Performance of Wireless Powered Amplify and ...

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Performance of Wireless Powered Amplify and Forward Relaying Over Nakagami-m Fading Channels With Nonlinear Energy Harvester Yanjie Dong, Student Member, IEEE, Md. Jahangir Hossain, Member, IEEE, and Julian Cheng, Senior Member, IEEE

Abstract—Performance of wireless powered relay with amplifyand-forward protocol is studied for Nakagami-m fading channels. Different from the existing literature, we consider the nonlinearity of the energy harvester. An analytical expression is derived for the complementary cumulative distribution function (CCDF) of the end-to-end signal-to-noise ratio. Using the CCDF, outage capacity is calculated. Index Terms—Nakagami-m fading, nonlinear energy harvester, wireless powered relay.

I. I NTRODUCTION Energy harvesting (EH) has emerged as a promising technology to provide perpetual energy and to prolong lifetime of energy constrained networks [1], [2]. Due to the instability of harvesting energy from natural source such as solar and wind [1], an alternate approach has been proposed to enable wireless devices scavenge energy from radio frequency (RF) signals [2]. Since RF signals carry both information and energy, a joint wireless information and power transfer technique finds applications in dual-hop networks [3]–[5]. Hence, EH becomes economy friendly and vital to keep relays or sensors active without relaying on conventional power sources. The recent research progress on wireless powered relaying (WPR), where the relay scavenges energy from RF signals, is summarized as follows. The end-to-end signal-to-noise ratio (SNR) outage probability was first studied for the WPR with amplify-and-forward protocol [3]. The work in [3] was extended to multiple source-destination pairs scenario [4]. The performance of WPR that can scavenge energy from ambient RF signals was studied in [5]. The scheme in [6] combines the conventional full-duplex relay with the energy harvesting technique. By optimizing the time switching ratio, the authors in [6] maximized the system throughput. In [7], the outage and diversity performances of WPR were investigated using the theory of stochastic geometry. However, with the exception of [8], the current literature [3]–[7] studies the performance of WPR with conventional linear energy harvester. It has been reported in [8] that the linear energy harvester is not practical due to the nonlinearity of the diodes, inductors and capacitors. To the authors’ best knowledge, the performance of WPR with nonlinear energy harvester has not been reported in the literature. Y. Dong, Md. J. Hossain, and J. Cheng are with the School of Engineering, The University of British Columbia, Kelowna, BC, V1V 1V7, Canada (email:{jahangir.hossain, julian.cheng}@ubc.ca). This work was supported by the NSERC Discovery Grants.

While the aforementioned works provide a good understanding of WPR [3]–[7], all of them assumed Rayleigh fading. However, the Nakagami-m fading is a generalized model that matches the various empirically obtained measurement data better than the Rayleigh fading [9]. Field tests results showed that Nakagami-m fading provided the best matches to land mobile and indoor mobile multipath propagation [10]. On the other hand, a statistical analysis of the experiment data in [11] also showed that Nakagami distribution fits the urban multipath channel environment better than other distributions such as Rayleigh, Rician and log-normal distributions. Meanwhile, the beamforming technique qualifies the practically utilization of wireless power transfer [12]. Hence, a performance study of beamforming in WPR with Nakagami-m fading channels and nonlinear energy harvester is a valuable step towards the practical application of energy harvesting technology. In this letter, we study the performance of WPR over independent and non-identical Nakagami-m fading channels with nonlinear energy harvester. The relay with nonlinear energy harvester is solely powered by the harvested energy from the RF signals of the source node. As a result, the transmission power of the relay depends both on the transmission power of the source and the saturation threshold of the energy harvester, which makes the state-of-the-art method presented in [9] not applicable in the analysis of the complementary cumulative density function (CCDF) of the end-to-end SNR. We apply the Jacobian method to obtain an analytical expression for the received SNR’s distribution. To assess the system performance, we also derive the analytical expression of outage capacity. Notations and Functions: Vectors are shown with bold lowercase letters. E [·] denotes the expectation. ∥·∥F is the † Frobenius norm. (·) is the conjugate transpose operator. Ix is the x×1 identity vector. Γ (·) denotes the Gamma function [13, eq. 8.310.1]. Kµ (·) represents the µ-order modified Bessel function of second kind [13, eq. 9.6.22]. II. S YSTEM M ODEL A. Overall Description As shown in Fig. 1, we consider a WPR where a source S communicates with a destination D through a relay R. The nodes S and D are equipped with Ns and Nd antennas, respectively. We assume the relay is equipped with one antenna. Without loss of generality, we assume there is no correlation among these antennas1 . Following [3], [6], we 1 This corresponds to the antennas are physically separated by at least half of the wavelength.

Energy Reception

Information Reception

Information Transmission

aT

1 (1 - a ) T 2

1 (1 - a ) T 2

Output Power (mW)

2

h Pth

Pth

Input Power (mW)

R S D Fig. 1. An illustration of a wireless powered relay system that consists of source, relay and destination. The relay is equipped with a nonlinear energy harvester with input/output relation shown in the northwest of this figure. This figure also illustrates the three phase protocol used in the system.

assume that the direct link S → D does not exist due to severe link attenuation. The channel state information (CSI) is supposed to be available at nodes S, R and D. The channels are assumed to experience quasi-static independent and nonidentically distributed Nakagami-m fading, so that they remain constant during a frame. We separate each frame into three time slots for energy transmission, information transmission and information reception respectively. As shown in Fig. 1, the first time slot is used for energy harvesting with the duration as αT , α ∈ [0, 1]. The remaining frame is equally divided into two time slots for information processing such that the relay uses the half of that, (1 − α) T /2, for information reception and the remaining half, (1 − α) T /2, for information transmission. In the first and the second time slots, the source signal x is weighted with an Ns × 1 beamforming vector [ ]ws to 2 form the transmission vector, where the term E |x| = 1. Using maximum ratio transmission, the received signal (at R is) √ denoted as yr = ∥hs,r ∥F Ps x + nr where nr ∼ CN 0, σr2 is the noise vector; hs,r is the Ns × 1 Nakagami-m1 channel vector; Ps is the transmit power of the source. In the third time slot, the relay amplifies the received signal and forwards it to the destination. Using the maximum ratio combining, the received signal at D after the combiner is denoted by [14] √ yd = ψ∥hr,d ∥F ∥hs,r ∥F Ps x + ψ∥hr,d ∥F nr +

h†r,d ∥hr,d ∥F

nd

(1)

( ) where nd ∼ CN 0, σd2 INd is the noise vector; hr,d is the √ Pr √ Nd × 1 Nakagami-m2 channel vector; ψ = P ∥h ∥ is the s s,r F relay gain which corresponds the relay can invert the first-hop channel perfectly. B. Energy Harvesting Model In most literature, the total harvested energy at relay is formulated as a linear model [5], [7] 2

Linear Pout = ηPs ∥hs,r ∥F

(2)

where η is the energy conversion efficiency and η ∈ [0, 1]; 2 the term Ps ∥hs,r ∥F is the received signal power at the relay R. It has been noted that the linear model is not practical [8], as an energy harvesting circuit usually comprises diodes, inductors and capacitors. On the other hand, the nonlinear energy harvester proposed in [8] is not analytically tractable. As an improvement of conventional linear energy harvester,

we propose a piece-wise linear energy harvester model that captures the saturation character of practical circuit (c.f. Figure 1). As shown in Fig. 1, the energy harvester will output a constant power denoted by ηPth when the input power is beyond the threshold Pth . Hence, we can obtain the transmit power of the relay as { 2 2 2αηPs 1−α ∥hs,r ∥F , Ps ∥hs,r ∥F ≤ Pth Pr = (3) 2 2αηPth Ps ∥hs,r ∥F > Pth 1−α , where η is the energy conversion efficiency in the linear region of the energy harvester; Pth is the saturation threshold. III. E ND - TO -E ND SNR Performing some algebraic manipulations on (1), the endto-end SNR can be expressed as γs,r γr,d γeq = (4) γs,r + γr,d P ∥h ∥2F Ps ∥hs,r ∥2F and γr,d , r σr,d . The dis2 σr2 d 2 2 tribution of( ∥hs,r ∥F and ) ∥hr,d ∥F are( respectively) given m1 2 where by Gamma Ns m1 , Ωs,r and Gamma Nd m2 , Ωmr,d

where γs,r ,

Ωs,r and Ωr,d are the pathloss attenuation for the S → R link and R → D link respectively. With the assistance of (3), we can show the analytical expression of γr,d as  2 2 2 r,d ∥F  2αηPs ∥hs,r ∥F ∥h , Ps ∥hs,r ∥F ≤ Pth 1−α σd2 γr,d = (5) 2 2  2αηPth ∥hr,d2 ∥F , Ps ∥hs,r ∥F > Pth . 1−α σ d

Theorem 1: The CCDF of end-to-end SNR of the studied WPR can be expressed as Pr {γeq > γ} = I1 + I2 where I1 and I2 are respectively shown in (6) and (7) where ( )Ns m1 { } 1 m1 γ m1 γ ω= exp − (8a) Γ (Ns m1 ) Ωs,r a Ωs,r a ( )m ( ) 1 m2 a Ns m1 − 1 (8b) εm,n = n m! Ωr,d b ( )Ns m1 m1 γ { } Ωs,r a m1 γ m2 γ δ= exp − − (8c) Γ (Ns m1 ) Ωs,r a Ωr,d c ( )m ( )( ) 1 m2 γ Ns m1 − 1 m µm,n,k = (8d) n k m! Ωr,d c ( ) n−k+1 2 m2 Ωs,r a νm,n,k = 2µm,n,k (8e) m1 Ωr,d c 2αηPs 2αηPth Pth s with a = P σr2 , b = (1−α)σd2 , c = (1−α)σd2 and d = σr2 . Proof: See Appendix. Remark 1: Let Pth → ∞, we note that d = Pσth → ∞. 2 r Hence, eq. (7) equals to 0 when d → ∞. On the other hand, we can obtain the closed form expression for I1 by setting d → ∞ as ) n−m+1 ( Nd∑ m2 −1 Ns∑ m1 −1 2 Ωs,r a2 m2 I1 = 2ω εm,n Ωr,d bm1 γ m=0 n=0 ( √ ) m1 m2 γ Kn−m+1 2 . (9) Ωs,r Ωr,d b

3

  I1 =



Nd m ∑2 −1 Ns m ∑1 −1

ω

m=0

εm,n

∫

d −1 γ

0

n=0

{ 2a z n−m exp − Ωm − r,d bz

m1 γz Ωs,r a

m=0

n=0

In this section, we simulate the performance of the wireless powered relay with the proposed energy harvester. Hereinafter, unless otherwise specified, the source and destination are equipped with 2 antennas. The relay has only one antenna. The transmission power of the source is 30 dBm. The coverage threshold γ is 40 dB. The power of noise plus interference at both relay and destination are −60 dBm. The pathloss exponent is 2.5. The efficiency, time switching factor and saturation threshold of the energy harvester is 0.8, 0.5 and 10 dBm, respectively. 0

10

−1

CCDF of γeq

10

−2

10

m1=1, m2=2 m1=2, m2=1 m1=m2=2 m1=m2=4 −4

Fig. 2.

35

Simulation 40

45

γ (dB)

50

55

(6)

γ≥d m1 γz Ωs,r a

} dz,

γ γ, Ps ∥Gs,r ∥F ≥ Pth . Hereinafter, we derive the analytical expressions for I1 and I2 . 1) Derivation of the expression of I1 : If γ ≥ d, we note that I1 = 0. For the case γ < d, we can let U , V , X and Y , respectively, denote a∥h 1 ∥2 , b∥h ∥21∥h ∥2 , ∥h 1 ∥2 and 2

s,r F

s,r F

2

r,d F

s,r F

. As both ∥hs,r ∥F and ∥hr,d ∥F are Gamma random variables, the probability density functions (PDFs) for X (and) Y can be, respectively, written as fX (x) = x12 f∥hs,r ∥2 x1 F ( ) and fY (y) = y12 f∥hr,d ∥2 y1 . 1

∥hr,d ∥2F

F

From (4), the relation between (U, V ) and (X, Y ) is U = X a and V = XY b . Using a Jacobian determinant, we obtain the

Ps = 10 mW

1.6

Ps = 20 mW Ps = 30 mW

1.4

Ps = 40 mW

1.2

Simulation

1 0.8 0.6 0.4 0.2 0

0

0.2

0.6

α

0.8

1

0.16

Ps = 10 mW

0.14

Ps = 20 mW Ps = 30 mW

0.12

Ps = 40 mW

0.1

Simulation 0.08 0.06 0.04 0.02 0

0

0.2

0.4

α

0.6

0.8

1

1.5

Ps = 10 mW Ps = 20 mW Ps = 30 mW Ps = 40 mW

1

Simulation

0.5

0

0

(a) ds,r = 2m and dr,d = 24m (b) ds,r = 13m and dr,d = 13m Outage capacity versus time switching factor for different ds,r with ds,r + dr,d = 26 m.

( I1 =

m1 Ωs,r

)Ns Nr m1

=

m1 aΩs,r

γm1 aΩs,r

)Ns Nr m1

a

m=0

m!(au)Ns Nr m1 +1

1 d

Nr Nd m2 −1

Γ (Ns Nr m1 ) )Ns m1

∑

∑

m=0

γ Ns m1 −1

Γ (Ns m1 )

1 m!

(

am2 Ωr,d b

0.2

0.4

α

0.6

0.8

1

(c) ds,r = 24m and dr,d = 2m

    m m 1 )  exp − (2 )− du  Ω b 1 −1 Ωs,r au  −1 r,d a uγ m



1

Nr Nd m2 −1 ∫γ

Γ (Ns Nr m1 ) (

(

0.4

Outage Capacity (Bits/sec/Hz)

2 1.8

Fig. 4.

I2 =

Outage Capacity (Bits/sec/Hz)

Outage Capacity (Bits/sec/Hz)

4

 Ωr,d ab

)m Ns N∑ r m1 −1 ( n=0

m (2

1 uγ

(10)

d

{ } ) γ∫−1 m1 γ (z + 1) m2 a N s N r m1 − 1 z n−m exp − − dz n Ωr,d bz Ωs,r a 0

{ } Nd∑ ( )m m2 −1 m2 γ 1 m2 γ exp − Ωr,d c m! Ωr,d c m=0

∫∞

( 1+

( )Ns m1 −1 { } m1 (z + γ) γ )m z m2 γ 2 1+ exp − − dz z γ Ωr,d cz Ωs,r a

max(d−γ,0)

=

 ( )N m s 1 ) ( m1 γ { } aΩ Nd m  ∑2 −1 1 ( m2 γ )m Ns m ∑1 −1 Ns m1 − 1  s,r m2 γ m1 γ   exp − Ωr,d c − Ωs,r a  Γ(Ns m1 ) m! Ωr,d c  n  m=0 n=0 ) ∞ (  } {  m ∫ ∑  m m γz m γ  n−k 1 2  z exp − Ωr,d cz − Ωs,r a dz, γ < d   × k=0 k d γ

−1

( )N m s 1 ) (  m1 γ { } aΩ  Nd m  ∑2 −1 1 ( m2 γ )m Ns m ∑1 −1 Ns m1 − 1 s,r  m γ m γ 2 1  2 exp − −  Ωr,d c Ωs,r a Γ(Ns Nr m1 ) m! Ωr,d c  n  m=0  )( ( ( ) n=0  n−k+1 ) √ m  ∑  2 m m2 Ωs,r a m1 m2   × Kn−k+1 2γ Ωr,d ,γ ≥ d m1 Ωr,d c Ωs,r ac k k=0

( bv ) joint PDF of (U, V ) as fU V (u, v) = ub fX (au) fY au where fX (·) and fY (·) are the PDFs for X and Y respectively. The derivation for the expression of I1 is shown in (10). 2) Derivation of the expression of I2 : As γs,r ≥ d, the output of energy harvester is saturated. Hence, γs,r = 2 Ps ∥hs,r ∥2 th ∥hr,d ∥2 and γr,d = 2αηP are independent. Hence, σr2 1−α σd2 the derivation for the expression of I2 is shown in (11). We obtain the analytical expression of the CCDF by taking the summation over (10) and (11). R EFERENCES [1] S. Sudevalayam and P. Kulkarni, “Energy harvesting sensor nodes: Survey and implications,” IEEE Commun. Surveys & Tuts., vol. 13, no. 3, pp. 443–461, Sept. 2011. [2] L. Xiao, P. Wang, D. Niyato, D. I. Kim, and Z. Han, “Wireless networks with RF energy harvesting: A contemporary survey,” IEEE Commun. Surveys & Tuts., vol. 17, no. 2, pp. 757–789, Nov. 2014. [3] A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Relaying protocols for wireless energy harvesting and information processing,” IEEE Trans. Wireless Commun., vol. 12, no. 7, pp. 3622–3636, Jul. 2013. [4] Z. Ding, S. M. Perlaza, I. Esnaola, and H. V. Poor, “Power allocation strategies in energy harvesting wireless cooperative networks,” IEEE Trans. Wireless Commun., vol. 13, no. 2, pp. 846–860, Feb. 2014. [5] Y. Gu and S. A¨ıssa, “RF-based energy harvesting in decode-and-forward relaying systems: Ergodic and outage capacities,” IEEE Trans. Wireless Commun., vol. 14, no. 11, pp. 6425–6434, Nov. 2015. [6] C. Zhong, H. A. Suraweera, G. Zheng, I. Krikidis, and Z. Zhang, “Wireless information and power transfer with full duplex relaying,” IEEE Trans. Commun., vol. 62, no. 10, pp. 3447–3461, Oct. 2014.

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