Simultaneous Wireless Information and Power Transfer ... - IEEE Xplore

IEEE ICC2016-Workshops: W10-Eighth Workshop on Cooperative and Cognitive Networks (CoCoNet8)

Simultaneous Wireless Information and Power Transfer for Spectrum Sharing in Cognitive Radio Communication Systems Fatma Benkhelifa*, Kamel Tourki**, and Mohamed-Slim Alouini* * Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division King Abdullah University of Science and Technology (KAUST) Thuwal, Makkah Province, Saudi Arabia {fatma.benkhelifa, slim.alouini}@kaust.edu.sa ** Mathematical and Algorithmic Sciences Lab, France Research Center Huawei Technologies Co. Ltd, France [email protected]

Abstract—In this paper, we consider the simultaneous wireless information and power transfer for the spectrum sharing (SS) in cognitive radio (CR) systems with a multi-antenna energy harvesting (EH) primary receiver (PR). The PR uses the antenna switching (AS) technique that assigns a subset of the PR’s antennas to harvest the energy from the radio frequency (RF) signals sent by the secondary transmitter (ST), and assigns the rest of the PR’s antennas to decode the information data. In this context, the primary network allows the secondary network to use the spectrum as long as the interference induced by the secondary transmitter (ST)’s signals is beneficial for the energy harvesting process at the PR side. The objective of this work is to show that the spectrum sharing is beneficial for both the SR and PR sides and leads to a win-win situation. To illustrate the incentive of the spectrum sharing cognitive system, we evaluate the mutual outage probability (MOP) introduced in [1] which declares an outage event if the PR or the secondary receiver (SR) is in an outage. Through the simulation results, we show that the performance of our system in terms of the MOP is always better than the performance of the system in the absence of ST and improves as the ST-PR interference increases. Index Terms—Energy harvesting, antenna switching, spectrum sharing, cognitive radio, mutual outage probability.

I. Introduction Simultaneous wireless information and power transfer (SWIPT) is a promising technique to ensure the selfsustainability and perpetual operation of wireless communication systems. SWIPT technique allows the simultaneous use of the radio frequency (RF) signals for wireless energy transfer (WET), as well as wireless information transfer (WIT). The pioneering papers [2] and [3] have studied the SWIPT in singleinput single-output (SISO) through flat fading and frequency selective channels, respectively. The authors have investigated the optimal tradeoff between the information rate and the energy transfer for the co-located information decoding (ID) and energy harvesting (EH) receivers. The work done in [2] and [3] was extended in [4] where two practical SWIPT schemes were presented for the multiple-input multiple-output (MIMO) communication systems. The practical schemes are the power splitting (PS) and the time switching (TS) that separate the ID and EH transfer over the power domain and the time domain, respectively. The rate-energy (R-E) region was investigated for the co-located and separated ID and EH receivers. In [5], a MIMO decode-and-forward (DF) relay system was considered where the source and the destination

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have single antennas and the relay is an EH multiple-antenna node. The authors investigated a low complexity SWIPT technique which is the antenna switching (AS) policy where the strongest antennas are exploited for ID while the others are used for EH (or vice versa). In [6], the sum rate maximization problem was considered for the two-user multiple-input singleoutput (MISO) interference channel. The authors have shown that even though the interference limits the achievable rate, it helps to gain more energy at the EH receivers. On the other hand, cognitive radio (CR) networks is a promising technique introduced by Mitola to solve the spectrum scarcity of the wireless applications [7]. Indeed, CR networks allow the unlicensed users (or the secondary users (SUs)) to use the spectrum only when the licensed users (or the primary users (PUs)) are idle. Recently, more research interest has been conducted to spectrum sharing (SS) in CR networks where the licensed users allow the unlicensed users to share the spectrum bands as long as the interference of the latter does not harm the licensed users [8]. Many works have studied the secondary capacity with single/multiple antennas at the primary and secondary networks [9]–[13]. The secondary capacity gains were investigated for the spectrum sharing in SISO CR networks with imperfect channel-stateinformation (CSI) [9]. The overlay spectrum sharing in MIMO CR networks was studied in [10] in which exact expressions of the rate and bit error rate were derived, as well as the asymptotic analysis at high signal-to-noise ratio (SNR). In [11], a ratio selection scheme using a mean value(MV)-based power allocation strategy for MIMO CR networks was studied with average/peak interference power and peak transmit power constraints. When the receiver has multiple antennas, the maximum ratio combining (MRC) technique was investigated for CR networks in [12], [13]. The energy efficiency and the spectrum scarcity are two major issues in wireless communication systems. Hence, research interest has recently been conducted to study the SWIPT technique in CR networks. In [14], an opportunistic spectrum access scheme was considered in CR networks where secondary transmitters either harvest energy from ambient transmissions or transmit signals when primary transmitters are far away. In [15], the closed form expression and the high SNR approximation of the outage probability were derived

for underlay CR networks with one primary receiver, one cognitive transmitter-receiver, and one EH relay. In [16], the rate maximization problem is considered where the primary receivers have EH capabilities. In this paper, we propose to study the SWIPT for the spectrum sharing in CR networks where the primary receiver has multiple antennas and uses the AS technique to gather the required energy for its operation. The presence of ST helps to improve the EH requirements at the PR since the interference boosts the energy gains at the EH receivers. Consequently, less antennas at the PR are connected to the EH receivers, and more antennas at the PR are assigned to the ID receivers to compensate the data losses due to the interference. Hence, we have a win-to-win situation at both PR and SR ends. To illustrate the incentive of spectrum sharing in CR networks, we study the performance of our system in terms of the mutual outage probability (MOP) defined in [1] which declares an outage when either the PR or the SR is in an outage. Through the simulation results, we show the benefit from the spectrum sharing in CR networks at both PR and SR sides. II. System Model We consider a cognitive network consisting of a primary transmitter (PT), a primary receiver (PR), a secondary transmitter (ST) and a secondary receiver (SR). The PT, ST, and SR are all equipped with single antenna, while the PR have N p antennas. The channel between the PT and SR, the channel between the PT and the j’th antenna of PR, the channel between the ST and SR, and the channel between the ST and the j’th antenna of PR are denoted by h ps , h p j , h ss , and h s j , ∀ j = 1, . . . , N p , respectively. All the channels h ps , h p j , h ss , and h s j are modeled as flat fading with Rayleigh distribution, with variances λ ps , λ pp , λ ss , and λ sp , respectively. The ST is equipped with a single antenna. The signal-tointerference-plus-noise ratio (SINR) at the SR is written as Pm |h ss |2 σ2s , (1) γs = 2 Ps h ps + 1 σ2 s

where Pm is the transmit power at ST, P s is the transmit power at PT, and σ2s is the variance of the additive white Gaussian noise (AWGN) at the SR. The PR is powered by the energy harvested from the received signals. We assume that PR is equipped with a battery with infinite capacity. Since the PR is equipped with multiple antennas, we exploit the practical EH scheme which is the AS scheme [5]. In the antenna switching scheme, each receiving antenna is used for either the EH mode or the ID mode. Let K be the number of antennas at PR connected to the ID circuits and N p − K be the number of antennas at PR connected to the EH circuits. The antennas are randomly ordered [5]. The combined SINR at the PR is given by [17] 2 P s PK j=1 h p j σ2p γID (K) = , (2) 2 P K ∗ Pm j=1 h p j h s j 2 σ p PK |h p j |2

+1

j=1

where σ2p is the variance of the AWGN noise at the PR. The harvested energy at the PNPR is given by p    j=K+1 E j , if 0 ≤ K < N p Q (K) =  (3)  0, if K = N p

 2  2 where E j = ζ P s h p j + Pm h s j is the harvested energy at the j’th antenna of the PR, and ζ is the conversion efficiency. Remark 1: We can see that Q (K + 1) < Q (K). Hence, Q (K) is strictly decreasing with respect to K. Note also that if the ST is not present, the combined SNR at the PR is given by K0 Ps X h 2 , (4) γ0ID (K0 ) = 2 pj σ p j=1 and the harvested energy PN at the PR is given by p    j=K0 +1 E 0j , if 0 ≤ K0 < N p Q0 (K0 ) =  (5)  0, if K0 = N p 2 where E 0j = ζP s h p j is the harvested energy at the j’th antenna of the PR. A. EH Model: How to Choose K Let us consider the EH model explaining how to choose the number of antennas K out of N p at PR assigned to the ID circuits. Let us denote by • E 0 : the state of the battery at instant 0. We assume that E0 follows a predefined PDF fE0 (·). This assumption was taken only for the sake of simplicity and will be revised in the journal version. • E m : the minimum required energy to be available in the battery. • E p : the energy to be used by the ID and EH circuits during a period T . For a given E0 and a given K = 0, . . . , N p , we assume that E p is known and is constant over the time. Let us denote by Et = E p + Em . • E s : the state of the battery at instant T . For a given E0 and a given K = 0, . . . , N p , we have E s (K) = E0 + Q (K) − E p , which should verify E s (K) ≥ Em ⇔ E0 + Q (K) ≥ Et . (6) In fact, • If E 0 ≥ E t , no harvesting is needed. Hence K = N p and Q (K) = 0. • If E 0 < E t , K is such that 0 ≤ K < N p . The best choice of K should satisfy Q (K + 1) < Et − E0 ≤ Q (K) . (7) Remark 2: If K = 0, so there is a data outage when Q (1) < Et − E0 . In fact, – If Q (1) < Et − E0 ≤ Q (0), all antennas are used for EH and no antennas for ID. So, there is a data outage. – If Q(0) < Et − E0 , the condition in (6) is not satisfied. If all the antennas are used for energy harvesting, they are not enough to reach the minimum required energy Em . So, we have also a data outage. B. PMF of K As shown above, the value of K is random and depends on the value of E0 and the channels h p j and h s j . Hence, we need to derive its probability mass function (PMF). Let us denote by δ p = ζP s λ pp , δ s = ζPm λ sp , and δ ps = δ p − δ s . Let f (1) (·) and F (1) (·) be the PDF and CDF of E j , respectively, and f (N p −k) (·) and F (N p −k) (·) be the PDF and CDF of Q (k), respectively, ∀ j = 1, . . . , N p and ∀k = 0, . . . , N p − 1. For more

details, check Appendices A and B. In fact, we can write the PMF of K, for a given E0 = x > 0, as • If E 0 = x ≥ E t , then ( 1, if k = N p P (K = k|E0 = x) = (8) 0, if k = 0, . . . , N p − 1. •

If E0 = x < Et , then P (K = k|E0 = x)   0,         P Et − x ≤ Q N p − 1 , =   P (Q (k + 1) < Et − x ≤ Q (k)) ,     P (Q (1) < Et − x) ,   0, if     (1)   1 − F (Et − x) , if    R Et −x (N p −k−1) (y) = f  0      × 1 − F (1) (Et − x − y) dy, if     F (N p −1) (E − x) , if t

(9) if if if if

k = Np k = Np − 1 1 ≤ k ≤ Np − 2 k=0

k = Np k = Np − 1 1 ≤ k ≤ Np − 2 k = 0.

Based on the value of δ ps , the PMF of K, for E0 = x < Et , is given by • If δ ps = 0, P (K = k|E0 = x < Et ) (11)   0, if k = N p     − Et −x   Et −x  δ  s 1+ e , if k = N p − 1       Et −x 2(Nδps −k−1)  Et −x 1 = 1+ E −x t δs  (2(N p −k−1)+1) δ s   e− δs , if 1 ≤ k ≤ N p − 2   2(N p −k−1)Γ(2(N p −k−1))      1  , if k = 0, γ 2 N p − 1 , Eδt −x Γ(2(N p −1)) s where Γ (·) and γ (·, ·) are the Gamma function and the lower incomplete Gamma function [18], respectively. If δ ps , 0, P (K = k|E0 = x < Et ) (12)   0, if k = N p     E −x Et −x  − δt −   δ p e p −δ s e δ s   , if k = N p − 1   δ ps   !N p −k−1  √  2δ δ     2π 2p s   δ ps δ − Et −x    Γ(N p −k−1)δ ps δ p e δ p Dk+1 δ ppsδs Et2−x , 1 =   !     δ ps Et −x − Eδt −x   s −δ e D , −1 , if 1 ≤ k ≤ N p − 2  s k+1 δ p δ s 2      ! N p −1  √  2δ p δ s     2π   δ2ps δ ps Et −x δ p +δ s   D , , if k = 0,  1 Γ(N p −1)

δ p δs

2

  0,    E −x E −x  − t  − t  δ p e δ p −δ s e δ s    ,  δ  ps !   Et −x  √  3 − δ1 + δ1 N −k−  2 p p s  2 (E πδ δ −x) e p s t    1  N p −k− 2   Γ(N p −k) δ ps  "    δ p +δs E −x !   δ ps 1  t  1 2 N − k − + I  p N p −k− 2 δ p δ s  2 δ p δs 2      #    δ ps Et −x δ ps Et −x   + δ p δs 2 IN p −k+ 12 δ p δs 2 ,            √ N −1 p  δ ps Et −x δ p +δ s    Γ(N 2π−1) 2δδ2p δs , , D 1 δ p δs 2 δ ps p ps

(10)

•

where Iv (·) is the modified Bessel function of the first kind [18]. Consequently, we deduce P (K = k|E0 = x < Et ) = (14)

δ ps

Rx 1 with Dk (x, a) = 0 uN p −k− 2 e−au IN p −k− 21 (u) du. Using Mathematica, we have  N −k− 23 δ ps Et −x δ ps Et −x p ± ! e δ p δs 2 δ p δs 2 δ ps Et − x   Dk+1 , ±1 = δ p δs 2 2 Np − k − 1 " ! ! ! δ ps Et − x δ ps Et − x 1 × 2 Np − k − ± IN p −k− 12 2 δ p δs 2 δ p δs 2 !# δ ps Et − x δ ps Et − x 1 + I , (13) δ p δ s 2 N p −k+ 2 δ p δ s 2

if k = N p if k = N p − 1

Et −x 2



if 1 ≤ k ≤ N p − 2 if k = 0.

It is worth nothing that, if ST is not present, the same EH model is applicable and the PMF of K0 , for 0 ≤ K0 ≤ N − p−1, is given by P (K0 = k0 |E0 = x < Et ) (15)   0, if k0 = N p    (1)   (E  1 − F − x) , if k0 = N p − 1 t R 0   = Et −x (N p −k0 −1)  (1)  (y) 1 − F0 (Et − x − y) dy, if 1 ≤ k0 ≤ N p − 2 f0   0   F (N p −1) (E − x) , if k0 = 0 t 0   0, if k0 = N p     − Eδt −x   p  , if k0 = N p − 1  e  N p −k0 −1 Et −x  (16) = − E −x 1 t   e δ p , if 1 ≤ k0 ≤ N p − 2  Γ(N p −k0 )  δ p         1 γ N p − 1, Eδt −x , if k0 = 0. Γ(N p −1) p III. Performance Analysis In this section, we study the performance analysis of the described scheme in terms of the mutual outage probability (MOP) which is a metric that was first introduced in [1]. This metric declares an outage event when either the PR or the SR is in an outage. is given h The MOP expression i  by MOP = P γID < ξ p or γ s < ξ s = FγID ξ p + Fγs (ξ s ) , (17) where ξ p = 2R p −1, ξ s = 2Rs −1, R p and R s are the transmission rates at the PT and the ST, respectively, and F X (·) denotes the cumulative density function (CDF) of the random variable X. Note that the right hand side of (17) is obtained since γID and γ s are independent. A. CDF of SINR at PR The  outage  h probability at PR cani be expressed as FγID ξ p = P γID (K) < ξ p & K ≥ 1 + P [K = 0] =

Np X

(18)

h i P γID (K) < ξ p |K = k P [K = k] + P [K = 0] .

k=1

(19)

Let us denote by γ p1 = P 2 K h∗ h s j j=1 p j PK 2 , j=1 |h p j |

g sp =

Ps , σ2p

γ p2 =

Pm , σ2p

g pp =

PK 2 j=1 h p j ,

γ pp = γ p1 g pp , and γ sp = γ p2 g sp . The SINR

at PR is then given by γID (K) =

γ p1 g pp γ pp = . γ p2 g sp + 1 γ sp + 1

(20)

The CDF of γID (K) given as h K = 1, . . . , N p is expressed i FγID (K)|1≤K≤N p (ξ p ) = P γID (K) < ξ p |1 ≤ K ≤ N p (21) # Z ∞ "   γ pp ≤ ξp = Fγ pp |u ξ p (u + 1) fγsp (u) du, (22) =P γ sp + 1 0 where

!

fγsp (u) =

1 1 u − u fgsp = e γ p2 λsp , u ≥ 0, γ p2 γ p2 γ p2 λ sp

and 



Fγ pp |u ξ p (u + 1) =

Z 0

ξ p (u+1)

(23)

B. CDF of SINR at SR Let us denote by γ s1 = Pσm2 , γ s2 = σP2s , γ ss = γ s1 |h ss |2 and s s 2 γ ps = γ s2 h ps . The SINR at the SR can be then written as γ ss γ s1 |h ss |2 γs = = . (33) 2 γ ps + 1 γ s2 h ps + 1 The CDF of γ" s is given by# Z ∞ γ ss ≤ ξs = Fγss |u (ξ s (u + 1)) fγ ps (u) du, Fγs (ξ s ) = P γ ps + 1 0 (34) where ! 1 1 u − u f|h ps |2 = e γs2 λ ps , u ≥ 0, fγ ps (u) = γ s2 γ s2 γ s2 λ ps and

ξ p (u + 1) 1 fγ pp (x) dx = γ K, . Γ(K) γ p1 λ pp (24) !

Fγss |u (ξ s (u + 1)) =

Z

ξ s (u+1)

− ξγs (u+1) λ ss

fγss (x) dx = 1 − e

s1

(35)

.

(36)

0

Consequently, we have Z ∞  − ξ s (u+1) Fγs (ξ s ) = 1 − e γs1 λss

1 − u e γs2 λ ps du (37) Note that we have used the fact that the PDF of g pp is fg pp (g) = γ s2 λ ps 0 w K−1 − g − 1 g λ pp −ξ s  Z ∞  ξs , and the PDF of g sp is fgsp (w) = λ1sp e λsp [17], Γ(K) λKpp e e γs1 λss − γ λ ss + γ 1λ ps u s1 s2 = 1 − e du (38) for g ≥ 0 and w ≥ 0. Consequently, we can write ! Z ∞ γ s2 λ ps 0 ξ p (u + 1) − γ uλsp 1 FγID (K)|1≤K≤N p (ξ p ) = γ K, e p2 du −ξ s γ s1 λ ss Γ(K)γ p2 λ sp 0 γ p1 λ pp (39) e γs1 λss . =1− γ s2 λ ps ξ s + γ s1 λ ss (25) 1 IV. Simulation Results Z ∞ γ λ pp γ p1 λ pp e γ p2 λsp − γp1 λ sp ξtp p2 (K, = dt, γ t) e In this section, we present some simulation results to show ξp Γ(K)γ p2 λ sp ξ p γ p1λ pp the accuracy of the closed form expression of MOP derived (26) in Section III. We assume that the number of simulations is N simulations = 104 . The transmit power at the PT and the or ! Z ∞ Z ∞ Z t transmit power at the ST are both chosen equal to P = Pm = 0 γ (K, t) e−αt dt = yK−1 e−y dy e−αt dt (27) dBm. The number of antennas at PR is N = 10. Thes variances p a ! Za a Z 0∞ of the channels are chosen equal to λ pp = λ pd = λ sd = 30 dBm. The noise variances at PR and SR are both equal to = e−αt dt yK−1 e−y dy 0 a σ2p = σ2s = −110 dBm. The conversion efficiency of the EH ! Z ∞ Z ∞ circuits at PR is ζ = 100%. The minimum required energy + e−αt dt yK−1 e−y dy (28) at PR is Emin = −25 dBm. The distribution of E0 is assumed a y Z ∞ to be the exponential distribution with rate λE0 = 60 dBm. −αa Z a e 1 yK−1 e−y dy + yK−1 e−(1+α)y dy The outage thresholds ξ p and ξ s at PR and SR are equal to = α 0 α a ξ p = ξ s = 0 dBm. We have plotted in Fig.1 the performance (29) analysis of our proposed system in terms of the MOP versus e−αa γ (K, a) + (1 + α)−K Γ (K, (1 + α) a) the power used by the ID and EH circuits Pr = E p /T during a = , period T = 1 for different values of λ sp . We have compared the α (30) MOP expression in (17) to the one obtained by Monte Carlo where Γ (·, ·) is the upper incomplete gamma function [18]. simulation. We can see the agreement between the analytic expression and the Monte Carlo simulation. We have also Thus, we deduce ! compared the MOP to the outage probability at PR without ξp 1 FγID (K)|1≤K≤N p (ξ p ) = γ K, the presence of ST. We can see the gain due to the presence of Γ(K) γ p1 λ pp ST. We can see also that as we increase the variance λ sp of the !−K ! !! 1 ξp 1 γ p1 λ pp 1 γ p1 λ pp interference channel between ST and PR, the MOP increases. γ p2 λ sp +e 1+ Γ K, 1 + . Hence, the interference induced by the signals from the ST ξ p γ p2 λ sp ξ p γ p2 λ sp γ p1 λ pp (31) improves the MOP. This can be explained by the fact that the interference is beneficial for the EH task. Subsequently, Remark 3: If the ST is not present, the CDF of SNR at PR as interference increases, less number of antennas at PR are is given by   required for the EH task, and more number of antennas are  ξ0p  1 0 Fγ0ID (K)|1≤K≤N p (ξ p ) = γ K, (32) used for the ID task which improves the information decoding  . Γ(K)  γ p1 λ pp  at PR. In Fig. 2, we have plotted the CDF of Q(K) with and without the presence of ST versus the power used by the ID

1

P out at PR without ST MOP: λ sp=20 dBm MOP: λ sp=30 dBm

0.8

MOP: λ sp=40 dBm

MOP

0.6

0.4

0.2

------- Analytic expression o Monte Carlo simulation

0 -5

0

5

10 15 P r=Ep/T

20

25

30

Figure 1. The mutual outage probability MOP versus the used power Pr = E p /T for different values of λ sp , with independent observations of E0 obtained from exponential distribution.

1

F(N p-K)(Et-E0)

0.8

Power outage without ST Power outage: λ sp= 20 dBm Power outage: λ sp= 30 dBm Power outage: λ sp= 40 dBm

0.6

0.4

0.2

0 -5

0

5

10 15 P r=Ep/T

20

25

30

Figure 2. The power outage F (N p −K) (Et − E0 ) versus the used power Pr = E p /T for different values of λ sp , with independent observations of E0 obtained from exponential distribution.

and EH circuits Pr = E p /T during a period T normalized to 1. We can see that the presence of ST improves the power outage at the SR. We can see also that the power outage improves as we increase λ sp . V. Conclusion In this paper, we have considered the performance analysis of the simultaneous wireless information and power transfer for the spectrum sharing in the cognitive radio system with a multi-antenna primary receiver that has EH capabilities using the antenna switching technique. We have studied the incentive of the spectrum sharing by evaluating the exact expression of the mutual outage probability. We have validated our analytic expressions by comparing them to the Monte Carlo simulation, and we have shown the benefit due to the presence of ST through the simulation results. References [1] K. Tourki and M. Hasna, “Proactive spectrum sharing incentive for physical layer security enhancement,” in IEEE Global Communications Conference, Exhibition & Industry Forum (Globecom’2015), San Diego, CA, USA, Dec. 2015.

[2] L. Varshney, “Transporting information and energy simultaneously,” in IEEE International Symposium on Information Theory (ISIT’2008), Jul. 2008, pp. 1612–1616. [3] P. Grover and A. Sahai, “Shannon meets Tesla: Wireless information and power transfer,” in IEEE International Symposium on Information Theory Proceedings (ISIT’2010), Jun. 2010, pp. 2363–2367. [4] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” IEEE Transactions on Wireless Communications, vol. 12, no. 5, pp. 1989–2001, May 2013. [5] I. Krikidis, S. Sasaki, S. Timotheou, and Z. Ding, “A low complexity antenna switching for joint wireless information and energy transfer in MIMO relay channels,” IEEE Transactions on Communications, vol. 62, no. 5, pp. 1577–1587, May 2014. [6] C. Shen, W.-C. Li, and T.-H. Chang, “Simultaneous information and energy transfer: A two-user MISO interference channel case,” in IEEE Global Communications Conference (GLOBECOM’2012), Dec. 2012, pp. 3862–3867. [7] J. Mitola and J. Maguire, G.Q., “Cognitive radio: making software radios more personal,” IEEE Personal Communications, vol. 6, no. 4, pp. 13– 18, Aug. 1999. [8] A. Ghasemi and E. Sousa, “Fundamental limits of spectrum-sharing in fading environments,” IEEE Transactions on Wireless Communications, vol. 6, no. 2, pp. 649–658, Feb. 2007. [9] L. Musavian and S. Aissa, “Fundamental capacity limits of cognitive radio in fading environments with imperfect channel information,” IEEE Transactions on Communications, vol. 57, no. 11, pp. 3472–3480, Nov. 2009. [10] R. Manna, R. H. Louie, Y. Li, and B. Vucetic, “Cooperative spectrum sharing in cognitive radio networks with multiple antennas,” IEEE Transactions on Signal Processing, vol. 59, no. 11, pp. 5509–5522, Nov. 2011. [11] K. Tourki, F. Khan, K. Qaraqe, H.-C. Yang, and M.-S. Alouini, “Exact performance analysis of MIMO cognitive radio systems using transmit antenna selection,” IEEE Journal onSelected Areas in Communications, vol. 32, no. 3, pp. 425–438, Mar. 2014. [12] D. Li, “Performance analysis of MRC diversity for cognitive radio systems,” IEEE Transactions on Vehicular Technology, vol. 61, no. 2, pp. 849–853, Feb. 2012. [13] R. Duan, M. Elmusrati, R. Jantti, and R. Virrankoski, “Capacity for spectrum sharing cognitive radios with MRC diversity at the secondary receiver under asymmetric fading,” in IEEE Global Telecommunications Conference (GLOBECOM’2010), Dec. 2010, pp. 1–5. [14] S. Lee, R. Zhang, and K. Huang, “Opportunistic wireless energy harvesting in cognitive radio networks,” Wireless Communications, IEEE Transactions on, vol. 12, no. 9, pp. 4788–4799, Sept. 2013. [15] Z. Yang, Z. Ding, P. Fan, and G. Karagiannidis, “Outage performance of cognitive relay networks with wireless information and power transfer,” IEEE Transactions on Vehicular Technology, vol. PP, no. 99, pp. 1–1, 2015. [16] F. Zhu, F. Gao, and M. Yao, “A new cognitive radio strategy for SWIPT system,” in International Workshop onHigh Mobility Wireless Communications (HMWC’2014), Nov. 2014, pp. 73–77. [17] M. Kang and M.-S. Alouini, “A comparative study on the performance of MIMO MRC systems with and without co-channel interference,” in IEEE International Conference on Communications (ICC’2003), vol. 3, May 2003, pp. 2154–2158 vol.3. [18] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic, 1994.

Appendix A Proof of PDF and CDF of E j for 1 ≤ j ≤ N p We have

−

t1

(40) −

t2

f|h p j |2 (t1 ) = λ1pp e λ pp , and f|hs j |2 (t2 ) = λ1sp e λsp . Let X j = 2 2 p j and Y j = ζPm h s j . Then, we have 1 −x 1 t2 fX j (x) = e δ p and fY j (y) = e− δs . (41) δp δs

where ζP h s

 2  2 E j = ζ P s h p j + Pm h s j ,

Consequently,Zwe deduce z

f (1) (z) =

0

•

z

e− δs fX j (t) fY j (z − t) dt = δ p δs

z

Z

e

δ ps δ p δs

t

dt

(43)

•

δ ps

The CDF of E j is given by • If δ ps = 0,   Z t ze− δzs (1) F (t) = P E j ≤ t = dz (44) 2 0 (δ s ) ! ! Z t δs t t − δt ue−u du = γ 2, = =1− 1+ e s. δs δs 0 (45) If δ ps , 0, 



F (t) = P E j ≤ t = (1)

Z

z t − δp

e

0

δpe

=1−

− δtp

z

− e− δs dz δ ps

(46)

t

− δ s e− δ s . δ ps

(47)

To sum up, the CDF of E j is given by − δts t  ,     δs e 1 − 1 + (1) t t −δ F (t) = P E j ≤ t =  p −δ e− δ s  δ e p s  1 − ,

if δ ps = 0, if δ ps , 0.

δ ps

Remark 4: If ST is not present, the PDF F0(1) (·) of E 0j are given by 1 − z0 f0(1) (z0 ) = e δ p , δp F0(1) (t0 ) = 1 − e

t

− δ0p

f0(1) (·)

(48)

Q (k) = ζ

Np  X

2 2  P s h p j + Pm h s j = X + Y,

(51)

j=k+1

PN p 2 P p 2 h p j , and Y = ζPm Nj=k+1 h s j . The with X = ζP s j=k+1 PDFs of X and Y are given by 1 xN p −k−1 − δxp fX (x) = e , (52) Γ(N p − k) δNp p −k fY (y) =

1

yN p −k−1

Γ(N p − k)

N −k δs p

Then, we can write f (N p −k) (z) = fX+Y (z) =

Z

y

e− δ s .

=

e

Γ(N p − k)

2 

δ p δs

(z) =

√

π p N −k− 1 Γ(N p − k) δ p δ s δ psp 2

×z

N p −k− 12 −

e

1 δs

+ δ1p



z 2

IN p −k− 12

(58) ! δ ps z . δ p δs 2

Consequently, the PDF of Q (k), for 0 ≤ k < N p , is given by (59) f (N p −k) (z) =  2(N p −k)−1 − z z 1  δs ,  if δ ps = 0,  2(N −k) e   δs p    Γ(2(N p −k))   √ z 1 1  1 − + δ π   zN p −k− 2 e δs δ p 2 IN p −k− 12 δ ppsδs 2z , if δ ps , 0.  √  N p −k− 1  Γ(N p −k)

δ p δ s δ ps

2

(60) Note here that we can verify that when k = N p − 1, the PDF of Q (k) is the same as f (1) (·). The CDF of Q (k), for 0 ≤ k < N p , is given by • If δ ps = 0, Z t

F (N p −k) (t) = P (Q (k) ≤ t) =

f (N p −k) (z) dz   t ! 1  γ 2 N p − k , =   . δs Γ 2 Np − k

(61)

0

(62)

If δ ps , 0,

Z t F (N p −k) (t) = P (Q (k) ≤ t) = f (N p −k) (z) dz (63) 0 √ π (64) = p N −k− 1 Γ(N p − k) δ p δ s δ psp 2   ! Z t 1 z 1 δ ps z N p −k− 21 − δ s + δ p 2 × IN p −k− 12 z e dz δ p δs 2 0 √  N −k ! δ ps t δ p + δ s 2π  2δ p δ s  p  2  , , (65) = Dk Γ(N p − k) δ ps δ p δ s 2 δ ps Rx 1 with Dk (x, a) = 0 uN p −k− 2 e−au IN p −k− 12 (u) du. To sum up, the CDF of Q (k),  for 0 ≤ k 0, that ! 1 Z u  βu  √ u µ− 2 βu µ−1 µ−1 βx x (u − x) e dx = π e 2 Γ(µ)Iµ− 12 . β 2 0 (57) Then, we deduce

and CDF

.

1 z2(N p −k)−1 z    2(N −k) e− δs . Γ 2 N p − k δs p

f (N p −k) (z) =

(42)

0

 − δz  ze s    (δs )2 , if δ ps = 0, = − z − z    e δ p −e δs , if δ ps , 0.

•

If δ ps = 0, we have

0

(55)

(66) Remark 5: If ST is not present, the PDF fQ0 (k0 ) (·) and CDF F Q0 (k0 ) (·) of Q0 (K) are given by N −k −1 z 0 p 0 − z0 1 (N p −k0 ) (z0 ) = f0 e δp , (67) Γ(N p − k0 ) δNp p −k0 ! 1 t0 (N −k ) F0 p 0 (t0 ) = γ N p − k, . (68) Γ(N p − k) δp