}{{\mathbb{E}\left\\{{{p_{k}}}\right\\}}}=\frac{{{p_{\max}}}}{p},$
(33)

where ${{p_{\max}}}$ is defined as (31) upon substituting ${\beta_{\min}}$
instead of ${\beta_{k}}$. Compared to the constant-envelope pilot symbols in
the identical pilot power allocation scheme, the optimal pilot power
allocation scheme will cause the PAPR increase. More explicitly, the PAPR
increase is quantitatively constrained by

$\displaystyle\sqrt{\frac{{{\beta_{\max}}}}{{{\beta_{\min}}}}}\geq{\text{PAPR}_{o}}\geq{\text{PAPR}_{i}}=0\
dB,$ (34)

which can be easily obtained by the scaling operation on (31),
${\beta_{\max}}$ and ${\beta_{\min}}$ denote the maximum and minimum value of
$\left\\{{{\beta_{1}},\cdots,{\beta_{K}}}\right\\}$, respectively. We note
that when ${\beta_{\max}}={\beta_{\min}}$, we have ${\text{PAPR}_{o}}=0$ dB,
which is consistent with our prior analysis.

##  VI Simulation Results

In this section, we will provide simulation results to verify our analysis.
The average noise power at the BS and user are set to be $\sigma_{z}^{2}=-110$
dBm and $\sigma_{n}^{2}=-90$ dBm, respectively. The average downlink transmit
power at the BS is set to be $q=40$ dBm. In our simulations, the location of
the BS is set to be $\left({0,0,{d_{h}}}\right)$ with the height of
${d_{h}}=10$ m. We consider that the direct link between the BS and user is
blocked throughout the user’s movement by assuming that the user is in the
vicinity of the building, as shown in Fig. 1. The path loss of the BS-IRS and
IRS-user links is modeled by ${\beta^{2}}={C_{0}}{d^{-\alpha}}$, where
${C_{0}}=-20$ dB is the path loss at the reference distance of ${d_{r}}=1$ m.
The path loss exponents of the BS-IRS and IRS-user links are set to be
${\alpha_{bi}}=2.2$ and ${\alpha_{iu}}=2.8$, respectively [28]. Furthermore,
the Rician fading channel model is adopted for both the BS-IRS link and the
IRS-user link, with the Rician factors of ${K_{bi}}=\infty$ and ${K_{iu}}=0$,
respectively. All simulation results are obtained by averaging $1000$ channel
realizations. For the sake of brevity, other specific parameters will be
specified in the following figures.

Figure 3: The simulation setup of the single-IRS assisted communication system
(from the top view). Figure 4: The ergodic achievable rate versus the
horizontal distance between the IRS and user.

First, let us verify the ergodic capacity performance with imperfect CSI.
Specifically, we consider a simple simulation setting as shown in Fig. 3,
where an IRS is deployed at the location $\left({{d_{0}},0,{d_{h}}}\right)$,
with the horizontal BS-IRS distance of ${d_{0}}=50$ m. The multiple-IRS
scenarios will be verified later by the proposed pilot power allocation
scheme. We consider that the user moves far away from the IRS along the line
of $\left({{d_{0}},d,0}\right)$, where the horizontal distance between the IRS
and user is increased from $d=4$ m to $d=26$ m. Fig. 4 plots the ergodic
achievable rate under different setups, considering the number of reflecting
elements of $M=100$ and $M=1000$, respectively; the uplink average pilot power
of $p=0$ dBm and $p=30$ dBm, respectively. It can be seen from Fig. 4 that the
achievable rate will be improved with the increase of the number of reflecting
elements or the average pilot power. In all cases, our theoretical analysis is
matched with the simulation results perfectly. In addition, one may note that,
as the number of reflecting elements increases from $M=100$ to $M=1000$, the
ergodic capacity is improved by about $6.6$ b/s/Hz, which is consistent with
the quadratic power law in [10], i.e., $2{\log_{2}}10\approx 6.6$ b/s/Hz.
Furthermore, we note that for the same number of reflecting elements, the
performance improvement caused by the increased average pilot power becomes
significant as the user moves far away from the IRS. For example, the rate
improvement is about $3$ b/s/Hz when increasing the average pilot power from
$p=0$ dBm to $p=30$ dBm for the user at the location of $d=26$ m. Since the
path loss becomes severe as the user moves far away from the IRS, it is
suggested to increase the average pilot power to obtain better channel
estimates and thus facilitate the coherent superposition between multiple
reflection channels for the user far away from the IRS.

Figure 5: The simulation setup of the two-IRS assisted communication system
(from the top view).

Next, we will consider a communication system, as shown in Fig. 5, with two
distributed IRSs to elaborate on the proposed pilot power allocation strategy
in detail. IRS-$1$ and IRS-$2$ are located at
$\left({{d_{0}},-{d_{v}},{d_{h}}}\right)$ and
$\left({{d_{0}},{d_{v}},{d_{h}}}\right)$, respectively, where ${d_{v}}$ is set
to be ${d_{v}}=10$ m. The user is located at the line parallel to the line
connecting IRS-$1$ and IRS-$2$, with the horizontal distance of ${d_{u}}=2$ m.
As a result, the user’s location is denoted by
$\left({{d_{0}}-{d_{u}},d,0}\right)$. The number of reflecting elements at
IRS-$1$ and IRS-$2$ are set to be ${M_{1}}={M_{2}}=100$. We consider the
y-coordinate of the user is increased from $d=-16$ m to $d=16$ m. The ergodic
capacity comparison of different power allocation strategies is shown in Fig.
6. For the sake of elaboration, the optimal ergodic capacity with perfect CSI
and the worst ergodic capacity with random phase shift configuration are also
plotted in Fig. 6. Specifically, we consider two cases of the average pilot
power of $p=-13$ dBm and $p=-23$ dBm, respectively. On one hand, the increased
average pilot power brings performance improvement, which is consistent with
our intuition. On the other hand, the proposed pilot power allocation strategy
further widens the performance gap with the identical pilot power allocation
strategy, especially for the user in the vicinity of an arbitrary one of two
IRSs. This uplifting result is consistent with our theoretical analysis, which
means that the ergodic capacity with imperfect CSI can be improved upon finely
allocating a different amount of power to the pilots used for estimating
different reflection channels.

Figure 6: The ergodic achievable rate versus the y-coordinate of the user
(symmetrical arrangement of two IRSs). Figure 7: The power and PAPR of pilot
signals versus the y-coordinate of the user (symmetrical arrangement of two
IRSs).

Moreover, the specific pilot power used for estimating the reflection channels
of the BS–IRS-$1$–user link and the BS–IRS-$2$–user link are shown as the left
y-axis in Fig. 7, where we consider the average pilot power of $p=-13$ dBm. It
can be seen from Fig. 7 that, when the user moves in the vicinity of the
IRS-$1$, the proposed pilot power allocation strategy suggests allocating more
amount of power to the pilots for estimating the reflection channels via the
IRS-$2$, which is counter-intuitive but effectively improves the achievable
rate of the user near the IRS-$1$. The same conclusions can be summarized as
the user moves near the IRS-$2$. In addition, the PAPR comparison of different
pilot power allocation strategies is shown in Fig. 7, which corresponds to the
right y-axis. It can be seen from Fig. 7 that the proposed pilot power
allocation strategy suffers from the PAPR increase.

Figure 8: The ergodic achievable rate versus the y-coordinate of the user
(asymmetrical arrangement of two IRSs). Figure 9: The power and PAPR of pilot
signals versus the y-coordinate of the user (asymmetrical arrangement of two
IRSs)

More specifically, when the user moves in the middle of IRS-$1$ and IRS-$2$,
the proposed pilot power allocation strategy is equivalent to the identical
pilot power allocation strategy, both of which have the PAPR of $0$ dB.
However, as the user moves toward the IRS-$1$ or IRS-$2$, the proposed pilot
power allocation strategy will incurs the PAPR increase. When the user moves
at the location nearest the IRS-$1$/IRS-$2$, the theoretical PAPR can be
obtained by
${{2\sqrt{d_{ui1}^{-{\alpha_{ui}}}}}\mathord{\left/{\vphantom{{2\sqrt{d_{ui1}^{-{\alpha_{ui}}}}}{\left({\sqrt{d_{ui1}^{-{\alpha_{ui}}}}+\sqrt{d_{ui2}^{-{\alpha_{ui}}}}}\right)}}}\right.\kern-1.2pt}{\left({\sqrt{d_{ui1}^{-{\alpha_{ui}}}}+\sqrt{d_{ui2}^{-{\alpha_{ui}}}}}\right)}}\approx
2.4$ dB, which is consistent with the simulation results as shown in Fig. 7.
Therefore, to be fair, the rate improvement for the user near the
IRS-$1$/IRS-$2$, as shown in Fig. 6, is gained at the compromise of the PAPR
increase of about $2.4$ dB.

Furthermore, let us consider the more general cases where IRS-$1$ and IRS-$2$
are equipped with a different number of reflecting elements due to the
specific surface area limit. In our simulations, the number of reflecting
elements at the IRS-$1$ and IRS-$2$ is set to be ${M_{1}}=1000$ and
${M_{2}}=100$, respectively. The simulation results are shown in Fig. 8, which
is significantly different from that in Fig. 6. As can be seen from 8, the
achievable rate of the user near the IRS-$1$ is higher than that near the
IRS-$2$, which is due to that more reflecting elements are equipped at the
IRS-$1$. Besides, compared to the identical pilot power allocation strategy,
the proposed pilot power allocation strategy is more effective in the vicinity
of the IRS-$2$, which means that the reflecting elements at IRS-$1$ play a
role in improving the capacity of the user near the IRS-$2$. However, for the
user in the vicinity of IRS-$1$, the smaller number of reflecting elements at
the IRS-$2$ hardly contribute to the rate improvement. In summary, the rate
improvement caused by the pilot power allocation is limited in the vicinity of
a large IRS.

Figure 10: The comparison of the two-IRS-working scheme and the one-IRS-
working scheme.

Likewise, the specific pilot power and the PAPR comparison for the asymmetric
number of reflecting elements are shown in Fig. 9. It can be seen from Fig. 9
that the qualitative conclusions that allocating more amount of power to the
pilots for estimating the weak reflection channels still holds. Besides, since
the number of reflecting elements at IRS-$1$ is much larger than that at
IRS-$2$, which makes the pilot power for estimating the reflection channels
via IRS-$1$ more similar as the identical pilot power allocation. Furthermore,
thanks to such a large IRS-$1$, the user near the IRS-$2$ suffers from slight
PAPR increase, e.g., only $0.3$ dB for the user nearest IRS-$2$, but gains
significant rate improvement. However, the unfair phenomena appear for the
user in the vicinity of the IRS-$1$, where a fairly high PAPR does not boost
the rate performance. For example, the PAPR of the pilot signals for the user
nearest the IRS-$1$ is as high as $6.5$ dB. Therefore, for the user near the
IRS-$1$, the proposed pilot power allocation strategy bring limited
performance improvement at the huge increase of PAPR.

In view of that the existence of IRS-$2$ is hardly helpful for the user near
the IRS-$1$, we next consider a benchmark scheme, where only a single IRS
works when the user moves into its vicinity. The specific simulation
parameters are shown in Fig. 10. As shown in Fig. 10, for the case that the
same number of reflecting elements are equipped at two IRSs, the proposed
optimal pilot power allocation strategy brings limited rate improvement than
the single-IRS assisted scheme. However, when the user moves in the middle of
two IRSs, the high power loss makes the combination of two IRSs a better
option. Furthermore, the asymmetry case with the number of reflecting elements
of ${M_{1}}=1000,\ {M_{2}}=100$ are also shown in Fig. 10. It can be seen from
Fig. 10 that, when the user is near the IRS-$1$, the single-IRS assisted
scheme almost approach the two-IRS assisted scheme with the optimal pilot
power allocation. Besides, it is interesting to note that the single-IRS
assisted scheme even outperforms the two-IRS assisted scheme with identical
pilot power allocation, which implies the worse estimates of the
BS–IRS-$2$–user link in the identical pilot power allocation strategy hardly
provides rate improvement for the user near the IRS-$1$. In addition, the
single-IRS assisted scheme also reduces the pilot overhead from
$\left({{M_{1}}+{M_{2}}}\right)$ to ${{M_{1}}}$ and remains the PAPR of $0$
dB. However, the situation is quite different for the user in the vicinity of
the IRS-$2$, where the single-IRS scheme suffers from severe rate loss than
the two-IRS scheme. Such a result reflects that a large number of reflecting
elements at the IRS-$1$ do help for improving the rate of the user near the
IRS-$2$. Therefore, the IRS controller needs to jointly perform the IRS
selection and optimal pilot power allocation to gain performance improvement
at the minimum cost. Basically, for the user near a large IRS, it is suggested
to choose the single-IRS scheme, where only the IRS nearest the user works
while other IRSs are switched off. By contrast, when the user moves near the
IRS with a small number of reflecting elements, it is better to switch on more
IRSs and perform the optimal pilot power allocation strategy to improve the
rate performance.

Figure 11: The simulation setup of the multi-IRS assisted communication system
(from the top view).

Finally, let us consider the general communication system assisted by multiple
IRSs, as shown in Fig. 11, where the user is located at
$\left({{d_{0}},0,0}\right)$ with the horizontal BS-user distance of
${d_{0}}=50$ m, while multiple IRSs are randomly located at the vicinity of
the user. Specifically, we consider that multiple IRSs are randomly deployed
in a circle with the user at the centre (from the top view), the radius is set
to be ${d_{r}}=20$ m. The height of all IRSs is set to be ${d_{h}}=10$ m. In
our simulations, we will consider the number of IRSs increases from $K=1$ to
$K=6$.

Figure 12: The ergodic achievable rate versus the number of IRSs (with the
constraint on the number of reflecting elements at each IRS). Figure 13: The
ergodic achievable rate versus the number of IRSs (with the constraint on the
total number of reflecting elements of all IRSs).

In Fig. 12, we consider that the number of reflecting elements at each IRS is
fixed at $1000$, i.e., ${M_{1}}=\cdots{M_{K}}=1000$. It can be seen from Fig.
12 that, with the increase of the number of IRSs, the performance improvement
of the optimal pilot power allocation strategy follows the quadratic power law
as that with perfect CSI. For example, when doubling the number of IRSs, the
rate improvement is about $2$ b/s/Hz, which is readily verified in Fig. 12. By
contrast, the identical pilot power allocation strategy suffers from a little
rate loss. As the number of IRSs increases, the performance advantages of the
proposed pilot power allocation strategy becomes more significant. Besides,
the rate performance can be further improved by increasing the average pilot
power.

Next, we consider the constraint on the total number of reflecting elements of
all IRSs. For the sake of elaboration, we still consider the same number of
reflecting elements at each IRS, and the total number of reflecting elements
of all IRSs is set to be ${M_{total}}=\sum\nolimits_{k=1}^{K}{{M_{k}}}=1440$
to make sure that the total number can be divisible by $K=1\sim 6$. Under this
constraint, the number of reflecting elements at each IRS is
${M_{1}}=\cdots={M_{K}}={{M_{total}}\mathord{\left/{\vphantom{{M_{total}}K}}\right.\kern-1.2pt}K}$.
The simulation results are shown in Fig. 13. It can be seen from Fig. 13 that,
due to the constraint of the total number of reflecting elements, the rate
improvement is limited with the increase of the number of IRSs. Nonetheless,
the rate improvement implies that, under the constraint of the total number of
reflecting elements, the distributed deployment of IRSs can obtain the
improved rate performance as the user moves in the vicinity of them. The
reason is that the user can only gain the rate benefit when it moves near one
of the IRSs, while the distributed deployment of IRSs can effectively enlarge
the effective area of the IRS swarm. In addition, when considering the
practical cases, distributed deployment of IRSs can improve the link quality
between the IRSs and user. More particularly, when one of the IRS-user links
are blocked, other IRSs can also work effectively. Besides, the proposed pilot
power allocation remains the most competitive option under all considered
simulation settings.

##  VII Conclusions

In this paper, we proposed the optimal pilot power allocation scheme for
multiple-IRS assisted communication systems. Specifically, we first derived
the ergodic achievable rate with imperfect CSI, which is dependent on the
specific pilot power for estimating different reflection channels. We then
formulated the ergodic capacity maximizing problem subject to the constraint
of average pilot power and solve the problem by leveraging the method of
Lagrange multipliers. The proposed pilot power scheme suggests allocating more
amount of power to the pilots used for estimating the weak channel helps to
improve the rate performance, which, however, increases the PAPR of pilot
signals. The simulation results verified the theoretical ergodic achievable
rate and the proposed pilot power allocation strategy. Some useful conclusions
are also summarized from our simulations. For example, for the user in the
vicinity of a large IRS, it is suggested to turn off other IRSs, while for
users near a small IRS, the IRS controller should switch on all IRSs and
perform the optimal pilot power allocation. Besides, our simulations have
demonstrated that the distributed deployment of IRSs can enlarge the effective
area of the IRS swarm, under the constraint of a total number of reflecting
elements. Finally, considering that the system model and the continuous phase-
shift assumption considered in this paper is relatively primary, the pilot
allocation in practical MIMO or multi-user MISO systems with the discrete
phase shift model will be left for our future research direction.

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