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Cooperative Spectrum Sensing with an Improved Energy Detector in Cognitive Radio Network Ajay Singh, Manav R. Bhatnagar, and Ranjan K. Mallik Department of Electrical Engineering Indian Institute of Technology Delhi Hauz Khas, IN-110016 New Delhi, India Email: [email protected],{manav,rkmallik}@ee.iitd.ac.in Abstractβ In this paper, performance of cooperative spectrum sensing with an improved energy detector is discussed. The Cognitive radios (CRs) utilize an improved energy detector for taking a decision of the presence of the primary user (PU). The improved energy detector uses an arbitrary positive power π of the amplitude of the received samples of the primary userβs signals. The decision of each CR is orthogonally forwarded to a fusion center (FC), which takes final decision of the presence of the PU. We minimize sum of the probability of false alarm and missed detection in cooperative spectrum sensing to obtain an optimized number of CRs for detecting a spectrum hole. An optimized value of π and sensing threshold of each CR is also obtained by minimizing the total probability of error. I. I NTRODUCTION Detecting an unused spectrum and sharing it without interfering the primary users (PUs) is a key requirement of the cognitive radio network. For detection of a spectrum hole, the cognitive communication network relies upon the cognitive radios (CRs) or secondary users [1]. Spectrum sensing is a procedure in which a CR captures the information of a band available for transmission and then shares the frequency band without interfering the primary users. Cooperation among the CRs provides improvement in detection of the primary signals [2]. Due to the random nature of the wireless channel, the signals sensed by the CRs are also random in nature. Therefore, sometimes a CR can falsely decide the presence or absence of the PU [3]. In [3], a cooperation based spectrum sensing scheme is proposed to detect the PU with an optimal linear combination of the received energies from the cooperating CRs in a fusion center (FC). The cooperative spectrum sensing scheme provides better immunity to fading, noise uncertainty, and shadowing as compared to a cognitive network that does not utilize cooperation among the CRs [4]. Probability of miss ππ and probability of false alarm ππ play important role in describing a reliable communication. We can fix either of the probabilities at a particular value and then optimize the other one for improving the probability of detection of the primary signals [5]. In [6], optimization of the total probability of error is performed by minimizing sum of the probability of false alarm and the probability of missed detection of an cooperative spectrum sensing scheme. It is explored in [6] that how many CRs are needed for optimum detection of the primary user in a cooperative spectrum sensing based scheme, such that the total error probability does not exceed a fixed value. For the detection of unknown deterministic signals corrupted by the additive white Gaussian noise (AWGN), an energy detector is derived in [7], which is also called conventional energy detector. The conventional energy detector [7] utilizes square of the magnitude of the received data sample, which is equal to the energy of the received sample. It is shown in [8] that the performance of a cognitive radio network can be improved by utilizing an improved energy detector in the CRs, where the conventional energy detector is modified by replacing the squaring operation of the received signal amplitude with an arbitrary positive power π. In this paper, we consider optimization of the cooperative spectrum sensing scheme with an improved energy detector to minimize the sum of the probability of false alarm and missed detection. We derive a closed form analytical expression to find an optimal number of the CRs required for cooperation while utilizing the improved energy detector in each CR. Moreover, we also perform optimization of the threshold and the energy detector to minimize the total error rate. II. S YSTEM M ODEL We consider a system consisting of π number of CRs, one primary user, and a fusion center. There are two hypothesis π»0 and π»1 in the π-th CR for the detection of the spectrum hole π»0 : π¦π (π‘) = π£π (π‘), if PU is absent, π»1 : π¦π (π‘) = π (π‘) + π£π (π‘), if PU is present, π© (0, ππ 2 ), (1) (2) ππ 2 where is the where π = 1, 2....π , π (π‘) βΌ average transmitted power of the PU, denotes a zero mean Gaussian signal transmitted by the PU, and π£π (π‘) βΌ π© (0, ππ2 ) is additive white Gaussian noise (AWGN) with zero mean and ππ2 variance. The variance of the signal received at each secondary user under π»1 will be ππ 2 + ππ2 . It is assumed that each CR contains an improved energy detector [8]. The π-th CR utilizes the following statistic for deciding of the presence of the PU [8] π =β£ π¦π β£π , π > 0. (3) It can be seen from (3) that for π = 2, π reduces to statistics corresponding to the conventional energy detector [7]. In a cooperative sensing scheme, multiple CRs exists in a cognitive radio network such that each CR makes independent decision regarding the presence or absence of PU. We consider a cooperative scheme in which each secondary user sends its binary decision ππ (0 or 1) to the fusion center by using an improved energy detector over error free orthogonal channels. The fusion center combines these binary decisions to find the presence or absence of the PU as follows : π·= π β ππ , (4) π=1 where D is the sum of the all 1-bit decisions from the CRs. Let π, π β€ π corresponds to a number of cooperating CRs out of π CRs. The FC uses a majority rule for deciding the presence or absence of the PU. As per the majority decision rule if π· is greater than π then hypothesis β1 holds and if π· is smaller than π then hypothesis β0 will be true. The hypothesis β0 and β1 can be written as β0 : π· < π, if PU is absent, (5) β1 : π· β₯ π, if PU is present. (6) III. P ROBABILITY OF FALSE A LARM AND MISSED DETECTION IN THE CR S The cumulative distribution function (c.d.f.) of the improved energy detector can be written as 1 Pr(β£π¦π β£π β€ π¦) =Pr(β£π¦π β£ β€ π¦ π ) 1 1 =Pr(π¦π β€ π¦ π ) β Pr(π¦π < βπ¦ π ), (7) where Pr(β ) denotes the probability. For finding the probability density function (p.d.f.) of the decision statistics π, we need to differentiate (8) with respect to (w.r.t.) π¦ to get 1 1 1 1βπ 1 1βπ (8) ππ (π¦) = π¦ π ππ¦π (π¦ π ) + π¦ π ππ¦π (βπ¦ π ), π π where ππ¦π (β ) is the p.d.f. of received signal at the FC under hypothesis π»0 or π»1 depending upon the presence or absence of the primary signal. Let ππ¦πβ£π»0 (β ) and ππ¦πβ£π»0 (β ) be the p.d.f.s of the received signal under hypothesis π»0 and π»1 , respectively. From (5) and (6), it can be deduced that ( ) 1 π₯2 exp β 2 , (9) ππ¦πβ£π»0 (π₯) = β 2ππ 2πππ2 1 ππ¦πβ£π»1 (π₯) = β 2π(ππ 2 + ππ2 ) ( exp β π₯2 + ππ2 ) 2(ππ 2 π1 where π1 is the decision region corresponding to hypothesis π»1 . Let us assume that the decision threshold in each CR is given as π. The probability of false alarm ππ in each CR is derived in Appendix I as ( ) 2 3 ππ 1 , , (14) ππ = β π€ 2 2ππ2 π where π€ (π) is the gamma function [9, Eq. (6.1.1)] and π€ (π, β« βπ₯) is the incomplete gamma function π€ (π, π₯)= π₯ π‘πβ1 exp(βπ‘)ππ‘ [9, Eq. (6.5.3)]. The probability of miss ππ is defined by [10, Eq. (41), Chapter 2] β« ππ β ππ β£π»1 (π¦)ππ¦, (15) π0 where π0 is the decision region corresponding to the hypothesis π»0 . In Appendix I, it is shown that the probability of miss ππ in each CR will be ( ) 2 ππ 3 1 , ππ = β πΎ , (16) 2 2(ππ 2 + ππ2 ) π where β«πΎ(π, π₯) is the incomplete gamma π₯ πΎ(π, π₯)= 0 π‘πβ1 exp(βπ‘)ππ‘ [9, Eq. (6.5.2)]. 1 =Pr(βπ¦ π β€ π¦π β€ π¦ π ) 1 The probability of false alarm ππ is defined as [10, Eq. (41), Chapter 2] β« ππ β ππ β£π»0 (π¦)ππ¦, (13) ) . (10) The p.d.f. of π under hypothesis π»0 is given from (8) and (9) as follows: 2 β 1βπ π¦π 2π¦ π exp(β 2π ) π β . (11) ππ β£π»0 (π¦) = π πππ From (8) and (10), the p.d.f. of π under hypothesis π»1 can be obtained as 2 β 1βπ π 2π¦ π exp(β 2(ππ¦2 +π2 ) ) π π . (12) ππ β£π»1 (π¦) = β β π π (ππ 2 + ππ2 ) function IV. O PTIMIZATION OF COOPERATIVE SPECTRUM SENSING SCHEME The probability of false alarm ππΉ of the FC for cooperative sensing will be ππΉ = Pr(β1 β£β0 ), (17) From (5), (6), and (17), the probability of false alarm of the FC can be written as ) π ( β π ππΉ = (18) πππ (1 β ππ )π βπ . π π=π In the FC, the probability of missed detection ππ will be ππ = Pr(β0 β£β1 ). (19) The probability of missed detection of the FC can be obtained from (5), (6), and (19) as follows: ) π ( β π ππ = 1 β (20) (1 β ππ )π (ππ )π βπ . π π=π Let us define π, which denotes the total error rate in the cooperative scheme as π βππΉ + ππ ) π ( β π =1 + πππ (1 β ππ )π βπ π π=π π β( π ) β (1 β ππ )π (ππ )π βπ , π π=π (21) 0 5 10 4.5 4 β1 10 Optimal number of CRs Total error probability 3.5 β2 10 n=1 n=2 n=3 n=4 n=5 n=6 β3 10 0 1 2 3 4 5 p 6 7 8 2.5 Ξ»=10 Ξ»=20 Ξ»=50 2 1.5 1 0.5 β4 10 3 9 10 0 0 1 2 3 4 5 p 6 7 8 9 10 Fig. 1. Total probability of error versus π for different number of CRs for π = 10 and SNR=10 dB. Fig. 2. Optimal number of CRs versus π for π = 10, 20, 50, and SNR=10 dB. It can be seen from (21) that π denotes the total error rate in the case when ππΉ and ππ are equiprobable. B. Optimization of the Improved Energy Detector A. Optimization of Number of Cooperating CRs In a cognitive network with very large number of cooperative CRs, large delays are incurred in deciding the presence of the spectrum hole. Therefore, it is expedient to find an optimized number of CRs which significantly contribute in deciding the presence of the PU. Theorem 1: Optimal number of CRs (n*) for a fixed value of the sensing threshold π, the positive power π of the amplitude of received sample of PU, and the signal-to-noise ratio (SNR) between the PU-CR link will be ( β β) π , (22) πβ = min π, 1+πΌ In order to get an optimized value of π we need to differentiate (21) w.r.t. π keeping π, π, and SNR fixed and set the result to zero. After some algebra we get π β βπ =β βπ π=π β1 π€ π ( ln πΌ= 1β β1π πΎ ( β1 πΎ π ln 2 3 ππ 2 , 2π 2 π 2 3 ππ 2 , 2(π 2 +π 2 ) π π 2 3 ππ 2 , 2(π 2 +π 2 ) π π 1β β1π π€ ( 2 3 ππ 2 , 2π 2 π ) ) ) (π β π)ππ π (1 β ππ )π βπβ1 βππ βπ and βππ βπ are given as 3 , (23) βππ βπ π=π where ) π π ) π ( β βππ π πππ π (1 β ππ )π βπ + π βπ π=π ( ) π β βππ π (π β π)ππ π βπβ1 (1 β ππ )π β π βπ π=π ) π ( β βππ π , (25) πππ π βπ (1 β ππ )πβ1 + π βπ where ( ( 2 π π π π ππππ exp(β 2π 2 ) βππ π β = , βπ 2ππ2 ππ3 (26) ) 3 2 ππ π π ππππ exp(β 2(π2 +π2 ) ) βππ π π =β β . 3 βπ 2ππ2 (ππ2 + ππ 2 ) 2 (27) and β.β denotes the ceiling function. Proof: In order to obtain an optimized number of the CRs, we need to differentiate (21) w.r.t. π and set the result equal to zero, to get ( ) ] [ π βπ π (1 β ππ )πβππ π (1 β ππ )π βπ = 0. (24) ππ π It is difficult to obtain a closed form solution of π, however, we can obtain an optimized value of π numerically by using (25). After some algebra and with the help of results given in [6] we get (22) from (24). An optimal value of π for given π, π, and SNR is obtained by differentiating (21) w.r.t. π and setting the derivative equal C. Optimization of Sensing Threshold β1 10 β0.001 10 β0.002 10 β2 Total error rate Probability of detection 10 β0.003 10 β0.004 10 β3 10 p=1.75 p=2.00 p=2.25 p=2.50 β0.005 p=1.75 p=2.00 p=2.25 p=2.50 10 β0.006 10 β4 10 0 1 2 3 4 5 SNR (dB) 6 7 8 9 10 Fig. 3. Total error rate versus SNR for π = 1.75, 2.00, 2.25, 2.50, π = 1, and sensing threshold π = 50. to zero. After some algebra we get π ( β βπ π =β π βπ π=π π ( β π + π π=π ( π β π β π π=π π ( β π + π ) ) ) ) βππ βπ and βππ βπ (π β π)ππ π (1 β ππ )π βπβ1 πππ π (1 β ππ )π βπ βππ βπ βππ βπ (π β π)ππ π βπβ1 (1 β ππ )π πππ π βπ (1 β ππ )πβ1 βππ , βπ βππ βπ (28) can be obtained as follow: 3 2 π π π π β1 exp(β 2π 2 ) βππ π β =β , 2 3 βπ 2ππ ππ 3 2 4 6 8 10 SNR (dB) 12 14 16 18 20 Fig. 4. Probability of detection versus SNR for π = 1.75, 2.00, 2.25, 2.50, π = 1, and sensing threshold π = 50. V. N UMERICAL R ESULTS π=π where 0 (29) 2 π π π β1 exp(β 2(ππ2 +π2 ) ) βππ π π =β β . 3 βπ 2ππ(ππ2 + ππ 2 ) 2 (30) The optimized value of π can be found numerically from (28). D. Probability of Detection of Spectrum Hole Probability of detection of spectrum hole ππ· in the FC can be obtain from (20) as follows: ππ· =1 β ππ ( )]π )[ 2 π ( β ππ 3 1 π , = 1β β πΎ π 2 2(ππ 2 + ππ2 ) π π=π )]π βπ ( [ 2 ππ 3 1 , × β πΎ . (31) 2 2(ππ 2 + ππ2 ) π In Fig. 1, we have plotted the total error rate versus π plots for different number of cooperative CRs π = 1, 2, 3, 4, 5, 6, π = 10, and SNR=10 dB. It can be seen from Fig. 1 that the total error rate is a convex function of π for a fixed value of π, π, and SNR. It can be seen from Fig. 1 that for π = 10 and SNR=10 dB, the total error rate is minimum for π = 2. It implies that for π = 10 and SNR=10 dB only two CRs are required for deciding the presence of the PU by using an improved energy detector. We have shown plots of the optimal number of CRs required for cooperation versus π for π = 10, 20, 50, and SNR=10 dB in Fig. 2. It can be seen from Fig. 2 that the optimal number of the cooperative CRs varies with the value of π and π at SNR=10 dB. For example, for π = 3 the optimal number of the CRs is 2, 3, and 4 for π = 50, 20, and 10, respectively. Fig. 3 shows that the total error rate versus SNR plot with π = 1.75, 2, 2.25, 2.50, π = 1, and π = 50. It can be seen from Fig. 3 that the total error rate decreases by increasing the SNR and the value of π. Further, it can be seen from Fig. 3 that the improved energy detector with π = 2.50 significantly outperforms the conventional energy detector, i.e., when π = 2. For π = 1.75, 2, 2.25, 2.50, π = 1, and π = 50, the probability of detection of a spectrum hole versus SNR curve is plotted in Fig. 4. It is shown in Fig. 4 that the probability of detection of a spectrum hole increases with increase in the value of π. Fig. 5 illustrates that for a fixed range of π and π there exists an optimal number of CRs which minimizes the total error rate. In Fig. 6, we have plotted the total error rate versus π and π plots for different number of cooperative CRs at 10 dB SNR. It can be seen from Fig. 6 that there exists an optimal value of π and π for which the total error rate is minimized for a fixed number of CRs. The plots of Figs. 5 and 6 suggest that in order to optimize the overall performance of the cooperative cognitive system utilizing the improved energy detector, we need to jointly optimize the number of cooperative CRs, the energy detector, and the sensing threshold in the CR. Fig. 5. Optimal number of CRs versus π and π at SNR=10 dB. VI. C ONCLUSIONS We have discussed how to choose an optimal number of cooperating CRs in order to minimize the total error rate of a cooperative cognitive communication system by utilizing an improved energy detector. It is shown that the performance of the cooperative spectrum sensing scheme depends upon the choice of the power of the absolute value of the received data sample and the sensing threshold of the cooperating CRs as well. We have obtained an analytical expression of the optimal number of CRs required for taking the decision of the presence of the primary user when the CRs utilize the improved energy detector. It is shown by simulations that an optimal value of the total error rate can be obtained by utilizing a joint optimization of the number of cooperating CRs, threshold in each CR, and the energy detector. A PPENDIX I P ROOF OF (14) AND (16) From (11) and (13), the probability of false alarm ππ in each CR can be obtained as β« β ππ/π»0 (π¦)ππ¦ ππ = β« π β 1βπ π 2 π π¦ exp(β 2π ) π β ππ¦. = π πππ π After applying some algebra in (32) we get β« β β 1 π‘ exp(βπ‘)ππ‘. ππ = β 2 π π π2 β 2π¦ (32) (33) 2ππ By comparing (33) with the expression of the incomplete gamma function [9, Eq. (6.5.3)] we get (14). From (12) and (15), the probability of miss ππ in each CR will be β« β ππ/π»1 (π¦)ππ¦ ππ = π β« = β π 2 β 1βπ π 2π¦ π exp(β 2(ππ¦2 +π2 ) ) π π ππ¦. β β π π (ππ 2 + ππ2 ) (34) Fig. 6. Total error rate versus π and π for different number cooperating users π = 1, 2, 3, 4, 5, 6 at SNR=10 dB. After some algebra we get ππ 1 =β π 2 ππ 2 +π 2 ) 2(ππ π β« 0 β π exp(βπ)ππ. (35) From (35) and [9, Eq. (6.5.2)] we get (16). R EFERENCES [1] S. Haykin, βCognitive Radio: Brain-empowered wireless communications,β IEEE J. Sel. Areas Commun., vol. 23, pp. 201-220, Feb. 2005. [2] G. 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