# Interference Exploitation in Full Duplex Communications: Trading
Interference Power for Both Uplink and Downlink Power Savings

Mahmoud T. Kabir, Muhammad R. A. Khandaker, and Christos Masouros

###### Abstract

This paper considers a multiuser full-duplex (FD) wireless communication
system, where a FD radio base station (BS) serves multiple single-antenna
half-duplex (HD) uplink and downlink users simultaneously. Unlike conventional
interference mitigation approaches, we propose to use the knowledge of the
data symbols and the channel state information (CSI) at the FD radio BS to
exploit the multi-user interference constructively rather than to suppress it.
We propose a multi-objective optimisation problem (MOOP) via the weighted
Tchebycheff method to study the trade-off between the two desirable system
design objectives namely the total downlink transmit power minimisation and
the total uplink transmit power minimisation problems at the same time
ensuring the required quality-of-service (QoS) for all users. In the proposed
MOOP, we adapt the QoS constraints for the downlink users to accommodate
constructive interference (CI) for both generic phase shift keying (PSK)
modulated signals as well as for quadrature amplitude modulated (QAM) signals.
We also extended our work to a robust design to study the system with
imperfect uplink, downlink and self-interference CSI. Simulation results and
analysis show that, significant power savings can be obtained. More
importantly, however, the MOOP approach here allows for the power saved to be
traded off for both uplink and downlink power savings, leading to an overall
energy efficiency improvement in the wireless link.

###### Index Terms:

full-duplex, multi-objective optimization, constructive interference, power
minimization, robust design.

##  I Introduction

The ever-increasing need for improved spectrum-efficiency in wireless links
has brought FD at the forefront of research attention. By allowing
simultaneous transmission and reception, FD since it has the potential to
drastically improve the spectral efficiency of the HD communication networks
[1, 2, 3, 4, 5, 6]. One major hurdle with the FD communication systems is the
self-interference (SI) from the transmit antennas to the receive antennas of
the wireless transceiver. This interference raises the noise floor and it
becomes a dominant factor in the performance of the FD system. However, major
breakthroughs have been made in practical FD system setups [1] and [2] that
show that the SI can be partially cancelled to within a few dB of the noise
floor. While others focused on resource management, in [3], the authors
investigated the spectral efficiency of FD small cell wireless systems by
considering a joint beamformer design to maximize the spectral efficiency
subject to power constraints. In [4], the authors discussed the resource
allocation problems in FD-MIMO, FD-Relay, FD-OFDMA and FD-HetNet systems
including power control, interference-aware beamforming, e.t.c. Also, resource
allocation and scheduling in FD-MIMO-OFDMA relaying systems was studied in
[5]. In [6], the authors used massive arrays at the FD relay station to cancel
out loop interference and as a result increase the sum spectral efficiency of
the system.

Many of the above FD solutions build upon existing beamforming solutions in
the literature, that have been extensively developed for the downlink channel,
moving from the sophisticated but capacity achieving non-linear beamforming
techniques [7, 8, 9, 10, 11] to the less complex linear beamforming techniques
[12, 13, 14, 15, 16]. Several optimization based schemes that provide optimal
solutions subject to required quality of service (QoS) constraints have been
proposed for multi-input single-output (MISO) systems in [17, 18, 19, 20]. In
[21, 22], the authors addressed the problem of robust designs in downlink
multiuser MISO systems with respect to erroneous channel state information
(CSI). The work in [23] focused on addressing both max-min signal-to-
interference (SINR) balancing problem and power minimisation problem with SINR
constraints. More recently, it has been shown in [13, 14, 24, 25] that with
the knowledge of the users’ data symbols and the CSI, the interference can be
classified into constructive and destructive interference. And further
findings in [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] show that
tremendous gains can be achieved by exploiting the constructive interference
based on symbol level optimization for both PSK and QAM modulations. However,
these findings are all based on MISO HD systems.

Our work extends the above interference exploitation concept to the FD
transmission, by employing multi-objective optimization, as most recently
studied for FD in [38, 39, 40]. The authors in [38] investigated the power
efficient resource allocation for a MU-MIMO FD system. They proposed a multi-
objective optimisation problem (MOOP) to study the total uplink and downlink
transmit power minimization problems jointly via the weighed Tchebycheff
method. They extended their work to a robust and secure FD systems model in
the presence of roaming users (eavesdroppers) in [39]. Similarly, in [40] the
authors used a similar model to investigate the resource allocation for FD
simultaneous wireless information and power transfer (SWIPT) systems.
Accordingly, in this work we aim to further reduce the power consumption in FD
MU-MIMO wireless communication systems by adopting the concept of constructive
interference in the literature to the downlink channel for both PSK and QAM
modulation. By exploiting interference constructively, useful signal power
from interference, we can provide a truly power efficient resource allocation
for a FD MU-MIMO system. The interference exploitation concept is yet to be
explored in the realm of FD transmission, where FD offers the unique
opportunity to trade-off the harvested interference power for both uplink and
downlink power savings through the MOOP designs. Against the state-of-the-art,
we summarize our contributions below:

  1. 1.

We first formulate the FD beamforming design problem that minimizes (a)the
total downlink transmit power and, (b)the total uplink transmit power problem,
for PSK and QAM modulation separately. Both problems are subject to downlink
users SINR requirement based on the constructive interference regions and
uplink users SINR requirement. Unlike conventional FD beamformers, we show
that the proposed optimizations are convex and can be easily solved by
conventional solvers.

  2. 2.

Building on the above single-objective problems, we then formulate a multi-
objective problem to study the trade-off between the total uplink and downlink
transmit power minimization problems jointly via the weighed Tchebycheff
method. Again, unlike the conventional FD beamformers, we show that the
proposed optimization is convex.

  3. 3.

We further derive robust MOOP for both the conventional and the proposed
interference exploitation approach by recasting the MOOP into a virtual
multicast problem for erroneous downlink, uplink and SI CSI with bounded
errors.

The rest of the paper is organised as follows. Section II introduces the
system model that is considered in this paper. Section III describes the two
conventional power minimisation problems of interest to the system operator
and then briefly describes the MOOP formulation based on the two problems. In
Section IV, the proposed power minimization optimisation problems based on
constructive interference regions are presented for PSK and QAM modulations.
Then in Section V, we present the robust version of the optimisation problem
presented in Section IV. In Section VI, we provide a computational complexity
analysis of the MOOP formulations. Section VII illustrates the important
results and discussions. And finally we conclude in Section VIII.

##  II System Model

We consider a FD multiuser communication system as shown in Fig. 1. The system
consists of a FD radio BS with N antennas serving K HD downlink users and J HD
uplink users. Each user is equipped with a single antenna to reduce hardware
complexity. Let ${\textbf{h}_{i}}\in\mathbb{C}^{N\times 1}$ be the channel
vector between the FD radio BS and the i-th downlink user, and
${\textbf{f}_{j}}\in\mathbb{C}^{N\times 1}$ be the channel vector between the
FD radio BS and the j-th uplink user. We denote the transmit signal vector
from the FD radio BS to the i-th downlink user as

$\displaystyle{\textbf{t}_{i}}$ $\displaystyle={\textbf{w}_{i}}d_{i}$ (1)

where ${\textbf{w}_{i}}\in\mathbb{C}^{N\times 1}$ and $d_{i}$ denote the
beamforming vector and the unit data symbol for the i-th downlink user. The
received signal at the i-th downlink user is:

$\displaystyle y_{i}$ $\displaystyle=\underset{\textrm{desired
signal}}{\underbrace{{\textbf{h}_{i}^{H}}{\textbf{t}_{i}}}}+\underset{\textrm{interference
plus noise}}{\underbrace{\sum_{k\neq
i}^{K}{\textbf{h}_{i}^{H}}{\textbf{t}_{k}}+n_{i}}}$ (2)

where $n_{i}\sim{\mathcal{CN}}\left(0,\sigma_{i}^{2}\right)$ represents the
additive white Gaussian noise AWGN at the i-th downlink user. For each time
slot the FD radio BS transmits K independent unit data symbols d
simultaneously at the same frequency to the K downlink users. The first term
in (2) represents the desired signal while the second term is the multiuser
interference signal. The received signal from the J uplink users at the FD
radio BS is:

Figure 1: System model with a FD radio BS with N antennas, K HD downlink users
and J HD uplink users.

$\displaystyle{\textbf{y}^{BS}}$
$\displaystyle=\sum_{j=1}^{J}{\sqrt{P}_{j}}{\textbf{f}_{j}}x_{j}+\underset{\textrm{residual
self-
interference}}{\underbrace{{\textbf{G}}\sum_{k=1}^{K}{\textbf{t}_{k}}}}+{\textbf{z}}$
(3)

where ${\textit{P}_{j}}$ and ${\textit{x}_{j}}$ denotes the uplink transmit
power and the data symbol from the j-th uplink user respectively. The vector
${\textbf{z}}\sim{\mathcal{CN}}(0,{\sigma}_{N}^{2})$ represents the additive
white Gaussian noise AWGN at the FD radio BS. The matrix
${\textbf{G}}\in\mathbb{C}^{N\times N}$ denotes the self-interference (SI)
channel at the FD radio BS. In the literature, different SI mitigation
techniques have been proposed [41, 42] to reduce the effect of self-
interference. In order to isolate our proposed scheme from the specific
implementation of a SI mitigation technique, since the SI cannot be cancelled
perfectly in FD systems due to limited dynamic range at the receiver even if
the SI channel is known perfectly [39, 42], we model the residual SI after
cancellation as $\left({\textbf{G}}\sum_{k=1}^{K}{\textbf{t}_{k}}\right)$ as
in [38, 39]. Accordingly, the first term of (3) represents the desired signal
from the j-th uplink user and the second term represents the residual SI.

Before we formulate the problem, we first define the signal-to-interference
ratio (SINR) at the i-th downlink user and at the FD radio BS respectively as

$\displaystyle SINR_{i}^{DL}$
$\displaystyle=\frac{{\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{i}}\mid}^{2}}{\sum_{k\neq
i}^{K}\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{k}}\mid^{2}+\sigma_{i}^{2}}$ (4)
$\displaystyle SINR_{j}^{UL}$
$\displaystyle=\frac{{P_{j}\mid{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\mid}^{2}}{\sum_{n\neq
j}^{J}P_{n}\mid{\textbf{f}_{n}^{H}}{\textbf{u}_{j}}\mid^{2}+\sum_{k=1}^{K}\mid{\textbf{u}_{j}^{H}}{\textbf{G}}{\textbf{w}_{k}}\mid^{2}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}$
(5)

where ${\textbf{u}_{j}}\in^{N\times 1}$ is the receive beamforming vector for
detecting the receivied symbol from the j-th uplink user. To reduce
complexity, we assume a zero-forcing receiver at the BS. Hence, the receive
beamformer for the j-th uplink user is given as

$\displaystyle{\textbf{u}_{j}}$
$\displaystyle=({\textbf{r}_{j}}{\textbf{F}^{\dagger}})^{H}$ (6)

where
${\textbf{r}_{j}}=[\underset{j-1}{\underbrace{0,\ldots,0,}}1,\underset{J-j}{\underbrace{0,\ldots,0}}]$,
${\textbf{F}^{\dagger}}=({\textbf{F}}^{H}{\textbf{F}})^{-1}{\textbf{F}}^{H},^{\dagger}$
denotes the pseudo-inverse operation and
${\textbf{F}}=[{\textbf{f}}_{1},\dots,{\textbf{f}}_{J}]$.

##  III Conventional Power Minimization Problem

In this section, we study the conventional power minimization (PM) problem
where all the interferences are treated as undesired signals. We first
formulate the downlink and uplink power minimization problems, which aim to
minimize the total average downlink and uplink transmit power, respectively,
subject to the downlink users SINR and uplink users SINR. Then we formulate a
multi-objective PM problem that aims to investigate the two system’s
objectives (downlink and uplink) jointly.

### Problem 1: Total Downlink Transmit PM Problem

The downlink PM problem for FD optimisation is typically formulated as [38,
39]:

$\begin{split}\mathcal{P}1:\quad\underset{{\textbf{w}_{i}},P_{j}}{\text{min}}\quad&\sum_{i=1}^{K}{\left\|{\textbf{w}_{i}}\right\|}^{2}\\\
\text{s.t.}\quad&A1:\frac{{\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{i}}\mid}^{2}}{\sum_{k\neq
i}^{K}\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{k}}\mid^{2}+\sigma_{i}^{2}}\geq\Gamma_{i}^{DL},\forall
i,\\\
&A2:\frac{{P_{j}\mid{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\mid}^{2}}{I_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j\end{split}$ (7)

where, $I_{j}=\sum_{n\neq
j}^{J}P_{n}\mid{\textbf{f}_{n}^{H}}{\textbf{u}_{j}}\mid^{2}+\sum_{k=1}^{K}\mid{\textbf{u}_{j}^{H}}{\textbf{G}}{\textbf{w}_{k}}\mid^{2}$,
we define $\Gamma_{i}^{DL}$ and $\Gamma_{j}^{UL}$ as the minimum required
SINRs for the i-th downlink user and the j-th uplink user, respectively. This
problem aims to minimize the total downlink transmit power with no regards to
the consumed uplink transmit power. This problem is non-convex and it is
commonly solved via semidefinite relaxation as in [38, 39].

### Problem 2: Total Uplink Transmit PM Problem

The uplink PM problem for FD optimisation is typically formulated as [38, 39]:

$\begin{split}\mathcal{P}2:\quad\underset{{\textbf{w}_{i}},P_{j}}{\text{min}}\quad&\sum_{j=1}^{J}{P_{j}}\\\
\text{s.t.}\quad&A1:\frac{{\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{i}}\mid}^{2}}{\sum_{k\neq
i}^{K}\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{k}}\mid^{2}+\sigma_{i}^{2}}\geq\Gamma_{i}^{DL},\forall
i,\\\
&A2:\frac{{P_{j}\mid{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\mid}^{2}}{I_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j\end{split}$ (8)

where, $\Gamma_{i}^{DL}$ and $\Gamma_{j}^{UL}$ are the minimum required SINRs
for the i-th downlink user and the j-th uplink user, respectively. This
problem unlike problem $\mathcal{P}1$ aims to minimize the total uplink
transmit power with no regards to the consumed downlink transmit power.
Problem $\mathcal{P}2$ is non-convex and it is commonly solved via
semidefinite relaxation as in [38, 39].

### Problem 3: Multi-objective PM Problem

This formulation combines the two objectives of problem $\mathcal{P}1$ and
$\mathcal{P}2$ since both objectives are very important to both the users and
system operator. The multi-objective optimization is employed when there is
need to study jointly the trade-off between two desirable objectives via the
concept of Pareto optimality. A point is said to be Pareto optimal if there is
no other point that improves any of the objectives without decreasing the
others [43]. [43] did a survey of multi-objective optimization methods in
engineering. By using the weighted Tchebycheff method [43] which can acheive
the complete Pareto optimal set with lower computational complexity, the
multi-objective PM problem for FD optimisation is typically formulated as [38,
39],

$\begin{split}\mathcal{P}3:\quad\underset{{\textbf{w}_{i}},P_{j}}{\text{min}}\quad&\underset{a=1,2}{\text{max}}\left\\{{\lambda_{a}}\left(R_{a}^{*}-R_{a}({\textbf{w}_{i}},P_{j})\right)\right\\}\\\
\text{s.t.}\quad&A1:\frac{{\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{i}}\mid}^{2}}{\sum_{k\neq
i}^{K}\mid{\textbf{h}_{i}^{H}}{\textbf{w}_{k}}\mid^{2}+\sigma_{i}^{2}}\geq\Gamma_{i}^{DL},\forall
i,\\\
&A2:\frac{{P_{j}\mid{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\mid}^{2}}{I_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j\end{split}$ (9)

where $R_{a}$ and $R_{a}^{*}$ denote the objective value and the optimal
objective value of the a-th optimisation problem, respectively. The variable
${\lambda_{a}}\geq 0$, $\sum{\lambda_{a}}=1$, specifies the priority given to
the a-th objective i.e. for a given ${\lambda_{1}}=0.8$ means $80\%$ priority
is given to the objective of problem $\mathcal{P}1$ and $20\%$ priority to the
objective of problem $\mathcal{P}2$. By varying ${\lambda_{a}}$ we can obtain
the complete Pareto optimal set. Problem $\mathcal{P}3$ is a non-convex
problem due to the SINR constraints A1 and A2, and it is commonly solved via
semidefinite relaxation as in [38, 39].

$\begin{split}\mathcal{P}4:\quad\underset{{\textbf{w}_{k}},{P_{j}}}{\text{min}}\quad&{\left\|\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{1})}}\right\|}^{2}\\\
\text{s.t.}\quad&B1:\left|Im\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)\right|\leq\left(Re\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)-{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\right)\tan\theta,\forall
i,\\\
&B2:\frac{{P_{j}\left|{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\right|}^{2}}{\sum_{n\neq
j}^{J}P_{n}\left|{\textbf{f}_{n}^{H}}{\textbf{u}_{j}}\right|^{2}+\sum_{k=1}^{K}\left|{\textbf{u}_{j}^{H}}{\textbf{G}}{\textbf{w}_{k}{e^{j({\phi}_{k}-{\phi}_{1})}}}\right|^{2}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j\end{split}$ (16)

##  IV Power Minimization Problem based on Constructive Interference

In this section, we study the PM optimization problems based on constructive
interference. With prior knowledge of the CSI and users’ data symbols for the
downlink users, the instantaneous interference can be exploited rather than
suppressed [32]. To be precise, constructive interference is the interference
that pushes the received signal further into the detection region of the
constellation and away from the detection threshold [32]. This concept has
been thoroughly studied in the literature for both PSK and QAM modulation. We
refer the reader to [26, 27, 28, 29, 30, 31, 32, 33] for further details of
this topic. Motivated by this idea, here, we apply this concept to the PM
problems in Section III for both PSK and QAM modulations. We note that
constructive interference is only applied to the downlink users and not the
uplink users following that only the prior knowledge of the CSI and users’
data symbols for the downlink users are available at the BS. Nevertheless, we
show in the following that power savings can be obtained for both uplink and
downlink transmission, by means of the MOOP design.

###  IV-A Constructive Interference for PSK modulation

To illustrate this concept, we provide a geometric illustration of the
constructive interference regions for a QPSK constellation in Fig. 2. We can
define the total transmit signal vector as

$\displaystyle\sum_{k=1}^{K}{\textbf{w}_{k}}d_{k}=\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}d_{i}$
(10)

where $d_{i}=de^{{\phi}_{i}}$ is the desired symbol for the i-th downlink
user. Therefore, the received signal (2) at the i-th downlink user can be
redefined as

$\displaystyle y_{i}$
$\displaystyle={\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}d_{k}+m_{i}$
(11)
$\displaystyle={\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}d_{i}+m_{i}$
(12)

Accordingly, since the interference contributes constructively to the received
signal, it has been shown in [14] that the downlink SNR at the i-th downlink
user (4) can be rewritten as

$\displaystyle SNR_{i}^{DL}$
$\displaystyle=\frac{{\left|{\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}d_{k}\right|}^{2}}{\sigma_{i}^{2}}$
(13)

Without loss of generality, by taking user 1 as reference the instantaneous
transmit power for a unit symbol is

$\displaystyle
P_{total}={\left\|\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{1})}}\right\|}^{2}$
(14)

As detailed in [32], the shaded area in Fig. 2 is the region of constructive
interference. If the recieved signal $y_{i}$ falls within this region, then
interference has been exploited constructively. The angle
$\theta=\pm\frac{\pi}{M}$ determines the maximum angle shift of the
constructive interference region for a modulation order $M$, $a_{I}$ and
$a_{R}$ are the imaginary and real parts of the received signal $y_{i}$
without the noise, respectively. The detection threshold is determined by
$\gamma=\sqrt{\Gamma_{i}^{DL}\sigma_{i}}$.

Therefore, by applying these definitions and basic geometry from Fig. 2 it can
be seen that for the received signal to fall in the constructive region of the
constellation we need to have $a_{I}\leq(a_{R}-\gamma)\tan\theta$.
Accordingly, we can define the downlink SINR constraint that guarantees
constructive interference at the i-th downlink user by

$\left|Im\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)\right|\leq\\\
\left(Re\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)-{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\right)\tan\theta$
(15)

Figure 2: Constructive interference region for a QPSK constellation point

####  IV-A1 Total Downlink Transmit PM Problem

Based on the analysis above, we can modify the SINR constraints for the
downlink users to accommodate CI. The optimisation problem for the total
downlink transmit PM is expressed in $\mathcal{P}4$. The total minimum
downlink transmit power is minimised subject to constraint B1, which
guarantees constructive interference for the downlink users for minimum
required SINR $\Gamma_{i}^{DL}$ while the constraint B2 guarantees that the
uplink users their minimum required SINR $\Gamma_{j}^{UL}$. Unlike its
conventional counterpart $\mathcal{P}1$, it can be seen that $\mathcal{P}4$ is
convex due to the substitution of the conventional downlink SINR constraint
with the CI SNR constraints and can be tackled with standard solvers.

####  IV-A2 Total Uplink Transmit PM Problem

On the other hand, we formulate the uplink transmit PM problem by minimising
the total uplink transmit power with no regards to the downlink transmit
power.

$\begin{split}\mathcal{P}5:\,\underset{{\textbf{w}_{i}},P_{j}}{\text{min}}\,&\sum_{j=1}^{J}{P_{j}}\\\
\text{s.t.}\quad&B1:\left|Im\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)\right|\\\
&\leq\left(Re\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)-{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\right)\tan\theta,\forall
i,\\\
&B2:\frac{{P_{j}\left|{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\right|}^{2}}{I^{PSK}_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j\end{split}$ (17)

where, $I^{PSK}_{j}=\sum_{n\neq
j}^{J}P_{n}\left|{\textbf{f}_{n}^{H}}{\textbf{u}_{j}}\right|^{2}+\sum_{k=1}^{K}\left|{\textbf{u}_{j}^{H}}{\textbf{G}}{\textbf{w}_{k}{e^{j({\phi}_{k}-{\phi}_{1})}}}\right|^{2}$.

Again, it can be seen that the above problem is convex and can be tackled with
standard solvers.

####  IV-A3 Multi-objective PM Problem

By adapting the downlink SINR constraints in $\mathcal{P}2$, we can further
obtain the MOOP for interference exploitation in the FD scenario under study
as

$\begin{split}\mathcal{P}6:\,\underset{{\textbf{w}_{i}},P_{j},t}{\text{min}}\quad&t\\\
\text{s.t.}\quad&B1:\left|Im\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)\right|\\\
&\leq\left(Re\left({\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{i})}}\right)-{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\right)\tan\theta,\forall
i,\\\
&B2:\frac{{P_{j}\left|{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\right|}^{2}}{I^{PSK}_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j,\\\
&B3:{\lambda_{a}}\left(R_{a}^{*}-R_{a}({\textbf{w}_{i}},P_{j})\right)\leq
t,\forall a\in\left\\{1,2\right\\}.\end{split}$ (18)

where $t$ is an auxiliary variable.

It can be observed that, due to the substitution of the conventional downlink
SINR constraint with the CI SNR constraints, this formulation unlike the
conventional problem in $\mathcal{P}3$ is convex and thus can be optimally
solved using standard convex softwares like CVX [44].

###  IV-B Constructive Interference for QAM modulation

Figure 3: Schematic representation of 16QAM constellation points

To illustrate the concept of constructive interference for QAM modulation we
provide a schematic representation for 16QAM constellation points in Fig. 3.
Based on [31], to guarantee constructive interference for the constellation
points, we rewrite the SINR constraints for the downlink users to exploit the
specific detection regions for each group of constellation points separately
as detailed below. First, we redefine the received signal without noise (12)
and the instantaneous transmit power (14) in terms of amplitude not phase as

$\displaystyle
y_{i}={\textbf{h}_{i}^{H}}\sum_{k=1}^{K}{\textbf{w}_{k}}d_{k},\forall i$ (19)

and,

$\displaystyle
P_{total}={\left\|\sum_{k=1}^{K}{\textbf{w}_{k}}d_{k}\right\|}^{2}$ (20)

From Fig. 3, to ensure constructive interference is achieved and the
constellation points are received in the correct detection region for the
downlink users, the following constraints are adopted. Note that the dotted
lines in Fig. 3 represent the decision boundaries.

  * •

For the group of constellation points in the box labelled ”1” in Fig. 3, since
they are all surrounded by the decision boundaries, the constraints should
guarantee that the received signals achieve the exact constellation point so
as not to exceed the decision boundaries. The constraints are

$\displaystyle C1:$ $\displaystyle
Re(y_{i})={\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Re(d_{i})$ $\displaystyle C2:$
$\displaystyle Im(y_{i})={\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Im(d_{i})$

  * •

For the group of constellation points labelled ”2” in Fig. 3, the constraints
should guarantee that the received signals fall in the detection region away
from the decision boundaries, which is the real axis. The constraints are

$\displaystyle C1:$ $\displaystyle
Re(y_{i})={\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Re(d_{i})$ $\displaystyle C2:$
$\displaystyle Im(y_{i})\gtreqless{\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Im(d_{i})$

  * •

For the group of constellation points labelled ”3” in Fig. 3, the constraints
should guarantee that the received signals fall in the detection region away
from the decision boundaries, which is the imaginary axis. The constraints are

$\displaystyle C1:$ $\displaystyle
Re(y_{i})\gtreqless{\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Re(d_{i})$ $\displaystyle
C2:$ $\displaystyle Im(y_{i})={\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Im(d_{i})$

  * •

For the group of constellation points labelled ”4” in Fig. 3, the constraints
should guarantee that the received signals fall in the detection region away
from the decision boundaries. Here, the constellation points are not
surrounded by the decision boundaries and therefore have a larger detection
region that extend infinitely. The constraints are

$\displaystyle C1:$ $\displaystyle
Re(y_{i})\gtreqless{\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Re(d_{i})$ $\displaystyle
C2:$ $\displaystyle
Im(y_{i})\gtreqless{\sqrt{\Gamma_{i}^{DL}}}\sigma_{i}Im(d_{i})$

####  IV-B1 Total Downlink Transmit PM Problem

By adopting the required constraints C1 and C2 for the corresponding group
constellation points, the total downlink transmit PM optimisation problem is
expressed as

$\begin{split}\mathcal{P}7:\,\underset{{\textbf{w}_{k}},P_{j}}{\text{min}}\quad&{\left\|\sum_{k=1}^{K}{\textbf{w}_{k}}d_{k}\right\|}^{2}\\\
\text{s.t.}\quad&\textrm{Constraints C1 and C2,}\forall i,\\\
&C3:\frac{{P_{j}\mid{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\mid}^{2}}{I^{QAM}_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j.\end{split}$ (21)

where $I^{QAM}_{j}=\sum_{n\neq
j}^{J}P_{n}\mid{\textbf{f}_{n}^{H}}{\textbf{u}_{j}}\mid^{2}+\sum_{k=1}^{K}\mid{\textbf{u}_{j}^{H}}{\textbf{G}}{\textbf{w}_{k}{d_{k}}}\mid^{2}$.

####  IV-B2 Total Uplink Transmit PM Problem

Similarly, the uplink PM problem can be written for the case of QAM as

$\begin{split}\mathcal{P}8:\,\underset{{\textbf{w}_{i}},P_{j}}{\text{min}}\quad&\sum_{j=1}^{J}{P_{j}}\\\
\text{s.t.}\quad&\textrm{Constraints C1 and C2,}\forall i,\\\
&C3:\frac{{P_{j}\mid{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\mid}^{2}}{I^{QAM}_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j.\end{split}$ (22)

####  IV-B3 Multi-objective PM Problem

Finally, we can design the MOOP for the case of QAM by employing the above
constraints C1 and C2 as

$\begin{split}\mathcal{P}9:\,\underset{{\textbf{w}_{i}},P_{j},t}{\text{min}}\quad&t\\\
\text{s.t.}\quad&\textrm{Constraints C1 and C2,}\forall i,\\\
&C3:\frac{{P_{j}\mid{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\mid}^{2}}{I^{QAM}_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j,\\\
&C4:{\lambda_{a}}\left(R_{a}^{*}-R_{a}({\textbf{w}_{i}},P_{j})\right)\leq
t,\forall a\in\left\\{1,2\right\\}.\end{split}$ (23)

Again, it can be observed that unlike their conventional counterparts, all
three optimizations above are convex and can be optimally solved using
standard convex softwares like CVX [44].

##  V Multi-objective Optimization Problem with Imperfect CSI

###  V-A Conventional Robust MOOP

In this section we study the robustness of the system when the downlink, the
uplink and the SI CSI is not perfectly known. For each channel, the actual CSI
is modeled as

$\displaystyle{\textbf{h}}_{i}={\check{\textbf{h}}}_{i}+{\Delta{\textbf{h}}}_{i},\forall
i,$ (24)
$\displaystyle{\textbf{f}}_{j}={\check{\textbf{f}}}_{j}+{\Delta{\textbf{f}}}_{j},\forall
j,$ (25)

and,

$\displaystyle{\textbf{G}}={\check{\textbf{G}}}+{\Delta{\textbf{G}}}.$ (26)

where ${\check{\textbf{h}}}_{i},{\check{\textbf{f}}}_{j}$ and
${\check{\textbf{G}}}$ denote the downlink, the uplink and the SI CSI
estimates known to the FD BS, respectively. And
${\Delta{\textbf{h}}}_{i},\forall i{\Delta{\textbf{f}}}_{j},\forall j$ and
${\Delta{\textbf{G}}}$ represent the downlink, the uplink and the SI CSI
uncertainties, respectively, which are assumed to be bounded such that

$\displaystyle\left\|{\Delta{\textbf{h}}}_{i}\right\|^{2}\leq\epsilon_{h,i}^{2},\,\textrm{for
some}\,\epsilon_{h,i}\geq 0,$ (27)
$\displaystyle\left\|{\Delta{\textbf{f}}}_{j}\right\|^{2}\leq\epsilon_{f,j}^{2},\,\textrm{for
some}\,\epsilon_{f,i}\geq 0,$ (28)
$\displaystyle\left\|{\Delta{\textbf{G}}}\right\|^{2}\leq\epsilon_{G}^{2},\,\textrm{for
some}\,\epsilon_{G}\geq 0.$ (29)

We assume that the FD BS has no knowledge of
${\Delta{\textbf{h}}}_{i},{\Delta{\textbf{f}}}_{j}$ and ${\Delta{\textbf{G}}}$
except for the error bounds, hence, we take the worst-case approach for our
problem design.

Henceforth, we focus on the multi-objective problem formulation since it is a
generalisation of both the downlink and uplink optimisation problems.
Therefore, the multi-object formulation of problem $\mathcal{P}3$ for
imperfect CSI is

$\begin{split}\mathcal{P}10:\quad\underset{{\textbf{w}_{i}},P_{j},t}{\text{min}}\quad&t\\\
\text{s.t.}\quad&\frac{{\left|\left(\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i}\right)^{H}{\textbf{w}_{i}}\right|}^{2}}{\sum_{k\neq
i}^{K}{\left|\left(\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i}\right)^{H}{\textbf{w}_{k}}\right|}^{2}+\sigma_{i}^{2}}\geq\Gamma_{i}^{DL},\\\
&\forall\left\|{\Delta{\textbf{h}}}_{i}\right\|^{2}\leq\epsilon_{h,i}^{2},\forall
i,\\\
&\frac{{P_{j}\left|\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)^{H}{\textbf{u}_{j}}\right|}^{2}}{\sum_{n\neq
j}^{J}P_{n}\left|\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)^{H}{\textbf{u}_{j}}\right|^{2}+C_{j}}\geq\Gamma_{j}^{UL},\\\
&\forall\left\|{\Delta{\textbf{G}}}\right\|^{2}\leq\epsilon_{G}^{2},\forall\left\|{\Delta{\textbf{f}}}_{j}\right\|^{2}\leq\epsilon_{f,j}^{2},\forall
j,\\\ &{\lambda_{a}}\left(R_{a}^{*}-R_{a}\right)\leq t,\forall
a\in\left\\{1,2\right\\}.\end{split}$ (30)

where
$C_{j}=\sum_{k=1}^{K}\left|{\textbf{u}_{j}^{H}}\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)\textbf{w}_{k}\right|^{2}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}$

In the downlink and uplink SINR constraints, there are infinitely many
inequalities which make the worst-case design particularly challenging. To
make $\mathcal{P}10$ more tractable, we formulate the problem as a
semidefinite program (SDP) then transform the constraints into linear matrix
inequalities (LMI), which can be efficiently solved by existing solvers like
CVX [44]. The SDP transformation of problem $\mathcal{P}10$ is

$\begin{split}&\underset{{\textbf{W}_{i}},P_{j},t}{\text{min}}\quad t\\\
\text{s.t.}\quad&\widetilde{\textrm{D1}}:\frac{\left(\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i}\right)^{H}{\textbf{W}_{i}}\left(\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i}\right)}{\sum_{k\neq
i}^{K}\left((\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i})^{H}{\textbf{W}_{k}}(\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i})\right)+\sigma_{i}^{2}}\geq\Gamma_{i}^{DL},\\\
&\forall\left\|{\Delta{\textbf{h}}}_{i}\right\|^{2}\leq\epsilon_{h,i}^{2},\forall
i,\\\
&\widetilde{\textrm{D2}}:\frac{P_{j}\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)^{H}{\textbf{U}_{j}}\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)}{\sum_{n\neq
j}^{J}P_{n}\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)^{H}{\textbf{U}_{j}}\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)+\widetilde{C}_{j}}\geq\Gamma_{j}^{UL},\\\
&\forall\left\|{\Delta{\textbf{G}}}\right\|^{2}\leq\epsilon_{G}^{2},\forall\left\|{\Delta{\textbf{f}}}_{j}\right\|^{2}\leq\epsilon_{f,j}^{2},\forall
j,\\\ &\widetilde{\textrm{D3}}:{\lambda_{a}}\left(R_{a}^{*}-R_{a}\right)\leq
t,\forall a\in\left\\{1,2\right\\}.\\\
&\widetilde{\textrm{D4}}:{\textbf{W}_{i}}\succeq 0,\forall i.\end{split}$ (31)

where,
$\widetilde{C}_{j}=\textrm{Tr}\left\\{\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)\sum_{k=1}^{K}{\textbf{W}_{k}}\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)^{H}{\textbf{U}}_{j}\right\\}+\sigma_{N}^{2}\textrm{Tr}\left\\{{\textbf{U}}_{j}\right\\}$
and we define $\textbf{W}_{i}=\textbf{w}_{i}\textbf{w}_{i}^{H}$ and
$\textbf{U}_{j}=\textbf{u}_{j}\textbf{u}_{j}^{H}$.

By applying the S-procedure as in [45] we can convert these constraints into
LMIs. First, we can rearrange constraint $\widetilde{\textrm{D1}}$ into

$\displaystyle\left(\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i}\right)^{H}{\textbf{Q}_{i}}\left(\check{\textbf{h}}_{i}+{\Delta{\textbf{h}}}_{i}\right)-\Gamma_{i}^{DL}\sigma_{i}^{2}\geq
0,\forall i,$ (32)

where, we introduce

$\displaystyle{\textbf{Q}_{i}}\triangleq{\textbf{W}_{i}}-\Gamma_{i}^{DL}\sum_{k\neq
i}^{K}{\textbf{W}_{k}},\forall i$

and then for constraint $\widetilde{\textrm{D2}}$, let’s define two vectors
$\widetilde{\textbf{f}}$ and $\widetilde{\Delta{\textbf{f}}}$ as

$\displaystyle\widetilde{\textbf{f}}={\begin{bmatrix}\check{\textbf{f}}_{j}\\\
\vdots\\\ \check{\textbf{f}}_{J}\end{bmatrix}}\in\mathbb{C}^{NJ\times
1},\widetilde{\Delta{\textbf{f}}}={\begin{bmatrix}{\Delta{\textbf{f}}}_{j}\\\
\vdots\\\ {\Delta{\textbf{f}}}_{J}\end{bmatrix}}\in\mathbb{C}^{NJ\times 1}$
(33)

hence, we can define any
$\check{\textbf{f}}_{j}=\textbf{B}_{j}\widetilde{\textbf{f}}$ and
${\Delta{\textbf{f}}}_{j}=\textbf{B}_{j}\widetilde{\Delta{\textbf{f}}},$ for
$j=1,\ldots,J,$ with $\textbf{B}_{j}\in\mathbb{R}^{N\times NJ}$ defined as
$\textbf{B}_{j}={\begin{bmatrix}\textbf{B}_{j,1},\ldots,\textbf{B}_{j,J}\end{bmatrix}},$
where $\textbf{B}_{j,j}=\textbf{I}_{N}$ and $\textbf{B}_{j,n}=\textbf{0}_{N},$
for $n=1,\ldots,J,n\neq j.$ We have $\textbf{I}_{N}$ and $\textbf{0}_{N}$ to
be an $N\times N$ identity matrix and zero matrix, respectively.

Now constraint $\widetilde{\textrm{D2}}$ can be rewritten as

$\displaystyle\frac{P_{j}\left((\textbf{B}_{j}\widetilde{\textbf{f}}+\textbf{B}_{j}\widetilde{\Delta{\textbf{f}}})^{H}{\textbf{U}_{j}}(\textbf{B}_{j}\widetilde{\textbf{f}}+\textbf{B}_{j}\widetilde{\Delta{\textbf{f}}})\right)}{\sum_{n\neq
j}^{J}P_{n}\left((\textbf{B}_{n}\widetilde{\textbf{f}}+\textbf{B}_{n}\widetilde{\Delta{\textbf{f}}})^{H}{\textbf{U}_{j}}(\textbf{B}_{n}\widetilde{\textbf{f}}+\textbf{B}_{n}\widetilde{\Delta{\textbf{f}}})\right)+\widetilde{C}_{j}}\geq\Gamma_{j}^{UL},\forall
j$ (34)

and can be simplified to give

$\displaystyle\frac{\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)^{H}{\textbf{Z}_{j}}\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)}{\textrm{Tr}\left\\{\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)\sum_{k=1}^{K}{\textbf{W}_{k}}\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)^{H}{\textbf{U}}_{j}\right\\}+\sigma_{N}^{2}\textrm{Tr}\left\\{{\textbf{U}}_{j}\right\\}}\geq\Gamma_{j}^{UL}$
(35)

where we introduce

$\displaystyle{\textbf{Z}_{j}}\triangleq
P_{j}\textbf{B}_{j}^{T}{\textbf{U}_{j}}\textbf{B}_{j}-\Gamma_{j}^{UL}\sum_{n\neq
j}^{J}P_{n}\textbf{B}_{n}^{T}{\textbf{U}_{j}}\textbf{B}_{n},\forall j$

we further simplify (35) by introducing slack variables $s_{j}>0,\forall{j}$
[45], such that (35) can be written as the following two constraints

$\displaystyle\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)^{H}{\textbf{Z}_{j}}\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)\geq
s_{j}\Gamma_{j}^{UL},\forall{j},$ (36)
$\displaystyle\textrm{Tr}\left\\{\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)\sum_{k=1}^{K}{\textbf{W}_{k}}\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)^{H}{\textbf{U}}_{j}\right\\}+\sigma_{N}^{2}\textrm{Tr}\left\\{{\textbf{U}}_{j}\right\\}\leq
s_{j},\forall j.$ (37)

Next, we review the definitions of the S-procedure.

Lemma 1. (S-procedure [45]): Let $g_{l}(\textbf{x}),$ $l=1,2,$ be defined as

$\displaystyle
g_{l}(\textbf{x})=\textbf{x}^{H}\textbf{A}_{l}\textbf{x}+2Re\left\\{\textbf{b}_{l}^{H}\textbf{x}\right\\}+c_{l}$

where $\textbf{A}_{l}\in\mathbb{C}^{n\times
n},\textbf{b}_{l}\in\mathbb{C}^{n}$ and $c_{l}\in\mathbb{R}$. Then, the
implication of $g_{1}(\textbf{x})\geq 0\Rightarrow g_{2}(\textbf{x})\geq 0$
holds if and only if there exists a $\lambda\geq 0$ such that

$\displaystyle\delta{\begin{bmatrix}\textbf{A}_{1}&\textbf{b}_{1}\\\
\textbf{b}_{1}^{H}&c_{1}\end{bmatrix}}-{\begin{bmatrix}\textbf{A}_{2}&\textbf{b}_{2}\\\
\textbf{b}_{2}^{H}&c_{2}\end{bmatrix}}\succeq 0$

provided there exists a point $\hat{\textbf{x}}$ with
$g_{1}(\hat{\textbf{x}})>0.$

Following Lemma 1 and using the fact that
$\textrm{Tr}\left\\{\textbf{ABCD}\right\\}=\textrm{vec}\left(\textbf{A}^{H}\right)^{H}\left(\textbf{D}^{H}\otimes\textbf{B}\right)\textrm{vec}\left(\textbf{C}\right)$,
constraints (32), (36) and (37) can be expanded as

$\displaystyle{\Delta{\textbf{h}}}_{i}^{H}{\textbf{Q}_{i}}{\Delta{\textbf{h}}}_{i}+2Re\left\\{\check{\textbf{h}}_{i}^{H}{\textbf{Q}_{i}}{\Delta{\textbf{h}}}_{i}\right\\}+\check{\textbf{h}}_{i}^{H}{\textbf{Q}_{i}}\check{\textbf{h}}_{i}-\Gamma_{i}^{DL}\sigma_{i}^{2}\geq
0,\forall i$ (38)
$\displaystyle\widetilde{\Delta{\textbf{f}}}^{H}{\textbf{Z}_{j}}\widetilde{\Delta{\textbf{f}}}+2Re\left\\{\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}\widetilde{\Delta{\textbf{f}}}\right\\}+\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}\widetilde{\textbf{f}}-s_{j}\Gamma_{j}^{UL}\geq
0,\forall j,$ (39)
${\Delta{\textbf{g}}}^{H}\left({\textbf{U}}_{j}\otimes\sum_{k=1}^{K}{\textbf{W}_{k}}\right){\Delta{\textbf{g}}}+2Re\left\\{\check{\textbf{g}}^{H}\left({\textbf{U}}_{j}\otimes\sum_{k=1}^{K}{\textbf{W}_{k}}\right){\Delta{\textbf{g}}}\right\\}\\\
+\check{\textbf{g}}^{H}\left({\textbf{U}}_{j}\otimes\sum_{k=1}^{K}{\textbf{W}_{k}}\right)\check{\textbf{g}}+\sigma_{N}^{2}\textrm{Tr}\left\\{{\textbf{U}}_{j}\right\\}-s_{j}\leq
0,\forall j$ (40)

$\begin{split}{\mathcal{P}11}:\quad\underset{{\textbf{W}_{i}},P_{j},t}{\text{min}}\quad&t\\\
\text{s.t.}\quad&{\begin{bmatrix}{\delta_{i}{\textbf{I}}+\textbf{Q}_{i}}&{\textbf{Q}_{i}}\check{\textbf{h}}_{i}\\\
\check{\textbf{h}}_{i}^{H}{\textbf{Q}_{i}}&\check{\textbf{h}}_{i}^{H}{\textbf{Q}_{i}}\check{\textbf{h}}_{i}-\Gamma_{i}^{DL}\sigma_{i}^{2}-\delta_{i}\epsilon_{h,i}^{2}\end{bmatrix}}\succeq
0,\forall i,\\\
&{\begin{bmatrix}{\mu_{j}{\textbf{I}}+\textbf{Z}_{j}}&{\textbf{Z}_{j}}\widetilde{\textbf{f}}\\\
\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}&\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}\widetilde{\textbf{f}}-s_{j}\Gamma_{j}^{UL}-\mu_{j}\epsilon_{f,j}^{2}\end{bmatrix}}\succeq
0,\forall j,\\\
&{\begin{bmatrix}{\rho{\textbf{I}}-\left({\textbf{U}}_{j}\otimes\sum_{k=1}^{K}{\textbf{W}_{k}}\right)}&-\left({\textbf{U}}_{j}\otimes\sum_{k=1}^{K}{\textbf{W}_{k}}\right)\check{\textbf{g}}\\\
-\check{\textbf{g}}^{H}\left({\textbf{U}}_{j}\otimes\sum_{k=1}^{K}{\textbf{W}_{k}}\right)&s_{j}-\check{\textbf{g}}^{H}\left({\textbf{U}}_{j}\otimes\sum_{k=1}^{K}{\textbf{W}_{k}}\right)\check{\textbf{g}}-\sigma_{N}^{2}\textrm{Tr}\left\\{{\textbf{U}}_{j}\right\\}-\rho\epsilon_{G}^{2}\end{bmatrix}}\succeq
0,\forall j,\\\ &{\lambda_{a}}\left(R_{a}^{*}-R_{a}\right)\leq t,\forall
a\in\left\\{1,2\right\\},\\\ &{\textbf{W}_{i}}\succeq 0,\delta_{i}\geq
0,\mu_{j}\geq 0,\rho\geq 0,s_{j}>0,\forall i,j.\end{split}$ (41)

we define
${\check{\textbf{g}}}=\textrm{vec}\left({\check{\textbf{G}}}^{H}\right)$ and
${\Delta{\textbf{g}}}=\textrm{vec}\left({\Delta{\textbf{G}}}^{H}\right)$
where, $\textrm{vec}\left(\cdot\right)$ stacks the columns of a matrix into a
vector and $\otimes$ stands for Kronecker product.

Hence, by exploiting the S-procedure in Lemma 1, (38), (39) and (40) can be
formulated as LMIs and the conventional robust optimisation problem
$\mathcal{P}10$ can be reformulated as shown in (41).

The problem ${\mathcal{P}11}$ is convex, and can be efficiently solved using
CVX [44]. The resulting optimal values obtained from ${\mathcal{P}11}$ provide
a lower bound for the conventional power minimisation problem.

Note that the problem ${\mathcal{P}11}$ is a relaxed form of
${\mathcal{P}10}$. When the relaxation in ${\mathcal{P}11}$ is tight, i.e.
${\mathcal{P}11}$ returns all rank-one solutions $({\textbf{W}_{i}})$, then
the optimal solution $({\textbf{w}_{i}})$ to solve ${\mathcal{P}10}$ can be
obtained by matrix decomposition or randomisation as in [46], such that
${\textbf{W}_{i}}=\textbf{w}_{i}\textbf{w}_{i}^{H},\forall i.$ Otherwise, the
required power in original problem ${\mathcal{P}10}$ is always higher than
that in ${\mathcal{P}11}$.

###  V-B Robust MOOP based on Constructive Interference

To study the robustness of the proposed system for the case of constructive
interference, we first formulate $\mathcal{P}6$ as a virtual multicast problem
[47]. To facilitate this, we simply incorporate each user’s channel with its
respective data symbol i.e.
${\widetilde{\textbf{h}}_{i}}={\textbf{h}_{i}}{e^{j({\phi}_{1}-{\phi}_{i})}}$
and let
$\textbf{w}=\sum_{k=1}^{K}{\textbf{w}_{k}}{e^{j({\phi}_{k}-{\phi}_{1})}}$.
Following this the multicast formulation of problem $\mathcal{P}6$ can be
written as

$\begin{split}{\mathcal{P}12}:\quad&\underset{\textbf{w},P_{j},t}{\text{min}}\quad
t\\\
\text{s.t.}\quad&\left|Im\left({{\widetilde{\textbf{h}}_{i}}^{H}}\textbf{w}\right)\right|\leq\left(Re\left({{\widetilde{\textbf{h}}_{i}}^{H}}\textbf{w}\right)-{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\right)\tan\theta,\forall
i,\\\
&\frac{{P_{j}\left|{\textbf{f}_{j}^{H}}{\textbf{u}_{j}}\right|}^{2}}{\sum_{n\neq
j}^{J}P_{n}\left|{\textbf{f}_{n}^{H}}{\textbf{u}_{j}}\right|^{2}+\left|{\textbf{u}_{j}^{H}}{\textbf{G}}\textbf{w}\right|^{2}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\forall
j,\\\ &{\lambda_{a}}\left(R_{a}^{*}-R_{a}\right)\leq t,\forall
a\in\left\\{1,2\right\\}.\end{split}$ (42)

Based on the multicast formulation ${\mathcal{P}12}$, for the worst-case
design we model the imperfect CSI as

$\displaystyle\widetilde{\textbf{h}}_{i}={\widetilde{\textbf{h}}}_{i}+\Delta\widetilde{\textbf{h}}_{i},\forall
i$ (43)

where ${\widetilde{\textbf{h}}}_{i}$ denotes the downlink CSI estimate known
to the FD BS. And ${\Delta\widetilde{\textbf{h}}}_{i}$ is the downlink CSI
uncertainty which is bounded such that
$\left\|{\Delta\widetilde{\textbf{h}}}_{i}\right\|^{2}\leq\epsilon_{h,i}^{2}$.
And we model the uplink and the SI CSI as in Section V-A, respectively. The
robust formulation of problem ${\mathcal{P}12}$ is

$\begin{split}{\mathcal{P}13}:\,\underset{\textbf{w},P_{j},t}{\text{min}}\quad&t\\\
\text{s.t.}\quad&\left|Im\left(({\widetilde{\textbf{h}}}_{i}+\Delta\widetilde{\textbf{h}}_{i})^{H}\textbf{w}\right)\right|\\\
&\leq\left(Re\left(({\widetilde{\textbf{h}}}_{i}+\Delta\widetilde{\textbf{h}}_{i})^{H}\textbf{w}\right)-{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\right)\tan\theta,\\\
&\forall\left\|{\Delta{\textbf{h}}}_{i}\right\|^{2}\leq\epsilon_{h,i}^{2},\forall
i,\\\
&\frac{{P_{j}\left|\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)^{H}{\textbf{u}_{j}}\right|}^{2}}{\sum_{n\neq
j}^{J}P_{n}\left|\left(\check{\textbf{f}}_{j}+{\Delta{\textbf{f}}}_{j}\right)^{H}{\textbf{u}_{j}}\right|^{2}+I_{j}}\geq\Gamma_{j}^{UL},\\\
&\forall\left\|{\Delta{\textbf{G}}}\right\|^{2}\leq\epsilon_{G}^{2},\forall\left\|{\Delta{\textbf{f}}}_{j}\right\|^{2}\leq\epsilon_{f,j}^{2},\forall
j,\\\ &{\lambda_{a}}\left(R_{a}^{*}-R_{a}\right)\leq t,\forall
a\in\left\\{1,2\right\\}.\end{split}$ (44)

where
$I_{j}=\left|{\textbf{u}_{j}^{H}}\left(\check{\textbf{G}}+{\Delta{\textbf{G}}}\right)\textbf{w}\right|^{2}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}$.

First, let’s consider the downlink SINR constraint. For convenience we
separate the real and imaginary part of the complex notations and represent
them as real valued numbers. Let

$\displaystyle{\underline{\textbf{w}}}$
$\displaystyle\triangleq{\begin{bmatrix}Re({\textbf{w}})\\\
Im({\textbf{w}})\end{bmatrix}},$ (45)
$\displaystyle{\underline{\widetilde{\textbf{h}}}}_{i}$
$\displaystyle\triangleq{\begin{bmatrix}Im({\widetilde{\textbf{h}}_{i}})^{H}&Re({\widetilde{\textbf{h}}_{i}})^{H}\end{bmatrix}},$
(46) $\displaystyle{\Delta\underline{\widetilde{\textbf{h}}}}_{i}$
$\displaystyle\triangleq{\begin{bmatrix}Im(\Delta{\widetilde{\textbf{h}}_{i}})^{H}&Re(\Delta{\widetilde{\textbf{h}}_{i}})^{H}\end{bmatrix}},$
(47) $\displaystyle{\boldsymbol{\Pi}}$
$\displaystyle\triangleq{\begin{bmatrix}{\textbf{0}}_{N}&-{\textbf{I}}_{N}\\\
{\textbf{I}}_{N}&{\textbf{0}}_{N}\end{bmatrix}}.$ (48)

Where, ${\textbf{0}}_{N}$ and ${\textbf{I}}_{N}$ denote $N$ x $N$ all-zero
matrix and identity matrix, respectively.

With the new notations we can express the real and imaginary terms of downlink
SINR constraint in ${\mathcal{P}13}$ as:

$\displaystyle\textrm{Im}({\widetilde{\textbf{h}}_{i}^{H}}{\textbf{w}})=({\underline{\widetilde{\textbf{h}}}}_{i}+{\Delta\underline{\widetilde{\textbf{h}}}}_{i}){\underline{\textbf{w}}},$
$\displaystyle\textrm{Re}({\widetilde{\textbf{h}}_{i}^{H}}{\textbf{w}})=({\underline{\widetilde{\textbf{h}}}}_{i}+{\Delta\underline{\widetilde{\textbf{h}}}}_{i}){\boldsymbol{\Pi}}{\underline{\textbf{w}}}$
(49)

From the definition of the error bound, we have
$\left\|{\Delta\widetilde{\textbf{h}}}_{i}\right\|^{2}\leq\epsilon_{h,i}^{2}$,
the downlink SINR constraint can be guaranteed by the following constraint

$\underset{\lVert{\Delta\widetilde{\textbf{h}}}_{i}\rVert^{2}\leq\epsilon_{h,i}^{2}}{\text{max}}\quad\left|\left({\underline{\widetilde{\textbf{h}}}}_{i}+{\Delta\underline{\widetilde{\textbf{h}}}}_{i}\right){\underline{\textbf{w}}}\right|\\\
-\left(\left({\underline{\widetilde{\textbf{h}}}}_{i}+{\Delta\underline{\widetilde{\textbf{h}}}}_{i}\right){\boldsymbol{\Pi}}{\underline{\textbf{w}}}-{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\right)\tan\theta\leq
0,\forall i$ (50)

Hence, by considering the absolute value term, (50) is equivalent to the
following two constraints

$\underset{\lVert{\Delta\widetilde{\textbf{h}}}_{i}\rVert^{2}\leq\epsilon_{h,i}^{2}}{\text{max}}\quad{\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}+{\Delta\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}-\left({\underline{\widetilde{\textbf{h}}}}_{i}+{\Delta\underline{\widetilde{\textbf{h}}}}_{i}\right){\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta\\\
+{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\tan\theta\leq 0,\forall i$ (51)

$\underset{\lVert{\Delta\widetilde{\textbf{h}}}_{i}\rVert^{2}\leq\epsilon_{h,i}^{2}}{\text{max}}\quad-{\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}-{\Delta\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}-\left({\underline{\widetilde{\textbf{h}}}}_{i}+{\Delta\underline{\widetilde{\textbf{h}}}}_{i}\right){\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta\\\
+{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\tan\theta\leq 0,\forall i$ (52)

Therefore, the the robust formulations of (51) and (52) are given by

${\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}-{\underline{\widetilde{\textbf{h}}}}_{i}{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta+\epsilon_{h,i}\left\|{\underline{\textbf{w}}}-{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta\right\|\\\
+{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\tan\theta\leq 0,\forall i$ (53)
$-{\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}-{\underline{\widetilde{\textbf{h}}}}_{i}{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta+\epsilon_{h,i}\left\|{-\underline{\textbf{w}}}-{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta\right\|\\\
+{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\tan\theta\leq 0,\forall i$ (54)

Next, we consider the uplink SINR constraint in problem (44). Following
equations (33) and (34) in Section V-A, the uplink SINR constraint can be
rewritten as

$\frac{\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)^{H}{\textbf{Z}_{j}}\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)}{\left|{\textbf{u}_{j}^{H}}\check{\textbf{G}}\textbf{w}+{\textbf{u}_{j}^{H}}{\Delta{\textbf{G}}}\textbf{w}\right|^{2}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}}\geq\Gamma_{j}^{UL},\\\
\forall\left\|{\Delta{\textbf{G}}}\right\|^{2}\leq\epsilon_{G}^{2},\forall\left\|{\Delta{\textbf{f}}}_{j}\right\|^{2}\leq\epsilon_{f,j}^{2},\forall
j.$ (55)

and we note that (55) can be guaranteed by the following constraints

$\underset{\lVert{\Delta\widetilde{\textbf{f}}}_{j}\rVert^{2}\leq\epsilon_{f,j}^{2}}{\text{max}}\quad\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)^{H}{\textbf{Z}_{j}}\left(\widetilde{\textbf{f}}+\widetilde{\Delta{\textbf{f}}}\right)-\Gamma_{j}^{UL}\left({c}_{j}+\sigma_{N}^{2}{\lVert{\textbf{u}_{j}}\rVert}^{2}\right)\\\
\geq 0,\forall j$ (56)
$\displaystyle\underset{\lVert{\Delta\widetilde{\textbf{G}}}\rVert^{2}\leq\epsilon_{G}^{2}}{\text{max}}\quad\left|{\textbf{u}_{j}^{H}}\check{\textbf{G}}\textbf{w}+{\textbf{u}_{j}^{H}}{\Delta{\textbf{G}}}\textbf{w}\right|^{2}\leq
c_{j},\forall j$ (57)

where $c_{j}>0,\forall{j}$ are introduced as slack variables [45].

Similar procedure as in Section V-A can be applied to (56). By exploiting the
S-procedure in Lemma 1, (56) can be expanded and converted into a LMI as shown
below

${\begin{bmatrix}{\mu_{j}{\textbf{I}_{N}}+\textbf{Z}_{j}}&{\textbf{Z}_{j}}\widetilde{\textbf{f}}\\\
\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}&\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}\widetilde{\textbf{f}}-\Gamma_{j}^{UL}c_{j}-\Gamma_{j}^{UL}\sigma_{N}^{2}\textrm{Tr}({\textbf{U}_{j}})-\mu_{j}\epsilon_{f,j}^{2}\end{bmatrix}}\\\
\succeq 0,\forall j,$ (58)

We note that by using the fact that
$\left\|x+y\right\|^{2}\leq\left(\left\|x\right\|+\left\|y\right\|\right)^{2}$,
(57) can always be guaranteed by the following constraint

$\displaystyle\underset{\lVert{\Delta\widetilde{\textbf{G}}}\rVert^{2}\leq\epsilon_{G}^{2}}{\text{max}}\quad\left(\left|{\textbf{u}_{j}^{H}}\check{\textbf{G}}\textbf{w}\right|+\left|{\textbf{u}_{j}^{H}}{\Delta{\textbf{G}}}\textbf{w}\right|\right)^{2}\leq
c_{j},\forall{j}$ (59)

whose robust formulation is given by

$\displaystyle\left(\left|{\textbf{u}_{j}^{H}}\check{\textbf{G}}\textbf{w}\right|+\epsilon_{G}\left|{\textbf{u}_{j}^{H}}\textbf{w}\right|\right)^{2}\leq
c_{j},\forall{j}$ (60)

Futhermore, we define
${\underline{\textbf{Y}}}_{j}\triangleq{\begin{bmatrix}Re({\textbf{u}_{j}^{H}}{\textbf{G}})&-Im({\textbf{u}_{j}^{H}}{\textbf{G}})\\\
Im({\textbf{u}_{j}^{H}}{\textbf{G}})&Re({\textbf{u}_{j}^{H}}{\textbf{G}})\end{bmatrix}}$
and
${\underline{\textbf{U}}}_{j}\triangleq{\begin{bmatrix}Re({\textbf{u}_{j}^{H}})&-Im({\textbf{u}_{j}^{H}})\\\
Im({\textbf{u}_{j}^{H}})&Re({\textbf{u}_{j}^{H}})\end{bmatrix}}$, therefore,
the constraint (60) can be written in terms of real valued numbers as

$\displaystyle\left(\left|{\underline{\textbf{Y}}}_{j}{\underline{\textbf{w}}}\right|+\epsilon_{G}\left|{\underline{\textbf{U}}}_{j}{\underline{\textbf{w}}}\right|\right)^{2}\leq
c_{j},\forall{j}$ (61)

Therefore, the robust optimisation problem based on CI is

$\begin{split}{\mathcal{P}14}:&\,\underset{{\underline{\textbf{w}}},P_{j},t}{\text{min}}\quad
t\\\ \text{s.t.}\\\
&{\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}-{\underline{\widetilde{\textbf{h}}}}_{i}{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta+\epsilon_{h,i}\left\|{\underline{\textbf{w}}}-{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta\right\|\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\tan\theta,\forall
i\\\
&-{\underline{\widetilde{\textbf{h}}}}_{i}{\underline{\textbf{w}}}-{\underline{\widetilde{\textbf{h}}}}_{i}{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta+\epsilon_{h,i}\left\|{-\underline{\textbf{w}}}-{\boldsymbol{\Pi}}{\underline{\textbf{w}}}\tan\theta\right\|\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq{\sqrt{\Gamma_{i}^{DL}\sigma_{i}^{2}}}\tan\theta,\forall
i,\\\
&{\begin{bmatrix}{\mu_{j}{\textbf{I}_{N}}+\textbf{Z}_{j}}&{\textbf{Z}_{j}}\widetilde{\textbf{f}}\\\
\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}&\widetilde{\textbf{f}}^{H}{\textbf{Z}_{j}}\widetilde{\textbf{f}}-\Gamma_{j}^{UL}c_{j}-\Gamma_{j}^{UL}\sigma_{N}^{2}\textrm{Tr}({\textbf{U}_{j}})-\mu_{j}\epsilon_{f,j}^{2}\end{bmatrix}}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\succeq 0,\forall j,\\\
&\left(\left|{\underline{\textbf{Y}}}_{j}{\underline{\textbf{w}}}\right|+\epsilon_{G}\left|{\underline{\textbf{U}}}_{j}{\underline{\textbf{w}}}\right|\right)^{2}\leq
c_{j},\forall{j},\\\ &{\lambda_{a}}\left(R_{a}^{*}-R_{a}\right)\leq t,\forall
a\in\left\\{1,2\right\\},\\\ &\mu_{j}\geq 0,\,c_{j}>0,\forall j.\end{split}$
(62)

Note that problem ${\mathcal{P}14}$ is a convex problem and thus can be
optimally solved using standard convex softwares like CVX [44]. After we
obtain the optimal ${\underline{\textbf{w}}}^{*}$ and $P_{j}^{*}$, the robust
solution ${\textbf{w}}^{*}$ can be obtained from the relation in (45).

##  VI Computational Complexity Analysis

In this Section, we mathematically characterize the computational complexity
of the conventional and proposed schemes based on MOOP formulations.

###  VI-A Transmit Complexity

We note that the convex MOOP formulations
${\mathcal{P}3},{\mathcal{P}6},{\mathcal{P}11}$ and ${\mathcal{P}14}$ involve
only LMI and second-order cone (SOC) constraints. As such, the problems can be
sovled by a standard interior-point method (IPM) [48]. Therefore we can use
the worst-case runtime to analyse the complexity of the conventional and the
proposed CI schemes.

Following [49] and [50], the complexity of a generic IPM for solving problems
like ${\mathcal{P}3},{\mathcal{P}6},{\mathcal{P}11}$ and ${\mathcal{P}14}$
involve the computation of a per-optimization cost. In each iteration, the
computation cost is dominated by (i) the formation of the coefficient matrix
of the linear system, and (ii) the factorization of the coefficient matrix.
The cost of formation of the coefficient ($C_{form}$) matrix is on the order
of

$\displaystyle C_{form}=\underset{\textrm{due to the
LMI}}{\underbrace{n\sum_{a=1}^{A}k_{a}^{3}+n^{2}\sum_{a=1}^{A}k_{a}^{2}}}+\underset{\textrm{due
to the SOC}}{\underbrace{n\sum_{a=A+1}^{B}k_{a}^{2}}}$

while the cost of factorizing ($C_{fact}$) is on the order of $C_{fact}=n^{3}$
($n=$ number of decision variables). Hence, the total computation cost per
optimization is on the order of $C_{form}+C_{fact}$ [49]. We assume for the
sake of simplicity that the decision variables in
${\mathcal{P}3},{\mathcal{P}6},{\mathcal{P}11}$ and ${\mathcal{P}14}$ are
real-valued.

Hence, using these concepts, we now analyse the comutational complexity of
${\mathcal{P}3},{\mathcal{P}6},{\mathcal{P}11}$ and ${\mathcal{P}14}$. First
we consider SDP formulation of ${\mathcal{P}3}$, which has $K$ LMI (trace)
constraints of size 1, $J$ LMI (trace) constraints of size 1, $K$ SOC
constraints of size $N$, $J$ LMI (trace) constraints of size 1 and $K$ LMI
(trace) constraints of size $N$. Therefore, the complexity of the SDP
formulation of ${\mathcal{P}3}$ is on the order shown in the first row of
Table I. Similarly, we can determine the complexity order of the formulations
${\mathcal{P}6},{\mathcal{P}11}$ and ${\mathcal{P}14}$ as shown in Table I,
respectively. From Table I, we can show that the proposed MOOP formulation
${\mathcal{P}6}$ has lower complexity than the SDP formulation of
${\mathcal{P}3}$ since it has lower order of variables to compute i.e lower
cost of factorization ($C_{fact}$). Also, we can straightforwardly show that
for the robust MOOP, the proposed formulation ${\mathcal{P}14}$ has a lower
complexity than the conventional formulation ${\mathcal{P}11}$ since
${\mathcal{P}11}$ involves a more complicated set of constraints (5 LMI
constraints and 1 SOC constrsint). This is also consistent with our simulation
results in the following Section.

Table I: Complexity Analysis of the MOOP Formulations MOOP | Complexity Order
---|---
${\mathcal{P}3}$(SDP) | $\mathcal{O}((KN^{2}+J)[K(1+N^{3})+2J+(KN^{2}+J)(K(1+N^{2})$
| $+2J)+KN^{2}+(KN^{2}+J)^{2}])$
${\mathcal{P}6}$ | $\mathcal{O}((KN+J)[2J(1+(KN+J))+2KN^{2}+(KN+J)^{2}])$
${\mathcal{P}11}$ | $\mathcal{O}((KN^{2}+J)[K(N+1)^{2}+J(NJ+1)^{3}+J(N^{2}+1)^{3}$
| $+J+KN^{3}+(KN^{2}+J)(K(N+1)^{2}+J(NJ+1)^{2}$
| $+J(N^{2}+1)^{2}+J+KN^{2})+(KN^{2}+J)(KN^{2})$
| $+(KN^{2}+J)^{2}])$
${\mathcal{P}14}$ | $\mathcal{O}((2N+J)[J(NJ+1)^{3}+J(N+1)^{3}+J$
| $+(2N+J)(J(NJ+1)^{2}+J+12N^{2})+(2N+J)^{2})])$

At this point, we emphasize that as the MOOP formulations in ${\mathcal{P}3}$
and ${\mathcal{P}11}$ are data independent, they only need to be applied once
during each channel coherence time. While as the proposed MOOP formulations in
${\mathcal{P}6}$ and ${\mathcal{P}14}$ are data dependent, they need to be run
on a symbol by symbol basis. In the following section we compare the resulting
transmit complexity of conventional and proposed MOOP approaches for both slow
and fast fading scenarios, and show that the average execution time per
downlink frames is comparable for both techniques.

###  VI-B Receiver Complexity

At the receiver side, for the case of the conventional beamforming, the
downlink users in our FD system scenario need to equalize the composite
channel ${\textbf{h}_{i}^{H}}{\textbf{w}_{i}}^{*}$ to recover their data
symbols, where $\left\\{{\textbf{w}_{i}}^{*}\right\\}_{i=1}^{K}$ is the
optimal solution of ${\mathcal{P}3}$. For the case of the proposed CI scheme,
since the received symbols already lie in the constructive region of the
constellation as shown in Fig. 2 and Fig. 3, equalization is not required by
the downlink users. This automatically translates to reduced complexity at the
receiver. Accordingly, this implies that CSI is not required for detection at
the downlink users for the proposed CI scheme. Thus, depending on the
signaling and pilots already involved for the SINR estimation, the proposed CI
scheme may lead to further savings in training time and overhead. Most
importantly, this makes the proposed scheme resistant to any quantization
errors from the CSI acquisition at the receiver.

##  VII Results

In this section, we investigate the performance of our proposed system through
simulations. We model all channels as independent and identically distributed
Rayleigh fading for both the perfect and imperfect CSI cases. Systems with
QPSK and 16QAM modulation are considered while it is clear that the benefit
extends to any lower or higher order modulation. For comparison in every
scenario, we compare the proposed technique, constructive interference (CI)
with the conventional case i.e. when all interference is treated as harmful
signal [38, 39]. We use $N\times K\times J$ to denote an FD radio BS with $N$
antennas, $K$ downlink users and $J$ uplink users, respectively.

Figure 4: Average system object trade-off region achieved by the proposed
scheme versus the conventional scheme N = 9, K = 6, J = 3.

###  VII-A Uplink-Downlink Power Trade-off

In Fig. 4, we investigate the trade-off between the downlink and uplink total
transmit power for the case of $N=9,K=6,J=3$ antennas. The trade-off region is
obtained by solving problem ${\mathcal{P}3}$ and ${\mathcal{P}6}$ for the
conventional and CI case, respectively, for $0\leq{\lambda_{a}}\leq
1,a\in(1,2)$ with a step size of 0.1. Note that ${\lambda_{a}}$ determines the
priority of the a-th objective. We assume the same required SINR for all
downlink users to be $\Gamma_{i}^{DL}=10dB$ and $\Gamma_{i}^{UL}=0dB$ for all
uplink users. It can be seen from the plot that there is a trade-off between
the two objectives (downlink and uplink) i.e. an increase in one leads to a
decrease in the other and vice versa.

Figure 5: Average system object trade-off region achieved by the proposed
scheme versus the conventional scheme N = 8, K = 6, J = 3. Figure 6: Average
system object trade-off region achieved by the proposed scheme versus the
conventional scheme N = 6, K = 6, J = 6.

We compare the trade-off plot for the conventional scheme and the CI schemes
when applied to QPSK and 16QAM modulations. We study the trade-off plots when
the total number of antennas at the users is equal to the number of antennas
at the FD radio BS. Thus, we can observe that the CI scheme has power savings
of about 7dB for the uplink users in both QPSK and 16QAM modulations and about
2dB and 1.2dB power savings for the downlink users in QPSK and 16QAM
modulations, respectively. Note that the proposed scheme is only outperformed
by the conventonal beamforming for the case $\lambda_{1}=0,\lambda_{2}=1$,
where all priority is given to the uplink PM problem, where interference
exploitation does not apply.

In Fig. 5, we plot the case when we have $N=8,K=6,J=3$. The same trend can be
seen with Fig. 4, where we have for QPSK modulation power savings of about 6dB
and 2dB for the uplink and downlink users, respectively. And for 16QAM
modulation, we have power savings of about 6dB and 1.8dB for the uplink and
downlink users, respectively. This two scenarios $N=9,K=6,J=3$ and
$N=8,K=6,J=3$, show a practical perspective in the sense that there is usually
more antennas at the FD radio BS than the number of antennas at the users and
the optimisation problems are always feasible.

In Fig. 6, we show a scenario where we have equal number of antennas at the FD
radio BS and at the users $N=K=J=6$. With this setup we can see for QPSK
modulation uplink and downlink user power savings of about 12dB and 4dB,
respectively, and about 10dB and 2dB, respectively, for 16QAM modulation. The
reason is because for $N=K=J=6$ the problem is more restricted in the
optimisation variable dimensions and the conventional scheme in this scenario
leads to greatly increased uplink and downlink powers while for the CI scheme
this scenario can be accommodated and has higher feasibility so consumes lower
power. These results highlight a key advantage of the proposed scheme over the
conventional approaches.

Figure 7: Average power consumption versus minimum required downlink SINR when
$\lambda_{1}=0.9,\lambda_{2}=0.1$ and $\Gamma^{UL}=0$dB for QPSK modulation
Figure 8: Average power consumption versus minimum required downlink SINR when
$\lambda_{1}=0.1,\lambda_{2}=0.9$ and $\Gamma^{UL}=0$dB for QPSK modulation
Figure 9: Average power consumption versus minimum required downlink SINR when
$\lambda_{1}=0.9,\lambda_{2}=0.1,\Gamma^{UL}=0$dB and
$\epsilon_{h}=\epsilon_{f}=\epsilon_{G}=0.1$ for QPSK modulation

###  VII-B Average Transmit Power versus Minimum Required SINR

In Fig. 7 and Fig. 8, we investigate the power consumption of the downlink and
uplink users for different minimum required downlink SINR ($\Gamma_{i}^{DL}$).
For both plots we assume a minimum required uplink SINR $\Gamma_{j}^{UL}=0dB$
for all uplink users. In Fig. 7, we select $\lambda_{1}=0.9$ and
$\lambda_{2}=0.1.$ which gives higher priority to the total downlink transmit
power minimisation problem. It can be observed that both the uplink and
downlink power consumption increases with increase in $\Gamma_{i}^{DL}$. This
is because an increase in the downlink SINR requirment translates to increace
in downlink transmit power and hence increase in the SI power. Therefore, the
uplink users have to transmit with a higher power to meet their QoS
requirement ($\Gamma_{j}^{UL}$). However, we can still see power savings of
12dB and 5dB for the uplink and downlink users, respectively, for the CI
scheme compared to the conventional scheme. Also, we note that while CI is
applied to only the downlink users, more power is saved for the uplink users
than the downlink users. This is because with CI the total downlink transmit
power is reduced and this directly reduces the residual SI power at the FD BS.
Accordingly, the constructive interference power has been traded off for both
uplink and downlink power savings. The same trend can be seen in the Fig. 8,
where $\lambda_{1}=0.1$ and $\lambda_{2}=0.9$. It can be observed that in this
scenario since we give higher priority to the uplink power minimisation
problem, we have higher power savings for the uplink users and lower power
savings for the downlink users compared to the Fig. 7.

###  VII-C MOOP with Imperfect CSI

In Fig. 9 and 10, we investigate the performance of the proposed CSI-robust CI
scheme for $N=K=J=6$, we select $\lambda_{1}=0.9$ and $\lambda_{2}=0.1$.

Figure 10: Average power consumption versus error bounds when
$\lambda_{1}=0.9,\lambda_{2}=0.1,\Gamma^{UL}=0dB$ and $\Gamma^{DL}=10dB$ for
QPSK modulation  Figure 11: Average execution time per optimisation versus
number of downlink users with $N=J=6$ when
$\lambda_{1}=0.9,\lambda_{2}=0.1,\Gamma^{UL}=0$dB, $\Gamma^{DL}=5$dB and
$\epsilon_{h}=\epsilon_{f}=\epsilon_{G}=0.01$

Fig. 9 shows the Average power consumption for the uplink and downlink users
when the error bounds $\epsilon_{h}=\epsilon_{f}=\epsilon_{G}=0.1$. It can be
seen that the CI scheme shows better performance than the conventional scheme
with power savings of 6dB and 4dB for the uplink and downlink users,
respectively. In addition, for the conventional cases, feasible solutions can
only be found for minimum required downlink SINR $\Gamma_{i}^{DL}\leq 20$dB.
This indicates that the channel error tolerance of the conventional scheme is
much lower than that of the proposed CI scheme. This is also shown in Fig. 10,
which shows the average power consumption with increasing error bounds. It can
be seen that feasible solutions can only be found for
$\epsilon_{h}=\epsilon_{f}=\epsilon_{G}\leq 0.2$. Besides, even if feasible
results could be found, significant amount of power will be consumed as can be
seen for error bound values between 0.15 and 0.2 for both uplink and downlink
users.

###  VII-D Complexity

In Fig. 11, we compare the Average execution time per optimisation of the
conventional scheme and the proposed CI scheme for different number of
downlink users ($K$) with $N=J=6$. We fixed
$\lambda_{1}=0.9,\lambda_{2}=0.1,\Gamma^{UL}=0$dB, $\Gamma^{DL}=5$dB and
$\epsilon_{h}=\epsilon_{f}=\epsilon_{G}=0.01.$ It can be seen that for the
perfect CSI case, the proposed CI scheme takes 83% of time taken by the
conventional scheme. While for the imperfect CSI case, the proposed CI scheme
takes about 28% of the time taken by the conventional scheme. This is because
the conventional approach involves a more complicated set of constraints,
hence, more computational cost as shown in Section VI-A above. Besides, the
proposed MOOP ${\mathcal{P}14}$ formulation involves a multicast approach
which reduces the number variables to compute.

As we have noted above however, the proposed data dependent optimization needs
to be run on a symbol-by-symbol basis. To obtain a fairer comparison, we plot
in Fig. 12 the average execution time per frame versus the number of downlink
users for slow and fast fading channels. Here, we assume the LTE Type 2 TDD
frame structure [51], where each frame is subdivided to 10 subframes each with
a duration $1ms$ and containing 14 symbol-time slots. Accordingly, we assume
that for fast fading the channel is constant for the duration of a subframe
with a number of symbols per coherence time $N_{coh}=14$, while for slow
fading we assume a coherence time equal to 5 subframes with $N_{coh}=70$ [51].
The results for both slow and fast fading channels show the end complexity of
the proposed CI approaches are comparable to those with the conventional
approaches. Accordingly, and in conjunction with the performance improvements
shown in the previous results, it can be seen that the proposed schemes
provide a much more favorable performance complexity trade-off w.r.t.
conventional interference mitigation.

Figure 12: Average execution time versus number of downlink users for
slow/fast fading channels with $N=J=6$ when
$\lambda_{1}=0.9,\lambda_{2}=0.1,\Gamma^{UL}=0$dB, $\Gamma^{DL}=5$dB and
$\epsilon_{h}=\epsilon_{f}=\epsilon_{G}=0.01.$

##  VIII Conclusion

In this paper we studied the application of the interference exploitation
concept to a MU-MIMO system with a FD radio BS. The optimisation problem was
formulated as a convex Multi-Objective Optimisation problem (MOOP) via the
weighted Tchebycheff method. The MOOP was formulated for both PSK and QAM
modulations by adapting the decision thresholds in both cases to accommodate
for constructive interference. The CI scheme was also extended to robust
designs for imperfect downlink, uplink and SI CSI with bounded CSI errors.
Simulation results proved the significant power savings of the CI scheme over
the conventional scheme in every scenario. More importantly, we have shown
that through the FD MOOP formulation, constructive interference power can be
traded off for both uplink and downlink power savings.

## Acknowledgment

The author would like to thank the Federal Republic of Nigeria and the
Petroleum Technology Development Fund (PTDF) for funding his PhD.

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