#  Joint Beamforming Design and Power Allocation for Full-Duplex NOMA
Cognitive Relay Systems

Mohammadali Mohammadi†, Batu K. Chalise∗, Azar Hakimi†, Himal A. Suraweera‡,
and Zhiguo Ding§
†Faculty of Engineering, Shahrekord University, Iran
∗Department of Electrical Engineering and Computer Science, Cleveland State
University, USA
‡Department of Electrical and Electronic Engineering, University of
Peradeniya, Sri Lanka
§School of Computing and Communications, Lancaster University, United Kingdom
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we consider a non-orthogonal multiple access cognitive radio
network, where a full-duplex multi-antenna relay assists transmission from a
base station (BS) to a cognitive far user, whereas, at the same time, the BS
transmits to a cognitive near user. Our objective is to enlarge the far-near
user rate region by maximizing the rate of the near user under a constraint
that the rate of the far user is above a certain threshold. To this end, a
non-convex joint optimization problem of relay beamforming and the transmit
powers at the BS and cognitive relay is solved as a semi-definite relaxation
problem, in conjunction with an efficiently solvable line-search approach. For
comparisons, we also consider low complexity fixed beamformer design, where
the optimum power allocation between the BS and cognitive relay is solved. Our
results demonstrate that the proposed joint optimization can significantly
reduce the impact of the residual self-interference at the FD relay and inter-
user interference in the near user case. ††footnotetext: The work of Z. Ding
was supported by the UK EPSRC under grant number EP/N005597/1. It was also
supported by the European Union H2020-MSCA-RISE-2015 under grant number
690750.

##  I Introduction

Non-orthogonal multiple access (NOMA) and full-duplex (FD) communication are
foreseen as two independent key technology components of fifth generation (5G)
wireless. NOMA exploits power domain to serve multiple users at the same time,
frequency and spreading codes [1]. NOMA transmitter sends a superimposed
signal with different power levels to the multiple users, where successive
interference cancellation (SIC) is utilized to separate superimposed signals
at the receiver side and to mitigate the inter-user interference. Therefore,
compared to conventional orthogonal multiple access schemes, NOMA can offer a
significant improvement in spectrum efficiency [2, 3].

On the other hand, FD technology has been recently received a lot of research
interest due to its potential to double the spectrum efficiency and
subsequently increase the data rate compared to half-duplex (HD) mode [4].
However, the main limitation in FD operation is self-interference (SI) caused
by the signal leakage from the transceiver output to the input [5].
Nevertheless, recent progress on FD radio implementations shows great
potential for doubling capacity through SI cancellation techniques [6, 7].

In the literature, the FD and NOMA combination has been invoked to further
enhance the spectral efficiency of the communication systems [8, 9, 10]. The
authors of [8] investigated the resource allocation algorithm design for a FD
multicarrier NOMA system, where a FD base station (BS) is simultaneously
serving multiple HD downlink and uplink users. In [9] a diversity analysis for
cooperative FD NOMA systems was provided to prove that the use of the direct
link overcomes the lack of diversity for the far user which otherwise serves
as a limitation of FD relaying. In [10] a dual-user NOMA system has been
studied, where a dedicated FD dual-antenna relay assists information
transmission to the far user with the weaker channel condition. The proposed
FD cooperative NOMA system [10] achieves a higher ergodic sum capacity
compared to the HD cooperative NOMA counterpart in the low to moderate signal-
to-noise (SNR) regimes.

Cognitive radio (CR) is another technology that has received wide attention
well over a decade now to improve the spectrum utilization. Under CR paradigm,
each cognitive user (CU) is allowed to access the spectrum of the primary
users (PUs) as long as the CU meets a certain interference threshold in the
primary network (PN) [11]. Despite the promise of FD and NOMA for cognitive
radio, to the best of our knowledge, current literature has not analyzed such
systems. In this work, we investigate a relay assisted cooperative NOMA system
at the cognitive network of the CR network, where a cognitive BS communicates
with a near and far NOMA CU. The main motivation for the adoption of multiple
antennas at the FD relay is that the SI cancellation can be performed in the
spatial domain using efficient beamforming design at the cognitive relay [5].
However, beamforming design also influences the achievable NOMA performance of
near and far user. We obtain dual-user rate region by designing the receive
and transmit beamformers at the relay and allocating optimal power to the BS
and cognitive relay. Speciﬁcally, the near user rate is maximized by ensuring
that the far user rate is above a certain threshold.

The main contributions of this paper are twofold.

  * •

A complicated non-convex optimization problem of joint cognitive relay
beamforming and power allocation (at the BS and relay) is transformed to a
semi-definite relaxation (SDR) problem consisting of relay transmit
beamforming matrix and BS power allocation parameter. The optimum solutions of
the joint optimization are obtained by solving the SDR problem in conjunction
with a line search over the BS power allocation parameter. The computational
complexity of the proposed approach is minimal, since this line search is
confined to a finite region and the SDR problem can be solved as a feasibility
problem. As compared to traditional HD operation, our new results show that
the proposed joint optimization significantly improves the rate region.

  * •

As a suboptimum approach, we also consider a power allocation problem with
fixed beamformer design. We compare the far and near user rate region for the
optimum and suboptimum methods to highlight the gains of the proposed optimum
design for different system parameters, such as the number of relay
transmit/receive antennas, level of the residual SI at the relay and the peak
power constraint at both BS and relay.

_Notation:_ We use bold upper case letters to denote matrices, bold lower case
letters to denote vectors. The superscripts $(\cdot)^{T}$, $(\cdot)^{*}$,
$(\cdot)^{{\dagger}}$, and $(\cdot)^{-1}$ stand for transpose, conjugated,
conjugate transpose, and matrix inverse respectively; the Euclidean norm of
the vector, the trace, and the expectation are denoted by $\|\cdot\|$, ${\rm
tr}(\cdot)$, and ${\tt E}\left\\{\cdot\right\\}$ respectively; and
$\mathcal{CN}(\mu,\sigma^{2})$ denotes a circularly symmetric complex Gaussian
RV $x$ with mean $\mu$ and variance $\sigma^{2}$.

##  II System Model

We consider a FD cognitve relay network as shown in Fig. 1, where the PUs and
CUs share the same spectral band. The cognitive network consists of a BS, a
decode-and-forward relay and two CUs, denoted by $\text{CU}_{1}$ and
$\text{CU}_{2}$. The BS communicates with the two CUs, by applying the NOMA
concept, where the near user, $\text{CU}_{1}$ directly communicates with the
BS, while the far user, $\text{CU}_{2}$ requires the assistance of the relay.
We assume that the BS, $\text{CU}_{1}$ and $\text{CU}_{2}$ are each equipped
with a single antenna [10]. To enable FD operation, the cognitive relay is
equipped with two sets of antennas, i.e., ${\mathsf{N_{R}}}$ receiveing
antennas and ${\mathsf{N_{T}}}$ transmitting antennas. We assume that no
direct link between the BS and $\text{CU}_{2}$ exists, similar to [12, 10].

In a spectrum sharing CR system, a CU can share the PU’s spectrum as long as
the interference inflicted on the primary receiver is below a predetermined
maximum tolerable interference level at the PU, $I_{\mathsf{th}}$ [11]. Since
the BS and relay transmit their signals at the same time using the same
spectrum, the primary receiver suffers interference from the BS and cognitive
relay simultaneously. Hence, the transmission powers of the BS and cognitive
relay must be constrained as [11]

$\displaystyle\beta_{BP}P_{S}|h_{BP}|^{2}+\beta_{RP}P_{R}|{\bf h}_{RP}^{T}{\bf
w}_{t}|^{2}\leq I_{\mathsf{th}},$ (1)

where $P_{S}$ and $P_{R}$ are the transmission powers of the BS and cognitive
relay, $h_{BP}$ and ${\bf h}_{RP}\in\mathcal{C}^{{\mathsf{N_{T}}}\times 1}$
denote the BS-primary receiver channel and the cognitive relay-primary
receiver channel respectively, $\beta_{BP}$ and $\beta_{RP}$ model the
corresponding path loss effects, and ${\bf
w}_{t}\in\mathcal{C}^{{\mathsf{N_{T}}}\times 1}$ denotes the transmit
beamforming vector at the cognitive relay.

Furthermore, similar to the model used in [13, 14, 15], we focus on the
coexistence of a long-range primary system and short range CR network. There
is a direct link in this set up, however the primary transmitter is far away
from the CUs and thus the interference inflicted at the CUs is negligible.

###  II-A Transmission Protocol

According to the NOMA concept, the BS transmits a combination of intended
messages to both CUs as

$\displaystyle s[n]=\sqrt{P_{S}a_{1}}x_{1}[n]+\sqrt{P_{S}a_{2}}x_{2}[n],$ (2)

where $x_{i}$, $i\in\\{1,2\\}$ denotes the information symbol intended for
$\text{CU}_{i}$, and $a_{i}$ denotes the power allocation coefficient, such
that $a_{1}+a_{2}=1$ and $a_{1}<a_{2}$.

The received signal at $\text{CU}_{1}$ can be written as

$\displaystyle
y_{1}[n]\\!=\\!\sqrt{\beta_{h_{1}}}h_{1}s[n]\\!+\\!\sqrt{\beta_{f_{1}}P_{R}}{\bf
f}_{1}^{T}{\bf w}_{t}x_{2}[n\\!-\\!\tau]\\!+\\!n_{1}[n],$ (3)

where $h_{1}$ is the channel between the BS and $\text{CU}_{1}$, ${\bf
f}_{1}\in\mathcal{C}^{{\mathsf{N_{R}}}\times 1}$ denotes the channel between
the cognitive relay and $\text{CU}_{1}$, the respective path losses of the
BS-$\text{CU}_{1}$ and relay-$\text{CU}_{1}$ links are denoted by
$\beta_{h_{1}}$ and $\beta_{f_{1}}$, respectively. Further, $\tau$ accounts
for the time delay caused by FD relay processing [5], and
$n_{1}[n]\sim\mathcal{CN}(0,\sigma_{n_{1}}^{2})$ is the additive white
Gaussian noise (AWGN) at $\text{CU}_{1}$.

Figure 1: Cognitive NOMA system model with FD relaying.

By invoking (3), the effective signal-to-interference-plus-noise ratio (SINR)
of $\text{CU}_{2}$ observed at $\text{CU}_{1}$ can be written as

$\displaystyle\gamma_{1,2}$
$\displaystyle=\frac{\beta_{h_{1}}P_{S}a_{2}|h_{1}|^{2}}{\beta_{h_{1}}P_{S}a_{1}|h_{1}|^{2}+\beta_{f_{1}}P_{R}|{\bf
f}_{1}^{T}{\bf w}_{t}|^{2}+\sigma_{n_{1}}^{2}}.$ (4)

It is assumed that symbol $x_{2}[n-\tau]$ is priory known to $\text{CU}_{1}$
and thus $\text{CU}_{1}$ can remove it via interference cancellation [10].
However, by considering realistic imperfect interference cancellation wherein
$\text{CU}_{1}$ cannot perfectly remove $x_{2}[n-\tau]$, we model ${\bf
f}_{1}\sim\mathcal{CN}(0,k_{1})$ as the inter-user interference channel where
the parameter $k_{1}$ presents the strength of inter-user interference [10].
Specifically, $k_{1}=0$ implies perfect interference cancellation at
$\text{CU}_{1}$. If $\text{CU}_{1}$ cancels the $\text{CU}_{2}$’s signal, the
SINR at $\text{CU}_{1}$ is given by

$\displaystyle\gamma_{1}$
$\displaystyle=\frac{\beta_{h_{1}}P_{S}a_{1}|h_{1}|^{2}}{\beta_{f_{1}}P_{R}|{\bf
f}_{1}^{T}{\bf w}_{t}|^{2}+\sigma_{n_{1}}^{2}}.$ (5)

The received signal at the cognitive relay can be written as

$\displaystyle y_{R}[n]$ $\displaystyle=\sqrt{\beta_{h_{2}}}{\bf
w}_{r}^{\dagger}{\bf h}_{2}s[n]+\sqrt{P_{R}}{\bf w}_{r}^{\dagger}{\bf
H}_{RR}{\bf w}_{t}x_{2}[n-\tau]$ $\displaystyle+\sqrt{\beta_{PR}P_{U}}{\bf
w}_{r}^{\dagger}{\bf h}_{PR}x_{P}[n]+{\bf w}_{r}^{\dagger}{\bf n}_{R}[n],$ (6)

where ${\bf w}_{r}\in\mathcal{C}^{{\mathsf{N_{R}}}\times 1}$ is the combining
receiver at the cognitive relay, ${\bf
h}_{2}\in\mathcal{C}^{{\mathsf{N_{R}}}\times 1}$ is the channel between the BS
and cognitive relay, $P_{U}$ is transmit power of the primary transmitter,
${\bf h}_{PR}\in\mathcal{C}^{{\mathsf{N_{R}}}\times 1}$ is the channel between
the primary transmitter and cognitive relay, $\beta_{h_{2}}$ and $\beta_{PR}$
model the path loss effect in the BS-cognitive relay channel and primary
transmitter-cognitive relay channel, $x_{p}[n]$ is the primary transmit
signal, and ${\bf n}_{R}[n]$ is the AWGN at the cognitive relay with
$\mathbb{E}\left\\{{\bf n}_{R}{\bf
n}_{R}^{\dagger}\right\\}=\sigma_{R}^{2}{\bf I}$. We assume imperfect SI
cancellation at the cognitive relay and similar to [5] model the elements of
the ${\mathsf{N_{R}}}\times{\mathsf{N_{T}}}$ residual SI channel ${\bf
H}_{RR}$ as independent identically distributed (i.i.d)
$\mathcal{CN}(0,\sigma^{2}_{\mathsf{SI}})$ RVs.

The cognitive relay decodes the information intended for $\text{CU}_{2}$
treating the symbol of $\text{CU}_{1}$ as interference. Hence, the SINR at the
cognitive relay can be expressed as

$\displaystyle\gamma_{R}=$ (7)
$\displaystyle\frac{\beta_{h_{2}}P_{S}a_{2}|{\bf w}_{r}^{\dagger}{\bf
h}_{2}|^{2}}{\beta_{h_{2}}P_{S}a_{1}|{\bf w}_{r}^{\dagger}{\bf
h}_{2}|^{2}\\!+\\!\\!P_{R}|{\bf w}_{r}^{\dagger}{\bf H}_{RR}{\bf
w}_{t}|^{2}\\!+\\!\beta_{PR}P_{U}|{\bf w}_{r}^{\dagger}{\bf
h}_{PR}|^{2}\\!\\!+\\!\sigma_{R}^{2}}.$

Moreover, the received signal at $\text{CU}_{2}$, transmitted by the cognitive
relay can be written as

$\displaystyle y_{2}[n]=\sqrt{\beta_{f_{2}}P_{R}}{\bf f}_{2}^{T}{\bf
w}_{t}x_{2}[n-\tau]+n_{2}[n],$ (8)

where ${\bf f}_{2}\in\mathcal{C}^{{\mathsf{N_{T}}}\times 1}$ denotes the
channel between the cognitive relay and $\text{CU}_{2}$, $\beta_{f_{2}}$ model
the path loss effect of the cognitive relay-$\text{CU}_{2}$ channel, and
$n_{2}[n]\sim\mathcal{CN}(0,\sigma_{n_{2}}^{2})$ denotes the AWGN at the
$\text{CU}_{2}$. Hence, the SNR at $\text{CU}_{2}$ is given by

$\displaystyle\gamma_{R,2}$
$\displaystyle=\frac{\beta_{f_{2}}P_{R}}{\sigma_{n_{2}}^{2}}|{\bf
f}_{2}^{T}{\bf w}_{t}|^{2}.$ (9)

##  III Beamforming Design and Power Allocation

In this section, we consider joint optimization of receive/transmit
beamformers at the cognitive relay and power allocation at the BS and
cognitive relay. We also propose a power allocation scheme between the BS and
cognitive relay when fixed beamformers are assumed at the relay. Specifically,
we consider maximum ratio transmit (MRT)/maximum ratio combining (MRC) as
transmit/receive beamformers, while results for other possible beamformers
such as transmit/receive are left out as future work.

###  III-A Optimum Scheme

Let us consider the joint design of transmit/receive beamformers and
allocation of BS and cognitive relay power such that achievable rate of
$\text{CU}_{1}$ is maximized, while the far user’s rate is guaranteed to be
above a certain value $\bar{r}$. As such, the optimization problem can be
formulated as

$\displaystyle\max_{{\bf w}_{t},{\bf w}_{r},P_{S},P_{R}}\hskip
8.00003ptC_{1}({\bf w}_{t},P_{S},P_{R}),$ $\displaystyle\hskip
20.00003pt\text{s.t}\hskip 30.00005ptC_{2}({\bf w}_{t},{\bf
w}_{r},P_{S},P_{R})\geq\bar{r},$ $\displaystyle\hskip
60.00009pt\beta_{BP}P_{S}|h_{BP}|^{2}+\beta_{RP}P_{R}|{\bf h}_{RP}^{T}{\bf
w}_{t}|^{2}\leq I_{\mathsf{th}},$ $\displaystyle\hskip 60.00009pt\|{\bf
w}_{r}\|=\|{\bf w}_{t}\|=1,\quad~{}P_{S},P_{R}\geq 0,$ (10)

where [10]

$\displaystyle C_{1}({\bf w}_{t},P_{S},P_{R})=\log_{2}\left(1+\gamma_{1}({\bf
w}_{t},P_{S},P_{R})\right),$ $\displaystyle C_{2}({\bf w}_{t},{\bf
w}_{r},P_{S},P_{R})=\log_{2}\left(1+\min\left(\gamma_{1,2}({\bf
w}_{t},P_{S},P_{R}),\right.\right.$ $\displaystyle\hskip
50.00008pt\left.\left.\gamma_{R}({\bf w}_{t},{\bf
w}_{r},P_{S},P_{R}),\gamma_{R,2}({\bf w}_{t},P_{R})\right)\right).$ (11)

The problem in (III-A) can be reformulated as

$\displaystyle\max_{{\bf w}_{t},{\bf w}_{r},P_{S},P_{R}}\hskip
10.00002pt\log_{2}\left(1\\!+\\!\frac{\beta_{h_{1}}P_{S}a_{1}|h_{1}|^{2}}{\beta_{f_{1}}P_{R}|{\bf
f}_{1}^{T}{\bf w}_{t}|^{2}+\sigma_{n_{1}}^{2}}\right),$ $\displaystyle\hskip
20.00003pt\text{s.t}\hskip 30.00005pt\min\left(\gamma_{1,2}({\bf
w}_{t},P_{S},P_{R}),\gamma_{R}({\bf w}_{t},{\bf w}_{r},P_{S},P_{R}),\right.$
$\displaystyle\hskip 90.00014pt\left.\gamma_{R,2}({\bf
w}_{t},P_{R})\right)\geq\tilde{r},$ $\displaystyle\hskip
60.00009pt\beta_{BP}P_{S}|h_{BP}|^{2}+\beta_{RP}P_{R}|{\bf h}_{RP}^{T}{\bf
w}_{t}|^{2}\leq I_{\mathsf{th}},$ $\displaystyle\hskip 60.00009pt\|{\bf
w}_{r}\|=\|{\bf w}_{t}\|=1,\quad P_{S},P_{R}\geq 0,$ (12)

where $\tilde{r}\triangleq 2^{\bar{r}}-1$. Moreover, the first constraint in
(III-A) can be expressed using the following inequalities:

$\displaystyle\beta_{h_{1}}P_{S}a_{2}|h_{1}|^{2}\geq$ $\displaystyle\hskip
10.00002pt\tilde{r}\left(\beta_{h_{1}}P_{S}a_{1}|h_{1}|^{2}+\beta_{f_{1}}P_{R}{\bf
w}_{t}^{\dagger}{\bf f}_{1}^{\ast}{\bf f}_{1}^{T}{\bf
w}_{t}+\sigma_{n_{1}}^{2}\right),$ (13a)
$\displaystyle\frac{\beta_{h_{2}}P_{S}a_{2}{\bf w}_{r}^{\dagger}{\bf
h}_{2}{\bf h}_{2}^{\dagger}{\bf w}_{r}}{{\bf w}_{r}^{\dagger}{\bf A}{\bf
w}_{r}}\geq\tilde{r},$ (13b) $\displaystyle\beta_{f_{2}}P_{R}{\bf
w}_{t}^{\dagger}{\bf f}_{2}^{\ast}{\bf f}_{2}^{T}{\bf
w}_{t}\geq\sigma_{n_{2}}^{2}\tilde{r},$ (13c)

where ${\bf A}=\beta_{h_{2}}P_{S}a_{1}{\bf h}_{2}{\bf
h}_{2}^{\dagger}+P_{R}{\bf H}_{RR}{\bf w}_{t}{\bf w}_{t}^{\dagger}{\bf
H}_{RR}^{\dagger}+\beta_{PR}P_{U}{\bf h}_{PR}{\bf
h}_{PR}^{\dagger}+\sigma_{R}^{2}{\bf I}$.

By inspecting the optimization problem in (III-A) we see that only
$\gamma_{R}({\bf w}_{t},{\bf w}_{r},P_{S},P_{R})$ depends on ${\bf w}_{r}$.
Therefore, it is obvious that the optimum ${\bf w}_{r}$ is one that maximizes
$\gamma_{R}({\bf w}_{t},{\bf w}_{r},P_{S},P_{R})$. Define
$g\triangleq\frac{{\bf w}_{r}^{\dagger}{\bf h}_{2}{\bf h}_{2}^{\dagger}{\bf
w}_{r}}{{\bf w}_{r}^{\dagger}{\bf A}{\bf w}_{r}}$. The optimum ${\bf w}_{r}$
is given by

$\displaystyle~{}{\bf w}_{r}=\frac{{\bf A}^{-1}{\bf h}_{2}}{\|{\bf A}^{-1}{\bf
h}_{2}\|}.$ (14)

Let ${\bf B}=(\beta_{h_{2}}P_{S}a_{1}{\bf h}_{2}{\bf
h}_{2}^{\dagger}+\beta_{PR}P_{U}{\bf h}_{PR}{\bf
h}_{PR}^{\dagger}+\sigma_{R}^{2}{\bf I})$. By substituting ${\bf w}_{r}$ from
(14) into $g$, we get

$\displaystyle g$ $\displaystyle={\bf h}_{2}^{\dagger}\left[P_{R}{\bf
H}_{RR}{\bf w}_{t}{\bf w}_{t}^{\dagger}{\bf H}_{RR}^{\dagger}+{\bf
B}\right]^{-1}{\bf h}_{2}$ (15) $\displaystyle={\bf h}_{2}^{\dagger}{\bf
B}^{-1}{\bf h}_{2}-\frac{\left({\bf h}_{2}^{\dagger}{\bf B}^{-1}{\bf
H}_{RR}{\bf w}_{t}{\bf w}_{t}^{\dagger}{\bf H}_{RR}^{\dagger}{\bf B}^{-1}{\bf
h}_{2}\right)P_{R}}{1+P_{R}{\bf w}_{t}^{\dagger}{\bf H}_{RR}^{\dagger}{\bf
B}^{-1}{\bf H}_{RR}{\bf w}_{t}},$

where we have used the Sherman-Morrison formula, $({\bf
B}+\bm{u}\bm{v}^{{\dagger}})^{-1}={\bf B}^{-1}-({\bf B}^{-1}{\bf u}{\bf
v}^{{\dagger}}{\bf B}^{-1})/(1+{\bf v}^{{\dagger}}{\bf B}^{-1}{\bf u})$ with
${\bf u}={\bf v}=\sqrt{P_{R}}{\bf H}_{RR}{\bf w}_{t}$.

By substituting (15) into (13b), the optimization problem (III-A) is reduced
w.r.t only ${\bf w}_{t}$, $P_{S}$, and $P_{R}$, as follows:

$\displaystyle\max_{{\bf w}_{t},P_{S},P_{R}}\hskip
5.0pt\log_{2}\left(1+\frac{\beta_{h_{1}}P_{S}a_{1}|h_{1}|^{2}}{\beta_{f_{1}}P_{R}{\bf
w}_{t}^{\dagger}{\bf f}_{1}^{\ast}{\bf f}_{1}^{T}{\bf
w}_{t}+\sigma_{n_{1}}^{2}}\right),$ $\displaystyle\hskip
15.00002pt\text{s.t}\hskip 10.00002pt\hskip 10.00002ptP_{R}{\bf
w}_{t}^{\dagger}{\bf f}_{1}^{\ast}{\bf f}_{1}^{T}{\bf w}_{t}\leq q_{1},$
$\displaystyle\hskip 42.50006pt\frac{\left({\bf h}_{2}^{\dagger}{\bf
B}^{-1}{\bf H}_{RR}{\bf w}_{t}{\bf w}_{t}^{\dagger}{\bf H}_{RR}^{\dagger}{\bf
B}^{-1}{\bf h}_{2}\right)P_{R}}{1\\!+\\!P_{R}{\bf w}_{t}^{\dagger}{\bf
H}_{RR}^{\dagger}{\bf B}^{-1}{\bf H}_{RR}{\bf w}_{t}}\leq q_{2},$
$\displaystyle\hskip 42.50006ptP_{R}{\bf w}_{t}^{\dagger}{\bf
f}_{2}^{\ast}{\bf f}_{2}^{T}{\bf w}_{t}\geq q_{3}$ $\displaystyle\hskip
42.50006ptP_{R}|{\bf h}_{RP}^{T}{\bf w}_{t}|^{2}\leq q_{4},$
$\displaystyle\hskip 42.50006pt\|{\bf w}_{t}\|=1,\quad P_{S},P_{R}\geq 0,$
(16)

where
$q_{1}\\!=\\!\left(\beta_{h_{1}}P_{S}a_{2}|h_{1}|^{2}\\!-\\!\tilde{r}\beta_{h_{1}}P_{S}a_{1}|h_{1}|^{2}\\!-\\!\tilde{r}\sigma_{n_{1}}^{2}\right)\\!/(\tilde{r}\beta_{f_{1}})$,
$q_{2}={\bf h}_{2}^{\dagger}{\bf B}^{-1}{\bf
h}_{2}-\tilde{r}/(\beta_{h_{2}}P_{S}a_{2})$,
$q_{3}=\sigma_{n_{2}}^{2}\tilde{r}/\beta_{f_{2}}$, and
$q_{4}=(I_{\mathsf{th}}-\beta_{BP}P_{S}|h_{BP}|^{2})/\beta_{RP}$. From the
optimization problem (III-A), we see that $\sqrt{P_{R}}{\bf w}_{t}$ can be
considered together as a single optimization variable $\bar{{\bf w}_{t}}$,
i.e, $\bar{{\bf w}_{t}}=\sqrt{P_{R}}{\bf w}_{t}$. Then (III-A) reduces to

$\displaystyle\max_{\bar{{\bf w}_{t}},P_{S},P_{R}}$ $\displaystyle\hskip
10.00002pt\log_{2}\left(1+\frac{\beta_{h_{1}}P_{S}a_{1}|h_{1}|^{2}}{\beta_{f_{1}}\bar{{\bf
w}_{t}}^{\dagger}{\bf f}_{1}^{\ast}{\bf f}_{1}^{T}\bar{{\bf
w}_{t}}+\sigma_{n_{1}}^{2}}\right),$ (17a) s.t $\displaystyle\hskip
10.00002pt\bar{{\bf w}_{t}}^{\dagger}{\bf f}_{1}^{\ast}{\bf
f}_{1}^{T}\bar{{\bf w}_{t}}\leq q_{1}\,$ (17b) $\displaystyle\hskip
10.00002pt\frac{{\bf h}_{2}^{\dagger}{\bf B}^{-1}{\bf H}_{RR}\bar{{\bf
w}_{t}}\bar{{\bf w}_{t}}^{\dagger}{\bf H}_{RR}^{\dagger}{\bf B}^{-1}{\bf
h}_{2}}{1+\bar{{\bf w}_{t}}^{\dagger}{\bf H}_{RR}^{\dagger}{\bf B}^{-1}{\bf
H}_{RR}\bar{{\bf w}_{t}}}\leq q_{2},$ (17c) $\displaystyle\hskip
10.00002pt\bar{{\bf w}_{t}}^{\dagger}{\bf f}_{2}^{\ast}{\bf
f}_{2}^{T}\bar{{\bf w}_{t}}\geq q_{3},$ (17d) $\displaystyle\hskip
10.00002pt\bar{{\bf w}_{t}}^{\dagger}{\bf h}_{RP}^{\ast}{\bf
h}_{RP}^{T}\bar{{\bf w}_{t}}\leq q_{4},$ (17e) $\displaystyle\hskip
10.00002pt\bar{{\bf w}_{t}}^{\dagger}\bar{{\bf w}_{t}}=P_{R},\quad P_{S}\geq
0.$ (17f)

The above problem is a complicated non-convex optimization problem, which to
the best of our knowledge, does not admit a closed-form solution. However, by
fixing $P_{S}$, optimum $\bar{{\bf w}_{t}}$ and $P_{R}$ can be efficiently
obtained. Then the joint optimization over $P_{S}$, $\bar{{\bf w}_{t}}$ and
$P_{R}$ can be solved by using one-dimensional search over a finite (also
small) region of $P_{S}$.

####  III-A1 Optimization over $\bar{{\bf w}_{t}}$ and $P_{R}$ for a given
$P_{S}$

Let us find the optimum $\bar{{\bf w}_{t}}$ and $P_{R}$ for a given $P_{S}$.
Problem (17) can be alternatively re-expressed as

$\displaystyle\min_{\bar{{\bf w}_{t}},P_{R}}$ $\displaystyle\hskip
20.00003pt\bar{{\bf w}_{t}}^{\dagger}{\bf f}_{1}^{\ast}{\bf
f}_{1}^{T}\bar{{\bf w}_{t}},$ (18a) s.t $\displaystyle\hskip
20.00003pt\bar{{\bf w}_{t}}^{\dagger}{\bf f}_{1}^{\ast}{\bf
f}_{1}^{T}\bar{{\bf w}_{t}}\leq q_{1},$ (18b) $\displaystyle\hskip
20.00003pt{\bf h}_{2}^{\dagger}{\bf B}^{-1}{\bf H}_{RR}\bar{{\bf
w}_{t}}\bar{{\bf w}_{t}}^{\dagger}{\bf H}_{RR}^{\dagger}{\bf B}^{-1}{\bf
h}_{2}\leq$ $\displaystyle\hskip 50.00008ptq_{2}\left(1+\bar{{\bf
w}_{t}}^{\dagger}{\bf H}_{RR}^{\dagger}{\bf B}^{-1}{\bf H}_{RR}\bar{{\bf
w}_{t}}\right),$ (18c) $\displaystyle\hskip 20.00003pt\bar{{\bf
w}_{t}}^{\dagger}{\bf f}_{2}^{\ast}{\bf f}_{2}^{T}\bar{{\bf w}_{t}}\geq
q_{3},$ (18d) $\displaystyle\hskip 20.00003pt\bar{{\bf w}_{t}}^{\dagger}{\bf
h}_{RP}^{\ast}{\bf h}_{RP}^{T}\bar{{\bf w}_{t}}\leq q_{4},$ (18e)
$\displaystyle\hskip 20.00003pt\bar{{\bf w}_{t}}^{\dagger}\bar{{\bf
w}_{t}}=P_{R}.$ (18f)

The minimum value in (18) will be less than or equal to $q_{1}$. Also (18) is
a non-convex optimization problem due to the fact that it is the minimization
of a quadratic function with non-convex quadratic inequality constraints (18c)
and (18d). However, it can be solved using an SDR approach. Introducing
$\bar{{\bf W}_{t}}\triangleq\bar{{\bf w}_{t}}\bar{{\bf w}_{t}}^{\dagger}$ and
relaxing the rank-one constraint of $\text{rank}({{\bf W}_{t}})=1$, (18) can
be expressed as an SDR problem:

$\displaystyle\min_{\bar{{\bf W}_{t}},P_{R}}$ $\displaystyle\hskip
20.00003pt\mathsf{tr}\left(\bar{{\bf W}_{t}}{\bf f}_{1}^{\ast}{\bf
f}_{1}^{T}\right),$ (19a) s.t $\displaystyle\hskip
20.00003pt\mathsf{tr}\left(\bar{{\bf W}_{t}}{\bf f}_{1}^{\ast}{\bf
f}_{1}^{T}\right)\leq q_{1},$ (19b) $\displaystyle\hskip
20.00003pt\mathsf{tr}\left(\bar{{\bf W}_{t}}{\bf H}_{RR}^{\dagger}{\bf
B}^{-1}{\bf h}_{2}{\bf h}_{2}^{\dagger}{\bf B}^{-1}{\bf H}_{RR}\right)\leq$
$\displaystyle\hskip 30.00005ptq_{2}\left(1+\mathsf{tr}\left(\bar{{\bf
W}}_{t}{\bf H}_{RR}^{\dagger}{\bf B}^{-1}{\bf H}_{RR}\right)\right),$ (19c)
$\displaystyle\hskip 20.00003pt\mathsf{tr}\left(\bar{{\bf W}_{t}}{\bf
f}_{2}^{\ast}{\bf f}_{2}^{T}\right)\geq q_{3},$ (19d) $\displaystyle\hskip
20.00003pt\mathsf{tr}\left(\bar{{\bf W}_{t}}{\bf h}_{RP}^{\ast}{\bf
h}_{RP}^{T}\right)\leq q_{4},$ (19e) $\displaystyle\hskip
20.00003pt\mathsf{tr}\left(\bar{{\bf W}_{t}}\right)=P_{R},\bar{{\bf
W}_{t}}\geq 0.$ (19f)

This problem can be solved using CVX software [16]. Problem (19) can be solved
without (19b), however if the optimum $\bar{{\bf W}_{t}}$ does not satisfy
(19b), the problem is infeasible. Moreover, $P_{S}$ and ${\tilde{r}}$ have to
be chosen such that (19) is feasible. To this end, initialize $P_{S}$ so that
$q_{1}$, $q_{2}$ and $q_{4}$ are positive. From $q_{1}\geq 0$, we get

$\displaystyle
P_{S}\geq\frac{\tilde{r}\sigma_{n_{1}}^{2}}{\beta_{h_{1}}a_{2}|h_{1}|^{2}\\!-\\!\tilde{r}\beta_{h_{1}}a_{1}|h_{1}|^{2}},$
(20)

which is established when
$\beta_{h_{1}}a_{2}|h_{1}|^{2}\\!-\\!\tilde{r}\beta_{h_{1}}a_{1}|h_{1}|^{2}>0$,
or equivalently when $\tilde{r}<\frac{a_{2}}{a_{1}}$. Moreover, from
$q_{2}\geq 0$ we have

$\displaystyle q_{2}$ $\displaystyle={\bf h}_{2}^{\dagger}{\bf B}^{-1}{\bf
h}_{2}-\frac{\tilde{r}}{\beta_{h_{2}}P_{S}a_{2}}\geq 0.$ (21)

Recall that ${\bf B}=(\beta_{h_{2}}P_{S}a_{1}{\bf h}_{2}{\bf
h}_{2}^{\dagger}+\beta_{PR}P_{U}{\bf h}_{PR}{\bf
h}_{PR}^{\dagger}+\sigma_{R}^{2}{\bf I})$ is a function of $P_{S}$. Define
${\bf E}\triangleq\left(\beta_{PR}P_{U}{\bf h}_{PR}{\bf
h}_{PR}^{\dagger}+\sigma_{R}^{2}{\bf I}\right)$ and $u\triangleq{\bf
h}_{2}^{\dagger}\left({\bf E}+\beta_{h_{2}}P_{S}a_{1}{\bf h}_{2}{\bf
h}_{2}^{\dagger}\right)^{-1}{\bf h}_{2}$. Then, using Sherman-Morrison formula
we get

Algorithm 1 The proposed optimization scheme

0: Define a fine grid of $P_{S}$, where
$P_{S}\in\left[v,\frac{I_{\mathsf{th}}}{\beta_{BP}|h_{BP}|^{2}}\right]$, in
steps of $\delta P_{S}$. Set
$P_{S}=\frac{I_{\mathsf{th}}}{\beta_{BP}|h_{BP}|^{2}}$.

0: Solve (19).

0: If feasible, stop and output $P_{S}$, $P_{R}$, and $\bar{{\bf W}_{t}}$.

0: If not, go to Step 2 with the decrement of $\delta P_{S}$.

$\displaystyle u$ $\displaystyle={\bf h}_{2}^{\dagger}\left[{\bf
E}^{-1}-\frac{{\bf E}^{-1}{\bf h}_{2}{\bf
h}_{2}^{\dagger}\beta_{h_{2}}P_{S}a_{1}{\bf
E}^{-1}}{1+\beta_{h_{2}}P_{S}a_{1}{\bf h}_{2}^{\dagger}{\bf E}^{-1}{\bf
h}_{2}}\right]{\bf h}_{2}$ $\displaystyle=\frac{{\bf h}_{2}^{\dagger}{\bf
E}^{-1}{\bf h}_{2}}{1+\beta_{h_{2}}P_{S}a_{1}\left({\bf h}_{2}^{\dagger}{\bf
E}^{-1}{\bf h}_{2}\right)}.$ (22)

Next, by substituting $u$ into $q_{2}$, it is clear that $q_{2}\geq 0$ if

$\displaystyle
P_{S}\geq\frac{{\tilde{r}}}{(a_{2}-{\tilde{r}}a_{1})\beta_{h_{2}}{\bf
h}_{2}^{\dagger}{\bf E}^{-1}{\bf h}_{2}}.$ (23)

Finally, from $q_{4}\geq 0$, we get

$\displaystyle P_{S}\leq\frac{I_{\mathsf{th}}}{\beta_{BP}|h_{BP}|^{2}}.$ (24)

From the conditions (20), (23), and (24) on $P_{S}$, it is clear that the
optimization problem (19) is feasible if

$\displaystyle
v\triangleq\max\limits\left\\{\frac{{\tilde{r}}}{(a_{2}-{\tilde{r}}a_{1})\beta_{h_{2}}{\bf
h}_{2}^{\dagger}{\bf E}^{-1}{\bf
h}_{2}},\frac{\tilde{r}\sigma_{n_{1}}^{2}}{\beta_{h_{1}}(a_{2}-\tilde{r}a_{1})|h_{1}|^{2}}\right\\}$
$\displaystyle\hskip
10.00002pt\leq\frac{I_{\mathsf{th}}}{\beta_{BP}|h_{BP}|^{2}},$ (25)

which also means that
$P_{S}\in\left[v,\frac{I_{\mathsf{th}}}{\beta_{BP}|h_{BP}|^{2}}\right]$.

####  III-A2 Joint Optimization of $P_{S}$, $P_{R}$, and $\bar{{\bf w}_{t}}$

The joint optimization problem (17) can be solved by solving the SDR problem
(19) for different values of $P_{S}$ (i.e., performing line search over
$P_{S}$), where
$P_{S}\in\left[v,\frac{I_{\mathsf{th}}}{\beta_{BP}|h_{BP}|^{2}}\right]$, and
taking those values of $\bar{{\bf w}_{t}}$ and $P_{S}$ that maximize the
objective function in (17). However, it is clear that this objective function
monotonically increases with $P_{S}$. This means that the optimum $P_{S}$ is
its largest value, for which the problem (19) is feasible. As such, starting
with $P_{S}=\frac{I_{\mathsf{th}}}{\beta_{BP}|h_{BP}|^{2}}$, the joint
optimization problem (17) can be solved by solving (19) until it turns to be
feasible. This leads to an iterative approach, which is outlined in Algorithm
1.

We end this subsection with the following remark.
Remark: Applying Shapiro-Barvinok-Pataki rank reduction result, it can be
shown that rank-one optimum solution of $\bar{{\bf W}_{t}}$ exists for the SDR
problem (19) [17]. This allows us to recover $\bar{{\bf w}_{t}}$ from
$\bar{{\bf W}_{t}}$, without any loss of optimality. Moreover, if the optimum
$\bar{{\bf W}_{t}}$ is rank-one, $\bar{{\bf w}_{t}}$ is given by $\bar{{\bf
w}_{t}}=\lambda_{\rm max}{\bf v}_{\rm max}$, where ${\bf v}_{\rm max}$ is the
eigenvector corresponding to the largest eigenvalue, $\lambda_{\rm max}$, of
$\bar{{\bf W}_{t}}$. Due to these reasons, relaxation in (19) is optimum. As
such, the proposed iterative algorithm finds the optimum solutions of $P_{S}$,
$P_{R}$, and ${\bf w}_{t}$. Moreover, the computational cost of implementing
the algorithm is minimal, since the search region of $P_{S}$ is found to be
finite and the algorithm can be stopped as soon as the optimization (19) is
feasible.

###  III-B Power Allocation for fixed Beamforming Design

In this subsection, we further investigate the power allocation problem by
considering fixed ${\bf w}_{t}$ and ${\bf w}_{r}$. The motivation for
considering a fixed choice for ${\bf w}_{t}$ and ${\bf w}_{r}$ is as follows:
Fixed beamformers constitute to low complex implementation. For example,
MRT/MRC beamformers are suitable for low complexity FD systems as they do not
need to estimate the SI channel. Moreover, MRT/MRC beamformers are preferred
for HD operation and hence it is interesting to characterize the achievable
performance in the FD case.

For a given ${\bf w}_{t}$ and ${\bf w}_{r}$, the optimization problem (III-A)
is expressed as

$\displaystyle\max_{P_{S},P_{R}}\hskip 8.00003ptC_{1}({\bf
w}_{t},P_{S},P_{R}),$ $\displaystyle\hskip 20.00003pt\text{s.t}\hskip
30.00005ptC_{2}({\bf w}_{t},{\bf w}_{r},P_{S},P_{R})\geq\bar{r},$
$\displaystyle\hskip
60.00009pt\beta_{BP}P_{S}|h_{BP}|^{2}+\beta_{RP}P_{R}|{\bf h}_{RP}^{T}{\bf
w}_{t}|^{2}\leq I_{\mathsf{th}},$ $\displaystyle\hskip
60.00009ptP_{S},P_{R}\geq 0.$ (26)

The problem (III-B) can be reformulated as

$\displaystyle\max_{P_{S},P_{R}}\hskip
10.00002pt\log_{2}\left(1\\!+\\!\frac{P_{S}\beta_{h_{1}}a_{1}|h_{1}|^{2}}{P_{R}\beta_{f_{1}}|{\bf
f}_{1}^{T}{\bf w}_{t}|^{2}+\sigma_{n_{1}}^{2}}\right),$ $\displaystyle\hskip
10.00002pt\text{s.t}\hskip
20.00003pt\frac{P_{S}\beta_{h_{1}}a_{2}|h_{1}|^{2}}{P_{S}\beta_{h_{1}}a_{1}|h_{1}|^{2}+P_{R}\beta_{f_{1}}|{\bf
f}_{1}^{T}{\bf w}_{t}|^{2}+\sigma_{n_{1}}^{2}}\geq\tilde{r},$
$\displaystyle\hskip 40.00006pt\frac{P_{S}\beta_{h_{2}}a_{2}|{\bf
w}_{r}^{{\dagger}}{\bf h}_{2}|^{2}}{P_{S}\beta_{h_{2}}a_{1}|{\bf
w}_{r}^{{\dagger}}{\bf h}_{2}|^{2}\\!+P_{R}|{\bf w}_{r}^{\dagger}{\bf
H}_{RR}{\bf w}_{t}|^{2}+\mu_{1}}\geq\tilde{r},$ $\displaystyle\hskip
40.00006ptP_{R}\frac{\beta_{f_{2}}}{\sigma_{n_{2}}^{2}}|{\bf f}_{2}^{T}{\bf
w}_{t}|^{2}\geq\tilde{r},$ $\displaystyle\hskip
40.00006ptP_{S}\beta_{BP}|h_{BP}|^{2}+P_{R}\beta_{RP}|{\bf h}_{RP}^{T}{\bf
w}_{t}|^{2}\leq I_{th},$ $\displaystyle\hskip 40.00006ptP_{S},P_{R}\geq 0,$
(27)

where $\mu_{1}\triangleq\beta_{PR}P_{U}|{\bf w}_{r}^{{\dagger}}{\bf
h}_{PR}|^{2}\\!+\\!\sigma_{R}^{2}$. Accordingly, when
$\tilde{r}<\frac{a_{2}}{a_{1}}$, the problem (III-B) can be expressed as

$\displaystyle\max_{P_{S},P_{R}}\hskip
10.00002pt\frac{P_{S}\beta_{h_{1}}a_{1}|h_{1}|^{2}}{P_{R}\beta_{f_{1}}|{\bf
f}_{1}^{T}{\bf w}_{t}|^{2}+\sigma_{n_{1}}^{2}},$ $\displaystyle\hskip
10.00002pt\text{s.t}\hskip 20.00003ptP_{S}\geq\frac{P_{R}\beta_{f_{1}}|{\bf
f}_{1}^{T}{\bf
w}_{t}|^{2}+\sigma_{n_{1}}^{2}}{\frac{\beta_{h_{1}}a_{2}|h_{1}|^{2}}{\tilde{r}}-\beta_{h_{1}}a_{1}|h_{1}|^{2}},$
$\displaystyle\hskip 40.00006ptP_{S}\leq\frac{I_{th}-P_{R}\beta_{RP}|{\bf
h}_{RP}^{T}{\bf w}_{t}|^{2}}{\beta_{BP}|h_{BP}|^{2}},$ $\displaystyle\hskip
40.00006ptP_{S}\geq\frac{P_{R}|{\bf w}_{r}^{\dagger}{\bf H}_{RR}{\bf
w}_{t}|^{2}+\mu_{1}}{\frac{\beta_{h_{2}}a_{2}|{\bf w}_{r}^{{\dagger}}{\bf
h}_{2}|^{2}}{\tilde{r}}-\beta_{h_{2}}a_{1}|{\bf w}_{r}^{{\dagger}}{\bf
h}_{2}|^{2}},$ $\displaystyle\hskip
40.00006ptP_{R}\geq\frac{\sigma_{n_{2}}^{2}\tilde{r}}{{\beta_{f_{2}}}|{\bf
f}_{2}^{T}{\bf w}_{t}|^{2}},,$ $\displaystyle\hskip
40.00006ptP_{R}\leq\frac{I_{th}}{\beta_{RP}|{\bf h}_{RP}^{T}{\bf
w}_{t}|^{2}}.$ (28)

Note that a feasible solution of $P_{R}$ exists in (III-B) if
$\frac{\sigma_{n_{2}}^{2}\tilde{r}}{{\beta_{f_{2}}}|{\bf f}_{2}^{T}{\bf
w}_{t}|^{2}}\leq\frac{I_{th}}{\beta_{RP}|{\bf h}_{RP}^{T}{\bf w}_{t}|^{2}}$.
Similarly, a feasible solution of $P_{S}$ exists if
${\tilde{v}}(P_{R})\leq\frac{I_{th}-P_{R}\beta_{RP}|{\bf h}_{RP}^{T}{\bf
w}_{t}|^{2}}{\beta_{BP}|h_{BP}|^{2}}$, where

$\displaystyle{\tilde{v}}(P_{R})$
$\displaystyle\triangleq\max\biggl{\\{}\frac{P_{R}\beta_{f_{1}}|{\bf
f}_{1}^{T}{\bf
w}_{t}|^{2}+\sigma_{n_{1}}^{2}}{\beta_{h_{1}}|h_{1}|^{2}\left(\frac{a_{2}}{\tilde{r}}-a_{1}\right)},$
$\displaystyle\hskip 50.00008pt\frac{P_{R}|{\bf w}_{r}^{\dagger}{\bf
H}_{RR}{\bf w}_{t}|^{2}+\mu_{1}}{\beta_{h_{2}}|{\bf w}_{r}^{{\dagger}}{\bf
h}_{2}|^{2}\left(\frac{a_{2}}{\tilde{r}}-a_{1}\right)}\biggr{\\}}.$ (29)

This also means that
$P_{S}\in\left[{\tilde{v}}(P_{R}),\frac{I_{th}-P_{R}\beta_{RP}|{\bf
h}_{RP}^{T}{\bf w}_{t}|^{2}}{\beta_{BP}|h_{BP}|^{2}}\right]$. On the other
hand, the objective function in (III-B) is maximized with the minimum value of
$P_{R}$ and the maximum value of $P_{S}$. Clearly, the minimum value of
$P_{R}$, i.e., $P_{R}=\frac{\sigma_{n_{2}}^{2}\tilde{r}}{{\beta_{f_{2}}}|{\bf
f}_{2}^{T}{\bf w}_{t}|^{2}}$ provides the largest region for feasible
solutions of $P_{S}$. Any other
$P_{R}>\frac{\sigma_{n_{2}}^{2}\tilde{r}}{{\beta_{f_{2}}}|{\bf f}_{2}^{T}{\bf
w}_{t}|^{2}}$ contracts this region. For
$P_{R}=\frac{\sigma_{n_{2}}^{2}\tilde{r}}{{\beta_{f_{2}}}|{\bf f}_{2}^{T}{\bf
w}_{t}|^{2}}$, the maximum possible value of $P_{S}$ is given by
$P_{S}=(I_{th}-\frac{\sigma_{n_{2}}^{2}\tilde{r}}{{\beta_{f_{2}}}|{\bf
f}_{2}^{T}{\bf w}_{t}|^{2}}\beta_{RP}|{\bf h}_{RP}^{T}{\bf
w}_{t}|^{2})/\beta_{BP}|h_{BP}|^{2}$. Consequently, the optimum solutions of
$P_{R}$ and $P_{S}$ are given by

$\displaystyle P_{R}$ $\displaystyle=$
$\displaystyle\frac{\sigma_{n_{2}}^{2}\tilde{r}}{{\beta_{f_{2}}}|{\bf
f}_{2}^{T}{\bf w}_{t}|^{2}},$ $\displaystyle P_{S}$ $\displaystyle=$
$\displaystyle\frac{I_{th}-\frac{\sigma_{n_{2}}^{2}\tilde{r}}{{\beta_{f_{2}}}|{\bf
f}_{2}^{T}{\bf w}_{t}|^{2}}\beta_{RP}|{\bf h}_{RP}^{T}{\bf
w}_{t}|^{2}}{\beta_{BP}|h_{BP}|^{2}}.$ (30)

Figure 2: Rate-region of the optimum and suboptimum schemes for different
antenna configurations at the relay. ($I_{\mathsf{th}}=15$ dBW, $k_{1}=0.01$)

##  IV Numerical and Simulation Results

In this section, we present numerical results to evaluate the rate region of
$\text{CU}_{1}$ and $\text{CU}_{2}$ due to the optimum and fixed beamformer
designs. Without loss of generality, the noise variance are set to $1$ dBW,
$P_{U}=10$ dBW, $a_{1}=0.05$ and $a_{2}=0.95$. We also adopt the same channel
parameters as in [10]. Hence, we set
$\beta_{BP}=\beta_{RP}=\beta_{h_{2}}=\beta_{f_{2}}=0.5$ and $\beta_{h_{1}}=1$.
We also show results for the HD mode where comparisons between FD and HD were
performed under the “RF-chain preserved” condition [7].

Fig. 2 shows the rate-region of the optimum and suboptimum schemes for
different antenna configurations at the relay. The rate-region of the HD mode
is shown with the BS and cognitive relay transmit powers set as
$\frac{I_{\mathsf{th}}}{\beta_{BP}|h_{BP}|^{2}}$ and
$\frac{I_{\mathsf{th}}}{\beta_{RP}|h_{RP}^{T}{\bf w}_{t}|^{2}}$, respectively.
Moreover, MRC/MRT processing is shown as an example of fixed beamforming
design case. From the figure, we can observe that the rate of both near and
far users with optimum scheme is improved when the number of transmit or
receive antenna is increased. Specifically, this increase is more pronounced
when the number of the receive antennas increases. However, in case of fixed
beamforming design, when only ${\mathsf{N_{R}}}$ is increased from 2 to 5, the
rate of the far user decreases, at higher achievable rates for the near user.
This is quite intuitive since to achieve higher rate at the near user, the
relay transmit power must be increased. However, an increase in relay transmit
power results in strong SI at the relay input which will degrade the
performance. With more transmit antennas at the relay, transmit power of the
relay can be controlled more precisely and hence both near and far user rates
are increased. Moreover, comparing FD and HD modes of operation, we see that
the FD mode with the optimum and suboptimum schemes provides superior rates
for both the near and far users.

Fig. 3 shows the impact of the residual SI on the rate region of the optimum
and suboptimum schemes. As expected, the rate of the far user, with the
optimum and suboptimum schemes, degrades when the residual SI becomes
stronger, while the rate of the far user with HD mode remains the same
regardless of the SI power level. More specifically it can be seen that, the
decrease of the far user’s rate associated with the optimum scheme is strictly
smaller than that of the suboptimum scheme which indicates that our proposed
joint beamforming design and power allocation scheme at the FD relay could
significantly suppress the residual SI and consequently can improve the rate
of the far user.

Figure 3: Rate-region of the optimum and suboptimum schemes for different
levels of SI strength. (${\mathsf{N_{T}}}={\mathsf{N_{R}}}=2$,
$I_{\mathsf{th}}=15$ dBW, $k_{1}=0.01$)

Fig. 4 compares the rate region of the optimum and suboptimum schemes for
different levels of $I_{\mathsf{th}}$ at the primary receiver. It can be
readily observed that the gap between the achievable rate of the far user for
optimum and suboptimum schemes increases when $I_{\mathsf{th}}$ decreases.
This is because that the BS and relay transmit power are decreased and hence
the rate of the far user is decreased. On the other hand, employing the joint
beamforming design and power allocation improves the far user rate
significantly and hence there is a slight gap between the far user’s rate with
low and moderate values of $I_{\mathsf{th}}$.

##  V Conclusion

In this paper, we have investigated the rate region of near and far users in a
FD relay assisted NOMA cognitive radio network. An optimum scheme was proposed
to maximize the rate of the near user through the joint receive and transmit
beamforming and power allocation design at the cognitive relay, by ensuring
that the rate of the far user is above a certain threshold. In addition we
considered suboptimal design in which a power allocation solution was derived
for any fixed receive and transmit beamforming design at the FD relay. Our
result indicate that FD relaying with proposed optimum and suboptimum schemes
can substantially boost both near and far user rates as compared to the HD
mode.

Figure 4: Rate-region of the optimum and suboptimum schemes for different
values of $I_{\mathsf{th}}$. (${\mathsf{N_{T}}}={\mathsf{N_{R}}}=2$,
$\sigma^{2}_{\mathsf{SI}}=-30$ dBW, $k_{1}=0.01$)

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