# Resource Allocation for Channel Estimation in Reconfigurable Intelligent
Surface-Aided Multi-Cell Networks

Yining Xu, Sheng Zhou TThis work was supported in part by the National Natural
Science Foundation of China under Grants 62341108, 62022049 and 62111530197.Y.
Xu and S. Zhou are with Department of Electronic Engineering, Tsinghua
University, Beijing 100084, China, and the the Beijing National Research
Center for Information Science and Technology (e-mail: xu-
<EMAIL_ADDRESS>sheng.zhou@tsinghua.edu.cn).

###### Abstract

Reconfigurable intelligent surface (RIS) is a promising solution to deal with
the blockage-sensitivity of millimeter wave band and reduce the high energy
consumption caused by network densification. However, deploying large scale
RISs may not bring expected performance gain due to significant channel
estimation overhead and non-negligible reflected interference. In this paper,
we derive the analytical expressions of the coverage probability, area
spectrum efficiency (ASE) and energy efficiency (EE) of a downlink RIS-aided
multi-cell network. In order to optimize the network performance, we
investigate the conditions for the optimal number of training symbols of each
antenna-to-antenna and antenna-to-element path (referred to as _the optimal
unit training overhead_) in channel estimation. Our study shows that: 1) RIS
deployment is not ‘the more, the better’, only when blockage objects are dense
should one deploy more RISs; 2) the coverage probability is maximized when the
unit training overhead is designed as large as possible; 3) however, the ASE-
and-EE-optimal unit training overhead exists. It is a monotonically increasing
function of the frame length and a monotonically decreasing function of the
average signal-to-noise-ratio (in the high signal-to-noise-ratio region).
Additionally, the optimal unit training overhead is smaller when communication
ends deploy particularly few or many antennas.

###### Index Terms:

Reconfigurable intelligent surface, multi-cell networks, directional
transmissions, channel estimation, resource allocation, stochastic geometry.

##  I Introduction

To satisfy the ever growing needs for data rate, seamless coverage and energy
efficiency (EE) of wireless communication networks, new technologies emerge,
including millimeter wave (mmWave), ultra-dense networks (UDNs) and massive
multiple-input multiple-output (MIMO). Yet the coverage demand can not always
be met since mmWave is sensitive to blockages due to its short wavelength [1].
In addition, the high energy consumption owing to network densification is
still one of the vital problems to be solved. As a promising way to circumvent
the high susceptibility of mmWave to blockages and lower down the network
energy consumption, reconfigurable intelligent surface (RIS) is slated to play
an important role in the next-generation mobile systems [2]. An RIS is
constructed with man-made surfaces of electromagnetic materials, e.g.,
conventional reflect-arrays, liquid crystal surfaces and software-defined
meta-surfaces, and thus the flexible control of electromagnetic propagation
environment can be realized [3]. Moreover, RISs can offer significant economic
and energy advantages of low-cost. At the same time, unlike relays, RISs do
not require complex processing and encoding/decoding, leading to ‘zero-delay’
reflection [4].

Despite these appealing potentials, RISs can reflect and so enlarge the
interference, especially under the dense deployment of RISs [5]. Although
beamforming reduces the interference in the UDN, how directional transmissions
behave in the RIS-aided multi-cell network and how many RISs should be
deployed are to be revealed. On top of that, the acquisition of channel state
information (CSI) in the RIS-aided multi-cell network becomes a serious
expense, in particular when RISs are equipped with a vast number of passive
elements [6]. Allocating appropriate resources to balance the CSI accuracy and
the overhead so as to promote the network performance, is of great importance.

###  I-A Related works

For RIS channel estimation, several schemes have been proposed and the
corresponding overhead has been estimated. An elements grouping method to
reduce the training overhead is proposed in [7][8], where only the combined
channels of each group of elements are estimated and the overhead in terms of
pilot transmission time is shown to be proportional to the number of groups.
Transmit power allocation and RIS reflection coefficients are optimized to
maximize the achievable rate in [7]. Furthermore, RIS reflection patterns to
aid channel estimation are designed and a closed-form expression of the
channel estimation error is derived in [8]. In [9], the authors propose a
joint beam training and positioning scheme, in which random beamforming and
maximum likelihood estimation are performed to acquire angle-of-arrival (AoA)
and angle-of-departure (AoD). Then an iterative positioning algorithm is
applied and the location information can further cross verify the estimation
of AoA and AoD. The pairwise error probability of AoA/AoD is proved to be
inversely proportional to the training overhead, i.e., the number of channel
measurements, in [9]. Ref. [10] develops an overhead-aware resource allocation
framework and optimizes the rate and EE w.r.t. RIS phase shifters, transmit
and receive filters, as well as power and bandwidth allocation. The time and
power overhead of channel estimation are proportional to the product of the
number of antennas/elements in base stations (BSs), user equipments (UEs) and
RISs. While the length of feedback phase depends on the number of feedback
bits for each RIS element and the number of RIS elements [10]. Furthermore,
based on [10], the number of RIS elements is optimized when BSs and UEs are
equipped with a single antenna [11]. The results show that rate increases with
the number of RIS elements increases, and EE is a concave function of the
number of RIS elements[11]. In [12], the strong correlation of UE-RIS-BS
uplink reflected channels between different UEs is utilized to reduce the
channel estimation time. Total $KMN+KM$ channel coefficients can be perfectly
recovered by $K+N+\max(K-1,\lceil(K-1)N/M)\rceil$ pilot symbols, where $K$,
$N$ and $M$ are the number of UEs, RIS elements and BS antennas, respectively
[12]. These works have effectively reduced the channel estimation overhead,
nonetheless, the limited training/feedback resources and the
noise/interference still result in imperfect CSI in practice [6].
Consequently, optimizing wireless resources for training/feedback/transmission
to balance the accuracy and the overhead of CSI is still of great importance.
Most works focus on the performance optimization for a single cell or several
adjacent cells, and thus a system-level analysis with large-scale deployed BSs
and RISs is needed.

Among the works on system-level performance analysis of RIS-aided networks
using stochastic geometry, ref. [13] derives the expressions of the average
ratio of blind-spot area and the probability distribution of path loss. The
results indicate that deploying RISs notably improves the coverage performance
when the interference is neglected [13]. In [14], the signal-to-interference-
ratio (SIR) coverage probability and the peak reflection power of RISs are
derived in closed-forms. The results show that deploying RISs are as effective
as equipping BSs with more antennas. Whereas, considering the reflection of
interference by RISs, there exists an optimal density of RISs, suggesting that
RIS deployment is not ‘the more, the better’, especially when BSs and UEs are
equipped with multi-antennas, as revealed in our previous work [5]. Ref. [15]
mainly characterizes the achievable throughput in a hybrid wireless network
comprising both active BSs and passive RISs. It is demonstrated that deploying
distributed RISs significantly boosts signal power but with marginal
interference increases under single antenna BSs and UEs. Apart from that, with
a total deployment cost constraint, the optimal RIS-to-BS density ratio that
maximizes the network throughput is observed. The authors in [16] study a
single-cell scenario with multi-users, where the high signal-to-noise-ratio
(SNR) slopes of ergodic rate and the diversity order of outage probability are
derived, showing that increasing the number of RIS elements improves the
spectrum efficiency and EE. Ref. [17] targets at comparing the performance of
different UE-RIS association policies, e.g., random association, the closest
association and all available association. Results show that the performance
comparison heavily relies on the number of RIS elements, cell radius and
blockage density. An RIS-aided multi-cell non-orthogonal multiple access
(NOMA) network is investigated in [18] and [19]. The distribution of angle-of-
incident and angle-of-reflection are studied, and the coverage probability of
paired NOMA users is derived in [18]. It is concluded that enlarging the size
of RIS increases the achievable rate till an upper limit. Besides, ref. [19]
considers inter-cell interference and derives the closed-form coverage
probability for paired NOMA users. The results evidence that strengthened
channel quality overtakes the interference introduced by RISs. Although the
performance of RIS-aided networks has been investigated in these previous
works, the studies on reflected interference and its impacts on large-scale
networks are still in its infancy, especially in terms of the coverage
probability, area spectrum efficiency (ASE) and EE.

###  I-B Contributions

In this paper, we study the large-scale network performance in the light of
the coverage probability, the ASE and the EE in an RIS-aided multi-cell
network with downlink directional transmissions. In particular, via stochastic
geometry, the impact of reflected interference from RISs is analyzed. The
optimal number of training symbols per path, referred to as _the optimal unit
training overhead_ , to maximize the coverage probability, the ASE and the EE
is also studied. Our main contributions are summarized as follows:

  * •

We utilize a general model of resource allocation for channel estimation and
data transmission. The beam alignment error, due to limited channel estimation
resources and thereby imperfect CSI, is also characterized. Particularly, the
impact of additional interference, from the reflection of RISs, is modeled and
studied.

  * •

We derive the analytical expressions of the coverage probability, the ASE and
the EE for a typical user in the network. Moreover, the monotonicity of the
coverage probability w.r.t. unit training overhead in channel estimation is
proved. And the condition for the ASE-optimal (also proved to be the EE-
optimal) unit training overhead is derived. As a result, the performance of
the network can be optimized with the proposed resource allocation scheme.

  * •

We adopt extensive numerical studies and find that: 1) The optimal RIS
deployment fraction, denoted as the ratio of RIS density to blockage density,
is a monotonically increasing function of blockage density and a monotonically
decreasing function of BS density. These results indicate that RIS deployment
is not ‘the more, the better’, only when blockage objects are dense should one
deploy more RISs. 2) The optimal unit training overhead to maximize the ASE
and the EE is a monotonically increasing function of the frame length and a
monotonically decreasing function of the average SNR (in the high SNR region).
Then, the optimal unit training overhead is significantly smaller under
especially few or many antennas deployment than that under medium scale of
antennas deployment.

###  I-C Organization

The organization of the rest is as follows. System model is introduced in
Section II. The expressions of the coverage probability, the ASE and the EE,
together with the condition for the optimal unit training overhead, are
derived in Section III. The numerical results in Section IV verify the
theoretical results. Finally, Section V concludes our work.

##  II System Model

###  II-A Network Deployment

Figure 1: An illustration of network deployment, communication links and
stochastic geometry models.

Consider the downlink of a multi-cell network with directional transmissions.
We assume that the locations of BSs and UEs follow two independent homogeneous
Poisson point processes (HPPPs) $\phi_{\text{B}}=\\{B_{i}\\}\in\mathbb{R}_{2}$
and $\phi_{\text{U}}=\\{U_{i}\\}\in\mathbb{R}_{2}$ with density
$\lambda_{\text{B}}$ and $\lambda_{\text{U}}$, respectively. The attributes of
blockages, such as location, orientation and size, are determined by a
simplified Boolean model [20]. The blockages are assumed to be line segments
with equal length $l$. The center points of the blockages follow another HPPP
$\phi_{\text{L}}=\\{L_{i}\\}\in\mathbb{R}_{2}$ with density
$\lambda_{\text{L}}$, which is independent of $\phi_{\text{B}}$ and
$\phi_{\text{U}}$. The orientation of the blockage $L_{i}$ is an uniformly
distributed random variable $\omega_{i}$ in the range of $[0,2\pi)$, which is
independent and identically distributed (i.i.d.). We consider an RISs
deployment strategy that a fraction of blockages are co-located with (a.k.a.
replaced by) RISs, as shown in Fig.1. Therefore, the center points of RISs,
denoted by $\phi_{\text{R}}=\\{R_{i}\\}\in\mathbb{R}_{2}$, actually follow a
thinning process of $\phi_{\text{L}}$ with density
$\lambda_{\text{R}}=\mu\lambda_{\text{L}}$, where $\mu\in[0,1]$ is the RIS
deployment fraction. We assume that RISs and blockages are of the same size,
and only one of the two surfaces of an RIS is the reflection surface.

###  II-B Channel Model: Blockage, Path Loss and Small-Scale Fading

For any communication link with distance $r$, the line-of-sight (LoS)
probability is $p_{\text{L}}(r)=\text{exp}(-\eta r)$, where
$\eta=\frac{2l\lambda_{\text{L}}}{\pi}$ under the aforementioned Boolean
blockage model [20]. We denote the LoS indicator as
$\mathbb{I}_{\text{L}}(r)$, where $\mathbb{I}_{\text{L}}(r)=1$ represents LoS
communication link and $\mathbb{I}_{\text{L}}(r)=0$ indicates the occurrence
of blockage. Therefore, the distribution of $\mathbb{I}_{\text{L}}(r)$ is

$\displaystyle\mathbb{I}_{\text{L}}(r)=\begin{cases}1&\text{w.p.
}p_{\text{L}}(r)\\\ 0&\text{w.p. }1-p_{\text{L}}(r).\end{cases}$ (1)

We consider two types of communication links, as shown in Fig.1. When the BS-
UE link is LoS, i.e., the UE associates to the BS directly, the path loss of a
direct link with distance $r_{\text{BU}}$ is
$P\\!L_{\text{D}}(r_{\text{BU}})=r_{\text{BU}}^{-\alpha}$, where $\alpha$ is
the path loss exponent. When the direct link is blocked, the RIS can provide
an reflected link to the blocked UE. The sum-distance path loss model is used
since mmWave wavelength is sufficiently small as compared with RIS element
size [21]. In addition, the energy loss after the reflection of an RIS is
modeled by a reflection power attenuation coefficient $\gamma$. In that case,
the path loss of a reflected link, with distance $r_{\text{BR}}$ between the
BS and the RIS and distance $r_{\text{RU}}$ between the RIS and the UE, is
$P\\!L_{\text{R}}(r_{\text{BR}}+r_{\text{RU}})=\gamma(r_{\text{BR}}+r_{\text{RU}})^{-\alpha}$,
where $r_{\text{BR}}+r_{\text{RU}}$ is the equivalent reflected link distance.

We assume a Rayleigh small-scale fading and the channel fading coefficient is
denoted as $h\sim\text{exp}(1)$. The channel fading coefficient $h$ is i.i.d.
for direct links and reflected links.

###  II-C Feasibility of Reflection

Figure 2: An illustration of feasible reflection.

As illustrated in Fig. 2, the feasibility of the reflected link depends on
whether the BS and the UE are on the same side of the RIS reflection surface.
We denote the probability of feasible reflection as
$p_{\text{F}}(r_{\text{BU}},r_{\text{RU}},\angle{\text{BUR}})$, where
$\angle{\text{BUR}}$ is the angle between the UE-BS path and the UE-RIS path.
The expression of
$p_{\text{F}}(r_{\text{BU}},r_{\text{RU}},\angle{\text{BUR}})$ is derived in
[13] as

$\displaystyle\begin{split}p_{\text{F}}(r_{\text{BU}},r_{\text{RU}},\angle{\text{BUR}})=\frac{1}{2}\left(\\!1\\!-\\!\frac{1}{\pi}\cos^{-1}\\!\\!\left(\frac{r_{\text{RU}}-r_{\text{BU}}\cos(\angle{\text{BUR}})}{\\!\\!\sqrt{\\!r_{\text{BU}}^{2}\\!+\\!r_{\text{RU}}^{2}\\!-\\!2r_{\text{BU}}r_{\text{RU}}\cos(\angle{\text{BUR}})}}\right)\\!\\!\right).\\\
\end{split}$ (2)

Similarly, we also denote a feasible reflection indicator
$\mathbb{I}_{\text{F}}(r_{\text{BU}},r_{\text{RU}},\angle{\text{BUR}})$ to
indicate whether the reflected link is feasible or not. The distribution of
$\mathbb{I}_{\text{F}}(r_{\text{BU}},r_{\text{RU}},\angle{\text{BUR}})$ is
given by

$\displaystyle\mathbb{I}_{\text{F}}(r_{\text{BU}},r_{\text{RU}},\angle{\text{BUR}})=\begin{cases}1&\text{\\!\\!w.p.
}p_{\text{F}}(r_{\text{BU}},r_{\text{RU}},\angle{\text{BUR}})\\\
0&\text{\\!\\!w.p.
}1\\!-\\!p_{\text{F}}(r_{\text{BU}},r_{\text{RU}},\angle{\text{BUR}}).\end{cases}$
(3)

###  II-D Beamforming Pattern

Figure 3: An illustration of beamforming patterns and beam alignment. In (a)
and (b), the beam alignment error is no larger than half beamwidth, therefore
the BS and the UE are aligned through a direct link and a reflected link,
respectively. In (c) and (d), the beam alignment error exceeds half beamwidth,
then the BS and the UE are misaligned in both the direct link and the
reflected link, respectively.

BSs and UEs are equipped with uniform linear arrays (ULAs) with $M_{\text{B}}$
and $M_{\text{U}}$ antennas, respectively. The beamforming pattern is
approximated to a sectored antenna model for tractability, as shown in Fig. 3.
The side lobe gain can be neglected under high front-back beamforming gain
ratio. As a result, the beamforming pattern is defined by two parameters: main
lobe gain $N_{j}$ and beamwidth $\theta_{j}$, where
$j\in\\{\text{B},\text{U}\\}$ represents BSs or UEs, respectively. The total
radiation gain constraint $N_{j}\theta_{j}=2\pi$ is satisfied under varying
beamforming patterns. For the ULAs with half-wavelength antenna separation, we
have $\theta_{j}=\frac{4}{M_{j}}$ in radians.

We also assume that the number of reflection elements in an RIS is
$M_{\text{R}}$. Since RISs are not equipped with power amplifier, there is no
power gain after the reflection of RISs. And the constant transmit (or
reflect) power consumption of a BS and an RIS are denoted as $P_{\text{B}}$
and $P_{\text{R}}$, respectively.

###  II-E User Association

Without loss of generality, we focus on a typical user located at the origin
$o$. The typical user associates to the BS with the strongest average received
signal power. The UEs, which associate to the same BS, are allocated with
orthogonal time-frequency resource blocks (RBs). Therefore, intra-cell
interference is zero. When the typical user associates to a BS $B_{i}$ through
a direct link, the average received signal power is

$\displaystyle
S_{B_{i}}(r_{B_{i}})=P_{\text{B}}\mathbb{I}_{\text{L}}(r_{B_{i}})P\\!L_{\text{D}}(r_{B_{i}}),$
(4)

where $r_{B_{i}}$ is the distance between the BS $B_{i}$ and the typical user.
While for an association BS $B_{i}$ with a reflected link through an RIS
$R_{k}$, the average received signal power of the typical user is

$\displaystyle\begin{split}S_{B_{i}\\!R_{k}}(r_{B_{i}},r_{R_{k}},r_{B_{i}\\!R_{k}},\angle{B_{i}oR_{k}})=P_{\text{B}}\mathbb{I}_{\text{L}}(r_{B_{i}\\!R_{k}})\mathbb{I}_{\text{L}}(r_{R_{k}})\mathbb{I}_{\text{F}}(r_{B_{i}},r_{R_{k}},\angle{B_{i}oR_{k}})P\\!L_{\text{R}}(r_{B_{i}\\!R_{k}}+r_{R_{k}}),\end{split}$
(5)

where $r_{R_{k}}$ is the distance between the RIS $R_{k}$ and the typical
user, and $\angle{B_{i}oR_{k}}$ is the angle between the direction from the
origin $o$ to the BS $B_{i}$ and the direction from the origin $o$ to the RIS
$R_{k}$, and
$r_{B_{i}\\!R_{k}}=\sqrt{r_{B_{i}}^{2}+r_{R_{k}}^{2}-2r_{B_{i}}r_{R_{k}}\cos(\angle{B_{i}oR_{k}})}$
is the distance between the BS $B_{i}$ and the RIS $R_{k}$, as illustrated in
Fig. 2 with $B\rightarrow B_{i}$, $R\rightarrow R_{k}$, $U\rightarrow o$,
$r_{\text{BU}}\rightarrow r_{B_{i}}$ and $r_{\text{RU}}\rightarrow r_{R_{k}}$,
specifically.

Therefore, the association BS $B^{*}$ is determined by

$\displaystyle\begin{split}B^{*}=\mathop{\arg\max}_{B_{i}\in\phi_{\text{B}}}\max\left(\\!S_{B_{i}}(r_{B_{i}}),\mathop{\max}_{R_{k}\in\phi_{\text{R}}}\\!\\!S_{B_{i}\\!R_{k}}(r_{B_{i}},r_{R_{k}},r_{B_{i}\\!R_{k}},\angle{B_{i}oR_{k}})\\!\\!\right).\end{split}$
(6)

This association policy attempts to find an LoS and feasible communication
link with the lowest path loss. In spite of that, UEs may experience
association failure when all the communication links are blocked by
obstructions or infeasible for reflection. In that case, the average received
signal power of the typical user is zero and a covering hole appears.

###  II-F Frame Structure and Resource Allocation

Figure 4: Frame structure with channel estimation phase and data transmission
phase.

We divide a transmission frame of fixed length $T$ into two sub-phases: the
channel estimation phase of length $T_{\text{E}}$ and the data transmission
phase of length $T_{\text{D}}$, as shown in Fig. 4.

In the channel estimation phase, a BS transmits $T_{\text{E}}$ shared pilots
to a UE and CSI is acquired at the UE through downlink training. After that,
CSI is provided to the BS by feedback, and the overhead of feedback is
regarded as a constant. Here we neglect the feedback overhead to simplify our
analyses. Both the direct link and the reflected link of the association BS
are estimated. Overall, the channel estimation overhead is modeled as

$\displaystyle
T_{\text{E}}=\beta(M_{\text{B}}M_{\text{R}}M_{\text{U}}+M_{\text{B}}M_{\text{U}}),$
(7)

where $\beta$ is the overhead of estimating a path from a single antenna
(element) to another in the corresponding communication end. The parameter
$\beta$ is the number of training symbols per path, and we refer to it as the
unit training overhead. Note that, $\beta$ is in the range of
$[0,\frac{T}{M_{\text{B}}M_{\text{R}}M_{\text{U}}+M_{\text{B}}M_{\text{U}}})$.
A non-integer $\beta$ can be regarded as a result of applying channel
estimation overhead reduction schemes, e.g., the RIS elements grouping scheme
[7][8]. Despite of that, the proposed channel estimation overhead model can
work as a worst case analysis [12]. Furthermore, $\beta=0$ means that we do
not transmit any pilot and choose the beam direction randomly. As a result,
the length of the data transmission phase is $T_{\text{D}}=T-T_{\text{E}}$.

On the other hand, the accuracy of CSI influences the precision of beam
alignment [22]. Under minimum mean square error (MMSE) estimation, the channel
estimation error is Gaussian distributed with variance
$\sigma_{\text{E}}^{2}=\frac{1}{1+\beta\text{SNR}}$, where SNR denotes the
average SNR of channels [23]. We denote the beam alignment error
$\varepsilon_{j}$ with $j\in\\{\text{B},\text{U}\\}$ as the angular difference
between the boresight direction and the actual pointing direction of the BS
and the UE, respectively, as shown in Fig. 3. Assume that the beam alignment
error is a truncated Gaussian distributed random variable with zero mean
[24][25], i.e.,
$\varepsilon_{j}\sim\mathcal{N}_{\text{T}}(0,\sigma_{j}^{2},-\pi,\pi)$, where
$\sigma_{j}^{2}$ is the variance and the variable is within the range of
$[-\pi,\pi]$. A natural idea is to tie the channel estimation error and the
beam alignment error together. Consequently, we assume that the variance of
the beam alignment error $\sigma_{j}^{2}$ and the variance of the channel
estimation error $\sigma_{\text{E}}^{2}$ satisfy the function that
$\sigma_{j}^{2}=k_{j}\pi^{2}\sigma_{\text{E}}^{2}$ with $k_{j}\in(0,1]$, where
$k_{j}\pi^{2}$ is the variance of the beam alignment error without channel
estimation. We can revisit this function from two aspects: 1) when
$\beta\rightarrow 0$, meaning that we almost choose the beam direction
randomly without channel estimation, therefore the variance of the beam
alignment error $\sigma_{j}^{2}$ approaches $k_{j}^{2}\pi^{2}$; 2) when
$\beta\rightarrow\infty$, meaning that we use enough pilots to estimate the
channel, therefore the variance of the beam alignment error $\sigma_{j}^{2}$
approaches $0$ and perfect beam alignment is almost reached.

In summary, a larger $\beta$ results in more accurate CSI and a smaller beam
alignment error, but the time left for data transmission is reduced
simultaneously. How to allocate the resources (either over time or bandwidth)
for channel estimation and data transmission is our focus in this paper.

###  II-G Performance Metrics

First, we will discuss the distribution of the total beamforming gain. We
denote the probability that the beam alignment error is no larger than half
beamwidth as $p_{\text{E}_{j}}(\sigma_{j},\theta_{j})$ with
$j\in\\{\text{B},\text{U}\\}$. The expression of
$p_{\text{E}_{j}}(\sigma_{j},\theta_{j})$ is obtained according to the
truncated Gaussian distribution of $\varepsilon_{j}$, which is

$\displaystyle
p_{\text{E}_{j}}(\sigma_{j},\theta_{j})=\mathbb{P}\left[|\varepsilon_{j}|\leq\frac{\theta_{j}}{2}\right]=\frac{\text{erf}\left(\frac{\theta_{j}}{2\sqrt{2\sigma_{j}^{2}}}\right)}{\text{erf}\left(\frac{\pi}{\sqrt{2\sigma_{j}^{2}}}\right)}.$
(8)

When the beam alignment error is no larger than half beamwidth, the main lobe
gain can still be obtained; otherwise, the corresponding communication end
only receives zero gain, i.e., the side lobe gain, named misalignment. Fig. 3
illustrates the situations of beam alignment and misalignment. Then, the
distribution of the total beamforming gain of the link between the typical
user and its association BS is expressed as

$\displaystyle G_{0}=\begin{cases}N_{\text{B}}N_{\text{U}}&\text{w.p.
}p_{\text{E}_{\text{B}}}(\sigma_{\text{B}},\theta_{\text{B}})p_{\text{E}_{\text{U}}}(\sigma_{\text{U}},\theta_{\text{U}})\\\
0&\text{w.p.
}1-p_{\text{E}_{\text{B}}}(\sigma_{\text{B}},\theta_{\text{B}})p_{\text{E}_{\text{U}}}(\sigma_{\text{U}},\theta_{\text{U}}).\end{cases}$
(9)

However, it is not the case for links between the typical user and interfering
BSs. The statistics of interference is invariant with beam misalignment [25].
Consequently, the distribution of the total beamforming gain of interfering
links is actually the distribution without beam alignment error, which is

$\displaystyle G_{i}=\begin{cases}N_{\text{B}}N_{\text{U}}&\text{w.p.
}\frac{\theta_{\text{B}}\theta_{\text{U}}}{4\pi^{2}}\\\ 0&\text{w.p.
}1-\frac{\theta_{\text{B}}\theta_{\text{U}}}{4\pi^{2}},\end{cases}$ (10)

and $G_{i}$ is i.i.d. for each interfering BS $B_{i}$.

Finally, the expression of received signal-to-interference-and-noise-ratio
(SINR) of the typical user is

$\displaystyle\begin{split}\text{SINR}=\frac{G_{0}h\left[\mathbb{I}_{\text{L}}(r_{\\!B^{*}}\\!)P\\!L_{\text{D}}(r_{B^{*}}\\!)+\\!(1\\!-\\!\mathbb{I}_{\text{L}}(r_{\\!B^{*}}\\!)\\!)P\\!L_{\text{R}}(r_{B^{*}\\!R^{*}}\\!+\\!r_{R^{*}})\right]}{N_{0}+I_{\text{D}}+I_{\text{R}}},\end{split}$
(11)

where $N_{0}$ is the noise power normalized by BS transmits power. And $R^{*}$
denotes the association RIS when the association link is a reflected link,
obtaining from

$\displaystyle\begin{split}R^{*}=\mathop{\arg\max}_{R_{k}\in\phi_{\text{R}}}S_{B^{*}\\!\\!R_{k}}(r_{B^{*}},r_{R_{k}},r_{B^{*}\\!\\!R_{k}},\angle{B^{*}\\!oR_{k}}).\end{split}$
(12)

While $I_{\text{D}}$ and $I_{\text{R}}$ are normalized direct interference
power and normalized reflected interference power, respectively. Their
expressions are listed as follows

$\displaystyle
I_{\text{D}}=\mathop{\sum}_{i:B_{i}\in\phi_{\text{B}}\backslash\\{B^{*}\\!\\}}\\!\\!\\!S_{B_{i}}(r_{B_{i}})G_{i}h/P_{\text{B}},$
(13) $\displaystyle
I_{\text{R}}=\mathop{\sum}_{i:B_{i}\in\phi_{\text{B}}\\!\backslash\\!\\{B^{*}\\!\\}}\mathop{\sum}_{k:R_{k}\in\phi_{\text{R}}}\\!\\!\\!S_{B_{i}\\!R_{k}}(r_{B_{i}},r_{R_{k}},r_{B_{i}\\!R_{k}},\angle{B_{i}oR_{k}})G_{i}h/P_{\text{B}}.$
(14)

There are three main performance metrics:

  1. 1.

Coverage probability, defined as $\mathcal{P}=\mathbb{P}[\text{SINR}>\tau]$,
is the probability that the received SINR of the typical user is larger than
the threshold $\tau$. This metric indicates the ability of providing the basic
coverage service.

  2. 2.

ASE, defined as
$\mathcal{A}=\frac{T_{\text{D}}}{T}\lambda_{\text{B}}\mathbb{E}[\text{log}_{2}(1+\text{SINR})\mathbb{I}\\{\text{SINR}>\tau\\}]$
with indication function $\mathbb{I}\\{\cdot\\}$, is the average throughput
per unit frequency and area. This metric measures how well the network
utilizes bandwidth resources.

  3. 3.

EE, defined as
$\mathcal{E}=\mathcal{A}/(\lambda_{\text{B}}P_{\text{B}}+\lambda_{\text{R}}P_{\text{R}})$,
is the throughput per unit frequency and power. EE represents how well the
network utilizes bandwidth and energy resources. The power consumption of UEs
are neglected since it is sufficiently small comparing with the power
consumption of BSs and RISs.

##  III Main Results

Before the derivation of the coverage probability, the ASE and the EE, we
first present some preliminary knowledge on link distance distributions and
association probability. Detailed proof can be found in [13].

###  III-A Link Distance Distribution

The distance of the shortest LoS direct link is denoted by a random variable
$r_{\text{D}}$ and its cumulative distribution function (CDF) is
$F_{\text{D}}(x)$. The conditional CDF of the distance of the shortest LoS
reflected link between the typical user and the BS at distance $u$ is denoted
by $F_{\text{R}|u}(x)$ with $x\geq u$. The distance of the shortest LoS
reflected link is denoted by a random variable $r_{\text{R}}$ and its CDF is
$F_{\text{R}}(x)$.

####  III-A1 The Distribution of the Shortest LoS Direct Link Distance

We have following expressions

$\displaystyle\begin{split}F_{\text{D}}(x)=&\mathbb{P}[r_{\text{D}}\leq x]\\\
=&1-\mathbb{P}[\mathcal{N}_{\phi_{\text{B,L}}}(\mathcal{B}(o,x))=0]\\\
=&1-\exp\left(-\int_{\mathcal{B}(o,x)}\lambda_{\text{B}}p_{\text{L}}(r)\text{d}r\right)\\\
=&1-\exp\left(-2\pi\lambda_{\text{B}}\frac{1-(\eta x+1)\exp(-\eta
x)}{\eta^{2}}\right),\end{split}$ (15)

where $\mathcal{N}_{\phi}(\mathcal{C})$ represents the number of points in the
region $\mathcal{C}$ following the PPP $\phi$. And $\phi_{\text{B,L}}$ is a
location-dependent thinning process of $\phi_{\text{B}}$ with density
$\lambda_{\text{B}}p_{\text{L}}(r)$, representing an inhomogeneous PPP of the
locations of LoS BSs. And $\mathcal{B}(o,x)$ is a disc centered at the origin
$o$ with radius $x$. Finally, the corresponding probability density function
(PDF) is expressed as

$\displaystyle\begin{split}f_{\text{D}}(x)=\frac{\text{d}F_{\text{D}}(x)}{\text{d}x}=2\pi\lambda_{\text{B}}x\exp\left(-\eta
x-2\pi\lambda_{\text{B}}\frac{1-(\eta x+1)\exp(-\eta
x)}{\eta^{2}}\right).\end{split}$ (16)

####  III-A2 The Conditional Distribution of the Shortest LoS Reflected Link
Distance

Similar to the derivation of $F_{\text{D}}(x)$, we can write the expression of
$F_{\text{R}|u}(x)$ as

$\displaystyle\begin{split}F_{\text{R}|u}(x)=&\mathbb{P}[r_{\text{R}}\leq
x|u]\\\ =&1-\mathbb{P}[\mathcal{N}_{\phi_{\text{R,L}}}(\mathcal{C}_{1})=0]\\\
=&1\\!-\\!\exp\\!\left(\\!\\!-\\!\\!\int_{\mathcal{C}_{1}}\\!\\!\lambda_{\text{R}}p_{\text{L}}\\!(t)p_{\text{L}}\\!\\!\left(\\!\\!\sqrt{u^{2}\\!+\\!t^{2}\\!-\\!\\!2ut\cos\psi}\right)\\!p_{\text{F}}(u,t,\psi)t\text{d}t\text{d}\psi\\!\\!\right)\\\
\overset{(a)}{=}&1\\!-\\!\exp\\!\Bigg{(}\\!\\!-\lambda_{\text{R}}\\!\\!\int_{\\!-\\!\pi}^{\pi}\\!\\!\int_{0}^{\frac{x^{2}-u^{2}}{2(\\!x\\!-\\!u\\!\cos\\!\psi\\!)}}\\!p_{\text{L}}\\!(t)p_{\text{L}}\\!\\!\left(\\!\\!\sqrt{u^{2}\\!+\\!t^{2}\\!-\\!\\!2ut\cos\psi}\right)\\!p_{\text{F}}(u,t,\psi)t\text{d}t\text{d}\psi\\!\\!\Bigg{)},\end{split}$
(17)

where $\phi_{\text{R,L}}$ is an inhomogeneous PPP representing the locations
of RISs that provide LoS feasible reflected links from the BS at $u$ to the
typical user. The density of $\phi_{\text{R,L}}$ is expressed as
$\lambda_{\text{R}}p_{\text{L}}(t)p_{\text{L}}\\!\left(\\!\\!\sqrt{\\!u^{2}\\!\\!+\\!t^{2}\\!-\\!2ut\\!\cos\\!\psi}\right)p_{\text{F}}(u,\\!t,\\!\psi)$,
where $t$ and $\psi$ indicate $r_{\text{RU}}$ and $\angle{\text{BUR}}$ for
simplicity. The points in the region $\mathcal{C}_{1}$ satisfy that the BS-
RIS-UE reflected link distance is no larger than $x$, i.e.,
$\mathcal{C}_{1}=\\{t,\psi:t+\sqrt{u^{2}\\!+\\!t^{2}\\!-\\!\\!2ut\cos\psi}\leq
x\\}$, resulting in the range of the integral in (a).

####  III-A3 The Distribution of the Shortest LoS Reflected Link Distance

We denote that the locations of the BSs, which are blocked in direct links and
have feasible LoS reflected links with link distance no larger than $x$,
follow an inhomogeneous PPP $\phi_{\text{B,R}}$. The density of
$\phi_{\text{B,R}}$ is
$\lambda_{\text{B}}(1-p_{\text{L}}(u))F_{\text{R}|u}(x)$. Therefore, the CDF
of the shortest reflected link distance $r_{\text{R}}$ is

$\displaystyle\begin{split}F_{\text{R}}(x)=\mathbb{P}[r_{\text{R}}\\!\leq
x]=1-\mathbb{P}[\mathcal{N}_{\phi_{\text{B,R}}}(\mathbb{R}^{2})=0]=1\\!-\\!\exp\\!\left(\\!\\!-2\pi\lambda_{\text{B}}\\!\\!\int_{0}^{x}(1-p_{\text{L}}\\!(u))F_{\text{R}|u}(x)u\text{d}u\\!\\!\right).\end{split}$
(18)

And the corresponding PDF is

$\displaystyle\begin{split}f_{\text{R}}(x)=\frac{\text{d}F_{\text{R}}(x)}{\text{d}x}=2\pi\lambda_{\text{B}}\exp\\!\\!\left(\\!\\!-2\pi\lambda_{\text{B}}\\!\\!\int_{0}^{x}\\!\\!\\!(1\\!-\\!p_{\text{L}}(u)\\!)F_{\text{R}|u}(x)u\text{d}u\\!\\!\right)\\!\\!\int_{0}^{x}\\!(1\\!-\\!p_{\text{L}}(u)\\!)\frac{\text{d}F_{\text{R}|u}(x)}{\text{d}x}u\text{d}u,\end{split}$
(19)

where

$\displaystyle\begin{split}\frac{\text{d}F_{\text{R}|u}(x)}{\text{d}x}=&\lambda_{\text{R}}\text{exp}\\!\left(\\!-\\!\lambda_{\text{R}}\\!\\!\\!\int_{\\!-\\!\pi}^{\pi}\\!\int_{0}^{\frac{x^{2}-u^{2}}{2\left(\\!x\\!-\\!u\\!\cos\\!\psi\\!\right)}}\\!\\!p_{\text{L}}\\!(t)p_{\text{L}}\\!\\!\left(\\!\\!\sqrt{u^{2}\\!\\!+\\!t^{2}\\!\\!-\\!\\!2ut\cos\psi}\right)\\!p_{\text{F}}(u,t,\psi)t\text{d}t\text{d}\psi\\!\\!\right)\\!\cdot\\\
&\quad\\!\\!\\!\\!\\!\int_{\\!-\\!\pi}^{\pi}\\!\\!\\!\\!p_{\text{L}}\\!\\!\left(\\!\\!\frac{x^{2}-u^{2}}{2\\!\left(\\!x\\!\\!-\\!u\\!\cos\\!\psi\right)}\\!\\!\right)\\!\\!p_{\text{L}}\\!\\!\\!\left(\\!\\!\sqrt{u^{2}\\!\\!+\\!\\!{\left(\\!\frac{x^{2}-u^{2}}{2\\!\left(\\!x\\!-\\!u\\!\cos\\!\psi\\!\right)}\\!\right)}^{\\!\\!2}\\!\\!\\!-\\!\\!\frac{u(x^{2}\\!-\\!u^{2})}{\\!x\\!-\\!u\\!\cos\\!\psi\\!}\\!\cos\psi}\right)\\!\\!p_{\text{F}}\\!\\!\left(\\!\\!u,\\!\frac{x^{2}-u^{2}}{2\\!\left(\\!x\\!-\\!u\\!\cos\\!\psi\\!\right)},\psi\\!\\!\right)\\!\cdot\\\
&\quad\quad\quad\\!\\!\frac{(x^{2}\\!\\!-\\!2ux\cos\psi\\!+\\!u^{2})(\\!x^{2}\\!-\\!u^{2})}{4(x-u\cos\psi)^{3}}\text{d}\psi.\end{split}$
(20)

###  III-B Association Probability

According to our user association policy introduced in Section II-E, the
typical user may have three kinds of association states:

  1. 1.

The typical user associates to an LoS BS through a direct link, denoted as
event $\mathcal{D}$;

  2. 2.

The typical user associates to a BS through an RIS reflected link, denoted as
event $\mathcal{R}$;

  3. 3.

The typical user has no BS to associate, denoted as event $\mathcal{O}$.

The probability of these three states are expressed as
$\mathbb{P}_{\mathcal{D}}$, $\mathbb{P}_{\mathcal{R}}$ and
$\mathbb{P}_{\mathcal{O}}$, respectively. Naturally,
$\mathbb{P}_{\mathcal{D}}+\mathbb{P}_{\mathcal{R}}+\mathbb{P}_{\mathcal{O}}=1$
should be satisfied.

First, the expression of $\mathbb{P}_{\mathcal{O}}$ is obtained from

$\displaystyle\begin{split}\mathbb{P}_{\mathcal{O}}\overset{\text{(a)}}{=}&\mathbb{E}\left[\prod_{B_{i}\in\phi_{\text{B}}}\\!\\!\\!\left(\\!1\\!-\\!\left(\mathbb{I}_{\text{L}}(\\!r_{B_{i}}\\!)+(1-\mathbb{I}_{\text{L}}(\\!r_{B_{i}}\\!))\\!\left(\\!\\!1\\!-\\!\\!\\!\\!\\!\prod_{R_{k}\in\phi_{\text{R}}}\\!\\!\\!\\!\left(1\\!-\\!\mathbb{I}_{\text{L}}(\\!r_{B_{i}\\!R_{k}}\\!)\mathbb{I}_{\text{L}}(\\!r_{R_{k}}\\!)\mathbb{I}_{\text{F}}(\\!r_{B_{i}},r_{R_{k}},\angle{B_{i}oR_{k}}\\!)\\!\right)\\!\\!\right)\\!\\!\\!\right)\\!\\!\\!\right)\\!\\!\right]\\\
\overset{\text{(b)}}{=}&\mathbb{E}_{\phi_{\text{B}}}\\!\\!\left[\prod_{B_{i}\in\phi_{\text{B}}}\\!\\!\\!\left(\\!1\\!-\\!\left(\\!\\!p_{\text{L}}(\\!r_{B_{i}}\\!)\\!+\\!(1\\!-\\!p_{\text{L}}(\\!r_{B_{i}}\\!)\\!)\mathbb{E}_{\phi_{\text{R}}}\\!\\!\left[\\!1\\!-\\!\\!\\!\\!\\!\prod_{R_{k}\in\phi_{\text{R}}}\\!\\!\\!\\!\left(1\\!-\\!p_{\text{L}}(\\!r_{B_{i}\\!R_{k}}\\!)p_{\text{L}}(\\!r_{R_{k}}\\!)p_{\text{F}}(\\!r_{B_{i}},r_{R_{k}},\angle{B_{i}oR_{k}}\\!)\\!\right)\\!\right]\\!\right)\\!\\!\\!\right)\\!\\!\right]\\\
\overset{\text{(c)}}{=}&\text{exp}\\!\Bigg{(}\\!\\!\\!-\\!2\pi\lambda_{\text{B}}\\!\\!\int_{0}^{\infty}\\!\\!\Bigg{(}\\!p_{\text{L}}\\!(u)+(1\\!-\\!p_{\text{L}}\\!(u)\\!)\cdot\Bigg{.}\Bigg{.}\\\
&\Bigg{.}\Bigg{.}\quad\quad\\!\\!\Bigg{(}\\!1-\exp\\!\Bigg{(}\\!-\\!\lambda_{\text{R}}\\!\\!\int_{-\pi}^{\pi}\\!\int_{0}^{\infty}\\!\\!p_{\text{L}}\\!\left(\\!\sqrt{t^{2}\\!+\\!u^{2}\\!-\\!2ut\cos\psi}\right)\\!p_{\text{L}}\\!(t)p_{\text{F}}(u,t,\psi)t\text{d}t\text{d}\psi\\!\Bigg{)}\\!\\!\Bigg{)}\\!\\!\Bigg{)}\\!u\text{d}u\\!\Bigg{)},\end{split}$
(21)

where (a) results from the fact that neither a direct link nor a reflected
link is available for the typical user in a covering hole. Equation (b)
results from the independent blockages for each link and the distributions of
$\mathbb{I}_{\text{L}}(r)$ and
$\mathbb{I}_{\text{F}}(r_{\text{BU}},r_{\text{RU}},\angle{\text{BUR}})$
proposed in (1) and (3), respectively. Equation (c) follows by the application
of probability generating functional (PGFL) of HPPP with variables
substitution.

Then, we turn to the expression of $\mathbb{P}_{\mathcal{D}}$. The typical
user associates to a BS through a direct link, that is to say, the association
link is the nearest LoS direct link and any other reflected link has higher
path loss. Conditioned on that the nearest LoS BS locates at distance
$r_{\text{D}}$, the locations of BSs form an inhomogeneous PPP
$\phi_{\text{B,R}|r_{\text{D}}}$, where the BSs are blocked in direct links
and provide reflected links with stronger average received signal power than
the nearest LoS BS. Therefore, the density of $\phi_{\text{B,R}|r_{\text{D}}}$
is
$\lambda_{\text{B}}(1-p_{\text{L}}(u))F_{\text{R}|u}(r_{\text{D}}\gamma^{\frac{1}{a}})$.
According to the definition of $\mathbb{P}_{\mathcal{D}}$, we have that

$\displaystyle\begin{split}\mathbb{P}_{\mathcal{D}}=&\mathbb{E}_{r_{\text{D}}}\\!\Big{[}\mathbb{P}[\mathcal{N}_{\phi_{\text{B,R}|r_{\text{D}}}}(\mathbb{R}^{2})=0|r_{\text{D}}]\Big{]}\\\
=&\mathbb{E}_{r_{\text{D}}}\\!\\!\left[\exp\\!\left(\\!-2\pi\lambda_{\text{B}}\\!\\!\int_{0}^{\infty}\\!\\!\\!(1-p_{\text{L}}(u))F_{\text{R}|u}(r_{\text{D}}\gamma^{\frac{1}{a}})u\text{d}u\\!\right)\\!\right]\\\
=&\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!f_{\text{D}}(x)\exp\\!\left(\\!-2\pi\lambda_{\text{B}}\\!\\!\int_{0}^{\infty}\\!\\!\\!(1-p_{\text{L}}(u))F_{\text{R}|u}(x\gamma^{\frac{1}{a}})u\text{d}u\\!\right)\\!\text{d}x.\end{split}$
(22)

Eventually, the probability of reflected links association is obtained

$\displaystyle\mathbb{P}_{\mathcal{R}}=1-\mathbb{P}_{\mathcal{O}}-\mathbb{P}_{\mathcal{D}}.$
(23)

In the next step, we derive the expressions of the coverage probability, the
ASE and the EE based on these preliminary knowledge on link distance
distributions and association probability.

###  III-C Coverage Probability

######  Theorem 1.

The coverage probability of the downlink RIS-aided multi-cell network with
directional transmissions is
$\displaystyle\begin{split}\mathcal{P}=&p_{\emph{E}_{\emph{B}}}\\!(\sigma_{\emph{B}},\theta_{\emph{B}})p_{\emph{E}_{\emph{U}}}\\!(\sigma_{\emph{U}},\theta_{\emph{U}})\\!\\!\left[\mathbb{P}_{\mathcal{D}}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\\!\exp\\!\left(\\!\\!-\frac{\tau
N_{0}}{N_{\emph{B}}N_{\emph{U}}P\\!L_{\emph{D}}(x)}\\!\\!\right)\\!\mathcal{L}_{I_{\emph{D}}}^{\emph{d}}\\!\\!\left(\\!\\!-\frac{\tau}{N_{\emph{B}}N_{\emph{U}}P\\!L_{\emph{D}}(x)}\\!\\!\right)\\!\mathcal{L}_{I_{\emph{R}}}^{\emph{d}}\\!\\!\left(\\!\\!-\frac{\tau}{N_{\emph{B}}N_{\emph{U}}P\\!L_{\emph{D}}(x)}\\!\\!\right)\\!\\!f_{\emph{D}}(x)\emph{d}x\right.\\\
&\left.+\mathbb{P}_{\mathcal{R}}\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\exp\\!\left(\\!\\!-\frac{\tau
N_{0}}{N_{\emph{B}}N_{\emph{U}}P\\!L_{\emph{R}}(x)}\\!\right)\\!\mathcal{L}_{I_{\emph{D}}}^{\emph{r}}\\!\\!\left(\\!\\!-\frac{\tau}{N_{\emph{B}}N_{\emph{U}}P\\!L_{\emph{R}}(x)}\\!\right)\\!\mathcal{L}_{I_{\emph{R}}}^{\emph{r}}\\!\\!\left(\\!\\!-\frac{\tau}{N_{\emph{B}}N_{\emph{U}}P\\!L_{\emph{R}}(x)}\\!\right)\\!\\!f_{\emph{R}}(x)\emph{d}x\right],\end{split}$
(24) where the functions $\mathcal{L}_{I_{\emph{D}}}^{\emph{d}}(\cdot)$,
$\mathcal{L}_{I_{\emph{R}}}^{\emph{d}}(\cdot)$,
$\mathcal{L}_{I_{\emph{D}}}^{\emph{r}}(\cdot)$ and
$\mathcal{L}_{I_{\emph{R}}}^{\emph{r}}(\cdot)$ are given by
$\displaystyle\begin{split}\mathcal{L}_{I_{\emph{D}}}^{\emph{d}}\\!\\!\left(\cdot\right)=\exp\\!\left(\\!\\!-\frac{\lambda_{\emph{B}}\theta_{\emph{B}}\theta_{\emph{U}}}{2\pi}\\!\\!\int_{x}^{\infty}\\!\\!\\!\frac{\tau
P\\!L_{\emph{D\\!}}(u)p_{\emph{L\\!}}(u)}{P\\!L_{\emph{D\\!}}(r)+\tau
P\\!L_{\emph{D\\!}}(u)}u\emph{d}u\\!\\!\right),\end{split}$ (25)
$\displaystyle\begin{split}&\mathcal{L}_{I_{\emph{R}}}^{\emph{d}}\\!\\!\left(\cdot\right)=\\\
&\exp\\!\\!\Bigg{(}\\!\\!\\!-\\!2\pi\lambda_{\emph{B}}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\Bigg{(}\\!\\!1\\!-\\!\exp\\!\\!\Big{(}\\!\\!\\!-\\!\frac{\lambda_{\emph{R}}\theta_{\emph{B}}\theta_{\emph{U}}}{4\pi^{2}}\\!\\!\\!\int_{\mathcal{C}_{2}}\\!\\!\\!\\!\frac{\tau\\!P\\!L_{\emph{R\\!}}(\\!\sqrt{\\!u^{2}\\!+\\!t^{2}\\!-\\!2ut\\!\cos\\!\psi}\\!+\\!t)p_{\emph{L\\!}}(\\!\sqrt{\\!u^{2}\\!+\\!t^{2}\\!-\\!2ut\\!\cos\\!\psi})p_{\emph{L\\!}}(t)p_{\emph{F}}(u,\\!t,\\!\psi)}{P\\!L_{\emph{D\\!}}(x)+\tau
P\\!L_{\emph{R\\!}}(\sqrt{\\!u^{2}+t^{2}-2ut\cos\psi}\\!+\\!t)}t\emph{d}t\emph{d}\psi\\!\Big{)}\\!\\!\\!\Bigg{)}\\!u\emph{d}u\\!\\!\Bigg{)},\end{split}$
(26)
$\displaystyle\begin{split}\mathcal{L}_{I_{\emph{D}}}^{\emph{r}}\\!\\!\left(\cdot\right)=\exp\\!\left(\\!\\!-\frac{\lambda_{\emph{B}}\theta_{\emph{B}}\theta_{\emph{U}}}{2\pi}\\!\\!\int_{x\gamma^{-\frac{1}{\alpha}}}^{\infty}\\!\frac{\tau
P\\!L_{\emph{D\\!}}(u)p_{\emph{L\\!}}(u)}{P\\!L_{\emph{R\\!}}(x)+\tau
P\\!L_{\emph{D}}(u)}u\emph{d}u\\!\\!\right),\end{split}$ (27)
$\displaystyle\begin{split}&\mathcal{L}_{I_{\emph{R}}}^{\emph{r}}\\!\\!\left(\cdot\right)=\\\
&\\!\exp\\!\\!\Bigg{(}\\!\\!\\!-\\!2\pi\lambda_{\emph{B}}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\Bigg{(}\\!\\!1\\!-\\!\exp\\!\\!\Big{(}\\!\\!\\!-\\!\frac{\lambda_{\emph{R}}\theta_{\emph{B}}\theta_{\emph{U}}}{4\pi^{2}}\\!\\!\\!\int_{\mathcal{C}_{3}}\\!\\!\\!\\!\frac{\tau\\!P\\!L_{\emph{R\\!}}(\\!\sqrt{\\!u^{2}\\!+\\!t^{2}\\!-\\!2ut\\!\cos\\!\psi}\\!+\\!t)p_{\emph{L\\!}}(\\!\sqrt{\\!u^{2}\\!+\\!t^{2}\\!-\\!2ut\\!\cos\\!\psi})p_{\emph{L\\!}}(t)p_{\emph{F}}(u,\\!t,\\!\psi)}{P\\!L_{\emph{R\\!}}(x)+\tau
P\\!L_{\emph{R\\!}}(\sqrt{\\!u^{2}+t^{2}-2ut\cos\psi}\\!+\\!t)}t\emph{d}t\emph{d}\psi\Big{)}\\!\\!\\!\Bigg{)}\\!u\emph{d}u\\!\\!\Bigg{)},\end{split}$
(28) where the regions $\mathcal{C}_{2}$ and $\mathcal{C}_{3}$ consist of two
sub-regions $\mathcal{C}_{21}$ and $\mathcal{C}_{22}$, $\mathcal{C}_{31}$ and
$\mathcal{C}_{32}$, respectively.
$\displaystyle\begin{split}\mathcal{C}_{21}\\!=\\!&\left\\{\\!t,\psi\\!:t\\!\in\\!\\!\left(\\!\\!\max\\!\left\\{\\!\frac{u^{2}-(r_{\text{D}}\gamma^{\frac{1}{\alpha}})^{2}}{2(u\cos\psi\\!-\\!r_{\text{D}}\gamma^{\frac{1}{\alpha}}\\!)},0\\!\right\\}\\!,\infty\\!\\!\right)\\!\\!,\psi\\!\in\\!\\!\left(\\!\\!-\pi,-\\!\arccos\\!\\!\left(\\!\\!\frac{r_{\text{D}}\gamma^{\frac{1}{\alpha}}}{u}\\!\\!\right)\\!\\!\\!\right)\\!\cup\\!\left(\\!\\!\arccos\\!\\!\left(\\!\\!\frac{r_{\text{D}}\gamma^{\frac{1}{\alpha}}}{u}\\!\\!\right)\\!\\!,\pi\\!\right)\\!\\!\right\\},\end{split}$
(29)
$\displaystyle\begin{split}\mathcal{C}_{22}\\!=\\!&\left\\{\\!t,\psi\\!:t\\!\in\\!\\!\left(0,\frac{u^{2}-(r_{\text{D}}\gamma^{\frac{1}{\alpha}})^{2}}{2(u\cos\psi\\!-\\!r_{\text{D}}\gamma^{\frac{1}{\alpha}}\\!)}\\!\right)\\!\\!,\psi\\!\in\\!\\!\left(\\!-\arccos\\!\left(\\!\frac{r_{\text{D}}\gamma^{\frac{1}{\alpha}}}{u}\\!\right),\arccos\\!\left(\\!\frac{r_{\text{D}}\gamma^{\frac{1}{\alpha}}}{u}\\!\right)\\!\\!\right)\\!\right\\},\end{split}$
(30)
$\displaystyle\begin{split}\mathcal{C}_{31}\\!=\\!&\left\\{t,\psi\\!:t\\!\in\\!\\!\left(\\!\max\\!\left\\{\\!\frac{u^{2}-r_{\text{R}}^{2}}{2(u\cos\psi\\!-\\!r_{\text{R}})},0\\!\right\\}\\!,\infty\\!\right)\\!,\psi\in\\!\left(\\!-\pi,-\\!\arccos\\!\left(\frac{r_{\text{R}}}{u}\right)\\!\right)\\!\cup\\!\left(\\!\arccos\\!\left(\frac{r_{\text{R}}}{u}\right)\\!,\pi\\!\right)\\!\\!\right\\},\end{split}$
(31)
$\displaystyle\begin{split}\mathcal{C}_{32}\\!=\\!&\left\\{t,\psi\\!:t\\!\in\\!\left(0,\frac{u^{2}-r_{\text{R}}^{2}}{2(u\cos\psi\\!-\\!r_{\text{R}})}\\!\right)\\!,\psi\\!\in\\!\left(\\!-\arccos\\!\left(\frac{r_{\text{R}}}{u}\right),\arccos\\!\left(\frac{r_{\text{R}}}{u}\right)\\!\right)\\!\right\\}.\end{split}$
(32) The functions $\mathbb{P}_{\mathcal{D}}$, $\mathbb{P}_{\mathcal{R}}$,
$f_{\emph{D}}(\cdot)$ and $f_{\emph{R}}(\cdot)$ are given in Eq. (22), (23),
(16) and (19), respectively.

###### Proof.

See Appendix A. ∎

The coverage probability of the network consists of three terms

$\displaystyle\mathcal{P}=$
$\displaystyle\mathbb{P}_{\mathcal{D}}\cdot\mathbb{P}[\text{SINR}\\!>\\!\tau|\mathcal{D}]\\!+\\!\mathbb{P}_{\mathcal{R}}\cdot\mathbb{P}[\text{SINR}\\!>\\!\tau|\mathcal{R}]\\!+\\!\mathbb{P}_{\mathcal{O}}\cdot
0,$ (33)

where $\mathbb{P}[\text{SINR}\\!>\\!\tau|\mathcal{D}]$ and
$\mathbb{P}[\text{SINR}\\!>\\!\tau|\mathcal{R}]$ are the coverage probability
conditioned on that the typical user associates to a BS through a direct link
and a reflected link, respectively. Specifically, when the typical user has no
BS to associate, the conditional coverage probability is zero. The full
expression of the coverage probability is displayed in Theorem 1 on the next
page.

We observe that the amount of the resources used in channel estimation, i.e.,
the value of the unit training overhead $\beta$, primarily affects the
coverage probability on the
$p_{\text{E}_{\text{B}}}(\sigma_{\text{B}},\theta_{\text{B}})p_{\text{E}_{\text{U}}}(\sigma_{\text{U}},\theta_{\text{U}})$
term. In this reason, we can simplify the relationship between the coverage
probability $\mathcal{P}$ and the unit training overhead $\beta$, leading to
the following Corollary.

######  Corollary 1.

The coverage probability $\mathcal{P}$ in Theorem 1 is a monotonically
increasing function of the unit training overhead $\beta$.

###### Proof.

See Appendix B. ∎

On the one hand, this result reveals that, to achieve the optimal coverage
performance, the resources allocated to channel estimation should be as much
as possible. Owing to that the penalty of redundant channel estimation
resources only affects the ASE and the EE, the coverage performance merely
enjoys the benefits of abundant channel estimation resources. On the other
hand, the coverage-optimal resource allocation scheme has nothing to do with
network deployment, i.e., $\lambda_{\text{B}}$ and $\lambda_{\text{R}}$, which
enlightens a separated design of network deployment and resource allocation.

###  III-D Area Spectrum Efficiency

As defined in Section II-G, the ASE $\mathcal{A}$ is expressed as

$\displaystyle\begin{split}\mathcal{A}=&\frac{T_{\text{D}}}{T}\lambda_{\text{B}}\mathbb{E}[\text{log}_{2}(1+\text{SINR})\mathbb{I}\\{\text{SINR}>\tau\\}]\\\
=&\frac{T\\!-\\!\beta(M_{\text{B}}M_{\text{R}}M_{\text{U}}\\!+\\!M_{\text{B}}M_{\text{U}})}{T}\lambda_{\text{B}}\\!\\!\left(\\!\frac{1}{\ln\\!2}\\!\\!\int_{\ln(1\\!+\tau)}^{\infty}\\!\\!\\!\\!\\!\\!\mathbb{P}[\text{SINR}\\!>\\!e^{t}\\!\\!-\\!1]\text{d}t\\!+\\!\text{log}_{2}(1\\!+\\!\tau)\mathbb{P}[\text{SINR}\\!>\\!\tau]\\!\\!\right).\end{split}$
(34)

The full expression of $\mathcal{A}$ can be obtained by applying the results
in Theorem 1 to Eq. (34). We neglect it due to page limitation. Similar to the
conclusions on the coverage probability, we propose the following Corollary on
the ASE-optimal unit training overhead $\beta^{*}_{\mathcal{A}}$.

######  Corollary 2.

The optimal unit training overhead $\beta^{*}_{\mathcal{A}}$ that maximizes
the ASE $\mathcal{A}$ satisfies the following condition

$\displaystyle\begin{split}&\frac{\emph{SNR}}{\sqrt{\pi}f(\beta^{*}_{\mathcal{A}})}\\!\Big{(}\frac{T}{M_{\emph{B}}M_{\emph{R}}M_{\emph{U}}\\!+\\!M_{\emph{B}}M_{\emph{U}}}\\!-\\!\beta^{*}_{\mathcal{A}}\\!\Big{)}\\!\\!\Bigg{(}\\!\\!\frac{ae^{-a^{2}\\!f^{2}\\!(\beta^{*}_{\mathcal{A}})}}{\emph{erf}(af\\!(\beta^{*}_{\mathcal{A}})\\!)}\\!+\\!\frac{be^{-b^{2}\\!f^{2}\\!(\beta^{*}_{\mathcal{A}})}}{\emph{erf}(bf\\!(\beta^{*}_{\mathcal{A}})\\!)}\\!-\\!\frac{ce^{-c^{2}\\!f^{2}\\!(\beta^{*}_{\mathcal{A}})}}{\emph{erf}(cf\\!(\beta^{*}_{\mathcal{A}})\\!)}\\!-\\!\frac{de^{-d^{2}\\!f^{2}\\!(\beta^{*}_{\mathcal{A}})}}{\emph{erf}(df\\!(\beta^{*}_{\mathcal{A}})\\!)}\\!\\!\Bigg{)}\\!\\!=\\!1,\end{split}$
(35)

where $f(\beta^{*}_{\mathcal{A}})=\sqrt{1+\beta^{*}_{\mathcal{A}}\emph{SNR}}$,
$a=\frac{\theta_{\emph{B}}}{2\pi\sqrt{2k_{\emph{B}}}}$,
$b=\frac{\theta_{\emph{U}}}{2\pi\sqrt{2k_{\emph{U}}}}$,
$c=\frac{1}{\sqrt{2k_{\emph{B}}}}$ and $d=\frac{1}{\sqrt{2k_{\emph{U}}}}$.

###### Proof.

See Appendix C. ∎

Although we are not able to derive a simpler expression of the ASE-optimal
unit training overhead $\beta^{*}_{\mathcal{A}}$, the numerical result of
$\beta^{*}_{\mathcal{A}}$ can be obtained with low complexity when the value
of other parameters are given. We will discuss the influence of other
parameters on $\beta^{*}_{\mathcal{A}}$ in the next section.

From Corollary 1 and Corollary 2, we notice that the optimal unit training
overhead to maximize the coverage probability and the ASE are different. As a
consequence, how to choose a proper unit training overhead to balance the
coverage probability (coverage performance) and the ASE (rate performance)
should be considered.

###  III-E Energy Efficiency

The expression of the EE, expressed as
$\mathcal{E}=\mathcal{A}/(\lambda_{\text{B}}P_{\text{B}}+\lambda_{\text{R}}P_{\text{R}})$,
can be directly obtained from the expression of $\mathcal{A}$. And the EE-
optimal unit training overhead is the same as the ASE-optimal unit training
overhead, i.e., $\beta^{*}_{\mathcal{E}}=\beta^{*}_{\mathcal{A}}$.

TABLE I: Default Values of Parameters Parameter | Notation and Value | Parameter | Notation and Value
---|---|---|---
Path loss exponent | $\alpha=4$ | Reflection power attenuation | $\gamma=0.85$
BS density | $\lambda_{\text{B}}=10\text{ BSs/km}^{2}$ | Blockage density | $\lambda_{\text{L}}=500\text{ Blockages/km}^{2}$
Number of antennas or elements | $M_{\text{B}}=16$ | RIS deployment fraction | $\mu=0.6$
$M_{\text{R}}=16$ | Size of blockages | $l=15\text{ m}$
$M_{\text{U}}=4$ | SINR threshold | $\tau=3\text{ dB}$
Beam alignment error parameters | $k_{\text{B}}=0.02$ | Frame length | $T=4480\text{ Symbols}$
$k_{\text{U}}=0.08$ | Unit training overhead | $\beta=1\text{ Symbols/path}$
Transmit or reflect power | $P_{\text{B}}=40\text{ dBm}$ | Normalized noise power | $N_{0}=-90\text{ dB}$
$P_{\text{R}}=15\text{ dBm}$ | Average SNR of the channel | $\text{SNR}=16\text{ dB}$

##  IV Numerical Results

In this section, we focus on the numerical results of the coverage
probability, the ASE and the EE with respect to the unit training overhead
$\beta$ and the RIS deployment fraction $\mu$. The properties of the ASE-
optimal (also the EE-optimal) unit training overhead $\beta^{*}_{\mathcal{A}}$
will be discussed under different system settings. The default values of
parameters are listed in Table I. In particular, the optimal performance
metrics are marked with stars in the following figures.

###  IV-A Performance w.r.t Resource Allocation

In this subsection, we study the performance metrics as functions of the unit
training overhead $\beta$ under different system settings, i.e., the number of
BS antennas $M_{\text{B}}$, beam alignment error parameters $k_{\text{B}}$ and
$k_{\text{U}}$, frame length $T$ and average SNR. In all the cases shown in
Fig. 5, 6, 7 and 8, the coverage probability increases monotonically w.r.t.
the unit training overhead $\beta$, as we have proved in Corollary 1. Since
using more resources on channel estimation brings a smaller beamforming error,
the association link has a lower probability of beam misalignment and the
coverage probability is promoted. As we have mentioned, the feasible region of
the unit training overhead is
$\beta\in[0,\frac{T}{M_{\text{B}}M_{\text{R}}M_{\text{U}}+M_{\text{B}}M_{\text{U}}})$,
and as a consequence, the maximum value of $\beta$ varies with the number of
BS antennas $M_{\text{B}}$ and frame length $T$ in Fig. 5 and 7, respectively.

(a) Coverage probability

(b) ASE and EE

Figure 5: Coverage probability, ASE and EE scaling with the unit training
overhead $\beta$ under different number of BS antennas $M_{\text{B}}$.

We first evaluate the performance metrics w.r.t the unit training overhead
$\beta$ affected by the number of BS antennas $M_{\text{B}}$ in Fig. 5. Since
the parameters of BSs and UEs are symmetric in our model, the influence of the
number of UE antennas is analogous. Generally, more antennas result in a
narrower beam and a higher probability of beam alignment error, while the
beamforming gain is also higher. When the number of BS antennas is relatively
small, e.g., $M_{\text{B}}=2,4,8$, the increasing beamforming gain dominates
the performance when $M_{\text{B}}$ increases, leading to increasing coverage
probability in Fig. 5(a). However, when the number of BS antennas is large,
e.g., $M_{\text{B}}=16,32$, the severe beam misalignment dominates the
performance and increasing $M_{\text{B}}$ leads to overall decreasing coverage
probability in Fig. 5(a). In Fig. 5(b), the ASE and the EE have the same
tendency, since the ratio of ASE to EE, i.e.,
$\mathcal{A}/\mathcal{E}=\lambda_{\text{B}}P_{\text{B}}+\lambda_{\text{R}}P_{\text{R}}$,
merely relies on network deployment and energy consumption. Similar to the
influence on the coverage probability, increasing the number of BS antennas
$M_{\text{B}}$ first increases and then decreases the ASE and the EE.
Moreover, a system equipped with more BS antennas is more sensitive to the
change of resource allocation, i.e., the variations of the ASE and the EE are
more dramatical w.r.t unit training overhead $\beta$ under large
$M_{\text{B}}$. The optimal unit training overhead $\beta^{*}_{\mathcal{A}}$
to maximize the ASE and the EE, marked with stars, first decreases, then
increases and finally decreases w.r.t the number of BS antennas
$M_{\text{B}}$, resulting from the trade-off between beam alignment error and
beamforming gain.

(a) Coverage probability

(b) ASE and EE

Figure 6: Coverage probability, ASE and EE scaling with the unit training
overhead $\beta$ under different beam alignment error parameters
$k_{\text{B}}$ and $k_{\text{U}}$.

(a) Coverage probability

(b) ASE and EE

Figure 7: Coverage probability, ASE and EE scaling with the unit training
overhead $\beta$ under different frame length $T$.

The performance under different levels of beam alignment errors of BSs and
UEs, i.e., $k_{\text{B}}$ and $k_{\text{U}}$, are displayed in Fig. 6.
Recalling that, the terms $k_{\text{B}}\pi^{2}$ and $k_{\text{U}}\pi^{2}$ are
the variances of the beam alignment error without channel estimation for BSs
and UEs, respectively. It is shown that the beam alignment error has more
impact on system performance than the number of antennas. Larger
$k_{\text{B}}$ and $k_{\text{U}}$ represent larger variances of the beam
alignment error. Consequently, the coverage probability, the ASE and the EE
have huge degradation as beam alignment error parameters $k_{\text{B}}$ and
$k_{\text{U}}$ increase. In addition, larger $k_{\text{B}}$ and $k_{\text{U}}$
lead to larger $\beta^{*}_{\mathcal{A}}$, indicating that more resources for
channel estimation should be allocated in order to compensate for the loss
brought by the beam alignment error.

Fig. 7 illustrates the performance under different frame length $T$. The
results reveal that increasing frame length does not influence the coverage
performance in Fig. 7(a), except for the region of $\beta$. On the contrary, a
longer frame length brings better ASE and EE performance in Fig. 7(b). And the
optimal unit training overhead $\beta^{*}_{\mathcal{A}}$ that maximizes the
ASE and the EE also increases monotonically with frame length $T$. The reason
is that a longer frame length $T$ leads to wider space for data transmission,
given the same channel estimation overhead. Moreover, the ASE and the EE have
the same tendency at the same $T$.

(a) Coverage probability

(b) ASE and EE

Figure 8: Coverage probability, ASE and EE scaling with the unit training
overhead $\beta$ under different average SNR.

The impact of average SNR is displayed in Fig. 8. Naturally, a higher average
SNR results in a smaller channel estimation error and thereby better
performance in all the cases. However, further increasing average SNR provides
marginal gain and the performance is approaching its limit, especially for the
coverage probability shown in Fig. 8(a). Besides, in Fig. 8(b), we find that
the ASE-optimal unit training overhead $\beta_{\mathcal{A}}^{*}$ decreases as
average SNR increases, because better channel environment consumes less
channel estimation resources to reach its maximum.

###  IV-B The Properties of the Optimal Unit Training Overhead
$\beta_{\mathcal{A}}^{*}$

Then we focus on the ASE-optimal (as well as the EE-optimal) unit training
overhead $\beta_{\mathcal{A}}^{*}$ derived in Corollary 2. The influence of
the number of BS antennas $M_{\text{B}}$, beam alignment error parameter
$k_{\text{B}}$, frame length $T$ and average SNR are discussed.

(a) The impact of the number of BS antennas $M_{\text{B}}$ and BS beam
alignment error parameter $k_{\text{B}}$

(b) The impact of average SNR and frame length $T$

Figure 9: The impact of some system parameters, i.e., the number of BS
antennas $M_{\text{B}}$, BS beam alignment error parameter $k_{\text{B}}$,
frame length $T$ and average SNR, to the optimal unit training overhead
$\beta_{\mathcal{A}}^{*}$.

Fig. 9(a) shows that two parameters of BS beamforming, i.e., $M_{\text{B}}$
and $k_{\text{B}}$, as influencing factors of $\beta_{\mathcal{A}}^{*}$. When
$k_{\text{B}}$ is quite small, e.g., $k_{\text{B}}\leq 0.08$, the optimal unit
training overhead $\beta_{\mathcal{A}}^{*}$ first decreases, then increases
and finally decreases as the number of BS antennas $M_{\text{B}}$ grows
(verified in Fig. 5). In contrast, the optimal unit training overhead
$\beta_{\mathcal{A}}^{*}$ first increases and then decreases when the number
of BS antennas $M_{\text{B}}$ increases under a relatively large
$k_{\text{B}}$, e.g., $0.08<k_{\text{B}}\leq 1$ in Fig. 9(a). In summary, the
optimal unit training overhead $\beta_{\mathcal{A}}^{*}$ as a function of the
number of BS antennas $M_{\text{B}}$ shows different tendencies in the regions
of high beam alignment error (approximately $0.08<k_{\text{B}}\leq 1$) and low
beam alignment error (approximately $k_{\text{B}}\leq 0.08$). However, in the
region of low beam alignment error, the variation of $\beta_{\mathcal{A}}^{*}$
between different $M_{\text{B}}$ is very small, i.e.,
$\beta_{\mathcal{A}}^{*}$ roughly falls in the rang of $(0.2,1.2)$.
Conversely, as $k_{\text{B}}$ increases, the probability of beam misalignment
increases, and the variation of $\beta_{\mathcal{A}}^{*}$ between different
$M_{\text{B}}$ becomes more drastic.

In Fig. 9(a), the red stars mark the maximum $\beta_{\mathcal{A}}^{*}$ and its
corresponding $M_{\text{B}}$ for each $k_{\text{B}}$. Firstly, it is suggested
that BSs with medium scale of antennas, e.g., $4\leq M_{\text{B}}\leq 16$,
should be allocated more resources on channel estimation, no matter the degree
of the beam alignment error. Since in this situation, beam misalignment
dominates the performance over beamforming gain. Secondly, when the number of
BS antennas is relatively large, e.g., $M_{\text{B}}>16$, beamwidth is quite
narrow and beam misalignment is always critical, which makes resource
allocation less useful, resulting in relatively few channel estimation
resources allocation. Finally, when the number of BS antennas is quite small,
e.g., $M_{\text{B}}=2$, channel estimation resources can be shrunk, because
the particularly wide beam is insensitive to beam misalignment. From another
point of view, the black diamonds mark the maximum $\beta_{\mathcal{A}}^{*}$
and the corresponding $k_{\text{B}}$ for each $M_{\text{B}}$. For most cases,
e.g., $M_{\text{B}}\leq 48$, a larger $k_{\text{B}}$ corresponds to allocating
more resources on channel estimation, in order to get a more accurate CSI
(verified in Fig. 6). While when the number of BS antennas is very large,
e.g., $M_{\text{B}}\geq 48$, a larger $k_{\text{B}}$ may not lead to a larger
$\beta_{\mathcal{A}}^{*}$ due to the resource saving concern. Yet
$\beta_{\mathcal{A}}^{*}$ for $M_{\text{B}}\geq 48$ is insensitive to both
$M_{\text{B}}$ and $k_{\text{B}}$.

Fig. 9(b) illustrates the impact of frame length $T$ and average SNR on the
optimal unit training overhead $\beta_{\mathcal{A}}^{*}$. In the left figure
of Fig. 9(b), the optimal unit training overhead $\beta_{\mathcal{A}}^{*}$
first increases and then decreases w.r.t average SNR. In the low SNR region,
e.g., $\text{SNR}<4\text{dB}$, the unit training overhead $\beta$ dominates
the variance of beam alignment error
($\sigma_{\text{B}}^{2}=k_{\text{B}}\pi^{2}/\sqrt{1+\beta\text{SNR}}$).
Therefore, higher SNR requires a larger $\beta_{\mathcal{A}}^{*}$ in the low
SNR region. On the contrary, in the high SNR region, beam alignment error is
low enough and a smaller $\beta_{\mathcal{A}}^{*}$ is preferred to reserve
more resources for data transmission (verified in Fig. 8). In the right figure
of Fig. 9(b), it is shown that longer frame length $T$ corresponds to more
resources to be allocated and a larger $\beta_{\mathcal{A}}^{*}$ should be
chosen (verified in Fig. 7).

###  IV-C Performance w.r.t Network Deployment

(a) Coverage probability

(b) ASE and EE

Figure 10: Coverage probability, ASE and EE scaling with the RIS deployment
fraction $\mu$ under different blockage density $\lambda_{\text{L}}$.

In this subsection, the performance metrics w.r.t the RIS deployment fraction
$\mu$ are evaluated under different blockage densities $\lambda_{\text{L}}$
and BS densities $\lambda_{\text{B}}$ to study the optimal network deployment
scheme.

Fig. 10 shows the performance under different blockage densities
$\lambda_{\text{L}}$. Overall, high density of blockages harms the coverage
probability, the ASE and the EE. And the optimal RIS deployment fraction
increases as the blockage density increases. When the blockage density is
small, e.g. $\lambda_{\text{L}}=100\text{ Blockages/km}^{2}$, most of the UEs
associate to serving BSs through direct links. Consequently, deploying more
RISs provides minor gain on desired signal enhancement, but the total
interference is greatly enlarged by reflection. Accordingly, the overall
performance even degrades as more RISs are deployed. However, when the network
has intensive blockages and most of the BS-UE links are blocked, e.g.
$\lambda_{\text{L}}=500\text{ Blockages/km}^{2}$ and
$\lambda_{\text{L}}=800\text{ Blockages/km}^{2}$, deploying appropriate amount
of RISs enhances the performance notably as RISs provide additional
communication links to NLoS UEs, and the impact of reflected interference is
well compensated. Fig. 10(b) indicates that the ASE-optimal RIS deployment
fraction may not equals to the EE-optimal RIS deployment fraction. And this
result reveals that trade-offs between the ASE and the EE performance should
be considered when designing RIS deployment schemes. In this case, the ASE-
optimal RIS deployment fraction significantly exceeds the EE-optimal RIS
deployment fraction. Especially, the EE performance highly depends on the
ratio $P_{\text{B}}/P_{\text{R}}$, and thus we should deploy more RISs when
deploying RISs is ‘energy-economical’.

(a) Coverage probability

(b) ASE and EE

Figure 11: Coverage probability, ASE and EE scaling with the RIS deployment
fraction $\mu$ under different BS density $\lambda_{\text{B}}$.

Finally, the performance under varying BS density $\lambda_{\text{B}}$ is
displayed in Fig. 11. Deploying denser BSs enhances the coverage probability,
the ASE and the EE as the average UE-BS distance is shortened. However,
deploying more RISs brings performance degradation under dense BS deployment,
especially for the EE in Fig. 11(b), since BS-UE direct links dominate the
association links. The optimal RIS deployment fraction is a monotonically
decreasing function of the BS density. This inspires a potential way of
replacing part of BSs with RISs for a lower cost, if the performance loss is
within tolerance. To summarize Fig.10 and 11, the greater the ratio of
blockage density to BS density, the better the performance of deploying RISs,
and the denser RISs should be.

##  V Conclusion

This paper has investigated the coverage probability, the ASE and the EE of a
downlink RIS-aided multi-cell network with directional transmissions, where
the optimal resource allocation and RIS deployment for network performance
have been studied. The reflection of interference by RISs, which has not
received enough attention, is especially considered. A general model on
resource allocation between channel estimation and data transmission is
introduced, and the relationship between the beam alignment error and the
channel estimation overhead is characterized. Based on these models, the
analytical expressions of the coverage probability, the ASE and the EE are
derived. Moreover, the monotonicity of the coverage probability w.r.t. the
unit training overhead is proved. And the properties of the optimal unit
training overhead w.r.t other system parameters, i.e., the number of antennas,
beam alignment error parameters, frame length and average SNR, are revealed.
Numerical results indicate that the reflection of interference is notable and
more RISs are needed only when the ratio of blockage density to BS density is
large. However, replacing part of BSs with RISs may result in performance
loss, even with lower cost. In the future, we will design more practical
interference cancellation and beamforming schemes to alleviate the
interference brought by RISs. Methods to reduce the channel estimation
overhead in the RIS-aided multi-cell networks with directional transmissions
are also worth studying.

## Appendix A: Proof of Theorem 1

For the first conditional coverage probability term
$\mathbb{P}[\text{SINR}\\!>\\!\tau|\mathcal{D}]$ in Eq. (33), we have the
following expressions

$\displaystyle\begin{split}\mathbb{P}[\text{SINR}\\!>\\!\tau|\mathcal{D}]=&\mathbb{E}_{r_{\text{D}}}\Bigg{[}\mathbb{P}\\!\left[\frac{G_{0}hP\\!L_{\text{D}}(r_{\text{D}})}{N_{0}+I_{\text{D}}^{\text{d}}+I_{\text{R}}^{\text{d}}}>\tau\Bigg{|}r_{\text{D}}\right]\Bigg{]}\\\
\overset{\text{(a)}}{=}&\int_{0}^{\infty}\\!\mathbb{P}\\!\left[\frac{G_{0}hP\\!L_{\text{D}}(r_{\text{D}})}{N_{0}+I_{\text{D}}^{\text{d}}+I_{\text{R}}^{\text{d}}}>\tau\Bigg{|}r_{\text{D}}\right]\\!f_{\text{D}}(r_{\text{D}})\text{d}r_{\text{D}}\\\
\overset{\text{(b)}}{=}&\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!p_{\text{E}_{\text{B}}}\\!(\sigma_{\text{B}},\theta_{\text{B}})p_{\text{E}_{\text{U}}}\\!(\sigma_{\text{U}},\theta_{\text{U}})\mathbb{P}\\!\left[\frac{N_{\text{B}}N_{\text{U}}hP\\!L_{\text{D}}(r_{\text{D}})}{N_{0}+I_{\text{D}}^{\text{d}}+I_{\text{R}}^{\text{d}}}\\!\\!>\\!\tau\Bigg{|}r_{\text{D}}\\!\right]\\!\\!f_{\text{D}}(r_{\text{D}})\text{d}r_{\text{D}}\\\
\overset{\text{(c)}}{=}&\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!p_{\text{E}_{\text{B}}}\\!(\sigma_{\text{B}},\theta_{\text{B}})p_{\text{E}_{\text{U}}}\\!(\sigma_{\text{U}},\theta_{\text{U}})\exp(-s_{1}N_{0})\mathcal{L}_{I_{\text{D}}^{\text{d}}}(s_{1})\mathcal{L}_{I_{\text{R}}^{\text{d}}}(s_{1})f_{\text{D}}(r_{\text{D}})\text{d}r_{\text{D}},\end{split}$
(36)

where
$s_{1}=\frac{\tau}{N_{\text{B}}N_{\text{U}}P\\!L_{\text{D}}(r_{\text{D}})}$
for simplicity, and $r_{\text{D}}$ is the link distance between the nearest
LoS BS and the typical user, while $I_{\text{D}}^{\text{d}}$ and
$I_{\text{R}}^{\text{d}}$ are the normalized direct interference and the
normalized reflected interference under a direct association link,
respectively. The variables $\mathcal{L}_{I_{\text{D}}^{\text{d}}}(s_{1})$ and
$\mathcal{L}_{I_{\text{R}}^{\text{d}}}(s_{1})$ indicate the Laplace transforms
of $I_{\text{D}}^{\text{d}}$ and $I_{\text{R}}^{\text{d}}$ evaluated at
$s_{1}$, respectively. Equation (a) follows from the total probability theorem
and the PDF of $r_{\text{D}}$ is given in Eq.(16). Equation (b) follows from
the fact that $G_{0}$ is i.i.d. for each link and the distribution of $G_{0}$
is shown in Eq.(9). Equation (c) follows from the exponential distribution of
$h$.

Then we focus on the Laplace transform term
$\mathcal{L}_{I_{\text{D}}^{\text{d}}}(s_{1})$, which is expressed as

$\displaystyle\begin{split}\mathcal{L}_{I_{\text{D}}^{\text{d}}}(s_{1})=&\mathbb{E}_{I_{\text{D}}^{\text{d}}}\\!\left[\exp\\!\left(-s_{1}I_{\text{D}}^{\text{d}}\right)\right]\\\
\overset{\text{(a)}}{=}&\mathbb{E}_{\phi_{\text{B}}}\\!\\!\left[\prod_{i:B_{i}\in\phi_{\text{B}}\\!\backslash\\{\\!B^{*}\\!\\}}\\!\\!\\!\\!\\!\mathbb{E}_{G_{i},h,\mathbb{I}_{\text{L}}}\\!\Big{[}\\!\exp\\!\left(-s_{1}\mathbb{I}_{\text{L}}(r_{B_{i}})P\\!L_{\text{D}}(r_{B_{i}})G_{i}h\right)\\!\Big{]}\\!\right]\\\
\overset{\text{(b)}}{=}&\mathbb{E}_{\phi_{\text{B}}}\\!\Bigg{[}\\!\prod_{i:B_{i}\in\phi_{\text{B}}\\!\backslash\\{\\!B^{*}\\!\\}}\\!\\!\\!\\!\\!p_{\text{L}}(r_{B_{i}})\mathbb{E}_{G_{i},h}\Big{[}\\!\exp(-s_{1}P\\!L_{\text{D}}(r_{B_{i}})G_{i}h)\Big{]}+1\\!-\\!p_{\text{L}}(r_{B_{i}})\Bigg{]}\\\
\overset{\text{(c)}}{=}&\mathbb{E}_{\phi_{\text{B}}}\\!\Bigg{[}\\!\prod_{i:B_{i}\in\phi_{\text{B}}\\!\backslash\\{\\!B^{*}\\!\\}}\\!\\!\\!\frac{\theta_{\text{B}}\theta_{\text{U}}}{4\pi^{2}}p_{\text{L}}(r_{B_{i}})\mathbb{E}_{h}\Big{[}\\!\exp\\!\left(\\!-s_{1}P\\!L_{\text{D\\!}}(r_{B_{i}})N_{\text{\\!B}}N_{\text{\\!U}}h\right)\\!\Big{]}+1-\frac{\theta_{\text{B}}\theta_{\text{U}}}{4\pi^{2}}p_{\text{L}}(r_{B_{i}})\\!\Bigg{]}\\\
\overset{\text{(d)}}{=}&\mathbb{E}_{\phi_{\text{B}}}\\!\\!\left[\\!\prod_{i:B_{i}\in\phi_{\text{B}}\\!\backslash\\{\\!B^{*}\\!\\}}\left(1-\frac{\theta_{\text{B}}\theta_{\text{U}}}{4\pi^{2}}\frac{s_{1}p_{\text{L\\!}}(r_{B_{i}})P\\!L_{\text{D\\!}}(r_{B_{i}})N_{\text{\\!B}}N_{\text{\\!U}}}{1+s_{1}P\\!L_{\text{D\\!}}(r_{B_{i}})N_{\text{\\!B}}N_{\text{\\!U}}}\right)\right]\\\
\overset{\text{(e)}}{=}&\exp\\!\left(\\!-\frac{\lambda_{\text{B}}\theta_{\text{B}}\theta_{\text{U}}}{2\pi}\\!\\!\int_{r_{\text{D}}}^{\infty}\\!\frac{s_{1}p_{\text{L\\!}}(u)P\\!L_{\text{D\\!}}(u)N_{\text{\\!B}}N_{\text{\\!U}}}{1+s_{1}P\\!L_{\text{D\\!}}(u)N_{\text{\\!B}}N_{\text{\\!U}}}u\text{d}u\\!\right),\end{split}$
(37)

where (a) holds because $G_{i},h$ and $\mathbb{I}_{\text{L}}(r)$ are i.i.d.
for each link. Equations (b) and (c) follow by the distributions of
$\mathbb{I}_{\text{L}}(r)$ and $G_{i}$ given in Eq. (1) and Eq. (10). Equation
(d) follows from the exponential distribution of $h$ and (e) follows from the
PGFL of PPP with variables substitution.

The Laplace transform term $\mathcal{L}_{I_{\text{R}}^{\text{d}}}(s_{1})$ can
be derived with a similar process and we stretch the major steps

$\displaystyle\begin{split}&\mathcal{L}_{I_{\text{R}}^{\text{d}}}(s_{1})\\\
=&\mathbb{E}_{\widehat{\phi}_{\text{B}},\widehat{\phi}_{\text{R}},G_{i},h,\mathbb{I}_{\text{L}},\mathbb{I}_{\text{F}}}\\!\Bigg{[}\exp\\!\Bigg{(}\\!\\!\\!-\\!s_{1}\\!\\!\\!\sum_{i:B_{i}\in\widehat{\phi}_{\text{B}}}\sum_{k:R_{k}\in\widehat{\phi}_{\text{R}}}\\!\\!\\!\mathbb{I}_{\text{L}}(\\!r_{B_{i}\\!R_{k}}\\!)\mathbb{I}_{\text{L}}(\\!r_{R_{k}}\\!)\mathbb{I}_{\text{F}}(\\!r_{B_{i}},r_{R_{k}},\angle{B_{i}oR_{k}}\\!)P\\!L_{\text{R}}(\\!r_{B_{i}\\!R_{k}}\\!\\!+\\!r_{R_{k}}\\!)G_{i}h\\!\Bigg{)}\\!\Bigg{]}\\\
=&\mathbb{E}_{\widehat{\phi}_{\text{B}},\widehat{\phi}_{\text{R}}}\\!\Bigg{[}\prod_{i:B_{i}\in\widehat{\phi}_{\text{B}}}\prod_{k:R_{k}\in\widehat{\phi}_{\text{R}}}\\!\\!\\!\Bigg{(}\\!\\!1\\!-\\!\frac{\theta_{\text{B}}\theta_{\text{U}}}{4\pi^{2}}\frac{s_{1}P\\!L_{\text{R\\!}}(\\!r_{B_{i}\\!R_{k}}\\!\\!+\\!r_{R_{k}}\\!)N_{\text{\\!B}}N_{\text{\\!U}}}{1\\!+\\!s_{1}\\!P\\!L_{\text{R\\!}}(\\!r_{B_{i}\\!R_{k}}\\!\\!+\\!r_{R_{k}}\\!)N_{\text{\\!B}}N_{\text{\\!U}}}p_{\text{L\\!}}(\\!r_{B_{i}\\!R_{k}}\\!)p_{\text{L\\!}}(\\!r_{R_{k}}\\!)p_{\text{F}}(r_{B_{i}},r_{R_{k}},\angle{B_{i}oR_{k}})\\!\Bigg{)}\\!\\!\Bigg{]}\\\
\overset{\text{(a)}}{=}&\exp\\!\Bigg{(}\\!\\!-\\!2\pi\\!\lambda_{\text{B}}\\!\\!\int_{0}^{\infty}\\!\\!\Bigg{(}\\!1\\!-\\!\exp\\!\Big{(}\\!\\!-\\!\\!\frac{\lambda_{\text{R}}\theta_{\text{B}}\theta_{\text{U}}}{4\pi^{2}}\\!\\!\int_{\mathcal{C}_{2}}\frac{s_{1}P\\!L_{\text{R\\!}}\\!\left(\\!\\!\sqrt{\\!u^{2}\\!+\\!t^{2}\\!-\\!2ut\cos\psi}\\!+\\!t\\!\right)\\!N_{\text{\\!B}}N_{\text{\\!U}}}{1\\!+\\!s_{1}P\\!L_{\text{R\\!}}\\!\left(\\!\\!\sqrt{\\!u^{2}\\!+\\!t^{2}\\!-\\!2ut\cos\psi}\\!+\\!t\\!\right)\\!N_{\text{\\!B}}N_{\text{\\!U}}}\cdot\\\
&\quad\quad
p_{\text{L\\!}}\\!\left(\\!\\!\sqrt{\\!u^{2}\\!+\\!t^{2}\\!-\\!2ut\cos\psi}\right)\\!p_{\text{L\\!}}(t)p_{\text{F}}(u,t,\psi)t\text{d}t\text{d}\psi\\!\Big{)}\\!\\!\Bigg{)}u\text{d}u\\!\\!\Bigg{)},\end{split}$
(38)

where in (a), the BSs and the RISs in $\widehat{\phi}_{\text{B}}$ and
$\widehat{\phi}_{\text{R}}$ are located in the region
$\mathcal{C}_{2}=\\{t,\psi:r_{\text{D}}^{-\alpha}\geq\gamma(\\!\sqrt{\\!u^{2}\\!+\\!t^{2}\\!-\\!2ut\cos\psi}+t)^{-\alpha}\\}$,
indicating that interfering links should have weaker average received signal
power than the association link. The region $\mathcal{C}_{2}$ can be rewritten
as the union of two sub-regions in Theorem 1. As a consequence, the full
expression of $\mathbb{P}[\text{SINR}\\!>\\!\tau|\mathcal{D}]$ is obtained by
combining Eq. (36) with Eq. (8), (16), (37) and (38).

For the second conditional coverage probability term
$\mathbb{P}[\text{SINR}\\!>\\!\tau|\mathcal{R}]$, we have

$\displaystyle\begin{split}\mathbb{P}[\text{SINR}\\!>\\!\tau|\mathcal{R}]=&\mathbb{E}_{r_{\text{R}}}\Bigg{[}\mathbb{P}\\!\left[\frac{G_{0}hP\\!L_{\text{R}}(r_{\text{R}})}{N_{0}+I_{\text{D}}^{\text{r}}+I_{\text{R}}^{\text{r}}}>\tau\Bigg{|}r_{\text{R}}\right]\Bigg{]}\\\
=&\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!p_{\text{E}_{\text{B}}}\\!(\sigma_{\text{B}},\theta_{\text{B}})p_{\text{E}_{\text{U}}}\\!(\sigma_{\text{U}},\theta_{\text{U}})\exp(-s_{2}N_{0})\mathcal{L}_{I_{\text{D\\!}}^{\text{r}}}(s_{2})\mathcal{L}_{I_{\text{R\\!}}^{\text{r}}}(s_{2})f_{\text{R}}(r_{\text{R}})\text{d}r_{\text{R}},\end{split}$
(39)

where
$s_{2}=\frac{\tau}{N_{\text{B}}N_{\text{U}}P\\!L_{\text{R}}(r_{\text{R}})}$,
while $\mathcal{L}_{I_{\text{D}}^{\text{r}}}(s_{2})$ and
$\mathcal{L}_{I_{\text{R}}^{\text{r}}}(s_{2})$ indicate the Laplace transforms
of $I_{\text{D}}^{\text{r}}$ and $I_{\text{R}}^{\text{r}}$ evaluated at
$s_{2}$, respectively. For the limited pages, we neglect the derivation of
$\mathcal{L}_{I_{\text{D}}^{\text{r}}}(s_{2})$ and
$\mathcal{L}_{I_{\text{R}}^{\text{r}}}(s_{2})$. The results are

$\displaystyle\begin{split}\mathcal{L}_{I_{\text{D}}^{\text{r}}}(s_{2})=&\exp\\!\left(\\!\\!-\frac{\lambda_{\text{B}}\theta_{\text{B}}\theta_{\text{U}}}{2\pi}\\!\\!\int_{r_{\text{R}}\gamma^{-\frac{1}{\alpha}}}^{\infty}\\!\frac{s_{2}p_{\text{L\\!}}(u)P\\!L_{\text{D\\!}}(u)N_{\text{B}}N_{\text{U}}}{1+s_{2}P\\!L_{\text{D\\!}}(u)N_{\text{B}}N_{\text{U}}}u\text{d}u\\!\\!\right),\end{split}$
(40)
$\displaystyle\begin{split}\mathcal{L}_{I_{\text{R}}^{\text{r}}}(s_{2})=&\exp\\!\Bigg{(}\\!\\!-\\!2\pi\\!\lambda_{\text{B}}\\!\\!\int_{0}^{\infty}\\!\\!\Bigg{(}\\!1\\!-\\!\exp\\!\Big{(}\\!\\!-\\!\\!\frac{\lambda_{\text{R}}\theta_{\text{B}}\theta_{\text{U}}}{4\pi^{2}}\\!\\!\int_{\mathcal{C}_{3}}\frac{s_{2}P\\!L_{\text{R\\!}}\\!\left(\\!\\!\sqrt{\\!u^{2}\\!+\\!t^{2}\\!-\\!2ut\cos\psi}\\!+\\!t\\!\right)\\!N_{\text{\\!B}}N_{\text{\\!U}}}{1\\!+\\!s_{2}P\\!L_{\text{R\\!}}\\!\left(\\!\\!\sqrt{\\!u^{2}\\!+\\!t^{2}\\!-\\!2ut\cos\psi}\\!+\\!t\\!\right)\\!N_{\text{\\!B}}N_{\text{\\!U}}}\cdot\\\
&\quad\quad
p_{\text{L\\!}}\\!\left(\\!\\!\sqrt{\\!u^{2}\\!+\\!t^{2}\\!-\\!2ut\cos\psi}\right)\\!p_{\text{L\\!}}(t)p_{\text{F}}(u,t,\psi)t\text{d}t\text{d}\psi\\!\Big{)}\\!\\!\Bigg{)}u\text{d}u\\!\\!\Bigg{)},\end{split}$
(41)

where
$\mathcal{C}_{3}=\\{t,\psi:\sqrt{\\!u^{2}\\!+\\!t^{2}\\!-\\!2ut\cos\psi}+t>r_{\text{R}}\\}$
based on the same thought of $\mathcal{C}_{2}$. And $\mathcal{C}_{3}$ can also
be rewritten as the union of two sub-regions. The full expression of
$\mathbb{P}[\text{SINR}\\!>\\!\tau|\mathcal{R}]$ is obtained by combining Eq.
(39) with (8), (19), (40) and (41). Finally, the expression of $\mathcal{P}$
in Theorem 1 is obtained with variables substitution.

## Appendix B: Proof of Corollary 1

Since the unit training overhead $\beta$ affects the coverage probability in
$p_{\text{E}_{\text{B}}}\\!(\sigma_{\text{B}},\\!\theta_{\text{B}}\\!)p_{\text{E}_{\text{U}}}\\!(\sigma_{\text{U}},\\!\theta_{\text{U}}\\!)$,
which is a multiplier of $\mathcal{P}$ in Theorem 1. We have following
equations

$\displaystyle\begin{split}\frac{\text{d}\mathcal{P}}{\text{d}\beta}=&K\frac{\text{d}(p_{\text{E}_{\text{B}}}(\sigma_{\text{B}},\theta_{\text{B}})p_{\text{E}_{\text{U}}}(\sigma_{\text{U}},\theta_{\text{U}}))}{\text{d}\beta}\\\
\overset{(\text{a})}{=}&K\frac{\text{d}\left(\frac{\text{erf}\left(\frac{\theta_{\text{B}}}{2\pi}\frac{\sqrt{1+\beta\text{SNR}}}{\sqrt{2k_{\text{B}}}}\right)\text{erf}\left(\frac{\theta_{\text{U}}}{2\pi}\frac{\sqrt{1+\beta\text{SNR}}}{\sqrt{2k_{\text{U}}}}\right)}{\text{erf}\left(\frac{\sqrt{1+\beta\text{SNR}}}{\sqrt{2k_{\text{B}}}}\right)\text{erf}\left(\frac{\sqrt{1+\beta\text{SNR}}}{\sqrt{2k_{\text{U}}}}\right)}\right)}{\text{d}\beta}\\\
\overset{(\text{b})}{=}&K\frac{\text{d}\left(\frac{\text{erf}\left(af(\beta)\right)\text{erf}\left(bf(\beta)\right)}{\text{erf}\left(cf(\beta)\right)\text{erf}\left(df(\beta)\right)}\right)}{\text{d}f(\beta)}\frac{\text{d}f(\beta)}{\text{d}\beta}\\\
\overset{(\text{c})}{=}&\frac{K\text{SNR}}{2\sqrt{1+\beta\text{SNR}}}\frac{\text{d}\left(\frac{\text{erf}\left(af(\beta)\right)\text{erf}\left(bf(\beta)\right)}{\text{erf}\left(cf(\beta)\right)\text{erf}\left(df(\beta)\right)}\right)}{\text{d}f(\beta)},\end{split}$
(42)

where $K$ is a positive constant that represents the multiplier in the
coverage probability $\mathcal{P}$ besides the term
$p_{\text{E}_{\text{B}}}(\sigma_{\text{B}},\theta_{\text{B}})p_{\text{E}_{\text{U}}}(\sigma_{\text{U}},\theta_{\text{U}})$.
Equation (a) follows from the definitions of $\sigma_{B}^{2}$ and
$\sigma_{U}^{2}$ in Section II-F. In (b), we use
$f(\beta)=\sqrt{1+\beta\text{SNR}}$,
$a=\frac{\theta_{\text{B}}}{2\pi\sqrt{2k_{\text{B}}}}$,
$b=\frac{\theta_{\text{U}}}{2\pi\sqrt{2k_{\text{U}}}}$,
$c=\frac{1}{\sqrt{2k_{\text{B}}}}$ and $d=\frac{1}{\sqrt{2k_{\text{U}}}}$ for
simplicity. Note that, the scope of parameters and variables are: $a\in(0,c]$,
$b\in(0,d]$, $c\in(0,+\infty)$, $d\in(0,+\infty)$, $\text{SNR}\in(0,+\infty)$,
$\beta\in[0,\frac{T}{M_{\text{B}}M_{\text{R}}M_{\text{U}}+M_{\text{B}}M_{\text{U}}})$
and $f(\beta)\in[1,+\infty)$.

Since the first term in (c) is positive, i.e.,
$\frac{K\text{SNR}}{2\sqrt{1+\beta\text{SNR}}}>0$, we focus on the second term

$\displaystyle\begin{split}&\frac{\text{d}\left(\frac{\text{erf}\left(af(\beta)\right)\text{erf}\left(bf(\beta)\right)}{\text{erf}\left(cf(\beta)\right)\text{erf}\left(df(\beta)\right)}\right)}{\text{d}f(\beta)}\\\
=&\frac{2\text{erf}(af(\beta)\\!)\text{erf}(bf(\beta)\\!)}{\sqrt{\pi}f(\beta)\text{erf}(cf(\beta)\\!)\text{erf}(df(\beta)\\!)}\\!\\!\left(\\!\frac{af\\!(\beta)e^{-a^{2}\\!f^{2}\\!(\beta)}}{\text{erf}(af\\!(\beta)\\!)}\\!-\\!\frac{cf\\!(\beta)e^{-c^{2}\\!f^{2}\\!(\beta)}}{\text{erf}(cf\\!(\beta)\\!)}\\!+\\!\frac{bf\\!(\beta)e^{-b^{2}\\!f^{2}\\!(\beta)}}{\text{erf}(bf\\!(\beta)\\!)}\\!-\\!\frac{df\\!(\beta)e^{-d^{2}\\!f^{2}\\!(\beta)}}{\text{erf}(df\\!(\beta)\\!)}\\!\right).\end{split}$
(43)

Obviously, the first term outside the brackets is positive. And each term in
the formula inside the brackets has the form of
$g(x)=xe^{-x^{2}}/\text{erf}(x)$ with $x>0$. Because $g(x)$ is a monotonically
decreasing function of $x$, then $g(af(\beta))-g(cf(\beta))\geq 0$ and
$g(bf(\beta))-g(df(\beta))\geq 0$ hold. Therefore, we have proved that
$\frac{\text{d}\mathcal{P}}{\text{d}\beta}>0$ and $\mathcal{P}$ is a
monotonically increasing function of $\beta$.

## Appendix C: Proof of Corollary 2

Since the unit training overhead $\beta$ only influences the ASE on its
multiplier term

$\displaystyle\frac{T\\!-\\!\beta(M_{\text{B}}M_{\text{R}}M_{\text{U}}\\!+\\!M_{\text{B}}M_{\text{U}})}{T}p_{\text{E}_{\text{B}}}(\sigma_{\text{B}},\theta_{\text{B}})p_{\text{E}_{\text{U}}}(\sigma_{\text{U}},\theta_{\text{U}}).$
(44)

Based on similar thoughts of the proof for Corollary 1, letting
$\frac{\text{d}\mathcal{A}}{\text{d}\beta}=0$ leads to

$\displaystyle\begin{split}&\frac{\text{d}\left(\left(\\!\frac{T}{M_{\text{B}}M_{\text{R}}M_{\text{U}}\\!+\\!M_{\text{B}}M_{\text{U}}}\\!-\\!\beta\\!\right)\\!p_{\text{E}_{\text{B}}}(\sigma_{\text{B}},\theta_{\text{B}})p_{\text{E}_{\text{U}}}(\sigma_{\text{U}},\theta_{\text{U}})\right)}{\text{d}\beta}=0,\end{split}$
(45)

and thus the final expression in Corollary 2 holds.

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