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Cognitive Multiple-Antenna Network with Outage and Rate Margins at the Primary System Behrouz Maham, Member, IEEE, and Petar Popovski, Senior Member, IEEE

Abstract— In the common model for spectrum sharing, cognitive users can access the spectrum as long as the target performance in the legitimate primary system is not violated. In this paper, we consider a downlink primary multiple-inputsingle-output (MISO) system which operates under a controlled interference from the downlink MISO cognitive radio, also called secondary system. We derive exact expressions for outage probability of the primary user under Rayleigh fading, when the primary system is exposed to interference from a secondary base station. We treat three different operating modes for the primary base station (BS): space-time coding, antenna selection, and beamforming, each of them with different channel information requirements. We first consider the case in which the primary BS uses a fixed rate and we analyze the outage probability. In high-SNR scenario, we derive closed-form asymptotic formulas for the outage probability. Furthermore, the optimum transmit power in the secondary system is investigated for maximizing the ergodic capacity when there is an outage constraint at the primary system, and simple solutions are proposed. We then consider the case with rate adaptation at the primary BS and introduce a suitable rate margin and a consistent requirement for primary throughput, for which we determine the outage probability. To be able to accommodate the secondary network, a rate margin is assumed at the primary link. We calculate the exact outage probabilities and average throughput of the adaptive-rate transmit-beamforming primary system and the adaptive-rate transmit antenna-selection primary system. The analytical results are confirmed by simulations, in which we analyze the impact of different parameters, such as the number of antennas and the amount of the interference on the system performance. Index Terms— Cognitive radio, multiple-antenna nodes, outage probability, adaptive-rate transmission.

I. I NTRODUCTION Cognitive radio is a promising method to solve the spectrum scarcity problem [1], [2]. Certain radio resources could be employed by cognitive radio network, i. e. the secondary system, provided that it does not cause an adverse interference to the primary system, a.k.a. spectrum owner or licensee. In this paper, we focus on concurrent cognitive radio network (or commons spectrum usage model [3]), in which secondary users are allowed to use the spectrum even when the primary system is active, provided that the amount of interference This work was supported by the Iran National Science Foundation (INSF) through the project 91000529. Preliminary version of a portion of this work is appeared in Proc. IEEE International Conference on Communications (ICC’13). Copyright (c) 2013 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. Behrouz Maham is with School of ECE, College of Engineering, University of Tehran, Iran. Petar Popovski is with Department of Electronic Systems, Aalborg University, Aalborg, Denmark. Emails: [email protected]. ir,[email protected].

to the primary receiver is kept below a particular threshold. As elaborated in [2], there are two types of interference in this type of system: from the primary to the secondary users and vice versa. Peaceful coexistence of secondary users with primary users requires that the secondary interference at a primary receiver is below a certain threshold [2]. The key element that allows cognitive (secondary) operation for the commons spectrum usage model is the operational margin in the primary system. For example, a time margin implies that the primary system leaves some time intervals unused, such that the secondary user can transmit in those intervals. Specifically, in this paper we consider an outage margin and rate margin. If the primary operates with an outage margin, then its maximal allowed outage probability is higher than the best possible outage probability that can be achieved for the primary link in absence of any secondary interference. Similarly, the primary link applies a rate margin if the transmission rate is discounted relative to the best rate that is achievable in absence of secondary interference. Various works [4]–[8] have discussed achievable rates in cognitive radio from the viewpoint of information theory. The seminal work [5] on the achievable rate of a single cognitive radio user set the constraint of zero interference to the primary user, while the primary system is oblivious to the presence of cognitive radios. In [6], [9] the achievable rate region for cognitive Gaussian multiple-access channels has been characterized. In [8], [10], opportunistic interference cancelation is used in a multiuser uplink cognitive radio system with single antenna nodes in the presence of a single primary link. The performance of cognitive radio multi-input multi-output system under instantaneous interference power constraint at the primary receiver is studied in [11]. The outage and ergodic capacity of single-antenna primary and secondary nodes have been studied in [12], [13]. In [12], the CDF of interference (interference temperature) at the primary receiver is defined as outage and it is assumed that it should be lower than a given threshold. In contrast to our work that is focused on the outage at the primary receiver, the authors in [13] have derived the outage probability at the secondary receiver. Another important problem that has received a significant attention is power allocation in cognitive radio networks. For instance, in [14], the authors proposed mixed distributed-centralized power control for multiuser cognitive radio to maximize the total throughput while maintaining a required signal to interference plus noise ratio (SINR) for primary users. In [15], an energyconstrained cognitive radio network is considered, where each node has a limited energy. Given the data rate requirement and the maximal power limit, a constrained optimization problem

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is formulated in [15] to minimize the energy consumption, while avoiding interference to the existing users. A power control scheme for maximum sum-rate of fading multiple access network is proposed in [16] under instantaneous interference power constraint at the primary network. Finally, [17] assumes perfect channel state information (CSI) of the fading channels from the secondary transmitter to the secondary and primary receivers and derives the optimal power allocation strategies to achieve the ergodic/outage capacities of a single secondary user. This paper investigates the problem of power control in a multiple-input-single-output (MISO) cognitive radio network in which the secondary user coexists with an outage-restricted single-cell MISO primary system. Note that [18] and [19] studied the same scenario when the instantaneous interference power at the primary receiver should be less than a threshold. The preliminary results of our approach have been presented in [20]. In that work, we have considered a fixed-rate primary system that operates with outage margin, where we have derived a closed-form expression of the outage probability and analyzed its behavior as the SNR grows asymptotically. The primary MISO system from [20] does not have instantaneous CSI. This paper contains at least two significant contributions with respect to [20]. First, still assuming fixed rate and outage margin, we consider the availability of full and partial CSI at the primary transmitter, such that it can operate with beamforming and antenna selection, respectively. Second, we introduce the concept of rate margin at the primary system and consider the case of a primary system with beamforming and rate adaptation; yet, due to the unpredictable secondary interference, the primary system needs to operate with a certain rate margin. Note that a similar concept has been introduced in [21] named "rate loss", and the authors have studied its effect on the secondary network. However, in this paper, we study the impact of rate margin on the primary system. Our main contributions can be summarized as follows. For the case in which the primary BS uses a fixed transmission rate: We derive a closed-form expression for the outage probability of MISO primary system in presence of interference from the multiple-antenna secondary transmitter over Rayleigh fading channels. The simplicity of the expression allows optimization of the system performance. The analysis of the asymptotic behavior of the outage probability reveals that in a primary system with Mp transmit antennas, full spatial diversity is still achievable in presence of interference from the space-time coded secondary user. We analyzed three cases of space-time coded primary BS, beamformed primary BS, and antenna selection primary BS. We formulate the problem of maximizing the secondary downlink ergodic capacity for an outage–restricted primary system under the the assumption of no instantaneous CSI knowledge at the secondary transmitter. We propose simple power control schemes to maximize the secondary downlink capacity given the outage probability constraint at the primary receiver under different transmit CSI assumptions at the primary BS. We then treat the case of adaptive rate in beamformed primary system, where a secondary network can be accommodated by choosing an appropriate rate

Fig. 1.

Wireless network with multiple cognitive users access.

margin. We derive a closed-form expression for the outage probability of the primary user under the interference from the multiple-antenna secondary system. Moreover, the average throughput of the adaptive-rate primary user is derived. The remainder of this paper is organized as follows: Section II describes the system model and protocol. The closed–form expressions for the outage probability of the space-time coded MISO primary link are presented in Section III, which are utilized for optimizing the system. Then, the problem of maximization of the secondary capacity through power control of the secondary device and under interference constraints at the primary system is studied. Section IV presents the contents of Section III for the case of antenna-selection primary in which CSI is available at the primary BS. The performance analysis of an adaptive-rate primary system is studied in Section V, when CSI is available at the primary BS. In Section VI, the overall system performance is presented for different numbers of antennas and channel conditions, and the correctness of the analytical formulas is confirmed by simulation results. Conclusions are presented in Section VII. II. S YSTEM M ODEL AND P ROTOCOL D ESCRIPTION We consider a cellular based primary network co-existing with a secondary network, both of them operating in downlink mode. Fig. 1 shows the system model considered in this work. In this model, a primary base station (BS) is communicating with the primary mobile station (MS) and there is an infrastructure-based secondary network. Based on the spectrum-sharing paradigm, the secondary BS is allowed to transmit to the secondary MS using primary frequencies while the service quality in the primary network is still guaranteed. The secondary transmitter (secondary BS) has Ms transmit antennas. Similarly, the primary transmitter (primary BS) has Mp transmit antennas. All the receiving MS are equipped with a single antenna. Note that our results can be also applied in the context of femtocell networks. In the latter setting, we replace the secondary networks by a femtocell network that coexists with the primary macrocell system. As it can be seen from [22, Eq. (6.29)], the boundary on sum-rate capacity of downlink Gaussian network has inter-user interference terms, and thus, its optimization is not straightforward. To avoid inter-user interference, here we assume an orthogonal multiple access, like time-division multiple access (TDMA) or orthogonal frequency division multiple access (OFDMA), is used for primary/secondary systems. Therefore, by orthogonal (in time, frequency, code, or signal domain)

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resource slots inside each cell, intra-cell interferences do not exist in primary and secondary networks, and thus, the system is equivalent to a single user primary/secondary networks. It is assumed that g p = [gp,1 , gp,2 , . . . , gp,Mp ] is the vector of channel coefficients from multiple-antenna primary BS to primary MS, and g = [g1 , g2 , . . . , gMp ] is the vector of channel coefficients of the interference link from the multipleantenna primary BS to the secondary MS. In addition, h = [h1 , h2 , . . . , hMs ], is the vector channel coefficients from the multiple-antenna secondary BS to the secondary MS and hp = [hp,1 , hp,2 , . . . , hp,Ms ] is the vector of interference link from the multiple-antenna secondary BS to the primary MS. Throughout this paper, we assume that all channels are modeled as independent Rayleigh fading, and the primary and secondary receivers have additive white Gaussian noise with variance Np and Ns , respectively. The total average power of the primary BS is P0 and the average power of each antenna of the secondary BS is assumed to be Ps . In the presence of interference from the secondary network, the received signal at the primary MS can be represented as yp =

√

P0

Mp ∑

Ms √ ∑ αn gp,n xp,n + Ps hp,m xm + vp , (1)

n=1

m=1

where xp,n is the signal sent by the n-th antenna of primary BS, normalized as E{|xp,n |2 } = 1, for n = 1, 2, . . . , Mp , ∑Mp 2 n=1 |αn | = 1, 0 ≤ |αn | ≤ 1, hp,m is the interference channel coefficient between the mth antenna of the secondary BS and the primary MS, and vp is the Gaussian noise at the primary MS with variance Np . Moreover, we assume that the signal √ transmitted from the m-th antenna of the secondary BS is Ps xm , where E{|xm |2 } = 1, for m = 1, 2, . . . , Ms . Note that the choice of the coefficients αn , = 1, . . . , Mp , depends on the type of transmission scheme. In the next section, we consider three transmission schemes, i.e., space-time coding, antenna selection, and beamforming. The secondary BS sends the signals simultaneously from all the antennas on the same resource (frequency and time). Thus, the received signal at the secondary MS is given as Mp Ms √ ∑ √ ∑ ys = Ps hm xm + P0 αn gn xp,n + vs , m=1

(2)

n=1

where vs is the Gaussian noise at the secondary MS with variance Ns . Since here we assume the instantaneous channel state information (CSI) is not available at the secondary transmitter, the optimal downlink capacity could be achieved by using full-rate space-time codes [22]. III. F IXED R ATE T RANSMISSION WITH O UTAGE M ARGIN AT THE P RIMARY S YSTEM The primary BS uses a fixed transmission rate of Rp in the downlink. The minimum SNR to support the rate Rp is denoted by γth = 2Rp − 1 and outage occurs if the instantaneous achievable rate is lower than Rp . Let the maximal allowed outage probability be ρmar . If a primary MS has an outage probability of ρout < ρmar , then the primary receiver has an outage margin and can tolerate additional interference from the secondary without violating its target operation regime.

A. Permissible Power Allocation with Unknown CSI at Transmitters In this subsection, we investigate the power levels that can be applied by the secondary BS, such that the spacetime coded primary system operates below a certain outage threshold. Moreover, we assume that instantaneous CSI is not available at the primary and secondary transmitters. Since the CSI is not known at the transmitters, we adopt space-time coding as a feasible transmit diversity technique to be employed at the primary BS deployed with multiple antennas. In the following, we derive the outage probability for the space-time coded MISO primary system when there is an outage margin at the primary receiver. Then, the results are utilized to find the permissible transmit power for the secondary system. 1) Outage Probability at the Primary Receiver: We now consider the case that full-diversity space-time codes, like GABBA codes [23]–[28], [28]–[31], are used at the primary BS, and we calculate a closed-form expression for the outage probability of primary system in presence of interference from a multiple-antenna secondary BS. As stated above, the interference from the secondary BS should be kept below a threshold in order to coexist with the primary system. Thus, the secondary BS chooses a transmit power of Ps such that the outage performance for the primary system is not violated. In a typical space-time code, power is equally distributed among the transmit antennas. Thus, for this case, the expressions in (1) and (2) can be modified by setting αm = √ 1/ Mp , m = 1, . . . , Mp . And, after space-time processing at primary user receiver, the equivalent single-input-single-output (SISO)√model of (1) becomes Mp Ms √ ∑ P0 ∑ |gp,n | xp,n + Ps hp,m xm + vp . (3) Mp n=1 m=1 Considering normalized bandwidth, from (3), the achievable instantaneous rate of the primary user is   P0 ∑Mp 2 n=1 |gp,n | Mp . rp = log2 1 + (4) ∑Ms Np + Ps m=1 |hp,m |2 We derive the outage probability ρout , Pr{rp < Rp } of the primary MS as the probability that the transmit rate Rp is larger than the supported rate rp in (4). This probability, expressed as cumulative distribution function (CDF), depends on the transmission parameters and the channel conditions in both the primary and the secondary system. Thus, from (4), the outage probability at the primary receiver is written as   P0 ∑Mp 2   |g | p,n n=1 M p < γth . (5) ρST ∑Ms out = Pr 2   N p + Ps m=1 |hp,m | Proposition 1: Given two finite sets of independent exponential random variables X = {X1 , . . . , XMp } and Y = {Y1 , . . . , YMs } with means of σx2 and σy2 , respectively, the CDF of ∑Mp n=1 Xn SINRST = ∑Ms 1 + m=1 Ym can be calculated as

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Pr {SINRST < γ} Mp −1

=1−

γ 2 n − σx

∑

σx2n n=0

γ e n! (Ms − 1)!

n ( ∑

)

n (i + Ms − 1)! σy2i ( )i+Ms . σy2 i 1 + σ2 γ x (6)

i=0

Proof: The proof is given in Appendix I. To guarantee that the interference from the secondary system does not reduce the quality of primary MS, the outage probability at the primary receiver should be lower than a P0 |gp,n |2 , threshold. From Proposition 1 and by defining Xn = M p Np P |h

|2

n = 1, . . . , Mp , and Ym = s Np,m , m = 1, . . . , Ms , the p outage probability in (5) can be written as Mp −1

ρST out

−

Mp γth Np

∑ Mpn γthn Npn e P0 σg2p =1 − (P0 σg2p )n n! (Ms − 1)! n=0 n ( ) ∑ (i + Ms − 1)! (Ps σh2 p )i n , × ( )i+Ms 2 i Mp Ps σh p i=0 i 1 + P0 σ2 γth Np

SNRp

) ( )n Mp ( M M SNRsp γth p Mp p ∑ Mp (n + Ms − 1)! ∆= , Mp ! (Ms − 1)! n=0 n Ms P0 σg2

where SNRp = Np p is the received SNR of the primary link. Proof: The proof is given in Appendix II. 2) Power Control at the Secondary System: If we assume instantaneous channels are unknown at the secondary transmitter, ergodic capacity can be used as an objective function for our power allocation scheme which is independent of the instantaneous channel variations. For the ergodic capacity of the secondary system, given as C s = E{Cs }, where E{·} denotes the expectation operation, from (2) we have { C s = E log2

(7)

gp

σg2p

outage probability of the primary receiver in presence of space∆ time coded secondary system can be stated as ρST Mp , out ≈

σh2 p

where and are the means of the channel coefficients |gp,n |2 , n = 1, . . . , Mp and |hp,m |2 , m = 1, . . . , Ms , respectively. To compare the derived outage probability with the outage probability in absence of interference, i.e., ρ0 , we have { ∑M } p P0 n=1 |gp,n |2 ρ0 = Pr < γth Mp Np ( )n M γ N Mp −1 ∑ 1 Mp γth Np − Pp σth2 p 0 gp =1− e 2 n! P σ 0 g p (n=0 ) M γ N γinc Pp0 σth2 p , Mp gp = , (8) (Mp − 1)! where γinc (x, k) is incomplete gamma function of order k [32]. Corollary 1: In the MISO primary link with space-time coding, the minimum outage margin requirement for the primary receiver which allows cognitive system to operate is given in (8), i.e., ρmar > ρ0 . Furthermore, from (7) and (8), the target outage probability at the primary receiver can be expressed in terms of ρ0 as ρmar

×

n ( ) ∑ n i=0

i

(i + Ms − 1)!

( 1+

2 Ms P s σh

SNRsp −1 Ms γinc ((Mp

SNRisp Msi

)i+Ms , − 1)!ρ0 , Mp ) (9)

p where SNRsp = is the average interference power Np −1 received from the secondary BS, and γinc (·, k) denotes the inverse function of γinc (·, k), which for example, could be found by the built-in function "gammaincinv(x, k)" in MATLAB. Proposition 2: In high SNRp scenario, for the space-time coded primary system over Rayleigh fading channels, the

Ms Ps ∑ 1+ |hm |2 Ns,I m=1

)} ,

(10)

where Ps is the transmit power per antenna and Ns,I = Ns when the primary signal is decoded removed at the ∑Mp Pand 0 |g |2 if the primary secondary user and Ns,I = Ns + n=1 n Mp signal is treated as noise at the secondary receiver. Since |hm |2 are i.i.d. random variables, a closed-form lowerbound for the expression in (10) is given by [33, Prop.1] ( (M −1 )) s ∑ Ps σh2 1 C s ≥ log2 1 + exp −κ , (11) NST m m=1 where κ ≈ 0.577 is Euler’s constant, σh2 is the mean of the channel coefficients |hm |2 , m = 1, . . . , Ms , and NST = Ns when the primary signal is decoded at the secondary user and NST = Ns + P0 σg2 if the primary signal is treated as noise at the secondary receiver. Now, using (7) and (11), we formulate the problem of power control in cognitive MISO system (or downlink cognitive network). Therefore, the power allocation problem, which has a constraint on the outage probability at the primary receiver node (MS), can be formulated as ( (M −1 )) s ∑ Ps σh2 1 max log2 1 + exp −κ , Ps NST m m=1 s.t. ρST out ≤ ρmar , Ps ≥ 0.

Mp −1

−1 ((Mp −1)!ρ0 ,Mp ) ∑ γ −n ((Mp − 1)!ρ0 , Mp )e−γinc inc =1 − n! (Ms − 1)! n=0

(

(12)

where ρST out is defined in (7). Since both the objective function in (12) and ρST out are increasing functions of the power coefficient Ps for Ps ≥ 0, to find the optimal value of the problem in (12), the first constraint is turned into the equality. Thus, the single positive root of the increasing function ρST out (Ps ) − ρmar = 0 should be calculated. Hence, for a given initial value, Ps∗ can be calculated using the following recursive equation:   1/Ms Mp γth Np − 2  P0 σg2p  e P0 σgp   , − 1 (13) Ps(t+1) =  2  (t) Mp σhp γth  Φ(Ps ) where

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Mp −1

Φ(Ps(t) ) =1 − ρmar − ×

n ( ∑ i=0

(t) Ps

)

∑

n=1

−

Mp γth Np P σ2

0 gp Mpn γthn Npn e (P0 σg2p )n n! (Ms − 1)!

(t) 1)! (Ps σh2 p )i

(i + Ms − n , (14) ( )i+Ms (t) 2 i Mp Ps σh p i 1 + P0 σ2 γth Np gp

and is the updated version of the power coefficient in the t-th iteration. For the case of single-antenna primary BS, the closed-form solution is obtained in the following proposition. Proposition 3: The optimum transmit power Ps∗ from each secondary BS antenna for maximizing the secondary capacity given an outage probability requirement ρmar at the primary MS, when Mp = 1, can be expressed as [ M γ N ] 2 − M p Pth σ2p P σ 0 g p Ps∗ = e s 0 gp (1 − ρmar )−1/Ms − 1 , (15) Mp σh2 p γth Proof: The proof follows from (13) and (14). B. Permissible Power Allocation with Known CSI at the Primary BS In this subsection, we investigate the permissible power level of the secondary system such that the primary system operates below a certain outage threshold when partial CSI or full CSI is available at the primary BS. By partial CSI, we mean that only the magnitudes of the channel gains g p are available at the primary transmitter and channels are independent. It is shown in [34] that the diversity benefit through antenna selection which only needs the channel magnitude is the same as that with transmit beamforming in which requires full CSI at the transmitter. 1) Partial CSI at the Primary BS: Antenna Selection: The beamforming techniques that depend on the estimation of channel phase parameters, such as angle-of-arrival (AOA), could only be useful if the LOS component dominates [35], which is not the case in pure Rayleigh fading channels. Therefore, in this subsection, we focus on antenna selection technique to get transmit diversity at the primary system. By defining r = arg max |gp,i |, (16) i=1,...,Mp

where | · | is the absolute value operation, the expressions in (1) and (2) can be modified by setting αr = 1, αj = 0, j ̸= r. We derive the outage probability at the primary MS and analyze the permissible power levels at the secondary BS. Similar to the previous subsection, here we assume the fixed transmit rate Rp at the primary system. From (1), the outage probability at the primary user can be represented as { } P0 |gp,r |2 AS ρout = Pr < γth . (17) ∑Ms Np + Ps m=1 |hp,m |2 where r is defined in (16). Proposition 4: Consider a finite set of independent exponential random variables X = {X1 , . . . , XMp } and Y = {Y1 , . . . , YMs }, with means of σx2 and σy2 , respectively. The CDF of the signal-to-interference-plus-noise (SINR) ratio max Xn n∈{1,...,Mp } , (18) SINRAS = ∑Ms 1 + m=1 Ym

can be calculated as Pr {SINRAS < γ}

)−Ms ( ) nγ ( σy2 Mp − σ 2 =1− (−1) e x 1+n 2γ . (19) n σx n=1 Proof: The proof is given in Appendix III. To guarantee that the interference from secondary system does not reduce the quality of primary MS, the outage probability at the primary receiver should be less than its outage margin requirement. From Proposition 1 and by defining P |g |2 P |h |2 Xn = 0 Np,n , n = 1, . . . , Mp , and Ym = s Np,m , p p m = 1, . . . , Ms , the outage probability in (17) can be written as )−Ms ( ) γth Np n ( Mp ∑ Ps σh2 p − P σ2 n−1 Mp 0 gp 1+ n γth . ρout = 1− (−1) e P0 σg2p n n=1 (20) where σg2p and σh2 p are the mean of the channel coefficients |gp,n |2 , n = 1, . . . , Mp and |hp,m |2 , m = 1, . . . , Ms , respectively. Proposition 5: In high SNR scenario for the primary link with the antenna selection transmitter, the outage probability of the system in presence of multiple antenna secondary system can be written as lim ρout = Mp ∑

n−1

SNRp →∞

)( )n Mp ( M ∑ SNRsp γth p Mp (Ms + n − 1)!, p n Ms SNRM p (Ms − 1)! n=0 (21) 2 P0 σg2 Ms Ps σh p where SNRp = Np p and SNRsp = . Np Proof: The proof is given in Appendix IV. As stated in the previous subsection, the performance metric for network optimization is the ergodic capacity, or more precisely, its lower bound expression. From (2), we have { ( )} Ms Ps ∑ AS C s = E log2 1 + |hm |2 , (22) Ns,I m=1 where Ns,I = Ns when the primary signal is decoded and removed at the secondary user and Ns,I = Ns + P0 |gr |2 if the primary signal is treated as noise at the secondary receiver. Similar to (11), a closed-form tight lower-bound for the ergodic capacity ( of the secondary ( system is given by )) M s −1 ∑ Ps σh2 1 AS C s ≥ log2 1 + exp −κ (23) NAS m m=1 where NAS = Ns when the primary signal is decoded at the secondary user and NAS = Ns + P0 σg2 if the primary signal is treated as noise at the secondary receiver. Now, using (20) and (23), we formulate the problem of power control in the cognitive MISO downlink system. Therefore, the power allocation problem, which has a constraint on the outage probability at the primary receiver, can be formulated (as (M −1 )) s ∑ 1 Ps σh2 exp −κ , max log2 1 + Ps NAS m m=1 )−Ms ( ) γth Np n ( Mp ∑ Ps σh2 p − P σ2 n−1 Mp 0 gp s.t. 1− 1+n (−1) e γth ≤ ρmar , n P0 σg2p n=1 Ps ≥ 0.

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(24)

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The objective function in (24) is an increasing function of the power coefficient Ps . Thus, for simplicity, we modify the objective function as Ps . Considering the first constraint in (24), we define ( ) γth Np n Mp ∑ − P σ2 n−1 Mp f (Ps ) =1 − (−1) e 0 gp n n=1 ( )−Ms Ps σh2 p × 1+n γth − ρmar , (25) P0 σg2p where f (Ps ) ≤ 0. Then, using the second constraint in (24), the constraint set becomes Df = {Ps ∈ [0, ∞) | f (Ps ) ≤ 0}. It can be shown that for Ps ≥ 0, the first derivative of f (Ps ) is positive, and thus, f (Ps ) is an increasing function of Ps . Hence, to find the optimal value of the problem in (24), the first constraint is turned into the equality. Thus, the single positive root of the increasing function f (Ps ) = 0 should be found. For this purpose, for some initial value of Ps , the following iterative equation can be used:   1/Ms γ N − P thσ2p 2   P0 σg  Mp e 0 gp  − 1 (26) Ps(t+1) = 2 p    ,  (t) σhp γth Ψ (Ps ) where ( ) − γPth0Nσ2p n Mp n−2 Mp gp ∑ (−1) n e Ψ (Ps(t) ) = 1 − ρmar + ) ( M s 2 (t) σh γth n=2 1 + nPs P0pσ2 gp

(t)

) γ N n Mp − Pth0 σg2pp e , ρ0 . (28) n n=1 Proof: From (25), and by the fact that f (Ps ) is an increasing function of Ps , we put Ps = 0 to obtain (28). To compare the derived outage probability with the outage probability in absence of { interference, }) ρ0 in (28) can be ( Mp ∑

(

(−1)n−1

rewritten as ρ0 = 1 − exp

γ N − P0thσ2p gp

Mp

, and thus, we have

) ( −γth Np 1/Mp = ln 1 − ρ 0 P0 σg2p

(29)

Then, combining (20) and (29), the target outage probability at the primary receiver can be expressed in terms of ρ0 as ( )( Mp )n ∑ Mp 1/M ρmar =1− (−1)n−1 1 − ρ0 p n n=1 [ )−n]−Ms ( SNRsp 1/M (30) ln 1 − ρ0 p × 1+ Ms 2 Ms Ps σh

yp =

Mp Ms √ ∑ √ ∑ |gp,n |2 P0 xp + Ps hp,m xm + vp . (31) ∥g p ∥ n=1 m=1

Thus, if CSI is available at the primary transmitter and optimal beamforming technique is employed, the outage probability in (5) can be modified by simply multiplying the SINR term with Mp . Hence, the analysis in Section III is also valid for the case of beamforming by simply scaling up a constant as stated in a paragraph after (5). Thus, the outage probability in (7) can be modified as γ N Mp −1

ρBF out = 1 −

∑

n=0

− Pthσ2p

γthn Npn e 0 gp (P0 σg2p )n n! (Ms − 1)! n ( ) 2 i ∑ n (i + Ms − 1)! (Ps σhp ) × , (32) ( )i+Ms 2 i Ps σh p i=0 i 1 + P0 σ2 γth Np gp

(27)

and Ps is the updated version of the power coefficient in the t-th iteration. Proposition 6: In the MISO primary link with transmit antenna selection, the minimum outage margin requirement for the primary receiver which allows cognitive system to operate is given by ρmar > 1 −

2) Full CSI at the Primary BS: Transmit Beamforming: Note that the constant M1p in the SINR term in (5) is due to the fact that transmit array gain is not achievable by using spacetime codes. Transmit array gain requires channel knowledge in the transmitter and is achieved by multiplying the transmit signal with the beamforming vector g ∗p /∥g p ∥ [36]. In this case, from (1), the received signal at the primary MS can be written as

p where SNRsp = is the total average interference to Np noise power received from the secondary BS deployed with full-rate, full-diversity space-time codes.

Similarly, the optimal transmit power at the secondary system can be calculated using (13) and (14) by replacing P0 with M p P0 . IV. A DAPTIVE R ATE T RANSMISSION WITH R ATE M ARGIN AND K NOWN CSI AT THE P RIMARY BS So far, we have assumed that the primary system uses a fixed transmit rate of Rp . If the primary transmitter knows the instantaneous SNR of its primary links, which is equivalent to the knowledge of channel gains |gp,n |, n = 1, . . . , Mp , the rate can be adjusted to achieve the maximum capacity. In the cases of transmit beamforming and antenna selection discussed in the previous section, such knowledge is available at the primary BS and can be used for the adaptive rate transmission. A. Transmit Beamforming Primary BS Assuming transmit beamforming employed at the primary system, in the interference-free environment, the rate ( case of ) P 0 ∑Mp 2 is chosen at primary BS. Rp ≤ log2 1 + N |g | n=1 p,n p Note that in contrast to transmit beamforming described in Subsection IV.A which was only used to improve the received power of the useful signal, here the transmission rate is adapted to its maximum possible value. In order to allow the secondary to operate, which introduce some interference that is not directly predictable, we should put a margin for the selected rate such that to have an outage probability of ρmar . Note that, if the rate in the i-th scheduling epoch is selected ∑ to be Rp (i), then the average throughput is τ = (1 − ρmar ) i Rp (i). This gives the idea that the primary could have a requirement on the target throughput. Then, the question is how to dynamically set the requirements on the outage probability ρmar and the

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selection of Rp to attain that throughput. Thus, the constrained optimization problem is formulated as follows. Let us define Rmar as the rate margin to accommodate secondary transmission. Thus, the primary BS decides to allocate the following data rate (33) Rp = log2 (1 + SNRsum ) − Rmar ∑ Mp P0 2 where SNRsum = N n=1 |gp,n | . Assuming adaptive prip mary rate, the outage probability at the primary receiver is given by ρout = Pr {log2 (1 + SINRsum ) < Rp } = Pr {log2 (1 + SINRsum ) < log2 (1 + SNRsum )−Rmar } . (34) The closed-form solution for the outage probability in (34) is presented in the following proposition. Proposition 7: For an adaptive rate MISO primary system with known full CSI at the transmitter, the outage probability in presence of interference from a multiple-antenna secondary system can be written as Mp n Ms −1 ( )( ∑ ∑ ∑ n Ms − 1) AR ρout = i m n=0 i=0 m=0 ( ) M +n−m−1 s 2Rmar (n−i)+1 2Rmar −1 NpMs +n × M 2M P0n σh2np n! Ps s σhp s (Ms − 1)! ( ) )m−n+i+1 (R (R ) N Np 2 −(2Rmar −1) P σp2 + 2 mar 2 mar −1 Ps σh2 p 2 Ps σ 0 gp hp ×e P0 σg2p ( ) √ 2Rmar (2Rmar − 1) × Km−n+i+1 2Np , Υ (Rmar ). (35) P0 σg2p Ps σh2 p The optimal rate margin in the primary link is calculated by Rmar = Υ −1 (ρmar ) for a given outage probability of ρmar , where Υ −1 (·) is an inverse of the function given in (35). Proof: The proof is given in Appendix V. As it can be seen from Proposition 7, the rate margin Rmar is not dependent on the instantaneous knowledge of channels and can be easily estimated at the primary BS. For adaptive rate transmission, when transmit CSI is known at the primary system, the throughput can be written by τ = Rp (1 − ρAR out ), where Rp and ρout are given in (33) and (35), respectively. Moreover, the average throughput can be found as τ = E{Rp } (1 − ρAR (36) out ) where E{Rp } = E{log2 (1 + SNRsum )} − Rmar ∫ ∞ = log2 (1 + x) pxs (x)dx − Rmar (37) 0 2 ∑Mp P0 |g | The Xs = n=1 Np,n has gamma distribution and its PDF p is represented as pxs (x) =

− σx2 xMp −1 x 2Mp e (M −1)! σx

From the secondary network point of view, we assume that the rate margin Rmar should be known to calculate the power control coefficients. If we assume an optimization problem similar to (12), the optimal power coefficients are only dependent on the statistics of the links, and thus, the signaling overhead is similar to the fixed-rate primary transmission studied in previous sections. This is because of the fact that, from Proposition 7, Rmar is only dependent on the statistical CSI of links toward the primary MS. B. Antenna Selection Primary BS In the case of antenna selection discussed in the previous section, for the interference-free environment, we should ) ( P |g |2 choose the rate RpAS ≤ log2 1 + 0 Np,r where r is the p index of the selected antenna at the primary BS as stated in (16). To accommodate the secondary transmission, the rate marAS gin Rmar is used as the tolerance of the interference at the primary MS. Thus, the primary BS decides to allocate the following data rate AS RpAS = log2 (1 + SNRAS ) − Rmar P |g

. Assuming adaptive primary rate, where SNRAS = 0 Np,r p the outage probability at the primary receiver is given by } { = Pr log2 (1 + SINRAS ) < RpAS ρARAS out } { AS . = Pr log2 (1 + SINRAS ) < log2 (1 + SNRAS ) − Rmar (40) The closed-form solution for the outage probability in (40) is presented in the following proposition. Proposition 8: For a MISO primary system with antenna selection transmitter and adaptive rate, the outage probability in presence of interference from a space-time coded secondary system can be written as )Ms−m−1 ( )( )( RAS Mp Ms −1(−1)n−1 Mp Ms −1 mar −1 2 NpMs ∑ ∑ n m ARAS ρout = s (Ms − 1)! PsMs σh2M n=1 m=0 p ( ) ( ) m+1 ( AS ) nN AS 2 Np Rmar 2 p Rmar n 2 − 1 P σ − 2 −1 s hp 2 + P σ2 P0 σg s h p p   × 2e AS 2Rmar P0 σg2p √ ( ( AS )) AS n 2Rmar 2Rmar − 1 AS × Km+1 2Np , Θ(Rmar ). (41) P0 σg2p Ps σh2 p The optimal rate margin in the primary link is calculated by AS AS Rmar = Υ −1 (ρAS mar ) for a given outage probability of ρmar , −1 where Θ (·) is an inverse of the function given in (35). Proof: The proof is given in Appendix VI. The average throughput can be found as τ AS = E{RpAS } (1 − ρARAS out )

. Then, using [32,

Eq. (4.337)], the closed-form solution for E{Rp } is obtained as follows 1 E{Rp } = 2M (Mp − 1)! σx p ∫ ∞ − x × xMp −1 log2 (1 + x) e σx2 dx − Rmar (38)

|

(39)

2

(42)

where AS E{RpAS } = E{log2 (1 + SNRAS )} − Rmar ∫ ∞ AS = log2 (1 + x) pxr (x)dx − Rmar 0

0

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(43)

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8

SNR at the primary receiver from the primary transmitter,

0

10

P0 σg2

Outage Probability with CR, Ms=2, Analytic Result

i.e., SNRp = Np p , and the average received SINR at the primary receiver from the secondary transmitter, i.e., SNRsp =

Outage Probability with CR, Ms=2, Simulations Outage Probability with CR, Ms=4, Analytic Result

2 Ms Ps σh p

Outage Probability ρout

Outage Probability with CR, Ms=4, Simulations Outage Probability without CR, Analytic Result Outage Probability without CR, Simulations −1

10

Mp=1

M =2 p

−2

10

10

20

30

40

50 60 SNRp/SNRsp

70

80

90

100

Fig. 2. Outage probability in the primary system as a function of the ratio between the average diffuse component of the primary signal and the average SNR of the interfering signal from the secondary at the primary receiver. The systems with different number of antennas Ms and Mp are compared by assuming the fixed SNRp = 10 dB.

{ }Mp P |g |2 The PDF of Xr = max 0 Np,n can be derived by p n=1 differentiating its CDF in (59) as ( ) Mp ∑ (−1)n+1 n Mp − σnx2 (44) pxr (x) = e x. 2 σ n x n=1 Then, using [32, Eq. (4.337)], the closed-form solution for E{Rp } is obtained as follows ( ) Mp ∑ (−1)n+1 n log2 (e) Mp E{RpAS } = n σx2 ∫ ∞ n=1 nx − AS × ln(1 + x)e σx2 dx − Rmar 0 ) ( ) nNp ( Mp ∑ Mp nN 2 p P0 σg n AS p E = (−1) log2 (e) e − Rmar i − 2 n P σ 0 gp n=1 (45) ∫x t where Ei(x) = −∞ e /t dt is exponential integral that, for example, could be computed by the built-in function "expint(x)" in MATLAB. From the secondary network point of view, we assume that AS the rate margin Rmar should be known to calculate the power control coefficients. If we assume an optimization problem similar to (24), the optimal power coefficients are only dependent on the statistics of the links, and thus, the signaling overhead is similar to the fixed-rate primary transmission studied in previous sections. This is because of the fact that, AS from Proposition 8, Rmar is only dependent on the statistical CSI of links toward the primary MS. V. N UMERICAL A NALYSIS In this section, we provide numerical results, calculated by assuming channels that are independent Rayleigh distributed with normalized variance. In Fig. 2, the outage probability experienced in the primary receiver as a function of the ratio between the average received

. In addition, the primary BS uses the space-timed Np coded transmission and the fixed transmission rate of Rp = 1 bits/sec/Hz. From Corollary 1, it can be seen when the power ratio goes to infinity, the outage probability is converging to the outage probability of the primary system without cognitive radio, i.e., ρ0 in (8). The curves are shown for a fixed primary transmit power of SNRp = 10 dB, and different BS antenna number of Mp = 1, 2 and Ms = 2, 4. For a fixed amount of interference from the secondary BS, larger number Mp of primary antennas reduces the outage probability at the primary node. However, it is shown that the number Ms of secondary BS antennas has almost no impact on the primary outage probability. In other words, it is shown that by changing the number of the secondary antennas from Ms = 2 to Ms = 4, the outage probability does not vary much for all cases depicted in Fig. 2. This result can be justified by the fact that interference emitted from the secondary antennas is treated as noise at the primary receiver, and the received interference power from the secondary transmitter has been fixed. Another interesting observation from Fig. 2 is that the outage margin could be more sensitive when Mp increases, i.e., the difference between the target outage probability ρmar and ρ0 (no interference) is higher for a larger Mp . Furthermore, the outage probability formula obtained in (7) is confirmed by simulations. Fig. 3 confirms that the analytical expressions in Subsection III-A about the average outage probability are aligned with the simulation results. We consider a cognitive network with Ms = 2 and the primary links with Mp = 2, 4 and Rp = 1 bits/sec/Hz. One observe that the closed-form analytical results based on (7) are similar to the the simulated results. Furthermore, we have sketched the asymptotic outage probability derived in Proposition 2. It can be seen that the asymptotic expression well approximates the simulations in high SNR conditions. In addition, one can observe that Fig. 3 shows that for obtaining an outage probability of 10−5 at the primary MS, around 4 dB more power is required when SNRsp = 0 dB and Mp = 4, compared to interference-free case. Moreover, from observing the behavior of curves in high SNR conditions, it can be observed that full-spatial diversity is achievable in presence of the secondary interference in expense of less coding gain. In Fig. 4, we compare the performance of antenna selection primary system discussed in Section IV with the space-timed coded primary system. It is shown that using transmit CSI knowledge at the primary system can bring us more than 2 dB SNR gain in medium and high SNR conditions in a network with Ms = 2, Mp = 4, and Rp = 1 bits/sec/Hz. Furthermore, the analytical expressions in Subsection IV-A for finding the average outage probability have been confirmed. In addition, one can observe from Fig. 4 that for obtaining an outage probability of 10−5 at the primary MS, around 3.5 dB more power is required when SNRsp = 0 dB, compared to interference-free case. Moreover, from observing the behavior of curves in high SNR conditions, it can be observed that full-

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9

0

0

10

10

−1

10

−2

Mp=4, SNRsp= 0 dB

Mp=2

−3

10

−1

10

−4

10

SNRsp= 0dB, Sim.

ρm

Outage Probability ρout

10

SNRsp= 0dB, Ana.

−5

10

sp

10

M =1, SNR = −10 dB p

SNRsp= −10dB, Sim.

−6

Mp=1, SNRsp= 0 dB

Mp=4

SNR = 0dB, Asy.

10

SNRsp= −10dB, Ana.

sp

M =4, SNR = −10 dB

−2

p

sp

SNR = −10dB, Asy.

−7

sp

10

Primary w/o CR −8

10

0

5

10

15 SNRp [dB]

20

25

Ms=2

30

Ms=10 −3

10

Fig. 3. Outage probability in the space-time coded primary system as a function of the received primary SNR, i.e., SNRp when Ms = 2 and Rp = 1 bits/s/Hz. The systems with different number of Mp and SNRsp are compared.

0

10

Antenna−selection primary, Sim. Antenna−selection primary, Ana. Antenna−selection primary, Asy. Antenna−selection primary, w/o CR Space−time coded primary

−1

10

−3

10

−2

10

−1

ρ0

10

0

10

Fig. 5. The target outage probability at primary node ρmar in presence of cognitive radio versus the outage probability in absence of the secondary system for different values of average interference SNR at primary receiver and different number of primary BS and secondary BS antennas Mp and Ms , respectively.

−2

Outage probability ρout

10

SNRsp = 0 dB

−3

10

−4

10

−5

10

SNRsp = −10 dB

−6

10

−7

10

0

5

10

15 SNRp [dB]

20

25

30

Fig. 4. Outage probability in the primary system as a function of the received primary SNR, i.e., SNRp when Ms = 2, Mp = 4, and Rp = 1 bits/s/Hz. The systems with different number of and SNRsp are compared.

spatial diversity is achievable in presence of the secondary interference in expense of less coding gain, which confirms the result stated in Corollary 3. Fig. 5 compares the target outage probability at primary user in presence of the secondary transmission versus the outage probability in absence of the secondary system, i.e., ρ0 , for different values of average interference SNR at primary receiver, i.e., SNRsp = −10 dB, 0 dB, and different number of primary BS antenna Mp = 1, 4 and secondary BS antenna Ms = 2, 10. In addition, the primary BS uses the spacetimed coded transmission and the fixed transmission rate of Rp = 1 bits/sec/Hz. It can be seen that as interference parameter SNRsp or the number of primary BS antenna Mp go down, the outage probability gets closer to ρ0 . Moreover, it is also observable that the relationship between ρmar and ρ0 is not sensitive to the number of secondary BS antennas Ms , specially for lower interference powers from secondary nodes. Fig. 6 demonstrates the ergodic capacity of the secondary system for different primary outage targets ρmar = 0.01, 0.1,

the fixed transmission rate of Rp = 1 bits/sec/Hz, and different number antennas for the space-time coded primary BS, i.e., Mp = 1, 4, and secondary BS antenna Ms = 2, 10. It is also assumed that the interference from the primary BS is strong, and thus, it can be canceled at the secondary receiver by using successive interference cancelation (SIC). For calculating the achievable capacity, the maximum allowable power is found using the results given in Subsection III-B. We have also assumed that distance of secondary BS to the primary MS is σ2 two times of its distance to the secondary MS, i.e., σ2h = 8 hp when path-loss exponent is equal to 3. It can be seen that when SNR of primary system is low, the secondary system should be turned off. For example, the threshold SNRp for operating point of the secondary system is around 10 dB when ρmar = 10−2 , Mp = 4, and Rp = 1 bits/s/HZ. Furthermore, from Fig. 6, it can be observed that for higher target outage ρmar and primary BS antenna Mp , the secondary capacity is increased. In this numerical example, we have also observed that when the outage probability ρmar = 10−2 is required at primary receiver, SNRp = 25 dB, and Ms = 2 or 10, by increasing Mp from 1 to 4 antenna, capacity of the secondary system is increased close to 2 bits/s/Hz. Finally, we observe that the ergodic capacity of the secondary system is not very sensitive to the number of the secondary BS antennas Ms . When ρmar = 10−1 , Ms = 10 leads to a slightly higher secondary rate gain than a network with Ms = 2. In Fig. 7, we compare the ergodic capacity of the secondary system for different primary systems, i.e., the antenna selection strategy with partial CSI and the space-time coded transmission with unknown CSI. We assume the outage targets of ρmar = 0.01, 0.1, the fixed transmission rate of Rp = 1 bits/sec/Hz, and the primary BS with Mp = 4 and the secondary BS with Ms = 10. We have also assumed that distance of secondary BS to the primary MS is two times of its distance

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10

14

12

Antenna−selection primary, ρ=10−1

Ms=2

Space−time coded primary, ρ=10−1 Spectral efficiency [bits/s/Hz]

Ms=10

10

Spectral efficiency [bits/s/Hz]

Antenna−selection primary, ρ=10−2

12

ρm=0.1, Mp=4

8

ρm=0.1, Mp=1

6

ρm=0.01, Mp=1

Space−time coded primary, ρ=10−2

10

8

6

4

4 2

ρ =0.01, M =4 m

p

0

2

0

0

5

10

15 SNRp [dB]

20

25

30

Fig. 6. Ergodic capacity of the secondary system as a function of the average SNR of the space-time coded primary system in a network with different number of primary BS and secondary BS antennas, outage targets of ρmar = 10−1 , 10−2 , and the primary rate Rp = 1 bits/s/Hz.

0

15 SNRp [dB]

20

25

30

5

10

Ms = 1 4

Ms = 2 Ms = 10

3

10

2

Spectral efficiency [bits/s/Hz]

to the secondary MS, i.e., = 8 when path-loss exponent is equal to 3. It can be seen that a higher secondary rate gain is achievable in a system with the antenna-selection primary BS compared to the space-time coded primary BS. For example, in SNRp = 20 dB, around 1 bits/sec/Hz more spectral efficiency gain can be achieved in the secondary system with antennaselection primary BS for both ρmar = 0.01, 0.1 cases. Fig. 8 shows the effects of the primary outage margin on the performance of the secondary system. In Fig. 8, we compare the ergodic capacity of the secondary system for different outage margin at the primary system with the spacetime coded transmission and unknown CSI. We assume the fixed transmission rate of Rp = 1 bits/sec/Hz, and the primary BS with Mp = 1 and the secondary BS with Ms = 1, 2, 10. It can be seen that, as the outage margin ρmar increases from its minimum value, i.e., ρ0 , we can achieve a higher secondary capacity. Specially, when outage margin is getting close to the extreme values of ρ0 and 1, we have steep increases in the secondary capacity curves. Fig. 9 and Fig. 10 demonstrate outage probabilities of the rate-adaptive beamforming and rate-adaptive antenna-selection primary systems, respectively, studied in Section V, versus the rate margin required to accommodate the secondary network. We have compared outage probability curves for different values of SNRp = 10, 20 dB, SNRsp = −10, 0 dB, Mp = 1, 4, and Ms = 1, 2. In Fig. 9 and and Fig. 10, one can observe that the parameter Rmar should be selected in the range of 0 to 4 for the typical system characteristics. Furthermore, it can be seen that Rmar is more sensitive the to interference emitted from the secondary system, i.e., SNRsp , compared to other parameters such as SNRp and antenna numbers of Mp and Ms . Finally, Fig. 11 compares the throughput of the primary systems as a function of the average transmit SNR of the primary system in a network with a fixed interference of

10

Fig. 7. Comparison of ergodic capacity of the secondary systems as a function of the average SNR of the primary system in a network with Mp = 4 and Ms = 2, outage targets of ρmar = 10−1 , 10−2 , and the primary rate Rp = 1 bits/s/Hz.

10

2 σh 2 σh p

5

10

SNRp = 30 dB, ρ0 ≈ 10−3

1

10

0

10

−1

10

−2

10

SNRp = 20 dB, ρ0 ≈ 10−2

−3

10

−4

10

−5

10

−3

10

−2

10

−1

ρmar

10

0

10

Fig. 8. Ergodic capacity of the secondary system as a function of the outage margin at the primary receiver in a network with different number of secondary BS antennas, Mp = 1, and the primary rate Rp = 1 bits/s/Hz.

SNRsp = −10 dB, antenna numbers of Mp = 4 and Ms = 2, and the outage target of ρmar = 10−2 . As expected, the rate adaptive transmissions which are studied in Section V have larger throughputs than the space-time coded system. Moreover, it can be observed that in medium and high SNR conditions, using beamformed rate-adaptation, around 1.6 bits/sec/Hz and 2.6 bits/sec/Hz gains in spectral efficiency are achievable compared to the fixed rate beamforming and antenna-selection primary systems, respectively. VI. C ONCLUSION In this paper, we consider the concurrent downlink cognitive radio network with multiple antenna transmitters. The outage probability at the primary receiver and permissible power level in the secondary system are investigated for three cases of space-time coded primary BS, transmit beamformed primary BS, and antenna-selection primary BS. In particular, we have derived simple closed-form expressions for the outage

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11

0

10

10

Mp=1, Ms=1, SNRp=10 dB Mp=1, Ms=1, SNRp= 20 dB

Space−time coded primary system Beamformed primary system Antenna−selection primary system Adaptive−rate beamformed primary system Adaptive−rate antenna−selection primary system

9

Mp=4, Ms=2, SNRp= 10 dB

−1

10

8

Mp=4, Ms=2, SNRp= 20 dB

Throughput [bits/sec/Hz]

Outage Probability ρAR out

7 −2

SNRsp=0 dB

10

−3

10

6 5 4 3

−4

10

SNRsp=0 dB

2 1

0

0.5

1

1.5 R

mar

2 2.5 [bits/sec/Hz]

3

3.5

0

4

Fig. 9. Outage probability curves in the rate-adaptive beamforming primary system as a function of the rate margin Rmar in a network with different primary/secondary SNR and antenna numbers.

0

5

10

15 SNRp [dB]

20

25

30

Fig. 11. Throughput of the primary systems as a function of the average transmit SNR of the primary system in a network with the fixed interference of SNRsp = −10 dB, antenna numbers of Mp = 4 and Ms = 2, and the outage target of ρmar = 10−2 .

0

10

Mp=1, Ms=1, SNRp=10 dB

Mp could have a severe impact on the system performance. For instance, it was also shown that when SNRp = 25 dB and ρmar = 10−2 , the spectral efficiency of the cognitive radio network can be increased up to 2 bits/s/HZ if 4 transmit antenna is deployed at the primary link compared to the single antenna case. Interesting extension of the presented work is power control strategy in the primary system, i.e. some water filling in time, in addition to the rate adaptation. With such a strategy the primary can still operate with certain outage, but the average rate should be improved.

Mp=1, Ms=1, SNRp=20 dB Mp=4, Ms=2, SNRp=10 dB

−1

Outage probability

10

Mp=4, Ms=2, SNRp=20 dB SNRsp= 0 dB

−2

10

−3

10

SNRsp= −10 dB

−4

10

0

0.5

1

1.5

2 2.5 Rmar [bits/sec/Hz]

3

3.5

4

Fig. 10. Outage probability curves in the rate-adaptive antenna selection primary system as a function of the rate margin Rmar in a network with different primary/secondary SNR and antenna numbers.

probability at the primary system when there is an interference from a cognitive multiple antenna system. Furthermore, by analyzing the system performance in high SNR regime, it is shown that full spatial diversity is achievable in the primary system under Rayleigh fading and in presence of the transmission from the secondary system. We formulated the problem of finding the maximum transmit power at the multiple antenna secondary system when there is an outage probability target at the primary receiver. Furthermore, we proposed an adaptive-rate antenna-selection primary system to increase the throughput. To be able to accommodate the secondary network, a rate margin was assumed at the primary link. We calculated the exact outage probability and average throughput of the adaptive-rate transmit beamforming primary system. We have shown in the simulations results that the spectral efficiency of both the primary and secondary networks are improved by using adaptive rate transmission at the primary BS. Simulations were in accordance with analytic results. Furthermore, numerical results showed that the performance of the system is not much sensitive to the number of secondary antenna Ms , while the number of primary transmit antenna

A PPENDIX I P ROOF OF P ROPOSITION 1 ∑Ms Ym which has a gamma distribution We define Y = m=1 Ms −1

−

y 2

Ms degrees of freedom with PDF py (y) = σ2Mys (M −1)! e σy . s ∑Myp X Moreover, the random variable X = n=1 n is defined which has Erlang distribution with CDF of Pr {X < x} = ∑Mp −1 xn − σx2 x 1 − n=0 σ2n n! e . By marginalizing over the random x X variable Y , the CDF of the SINRST = 1+Y can be calculated as ∫ ∞ Pr {SINRST < γ} = Pr {X < γ(1 + y)} py (y) dy Mp −1∫ ∞ ∑

= 1−

n=0 0 Mp −1

=1−

∑ ∫

n=0

×

0 − σy2 (γ + γ y)n − γ(1+y) y Ms −1 2 σx y dy e e σx2n n! σy2Ms (Ms − 1)! −

γ

γ n e σx2 σx2n n! σy2Ms (Ms − 1)!

∞

(1 + y)n e

− γσ2y x

y Ms −1 e

− σy2

y

dy.

(46)

0

Using Taylor series for expansion of (1 + y)n , the closed-form solution for integral in (46) is obtained as (6). A PPENDIX II P ROOF OF P ROPOSITION 2 ∑Mp We express the CDF of X = n=1 Xn in Proposition 1 in terms of incomplete gamma function as Pr {X < x} =

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12 (

∫

) x 2 σx

γinc

,Mp

. From [37, Eq. (8.7.1)], the series expan) ( ) ( is given by γinc σx2 , Mp = sion of γinc σx2 , Mp x (x )k+Mp k ∑∞ (−1) x , and thus, for a given x and σx2 ≫ 2 k=0 (Mp +k) k! σx 1, we have ( )Mp 1 x . (47) Pr {X < x} ≈ Mp ! σx2 By marginalizing over the random variable Y and using (47), X the CDF of the SINRST = 1+Y in (46) can be rewritten as ∫ ∞ Pr {SINRST < γ} = Pr {X < γ(1 + y)} py (y) dy 0 ∫ ∞ Mp − σy2 γ (1 + y)Mp y Ms −1 y dy ≈ e 2M 2M σx p Mp ! σy s (Ms − 1)! 0 ) Mp ( ∑ Mp γ Mp (n + Ms − 1)!σy2n , = 2Mp 2Mp n σx Mp ! σx (Ms − 1)! n=0 (48) where in the third equality, the binomial series expansion of (1 + y)n is used and the closed-form solution for integral is obtained. Then, by the fact that σx2 → ∞ is equivalent to SNRp → ∞, the result in Proposition 2 is obtained. (Mp −1)!

A PPENDIX III P ROOF OF P ROPOSITION 4 Using the order statistics, we have Pr {X < x} = ) − x Mp 1 − e σx2 . By marginalizing over the set of independent random variables Y, the CDF of the SINR can be calculated as Pr {SINRAS < γ} { } M ∫ ∞ Ms ∑ ∏s = Pr X < γ + γ yn pm (ym ) dym

∞

P {SINRAS < γ} = Pr {X < γ(1 + y)} py (y) dy 0 ) ∫ ∞( Mp γ(1+y) − σy2 y Ms −1 − y dy. (50) = 1 − e σx2 e σy2Ms (Ms − 1)! 0 Now, by assuming σx2 → ∞, we have lim P {SINRAS < γ} 2 →∞ σx ∫ ∞ Mp − σy2 γ (1 + y)Mp y Ms −1 y dy = e 2M σy2Ms (Ms − 1)! σx p 0 )∫ ∞ Mp ( ∑ − y2 γ Mp Mp = 2Mp 2M y Ms +n−1 e σy dy σx σy s (Ms − 1)! n=0 n 0 ( ) M p ∑ Mp γ Mp = 2Mp σy2n (Ms + n − 1)!, (51) σx (Ms − 1)! n=0 n where in the first and second equalities, we used Taylor series for expansion of ey and (1 + y)n , respectively. By replacing P0 σg2

σx2 = Np p and σy2 = can be written as

2 P s σh p

Np

, the outage probability in (51)

lim P {SINRAS < γth } = 2

M

γth p M

2Mp

(Ms − 1)! ( ) ∑ Mp × Psn σh2np NpMp −n (Ms + n − 1)!. (52) n n=0 Thus, the outage probability of the primary system in the presence of the secondary system is obtained as (21). σx →∞

P0 p σgp

Mp

(

0;Ms −fold

∫

m=1

[

]Mp

m=1 ym

− 2 Ms ∏ e σy dym = 1−e σy2 0;Ms −fold m=1   ym ( ) nγ(1+∑Ms ym ) ∏ ∫ ∞ Mp Ms − σ2 ∑ m=1 M e y − p 2 n σx   (−1) e = dym n σy2 0;Ms−fold n=0 m=1 ∞

−

γ(1+

∑M s m=1 ym ) 2 σx

n γym ym ( ) nγ ∏ Ms ∫ ∞ − σ2 − σ2 x y e Mp − σ 2 = dym (−1) e x 2 σy n m=1 0 n=0 )−Ms ( ) nγ ( Mp ∑ σy2 − σ2 n Mp = (−1) e x 1+n 2γ . (49) σx n n=0 where in the third equality, we used Binomial series expansion. Thus, the closed-form solution for integral in (49) is obtained as (19).

Mp ∑

n

A PPENDIX IV P ROOF OF P ROPOSITION 5 ∑Ms We define Y = m=1 Ym which has a gamma distribution with Ms degrees of freedom and PDF of py (y) = y Ms −1 e σy2Ms (Ms −1)!

− σy2 y

. By marginalizing over the random variable Y , another form of the CDF of SINR in (18) can be calculated as

A PPENDIX V P ROOF OF P ROPOSITION 7 ∑Ms Ps |hp,m |2 By defining Y = , which has a gamma m=1 Np Ms −1

−

y 2

distribution with PDF of py (y) = σ2Mys (M −1)! e σy where s y 2 ∑Mp P0 |gp,n |2 Ps σh σy2 = Np p , and Xs = n=1 , we have SNRsum = Np Xs Xs and SINRsum = 1+Y . Thus, from (34), the outage probability { at the ( primary user)is written as } Xs ρout = Pr log2 1 + < log2 (1 + Xs ) − Rmar . 1+Y (53) By marginalizing over the random variable Y , another form of the outage in (60) )}as { ( can be calculated ∫ ∞ probability 1 py (y) dy ρout = Pr 2−Rm λ < Xs 2−Rm − 1+y 0 } ∫ ∞ { − σy2 λ(1 + y) y Ms −1 y dy = Pr Xs > e y−λ σy2Ms (Ms − 1)! λ (54) ∑M where λ = 2Rmar − 1. Moreover, Xs = n=1 Xn is Erlang ∑M −1 xn − σx2 x , distributed with CDF Pr {Xs < x} = 1 − n=0 σ2n n! e x

P0 σg2

where σx2 = Np p . Thus, we have − σy2 ∫ ∞∑ Mp Ms −1 y y e λn (1 + y)n − σλ(1+y) e x2 (y−λ) 2Ms ρout = dy n σ 2n n! (y − λ) σy (Ms − 1)! λ n=0 x =

Mp ∑

λn (Ms − 1)!

σ 2n n! σy2Ms n=0 x ∫ ∞ n

×

λ

− σy2 (1 + y) Ms −1 − σλ(1+y) 2 x (y−λ) y dy y e (y − λ)n

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(55)

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13

By changing of the variable z = y − λ, (61) can be simplified to Mp ∑ λn ρout = σ 2n n! σy2Ms (Ms − 1)! n=0 x ∫ ∞ λ(1+λ+z) − σ2 z − λ+z (1 + λ + z)n 2 σy x × (λ + z)Ms −1 e dz. n z λ (56) Using Taylor series for expansion of (1 + z)n , the outage probability can be written as Mp ∑ λn ρout = σ 2n n! σy2Ms (Ms − 1)! n=0 x ) ∫ ∞ (∑ n ( ) n n−i i−n × (1 + λ) z i λ i=0 ) (M −1 ( ) s ∑ − λ+z Ms − 1 M s−1−m m − λ(1+λ+z) 2z 2 σx σy × λ z e dz m m=0 Mp n Ms −1 (n)(Ms −1) ∑ ∑ ∑ (1 + λ)n−i λMs +n−m−1 i m = σx2n n! σy2Ms (Ms − 1)! n=0 i=0 m=0 ( ) ∫ ∞ λ(1+λ) −λ σ12 + σ12 − z2 − 2 x y ×e (57) z m−n+i e σx z σy dz

{ ( )} 1 −Rm −Rm = Pr 2 λAS < Xr 2 − py (y) dy 1+y } ∫0 ∞ { − σy2 λAS (1 + y) y Ms −1 y dy e = Pr Xr > 2M y − λAS σy s (Ms − 1)! λAS − σy2 ( ) nλ (1+y) ∫ ∞∑ Mp Ms −1 AS y M y e − p (−1)n−1 e σx2 (y−λAS ) 2Ms = dy n σy (Ms − 1)! λAS n=1 ( ) ∫ ∞ nλ (1+y) Mp ∑ − y2 (−1)n−1 Mnp − AS = e σx2 (y−λAS ) y Ms −1 e σy dy 2Ms σ (Ms − 1)! λAS n=1 y (61) AS where λAS = 2Rmar − 1. By changing of the variable z = y − λAS , (61) can be simplified to ( ) ∫ ∞ Mp ∑ (−1)n−1 Mnp ρAS = (λAS + z)Ms −1 out 2Ms σ (Ms − 1)! 0 n=1 y ∫

ρAS out

∞

×e

−

∑ (−1) Mp

=

nλAS (1+λAS +z) λ +z − AS 2z 2 σx σy

( ) n−1 Mp

dz

(

−λAS

) n 2 σx

+ σ12

n y e 2Ms σ (M − 1)! y s n=1 ∫ ∞ nλ (1+λ ) − ASσ2 z AS − σz2 x y dz × (λAS + z)Ms −1 e 0 Mp Ms −1 ∑ ∑ (−1)n−1

(Mp )(Ms −1)

s −m−1 λM AS n m Using [32, Eq. (3.471.9)], the closed-form solution for = σy2Ms (Ms − 1)! integral in (57) is obtained, and thus, the outage probability n=1 m=0 ( ) ∫ ∞ nλ (1+λ ) becomes ) ( −λAS σn2 + σ12 − ASσ2 z AS − σz2 x y x y dz. 1 1 ×e zme (62) −λ σ2 + σ2 Mp n Ms −1 (Ms−1) n−i Ms +n−m−1 x y ∑∑∑ 0 (1+λ) λ e m ρout = Using [32, Eq. (3.471.9)], we have σx2n n! σy2Ms (Ms − 1)! n=0 i=0 m=0 ( ) ( )( ) s −m−1 Mp Ms −1 )m−n+i+1 ( √ ) ∑ ∑ (−1)n−1 Mp Ms −1 λM −λAS σn2 + σ12 ( )( 2 2 AS AS n m x y λ(1 + λ)σy 2 λ(1 + λ) e ρ = n ×2 Km−n+i+1 . out n=1 m=0 σy2Ms (Ms − 1)! 2 i σx σx σy ( ( √ ) ) m+1 2 (58) nλ(1 + λAS )σy2 2 nλAS (1 + λAS ) ×2 Km+1 . Then, by replacing λ, σx2 , and σy2 into (58) with their correσx2 σx σy sponding values, we get (35). (63) Then, by replacing λAS , σx2 , and σy2 into (63) with their A PPENDIX VI corresponding values, we get (41). P ROOF OF P ROPOSITION 8 ∑Ms Ps |hp,m |2 By defining Y = , which has a gamma m=1 Np R EFERENCES − y2 Ms −1 distribution with PDF of py (y) = σ2Mys (M −1)! e σy where [1] S. Haykin, “Cognitive radio: Brain-empowered wireless communicas }Mp {y 2 Ps σh tions,” IEEE Trans. Inf. Theory, vol. 23, no. 2, pp. 1971–1988, Feb. P0 |gp,n |2 p 2 σy = Np , and Xr = max , we have 2005. Np

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SNRAS = Xr and SINRAS = by

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n=1

. The CDF of Xr is given

( ) nx Mp ) ( ∑ Mp − σ2 − x Mp = 1− Pr {Xr < x} = 1−e σx2 (−1)n−1 e x, n n=1 (59) P0 σg2

where σx2 = Np p . Thus, from (34), the outage probability at the primary user is written as ( { ) } Xr AS AS ρout = Pr log2 1 + < log2 (1 + Xr ) − Rmar . 1+Y (60) By marginalizing over the random variable Y , another form of the outage probability in (60) can be calculated as

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Behrouz Maham (S’07, M’10) received the B.Sc. and M.Sc in electrical engineering, from University of Tehran, in 2005 and 2007, respectively, and his PhD from the University of Oslo, Norway, in April 2010. He worked as a system engineer from 2006 till 2007 at Iran Telecommunication Research Center. From September 2008 to August 2009 he was with the Dept. of Electrical Engineering at Stanford University, USA. Currently, he is an Assistant Professor in the School of Electrical and Computer Engineering at the University of Tehran. He has held visiting appointments at Aalto University (formerly Helsinki University of Technology) and University of Oulu, Finland, the Alcatel-Lucent Chair at SUPÉLEC in France, and University of Toronto in Canada. His fields of interest span the broad area of scalable wireless communication and networking, with emphasis on relay techniques, cognitive radio, interference management, optimization theory, and nanoscale and molecular communications. He has over 70 publications in major technical journals and conferences. Since April 2011, he has been serving as an Editor for the Transactions on Emerging Telecommunications Technologies (ETT). He served as a Technical Program Chair and technical program committee (TPC) member of several major IEEE conferences.

Petar Popovski (S’97-A’98-M’04-SM’10) received Dipl.-Ing. in electrical engineering (1997) and Magister Ing. in communication engineering (2000) from Sts. Cyril and Methodius University, Skopje, Macedonia, and Ph. D. from Aalborg University, Denmark, in 2004. He is currently a Professor at Aalborg University. He has more than 190 publications in journals, conference proceedings and books and has more than 30 patents and patent applications. He has received the Young Elite Researcher award and the SAPERE AUDE career grant from the Danish Council for Independent Research. He has received six best paper awards, including three from IEEE. Dr. Popovski serves on the editorial board of IEEE Transactions on Communications and IEEE JSAC Cognitive Radio Series. He is a Steering Committee member for IEEE Internet of Things Journal and Chair of the ComSoc subcommittee on Smart Grid Communications. His research interests are in the broad area of wireless communication and networking, communication theory and protocol design.

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