arX

iv:2

012.

0316

9v1

[cs

.IT

] 6

Dec

202

01

Low-complexity and High-performance

Receive Beamforming for Secure Directional

Modulation Networks against an

Eavesdropping-enabled Full-duplex Attacker

Yin Teng, Jiayu Li, Lin Liu, Guiyang Xia, Xiaobo Zhou, Feng Shu,

Jiangzhou Wang, Fellow, IEEE, and Xiaohu You, Fellow, IEEE

Abstract

In this paper, we present a novel scenario for directional modulation (DM) networks with a full-

duplex (FD) malicious attacker (Mallory), where Mallory can eavesdrop the confidential message from

Alice to Bob and simultaneously interfere Bob by sending a jamming signal. Considering that the

jamming plus noise at Bob is colored, an enhanced receive beamforming (RBF), whitening-filter-based

maximum ratio combining (MRC) (WFMRC), is proposed. Subsequently, two RBFs of maximizing

the secrecy rate (Max-SR) and minimum mean square error (MMSE) are presented to show the same

performance as WFMRC. To reduce the computational complexity of conventional MMSE, a low-

complexity MMSE is also proposed. Eventually, to completely remove the jamming signal from Mallory

and transform the residual interference plus noise to a white one, a new RBF, null-space projection (NSP)

based maximizing WF receive power, called NSP-based Max-WFRP, is also proposed. From simulation

Yin Teng, Jiayu Li, Lin Liu, Guiyang Xia, and Feng Shu are with the School of Electronic and Optical Engineering, Nanjing

University of Science and Technology, 210094, CHINA. (e-mail: shufeng0101@163.com).

Xiaobo Zhou is with the School of Physics and Electronic Engineering, Fuyang Normal University, Fuyang 236037, China.

(e-mail:zxb@fynu.edu.cn).

Feng Shu is also with the School of Information and Communication Engineering, Hainan University, Haikou, 570228, China.

Jiangzhou Wang is with the School of Engineering and Digital Arts, University of Kent, Canterbury CT2 7NT, U.K. (e-mail:

j.z.wang@kent.ac.uk).

Xiaohu You is with the School of Information Science and Engineering, Southeast University, 210094, CHINA. (e-mail:

xhyu@seu.edu.cn).

2

results, we find that the proposed Max-SR, WFMRC, and low-complexity MMSE have the same SR

performance as conventional MMSE, and achieve the best performance while the proposed NSP-based

Max-WFRP performs better than MRC in the medium and high signal-to-noise ratio regions. Due to its

low-complexity,the proposed low-complexity MMSE is very attractive. More important, the proposed

methods are robust to the change in malicious jamming power compared to conventional MRC.

Index Terms

Malicious attacker, secure, directional modulation, secrecy rate, receive beamforming, null-space

projection (NSP).

I. INTRODUCTION

In the past decade, physical layer security (PLS) has attracted wide attention from academia

[1]–[8]. Directional modulation (DM), as an advanced PLS transmission technique, is suitable

for line-of-sight (LoS) propagation channel. Beamforming technology with artificial noise (AN)

has a capability of creating a secure transmission in the LoS propagation channel [4], [9]–[11].

In [12], the authors presented a DM technique that uses the phased arrays to generate mod-

ulation and provides security by deliberately distorting signals in other directions and making

the signal independent of direction. An orthogonal vector approach was proposed in [9] for the

synthesis of multi-beam DM transmitters. In [10], [13], the authors developed robust synthesis

schemes, in which DM is able to enhance the security performance of desired directions and

distort the constellation points of undesired directions. These systems have the capability of

concurrently projecting independent data streams along different specified spatial directions while

simultaneously distorting signal constellations in all other directions. A directional modulation

technique using frequency diverse array (FDA) for secure communications was proposed in

[14]–[20]. In [14], [15], the authors developed a novel DM scheme based on random frequency

diverse arrays with artificial noise (RFDA-DM-AN) to enhance physical layer security of wireless

communications. In order to address the limitations of the previous works and further enhance

physical layer (PHY) security, an AN-aided index modulation (IM) scheme with cooperative

LUs based on FDA beamforming was proposed in [16]. Multi-beam directional modulation

schemes based on FDA were investigated in [16]–[18]. Different from the traditional directional

modulation that only implements angle dependent directional modulation, the authors proposed

3

some precise secure transmission schemes of achieving two-dimensional dependence on angle

and range dimensions in [14], [19], [20]. In the multi-carrier based DM antenna array system

design, to solve the problems of high peak-to-average-power ratio (PAPR) and phase pattern

formation simultaneously, a method, called wideband beam and phase pattern formation by

Newton’s (WBPFN), was proposed in [21]. In [22], the authors developed a multi-carrier based

DM framework using antenna arrays, which can realize data transmission on multiple frequencies

simultaneously, so as to obtain a higher data rate. A practical wireless transmission scheme was

proposed in [11] to transmit the confidential message (CM) to the desired user securely and

precisely by jointly exploiting multiple techniques, including artificial noise (AN) projection,

phase alignment/beamforming, and random subcarrier selection (RSCS) based on orthogonal

frequency division multiplexing (OFDM) [23], [24], and DM. Particularly, if a DM transmitter

intends to transmit CM to Bob and to interfere eavesdropper (Eve) with AN, she needs to know

the directions of Bob and Eve in advance. In [25], two phase alignment (PA) methods, hybrid

analog and digital PA and hybrid digital and analog PA, were proposed to estimate direction

of arrival (DOA) based on the parametric method. In [26], the authors proposed an improved

hybrid analog-digital (HAD) estimation of signal parameters via rotational invariance techniques

(ESPRIT) at HAD transceiver to measure the DOA of a desired user.

However, the aforementioned existing works focused on the scenarios with only a passive

eavesdropper Eve having no active malicious attacking capacity. In such a situation, it is hard

for Alice to obtain the channel state information (CSI) from Alice to Eve. This is an embarrassing

issue usually raised by secure expert. How to address this challenging issue? If Eve behaves

like Mallory, with an active malicious attacking ability, this problem naturally disappears. By

channel estimating, Alice may obtain the CSI from Alice to Mallory and Bob can obtain the

CSI from Mallory to Bob. In [27], the authors proposed a jamming detection method for non-

coherent single input multiple output (SIMO) systems without CSI. A hybrid wiretapping wireless

system with a half-duplex adversary was proposed in [28], where the adversary can decide to

either jam or eavesdrop the transmitter-to-receiver channel. In the presence of a full-duplex

eavesdropper having both eavesdropping and jamming capabilities, the authors in [29] proposed

a novel secure transmission strategy against eavesdropping by the sophisticated adversary. The

authors in [30] studied the secrecy outage probability and mean SR of the system to evaluate the

4

security performance of the system. In this paper, our focus is on how to suppress the malicious

jamming and improve the secure performance by designing receive beamforming (RBF) at Bob

in the presence of the malicious jamming.

In this paper, we present a novel DM scenario with a full-duplex (FD) malicious attacker

Mallory. Moreover, Mallory also eavesdrops the CM conveyed from Alice to Bob. To reduce

the impact of jamming from Mallory on Bob and improve the secure performance, four RBF

methods are proposed. Our main contributions are summarized as follows:

1) To alleviate the jamming from Mallory on Bob and improve the secure performance,

the conventional MRC RBF at Bob is presented to strengthen the confidential signal.

To enhance its performance, a WFMRC RBF is proposed by converting the colored

interference plus noise at Bob into a white one with its covariance matrix being a multiple

of identity matrix. Then, two RBFs Max-SR and conventional MMSE are derived. From

simulation results, the proposed two RBFs WFMRC and Max-SR obviously outperform the

conventional MRC in the medium and high SNR regions and achieve the same performance

as conventional MMSE in terms of SR and BER.

2) To completely remove the jamming from Mallory, a new RBF of maximizing the WF-

based receive power at Bob is proposed to force the malicious jamming onto the null-space

(NSP) of the channel spanning from Alice to Bob. By exploiting the rank-one property

of channel steering vector, a low-complexity MMSE is proposed to reduce the complexity

of conventional MMSE without performance loss. According to simulation, we find that

the proposed NSP-based Max-WFRP is slightly worse than Max-SR, WFMRC and low-

complexity MMSE in terms of the SR performance. Its main advantage is that its SR

performance is independent of the change in malicious jamming power. Moreover, the

proposed low-complexity MMSE has achieved the same performance as three proposed

high-performance RBFs WFMRC, conventional MMSE, and Max-SR with an extremely

low-complexity.

The remainder of this paper is organized as follows. Section II presents the DM system

model. In Section III, four schemes of design RBF vector vBR is proposed and its closed-

form expression is given. Simulation and numerical results are shown in Section IV. Finally, we

draw our conclusions in Section V.

5

Notations: Throughout the paper, matrices, vectors, and scalars are denoted by letters of bold

upper case, bold lower case, and lower case, respectively. Signs (·)T , (·)−1, (·)†, (·)H , ‖ · ‖and tr(·) represent transpose, inverse, Moore-Penrose, conjugate transpose, norm and trace,

respectively. IN denotes the N×N identity matrix. The notation E{·} represents the expectation

operation.

II. SYSTEM MODEL

Alice.

Mallory

Bob

1

NA

1

1

NB

H(θAB)

Self-interference

H(θAM)

H(θMB)

HM

.

.

.

.

.

θr,AM

θt,AB

θt,AM

θt,MB

θr,MB

θr,AB

. . .NM

Fig. 1. Block diagram of the DM network with FD malicious attacker.

The proposed DM system is illustrated in Fig. 1. It consists of an NA-antennas base station

(Alice), a legitimate NB-antennas user (Bob) and an illegal NM -antennas malicious attacker

(Mallory). Here, the Alice sends CM to the Bob. In addition, there is an illegal malicious

attacker Mallory trying to intercept the CM. Mallory operates on the FD model. That is, he/she

can eavesdrop on CM, and simultaneously initiate an active attack on Bob. The baseband transmit

signal can be expressed as

sA =√

β1PAvAdA +√

(1− β1)PATA,ANzA,AN , (1)

where PA is the total transmit power, β1 stands for the power allocation (PA) factor, vA ∈ CNA×1

denotes the transmit beamforming vector of the CM, and TA,AN ∈ CNA×NA is the projection

matrix for controlling AN to the undesired direction, where vHAvA = 1 and tr[TA,ANT

HA,AN] = 1.

In addition, vA is defined as the transmit beamforming vector for the CM. In (1), dA is the CM

with average power E[‖dA‖2] = 1 and zA,AN ∈ CNA×1 denotes the AN vector with complex

6

Gaussian distribution, i.e., zA,AN ∼ CN (0, INA). The malicious attacking signal at Mallory can

be expressed as

sM =√PMTM,ANzM,AN , (2)

where TM,AN ∈ CNM×NJ is the projection matrix for forcing the jamming to Bob, NJ ∈{1, 2, ...., NM − 1}. Herein, PM is the transmit power at Mallory, zM,AN ∈ CNJ×1 denotes the

AN vector with complex Gaussian distribution, i.e., zM,AN ∼ CN (0, INJ).

The corresponding received signal at Bob can be written as

rB = vHBR(

√gABH

H(θAB)sA +√gMBH

H(θMB)sM + nB)

= vHBR(

√gABβ1PAH

H(θAB)vAdA +√

gAB(1− β1)PAHH(θAB)TA,ANzA,AN︸ ︷︷ ︸

nA

+√gMBH

H(θMB)sM︸ ︷︷ ︸nM

+nB)

= vHBR(

√gABβ1PAH

H(θAB)vAdA + nA + nM + nB︸ ︷︷ ︸nB

), (3)

where HH(θAB) = h(θr,AB)hH(θt,AB) and HH(θMB) = h(θr,MB)h

H(θt,MB). HH(θAB) ∈C

NB×NA denotes the channel matrix from Alice to Bob. HH(θMB) ∈ CNB×NM denotes the

channel matrix from Mallory to Bob. The normalized steering vector h(θ) as

h(θ) =1√N[ej2πΨθ(1), ..., ej2πΨθ(n), ..., ej2πΨθ(N)]T , (4)

where the phase function Ψθ(n) is defined by [31]

Ψθ(n) , −(n− N + 1

2

)d cos θ

λ, n = 1, · · · , N, (5)

where θ is the direction of arrival or departure, n is the index of antenna, d represents the

element spacing in the transmit antenna array, and λ is the wavelength. In (3), nB ∈ CNB×1 is

the complex additive white Gaussian noise (AWGN) vector, distributed as nB ∼ CN (0, σ2BINB

).

Notably, gAB = αdcAB

represents the path loss from Alice to Bob. Here, dAB is the distance

between Alice and Bob, c denotes the path loss exponent and α means the path loss at reference

distance d0. Moreover, vBR ∈ CNB×1 represents the receive beamforming vector of Bob. Using

the concept of the null-space projection, we can design

TA,AN = INA−H(θAB)[H

H(θAB)H(θAB)]−1HH(θAB). (6)

7

Similarly, the receive signal at Mallory is given by

rM = vHMR(

√gAMHH(θAM )sA +

√ρHH

MsM + nM )

= vHMR(

√gAMβ1PAH

H(θAM )vAdA +√gAM(1− β1)PAH

H(θAM )TA,ANzA,AN

+√ρPMHH

MTM,ANzM,AN + nM ), (7)

where HH(θAM) = h(θr,AM)hH(θt,AM). HH(θAM) ∈ CNM×NA denotes the channel matrix from

Alice to Mallory. gAM = αdcAM

represents the corresponding path loss, and dAM is the distance

between Alice and Mallory. Additionally, vMR represents the receive beamforming vector at

Mallory. The complex AWGN at Mallory is denoted by nM ∼ CN (0, σ2MINM

).√ρHH

M ∈CNM×NM is the residual self-interference (RSI) channel matrix of Mallory, ρ ∈ [0, 1] denotes the

residual self-interference parameter of the Mallory after self-interference cancelation [29].

As per (3) and (7), we can derive the achievable rate along Bob and Mallory as

RAB(vBR) = log2

(1 +

vHBRAvBR

vHBRBvBR + vH

BRDvBR + σ2B

), (8)

and

RAM = log2

(1 +

vHMREvMR

vHMRFvMR + vH

MRRMvMR + σ2M

), (9)

respectively, where RM is the covariance matrix of RSI channel,

RM = ρPMHHMTM,ANT

HM,ANHM , (10)

A = gABβ1PAHH(θAB)vAv

HAH(θAB), (11)

B = gAB(1− β1)PAHH(θAB)TA,ANT

HA,ANH(θAB), (12)

D = gMBPMHH(θMB)TM,ANTHM,ANH(θMB), (13)

E = gAMβ1PAHH(θAM)vAv

HAH(θAM), (14)

F = gAM(1− β1)PAHH(θAM )TA,ANT

HA,ANH(θAM). (15)

Then we arrive the achievable SR, Rs, as follows:

Rs(vBR) = max {0, RAB(vBR)− RAM} . (16)

III. PROPOSED FOUR RBF METHODS

In this section, to improve the security rate performance and reduce the effect of jamming

from Mallory on Bob, four RBF schemes are proposed. Additionally, the conventional MRC is

also presented as a performance benchmark for the future comparison.

8

A. Conventional MRC and Proposed WFMRC

From the definition of conventional MRC, we have

vHBR =

(HH(θAB)vA

)H

‖(HH(θAB)vA)H‖2, (17)

under the approximation that nB is viewed as a white Gaussian noise vector with zero mean and

covariance matrix being identity matrix multiplied by a scalar. However, nB is a sum of three

terms including malicious interference from Mallory, and is colored. Its covariance matrix is

CnB= E

{nBn

HB

}= B+D+ σ2

BINB(18)

which is a positive definite Hermitian matrix, i.e., a normal matrix, and has the eigenvalue-

decomposition (EVD) form [32]

CnB= QΛQH , (19)

where Q is a unitary matrix, Λ is a diagonal matrix diag (d1, ..., dNB) with di being the i-th

eigenvalue of matrix CnB. We can construct the following WF matrix

WWF = (QΛ1

2 )−1 = Λ− 1

2QH . (20)

Left multiplication of rB by WWF leads to

rB = WWFrB

=√

gABβ1PAWWFHH(θAB)vAdA +WWFnB, (21)

where

rB =√

gABβ1PAHH(θAB)vAdA + nB. (22)

Obviously, the new noise vector WWFnB becomes a white Gaussian vector with convariance

matrix being an identity. Now, similar to (17), the WFMRC is directly given as

vHBR =

(WWFHH(θAB)vA)

H

‖(WWFHH(θAB)vA)H‖2. (23)

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B. Proposed Max-SR Method

Now, we turn to the method of maximizing the SR in (16). Observing the SR in (16), it is

evident that the available rate at Mallory is independent of vBR. Thus, the optimization problem

of the Max-SR

maxvBR

Rs(vBR)

s.t. vHBRvBR = 1, (24)

reduces to

maxvBR

RAB(vBR)

s.t. vHBRvBR = 1, (25)

which can be rewritten as

maxvBR

vHBRAvBR

vHBRCnB

vBR

s.t. vHBRvBR = 1, (26)

which is the Max-SJNR. Let us define

v′HBR = vH

BR(C1

2

nB)H , (27)

then SJNR can be expressed as follows

SJNR =v′HBR

R︷ ︸︸ ︷(C

− 1

2

nB)HA(C

− 1

2

nB)v′

BR

v′HBRv

′BR

, (28)

Max-SJNR is rewritten as

maxv′

BR

v′HBRRv′

BR

s.t. v′HBRv

′BR = 1. (29)

Let us differentiate the SJNR expression with respect to v′HBR and set it to zero

∂

∂v′BR

(v′HBRRv′

BR

v′HBRv

′BR

) =Rv′

BR · v′HBRv

′BR − v′H

BRRv′BR · v′

BR

(v′HBRv

′BR)

2= 0, (30)

which means that the numerator equals zero and

Rv′BR =

v′HBRRv′

BR

v′HBRv

′BR︸ ︷︷ ︸

λ

· v′BR︸︷︷︸v

. (31)

10

Observing the above equation, λ is the eigenvalue and v is the eigenvector associated with λ. In

other words, the maximum SJNR is achieved as v′BR is taken to be the eigenvector corresponding

to the largest eigenvalue λmax of R. In particular, the semi-definite positive matrix R is rank

one, written as

R = aaH (32)

with

a =√gABβ1PA(C

− 1

2

nB)HHH(θAB)vA. (33)

Let us define

v =a

‖a‖2, (34)

and multiplying the equality R = aaH from right by v yields

Ra

‖a‖2︸ ︷︷ ︸v

=(aHa

)︸ ︷︷ ︸λmax

a

‖a‖2︸ ︷︷ ︸v

, (35)

which means

v =a

‖a‖2(36)

is the eigenvector corresponding to the largest eigenvalue of matrix R. Clearly, we have

v′∗BR = vmax = v =

a

‖a‖2, (37)

which is in agreement with the expression in (23). Substituting the above back into (27), we

have

v∗BR = C

− 1

2

nBv′∗BR = C

− 1

2

nB

a

‖a‖2. (38)

C. Proposed low-complexity MMSE Method

In the following, we propose a low-complexity MMSE algorithm based on the minimum mean

square error criterion to design receive beamforming. We can derive the vBR by the following

optimization formula

11

minvBR

f(vBR) = E {(rB − dA)(rB − dA)∗}

= tr[vHBR(gABβ1PAH

H(θAB)vAvHAH(θAB) +RnA

+RnM+RnB

)vBR

−√gABβ1PAv

HBRH

H(θAB)vA −√

gABβ1PAvHAH(θAB)vBR + 1], (39)

where RnA= E

{nAn

HA

}= B, RnM

= E{nMnH

M

}= D and RnB

= E{nBn

HB

}= σ2

BINB.

To obtain the optimal receive beamforming, we need to compute the derivative of m(vBR)

with respect to vBR,

∂f(vBR)

∂vBR

= 2(gABβ1PAHH(θAB)vAv

HAH(θAB) +RnA

+RnM+RnB

)vBR

− 2√

gABβ1PAHH(θAB)vA. (40)

It is easy to see from Eq.(40) that matrix gABβ1PAHH(θAB)vAv

HAH(θAB)+RnA

+RnM+RnB

is positive definite. Let∂f(vBR)∂vBR

= 0, we have the conventional MMSE

vBR =√gABβ1PA(gABβ1PAH

H(θAB)vAvHAH(θAB) +RnA

+RnM+RnB︸ ︷︷ ︸

O

)−1HH(θAB)vA.

(41)

However, the complexity of the conventional MMSE method mentioned above is the cubic

function of NB , which will become high as NB tends to large-scale. Interestingly, since the rank

of matrix HH(θAB)vAvHAH(θAB), RnA

, and RnMin matrix O are one, the Sherman-Morrison

formula can be applied to provide a low-complexity way of computing the inverse of matrix

O. By exploiting the rank-one property of channel steering vector, a low-complexity MMSE is

proposed in what follows.

By repeatedly making use of the Sherman-Morrison formula

(Z+ uvT )−1 = Z−1 − Z−1uvTZ−1

1 + vTZ−1u(42)

where Z ∈ Rn×n, u,v ∈ Rn×1, Z and Z + uvT are invertible, and 1 + vTZ−1u 6= 0, we have

the low-complexity MMSE method as follows

vBR =√gABβ1PAO

−1HH(θAB)vA, (43)

where

O−1 = K−1 − gMBPMK−1HH(θMB)TM,ANTHM,ANH(θMB)K

−1

gMBPMTHM,ANH(θMB)K−1HH(θMB)TM,AN + 1

, (44)

12

where

K−1 = L−1 − gAB(1− β1)PAL−1HH(θAB)h(θt,AB)h

H(θr,AB)L−1

1 + gAB(1− β1)PAhH(θr,AB)L−1HH(θAB)h(θt,AB), (45)

where

L−1 = M−1 +2gAB(1− β1)PAM

−1HH(θAB)h(θt,AB)hH(θr,AB)M

−1

1− 2gAB(1− β1)PAhH(θr,AB)M−1HH(θAB)h(θt,AB), (46)

where

M−1 = N−1 − gAB(1− β1)PAN−1HH(θAB)h(θt,AB)h

H(θr,AB)N−1

1 + gAB(1− β1)PAhH(θr,AB)N−1HH(θAB)h(θt,AB), (47)

where

N−1 = σ−2B INB

− σ−4B gABβ1PAH

H(θAB)vAvHAH(θAB)

1 + gABβ1PAvHAH(θAB)HH(θAB)vA

. (48)

Proof : See Appendix for detailed deriving process.

After we complete the above derivation, the complexity of the proposed low-complexity MMSE

is reduced to the quadratic function of NB .

D. Proposed NSP-based Max-WFRP Method

We now trun our attention on solving the Max-RP problem, which can be cast as

maxvBR

vHBRH

H(θAB)vAvHAH(θAB)vBR

s.t. (C1) vHBRH

H(θMB) = 01×NM

(C2) vHBRvBR = 1, (49)

where constraint (C1) ensures the malicious jamming lying in the null-space of Bob. To simplify

the above optimization problem, constraint (C1) implies vBR = GvBR, and

G = INB−HH(θMB)[H(θMB)H

H(θMB)]−1H(θMB), (50)

where vBR is defined as a new optimization variable. Since (C1) has projected Malicious jamming

into the null-space of Bob’s channel, we obtain the new model,

rB =√

gABβ1PAGHHH(θAB)vAdA +GH(nA + nM + nB)

=√

gABβ1PAGHHH(θAB)vAdA +GH(nA + nB). (51)

13

Similar to (21), the left multiplication of (51) by WWF yields

rB =√

β1PAgABWWFGHHH(θAB)vAdA + WWF GH(nA + nB)︸ ︷︷ ︸

nB

, (52)

where WWF is defined as

WWF = (QΛ1

2 )−1 = Λ− 1

2 QH , (53)

and

QΛQH = CnB= E

{nBn

HB

}= GH

(B+ σ2

BINB

)G. (54)

As such, the optimization problem of NSP-based Max-WFRP can be written as

maxvBR

vHBRJvBR

s.t. vHBRvBR = 1, (55)

where

J = WWFGHHH(θAB)vAv

HAH(θAB)GWH

WF . (56)

Considering that the rank of the semi-definite positive matrix J is unit, J can be written as

J = bbH , with b = WWFGHHH(θAB)vA. Similar to (29), the optimal solution of vBR is

directly given by

v∗BR =

b

‖b‖2=

WWFGHHH(θAB)vA

‖WWFGHHH(θAB)vA‖2. (57)

E. Computational Complexity Analysis

In this subsection, we turn to analyze the computational complexities of the above-mentioned

schemes. The computational complexities of MRC, WFMRC, Max-SR, NSP-based Max-WFRP,

14

low-complexity MMSE and conventional MMSE are respectively given by

CMRC = O(3NANB + 2NB), (58)

CWFMRC = O(N3B + 4N2

A + 7N2B + 5NANB + 3NBNM − 2NA −NB − 1), (59)

CMax−SR = O(N3B + 4N2

A + 8N2B + 5NANB + 3NBNM − 2NA −NB − 1), (60)

CMax−WFRP = O(4N3B +N3

M + 4N2A + 7N2

B + 4NANB + 3NBNM + 3N2M

− 2NA −NM − 2), (61)

CLow−complexity MMSE = O(36N2B + 12NANB + 6NBNM + 3NA +NM − 14NB − 6), (62)

CConventional MMSE = O(N3B + 2N2

ANB + 2N2BNA + 7N2

B +NANB + 2NBNM −NB − 1).

(63)

float-point operations (FLOPs). Obviously, their complexities have a decreasing order: NSP-

based Max-WFRP>Conventional MMSE>Max-SR>WFMRC>Low-complexity MMSE>MRC.

Observing equations from (56) to (62), we find the fact: as NB tends to large-scale, the complex-

ities of MRC, low-complexity MMSE , and remaining methods have the orders O(NB),O(N2B),

and O(N3B). Clearly, the conventional MRC has the lowest complexity among six methods

because its complexity is a linear function of NB . The proposed MMSE, Max-SR, and WFMRC

have the highest same order complexity O(N3B) FLOPs. The proposed low-complexity MMSE

is in between them because its complexity is a quadratic function of NB.

IV. SIMULATION RESULTS AND DISCUSSIONS

In this section, simulations are done to evaluate the performance of the proposed RBF schemes.

System parameters are set as follows: quadrature phase shift keying (QPSK) modulation, PA =

10W , NA = 4, NB = 4, NM = 4, ρ = 10−11, β1 = 0.9, θt,AB = 90◦, θt,AM = 125◦, θt,MB = 45◦,

dAB = 1km, dAM = 4km, dMB = 3km, σ2B = σ2

M .

Fig. 2 plots the curves of SR versus SNR of the four proposed methods with PM = 10W,

where conventional MRC and MMSE are used as a performance benchmark. It can be seen

from this figure that the proposed Max-SR, WFMRC, and low-complexity MMSE have the

same SR performance as conventional MMSE, and achieve the best SR performance among

all six methods. In the low SNR region, the SR performance of conventional MRC is close to

those of the proposed Max-SR, WFMRC, and low-complexity MMSE and better than that of

15

-10 -5 0 5 10 15 20 25SNR (dB)

0

1

2

3

4

5

6

Sec

recy

Rat

e (b

its/

s/H

z)

Proposed Max-SRConventional MRCProposed WFMRCProposed NSP-based Max-WFRPConventional MMSEProposed Low-complexity MMSE

Fig. 2. Curves of SR versus SNR with PM = 10W .

0 20 40 60 80 100Jamming power P

M(W)

0

1

2

3

4

5

Sec

recy

Rat

e (b

its/

s/H

z)

Proposed Max-SRConventional MRCProposed WFMRCProposed NSP-based Max-WFRPConventional MMSEProposed Low-complexity MMSE

Fig. 3. Curves of SR versus PM with SNR = 15dB and fixed PA=10W.

NSP-based Max-WFRP. The proposed NSP-based Max-WFRP performs much better than MRC

and worse than the proposed WFMRC, Max-SR, and low-complexity MMSE in the medium

and high SNR regions.

Fig. 3 demonstrates the curves of SR versus PM for the proposed five different methods

with SNR = 15dB and fixed PA=10W, where conventional MRC and MMSE are still used

as a performance benchmark. From Fig. 3, it is seen that regardless of the value of PM , the

proposed WFMRC, Max-SR, and low-complexity MMSE still perform much better than the

proposed NSP-based Max-WFRP and conventional MRC in terms of SR. As PM increases,

the SR performance of the proposed WFMRC, Max-SR , and low-complexity MMSE degrade

gradually and finally converge to a SR floor. However, this SR floor is still larger than that of

16

NSP-based Max-WFRP with NSP. Interestingly, the SR of NSP-based Max-WFRP with NSP

keeps constant and is independent of the change in value of PM due to the use of NSP operation.

Additionally, the conventional MRC shows a dramatic reduction on SR as PM increases. This

means that the proposed NSP-based Max-WFRP with NSP is the most robust one among six

methods. The conventional MRC becomes non-robust as the jamming power PM increases. The

proposed remaining methods are in between NSP-based Max-WFRP and MRC in accordance

with robustiness.

-10 -5 0 5 10 15 20SNR (dB)

10-6

10-5

10-4

10-3

10-2

10-1

100

BE

R

Proposed Max-SRConventional MRCProposed WFMRCProposed NSP-based Max-WFRPConventional MMSEProposed Low-complexity MMSE

Fig. 4. Curves of BER versus SNR with PM=10W.

Fig. 4 demonstrates the curves of BER versus SNR of the proposed five methods with PM =

10W, where conventional MRC is still used as a BER performance benchmark. From Fig. 4,

we can obtain the same performance tendency as Fig. 2. The proposed Max-SR, WFMRC, and

low-complexity MMSE still have the best performance in our BER simulation. In the low SNR

region, the NSP-based Max-WFRP has the worst BER performance, however when the SNR

increases up to a certain level, for example SNR=20dB, its performance will be better than that

of conventional MRC. Six RBF methods have an increasing order on BER performance: MRC,

NSP-based Max-WFRP, Max-SR ≈ WFMRC ≈ conventional MMSE≈ low-complexity MMSE.

V. CONCLUSION

In this paper, we have investigated RBF schemes in a DM network with a FD attacker

Mallory. Four high-performance RBF schemes, WFMRC, Max-SR, NSP-based Max-WFRP, and

low-complexity MMSE were proposed to mitigate the impact of the jamming signal on Bob.

17

First, the conventional MRC was presented to strengthen CM. Due to the colored the jamming

plus noise at Bob, the WFMRC was proposed to convert the colored noise to a white one.

Then, the Max-SR and low-complexity MMSE were proposed and shown to be equivalent to

WFMRC. Eventually, to completely remove the jamming from Mallory, a NSP-based Max-

WFRP was proposed. Moreover, their closed-form expressions were derived with extremely

low-complexities. Simulation results show that the proposed four methods perform much better

than the conventional MRC in the medium and high SNR regions, and their SR performances are

in an increasing order: MRC, NSP-based Max-WFRP, and Max-SR=WFMRC=low-complexity

MMSE. Compared with conventional MRC, the proposed five methods are more robust against

the change of jamming power at Mallory. In particular, the proposed NSP-based Max-WFRP

is independent of the change in jamming power at Mallory, and the most robust one among

six methods. More importantly, the proposed low-complexity MMSE achieve the same optimal

performance as WFMRC, Max-SR, and conventional MMSE while its complexity is one order

of magnitude lower than those of the latter methods. This is very attractive.

APPENDIX

APPENDIX: DERIVATION OF LOW-COMPLEXITY MMSE

In order to reduce the complexity of the conventional MMSE in (41), let us first define a new

matrix

O = gABβ1PAHH(θAB)vAv

HAH(θAB) +RnA

+RnM+RnB

= gABβ1PAHH(θAB)vAv

HAH(θAB) + gAB(1− β1)PAH

H(θAB)TA,ANTHA,ANH(θAB)

+ gMBPMHH(θMB)TM,ANTHM,ANH(θMB) + σ2

BINB, (64)

where the rank of matrix HH(θAB)vAvHAH(θAB), RnA

, and RnMin matrix O are one. The

receive beamforming at Bob can be rewritten as

vBR =√gABβ1PAO

−1HH(θAB)vA. (65)

18

In order to compute the inverse of matrix O, we rearrange its inverse in the following form

O−1 = [σ2BINB

+A+ gAB(1− β1)PAHH(θAB)TA,ANT

HA,ANH(θAB)︸ ︷︷ ︸

K

+ gMBPMHH(θMB)TM,AN︸ ︷︷ ︸uO

THM,ANH(θMB)︸ ︷︷ ︸

vT

O

]−1

= (K+ uOvTO)−1

(66)

Observing the above expression, it is clear that uO and vO are the rank-one column vectors,

using the Sherman-Morrison formula, we directly have

O−1 = K−1 − K−1uOvTOK−1

1 + vTOK−1uO

= K−1 − gMBPMK−1HH(θMB)TM,ANTHM,ANH(θMB)K

−1

1 + gMBPMTHM,ANH(θMB)K−1HH(θMB)TM,AN

, (67)

Similarly, in the following, by repeatedly making use of the Sherman-Morrison formula four

times, we get the inverse matrix

K−1 = [σ2BINB

+A+ gAB(1− β1)PAHH(θAB)H(θAB)− 2gAB(1− β1)PAH

H(θAB)H(θAB)︸ ︷︷ ︸L

+ gAB(1− β1)PAHH(θAB)h(θt,AB)︸ ︷︷ ︸

uK

hH(θr,AB)︸ ︷︷ ︸vT

K

]−1

= (L + uKvTK)−1

= L−1 − L−1uKvTKL−1

1 + vTKL−1uK

= L−1 − gAB(1− β1)PAL−1HH(θAB)h(θt,AB)h

H(θr,AB)L−1

1 + gAB(1− β1)PAhH(θr,AB)L−1HH(θAB)h(θt,AB), (68)

where uK and vK are the rank-one column vectors, which yields

L−1 = [σ2BINB

+A+ gAB(1− β1)PAHH(θAB)H(θAB)︸ ︷︷ ︸

M

−2gAB(1− β1)PAHH(θAB)h(θt,AB)︸ ︷︷ ︸

uL

• hH(θr,AB)︸ ︷︷ ︸vT

L

]−1

= (M+ uLvTL)−1

= M−1 − M−1uLvTLM−1

1 + vTLM−1uL

= M−1 +2gAB(1− β1)PAM

−1HH(θAB)h(θt,AB)hH(θr,AB)M

−1

1− 2gAB(1− β1)PAhH(θr,AB)M−1HH(θAB)h(θt,AB), (69)

19

where uL and vL are the rank-one column vectors, which yields

M−1 = [σ2BINB

+A︸ ︷︷ ︸N

+ gAB(1− β1)PAHH(θAB)h(θt,AB)︸ ︷︷ ︸

uM

hH(θr,AB)︸ ︷︷ ︸vT

M

]−1

= (N+ uMvTM)−1

= N−1 − N−1uMvTMN−1

1 + vTMN−1uM

= N−1 − gAB(1− β1)PAN−1HH(θAB)h(θt,AB)h

H(θr,AB)N−1

1 + gAB(1− β1)PAhH(θr,AB)N−1HH(θAB)h(θt,AB), (70)

where uM and vM are the rank-one column vectors, which yields

N−1 = [σ2BINB

+ gABβ1PAHH(θAB)vA︸ ︷︷ ︸

uN

vHAH(θAB)︸ ︷︷ ︸

vT

N

]−1

= (σ2BINB

+ uNvTN)−1

= σ−2B INB

− σ−4B uNv

TN

1 + σ−2B vT

NuN

= σ−2B INB

− σ−4B gABβ1PAH

H(θAB)vAvHAH(θAB)

1 + gABβ1PAvHAH(θAB)HH(θAB)vA

, (71)

where uN and vN are still the rank-one column vectors.

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