#  Direction-Based Jamming Detection and Suppression in mmWave Massive MIMO
Systems

Saeed Bagherinejad and S. Mohammad Razavizadeh

Iran University of Science and Technology (IUST)
Narmak, Tehran 16846-13114, IRAN

###### Abstract

In this paper, we study the problem of physical layer security in the uplink
of millimeter-wave massive multiple-input multiple-output (MIMO) networks and
propose a jamming detection and suppression method. The proposed method is
based on directional information of the received signals at the base station
antenna array. The proposed jamming detection method can accurately detect
both the existence and direction of the jammer using the received pilot
signals in the training phase. The obtained information is then exploited to
develop a channel estimator that excludes the jammer’s angular subspace from
received training signals. The estimated channel information is then used for
designing a combiner at the base station that is able to effectively cancel
out the deliberate interference of the jammer. By numerical simulations, we
evaluate the performance of the proposed jamming detection method in terms of
correct detection probability and false alarm probability and show its
effectiveness when the jammer’s power is substantially lower than the user’s
power. Also, our results show that the proposed jamming suppression method can
achieve a very close spectral efficiency as the case of no jamming in the
network.

###### Index Terms:

mmWave, Massive MIMO, Physical layer security, Jamming, Directional
Information

##  I Introduction

Recently, the advent of new mobile and wireless applications has caused a
fast-growing demand for very high data rate communications in the fifth-
generation (5G) and beyond wireless networks. There are many techniques that
have been proposed to address these demands among them massive multiple-input
multiple-output (MIMO) and millimeter-wave (mmWave) are the most promising [1,
2]. In addition to higher bandwidth in the mmWave bands, their shorter
wavelength enables us to employ a large number of antennas at the base
stations (BSs) and makes it appropriate for combination with the massive MIMO
systems [3]. One of the inherent weaknesses of wireless networks is their
vulnerability to security attacks at the physical channels including jamming
and eavesdropping. Security can be provided in different layers of the network
in which the physical layer security is a powerful technique that has
attracted much attention in the recent years[4]. Massive MIMO systems are
naturally immune to a _passive eavesdropping_ attacks due to their ability to
create very narrow beams toward the legitimate users which reduces any signal
leakage to the illegitimate terminals. However, an _active eavesdropping_
attack that disrupts the training phase by transmitting a jamming signal can
reduce the secrecy rate of the massive MIMO systems [5, 6]. In addition to the
training phase, a jamming attack can also occur at the data transmission phase
with the goal of decreasing spectral efficiency of the system [7].
Consequently, one of the crucial problems in the massive MIMO systems is
detecting jamming attacks and then using techniques to suppress them or
alleviate their effects. This is of high importance especially for operation
in hostile environments.

The problems of jamming detection and suppression are discussed in many of the
prior works in the field of physical layer security in massive MIMO systems
[8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. In [8], a jamming attack
detection scheme is proposed based on pilots drawn randomly from a known
constellation. The authors in [9, 10] use additional signaling and cooperation
between the BS and users to suggest jamming attack detection methods. Two
approaches based on random matrix theory for jamming detection have been
proposed in [11, 12] in which the authors use the rank of the received
signal’s covariance matrix and eigenvalues of the received signal matrix. Also
in [13], the intentionally unallocated pilots are leveraged by the authors to
propose a jamming detection technique. Authors of [14, 15] propose two similar
approaches based on double channel training where the received signals during
two training phases in a single coherence block are compared to determine the
presence of pilot spoofing attack. None of the aforementioned researches or
other related works utilize directional information for detecting the presence
of the jamming attack. On the jamming suppression problem, a jamming attack
mitigation method is proposed in [11] in which, the users’ eigen-subspace is
estimated and then the received signal are projected onto this subspace. Zhao
et. al. in [16] propose a jamming rejection technique using cooperation
between transmitter and receiver and using an interference alignment
technique. In [17] and [18], two approaches are suggested that leverage unused
pilots to estimate the jammer’s channel information and then utilize that
information to null out the jammer’s signal. Also, a framework for estimating
legitimate users’ channels and maximizing spectral efficiency under jamming
attack is proposed in [19] which is based on statistical information of the
channels and can be implemented in spatially correlated channels.

High angular resolution is one of the most important properties of the massive
MIMO systems which can be used for combating different challenges of these
networks [20, 21]. Accurate directional information provided by high angular
resolution can be also utilized to mitigate the undesirable effects of the
jamming and eavesdropping attacks [22, 23, 24, 25]. For example, a secure
downlink transmission scheme exploiting angular information is suggested in
[22] where authors assume the BS has perfect knowledge of the eavesdropper’s
directional information and propose three precoding methods based on this
information. In [23], a beam domain secure transmission is proposed which
optimizes allocated power to each of angular paths to maximize the secrecy
sum-rate of the downlink transmission in a massive MIMO system where a passive
multi-antenna eavesdropper is present. Authors in [24] propose a technique to
estimate the angular information of users and the eavesdropper in airborne
massive MIMO systems and use this information to enhance channel estimation
performance. Xu et. al. in [25] suggest a hybrid beamforming technique that
transmits confidential data towards a legitimate user’s dominant directions
and an artificial noise signal towards all other directions using the
statistical angular information. All the abovementioned works that rely on
directional information have assumed that the BS is aware of the presence of
the adversarial terminal, therefore, they do not discuss the detection of the
adversary. Also, in none of the above and other related works, the problem of
suppressing a jamming attack on the uplink transmission of a massive MIMO
system using directional information is addressed.

Motivated by the above, in this paper, we address the problem of jamming
detection and suppression in mmWave massive MIMO networks using _directional
(angular) information_ of the users and jammer. The network consists of a BS
with a large number of antennas that serves a number of single-antenna users.
There is also a jammer in the network that transmits interfering signals in
both training and data transmission phases to sabotage the system’s
performance. A discrete channel model comprising a number of spatially
_resolvable paths_ (RP) is presented to describe the mmWave massive MIMO
channel in the angular domain. Because of the limited scattering in the
environment, the received signal from each terminal (i.e. a user or the
jammer) arrives the BS array from only a few RPs which are referred to as
_active_ RPs of that terminal. First, for the jamming detection purpose, based
on the received pilot signals in the training phase, we propose a method that
estimates the set of RPs through which a particular pilot is received. Then,
we check whether there are active RPs that are common among a large number of
pilots and accordingly detect the presence as well as the directional
information of the jammer. This information along with other information
obtained about the users’ directional information is then used for jamming
suppression in the next phase. In the next part of the paper and for the
jamming suppression purpose, utilizing the above directional information, we
propose a channel estimation scheme which is based on projecting received
pilot signals onto the orthogonal complement of the jammer’s angular subspace.
Then, the estimated channel information is used for designing the combining
vectors which are orthogonal to the jammer’s channel and cancels out the
jamming signal. Note that the directional information obtained in the channel
training phase can be used for the jamming suppression in several consecutive
intervals of coherence time. The reason is that the spatial characteristics,
such as active RPs of a terminal’s channel, change slower than the small scale
fading parameters such as the gains along with the active RPs. The key
contributions of this paper can be summarized as follows:

  * •

A jamming detection and suppression method in a mmWave massive MIMO network is
proposed which relies on directional (angular) information obtained during the
channel training phase. Received energy along with each RP over several sub-
carriers (sub-channels) is leveraged to determine if a RP is an active RP or
not.

  * •

In contrast to other works, our proposed jamming detection technique does not
require any prior knowledge of the users’ and jammer’s channel state
information or any necessary precondition such as unused pilots in the
network. Also, it does not introduce any additional signaling overhead.

  * •

The proposed detection technique is capable of detecting the jammer even if
its power is very low. Besides, the false alarm probability of this detection
method is substantially lower than other similar techniques. The performance
of the proposed scheme can be further enhanced by increasing the number of
antennas at the BS or the number of sub-carriers (sub-channels) at the system.
The jamming detection method directly yields directional information of the
jammer which can be utilized to suppress the jamming attack over several
intervals of coherence time.

  * •

The proposed jamming suppression also can efficiently cancel out the jamming
attacks. We show that this method performs very well and similar to the case
where no jamming is present in the network. Also, its performance does not
depend on the jammer’s power and is effective even with relatively high-power
jamming attacks.

Figure 1: System model of a massive MIMO network attacked by a jammer.

##  II System Model

As depicted in Fig. 1, we consider uplink of a single-cell massive MIMO system
consisting of a BS with a large array of $M$ antennas that serves $K$ single-
antenna legitimate users in the presence of a single-antenna jammer, which all
are randomly located in the cell. The network operates in the mmWave bands and
a multi-carrier transmission with $N$ sub-carriers (or sub-channels) is
considered. We also consider a block-fading channel model where channel gains
are constant within a coherence block of length $T=T_{c}N_{c}$ symbols and
varies independently from one block to another. $T_{c}$ and $N_{c}$ denote the
number of time symbols in a coherence time and the number of sub-carriers in a
coherence bandwidth, respectively.

###  II-A Angular Domain Channel Modeling

Since our proposed scheme is based on the directional information of the
channels, we need a channel model that properly describes signals in the
angular domain. We assume an uniform linear array (ULA) at the BS and for each
direction of arrival (DOA), $\theta\in[-\pi/2,\pi/2]$, the BS has the
following array steering vector

$\displaystyle\mathbf{a}(\theta)=\frac{1}{\sqrt{M}}[1,e^{-j\frac{2{\pi}d}{\lambda}sin(\theta)},e^{-j2\frac{2{\pi}d}{\lambda}sin(\theta)},$
$\displaystyle\ldots,e^{-j(M-1)\frac{2{\pi}d}{\lambda}sin(\theta)}]^{T},$ (1)

where $d$ and $\lambda$ are array element spacing and carrier wavelength,
respectively. It is assumed that $d=\frac{\lambda}{2}$. Also,
$\Theta=sin(\theta)$ denotes the directional sine and since $\theta$ is in the
interval of $[-\pi/2,\pi/2]$, there is a one-to-one mapping between $\theta$
and $\Theta$. We assume that the DOAs from which signals of the $k$th user are
received are uniformly distributed in the interval of
$\mathcal{I}_{k}=[\bar{\theta}_{k}-\Delta_{k}/2,\bar{\theta}_{k}+\Delta_{k}/2]$,
where $\bar{\theta}_{k}$ and $\Delta_{k}$ are average incident angle and
angular spread, respectively. We call the interval $\mathcal{I}_{k}$ as the
_angular span_ of the $k$th user.

The angular resolution of massive MIMO depends on the _array’s length_ , $L$
which is equal to $L=M\frac{d}{\lambda}$ [26]. If the directional sine of two
paths differ less than $1/L$, these paths would not be resolvable by the array
[26]. Hence, the angular domain can be sampled at fixed angles with a spacing
of $1/L$ in the directional sine. The steering vectors with angular spacing of
$1/L$ at their $\Theta$ are orthonormal, thus they can form an orthogonal
basis for channel expansion that can be represented as [27]

$\mathbf{U}\triangleq[\mathbf{a}(\phi_{1}),\mathbf{a}(\phi_{2})),\ldots,\mathbf{a}(\phi_{M})],$
(2)

where $\phi_{i}\triangleq sin^{-1}(\frac{1}{L}(i-1-(M-1)/2)),i=1,2,\ldots,M$
are the sampled angles.

By these definitions, we can decompose the channels to $M$ different physical
directions which we refer to them as resolvable path (RP). The gain of the
$i$th RP can be represented as the superposition of all paths whose $\Theta$
are located within a window of width $1/L$ around $sin(\phi_{i})$.
Subsequently, we define a virtual channel representation (VCR) model which
samples the angular domain using spatial orthogonal basis $\mathbf{U}$. The
VCR channel vector of the $k$th user in the $n$th sub-carrier can be denoted
by

$\mathbf{h}_{k}^{n}=\sqrt{\frac{M\beta_{k}}{C_{k}}}\mathbf{U}\tilde{\mathbf{g}}_{k}^{n},\;n=1,\ldots,N;\;k=1,\ldots,K,$
(3)

where
$\tilde{\mathbf{g}}_{k}^{n}=[\tilde{g}_{k,1}^{n},\tilde{g}_{k,2}^{n},\ldots,\tilde{g}_{k,M}^{n}]$
is a complex gain vector and $\tilde{g}_{k,i}^{n}$ indicates the small scale
gain of $i$th RP. If a RP is located in the angular span of the $k$th user,
i.e. $\phi_{i}\in\mathcal{I}_{k}$, we call it as an active RP of that user.
Since each active RP includes several physical paths, we assume that the small
scale random modelling of an active RP is a complex Gaussian random
coefficient with zero mean and unit variance, i.e.
$\tilde{g}_{k,i}^{n}\sim\mathcal{CN}(0,1)$. If a RP is not an active RP, i.e.
$\phi_{i}\notin\mathcal{I}_{k}$, its gain would be equal to zero, i.e.
$\tilde{g}_{k,i}^{n}=0$. Furthermore, $\beta_{k}$ and $C_{k}$ are large scale
fading coefficient and the number of active RPs of the $k$th user,
respectively. Due to limited scattering characteristic of the mmWave channels,
the received power is concentrated on a narrow interval in the angular domain.
Thus, the angular spread is relatively small and only a few RPs are located
within it. Thus, $C_{k}$ is considerably smaller than the number of array
antennas, i.e. $C_{k}\ll M$ [22].

The angular span of each user is constant over different sub-carriers since
the spatial propagation characteristics are unchanged within system’s
bandwidth. Therefore

$\Omega_{k}=supp({\tilde{\mathbf{g}}_{k}^{1}})=supp(\tilde{\mathbf{g}}_{k}^{2})=\ldots=supp(\tilde{\mathbf{g}}_{k}^{N}),$
(4)

where $supp{}$ denotes the support set of a vector. This equation implies the
active RPs of each user are unchanged over different sub-carriers. The set of
$\Omega_{k}$ is also referred to as _spatial signature_.

Figure 2: An example of a virtual channel representation (VCR) model for an
array with $M=18$ antennas.

Fig. 2 illustrates an example of the described channel model in angular domain
for $M=18$ antennas at the BS. Solid arrows show the spatially resolvable
paths that three of them are located within the angular span of the user and
in fact are the active RPs. These active RPs are demonstrated by red arrows.
Thus, in this example, the set of active RPs for this user is
$\Omega_{k}=\\{5,6,7\\}$.

Similar to the users’ channels, the jammer’s channel in the $n$th sub-carrier
can also be denoted by
$\mathbf{h}_{w}^{n}=\sqrt{{M\beta_{w}}/{C_{w}}}\mathbf{U}\tilde{\mathbf{g}}_{w}^{n}$
where $\beta_{w}$ and $C_{w}$ denote the large scale fading coefficient and
the number of active RPs of the jammer, respectively. The angular span of
jammer is also denoted by
$\mathcal{I}_{w}=[\bar{\theta}_{w}-\Delta_{w}/2,\bar{\theta}_{w}+\Delta_{w}/2]$,
where $\bar{\theta}_{w}$ and $\Delta_{w}$ are the average incident angle and
angular spread of the jammer, respectively. Also, similar to (4), the set of
indices of active RPs (spatial signature) for the jammer is denoted by
$\Omega_{w}$.

We assume that the users and the jammer are randomly distributed in the cell,
thus we consider average incident angle of the $k$th user and jammer to be
uniformly distributed in the intervals of
$[-\pi/2+\Delta_{k}/2,\pi/2-\Delta_{k}/2]$ and
$[-\pi/2+\Delta_{w}/2,\pi/2-\Delta_{w}/2]$, respectively. The $k$th user’s and
the jammer’s channels can be alternatively defined as

$\displaystyle\mathbf{h}_{k}^{n}=\sqrt{\frac{M\beta_{k}}{C_{k}}}\sum_{i\in\Omega_{k}}\tilde{g}_{k,i}^{n}\mathbf{a}(\phi_{i})$
(5)
$\displaystyle\mathbf{h}_{w}^{n}=\sqrt{\frac{M\beta_{w}}{C_{w}}}\sum_{i\in\Omega_{w}}\tilde{g}_{w,i}^{n}\mathbf{a}(\phi_{i})$
(6)

###  II-B Pilot and Data Transmission Phases

Transmission from the users to the BS in the uplink occurs in two phases.
First, in the channel training pahse, each user sends its allocated orthogonal
pilot sequence to the BS for estimating that user’s channel. In the second
phase which is data transmission phase, the user sends its data to the BS. The
jammer also transmits jamming signals during both training and data phases
with different powers $q_{t}$ and $q_{d}$, respectively.

####  II-B1 Channel Training

Assuming a block fading channel with $N_{c}$ sub-channels that are placed
within its coherence bandwidth. As a result, only one sub-carrier out of
$N_{c}$ needs to be estimated using orthogonal pilot sequences [3, 1]. We
denote the number of estimated sub-carriers by $N_{e}=N/N_{c}$. The set of
orthogonal pilots is
$\mathbf{S}=[\mathbf{s}_{1},\mathbf{s}_{2},\ldots,\mathbf{s}_{\tau}]\in\mathbb{C}^{\tau\times\tau}$,
where $\tau$ is length of orthogonal pilots, and we assume $\tau=K$. Also,
pilots are considered to have unit power, i.e.
$\mathbf{S}^{H}\mathbf{S}=\mathbf{I}$. The $k$th user transmits the $k$th
pilot over $N_{e}$ predefined sub-carriers. At this phase, the received signal
in the $n$th sub-carrier can be represented as

$\mathbf{Y}_{t}^{n}=\sum_{k=1}^{K}\sqrt{\tau
p_{t,k}}\mathbf{h}_{k}^{n}\mathbf{s}_{k}^{H}+\sqrt{\tau
q_{t}}\mathbf{h}_{w}^{n}(\mathbf{s}_{w}^{n})^{H}+\mathbf{Z}_{t}^{n},$ (7)

where $p_{t,k}$ is the pilot transmission power of the $k$th user and
$\mathbf{s}_{w}^{n}\in\mathbb{C}^{\tau\times 1}$ is the jamming pilot signal
sent by the jammer in the $n$th sub-carrier.
$\mathbf{Z}_{t}^{n}\in\mathbb{C}^{M\times\tau}$ is also additive white
Gaussian noise (AWGN) whose elements are
$[\mathbf{Z}_{t}^{n}]_{i,j}\sim\mathcal{CN}(0,\sigma_{z}^{2})$. In order to
estimate the $k$th user’s channel, the BS de-spread the received signal in (7)
with the pilot $\mathbf{s}_{k}$ as

$\mathbf{y}_{t,k}^{n}=\mathbf{Y}_{t}^{n}\mathbf{s}_{k}=\sqrt{\tau
p_{t,k}}\mathbf{h}_{k}^{n}+\sqrt{\tau
q_{t}}\gamma_{k}^{n}\mathbf{h}_{w}^{n}+\mathbf{z}_{t,k}^{n},$ (8)

where $\mathbf{z}_{t,k}^{n}=\mathbf{Z}_{t}^{n}\mathbf{s}_{k}$ is the
correlation between noise and the $k$th pilot. Also,
$\gamma_{k}^{n}=(\mathbf{s}_{w}^{n})^{H}\mathbf{s}_{k}$ indicates the
correlation between the jamming signal and the $k$th pilot. It is clear that
the interference exerted by other users have been eliminated by orthogonality
of pilots; however, the deliberate interference by the jammer signal still
remains.

A jammer usually aims to limit the SE of the massive MIMO network rather than
targeting a special user. The reason is that, in the massive MIMO systems, if
a random pilot hopping method is used for assigning pilots to the users [17,
18, 6], it is difficult for a jammer to acquire a particular user’s pilot.
However, it is reasonable to assume that the jammer has knowledge about the
pilot set $\mathbf{S}$.

In addition, a necessary condition for jamming signal with the purpose of
mitigating performance of the system, is that the $Prob(\gamma_{k}^{n}=0)=0$
which can be achieved by spreading jamming power over all the pilots. A known
method for designing jamming signal in the training phase is that
$E(|(\mathbf{s}_{w}^{n})^{H}\mathbf{s}_{i}|^{2})=1/\tau$. One of the signals
that satisfies this property is [17]

$\mathbf{s}_{w}^{n}\thicksim\mathcal{CN}(0,\frac{1}{\tau}\mathbf{I}),\;n=1,\ldots,N_{e}.$
(9)

To restrain the jamming attack on the training phase, some extra information
about the users’ and jammer’s channels is needed which is the directional
information in our work. In Sec. III-A and III-B, we will investigate how to
obtain this information and then utilize it in Sec. IV to propose the channel
estimation technique.

####  II-B2 Data Transmission

In the uplink data transmission phase, each user $k$ sends its data with power
$p_{k,d}$. Simultaneously, the jammer sends its jamming signal to the BS. The
received signal at the BS over the $n$th sub-carrier can be expressed as,

$\mathbf{y}_{d}^{n}=\sum_{k=1}^{K}\sqrt{p_{d,k}}\mathbf{h}_{k}^{n}x_{k}^{n}+\sqrt{q_{d}}\mathbf{h}_{w}^{n}x_{w}^{n}+\mathbf{z}_{d}^{n}$
(10)

where
$\mathbf{z}_{d}^{n}\thicksim\mathcal{(}\mathbf{0},\sigma_{z}^{2}\mathbf{I})$
is the AWGN noise, $x_{k}^{n}$ and $x_{w}^{n}$ are the $k$th user’s data and
jamming signal, respectively. These signals are considered to be i.i.d
Gaussian random variable with zero mean and unit variance, i.e.
$x_{k}^{n},x_{w}^{n}\sim\mathcal{CN}(0,1)$. Then the BS receives this signal
and decodes it by the use of information obtained in the training phase.
Suppose $\mathbf{V}^{n}$ is the decoding (combining) matrix at the BS and
$\mathbf{v}_{k}^{n}$ is its $k$th column which is the receive combining vector
of the $k$th user. By calculating the inner product of $\mathbf{v}_{k}^{n}$
and $\mathbf{y}_{d}^{n}$, the signal from the $k^{th}$ user in the $n$th sub-
carrier can be decoded as

$\displaystyle y_{d,k}^{n}={\mathbf{v}_{k}^{n}}^{H}\mathbf{y}_{d}^{n}$
$\displaystyle=\sqrt{p_{d,k}}{\mathbf{v}_{k}^{n}}^{H}\mathbf{h}_{k}^{n}x_{k}^{n}+\sum_{\stackrel{{\scriptstyle
l=1}}{{l\neq
k}}}^{K}\sqrt{p_{d,l}}{\mathbf{v}_{k}^{n}}^{H}\mathbf{h}_{l}^{n}x_{l}^{n}$
$\displaystyle+\sqrt{q_{d}}{\mathbf{v}_{k}^{n}}^{H}\mathbf{h}_{w}^{n}x_{w}^{n}+{\mathbf{v}_{k}^{n}}^{H}\mathbf{z}_{d}^{n}.$
(11)

Since the BS has an imperfect CSI of user’s channel, we utilize a lower bound
of the capacity called use-and-then-forget which is widely used in massive
MIMO [1, 3]. The name originates from the fact that estimated channels are
used for the design of the receive combining vectors and then disregarded in
the signal detection. This achievable rate is defined as follows

$\displaystyle
R_{k}^{n}=(1-\frac{\tau}{T})\mathbb{E}_{\Psi}\\{\log_{2}(1+\rho_{k}^{n})\\},$
(12)

where $\rho_{k}^{n}$ is the effective SINR defined in (13) at the top of the
next page. Note that this SINR is conditioned on $\Psi$ which is the
directional information and will be discussed in the following sections.

$\displaystyle\rho_{k}^{n}=\dfrac{p_{d,k}|\mathbb{E}\\{{\mathbf{v}_{k}^{n}}^{H}\mathbf{h}_{k}^{n}|\Psi\\}|^{2}}{p_{d,k}\mathtt{var}\\{{\mathbf{v}_{k}^{n}}^{H}\mathbf{h}_{k}^{n}|\Psi\\}+\sum_{\stackrel{{\scriptstyle
l=1}}{{l\neq
k}}}^{K}p_{d,l}\mathbb{E}\\{|{\mathbf{v}_{k}^{n}}^{H}\mathbf{h}_{l}^{n}|^{2}|\Psi\\}+q_{d}\mathbb{E}\\{|{\mathbf{v}_{k}^{n}}^{H}\mathbf{h}_{w}^{n}|^{2}|\Psi\\}+\mathbb{E}\\{|{\mathbf{v}_{k}^{n}}^{H}\mathbf{z}_{d}^{n}|^{2}|\Psi\\}}$
(13)

##  III Jamming Detection

In this section, we discuss the proposed jamming detection scheme which
consist of two stages. In the first stage, the signals in (8) are used to
estimate the active RPs from which any particular pilot is received. To this
end, we exploit the received training signals in several sub-carriers to
distinguish the active RPs (i.e those that include a signal) from the RPs that
only include noise. Then, in the second stage, we present our proposed
detection method. In this method, after determining the active RPs, we
evaluate the number of pilots that are received through each RP and in this
way we determine if there is a jammer in the network.

###  III-A Active RP Estimation

As we explained before, the $k$th pilot is transmitted by both the $k$th user
and the jammer. The union set of active RP indices for the jammer and the
$k$th user is defined as $\Omega_{k,w}=\Omega_{k}\cup\Omega_{w}$. In fact,
this set shows the spatial directions from which the $k$th pilot is received.

Calculating the inner product of $\mathbf{U}$ in (2) with $\mathbf{y}_{k}^{n}$
in (8) maps the received signals to the angular domain. Therefore,

$\tilde{\mathbf{y}}_{t,k}^{n}=\mathbf{U}^{H}\mathbf{y}_{t,k}^{n}=\sqrt{\tau
p_{t,k}\mu_{k}}\tilde{\mathbf{g}}_{k}^{n}+\sqrt{\tau
q_{t}\mu_{w}}\gamma_{k}^{n}\tilde{\mathbf{g}}_{w}^{n}+\tilde{\mathbf{z}}_{t,k}^{n},$
(14)

where $\tilde{\mathbf{z}}_{k}^{n}=\mathbf{U}^{H}\mathbf{z}_{k}^{n}$ whose
distribution is the same as $\mathbf{z}_{k}^{n}$ since $\mathbf{U}$ is an
unitary matrix, i.e.
$\tilde{\mathbf{z}}_{k}^{n}\thicksim\mathcal{CN}(\mathbf{0},\sigma_{z}^{2}\mathbf{I})$.
We need a sensing scheme that can distinguish between the active RPs and the
RPs that only consist of noise. For this purpose, we first define a hypothesis
test for the $i$th RP as

$\displaystyle\mathcal{H}_{0,i}^{RP}:[\tilde{\mathbf{y}}_{t,k}^{n}]_{i}=[\tilde{\mathbf{z}}_{t,k}^{n}]_{i}$
$\displaystyle\mathcal{H}_{1,i}^{RP}:[\tilde{\mathbf{y}}_{t,k}^{n}]_{i}=\begin{cases}\sqrt{\tau
p_{t,k}\mu_{k}}\tilde{g}_{k,i}^{n}+\sqrt{\tau
q_{t}\mu_{w}}\gamma_{k}^{n}\tilde{g}_{w,i}^{n}+[\tilde{\mathbf{z}}_{t,k}^{n}]_{i}\\\
\sqrt{\tau
p_{t,k}\mu_{k}}\tilde{g}_{k,i}^{n}+[\tilde{\mathbf{z}}_{t,k}^{n}]_{i}\\\
\sqrt{\tau
q_{t}\mu_{w}}\gamma_{k}^{n}\tilde{g}_{w,i}^{n}+[\tilde{\mathbf{z}}_{t,k}^{n}]_{i}\end{cases}$
(15)

where $\mathcal{H}_{1,i}^{RP}$ denotes three cases of (a) both the $k$th user
and the jammer transmit the signal, (b) the user transmits the signal and (c)
the jammer transmit the signal all along with the $i$th RP.
$\mathcal{H}_{0,i}^{RP}$ denotes the opposite case in which no signal is
received from the jammer or the $k$th user and only a noise signal is present
along with the $i$th RP.

Furthermore, the energy received along a certain RP can be utilized to
determine if it is a active RP or not. This stems from the fact that receiving
signal along a RP increases the energy level comparing to the case that it
only receiving noise. Thus, it is judicious to infer that a RP is active if
its energy is more than a predefined threshold. To this end, for the $i$th RP,
we compute the sum received energy from $N_{d}$ sub-carriers as

$W_{i,k}\triangleq\sum_{n=1}^{N_{d}}|[\tilde{\mathbf{y}}_{t,k}^{n}]_{i}|^{2}.$
(16)

After computing the received energy, we can use it for determining the active
RPs according to what is mentioned above. Therefore, we define the decision
rule as

$W_{i,k}\underset{\mathcal{H}_{0,i}^{RP}}{\overset{\mathcal{H}_{1,i}^{RP}}{\gtrless}}\epsilon_{k},$
(17)

where $\epsilon_{k}$ is the decision threshold which can be chosen to satisfy
various criteria. Here, we propose a condition for selecting the threshold
that guarantees the probability of false alarm (i.e. declaring a non-active RP
as an active RP) to be less than a given value $\eta$. The condition is
defined as

$Prob\\{W_{i,k}>\epsilon_{k}|\mathcal{H}_{0,i}^{RP}\\}\leq\eta.$ (18)

By choosing $\eta$ close to zero, e.g. $10^{-3}$, and determining
$\epsilon_{k}$ accordingly, it will be assured that the RPs selected by the
decision rule in (17) are always included in $\Omega_{k,w}$. Note that,
$W_{i,k}$ when the $i$th RP only comprises of noise, is a Gamma distributed
random variable, i.e $W_{i,k}\thicksim Gamma(N_{d},\sigma_{z}^{2})$, which can
be used to calculate $\epsilon_{k}$.

In order to efficiently estimate the active RPs, the user’s and jammer’s
powers have to be sufficiently large compared to the noise power. It is
rational to assume that the users’ powers have been adjusted to be adequately
more than noise power. However, there is no control over the jammer’s power.
In the case of low power jamming, the BS can use a larger number of sub-
carriers. It enhances the RP detection performance since the gains of active
RPs in different sub-channels are independent and it is highly probable that
at least some of these gains are large enough to make the RP detectable. In
general, increasing the number of received samples in the summation (16) would
always help to improve the detection performance. It can be done either by
increasing the number of considered sub-channels or considering the received
training signal over several consecutive intervals of coherence time since the
directional characteristics of channels change more slowly in time. This means
we can sum the energy received along a RP over several sub-channels and in
some consecutive intervals of time coherence. The impact of increasing number
of sub-carriers or number of coherence time intervals is the same, thus, we
only consider one coherence time and adjust the number of sub-channels.

Finally, the estimated set of RPs for $k$th pilot is described as follows

$\hat{\Omega}_{k,w}=\\{i\in\\{1,\ldots,M\\}|W_{i,k}>\epsilon_{k}\\}.$ (19)

Having estimated the active RP sets, we exploit them to suggest a jamming
detection scheme in the next section.

###  III-B Jamming Detection

In this section, we propose a jamming detection scheme based on the estimated
RP sets in the previous section. Initially, we discuss a principal
characteristic of users’ active RP sets
$\\{\Omega_{1},\Omega_{2},\ldots,\Omega_{K}\\}$, which we are going to use to
propose a jamming detection method.

######  Remark 1.

Suppose that average incident angles of all $K$ user are uniformly distributed
in the interval of
$(\frac{\pi}{2}-\frac{\Delta}{2},-\frac{\pi}{2}+\frac{\Delta}{2})$, and the
angular spread is equal for all user, i.e.
$\Delta_{1}=\ldots=\Delta_{K}=\Delta$. Then, the probability that $g$ out of
$K$ users have at least one common active path, $P_{g}^{K}$, is upper-bounded
by

$P_{g}^{K}\leqslant\bar{P}_{g}^{K},$ (20)

where
$\bar{P}_{g}^{K}=\min(1,\binom{K}{g}(1-(\frac{\pi-2\Delta}{\pi-\Delta})^{2})^{g-1})$.

###### Proof.

Please refer to Appendix VII-A. ∎

It should be noted that when $g$ has a low value, e.g. $g=2$ the upper-bound
is close to one, i.e. $\bar{P}_{g}^{K}\sim 1$. However, as $g$ increases to
$K$, $\bar{P}_{g}^{K}$ exponentially decreases to a very low value. For
instance, for $K=10$, $\Delta=\pi/18$ and $g=6,8,10$, the upper-bound is
proportional to $\bar{P}_{6}^{10}\sim 10^{-3},\;\bar{P}_{8}^{10}\sim 10^{-5}$
and $\bar{P}_{10}^{10}\sim 10^{-9}$. Thus, it is rational to assume that it is
not likely for a large number of users to have common active RPs. Another
conclusion that can be drawn is that a common active RP can indicate the
existence of a jammer.

Having this property, we define another hypothesis test as

$\displaystyle\mathcal{H}_{0}^{JD}:\mathbf{Y}_{t}^{n}=\sum_{k=1}^{K}\sqrt{\tau
p_{t,k}}\mathbf{h}_{k}^{n}\mathbf{s}_{k}^{H}+\mathbf{Z}_{t}^{n}$
$\displaystyle\mathcal{H}_{1}^{JD}:\mathbf{Y}_{t}^{n}=\sum_{k=1}^{K}\sqrt{\tau
p_{t,k}}\mathbf{h}_{k}^{n}\mathbf{s}_{k}^{H}+\sqrt{\tau
q_{t}}\mathbf{h}_{w}^{n}(\mathbf{s}_{w}^{n})^{H}+\mathbf{Z}_{t}^{n},$ (21)

where the hypothesis $\mathcal{H}_{0}^{JD}$ and $\mathcal{H}_{1}^{JD}$ denote
absence and presence of the jammer, respectively. Then, we define the
following set that contains indices of common active RPs between at least $g$
pilots

$\mathcal{Q}_{g}=\\{i\in\\{1,\ldots,M\\}|\mathcal{R}(i)\geq
g\\},\;g=2,\ldots,K$ (22)

where $\mathcal{R}(i)$ is a function that returns the number of pilots that is
received along $i$th RP according to the estimated sets
$\\{\hat{\Omega}_{1,w},\hat{\Omega}_{2,w},\ldots,\hat{\Omega}_{K,w}\\}$. In
addition, as illustrated in Remark 1, in the absence of the jammer, we
anticipate only a small subset of pilots to have common active RPs since then
pilots would be only received from the users’ RPs. On the other hand, when a
jammer is present in the network, its active RPs would be common for many of
the pilots and hence, the number of pilots with common RPs would be relatively
large. Based on this fact, we infer that if the number of pilots that have
common RPs is more than a predefined threshold, a jammer is present in the
network and vice versa. Therefore, we can rewrite the detector as

$\displaystyle\mathcal{H}_{0}^{JD}:\mathcal{Q}_{g}=\emptyset$
$\displaystyle\mathcal{H}_{1}^{JD}:\mathcal{Q}_{g}\neq\emptyset.$ (23)

This decision rule states that if $\mathcal{Q}_{g}$ is an empty set where $g$
is the predefined threshold, the number of pilots which have common RPs is
less than $g$ and thus, there is no jammer present in the network. On the
other hand, when $\mathcal{Q}_{g}$ is a non-empty set, there are active RPs
common between at least $g$ pilots and therefore, there is a jammer present in
the network. Furthermore, the parameter $g$ should be selected properly to
avoid a high false alarm probability (FAP). If $g$ is not large enough, the
set $\mathcal{Q}_{g}$ can be non-empty even when there is no jammer, which
leads to a high FAP. If $g$ is large, FAP would be low but correct detection
probability (CDP) would be lower, especially when the jammer’s power is low.
The reason is that the jammer’s RPs may not appear in a large number of
estimated pilot RP sets. Finally, note that the set $\mathcal{Q}_{g}$ gives us
an estimate of the jammer’s RPs since the common RPs are jammer’s paths with a
high probability.

##  IV Jamming Suppression

In this section, we utilize the directional information attained during the
jamming detection phase to design a signal detection scheme which mitigates
the intentional interference of the jammer in the data transmission phase. The
proposed method relies on a modified channel estimation which takes into
account the effect of the jammer. This channel estimation is implemented by
projecting the received pilot signals onto the orthogonal complement of the
jammer’s angular subspace and is based on a linear minimum mean squared error
(LMMSE) estimation technique in the training phase. Afterward, in the data
transmission phase, a maximum ratio combining (MRC) is constructed by the use
of the aforementioned estimated channels. We show that the combining vector is
orthogonal to the jammer’s channel and can effectively suppress the effect of
jamming.

###  IV-A Channel Estimator

As discussed in section II, the channel of a user is constituted of two types
of information, 1) a set of active RPs, 2) the gains of these active RPs. We
obtained an estimate of RPs along which pilot signals were received,
$\hat{\Omega}_{k,w}$, which also included paths of the jammer. The proposed
jamming detection scheme yields an estimate of the jammer’s spatial signature.
By subtracting the jammer’s paths from $\hat{\Omega}_{k,w}$, we obtain a set
that only comprises the $k$th user’s RPs, i.e.
$\hat{\Omega}_{k}=\hat{\Omega}_{k,w}\setminus\mathcal{Q}_{g}$. Thus, by
estimating the gain along these paths, an estimation of the users’ channel
will be attained.

The signals received in the training phase along with $i$th RP for
$i\in\hat{\Omega}_{k}$ can be written as

$[\tilde{\mathbf{y}}_{t,k}^{n}]_{i}=\mathbf{a}(\phi_{i})^{H}\mathbf{y}_{t,k}^{n}=\sqrt{\tau
p_{t,k}\mu_{k}}\tilde{g}_{k,i}^{n}+[\tilde{\mathbf{z}}_{t,k}^{n}]_{i}.$ (24)

Therefore, according to [28], we can define the LMMSE estimation of the $i$th
RP’s gain as

$\hat{\tilde{g}}_{k,i}^{n}=\frac{\mathbb{E}\\{\tilde{g}_{k,i}^{n}[\tilde{\mathbf{y}}_{t,k}^{n}]_{i}\\}}{\mathbb{E}\\{|[\tilde{\mathbf{y}}_{t,k}^{n}]_{i}|^{2}\\}}[\tilde{\mathbf{y}}_{t,k}^{n}]_{i}=\dfrac{\sqrt{\tau
p_{k,t}\mu_{k}}}{\sigma_{z}^{2}+\tau
p_{k,t}\mu_{k}}\mathbf{a}(\phi_{i})^{H}\mathbf{y}_{t,k}^{n},$ (25)

where $\mu_{k}=\frac{M\beta_{k}}{C_{k}}$. The mean-square of the estimated
gain are

$\xi_{k}=\mathbb{E}\\{|\hat{\tilde{g}}_{k,i}^{n}|^{2}\\}=\frac{|\mathbb{E}\\{\tilde{g}_{k,i}^{n}[\tilde{\mathbf{y}}_{t,k}^{n}]_{i}\\}|^{2}}{\mathbb{E}\\{|[\tilde{\mathbf{y}}_{t,k}^{n}]_{i}|^{2}\\}}=\dfrac{\tau
p_{k,t}\mu_{k}}{\sigma_{z}^{2}+\tau p_{k,t}\mu_{k}}.$ (26)

Note that this estimator is the same for all the RPs of a user, since they all
have the same distribution. Moreover, the estimation error is defined as
$e_{k,i}^{n}=\hat{\tilde{g}}_{k,i}^{n}-\tilde{g}_{k,i}^{n}$ and its mean-
square is
$\mathbb{E}\\{|e_{k,i}^{n}|^{2}\\}=\dfrac{\sigma_{z}^{2}}{\sigma_{z}^{2}+\tau
p_{k,t}\mu_{k}}$. By having the RP set of a user and the RPs’ gains, the
channel of the $k$th user can be estimated as

$\hat{\mathbf{h}}_{k}^{n}\triangleq\sqrt{\mu_{k}}\sum_{i\in\hat{\Omega}_{k}}\hat{\tilde{g}}_{k,i}^{n}\mathbf{a}(\phi_{i}).$
(27)

This estimation technique has a great advantage that makes the estimated
channels orthogonal to the jammer’s channel, i.e.
$(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{w}^{n}\simeq 0$. It stems from the
fact that by eliminating the jammer’s active RPs, the angular subspace of the
jammer’s channel is excluded and the users’ channels lie in its null space.
Although some of the jammer’s RPs which have insignificant gains may remain,
especially when the jammer’s power is very low, their impact on the estimation
accuracy and orthogonality to the jammer’s channel will be negligible due to
their small gain. We should also note that for the users that have some common
RPs with the jammer, the accuracy of channel estimation decreases. However, it
is unlikely and would not affect the majority of users.

###  IV-B MRC Decoder

As mentioned before, in this paper we adopt an MRC combining method which is a
simple and efficient precoder for massive MIMO systems. Hence, in the uplink
data transmission phase, the decoding matrix will be
$\mathbf{V}^{n}=\hat{\mathbf{H}}^{n}$, where
$\hat{\mathbf{H}}^{n}=\\{\hat{\mathbf{h}}_{1}^{n},\hat{\mathbf{h}}_{2}^{n},\ldots,\hat{\mathbf{h}}_{K}^{n}\\}$.
As mentioned before, canceling out the jammer’s RPs would eliminate its
deliberate interference due to the orthogonality of the combining vectors to
the jammer’s channel which can be interpreted as a kind of spatial filtering.
Using MRC, terms of SINR in (13) for the $k$th user in the $n$th subcarrier
will be obtained as

$\displaystyle|\mathbb{E}\\{(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{k}^{n}\\}|^{2}=\mu_{k}^{2}C_{k/w}^{2}\xi_{k}^{2},$
(28)
$\displaystyle\mathtt{var}\\{(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{k}^{n}\\}=\mu_{k}^{2}C_{k/w}\xi_{k}^{2},$
(29)
$\displaystyle\mathbb{E}\\{|(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{l}^{n}|^{2}\\}=C_{k,l}\xi_{k}\mu_{k}\mu_{l},$
(30)
$\displaystyle\mathbb{E}\\{|(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{w}^{n}|^{2}\\}\simeq
0,$ (31)
$\displaystyle\mathbb{E}\\{|(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{z}_{d}^{n}|^{2}\\}=\mu_{k}C_{k/w}\xi_{k}\sigma_{z}^{2},$
(32)

where $C_{k/w}=card\\{\hat{\Omega}_{k}\\}$ and
$C_{k,l}=C_{l,k}=card\\{\hat{\Omega}_{k}\cap\hat{\Omega}_{l}\\}$. The
derivation of these terms is included in Appendix VII-B with more details.
Now, after calculating these terms, the effective SINR of the $k$th user in
the $n$th sub-carrier is described as

$\rho_{k}^{n,MRC}=\dfrac{p_{d,k}\mu_{k}C_{k/w}^{2}\xi_{k}}{p_{d,k}\mu_{k}C_{k/w}\xi_{k}+\sum_{\stackrel{{\scriptstyle
l=1}}{{l\neq k}}}^{K}p_{d,l}\mu_{l}C_{k,l}+C_{k/w}\sigma_{z}^{2}}.$ (33)

##  V Simulation Results

In this section, we evaluate the proposed jamming detection and suppression
scheme by numerical simulations. In our simulations, we assumed $K=10$ and
equal user transmit power in both training and data transmission phases with
$p_{k,d}=p_{k,t}=0$ dBw for all users and noise power is $\sigma_{z}^{2}=-25$
dBw. Also, the large scale fading coefficients are assumed to be equal to $0$
dB for users and the jammer, i.e. $\beta_{k}=\beta_{w}=1$. The length of pilot
signals is taken as $\tau=K=10$ and the length of coherence block is set to
$T=200$. The users’ and jammer’s average incident angles are considered to be
uniformly distributed in the interval $[-\pi/2+\Delta/2,\pi/2-\Delta/2]$ and
angular spread is constant for all terminals. The number of sub-carriers is
$N_{d}=1,20$ and the threshold in (17) is selected as
$\epsilon_{k}=0.02,0.11$.

Figure 3: CDP of the proposed jamming detection method versus the jammer’s
power in the training phase (for $M=200$ and $\Delta=\pi/18$)

At first, we evaluate the performance of the proposed jamming detection method
by evaluating its CDP and FAP. Fig. 3 shows the CDP versus the jammer’s power
$q_{t}$ for different values of $g=6,8,10$. It is apparent that CDP improves
with the jammer’s power as the jammer’s RPs become easier to estimate.
Furthermore, this figure shows that using a larger number of sub-carriers
$N_{d}$ leads to a better CDP performance. For example, using $N_{d}=20$ sub-
carriers helps the BS to detect the jammer with 10dB lower power. Besides, a
lower value of $g$ results in a better detection probability.

Figure 4: FAP of the proposed jamming detection method versus angular spread
$\Delta$ (for $M=200$)

Fig. 4 demonstrates the relation between the FAP and angular spread $\Delta$.
As shown in this figure, FAP increases with the angular spread $\Delta$. It is
due to the fact that with a larger angular spread, the probability that an RP
is common among a large subset of users would rise. In addition, a smaller
value of $g$ would result in a higher FAP. In other words, the FAP of the
proposed detection method can be decreased by a more careful selection of $g$.

Figure 5: Sum SE versus the jammer’s power (for $K=10$, $M=200$,
$\Delta=\pi/18$, $g=6$, $N_{d}=20$ and $T=200$)

Fig. 5 depicts how the jammer’s power affects sum-SE of the network for three
different scenarios, i) no jammer in the network, ii) jamming with the
proposed jamming detection and suppression scheme and iii) jamming when no
suppression. In this simulation, we considered $M=200$, $\Delta=\pi/18$, $g=6$
and $N_{d}=20$. We can see that the proposed suppression technique is able to
effectively cancel out the jamming attack and the resulting sum-SE is very
close to the case that there is no jammer in the network. However, it is worth
mentioning that for jamming power below $-5$ dBw, using the proposed
suppression scheme is undesirable. However, in this case, the effect of the
jamming on the system’s performance is negligible.

Figure 6: Sum SE versus the number of BS’s antenna $M$ (for $q_{t}=q_{d}=0$
dB, $K=10$, $\Delta=\pi/18$, $g=6$, $N_{d}=20$ and $T=200$)

Fig. 6 illustrates how sum-SE changes with the number of array antennas. In
this figure we also consider three aforementioned scenarios and set the
parameters as $q_{t}=q_{d}=0$ dB, $g=6$, $N_{d}=20$ and $\Delta=\pi/18$. As
expected, the sum-SE of the system increases with a higher number of array
antennas. It can also be deducted from this figure that the proposed strategy
can successfully reject the jammer’s attack and be close to the case that
there is no jamming in the network. The insignificant loss of SE is the result
of excluding the common RPs between the jammer and the users.

##  VI Conclusion

In this paper, we investigated a direction (angular)-based strategy for
detecting and mitigating jamming attacks on the uplink transmission of mmWave
massive MIMO networks. The proposed scheme first utilized the received signals
in the training phase to estimate the active RPs (i.e. directions) from which
a pilot signal is received. Then, by using this information, the presence of
the jammer and its direction were detected. Finally, using the information
obtained in the previous step, a suppression technique was proposed which was
able to countermeasure the jammer’s attacks in the data transmission phase.
Our numerical analysis showed the effectiveness of the proposed strategy
against the jamming attacks under different scenarios.

##  VII Appendix

###  VII-A _Proof of_ Remark 1

There are $\binom{K}{g}$ combinations of $g$ user from $K$ users. We denote
the set of chosen users by $\mathcal{M}_{i},i=1,...,\binom{K}{g}$. Therefore,
we can define

$\displaystyle P_{g}^{K}=$ $\displaystyle
Prob(\bigcup_{i=1}^{\binom{K}{g}}(\bigcap_{j\in\mathcal{M}_{i}}\Omega_{j}\neq\emptyset))$
$\displaystyle\stackrel{{\scriptstyle
a}}{{\leq}}\min(1,\sum_{i=1}^{\binom{K}{g}}Prob(\bigcap_{j\in\mathcal{M}_{i}}\Omega_{j}\neq\emptyset))$
$\displaystyle\stackrel{{\scriptstyle
b}}{{=}}\min(1,\binom{K}{g}Prob(\bigcap_{j\in\mathcal{M}_{1}}\Omega_{j}\neq\emptyset)),$
(34)

where $\mathcal{M}_{1}=\\{1,\ldots,g\\}$ and ($a$) is derived from Frechet
inequality for union of events and ($b$) holds from the fact that
$Prob(\bigcap_{j\in\mathcal{M}_{i}}\Omega_{j})$ is constant for different
combinations of users since no distinctive characteristics are considered for
different users. Now, we investigate the probability
$Prob(\bigcap_{j=1}^{g}\Omega_{j})$. First, we define the $\mathcal{I}_{int}$
as intersection of $\mathcal{I}_{j},j=1,\ldots,g$. Now, we can write

$Prob(\bigcap_{j=1}^{g}\Omega_{j}\neq\emptyset)=P_{\Psi}Prob(\mathcal{I}_{int}\neq\emptyset)\stackrel{{\scriptstyle
c}}{{\leq}}Prob(\mathcal{I}_{int}\neq\emptyset),$ (35)

where $P_{\Psi}$ is the probability that there is at least one RP in
$\mathcal{I}_{int}$ and ($c$) holds since $P_{\Psi}\leq 1$. In order to
$\mathcal{I}_{int}$ to be non-empty, all $\mathcal{I}_{k}$ should overlap with
each other, which would be achieved if for any $i,j\in\mathcal{M}_{1},i<j$,
the distance between $\bar{\theta}_{i}$ and $\bar{\theta}_{j}$ is less than
$\Delta$. We define these events as

$E_{i,j}\triangleq(|\bar{\theta}_{i}-\bar{\theta}_{j}|\leq\Delta)\equiv(\mathcal{I}_{i}\cap\mathcal{I}_{j}\neq\emptyset)$
(36)

and $Prob(E_{i,j})=1-(\frac{\pi-2\Delta}{\pi-\Delta})^{2}$ since
$\bar{\theta}_{i},i=1,\ldots,K$ are independent uniformly distributed in
interval $[\frac{\pi}{2}-\frac{\Delta}{2},-\frac{\pi}{2}+\frac{\Delta}{2}]$.
Now, we can write

$Prob(\mathcal{I}_{int}\neq\emptyset)=Prob(\bigcap_{\stackrel{{\scriptstyle
i,j=1}}{{i<j}}}^{g}E_{i,j}).$ (37)

Although these events are not independent, but they can be divided into
independent events, e.g. $E_{i,j}\perp\\!\\!\\!\perp
E_{i,k},\;i,j,k=1,\ldots,g;\;j\neq k;\;i<j,k$. We define new events as
intersection of independent events

$E_{i}=\bigcap_{j=i+1}^{g}E_{i,j},\;i=1,\ldots,g-1.$ (38)

These new events are not independent but their probability can be attained by
$Prob(E_{i})=(1-(\frac{\pi-2\Delta}{\pi-\Delta})^{2})^{g-i}$. By substituting
(38) to (37), we can obtain

$Prob(\bigcap_{\stackrel{{\scriptstyle
i,j=1}}{{i<j}}}^{g}E_{i,j})=Prob(\bigcap_{i=1}^{g-1}E_{i})\stackrel{{\scriptstyle
d}}{{\leq}}\min_{i}(Prob(E_{i})),$ (39)

where ($d$) is derived according to Frechet inequality for intersection of
events and
$\min_{i}(Prob(E_{i}))=Prob(E_{1})=(1-(\frac{\pi-2\Delta}{\pi-\Delta})^{2})^{g-1}$
since it has the largest exponent. Thus, according to (VII-A-39), the upper
bound of (20) is derived.

###  VII-B The derivation of SINR terms using MRC decoder

When MRC is utilized, the terms of achievable rate mentioned in (13) can be
evaluated as follows.

  * •

The desired signal
$|\mathbb{E}\\{(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{k}^{n}|\Psi\\}|^{2}$:
By substituting (5) and (27) in
$\mathbb{E}\\{(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{k}^{n}|\Psi\\}$, we
have

$\displaystyle\mathbb{E}\\{(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{k}^{n}|\Psi\\}$
$\displaystyle=\mathbb{E}\\{(\sqrt{\mu_{k}}\sum_{i\in\hat{\Omega}_{k}}\hat{\tilde{g}}_{k,i}^{n}\mathbf{a}(\phi_{i}))^{H}\sqrt{\mu_{k}}\sum_{i\in\Omega_{k}}\tilde{g}_{k,i}^{n}\mathbf{a}(\phi_{i})\\}$
$\displaystyle\stackrel{{\scriptstyle
e}}{{=}}\mu_{k}\mathbb{E}\\{\sum_{i\in\hat{\Omega}_{k}}{\hat{\tilde{g}}_{k,i}^{n}}^{*}\tilde{g}_{k,i}^{n}\\},$
(40)

where ($e$) holds since $a(\phi_{i})^{H}a(\phi_{j})=0$ for $i\neq j$ and
$\hat{\Omega}_{k}\subseteq\Omega_{k}$. We know
$e_{k,i}^{n}=\hat{\tilde{g}}_{k,i}^{n}-\tilde{g}_{k,i}^{n}$, and hence, we can
rewrite the (40) as

$\displaystyle\mathbb{E}\\{(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{k}^{n}|\Psi\\}$
$\displaystyle=\mu_{k}\mathbb{E}\\{\sum_{i\in\hat{\Omega}_{k}}|\hat{\tilde{g}}_{k,i}^{n}|^{2}\\}-\mu_{k}\mathbb{E}\\{\sum_{i\in\hat{\Omega}_{k}}{\hat{\tilde{g}}_{k,i}^{n}}^{*}e_{k,i}^{n}\\}$
$\displaystyle\stackrel{{\scriptstyle
f}}{{=}}\mu_{k}\mathbb{E}\\{\sum_{i\in\hat{\Omega}_{k}}|\hat{\tilde{g}}_{k,i}^{n}|^{2}\\},$
(41)

where ($f$) holds since estimated RP gain $\hat{\tilde{g}}_{k,i}^{n}$ is
uncorrelated with the estimation error $e_{k,i}^{n}$ in LMMSE [28], i.e.
$\mathbb{E}\\{{\hat{\tilde{g}}_{k,i}^{n}}^{*}e_{k,i}^{n}\\}=0$. The estimated
gain $\hat{\tilde{g}}_{k,i}^{n}$ is a Gaussian variable with a known variance
$\xi_{k}$, thus
$\sum_{i\in\hat{\Omega}_{k}}|\hat{\tilde{g}}_{k,i}^{n}|^{2}\thicksim
Gamma(C_{k/w},\xi_{k})$. Therefore

$\mathbb{E}\\{\sum_{i\in\hat{\Omega}_{k}}|\hat{\tilde{g}}_{k,i}^{n}|^{2}\\}=C_{k/w}\xi_{k},$
(42)

Considering (41) and (42), we can conclude
$\mathbb{E}\\{(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{k}^{n}|\Psi\\}=\mu_{k}C_{k/w}\xi_{k}$.
Therefore,

$|\mathbb{E}\\{(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{k}^{n}|\Psi\\}|^{2}=\mu_{k}^{2}C_{k/w}^{2}\xi_{k}^{2}.$
(43)

  * •

The signal gain uncertainty
$\mathtt{var}\\{(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{k}^{n}\\}$: As
mentioned above
$\sum_{i\in\hat{\Omega}_{k}}|\hat{\tilde{g}}_{k,i}^{n}|^{2}\thicksim
Gamma(C_{k/w},\xi_{k})$, hence

$\mathtt{var}\\{(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{k}^{n}\\}=\mu_{k}^{2}\mathtt{var}\\{\sum_{i\in\hat{\Omega}_{k}}|\hat{\tilde{g}}_{k,i}^{n}|^{2}\\}=\mu_{k}^{2}C_{k/w}\xi_{k}^{2}.$
(44)

  * •

Interference exerted by other users
$\mathbb{E}\\{|(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{l}^{n}|^{2}\\}$:

$\mathbb{E}\\{|(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{l}^{n}|^{2}\\}=\mathbb{E}\\{((\mathbf{h}_{l}^{n})^{H}\hat{\mathbf{h}}_{k}^{n})((\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{l}^{n})\\}$
(45)

By substituting
$\hat{\mathbf{h}}_{k}^{n}=\sqrt{\mu_{k}}\sum_{i\in\hat{\Omega}_{k}}\hat{\tilde{g}}_{k,i}^{n}\mathbf{a}(\phi_{i})$
and $\mathbf{h}_{l}^{n}=\\\
\sqrt{\mu_{l}}\sum_{i\in\Omega_{l}}\tilde{g}_{l,i}^{n}\mathbf{a}(\phi_{i})$ in
(45), we obtain

$\displaystyle\mu_{k}\mu_{l}\mathbb{E}\\{(\sum_{i\in\Omega_{k,l}}\tilde{g}_{l,i}^{n^{*}}\hat{\tilde{g}}_{k,i}^{n})(\sum_{i\in\Omega_{k,l}}\hat{\tilde{g}}_{k,i}^{n^{*}}\tilde{g}_{l,i}^{n})\\}$
$\displaystyle\stackrel{{\scriptstyle
g}}{{=}}\mu_{k}\mu_{l}\sum_{i\in\Omega_{k,l}}\mathbb{E}\\{|\hat{\tilde{g}}_{k,i}^{n}|^{2}|\tilde{g}_{l,i}^{n}|^{2}\\},$
(46)

where the equality ($g$) holds since
$\mathbb{E}\\{\hat{\tilde{g}}_{k,i}^{n}\hat{\tilde{g}}_{k,j}^{n^{*}}\\}=0,\;i\neq
j$, $\mathbb{E}\\{\tilde{g}_{l,i}^{n}\tilde{g}_{l,j}^{n^{*}}\\}=0,\;i\neq j$
and the fact that $\hat{\tilde{g}}_{k,i}^{n}$ is uncorrelated with
$\tilde{g}_{l,i}^{n}$. Therefore,

$\displaystyle\mathbb{E}\\{|(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{l}^{n}|^{2}\\}$
$\displaystyle=\mu_{k}\mu_{l}\sum_{i\in\Omega_{k,l}}\mathbb{E}\\{|\hat{\tilde{g}}_{k,i}^{n}|^{2}|\\}\mathbb{E}\\{|\tilde{g}_{l,i}^{n}|^{2}\\}$
$\displaystyle=C_{k,l}\xi_{k}\mu_{k}\mu_{l}.$ (47)

  * •

Interference exerted by the jammer
$\mathbb{E}\\{|(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{w}^{n}|^{2}\\}$: As
mentioned in section IV, since the subspace of the jammer is excluded from
estimated channels, they would be approximately orthogonal. Thus

$\mathbb{E}\\{|(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{h}_{w}^{n}|^{2}\\}\simeq
0.$ (48)

  * •

The noise term
$\mathbb{E}\\{|(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{z}_{d}^{n}|^{2}\\}$:

$\mathbb{E}\\{|(\hat{\mathbf{h}}_{k}^{n})^{H}\mathbf{z}_{d}^{n}|^{2}\\}\stackrel{{\scriptstyle
h}}{{=}}\mathbb{E}\\{||\hat{\mathbf{h}}_{k}^{n}||^{2}\\}\mathbb{E}\\{||\mathbf{z}_{d}^{n}||^{2}\\}=\mu_{k}C_{k/w}\xi_{k}\sigma_{z}^{2},$
(49)

where the first equality ($h$) is correct since noise and the estimated
channel are uncorrelated.

Finally, by substituting (43), (44), (• ‣ VII-B), (48) and (49) in (13), we
can obtain (33).

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