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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, MONTH YEAR
1
Disrupting MIMO Communications with Optimal
Jamming Signal Design
Qian Liu, Member, IEEE, Ming Li, Member, IEEE, Xiangwei Kong, Member, IEEE, and
Nan Zhao, Member, IEEE
Abstract—This paper considers the problem of intelligent
jamming attack on a MIMO wireless communication link with a
transmitter, a receiver, and an adversarial jammer, each equipped
with multiple antennas. We present an optimal jamming signal
design which can maximally disrupt the MIMO transmission
when the transceiver adopts an anti-jamming mechanism. In
particular, signal-to-jamming-plus-noise ratio (SJNR) at the receiver is used as the anti-jamming reliability metric of the
legitimate MIMO transmission. The jamming signal design is
developed under the most crucial scenario for the jammer where
the legitimate transceiver adopt jointly designed maximum-SJNR
transmit beamforming and receive filter to suppress/mitigate the
disturbance from the jammer. Under this best anti-jamming
scheme, we aim to optimize the jamming signal to minimize
the receiver’s maximum-SJNR under a given jamming power
budget. The optimal jamming signal designs are developed in
different cases with accordance to the availability of channel state
information (CSI) at the jammer. The analytical approximations
of the jamming performance in terms of average maximumSJNR are also provided. Extensive simulation studies confirm our
analytical predictions and illustrate the efficiency of the designed
optimal jamming signal on disrupting MIMO communications.
Index Terms—Artificial interference, beamforming, jamming,
multiple-input multiple-output (MIMO), power allocation, signalto-jamming-plus-noise ratio (SJNR).
I. I NTRODUCTION
HE past decade witnesses the rapid worldwide deployment of wireless multiple-input multiple-output (MIMO)
systems. Consequently, there is an urgent request of MIMO
jamming technology because of its vital usage in military
networks for adversary defense and proactive attack to hostile
forces. The optimization designs of jamming signals and
strategies to maximally impair legitimate MIMO communication links have been becoming an important research topic
and received significant attention in recent years.
The goal of a jammer is to intentionally disturb legitimate
wireless transmissions by degrading the receiving performance
T
Manuscript received October 16, 2014; revised March 27, 2015; accepted
May 27, 2013. This work was supported in part by the Fundamental Research
Funds for the Central Universities (Grant No. DUT14RC(3)103 and Grant No.
DUT14QY44) and the National Natural Science Foundation of China (NSFC)
under Grant 61201224, and the Foundation for Innovative Research Groups
of the NSFC (Grant No. 71421001). The associate editor coordinating the
review of this paper and approving it for publication was R. Zhang. The
corresponding author is Nan Zhao
Qian Liu is with the Department of Computer Science and Engineering,
State University of New York at Buffalo, Buffalo, NY, 14260, USA (E-mail:
[email protected]).
Ming Li, Xiangwei Kong, and Nan Zhao are with the School of Information
and Communication Engineering, Dalian University of Technology, Dalian,
Liaoning, 116024, P. R. China (E-mail: {mli,kongxw,zhaonan}@dlut.edu.cn).
Digital Object Identifier XXXXX
in order to trigger reliability outage and service disruption [1].
The jammer’s ability of disrupting the legitimate communications can be substantially enhanced if certain priori knowledge
(for example transmit signal, channel state information (CSI),
etc.) is available at the jammer side. When a jammer can fully
or partially acquire the transmit signal (by eavesdropping for
example), it can efficiently disrupt the legitimate communication by deliberately generating jamming signal correlated
to the transmit signal. Such “correlated jamming” approaches
have been investigated under various assumptions [2]-[9].
In [2], [3], a so-called Gaussian test channel was considered
with the jamming signals correlated with the output of the
encoder [2] and the input to the encoder [3]. A more general
class of jammers was studied in [4] where the jammer chooses
the transition probability from a set of allowed channels
to minimize the capacity. Another popular approach is to
model the legitimate transmitter and the adversary jammer
as players in a game-theoretic formulation to identify the
optimal transmit strategies for both parties. In [5], the best
transmitter/jammer strategies were found for an additive white
Gaussian noise (AWGN) channel with one user and one
jammer who participate in a zero-sum mutual information
game. In [6], Kashyap et al. pursued related strategies for a
single-user MIMO fading channel where the jammer has full
knowledge of the source signal. The studies of mutual information games with correlated jamming were then extended to
relay channels [7], multiple access channels [8], and multiuser channels [9]. In [10], Diggavi and Cover showed that for
high signal power, the maximum entropy noise is the worst
additive noise with a banded covariance constraint.
On the other hand, in [11]-[14] several “uncorrelated jamming” designs are developed where the jammer attempts to
launch more effective jamming attacks by disrupting the channel estimation procedure (period) of MIMO communications.
However, such approaches require protocol knowledge and accurate system synchronization to target channel estimation and
CSI feedback procedure. In [15]-[17], uncorrelated jamming
on MIMO channels is investigated in information-theoretic
context which aims to minimize the capacity (or mutual
informatin) of the legitimate MIMO channels by injecting
a spatially-correlated Gaussian interference signal. In these
studies, full channel state information (CSI) of both legitimate
and jamming channels is assumed, but the jammer has no
knowledge of the legitimate users’ signals. In [18], instead
of capacity, Jorswieck et al. used the expectation of the trace
of some arbitrary matrix-monotone function as the average
performance metric for the jamming design problems. Two
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, MONTH YEAR
common performance metrics (mutual information and meansquare-error (MSE)) can be obtained by selecting different
matrix-monotone functions. The optimal jamming strategies
were proposed for the cases of no CSI, statistical knowledge,
and perfect CSI.
In addition to the usage in electronic warfare area, jamming
signal can also be utilized to improve the physical layer
security of wireless transmission in presence of eavesdroppers
[19]-[31]. In these works, the legitimate users transmit secret
messages along with the sophisticatedly designed jamming
signal (artificial interference) which can disturb the eavesdropper’s reception but has no harm to the legitimate receivers.
In this paper, we study that how an intelligent adversary
jammer can maximally disrupt MIMO communications by
generating optimal jamming signal. We consider a MIMO
communication link in the presence of an adversary jammer
equipped with multiple antennas. We are interested in an
uncorrelated jamming problem in which the jammer does
not have any knowledge of signals transmitted by the legitimate user. But the jammer may have full or partial CSI
about jamming channel and legitimate channel. Under such
assumption, we seek an optimal jamming approach for the
adversary jammer to maximally disrupt the legitimate MIMO
transmission.
Previous uncorrelated jamming works [15]-[18] utilize
information-theoretic metric. However, the channel capacity
(or mutual information) can only provide an upper bound of
the achievable throughput of the legitimate transmission link.
With the attack of a malicious jammer, we need to evaluate
the maximum reliable transmission rate of the legitimate link
equipped with a practical anti-jamming mechanism. Furthermore, in order to evaluate the channel capacity, we need the
full CSI knowledge of both legitimate and jamming channels,
which may not be always available in practice. Therefore,
another practical reliability metric needs to be considered for
the jamming signal design.
In this paper, we consider a more practical signal processing
context and use legitimate receiver’s signal-to-jamming-plusnoise ratio (SJNR) as our metric of the reliability of the
legitimate MIMO transmission. Particularly, we explore the
jammer design under the most crucial scenario for the jammer
where the legitimate transmitter and receiver adopt jointly
designed maximum-SJNR transmit beamforming and receiver
filter to suppress/mitigate the disturbance from the jammer.
With such best anti-jamming scheme employed by the legitimate transceiver, we aim to optimize the jamming signal for
the jammer to minimize the legitimate receiver’s maximumSJNR under a given jamming power budget constraint. The
optimal jamming signal designs are developed in different
cases with accordance to the availability of CSI for the jammer.
Comparing to the information-theoretic metric [15]-[18],
maximum-SJNR is a practical measure to evaluate the legitimate transmission link under a jamming attack. It can reflect
the achievable quality of service (QoS) of the legitimate transmission link with practical transmit and receive implementation (e.g. beamforming, simple coding, etc.). Therefore, using
SJNR as the jamming design metric, the performance of the
proposed jammer is practically achievable in wireless MIMO
networks. It is also worth pointing out that in [18] the objective
metric can be set as MSE or sum-SJNR by selecting an
appropriate matrix-monotone function. However, these metrics
can only reflect the average performance of the legitimate link,
while maximum-SJNR indicates the best achievable QoS at
the legitimate receiver. To evaluate and optimize the jamming
attack, we are more interested in how to maximally degrade
the best legitimate QoS. Therefore, minimizing the maximumSJNR is more meaningful in practice. In addition, unlike [11][14], the proposed jamming scheme aims at disrupting the
data transmission procedure of MIMO communications rather
than the channel estimation period. Since the data transmission
procedure is the dominant component of the entire communications, delicate synchronization is not a necessity. As a result,
the proposed jamming approach is easier to implement than
exiting approaches [11]-[14].
The rest of the paper is organized as follows. The jamming signal optimization problem is formulated in Section
II. Optimal jamming signal designs are developed in Section
III under various assumptions of CSI availability. In Section
IV, simulation results illustrate the jamming performance of
our developments and, finally, a few conclusions are drawn in
Section V.
The following notation is used throughout the paper. Boldface lower-case letters indicate column vectors and boldface
upper-case letters indicate matrices; C is the set of all complex
numbers; (·)H denotes transpose-conjugate operation; IL is
the L × L identity matrix; and E{·} represents statistical
expectation. X ≽ 0 states that X is positive semidefinite;
Tr{X} is the trace of X; X† denotes the Moore-Penrose
pseudo inverse of X.
II. S YSTEM M ODEL
We consider a wireless communication system from a legitimate transmitter to a legitimate receiver in the presence of an
adversarial jammer who attempts to disrupt the transmission.
For convenience, we follow the common language in the field
and name the transmitter, receiver, and jammer, Alice, Bob,
and Jeff, respectively. A simple diagram is shown in Fig. 1 to
illustrate this basic communication scenario.
Alice, Bob, and Jeff are equipped with Na , Nb , and Nj
antennas, respectively. The channel between Alice and Bob
is assumed to be flat Rayleigh fading and is represented by
Hab ∈ CNb ×Na where Hab (i, j) is the channel coefficient
from the jth transmit antenna at Alice to the ith receive
antenna at Bob. Similarly, let Hjb ∈ CNb ×Nj be the flat
Rayleigh fading channel matrix between Jeff and Bob.
Alice transmits a symbol multiplied by the corresponding
transmit beamforming vector (also referred to as spatial signature or linear precoder in the literature). The transmitted signal
has a form of
√
(1)
x = Pa s t
where Pa > 0 denotes Alice’s transmit power, s ∈ C, E{s} =
0, E{|s|2 } = 1, denotes the unit energy information symbol,
and t ∈ CNa , ∥t∥ = 1, denotes the beamforming vector,
respectively.
LI et al.: DISRUPTING MIMO COMMUNICATIONS WITH OPTIMAL JAMMING SIGNAL DESIGN
3
where λab,1 is the largest eigenvalue of matrix Hab HH
ab and
2
λab,1 = σab,1
. The statistics of λab,1 of a Wishart matrix
Hab HH
ab has been intensively investigated [32], [33]. We also
define the average maximum-SNR as
SNRmax , EHab {SNRmax } =
Pa λab,1
σn2
(8)
where λab,1 , EHab {λab,1 }. In the rest of this paper, we will
use SNRmax and SNRmax as the performance benchmark.
B. Jamming Signal Present
Fig. 1.
Now we consider the case in which the jammer attempts to
disturb Alice’s signal by emitting jamming signal z ∈ CNj via
Nj antennas. Let Rz , E{zzH } be the autocorrelation matrix
of jamming signal z and the power of the emitted jamming
signal is Pj = Tr{Rz }.
When the jamming signal is present, the signal vector y ∈
CNb received by Bob can be expressed as
√
y = Pa sHab t + Hjb z + n.
(9)
MIMO channel in presence of an adversary jammer.
A. Jamming Signal Absent
We first investigate the case in which no jamming is
presented and consider the performance in such a case as a
benchmark. Without any jamming signal from jammer, the
signal vector y ∈ CNb received by Bob can be expressed as
√
y = Pa sHab t + n
(2)
where n ∼ CN (0, σn2 I) represents circularly symmetric complex AWGN vector with power σn2 . Bob utilizes a normalized
receive beamforming vector (filter) w ∈ CNb , ∥w∥ = 1, to
retrieve information symbol s. The signal after filtering is
√
sb = wH y = Pa swH Hab t + wH n.
(3)
The two terms in (3) are desired signal and noise, respectively.
The output signal-to-noise ratio (SNR) of filtering is
{√
}
E | Pa swH Hab t|2
SNR =
E {|wH n|2 }
Pa |wH Hab t|2
=
.
(4)
σn2
Let Hab ,
be the singular value decomposition (SVD) of Hab where Uab ∈ CNb ×Nb and Vab ∈ CNa ×Na
are two unitary matrices, Σab is an Nb × Na diagonal matrix
with nonnegative real numbers σab,1 ≥ . . . ≥ σab,m on
the diagonal, m = min{Na , Nb }. In order to maximize the
output SNR in (4), Alice and Bob adopt the optimal transmit
beamformer
tmaxSNR = Vab (1)
(5)
H
Uab Σab Vab
and receive filter
wmaxSNR = Uab (1),
(6)
where Vab (1) and Uab (1) are the first (leftmost) vector of
Vab and Uab corresponding to the largest singular value σab,1 .
With those optimal transmit beamformer (5) and receive filter
(6), the maximum-SNR of Bob is
SNRmax =
2
Pa σab,1
Pa λab,1
=
2
σn
σn2
(7)
The retrieved signal after filtering by w is
√
sb = wH y = Pa swH Hab t + wH Hjb z + wH n.
(10)
The three terms in (10) are desired signal, interference from
jammer, and noise, respectively. The output SJNR of filtering
is
{√
}
E | Pa swH Hab t|2
SJNR =
E {|wH Hjb z + wH n|2 }
Pa wH Hab ttH HH
ab w
.
=
(11)
H
H
2
w (Hjb Rz Hjb + σn INb )w
In order to mitigate/suppress the disturbance from jammer
Alice and Bob take the best anti-jamming effort and adopt joint
transmit beamformer and receiver filter design to maximize
the output SJNR. It is well known that, for any given transmit
beamformer t, the maximum-SJNR filter wmaxSJNR ∈ CNb
can be obtained by [34]
2
−1
wmaxSJNR = c (Hjb Rz HH
Hab t
jb + σn INb )
(12)
where a scale c > 0 guarantees the normalization of the filter
[34]. We need to point out that Hjb and Rz may be not
available for the legitimate users. But the disturbance autocor2
relation matrix Hjb Rz HH
jb + σn INb can be estimated at Bob’s
e = Hjb z + n when
by observing interference and noise only y
signal of interest is absent and then averaging
the observed
∑N
1
2
e
samples as Hjb Rz HH
+
σ
I
≈
y
(n)e
y(n)H .
N
n
b
jb
n=1
N
With the SJNR-optimum filter wmaxSJN R , the maximized
output SJNR is
H
2
−1
SJNRmax = Pa tH HH
Hab t (13)
ab (Hjb Rz Hjb + σn INb )
which is a direct function of the transmit beamforming vector
t. The SJNR-maximization beamforming vector t, which
maximizes output SJNR in (13), can be obtained by
(
)
H
2
−1
tmaxSJNR = Umax HH
Hab (14)
ab (Hjb Rz Hjb + σn INb )
where Umax (A) denotes the right eigenvector of matrix
A corresponding to the largest eigenvalue λmax (A), i.e.
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, MONTH YEAR
AUmax (A) = λmax (A)Umax (A). The attained maximum-SJNR
by such joint optimal transmit beamformer (12) and receive
filter (14) design is
(
)
H
2
−1
SJNRmax = Pa λmax HH
Hab .
ab (Hjb Rz Hjb + σn INb )
(15)
While the anti-jamming technologies have been intensively
studied for the legitimate transmissions, in this work we stand
at jammer’s side and consider the problem from jammer’s
design perspective. We focus our attention on the problem
of jamming signal design to maximize jammer’s impairment
on the Alice-to-Bob transmission even when the maximumSJNR transmit beamforming and receive filter are adopted by
Alice and Bob. In particular, our technical goal is to adaptively
design the correlation matrix Rz of the jamming signal z
to minimize Bob’s maximum-SJNR under a given jamming
power budget. The optimal jamming signal design will be
developed in the next section in different scenarios according
to the availability of Hjb and Hab for the jammer.
III. O PTIMAL JAMMING S IGNAL D ESIGNS
We first consider the case that the jammer has the knowledge
of both Hjb and Hab . The knowledge of Hab may also be
obtained by the jammer if the locations of Alice and Bob are
known or projected/anticipated. Another scenario, in which
such assumption will be valid, is that the jammer can also
obtain the knowledge of Hab by eavesdropping the feedback
channel in which Alice and Bob share Hab . Regarding to
Hjb , jammer can estimate it by analyzing the observed signal
transmitted by Bob when Bob is also a transmitter for other
transmission tasks. For example, Alice-Bob may be a timedivision duplex (TDD) communication link and Bob will send
data to Alice in other time slots.
Our objective is to design the autocorrelation matrix Rz of
jamming signal z to minimize Bob’s maximum-SJNR with
a jamming power budget constraint Pj . This optimization
problem can be formulated in the following form
=
s. t.
arg
min
Rz ∈CNj ×Nj
where λmax (A, B) is the maximum generalized eigenvalue of
matrices A and B.
To successfully disturb the intended signal, the jamming
signal should be very strong and overwhelm the noise signal.
Therefore, for development convenience, we neglect the noise
component σn2 INb in (20) and approximate the output SJNR
as1
(
)
H
SJNRmax ≈ Pa λmax Hab HH
(21)
ab , Hjb Rz Hjb .
Plugging (21) into our original optimization problem (16)(18) and ignoring the constant power Pa , we can (approximately) re-formulate the objective as
(
)
H
Ropt
= arg min
λmax Hab HH
z
ab , Hjb Rz Hjb (22)
Rz ∈CNj ×Nj
A. Known both Hjb and Hab
Ropt
z
By Theorems 1 and 2, we can re-formulate SJNRmax in
(15) as
(
)
H
2
−1
SJNRmax = Pa λmax HH
Hab
ab (Hjb Rz Hjb + σn INb )
(
)
2
−1
= Pa λmax (Hjb Rz HH
Hab HH
jb + σn INb )
ab
(
)
H
2
= Pa λmax Hab HH
ab , Hjb Rz Hjb + σn INb (20)
SJNRmax
Tr{Rz } ≤ Pj ,
Rz = RH
z , Rz ≽ 0.
(16)
(17)
(18)
Note that constraint (18) assures that the optimized matrix Rz
is an autocorrelation matrix which should be Hermitian and
positive semidefinite. Before developing the jamming signal
design, we first recall following two theorems [35].
Theorem 1: Suppose that A ∈ Cm×n and B ∈ Cn×m .
Then AB has the same non-zero eigenvalues as BA.
Theorem 2: Suppose that A ∈ Cm×m and B ∈ Cm×m , B
is invertible. Let λi and qi , i = 1, . . . , m, be the generalized
eigenvalues and corresponding eigenvectors of matrices A and
B, respectively. Then, we have
Aqi = λi Bqi ⇒ B−1 Aqi = λi qi .
(19)
−1
Therefore, the eigenvalues and eigenvectors of matrix B A
are the same as the generalized eigenvalues and eigenvectors
of matrices A and B.
s. t. Tr{Rz } ≤ Pj ,
Rz =
RH
z , Rz
(23)
≽ 0.
(24)
Note that the true SJNRmax in (20) contains the noise component and is always less than the noise-excluded approximated
form in (21) with σn > 0. Then, the approximated optimization problem (22)-(24) is essentially aiming to minimize the
upper bound of SJNRmax .
To solve the optimization problem in (22)-(24), we first
consider the case Nj ≥ Nb and transform it into an equivalent
form as described in Proposition 1 whose proof is offered in
Appendix A.
Proposition 1: When Nj ≥ Nb , the optimization problem
in (22)-(24) is equivalent to
(
)
† H
Ropt
= arg min
λmax H†jb Hab HH
z
ab (Hjb ) , Rz (25)
Rz ∈CNj ×Nj
s. t. Tr{Rz } ≤ Pj ,
Rz = RH
z , Rz ≽ 0.
(26)
(27)
The autocorrelation matrix Rz can be constructed by eigendecomposition
Rz := Uz Λz UH
(28)
z
where Uz ∈ CNj ×Nj is a unitary matrix and Λz =
diag{λz,1 , . . . , λz,Nj } ≽ 0. After obtaining the equivalent
form of the optimization problem, the optimal correlation
matrix Rz of jamming signal can be designed by following
proposition whose proof is offered in Appendix B.
Proposition 2: Consider the optimization problem in (25)(27). Let q1 , q2 , . . . , qNj be the eigenvectors of matrix Rh ,
† H
H†jb Hab HH
with corresponding positive eigenvalues
ab (Hjb )
λh,1 ≥ λh,2 ≥ . . . ≥ λh,Nj . When Nj ≥ Nb , the optimal
1 The noise component will be considered and accounted for SJNR evaluation in our simulation studies.
LI et al.: DISRUPTING MIMO COMMUNICATIONS WITH OPTIMAL JAMMING SIGNAL DESIGN
correlation matrix Rz of jamming signal that minimizes Bob’s
SJNRmax can be constructed by
opt opt
Ropt
:= Uopt
z
z Λz Uz
H
(29)
and the optimal Uopt
and Λopt
are
z
z
Uopt
z
=
[q1 , . . . , qNj ],
(30)
Λopt
z
=
opt
diag{λopt
z,1 , . . . , λz,Nj },
(31)
where λopt
z,i
=
Pj λh,i
, i = 1, . . . , Nj .
∑Nj
d=1 λh,d
(32)
5
the degrees-of-freedom of the legitimate transmission from
Nb to Nb − Nj , the reliability of the legitimate transmission
can still be degraded by the jamming signal to a certain
level. Therefore, in the case of Nj < Nb , we still suggest
to use the jamming signal design in Proposition 2 which can
evenly disturb the jamming subspace and can force Alice and
Bob to use the non-jamming subspace only. The jamming
e ab ). These findings are
performance SJNRmax ≈ SNRmax (H
also available for unknown Hab (and unknown Hjb ) cases as
long as Nj < Nb and would not be further discussed in the
next two subsections.
With Ropt
optimally designed in (29), the maximum generz
∑Nj
λ
d=1 h,d
alized eigenvalue λmax (Rh , Ropt
z ) is minimized at
Pj
and the maximum output SJNR of Bob is minimized at
∑Nj
Pa d=1
λh,d
SJNRmax =
.
(33)
Pj
The closed-form expression of the average maximum-SJNR,
i.e. SJNRmax , EHab ,Hjb {SJNRmax }, is very difficult to be
derived for all possible Pj , Nj , Na , and Nb . Therefore, we
turn to find out a practical approximated expression which is
summarized in Proposition 3. The derivation of Proposition
3 can be found in Appendix C and the accuracy of the
approximation expression is verified by simulation studies
shown in Section IV.
Proposition 3: Consider jamming with known Hjb and
Hab , Nj > Nb . When the optimal jamming signal design
in Proposition 2 is used, the jamming performance in terms
of SJNRmax can be approximated as
SJNRmax ≈
Pa Na Nb
.
Pj (Nj − Nb )
(34)
The above results illustrate that SJNRmax is inversely
proportional to Nj and the larger Nj can enhance the jamming
performance. On the other side, SJNRmax is directly proportional to Na and Nb . The larger Na and/or Nb can mitigate
the disturbance due to jamming signal. These findings also
coincide with our intuition.
While the jamming design for the case Nj ≥ Nb is developed and summarized in Propositions 1-3, now we attempt to
investigate the jamming design when Nj < Nb . Let Hjb ,
H
Ujb Σjb Vjb
be the SVD of Hjb . When Nj < Nb , we can
e jb , 0N ×(N −N ) ]T , Σ
e jb is a Nj × Nj diagonal
form Σjb , [Σ
j
j
b
e jb , Ujb ], U
e jb ∈ CNb ×Nj is the jamming
matrix, and Ujb , [U
Nb ×(Nb −Nj )
subspace, Ujb ∈ C
is the non-jamming subspace.
If Bob uses a jamming zero forcing filter w ∈ span(Ujb ),
then the jamming signal can be totally suppressed. When
jamming power is sufficient and jamming signal is sophisticatedly designed to evenly disturb all basis of the jamming
subspace, such jamming zero forcing filter is the best choice
for Alice and Bob. Then, in such case, the transmission over a
jamming channel is statistically equivalent to the transmission
e ab ∈ C(Nb −Nj )×Na with a
over a jamming free channel H
maximum-SNR transmitter and receiver designs as discussed
in Section II-A. Since this effort by the jammer can reduce
B. Known Hjb but unknown Hab
Now we consider the scenario that the jammer has only
the knowledge of the channel Hjb between himself and Bob,
but the channel Hab between Alice and Bob is unknown.
The jamming signal design solution developed in the previous
section cannot be adopted due to the lack of access to Bob’s
SJNR. Therefore, we turn to evaluate the power of post-filtered
jamming signal which can be expressed as
σj2
H
H H
= E{wmaxSJN
R Hjb zz Hjb wmaxSJN R }
H
H
= wmaxSJN
R Hjb Rz Hjb wmaxSJN R
(35)
where Hjb Rz HH
jb is the autocorrelation matrix of Bob’s
received jamming signal Hjb z.
Let n = min{Nj , Nb } denote the rank of the received
jamming signal autocorrelation matrix Hjb Rz HH
jb and let ui ,
i = 1, . . . , n, denote orthnormal spatial basis of the received
jamming signal Hjb z with corresponding power αi . Then,
the autocorrelation matrix Hjb Rz HH
jb can be expressed in a
∑n
H
α
u
u
form of Hjb Rz HH
=
jb
i=1 i i i . To suppress the postfiltered jamming signal power σj2 , the maximum-SJNR receive
filter wmaxSJN R is adaptively designed to avoid the spatial
subspace ui with large αi . However, the filter wmaxSJN R
calculated by (12) is unkown by the jammer when Hab is
not available. Therefore, by common intuition, the best effort
by the jammer is to isotropically/equally maximize the power
of each spatial basis ui of Bob’s received jamming signal in
such a way to statistically maximize the post-filtered jamming
signal power. Specifically, we aim to design Rz to maximize
the received jamming powers α1 , . . . , αn among different
spatial basis subject to α1 = . . . = αn . The optimal design
problem can be formulated as follows
Ropt
z
=
s. t.
arg
max
Rz ∈CNj ×Nj
Hjb Rz HH
jb
α
=α
(36)
n
∑
ui uH
i ,
(37)
i=1
Tr{Rz } ≤ Pj ,
Rz =
RH
z , Rz
(38)
≽ 0.
(39)
The optimal design of Rz for the problem (36)-(39) is based
on the waterfilling principle and is described in Proposition 4.
The proof is provided in Appendix D.
Proposition 4: Consider the optimization problem (36)H
(39). Let Hjb , Ujb Σjb Vjb
be the SVD of Hjb . Let
Nj ×n
e
Vjb ∈ C
, n = min{Nj , Nb }, be a matrix containing the
6
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, MONTH YEAR
n most-left vectors in Vjb . The optimal correlation matrix Rz
of jamming signal to the optimization problem (36)-(39) can
opt opt H
be constructed by Ropt
= Uopt
and the optimal
z
z Λz Uz
Uopt
and Λopt
are
z
z
Uopt
z
Λopt
z
where λopt
z,i
e jb ,
= V
opt
= diag{λopt
z,1 , . . . , λz,n },
=
Pj
∑n
(40)
(41)
1
2
σjb,i
1
2
p=1 σjb,p
, i = 1, . . . , n.
(42)
With∑
the optimal Ropt
z , α is maximized to a value αmax =
n
Pj /( p=1 σ21 ).
jb,p
When Nj ≥ Nb , constraint (37) can be simplified as
Hjb Rz HH
jb = αINb . In such case, the optimal jamming signal
received by Bob behaves like AWGN and the power of filtered
jamming signal
∑n can be always maintained at a maximum level
σJ2 = Pj /( p=1 σ21 ) with any receive filter. Therefore, with
jb,p
a transmit beamformer t and receive filter w, the output SJNR
of Bob can be expressed as
SJNR =
Pa |wH Hab t|2
Pa |wH Hab t|2
∑n
=
.
1
σJ2 + σn2
Pj /( p=1 σ2 ) + σn2
(43)
jb,p
While the power of filtered jamming signal is independent
of transmit beamformer t and receive filter w, in an effort to
maximize the SJNR, Alice and Bob adopt the SNR-optimum
transmit beamformer tmaxSN R = Vab (1) and receive filter
wmaxSJR = Uab (1) (See Section II-A). The jammingdegraded maximum-SJNR of Bob is
SJNRmax =
Pa λab,1
.
σJ2 + σn2
(44)
With the maximum-SINR result in (44), the average jamming performance SJNRmax is evaluated and shown in Proposition 5 whose proof is provided in Appendix E.
Proposition 5: Consider jamming case with known Hjb ,
but unknown Hab , Nj > Nb . When the optimal jamming
design in Propostion 4 is used, then the jamming performance
SJNRmax has an expression as
SJNRmax =
Pa λab,1
Nj −Nb
Pj Nb +
σn2
.
(45)
C. Unknown Hjb and unknown Hab
If both Hjb is unknown, Bob’s received jamming signal
Hjb z cannot be predicted and amended by appropriately
adjusting the jammer’s emitted signal z. Therefore, no matter
Hab is known or not, the best strategy for jammer is to
inject individual jamming signal via each antenna with equal
power. The autocorrelation matrix Rz of the jamming signal
generated by such approach is
Pj
IN .
Rz =
Nj j
(46)
The jamming performance with this jamming design is evaluated in the following Proposition whose detailed derivation is
provided in Appendix F.
Proposition 6: Consider jamming case with unknown Hjb ,
unknown Hab , and Nj > Nb . When the optimal jamming
design in (46) is used and the jamming power Pj is sufficient,
then the jamming performance SJNRmax has an approximated
expression as
SJNRmax ≈
Pa λab,1 Nj
.
(Pj + σn2 )(Nj − Nb )
(47)
It is also worth emphasizing that this interference injection
strategy will spread the jamming signal over all spatial basis of
Bob’s received signal and can cause more difficulty for Alice
and Bob to avoid/suppress those multiple-spatial interference.
It can provide better jamming performance than using only
single jamming antenna or multiple jamming antennas with
the same jamming signal. In those jamming approaches, Bob’s
received jamming signal has only one spatial subspace and is
easy to be avoided and suppressed. The average maximumSJNR of a MIMO transmission in presence of jamming signal
from single antenna is shown in the following Proposition
whose proof is offered in Appendix G.
Proposition 7: When only single antenna is used at the
jamming side, the average jamming performance has a lower
bound as
(
)
1
Pa λab,1
1−
SJNRmax ≥
(48)
Pj
Nb
(
)
1
= SNRmax 1 −
(49)
.
Nb
Compare the findings in Propositions 3, 5, 6, and 7, we can
conclude that:
i With single-antenna jamming, jamming performance is
independent of Pj and increasing jamming power would
not enhance the jamming performance.
ii With multi-antenna jamming, for all three cases, jamming
performance is strictly proportional to Pj and Nj , and
inversely proportional to Nb (and Na ).
iii Multi-antenna jamming will provide much better jamming performance than single-antenna jamming.
These discussions will be verified by intensive simulation
results in Section IV.
D. Jamming Signal Generation
The optimal jamming signal designs developed in the previous three subsections focused on the second-order statistics
optimization and provided the optimal jamming signal autoopt
correlation matrix Ropt
, the jamming signal z
z . With Rz
can be simply generated by
z=
Nj
∑
zi λz,i uz,i
(50)
i=1
where λz,i and uz,i are eigenvalue and eigenvector of Ropt
z , zi ,
i = 1, . . . , Nj , are independent random variables with zeromean and unit-variance. The jammer can emit the jamming
LI et al.: DISRUPTING MIMO COMMUNICATIONS WITH OPTIMAL JAMMING SIGNAL DESIGN
IV. S IMULATION S TUDIES
In the following, we present extensive simulation studies
that we obtained from the implementation of the optimal jamming signal designs. In all simulations, the channels Hab and
Hjb are modeled as Rayleigh fading with the channel matrix
comprising independent and identically distributed samples of
a complex Gaussian random variable with zero mean and unit
variance. The noise power is fixed at σn2 = 1.
We first set the number of antennas for each node as
Na = 4, Nb = 4, Nj = 6, respectively. Alice’s transmit power
is fixed at Pa = 10dB. The average SJNRmax of Bob over
106 channel realizations is plotted in Fig. 2 as a function of
jammer’s power budget Pj which varies from 10dB to 30dB.
Three multi-antenna jamming strategies are examined under
various assumptions about the knowledge of CSI: i) known
both Hjb and Hab ; ii) known Hjb but unknown Hab ; and
iii) unknown Hjb and unknown Hab . As a reference, average
SJNRmax of Bob in presence of a single-antenna jammer
using the same jamming power budget is also included. In
addition to the simulation results (blue solid curves), the
analytical approximation SJNRmax of each jamming case are
plotted (red dash curves). Finally, as the jamming performance
benchmark, average SNRmax with no jamming signal is also
illustrated in the same figure.
It can be observed from Fig. 2 that single-antenna jamming can just merely degrade Alice-to-Bob transmission and
increasing the jamming power would not further affect the
quality of the legitimate link. Multiple-antenna jamming, all
three cases, can offer significant better jamming performance
than single-antenna jamming and the jamming performance is
proportional to the jamming power. Particularly, the optimal
jamming signal design with known Hjb and Hab can keep
the SJNRmax of Bob at the lowest values and provides best
jamming performance. In addition, the developed analytical
jamming performance approximation SJNRmax can almost
perfectly predict the jamming performance for the known Hjb
cases while the accuracy becomes a little worse but is still
satisfactory for the unknown Hjb and Hab case.
20
Known Hjb, known Hab
Average SINR (dB)
15
Known Hjb, unknown Hab
Unknown Hjb, unknown Hab
10
Single−antenna jamming
No jamming
5
0
−5
Known Hjb, known Hab, theo
Known Hjb, unknown Hab, theo
−10
Unknown Hjb, unknown Hab, theo
Single−antenna jamming, theo
−15
10
15
20
Jamming power Pj (dB)
25
30
Fig. 2. Average SINRmax of Bob versus jamming power Pj (Na = 4,
Nb = 4, Nj = 6, Pa = 10dB).
1
0.9
0.8
Probability of outage
signal z via its antennas to maximally interrupt the MIMO
communication between Alice and Bob.
While SJNR is the second-order statistics metric of the
signal, any type of distribution of jamming signal zi can
be adopted to pursue optimal jamming performance in terms
of SJNR. Gaussian jamming distribution may be the most
common choice since it can minimize the mutual information
of an additive noise/interference channel with Gaussian inputs
[36]. However, the Gaussian distribution is not necessarily
optimal on minimizing the mutual information when the inputs
are not Gaussian. It is shown in [37] that the best jamming
distribution for additive noise channels with binary inputs
is a mixture of two lattice probability mass functions. This
suggests to further impair the reliable transmission rate by
optimizing the jamming signal distribution in conjugation with
the autocorrelation matrix Rz (i.e. the second-order statistics).
The jamming signal distribution optimization is beyond the
scope of this paper and we will consider this problem in future
studies.
7
0.7
0.6
0.5
0.4
Known H , known H
jb
0.3
ab
Known H , unknown H
jb
ab
0.2
Unknown Hjb, unknown Hab
0.1
Single−antenna jamming
0
10
15
20
Jamming power P (dB)
25
30
j
Fig. 3. Probability of outage versus jamming power Pj (Na = 4, Nb = 4,
Nj = 6, Pa = 10dB, γ = 3).
To further illustrate the impact on Bob’s reception due to
the jamming signal, we also investigate the outage probability
of Bob, which is defined as the probability that Bob’s instantaneous SJNRmax is less than a certain outage threshold
γ = 3. The outage probabilities under different jamming
signal designs are plotted in Fig. 3 as a function of jammer’s
power budget Pj . The single-antenna jamming always has zero
outage probability no matter how large the jamming power is.
This means that the single-antenna jamming cannot destroy the
MIMO transmission if the anti-jamming scheme is employed.
Therefore, multi-antenna jamming is always suggested if the
legitimate transmission is over an MIMO channel. Moreover,
the multi-antenna jamming design with known Hjb always
provide higher outage probability of Bob (i.e. better jamming
performance) than unknown Hjb case.
In Figs. 4 and 5, we repeat the same study with a larger
number of jamming antennas Nj = 8. Comparing to Figs. 2
and 3, we notice the jamming performance can be improved
8
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. X, NO. X, MONTH YEAR
20
20
Known Hjb, known Hab
ab
Unknown Hjb, unknown Hab
10
5
0
−5
−10
jb
ab
Known Hjb, unknown Hab
Unknown H
jb
10
Single−antenna jamming
max
Single−antenna jamming
No jamming
Average SJNR
Average SINR (dB)
jb
Known H , known H
15
Known H , unknown H
of Bob (dB)
15
Known Hjb, known Hab, theo
5
0
Known Hjb, unknown Hab, theo
−5
Unknown Hjb, unknown Hab, theo
Single−antenna jamming, theo
−15
10
15
20
Jamming power Pj (dB)
25
Fig. 4. Average SINRmax of Bob versus jamming power Pj (Na = 4,
Nb = 4, Nj = 8, Pa = 10dB).
4
6
8
10
12
14
Number of jamming antennas Nj
16
18
20
Fig. 6. Average SINRmax of Bob versus number of jamming antennas Nj
(Na = 4, Nb = 4, Pa = 10dB, Pj = 16dB).
1
1
0.9
0.9
Known H , known H
0.8
0.8
Known H , unknown H
0.7
0.7
Unknown Hjb, unknown Hab
jb
Probability of outage
Probability of outage
2
30
0.6
0.5
0.4
Known H , known H
jb
0.3
ab
Known Hjb, unknown Hab
0.2
ab
Single−antenna jamming
0.6
0.5
0.4
0.3
0.2
Unknown H , unknown H
jb
0.1
ab
0.1
Single−antenna jamming
0
0
10
ab
jb
15
20
Jamming power P (dB)
25
30
2
4
6
8
10
12
14
Number of jamming antennas N
Fig. 5. Probability of outage versus jamming power Pj (Na = 4, Nb = 4,
Nj = 8, Pa = 10dB, γ = 3).
by using a larger number of jamming antennas.
Then, we turn to evaluate the effect of the number of
jamming antennas on the jamming performance. Jammer’s
power and Alice’s power are fixed at Pj = 16dB and
Pa = 10dB, respectively. The numbers of Alice’s antennas
and Bob’s antennas are Na = 4 and Nb = 4, respectively.
The jamming performances in terms of average SJNRmax and
outage probability are illustrated in Figs. 6 and 7, respectively,
as a function of the number of jamming antennas which varies
from 1 to 20. Clearly, when Nj < Nb , the jamming signal
can be drastically suppressed by Alice and Bob’s optimal
anti-jamming beamforming and filtering. When Nj ≥ Nb ,
the impact of the adversary jammer on the quality of the
MIMO communication link becomes notable and significant.
The disruption due to the jamming signal will be more severe
when the number of jamming antennas increases.
Finally, we investigate the jamming performance with different number of Bob’s antennas. Jammer’s power and Alice’s
16
18
20
j
j
Fig. 7. Probability of outage versus number of jamming antennas Nj (Na =
4, Nb = 4, Pa = 10dB, Pj = 16dB, γ = 3).
power are fixed at Pj = 30dB and Pa = 10dB, respectively.
The numbers of Alice’s antennas and jammer’s antennas are
Na = 4 and Nj = 8, respectively. The jamming performances
in terms of average SJNRmax and outage probability are
illustrated in Figs. 8 and 9, respectively, as a function the
number of Bob’s antennas which varies from 2 to 16. Jammer
with optimal signal design can successfully disrupt the MIMO
communication link as long as Nj ≥ Nb . When Nj < Nb ,
all three multi-antenna jamming strategies have the same
performance. This simulation result verifies our discussion in
the end of Section III-A.
V. C ONCLUSIONS
We investigated the problem of optimal jamming signal
design to intelligently attack a MIMO communication system.
With a given jamming power budget constraint, our goal was
to optimize the jamming signal for the jammer to minimize receiver’s maximum-SJNR. The optimal jamming signal designs
LI et al.: DISRUPTING MIMO COMMUNICATIONS WITH OPTIMAL JAMMING SIGNAL DESIGN
When Nj = Nb , Hjb is an invertible square matrix and
†
H−1
jb = Hjb . Then, the generalized eigen decomposition (51)
can be re-formulated as
25
Average SJNRmax of Bob (dB)
20
† H
H†jb Hab HH
ab (Hjb ) ui = λi Rz ui , i = 1, . . . , Nb ,
15
(52)
where ui , HH
jb qi . The equivalence is proven for the case
Nj = Nb .
Now, we consider the case that Nj > Nb . Let Hjb ,
H
Ujb Σjb Vjb
be the SVD of Hjb . Then, we can rewrite (51)
as
10
5
0
−5
Known Hjb, known Hab
Known Hjb, unknown Hab
−10
Unknown H
jb
−15
Single−antenna jamming
2
4
6
8
10
12
Number of Bob’s antennas Nb
14
16
Fig. 8. Average SINRmax of Bob versus number of Bob’s antennas Nb
(Na = 4, Nj = 8, Pa = 10dB, Pj = 30dB).
1
Known H , known H
jb
ab
Known Hjb, unknown Hab
0.9
Unknown Hjb, unknown Hab
0.8
Probability of outage
9
Single−antenna jamming
0.7
0.6
0.5
H
T
H
Hab HH
ab qi = λi Ujb Σjb Vjb Rz Vjb Σjb Ujb qi ,
(53)
H
H
T
⇒ UH
jb Hab Hab Ujb pi = λi Σjb Vjb Rz Vjb Σjb pi ,
(54)
where pi , UH
= 1, . . . , Nb . Form Σjb ,
jb qi , i
e
e
[Σjb , 0Nb ×(Nj −Nb ) ], Σjb is an Nb × Nb diagonal matrix, and
e jb , Vjb ], V
e jb ∈ CNj ×Nb , Vjb ∈ CNj ×(Nj −Nb ) .
Vjb , [V
Then, we have
H
Σjb Vjb
Rz Vjb ΣTjb
][
]
[
][
e H Rz V
e jb
e jb
V
0Nb ×(Nj −Nb )
Σ
jb
e
= Σjb , 0
.
0
0(Nj −Nb )×(Nj −Nb ) 0(Nj −Nb )×(Nb )
(55)
Note that the component of Rz in subspace Vjb always
results 0 in (55) and will not affect the eigenvalues. Thus,
H
=
the optimal design Ropt
must have property that Vjb Ropt
z
z
0(Nj −Nb )×Nb . After plugging (55) into (54), we obtain
0.4
H
UH
jb Hab Hab Ujb pi =
[
][ eH
e jb
Vjb Rz V
e
λi Σjb , 0
0
0.3
0.2
0.1
][
0
0
e jb
Σ
0
]
pi .
(56)
Left multiply Σ†jb on both sides of (56), we obtain
0
2
4
6
8
10
12
Number of Bob’s antennas N
14
16
b
Fig. 9. Probability of outage versus number of Bob’s antennas Nb (Na = 4,
Nj = 8, Pa = 10dB, Pj = 30dB, γ = 3).
were developed in different cases according to the availability
of CSI for the jammer. The analytical approximations of the
jamming performance in terms of the average maximumSJNR were also provided. Extensive simulation studies illustrated the importance of using multiple jamming antennas and
demonstrated the benefits of the designed jamming signal on
degrading the performances of MIMO transmission in terms
of maximum-SJNR as well as outage probability.
As a natural next step in future work, we will investigate the
case of imperfect CSI of the jamming channel which is more
practical in realistic systems. We plan to study the performance
degradation due to the presence of imperfect CSI and develop
robust jamming signal designs.
A PPENDIX A - P ROOF OF P ROPOSITION 1
Let λi and qi , i = 1, . . . , Nb , be the generalized eigenvalue
H
and eigenvector pairs of matrices Hab HH
ab and Hjb Rz Hjb :
H
Hab HH
ab qi = λi Hjb Rz Hjb qi , i = 1, . . . , Nb .
(51)
H
Σ†jb UH
jb Hab Hab Ujb pi =
[ H
e Rz V
e jb
V
jb
λi
0
⇒
† H
H
Σ†jb UH
jb Hab Hab Ujb (Σjb ) gi
0
0
][
[
= λi
e jb
Σ
0
]
pi
e H Rz V
e jb
V
jb
0
(57)
0
0
]
gi
(58)
H
where gi , ΣTjb pi . Plugging the properties Vjb Rz Vjb = 0,
H
e jb = 0, and V
e H Rz Vjb = 0 for the optimal Rz into
Vjb Rz V
jb
(58), we have
† H
H
Σ†jb UH
jb Hab Hab Ujb (Σjb ) gi =
[
]
e jb V
e H Rz Vjb
e H Rz V
V
jb
jb
λi
gi
(59)
H
e jb VH Rz Vjb
V Rz V
jb
jb
† H
H
H
⇒ Σ†jb UH
jb Hab Hab Ujb (Σjb ) gi = λi Vjb Rz Vjb gi
(60)
† H H
H
⇒ Vjb Σ†jb UH
jb Hab Hab Ujb (Σjb ) Vjb vi = λi Rz vi
(61)
H †
where vi , Vjb gi . Since H†jb = Vjb
Σjb Ujb , we finally
obtain
† H
H†jb Hab HH
(62)
ab (Hjb ) vi = λi Rz vi .
10
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A PPENDIX B -P ROOF OF P ROPOSITION 2
The min-max optimization problem in (25) is equivalent to
H
hs
minimize the maximum value of Rayleigh ratio ssH R
for
Rz s
H
hs
any s ∈ CNj , i.e. minimize max{ ssH R
, s ∈ CNj }. Let
Rz s
A PPENDIX D - P ROOF OF P ROPOSITION 4
∑Nb
When Nj ≥ Nb , α i=1
ui uH
i = αINb . Then, with Hjb ,
H
Ujb Σjb Vjb , the constrain (37) can be further rewritten as
H
H
Ujb Σjb Vjb
Rz Vjb ΣH
jb Ujb = αINb
Rz ∈CNj ×Nj
si ∈ CNj , i = 1, . . . , Nj , be orthonormal vectors and s1 is
H
hs
the vector which has maximum value of ssH R
for a given
Rz s
H
Rz . For convenience, we define αi , si Rh si , βi , sH
i Rz si ,
sH R s
and siH Rhz sii = αβii , i = 1, . . . , Nj . Then, we can obtain αβ11 ≥
i
∑Nj
∑Nj
αNj ∑Nj
α2
i=1 αi =
i=1 λh,i , and
i=1 βi =
β2 ≥ . . . ≥ βNj ,
∑Nj
i=1 λz,i = Tr{Rz }.
If the design in Proposition 2 is not optimal, then there exist
a Rz and corresponding s1 such that
∑Nj
d=1
λh,d
Pj
>
sH
1 Rh s1
,
sH
1 Rz s1
(63)
† H
⇒ Rz = αVjb Σ†jb (ΣH
jb ) Vjb .
(68)
1
†
Note that Σ†jb (ΣH
, . . . , σ2 1 , 0, . . . , 0} and
2
jb ) = Diag{ σjb,1
jb,Nb
then rewrite (68) as
e jb Diag{ 1 , . . . , 1 }V
eH
Rz = α V
(69)
jb
2
2
σjb,1
σjb,N
b
e jb ∈ CNj ×Nb is a matrix containing the Nb most-left
where V
vectors in Vjb . Therefore, the design in Proposition 4 satisfies
this constrain and is also optimal based on the waterfilling
principle. The proof for the case Nj < Nb is similar and
omitted.
A PPENDIX E - P ROOF OF P ROPOSITION 5
∑Nj
⇒
i=1
Pj
λh,i
>
αNj
α1
α2
≥
≥ ... ≥
.
β1
β2
βNj
(64)
∑Nj
∑Nj
To satisfy (64) with i=1
αi = i=1
λh,i , we need to have
∑Nj
β
=
Tr{R
}
>
P
which
is
a
contradiction
with the
i
z
j
i=1
∑Nj
=
λh,d
must be the
constrain = Tr{Rz } ≤ Pj . Therefore,
Pj
optimal result of the optimization problem and the design in
Proposition 2 is optimal.
d=1
With maximum-SJNR (44), the jamming performance can
be expressed as
}
{
Pa λ2ab,1
SJNRmax = EHab ,Hjb
σJ2 + σn2
A PPENDIX C - D ERIVATION OF P ROPOSITION 3
Recall that λab,1 , EHab {λ
{ ab,1 } (see Section
} II-A) and
∑Nb 1
H −1
b
EHjb { p=1 σ2 } = EHjb Tr{(Hjb Hjb ) } = NjN−N
b
jb,p
(by random matrix theory [32]). Then, we apply these results
into (70) and obtain
SJNRmax =
Pa
† H
EHab ,Hjb {Tr{H†jb Hab HH
ab (Hjb ) }}.(65)
Pj
Loosely speaking, if we can consider H†jb as a random
matrix, then H†jb has elements with variance Var(H†jb ) =
†
Nb
Nb
1
(Nj −Nb )Nj Nb = (Nj −Nb )Nj and Var(Hjb Hab ) = (Nj −Nb )Nj .
With these results, we have
{
}
† H
EHab ,Hjb Tr{H†jb Hab HH
ab (Hjb ) }
=
Nb Na
Nb
Na Nj =
.
(Nj − Nb )Nj
(Nj − Nb )
(66)
Applying (66) into (65), we finally have
SJNRmax =
Pa Na Nb
.
Pj (Nj − Nb )
Pa λab,1
Nj −Nb
Pj Nb +
σn2
.
(71)
{
}
H
By random matrix theory [32], E Tr{(Hjb HH
)
}
=
jb
Nb
Nj −Nb .
(70)
jb,p
With the maximum-SJNR in (33), the SJNRmax is
}
{
∑Nj
λh,d
Pa d=1
SJNRmax = EHab ,Hjb
Pj
=
Pa EHab {λ2ab,1 }
.
∑Nb 1
Pj /EHjb { p=1
} + σn2
σ2
A PPENDIX F - D ERIVATION OF P ROPOSITION 6
When jamming channel is Hjb and the jamming signal
P
has Rz = Njj INj , the maximized output SJNR of the filter
wmaxSJNR is
Pj
2
−1
SJNRmax = Pa tH HH
Hjb HH
Hab t. (72)
ab (
jb + σn INb )
Nj
If the jamming power Pj is sufficient large and Pj ≫ σn2 , the
maximum-SJNR in (72) can be approximated in a form of
SJNRmax =
(73)
If Alice and Bob adopt the transmit beamformer t = Vab (1)
which is optimal for the jamming-absent case but not for
jamming-present case, SJNRmax has a lower bound:
Pa Nj
H −1
Vab (1)H HH
Hab Vab (1)
ab (Hjb Hjb )
Pj
H
Uab Σab Vab
Vab (1)
Pa λab,1 Nj
−1
=
Uab (1)H (Hjb HH
Uab (1).(74)
jb )
Pj
SJNRmax ≥
(67)
Pa Nj H H
−1
t Hab (Hjb HH
Hab t.
jb )
Pj
LI et al.: DISRUPTING MIMO COMMUNICATIONS WITH OPTIMAL JAMMING SIGNAL DESIGN
Recall that Uab (1) and Hjb are independent and
select a transmit beamformer t = Vab (1) which is optimal
for no jamming case but not for jamming case, we can obtain
SJNR performance lower bound as
−1
(Hjb HH
Uab (1}
jb )
EHab ,Hjb {Uab (1)
1
=
E{Tr{Hjb HH
jb }}
Nb
Nb
1
1
=
=
.
Nb Nj − Nb
Nj − Nb
H
Pa λab,1 Nj
.
Pj (Nj − Nb )
(75)
(76)
When Nj → ∞, we have Hjb HH
jb ≈ Nj INb and
SJNRmax
Pa λab,1
Pa
H
(86)
− 2 Vab (1)H HH
ab vv Hab Vab (1)
σn2
σn
)
Pa λab,1 (
=
1 − |Uab (1)H v|2 .
(87)
2
σn
SJNRmax ≥
Applying (75) into (74), we obtain an approximation
SJNRmax ≥
11
2
−1
= Pa λmax (HH
Hab )
ab (Pj INb + σn INb )
Pa λab,1
=
(77)
Pj + σn2
Now we turn to evaluate the statistics of random variable
|Uab (1)H v|2 . Recall that Uab (1) is the basis of channel
matrix Hab and v , hjb /∥hjb ∥. Since Hab and hjb are two
independent Rayleigh fading channels, Uab (1) and v are two
independent normalized random vectors of length Nb and have
statistic property:
EHab ,hjb {|Uab (1)H v|2 } =
and consequently
SJNRmax (Nj → ∞) =
Pa λab,1
.
Pj + σn2
(78)
With the asymptotic performance bound (78), we reformulate the approximation expression in (76) in order to
compensate the noise term which is ignored during the derivation:
Pa λab,1 Nj
SJNRmax ≈
.
(79)
(Pj + σn2 )(Nj − Nb )
Appendix G - Proof of Proposition 7
When jamming signal is present but only single antenna is
used at the jamming side, the signal vector y ∈ CNb received
by Bob can be expressed as
√
y = Pa sHab t + hjb z + n
(80)
where hjb ∈ CNb is the channel vector between jammer and
Bob, E{|z|2 } = Pj . Similarly, the maximized output SJNR of
the filter wmaxSJNR is
−1
2
H
Hab t.
SJNRmax = Pa tH HH
ab (Pj hjb hjb + σn INb )
(81)
H
e
Rewrite Pj hjb hH
where Pej = Pj ∥hjb ∥2 and v =
jb as Pj vv
hjb /∥hjb ∥. By matrix inversion lemma, we have
−1
2
(Pj hjb hH
= (Pej vvH + σn2 INb )−1
(82)
jb + σn INb )
e
1
Pj
= 2 INb −
vvH . (83)
2
2
σn
σn (σn + Pej )
Applying (83) into (81), we can obtain
Pa H H
Pa Pej
H
tH HH
t Hab Hab t−
ab vv Hab t.
2
σn
σn2 (σn2 + Pej )
(84)
e
P
In jamming scenario, Pej ≫ σn2 and (σ2 +jPe ) ≈ 1. Then, (84)
j
n
can be rewritten as
Pa H H H
Pa
SJNRmax ≈ 2 tH HH
ab Hab t − 2 t Hab vv Hab t. (85)
σn
σn
SJNRmax =
Clearly, the optimal transmit beamformer t is the largest
H
2
−1
eigenvector of matrix HH
Hab . If we
ab (Pj hjb hjb + σn INb )
1
.
Nb
Finally, we apply the results (88) into (87) and obtain
(
)
Pa λab,1
1
SJNRmax ≥
1
−
.
σn2
Nb
(88)
(89)
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Qian Liu (S’09, M’14) received B.S. and M.S.
degrees from Dalian University of Technology, China, in 2006 and 2009, and Ph.D. degree from State University of New York at Buffalo (SUNYBuffalo) in 2013. She is now a postdoctoral fellow
in Ubiquitous Multimedia Lab at SUNY-Buffalo.
She received the Best Paper Runnerup award in
ICME 2012, and the Best Student Paper Finalist in
ISCAS 2011. Her current research interests include
multimedia transmission over MIMO systems, IEEE
802.11 wireless networks and LTE networks.
Ming Li (S’05, M’11) received the M.S. and Ph.D.
degrees in electrical engineering from the State
University of New York at Buffalo, Buffalo, in 2005
and 2010, respectively. From Jan. 2011 to Aug.
2013, he was a Post-Doctoral Research Associate
with the Signals, Communications, and Networking
Research Group, Department of Electrical Engineering, State University of New York at Buffalo. From
Aug. 2013 to June 2014, Dr. Li joined Qualcomm
Technologies Inc. as a Senior Engineer. Since June
2014, he has been with the School of Information
and Communication Engineering, Dalian University of Technology, Dalian,
China, where he is presently an Associate Professor. His research interests
are in the general areas of communication theory and signal processing with
applications to interference channels and signal waveform design, secure
wireless communications, cognitive radios and networks, data hiding and
steganography, and compressed sensing.
Xiangwei Kong (M’06) received the B.E. and M.S.
degrees from Harbin Shipbuilding Engineering Institute, Harbin, China, in 1985 and 1988, respectively, and the Ph.D. degree from Dalian University of Technology, Dalian, China, in 2003. She
is a Professor with the School of Information and
Communication Engineering and the vice director
of the Information Security Research Center, Dalian
University of Technology, Dalian, China. She is also
the vice director of the Multimedia Security Session
of the Chinese Institute of Electronics. Her research
interests include multimedia security and forensics, digital image processing,
and pattern recognition.
Nan Zhao (S’08-M’11) is currently an associate
professor in the School of Information and Communication Engineering at Dalian University of
Technology, China. He received the B.S. degree in
electronics and information engineering in 2005, the
M.E. degree in signal and information processing
in 2007, and the Ph.D. degree in information and
communication engineering in 2011, from Harbin
Institute of Technology, Harbin, China. From Jun.
2011 to Jun. 2013, Nan Zhao did postdoctoral research in Dalian University of Technology, Dalian,
China. His recent research interests include Interference Alignment, Physical Layer Security, Cognitive Radio, Wireless Power Transfer, and Optical
Communications. He has published nearly 50 papers in refereed journals and
international conferences.
Dr. Zhao is a member of the IEEE. He serves as an Area Editor of AEUInternational Journal of Electronics and Communications, and an Editor of
Ad Hoc & Sensor Wireless Networks. Additionally, he served as a technical
program committee (TPC) member for many interferences, e.g., Globecom,
VTC, WCSP. He is also a peer-reviewer for a number of international journals,
as IEEE Trans. Commun., IEEE Trans. Wireless Commun., IEEE Trans. Veh.
Tech., etc.