# Performance Analysis of In-Band-Full-Duplex Multi-Cell Wideband IAB Networks

Junkai Zhang, and Tharmalingam Ratnarajah J. Zhang is with School of
Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China
(e-mail: jk.zhang@xjtu.edu.cn).T. Ratnarajah is with the Institute for
Imaging, Data and Communications (IDCOM), The University of Edinburgh,
Edinburgh, EH9 3BF, UK (e-mail: t.ratnarajah@ed.ac.uk).Junkai Zhang was with
the Institute for Imaging, Data and Communications, The University of
Edinburgh and he is now working with the School of Mathematics and Statistics,
Xi’an Jiaotong University. This work was supported by the U.K. Engineering and
Physical Sciences Research Council (EPSRC) under Grant EP/T021063/1.This work
is based on the Ph.D. dissertation [1] of Junkai Zhang.Corresponding author:
Junkai Zhang (e-mail: jk.zhang@xjtu.edu.cn).

###### Abstract

This study analyzes the performance of 3rd Generation Partnership Project
(3GPP)-inspired multi-cell wideband single-hop backhaul millimeter-wave-in-
band-full-duplex (IBFD)-integrated access and backhaul (IAB) networks using
stochastic geometry. We modeled the wired-connected Next Generation NodeBs
(gNBs) as the Matérn hard-core point process (MHCPP) to meet real-world
deployment requirements and reduce the cost caused by wired connections in the
network. We first derive association probabilities that reflect the likelihood
that typical user-equipment is served by a gNB or an IAB-node based on the
maximum long-term averaged biased-received-desired-signal power criteria.
Furthermore, by leveraging the composite Gamma-Lognormal distribution, we
derived the closed-form signal to interference plus noise ratio coverage,
capacity with outage, and ergodic capacity of the network. To avoid
underestimating the noise, we consider the sidelobe gain on the inter-cell
interference links and analog-to-digital converter quantization noise.
Compared with half-duplex transmission, the numerical results show an enhanced
capacity with outage and ergodic capacity provided by the IBFD under
successful self-interference cancellation. We also study how the power bias
and density ratio of the IAB-node to gNB and the hard-core distance can affect
system performance.

###### Index Terms:

Millimeter-wave, integrated access and backhaul, in-band-full-duplex, Matérn
hard-core point process, stochastic geometry.

##  I Introduction

Millimeter-wave (mmWave) communications have been provided to satisfy the
urgent demand for large available bandwidth in beyond fifth-generation (B5G)
communications, as the number of wireless devices increases sharply in current
networks [2, 3]. Although a large available bandwidth of 30-300 GHz can
significantly increase the capacity of the wireless network, the path loss
becomes more severe than that in microwave communications because the
effective antenna aperture is proportional to the wavelength. Therefore, dense
base station (BS) deployment with large-scale antenna arrays is required. For
hardware and power efficiency, fully-connected/subarray hybrid beamforming is
used for mmWave communication [4, 5].

To further explore the benefits of the mmWave spectrum, in-band-full-duplex
(IBFD) transmission, where transmission and reception occur at the same time
and frequency band, has been proposed [6, 7]. Compared with half-duplex (HD)
transmission, the IBFD scheme provides twice the spectral efficiency (SE) and
half the latency [8]. However, dealing with self-interference (SI), which can
be more than 100 dB higher than the desired signal, is the biggest challenge
in realizing IBFD transmission. Fortunately, a combination of antenna [9],
analog [8, 10], and digital [11] SI cancellation (SIC) can achieve
satisfactory performance in mmWave IBFD networks.

Moreover, integrated access and backhaul (IAB) networks have been studied in
the technical specifications–TR 38.874 (Rel. 16) provided by the 3rd
Generation Partnership Project (3GPP) [12], where wireless backhaul is
introduced instead of fiber connections. Plug-and-play deployment [12], multi-
hop networks [13], and performance analysis [14] of IAB networks have been
used in recent research. Although the authors in [15] concluded that all-wired
networks (i.e., all BSs connected to the core network by fiber directly)
provide better performance than IAB networks, IAB networks provide a good
solution for reducing the deployment cost in mmWave communications. Although
we designed single-cell wideband single-hop backhaul mmWave-IBFD-IAB networks
in [2, 11], more profound studies on the performance analysis of multi-cell
wideband single-hop backhaul mmWave-IBFD-IAB networks are lacking in the
literature, which is the main subject of this work.

In previous studies [16, 3, 14, 17], stochastic geometry was used to analyze
the performance of IAB networks. However, none of these studies considered
IBFD transmission and hybrid beamforming, except for [3]. In addition, in
terms of the distribution of small fading, only [14] considered the
Nakagami-$M$ distribution, which is more suitable for mmWave communications
than the Rayleigh distribution because the Rayleigh fading model for sub-6 GHz
communications is predicated on a large amount of local scattering, which is
not the case for mmWave communications [18]. Moreover, the shadowing effect
was lacking in most studies. Only [17] considers both small fading and path
loss with the shadowing effect; however, the Rayleigh distribution is used to
model the small fading rather than the Nakagami-$M$ distribution. Therefore,
to analyze a more general mmWave-IAB network suitable for beyond 5G
communications, we consider IBFD, hybrid beamforming, Nakagami-$M$ small
fading, and lognormal shadowing effects in this work.

Stochastic geometry is a powerful tool for analyzing the performance of
wireless networks such as mmWave cellular networks [19], heterogeneous
cellular networks [20], and unmanned aerial vehicle (UAV) networks [21].
However, to the best of our knowledge, work on IAB systems with stochastic
geometry is insufficient. For instance, the performance of mmWave-HD-IAB
systems was analyzed in [16] with two resource allocation strategies at Next
Generation NodeBs (gNBs), i.e., integrated resource allocation and orthogonal
resource allocation. The authors in [3] studied the performance of mmWave-
IBFD-IAB systems in wireless edge caching. Bandwidth partitioning for mmWave-
HD-IAB systems with performance analysis by stochastic geometry was studied in
[14]. A multi-cell model of a single-tier mmWave-HD-IAB system was proposed in
[17]. However, in the aforementioned stochastic geometry-related IAB networks
studies, only the work in [17] considered the shadowing effect in mmWave
communications. Moreover, Nakagami-$M$ small fading was only introduced in
[14].

Meanwhile, in the real world, BSs are not entirely deployed randomly or
regularly, which results in the deterministic or Poisson point process (PPP)
model becoming insufficient. Therefore, non-Poisson models are preferred for
modeling real-world BS deployment. For analytical tractability, the authors of
[22] suggested a method that uses PPP to approximate the signal-to-
interference-ratio meta-distribution for non-Poisson networks, which can
provide good accuracy. In this work, to model the deployment of wired-
connected gNBs, that is, IAB donors, the Matérn hard-core point process
(MHCPP) was applied, where a hard-core distance (i.e., the minimum distance
between two gNBs separated in the network) was utilized. The MHCPP model
reflects the repulsion between gNBs because gNBs act as wired-anchored BSs.
The two gNBs should not be too close or overlapping. Moreover, while providing
satisfactory service quality, MHCPP generates a lower density of gNBs than
PPP, which avoids the high cost of wired connections. Although MHCPP has been
applied to the modeling of UAVs [23], multi-cell IBFD cellular networks [24],
and cloud radio access networks [25], to the best of our knowledge, this is
the first study to introduce MHCPP into the analysis of multi-cell mmWave-
IBFD-IAB networks.

In this study, we evaluate the performance of multi-cell wideband single-hop
backhaul mmWave-IBFD-IAB networks using stochastic geometry. The main
contributions of this study are summarized as follows. A comparison between
the existing stochastic geometry-based studies and our work is summarized in
Table. I.

TABLE I: Comparison between existing stochastic geometry-based studies and our work | mmWave | IBFD | IAB | Hybrid Beamforming | Nakagami-$M$ | Lognormal | MHCPP | Sidelobe Gain | Quantization Noise
---|---|---|---|---|---|---|---|---|---
[3] | $\surd$ | $\surd$ | $\surd$ | $\surd$ |  |  |  | $\surd$ |
[5] | $\surd$ |  |  | $\surd$ |  | $\surd$ |  | $\surd$ |
[14] | $\surd$ |  | $\surd$ |  | $\surd$ |  |  |  |
[16] | $\surd$ |  | $\surd$ |  |  |  |  |  |
[17] | $\surd$ |  | $\surd$ |  |  | $\surd$ |  |  |
[19] | $\surd$ |  |  |  | $\surd$ |  |  |  |
[21] | $\surd$ |  |  |  | $\surd$ |  | $\surd$ |  |
[24] |  | $\surd$ |  |  |  |  | $\surd$ |  |
[26] |  | $\surd$ | $\surd$ |  |  |  |  |  |
[27] | $\surd$ |  |  |  | $\surd$ |  |  | $\surd$ |
[28] | $\surd$ |  |  | $\surd$ |  | $\surd$ |  | $\surd$ |
[29] |  |  |  |  | $\surd$ | $\surd$ |  |  |
Our work | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$

  * •

Unlike previous studies on mmWave communications using stochastic geometry,
which do not consider hybrid beamforming, we analyzed the performance with
energy-efficient subarray-based hybrid beamforming because mmWave
communications need to adopt large-scale antenna arrays. Hybrid beamforming
has lower hardware cost and power consumption than transitional full digital
beamforming because the number of RF chains required is reduced.

  * •

Unlike prior studies on IAB networks, which use PPP to model the location of
gNBs, we use MHCPP to meet the real-world deployment scenario in which BSs are
deployed in the non-Poisson model. In this manner, wired-connected gNBs, which
act as anchored BSs, can be deployed without overlapping or being too close.
In addition, MHCPP produces fewer gNBs than PPP, which reduces the cost of
deploying wired gNBs. We use the probability generating functional (PGFL) of
PPP to estimate the mean interference from gNBs in the backhaul link and gNB-
associated access link, which makes the analysis more tractable.

  * •

Instead of analyzing the performance of mmWave-HD-all-wired networks, we
consider mmWave-IBFD-IAB networks, which satisfy the requirement of enhanced
communication performance and dense deployment with less cost in beyond 5G
communications. The IAB-nodes operate in the IBFD mode for simultaneous
transmission and reception. Assuming successful SIC, only the residual SI
(RSI), modeled as Gaussian noise, is considered, whose power is modeled as a
fraction of the transmit power [3, 26, 11].

  * •

To accurately model the channel and path loss for mmWave communications, we
consider Nakagami-$M$ small fading and include the lognormal shadowing effect
in the path loss model, which is different from most of the existing
literature on mmWave communications using stochastic geometry. For
tractability, we introduced the closed-form statistical property of the
composite Gamma-Lognormal (GL) distribution to derive the analytical results.

  * •

To avoid underestimation of noise and interference, we consider the sidelobe
gain for the inter-cell interference links and analog-to-digital converter
(ADC) quantization noise.

Numerical results show that by tuning the bias ratio of the IAB-node to gNB,
signal-to-interference-plus-noise-ratio ($\mathrm{SINR}$) coverage yields a
convex-like curve at a fixed $\mathrm{SINR}$ threshold. With a selected bias
ratio (e.g., 0 dB), the capacity with outage of the IBFD scheme outperforms
that of the HD scheme, regardless of the RSI power. In addition, for our
default simulation setting, at $\mathrm{SINR}=0$ dB, the ADC quantization
noise effect becomes negligible at a resolution of approximately five bits,
which emphasizes the feasibility of using a low resolution ADC in our proposed
network. Moreover, under the same gNB density, the network ergodic capacity
obtained by deploying gNBs with the PPP-based model is less than that obtained
by deploying gNBs with the MHCPP-based model; the larger the hard-core
distance, the higher the ergodic capacity. In addition, with successful SIC,
the ergodic capacity increases as the density ratio of the IAB-node to the gNB
increases.

Notations: $\mathcal{B},\mathbf{B},\mathbf{b}$, $b$ represent the set, the
matrix, the vector, and the scalar, respectively.
$\mathbf{B}^{H},\mathbf{B}^{-1}$, and $\mathbf{B}^{T}$ are the Hermitian,
inverse, and transpose of $\mathbf{B}$, respectively.
$\mathrm{blkdiag}\left[\mathbf{B}_{1},\mathbf{B}_{2}\right]$ is the block
diagonal matrix formed by matrices $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$.
$\left\|\cdot\right\|_{F}$ denotes the Frobenius norm. $|b|$ is the norm of
$b$. $||\mathbf{b}||$ is the norm of $\mathbf{b}$. $C_{n}^{k}$ is the possible
combinations of $k$ elements out of a group of $n$ elements where order does
not matter. $\mathcal{N}(m,n)$ represents a normal distribution with mean $m$
and variance $n$. $\mathrm{ln}\mathcal{N}(m,n)$ denotes a lognormal
distribution with parameters $m$ and $n$. $\mathcal{U}[a,b]$ represents a
uniform distribution between $a$ and $b$.

##  II System Model

###  II-A IAB Network Topology

We consider the downlink of a standalone single-hop backhaul IAB network with
a spanning tree (ST) topology consisting of the following nodes:

  * •

IAB donor: This is also called a gNB, which is a single logical node. The gNB
connects to the next-generation core network using fibers and wirelessly
transmits signals to other types of nodes in the network.

  * •

IAB-node: This node operates in the IBFD manner, contributes SI from its
transmitter to its receiver, and uses wireless links to communicate with
others. Owing to the single-hop backhaul and ST topology, an IAB-node has only
one parent gNB for backhauling.

  * •

User-equipment (UE): It receives the desired signal from its serving gNB or
IAB-node.

The backhaul link is the wireless link between the gNB and the IAB-node,
whereas the access link is between the UE and the gNB or the IAB-node. More
details of the 3GPP architecture can be found in our recent study [2]. An
illustration of the proposed model is shown in Fig. 1. In this study, the gNBs
and IAB-nodes formed two different tiers. Multi-hop backhaul IAB networks will
not be considered because of the challenge of network configuration, and the
feasibility of using stochastic geometry in this scenario [16].

###  II-B Spatial Arrangement

There are two types of MHCPP: both are generated by thinning the parent PPP.
In Type I, a point and its neighbors whose distance is less than the hard-core
distance are removed from the parent PPP. In type II, each point is assigned a
random number between 0 and 1 as their marks; only points that have larger
marks and are within the hard-core region of their neighbors are removed [30].
In this study, we utilize MHCPP Type II111Under the same parent PPP density,
the thinning rule of MHCPP type II is more relaxed than that of MHCPP Type I,
which can provide a higher density of gNB to satisfy the dense deployment
condition for mmWave communications. to capture the location of gNBs. Consider
that gNBs are distributed according to an MHCPP $\Phi_{\mathrm{m}}$ with
density $\lambda_{\mathrm{m}}$ and hard-core distance $\xi$ on
$\mathbb{R}^{2}$, where the density is defined by the dependent thinning of a
parent homogeneous PPP $\widetilde{\Phi}_{\mathrm{m}}$ with density of
$\widetilde{\lambda}_{\mathrm{m}}$. Given that the probability of an arbitrary
point in the parent PPP is retained for MHCPP Type II as

$\rho=\frac{1-e^{-\pi\xi^{2}\widetilde{\lambda}_{\mathrm{m}}}}{\pi\xi^{2}\widetilde{\lambda}_{\mathrm{m}}},$
(1)

where $\lambda_{\mathrm{m}}=\rho\widetilde{\lambda}_{\mathrm{m}}$ [31]. The
IAB-nodes and UE are drawn from two independent homogeneous PPP
$\Phi_{\mathrm{s}}$ and $\Phi_{\mathrm{u}}$ with densities
$\lambda_{\mathrm{s}}$ and $\lambda_{\mathrm{u}}$ on $\mathbb{R}^{2}$,
respectively. Our analysis was performed on a circular region with a radius
$R_{0}$. A realization of such a network is illustrated in Fig. 1.

Figure 1: (a) Illustration of a multi-cell single-hop backhaul mmWave-IBFD-IAB
network. (b) A realized mmWave-IBFD-IAB network. $R_{0}=1000$m,
$\lambda_{\mathrm{m}}=1\times 10^{-5}/\text{m}^{2}$,
$\lambda_{\mathrm{s}}=4\times 10^{-5}/\text{m}^{2}$,
$\lambda_{\mathrm{u}}=2\times 10^{-4}/\text{m}^{2}$, $\xi=100$m.

###  II-C Propagation Model

####  II-C1 Blockage Model

At mmWave frequencies, the signals are easily blocked by obstacles in the
propagation environment, which results in the coexistence of line-of-sight
(LoS) and non-line-of-sight (NLoS) propagations. Blockages typically form a
process of random shapes, for example, a Boolean scheme of rectangles [32]. In
this section, the probability of being LoS $p_{\mathrm{L}}(r)$ or NLoS
$p_{\mathrm{N}}(r)$ transmission is modeled by a two-state exponential
blockage model with respect to the blockage density constant $\epsilon$ and
the link distance $r$ [27], which is expressed as

$\left\\{\begin{aligned} \quad&p_{\mathrm{L}}(r)=e^{-\epsilon r},\\\
&p_{\mathrm{N}}(r)=1-e^{-\epsilon r}.\end{aligned}\right.$ (2)

Note that we adopted the above model for IAB-nodes because gNBs can be located
well above ground level because of the role of anchored BSs and are usually
placed on open ground with fewer obstacles to increase the likelihood of LoS
links. For tractability, we assume LoS transmission only for the gNB to ease
the lack of transformation properties of MHCPP, such as the PGFL. Moreover, we
consider the independent blockage effects between the links in this study.

Given the blockage model, we can treat the PPP IAB-nodes $\Phi_{\mathrm{s}}$
as the superposition of PPP LoS IAB-nodes $\Phi_{\mathrm{s},\mathrm{L}}$ and
PPP NLoS IAB-nodes $\Phi_{\mathrm{s},\mathrm{N}}$ with densities of
$\lambda_{\mathrm{s},\mathrm{L}}(r)=\lambda_{\mathrm{s}}p_{\mathrm{L}}(r)$ and
$\lambda_{\mathrm{s},\mathrm{N}}(r)=\lambda_{\mathrm{s}}p_{\mathrm{N}}(r)$,
respectively.

####  II-C2 Path Loss Model

The path loss model utilized in [17] was adopted in this study to measure the
path loss between two nodes located at $\mathbf{x}$ and $\mathbf{y}$. The gain
for a single antenna element is given by

${L}(\mathbf{x},\mathbf{y})[\text{dB}]=\beta+10\alpha\log_{10}R_{\mathbf{x}\mathbf{y}}+\chi,$
(3)

where $\beta=20\log_{10}\left(\tfrac{4\pi f_{\mathrm{c}}}{c}\right)$.
$f_{\mathrm{c}}$, $c$, $\alpha$, and $\chi\sim\mathcal{N}(0,\zeta^{2})$ denote
the carrier frequency, light speed, path loss exponent, and shadowing effect
random variable, respectively. $R_{\mathbf{x}\mathbf{y}}$ represents the
Euclidean distance between $\mathbf{x}$ and $\mathbf{y}$. Therefore, we have
${L}(\mathbf{x},\mathbf{y})=\tfrac{R^{\alpha}_{\mathbf{x}\mathbf{y}}}{10^{-0.1(\beta+\chi)}}$,
where
$10^{-0.1(\beta+\chi)}\sim\mathrm{ln}\mathcal{N}\left(-0.1\beta\mathrm{ln}10,\left({0.1\zeta\mathrm{ln}10}\right)^{2}\right)$
is a lognormal distributed random variable222If $Z$ is a normally distributed
random variable with zero mean and variance $\sigma^{2}$, then $X=e^{m+nZ}$ is
a lognormal random variable with parameters $\hat{\mu}=m$ and
$\hat{\sigma}=n\sigma$, denoted as
$X\sim\mathrm{ln}\mathcal{N}(\hat{\mu},\hat{\sigma}^{2})$ with
$\mathbb{E}\\{X\\}=e^{\hat{\mu}+0.5\hat{\sigma}^{2}}$.. Note that the path
loss exponent and Gaussian distributed shadowing effect random variable for
the transmissions from gNB are denoted as $\alpha_{\mathrm{m}}$ and
$\chi_{\mathrm{m}}\sim\mathcal{N}(0,\zeta_{\mathrm{m}}^{2})$, respectively.
For the IAB-node, we have the path loss exponent $\alpha_{\mathrm{s},i}$ and
shadowing effect random variables
$\chi_{\mathrm{s},i}\sim\mathcal{N}(0,\zeta_{\mathrm{s},i}^{2})$, where
$i=\mathrm{L}$ for the LoS IAB-node and $i=\mathrm{N}$ for the NLoS IAB-node.

####  II-C3 Channel Model

Assume an OFDM single-path (i.e., flat fading) channel for all blockage
scenarios, which leverages the sparsity of the FR2 band channel. A multi-path
case will be left in the future. At subcarrier $k=1,2,\ldots,K$, the wideband
mmWave channel from the node at $\mathbf{x}$ with $N_{\mathrm{T}}$ transmit
antennas to the node at $\mathbf{y}$ with $N_{\mathrm{R}}$ receive antennas is
given as [4]

$\displaystyle\mathbf{{H}}_{\mathbf{yx}}[k]=$
$\displaystyle\sqrt{\tfrac{{N}_{\mathrm{T}}{N}_{\mathrm{R}}}{{L}(\mathbf{x},\mathbf{y})}}h^{\mathbf{yx}}\gamma[k]\mathbf{a}_{\mathrm{R}}(\bar{\theta}_{\mathrm{R}}^{\mathbf{yx}},\bar{\phi}_{\mathrm{R}}^{\mathbf{yx}})\mathbf{a}^{H}_{\mathrm{T}}(\bar{\theta}_{\mathrm{T}}^{\mathbf{yx}},\bar{\phi}_{\mathrm{T}}^{\mathbf{yx}}),$
(4)

where $h^{\mathbf{yx}}$ is the small fading gain with a Nakagami-$M$
distribution [33, 34].
$M\in\\{M_{\mathrm{m}},M_{\mathrm{s},\mathrm{L}},M_{\mathrm{s},\mathrm{N}}\\}\in\mathbb{N}^{+}$
are Nakagami parameters for the gNB, LoS IAB-node, and NLoS IAB-node,
respectively.
$\gamma[k]=\sum_{d=0}^{D-1}p(d-\widetilde{\tau})e^{-j\tfrac{2{\pi}kd}{K}}$
with $p(\cdot)$ is the raised-cosine pulse-shaping filter,
$\widetilde{\tau}\sim\mathcal{U}[0,D]$ being the normalized path delay, and
$D$ denotes the number of delay taps.

For simplicity, we leverage the virtual transmit and receive steering vectors
$\mathbf{a}_{\mathrm{T}}(\bar{\theta}_{\mathrm{T}}^{\mathbf{yx}},\bar{\phi}_{\mathrm{T}}^{\mathbf{yx}})$
and
$\mathbf{a}_{\mathrm{R}}(\bar{\theta}_{\mathrm{R}}^{\mathbf{yx}},\bar{\phi}_{\mathrm{R}}^{\mathbf{yx}})$
for the uniform planar array (UPA), respectively, where the azimuth
$\bar{\theta}_{\mathrm{T}}^{\mathbf{yx}}$/$\bar{\theta}_{\mathrm{R}}^{\mathbf{yx}}$
and elevation
$\bar{\phi}_{\mathrm{T}}^{\mathbf{yx}}$/$\bar{\phi}_{\mathrm{R}}^{\mathbf{yx}}$
virtual angles correspond to the angle of departure/arrival (AoD/AoA) [35].
With half-wavelength spaced elements, we have the relationship between the
virtual and physical azimuth (elevation) AoD/AoA as
$\bar{\theta}=\pi\cos\theta\cos\phi$ ($\bar{\phi}=\pi\sin\theta\cos\phi$),
where $\theta\sim\mathcal{U}[0,\pi]$ and
$\phi\sim\mathcal{U}[-\tfrac{\pi}{2},\tfrac{\pi}{2}]$ are the physical azimuth
and elevation AoD/AoA, respectively. By extending the model in [35], we
quantize the virtual azimuth and elevation AoD/AoA to set
$\bar{\theta}\in\left\\{-\pi,-\pi+\tfrac{2\pi}{N_{\mathrm{x}}},\ldots,-\pi+\tfrac{2\pi(N_{\mathrm{x}}-1)}{N_{\mathrm{x}}}\right\\}$
and
$\bar{\phi}\in\left\\{0,\tfrac{\pi}{N_{\mathrm{y}}},\ldots,\tfrac{\pi(N_{\mathrm{y}}-1)}{N_{\mathrm{y}}}\right\\}$
with ${N_{\mathrm{x}}}$ and ${N_{\mathrm{y}}}$ being the number of antennas
along $x$\- and $y$-axis of the UPA. Moreover, we assume that virtual angles
are uniformly distributed in their quantized sets for tractability [28]. From
the orthogonality of the virtual steering vector, we have
$\mathbf{a}^{H}(\bar{\theta}_{1},\bar{\phi}_{1})\mathbf{a}(\bar{\theta}_{2},\bar{\phi}_{2})=1$
for $\bar{\theta}_{1}=\bar{\theta}_{2},\bar{\phi}_{1}=\bar{\phi}_{2}$ and 0
otherwise [36, Corollary 2], which gives us an ON/OFF model. Using this ON/OFF
model, $\mathrm{SINR}$ expressions can be simplified.

Because this work aims to evaluate the performance of IBFD-IAB networks after
the SI has been successfully suppressed, only the residual SI, treated as
Gaussian noise, whose power is modeled as a fraction of the transmit power,
will be considered. Readers are directed to [11] for a hypothetical Rician-
like SI channel model and staged SIC strategy.

##  III Antenna Configuration and Beamforming Design

###  III-A Antenna Configuration

The gNB and IAB-node transmitters are adopted with subarray-based hybrid
beamforming (i.e., one RF chain connects with only a portion of the antennas),
as proposed in [11, 4]. The gNB has $N_{\mathrm{m}}$ subarrays, each having
$\tfrac{N_{\mathrm{T}}^{\mathrm{m}}}{N_{\mathrm{m}}}$ transmit antennas and
one RF chain sending a single data stream to serve one UE or IAB-node.
Similarly, the IAB-node is equipped with $N_{\mathrm{s}}$ subarrays, each with
$\tfrac{N_{\mathrm{T}}^{\mathrm{s}}}{N_{\mathrm{s}}}$ transmitting antennas
and a single RF chain sending one data stream to serve a UE. The IAB-node
receiver is equipped with $N_{\mathrm{R}}^{\mathrm{s}}$ receiving antennas
adopting analog combining to receive information from its serving gNB. The UE
receiver has $N_{\mathrm{R}}^{\mathrm{u}}$ antennas with analog combining for
communicating with its serving BS.

###  III-B Beamforming Design

According to the subarray hybrid beamforming structure given in [4], the RF
and BB precoders at the gNBs and IAB-nodes have the form of $\mathbf{F}_{{\rm
RF}}^{(\cdot)}=\mathrm{blkdiag}\big{[}\mathbf{f}_{{\rm RF},1}^{(\cdot)},$
$\mathbf{f}_{{\rm RF},2}^{(\cdot)},\ldots,\mathbf{f}_{{\rm
RF},N_{i}}^{(\cdot)}\big{]}\in\mathbb{C}^{N_{\mathrm{T}}^{i}\times N_{i}}$ and
$\mathbf{F}_{{\rm BB}}^{(\cdot)}[k]=\big{[}\mathbf{f}_{{\rm
BB},1}^{(\cdot)}[k],$ $\mathbf{f}_{{\rm
BB},2}^{(\cdot)}[k],\ldots,\mathbf{f}_{{\rm
BB},N_{i}}^{(\cdot)}[k]\big{]}\in\mathbb{C}^{N_{i}\times N_{i}}$,
respectively, where $(\cdot)$ is the location of the gNB or IAB-node;
$i=\mathrm{m}$ represents the precoders at the gNB and $i=\mathrm{s}$ are
those at the IAB-node. The RF combiner at the IAB-node and UE are in the form
of $\mathbf{w}_{{\rm RF}}^{(\cdot)}\in\mathbb{C}^{N_{\mathrm{R}}^{i}\times
1}$, where $i=\mathrm{s}$ denotes the combiner at the IAB-node and
$i=\mathrm{u}$ is that at the UE.

RF beamformers are realized using phase shifters (PSs) with unit norm entries.
According to [28], the optimal RF precoder/combiner that maximizes the desired
signal power is to steer the beam in the direction of the AoD/AoA of the
desired signal channel. Given the channel from the transmitter at $\mathbf{x}$
to the receiver at $\mathbf{y}$ in (4), owing to the use of subarray-based
hybrid beamforming, a subset of the virtual transmit steering vector is used
to design the $v$th column of the RF precoder matrix $\mathbf{F}_{{\rm
RF}}^{\mathbf{x}}$, that is,

$\mathbf{f}^{\mathbf{x}}_{{\rm
RF},v}=\sqrt{N_{\mathrm{T}}^{i}}\left[\mathbf{a}_{\mathrm{T}}(\bar{\theta}_{\mathrm{T}}^{\mathbf{yx}},\bar{\phi}_{\mathrm{T}}^{{\mathbf{yx}}})\right]_{(v-1)\tfrac{N_{\mathrm{T}}^{i}}{N_{i}}+\left(1:\tfrac{N_{\mathrm{T}}^{i}}{N_{i}}\right),:},$
(5)

where $i=\mathrm{m}$ represents the RF precoder at the gNB, and $i=\mathrm{s}$
denotes that at the IAB-node.

Similarly, the RF combiner is given by the virtual receiver steering vector,
that is,

$\mathbf{w}_{{\rm
RF}}^{\mathbf{y}}=\sqrt{N_{\mathrm{R}}^{i}}\mathbf{a}_{\mathrm{R}}(\bar{\theta}_{\mathrm{R}}^{\mathbf{yx}},\bar{\phi}_{\mathrm{R}}^{\mathbf{yx}}),$
(6)

where $i=\mathrm{s}$ denotes the combiner at the IAB-node, and $i=\mathrm{u}$
denotes that at the UE.

To mitigate the intra-cell interference, the zero-forcing (ZF) BB precoder is
utilized at both the gNB and IAB-node. For a transmitter at $\mathbf{x}$,
given all its serving receivers at
$\mathbf{y}_{1},\mathbf{y}_{2},\ldots,\mathbf{y}_{N_{i}}$ (for $i=\mathrm{m}$
if $\mathbf{x}$ is a gNB and $i=\mathrm{s}$ if $\mathbf{x}$ is an IAB-node),
the ZF BB precoder matrix can be given as

$\mathbf{F}_{{\rm
BB}}^{\mathbf{x}}[k]=\widetilde{\mathbf{H}}_{\mathbf{yx}}^{H}[k]\left(\widetilde{\mathbf{H}}_{\mathbf{yx}}[k]\widetilde{\mathbf{H}}_{\mathbf{yx}}^{H}[k]\right)^{-1},$
(7)

where $\widetilde{\mathbf{H}}_{\mathbf{yx}}[k]=\left(\mathbf{W}_{{\rm
RF}}^{\mathbf{y}}\right)^{H}{\hat{\mathbf{H}}}_{\mathbf{yx}}[k]\mathbf{F}_{{\rm
RF}}^{\mathbf{x}}$ with $\mathbf{W}_{{\rm
RF}}^{\mathbf{y}}=\mathrm{blkdiag}[\mathbf{w}_{{\rm
RF}}^{\mathbf{y}_{1}},\mathbf{w}_{{\rm
RF}}^{\mathbf{y}_{2}},\ldots,\mathbf{w}_{{\rm RF}}^{\mathbf{y}_{N_{i}}}]$ and
$\hat{\mathbf{H}}_{\mathbf{yx}}[k]=[\mathbf{H}^{T}_{\mathbf{y}_{1}\mathbf{x}}[k],$
$\mathbf{H}^{T}_{\mathbf{y}_{2}\mathbf{x}}[k],\ldots,\mathbf{H}^{T}_{\mathbf{y}_{N_{i}}\mathbf{x}}[k]]^{T}$.

##  IV $\mathrm{SINR}$ Characterization

In this section, we derive $\mathrm{SINR}$ expressions for the $k$th
subcarrier. According to Slivnyak’s theorem [30], given that both IAB-nodes
and UEs are homogeneous PPP, we consider typical backhaul and access links
without loss of generalization, that is, we derive the typical backhaul
(access) link $\mathrm{SINR}$ by assuming that the typical IAB-node (UE) is
located at the origin $\hat{\mathbf{0}}$ ($\mathbf{0}$). Note that
${\mathbf{0}}$ and $\hat{\mathbf{0}}$ represent the origins of the same
coordinate system.

Assume that we know the channel state information perfectly for all channels,
and that all signals are uncorrelated. Let the average total transmit power at
the gNB (IAB-node) across all subcarriers be $P_{\mathrm{m}}$
$\left(P_{\mathrm{s}}\right)$. With equal power allocation, each data stream
at subcarrier $k$ transmitted from the gNB (IAB-node) has the power of
$\tfrac{P_{\mathrm{m}}}{KN_{\mathrm{m}}}$
$\left(\tfrac{P_{\mathrm{s}}}{KN_{\mathrm{s}}}\right)$. Moreover, as the ZF
precoder is applied, the intra-cell interference can be canceled out; thus, is
ignored in the following equations. In addition, the following assumption was
made simplify the analysis.

Assumption 1: The channel model in Eq. (4) is a function of $k$, where $k$
appears only in $\gamma[k]$. Because our work is based on a probabilistic
perspective, for tractability, we assume the normalized path delay to be
$\widetilde{\tau}=\tfrac{D}{2}$, which is the mean of the random variable
$\widetilde{\tau}$. Given that the channel is flat fading, all subcarriers
experience the same channel conditions. Thus, we have
$\mathbb{E}\\{|\gamma[k]|^{2}\\}=1\forall k$, which makes the channel
independent of $k$ [3].

####  IV-1 Backhaul link

Assuming that the gNB at $\mathbf{x}_{\mathrm{m}}$ uses its $v$th subarray to
provide backhaul communication with the typical IAB-node at
$\hat{\mathbf{0}}$, the $\mathrm{SINR}$ of the typical backhaul link at the
$k$th subcarrier is written as

$\displaystyle\mathrm{SINR}_{\hat{\mathbf{0}}\mathbf{x}_{\mathrm{m}}}[k]=\tfrac{G_{\mathrm{b}}[k]}{I_{\mathrm{m,b}}[k]+I_{\mathrm{s,b}}[k]+\underbrace{\tfrac{\eta
P_{\mathrm{s}}}{K}}_{\text{RSI}}+\mathbb{E}\left\\{\left|\left(\mathbf{w}_{{\rm
RF}}^{\hat{\mathbf{0}}}\right)^{H}\mathbf{n}_{\mathrm{b}}[k]\right|^{2}\right\\}+\mathbb{E}\left\\{\left|e_{\mathrm{b}}[k]\right|^{2}\right\\}},$
(8a) $\displaystyle
G_{\mathrm{b}}[k]=\frac{P_{\mathrm{m}}}{KN_{\mathrm{m}}}\left|\left(\mathbf{w}_{{\rm
RF}}^{\hat{\mathbf{0}}}\right)^{H}\mathbf{H}_{\hat{\mathbf{0}}\mathbf{x}_{\mathrm{m}}}[k]\mathbf{F}_{{\rm
RF}}^{\mathbf{x}_{\mathrm{m}}}\mathbf{f}_{{\rm
BB},v}^{\mathbf{x}_{\mathrm{m}}}[k]\right|^{2}$
$\displaystyle\qquad\;\approx\frac{P_{\mathrm{m}}N_{\mathrm{T}}^{\mathrm{m}}\left(N_{\mathrm{R}}^{\mathrm{s}}\right)^{2}\widetilde{h}^{\hat{\mathbf{0}}\mathbf{x}_{\mathrm{m}}}p_{\mathrm{ZF}}(N_{\mathrm{T}}^{\mathrm{m}},N_{\mathrm{m}})}{KN^{2}_{\mathrm{m}}R_{\mathbf{x}_{\mathrm{m}}\hat{\mathbf{0}}}^{\alpha_{\mathrm{m}}}},$
(8b) $\displaystyle
I_{\mathrm{m,b}}[k]=\frac{P_{\mathrm{m}}}{KN_{\mathrm{m}}}\left\lVert\sum\limits_{{\omega}\in\atop\Phi_{\mathrm{m}}\backslash{\mathbf{x}_{\mathrm{m}}}}\left(\mathbf{w}_{{\rm
RF}}^{\hat{\mathbf{0}}}\right)^{H}\mathbf{H}_{\hat{\mathbf{0}}{\omega}}[k]\mathbf{F}_{{\rm
RF}}^{{\omega}}\mathbf{F}_{{\rm BB}}^{{\omega}}[k]\right\rVert^{2}$
$\displaystyle\qquad\;\;\;\approx\sum\limits_{\omega\in\atop\Phi_{\mathrm{m}}\backslash{\mathbf{x}_{\mathrm{m}}}}\frac{P_{\mathrm{m}}N^{\mathrm{m}}_{\mathrm{T}}\left(N^{s}_{\mathrm{R}}\right)^{2}\widetilde{h}^{\hat{\mathbf{0}}\omega}p_{\mathrm{BF}}(N^{\mathrm{m}}_{\mathrm{T}},N^{\mathrm{s}}_{\mathrm{R}},N_{\mathrm{m}})}{KN_{\mathrm{m}}R_{\omega\hat{\mathbf{0}}}^{\alpha_{\mathrm{m}}}},$
(8c) $\displaystyle
I_{\mathrm{s,b}}[k]=\sum\limits_{i\in\\{\mathrm{L},\mathrm{N}\\}}I_{\mathrm{s,b}}^{i}[k]$
$\displaystyle\qquad\;=\frac{P_{\mathrm{s}}}{KN_{\mathrm{s}}}\left\lVert\sum\limits_{i\in\\{\mathrm{L},\mathrm{N}\\}}\sum\limits_{\hat{{\omega}}\in\Phi_{\mathrm{s},i}\backslash{\hat{\mathbf{0}}}}\left(\mathbf{w}_{{\rm
RF}}^{\hat{\mathbf{0}}}\right)^{H}\mathbf{H}_{\hat{\mathbf{0}}\hat{\omega}}[k]\mathbf{F}_{{\rm
RF}}^{\hat{\omega}}\mathbf{F}_{{\rm BB}}^{\hat{\omega}}[k]\right\rVert^{2}$
$\displaystyle\qquad\;\approx\sum\limits_{i\in\\{\mathrm{L},\mathrm{N}\\}}\sum\limits_{\hat{\omega}\in\Phi_{\mathrm{s},i}\backslash{\hat{\mathbf{0}}}}\frac{P_{\mathrm{s}}N^{\mathrm{s}}_{\mathrm{T}}\left(N^{\mathrm{s}}_{\mathrm{R}}\right)^{2}\widetilde{h}^{\hat{\mathbf{0}}\hat{\omega}}p_{\mathrm{BF}}(N^{\mathrm{s}}_{\mathrm{T}},N^{\mathrm{s}}_{\mathrm{R}},N_{\mathrm{s}})}{KN_{\mathrm{s}}R_{\hat{\omega}\hat{\mathbf{0}}}^{\alpha_{\mathrm{s},i}}},$
(8d)

where in (8a), $G_{\mathrm{b}}[k]$, $I_{\mathrm{m,b}}[k]$, and
$I_{\mathrm{s,b}}[k]$ denote the power of the desired signal, the power of the
interference from non-target gNBs, and that of the interference from IAB-nodes
at subcarrier $k$, respectively; $0<\eta<1$ is the RSI controlling factor,
which controls the amount of RSI;
$\mathbb{E}\left\\{\left|\left(\mathbf{w}_{{\rm
RF}}^{\hat{\mathbf{0}}}\right)^{H}\mathbf{n}_{\mathrm{b}}[k]\right|^{2}\right\\}=\sigma_{\mathrm{n}}^{2}N^{\mathrm{s}}_{\mathrm{R}}$
is the Gaussian noise power with
$\mathbf{n}_{\mathrm{b}}[k]\sim\mathcal{CN}(\mathbf{0},\sigma_{\mathrm{n}}^{2}\mathbf{I}_{N^{\mathrm{s}}_{\mathrm{R}}})$;
$\mathbb{E}\left\\{\left|e_{\mathrm{b}}[k]\right|^{2}\right\\}$ represents the
quantization noise power owing to $q$ bits ADC uniform quantization. In
(8b)-(8d),
$\widetilde{h}^{\hat{\mathbf{0}}\mathbf{x}_{\mathrm{m}}}=10^{-0.1(\beta+\chi_{\mathrm{m}})}\left|h^{\hat{\mathbf{0}}\mathbf{x}_{\mathrm{m}}}\right|^{2}$,
$\widetilde{h}^{\hat{\mathbf{0}}\omega}=10^{-0.1(\beta+\chi_{\mathrm{m}})}\left|h^{\hat{\mathbf{0}}\omega}\right|^{2}$,
and
$\widetilde{h}^{\hat{\mathbf{0}}\hat{\omega}}=10^{-0.1(\beta+\chi_{\mathrm{s},i})}\left|h^{\hat{\mathbf{0}}\hat{\omega}}\right|^{2}$
follow the composite GL distribution (see Appendix-A) because
$\left|h^{(\cdot)}\right|^{2}$ is a Gamma distributed random variable333If $X$
is a Nakagami-$M$ distributed random variable, then
$X^{2}\sim\mathrm{Gamma}(M,1/M)$ is a unit mean Gamma-distributed random
variable.; $p_{\mathrm{ZF}}(N_{\mathrm{T}}^{\mathrm{m}},N_{\mathrm{m}})$ is
the ZF penalty, which is a Bernoulli random variable with success probability
$(1-\tfrac{1}{N_{\mathrm{T}}^{\mathrm{m}}})^{N_{\mathrm{m}}-1}$ (see Remark
1);
$p_{\mathrm{BF}}(N^{\mathrm{m}}_{\mathrm{T}},N^{\mathrm{s}}_{\mathrm{R}},N_{\mathrm{m}})$
and
$p_{\mathrm{BF}}(N^{\mathrm{s}}_{\mathrm{T}},N^{\mathrm{s}}_{\mathrm{R}},N_{\mathrm{s}})$
are beamforming gains that take into account the sidelobe gains (see
Proposition 1) because using the ON/OFF model given by the orthogonality of
the steering vector can result in underestimation of the inter-cell
interference as it gives zero beamforming gain when the azimuth and/or
elevation angles are not the same. The approximation in (8b) is owing to the
ZF penalty. The approximation in (8c) is obtained by assuming
$\mathbf{F}_{{\rm
BB}}^{{\omega}}[k]=\sqrt{\tfrac{N_{\mathrm{T}}^{\mathrm{m}}}{N_{\mathrm{m}}}}\mathbf{I}_{N_{\mathrm{m}}}$,
where $\sqrt{\tfrac{N_{\mathrm{T}}^{\mathrm{m}}}{N_{\mathrm{m}}}}$ ensures
$\left\|\mathbf{F}_{{\rm RF}}^{\omega}\mathbf{F}_{{\rm
BB}}^{{\omega}}\right\|_{F}^{2}=N_{\mathrm{m}}$. For a large number of
antennas, the ZF penalty and the effect of the ZF on the inter-cell
interference link can be neglected, as evaluated in [5], and (8d) follows a
similar approximation.

Remark 1: For notation simplicity, we assume that the transmitter at
$\mathbf{x}$ uses its $1$st subarray to communicate with the typical receiver
at $\mathbf{y}$. By adopting the orthogonality of the steering vector, given
the transmit steering vector
$\mathbf{a}_{\mathrm{T}}(\bar{\theta}_{\mathrm{T}}^{\mathbf{yx}},\bar{\phi}_{\mathrm{T}}^{\mathbf{yx}})$,
we have
$\mathbf{a}_{\mathrm{T}}(\bar{\theta}_{\mathrm{T}}^{\mathbf{yx}},\bar{\phi}_{\mathrm{T}}^{\mathbf{yx}})\mathbf{F}_{{\rm
RF}}^{\mathbf{x}}=\left[\tfrac{\sqrt{N_{\mathrm{T}}^{i}}}{N_{i}},0,\ldots,0\right]\in\mathbb{R}^{1\times
N_{i}}$ only occurs when the AoD for the typical link is different from that
for others served by the transmitter at $\mathbf{x}$, which has a probability
of $\left(1-\tfrac{1}{N_{\mathrm{T}}^{i}}\right)^{N_{i}-1}$ for
$p_{\mathrm{ZF}}(N_{\mathrm{T}}^{i},N_{i})=1$. Accordingly, we have
$\mathbf{f}_{{\rm
BB},1}^{\mathbf{x}}[k]=\left[\sqrt{\tfrac{N_{i}}{N_{\mathrm{T}}^{i}}},0,\ldots,0\right]^{T}$
satisfies $\left\|\mathbf{F}_{{\rm RF}}^{\mathbf{x}}\mathbf{f}_{{\rm
BB},1}^{\mathbf{x}}[k]\right\|_{F}^{2}=1$. For a large number of antennas,
other cases occur with very low probability, that is,
$1-\left(1-\tfrac{1}{N_{\mathrm{T}}^{i}}\right)^{N_{i}-1}$, for tractability,
the signal power is approximated to be 0 by setting
$p_{\mathrm{ZF}}(N_{\mathrm{T}}^{i},N_{i})=0$. Note that $i=\mathrm{m}$
denotes the transmitter at the gNB and $i=\mathrm{s}$ denotes that at the IAB-
node [28].

Proposition 1: The beamforming gain (including sidelobe gains) of the inter-
cell interference links is given by
$p_{\mathrm{BF}}(N_{\mathrm{T}}^{i},N_{\mathrm{R}}^{j},N_{i})$, which can be
chosen from one of the values with the corresponding probability of occurrence
in (9) shown on top of the next page,

$\displaystyle
p_{\mathrm{BF}}(N_{\mathrm{T}}^{i},N_{\mathrm{R}}^{j},N_{i})=\left\\{\begin{aligned}
&p+qg_{\mathrm{T},i}^{2}+(N_{i}-p-q){\hat{g}_{\mathrm{T},i}^{2}}\qquad\qquad\text{w.p.}\;\tfrac{1}{N_{\mathrm{R}}^{j}}C_{N_{i}}^{p}C_{N_{i}-p}^{q}\left(\tfrac{1}{N_{\mathrm{T}}^{i}}\right)^{p}\left(1-\tfrac{1}{N_{\mathrm{T,x}}^{i}}-\tfrac{1}{N_{\mathrm{T,y}}^{i}}+\tfrac{1}{N_{\mathrm{T}}^{i}}\right)^{q}\\\
&\;\qquad\;\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\left(\tfrac{1}{N_{\mathrm{T,x}}^{i}}+\tfrac{1}{N_{\mathrm{T,y}}^{i}}-\tfrac{2}{N_{\mathrm{T}}^{i}}\right)^{N_{i}-p-q}\\\
&g_{\mathrm{R},j}^{2}\left[p+qg_{\mathrm{T},i}^{2}+(N_{i}-p-q){\hat{g}_{\mathrm{T},i}}^{2}\right]\quad\;\text{w.p.}\;\left(1-\tfrac{1}{N_{\mathrm{R,x}}^{j}}-\tfrac{1}{N_{\mathrm{R,y}}^{j}}+\tfrac{1}{N_{\mathrm{R}}^{j}}\right)C_{N_{i}}^{p}C_{N_{i}-p}^{q}\left(\tfrac{1}{N_{\mathrm{T}}^{i}}\right)^{p}\\\
&\;\qquad\;\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\left(1-\tfrac{1}{N_{\mathrm{T,x}}^{i}}-\tfrac{1}{N_{\mathrm{T,y}}^{i}}+\tfrac{1}{N_{\mathrm{T}}^{i}}\right)^{q}\left(\tfrac{1}{N_{\mathrm{T,x}}^{i}}+\tfrac{1}{N_{\mathrm{T,y}}^{i}}-\tfrac{2}{N_{\mathrm{T}}^{i}}\right)^{N_{i}-p-q}\\\
&\hat{g}_{\mathrm{R},j}^{2}\left[p+qg_{\mathrm{T},i}^{2}+(N_{i}-p-q){\hat{g}_{\mathrm{T},i}}^{2}\right]\quad\;\text{w.p.}\;\left(\tfrac{1}{N_{\mathrm{R,x}}^{j}}+\tfrac{1}{N_{\mathrm{R,y}}^{j}}-\tfrac{2}{N_{\mathrm{R}}^{j}}\right)C_{N_{i}}^{p}C_{N_{i}-p}^{q}\left(\tfrac{1}{N_{\mathrm{T}}^{i}}\right)^{p}\\\
&\;\qquad\;\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\left(1-\tfrac{1}{N_{\mathrm{T,x}}^{i}}-\tfrac{1}{N_{\mathrm{T,y}}^{i}}+\tfrac{1}{N_{\mathrm{T}}^{i}}\right)^{q}\left(\tfrac{1}{N_{\mathrm{T,x}}^{i}}+\tfrac{1}{N_{\mathrm{T,y}}^{i}}-\tfrac{2}{N_{\mathrm{T}}^{i}}\right)^{N_{i}-p-q},\end{aligned}\right.$
(9)

for $p,q\in\mathbb{N}$, $p+q\leq{N_{i}}$, where $N_{\mathrm{T,x}}^{i}$ and
$N_{\mathrm{T,y}}^{i}$ ($N_{\mathrm{R,x}}^{j}$ and $N_{\mathrm{R,y}}^{j}$)
denote the number of antennas along $x$\- and $y$-axis of the transmit
(receive) UPA. $g_{\mathrm{T},i}$ and $\hat{g}_{\mathrm{T},i}$
($g_{\mathrm{R},j}$ and $\hat{g}_{\mathrm{R},j}$) are the sidelobe gains at
the transmitter (receiver). $i=\mathrm{m}$, $j=\mathrm{s}$; $i=\mathrm{s}$,
$j=\mathrm{s}$; $i=\mathrm{m}$, $j=\mathrm{u}$; and $i=\mathrm{s}$,
$j=\mathrm{u}$ denote the beamforming gain of the interference from gNBs to
the typical IAB-node, IAB-nodes to the typical IAB-node, gNBs to the typical
UE, and IAB-nodes to the typical UE, respectively.

###### Proof:

Assume squared UPA, the inner product between transmit steering vectors is
$\mathbf{a}_{\mathrm{T},i}^{H}(\bar{\theta}_{1},\bar{\phi}_{1})\mathbf{a}_{\mathrm{T},i}(\bar{\theta}_{2},\bar{\phi}_{2})=$

$\displaystyle\left\\{\begin{aligned}
\quad&1\quad\;\;\;\;\;\qquad\qquad\qquad\qquad\;\text{if}\;\bar{\theta}_{1}=\bar{\theta}_{2},\bar{\phi}_{1}=\bar{\phi}_{2}\\\
&g_{\mathrm{T},i}=\tfrac{1}{\sin^{2}(0.244)N_{\mathrm{T,x}}^{i}N_{\mathrm{T,y}}^{i}}\quad\text{if}\;\bar{\theta}_{1}\neq\bar{\theta}_{2},\bar{\phi}_{1}\neq\bar{\phi}_{2}\\\
&\hat{g}_{\mathrm{T},i}=\tfrac{1}{\sin(0.244)N_{\mathrm{T,x}}^{i}}\quad\qquad\text{otherwise},\end{aligned}\right.$
(10)

where the sidelobe gains $g_{\mathrm{T},i}$ and $\hat{g}_{\mathrm{T},i}$ are
derived according to [37, 5]. Similarly, we can model the inner product
between the receiver steering vectors with sidelobe gains $g_{\mathrm{R},j}$
and $\hat{g}_{\mathrm{R},j}$. Recall that the virtual angles are uniformly
distributed in their quantization sets, and according to the beamformer design
in Section III-B, this proposition can be easily proven. ∎

According to [38], the ADC quantization noise power can be approximated as

$\mathbb{E}\left\\{\left|e_{\mathrm{b}}[k]\right|^{2}\right\\}\approx\frac{1}{K}\sum_{k^{\prime}=0}^{K-1}\tfrac{G_{\mathrm{b}}[k^{\prime}]+\eta_{\mathrm{dig}}\tfrac{\eta
P_{\mathrm{s}}}{K}+I_{\mathrm{m,b}}[k^{\prime}]+I_{\mathrm{s,b}}[k^{\prime}]+\sigma_{\mathrm{n}}^{2}N^{\mathrm{s}}_{\mathrm{R}}}{1.5\cdot
2^{2q}},$ (11)

where $\eta_{\mathrm{dig}}>1$ is the amount of digital cancellation,
introduced to reproduce the RSI power before the ADC (i.e., after analog SIC)
because $\tfrac{\eta P_{\mathrm{s}}}{K}$ is the RSI power after the ADC (i.e.,
after digital SIC). $q$ denotes the ADC quantization bits. Owing to Assumption
1, the value of the numerator of (11) is the same for all subcarriers. Thus,
(11) can be simplified as

$\mathbb{E}\left\\{\left|e_{\mathrm{b}}[k]\right|^{2}\right\\}\approx\frac{G_{\mathrm{b}}[k]+\eta_{\mathrm{dig}}\tfrac{\eta
P_{\mathrm{s}}}{K}+I_{\mathrm{m,b}}[k]+I_{\mathrm{s,b}}[k]+\sigma_{\mathrm{n}}^{2}N^{\mathrm{s}}_{\mathrm{R}}}{1.5\cdot
2^{2q}}.$ (12)

####  IV-2 Access link

Next, assuming that gNB at $\mathbf{x}_{\mathrm{m}}$ uses its $u$th subarray
to communicate with the typical UE at $\mathbf{0}$, we derive the
$\mathrm{SINR}$ expression of the gNB-associated typical access link at
subcarrier $k$, which is cast as

$\displaystyle\mathrm{SINR}_{{\mathbf{0}}\mathbf{x}_{\mathrm{m}}}[k]=\tfrac{G_{\mathrm{a}}[k]}{I_{\mathrm{m,a}}[k]+I_{\mathrm{s,a}}[k]+\mathbb{E}\left\\{\left|\left(\mathbf{w}_{{\rm
RF}}^{{\mathbf{0}}}\right)^{H}\mathbf{n}_{\mathrm{a,m}}[k]\right|^{2}\right\\}+\mathbb{E}\left\\{\left|e_{\mathrm{a,m}}[k]\right|^{2}\right\\}},$
(13a) $\displaystyle
G_{\mathrm{a}}[k]=\frac{P_{\mathrm{m}}}{KN_{\mathrm{m}}}\left|\left(\mathbf{w}_{{\rm
RF}}^{{\mathbf{0}}}\right)^{H}\mathbf{H}_{{\mathbf{0}}\mathbf{x}_{\mathrm{m}}}[k]\mathbf{F}_{{\rm
RF}}^{\mathbf{x}_{\mathrm{m}}}\mathbf{f}_{{\rm
BB},u}^{\mathbf{x}_{\mathrm{m}}}[k]\right|^{2}$
$\displaystyle\qquad\;\approx\frac{P_{\mathrm{m}}N_{\mathrm{T}}^{\mathrm{m}}\left(N_{\mathrm{R}}^{\mathrm{u}}\right)^{2}\widetilde{h}^{{\mathbf{0}}\mathbf{x}_{\mathrm{m}}}p_{\mathrm{ZF}}(N_{\mathrm{T}}^{\mathrm{m}},N_{\mathrm{m}})}{KN^{2}_{\mathrm{m}}R_{\mathbf{x}_{\mathrm{m}}{\mathbf{0}}}^{\alpha_{\mathrm{m}}}},$
(13b) $\displaystyle
I_{\mathrm{m,a}}[k]=\frac{P_{\mathrm{m}}}{KN_{\mathrm{m}}}\left\lVert\sum\limits_{{\omega}\in\Phi_{\mathrm{m}}\backslash{\mathbf{x}_{\mathrm{m}}}}\left(\mathbf{w}_{{\rm
RF}}^{{\mathbf{0}}}\right)^{H}\mathbf{H}_{{\mathbf{0}}{\omega}}[k]\mathbf{F}_{{\rm
RF}}^{{\omega}}\mathbf{F}_{{\rm BB}}^{{\omega}}[k]\right\rVert^{2}$
$\displaystyle\qquad\;\;\;\approx\sum\limits_{\omega\in\Phi_{\mathrm{m}}\backslash{\mathbf{x}_{\mathrm{m}}}}\frac{P_{\mathrm{m}}N^{\mathrm{m}}_{\mathrm{T}}\left(N^{u}_{\mathrm{R}}\right)^{2}\widetilde{h}^{{\mathbf{0}}\omega}p_{\mathrm{BF}}(N^{\mathrm{m}}_{\mathrm{T}},N^{\mathrm{u}}_{\mathrm{R}},N_{\mathrm{m}})}{KN_{\mathrm{m}}R_{\omega{\mathbf{0}}}^{\alpha_{\mathrm{m}}}},$
(13c) $\displaystyle
I_{\mathrm{s,a}}[k]=\sum\limits_{i\in\\{\mathrm{L},\mathrm{N}\\}}I_{\mathrm{s,a}}^{i}[k]$
$\displaystyle\qquad\;=\frac{P_{\mathrm{s}}}{KN_{\mathrm{s}}}\left\lVert\sum\limits_{i\in\\{\mathrm{L},\mathrm{N}\\}}\sum\limits_{\hat{{\omega}}\in\Phi_{\mathrm{s},i}\backslash{{\mathbf{0}}}}\left(\mathbf{w}_{{\rm
RF}}^{{\mathbf{0}}}\right)^{H}\mathbf{H}_{{\mathbf{0}}\hat{\omega}}[k]\mathbf{F}_{{\rm
RF}}^{\hat{\omega}}\mathbf{F}_{{\rm BB}}^{\hat{\omega}}[k]\right\rVert^{2}$
$\displaystyle\qquad\;\approx\sum\limits_{i\in\\{\mathrm{L},\mathrm{N}\\}}\sum\limits_{\hat{\omega}\in\Phi_{\mathrm{s},i}\backslash{{\mathbf{0}}}}\frac{P_{\mathrm{s}}N^{\mathrm{s}}_{\mathrm{T}}\left(N^{\mathrm{u}}_{\mathrm{R}}\right)^{2}\widetilde{h}^{{\mathbf{0}}\hat{\omega}}p_{\mathrm{BF}}(N^{\mathrm{s}}_{\mathrm{T}},N^{\mathrm{u}}_{\mathrm{R}},N_{\mathrm{s}})}{KN_{\mathrm{s}}R_{\hat{\omega}{\mathbf{0}}}^{\alpha_{\mathrm{s},i}}},$
(13d)

where in (13a), $G_{\mathrm{a,m}}[k]$, $I_{\mathrm{m,a}}[k]$, and
$I_{\mathrm{s,a}}[k]$ denote the power of the desired signal, that of the
interference from non-target gNBs, and that of the interference from IAB-nodes
at subcarrier $k$, respectively;
$\mathbb{E}\left\\{\left|\left(\mathbf{w}_{{\rm
RF}}^{{\mathbf{0}}}\right)^{H}\mathbf{n}_{\mathrm{a,m}}[k]\right|^{2}\right\\}=\sigma_{\mathrm{n}}^{2}N^{\mathrm{u}}_{\mathrm{R}}$
is the Gaussian noise power with
$\mathbf{n}_{\mathrm{a,m}}[k]\sim\mathcal{CN}(\mathbf{0},\sigma_{\mathrm{n}}^{2}\mathbf{I}_{N^{\mathrm{u}}_{\mathrm{R}}})$;
$\mathbb{E}\left\\{\left|e_{\mathrm{a,m}}[k]\right|^{2}\right\\}$ represents
the ADC quantization noise power with a definition similar to that in (12). In
(13b)-(13d),
$\widetilde{h}^{\mathbf{0}\mathbf{x}_{\mathrm{m}}}=10^{-0.1(\beta+\chi_{\mathrm{m}})}\left|h^{\mathbf{0}\mathbf{x}_{\mathrm{m}}}\right|^{2}$,
$\widetilde{h}^{\mathbf{0}\omega}=10^{-0.1(\beta+\chi_{\mathrm{m}})}\left|h^{\mathbf{0}\omega}\right|^{2}$,
and
$\widetilde{h}^{\mathbf{0}\hat{\omega}}=10^{-0.1(\beta+\chi_{\mathrm{s},i})}\left|h^{\mathbf{0}\hat{\omega}}\right|^{2}$
follows from the composite GL distribution;
$p_{\mathrm{ZF}}(N_{\mathrm{T}}^{\mathrm{m}},N_{\mathrm{m}})$ is the ZF
penalty following Remark 1;
$p_{\mathrm{BF}}(N_{\mathrm{T}}^{\mathrm{m}},N_{\mathrm{R}}^{\mathrm{u}},N_{\mathrm{m}})$
and
$p_{\mathrm{BF}}(N_{\mathrm{T}}^{\mathrm{s}},N_{\mathrm{R}}^{\mathrm{u}},N_{\mathrm{s}})$
are beamforming gains according to Proposition 1.

Finally, assuming the IAB-node at $\mathbf{x}_{\mathrm{s}}^{j}$ with
$j\in\\{\mathrm{L},\mathrm{N}\\}$ uses its $v$th subcarrier to provide
communications to the typical UE at $\mathbf{0}$, the $\mathrm{SINR}$
expression of the IAB-node associated typical access link at the subcarrier
$k$ is expressed as

$\displaystyle\mathrm{SINR}_{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{j}}[k]=\tfrac{G_{\mathrm{a,s}}[k]}{\hat{I}_{\mathrm{m,a}}[k]+\hat{I}_{\mathrm{s,a}}[k]+\mathbb{E}\left\\{\left|\left(\mathbf{w}_{{\rm
RF}}^{\mathbf{0}}\right)^{H}\mathbf{n}_{\mathrm{a,s}}[k]\right|^{2}\right\\}+\mathbb{E}\left\\{\left|e_{\mathrm{a,s}}[k]\right|^{2}\right\\}}$
(14a) $\displaystyle
G_{\mathrm{a,s}}[k]=\frac{P_{\mathrm{s}}}{KN_{\mathrm{s}}}\left|\left(\mathbf{w}_{{\rm
RF}}^{\mathbf{0}}\right)^{H}\mathbf{H}_{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{j}}[k]\mathbf{F}_{{\rm
RF}}^{\mathbf{x}_{\mathrm{s}}^{j}}\mathbf{f}_{{\rm
BB},v}^{\mathbf{x}_{\mathrm{s}}^{j}}\right|^{2}$
$\displaystyle\quad\quad\;\;\;\approx\frac{P_{\mathrm{s}}N_{\mathrm{T}}^{\mathrm{s}}\left(N_{\mathrm{R}}^{\mathrm{u}}\right)^{2}\widetilde{h}^{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{j}}p_{\mathrm{ZF}}(N^{\mathrm{s}}_{\mathrm{T}},N_{\mathrm{s}})}{KN^{2}_{\mathrm{s}}R_{\mathbf{x}_{\mathrm{s}}^{j}\mathbf{0}}^{\alpha_{\mathrm{s},j}}},$
(14b)
$\displaystyle\hat{I}_{\mathrm{m,a}}[k]=\frac{P_{\mathrm{m}}}{KN_{\mathrm{m}}}\left\lVert\sum\limits_{\omega\in\Phi_{\mathrm{m}}}\left(\mathbf{w}_{{\rm
RF}}^{\mathbf{0}}\right)^{H}\mathbf{H}_{\mathbf{0}\omega}[k]\mathbf{F}_{{\rm
RF}}^{\omega}\mathbf{F}_{{\rm BB}}^{\omega}\right\rVert^{2}$
$\displaystyle\quad\quad\;\;\;\approx\sum\limits_{\omega\in\Phi_{\mathrm{m}}}\frac{P_{\mathrm{m}}N_{\mathrm{T}}^{\mathrm{m}}\left(N_{\mathrm{R}}^{\mathrm{u}}\right)^{2}\widetilde{h}^{\mathbf{0}\omega}p_{\mathrm{BF}}(N_{\mathrm{T}}^{\mathrm{m}},N_{\mathrm{R}}^{\mathrm{u}},N_{\mathrm{m}})}{KN_{\mathrm{m}}R_{\omega\mathbf{0}}^{\alpha_{\mathrm{m}}}},$
(14c)
$\displaystyle\hat{I}_{\mathrm{s,a}}[k]=\sum\limits_{i\in\\{\mathrm{L},\mathrm{N}\\}}\hat{I}_{\mathrm{s,a}}^{i}[k]$
$\displaystyle\quad\quad\;\;=\frac{P_{\mathrm{s}}}{KN_{\mathrm{s}}}\left\lVert\sum\limits_{i\in\\{\mathrm{L},\mathrm{N}\\}}\sum\limits_{\hat{\omega}\in\Phi_{\mathrm{s},i}\backslash{\mathbf{x}_{\mathrm{s}}^{j}}}\left(\mathbf{w}_{{\rm
RF}}^{\mathbf{0}}\right)^{H}\mathbf{H}_{\mathbf{0}\hat{\omega}}[k]\mathbf{F}_{{\rm
RF}}^{\hat{\omega}}\mathbf{F}_{{\rm BB}}^{\hat{\omega}}\right\rVert^{2}$
$\displaystyle\quad\quad\;\;\approx\sum\limits_{i\in\\{\mathrm{L},\mathrm{N}\\}}\sum\limits_{\hat{\omega}\in\Phi_{\mathrm{s},i}\backslash{\mathbf{x}_{\mathrm{s}}^{j}}}\frac{P_{\mathrm{s}}N_{\mathrm{T}}^{\mathrm{s}}\left(N_{\mathrm{R}}^{\mathrm{u}}\right)^{2}\widetilde{h}^{\mathbf{0}\hat{\omega}}p_{\mathrm{BF}}(N_{\mathrm{T}}^{\mathrm{s}},N_{\mathrm{R}}^{\mathrm{u}},N_{\mathrm{s}})}{KN_{\mathrm{s}}R_{\hat{\omega}\mathbf{0}}^{\alpha_{\mathrm{s},i}}},$
(14d)

where in (14a), $G_{\mathrm{a,s}}[k]$, $\hat{I}_{\mathrm{m,a}}[k]$, and
$\hat{I}_{\mathrm{s,a}}[k]$ denote the power of the desired signal, that of
the interference from gNBs, and that of the interference from non-target IAB-
nodes at subcarrier $k$, respectively;
$\mathbb{E}\left\\{\left|\left(\mathbf{w}_{{\rm
RF}}^{\mathbf{0}}\right)^{H}\mathbf{n}_{\mathrm{a,s}}[k]\right|^{2}\right\\}=\sigma_{\mathrm{n}}^{2}N^{\mathrm{u}}_{\mathrm{R}}$
is the Gaussian noise power with
$\mathbf{n}_{\mathrm{a,s}}[k]\sim\mathcal{CN}(\mathbf{0},\sigma_{\mathrm{n}}^{2}\mathbf{I}_{N^{\mathrm{u}}_{\mathrm{R}}})$;
$\mathbb{E}\left\\{\left|e_{\mathrm{a,s}}[k]\right|^{2}\right\\}$ represents
the ADC quantization noise power similar to (12). In (14b)-(14d),
$\widetilde{h}^{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{j}}=10^{-0.1(\beta+\chi_{\mathrm{s},j})}\left|h^{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{j}}\right|^{2}$,
$\widetilde{h}^{\mathbf{0}\omega}=10^{-0.1(\beta+\chi_{\mathrm{m}})}\left|h^{\mathbf{0}\omega}\right|^{2}$,
and
$\widetilde{h}^{\mathbf{0}\hat{\omega}}=10^{-0.1(\beta+\chi_{\mathrm{s},i})}\left|h^{\mathbf{0}\hat{\omega}}\right|^{2}$
are composite GL distributed random variables;
$p_{\mathrm{ZF}}(N^{\mathrm{s}}_{\mathrm{T}},N_{\mathrm{s}})$ is the ZF
penalty following Remark 1;
$p_{\mathrm{BF}}(N_{\mathrm{T}}^{\mathrm{m}},N_{\mathrm{R}}^{\mathrm{u}},N_{\mathrm{m}})$
and
$p_{\mathrm{BF}}(N_{\mathrm{T}}^{\mathrm{s}},N_{\mathrm{R}}^{\mathrm{u}},N_{\mathrm{s}})$
are beamforming gains following Proposition 1.

##  V Performance Analysis

For a standard OFDM system, owing to different channel gains occurring on
different subcarriers, the effective $\mathrm{SINR}$, whose statistics can be
obtained by Lognormal or Gaussian approximations, is used to derive the
performance metrics [39, 40], that is

$\displaystyle\mathrm{SINR}_{eff}=\widetilde{\vartheta}\mathrm{ln}\left(\frac{1}{K}\sum_{k=0}^{K}e^{\frac{\mathrm{SINR}[k]}{\widetilde{\vartheta}}}\right),$
(15)

where $\widetilde{\vartheta}$ is a parameter that depends on the coding scheme
and number of coded bits in a block. However, in this work, thanks to
Assumption 1, the value of $\mathrm{SINR}$ is the same for all subcarriers,
which means $\mathrm{SINR}[k]\;\forall\;k$ equals to the effective
$\mathrm{SINR}$. Therefore, $\mathrm{SINR}[k]$ was utilized for the analysis,
and the subcarrier index was omitted for simplicity.

###  V-A Association Probabilities

This subsection introduces the association probabilities; that is, the
probabilities of a typical UE served by a gNB and an IAB-node, based on the
maximum long-term averaged biased-received-desired-signal power (i.e.,
$\mathbb{E}\\{\text{received-desired-signal power}\\}\times\text{bias
factor}\;T_{i}$, for $i=\mathrm{m}$ if the desired signal comes from the gNB
and $i=\mathrm{s}$ if it comes from the IAB-node) criteria [20]. The bias
factor plays the role of offloading the traffic from gNB to IAB-node (or from
IAB-node to gNB) to satisfy network-wide performance requirement [41]. The
$\mathrm{SINR}$ coverage could be changed by tuning the bias ratio
$\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}$, as shown in Section VI.

To derive the association probabilities, in the following lemmas, we first
show the probability density function (PDF) and cumulative distribution
function (CDF) of the contact distance (i.e., the distance of a typical point
in a point process to its nearest point in an independent point process) for
three different cases:

  * (i)

The typical UE in PPP to its nearest LoS IAB-node in an independent PPP.

  * (ii)

The typical UE in PPP to its nearest NLoS IAB-node in an independent PPP.

  * (iii)

The typical UE in PPP to its nearest gNB in an independent MHCPP.

Lemma 1: The CDF of the contact distance from the PPP typical UE to the PPP
LoS/NLoS IAB-node with density
$\lambda_{\mathrm{s},\mathrm{L}}(x)/\lambda_{\mathrm{s},\mathrm{N}}(x)$ is
cast as

$\displaystyle
F_{R_{\mathbf{x}_{\mathrm{s}}^{i}\mathbf{0}}}(r_{\mathrm{s},i})=1-e^{-\int_{0}^{r_{\mathrm{s},i}}2\pi
x\lambda_{\mathrm{s},i}(x)\mathrm{d}x}.$ (16)

The corresponding PDF is written as

$\displaystyle
f_{R_{\mathbf{x}_{\mathrm{s}}^{i}\mathbf{0}}}(r_{\mathrm{s},i})=2\pi\lambda_{\mathrm{s},i}(r_{\mathrm{s},i})e^{-\int_{0}^{r_{\mathrm{s},i}}2\pi
x\lambda_{\mathrm{s},i}(x)\mathrm{d}x},$ (17)

where $r_{\mathrm{s},i}>0$ denotes the distance from the PPP typical UE to the
PPP LoS IAB-node for $i=\mathrm{L}$; and that to the PPP NLoS IAB-node for
$i=\mathrm{N}$.

###### Proof:

The similar proof can be found in [42]. ∎

Regarding the PDF and CDF of the PPP to MHCPP contact distance, it is
impossible to obtain exact expressions because of the correlations among the
MHCPP points. Fortunately, the authors in [43] provided a piece-wise model, as
shown in the following lemma, which has been verified to provide a very close
solution compared with the empirical data.

Lemma 2: The CDF of the contact distance from the PPP typical UE to the MHCPP
gNB with density $\lambda_{\mathrm{m}}$ and hard-core distance $\xi$ is
approximated as

$\displaystyle
F_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}}}(r_{\mathrm{m}})=\left\\{\begin{aligned}
F_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}},1}(r_{\mathrm{m}})&=\pi\lambda_{\mathrm{m}}r_{\mathrm{m}}^{2}\quad\;\;{0\leq
r_{\mathrm{m}}\leq\tfrac{\xi}{2}},\\\
F_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}},2}(r_{\mathrm{m}})&=1-\frac{4-\pi\lambda_{\mathrm{m}}\xi^{2}}{4}\\\
&\times
e^{\tfrac{2\pi\lambda_{\mathrm{m}}\xi^{2}\left[1-\left(\tfrac{2r_{\mathrm{m}}}{\xi}\right)^{\varrho}\right]}{\varrho\left(4-\pi\lambda_{\mathrm{m}}\xi^{2}\right)}}\\\
&\qquad\qquad\qquad\;\;\quad{r_{\mathrm{m}}>\tfrac{\xi}{2}}.\end{aligned}\right.$
(18)

Taking the first derivative with respect to $r_{\mathrm{m}}$, we obtain the
PDF of the contact distance as

$\displaystyle
f_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}}}(r_{\mathrm{m}})=\left\\{\begin{aligned}
f_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}},1}(r_{\mathrm{m}})&=2\pi\lambda_{\mathrm{m}}r_{\mathrm{m}}\quad\qquad\;{0\leq
r_{\mathrm{m}}\leq\tfrac{\xi}{2}},\\\
f_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}},2}(r_{\mathrm{m}})&=2\pi\lambda_{\mathrm{m}}r_{\mathrm{m}}\left(\tfrac{2r}{\xi}\right)^{\varrho-2}\\\
&\times
e{\tfrac{2\pi\lambda_{\mathrm{m}}\xi^{2}\left[1-\left(\tfrac{2r_{\mathrm{m}}}{\xi}\right)^{\varrho}\right]}{\varrho\left(4-\pi\lambda_{\mathrm{m}}\xi^{2}\right)}}\;\;{r_{\mathrm{m}}>\tfrac{\xi}{2}},\end{aligned}\right.$
(19)

where $r_{\mathrm{m}}>0$ denotes the distance from the PPP typical UE to MHCPP
gNB; $\varrho\approx
0.3686(\lambda_{\mathrm{m}}\pi\xi^{2})^{2}+0.0985\lambda_{\mathrm{m}}\pi\xi^{2}+2$
based on the simulation in [43].

###### Proof:

A detailed proof can be found in [43]. ∎

Given the statistics of the contact distance, we now derive the probabilities
of the typical UE associated with an LoS IAB-node, an NLoS IAB-node, and a gNB
in the following propositions.

Proposition 2: The probability of a typical UE associated with an LoS/NLoS
IAB-node at
$\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}$/$\mathbf{x}_{\mathrm{s}}^{\mathrm{N}}$,
denoted as
$\mathcal{A}_{\mathrm{s},\mathrm{L}}$/$\mathcal{A}_{\mathrm{s},\mathrm{N}}$,
is expressed as:

$\displaystyle\mathcal{A}_{\mathrm{s},i}$
$\displaystyle=\int_{0}^{\left(\tfrac{\xi}{2\Delta_{i,1}}\right)^{\tfrac{\alpha_{\mathrm{m}}}{\alpha_{\mathrm{s},i}}}}\left(1-F_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}},1}\left(r_{\mathrm{s},i}^{\tfrac{\alpha_{\mathrm{s},i}}{\alpha_{\mathrm{m}}}}\Delta_{i,1}\right)\right)$
$\displaystyle\times\left(1-F_{R_{\mathbf{x}_{\mathrm{s}}^{j}\mathbf{0}}}\left(r_{\mathrm{s},i}^{\tfrac{\alpha_{\mathrm{s},i}}{\alpha_{\mathrm{s},j}}}\Delta_{i,2}\right)\right)f_{R_{\mathbf{x}_{\mathrm{s}}^{i}\mathbf{0}}}(r_{\mathrm{s},i})\mathrm{d}r_{\mathrm{s},i}$
$\displaystyle+\int^{\infty}_{\left(\tfrac{\xi}{2\Delta_{i,1}}\right)^{\tfrac{\alpha_{\mathrm{m}}}{{\alpha_{\mathrm{s},i}}}}}\left(1-F_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}},2}\left(r_{\mathrm{s},i}^{\tfrac{\alpha_{\mathrm{s},i}}{\alpha_{\mathrm{m}}}}\Delta_{i,1}\right)\right)$
$\displaystyle\times\left(1-F_{R_{\mathbf{x}_{\mathrm{s}}^{j}\mathbf{0}}}\left(r_{\mathrm{s},i}^{\tfrac{\alpha_{\mathrm{s},i}}{\alpha_{\mathrm{s},j}}}\Delta_{i,2}\right)\right)f_{R_{\mathbf{x}_{\mathrm{s}}^{i}\mathbf{0}}}(r_{\mathrm{s},i})\mathrm{d}r_{\mathrm{s},i},$
(20)

where $i=\mathrm{L}$, $j=\mathrm{N}$; $i=\mathrm{N}$, $j=\mathrm{L}$.
$\Delta_{i,1}=\left(\tfrac{P_{\mathrm{m}}N_{\mathrm{T}}^{\mathrm{m}}N^{2}_{\mathrm{s}}T_{\mathrm{m}}}{P_{\mathrm{s}}N_{\mathrm{T}}^{\mathrm{s}}N_{\mathrm{m}}^{2}T_{\mathrm{s}}}\right)^{\tfrac{1}{\alpha_{\mathrm{m}}}}e^{{\tfrac{\left(0.1\zeta_{\mathrm{m}}\mathrm{ln}10\right)^{2}-\left(0.1\zeta_{\mathrm{s},i}\mathrm{ln}10\right)^{2}}{2\alpha_{\mathrm{m}}}}}$
and $\Delta_{i,2}=$
$e^{{\tfrac{\left(0.1\zeta_{\mathrm{s},j}\mathrm{ln}10\right)^{2}-\left(0.1\zeta_{\mathrm{s},i}\mathrm{ln}10\right)^{2}}{2\alpha_{\mathrm{s},j}}}}$.
The integration bound is given by solving
$r_{\mathrm{s},i}^{\tfrac{\alpha_{\mathrm{s},i}}{\alpha_{\mathrm{m}}}}\Delta_{i,1}=\tfrac{\xi}{2}$.

###### Proof:

The proof can be found in Appendix-B. ∎

Proposition 3: The probability of a typical UE associated with a gNB at
$\mathbf{x}_{\mathrm{m}}$, represented as $\mathcal{A}_{\mathrm{m}}$, is
expressed as:

$\displaystyle\mathcal{A}_{\mathrm{m}}=$
$\displaystyle\int_{0}^{\tfrac{\xi}{2}}\left(1-F_{R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}\mathbf{0}}}\left(r_{\mathrm{m}}^{\tfrac{\alpha_{\mathrm{m}}}{\alpha_{\mathrm{s},\mathrm{L}}}}\Delta_{\mathrm{L}}\right)\right)$
$\displaystyle\times\left(1-F_{R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{N}}\mathbf{0}}}\left(r_{\mathrm{m}}^{\tfrac{\alpha_{\mathrm{m}}}{\alpha_{\mathrm{s},\mathrm{N}}}}\Delta_{\mathrm{N}}\right)\right)f_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}},1}(r_{\mathrm{m}})\mathrm{d}r_{\mathrm{m}}$
$\displaystyle+\int^{\infty}_{\tfrac{\xi}{2}}\left(1-F_{R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}\mathbf{0}}}\left(r_{\mathrm{m}}^{\tfrac{\alpha_{\mathrm{m}}}{\alpha_{\mathrm{s},\mathrm{L}}}}\Delta_{\mathrm{L}}\right)\right)$
$\displaystyle\times\left(1-F_{R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{N}}\mathbf{0}}}\left(r_{\mathrm{m}}^{\tfrac{\alpha_{\mathrm{m}}}{\alpha_{\mathrm{s},\mathrm{N}}}}\Delta_{\mathrm{N}}\right)\right)f_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}},2}(r_{\mathrm{m}})\mathrm{d}r_{\mathrm{m}},$
(21)

where
$\Delta_{i}=\left(\tfrac{P_{\mathrm{s}}N_{\mathrm{T}}^{\mathrm{s}}N_{\mathrm{m}}^{2}T_{\mathrm{s}}}{P_{\mathrm{m}}N_{\mathrm{T}}^{\mathrm{m}}N_{\mathrm{s}}^{2}T_{\mathrm{m}}}\right)^{\tfrac{1}{\alpha_{\mathrm{s},i}}}e^{\tfrac{\left(0.1\zeta_{s,i}\mathrm{ln}10\right)^{2}-\left(0.1\zeta_{m}\mathrm{ln}10\right)^{2}}{2\alpha_{\mathrm{s},i}}}$
for $i\in\\{\mathrm{L},\mathrm{N}\\}$.

###### Proof:

The proof follows similar steps to Proposition 2 given the condition that the
nearest gNB provides higher biased-received-desired-signal power than the
nearest LoS and NLoS IAB-nodes. ∎

Having obtained the association probabilities, we can obtain the PDFs of the
serving distance (i.e., the distance of the typical UE to its serving gNB or
IAB-node) in the following proposition.

Proposition 4: Conditioned on the typical UE is served by its nearest gNB at
$\mathbf{x}_{\mathrm{m}}$, and the corresponding PDF of the serving distance
is written as

$\displaystyle\hat{f}_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}}}(r_{\mathrm{m}})$
$\displaystyle=\tfrac{1}{\mathcal{A}_{\mathrm{m}}}\left(1-F_{R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}\mathbf{0}}}\left(r_{\mathrm{m}}^{\tfrac{\alpha_{\mathrm{m}}}{\alpha_{\mathrm{s},\mathrm{L}}}}\Delta_{\mathrm{L}}\right)\right)$
$\displaystyle\times\left(1-F_{R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{N}}\mathbf{0}}}\left(r_{\mathrm{m}}^{\tfrac{\alpha_{\mathrm{m}}}{\alpha_{\mathrm{s},\mathrm{N}}}}\Delta_{\mathrm{N}}\right)\right)f_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}}}(r_{\mathrm{m}}),$
(22)

where $f_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}}}(r_{\mathrm{m}})$ is piece-
wise at point $r_{\mathrm{m}}=\tfrac{\xi}{2}$ as stated in Lemma 2.

Similarly, conditioned on the typical UE being served by its nearest LoS (or
NLoS) IAB-node at $\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}$ (or
$\mathbf{x}_{\mathrm{s}}^{\mathrm{N}}$), the corresponding PDF of the serving
distance is given as

$\displaystyle\hat{f}_{R_{\mathbf{x}_{\mathrm{s}}^{i}\mathbf{0}}}(r_{\mathrm{s},i})$
$\displaystyle=\tfrac{1}{\mathcal{A}_{\mathrm{s},i}}\left(1-F_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}}}\left(r_{\mathrm{s},i}^{\tfrac{\alpha_{\mathrm{s},i}}{\alpha_{\mathrm{m}}}}\Delta_{i,1}\right)\right)$
$\displaystyle\times\left(1-F_{R_{\mathbf{x}_{\mathrm{s}}^{j}\mathbf{0}}}\left(r_{\mathrm{s},i}^{\tfrac{\alpha_{\mathrm{s},i}}{\alpha_{\mathrm{s},j}}}\Delta_{i,2}\right)\right)f_{R_{\mathbf{x}_{\mathrm{s}}^{i}\mathbf{0}}}(r_{\mathrm{s},i}).$
(23)

When $i=\mathrm{L}$, we have $j=\mathrm{N}$; and when $i=\mathrm{N}$, we have
$j=\mathrm{L}$. According to Lemma2,
$F_{R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}}}\left(r_{\mathrm{s},i}^{\tfrac{\alpha_{\mathrm{s},i}}{\alpha_{\mathrm{m}}}}\Delta_{i,1}\right)$
is piece-wise at point
$r_{\mathrm{s},i}={\left(\tfrac{\xi}{2\Delta_{i,1}}\right)^{\tfrac{\alpha_{\mathrm{m}}}{\alpha_{\mathrm{s},i}}}}$
by solving
$r_{\mathrm{s},i}^{\tfrac{\alpha_{\mathrm{s},i}}{\alpha_{\mathrm{m}}}}\Delta_{i,1}=\tfrac{\xi}{2}$.

###### Proof:

A simple proof can be found in [16, Lemma 4]. ∎

In the remaining subsections, we derive some performance metrics, such as
$\mathrm{SINR}$ coverage, capacity with outage, and ergodic capacity for
single-hop backhaul mmWave-IBFD/HD-IAB networks. We set the available
bandwidth for IBFD communications as $W$. Assume that the HD transmission is
in a frequency-division-duplexing (FDD) manner, whose available bandwidth is
equally divided into two orthogonal portions (i.e., $W/2$): one for the gNB
tier and the other for the IAB-node tier.

###  V-B $\mathrm{SINR}$ Coverage

We first give the expression on the $\mathrm{SINR}$ coverage in the following
theorem.

Theorem 1: The $\mathrm{SINR}$ coverage of the single-hop backhaul mmWave-
IBFD-IAB network is defined as

$\displaystyle\mathcal{P}_{\mathrm{FD}}(\tau)$
$\displaystyle=\mathcal{A}_{\mathrm{m}}\mathbb{P}\left(\mathrm{SINR}_{\mathbf{0}\mathbf{x}_{\mathrm{m}}}>\tau\right)$
$\displaystyle+\sum_{i\in\\{\mathrm{L},\mathrm{N}\\}}\mathcal{A}_{\mathrm{s},i}\mathbb{P}\left(\mathrm{SINR}_{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{i}}>\tau,\mathrm{SINR}_{\mathbf{x}_{\mathrm{s}}^{i}\mathbf{x}_{\mathrm{m}}}>\tau\right),$
(24)

where $\tau$ denotes the target $\mathrm{SINR}$.
$\mathrm{SINR}_{\mathbf{x}_{\mathrm{s}}^{i}\mathbf{x}_{\mathrm{m}}}$ denotes
the backhaul link $\mathrm{SINR}$ of the IAB-node at
$\mathbf{x}_{\mathrm{s}}^{i}$ for serving a typical UE. However, solving
$\mathbb{P}\left(\mathrm{SINR}_{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{i}}>\tau,\mathrm{SINR}_{\mathbf{x}_{\mathrm{s}}^{i}\mathbf{x}_{\mathrm{m}}}>\tau\right)$
is intractable. Fortunately, since IAB-nodes and UEs are drawn from
homogeneous PPP, according to Slivnyak’s theorem, we have [16, 17]

$\displaystyle\mathbb{P}\left(\mathrm{SINR}_{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{i}}>\tau,\mathrm{SINR}_{\mathbf{x}_{\mathrm{s}}^{i}\mathbf{x}_{\mathrm{m}}}>\tau\right)$
$\displaystyle=\mathbb{P}\left(\mathrm{SINR}_{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{i}}>\tau\right)$
$\displaystyle\times\mathbb{P}\left(\mathrm{SINR}_{\hat{\mathbf{0}}\mathbf{x}_{\mathrm{m}}}>\tau\right).$
(25)

The solution of this theorem can be found in Appendix-D.

The $\mathrm{SINR}$ coverage of the single-hop backhaul mmWave-HD-IAB network
$\mathcal{P}_{\mathrm{HD}}(\tau)$ can be derived similarly to Theorem 1;
however, owing to the FDD, the interference from the other tier and the RSI in
the $\mathrm{SINR}$ expressions should be removed. Moreover, the Gaussian
noise power of HD is reduced by half compared to that of IBFD because the
bandwidth of IBFD is twice as large as that of HD.

###  V-C Capacity with Outage

In the following theorem, we derive the expression on the capacity with
outage.

Theorem 2: The capacity with outage, that is, the average rate correctly
received over many transmission bursts, of the single-hop backhaul mmWave-
IBFD-IAB network is cast as

$\displaystyle\mathcal{C}_{\mathrm{FD}}(\tau_{\mathrm{min}})=W\log_{2}\left(1+\tau_{\mathrm{min}}\right)\mathcal{P}_{\mathrm{FD}}(\tau_{\mathrm{min}}),$
(26)

where $\tau_{\mathrm{min}}$ is the minimum received $\mathrm{SINR}$ fixed at
the transmitter because the transmitter does not know the instantaneous
received $\mathrm{SINR}$; thus, the transmission rate should be fixed
independently.

Similarly, the capacity with outage of the single-hop backhaul mmWave-HD-IAB
network $\mathcal{C}_{\mathrm{HD}}(\tau_{\mathrm{min}})$ is given by replacing
$W$ with $W/2$ and $\mathcal{P}_{\mathrm{FD}}(\tau_{\mathrm{min}})$ with
$\mathcal{P}_{\mathrm{HD}}(\tau_{\mathrm{min}})$ in Theorem 2.

###  V-D Ergodic Capacity

The expression on the ergodic capacity is presented in the following theorem.

Theorem 3: The ergodic capacity for single-hop backhaul mmWave-IBFD-IAB
networks, defined as the average of the instantaneous capacity, is written as

$\displaystyle\mathcal{E}_{\mathrm{FD}}$
$\displaystyle=\mathcal{A}_{\mathrm{m}}{\bar{\mathcal{R}}_{\mathrm{m}}}+\mathcal{A}_{\mathrm{s},\mathrm{L}}\min\left({\bar{\mathcal{R}}_{\mathrm{s},\mathrm{L}}},{\bar{\mathcal{R}}_{\mathrm{b}}}\right)$
$\displaystyle+\mathcal{A}_{\mathrm{s},\mathrm{N}}\min\left({\bar{\mathcal{R}}_{\mathrm{s},\mathrm{N}}},{\bar{\mathcal{R}}_{\mathrm{b}}}\right),$
(27)

where
$\bar{\mathcal{R}}_{\mathrm{m}}=\mathbb{E}\\{W\log_{2}(1+\mathrm{SINR}_{\mathbf{0}\mathbf{x}_{\mathrm{m}}})\\}$,
$\bar{\mathcal{R}}_{\mathrm{s},\mathrm{L}}=\mathbb{E}\\{W\log_{2}(1+\mathrm{SINR}_{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}})\\}$,
$\bar{\mathcal{R}}_{\mathrm{s},\mathrm{N}}=\mathbb{E}\\{W\log_{2}(1+\mathrm{SINR}_{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{\mathrm{N}}})\\}$
and
$\bar{\mathcal{R}}_{\mathrm{b}}=\mathbb{E}\\{W\log_{2}(1+\mathrm{SINR}_{\hat{\mathbf{0}}\mathbf{x}_{\mathrm{m}}})\\}$
represent ergodic capacities of the gNB associated access link, LoS IAB-node
associated access link, NLoS IAB-node associated access link, and backhaul
link, respectively, which are derived in Appendix-E.

Likewise, the ergodic capacity of the corresponding HD network
$\mathcal{E}_{\mathrm{HD}}$ is given by scaling $W$ in Theorem 3 with $1/2$
and removing the RSI and interference from the other tier in $\mathrm{SINR}$
expressions.

Note that we choose the minimum ergodic capacity between the IAB-node
associated access link and the backhaul link as the final ergodic capacity at
the UE associated with the IAB-node. This is because, in the IAB system, the
maximum ergodic capacity that an IAB-node can provide is limited by its
corresponding backhaul link [17, 14].

TABLE II: System parameters and default values Notation | Physical Meaning | Values
---|---|---
$K$ | Number of subcarriers | 512 [44]
$W$ | Bandwidth | 800 MHz
$f_{\mathrm{c}}$ | Carrier frequency | 38 GHz
$M_{\mathrm{m}},M_{\mathrm{s},\mathrm{L}},M_{\mathrm{s},\mathrm{N}}$ | Nakagami factor | 4, 3, 2
$\sigma_{\mathrm{n}}^{2}$ | Gaussian noise power |  |  $-174\;\mathrm{dBm}+10\log_{10}W+10$ dB (IBFD)
---
$-174\;\mathrm{dBm}+10\log_{10}\tfrac{W}{2}+10$ dB (HD)
$\epsilon$ | Blockage density constant | 0.0071[19]
$R_{0}$ | Radius of analysis region | 1000m
$\xi$ | Hard-core distance | 100m
${\lambda}_{\mathrm{m}},\lambda_{\mathrm{s}},{\lambda}_{\mathrm{u}}$ | gNB, IAB-node, and UE density |  $1\times 10^{-5}/\mathrm{m}^{2}$, $4\times 10^{-5}/\mathrm{m}^{2}$, $2\times 10^{-4}/\mathrm{m}^{2}$
$\eta$ | RSI controlling factor |  $80$ dB [11]
$\eta_{\mathrm{dig}}$ | The amount of digital SIC |  $25$ dB [11]
$T$ | Gamma distribution samples |  $5$ [29]
${P_{\mathrm{m}}},{P_{\mathrm{s}}}$ | Transmit power |  $40$ dBm, $33$ dBm
${T_{\mathrm{s}}}/{T_{\mathrm{m}}}$ | Bias factor ratio | 0 dB
${N_{\mathrm{m}}},{N_{\mathrm{s}}}$ | Number of subarrays |  $8$, $4$
${N_{\mathrm{T}}^{\mathrm{m}}},{N_{\mathrm{T}}^{\mathrm{s}}}$ | Number of transmit antennas |  $32\times 32$, $16\times 16$
${N_{\mathrm{R}}^{\mathrm{s}}},{N_{\mathrm{R}}^{\mathrm{u}}}$ | Number of receive antennas |  $16\times 16$, $8\times 8$
$\alpha_{\mathrm{m}},\alpha_{\mathrm{s},\mathrm{L}},\alpha_{\mathrm{s},\mathrm{N}}$ | Path loss exponent | 1.9, 2, 3.3 [45]
$\zeta_{\mathrm{m}},\zeta_{\mathrm{s},\mathrm{L}},\zeta_{\mathrm{s},\mathrm{N}}$ | Shadowing effect standard deviation | 3.7, 4.3, 10.7 [45]
$q$ | ADC quantization bits | 8 bits
$\hat{g}_{\mathrm{T,m}},\hat{g}_{\mathrm{T,s}},{g}_{\mathrm{T,m}},{g}_{\mathrm{T,s}}$ | Transmitter sidelobe gain | -8.9 dB, -5.9 dB, -17.8 dB, -11.7 dB [37, 5]
$\hat{g}_{\mathrm{R,s}},\hat{g}_{\mathrm{R,u}},{g}_{\mathrm{R,s}},{g}_{\mathrm{R,u}}$ | Receiver sidelobe gain | -5.9 dB, -2.9 dB, -11.7 dB, -5.7 dB [37, 5]

##  VI Numerical Results

In this section, we verify the theoretical results derived in this study and
evaluate the performance of the single-hop backhaul mmWave-IBFD-IAB networks.
Furthermore, to better understand the advantages and disadvantages of IBFD
transmission, comparisons are made between IBFD and HD schemes. Monte Carlo
simulations with $10^{6}$ iterations are performed to verify the accuracy of
the derived theoretical results. Unless otherwise specified, the values of the
simulation parameters are presented in Table II.

###  VI-A Association Probabilities and $\mathrm{SINR}$ Coverage

Both figures in Fig. 2 show a close match between the simulation and
analytical results, which indicates good accuracy given by our approximated
analytical results derived in this study. In Fig. 2, we evaluate how the
association probabilities can be affected by changing the bias ratio of the
IAB-node to the gNB, that is, $\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}$. We can
find that the association probability of a typical UE associates with an IAB-
node, that is, $\mathcal{A}_{\mathrm{s}}$, increases as the ratio increases,
and that of the typical UE associates with a gNB, that is,
$\mathcal{A}_{\mathrm{m}}$, shows an opposite trend because as
$\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}$ increases, more traffic is offloaded
from the gNB to the IAB-node.

In Fig. 2, we study the impact of $\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}$ on
$\mathrm{SINR}$ coverage. The $\mathrm{SINR}$ coverage at a fixed threshold
first increases as $\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}$ increases; after
reaching the maximum at $\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}=0$ dB, the
$\mathrm{SINR}$ coverage decreases as the ratio increases. The reasons for
this decrease are as follows. As $\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}$
increases, $\mathcal{A}_{\mathrm{s}}$ becomes dominant, as shown in Fig. 2.
Thus, according to Theorem 1, the $\mathrm{SINR}$ coverage of the network can
be dominated by that of the IAB-node tier at a high bias ratio. Even if the
gNB is tower-mounted, it has a direct fiber connection but a limited quantity
owing to the high deployment cost of fiber-connected BSs. Biasing towards
wireless IAB-nodes has the potential benefit of reducing deployment costs.
However, because i) the transmit power and propagation properties of the IAB-
node are worse than those of the gNB; ii) the performance of the IAB-node
associated access link is restricted by the backhaul link, and iii) the
backhaul link suffers from RSI, the network coverage becomes distorted.
Moreover, it is interesting to note that the rate of increase in coverage is
much lower than the decrement rate.

Figure 2: Verification of theoretical results for (a) association
probabilities and (b) $\mathrm{SINR}$ coverage v.s. bias ratio
$\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}$ at $\mathrm{SINR}$ threshold
$\tau=0,5,10$ dB.

###  VI-B Capacity with Outage

We assume that the transmitter fixes the minimum received $\mathrm{SINR}$ at
$\tau_{\mathrm{min}}=0$ dB. In Fig. 3, we compare the capacity with outage of
single-hop backhaul mmWave-IBFD- and HD-IAB networks in terms of different
bias ratios $\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}$ with various values of
the RSI controlling factor $\eta$, which denotes the level of SIC (higher
values denote poorer SIC). In general, as
$\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}$ increases, the capacity with coverage
decreases, which can be explained by Theorem 2 and the analysis in previous
subsection. When $\eta>-50$ dB, the effect of the RSI on the capacity with
outage becomes obvious. However, when $\eta>-20$ dB, the capacity with outage
for different values of $\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}$ tends to be
stable. This is because a high RSI makes the $\mathrm{SINR}$ coverage of the
IAB-node tier approach 0; hence, the $\mathrm{SINR}$ coverage of the network
is nearly dominated by that of the gNB tier. In addition, a trade-off between
using IBFD and HD can be observed in the figure because the capacity with
outage ratio of IBFD to HD decreased as $\eta$ and/or
$\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}$ increased. In some circumstances, HD
can outperform the IBFD. For example, for
$\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}=20$ dB, HD provides a higher capacity
with outage than IBFD when $\eta>-40$ dB. Interestingly, with efficient SIC,
for $\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}=0$ dB, using IBFD transmission can
nearly double the capacity with outage compared to using HD; even if a high
RSI is reserved, the capacity with outage of IBFD still outperforms that of
HD. This is an attractive result because one can benefit from IBFD with high
RSI power by tuning $\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}$ rather than
exploring difficult and expensive SIC techniques to achieve higher levels of
SIC. In addition, the capacity with outage ratio of IBFD to HD decreased as
$\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}$ increased.

In Fig. 3, we also study how the capacity with outage of the single-hop
backhaul mmWave-IBFD-IAB network is affected when using an ADC with different
resolutions at various bias ratios at $\tau_{\mathrm{min}}=0$ dB. Evidently,
the capacity with outage can be improved by increasing the ADC quantization
bits and reducing the bias ratio. Meanwhile, the capacity with outage
saturates when the ADC resolution is higher than 5 bits, indicating that the
ADC quantization noise effect becomes negligible compared to other
interference and noise in our analyzed network. This result emphasizes the
feasibility of using a low-resolution ADC in a network. Low-resolution ADCs
are preferred in mmWave communications because their low power consumption can
compensate for the high power consumption imposed by numerous RF chains [46].

The simulation results for verifying the analytical results of capacity with
outage are omitted because they depend on the $\mathrm{SINR}$ coverage, which
shows a close match between the simulation and analytical results in Fig. 2.

Figure 3: Single-hop backhaul mmWave-IBFD-IAB networks capacity with outage at
a minimum received $\mathrm{SINR}=0$ dB (i.e., $\tau_{\mathrm{min}}=0$ dB) in
terms of $\tfrac{T_{\mathrm{s}}}{T_{\mathrm{m}}}=0,10,20$ dB (a) v.s.
different values of RSI factors; (b) v.s. different values of ADC quantization
bits.

###  VI-C Ergodic Capacity

Fig. 4 compares the network ergodic capacity with varying RSI factor $\eta$
and hard-core distance $\xi$ between single-hop backhaul FR2-IBFD- and HD-IAB
networks in terms of different density ratios of the IAB-node to gNB
$\tfrac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{m}}}$. Fig. 4 shows how the RSI
affects the network ergodic capacity. Note that when
$\tfrac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{m}}}$ increases, the effective
cell radius of the IAB-node tier decreases, resulting in a higher ergodic
capacity of the corresponding link. In contrast, the ergodic capacity of the
gNB tier is reduced because the UEs are more likely to be served by the IAB-
nodes as $\tfrac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{m}}}$ increases. When
$\eta\leq-50$ dB, the ergodic capacity increment of the IAB-node tier is
higher than the decrement of the gNB tier as the ratio increases. Thus, the
overall ergodic capacity increases as the ratio increases. However, for
$\eta>-50$ dB, the growth in the ergodic capacity of the IAB-node tier is less
than that of the gNB tier as the ratio increases; hence, the overall ergodic
capacity decreases as the ratio increases. This result indicates that with
successful SIC, the network ergodic capacity can be improved by increasing
$\tfrac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{m}}}$. Compared with HD
transmission, if $\eta\leq-50$ dB, the ergodic capacity provided by the IBFD
scheme is around 1.6 times higher than that provided by HD transmission,
regardless of the density ratio. However, in the high RSI factor $\eta$
region, the higher the density ratio, the lower the ergodic capacity provided
by the IBFD, and thus, the lower the ergodic capacity ratio of IBFD to HD
$\tfrac{\mathcal{E}_{\mathrm{FD}}}{\mathcal{E}_{\mathrm{HD}}}$ is. Moreover, a
similar trade-off between using HD and IBFD, as shown in Fig. 3, can be
concluded here, i.e., the ergodic capacity of HD can outperform IBFD for a
high $\tfrac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{m}}}$ and/or high $\eta$.
Since the close match between the simulation and analytical results is shown
in the figure, which indicates good accuracy given by our analytical results
derived for the network ergodic capacity.

In Fig. 4, by fixing the density of the gNB, the interference given by the
gNBs is suppressed by increasing the hard-core distance $\xi$, and as $\xi$
increases, the network ergodic capacity increases, which is the case for both
the IBFD and HD schemes. In particular, as $\xi$ increases, the IBFD ergodic
capacity gap between different node density ratios is reduced. Interestingly,
when $\xi$ is equal to 0, MHCPP converges to PPP, which yields the lowest
ergodic capacity. This result highlights the necessity of imposing a suitable
distance between the gNBs. Moreover, with successful SIC, IBFD provides a
higher ergodic capacity than HD, regardless of $\xi$ is. The simulation
results are omitted here because the accuracy of our analytical results on the
network ergodic capacity is verified in Fig. 4.

Figure 4: Single-hop backhaul mmWave-IBFD-IAB networks ergodic capacity in
terms of $\tfrac{\lambda_{\mathrm{s}}}{\lambda_{\mathrm{m}}}=4,6,8$
(${\lambda_{\mathrm{m}}}=1\times 10^{-5}/\mathrm{m}^{2}$) (a) v.s. different
RSI factor; (b) v.s. different hard-core distance.

###  VI-D Effect of Assumption 1

In Fig. 5, we simulate the ergodic capacity for different links per subcarrier
to analyze the effect of Assumption 1 by simulations and provide the
corresponding ergodic capacity for the multi-path scenario with three paths
for both the LoS and NLoS. For the multi-path scenario, we relax Assumption 1,
and the power of the desired signal is calculated based on Lemma 1 in [5].
This lemma states that for a large number of antennas, when the number of
paths is much smaller than the number of antennas, the power of the desired
signal converges to the power given by the strongest path, where the optimal
RF beamformers are given by the AoA and AoD of the strongest path. With
Assumption 1, the single path channel is assumed to exhibit flat fading, that
is, the channel gain is the same for all subcarriers. Therefore, we obtain a
constant ergodic capacity across all subcarriers. By relaxing Assumption 1,
$\gamma[k]$ in (4) varies for each subcarrier, which gives us the convex
curves in Fig. 5. It can be seen that Assumption 1 is a strong assumption,
which gives us an upper band on ergodic capacity. Without Assumption 1, the
performance analyzed in the previous subsections is degraded. Moreover, by
adding a multi-path to the picture, we can observe further degradation of
ergodic capacity. This is because multi-path propagation introduces additional
interference, which reduces the $\mathrm{SINR}$ compared with that in the
single path scenario. However, even if Assumption 1 is a strong assumption, it
provides tractability in stochastic geometry analysis for OFDM modulation,
because the statistics of raised cosine pulse shaping filters are still
unknown in the literature and are difficult to derive owing to the complex
impulse response of the filter [47, 48].

Figure 5: Ergodic capacity comparison for different links per subcarrier in
terms of single path and multi-path scenario.

##  VII Conclusion

In this study, the performance of multi-cell wideband single-hop backhaul
mmWave-IBFD-IAB networks using stochastic geometry, where the gNBs are
deployed in MHCPP, was investigated. For the Nakagami-$M$ small fading and
lognormal shadowing effect, we leveraged the approximated composite GL
distribution for network performance analysis. Moreover, for simplicity, the
PGFL of PPP was adopted to approximate the mean interference given by the
gNBs. Consequently, our closed-form performance metrics showed a close match
with the simulations. Furthermore, comparisons were made with the performance
of the single-hop backhaul mmWave-HD-IAB networks. The numerical results
demonstrated that tuning the bias ratio of the IAB-node to gNB, the
$\mathrm{SINR}$ coverage at a fixed $\mathrm{SINR}$ threshold shows a convex
trend. Interestingly, we found that by selecting a suitable bias ratio, the
IBFD scheme can always perform better than HD, even with a high RSI power,
relaxing the requirement for high-quality SIC techniques. Moreover, at
$\mathrm{SINR}=0$ dB, the effect of ADC quantization noise can be neglected at
a lower resolution. As for the network ergodic capacity, fixing the density of
gNB, the ergodic capacity was enhanced by deploying gNBs with MHCPP compared
to deploying gNBs with PPP, emphasizing the advantage of utilizing MHCPP.
Moreover, the larger the hard-core distance, the higher the ergodic capacity.
Additionally, with a small RSI power, the ergodic capacity can be improved by
increasing the ratio of the density of the IAB-node to gNB.

Further work can be focused on determing the transformation properties of the
MHCPP and deriving the system performance with the impact of hardware
impairments and channel estimation error. Moreover, the design and analysis of
a multi-hop backhaul scenario are worth studying.

##  Appendix A Composite Gamma-Lognormal Distribution

Given that $X\sim\mathrm{Gamma}(M,1/M)$ and
$Y\sim\mathrm{ln}\mathcal{N}(\hat{\mu},\hat{\sigma}^{2})$ are independent
Gamma and Lognormal distributed random variables, respectively, then $Z=XY\sim
GL(M,\tfrac{1}{M},\hat{\mu},\hat{\sigma}^{2})$ is defined as a composite GL
distributed random variable with
$\mathbb{E}\\{Z\\}=e^{\hat{\mu}+0.5\hat{\sigma}^{2}}$. As the exact PDF is
intractable for deriving the corresponding CDF and Laplace transform, a good
approximation is provided by [29, Lemma 1].

The corresponding approximated CDF is given as

$\displaystyle
F_{Z}(z)\approx\tfrac{1}{\Gamma(M)\sum_{t=1}^{T}w_{t}}\sum_{t=1}^{T}w_{t}\gamma\left(M,zMe^{-(\sqrt{2}\hat{\sigma}\nu_{t}+\hat{\mu})}\right),$
(28)

where $\gamma(s,x)=\int_{0}^{x}t^{s-1}e^{-t}\mathrm{d}t$ is the incomplete
gamma function. $T$, $w_{t}$, and $\nu_{t}$ are the number of gamma
distribution samples, weight, and abscissas factors for the Gaussian-Hermite
integration, respectively, whose values can be obtained from [49].

The Laplace transform is approximated as

$\displaystyle\mathscr{L}_{Z}(s)=\mathbb{E}\left\\{e^{-sz}\right\\}\approx\tfrac{1}{\sum_{t=1}^{T}w_{t}}\sum_{t=1}^{T}\tfrac{w_{t}M^{M}e^{-M(\sqrt{2}\hat{\sigma}\nu_{t}+\hat{\mu})}}{\left(s+Me^{-(\sqrt{2}\hat{\sigma}\nu_{t}+\hat{\mu})}\right)^{M}}.$
(29)

$\displaystyle\mathbb{E}\left\\{e^{-\mathcal{G}_{\mathrm{s},t}r_{\mathrm{s,L}}^{\alpha_{\mathrm{s},\mathrm{L}}}\hat{I}_{\mathrm{s,a}}^{\mathrm{L}}}\right\\}\overset{(c)}{\approx}\mathbb{E}\left\\{\prod\limits_{r_{\mathrm{a,\mathrm{L}}}>r_{\mathrm{s,L}}}\tfrac{1}{\sum_{p=1}^{T}w_{p}}\sum_{p=1}^{T}\tfrac{w_{p}M_{\mathrm{s},\mathrm{L}}^{M_{\mathrm{s},\mathrm{L}}}e^{-M_{\mathrm{s},\mathrm{L}}(\sqrt{2}\hat{\sigma}_{\mathrm{s},\mathrm{L}}\nu_{p}+\hat{\mu})}}{\left(\mathcal{G}_{\mathrm{s},t}\tfrac{P_{\mathrm{s}}N^{\mathrm{s}}_{\mathrm{T}}\left(N^{\mathrm{u}}_{\mathrm{R}}\right)^{2}r_{\mathrm{s,L}}^{\alpha_{\mathrm{s},\mathrm{L}}}}{KN_{\mathrm{s}}r^{\alpha_{\mathrm{s},\mathrm{L}}}_{\mathrm{a,L}}}p_{\mathrm{BF}}(N^{\mathrm{s}}_{\mathrm{T}},N^{\mathrm{u}}_{\mathrm{R}},N_{\mathrm{s}})+M_{\mathrm{s},\mathrm{L}}e^{-(\sqrt{2}\hat{\sigma}_{\mathrm{s},\mathrm{L}}\nu_{p}+\hat{\mu})}\right)^{M_{\mathrm{s},\mathrm{L}}}}\right\\}$
$\displaystyle\overset{(d)}{=}e^{-2\pi\int_{r_{\mathrm{s,L}}}^{R_{0}}\lambda_{\mathrm{s},\mathrm{L}}(r_{\mathrm{a,\mathrm{L}}})r_{\mathrm{a,\mathrm{L}}}\left[1-\tfrac{1}{\sum_{p=1}^{T}w_{p}}\sum_{p=1}^{T}\tfrac{w_{p}M_{\mathrm{s},\mathrm{L}}^{M_{\mathrm{s},\mathrm{L}}}e^{-M_{\mathrm{s},\mathrm{L}}(\sqrt{2}\hat{\sigma}_{\mathrm{s},\mathrm{L}}\nu_{p}+\hat{\mu})}}{\left(\mathcal{G}_{\mathrm{s},t}\tfrac{P_{\mathrm{s}}N^{\mathrm{s}}_{\mathrm{T}}\left(N^{\mathrm{u}}_{\mathrm{R}}\right)^{2}r_{\mathrm{s,L}}^{\alpha_{\mathrm{s},\mathrm{L}}}}{KN_{\mathrm{s}}r^{\alpha_{\mathrm{s},\mathrm{L}}}_{\mathrm{a,L}}}p_{\mathrm{BF}}(N^{\mathrm{s}}_{\mathrm{T}},N^{\mathrm{u}}_{\mathrm{R}},N_{\mathrm{s}})+M_{\mathrm{s},\mathrm{L}}e^{-(\sqrt{2}\hat{\sigma}_{\mathrm{s},\mathrm{L}}\nu_{p}+\hat{\mu})}\right)^{M_{\mathrm{s},\mathrm{L}}}}\right]\mathrm{d}r_{\mathrm{a,\mathrm{L}}}}$
$\displaystyle\overset{(e)}{=}\prod\limits_{i\in\atop
p_{\mathrm{BF}}(N^{\mathrm{s}}_{\mathrm{T}},N^{\mathrm{u}}_{\mathrm{R}},N_{\mathrm{s}})}e^{-2\pi\int_{r_{\mathrm{s,L}}}^{R_{0}}\lambda_{\mathrm{s},\mathrm{L}}(r_{\mathrm{a,\mathrm{L}}})r_{\mathrm{a,\mathrm{L}}}\left[1-\tfrac{1}{\sum_{p=1}^{T}w_{p}}\sum_{p=1}^{T}\tfrac{w_{p}M_{\mathrm{s},\mathrm{L}}^{M_{\mathrm{s},\mathrm{L}}}e^{-M_{\mathrm{s},\mathrm{L}}(\sqrt{2}\hat{\sigma}_{\mathrm{s},\mathrm{L}}\nu_{p}+\hat{\mu})}}{\left(\mathcal{G}_{\mathrm{s},t}\tfrac{P_{\mathrm{s}}N^{\mathrm{s}}_{\mathrm{T}}\left(N^{\mathrm{u}}_{\mathrm{R}}\right)^{2}r_{\mathrm{s,L}}^{\alpha_{\mathrm{s},\mathrm{L}}}}{KN_{\mathrm{s}}r^{\alpha_{\mathrm{s},\mathrm{L}}}_{\mathrm{a,L}}}b_{i}+M_{\mathrm{s},\mathrm{L}}e^{-(\sqrt{2}\hat{\sigma}_{\mathrm{s},\mathrm{L}}\nu_{p}+\hat{\mu})}\right)^{M_{\mathrm{s},\mathrm{L}}}}\right]c_{i}\mathrm{d}r_{\mathrm{a,\mathrm{L}}}},$
(33)

##  Appendix B Proof of Proposition 2

A typical UE is associated with its nearest LoS IAB-node rather than the
nearest gNB and NLoS IAB-node if the nearest LoS IAB-node can provide the
highest long-term averaged biased-received-desired-signal power, given as

$\displaystyle\mathcal{A}_{\mathrm{s},\mathrm{L}}=\mathbb{P}\left(\tfrac{P_{\mathrm{s}}N_{\mathrm{T}}^{\mathrm{s}}\left(N_{\mathrm{R}}^{\mathrm{u}}\right)^{2}T_{\mathrm{s}}}{KN^{2}_{\mathrm{s}}}\mathbb{E}\left\\{\widetilde{h}^{\mathbf{0}\mathbf{x}^{\mathrm{L}}_{\mathrm{s}}}\right\\}R_{\mathbf{x}^{\mathrm{L}}_{\mathrm{s}}\mathbf{0}}^{-\alpha_{\mathrm{s},\mathrm{L}}}\geq\tfrac{P_{\mathrm{m}}N_{\mathrm{T}}^{\mathrm{m}}\left(N_{\mathrm{R}}^{\mathrm{u}}\right)^{2}T_{\mathrm{m}}}{KN_{\mathrm{m}}^{2}}\right.$
$\displaystyle\;\quad\;\;\quad\times\mathbb{E}\left\\{\widetilde{h}^{\mathbf{0}\mathbf{x}_{\mathrm{m}}}\right\\}R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}}^{-\alpha_{\mathrm{m}}}\bigcap\tfrac{P_{\mathrm{s}}N^{\mathrm{s}}_{\mathrm{T}}\left(N_{\mathrm{R}}^{\mathrm{u}}\right)^{2}T_{\mathrm{s}}}{KN^{2}_{\mathrm{s}}}\mathbb{E}\left\\{\widetilde{h}^{\mathbf{0}\mathbf{x}^{\mathrm{L}}_{\mathrm{s}}}\right\\}R_{\mathbf{x}^{\mathrm{L}}_{\mathrm{s}}\mathbf{0}}^{-\alpha_{\mathrm{s},\mathrm{L}}}$
$\displaystyle\left.\;\quad\;\;\quad\geq\tfrac{P_{\mathrm{s}}N_{\mathrm{T}}^{\mathrm{s}}\left(N_{\mathrm{R}}^{\mathrm{u}}\right)^{2}T_{\mathrm{s}}}{KN^{2}_{\mathrm{s}}}\mathbb{E}\left\\{\widetilde{h}^{\mathbf{0}\mathbf{x}^{\mathrm{N}}_{\mathrm{s}}}\right\\}R_{\mathbf{x}^{\mathrm{N}}_{\mathrm{s}}\mathbf{0}}^{-\alpha_{\mathrm{s},\mathrm{N}}}\right)$
$\displaystyle\overset{(a)}{=}\mathbb{P}\left(\tfrac{P_{\mathrm{s}}N^{\mathrm{s}}_{\mathrm{T}}\left(N_{\mathrm{R}}^{\mathrm{u}}\right)^{2}T_{\mathrm{s}}}{KN^{2}_{\mathrm{s}}}e^{-0.1\beta\mathrm{ln}10+\tfrac{(0.1\zeta_{\mathrm{s},\mathrm{L}}\mathrm{ln}10)^{2}}{2}}R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}\mathbf{0}}^{-\alpha_{\mathrm{s},\mathrm{L}}}\right.\geq
R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}}^{-\alpha_{\mathrm{m}}}$
$\displaystyle\;\quad\times\tfrac{P_{\mathrm{m}}N^{\mathrm{m}}_{\mathrm{T}}\left(N_{\mathrm{R}}^{\mathrm{u}}\right)^{2}T_{\mathrm{m}}}{KN_{\mathrm{m}}^{2}}e^{-0.1\beta\mathrm{ln}10+\tfrac{(0.1\zeta_{\mathrm{m}}\mathrm{ln}10)^{2}}{2}}\bigcap\tfrac{P_{\mathrm{s}}N^{\mathrm{s}}_{\mathrm{T}}\left(N_{\mathrm{R}}^{\mathrm{u}}\right)^{2}T_{\mathrm{s}}}{KN^{2}_{\mathrm{s}}}$
$\displaystyle\quad\;\times
e^{-0.1\beta\mathrm{ln}10+\tfrac{(0.1\zeta_{\mathrm{s},\mathrm{L}}\mathrm{ln}10)^{2}}{2}}R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}\mathbf{0}}^{-\alpha_{\mathrm{s},\mathrm{L}}}\geq\tfrac{P_{\mathrm{s}}N^{\mathrm{s}}_{\mathrm{T}}\left(N_{\mathrm{R}}^{\mathrm{u}}\right)^{2}T_{\mathrm{s}}}{KN^{2}_{\mathrm{s}}}R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{N}}\mathbf{0}}^{-\alpha_{\mathrm{s},\mathrm{N}}}$
$\displaystyle\quad\;\left.\times
e^{-0.1\beta\mathrm{ln}10+\tfrac{(0.1\zeta_{\mathrm{s},\mathrm{N}}\mathrm{ln}10)^{2}}{2}}\right)$
$\displaystyle\overset{(b)}{=}\int_{0}^{\infty}\mathbb{P}\left(R_{\mathbf{x}_{\mathrm{m}}\mathbf{0}}\geq
r_{\mathrm{s},\mathrm{L}}^{\tfrac{\alpha_{\mathrm{s},\mathrm{L}}}{\alpha_{\mathrm{m}}}}\Delta_{\mathrm{L},1}\right)\mathbb{P}\left(R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{N}}\mathbf{0}}\geq
r_{\mathrm{s},\mathrm{L}}^{\tfrac{\alpha_{\mathrm{s},\mathrm{L}}}{\alpha_{\mathrm{s},\mathrm{N}}}}\Delta_{\mathrm{L},2}\right)$
$\displaystyle\quad\;\times
f_{R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}\mathbf{0}}}(r_{\mathrm{s},\mathrm{L}})\mathrm{d}r_{\mathrm{s},\mathrm{L}},$
(1)

where $(a)$ is derived by substituting the mean power of the composite fading.
$(b)$ is obtained by taking the average of
$R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}\mathbf{0}}=r_{\mathrm{s},\mathrm{L}}$.
By using Lemma 1 and Lemma 2, we can complete the proof. The proof of
$\mathcal{A}_{\mathrm{s},\mathrm{N}}$ can be provided in a similar manner.

##  Appendix C A Useful Lemma

A useful lemma shown below can support the derivation of $\mathrm{SINR}$
coverage. With the following lemma, one can easily deploy the PGFL of PPP to
estimate the mean interference from gNBs in the backhaul link and the gNB-
associated access link.

Lemma 3: Given the density of the MHCPP gNB as $\lambda_{\mathrm{m}}$ with
hard-core distance $\xi$, it can be approximated by thinning the density of
its parent PPP $\widetilde{\lambda}_{\mathrm{m}}$ with
$\rho_{\mathrm{M}}(r_{0})$, where $\rho_{\mathrm{M}}(r_{0})$ is the
conditional thinning Palm probability with $r_{0}>0$ being the distance
between the two gNBs, which is expressed as

$\displaystyle\rho_{\mathrm{M}}(r_{0})=\left\\{\begin{aligned}
&\tfrac{2}{\widetilde{\lambda}_{\mathrm{m}}(\kappa-\pi\xi^{2})}\left[1-\tfrac{\pi\xi^{2}\widetilde{\lambda}_{\mathrm{m}}\left(1-e^{-\widetilde{\lambda}_{\mathrm{m}}\kappa}\right)}{\widetilde{\lambda}_{\mathrm{m}}\kappa\left(1-e^{-\pi\xi^{2}\widetilde{\lambda}_{\mathrm{m}}}\right)}\right]\;\xi\leq
r_{0}<2\xi\\\ &\rho\quad\;r_{0}\geq 2\xi\\\
&0\quad\text{otherwise},\end{aligned}\right.$ (31)

with
$\kappa=2\pi\xi^{2}-2\xi^{2}cos^{-1}\left(\tfrac{r_{0}}{2\xi}\right)+r_{0}\sqrt{\xi^{2}-\tfrac{r_{0}^{2}}{4}}$.

###### Proof:

A proof can be found in [23]. ∎

##  Appendix D Solution of Theorem 1

###  D-A Solution of
$\mathbb{P}\left(\mathrm{SINR}_{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}}>\tau\right)$

The $\mathrm{SINR}$ coverage for the LoS IAB-node associated access link is
expressed as

$\displaystyle\mathbb{P}\left(\mathrm{SINR}_{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}}>\tau\right)=\left(1-\tfrac{1}{N^{\mathrm{s}}_{\mathrm{T}}}\right)^{N_{\mathrm{s}}-1}\int_{0}^{R_{0}}\mathbb{P}\left(\widetilde{h}^{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}}>\mathcal{G}_{\mathrm{s}}\right.$
$\displaystyle\left.\quad\;\times\underbrace{r_{\mathrm{s,L}}^{\alpha_{\mathrm{s},\mathrm{L}}}\left(\hat{I}_{\mathrm{m,a}}+\hat{I}_{\mathrm{s,a}}^{\mathrm{L}}+\hat{I}_{\mathrm{s,a}}^{\mathrm{N}}+\sigma_{\mathrm{n}}^{2}N^{\mathrm{u}}_{\mathrm{R}}\right)}_{I}\right)\hat{f}_{R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}\mathbf{0}}}(r_{\mathrm{s,L}})\mathrm{d}r_{\mathrm{s,L}}$
$\displaystyle\overset{(a)}{\approx}\left(1-\tfrac{1}{N^{\mathrm{s}}_{\mathrm{T}}}\right)^{N_{\mathrm{s}}-1}\int_{0}^{R_{0}}\left[1-\mathbb{E}\left\\{\tfrac{1}{\Gamma(M_{\mathrm{s},\mathrm{L}})\sum_{t=1}^{T}w_{t}}\sum_{t=1}^{T}w_{t}\right.\right.$
$\displaystyle\quad\;\times\left.\left.\gamma\left(M_{\mathrm{s},\mathrm{L}},{\mathcal{G}_{\mathrm{s},t}I}\right)\right\\}\right]\hat{f}_{R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}\mathbf{0}}}(r_{\mathrm{s,L}})\mathrm{d}r_{\mathrm{s,L}}$
$\displaystyle\overset{(b)}{=}\left(1-\tfrac{1}{N^{\mathrm{s}}_{\mathrm{T}}}\right)^{N_{\mathrm{s}}-1}\sum_{t=1}^{T}\sum_{n=0}^{M_{\mathrm{s},\mathrm{L}}-1}\frac{w_{t}}{n!\sum_{t=1}^{T}w_{t}}$
$\displaystyle\quad\;\times\int_{0}^{R_{0}}\mathbb{E}\left\\{\left(\mathcal{G}_{\mathrm{s},t}I\right)^{n}e^{-\mathcal{G}_{\mathrm{s},t}I}\right\\}\hat{f}_{R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}\mathbf{0}}}(r_{\mathrm{s,L}})\mathrm{d}r_{\mathrm{s,L}}$
$\displaystyle=\left(1-\tfrac{1}{N^{\mathrm{s}}_{\mathrm{T}}}\right)^{N_{\mathrm{s}}-1}\sum_{t=1}^{T}\sum_{n=0}^{M_{\mathrm{s},\mathrm{L}}-1}\frac{w_{t}(-\mathcal{G}_{\mathrm{s},t})^{n}}{n!\sum_{t=1}^{T}w_{t}}\frac{\mathrm{d}^{n}}{\mathrm{d}\mathcal{G}_{\mathrm{s},t}^{n}}$
$\displaystyle\quad\;\times\left(\int_{0}^{R_{0}}\mathbb{E}\left\\{e^{-\mathcal{G}_{\mathrm{s},t}r_{\mathrm{s,L}}^{\alpha_{\mathrm{s},\mathrm{L}}}\hat{I}_{\mathrm{s,a}}^{\mathrm{L}}}\right\\}\mathbb{E}\left\\{e^{-\mathcal{G}_{\mathrm{s},t}r_{\mathrm{s,L}}^{\alpha_{\mathrm{s},\mathrm{L}}}\hat{I}_{\mathrm{s,a}}^{\mathrm{N}}}\right\\}\right.$
$\displaystyle\quad\;\left.\times\mathbb{E}\left\\{e^{-\mathcal{G}_{\mathrm{s},t}r_{\mathrm{s,L}}^{\alpha_{\mathrm{s},\mathrm{L}}}\hat{I}_{\mathrm{m,a}}}\right\\}e^{-\mathcal{G}_{\mathrm{s},t}r_{\mathrm{s,L}}^{\alpha_{\mathrm{s},\mathrm{L}}}\sigma_{\mathrm{n}}^{2}N^{\mathrm{u}}_{\mathrm{R}}}\hat{f}_{R_{\mathbf{x}_{\mathrm{s}}^{\mathrm{L}}\mathbf{0}}}(r_{\mathrm{s,L}})\mathrm{d}r_{\mathrm{s,L}}\right),$
(32)

where $\mathcal{G}_{\mathrm{s}}=\tfrac{\tau
KN_{\mathrm{s}}^{2}\left(1+\tfrac{1}{1.5\cdot
2^{2q}}\right)}{P_{\mathrm{s}}N^{\mathrm{s}}_{\mathrm{T}}\left(N^{\mathrm{u}}_{\mathrm{R}}\right)^{2}\left(1-\tfrac{\tau}{1.5\cdot
2^{2q}}\right)}$.
$\mathcal{G}_{\mathrm{s},t}=\mathcal{G}_{\mathrm{s}}M_{\mathrm{s},\mathrm{L}}e^{-(\sqrt{2}\hat{\sigma}_{\mathrm{s},\mathrm{L}}\nu_{t}+\hat{\mu})}$
with $\hat{\mu}=-0.1\beta\mathrm{ln}10$ and
$\hat{\sigma}_{\mathrm{s},\mathrm{L}}=0.1\zeta_{\mathrm{s},\mathrm{L}}\mathrm{ln}10$.
$(a)$ is derived from the approximated CDF of the composite GL distribution
and $(b)$ can be referred to [50, Eq.(8.4.8)].

Next, we present the mean interference from LoS IAB-nodes in (33) shown at the
top of the page, where $(c)$ is derived according to the approximated Laplace
transform of the composite GL distribution. After applying the PGFL of PPP, we
obtain $(d)$. In $(e)$, we explore the independence of different beamforming
gain scenarios, $b_{i}$ takes one of the beamforming gains from (9), and
$c_{i}$ takes the corresponding probability.

Then, the mean interference from NLoS IAB-nodes
$\mathbb{E}\left\\{e^{-\mathcal{G}_{\mathrm{s},t}r_{\mathrm{s,L}}^{\alpha_{\mathrm{s},\mathrm{L}}}\hat{I}_{\mathrm{s,a}}^{\mathrm{N}}}\right\\}$
can be obtained by a similar step in (33) with integration from
${r_{\mathrm{s,L}}^{\tfrac{\alpha_{\mathrm{s},\mathrm{L}}}{\alpha_{\mathrm{s},\mathrm{N}}}}\Delta_{\mathrm{L},2}}$
to ${R_{0}}$.

Finally, we derive the mean interference from the gNBs. Because of the lack of
knowledge of the PGFL of MHCPP, we assume gNBs as a virtual PPP with a density
of ${\lambda}_{\mathrm{m}}$ to approximate the mean interference, which has
been verified to provide a good approximation in [31]. Consequently, with the
help of the PGFL of PPP, we have (34) shown at the top of the next page,

$\displaystyle\mathbb{E}\left\\{e^{-\mathcal{G}_{\mathrm{s},t}r_{\mathrm{s,L}}^{\alpha_{\mathrm{s},\mathrm{L}}}\hat{I}_{\mathrm{m,a}}}\right\\}\approx\prod\limits_{i\in\atop
p_{\mathrm{BF}}(N^{\mathrm{m}}_{\mathrm{T}},N^{\mathrm{u}}_{\mathrm{R}},N_{\mathrm{m}})}e^{-2\pi\int_{r_{\mathrm{s,L}}^{\tfrac{\alpha_{\mathrm{s},\mathrm{L}}}{\alpha_{\mathrm{m}}}}\Delta_{\mathrm{L},1}}^{R_{0}}{\lambda}_{\mathrm{m}}\left[1-\tfrac{1}{\sum_{p=1}^{T}w_{p}}\sum_{p=1}^{T}\tfrac{w_{p}M_{\mathrm{m}}^{M_{\mathrm{m}}}e^{-M_{\mathrm{m}}(\sqrt{2}\hat{\sigma}_{\mathrm{m}}\nu_{p}+\hat{\mu})}}{\left(\mathcal{G}_{\mathrm{s},t}\tfrac{P_{\mathrm{m}}N_{\mathrm{T}}^{\mathrm{m}}\left(N^{\mathrm{u}}_{\mathrm{R}}\right)^{2}r_{\mathrm{s,L}}^{\alpha_{\mathrm{s},\mathrm{L}}}}{KN_{\mathrm{m}}r^{\alpha_{\mathrm{m}}}_{\mathrm{a}}}b_{i}+M_{\mathrm{m}}e^{-(\sqrt{2}\hat{\sigma}_{\mathrm{m}}\nu_{p}+\hat{\mu})}\right)^{M_{\mathrm{m}}}}\right]c_{i}r_{\mathrm{a}}\mathrm{d}r_{\mathrm{a}}},$
(34)
$\displaystyle\mathbb{E}\left\\{e^{-\mathcal{G}_{\mathrm{b},t}r^{\alpha_{\mathrm{m}}}{I}_{\mathrm{m,b}}}\right\\}\approx\prod\limits_{i\in\atop
p_{\mathrm{BF}}(N^{\mathrm{m}}_{\mathrm{T}},N^{\mathrm{s}}_{\mathrm{R}},N_{\mathrm{m}})}e^{-\int_{0}^{2\pi}\int_{\xi}^{R_{0}}\widetilde{\lambda}_{\mathrm{m}}\rho_{\mathrm{M}}(r_{0})\left[1-\tfrac{1}{\sum_{p=1}^{T}w_{p}}\sum_{p=1}^{T}\tfrac{w_{p}M_{\mathrm{m}}^{M_{\mathrm{m}}}e^{-M_{\mathrm{m}}(\sqrt{2}\hat{\sigma}_{\mathrm{m}}\nu_{p}+\hat{\mu})}}{\left(\mathcal{G}_{\mathrm{b},t}\tfrac{P_{\mathrm{m}}N^{\mathrm{m}}_{\mathrm{T}}\left(N^{\mathrm{s}}_{\mathrm{R}}\right)^{2}r^{\alpha_{\mathrm{m}}}}{KN_{\mathrm{m}}r_{\mathrm{b}}^{\alpha_{\mathrm{m}}}}b_{i}+M_{\mathrm{m}}e^{-(\sqrt{2}\hat{\sigma}_{\mathrm{m}}\nu_{p}+\hat{\mu})}\right)^{M_{\mathrm{m}}}}\right]c_{i}r_{0}\mathrm{d}r_{0}\mathrm{d}\vartheta},$
(35)

where $\hat{\sigma}_{\mathrm{m}}=0.1\zeta_{\mathrm{m}}\mathrm{ln}10$. The
solution for
$\mathbb{P}\left(\mathrm{SINR}_{\mathbf{0}\mathbf{x}_{\mathrm{s}}^{\mathrm{N}}}>\tau\right)$
can be similarly derived.

###  D-B Solution of
$\mathbb{P}\left(\mathrm{SINR}_{\hat{\mathbf{0}}\mathbf{x}_{\mathrm{m}}}>\tau\right)$

Likewise, the $\mathrm{SINR}$ coverage for the typical backhaul link
$\mathbb{P}\left(\mathrm{SINR}_{\hat{\mathbf{0}}\mathbf{x}_{\mathrm{m}}}>\tau\right)$
can be approximated in a manner similar to (32). The mean interference from
LoS/NLoS IAB-nodes can be derived from a similar step in (33) with an
integration limit from 0 to ${R_{0}}$ because the typical IAB-node can only be
served by the gNB.

As for the mean interference from other gNBs, we still use the PGFL of PPP to
make the approximation, which is given in (35) shown at the top of the next
page, where $\mathcal{G}_{\mathrm{b}}=\tfrac{\tau
KN_{\mathrm{m}}^{2}\left(1+\tfrac{1}{1.5\cdot
2^{2q}}\right)}{P_{\mathrm{m}}N^{\mathrm{m}}_{\mathrm{T}}\left(N^{\mathrm{s}}_{\mathrm{R}}\right)^{2}\left(1-\tfrac{\tau}{1.5\cdot
2^{2q}}\right)}$,
$\mathcal{G}_{\mathrm{b},t}=\mathcal{G}_{\mathrm{b}}M_{\mathrm{m}}e^{-(\sqrt{2}\hat{\sigma}_{\mathrm{m}}\nu_{t}+\hat{\mu})}$,
$r_{\mathrm{b}}=\sqrt{r^{2}+r_{0}^{2}-2rr_{0}\cos(\vartheta)}$,
$\rho_{\mathrm{M}}(r_{0})$ is the thinning probability derived in [23, Lemma
1].

###  D-C Solution of
$\mathbb{P}\left(\mathrm{SINR}_{\mathbf{0}\mathbf{x}_{\mathrm{m}}}>\tau\right)$

Similarly, the $\mathrm{SINR}$ coverage for the gNB-associated access
$\mathbb{P}\left(\mathrm{SINR}_{{\mathbf{0}}\mathbf{x}_{\mathrm{m}}}>\tau\right)$
can be approximated by following the steps in (32). The mean interference from
LoS/NLoS IAB-nodes follows the similar way as in (33), with the integration
limit from
${r_{\mathrm{m}}^{\tfrac{\alpha_{\mathrm{m}}}{\alpha_{\mathrm{s},j}}}\Delta_{j}}$
to ${R_{0}}$, where $j\in\\{\mathrm{L},\mathrm{N}\\}$. The mean interference
from other gNBs can be approximated in a manner similar to that in (35).

##  Appendix E Solution of Theorem 3

According to [24, Lemma 2], we have

$\displaystyle\bar{\mathcal{R}}_{\mathrm{m}}=\mathbb{E}\\{W\log_{2}(1+\mathrm{SINR}_{\mathbf{0}\mathbf{x}_{\mathrm{m}}})\\}=\left(1-\tfrac{1}{N^{\mathrm{m}}_{\mathrm{T}}}\right)^{N_{\mathrm{m}}-1}$
$\displaystyle\times\int_{0}^{R_{0}}\frac{W}{\ln{2}}\int_{0}^{\infty}\mathbb{E}\left\\{e^{-z\left(\tfrac{{I}_{\mathrm{m,a}}+{I}_{\mathrm{s,a}}^{\mathrm{L}}+{I}_{\mathrm{s,a}}^{\mathrm{N}}}{\sigma_{\mathrm{n}}^{2}N^{\mathrm{u}}_{\mathrm{R}}}+\tfrac{\tfrac{1}{1.5\cdot
2^{2q}}G_{\mathrm{a,m}}}{\left(1+\tfrac{1}{1.5\cdot
2^{2q}}\right)\sigma_{\mathrm{n}}^{2}N^{\mathrm{u}}_{\mathrm{R}}}\right)}\right\\}$
$\displaystyle\times\left(1-\mathbb{E}\left\\{e^{-z\tfrac{G_{\mathrm{a,m}}}{\left(1+\tfrac{1}{1.5\cdot
2^{2q}}\right)\sigma_{\mathrm{n}}^{2}N^{\mathrm{u}}_{\mathrm{R}}}}\right\\}\right)\frac{e^{-z}}{z}\mathrm{d}z\hat{{f}}_{R_{\mathbf{x}_{\mathrm{m}}{\mathbf{0}}}}(r)\mathrm{d}r.$
(36)

Similar to the process in Appendix-D, we can obtain the solution by leveraging
the approximated Laplace transform of the composite GL distribution in (29)
and the PGFL of PPP. Similarly, $\bar{\mathcal{R}}_{\mathrm{s},\mathrm{L}}$,
$\bar{\mathcal{R}}_{\mathrm{s},\mathrm{N}}$ and
$\bar{\mathcal{R}}_{\mathrm{b}}$ can be derived easily.

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