# Design and Analysis of Wideband In-Band-Full-Duplex FR2-IAB Networks

Junkai Zhang, , Haifeng Luo, Navneet Garg, Abhijeet Bishnu, Mark Holm, and
Tharmalingam Ratnarajah Manuscript received March 30, 2021; revised August 26,
2021, and November 3, 2021; accepted November 4, 2021. The work of J. Zhang,
H. Luo, N. Garg, and A. Bishnu was supported by the research grant from Huawei
Technologies (Sweden) AB. The work of T. Ratnarajah was supported by the U.K.
Engineering and Physical Sciences Research Council (EPSRC) under Grant
EP/P009549/1. The associate editor coordinating the review of this article and
approving it for publication was Trung Q. Duong. (Corresponding author: Junkai
Zhang.)J. Zhang, H. Luo, N. Garg, A. Bishnu and T. Ratnarajah are with
Institute for Digital Communications, The University of Edinburgh, Edinburgh,
EH9 3FG, UK (e-mail: {jzhang15, hluo2, ngarg, abishnu, T.
Ratnarajah}@ed.ac.uk.)M. Holm is with Radio Basestation Systems Department,
Huawei Technologies (Sweden) AB, Gothenburg, Sweden. (e-mail:
mark.holm@huawei.com).

###### Abstract

This paper develops a 3GPP-inspired design for the in-band-full-duplex (IBFD)
integrated access and backhaul (IAB) networks in the frequency range 2 (FR2)
band, which can enhance the spectral efficiency (SE) and coverage while
reducing the latency. However, the self-interference (SI), which is usually
more than 100 dB higher than the signal-of-interest, becomes the major
bottleneck in developing these IBFD networks. We design and analyze a
subarray-based hybrid beamforming IBFD-IAB system with the RF beamformers
obtained via RF codebooks given by a modified Linde-Buzo-Gray (LBG) algorithm.
The SI is canceled in three stages, where the first stage of antenna isolation
is assumed to be successfully deployed. The second stage consists of the
optical domain (OD)-based RF cancellation, where cancelers are connected with
the RF chain pairs. The third stage is comprised of the digital cancellation
via successive interference cancellation followed by minimum mean-squared
error baseband receiver. Multiuser interference in the access link is canceled
by zero-forcing at the IAB-node transmitter. Simulations show that under 400
MHz bandwidth, our proposed OD-based RF cancellation can achieve around 25 dB
of cancellation with 100 taps. Moreover, the higher the hardware impairment
and channel estimation error, the worse digital cancellation ability we can
obtain.

###### Index Terms:

Wideband in-band-full-duplex millimeter wave (FR2 band), subarray hybrid
beamforming, integrated access and backhaul, codebook design, self-
interference cancellation.

##  I Introduction

Frequency range 2 (FR2) band (i.e., millimeter wave) communications have been
identified as the key technology for the beyond fifth-generation (5G) wireless
communications to provide much larger bandwidth, narrower beam, and high data
rate services. Different from the FR1 band ($\leq 7.225$ GHz), in the FR2 band
($\geq 24.250$ GHz), high path loss and blockages become the major obstacles
for broader coverage. However, the short wavelengths at the FR2 frequencies
facilitate the deployment of large-scale antenna arrays, which could
compensate for such high losses with highly directional narrow beamforming and
provide reliable transmission quality [1, 2].

In the recent 3rd Generation Partnership Project (3GPP) technical report TR
38.874 (Rel. 16) [3], the integrated access and backhaul (IAB) networks have
been proposed for the FR2 band communications, where only IAB donors connect
with the core network by fiber. IAB-nodes can wirelessly communicate with both
the access and the backhaul links as well as perform IAB-specific tasks such
as resource allocation, route selection, and optimization [4]. This novel
architecture enables cheap and dense deployment while extending the coverage
in FR2 bands. Despite the visible advantages of this architecture, the study
of IAB networks is still in its infancy.

In-band-full-duplex (IBFD) transmission, which has been treated as another
breakthrough for beyond 5G wireless communications, breaks the rule that
downlink and uplink communications should occur in different time/frequency
slots. In the IAB networks, IAB-nodes are preferred to run under the IBFD mode
[5]. Compared with the half-duplex (HD), thanks to simultaneous transmissions,
the IBFD mode can almost double the spectral efficiency (SE) without the need
for the large guard time/band arranged in standard time-division duplex and
frequency-division duplex systems [6, 7]. However, the major obstacle of IBFD
communications is the existence of strong self-interference (SI), which is
usually seen as more than 100 dB stronger than the signal of interest [8].
Therefore, finding efficient SI cancellation (SIC) techniques is important for
IBFD operation and has recently been a popular research topic. Through
hardware prototype measurements in 28 GHz, authors in [9] evaluate the
framework’s link-level SI reduction in the propagation domain and system-level
performance to verify the feasibility of IBFD-IAB systems; however, the large-
scale antenna array and hybrid precoding were not considered.

For the wideband IBFD-FR2 communications, we propose a three-stage SIC, which
consists of the antenna isolation stage (i.e., by isolating the transceiver
antennas electromagnetically for passive cancellation) [10], the analog
cancellation (A-SIC) stage (i.e., by establishing a circuit canceler between
each transceiver pair to replicate the SI channel as accurately as possible)
[11], and the digital cancellation (D-SIC) stage (i.e., handling of the
residual SI (RSI) left by previous stages by designing efficient beamformers)
[8, 12, 13]. In the A-SIC, the conventional micro-strip analog canceler
requires a huge number of taps for wideband SIC. However, due to the large
insertion losses and realization of hundreds of taps, wideband SIC becomes
infeasible in practice. Besides, it is challenging for the micro-strip analog
canceler to be directly extended to FR2 band scenarios due to the hardware
limitation (i.e., RF components usually do not have such processing properties
at the FR2 band). Thus, a hardware efficient optical domain (OD)-based analog
canceler has been investigated in [14] for the single antenna system. However,
OD-based A-SIC for multi-antenna systems or IAB networks is lacking in the
literature.

Due to the use of large-scale array systems, the traditional full digital
beamforming scheme for the FR1 band becomes expensive to implement for the FR2
band. Thus, towards the need for cost-friendly system design, hybrid
beamforming has become a powerful and economical tool in large-scale array
systems, which reduces the requirement on the number of RF chains and
simplifies the system complexity [15]. Based on the extension of the standard
Orthogonal Matching Pursuit (OMP) algorithm, a novel hybrid beamforming design
was proposed in [16]. Compared with the fully connected hybrid beamforming
structure [2], to improve the deployment cost and guarantee the similar
performance of the system, authors in [1, 17], and [18] develop a subarray
hybrid beamforming structure, where one RF chain only connects with a portion
of antenna arrays. However, the works that consider the wideband IBFD multi-
user IAB networks with subarray hybrid beamforming in the FR2 band still need
more investigation.

The hybrid beamforming design algorithms in [1, 2, 15, 16, 17] need to access
the large and sparse channel matrix, which is hard to acquire in reality.
Although the compressed sensing-based channel estimation approaches are
presented in [19], it is difficult to realize in practical scenarios. Instead,
the RF effective channel is estimated using standard estimation methods in
practice, where the RF precoding and combining matrices are selected from the
pre-defined codebooks. In [2], the RF codebook is designed by the Lloyd type
algorithm. A $K$-means-based beam codebook is proposed by Mo et al., whose
codewords are defined by maximizing the beamforming gain [20]. Unfortunately,
their vector-wise codebooks may lead to a low-rank beamforming matrix, which
directly amounts to a loss in the degrees of freedom, especially when the
number of RF chains is more than one.

Further, the hardware impairments (HWI), which takes into account the
imperfection in the hardware, such as oscillators noise, amplifiers noise,
non-linearities in the digital-to-analog converters (DACs) and the analog-to-
digital converters (ADCs), and etc., have not been considered in most of the
studies yet. Authors in [21] have mentioned that the independent Gaussian
model can optimally capture those combined non-ideal hardware effects.

Based on the above motivations, in this paper, we investigate the design and
analysis of multiuser FR2-IBFD-IAB networks with subarray-based hybrid
beamforming. The contributions of this work are given as follows:

  * •

RF codebook design and RF effective channel estimation: For the subarray
hybrid beamforming scheme, the RF precoders and combiners are selected by
scanning from the matrix-wise codebooks, designed with our modified mean
squared error (MSE)-based Linde-Buzo-Gray (LBG) algorithm, and the RF
effective channel can be then estimated with standard estimation methods.
Simulations show that, with the proposed codebooks, we can achieve a similar
SE as that with infinite resolution phase shifters (PSs) without suffering
from low rank quantized beamforming matrices.

  * •

Staged SIC: We propose a staged SIC scheme in this paper, where the A-SIC is
realized by the OD-based canceler connected with the RF chain pairs on the
IAB-node to reduce the space and cost. Compared with the conventional micro-
strip analog canceler, our canceler can provide a significant number of true
delay lines for wideband operations and have good frequency-flatness.
Simulations show that with our OD-based canceler, 25 dB of A-SIC can be
achieved with about 100 taps over 400 MHz bandwidth.

  * •

System Analysis with RSI: In order to explore how the RSI caused by the HWI
and RF effective channel uncertainties can affect the performance of the IBFD
system, we analyze the SE of the backhaul link by varying the HWI factors and
SI RF effective channel estimation errors. Simulation results show that as SNR
increases, the system becomes more vulnerable to the RSI; however, the
tolerance is improved when increasing the codebook size. It is also shown that
at lower RSI values, IBFD operation doubles the SE compared to that of the HD.

The rest of the paper is organized as follows. In Section II, the system model
and channel models are identified, followed by introducing the OD-based analog
canceler design in Section III. Then, the modified LBG algorithm for the RF
codebook design is proposed in Section IV, where RF effective channels are
estimated with selected RF beamformer pairs. Next, D-SIC is processed in
Section V. In Section VI, the SE expressions are evaluated, followed by the
design of BB beamformers for both backhaul and access links. Finally, some
simulation results and a brief conclusion are shown in Section VII and Section
VIII, respectively.

Notations: $\mathcal{B},\mathbf{B},\mathbf{b}$, $b$ represent a set, a matrix,
a vector, and a scalar, respectively. $\mathbf{B}^{H},\mathbf{B}^{-1}$, and
$\mathbf{B}^{T}$ are the Hermitian, inverse, and transpose of $\mathbf{B}$,
respectively. $|\mathcal{B}|$ is the cardinality of $\mathcal{B}$.
$\lVert\mathbf{B}\rVert_{F}$, $|\mathbf{B}|_{mn}$,
$\mathrm{det}\\{\mathbf{B}\\}$, and $\mathrm{tr}[\mathbf{B}]$ are the
Frobenius norm, absolute value of the ($m,n$)th entry, determinant, and trace
of $\mathbf{B}$, respectively. $\lVert\mathbf{b}\rVert_{2}$ is the L2-norm of
$\mathbf{b}$. $\lVert b\rVert$ is the norm of $b$. $arg(\mathbf{B})$ takes the
angle of each entry of $\mathbf{B}$. $\mathrm{diag}[\mathbf{B}]$ takes the
diagonal elements of the matrix.
$\mathrm{blkdiag}[\mathbf{B}_{1},\mathbf{B}_{2}]$ is the block diagonal matrix
formed by matrix $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$. $[\mathbf{B}]_{:,1:n}$
and $[\mathbf{B}]_{m,n}$ denote the first $n$ columns and the $(m,n)$th entry
of $\mathbf{B}$, respectively. $\mathrm{Cov}[\mathbf{B}]$ is the covariance
matrix, i.e., $\mathbb{E}\\{\mathbf{B}\mathbf{B}^{H}\\}$. $\odot$ indicates
the Hadamard product. $d(\cdot,\cdot)$ is the distance measurement.
$\mathcal{CN}(m,n)$ denotes a complex Gaussian distribution with mean value of
$m$ and variance $n$, and $\mathbf{I}_{K}$ is the $K\times K$ identity matrix.

##  II System and Channel Models

###  II-A System Model

In this subsection, the system model is described for the wideband FR2-IBFD-
IAB multiuser networks. According to the technical specifications–TR 38.874
(Rel. 16) provided by the 3GPP, standalone (SA) and non-standalone (NSA) are
two typical deployments considered for IAB networks [3]. In this work, we
consider the downlink of a single-cell FR2-IBFD-IAB multiuser network with SA
deployment111The reason why SA structure is considered in this work is that
the NSA architecture permits IAB-nodes and UEs to communicate with both 4G
base stations (i.e., eNBs) as well as 5G base stations (i.e., gNBs); however,
SA only allows connections with 5G base stations, which is considered for
future wireless communication network environment. With minor modifications,
the present design and analysis can be used for NSA as well., which consists
of the following parts, that are

  * •

an IAB donor, also called gNB, which is a single logical node and acts as the
base station;

  * •

an IBFD-IAB-node, which contributes SI from its transmitter to its receiver;

  * •

$U$ downlink user-equipments (UEs).

The IAB donor connects to the 5G next-generation core (NGC) network by fiber
and communicates with the IAB-node through a wireless backhaul link. The IAB-
node serves the users by wireless access links. Note that, in this work, the
IAB donor only provides backhaul link service. An illustration of this IBFD-
IAB multiuser network used in this work is depicted in Fig. 1, and more
information about the 3GPP architecture can be found in our recent work [4].

Figure 1: Illustration of a single-cell IBFD-IAB multiuser network under the
SA deployment.

The IAB donor and IAB-node are equipped with the subarray-based hybrid
beamforming structure [17], where each RF chain only connects with a portion
of antenna elements. Compared with the fully connected structure [17], the
subarray structure provides a cost-efficient solution for connecting RF chains
to the antenna arrays. The number of subarrays (RF chains) at the IAB-donor
and IAB-node is assumed to be the same as the number of devices at the UEs
node, i.e., $U$. Meanwhile, the number of data streams transmitted from the
IAB donor and IAB-node is assumed to be $U$ as well. However, since each user
is assumed to have one RF chain receiving one data stream, only analog
beamforming is required. For the FR2 band, the Orthogonal Frequency Division
Multiplexing (OFDM) system is adopted, where we assume i) the length of the
data block is the same as the number of subcarriers, i.e., ${K}$; ii) the RF
beamformers are frequency-flat and the same for all subcarriers. In contrast,
the baseband (BB) beamformers are different for different subcarriers [2]. The
beamforming structure for this wideband FR2-IBFD-IAB network is shown in Fig.
2.

At the IAB donor, $U$ data streams are transmitted through $U$ RF chains and
${N_{T}}$ transmit antenna arrays. The total number of antenna arrays are
equally divided into $U$ subarrays, each with one RF chain. Hence, the
transmitted signal at the $k=1,2,\dotsc,{K}$th subcarrier from the IAB donor
is given by

$\mathbf{x}_{\mathrm{D}}[k]=\mathbf{F}_{{\rm
RF}\mathrm{D}}\left(\underbrace{\mathbf{F}_{{\rm
BB}\mathrm{D}}[k]\mathbf{s}_{\mathrm{D}}[k]}_{\widetilde{\mathbf{x}}_{\mathrm{D}}[k]}+\mathbf{e}_{\mathrm{D}}[k]\right),$
(1)

where $\mathbf{F}_{{\rm RF}\mathrm{D}}=\mathrm{blkdiag}\left[\mathbf{f}_{{\rm
RF}\mathrm{D},1},\mathbf{f}_{{\rm RF}\mathrm{D},2},\ldots,\mathbf{f}_{{\rm
RF}\mathrm{D},U}\right]\in\mathbb{C}^{N_{T}\times U}$ is the block diagonal RF
precoder matrix with $\mathbf{f}_{{\rm
RF}\mathrm{D},u}\in\mathbb{C}^{\frac{N_{T}}{U}\times 1},\forall
u\in\\{1,2,\ldots,U\\}$ representing the RF precoder vector of the $u$th
subarray. $\mathbf{F}_{{\rm BB}\mathrm{D}}[k]\in\mathbb{C}^{U\times U}$
represents the BB precoder matrix. The transmit data vector
$\mathbf{s}_{\mathrm{D}}[k]\in\mathbb{C}^{U\times 1}$ at the subcarrier $k$
has the covariance matrix of
$\mathbb{E}\left\\{\mathbf{s}_{\mathrm{D}}[k]\mathbf{s}^{H}_{\mathrm{D}}[k]\right\\}=\frac{P_{t}}{KU}\mathbf{I}_{U}$,
where $P_{t}$ is the average total transmit power across all subcarriers. By
applying the transmit power constraint with equal power allocation, we get the
constraint on the precoder as $\lVert\mathbf{F}_{{\rm
RF}\mathrm{D}}\mathbf{F}_{{\rm BB}\mathrm{D}}[k]\rVert_{F}^{2}=U$ for all
subcarriers. The vector $\mathbf{e}_{\mathrm{D}}[k]\in\mathbb{C}^{U\times
1}\sim\mathcal{CN}\left(\mathbf{0},\rho\mathrm{diag}\left[\mathrm{Cov}\left[\widetilde{\mathbf{x}}_{\mathrm{D}}[k]\right]\right]\right)$
captures the transmitter HWI at the IAB donor with $\rho<<1$, where the
transmitter HWI is uncorrelated with the transmit signal.

At the IBFD-IAB-node, separate antennas are configured for transmission and
reception (i.e., there are ${n}_{{T}}$ transmit antenna arrays and $U$ RF
chains for transmitting to the UEs node; and ${n}_{{R}}$ antenna arrays with
$U$ RF chains for receiving data from the IAB donor). Similarly, the subarray
structure divides those antenna arrays into $U$ equal panels, each with one RF
chain. Without SIC, the decoded signal at the IAB-node for subcarrier $k$ is
expressed in (2), shown on the top of next page,

$\displaystyle\mathbf{y}_{\mathrm{N}}[k]=\mathbf{W}_{{\rm
BB}\mathrm{N}}^{H}[k]\left[\underbrace{\mathbf{W}_{{\rm
RF}\mathrm{N}}^{H}\left(\mathbf{H}_{\mathrm{ND}}[k]\mathbf{x}_{\mathrm{D}}[k]+\mathbf{H}_{\mathrm{SI}}[k]\mathbf{x}_{\mathrm{N}}[k]+\mathbf{z}_{\mathrm{N}}[k]\right)}_{\widetilde{\mathbf{y}}_{\mathrm{N}}[k]}+\mathbf{g}_{\mathrm{N}}[k]\right]$
(2)

where $\mathbf{W}_{{\rm RF}\mathrm{N}}=\mathrm{blkdiag}\left[\mathbf{w}_{{\rm
RF}\mathrm{N},1},\mathbf{w}_{{\rm RF}\mathrm{N},2},\ldots,\mathbf{w}_{{\rm
RF}\mathrm{N},U}\right]\in\mathbb{C}^{{n}_{R}\times U}$ represents the RF
combiner matrix with $\mathbf{w}_{{\rm
RF}\mathrm{N},u}\in\mathbb{C}^{\frac{n_{R}}{U}\times 1},\forall
u\in\\{1,2,\ldots,U\\}$ denoting the RF combiner vector of subarray $u$.
$\mathbf{W}_{{\rm BB}\mathrm{N}}[k]\in\mathbb{C}^{U\times U}$ is the BB
combiner matrix.
$\mathbf{H}_{\mathrm{ND}}[k]\in\mathbb{C}^{{n}_{{R}}\times{N}_{{T}}}$ and
$\mathbf{H}_{\mathrm{SI}}[k]\in\mathbb{C}^{{n}_{{R}}\times{n}_{{T}}}$ are the
ideal backhaul channel matrix and SI channel matrix at the subcarrier $k$,
respectively. $\mathbf{z}_{\mathrm{N}}[k]\in\mathbb{C}^{n_{R}\times
1}\sim\mathcal{CN}(\mathbf{0},\sigma^{2}_{\mathrm{N}}\mathbf{I}_{n_{R}})$ is
the circularly symmetric Gaussian noise. The vector
$\mathbf{g}_{\mathrm{N}}[k]\in\mathbb{C}^{U\times
1}\sim\mathcal{CN}(\mathbf{0},\beta\mathrm{diag}[\mathrm{Cov}[\widetilde{\mathbf{y}}_{\mathrm{N}}[k]]])$
accounts for the receiver HWI at the IAB-node, which is uncorrelated with the
received signal, with $\beta<<1$.

Figure 2: Illustration of a wideband FR2-IBFD-IAB multiuser system with
subarray hybrid beamforming.

The vector $\mathbf{x}_{\mathrm{N}}[k]$ in (2) denotes the signal transmitted
from the IAB-node at the $k$th subcarrier, given as

$\mathbf{x}_{\mathrm{N}}[k]=\mathbf{F}_{{\rm
RF}\mathrm{N}}\left(\underbrace{\mathbf{F}_{{\rm
BB}\mathrm{N}}[k]\mathbf{s}_{\mathrm{N}}[k]}_{\widetilde{\mathbf{x}}_{\mathrm{N}}[k]}+\mathbf{e}_{\mathrm{N}}[k]\right),$
(3)

where $\mathbf{F}_{{\rm RF}\mathrm{N}}=\mathrm{blkdiag}\left[\mathbf{f}_{{\rm
RF}\mathrm{N},1},\mathbf{f}_{{\rm RF}\mathrm{N},2},\ldots,\mathbf{f}_{{\rm
RF}\mathrm{N},U}\right]\in\mathbb{C}^{n_{T}\times U}$ is the RF precoder
matrix with $\mathbf{f}_{{\rm
RF}\mathrm{N},u}\in\mathbb{C}^{\frac{n_{T}}{U}\times 1},\forall
u\in\\{1,2,\ldots,U\\}$ denoting the RF precoder vector of the $u$th subarray.
$\mathbf{F}_{{\rm BB}\mathrm{N}}[k]=\left[\mathbf{f}_{{\rm
BB}\mathrm{N},1}[k],\mathbf{f}_{{\rm
BB}\mathrm{N},2}[k],\ldots,\mathbf{f}_{{\rm
BB}\mathrm{N},U}[k]\right]\in\mathbb{C}^{U\times U}$ represents the BB
precoder matrix with $\mathbf{f}_{{\rm
BB}\mathrm{N},u}[k]\in\mathbb{C}^{U\times 1},\forall u\in\\{1,2,\ldots,U\\}$.
$\mathbf{s}_{\mathrm{N}}[k]\in\mathbb{C}^{U\times 1}$ is the transmit data
vector with covariance matrix of
$\mathbb{E}\left\\{\mathbf{s}_{\mathrm{N}}[k]\mathbf{s}^{H}_{\mathrm{N}}[k]\right\\}=\frac{P_{t}}{KU}\mathbf{I}_{U}$
and is uncorrelated with $\mathbf{s}_{\mathrm{D}}[k]$. The vector
$\mathbf{e}_{\mathrm{N}}[k]\in\mathbb{C}^{U\times
1}\sim\mathcal{CN}\left(\mathbf{0},\rho\mathrm{diag}\left[\mathrm{Cov}\left[\widetilde{\mathbf{x}}_{\mathrm{N}}[k]\right]\right]\right)$
denotes the transmitter HWI at the IAB node, which is uncorrelated with the
transmit signal. In addition, for all subcarriers, the precoder per subarray
has to satisfy the constraint of $\lVert\mathbf{F}_{{\rm
RF}\mathrm{N}}\mathbf{f}_{{\rm BB}\mathrm{N},u}[k]\rVert_{F}^{2}=1,\forall
u\in\\{1,2,\ldots,U\\}$ for sending data stream to the $u$th UE.

At the UEs node, there are $U$ devices, each is equipped with $N_{R}$ receive
antennas and a single RF chain. Thus, the received signal at all UEs can be
jointly written as

$\mathbf{y}_{\mathrm{E}}[k]=\underbrace{\mathbf{W}_{{\rm
RF}\mathrm{E}}^{H}\left(\mathbf{H}_{\mathrm{EN}}[k]\mathbf{x}_{\mathrm{N}}[k]+\mathbf{z}_{\mathrm{E}}[k]\right)}_{\widetilde{\mathbf{y}}_{\mathrm{E}}[k]}+\mathbf{g}_{\mathrm{E}}[k],$
(4)

where
$\mathbf{y}_{\mathrm{E}}[k]=\left[y_{\mathrm{E},1}[k],y_{\mathrm{E},2}[k],\ldots,y_{\mathrm{E},U}[k]\right]^{T}\in\mathbb{C}^{U\times
1}$ with $y_{\mathrm{E},u}[k],\forall u\in\\{1,2,\ldots,U\\}$ denoting the
decoded signal at the $u$th UE. $\mathbf{W}_{{\rm
RF}\mathrm{E}}=\mathrm{blkdiag}\left[\mathbf{w}_{{\rm
RF}\mathrm{E},1},\mathbf{w}_{{\rm RF}\mathrm{E},2},\ldots,\mathbf{w}_{{\rm
RF}\mathrm{E},U}\right]\in\mathbb{C}^{UN_{R}\times U}$ is the RF combiner
matrix with $\mathbf{w}_{{\rm RF}\mathrm{E},u}\in\mathbb{C}^{{N_{R}}\times
1},\forall u\in\\{1,2,\ldots,U\\}$ being the RF combiner vector of the $u$th
UE.
$\mathbf{H}_{\mathrm{EN}}[k]=\left[\mathbf{H}_{\mathrm{EN},1}^{T}[k],\mathbf{H}_{\mathrm{EN},2}^{T}[k],\ldots,\mathbf{H}_{\mathrm{EN},U}^{T}[k]\right]^{T}\in\mathbb{C}^{UN_{R}\times{n_{T}}}$
is the ideal access link channel matrix, where
$\mathbf{H}_{\mathrm{EN},u}[k]\in\mathbb{C}^{{N}_{R}\times{n}_{T}}$ represents
the access link channel matrix from the IAB-node to the $u$th UE.
$\mathbf{z}_{\mathrm{E}}[k]=\left[\mathbf{z}_{\mathrm{E},1}^{T}[k],\mathbf{z}_{\mathrm{E},2}^{T}[k],\ldots,\mathbf{z}_{\mathrm{E},U}^{T}[k]\right]^{T}\in\mathbb{C}^{{UN}_{R}\times
1}\sim\mathcal{CN}(\mathbf{0},\sigma^{2}_{\mathrm{E}}\mathbf{I}_{UN_{R}})$ is
the Gaussian noise vector with
$\mathbf{z}_{\mathrm{E},u}[k]\in\mathbb{C}^{n_{R}\times 1},\forall
u\in\\{1,2,\ldots,U\\}$ being the Gaussian noise vector at the $u$th UE. The
receiver HWI vector
$\mathbf{g}_{\mathrm{E}}[k]=[g_{\mathrm{E},1}[k],g_{\mathrm{E},2}[k],\ldots,$
$g_{\mathrm{E},U}[k]]^{T}\in\mathbb{C}^{U\times
1}\sim\mathcal{CN}(\mathbf{0},\beta\mathrm{diag}[\mathrm{Cov}[\widetilde{\mathbf{y}}_{\mathrm{E}}[k]]])$
with $g_{\mathrm{E},u}[k],\forall u\in\\{1,2,\ldots,U\\}$ denoting the
receiver HWI at the $u$th UE, which is uncorrelated with the received signal.

###  II-B General Channel

For the wideband FR2 communications with the OFDM system, a cyclic prefix of
length $D$ is added to each OFDM symbol, which is equal to the number of delay
taps for the wideband channel. Due to the scattering effect, the FR2 signals
are likely to arrive in ${N_{C}}$ clusters, with ${N_{L}}$ paths reflected by
different obstacles in each cluster. A raised-cosine pulse shaping filter
$p(dT_{s}-\tau_{c,l})$, for $d=0,1,\ldots,D-1$, with $T_{s}$-spaced signaling
is utilized, where the delay $\tau_{c,l}$ is defined for the $l$th path in the
$c$th cluster [22]. Assuming uniform planar arrays (UPAs) with half-wavelength
spaced elements, the transmit and receive steering vectors can be written as
$\mathbf{a}_{\mathrm{t}}(\theta_{c,l}^{t},\phi_{c,l}^{t})$ and
$\mathbf{a}_{\mathrm{r}}(\theta_{c,l}^{r},\phi_{c,l}^{r})$, respectively,
where the azimuth $\theta_{c,l}^{r}/\theta_{c,l}^{t}$ and elevation
$\phi_{c,l}^{r}/\phi_{c,l}^{t}$ angles correspond to the angles of
arrival/departure (AoAs/AoDs) for each path in their clusters. Hence, at
subcarrier $k$, a typical FR2 channel model between two nodes can be expressed
as

$\mathbf{H}[k]=\mathbf{A}_{r}\mathbf{\Pi}[k]\mathbf{A}_{t}^{H},$ (5)

where

$\mathbf{A}_{r}=[\mathbf{a}_{\mathrm{r}}(\theta_{1,1}^{r},\phi_{1,1}^{r}),\ldots,\mathbf{a}_{\mathrm{r}}(\theta_{c,l}^{r},\phi_{c,l}^{r}),\ldots,\mathbf{a}_{\mathrm{r}}(\theta_{{N}_{C},{N}_{L}}^{r},\phi_{{N}_{C},{N}_{L}}^{r})],$
(6)
$\mathbf{A}_{t}=[\mathbf{a}_{\mathrm{t}}(\theta_{1,1}^{t},\phi_{1,1}^{t}),\ldots,\mathbf{a}_{\mathrm{t}}(\theta_{c,l}^{t},\phi_{c,l}^{t}),\ldots,\mathbf{a}_{\mathrm{t}}(\theta_{{N}_{C},{N}_{L}}^{t},\phi_{{N}_{C},{N}_{L}}^{t})],$
(7)
$\mathbf{\Pi}[k]=\sqrt{\tfrac{{N}_{r}{N}_{t}}{{N}_{C}{N}_{L}\bar{PL}}}\begin{bmatrix}\begin{smallmatrix}\alpha_{1,1}\chi_{1,1}[k]&&\ldots&&0\\\
\vdots&&\ddots&&\vdots\\\ 0&\ldots&\alpha_{c,l}\chi_{c,l}[k]&\ldots&0\\\
\vdots&&\ddots&&\vdots\\\
0&&\ldots&&\alpha_{{N}_{C},{N}_{L}}\chi_{{N}_{C},{N}_{L}}[k]\end{smallmatrix}\end{bmatrix},$
(8)

and
$\chi_{c,l}[k]=\sum_{d=0}^{D-1}p(dT_{s}-\tau_{c,l})e^{(-j\frac{2{\pi}kd}{K})}$.
${N}_{t}$ and ${N}_{r}$ denote the number of transmit and receive antennas,
respectively. $\alpha_{c,l}$ is the complex gain. $\bar{PL}$ denotes the
average path loss due to the high attenuation of FR2 band channel. The close-
in path loss model is adopted rather than the free space path loss [23], given
as

$\displaystyle\bar{PL}=\left(\frac{4\pi
r_{0}}{\lambda}\right)^{2}\left(\frac{r}{r_{0}}\right)^{\mu},$ (9)

where $r_{0}$, $r$, $\lambda$, and $\mu$ represent the reference distance,
distance between transceiver, wavelength, and path loss exponent,
respectively. Moreover, since for arbitrary transmission networks, the line-
of-sight (LOS) component has a high probability of being blocked by obstacles.
Therefore, an non-line-of-sight (NLOS) path loss exponent is preferred.
Furthermore, the steering vector is defined as

$\mathbf{a}(\theta,\phi)=\tfrac{1}{\sqrt{N}}\big{[}1,a_{1}(\theta,\phi),\dotsc,a_{N-1}(\theta,\phi)]^{T},$
(10)

where
$a_{n}(\theta,\phi)=e^{j\frac{2\pi}{\lambda}\mathbf{r}_{n}^{T}\mathbf{u}(\theta,\phi)}$;
$N$ is the number of antenna arrays in the UPA;
$\mathbf{r}_{n}=[x_{n},y_{n},z_{n}]^{T}$ is the coordinate of the $n$th
antenna element;
$\mathbf{u}(\theta,\phi)=[\cos\theta\cos\phi,\sin\theta\cos\phi,\sin\phi]^{T}$
is the unit-norm direction vector. In this work, the arrays are placed in the
XY-plane, and the elevation angles are measured from the XY-plane. Besides,
the $z$-axis indicates the array height measured from the UPA plane, which is
assumed to be negligible, i.e., $z_{n}\approx 0$.

###  II-C Self-Interference Channel

The most important issue in the IBFD transmission is the introduction of the
SI on the IAB-node. Due to the proximity of the transceiver on the IAB-node,
the attenuation of the SI channel is significantly less than that of the
typical communication channels, which contributes high power SI to the
backhaul link and degrades its SE. In order to reduce the effect of the SI, a
staged SIC scheme will be introduced in later sections.

Since the distinct SI channel model for the FR2 band is still unknown, most of
the works have considered the hypothetical SI channel for narrowband
communications [6, 24]. Fortunately, a hypothetical model is proposed for the
wideband SI channel in [25]. According to [24, 25], after some minor
modifications, we model the hypothesis wideband SI channel as follows. Unlike
the general channel in the previous subsection, the SI channel is likely to be
modeled as a Rician-alike channel with Rician factor $\kappa$. The LOS part,
$\mathbf{H}_{\mathrm{SI,L}}$, is adopted to a near-field model with spherical
waveform and is assumed to be frequency flat. The frequency response of the
LOS component is given as

$\displaystyle\mathbf{H}_{\mathrm{SI,L}}=\left[\mathbf{a}_{r}(\theta^{r},\phi^{r})\mathbf{a}^{H}_{t}(\theta^{t},\phi^{t})\right]\odot\mathbf{R},$
(11)

where only one AoA/AoD is assumed for the LOS link. The entries of
$\mathbf{R}$ is
$[\mathbf{R}]_{p,q}=\frac{\gamma}{{r}_{pq}}e^{-j2\pi\frac{{r}_{pq}}{\lambda}}$
with ${r}_{pq}$ denoting the distance between the $p$th element of the receive
antenna and the $q$th element of the transmit antenna at the IAB-node.
$\gamma=\sqrt{n_{R}n_{T}}$ is the normalization factor ensuring that the norm
of $\mathbf{H}_{\mathrm{SI,L}}$ remains the same before and after multiplying
with the steering vectors.

The NLOS part, $\mathbf{H}_{\mathrm{SI,N}}$, is expressed similar to the
general channel model in (5), but with a few clusters and rays. Consequently,
the entire SI channel for subcarrier $k$ can be expressed as

$\mathbf{H}_{\mathrm{SI}}[k]=\sqrt{\frac{\kappa}{\kappa+1}}\mathbf{H}_{\mathrm{SI,L}}+\sqrt{\frac{1}{\kappa+1}}\mathbf{H}_{\mathrm{SI,N}}[k].$
(12)

##  III Analog Self-Interference Cancellation

In this section, the working principle and limiting factor of the conventional
A-SIC idea are presented first. Then, the OD-based canceler is described,
followed by the implementation details of such canceler design for the
FR2-IBFD-IAB networks. In this work, we assume the antenna isolation has
already been deployed before A-SIC.

###  III-A Working Principle and Limitations

A-SIC is essential to avoid receiver saturation. Otherwise, the signal-of-
interest cannot be quantized precisely [26, 11]. Active A-SIC is based on a
subtraction idea, i.e., a replica of the received SI signal generated by the
analog canceler is inserted into the receiver chain to subtract the received
SI. The canceler is made up of limited number of tunable delay lines to
capture the multi-path nature of the SI channel, where passive components are
utilized to construct tunable delay lines to minimize the non-linearity
effects. With multi-tap RF canceler, one can cancel the SI from reflection
paths in addition to the direct path. By considering the hardware insertion
losses, the frequency response of a single multi-tap RF canceler can be given
as

$h_{\mathrm{can}}[\omega]=\hat{\alpha}\sum_{m=1}^{M}\alpha_{m}\beta_{m}\left(w_{I,m}+jw_{Q,m}\right)e^{-j\omega\tau_{m}},$
(13)

where $\hat{\alpha}$ is the attenuation introduced by coupling the RF signal
into the canceler; $\alpha_{m}$ is the propagation loss of each delay line;
$\beta_{m}$ denotes the tap coupling factor [14, (4)]; $w_{I,m}$ and $w_{Q,m}$
are tunable weights; and $\tau_{m}$ is the delay. The optimal weights are
tuned to minimize the difference between the frequency components of the
canceler and the SI channel within the band of interest (BoI) (for details,
see [11]). Equation (13) suggests that the number of taps $M$ decides the
available degrees of freedom for this optimization. The key factor for
efficient wideband A-SIC is the realization of a sufficient number of taps
(i.e., delay lines) [27]. For wider operational bandwidth, more frequency
components need to be optimized, and more degrees of freedom, i.e., taps, are
required.

Figure 3: Illustration of the OD-based analog canceler.

###  III-B OD-Based A-SIC

For the conventional canceler, the insertion losses increase with an
increasing number of taps (i.e., $\alpha_{m}$ and $\beta_{m}$ are small for
large $m$ in (13)), which results in a large difference between the signal
power at the first and the later taps. Therefore, the signals coupled into the
later taps cannot replicate the desired signal level and degrade the
cancellation performance. Conventionally, electrical attenuators and micro-
strips or cables can be used for constructing the tunable delay lines.
However, it is demonstrated that these electrical components have significant
propagation loss and coupling loss that limit the number of effective taps
[14], thus limiting the operational bandwidth and cancellation performance.
Therefore, to overcome these drawbacks, an OD-based analog canceler has
recently been investigated in [14], whose structure is illustrated in Fig. 3.

Regarding the OD-based canceler mechanism, the RF reference signal is first
converted to the optical domain by modulating onto optical carriers through
the Mach–Zehnder modulator (MZM). These optical carriers are generated by
tunable lasers according to the grating wavelengths, and the power of these
carriers is adjusted by variable optical attenuators (VOAs). Then, $M$ optical
carriers are combined by a multiplexer (MUX) for propagating into a single
fiber according to the obtained weights. The reference signal modulated on the
optical carrier at wavelength $\lambda_{B,m}$ will be reflected at the $m$th
grating while propagating through the fiber-Bragg-grating (FBG). This
reflection happens at different gratings causes different time delays to the
coupled reference signal. Next, the reflected signals are detected by photo-
diodes to remove the optical carriers. Finally, the canceler yields an
accumulation of multiple weighted and delayed versions of the input reference
signal as the canceler output [14]. Since the weights are achieved by
attenuators, which can only be real and non-negative; however, the SI channel
is complex. Thus, four FBGs are needed to realize the complex response of the
canceler.

The OD-based canceler can also be described by (13). Compared with
conventional canceler, OD-based canceler has smaller insertion losses, i.e.,
$\alpha_{m}$ and $\beta_{m}$ are almost constant with increasing $m$.
Theoretically, almost constant insertion losses in the OD-based canceler allow
hundreds of effective taps to be implemented to enlarge the operational
bandwidth.

###  III-C Proposed OD-Based A-SIC

In order to realize the OD-based canceler design in the MIMO system,
${n_{R}}\times{n_{T}}$ cancelers are traditionally required to match the
${n_{R}}\times{n_{T}}$ SI channel matrix, where each canceler is constructed
and tuned as described above. However, such a canceler deployment will be
extremely costly for the FR2 communications, especially for the OD-based
canceler. In order to reduce the cost, we tap off the SI signal from the RF
chains before the RF precoder at the IAB-node transmitter and insert the
outputs of these analog cancelers back to the RF chains at the IAB-node
receiver after the RF combiner (see Fig 2) [28]. With this architecture, the
required number of analog cancelers can be reduced from ${n_{R}}\times{n_{T}}$
to $U\times U$, which is of great benefit to the cost and practical
implementation. Since a single canceler can be tuned by adjusting the weights
to imitate the estimated RF SI channel $\hat{h}_{\mathrm{SI},pq}[\omega]$
between the $p$th transmitter’s RF chain to the $q$th receiver’s RF chain,
where $p,q\in\\{1,2,\ldots,U\\}$, the following optimization problem will need
to be run for each canceler established between the $pq$th RF chain pair over
the BoI, which is cast as

$\displaystyle\arg\min_{\left\\{w_{I,m}^{pq},\;w_{Q,m}^{pq}\right\\}_{m=1}^{M}}\,\,\sum_{p=1}^{U}\sum_{q=1}^{U}\left\\{\left\|\hat{h}_{\mathrm{SI},pq}[\omega]-h_{\mathrm{can},pq}[\omega]\right\|^{2}\right\\}_{\omega=\omega_{0}}^{\omega_{1}}$
(14) $\displaystyle\text{ s.t.}\quad-1\leq w_{I,m}^{pq}\leq 1,\,-1\leq
w_{Q,m}^{pq}\leq 1,$

where $[{\omega_{0}},{\omega_{1}}]$ spans the BoI, i.e., [27.8 GHz, 28.2 GHz]
in this work. The operator
$\\{\left\|\cdot\right\|^{2}\\}_{\omega=\omega_{0}}^{\omega_{1}}$ means the
sum of the squared error across all frequency components within
$[{\omega_{0}},{\omega_{1}}]$ since the sampled version of the BoI is
considered [27]. $h_{\mathrm{can},pq}[\omega]$ is the canceler response for
mitigating the SI between the $pq$th transceiver RF chain pair, which is
represents by (13). The constraints come from passive VOAs. With the channel
state information (CSI) of the estimated RF SI channel and the frequency
response of the canceler without VOA effects being known as a prior, the
optimal weights can be obtained by the least-squares (LS) method.

Since the A-SIC performance mainly depends on the frequency selectivity of the
SI channel and the number of taps in the canceler, and due to the fact that
the RF beamformers do not affect the frequency selectivity of the SI channel,
we assume the amount of cancellation for the RF SI channel to be the same as
that for the SI channel. Thus, we obtain the A-SIC performance through
simulating with the SI channel instead of the RF SI channel and reflect the
A-SIC effect by simply scaling the SI signal with a power attenuation factor.

In this work, we assume antenna isolation also attenuates the SI signal in a
frequency-flat manner [29]. Thus, after A-SIC, the term
$\mathbf{H}_{\mathrm{SI}}[k]\mathbf{x}_{\mathrm{N}}[k]$ in (2) is scaled by
$\sqrt{\eta}$ with the scalar $\eta$ being the amount of SI signal strength
attenuated by both the antenna isolation and A-SIC.

##  IV RF Codebook Design and RF Effective Channel Estimation

In practice, the RF precoders/combiners are usually implemented using finite
resolution PSs, i.e., they are selected from the pre-defined RF codebooks.
Besides, the estimation of the large and sparse mmWave channel is difficult in
reality. Motivated by these, in this section, a modified LBG algorithm will be
introduced for designing the RF codebook, followed by the estimation of the RF
effective channels after A-SIC.

###  IV-A Modified MSE-Based LBG Algorithm for RF Codebook Design

The LBG algorithm is a popular vector quantization scheme and is treated as an
extension of the Lloyd-Max scalar quantization algorithm [30]. Conventionally,
for a matrix quantization, the existing codebooks work by vector-wise
comparison can lead to a low-rank behavior on the quantized matrix222Suppose
each subarray has multiple RF chains. With a vector-wise codebook, likely, the
columns for the RF beamforming matrix of a certain subarray may be assigned to
the same vector codeword, which can result in a low-rank matrix and the loss
of degrees of freedom.. Therefore, to avoid that, we modify the LBG algorithm
to yield the $B$ bits codebook with matrix codewords directly, whose steps are
described as follows.

  * •

Step 1 (Initialization):
Given the training set $\mathcal{F}=\big{\\{}\mathbf{F}_{{\rm
RF},t}|t=1,2,\ldots,T,\left|\mathbf{F}_{{\rm RF},t}\right|_{pq}=1$ if
$\left[\mathbf{F}_{{\rm RF},t}\right]_{p,q}\neq 0\big{\\}}$ with $T$ entries,
whose each entry is a block diagonal matrix with each block denoted by the
angle of complex Gaussian random numbers with zero mean and unit
variance333Optimally, the training set should have consisted of the optimal RF
precoders/combiners, which are derived by the angle of the dominant
eigenvector(s) corresponding to the eigenvalue decomposition (EVD) of the
channel correlation matrix (i.e., the sample covariance matrix) [17]. However,
as aforementioned, the mmWave channel is hard to be estimated. Therefore, by
exploring the distribution of the RF precoders/combiners, i.e., the values in
the RF precoder/combiner matrix are isotropically (uniformly) distributed [31,
32, Lemma 1, 2], we construct the entries of the training set by the angle of
$\mathcal{CN}(0,1)$ random numbers. The codebook $\mathcal{C}$ is initialized
with an entry $\mathbf{C}_{1}(0)$, obtained by the angle of the mean value of
the training set as

$\mathbf{C}_{1}(0)=e^{j\arg\left(\frac{\sum_{t=1}^{T}\mathbf{F}_{{\rm
RF},t}}{T}\right)}.$ (15)

  * •

Step 2 (Splitting):
This step splits each entry of the $b$ bits codebook $\mathcal{C}$ into two
new ones to initialize the $b+1$ bits codebook, where $b=0,1,\ldots,B-1$. To
achieve that, we perturb each entry $\mathbf{C}_{i}(b)$ as,

$\displaystyle\mathbf{C}^{(0)}_{i+2^{b}}(b+1)=e^{j\arg\left(\sqrt{1-\epsilon^{2}}\mathbf{C}_{i}(b)+\epsilon\mathbf{P}_{i}(b)\right)},$
$\displaystyle\mathbf{C}^{(0)}_{i}(b+1)=e^{j\arg\left(\sqrt{1-\epsilon^{2}}\mathbf{C}_{i}(b)-\epsilon\mathbf{P}_{i}(b)\right)},$
(16)

where $i=1,2,\ldots,2^{b}$, $\epsilon$ is a small positive value (e.g.,
$10^{-3}$), $\mathbf{P}_{i}(b)$ is a block diagonal matrix, whose each block
is drawn from the angle of $\mathcal{CN}(0,1)$ random numbers.

  * •

Step 3 (Cluster Assignment):
In this step, using the nearest neighbor routine based on MSE, the training
set is divided into $2^{b+1}$ (i.e., $|\mathcal{C}|$) clusters, the centroid
of cluster $j$ is given by $\mathbf{C}_{j}^{(v)}(b+1)$, where
$v=0,1,\ldots,V-1$ with $V$ being the maximum number of iterations of Step 5.
E.g., $\mathbf{F}_{{\rm RF},t}$ is in the cluster 1 if
$d\left(\mathbf{F}_{{\rm RF},t},\mathbf{C}_{1}^{(v)}(b+1)\right)\leq
d\left(\mathbf{F}_{{\rm RF},t},\mathbf{C}_{j}^{(v)}(b+1)\right)$, $\forall
j=1,2,\ldots,|\mathcal{C}|$, where
$d\left(\mathbf{X},\mathbf{Y}\right)=\frac{1}{PQ}\sum_{p=1}^{P}\sum_{q=1}^{Q}\left([\mathbf{X}]_{p,q}-[\mathbf{Y}]_{p,q}\right)^{2}$,
and $P$, $Q$ denote the number of rows and columns of the matrix,
respectively.

  * •

Step 4 (Centroid Update):
Each entry of the codebook is updated with the centroid of the corresponding
cluster. The centroid is computed via the solution of the following
optimization problem, that is

$\mathbf{{C}}_{j}^{(v)}(b+1)=\arg\underset{\mathbf{C}_{j}^{(v)}(b+1)}{\min}{\underset{\mathbf{F}_{{\rm
RF},t}\in j}{\sum}d\left(\mathbf{F}_{{\rm
RF},t},\mathbf{C}_{j}^{(v)}(b+1)\right)}.$ (17)

Thus, the new centroid $\mathbf{{C}}_{j}^{(v)}(b+1)$ is given by the angle of
the mean value of all $\mathbf{F}_{{\rm RF},t}$ in the $j$th cluster.

  * •

Step 5 (Inner loop):
Go to step 3 until the maximum number of iterations $V$ is reached (e.g.,
$V=50$).

  * •

Step 6 (Outer loop):
Go to step 2 until the length of the codebook $b+1$ is equal to the desired
codebook length $B$.

###  IV-B RF Effective Channel Estimation

Given the RF codebooks, we can estimate the RF effective channels for
designing the BB beamformers. Note that the RF effective channels estimated in
this section are those after A-SIC by assuming BB beamformers to be identity
matrices [13].

There are two phases in the RF effective channel estimation: i) RF precoder-
combiner pair selection; ii) RF effective channel estimation. The RF
beamformers are designed to maximize the desired signal in their corresponding
links. We treat the whole OFDM symbols as pilots and assume only the IAB donor
or the IAB node can transmit data in a time slot. Moreover, the identity BB
beamformer matrices are omitted here.

####  IV-B1 Phase 1 (RF precoder-combiner pair selection)

The received backhaul link signal of the $k$th pilot subcarrier at the IAB-
node, which uses the $p$th codeword of the codebook $\mathcal{F}_{\mathrm{D}}$
as the RF precoder and the $q$th codeword of the codebook
$\mathcal{W}_{\mathrm{N}}$ as the RF combiner, is given by

$\displaystyle\mathbf{Y}_{\mathrm{N}}[k](p,q)$ $\displaystyle=\mathbf{W}_{{\rm
RF}\mathrm{N},q}^{H}[\mathbf{H}_{\mathrm{ND}}[k]\mathbf{F}_{{\rm
RF}\mathrm{D},p}\left(\mathbf{S}_{\mathrm{D}}[k]+\mathbf{E}_{\mathrm{D}}[k]\right)$
$\displaystyle+\mathbf{Z}_{\mathrm{N}}[k]]+\mathbf{G}_{\mathrm{N}}[k],$ (18)

where $\mathbf{S}_{\mathrm{D}}[k]\in\mathbb{C}^{U\times U}$ is the matrix of
orthogonal pilot signal with
$\mathbf{S}_{\mathrm{D}}[k]\mathbf{S}_{\mathrm{D}}^{H}[k]=\frac{P_{t}}{KU}\mathbf{I}_{U}$.
$\mathbf{E}_{\mathrm{D}}[k]\in\mathbb{C}^{U\times U}$,
$\mathbf{G}_{\mathrm{D}}[k]\in\mathbb{C}^{U\times U}$, and
$\mathbf{Z}_{\mathrm{N}}[k]\in\mathbb{C}^{U\times U}$ are the noise matrices
caused by the transmitter HWI, receiver HWI, and Gaussian noise, respectively,
following the same statistics in (1) and (2).

Similarly, the jointly received access link signal of the $k$th pilot
subcarrier across all UEs, which uses the $p$th codeword of the codebook
$\mathcal{F}_{\mathrm{N}}$ as the RF precoder and the $q$th codeword of the
codebook $\mathcal{W}_{\mathrm{E}}$ as the RF combiner, is cast as

$\displaystyle\mathbf{Y}_{\mathrm{E}}[k](p,q)$ $\displaystyle=\mathbf{W}_{{\rm
RF}\mathrm{E},q}^{H}[\mathbf{H}_{\mathrm{EN}}[k]\mathbf{F}_{{\rm
RF}\mathrm{N},p}\left(\mathbf{S}_{\mathrm{N}}[k]+\mathbf{E}_{\mathrm{N}}[k]\right)$
$\displaystyle+\mathbf{Z}_{\mathrm{E}}[k]]+\mathbf{G}_{\mathrm{E}}[k],$ (19)

where the matrix of orthogonal pilot signal
$\mathbf{S}_{\mathrm{N}}[k]\in\mathbb{C}^{U\times U}$ has
$\mathbf{S}_{\mathrm{N}}[k]\mathbf{S}_{\mathrm{N}}^{H}[k]=\frac{P_{t}}{KU}\mathbf{I}_{U}$.
$\mathbf{E}_{\mathrm{N}}[k]\in\mathbb{C}^{U\times U}$,
$\mathbf{G}_{\mathrm{E}}[k]\in\mathbb{C}^{U\times U}$, and
$\mathbf{Z}_{\mathrm{E}}[k]\in\mathbb{C}^{U\times U}$ are the transmitter HWI,
receiver HWI, and Gaussian noise matrix, respectively, with the same
statistics in (3) and (4).

According to the beam management [33], each time, a codeword is chosen from
their corresponding codebook and the RF precoder and combiner pairs that can
maximize the received power among all pilot subcarriers are selected, given as

$\displaystyle\left\\{\mathbf{F}_{{\rm RF}\mathrm{D}},\mathbf{W}_{{\rm
RF}\mathrm{N}}\right\\}=\arg\underset{p,q}{\max}{\sum_{k=1}^{K}\left\|\mathbf{Y}_{\mathrm{N}}[k](p,q)\right\|_{F}^{2}}$
(20a) $\displaystyle\text{subject to}\quad\mathbf{F}_{{\rm
RF}\mathrm{D},p}\in\mathcal{F}_{\mathrm{D}},\quad\mathbf{W}_{{\rm
RF}\mathrm{N},q}\in\mathcal{W}_{\mathrm{N}}.$ (20b)
$\displaystyle\left\\{\mathbf{F}_{{\rm RF}\mathrm{N}},\mathbf{W}_{{\rm
RF}\mathrm{E}}\right\\}=\arg\underset{p,q}{\max}{\sum_{k=1}^{K}\left\|\mathbf{Y}_{\mathrm{E}}[k](p,q)\right\|_{F}^{2}}$
(21a) $\displaystyle\text{subject to}\quad\mathbf{F}_{{\rm
RF}\mathrm{N},p}\in\mathcal{F}_{\mathrm{N}},\quad\mathbf{W}_{{\rm
RF}\mathrm{E},q}\in\mathcal{W}_{\mathrm{E}}.$ (21b)

In this work, the RF beamformers for all nodes can be selected from the same
isotropic RF codebook derived from the last subsection.

####  IV-B2 Phase 2 (RF effective channel estimation)

Given the RF precoder/combiner, one can estimate the RF effective channel with
the help of pilot signal by standard estimation methods, such as, the LS.

Consequently, after estimation, we can write the ideal RF effective channel
matrix as the sum of the estimated RF effective channel matrix
$\hat{\mathbf{H}}^{eff}_{(\cdot)}[k]$ and the estimation error matrix
$\mathbf{\Delta}_{(\cdot)}[k]$, given as

$\mathbf{W}_{{\rm
RF}\mathrm{N}}^{H}\mathbf{H}_{\mathrm{ND}}[k]\mathbf{F}_{{\rm
RF}\mathrm{D}}=\hat{\mathbf{H}}^{eff}_{\mathrm{ND}}[k]+\mathbf{\Delta}_{\mathrm{ND}}[k],$
(22) $\sqrt{\eta}\mathbf{W}_{{\rm
RF}\mathrm{N}}^{H}\mathbf{H}_{\mathrm{SI}}[k]\mathbf{F}_{{\rm
RF}\mathrm{N}}=\hat{\mathbf{H}}^{eff}_{\mathrm{SI}}[k]+\mathbf{\Delta}_{\mathrm{SI}}[k],$
(23) $\mathbf{W}_{{\rm
RF}\mathrm{E}}^{H}\mathbf{H}_{\mathrm{EN}}[k]\mathbf{F}_{{\rm
RF}\mathrm{N}}=\hat{\mathbf{H}}^{eff}_{\mathrm{EN}}[k]+\mathbf{\Delta}_{\mathrm{EN}}[k],$
(24)

where we assume the channel estimation errors
$\mathbf{\Delta}_{\mathrm{ND}}[k]$, $\mathbf{\Delta}_{\mathrm{SI}}[k]$, and
$\mathbf{\Delta}_{\mathrm{EN}}[k]$ have the covariance matrices of
$\mathrm{Cov}\left[\mathbf{\Delta}_{\mathrm{ND}}[k]\right]=\sigma_{e,\mathrm{ND}}^{2}\mathbf{I}_{M}$,
$\mathrm{Cov}\left[\mathbf{\Delta}_{\mathrm{SI}}[k]\right]=\sigma_{e,\mathrm{SI}}^{2}\mathbf{I}_{M}$,
and
$\mathrm{Cov}\left[\mathbf{\Delta}_{\mathrm{EN}}[k]\right]=\sigma_{e,\mathrm{EN}}^{2}\mathbf{I}_{M}$
[34, 35].

##  V Digital Self-Interference Cancellation

After A-SIC, the RSI left by previous stages will be processed in the digital
domain of the IAB-node receiver. In practice, since the IAB-node knows its
transmitted codeword $\mathbf{s}_{\mathrm{N}}[k]$ and we can know the
estimated RF effective SI channel $\hat{\mathbf{H}}^{eff}_{\mathrm{SI}}[k]$ by
the process in Section IV. Then, with the help of successive interference
cancellation, we can cancel out
$\hat{\mathbf{H}}^{eff}_{\mathrm{SI}}[k]\mathbf{F}_{\mathrm{{\rm
BB}\mathrm{N}}}[k]\mathbf{s}_{\mathrm{N}}[k]$.

Consequently, after subtraction, the decoded signal at the IAB-node in (2) can
be reconstructed as

$\displaystyle\hat{\mathbf{y}}_{\mathrm{N}}[k]=\mathbf{W}_{{\rm
BB}\mathrm{N}}^{H}[k]\left({\widetilde{\hat{\mathbf{y}}}_{\mathrm{N}}[k]}+\mathbf{g}_{\mathrm{N}}[k]\right),$
(25)

where $\mathbf{W}_{{\rm BB}{\mathrm{N}}}[k]$ is designed to act as the minimum
mean-squared error (MMSE) BB combiner, which will be described in the next
section. $\widetilde{\hat{\mathbf{y}}}_{\mathrm{N}}[k]=\mathbf{W}_{{\rm
RF}\mathrm{N}}^{H}\left(\mathbf{H}_{\mathrm{ND}}[k]\mathbf{x}_{\mathrm{D}}[k]+\sqrt{\eta}\mathbf{H}_{\mathrm{SI}}[k]\mathbf{F}_{{\rm
RF}\mathrm{N}}\mathbf{e}_{\mathrm{N}}[k]+\mathbf{z}_{\mathrm{N}}[k]\right)+\mathbf{\Delta}_{\mathrm{SI}}[k]\mathbf{F}_{{\rm
BB}\mathrm{N}}[k]\mathbf{s}_{\mathrm{N}}[k]$

##  VI Spectral Efficiency and Baseband Beamforming Design

###  VI-A Spectral Efficiency

Define $\zeta=\frac{P_{t}}{KU}$ and substitute (1), (22), and (23) into (25),
the SE of the backhaul link is expressed according to (25), given as

$\displaystyle\mathcal{R}_{b}$
$\displaystyle=\frac{1}{K}\sum_{k=1}^{K}\log_{2}\mathrm{det}\Big{\\{}\mathbf{I}_{U}+{\mathbf{W}_{{\rm
BB}{\mathrm{N}}}^{H}[k]\mathbf{\Phi}_{b}[k]\mathbf{W}_{{\rm
BB}{\mathrm{N}}}[k]}$ $\displaystyle\times\left(\mathbf{W}_{{\rm
BB}{\mathrm{N}}}^{H}[k]\mathbf{\Omega}_{b}[k]\mathbf{W}_{{\rm
BB}{\mathrm{N}}}[k]\right)^{-1}\Big{\\}},$ (26)

where $\mathbf{\Phi}_{b}[k]$ is the covariance matrix for the known part of
the desired signal. $\mathbf{\Omega}_{b}[k]$ represents the covariance matrix
consisting of the noise given by the channel estimation error, the transceiver
HWI, and the Gaussian noise.

$\displaystyle\mathbf{\Phi}_{b}[k]=\zeta\hat{\mathbf{H}}_{\mathrm{ND}}^{eff}[k]\mathbf{F}_{\mathrm{{\rm
BB}\mathrm{D}}}[k]\mathbf{F}_{\mathrm{{\rm
BB}\mathrm{D}}}^{H}[k]\left(\hat{\mathbf{H}}_{\mathrm{ND}}^{eff}[k]\right)^{H}.$
(27) $\displaystyle\mathbf{\Omega}_{b}[k]$
$\displaystyle=\mathbf{\Omega}_{b}^{(1)}[k]+\mathbf{\Omega}_{b}^{(2)}[k]+\mathbf{\Omega}_{b}^{(3)}[k]+\underbrace{\sigma_{\mathrm{N}}^{2}\mathbf{W}_{{\rm
RF}\mathrm{N}}^{H}\mathbf{W}_{{\rm RF}\mathrm{N}}}_{\text{Gaussian noise}}$
$\displaystyle\overset{(a)}{=}\mathbf{\Omega}_{b}^{(1)}[k]+\mathbf{\Omega}_{b}^{(2)}[k]+\mathbf{\Omega}_{b}^{(3)}[k]+\sigma_{\mathrm{N}}^{2}\frac{n_{R}}{U}\mathbf{I}_{U},$
(28)

where $(a)$ is derived according to the property of RF beamformers, i.e.,
$\mathbf{W}_{{\rm RF}\mathrm{N}}^{H}\mathbf{W}_{{\rm
RF}\mathrm{N}}=\frac{n_{R}}{U}\mathbf{I}_{U}$.

$\displaystyle\mathbf{\Omega}_{b}^{(1)}[k]$
$\displaystyle=\mathrm{Cov}\bigg{[}\underbrace{\hat{\mathbf{H}}_{\mathrm{ND}}^{eff}[k]\mathbf{e}_{\mathrm{D}}[k]+\mathbf{\Delta}_{\mathrm{ND}}[k]\mathbf{e}_{\mathrm{D}}[k]}_{\text{backhaul
channel transmitter HWI}}$
$\displaystyle\quad+\underbrace{\mathbf{\Delta}_{\mathrm{ND}}[k]\mathbf{F}_{{\rm
BB}\mathrm{D}}[k]\mathbf{s}_{\mathrm{D}}[k]}_{\text{backhaul channel
estimation error}}\bigg{]}$
$\displaystyle\overset{(b)}{=}\zeta\rho\hat{\mathbf{H}}_{\mathrm{ND}}^{eff}[k]\mathrm{diag}\left[\mathbf{F}_{{\rm
BB}\mathrm{D}}[k]\mathbf{F}_{{\rm
BB}\mathrm{D}}^{H}[k]\right]\left(\hat{\mathbf{H}}_{\mathrm{ND}}^{eff}[k]\right)^{H}$
$\displaystyle\quad+\sigma_{e,\mathrm{ND}}^{2}\zeta(\rho+1)\mathrm{tr}\left[\mathbf{F}_{{\rm
BB}\mathrm{D}}[k]\mathbf{F}_{{\rm BB}\mathrm{D}}^{H}[k]\right]\mathbf{I}_{U},$
(29a) where $(b)$ is obtained by following simplifications:
$\displaystyle\left[\mathrm{Cov}\left[\mathbf{\Delta}_{\mathrm{ND}}[k]\mathbf{e}_{\mathrm{D}}[k]\right]\right]_{m,n}$
$\displaystyle=\sum_{p}\left[\mathbb{E}\left\\{\left[\mathbf{\Delta}_{\mathrm{ND}}[k]\right]_{m,p}\left[\mathbf{\Delta}_{\mathrm{ND}}^{H}[k]\right]_{p,n}\left[\left\|\mathbf{e}_{\mathrm{D}}[k]\right\|^{2}\right]_{p}\right\\}\right]_{m,n}$
$\displaystyle=\sigma_{e,\mathrm{ND}}^{2}\sum_{p}\left[\mathbb{E}\left\\{\left[\left\|\mathbf{e}_{\mathrm{D}}[k]\right\|^{2}\right]_{p}\right\\}\right]_{m,n}\delta_{m,n}$
$\displaystyle=\sigma_{e,\mathrm{ND}}^{2}\delta_{m,n}\mathrm{tr}\left[\mathbb{E}\left\\{\mathbf{e}_{\mathrm{D}}[k]\mathbf{e}_{\mathrm{D}}^{H}[k]\right\\}\right]$
$\displaystyle=\sigma_{e,\mathrm{ND}}^{2}\zeta\rho\mathrm{tr}\left[\mathrm{diag}\left[\mathbf{F}_{{\rm
BB}\mathrm{D}}[k]\mathbf{F}_{{\rm
BB}\mathrm{D}}^{H}[k]\right]\right]\delta_{m,n}$
$\displaystyle=\sigma_{e,\mathrm{ND}}^{2}\zeta\rho\mathrm{tr}\left[\mathbf{F}_{{\rm
BB}\mathrm{D}}[k]\mathbf{F}_{{\rm BB}\mathrm{D}}^{H}[k]\right]\delta_{m,n},$
(29b)
$\displaystyle\left[\mathrm{Cov}\left[\mathbf{\Delta}_{\mathrm{ND}}[k]\mathbf{F}_{{\rm
BB}\mathrm{D}}[k]\mathbf{s}_{\mathrm{D}}[k]\right]\right]_{m,n}$
$\displaystyle=\zeta\sum_{p,q}\left[\mathbb{E}\left\\{\left[\mathbf{\Delta}_{\mathrm{ND}}[k]\right]_{m,p}\left[\mathbf{F}_{{\rm
BB}\mathrm{D}}[k]\mathbf{F}_{{\rm
BB}\mathrm{D}}^{H}[k]\right]_{p,q}\left[\mathbf{\Delta}_{\mathrm{ND}}^{H}[k]\right]_{q,n}\right\\}\right]_{m,n}$
$\displaystyle=\sigma_{e,\mathrm{ND}}^{2}\zeta\sum_{p,q}\left[\mathbf{F}_{{\rm
BB}\mathrm{D}}[k]\mathbf{F}_{{\rm
BB}\mathrm{D}}^{H}[k]\right]_{p,q}\delta_{m,n}\delta_{p,q}$
$\displaystyle=\sigma_{e,\mathrm{ND}}^{2}\zeta\mathrm{tr}\left[\mathbf{F}_{{\rm
BB}\mathrm{D}}[k]\mathbf{F}_{{\rm BB}\mathrm{D}}^{H}[k]\right]\delta_{m,n}.$
(29c) $\displaystyle\mathbf{\Omega}_{b}^{(2)}[n]$
$\displaystyle=\mathrm{Cov}\bigg{[}\underbrace{\hat{\mathbf{H}}_{\mathrm{SI}}^{eff}[k]\mathbf{e}_{\mathrm{N}}[k]+\mathbf{\Delta}_{\mathrm{SI}}[k]\mathbf{e}_{\mathrm{N}}[k]}_{\text{SI
channel transmitter HWI}}$
$\displaystyle\quad+\underbrace{\mathbf{\Delta}_{\mathrm{SI}}[k]\mathbf{F}_{{\rm
BB}\mathrm{N}}[k]\mathbf{s}_{\mathrm{N}}[k]}_{\text{SI channel estimation
error}}\bigg{]}$
$\displaystyle\overset{(c)}{=}\zeta\rho\hat{\mathbf{H}}_{\mathrm{SI}}^{eff}[k]\mathrm{diag}\left[\mathbf{F}_{{\rm
BB}\mathrm{N}}[k]\mathbf{F}_{{\rm
BB}\mathrm{N}}^{H}[k]\right]\left(\hat{\mathbf{H}}_{\mathrm{SI}}^{eff}[k]\right)^{H}$
$\displaystyle\quad+\sigma_{e,\mathrm{SI}}^{2}\zeta(\rho+1)\mathrm{tr}\left[\mathbf{F}_{{\rm
BB}\mathrm{N}}[k]\mathbf{F}_{{\rm BB}\mathrm{N}}^{H}[k]\right]\mathbf{I}_{U},$
(30)

where $(c)$ is derived by using the similar simplification processes shown in
(29b) and (29c).

$\displaystyle\mathbf{\Omega}_{b}^{(3)}[k]$
$\displaystyle=\underbrace{\beta\mathrm{diag}\left[\mathrm{Cov}\left[\widetilde{\hat{\mathbf{y}}}_{\mathrm{N}}[k]\right]\right]}_{\text{receiver
HWI}}$
$\displaystyle=\beta\mathrm{diag}\left[\mathbf{\Phi}_{b}[k]+\mathbf{\Omega}_{b}^{(1)}[k]+\mathbf{\Omega}_{b}^{(2)}[k]+\sigma_{\mathrm{N}}^{2}\frac{n_{R}}{U}\mathbf{I}_{U}\right].$
(31)

Next, we will derive the sum SE expression of the access link across all
users. The decoded signal at the $u$th user is given as

${y}_{\mathrm{E},u}[k]=\underbrace{\mathbf{w}_{{\rm
RF}\mathrm{E},u}^{H}\left(\mathbf{H}_{\mathrm{EN,u}}[k]\mathbf{x}_{\mathrm{N}}[k]+\mathbf{z}_{\mathrm{E,u}}[k]\right)}_{\widetilde{{y}}_{\mathrm{E},u}[k]}+g_{\mathrm{E},u}[k].$
(32)

By substituting (3) and (24) into (32), we can have the sum SE expression of
the access link as follows, that is

$\mathcal{R}_{a}=\sum_{u=1}^{U}\frac{1}{K}\sum_{k=1}^{K}\log_{2}\left(1+\frac{{\Phi}_{a,u}[k]}{{\Omega}_{a,u}[k]}\right),$
(33)

where ${\Phi}_{a,u}[k]$ denotes the covariance for the known part of the $u$th
user’s desired signal and ${\Omega}_{a,u}[k]$ represents the covariance of the
noise given by the multiuser interference, the channel estimation error, the
transceiver HWI, and the Gaussian noise at the $u$th user.

${\Phi}_{a,u}[k]=\zeta\hat{\mathbf{h}}_{\mathrm{EN},u}^{eff}[k]\mathbf{f}_{{\rm
BB}\mathrm{N},u}[k]\mathbf{f}_{{\rm
BB}\mathrm{N},u}^{H}[k]\left(\hat{\mathbf{h}}_{\mathrm{EN},u}^{eff}[k]\right)^{H},$
(34)

where
$\hat{\mathbf{H}}_{\mathrm{EN}}^{eff}[k]=\left[\left(\hat{\mathbf{h}}_{\mathrm{EN},1}^{eff}[k]\right)^{T},\left(\hat{\mathbf{h}}_{\mathrm{EN},2}^{eff}[k]\right)^{T},\ldots,\left(\hat{\mathbf{h}}_{\mathrm{EN},U}^{eff}[k]\right)^{T}\right]^{T}$
with
$\left\\{\hat{\mathbf{h}}_{\mathrm{EN},u}^{eff}[k]\right\\}_{u=1}^{U}\in\mathbb{C}^{1\times
U}$.

$\displaystyle{\Omega}_{a,u}[k]$
$\displaystyle={\Omega}_{a,u}^{(1)}[k]+{\Omega}_{a,u}^{(2)}[k]+{\Omega}_{a,u}^{(3)}[k]+\underbrace{\sigma_{\mathrm{E}}^{2}\mathbf{w}_{{\rm
RF}\mathrm{E},u}^{H}\mathbf{w}_{{\rm RF}\mathrm{E},u}}_{\text{Gaussian
noise}}$
$\displaystyle\overset{(d)}{=}{\Omega}_{a,u}^{(1)}[k]+{\Omega}_{a,u}^{(2)}[k]+{\Omega}_{a,u}^{(3)}[k]+\sigma_{\mathrm{E}}^{2}N_{R},$
(35)

where $(d)$ comes from the property of RF beamformers, i.e., $\mathbf{w}_{{\rm
RF}\mathrm{E},u}^{H}\mathbf{w}_{{\rm RF}\mathrm{E},u}=N_{R}$.

$\displaystyle{\Omega}_{a,u}^{(1)}[k]$
$\displaystyle=\mathrm{Cov}\bigg{[}\underbrace{\sum_{v=1,v\neq
u}^{U}\hat{\mathbf{h}}_{\mathrm{EN},u}^{eff}[k]\mathbf{f}_{{\rm
BB}\mathrm{N},v}[k]{s}_{\mathrm{N},v}[k]}_{\text{multiuser
interference}}+\underbrace{\hat{\mathbf{h}}_{\mathrm{EN},u}^{eff}[k]\mathbf{e}_{\mathrm{N}}[k]}_{\text{transmitter
HWI}}\bigg{]}$ $\displaystyle=\zeta\sum_{v=1,v\neq
u}^{U}\hat{\mathbf{h}}_{\mathrm{EN},u}^{eff}[k]\mathbf{f}_{{\rm
BB}\mathrm{N},v}[k]\mathbf{f}_{{\rm
BB}\mathrm{N},v}^{H}[k]\left(\hat{\mathbf{h}}_{\mathrm{EN},u}^{eff}[k]\right)^{H}$
$\displaystyle\quad+\zeta\rho\hat{\mathbf{h}}_{\mathrm{EN},u}^{eff}[k]\mathrm{diag}\left[\mathbf{F}_{{\rm
BB}\mathrm{N}}[k]\mathbf{F}_{{\rm
BB}\mathrm{N}}^{H}[k]\right]\left(\hat{\mathbf{h}}_{\mathrm{EN},u}^{eff}[k]\right)^{H},$
(36)

where
$\mathbf{s}_{\mathrm{N}}[k]=\left[{s}_{\mathrm{N},1}[k],{s}_{\mathrm{N},2}[k],\ldots,{s}_{\mathrm{N},U}[k]\right]^{T}$.

$\displaystyle{\Omega}_{a,u}^{(2)}[k]$
$\displaystyle=\mathrm{Cov}\bigg{[}\underbrace{\mathbf{\Delta}_{\mathrm{EN},u}[k]\mathbf{e}_{\mathrm{N}}[k]}_{\text{transmitter
HWI}}+\underbrace{\mathbf{\Delta}_{\mathrm{EN},u}[k]\mathbf{F}_{{\rm
BB}\mathrm{N}}[k]\mathbf{s}_{\mathrm{N}}[k]}_{\text{channel estimation
error}}\bigg{]}$
$\displaystyle\overset{(e)}{=}\sigma_{e,\mathrm{EN}}^{2}\zeta(\rho+1)\mathrm{tr}\left[\mathbf{F}_{{\rm
BB}\mathrm{N}}[k]\mathbf{F}_{{\rm BB}\mathrm{N}}^{H}[k]\right],$ (37)

where
$\mathbf{\Delta}_{\mathrm{EN}}[k]=\left[\left(\mathbf{\Delta}_{\mathrm{EN},1}[k]\right)^{T},\left(\mathbf{\Delta}_{\mathrm{EN},2}[k]\right)^{T},\ldots,\left(\mathbf{\Delta}_{\mathrm{EN},U}[k]\right)^{T}\right]^{T}$
with
$\left\\{\mathbf{\Delta}_{\mathrm{EN},u}[k]\right\\}_{u=1}^{\mathrm{U}}\in\mathbb{C}^{1\times
U}$ and $(e)$ is obtained by adopting the similar simplifications in (29b) and
(29c).

${\Omega}_{a,u}^{(3)}[k]=\underbrace{\beta\left\|\widetilde{{y}}_{\mathrm{E},u}[k]\right\|^{2}}_{\text{receiver
HWI}}=\beta\left({\Phi}_{a,u}[k]+{\Omega}_{a,u}^{(1)}[k]+{\Omega}_{a,u}^{(2)}[k]+\sigma_{\mathrm{E}}^{2}N_{R}\right).$
(38)

###  VI-B Baseband Beamforming Design

Given the RF beamformers and RF effective channels derived from Section IV, we
aim to design the BB beamformers for both the backhaul and access links.

For the backhaul link, the $k$th BB precoder which maximizes the SE is
obtained using the right singular vectors $\mathbf{V}_{\mathrm{ND}}[k]$ of the
$k$th estimated RF effective backhaul link channel matrix
$\hat{\mathbf{H}}^{eff}_{\mathrm{ND}}[k]$, that is

$\mathbf{F}_{{\rm
BB}\mathrm{D}}[k]=\left[\mathbf{V}_{\mathrm{ND}}[k]\right]_{:,1:U}.$ (39)

Due to the precoder constraint, the BB precoder is updated as
$\mathbf{F}_{{\rm BB}\mathrm{D}}[k]\leftarrow\frac{\sqrt{U}\mathbf{F}_{{\rm
BB}\mathrm{D}}[k]}{\left|\left|\mathbf{F}_{{\rm RF}\mathrm{D}}\mathbf{F}_{{\rm
BB}\mathrm{D}}[k]\right|\right|_{F}}$.

Next, the design of the BB precoder $\mathbf{F}_{{\rm BB}\mathrm{N}}[k]$ at
the IAB-node transmitter aims to null the multiuser interference by the zero
forcing, which is

$\mathbf{F}_{{\rm
BB}\mathrm{N}}[k]=\left(\hat{\mathbf{H}}_{\mathrm{EN}}^{eff}[k]\right)^{H}\left[\hat{\mathbf{H}}_{\mathrm{EN}}^{eff}[k]\left(\hat{\mathbf{H}}_{\mathrm{EN}}^{eff}[k]\right)^{H}\right]^{-1}.$
(40)

Similarly, the BB precoder should be normalized as $\mathbf{f}_{{\rm
BB}\mathrm{N},u}[k]\leftarrow\frac{\mathbf{f}_{{\rm
BB}\mathrm{N},u}[k]}{\left|\left|\mathbf{F}_{{\rm
RF}\mathrm{N}}\mathbf{f}_{{\rm BB}\mathrm{N},u}[k]\right|\right|_{F}}\;\forall
u\in\\{1,2,\ldots,U\\}$.

Finally, with the fact that the channel estimation error is uncorrelated with
the data vector, and we assume the strength of HWI and channel estimation
error are known as a prior. The MMSE BB combiner for the $k$th subcarrier
$\mathbf{W}_{{\rm BB}\mathrm{N}}[k]$ is designed by solving the following
optimization problem, which is

$\arg\underset{\mathbf{W}_{{\rm
BB}\mathrm{N}}[k]}{\min}{\mathbb{E}\left\\{\left\|\mathbf{s}_{\mathrm{D}}[k]-\hat{\mathbf{y}}_{\mathrm{N}}[k]\right\|^{2}_{2}\right\\}}.$
(41)

By solving
$\frac{\partial\mathbb{E}\left\\{\left\|\mathbf{s}_{\mathrm{D}}[k]-\hat{\mathbf{y}}_{\mathrm{N}}[k]\right\|^{2}_{2}\right\\}}{\partial\mathbf{W}_{{\rm
BB}\mathrm{N}}^{H}[k]}=0$ (see Appendix-A), we have

$\displaystyle\mathbf{W}_{{\rm BB}\mathrm{N}}[k]$
$\displaystyle=\mathbb{E}\left\\{\left({\widetilde{\hat{\mathbf{y}}}_{\mathrm{N}}[k]}+\mathbf{g}_{\mathrm{N}}[k]\right)\left({\widetilde{\hat{\mathbf{y}}}_{\mathrm{N}}[k]}+\mathbf{g}_{\mathrm{N}}[k]\right)^{H}\right\\}^{-1}$
$\displaystyle\quad\times\mathbb{E}\left\\{\left({\widetilde{\hat{\mathbf{y}}}_{\mathrm{N}}[k]}+\mathbf{g}_{\mathrm{N}}[k]\right)\mathbf{s}_{\mathrm{D}}^{H}[k]\right\\}$
$\displaystyle=\zeta\left(\mathbf{\Phi}_{b}[k]+\mathbf{\Omega}_{b}[k]\right)^{-1}\hat{\mathbf{H}}_{\mathrm{ND}}^{eff}[k]\mathbf{F}_{{\rm
BB}\mathrm{D}}[k].$ (42)

TABLE I: System Parameters and Default Values

Notation | Physical Meaning | Values
---|---|---
$K$ | Number of subcarriers | 512
$D$ | Number of cyclic prefixes | 128
$W$ | Bandwidth | 400 MHz
$f_{c}$ | Carrier frequency | 28 GHz
$U$ | Number of users (subarrays, RF chains, data streams) | 4
${N_{T}},{n_{T}}$ | Number of transmit antennas at the IAB donor and the IAB-node, respectively |  $16\times 16$, $16\times 16$
${n_{R}},{N_{R}}$ | Number of receive antennas at the IAB-node and each user, respectively |  $16\times 16$, $16\times 4$
$\sigma_{\mathrm{N}},\sigma_{\mathrm{E}}$ | Gaussian noise power at the IAB-node and each user, respectively |  | $-174\;\text{dBm}+10\log_{10}W$
---
$+10$ dB
$T_{s}$ | Sampling time | $1/W$
$\tau_{c,l}$ | Path delay | $\mathcal{U}(0,DT_{s})$
$\alpha_{c,l}$ | Complex gain | $\mathcal{CN}(0,1)$
$r_{0}$ | Reference distance | 1 m
$r$ | Distance between transceiver, respectively | 100 m (0.1 m)4
$\mu$ | Path loss exponent | 3.4 [23]
$\lambda$ | Wavelength | $3\times 10^{8}/f_{c}$
$\kappa$ | Rician factor | 10 dB
$\eta$ | Power of the SI signal attenuated by antenna isolation and A-SIC | -80 dB

  * 4

$r=100$ m for backhaul and access link channels; $r=0.1$ m ($\approx
10\lambda$ [36]) for SI channel.

##  VII Simulations

In this section, simulation results will be shown to analyze the performance
of our designed networks. Each subarray (users) has 16$\times$4 UPA with 1 RF
chain. The rolling factor of the pulse shaping filter is 1. Both communication
links have $\mathrm{N_{C}}=8$ clusters, each with $\mathrm{N_{L}}=10$ rays,
whereas the NLOS component of the SI channel has $\mathrm{N_{C}}=2$ clusters,
each with $\mathrm{N_{L}}=8$ rays. Both azimuth and elevation AOAs/AODs can be
expressed as the sum of the mean angle of each cluster and the angle shifts in
the cluster. The mean azimuth and elevation AOAs/AODs of each cluster are
assumed as uniformly distributed in $[-\pi,\pi]$, and
$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, respectively. In each cluster,
the AOAs/AODs have Laplacian distribution with an angle spread of $5^{\circ}$.
The transceiver arrays at the IBFD-IAB-node have a separation angle of
$\frac{\pi}{6}$. Assume
$\sigma^{2}_{\mathrm{N}}=\sigma^{2}_{\mathrm{E}}=\sigma^{2}$, we define
$\mathrm{SNR}\triangleq\frac{P_{\mathrm{r}}}{\sigma^{2}KU}$, where
$P_{\mathrm{r}}=\frac{P_{\mathrm{t}}}{\bar{PL}}$ is the ratio between transmit
power and average path loss according to the Friis’ law. We let HWI factors
$\rho=\beta$ be the same for all channels. The backhaul link SE of the HD
scheme is given by removing the part relevant to the SI in (26) due to non-
simultaneous transmission and reception. Moreover, for both links, the (sum)
SE expressions for HD transmission need to be scaled by 0.5 since separate
time-frequency signaling channels are used for backhaul and access link. Other
parameters and their default values used in the simulations are summarized in
Table I.

Figure 4: Comparison between the performance of (a) traditional micro-strip
analog canceler; (b) OD-based analog canceler (SI channel has a delay spread
of 200 ns).

###  VII-A Performance of OD-Based Analog Canceler

Assume the propagation loss of the FBG (coiled into 2 cm) is 0.461 dB/m, and
that of the micro-strip is 2.967 dB/m [14]. The OD-based design uses a 20 dB
hybrid coupler to couple the RF reference signal into the OD-based canceler,
while the conventional electrical canceler uses a 0 dB coupler. Besides, to
explore the best performance, the tap delay varies according to the number of
taps to cover the delay spread. Fig. 4 shows the A-SIC abilities (in dB) of
the traditional micro-strip canceler (see Fig. 4) and the OD-based canceler
(see Fig. 4) for different bandwidths and numbers of taps. Simulations are run
with 200 ns of significant delay spread for the SI channel, which reflects a
bad channel condition. Although a measurement for the SI channel delay spread
is done in [37], a general delay spread value is still lacking in the
literature. Fig. 4 shows that creating a large number of taps with
conventional electrical components (e.g., cables or micro-strips) degrades the
performance rather than improving it due to significant insertion losses. It
can be seen that less than 15 dB of cancellation is achieved under 200 MHz
bandwidth. Fig. 4 shows that under 400 MHz bandwidth, OD-based canceler can
achieve around 25 dB of cancellation in FR2 wideband with 100 taps, which is
also proved in [14]. Note that this result shows the cancellation ability that
can be achieved between a single RF chain pair. In this work, we assume
antenna isolation and our A-SIC can attenuate the SI signal power by 55 dB
[10] and 25 dB, respectively.

Figure 5: Comparison on the (sum) SE of 4-subarray hybrid beamforming
structure with different kinds of codebooks for (a) backhaul link; (b) access
link with 4 users. Each subarray (user) is equipped with $16\times 4$ UPA and
1 RF chain (perfect CSI without HWI).

###  VII-B Performance of the Proposed Codebook Design

The comparison on the (sum) SE of the backhaul and access links with RF
precoders/combiners selected from our proposed matrix-wise codebooks and
vector-wise codebooks designed by conventional MSE-based LBG algorithm in
[30], respectively, for the subarray structure is plotted in Fig. 5. In order
to get a fair comparison, a $b$-bit vector codebook should be compared with an
$N_{{\rm RF}}b$-bit matrix codebook, where $N_{\rm RF}$ is the number of RF
chains555For vector quantization, since each column of the RF beamformer
matrix selects one codeword from a $b$-bit vector codebook, we can get
$2^{N_{{\rm RF}}b}$ different candidate matrices, which is equal to the number
of codeworks in a $N_{{\rm RF}}b$-bit matrix codebook.. We assume perfect CSI
and hardware. The RF precoders/combiners with infinite resolution PSs are
designed according to [17]. It can be seen that, for both kinds of codebooks,
as the number of codebook size increases, the performance becomes closer to
the ideal one (i.e., infinite resolution). Obviously, our matrix codebook can
provide better performance than the vector codebook designed by the
conventional LBG algorithm, which shows the successful applicability of our
modified MSE-based LBG codebook design. However, there is still a small gap
between the ideal one and the curves derived with 8 bits matrix codebook for
both links. A large size of codebook can be used to reduce the gap. Moreover,
the HD operation yields lower (sum) SE than that of the IBFD scheme.

Figure 6: SE of backhaul link for different beamforming schemes. Each subarray
(user) has $16\times 4$ UPA. The IAB donor and IAB-node have $16\times 16$ UPA
for fully connected structure. ($\rho=\beta=-80$ dB,
$\sigma_{e,\mathrm{ND}}^{2}=\sigma_{e,\mathrm{EN}}^{2}=\sigma_{e,\mathrm{SI}}^{2}=-120$
dB)

###  VII-C Performance of Different Beamforming Schemes

Fig. 6 shows the SE of the backhaul link for different beamforming schemes.
The ideal curves are plotted by assuming perfect CSI and SIC without HWI. The
design of the RF precoders/combiners for the ideal fully connected and
subarray structures follows the process in [17], which have infinite
resolution. The non-ideal curves are plotted by our proposed design algorithm
with 8 bits RF codebook and setting $\rho=\beta=-80$ dB,
$\sigma_{e,\mathrm{ND}}^{2}=\sigma_{e,\mathrm{EN}}^{2}=\sigma_{e,\mathrm{SI}}^{2}=-120$
dB. It can be observed that for the IBFD scheme, these three beamforming
schemes evaluated in the figure are separated by a significant rate loss.
Although the rate loss is evident, the subarray structure can significantly
reduce the hardware complexity and provide low-computationally intensive
precoders, which is beneficial for industrial implementations. Further, with
our staged SIC, the SE of the subarray structure is very close to its ideal
one; however, it shows some degrees of freedom loss at high SNR due to RSI
caused by HWI and RF effective channel uncertainties. Fortunately, the losses
on degrees of freedom and SE are further reduced by increasing the number of
RF chains at the IAB-node receiver from 4 to 8 (see the green and orange
curves in Fig. 6).

Figure 7: SE of the backhaul link at $\text{SNR}=-5,0,5$ dB with 4-subarray
hybrid beamforming structure, where RF beamformers are selected from different
size of codebooks, in the presence of different values of (a) SI RF effective
channel estimation error ($\rho=\beta=-80$ dB,
$\sigma_{e,\mathrm{ND}}^{2}=\sigma_{e,\mathrm{EN}}^{2}=-120$ dB); (b) HWI
($\rho=\beta$,
$\sigma_{e,\mathrm{ND}}^{2}=\sigma_{e,\mathrm{EN}}^{2}=\sigma_{e,\mathrm{SI}}^{2}=-120$
dB).

###  VII-D Effect of RSI on the SE of the Backhaul Link

In Fig. 7, with RF precoders/combiners selected from 1, 4, 8 bits codebooks,
respectively, we would like to study how the RSI caused by RF effective SI
channel estimation error and HWI can affect the SE performance of the backhaul
link at different SNR values. With $\rho=\beta=-80$ dB and
$\sigma_{e,\mathrm{ND}}^{2}=\sigma_{e,\mathrm{EN}}^{2}=-120$ dB, we plot the
SE performance of the backhaul link in Fig. 7 by varying the channel
estimation error of the SI RF effective channel. Interestingly, it is worth
noting that as the size of the RF codebook increases, the intersection point
(i.e., the point where both the IBFD and HD have the same performance) shifts
to the right at a fixed SNR, which means the system can tolerate more RSI
caused by channel estimation error. On the contrary, when the codebook size is
fixed, as SNR increases, the intersection point shifts to the left. By
assuming all effective channels have the same estimation error of -120 dB,
Fig. 7 shows the backhaul link SE performance with varying HWI factors.
Similar to the trend in Fig. 7, with the same codebook size, as the SNR
increases, the system can tolerate less RSI caused by HWI. The tolerance is
improved at a fixed SNR when the codebook size increases. Moreover, an almost
doubled SE can be achieved by the IBFD compared to that of the HD when HWI
factors (and channel estimation errors) are small enough, as can be seen in
Fig. 7.

##  VIII Conclusion

In this paper, we have studied FR2 wideband IBFD-IAB networks under subarray
structures, which are simpler to deploy and more cost-effective than fully-
connected ones. For this system, we have proposed the RF codebook design for
the subarray structure with hybrid beamforming. Compared with the traditional
vector-wise codebook, our matrix-wise codebook can avoid low-rank matrix and
loss of degrees of freedom. We also introduced the staged SIC scheme. In order
to reduce the deployment cost, we have established the canceler on each RF
chain pair and utilized the OD-based analog canceler to reduce the effect of
insertion loss. The RSI left by the A-SIC was handled in the digital domain by
successive interference cancellation and MMSE BB combiner. Simulations have
shown that under 400 MHz bandwidth, our OD-based canceler can achieve about 25
dB cancellation with 100 taps as well as experiencing constant insertion loss,
which can not be realized by the traditional micro-strip canceler. With large
HWI and RF effective SI channel uncertainties, the IBFD transmission
experiences performance limitation in the backhaul link; however, for small
HWI and uncertainties, the IBFD promises almost doubled SE compared with that
of the HD.

Further work will include investigating multicell IBFD-IAB systems, optimal
power allocation, and efficient antenna cancellation. Besides, the SI channel
model will also be studied by real-world measurements or other reliable
mathematics models.

##  Appendix A MMSE BB Combiner

$\displaystyle\frac{\partial\mathbb{E}\left\\{\left\|\mathbf{s}_{\mathrm{D}}[k]-\hat{\mathbf{y}}_{\mathrm{N}}[k]\right\|^{2}_{2}\right\\}}{\partial\mathbf{W}_{{\rm
BB}\mathrm{N}}^{H}[k]}$
$\displaystyle=\frac{\partial\mathbb{E}\left\\{(\mathbf{s}_{\mathrm{D}}[k]-\hat{\mathbf{y}}_{\mathrm{N}}[k])(\mathbf{s}_{\mathrm{D}}[k]-\hat{\mathbf{y}}_{\mathrm{N}}[k])^{H}\right\\}}{\partial\mathbf{W}_{{\rm
BB}\mathrm{N}}^{H}[k]}$
$\displaystyle=\frac{\partial\mathbb{E}\left\\{\mathbf{s}_{\mathrm{D}}[k]\mathbf{s}^{H}_{\mathrm{D}}[k]-\mathbf{s}_{\mathrm{D}}[k]\hat{\mathbf{y}}_{\mathrm{N}}^{H}[k]-\hat{\mathbf{y}}_{\mathrm{N}}[k]\mathbf{s}^{H}_{\mathrm{D}}[k]+\hat{\mathbf{y}}_{\mathrm{N}}[k]\hat{\mathbf{y}}_{\mathrm{N}}^{H}[k]\right\\}}{\partial\mathbf{W}_{{\rm
BB}\mathrm{N}}^{H}[k]}.$ (43)

By substituting (25) into (43), we have

$\displaystyle\frac{\partial\mathbb{E}\left\\{\left\|\mathbf{s}_{\mathrm{D}}[k]-\hat{\mathbf{y}}_{\mathrm{N}}[k]\right\|^{2}_{2}\right\\}}{\partial\mathbf{W}_{{\rm
BB}\mathrm{N}}^{H}[k]}=-\left({\widetilde{\hat{\mathbf{y}}}_{\mathrm{N}}[k]}+\mathbf{g}_{\mathrm{N}}[k]\right)\mathbf{s}^{H}_{\mathrm{D}}[k]$
$\displaystyle+\left({\widetilde{\hat{\mathbf{y}}}_{\mathrm{N}}[k]}+\mathbf{g}_{\mathrm{N}}[k]\right)\left({\widetilde{\hat{\mathbf{y}}}_{\mathrm{N}}[k]}+\mathbf{g}_{\mathrm{N}}[k]\right)^{H}\mathbf{W}_{{\rm
BB}\mathrm{N}}[k].$ (44)

Let (44) equal to 0, we can have the MMSE BB combiner given in (42).

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|  Junkai Zhang (Student Member, IEEE) received the B.Eng. degree in
communication engineering from Shenyang Ligong University, Shenyang, China, in
2018, and the M.Sc. degree in 2019, in signal processing and communications
(with Distinction) from The University of Edinburgh, Edinburgh, United
Kingdom, where he is currently working toward the Ph.D. degree with the
Institute for Digital Communications. His research interests include 5G and
beyond wireless networks, millimeter-wave communications, in-band-full-duplex
radio, stochastic geometry, integrated sensing and communications, and massive
MIMO.
---|---
|  Haifeng Luo received the Bachelor’s degree from the Civil Aviation
University of China, Tianjin, China, in 2018, and received the Master’s degree
from The University of Edinburgh, Edinburgh, United Kingdom, in 2019. He is
currently pursuing the Ph.D. degree in the Institute for Digital
Communications, The University of Edinburgh, Edinburgh, United Kingdom. His
research interests include signal processing aspects of beyond 5G wireless
networks, full-duplex radios, and machine learning aided communications.
---|---
|  Navneet Garg (Member, IEEE) received the B.Tech. degree in electronics and
communication engineering from College of Science & Engineering, Jhansi,
India, in 2010, and the M.Tech. degree in digital communications from ABV-
Indian Institute of Information Technology and Management, Gwalior, in 2012.
He has completed the Ph.D. degree in June 2018 from the department of
electrical engineering at the Indian Institute of Technology Kanpur, India.
From July 2018-Jan 2019, he visited The University of Edinburgh, Edinburgh,
United Kingdom. From February 2019-2020, he is employed as a research
associate in Heriot-Watt university, Edinburgh, United Kingdom. Since February
2020, he is working as a research associate in The University of Edinburgh,
Edinburgh, United Kingdom. His main research interests include wireless
communications, signal processing, optimization, and machine learning.
---|---
|  Abhijeet Bishnu (Member, IEEE) received his B.Eng. degree in Electronics
and Communication Engineering from Technocrat Institute of Technology, Bhopal,
India in 2010. He also received the M.Eng. degree in Electronics and
Telecommunication Engineering from S.G.S.I.T.S. Indore, India in 2013, and the
Ph.D. degree in Electrical Engineering from Indian Institute of Technology
Indore, India, in 2019. He was also a visiting research scholar in The
University of Edinburgh, Edinburgh, United Kingdom in 2019. He is currently a
postdoctoral research associate in The University of Edinburgh, Edinburgh,
United Kingdom. His research interests include channel estimation, cognitive
radio, MIMO-OFDM system, full-duplex communication and 5G and beyond
communication. He is served as reviewer for many IEEE and Springer journals.
---|---
|  Mark Holm (Member, IEEE) received the B.S. degree (Hons.) in laser physics
and optoelectronics and the Ph.D. degree in physics from the University of
Strathclyde, in 1997 and 2001, respectively. He currently works as the
Technical Lead and a Hardware System Architect with Huawei Technologies
(Sweden) AB, with interest in microwave radio, phased array antennas, full
duplex radio systems, and photonic radios. In the past, he was the Microwave
Lead on AESA radar systems, a Senior Engineer responsible for GaAs pHemt
modeling, and a Laser and Package Design Engineer for SFP/XENPACK fiber
modules. He has published in the fields of laser design and GaAs device
modeling.
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|  Tharmalingam Ratnarajah (Senior Member, IEEE) is currently with the
Institute for Digital Communications, The University of Edinburgh, Edinburgh,
UK, as a Professor in Digital Communications and Signal Processing. He was a
Head of the Institute for Digital Communications during 2016-2018. His
research interests include signal processing and information theoretic aspects
of beyond 5G wireless networks, full-duplex radio, mmWave communications,
random matrices theory, interference alignment, statistical and array signal
processing and quantum information theory. He has published over 400
publications in these areas and holds four U.S. patents. He has supervised 16
PhD students and 21 post-doctoral research fellows and raised $\$$11+ million
USD of research funding. He was the coordinator of the EU projects ADEL (3.7M
€) in the area of licensed shared access for 5G wireless networks, HARP (4.6M
€) in the area of highly distributed MIMO, as well as EU Future and Emerging
Technologies projects HIATUS (3.6M €) in the area of interference alignment
and CROWN (3.4M €) in the area of cognitive radio networks. Dr Ratnarajah was
an associate editor IEEE Transactions on Signal Processing, 2015-2017 and
Technical co-chair, The 17th IEEE International workshop on Signal Processing
advances in Wireless Communications, Edinburgh, UK, 3-6, July 2016. Dr
Ratnarajah is a Fellow of Higher Education Academy (FHEA).
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