# Performance Analysis of IOS-Assisted NOMA System with Channel Correlation
and Phase Errors

Tianxiong Wang, Mihai-Alin Badiu, Gaojie Chen,
and Justin P. Coon This work was supported by EPSRC grant number EP/T02612X/1.
For the purpose of Open Access, the author has applied a CC BY public
copyright licence to any Author Accepted Manuscript (AAM) version arising from
this submission. (Corresponding author: Gaojie Chen)T. Wang, M. A. Badiu and
J. P. Coon are with the Department of Engineering Science, University of
Oxford, Oxford, OX1 3PJ, U.K. (e-mail: {tianxiong.wang, mihai.badiu,
justin.coon}@eng.ox.ac.uk).G. Chen is with 5GIC & 6GIC, Institute for
Communication Systems (ICS), University of Surrey, Guildford, GU2 7XH, United
Kingdom (e-mail: gaojie.chen@surrey.ac.uk).

###### Abstract

In this paper, we investigate the performance of an intelligent omni-surface
(IOS) assisted downlink non-orthogonal multiple access (NOMA) network with
phase quantization errors and channel estimation errors, where the channels
related to the IOS are spatially correlated. First, upper bounds on the
average achievable rates of the two users are derived. Then, channel hardening
is shown to occur in the proposed system, based on which we derive
approximations of the average achievable rates of the two users. The
analytical results illustrate that the proposed upper bound and approximation
on the average achievable rates are asymptotically equivalent in the number of
elements. Furthermore, it is proved that the asymptotic equivalence also holds
for the average achievable rates with correlated and uncorrelated channels.
Additionally, we extend the analysis by evaluating the average achievable
rates for IOS assisted orthogonal multiple access (OMA) and IOS assisted
multi-user NOMA scenarios. Simulation results corroborate the theoretical
analysis and demonstrate that: i) low-precision elements with only two-bit
phase adjustment can achieve the performance close to the ideal continuous
phase shifting scheme; ii) The average achievable rates with correlated
channels and uncorrelated channels are asymptotically equivalent in the number
of elements; iii) IOS-assisted NOMA does not always perform better than OMA
due to the reconfigurability of IOS in different time slots.

###### Index Terms:

Intelligent omni-surface (IOS), NOMA, spatial correlation, average achievable
rate

##  I Introduction

The future sixth-generation (6G) is expected to support billions of connected
devices with significant demands in spectrum efficiency, reliability, latency,
and connectivity [1]. To handle these challenging requirements, a wide range
of technologies, including massive multiple-input multiple-output (MIMO),
network densification, and millimeter-wave (mm-wave) communications, have been
investigated extensively in recent years [2]. Among various technologies, non-
orthogonal multiple access (NOMA) has been regarded as a superior multiple
access technology in the future wireless networks due to its promising
advantages in terms of spectrum efficiency [3, 4, 5]. For example, multiple
users could share the same resource block simultaneously in NOMA [6]. As
pointed in [7], NOMA has better performance than orthogonal multiple access
(OMA) when the channels gains of users are remarkably different.
Conventionally, the channel gains of the users are determined by the
uncontrollable wireless propagation environment [8, 9]. Hence, the
applications of NOMA are limited in conventional wireless networks [10].

With the rapid development of metasurfaces, reconfigurable intelligent
surfaces (RISs) have been recognized as a promising technology in the next-
generation wireless communications networks [11]. Physically, a RIS consists
of a large number of passive reflecting elements, each of which can shift the
phases of the incident electromagnetic (EM) waves. Through intelligent phase
shifting, a RIS could increase or decrease the composite channel gains of
different users [12, 13]. Due to the advantage of RIS in reconfiguring the
channels, many existing works investigated the potential of applying RIS in
NOMA systems to broaden the applications of NOMA. The authors in [14] and [15]
analyzed the performance of RIS assisted NOMA networks where each element of
the RIS applies 1-bit coding scheme. The authors in [16] proposed a priority
oriented design where the RIS is controlled to boost the channel of the
prioritized NOMA user and evaluated the system performance. The uplink outage
probability of a RIS-aided NOMA network was derived in [17]. Considering
coherent and random phase shifting, the authors in [18] examined the impact of
these two phase-shifting schemes on the outage performance of a RIS-assisted
NOMA system. The capacity of a RIS-empowered NOMA network with multiple users
was analyzed in [19]. In [20], the passive beamforming at RIS were optimized
for minimizing the transmit power and the performance between RIS-assisted
NOMA and RIS-assisted OMA were compared.

However, the above works considered the reflecting-only RIS, which requires
users located on the same side as the transmitter. As a result, the users
located on the other side of the reflecting-only RIS are blocked, which limits
the service coverage of RIS. The concept of intelligent omni-surface (IOS)
with dual signal refraction and reflection functions was proposed to tackle
this problem in some recent works. In contrast to the conventional reflecting-
only RIS, signals impinging on one side of the IOS can be reflected and
transmitted to users on the same and opposite sides of the surface as the
transmitter, as illustrated in Fig. 1. A prototype of IOS was developed by NTT
DOCOMO, Japan [21]. By dynamically adjusting the space between the metasurface
and the substrate, the incident radio waves can be entirely reflected,
entirely transmitted, or simultaneously reflected and transmitted without
attenuation. The authors in [22] optimized the spectral efficiency of a
single-input single-output (SISO) communication link assisted by an IOS where
the transmitting and reflecting signals share the same phase shift and showed
the deployment of IOS can significantly enlarge the wireless coverage. As a
step further, the joint active and passive beamforming design in a downlink
IOS-assisted communication network with a multiple-antenna access point (AP)
and multiple users was studied in [23, 24]. The authors in [25] proposed to
use an IOS to alleviate the inter-cell interference between two independent
access points (APs). IOS-empowered physical layer security was investigated in
[26]. The authors in [27] proposed a new structure of IOS, also referred to as
simultaneous transmitting and reflecting reconfigurable intelligent surface
(STAR-RIS), where the phase shifts of the reflecting and transmitting signals
can be configured independently. The authors in [28] studied the near-field
and far-field channel models of an IOS-assisted communication network. In
[29], active beamforming at the base station (BS) and passive transmitting and
reflecting beamforming at the IOS were jointly optimized to minimize the power
consumption of the network. IOS-aided NOMA systems were also studied in
several recent works. Under the constraint on the communication requirement of
each user, the coverage range of an IOS-assisted NOMA network was optimized in
[30]. The authors in [31] proposed a multi-user IOS-assisted NOMA system and
optimized the sum-rate with requirements on the quality-of-service of
individual users. The outage probability and diversity gain of an IOS
empowered NOMA system was derived in [32].

Although IOS-assisted communications have attracted significant research
interests recently, the existing works have not covered several aspects of
performance analysis and system design. First, most of the current works do
not consider the effect of channel correlation in the performance analysis of
IOS-assisted systems. However, as pointed in [33], the assumption of
independent channels is valid only when the half-wavelength spaced elements
are placed in a linear array. Hence, the performance of IOS-assisted networks
with correlated channels requires more investigation. Second, the existing
works on the performance analysis of IOS-assisted networks usually assume the
channel estimation and phase adjustment are perfect. However, imprecise
channel estimation and phase quantization lead to phase errors in practice. It
is essential to include phase errors in the performance analysis.

This paper presents novel results on the average achievable rates of an
energy-splitting IOS-assisted NOMA network with correlated channels and
imperfect phase adjustment. The main contributions of this paper are
summarized as follows:

  1. 1.

We analyze a downlink energy-splitting IOS-assisted NOMA communication system
with spatially correlated channels and phase errors. Upper bounds on the
average achievable rates of the users with strong and weak channel conditions
are derived by using Jensen’s inequality.

  2. 2.

We reveal the channel hardening effect in the IOS-assisted NOMA system with
correlated channels and phase errors. Then, using the results of the channel
hardening effect, two approximate expressions of the average achievable rates
are obtained. We also analyze the impact of discrete phase shifts on the
performance and show that the improvement brought by having more bits for
phase adjustment decreases with the increase of phase adjustment bits. Hence,
it is not beneficial to build high-precision elements in practice.

  3. 3.

The upper bound derived from Jensen’s inequality and the approximation derived
from the channel hardening effect are both proved to be asymptotically
equivalent. Moreover, the average achievable rate with correlated channels is
asymptotically equivalent to that with uncorrelated channels for a large
number of elements.

  4. 4.

We extend the analysis and study the performance of IOS assisted multi-user
NOMA and IOS assisted OMA. We show that, except the user with the last
decoding order, the average achievable rates of all other users are determined
by the power allocation scheme of the transmitter in the large transmit signal
to noise ratio (SNR) regime in a multi-user NOMA system. We also show that
IOS-assisted NOMA may not always have higher sum rate than OMA since IOS can
reconfigure the channels of users in different time slots of OMA.

Figure 1: System model of the IOS-assisted NOMA system.

Figure 2: IOS structure. TABLE I: Notations of Rates in Different Scenarios of User $M$. $M\in\\{T,R\\}$. $m\in\\{t,r\\}$. $R_{m}$ | Average achievable rate of $M$ under NOMA.
---|---
$R_{m}^{*}$ | An Upper bound on $R_{m}$ by Jensen’s inequality.
$R_{m}^{+}$ | An approximation to $R_{m}$ by large array analysis.
$R_{m,u}$ |  $R_{m}$ with uncorrelated channels.
$\tilde{R}_{m}$ | Average achievable rate of $M$ under OMA.
$\tilde{R}_{m}^{*}$ | An Upper bound on $\tilde{R}_{m}$ by Jensen’s inequality.
$\tilde{R}_{m}^{+}$ | An approximation to $\tilde{R}_{m}$ by large array analysis.

This paper is organized as follows. In Section II, we introduce the system
model used in this paper. Upper bounds on the average achievable rates of
different users are analyzed in Section III. Section IV discusses the channel
hardening effect, the asymptotic behavior of the average achievable rates, and
the impact of quantization bits. IOS assisted multi-user NOMA is analyzed in
Section V. Section VI presents the numerical results, followed by the
Conclusion in Section VII. For clarity, Table. I lists some key notations of
rates in different scenarios.

Notations: $\mathbb{E}[X]$ and ${\rm Var}[X]$ denote the expectation and
variance of a random variable $X$. ${\rm Cov}\\{X,Y\\}$ denotes the covariance
of random variables $X$ and $Y$. Bold lower-case and bold upper-case letters
represent vectors and matrices, respectively. ${\rm mod}(a,b)$ denotes the
remainder of division $a$ by $b$. $\lfloor a\rfloor$ represents the integer
closet to $a$ that is less than or equal to $a$. $\mathbb{C}^{N\times 1}$
denotes the $N\times 1$ complex vectors space. $\mathbf{A}^{T}$ and
$\mathbf{A}^{H}$ respectively stand for the transpose and Hermitian transpose
of matrix $\mathbf{A}$. ${\rm diag(\mathbf{a})}$ represents a diagonal matrix
with the elements of $\mathbf{a}$ on its main diagonal. ‘$\to$’,
‘$\overset{P}{\to}$’ and ‘$\triangleq$’ stand for ‘approaching to a particular
value’, ‘convergence in probability’ and ‘definition’, respectively.

##  II System Model

In this paper, we consider an IOS-assisted communication network, where a
single-antenna transmitter (TX) communicates with two single-antenna users.
Two users are located on different sides of the IOS. As shown in Fig. 1, the
reflecting user $R$ is located on the same side of the IOS as TX, while the
transmitting user $T$ is on the other side. It is assumed that obstacles block
the direct links from the TX to the users. This set-up reflects scenarios
where IOSs are deployed on the windows or building facades, serving users
located inside and outside of the buildings simultaneously.

###  II-A IOS Implementations

We consider an IOS with $N=N_{h}N_{v}$ elements placed in a planar rectangular
array. As illustrated in Fig. 2, the IOS is on the $yOz$ plane with $N_{h}$
elements per row and $N_{v}$ elements per column. Without loss of generality,
we assume each element is of size $S=l\times w$, where $l$ and $w$ are the
horizontal and vertical lengths of each element. The IOS is equipped on a
building of height $l_{0}$, and the coordinate of the $n$th element can be
written as

$\mathbf{a}_{n}=\left[0,y(n)l,z(n)w+l_{0}\right],$ (1)

where

$y(n)={\rm mod}(n-1,N_{h}),~{}~{}~{}z(n)=\lfloor\frac{n-1}{N_{h}}\rfloor.$ (2)

$y(n)$ and $z(n)$ are the indexes of the $n$th element on the $y$-axis and
$z$-axis.

Each IOS element is able to adjust phase shifts for the user $T$ and user $R$
independently, as introduced in [28, 29, 30]. Denoting the phase shifts
imposed on the incident signals by the $n$th element in the transmitting and
reflecting modes as $\theta_{n}$ and $\psi_{n}$, the transmitting and
reflecting coefficients matrices can be written as

$\displaystyle\mathbf{\Theta}$ $\displaystyle=\alpha\,{\rm
diag}\left(e^{j\theta_{1}},...,e^{j\theta_{N}}\right),$ (3a)
$\displaystyle\mathbf{\Psi}$ $\displaystyle=\beta\,{\rm
diag}\left(e^{j\psi_{1}},...,e^{j\psi_{N}}\right),$ (3b)

where $\alpha$ and $\beta$ are the transmitting and reflecting amplitude
coefficients of the IOS. The IOS is assumed to be passive with no energy
dissipation. Hence, the amplitude coefficients of each element satisfy [32]

$\alpha^{2}+\beta^{2}=1.$ (4)

###  II-B Channel and Signal Models

####  II-B1 Small Scale Fading

Let $\mathbf{h}=\left(h_{1},...,h_{N}\right)^{T}\in\mathbb{C}^{N\times 1}$,
$\mathbf{g}=\left(g_{1},...,g_{N}\right)^{T}\in\mathbb{C}^{N\times 1}$ and
$\mathbf{r}=\left(r_{1},...,r_{N}\right)^{T}\in\mathbb{C}^{N\times 1}$ denote
the normalized small scale fading of the TX-IOS, IOS-user $T$ and IOS-user $R$
channels, respectively. IOS operates in the far-field of TX and users, thus
the EM waves impinging on the IOS are plane waves with the wave vector [34]

$\mathbf{v}(\varrho,\varphi)=\frac{2\pi}{\lambda}\left[\cos(\varrho)\cos(\varphi),\sin(\varrho)\cos(\varphi),\sin(\varphi)\right],$
(5)

where $\varrho$ and $\varphi$ are the azimuth and elevation angles; $\lambda$
is the wavelength.

Due to the multi-path propagation, the small scale fading $\mathbf{h}$ is a
superposition of the independent array response of each multi-path component
$\mathbf{c}_{k}$ as [35]

$\mathbf{h}=\sum\limits_{k=1}^{K}\mathbf{c}_{k}=\sum\limits_{k=1}^{K}\frac{l_{k}}{\sqrt{K}}\hat{\mathbf{c}}_{k},$
(6)

where $l_{k}/\sqrt{K}$ is a complex random variable that stands for the
attenuation and phase-rotation of the $k$th multi-path component and
$\hat{\mathbf{c}}_{k}=\left[\exp\left({j\mathbf{v}\mathbf{a}_{1}^{T}}\right),\exp\left({j\mathbf{v}\mathbf{a}_{2}^{T}}\right),...,\exp\left({j\mathbf{v}\mathbf{a}_{N}^{T}}\right)\right]^{T}$.
In non-LoS scenarios, $l_{k}$, $k\in\\{1,2,...,K\\}$, are independent and
identical (i.i.d.) random variables with zero mean and unit variance. In a
rich scattering environment, there are infinite number of multi-path
components ($K\to\infty$). Hence, with the help of central limit theorem [33],
we have111 This model is valid for rich scattering urban environments without
a stable LoS path.

$\mathbf{h}\sim\mathcal{CN}\left(\mathbf{0}_{N},\mathbf{R}\right),$ (7)

where $\mathbf{R}$ is the covariance matrix of $\mathbf{h}$, defined as

$\mathbf{R}=\mathbb{E}\left[\mathbf{h}\mathbf{h}^{H}\right]=\mathbb{E}\left[\sum\limits_{k=1}^{K}\mathbf{c}_{k}\mathbf{c}_{k}^{H}\right]=\frac{1}{K}\mathbb{E}\left[\sum\limits_{k=1}^{K}\hat{\mathbf{c}}_{k}\hat{\mathbf{c}}_{k}^{H}\right].$
(8)

Using the result in [33, Prop. 1], each element in $\mathbf{R}$ can be written
as

$\left[\mathbf{R}\right]_{m,n}=\frac{\sin\left(\frac{2\pi}{\lambda}\parallel\mathbf{a}_{m}-\mathbf{a}_{n}\parallel\right)}{\frac{2\pi}{\lambda}\parallel\mathbf{a}_{m}-\mathbf{a}_{n}\parallel}.$
(9)

In similar ways, we can have

$\displaystyle\mathbf{g}$
$\displaystyle\sim\mathcal{CN}\left(\mathbf{0}_{N},\mathbf{R}\right),$ (10a)
$\displaystyle\mathbf{r}$
$\displaystyle\sim\mathcal{CN}\left(\mathbf{0}_{N},\mathbf{R}\right).$ (10b)

####  II-B2 Large Scale Fading

The pathloss of the TX-IOS-user $T$ and TX-IOS-user $R$ links can be written
as [36]

$\eta_{t}=\frac{\Lambda_{t}}{d_{b}^{\chi}d_{t}^{\chi}},\,\eta_{r}=\frac{\Lambda_{r}}{d_{b}^{\chi}d_{r}^{\chi}},$
(11)

where $\Lambda_{t}$ and $\Lambda_{r}$ stand for the intercepts at a reference
distance of $1{\rm m}$ of the TX-IOS-user $T$ and TX-IOS-user $R$ links,
respectively; $\chi$ is the pathloss exponent; $d_{b}$, $d_{t}$ and $d_{r}$
denote the distances of the TX-IOS, IOS-user $T$ and IOS-user $R$ links,
respectively.

####  II-B3 Signal Models

In the NOMA transmission scheme, TX broadcasts the superimposed signals of the
two users222Note that more than two users can be selected to perform NOMA, but
the performance might be degraded since the co-channel interference of the
NOMA scheme can be severe and the SIC can be very complex. To optimize the
performance of NOMA, hybrid NOMA can be implemented, by combining NOMA with
conventional OMA, such as TDMA, FDMA. as

$s=\sqrt{P}\left(q_{t}s_{t}+q_{r}s_{r}\right),$ (12)

where $s_{t}$ and $s_{r}$ denote the signals of user $T$ and user $R$ with
unit power, i.e.,
${\mathbb{E}}[\left|s_{t}\right|^{2}]={\mathbb{E}}[\left|s_{r}\right|^{2}]=1$.
$P$ is the total transmit power of TX. $q_{t}$ and $q_{r}$ are the transmit
amplitude coefficients for user $T$ and user $R$ at TX, satisfying
$q_{t}^{2}+q_{r}^{2}=1$. Hence, the received signals at user $T$ and user $R$
can be written as

$\displaystyle y_{t}$
$\displaystyle=\sqrt{\eta_{t}}\mathbf{g}^{T}\mathbf{\Theta}\mathbf{h}\sqrt{P}\left(q_{t}s_{t}+q_{r}s_{r}\right)+n_{t},$
(13a) $\displaystyle y_{r}$
$\displaystyle=\sqrt{\eta_{r}}\mathbf{r}^{T}\mathbf{\Psi}\mathbf{h}\sqrt{P}\left(q_{t}s_{t}+q_{r}s_{r}\right)+n_{r},$
(13b)

where $n_{t}$ and $n_{r}$ are the additive white Gaussian noise at user $T$
and user $R$ with zero mean and variance $\sigma_{0}^{2}$.

In the NOMA transmission scheme, one of the two users performs successive
interference cancellation (SIC) by decoding the other user’s signal,
subtracting it from the received signal, and decoding its own signal. The
optimal decoding order corresponds to the ascending order of the channel gains
of the two users. However, the order of the channel gains is dictated by the
small scale fading which fluctuates quickly and renders the analysis
complicated. It is worth highlighting that we do not aim to investigate the
optimal decoding order in this work. Alternatively, we assume the distance of
the IOS-user $T$ channel is smaller than that of the IOS-user $R$ channel333
In practice, the locations of different users can be readily obtained from a
location information system, such as global positioning system (GPS) [37].,
and the decoding order is fixed as ($R$, $T$). Specifically, user $R$ decodes
its signal directly by treating the signal of user $T$ as interference, and
user $T$ performs SIC. A typical application scenario of IOS is to deploy IOS
on the facades of buildings to serve indoor user ($T$) and outdoor user ($R$)
simultaneously. Since the indoor user ($T$) is closer to the IOS than the
outdoor user ($R$) in general, the large scale fading of the indoor user ($T$)
is less severe than that of the outdoor user ($R$), which supports the
practicability of the assumption. Furthermore, this assumption is also adopted
in many other similar works, such as [6, 14, 10, 38]. Based on the principles
of NOMA transmission scheme, user $R$ is allocated with more transmit power to
ensure that user $T$ could perform SIC successfully, i.e., $q_{t}<q_{r}$.
Hence, signal to interference plus noise ratio (SINR) of user $T$ decoding the
signal of user $R$ in the SIC process can be written as

$\gamma_{t\to r}=\frac{\gamma_{t\to r}^{(s)}}{\gamma_{t\to
r}^{(i)}}=\frac{Pq_{r}^{2}\eta_{t}\left|\mathbf{g}^{T}\mathbf{\Theta}\mathbf{h}\right|^{2}}{Pq_{t}^{2}\eta_{t}\left|\mathbf{g}^{T}\mathbf{\Theta}\mathbf{h}\right|^{2}+\sigma_{0}^{2}}.$
(14)

After subtracting the signal of user $R$ from the received signal, user $T$ is
able to decode its own signal with SNR

$\gamma_{t}=\gamma_{0}q_{t}^{2}\eta_{t}\left|\mathbf{g}^{T}\mathbf{\Theta}\mathbf{h}\right|^{2},$
(15)

where $\gamma_{0}=P/\sigma_{0}^{2}$ is the transmit SNR at TX.

On the other hand, user $R$ decodes its own signal directly with SINR

$\gamma_{r}=\frac{\gamma_{r}^{(s)}}{\gamma_{r}^{(i)}}=\frac{Pq_{r}^{2}\eta_{r}\left|\mathbf{r}^{T}\mathbf{\Psi}\mathbf{h}\right|^{2}}{Pq_{t}^{2}\eta_{r}\left|\mathbf{r}^{T}\mathbf{\Psi}\mathbf{h}\right|^{2}+\sigma_{0}^{2}}.$
(16)

####  II-B4 Phase Adjustments

It can be proved that $\gamma_{t\to r}$, $\gamma_{t}$ and $\gamma_{r}$ in
(14), (15) and (16) are maximized when the phase shifts $\theta_{n}$ and
$\psi_{n}$ are controlled to co-phase the transmitting and reflecting links,
respectively [30]. Thus, the optimal phase shifts can be written as

$\displaystyle\bar{\theta}_{n}$
$\displaystyle=-\arg\left(h_{n}\right)-\arg\left(g_{n}\right),\,n\in\\{1,...,N\\},$
(17a) $\displaystyle\bar{\psi}_{n}$
$\displaystyle=-\arg\left(r_{n}\right)-\arg\left(g_{n}\right),\,n\in\\{1,...,N\\}.$
(17b)

However, the phase adjustment may not be perfect in practice due to imperfect
channel knowledge and phase quantization, which leads to the phase error
$\\{\phi_{n}^{u}\\}_{n=1}^{N}$, $u\in\\{t,r\\}$ at each IOS element [8]. These
two types of errors are introduced as follows:

  * •

Channel estimation errors: Channel estimation may not be precise in practice.
And the phase error $\\{\phi_{n}^{u}\\}_{n=1}^{N}$, $u\in\\{t,r\\}$ due to
inaccurate channel estimation is modeled by i.i.d. Von Mises random variables
with zero mean and concentration parameter $\kappa_{u}$ [8, 39], of which the
probability density function (PDF) can be written as

$f_{\phi_{n}^{u}}(\theta)=\frac{e^{\kappa_{u}\cos(\theta)}}{2\pi
I_{0}(\kappa_{u})},\qquad-\pi<\theta<\pi,$ (18)

where $I_{0}(\cdot)$ is the modified Bessel function of the first kind of
order zero. PDF of $\phi_{n}^{u}$ is symmetric around zero and gets more
concentrated with the increase of $\kappa_{u}$, which also means the channel
estimation is more accurate.

  * •

Phase quantization errors: Assuming that each element of the IOS is a $b$-bit
phase shifter, the phase error can be modeled by i.i.d. random variables
uniformly distributed over $\\{-\frac{\pi}{2^{b}},\frac{\pi}{2^{b}}\\}$, whose
distribution is symmetric around zero [40].

Due to phase errors, the realistic phase shifts at each element can be written
as

$\hat{\theta}_{n}=\bar{\theta}_{n}+\phi_{n}^{t},\qquad\hat{\psi}_{n}=\bar{\psi}_{n}+\phi_{n}^{r}.$
(19)

####  II-B5 Average Achievable Rate

After phase adjustment, $\gamma_{t\to r}$, $\gamma_{t}$ and $\gamma_{r}$ can
be rewritten as

$\displaystyle\gamma_{t}$
$\displaystyle=\gamma_{0}q_{t}^{2}\eta_{t}\alpha^{2}H_{t},$ (20a)
$\displaystyle\gamma_{t\to r}$ $\displaystyle=\frac{\gamma_{t\to
r}^{(s)}}{\gamma_{t\to
r}^{(i)}}=\frac{Pq_{r}^{2}\eta_{t}\alpha^{2}H_{t}}{Pq_{t}^{2}\eta_{t}\alpha^{2}H_{t}+\sigma_{0}^{2}},$
(20b) $\displaystyle\gamma_{r}$
$\displaystyle=\frac{\gamma_{r}^{(s)}}{\gamma_{r}^{(i)}}=\frac{Pq_{r}^{2}\eta_{r}\beta^{2}H_{r}}{Pq_{t}^{2}\eta_{r}\beta^{2}H_{r}+\sigma_{0}^{2}},$
(20c)

where

$H_{t}=\left|\sum\limits_{n=1}^{N}\left|g_{n}\right|\left|h_{n}\right|e^{j\phi_{n}^{t}}\right|^{2},\quad
H_{r}=\left|\sum\limits_{n=1}^{N}\left|r_{n}\right|\left|h_{n}\right|e^{j\phi_{n}^{r}}\right|^{2}.$
(21)

The average achievable rates of user $T$ and user $R$ can be written as [6]

$\displaystyle R_{t}$ $\displaystyle=\mathbb{E}[\log_{2}(1+\gamma_{t})],$ (22)
$\displaystyle R_{r}$
$\displaystyle=\mathbb{E}\left[\min\left\\{\log_{2}(1+\gamma_{t\to
r}),\log_{2}(1+\gamma_{r})\right\\}\right].$ (23)

###  II-C IOS-Assisted OMA

For the purpose of comparison, the OMA scheme using time division multiple
access (TDMA) is introduced here. In the OMA scheme, TX communicates to user
$T$ and user $R$ consecutively over two identical time slots. In each time
slot, the phase shifts and amplitude coefficients of the IOS are adjusted to
enhance the link of one of the users. Hence, the average achievable rate of
user $T$ in the OMA scenario are

$\tilde{R}_{t}=\frac{1}{2}\mathbb{E}[\log_{2}(1+\tilde{\gamma}_{t})],$ (24)

where $\tilde{\gamma}_{t}=\gamma_{0}\eta_{t}H_{t}$. Similarly, the average
achievable rate of use $R$ is

$\tilde{R}_{r}=\frac{1}{2}\mathbb{E}[\log_{2}(1+\tilde{\gamma}_{r})],$ (25)

where $\tilde{\gamma}_{r}=\gamma_{0}\eta_{r}H_{r}$.

##  III Average Achievable Rate Analysis

In this section, we analyze the average achievable rates of user $T$ and user
$R$, respectively.

###  III-A Average Achievable Rate of User $T$

It is mathematically intractable to derive the exact close-form expression of
$R_{t}$. Alternatively, we use Jensen’s inequality to obtain an upper bound
$R_{t}^{*}$ as

$\begin{split}R_{t}^{*}&=\log_{2}\left(1+\mathbb{E}[\gamma_{t}]\right)\\\\[5.0pt]
&=\log_{2}\left(1+\gamma_{0}q_{t}^{2}\eta_{t}\alpha^{2}\mathbb{E}\left[H_{t}\right]\right),\end{split}$
(26)

where $\mathbb{E}\left[H_{t}\right]$ can be further decomposed as (27) shown
at the top of the next page.

$\begin{split}\mathbb{E}\left[H_{t}\right]=&\mathbb{E}\left[\sum\limits_{n=1}^{N}\left|g_{n}\right|^{2}\left|h_{n}\right|^{2}\right]+2\,\mathbb{E}\left[\sum\limits_{n=1}^{N-1}\sum\limits_{i=n+1}^{N}\left|g_{n}\right|\left|h_{n}\right|\cos(\phi_{n}^{t})\left|g_{i}\right|\left|h_{i}\right|\cos(\phi_{i}^{t})\right]\\\
&+2\,\mathbb{E}\left[\sum\limits_{n=1}^{N-1}\sum\limits_{i=n+1}^{N}\left|g_{n}\right|\left|h_{n}\right|\sin(\phi_{n}^{t})\left|g_{i}\right|\left|h_{i}\right|\sin(\phi_{i}^{t})\right]\end{split}$
(27)

Since the channels $\mathbf{h}$ and $\mathbf{g}$ are correlated, we need to
figure out $\mathbb{E}\left[\left|h_{n}\right|\left|h_{i}\right|\right]$ and
$\mathbb{E}\left[\left|g_{n}\right|\left|g_{i}\right|\right]$. The result is
summarized in the following proposition.

######  Proposition 1.

For $n\neq i$, $\mathbb{E}\left[\left|w_{n}\right|\left|w_{i}\right|\right]$,
$w\in\\{h,g,r\\}$ can be expressed in terms of
$\mathbb{E}\left[w_{n}w_{i}^{*}\right]$ as

$\displaystyle\mathbb{E}\left[\left|w_{n}\right|\left|w_{i}\right|\right]=$
$\displaystyle\left(\frac{\left|\mathbb{E}[w_{n}w_{i}^{*}]\right|^{2}}{2}-\frac{1}{2}\right)K\left({\left|\mathbb{E}[w_{n}w_{i}^{*}]\right|^{2}}\right)+E\left({\left|\mathbb{E}[w_{n}w_{i}^{*}]\right|^{2}}\right).$
(28)

For $n=i$,

$\mathbb{E}\left[\left|w_{n}\right|\left|w_{i}\right|\right]=1.$ (29)

$K(\cdot)$ and $E(\cdot)$ are the complete elliptic integral of the first kind
and the second kind, respectively.

###### Proof:

See Appendix A. ∎

######  Corollary 1.

Under the proposed system settings,
$\frac{\pi}{4}\leq\mathbb{E}\left[\left|w_{n}\right|\left|w_{i}\right|\right]\leq
1$, for all $n,i$.

###### Proof:

See Appendix B. ∎

Since the covariance matrices of $\mathbf{h}$ and $\mathbf{g}$ are equal, it
can be learned that the matrices $\mathbb{E}[|\mathbf{h}||\mathbf{h}|^{T}]$
and $\mathbb{E}[|\mathbf{g}||\mathbf{g}|^{T}]$ are equal, and we denote each
by $\bar{\mathbf{R}}$. Applying Proposition 1, an upper bound on the average
achievable rate of user $T$ can be obtained, which is summarized in the
following proposition.

######  Proposition 2.

An upper bound on the average achievable rate of user $T$ can be written as

$R_{t}^{*}=\log_{2}\left(1+\gamma_{0}q_{t}^{2}\eta_{t}\alpha^{2}\left(N(1-\epsilon_{t}^{2})+\epsilon_{t}^{2}\mathbf{tr}\left(\mathbf{\bar{R}\bar{R}}\right)\right)\right),$
(30)

where

$\epsilon_{t}=\mathbb{E}{\left[\cos(\phi_{n}^{t})\right]},\,n\in\\{1,2,...,N\\}.$
(31)

For phase estimation errors,

$\epsilon_{t}=\frac{I_{1}(\kappa_{t})}{I_{0}(\kappa_{t})}.$ (32)

For phase quantization errors,

$\epsilon_{t}=\frac{2^{b}\sin{\left(\frac{\pi}{2^{b}}\right)}}{\pi}.$ (33)

###### Proof:

See Appendix C. ∎

######  Remark 1.

For perfect phase adjustment, i.e., $\phi_{n}^{t}=0$, $n\in\\{1,2,...,N\\}$,
the upper bound on the average achievable rate of user $T$ is

$R_{t}^{*}=\log_{2}\left(1+\gamma_{0}q_{t}^{2}\eta_{t}\alpha^{2}\mathbf{tr}\left(\mathbf{\bar{R}\bar{R}}\right)\right).$
(34)

######  Remark 2.

When the phase errors are uniformly distributed over $[-\pi,\pi)$, i.e.,
$\epsilon_{t}=0$ , the upper bound on the average achievable rate of user $T$
is

$R_{t}^{*}=\log_{2}\left(1+\gamma_{0}q_{t}^{2}\eta_{t}\alpha^{2}N\right).$
(35)

It can be seen that $R_{t}^{*}$ scales with $\log_{2}(N)$ when the phase
errors are uniformly distributed over $[-\pi,\pi)$, which coincides with the
proposition in [41].

######  Corollary 2.

Under the proposed system settings, $\mathbb{E}[H_{1}]$ satisfies

$N+\frac{\pi^{2}N(N-1)}{16}\epsilon_{t}^{2}\,{\leq}\,\mathbb{E}[H_{1}]\,{<}\,N+N(N-1)\epsilon_{t}^{2}.$
(36)

###### Proof:

The conclusion is obtained by substituting the result in Corollary 1 into
(27). ∎

######  Remark 3.

It can be learned from Corollary 2 that $R_{t}^{*}$ scales with
$\log_{2}(N^{2})$ when $\epsilon_{t}\neq 0$. Compared with Remark 2, it can be
seen that even imperfect phase adjustment is able to provide performance gain,
which scales with $\log_{2}(N)$. This is a known result for the system with
uncorrelated channels [8], and it is shown here that it still works for
correlated channels. Simulation results will illustrate this point in Section
V.

######  Corollary 3.

Following the similar procedures in the Propositions 1 and 33, upper bounds on
the average achievable rates of user $T$ and user $R$ by using OMA scheme is
given by

$\displaystyle\tilde{R}_{t}^{*}$
$\displaystyle=\frac{1}{2}\log_{2}\left(1+\gamma_{0}\eta_{t}\left(N(1-\epsilon_{t}^{2})+\epsilon_{t}^{2}\mathbf{tr}\left(\mathbf{\bar{R}\bar{R}}\right)\right)\right),$
(37a) $\displaystyle\tilde{R}_{r}^{*}$
$\displaystyle=\frac{1}{2}\log_{2}\left(1+\gamma_{0}\eta_{r}\left(N(1-\epsilon_{r}^{2})+\epsilon_{r}^{2}\mathbf{tr}\left(\mathbf{\bar{R}\bar{R}}\right)\right)\right).$
(37b)

###  III-B Average Achievable Rate of User $R$

It appears that obtaining the close-form expression of $R_{r}$ in (23) is
intricate. Alternatively, we first analyze $R_{r}$ for large transmit SNR
$\gamma_{0}$ and then seek for an upper bound based on Jensen’s inequality.

####  III-B1 Analysis of Large Transmit SNR

$\gamma_{t\to r}$, defined in (20b), can be rewritten as

$\gamma_{t\to
r}=\frac{q_{r}^{2}}{q_{t}^{2}+\left(\gamma_{0}\eta_{t}\alpha^{2}H_{t}\right)^{-1}}.$
(38)

As $\gamma_{0}\to\infty$, we can have

$\gamma_{t\to r}\to\frac{q_{r}^{2}}{q_{t}^{2}}.$ (39)

The same conclusion can be derived for $\gamma_{r}$:
$\gamma_{r}\to{q_{r}^{2}}/{q_{t}^{2}}$ as $\gamma_{0}\to\infty$. Hence,

$R_{r}\to\log_{2}\left(1+\frac{q_{r}^{2}}{q_{t}^{2}}\right),~{}~{}~{}\gamma_{0}\to\infty.$
(40)

####  III-B2 Analysis of Finite Transmit SNR

The $\min$ function is concave [42]. Therefore, Jensen’s inequality can be
applied on $R_{r}$ to obtain an upper bound on $R_{r}$ as

$R_{r}\leq\min\left\\{\log_{2}(1+\mathbb{E}[\gamma_{t\to
r}]),\log_{2}(1+\mathbb{E}[\gamma_{r}])\right\\}.$ (41)

Referring to (20b) and (20c), (41) is hard to deal with due to the common
terms in the numerators and denominators of $\gamma_{t\to r}$ and
$\gamma_{r}$. Specifically, the common term in the numerator and denominator
of $\gamma_{t\to r}$ is $H_{t}$, and the common term of $\gamma_{r}$ is
$H_{r}$. Fortunately, it can be proved that $\gamma_{t\to r}$ and $\gamma_{r}$
are concave to their respective common terms. Thus, Jensen’s inequality can be
applied again, which gives

$R_{r}\leq\min\left\\{\log_{2}\left(1+\frac{\mathbb{E}[\gamma_{t\to
r}^{s}]}{\mathbb{E}[\gamma_{t\to
r}^{i}]}\right),\log_{2}\left(1+\frac{\mathbb{E}[\gamma_{r}^{s}]}{\mathbb{E}[\gamma_{r}^{i}]}\right)\right\\}.$
(42)

Following the similar procedures in the Propositions 1 and 33, an upper bound
$R_{r}^{*}$ can be written as

$\begin{split}&R_{r}^{*}=\\\
&\min\left\\{\log_{2}\left(1+\frac{q_{r}^{2}}{q_{t}^{2}+f_{t}^{-1}}\right),\log_{2}\left(1+\frac{q_{r}^{2}}{q_{t}^{2}+f_{r}^{-1}}\right)\right\\}\\\\[5.0pt]
&=\left\\{\begin{split}&\quad\log_{2}\left(1+\frac{q_{r}^{2}}{q_{t}^{2}+f_{t}^{-1}}\right),\quad
f_{t}<f_{r},\\\\[5.0pt]
&\quad\log_{2}\left(1+\frac{q_{r}^{2}}{q_{t}^{2}+f_{r}^{-1}}\right),\quad
f_{t}\geq f_{r},\end{split}\right.\end{split}$ (43)

where

$\displaystyle\epsilon_{r}$
$\displaystyle=\mathbb{E}{\left[\cos(\phi_{n}^{r})\right]},\,n\in\\{1,2,...,N\\},$
(44a) $\displaystyle f_{t}$
$\displaystyle=\gamma_{0}\eta_{t}\alpha^{2}\left(N(1-\epsilon_{t}^{2})+\epsilon_{t}^{2}\mathbf{tr}\left(\mathbf{\bar{R}\bar{R}}\right)\right),$
(44b) $\displaystyle f_{r}$
$\displaystyle=\gamma_{0}\eta_{r}\beta^{2}\left(N(1-\epsilon_{r}^{2})+\epsilon_{r}^{2}\mathbf{tr}\left(\mathbf{\bar{R}\bar{R}}\right)\right).$
(44c)

##  IV Large Array Analysis

In this section, we show that the channel hardening effect appears in the
proposed system and revisit the average achievable rates for a large number of
IOS elements.

###  IV-A Channel Hardening Analysis

As defined in [33], the channel hardening effect in a RIS-assisted network
refers that the received SNR variations average out and the received SNR is
approximately equal to $N^{2}$ times a constant as $N$ goes large. The channel
hardening effect in the proposed system model with correlated channels and
imperfect phase adjustment is presented in the following proposition.

######  Proposition 3.

The composite channel gains associated with user $T$ and user $R$, namely
$H_{t}$ and $H_{r}$, satisfy

$\displaystyle\frac{1}{N^{2}}H_{t}\overset{P}{\to}\frac{\pi^{2}}{16}\epsilon_{t}^{2},\,\,N\to\infty,$
(45a)
$\displaystyle\frac{1}{N^{2}}H_{r}\overset{P}{\to}\frac{\pi^{2}}{16}\epsilon_{r}^{2},\,\,N\to\infty,$
(45b)

with the convergence in probability. Hence, for non-uniform phase errors over
$[-\pi,\pi)$, $H_{t}$ and $H_{r}$ can be approximated by a constant as

$H_{t}\approx\frac{\pi^{2}}{16}N^{2}\epsilon_{t}^{2},~{}~{}~{}H_{r}\approx\frac{\pi^{2}}{16}N^{2}\epsilon_{r}^{2},$
(46)

when the number of IOS elements is large.

###### Proof:

See Appendix D. ∎

######  Remark 4.

Under the correlation profile proposed in (9), the channel correlation does
not play a role in the conclusion of Proposition 3. And, in the case of
uncorrelated channels, the same approximations on $H_{t}$ and $H_{r}$ can
derived by using the conventional law of large numbers.

###  IV-B Average Achievable Rates for Large Number of Elements

We revisit the average achievable rates when the number of elements is large.
By using (46), the average achievable rates of user $T$ and user $R$ with non-
uniform phase errors over $[-\pi,\pi)$ can be approximated as

$R_{t}\approx
R_{t}^{+}=\log_{2}\left(1+\frac{\pi^{2}N^{2}\gamma_{0}\epsilon_{t}^{2}q_{t}^{2}\eta_{t}\alpha^{2}}{16}\right),$
(47) $R_{r}\approx
R_{r}^{+}=\left\\{\begin{split}&\log_{2}\left(1+\frac{q_{r}^{2}}{q_{t}^{2}+16(\pi^{2}N^{2}\gamma_{0}\epsilon_{t}^{2}\eta_{t}\alpha^{2})^{-1}}\right),\\\\[5.0pt]
&\qquad\qquad\qquad\qquad\qquad~{}~{}\epsilon_{t}^{2}\eta_{t}\alpha^{2}<\epsilon_{r}^{2}\eta_{r}\beta^{2},\\\\[5.0pt]
&\log_{2}\left(1+\frac{q_{r}^{2}}{q_{t}^{2}+16(\pi^{2}N^{2}\gamma_{0}\epsilon_{r}^{2}\eta_{r}\beta^{2})^{-1}}\right),\\\\[5.0pt]
&\qquad\qquad\qquad\qquad\qquad~{}~{}\epsilon_{t}^{2}\eta_{t}\alpha^{2}\geq\epsilon_{r}^{2}\eta_{r}\beta^{2}.\end{split}\right.$
(48)

######  Remark 5.

The impact of phase quantization bits on the average achievable rate can be
analyzed by using the result in (47). The average achievable rate improvement
of user $T$ brought by adding one bit for phase quantization can be expressed
as

$R_{b}=\log_{2}\left(\frac{16+\pi^{2}N^{2}\gamma_{0}\epsilon_{b+1}^{2}q_{t}^{2}\eta_{t}\alpha^{2}}{16+\pi^{2}N^{2}\gamma_{0}\epsilon_{b}^{2}q_{t}^{2}\eta_{t}\alpha^{2}}\right),$
(49)

where

$\epsilon_{b}=\frac{2^{b}\sin{\left(\frac{\pi}{2^{b}}\right)}}{\pi}.$ (50)

For large number of elements, we can have

$R_{b}\sim f(b),~{}~{}~{}N\to\infty,$ (51)

where

$f(b)=\log_{2}\left(\frac{4\sin^{2}{\left(\frac{\pi}{2^{b+1}}\right)}}{\sin^{2}{\left(\frac{\pi}{2^{b}}\right)}}\right).$
(52)

$f(b)$ is always positive. However,

$f^{\prime}(b)=-\frac{\pi}{2^{b}}\tan\left(\frac{\pi}{2^{b+1}}\right)<0.$ (53)

Therefore, adding the number of quantization bits can always improve the
average achievable rate of user $T$, but the rate of improvement decreases.

The impact of inaccurate channel estimation can also be studied. According to
(32), $\epsilon_{u}$, $u\in\\{t,r\\}$ is an increasing function to
$\kappa_{u}$. Further, $R_{u}^{+}$ is also an increasing function to
$\epsilon_{u}$. Therefore, we can learn that more accurate channel estimation
(larger $\kappa_{u}$) will lead to larger average achievable rates of both
users.

######  Corollary 4.

In the OMA scheme, the average achievable rates of user $T$ and user $R$ are
approximated by

$\displaystyle\tilde{R}_{t}\approx\tilde{R}_{t}^{+}=\frac{1}{2}\log_{2}\left(1+\frac{\pi^{2}N^{2}\gamma_{0}\epsilon_{t}^{2}\eta_{t}}{16}\right),$
(54a)
$\displaystyle\tilde{R}_{r}\approx\tilde{R}_{r}^{+}=\frac{1}{2}\log_{2}\left(1+\frac{\pi^{2}N^{2}\gamma_{0}\epsilon_{r}^{2}\eta_{r}}{16}\right).$
(54b)

Compared (47) with (54a), we can see that $T$ obtains a higher SNR in OMA at
the penalty of a $1/2$ pre-log factor. It can be computed that
${R}_{t}^{+}>\tilde{R}_{t}^{+}$ if

$\gamma_{0}>\frac{16-32q_{t}^{2}\alpha^{2}}{\pi^{2}N^{2}\epsilon_{t}^{2}\eta_{t}q_{t}^{4}\alpha^{4}}.$
(55)

Compared (48) with (54b), it can be learned that
$\tilde{R}_{r}^{+}>{R}_{r}^{+}$ in the large transmit SNR regime, since
${R}_{r}^{+}$ converges to a constant as $\gamma_{0}$ goes large.

Due to the reconfigurability of IOS in different time slots of OMA, IOS
assisted NOMA may not always achieve better performance than NOMA in terms of
the system sum rate. Actually, which technique is preferable depends on the
system setup. The approximate sum rates of NOMA and OMA, i.e, $S_{NOMA}$ and
$S_{OMA}$, is given by

$\displaystyle S_{NOMA}$ $\displaystyle=R_{t}^{+}+R_{r}^{+},$ (56a)
$\displaystyle S_{OMA}$ $\displaystyle=\tilde{R}_{t}^{+}+\tilde{R}_{r}^{+}.$
(56b)

After some mathematical manipulations, it can be derived that $S_{NOMA}$ and
$S_{OMA}$ comply with the following relationship in the large transmit SNR
regime:

$\left\\{\begin{split}&S_{NOMA}>S_{OMA},~{}~{}~{}\alpha^{4}\epsilon_{t}^{2}\eta_{t}>\epsilon_{r}^{2}\eta_{r},\\\\[5.0pt]
&S_{NOMA}<S_{OMA},~{}~{}~{}\alpha^{4}\epsilon_{t}^{2}\eta_{t}<\epsilon_{r}^{2}\eta_{r}.\end{split}\right.$
(57)

Contrary to the conventional idea that NOMA has superiority over OMA in the
conventional wireless system without IOS, NOMA does not always perform better
than OMA in the IOS-assisted system due to the fact that IOS is able to
reconfigure the channel in each time slot of OMA. Namely, in the time slot for
$T$, the amplitude coefficients of IOS is set to be $\alpha=1$ and $\beta=0$,
and in the time slot for $R$, $\alpha=0$ and $\beta=1$. Our analysis shows
that, for large transmit SNR, which transmission scheme is better depends on
the pathloss, amplitude coefficients at the IOS, and the phase errors
distribution. NOMA performs better when the channel of $T$ is remarkably
stronger than the channel of $R$. This corresponds to the conclusion in [20].
We will corroborate this analysis with numerical results later.

###  IV-C Asymptotic Analysis

Relations between $R_{t}^{*}$ and $R_{t}^{+}$ as well as between $R_{r}^{*}$
and $R_{r}^{+}$ are analyzed and summarized in the following proposition.

######  Proposition 4.

For large number of IOS elements with non-uniform phase errors over
$[-\pi,\pi)$, the following asymptotic equivalence relations hold:

$\displaystyle R_{t}^{*}$ $\displaystyle\sim R_{t}^{+},~{}~{}~{}N\to\infty,$
(58a) $\displaystyle R_{r}^{*}$ $\displaystyle\sim
R_{r}^{+},~{}~{}~{}N\to\infty.$ (58b)

###### Proof:

See Appendix E. ∎

Next, we consider the relation between $R_{t}$ and $R_{t}^{*}$.

######  Proposition 5.

For large number of IOS elements with non-uniform phase errors over
$[-\pi,\pi)$, $R_{t}$ and $R_{t}^{*}$ satisfy the asymptotic equivalence,
i.e.,

$R_{t}\sim R_{t}^{*},~{}~{}~{}N\to\infty.$ (59)

Further, due to the transitive property of asymptotic analysis and Proposition
58, $R_{t}$ and $R_{t}^{+}$ are also asymptotically equivalent, i.e.,

$R_{t}\sim R_{t}^{+},~{}~{}~{}N\to\infty.$ (60)

###### Proof:

See Appendix F. ∎

######  Remark 6.

Eq.(59) in Proposition 60 does not hold when the phase errors are uniformly
distributed over $[-\pi,\pi)$. In this case, $R_{t}^{*}$ only acts as an upper
bound to $R_{t}$. The reason is clarified in Appendix F.

######  Remark 7.

Due to changing the order of computing expectation and minimum, the relation
between $R_{r}$ and $R_{r}^{*}$ cannot be analyzed similarly. However, for
large number of elements, $R_{r}$ can be well approximated by
$\log_{2}\left(1+q_{r}^{2}/q_{t}^{2}\right)$.

With the conclusions in Propositions 58 and 60, the relation between the
average achievable rates of user $T$ with correlated channels, $R_{t}$, and
uncorrelated channels, $R_{t,u}$, is presented in the following proposition.

######  Proposition 6.

For large number of IOS elements with non-uniform phase errors over
$[-\pi,\pi)$, $R_{t}$ is asymptotically equivalent to $R_{t,u}$, i.e.,

$R_{t}\sim R_{t,u},~{}~{}~{}N\to\infty.$ (61)

###### Proof:

Denote the approximate expression of $R_{t,u}$ derived from the channel
hardening effect as $R_{t,u}^{+}$. Referring to Remark 4, we have
$R_{t}^{+}=R_{t,u}^{+}$. Then, following the similar procedures in
Propositions 58 and 60, we can obtain $R_{t}\sim R_{t,u}$ as $N\to\infty$. ∎

######  Remark 8.

Since $\tilde{R}_{t}$ and $\tilde{R}_{r}$ have the same form as $R_{t}$, the
conclusions in the Propositions 58, 60 and 61 can also be derived for the OMA
scenario.

##  V Discussion on Multiple Users Scenario

The above analysis can be extended to the multi-user NOMA scenario, where
multiple users are located on both sides of the IOS. In this section, we
consider a four-user scenario, with two users located on each side of the IOS,
i.e., $\mathcal{U}_{t}\in\\{T,T^{\prime}\\}$ and
$\mathcal{U}_{r}\in\\{R,R^{\prime}\\}$, where $\mathcal{U}_{t}$ and
$\mathcal{U}_{r}$ are the collections of users in the transmitting and
reflecting regions, respectively. Similar results can be derived using the
same method for more users. Without loss of generality, assuming the path loss
coefficients of the four users are ordered as

$\eta_{r^{\prime}}<\eta_{t^{\prime}}<\eta_{r}<\eta_{t},$ (62)

the decoding order is fixed as $(R^{\prime},T^{\prime},R,T)$. The transmit
amplitude coefficients at TX are $q_{r^{\prime}}$, $q_{t^{\prime}}$, $q_{r}$,
$q_{t}$, with $q_{r^{\prime}}^{2}+q_{t^{\prime}}^{2}+q_{r}^{2}+q_{t}^{2}=1$.
By using the priority oriented NOMA scheme proposed in [17, 16], the
transmitting and reflecting phase shifts are adjusted to boost the channels of
$T$ and $R$, respectively. The average achievable rates of the four users are
given by

$\displaystyle R_{t^{\prime}}$
$\displaystyle=\mathbb{E}[\min\\{\log_{2}(1+\gamma_{t^{\prime}}),\log_{2}(1+\gamma_{r\to
t^{\prime}}),$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\log_{2}(1+\gamma_{t\to
t^{\prime}})\\}]$ (63a) $\displaystyle R_{r^{\prime}}$
$\displaystyle=\mathbb{E}[\min\\{\log_{2}(1+\gamma_{r^{\prime}}),\log_{2}(1+\gamma_{t^{\prime}\to
r^{\prime}}),$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\log_{2}(1+\gamma_{r\to
r^{\prime}}),\log_{2}(1+\gamma_{t\to r^{\prime}})\\}],$ (63b)

and $R_{t}$ and $R_{r}$ are given in (22) and (23), respectively. Following
the similar procedures in Sec. III, closed-form upper bounds of
$R_{r^{\prime}}$ and $R_{t^{\prime}}$ is formulated in the following
proposition.

######  Proposition 7.

Upper bounds on the average achievable rates of $T^{\prime}$ and $R^{\prime}$
are given by

$\begin{split}R_{t^{\prime}}^{*}=\left\\{\begin{split}&\log_{2}\left(1+\frac{q_{t^{\prime}}^{2}}{q_{t}^{2}+q_{r}^{2}+f_{t^{\prime}}^{-1}}\right),\quad
f_{t^{\prime}}<f_{r},\\\\[5.0pt]
&\log_{2}\left(1+\frac{q_{t^{\prime}}^{2}}{q_{t}^{2}+q_{r}^{2}+f_{r}^{-1}}\right),\quad
f_{t^{\prime}}\geq f_{r},\end{split}\right.\end{split}$ (64)
$R_{r^{\prime}}^{*}=\left\\{\begin{split}&\log_{2}\left(1+\frac{q_{r^{\prime}}^{2}}{q_{t}^{2}+q_{r}^{2}+q_{t^{\prime}}^{2}+f_{t^{\prime}}^{-1}}\right),\quad
f_{t^{\prime}}<f_{r^{\prime}},\\\\[5.0pt]
&\log_{2}\left(1+\frac{q_{r^{\prime}}^{2}}{q_{t}^{2}+q_{r}^{2}+q_{t^{\prime}}^{2}+f_{r^{\prime}}^{-1}}\right),\quad
f_{t^{\prime}}\geq f_{r^{\prime}},\end{split}\right.$ (65)

where $f_{t}$ and $f_{r}$ is defined in (44b)and (44c);
$f_{t^{\prime}}=\gamma_{0}\eta_{t^{\prime}}\alpha^{2}N$ and
$f_{r^{\prime}}=\gamma_{0}\eta_{r^{\prime}}\beta^{2}N$.

###### Proof:

See Appendix G. ∎

######  Remark 9.

Referring to Appendix G, it can be learned that

$\displaystyle R_{t^{\prime}}$
$\displaystyle\to\log_{2}\left(1+\frac{q_{t^{\prime}}^{2}}{q_{t}^{2}+q_{r}^{2}}\right),~{}~{}~{}\gamma_{0}\to\infty,$
(66a) $\displaystyle R_{r^{\prime}}$
$\displaystyle\to\log_{2}\left(1+\frac{q_{r^{\prime}}^{2}}{q_{t}^{2}+q_{r}^{2}+q_{t^{\prime}}^{2}}\right),~{}~{}~{}\gamma_{0}\to\infty.$
(66b)

######  Remark 10.

With (30), (40), (66a) and (66b), we can conclude that, in NOMA transmission
scheme, except the user of the last order (which is $T$ in this paper), the
average achievable rates of all other users are determined by the transmit
amplitude coefficients of TX as $\gamma_{0}$ goes large.”

##  VI Numerical Results

In this section, we present numerical results to evaluate the accuracy of the
proposed analysis by Monte Carlo (MC) simulations and show the impacts of
phase errors and channel correlations on the system performance.

###  VI-A Simulation Setup

Unless otherwise stated, the simulation parameters are set as follows. The
distances of the TX-IOS, IOS-user $T$ and IOS-user $R$ links are set to be
$d_{b}=10\,{\rm m}$, $d_{t}=5\,{\rm m}$ and $d_{r}=10\,{\rm m}$. The pathloss
exponent $\chi$ is $2.4$ and the carrier frequency is $3\,{\rm GHz}$ with the
wavelength $\lambda=0.1\,{\rm m}$. Each IOS element is of size
$l=\frac{\lambda}{2}$ and $w=\frac{\lambda}{2}$. The transmitting and
reflecting amplitude coefficients of the IOS are given as $\alpha=0.8$ and
$\beta=0.6$, such that $\alpha^{2}+\beta^{2}=1$. The transmit amplitude
coefficients at TX are $q_{t}=0.6$ and $q_{r}=0.8$, such that
$q_{t}^{2}+q_{r}^{2}=1$.444 In practice, $\alpha$, $\beta$, $q_{t}$ and
$q_{r}$ can be adjusted to meet different performance requirement of each
user. For example, if $R$ needs better performance, we can increase $q_{r}$
and $\beta$. The intercepts at unit distance of the transmitting and
reflecting channels are $\Lambda_{t}=\Lambda_{r}=-30\,{\rm dB}$. The noise
power is $\sigma_{0}^{2}=-50\,{\rm dBm}$.

###  VI-B Average Achievable Rates versus Number of IOS Elements

Figure 3: Average achievable rate of user $T$ versus number of elements.

Figure 4: Average achievable rate of user $R$ versus number of elements.

We investigate the average achievable rates versus the number of IOS elements
with different phase quantization bits. IOS elements are assumed to be placed
in a uniform rectangular array with four elements per column, i.e., $N_{v}=4$.
The number of elements per row varies in $N_{h}\in[1,25]$. The transmit power
is set to be $20\,{\rm dBm}$. As benchmarks, the systems with perfect phase
adjustment, i.e., $\phi_{n}^{u}=0$, and with uniform phase error over
$[-\pi,\pi)$, are both shown. In the legend, ‘Jensen’ and ‘Hardening’
represent the upper bound and approximation derived in Sec. III and Sec. IV;
‘MC Simulations’ denotes the MC simulation results.

As shown in Fig. 3, the proposed upper bound and approximation on $R_{t}$ are
very accurate even for a small number of elements. Increasing the number of
elements can increase the average achievable rate of user $T$, but the slope
of the curve becomes flat, which can be concluded from the log relation
between $R_{t}$ and $N^{2}$ in Remark 3. As expected in Remark 5, adding the
number of quantization bits is able to improve system performance. However,
the performance gain diminishes with the increase of quantization levels. For
example, when the number of elements is 60, the average achievable rates of
user $T$ are 5 bit/s/Hz and 9.7 bit/s/Hz for uniform phase errors and 1-bit
phase shifting, respectively. However, compared with 1-bit phase shifting, the
performance gain brought by 2-bit phase shifting is much smaller with 10.7
bit/s/Hz, and it is very close to the performance of perfect phase shifting
with 11 bit/s/Hz. Besides, as expected in Remark 6, $R_{t}^{*}$ can only act
as an upper bound when the phase errors are uniformly distributed over
$[-\pi,\pi)$.

Fig. 4 illustrates the average achievable rate of user $R$ against the number
of elements. The proposed upper bound is relatively loose for a small number
of elements and becomes tight as the number of elements grows large. ‘Upper
bound’ in the legend stands for $\log_{2}(1+q_{r}^{2}/q_{t}^{2})$. As can be
seen from Fig. 4, the average achievable rate of user $R$ converges to
$\log_{2}(1+q_{r}^{2}/q_{t}^{2})$ with increasing numbers of elements.

###  VI-C Average Achievable Rates versus Transmit SNR

Figure 5: Average achievable rate of user $T$ versus transmit SNR.

Figure 6: Average achievable rate of user $R$ versus transmit SNR.

Figure 7: Sum rate versus transmit SNR, $\kappa_{t}=\kappa_{r}=2$.

The relation between average achievable rates and transmit SNR is explored in
Fig. 5, where a rectangular IOS with $N_{v}=4$ elements per column and
$N_{h}=10$ elements per row is used. The transmit SNR lies in the interval
$\gamma_{0}\in[20,90]$ dB. Channel estimation errors are considered by setting
the concentration parameter of the Von Mises distribution to
$\kappa_{u}\in\\{1,2\\}$, $u\in\\{t,r\\}$. It can be observed from Fig. 5 that
the average achievable rate of $T$ increases with the transmit SNR, as
expected. The proposed upper bound and approximation are very close to each
other. In addition, in Fig. 6, the average achievable rate of $R$ grows and
converges to the upper bound $\log_{2}(1+q_{r}^{2}/q_{t}^{2})$ with increasing
SNR. This observation corroborates our discussion in Sec. III. The results of
OMA are also given for comparison. We can see that the average achievable rate
of $T$ in NOMA outperforms that of OMA in the large transmit SNR regime.
However, the average achievable rate of $R$ in NOMA is inferior to that of OMA
because it converges to a constant value. As a step further, we compare the
system sum rates of NOMA and OMA in Fig. 7. Two locations of $R$, i.e.,
$d_{r}=6\,{\rm m}$ and $d_{r}=15\,{\rm m}$, are considered. The large transmit
SNR regime is focused. $\kappa_{t}=\kappa_{r}=2$. For $d_{r}=6\,{\rm m}$,
$\alpha^{4}\epsilon_{t}^{2}\eta_{t}<\epsilon_{r}^{2}\eta_{r}$, and OMA is
better; For $d_{r}=15\,{\rm m}$,
$\alpha^{4}\epsilon_{t}^{2}\eta_{t}>\epsilon_{r}^{2}\eta_{r}$, and NOMA has
larger sum rate. Numerical results confirm our analysis.

###  VI-D Correlated Channels versus Uncorrelated Channels

Figure 8: Average achievable rate of user $T$ versus transmit SNR for
different channel conditions.

Figure 9: Average achievable rates of $T$, $R$, $T^{\prime}$ and $R^{\prime}$
versus the transmit SNR.

The impact of channel correlations on the system performance is investigated
in Fig. 8, where a 1-bit phase adjustment is performed at the IOS. We consider
a rectangular IOS, where each IOS element is of size $5\,{\rm cm}\times
5\,{\rm cm}$ and is placed in a $5\times N_{h}$ rectangular array, with
$N_{h}$ varying in the interval $[1,20]$. We assume the wavelength
$\lambda=20\,{\rm cm}$. According to (9), the channels are correlated in this
case. The average achievable rate with uncorrelated channels is provided as a
benchmark555In practice, the channels associated with the IOS are uncorrelated
when the elements are placed in a linear array, and the spacing of elements
equals $\lambda/2$ times integers. Besides, the covariance matrix $R$ of
uncorrelated channels in (9) is the identity matrix.. As shown in Fig. 8, the
average achievable rates of uncorrelated and correlated channels are close to
each other, even for a small number of elements. For example, when $N=15$, the
performance gap is $0.09\,{\rm bit/s/Hz}$. Moreover, the performance gap of
the average achievable rates between uncorrelated and correlated channels
decreases with an increase in the number of elements, as expected from
Proposition 61. For example, for $N=90$, the performance gap decreases to
$0.02\,{\rm bit/s/Hz}$.

###  VI-E Multiple Users NOMA

Two more users $T^{\prime}$ and $R^{\prime}$ are added to the system. The
distances from the IOS to $T^{\prime}$ and $R^{\prime}$ are
$d_{t^{\prime}}=12\,{\rm m}$ and $d_{r^{\prime}}=15\,{\rm m}$, respectively.
The transmit amplitude coefficients at TX are adjusted to be
$q_{r^{\prime}}^{2}=0.4$, $q_{t^{\prime}}^{2}=0.3$, $q_{r}^{2}=0.2$ and
$q_{t}^{2}=0.1$. We use a rectangular IOS with $N_{v}=4$ elements per column
and $N_{h}=10$ elements per row and 1-bit phase adjustment is performed. Other
simulation parameters are not changed. Fig. 9 shows the average achievable
rates of $T$, $R$, $T^{\prime}$ and $R^{\prime}$ versus the transmit SNR. As
expected, the average achievable rates of $R$, $T^{\prime}$ and $R^{\prime}$
respectively converge to $\log_{2}\left(1+{q_{r}^{2}}/{q_{t}^{2}}\right)$,
$\log_{2}\left(1+{q_{t^{\prime}}^{2}}/{(q_{t}^{2}+q_{r}^{2})}\right)$ and
$\log_{2}\left(1+{q_{r^{\prime}}^{2}}/({q_{t}^{2}+q_{r}^{2}+q_{t^{\prime}}^{2}})\right)$
as the transmit SNR goes large. In a multi-user NOMA system, the performance
of users with higher decoding orders, e.g., $T^{\prime}$ and $R^{\prime}$ in
this system, is limited and highly determined by the transmit amplitude
coefficients at TX. Therefore, in existing literatures, it is more common to
consider two-user NOMA, in which two paired users are grouped into the same
resource block and different NOMA pairs are allocated with orthogonal resource
blocks.

##  VII Conclusions

This paper proposed an IOS-assisted downlink NOMA network with spatially
correlated channels and phase errors and analyzed the average achievable rates
of two users. Specifically, we investigated an upper bound on the average
achievable rates using Jensen’s inequality. The channel hardening effect in
the proposed system was exploited, from which an approximation on the average
achievable rates was derived. The proposed upper bound and approximation were
proved to be asymptotically equivalent. We also proved that the average
achievable rates with correlated and uncorrelated channels are asymptotically
equivalent for a large number of elements. IOS assisted multi-user NOMA and
IOS assisted OMA were also analyzed with the same framework. Numerical results
illustrated that the performance gap between low-resolution elements with two-
bit phase shifting and the ideal elements with continuous phase shifting is
narrow and confirmed that NOMA is not always superior to OMA due to the
reconfigurability of IOS over different time slots.

## Appendix A

In the proposed system model, $h_{n}$ can be written as

$h_{n}=R_{n}\exp\left(j\vartheta_{n}\right)=X_{n}+jY_{n},\,n\in\\{1,2,...,N\\},$
(A.1)

where $R_{n}$, $\vartheta_{n}$, $X_{n}$ and $Y_{n}$ denote the amplitude,
phase angle, real part and imaginary part of $h_{n}$, respectively; $X_{n}$
and $Y_{n}$ are uncorrelated normal random variables, each with zero mean and
variance as

$\tau\triangleq{\rm Var}[X_{n}]={\rm Var}[Y_{n}]=\frac{1}{2}.$ (A.2)

For the channels associated with two different IOS elements, i.e., $h_{n}$ and
$h_{i}$, it can be obtained that

$\displaystyle\mathbb{E}\left[X_{n}X_{i}\right]$
$\displaystyle=\mathbb{E}\left[|h_{n}||h_{i}|\cos(\vartheta_{n})\cos(\vartheta_{i})\right],$
(A.3a) $\displaystyle\mathbb{E}\left[Y_{n}Y_{i}\right]$
$\displaystyle=\mathbb{E}\left[|h_{n}||h_{i}|\sin(\vartheta_{n})\sin(\vartheta_{i})\right]$
$\displaystyle=\mathbb{E}\left[|h_{n}||h_{i}|\cos(\vartheta_{n}+\pi/2)\cos(\vartheta_{i}+\pi/2)\right].$
(A.3b)

Since $\vartheta_{n}$, $n\in\\{1,2,...,N\\}$, are uniformly distributed over
$[-\pi,\pi)$ and the above functions are periodic, we can have

$\tau_{1}\triangleq\mathbb{E}\left[X_{n}X_{i}\right]=\mathbb{E}\left[Y_{n}Y_{i}\right].$
(A.4)

Similarly,

$\displaystyle\mathbb{E}\left[X_{n}Y_{i}\right]$
$\displaystyle=\mathbb{E}\left[|h_{n}||h_{i}|\cos(\vartheta_{n})\sin(\vartheta_{i})\right],$
(A.5a) $\displaystyle\mathbb{E}\left[Y_{n}X_{i}\right]$
$\displaystyle=\mathbb{E}\left[|h_{n}||h_{i}|\sin(\vartheta_{n})\cos(\vartheta_{i})\right]$
$\displaystyle=-\mathbb{E}\left[|h_{n}||h_{i}|\cos(\vartheta_{n}+\pi/2)\sin(\vartheta_{i}+\pi/2)\right].$
(A.5b)

Hence,

$\tau_{2}\triangleq\mathbb{E}\left[X_{n}Y_{i}\right]=-\mathbb{E}\left[Y_{n}X_{i}\right].$
(A.6)

For $n\neq i$, the joint distribution of $(X_{n},Y_{n},X_{i},Y_{i})$ can be
written as

$f(x_{n},y_{n},x_{i},y_{i})=\frac{1}{(2\pi)^{2}\sqrt{{\rm
det}\mathbf{C}}}\exp\left(-\frac{1}{2}\mathbf{v}\mathbf{C}^{-1}\mathbf{v}^{T}\right),$
(A.7)

where $\mathbf{v}=[x_{n},y_{n},x_{i},y_{i}]$; $\mathbf{C}$ is the covariance
matrix of $\\{X_{n},Y_{n},X_{i},Y_{i}\\}$, given by

$\mathbf{C}=\begin{pmatrix}\tau&0&\tau_{1}&\tau_{2}\\\
0&\tau&-\tau_{2}&\tau_{1}\\\ \tau_{1}&-\tau_{2}&\tau&0\\\
\tau_{2}&\tau_{1}&0&\tau\end{pmatrix}.$ (A.8)

The determinant and inverse of $\mathbf{C}$ can be calculated as

${\rm det}\\{\mathbf{C}\\}=(\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2})^{2},$ (A.9)
$\mathbf{C}^{-1}=\begin{pmatrix}\frac{\tau}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&0&\frac{-\tau_{1}}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&\frac{-\tau_{2}}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}\\\\[5.0pt]
0&\frac{\tau}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&\frac{\tau_{2}}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&\frac{-\tau_{1}}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}\\\\[5.0pt]
\frac{-\tau_{1}}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&\frac{\tau_{2}}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&\frac{\tau}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&0\\\\[5.0pt]
\frac{-\tau_{2}}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&\frac{-\tau_{1}}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}&0&\frac{\tau}{\tau^{2}-\tau_{1}^{2}-\tau_{2}^{2}}\end{pmatrix}.$
(A.10)

Taking (A.9) and (A.10) into (A.7), the joint distribution of
$(X_{n},Y_{n},X_{i},Y_{i})$ can be obtained. Then, using the theorem of the
transformation of random variables:

$f(r_{n},r_{i},\vartheta_{n},\vartheta_{i})=\left|{\rm
det}(J(r_{n},r_{i},\vartheta_{n},\vartheta_{i}))\right|f(x_{n},y_{n},x_{i},y_{i}),$
(A.11)

where

$J(r_{n},r_{i},\vartheta_{n},\vartheta_{i})=\frac{\partial(x_{n},y_{n},x_{i},y_{i})}{\partial(r_{n},r_{i},\vartheta_{n},\vartheta_{i})}$
(A.12)

is the Jacobian matrix for the transformation

$\begin{split}\\{&x_{n}=r_{n}\cos(\vartheta_{n}),y_{n}=r_{n}\sin(\vartheta_{n}),\\\
&x_{i}=r_{i}\cos(\vartheta_{i}),y_{i}=r_{i}\sin(\vartheta_{i})\\},\end{split}$
(A.13)

we can derive the joint distribution of
$(R_{n},R_{i},\vartheta_{n},\vartheta_{i})$ as

$\begin{split}&f(r_{n},r_{i},\vartheta_{n},\vartheta_{i})=\frac{r_{n}r_{i}}{(2\pi)^{2}(\tau^{2}-q^{2})}\\\
&~{}~{}\times\exp\left[-\frac{\tau\left(r_{n}^{2}+r_{i}^{2}\right)-2r_{n}r_{i}q\cos\left(\vartheta_{i}-\vartheta_{n}-\xi\right)}{2\left(\tau^{2}-{q^{2}}\right)}\right],\end{split}$
(A.14)

where

$\displaystyle\xi$
$\displaystyle=\arctan\left(\frac{\tau_{2}}{\tau_{1}}\right),$ (A.15a)
$\displaystyle q^{2}$
$\displaystyle=\tau_{1}^{2}+\tau_{2}^{2}=\frac{\left|\mathbb{E}[h_{n}h_{i}^{*}]\right|^{2}}{4}.$
(A.15b)

Using [43, eq.6.631.1, eq.7.621.4] and after some mathematical manipulation,
$\mathbb{E}[R_{n}R_{i}]$ can be expressed as

$\begin{split}&\mathbb{E}[R_{n}R_{i}]\\\
=&\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{2\pi}\int_{0}^{2\pi}r_{n}r_{i}f(r_{n},r_{i},\vartheta_{n},\vartheta_{i}){\,\mathrm{d}}\vartheta_{n}{\,\mathrm{d}}\vartheta_{i}{\,\mathrm{d}}r_{n}{\,\mathrm{d}}r_{i}\\\\[5.0pt]
=&\left(\frac{q^{2}}{\tau}-\tau\right)K\left(\frac{q^{2}}{\tau^{2}}\right)+2\tau
E\left(\frac{q^{2}}{\tau^{2}}\right).\end{split}$ (A.16)

For $n=i$, it is easy to show that

$\mathbb{E}[R_{n}R_{i}]=2\tau=1.$ (A.17)

The same conclusions can be derived for the channels $\mathbf{g}$ and
$\mathbf{r}$ following the framework. Thus, the statement in Proposition 1 is
proved.

## Appendix B

Define

$f(x)=\left(\frac{x}{2}-\frac{1}{2}\right)K\left({x}\right)+E\left({x}\right).$
(B.1)

It can be proved that $f(x)$ is an increasing function for $x\geq 0$. Since

$0\leq\left|\mathbb{E}[w_{n}w_{i}^{*}]\right|^{2}\leq 1,$ (B.2)

taking (B.2) into (28), we can have

$\frac{\pi}{4}\leq\mathbb{E}\left[\left|w_{n}\right|\left|w_{i}\right|\right]\leq
1.$ (B.3)

## Appendix C

Since

$\mathbb{E}\left[\sum\limits_{n=1}^{N}\left|g_{n}\right|^{2}\left|h_{n}\right|^{2}\right]=N,$
(C.1)

we can have

$\small\mathbb{E}\left[\sum\limits_{n=1}^{N-1}\sum\limits_{i=n+1}^{N}\left|g_{n}\right|\left|h_{n}\right|\left|g_{i}\right|\left|h_{i}\right|\right]=\frac{\mathbf{tr}\left(\mathbf{\bar{R}\bar{R}}\right)-N}{2}.$
(C.2)

Therefore,

$\begin{split}&\mathbb{E}\left[\sum\limits_{n=1}^{N-1}\sum\limits_{i=n+1}^{N}\left|g_{n}\right|\left|h_{n}\right|\cos(\phi_{n}^{t})\left|g_{i}\right|\left|h_{i}\right|\cos(\phi_{i}^{t})\right]\\\\[5.0pt]
&=\frac{\epsilon_{t}^{2}}{2}\left(\mathbf{tr}\left(\mathbf{\bar{R}\bar{R}}\right)-N\right).\end{split}$
(C.3)

Moreover,

$\mathbb{E}\left[\sum\limits_{n=1}^{N-1}\sum\limits_{i=n+1}^{N}\left|g_{n}\right|\left|h_{n}\right|\sin(\phi_{n}^{t})\left|g_{i}\right|\left|h_{i}\right|\sin(\phi_{i}^{t})\right]=0,$
(C.4)

due to the symmetrical distribution around zero of $\phi_{n}^{t}$,
$n\in\\{1,...,N\\}$. Substituting (C.1), (C.3) and (C.4) into (27), we can
obtain (30). Thus, the proof is complete.

## Appendix D

To begin with, a Lemma regarding the sum of correlated random variables is
presented as follows.

######  Lemma 1 ([44]).

Consider a sequence of random variables $\\{W_{k}\\}$. If $\\{W_{k}\\}$
satisfies

  * •

$\mathbb{E}\left[W_{k}\right]=a$,

  * •

${\rm Var}\left[W_{k}\right]$ is bounded,

  * •

${\rm Cov}\\{W_{i},W_{j}\\}\to 0$ as $|i-j|\to\infty$,

then,

$\frac{1}{K}\sum\limits_{k=1}^{K}W_{k}\overset{P}{\to}a,\,\,K\to\infty,$ (D.1)

with the convergence in probability.

###### Proof:

Even though this is a known result, we include a proof because the obtained
scaling of the variance with $K$ is used in this paper in the proof of
Proposition 58.

Since the variance is bounded, the covariance of $W_{i}$ and $W_{j}$ is
bounded as

${\rm Cov}\left\\{W_{i},W_{j}\right\\}\leq c_{0},$ (D.2)

where $c_{0}$ is a constant. Denoting $V_{K}=\sum\limits_{k=1}^{K}W_{k}$, the
variance of $V_{K}$ can be written as

$\displaystyle{\rm var}[V_{K}]$ $\displaystyle=\sum\limits_{k=1}^{K}{\rm
Cov}\left\\{W_{k},W_{k}\right\\}+2\sum\limits_{i=1}^{K-1}\sum\limits_{j=i+1}^{K}{\rm
Cov}\left\\{W_{i},W_{j}\right\\}$ $\displaystyle\leq
Kc_{0}+2\sum\limits_{i=1}^{K-1}\sum\limits_{j=i+1}^{K}{\rm
Cov}\left\\{W_{i},W_{j}\right\\}.$ (D.3)

Since ${\rm Cov}\left\\{W_{i},W_{j}\right\\}\to 0$ when $|i-j|\to\infty$, we
can find $K_{0}$ such that for $|i-j|>K_{0}$, ${\rm
Cov}\left\\{W_{i},W_{j}\right\\}<\varepsilon$, for any $\varepsilon>0$. Hence,

$\displaystyle\left|\sum\limits_{i=1}^{K-1}\sum\limits_{j=i+1}^{K}{\rm
Cov}\left\\{W_{i},W_{j}\right\\}\right|$ $\displaystyle\leq$
$\displaystyle\left|\sum\limits_{i=1}^{K-1}\sum\limits_{j=i+1}^{i+K_{0}}{\rm
Cov}\left\\{W_{i},W_{j}\right\\}\right|+\left|\sum\limits_{i=1}^{K-1}\sum\limits_{j=i+K_{0}+1}^{K}{\rm
Cov}\left\\{W_{i},W_{j}\right\\}\right|$ $\displaystyle\leq$ $\displaystyle
KK_{0}c_{0}+K^{2}\varepsilon.$ (D.4)

Therefore, the variance of $V_{K}/K$ satisfies

${\rm
Var}\left[\frac{V_{K}}{K}\right]\leq\frac{c_{0}}{K}+\frac{2K_{0}c_{0}}{K}+2\varepsilon.$
(D.5)

It can be seen that ${\rm Var}\left[\frac{V_{K}}{K}\right]\to 0$, as
$K\to\infty$. Thus, the proof of Lemma 1 is complete. ∎

The composite channel associated with user $T$ is analyzed first. Let
$Y_{n}=|g_{n}||h_{n}|e^{j\phi_{n}^{t}}$. We can have

$\mathbb{E}[Y_{n}]=\frac{\pi}{4}\epsilon_{t},$ (D.6)

and the variance of $Y_{n}$ is bounded. Covariance of $\\{Y_{n},Y_{i}\\}$ can
be written as

$\begin{split}{\rm
Cov}\\{Y_{n},Y_{i}\\}=&\mathbb{E}\left[|g_{n}||h_{n}|e^{j\phi_{n}^{t}}|g_{i}||h_{i}|e^{-j\phi_{i}^{t}}\right]\\\
&-\mathbb{E}\left[|g_{n}||h_{n}|e^{j\phi_{n}^{t}}\right]\mathbb{E}\left[|g_{i}||h_{i}|e^{-j\phi_{i}^{t}}\right].\end{split}$
(D.7)

From (9) and (28), it can be learned that

$\mathbb{E}\left[|w_{n}||w_{i}|\right]\to\mathbb{E}\left[|w_{n}|\right]\mathbb{E}\left[|w_{i}|\right]=\frac{\pi}{4},\,\,|n-i|\to\infty,$
(D.8)

where $w\in\\{h,g,r\\}$. Given that the phase errors are i.i.d. with symmetric
distribution around zero, it can be proved that

$\begin{split}{\rm
Cov}\\{Y_{n},Y_{i}\\}=&\,\mathbb{E}\left[|g_{n}||g_{i}|\right]\mathbb{E}\left[|h_{n}||h_{i}|\right]\epsilon_{t}^{2}\\\
&-\mathbb{E}\left[|g_{n}|\right]\mathbb{E}\left[|g_{i}|\right]\mathbb{E}\left[|h_{n}|\right]\mathbb{E}\left[|h_{i}|\right]\epsilon_{t}^{2}\\\
\to&\,0,\,\,|n-i|\to\infty.\end{split}$ (D.9)

Hence, using Lemma 1, we obtain

$\frac{1}{N}\sum\limits_{n=1}^{N}\left|g_{n}\right|\left|h_{n}\right|e^{j\phi_{n}^{t}}\overset{P}{\to}\frac{\pi}{4}\epsilon_{t},\,\,N\to\infty.$
(D.10)

With the continuous mapping theorem, (45a) in Proposition 3 can be obtained.
Following the same framework, (45b) can be derived.

## Appendix E

Let

$\displaystyle A_{N}$
$\displaystyle=\sum\limits_{n=1}^{N}\left|g_{n}\right|\left|h_{n}\right|\cos(\phi_{n}^{t}),$
(E.1a) $\displaystyle B_{N}$
$\displaystyle=\sum\limits_{n=1}^{N}\left|g_{n}\right|\left|h_{n}\right|\sin(\phi_{n}^{t}).$
(E.1b)

The expectations of $A_{N}$ and $B_{N}$ are
$\mathbb{E}[A_{N}]=\frac{\pi}{4}N\epsilon_{t}$ and $\mathbb{E}[B_{N}]=0$.
$\mathbb{E}[A_{N}^{2}+B_{N}^{2}]$ can be written as

$\mathbb{E}[A_{N}^{2}+B_{N}^{2}]=\mathbb{E}^{2}[A_{N}]+{\rm
var}[A_{N}]+\mathbb{E}^{2}[B_{N}]+{\rm var}[B_{N}].$ (E.2)

As shown in Lemma 1, the variances of $A_{N}$ and $B_{N}$ scale with $N$.
Thus, when the phase errors are non-uniform over $[-\pi,\pi)$,
$\mathbb{E}[A_{N}^{2}+B_{N}^{2}]$ satisfies the asymptotic equivalence as

$\mathbb{E}[A_{N}^{2}+B_{N}^{2}]\sim\frac{\pi^{2}}{16}N^{2}\epsilon_{t}^{2},~{}~{}~{}N\to\infty.$
(E.3)

Thus, with the properties of asymptotic analysis, we can obtain (58a) in
Proposition 58. Following the similar procedures, (58b) can be derived as
well.

## Appendix F

As shown in Proposition 3, the random variable $G_{N}=\frac{1}{N^{2}}H_{t}$
satisfies

$G_{N}\overset{P}{\to}\frac{\pi^{2}}{16}\epsilon_{t}^{2},~{}~{}~{}N\to\infty,$
(F.1)

with the convergence in probability. As shown in Proposition 58 and its proof,
the expectation and variance of $G_{N}$ satisfy

$\mathbb{E}[G_{N}]\to\frac{\pi^{2}}{16}\epsilon_{t}^{2},~{}~{}{\rm
Var}[G_{N}]\to 0,~{}~{}~{}N\to\infty.$ (F.2)

Thus, when the phase errors are non-uniform over $[-\pi,\pi)$, the expectation
and variance of $H_{t}$ satisfy

$\frac{{\rm Var}\left[H_{t}\right]}{\mathbb{E}^{2}\left[H_{t}\right]}\to
0,~{}~{}~{}N\to\infty.$ (F.3)

Invoking [45, Theorem 4], the asymptotic equivalence of $R_{t}$ and
$R_{t}^{*}$ can be obtained. With the transitive property of asymptotic
analysis, the asymptotic equivalence of $R_{t}$ and $R_{t}^{+}$ also holds.

When the phase errors are uniform over $[-\pi,\pi)$, the expectation of
$G_{N}$ goes to zero as $N$ goes large. The asymptotic equivalence of $R_{t}$
and $R_{t}^{*}$ does not hold since the relation in (F.3) is not satisfied.

## Appendix G

$R_{r^{\prime}}$ is analyzed first. $\gamma_{r^{\prime}}$,
$\gamma_{t^{\prime}\to r^{\prime}}$, $\gamma_{r\to r^{\prime}}$ and
$\gamma_{t\to r^{\prime}}$ are given by

$\displaystyle\gamma_{r^{\prime}}$
$\displaystyle=\frac{P\eta_{r^{\prime}}\beta^{2}H_{r^{\prime}}q_{r^{\prime}}^{2}}{P\eta_{r^{\prime}}\beta^{2}H_{r^{\prime}}\left(q_{t}^{2}+q_{r}^{2}+q_{t^{\prime}}^{2}\right)+\sigma_{0}^{2}},$
(G.1a) $\displaystyle\gamma_{t^{\prime}\to r^{\prime}}$
$\displaystyle=\frac{P\eta_{t^{\prime}}\alpha^{2}H_{t^{\prime}}q_{r^{\prime}}^{2}}{P\eta_{t^{\prime}}\alpha^{2}H_{t^{\prime}}\left(q_{t}^{2}+q_{r}^{2}+q_{t^{\prime}}^{2}\right)+\sigma_{0}^{2}},$
(G.1b) $\displaystyle\gamma_{r\to r^{\prime}}$
$\displaystyle=\frac{P\eta_{r}\beta^{2}H_{r}q_{r^{\prime}}^{2}}{P\eta_{r}\beta^{2}H_{r}\left(q_{t}^{2}+q_{r}^{2}+q_{t^{\prime}}^{2}\right)+\sigma_{0}^{2}}$
(G.1c) $\displaystyle\gamma_{t\to r^{\prime}}$
$\displaystyle=\frac{P\eta_{t}\alpha^{2}H_{t}q_{r^{\prime}}^{2}}{P\eta_{t}\alpha^{2}H_{t}\left(q_{t}^{2}+q_{r}^{2}+q_{t^{\prime}}^{2}\right)+\sigma_{0}^{2}},$
(G.1d)

respectively, where $H_{t}$ and $H_{r}$ is defined in (21). $H_{t^{\prime}}$
and $H_{r^{\prime}}$ can be written as

$H_{t^{\prime}}=\left|\sum\limits_{n=1}^{N}\left|g^{\prime}_{n}\right|\left|h_{n}\right|e^{j\phi_{n}^{t^{\prime}}}\right|^{2},H_{r^{\prime}}=\left|\sum\limits_{n=1}^{N}\left|r^{\prime}_{n}\right|\left|h_{n}\right|e^{j\phi_{n}^{r^{\prime}}}\right|^{2}.$
(G.2)

where $g^{\prime}_{n}$ and $r^{\prime}_{n}$ are the channels from the $n$th
element to $T^{\prime}$ and $R^{\prime}$, respectively. Since the priority
oriented scheme is used and phase shifts are adjusted to boost $H_{t}$ and
$H_{r}$, $\phi_{n}^{t^{\prime}}$ and $\phi_{n}^{r^{\prime}}$ can be regarded
as uniformly distributed random variables over $[-\pi,\pi)$. Following the
similar procedures in Propositions 1 and 33, an upper bound of
$R_{r^{\prime}}$ is derived by

$R_{r^{\prime}}^{*}=\underset{\zeta\in\\{t,r,t^{\prime},r^{\prime}\\}}{\min}\left\\{\log_{2}\left(1+\frac{q_{r^{\prime}}^{2}}{q_{t}^{2}+q_{r}^{2}+q_{t^{\prime}}^{2}+f_{\zeta}^{-1}}\right)\right\\},$
(G.3)

Since $f_{t^{\prime}}<f_{t}$ and $f_{r^{\prime}}<f_{r}$, $R_{r^{\prime}}^{*}$
can be simplified as in (65). Following the same method, $R_{t^{\prime}}^{*}$
can be also derived.

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