# MDD-Enabled Two-Tier Terahertz Fronthaul in Indoor Industrial Cell-Free
Massive MIMO

Bohan Li, Diego Dupleich, Guoqing Xia, Huiyu Zhou, Yue Zhang, Pei Xiao, Senior
Member, IEEE, Lie-Liang Yang, Fellow, IEEE B. Li and H. Zhou are with the
School of CMS, Uni. of Leicester, LE1 7RH Leicester, UK (E-mail:
bl204,hz143@leicester.ac.uk). D. Dupleich is with Fraunhofer Institute for
Integrated Circuits IIS, 91058 Erlangen, Germany (E-mail:
diego.dupleich@iis.fraunhofer.de). G. Xia is with the School of Engineering,
Uni. of Leicester, LE1 7RH Leicester, UK (E-mail: gx21@leicester.ac.uk). Y.
Zhang is with the Institute of Communication Measurement Technology, 610095
Chengdu, China (e-mail: yuezhang@icsmcn.cn). P. Xiao is with 5GIC & 6GIC, Uni.
of Surrey, GU2 7XH Guildford, UK (Email: p.xiao@surrey.ac.uk). L. Yang is with
the School of ECS, Univ. of Southampton, SO17 1BJ Southampton, UK. (E-mail:
<EMAIL_ADDRESS>
###### Abstract

To make indoor industrial cell-free massive multiple-input multiple-output
(CF-mMIMO) networks free from wired fronthaul, this paper studies a
multicarrier-division duplex (MDD)-enabled two-tier terahertz (THz) fronthaul
scheme. More specifically, two layers of fronthaul links rely on the mutually
orthogonal subcarreir sets in the same THz band, while access links are
implemented over sub-6G band. The proposed scheme leads to a complicated
mixed-integer nonconvex optimization problem incorporating access point (AP)
clustering, device selection, the assignment of subcarrier sets between two
fronthaul links and the resource allocation at both the central processing
unit (CPU) and APs. In order to address the formulated problem, we first
resort to the low-complexity but efficient heuristic methods thereby relaxing
the binary variables. Then, the overall end-to-end rate is obtained by
iteratively optimizing the assignment of subcarrier sets and the number of AP
clusters. Furthermore, an advanced MDD frame structure consisting of three
parallel data streams is tailored for the proposed scheme. Simulation results
demonstrate the effectiveness of the proposed dynamic AP clustering approach
in dealing with the varying sizes of networks. Moreover, benefiting from the
well-designed frame structure, MDD is capable of outperforming TDD in the two-
tier fronthaul networks. Additionally, the effect of the THz bandwidth on
system performance is analyzed, and it is shown that with sufficient frequency
resources, our proposed two-tier fully-wireless fronthaul scheme can achieve a
comparable performance to the fiber-optic based systems. Finally, the
superiority of the proposed MDD-enabled fronthaul scheme is verified in a
practical scenario with realistic ray-tracing simulations.

###### Index Terms:

cell-free massive MIMO, multicarrier-division duplex, indoor industrial
networks, terahertz communications

##  I Introduction

Compared with the conventional co-located massive multiple-input multiple-
output (mMIMO), cell-free mMIMO (CF-mMIMO), reaping the merits of coordinated
multipoint (CoMP) transmission and cloud radio access network (C-RAN), is
capable of achieving more consistent and robust quality of services (QoS) with
densely distributed access points (APs) [1]. In order to facilitate CF-mMIMO,
there are two main operation modes, namely centralized mode, where central
processing unit (CPU) carries out all the signal processing tasks such as
channel estimation, coordinated beamforming and data detection, and
distributed mode, where each AP equipped with simple baseband processors
operates independently and processes everything except the final data
detection. Unsurprisingly, the former catches more attention due to its higher
achievable rates, but it heavily relies on the fronthaul capacity for sending
(collecting) pre-processed data (raw data) to (from) APs. Additionally, no
matter which operation mode is applied, the synchronization between APs and
users has to be conducted over fronthaul links. Therefore, fronthaul links are
of paramount importance in CF-mMIMO systems.

The classic architecture of centralized CF-mMIMO was studied in [2], where all
densely distributed APs are connected with the CPU via wired fronthaul links
and each AP simultaneously serves all users. Although it was shown to
outperform the conventional small-cell systems, the wired fronthaul incurs
high cost, rendering it impractical especially when the number of APs and
users are significantly large. In order to relieve the signaling burden of
fronthaul links, the authors in [3] proposed a scalable solution based on
user-centric dynamic cooperation clustering for both centralized and
distributed CF-mMIMO systems. In addition, various user-centric clustering
(UCC) approaches for mitigating the fronthaul overhead have been also
proposed, e.g, in [4] and [5]. However, the UCC only considers addressing the
problem of low fronthaul capacity, while the huge infrastructure cost of wired
fronthaul like fiber-optic cables are neglected. In order to avoid the
excessive use of cables, authors in [6] proposed a hierarchical wire-based
topology, where the CPU only connects with a leading AP within each AP
cluster, and the leading AP then forwards the fronthaul signal to the
remaining APs in the cluster. Although this topology can avoid overlength
cables and reduce CPU overhead, it exhibits the poor flexibility when the
extra APs or users have to be installed or served. Moreover, the conception of
radio stripe systems was recently proposed in [7, 8]. Different from the
conventional wire-based star topology, radio stripe systems can accommodate
all the APs using one stripe, and therefore significantly lower the cost of
cabling. However, the installation of radio stripes are restricted by the
physical structure of network scenarios, leading to the fact that the APs in
the CF systems cannot be freely deployed. In addition, the radio stripes or
any other wired cables are not easily built up/tear down, especially in a
complicated indoor scenario.

In order to provide high flexibility without sacrificing the end-to-end system
performance, the high-band techniques relied wireless fronthaul have recently
caught the attention of researchers. For instance, due to the small wavelength
and large bandwidth of millimeter-wave (mmWave), the mmWave-based fronthaul is
capable of providing the sufficient fronthaul capacity for ultra-dense
networks [9, 10]. In [11], the authors applied mmWave-enabled point-to-point
fronthaul between the CPU and all the APs in CF-mMIMO. To further reduce the
expenditure of infrastructure, similar to [6], a hybrid two-tier fronthaul
scheme was also proposed in [11], where the CPU transmits fronthaul signal to
the leading AP of each cluster over the mmWave band, and the set of APs within
cluster are connected with wired cables. However, the wired connections still
inevitably limit the flexibility of AP clustering. That is to say, once the AP
clusters are determined, it is not easy to readjust them when the number and
also the locations of users have changed.

###  I-A Motivations

To date, the works on wireless fronthaul in CF-mMIMO are very limited,
possibly due to the following two challenges: (i) As APs are densely
distributed, the wireless fronthaul links between the CPU and APs should
provide sufficiently high capacity and wide coverage; (ii) The conventional
TDD-based fronthaul fails to fully exploit the time-frequency resources, and
hence leading to the degradation of end-to-end performance. On the other hand,
compared with outdoor CF-mMIMO, the indoor industrial CF-mMIMO systems are
even more in need of wireless fronthaul, from the perspective of convenient
upgrade and low deployment cost. In fact, with the evolution of smart
industry, CF-mMIMO has been deemed as the most promising technique to realize
the reliable and fast wireless services for industrial communications [12]. In
[6, 13, 14, 15], the indoor industrial CF-mMIMO with wired fronthaul have been
investigated, but to the best of our knowledge, no previous research has
studied the indoor industrial CF-mMIMO employed with wireless fronthaul.

Against this background, in this paper, we propose a multicarrier-division
duplex (MDD)-enabled two-tier terahertz (THz) fronthaul scheme, aiming to
address the above-mentioned challenges for indoor industrial CF-mMIMO systems.
More specifically, inspired by [6] and [11], a two-tier fully-wireless
fronthaul scheme, consisting of the links between the CPU and leading APs, and
between the leading APs and APs, is proposed so as to realize the dynamic AP
clustering and achieve high end-to-end rates. To facilitate the proposed
scheme, MDD is leveraged to activate the full-duplex (FD) operation among two
wireless fronthaul links over orthogonal subcarrier sets [16]. Compared with
its time-division duplex (TDD) and in-band full-duplex (IBFD) counterparts,
MDD is able to largely exploit the time-frequency resources at the little
expense of self-interference (SI) and cross-layer interference (CLI) in CF-
mMIMO systems [17]. The term CLI refers to the interference caused by the
node’s transmitted signal onto its neighboring nodes during signal reception
at the same frequency band and time slot. On the other hand, with the
increasing demands for higher rates and better quality-of-service (QoS) in the
industrial networks, THz technique could provide over Gigabit-per-second
(Gbps) rates with abundant unoccupied bandwidth [18], and achieve the high
synchronization accuracy between the CPU and all the APs [11]. Hence, THz-
based fronthaul is expected to fulfill the goals of replacing the conventional
wired fronthaul. In addition, the combination of THz and MDD has the synergy
to mitigate the effect of SI and CLI. More specifically, in the systems
operated in the FD mode, the interference cancellation depends on the
propagation-, analog-, and digital-domain approaches [19]. According to the
principle of MDD, digital-domain interference can be thoroughly canceled by
the built-in analog-to-digital converter (ADC) of receiver chains with the FFT
operation [16]. Therefore, with the aid of extremely large path loss in the
propagation domain and the FFT operation in the digital domain, MDD-based
systems operated over the THz band are expected to be completely free from SI
and CLI.

###  I-B Contributions

The main contributions of this paper are as follows:

  * •

In order to realize the fully-wireless fronthaul for indoor industrial
networks, an MDD-enabled two-tier THz scheme is proposed, which is capable of
providing high data rates and wide converge for the distributed APs.
Additionally, the proposed scheme can significantly save the cost of wired
fronthaul cables and dynamically implement AP clustering to adapt to the
variations in the network with different numbers of APs and devices.

  * •

The overall end-to-end optimization considering two THz fronthaul and sub-6
GHz access links is formulated as a max-min fairness problem. The problem
jointly considers the AP clustering, device selection, the assignment of
subcarrier sets between two fronthaul links, power and subcarrier allocation
at the CPU and CAPs, and power allocation at APs. To deal with the complicated
problem, heuristic methods are first leveraged to relax the binary variables
with regard to AP clustering, device selction and the assignment of subcarrier
sets. Then, the original problem is transformed into two individual fronthaul
and access max-min sub-problems. The overall end-to-end rate maximization is
solved by iteratively optimizing the assignment of subcarrier sets and the
number of AP clusters.

  * •

To further facilitate the proposed fronthaul scheme, an advanced MDD frame
structure enabling three parallel data streams is proposed. For comparison,
the frame structure in TDD-based two-tier THz fronthaul scheme is also
specifically designed for the end-to-end rate maximization.

  * •

To verify the performance of the proposed scheme, we apply it to a practical
indoor industrial scenario relying on the ray-tracing (RT) channel
measurements. Simulation results show that MDD outperforms TDD in the two-tier
fully-wireless fronthaul scheme. Moreover, when provided with sufficient THz
bandwidth, indoor industrial CF-mMIMO systems can be free from the wired
fronthaul with the aid of our proposed MDD-enabled two-tier THz fronthaul
scheme.

##  II System Model

We study the downlink (DL) transmission in an indoor industrial cell-free
massive MIMO (CF-mMIMO) network, where the CPU is equipped with a
$\left(N_{\text{CPU}}^{{}^{\prime}}\times
N_{\text{CPU}}^{{}^{\prime\prime}}\right)$ uniform planar array (UPA) having
$N_{\text{CPU}}=N_{\text{CPU}}^{{}^{\prime}}N_{\text{CPU}}^{{}^{\prime\prime}}$
antennas, and sends data to APs via wireless fronthaul channels over THz
frequency-band, while $Q$ APs, thereafter, transmit the received data to $U$
single-antenna devices via access channels over sub-6 GHz band. To enable such
a dual-band architecture, every AP employs both the high- and low-frequency RF
chains [9, 11], and each RF chain is connected with a
$\left(N_{\text{AP}}^{{}^{\prime}}\times
N_{\text{AP}}^{{}^{\prime\prime}}\right)$ UPA of
$N_{\text{AP}}=N_{\text{AP}}^{{}^{\prime}}N_{\text{AP}}^{{}^{\prime\prime}}$
antennas at transmitter and single antenna at receiver111In this paper, we
leverage two separate antenna arrays at AP to realize simultaneous signal
transmission over the THz and sub-6 GHz bands, respectively. Furthermore,
considering the system’s overhead and space usage, an integrated low-band and
high-band MIMO antenna module proposed in [20] can be applied to make our
dual-band architecture more efficient..

In general, it is straightforward to connect every AP with CPU through
wireless fronthaul links [21]. However, as APs are densely distributed in CF-
mMIMO systems, it would be a demanding task to compute all the beamforming
matrices at CPU, while the management of inter-beam interference is also
highly challenging. To circumvent this problem, we propose a two-tier fully-
wireless fronthaul architecture, as shown in Fig. 1, where $Q$ APs are
classified into $L$ distinct clusters, expressed as $\mathcal{C}_{l}$,
$\forall l\in\left\\{1,...,L\right\\}$, and
$\left|\bigcup_{l=1}^{L}\mathcal{C}_{l}\right|=Q$. The CPU only establishes
fronthaul link with a leading AP, namely computational AP (CAP), in each
cluster. The CAP of each cluster takes charge of both the fronthaul
transmissions with the remaining APs and the signal processing tasks, such as
DL precoding and channel estimation222In order to achieve the dynamic AP
clustering and CAP selection as discussed in Section III-A, each AP is
equipped with a basic baseband processor such that any AP within a cluster can
be chosen as the CAP. This kind of configuration, similar to that in the
distributed CF-mMIMO [1], inevitably increases the overhead of system, but it
can significantly reduce the cost on fiber-optic cables, and its budget may be
further reduced by the developed hardware techniques in the future.. This
architecture enables CF-mMIMO to get rid of the wired fronthaul connections,
thereby exhibiting high flexibility and scalability. Furthermore, it conforms
to the principle of the user-centric CF-mMIMO, where CAPs act as
geographically distributed edge-cloud processors to divide the tasks of
CPU333Although the CAPs employed with low-frequency RF chains can implement DL
transmissions over sub-6 GHz band like other APs, it is assumed to only
transmit and receive fronthaul signal for the sake of saving computational
overhead.. Moreover, we assume that all the APs in one cluster serve the same
device, and the group of APs serving device $u$, expressed as
$\mathcal{G}_{u}$, is determined via selecting the set of clusters, such that
$\mathcal{G}_{u}\subseteq\left\\{1,..,L\right\\}$. Let $\mathcal{U}_{l}$
denote the set of devices concurrently served by cluster $l$. We further
assume that each cluster can serve at most $U_{\text{max}}$ devices, i.e.,
$\left|\mathcal{U}_{l}\right|\leq U_{\text{max}},\forall
l\in\left\\{1,...,L\right\\}$, while each device can choose up to
$C_{\text{max}}$ serving clusters, meaning that
$\left|\mathcal{G}_{u}\right|\leq C_{\text{max}},\forall
u\in\left\\{1,...,U\right\\}$.

Figure 1: The proposed MDD-Enabled THz Fronthaul in indoor industrial CF-
mMIMO.

To overcome multi-path fading, especially in wideband THz communications, we
resort to OFDM for signal transmission. In particular, we assume that CPU and
CAPs transmit MDD-based fronthaul signals relying on two mutually orthogonal
sets of subcarriers in the same THz band, i.e., $\mathcal{M}_{\text{CC}}$ and
$\mathcal{M}_{\text{CA}}$, such that
$\mathcal{M}_{\text{CC}}\bigcap\mathcal{M}_{\text{CA}}=\emptyset$ and
$\mathcal{M}_{\text{CC}}\bigcup\mathcal{M}_{\text{CA}}=\mathcal{M}_{\text{FH}}$,
while the data transmitted from APs to devices are modulated on the subcarrier
set $\mathcal{M}_{\text{AD}}$ over the sub-6 GHz band. In CF-mMIMO, it is
intuitive that each cluster of APs may simultaneously serve more than one
device, which requires CAPs to receive (transmit) the multi-device signals
from (to) CPU (APs). To this end, in the proposed multicarrer system, the CPU
and CAPs adopt precoders to map the different devices’ signals within each
cluster onto different subcarriers of $\mathcal{M}_{\text{CC}}$ and
$\mathcal{M}_{\text{CA}}$, respectively.

###  II-A Fronthaul Transmissions

The fronthaul in our proposed architecture consists of CPU-to-CAP and CAP-to-
AP links. As these links rely on THz communications, we first establish the
delay-$t$ THz channel model between the $\left(N^{{}^{\prime}}\times
N^{{}^{\prime\prime}}\right)$ UPA transmitter and the single-antenna receiver
as [22]

$\displaystyle\boldsymbol{h}_{t}(f_{m},d)=\boldsymbol{h}_{t}^{\text{LoS}}(f_{m},d)+\boldsymbol{h}_{t}^{\text{NLoS}}(f_{m},d)=\sqrt{N^{{}^{\prime}}N^{{}^{\prime\prime}}}\alpha^{\text{LoS}}(f_{m},d)p_{\text{rc}}(tT_{\text{s}}-\delta)G_{\text{tx}}G_{\text{rx}}\boldsymbol{a}_{\text{tx}}\left(\phi^{\text{tx}},\theta^{\text{tx}}\right)$
$\displaystyle+\sqrt{N^{{}^{\prime}}N^{{}^{\prime\prime}}/N_{\text{ray}}}\sum_{i=1}^{N_{\text{ray}}}\alpha_{i}(f_{m},d)G_{\text{tx}}G_{\text{rx}}p_{\text{rc}}(tT_{\text{s}}-\delta_{i})\boldsymbol{a}_{\text{tx}}\left(\phi_{i}^{\text{tx}},\theta_{i}^{\text{tx}}\right),$
(1)

where $f_{m}$ is the center frequency of the $m$-th subcarrier, $d$ is the
path distance, $G_{\text{tx}}$ and $G_{\text{rx}}$ denote the antenna gains of
transmitter and receiver, respectively, and $p_{\text{rc}}(\cdot)$ denotes the
combined effect of pulse shaping and analog filtering, with $T_{\text{s}}$ and
$\delta$ denoting the sampling interval and delay. Moreover,
$\alpha^{\text{LoS}}(f_{m},d)$, dependent on frequency and path distance, is
the complex gain of the LoS path component, and its squared norm is given by
$\left|\alpha^{\text{LoS}}(f_{m},d)\right|^{2}=L_{\text{spread}}(f_{m},d)L_{\text{abs}}(f_{m},d),$
where $L_{\text{spread}}(f_{m},d)=\left(\frac{c}{4\pi f_{m}d}\right)^{2}$ and
$L_{\text{abs}}(f_{m},d)=e^{-k_{\text{abs}}(f_{m})d}$ with
$k_{\text{abs}}(f_{m})$ denoting the frequency-related absorption coefficient,
which is determined by the propagation medium [18]. By contrast, the complex
gain of the NLoS path is subject to not only the distance and frequency, but
also the reflection coefficient, which is given by
$\left|\alpha_{i}(f_{m},d)\right|^{2}=\Gamma_{i}^{2}(f_{m})\left|\alpha^{\text{LoS}}(f_{m},d)\right|^{2}$,
where $\Gamma_{i}^{2}(f_{m})$ is the product of the Fresnel reflection
coefficient and the Rayleigh roughness factor [18]. Furthermore,
$\boldsymbol{a}_{\text{tx}}(\phi^{\text{tx}},\theta^{\text{tx}})$ denotes the
UPA response vector at the transmitter [18], where $\phi$, $\theta$ denote
azimuth angle of departure (AoD) and elevation AoD, receptively.

Given the delay-$t$ THz channel model in (II-A), the $m$-th subcarrier channel
can be expressed as444Note that as the main objectives of this paper are to
design a two-tier fully-wireless fronthaul architecture and jointly implement
resource allocation among fronthaul and access links, the problems of beam
squint and carrier frequency offset arising from THz channels have not been
considered [22, 23], which are left to our future study.

$\boldsymbol{h}[m]=\sum_{t=0}^{T_{\text{delay}}-1}\boldsymbol{h}_{t}(f_{m},d)e^{-j\frac{2\pi
mt}{\left|\mathcal{M}_{\text{FH}}\right|}}.$ (2)

####  II-A1 CPU-to-CAP Link

The CPU beamforms the signal toward CAPs over the subcarriers selected from
$\mathcal{M}_{\text{CC}}$. Let $x_{l,u}[m]$ denote the data intended for
device $u$ associated with the $l$-th cluster over the $m$-th subcarrier, with
$\mathbb{E}\left[\left|x_{l,u}[m]\right|^{2}\right]=1$. Then, the $m$-th
subcarrier signal received at the $l$-th CAP can be expressed as

$\displaystyle
y_{l}[m]=\underbrace{\boldsymbol{h}_{l}^{H}[m]\sum_{u\in\mathcal{U}_{l}}\gamma_{l,u,m}\sqrt{p_{l,u}[m]}\boldsymbol{f}_{l,u}[m]x_{l,u}[m]}_{\text{Desired
signal}}+\underbrace{\boldsymbol{h}_{l}^{H}[m]\sum_{l^{\prime}\neq
l}\sum_{u^{\prime}\in\mathcal{U}_{l^{\prime}}}\gamma_{l^{\prime},u^{\prime},m}\sqrt{p_{l^{\prime},u^{\prime}}[m]}\boldsymbol{f}_{l^{\prime},u^{\prime}}[m]x_{l^{\prime},u^{\prime}}[m]}_{\text{Inter-
CAP interference}}$ $\displaystyle+\text{SI}_{l}+n_{l}[m],\ \ \forall
m\in\mathcal{M}_{\text{CC}},$ (3)

where $\gamma_{l,u,m}=1$ denotes that the CPU sends the data of device $u$ to
the $l$-th CAP over the $m$-th subcarrier of $\mathcal{M}_{\text{CC}}$, and
$\gamma_{l,u,m}=0$, otherwise, $p_{l,u}[m]$ is the power coefficient with
respect to the device $u$ within the $l$-th cluster over subcarrier $m$,
$\boldsymbol{h}_{l}[m]\in\mathbb{C}^{N_{\text{CPU}}\times 1}$ is the channel
impulse response of fronthaul channel from CPU to CAP $l$, which follows the
expression of (2),
$\boldsymbol{f}_{l,u}[m]\in\mathbb{C}^{{N}_{\text{CPU}}\times 1}$ is the
precoder specific for device $u$ associated with the $l$-th cluster over
subcarrier $m$, $n_{l}[m]\sim\mathcal{CN}(0,\sigma_{\text{l}}^{2})$ is the
additive white Gaussian noise. Note that, as CAP leverages the subcarrier
diversity to simultaneously receive the data of different devices, one
subcarrier cannot be assigned to multiple devices within one cluster, i.e.,
$\gamma_{l,u_{1},m}\gamma_{l,u_{2},m}=0$ when $u_{1}\neq u_{2}$. Furthermore,
$\text{SI}_{l}\sim\mathcal{CN}(0,\sigma_{\text{SI}}^{2})$ denotes the residual
SI caused by the CAP’s own transmitter.

######  Remark 1.

Compared with the conventional in-band full-duplex, MDD enables simultaneous
DL and UL transmissions over orthogonal subcarrier sets in the same band.
Owing to this feature, the SI is orthogonal to the UL signal in the digital
domain at the CAP’s receiver, and hence can be readily canceled with the FFT
operation. However, as SI is strongly coupled with the UL signal in the analog
domain, the analog-domain SIC is still indispensable. Otherwise, the
interference of overlarge power may overwhelm the desired signal of relatively
small power, leading to the severe quantization noise after its passing
through ADC. It is noteworthy that the suppression of the analog-domain SI in
high-frequency bands is much easier than that of the digital-domain SI, due to
the highly directional beam and large path-loss [24]. To this end, the SI
mitigation in MDD-based THz systems can be achieved at less expenses. Hence,
without deviation of the core of this paper, the detailed discussion of SI
cancellation is omitted. Instead, we model the residual interference as
Gaussian noise consisting of the combined effect of the additive noise
introduced by automatic gain control, non-linearity of ADC and the phase noise
arose from oscillator due to RF imperfection [25].

Based on (II-A1), the achievable rate of CAP $l$ corresponding to device $u$
can be expressed as

$\small
C_{l,u}^{\text{CC}}=b^{\text{FH}}\sum_{m\in\mathcal{M}_{\text{CC}}}\log\left(1+\frac{\gamma_{l,u,m}p_{l,u}[m]\left|\boldsymbol{h}_{l}^{H}[m]\boldsymbol{f}_{l,u}[m]\right|^{2}}{\sum_{l^{\prime}\neq
l}\sum_{u^{\prime}\in\mathcal{U}_{l^{\prime}}}\gamma_{l^{\prime},u^{\prime},m}p_{l^{\prime},u^{\prime}}[m]\left|\boldsymbol{h}_{l}^{H}[m]\boldsymbol{f}_{l^{\prime},u^{\prime}}[m]\right|^{2}+\sigma_{\text{SI}}^{2}+\sigma_{\text{l}}^{2}}\right),$
(4)

where $b^{\text{FH}}$ denotes the subcarrier bandwidth in THz band. Since
device $u$ may have $\left|\mathcal{G}_{u}\right|$ serving clusters, its
message will be received by multiple CAPs. In this case, the effective rate of
device $u$ in the CPU-to-CAP links can be defined as the minimum rate of the
CAPs in the serving cluster $\mathcal{G}_{u}$, which can be written as
$C_{u}^{\text{CC}}=\min_{l\in\mathcal{G}_{u}}\
\left\\{C_{l,u}^{\text{CC}}\right\\}$.

####  II-A2 CAP-to-AP Link

Each CAP $l$ forwards the CPU signal, relying on $\mathcal{M}_{\text{CA}}$
subcarriers, to the remaining APs within its serving cluster
$\mathcal{C}_{l}$. Let $\tilde{x}_{l,q,u}[\bar{m}]$ denote the data intended
for device $u$ associated with the $l$-th cluster and $q$-th AP over the
$\bar{m}$-th subcarrier. Then, the signal to be transmitted at the CAP $l$ can
be expressed as
$\boldsymbol{s}_{l}=\sum_{\bar{m}\in\mathcal{M}_{\text{CA}}}\boldsymbol{s}_{l}[\bar{m}]$,
where
$\boldsymbol{s}_{l}[\bar{m}]=\sum_{q\in\mathcal{C}_{l}}\sum_{u\in\mathcal{U}_{l}}\gamma_{l,q,u,\bar{m}}\sqrt{p_{l,q,u}[\bar{m}]}\boldsymbol{w}_{l,q,u}[\bar{m}]\tilde{x}_{l,q,u}[\bar{m}]$,
$\gamma_{l,q,u,\bar{m}}=1$ denotes that the $l$-th CAP sends the data of
device $u$ to the $q$-th AP over the $\bar{m}$-th subcarrier in
$\mathcal{M}_{\text{CA}}$, and $\gamma_{l,q,u,\bar{m}}=0$, otherwise,
$p_{l,q,u}[\bar{m}]$ is the power coefficient associated with the device $u$
at AP $q$ over subcarrier $\bar{m}$,
$\boldsymbol{w}_{l,q,u}[\bar{m}]\in\mathbb{C}^{{N}_{\text{AP}}\times 1}$ is
the precoder specific for device $u$ associated with AP $q$ over subcarrier
$\bar{m}$, $n_{l}[\bar{m}]\sim\mathcal{CN}(0,\sigma_{\text{n}}^{2})$ is the
additive white Gaussian noise. Analogously, as AP only has one receiver
antenna, the receiving of different devices’ data depends on subcarrier
diversity, leading to
$\gamma_{l,q,u_{1},\bar{m}}\gamma_{l,q,u_{2},\bar{m}}=0$, when $u_{1}\neq
u_{2}$.

Then, the signal received at the $q$-th AP of $\mathcal{C}_{l}$ over
subcarrier $\bar{m}$ can be written as

$\displaystyle
y_{l,q}[\bar{m}]=\underbrace{\boldsymbol{h}_{l,q}^{H}\\!\sum_{u\in\mathcal{U}_{l}}\\!\gamma_{l,q,u,\bar{m}}\sqrt{p_{l,q,u}[\bar{m}]}\boldsymbol{w}_{l,q,u}[\bar{m}]\tilde{x}_{l,q,u}[\bar{m}]}_{\text{Desired
signal}}\\!+\\!\underbrace{\boldsymbol{h}_{l,q}^{H}\\!\sum_{q^{\prime}\neq
q}\\!\sum_{u^{\prime}\in\mathcal{U}_{l}}\\!\gamma_{l,q^{\prime},u^{\prime},\bar{m}}\sqrt{p_{l,q^{\prime},u^{\prime}}[\bar{m}]}\boldsymbol{w}_{l,q^{\prime},u^{\prime}}[\bar{m}]\tilde{x}_{l,q^{\prime},u^{\prime}}[\bar{m}]}_{\text{Intra-
cluster interference}}$ $\displaystyle+\underbrace{\sum_{l^{\prime}\neq
l}\boldsymbol{h}_{l^{\prime},q}^{H}\boldsymbol{s}_{l^{\prime}}[\bar{m}]}_{\text{Inter-
cluster interference}}+n_{l,q}[\bar{m}],\ \
\forall\bar{m}\in\mathcal{M}_{\text{CA}},$ (5)

where $\boldsymbol{h}_{l,q}[\bar{m}]\in\mathbb{C}^{N_{\text{AP}}\times 1}$ is
the fronthaul channel between CAP $l$ and AP $q$ following the expression of
(2).

######  Remark 2.

Similar to the SI at CAP receiver, as two fronhaul links operate over
orthogonal subcarriers, the cross-link interference (CLI) imposed by CPU at
each AP receiver is orthogonal to the CAP-to-AP fronthaul signal in the
digital domain. Moreover, due to the employment of beamforming at CPU and
extremely high large-scale fading between CPU and APs, the analog-domain CLI
significantly deteriorates before arriving at ADC of APs’ receiver.
Accordingly, the CLI can be neglected in this case.

Consequently, based on (II-A2), the achievable rate of AP $q$ in the cluster
$\mathcal{C}_{l}$ corresponding to device $u$ can be expressed as

$\displaystyle C_{l,q,u}^{\text{CA}}=b^{\text{FH}}\cdot$
$\displaystyle{\scriptsize\sum_{m\in\mathcal{M}_{\text{CC}}}\\!\log\\!\left(\\!1\\!+\\!\frac{\gamma_{l,q,u,\bar{m}}p_{l,q,u}[\bar{m}]\\!\left|\boldsymbol{h}_{l,q}^{H}[\bar{m}]\boldsymbol{w}_{l,q,u}[\bar{m}]\right|^{2}}{\sum_{q^{\prime}\neq
q}\\!\sum_{u^{\prime}\in\mathcal{U}_{l}}\\!\gamma_{l,q^{\prime},u^{\prime},\bar{m}}p_{l,q^{\prime},u^{\prime}}[\bar{m}]\\!\left|\boldsymbol{h}_{l,q}^{H}[\bar{m}]\boldsymbol{w}_{l,q^{\prime},u^{\prime}}[\bar{m}]\right|^{2}\\!+\\!\sum_{l^{\prime}\neq
l}\\!\left|\mathbb{E}\left\\{\boldsymbol{h}_{l^{\prime},q}^{H}\boldsymbol{s}_{l^{\prime}}[\bar{m}]\right\\}\right|^{2}\\!+\\!\sigma_{q}^{2}}\right).}$
(6)

Therefore, the effective rate in cluster $\mathcal{C}_{l}$ corresponding to
device $u$ can be written as
$C_{u}^{\text{CA}}=\min_{l\in\mathcal{G}_{u},q\in\mathcal{C}_{l}}\
\left\\{C_{l,q,u}^{\text{CA}}\right\\}$.

###  II-B Device Communications via AP-to-Device Link

Once the APs decode the fronthaul signal received from the CAP, it will
communicate with devices via the sub-6 GHz access channels, which are assumed
to experience flat fading with $|\mathcal{M}_{\text{AD}}|=1$, for ease of
discussion. In particular, the channel between the $q$-th AP in
$\mathcal{C}_{l}$ and device $u$, i.e.,
$\tilde{\boldsymbol{h}}_{l,q,u}\in\mathbb{C}^{N_{\text{AP}}\times 1}$, is
modeled as $(\tilde{\boldsymbol{h}}_{l,q,u})_{i}=\sqrt{\beta_{l,q,u}}\
\alpha_{\text{s}}$ with $\beta_{l,q,u}$ and $\alpha_{\text{s}}$ denoting the
large and small-scale fading coefficients, respectively.

The transmitted signal from the $q$-th AP within cluster $\mathcal{C}_{l}$ can
be written as
$\boldsymbol{s}_{l,q}=\sum_{u\in\mathcal{U}_{l}}\sqrt{p_{l,q,u}}\boldsymbol{v}_{l,q,u}x_{u}$,
where $p_{l,q,u}$ denotes the transmit power coefficient for device $u$ at the
$q$-th AP, $\boldsymbol{v}_{l,q,u}\in\mathbb{C}^{{N}_{\text{AP}}\times 1}$ is
the precoder specific for device $u$ at the $q$-th AP. Then, the received
signal at device $u$ can be expressed as

$\small
y_{u}=\underbrace{\sum_{l\in\mathcal{G}_{u}}\boldsymbol{h}_{l,u}^{H}\boldsymbol{P}_{l,u}\boldsymbol{v}_{l,u}x_{u}}_{\text{Desired
signal}}+\underbrace{\sum_{l\in\mathcal{G}_{u}}\sum_{u^{\prime}\in\mathcal{U}_{l},u^{\prime}\neq
u}\boldsymbol{h}_{l,u}^{H}\boldsymbol{P}_{l,u^{\prime}}\boldsymbol{v}_{l,u^{\prime}}x_{u^{\prime}}}_{\text{Inter-
device
interference}}+\underbrace{\sum_{l^{\prime}\notin\mathcal{G}_{u}}\sum_{q\in\mathcal{C}_{l^{\prime}}}\tilde{\boldsymbol{h}}_{l^{\prime}q,u}^{H}\boldsymbol{s}_{l^{\prime},q}}_{\text{Inter-
cluster interference}}+n_{u},$ (7)

where
$\small\boldsymbol{h}_{l,u}\\!=\\!\left[\tilde{\boldsymbol{h}}_{l,1,u}^{T},...,\tilde{\boldsymbol{h}}_{l,\left|\mathcal{C}_{l}\right|,u}^{T}\right]^{T}$,
$\small\tilde{\boldsymbol{P}}_{l,u}\\!=\\!\left(\boldsymbol{P}_{l,u}\\!\otimes\\!\boldsymbol{I}_{N_{\text{AP}}}\right)$
with
$\small\boldsymbol{P}_{l,u}\\!=\\!\text{diag}\left(\left[\sqrt{p_{l,1,u}},...,\sqrt{p_{l,{\left|\mathcal{C}_{l}\right|},u}}\right]\right)$,
and
$\small\boldsymbol{v}_{l,u}=\left[\left(\boldsymbol{v}_{l,1,u}\right)^{T},...,\left(\boldsymbol{v}_{l,\left|\mathcal{C}_{l}\right|,u}\right)^{T}\right]^{T}$.
The achievable rate of the AP-to-Device link corresponding to device $u$ can
be written as

$\small{\scriptsize
C_{u}^{\text{AD}}\\!=\\!B^{\text{AD}}\log\\!\left(\\!1+\frac{\left|\sum_{l\in\mathcal{G}_{u}}\boldsymbol{h}_{l,u}^{H}\tilde{\boldsymbol{P}}_{l,u}\boldsymbol{v}_{l,u}\right|^{2}}{\sum_{l\in\mathcal{G}_{u}}\sum_{u^{\prime}\in\mathcal{U}_{l},u^{\prime}\neq
u}\left|\boldsymbol{h}_{l,u}^{H}\tilde{\boldsymbol{P}}_{l,u^{\prime}}\boldsymbol{v}_{l,u^{\prime}}\right|^{2}+\sum_{l^{\prime}\notin\mathcal{G}_{u}}\sum_{q\in\mathcal{C}_{l^{\prime}}}\left|\mathbb{E}\left\\{\tilde{\boldsymbol{h}}_{l^{\prime}q,u}^{H}\boldsymbol{s}_{l^{\prime},q}\right\\}\right|^{2}+\sigma^{2}_{u}}\\!\right),}$
(8)

where $B^{\text{AD}}$ denotes the bandwidth of the sub-6 GHz band.

###  II-C Optimization Problem

Considering the fact that the data flows from CPU to each device, as shown in
Fig. 1, the achievable rate of device $u$ in $bits/s$ can be written as
$C_{u}=\text{min}\left\\{C_{u}^{\text{CC}},C_{u}^{\text{CA}},C_{u}^{\text{AD}}\right\\}$.
Then, to guarantee the fairness among all the devices, the optimization
problem can be stated as:

$\displaystyle\underset{\mathcal{C},\mathcal{G},\mathcal{M},\left\\{\boldsymbol{f}\right\\},\left\\{\boldsymbol{w}\right\\},\left\\{\boldsymbol{v}\right\\},\left\\{\gamma\right\\}\left\\{p\right\\}}{\text{maximize}}\
\min_{u\in\left\\{1,...,U\right\\}}C_{u}$ (9a) s.t.
$\displaystyle\sum_{l=1}^{L}\left|\mathcal{C}_{l}\right|=Q,\
\mathcal{C}_{l_{1}}\cap\mathcal{C}_{l_{2}}=\emptyset,\text{if}\ l_{1}\neq
l_{2},$ (9b) $\displaystyle\left|\mathcal{U}_{l}\right|\leq
U_{\text{max}},\forall l\in\left\\{1,...,L\right\\},$ (9c)
$\displaystyle\left|\mathcal{G}_{u}\right|\leq C_{\text{max}},\forall
u\in\left\\{1,..,U\right\\},$ (9d)
$\displaystyle\mathcal{M}_{\text{CC}}\cup\mathcal{M}_{\text{CA}}=\mathcal{M}_{\text{FH}},\mathcal{M}_{\text{CC}}\cap\mathcal{M}_{\text{CA}}=\emptyset,$
(9e) $\displaystyle\gamma_{l,u,m}\in\left\\{0,1\right\\},\
\gamma_{l,u_{1},m}\gamma_{l,u_{2},m}=0,\forall u_{1}\neq u_{2},$ (9f)
$\displaystyle\gamma_{l,q,u,\bar{m}}\in\left\\{0,1\right\\},\
\gamma_{l,q,u_{1},\bar{m}}\gamma_{l,q,u_{2},\bar{m}}=0,\forall u_{1}\neq
u_{2},$ (9g)
$\displaystyle\sum_{m\in\mathcal{M}_{\text{CC}}}\sum_{l=1}^{L}\sum_{u=1}^{U}\gamma_{l,u,m}p_{l,u}[m]\left\|\boldsymbol{f}_{l,u}[m]\right\|_{2}^{2}\leq
P_{\text{CPU}},$ (9h)
$\displaystyle\sum_{\bar{m}\in\mathcal{M}_{\text{CA}}}\sum_{q\in\mathcal{C}_{l}}\sum_{u\in\mathcal{U}_{l}}\gamma_{l,q,u,\bar{m}}p_{l,q,u}[\bar{m}]\left\|\boldsymbol{w}_{q,u}[\bar{m}]\right\|_{2}^{2}\leq
P_{\text{AP}},\forall l\in\left\\{1,...,L\right\\},$ (9i)
$\displaystyle\sum_{u\in\mathcal{U}_{l}}p_{l,q,u}\left\|\boldsymbol{v}_{l,q,u}\right\|_{2}^{2}\leq
P_{\text{AP}},\forall
l\in\left\\{1,...,L\right\\},q\in\left\\{1,...,Q\right\\},$ (9j)

It can be observed that the overall optimization problem is non-convex and
involved with different types of variables. Hence, it is extremely time-
consuming and high-complexity to find an optimal solution. To work around this
issue, in the next section, we aim to relieve the constraints (9b)-(9e) so as
to find the approximate solution with high computational efficiency.

##  III Relaxation of Optimization Problem

According to formulation (9), the AP clustering, device selection and the
assignment of subcarrier sets consist of a large number of binary variables.
It is impractical to simultaneously optimize all the binary variables,
especially for the networks of large size. Therefore, we will leverage the
heuristic but efficient algorithms to deal with these problems, provided that
the number of clusters $L$ and the sizes of $\mathcal{M}_{\text{CC}}$ and
$\mathcal{M}_{\text{CA}}$ are already determined.

###  III-A AP Clustering

In the light of the fact that the indoor THz-based communications are
distance-sensitive due to the extremely high path loss, it is reasonable to
implement AP clustering mainly based on the proximity principle. In other
words, a group of APs situated closely to each other are most likely to be
arranged in a cluster. In doing so, the high-rate fronthaul transmissions
between CAP and APs within the cluster can be achieved, and the centralized
beamforming design can produce more focused beams toward devices. Apart from
AP clustering, the selection of CAP in a cluster is critical to the proposed
cluster-based wireless fronthaul scheme. Specifically, each CAP acts as the
pivot of two fronthaul links, simultaneously receiving data from CPU and
transmitting data to the APs within the cluster. Hence, the potential CAP
candidates should have the capability to build the reliable wireless fronthaul
connections with both CPU and APs. Additionally, as THz communications are
leveraged for fronthaul, the selection of CAP should consider mainly the
large-scale fading between CPU and APs. In this paper, three AP clustering
approaches, namely Dynamic clustering (DC), Static clustering (SC) and
integrated DC and SC (IDSC), are taken into account and presented as follows:

  1. (i)

DC: The DC algorithm consists of two steps. In the first step, the AP
clustering can be implemented by K-medoids method [26], and
$\left\\{\mathcal{C}_{l}\right\\}_{l=1}^{L}$ is then obtained. In the second
step, the CAP of each cluster needs to be determined. Following (II-A), let
$\alpha^{\text{CC-LoS}}_{q}$ denote the LoS complex gain between CPU and the
$q$-th AP, while $\alpha^{\text{CA-LoS}}_{q,q^{{}^{\prime}}}$ denote the LoS
complex gain between the $q$-th AP and the $q^{{}^{\prime}}$-th AP in the same
cluster. Then, the total LoS channel gain corresponding to AP $q$ is
calculated as $\Psi_{q}=|\alpha^{\text{CC-
LoS}}_{q}|+\sum_{q^{{}^{\prime}}\in\mathcal{C}_{l},q^{{}^{\prime}}\neq
q}|\alpha^{\text{CA-LoS}}_{q,q^{{}^{\prime}}}|$, and the CAP in cluster $l$
can be selected as $\text{CAP}_{l}=\operatorname*{argmax}_{q}\Psi_{q}$.

  2. (ii)

SC: As we can see from DC that the complexity of K-medoids increases with the
number of APs. Moreover, the selected CAPs of different AP clusters may be
close to each other, leading to high inter-cluster interference. Hence, from
the perspective of practical implementation, in SC, we can simply divide the
whole communication area into $L$ sub-areas, and the APs located within a sub-
area are classified as one AP cluster. Instead of dynamically selecting CAP,
each sub-area manually deploys one CAP at its central point.

  3. (iii)

IDSC: Followed by DC and SC methods, it is intuitive to propose an integrated
DC and SC method. Specifically, in IDSC, the AP clustering is achieved in the
same way as the SC method, while the CAP selection follows the approach in the
DC method.

######  Remark 3.

As $L$ increases, some AP clusters may only have a single AP, i.e., CAP. In
this case, it is assumed that the CAP receives signal via CPU-to-CAP
fronthaul, and then directly communicates with devices. Furthermore, when $L$
is increased to $Q$, i.e., $L=Q$, the proposed MDD-based wireless fronthaul
degrades to the conventional single-tier fronthaul network, which can be
deemed as a benchmark for comparison in Section VI.

###  III-B Device Selection

The device selection here is also known as the user-centric clustering [4, 3].
After AP clustering, since all the APs within a cluster are in vicinity to
each other and also receive the fronthaul signals from the same CAP, we assume
that all the APs in one cluster serve the same set of devices. In other words,
in our proposed user-centric CF-mMIMO, devices actively select AP clusters
rather than individual APs. To implement the device selection, let
$v_{l,u}=\sum_{q\in\mathcal{C}_{l}}\left\|\tilde{\boldsymbol{h}}_{l,q,u}\right\|_{2}^{2}$
denote the channel gain between the $l$-th cluster and $u$-th device. The
selection process consists of several steps: (i) To begin with, we set
$\mathcal{G}_{u}=\mathcal{U}_{l}=\emptyset$ and
$\mathcal{L}=\left\\{1,...,L\right\\}$, $\forall u,l$, and sequentially assign
an initial cluster to each device $u$, following the rule of
$l^{\ast}=\operatorname*{argmax}_{l\in\mathcal{L}}{v_{l,u}}$. Then, we update
$\mathcal{G}_{u}=\mathcal{G}_{u}\bigcup\left\\{l^{\ast}\right\\}$ and
$\mathcal{U}_{l}=\mathcal{U}_{l}\bigcup\left\\{u\right\\}$. During this step,
if $\left|\mathcal{U}_{l}\right|>U_{\text{max}}$, the $l$-th cluster can no
longer be chosen, and $\mathcal{L}=\mathcal{L}\backslash l$; (ii) Considering
the fairness among devices, we re-rank the devices in the order of total
channel gain, i.e.,
$v_{u}=\sum_{l\in\mathcal{G}_{u}}\sum_{q\in\mathcal{C}_{l}}\left\|\tilde{\boldsymbol{h}}_{l,q,u}\right\|_{2}^{2}$.
The device having the smallest total channel gain is allowed to first choose
the optimal $l^{\ast}$ from the remaining clusters. The corresponding sets are
then updated; (iii) Step (ii) is repeated until
$\left|\mathcal{U}_{l}\right|=U_{\text{max}}$ or
$\left|\mathcal{G}_{u}\right|=C_{\text{max}}$ for $\forall
l\in\left\\{1,...,L\right\\}$ and $\forall u\in\left\\{1,...,U\right\\}$.

###  III-C $\mathcal{M}_{\text{CC}}\ \&\ \mathcal{M}_{\text{CA}}$ Assignment

Intuitively, the CAP-to-AP link has the priority over the CPU-to-CAP link to
choose subcarriers. The reason can be explained as follows: (i) The CPU having
more antennas and larger transmit power is capable of mitigating the
interference between CAPs and overcoming the high path loss; (ii) On the
contrary, CAP has to serve multiple APs in the cluster with relatively less
antennas and transmit power, while the quality of CAP-to-AP link is also
subject to the inter-cluster interference. Therefore, during the
initialization process, it is reasonable to allocate the subcarriers with
better channel quality to CAP-to-AP link, thereby guaranteeing the reliable
fronthaul rate.

In this regard, we adopt the unfair greedy algorithm to first allocate the
subcarriers to CAP-to-AP link, and the remaining subcarriers are then assigned
to CPU-to-CAP link. Note that the maximum ratio transmission (MRT) precoding
strategy is applied to reduce the computational complexity during the
subcarrier selection. Next, we mathematically describe the process of the
proposed approach. Let
$\mathcal{M}_{\text{CC}}=\mathcal{M}_{\text{CA}}=\emptyset$. Given that CAP
$l$ can obtain $\boldsymbol{h}_{l,q}[\tilde{m}],\ \forall
q\in\mathcal{C}_{l},\tilde{m}\in\mathcal{M}_{\text{FH}}$, the channel gain
between CAP $l$ and AP $q$ over the $\tilde{m}$-th subcarrier is derived as
$w_{l,q}[\tilde{m}]=\left\|\boldsymbol{h}_{l,q}[\tilde{m}]\right\|_{2}^{2}$.
Then, we find the subcarrier $\bar{m}$ having the maximum channel gain in
$\mathcal{C}_{l}$, i.e.,
$\bar{m}=\operatorname*{argmax}_{\tilde{m}}\sum_{q\in\mathcal{C}_{l}}w_{l,q}[\tilde{m}]$,
and update $\mathcal{M}_{\text{CA}}$ with
$\mathcal{M}_{\text{CA}}=\mathcal{M}_{\text{CA}}\bigcup\left\\{\bar{m}\right\\}$.
All the CAPs sequentially select their optimal subcarriers and incorporate
them into $\mathcal{M}_{\text{CA}}$, until the size of
$\mathcal{M}_{\text{CA}}$ is satisfied.

##  IV Proposed Solution for the End-to-End Rate Optimization

With the methods proposed in Section III, the results of AP clustering, device
selection and subcarrier sets assignment can be obtained for the particular
$L$ and sizes of $\mathcal{M}_{\text{CC}}$ and $\mathcal{M}_{\text{CA}}$.
Then, the original problem in (9) is divided into two independent sub-
problems, that is, the optimization of two fronthaul links relying on the
power allocation and subcarrier distribution at the CPU and CAPs for conveying
different devices’ data, and the optimization of access links related to the
power allocation at APs, which can be expressed as follows:

$\displaystyle\left(\mathcal{P}_{1}\right)\ \ \ $
$\displaystyle\underset{\left\\{\boldsymbol{v}\right\\},\left\\{p_{l,q,u}\right\\}}{\text{maximize}}\
\min_{u\in\left\\{1,...,U\right\\}}\left\\{C_{u}^{\text{AD}}\right\\},\
\text{s.t.}\ ~{}~{}(\ref{eq:MDD-THz:opt}\text{j}),$ (10)

and

$\displaystyle\left(\mathcal{P}_{2}\right)\ \ \ $
$\displaystyle\underset{\left\\{\boldsymbol{f}\right\\},\left\\{\boldsymbol{w}\right\\},\left\\{\gamma\right\\},\left\\{p_{l,u}[m]\right\\},\left\\{p_{l,q,u}[\bar{m}]\right\\}}{\text{maximize}}\
\min_{u\in\left\\{1,...,U\right\\}}\left\\{C_{u}^{\text{CC}},C_{u}^{\text{CA}}\right\\}$
(11a) s.t. $\displaystyle~{}~{}(\ref{eq:MDD-THz:opt}\text{f}),(\ref{eq:MDD-
THz:opt}\text{g}),(\ref{eq:MDD-THz:opt}\text{h}),(\ref{eq:MDD-
THz:opt}\text{i}).$ (11b)

###  IV-A The Optimization of DL Transmissions

During the optimization of resource allocation, we assume that the regularized
zero-forcing (RZF) precoding is applied. Then, we have
$\boldsymbol{V}^{\text{RZF}}_{l}=\left[\boldsymbol{v}^{\text{RZF}}_{l,1},...,\boldsymbol{v}^{\text{RZF}}_{l,\left|\mathcal{U}_{l}\right|}\right]=\boldsymbol{H}_{l}\left(\boldsymbol{H}_{l}^{H}\boldsymbol{H}_{l}+\epsilon\boldsymbol{I}_{\left|\mathcal{U}_{l}\right|}\right)^{-1}$,
where
$\small\boldsymbol{v}^{\text{RZF}}_{l,u}\in\mathbb{C}^{\left|\mathcal{C}_{l}\right|N^{\text{AP}}\times
1}=\left[\left(\boldsymbol{v}_{l,1,u}^{\text{RZF}}\right)^{T},...,\left(\boldsymbol{v}_{l,\left|\mathcal{C}_{l}\right|,u}^{\text{RZF}}\right)^{T}\right]^{T}$,
$\small\boldsymbol{H}_{l}\in\mathbb{C}^{\left|\mathcal{C}_{l}\right|N^{\text{AP}}\times\left|\mathcal{U}_{l}\right|}=\left[\boldsymbol{h}_{l,1},...,\boldsymbol{h}_{l,\left|\mathcal{U}_{l}\right|}\right]$,
$\epsilon>0$ is the regularization factor. Note that
$\boldsymbol{v}_{l,1,u}^{\text{RZF}}$ is the normalized precoder following
$\small\boldsymbol{v}_{l,q,u}^{\text{RZF}}\leftarrow{\boldsymbol{v}_{l,q,u}^{\text{RZF}}}/{\left\|\boldsymbol{v}^{\text{RZF}}_{l,u}\right\|_{2}}$.
Correspondingly, the optimization $\left(\mathcal{P}_{1}\right)$ can be
reformed as

$\displaystyle\underset{\left\\{p_{l,q,u}\right\\}}{\text{maximize}}\min_{u\in\left\\{1,...,U\right\\}}$
$\displaystyle{\scriptsize\left\\{\text{SINR}_{u}=\frac{\left|\sum_{l\in\mathcal{G}_{u}}\boldsymbol{h}_{l,u}^{H}\tilde{\boldsymbol{P}}_{l,u}\boldsymbol{v}_{l,u}^{\text{RZF}}\right|^{2}}{\sum_{l\in\mathcal{G}_{u}}\sum_{u^{\prime}\in\mathcal{U}_{l},u^{\prime}\neq
u}\left|\boldsymbol{h}_{l,u}^{H}\tilde{\boldsymbol{P}}_{l,u^{\prime}}\boldsymbol{v}^{\text{RZF}}_{l,u^{\prime}}\right|^{2}+\sum_{l^{\prime}\notin\mathcal{G}_{u}}\sum_{u^{\prime}\in\mathcal{U}_{l^{\prime}}}\left|\boldsymbol{h}_{l^{\prime},u^{\prime}}^{H}\tilde{\boldsymbol{P}}_{l^{\prime},u^{\prime}}\boldsymbol{v}^{\text{RZF}}_{l^{\prime},u^{\prime}}\right|^{2}+\sigma^{2}_{u}}\right\\}}$
(12a) $\displaystyle\text{s.t.}\
\sum_{u\in\mathcal{U}_{l}}p_{l,q,u}\left\|\boldsymbol{v}^{\text{RZF}}_{l,q,u}\right\|_{2}^{2}\leq
P_{\text{AP}},\forall
l\in\left\\{1,...,L\right\\},q\in\left\\{1,...,Q\right\\}.$ (12b)

Upon introducing a slack variable $\chi$ associated with SINR, an alternative
optimization problem can be formulated as

$\displaystyle\underset{\left\\{p_{l,q,u}\right\\}}{\text{minimize}}\
\sum_{l=1}^{L}\sum_{u\in\mathcal{U}_{l}}\left\|\boldsymbol{P}_{l,u}\right\|_{\text{F}},$
(13a) $\displaystyle\text{s.t.}\
{\scriptsize\left\|\left[\underbrace{...,\boldsymbol{h}_{l,u}^{H}\tilde{\boldsymbol{P}}_{l,u^{\prime}}\boldsymbol{v}^{\text{RZF}}_{l,u^{\prime}},...,}_{\sum_{l\in\mathcal{G}_{u}}(\left|\mathcal{U}_{l}\right|-1)\
\text{elements}}\underbrace{...,\boldsymbol{h}_{l^{\prime},u^{\prime}}^{H}\tilde{\boldsymbol{P}}_{l^{\prime},u^{\prime}}\boldsymbol{v}^{\text{RZF}}_{l^{\prime},u^{\prime}},...,}_{\sum_{l^{\prime}\notin\mathcal{G}_{u}}\left|\mathcal{U}_{l^{\prime}}\right|\
\text{elements}}\sqrt{\sigma^{2}_{u}}\right]\right\|\leq\left|\sum_{l\in\mathcal{G}_{u}}\boldsymbol{h}_{l,u}^{H}\tilde{\boldsymbol{P}}_{l,u}\boldsymbol{v}_{l,u}^{\text{RZF}}\right|\sqrt{\frac{1}{\chi}}},\
\forall u\in\left\\{1,...,U\right\\},$ $\displaystyle\text{and}\ (\ref{eq:MDD-
THz:SE_resub1}\text{b}),$ (13b)

where the introduction of (13a) aims to accelerate the convergence rate.
Accordingly, the problem (12) can be equivalently solved by finding the
maximum value within the set $\left\\{\chi\right\\}$, which makes the second-
order cone optimization in (13) feasible. In order to search the optimal
$\chi$, the upper bound of $\chi$ is required to be determined first, which is
described as

$\small\chi_{\text{upper}}=\underset{{u\in\left\\{1,...,U\right\\}}}{\min}\left\\{\underset{\left\\{p_{l,q,u}\right\\}}{\text{maximize}}{\left|\sum_{l\in\mathcal{G}_{u}}\boldsymbol{h}_{l,u}^{H}\tilde{\boldsymbol{P}}_{l,u}\boldsymbol{v}_{l,u}^{\text{RZF}}\right|^{2}}/{\sigma^{2}_{u}},\
\text{s.t.}\ (\ref{eq:MDD-THz:SE_resub1}\text{b})\right\\}.$ (14)

Then, the bisection method (e.g., [1, Algorithm 7.5]) can be applied to derive
the solution.

###  IV-B The Optimization of Fronthaul Transmissions

The objective of $\left(\mathcal{P}_{2}\right)$ is to maximize the fronthaul
rates under the fairness constraint. As seen in (11), given that the
assignment of the subcarrier sets for two MDD-based fronthaul links is fixed,
the optimization of $C^{\text{CC}}$ and $C^{\text{CA}}$ can be implemented in
a parallel way. The RZF precoding is assumed to be adopted during resource
allocation over two fronthaul links. More specifically, we have
$\small\boldsymbol{F}^{\text{RZF}}[m]\in\mathbb{C}^{N^{\text{CPU}}\times
L}=\left[\boldsymbol{f}^{\text{RZF}}_{1,u}[m],...,\boldsymbol{f}^{\text{RZF}}_{L,u}[m]\right]=\boldsymbol{H}[m]\left(\boldsymbol{H}^{H}[m]\boldsymbol{H}[m]+\epsilon\boldsymbol{I}_{L}\right)^{-1}$
and
$\small\boldsymbol{W}^{\text{RZF}}_{l}[\bar{m}]\in\mathbb{C}^{N^{\text{AP}}\times\left|\mathcal{C}_{l}\right|}=\left[\boldsymbol{w}^{\text{RZF}}_{l,1,u}[\bar{m}],...,\boldsymbol{w}^{\text{RZF}}_{l,\left|\mathcal{C}_{l}\right|,u}[\bar{m}]\right]=$
$\small\boldsymbol{H}_{l}[\bar{m}]\left(\boldsymbol{H}_{l}^{H}[\bar{m}]\boldsymbol{H}_{l}[\bar{m}]+\epsilon\boldsymbol{I}_{\left|\mathcal{C}_{l}\right|}\right)^{-1}$,
where $\small\boldsymbol{H}[m]\in\mathbb{C}^{N^{\text{CPU}}\times
L}=\left[\boldsymbol{h}_{1}[m],...,\boldsymbol{h}_{L}[m]\right]$,
$\small\boldsymbol{H}_{l}[\bar{m}]\in\mathbb{C}^{N^{\text{AP}}\times\left|\mathcal{C}_{l}\right|}=\left[\boldsymbol{h}_{l,1}[\bar{m}],...,\boldsymbol{h}_{l,\left|\mathcal{C}_{l}\right|}[\bar{m}]\right]$.
Note that each column of $\boldsymbol{F}^{\text{RZF}}[m]$ and
$\boldsymbol{W}^{\text{RZF}}_{l}[\bar{m}]$ has been normalized, i.e.,
$\small\boldsymbol{f}^{\text{RZF}}_{l,u}[m]\leftarrow\boldsymbol{f}^{\text{RZF}}_{l,u}[m]/\left\|\boldsymbol{f}^{\text{RZF}}_{l,u}[m]\right\|_{2}$
and
$\small\boldsymbol{w}^{\text{RZF}}_{l,q,u}[\bar{m}]\leftarrow\boldsymbol{w}^{\text{RZF}}_{l,q,u}[\bar{m}]/\left\|\boldsymbol{w}^{\text{RZF}}_{l,q,u}[\bar{m}]\right\|_{2}$.
Then, the optimization of CPU-to-CAP link can be formulated as

$\displaystyle\left(\mathcal{P}_{3}\right)\
\underset{\left\\{p_{l,u}[m]\right\\},\left\\{\gamma_{l,u,m}\right\\}}{\text{maximize}}\min_{u\in\left\\{1,...,U\right\\},l\in\mathcal{G}_{u}}\left\\{C_{l,u}^{\text{CC}}\right\\}$
(15a) $\displaystyle\text{s.t.}\
\sum_{m\in\mathcal{M}_{\text{CC}}}\sum_{l=1}^{L}\sum_{u=1}^{U}\gamma_{l,u,m}p_{l,u}[m]\leq
P_{\text{CPU}},\ \text{and}~{}~{}(\ref{eq:MDD-THz:opt}\text{f}).$ (15b)

Obviously, $\left(\mathcal{P}_{3}\right)$ is a non-convex optimization problem
due to the nature of objective function and binary variables
$\left\\{\gamma_{l,u,m}\right\\}$ along with constraint (9f). To this end, in
what follows, we present a tractable form of (15) by exploiting the
relationship between continuous and binary variables, based on which the
combination of quadratic transform (QT) and bisection method can be leveraged
to solve it in an iterative way.

####  IV-B1 Binary Reduction of $\gamma$

The binary variable $\gamma$ and the continuous variable $p$ are strongly
coupled, and their relationship can be demonstrated by the following Lemma:

######  Lemma 1.

There exist only two potential combinations of $\gamma_{l,u,m}$ and
$p_{l,u}[m]$ in the optimal solution to $\left(\mathcal{P}_{3}\right)$, which
are
$(\gamma_{l,u,m}^{\ast},p_{l,u}^{\ast}[m])\in\left\\{(0,p_{l,u}[m]=0),(1,p_{l,u}[m]\neq
0)\right\\}$, where $p_{l,u}[m]\neq 0$.

###### Proof.

Refer to [17, Lemma 1] ∎

Based on Lemma 1, only $p_{l,u}[m]$ is needed during the optimization process.
The feasible $\gamma_{l,u,m}^{\ast}$ can be subsequently obtained from the
optimal $p_{l,u}^{\ast}[m]$, which is given as

$\gamma_{l,u,m}^{\ast}=\begin{cases}0,&\frac{p_{l,u}^{\ast}[m]}{P_{\text{CPU}}}<\zeta\\\
1,&\frac{p_{l,u}^{\ast}[m]}{P_{\text{CPU}}}\geq\zeta\end{cases},$ (16)

where $\zeta$ is a very small number, implying that a small value of
$p_{l,u}^{\ast}[m]$ can be deemed as zero.

####  IV-B2 Tractable Constraint

According to Lemma 1, (9f) can be substituted with the new constraint only
related to $p_{l,u}[m]$, which is given by

$\left\|\boldsymbol{p}_{l}[m]\right\|_{0}\leq 1,\forall
l\in\left\\{1,...,L\right\\},m\in\mathcal{M}_{\text{CC}},$ (17)

where
$\boldsymbol{p}_{l}[m]=\left[p_{l,1}[m],...,p_{l,\left|\mathcal{U}_{l}\right|}[m]\right]^{T}$.
As $L^{0}$-norm is non-convex, it has to be approximated by the tractable
convex functions. In particular, we resort to the exp-based surrogate function
for the sake of better performance, which can be expressed as [27, 28]

$\vspace{-0.3cm}\left\|\boldsymbol{p}_{l}[m]\right\|_{0}\approx\sum_{u=1}^{\left|\mathcal{U}_{l}\right|}\left(1-\exp\left(-\psi
p_{l,u}[m]\right)\right):=\Xi(\boldsymbol{p}_{l}[m]),$ (18)

where $\psi\gg 0$ is a sufficiently large value to guarantee the
approximation. Then, since $\Xi(\boldsymbol{p}_{l}[m])$ is concave with
respect to $p_{l,u}[m]\in\mathbb{R}^{+}$, based on its subgradient inequality,
we have

$\displaystyle\Xi(\boldsymbol{p}_{l}[m])\leq\Xi(\boldsymbol{p}_{l}^{(i)}[m])+\sum_{u=1}^{\left|\mathcal{U}_{l}\right|}\left[\psi\exp\left(-\psi
p_{l,u}^{(i)}[m]\right)\left(p_{l,u}[m]-p_{l,u}^{(i)}[m]\right)\right]:=\tilde{\Xi}^{(i)}(\boldsymbol{p}_{l}[m]),$
(19)

where $p_{l,u}^{(i)}[m]$ is the feasible value of $p_{l,u}[m]$ at the $i$-th
iteration. Consequently, $\tilde{\Xi}(\boldsymbol{p}_{l}[m])$ is the convex
function with respect to $p_{l,u}[m]$, and the constraint

$\displaystyle\tilde{\Xi}^{(i)}(\boldsymbol{p}_{l}[m])\leq 1,\forall
l\in\left\\{1,...,L\right\\},m\in\mathcal{M}_{\text{CC}}$ (20)

can substitute for (17).

####  IV-B3 Tractable Objective Function

$C_{l,u}^{\text{CC}}\\!=\\!b^{\text{CC}}\\!\sum_{m\in\mathcal{M}_{\text{CC}}}\\!\log\left(\\!1\\!+\\!{A_{l,u,m}\left(\boldsymbol{p}[m]\right)}/{B_{l,u,m}\left(\boldsymbol{p}[m]\right)}\\!\right)$
in the objective function (15a) belongs to the multiple-ratio fractional
programming (MRFP) problem, where
$A_{l,u,m}\left(\boldsymbol{p}[m]\right)=p_{l,u}[m]\left|\boldsymbol{h}_{l}^{H}[m]\boldsymbol{f}^{\text{RZF}}_{l,u}[m]\right|^{2}$,
$B_{l,u,m}\left(\boldsymbol{p}[m]\right)\\!=\\!\sum_{l^{\prime}\neq
l}\\!\sum_{u^{\prime}\in\mathcal{U}_{l^{\prime}}}\\!\times$
$p_{l^{\prime},u^{\prime}}[m]\\!\left|\boldsymbol{h}_{l}^{H}[m]\boldsymbol{f}^{\text{RZF}}_{l^{\prime},u^{\prime}}[m]\right|^{2}\\!+\\!\left|\mathbb{E}\left\\{\text{SI}_{l}\right\\}\right|^{2}\\!+\\!\sigma_{\text{n}}^{2}$,
and
$\boldsymbol{p}[m]\in\mathbb{R}^{\sum_{l}\left|\mathcal{U}_{l}\right|\times
1}$ denotes the power allocation vector for the $m$-th subcarrier. Since
$\log(\cdot)$ function is non-decreasing and
$\frac{A_{l,u,m}\left(\boldsymbol{p}[m]\right)}{B_{l,u,m}\left(\boldsymbol{p}[m]\right)}$
satisfies the convex-over-concave form, $\left(\mathcal{P}_{3}\right)$ can be
equivalently formulated as

$\displaystyle\left(\mathcal{P}_{3-1}\right)\
\underset{\left\\{p_{l,u}[m]\right\\},\left\\{z_{l,u,m}^{(i)}\right\\}}{\text{maximize}}\tilde{\chi}$
(21a) $\displaystyle\text{s.t.}\ z_{l,u,m}^{(i)}\in\mathbb{R},\
p_{l,u}^{(i)}[m]\in\mathbb{R}^{+},\ \forall
l\in\left\\{1,...,L\right\\},u\in\left\\{1,...,U\right\\},m\in\mathcal{M}_{\text{CC}},$
(21b) $\displaystyle
b^{\text{CC}}\sum_{m\in\mathcal{M}_{\text{CC}}}\log\left(1+2z_{l,u,m}^{(i)}\sqrt{A_{l,u,m}\left(\boldsymbol{p}[m]\right)}-\left(z_{l,u,m}^{(i)}\right)^{2}B_{l,u,m}\left(\boldsymbol{p}[m]\right)\right)\geq\tilde{\chi},$
$\displaystyle\forall u\in\left\\{1,...,U\right\\},l\in\mathcal{G}_{u},$ (21c)
$\displaystyle\sum_{m\in\mathcal{M}_{\text{CC}}}\sum_{l=1}^{L}\sum_{u=1}^{U}p_{l,u}[m]\leq
P_{\text{CPU}},\ \text{and}~{}~{}(\ref{eq:MDDTHz:finacon}),$ (21d)

where $z_{l,u,m}^{(i)}$ is the feasible value at the $i$-th iteration.

To verify this, we can first rewrite $\left(\mathcal{P}_{3}\right)$ as the
maximization of $\tilde{\chi}$ subject to (21d) and $\tilde{\chi}\leq
b^{\text{CC}}\sum_{m\in\mathcal{M}_{\text{CC}}}\log\left(1+{A_{l,u,m}\left(\boldsymbol{p}[m]\right)}/{B_{l,u,m}\left(\boldsymbol{p}[m]\right)}\right)$.
According to the principle of QT [29], the latter constraint can be reformed
as

$\small\tilde{\chi}\leq{\text{maximize}}_{\left\\{z_{l,u,m}^{(i)}\right\\}}\ \
b^{\text{CC}}\sum_{m\in\mathcal{M}_{\text{CC}}}\log\left(1+2z_{l,u,m}^{(i)}\sqrt{A_{l,u,m}\left(\boldsymbol{p}[m]\right)}-\left(z_{l,u,m}^{(i)}\right)^{2}B_{l,u,m}\left(\boldsymbol{p}[m]\right)\right).$
(22)

As this new constraint is a less-than-max inequality,
${\text{maximize}}_{\left\\{z_{l,u,m}^{(i)}\right\\}}$ can be integrated into
${\text{maximize}}_{\left\\{p_{l,u}[m]\right\\}}$, as in (21a).

Input :
$\mathcal{M}_{\text{CC}},\left\\{\mathcal{C}_{l}\right\\}_{l=1}^{L},\left\\{\mathcal{U}_{l}\right\\}_{l=1}^{L},\left\\{\boldsymbol{F}^{\text{RZF}}[m]\right\\}_{m=1}^{\left|\mathcal{M}_{\text{CC}}\right|},\left\\{\boldsymbol{H}[m]\right\\}_{m=1}^{\left|\mathcal{M}_{\text{CC}}\right|}$;

1 Initialization: Randomly set $\left\\{\gamma_{l,u,m}^{(0)}\right\\}$ under
the constraint (15c);

2 Initialize
$p_{l,u}^{(0)}[m]=\frac{P_{\text{CPU}}}{\left|\mathcal{M}_{\text{CC}}\right|L},\forall
l\in\left\\{1,...,L\right\\},u\in\left\\{1,...,U\right\\},m\in\mathcal{M}_{\text{CC}}$;

3 Compute
$z_{l,u,m}^{(0)}=\frac{\sqrt{A_{l,u,m}\left(\boldsymbol{p}^{(0)}[m]\right)}}{B_{l,u,m}\left(\boldsymbol{p}^{(0)}[m]\right)}$
and set $i=0$;

4 Set $\kappa$, $\tilde{\chi}_{\text{lower}}=0$ and
$\tilde{\chi}_{\text{upper}}=\underset{u\in\left\\{1,...,U\right\\},l\in\mathcal{G}_{u}}{\min}\underset{\left\\{p_{l,u}[m]\right\\}}{\text{maximize}}\left\\{b^{\text{CC}}\sum_{m\in\left|\mathcal{M}_{\text{CC}}\right|}\log\left(1+\frac{p_{l,u}[m]\left|\boldsymbol{h}_{l}^{H}[m]\boldsymbol{f}^{\text{RZF}}_{l,u}[m]\right|^{2}}{\sigma^{2}_{n}}\right),\text{s.t.
}\sum_{m\in\mathcal{M}_{\text{CC}}}p_{l,u}[m]\leq P_{\text{CPU}}\right\\}$;

5 [QT and Bisection Iteration]  __

6 repeat

7 Set
$\tilde{\chi}=\frac{\tilde{\chi}_{\text{upper}}+\tilde{\chi}_{\text{lower}}}{2}$;

8 For the fixed $\left\\{z_{l,u,m}^{(i)}\right\\}$, update
$\left\\{p_{l,u}^{(i+1)}[m]\right\\}$ by solving the following optimization
problem:

9
$\displaystyle\underset{\left\\{p_{l,u}[m]\right\\}}{\text{minimize}}\sum_{m\in\mathcal{M}_{\text{CC}}}\left\|\boldsymbol{p}[m]\right\|_{1},\text{s.t.}\
(\ref{eq:MDD-THz:SE_sub3_1}c)\ \text{and}\ (\ref{eq:MDD-
THz:SE_sub3_1}d)\vspace{-0.3cm}$ (23) if  _( 23) is feasible_ then

10 Update $\tilde{\chi}_{\textup{lower}}=\tilde{\chi}$ and
$\left\\{p_{l,u}^{\ast}[m]\right\\}=\left\\{p_{l,u}^{(i+1)}[m]\right\\}$;

11 Compute $\left\\{\gamma_{l,u,m}^{(i+1)}\right\\}$ as in (16) and update
$\left\\{\gamma_{l,u,m}^{\ast}\right\\}=\left\\{\gamma_{l,u,m}^{(i+1)}\right\\}$;

12 Update
$z_{l,u,m}^{(i+1)}=\frac{\sqrt{A_{l,u,m}\left(\boldsymbol{p}^{(i+1)}[m]\right)}}{B_{l,u,m}\left(\boldsymbol{p}^{(i+1)}[m]\right)}$
and set $i=i+1$;

13 if  _$\tilde{\chi}_{\textup{upper}}-\tilde{\chi}_{\textup{lower}} <\kappa$_
then

14
$\tilde{\chi}_{\text{upper}}=\tilde{\chi}_{\text{upper}}+\chi_{\text{com}}$;

15 end if

16

17 else

18 $\tilde{\chi}_{\textup{upper}}=\tilde{\chi}$;

19

20 end if

21

22 until  _convergence_ ;

23 end

Output :
$C^{\text{CC}}=\min_{u\in\left\\{1,...,U\right\\},l\in\mathcal{G}_{u}}\left\\{b^{\text{CC}}\sum_{m\in\mathcal{M}_{\text{CC}}}\log\left(1+\frac{A_{l,u,m}\left(\boldsymbol{p}[m]\right)}{B_{l,u,m}\left(\boldsymbol{p}[m]\right)}\right)|\left\\{p_{l,u}^{\ast}[m]\right\\}\right\\}$
and $\left\\{\gamma_{l,u,m}^{\ast}\right\\}$

Algorithm 1 The Optimization of CPU-to-CAP fronthaul

Accordingly, $\left(\mathcal{P}_{3-1}\right)$ can be solved by applying the QT
and bisection methods to iteratively optimize the primal variable
$\left\\{p_{l,u}[m]\right\\}$ and the auxiliary variables $z_{l,u,m}^{(i)}$
and $p_{l,u}^{(i)}[m]$. To be more specific, we can initialize the set
$\left\\{p_{l,u}^{(0)}[m]\right\\}$ by equally allocating the power and
subcarriers among devices, and then the optimal set
$\left\\{z_{l,u,m}^{(0)}\right\\}$ can be obtained based on the QT principle,
i.e.,
$z_{l,u,m}^{(0)}=\frac{\sqrt{A_{l,u,m}\left(\boldsymbol{p}^{(0)}[m]\right)}}{B_{l,u,m}\left(\boldsymbol{p}^{(0)}[m]\right)}$.
Next, we solve the optimization problem, i.e.,
${\text{minimize}}_{\left\\{p_{l,u}[m]\right\\}}\sum_{m\in\mathcal{M}_{\text{CC}}}\left\|\boldsymbol{p}[m]\right\|_{1}$,
subject to (21c)-(21e), with
$\tilde{\chi}=\frac{\tilde{\chi}_{\text{upper}}+\tilde{\chi}_{\text{lower}}}{2}$,
to obtain the optimal set $\left\\{p_{l,u}^{(1)}[m]\right\\}$. It is
noteworthy that after the $i$-th iteration with the $\tilde{\chi}$ limited to
$\tilde{\chi}^{(i)}_{\text{upper}}$ and new
$\left\\{p_{l,u}^{(i)}[m]\right\\}$, the updated
$\left\\{z_{l,u,m}^{(i)}\right\\}$ may lead to that the maximum of the left-
hand side of (21c) is larger than $\tilde{\chi}^{(i)}_{\text{upper}}$, due to
the nondecreasing feature of QT [29]. Hence, we introduce a compensation
factor $\chi_{\text{com}}$ to overcome this problem, that is, after each
iteration, if
$\tilde{\chi}_{\textup{upper}}-\tilde{\chi}_{\textup{lower}}<\kappa$, we set
$\tilde{\chi}^{(i)}_{\text{upper}}=\tilde{\chi}^{(i)}_{\text{upper}}+\chi_{\text{com}}$,
where $\kappa$ is a predefined very small value. The above-mentioned processes
can be iteratively implemented until convergence is achieved. In summary, the
overall optimization is stated as Algorithm 1.

Analogously, the optimization of CAP-to-AP fronthaul can be solved using the
same method proposed in Algorithm 1, since the optimization of these two
fronthaul links have the similar formulation and constraints as shown in (4),
(II-A2) and (9).

###  IV-C Overall End-to-End Optimization

The above-proposed algorithms for the optimization of access and fronthaul
links are based on the given results of AP clustering, device selection and
the assignment of subcarrier sets. Next, we aim to iteratively assign
different numbers of subcarriers between two MDD-based fronthaul links, and
select the optimal number of AP clusters in order to achieve the end-to-end
optimization in (9).

1 Initialization: Set
$\left|\mathcal{M}_{\text{CA}}\right|=\left|\mathcal{M}_{\text{CC}}\right|=\frac{1}{2}\left|\mathcal{M}_{\text{FH}}\right|$,
$L_{\text{ini}}=\frac{U}{U_{\text{max}}}$, $i=1$, $l_{\text{step}}$ and
$m_{\text{step}}$;

2 while  _$L_{\text{ini}}+(i-1)l_{\textup{step}}\leq Q$_ do

3 Obtain
$\left\\{\mathcal{C}_{l}\right\\}_{l=1}^{L},\left\\{\mathcal{U}_{l}\right\\}_{l=1}^{L}$
and $\left\\{\mathcal{G}_{u}\right\\}_{u=1}^{U}$ using the heuristic methods
in Section III-A;

4 Compute $C_{i}^{\text{AD}}$ by solving (13);

5 repeat

6 Set $j=1$;

7 Obtain $\mathcal{M}_{\text{CC}}$ and $\mathcal{M}_{\text{CA}}$ using the
heuristic method in Section III-C;

8 Compute $C_{i}^{\text{CC}}$ and $C_{i}^{\text{CA}}$ by Algorithm 1;

9 if  _$C_{i}^{\text{CC}} >C_{i}^{\text{CA}}$_ then

10
$\left|\mathcal{M}_{\text{CA}}\right|=\left|\mathcal{M}_{\text{CA}}\right|+\vartheta^{(j-1)}\cdot
m_{\text{step}}$;

11 else

12
$\left|\mathcal{M}_{\text{CA}}\right|=\left|\mathcal{M}_{\text{CA}}\right|-\vartheta^{(j-1)}\cdot
m_{\text{step}}$;

13 end if

14 Update $j=j+1$;

15

16 until
_$\left|C^{\text{CC}}_{i}-C^{\text{CA}}_{i}\right|\leq\kappa^{{}^{\prime}}$_ ;

17 Obtain
$C_{i}^{\text{FH}}=\min\left\\{C_{i}^{\text{CC}},C_{i}^{\text{CA}}\right\\}$;

18 Update $i=i+1$;

19

20 end while

Output :
$C^{\ast}=\underset{i}{\text{maximize}}\min\left\\{C_{i}^{\text{AD}},C_{i}^{\text{FH}}\right\\}$

Algorithm 2 Overall End-to-End Rate Optimization

####  IV-C1 Iterative Assignment of Subcarrier Sets

In Section IV-B, given the fixed assignment of subcarrier sets, the
optimization of each fronthaul can be solved by Algorithm 1. However,
considering that the total fronthaul rates is calculated as
$C^{\text{FH}}=\min\left\\{C^{\text{CC}},C^{\text{CA}}\right\\}$, it is
intuitive to highlight the following trade-off: Provided that the transmit
power is fixed, the more subcarriers are allocated, the higher rates can be
achieved, owing to the subcarrier diversity gain. Therefore, in order to
maximize $C^{\text{FH}}$, the sizes of two subcarriers sets (i.e.,
$\mathcal{M}_{\text{CC}}$ and $\mathcal{M}_{\text{CA}}$) can be iteratively
increased or decreased by following a pre-defined step namely
$m_{\text{step}}$, according to the relationship between two fronthaul rates.
To be more specific, after optimizing the two fronthaul links separately for
given
$\left|\mathcal{M}_{\text{CA}}\right|=\frac{1}{2}\left|\mathcal{M}_{\text{FH}}\right|$,
the iterative assignment of subcarrier sets commences. If
$C^{\text{CC}}>C^{\text{CA}}$, the size of $\mathcal{M}_{\text{CA}}$ is
increased to
$\left|\mathcal{M}_{\text{CA}}\right|=\left|\mathcal{M}_{\text{CA}}\right|+\vartheta^{(j-1)}\cdot
m_{\text{step}}$, where $\vartheta$ is a decay factor, $j$ is the current
iteration index. Then, the subcarrier assignment method proposed in Section
III-C is re-calculated. Otherwise, if $C^{\text{CC}}<C^{\text{CA}}$, the
subcarrier assignment process is re-run with the size of
$\mathcal{M}_{\text{CA}}$ decreased to
$\left|\mathcal{M}_{\text{CA}}\right|=\left|\mathcal{M}_{\text{CA}}\right|-\vartheta^{(j-1)}\cdot
m_{\text{step}}$. This process ends up with
$\left|C^{\text{CC}}-C^{\text{CA}}\right|\leq\kappa^{{}^{\prime}}$.

####  IV-C2 Iterative Optimization of Cluster Number

Given that the AP clustering is accomplished by the heuristic method proposed
in Section III-A, the trade-off between access and two fronthaul links only
depends on the number of AP clusters $L$, which can be highlighted as follows:
(i) Changing the number of AP clusters may impose distinct impact on two
fronthaul links. In particular, a smaller $L$ results in more power received
at each CAP and hence the CPU-to-CAP fronthaul rates are increased. By
contrast, a smaller $L$ means that the average number of APs in each AP
cluster is increased, which leads to less power received at APs and causes
lower CAP-to-AP fronthaul rates. (ii) On the other hand, the number of AP
clusters has an ambiguous impact on the AP-to-Device access links. To be more
specific, when $L$ is smaller, the total number of APs serving a particular
device $u$ is increased. However, it is possible that most of the APs have
negligibly small channel gains to the device $u$, and hence the power
transmitted to the device $u$ may cause extra interference to the nearby
devices. By contrast, when $L$ becomes larger, the number of APs within each
cluster is decreased. Then, with the constraints of $C_{\text{max}}$ and
$U_{\text{max}}$, each device may only be served by its nearby AP clusters,
and each AP cluster only allocates power to the devices with good channel
conditions. In this case, the achievable rate of access links relies on the
specific distributions of APs and devices, rather than the number of AP
clusters.

According to the above analysis, the overall end-to-end optimization can be
implemented as follows: (i) We firstly determine the range of $L$ to be
$\frac{U}{U_{\text{max}}}\leq L\leq Q$, where $\frac{U}{U_{\text{max}}}\leq L$
guarantees that every device is served by at least one AP cluster555Note that,
this value range is only applicable to DC, as it is capable of dynamically
adjusting the AP clustering. By contrast, the implementation of AP grouping in
SC and IDSC rely on the sub-area division, and for the sake of practical
application, only few possible numbers of AP clusters will be considered
during simulations in Section VI. . Then, $L$ is initialized to
$L_{\text{ini}}=\frac{U}{U_{\text{max}}}$ and
$\left|\mathcal{M}_{\text{CA}}\right|=\frac{1}{2}\left|\mathcal{M}_{\text{FH}}\right|$;
(ii) Next, we set $i=1$, and $L_{i}=L_{\text{ini}}+(i-1)l_{\text{step}}$,
where $l_{\text{step}}$ is a pre-defined step. Then,
$C_{i}^{\text{CC}},C_{i}^{\text{CA}}$ and $C_{i}^{\text{AD}}$ can be obtained,
followed by the proposed iterative assignment of subcarrier sets. The optimal
fronthaul rate is derived as
$C_{i}^{\text{FH}}=\operatorname*{argmax}_{\left\\{\mathcal{M}_{\text{CC}},\mathcal{M}_{\text{CA}}\right\\}}\min\left\\{C_{i}^{\text{CC}},C_{i}^{\text{CA}}\right\\}$;
(iii) The process continues until $L_{i}=Q$, and the optimal end-to-end rate
can be obtained as
$C^{\ast}=\underset{i}{\text{maximize}}\min\left\\{C_{i}^{\text{AD}},C_{i}^{\text{FH}}\right\\}$.
Algorithm 2 summarizes the overall end-to-end optimization process.

##  V The Advanced Design of MDD Frame Structure

In this section, followed by the proposed solution for jointly optimizing two
fronthaul and access links, we integrate it into the specifically designed MDD
frame structure so as to further improve the end-to-end rate. In order to
exhibit the advantage of MDD, the TDD-based frame structure compatible with
our designed two-tier wireless fronthaul architecture is also presented as a
benchmark.

Figure 2: Frame structure of MDD-enbaled THz fronthaul.

Figure 3: Frame structure of TDD-enbaled THz fronthaul.

###  V-A Frame Structure for MDD-Enabled Fronthaul System

Since two fronthaul transmissions are implemented in the MDD mode over the THz
band and DL communications take place over the sub-6 GHz band, the MDD frame
structure can be designed in a parallel way, as shown in Fig. 3. Specifically,
we consider a frame of transmission period within the coherence time, which is
split into three parallel streams and each of them includes $Z$ subframes. Let
the duration of one subframe be $T_{\text{sub}}$. The CPU-to-CAP, CAP-to-AP
and AP-to-Device transmission durations are assumed to account for the same
proportion within one subframe, i.e.,
$\tau_{\text{CC}}=\tau_{\text{CA}}=\tau_{\text{AD}}=\frac{1}{3}$, and the
total duration of transmission frame is equal to $(Z+{2}/{3})T_{\text{sub}}$.
Then, when $Z\rightarrow\infty$, the rates of two fronthaul and access links
within one transmission frame are $\frac{ZC^{\text{CC}}}{(Z+2/{3})}\approx
C^{\text{CC}},\frac{ZC^{\text{CA}}}{(Z+{2}/{3})}\approx C^{\text{CA}}$ and
$\frac{ZC^{\text{AD}}}{(Z+{2}/{3})}\approx C^{\text{AD}}$, respectively.
Therefore, the end-to-end rate optimization within one transmission frame is
equal to the problem in (9), and can be solved by Algorithm 2.

Notably, in order to avoid the transmission collision, the assumption of
$\tau_{\text{CC}}=\tau_{\text{CA}}=\tau_{\text{AD}}$ is essential to our
proposed MDD frame structure. In this case, although the rate optimization
between fronthaul and access links can not be further tuned via optimizing the
proportion of their durations after solving (9), the proposed frame structure
can significantly improve the end-to-end rates with the aid of three
simultaneous data streams.

1 Initialization: Set
$\left|\mathcal{M}_{\text{CA}}\right|=\left|\mathcal{M}_{\text{CC}}\right|=\left|\mathcal{M}_{\text{FH}}\right|$,
$L_{\text{ini}}=\frac{U}{U_{\text{max}}}$, $i=1$ and $l_{\text{step}}$;

2 while  _$L_{\text{ini}}+(i-1)l_{\textup{step}}\leq Q$_ do

3 Obtain
$\left\\{\mathcal{C}_{l}\right\\}_{l=1}^{L},\left\\{\mathcal{U}_{l}\right\\}_{l=1}^{L}$
and $\left\\{\mathcal{G}_{u}\right\\}_{u=1}^{U}$ using the heuristic methods
in Section III-A;

4 Compute $C_{i}^{\text{AD}}$ by solving (13);

5 Compute $C_{i}^{\text{CC}}$ and $C_{i}^{\text{CA}}$ by Algorithm 1;

6 Solve the convex problem (provided that $Z\rightarrow\infty$):
$\displaystyle
C_{i}=\underset{\tau_{\text{CC}},\tau_{\text{CA}},\tau_{\text{AD}}}{\text{maximize}}\
\min\left\\{\frac{\tau_{\text{CC}}}{\tau_{\text{CC}}+\tau_{\text{CA}}+2\tau_{\text{GP}}}C_{i}^{\text{CC}},\frac{\tau_{\text{CA}}}{\tau_{\text{CC}}+\tau_{\text{CA}}+2\tau_{\text{GP}}}C_{i}^{\text{CA}},\frac{\tau_{\text{AD}}}{\tau_{\text{CC}}+\tau_{\text{CA}}+2\tau_{\text{GP}}}C_{i}^{\text{AD}}\right\\}$
$\displaystyle\text{s.t.}\
\tau_{\text{CC}}+\tau_{\text{CA}}+\tau_{\text{AD}}+\tau_{\text{GP}}\leq 1\
\text{and}\
\tau_{\text{AD}}\leq\tau_{\text{CC}}+\tau_{\text{CA}}+2\tau_{\text{GP}}.$ (24)

7 Set i = i+1;

8

9 end while

Output : $C^{\ast}=\underset{i}{\text{maximize}}\left\\{C_{i}\right\\}$

Algorithm 3 The Overall End-to-End Rate Optimization for TDD-Enabled Fronthaul
System

###  V-B Frame Structure for TDD-Enabled Fronthaul System

As for comparison, we also present the TDD frame structure for our proposed
fronthaul system, which is shown in Fig. 3. Specifically, the TDD transmission
frame is divided into $Z$ subframes. Each subframe accounts for
$T_{\text{sub}}$ duration including two fronthauls, DL transmission and guard
period. Different from MDD-enabled fronthaul, since both two fronthauls in TDD
happen over all the subcarriers of the same THz band, they need to be
accurately separated for avoiding collision and implemented in a sequential
way. Moreover, as DL signals are transmitted over sub-6 GHz band, as shown in
Fig. 3, the DL transmission of the previous subframe can coexist with the
fronthaul transmissions of the current subframe. However, in order to avoid
the transmission delay caused by the previous subframe, we assume
$\tau_{\text{AD}}\leq\tau_{\text{CC}}+\tau_{\text{CA}}+2\tau_{\text{GP}}$ such
that the previous DL transmission can be finished before the current DL
transmission starts. The overall end-to-end rate optimization of the TDD-
enabled fronthaul system is summarized in Algorithm 3. Compared with MDD,
although TDD is not able to balance $C^{\text{CC}}$ and $C^{\text{CA}}$
through adjusting the sizes of $\mathcal{M}_{\text{CC}}$ and
$\mathcal{M}_{\text{CA}}$, it can allocate the time durations for two
fronthaul and access links to achieve the optimal end-to-end rate.

TABLE I: Simulation parameters Default parameters | Value
---|---
CPU’s and AP’s Power budget ($P_{\text{CPU}},P_{\text{AP}}$) |  $(35,45)$ dBm
Transmit and Receive antenna gain ($G_{t},G_{r}$) | (20,20) dBi
Absorption coefficient ($k_{\text{abs}}$) | $0.0033\ \text{m}^{-1}$
Fresnel reflection coefficient | 0.15
Roughness factor | $0.088\times 10^{-3}$
Number of delay taps ($T_{\text{delay}}$) | 6
Number of NLoS path ($N_{\text{ray}}$) | 3
Cyclic prefix length ($N_{\text{CP}}$) | 16
Sampling rate ($T_{\text{s}}$) |  $7.8\times 10^{-12}$ s
NLoS path delay ($\delta$) | $\delta\sim\mathcal{U}[0,N_{\text{CP}}T_{\text{s}}]$
Azimuth AoD ($\phi$) | $\phi\sim\mathcal{U}(-\pi,\pi)$
Elevation AoD ($\theta$) | $\theta\sim\mathcal{U}(-\frac{\pi}{2},\frac{\pi}{2})$

##  VI Simulation Results

###  VI-A Parameters and Setup

In the simulation, we employ the proposed two-tier THz fronthaul for MDD-CF
systems, as shown in Fig. 1, where an indoor industrial workshop with a square
area of $(100\ \text{m}\times 100\ \text{m})$ and a roof height of $10$ m is
considered. The CPU equipped with a $(8\times 8)$ UPA antenna array is
installed on the roof at the central point. The $Q=32$ APs equipped with a
$(4\times 4)$ UPA antenna array at transmitter and single antenna at receiver
are installed on the wall or pillars, and follow uniform distribution within
the $xy$-plane, while their height are uniformly distributed in $[4,6]$ m. The
$U=8$ single-antenna devices are uniformly distributed on the ground with a
fixed height of $1$ m. $U_{\text{max}}$ and $C_{\text{max}}$ are assumed to be
2 and 4, respectively.

For THz fronthaul channels, we assume that the central carrier frequency is
$200$ GHz and the allocated bandwidth for fronthaul is $B^{\text{FH}}=1$ GHz
including $\left|\mathcal{M}_{\text{FH}}\right|=32$ subcarriers. Then, each
subcarrier’s bandwidth can be computed as
$b^{\text{FH}}=\frac{B^{\text{FH}}}{\left|\mathcal{M}_{\text{FH}}\right|}$.
The pulse shaping function used in (II-A) is given by [30, Eq. (61)], where
the roll-off factor is set to 1. For sub-6 GHz access channels, the central
carrier frequency is assumed to be $5$ GHz with $B^{\text{AD}}=100$ MHz
bandwidth. The large-scale fading coefficient
$\left\\{\beta_{l,q,u}\right\\},\forall
l\in\mathcal{L},q\in\mathcal{C}_{l},u\in\mathcal{U}_{l}$ is given by
$\beta_{l,q,u}[\text{dB}]=-30.5-36.7\log_{10}(d)+\sigma_{\text{sh}}z$ [1],
where $d$ denotes the distance between the $q$-th AP and the $u$-th device,
the shadowing is characterized by $\sigma_{\text{sh}}z$ with a standard
deviation of $\sigma_{\text{sh}}=4$ dB and $z\sim\mathcal{N}(0,1)$. The noise
power is determined based on the bandwidth of fronthaul and access channels
following $-174$ dBm/Hz, and hence
$\sigma^{2}_{l}=\sigma^{2}_{q}\neq\sigma^{2}_{u}$. We model the SI power as
$\sigma^{2}_{\text{SI}}=\Delta\sigma^{2}_{l}$, where the default value of
$\Delta$ is -10 dB. Unless otherwise specified, the other parameters are
listed in Table I.

###  VI-B Performance Comparison among DC, SC and IDSC

In this subsection, We will investigate the performance comparison among three
AP clustering methods. In Fig. 4, one random network realization along with
the results of AP clustering is presented, where the total numbers of APs and
MSs are $Q=32$ and $U=8$, respectively. To be more specific, Fig. 4(a) shows
the 3D distributions of CPU, APs and devices, where the AP grouping and CAP
selection are implemented by the DC method. In Fig. 4(b) and Fig. 4(c), for
the ease of illustration, only the $xy$-plane is plotted without the
distribution of devices. As we can observe from Fig. 4(a) that, with the aid
of the DC method, APs can be divided into any number of clusters without the
space limitation666Here, eight AP clusters are used for demonstration. In
fact, the DC method can dynamically adjust the AP clusters and finally find
the optimal number of AP cluster., while SC and IDSC carry out AP clustering
relying on sub-areas division. Furthermore, as shown in Fig. 4(b), since CAPs
are fixed at the central points of each sub-area, the AP clustering can not be
dynamically changed. As for IDSC, although the placement of CAPs are flexible,
it can hardly enable fully dynamic AP clustering like DC. The reason behind is
that as the number of sub-areas significantly increases, most of clusters may
not have any APs, and hence fail to serve devices. In this case, for the
practical implementation, five cases of sub-area division during SC and IDSC
are considered in the following simulations, i.e.,
$L=\left\\{4,8,12,16,20\right\\}$.

(a) DC (b) SC (c) IDSC

Figure 4: The random network realization along with three methods of AP
clustering, where $Q=32$, $U=8$ and $L=8$.

The performance comparison among DC, SC and IDSC in terms of achievable rates
per-device is presented in Fig. 5, where SC-$l$ denote the SC method is
implemented with $l$ AP clusters. It can be observed from Fig. 5(a) that, when
$Q=32$ and $U=8$, DC achieves the similar 90% likely rates (when the CDF curve
is 0.1) with SC-4 and outperforms other methods in terms of median rates (when
the CDF curve is 0.5). Especially, DC is more likely to attain the higher
rates, with at least 0.2 Gbps rate improvement over other methods. The
rationale behind is that with the aid of the K-medoids method, APs in close
proximity can be classified into one cluster, and hence leading to high
beamforming gain. However, the situation becomes different for densely
distributions APs and devices. e.g., $Q=48$ and $U=16$. As shown in Fig. 5(b),
although DC can still outperform other methods in terms of 90% likely rates,
its 30% likely rates lags behind SC-16. This is because, with the increasing
number of APs, the CAP selection in the DC method may cause the high inter-
cluster interference (ICI), when APs belonging to different clusters and
staying in close proximity to each other are selected as CAPs777In the DC
method, the CAP selection only considers the LoS complex gains of CPU-to-CAP
and CAP-to-AP links, and the potential ICI is neglected. In fact, if ICI is
also considered, the weights among two gains and ICI during the CAP selection
is of importance, and could be efficiently assigned by deep reinforcement
learning method, which is left for our future research.. Since DC and IDSC use
the same method of CAP selection, we can leverage Fig. 4(c) to further
illustrate. As shown in Fig. 4(c), after the CAP selection, the CAPs of upper-
right two clusters situate close to each other, leading to the severe ICI
problem. In contrast, CAPs in SC method are always placed at the central
points of each sub-area, which can mitigate the ICI to some extent. In
addition, since IDSC is not only subject to the rigid AP clustering method,
but also has the same ICI problem with DC, it achieves a poor performance in
both cases, as observed from Fig. 5(a) and Fig. 5(b). It can be concluded
that, although SC method with a certain pattern of sub-area division is more
likely to attain the higher per-device rates, it fails to be adaptive to the
varying sizes of networks. Consequently, the DC method can be deemed as a more
promising approach for our proposed two-tier wireless fronthaul systems.

Figure 5: Performance comparison between DC, SC and IDSC in terms of
achievable rates per-device. (a): $Q=32$ and $U=8$; (b): $Q=48$, $U=16$, and
note that as $U_{\text{max}}=2$, the minimum $L$ is set to $8$ for SC method
so as to guarantee that each device can be served with at least one AP
cluster. Moreover, for fair comparison, SC-20 is added for case (b).

Fig. 6(a)-(b) shows the end-to-end and fronthaul/access rates per-device
achieved by the DC method under different numbers of APs and THz bandwidth. To
be more specific, in Fig. 6(a), two-tier fronthaul with 1 GHz bandwidth
achieve much lower rates than the AP-to-Device access link. In this case, the
end-to-end rates mainly depends on the fronthaul rates. Accordingly, as
observed from Fig. 6(b), the increasing APs may impose the burden on the
fronthaul links, and hence lead to the decreased end-to-end rates. In order to
lift the end-to-end rate, we increase the THz bandwidth to 4 GHz, and it can
be seen from Fig. 6(a) that, equipped with the sufficient bandwidth, the
fronthaul rates are larger than the access rates, and the end-to-end rates of
the system can be further enhanced by increasing the number of APs, owing to
the extra beamforming gains, as shown in Fig. 6(b).

Fig. 6(c)-(d) shows the end-to-end and fronthaul/access rates per-device
achieved by the DC method under different numbers of devices and THz
bandwidth. Similar to Fig. 6(a)-(b), the system employed with larger THz
bandwidth is able to achieve better performance in terms of fronthaul/access
and end-to-end rates. However, a different observation is that with the
increasing number of devices, both the fronthaul and access rates decrease,
since the available resource such as subcarriers and power for each device are
reduced. In particular, when $U$ is increased to 20, it can be observed from
Fig. 6(c) that although the larger THz bandwidth can significantly lift the
performance of fronthaul link, the access link is highly affected by the dense
distribution of devices due to the severe interference, and hence the
improvement of overall end-to-end rates by allocating more THz bandwidth is
limited when the number of devices is very large, as shown in Fig. 6(d). In
this regard, we can increase the number of APs to provide more beamforming
gains, or apply NOMA to mitigate the inference caused by the neighboring AP
clusters, which will be left for our future research.

(a) Fronthaul and access rates, where $U=8$ (b) End-to-end rates, where $U=8$
(c) Fronthaul and access rates, where $Q=32$ (d) End-to-end rates, where
$Q=32$

Figure 6: The fronthaul/access and overall end-to-end rates achieved by the
proposed DC method with different numbers of APs/devices and THz bandwidth.

###  VI-C Performance Comparison among Different Fronthaul Schemes

Firstly, we compare MDD and TDD modes in our proposed two-tier wireless
fronthaul system. The default value of $\tau_{\text{GP}}$ in the TDD frame
structure is assumed to be 0.05 according to [31, 32]888The proportion of
guard period in the TDD frame structure depends on the subframe configuration
[32]. Here, we set $\tau_{\text{GP}}=0.05$ to minimize the influence of the
guard period on the evaluation of TDD system’s performance.. From Fig. 7(a),
when $\Delta\leq 20$ dB meaning that the power of the residual SI is 100 times
less than that of noise, MDD is capable of outperforming TDD, owing to the
well-designed parallel frame pattern and the non-GP configuration. Moreover,
if $\Delta$ is increased to 30 dB, the large power of residual SI makes MDD
lose its advantages over TDD. It is worth mentioning that compared with the SI
cancellation in the conventional IBFD-based sub-6G systems, the mitigation of
SI could be much easier in MDD-based THz systems. The reason is that the path
loss of LoS in the THz band is extremely large, not to mention that of NLoS.
Hence, the most of SI over THz band can be significantly suppressed in the
propagation domain by using the classic methods such as adding absorber or
blockages between transmitter and receiver [33]. Then, as desired signal and
SI are over orthogonal subcarriers, MDD is able to further cancel the residual
SI in the digital domain. In this case, SI is expected to be suppressed under
the noise floor (i.e., $\Delta\leq 0$ dB), which makes MDD a promising
solution for two-tier wireless fronthaul systems. We further study the impact
of $\tau_{\text{GP}}$ in TDD-based systems in Fig. 7(b), as expected, TDD is
heavily subject to the proportion of guard period. In particular, when
$\tau_{\text{GP}}=0.1$, the 90% likely rates per device of TDD is 0.1 Gbps
less than that of MDD. Moreover, it can be also observed that if TDD removes
guard period, it may achieve a comparable performance to MDD.

Figure 7: Performance comparison between MDD and TDD in our proposed two-tier
wireless fronthaul architecture.

Obviously, compared with the traditional fiber-optic or radio stripes
fronthaul, the proposed THz-enabled two-tier wireless fronthaul is capable of
providing the highest flexibility in deploying APs in an indoor industrial
scenario. Nevertheless, the large path loss and interference may cause the
reduced fronthaul rates. To address this problem, the abundant bandwidth in
THz band can be leveraged to close the performance gap with wired fronthaul
schemes. In this regard, we will make a comprehensive performance comparison
among different fronthaul schemes, and verify if our proposed scheme can be
comparable to the wired fronthaul schemes in terms of achievable rates in
cell-free indoor industrial systems. The following benchmarks are considered:

  * •

TDD-TTWL: This scheme denotes the TDD-based two-tier wireless fronthaul.

  * •

CC-HY: According to [11], CC-HY denotes a hybrid fronthaul scheme that the
CPU-to-CAP links are operated over the THz band, while each AP within the
cluster is connected with the CAP via fiber-optic cables.

  * •

CA-HY: Inspired by [5], in the CA-HY scheme, the CAP-to-AP links within each
cluster are based on THz fronthaul, and the wired links connect CPU and CAPs.

  * •

TTW: This scheme denotes the two-tier wired fronthaul proposed in [6], where
both CPU-to-CAP and CAP-to-AP links are implemented over fiber-optic cables.

  * •

STWL: This scheme denotes the single-tier wireless fronthaul, where CPU
directly sends fronthaul signal to every AP over THz band and the AP
clustering is no longer needed.

  * •

STW: Contrary to STWL, STW enables the single-tier wired connections between
CPU and all the APs.

Note that apart from STWL and STW, all the other schemes implement AP
clustering, and the optimal $L$ is determined by following step 2-18 in
Algorithm 2 without the process of subcarrier assignment (that is to say, the
$\mathcal{M}_{\text{CC}}$ and $\mathcal{M}_{\text{CA}}$ assignment is
exclusive in the MDD-based schemes).

Figure 8: Performance comparison among different fronthaul schemes versus the
THz bandwidth.

From Fig. 8, as expected, since TTW scheme applies fiber-optic cables to both
CPU-to-CAP and CAP-to-AP links, it achieves the highest rates per device. As
the performance of wireless CAP-to-AP links are affected by not only the
inter-AP interference within the cluster, but also the inter-cluster
interference, and hence CA-HY lags behind CC-HY. Furthermore, MDD-TTWL
outperforms TDD-TTWL owing to the parallel frame structure and free of the
guard period, as we analyzed in Fig. 7. In particular, MDD-TTWL is able to
attain the similar rates as TTW when $B^{\text{FH}}$ is increased to 4 GHz.
The reason is that as the THz bandwidth increases, the overall system
performance mainly depends on the AP-to-Device access links over sub-6 GHz
band. In this case, it was shown that the sufficient THz bandwidth makes the
proposed two-tier wireless fronthaul scheme feasible in indoor industrial CF
systems. On the other hand, without the optimization of AP clustering, STW and
STWL cannot achieve a comparable performance to other fronthaul schemes. More
specifically, in the STWL scheme, since the CPU communicates with all the APs
over the THz band, the AP far away from the CPU experiences the undesirable
fronthaul connection leading to the low overall end-to-end rates.

###  VI-D Ray-Tracing Based Simulations in Practical Scenario

We further evaluate our proposed MDD-enabled two-tier THz fronthaul in a
practical indoor industrial scenario, as shown in Fig. 9(a)-(b). This scenario
corresponds to the digital twin obtained from LiDAR scans in a BOSCH factory
in Blaichach, Germany. Electromagnetic properties have been assigned to the
different objects in the scenario to obtain realistic frequency dependent
simulations. The ray-tracing simulations were performed with the software
WinProp. The height of CPU is 6.9 m, while the heights of all the APs and
devices are 3 m and 1.5 m, respectively. A rich multipath propagation
environment can be observed in the isotropic power delay profiles (PDP)
depicted in Fig. 9(c) (for sake of simplicity only the results observed at the
AP 8 receiver are displayed). The difference of bandwidth between access and
THz channels is observed in the resolution in the delay domain. From the
figure, it can be observed that subject to the feature of high frequency, the
received power over the THz channels attenuate heavily, especially when two
nodes stay away from each other and there is no LoS path between them.

Figure 9: (a) A practical indoor industrial scenario, where the CPU is
installed on the roof, while all the APs are installed on the wall or pillars.
(b) The corresponding ray-tracing environment. (c) Power delay profile
measured at AP 8 receiver. (d) Performance comparison among different
fronthaul schemes versus the THz bandwidth in practical indoor industrial
scenario.

Furthermore, we obtain the CIR of both access and THz channels based on ray-
tracing channel measurements. Except the channel related parameters, all the
other parameters (such as power budget, antenna gain and noise/SI power) are
kept the same as in Section VI-A, and then a performance comparison among
various fronthaul schemes in this practical scenario is presented in Fig.
9(d). As expected, the results match with that in Fig. 8. Specifically, the
MDD-TTWL outperforms TDD-TTWL when the THz bandwidth is less than 2 GHz, and
CC-HY and TTW achieve the upper bound of achievable rates. Moreover, when the
available bandwidth of THz-based fronthaul channels increases to 2 GHz, MDD-
TTWL can attain the same rates as CC-HY and TTW. This observation implies that
for the consideration of ease deployment and robust achievable rates, our
proposed MDD-enabled two-tier THz scheme can be deemed as a promising approach
to enabling fully-wireless fronthaul in practical indoor industrial scenarios.

##  VII Conclusion

In this paper, we proposed an MDD-enabled two-tier THz fronthaul scheme, which
can make the indoor industrial CF-mMIMO systems free from the costly wired
fronthaul links. To maximize the achievable rates in such a complicated
scheme, we firstly resorted to the low-complexity but efficient heuristic
methods to relax binary variables. Then, the iterative optimization of the
assignment of subcarrier sets and the number of AP clusters was applied to
obtain the optimal overall end-to-end rates. Furthermore, an advanced MDD
frame structure was tailored for the proposed scheme. Our studies and
numerical results showed that compared with TDD, MDD is a potentially
promising solution to enable two-tier THz fronthaul scheme. Moreover, with the
aid of the sufficient THz bandwidth, our proposed scheme is capable of not
only achieving the same rates with the existing fiber-optic based CF-mMIMO
systems, but also imposing less burden on the system overhead. In order to
further demonstrate the practicality of the proposed scheme, we applied it to
a real indoor industrial scenario relying on ray-tracing based channel
measurements, and the simulation results showed its superior over other
methods.

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