# Cell-Free Massive MIMO Surveillance Systems

Zahra Mobini†, Hien Quoc Ngo†, Michail Matthaiou†, and Lajos Hanzo‡
†Centre for Wireless Innovation (CWI), Queen’s University Belfast, U.K.
‡Department of Electronics and Computer Science, University of Southampton,
SO17 1BJ Southampton, U.K.
Email:{zahra.mobini, hien.ngo<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Wireless surveillance, in which untrusted communications links are proactively
monitored by legitimate agencies, has started to garner a lot of interest for
enhancing the national security. In this paper, we propose a new cell-free
massive multiple-input multiple-output (CF-mMIMO) wireless surveillance
system, where a large number of distributed multi-antenna aided legitimate
monitoring nodes (MNs) embark on either observing or jamming untrusted
communication links. To facilitate concurrent observing and jamming, a subset
of the MNs is selected for monitoring the untrusted transmitters (UTs), while
the remaining MNs are selected for jamming the untrusted receivers (URs). We
analyze the performance of CF-mMIMO wireless surveillance and derive a closed-
form expression for the monitoring success probability of MNs. We then propose
a greedy algorithm for the observing vs, jamming mode assignment of MNs,
followed by the conception of a jamming transmit power allocation algorithm
for maximizing the minimum monitoring success probability concerning all the
UT and UR pairs based on the associated long-term channel state information
knowledge. In conclusion, our proposed CF-mMIMO system is capable of
significantly improving the performance of the MNs compared to that of the
state-of-the-art baseline. In scenarios of a mediocre number of MNs, our
proposed scheme provides an 11-fold improvement in the minimum monitoring
success probability compared to its co-located mMIMO benchmarker.

††footnotetext: 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). The
work of H. Q. Ngo was supported by the U.K. Research and Innovation Future
Leaders Fellowships under Grant MR/X010635/1. The work of 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).

##  I Introduction

While the development of wireless communication systems has been dramatically
improved throughout the wireless generations, there is an increased demand for
further improved wireless security. More specifically, infrastructure-free or
user-controlled wireless communication networks, such as device-to-device
(D2D) and mobile ad-hoc communications, make public safety more vulnerable to
threats. Unauthorised or malicious users may misuse the available wireless
infrastructure-free networks to perform illegal activities, commit cyber
crime, and jeopardize public safety. This has motivated researchers to propose
new surveillance methods, termed as proactive monitoring, to allow the
legitimate monitor to observe or even degrade the quality of untrusted
communication [1].

In proactive monitoring, the legitimate monitor operates in a full-duplex (FD)
mode and sends a jamming signal to interfere with the reception of the UR,
thereby degrading the rate of the untrusted link. This improves the monitoring
performance [2]. The applications and different features of proactive
monitoring in wireless information surveillance have been studied in the
literature in various scenarios, such as MIMO systems [2, 3], unmanned aerial
vehicle (UAV) networks [4], cognitive radio networks [5], relay-aided
monitoring links [6, 7], and non-orthogonal multiple access (NOMA) networks
[8]. However, current studies tend to investigate the simple scenario of a
single untrusted link and/or a single legitimate monitor for direct
monitoring. These assumptions are optimistic, because realistic systems are
likely to have more than one untrusted communication links in practice. But in
the face of a distributed deployment of untrusted pairs, it is impractical to
arrange for the direct monitoring of each and every untrusted pair by relying
on a single monitor. Xu and Zhu [9] studied proactive monitoring using a
single monitor observing multiple untrusted pairs in scenarios associated with
quality-of-service-constrained untrusted links, while Li _et al._ [10] used
proactive monitoring with relaying features to increase the signal-to-
interference-plus-noise ratio (SINR) of the untrusted link, which results in
higher rate of the untrusted link, and hence higher observation rate.
Moreover, Moon _et al._ [11] looked into proactive monitoring, relying on a
group of single-antenna intermediate relay nodes harnessed for supporting a
distant legitimate monitor, which act either as a jammer or as an observer
node. However, Moon _et al._ [11] only focused on the single-untrusted-link
scenario under the overly optimistic assumption of having perfect global
channel state information (CSI) knowledge of all links at the monitor.
Therefore, the study of how to efficiently perform a realistic surveillance
operation using multiple monitors in the presence of multiple untrusted pairs
is meaningful and important, but is still an open problem at the time of
writing.

To address the need for reliable information surveillance in complex practical
scenarios, we are inspired by the emerging concept of CF-mMIMO to propose a
new proactive monitoring system, termed as _CF-mMIMO surveillance_. CF-mMIMO
breaks the concept of cells or cell boundaries by using numerous distributed
access points for simultaneously serving a smaller number of users [12]. It
enjoys all benefits granted by the mMIMO and coordinated multi-point concepts
relying on joint transmission, and hence, can guarantee ubiquitous coverage
for all the users. Our CF-mMIMO information surveillance system compromises a
large number of spatially distributed legitimate MNs, which coherently perform
surveillance of multiple untrusted pairs distributed over a geographically
wide area.

In our system, for any given untrusted pair, there are typically several MNs
in close proximity, and hence, high macro diversity gain can be achieved. As a
consequence, CF-mMIMO surveillance is expected to offer an improved monitoring
performance compared to its single-monitor (co-located) mMIMO based
counterpart. This is achieved by CF-mMIMO, because a virtual FD mode can be
realized by relying on half-duplex MNs. More precisely, in the system, two
types of MNs are relied upon: 1) a specifically selected subset of the MNs is
purely used for observing the UTs; 2) the rest of the MNs cooperatively jam
the URs. Since the MNs are distributed across the area, the node-to-node
interference in our proposed system is significantly reduced compared to a
conventional FD monitoring/jamming system. The observing mode or jamming mode
and MN transmit power can be dynamically adjusted for maximizing the overall
monitoring performance based on only long-term CSI.

The main technical contributions and key novelty of this paper are summarized
as follows:

  * •

We propose a novel wireless surveillance system, which is based on the CF-
mMIMO technology and either observing or jamming mode assignment. In
particular, by assuming realistic imperfect CSI knowledge, we characterize the
monitoring success probability of CF-mMIMO surveillance system over multiple
untrusted pairs.

  * •

We formulate an optimization problem for maximizing the minimum monitoring
success probability of all the untrusted pairs subject to a realistic per-MN
average transmit power constraint. The proposed power control solution can be
found by solving a sequence of linear programs. We also propose a greedy
algorithm for MN mode assignment.

  * •

Numerical results show that the proposed CF-mMIMO surveillance system
substantially improves the monitoring performance over that of mMIMO based
proactive monitoring systems relying on FD operation, where all MNs are co-
located as an antenna array, which simultaneously perform observation and
jamming.

Notation: We use bold upper case letters to denote matrices, and lower case
letters to denote vectors. The superscripts $(\cdot)^{*}$, $(\cdot)^{T}$ and
$(\cdot)^{\dagger}$ stand for the conjugate, transpose, and conjugate-
transpose (Hermitian), respectively. The zero mean circular symmetric complex
Gaussian distribution having variance $\sigma^{2}$ is denoted by
$\mathcal{CN}(0,\sigma^{2})$, while $\mathbf{I}_{N}$ denotes the $N\times N$
identity matrix. Finally, $\mathbb{E}\\{\cdot\\}$ denotes the statistical
expectation.

##  II System model

We consider the surveillance scenario of multiple untrusted communication
links, including $M$ MNs and $K$ untrusted communication pairs. Each UT and UR
is equipped with a single antenna, while each MN is equipped with $N$
antennas. All MNs, UTs, and URs are half-duplex devices. We assume that all
MNs are connected to the central processing unit (CPU) via fronthaul links.
The MNs can switch between observing mode, where they receive untrusted
messages, and jamming mode, where they send jamming signals to disrupt URs.
The assignment of each mode to its corresponding MN is designed to maximize
the monitoring success probability (see Section III). The binary variable
representing the mode assignment for each MN $m$ is expressed as $a_{m}=1$
($a_{m}=0$) if MN $m$ operates in the jamming mode (observing mode).

Denote the jamming channel (observing channel) between the $m$-th MN and the
$k$-th UR ($k$-th UT) by ${\bf g}_{mk}^{\mathtt{J}}\in\mathbb{C}^{N\times 1}$
(${\bf g}_{mk}^{\mathtt{O}}\in\mathbb{C}^{N\times 1}$), $\forall
k\in\operatorname{\mathcal{K}}\triangleq\\{1,\dots,K\\},m\in\operatorname{\mathcal{M}}\triangleq\\{1,\dots,M\\}$,
respectively. We model it by ${\bf
g}_{mk}^{\mathtt{J}}=\sqrt{\beta_{mk}^{\mathtt{J}}}\tilde{{\bf
g}}_{mk}^{\mathtt{J}},~{}({\bf
g}_{mk}^{\mathtt{O}}=\sqrt{\beta_{mk}^{\mathtt{O}}}\tilde{{\bf
g}}_{mk}^{\mathtt{O}})$, where $\beta_{mk}^{\mathtt{J}}$
($\beta_{mk}^{\mathtt{O}}$) is the large-scale fading coefficient and
$\tilde{{\bf g}}_{mk}^{\mathtt{J}}\in\mathbb{C}^{N\times 1}$ ($\tilde{{\bf
g}}_{mk}^{\mathtt{O}}\in\mathbb{C}^{N\times 1}$) is the small-scale fading
vector associated with independent and identically distributed (i.i.d.)
$\mathcal{CN}(0,1)$ random variables (RVs). Also, the channel gain between the
$l$-th UT and the $k$-th UR is $h_{\ell k}=(\beta_{\ell
k}^{\mathtt{U}})^{1/2}\breve{h}_{\ell k}$, where $\beta_{\ell k}^{\mathtt{U}}$
is the large-scale fading coefficient and $\breve{h}_{\ell k}$ represents
small-scale fading, distributed as $\mathcal{CN}(0,1)$. We note that $h_{kk}$
represents the channel coefficient of the $k$-th untrusted link spanning from
the $k$-th UT to the $k$-th UR, $\forall k\in\operatorname{\mathcal{K}}$.
Finally, the channel matrix between MN $m$ and MN $i$, $\forall
m,i\in\operatorname{\mathcal{M}}$, is denoted by ${\bf
F}_{mi}\in\mathbb{C}^{N\times N}$. We assume that the elements of ${\bf
F}_{mi}$, for $i\neq m$, are i.i.d. $\mathcal{CN}(0,\beta_{mi})$ RVs and
$\operatorname{\mathbf{F}}_{mm}=\mathbb{0},\forall m$. Note that the channels
${\bf g}_{mk}^{\mathtt{J}}$ and ${\bf g}_{mk}^{\mathtt{O}}$ may be estimated
at the legitimate MN by overhearing the pilot signals sent by UR $k$ and UT
$k$, respectively [6]. We denote the estimates of ${\bf g}_{mk}^{\mathtt{J}}$
and ${\bf g}_{mk}^{\mathtt{O}}$ by $\hat{{\bf g}}_{mk}^{\mathtt{J}}$ and
$\hat{{\bf g}}_{mk}^{\mathtt{O}}$, respectively where $\hat{{\bf
g}}_{mk}^{\mathtt{J}}\sim\mathcal{CN}(\mathbb{0},\gamma_{mk}^{\mathtt{J}}\mathbf{I}_{N})$,
and $\hat{{\bf
g}}_{mk}^{\mathtt{O}}\sim\mathcal{CN}(\mathbb{0},\gamma_{mk}^{\mathtt{O}}\mathbf{I}_{N})$.
Since it is difficult for the legitimate MNs to obtain the CSI of untrusted
links, we assume that $h_{k\ell}$ is unknown to the MNs.

All the UTs simultaneously send independent untrusted messages to their
corresponding receivers over the same frequency band. The signal transmitted
from UT $k$ is denoted by
$x_{k}^{\mathtt{UT}}=\sqrt{\rho_{\mathtt{UT}}}s_{k}^{\mathtt{UT}}$, where
$s_{k}^{\mathtt{UT}}$, with
$\mathbb{E}\left\\{|s_{k}^{\mathtt{UT}}|^{2}\right\\}=1$, and
$\rho_{\mathtt{UT}}$ represent the transmitted symbol and the normalized
transmit power at each UT, respectively. In addition, at the same time, the
MNs in jamming mode intentionally send jamming signals to interfere with the
untrusted transmission links. This reduces the achievable data rate at the
URs, while improving the monitoring success probability. In particular, the
MNs in jamming mode use the maximum-ratio (MR) transmission technique (a.k.a.
conjugate beamforming) to jam the reception of the URs. Denote the jamming
symbol intended for the untrusted link $k$ by $s_{k}^{\mathtt{J}}$, which is a
RV with zero mean and unit variance. With MR precoding, the $N\times 1$ signal
vector transmitted by MN $m$ can be expressed as ${\bf
x}_{m}^{\mathtt{J}}=a_{m}\sqrt{\rho_{\mathtt{J}}}\sum_{k\in\mathcal{K}}\sqrt{\theta_{mk}}\left(\hat{{\bf
g}}_{mk}^{\mathtt{J}}\right)^{*}s_{k}^{\mathtt{J}}$, where $\rho_{\mathtt{J}}$
is the maximum normalized transmit power at each MN in the jamming mode.
Moreover, $\theta_{mk}$ is the power control coefficient chosen to satisfy the
power constraint $\mathbb{E}\left\\{\|{\bf
x}_{m}^{\mathtt{J}}\|^{2}\right\\}\leq\rho_{\mathtt{J}}$ at each MN, which can
be further written as

$\displaystyle
a_{m}\sum_{k\in\mathcal{K}}\gamma_{mk}^{\mathtt{J}}\theta_{mk}\leq\frac{1}{N},\forall
m.$ (1)

Accordingly, the signal received by UR $k$ can be written as

$\displaystyle
y_{k}^{\mathtt{UR}}=h_{kk}x_{k}^{\mathtt{UT}}+\sum_{\ell\in\mathcal{K},\ell\neq
k}h_{\ell k}x_{\ell}^{\mathtt{UT}}+\\!\sqrt{\rho_{\mathtt{J}}}\times$
$\displaystyle\sum_{m\in\mathcal{M}}\sum_{k^{\prime}\in\mathcal{K}}a_{m}\sqrt{\theta_{mk^{\prime}}}\left({\bf
g}_{mk}^{\mathtt{J}}\right)^{T}\left(\hat{{\bf
g}}_{mk^{\prime}}^{\mathtt{J}}\right)^{*}s_{k^{\prime}}^{\mathtt{J}}+w_{k}^{\mathtt{UR}},$
(2)

where $w_{k}^{\mathtt{UR}}\sim\mathcal{CN}(0,1)$ is the additive white
Gaussian noise (AWGN) at UR $k$. We notice that the second term in (II), is
the interference caused by other UTs due to their concurrent transmissions
over the same frequency band while the third term is the interference
emanating from the MNs in the jamming mode.

The MNs in the observing mode, i.e., MNs with $a_{m}=0,\forall m$, receive the
transmit signals from all UTs. The received signal ${\bf
y}_{m}^{\mathtt{O}}\in\mathbb{C}^{N\times 1}$ at MN $m$ in the observing mode
is given by

$\displaystyle{\bf
y}_{m}^{\mathtt{O}}=\sqrt{\rho_{\mathtt{UT}}}\sum_{k\in\mathcal{K}}(1-a_{m}){\bf
g}_{mk}^{\mathtt{O}}s_{k}^{\mathtt{UT}}+\sqrt{\rho_{\mathtt{J}}}\times$ (3)
$\displaystyle~{}\sum_{i\in\mathcal{M}}\sum_{k\in\mathcal{K}}(1-a_{m})a_{i}\sqrt{\theta_{ik}}{\bf
F}_{mi}(\hat{{\bf g}}_{ik}^{\mathtt{J}})^{*}s_{k}^{\mathtt{J}}+(1-a_{m}){\bf
w}_{m}^{\mathtt{O}},$

where ${\bf w}_{m}^{\mathtt{O}}$ is the
$\mathcal{CN}(\mathbb{0},\mathbf{I}_{N})$ AWGN vector. We note from (3) that
if MN $m$ does not operate in the observing mode, i.e., $a_{m}=1$, it does not
receive any signal, i.e., ${\bf y}_{m}^{\mathtt{O}}=\bm{0}$.

Then, MN $m$ in the observing mode performs MR combining (MRC) reception by
partially equalizing the received signal in (3) based on the Hermitian of the
local channel estimates as $(\hat{{\bf g}}_{mk}^{\mathtt{O}})^{\dagger}{\bf
y}_{m}^{\mathtt{O}}$. The resultant signal is then forwarded to the CPU for
detecting the untrusted signals, where the receive combiner sums up the
equalized signals. The aggregated received signal for UT $k,\forall k,$ at the
CPU can be written as

$\displaystyle r_{k}^{\mathtt{O}}=\sum_{m=1}^{M}(\hat{{\bf
g}}_{mk}^{\mathtt{O}})^{\dagger}{\bf y}_{m}^{\mathtt{O}}.$ (4)

Finally, the observed information $s_{k}^{\mathtt{UT}}$ can be detected from
${\bf r}_{k}^{\mathtt{O}}$.

####  II-1 Effective SINR of the Untrusted Communication Links

We define the effective noise as

$\displaystyle\tilde{w}_{k}^{\mathtt{UR}}=\sqrt{\rho_{\mathtt{UT}}}\sum_{\ell\in\mathcal{K},\ell\neq
k}h_{\ell k}s_{\ell}^{\mathtt{UT}}$
$\displaystyle+\sqrt{\rho_{\mathtt{J}}}\sum_{m\in\mathcal{M}}\\!\sum_{k^{\prime}\in\mathcal{K}}\\!\\!a_{m}\sqrt{\theta_{mk^{\prime}}}\left({\bf
g}_{mk}^{\mathtt{J}}\right)^{T}\left(\hat{{\bf
g}}_{mk^{\prime}}^{\mathtt{J}}\right)^{*}s_{k^{\prime}}^{\mathtt{J}}\\!+w_{k}^{\mathtt{UR}},$
(5)

and reformulate the signal received at UR $k$ in (II) as

$\displaystyle~{}y_{k}^{\mathtt{UR}}=\sqrt{\rho_{\mathtt{UT}}}h_{kk}s_{k}^{\mathtt{UT}}+\tilde{w}_{k}^{\mathtt{UR}}.$
(6)

Since $s_{\ell}^{\mathtt{UT}}$ is independent of $s_{k}^{\mathtt{UT}}$ for any
$\ell\neq k$, the first term of the effective noise (II-1) is uncorrelated
with the first term in (6). Moreover, the second and third terms of (II-1) are
uncorrelated with the first term of (6). Therefore, the effective noise
$\tilde{w}_{k}^{\mathtt{UR}}$ and the input RV $x_{k}^{\mathtt{UT}}$ are
uncorrelated. Accordingly, we obtain the closed-form expression for the
effective SINR of the untrusted link $k$ as in the following theorem.

######  Theorem 1.

The effective SINR of the untrusted link $k$ can be formulated as

$\displaystyle\mathsf{SINR}_{k}^{\mathtt{UR}}({\bf
a},\bm{\theta})=\frac{\rho_{\mathtt{UT}}|h_{kk}|^{2}}{\xi_{k}({\bf
a},\bm{\theta})},$ (7)

where

$\displaystyle\xi_{k}({\bf a},\bm{\theta})=$
$\displaystyle\rho_{\mathtt{UT}}\\!\\!\\!\sum_{\ell\in\mathcal{K}\setminus
k}\\!\beta_{\ell
k}^{\mathtt{U}}\\!+{\rho_{\mathtt{J}}}N\sum_{k^{\prime}\in\mathcal{K}}\sum_{m\in\mathcal{M}}a_{m}\theta_{mk^{\prime}}\beta_{mk}^{\mathtt{J}}\gamma_{mk^{\prime}}^{\mathtt{J}}$
$\displaystyle\quad+{\rho_{\mathtt{J}}}N^{2}\Big{(}\sum_{m\in\mathcal{M}}a_{m}\sqrt{\theta_{mk}}\gamma_{mk}^{\mathtt{J}}\\!\Big{)}^{2}+1,$
(8)

with ${\bf a}\triangleq\\{a_{m}\\}$ and
$\bm{\theta}\triangleq\\{\theta_{mk}\\}$, $\forall m,k$, respectively.

Proof: See Appendix A.$\hskip 140.00021pt\blacksquare$

####  II-2 Effective SINR for Observing

The CPU detects the observed information $s_{k}^{\mathtt{UT}}$ from ${\bf
r}_{k}^{\mathtt{O}}$ in (4). We assume that it does not have instantaneous CSI
knowledge of the observing, jamming, and untrusted channels and uses only
statistical knowledge when performs detection. By employing the use-and-then-
forget capacity-bounding methodology [12], the received SINR for observing the
untrusted link $k$ can be obtained as in the following theorem.

######  Theorem 2.

The received SINR for the $k$-th untrusted link at the CPU is given by

$\displaystyle\mathsf{SINR}_{k}^{\mathtt{O}}({\bf a},\bm{\theta})=$
$\displaystyle\frac{N\rho_{\mathtt{UT}}\Big{(}\sum\limits_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}(1-a_{m})\gamma_{mk}^{\mathtt{O}}\Big{)}^{2}}{\mu_{k}+\rho_{\mathtt{J}}N\\!\sum\limits_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\sum\limits_{\begin{subarray}{c}i\in\mathcal{M}\end{subarray}}\sum\limits_{\ell\in\mathcal{K}}\\!(1-a_{m})a_{i}\theta_{i\ell}\gamma_{mk}^{\mathtt{O}}\beta_{mi}\gamma_{i\ell}^{\mathtt{J}}},$
(9)

with

$\displaystyle\mu_{k}\triangleq$
$\displaystyle\rho_{\mathtt{UT}}\\!\sum\limits_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\sum\limits_{\ell\in\mathcal{K}}\\!(1-a_{m})\beta_{m\ell}^{\mathtt{O}}\gamma_{mk}^{\mathtt{O}}+\sum\limits_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}(1\\!\\!-\\!a_{m})\gamma_{mk}^{\mathtt{O}}.$

Proof: See Appendix B. $\hskip 140.00021pt\blacksquare$

####  II-3 Monitoring Success Probability

To achieve reliable detection at UR $k$, UT $k$ varies its transmission rate
according to $\mathsf{SINR}_{k}^{\mathtt{UR}}$. Hence, if
$\mathsf{SINR}_{k}^{\mathtt{O}}\geq\mathsf{SINR}_{k}^{\mathtt{UR}}$, the CPU
can also reliably detect the information of the untrusted link $k$. On the
other hand, if
$\mathsf{SINR}_{k}^{\mathtt{O}}\leq\mathsf{SINR}_{k}^{\mathtt{UR}}$, the CPU
may detect this information at a high probability of error. Therefore, the
following indicator function can be designed for characterising the event of
successful monitoring at the CPU [2]

$\displaystyle X_{k}=\begin{cases}\displaystyle 1&\text{if
}\mathsf{SINR}_{k}^{\mathtt{O}}\geq\mathsf{SINR}_{k}^{\mathtt{UR}},\\\
\displaystyle 0&\text{otherwise},\end{cases}$

where $X_{k}=1$ and $X_{k}=0$ indicate monitoring success and failure events
for the untrusted link $k$, respectively. Thus, a suitable performance metric
for monitoring each untrusted communication link $k$ is the monitoring success
probability $\mathbb{E}\\{X_{k}\\}$, defined as

$\mathbb{E}\\{X_{k}\\}=\text{Pr}\left(\mathsf{SINR}_{k}^{\mathtt{O}}\geq\mathsf{SINR}_{k}^{\mathtt{UR}}\right)\\!.$
(10)

From (7), (2), and (10) we have

$\mathbb{E}\\{X_{k}\\}=\mathbb{P}\Big{(}|h_{kk}|^{2}\leq\frac{\mathsf{SINR}_{k}^{\mathtt{O}}\xi_{k}}{\rho_{\mathtt{UT}}}\Big{)}.$
(11)

Using the cumulative distribution function (CDF) of the exponentially
distributed RV $|h_{kk}|^{2}$, the monitoring success probability of our CF-
mMIMO surveillance system can be expressed in closed form as

$\mathbb{E}\\{X_{k}\\}=1-\exp\left(-\frac{\mathsf{SINR}_{k}^{\mathtt{O}}\xi_{k}}{\beta_{kk}^{\mathtt{U}}\rho_{\mathtt{UT}}}\right).$
(12)

##  III Max Min Optimization

In this section, we aim for maximizing the lowest probability of successful
monitoring by optimizing the observing and jamming mode assignment vector
$\operatorname{\mathbf{a}}$ and power control coefficient vector
$\operatorname{\bm{\theta}}$ under the constraint of the transmit power at
each MN in (1). More precisely, we formulate an optimization problem as

$\displaystyle\underset{\\{{\bf a},\bm{\theta}\\}}{\mathrm{max}}\,\,$
$\displaystyle\hskip
10.00002pt\underset{k\in\mathcal{K}}{\min}\,\,\mathbb{E}\\{X_{k}({\bm{{\bf
a},\theta}})\\}$ (13a) $\displaystyle\mathrm{s.t.}\,\,$ $\displaystyle\hskip
10.00002pta_{m}\sum_{k\in\mathcal{K}}\gamma_{mk}^{\mathtt{J}}\theta_{mk}\leq\frac{1}{N},~{}~{}m\in\mathcal{M},$
(13b) $\displaystyle\hskip 10.00002pt\theta_{mk}\geq
0,~{}~{}m\in\mathcal{M},~{}k\in\mathcal{K},$ (13c) $\displaystyle\hskip
10.00002pta_{m}\in\\{0,1\\},~{}~{}m\in\mathcal{M}.$ (13d)

Since $1-\exp(-x)$ is a monotonically increasing function of $x$, the problem
(13) is equivalent to the following problem

$\displaystyle\underset{\\{{\bf a},\bm{\theta}\\}}{\mathrm{max}}\,\,\hskip
10.00002pt\underset{k\in\mathcal{K}}{\min}\,\,\mathsf{SINR}_{k}^{\mathtt{O}}({\bf
a},\bm{\theta})\xi_{k}({\bf a},\bm{\theta})$ (14a)
$\displaystyle~{}\text{s.t.}\hskip
20.00003pt~{}\eqref{opt:cons1},~{}\eqref{opt:cons2}.$ (14b)

Problem (14) is a challenging combinatorial problem. Therefore, for MN mode
assignment, we only focus on a heuristic greedy method which simplifies the
computation, while providing a significant successful monitoring performance
gain.

###  III-A Greedy MN Mode Assignment for Fixed Power Control

Let $\mathcal{A}_{\mathtt{O}}$ and $\mathcal{A}_{\mathtt{J}}$ denote the sets
containing the indices of MNs in observing mode, i.e., MNs with $a_{m}=0$, and
MNs in jamming mode, i.e., MNs with $a_{m}=1$, respectively. In addition,
$\mathbb{E}\\{X_{k}({A}_{\mathtt{O}},\mathcal{A}_{\mathtt{J}})\\}$ presents
the dependence of the monitoring success probability on the different choices
of MN mode assignments. Our greedy algorithm of MN mode assignment is shown in
Algorithm 1. All MNs are initially assigned to observing mode, i.e.,
$\mathcal{A}_{\mathtt{O}}=\mathcal{M}$ and
$\mathcal{A}_{\mathtt{J}}=\emptyset$. Then, in each iteration, one MN switches
into jamming mode for maximizing the minimum monitoring success probability
(12) among the untrusted links, until there is no more improvement.

###  III-B Power Control

For a given MN mode assignment, the optimization problem (14) reduces to the
power control problem. Using (1), (2), and (14) the max-min fairness problem
is now formulated as

$\displaystyle\max_{\bm{\theta}}~{}\min\limits_{\forall
k\in\mathcal{K}}\frac{\xi_{k}(\bm{\theta})}{\\!\mu_{k}\\!+\\!\rho_{\mathtt{J}}N\\!\\!\\!\\!\sum\limits_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\\!\sum\limits_{\begin{subarray}{c}i\in\mathcal{M}\end{subarray}}\sum\limits_{\ell\in\mathcal{K}}\\!(\\!1\\!-\\!a_{m}\\!)a_{i}\theta_{i\ell}\gamma_{mk}^{\mathtt{O}}\beta_{mi}\gamma_{i\ell}^{\mathtt{J}}}$
(15a) $\displaystyle\text{s.t.}\hskip
20.00003pt~{}\eqref{opt:cons1},~{}\eqref{opt:cons2}.$ (15b)

By introducing the slack variable $\varrho$, we reformulate (15) as

$\displaystyle\max_{\\{\bm{\theta},\varrho\\}}~{}\varrho$ (16a)
$\displaystyle\text{s.t.}\hskip
20.00003pt\rho_{\mathtt{J}}N\\!\sum\limits_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\sum\limits_{\begin{subarray}{c}i\in\mathcal{M}\end{subarray}}\sum\limits_{\ell\in\mathcal{K}}(1-a_{m})a_{i}\theta_{i\ell}\gamma_{mk}^{\mathtt{O}}\beta_{mi}\gamma_{i\ell}^{\mathtt{J}}$
$\displaystyle\hskip
80.00012pt-\frac{1}{\varrho}{\xi}_{k}(\bm{\theta})+\mu_{k}\leq 0,~{}~{}\forall
k\in\mathcal{K},$ (16b) $\displaystyle\hskip
20.00003pt~{}\eqref{opt:cons1},~{}\eqref{opt:cons2}.$ (16c)

To arrive at a computationally more efficient formulation, we use the
inequality
$\left(\sum_{m\in\mathcal{M}}\sqrt{\theta_{mk}}\gamma_{mk}^{\mathtt{J}}\right)^{2}\geq\sum_{m\in\mathcal{M}}\theta_{mk}(\gamma_{mk}^{\mathtt{J}})^{2}$
and replace the constraint (16b) by

$\displaystyle\rho_{\mathtt{J}}N\\!\sum\limits_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\sum\limits_{\begin{subarray}{c}i\in\mathcal{M}\end{subarray}}\sum\limits_{\ell\in\mathcal{K}}(1-a_{m})a_{i}\theta_{i\ell}\gamma_{mk}^{\mathtt{O}}\beta_{mi}\gamma_{i\ell}^{\mathtt{J}}$
$\displaystyle-\frac{1}{\varrho}\tilde{\xi}_{k}(\bm{\theta})+\mu_{k}\leq
0,~{}~{}\forall k\in\mathcal{K},$ (17)

where

$\displaystyle\tilde{\xi}_{k}\\!(\bm{\theta})\\!=\\!\rho_{\mathtt{UT}}\\!\\!\\!\sum_{\ell\in\mathcal{K}\setminus
k}\\!\beta_{\ell
k}^{\mathtt{U}}\\!+\\!{\rho_{\mathtt{J}}}N\\!\sum_{k^{\prime}\in\mathcal{K}}\sum_{m\in\mathcal{M}}\\!a_{m}\theta_{mk^{\prime}}\beta_{mk}^{\mathtt{J}}\gamma_{mk^{\prime}}^{\mathtt{J}}$
$\displaystyle\qquad\quad+{\rho_{\mathtt{J}}}N^{2}\sum_{m\in\mathcal{M}}a_{m}\theta_{mk}(\gamma_{mk}^{\mathtt{J}})^{2}+1.$
(18)

Now, for a fixed $\varrho$, all inequalities involved in (16) are linear,
hence the program (16) is quasi–linear. Since the second constraint in (16) is
an increasing functions of $\varrho$, the solution to the optimization problem
is obtained by harnessing a line-search over $\varrho$ to find the maximal
feasible value. As a consequence, we use the bisection method in Algorithm 2
to obtain the solution.

Algorithm 1 Greedy MN Mode Assignment

1: Initialize: Set $\mathcal{A}_{\mathtt{O}}=\mathcal{M}$ and
$\mathcal{A}_{\mathtt{J}}=\emptyset$. Set iteration index $i=0$.

2: Calculate
$\Pi^{\star}[i]=\underset{k\in\mathcal{K}}{\min}\,\,\mathbb{E}\\{X_{k}(\mathcal{A}_{\mathtt{O}},\mathcal{A}_{\mathtt{J}})\\}$

3: repeat

4: for all $m\in\mathcal{A}_{\mathtt{O}}$ do

5: Set $\mathcal{A}_{s}=\mathcal{A}_{\mathtt{O}}\setminus m$

6: Calculate
$\Pi_{m}=\underset{k\in\mathcal{K}}{\min}\,\,\mathbb{E}\\{X_{k}(\mathcal{A}_{s},\mathcal{A}_{\mathtt{J}}\bigcup
m)\\}$

7: end for

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

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

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

11: Select MN
$m^{\star}=\operatorname*{arg\,max}_{m\in\mathcal{A}_{\mathtt{O}}}\\{\Pi_{m}\\}$

12: Update $\mathcal{A}_{\mathtt{J}}=\\{\mathcal{A}_{\mathtt{J}}\bigcup
m^{\star}\\}$ and $\mathcal{A}_{\mathtt{O}}=\mathcal{A}_{\mathtt{O}}\setminus
m^{\star}$

13: end if

14: Set $i=i+1$

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

16: return $\mathcal{A}_{\mathtt{O}}$ and $\mathcal{A}_{\mathtt{J}}$, i.e.,
the indices of MNs in observing mode and jamming mode, respectively.

Algorithm 2 Bisection Method for Max-Min Power Control

1: Initialization of $\varrho_{\min}$ and $\varrho_{\max}$, where
$\varrho_{\min}$ and $\varrho_{\max}$ define a range of relevant values of the
objective function in (15). Initial line-search accuracy $\epsilon$.

2: repeat

3: Set $\varrho:=\frac{\varrho_{\min}+\varrho_{\max}}{2}$. Solve the following
convex feasibility program
$\displaystyle\begin{cases}&\\!{{\mu_{k}\\!+\\!\rho_{\mathtt{J}}N\\!\\!\\!\sum\limits_{\begin{subarray}{c}m\in\mathcal{M}\end{subarray}}\sum\limits_{\begin{subarray}{c}i\in\mathcal{M}\end{subarray}}\sum\limits_{\ell\in\mathcal{K}}\\!(\\!1\\!-\\!a_{m}\\!)a_{i}\theta_{i\ell}\gamma_{mk}^{\mathtt{O}}\beta_{mi}\gamma_{i\ell}^{\mathtt{J}}}}\\\
&~{}~{}-{\frac{1}{\varrho}\tilde{\xi}_{k}(\bm{\theta})\leq 0~{}~{}\forall
k\in\mathcal{K}},\\\
&a_{m}\sum_{k\in\mathcal{K}}\gamma_{mk}^{\mathtt{J}}\theta_{mk}\leq\frac{1}{N},~{}\forall
m\in\mathcal{M},\\\ &\\!\theta_{mk}\geq 0,~{}\forall
k\in\mathcal{K},~{}\forall m\in\mathcal{M}.\end{cases}$ (19)

4: If problem (19) is feasible, then set $\varrho_{\min}:=\varrho$, else set
$\varrho_{\max}:=\varrho$.

5: until $\varrho_{\max}-\varrho_{\min}<\epsilon$

##  IV Numerical Results

In this section, numerical results are presented for studying the performance
of the proposed CF-mMIMO surveillance system. Moreover, the performance of CF-
mMIMO is compared to that of a co-located mMIMO surveillance system, where all
the $M$ MNs are located at the centre of the coverage area as a single virtual
FD large MN.

We consider a CF-mMIMO surveillance system, where the MNs, UTs, and URs are
randomly distributed in an area of $1\times 1$ km2 having wrapped around edges
to reduce the boundary effects. The distances between adjacent MNs are at
least $80$ m and between each untrusted pair is within the range of $80-160$
m. Moreover, we set the number of antennas to $N=2$ and the channel bandwidth
to $B=50$ MHz. The maximum transmit power for each MN and each UT is $1$ W and
$250$ mW, respectively, while the corresponding normalized maximum transmit
powers ${\rho}_{\mathtt{J}}$, ${\rho}_{\mathtt{UT}}$, can be calculated upon
dividing these powers by the noise power of $-92$ dBm. The large-scale fading
coefficient $\beta_{mk}$ is represented by
$\beta_{mk}=10^{{\text{PL}_{mk}^{d}}/{10}}10^{{A_{mk}}/{10}}$, where the first
term models the path loss, and the second term models the shadow fading with
$A_{mk}\in\mathcal{N}(0,4^{2})$ (in dB), respectively; $\text{PL}_{mk}^{d}$
(in dB) is calculated as [13]
$\text{PL}_{mk}^{d}=-30.5-36.7\log_{10}\left(\frac{d_{mk}}{1\,\text{m}}\right),$
and the correlation between the shadowing terms from the MN $m,\forall
m\in\mathcal{M}$ to the untrusted users $k\in\mathcal{K}$ is given by
$\mathbb{E}\\{A_{mk}A_{jk^{\prime}}\\}=4^{2}2^{-\delta_{kk^{\prime}}/9\,\text{m}}$,
if $j=m$, and $\mathbb{E}\\{A_{mk}A_{jk^{\prime}}\\}=0$, if $j\neq m$,
$\forall j\in\mathcal{M},$ where $\delta_{kk^{\prime}}$ is the distance
between UTs $k$ and $k^{\prime}$.

Figure 1 illustrates the minimum monitoring success probability achieved by
the CF-mMIMO surveillance system for different number of MNs $M$, where
$K=10$. Our results verify the advantage of our proposed power allocation
(PPA) and greedy mode assignment over equal power allocation (EPA) and random
mode assignment, respectively. More specifically, PPA provides performance
gains of up to $40\%$ and $100\%$ with respect to EPA for the system relying
on greedy mode assignment and random mode assignment, respectively. In
addition, the remarkable performance gap between the greedy and random mode
assignment verifies the importance of an adequate mode selection in terms of
monitoring performance in the CF-mMIMO surveillance system. In Fig. 1, the
monitoring success probability of the CF-mMIMO and co-located mMIMO is also
compared. It can be observed that the CF-mMIMO surveillance system
significantly outperforms its co-located mMIMO counterpart. For example, for
$M=30$, the CF-mMIMO system provides around $11$-fold improvement in the
minimum monitoring success performance over the co-located system. The reason
is twofold: primarily due to the ability of the former to surround each UT and
each UR by MNs in observing and jamming mode, respectively; secondarily, in
contrast to our CF-mMIMO with distributed MNs, a co-located mMIMO suffers from
excessive self-interference due to the short distance between the transmit and
receive antennas located at a single large MN.

Next, in Fig. 2, we investigate the impact of the number of untrusted pairs on
the monitoring success probability performance. We observe that upon
increasing $K$, the monitoring performance of all schemes deteriorates.
Interestingly, the relative performance gap between our CF-mMIMO system using
the PPA and greedy mode assignment and other schemes considerably increases
with $K$. This result shows that the CF-mMIMO surveillance system along
employing PPA and suitable mode assignment guarantees excellent monitoring
performance in practical multiple-untrusted-pair scenarios.

Figure 1: Average minimum monitoring success probability versus the number of
MNs, $M$, where $K=10$. Figure 2: Average minimum monitoring success
probability versus the number of untrusted pairs, $K$, where $M=30$.

##  V Conclusion

We designed CF-mMIMO surveillance systems for monitoring multiple untrusted
pairs and analyzed the performance. We proposed a long-term-based greedy
algorithm for designing the MN mode assignment and a power control technique
for the MNs that are in jamming mode in order to maximize the minimum
monitoring success probability across all untrusted pairs under a realistic
transmit power constraint. We showed that our proposed schemes provide
significant monitoring gains over the random mode assignment and equal-power
based schemes. Our results also confirmed that our CF-mMIMO surveillance
system significantly outperforms conventional co-located surveillance systems
in terms of monitoring, even for relatively small number of MNs.

##  Appendix A Proof of Theorem 1

To derive the closed-form expression for the effective SINR of the untrusted
link, we have to calculate $\xi_{k}({\bf
a},\bm{\theta})=\mathbb{E}\Big{\\{}\big{|}\tilde{w}_{k}^{\mathtt{UR}}\big{|}^{2}\Big{\\}}$.
Denote the jamming channel estimation error by
$\bm{\varepsilon}_{mk}^{\mathrm{J}}={\bf g}_{mk}^{\mathtt{J}}-\hat{{\bf
g}}_{mk}^{\mathtt{J}}$. Therefore, we have

$\displaystyle\xi_{k}({\bf
a},\bm{\theta})=\mathbb{E}\Big{\\{}\big{|}\tilde{w}_{k}^{\mathtt{UR}}\big{|}^{2}\Big{\\}}=\rho_{\mathtt{UT}}\sum_{\ell\in\mathcal{K}\setminus
k}\beta_{\ell k}^{\mathtt{U}}+{\rho_{\mathtt{J}}}\mathcal{I}+1,$ (20)

where

$\displaystyle\mathcal{I}\triangleq$
$\displaystyle\sum_{k^{\prime}\in\mathcal{K}}\mathbb{E}\Big{\\{}\Big{|}\sum_{m\in\mathcal{M}}a_{m}\sqrt{\theta_{mk^{\prime}}}({\bf
g}_{mk}^{\mathtt{J}})^{T}\left(\hat{{\bf
g}}_{mk^{\prime}}^{\mathtt{J}}\right)^{*}\Big{|}^{2}\Big{\\}}.$

To calculate $\mathcal{I}$, owing to the fact that the variance of a sum of
independent RVs is equal to the sum of the variances, we have

$\displaystyle\mathcal{I}\stackrel{{\scriptstyle(a)}}{{=}}\sum_{k^{\prime}\in\mathcal{K}\setminus
k}\mathbb{E}\Big{\\{}\Big{|}\\!\sum_{m\in\mathcal{M}}\\!\\!a_{m}\sqrt{\theta_{mk^{\prime}}}\left({\bf
g}_{mk}^{\mathtt{J}}\right)^{T}\left(\hat{{\bf
g}}_{mk^{\prime}}^{\mathtt{J}}\right)^{*}\Big{|}^{2}\Big{\\}}$
$\displaystyle\qquad+\mathbb{E}\Big{\\{}\Big{|}\sum_{m\in\mathcal{M}}a_{m}\sqrt{\theta_{mk}}\left(\bm{\varepsilon}_{mk}^{\mathrm{J}}+\hat{{\bf
g}}_{mk}^{\mathtt{J}}\right)^{T}\left(\hat{{\bf
g}}_{mk}^{\mathtt{J}}\right)^{*}\Big{|}^{2}\Big{\\}}$
$\displaystyle\\!\\!\\!\stackrel{{\scriptstyle(b)}}{{=}}\\!\\!\\!\\!\sum_{k^{\prime}\in\mathcal{K}\setminus
k}\sum_{m\in\mathcal{M}}\\!\\!\\!a_{m}{\theta_{mk^{\prime}}}\mathbb{E}\\!\Big{\\{}\\!\left({\bf
g}_{mk}^{\mathtt{J}}\right)^{\\!T}\\!\\!\mathbb{E}\Big{\\{}\\!\left(\hat{{\bf
g}}_{mk^{\prime}}^{\mathtt{J}}\right)^{*}\\!\\!\left(\hat{{\bf
g}}_{mk^{\prime}}^{\mathtt{J}}\right)^{\\!T}\\!\\!\Big{\\}}\left({\bf
g}_{mk}^{\mathtt{J}}\right)^{*}\\!\\!\Big{\\}}$
$\displaystyle\qquad+\sum_{m\in\mathcal{M}}a_{m}\theta_{mk}\big{(}\mathbb{E}\big{\\{}\big{\|}\hat{{\bf
g}}_{mk}^{\mathtt{J}}\big{\|}^{4}\big{\\}}+\mathbb{E}\big{\\{}\big{|}{(\bm{\varepsilon}}_{mk}^{\mathrm{J}})^{T}(\hat{{\bf
g}}_{mk}^{\mathtt{J}})^{*}\big{|}^{2}\\!\big{\\}}\\!\big{)}$
$\displaystyle~{}~{}+\sum_{m\in\mathcal{M}}\sum_{n\in\mathcal{M}\setminus
m}a_{m}a_{n}\sqrt{\theta_{mk}\theta_{nk}}\mathbb{E}\big{\\{}\|\hat{{\bf
g}}_{mk}^{\mathtt{J}}\|^{2}\\!\big{\\}}\mathbb{E}\big{\\{}\|\hat{{\bf
g}}_{nk}^{\mathtt{J}}\|^{2}\\!\big{\\}}$
$\displaystyle\\!\stackrel{{\scriptstyle(c)}}{{=}}\\!N\\!\\!\sum_{k^{\prime}\in\mathcal{K}}\\!\sum_{m\in\mathcal{M}}\\!\\!\\!\\!a_{m}\theta_{mk^{\prime}}\beta_{mk}^{\mathtt{J}}\gamma_{mk^{\prime}}^{\mathtt{J}}\\!\\!+\\!\\!N^{2}\big{(}\\!\\!\sum_{m\in\mathcal{M}}\\!\\!\\!a_{m}\sqrt{\theta_{mk}}\gamma_{mk}^{\mathtt{J}}\\!\big{)}^{\\!2}\\!\\!,$
(21)

where $(a)$ follows from the fact that $\hat{{\bf
g}}_{mk^{\prime}}^{\mathtt{J}}$ has zero mean and it is independent of ${\bf
g}_{mk}^{\mathtt{J}}$ and $\hat{{\bf g}}_{mk}^{\mathtt{J}}$, $(b)$ follows
from the fact that $\bm{\varepsilon}_{mk}^{\mathrm{J}}$ is independent of
$\hat{{\bf g}}_{mk^{\prime}}^{\mathtt{J}}$ and it is a zero-mean RV and $(c)$
follows from the fact that $\mathbb{E}\big{\\{}\|\hat{{\bf
g}}_{mk}^{\mathtt{J}}\|^{4}\big{\\}}=N(N+1)(\gamma_{mk}^{\mathtt{J}})^{2}$.
Substituting (A) into (20) completes the proof.

##  Appendix B Proof of Theorem 2

By substituting (3) into (4) and then employing the use-and-then-forget
capacity-bounding methodology [12], the SINR observed for the untrusted link
$k$ can be written as

$\mathsf{SINR}_{k}^{\mathtt{O}}=\frac{{|{\text{DS}}_{k}^{\mathtt{O}}{|^{2}}}}{{{\text{BU}}_{k}^{\mathtt{O}}+{\text{UI}}_{k}^{\mathtt{O}}+{\text{MI}}_{k}^{\mathtt{O}}+{\text{AN}}_{k}^{\mathtt{O}}}},$
(22)

where
$\text{DS}_{k}^{\mathtt{O}},\text{BU}_{k}^{\mathtt{O}},\text{UI}_{k}^{\mathtt{O}},\text{MI}_{k}^{\mathtt{O}}$,
and $\text{AN}_{k}^{\mathtt{O}}$ are the desired signal, beamforming gain
uncertainty, inter-untrusted user interference, inter-MN interference, and
additive noise, respectively, which can be written as

$\displaystyle\text{DS}_{k}^{\mathtt{O}}\triangleq\sqrt{\rho_{\mathtt{UT}}}\mathbb{E}\Big{\\{}\sum\limits_{m\in{\mathcal{M}}}(1-a_{m}){(\hat{{\bf
g}}_{mk}^{\mathtt{O}})^{\dagger}}{\bf g}_{mk}^{\mathtt{O}}\Big{\\}},$
$\displaystyle\text{BU}_{k}^{\mathtt{O}}\triangleq\mathbb{E}\Big{\\{}\Big{|}\sqrt{{\rho_{\mathtt{UT}}}}\Big{(}\sum\limits_{m\in{\mathcal{M}}}(1-a_{m}){(\hat{{\bf
g}}_{mk}^{\mathtt{O}})^{\dagger}}{\bf g}_{mk}^{\mathtt{O}}$
$\displaystyle\qquad\quad-\mathbb{E}\Big{\\{}\sum\limits_{m\in{\mathcal{M}}}(1-a_{m}){(\hat{{\bf
g}}_{mk}^{\mathtt{O}})^{\dagger}}{\bf
g}_{mk}^{\mathtt{O}}\Big{\\}}\Big{)}{\Big{|}^{2}}\Big{\\}},$
$\displaystyle{\text{UI}}_{k}^{\mathtt{O}}\triangleq{\rho_{\mathtt{UT}}}\sum\limits_{{\ell\in{\mathcal{K}}\backslash
k}}{\mathbb{E}\Big{\\{}\Big{|}\sum\limits_{m\in{\mathcal{M}}}(1-a_{m}){(\hat{{\bf
g}}_{mk}^{\mathtt{O}})^{\dagger}}{\mathbf{g}}_{m\ell}^{\mathtt{O}}{\Big{|}^{2}}\Big{\\}},}$
$\displaystyle{\text{MI}}_{k}^{\mathtt{O}}\triangleq{\rho_{\mathtt{J}}}\sum\limits_{\ell\in{{\mathcal{K}}}}\mathbb{E}\Big{\\{}\Big{|}\sum\limits_{m\in{\mathcal{M}}}\sum\limits_{i\in{\mathcal{M}}}(1-a_{m})a_{i}\sqrt{\theta_{i\ell}}$
$\displaystyle\hskip
30.00005pt\times{({\mathbf{\hat{g}}}_{mk}^{\mathtt{O}})^{\dagger}}{{\bf
F}_{mi}}{({\mathbf{\hat{g}}}_{i\ell}^{\mathtt{J}})^{\ast}}{\Big{|}^{2}}\Big{\\}},$
$\displaystyle{\text{AN}}_{k}^{\mathtt{O}}\triangleq\sum\limits_{m\in{\mathcal{M}}}(1-a_{m})\mathbb{E}\big{\\{}|{(\hat{{\bf
g}}_{mk}^{\mathtt{O}})^{\dagger}}{|^{2}}\big{\\}}.$

Denote the observing channel estimation error which is independent of
$\hat{\bf g}_{mk}^{\mathtt{O}}$ and zero-mean RV by
$\bm{\varepsilon}_{mk}^{\mathtt{O}}={\bf g}_{mk}^{\mathtt{O}}-\hat{\bf
g}_{mk}^{\mathtt{O}}$. By exploiting the independence between the channel
estimation errors and the estimates, we have

$\displaystyle{\text{DS}}_{k}^{\mathtt{O}}$
$\displaystyle=\sqrt{{\rho_{\mathtt{UT}}}}\sum\limits_{m\in{\mathcal{M}}}(1-a_{m})\mathbb{E}\left\\{{(\hat{{\bf
g}}_{mk}^{\mathtt{O}})^{\dagger}}(\hat{{\bf
g}}_{mk}^{\mathtt{O}}+{\bm{\varepsilon}}_{mk}^{\mathtt{O}})\right\\}$
$\displaystyle=\sqrt{{\rho_{\mathtt{UT}}}}\sum\limits_{m\in{\mathcal{M}}}(1-a_{m})N\gamma_{mk}^{\mathtt{O}}.$
(23)

Additionally, ${\text{BU}}_{k}^{\mathtt{O}}$ can be written as

$\displaystyle\text{BU}_{k}^{\mathtt{O}}=\\!{\rho_{\mathtt{UT}}}\sum\limits_{m\in{\mathcal{M}}}\\!(1\\!-\\!a_{m})\mathbb{E}\Big{\\{}|{(\hat{{\bf
g}}_{mk}^{\mathtt{O}})^{\dagger}}{\bf
g}_{mk}^{\mathtt{O}}-\mathbb{E}\\{{(\hat{{\bf
g}}_{mk}^{\mathtt{O}})^{\dagger}}{\bf g}_{mk}^{\mathtt{O}}\\}{|^{2}}\Big{\\}}$
$\displaystyle=\\!\rho_{\mathtt{UT}}\\!\\!\\!\sum\limits_{m\in{\mathcal{M}}}\\!(1-a_{m})\Big{(}\\!{\mathbb{E}}\Big{\\{}\big{|}{(\hat{{\bf
g}}_{mk}^{\mathtt{O}})^{\dagger}}\bm{\varepsilon}_{mk}^{\mathtt{O}}\\!+\\!\|\hat{{\bf
g}}_{mk}^{\mathtt{O}}{\|^{2}}{\big{|}^{2}}\\!\Big{\\}}\\!-\\!N^{2}(\gamma_{mk}^{\mathtt{O}})^{2}\\!\Big{)}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\rho_{\mathtt{UT}}\sum\limits_{m\in{\mathcal{M}}}(1-a_{m})\big{(}\mathbb{E}\big{\\{}|{(\hat{{\bf
g}}_{mk}^{\mathtt{O}})^{\dagger}}\bm{\varepsilon}_{mk}^{\mathtt{O}}{|^{2}}\big{\\}}$
$\displaystyle\hskip 100.00015pt+\mathbb{E}\big{\\{}\|\hat{{\bf
g}}_{mk}^{\mathtt{O}}{\|^{4}}\big{\\}}-N^{2}(\gamma_{mk}^{\mathtt{O}})^{2}\big{)}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\rho_{\mathtt{UT}}\sum\limits_{m\in{\mathcal{M}}}(1-a_{m})N\beta_{mk}^{\mathtt{O}}\gamma_{mk}^{\mathtt{O}},$
(24)

where we have exploited: in $(a)$ $\bm{\varepsilon}_{mk}^{\mathtt{O}}$ is
independent of $\hat{\bf g}_{mk}^{\mathtt{O}}$ and a zero-mean RV; in $(b)$
$\mathbb{E}\big{\\{}\|\hat{{\bf
g}}_{mk}^{\mathtt{O}}\|^{4}\big{\\}}=N(N+1)(\gamma_{mk}^{\mathtt{O}})^{2}$.

By exploiting the fact that $\hat{{\bf g}}_{mk}^{\mathtt{O}}$ is independent
of ${\mathbf{g}}_{m\ell}^{\mathtt{O}}$ for $k\neq\ell$, while $\hat{{\bf
g}}_{mk}^{\mathtt{O}}$, ${{\bf F}_{mi}}$, and
${\mathbf{\hat{g}}}_{i\ell}^{\mathtt{J}}$ are independent, we can formulate
$\text{UI}_{k}^{\mathtt{O}}$ and $\text{MI}_{k}^{\mathtt{O}}$, respectively,
as

$\displaystyle{\text{UI}}_{k}^{\mathtt{O}}$
$\displaystyle={\rho_{\mathtt{UT}}}\sum\limits_{\ell\in{{\mathcal{K}}}\backslash
k}{\sum\limits_{m\in{\mathcal{M}}}}(1-a_{m})N\gamma_{mk}^{\mathtt{O}}\beta_{m\ell}^{\mathtt{O}},$
(25) $\displaystyle{\text{MI}}_{k}^{\mathtt{O}}$
$\displaystyle={\rho_{\mathtt{J}}}\sum\limits_{\ell\in{\mathcal{K}}}\sum\limits_{m\in{\mathcal{M}}}\sum\limits_{i\in{\mathcal{M}}}(1-a_{m})a_{i}\theta_{i\ell}$
$\displaystyle\hskip
50.00008pt\times\mathbb{E}\\{|{({\mathbf{\hat{g}}}_{mk}^{\mathtt{O}})^{\dagger}}{{\mathbf{F}}_{mi}}{({\mathbf{\hat{g}}}_{i\ell}^{\mathtt{J}})^{\ast}}{|^{2}}\\}$
$\displaystyle~{}={\rho_{\mathtt{J}}}\sum\limits_{\ell\in{\mathcal{K}}}{\sum\limits_{m\in{\mathcal{M}}}{\sum\limits_{i\in{\mathcal{M}}}}}N^{2}(1-a_{m})a_{i}\theta_{i\ell}\gamma_{mk}^{\mathtt{O}}\beta_{mi}\gamma_{i\ell}^{\mathtt{J}}.$
(26)

The substitution of (B), (B), (25), and (26) into (22) and inserting
$\text{AN}_{k}^{\mathtt{O}}\\!=\\!N\\!\\!\\!\\!\sum\limits_{m\in{\mathcal{M}}}(1-a_{m})\gamma_{mk}^{\mathtt{O}}$,
yields (2).

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