# Full-Duplex mmWave Massive MIMO Systems:
A Joint Hybrid Precoding/Combining and
Self-Interference Cancellation Design

Asil Koc, Tho Le-Ngoc  This work was supported in part by Huawei Technologies
Canada and in part by the Natural Sciences and Engineering Research Council of
Canada. A. Koc and T. Le-Ngoc are with the Department of Electrical and
Computer Engineering, McGill University, Montréal, QC H3A 0E9, Canada (e-mail:
<EMAIL_ADDRESS>tho.le-ngoc@mcgill.ca). (Corresponding author: Asil
Koc)

###### Abstract

Millimeter-wave (mmWave) massive multiple-input multiple-output (MIMO) systems
have been considered as one of the primary candidates for the fifth generation
(5G) and beyond 5G wireless communication networks to satisfy the ever-
increasing capacity demands. Full-duplex technology can further enhance the
advantages of mmWave massive MIMO systems. However, the strong self-
interference (SI) is the major limiting factor in the full-duplex technology.
Hence, this paper proposes a novel angular-based joint hybrid
precoding/combining (AB-JHPC) technique for the full-duplex mmWave massive-
MIMO systems. Our primary goals are listed as: (i) improving the self-
interference cancellation (SIC), (ii) increasing the intended signal power,
(iii) decreasing the channel estimation overhead, (iv) designing the massive
MIMO systems with a low number of RF chains. First, the RF-stage of AB-JHPC is
developed via slow time-varying angle-of-departure (AoD) and angle-of-arrival
(AoA) information. A joint transmit/receive RF beamformer design is proposed
for covering (excluding) the AoD/AoA support of intended (SI) channel. Second,
the BB-stage of AB-JHPC is constructed via the reduced-size effective intended
channel. After using the well-known singular value decomposition (SVD)
approach at the BB-stage, we also propose a new semi-blind minimum mean square
error (S-MMSE) technique to further suppress the residual SI power by using
AoD/AoA parameters. Thus, the instantaneous SI channel knowledge is not needed
in the proposed AB-JHPC technique. Finally, we consider a transfer block
architecture to minimize the number of RF chains. The numerical results
demonstrate that the SI signal is remarkably canceled via the proposed AB-JHPC
technique. It is shown that AB-JHPC achieves $85.7$ dB SIC and the total
amount of SIC almost linearly increases via antenna isolation techniques. We
observe that the proposed full-duplex mmWave massive MIMO systems double the
achievable rate capacity compared to its half-duplex counterpart as the
antenna array size increases and the transmit/receive antenna isolation
improves. Moreover, the proposed S-MMSE algorithm provides considerably high
capacity than the conventional SVD approach.

###### Index Terms:

Full-duplex, massive MIMO, millimeter wave communications, hybrid precoding,
hybrid combining, low CSI overhead, RF chain, semi-blind MMSE, energy
efficiency, imperfect angle information.

##  I Introduction

Massive multiple-input multiple-output (MIMO) systems operating in millimeter
wave (mmWave) frequency bands are the primary candidates for fifth-generation
(5G) and beyond 5G (B5G) wireless communication networks [1, 2, 3, 4]. Massive
MIMO systems with excessively large antenna arrays is a key technology to
address the massive machine-type communications (mMTC) and enhanced mobile
broadband (eMBB) requirements to support the development of various emerging
applications (e.g., virtual reality, augmented reality, autonomous driving,
internet of things, smart cities, etc.) [3]. Furthermore, the wide range of
available bandwidth in the mmWave frequency bands does not only address the
bandwidth shortage issue in the current wireless networks but also provides
greatly enhanced capacity. However, different from to the conventional rich
scattering multipath channels, a limited-scattering propagation is experienced
in the mmWave frequencies. For ensuring the sufficient received signal power
and compensating the severe path loss, the massive MIMO technology is widely
considered as a useful application in the mmWave communications [4, 5, 6, 7].
Because the high beamforming gain in the massive MIMO systems can focus the
signal energy through the desired limited-scattering regions [5].
Additionally, the shorter wavelengths in the mmWave frequency enable the
utilization large antenna arrays under the area requirements in practical
applications [6].

Massive MIMO systems have been originally considered for the half-duplex
communications (i.e., time-division duplex (TDD) [8, 9, 10] and frequency-
division duplex (FDD) [10, 11, 12]). The expected impacts of mmWave massive
MIMO systems can be further extended via full-duplex communications, which
theoretically doubles the capacity via more efficient utilization of the
limited resources (i.e., time and frequency). However, the performance of
full-duplex communications is severely affected by the strong self-
interference (SI) occurred due to the simultaneous transmission and reception
over the same frequency band, which is not present in the half-duplex
communications. On the other hand, the recent developments in antenna
technology and signal processing techniques make full-duplex communications
more applicable via advanced SI cancellation (SIC) techniques, e.g., passive
(propagation domain) antenna isolation, analog cancellation and digital
cancellation [13, 14, 15, 16, 17, 18]. The recent practical measurements in
[17] show that the passive antenna isolation achieves $60-65$ dB SIC.
Furthermore, in [18], the antenna isolation based SIC is enhanced up to $74$
dB via absorptive shielding, cross polarization and directional isolation. The
quality of SIC can be further improved via analog and digital cancellation
techniques [13, 14, 15, 16], however, the SI component cannot be completely
removed. Even though the residual SI limits the performance of full-duplex
communications, the experimental and theoretical studies in the literature
demonstrate that full-duplex transmission can outperform the conventional
half-duplex as the quality of the SIC increases [13]. Additionally, the full-
duplex massive MIMO systems enable enhanced estimation and cancellation of the
strong SI channel by means of high degrees of freedom provided by the large
antenna arrays [19, 20, 21, 22, 23, 24, 25, 26, 27].

In order to ensure reliable transmission quality for the massive MIMO systems,
the precoding (combining) at the transmitter (receiver) is a crucial signal
processing procedure for the downlink (uplink) transmission. The conventional
MIMO systems widely consider the single-stage fully-digital
precoding/combining (FDPC) architecture [28]. However, the large antenna
arrays in the massive MIMO systems induce two interesting challenges: (i)
higher hardware cost/complexity as well as the power consumption (i.e., one
dedicated power-hungry radio frequency (RF) chain per each antenna), (ii)
longer channel estimation overhead (i.e., the requirement of full
instantaneous channel state information (CSI) in FDPC). Hence, even though the
large antenna arrays can improve the capacity, the energy-efficiency sharply
decreases with the utilization of more power-hungry RF chains. The single-
stage fully analog precoding/combining (FAPC) technique significantly reduces
the hardware cost/complexity by minimizing the RF chain utilization [29, 30,
31], however, FAPC achieves poor capacity compared to FDPC. As a promising
solution, the two-stage hybrid precoding/combining (HPC) technique has been
proposed to address the aforementioned trade-off between FDPC and FAPC, while
providing the comparable capacity as in FDPC [32, 31, 33]. In the HPC
architecture, the analog RF-stage and digital baseband(BB)-stage are connected
to each other via RF chains, where the number of RF chains is chosen between
the number of antennas and the independent data streams. Both half-duplex and
full-duplex transmission schemes have been investigated for the HPC design in
the massive MIMO systems in [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]
and in [19, 20, 21, 22, 23, 24, 25, 26, 27], respectively.

###  I-A Related Works

Preliminary studies in the HPC technique employ the half-duplex
communications. There are two major approaches regarding the RF-stage design
in the hybrid architectures: (i) utilizing full instantaneous CSI bringing in
large channel estimation overhead [34, 35, 37, 38, 36], (ii) using the slow
time-varying channel characteristics (e.g., channel covariance matrix, angle-
of-departure (AoD), angle-of-arrival (AoA)) [39, 40, 41, 44, 42, 43, 45].
Thus, the latter approach only uses the reduced-size fast time-varying
effective CSI at the BB-stage design. Therefore, the hybrid architectures in
[39, 40, 41, 42, 43, 44, 45] can address two interesting challenges in the
massive MIMO systems: (i) the reduced number of RF chains for low power
consumption and the hardware cost/complexity, (ii) the low CSI overhead for
shorter training sequence. In [41], a two-stage angular-based hybrid-precoding
(HP) technique is proposed for the downlink massive MIMO systems, where the
CSI overhead is greatly reduced by designing the RF-stage via AoD parameters.
The numerical results in [41] show that the angular-based HP significantly
enhances the energy efficiency compared to the single-stage fully-digital
precoding (FDP) by means of the reduced number of RF chains. Moreover, the
angular-based HP improves the sum-rate capacity with respect to its two-stage
HP counterparts as eigen-beamforming based HP [40], block-diagonalization
based HP [39], non-orthogonal angle space based HP [35]. In [42], the fully-
connected and sub-connected hybrid architectures are analyzed for the angular-
based HP technique. Then, a swarm intelligence based power allocation
algorithm is proposed to further enhance the sum-rate capacity in [43].
Afterwards, the angular-based HP technique is extended for the multi-cell
systems in [44], where the downlink cooperation strategies are proposed to
mitigate the inter-cell interference. Considering both downlink and uplink
transmission in the massive MIMO systems, an angular-based HPC technique is
also proposed in [45], where the effect of low-resolution digital-to-analog
and analog-to-digital converters is also investigated.

Recently, the precoding/combining techniques are also studied for the full-
duplex communication systems in [19, 20, 21, 22, 23, 24, 25, 26, 27]. First,
the performance of full-duplex MIMO and massive MIMO systems is investigated
for the FDPC architecture in [19, 20, 21], where the promising results imply
that full-duplex systems can outperform its half-duplex counterpart. Then, the
FAPC architecture is designed for the full-duplex massive MIMO systems in
[22], where the performance gap between the fully-digital and fully-analog
structure is represented. Afterwards, the HPC structure is built for the full-
duplex massive MIMO systems in [23, 24], where the authors assume the
availability of perfect SI channel knowledge for the SIC algorithm. The full-
duplex relay backhaul system is considered in [25], where the SIC algorithm
aims to mitigate the SI component via the HPC design based on the perfect SI
channel knowledge. Similarly, [26] also investigates the full-duplex relay
system, whereas, the residual SI channel is modeled as Gaussian noise. The
effect of low-resolution phase-shifters at the RF-stage of HPC is analyzed in
[27], where the numerical results show that when 25 dB SIC is provided by
antenna isolation, the HPC with 1-bit phase-shifters can improve the total SIC
up to 44 dB. It is worthwhile to note that the availability of full
instantaneous CSI is needed for all above-mentioned FDPC, FAPC and HPC
architectures developed for the full-duplex massive MIMO systems.

###  I-B Contribution and Organization

This paper proposes a novel angular-based joint HPC (AB-JHPC) technique for
the full-duplex mmWave massive MIMO systems. The proposed AB-JHPC mainly
targets to (i) improve the quality of SIC, (ii) enhance the power of
desired/intended signal, (iii) decrease the CSI overhead size, (iv) reduce the
hardware cost/complexity and power consumption by employing less number of RF
chains. Table I provides a detailed comparison of this work with respect to
the state-of-the-art. The main contributions of this paper are summarized as
follows:

TABLE I: State-of-the-art for the precoding/combining techniques in full-duplex massive MIMO systems. Reference | Transmission Mode | Precoder/Combiner Design | CSI Overhead | SIC without Instantaneous
---|---|---|---|---
Full-Duplex | Half-Duplex | Hybrid | Digital | Analog | Full | Low | SI Channel Knowledge
[34, 35, 37, 38, 36] |  | ✓ | ✓ |  |  | ✓ |  |
[39, 40, 41, 44, 42, 43, 45] |  | ✓ | ✓ |  |  |  | ✓ |
[19, 20, 21] | ✓ |  |  | ✓ |  | ✓ |  |
[22] | ✓ |  |  |  | ✓ | ✓ |  |
[23, 24, 25, 26, 27] | ✓ |  | ✓ |  |  | ✓ |  |
This Paper | ✓ |  | ✓ |  |  |  | ✓ | ✓

  * •

Proposed Hybrid Precoding/Combining Technique: Considering both
transmission/reception and hybrid architecture, we aim to design four sub-
blocks in the AB-JHPC technique: (i) BB precoder, (ii) BB combiner, (iii)
transmit RF beamformer, (iv) receive RF beamformer. The transmit/receive RF
beamformer are built by the low-cost phase-shifters, which induces the
constant modulus (CM) constraint at the RF-stage design. Therefore, the
optimization problem for the total achievable-rate becomes non-convex due to
the CM constraint. Thus, the optimization problem in the proposed AB-JHPC is
divided into two steps: (i) joint RF beamformer, (ii) BB precoder/combiner.

  * •

RF Beamformer: In order to maximize the desired/intended signal power while
suppressing the SI power, the transmit/receive RF beamformers are constructed
via the slow time-varying AoD/AoA information. Hence, the RF-stage design does
not require any instantaneous channel knowledge. For improving the quality of
SIC, the joint RF beamformer generates the orthogonal transmit/receive beams
spanning (excluding) the AoD/AoA support of intended (SI) channel. As
presented in Section VI and in Table IV, the proposed RF-stage design is
capable of decreasing the CSI overhead size by $99.8\%$.

  * •

BB Precoder & BB Combiner: After developing the transmit/receive RF
beamformers, the BB-stage is designed via the reduced-size effective intended
channel seen from the BB-stage. First, the BB precoder and BB combiner are
constructed via the well-known singular value decomposition (SVD). Also, a new
semi-blind minimum mean square error (S-MMSE) algorithm is proposed for the BB
combiner. It is classified as semi-blind because the S-MMSE algorithm uses
only the fast time-varying effective intended channel coefficients. In other
words, the S-MMSE algorithm does not depend on the fast time-varying SI
channel coefficients, which might be impractical to be perfectly estimated.
Instead, similar to the RF-stage design, the S-MMSE-based BB combiner employs
the AoA support of the SI channel as the slow time-varying channel parameters.
Numerical results demonstrate that the proposed S-MMSE-based BB combiner
enhances the quality of SIC and provides superior achievable-rate performance
compared to the SVD-based BB combiner.

  * •

Self-Interference Cancellation: The illustrative results show that the SI
channel is greatly suppressed via the proposed joint transmit/receive RF
beamformer design. It is seen that the power of far-field (near-field) SI
channel is canceled by $81.5$ dB ($44.5$ dB) after the joint RF beamformer
design, while it preserves most of the intended channel power with only $2$ dB
degradation. Considering the SI occurred at a given data stream, we observe
$85.7$ dB SIC is achieved via only the AB-JHPC technique without applying any
antenna isolation. Moreover, the total amount of SIC can be enhanced further
enhanced via the advanced antenna isolation technique. After applying the
proposed AB-JHPC technique, we monitor a near-linear relationship between the
antenna isolation and the total amount of SIC.

  * •

Full-Duplex vs. Half-Duplex: The proposed full-duplex mmWave massive MIMO
systems remarkably outperform its half-duplex counterpart in terms of total
achievable rate capacity as the number of antennas and the antenna isolation
based SIC increase. However, the full-duplex to half-duplex ratio gain
slightly decreases as the number of data streams increases due to the enhanced
SI power.

The rest of this paper is organized as follows. The system and channel models
are introduced in Section II and Section III, respectively In Section IV, we
develop the transmit/receive RF beamformer and BB precoder/combiner in the
proposed AB-JHPC technique for the full-duplex mmWave massive MIMO systems.
Section V expresses the transfer block architecture to further reduce the
number of RF chains. The illustrative results are provided in Section VI.
Finally, the paper is concluded in Section VII. Table II summarizes the
various abbreviations used in this paper. For improving the clarity of
mathematical exposition, the frequently-used symbols are listed in Table III.

TABLE II: List of abbreviations.

2D | Two-dimensional
---|---
3D | Three-dimensional
3GPP | Third generation partnership project
5G | Fifth generation
AB-HPC | Angular-based HPC
AB-JHPC | Angular-based joint HPC
AoA | Angle-of-arrival
AoD | Angle-of-departure
B5G | Beyond 5G
BB | Baseband
CM | Constant modulus
CSI | Channel state information
eMBB | Enhanced mobile broadband
FAPC | Fully analog precoding/combining
FDP | Fully digital precoding
FDPC | Fully digital precoding/combining
HP | Hybrid precoding

HPC | Hybrid precoding/combining
---|---
ISI | Inter-symbol interference
LoS | Line-of-sight
MIMO | Multiple-input multiple-output
MMSE | Minimum mean square error
mMTC | Massive machine-type communications
mmWave | Millimeter wave
MSE | Mean square error
NLoS | Non-line-of-sight
PSD | Power spectral density
RF | Radio frequency
S-MMSE | Semi-blind minimum mean square error
SI | Self-interference
SIC | Self-interference cancellation
SVD | Singular value decomposition
ULA | Uniform linear array
URA | Uniform rectangular array

TABLE III: List of frequently-used symbols.

$i,j$ | Node index ($i,j\in\left\\{{1,2}\right\\}$, $i\neq j$)
---|---
${M_{t,i}}$ | # of transmit antenna
${M_{r,i}}$ | # of receive antenna
$N_{t,i}$ | # of transmit RF chains
$N_{r,i}$ | # of receive RF chains
${S_{i}}$ | # of data streams
$P_{T}$ | Transmission power
${{\bf{H}}_{i}}\in\mathbb{C}^{{M_{r,j}}\times M_{t,i}}$ | Intended channel
$\bm{\mathcal{H}}_{i}\in\mathbb{C}^{N_{r,j}\times N_{t,i}}$ | Effective intended channel
${{\bf{H}}_{\textrm{SI},i}\in{\mathbb{C}^{M_{r,j}\times M_{t,j}}}}$ | SI channel
$\bm{\mathcal{H}}_{\textrm{SI},i}\in\mathbb{C}^{N_{r,i}\times N_{t,i}}$ | Effective SI channel
${\bf{H}}_{\textrm{LoS},i}\in\mathbb{C}^{N_{r,i}\times N_{t,i}}$ | Near-field SI channel
${\bf{H}}_{\textrm{NLoS},i}\in\mathbb{C}^{N_{r,i}\times N_{t,i}}$ | Far-field SI channel

${\bf{B}}_{t,i}\in{\mathbb{C}^{N_{t,i}\times{S_{i}}}}$ | BB precoder
---|---
${\bf{B}}_{r,i}\in{\mathbb{C}^{{S_{j}}\times N_{r,i}}}$ | BB combiner
${\bf{F}}_{t,i}\in{\mathbb{C}^{M_{t,i}\times N_{t,i}}}$ | Transmit RF beamformer
${\bf{F}}_{r,i}\in{\mathbb{C}^{N_{r,i}\times M_{r,i}}}$ | Receive RF beamformer
${\bf G}_{i}\in\mathbb{C}^{L_{i}\times L_{i}}$ | Diagonal path gain matrix
${{\bf{\Phi}}_{t,i}}\in\mathbb{C}^{L_{i}\times M_{t,i}}$ | Transmit phase response matrix
${{\bf{\Phi}}_{r,j}}\in\mathbb{C}^{M_{r,j}\times L_{i}}$ | Receive phase response matrix
$L_{i}=\sum\nolimits_{c=1}^{C_{i}}L_{c,i}$ | # of paths
$C_{i}$ | # of scattering-clusters
${\theta_{r,c}^{\left(j\right)}}$, ${\psi_{r,c}^{\left(j\right)}}$ | Mean elevation and azimuth AoA
${\theta_{t,c}^{\left(i\right)}}$, ${\psi_{t,c}^{\left(i\right)}}$ | Mean elevation and azimuth AoD
${\delta_{r,c}^{\theta,\left(j\right)}}$, ${\delta_{r,c}^{\psi,\left(j\right)}}$ | Elevation and azimuth AoA spread
${\delta_{t,c}^{\theta,\left(i\right)}}$, ${\delta_{t,c}^{\psi,\left(i\right)}}$ | Elevation and azimuth AoD spread

Notation: Bold upper/lower case letters denote matrices/vectors.
$\left(\cdot\right)^{*}$, $\left(\cdot\right)^{T}$, $\left(\cdot\right)^{H}$
$\left\|\cdot\right\|_{2}$ and $\left\|\cdot\right\|_{F}$ represent the
complex conjugate, the transpose, the conjugate transpose, the $2$-norm and
the Frobenius norm of a vector or matrix, respectively. $\mathbf{I}_{K}$,
${\mathbb{E}}\left\\{\cdot\right\\}$, $\rm{tr}\left(\cdot\right)$ and
$\angle\left(\cdot\right)$ stand for $K\times K$ identity matrix, the
expectation operator, the trace operator and the argument of a complex number,
respectively. ${\bf X}\left(m,n\right)$ denotes the element at the
intersection of $m^{th}$ row and $n^{th}$ column.
$\mathbf{X}\otimes\mathbf{Y}$ denotes the Kronecker product of two matrices
$\mathbf{X}$ and $\mathbf{Y}$. We use ${{x}}\sim{\cal C}{\cal
N}\left({0,{\sigma}}\right)$ when ${x}$ is a complex Gaussian random variable
with zero-mean and variance ${\sigma}$. We use the ramp function as
$\left(x\right)^{+}=\max\left(0,x\right)$.

##  II System Model

Fig. 1 illustrates the system model for the full-duplex mmWave massive MIMO
systems, where both nodes employ the HPC architecture and operate in full-
duplex transmission mode. Hence, two-way data transmission and reception
operations are simultaneously performed over the same frequency band. As one
of the practical applications of the proposed system model, the wireless
backhaul links can be considered for connecting the multiple macro-cell base
stations with large antenna arrays [46, 25]. The wireless backhaul links can
also connect the macro-cell and small-cell base stations to enhance the cell-
edge performance [47]. Moreover, as specified in 3GPP standards, the massive
MIMO cellular systems have the ability to switch between point-to-point
(single-user) and multi-user transmission [48]. Therefore, the proposed system
model has also practical applicability considering the point-to-point massive
MIMO cellular systems.

Figure 1: Full-duplex mmWave massive MIMO systems with HPC architecture.

As seen from Fig. 1, when the node $i$ transmits data to the node $j$
($i,j\in\left\\{{1,2}\right\\}$ and $i\neq j$), the corresponding intended
channel matrix between them is denoted as
${{\bf{H}}_{i}}\in\mathbb{C}^{{M_{r,j}}\times M_{t,i}}$. Moreover,
${{\bf{H}}_{\textrm{SI},i}\in{\mathbb{C}^{M_{r,j}\times M_{t,j}}}}$ represents
the SI channel at the node $i$ due to the full-duplex transmission. According
to the HPC architecture, four sub-blocks are placed at each node: (i) the BB
precoder ${\bf{B}}_{t,i}\in{\mathbb{C}^{N_{t,i}\times{S_{i}}}}$, (ii) the BB
combiner ${\bf{B}}_{r,i}\in{\mathbb{C}^{{S_{j}}\times N_{r,i}}}$, (iii) the
transmit RF beamformer ${\bf{F}}_{t,i}\in{\mathbb{C}^{M_{t,i}\times N_{t,i}}}$
and (iv) the receive RF beamformer
${\bf{F}}_{r,i}\in{\mathbb{C}^{N_{r,i}\times M_{r,i}}}$. At the node $i$, the
transmitter and receiver are equipped with a uniform rectangular array
(URA)111 It is worthwhile to note that although uniform linear arrays (ULA) is
widely investigated in the literature due to its simple structure (e.g., in
[20, 21, 22, 24, 25, 26, 27, 34, 35, 36, 23]), URA is more advantageous in
practical deployment by fitting antennas on two-dimensional (2D) grid as shown
in [38, 40, 41, 42, 43, 44, 45, 49]. Moreover, the URA structure enables
three-dimensional (3D) beamforming by utilizing both azimuth and elevation
angles, which is not possible for ULA [50].  having
${M_{t,i}}={M_{t,i}^{\left(x\right)}}\times{M_{t,i}^{\left(y\right)}}$ and
${M_{r,i}}={M_{r,i}^{\left(x\right)}}\times{M_{r,i}^{\left(y\right)}}$
antennas, respectively. Here,
${M_{t,i}^{\left(x\right)}}~{}\big{(}{{M_{r,i}^{\left(x\right)}}}\big{)}$ and
${M_{t,i}^{\left(y\right)}}~{}\big{(}{{M_{r,i}^{\left(y\right)}}}\big{)}$
represent the transmit (receive) antennas along $x$-axis and $y$-axis,
respectively. As shown in Fig. 1, an antenna isolation block is placed between
the transmit and receive URAs for the propagation domain SIC techniques222 The
antenna isolation based SIC techniques have been investigated in the
literature (e.g., in [13, 14, 15, 16, 17, 18]). For instance, the numerical
results in [17] and [18] show that the antenna isolation can provide $60$ dB
and $74$ dB SIC, respectively. The design of antenna isolation block is beyond
the scope of this paper. . The BB precoder and transmit RF beamformer are
connected via $N_{t,i}$ RF chains at the node $i$. Similarly, $N_{r,i}$ RF
chains connect the BB combiner and receive RF beamformer. Therefore, there are
${M_{i}}=M_{t,i}+M_{r,i}$ antennas and $N_{i}=N_{t,i}+N_{r,i}$ RF chains at
the node $i$. Additionally, ${S_{i}}$ denotes the number of parallel data
streams transmitted from the node $i$ to the node $j$. In order to support
${S_{i}}$ parallel data streams, it is necessary to satisfy that
${S_{i}}\leq\min\big{(}{N_{t,i},N_{r,j}}\big{)}$. On the other hand, to reduce
the hardware cost/complexity of the full-duplex mmWave massive MIMO systems
with large antenna arrays, the number of RF chains at each node is
significantly smaller than the number of antennas (i.e., $N_{t,i}\ll M_{t,i}$
and $N_{r,i}\ll M_{r,i}$). It is important to note that when the point-to-
point massive MIMO cellular system is considered for the proposed system
model, the user node with few antenna elements has the flexibility of using
either FDPC or HPC. In this special case, the user node using FDPC simply
chooses the transmit/receive RF beamformers as identity matrices (i.e.,
${\bf{F}}_{t,i}={\bf{I}}_{M_{t,i}}$ and ${\bf{F}}_{r,i}={\bf{I}}_{M_{r,i}}$).

According to the HPC architecture, the precoded signal vector at the node $i$
is defined as
${{{\bf{s}}_{i}}={\bf{F}}_{t,i}{\bf{B}}_{t,i}{{\bf{d}}_{i}}\in\mathbb{C}^{M_{t,i}}}$,
where ${{\bf{d}}_{i}}\in\mathbb{C}^{S_{i}}$ is the data vector encoded by
i.i.d. Gaussian codebooks (i.e., i.i.d. entries of ${{\bf{d}}_{i}}$ follow the
distribution of $\mathcal{CN}\left(0,1\right)$, so we have
$\mathbb{E}\left\\{{\bf d}_{i}{\bf d}_{i}^{H}\right\\}={\bf I}_{S_{i}}$). The
precoded signal is transmitted by the node $i$, which satisfies a transmit
power constraint $P_{T}$ (i.e.,
$\mathbb{E}\left\\{\left\|{\bf{s}}_{i}\right\|_{2}^{2}\right\\}\leq P_{T}$).
Then, the received signal by the node $j$ is given by:

$\displaystyle{{\bf{r}}_{j}}$
$\displaystyle={{\bf{H}}_{i}}{{\bf{s}}_{i}}+{\bf{H}}_{\textrm{SI},j}{{\bf{s}}_{j}}+{{\bf{w}}_{j}}$
(1)
$\displaystyle=\underbrace{{{\bf{H}}_{i}}{\bf{F}}_{t,i}{\bf{B}}_{t,i}{{\bf{d}}_{i}}}_{{\rm{Intended}}{\rm{~{}Signal}}}+\underbrace{{\bf{H}}_{\textrm{SI},j}{\bf{F}}_{t,j}{\bf{B}}_{t,j}{{\bf{d}}_{j}}}_{{\rm{Self-
Interference}}({\rm{SI}})}+\underbrace{{{\bf{w}}_{j}}}_{{\rm{Noise}}},$

where ${{\bf{w}}_{j}}\sim\mathcal{CN}\big{(}{\bf
0},\sigma_{w}^{2}{\bf{I}}_{M_{r,j}}\big{)}$ is the complex circularly
symmetric Gaussian noise vector. After the hybrid combiner at the node $j$,
the combined signal can be written as follows:

$\displaystyle{{{\tilde{\bf r}}}_{j}}$
$\displaystyle={\bf{B}}_{r,j}{{\bf{F}}_{r,j}{{\bf{r}}_{j}}}$ (2)
$\displaystyle=\underbrace{{\bf{B}}_{r,j}\bm{\mathcal{H}}_{i}{\bf{B}}_{t,i}{{\bf{d}}_{i}}}_{{\rm{Intended~{}Signal}}}+\underbrace{{\bf{B}}_{r,j}\bm{\mathcal{H}}_{\textrm{SI},j}{\bf{B}}_{t,j}{{\bf{d}}_{j}}}_{{\rm{Self-
Interference}}({\rm{SI}})}+\underbrace{{\bf{B}}_{r,j}{\bf{F}}_{r,j}{{\bf{w}}_{j}}}_{{\rm{Modified~{}Noise}}},$

where $\bm{\mathcal{H}}_{i}={\bf F}_{r,j}{\bf H}_{i}{\bf
F}_{t,i}\in\mathbb{C}^{N_{r,j}\times N_{t,i}}$ and
$\bm{\mathcal{H}}_{\rm{SI},j}={{\bf{F}}_{r,j}{\bf{H}}_{\rm{SI},j}{\bf{F}}_{t,j}}\in\mathbb{C}^{N_{r,j}\times
N_{t,j}}$ are respectively the reduced-size effective intended and SI channels
seen from the BB-stages. Particularly, the combined signal at the $k^{th}$
data stream with $k=1,\cdots,S_{i}$ is given by:

$\displaystyle{{{\tilde{r}}}_{j,k}}$
$\displaystyle=\underbrace{{\bf{b}}_{r,j,k}^{H}\bm{\mathcal{H}}_{i}{\bf{b}}_{t,i,k}{{{d}}_{i,k}}}_{{\rm{Intended~{}Signal}}}+\underbrace{\sum\nolimits_{q\neq
k}^{S_{i}}{\bf{b}}_{r,j,k}^{H}\bm{\mathcal{H}}_{i}{\bf{b}}_{t,i,q}{{{d}}_{i,q}}}_{{\rm{Inter-
Symbol~{}Interference(ISI)}}}$ (3)
$\displaystyle+\underbrace{{\bf{b}}_{r,j,k}^{H}\bm{\mathcal{H}}_{\textrm{SI},j}{\bf{B}}_{t,j}{{\bf{d}}_{j}}}_{{\rm{Self-
Interference}}({\rm{SI}})}+\underbrace{{\bf{b}}_{r,j,k}^{H}{\bf{F}}_{r,j}{{\bf{w}}_{j}}}_{{\rm{Modified~{}Noise}}},$

where ${\bf{b}}_{t,i,k}\in\mathbb{C}^{N_{i}}$ is the $k^{th}$ column of BB
precoder ${\bf{B}}_{t,i}$ and ${\bf{b}}_{j,r,k}\in\mathbb{C}^{N_{j}}$ is the
$k^{th}$ row of BB combiner ${\bf{B}}_{j,r}$. Here, the inter-symbol
interference (ISI) can be well mitigated via the BB-stage design [36].
However, the strong SI term given in (2) and (3) can be classified as the
major limiting factor for the full-duplex communications. Thus, during the HPC
design, we aim to both maximize the intended signal power (i.e.,
$\big{\|}{{\bf{B}}_{r,j}{\bf{F}}_{r,j}{{\bf{H}}_{i}}{\bf{F}}_{t,i}{\bf{B}}_{t,i}{{\bf{d}}_{i}}}\big{\|}_{2}^{2}$)
and suppress the SI power (i.e.,
$\big{\|}{{\bf{B}}_{r,j}{\bf{F}}_{r,j}{\bf{H}}_{\textrm{SI},j}{\bf{F}}_{t,j}{\bf{B}}_{t,j}{{\bf{d}}_{j}}}\big{\|}_{2}^{2}$).
Therefore, a joint HPC design is needed to improve the performance of full-
duplex mmWave massive MIMO systems. Regarding the SIC algorithm, the RF
beamformers should satisfy the following condition as:

$\bm{\mathcal{H}}_{\textrm{SI},j}={{\bf{F}}_{r,j}{\bf{H}}_{\textrm{SI},j}{\bf{F}}_{t,j}}\approx{\bf
0}.$ (4)

The above approximate zero-condition can be achieved via the joint
transmit/receive RF beamformer design. It is worthwhile to remark that the SI
terms given in (2) and (3) do not exist, when the half-duplex communications
is employed at each node. By using the combined signal at the node $j$ given
in (2), the covariance matrix of the combined SI and modified noise is given
by:

$\displaystyle{{\bf{C}}_{tot,j}}$
$\displaystyle=\mathbb{E}\left\\{\big{(}{\tilde{\bf{r}}_{j}}-{\bf{B}}_{r,j}{\bm{\mathcal{H}}_{i}}{\bf{B}}_{t,i}{{\bf{d}}_{i}}\big{)}\big{(}{\tilde{\bf{r}}_{j}}-{\bf{B}}_{r,j}{\bm{\mathcal{H}}_{i}}{\bf{B}}_{t,i}{{\bf{d}}_{i}}\big{)}^{H}\right\\}$
(5)
$\displaystyle={\bf{B}}_{r,j}{\bm{\mathcal{H}}_{\textrm{SI},j}{\bf{B}}_{t,j}}{\bf{B}}_{t,j}^{H}\bm{\mathcal{H}}_{\textrm{SI},j}^{H}{\bf{B}}_{r,j}^{H}+\sigma_{w}^{2}{\bf{B}}_{r,j}{\bf{F}}_{r,j}{\bf{F}}_{r,j}^{H}{\bf{B}}_{r,j}^{H}.$

By using (2) and (5), the average achievable rate at the node $j$ for the
proposed full-duplex mmWave massive MIMO systems is derived as follows:

$\displaystyle R_{j}\hskip 8.61108pt$
$\displaystyle=\mathbb{E}\left\\{{\log_{2}}\left|{{{\bf{I}}_{{S_{j}}}}+{\bf{C}}_{tot,j}^{-1}{\bf{B}}_{r,j}{\bm{\mathcal{H}}}_{i}{\bf{B}}_{t,i}{\bf{B}}_{t,i}^{H}{\bm{\mathcal{H}}}_{i}^{H}{\bf{B}}_{r,j}^{H}}\right|\right\\},$
(6a) $\displaystyle\rm{s.t.}$ $\displaystyle~{}\bm{\mathcal{H}}_{i}={\bf
F}_{r,j}{\bf H}_{i}{\bf F}_{t,i}\in\mathbb{C}^{N_{r,j}\times
N_{t,i}},~{}i,j\in\left\\{{1,2}\right\\},~{}i\neq j,$ (6d)
$\displaystyle~{}\left|{\bf{F}}_{t,i}\left(m,n\right)\right|={\frac{1}{\sqrt{M_{t,i}}}},\left|{\bf{F}}_{r,j}\left(m,n\right)\right|={\frac{1}{\sqrt{M_{r,j}}}},\forall
m,n,$
$\displaystyle~{}\mathbb{E}\big{\\{}{\big{\|}{\bf{s}}_{i}\big{\|}_{2}^{2}}\big{\\}}={\rm{tr}}\big{(}{{\bf{F}}_{t,i}}{{\bf{B}}_{t,i}}{{\bf{B}}_{t,i}^{H}}{{\bf{F}}_{t,i}^{H}}\big{)}\leq{P_{T}},$

where (6d) is the effective channel matrix from the node $i$ to the node $j$,
(6d) indicates the CM property due to the utilization of phase-shifters at the
RF-stages as shown in Fig. 1, (6d) implies the maximum transmit power
constraint $P_{T}$. Then, the total achievable rate in the proposed full-
duplex mmWave massive MIMO system is given by:

$R_{{total}}=R_{1}+R_{2}\textrm{~{}~{}[bps/Hz]}.$ (7)

As seen from (5), (6a) and (7), it is necessary to jointly construct the BB
precoder, transmit RF beamformer, receive RF beamformer and BB combiner during
the HPC design for enhancing the overall system capacity while improving the
quality of SIC. The proposed joint HPC technique is presented in Section IV.

##  III Channel Model

As illustrated in Fig. 1, it is necessary to individually model both intended
channel ${\bf{H}}_{i}$ and SI channel ${\bf{H}}_{\textrm{SI},j}$. Hence, we
model both ot them throughout this section.

###  III-A Intended Channel

Owing to the limited-scattering propagation environment, the mmWave channels
experience only a few spatial paths different from the conventional rich-
scattering channels [7]. Thus, the intended channel from the node $i$ to the
node $j$ is modeled by $C_{i}$ scattering-clusters and $L_{c,i}$ paths in the
$c^{th}$ cluster with $c=1,2,\cdots,C_{i}$. In total, there are
$L_{i}=\sum\nolimits_{c=1}^{C_{i}}L_{c,i}$ paths between the transmitter and
receiver. Considering the 3D geometry-based mmWave channel model [51] and the
URA structure [52], the channel matrix from the node $i$ to the node $j$ can
be written as:

$\displaystyle{\bf{H}}_{i}$
$\displaystyle=\sum\limits_{c=1}^{{C}}\sum\limits_{l=1}^{{L_{c}^{\left(i\right)}}}\tau_{i,c_{l}}^{-\eta}{{g_{i,c_{l}}}}{{{\bm{\phi}}}_{r,j}}\left({{\gamma_{r,c_{l}}^{\left(x\right)}},{\gamma_{r,c_{l}}^{\left(y\right)}}}\right){\bm{\phi}}_{t,i}^{H}\big{(}{{\gamma_{t,c_{l}}^{\left(x\right)}},{\gamma_{t,c_{l}}^{\left(y\right)}}}\big{)}$
(8) $\displaystyle={{\bf{\Phi}}_{r,j}}{\bf{G}}_{i}{{\bf{\Phi}}_{t,i}},$

where $\tau_{i,c_{l}}$ and
${g_{i,c_{l}}\sim\mathcal{CN}\big{(}0,\frac{1}{L_{i}}\big{)}}$ are
respectively the distance and complex path gain of the $l^{th}$ path in the
$c^{th}$ cluster with the path index of
$c_{l}=l+\sum\nolimits_{q=1}^{c-1}L_{q}^{\left(i\right)}$, $\eta$ is the path
loss exponent,
${\bm{\phi}_{u,i}}\big{(}{\gamma_{u,c_{l}}^{\left(x\right)},\gamma_{u,c_{l}}^{\left(y\right)}}\big{)}\in\mathbb{C}^{M_{u,i}}$
with $u\in\left\\{t,r\right\\}$ is either the receive phase response vector
for $u=r$ or the transmit phase response vector for $u=t$, ${\bf
G}_{i}=\rm{diag}\big{(}\tau_{i,1}^{-\eta}g_{i,1},\cdots,\tau_{i,L_{i}}^{-\eta}g_{i,L_{i}}\big{)}\in\mathbb{C}^{L_{i}\times
L_{i}}$ is the diagonal path gain matrix,
${{\bf{\Phi}}_{r,j}}\in\mathbb{C}^{M_{r,j}\times L_{i}}$ and
${{\bf{\Phi}}_{t,i}}\in\mathbb{C}^{L_{i}\times M_{t,i}}$ are the receive and
transmit phase response matrices, respectively. Then, the phase response
vector is defined as:

$\displaystyle{\bm{\phi}_{u,i}}\left({{\gamma_{x},\gamma_{y}}}\right)$
$\displaystyle=\big{[}{1,{e^{j2\pi d{{\gamma_{x}}}}},\cdots,{e^{j2\pi
d({{M_{u,i}^{\left(x\right)}}-1}){{\gamma_{x}}}}}}\big{]}^{T}$ (9)
$\displaystyle\otimes\big{[}{1,{e^{j2\pi d{{\gamma_{y}}}}},\cdots,{e^{j2\pi
d({{M_{u,i}^{\left(y\right)}}-1}){{\gamma_{y}}}}}}\big{]}^{T},~{}u\in\left\\{t,r\right\\},$

where $d$ is the distance between two antenna elements normalized by the
wavelength. By using (8) and (9), the receive phase response matrix at node
$j$ and transmit phase response matrix at node $i$ are respectively given by:

$\displaystyle{{\bf{\Phi}}_{r,j}}=\left[{\begin{array}[]{*{20}{c}}{{\bm{\phi}_{r,j}^{H}}\big{(}{{\gamma_{r,1}^{\left(x\right)}}},{{\gamma_{r,1}^{\left(y\right)}}}\big{)}}\\\
{\vdots}\\\
{{\bm{\phi}_{r,j}^{H}}\big{(}{{\gamma_{r,L_{i}}^{\left(x\right)}}},{{\gamma_{r,L_{i}}^{\left(y\right)}}}\big{)}}\\\
\end{array}}\right]^{H},~{}{{\bf{\Phi}}_{t,i}}=\left[{\begin{array}[]{*{20}{c}}{{\bm{\phi}_{t,i}^{H}}\big{(}{{\gamma_{t,1}^{\left(x\right)}}},{{\gamma_{t,1}^{\left(y\right)}}}\big{)}}\\\
{\vdots}\\\
{{\bm{\phi}_{t,i}^{H}}\big{(}{{\gamma_{t,L_{i}}^{\left(x\right)}}},{{\gamma_{t,L_{i}}^{\left(y\right)}}}\big{)}}\\\
\end{array}}\right].$ (10)

At the node $j$ as the receiver,
${\gamma_{r,c_{l}}^{\left(x\right)}}=\sin\big{(}{{\theta_{r,c_{l}}^{\left(j\right)}}}\big{)}\cos\big{(}{{\psi_{r,c_{l}}^{\left(j\right)}}}\big{)}$
and
${\gamma_{r,c_{l}}^{\left(y\right)}}=\sin\big{(}{{\theta_{r,c_{l}}^{\left(j\right)}}}\big{)}\sin\big{(}{{\psi_{r,c_{l}}^{\left(j\right)}}}\big{)}$
are the coefficients carrying the elevation AoA and azimuth AoA parameters of
the $l^{th}$ path in the $c^{th}$ cluster, where
${\theta_{r,c_{l}}^{\left(j\right)}}\in\big{[}{{\theta_{r,c}^{\left(j\right)}}-{\delta_{r,c}^{\theta,\left(j\right)}}},{{\theta_{r,c}^{\left(j\right)}}+{\delta_{r,c}^{\theta,\left(j\right)}}}\big{]}$
with mean elevation AoA ${\theta_{r,c}^{\left(j\right)}}$ and elevation AoA
spread ${\delta_{r,c}^{\theta,\left(j\right)}}$, and
${\psi_{r,c_{l}}^{\left(j\right)}}\in\big{[}{{\psi_{r,c}^{\left(j\right)}}-{\delta_{r,c}^{\psi,\left(j\right)}}},{{\psi_{r,c}^{\left(j\right)}}+{\delta_{r,c}^{\psi,\left(j\right)}}}\big{]}$
with mean azimuth AoA ${\psi_{r,c}^{\left(j\right)}}$ and azimuth AoA spread
${\delta_{r,c}^{\psi,\left(j\right)}}$. Similarly, at the node $i$ as the
transmitter,
${\gamma_{t,c_{l}}^{\left(x\right)}}=\sin\big{(}{{\theta_{t,c_{l}}^{\left(i\right)}}}\big{)}\cos\big{(}{{\psi_{t,c_{l}}^{\left(i\right)}}}\big{)}$
and
${\gamma_{t,c_{l}}^{\left(y\right)}}=\sin\big{(}{{\theta_{t,c_{l}}^{\left(i\right)}}}\big{)}\sin\big{(}{{\psi_{t,c_{l}}^{\left(i\right)}}}\big{)}$
reflect the elevation AoD and azimuth AoD information of the $l^{th}$ path in
the $c^{th}$ cluster, where
${\theta_{t,c_{l}}^{\left(i\right)}}\in\big{[}{{\theta_{t,c}^{\left(i\right)}}-{\delta_{t,c}^{\theta,\left(i\right)}}},{{\theta_{t,c}^{\left(i\right)}}+{\delta_{t,c}^{\theta,\left(i\right)}}}\big{]}$
with mean elevation AoD ${\theta_{t,c}^{\left(i\right)}}$ and elevation AoD
spread ${\delta_{t,c}^{\theta,\left(i\right)}}$, and
${\psi_{t,c_{l}}^{\left(i\right)}}\in\big{[}{{\psi_{t,c}^{\left(i\right)}}-{\delta_{t,c}^{\psi,\left(i\right)}}},{{\psi_{t,c}^{\left(i\right)}}+{\delta_{t,c}^{\psi,\left(i\right)}}}\big{]}$
with mean azimuth AoD ${\psi_{t,c}^{\left(i\right)}}$ and azimuth AoD spread
${\delta_{t,c}^{\psi,\left(i\right)}}$.

As in [39, 40, 41, 44, 45], we consider that the intended channel has two
parts: (i) fast time-varying instantaneous CSI as given in (8) and (ii) slow
time-varying phase response matrices based on AoD/AoA information expressed in
(10).

###  III-B Self-Interference Channel

According to the two-way full-duplex transmission, the SI channel matrix
${\bf{H}}_{\textrm{SI}}^{\left(i\right)}$ is present at the node $i$. The
transmit and receive URAs at each node are placed next to each other as
illustrated in Fig. 2, where $D_{1}$ ($D_{2}$) is the distance between the
transmit and receive URAs along $x$-axis ($z$-axis) normalized by the
wavelength, $\Theta$ is the rotation angle of URA around $y$-axis.

Figure 2: Illustration of transmit and receive URAs with the near-field and
far-field SI channels based on LoS and NLoS paths.

As discussed earlier, the antenna isolation block is also included to suppress
the SI channel power, especially the strong near-field effect (see Fig. 1).
However, it cannot completely remove the near-field SI channel [18]. Hence, as
demonstrated in Fig. 2, the SI channel is composed of two parts: (i) the
residual near-field SI channel representing the direct line-of-sight (LoS)
paths after the antenna isolation based SIC, (ii) the far-field SI channel
representing the reflected non-line-of-sight (NLoS) paths [23, 24, 25]. Here,
we first define the complete SI channel matrix at the node $i$ as:

${\bf{H}}_{\textrm{SI},i}={\bf{H}}_{\textrm{LoS},i}+{\bf{H}}_{\textrm{NLoS},i}\in\mathbb{C}^{M_{r,i}\times
M_{t,i}},$ (11)

where ${\bf{H}}_{\textrm{LoS},i}\in\mathbb{C}^{M_{r,i}\times M_{t,i}}$ is the
residual near-field SI component and
${\bf{H}}_{\textrm{NLoS},i}\in\mathbb{C}^{M_{r,i}\times M_{t,i}}$ is the far-
field SI component. The channel model given in (8) assumes the planar
wavefront, which requires two communication nodes to be far apart. It makes
the aforementioned channel model given in (8) an unrealistic model for
${\bf{H}}_{\textrm{LoS},i}$. Based on the spherical wave propagation model,
the near-field SI channel is modeled for the ULA structure in [23, 24, 25].
According to the transmit and receive URAs demonstrated in Fig. 2, the
residual near-field SI channel between the $\left(m,n\right)^{th}$ transmit
and $\left(u,v\right)^{th}$ receive antenna pair at the node $i$ is defined as
in (12),

${\bf{H}}_{\textrm{LoS},i}\big{(}{\big{[}{\left({u-1}\right)M_{r,i}^{\left(y\right)}+v}\big{]},\big{[}{\left({m-1}\right)M_{t,i}^{\left(y\right)}+n}\big{]}}\big{)}=\frac{\kappa}{{{\Delta_{\left({m,n}\right)\rightarrow\left({u,v}\right)}}}}{e^{-j2\pi{{{\Delta_{\left({m,n}\right)\rightarrow\left({u,v}\right)}}}}}}.$
(12)

${\Delta_{\left({m,n}\right)\rightarrow\left({u,v}\right)}}=\sqrt{{\left[{\left({u-1}\right)d\sin\left(\Theta\right)+{D_{2}}}\right]}^{2}+{\left[{\left({m-1}\right)d+\left({u-1}\right)d\cos\left(\Theta\right)+{D_{1}}}\right]}^{2}+{\left[{\left({n-v}\right)d}\right]}^{2}}.$
(13)

where $\kappa$ is the normalization scalar so that
$10\log_{10}\left(\mathbb{E}\big{\\{}\|{{\bf{H}}_{\textrm{LoS},i}}\|_{F}^{2}\big{\\}}\right)=-P_{\textrm{IS,dB}}$,
$P_{\textrm{IS,dB}}$ is the SIC in dB achieved by the antenna isolation333When
$P_{\textrm{IS,dB}}=0$ dB inferring no antenna isolation, the near-field SI
channel has the unity power on average as
$\mathbb{E}\big{\\{}\|{{\bf{H}}_{\textrm{LoS},i}}\|_{F}^{2}\big{\\}}=1$.
Additionally, if $P_{\textrm{IS,dB}}=30$ dB SIC is achieved by the antenna
isolation, the average power of the residual near-field SI channel is
calculated as
$\mathbb{E}\big{\\{}\|{{\bf{H}}_{\textrm{LoS},i}}\|_{F}^{2}\big{\\}}=0.001$.,
${{\Delta_{\left({m,n}\right)\rightarrow\left({u,v}\right)}}}$ is the distance
between the $\left(m,n\right)^{th}$ transmit and $\left(u,v\right)^{th}$
receive antenna pair normalized by the wavelength. By using Fig. 2, the
distance between the antenna elements is obtained as in (13).

Regarding the far-field SI channel ${\bf H}_{\textrm{NLoS},i}$ given in (11),
we utilize the 3D geometry-based mmWave channel model because the plane wave
propagation model is suitable for the NLoS paths. By using (8) and (10), the
far-field SI channel at the node $i$ can be written as:

$\displaystyle{\bf{H}}_{\textrm{{NLoS}},i}={{\bf{\Phi}}_{\textrm{SI},r,i}}{\bf{G}}_{\textrm{SI},i}{{\bf{\Phi}}_{\textrm{SI},t,i}},$
(14)

where ${\bf
G}_{\textrm{SI},i}=\rm{diag}\big{(}\tau_{\textrm{SI},i,1}^{-\eta}g_{\textrm{SI},i,1},\cdots,\tau_{\textrm{SI},i,L_{\textrm{SI}}}^{-\eta}g_{\textrm{SI},i,L_{\textrm{SI}}}\big{)}\in\mathbb{C}^{L_{\textrm{SI}}\times
L_{\textrm{SI}}}$ is the diagonal path gain matrix with $L_{\textrm{SI}}$
paths, $\tau_{\textrm{SI},i,l}$ and
$g_{\textrm{SI},i,l}\sim\mathcal{CN}\Big{(}0,\frac{1}{L_{\textrm{SI}}}\Big{)}$
are the distance and complex path gain of the $l^{th}$ NLoS path,
respectively, ${{\bf{\Phi}}_{\textrm{SI},r,i}}\in\mathbb{C}^{M_{r,i}\times
L_{\textrm{SI}}}$ and
${{\bf{\Phi}}_{\textrm{SI},t,i}}\in\mathbb{C}^{L_{\textrm{SI}}\times M_{t,i}}$
are the receive and transmit phase response matrices for the far-field SI
channel, respectively. In order to generate the phase response matrices as
expressed in (10), we first consider $C_{\textrm{SI}}$ scattering-clusters for
the NLoS paths. Then, the mean elevation and azimuth AoA (AoD) of paths inside
the $c^{th}$ cluster are defined as
${\theta_{\textrm{SI},r,c}^{\left(i\right)}}$
(${\theta_{\textrm{SI},t,c}^{\left(i\right)}}$) and
${\psi_{\textrm{SI},r,c}^{\left(i\right)}}$
(${\psi_{\textrm{SI},t,c}^{\left(i\right)}}$), respectively. Also, the
corresponding elevation and azimuth AoA (AoD) spread are given as
$\delta_{\textrm{SI},r,c}^{\theta,\left(i\right)}$
($\delta_{\textrm{SI},t,c}^{\theta,\left(i\right)}$) and
$\delta_{\textrm{SI},r,c}^{\psi,\left(i\right)}$
($\delta_{\textrm{SI},t,c}^{\psi,\left(i\right)}$), respectively.

##  IV Joint Hybrid Precoding/Combining

In this section, the proposed angular-based joint hybrid precoding/combining
(AB-JHPC) technique is developed for the full-duplex mmWave massive MIMO
systems. Here, we have four ultimate objectives: (i) enhance the quality of
SIC, (i) maximize the intended signal power, (iii) decrease the CSI overhead
size, (iv) reduce the hardware cost/complexity by employing less number of RF
chains. Thus, we first introduce the transmit and receive RF beamformer design
based on the slow time-varying AoD/AoA information, where we aim to maximize
the beamforming gain for the intended signal and suppress the SI signal
(please see (4)). Afterwards, the transmit BB precoder and receive BB combiner
are built via the reduced-size effective CSI seen from the BB-stage, where we
employ SVD and S-MMSE based algorithms.

###  IV-A RF-Stage Design

The primary target at the RF-stage design is to maximize the beamforming gain
for the intended signal expressed in (2). By using (8), the effective intended
channel is rewritten as:

$\bm{\mathcal{H}}_{i}={\bf F}_{r,j}{\bf H}_{i}{\bf F}_{t,i}={\bf
F}_{r,j}{{\bf{\Phi}}_{r,j}}{\bf{G}}_{i}{{\bf{\Phi}}_{t,i}}{\bf
F}_{t,i}.\vspace{-1ex}$ (15)

In order to maximize the transmit (receive) beamforming gain and exploit all
degrees of freedom provided by the intended channel, the columns of ${\bf
F}_{t,i}$ (${\bf F}_{r,j}$) should belong to the subspace spanned by
${{\bf{\Phi}}_{t,i}}$ (${{\bf{\Phi}}_{r,j}}$). Thus, we should satisfy
$\rm{Span}\left({\bf
F}_{t,i}\right)\subset\rm{Span}\left({{\bf{\Phi}}_{t,i}}\right)$ and
$\rm{Span}\left({\bf
F}_{r,j}\right)\subset\rm{Span}\left({{\bf{\Phi}}_{r,j}}\right)$. Here, it is
worthwhile to mention that the transmit (receive) phase response matrix
${{\bf{\Phi}}_{t,i}}$ (${{\bf{\Phi}}_{r,j}}$) is a function of slow time-
varying AoD (AoA) information (please see (10)). For assuring the primary
target, AoD and AoA support for the intended channel from the node $i$ to the
node $j$ are respectively defined as follows:

$\displaystyle{\rm{Ao}}{{\rm{D}}}_{i}$ $\displaystyle=$
$\displaystyle\Big{\\{}{{{\left[{{\gamma_{x}},{\gamma_{y}}}\right]}}=\sin\left(\theta\right){{\left[{\cos\left(\psi\right),\sin\left(\psi\right)}\right]}}}\hskip
0.86108pt\Big{|}\hskip
0.86108pt\theta\in{\bm{\theta}}_{t,i},\psi\in{\bm{\psi}}_{t,i}\Big{\\}},$
(16a) $\displaystyle{\rm{Ao}}{{\rm{A}}}_{j}$ $\displaystyle=$
$\displaystyle\Big{\\{}{{{\left[{{\gamma_{x}},{\gamma_{y}}}\right]}}=\sin\left(\theta\right){{\left[{\cos\left(\psi\right),\sin\left(\psi\right)}\right]}}}\hskip
0.86108pt\Big{|}\hskip
0.86108pt\theta\in{\bm{\theta}}_{r,j},\psi\in{\bm{\psi}}_{r,j}\Big{\\}},$
(16b)

where
${\bm{\theta}}_{t,i}=\cup_{c=1}^{C}\big{[}{{\theta_{t,c}^{\left(i\right)}}\hskip
4.30554pt-\hskip
4.30554pt{\delta_{t,c}^{\theta,\left(i\right)}}},{{\theta_{t,c}^{\left(i\right)}}\hskip
4.30554pt+\hskip 4.30554pt{\delta_{t,c}^{\theta,\left(i\right)}}}\big{]}$ and
${\bm{\psi}}_{t,i}=\cup_{c=1}^{C}\big{[}{{\psi_{t,c}^{\left(i\right)}}\hskip
4.30554pt-\hskip
4.30554pt{\delta_{t,c}^{\psi,\left(i\right)}}},{{\psi_{t,c}^{\left(i\right)}}\hskip
4.30554pt+\hskip 4.30554pt{\delta_{t,c}^{\psi,\left(i\right)}}}\big{]}$ denote
the elevation and azimuth AoD supports of the intended channel at the node
$i$, respectively. Similarly,
${\bm{\theta}}_{r,j}=\cup_{c=1}^{C}\big{[}{{\theta_{r,c}^{\left(j\right)}}\hskip
3.22916pt-\hskip
3.22916pt{\delta_{r,c}^{\theta,\left(j\right)}}},{{\theta_{r,c}^{\left(j\right)}}\hskip
3.22916pt+\hskip 3.22916pt{\delta_{r,c}^{\theta,\left(j\right)}}}\big{]}$ and
${\bm{\psi}}_{r,j}=\cup_{c=1}^{C}\big{[}{{\psi_{r,c}^{\left(j\right)}}\hskip
1.07639pt-\hskip
1.07639pt{\delta_{r,c}^{\psi,\left(j\right)}}},{{\psi_{r,c}^{\left(j\right)}}\hskip
1.07639pt+\hskip 1.07639pt{\delta_{r,c}^{\psi,\left(j\right)}}}\big{]}$
represent the elevation and azimuth AoA supports of the intended channel at
the node $j$, respectively.

On the other hand, the secondary target at the RF-stage design is to suppress
the effective SI channel power by satisfying the approximate zero condition
given in (4). By substituting (12) and (11) into (4), the effective SI channel
at the node $j$ is also rewritten as:

$\displaystyle\bm{\mathcal{H}}_{\textrm{SI},j}$
$\displaystyle={{\bf{F}}_{r,j}{\bf{H}}_{\textrm{SI},j}{\bf{F}}_{t,j}}={\bf{F}}_{r,j}\left({\bf{H}}_{\textrm{NLoS},j}+{\bf{H}}_{\textrm{LoS},j},\right){\bf{F}}_{t,j}$
(17)
$\displaystyle=\underbrace{{\bf{F}}_{r,j}{{\bf{\Phi}}_{\textrm{SI},r,j}}}_{\scriptstyle\rm{Receive}\atop\scriptstyle\rm{Beamforming}}{\bf{G}}_{\textrm{SI},j}\underbrace{{{\bf{\Phi}}_{\textrm{SI},t,j}}{\bf{F}}_{t,j}}_{\scriptstyle\rm{Transmit}\atop\scriptstyle\rm{Beamforming}}+\underbrace{{\bf{F}}_{r,j}{\bf{H}}_{\textrm{LoS},j}{\bf{F}}_{t,j}}_{\scriptstyle\rm{Suppressed~{}via}\atop\scriptstyle\rm{Antenna~{}Isolation}}\approx{\bf{0}},$

where the approximate zero condition can be achieved via the joint design of
the receive RF beamformer ${\bf{F}}_{r,j}$ and the transmit RF beamformer
${\bf{F}}_{t,j}$.

$\displaystyle{{\bf{\Phi}}_{\textrm{SI},t,j}}{{{\bf{e}}_{t,i}}\big{(}\lambda_{t,i}^{\left(x\right),k},\lambda_{t,i}^{\left(y\right),n}\big{)}}=\frac{1}{\sqrt{{M_{t,i}}}}\left[{\begin{array}[]{*{20}{c}}{\sum\limits_{{m_{x}}}^{M_{t,i}^{\left(x\right)}}{\sum\limits_{{m_{y}}}^{M_{t,i}^{\left(y\right)}}{{e^{j2\pi
d\left[{m{{}_{x}}\left({\lambda_{t,i}^{\left(x\right),k}-\gamma_{t,1}^{\left(x\right)}}\right)+{m_{y}}\left({\lambda_{t,i}^{\left(y\right),k}-\gamma_{t,1}^{\left(y\right)}}\right)}\right]}}}}}\\\
{\vdots}\\\
{\sum\limits_{{m_{x}}}^{M_{t,i}^{\left(x\right)}}{\sum\limits_{{m_{y}}}^{M_{t,i}^{\left(y\right)}}{{e^{j2\pi
d\left[{m{{}_{x}}\left({\lambda_{t,i}^{\left(x\right),k}-\gamma_{t,{L_{SI}}}^{\left(x\right)}}\right)+{m_{y}}\left({\lambda_{t,i}^{\left(y\right),k}-\gamma_{t,{L_{SI}}}^{\left(y\right)}}\right)}\right]}}}}}\end{array}}\right].$
(20)

Here, the effective SI channel has two components representing the LoS and
NLoS paths. As discussed in Section III-B, the LoS path reflecting the strong
near-field SI channel is suppressed via the antenna isolation based SIC
techniques [13, 14, 15, 16, 17, 18]. However, after the canceling strong near-
field SI channel, the NLoS paths reflecting the far-field SI channel become
more dominant444The experimental results in [18] show that 74 dB SIC is
achieved by antenna isolation measured in the anechoic chamber as a low-
reflection environment. However, the SIC is reduced to 44 dB in the reflective
room. The main reason considering 30 dB decrease of SIC is owing to the
dominant reflected paths (i.e., NLoS paths).. Hence, the secondary target can
be accomplished when the columns of ${\bf F}_{t,j}$ (${\bf F}_{r,j}$) should
belong to the null space of ${{\bf{\Phi}}_{\textrm{SI},t,j}}$
(${{\bf{\Phi}}_{\textrm{SI},r,j}}$). In other words, the secondary target
requires $\rm{Span}\left({\bf
F}_{t,j}\right)\subset\rm{Null}\left({{\bf{\Phi}}_{\rm{SI},t,j}}\right)$ and
$\rm{Span}\left({\bf
F}_{r,j}\right)\subset\rm{Null}\left({{\bf{\Phi}}_{\textrm{SI},r,j}}\right)$.
For addressing the secondary target, we also define the AoD and AoA supports
of the SI channel at the node $j$:

(18a) (18b)

where
${\bm{\theta}}_{\textrm{SI},t,j}=\cup_{c=1}^{C}\big{[}{{\theta_{\textrm{SI},t,c}^{\left(j\right)}}\hskip
4.30554pt-\hskip
4.30554pt{\delta_{\textrm{SI},t,c}^{\theta,\left(j\right)}}},{{\theta_{\textrm{SI},t,c}^{\left(j\right)}}\hskip
4.30554pt+\hskip
4.30554pt{\delta_{\textrm{SI},t,c}^{\theta,\left(j\right)}}}\big{]}$ and
${\bm{\psi}}_{\textrm{SI},t,j}=\cup_{c=1}^{C}\big{[}{{\psi_{\textrm{SI},t,c}^{\left(j\right)}}\hskip
4.30554pt-\hskip
4.30554pt{\delta_{\textrm{SI},t,c}^{\psi,\left(j\right)}}},{{\psi_{\textrm{SI},t,c}^{\left(j\right)}}\hskip
4.30554pt+\hskip
4.30554pt{\delta_{\textrm{SI},t,c}^{\psi,\left(j\right)}}}\big{]}$ denote the
elevation and azimuth AoD supports of the SI channel at the node $j$,
respectively. Also,
${\bm{\theta}}_{\textrm{SI},r,j}=\cup_{c=1}^{C}\big{[}{{\theta_{\textrm{SI},r,c}^{\left(j\right)}}\hskip
4.30554pt-\hskip
4.30554pt{\delta_{\textrm{SI},r,c}^{\theta,\left(j\right)}}},{{\theta_{\textrm{SI},r,c}^{\left(j\right)}}\hskip
4.30554pt+\hskip
4.30554pt{\delta_{\textrm{SI},r,c}^{\theta,\left(j\right)}}}\big{]}$ and
${\bm{\psi}}_{\textrm{SI},r,j}=\cup_{c=1}^{C}\big{[}{{\psi_{\textrm{SI},r,c}^{\left(j\right)}}-{\delta_{\textrm{SI},r,c}^{\psi,\left(j\right)}}},{{\psi_{\textrm{SI},r,c}^{\left(j\right)}}+{\delta_{\textrm{SI},r,c}^{\psi,\left(j\right)}}}\big{]}$
represent the elevation and azimuth AoA supports of the SI channel at the node
$j$, respectively.

For satisfying both the primary and secondary targets, the transmit/receive RF
beamformers should cover the angular support of the intended channel given in
(16a) while excluding the angular support of the SI channel given in (18a).
For this purpose, according to the phase response vectors expressed in (9), we
first define the transmit and receive steering vectors as
${{\bf{e}}_{t,i}}\left({{\gamma_{x}},{\gamma_{y}}}\right)=\frac{1}{\sqrt{M_{t,i}}}{\bm{\phi}_{t,i}}\left({{\gamma_{x},\gamma_{y}}}\right)$
and
${{\bf{e}}_{r,i}}\left({{\gamma_{x}},{\gamma_{y}}}\right)=\frac{1}{\sqrt{M_{r,i}}}{\bm{\phi}_{r,i}^{*}}\left({{\gamma_{x},\gamma_{y}}}\right)$,
respectively. Then, we define the orthogonal quantized angle-pairs
$\big{(}\lambda_{u,i}^{\left(x\right),k},\lambda_{u,i}^{\left(y\right),n}\big{)}$
as:

$\displaystyle{\lambda_{u,i}^{\left(x\right),k}}$ $\displaystyle=$
$\displaystyle-1+\frac{2k-1}{{{M_{u,i}^{\left(x\right)}}}}\textrm{,~{}~{}}\forall
k=1,\cdots,{M_{u,i}^{\left(x\right)}},$ (19a)
$\displaystyle{\lambda_{u,i}^{\left(y\right),n}}$ $\displaystyle=$
$\displaystyle-1+\frac{2n-1}{{{M_{u,i}^{\left(y\right)}}}}\textrm{,~{}~{}}\forall
n=1,\cdots,{M_{u,i}^{\left(y\right)}},$ (19b)

where $u\in\left\\{t,r\right\\}$ represents either the transmitter side for
$u=t$ or the receiver side for $u=r$. The orthogonality property of the
quantized angle-pairs generates $M_{t,i}$ orthogonal transmit steering vectors
and $M_{r,i}$ orthogonal receive steering vectors (i.e.,
${{{\bf{e}}_{u,i}^{H}}\big{(}{\lambda_{u,i}^{\left(x\right),k}}{\lambda_{u,i}^{\left(x\right),n}}\big{)}{{\bf{e}}_{u,i}}\big{(}{\lambda_{u,i}^{\left(x\right),k^{\prime}}}{\lambda_{u,i}^{\left(x\right),n^{\prime}}}\big{)}=0}$
for $u\in\left\\{t,r\right\\}$, $\forall(k,n)\neq(k^{\prime},n^{\prime})$).
Furthermore, the quantized angle-pairs defined in (19a) produces the minimum
number of angle-pairs in order to cover the complete 3D angular support [41].
It also reduces the number of RF chain utilization, while exploiting all
degrees of freedom provided by the intended channel (i.e.,
$\textrm{Span}\left({\bf
F}_{t,i}\right)\subset\textrm{Span}\left({{\bf{\Phi}}_{t,i}}\right)$ and
$\textrm{Span}\left({\bf
F}_{r,j}\right)\subset\textrm{Span}\left({{\bf{\Phi}}_{r,j}}\right)$). Hence,
the primary target is satisfied, when we develop the transmit and receive RF
beamformers by utilizing the corresponding steering vector with the quantized
angle-pairs as
$\big{(}\lambda_{t,i}^{\left(x\right),k},\lambda_{t,i}^{\left(y\right),n}\big{)}\in{\rm{AoD}}_{i}$
and
$\big{(}\lambda_{r,i}^{\left(x\right),k},\lambda_{r,i}^{\left(y\right),n}\big{)}\in{\rm{AoA}}_{i}$,
respectively.

On the other hand, when the transmit RF beamformer is built via the transmit
steering vector ${\bf{e}}_{t,i}\left(\cdot,\cdot\right)$ with any quantized
angle-pairs satisfying both
$\big{(}\lambda_{t,i}^{\left(x\right),k},\lambda_{t,i}^{\left(y\right),n}\big{)}\in{\rm{AoD}}_{i}$
and
$\big{(}\lambda_{t,i}^{\left(x\right),k},\lambda_{t,i}^{\left(y\right),n}\big{)}\notin{\rm{AoD}}_{\textrm{SI},i}$,
one can show that each element of the multiplication given in (20)
asymptotically converges to 0 for large transmit URAs as in the massive MIMO
systems [41, eq. (16)]. Similarly, this behavior is also valid for the receive
RF beamformer design. Thus, the approximate zero condition given in (17) can
be satisfied by choosing the quantized angle-pairs outside of the AoD/AoA
support of SI channel, which also ensures the secondary target. In other
words, as the antenna array size increases, the corresponding quantized angle-
pairs assures both ${\textrm{Span}\left({\bf
F}_{t,j}\right)\subset\textrm{Null}\left({{\bf{\Phi}}_{\textrm{SI},t,j}}\right)}$
and ${\textrm{Span}\left({\bf
F}_{r,j}\right)\subset\textrm{Null}\left({{\bf{\Phi}}_{\textrm{SI},r,j}}\right)}$,
which are earlier defined as the necessary conditions for the secondary
target.

In order to meet the primary and secondary targets, a joint design of
transmit/receive RF beamformers is required based on the AoD/AoA support of
the intended and SI channels. For the transmit RF beamformer at the node $i$,
the quantized angle-pairs covering the AoD support of the intended channel
${\bf H}_{i}$ given in (16a) and excluding the AoD support of the SI channel
${\bf H}_{\textrm{SI},i}$ given in (18a) are found as:

$\big{(}\lambda_{t,i}^{\left(x\right),k},\lambda_{t,i}^{\left(y\right),n}\big{)}\hskip
2.15277pt\left|\begin{array}[]{l}{\gamma_{x}}\in{\bm{\lambda}}_{t,i}^{\left(x\right),k},{\gamma_{y}}\in{\bm{\lambda}}_{t,i}^{\left(y\right),n},\\\
\\\
\left({{\gamma_{x}},{\gamma_{y}}}\right)\in{\rm{Ao}}{{\rm{D}}_{i}},\left({{\gamma_{x}},{\gamma_{y}}}\right)\notin{\rm{Ao}}{{\rm{D}}_{\textrm{SI},i}},\end{array}\right.$
(21)

where
${\bm{\lambda}}_{t,i}^{\left(x\right),k}=\big{[}{\lambda_{t,i}^{\left(x\right),k}\hskip
4.30554pt-\hskip
4.30554pt\frac{1}{{{M_{t,i}^{\left(x\right)}}}},\lambda_{t,i}^{\left(x\right),k}\hskip
4.30554pt+\hskip 4.30554pt\frac{1}{{{M_{t,i}^{\left(x\right)}}}}}\big{]}$ and
${{\bm{\lambda}}_{t,i}^{\left(y\right),n}\hskip 4.30554pt=\hskip
4.30554pt\big{[}{\lambda_{t,i}^{\left(y\right),n}-\frac{1}{{{M_{t,i}^{\left(y\right)}}}},\lambda_{t,i}^{\left(y\right),n}+\frac{1}{{{M_{t,i}^{\left(y\right)}}}}}\big{]}}$
represent the boundaries of $\lambda_{t,i}^{\left(x\right),k}$ and
$\lambda_{t,i}^{\left(y\right),n}$, respectively. Here, (21) implies that if
any $\left(\gamma_{x},\gamma_{y}\right)$ angle-pairs inside the boundaries of
$\lambda_{t,i}^{\left(x\right),k}$ and $\lambda_{t,i}^{\left(y\right),n}$ is
both within ${\rm{AoD}}_{i}$ and outside of ${\rm{AoD}}_{\textrm{SI},i}$, then
the corresponding quantized angle-pair
$\big{(}\lambda_{t,i}^{\left(x\right),k},\lambda_{t,i}^{\left(y\right),n}\big{)}$
is utilized for the transmit RF beamformer design. We consider that there are
$N_{t,i}$ quantized angle-pairs satisfying (21), which also represents the
number of orthogonal-beams and the number of RF chains at the transmitter of
the node $i$ (please see Fig. 1). Finally, by using (19a) and (21), the
transmit RF beamformer at the node $i$ is obtained as follows:

${{\bf{F}}_{t,i}}=\Big{[}{{\bf{e}}_{t,i}\big{(}\lambda_{t,i}^{\left(x\right),k_{1}},\lambda_{t,i}^{\left(y\right),n_{1}}\big{)},\cdots,{\bf{e}}_{t,i}\big{(}\lambda_{t,i}^{\left(x\right),k_{N_{t,i}}},\lambda_{t,i}^{\left(y\right),n_{N_{t,i}}}\big{)}}\Big{]}.$
(22)

Similarly, regarding the receive RF beamformer at the node $j$, the quantized
angle-pairs covering the AoA support of the intended channel ${\bf H}_{i}$
given in (16b) and excluding the AoA support of the SI channel ${\bf
H}_{\textrm{SI},j}$ given in (18b) are found as:

$\big{(}\lambda_{r,j}^{\left(x\right),k},\lambda_{r,j}^{\left(y\right),n}\big{)}\hskip
2.15277pt\left|\begin{array}[]{l}{\gamma_{x}}\in{\bm{\lambda}}_{r,j}^{\left(x\right),k},{\gamma_{y}}\in{\bm{\lambda}}_{r,j}^{\left(y\right),n},\\\
\\\
\left({{\gamma_{x}},{\gamma_{y}}}\right)\in{\rm{Ao}}{{\rm{A}}_{j}},\left({{\gamma_{x}},{\gamma_{y}}}\right)\notin{\rm{Ao}}{{\rm{A}}_{\textrm{SI},j}},\end{array}\right.$
(23)

where
${\bm{\lambda}}_{r,i}^{\left(x\right),k}=\big{[}{\lambda_{r,i}^{\left(x\right),k}\hskip
4.30554pt-\hskip
4.30554pt\frac{1}{{{M_{r,i}^{\left(x\right)}}}},\lambda_{r,i}^{\left(x\right),k}\hskip
4.30554pt+\hskip 4.30554pt\frac{1}{{{M_{r,i}^{\left(x\right)}}}}}\big{]}$ and
${{\bm{\lambda}}_{r,i}^{\left(y\right),n}\hskip 4.30554pt=\hskip
4.30554pt\big{[}{\lambda_{r,i}^{\left(y\right),n}-\frac{1}{{{M_{r,i}^{\left(y\right)}}}},\lambda_{r,i}^{\left(y\right),n}+\frac{1}{{{M_{r,i}^{\left(y\right)}}}}}\big{]}}$
indicate the boundaries of $\lambda_{r,i}^{\left(x\right),k}$ and
$\lambda_{t,i}^{\left(y\right),n}$, respectively. By assuming $N_{r,j}$
quantized angle-pairs satisfying (23), the receive RF beamformer at the node
$j$ is derived as:

${{\bf{F}}_{r,j}}=\Big{[}{{\bf{e}}_{r,j}\big{(}\lambda_{r,j}^{\left(x\right),k_{1}},\lambda_{r,j}^{\left(y\right),n_{1}}\big{)},\cdots,{\bf{e}}_{r,j}\big{(}\lambda_{r,j}^{\left(x\right),k_{N_{r,j}}},\lambda_{r,j}^{\left(y\right),n_{N_{r,j}}}\big{)}}\Big{]}^{T}.$
(24)

The RF-stage design for the proposed AB-JHPC technique is summarized in
Algorithm 1. It is important to mention that the transmit and receive RF
beamformers developed in this section follow the CM property given in (6d),
which makes it possible to design the RF-stage via the low-cost phase-shifters
as illustrated in Fig. 1. By means of the quantized angle-pairs defined in
(19a), the transmit and receive RF beamformers at the node $i$ can be
constructed via the phase-shifters with $\log_{2}\left(M_{t,i}\right)$ and
$\log_{2}\left(M_{r,i}\right)$ bit resolution, respectively. Furthermore, as
expressed in Algorithm 1, the RF-stage design does not require the fast time-
varying instantaneous CSI and it is only based on the slow time-varying
AoD/AoA information. Particularly, the design of transmit (receive) RF
beamformer requires only four angular parameters, which are the mean of
elevation and azimuth AoD (AoA) and their spread555Standard angle estimation
techniques requiring online channel sounding can be utilized to track the slow
time-varying AoD/AoA parameters [50, 53]. Recently, an offline channel
estimation technique is proposed in [54], where the AoD/AoA parameters are
tracked via the geospatial data, fuzzy c-means algorithm and ray tracing..
Also, ${\bf F}_{t,i}$ and ${\bf F}_{r,i}^{H}$ are tall unitary matrices (i.e.,
${\bf F}_{t,i}^{H}{\bf F}_{t,i}={\bf I}_{N_{t,i}}$, ${\bf F}_{r,i}{\bf
F}_{r,i}^{H}={\bf I}_{N_{r,i}}$).

Algorithm 1 RF-stage design

1:$M_{t,i}$, $M_{r,i}$,
$\big{\\{}\theta_{t,c}^{\left(i\right)},\psi_{t,c}^{\left(i\right)},\delta_{t,c}^{\theta,\left(i\right)},\delta_{t,c}^{\psi,\left(i\right)},\big{\\}}$,
$\big{\\{}\theta_{r,c}^{\left(i\right)},\psi_{r,c}^{\left(i\right)},\delta_{r,c}^{\theta,\left(i\right)},\delta_{r,c}^{\psi,\left(i\right)}\big{\\}}$,
for $i=1:2$ and $c=1:C$.

2:Define the node index as $j\in\left\\{{1,2}\right\\}$, $j\neq i$.

3:for $i=1:2$, do

4: Build ${\rm{AoD}}_{i}$ for the intended channel ${\bf H}_{i}$ via (16a).

5: Build ${\rm{AoA}}_{i}$ for the intended channel ${\bf H}_{j}$ via (16b).

6: Build ${\rm{AoD}}_{\textrm{SI},i}$ and ${\rm{AoA}}_{\textrm{SI},i}$ for the
SI channel ${\bf H}_{\textrm{SI},i}$ via (18a).

7: Construct the quantized angle-pairs
$\big{(}{\lambda_{t,i}^{\left(x\right),k}}{\lambda_{t,i}^{\left(x\right),n}}\big{)}$
and
$\big{(}{\lambda_{r,i}^{\left(x\right),k}}{\lambda_{r,i}^{\left(x\right),n}}\big{)}$
given in (19a).

8: Find
$\big{(}\lambda_{t,i}^{\left(x\right),k},\lambda_{t,i}^{\left(y\right),n}\big{)}$
pairs covering ${\rm{AoD}}_{i}$ and excluding ${\rm{AoD}}_{\textrm{SI},i}$ as
in (21).

9: Generate the transmit RF beamformer ${{\bf{F}}_{t,i}}$ via (22).

10: Find
$\big{(}\lambda_{r,i}^{\left(x\right),k},\lambda_{r,i}^{\left(y\right),n}\big{)}$
pairs covering ${\rm{AoA}}_{i}$ and excluding ${\rm{AoA}}_{\textrm{SI},i}$ as
in (23).

11: Generate the receive RF beamformer ${{\bf{F}}_{r,i}}$ via (24).

12:end for

13:${\bf{F}}_{t,1}$, ${\bf{F}}_{t,2}$, ${\bf{F}}_{r,1}$, ${\bf{F}}_{r,2}$

###  IV-B BB-Stage Design

According to the proposed RF-stage design, the reduced-size effective intended
channel $\bm{\mathcal{H}}_{i}$ is utilized for the BB-stage design. In
contrast to [23, 24], the proposed BB-stage design does not require the
instantaneous SI channel knowledge because the perfect estimation of SI
channel could be impractical for the full-duplex communications666 According
to the received signal expression given in (1), when the full-size or reduced-
size perfect SI channel knowledge (i.e., ${\bf H}_{\textrm{SI},j}$ or
$\bm{\mathcal{H}}_{\textrm{SI},j}$) is available at the node $j$, any of
passive antenna isolation, digital or analog SIC techniques is not necessary
for the full-duplex communications. Because the transmit RF beamformer ${\bf
F}_{t,j}$, the BB precoder ${\bf B}_{t,j}$ and the data signal ${\bf d}_{j}$
is already available at the node $j$ in addition to the assumption of the
perfect SI channel ${\bf H}_{\textrm{SI},j}$. Then, we could simply subtract
the SI component ${\bf H}_{\textrm{SI},j}{\bf F}_{t,j}{\bf B}_{t,j}{\bf
d}_{j}$ from the received signal given in (1), which could imply to doubling
the capacity via the full-duplex communications with respect to the
conventional half-duplex communications without any SIC techniques. However,
the SI channel must be estimated through the pilot signals in practice,
therefore, there exists the estimation errors due to the noise as well as the
hardware imperfections [13, 14, 15, 55]. Thus, it motivates the development of
advanced SIC techniques in the literature [14]. . Here, we first develop the
BB precoder and BB combiner via the well-known singular value decomposition
(SVD). Then, we also propose a novel semi-blind minimum mean square error
(S-MMSE) algorithm to further enhance the quality of SIC provided by the RF-
stage.

Let the SVD of $\bm{\mathcal{H}}_{i}$ be defined as $\bm{\mathcal{H}}_{i}={\bf
U}_{j}{\bf\Sigma}_{i}{\bf V}_{i}^{H}$, where ${\bf
U}_{j}\in\mathbb{C}^{N_{r,j}\times\textrm{rank}\left(\bm{\mathcal{H}}_{i}\right)}$
is a tall unitary matrix,
${\bf\Sigma}_{n}=\textrm{diag}\big{(}\sigma_{n,1}^{2},\cdots,\sigma_{n,\textrm{rank}\left(\bm{\mathcal{H}}_{i}\right)}^{2}\big{)}\in\mathbb{R}^{\textrm{rank}\left(\bm{\mathcal{H}}_{i}\right)\times\textrm{rank}\left(\bm{\mathcal{H}}_{i}\right)}$
is a diagonal matrix with the singular values in the decreasing order, ${\bf
V}_{i}\in\mathbb{C}^{N_{t,i}\times\textrm{rank}\left(\bm{\mathcal{H}}\right)}$
is a tall unitary matrix. Assuming
$\textrm{rank}\left(\bm{\mathcal{H}_{i}}\right)\geq S_{i}$, we can partition
${\bf V}_{i}$ and ${\bf U}_{j}$ as ${\bf V}_{i}=\left[{\bf V}_{i,1},{\bf
V}_{i,2}\right]$ with ${\bf V}_{i,1}\in\mathbb{C}^{N_{t,i}\times S_{i}}$. and
${\bf U}_{j}=\left[{\bf U}_{j,1},{\bf U}_{j,2}\right]$ with ${\bf
U}_{j,1}\in\mathbb{C}^{N_{r,j}\times S_{i}}$, respectively According to the
well-known SVD approach [38, 37, 36], the BB precoder at the node $i$ and the
BB combiner at the node $j$ are respectively derived as follows:

$\displaystyle{\bf B}_{t,i}^{\textrm{SVD}}$ $\displaystyle=$
$\displaystyle{\bf V}_{i,1}{\bf
P}_{i}^{\frac{1}{2}}\in\mathbb{C}^{N_{t,i}\times S_{i}},$ (25a)
$\displaystyle{\bf B}_{r,j}^{\textrm{SVD}}$ $\displaystyle=$
$\displaystyle{\bf U}_{j,1}^{H}\in\mathbb{C}^{S_{i}\times N_{r,j}},$ (25b)

where ${\bf
P}_{i}=\textrm{diag}\left(P_{i,1},\cdots,P_{i,S_{i}}\right)\in\mathbb{R}^{S_{i}\times
S_{i}}$ is a diagonal power allocation matrix with
$\sum\nolimits_{n=1}^{S_{i}}P_{i,n}\leq{P_{T}}$ for the transmit power
constraint given in (6d). By using the well-known water-filling technique
[36], the allocated powers are obtained as:

${P_{i,n}}={\left({\mu-\frac{\sigma_{w}^{2}}{{P_{T}\sigma_{n}^{2}}}}\right)^{+}},{\textrm{
s.t.}}\sum\limits_{n=1}^{S_{i}}{\left({\mu-\frac{\sigma_{w}^{2}}{{P_{T}\sigma_{n}^{2}}}}\right)^{+}}={P_{T}}.$
(26)

For maximizing the power of intended signal and mitigating the power of ISI
expressed in (3), the BB precoder ${\bf{B}}_{t,i}^{\textrm{SVD}}$ is optimal
as proven in [36, 38]. However, the BB combiner
${\bf{B}}_{r,j}^{\textrm{SVD}}$ is not optimal because it does not consider
the effect of SI and noise present at the receiver node. Hence, we aim to
develop the optimal BB combiner by minimizing the mean square error (MSE). By
using (2), the MSE expression at the node $j$ is written as follows:

$\displaystyle{{\rm{MSE}}}_{j}$
$\displaystyle=\mathbb{E}\big{\\{}{{{\big{\|}{\tilde{\bf{r}}_{j}-{\bf{d}}_{i}}\big{\|}}_{2}^{2}}}\big{\\}}$
(27)
$\displaystyle\mathop{=}\limits^{\left(a\right)}{\textrm{tr}}\big{(}\mathbb{E}{\big{\\{}{\tilde{\bf{r}}_{j}{{\tilde{\bf{r}}}_{j}^{H}}-{\bf{d}}_{i}{{\tilde{\bf{r}}}_{j}^{H}}-\tilde{\bf{r}}_{j}{{\bf{d}}_{i}^{H}}+{\bf{d}}_{i}{{\bf{d}}_{i}^{H}}}\big{\\}}}\big{)}$
$\displaystyle\mathop{=}\limits^{\left(b\right)}\textrm{tr}\Big{(}{{\bf{B}}_{r,j}}\big{[}{{{\bm{\mathcal{H}}}_{i}}{{\bf{B}}_{t,i}}{\bf{B}}_{t,i}^{H}{\bm{\mathcal{H}}}_{i}^{H}+{{\bf{W}}_{j}}+\sigma_{w}^{2}{{\bf{I}}_{{N_{r,j}}}}}\big{]}{\bf{B}}_{r,j}^{H}$
$\displaystyle\Big{.}-{{\bf{B}}_{r,j}}{{\bm{\mathcal{H}}}_{i}}{{\bf{B}}_{t,i}}-{\bf{B}}_{t,i}^{H}{\bm{\mathcal{H}}}_{i}^{H}{\bf{B}}_{r,j}^{H}+{{\bf{I}}_{{S_{i}}}}\Big{)},\vspace{-1ex}$

where ($a$) is the direct consequence of the linearity property of trace
operator, ($b$) applies the unitary proper of the transmit RF beamformer
(i.e., ${\bf F}_{t,i}^{H}{\bf F}_{t,i}={\bf I}_{N_{t,i}}$) and
$\mathbb{E}\left\\{{\bf d}_{i}{\bf d}_{i}^{H}\right\\}={\bf I}_{S_{i}}$. Here,
${{\bf{W}}_{j}}=\mathbb{E}\big{\\{}\bm{\mathcal{H}}_{{\textrm{SI}},j}{\bf{B}}_{t,j}{\bf{B}}_{t,j}^{H}\bm{\mathcal{H}}_{{\textrm{SI}},j}^{H}\big{\\}}$
is a function of the effective SI channel
$\bm{\mathcal{H}}_{{\textrm{SI}},j}$. Nonetheless, the minimization problem
for the current form of $\textrm{MSE}_{j}$ depends on the effective SI channel
$\bm{\mathcal{H}}_{{\textrm{SI}},j}$, which is not available at the node $j$.
According to the SI channel model given in (11), the SI channel has two parts
indicating the LoS paths (i.e., the near-field SI channel) and NLoS paths
(i.e., far-field SI channel). After applying the antenna separation based SIC,
the NLoS paths become more dominant compared to the LoS paths (i.e.,
$\bm{\mathcal{H}}_{\textrm{SI},j}={{\bf{F}}_{r,j}{\bf{H}}_{\textrm{SI},j}{\bf{F}}_{t,j}}\approx{\bf{F}}_{r,j}{\bf
H}_{\textrm{NLoS},j}{\bf{F}}_{t,j}$). Hence, in order to remove the dependency
of the instantaneous effective SI channel, we derive an approximation of
${{\bf{W}}_{j}}$ as:

$\displaystyle{{\bf{W}}_{j}}$
$\displaystyle=\mathbb{E}\Big{\\{}\bm{\mathcal{H}}_{{\textrm{SI}},j}{\bf{B}}_{t,j}{\bf{B}}_{t,j}^{H}\bm{\mathcal{H}}_{{\textrm{SI}},j}^{H}\Big{\\}}$
(28)
$\displaystyle\mathop{\approx}\limits^{\left(a\right)}\mathbb{E}\Big{\\{}{\bf{F}}_{r,j}{\bf{H}}_{{\textrm{LoS}},j}{\bf{F}}_{t,j}{\bf{B}}_{t,j}{\bf{B}}_{t,j}^{H}{\bf{F}}_{t,j}^{H}{\bf{H}}_{{\textrm{LoS}},j}^{H}{\bf{F}}_{r,j}^{H}\Big{\\}}$
$\displaystyle\mathop{=}\limits^{\left(b\right)}\mathbb{E}\Big{\\{}{\bf{F}}_{r,j}{{\bf{\Phi}}_{{\textrm{SI}},j,r}}{{\bf{G}}_{{\textrm{SI}},j}}{{\bf{\Phi}}_{{\textrm{SI}},j,t}}{\bf{F}}_{t,j}{\bf{B}}_{t,j}\Big{.}$
$\displaystyle\Big{.}\times{\bf{B}}_{t,j}^{H}{\bf{F}}_{t,j}^{H}{\bf{\Phi}}_{{\textrm{SI}},j,t}^{H}{\bf{G}}_{{\textrm{SI}},j}^{H}{\bf{\Phi}}_{{\textrm{SI}},j,r}^{H}{\bf{F}}_{r,j}^{H}\Big{\\}}$
$\displaystyle\mathop{\approx}\limits^{\left(c\right)}\frac{1}{\hat{\tau}_{\textrm{SI},j}^{2\eta}L_{\textrm{SI}}}{\bf{F}}_{r,j}{\hat{\bf{\Phi}}_{{\textrm{SI}},j,r}}{\hat{\bf{\Phi}}_{{\textrm{SI}},j,t}}{\bf{F}}_{t,j}{\bf{B}}_{t,j}$
$\displaystyle\times{\bf{B}}_{t,j}^{H}{\bf{F}}_{t,j}^{H}\hat{\bf{\Phi}}_{{\textrm{SI}},j,t}^{H}\hat{\bf{\Phi}}_{{\textrm{SI}},j,r}^{H}{\bf{F}}_{r,j}^{H}$
$\displaystyle\mathop{=}\limits^{\left(d\right)}\hat{\bf W}_{j},$

where ($a$) is obtained via the assumption of dominant NLoS paths after the
antenna separation based SIC, ($b$) is written by using (14), ($c$) is
approximated by applying the expectation operator on the fast time-varying
complex path gain matrix ${\bf G}_{\textrm{SI},j}$, ($d$) represents the
approximation of ${\bf W}_{j}$ as $\hat{\bf W}_{j}$. Here, $\hat{\tau}_{SI,j}$
is the average distance for the NLoS paths,
$\hat{\bf{\Phi}}_{{\textrm{SI}},j,t}$ and
$\hat{\bf{\Phi}}_{{\textrm{SI}},j,r}$ are the approximated transmit and
receive phase response matrices, respectively. It is worthwhile to remark that
the approximated phase response matrices does not require the knowledge of
exact AoD/AoA pairs. Instead, they require the knowledge of AoD/AoA support of
the SI channel defined in (18a) as in the RF-stage design (please see
Algorithm 1). Therefore, we generate $\hat{L}_{SI}$ random AoD/AoA pairs
inside the AoD/AoA support. Then, we build the approximated phase response
matrices by using (10) and the corresponding AoD/AoA pairs. By replacing
$\hat{\bf W}_{j}$ with ${\bf W}_{j}$ in (27), the MSE expression at the node
$j$ can be also approximated as follows:

$\displaystyle\hat{{\rm{MSE}}}_{j}$
$\displaystyle=\textrm{tr}\Big{(}{{\bf{B}}_{r,j}}\big{[}{{{\bm{\mathcal{H}}}_{i}}{{\bf{B}}_{t,i}}{\bf{B}}_{t,i}^{H}{\bm{\mathcal{H}}}_{i}^{H}+{\hat{\bf{W}}_{j}}+\sigma_{w}^{2}{{\bf{I}}_{{N_{r,j}}}}}\big{]}{\bf{B}}_{r,j}^{H}$
(29)
$\displaystyle\Big{.}-{{\bf{B}}_{r,j}}{{\bm{\mathcal{H}}}_{i}}{{\bf{B}}_{t,i}}-{\bf{B}}_{t,i}^{H}{\bm{\mathcal{H}}}_{i}^{H}{\bf{B}}_{r,j}^{H}+{{\bf{I}}_{{S_{i}}}}\Big{)}.$

Now, in addition to the effective intended channel $\bm{\mathcal{H}}_{i}$, the
approximated $\hat{\textrm{MSE}}_{j}$ only depends the AoD/AoA support of SI
channel. In other words, it is blind to the effective SI channel
$\bm{\mathcal{H}}_{\textrm{SI},j}$. Therefore, the optimization problem for
minimizing $\hat{\textrm{MSE}}_{j}$ is classified as semi-blind minimum mean
square error (S-MMSE). According to the expressed transmit RF beamformer ${\bf
F}_{t,i}$ in (22), receive RF beamformer ${\bf F}_{r,j}$ in (24), and BB
precoder ${\bf B}_{t,i}$ in (25a), the BB combiner problem for the proposed
S-MMSE algorithm is defined as:

${\bf B}_{r,j}^{\textrm{S-MMSE}}=\arg\min_{\left\\{{\bf
B}_{r,j}\right\\}}\hat{{\rm{MSE}}}_{j}.$ (30)

In order to find the optimal solution (30), we apply the differentiation rules
for the complex-valid matrices. Then, the derivative of
$\hat{\textrm{MSE}}_{j}$ with respect to ${\bf B}_{r,j}$ is obtained as:

$\displaystyle\frac{{\partial\hat{{\rm{MSE}}}_{j}}}{{\partial{{\bf{B}}_{r,j}}}}$
$\displaystyle=2{{\bf{B}}_{r,j}}\big{(}{{{\bm{\mathcal{H}}}_{i}}{{\bf{B}}_{t,i}}{\bf{B}}_{t,i}^{H}{\bm{\mathcal{H}}}_{i}^{H}+{\hat{\bf{W}}_{j}}+\sigma_{w}^{2}{{\bf{I}}_{{N_{r,j}}}}}\big{)}$
(31) $\displaystyle-2{{\bf{B}}_{t,i}^{H}}{{\bm{\mathcal{H}}}_{i}^{H}}.$

Finally, we derive the BB combiner ${\bf B}_{r,j}$ satisfying the S-MMSE
criterion defined in (30) (i.e.,
$\frac{{\partial\hat{{\rm{MSE}}}_{j}}}{{\partial{{\bf{B}}_{r,j}}}}=0$) as:

${\bf
B}_{r,j}^{\textrm{S-MMSE}}={{\bf{B}}_{t,i}^{H}}{{\bm{\mathcal{H}}}_{i}^{H}}\big{(}{{\bm{\mathcal{H}}}_{i}}{{\bf{B}}_{t,i}}{\bf{B}}_{t,i}^{H}{\bm{\mathcal{H}}}_{i}^{H}+{\hat{\bf{W}}_{j}}+\sigma_{w}^{2}{{\bf{I}}_{{N_{r,j}}}}\big{)}^{-1}.$
(32)

Algorithm 2 outlines the BB-stage design for the proposed AB-JHPC technique.

Algorithm 2 BB-stage design

1:$\bm{\mathcal{H}}_{i}$, ${\bf F}_{t,i}$, ${\bf F}_{r,i}$,
$\big{\\{}\theta_{t,c}^{\left(i\right)},\psi_{t,c}^{\left(i\right)},\delta_{t,c}^{\theta,\left(i\right)},\delta_{t,c}^{\psi,\left(i\right)},\big{\\}}$,
$\big{\\{}\theta_{r,c}^{\left(i\right)},\psi_{r,c}^{\left(i\right)},\delta_{r,c}^{\theta,\left(i\right)},\delta_{r,c}^{\psi,\left(i\right)}\big{\\}}$,
for $i=1:2$ and $c=1:C$.

2:for $i=1:2$ do

3: Perform SVD as $\bm{\mathcal{H}}_{i}={\bf U}_{j}{\bf\Sigma}_{i}{\bf
V}_{i}^{H}$ with ${\bf V}_{i}=\left[{\bf V}_{i,1},{\bf V}_{i,2}\right]$. and
${\bf U}_{i}=\left[{\bf U}_{i,1},{\bf U}_{i,2}\right]$.

4: Calculate the power allocation matrix ${\bf P}_{i}$ via (26).

5: Generate SVD-based BB precoder ${\bf B}_{t,i}$ via (25a).

6: if SVD-based BB combiner, then

7: Generate BB combiner ${\bf B}_{r,i}$ via (25b).

8: else if S-MMSE-based BB combiner, then

9: Find $\hat{\bf W}_{j}$ via (28).

10: Form $\hat{\bf{\Phi}}_{{\textrm{SI}},j,t}$ and
$\hat{\bf{\Phi}}_{{\textrm{SI}},j,r}$ via (10) with random AoD/AoA pairs
according to (18a).

11: Generate BB combiner ${\bf B}_{r,i}$ via (32).

12: end if

13:end for

14:${\bf{B}}_{t,1}$, ${\bf{B}}_{t,2}$, ${\bf{B}}_{r,1}$, ${\bf{B}}_{r,2}$

Ultimately, the RF-stage design explained in Algorithm 1 requires only the
slow time-varying AoD/AoA parameters of the intended and SI channels. Then,
the BB-stage design needs only the reduced-size effective intended channel
$\bm{\mathcal{H}}_{t,i}$ in addition to the AoD/AoA parameters as seen in
Algorithm 2. Therefore, the proposed AB-JHPC technique reduces the CSI
overhead size from $M_{t,i}\times M_{r,j}$ to $N_{t,i}\times N_{r,j}$ in
comparison to the conventional FDPC.

##  V RF Chain Reduction with Transfer Block Architecture

As shown in Fig. 1, the proposed AB-JHPC technique utilizes $N_{t,i}+N_{r,i}$
RF chains at the node $i$ for exploiting all degrees of freedom provided by
the intended channel while canceling the strong SI channel. Thus, in
comparison to the conventional FDPC, the number of RF chains is decreased from
$M_{t,i}+M_{r,j}$ to $N_{t,i}+N_{r,j}$, where $M_{t,i}+M_{r,j}$ is a
significantly large value for the massive MIMO systems (i.e., $N_{t,i}\ll
M_{t,i}$ and $N_{r,i}\ll M_{r,i}$). In order to further reduce the hardware
cost/complexity and power consumption, we employ a transfer block architecture
as demonstrated in Fig. 3, where the number of power-hungry RF chains is
reduced to $S_{i}+S_{j}$ as the minimum value for simultaneously transmitting
$S_{i}$ data streams and receiving $S_{j}$ data streams. For the hybrid
precoding shown in Fig. 3, the transmit transfer block ${\bf
T}_{t,i}\in\mathbb{C}^{N_{t,i}\times S_{i}}$ is placed between the transmit RF
beamformer ${\bf F}_{t,i}\in\mathbb{C}^{M_{t,i}\times N_{t,i}}$ and the
reduced-size BB precoder ${\bf B}_{\textrm{red},t,i}\in\mathbb{C}^{S_{i}\times
S_{i}}$. Similarly, the hybrid combining design shown in Fig. 3 utilizes the
receive transfer block ${\bf T}_{r,i}\in\mathbb{C}^{S_{j}\times N_{r,i}}$ for
connecting the receive RF beamformer ${\bf
F}_{r,i}\in\mathbb{C}^{N_{r,i}\times M_{r,i}}$ and the reduced-size BB
combiner ${\bf B}_{\textrm{red},r,i}\in\mathbb{C}^{S_{j}\times S_{j}}$. It is
seen that each input of ${\bf T}_{t,i}$ and ${\bf T}_{r,i}$ is passed through
two-pair of phase-shifters and their summation is transferred to the
corresponding output. Therefore, the modulus of each element in ${\bf
T}_{t,i}$ and ${\bf T}_{r,i}$ varies between $0$ and $2$.

Figure 3: Transfer block architecture at node $i$: (a) hybrid precoding with
$S_{i}$ RF chains, (b) hybrid combining with $S_{j}$ RF chains.

When the BB precoder ${\bf B}_{t,i}$ is obtained by using (25a), the
optimization problem for the transmit transfer block ${\bf T}_{t,i}$ and the
reduced-size BB precoder ${\bf B}_{\textrm{red},t,i}$ is written as:

$\displaystyle\left[{\bf T}_{t,i},{\bf B}_{\textrm{red},t,i}\right]$
$\displaystyle=\arg\min_{\left\\{{\bf T},{\bf
B}_{\textrm{red}}\right\\}}\left\|{\bf B}_{t,i}-{\bf T}{\bf
B}_{\textrm{red}}\right\|_{F}^{2},$ (33) $\displaystyle\rm{s.t.}$
$\displaystyle~{}|{\bf{T}}\left(m,n\right)|\in\left[0,2\right],\forall m,n$
$\displaystyle~{}{\rm{tr}}\left({\bf B}_{\textrm{red}}^{H}{\bf T}^{H}{\bf
T}{\bf B}_{\textrm{red}}\right)\leq P_{T},$

where the constraints refer the modulus of the transmit transfer block and the
maximum transmit power at the transmitter, respectively. By using [41, Lemma
1], the optimal closed-form solutions are derived as follows:

$\displaystyle{\bf T}_{t,i}\left(m,n\right)$ $\displaystyle=e^{j\angle{\bf
B}_{t,i}\left(m,n\right)}\left[e^{j\beta_{t,i}^{\left(m,n\right)}}+e^{-j\beta_{t,i}^{\left(m,n\right)}}\right],$
(34) $\displaystyle{\bf B}_{\textrm{red},t,i}$ $\displaystyle=\left({\bf
T}_{t,i}^{H}{\bf T}_{t,i}\right)^{-1}{\bf T}_{t,i}^{H}{\bf B}_{t,i},$

where $\beta_{t,i}^{\left(m,n\right)}=\cos^{-1}\left(\frac{|{\bf
B}_{t,i}\left(m,n\right)|}{\max_{u,v}|{\bf B}_{t,i}\left(u,v\right)|}\right)$.
Here, it is worthwhile to remark that the derived closed-form solutions
guarantees the optimality by satisfying ${\bf B}_{t,i}={\bf T}_{t,i}{\bf
B}_{\textrm{red},t,i}$.

Similarly, after acquiring the BB combiner ${\bf B}_{r,i}$ via (25b) for the
SVD approach or via (32) for the proposed S-MMSE algorithm, the optimization
problem for the receive transfer block architecture is defined as:

TABLE IV: AB-JHPC vs. FDPC: CSI Overhead Size, RF Chains, Phase-Shifters. | CSI Overhead Size | # of RF Chains | # of Phase Shifters
---|---|---|---
FDPC | $\sum\limits_{i=1}^{2}\sum\limits_{j\neq i}M_{r,j}\times M_{t,i}$ | $\sum\limits_{i=1}^{2}M_{t,i}+M_{r,i}$ | -
| AB-JHPC w/out
---
Transfer Block
$\sum\limits_{i=1}^{2}\sum\limits_{j\neq i}N_{r,j}\times N_{t,i}$ | $\sum\limits_{i=1}^{2}N_{t,i}+N_{r,i}$ | $\sum\limits_{i=1}^{2}N_{t,i}\times M_{t,i}+N_{r,i}\times M_{r,i}$
| AB-JHPC using
---
Transfer Block
$\sum\limits_{i=1}^{2}\sum\limits_{j\neq i}N_{r,j}\times N_{t,i}$ | $\sum\limits_{i=1}^{2}2S_{i}$ | $\sum\limits_{i=1}^{2}\sum\limits_{j\neq i}N_{t,i}\times\left(M_{t,i}+2S_{i}\right)+N_{r,i}\times\left(M_{r,i}+2S_{j}\right)$
TABLE V: Simulation parameters. # of antennas[56] | $M_{t,i}=M_{r,i}=16\times 16=256$
---|---
Data Streams | $S_{1}=S_{2}=4$
Path loss exponent [57] | $\eta=3.76$
Noise PSD [57] |  $-174$ dBm/Hz
Channel bandwidth [57] |  $10$ MHz
Antenna spacing (in wavelength)[56] | ${d=0.5}$
# of network realizations | $2.000$
${\bf H}_{i}$: Distance for paths | $\tau_{i}\in\left[35\rm{m},50\rm{m}\right]$
${\bf H}_{i}$: # of paths[56] | $L_{i}=20$
${\bf H}_{i}$: Mean azimuth AoD/AoA | $\psi_{t}^{\left(1\right)}=315^{\circ},\psi_{t}^{\left(2\right)}=355^{\circ}~{}\left|\right|~{}\psi_{r}^{\left(1\right)}=205^{\circ},\psi_{r}^{\left(2\right)}=245^{\circ}$
${\bf H}_{i}$: Mean elevation AoD/AoA | $\theta_{t}^{\left(i\right)}=\theta_{r}^{\left(i\right)}=40^{\circ}$
${\bf H}_{i}$: Azimuth/elevation spread[56] | ${\delta_{t}^{\psi,\left(i\right)}}={\delta_{r}^{\psi,\left(i\right)}}={\delta_{t}^{\theta,\left(i\right)}}={\delta_{r}^{\theta,\left(i\right)}}=10^{\circ}$
${\bf H}_{\textrm{NLoS},i}$: Distance for paths | $\tau_{\textrm{SI},i}\in\left[5\rm{m},15\rm{m}\right]$
${\bf H}_{\textrm{NLoS},i}$: # of paths[56] | $L_{\textrm{SI}}=20$
${\bf H}_{\textrm{NLoS},i}$: Mean azimuth AoD/AoA | $\psi_{\textrm{SI},t}^{\left(i\right)}=150^{\circ}~{}\left|\right|~{}\psi_{\textrm{SI},r}^{\left(i\right)}=75^{\circ}$
${\bf H}_{\textrm{NLoS},i}$: Mean elevation AoD/AoA | $\theta_{\textrm{SI},t}^{\left(i\right)}=\theta_{\textrm{SI},r}^{\left(i\right)}=40^{\circ}$
${\bf H}_{\textrm{NLoS},i}$: Azimuth/elevation spread[56] | ${\delta_{\textrm{SI},t}^{\psi,\left(i\right)}}={\delta_{\textrm{SI},r}^{\psi,\left(i\right)}}={\delta_{\textrm{SI},t}^{\theta,\left(i\right)}}={\delta_{\textrm{SI},r}^{\theta,\left(i\right)}}=10^{\circ}$

$\displaystyle\left[{\bf B}_{\textrm{red},r,i},{\bf T}_{r,i}\right]$
$\displaystyle=\arg\min_{\left\\{{\bf B}_{\textrm{red}},{\bf
T}\right\\}}\left\|{\bf B}_{r,i}-{\bf B}_{\textrm{red}}{\bf
T}\right\|_{F}^{2},$ (35) $\displaystyle\rm{s.t.}$
$\displaystyle~{}|{\bf{T}}\left(m,n\right)|\in\left[0,2\right],\forall m,n,$

By using (33) and (34), we obtain the optimal closed-form solutions as
follows:

$\displaystyle{\bf T}_{r,i}\left(m,n\right)$ $\displaystyle=e^{j\angle{\bf
B}_{r,i}\left(m,n\right)}\left[e^{j\beta_{r,i}^{\left(m,n\right)}}+e^{-j\beta_{r,i}^{\left(m,n\right)}}\right],$
(36) $\displaystyle{\bf B}_{\textrm{red},r,i}$ $\displaystyle={\bf
B}_{r,i}{\bf T}_{r,i}^{H}\left({\bf T}_{r,i}{\bf T}_{r,i}^{H}\right)^{-1},$

where $\beta_{r,i}^{\left(m,n\right)}=\cos^{-1}\left(\frac{|{\bf
B}_{r,i}\left(m,n\right)|}{\max_{u,v}|{\bf B}_{r,i}\left(u,v\right)|}\right)$.
The obtained closed-form solutions also maintain the optimality by satisfying
optimal solutions to (35) (i.e., ${\bf B}_{r,i}={\bf B}_{\textrm{red},r,i}{\bf
T}_{r,i}$). Therefore, the optimal solutions derived in (34) and (36) enable
the proposed transfer block architecture to minimize the RF chain utilization
without penalizing the achievable rate performance defined in (6a). Moreover,
the reduced number of power-hungry RF chains enhances the energy efficiency,
which is illustrated in Section VI.

Finally, Table IV summarizes the CSI overhead size, the number of RF chains
and the number of phase-shifters requirements for the proposed full-duplex
mmWave massive MIMO systems. We consider the proposed AB-JHPC technique
without the transfer block as shown in Fig. 1 and with the transfer block as
shown in Fig. 3. Additionally, as a benchmark, the CSI overhead size and the
number of RF chains for the conventional FDPC is given in Table IV [32].

##  VI Illustrative Results

In this section, we evaluate the performance of the proposed AB-JHPC technique
for the full-duplex mmWave massive MIMO systems. Based on the recent 3GPP
Release 16 specifications in [56, 57], Table V summarizes the numerical values
of the parameters used in the simulation setup, unless otherwise stated.
Additionally, we assume that the transmit and receive URAs at each node are
placed on the same surface with the separation by two wavelengths (i.e.,
$D_{1}=2$, $D_{2}=0$, $\Theta=0^{\circ}$ as shown in Fig. 2). Here, we first
demonstrate how the transmit/receive RF beamformers are generated according to
the given AoD/AoA parameters. Then, the amount of SIC achieved by the proposed
AB-JHPC design is presented. Afterwards, the total achievable-rate and energy
efficiency results are provided to compare the advantages of the proposed
full-duplex mmWave massive MIMO systems compared to its half-duplex
counterparts.

###  VI-A RF-Stage Design via AoD/AoA Parameters

Fig. 4 demonstrates the AoD/AoA supports for the intended channels (i.e.,
${\bf H}_{1}$ and ${\bf H}_{2}$) and the SI channels (i.e., ${\bf
H}_{\textrm{SI},1}$ and ${\bf H}_{\textrm{SI},2}$) based on the simulation
parameters given in Table V, where we assume a single scattering cluster with
$L_{i}=L_{\textrm{SI}}=20$ paths [56].

Figure 4: RF-stage design and AoD/AoA support for the intended and SI channels
(${\bf H}_{1}$, ${\bf H}_{2}$, ${\bf H}_{\textrm{SI},1}$, ${\bf
H}_{\textrm{SI},2}$).

Here, it is worthwhile to recall that
$\gamma_{x}=\sin\left(\theta\right)\cos\left(\psi\right)$ and
$\gamma_{y}=\sin\left(\theta\right)\sin\left(\psi\right)$ are the coefficients
reflecting both elevation angle ($\theta$) and azimuth angle ($\psi$). As
summarized in Algorithm 1, the RF-stage design starts by building the AoD/AoA
supports for each channel via (16a) and (18a). Then, we construct the
quantized angle-pairs by using the transmit and receive antenna elements at
each node. Considering transmit/receive URA size as $16\times 16=256$ [56],
there are $256$ possible quantized angle-pairs represented via black circles
in Fig. 4. Afterwards, we find the quantized angle-pairs covering the
corresponding AoD/AoA support of the intended channel and excluding that of
the SI channel. For instance, $N_{t,1}=11$ quantized angle-pairs represented
via blue cross symbols are selected for the transmit RF beamformer ${\bf
F}_{t,1}$, which are inside the AoD support of ${\bf H}_{1}$ and also
orthogonal to the AoD support of ${\bf H}_{\textrm{SI},1}$. Thus, in this
simulation setup, the transmit RF beamformer ${\bf F}_{t,1}$ generates
$N_{t,1}=11$ orthogonal-beams for sending data streams to the node $2$ and
suppressing the SI signal power occurred at the node $1$. Similarly, the
receive RF beamformer ${\bf F}_{r,1}$ is generated by using $N_{r,1}=8$
quantized angle-pairs illustrated with red star symbols, which covers the AoA
support of ${\bf H}_{2}$ and excludes that of ${\bf H}_{\textrm{SI},1}$.
Finally, the proposed hybrid architecture shown in Fig. 1 requires only
$N_{1}=N_{t,1}+N_{r,1}=19$ RF chains to develop transmit/receive RF
beamformers at the node $1$. Therefore, the hybrid architecture reduces the
number of RF chains by $96.2\%$ compared to its fully-digital counterparts.
Furthermore, the number of RF chains can be further reduced via the propsoed
transfer block architecture expressed in Section V (please see Table IV).

###  VI-B Self-Interference Cancellation

After designing the RF-stage, Fig. 5 demonstrates the SIC achieved by the
joint transmit/receive RF beamforming, where we consider that the power of
near-field SI channel reflecting the strong LoS paths is reduced by
$P_{\textrm{IS,dB}}\in\left[0\textrm{~{}dB},120\textrm{~{}dB}\right]$ via
antenna isolation.

Figure 5: Self-interference cancellation (SIC) achieved by the joint
transmit/receive RF beamforming versus antenna isolation.

According to the SI occurred at the node $2$, we compare the far-field SI
channel as ${\bf{H}}_{\textrm{NLoS},2}$, the near-field SI channel as
${\bf{H}}_{\textrm{LoS},2}$ and complete SI channel as
${\bf{H}}_{\textrm{SI},2}={\bf{H}}_{\textrm{LoS},2}+{\bf{H}}_{\textrm{NLoS},2}$
in terms of their powers777For the sake of simplicity, Fig. 5 only plots the
SI channel powers at the node $2$. Equivalently, the same behaviors are also
observed for the SI channel at the node $1$.. As expressed in Section III, the
power of near-field SI channel is inversely proportional to the antenna
isolation (i.e.,
$10\log_{10}\left(\mathbb{E}\big{\\{}\|{{\bf{H}}_{\textrm{LoS},i}}\|_{F}^{2}\big{\\}}\right)=-P_{\textrm{IS,dB}}$).
In order to precisely display the benefits of the proposed joint
transmit/receive RF beamforming, the average SI channel power at the node $2$
is plotted for four cases: (i) without RF beamforming as
$\mathbb{E}\left\\{\left\|{\bf H}_{\textrm{SI},2}\right\|_{F}^{2}\right\\}$
(ii) with only transmit RF beamforming as $\mathbb{E}\left\\{\left\|{\bf
H}_{\textrm{SI},2}{\bf F}_{t,2}\right\|_{F}^{2}\right\\}$ (iii) with only
receive RF beamforming as $\mathbb{E}\left\\{\left\|{\bf F}_{r,2}{\bf
H}_{\textrm{SI},2}\right\|_{F}^{2}\right\\}$ (iv) with joint transmit/receive
RF beamforming as
$\mathbb{E}\left\\{\left\|\bm{\mathcal{H}}_{\textrm{SI},2}\right\|_{F}^{2}\right\\}=\mathbb{E}\left\\{\left\|{{\bf{F}}_{r,2}{\bf{H}}_{\textrm{SI},2}{\bf{F}}_{t,2}}\right\|_{F}^{2}\right\\}$.

  * •

Case 1: We observe that when $P_{\textrm{IS,dB}}$ is very poor (e.g.,
$P_{\textrm{IS,dB}}<15$ dB), the near-field SI channel represented with black
dotted curve is dominant as expected. On the other hand, when
$P_{\textrm{IS,dB}}$ is higher than $20$ dB, the far-field SI channel
represented with black dashed curve becomes dominant. Moreover, even though
the quality of SIC is enhanced via more advanced antenna isolation techniques,
the complete SI channel remains constant due to the dominant NLoS paths.
Because the power of far-field SI channel expressed in (14) is independent
from $P_{\textrm{IS,dB}}$ and it is a function of the distance, path loss
exponent and complex path gains. Therefore, the complete SI channel power
without RF beamforming is only reduced from $0$ dB to $-18$ dB, when
$P_{\textrm{IS,dB}}$ is greater than 30 dB. It implies that $18$ dB SIC is
accomplished at most by only applying passive antenna isolation, which does
not aim to suppress the far-field SI channel. This behavior is also
independently observed in the experiment results in [18], where the authors
measure the SIC achieved by antenna isolation as $74$ dB in the anechoic
chamber as a low-reflection environment. However, the measurements inside a
reflective room in [18] shows that the SIC achieved by antenna isolation is
reduced to $44$ dB due to the dominant NLoS paths.

  * •

Cases 2 & 3: We apply only transmit (receive) RF beamforming for reducing the
power of far-field SI channel and enhancing the quality of SIC. The numerical
results show that the power of far-field SI channel is reduced from $-18$ dB
to $-56$ dB ($-60$ dB) as represented with green (blue) dashed curves.
Furthermore, when there is no antenna isolation (i.e., $P_{\textrm{IS,dB}}=0$
dB), only transmit (receive) RF beamforming reduces the power of near-field SI
signal from $0$ dB to $-14$ dB ($-24$ dB). Eventually, when
$P_{\textrm{IS,dB}}\geq 50$ dB SIC is satisfied via antenna isolation, the
power of the complete SI channel is $-56$ dB (${-60~{}\textrm{dB}}$), which
means an additional $38$ dB ($42$ dB) reduction via only transmit (receive) RF
beamforming compared to the first case without RF beamforming.

  * •

Case 4: The proposed joint transmit/receive RF beamformer design in Section IV
further suppresses the complete SI channel power compared to the previous
three cases as illustrated in Fig. 5. The first important observation is that
the far-field and near-field SI channel powers are reduced by $81.5$ dB and
$45.5$ dB, respectively. In comparison to Cases 2 & 3, the joint
transmit/receive RF beamformer design achieves higher suppression than the
summation of only transmit RF beamforming and only receive RF beamforming
(i.e., $81.5\textrm{ dB}>38\textrm{ dB}+42\textrm{ dB}$ for NLoS paths and
$45.5\textrm{ dB}>14\textrm{ dB}+24\textrm{ dB}$ for LoS paths). Moreover,
when $P_{\textrm{IS,dB}}\leq 50$ dB, there is a near-linear relationship
between the complete SI channel power and $P_{\textrm{IS,dB}}$. Thus,
according to the proposed joint transmit/receive RF beamforming, the complete
SI channel power in this simulation setup can be modeled as follows:

$\displaystyle\leavevmode\resizebox{375.80542pt}{}{$10\log_{10}\left(\mathbb{E}\left\\{\left\|\bm{\mathcal{H}}_{\textrm{SI},2}\right\|_{F}^{2}\right\\}\right)=10\log_{10}\left(\mathbb{E}\left\\{\left\|{{\bf{F}}_{r,2}{\bf{H}}_{\textrm{SI},2}{\bf{F}}_{t,2}}\right\|_{F}^{2}\right\\}\right)$}$
(37) $\displaystyle\hskip 17.22217pt\approx-P_{\textrm{IS,dB}}-45.5,~{}\forall
P_{\textrm{IS,dB}}\in\left[0\textrm{~{}dB},50\textrm{~{}dB}\right].$

The above equation affirms the importance of utilizing both joint
transmit/receive RF beamforming and advanced antenna isolation techniques in
order to cancel the strong SI channel. In other words, they works well
together. For example, in the case 1 (without RF beamforming), the residual SI
channel power is saturated at $-18$ dB, even if we achieve
${74\textrm{~{}dB}}$ antenna isolation as in [18]. On the other hand, in the
case 4 (joint transmit/receive RF beamforming), we have $45.5$ dB SIC, when
the antenna isolation is omitted (i.e.,
${P_{\textrm{IS,dB}}=0\textrm{~{}dB}}$). However, when we combine both of the
techniques, the complete SI channel power can be suppressed up to $99.5$ dB as
shown in Fig. 5.

Fig. 6 analyzes the SIC achieved by the proposed AB-JHPC technique, where we
consider the effects of both RF-stage and BB-stage design. The transmit power
is chosen as ${P_{T}=30}$ dBm for the simultaneous transmission of
$S_{1}=S_{2}=4$ data streams at both nodes.

Figure 6: Self-interference cancellation (SIC) achieved by the proposed AB-
JHPC technique versus antenna isolation, where the transmit power is
$P_{T}=30$ dBm.

By using the combined signal expression given in (3), Fig. 6 plots the powers
of intended signal, SI signal and noise for the first data stream at the node
2888The power of ISI given in (3) is not plotted in Fig. 6 because it is well
mitigated via the BB-stage design. Hence, ISI is not a limiting factor for the
full-duplex mmWave massive MIMO systems compared to the noise and strong SI..
According to the given noise power spectral density (PSD) as $-174$ dBm/Hz and
the channel bandwidth as $10$ MHz [57], the noise power is calculated as
$\sigma_{w}^{2}=-104$ dBm. Also, the power of modified noise given in (3) is
kept as $\sigma_{w}^{2}=-104$ dBm by normalizing the BB combiner999By using
${{{\bf{F}}_{r,2}}{\bf{F}}_{r,2}^{H}={{\bf{I}}_{{N_{r,2}}}}}$ and
${{\bf{w}}_{2}}\sim\mathcal{CN}\big{(}0,\sigma_{w}^{2}{\bf{I}}_{M_{r,2}}\big{)}$,
the power of modified noise in (3) is calculated as
$\mathbb{E}\big{\\{}{\big{|}{{\bf{b}}_{r,2,k}^{H}{{\bf{F}}_{r,2}}{{\bf{w}}_{2}}}\big{|}^{2}}\big{\\}}=\sigma_{w}^{2}{\bf{b}}_{r,2,k}^{H}{{\bf{b}}_{r,2,k}}$.
For keeping the modified noise power as $\sigma_{w}^{2}$, the $k^{th}$ row of
BB combiner is normalized as ${{{\bar{\bf
b}}}_{r,2,k}}=\frac{{{{\bf{b}}_{r,2,k}}}}{{\sqrt{{\bf{b}}_{r,2,k}^{H}{{\bf{b}}_{r,2,k}}}}}$.
. Therefore, the noise floor represented with black dotted curve is placed at
$-104$ dBm. Regarding the intended signal received at the node $2$, we first
consider the power of full-size intended channel ${\bf H}_{1}$ before applying
AB-JHPC. In that case, the transmit power $P_{T}$ is assumed to be equally
divided among four data streams, then the corresponding intended signal power
at the first data stream is calculated as
$\mathbb{E}\left\\{\frac{P_{T}}{4}\left\|{\bf
H}_{1}\right\|_{F}^{2}\right\\}$. After applying the proposed AB-JHPC with
S-MMSE and SVD, we also plot the power of intended signal given in (3). It is
seen that both S-MMSE and SVD approaches used in the BB combiner of AB-JHPC
provides almost the same intended signal power. Moreover, approximately $2$ dB
degradation is observed compared to the full-size intended channel power
before applying AB-JHPC, which means that the proposed AB-JHPC scheme
preserves most of the power provided by the channel. Considering the SI signal
occurred at the node $2$, we also plot the power of full-size SI channel ${\bf
H}_{\textrm{SI,2}}$ with the transmit power $P_{T}$ before applying AB-JHPC,
which is calculated as $\mathbb{E}\left\\{{P_{T}}\left\|{\bf
H}_{\textrm{SI},2}\right\|_{F}^{2}\right\\}$. As expected, when there is no
antenna isolation, the power of corresponding received SI is $30$ dBm. For
larger antenna isolation, the corresponding received SI is reduced to $12$
dBm, when $P_{\textrm{IS,dB}}$ is greater than 30 dB. Therefore, the antenna
isolation without applying AB-JHPC only provides up to $18$ dB SIC similar to
Fig. 5. Then, we employ the proposed AB-JHPC technique to significantly
enhance the achievable SIC. The residual SI is reduced from $30$ dBm to
$-55.7$ dBm, even when there is no antenna isolation (i.e.,
${P_{\textrm{IS,dB}}=0}$ dB). This represents $85.7$ dB SIC provided for the
first data stream by only employing the proposed AB-JHPC. Moreover, when
$P_{\textrm{IS,dB}}$ increases from $0$ dB to $70$ dB, the total SIC achieved
via antenna isolation and AB-JHPC with S-MMSE is a near-linear function of
$P_{\textrm{IS,dB}}$ (i.e., the total SIC is $P_{\textrm{IS,dB}}+85.7$ dB).
Additionally, the power of residual SI signal is reduced below noise floor
level for $P_{\textrm{IS,dB}}\geq 50$ dB. On the other hand, the residual SI
power is the major limiting factor for $P_{\textrm{IS,dB}}<50$ dB.
Furthermore, regarding the BB combiner of AB-JHPC, the proposed S-MMSE
algorithm further suppresses the SI signal power compared to the SVD approach
for $P_{\textrm{IS,dB}}\geq 50$ dB, where the NLoS paths are dominant in the
complete SI channel. For instance, when $P_{\textrm{IS,dB}}=60$ dB, AB-JHPC
with S-MMSE reduces the residual SI signal power to $-115.4$ dB, which is
$6.1$ dB lower compared to AB-JHPC with SVD.

Fig. 7 illustrates the signal power versus the square URA size, where the
transmit power is $P_{T}=30$ dBm and the antenna isolation is
$P_{\textrm{IS,dB}}=60$ dB. Before applying AB-JHPC, we observe that both the
intended signal and SI powers increase as the larger URAs are utilized by
means of the antenna array gain.

Figure 7: Self-interference cancellation (SIC) achieved by the proposed AB-
JHPC versus the square transmit/receive URA sizes, where the transmit power is
$P_{T}=30$ dBm and the antenna isolation is $P_{\textrm{IS,dB}}=60$ dB.

On the other hand, when more antenna elements are utilized, the power of SI
signal declines after applying the proposed AB-JHPC. The main reason for this
behavior is the enhancement of beamforming gain in the desired direction,
which also suppressing the side lobes affecting the undesired directions, such
as the SI signal. Moreover, it is shown that the quality of SIC also improves
for the larger URAs. To illustrate, after applying AB-JHPC with S-MMSE, the SI
signal power is $29.8$ dB below the noise floor $M_{t,i}=M_{r,i}=1024$ antenna
elements However, when there are transmit and receive $M_{t,i}=M_{r,i}=64$
antenna elements, the SI power is $11.4$ dB above the noise floor.
Furthermore, the proposed AB-JHPC with S-MMSE algorithm offers higher amount
of SIC for all URA sizes.

###  VI-C Total Achievable Rate: Full-Duplex vs. Half-Duplex

The total achievable rate performance of the proposed full-duplex mmWave
massive MIMO systems is investigated in Fig. 8, where the antenna isolation is
assumed as
${P_{\textrm{IS,dB}}=40\textrm{~{}dB},60\textrm{~{}dB},80\textrm{~{}dB},100\textrm{~{}dB}}$.
It is important to highlight that each node is equipped with $16\times 16=256$
transmit and receive antenna elements to simultaneously support
$S_{1}=S_{2}=4$ data streams. Also, we consider two half-duplex schemes as a
benchmark to compare the performance of full-duplex communications via the
proposed AB-JHPC technique: (i) AB-HPC as a hybrid architecture reducing the
number of RF chains and CSI overhead size as in the proposed scheme [45], (ii)
FDPC as the fully-digital architecture requiring large CSI overhead and a
single RF chain per each antenna element101010It is worthwhile to note that
the total achievable rate for the full-duplex communications is obtained by
using (6a) and (7). On the other hand, when both communication nodes employ
half-duplex, it is necessary to use separate time/frequency resources in order
to establish two-way data transmission and reception operations. Moreover, in
the case of half-duplex, there is no SI component expressed in (3). Therefore,
the total achievable rate for the half-duplex mmWave massive MIMO systems is
calculated via
${R_{total}^{\left(\textrm{HD}\right)}=\frac{1}{2}R_{1}^{\left(\textrm{HD}\right)}+\frac{1}{2}R_{2}^{\left(\textrm{HD}\right)}}$,
where $R_{1}^{\left(\textrm{HD}\right)}$ and
$R_{2}^{\left(\textrm{HD}\right)}$ can be obtained by adopting (3) and (6a)
without the SI component.. As presented in Fig. 4, the proposed AB-JHPC
requires $N_{1}=19$ and $N_{2}=20$ RF chains at the node 1 and node 2,
respectively. Therefore, in order to compare the hybrid architecture under the
same hardware cost/complexity, AB-HPC is also designed by using the same
number of RF chains. In Fig. 8, when we first analyze the half-duplex
communications, the performance gap between FDPC and AB-HPC is observed as
only $1.4$ dB. On the other hand, the hybrid architectures significantly
reduce the hardware cost/complexity by placing only $N_{1}=19$ ($N_{2}=20$) RF
chains at the node $1$ (node $2$), whereas FDPC requires $M_{1}=M_{2}=512$ RF
chains at each node. It implies a remarkably reduction in the hardware
cost/complexity by $96.2\%$. Also, the CSI overhead size is remarkably lowered
by $99.8\%$ (please see Table IV and Fig. 4). Regarding the achievable rate
performance of the proposed AB-JHPC technique, we both utilize the S-MMSE and
SVD approaches for the BB combiner design. Also, by using the total achievable
rate curves for AB-JHPC using full-duplex and AB-HPC using half-duplex plotted
in Fig. 8, the gain ratio shown in Fig. 8 is simply calculated by taking the
ratio of them (i.e.,
${R_{total}^{\left(\textrm{FD}\right)}}/{R_{total}^{\left(\textrm{HD}\right)}}$).
In Fig. 8, when the transmit power increases, the performance floor behavior
is monitored for the full-duplex transmission due to the residual SI power. We
observe that as the quality of antenna isolation improves, the total
achievable rate capacity reaches to a higher performance floor. For instance,
as shown in Fig. 6, the residual SI signal power is higher than the noise
floor, when $P_{\textrm{IS,dB}}=40$ dB and $P_{T}=30$ dBm. It is also visible
in Fig. 8, where AB-JHPC reaches to interference floor at the transmit power
of $P_{T}=30$ dBm for $P_{\textrm{IS,dB}}=40$ dB. On the other hand, when the
antenna isolation accomplishes $P_{\textrm{IS,dB}}=80$ dB SIC, the noise floor
is above the residual SI power in Fig. 6, especially for AB-JHPC with S-MMSE.
It is also verified by the achievable rate results in Fig. 8 (e.g., no
performance floor at $P_{T}=30$ dBm and $P_{\textrm{IS,dB}}=80$ dB) and in
Fig. 8 (e.g., the gain ratio is found as $1.98$ at $P_{T}=30$ dBm and
$P_{\textrm{IS,dB}}=80$ dB). Then, Fig. 8 shows that the proposed full-duplex
scheme doubles the achievable rate capacity for the low and medium transmit
power regime. Furthermore, the doubled-capacity property remains resistant for
the high transmit power regime as the antenna isolation based SIC is enhanced.
Regarding the BB combiner design of AB-JHPC, the proposed S-MMSE algorithm
provides higher achievable rate and converges to the higher performance floor
compared to the conventional SVD approach.

Figure 8: Achievable rate performance versus the transmit power ($P_{T}$) for
the proposed AB-JHPC technique using full-duplex compared to AB-HPC [45] and
FDPC using half-duplex: (a) total achievable rate, (b) full-duplex to half-
duplex gain ratio.

Fig. 9 illustrates the full-duplex to half-duplex gain ratio in terms of
achievable rate versus the square URA sizes ($M=M_{t,i}=M_{r,i}$) and number
of data streams ($S=S_{1}=S_{2}$), where the transmit power is $P_{T}=30$ dBm
and the antenna isolation is $P_{\textrm{IS,dB}}=74$ dB as in [18]. Although
the number of data streams is selected between $S=1$ and $S=6$, the gain ratio
for $M=64$ antennas is plotted until $S=4$ data streams because we obtain
$\min\left(N_{t,1},N_{r,1},N_{t,2},N_{r,2}\right)=4$ for $M=64$. It is seen
that the gain ratio almost approaches to $2$ for the larger URAs utilized at
the transmitter/receiver. However, when more data streams are transmitted, the
gain ratio degrades due to the increased residual SI power. Moreover, AB-JHPC
with S-MMSE has always the gain ratio higher than unity as illustrated in Fig.
9(a). By comparing Fig. 9(a) and Fig. 9(b), we observe that the proposed
S-MMSE technique outperforms SVD. For instance, when the URA size is chosen as
$M=144$ for serving $S=3$ data streams, AB-JHPC with S-MMSE provides the gain
ratio of $1.83$, while it is only $1.55$ for AB-JHPC with SVD.

(a) AB-JHPC with S-MMSE

(b) AB-JHPC with SVD

Figure 9: Full-duplex to half-duplex gain ratio versus the square URA sizes
($M=M_{t,i}=M_{r,i}$) and number of data streams ($S=S_{1}=S_{2}$): (a) AB-
JHPC with S-MMSE and (b) AB-JHPC with SVD, where the transmit power is
$P_{T}=30$ dBm and the antenna isolation is $P_{\textrm{IS,dB}}=74$ dB [18].

###  VI-D Energy Efficiency

In addition to improving the quality of SIC and reducing the hardware
cost/complexity, another primary objective in the proposed hybrid architecture
is decreasing the total power consumption for the massive MIMO systems. By
using the models given in [58, 59, 60], the total power consumption for the
two-way data transmission is defined as:

$P_{total}=2P_{T}+N_{RF}P_{RF}+N_{PS}P_{PS}\textrm{~{}~{}[W]},$ (38)

where $P_{T}$ is the total transmission power at each node, $N_{RF}$ and
$N_{PS}$ are the number of RF chains and phase-shifters, respectively,
$P_{RF}$ and $P_{PS}$ are the power consumption by each RF chain and phase-
shifter, respectively. Table IV summarizes the number of hardware components
for the proposed scheme using transfer block architecture (please see Fig. 3)
or without transfer block architecture (please see Fig. 1). For instance, when
no transfer block is utilized, there are $N_{RF}=39$ RF chains as shown in
Fig. 4. On the other hand, only $N_{RF}=16$ RF chains are required in the
transfer block architecture to support $S_{1}=S_{2}=4$ data streams. Also, we
consider ${P_{RF}}=250$ mW and ${P_{PS}}=1$ mW as in [58, 59]. Afterwards, the
energy efficiency is calculated by taking the ratio of total achievable rate
given in (7) with respect to the total power consumption given in (38) (i.e.,
$\frac{R_{total}}{P_{total}}$).

Fig. 10 shows the energy efficiency results for both full-duplex and half-
duplex schemes. Moreover, we consider the utilization of transfer block
architecture to enhance the energy efficiency results by reducing the number
of RF chains111111AB-HPC using the half-duplex transmission does not consider
the transfer block architecture [45]..

Figure 10: Energy efficiency (a) versus the transmit power ($P_{T}$) for
$P_{\textrm{IS,dB}}=60$ dB and (b) versus the antenna isolation
($P_{\textrm{IS,dB}}$) for $P_{T}=30$ dBm. Comparison of the proposed AB-JHPC
with S-MMSE technique using full-duplex with respect to AB-HPC [45] and FDPC
using half-duplex.

As proven in Section V, the total achievable rate performance remains the same
whether the transfer block architecture is utilized or not. First, Fig. 10
plots the energy efficiency versus the transmit power $P_{T}$, where
$P_{\textrm{IS,dB}}=60$ dB. The numerical results show that using the transfer
block improves the energy efficiency (e.g., at $P_{T}=30$ dB, we achieve $5.4$
bps/Hz/W by using the transfer block, whereas it reduces to $4.3$ bps/W/Hz
without transfer block). On the other hand, the energy efficiency of FDPC is
remarkably low compared to the hybrid schemes due to the utilization of power-
hungry RF chains. As explained in Table IV, the proposed AB-JHPC technique
requires significantly less number of RF chains than FDPC, which results in
the reduced power consumption. Second, Fig. 10 depicts the energy efficiency
versus the antenna isolation ($P_{\textrm{IS,dB}}$) for $P_{T}=30$ dBm. We
observe that the energy efficiency improves for the higher values of
$P_{\textrm{IS,dB}}$. However, the energy efficiency is almost saturated when
$P_{\textrm{IS,dB}}$ is higher than 80 dB.

Figure 11: Achievable rate performance of the proposed AB-JHPC with S-MMSE
technique versus the transmit power ($P_{T}$) for perfect and imperfect angle
information, where the antenna isolation is $P_{\textrm{IS,dB}}=74$ dB [18].

###  VI-E Imperfect Angle Information (AoD/AoA)

As discussed earlier, the RF beamformer design is based on the slow time-
varying AoD/AoA information (i.e., the mean of elevation angle ($\theta$) and
azimuth angle ($\psi$), and their spread). Therefore, according to the perfect
angle information, the transmit/receive RF beamformers are developed via
$\gamma_{x}=\sin\left(\theta\right)\cos\left(\psi\right)$ and
$\gamma_{y}=\sin\left(\theta\right)\sin\left(\psi\right)$ coefficients as
illustrated in Fig. 4. However, considering the imperfect angle information,
the estimated coefficients are defined as
$\hat{\gamma}_{x}=\gamma_{x}+\Delta\hat{\gamma}_{x}$ and
$\hat{\gamma}_{y}=\gamma_{y}+\Delta\hat{\gamma}_{y}$, where
$\Delta\hat{\gamma}_{x}$ and $\Delta\hat{\gamma}_{y}$ are the estimation
errors on $\gamma_{x}$ and $\gamma_{y}$ [61, 62, 63]. Here,
$\Delta\hat{\gamma}_{x}\in\left[-\frac{\epsilon_{x}}{2\pi},+\frac{\epsilon_{x}}{2\pi}\right]$
and
$\Delta\hat{\gamma}_{y}\in\left[-\frac{\epsilon_{y}}{2\pi},+\frac{\epsilon_{y}}{2\pi}\right]$
are uniformly distributed random variables, where $\epsilon_{x}$ and
$\epsilon_{y}$ denote the maximum angle estimation errors up to the beamwidth
between the first nulls [61]. When URA has $M=M_{x}\times M_{y}$ antennas
(i.e., $M_{x}$ and $M_{y}$ antennas along $x$-axis and $y$-axis as shown in
Fig. 2), we have $\epsilon_{x}=\frac{2\pi}{M_{x}}$ and
$\epsilon_{y}=\frac{2\pi}{M_{y}}$ [63].

Fig. 11 demonstrates the effect of imperfect angle information on the
achievable rate performance of the proposed AB-JHPC with S-MMSE technique,
where the antenna isolation is $P_{\textrm{IS,dB}}=74$ dB as in [18]. By
assuming the same size transmit and receive URAs at each node
($M=M_{t,i}=M_{r,i}$), three square URA cases are considered as ${M=144}$,
${M=256}$, and ${M=400}$. Also, we assume either ${S=1}$ or ${S=4}$ data
streams transmitted by each node for two-way communications. The numerical
results imply that the proposed AB-JHPC technique for the full-duplex massive
MIMO systems is robust to the imperfect angle information. The achievable rate
performance with the imperfect angle information closely approaches the
perfectly estimated angle information case, as the URA size increases. The
main reason for this behavior is the reduced angle estimation error with
larger URAs (i.e., $\epsilon_{x}=\frac{2\pi}{M_{x}}$ and
$\epsilon_{y}=\frac{2\pi}{M_{y}}$). For instance, when the URA size is chosen
as $M=144$ to support $S=4$ data streams, the performance gap between the
perfect and imperfect angle information is observed as $4.7$ bps/Hz at
$P_{T}=30$ dBm, where the perfect angle information provides $84.4$ bps/Hz
achievable rate capacity. On the other hand, when the URA size is increased to
$M=256$ ($M=400$), the performance gap reduces to only $1.2$ bps/Hz ($0.9$
bps/Hz) at $P_{T}=30$ dBm, while the total achievable rate capacity with
perfectly estimated angle information is also enhanced to $113.2$ bps/Hz
($129.7$ bps/Hz). In other words, in the case of imperfect angle information
utilized at the RF beamformer design, we achieve $99.1\%$ ($99.3\%$) of the
capacity obtained with the perfect angle information, when the URA size is
considered as $M=256$ ($M=400$).

##  VII Conclusions

In this paper, a novel angular-based joint hybrid precoding/combining (AB-
JHPC) has been proposed for the full-duplex mmWave massive-MIMO systems. Our
primary objectives in the proposed AB-JHPC technique are to (i) enhance the
quality of self-interference cancellation (SIC), (ii) maximize the power of
desired/intended signal, (iii) reduce the channel estimation overhead size,
(iv) decrease the hardware cost/complexity via a low number of RF chains.
First, the RF-stage of AB-JHPC has been constructed via slow time-varying
AoD/AoA information, where the transmit/receive RF beamformers aim to maximize
the desired/intended signal power and cancel the strong SI power. Therefore, a
joint RF beamformer structure has been designed for generating orthogonal
transmit/receive beams covering AoD/AoA support of intended channel and
excluding AoD/AoA support of SI channel. Then, the BB-stage of AB-JHPC has
been built via the reduced-size effective intended channel. Thus, the proposed
AB-JHPC technique does not require any instantaneous SI channel knowledge
neither in the RF-stage nor in the BB-stage. We have primarily considered the
well-known SVD approach to develop the BB precoder and BB combiner. In
addition to SVD, we have also proposed a new semi-blind minimum mean square
error (S-MMSE) technique in the BB-stage design, which seeks to suppress the
power of far-field SI component by using the slow time-varying channel
characteristics. Afterwards, a transfer block architecture has been employed
to minimize the RF chain utilization without sacrificing the achievable rate
performance. The promising numerical results show that the joint
transmit/receive RF beamformer suppresses the power of far-field (near-field)
SI channel by $81.5$ dB ($44.5$ dB). On the other hand, the SI signal is
considerably mitigated via the proposed AB-JHPC technique. We observe that
$85.7$ dB SIC is achieved via only AB-JHPC and the total amount of SIC can be
linearly increased via the advanced antenna isolation techniques. In terms of
the total achievable rate capacity, the proposed full-duplex mmWave massive
MIMO systems can double the total achievable rate capacity compared to its
half-duplex counterpart. Also, the proposed S-MMSE algorithm remarkably
outperforms the conventional SVD approach, especially in the high transmit
power regime. It is shown that the transfer block architecture utilizing less
number of RF chains further improves the energy efficiency of the full-duplex
mmWave massive MIMO systems.

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