# Higgs-Portal Scalar Dark Matter: Scattering Cross Section and Observable
Limits

Huayong Han and Sibo Zheng
Department of Physics, Chongqing University, Chongqing 401331, P.R. China

Abstract

The simplest Higgs-portal dark matter model is studied in the light of dark
matter self-interacting effects on the formation of large scale structures. We
show the direct detection limits in both the resonant and large mass region.
Finally, we also compare these limits with those at the LHC and Xenon 1T
experiments.

##  1 Introduction

The observed dark matter (DM) from galaxy rotation curves requires extension
beyond the Standard Model (SM), in which there is no viable candidate. Among
other things, one of the simplest DM models corresponds to coupling the DM
sector to SM sector, with the SM Higgs scalar as the interaction mediator.
This scenario is known as the Higgs-portal DM. The direct detection limits at
the LUX experiment have excluded a fermion-like but still allow a scalar-like
DM within mass range between 1 GeV and 10 TeV.

In the minimal version of Higgs-portal scalar DM [1, 2, 3, 4] there are only
three model parameters, which include the DM mass $m_{s}$, the Yukawa coupling
constant $\kappa_{s}$ between DM and the SM Higgs, and the DM self-interaction
coupling constant $\lambda_{s}$. The signals of direct or indirect detection
in this model are very predictive.

  * •

$\mathbf{Indirect~{}detection}$ mainly includes limits on DM annihilation into
$e^{+}e^{-}$ at PAMELA [5, 6, 7], into $\gamma$ rays at Fermi-LAT [8, 9, 10],
neutrinos in the sun [11, 12, 13], and Higgs invisible decay for the DM mass
below half of the Higgs mass $m_{h}$.

  * •

$\mathbf{Direct~{}detection}$ mainly includes limits on the DM-nucleon spin-
independent scattering at Xenon 100 [14] and LUX [15, 16], and the direct
production at hadron [17, 18, 19, 20] and lepton [21] colliders.

Summarizing experimental limits above, the scalar DM mass is tightly
constrained to two regions 111If the Hubble parameter $H$ during inflation is
above $10^{16}$ GeV, the resonant mass region is totally excluded [22]. In
contrast, these two regions are both consistent with present experimental
limits if $H$ is small enough. In this letter, we take the later assumption.,

resonant mass region $\displaystyle:$ $\displaystyle 62.5~{}\text{GeV}\leq
m_{s}\leq 66~{}\text{GeV},$ large mass region $\displaystyle:$ $\displaystyle
m_{s}\geq 185~{}\text{GeV}.$ (1.1)

In this paper we explore direct detection on this model via DM self-
interacting effects on the formation of large scale structures (LSS) [23],
which is less studied in comparison with the DM-nucleon spin-independent
scattering. As firstly described by Spergel and Steinhardt [24], self-
interacting DM may be used to explain the constant core problem [25, 26, 27,
28, 29] and missing satellites in DM halos at the dwarf scale 222The number of
DM halos at this scale is roughly about $\sim 1000$ as inferred either from
simulation [30] or analytic theory [31], but less than $\sim 100$ galaxies are
observed [32].. In the former one, kinetic energy is transmitted from the hot
outer halo inward because of DM self-interaction, with suitable strength (as
described by $\sigma/m_{s}$, here $\sigma$ denotes the scattering cross
section for $ss\rightarrow ss$ ). In the later case, DM self-interaction could
lead to satellite evaporation due to the DM particles within the satellites
being kicked out by high-velocity encounters with DM particles from the
surrounding dark halo of the parent galaxy.

Table 1 shows present limits on $\sigma/m_{s}$ based on astrophysical
observations at different galactic scales. The studies of DM self-interacting
effects on the formation of LSS will shed light on two aspects. At first, the
DM self-interaction coupling constant $\lambda_{s}$ is constrained more
efficiently, in compared with constraints arising from the DM relic density,
direct detection limits at the LUX or LHC, which have little relevance to
$\lambda_{s}$. Also, it provides the limits for discovery of DM in terms of
astrophysical observations.

Galactic Scale | limit ($\text{cm}^{2}/\text{g}$) | velocity (km/s) | Refs.
---|---|---|---
Milky Way | $\sigma/m_{s}\leq 1.0$ | $\sim 10^{2}$ | [33]
Cluster | $\sigma/m_{s}\leq 1.25$ | $\sim 10^{3}$ | [34]
Table 1: Upper bounds on $\sigma/m_{s}$ at different galactic scales inferred
from DM self-interacting effects on the formation of LSS. In comparison with
[34], recent observations of cluster collisions [35] give rise to a slightly
stronger upper bound $\sigma/m_{s}\leq 0.47$.

The rest of this paper is organized as follows. In Sec. 2, we calculate the
tree-level value for DM scattering cross section $\sigma_{0}$ in terms of
Madgraph5 [36] and Feynman rules generator [37] . We eliminate parameter
$\kappa_{s}$ via the constraint from DM relic abundance, therefore
$\sigma_{0}$ only depends on the remaining parameters $m_{s}$ and
$\lambda_{s}$. In Sec. 3, we consider the Sommerfeld effect [38] on DM
scattering cross section due to the DM self-interaction [39, 40, 41]. In
comparison with a massless or light-mass mediator, the Higgs mass upper bounds
the enhancement factor significantly. The enhancement on the DM scattering
cross section is verified to be mild in the resonant mass region, and less
than $\sim 10^{4}-10^{5}$ in the large mass region for $m_{s}$ above $\sim 2$
TeV. In Sec. 4 we compare the experimental limits with those at the LHC and
Xenon experiments. Finally, we conclude in Sec. 5.

Figure 1: The dependence of tree-level scattering cross section $\sigma_{0}$
on DM mass without DM self-interaction ($\lambda_{s}=0$). Note that the
dependence on $\kappa_{s}$ is eliminated via the measured DM relic abundance
[42], which is numerically calculated via MicrOMEGAs [43].

##  2 Tree-Level Scattering cross section

The relevant Lagrangian for our model is given by,

$\displaystyle{}\mathcal{L}=-\frac{1}{2}\left(\partial
s\right)^{2}+\frac{\mu^{2}_{s}}{2}s^{2}+\frac{\lambda_{s}}{2}s^{4}+\frac{\kappa_{s}}{2}s^{2}\mid
H\mid^{2}.$ (2.1)

Expand DM field $s$ and Higgs field $h$ along their vacuum expectation value
$\left<s\right>=0$ and $\left<H\right>=(\upsilon_{\text{EW}}+h)/\sqrt{2}$,
respectively, we obtain

$\displaystyle{}\mathcal{L}=-\frac{1}{2}\left(\partial
s\right)^{2}+\frac{1}{2}m_{s}^{2}s^{2}+\frac{\lambda_{s}}{2}s^{4}+\frac{\kappa_{s}\upsilon_{\text{EW}}}{2}s^{2}h+\frac{\kappa_{s}}{4}s^{2}h^{2},$
(2.2)

where $m^{2}_{s}=\mu_{s}^{2}+\kappa_{s}\upsilon_{\text{EW}}^{2}/2$, and the
electroweak scale $\upsilon_{\text{EW}}=246$ GeV.

The contributions to DM scattering cross section include two types of Feynman
diagrams \- one with intermediate Higgs scalar field and the other with
contact interaction. The tree-level value for $\sigma_{0}$ without and with
quartic interaction is shown in Fig.1 and Fig.2, respectively, by using
Madgraph5 [36]. In Fig.1 the dependence of $\sigma_{0}$ on $\kappa_{s}$ is
eliminated in terms of the measured DM relic abundance. Consequently, the
total contribution to $\sigma_{0}$, as shown in Fig.2, can be presented in the
parameter space of $m_{s}$ and $\lambda_{s}$. These numerical values are
compatible with analytic approximations in different mass limits [44],

${}\frac{\sigma}{10^{4}}[\text{pb}]\sim\left\\{\begin{array}[]{lcl}1.4\cdot\lambda_{s}^{2}\cdot\left(\frac{100~{}\text{GeV}}{m_{s}}\right)^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}m_{s}>>m_{h},\\\
5.6\cdot\left(\frac{\lambda_{s}}{2}-\frac{\kappa^{2}_{s}\upsilon^{2}}{8m^{2}_{h}}\right)^{2}\cdot\left(\frac{100~{}\text{GeV}}{m_{s}}\right)^{2},~{}m_{s}<<m_{h}.\end{array}\right.$
(2.3)

Figure 2: Similar to Fig.1 but with DM self-interaction included. Comparison
with Fig.1 indicates that large $\lambda_{s}$ dominates the contribution to
$\sigma_{0}$ in the large mass region.

Fig.2 indicates that $\sigma_{0}$ is upper bounded as $\sigma\leq 10^{12}$ pb
in the whole range $0\leq\lambda_{s}\leq 2$, which implies 333There is a
useful relation among different units: 1 $\text{cm}^{2}/g$=$1.8\times
10^{12}\text{pb/GeV}=4.62\times 10^{3}\text{GeV}^{-3}$. that $\sigma/m_{s}\leq
1~{}\text{cm}^{2}/\text{g}$ for $m_{s}$ above $1$ GeV. Although small $\sigma$
is compatible with the limits shown in Table 1, relative larger $\sigma$ is
more favored in the light of direct detection at further astrophysical
observations. As we will show in the next section, the Sommerfeld effect
enhaces the magnitude of $\sigma_{0}$, which is as large as of order $\sim
10^{4}-10^{5}$ in the large mass region. It seems that the discovery potential
for large mass region can be improved. This issue will be discussed in detail
in Sec. 4.

##  3 Sommerfeld Effects

The $S$-wave annihilation cross section for two DM particles moving at small
relative velocities, is enhanced by a factor ($S$) depending on the inverse
velocity $v\sim 10^{-3}$, in compared with $v\sim 0.3$ at the freeze-out time.
This enhancement is known as the Sommerfeld effect, which corresponds to the
summation of a series of ladder diagrams with the mediator repeatedly
exchanged. Since firstly applied to the wino dark matter [45], it has been
clear that the DM annihilation cross section may be significantly differs from
the DM scattering cross section when these two cross sections are both
$S$-wave dominated.

Figure 3: Sommerfeld enhancement factor in the parameter space of $m_{s}$ and
$v$. Note that the dependence of S factor on parameter $\kappa_{s}$ is
eliminated by the requirement of DM relic abundance similar to previous
treatments.

This mediator is the Higgs scalar in our model. By following the works in [39,
40, 41], one obtains the enhancement factor in terms of solving the non-
relativistic schrodinger equation,

$\displaystyle{}-\frac{1}{m_{s}}\frac{d^{2}\chi}{dr^{2}}+V(r)\chi=m_{s}v^{2}\chi$
(3.1)

where in our case the Yukawa potential 444Ref. [46] has considered a similar
model. The Yukawa potential therein differs from ours due to different
conventions.,

$\displaystyle{}V(r)=-\frac{\kappa^{2}_{s}}{4\pi r}\exp{(-m_{h}r)}$ (3.2)

The boundary conditions are given by

$\displaystyle\chi^{\prime}(r)$ $\displaystyle=$ $\displaystyle
im_{s}v\chi(r),$ $\displaystyle\chi(r)\mid_{r\rightarrow\infty}$
$\displaystyle\rightarrow$ $\displaystyle\exp{(im_{s}vr)}.$ (3.3)

Under such notation, the Sommerfeld enhancement factor $S$ reads as,

$\displaystyle{}S=\frac{\mid\chi(\infty)\mid^{2}}{\mid\chi(0)\mid^{2}}$ (3.4)

$S$ depends only on parameters $v$ and $m_{s}$, as its dependence on
$\kappa_{s}$ can be eliminated by the requirement of DM relic abundance. Fig.3
shows our numerical solution to the Sommerfeld enhancement factor $S$ in the
parameter space of $m_{s}$ and $v$. $S$ is around unity for DM mass below $2$
TeV, and its maximal value is about $\sim 10^{5}-10^{7}$ for $m_{s}\geq 3$
TeV. These numerical results agree with the analytic approximation [40, 41,
7],

$\displaystyle{}S=\frac{\pi}{\epsilon_{v}}\frac{\sinh\left(\frac{2\epsilon_{v}}{\pi^{2}\epsilon_{s}/6}\right)}{\cosh\left(\frac{2\epsilon_{v}}{\pi\epsilon_{s}/6}\right)-\cos\left(2\pi\sqrt{\frac{1}{\pi^{2}\epsilon_{s}/6}-\frac{\epsilon^{2}_{v}}{(\pi^{2}\epsilon_{s}/6)^{2}}}\right)},$

where $\epsilon_{v}=v/\alpha_{\kappa_{s}}$ and
$\epsilon_{s}=m_{h}/(\alpha_{\kappa_{s}}m_{s})$. Although it is not obvious in
Fig.3, we have also verified that $S$ decreases as the velocity $v$ increases.

Figure 4: $\sigma/m_{s}$ as function of $m_{s}$ for $\lambda_{s}$ between
$0.1$ and $2$ and $v=\\{10,10^{2},10^{3}\\}$ km/s. Similar to Fig.3 the
dependence on parameter $\kappa_{s}$ is eliminated. Note that both the red and
green lines overlap with the blue ones for DM mass below $\sim 3$ TeV.

##  4 Comparison with LHC and Xenon 1T

Combining the Sommerfeld effect in the previous section gives rise to our
final result on the DM scattering cross section,

$\displaystyle{}\sigma=S(\upsilon,m_{s})\sigma_{0}(\lambda,m_{s}).$ (4.1)

In terms of Fig.3 we plot $\sigma/m_{s}$ as function of $m_{s}$ for different
$\lambda_{s}$ and velocity $v$ in the range of $10-10^{3}$ km/s in Fig.4. It
is shown that the resonant mass region for large $\lambda_{s}\sim 2$ can be
probed for $\sigma/m_{s}$ of order $\sim 10^{-7}$ $\text{cm}^{2}/g$; and
$\sigma/m_{s}$ of order $\sim 10^{-11}$ $\text{cm}^{2}/g$ is required for
small $\lambda_{s}\sim 0.1$. Smaller limits on $\sigma/m_{s}$ are required for
the detection for either smaller $\lambda_{s}$ or larger DM mass.

Obviously, the value of $\sigma/m_{s}$ is consistent with present
astrophysical limits shown in Table 1 in the whole mass region. It is also
obvious that the simplest Higgs-portal DM model can not provide $\sigma$ large
enough to explain the puzzles at the dwarf scale as mentioned in the
introduction.

The required limits on $\sigma/m_{s}$ for detection seems too small in
compared with present limits (of order $\sim 10^{-1}$ $\text{cm}^{2}/g$ ).
Does it imply that the astrophysical observations on LSS are less efficient in
compared with other direct detection facilities ? Let us compare these limits
with those at the future LHC and Xenon 1T experiments as required for
discovery. The main observations are summarized in Table 2. See what follows
for explanation.

DM Mass (GeV) | LHC | Xenon 1T | LSS
---|---|---|---
Resonant mass region | $\surd$ | $\times$ | $\surd$
Large mass region ($185\leq m_{s}<3000$) | $\times$ | $\surd$ | $\times$
Large mass region ($m_{s}\geq 3000$) | $\times$ | $\times$ | $\times$
Table 2: Prospect for the discovery of DM at different experimental
facilities. We have assumed that the limit on $\sigma/m_{s}$ of order $\sim
10^{-7}$ $\text{cm}^{2}/g$ can be reached in the further astrophysical
observations on LSS. In comparison with the required integrated luminosity
$\mathcal{L}$ at least of order $10^{3}$ $\text{fb}^{-1}$ at the 14-TeV LHC,
LSS provides a complementary way to detect the resonant mass region.

$(i)$ As shown in Fig.5, the production cross section for DM at the 13-TeV LHC
is less than $\sim 10^{-1}$ fb and $10^{-4}$ fb in the resonant mass region
and the large mass region respectively. Therefore the later case is beyond the
reach of LHC, and the former case can be detected only for extremely large
integrated luminosity $\mathcal{L}$ at least of order $\sim 10^{3}$
$\text{fb}^{-1}$ if one takes care of the SM background [20].

$(ii)$ The DM-nucleon spin-independent scattering cross section are less than
$\sim 10^{-13}$ pb and $\sim 10^{-8}$ pb in the resonant mass region and large
mass region with $m_{s}\geq 3$ TeV, respectively, which are both beyond the
reach of Xenon 1T experiment [22].

$(iii)$ With the assumption that the limit on $\sigma/m_{s}$ of order $\sim
10^{-7}$ $\text{cm}^{2}/g$ can be reached in the further astrophysical
observations on LSS, the resonant mass region with large $\lambda\sim 2$ can
be totally detected. Therefore, in comparison with the required integrated
luminosity $\mathcal{L}$ of order $10^{3}$ $\text{fb}^{-1}$ at the 14-TeV LHC,
LSS provides a complementary way to detect the resonant mass region.

Figure 5: Contributions to the production cross section for DM at the 13-TeV
LHC, which are dominated by the vector boson fusion processes. The dependence
on $\kappa_{s}$ is eliminated similar to Fig. 1.

##  5 Conclusions

In this letter we have studied the DM scattering cross section in the simplest
Higgs-portal DM model. We have also discussed the limits required for direct
detection in terms of the astrophysical observations on LSS. We observe that
$(a)$ in compared with the future LHC with extremely large integrated
luminosity $\mathcal{L}$ (at least of order $10^{3}$ $\text{fb}^{-1}$)
astrophysical observations on LSS provides a complementary way to detect the
resonant mass region, which is beyond the reach of Xenon 1T experiment; $(b)$
the large mass region with $m_{s}$ above 3 TeV is beyond the reaches of all
direct detections from the LSS, LHC and Xenon 1T.

$\mathbf{Acknowledgments}$
We would like to thank the referee for comments. This work is supported in
part by Natural Science Foundation of China under Grant No.11247031 and
11405015.

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# Full Duplex Hybrid A/D Beamforming with Reduced Complexity Multi-Tap Analog
Cancellation

George C. Alexandropoulos1, Md Atiqul Islam2, and Besma Smida2 This work was
partially funded by the National Science Foundation CAREER award #1620902.
1Department of Informatics and Telecommunications, National and Kapodistrian
University of Athens, Greece
2Department of Electrical and Computer Engineering, University of Illinois at
Chicago, USA
emails<EMAIL_ADDRESS>{mislam23<EMAIL_ADDRESS>
###### Abstract

Although the hardware complexity of the analog self-interference canceller in
full duplex Multiple Input Multiple Output (MIMO) designs does not necessarily
scale with the number of transceiver antennas, exploiting the benefits of
analog cancellation in massive MIMO systems with hundreds of antenna elements
is still quite impractical. Hybrid Analog and Digital (A/D) beamforming
architectures have been lately considered as a candidate technology for
realizing massive MIMO transceivers with very large number of antenna
elements, but with much fewer numbers of Radio Frequency (RF) chains. In this
paper, we present a novel architecture for full duplex hybrid A/D beamforming
transceivers including multi-tap analog cancellation with reduced number of
taps and simple multiplexers for efficient signal routing among the
transceiver RF chains. Capitalizing on the proposed transceiver architecture,
we present a joint design of analog cancellation and A/D beamforming with the
objective to maximize the achievable full duplex rate performance.
Representative millimeter wave simulation results demonstrate the
effectiveness of the proposed architecture and algorithmic framework for
enabling simultaneous uplink and downlink communications with reduced
complexity analog self-interference cancellation.

###### Index Terms:

Analog cancellation, full duplex, hybrid beamforming, joint optimization,
multi-user communication, massive MIMO.

##  I Introduction

In band full duplex, also known shortly as Full Duplex (FD), is a candidate
technology for the Release $17$ of the fifth Generation (5G) New Radio (NR)
standard enabling simultaneous UpLink (UL) and DownLink (DL) communication
within the entire frequency band [1]. An FD radio can transmit and receive at
the same time and frequency resource units, consequently, it can double the
spectral efficiency achieved by a Half Duplex (HD) radio. Current wireless
systems exploit Multiple Input Multiple Output (MIMO) communication, where
increasing the number of Transmitter (TX) and Receiver (RX) antennas can
increase the spatial Degrees of Freedom (DoF), hence boosting rate
performance. Combining FD with MIMO operation can provide further spectral
efficiency gains [2, 3, 4, 5, 6, 7, 8].

FD radios suffer from Self Interference (SI), a term referring to the signal
transmitted by the FD radio TX that leaks to the FD radio RX. At the RX of the
FD radio, the SI power can be many times stronger than the power of the
received signal of interest. Consequently, SI can severely degrade the
reception of the signal of interest, and thus SI mitigation is required in
order to maximize the spectral efficiency gain of the FD operation. As the
number of antennas increases, mitigating SI becomes more challenging, since
more antennas naturally result in more SI components. Conventional SI
suppression techniques in Single-Input Single-Output (SISO) systems include
propagation domain isolation, analog domain suppression, and digital
cancellation[4, 9]. Although analog SI cancellation in FD MIMO systems can be
implemented through SISO replication, its hardware requirements scale with the
number of TX/RX antennas. The authors in [2, 5] presented spatial suppression
techniques that alleviate the need for analog SI cancellation, which was
replaced solely by digital TX/RX beamforming. In [10], a joint design of
multi-tap analog cancellation and TX/RX beamforming, where the number of taps
does not scale with the product of TX and RX antenna elements, was proposed.
The FD technology has been lately theoretically combined with Hybrid analog
and digital BeamForming (HBF) [11] to enable simultaneous UL and DL
communications in massive MIMO systems operating in the millimeter wave
frequency bands [12, 13, 14, 15]. These works mainly assigned the role of SI
mitigation to the hybrid beamformers and/or deployed analog SI cancellation
that scales with the number of TX/RX antennas.

In this paper, we present a novel hardware architecture for FD HBF systems
enabling the joint design of A/D TX/RX beamformers with reduced complexity
tap-based analog cancellation. The proposed analog canceller interconnects a
subset of the outputs of the TX Radio Frequency (RF) chains to a subset of the
inputs to the RX RF chains in order to ensure that the residual SI signal
after A/D TX precoding and analog RX combining remains below the noise floor.
Our indicative simulation results with the proposed architecture and an
example FD HBF algorithmic framework showcase a $1.7$ times rate improvement
over HD HBF communication.

Figure 1: The considered bidirectional communication system with the proposed
FD HBF architecture in the MIMO node $k$ including $N$-tap analog cancellation
and A/D TX/RX beamforming. The HD multi-antenna nodes $q$ and $m$ communicate
with node $k$ in the DL and UL directions, respectively. Each TX RF chain
consists of a Digital to Analog Converter (DAC), a mixer upconverting the
signal from BaseBand (BB) to RF, and a Power Amplifier (PA). A RX RF chain
consists of a Low Noise Amplifier (LNA), a mixer downconverting the signal
from RF to BB, and an Analog to Digital Converter (ADC). Upsampling and pulse
shaping are used to prepare the BB signal for DAC and RF transmission at the
TX side, whereas matched filtering and downsampling are used at the RX side
before BB processing of the received RF signal.

Notation: Vectors and matrices are denoted by boldface lowercase and boldface
capital letters, respectively. The transpose and Hermitian transpose of
$\mathbf{A}$ are denoted by $\mathbf{A}^{\rm T}$ and $\mathbf{A}^{\rm H}$,
respectively, and $\det(\mathbf{A})$ is the determinant of $\mathbf{A}$, while
$\mathbf{I}_{n}$ ($n\geq 2$) is the $n\times n$ identity matrix and
$\mathbf{0}_{n\times m}$ ($n,m\geq 2$) is a $n\times m$ matrix with zeros.
$\|\mathbf{A}\|_{\rm F}$ is the Frobenius norm of $\mathbf{A}$,
$\|\mathbf{a}\|$ stands for $\mathbf{a}$’s Euclidean norm, and ${\rm
diag}\\{\mathbf{a}\\}$ denotes a square diagonal matrix with $\mathbf{a}$’s
elements in its main diagonal. $[\mathbf{A}]_{i,j}$ represents $\mathbf{A}$’s
$(i,j)$-th element, while $[\mathbf{a}]_{i}$ denotes the $i$-th element of
$\mathbf{a}$. $\mathbb{R}$ and $\mathbb{C}$ represent the real and complex
number sets, respectively, and $|\cdot|$ denotes the amplitude of a complex
number.

##  II System Model and Proposed Architecture

###  II-A System Model

We consider the $3$-user bidirectional communication system in Fig. 1
comprising of a FD MIMO node $k$ equipped with $N_{k}$ TX and $M_{k}$ RX
antenna elements, and two HD multi-antenna nodes $q$ and $m$ having $M_{q}$
and $N_{m}$ antennas, respectively. It is assumed that node $k$ communicates
simultaneously (in the same time and frequency resources) with node $q$ in the
DL and node $m$ in the UL. All nodes are considered capable of performing
digital beamforming, which for simplicity we assume to be realized with linear
filters. Node $k$ is also capable of analog TX/RX beamforming using the
partially connected HBF architecture [11], as will be detailed in the sequel.

It is assumed that node $m$ makes use of the digital precoding matrix
$\mathbf{V}_{m}^{(\mathrm{BB})}\in\mbox{$\mathbb{C}$}^{N_{m}\times d_{m}}$ for
processing in BaseBand (BB) its unit power symbol vector
$\mathbf{s}_{m}\in\mbox{$\mathbb{C}$}^{d_{m}\times 1}$ (chosen in practice
from a discrete modulation set) before UL transmission. The dimension of
$\mathbf{s}_{m}$ satisfies $d_{m}\leq\min\\{M_{k}^{(\mathrm{RF})},N_{m}\\}$
with $M_{k}^{(\mathrm{RF})}$ denoting the number of RX RF chains at node $k$.
It holds $M_{k}^{(\mathrm{RF})}\leq M_{k}$, although in practical systems it
can be $M_{k}^{(\mathrm{RF})}\ll M_{k}$. The constraint for $d_{m}$ certifies
data decodability for the considered UL communication. It additionally holds
that
$\mathbb{E}\\{\|\mathbf{V}_{m}^{(\mathrm{BB})}\mathbf{s}_{m}\|^{2}\\}\leq{\rm
P}_{m}$, where ${\rm P}_{m}$ is the total TX power of node $m$. On the DL, the
reception node $q$ applies the digital combining matrix
$\mathbf{U}_{q}^{(\mathrm{BB})}\in\mbox{$\mathbb{C}$}^{N_{q}\times d_{k}}$ in
the BB received signal that includes the unit power symbol vector
$\mathbf{s}_{k}\in\mbox{$\mathbb{C}$}^{d_{k}\times 1}$ (again chosen from a
discrete modulation set) transmitted from node $k$ such that
$d_{k}\leq\min\\{M_{q},N_{k}^{(\mathrm{RF})}\\}$ with $N_{k}^{(\mathrm{RF})}$
($N_{k}^{(\mathrm{RF})}\leq N_{k}$, but practically it can be
$N_{k}^{(\mathrm{RF})}\ll N_{k}$) denoting the number of TX RF chains at node
$k$. Similarly, the latter constraint verifies the spatial DoF of the
effective $M_{q}\times N_{k}^{(\mathrm{RF})}$ DL MIMO channel between the TX
RF chains of node $k$ and the RX RF chains of node $q$. It is noted that each
antenna at node $q$ is connected to a dedicated RF chain.

###  II-B Proposed FD HBF Hardware Architecture

The proposed FD HBF hardware architecture, comprising of $N$-tap analog
cancellation for the SI signal as well as A/D precoding and combining for the
outgoing and incoming signals, is adopted for the MIMO node $k$, as depicted
in the left part of Fig. 1. Differently from [10]’s architecture that
considered only fully digital TX/RX beamforming, node $k$ is capable of HBF
through its partially connected beamforming architecture. As shown in the
figure, the analog canceller interconnects the $N_{k}^{(\mathrm{RF})}$ inputs
of the analog TX precoder to the $M_{k}^{(\mathrm{RF})}$ outputs of the analog
RX combiner. The complexity of the analog canceller expressed in the number of
taps $N$ is independent of the numbers $N_{k}$ and $M_{k}$ of the TX and RX
antennas, respectively, and as it will be shown next, it scales with the
product $\xi N_{k}^{(\mathrm{RF})}M_{k}^{(\mathrm{RF})}$ with $\xi<1$. This is
in contrast to [10] where the analog canceller interconnects the $N_{k}$ TX
antenna inputs to the $M_{k}$ RX antenna outputs.

####  II-B1 Partially Connected HBF

Each of the $N_{k}^{(\mathrm{RF})}$ TX RF chains of node $k$ is connected to a
separate subset of the available TX antenna elements. As shown in Fig. 1, the
$i$-th TX RF chain with $i=1,2,\ldots,N_{k}^{(\mathrm{RF})}$ is connected via
phase shifters with $N_{k}^{({\rm A})}$ TX antenna elements, each denoted as
${\rm TX}(i,j)$ $\forall$$j=1,2,\ldots,N_{k}^{({\rm A})}$. Clearly, it holds
$N_{k}=N_{k}^{(\mathrm{RF})}N_{k}^{({\rm A})}$ for the total number of TX
antennas at node $k$. Stacking the values of the $N_{k}^{({\rm A})}$ phase
shifters that connect each $i$-th TX RF chain with its antenna elements in a
complex-valued $N_{k}^{({\rm A})}\times 1$ vector $\mathbf{v}_{i}$, we can
formulate the complex-valued $N_{k}\times N_{k}^{(\mathrm{RF})}$ analog TX
precoder, as follows:

$\mathbf{V}_{k}^{(\mathrm{RF})}=\left[\begin{matrix}\mathbf{v}_{1}&\mathbf{0}_{N_{k}^{({\rm
A})}\times 1}&\cdots&\mathbf{0}_{N_{k}^{({\rm A})}\times 1}\\\
\mathbf{0}_{N_{k}^{({\rm A})}\times
1}&\mathbf{v}_{2}&\cdots&\mathbf{0}_{N_{k}^{({\rm A})}\times 1}\\\
\vdots&\vdots&\ddots&\vdots\\\ \mathbf{0}_{N_{k}^{({\rm A})}\times
1}&\mathbf{0}_{N_{k}^{({\rm A})}\times 1}&\cdots&\mathbf{v}_{N_{k}^{({\rm
RF})}}\end{matrix}\right].$ (1)

The elements of each $\mathbf{v}_{i}$ are assumed to have constant magnitude,
i.e., $|[\mathbf{v}_{i}]_{n}|^{2}=1/N_{k}^{({\rm A})}$
$\forall$$n=1,2,\ldots,N_{k}^{({\rm A})}$. We also assume that
$\mathbf{v}_{i}\in\mathbb{F}_{\rm TX}$
$\forall$$i=1,2,\ldots,N_{k}^{(\mathrm{RF})}$, which means that all analog TX
precoding vectors belong in a predefined beam codebook $\mathbb{F}_{\rm TX}$
including ${\rm card}(\mathbb{F}_{\rm TX})$ distinct vectors (or analog
beams). Apart from applying $\mathbf{V}_{k}^{(\mathrm{RF})}$ in the analog
domain to the signal before transmission, the symbol vector $\mathbf{s}_{k}$
is also processed in BB with the digital TX precoder
$\mathbf{V}_{k}^{(\mathrm{BB})}\in\mbox{$\mathbb{C}$}^{N_{k}^{(\mathrm{RF})}\times
d_{k}}$ (recall that $d_{k}\leq\min\\{M_{q},N_{k}^{(\mathrm{RF})}\\}$) before
entering into the $N_{k}^{(\mathrm{RF})}$ TX RF chains, as shown in Fig. 1.
Similar to the UL communication from node $m$ to $k$, we assume that the DL
transmission from node $k$ to $q$ is power limited according to
$\mathbb{E}\\{\|\mathbf{V}_{k}^{(\mathrm{RF})}\mathbf{V}_{k}^{(\mathrm{BB})}\mathbf{s}_{k}\|^{2}\\}\leq{\rm
P}_{k}$ with ${\rm P}_{k}$ being the total available TX power at node $k$.

The RX of node $k$ is composed of an analog combiner connecting the RX antenna
elements to the inputs of the RX RF chains, and a digital combiner that
processes the outputs of the RX RF chains in BB before signal decoding. In
particular, the $n$-th RX RF chain with $n=1,2,\ldots,M_{k}^{(\mathrm{RF})}$
is connected through phase shifters with $M_{k}^{({\rm A})}$ distinct RX
antennas; these phase shifters are denoted as ${\rm RX}(n,\ell)$
$\forall$$\ell=1,2,\ldots,M_{k}^{({\rm A})}$. It should hold that
$M_{k}=M_{k}^{(\mathrm{RF})}M_{k}^{({\rm A})}$ for the total number of RX
antennas at node $k$. We define the complex-valued $M_{k}\times
M_{k}^{(\mathrm{RF})}$ analog RX combiner $\mathbf{U}_{k}^{(\mathrm{RF})}$
having a similar block diagonal structure to (1). In particular,
$\mathbf{U}_{k}^{(\mathrm{RF})}$ contains $\mathbf{u}_{n}$’s with
$n=1,2,\ldots,M_{k}^{(\mathrm{RF})}$ in the diagonal, where each
$\mathbf{u}_{n}$ contains the constant magnitude values of the $M_{k}^{({\rm
A})}$ phase shifters (i.e., $|[\mathbf{u}_{n}]_{j}|^{2}=1/M_{k}^{({\rm A})}$
$\forall$$j=1,2,\ldots,M_{k}^{({\rm A})}$) connecting each $n$-th RX RF chain
with its antenna elements. We also assume that
$\mathbf{u}_{n}\in\mathbb{F}_{\rm RX}$ $\forall$$n$, i.e., all analog RX
combiners belong in a predefined beam codebook $\mathbb{F}_{\rm RX}$ having
${\rm card}(\mathbb{F}_{\rm RX})$ vectors. Finally,
$\mathbf{U}_{k}^{(\mathrm{BB})}\in\mbox{$\mathbb{C}$}^{M_{k}^{(\mathrm{RF})}\times
d_{m}}$ with $d_{m}\leq\min\\{M_{k}^{(\mathrm{RF})},N_{m}\\}$ represents the
digital RX combiner at node $k$.

####  II-B2 Multi-Tap Analog Cancellation

The analog canceller at node $k$ consists of $N$ taps with each tap connected
via a $N_{k}^{\mathrm{(RF)}}$-to-$1$ MUltipleXer (MUX) to all
$N_{k}^{\mathrm{(RF)}}$ outputs of the respective TX RF chains. A tap includes
a fixed delay, a variable phase shifter, and a variable attenuator [16, 10].
To route the cancellation signal to one of the adders located just before the
RX RF chains, the output of each tap is connected to a
$1$-to-$M_{k}^{\mathrm{(RF)}}$ DEMUltipleXer (DEMUX). There is a total of
$NM_{k}^{\mathrm{(RF)}}$ such adders and we use the notation “Adder ($i,j$)”
to label the adder that connects DEMUX $j$ to RX RF chain $i$, where
$i=1,2,\ldots,M_{k}^{\mathrm{(RF)}}$ and $j=1,2,\ldots,N$. The adders before
the RX RF chains can be implemented via power combiners or directional
couplers, while the analog RF MUXs/DEMUXs can be implemented with RF switches.
Clearly, the proposed analog canceller interconnects the outputs of some of
the available TX RF chains to the inputs of some of the RX RF chains, and in
contrast to [10], the size of each MUX/DEMUX depends on the number of TX/RF RF
chains and not on the number of TX/RX antennas. Similar to [10], we model in
BB the analog processing realized by the analog canceller as
$\mathbf{C}_{k}\triangleq\mathbf{L}_{3}\mathbf{L}_{2}\mathbf{L}_{1}\in\mbox{$\mathbb{C}$}^{M_{k}^{\mathrm{(RF)}}\times
N_{k}^{\mathrm{(RF)}}}$, where $\mathbf{L}_{1}\in\mbox{$\mathbb{R}$}^{N\times
N_{k}^{\mathrm{(RF)}}}$, $\mathbf{L}_{2}\in\mbox{$\mathbb{C}$}^{N\times N}$,
and $\mathbf{L}_{3}\in\mbox{$\mathbb{R}$}^{M_{k}^{\mathrm{(RF)}}\times N}$.
The elements $[\mathbf{L}_{1}]_{j,\ell}$ and $[\mathbf{L}_{3}]_{i,j}$ with
$j=1,2,\ldots,N$, $\ell=1,2,\ldots,N_{k}^{\mathrm{(RF)}}$, and
$i=1,2,\ldots,M_{k}^{\mathrm{(RF)}}$ take the binary values $0$ or $1$, and it
must hold that

$\sum_{\ell=1}^{N_{k}^{\mathrm{(RF)}}}[\mathbf{L}_{1}]_{j,\ell}=\sum_{i=1}^{M_{k}^{\mathrm{(RF)}}}[\mathbf{L}_{3}]_{i,j}=1\,\,\,\forall
j=1,2,\ldots,N.$ (2)

The $\mathbf{L}_{2}$ in $\mathbf{C}_{k}$ is a diagonal matrix whose complex
entries represent the attenuation and phase shift of the canceller taps; the
magnitude and phase of the element $[\mathbf{L}_{2}]_{i,i}$ with
$i=1,2,\ldots,N$ specify the attenuation and phase of the $i$-th tap. Recall
that the tap delays in each canceller tap are fixed, hence, we model the
effects of the $i$-th tap delay as a phase shift that is incorporated to the
phase of $[\mathbf{L}_{2}]_{i,i}$.

###  II-C Received Signal Models

Using the previously described system configuration, the BB received signal
$\mathbf{y}_{q}\in\mbox{$\mathbb{C}$}^{M_{q}\times 1}$ at node $q$ in the DL
communication can be mathematically expressed as

$\mathbf{y}_{q}\triangleq\mathbf{H}_{q,k}\mathbf{V}_{k}^{(\mathrm{RF})}\mathbf{V}_{k}^{(\mathrm{BB})}\mathbf{s}_{k}+\mathbf{H}_{q,m}\mathbf{V}_{m}\mathbf{s}_{m}+\mathbf{n}_{q},$
(3)

where $\mathbf{H}_{q,k}\in\mbox{$\mathbb{C}$}^{M_{q}\times N_{k}}$ is the DL
channel gain matrix (i.e., between nodes $q$ and $k$),
$\mathbf{H}_{q,m}\in\mbox{$\mathbb{C}$}^{M_{q}\times N_{m}}$ denotes the
channel gain matrix for inter-node interference (i.e., between nodes $q$ and
$m$), and $\mathbf{n}_{q}\in\mbox{$\mathbb{C}$}^{M_{q}\times 1}$ represents
the Additive White Gaussian Noise (AWGN) at node $q$ with variance
$\sigma_{q}^{2}$. In the UL communication, the symbol vector
$\hat{\mathbf{s}}_{m}\in\mbox{$\mathbb{C}$}^{d_{m}\times 1}$ used for the
estimation of $\mathbf{s}_{m}$ at the FD HBF node $k$ is derived as

$\displaystyle\hat{\mathbf{s}}_{m}\triangleq$
$\displaystyle\left(\mathbf{U}_{k}^{(\mathrm{BB})}\right)^{\rm
H}\left(\left(\mathbf{U}_{k}^{(\mathrm{RF})}\right)^{\rm
H}\mathbf{H}_{k,k}\mathbf{V}_{k}^{(\mathrm{RF})}+\mathbf{C}_{k}\right)\mathbf{V}_{k}^{(\mathrm{BB})}\mathbf{s}_{k}$
$\displaystyle+\left(\mathbf{U}_{k}^{(\mathrm{BB})}\right)^{\rm
H}\left(\mathbf{U}_{k}^{(\mathrm{RF})}\right)^{\rm
H}\left(\mathbf{H}_{k,m}\mathbf{V}_{m}\mathbf{s}_{m}+\mathbf{n}_{k}\right),$
(4)

where $\mathbf{H}_{k,k}\in\mbox{$\mathbb{C}$}^{M_{k}\times N_{k}}$ denotes the
SI channel seen at the RX antennas of node $k$ due to its own DL transmission,
$\mathbf{H}_{k,m}\in\mbox{$\mathbb{C}$}^{M_{k}\times N_{m}}$ is the UL channel
gain matrix (i.e., between nodes $k$ and $m$), and
$\mathbf{n}_{k}\in\mbox{$\mathbb{C}$}^{M_{k}\times 1}$ denotes the received
AWGN at node $k$ with variance $\sigma_{k}^{2}$. The first term in (II-C)
describes the residual SI signal after analog cancellation and A/D TX/RX
beamforming, while its second term contains the A/D RX combined signal
transmitted from node $m$ plus AWGN. In contrast to [10], $\mathbf{C}_{k}$
needs to cancel the SI channel $(\mathbf{U}_{k}^{(\mathrm{RF})})^{\rm
H}\mathbf{H}_{k,k}\mathbf{V}_{k}^{(\mathrm{RF})}$, which is a matrix of
dimension $M_{k}^{\mathrm{(RF)}}\times N_{k}^{\mathrm{(RF)}}$ and not the
actual $M_{k}\times N_{k}$ SI channel $\mathbf{H}_{k,k}$.

##  III Joint Design Problem Formulation

We focus on the FD HBF node $k$ in Fig. 1 and present a sum-rate optimization
framework for the joint design of $\mathbf{C}_{k}$,
$\mathbf{V}_{k}^{(\mathrm{RF})}$, $\mathbf{V}_{k}^{(\mathrm{BB})}$,
$\mathbf{U}_{k}^{(\mathrm{RF})}$, and $\mathbf{U}_{k}^{(\mathrm{BB})}$. Using
the notation
$\mathbf{V}_{k}\triangleq\mathbf{V}_{k}^{(\mathrm{RF})}\mathbf{V}_{k}^{(\mathrm{BB})}$
and assuming Gaussian signaling and capacity-achieving combining at node $q$,
the achievable DL rate that is a function of the A/D TX precoding matrices
$\mathbf{V}_{k}^{(\mathrm{RF})}$ and $\mathbf{V}_{k}^{(\mathrm{BB})}$ of node
$k$ as well as the digital TX precoder $\mathbf{V}_{m}$ of node $m$, is given
by

$\mathcal{R}_{\rm DL}=\log_{2}\left({\rm
det}\left(\mathbf{I}_{M_{q}}+\mathbf{H}_{q,k}\mathbf{V}_{k}\mathbf{V}_{k}^{\rm
H}\mathbf{H}_{q,k}^{\rm H}\mathbf{Q}_{q}^{-1}\right)\right),$ (5)

where $\mathbf{Q}_{q}\in\mathbb{C}^{M_{q}\times M_{q}}$ denotes the covariance
matrix of the Interference-plus-Noise (IpN) at node $q$ that is obtained as

$\mathbf{Q}_{q}\triangleq\mathbf{H}_{q,m}\mathbf{V}_{m}\mathbf{V}_{m}^{\rm
H}\mathbf{H}_{q,m}^{\rm H}+\sigma_{q}^{2}\mathbf{I}_{M_{q}}.$ (6)

We hereinafter assume that there is no inter-node interference between the HD
multi-antenna nodes $q$ and $m$ due to, for example, appropriate node
scheduling [7] for the FD operation at node $k$. The latter assumption
translates to setting the channel matrix between those involved nodes in (3)
as $\mathbf{H}_{q,m}=\mathbf{0}_{M_{q}\times N_{k}}$, which means that (6)
simplifies to $\mathbf{Q}_{q}=\sigma_{q}^{2}\mathbf{I}_{M_{q}}$.

For the computation of the achievable UL rate, we use the notation
$\mathbf{U}_{k}\triangleq\mathbf{U}_{k}^{(\mathrm{RF})}\mathbf{U}_{k}^{(\mathrm{BB})}$
to express this rate as a function of the A/D RX combiners
$\mathbf{U}_{k}^{(\mathrm{RF})}$ and $\mathbf{U}_{k}^{(\mathrm{BB})}$, the A/D
TX precoders $\mathbf{V}_{k}^{(\mathrm{RF})}$ and
$\mathbf{V}_{k}^{(\mathrm{BB})}$, and the analog cancellation matrix
$\mathbf{C}_{k}$ of node $k$ as well as of the digital TX precoder
$\mathbf{V}_{m}$ of node $m$. Using (II-C), the UL rate is given by

$\mathcal{R}_{\rm UL}=\log_{2}\left({\rm
det}\left(\mathbf{I}_{d_{m}}+\mathbf{U}_{k}^{\rm
H}\mathbf{H}_{k,m}\mathbf{V}_{m}\mathbf{V}_{m}^{\rm H}\mathbf{H}_{k,m}^{\rm
H}\mathbf{U}_{k}\mathbf{Q}_{k}^{-1}\right)\right).$ (7)

where $\mathbf{Q}_{k}\in\mathbb{C}^{d_{m}\times d_{m}}$ denotes the IpN
covariance matrix after A/D RX combining at node $q$, which can be expressed
as

$\begin{split}\mathbf{Q}_{k}\triangleq&\left(\mathbf{U}_{k}^{(\mathrm{BB})}\right)^{\rm
H}\tilde{\mathbf{H}}_{k,k}\mathbf{V}_{k}^{(\mathrm{BB})}\left(\mathbf{V}_{k}^{(\mathrm{BB})}\right)^{\rm
H}\tilde{\mathbf{H}}_{k,k}^{\rm H}\mathbf{U}_{k}^{(\mathrm{BB})}\\\
&+\sigma_{k}^{2}\left(\mathbf{U}_{k}^{(\mathrm{BB})}\right)^{\rm
H}\left(\mathbf{U}_{k}^{(\mathrm{RF})}\right)^{\rm
H}\mathbf{U}_{k}^{(\mathrm{RF})}\mathbf{U}_{k}^{(\mathrm{BB})}.\end{split}$
(8)

In the latter expression,
$\tilde{\mathbf{H}}_{k,k}\in\mbox{$\mathbb{C}$}^{M_{k}^{\mathrm{(RF)}}\times
N_{k}^{\mathrm{(RF)}}}$ denotes the effective SI channel after performing
analog TX/RX beamforming and analog cancellation, which is defined as

$\tilde{\mathbf{H}}_{k,k}\triangleq\left(\mathbf{U}_{k}^{(\mathrm{RF})}\right)^{\rm
H}\mathbf{H}_{k,k}\mathbf{V}_{k}^{(\mathrm{RF})}+\mathbf{C}_{k}.$ (9)

Using the expressions (5) and (7) for the achievable DL and UL rates,
respectively, the sum-rate optimization problem for the joint design of the
analog canceller and the A/D TX/RX beamformers is mathematically expressed as

$\begin{split}&\mathcal{OP}:\max_{\mathbf{C}_{k},\mathbf{V}_{k}^{(\mathrm{RF})},\mathbf{V}_{k}^{(\mathrm{BB})},\mathbf{U}_{k}^{(\mathrm{RF})},\mathbf{U}_{k}^{(\mathrm{BB})}}\mathcal{R}_{\rm
DL}+\mathcal{R}_{\rm UL}\\\ &\textrm{s.t.}\leavevmode\nobreak\
\leavevmode\nobreak\
\mathrm{tr}\\{\mathbf{V}_{k}^{(\mathrm{RF})}\mathbf{V}_{k}^{(\mathrm{BB})}\left(\mathbf{V}_{k}^{(\mathrm{BB})}\right)^{\rm
H}\left(\mathbf{V}_{k}^{(\mathrm{RF})}\right)^{\rm H}\\}\leq{\rm
P}_{k},\,\,({\rm C1})\\\
&\mathbf{C}_{k}=\mathbf{L}_{3}\mathbf{L}_{2}\mathbf{L}_{1}\,\,{\rm
with}\,\,\eqref{Eq:L_1_L_3}\,\,{\rm and}\,\,[\mathbf{L}_{2}]_{i,j}=0\,\,{\rm
for}\,\,i\neq j,\hskip 5.97527pt({\rm C2})\\\
&\left\|\left[\tilde{\mathbf{H}}_{k,k}\mathbf{V}_{k}^{(\mathrm{BB})}\right]_{(j,:)}\right\|^{2}\leq\rho_{\rm
A}\,\,\forall j=1,2,\ldots,M_{k}^{\mathrm{(RF)}},\hskip 4.97931pt({\rm C3})\\\
&\mathbf{u}_{j}\in\mathbb{F}_{\rm RX}\,\,\forall j\,\,{\rm
and}\,\,\mathbf{v}_{n}\in\mathbb{F}_{\rm TX}\,\,\forall
n=1,2,\ldots,N_{k}^{({\rm RF})},({\rm C4})\end{split}$

where constraint $({\rm C1})$ relates to the average TX power at node $k$ and
constraint $({\rm C2})$ refers to the hardware capabilities of the analog
canceller. Constraint $({\rm C3})$ imposes the threshold $\rho_{\rm
A}\in\mbox{$\mathbb{R}$}$ on the average power of the residual SI signal after
analog cancellation and analog TX/RX beamforming. Finally, constraint $({\rm
C4})$ refers to the predefined TX and RX beam codebooks. To tackle
$\mathcal{OP}$, which is a nonconvex problem with nonconvex constraints, we
adopt a similar to [10] decoupled way that in this case requires at most
$\alpha_{\max}\triangleq N_{k}^{(\mathrm{RF})}-1$ iterations including closed
form expressions for the design parameters. We first solve for
$\mathbf{C}_{k}$, $\mathbf{V}_{k}^{(\mathrm{RF})}$,
$\mathbf{V}_{k}^{(\mathrm{BB})}$, and $\mathbf{U}_{k}^{(\mathrm{RF})}$
maximizing the DL rate, and then find $\mathbf{U}_{k}^{(\mathrm{BB})}$
maximizing the UL rate. Specifically, we formulate the following optimization
subproblem for the design of $\mathbf{C}_{k}$,
$\mathbf{V}_{k}^{(\mathrm{RF})}$, $\mathbf{V}_{k}^{(\mathrm{BB})}$, and
$\mathbf{U}_{k}^{(\mathrm{RF})}$:

$\begin{split}\mathcal{OP}1:&\max_{\mathbf{C}_{k},\mathbf{V}_{k}^{(\mathrm{RF})},\mathbf{V}_{k}^{(\mathrm{BB})},\mathbf{U}_{k}^{(\mathrm{RF})}}\mathcal{R}_{\rm
DL}\leavevmode\nobreak\ \leavevmode\nobreak\ \textrm{s.t.}\leavevmode\nobreak\
\leavevmode\nobreak\ ({\rm C1}),\,({\rm C2}),\,\text{and}\,({\rm
C3}).\end{split}$

Algorithm 1 Digital TX Precoder Design

1:Input: ${\rm P}_{k}$, $\mathbf{V}_{k}^{(\mathrm{RF})}$ and
$\mathbf{U}_{k}^{(\mathrm{RF})}$ solving $\mathcal{OP}2$, $\mathbf{H}_{k,k}$,
and $\mathbf{H}_{q,k}$ as well as a realization of $\mathbf{C}_{k}$ for a
given $N$ satisfying constraint $({\rm C2})$.

2:Set
$\tilde{\mathbf{H}}_{k,k}=\left(\mathbf{U}_{k}^{(\mathrm{RF})}\right)^{\rm
H}\mathbf{H}_{k,k}\mathbf{V}_{k}^{(\mathrm{RF})}+\mathbf{C}_{k}$.

3:Obtain $\mathbf{D}_{k}$ with the $N_{k}^{\mathrm{(RF)}}$ right-singular
vectors of $\widetilde{\mathbf{H}}_{k,k}$ corresponding to the singular values
in descending order.

4:for $\alpha=\alpha_{\max},\alpha_{\max}-1,\ldots,2$ do

5: Set
$\mathbf{F}_{k}=[\mathbf{D}_{k}]_{(:,N_{k}^{\mathrm{(RF)}}-\alpha+1:N_{k}^{\mathrm{(RF)}})}$.

6: Set $\mathbf{G}_{k}$ as the optimum precoding for the effective

7: DL MIMO channel
$\mathbf{H}_{q,k}\mathbf{V}_{k}^{(\mathrm{RF})}\mathbf{F}_{k}$ given ${\rm
P}_{k}$.

8: if
$\|[\widetilde{\mathbf{H}}_{k,k}\mathbf{F}_{k}\mathbf{G}_{k}]_{(i,:)}\|^{2}\leq\rho_{\rm
A}$ $\forall i=1,\ldots,M_{k}^{\mathrm{(RF)}}$, then

9: Output $\mathbf{V}_{k}^{(\mathrm{BB})}=\mathbf{F}_{k}\mathbf{G}_{k}$ and
stop the algorithm.

10: end if

11:end for

12:Set $\mathbf{F}_{k}=[\mathbf{D}_{k}]_{(:,N_{k}^{\mathrm{(RF)}})}$ and
$\mathbf{G}_{k}={\rm P}_{k}^{1/2}$.

13:if
$|[\widetilde{\mathbf{H}}_{k,k}\mathbf{F}_{k}\mathbf{G}_{k}]_{i}|^{2}\leq\rho_{\rm
A}$ $\forall i=1,\ldots,M_{k}^{\mathrm{(RF)}}$, then

14: Output $\mathbf{V}_{k}^{(\mathrm{BB})}=\mathbf{F}_{k}\mathbf{G}_{k}$ and
stop the algorithm.

15:else

16: Output that the $\mathbf{C}_{k}$ realization does not meet

17: the residual SI constraint.

18:end if

For the solution of the latter problem we use an alternating optimization
approach. First, we find $\mathbf{V}_{k}^{(\mathrm{RF})}$ and
$\mathbf{U}_{k}^{(\mathrm{RF})}$ constrained as $({\rm C4})$ that boost the DL
rate, while minimizing the SI signal before any other form of cancellation.
Particularly, we perform the following exhaustive search:

$\begin{split}\mathcal{OP}2:&\max_{\mathbf{V}_{k}^{(\mathrm{RF})},\mathbf{U}_{k}^{(\mathrm{RF})}}\frac{\left\|\mathbf{H}_{q,k}\mathbf{V}_{k}^{(\mathrm{RF})}\right\|_{\rm
F}}{\left\|\left(\mathbf{U}_{k}^{(\mathrm{RF})}\right)^{\rm
H}\mathbf{H}_{k,k}\mathbf{V}_{k}^{(\mathrm{RF})}\right\|_{\rm
F}}\leavevmode\nobreak\ \leavevmode\nobreak\ \textrm{s.t.}\leavevmode\nobreak\
\leavevmode\nobreak\ ({\rm C4}).\end{split}$

In the sequel, given the solution for $\mathcal{OP}2$ and supposing that the
available number of analog canceller taps $N$ and a realization of
$\mathbf{C}_{k}$ satisfying $({\rm C2})$ are given, we seek for
$\mathbf{V}_{k}^{(\mathrm{BB})}$ maximizing the DL rate while meeting $({\rm
C1})$ and $({\rm C3})$. The latter procedure is repeated for all allowable
realizations of $\mathbf{C}_{k}$ for the given $N$ in order to find the best
$\mathbf{V}_{k}^{(\mathrm{BB})}$ solving $\mathcal{OP}1$; this procedure is
summarized in Algorithm 1. The values for $\mathbf{C}_{k}$,
$\mathbf{V}_{k}^{(\mathrm{RF})}$, $\mathbf{V}_{k}^{(\mathrm{BB})}$, and
$\mathbf{U}_{k}^{(\mathrm{RF})}$ solving $\mathcal{OP}1$ are finally
substituted into the achievable UL rate expression $\mathcal{R}_{\rm UL}$ in
(7). The $\mathbf{U}_{k}^{(\mathrm{BB})}$ maximizing this point-to-point MIMO
rate is obtained in closed form using [17, Sec. 4.2].

##  IV Simulation Results and Discussion

In this section, we investigate the performance of the considered $3$-user
bidirectional communication system for the case where
$N_{k}^{(\mathrm{RF})}=M_{q}=4$, $M_{k}^{(\mathrm{RF})}=2$, $N_{m}=1$, and
$N_{k}^{(\mathrm{A})}=M_{k}^{(\mathrm{A})}=\\{16,128\\}$. We have assumed that
the antennas at the FD HBF node $k$ are arranged in Uniform Linear Arrays
(ULAs) with $\lambda/2$ space separation between adjacent elements, where
$\lambda$ is the wavelength. The distance and angle between the TX and RX ULAs
at node $k$ were set respectively to $d=2\lambda$ and $\omega=\pi/6$ [12]. The
DL and UL channels were simulated as millimeter wave clustered channels, as
described in [13, eq. (2)] with pathloss $110$dB. In addition, the SI channel
was modeled as Rician [13, eq. (7)] with $K$-factor $35$dB and pathloss
$40$dB. The RX noise floors at both nodes $k$ and $q$ were set to $-110$dBm,
resulting in an effective dynamic range of $62$dB for $14$-bit ADCs with a
$10$dB peak-to-average power ratio. Hence, to avoid saturation, the residual
SI power after analog cancellation at the input of each RX RF chain needs to
be below $-47$dBm. Non-ideal $N$-tap analog cancellation has been considered
as in [10], and for the analog TX/RX beamformers, we have used beam codebooks
based on the discrete Fourier transform matrix.

The achievable FD rate as a function of the transmit powers of nodes $k$ and
$m$ is illustrated in Fig. 2 for $N=4$ taps for the proposed analog canceller,
which translates to $50\%$ reduction in the number of taps compared to a case
that connects all outputs of the TX RF chains to every input to the RX RF
chains. For the rate results, we averaged over $1000$ independent channel
realizations and calculated the FD rate with the proposed Algorithm 1, as well
as the achievable HD rate. It is shown in the figure that all rates increase
with increasing transmit power, and no rate saturation is exhibited for the
proposed FD HBF technique. The latter trend witnesses the effective
adaptability of Algorithm 1 to the SI conditions for both considered pairs of
$N_{k}^{(\mathrm{A})}$ and $M_{k}^{(\mathrm{A})}$. As an indicative example,
it is shown that for $40$dBm transmit powers, the proposed approach results in
a $52$bps/Hz achievable rate, which is around $1.7$ times more than the
achievable HD rate.

Figure 2: Average FD and HD rates vs the transmit powers in dBm for
$N_{k}^{(\mathrm{RF})}=M_{q}=N=4$, $M_{k}^{(\mathrm{RF})}=2$, $N_{m}=1$, and
different values for $N_{k}^{(\mathrm{A})}$ and $M_{k}^{(\mathrm{A})}$ at node
$k$.

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