Cooperative spectrum sensing against noise uncertainty using ...
Applied Soft Computing 13 (2013) 3307–3313
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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Cooperative spectrum sensing against noise uncertainty using Neyman–Pearson lemma on fuzzy hypothesis test Abdolreza Mohammadi a , Mohammad Reza Taban a,∗ , Jamshid Abouei a , Hamzeh Torabi b a b
Department of Electrical and Computer Engineering, Yazd University, 89195-741, Yazd, Iran Department of Mathematics, Yazd University, 89195-741, Yazd, Iran
a r t i c l e
i n f o
Article history: Received 8 August 2012 Received in revised form 19 December 2012 Accepted 14 February 2013 Available online 5 March 2013 Keywords: Cognitive radio Fuzzy hypothesis test Neyman–Pearson lemma Noise power uncertainty Cooperative spectrum sensing
a b s t r a c t In this paper, we consider the problem of cooperative spectrum sensing in the presence of the noise power uncertainty. We propose a new spectrum sensing method based on the fuzzy hypothesis test (FHT) that utilizes membership functions as hypotheses for the modeling and analyzing such uncertainty. In particular, we apply the Neyman–Pearson lemma on the FHT and propose a threshold-based local detector at each secondary user (SU) in which the threshold depends on the noise power uncertainty. In the proposed scheme, a centralized manner in the cooperative spectrum sensing is deployed in which each SU sends its one bit decision to a fusion center. The fusion center makes a final decision about the absence/presence of a primary user (PU). The performance of the PU’s signal detection is evaluated by the probability of signal detection for a specific signal to noise ratio when the probability of false alarm is set to a fixed value. The performance of the proposed algorithm is compared numerically with two classical threshold-based energy detectors. Simulation results show that the proposed algorithm considerably outperforms the methods with a bi-thresholds energy detector and a simple energy detector in the presence of the noise power uncertainty. © 2013 Elsevier B.V. All rights reserved.
1. Introduction In recent years, the spectrum scarcity has been a challenging problem in wireless communication systems. The main reasons are the development of various wireless technologies and an increase in the demand for higher rate wireless services. Moreover, the spectrum allocation policy is such inflexible and inefficient so that many portions of the licensed spectrum are not utilized during significant time periods [1]. One heuristic solution to improve the efficiency of the spectrum utilization is the use of Cognitive Radio (CR) technology [2,3]. In this technology, the unlicensed users known as secondary users (SUs), are allowed to use the licensed spectrum bands in an opportunistic manner. One challenge faced in a CR system is that the SUs must accurately sense the spectrum, realize the spectrum holes for their transmissions and vacate the frequency band as soon as the primary users (PUs) start their transmissions [4,5]. Therefore, the spectrum sensing is an essential task to detect the presence of the PU that can be performed either individually or
∗ Corresponding author at: Department of Electrical and Computer Engineering, Yazd University, Daneshgah Boulevard, P.O. Box: 89195-741, Yazd, Iran. Tel.: +98 9131519702; fax: +98 351 8200144. E-mail addresses: a [email protected] (A. Mohammadi), [email protected] (M.R. Taban), [email protected] (J. Abouei), [email protected] (H. Torabi). 1568-4946/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2013.02.009
cooperatively. The decision on the presence or absence of the PU in the non-cooperative spectrum sensing is performed individually by each SU, while in the cooperative case, the spectrum sensing is performed by collaboration between a group of SUs to mitigate some problems of spectrum sensing such as multipath fading, shadowing effects, and hidden PUs [6]. In recent years, many studies have been devoted on the cooperative spectrum sensing and many approaches are provided for detecting the PU signals including the cyclostationary feature detection [7] and the energy detection [8]. In most of the work, the power of the noise is assumed to be previously known for threshold setting. However, due to the limited sensing time and the fluctuation of the noise power, the power of the noise is not known precisely and the performance of some classical cooperative spectrum sensing techniques is susceptible to the noise power uncertainty. The main sources of the noise power uncertainty are the noise uncertainty of the receiver device due to the non-linearity and the thermal noise of the components, and the environment noise uncertainty caused by the transmissions of other users [9,10]. Several techniques have been proposed to mitigate the noise power uncertainty in cooperative-based CR systems [11–13]. The cooperative spectrum sensing method in the presence of the noise uncertainty in [11] is based on an energy detector that uses three thresholds for local sensing, while the proposed cooperative approach in [12], is based on a cooperative covariance and eigenvalue based detection approach. The methods proposed
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in [11] and [12] send local sensing results to a fusion center and hence occupy more bandwidth of the control channel. The authors in [13] utilize a bi-thresholds energy detector in each SU where one bit local decision is sent by the SU to a fusion center. In this paper, we investigate a centralized hard decision-based cooperative spectrum sensing problem in the presence of the noise power uncertainty using fuzzy set theory concepts as a mathematical framework to model such uncertainty. We propose a new spectrum sensing method based on the fuzzy hypothesis test (FHT) that utilizes membership functions as hypotheses for modeling and analyzing the noise power uncertainty. In particular, we apply the Neyman–Pearson lemma on the FHT and propose a thresholdbased local detector at each SU in which the threshold depends on the noise power uncertainty. In the proposed scheme, a centralized manner in the cooperative spectrum sensing is deployed in which each SU sends its one bit decision to a fusion center. The fusion center makes a final decision on the absence/presence of the PU. The performance of the PU’s signal detection is evaluated by the probability of the signal detection for a specific Signal to Noise Ratio (SNR) when the probability of the false alarm is set to a fixed value. We compare numerically the performance of the proposed algorithm with two classical threshold-based energy detectors. Simulation results show that the proposed algorithm considerably outperforms the methods with a bi-thresholds energy detector and a simple energy detector in the presence of the noise power uncertainty. The rest of the paper is organized as follows. In Section 2, the FHT background and the system model are provided. The sensing algorithm using the Neyman–Pearson lemma for the FHT is presented in Section 3. Simulation results are presented in Section 4. Finally, conclusions are drawn in Section 5. Notations: Throughout this paper, we use boldface lower case letters to denote vectors. A Gaussian random variable with mean m and variance 2 is represented by x∼N(m, 2 ). We use E[.] as the expectation operator, and P{.} for representing the probability of the given event. The signs “” and “” mean “almost smaller” and “almost greater”, respectively, and Q (.) is the complementary cumulative distribution function, which calculates the of a zero mean unit variance Gaussian variable, i.e. tail probability ∞ Q (x) = x √1 exp(−t 2 /2)dt. Also, (.)T stands for the transpose of 2
Definition 2.1. Any hypothesis of the form “H : is H()” is called a fuzzy hypothesis, implying that is in a fuzzy set of with the membership function H() which is a function from to [0, 1] [20]. Note that the crisp hypothesis “Hi : ∈i ” is a fuzzy hypothesis with the membership function “Hi = 1 at ∈ i , and zero otherwise”. 2.2. Neyman–Pearson lemma for FHT Let X = (X1 , . . . , Xn )T be a random sample vector, with the observed value x = (x1 , . . . , xn )T where Xi has the probability density function (pdf) f(xi ; ) with the unknown parameter ∈ , in which is the parameter space. Suppose the two membership functions H0 () and H1 () are known and we would like to test
H0 :
is H0 (),
H1 :
is H1 ().
(2)
The aim is to accept or reject H0 on the basis of x. In other words, we would like to make a test function (X) such that (x) is the probability of rejecting H0 if X = x is observed. We use the above FHT for our analysis in the subsequent section. Definition 2.2. For the fuzzy hypothesis test in (2.1) and the test function (X), the false alarm probability denoted by Pfa is defined by [22] Pfa =
1 M
H0 ()E [ (X)] d,
(3)
where
M=
H0 ()d,
(4)
and
E [(X)] =
(x)f (x; )dx,
(5)
where f(x ; ) is the joint pdf of X with the unknown parameter ∈ .
a matrix or vector.
In the next theorem, we present the Neyman–Pearson lemma for the above fuzzy hypothesis test.
2. FHT background and system model
Theorem 2.3. Let X = (X1 , . . ., Xn )T be a random sample vector, with the observed value x = (x1 , . . ., xn )T where Xi has the pdf f(xi ; ) with the unknown parameter ∈ . Defining (x) = ( f(x ; )H1 ()d)/( f(x ; )H0 ()d), for the fuzzy hypothesis test problem in (2.1)
2.1. Fuzzy hypothesis test The conventional binary detection problem is a decision between the two crisp hypotheses H0 , known as the null hypothesis versus the alternative hypothesis H1 . In some situations, we face many practical problems in which the observed data are associated with some uncertainty. Over the past years, there have been some efforts to analyze this uncertainty using the fuzzy set theory [14–16]. Taking the uncertainty into account in the hypothesis test introduces an interesting problem called Fuzzy Hypothesis Test (FHT). To introduce the FHT, suppose that we are interested to test the value of the mean parameter of a normal distribution denoted by based on the observations. In the ordinary case, we use the test H0 : = 0 against H1 : = / 0 . Due to the uncertainty on the model parameter, a more realistic hypothesis can be written as
H0 :
is close to 0
H1 :
is away from 0 .
(a) Any test with the test function (x) =
⎧ ⎨1 ⎩
(x) > ,
ı(x) (x) = , 0
(6)
(x) < ,
for some ≥ 0 and 0 ≤ ı(x) ≤ 1, is the best test of size Pfa . (b) For 0 ≤ ˛ ≤ 1, there exists a test of form (6) with ı(x) = ı (a constant), for which Pfa = ˛. See [22] for the proof. 2.3. System model
(1)
Such hypothesis test is a FHT which has been extensively investigated in the literature [17–21].
Suppose in a homogeneous cognitive radio network all the SUs use the same protocol for the local spectrum sensing. The network contains K secondary users indexed by K = {1, . . . , K} to exploit the spectrum holes. We use a centralized manner in the cooperative
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for N = 2TW, where W and T are the signal bandwidth and the detection time, respectively, and yk [n] is the sample of yk (t) at discrete time t = n/2W. Tk and k are the test statistic and the detection threshold at the kth SU, respectively. It is a well-known fact that if N is large enough, the statistics of Tk can be approximately written as [25]
Tk ∼
2 , 2N 4 ), N(Nw,k w,k
under H0 ,
2 N(N(w,k
2 ), 2N( 4 + s,k w,k
2 w,k
2 s,k
4 + s,k
2 2 )), + 2w,k s,k
under H1 , (9)
where and are the estimated noise and signal variances, respectively. Therefore, the probability of false alarm in the kth SU, Pfak is given by
Pfak = P Tk > k | H0
spectrum sensing in which each SU sends its one bit decision to a fusion center as shown in Fig. 1. The spectrum sensing problem for the kth SU, k ∈ K, can be represented by the following binary hypothesis test: H0 :
yk (t) = wk (t),
H1 :
yk (t) = sk (t) + wk (t),
3. Cooperative spectrum sensing using Neyman–Pearson lemma for FHT
One of the most common detectors that is largely used for spectrum sensing is the energy detector, because of a low computational and implementation complexities. Fig. 2 shows the block diagram of the energy detector. The frequency band of the interest is chosen by applying a Band Pass Filter (BPF) to the received signal and the power of the received signal is computed by the following equation at the kth SU:
n=0
H0
(10)
(11)
H0 :
k2 ≤ k ,
H1 :
k2 > k ,
(12)
where k2 is the variance of the received signal on the observation interval at the kth SU which can be written as
k2
=
2 , w,k
under H0
2 w,k
under H1 .
2 , + s,k
(13)
According to our discussion about using the membership function in the hypothesis test (2.1), we can write (12) using the crisp membership functions as follows:
H1 :
k2 is H1 (k2 ) = U(k2 − k ),
H0 :
k2 is H0 (k2 ) = 1 − H1 (k2 ),
(14)
where U(.) is the unit step function. 3.2. Local spectrum sensing using FHT
3.1. Energy detection
N−1
H yk [n] 2 ≷1 k ,
=Q
2 k − Nw,k √ 2 w,k 2N
Using the energy detector in (8), the local detection of the PU’s signal is equivalent to the comparison of the variance of the received signal at the SU with the threshold k , i.e.,
(7)
where H1 and H0 are the hypotheses of the presence and absence of the PU’s signal, respectively, yk (t) denotes the received signal at the kth SU and wk (t) represents an additive complex white Gaussian noise. In addition, we assume that sk (t) is a zero mean Gaussian signal from the PU at the kth SU’s receiver. This assumption is used in many literature related to the spectrum sensing in CR networks (e.g. [23,24]). We also assume that sk (t) and wk (t) are independent. The probability of detection and the probability of false alarm will be used as the criterion for the performance evaluation of our algorithm. In a CR network, a smaller false alarm probability indicates a higher spectrum efficiency, and a larger detection probability indicates a lower interference between the PU and the SUs.
Tk =
and the detection threshold in (8) is obtained as √ 2 k = w,k ( 2NQ −1 (Pfak ) + N).
Fig. 1. Centralized cooperative spectrum sensing in a cognitive radio network.
(8)
As previously mentioned, the energy detector computes the power of the input signal and compares it with a threshold which depends on the noise power. However, the power of the noise is not known precisely and the performance of the energy detector is susceptible to the noise power uncertainty. To represent this uncer2 belongs to the tainty, assume that the estimated noise variance w,k
2 , (1 + ) 2 ], where 2 is a known nominal interval [(1 − k )0,k k 0,k 0,k noise variance at the kth SU’s receiver, and 0 < k < 1 is a parameter that defines the noise power uncertainty bound [26]. In practice, the range of k is normally between 0.12 and 0.6 [23]. One heuristical approach to encounter the noise power uncertainty is to use the fuzzy hypothesis test that utilizes the membership functions as hypotheses. According to the previous discussion about the fuzzy
Fig. 2. Block diagram of the energy detector.
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distributed complex Gaussian random variables with zero-mean and variance k2 and with regard to (8), f (yk ; k2 ) can be written as
1 0.9
H0 Membership Function
Fuzzy Hypothesis, ρ = 0.6 Fuzzy Hypothesis, ρ = 0.26
0.8
f (yk ; k2 ) =
Crisp Hypothesis
Tk
−
.
k2
(19)
Thus, the test statistic is represented by
∞
0.6
0
0.5
(yk ) =
∞
0.4
0
(
exp −
1 )N 2
exp −
k
(
Observations
Fig. 3. Comparison between the membership functions of the crisp and fuzzy hypothesis models for different values of the uncertainty factor.
hypothesis test, the corresponding fuzzy hypothesis test can be implemented by converting (12) to the following equation: H0 :
k2 k ,
(15) k2 k , √ 2 ( 2NQ −1 (P ) + N), and 2 comes from (13). where k = 0,k fak k By extending the crisp membership functions in (14), we propose the membership functions for the fuzzy hypothesis test as follows: H1 :
⎧ ⎪ ⎨ H1 : k2 is H1 (k2 ) = H0 :
1 , 5 − (k2 − k ) 1 + e k k2 is H0 (k2 ) = 1 − H1 (k2 ).
− 5 ( 2 − )
(17)
H0
where yk = [yk (0) , yk (1) , . . . , yk (N − 1)]T is the received vector at the kth SU’s receiver, k ≥ 0 is the test threshold, and the test statistic (yk ) is given by Theorem 2.3 as
0
(yk ) =
1 5
2
− ( − ) 1+e k k k
5
dk2
5
2
( − ) − 1+e k k k
(20)
.
2
− ( − ) e k k k
dk2
1, Tk > k ,
(21)
Tk < k ,
0,
where k is determined based on the desired probability of the false alarm. In fact, the false alarm probability for the fuzzy hypothesis testing is given by (3) as follows: 1 M
(k)
Pfa =
∞
e
− 5 ( 2 − k )
1+e
0
k
k
(18)
For the above equation, f (yk ; k2 ) is the pdf of the received signal by the kth SU. Since yk consists of N independent and identically
E 2 [ (yk )] dk2 ,
− 5 ( 2 − k )
(22)
k
k
k
where from (4) we have
M= 0
∞
e
− 5 ( 2 − k )
1+e
k
k
− 5 ( 2 − k ) k
k
k
we have (k2 − k ) < 0 and e k k k → ∞, then the membership function values will be H1 (k2 ) = 0 and H0 (k2 ) = 1. Fig. 3 shows the fuzzy membership function of H0 for the crisp and fuzzy hypothesis tests for different values of uncertainty factor k . It is seen that the uncertainty in the fuzzy hypothesis model causes a smooth transition from H0 to H1 and vice versa. In addition, the fuzzy hypothesis model tends to the crisp hypothesis model when k decreases. The local spectrum sensing at each SU, using the Neyman–Pearson lemma for the fuzzy hypothesis test, in Theorem 2.3 can be performed as:
f (yk ; k )H0 (k )dk
Tk 2
k
E 2 [(yk )] = P Tk ≥ k
−5/ ( 2 − )
∞ f (yk ; k2 )H1 (k2 )dk2 (yk ) = 0∞ . 2 2 2
dk2 =
and applying (21) in (5)
k k k have (k2 − k ) > 0 and e → 0. Thus, from (16) we have H1 (k2 ) = 1 and H0 (k2 ) = 0, while, in the absence of the PU’s signal,
H1
k
k
(16)
The proposed membership functions yield the desirable properties H1 (k2 ) = 0 for k2 k and H1 (k2 ) = 1 for k2 k . In addition, in the ideal case that there is no noise power uncertainty, i.e.
k → 0, the membership functions tend to the crisp forms. More precisely, in the presence of the PU’s signal at the kth SU, we
(yk ) ≷ k ,
Tk 2
Taking the first-order derivative of (20) in terms of Tk , it is shown that (yk ) is an increasing function of Tk . Therefore, the best test function, based on the Neyman–Pearson lemma for the FHT, is obtained by comparing the energy of the received signal Tk with a new threshold denoted by k , i.e.:
Threshold
0
1 )N 2
0.3
0.1
⎪ ⎩
exp
k2
0.7
0.2
N
1
k ln (1 + e5/ k ), 5
(23)
∞
= k
(24)
fTk (tk )dt k ,
where fTk (tk ) is the pdf of the Tk . Using (8), since Tk is the square summation of N independent and identically distributed Gaussian random variables with zero-mean and variance k2 , where N is sufficiently large, Tk is approximated by N(Nk2 , 2Nk4 ) [27]. Therefore, we have
E 2 [(yk )] = Q k
k − Nk2 √ 2Nk2
.
(25)
Thus, the false alarm probability at the kth SU is (k) Pfa
5
= k ln (1 + e k ) 5
0
∞
e
− 5 ( 2 − k )
1+e
k
k
− 5 ( 2 − k ) k
k
Q
k − Nk2 √ 2Nk2
dk2 . (26)
In the proposed algorithm, k is determined according to the 2 . Based on (26), the noise condition and k depends on the 0,k threshold values k are calculated for different values of the local false alarm probability and are used by the local detectors. Although, the calculation of the threshold values in this method is complicated, the threshold values can be calculated offline. 3.3. Computational complexity The main complexity of the proposed fuzzy method comes from the computation of two parts: the energy detector and the test threshold. The energy detector needs N multiplications and N − 1 additions. Since the threshold can be calculated offline, tabulated, and used by the detector at each SU, its complexity does not affect
A. Mohammadi et al. / Applied Soft Computing 13 (2013) 3307–3313 Table 1 2 = 1 and different values of Pfa and . Testthreshold of the proposed method for 0,k
3311
1 0.9
Test threshold values Pfa = 0.1
Pfa = 0.15
195.65 200
180.78 183.01
169.166 170.32
the total complexity. For instance, Table 1 gives some threshold val2 = 1, and different values of and the desired local false ues for 0,k alarm probability. Therefore, the total complexity of the proposed fuzzy method is the same as that of the energy detector. In the next section, we use the results of the local spectrum sensing at each SU to perform the cooperative spectrum sensing.
0.8
Probability of Detection
= 0.12
= 0.26
Pfa = 0.05
0.7 0.6 0.5 Fuzzy Method ( ρ = 0.12)
0.4
Energy Detector ( ρ = 0.12) 0.3
Fuzzy Method ( ρ = 0.26) Energy Detector ( ρ = 0.26)
0.2
Fuzzy Method ( ρ = 0.6) Energy Detector ( ρ = 0.6)
0.1
3.4. Centralized cooperative spectrum sensing 0 0
Cooperative spectrum sensing improves the performance of the spectrum sensing at the low SNR and decreases the sensing time. It also reduces the effects of the hidden PU, multipath fading, and shadowing problems. In this paper, we consider the centralized CR networks in which each SU sends its sensing result to a fusion center via a dedicated control channel. The fusion center combines the received sensing results and makes a decision on the presence or absence of the PU signal. We use a hard decision fusion method to reduce the communication overhead at the control channel. In this method, the SUs send a binary local decision to the fusion center, and the fusion center decides about the presence of the PU with some hard rules such as “OR” and “AND” rules. Based on the AND rule, the fusion center decides the presence of the PU’s signal, if all SUs decide H1 , while, based on the OR rule, the fusion center decides about the H1 , if at least one SU decides H1 . The detection and the false alarm probabilities based on the AND and OR rules in the cooperative spectrum sensing can be written as follows [28]: Qf −AND =
K
(k)
(27)
(k)
(28)
Pfa ,
k=1
Qd−AND =
K
Pd ,
k=1
Qf −OR = 1 −
K
(k)
1 − Pfa
,
(29)
(k)
1 − Pd
,
(30)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 4. ROC curves of a single SU for the fuzzy method and the energy detector for N = 200, SNR = −10dB and various noise power uncertainties.
It is observed that by decreasing the noise power uncertainty the performance of both methods improves. Moreover, the proposed fuzzy method outperforms the energy detector in all cases. To improve the performance, we use the centralized cooperative spectrum sensing with the “OR” and “AND” combining rules. The performance of the proposed method is compared with the bi-thresholds detector proposed in [13] and the simple energy detector. In Fig. 5, we compare the proposed fuzzy method with the simple and bi-thresholds energy detectors for = 0.6 and SNR = −20 dB for each SU. As can be realized, the performance of the proposed fuzzy method with both rules is the best. As mentioned before, in the proposed method, we use membership functions as hypotheses which have a soft transition behavior from the H1 hypothesis to the H0 and vice versa that depends on the noise power uncertainty. In other words, the membership functions take different values, depending on the noise power uncertainty and the threshold is determined based on these values, while, the threshold in the simple energy detector is set regardless of the noise power uncertainty. The bi-thresholds energy detector utilizes only two thresholds that depend on the noise power uncertainty 0
10
0
k=1
where Qf and Qd represent the false alarm and the detection prob(k)
(k)
abilities at the fusion center, respectively, and Pfa and Pd denote the false alarm and the detection probabilities at the kth SU, respectively. 4. Simulation results In this section, we present some numerical results to evaluate the performance of the proposed FHT algorithm using the detection probability versus SNR and the Receiver Operating Characteristic (ROC) curves. In our simulation, k and SNRk indicate the noise power uncertainty bound and the SNR at the kth SU, respectively. Moreover, the number of samples (N) is equal to 200 and the number of SUs (K) is equal to 5. Fig. 4 shows the ROC curves of a single SU for the proposed fuzzy method compared to the energy detector for SNR = −10dB and different values of the noise power uncertainty.
ROC in logarithmic scale
10
Probability of Detection
K
0.2
Probability of False Alarm
k=1
Qd−OR = 1 −
0.1
−1
−1
10
10
−2
10
−3
−2
10
−3
10
0
10
Fuzzy Method (AND) 10−3 Fuzzy Method (OR) Bi−thresholds (AND) Bi−thresholds (OR) Simple Energy Detector (AND) Simple Energy Detector (OR) 0.2
0.4
−2
10
0.6
−1
10
0.8
0
10
1
Probability of False Alarm Fig. 5. ROC curves for comparison of the fuzzy method with a simple energy detector and a bi-thresholds energy detector in cooperative spectrum sensing (for each SU: N = 200, SNR = −20 dB, = 0.6).
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1
10
0.9
Probability of Detection
Probability of Detection
0.8 −1
10
Fuzzy Method (AND) Fuzzy Method (OR) Bi−thresholds (AND) Bi−thresholds (OR) Simple Energy Detector (AND) Simple Energy Detector (OR)
−2
10
−3
10
−3
10
−2
10
−1
10
0.7 0.6 0.5 ρ = 0.12 , Qfa FC = 0.001
0.4
ρ = 0.26 , Qfa FC = 0.001 ρ = 0.6 , Qfa FC = 0.001
0.3
ρ = 0.6 , Qfa FC = 0.01 ρ = 0.6 , Q
0.2 0
10
−20
fa FC
−18
bound. In Fig. 6, we consider the case that the noise power uncertainty factor for each SU is different. For this simulation, we set SNR = −20 dB, 1 = 0.26, 2 = 0.12, 3 = 0.6, 4 = 0.6, and 5 = 0.26. It is seen from Fig. 6 that the proposed fuzzy method along with the “AND” rule outperforms the other detectors. A more practical case is when both uncertainty factor and SNR are different for all SUs. For this case, Fig. 7 depicts the ROC curves of the aforementioned detectors in which we have set SNR1 = −12 dB, 1 = 0.12, SNR2 = −15 dB, 2 = 0.26, SNR3 = −20 dB, 3 = 0.12, SNR4 = −15 dB,
4 = 0.6, andSNR5 = −18 dB, 5 = 0.6. It is observed that the proposed fuzzy method with the “AND” rule outperforms the other detectors. Fig. 8 depicts the probability of detection at the fusion center with the “AND” rule fuzzy method (Qd−AND ) versus SNR for different values of uncertainty factor and probability of false alarm at the fusion center (Qfa−FC ). In this figure, we assume that all SUs take the same value of the uncertainty factor and the false alarm probability. As can be observed, by increasing the SNR, the performance of the
−14
−12
−10
−8
−6
−4
−2
Signal to Noise Ratio (SNR)
Probability of False Alarm Fig. 6. ROC curves for comparison of the fuzzy method with a simple energy detector and a bi-thresholds energy detector in cooperative spectrum sensing, for N = 200, SNR = −20 dB and different uncertainty factors 1 = 0.26, 2 = 0.12, 3 = 0.6, 4 = 0.6, and 5 = 0.26.
−16
= 0.05
Fig. 8. Pd versus SNR for the fuzzy method with the “AND” rule versus SNR at different uncertainty factors and Qfa−FC in cooperative spectrum sensing, when all SUs are the same.
detector improves and the proposed fuzzy method with the “AND” rule fulfills the spectrum sensing requirements mentioned in the IEEE 802.22 standard [29] at Qfa−FC = 0.05. 5. Conclusion In this paper, a new centralized cooperative spectrum sensing algorithm based on the fuzzy hypothesis test was proposed facing with the noise power uncertainty. We derived a detector using the Neyman–Pearson lemma for the fuzzy hypothesis test at each SU and showed that the proposed detector is an energy detector with a threshold depending on the noise power uncertainty. The local hard decisions from the SUs are sent to the fusion center and combined using the “AND” and “OR” rules for the final decision. Simulation results showed that the proposed algorithm performs superior than the cooperative spectrum sensing with the simple and bi-thresholds energy detectors and is preferable for spectrum sensing applications. References
0
Probability of Detection
10
−1
10
Fuzzy Method (AND) Fuzzy Method (AND) Bi−thresholds (AND) Bi−thresholds (OR) Simple Energy Detector (AND) Simple Energy Detector (OR)
−2
10
−3
10
−2
10
−1
10
0
10
Probability of False Alarm Fig. 7. ROC curves for comparison of the fuzzy method with a simple energy detector and a bi-thresholds energy detector in cooperative spectrum sensing, for SNR1 = −12 dB, 1 = 0.12, SNR2 = −15 dB, 2 = 0.26, SNR3 = −20 dB, 3 = 0.12, SNR4 = −15 dB, 4 = 0.6andSNR5 = −18 dB, 5 = 0.6.
[1] F.C. Commissions, Spectrum policy task force report (ET Docket No. 02-135), Washington, DC. [2] I.F. Akyildiz, W.Y. Lee, M.C. Vuran, S. Mohanty, NeXt generation/dynamic spectrum access/cognitive radio wireless networks: a survey, Computer Networks 50 (2006) 2127–2159. [3] B. Wang, K.J.R. Liu, Advances in cognitive radio networks: a survey, IEEE Journal Selected Topics in Signal Processing 5 (2011) 5–23. [4] S. Haykin, Cognitive radio: brain-empowered wireless communications, IEEE Journal Selected Areas on Communication 23 (2005) 201–220. [5] J.D. Ser, M. Matinmikko, S. Gil-López, M. Mustonen, Centralized and distributed spectrum channel assignment in cognitive wireless networks: a harmony search approach, Applied Soft Computing 12 (2012) 921–930. [6] I.F. Akyildiz, B.F. Lo, R. Balakrishnan, Cooperative spectrum sensing in cognitive radio networks: a survey, Physical Communication 4 (2011) 40–62. [7] J. Lunden, V. Koivunen, A. Huttunen, H. Poor, Collaborative cyclostationary spectrum sensing for cognitive radio systems, IEEE Transactions on Signal Processing 57 (2009) 4182–4195. [8] S. Atapattu, C. Tellambura, H. Jiang, Energy detection based cooperative spectrum sensing in cognitive radio networks, IEEE Transactions On Wireless Communications 10 (2011) 1232–1241. [9] A. Sahai, D. Cabric, Spectrum sensing: fundamental limits and practical challenges, in: Proc. IEEE International Symp. New Frontiers Dynamic Spectrum Access Networks (DySPAN’05), Baltimore, MD, USA., 2005. [10] Y. Zeng, Y.C. Liang, Eigenvalue-based spectrum sensing algorithms for cognitive radio, IEEE Transactions On Communications 57 (2009) 1784–1793. [11] C. Song, Y.D. Alemseged, H.N. Tran, G. Villardi, C. Sun, S. Filin, H. Harada, Adaptive two thresholds based energy detection for cooperative spectrum sensing, in: 7th IEEE Confernce on Consumer Communications and Networking (CCNC’2010), Las Vegas, NV, USA, 2010, pp. 1–6.
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Contents lists available at SciVerse ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Cooperative spectrum sensing against noise uncertainty using Neyman–Pearson lemma on fuzzy hypothesis test Abdolreza Mohammadi a , Mohammad Reza Taban a,∗ , Jamshid Abouei a , Hamzeh Torabi b a b
Department of Electrical and Computer Engineering, Yazd University, 89195-741, Yazd, Iran Department of Mathematics, Yazd University, 89195-741, Yazd, Iran
a r t i c l e
i n f o
Article history: Received 8 August 2012 Received in revised form 19 December 2012 Accepted 14 February 2013 Available online 5 March 2013 Keywords: Cognitive radio Fuzzy hypothesis test Neyman–Pearson lemma Noise power uncertainty Cooperative spectrum sensing
a b s t r a c t In this paper, we consider the problem of cooperative spectrum sensing in the presence of the noise power uncertainty. We propose a new spectrum sensing method based on the fuzzy hypothesis test (FHT) that utilizes membership functions as hypotheses for the modeling and analyzing such uncertainty. In particular, we apply the Neyman–Pearson lemma on the FHT and propose a threshold-based local detector at each secondary user (SU) in which the threshold depends on the noise power uncertainty. In the proposed scheme, a centralized manner in the cooperative spectrum sensing is deployed in which each SU sends its one bit decision to a fusion center. The fusion center makes a final decision about the absence/presence of a primary user (PU). The performance of the PU’s signal detection is evaluated by the probability of signal detection for a specific signal to noise ratio when the probability of false alarm is set to a fixed value. The performance of the proposed algorithm is compared numerically with two classical threshold-based energy detectors. Simulation results show that the proposed algorithm considerably outperforms the methods with a bi-thresholds energy detector and a simple energy detector in the presence of the noise power uncertainty. © 2013 Elsevier B.V. All rights reserved.
1. Introduction In recent years, the spectrum scarcity has been a challenging problem in wireless communication systems. The main reasons are the development of various wireless technologies and an increase in the demand for higher rate wireless services. Moreover, the spectrum allocation policy is such inflexible and inefficient so that many portions of the licensed spectrum are not utilized during significant time periods [1]. One heuristic solution to improve the efficiency of the spectrum utilization is the use of Cognitive Radio (CR) technology [2,3]. In this technology, the unlicensed users known as secondary users (SUs), are allowed to use the licensed spectrum bands in an opportunistic manner. One challenge faced in a CR system is that the SUs must accurately sense the spectrum, realize the spectrum holes for their transmissions and vacate the frequency band as soon as the primary users (PUs) start their transmissions [4,5]. Therefore, the spectrum sensing is an essential task to detect the presence of the PU that can be performed either individually or
∗ Corresponding author at: Department of Electrical and Computer Engineering, Yazd University, Daneshgah Boulevard, P.O. Box: 89195-741, Yazd, Iran. Tel.: +98 9131519702; fax: +98 351 8200144. E-mail addresses: a [email protected] (A. Mohammadi), [email protected] (M.R. Taban), [email protected] (J. Abouei), [email protected] (H. Torabi). 1568-4946/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2013.02.009
cooperatively. The decision on the presence or absence of the PU in the non-cooperative spectrum sensing is performed individually by each SU, while in the cooperative case, the spectrum sensing is performed by collaboration between a group of SUs to mitigate some problems of spectrum sensing such as multipath fading, shadowing effects, and hidden PUs [6]. In recent years, many studies have been devoted on the cooperative spectrum sensing and many approaches are provided for detecting the PU signals including the cyclostationary feature detection [7] and the energy detection [8]. In most of the work, the power of the noise is assumed to be previously known for threshold setting. However, due to the limited sensing time and the fluctuation of the noise power, the power of the noise is not known precisely and the performance of some classical cooperative spectrum sensing techniques is susceptible to the noise power uncertainty. The main sources of the noise power uncertainty are the noise uncertainty of the receiver device due to the non-linearity and the thermal noise of the components, and the environment noise uncertainty caused by the transmissions of other users [9,10]. Several techniques have been proposed to mitigate the noise power uncertainty in cooperative-based CR systems [11–13]. The cooperative spectrum sensing method in the presence of the noise uncertainty in [11] is based on an energy detector that uses three thresholds for local sensing, while the proposed cooperative approach in [12], is based on a cooperative covariance and eigenvalue based detection approach. The methods proposed
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in [11] and [12] send local sensing results to a fusion center and hence occupy more bandwidth of the control channel. The authors in [13] utilize a bi-thresholds energy detector in each SU where one bit local decision is sent by the SU to a fusion center. In this paper, we investigate a centralized hard decision-based cooperative spectrum sensing problem in the presence of the noise power uncertainty using fuzzy set theory concepts as a mathematical framework to model such uncertainty. We propose a new spectrum sensing method based on the fuzzy hypothesis test (FHT) that utilizes membership functions as hypotheses for modeling and analyzing the noise power uncertainty. In particular, we apply the Neyman–Pearson lemma on the FHT and propose a thresholdbased local detector at each SU in which the threshold depends on the noise power uncertainty. In the proposed scheme, a centralized manner in the cooperative spectrum sensing is deployed in which each SU sends its one bit decision to a fusion center. The fusion center makes a final decision on the absence/presence of the PU. The performance of the PU’s signal detection is evaluated by the probability of the signal detection for a specific Signal to Noise Ratio (SNR) when the probability of the false alarm is set to a fixed value. We compare numerically the performance of the proposed algorithm with two classical threshold-based energy detectors. Simulation results show that the proposed algorithm considerably outperforms the methods with a bi-thresholds energy detector and a simple energy detector in the presence of the noise power uncertainty. The rest of the paper is organized as follows. In Section 2, the FHT background and the system model are provided. The sensing algorithm using the Neyman–Pearson lemma for the FHT is presented in Section 3. Simulation results are presented in Section 4. Finally, conclusions are drawn in Section 5. Notations: Throughout this paper, we use boldface lower case letters to denote vectors. A Gaussian random variable with mean m and variance 2 is represented by x∼N(m, 2 ). We use E[.] as the expectation operator, and P{.} for representing the probability of the given event. The signs “” and “” mean “almost smaller” and “almost greater”, respectively, and Q (.) is the complementary cumulative distribution function, which calculates the of a zero mean unit variance Gaussian variable, i.e. tail probability ∞ Q (x) = x √1 exp(−t 2 /2)dt. Also, (.)T stands for the transpose of 2
Definition 2.1. Any hypothesis of the form “H : is H()” is called a fuzzy hypothesis, implying that is in a fuzzy set of with the membership function H() which is a function from to [0, 1] [20]. Note that the crisp hypothesis “Hi : ∈i ” is a fuzzy hypothesis with the membership function “Hi = 1 at ∈ i , and zero otherwise”. 2.2. Neyman–Pearson lemma for FHT Let X = (X1 , . . . , Xn )T be a random sample vector, with the observed value x = (x1 , . . . , xn )T where Xi has the probability density function (pdf) f(xi ; ) with the unknown parameter ∈ , in which is the parameter space. Suppose the two membership functions H0 () and H1 () are known and we would like to test
H0 :
is H0 (),
H1 :
is H1 ().
(2)
The aim is to accept or reject H0 on the basis of x. In other words, we would like to make a test function (X) such that (x) is the probability of rejecting H0 if X = x is observed. We use the above FHT for our analysis in the subsequent section. Definition 2.2. For the fuzzy hypothesis test in (2.1) and the test function (X), the false alarm probability denoted by Pfa is defined by [22] Pfa =
1 M
H0 ()E [ (X)] d,
(3)
where
M=
H0 ()d,
(4)
and
E [(X)] =
(x)f (x; )dx,
(5)
where f(x ; ) is the joint pdf of X with the unknown parameter ∈ .
a matrix or vector.
In the next theorem, we present the Neyman–Pearson lemma for the above fuzzy hypothesis test.
2. FHT background and system model
Theorem 2.3. Let X = (X1 , . . ., Xn )T be a random sample vector, with the observed value x = (x1 , . . ., xn )T where Xi has the pdf f(xi ; ) with the unknown parameter ∈ . Defining (x) = ( f(x ; )H1 ()d)/( f(x ; )H0 ()d), for the fuzzy hypothesis test problem in (2.1)
2.1. Fuzzy hypothesis test The conventional binary detection problem is a decision between the two crisp hypotheses H0 , known as the null hypothesis versus the alternative hypothesis H1 . In some situations, we face many practical problems in which the observed data are associated with some uncertainty. Over the past years, there have been some efforts to analyze this uncertainty using the fuzzy set theory [14–16]. Taking the uncertainty into account in the hypothesis test introduces an interesting problem called Fuzzy Hypothesis Test (FHT). To introduce the FHT, suppose that we are interested to test the value of the mean parameter of a normal distribution denoted by based on the observations. In the ordinary case, we use the test H0 : = 0 against H1 : = / 0 . Due to the uncertainty on the model parameter, a more realistic hypothesis can be written as
H0 :
is close to 0
H1 :
is away from 0 .
(a) Any test with the test function (x) =
⎧ ⎨1 ⎩
(x) > ,
ı(x) (x) = , 0
(6)
(x) < ,
for some ≥ 0 and 0 ≤ ı(x) ≤ 1, is the best test of size Pfa . (b) For 0 ≤ ˛ ≤ 1, there exists a test of form (6) with ı(x) = ı (a constant), for which Pfa = ˛. See [22] for the proof. 2.3. System model
(1)
Such hypothesis test is a FHT which has been extensively investigated in the literature [17–21].
Suppose in a homogeneous cognitive radio network all the SUs use the same protocol for the local spectrum sensing. The network contains K secondary users indexed by K = {1, . . . , K} to exploit the spectrum holes. We use a centralized manner in the cooperative
A. Mohammadi et al. / Applied Soft Computing 13 (2013) 3307–3313
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for N = 2TW, where W and T are the signal bandwidth and the detection time, respectively, and yk [n] is the sample of yk (t) at discrete time t = n/2W. Tk and k are the test statistic and the detection threshold at the kth SU, respectively. It is a well-known fact that if N is large enough, the statistics of Tk can be approximately written as [25]
Tk ∼
2 , 2N 4 ), N(Nw,k w,k
under H0 ,
2 N(N(w,k
2 ), 2N( 4 + s,k w,k
2 w,k
2 s,k
4 + s,k
2 2 )), + 2w,k s,k
under H1 , (9)
where and are the estimated noise and signal variances, respectively. Therefore, the probability of false alarm in the kth SU, Pfak is given by
Pfak = P Tk > k | H0
spectrum sensing in which each SU sends its one bit decision to a fusion center as shown in Fig. 1. The spectrum sensing problem for the kth SU, k ∈ K, can be represented by the following binary hypothesis test: H0 :
yk (t) = wk (t),
H1 :
yk (t) = sk (t) + wk (t),
3. Cooperative spectrum sensing using Neyman–Pearson lemma for FHT
One of the most common detectors that is largely used for spectrum sensing is the energy detector, because of a low computational and implementation complexities. Fig. 2 shows the block diagram of the energy detector. The frequency band of the interest is chosen by applying a Band Pass Filter (BPF) to the received signal and the power of the received signal is computed by the following equation at the kth SU:
n=0
H0
(10)
(11)
H0 :
k2 ≤ k ,
H1 :
k2 > k ,
(12)
where k2 is the variance of the received signal on the observation interval at the kth SU which can be written as
k2
=
2 , w,k
under H0
2 w,k
under H1 .
2 , + s,k
(13)
According to our discussion about using the membership function in the hypothesis test (2.1), we can write (12) using the crisp membership functions as follows:
H1 :
k2 is H1 (k2 ) = U(k2 − k ),
H0 :
k2 is H0 (k2 ) = 1 − H1 (k2 ),
(14)
where U(.) is the unit step function. 3.2. Local spectrum sensing using FHT
3.1. Energy detection
N−1
H yk [n] 2 ≷1 k ,
=Q
2 k − Nw,k √ 2 w,k 2N
Using the energy detector in (8), the local detection of the PU’s signal is equivalent to the comparison of the variance of the received signal at the SU with the threshold k , i.e.,
(7)
where H1 and H0 are the hypotheses of the presence and absence of the PU’s signal, respectively, yk (t) denotes the received signal at the kth SU and wk (t) represents an additive complex white Gaussian noise. In addition, we assume that sk (t) is a zero mean Gaussian signal from the PU at the kth SU’s receiver. This assumption is used in many literature related to the spectrum sensing in CR networks (e.g. [23,24]). We also assume that sk (t) and wk (t) are independent. The probability of detection and the probability of false alarm will be used as the criterion for the performance evaluation of our algorithm. In a CR network, a smaller false alarm probability indicates a higher spectrum efficiency, and a larger detection probability indicates a lower interference between the PU and the SUs.
Tk =
and the detection threshold in (8) is obtained as √ 2 k = w,k ( 2NQ −1 (Pfak ) + N).
Fig. 1. Centralized cooperative spectrum sensing in a cognitive radio network.
(8)
As previously mentioned, the energy detector computes the power of the input signal and compares it with a threshold which depends on the noise power. However, the power of the noise is not known precisely and the performance of the energy detector is susceptible to the noise power uncertainty. To represent this uncer2 belongs to the tainty, assume that the estimated noise variance w,k
2 , (1 + ) 2 ], where 2 is a known nominal interval [(1 − k )0,k k 0,k 0,k noise variance at the kth SU’s receiver, and 0 < k < 1 is a parameter that defines the noise power uncertainty bound [26]. In practice, the range of k is normally between 0.12 and 0.6 [23]. One heuristical approach to encounter the noise power uncertainty is to use the fuzzy hypothesis test that utilizes the membership functions as hypotheses. According to the previous discussion about the fuzzy
Fig. 2. Block diagram of the energy detector.
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distributed complex Gaussian random variables with zero-mean and variance k2 and with regard to (8), f (yk ; k2 ) can be written as
1 0.9
H0 Membership Function
Fuzzy Hypothesis, ρ = 0.6 Fuzzy Hypothesis, ρ = 0.26
0.8
f (yk ; k2 ) =
Crisp Hypothesis
Tk
−
.
k2
(19)
Thus, the test statistic is represented by
∞
0.6
0
0.5
(yk ) =
∞
0.4
0
(
exp −
1 )N 2
exp −
k
(
Observations
Fig. 3. Comparison between the membership functions of the crisp and fuzzy hypothesis models for different values of the uncertainty factor.
hypothesis test, the corresponding fuzzy hypothesis test can be implemented by converting (12) to the following equation: H0 :
k2 k ,
(15) k2 k , √ 2 ( 2NQ −1 (P ) + N), and 2 comes from (13). where k = 0,k fak k By extending the crisp membership functions in (14), we propose the membership functions for the fuzzy hypothesis test as follows: H1 :
⎧ ⎪ ⎨ H1 : k2 is H1 (k2 ) = H0 :
1 , 5 − (k2 − k ) 1 + e k k2 is H0 (k2 ) = 1 − H1 (k2 ).
− 5 ( 2 − )
(17)
H0
where yk = [yk (0) , yk (1) , . . . , yk (N − 1)]T is the received vector at the kth SU’s receiver, k ≥ 0 is the test threshold, and the test statistic (yk ) is given by Theorem 2.3 as
0
(yk ) =
1 5
2
− ( − ) 1+e k k k
5
dk2
5
2
( − ) − 1+e k k k
(20)
.
2
− ( − ) e k k k
dk2
1, Tk > k ,
(21)
Tk < k ,
0,
where k is determined based on the desired probability of the false alarm. In fact, the false alarm probability for the fuzzy hypothesis testing is given by (3) as follows: 1 M
(k)
Pfa =
∞
e
− 5 ( 2 − k )
1+e
0
k
k
(18)
For the above equation, f (yk ; k2 ) is the pdf of the received signal by the kth SU. Since yk consists of N independent and identically
E 2 [ (yk )] dk2 ,
− 5 ( 2 − k )
(22)
k
k
k
where from (4) we have
M= 0
∞
e
− 5 ( 2 − k )
1+e
k
k
− 5 ( 2 − k ) k
k
k
we have (k2 − k ) < 0 and e k k k → ∞, then the membership function values will be H1 (k2 ) = 0 and H0 (k2 ) = 1. Fig. 3 shows the fuzzy membership function of H0 for the crisp and fuzzy hypothesis tests for different values of uncertainty factor k . It is seen that the uncertainty in the fuzzy hypothesis model causes a smooth transition from H0 to H1 and vice versa. In addition, the fuzzy hypothesis model tends to the crisp hypothesis model when k decreases. The local spectrum sensing at each SU, using the Neyman–Pearson lemma for the fuzzy hypothesis test, in Theorem 2.3 can be performed as:
f (yk ; k )H0 (k )dk
Tk 2
k
E 2 [(yk )] = P Tk ≥ k
−5/ ( 2 − )
∞ f (yk ; k2 )H1 (k2 )dk2 (yk ) = 0∞ . 2 2 2
dk2 =
and applying (21) in (5)
k k k have (k2 − k ) > 0 and e → 0. Thus, from (16) we have H1 (k2 ) = 1 and H0 (k2 ) = 0, while, in the absence of the PU’s signal,
H1
k
k
(16)
The proposed membership functions yield the desirable properties H1 (k2 ) = 0 for k2 k and H1 (k2 ) = 1 for k2 k . In addition, in the ideal case that there is no noise power uncertainty, i.e.
k → 0, the membership functions tend to the crisp forms. More precisely, in the presence of the PU’s signal at the kth SU, we
(yk ) ≷ k ,
Tk 2
Taking the first-order derivative of (20) in terms of Tk , it is shown that (yk ) is an increasing function of Tk . Therefore, the best test function, based on the Neyman–Pearson lemma for the FHT, is obtained by comparing the energy of the received signal Tk with a new threshold denoted by k , i.e.:
Threshold
0
1 )N 2
0.3
0.1
⎪ ⎩
exp
k2
0.7
0.2
N
1
k ln (1 + e5/ k ), 5
(23)
∞
= k
(24)
fTk (tk )dt k ,
where fTk (tk ) is the pdf of the Tk . Using (8), since Tk is the square summation of N independent and identically distributed Gaussian random variables with zero-mean and variance k2 , where N is sufficiently large, Tk is approximated by N(Nk2 , 2Nk4 ) [27]. Therefore, we have
E 2 [(yk )] = Q k
k − Nk2 √ 2Nk2
.
(25)
Thus, the false alarm probability at the kth SU is (k) Pfa
5
= k ln (1 + e k ) 5
0
∞
e
− 5 ( 2 − k )
1+e
k
k
− 5 ( 2 − k ) k
k
Q
k − Nk2 √ 2Nk2
dk2 . (26)
In the proposed algorithm, k is determined according to the 2 . Based on (26), the noise condition and k depends on the 0,k threshold values k are calculated for different values of the local false alarm probability and are used by the local detectors. Although, the calculation of the threshold values in this method is complicated, the threshold values can be calculated offline. 3.3. Computational complexity The main complexity of the proposed fuzzy method comes from the computation of two parts: the energy detector and the test threshold. The energy detector needs N multiplications and N − 1 additions. Since the threshold can be calculated offline, tabulated, and used by the detector at each SU, its complexity does not affect
A. Mohammadi et al. / Applied Soft Computing 13 (2013) 3307–3313 Table 1 2 = 1 and different values of Pfa and . Testthreshold of the proposed method for 0,k
3311
1 0.9
Test threshold values Pfa = 0.1
Pfa = 0.15
195.65 200
180.78 183.01
169.166 170.32
the total complexity. For instance, Table 1 gives some threshold val2 = 1, and different values of and the desired local false ues for 0,k alarm probability. Therefore, the total complexity of the proposed fuzzy method is the same as that of the energy detector. In the next section, we use the results of the local spectrum sensing at each SU to perform the cooperative spectrum sensing.
0.8
Probability of Detection
= 0.12
= 0.26
Pfa = 0.05
0.7 0.6 0.5 Fuzzy Method ( ρ = 0.12)
0.4
Energy Detector ( ρ = 0.12) 0.3
Fuzzy Method ( ρ = 0.26) Energy Detector ( ρ = 0.26)
0.2
Fuzzy Method ( ρ = 0.6) Energy Detector ( ρ = 0.6)
0.1
3.4. Centralized cooperative spectrum sensing 0 0
Cooperative spectrum sensing improves the performance of the spectrum sensing at the low SNR and decreases the sensing time. It also reduces the effects of the hidden PU, multipath fading, and shadowing problems. In this paper, we consider the centralized CR networks in which each SU sends its sensing result to a fusion center via a dedicated control channel. The fusion center combines the received sensing results and makes a decision on the presence or absence of the PU signal. We use a hard decision fusion method to reduce the communication overhead at the control channel. In this method, the SUs send a binary local decision to the fusion center, and the fusion center decides about the presence of the PU with some hard rules such as “OR” and “AND” rules. Based on the AND rule, the fusion center decides the presence of the PU’s signal, if all SUs decide H1 , while, based on the OR rule, the fusion center decides about the H1 , if at least one SU decides H1 . The detection and the false alarm probabilities based on the AND and OR rules in the cooperative spectrum sensing can be written as follows [28]: Qf −AND =
K
(k)
(27)
(k)
(28)
Pfa ,
k=1
Qd−AND =
K
Pd ,
k=1
Qf −OR = 1 −
K
(k)
1 − Pfa
,
(29)
(k)
1 − Pd
,
(30)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 4. ROC curves of a single SU for the fuzzy method and the energy detector for N = 200, SNR = −10dB and various noise power uncertainties.
It is observed that by decreasing the noise power uncertainty the performance of both methods improves. Moreover, the proposed fuzzy method outperforms the energy detector in all cases. To improve the performance, we use the centralized cooperative spectrum sensing with the “OR” and “AND” combining rules. The performance of the proposed method is compared with the bi-thresholds detector proposed in [13] and the simple energy detector. In Fig. 5, we compare the proposed fuzzy method with the simple and bi-thresholds energy detectors for = 0.6 and SNR = −20 dB for each SU. As can be realized, the performance of the proposed fuzzy method with both rules is the best. As mentioned before, in the proposed method, we use membership functions as hypotheses which have a soft transition behavior from the H1 hypothesis to the H0 and vice versa that depends on the noise power uncertainty. In other words, the membership functions take different values, depending on the noise power uncertainty and the threshold is determined based on these values, while, the threshold in the simple energy detector is set regardless of the noise power uncertainty. The bi-thresholds energy detector utilizes only two thresholds that depend on the noise power uncertainty 0
10
0
k=1
where Qf and Qd represent the false alarm and the detection prob(k)
(k)
abilities at the fusion center, respectively, and Pfa and Pd denote the false alarm and the detection probabilities at the kth SU, respectively. 4. Simulation results In this section, we present some numerical results to evaluate the performance of the proposed FHT algorithm using the detection probability versus SNR and the Receiver Operating Characteristic (ROC) curves. In our simulation, k and SNRk indicate the noise power uncertainty bound and the SNR at the kth SU, respectively. Moreover, the number of samples (N) is equal to 200 and the number of SUs (K) is equal to 5. Fig. 4 shows the ROC curves of a single SU for the proposed fuzzy method compared to the energy detector for SNR = −10dB and different values of the noise power uncertainty.
ROC in logarithmic scale
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Probability of False Alarm Fig. 5. ROC curves for comparison of the fuzzy method with a simple energy detector and a bi-thresholds energy detector in cooperative spectrum sensing (for each SU: N = 200, SNR = −20 dB, = 0.6).
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0.7 0.6 0.5 ρ = 0.12 , Qfa FC = 0.001
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bound. In Fig. 6, we consider the case that the noise power uncertainty factor for each SU is different. For this simulation, we set SNR = −20 dB, 1 = 0.26, 2 = 0.12, 3 = 0.6, 4 = 0.6, and 5 = 0.26. It is seen from Fig. 6 that the proposed fuzzy method along with the “AND” rule outperforms the other detectors. A more practical case is when both uncertainty factor and SNR are different for all SUs. For this case, Fig. 7 depicts the ROC curves of the aforementioned detectors in which we have set SNR1 = −12 dB, 1 = 0.12, SNR2 = −15 dB, 2 = 0.26, SNR3 = −20 dB, 3 = 0.12, SNR4 = −15 dB,
4 = 0.6, andSNR5 = −18 dB, 5 = 0.6. It is observed that the proposed fuzzy method with the “AND” rule outperforms the other detectors. Fig. 8 depicts the probability of detection at the fusion center with the “AND” rule fuzzy method (Qd−AND ) versus SNR for different values of uncertainty factor and probability of false alarm at the fusion center (Qfa−FC ). In this figure, we assume that all SUs take the same value of the uncertainty factor and the false alarm probability. As can be observed, by increasing the SNR, the performance of the
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Probability of False Alarm Fig. 6. ROC curves for comparison of the fuzzy method with a simple energy detector and a bi-thresholds energy detector in cooperative spectrum sensing, for N = 200, SNR = −20 dB and different uncertainty factors 1 = 0.26, 2 = 0.12, 3 = 0.6, 4 = 0.6, and 5 = 0.26.
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Fig. 8. Pd versus SNR for the fuzzy method with the “AND” rule versus SNR at different uncertainty factors and Qfa−FC in cooperative spectrum sensing, when all SUs are the same.
detector improves and the proposed fuzzy method with the “AND” rule fulfills the spectrum sensing requirements mentioned in the IEEE 802.22 standard [29] at Qfa−FC = 0.05. 5. Conclusion In this paper, a new centralized cooperative spectrum sensing algorithm based on the fuzzy hypothesis test was proposed facing with the noise power uncertainty. We derived a detector using the Neyman–Pearson lemma for the fuzzy hypothesis test at each SU and showed that the proposed detector is an energy detector with a threshold depending on the noise power uncertainty. The local hard decisions from the SUs are sent to the fusion center and combined using the “AND” and “OR” rules for the final decision. Simulation results showed that the proposed algorithm performs superior than the cooperative spectrum sensing with the simple and bi-thresholds energy detectors and is preferable for spectrum sensing applications. References
0
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Probability of False Alarm Fig. 7. ROC curves for comparison of the fuzzy method with a simple energy detector and a bi-thresholds energy detector in cooperative spectrum sensing, for SNR1 = −12 dB, 1 = 0.12, SNR2 = −15 dB, 2 = 0.26, SNR3 = −20 dB, 3 = 0.12, SNR4 = −15 dB, 4 = 0.6andSNR5 = −18 dB, 5 = 0.6.
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