# Multiple-Target Detection in Cell-Free Massive MIMO-Assisted ISAC

Mohamed Elfiatoure, Mohammadali Mohammadi,
Hien Quoc Ngo, and Michail Matthaiou This work is a contribution by Project
REASON, a UK Government funded project under the Future Open Networks Research
Challenge (FONRC) sponsored by the Department of Science Innovation and
Technology (DSIT). It was also supported by the U.K. Engineering and Physical
Sciences Research Council (EPSRC) (grant No. EP/X04047X/1). The work of H. Q.
Ngo was supported by the U.K. Research and Innovation Future Leaders
Fellowships under Grant MR/X010635/1, and a research grant from the Department
for the Economy Northern Ireland under the US-Ireland R&D Partnership
Programme. The work of M. Mohammadi and M. Matthaiou was supported by the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 101001331).The authors
are with the Centre for Wireless Innovation (CWI), Queen’s University Belfast,
BT3 9DT Belfast, U.K., email:{melfiatoure01, m.mohammadi, hien.ngo,
<EMAIL_ADDRESS>Parts of this paper appeared at the 2023 IEEE GLOBECOM
conference [1].

###### Abstract

We propose a distributed implementation for integrated sensing and
communication (ISAC) backed by a massive multiple input multiple output (CF-
mMIMO) architecture without cells. Distributed multi-antenna access points
(APs) simultaneously serve communication users (UEs) and emit probing signals
towards multiple specified zones for sensing. The APs can switch between
communication and sensing modes, and adjust their transmit power based on the
network settings and sensing and communication operations’ requirements. By
considering local partial zero-forcing and maximum-ratio-transmit precoding at
the APs for communication and sensing, respectively, we first derive closed-
form expressions for the spectral efficiency (SE) of the UEs and the mainlobe-
to-average-sidelobe ratio (MASR) of the sensing zones. Then, a joint operation
mode selection and power control design problem is formulated to maximize the
SE fairness among the UEs, while ensuring specific levels of MASR for sensing
zones. The complicated mixed-integer problem is relaxed and solved via
successive convex approximation approach. We further propose a low-complexity
design, where AP mode selection is designed through a greedy algorithm and
then power control is designed based on this chosen mode. Our findings reveal
that the proposed scheme can consistently ensure a sensing success rate of
$100\%$ for different network setups with a satisfactory fairness among all
UEs.

###### Index Terms:

Cell-free massive multiple-input multiple-output (CF-mMIMO), integrated
sensing and communication (ISAC), mainlobe-to-average sensing ratio (MASR),
spectral efficiency (SE).

##  I Introduction

The advent of integrated sensing and communication (ISAC) marks a significant
leap forward for the sixth generation (6G) of wireless technology, introducing
a pivotal shift in the use of limited spectral resources [2, 3]. This
innovative approach aims at enhancing both the radar sensing capabilities and
communication efficiency, by assigning frequencies traditionally dedicated to
sensing for wireless communication purposes as well. Such dual functionality
supports high data-rate applications, contributing to the efficient
utilization of spectral resources. ISAC’s innovative framework has catalyzed
the development of two distinct system architectures: separated systems and
co-located systems. A separated system, also referred to as coexistence
communication and radar, relies on distinct devices to perform radar sensing
and communication within the same frequency band, generating sensing and
communication beams independently. This method has been thoroughly
investigated in our previous studies [4, 5] among others. In contrast, co-
located systems, or dual-functional radar-communication (DFRC), integrate
these functionalities within a single device. This allows for the simultaneous
detection of radar targets and communication with multiple users (UEs),
facilitated by an ISAC base station (BS) that utilizes shared hardware
components for concurrent beam generation [6]. Recognized for its efficiency
and cost-effectiveness, the DFRC architecture is heralded as a significant
breakthrough for the beyond 5G (B5G) era, poised to redefine the network
performance and optimization [7, 8, 9].

Transitioning from separated to co-located systems reflects an intention to
streamline operational efficiency and reduce the complexity inherent in
managing the exchange of side-information between radar and communication
technologies [8, 9]. This evolution is notably supported by the integration of
massive multiple-input multiple-output (mMIMO) technology, which particularly
within co-located architectures, can effectively minimize the inter-user
interference across the same time-frequency resources. Leveraging the
substantial spatial degrees of freedom and intrinsic sensing capabilities, the
fusion of mMIMO technology with ISAC is poised to deliver high-quality
wireless communication, while ensuring high-resolution and robust sensing.
Such advancements are instrumental in accelerating the development of various
emerging applications, including, but not limited to, autonomous driving in
intelligent transportation systems and unmanned aerial vehicle networks in
smart cities [10, 11].

Expanding on this groundwork, recent research endeavors, as highlighted in
studies by [12] and [13], have delved into the intricacies of a dual-
functional system that marries communication and radar functionalities within
an mMIMO orthogonal frequency division mutliplexing (OFDM) architecture. These
studies comprehensively address both downlink and uplink scenarios, offering
insightful analytical derivations of the achievable rate and detection
efficiency under various conditions of channel side information (CSI),
encompassing both perfect and imperfect scenarios. Moreover, the work in [14]
sought to optimize this dual-functional system. The focus was on maximizing
the sum-rate and energy efficiency, while ensuring compliance with essential
operational parameters, including a baseline target detection probability and
the satisfaction of individual UE rate demands.

Another powerful network topology is cell-free mMIMO (CF-mMIMO), which offers
great potential to significantly improve the network connectivity. Diverging
from traditional cellular setups that centralize all antennas at the BS, CF-
mMIMO distributes antennas via the use of multiple access points (APs). These
APs are then coordinated by several central processing units (CPUs), paving
the way for a more distributed and flexible network architecture [15, 16, 17,
18]. A CF-mMIMO ISAC topology has several advantages over the single-cell
ISAC, including larger monitoring areas, broader sensing coverage, and a wider
range of sensing angles. Nevertheless, in such systems, proper resource
allocation plays a key role to facilitate both communication and sensing
functionalities.

###  I-A Related Works

Some recent papers have focused on integrating ISAC capabilities within CF-
mMIMO. These studies aim to harness the benefits of both ISAC and CF-mMIMO
technologies to further enhance the network performance and service delivery,
as evidenced by works such as those in [19, 20, 21, 22, 23].

TABLE I: Contrasting our contributions to the mMIMO ISAC literature Contributions |  This paper |  [19] |  [20] |  [21] |  [22]
---|---|---|---|---|---
CF-mMIMO |  ✓ |  ✓ |  ✓ |  ✓ |  ✓
Power allocation |  ✓ |  ✓ |  ✓ |  ✓ |
Beamforming design |  ✓ |  ✓ |  ✓ |  ✓ |
AP operation mode selection |  ✓ |  |  |  |
Multiple targets |  ✓ |  |  |  |  ✓
Asymptotic analysis |  ✓ |  |  |  |

Among these contributions, Zeng et al.[19] made notable advancements in power
allocation techniques within CF-mMIMO ISAC systems, optimizing the balance
between communication and sensing functionalities. Behdad et al.[20] further
explored the operational dynamics of CF-mMIMO ISAC systems, demonstrating how
transmitting APs not only provide service to UEs but also partake in sensing
operations for target location identification. Their research introduced a
power distribution strategy that enhances the sensing signal-to-noise ratio
(SNR) while adhering to signal-to-interference-plus-noise ratio (SINR)
constraints for UEs. Building on these themes, Demirhan et al. [21] tackled
the challenges of beamforming design in CF-mMIMO ISAC systems. They proposed a
joint beamforming approach designed to maximize the sensing SNR without
compromising the communication SINR, highlighting the critical balance
required for effective sensing and communication within these networks.
Additionally, the papers by Mao et al.[22] and Da et al.[23] have been
instrumental in advancing precoder design for CF-mMIMO ISAC systems. These
works also delved into the assessment of privacy risks associated with the
inference of target locations by internal adversaries, marking a significant
exploration beyond the technical aspects to include security and privacy
considerations within ISAC networks.

###  I-B Contributions

Contrary to the above studies, which assume static AP operation modes, our
research introduces an innovative approach to CF-mMIMO ISAC networks by
adopting dynamic AP operation mode selection. This novel strategy aims to
optimize the spectral efficiency (SE) fairness among the UEs, while also
ensuring the system’s capability to detect multiple designated targets. By
leveraging long-term CSI, the APs are categorized into communication APs
(C-APs) and sensing APs (S-APs) to simultaneously support both downlink
communication and sensing tasks. Compared to our recent work [1], where a
single sensing zone area was considered and maximum ratio transmission (MRT)
was performed at all APs, we consider a multiple sensing zone scenario and
apply local partial zero-forcing (PZF) and MRT precoding at the C-APs and
S-APs, respectively. The principle behind the PZF design is that each AP only
suppresses the inter-UE interference it causes to the strongest UEs, namely
the UEs with the largest channel gain, while the inter-UE interference caused
to the weakest UEs is tolerated [24]. The main contributions of our paper can
be summarized as follows:

  * •

We develop a framework for analyzing the performance of a CF-mMIMO ISAC system
with multiple C-APs and S-APs employing PZF and MRT precodings, respectively,
under channel estimation errors. By leveraging the use-and-then-forget
strategy, we derive closed-form expressions for the downlink SE of the
communication UEs and mainlobe-to-average sensing ratio (MASR) of the sensing
zones. We further pursue an asymptotic SE analysis, which discloses that when
the number of C-APs and S-APs, denoted by $M_{c}$ and $M_{s}$ respectively,
are large, we can scale down the transmit powers at the C-APs and S-APs
proportionally to $1/M_{c}^{2}$ and $1/M_{s}^{2}$, respectively.

  * •

We formulate an interesting problem of joint AP operation mode selection and
power control design, considering per-AP power constraints and a MASR
constraint for target detection in a multi-target environment. A new algorithm
is developed to solve the challenging formulated mixed-integer non-convex
problem. In particular, we transform the formulated problem into a more
tractable problem with continuous variables only. Then, we solve the problem
using successive convex approximation (SCA) techniques.

  * •

To achieve a performance-complexity tradeoff, we propose a greedy algorithm
for AP operation mode selection. This algorithm iteratively selects the
optimal mode for each AP by considering the constraints for sensing operation,
while maximizing fairness among UEs. Then, we propose a power control design
algorithm for fixed AP operation mode design.

  * •

Our numerical results show that the proposed joint algorithm can provide
noticeable fairness among the UEs, while ensuring successful sensing
performance for all sensing zones. The greedy algorithm achieves an acceptable
level of success in the sensing rate.

A comparison of our contributions against the state of the art in the space of
CF-mMIMO ISAC is tabulated in Table I.

###  I-C Paper Organization and Notation

The rest of this paper is organized as follows: In Section II, we describe the
system model for the proposed CF-mMIMO ISAC system with the corresponding
communication and sensing metrics. In Section III, we analyze the per-UE SE in
the asymptotic regime. The proposed AP mode selection and power allocation
schemes are discussed in Section IV. Finally, the numerical results and some
discussions are provided in Section V, followed by the conclusion remarks in
Section VI.

Notation: We use bold lower (capital) case letters to denote vectors
(matrices). The superscripts $(\cdot)^{H}$ and $(\cdot)^{T}$ stand for the
Hermitian and transpose operation, respectively; $\mathbf{I}_{N}$ denotes the
$N\times N$ identity matrix; $\mathrm{tr}(\cdot)$ and $(\cdot)^{-1}$ denote
the trace operator and matrix inverse, respectively. A circular symmetric
complex Gaussian distribution having variance $\sigma^{2}$ is denoted by
$\mathcal{CN}(0,\sigma^{2})$. Finally, $\mathbb{E}\\{\cdot\\}$ denotes the
statistical expectation.

##  II System Model

We consider a CF-mMIMO ISAC system operating under time division duplex (TDD),
where $M$ APs serve $K$ downlink UEs, while concurrently emitting probing
signals towards $L$ specific sensing zones. Each UE is equipped with a single
antenna, and each AP deploys an array of $N$ antennas. All APs and UEs
function in a half-duplex mode. For the ease of exposition, we introduce the
sets $\mathcal{M}\triangleq\\{1,\ldots,M\\}$,
$\mathcal{L}\triangleq\\{1,\dots,L\\}$, and
$\mathcal{K}\triangleq\\{1,\dots,K\\}$ to represent the indices of the APs,
sensing areas, and UEs, respectively. As depicted in Fig. 1, downlink
communication and target detection are conducted concomitantly over the same
frequency band. To accommodate the varying network demands, a dynamic AP
operation mode selection strategy is implemented, determining the allocation
of APs for downlink information transmission or radar sensing. A subset of
APs, designated as communication-APs, termed as C-APs, is responsible for
delivering information to the UEs. In contrast, the rest, labeled as S-APs,
are employed for target detection. This categorization enables a specialized
functionality of each AP group, where the C-APs aim to optimize the
communication quality, while the S-APs enhance the radar sensing accuracy.

###  II-A Channel Model and Uplink Training

We assume a quasi-static channel model, with each channel coherence interval
spanning a duration of $\tau$ symbols. The duration of the training is denoted
as $\tau_{t}$, while the duration of downlink information transfer and target
detection is $(\tau-\tau_{t})$.

For the sensing channel model, we assume there is a line-of-sight (LoS) path
between the sensing area and each AP, which is a commonly adopted model in the
literature [20, 21]. The LoS channel between AP $m$ and senssing area $l$ is
given by

$\displaystyle~{}\bar{{\bf g}}_{ml}$ $\displaystyle={\bf
a}_{N}(\phi_{t,ml}^{a},\phi_{t,ml}^{e}),~{}\forall m\in\mathcal{M},$ (1)

where $\phi_{t,ml}^{a}$, $\phi_{t,ml}^{e}$ denote the azimuth and elevation
angles of departure (AoD) from AP $m$ towards the sensing area $l$. Moreover,
the $q$-th entry of the array response vector ${\bf
a}_{N}(\phi_{t,ml}^{a},\phi_{t,ml}^{e})\in\mathbb{C}^{N\times 1}$, is given by

$\displaystyle[{\bf a}_{N}(\phi_{t,ml}^{a},\phi_{t,ml}^{e})]_{q}$
$\displaystyle\hskip 30.00005pt=\exp\bigg{(}j\frac{2\pi
d}{\lambda}(q-1)\sin(\phi_{t,ml}^{a})\cos(\phi_{t,ml}^{e})\bigg{)},$ (2)

where $d$ and $\lambda$ denote the AP antenna spacing and carrier wavelength,
respectively.

The channel vector between the $m$-th AP and $k$-th UE is modeled as ${\bf
g}_{mk}=\sqrt{\beta_{mk}}{\bf h}_{mk}$, where $\beta_{mk}$ is the large scale
fading coefficient, while ${\bf h}_{mk}\in\mathbb{C}^{N\times 1}$ is the
small-scale fading vector, whose elements are independent and identically
distributed $\mathcal{CN}(0,1)$ random variables (RVs).

An uplink training process is implemented to acquire the local CSI between
each AP and all UEs. In each coherence block of length $\tau$, all UEs are
assumed to transmit their pairwisely orthogonal pilot sequence of length
$\tau_{t}$ to all APs, which requires $\tau_{t}\geq K$. At AP $m$, ${\bf
g}_{mk}$ is estimated by using the received pilot signals and applying the
minimum mean-square error (MMSE) estimation technique. By following [17], the
MMSE estimate $\hat{{\bf g}}_{mk}$ of ${\bf g}_{mk}$ is obtained as $\hat{{\bf
g}}_{mk}\sim\mathcal{CN}\left(\bm{0},\gamma_{mk}{\bf I}_{N}\right)$, where

$\displaystyle\gamma_{mk}$
$\displaystyle=\frac{\tau_{t}\rho_{t}\beta_{mk}^{2}}{\tau_{t}\rho_{t}\beta_{mk}+1},$
(3)

while $\rho_{t}$ represents the normalized transmit power of each pilot
symbol.

Figure 1: Illustration of the CF-mMIMO ISAC system.

###  II-B Probing and Data Signal Transmission

AP operation mode selection is performed by considering large-scale fading
effects and relying on the statistical CSI. The binary variables used to
indicate the operation mode for each AP $m$ are defined as

$\displaystyle a_{m}=\begin{cases}&\text{1, if AP $m$ operates as C-AP}\\\
&\text{0, if AP $m$ operates as S-AP.}\end{cases}$ (4)

The transmission phase comprises information transmission from the C-APs to
UEs and probing signal transmission from the S-APs to the target. Let
$\mathbf{x}_{c,m}$ and $\mathbf{x}_{r,m}$ denote the data and probing signals,
respectively. The signal vector transmitted from AP $m$ can be expressed as

$\displaystyle\mathbf{x}_{m}=\sqrt{a_{m}}{{\bf x}}_{c,m}+\sqrt{(1-a_{m})}{{\bf
x}_{r,m}}.$ (5)

The probing signal transmitted by the $m$-th S-AP, targeting specific sensing
zones, can be expanded as

$\displaystyle{\bf
x}_{r,m}=\sum\nolimits_{l\in\mathcal{L}}\sqrt{\rho\eta^{s}_{ml}}{\bf
t}_{ml}^{\mathrm{Sen}}x_{r,ml},$ (6)

where $\rho$ denotes the maximum normalized downlink power; $\eta^{s}_{ml}$ is
the power control coefficient at S-AP $m$ related to sensing zone $l$; ${\bf
t}_{ml}^{\mathrm{Sen}}\in\mathbb{C}^{N\times 1}$ denotes the beamforming
vector for sensing the $l$th sensing area, while $x_{r,ml}$, with
$\mathbb{E}\\{|x_{r,ml}|^{2}\\}=1$, is the radar sensing symbol from S-AP $m$
for sensing the $l$th zone. This directional beamforming approach allows each
S-AP to transmit a beam focused on a specific sector, facilitating spatially
distributed sensing across the network. Therefore, to detect the presence of a
target within its designated sector, each S-AP tailors its sensing beamforming
vector to concentrate the radiated energy towards a specific direction. This
is accomplished by configuring the beamforming vector to align with the
azimuth and elevation angles that correspond to the sector of interest.
Therefore, to detect the presence of a target in a certain location, we design
the sensing beamforming vector at the $m$th S-AP as

$\displaystyle{\bf t}_{ml}^{\mathrm{Sen}}={\bf
a}_{N}(\phi_{t,ml}^{a},\phi_{t,ml}^{e}).$ (7)

This focused beamforming ensures that the radar echo from the target is
sufficiently strong for reliable detection.

###  II-C Downlink Communication and SE

In this section, we investigate the performance of partial zero forcing (PZF)
precoding scheme. The conventional maximum ratio (MR) and ZF are special cases
of PZF, and hence, their performances can be directly obtained from PZF.

PZF precoding emerges as an innovative method with the potential to combine
the benefits of both ZF and MR approaches. The cornerstone of PZF lies in its
ability to suppress interference caused to the strongest UEs, i.e., UEs with
the highest channel gain, while tolerating the interference caused to the
weakest UEs. In more detail, for any given AP (denoted as AP $m$), the set of
active UEs is conceptually divided into two distinct subsets: (i) strong UEs,
and (ii) weak UEs. The sets of indices for strong and weak UEs are represented
as $\mathcal{S}_{m}\subset\\{1,\ldots,K\\}$, and
$\mathcal{W}_{m}\subset\\{1,\ldots,K\\}$ respectively. It should be noted that
$\mathcal{S}_{m}\cap\mathcal{W}_{m}=\phi$ and
$|\mathcal{S}_{m}|+|\mathcal{W}_{m}|=K$. The grouping strategy can be
determined using various criteria. For example, it may be based on the mean-
square of the channel gain: a UE $k$ is allocated to $\mathcal{S}_{m}$ if
$\beta_{mk}$ exceeds a predetermined threshold, otherwise, $k$ is placed in
$\mathcal{W}_{m}$. PZF operates locally in the following manner: AP $m$
transmits to all the UEs $k\in\mathcal{S}_{m}$ using ZF, and to all the UEs
$k\in\mathcal{W}_{m}$ using MR.

The UE grouping can be based on different criteria. Inspired by [25, 24], the
UE grouping strategy in PZF relies on the following rule

$\displaystyle~{}\sum\nolimits_{k=1}^{|\mathcal{S}_{m}|}\frac{\bar{\beta}_{mk}}{\sum\nolimits_{t=1}^{K}\beta_{mt}}\geq\varrho\%,$
(8)

where AP $m$ constructs its set $\mathcal{S}_{m}$ by selecting the UEs that
contribute to, at least, $\varrho\%$ of the overall channel gain. Note that in
(8), $\\{\bar{\beta}_{m1},\ldots,\bar{\beta}_{mK}\\}$ indicates the set of the
large-scale fading coefficients sorted in descending order.

Therefore, transmit signal from AP $m$ can be expressed as

$\displaystyle~{}{{\bf x}}_{c,m}=$
$\displaystyle\Big{(}\sum\nolimits_{k\in\mathcal{S}_{m}}\sqrt{\rho{\eta}_{mk}^{c}}{\bf
t}_{mk}^{\mathrm{ZF-Com}}$
$\displaystyle\\!+\\!\sum\nolimits_{k\in\mathcal{W}_{m}}\sqrt{\rho{\eta}_{mk}^{c}}{\bf
t}_{mk}^{\mathrm{MR-Com}}\Big{)}x_{c,k},$ (9)

where ${\eta}_{mk}^{c}$ denotes the power control coefficient between C-AP $m$
and UE $k$, $k\in\mathcal{K}$; $x_{c,k}$ is the communication symbol
satisfying $\mathbb{E}\\{|x_{c,k}|^{2}\\}=1$; ${\bf t}^{\mathrm{ZF-Com}}_{mk}$
and ${\bf t}^{\mathrm{MR-Com}}_{mk}$ represent the ZF and MR precoding
vectors, respectively, given by

$\displaystyle{\bf t}^{\mathrm{ZF-Com}}_{mk}$
$\displaystyle=\gamma_{mk}\hat{{\bf G}}_{\mathcal{S}_{m}}\left(\hat{{\bf
G}}_{\mathcal{S}_{m}}^{H}\hat{{\bf G}}_{\mathcal{S}_{m}}\right)^{-1}{\bf
e}_{k}$ (10) $\displaystyle{\bf t}^{\mathrm{MR-Com}}_{mk}$
$\displaystyle=\mathbf{\hat{g}}^{*}_{mk},$ (11)

where ${\bf e}_{k}$ is the $k$th column of ${\bf I}_{|\mathcal{S}_{m}|}$ and
$\mathbf{\hat{G}}_{\mathcal{S}_{m}}=[\mathbf{\hat{g}}^{T}_{mk}:k\in\mathcal{S}_{m}]$.
Therefore, for any pair of UEs $k$ and $k^{\prime}$ belonging to the
$\mathcal{S}_{m}$, we have

$\mathbf{\hat{g}}^{T}_{mk^{\prime}}{\bf t}^{\mathrm{ZF-
Com}}_{mk}=\begin{cases}\gamma_{mk}&\text{if }k=k^{\prime},\\\
0&\text{otherwise. }\end{cases}$ (12)

The power control coefficients at AP $m$ are chosen to satisfy the power
constraint at each S-AP and C-AP, respectively, i.e.,

$\displaystyle~{}{a_{m}}\mathbb{E}\\{\|\mathbf{x}_{c,m}\|^{2}\\}+({1-a_{m}})\mathbb{E}\\{\|\mathbf{x}_{r,m}\|^{2}\\}\leq\rho.$
(13)

For the sake of notation simplicity, we introduce $\mathcal{Z}_{k}$ and
$\mathcal{M}_{k}$ as the set of indices of APs that transmit to UE $k$ using
${\bf t}^{\mathrm{ZF-Com}}_{mk}$ and ${\bf t}^{\mathrm{MR-Com}}_{mk}$
respectively, given as

$\displaystyle\mathcal{Z}_{k}$
$\displaystyle\triangleq\big{\\{}m:k\in\mathcal{S}_{m},m=1,\ldots,M\big{\\}},$
(14a) $\displaystyle\mathcal{M}_{k}$
$\displaystyle\triangleq\big{\\{}m:k\in\mathcal{W}_{m},m=1,\ldots,M\big{\\}},$
(14b)

with $\mathcal{Z}_{k}\cap\mathcal{M}_{k}=\phi$ and
$\mathcal{Z}_{k}\cup\mathcal{M}_{k}=\mathcal{M}$. The PZF precoding approach
provides a sophisticated balance between ZF and MR, catering to interference
suppression for the strongest UEs and signal strength maximization for the
weakest ones.

The received signal at the $k$-th UE can be represented as

$\displaystyle{y}_{k}$
$\displaystyle=\Big{(}\sum\nolimits_{m\in{\mathcal{Z}_{k}}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{ZF-Com}}_{mk}$
$\displaystyle+\sum\nolimits_{m\in{\mathcal{M}_{k}}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{MR-Com}}_{mk}\Big{)}x_{c,k}$
$\displaystyle+\sum\nolimits_{k^{\prime}\in\mathcal{K}\setminus
k}\Big{(}\sum\nolimits_{m\in{\mathcal{Z}_{k}}}\sqrt{a_{m}\rho{\eta}_{mk^{\prime}}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{ZF-Com}}_{mk^{\prime}}$
$\displaystyle+\sum\nolimits_{m\in\mathcal{M}_{k}}\sqrt{a_{m}\rho{\eta}_{mk^{\prime}}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{MR-Com}}_{mk^{\prime}}\Big{)}x_{c,k^{\prime}}$
$\displaystyle+{\sum\nolimits_{m\in\mathcal{M}}\\!\sum\nolimits_{l\in\mathcal{L}}\\!\sqrt{\left(1\\!-\\!a_{m}\right)\rho\eta^{s}_{ml}}{\bf
g}^{T}_{mk}{\bf t}_{ml}^{\mathrm{Sen}}}x_{r,ml}\\!+\\!{n}_{k},$ (15)

where the second term is the inter-user interference, the third term
represents the interference from S-APs, and $n_{k}\sim\mathcal{CN}(0,1)$
denotes the additive white Gaussian noise at the UE $k$. In order to apply the
use-then-forget technique to derive the downlink SE at UE $k$, we rewrite
(II-C) as

$\displaystyle y_{k}$
$\displaystyle=\mathrm{DS}_{k}x_{c,k}+\mathrm{BU}_{k}x_{c,k}$
$\displaystyle+\sum\nolimits_{k^{\prime}\in\mathcal{K}\setminus
k}\mathrm{IUI}_{kk^{\prime}}x_{c,k^{\prime}}+\mathrm{IR}_{k}+n_{k},$ (16)

where

$\displaystyle\mathrm{DS}_{k}$
$\displaystyle\triangleq\mathbb{E}\Big{\\{}\sum\nolimits_{m\in\mathcal{Z}_{k}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{ZF-Com}}_{mk}$ $\displaystyle\hskip
10.00002pt+\sum\nolimits_{m\in\mathcal{M}_{k}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{MR-Com}}_{mk}\Big{\\}}$ (17a)
$\displaystyle\mathrm{BU}_{k}$
$\displaystyle\triangleq\Big{(}\sum\nolimits_{m\in\mathcal{Z}_{k}}\sqrt{a_{m}\rho{\eta}_{mk^{\prime}}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{ZF-Com}}_{mk^{\prime}}$
$\displaystyle+\sum\nolimits_{m\in\mathcal{M}_{k}}\sqrt{a_{m}\rho{\eta}_{mk^{\prime}}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{MR-Com}}_{mk^{\prime}}\Big{)}-\mathrm{DS}_{k},$
(17b) $\displaystyle\mathrm{IUI}_{kk^{\prime}}$
$\displaystyle\triangleq\sum\nolimits_{m\in\mathcal{Z}_{k}}\sqrt{a_{m}\rho{\eta}_{mk^{\prime}}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{ZF-Com}}_{mk^{\prime}}$ $\displaystyle\hskip
10.00002pt+\sum\nolimits_{m\in\mathcal{M}_{k}}\sqrt{a_{m}\rho{\eta}_{mk^{\prime}}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{MR-Com}}_{mk^{\prime}},$ (17c)
$\displaystyle\mathrm{IR}_{k}$
$\displaystyle\triangleq\sum\nolimits\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l\in\mathcal{L}}\sqrt{(1-a_{m})\rho\eta^{s}_{ml}}{\bf
g}_{mk}^{T}{\bf t}_{ml}^{\mathrm{Sen}}x_{r,ml},~{}$ (17d)

represent the strength of the desired signal ($\mathrm{DS}_{k}$), the
beamforming gain uncertainty ($\mathrm{BU}_{k}$), the interference caused by
the $k^{\prime}$-th UE ($\mathrm{IUI}_{kk^{\prime}}$) and the interference
caused by S-APs ($\mathrm{IR}_{k}$), respectively. By invoking (II-C), an
achievable downlink SE at the $k$-th UE can be expressed as

$\displaystyle~{}\mathrm{SE}_{k}=\Big{(}1-\frac{\tau_{t}}{\tau}\Big{)}\log_{2}\big{(}1+\mathrm{SINR}_{k}\big{)},$
(18)

where $\mathrm{SINR}_{k}\\!=$

$\displaystyle\frac{\big{|}\mathrm{DS}_{k}\big{|}^{2}}{\mathbb{E}\big{\\{}\\!\big{|}\mathrm{BU}_{k}\big{|}^{2}\\!\big{\\}}\\!+\\!\sum_{k^{\prime}\in\mathcal{K}\setminus
k}\\!\mathbb{E}\big{\\{}\\!\big{|}\mathrm{IUI}_{kk^{\prime}}\big{|}^{2}\\!\big{\\}}\\!+\\!\mathbb{E}\big{\\{}\\!\big{|}\mathrm{IR}_{k}\big{|}^{2}\\!\big{\\}}\\!+\\!1}\\!.$
(19)

For the simplicity of notation, let us introduce a pair of binary variables to
indicate the group assignment for each DL UE $k$ and AP $m$ in the PZF
combining scheme as

$\displaystyle\zeta_{m}^{\mathcal{Z}_{k}}$
$\displaystyle=\begin{cases}1&\text{if }m\in\mathcal{Z}_{k},\\\
0&\text{otherwise}.\end{cases}$ (20a)
$\displaystyle\zeta_{m}^{\mathcal{M}_{k}}$
$\displaystyle=\begin{cases}1&\text{if }m\in\mathcal{M}_{k},\\\
0&\text{otherwise}.\end{cases}$ (20b)

######  Proposition 1.

The SE achieved by the PZF scheme is represented by (18), where
$\mathrm{SINR}_{k}$ is given in (21) at the top of the next page.

###### Proof.

See Appendix B. ∎

######  Remark 1.

Setting $\zeta_{m}^{\mathcal{M}_{k}}=1$ and $\zeta_{m}^{\mathcal{Z}_{k}}=0$,
$\forall m\in\mathcal{M}$, results in the SINR given by (21) being reduced to
that achieved by the MR precoding scheme. Similarly, if
$\zeta_{m}^{\mathcal{M}_{k}}=0$ and $\zeta_{m}^{\mathcal{Z}_{k}}=1$, $\forall
m\in\mathcal{M}$, the SINR in (21) reduces to that achieved by the ZF
precoding scheme, provided that $N>K$.

$\displaystyle\mathrm{SINR}_{k}(\mathbf{a},\bm{\eta}^{c},\bm{\eta}^{s})=\frac{\rho\Big{(}\\!\sum_{m\in\mathcal{M}}\sqrt{a_{m}{\eta}_{mk}^{c}}\gamma_{mk}(\zeta_{m}^{\mathcal{Z}_{k}}+N\zeta_{m}^{\mathcal{M}_{k}})\Big{)}^{2}}{\rho\sum_{m\in\mathcal{M}}\sum_{k^{\prime}\in\mathcal{K}}\\!\\!\Big{(}a_{m}{\eta}_{mk^{\prime}}^{c}\gamma_{mk^{\prime}}\big{(}\zeta_{m}^{\mathcal{Z}_{k}}\frac{\beta_{mk}-\gamma_{mk}}{N-|\mathcal{S}_{m}|}+\zeta_{m}^{\mathcal{M}_{k}}N\beta_{mk}\big{)}\Big{)}\\!+\\!\rho\sum\nolimits_{m\in\mathcal{M}}\\!\sum\nolimits_{l\in\mathcal{L}}\\!\eta^{s}_{ml}N(1-a_{m})\beta_{mk}\\!+\\!1},$
(21)

###  II-D Sensing Operation and MASR

For a given angles $(\phi_{m\in\mathcal{M}}^{a},\phi_{m\in\mathcal{M}}^{e})$,
the average spatial power pattern is

$\displaystyle~{}P^{\mathrm{ave}}(\phi_{m\in\mathcal{M}}^{a},\phi_{m\in\mathcal{M}}^{e})$
$\displaystyle=\mathbb{E}\bigg{\\{}\Big{|}\sum\nolimits_{m\in\mathcal{M}}\mathbf{a}_{N}^{H}(\phi_{m}^{a},\phi_{m}^{e})\mathbf{x}_{m}\Big{|}^{2}\bigg{\\}}$
$\displaystyle=\underbrace{\mathbb{E}\bigg{\\{}\Big{|}\sum\nolimits_{m\in\mathcal{M}}\sqrt{a_{m}}\mathbf{a}_{N}^{H}(\phi_{m}^{a},\phi_{m}^{e}){{\bf
x}}_{c,m}\Big{|}^{2}\bigg{\\}}}_{\mathcal{T}_{1}}$
$\displaystyle+\underbrace{\mathbb{E}\bigg{\\{}\Big{|}\sum\nolimits_{m\in\mathcal{M}}\sqrt{(1-a_{m})}\mathbf{a}_{N}^{H}(\phi_{m}^{a},\phi_{m}^{e}){{\bf
x}}_{r,m}\Big{|}^{2}\bigg{\\}}}_{\mathcal{T}_{2}},$ (22)

where the expectation is taken over the transmitted signal ${\bf x}_{m}$ and
the small-sale fading.

Now, we proceed to derive $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$. By using
(II-C), we can express $\mathcal{T}_{1}$ as

$\displaystyle\mathcal{T}_{1}$
$\displaystyle\\!=\mathbb{E}\bigg{\\{}\Big{|}\\!\sum_{k\in\mathcal{K}}\Big{(}\sum_{m\in\mathcal{Z}_{k}}\\!\sqrt{a_{m}\rho{\eta}_{mk}^{c}}{\bf
a}_{N}^{H}(\phi_{m}^{a},\phi_{m}^{e}){\bf t}_{mk}^{\mathrm{ZF-Com}}$
$\displaystyle\\!+\\!\sum_{m\in\mathcal{M}_{k}}\\!\\!\\!\\!\sqrt{a_{m}\rho{\eta}_{mk}^{c}}{\bf
a}_{N}^{H}(\phi_{m}^{a},\phi_{m}^{e}){\bf t}_{mk}^{\mathrm{MR-
Com}}\Big{)}\Big{|}^{2}\\!\bigg{\\}},$
$\displaystyle=\sum_{k\in\mathcal{K}}\bigg{(}\sum_{m\in\mathcal{Z}_{k}}\\!\\!{a_{m}\rho{\eta}_{mk}^{c}}\mathbb{E}\bigg{\\{}\Big{|}{\bf
a}_{N}^{H}(\phi_{m}^{a},\phi_{m}^{e}){\bf t}_{mk}^{\mathrm{ZF-
Com}}\Big{|}^{2}\\!\bigg{\\}}$
$\displaystyle\\!+\\!\\!\\!\sum_{m\in\mathcal{M}_{k}}\\!\\!\\!{a_{m}\rho{\eta}_{mk}^{c}}\mathbb{E}\bigg{\\{}\Big{|}{\bf
a}_{N}^{H}(\phi_{m}^{a},\phi_{m}^{e}){\bf t}_{mk}^{\mathrm{MR-
Com}}\Big{|}^{2}\\!\bigg{\\}}\bigg{)}.$ (23)

By invoking Lemma 1 in Appendix A, we have

$\displaystyle~{}\mathcal{T}_{1}$
$\displaystyle=\sum\nolimits_{k\in\mathcal{K}}\Big{(}\sum\nolimits_{m\in\mathcal{Z}_{k}}\\!\\!{a_{m}\rho{\eta}_{mk}^{c}}\frac{\gamma_{mk}}{N-|\mathcal{S}_{m}|}$
$\displaystyle~{}\hskip
30.00005pt+\sum\nolimits_{m\in\mathcal{M}_{k}}\\!\\!{a_{m}\rho{\eta}_{mk}^{c}}N\gamma_{mk}\Big{)}.$
(24)

Moreover, $\mathcal{T}_{2}$ can be obtained as

$\displaystyle~{}\mathcal{T}_{2}=\rho\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l^{\prime}\in\mathcal{L}}(1-a_{m})\eta^{s}_{ml^{\prime}}\big{|}\mathbf{a}_{N}^{H}(\phi_{m}^{a},\phi_{m}^{e}){\bf
t}_{ml^{\prime}}^{\text{sen}}\big{|}^{2}.$ (25)

For the simplicity of notation, let us introduce a pair of binary variables to
indicate the group assignment for each AP $m$ and DL UE $k$, as

$\displaystyle\delta_{k}^{\mathcal{S}_{m}}$
$\displaystyle=\begin{cases}1&\text{if }k\in\mathcal{S}_{m},\\\
0&\text{otherwise}.\end{cases}$ (26a)
$\displaystyle\delta_{k}^{\mathcal{W}_{m}}$
$\displaystyle=\begin{cases}1&\text{if }k\in\mathcal{W}_{m},\\\
0&\text{otherwise}.\end{cases}$ (26b)

By invoking (II-D), the average spatial power pattern for sensing is given by

$\displaystyle P^{\mathrm{ave}}(\phi_{m}^{a},\phi_{m}^{e})$
$\displaystyle=P^{\mathrm{ave}}_{\mathrm{Com}}(\phi_{m}^{a},\phi_{m}^{e})$
$\displaystyle+\\!P^{\mathrm{ave}}_{\mathrm{Sen}}(\phi_{m}^{a},\phi_{m}^{e}),$
(27)

where
$P^{\mathrm{ave}}_{\mathrm{Sen}}(\phi_{m}^{a},\phi_{m}^{e})=\mathcal{T}_{2}$
represents power pattern related to $L$ sensing zones, and
$P^{\mathrm{ave}}_{\mathrm{Com}}(\phi_{m}^{a},\phi_{m}^{e})=\mathcal{T}_{1}$
represents the average sensing power distortion caused by simultaneous
communication symbol transmissions in the network, given by

$\displaystyle P^{\mathrm{ave}}_{\mathrm{Com}}(\phi_{m}^{a},\phi_{m}^{e})$
$\displaystyle=\rho\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{k\in\mathcal{K}}\\!\\!a_{m}{\eta}_{mk}^{c}\gamma_{mk}$
$\displaystyle\times\Big{(}\frac{\delta_{k}^{\mathcal{S}_{m}}}{N-|\mathcal{S}_{m}|}\\!+N\delta_{k}^{\mathcal{W}_{m}}\Big{)},$
(28)

which is independent of the angles.

Next, we consider the sensing performance associated with target $l$, i.e., at
angle $(\phi_{t,ml}^{a},\phi_{t,ml}^{e})$ for all $m\in\mathcal{M}$. By using
(25), $P^{\mathrm{ave}}_{\mathrm{Sen}}(\phi_{t,ml}^{a},\phi_{t,ml}^{e})$ can
be expressed as

$\displaystyle
P^{\mathrm{ave}}_{\mathrm{Sen}}(\phi_{t,ml}^{a},\phi_{t,ml}^{e})=\rho
N^{2}\sum\nolimits_{m\in\mathcal{M}}(1-a_{m})\eta^{s}_{ml}$
$\displaystyle\\!+\\!\rho\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l^{\prime}\in\mathcal{L}\setminus
l}(1-a_{m})\eta^{s}_{ml^{\prime}}\big{|}\mathbf{a}_{N}^{H}(\phi_{t,ml}^{a},\phi_{t,ml}^{e}){\bf
t}_{ml^{\prime}}^{\text{sen}}\big{|}^{2},$ (29)

where the first term denotes the desired power pattern for sensing at the
angles related to target $l$ and the second term is the average sensing power
pattern distortion related to the other sensing zones.

It is desirable to have the average sensing power pattern distortion as well
as $P^{\mathrm{ave}}_{\mathrm{Com}}(\phi_{m}^{a},\phi_{m}^{e})$,
$\forall~{}\phi_{m}^{a}$ and $\phi_{m}^{e}$, as small as possible to confine
the pattern distortion. Moreover, for illuminating a target angle
$(\phi_{t,ml}^{a},\phi_{t,ml}^{e})$, it is desirable that the mainlobe level
of desired term in
$P^{\mathrm{ave}}_{\mathrm{Sen}}(\phi_{t,ml}^{a},\phi_{t,ml}^{e})$ is higher
than the sum of sensing power pattern distortion and
$P^{\mathrm{ave}}_{\mathrm{Com}}(\phi_{m}^{a},\phi_{m}^{e})$ by a certain
minimum sensing level $\kappa$, $\forall~{}\phi_{m}^{a}$ and $\phi_{m}^{e}$,
which is referred to as the MASR [26]. Accordingly, the MASR for the $l$th
sensing zone is defined by (30) at the top of the next page. Note that, in
general, in the MASR, we need to choose angles that maximize the average
sensing power pattern distortion. However, this leads to an intractable MASR
form. To alleviate such difficulty and make the MASR more amenable to further
design, we choose $(\phi_{t,ml}^{a},\phi_{t,ml}^{e})$ for computing the
sensing power pattern distortion in MASR. This is reasonable since we assume
that we know the sensing zone $l$, and thus, we only need to search some
directions around the angles of target $l$.

$\displaystyle~{}\mathrm{MASR}_{l}(\mathbf{a},\bm{\eta}^{c},\bm{\eta}^{s})$
$\displaystyle\\!=\\!\frac{\sum\nolimits_{m\in\mathcal{M}}(1-a_{m})N^{2}\eta^{s}_{ml}}{\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{k\in\mathcal{K}}a_{m}{\eta}_{mk}^{c}\gamma_{mk}\Big{(}\frac{\delta_{k}^{\mathcal{S}_{m}}}{N-|\mathcal{S}_{m}|}\\!+\\!N\delta_{k}^{\mathcal{W}_{m}}\Big{)}\\!+\\!\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l^{\prime}\in\mathcal{L}\setminus
l}(1\\!-\\!a_{m})\eta^{s}_{ml^{\prime}}\big{|}\mathbf{a}_{N}^{H}(\phi_{t,ml}^{a},\phi_{t,ml}^{e}){\bf
t}_{ml^{\prime}}^{\text{sen}}\big{|}^{2}}.$ (30)

##  III Asymptotic Analysis

The system comprises a total of $M$ APs. The APs are divided into two
categories: $M_{c}$ APs designated for communication activities and $M_{s}$
APs dedicated to sensing tasks, ensuring the total AP count remains as
$M=M_{c}+M_{s}$. With the application of MR precoding across all communication
APs, we can pursue a detailed assessment of MR precoding’s impact on the
effectiveness of CF-mMIMO ISAC systems. In addition, we assume that equal
power control is applied at the C-APs and S-APs, where

$\displaystyle\eta^{c}_{mk}=\frac{1}{N\sum_{k^{\prime}\in\mathcal{K}}\gamma_{mk^{\prime}}},~{}\forall
k,$ (31a) $\displaystyle\eta^{s}_{ml}=\frac{1}{NL}.$ (31b)

We analyze the performance of two case studies: i) $M$ is large and $N$ is
fixed, and; ii) $N$ is large and $M$ is fixed.

###  III-A Case I ($M_{s},M_{c}\to\infty$ and $N$ is fixed)

In this section, we explore a scenario where both the numbers of APs
designated for $M_{s}$ and $M_{c}$ approach infinity. Importantly, during this
expansion, we maintain a constant ratio between $M_{c}$ and $M_{s}$,
symbolized as $c=\frac{M_{c}}{M_{s}}$. This examination sheds light on the
system’s asymptotic performance characteristics, when both the sensing and
communication resources are scaled infinitely but proportionally. A critical
aspect of this scenario involves the transmit power scaling of each AP. As
$M_{c}$ becomes very large, the transmit power per AP is adjusted according to
$\rho=\frac{E}{M_{c}^{2}}$, where $E$ is fixed. By applying (31) into (II-C),
we have

$\displaystyle
y_{k}=\sum\nolimits_{m\in{\mathcal{M}}}\sqrt{\frac{a_{m}E}{N\sum_{k\in\mathcal{K}}\gamma_{mk}}}\frac{1}{M_{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{MR-Com}}_{mk}x_{c,k}$ $\displaystyle\hskip
10.00002pt+\\!\\!\sum\nolimits_{k^{\prime}\in\mathcal{K}\setminus
k}\\!\sum\nolimits_{m\in\mathcal{M}}\\!\\!\sqrt{\\!\frac{a_{m}E}{N\\!\sum_{k\in\mathcal{K}}\gamma_{mk}}}\frac{1}{M_{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{MR-Com}}_{mk^{\prime}}\\!x_{c,k^{\prime}}$
$\displaystyle\hskip
10.00002pt+\\!\\!\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l\in\mathcal{L}}\sqrt{\frac{\left(1-a_{m}\right)E}{NL}}\frac{1}{cM_{s}}{\bf
g}^{T}_{mk}{\bf t}_{ml}^{\mathrm{Sen}}x_{r}+{n}_{k}.$ (32)

Then, by applying Tchebyshev’s theorem we have111Let $X_{1},\ldots,X_{n}$ be
independent RVs, such that $\mathbb{E}\Big{\\{}X_{i}\\}=\bar{x}_{i}$ and
$\text{Var}\leq c\leq\infty.$ Then, Tchebyshev’s theorem states
$\frac{1}{n}\sum_{n^{\prime}=1}^{n}X_{n^{\prime}}-\frac{1}{n}\sum_{n^{\prime}}\bar{x}_{n^{\prime}}\xrightarrow[n\to\infty]{P}0$
[27].

$\displaystyle\sum\nolimits_{m\in{\mathcal{M}}}\sqrt{\frac{a_{m}E}{N\sum_{k\in\mathcal{K}}\gamma_{mk}}}\frac{1}{M_{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{MR-Com}}_{mk}$
$\displaystyle-\sum\nolimits_{m\in\mathcal{M}}\sqrt{\frac{a_{m}NE}{\sum_{k\in\mathcal{K}}\gamma_{mk}}}\frac{\gamma_{mk}}{M_{c}}\xrightarrow[M_{c}\to\infty]{P}0.$
(33)

Now, we turn our attention on the interference terms. For inter-user
interference, by noting that the zero-mean channel vector ${\bf g}_{mk}$ is
independent of ${\bf t}^{\mathrm{MR-Com}}_{mk^{\prime}}$ for $k^{\prime}\neq
k$, we have

$\displaystyle~{}\sum_{m\in\mathcal{M}}\sqrt{\frac{a_{m}E}{N\sum_{k\in\mathcal{K}}\gamma_{mk}}}\frac{1}{M_{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{MR-
Com}}_{mk^{\prime}}\xrightarrow[M_{c}\to\infty]{P}0.$ (34)

Moreover, for the deterministic ${\bf t}_{ml}^{\mathrm{Sen}}$, we have

$\displaystyle~{}\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l\in\mathcal{L}}\sqrt{\frac{\left(1-a_{m}\right)E}{NL}}\frac{1}{cM_{s}}{\bf
g}^{T}_{mk}{\bf t}_{ml}^{\mathrm{Sen}}\xrightarrow[M_{s}\to\infty]{P}\\!0.$
(35)

Accordingly, by using (III-A), (34), and (35), we have

$\displaystyle{y_{k}}\\!\\!-\\!\\!\bigg{(}\\!\\!\sum\nolimits_{m\in\mathcal{M}}\\!\sqrt{\frac{a_{m}NE}{\sum_{k\in\mathcal{K}}\gamma_{mk}}}\frac{\gamma_{mk}}{M_{c}}\bigg{)}x_{c,k}\\!\\!-\\!n_{k}\xrightarrow[M_{c},M_{s}\to\infty]{P}\\!0.$
(36)

The result in (36) indicates that when both $M_{c}$ and $M_{s}$ approach
infinity, the received signal contains the desired signal plus noise. This
indicates that inter-user interference and interference caused by S-APs fade,
enhancing the signal quality. Most importantly, as $M$ grows large, while the
ratio of $M_{c}$ and $M_{s}$ is kept fixed, the transmit power at each AP can
be made inversely proportional to the square of the number of APs with no
degradation in performance.

###  III-B Case II ($N\to\infty$ and $M_{c}$ and $M_{s}$ are fixed)

We explore the scenario where $N$ approaches infinity, while $M_{c}$ and
$M_{s}$ remain constant. The scaling of the transmit power for each AP is
adjusted according to $\rho=\frac{E}{N}$, where $E$ remains a fixed value.

By implementing the power scaling rule $\rho=\frac{E}{N}$, considering (II-C)
and substituting ${\bf g}_{mk}=(\hat{{\bf g}}_{mk}+\tilde{{\bf g}}_{mk})$, the
desired received signal at the UE can be characterized, as follows

$\displaystyle\sum\nolimits_{m\in{\mathcal{M}}}\sqrt{\frac{a_{m}E}{\sum_{k\in\mathcal{K}}\gamma_{mk}}}\frac{1}{N}(\hat{{\bf
g}}_{mk}+\tilde{{\bf g}}_{mk})^{T}\mathbf{\hat{g}}^{*}_{mk}$
$\displaystyle\buildrel
a.s.\over{\longrightarrow}\sum\nolimits_{m\in{\mathcal{M}}}\sqrt{\frac{a_{m}E}{\sum_{k\in\mathcal{K}}\gamma_{mk}}}\gamma_{mk},$
(37)

where we have used the results on very long random vectors [28].

For the interference terms, noticing that ${\bf g}^{T}_{mk}$ and ${\bf
t}^{\mathrm{MR-Com}}_{mk^{\prime}}$ are independent zero-mean vectors, we have

$\displaystyle\sum\nolimits_{m\in\mathcal{M}}\sqrt{\frac{a_{m}E}{\sum_{k\in\mathcal{K}}\gamma_{mk}}}\frac{1}{N}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{MR-Com}}_{mk^{\prime}}\buildrel
a.s.\over{\longrightarrow}0$ (38a)
$\displaystyle\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l\in\mathcal{L}}\sqrt{\frac{\left(1-a_{m}\right)E}{L}}\frac{1}{N}{\bf
g}^{T}_{mk}{\bf t}_{ml}^{\mathrm{Sen}}\buildrel a.s.\over{\longrightarrow}0.$
(38b)

Then, by using (III-B) and (38) we have

$\displaystyle y_{k}$
$\displaystyle-\sum\nolimits_{m\in{\mathcal{M}}}\sqrt{\frac{a_{m}E}{\sum_{k\in\mathcal{K}}\gamma_{mk}}}\gamma_{mk}x_{c,k}$
$\displaystyle-n_{k}\buildrel
a.s.\over{\longrightarrow}0,~{}\text{as}~{}N\rightarrow\infty,$ (39)

where $\buildrel a.s.\over{\rightarrow}$ denotes almost sure convergence. Akin
to colocated mMIMO systems, the huge power gain (i.e. $\rho=\frac{E}{N}$) can
be achieved, and the channels between the UEs and APs tend toward
orthogonality as $N\to\infty$, elucidating the profound impact of deploying an
extensive number of antennas in the CF-mMIMO ISAC space. This tendency towards
channel orthogonality inherently mitigates the interference and enhances the
signal quality, thereby reinforcing the premise that increasing the antenna
count per AP not only avails of the advantages seen in traditional mMIMO
architectures, but also significantly boosts the overall system performance
and SE within the CF-mMIMO ISAC paradigm.

##  IV Proposed Design Problems and Solution

In this extended section, we formulate and solve the problem of joint AP mode
selection and power allocation to provide fairness across the UEs, subject to
quality of service (QoS) requirements of the sensing zone. More specifically,
we aim to optimize the AP operation mode selection vector (${\bf a}$) and
power control coefficients ($\bm{\eta}^{c}_{mk},\bm{\eta}^{s}_{ml}$) to
maximize the minimum per-UE SE subject to a prescribed MASR level for the
target detection and transmit power constraints at the APs.

In order to further simplify the optimization problem, we propose a greedy-
based algorithm for AP mode selection and power control design. More
specifically, the joint optimization problem is decomposed into two sub-
problems: 1) AP mode selection, which is performed via a greedy algorithm and
2) Power control design for fixed AP modes.

Note that AP $m$ is required to meet the average normalized power constraint,
i.e., $\mathbb{E}\left\\{\|{\bf x}_{c,m}\|^{2}\right\\}\leq\rho$. By invoking
(13), and noticing that $\mathbb{E}\Big{\\{}\big{\|}{\bf t}^{\mathrm{ZF-
Com}}_{mk}\big{\|}^{2}\Big{\\}}=\frac{\gamma_{mk}}{N-|\mathcal{S}_{m}|}$, we
have the following per-AP power constraint

$\displaystyle\sum\nolimits_{k\in\mathcal{S}_{m}}\eta^{c}_{mk}\frac{\gamma_{mk}}{N-|\mathcal{S}_{m}|}$
$\displaystyle\hskip
20.00003pt+\sum\nolimits_{k\in\mathcal{W}_{m}}N\gamma_{mk}\eta^{c}_{mk}\leq
a_{m}.$ (40)

###  IV-A Joint AP Mode Selection and Power Allocation Design

In this subsection, we formulate and solve the problem of joint AP mode
selection and power control design. Define
$\mathbf{a}\triangleq\\{a_{1},\ldots,a_{M}\\}$,
$\bm{\eta}^{c}\triangleq\\{\eta^{c}_{m1},\ldots,\eta^{c}_{mK}\\}$ and
$\bm{\eta}^{s}\triangleq\\{\eta_{m1}^{s},\ldots,\eta^{s}_{mL}\\}$ for all
$m\in\mathcal{M}$. The optimization problem is then formulated as

$\displaystyle\text{{(P1):}}~{}\underset{\mathbf{a},\bm{\eta}^{c},\bm{\eta}^{s}}{\max}\min_{k\in\mathcal{K}}\hskip
10.00002pt\,$
$\displaystyle\mathrm{SINR}_{k}(\mathbf{a},\bm{\eta}^{c},\bm{\eta}^{s})$ (41a)
$\displaystyle\mathrm{s.t.}\hskip 10.00002pt\,$
$\displaystyle\mathrm{MASR}_{l}(\mathbf{a},\bm{\eta}^{c},\bm{\eta}^{s})\geq\kappa,~{}\forall
l\in\mathcal{L},$ (41b)
$\displaystyle\sum\nolimits_{k\in\mathcal{S}_{m}}\eta^{c}_{mk}\frac{\gamma_{mk}}{N-|\mathcal{S}_{m}|}$
$\displaystyle+\sum\nolimits_{k\in\mathcal{W}_{m}}N\gamma_{mk}\eta^{c}_{mk}\leq
a_{m},\forall m\in\mathcal{M},$ (41c)
$\displaystyle\sum_{l\in\mathcal{L}}\\!\eta^{s}_{ml}\\!\leq\\!\frac{(1\\!-\\!a_{m})}{N},\forall
m\in\mathcal{M},$ (41d) $\displaystyle a_{m}\in\\{0,1\\},$ (41e)

where the first constraint (41b) is to explicitly control the level of the
MASR for all sensing zones, while the second and third constraint control the
transmit power at the C-APs and S-APs, respectively.

Before proceeding, we introduce an auxiliary variable
$t=\min_{k\in\mathcal{K}}\mathrm{SINR}_{k}(\mathbf{a},\bm{\eta}^{c},\bm{\eta}^{s})$.
We further relax the binary constraint (42f). Since at the optimal point
$a_{m}\in\\{0,1\\}$, we replace $a_{m}$ with $a_{m}^{2}$ in the second and
third constraint to accelerate the convergence speed of the optimization
problem. Accordingly, the optimization problem (41) is reformulated as
follows:

$\displaystyle\text{{(P2):}}~{}\underset{{\bf
a},\bm{\eta}^{c},\bm{\eta}^{s},t}{\max}\hskip 10.00002pt$ $\displaystyle
t\\!-\\!\lambda\sum\nolimits_{m\in\mathcal{M}}\\!\\!a_{m}-a_{m}^{(n)}\Big{(}2a_{m}\\!-\\!a_{m}^{(n)}\Big{)}$
(42a) $\displaystyle\mathrm{s.t.}\hskip 10.00002pt\,$
$\displaystyle\mathrm{SINR}_{k}(\mathbf{a},\bm{\eta}^{c},\bm{\eta}^{s})\geq
t,~{}\forall k\in\mathcal{K}$ (42b)
$\displaystyle\mathrm{MASR}_{l}(\mathbf{a},\bm{\eta}^{c},\bm{\eta}^{s})\geq\kappa,~{}\forall
l\in\mathcal{L},$ (42c)
$\displaystyle\sum\nolimits_{k\in\mathcal{S}_{m}}\eta^{c}_{mk}\frac{\gamma_{mk}}{N\\!-\\!|\mathcal{S}_{m}|}\\!+\\!\sum\nolimits_{k\in\mathcal{W}_{m}}\\!\\!N\gamma_{mk}\eta^{c}_{mk}$
$\displaystyle\leq{a_{m}^{(n)}\Big{(}2a_{m}\\!-\\!a_{m}^{(n)}\Big{)}},\forall
m\in\mathcal{M},$ (42d)
$\displaystyle\sum_{l\in\mathcal{L}}\eta^{s}_{ml}\leq\frac{(1-a_{m}^{2})}{N},~{}\forall
m\in\mathcal{M},$ (42e) $\displaystyle 0\leq a_{m}\leq 1,~{}\forall
m\in\mathcal{M}.$ (42f)

The non-convex nature of constraints (42b) and (42c) make the resulting
problem non-convex. To address the non-convexity, we employ the method of SCA.
To deal with the non-convexity of (42b), we first define

$\displaystyle\varrho_{mk}$
$\displaystyle\triangleq\zeta_{m}^{\mathcal{Z}_{k}}\frac{(\beta_{mk}-\gamma_{mk})}{N-|\mathcal{S}_{m}|}+\zeta_{m}^{\mathcal{M}_{k}}N\beta_{mk},$
(43a) $\displaystyle f_{mk}$
$\displaystyle\triangleq\zeta_{m}^{\mathcal{Z}_{k}}+N\zeta_{m}^{\mathcal{M}_{k}}.$
(43b)

Now, constraint (42b) can be expressed as

$\displaystyle\frac{\Big{(}\sum_{m\in\mathcal{M}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}\gamma_{mk}f_{mk}\Big{)}^{2}}{t}\geq$
$\displaystyle\rho\sum\nolimits_{m\in\mathcal{M}}a_{m}\bigg{(}\sum\nolimits_{k^{\prime}\in\mathcal{K}}{\eta}_{mk^{\prime}}^{c}\gamma_{mk^{\prime}}\varrho_{mk}-\sum\nolimits_{l\in\mathcal{L}}\eta^{s}_{ml}N\beta_{mk}\bigg{)}$
$\displaystyle+\rho\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l\in\mathcal{L}}\eta^{s}_{ml}N\beta_{mk}+1.$
(44)

For ease of description, let us denote

$\displaystyle\mu_{mk}$
$\displaystyle\triangleq\sum\nolimits_{k^{\prime}\in\mathcal{K}}{\eta}_{mk^{\prime}}^{c}\gamma_{mk^{\prime}}\varrho_{mk}-\sum\nolimits_{l\in\mathcal{L}}\eta^{s}_{ml}N\beta_{mk}.$
(45)

To this end, (IV-A) is equivalent to

$\displaystyle\frac{\Big{(}2\sum_{m\in\mathcal{M}}\sqrt{a_{m}{\eta}_{mk}^{c}}\gamma_{mk}f_{mk}\Big{)}^{2}}{t}\\!+\\!\sum\nolimits_{m\in\mathcal{M}}(a_{m}\\!-\\!\mu_{mk})^{2}$
$\displaystyle\\!\geq\\!\sum\nolimits_{m\in\mathcal{M}}(a_{m}\\!+\\!\mu_{mk})^{2}\\!\\!+\\!4\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l\in\mathcal{L}}\eta^{s}_{ml}N\beta_{mk}\\!+\\!4/\rho.$
(46)

It is clear that we need to find a concave lower bound of the left-hand side
of the above inequality. To this end, we note that the function $x^{2}/y$ is
convex for $y>0$, and thus, the following inequality holds

$\displaystyle~{}\frac{x^{2}}{y}\geq\frac{x_{0}}{y_{0}}\bigl{(}2x-\frac{x_{0}}{y_{0}}y\bigr{)},$
(47)

which is obtained by linearizing $x^{2}/y$ around $x_{0}$ and $y_{0}$. Let us
now define

$\displaystyle
q_{k}^{(n)}\triangleq\frac{2\sum_{m\in\mathcal{M}}\sqrt{a_{m}^{(n)}({\eta}_{mk}^{c})^{(n)}}\gamma_{mk}f_{mk}}{t^{(n)}}.$
(48)

From the inequality (47), we can recast (IV-A) as

$\displaystyle
q_{k}^{(n)}\Big{(}4\sum\nolimits_{m\in\mathcal{M}}\sqrt{a_{m}{\eta}_{mk}^{c}}\gamma_{mk}f_{mk}\\!-\\!q_{k}^{(n)}t\Big{)}$
$\displaystyle+\\!\sum\nolimits_{m\in\mathcal{M}}\\!\big{(}a_{m}^{(n)}\\!-\\!\mu_{mk}^{(n)}\big{)}\Big{(}2\big{(}a_{m}-\mu_{mk}\big{)}-\big{(}a_{m}^{(n)}-\mu_{mk}^{(n)}\big{)}\Big{)}$
$\displaystyle\geq\sum\nolimits_{m\in\mathcal{M}}(a_{m}+\mu_{mk})^{2}$
$\displaystyle\hskip
40.00006pt+\\!4\sum\nolimits_{m\in\mathcal{M}}\\!\sum\nolimits_{l\in\mathcal{L}}\\!\eta^{s}_{ml}N\beta_{mk}\\!+\\!4/\rho,$
(49)

where we have used the following inequality

$~{}x^{2}\geq x_{0}(2x-x_{0}),$ (50)

and replaced $x$ and $x_{0}$ by $a_{m}-\mu_{mk}$ and
$a_{m}^{(n)}-\mu_{mk}^{(n)}$, respectively.

Now, we focus on (42c), which can be expressed as

$\displaystyle{\sum\nolimits_{m\in\mathcal{M}}(1-a_{m})N^{2}\eta^{s}_{ml}}$
$\displaystyle\geq\kappa\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l^{\prime}\in\mathcal{L}\setminus
l}(1\\!-\\!a_{m})\eta^{s}_{ml^{\prime}}\big{|}\mathbf{a}_{N}^{H}(\phi_{t,ml}^{a},\phi_{t,ml}^{e}){\bf
t}_{ml^{\prime}}^{\text{sen}}\big{|}^{2}$
$\displaystyle+\kappa\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{k\in\mathcal{K}}a_{m}{\eta}_{mk}^{c}\gamma_{mk}\theta_{mk},$
(51)

where
$\theta_{mk}\triangleq\frac{\delta_{k}^{\mathcal{S}_{m}}}{N-|\mathcal{S}_{m}|}\\!+N\delta_{k}^{\mathcal{W}_{m}}$.
Now, we can recast (IV-A) as

$\displaystyle{\sum\nolimits_{m\in\mathcal{M}}N^{2}\eta^{s}_{ml}}\geq$
$\displaystyle\sum\nolimits_{m\in\mathcal{M}}a_{m}\Big{(}N^{2}\eta^{s}_{ml}+\kappa\sum\nolimits_{k\in\mathcal{K}}{\eta}_{mk}^{c}\gamma_{mk}\theta_{mk}$
$\displaystyle\\!-\\!\kappa\sum\nolimits_{l^{\prime}\in\mathcal{L}\setminus
l}\eta^{s}_{ml^{\prime}}\big{|}\mathbf{a}_{N}^{H}(\phi_{t,ml}^{a},\phi_{t,ml}^{e}){\bf
t}_{ml^{\prime}}^{\text{sen}}\big{|}^{2}\Big{)}$
$\displaystyle\\!+\\!\kappa\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l^{\prime}\in\mathcal{L}\setminus
l}\eta^{s}_{ml^{\prime}}\big{|}\mathbf{a}_{N}^{H}(\phi_{t,ml}^{a},\phi_{t,ml}^{e}){\bf
t}_{ml^{\prime}}^{\text{sen}}\big{|}^{2}.$ (52)

For ease of description let us denote

$\displaystyle\nu_{ml}$
$\displaystyle\triangleq\Big{(}N^{2}\eta^{s}_{ml}+\kappa\sum\nolimits_{k\in\mathcal{K}}{\eta}_{mk}^{c}\gamma_{mk}\theta_{mk}$
$\displaystyle\\!-\\!\kappa\sum\nolimits_{l^{\prime}\in\mathcal{L}\setminus
l}\eta^{s}_{ml^{\prime}}\big{|}\mathbf{a}_{N}^{H}(\phi_{t,ml}^{a},\phi_{t,ml}^{e}){\bf
t}_{ml^{\prime}}^{\text{sen}}\big{|}^{2}\Big{)}.$ (53)

Accordingly, (IV-A) can be written as

$\displaystyle
4\sum\nolimits_{m\in\mathcal{M}}N^{2}\eta^{s}_{ml}+\sum\nolimits_{m\in\mathcal{M}}(a_{m}-\nu_{ml})^{2}\geq$
$\displaystyle\sum\nolimits_{m\in\mathcal{M}}(a_{m}+\nu_{ml})^{2}$
$\displaystyle+4\kappa\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l^{\prime}\in\mathcal{L}\setminus
l}\eta^{s}_{ml^{\prime}}\big{|}\mathbf{a}_{N}^{H}(\phi_{t,ml}^{a},\phi_{t,ml}^{e}){\bf
t}_{ml^{\prime}}^{\text{sen}}\big{|}^{2}.$ (54)

To this end, by using (50), we obtain the concave lower bound of the left-hand
side of the above inequality. Then, we get

$\displaystyle
4\sum\nolimits_{m\in\mathcal{M}}N^{2}\eta^{s}_{ml}+\sum\nolimits_{m\in\mathcal{M}}\big{(}a_{m}^{(n)}-\nu_{ml}^{(n)}\big{)}$
$\displaystyle\times\Big{(}2(a_{m}-\nu_{ml})-(a_{m}^{(n)}-\nu_{ml}^{(n)})\Big{)}\geq\sum\nolimits_{m\in\mathcal{M}}(a_{m}+\nu_{ml})^{2}$
$\displaystyle\\!+\\!4\kappa\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l^{\prime}\in\mathcal{L}\setminus
l}\eta^{s}_{ml^{\prime}}\big{|}\mathbf{a}_{N}^{H}(\phi_{t,ml}^{a},\phi_{t,ml}^{e}){\bf
t}_{ml^{\prime}}^{\text{sen}}\big{|}^{2}.$ (55)

Now, the convex optimization problem is given as (56), at the top of the next
page. Problem (P3) is a convex optimization problem and can be efficiently
solved using CVX [29]. In Algorithm 1, we outline main steps to solve problem
(P3), where $\widetilde{{\bf x}}\triangleq\\{{\bf
a},\bm{\eta}^{c},\bm{\eta}^{s}\\}$ and
$\widehat{\mathcal{F}}\triangleq\\{\eqref{P4:a},~{}\eqref{P4:b},~{}\eqref{P4:c},~{}\eqref{P4:d},~{}\eqref{P4:e}\\}$
is a convex feasible set. Starting from a random point $\widetilde{{\bf
x}}\in\widehat{\mathcal{F}}$, we solve (56) to obtain its optimal solution
$\widetilde{{\bf x}}^{*}$, and use $\widetilde{{\bf x}}^{*}$ as an initial
point in the next iteration. The proof of this convergence property uses
similar steps as the proof of [30, Proposition 2], and hence, is omitted
herein due to lack of space.

Complexity of Algorithm 1: Algorithm 1 requires solving a series of convex
problems (56). For ease of presentation, if we let $K=L$, problem (56) can be
transformed to an equivalent problem that involves $A_{v}\triangleq(M+2MK+1)$
real-valued scalar variables, $A_{l}\triangleq 3M$ linear constraints,
$A_{q}\triangleq 2K$ quadratic constraints. Therefore, the algorithm for
solving problem (56) requires a complexity of
$\mathcal{O}(\sqrt{A_{l}+A_{q}}(A_{v}+A_{l}+A_{q})A_{v}^{2})$ in each
iteration [31, 32]. In Section V, we will show that this algorithm converges
to the optimal solution after a few iterations.

$\displaystyle\text{{(P3):}}~{}\underset{{\bf
a},\bm{\eta}^{c},\bm{\eta}^{s},t}{\max}\hskip 10.00002pt$ $\displaystyle
t-\lambda\sum\nolimits_{m\in\mathcal{M}}a_{m}-a_{m}^{(n)}\Big{(}2a_{m}\\!-\\!a_{m}^{(n)}\Big{)}$
(56a) $\displaystyle\mathrm{s.t.}\hskip 10.00002pt\,$ $\displaystyle
q_{k}^{(n)}\Big{(}4\sum\nolimits_{m\in\mathcal{M}}\sqrt{a_{m}{\eta}_{mk}^{c}}\gamma_{mk}f_{mk}\\!-\\!q_{k}^{(n)}t\Big{)}\\!+\\!\sum\nolimits_{m\in\mathcal{M}}\\!\big{(}a_{m}^{(n)}\\!-\\!\mu_{mk}^{(n)}\big{)}\Big{(}2\big{(}a_{m}-\mu_{mk}\big{)}-\big{(}a_{m}^{(n)}-\mu_{mk}^{(n)}\big{)}\Big{)}$
$\displaystyle\hskip
40.00006pt\geq\sum\nolimits_{m\in\mathcal{M}}(a_{m}+\mu_{mk})^{2}+4\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l\in\mathcal{L}}\eta^{s}_{ml}N\beta_{mk}+4/\rho,~{}$
(56b) $\displaystyle
4\sum\nolimits_{m\in\mathcal{M}}N^{2}\eta^{s}_{ml}+\sum\nolimits_{m\in\mathcal{M}}\big{(}a_{m}^{(n)}-\nu_{ml}^{(n)}\big{)}\Big{(}2(a_{m}-\nu_{ml})-(a_{m}^{(n)}-\nu_{ml}^{(n)})\Big{)}$
$\displaystyle\hskip
40.00006pt\geq\sum\nolimits_{m\in\mathcal{M}}(a_{m}+\nu_{ml})^{2}\\!+\\!4\kappa\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l^{\prime}\in\mathcal{L}\setminus
l}\eta^{s}_{ml^{\prime}}\big{|}\mathbf{a}_{N}^{H}(\phi_{t,ml}^{a},\phi_{t,ml}^{e}){\bf
t}_{ml^{\prime}}^{\text{sen}}\big{|}^{2},~{}$ (56c)
$\displaystyle\sum\nolimits_{k\in\mathcal{S}_{m}}\eta^{c}_{mk}\frac{\gamma_{mk}}{N-|\mathcal{S}_{m}|}+\sum\nolimits_{k\in\mathcal{W}_{m}}N\gamma_{mk}\eta^{c}_{mk}\leq{a_{m}^{(n)}\Big{(}2a_{m}\\!-\\!a_{m}^{(n)}\Big{)}},\forall
m\in\mathcal{M},~{}$ (56d)
$\displaystyle\sum_{l\in\mathcal{L}}\eta^{s}_{ml}\leq\frac{(1-a_{m}^{2})}{N},\forall
m\in\mathcal{M},~{}$ (56e) $\displaystyle 0\leq a_{m}\leq 1.~{}$ (56f)
Algorithm 1 Proposed algorithm for joint AP mode selection and power
allocation design (JAP-PA)

1: Initialize: $n\\!=\\!0$, $\lambda>1$, a random initial poin
$\widetilde{{\bf x}}^{(0)}\\!\in\\!\widehat{\mathcal{F}}$.

2: repeat

3: Update $n=n+1$

4: Solve (56) to obtain its optimal solution $\widetilde{{\bf x}}^{*}$

5: Update $\widetilde{{\bf x}}^{(n)}=\widetilde{{\bf x}}^{*}$

6: until convergence

###  IV-B Greedy AP Mode Selection and Optimized Power Allocation

In this subsection, to reduce the complexity of the joint optimization
problem, while maintaining an acceptable system performance, we propose a
greedy algorithm to determine the operation modes of the APs. Then, given the
operation modes for the APs, we jointly optimize the power control
coefficients at the C-APs and S-APs.

####  IV-B1 Greedy AP Mode Selection

Let $\mathcal{A}_{\mathrm{Sen}}$ and $\mathcal{A}_{\mathrm{Com}}$ denote the
sets containing the indices of APs operating as radar, i.e., APs with
$a_{m}=0$, and of APs operating in communication mode, i.e., APs with
$a_{m}=1$, respectively. In addition,
$\mathrm{MASR}_{l}(\mathcal{A}_{\mathrm{Sen}},\mathcal{A}_{\mathrm{Com}})$ and
$\mathrm{SINR}_{k}(\mathcal{A}_{\mathrm{Sen}},\mathcal{A}_{\mathrm{Com}})$
underline the dependence of the sensing MASR (of the $l$-th zone) and received
SINR (of the $k$-th UE) on the different choices of AP mode selection. Our
greedy algorithm of AP mode selection is shown in Algorithm 2. To guarantee
the sensing MASR requirement, all APs are initially assigned for sensing
operation, i.e., $\mathcal{A}_{\mathrm{Sen}}=\mathcal{M}$ and
$\mathcal{A}_{\mathrm{Com}}=\emptyset$. Then, in each iteration, one AP
switches into communication operation mode for maximizing the minimum SE (or
equivalently SINR), while the minimum MARSs, required for target sensing, in
all sensing zones are guaranteed. This process continues until there is no
more improvement in the minimum SINR across all UEs.

####  IV-B2 Power Allocation

For a given operation mode selection vector (${\bf a}$), we optimize the power
control coefficients ($\bm{\eta}^{c}_{mk},\bm{\eta}^{s}_{ml}$) to achieve
maximum fairness among the EUs. Therefore, the optimization problem (41) is
reduced to

$\displaystyle\text{{(P4):}}~{}\underset{\bm{\eta}^{c},\bm{\eta}^{s}}{\max}\hskip
10.00002pt$ $\displaystyle t$ (57a) $\displaystyle\mathrm{s.t.}\hskip
10.00002pt\,$ $\displaystyle\mathrm{SINR}_{k}(\bm{\eta}^{c},\bm{\eta}^{s})\geq
t,$ (57b)
$\displaystyle\mathrm{MASR}_{l}(\bm{\eta}^{c},\bm{\eta}^{s})\geq\kappa,~{}\forall
l\in\mathcal{L},$ (57c)
$\displaystyle\sum\nolimits_{k\in\mathcal{S}_{m}}\eta^{c}_{mk}\frac{\gamma_{mk}}{N-|\mathcal{S}_{m}|}$
$\displaystyle+\sum\nolimits_{k\in\mathcal{W}_{m}}N\gamma_{mk}\eta^{c}_{mk}\leq
a_{m},\forall m\in\mathcal{M},$ (57d)
$\displaystyle\sum\nolimits_{l\in\mathcal{L}}\eta^{s}_{ml}\leq\frac{(1-a_{m})}{N},\forall
m\in\mathcal{M}.$ (57e)

We note that constraints (57b) and (57c) are non-convex. To deal with this
non-convexity, we first introduce the auxiliary variables
$\xi_{mk}^{2}=\eta^{c}_{mk}$, which yields the optimization problem (58), at
the middle of the next page.

$\displaystyle\text{{(P5):}}~{}\underset{\bm{\xi},\bm{\eta}^{s},t}{\max}\hskip
10.00002pt$ $\displaystyle t$ (58a) $\displaystyle\mathrm{s.t.}\hskip
10.00002pt\,$ $\displaystyle\frac{\Big{(}\\!\sum_{m\in\mathcal{M}}\sqrt{\rho
a_{m}}\xi_{mk}\gamma_{mk}f_{mk}\Big{)}^{2}}{\rho\sum_{m\in\mathcal{M}}a_{m}\varrho_{mk}\sum_{k^{\prime}\in\mathcal{K}}\xi_{mk^{\prime}}^{2}\gamma_{mk^{\prime}}\\!+\\!\rho\sum\nolimits_{m\in\mathcal{M}}N(1-a_{m})\beta_{mk}\sum\nolimits_{l\in\mathcal{L}}\eta^{s}_{ml}\\!+\\!1}\geq
t,$ (58b)
$\displaystyle\frac{\sum\nolimits_{m\in\mathcal{M}}(1-a_{m})N^{2}\eta^{s}_{ml}}{\sum\nolimits_{m\in\mathcal{M}}a_{m}\sum\nolimits_{k\in\mathcal{K}}\xi_{mk}^{2}\gamma_{mk}\theta_{mk}\\!+\\!\sum\nolimits_{m\in\mathcal{M}}(1\\!-\\!a_{m})\sum\nolimits_{l^{\prime}\in\mathcal{L}\setminus
l}\eta^{s}_{ml^{\prime}}\big{|}\mathbf{a}_{N}^{H}(\phi_{t,ml}^{a},\phi_{t,ml}^{e}){\bf
t}_{ml^{\prime}}^{\text{sen}}\big{|}^{2}}\geq\kappa,~{}\forall
l\in\mathcal{L},$ (58c)
$\displaystyle\sum\nolimits_{k\in\mathcal{K}}\xi_{mk}^{2}\gamma_{mk}\theta_{mk}\leq
a_{m},\forall m\in\mathcal{M},$ (58d)
$\displaystyle\sum\nolimits_{l\in\mathcal{L}}\eta^{s}_{ml}\leq\frac{(1-a_{m})}{N},\forall
m\in\mathcal{M},$ (58e)

At the next step, we replace the convex function in the numerator of (58b)
with its concave lower bound, which results in the following constraint

$\displaystyle~{}\frac{z_{k}^{(n)}\Big{(}2\sum_{m\in\mathcal{M}}\sqrt{a_{m}}\xi_{mk}\gamma_{mk}f_{mk}-z_{k}^{(n)}\Big{)}}{\Delta(\bm{\xi},\bm{\eta}^{s})}\geq
t,$ (59)

where
$\Delta(\bm{\xi},\bm{\eta}^{s})\triangleq\rho\sum_{m\in\mathcal{M}}a_{m}\varrho_{mk}\sum_{k^{\prime}\in\mathcal{K}}\xi_{mk^{\prime}}^{2}\gamma_{mk^{\prime}}\\!+\\!\rho\sum\nolimits_{m\in\mathcal{M}}N(1-a_{m})\beta_{mk}\sum\nolimits_{l\in\mathcal{L}}\eta^{s}_{ml}\\!+\\!1$
and
$z_{k}^{(n)}\triangleq\sum_{m\in\mathcal{M}}\sqrt{a_{m}}\xi_{mk}^{(n)}\gamma_{mk}f_{mk}$.
Therefore, the power control design problem for fixed AP mode operation is
given by

$\displaystyle\text{{(P6):}}~{}\underset{\bm{\xi},\bm{\eta}^{s},t}{\max}\hskip
10.00002pt$ $\displaystyle t$ (60a) $\displaystyle\mathrm{s.t.}\hskip
10.00002pt\,$
$\displaystyle~{}\eqref{eq:zkn},~{}\eqref{OP:PA:f1b}-\eqref{OP:PA:f1d},$ (60b)

Algorithm 2 Greedy AP Mode Selection

1: Initialize: Set $\mathcal{A}_{\mathrm{Com}}=\emptyset$ and
$\mathcal{A}_{\mathrm{Sen}}=\mathcal{M}$. Set iteration index $i=0$.

2: Calculate
$\Pi^{\star}[i]=\min_{k\in\mathcal{K}}\mathrm{SE}_{k}({A}_{\mathrm{Sen}},\mathcal{A}_{\mathrm{Com}})$

3: repeat

4: for all $m\in\mathcal{A}_{\mathrm{Sen}}$ do

5: Set $\mathcal{A}_{s}=\mathcal{A}_{\mathrm{Sen}}\setminus m$.

6: if $\mathrm{MASR}_{l}(\mathcal{A}_{s},\mathcal{A}_{\mathrm{Com}}\bigcup
m)\\!\geq\\!\kappa$, $\forall l\in\mathcal{L}$ then

7: Calculate
$\Pi_{m}=\min_{k\in\mathcal{K}}\mathrm{SINR}_{k}(\mathcal{A}_{s},\mathcal{A}_{\mathrm{Com}}\bigcup
m)$

8: else

9: Set $\Pi_{m}=0$

10: end if

11: end for

12: Set
$\Pi^{\star}[i+1]=\underset{m\in\mathcal{A}_{\mathrm{Com}}}{\max}\,\,\Pi_{m}$

13: $e=|\Pi^{\star}[i+1]-\Pi^{\star}[i]|$

14: if $e\geq e_{\min}$ then

15: Update
$\mathcal{A}_{\mathrm{Com}}\\!=\\!\\{\mathcal{A}_{\mathrm{Com}}\bigcup
m^{\star}\\}$ and
$\mathcal{A}_{\mathrm{Sen}}\\!=\\!\mathcal{A}_{\mathrm{Sen}}\\!\setminus\\!m^{\star}$

16: end if

17: Set $i=i+1$

18: until $e<e_{\min}$

19: return $\mathcal{A}_{\mathrm{Sen}}$ and $\mathcal{A}_{\mathrm{Com}}$,
i.e., the indices of APs operating in radar mode and communication mode,
respectively.

Algorithm 3 Bisection algorithm for solving (60)

  * (1)

_Initialization_ : Choose the initial values of $t_{\rm max}$ and $t_{\rm
min}$, where $t_{\rm max}$ and $t_{\rm min}$ define a range of objective
function values. Set tolerance $\epsilon>0$.

  * (2)

Set $t:=\frac{t_{\rm max}+t_{\rm min}}{2}$ and solve the convex feasibility
problem (60).

  * (3)

If the problem in Step $2$ is feasible, set $t_{\min}:=t$; else set $t_{\rm
max}:=t$.

  * (5)

Stop if $t_{\rm max}-t_{\rm min}<\epsilon$. Otherwise, go to Step $2$.

The optimization problem in (60) is again convex, which we can solve directly
using the bisection technique and solving linear feasibility problems, as
shown in Algorithm 3.

######  Remark 2.

It is important to note that, for a given network realization, it is likely
that the greedy and/or random AP selection schemes cannot guarantee the
sensing requirement in the network design. Hence, we introduce a new metric
known as the success sensing rate, defined as the ratio of the number of
network realizations with successful sensing for all sensing zones to the
total number of channel realizations. For a fair comparison, in each channel
realization, if the MASR requirements are not met or the optimization problem
of a scheme is infeasible, we set the SE of that scheme to zero.

##  V Numerical Examples

In this section, we verify the correctness of our analytical results and the
performance of the proposed algorithms.

###  V-A Large-scale Fading Model and System Parameters

We assume that the $M$ APs and $K$ UEs are uniformly distributed in a square
of $0.5\times 0.5$ km2, whose edges are wrapped around to avoid the boundary
effects. The large-scale fading coefficients $\beta_{mk}$ are modeled as [33]

$\displaystyle\beta_{mk}[\text{dB}]=-30.5-36.7\log_{10}\left(\frac{d_{mk}}{1\,\text{m}}\right)+F_{mk},$
(61)

where $d_{mk}$ is the distance between UE $k$ and AP $m$ (computed as the
minimum over different wrap-around cases) and
$F_{mk}\sim\mathcal{CN}(0,4^{2})$ is the shadow fading. The shadowing terms
from an AP to different UEs are correlated, as

$\displaystyle\mathbb{E}\\{F_{mk}F_{jk^{\prime}}\\}\triangleq\begin{cases}4^{2}2^{-\delta_{kk^{\prime}}/9\,\text{m}},&\text{if
$j=m$},\\\ 0,&\mbox{otherwise},\end{cases}$ (62)

where $\delta_{kk^{\prime}}$ is the physical distance between UEs $k$ and
$k^{\prime}$.

The values of the network parameters are $\tau=200$, and $\tau_{t}=K+L$. We
further set the bandwidth $B=50$ MHz and noise figure $F=9$ dB. Thus, the
noise power $\sigma^{2}_{n}=k_{B}T_{0}BF$, where $k_{B}=1.381\times 10^{-23}$
Joules/oK is the Boltzmann constant, while $T_{0}=290^{o}$K is the noise
temperature. Let $\tilde{\rho}=1$ W, and $\tilde{\rho}_{t}=0.25$ W be the
maximum transmit power of the APs and uplink training pilot sequences,
respectively. The normalized maximum transmit powers ${\rho}$ and ${\rho}_{t}$
are calculated by dividing these powers by the noise power.

To evaluate the performance of proposed AP mode selection schemes, we consider
random AP selection with optimized power allocation (RAP-OPA). In the
numerical results, JAP-OPA denotes our proposed design in Algorithm 1, i.e.,
joint AP mode selection and power allocation design. Moreover, greedy AP
selection and optimized power allocation design is indicated by GAP-OPA.

Figure 2: CDF of the per-UE SE and sensing success rate ($\kappa=8$ dB,
$M=30$, $N=16$, $K=4$, and $L=2$).

###  V-B Results and Discussions

The empirical cumulative distributed functions (CDFs) of the per-UE SE for all
scenarios with $\kappa=8$ dB, $MN=480$, $K=4$ and $L=2$ are plotted in Fig. 2.
The bar chart corresponds to the sensing success rate of different schemes. It
can be observed that while JAP-OPA guarantees the sensing requirement for all
network realizations, RAP-OPA fails to meet this requirement, with a success
rate of $10\%$. In other words, the power allocation problem becomes
infeasible in $90\%$ of cases. On the other hand, GAP-OPA yields comparable
results, achieving a success rate of approximately $70\%$, which confirms the
effectiveness of the proposed greedy algorithm. For the infeasible cases,
according to our discussion in Remark 2, we set the SE of the UEs to zero. By
comparing the CDFs of the per-UE SE across all schemes, we observe that JAP-
OPA provides a satisfactory fairness among the UEs. Moreover, the SE achieved
by JAP-OPA is much higher than that of the GAP-OPA and RAP-OPA.

Figure 3 shows the worst SE (minimum SE) across the UEs for given realizations
of the large-scale fading coefficients and for two different number of sensing
zones: $L=3$ (dashed lines) and $L=2$ (solid lines). The proposed JAP-OPA
scheme significantly improves the worst SE over the benchmarks and provides
$100\%$ sensing success rate for both scenarios. Moreover, we observe that
joint optimization can guarantee nearly the same minimum SE per UE across all
different channel realizations, while random and greedy AP selection provides
varying levels of per-UE SE. For example, when $L=2$, in $60\%$ of channel
realizations, the minimum SE obtained via GAP-OPA is less than $3$ bit/s/Hz,
while JAP-OPA consistently provides a minimum SE greater than $4$ bit/s/Hz.
Moreover, by increasing the number of sensing zones from $2$ to $3$, the
sensing performance of both the GAP-OPA and RAP-OPA severely degraded,
resulting in success rates of $27\%$ and $1\%$ for GAP-OPA and RAP-OPA,
respectively. These findings confirm the importance of joint AP mode selection
and power allocation for both communication and sensing operations.

Figure 3: CDF of the per-UE minimum SE for different schemes and for different
number of sensing zones. The dashed lines depict results for $L=3$ while the
solid lines show results for $L=2$ ($\kappa=6$ dB, $M=30$, $N=16$, $K=4$).

Figure 4 shows the average of the minimum SE per-UE with a fixed number of
antenna units in the network versus the number of APs. By increasing the
number of APs, the proposed JAP-OPA outperforms all other schemes, while RAP-
OPA yields the worst performance in terms of SE. Moreover, the sensing success
rate of GAP-OPA and RAP-OPA decreases as the number of APs increases. It is
most likely for RAP-OPA to fail when $M>30$. This behavior is due to the fact
that the MASR heavily depends on $N$, (it is proportional to $N^{2}$ according
to (30)). Therefore, the MASR decreases significantly as $M$ increases, and
both RAP-OPA and GAP-OPA schemes cannot guarantee to meet the sensing
requirement under this condition. Conversely, the proposed JAP-OPA scheme
ensures successful sensing for all network realizations, while delivering much
higher per-UE minimum SE compared to the RAP-OPA and GAP-OPA schemes.
Interestingly, the SE performance of the the proposed JAP-OPA scheme is
improved by increasing the number of APs. These results reveal the importance
of joint AP mode selection and power control design in dense CF-mMIMO
networks.

Figure 4: Average of the per-UE minimum SE and sensing success rate versus the
number of APs ($\kappa=8$ dB, $MN=480$, $K=4$, and $L=2$).

Figure 5 illustrates the average of the minimum SE per-UE versus $\kappa$,
along with the sensing success rate of different schemes as a function of
$\kappa$. As the sensing requirements increase, the sensing success rate of
the RAP-OPA and GAP-OPA schemes decreases, particularly when $\kappa>8$ dB.
Additionally, the minimum SE per-UE sharply decreases for these two schemes.
However, the proposed JAP-OPA scheme continues to meet the MASR requirements
and delivers high values for the minimum per-UE . Figure 5 illustrates the
average of the minimum SE per-UE versus $\kappa$, along with the sensing
success rate of different schemes as a function of $\kappa$. As the sensing
requirements increase, the sensing success rate of the RAP-OPA and GAP-OPA
schemes decreases, particularly when $\kappa>8$ dB. Additionally, the minimum
SE per-UE sharply decreases for these two schemes. However, the proposed JAP-
OPA continues to meet the MASR requirements and delivers high values for the
minimum per-UE SE.

Figure 5: Average of the per-UE minimum SE and sensing success rate versus
$\kappa$ ($MN=480$, $M=30$, $K=4$, and $L=2$).

In Fig. 6, we compare the convergence rate of Algorithm 1 for different number
of APs in the network. To solve (56), we use the convex conic solver MOSEK and
set $\lambda=10$. We can see that with a small number of iterations (less than
$60$ iterations), Algorithm 1 returns the optimized solution. The transient
behavior stems from the penalty term
$\lambda\sum\nolimits_{m\in\mathcal{M}}a_{m}-a_{m}^{(n)}\big{(}2a_{m}\\!-\\!a_{m}^{(n)}\big{)}$
in the optimization problem. Furthermore, it is worth mentioning that the
resulting values of the parameters $a_{m}$ converge to $1$ and $0$ with high
accuracy.

Figure 6: Convergence behavior of Algorithm 1 ($MN=480$, $K=4$, $L=2$,
$\lambda=10$).

##  VI Conclusion

In this paper, we proposed a distributed ISAC implementation underpinned by a
CF-mMIMO architecture. We analyzed both the exact and asymptotic SE
performance of the downlink communication system and provided exact results
for the MASR metric of the sensing zones. The operation mode of the
distributed APs and their transmit power coefficients were jointly optimized
to maximize fairness among communication UEs, while ensuring a specific level
of MASR for different sensing zones in the network. To reduce the
computational complexity of the proposed optimal scheme, a low-complexity
design was also developed, wherein the operation mode of the APs was
determined through a greedy algorithm and then the transmit power of the APs
was optimized via a bisection algorithm. The proposed AP mode selection and
power control design has shown to provide significant performance over the
benchmark systems with random/greedy AP mode selection for both communication
and sensing operations.

##  Appendix A Useful Lemma

######  Lemma 1.

For the ZF beamforming vector ${\bf t}^{\mathrm{ZF-Com}}_{mk}$, defined in
(10), we have

$\displaystyle~{}\mathbb{E}\Big{\\{}{\bf t}_{mk}^{\mathrm{ZF-Com}}({\bf
t}_{mk}^{\mathrm{ZF-Com}})^{H}\Big{\\}}$
$\displaystyle=\frac{\gamma_{mk}}{N(N-|\mathcal{S}_{m}|)}{\bf I}_{N}.$ (63)

###### Proof.

We first define ${\bf Q}\\!\triangleq\\!\hat{{\bf
G}}_{\mathcal{S}_{m}}\big{(}\hat{{\bf G}}_{\mathcal{S}_{m}}^{H}\hat{{\bf
G}}_{\mathcal{S}_{m}}\big{)}^{\\!-1}$ and ${\bf q}_{k}={\bf Q}{\bf e}_{k}$,
which is the $k$th column of ${\bf
Q}\in\mathbb{C}^{N\times|\mathcal{S}_{m}|}$. Then, we have

$\displaystyle\gamma_{mk}^{2}\mathbb{E}\Big{\\{}{\bf q}_{k}{\bf
q}_{k}^{H}\Big{\\}}=\frac{\gamma_{mk}^{2}}{|\mathcal{S}_{m}|}\sum^{|\mathcal{S}_{m}|}_{k^{\prime}=1}\mathbb{E}\Big{\\{}{\bf
q}_{k^{\prime}}{\bf q}_{k^{\prime}}^{H}\Big{\\}}.$ (64)

Accordingly, we have

$\displaystyle\mathbb{E}\Big{\\{}{\bf t}_{mk}^{\mathrm{ZF-Com}}({\bf
t}_{mk}^{\mathrm{ZF-
Com}})^{H}\Big{\\}}=\frac{\gamma_{mk}^{2}}{|\mathcal{S}_{m}|}\mathbb{E}\big{\\{}{\bf
Q}{\bf Q}^{H}\big{\\}}$
$\displaystyle=\frac{\gamma_{mk}^{2}}{|\mathcal{S}_{m}|}\mathbb{E}\Big{\\{}\hat{{\bf
G}}_{\mathcal{S}_{m}}\\!\left(\hat{{\bf G}}_{\mathcal{S}_{m}}^{H}\\!\hat{{\bf
G}}_{\mathcal{S}_{m}}\right)^{\\!-1}\\!\left(\\!\hat{{\bf
G}}_{\mathcal{S}_{m}}^{\\!H}\\!\hat{{\bf
G}}_{\mathcal{S}_{m}}\\!\right)^{\\!-1}\hat{{\bf
G}}_{\mathcal{S}_{m}}^{H}\Big{\\}}.$ (65)

Let ${\bf B}=\mathbb{E}\big{\\{}\hat{{\bf
G}}_{\mathcal{S}_{m}}\big{(}\hat{{\bf G}}_{\mathcal{S}_{m}}^{H}\hat{{\bf
G}}_{\mathcal{S}_{m}}\big{)}^{-1}\big{(}\hat{{\bf
G}}_{\mathcal{S}_{m}}^{H}\hat{{\bf G}}_{\mathcal{S}_{m}}\big{)}^{-1}\hat{{\bf
G}}_{\mathcal{S}_{m}}^{H}\big{\\}}$, For any $N\times N$ unitary matrix
$\bm{\Omega}$, we have

$\displaystyle~{}\bm{\Omega}{\bf
B}\bm{\Omega}^{H}\\!=\\!\mathbb{E}\big{\\{}\bm{\Omega}\hat{{\bf
G}}_{\mathcal{S}_{m}}\big{(}\hat{{\bf G}}_{\mathcal{S}_{m}}^{H}\hat{{\bf
G}}_{\mathcal{S}_{m}}\big{)}^{\\!\\!-1}\\!\\!\big{(}\hat{{\bf
G}}_{\mathcal{S}_{m}}^{\\!\\!H}\hat{{\bf
G}}_{\mathcal{S}_{m}}\big{)}^{\\!\\!-1}\hat{{\bf
G}}_{\mathcal{S}_{m}}^{\\!H}\bm{\Omega}^{H}\big{\\}}.$ (66)

Now, we define $\bar{{\bf G}}_{\mathcal{S}_{m}}=\bm{\Omega}\hat{{\bf
G}}_{\mathcal{S}_{m}}$. Thus, using (66), we have

$\displaystyle\bm{\Omega}{\bf
B}\bm{\Omega}^{H}\\!=\\!\\!\mathbb{E}\big{\\{}\bar{{\bf
G}}_{\mathcal{S}_{m}}\big{(}\bar{{\bf G}}_{\mathcal{S}_{m}}^{H}\\!\\!\bar{{\bf
G}}_{\mathcal{S}_{m}}\big{)}^{-1}\big{(}\bar{{\bf
G}}_{\mathcal{S}_{m}}^{H}\\!\\!\bar{{\bf
G}}_{\mathcal{S}_{m}}\big{)}^{\\!-1}\bar{{\bf
G}}_{\mathcal{S}_{m}}^{H}\big{\\}},$ (67)

where we have used

$\displaystyle\bar{{\bf G}}^{H}_{\mathcal{S}_{m}}\bar{{\bf
G}}_{\mathcal{S}_{m}}$ $\displaystyle=\hat{{\bf
G}}^{H}_{\mathcal{S}_{m}}\bm{\Omega}^{H}\bm{\Omega}\hat{{\bf
G}}_{\mathcal{S}_{m}}=\hat{{\bf G}}^{H}_{\mathcal{S}_{m}}\hat{{\bf
G}}_{\mathcal{S}_{m}}.$ (68)

Since $\bar{{\bf G}}_{\mathcal{S}_{m}}=\bm{\Omega}\hat{{\bf
G}}_{\mathcal{S}_{m}}$ is statistically identical to $\hat{{\bf
G}}_{\mathcal{S}_{m}}$, we have $\bm{\Omega}{\bf B}\bm{\Omega}^{H}={\bf B}$,
for any unitary matrix $\bm{\Omega}$. By using an eigenvalue decomposition,
${\bf B}$ can be expressed as ${\bf B}={\bf W}{\bf D}_{\lambda}{\bf W}^{H}$,
where ${\bf W}$ is a unitary matrix and ${\bf D}_{\lambda}$ is a diagonal
matrix. Then, $\bm{\Omega}{\bf B}\bm{\Omega}^{H}={\bf B}$ is equivalent to

$\displaystyle\bm{\Omega}{\bf W}{\bf D}_{\lambda}\bm{\Omega}^{H}{\bf
W}^{H}={\bf U}{\bf D}_{\lambda}{\bf U}^{H}={\bf B}.$ (69)

Since (69) is true for any unitary ${\bf U}$, ${\bf B}$ must be a scaled
identity matrix. This implies that ${\bf B}=c_{1}{\bf I}_{N}$, where $c_{1}$
is a constant, given by

$\displaystyle c_{1}=$
$\displaystyle\frac{1}{N}\mathbb{E}\Big{\\{}\mathrm{tr}\Big{(}\hat{{\bf
G}}_{\mathcal{S}_{m}}\left(\hat{{\bf G}}_{\mathcal{S}_{m}}^{H}\hat{{\bf
G}}_{\mathcal{S}_{m}}\right)^{\\!-1}\left(\hat{{\bf
G}}_{\mathcal{S}_{m}}^{H}\hat{{\bf
G}}_{\mathcal{S}_{m}}\right)^{\\!-1}\hat{{\bf
G}}_{\mathcal{S}_{m}}^{H}\Big{)}\Big{\\}}$
$\displaystyle=\frac{1}{N}\mathbb{E}\Big{\\{}\mathrm{tr}(\hat{{\bf
G}}_{\mathcal{S}_{m}}^{H}\hat{{\bf G}}_{\mathcal{S}_{m}})^{-1}\Big{\\}}$
$\displaystyle=\frac{|\mathcal{S}_{m}|}{N(N-|\mathcal{S}_{m}|)\gamma_{mk}}.$
(70)

##  Appendix B Proof of Proposition 1

We first present the desired signal term as

$\displaystyle~{}\mathrm{DS}_{k}$
$\displaystyle=\mathbb{E}\Big{\\{}\sum\nolimits_{m\in\mathcal{Z}_{k}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}(\hat{{\bf
g}}_{mk}+\tilde{{\bf g}}_{mk})^{T}{\bf t}^{\mathrm{ZF-Com}}_{mk}$
$\displaystyle\hskip
10.00002pt+\sum\nolimits_{m\in\mathcal{M}_{k}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}(\hat{{\bf
g}}_{mk}+\tilde{{\bf g}}_{mk})^{T}{\bf t}^{\mathrm{MR-Com}}_{mk}\Big{\\}}$
$\displaystyle=\sum\nolimits_{m\in\mathcal{Z}_{k}}\\!\\!\\!\sqrt{a_{m}\rho{\eta}_{mk}^{c}}\gamma_{mk}\\!\\!+\\!\\!N\\!\sum\nolimits_{m\in\mathcal{M}_{k}}\\!\\!\sqrt{a_{m}\rho{\eta}_{mk}^{c}}\gamma_{mk},$
(71)

where we have used the fact that ${\bf t}^{\mathrm{ZF-Com}}_{mk}$, ${\bf
t}^{\mathrm{MR-Com}}_{mk}$ and $\tilde{{\bf g}}_{mk}$ are zero mean and
independent. In addition, the variance of the sum of independent RVs is equal
to the sum of the variances. We can now proceed with the following derivation

$\displaystyle\mathbb{E}\Big{\\{}\big{|}\mathrm{BU}_{k}\big{|}^{2}\Big{\\}}=\mathbb{E}\Big{\\{}\Big{|}\sum\nolimits_{m\in\mathcal{Z}_{k}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{ZF-Com}}_{mk}$ $\displaystyle\hskip
10.00002pt+\sum\nolimits_{m\in\mathcal{M}_{k}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{MR-
Com}}_{mk}\Big{|}^{2}\Big{\\}}-\mathrm{DS}_{k}.$ (72)

We notice that

$\displaystyle~{}\vartheta$
$\displaystyle=\mathbb{E}\bigg{\\{}\Big{|}\sum\nolimits_{m\in\mathcal{Z}_{k}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{ZF-Com}}_{mk}$ $\displaystyle\hskip
20.00003pt+\sum\nolimits_{m\in\mathcal{M_{\kappa}}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{MR-Com}}_{mk}\Big{|}^{2}\bigg{\\}}$
$\displaystyle=\vartheta_{Z}+\vartheta_{M}+2\Big{(}\sum\nolimits_{m\in\mathcal{Z}_{k}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}\gamma_{mk}\Big{)}$
$\displaystyle\hskip
20.00003pt\times\Big{(}N\sum\nolimits_{m\in\mathcal{\kappa^{M}}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}\gamma_{mk}\Big{)},$
(73)

where

$\displaystyle\vartheta_{Z}$
$\displaystyle=\mathbb{E}\Big{\\{}\big{|}{{\sum\nolimits_{m\in\mathcal{Z}_{k}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{ZF-Com}}_{mk}}}\big{|}^{2}\Big{\\}}$
$\displaystyle\vartheta_{M}$
$\displaystyle=\mathbb{E}\Big{\\{}\big{|}{{\sum\nolimits_{m\in\mathcal{M}_{k}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}{\bf
g}^{T}_{mk}{\bf t}^{\mathrm{MR-Com}}_{mk}}}\big{|}^{2}\Big{\\}}.$ (74)

We first focus on $\vartheta_{Z}$, which can be obtained as

$\displaystyle~{}\vartheta_{Z}$
$\displaystyle=\mathbb{E}\bigg{\\{}\Big{|}\sum\nolimits_{m\in\mathcal{Z}_{k}}\sqrt{a_{m}\rho{\eta}_{mk}^{c}}(\hat{{\bf
g}}_{mk}+\tilde{{\bf g}}_{mk})^{T}{\bf t}^{\mathrm{ZF-
Com}}_{mk}\Big{|}^{2}\bigg{\\}}$
$\displaystyle=\sum\nolimits_{m\in\mathcal{Z}_{k}}a_{m}\rho{\eta}_{mk}^{c}$
$\displaystyle\hskip 20.00003pt\times\mathbb{E}\Big{\\{}({\bf t}^{\mathrm{ZF-
Com}}_{mk})^{H}\mathbb{E}\Big{\\{}\tilde{{\bf g}}^{T}_{mk}(\tilde{{\bf
g}}^{T}_{mk})^{H}\Big{\\}}{\bf t}^{\mathrm{ZF-Com}}_{mk}\Big{\\}}$
$\displaystyle=\sum\nolimits_{m\in\mathcal{Z}_{k}}\\!\\!\\!\sqrt{a_{m}\rho{\eta}_{mk}^{c}}\gamma_{mk}\sum\nolimits_{m\in\mathcal{Z}_{k}}\\!\\!\\!\\!a_{m}\rho{\eta}_{mk}^{c}(\beta_{mk}\\!\\!-\\!\gamma_{mk})$
$\displaystyle\hskip
20.00003pt\times\gamma_{mk}^{2}\mathbb{E}\Big{\\{}\\!\big{[}({\bf
G}_{\mathcal{S}_{m}}^{H}{\bf
G}_{\mathcal{S}_{m}}\\!)^{\\!-1}\big{]}_{kk}\\!\Big{\\}}$
$\displaystyle=\sum\nolimits_{m\in\mathcal{Z}_{k}}\\!\\!\\!\sqrt{a_{m}\rho{\eta}_{mk}^{c}}\gamma_{mk}$
$\displaystyle\hskip
20.00003pt+\sum\nolimits_{m\in\mathcal{Z}_{k}}a_{m}\rho{\eta}_{mk}^{c}(\beta_{mk}-\gamma_{mk})\frac{\gamma_{mk}}{N-|\mathcal{S}_{m}|}.$
(75)

Then, the second term of (B), $\vartheta_{M}$, can be derived as

$\displaystyle~{}\vartheta_{M}$
$\displaystyle=\sum\nolimits_{m\in\mathcal{M}_{k}}\\!\\!\\!a_{m}\rho{\eta}_{mk}^{c}\text{var}\Big{(}(\|\hat{{\bf
g}}_{mk}\|^{2}+\tilde{{\bf g}}_{mk}^{T}\hat{{\bf g}}_{mk}\Big{)}$
$\displaystyle+\Big{|}\mathbb{E}\Big{\\{}\sum\nolimits_{m\in\mathcal{M}_{k}}a_{m}\sqrt{\rho{\eta}_{mk}^{c}}(\hat{{\bf
g}}_{mk}+\tilde{{\bf g}}_{mk})^{T}{\bf t}^{\mathrm{MR-
Com}}_{mk}\Big{\\}}\Big{|}^{2}$
$\displaystyle=\sum\nolimits_{m\in\mathcal{M}_{k}}a_{m}\rho{\eta}_{mk}^{c}N\gamma_{mk}\beta_{mk}$
$\displaystyle\hskip
40.00006pt+\Big{(}N\sum\nolimits_{m\in\mathcal{M}_{k}}a_{m}\sqrt{\rho{\eta}_{mk}^{c}}\gamma_{mk}\Big{)}^{2}.$
(76)

As a result, plugging (B) and (B) into (B), we have

$\displaystyle\mathbb{E}\Big{\\{}\big{|}\mathrm{BU}_{k}\big{|}^{2}\Big{\\}}=\rho\sum\nolimits_{m\in\mathcal{Z}_{k}}a_{m}{\eta}_{mk}^{c}(\beta_{mk}-\gamma_{mk})\frac{\gamma_{mk}}{N-|\mathcal{S}_{m}|}$
$\displaystyle\hskip
60.00009pt+\rho\sum\nolimits_{m\in\mathcal{M}_{k}}Na_{m}{\eta}_{mk}^{c}\gamma_{mk}\beta_{mk}.$
(77)

The same steps can be followed to compute $\mathrm{IUI}_{kk^{\prime}}$:

$\displaystyle~{}\mathrm{IUI}_{kk^{\prime}}=$
$\displaystyle\rho\sum\nolimits_{m\in\mathcal{Z}_{k}}a_{m}{\eta}_{mk^{\prime}}^{c}(\beta_{mk}-\gamma_{mk})\frac{\gamma_{mk^{\prime}}}{N-|\mathcal{S}_{m}|}$
$\displaystyle+N\sum\nolimits_{m\in\mathcal{M}_{k}}a_{m}{\eta}_{mk^{\prime}}^{c}\gamma_{mk}\beta_{mk^{\prime}}.$
(78)

Moreover, by substituting (7) into (17d), taking expectation over $x_{r,ml}$,
we get

$\displaystyle\mathbb{E}\Big{\\{}\big{|}\mathrm{IR}_{k}\big{|}^{2}\Big{\\}}$
$\displaystyle=\rho\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l\in\mathcal{L}}\rho\eta^{s}_{ml}(1-a_{m})$
$\displaystyle\times\mathbb{E}\Big{\\{}|{\bf g}_{mk}^{T}{\bf
a}_{N}(\phi_{t,ml}^{a},\phi_{t,ml}^{e})|^{2}\Big{\\}}.$ (79)

Letting ${\bf A}={\bf a}_{N}(\phi_{t,ml}^{a},\phi_{t,ml}^{e}){\bf
a}^{H}_{N}(\phi_{t,ml}^{a},\phi_{t,ml}^{e})$, we have

$\displaystyle\mathbb{E}\Big{\\{}\big{|}\mathrm{IR}_{k}\big{|}^{2}\Big{\\}}=\\!\rho\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l\in\mathcal{L}}\eta^{s}_{ml}(1-a_{m})\mathbb{E}\Big{\\{}|{\bf
g}_{mk}^{T}{\bf A}|^{2}\Big{\\}}$
$\displaystyle=\\!\rho\sum\nolimits_{m\in\mathcal{M}}\sum\nolimits_{l\in\mathcal{L}}\eta^{s}_{ml}(1-a_{m})\mathbb{E}\Big{\\{}{\bf
g}_{mk}^{T}{\bf A}{\bf g}_{mk}^{*}\Big{\\}}$
$\displaystyle=\\!\rho\sum\nolimits_{m\in\mathcal{M}}\\!\\!\sum\nolimits_{l\in\mathcal{L}}\\!\\!\eta^{s}_{ml}(1\\!-\\!a_{m})\beta_{mk}\mathrm{tr}({\bf
A}),$ (80)

where $\mathrm{tr}({\bf A})=N$. To this end, by substituting (B), (B), (B),
and (B) into (19), the desired result in (21) is obtained.

∎

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