The Impact of Sensing Range on Spatial-Temporal Opportunity

558 XIAN ZHANG, XIAOQIANG LI, ZI YUN, THE IMPACT OF SENSING RANGE ON SPATIAL-TEMPORAL OPPORTUNITY The Impact of Sensing Range on Spatial-Temporal Opportunity Xian ZHANG1, Xiaoqiang LI1, Zi YUN2 1Institute of Communications Engineering, PLA University of Science and Technology, Post 105 of BiaoYing 2, YuDao Street, 210007 NanJing, China 2Unit 94860 of PLA [email protected] Abstract. In this paper, we study the impact of secondary user (SU) sensing range on spectrum access opportunity in cognitive radio networks. We first derive a closed-form expression of spectrum access opportunity by taking into account the random variations in number, locations and transmitted powers of primary users (PUs). Then, we show how SU sensing range affects spectrum access opportunity, and the tradeoff between SU sensing range and spectrum access opportunity is formulated as an optimization problem to maximize spectrum access opportunity. Furthermore, we prove that there exists an optimal SU sensing range which yields the maximum spectrum access opportunity, and numerical results validate our theoretical analysis. Keywords Cognitive radio, spectrum access opportunity, spatial false alarm, optimal SU sensing range. 1. Introduction In cognitive radio networks, secondary user (SU) can utilize a licensed channel of primary user (PU) when there is no active PU within a certain sensing range. SU equipped with frequency-agile radio is capable of sensing a given channel locally and deciding whether the channel is available or not. Conventionally, it is considered that the occurrence of false alarm is caused by the intrinsic feature of radio channel and noise in temporal domain [1] [2]. However, when there is no active PU within SU sensing range, SU can still detect the presence of signal of PU locating outside of SU sensing range due to the signal characteristic in spatial domain. Accordingly, spectrum access opportunity will be lost. This phenomenon has been termed as spatial false alarm [3] [4]. For a given channel, spectrum access opportunity can be characterized as spatial and temporal. Recently, there are several studies devoted to the researches of spectrum sensing taking into account of spatial and temporal characteristics. [3] discussed spectrum sensing from a spatial-temporal domain perspective, and presented unified spatial-temporal metrics to evaluate the performance of spectrum sensing. [4] considered the scenario that a single PU locating in a circular region uniformly and quantified spatial false alarm by a closed-form expression. In [4], the occurrence of spatial false alarm is caused by an active PU locating outside SU sensing range. [5] investigated the occurrence of spatial spectrum opportunity through a careful examination of the definition of spectrum access opportunity, and demonstrated the difference between detecting the signal of PU and detecting spectrum access opportunity. However, most studies focused on the performance analysis of spectrum sensing taking into account spatial and temporal characteristics, but ignored the impact of SU sensing range. Specifically, SU sensing range not only affects SU coverage area, which can be considered as SU capacity, but also the performance analysis of spectrum access opportunity at SU. From the detection’s perspective, the physics meaning of SU sensing range is a certain configuration of detector for a given requirement on system performance. We will discuss it in detail in Section 3. Furthermore, the impact of SU sensing range on spectrum access opportunity will be more complicated especially for a random PU network with multiple PUs due to spatial false alarm. Therefore, it is critical to understand the impact of SU sensing range on spectrum access opportunity, which is the main work of this paper. In this paper, we first present the quantitative analysis of spectrum access opportunity taking into account the spatial and temporal characteristics with random variations in the number, locations and transmitted powers of PUs. Through modeling PU network in terms of stochastic geometry, a closed-form expression of spectrum access opportunity is derived. Modeling PU network in terms of stochastic geometry seems particularly tractable. Then, we show how SU sensing range affects spectrum access opportunity, and formulate the fundamental tradeoff between SU sensing range and spectrum access opportunity as an optimization problem. Finally, we prove that there indeed exists an optimal SU sensing range which yields the maximum spectrum access opportunity. A list of the key mathematical symbols used in this paper is given in Tab. 1. RADIOENGINEERING, VOL. 22, NO. 2, JUNE 2013 559 2. System Model 2.1 Stochastic Geometric Network Model for PU Network Generally, classical analysis methods are insufficient to analyze spectrum access opportunity with random PU networks for the following reasons: (i) It is impossible for SU to know or predict the number and locations of all but perhaps a few PUs. (ii) The received signal power at SU is a function of PU network geometry which the pass-loss and fading characteristics are dependent upon. Stochastic geometry [6] has been proved to be helpful in circumventing the above difficulties. Stochastic geometry provides a natural way of defining and computing macroscopic properties of random network. In this paper, we consider a single SU located at s∈R2, within a decentralized random PU network, as depicted in Fig. 1. Consider a marked Point Process (P.P) ˜ ∏Sall = {xi,pi}with points on the plane ∏Sall ={xi}∈R2and marks pi∈R+, where points represent the locations of active PUs and marks represent PUs transmitted powers at any given time instant in the area of PU network Sall [7]. Standard stochastic scenarios are considered for marked point process: (i) ˜ ∏Sall is a stationary independently marked point process with the location of PU {xi}and intensity λp; (ii) The mark (transmitted power) pidoes not depend on the location of PU [8]. The intensity λpmeans that the average number of PU for unit area is λp. For a given area of primary network, corresponding average number of PU can be obtained. −1500 −1000 −500 0 500 1000 1500 −1000 −500 0 500 1000 x (m) y (m) SU PU Sall Sin rs rp Fig. 1. A snapshot of network topology. 2.2 Channel Model The propagation power loss from location xiof PU ito sis modeled by gil(ks−xik), where giis a random variable that characterizes the cumulative effect of shadowing and fading. l(ks−xik)is the distance-dependent path-loss [9]. giis assumed to be independent and identically distributed (i.i.d.) across different users and also independent of PU’s location, with probability density function (PDF) fG(gi).lis modeled as a power law, l(ks−xik) = Cks−xik−α, where α>2 is the pass-loss exponent and Cis a constant [10]. Moreover, such model holds for ks−xik≥1, and in the remainder of this paper we will only consider the case of ks−xik≥1 [9] [11]. Therefore, the received signal power at SU from PU ican be denoted as Pi=Cpigiks−xik−α[9]. Symbol Definition PU Primary User SU Secondary User P.P Point Process PPP Poisson Point Process Sall Area of PU network Sin Area of SU sensing range Sout The rest area of Sall excluding Sin sLocation of SU xiLocation of PU i piTransmitted power of PU i λpIntensity of PU network giCumulative effect of shadowing and fading l(kk)Distance-dependent path-loss fG(gi)Probability density function of gi CConstant αPath-loss exponent PiReceived signal power at SU from PU i MNumber of samples WBandwidth of channel TSensing time B(s,rs)Disk centered at SU location swith radius rs POpp Probability of spectrum access opportunity P(s)Total received power from all PUs to SU at s γH+ 0Average received SNR at SU Tab. 1. List of symbols. 2.3 Energy Detection Energy detection is the most widely used method for detecting the presence of PU signal in a particular frequency channel [12]. Energy detector simply measures the energy received on the licensed channel during a sensing time and compares it with a sensing threshold. Energy detector will declare a spectrum access opportunity if the measured energy is less than the sensing threshold. Let M=WT be the number of samples (bandwidth time product) where Wis the bandwidth of the channle and Tis the sensing time (sample time) in energy detector. The test statistic for energy detector at SU is T(y) = ∑M m=1|y(m)|2.M, where y(m)is the received signal sample at SU. A binary hypothesis test for temporal-spatial spectrum sensing [4] at SU is denoted as follows: H0: opportunity,H− 0:∏Sall =φ H+ 0:∏Sin =φ∩∏Sout 6=φ H1: no opportunity, ∏Sin 6=φ (1) where φis the null set. Here, Sin =B(s,rs)∈R2is the area of SU sensing range, where B(s,rs)denotes the disk centered at SU location swith radius rs.Sall is the area of PU 560 XIAN ZHANG, XIAOQIANG LI, ZI YUN, THE IMPACT OF SENSING RANGE ON SPATIAL-TEMPORAL OPPORTUNITY network, Sout is the rest area of Sall excluding Sin, namely Sall =Sin +Sout .∏Sin is corresponding P.P consisting of PUs’ locations within Sin, and ∏Sall and ∏Sout are similarly defined [13]. Therefore, the received signal at SU y(m)can be represented as H0:y(m) = (H− 0:n(m), H+ 0:∑ i∈∏Sout 6=φ √Pisi(m)+n(m), H1:y(m) = ∑ i∈∏Sin 6=φ √Pisi(m)+n(m) (2) where m=1,2,...,Mand n(m)is AWGN with power spectral density N0. 3. Closed-Form Expression of Spectrum Access Opportunity SU is permitted to access the channel as long as there is no active PU within SU sensing range. However, for energy detection, false alarm which may result from noise or the signal of PU outside of SU sensing range will affect the sensing performance. Thus, it is necessary to take the impact of false alarm into account when analyzing the probability of spectrum access opportunity. Note that false alarm has temporal and spatial characteristics which are produced under different two hypotheses, i.e., H− 0and H+ 0. Thus, the probability of spectrum access opportunity POpp is the combination of spectrum opportunity probabilities under the hypothesis H− 0 and H+ 0. Consequently, the probability of spectrum access opportunity POpp can be denoted as POpp =P(H− 0)(1−P(H1|H− 0))+ P(H+ 0)(1−P(H1|H+ 0)) (3) where P(H− 0)and P(H+ 0)represent the probabilities of H− 0 and H+ 0, respectively. P(H1|H− 0)is the conventional (temporal) false alarm probability and P(H1|H+ 0)denotes the spatial false alarm probability. In this paper, we consider the case where underlying independently marked PP is Poisson (PPP). According to the proprieties of PPP [8], P(H− 0)is given as P(H− 0) = Pr{∏ Sall =φ}=e−Sall λp.(4) Similar to P(H− 0),P(H+ 0)can be denoted as P(H+ 0) = Pr{∏ Sin =φ,∏ Sout 6=φ}.(5) Note that PUs are independent of each other according to the property of PPP, thus we have P(H+ 0) =Pr{∏ Sin =φ}Pr{∏ Sout 6=φ} =e−Sinλp(1−e−Sout λp). (6) From (2), we know that under hypothesis H− 0, the received noise power is WN0. Thus, as applying central limit theorem (CLT) and the formula of false alarm probability in [12], for a given sensing threshold ε, we have P(H1|H− 0) = Q ε WN0−1√M(7) where Q(·)is the normal Q-function. For energy detection, under hypothesis H+ 0, quantitative analysis on detection performance needs the knowledge of the average total received signal power. From the definition of the received signal power pifrom PU i, we can have the total received signal power from all PUs to SU under hypothesis H+ 0denoted as P(s) = ∑Pi=∑ (xi,pi)∈∏Sout 6=φ pigil(ks−xik).(8) Applying Campbell’s theorem [8], the Laplace transform of P(s)at SU is LP(s)(t) = exp2πλpRGRrp rsr fG(g)(exp(jωppgr−α)−1)drdg (9) where pi=ppis the constant transmitted power of PU. To make the model concrete, we consider the disk model for PU network, namely Sall =B(s,rp)∈R2, so that it will lead to clean tractable solution that highlight the main characteristic regarding spectrum access opportunity. Thus, the average received signal power at SU under hypothesis H+ 0is obtained by E[P(s)] = 1 P(H+ 0)∂LP(t) ∂tt=0 =2πλpˆ C(r2−α s−r2−α p) (α−2)(1−eπr2 sλp−πr2 pλp) (10) where ˆ C=CppRGg fG(g)dg. Consequently, the average received signal-to-noise ratio (SNR) γH+ 0at SU is denoted as γH+ 0=E[P(s)] WN0 =2πλpˆ C(r2−α s−r2−α p) (α−2)(1−eπr2 sλp−πr2 pλp)WN0 .(11) Thus, based on the test static T(y),P(H1|H+ 0)is denoted as P(H1|H+ 0) = Q ε WN0−γH+ 0−1sM 2γH+ 0+1!.(12) Subsequently, substituting (4-6) and (12) into (3) yields the probability of spectrum access opportunity as POpp =e−πr2 pλp1−Q ε WN0−1√M+(e−πr2 sλp −e−πr2 pλp) 1−Q ε WN0−γH+ 0−1sM 2γH+ 0+1!! (13) Note that the probability of total false alarm including temporal (7) and spatial (12) has a direct relationship RADIOENGINEERING, VOL. 22, NO. 2, JUNE 2013 561 with SU sensing range. Thus, when designing energy detector to satisfy the requirement on the false alarm probability, we should take SU sensing range into account. From the perspective of energy detector, the detection performance of false alarm is determined by the sensing threshold. Thus, the physics meaning of SU sensing range can be considered as the sensing threshold in energy detector. In other words, the requirement on the detection performance can be achieved by adjusting the sensing threshold in energy detector to different SU sensing ranges. 4. Tradeoff Between SU Sensing Range and Spectrum Access Opportunity In this section, the characteristics of the impact of SU sensing range on spectrum access opportunity are identified as following. Proposition 1: Without considering the effect of false alarm, P(H0)decreases exponentially with r2 s. Proof: Spectrum access opportunity for SU means that no active PU is within SU sensing range. Obviously, for a given intensity λpof active PU, the larger rs, the larger the area of Sin, which corresponds to the case that no active PU is within SU sensing range with a smaller probability. Mathematically, the probability of spectrum access opportunity resulting from (1) without considering the effect of false alarm can be denoted as P(H0) = P(H− 0)+ P(H+ 0) = e−πr2 sλp,(14) which is exponentially decreasing with r2 s, as shown in Fig. 2. 0 200 400 600 800 1000 1200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SU sensing range rs(m) Probability of spatial opportunity P(H0) Fig. 2. P(H0)versus rswithout considering the effect of false alarm. To show all range of P(H0), the parameters of PU network are chosen as rp=1200 m and λp=6×10−7. In contrast, from the perspective of SU capacity, rs should be designed as large as possible (the area of Sin can be considered as SU capacity to a certain extent). Thus, there should exist a tradeoff between SU capacity and rs. Proposition 2: With the increase of rs, the impact of spatial false alarm P(H1|H+ 0)will be mitigated. Proof: For a given Sall,Sout decreases with the increase of Sin. Correspondingly, the received signal power from PUs in Sout , which induces spatial false alarms, will also be decreasing. We quantitatively characterize the probability of spatial false alarm (12) as a function of γH+ 0. From (12), we have P(H1|H+ 0)is an increasing function of γH+ 0. Moreover, the expression of γH+ 0(11) is a decreasing function of rs. Thus, we can conclude that the probability of spatial false alarm P(H1|H+ 0)is a decreasing function of rs. In other words, the impact of spatial false alarm will be mitigated with the increase of rs. Based on Proposition 1 and 2 mentioned above, we have the following theorem regarding the tradeoff between rsand spectrum access opportunity. Theorem 1. Under the disk propagation model, for λp1, there exists an optimal r∗ swhich yields the maximum spectrum access opportunity for SU. Proof: Note that P(H+ 0) = e−πr2 sλp−e−πr2 pλpis a decreasing function of rs. Furthermore, since only the second term of (13), denoted as PO=P(H+ 0)(1−P(H1|H+ 0)), is affected by rs, then we adopt POhere as the performance measure. Therefore, the problem of maximizing spectrum access opportunity can be formulated as follows: r∗ s=argmax rs{PO}, s.t.1≤rs≤rp. (15) Differentiating POwith respect to rsgives: P0 O=P(H+ 0)01−P(H1|H+ 0)+P(H+ 0)1−P(H1|H+ 0)0. (16) From (6), we have lim rs→1P(H+ 0)0=−2πλpe−πλp≈0 due to λp1. For rs→1, P(H+ 0) = e−πλp−e−πr2 pλp>0 and 0<1−P(H1|H+ 0)according to (12). From Proposition 2, we know lim rs→11−P(H1|H+ 0)0>0. Therefore, substituting into (16), we have lim rs→1P0 O>0.(17) Obviously, similar to the case of rs→1, lim rs→rp P(H+ 0)0<0, and lim rs→rp1−P(H1|H+ 0)>0. Furthermore, it can be verified that lim rs→rp P(H+ 0) = 0. Thus, we have lim rs→rp P0 O<0.(18) 562 XIAN ZHANG, XIAOQIANG LI, ZI YUN, THE IMPACT OF SENSING RANGE ON SPATIAL-TEMPORAL OPPORTUNITY In summary, (17) and (18) mean that POincreases when rsapproaches 1 and decreases when rsapproaches rp. Hence, there is a maximum point of POwithin interval (1,rp), as depicted in Fig. 3. 0 200 400 600 800 1000 1200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 rs (m) POpp λp=6×10−7 (Theory) λp=6×10−7 (Simulation) λp=1×10−6 (Theory) λp=1×10−6 (Simulation) λp=4×10−6 (Theory) λp=4×10−6 (Simulation) Fig. 3. The theory and simulation results on POpp versus rs. The parameters of network are T=2 ms, W=6 MHz, N0=−174 dBm, α=4 and ˆ C=1. The red markers xrepresent the maximum POpp with respect to the optimal rs. In Fig. 3, the experiment parameters are defined as following: the sensing time T=2 ms is chosen according to [12] and the bandwidth W=6 MHz is the standard bandwidth of DTV [12]. The noise power spectral density N0=−174 dBm is the most used value in common environment [7]. Furthermore, it can be further proved that POis concave for a certain range of rsin which P00 O<0. This make the maximum point of POunique in this range. In this case, efficient search algorithms can then be developed, like convex optimization. Otherwise, exhaustive search is needed in order to find the optimal sensing range. The detail of search algorithm is omitted here for the general case. Lemma 1. For a large λp, the impact of rson POpp can be ignored due to severe spatial false alarms. Accordingly, local energy detection will be insufficient to detect spectrum access opportunity. Fig. 3 also depicts POpp versus rsunder different values of λp. It is seen that for both theory and simulation quantities, there exists an optimal r∗ swhich yields the maximum POpp. We notice that for λp=4×10−6, the corresponding average number of PUs in Sall is πr2 pλp≈18.09, and POpp is so small that the impact of rscan be ignored. This is because when the number of PUs is large, local energy detection will always detect the presences of signals of PUs and the impact of spatial false alarm will result in severe loss of spectrum access opportunity. Thus, local energy detection is insufficient to detect spectrum access opportunity for a large λp, and other detection schemes should be used to improve the performance of spectrum sensing, for instance using the position information of PUs and SU or cooperative sensing, which are worthy to investigate in future study. 5. Conclusion In this paper, we quantified spectrum access opportunity by a closed-form expression through modeling PU network in terms of stochastic geometry. This will provide a metric to evaluate sensing performance. Moreover, the impact of SU sensing range on spectrum access opportunity was considered, and the existence of optimal SU sensing range which yields the maximum spectrum access opportunity was also proved. The optimal SU sensing range will provide a fundamental framework for designing SU network. Acknowledgements This work was supported by the National Basic Research Program (973) of China under grant No. 2009CB320400, the National Science Foundation of China (No.60932002), the National Science Fund of China (No. 61172062), and Natural Science Fund of Jiangsu (No. BK2011116). References [1] ZHAO, Q., SWAMI, A. A decision-theoretic framework for opportunistic spectrum access. IEEE Wireless Communications Magazine, 2007, vol. 14, no. 4, p. 14 - 20. [2] WU, Q., HAN, H., WANG, J., ZHAO, Z., ZHANG, Z. Sensing task allocation for heterogeneous channels in cooperative spectrum sensing. Radioengineering, 2010, vol. 19, no. 4, p. 544 - 551. [3] TANDRA, R., SAHAI, A., VEERAVALLI, V. Unified space-time metrics to evaluate spectrum sensing. IEEE Communications Magazine, 2011, vol. 49, no. 3, p. 54 - 61. [4] HAN, W., LI, J., LIU, Q., ZHAO, L. Spatial false alarms in cognitive radio. IEEE Communications Letters, 2011, vol. 15, no. 5, p. 518 - 520. [5] REN, W., ZHAO, Q., SWAMI, A. Power control in cognitive radio networks: How to cross a multi-lane highway. IEEE Journal on Selected Areas in Communications, 2009, vol. 27, no. 7, p. 1283 - 1296. [6] HAENGGI, M., ANDREWS, J. G., BACCELLI, F., DOUSSE, O., FRANCESCHETTI, M. Stochastic geometry and random graphs for the analysis and design of wireless networks. IEEE Journal on Selected Areas in Communications, 2009 vol. 27, no. 7, p. 1029 - 1046. [7] CHOI, K. W., HOSSAIN, E., KIM, D. I. Cooperative spectrum sensing under a random geometric primary user network model. IEEE Transactions on Wireless Communications, 2011 vol. 10, no. 6, p. 1932 - 1944. [8] BACCELLI, F., BLSZCZYSZYN, B. Stochastic Geometry and Wireless Networks, Volume I: Theory. Now Publishers Inc., 2009. [9] GHASEMI, A., SOUSA, E. S. Interference aggregation in spectrumsensing cognitive wireless networks. IEEE Journal of Selected Topics in Signal Processing, 2008, vol. 2, no. 1, p. 41 - 56. [10] LEE, C. H., HAENGGI, M. Interference and outage in Poisson cognitive networks. IEEE Transactions on Wireless Communications, 2012 vol. 11, no. 4, p. 1392 - 1401. [11] DERAKHASHANI, M., LE-NGOC, T. Aggregate interference and capacity-outage analysis in a cognitive radio network. IEEE Transactions on Vehicular Technology, 2012, vol. 61, no. 1, p. 196 - 207. RADIOENGINEERING, VOL. 22, NO. 2, JUNE 2013 563 [12] LIANG, Y.-C., ZENG, Y., PEH, E. C., HOANG, A. T. Sensingthroughput tradeoff for cognitive radio networks. In IEEE Transactions on Wireless Communications, 2008, vol. 7, no. 4, p. 1326 - 1337. [13] ALJUAID, M., YANIKOMEROGLU, H. Investigating the Gaussian convergence of the distribution of the aggregate interference power in large wireless networks. In IEEE Transactions on Vehicular Technology, 2010 vol. 59, no. 9, p. 4418 - 4426. About Authors . . . Xian ZHANG is working towards his Ph.D. in Institute of Communications Engineering of PLA University of Science and Technology. His research interests include cognitive radio and cooperative spectrum sensing. Xiaoqiang LI received his M.Sc. degree from PLA University of Science and Technology in 2007. Now he is working towards his Ph.D. at the Institute of Communications Engineering of the same university. His research interests include cognitive radio and resource allocation. Zi YUN received her M.Sc. degree from PLA University of Science and Technology in 2010. She is working at PLA Unit of 94860 and her research interests include algorithms and protocols.