#  Rate-Splitting Multiple Access for Multi-Antenna Broadcast Channel with
Imperfect CSIT and CSIR

Jihye An†, Onur Dizdar‡, Bruno Clerckx‡, and Wonjae Shin† †Department of
Electronics Engineering, Pusan National University, Busan, Korea
‡Department of Electrical and Electronic Engineering, Imperial College London,
United Kingdom
Email: †{jh_an<EMAIL_ADDRESS>‡{o.dizdar<EMAIL_ADDRESS>
###### Abstract

Rate-splitting multiple access (RSMA) has appeared as a powerful transmission
and multiple access strategy for multi-user multi-antenna communications.
Uniquely, this paper studies the optimization of the sum-rate of RSMA with
imperfect channel state information (CSI) at the transmitter (CSIT) and the
receivers (CSIR). Robustness of the RSMA approach against imperfect CSIT has
been investigated in the previous studies while there has been no
consideration for the effects of imperfect CSIR. This motivates us to develop
a robust design relying on RSMA in the presence of both imperfect CSIT and
CSIR. Since the optimization problem for the design of RSMA precoder and power
allocations to maximize the sum-rate is non-convex, it is hard to solve
directly. To tackle the non-convexity, we propose a novel alternating
optimization algorithm based on semidefinite relaxation (SDR) and concave-
convex procedure (CCCP) techniques. By comparing simulation results with
conventional methods, it turns out that RSMA is quite robust to imperfect CSIR
and CSIT, thereby improving the sum-rate performance.

###### Index Terms:

Rate-splitting multiple access (RSMA), sum-rate, muti-user multiple-input
single-output (MU-MISO).

††publicationid: pubid: 978-1-7281-4490-0/20/$31.00 © 2020 IEEE

##  I Introduction

Due to the increase in data traffic and the number of communicating devices,
there is an increasing need to design efficient communication strategies to
boost the data rate, spectral efficiency and manage the interference. To that
end, multi-antenna/multiple-input multi-output (MIMO) processing is a key
technology. To deal with the interference problem in multi-user multi-antenna
systems, the perfect channel state information (CSI) at receiver (CSIR) and
transmitter (CSIT) are essential. However, it is difficult to obtain accurate
CSI due to quantization error, channel mobility, and estimation error. Even
with the ideal assumption of perfect CSIR, it is questionable whether a base
station (BS) can obtain accurate CSIT.

Rate-splitting multiple access (RSMA) has recently emerged and has been found
to have multiple advantages over conventional multiple access methods in terms
of robustness against imperfect CSIT [1], and spectral and energy efficiencies
[2], [3]. The key feature of RSMA is the split of the messages into common and
private parts. The common parts are encoded in a common stream that can be
decoded by multiple users. On the other hand, each of the private messages is
encoded in a private stream which is decoded by its respective receiver. Each
receiver then decodes the common stream, retrieves its intended common part,
then removed the common stream from the received signal using successive
interference cancellation (SIC). After removing the common stream, each
receiver can decode its intended private stream by treating the remaining
private streams as interferences. From the common stream and the private
stream, each receiver can reconstruct the original message. The flexibility of
RSMA lies in adjusting the content and power allocated to the common and
private streams, so as to partially decode interference and partially treat
interference as noise [4]. Such flexibility leads to more robustness and
performance enhancements in various network and propagation conditions [4].

In [5],[6], it shown that RSMA can outperform conventional approaches in terms
of rate maximization under perfect CSI assumption. Especially in [5], it is
shown that RSMA unifies other four strategies (i.e, non-orthogonal multiple
access (NOMA), space-division multiple access (SDMA), orthogonal multiple
access (OMA) and multicasting) and outperforms them in a two user multiple-
input single-output (MISO) broadcast channel (BC) channel.

In case of imperfect CSIT and perfect CSIR scenario, the BS is unable to
calculate the achievable rates at the receivers accurately. Thus, the BS
should adjust precoding vector and power allocation by using estimated channel
and error information. In [7], sample average approximation combined with a
weighted minimum mean square error (WMMSE) algorithm is used for sum-rate
maximization in RSMA by generating channel error ensembles. In [1], max-min
fairness optimization using the worst case rate and WMMSE is proposed with
bounded channel error. Both studies show robust transmission of RSMA in multi-
user (MU) MISO compared to conventional approaches.

In this paper, we consider both imperfect CSIR and CSIT under the assumption
that the BS obtains CSI from the receiver through lossless channel feedbacks.
This is the first paper studying the design and optimization of RSMA with both
imperfect CSIT and CSIR. We formulate the sum-rate maximization problem in
RSMA based MU-MISO system. For converting non-convex problem to convex
problem, the algorithm using the two methods semidefinite relaxation (SDR) and
concave-convex procedure (CCCP) is proposed. By the proposed algorithm, we
jointly optimize precoding vectors and power allocation. In simulation, we
show the performance gains of the proposed RSMA over existing techniques.

The reminder of this paper is organized as follows. In section II, system
model and achievable rate in imperfect CSI are described. In section III, the
optimization problem for maximizing the sum-rate is formulated and joint
precoding vector and power allocation optimization is conducted for sum-rate
optimization by the proposed algorithm based on SDR and CCCP. Simulation
result are provided in section IV. The paper is concluded in V.

###  I-A Notaion

Standard letter indicates scalar, lower case boldface letter denotes vector,
and upper case boldface letter denotes matrix. Notation
$\mathbf{A}\succeq\mathbf{B}$ indicates that matrix $\mathbf{A}-\mathbf{B}$ is
positive semidefinite matrix. Superscript $(\cdot)^{H}$ denotes hermitian
(conjugate transpose). Trace of matrix $\mathbf{A}$ is denoted by
$\mathrm{tr}(\mathbf{A})$ and rank of matrix $\mathbf{A}$ is denoted by
$\mathrm{rank}(\mathbf{A})$. Notations of $|\cdot|$, $||\cdot||$, and
$\mathbb{E}[\cdot]$ refer to the absolute value, Euclidean norm, and
expectation operation, respectively. A matrix $\mathbf{I}_{n}$ denotes a $n$
by $n$ identity matrix.

Figure 1: System architecture of rate-splitting multiple access with imperfect
CSIR and CSIT in MU-MISO BC

##  II System Model

###  II-A Rate-Splitting Multiple Access Based System

We consider a single cell MU-MISO system operating in downlink where the BS
equipped with $N_{t}$ antennas serves $K$ single antenna users. As shown in
Fig. 1, the main idea of RSMA is to split a message for user-$k$ $W_{k}$,
$k=1,\dots,K$ into common and private parts, i.e.
$W_{k}=\\{W_{p,k},W_{c,k}\\}$. The common part can be decoded by all users and
the private part can be decoded by only the corresponding user. All common
parts of each user message are combined into one common message $W_{c}$, i.e.
$W_{c}=\\{W_{c,1},\dots,W_{c,K}\\}$. The common message $W_{c}$ is encoded
into the common stream $s_{c}$ by using a codebook known to all users and each
private message $W_{p,k}$ is encoded into the private stream $s_{k}$ by using
a codebook known to only the intended receiver. Each stream is assumed to be
independent zero mean unit variance Gaussian random variable, i.e.
$s_{i}\sim\mathcal{CN}(0,1),~{}i\in\mathcal{I}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptscriptstyle\Delta}}}\\{c,1,\dots,K\\}.$
These $K+1$ streams are linearly precoded by using precoding vector
$\mathbf{p}_{i}\in\mathbb{C}^{N_{t}\times 1},~{}i\in\mathcal{I}$. The
transmitted signal at the BS is expressed as

$\displaystyle\mathbf{x}$
$\displaystyle=\mathbf{p}_{\mathrm{c}}s_{\mathrm{c}}+\sum_{k=1}^{K}\mathbf{p}_{k}s_{k},$
(1)

where a transmitted signal power constraint with a total power $P_{t}$ is

$\displaystyle\sum\limits_{\begin{subarray}{c}i\in\mathcal{I}\end{subarray}}\left\lVert\mathbf{p}_{i}\right\rVert^{2}\leq
P_{t}.$ (2)

We refer to $\mathbf{h}_{k}\in\mathbb{C}^{N_{t}\times 1}$ as a downlink
channel vector from the BS to user-$k$ and a received signal at user-$k$ is
denoted by

$y_{k}=\mathbf{h}_{k}^{H}\bm{\mathrm{x}}+n,~{}k=1,\dots,K,$ (3)

where $n\sim\mathcal{CN}(0,\sigma_{{n}}^{2})$ is additive white Gaussian noise
(AWGN). Since the common stream can be decoded by all users, users can remove
the common stream by SIC. Thus, users decode the private stream after SIC.
When decoding the common stream, all private streams are treated as
interference. When decoding a private stream, only other private streams are
treated as interference, provided that the common stream is completely
removed. Each user reconstructs the original message after retrieving the part
of its message encoded in the common stream and the part encoded in the
private stream.

###  II-B Assumption on Channel State Information

We assume that users cannot accurately estimate the channel vector, i.e.
imperfect CSIR. The channel model is given by

$\mathbf{h}_{k}=\hat{\mathbf{h}}_{k}+\mathbf{e}_{k},$ (4)

where $\hat{\mathbf{h}}_{k}$ is an estimated channel and
$\mathbf{e}_{k}\sim\mathcal{CN}(0,\mathbf{\Phi}_{k})$ is a channel error.
Also, we assume that the BS has the same CSI with the users because of
lossless channel feedback. Thus, all users and the BS know the expectation of
the channel, $\mathbb{E}[\mathbf{h}_{k}]=\hat{\mathbf{h}}_{k}$, and the
covariance of the channel,
$\mathbb{E}[(\mathbf{h}_{k}-\mathbb{E}[\mathbf{h}_{k}])(\mathbf{h}_{k}-\mathbb{E}[\mathbf{h}_{k}])^{H}]=\mathbf{\Phi}_{k}$.
In this paper, it is assumed that the covariance matrix of channel error
$\mathbf{e}_{k}$ is $\mathbf{\Phi}_{k}=\sigma_{\mathrm{e},k}^{2}\mathbf{I}$.
In other words, the channel error is assumed as a vector of independent and
identically distributed (i.i.d) random variables.

###  II-C Achievable Rate

It is difficult to determine an explicit achievable rate under the imperfect
CSIR assumption, since the users do not know the actual channel. Thus, the
concept of generalized mutual information (GMI) is used in order to
characterize the achievable rate at a user with imperfect CSI [8],[9]. We
first introduce a general form of GMI by considering a point to point case for
simplicity. When the input has Gaussian distribution
$x\sim\mathcal{CN}(0,\epsilon_{x})$, the output signal is expressed by

$\displaystyle y=hx+n$ (5)

where $h$ is the fading channel and $n\sim\mathcal{CN}(0,N)$ is the noise.
When knowing expectation and variance of channel, $h$ can be broken into
$\hat{h}$ and $e$, i.e. ${h}=\hat{h}+e$ where $\mathbb{E}[h]=\hat{h}$ and
$\mathbb{E}[e]=0$. We can intuitively consider $\hat{h}$ as an estimate of the
channel and $e$ as a channel error having zero mean with variance
$\sigma_{h}^{2}$. GMI is defined by

$\displaystyle{I_{\mathrm{GMI}}}=\log_{2}\left(1+\frac{|\hat{h}|^{2}\epsilon_{x}}{\mathbb{E}[|e|^{2}]\epsilon_{x}+N}\right),$
(6)

where $\mathbb{E}[|e|^{2}]=\sigma^{2}_{h}$. In the case of imperfect CSIR, GMI
corresponds to an achievable rate when a user uses a nearest neighbor decoder
and the input is Gaussian distribution [10]. By using this property, we apply
GMI to RSMA based system and derive the achievable rate under imperfect CSI.

In RSMA approach, a user first decodes the common stream and then decodes the
corresponding private stream after SIC. By this feature, the rate of the
private stream is derived using the received signal after SIC. The received
signal at user-$k$ in (3) is rewritten as

$\displaystyle y_{k}$
$\displaystyle=\hat{\mathbf{h}}_{k}^{H}\mathbf{x}+\mathbf{e}_{k}^{H}{\mathbf{x}}+n,~{}k=1,\dots,K$
(7)
$\displaystyle=\hat{\mathbf{h}}_{k}^{H}\mathbf{p}_{\mathrm{c}}s_{\mathrm{c}}+\mathbf{e}_{k}^{H}\mathbf{p}_{\mathrm{c}}s_{\mathrm{c}}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\sum_{j=1}^{K}(\hat{\mathbf{h}}_{k}^{H}\mathbf{p}_{j}s_{j}+\mathbf{e}_{k}^{H}\mathbf{p}_{j}s_{j})+n.$
(8)

Considering the common stream, the signal received from user-$k$ in (8) can be
re-expressed in the form of (5) as

$\displaystyle y_{k}$
$\displaystyle=\hat{h}_{k,c}s_{\mathrm{c}}+e_{k,c}s_{\mathrm{c}}+z_{\mathrm{c}}$
(9) $\displaystyle={h}_{k,c}s_{\mathrm{c}}+z_{\mathrm{c}}.$ (10)

where ${h}_{k,c}=\hat{h}_{k,c}+e_{k,c}$,
$\hat{h}_{k,c}=\hat{\mathbf{h}}_{k}^{H}\mathbf{p}_{\mathrm{c}}$,
$e_{k,c}=\mathbf{e}_{k}^{H}\mathbf{p}_{\mathrm{c}}$, and
$z_{\mathrm{c}}=\sum_{j=1}^{K}(\hat{\mathbf{h}}_{k}^{H}\mathbf{p}_{j}s_{j}+\mathbf{e}_{k}^{H}\mathbf{p}_{j}s_{j})+n$.
Due to the independence between each stream and the noise, the expectation and
variance of each component are derived as follows:

$\displaystyle\mathbb{E}[{h}_{k,c}]=\hat{h}_{k,c}=\hat{\mathbf{h}}_{k}^{H}\mathbf{p}_{\mathrm{c}},$
(11) $\displaystyle\mathbb{E}[e_{k,c}]=0,~{}\mathbb{E}[z_{\mathrm{c}}]=0,$
(12)
$\displaystyle\mathbb{E}[|e_{k,c}|^{2}]=\mathbb{E}[|\mathbf{e}_{k}^{H}\mathbf{p}_{\mathrm{c}}|^{2}],$
(13)
$\displaystyle\mathbb{E}[|z_{\mathrm{c}}|^{2}]=\sum\limits_{j=1}^{K}({|\hat{\mathbf{h}}{}^{H}_{k}\mathbf{p}_{j}|}{}^{2}+\mathbb{E}[|\mathbf{e}_{k}^{H}\mathbf{p}_{j}|^{2}])+\sigma_{n}^{2}.$
(14)

Substituting the values in (11), (13) and (14) into (6), the GMI for the
common stream under imperfect CSIT can be obtained as

$\displaystyle R_{c,k}$

where
$\mathbb{E}[|\mathbf{e}^{H}_{k}\mathbf{p}_{j}|^{2}]={\mathbf{p}_{j}^{\mathrm{H}}\mathbf{\Phi}_{k}\mathbf{p}_{j}}$,
due to $\mathbb{E}[\mathbf{e}_{k}{\mathbf{e}_{k}}^{H}]=\mathbf{\Phi}_{k}$.
Note that $|\mathbf{e}^{H}_{k}\mathbf{p}_{c}|^{2}$ is associated with not only
the desired stream but also the channel error. Thus this term is considered as
interference when decoding the desired stream, since users do not have any
information on the channel error. The same phenomenon occurs when decoding the
private streams.

When operating with perfect CSIR, the common stream can be removed perfectly
by SIC. However, the common stream cannot be removed perfectly under imperfect
CSIR, since the users do not have accurate information the actual channel.
Thus, the part of the common stream associated with channel error still
remains after SIC. The received signal after SIC with imperfect CSIR is
expressed by

$\displaystyle y_{k,\mathrm{SIC}}$
$\displaystyle=y_{k}-\hat{\mathbf{h}}_{k}^{H}\mathbf{p}_{\mathrm{c}}s_{\mathrm{c}}$
(15)
$\displaystyle=\mathbf{e}_{k}^{H}\mathbf{p}_{\mathrm{c}}s_{\mathrm{c}}+\sum_{j=1}^{K}(\hat{\mathbf{h}}_{k}^{H}\mathbf{p}_{j}s_{j}+\mathbf{e}_{k}^{H}\mathbf{p}_{j}s_{j})+n.$
(16)

The rate of the private stream can be obtained in a similar manner as the
common part. The received signal after SIC is rewritten as

$\displaystyle y_{k,\mathrm{SIC}}$
$\displaystyle=\hat{h}_{k}s_{k}+e_{k}s_{k}+z_{k}$ (17)
$\displaystyle={h}s_{k}+z_{k}.$ (18)

where ${h}_{k}=\hat{h}_{k}+e_{k}$,
$\hat{h}_{k}=\hat{\mathbf{h}}_{k}^{H}\mathbf{p}_{k}$,
$e_{k}=\mathbf{e}_{k}^{H}\mathbf{p}_{k}$, and
$z_{k}=\mathbf{e}_{k}^{H}\mathbf{p}_{c}s_{\mathrm{c}}+\sum_{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}^{K}(\hat{\mathbf{h}}_{k}^{H}\mathbf{p}_{j}s_{j}+\mathbf{e}_{k}^{H}\mathbf{p}_{j}s_{j})+n$.
Thus, the achievable rate of private stream for user-$k$ with imperfect CSIR
is determined as

$\displaystyle R_{k}$
$\displaystyle=\log_{2}\bBigg@{3}(1+\frac{{|\mathbf{\hat{h}}{}^{H}_{k}\mathbf{p}_{k}|}{}^{2}}{\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}^{K}|\hat{\mathbf{h}}{}^{H}_{k}\mathbf{p}_{j}|^{2}\\!\\!+\\!\\!\sum\limits_{j\in\mathcal{I}}{\mathbb{E}[|\mathbf{e}^{H}_{k}\mathbf{p}_{j}|^{2}]}+\sigma_{{n}}^{2}}\bBigg@{3}).$
(19)

##  III  Sum-rate Maximization with imperfect CSI

In this section, a sum-rate maximization problem is formulated. We transform
the optimization problem and propose a novel algorithm for solving the
optimization problem that is non-convex.

###  III-A Problem Formulation

Our objective is to optimize precoding vectors consisting of power and
direction for maximizing the sum-rate. The sum-rate is expressed as

$\displaystyle R_{s}=R_{c}+\sum\limits_{k=1}^{K}R_{k},$ (20)

in which the common rate
$R_{c}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptscriptstyle\Delta}}}\min_{k}R_{c,k},~{}k=1,\dots,K$
because the common stream is decoded by all users. The optimization problem
for sum-rate maximization is formulated as: —s—[0] p_i,∀i R_c+∑_k=1^KR_k (P1):
R_c,k ≥R_c ∑_∀i ∈I ∥p_i ∥^2≤P_t. We first simplify expression of the objective
function of $\bm{\mathsf{(P1)}}$ using a stacking method introduced in [11].
First, we equivalently transform the expressions of the rate (LABEL:Rck1) and
(19) as

$\displaystyle R_{c,k}$
$\displaystyle=\log_{2}\bBigg@{4}(\frac{\sum\limits_{\begin{subarray}{c}j\in\mathcal{I}\end{subarray}}{\mathbf{p}_{j}^{H}(\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}{}^{H}_{k}+\mathbf{\Phi}_{k})\mathbf{p}_{j}}+\sigma_{{n}}^{2}}{\sum\limits_{j=1}^{K}{\mathbf{p}_{j}^{H}\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}{}^{H}_{k}\mathbf{p}_{j}}+\sum\limits_{\begin{subarray}{c}j\in\mathcal{I}\end{subarray}}{\mathbf{p}_{j}^{H}\mathbf{\Phi}_{k}\mathbf{p}_{j}}+\sigma_{{n}}^{2}}\bBigg@{4})$
(21)

and

$\displaystyle
R_{k}=\log_{2}\bBigg@{4}(\frac{\sum\limits_{\begin{subarray}{c}j=1\end{subarray}}^{K}\mathbf{p}_{j}^{H}\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}{}^{H}_{k}\mathbf{p}_{j}\\!\\!+\\!\\!\sum\limits_{j\in\mathcal{I}}{\mathbf{p}_{j}^{H}\mathbf{\Phi}_{k}\mathbf{p}_{j}}+\sigma_{{n}}^{2}}{\sum\limits_{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}^{K}\mathbf{p}_{j}^{H}\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}{}^{H}_{k}\mathbf{p}_{j}+\\!\\!\sum\limits_{j\in\mathcal{I}}{\mathbf{p}_{j}^{H}\mathbf{\Phi}_{k}\mathbf{p}_{j}}+\sigma_{{n}}^{2}}\bBigg@{4}).$
(22)

By using a combined precoding vector
$\mathbf{p}=[\mathbf{p}_{1}^{H},\dots,\mathbf{p}_{K}^{H},\mathbf{p}_{c}^{H}]^{H}\in\mathbb{C}^{N_{t}(K+1)\times
1}$, the numerator term in (21) can be expressed by

$\displaystyle{\sum\limits_{\begin{subarray}{c}j\in\mathcal{I}\end{subarray}}{\mathbf{p}_{j}^{H}(\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}{}^{H}_{k}+\mathbf{\Phi}_{k})\mathbf{p}_{j}}+\sigma_{{n}}^{2}}=\mathbf{p}^{H}\mathbf{A}_{k}\mathbf{p},$
(23)

where $\mathbf{A}_{k}\in\mathbb{C}^{N_{t}(K+1)\times N_{t}(K+1)}$ is a block
diagonal and positive definite matrix defined by

$\displaystyle\bm{\mathrm{A}}_{k}=\begin{bmatrix}\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}{}^{H}_{k}&0&\cdots&0\\\
0&\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}{}^{H}_{k}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}{}^{H}_{k}\end{bmatrix}$
$\displaystyle+\left(\frac{\sigma_{{n}}^{2}}{P_{t}}+\sigma_{\mathrm{e},k}^{2}\right)\mathbf{I}_{N_{t}(K+1)}$
(24)

under the assumption that all transmission power is used, i.e.,
$\left\lVert\mathbf{p}\right\rVert^{2}=P_{t}$, and
$\mathbf{\Phi}_{k}=\sigma_{\mathrm{e},k}^{2}\mathbf{I}$. We apply the same
approach to the denominator and the numerator terms of (21), (22). Each term
can be rewritten as:

$\displaystyle{\sum\limits_{j=1}^{M}{\mathbf{p}_{j}^{H}\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}{}^{H}_{k}\mathbf{p}_{j}}+\sum\limits_{\begin{subarray}{c}j\in\mathcal{I}\end{subarray}}{\mathbf{p}_{j}^{H}\mathbf{\Phi}_{k}\mathbf{p}_{j}}+\sigma_{{n}}^{2}}=\mathbf{p}^{H}\mathbf{B}_{k}\mathbf{p},$
(25) $\displaystyle{\sum\limits_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{M}\mathbf{p}_{j}^{H}\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}{}^{H}_{k}\mathbf{p}_{j}+\\!\\!\sum\limits_{j\in\mathcal{I}}{\mathbf{p}_{j}^{H}\mathbf{\Phi}_{k}\mathbf{p}_{j}}+\sigma_{{n}}^{2}}=\mathbf{p}^{H}\mathbf{D}_{k}\mathbf{p},$
(26)

where the positive definite matrix $\mathbf{B}_{k},\mathbf{D}_{k}$ are defined
as

$\displaystyle\bm{\mathrm{B}}_{k}=\bm{\mathrm{A}}_{k}-\begin{bmatrix}0&\cdots&\cdots&0\\\
\vdots&\ddots&\ddots&\vdots\\\ 0&\cdots&0&0\\\
0&\cdots&0&\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}{}^{H}_{k}\end{bmatrix},$ (27)
$\displaystyle\bm{\mathrm{D}}_{k}=\bm{\mathrm{B}}_{k}-\begin{bmatrix}{0}&\cdots&0&\cdots&0\\\
\vdots&\ddots&\vdots&\ddots&\vdots\\\
0&\cdots&{\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}{}^{H}_{k}}&\cdots&0\\\
\vdots&\ddots&\vdots&\ddots&\vdots\\\ 0&\cdots&0&\cdots&{0}\end{bmatrix}.$
(28)

The matrix in the second term of (28) is a block diagonal matrix formulated by
diag($\bm{0},\dots,\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}{}^{H}_{k},\dots,\bm{0}$)
in which $\hat{\mathbf{h}}_{k}\hat{\mathbf{h}}{}^{H}_{k}$ is a $k$th diagonal
sub-block. As the result, the achievable rates are simplified to

$\displaystyle
R_{c,k}=\log_{2}\left(\frac{\mathbf{p}^{H}\mathbf{A}_{k}\mathbf{p}}{\mathbf{p}^{H}\mathbf{B}_{k}\mathbf{p}}\right),~{}R_{k}=\log_{2}\left(\frac{\mathbf{p}^{H}\mathbf{B}_{k}\mathbf{p}}{\mathbf{p}^{H}\mathbf{D}_{k}\mathbf{p}}\right),$
(29)

and the objective function of the optimization problem $\bm{\mathsf{(P1)}}$
can also be simplified to

$\displaystyle
f(\mathbf{p})=\sum\limits_{k=1}^{K}\log_{2}\left(\frac{\mathbf{p}^{H}\mathbf{B}_{k}\mathbf{p}}{\mathbf{p}^{H}\mathbf{D}_{k}\mathbf{p}}\right)+\min_{j}\log_{2}\left(\frac{\mathbf{p}^{H}\mathbf{A}_{j}\mathbf{p}}{\mathbf{p}^{H}\mathbf{B}_{j}\mathbf{p}}\right).$
(30)

In this case, because $f(\mathbf{p})=f(\alpha\mathbf{p})$ with non-zero
parameter $\alpha$, the power constraint (III-A) can be ignored. Each rate is
written as the difference of concave functions, e.g.
$R_{k}=\log_{2}(\mathbf{p}^{H}\mathbf{B}_{k}\mathbf{p})-\log_{2}(\mathbf{p}^{H}\mathbf{D}_{k}\mathbf{p})$,
which is a non-convex function. Thus, finding the optimal solution is
difficult due to the non-convexity of the objective function. In order to
solve the non-convex problem, we propose an algorithm based on alternating
optimization.

###  III-B Proposed Optimization Algorithm

Before finding a solution, to transform the problem, we derive upper and lower
bounds of the denominator and numerator in (29) by using auxiliary variables
$a_{k},b_{k},c_{k},d_{k}$:

$\displaystyle\mathbf{p}^{H}\mathbf{A}_{k}\mathbf{p}\geq
e^{a_{k}},\,\mathbf{p}^{H}\mathbf{B}_{k}\mathbf{p}\geq e^{c_{k}},$ (31)
$\displaystyle\mathbf{p}^{H}\mathbf{B}_{k}\mathbf{p}\leq
e^{b_{k}},\,\mathbf{p}^{H}\mathbf{D}_{k}\mathbf{p}\leq e^{d_{k}}.$ (32)

Using these bounds, we can induce the lower bound of objective function
$f(\mathbf{p})$ by

$\displaystyle f(\mathbf{p})$
$\displaystyle\geq\frac{1}{\ln{2}}\left[\sum\limits_{k=1}^{K}\ln\left(\frac{e^{c_{k}}}{e^{d_{k}}}\right)+\min_{j}\ln\left(\frac{e^{a_{j}}}{e^{b_{j}}}\right)\right]$
$\displaystyle=\frac{1}{\ln{2}}\left[\sum\limits_{k=1}^{K}\left(c_{k}-d_{k}\right)+\min_{j}(a_{j}-b_{j})\right].$
(33)

Finally, by adding one more slack variable $l_{c}$ for term of
$\min_{j}(a_{j}-b_{j})$, the optimization problem $\bm{\mathsf{(P1)}}$ can be
transformed to $\bm{\mathsf{(P2)}}$ denoted as —s—[0] a,b,c,dp,l_c∑_k=1^K(c_k-
d_k)+l_c (P2): a_k-b_k≥l_c, k=1,…,K (31), (32), where
$\mathbf{a}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptscriptstyle\Delta}}}[a_{1},\dots,a_{K}]$,
$\mathbf{b}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptscriptstyle\Delta}}}[b_{1},\dots,b_{K}]$,
$\mathbf{c}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptscriptstyle\Delta}}}[c_{1},\dots,c_{K}]$,
and
$\mathbf{d}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptscriptstyle\Delta}}}[d_{1},\dots,d_{K}]$.
The problem is still non-convex, since the constraints (31), (32) are non-
convex. For constraint (31), we apply SDR technique which obtains a solution
that is close to the optimal solution in non-convex quadratically constrained
quadratic program (QCQP) [12]. SDR converts a non-convex problem into a convex
problem by removing the rank one constraint which causes non-convexity. To
apply SDR, we transform the quadratic term of (31) as

$\displaystyle\mathbf{p}^{H}\mathbf{A}_{k}\mathbf{p}=\mathrm{tr}(\mathbf{p}^{H}\mathbf{A}_{k}\mathbf{p})=\mathrm{tr}(\mathbf{A}_{k}\mathbf{p}\mathbf{p}^{H})=\mathrm{tr}(\mathbf{A}_{k}\mathbf{X})$
(34)

by converting $\mathbf{p}\mathbf{p}^{H}$ to $\mathbf{X}$ with constraints
$\mathbf{X}\succeq 0$ and $\mathrm{rank}(\mathbf{X})=1$. Then the constraint
(31) becomes convex when the rank constraint is removed. Generally, an optimal
solution of the relaxed problem may not satisfy the rank constraint, which
implies that an additional process is required to construct a genuine solution
that satisfies the rank constraint. This issue will be tackled after the
algorithm is described.

Algorithm 1 Alternating Optimization based on SDR and CCCP

1: Initialize $b_{k}^{(0)},d_{k}^{(0)}$ and set s=0 and $\epsilon$ to a small
value

2: repeat

3: $s\leftarrow s+1$

4: Given $b_{k}^{(s-1)},d_{k}^{(s-1)}$ , solve the problem $\mathsf{(P3)}$ and
obtain optimal
$\mathbf{X}^{*},a_{k}^{*},b_{k}^{*},c_{k}^{*},d_{k}^{*},l_{c}^{*}$

5: Update $b_{k}^{(s)}\leftarrow b_{k}^{*}$, $d_{k}^{(s)}\leftarrow d_{k}^{*}$

6: until convergence of $b_{k}^{(s)},d_{k}^{(s)}$

7: Decomposition $\mathbf{X}^{*}=\mathbf{U\Sigma U}^{H}$

8: Generate enough random vectors
$\mathbf{r}\sim\mathcal{CN}(0,\mathbf{I}_{N_{t}(K+1)})$

9: Choose the best $\mathbf{\bar{r}}=\mathbf{U\Sigma}^{1/2}\mathbf{r}$ as a
solution $\mathbf{p}^{*}$

The constraint functions in (32) are difference of convex (DC) functions that
are generally non-convex. For this problem, we approximate the DC function to
a convex function by the CCCP method, which guarantees a local optimal
solution of the DC problem [13]. For such approximation, we linearize the
exponential term in (32), which is concave, by using the first-order Taylor
series approximation. Finally, constraints at the $s$th iteration can be
denoted by

$\displaystyle\mathrm{tr}(\mathbf{A}_{k}\mathbf{X})\geq
e^{a_{k}},~{}\mathrm{tr}(\mathbf{B}_{k}\mathbf{X})\geq e^{c_{k}},$ (35)
$\displaystyle\mathrm{tr}(\mathbf{B}_{k}\mathbf{X})\leq
e^{b_{k}^{(s-1)}}(b_{k}-b_{k}^{(s-1)}+1),$ (36)
$\displaystyle\mathrm{tr}(\mathbf{D}_{k}\mathbf{X})\leq
e^{d_{k}^{(s-1)}}(d_{k}-d_{k}^{(s-1)}+1).$ (37)

The sub-problem at $s$th iteration $\bm{\mathsf{(P3)}}$ is a convex problem
expressed by —s—[0] a,b,c,dp,l_c∑_k=1^M(c_k-d_k)+l_c (P3): a_k-b_k≥l_c,
k=1,…,K X⪰0 (35), (36), (37). The value of $b_{k}^{(s-1)}$ and $d_{k}^{(s-1)}$
are updated by solving the sub-problem $\bm{\mathsf{(P3)}}$ and a local
optimal solution is obtained with a sufficient number of iterations. The
detailed process is expressed in Algorithm 1. The convex problem can be solved
using CVX toolbox [14].

It is noted that a obtained solution $\mathbf{X}^{*}$ does not satisfy the
constraint $\mathrm{rank}(\mathbf{X})=1$. Therefore, we refine
$\mathbf{X}^{*}$ to satisfy the rank constraint. Since $\mathbf{X}^{*}$ is a
Hermitian positive semidefinite matrix, it can be decomposed in the form
$\mathbf{X}^{*}=\mathbf{U\Sigma U}^{H}$ by the singular value decomposition.
Then, we generate sufficient number of random vectors
$\mathbf{r}\in\mathbb{C}^{N_{t}(K+1)\times 1}$ and obtain
$\mathbf{\bar{r}}=\mathbf{U\Sigma}^{1/2}\mathbf{r}$. Finally, we choose the
best $\mathbf{\bar{r}}$ maximizing the objective function $f(\mathbf{p})$ as a
final solution $\mathbf{p}^{*}$. It has been shown that SDR with sufficiently
large number of random vectors guarantees a solution close to the optimal
solution [12]. Overall step of the proposed algorithm is described in
Algorithm 1.

Figure 2: Convergence of proposed algorithm according to the number of
transmitter antennas and users

$\mathbf{Remark~{}(Convergence):}$ In Algorithm 1, we randomly initialize the
values of $b_{k}^{(0)}$ and $d_{k}^{(0)}$. In Fig. 2, it is numerically
confirmed that the proposed algorithm converges to finite point as the number
of iterations increases when SNR$=20$dB, $\sigma_{\mathrm{e},k}^{2}=0.1$. The
value of objective function gradually increased without fluctuation. We can
observe that as the number of transmitter antennas and users increases, the
number of iterations required for convergence increases.

##  IV simulation results

We assume $\mathbf{h}_{k}$ has i.i.d complex Gaussian distribution with zero
mean and unit variance, i.e. $\mathbf{h}_{k}\sim\mathcal{CN}(0,1)$. The
variance of AWGN is fixed as $\sigma_{n}^{2}=1$. The channel error is also
distributed by complex Gaussian distribution, i.e.,
$\mathbf{e}_{k}\sim\mathcal{CN}(0,\sigma_{\mathrm{e},k}^{2}\mathbf{I})$. The
estimated channel $\hat{\mathbf{h}_{k}}$ is independent from the channel error
and has complex Gaussian distribution with zero mean and variance
$\sigma_{\hat{\mathbf{h}}_{k}}^{2}=1-\sigma_{\mathrm{e},k}^{2}\mathbf{I}$,
i.e.
$\hat{\mathbf{h}}_{k}\sim\mathcal{CN}(0,1-\sigma_{\mathrm{e},k}^{2}\mathbf{I})$.
We also consider that all channel error has same covariance matrix
$\sigma_{\mathrm{e},k}^{2}\mathbf{I}=\sigma_{\mathrm{e}}^{2}\mathbf{I}$.

###  IV-A Comparison with Conventional Multiple Access Techniques

In this section, we consider a 2-user scenario and provide simulation results
to compare with existing multiple access strategies: SDMA, NOMA, and OMA. It
has been shown that these conventional strategies are special cases of RSMA in
a 2-user scenario [5]. As shown in TABLE I, RSMA can boil down to conventional
strategies depending on the power levels allocated to the streams. When user-1
has a stronger channel than user-2, the private stream of user-2 should be
turned off, resulting in NOMA. Regardless of the number of users, RSMA works
as SDMA when the common stream is turned off. Thus, the optimal power
allocation and precoding vectors in NOMA and SDMA can be carried out by
modifying Algorithm 1. Specifically, the precoding vector of the private
stream of user-2 is set to zero vector in case of NOMA, while that of the
common stream is set to zero vector in case of SDMA. Also, RSMA can be reduced
to OMA by allocating the total transmit power to one private stream within a
time slot. For OMA, we apply maximum ratio transmission (MRT) to precoding
vector and assume that the same time resource is allocated to user-1 and
user-2 for fairness.

Figure 3:  Sum-rate comparison between RSMA and convectional schemes, where $\sigma_{\mathrm{e}}^{2}=0.05$, $K=2$, and $N_{t}=2$ TABLE I: power assigned to each stream according to different multiple access schemes $\mathbf{Multiple~{}Access}$ |  $s_{1}$ |  $s_{2}$ |  $s_{c}$
---|---|---|---
$\mathbf{RSMA}$ | $P_{1}$ | $P_{2}$ | $P_{c}$
$\mathbf{NOMA}$ | $P_{1}$ | 0 | $P_{c}$
$\mathbf{SDMA}$ | $P_{1}$ | $P_{2}$ | 0
$\mathbf{OMA}$ | $P_{1}$ | 0 | 0

In order to confirm the usefulness of the channel error information, a
scenario in which the BS has no information about channel error, labeled as
no-info $\sigma_{e}$, is further considered with the schemes described above.
In other words, the BS optimizes the precoding vectors by considering
$\hat{\mathbf{h}}_{k}$ as the perfect channel estimate. Note that in OMA,
since the precoding vector and the power allocation are fixed, the sum-rate
performance is not changed depending on whether or not there is no information
about channel error in the BS.

We illustrate the achievable sum-rate of the proposed method according to SNR
when the BS has two antennas and $\sigma_{\mathrm{e}}^{2}=0.05$. As shown in
Fig. 3, RSMA has a better performance than other multiple accesses. The gap in
sum-rate between RSMA and the other schemes is notable in high SNR. It is
worth pointing out that these benefits come from the flexibility of RSMA that
generalizes and bridges the conventional schemes for 2-user scenario. Compared
to SDMA, the use of the common stream for RSMA, offers more design flexibility
in jointly optimizing its precoding vector and power allocation. Under perfect
CSIR, there is no increment of interference when a desired signal transmit
power is increased. However, under imperfect CSIR, the interference from the
channel error is also increased as the desired signal transmit power is
increased. Thus, GMI and the sum-rate are saturated at high SNR. Furthermore,
it is described in Fig. 3 that the performance is degraded in the absence of
information about channel error and RSMA is less sensitive to the knowledge on
channel errors than the other schemes.

Figure 4: Sum-rates of RSMA achieved by proposed and fixed precoding vectors according to SNR and $\sigma_{e}^{2}$, where $K=2$ and $N_{t}=2$ TABLE II: Precoding vector of private stream with ZF and MRT $\mathbf{Precoding}$ | Direction of precoding vector for private stream
---|---
$\mathbf{ZF}$ |  |  $\mathbf{v}_{k}=\mathbf{w}_{k}/\left\lVert\mathbf{w}_{k}\right\rVert$ where $\mathbf{W}=\hat{\mathbf{H}}^{H}(\hat{\mathbf{H}}\hat{\mathbf{H}}^{H})^{-1},$
---
$\mathbf{W}=[\mathbf{w}_{1},\cdot\cdot\cdot,\mathbf{w}_{K}]$, and
$\hat{\mathbf{H}}=[\hat{\mathbf{h}}_{1},\cdot\cdot\cdot,\hat{\mathbf{h}}_{K}]^{H}$
$\mathbf{MRT}$ | $\mathbf{v}_{k}=\hat{\mathbf{h}}_{k}/||{\hat{\mathbf{h}}}_{k}||$

###  IV-B Impact of Proposed Precoding in RSMA

In this section, performances of the precoding vectors are analyzed by
comparing simulation results of RSMA with the proposed precoding vectors and
existing fixed precoding vectors, zero-forcing (ZF) and MRT. ZF beamforming
aims to remove the interference by nulling, i.e.
$|\hat{\mathbf{h}}{}^{H}_{k}\mathbf{p}_{j}|=0,~{}j\neq k$ [15] and MRT refers
to precoding the stream in the same direction with the channel vector. ZF and
MRT are applied to the precoding vectors of the private streams. As shown in
TABLE II, ZF and MRT determine the direction of the precoding vector for the
private stream based on the estimated channel. As a result, the direction of
the precoding vectors for the private stream,
$\mathbf{v}_{k}=\mathbf{p}_{k}/\left\lVert\mathbf{p}_{k}\right\rVert,~{}k=1,\dots,K$,
are fixed. Thus, power allocation,
$P_{k}=\left\lVert\mathbf{p}_{k}\right\rVert$, and precoding vector for the
common stream ${\mathbf{p}_{c}}$ should be optimized. We optimize the sum-rate
of RSMA-ZF/MRT with the similar approach of the proposed RSMA by formulating
an optimization problem for $P_{k}$ and ${\mathbf{p}_{c}}$.

Fig. 4 demonstrates the changes in the performance of RSMA with respect to
channel error covariance $\sigma_{e}^{2}$ and SNR, where $K=2$ and $N_{t}=2$.
The results confirm that the rate under imperfect CSI has different
characteristics from that obtained under perfect CSI. Under perfect CSI, it is
well known that MRT is near optimal in low SNR and ZF is asymptotically
optimal in high SNR [16]. However, the imperfect knowledge of channel brings
out the residual interference caused by the inaccurate operations of precoding
(at the transmitter) as well as coherent detection including SIC (at the
receiver), which corresponds to low SNR scenarios. When the variance of
channel error, $\sigma_{e}^{2}$, is dominant, RSMA tends to operate as in low
SNR under perfect CSI so that the optimized beamformers approximate to MRT. On
the other hand, when channel error variance is very small, e.g.
$\sigma_{e}^{2}\approx 0$, ZF can provide a better sum-rate perforamnce than
that of MRT due to negligible residual interference at high SNR regime. As
shown in Fig. 4, RSMA with the proposed precoding vector performs stricly
better than RSMA with ZF and MRT precoders regardless of the variance of
channel error over the entire range of SNR.

##  V Conclusion

In this paper, we have studied a robust design of RSMA when perfect CSI is not
available at both transmitter and receiver. To tackle the sum-rate
maximization problem turned out to be non-convex, the proposed algorithm has
utilized the SDR and CCCP in jointly optimizing the precoding vectors and
power allocation. The simulations results have numerically shown that the
proposed RSMA achieves the enhanced sum-rate performance compared to the
conventional multiple access schemes. Also, RSMA with joint optimization of
power allocation and the precoding vectors provides the sum-rate improvement
over RSMA in which the fixed precoding schemes, ZF and MRT, which are applied
to the private streams. From these results, it can be seen that the proposed
RSMA is a powerful technique in terms of the sum-rate performance under
imperfect CSIR and CSIT.

## Acknowledgment

This work was supported by the Basic Science Research Programs under the
National Research Foundation of Korea (NRF) through the Ministry of Science
and ICT under Grants NRF-2019R1C1C1006806.

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