Wireless Energy Harvesting in Interference Alignment Networks with Adversarial Jammers ling Guo t , Nan Zhao t , F. Richard Yu+, Xin Liu t , and Victor C.M. Leung§ t School of Information and Communication Engineering, Dalian University of Technology, Dalian, China +Department of Systems and Computer Engineering, Carleton University, Ottawa, ON, KIS 5B6, Canada §Department of Electrical and Computer Engineering, the University of British Columbia, Vancouver, BC V6T lZ4 Canada Email:
[email protected];{zhaonan.liuxinstar1984}@dlut.edu.cn;
[email protected];
[email protected]
Abstract-Anti-jamming interference alignment (IA) algorithm is an effective method to battle with adversarial jammers. Nevertheless, the abundant power from the jammers and interferences can be harvested by the legitimate users as a natural power supply. In this paper, we propose a novel anti-jamming IA scheme with wireless energy harvesting (EH). In the scheme, the power partition coefficient and transmit power are jointly optimized to minimize the total transmit power of the network, with the requirements of snm rate and harvested energy guaranteed. To further reduce the complexity of the joint optimization, a suboptimal algorithm is also developed with lower complexity. Extensive simulation results are presented to show the effectiveness of the proposed anti-jamming IA scheme with wireless EH. Index Terms-Anti-jamming scheme, energy harvesting, interference alignment, power allocation, power splitting. I. INTRODUCTION
Recently, the emerging interference alignment (IA) has become an effective method to manage interference in wireless networks [1]. The closed-form solutions of IA are difficult to obtain when plenty of users exist in the network. Gomadam et al. proposed a distributed numerical approach in [2] to solve this intractable problem, and the altruistic principle and network reciprocity were utilized to develop some iterative algorithms for IA. The feasible conditions for IA were analyzed in [3], in which Yetis et al. utilized the algebraic geometry to relate the feasibility issue to the problem of determining the solvability of a multivariate polynomial system. On the other hand, it is a challenge to guarantee the security of information transmission in wireless networks [4], [5]. Eavesdropping and jamming are two main attacks at the physical layer of wireless networks. When jamming is considered, the legitimate transmission should not be disrupted by the adversarial jammers. To solve this problem, there exist plenty of anti-jamming techniques due to the serious threat of jamming to legitimate transmission [6]. In our previous work [7], the anti-jamming issue in IA networks is considered to eliminate the jamming signal together with the interferences. In addition, green communication is attracting more and more attentions these years, and energy harvesting (EH) is one of the key methods to realize green communication [8]. This work was supported in part by the Xinghai Scholars Program, the National Natural Science Foundation of China (NSFC) under Grant 61601221, and the Fundamental Research Funds for the Central Universities under DUT16RC(3)045. Xin Liu is the corresponding author.
Since radio-frequency (RF) signals can be leveraged as a carrier for transferring both information and energy in wireless networks, simultaneous wireless information and power transfer (SWIPT) has attracted great attentions [9], [10]. To disturb the legitimate transmission effectively, the power of jamming signals is usually extremely high, which can be reutilized for EH of the legitimate IA network. Thus in this paper, we propose a novel anti-jamming IA scheme with wireless EH. In the scheme, the power partition coefficient and transmit power are jointly optimized to minimize the total transmit power of the IA network. To further reduce the complexity, a suboptimal algorithm is also designed for the joint optimization. Extensive simulation results are presented to show the effectiveness of the proposed anti-jamming IA scheme with wireless EH. The remaining of this paper is arranged as follows. In Section II, the system model of lA-based networks is introduced, and the anti-jamming IA scheme is analyzed briefly. In Section III, a novel anti-jamming IA scheme with wireless EH is proposed, and its suboptimal algorithm is designed. Simulation results are discussed in Section IV. Finally, we conclude this work in Section V. Notation: Id represents the d x d identity matrix. At is the Hermitian transpose of matrix A. 11·11 is the Euclidean norm of a complex vector. I. I denotes the absolute value of a complex number. CN(a, A) is a circularly symmetric complex Gaussian distribution with mean a and covariance matrix A. II. SYSTEM MODEL
A. The IA Wireless Network
Consider a K-user interference network with no jamming signals. M[k] and N[k] antennas are equipped at each transmitter and receiver, respectively. When IA is adopted, the recovered signal at the kth receiver can be expressed as K
y[k] = U[kltH[kk]y[k]x[k] + 2:U[k]tH[ki]y[i]X[i]+U[kltn[k], (1) i=l,i#k [k']
where H Z E C x represents the channel matrIx from transmitter i to receiver k, with each of its elements i.i.d. and following CN(O, ap ), where < ap < 1 denotes the fading [k] M[kJ d[kJ extent caused b~ the path-loss exponent. Y E C x and U[k] E CN[k] xd k] are the precoding and zero-forcing matrices of the kth transmitter and receiver, respectively. d[k] is the
978-1-5090-2860-3/16/$31.00 ©2016 IEEE
N[k]
M[;]
•
°
number of data streams of user k. n[k] E C N1k ] x 1 is the vector of background noise at receiver k following eN (0, (J;;'INlk]). Xli] E Cd!'] x 1 is the vector of data streams of user i, and JE = p[i] is the transmit power of the ith user.
[llx[i]
In
When IA is feasible,
V[k]
and
U[kltH[kiIV[il =
U[k]
0, \:Ii =F k,
B. The Anti-jamming fA Scheme In an IA network with the presence of adversarial jamming
signals, N j is the number of antennas at the jammer, and the received signals of the kth receiver can be written as
+ U[k1tH)kl Zj +U[klt n[k],
(5)
i=l,i¥-k
where H)k] E CN!k xNj is the channel matrix from the janlffier to the kth receiver. Zj represents the vector of the jamming signal, with its power equal to Pj. Due to the disruption of jamming, the transmission rate of user k can be denoted as J
log2 I d1kJ
p!kJ U[kltH[kk]V[kIV[kJtH[kkltU[k]
+ dlkf
U[klt
1
fA receiver :
Fig. 1. A division of labor for the IA users with adversarial jamming signals.
i.e., information decoding (ID) and EH for SWIPT, and the power splitting and power allocation will be jointly optimized to improve the performance of the proposed scheme. A. Wireless Energy Harvesting with Adversarial Jammers
y[k] = U[kltH[kk]V[klx[k]
R[k] =
1
(3)
(4)
U[kltH[kiJv[ilx[il
:
I
(2)
The expression for received signals thus can be simplified as
+L
I D.. IU
should abide by
rank (U[kltH[kk] V[k l ) = d[kl.
K
r------------I
1t (H[k]H[k J J
+ (J21 ) n N[k]
U[k]
.
(6)
From (6), the performance of communication is degraded seriously due to the existence of the jamming signal. Based on the idea of lA, if we re-design the precoding matrices to combat with the jamming and perform the transmission free of interference. In our previous work [7], a novel antijamming scheme is proposed for IA networks. In the scheme, the interferences are aligned into the same subspace as the jamming signals, and thus jamming signals can be perfectly eliminated as well as the interferences. For simplicity, assume that M[k] = M, N[k] = N, and d[kl = d. Based on Theorem 2 in [7], the feasibility condition of the anti-jamming IA scheme can be expressed as
M +N ~ (K + l)d+Nj N ~ N j +d,
From the above analysis, each IA receiver can also perform as both ID and EH terminals through a power splitter to SWIPT. The division of labor for the IA users is shown in Fig. 1. In this anti-jamming IA network, each IA receiver is split in two parts, ID terminal and EH terminal, respectively. The harvested energy will be utilized for battery charging. Assume that an ID terminal and an EH terminal are equipped at each IA receiver in the anti-jamming IA network. Define a power partition coefficient as p[k] E [0,1], which determines the ratio of received power for information transmission of the kth user. Thus 1- p[k) is the portion of received power for EH. In consequence, the received information split to the ID of the kth receiver can be expressed as
yy~ = .JPTkT (~U[k]tH[ki]V[i]X[i] + u[k]tH)k]Zj + U[k]tn[k]) +
U[klt W[k),
(8)
where W[k] rv CN(O, b;;,IN ) is the additional AWGN caused by the procedure of ID. Accordingly, the received SINR at the ID terminal of the kth receiver can be given by (9)
,
(7)
M~d.
When the anti-jamming IA scheme is feasible, the interference and jamming signals can be suppressed completely.
On the other hand, the signal split to the EH of the kth receiver can be expressed as
III. WIRELESS EH IN ANTI-JAMMING IA NETWORKS In the traditional anti-jamming IA scheme [7], it is a huge waste of power to just eliminate the jamming signal. Contrary to the traditional view towards disturbance, it can be treated as a useful resource by IA users to perform wireless EH. Thus, a power splitter can be equipped at each IA receiver to split the received signals into two parts at each time slot,
Accordingly, the harvested power at the EH of the kth receiver can be given by
B. Optimal Power Splitting and Power Allocation Algorithm
Consider the requirement that the minimum transmission rate should be guaranteed to maintain communication, the ill terminal of the kth IA receiver should ensure its SINR to be equal to or greater than a certain value, ,[k]. Meanwhile, the EH terminal of the kth IA receiver should require the harvested power to be not less than a given threshold, elk], to sustain the operation of the IA network. Under the ill and EH constraints, the power partition coefficient and transmit power of all the IA users should be jointly optimized to minimize the total power transmitted by all the IA users. The mathematical optimization function can be expressed as K
min
plkl,P[kJ
s.t.
LP[k] k=1 P[k]p[k]lu[kltH[kk]v[k]
2 1
------,;.[k"--]--::2'--+-----:;;:7 2----'-
P ern
Un
~ ,[k] ,
((1 - p[kl)(~ p[;1 II H[kjlv [;fl-P
j
o~
p[k] ~
II Hlklll '2:e[kl , (12)
1.
Naturally, the SINR and the harvested power of the kth user are both greater than zero, i.e., the thresholds ,[k] > 0 and elk] > O. Thus the partition coefficient p[k] should satisfy o ~ p[k] ~ 1, which is in the constraints of objective function (12). Through the optimization function of (12), the total transmit power of IA users is minimized, with the SINR threshold ,[k] and power harvested threshold elk] guaranteed, k = 1,2, ... , K. However, the optimization problem (12) is non-convex, since the variables p[k] and p[k] are coupled together. To obtain the solutions, we convert (12) into a convex problem as follows
min
s.t.
LP[k] k=1
_l_ p [k] lu[k]tH[kk]v[k]1 ,~
2
> er2 -
n
+
J=1
o~
p[k] ~
1.
To further reduce the computational complexity of Algorithm 1, in this part, a suboptimal algorithm for power splitting and power allocation is proposed, which intends to optimize the the partition coefficient and the transmit power detachedly. The optimization problem (12) contains two coupled variables, which leads to the non-convex property. To make it much easier to solve, we can fix a certain variable first, then find the optimal solution of the other variable. Without considering the power splitting ratio p[k], i.e., set p[k] = 1, we first solve the optimization problem under only the SINR constraint. The SINR optimization problem can be expressed as K
m1n
LP[k]
p[kJ
k=1 p[k]
2 lu[kltH[kk]v[k] 1
----'--er-:::2-+--=672-----'- > - , [k] .
s.t.
n
n
(14)
Apparently, the optimal solution of the objective function
(14) can be expressed as
- [k] = P
6;
p~'
~ pU]IIH[kj]vU]112 >- ((1 elk] _ p.IIH[k] _ p[k]) J J
L..J
C. Suboptimal Power Splitting and Power Allocation Algorithm
,[k] (er;
+ 6;)
2'
(15)
lu[kltH[kk]v[k] 1
Then, we try to find the appropriate partition coefficient p[k]. We define a constant factor A to scale up the transmit power p[k]. Through jointly optimizing A and p[k] to minimize the total transmit power, the minimization problem can be written as
K
p[kJ,Plkl
Algorithm 1 Optimal algorithm for the proposed scheme 1: When time slot n starts, the anti-jamming IA network is formed by the proposed algorithm. 2: Obtain the precoding and decoding matrices V and U through using the anti-jamming IA scheme for the IA users. 3: Solve problem (13) by CVX, and obtain the optimal solutions of the partition coefficient p[k] and the transmit power p[k], k = 1,2, ... , K.
K
min
2 11
'
(13)
(13) is convex due to the fact that both pk and W-.!-ptkJ) are convex functions over p[k] with 0 < p[k] < 1. Thus the optimal solutions to problem (12) can be obtained via solving (13). The optimal solutions can be calculated by the software CVX, which is based on the interior point method. The optimization algorithm can be summarized in Algorithm 1. To further reduce the computational complexity of Algorithm 1, we design a suboptimal algorithm for the anti-jamming IA scheme with wireless EH in the next subsection.
plkJ,A
s.t.
LAP[k] k=1 AP[k]p[k]
IU[kltH[kk]v[k] 1
2
_ _---,+--::-------:,---_'-
«'-
P[k] ern2 +;;:2 un
>
-"
[k]
p[kl )(At, j>[;11I H[kjlv [;1 II
0< p[k]
A> 1.
'+13 II Hlklll ') 2:e[kl,
< 1, (16)
When A = 1 and p[k] = 1, p[k] can satisfy the function (14) for the SINR constraint. While we need to perform both information transmission and energy harvesting, i.e., the
partItion coefficient p[k] should meet 0 < p[k] < 1. As a consequence, >. > 1 in function (16) is constrained for ID and EH. To obtain the closed-form solution of the problem (16) conveniently, a proposition is presented as follows. Proposition 1: Define
40 35
----4-- No optimization, y=30dB, e=OdBm, p=0.5
2
P[k] IU[k]tH[kk]V[k] 1
---+- No optimization, y=20dB, e=OdBm, p=0.5
---e-- No optimization, y=10dB, e=OdBm, p=0.5 ----4-- Suboptimal algorithm, y=30dB, e=OdBm
,[k]
d[k]
L K
=
p[j]
---+- Suboptimal algorithm, y=20dB, e=OdBm
2
II H[kj] v[j] I
---e-- Suboptimal algorithm, y=10dB, e=OdBm
j=1
m[k]
IIH;k]
= Pj
2 11
>.[k] is the largest root of the following equation
J2 n
elk]
>.d k] -
0"; + ( (>'d[k] + m[k])
oLIJ:!!!~~~~ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-1-0
-.
Pj(mW)
Let):. = max>.[k]. Then the suboptimal solutions can be driven as follows p[k]* = 1-
e ( ():.d[kJ
[k]
+ m[k] )
.
(17)
Proof' According to the definition in Proposition 1, the problem (16) can be rewritten as
mm
plkJ,A
s.t.
[k]
J;
>
- >.d k] [k]
0"2n ' elk]
1- P
~ -(--'--(>'-d-'-'-[ko-]+-m-"-[k~]) ,
0
.
p[k]
After combining the first SINR constraint and the second EH constraint in function (18), the optimal problem can be converted to the following expression.
s.t.
fk (>')
A>1
J2 where h(>') = >.d k] ~
:::; 0, elk]
0"; + ((>'d[k] + m[k])
- 1. Consider
= 0, i.e., J;( (>'d[k] + m[k]) + (>.c[k] - 0";) = O. Since the second-order coefficient (_(c[k]d(kJ) < 0, the curve of h(>') =
0";) -
(>'d[k]
+ m[k])
o is a downward parabola.
((>'d[k]
+ m[k])
Obtain the optimal solution P[k] for problem (14). 2 P[k] IU[k]tH[kk]v[k]1 2: Set c[ k] = ----'-----;-:-,------',[k]
Assume that >.[kl] and >.[k2] are two roots of the equation fk(>') = 0, in addition, >.[kl] < >.[k2]. It can be easily observed that when>. = 1, fd>') > O. Since h(>') = 0 is a downward parabola, we can derive that >.[kl] < 1 < >.[k2]. Thus the optimal solutions for problem (19) must be ):. = max >. [k2] , k = 1,2, .. . ,K. On the other hand, we can easily obtain the suboptimal solutions of the partition coefficient based on the second
=
k~l prj] IIH[kj]V[j]
r
r
3:
Set d[k]
4:
Set m[k] = Pj IIH;k]
5:
Set >.[k] as the largest root of the following equation. J2 elk) >.dk] ~ 0"; + ((>'d[k] + m[k]) - 1 = O.
(19)
the quadratic equation h(>') elk] (>.c[k] -
[k]
1 _ -----o_=---e---
Algorithm 2 Suboptimal algorithm for the proposed scheme
(18)
>.
=
1:
> 1.
min
constraint of function (18), i.e., p[k]*
Thus the suboptimal solutions (17) is proved. • Based on Proposition 1, the suboptimal algorithm for the anti-jamming IA scheme with wireless EH can be summarized as Algorithm 2.
>. p
Fig. 2. Comparison of the total transmit power of the anti-jamming IA scheme between the cases when the proposed suboptimal algorithm is adopted and when no optimization is performed, with different values of the power of jammer and 'Y.
6: 7:
Set):. = max>.[k), k = 1,2, ... , K. According to the equation (17), obtain the suboptimal solutions for p[k]* and P[k]*.
IV. SIMULATION RESULTS AND DISCUSSIONS
In the simulations, there are 3 IA users in the wireless network according to the feasibility condition in (7), M = 3, N = 2, N j = 1, d = 1, and the loss exponent ap is set to 0.5. The transmit power of each IA user and the transmit power = - 70dBm, of the jammer is set to P and Pj , respectively. 0"; = -50dBm and ( = 0.5. ,[k] = , and elk] = e for all the IA users.
J;
60 r---'---'-"""'F::I::::c::::Jc::::::I::::I:::::r:::::I::I:::c::::::r::::I::::I:::::r:::~ ~ No optimization, "(=30dB, e=OdBm, p=0.5 55 ---+- No optimization, "(=30dB, e=-3dBm, p=0.5 ---e-- No optimization, "(=30dB, e=-10dBm, p=0.5 50 ~ Suboptimal algorithm, "(=30dB, e=OdBm ---+- Suboptimal algorithm, "(=30dB, e=-3dBm ~4 ---e-- Suboptimal algorithm, "(=30dB, e=-10dBm E '::' 40
~
