IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 4, FEBRUARY 15, 2014 993
Adaptive Limited Feedback for MISO Wiretap
Channels With Cooperative Jamming
Minyan Pei, A. Lee Swindlehurst, Fellow, IEEE, Dongtang Ma, and Jibo Wei
Abstract—This paper studies a multi-antenna wiretap channel
with a passive eavesdropper and an external helper, where only
quantized channel information regarding the legitimate receiver is
available at the transmitter and helper due to finite-rate feedback.
Given a fixed total bandwidth for the two feedback channels, the
receiver must determine how to allocate its feedback bits to the
transmitter and helper. Assuming zero-forcing transmission at the
helper and random vector quantization of the chann els, an ana-
lytic e xpression for the achievable ergodic secrecy rate due to the
resulting quantization errors is derived. While direct optimization
of the secrecy rate is difficult, an approximate upper bound for the
mean loss in secrecy rate is derived and a feedback bit allocation
method that minimizes the average upper bound on the secrecy
rate loss is studied. A closed-form solution is shown to be possible
if the integer constraint on the bit allocation is relaxed. Numerical
simulations indicate the significant advantage that can be a chieved
by adaptively allocating the available feedback bits.
Index Terms—Cooperative jamming, feedback bits allocation,
limited feedback, MISO wiretap channel.
I. INTRODUCTION
P
HYSICAL layer securi
ty has attracted considerable at-
tention recently as an alternative to or an augm entation
of traditional cry pto graphy-based security. The goal of such
methodsistoexp
loit the physical characteristics of the wireless
channel to enhance the security of wireless communicatio n
systems. The pioneering work of Wyner introduced the concept
of the wire
tap channel and secrecy capacity, and laid the basis
for informa tion-theoretic approaches for secure communication
[1]. More recently, a significant effort has been invested in the
study
of secrecy capacity in wiretap channels with multip le
antennas [2]–[8]. More detailed results are possib le for mul-
tiple-input single-output (MISO) wiretap channels, which have
Manuscript received May 30, 2013; revised September 11, 2013; accepted
November 27, 2013. Date of publication Decem ber 13, 2013; date of current
version January 21, 2014. The associate e ditor coordinating the review of this
manuscript and approving it for publication was Dr. Rong-Rong Chen. This
work was supported by the National Nature Science Foundation of China under
Grant 61002032 and by the U.S. National Science Foundation under Grant
CCF-1117983. This paper was presented in part at IEEE Workshop on Signal
Processing Advances in Wireless Communications, Darmstadt, Germany, June
2013.
M. Pei, D. Ma, and J. Wei are with the Colleg e of Electronic Science and Engi-
neering, National University of Defense Technology, Changsha 410073, China
(e-mail: mypei86@nudt.edu.cn; don gtangma@nudt.edu.cn; wjbhw@nudt.edu.
cn).
A. L. Swindlehurst is with the Center for Pervasive Communication s and
Computing (CPCC), University of California, Irvine, CA 92664 USA (e -mail:
swindle@uci.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2013.2295059
been studied in [2]–[5] under different assumption s on chann el
state information and fading. In particular, for the MISO case
where the channel to the legitimate receiver is known but only
trivial statistical information ab out the eavesdropper’s channel
is available, it was shown in [2] that the optimal communica-
tion strategy that achieves the highest secrecy rate, is based on
beamformin g. With the additional degrees of freedom available
in multi-antenna systems, the use of artificial noise (AN ) has
been proposed for selectively degrading the eavesdropper’s
channel, particularly in situations where no info rm ation or only
statistical inform ation is available about the eavesdropp er [ 9],
[10].
Recent work has also considered the use of friendly helpers to
provide jam m ing signals to confuse the eavesdropper [11]–[19].
The idea of a terminal helping another improve its secrecy rate
first appeared in 2006 [11], and later in [12], [13] the term coop-
erative jamming was introduced for this idea. It should be noted
that although in th is paper we w ill study cooperative jamming
with Gaussian noise, cooperative jammers can also improve se-
crecy using Gaussian or lattice codewords, as in [14], [16]. The
MISO wiretap scenario was first considered in [18], where it
was shown that zero-forcing (ZF) beam forming at the helper is
nearly optimal in the hig h signal- to- no ise ratio (SNR) regime.
The work of [19] showed how to obtain optimal transmit beam-
formers a t the transmitter a nd helper, and also demonstrated that
using a ZF beamformer at the helper is a near-optimal choice for
obtaining the secrecy capacity in this scenario, assuming that
the transmitter and helper h ave perfect channel state inform a-
tion (CSI) for the channels to the receiver. In practice however,
knowledge of the CSI at the transmitter and helper (referred to
as CSIT here) is destined to be in error . This is par ti cularly true
in frequency-division duplex (FDD) systems, where the legiti-
mate user quantizes the CSI using a finite-sized codebook that
is known to both the transmitter and receiver, and then feeds the
quantized information back to the transmitter.
The effects of quantized channel feedback on transceiver
design have been studied extensively for both single an d
multiuser downlink systems without secr ecy co nsider ati ons
[20]–[24]. Only the recent work of [25]–[27] has considered
the impact of limited feedback on secrecy for the simple
wiretap ch annel without a h elper. In [25], the d egradation in
secrecy rate for AN-assisted beamforming due to quantized
channel direction information was studied, and the optimal
power allocation between the message-bearing signal and the
AN for a given number of feedback bits was exam ine d. A
lower bound on the ergodic secrecy capacity was obtained
using num erical integration in [26] for AN-assisted wiretap
channels. In [27], the secrecy outage probability was charac-
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994 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 4, FEBRUARY 15, 2014
terized for cod eboo k-based transmit beamforming. However,
none of the prior work has a d dressed the impact of finite-rate
feedback in the context of multiple-antenna wiretap channels
with an external helper. This problem is interesting since each
receiver must feedback CSI to b oth the transmitter and the
helper. Errors in the CSI at the transmitter, which reduce the
gain available to the receiver, must be balanced against errors
in the helper CSI, which will lead to increased interference
due t o imperfectly nulled AN transmissions. Wh en the to
tal
bandwidth of the two feedback channels is fixed, the optimal
feedback bit allocation must address this trade-off given the
available feedback throughput, SNR, number o f a
ntennas and
the channel conditions.
Feedback bit allocation strategies have been proposed for
other applications where both d esired
and interfering transmit-
ters are presen t [28]–[30], although again not for t he physical
layer security problem. The two-user MI M O inter fer ence
channel was considered in [28], a
nd cooperative feedback to
the in ter fer ing transm itt er was proposed in addition to standard
feedback to the desired transmitter. Under a constraint on the
throughput loss caused b
y precoder quantization, the required
number of cooperative feedback bits w as derived. Cooperative
multi-cell sy ste ms with limited feedback are another relevant
application. The a
uthors of [29] proposed an approach where
the available bits assigned to the desired and interfering trans-
mitters were chosen to r educe the mean loss in sum-rate due to
quantizatio
n for a soft hand-off model. In their scheme, beam-
forming vectors were designed using a generalized eigenvector
approach to maximize the sum-rateassumingasingleinterferer
scenar
io, which leads to an objective different from the one
considered here. In [30], a multicell MISO system with joint
processing was considered, where the base stations exchange
bo
th CSI and data, and a feedback bit allocation scheme was
proposed to maximize the quantization accuracy.
In this paper, we consider cooperative jamm ing for the MISO
wiretap channel with finite-rate feedback, w here the transmitter
uses a maximum ratio transmission (MRT) strategy, and the
helper enhances the secrecy of the legitimate channels via a
ZF approach intended to produce AN that is ideally invisible to
the legitimate user . Assuming a fixed total number of feedback
bits available at the legitimate user, we study how to optimize
allocation of the bits for the two feedback channels assuming
random vector quantization (RVQ) codeb ook s, where t he quan-
tization vectors are independently chosen from an isotropic dis-
tribution on the unit hypersphere [21]. We first consider the gen-
eral problem of maximizing the ergodic secrecy rate. While we
show how to obtain an analytic expression for the ergodic se-
crecy rate that can be used for perform ance optimization, it is
cumbersome and requires num erical integration. As an alterna-
tive, we then derive an upper-b oun d on the mean loss in se-
crecy rate assuming a fixed power allo cation at the transmitter
and helper. We show that for the general scenario with a global
constraint on the feedback bandwidth, a closed-form solution
can be found by standard conv ex op timization techniques if the
integer constraint on the bits is relaxed. Our simulation results
show that proper allocation of t he feedback bits to the trans-
mitter and helper can have a significant impact on the secrecy
of the wiretap channel.
In the next section, the system model is presented together
with the proposed tra nsmission strategy. An analytical expres-
sion for the ergodic secrecy rate is developed in Section III, and
then an approximate upper bound for the secrecy rate loss is
derived in Section IV, together with the algorithm for adaptive
limited feedback that minimizes the upper bound under a con-
straint on the total number o f feedback bits. Numerical results
are p rovided in Section V and we conclude in Section VI.
Notations:
We use uppercase boldface for matrices and low
-
ercase boldface for vectors. The superscripts and de-
note conjugate transposition and m atrix inversion, r espectively.
, and denote the trace, column vectoriza
tion
and determinant operations, respectively. denotes expecta-
tion with respect to , denotes the Euclidean norm of vector
, and the function represents
. T he notation
means that is a vector of circularly sym m etric
complex Gaussian random variables with mean vector and co-
variance matrix
. denotes an
identity matrix (the
subscript is dropped when the dim ension is obvious).
II. S
YSTEM MODEL AND ASSUMPTIONS
A. System Model
We consider a MISO wiretap channel that includes a trans-
mitter (Alice), a legitimate receiver (Bob), a friendly jammer
(Helper) and an eavesdropper (Eve). We assume
antennas at
Alice,
antennas at Helper, antennas at Eve, and a single
antenna at Bob. In this model, the transmitter Alice wishes to
send a confid e ntial message to Bob in the presence of Eve, with
the aid of the Helper. We assum e that the Helper does not know
the c onfidential m essage and assists Alice by producing artifi-
cial Gaussian noise (jamming) to confuse Eve.
We assume the vector
is the confidential in-
formation-bearing signal transmitted by Alice,
is the Gaussian jamming signal generated by the Helper, and
the sig nals
satisfy the pow er constraints ,
. The received signal at B ob and Eve are respec-
tively given by
(1)
(2)
where
are the channel vectors
to Bob from Alice and the Helper respectively, and
are the corresponding channels for
Eve. The terms
and represent circularly symmetric unit-
variance Gaussian noise at Bob and Eve, respectively; their dis-
tributions are denoted by
and .
All channels are assumed to be m utually independent and each
composed of circularly symmetric complex Gaussian entries,
i.e.,
and .Theinstan-
taneous realization s o f both
and are known perfectly to
Bob, but Alice and the Helper only hav e quantized information
about them obtained via distinct finite-rate feedback channels
from the legitimate receiver. We also assum e that
and , and that these di s-
tributions are known to all legitimate parties (although we wil l
PEI et al. : MISO WIRETAP CH ANNELS WITH COOPERATIVE JAMMING 995
see that this assumption can be relaxed when dealing with se-
crecy rate loss). All channels are assumed to remain constant
during the tim e required for channel estimatio n and feedback.
In the finite-rate feedback model, the legitim ate r eceiver
first quantizes the channel direction information (CDI),
and , by exploiting two di s -
tinct quantization codebooks,
and , which consist of -and
-dimensional unit norm vectors and are of size and ,
respectively (
is the codebook at Alice, is the codebook
at the Helper). The codeb ook s are de signed off-line and known
to all par ti es. Using the m inimum chordal distance metric [20],
[23], the CDI i ndices are computed by Bob as
(3)
(4)
where
and are the quantized CDI of the two links for
Bob. After quantization, Bob informs Alice and the Help e r of
their respective codebook indices
and through distinct
error- and delay-free feedback channels. Note that only the CD I
is quantized, since the transmission strategies discussed below
do not req uire knowledge of the channel gain.
B. Transmission Strategies
We focus on MRT beamforming at Alice and a ZF trans-
mission scheme at the Helper, which is a reasonable approach
for the MISO case con sidered here. Thus, the unit-norm
beamforming vector for Bob at Alice is chosen as
.
The ZF constraint imposed on the jamming noise gener-
ated by the Helper can be expressed as
,where
is an orthonormal basis for the null-space
of
,and is a vecto r of independent and identically
distributed (i.i.d) Gaussian random variables of variance
. Due to the power constraint at the Helper, we have
. T he received
signal at Bob and Eve are thus
(5)
(6)
The fundamenta
l problem we are addressing is the following:
if the total nu
mber of feedback bits
from the legiti mat e r e -
ceiver is fixed
, h ow should
and be allocated among the
transmitte
r and helper for optimal secrecy? Besides optimiza-
tion of the f
eedback bit allocation, there is also an optimal power
level for
; if too much power is allocated to the Helper when
there are i
nsufficient bits available for good CSI, the AN pro-
vided by t
he Helper does more harm than good due to i nterfer-
ence leak
age. The choice of transmit power to use at Alice could
in princ
iple also be optimized in order to maximize secrecy per-
forman
ce, although our simulations indicate that the best per-
forman
ce is ach ieved when A lice transmits with full power
.
In the
absence of a form al proof o f this observation, we will as-
sume
that Al ice transmits with full power
, recognizing that
this
may not be optimal in all cases. In the next section, we at-
tem
pt a direct analysis using the ergodic secrecy rate. While our
res
ulting analytic expression can b e used to solve the desired
optimization problem, a complicated process involving numer-
ical integration is required. Thus, in S ection IV, we consider a
simpler approach based on secrecy rate loss, and we optimize
the bit allocation to minimize an upper bound on the loss for
fixed
and .
III. E
RGODIC SECRECY RAT E A NALYSIS WITH LIMITED
FEEDBACK
A. Achievable Secrecy Rate With Quantized CDI
To quantify the secrecy perform ance of the scenario de-
scribed above, we assume that the channels are ergodic
block-fading that remain constant over a sufficient amount
of time for signal transmission and feedback, and that the
messages are coded across multiple fading blocks (a similar
model was used in the related study of [25]). We further assume
a scenario with de lay- tolerant traffic, and use ergodic secrecy
rate as our performance metric rather than secrecy outage
rate [31]. For Gaussian signaling with
,the
achievable ergodic secrecy rate at B ob for the above ZF-based
transmission strategy is given b y [25]
(7)
where
To represent the effect of quantized CDI on the achievable
secrecy rate, we rewrite the actual channel direction vectors as
(8)
(9)
where
, ,and and are
unit-norm vectors orthogo nal to
and , respectively. Thus,
we have
(10)
The interference from the jamme r is implici tly defined by
the first term in the denominator of (10).
B. Auxiliary Results
In this s
ubsection, we provide three lemmas that will be used
in the e
rgodic secrecy rate analysis. The quantization cell ap-
proxim
ation used in the proofs is based on the assumption that
each q
uantization cell is a Voronoi region with surface area
equal
to
of the total area of the unit sphere f or a -bit code-
book
. Details for this model can be found in [20 ], [24]. As shown
in [
24], analysis based o n the quantization cell approximatio n is
996 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 4, FEBRUARY 15, 2014
close to the performance of random vector quantization, and so
we use this approach to analyze the achievable rate.
Lemma 1 (Signal Power Distribution): Define
,where is defined i n (8). Then under the quan-
tization cell app roximation model, the cumulative distribution
function (CDF) of
is given by
(11)
where .
Proof: See A ppen dix A .
Lemma 2 (Interference Power Distributio n): Under the quan-
tization cell ap proximation model, the interference term
is an
exponential random var iab le wi th p rob abil ity distribution func -
tion (PDF) given by
(12)
where
.
Proof: See A ppen dix B.
Lemma 3 (D istribution of SINR for Eve): Define
,where ,and
are independent zero-mean
complex Gaussian variables with
. T hen the
complementary CDF of
is
(13)
where
, ,
and is the coefficient of in .
Proof: This result is provided by eq. (11) in [32 ].
C. Ergodic Secrecy Rate
Using Lemma 1 and Lemma 2, we can derive an analytic
expression for the ergodic rate of the channel between Alice
and Bob averaged over the parameters of the quantization,
whichwedenoteby
,where
denotes expectation with respect to . Next we de-
fine
and , and note that
are independent zero-m ean
complex Gaussian variables. The SINR at Eve becomes
(14)
and we can derive an analytic expression for the ergodic rate
of the eavesdropper ’s channel using Lem ma 3. Putting these
results together, an analytic expression for the ergodic secrecy
rate can be obtained, w hich is provided below in Theorem 1.
Theorem 1: Denote
,
,and .Then
,Where
(15)
(16)
and
is the integral
(17)
which can be evaluated as explained in (47),
is the
integral
(18)
which can be evaluated as explained in (48).
Proof: See A ppen dix C.
The resulting expression is quite complicated in the general
case. In the following, we provide sim plifi ed or approximate
expressions for the ergodic secrecy rate in several special sce-
narios, where
. These expressions
will be used to obtain analytical results a n d useful insights on
the achievable average secrecy rate and allocation of the feed-
back bits. No te that the derived approx im ation may not be an
achievable secrecy rate, although it is useful for f eedback de-
sign.
1) High SNR Case: If we define
and let grow
large, the noise term is neglig ibl e with respect to the inter fer ence
term in the SINR expressions. In this case, the ergodic secrecy
rate drops to zero when
, since the eavesdropper can
PEI et al. : MISO WIRETAP CH ANNELS WITH COOPERATIVE JAMMING 997
theoretically null the interference from the Helper. For the case
where
, the ergodic secrecy rate can be expressed as
in Theorem 2.
Theorem 2: Denote
, ,
.If ,th
en
,where and are derived as (19)
and ( 20) at the bottom of the page.
Proof: See A ppen dix D .
2) Single-Antenna Eavesdropper: For the special case where
, , the achievable ergodic rates and
in Theorem 1 reduce to (21) and (22) at the botto m of the
page. This can be substituted into
to y ield a simplifi ed expression for the ergodic secrecy rate. For
high SNR with
and , the expressions for
and in (21) and (22) can be approximated as (23)
and ( 24) at the bottom of the page.
Fig. 1 shows the ergodic secrecy rate as a fun ction of the
transmit power
, with , and
for , 5. The dash -dotted lines are obtained using the
asymptotic expressions in (23) and (24). We see th at the secrecy
rate increases with
(higher relative power at the Helper) and
as the total number of feedback bits for Bob
is increased.
Also, for large
, the secrecy rate yielded by (21) and (22)
asymptotically converges to the limiting value deriv ed by (23)
and (24). Furtherm ore, the simplified expression for the ergodic
(19)
(20)
(21)
(22)
(23)
(24)
