# Reconfigurable Intelligent Surface-Aided NOMA with Limited Feedback

Mojtaba Ahmadi Almasi and Hamid Jafarkhani The authors are with Center for
Pervasive Communications and Computing, University of California, Irvine. This
work was supported in part by the NSF Award CNS-2008786.

###### Abstract

The design of feedback channels in frequency division duplex (FDD) systems is
a major challenge because of the limited available feedback bits. We consider
non-orthogonal multiple access (NOMA) systems that incorporate reconfigurable
intelligent surfaces (RISs). In limited feedback RIS-aided NOMA systems, the
RIS-aided channel and the direct channel gains should be quantized and fed
back to the transmitter. This paper investigates the rate loss of the overall
RIS-aided NOMA systems suffering from quantization errors. We first consider
random vector quantization for the overall RIS-aided channel and identical
uniform quantizers for the direct channel gains. We then obtain an upper bound
for the rate loss, due to the quantization error, as a function of the number
of feedback bits and the size of RIS. Our numerical results indicate the sum
rate performance of the limited feedback system approaches that of the system
with full CSI as the number of feedback bits increases.

##  I Introduction

Reconfigurable intelligent surfaces (RISs) are presumed as an attractive
solution to enhance the spectral, power efficiency, and coverage of wireless
communication systems [1]. These surfaces consist of many passive and cost-
effective elements capable of controlling the propagation environment by
properly adjusting the direction of coming signals. These distinctive
properties make RIS a promising solution for broad connectivity in the next
generation of wireless networks. Previously, intelligent surfaces that are not
reconfigurable [2, 3] and reconfigurable multiple-input multiple-output (MIMO)
systems [4, 5] have been proposed for orthogonal multiple access (OMA)
systems. Also, it is shown that RISs can have notable use cases and boost
performance when merged with other emerging technologies such as non-
orthogonal multiple access (NOMA) [6, 7].

NOMA has been a topic of research as a promising new technology for the next
generation of wireless communications. Specifically, in the downlink, power-
domain NOMA aims to serve two or more users by sharing the same
time/frequency/code resource block [8]. At the transmitter side, NOMA squeezes
the information signals using superposition coding. Before decoding the
intended signal, the stronger user applies successive interference
cancellation, i.e., first decodes the weaker user’s signal and then subtracts
it from the received signal.

In this paper, we incorporate the RIS in downlink NOMA to improve the quality
of the weak user’s channel. Unlike many other time division duplex (TDD)-based
RIS-assisted NOMA (RIS-NOMA) systems like [6, 7, 9, 10], we consider a
frequency division duplex (FDD) system. The FDD-based RIS-NOMA is more
challenging in the sense that the channel must be estimated at the receiver
and fed back to the transmitter via a limited feedback channel. The
availability of the quantized channel gains instead of perfect channel state
information (CSI) creates the following two major issues. First, the quantized
channel gains can result in a severe rate loss [11, 12]. This phenomenon is
more harmful when the quantized channel gains inaccurately change the order of
NOMA users. Second, the phase information obtained from the overall quantized
channel restricts the performance of the RIS [13]. Motivated by this, we
investigate the impact of quantizing the overall RIS-aided channel and the
direct channel gains on the system’s rate loss.

The limited feedback problem in RIS-aided systems is studied in [14, 15, 16,
17]. Ref. [14] proposes a cascaded codebook design and bit partitioning
strategy in the presence of line-of-sight (LoS) and non-LoS (NLoS) channels.
In [15], the feedback is divided into two parts, channel feedback and angle
information feedback. In particular, [15] uses a random vector quantizer (RVQ)
codebook to quantize the channel followed by feeding back the indices related
to the angle of arrival (AoA) and angle of departure (AoD) information of the
cascade channel matrix. Similarly, [16] designs a codebook-based limited
feedback protocol for RIS using learning methods. In [17], authors aim to
reconstruct the channel using the signal strength feedback and exploiting the
sparsity and low-rank properties. None of the above limited feedback methods
can be applied to our RIS-NOMA. Particularly, the feedback methods in [15, 14,
16] mainly try to send the normalized channel vectors to perform beamforming
at the transmitter. Further, [17] determines the channel direction while the
channel gains are estimated based on the distance and not precisely. However,
our RIS-NOMA system requires precise channel gains to accurately order the
users and perform the superposition coding. Also, it needs the overall RIS-
aided channel vector to adjust the RIS. The main contributions of this paper
are as follows:

  * •

We provide a limited feedback framework for RIS-NOMA systems and analyze its
performance.

  * •

We find an upper bound on the rate loss caused by quantization.

We conduct numerical simulations to evaluate the sum rate and the rate loss of
the limited feedback RIS-NOMA system. The results verify our theoretical
derivations.

Notations: In this paper, $j=\sqrt{-1}$. Small letters, bold letters, and bold
capital letters designate scalars, vectors, and matrices, respectively.
Superscripts $(\cdot)^{T}$ and $(\cdot)^{\dagger}$ denote, respectively, the
transpose and the transpose-conjugate operations. Further, $|x|$,
$\mathbb{E}[x]$, and $\mathbb{V}[x]$ denote the absolute, the expected, and
the variance of $x$, respectively. The operation $\angle\mathbf{x}$ calculates
the element-wise angle of the vector $\mathbf{x}$ and $\lfloor\cdot\rfloor$ is
the floor function. Finally, $\gamma(n,x)=\int_{0}^{x}t^{n-1}e^{-t}dt$ denotes
the lower incomplete gamma function.

##  II System Model

Our system model is shown in Fig. 1 in which a single-input single-output
system model similar to [18, 19] is considered. NOMA is a suitable multiple
access technique for a single-antenna setup because multi-user solutions are
not applicable. The base station (BS) uses NOMA to simultaneously serve two
users named User 1 and User 2.111User pairing is out of the scope of this
paper. We assume that the users are paired using one of the existing methods
in the literature such as [20, 21, 10]. The complexity of rate loss
calculation increases as the number of NOMA users grows. The distance from
User 2 to the BS is more than that of User 1. To determine which user is near,
the BS captures the distance information through a channel quality indicator
(CQI). The purpose of the RIS is to serve the far user to improve the channel
quality[6, 7]. The RIS is equipped with $N$ antenna elements that ideally can
direct the incident signal to any arbitrary directions in
$[-\frac{\pi}{2},\frac{\pi}{2}]$.

Like [22, 23], it is assumed that the perfect CSI is estimated at the users.
This assumption allows us to focus on studying the impact of the channel gain
and phase vector quantization errors. Recently, [24] has investigated the
impact of CSI impairments such as erroneous channel estimation or delay in
feedback on beamforming in a RIS-NOMA system without considering quantization
error. As a promising solution, our work on beamforming in relay networks with
channel statistics [25] and quantized feedback [26, 27] can be extended to the
underlying RIS-NOMA system to study the CSI impairments.

Figure 1: The limited feedback RIS-NOMA system model. $B_{1}$ bits and
$B_{2}+B^{\prime}$ bits are allocated to User 1 and User 2, respectively.

###  II-A Transmit Channel Model

We recall that both users are capable of estimating their channels. That is,
User 1 obtains $h_{1}\in\mathbb{C}$ and User 2 obtains
$\mathbf{h}_{2}\in\mathbb{C}^{N\times 1}$ and
$\mathbf{g}\in\mathbb{C}^{N\times 1}$ [28]. The channels capture the small-
scale fading and path loss effects. For User 1,
$h_{1}=\sqrt{{L}_{1}}h_{1}^{\prime}$, where $L_{1}=d_{1}^{-\alpha_{1}}$ is due
to the path loss. The parameters $d_{1}$ and $\alpha_{1}$ denote the distance
between the BS and User 1 and the path loss factor, respectively. Further,
$h_{1}^{\prime}\sim\mathcal{CN}(0,1)$ denotes the small-scale Rayleigh fading.
We define the channel gain $H_{1}=\left\lvert h_{1}\right\rvert^{2}$ with the
probability density function (pdf) of
$f_{H_{1}}(H_{1})=\frac{1}{L_{1}}e^{-\frac{H_{1}}{L_{1}}}$. Also, the channel
between the BS and the RIS is defined as
$\mathbf{h}_{2}=\sqrt{L_{2}}\mathbf{h}_{2}^{\prime}$, where
$L_{2}=d_{2}^{-\alpha_{2}}$ and
$\mathbf{h}_{2}^{\prime}\sim\mathcal{CN}(\boldsymbol{0},\mathbf{I})$. The
channel between the RIS and User 2 is given by
$\mathbf{g}=\sqrt{L_{g}}\mathbf{g}^{\prime}$, where
$L_{g}=d_{g}^{-\alpha_{g}}$ and
$\mathbf{g}^{\prime}\sim\mathcal{CN}(\boldsymbol{0},\mathbf{I})$. We note that
$\mathbf{I}$ represents the identity matrix of size $N\times N$. In fact, the
small-scale fading from the BS to User 2 is subject to the double-Rayleigh
fading [18, 29, 30]. Another possible channel model from the BS to the RIS is
Rician fading. Since the RIS is helping the far user, it is reasonable to
assume that its distance from the BS is large [6, 7]. Thus, it is likely that
the LoS channel is blocked by moving objects or buildings justifying a
Rayleigh fading model. The parameters $d_{2}$ and $d_{g}$ denote the distance
between the transmitter and the RIS and the distance from the RIS to User 2,
respectively. Further, the parameters $\alpha_{2}$ and $\alpha_{g}$ denote the
path loss factors. The effective overall channel between the BS and User 2 is
defined as $\tilde{h}_{2}={\mathbf{g}^{T}\boldsymbol{\Theta}\mathbf{h}_{2}}$.
Correspondingly, the channel gain is
$\tilde{H}_{2}=\left\lvert\mathbf{g}^{T}\boldsymbol{\Theta}\mathbf{h}_{2}\right\rvert^{2}$
in which $\boldsymbol{\Theta}=\text{diag}\left(\boldsymbol{\theta}\right)$,
where $\boldsymbol{\theta}=[e^{j\phi_{1}},\cdots,e^{j\phi_{N}}]$ and
$\phi_{i}\in[-\frac{\pi}{2},\frac{\pi}{2}]$ reflect the impact of the RIS. The
optimal values of $\boldsymbol{\Theta}$ result in the maximum gain of
$H_{2}=\left(\sum_{i=1}^{N}\left\lvert h_{2,i}\right\rvert\left\lvert
g_{i}\right\rvert\right)^{2}$. Deriving the exact pdf of $H_{2}$ is
complicated. For the sake of simplicity, we first use the following upper
bound on the pdf of the random variable $z=\sum_{i=1}^{N}|h_{2,i}||g_{i}|$
[7]:

$f_{z}(z)\leq\frac{C_{1}}{C_{2}}\left(\frac{z}{C_{2}}\right)^{\frac{3N-2}{2}}e^{-2\frac{z}{C_{2}}},$

(1)

where
$C_{1}=\frac{2^{N}\pi^{\frac{N}{2}}\Gamma^{N}\left(\frac{3}{2}\right)}{\left(\frac{3N-2}{2}\right)!}$
and $C_{2}=\sqrt{L_{2}L_{g}}$. Through extensive simulations, it is shown that
this bound is tight [7]. Without loss of generality, we assume $N$ is an even
number. Then, noting that $H_{2}=z^{2}$, we have
$f_{H_{2}}\left(H_{2}\right)=\frac{1}{2\sqrt{H_{2}}}f_{z}\left(\sqrt{H_{2}}\right)$.
Finally, an upper bound on the pdf of $H_{2}$ follows as:

$f_{H_{2}}\left(H_{2}\right)\leq\frac{C_{1}}{C_{2}}\frac{1}{2\sqrt{H_{2}}}\left(\frac{\sqrt{H_{2}}}{C_{2}}\right)^{\frac{3N-2}{2}}e^{-\frac{2\sqrt{H_{2}}}{C_{2}}}.$

(2)

###  II-B Feedback Channel

In FDD systems, the downlink channel is estimated at the user side and then
fed back to the BS and the other user using the limited feedback channel. In
our system model shown in Fig. 1, User 1 quantizes the channel gain $H_{1}$
and maps it to $q(H_{1})$. Then, the index of $q(H_{1})$ is fed back to the BS
using $B_{1}$ bits. Since User 2 communicates with the BS through the RIS, the
phase information should be sent to the RIS as well. In this regard, first,
User 2 maps the overall channel vector $\mathbf{G}\mathbf{h}_{2}$ to
$Q(\mathbf{G}\mathbf{h}_{2})$ using ${B}^{\prime}$ bits, where $Q(\cdot)$ is a
RVQ and $\mathbf{G}=\text{diag}(\mathbf{g})$. User 2 then determines the phase
vector $\boldsymbol{\theta}$ of the quantized channel
$\mathbf{G}\mathbf{h}_{2}$ denoted by $\boldsymbol{\theta}_{Q}$. The exact
structure of quantizers $q(\cdot)$ and $Q(\cdot)$ is discussed in the next
section, but does not change the overall characteristics of the feedback
channel model. Next, User 2 quantizes the channel gain
$H_{2,Q}=|\boldsymbol{\theta}_{Q}^{T}\mathbf{G}\mathbf{h}_{2}|^{2}$, i.e.,
$q(H_{2,Q})$, with $B_{2}$ bits. User 2 feeds the corresponding
${B_{2}}+{B}^{\prime}$ bits back to the BS. The feedback link from User 2 to
the BS is assumed to support $B_{2}+B^{\prime}$ bits, although, the direct
links might be blocked [14]. The same explanation holds for the feedback link
from User 1 to the BS.

In our system model, the BS, the RIS, and the users are fixed and the phase
vector $\boldsymbol{\theta}_{Q}$ and the channel gains $H_{1}$ and $H_{2}$ are
only required once for every channel coherence time.

###  II-C Sum Rate

To maximize the sum rate by efficiently allocating the transmit power and
subject to some minimum rate for each user, the following optimization problem
can be defined βR_1 + R_2 R_1, R_2≥R_th P_1 + P_2= P, where
$R_{1}=\text{log}_{2}\left(1+\beta PH_{1}\right)$ and
$R_{2}=\text{log}_{2}\left(1+\frac{\left(1-\beta\right)PH_{2}}{\beta
PH_{2}+1}\right)$ . The power allocation can be parameterized by a factor
$\beta$ such that $P_{1}=\beta P$ and $P_{2}=\left(1-\beta\right)P$. In [31],
given full CSI, Problem (II-C) is extended to an arbitrary number of users and
individual minimum rate constrains. Using the solution given in [31, Eq. 15],
we obtain the optimum factor $\beta$ for $H_{2}\leq H_{1}$ as
$\beta^{*}=\frac{PH_{2}-\epsilon}{\left(1+\epsilon\right)PH_{2}}$ where
$\epsilon=2^{R_{th}}-1$. Hence,
$R_{1}=\text{log}_{2}\left(1+\frac{PH_{2}H_{1}-\epsilon
H_{1}}{H_{2}\left(1+\epsilon\right)}\right)$ and $R_{2}=R_{th}$. The threshold
$R_{\textit{th}}$ is the same for all users to ease the formulation but the
approach works for arbitrary thresholds. Further, there are other useful
objective functions to be considered. For example, similar to Ref. [11], we
can study the rate fairness in our RIS-NOMA system with limited feedback and
quantization error.

##  III Uniform and Random vector Quantizers

In this section, we describe uniform quantizers and RVQs, used in our system.
We use uniform quantizers to compress the scalar channel gains and RVQs to
quantize the overall channel vector.

To quantize $H_{1}$, we define the following uniform quantizer
$q:\mathbb{R}\to\mathbb{R}$, shown in Fig. 2:

$\displaystyle
q(x)=\begin{cases}\lfloor\frac{x}{\delta}\rfloor\times\delta,&x\leq\left(2^{B_{1}}-1\right)\delta,\\\
\left(2^{B_{1}}-1\right)\delta,&x>\left(2^{B_{1}}-1\right)\delta,\end{cases}$
(3)

where $x$ is any non-negative real number and $\delta$ denotes the size of
quantization partitions. The index of $q(H_{1})$ is fed back by $B_{1}$ bits.
The method in (3) quantizes the channel gain to the left boundary of the
partition instead of the center point. When the gain is quantized to the
center point, the quantized value might be higher than the true channel gain.
This can frequently cause outage at the weak user, i.e., solving the
optimization problem in (II-C) may result in allocating insufficient power to
the weak user. Thus, Constraint (II-C) may not hold for the quantized channel
gain. To avoid the outage, we quantize the channel gain to the left boundary
which guarantees the power allocation to the weak user is more than the needed
optimal value.222The optimal power allocation, i.e., $P_{1}^{*}$ and
$P_{2}^{*}$, is obtained by determining $\beta^{*}$ in (II-C). The uniform
quantization is selected for simplicity but our approach works for non-uniform
quantization as well.

Figure 2: Applied uniform quantizer for quantizing channel gains ($i=1,2$).

In general, there are two options for quantizing the vector
$\mathbf{G}\mathbf{h}_{2}$: vector quantization and scalar quantization,
applied to vector’s elements. Since the number of elements in RISs can be
large, scalar quantization will require a huge number of feedback bits and may
not be practical. Inspired by this, we use random vector quantization in which
the feedback bits can be far less than the number of elements333RVQ is a
simple method but not the most efficient to quantize a vector with a limited
number of feedback bits [32]. Other techniques such as Lloyd Algorithm [33]
and variable-length limited feedback beamforming [34] which outperform the RVQ
can be used to enhance the performance of the underlying limited feedback
system although they increase the complexity.. We define the RVQ codebook
$\mathcal{W}=\\{\mathbf{w}_{1},\mathbf{w}_{2},\dots,\mathbf{w}_{M}\\}$, in
which the codeword $\mathbf{w}_{i}\in\mathbb{C}^{N\times 1}$, is the quantized
overall RIS-aided channel vector $\mathbf{G}\mathbf{h}_{2}$. The codebook
$\mathcal{W}$ is generated by selecting each of $M=2^{B^{\prime}}$ vectors
independently from a uniform distribution on the complex unit sphere [35].

We aim to maximize the channel gain using the codebook such that

$Q\left(\mathbf{G}\mathbf{h}_{2}\right)=\underset{\mathbf{w}\in\mathcal{W}}{\text{argmax}}\lvert\mathbf{w}^{\dagger}\mathbf{G}\mathbf{h}_{2}\rvert^{2}.$

(4)

Further, we let  $\boldsymbol{\phi}_{Q}=\angle Q(\mathbf{G}\mathbf{h}_{2})$
and  $\boldsymbol{\theta}_{Q}=e^{j\boldsymbol{\phi}_{Q}}$ where
$\boldsymbol{\phi}_{Q}=[\phi_{1,Q},\cdots,\phi_{N,Q}]^{T}$ . The channel gain
$H_{2,Q}=\left\lvert\boldsymbol{\theta}_{Q}^{T}\mathbf{G}\mathbf{h}_{2}\right\rvert^{2}$
takes any non-negative value and is mapped to $q\left(H_{2,Q}\right)$ using
the quantizer in (3). As mentioned before, a uniform quantizer is applied to
$H_{2,Q}$ instead of $H_{2}$. Since $H_{2}$ and $H_{2,Q}$ do not necessarily
belong to the same partition, their quantized values might be different, i.e.,
$q(H_{2,Q})\leq q(H_{2})$. The index of $q\left(H_{2,Q}\right)$ is sent using
$B_{2}$ feedback bits. The uniform quantizers applied to $H_{1}$ and $H_{2,Q}$
include the same number of partitions, i.e., $B_{1}=B_{2}=B$. Defining
$\eta=\frac{H_{2,Q}}{H_{2}}$, we have $\eta\in[0,1]$. Deriving the pdf of
$\eta$ is not straightforward, but needed in some analysis later. The exact
pdf of $\eta$ for a $2\times 1$ Rayleigh channel vector and a large number of
feedback bits is derived in [36]. However, each element in $\mathbf{Gh}_{2}$
is a double-Rayleigh variable and its pdf is different from that of Rayleigh
channels. Recently, the study of the pdf of
$\Big{|}\sum_{i=1}^{N}|h_{2,i}||g_{i}|e^{j\kappa_{i}}\Big{|}^{2}$ has been the
topic of research in RIS-aided systems [37, 18]. Note that $\kappa_{i}$ does
not necessarily equal to $\phi_{i,Q}$. In [18, Lemma 1], the pdf of the random
variable  $\Big{|}\sum_{i=1}^{N}|h_{2,i}||g_{i}|e^{j\kappa_{i}}\Big{|}^{2}$
where $\kappa_{i}$ is treated as a phase-noise is accurately approximated by a
Gamma random variable. Following the same approach, our extensive empirical
study shows that the pdf of the random variable $\eta$ can be approximated by
the pdf of a beta random variable with the shape parameters $r_{1}$ and
$r_{2}$, i.e.,

$f_{\eta}(\eta)\approx\frac{1}{\mathcal{B}(r_{1},r_{2})}\eta^{r_{1}-1}\left(1-\eta\right)^{r_{2}-1},$

(5)

where
$r_{1}=\left(\frac{\mathbb{E}[\eta]\left(1-\mathbb{E}[\eta]\right)}{\mathbb{V}[\eta]}\right)\mathbb{E}[\eta]$
,
$r_{2}=\left(\frac{\mathbb{E}[\eta]\left(1-\mathbb{E}[\eta]\right)}{\mathbb{V}[\eta]}\right)\left(1-\mathbb{E}[\eta]\right)$
, and
$\mathcal{B}\left(r_{1},r_{2}\right)=\frac{\Gamma(r_{1}+r_{2})}{\Gamma(r_{1})\Gamma(r_{2})}$
represents a normalization constant that ensures the total probability is 1.
The values of $\mathbb{E}[\eta]$ and $\mathbb{V}[\eta]$ depend on the RIS’s
size $N$ and the feedback bits $B^{\prime}$. The empirical results show a
close resemblance between the real pdf and the approximation, but we do not
have space to show them in this paper.

When the full CSI is available at the BS, the user ordering is always accurate
and the rates are calculated using (II-C). In a limited feedback system, an
inaccurate user ordering can impose severe rate loss. Even if the user
ordering is accurate, the quantization can reduce the achievable sum rate.

##  IV Rate Loss Analysis

Let us name the user with a higher channel gain the strong user. We calculate
the rate loss only for the strong user because the weak user will not
experience quantization rate loss. When instead of full CSI, the quantized
channel gains are used in Problem (II-C), we call the resulting power
allocation factor $\beta_{q}$.

In general, the strong user’s rate loss is obtained as

(6)

where $\Delta X=X_{i}-X_{i,q}$ denotes the normalized signal-to-noise ratio
(SNR) loss. When user ordering is accurate, we have  $X_{1}=\beta
H_{1}\leavevmode\nobreak\ (X_{2}=\beta H_{2})$ and
$X_{1,q}=\beta_{q}H_{1}\leavevmode\nobreak\ (X_{2,q}=\beta_{q}H_{2,Q})$ . For
an inaccurate user ordering, $X_{i}$ is similar to the accurate one while
$X_{1,q}=\frac{\left(1-\beta_{q}\right)Pq\left(H_{1}\right)}{\beta_{q}Pq\left(H_{1}\right)+1}$
and
$X_{2,q}=\frac{\left(1-\beta_{q}\right)Pq\left(H_{2,Q}\right)}{\beta_{q}Pq\left(H_{2,Q}\right)+1}$
. Based on (IV), an upper bound on the average rate loss is found as

$\displaystyle\mathbb{E}[\Delta R]$
$\displaystyle\leq\mathbb{E}[\text{log}_{2}\left(1+P\Delta
X\right)]\leq\text{log}_{2}\left(1+P\mathbb{E}[\Delta X]\right).$ (7)

The second inequality is due to Jensen’s inequality. Thus, to find the upper
bound, we first derive $\Delta X$ and then calculate the expectation of
$\Delta X$, i.e., $\mathbb{E}[\Delta X]$. However, the calculation of $\Delta
X$ heavily depends on the values of $H_{1}$ and $H_{2}$. The main three Super
Regions for this calculation, as shown in Fig. 3, are:

  * •

Super Region I: This consists of the conditions in which $q(H_{1})=0$ and/or
$q(H_{2,Q})=0$ which results in $\beta_{q}=\infty$.

  * •

Super Region II: This includes the main partitions of User 2’s uniform
quantizer, i.e., $\delta\leq q(H_{2,Q})<\left(2^{B}-1\right)\delta$.

  * •

Super Region III: This includes the upper marginal partition of User 2’s
uniform quantizer, i.e., $q(H_{2,Q})=\left(2^{B}-1\right)\delta$.

We denote $\Delta X$ in Super Region I as $\Delta X_{\text{I}}$. In what
follows, we calculate $\mathbb{E}[\Delta X_{\text{I}}]$, i.e., the expected
normalized SNR loss in each region.

Figure 3: Presentation of all the possible regions for the rate loss.

###  IV-A Super Region I

Since $\beta_{q}=\infty$, in this super region, NOMA is not feasible and we
let $\mathbb{E}[\Delta X_{\text{I}}]=0$.

###  IV-B Super Region II

######  Lemma 1.

The total average normalized SNR loss of Super Region II is

(8)

where  $C_{3}=C_{6}+C_{7}+C_{8}$ and  $C_{4}=C_{9}+C_{10}$ . In detail,
$C_{5}\geq C_{5}^{{}^{\prime}}\mathbb{E}\left[\frac{1}{\sqrt{\eta}}\right]$ ,
$C_{6}\geq
C_{6}^{{}^{\prime}}\mathbb{E}\left[\frac{1-\eta}{\sqrt{\eta}}\right]$ ,
$C_{7}\geq
C_{7}^{{}^{\prime}}\mathbb{E}\left[\frac{1-\eta}{\sqrt{\eta}}\right]$ ,
$C_{8}\geq C_{7}^{{}^{\prime}}\mathbb{E}\left[(1-\eta)\sqrt{\eta}\right]$ ,
$C_{9}\geq C_{9}^{{}^{\prime}}\mathbb{E}\left[\frac{1}{\sqrt{\eta}}\right]$ ,
and  $C_{10}\geq
C_{10}^{{}^{\prime}}\mathbb{E}\left[\frac{1}{\sqrt{\eta}}\right]$ in which
$C_{5}^{{}^{\prime}}=\frac{C_{1}C_{2}}{2^{\frac{3N+2}{2}}}\left(\frac{3N+2}{2}\right)!$
,
$C_{6}^{{}^{\prime}}=\frac{C_{1}C_{2}}{2^{\frac{3N+2}{2}}}\left(\frac{3N+2}{2}\right)!+\frac{C_{1}L_{1}}{2^{\frac{3N-2}{2}}C_{2}}\left(\frac{3N-2}{2}\right)!$
,
$C_{7}^{{}^{\prime}}=\frac{C_{1}C_{2}^{3}}{2^{\frac{3N+6}{2}}L_{1}}\left(\frac{3N+6}{2}\right)!$
,
$C_{9}^{{}^{\prime}}=\frac{C_{1}}{2^{\frac{3N-2}{2}}C_{2}}\left(\frac{3N-2}{2}\right)!+\frac{C_{1}L_{1}}{2^{\frac{3N-6}{2}}C_{2}^{3}}\left(\frac{3N-6}{2}\right)!$
, and
$C_{10}^{{}^{\prime}}=\frac{C_{1}C_{2}}{2^{\frac{3N+2}{2}}L_{1}}\left(\frac{3N+2}{2}\right)!$
. Also,  $\text{E}_{1}(x)=\int_{x}^{\infty}\frac{e^{-t}}{t}dt$ denotes the
exponential-integral function.

###### Proof.

Please see Appendix A. ∎

###  IV-C Super Region III

######  Lemma 2.

The total average normalized SNR loss for Super Region III is bounded by

$\displaystyle\scalebox{0.85}{$\mathbb{E}[\Delta X_{\text{III}}]\leq
e^{-2\frac{\sqrt{\left(2^{B}-1\right)\delta}}{C_{2}}}\Bigg{[}2C_{11}\left(1+\frac{\left(2^{B}-1\right)\delta}{L_{1}}\right)$}\times$
$\displaystyle\scalebox{0.85}{$\left(1+\left(2\sqrt{\frac{\left(2^{B}-1\right)\delta}{C_{2}^{2}}}\right)^{\frac{3N+2}{2}}\right)+C_{12}\left(1+\left(2\sqrt{\frac{\left(2^{B}-1\right)\delta}{C_{2}^{2}}}\right)^{\frac{3N-2}{2}}\right)\Bigg{]}$}.$
(9)

where
$C_{11}=\frac{C_{1}C_{2}^{2}}{2^{\frac{3N+4}{2}}}\left(\frac{3N+4}{2}\right)!$
and
$C_{12}=\frac{C_{1}C_{2}L_{1}}{2^{\frac{3N+2}{2}}}\left(\frac{3N}{2}\right)!$.

###### Proof.

Please see Appendix C. ∎

Finally, we have the following theorem on the expectation of the total rate
loss for the quantized RIS-NOMA.

######  Theorem 1.

The total average rate loss for the quantized RIS-NOMA system with limited
feedback is upper bounded by

$\mathbb{E}[\Delta R]\leq\text{log}_{2}(1+P\mathbb{E}[\Delta X]),$ (10)

where

$\displaystyle\quad\quad\quad\quad\quad\quad\quad\scalebox{0.85}{$+C_{12}\left(1+\left(2\sqrt{\frac{\left(2^{B}-1\right)\delta}{C_{2}^{2}}}\right)^{\frac{3N-2}{2}}\right)\Bigg{]}$}.$
(11)

###### Proof.

We know that $\mathbb{E}[\Delta X]=\mathbb{E}[\Delta
X_{\text{I}}]+\mathbb{E}[\Delta X_{\text{II}}]+\mathbb{E}[\Delta
X_{\text{III}}]$. Noting $\mathbb{E}[\Delta X_{\text{I}}]=0$ and replacing
$\mathbb{E}[\Delta X_{\text{II}}]$ and $\mathbb{E}[\Delta X_{\text{III}}]$
with (8) and (2), respectively, results in (1). ∎

To guarantee that the rate loss approaches zero as $B$ increases, one feasible
solution is to define $\delta=\zeta_{1}2^{-{\zeta_{2}}B}$ for
$\zeta_{1},\zeta_{2}\in(0,1)$. The parameters $\zeta_{1}$ and $\zeta_{2}$ are
design parameters and should be optimized. Such a parameter optimization is
out of the scope of this paper although the choice of $\zeta_{1}$ and
$\zeta_{2}$ will not affect the system model. In simulations, we will
intuitively select $\zeta_{1}$ and $\zeta_{2}$ to achieve good performance.

##  V Numerical Results

We compare the sum rate performance of the RIS-NOMA system with limited
feedback to that of the RIS-assisted orthogonal multiple access (RIS-OMA)
system. The RIS-NOMA system is described in Section II. For the RIS-OMA, we
consider the same system model in Section II and replace NOMA with OMA.
Furthermore, since power allocation is not required in the RIS-OMA, the
channel gains are not fed back although the RIS-aided channel vector
information should be fed back for beamforming at the RIS.

The parameters are set according to [9] as follows. The distances are selected
as $d_{1}$=10 m, $d_{2}$=40 m, and $d_{g}$=10 m. Further, the path loss
exponents are set as $\alpha_{1}=3.5$, $\alpha_{2}=2.5$, and $\alpha_{g}=2.5$.
The number of RIS elements is set to $N=10$.

(a)

(b)

Figure 4: Performance of the sum rate versus the transmit power $P$ for (a)
$\zeta_{1}=10^{-5}$ and $\zeta_{2}=0.95$ and (b) $\zeta_{1}=0.5\times 10^{-5}$
and $\zeta_{2}=0.95$.

We present the simulation results for the sum rate performance versus the
total transmit power $P$ for various feedback bits in Fig. 4. The transmit
power depends on the users’ path loss such that the power should compensate
for the propagation loss. Simulation is conducted for the full CSI RIS-NOMA,
full CSI RIS-OMA, limited feedback RIS-NOMA, and limited feedback RIS-OMA. To
study the impact of $\delta$ where $\delta=\zeta_{1}2^{-\zeta_{2}B}$, we set
$\zeta_{2}=0.95$ and consider two different values for $\zeta_{1}$. We set
$\zeta_{1}=10^{-5}$ and $0.5\times 10^{-5}$ in Figs. 4(a) and 4(b),
respectively. The full CSI RIS-NOMA achieves the highest sum rate. When $B$
and $B^{\prime}$ increase, the limited feedback RIS-NOMA’s sum rate and the
limited feedback RIS-OMA’s sum rate approach those of the full CSI RIS-NOMA
and the full CSI RIS-OMA, respectively. This is consistent with our findings
in Theorem 1. For instance, $B=6$ and $B^{\prime}=4$, the sum rate is almost
the same as that of the full CSI. Further, as we decrease the resolution of
the quantizer, the length of the region in which we quantize the channel gain
to 0 enlarges. Adopting a zero channel gain results in a zero sum rate as
indicated in the low power portion of Fig. 4. However, in RIS-OMA, we do not
impose any minimum rate constraint. This causes the RIS-OMA’s sum rate to be
equal or slightly higher than the RIS-NOMA’s sum rate at low power levels
although there is no guarantee for the minimum sum rate.

We also observe that for $B=2$ where $B=B_{1}=B_{2}$ and $B^{\prime}=2$, i.e.,
a total of 6 feedback bits, the limited feedback RIS-NOMA’s sum rate shows
different behavior compared to the limited feedback RIS-OMA with
$B^{\prime}=6$. At low transmit powers, the limited feedback RIS-OMA’s sum
rate is better than that of the limited feedback RIS-NOMA for both $\delta$
values. As we increase the power, the limited feedback RIS-NOMA improves the
sum rate in comparison to the limited feedback RIS-OMA. However, at very high
transmit power levels, the limited feedback RIS-NOMA’s slope is smaller than
that of the limited feedback RIS-OMA, as shown in Fig. 4(a). When we select a
smaller $\delta$, as in Fig. 4(b), for any given $B$ and $B^{\prime}$, the
limited feedback RIS-NOMA’s sum rate becomes higher than that of the limited
feedback RIS-OMA. Another important observation is the impact of allocating
$B$ and $B^{\prime}$ on the limited feedback RIS-NOMA’s sum rate. For
instance, let us assume the total number of feedback bits is 12. When $B=4$
and $B^{\prime}=4$, in Fig. 4(a), the sum rate is higher than that of $B=2$
and $B^{\prime}=8$. Whereas, given the same $B$ and $B^{\prime}$, in Fig.
4(b), these two schemes achieve almost the same sum rate.

Figure 5: The average rate loss versus the number of the feedback bits.

Fig. 5 compares the limited feedback RIS-NOMA’s average rate loss for
$\zeta_{1}=0.5\times 10^{-5}$ and $\zeta_{2}=0.95$. As the feedback bits and
the power increase, the rate loss reduces. When $B^{\prime}$ is fixed, by
increasing $B$, the rate loss at power $60$ dBm reduces faster than at power
$40$ dBm. In fact, increasing the power can compensate for the quantization
error.

##  VI Conclusion

In this paper, we studied an FDD-based RIS-NOMA system with a limited feedback
channel. We used a RVQ to quantize the RIS-aided channel vector. Also, we
considered a uniform quantizer for the channel gains. We then analyzed the
rate loss of the strong user under the accurate and inaccurate user ordering
conditions. Since the BS receives the quantized channel gains, inaccurate user
ordering can happen often. We derived the rate loss resulted from quantization
and showed that the rate loss essentially depends on the number of feedback
bits, $B$ and $B^{\prime}$. From the simulations, we observed that the
parameters $B$ and $B^{\prime}$ affect the sum-rate, the rate loss, and the
probability that NOMA is not useful. As the number of feedback bits increases,
the quantized RIS-NOMA’s performance approaches that of the full CSI RIS-NOMA.

##  Appendix A Proof of Lemma 1

We divide this super region into Regions II.A-II.D, as shown in Fig. 3. Due to
space limitations, we only provide the detailed calculation of the upper bound
and constants for Region II.A. The upper bound and constants for other regions
can be found similarly.

Region II.A: In this region, $H_{2}\leq H_{1}$ which means User 1 is the
strong user. It is clear that the output of the RVQ results in $H_{2,Q}\leq
H_{2}\leq H_{1}$ and the uniform quantizer results in $q(H_{2,Q})\leq
q(H_{1})$. Thus, the user ordering is accurate and the BS recognizes User 1 as
the strong user. The average normalized SNR loss is upper bounded as

(12)

The proof of (12) and the values of $C_{6}$ and $C_{9}$ are provided in
Appendix B.

Region II.B: In this region, $H_{1}\leq H_{2}$ indicates User 2 is stronger
than User 1. It is possible that the RVQ leads to $H_{2,Q}\leq H_{1}\leq
H_{2}$. Obviously, the uniform quantizer results in $q(H_{1})\geq q(H_{2,Q})$.
Thus, the BS recognizes User 1 as the strong user which is not an accurate
user ordering. The average normalized SNR loss is expressed as

(13)

It should be mentioned that $C_{7}$ is a constant and bounded since
$\mathbb{E}\left[\frac{1-\eta}{\sqrt{\eta}}\right]$ is bounded. Also,
$\mathbb{E}\left[\frac{1-\eta}{\sqrt{\eta}}\right]$ can be approximated as
$\frac{\mathcal{B}\left(r_{1}-\frac{1}{2},r_{2}+1\right)}{\mathcal{B}\left(r_{1},r_{2}\right)}$
using (5).

Region II.C: In this region, $H_{1}\leq H_{2}$ and using the RVQ results in
$H_{1}<H_{2,Q}\leq H_{2}$. For $H_{2,Q}-H_{1}<\delta$, the uniform quantizer
leads to $q(H_{1})=q(H_{2,Q})$. In such a region, the user ordering is
inaccurate and the BS picks User 1 as the strong user. Hence, the average
normalized SNR loss is obtained as

(14)

Region II.D: In this region, $H_{1}\leq H_{2}$ and the RVQ leads to
$H_{1}<H_{2,Q}\leq H_{2}$. If the uniform quantizer results in
$q(H_{1})<q(H_{2,Q})$, the user ordering is accurate and the BS selects User 2
as the strong user. Then, the average normalized SNR loss is bounded by

(15)

The constant $C_{8}$ is bounded, and using (5) we obtain the approximated
value of $\mathbb{E}\left[(1-\eta)\sqrt{\eta}\right]$ as
$\frac{\mathcal{B}\left(r_{1}+\frac{1}{2},r_{2}+1\right)}{\mathcal{B}\left(r_{1},r_{2}\right)}$
which is finite.

##  Appendix B Proof of (12)

In Region II.A, the normalized SNR loss is obtained as

(16)

The inequality (a) follows from the fact that for any $\delta\leq
q(H_{2,Q})<\left(2^{B}-1\right)\delta$, the inequalities $q(H_{2,Q})\geq
H_{2,Q}-\delta$ and $H_{2,Q}\geq q(H_{2,Q})$ hold. The inequality (b) is due
to $\epsilon\geq 0$. Further, (c) is true because $H_{2,Q}=\eta H_{2}$. For a
given constant $\eta$, the expectation of $\Delta X_{\text{II.A}}$ over the
super region defined by $\frac{\delta}{\eta}\leq
H_{2}<\frac{\left(2^{B}-1\right)\delta}{\eta}$ and $H_{2}\leq H_{1}$ is given
as

$\displaystyle\quad\scalebox{0.9}{$+\delta\underbrace{\int_{\frac{\delta}{\eta}}^{\frac{\left(2^{B}-1\right)\delta}{\eta}}\int_{H_{2}}^{\infty}\frac{H_{1}}{H_{2}}f_{H_{1}}\left(H_{1}\right)f_{H_{2}}\left(H_{2}\right)dH_{1}dH_{2}}_{\equiv
I_{2}}$}.$ (17)

Next, we compute the integrals $I_{1}$ and $I_{2}$. For $I_{1}$, we have

(18)

The integral $I_{1,1}$ is obtained as

(19)

The inequality $(a)$ follows from the fact that $e^{-\frac{H_{2}}{L_{1}}}\leq
1$. The inequality $(b)$ is due to the definition of the lower incomplete
gamma function and using the upper bound  $\gamma(n,x)\leq(n-1)!x$ .

Likewise, we obtain an upper bound on $I_{1,2}$ as

(20)

Substituting (B) and (20) into (B) gives

(21)

Also, we calculate $I_{2}$ as

(22)

Thus, the upper bound on $\mathbb{E}[\Delta X_{\text{II.A}}|\eta]$ is found as

(23)

where the upper bounds on $I_{1}$ and $I_{2}$ are derived in (B) and (22),
respectively. In order to calculate $\mathbb{E}[\Delta X_{\text{II.A}}]$, we
have

$\mathbb{E}[\Delta X_{\text{II.A}}]$

$\displaystyle\leq\scalebox{0.9}{$\int_{0}^{1}(1-\eta)I_{1}f_{\eta}(\eta)d\eta+\delta\int_{0}^{1}I_{2}f_{\eta}(\eta)d\eta$}$
$\displaystyle\leq\scalebox{0.9}{$C_{6}^{{}^{\prime}}\sqrt{\left(2^{B}-1\right)\delta}\int_{0}^{1}\frac{1-\eta}{\sqrt{\eta}}f_{\eta}(\eta)d\eta$}$
$\displaystyle\leq\scalebox{0.9}{$C_{6}\sqrt{\left(2^{B}-1\right)\delta}+C_{9}\delta\sqrt{\left(2^{B}-1\right)\delta},$}$
(24)

Note that  $\mathbb{E}\left[\frac{1-\eta}{\sqrt{\eta}}\right]$ and
$\mathbb{E}\left[\frac{1}{\sqrt{\eta}}\right]$ are finite and for the
approximation in (5) are, respectively, equal to
$\frac{\mathcal{B}\left(r_{1}-\frac{1}{2},r_{2}+1\right)}{\mathcal{B}\left(r_{1},r_{2}\right)}$
and
$\frac{\mathcal{B}\left(r_{1}-\frac{1}{2},r_{2}\right)}{\mathcal{B}(r_{1},r_{2})}$
.

##  Appendix C Proof of Lemma 2

We divide Super Region III into Regions III.A-III.C, as shown in Fig. 3.

Region III.A: In this region, User 1 has a better channel compared to User 2,
i.e., $H_{2}\leq H_{1}$. If the RVQ results in $\left(2^{B}-1\right)\delta\leq
H_{2,Q}\leq H_{2}\leq H_{1}$, then uniform quantizer’s outcome will be
$q(H_{1})=q(H_{2,Q})=\left(2^{B}-1\right)\delta$ and the ordering is accurate.
In this region, the average normalized SNR loss is bounded by

$\displaystyle\quad\scalebox{0.9}{$+C_{12}\left(1+\left(2\sqrt{\frac{\left(2^{B}-1\right)\delta}{C_{2}^{2}}}\right)^{\frac{3N-2}{2}}\right)$}\Bigg{]},$
(25)

Region III.B: In this region, User 2 has a better channel and
$q(H_{1})=\left(2^{B}-1\right)\delta$, i.e., $\left(2^{B}-1\right)\delta\leq
H_{1}\leq H_{2}$. In such a region, the RVQ will result in either
$\left(2^{B}-1\right)\delta\leq H_{1}\leq H_{2,Q}$ or
$\left(2^{B}-1\right)\delta\leq H_{2,Q}\leq H_{1}$. Also, the uniform
quantizer will provide $q(H_{1})=q(H_{2,Q})=\left(2^{B}-1\right)\delta$.
Hence, the quantizers will inaccurately change the users’ order. The average
normalized SNR loss for this region is bounded as

$\displaystyle\scalebox{0.9}{$\mathbb{E}[\Delta X_{\text{III.B}}]\leq
C_{11}e^{-2\frac{\sqrt{\left(2^{B}-1\right)\delta}}{C_{2}}}\left(1+\left(2\sqrt{\frac{\left(2^{B}-1\right)\delta}{C_{2}^{2}}}\right)^{\frac{3N+2}{2}}\right)$}.$
(26)

Region III.C: In this region, User 1’s channel gain is lower than that of User
2 such that $\delta\leq q(H_{1})<q(H_{2,Q})=\left(2^{B}-1\right)\delta$. Thus,
User 2 is the strong user and the ordering is accurate. We obtain the
expectation of the normalized SNR loss as

$\displaystyle\scalebox{0.85}{$\mathbb{E}[\Delta X_{\text{III.C}}]\leq
2\frac{C_{11}}{L_{1}}\left(2^{B}-1\right)\delta
e^{-2\frac{\sqrt{\left(2^{B}-1\right)\delta}}{C_{2}}}\left(1+\left(2\sqrt{\frac{\left(2^{B}-1\right)\delta}{C_{2}^{2}}}\right)^{\frac{3N+2}{2}}\right)$}.$
(27)

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