A Jarque-Bera Test Based Spectrum Sensing for Cognitive Radio Agus Subekti∗# ,Sugihartono∗ , Nana Rachmana S.∗ , Andriyan B. Suksmono∗ ∗
School of Electrical Engineering & Informatics Bandung Institute of Technology (ITB), Indonesia # Research Center for Informatics Indonesian Institute of Sciences (LIPI),Indonesia
[email protected]
Abstract—A cognitive radio has to perform spectrum sensing to detect the vacant channel. The detection faces some challenges due to the required performance and limited knowledge on the primary signals and the channel. In this paper we proposed a blind spectrum sensing method for cognitive radio network. The proposed method based on the difference on distribution of the condition between when the transmission from primary user is active and when it’s inactive. If it is inactive, the received signal will contain only noise. The distribution of noise which is Gaussian, differs from the distribution of signal which is contaminated with noise. The Jarque-Bera (JB) test is used for the detection of the event. The target of the analysis is both the real and the imaginary parts of FFT’s output. The proposed algorithm was tested to detect the DTV signal. Results show that our method is performed better than previous similar method.
I. I NTRODUCTION Cognitive radio is defined as a radio that understands the context in which it finds itself and as a result can tailor the communication process in line with that understanding [1]. Cognitive radio has several potential applications. One of the most popular today is to use the cognitive radio for communication in the white space. The white space term is used for segments of licensed spectrum which is not being used by the license holder and may be used by unlicensed users. This application comes from the fact that the recent advance in wireless networks results in frequency scarcity problem. One of current available new radio access standard based on cognitive radio concept has been issued lately such as standardized by IEEE 802 and IEEE 1900 [2]. In the IEEE std. 802.22 for example, a cognitive radio platform may transmit at the spectrum holes in the TV bands as long as the channel is not being used by primary user or the owner of spectrum license. In cognitive radio networks there are primary users as the owner of license and secondary users. Because the secondary user may use only unused spectrum, it has to ensure that its transmission will not cause harmful interference to the primary user. This task is performed by its sensing capability before transmitting. The spectrum sensing performance have to meet
a required sensitivity of detection. It should not cause harmful interference due to occupied spectrum is falsely detected as vacant. On the other hand, if the channel that is not being used by primary user detected as occupied, it will give lower spectrum holes utilization. It has to be able to detect very weak signal. This quite challenging because it has to deal with some constrains such as wireless fading, noise fluctuation and limited knowledge on signal’s parameters. Several sensing methods have been proposed to combat the above mentioned challenges such as matched filtering, feature detection approach and the energy detection [3]. Among other methods, matched filtering gives better performance as it can detect signal at very low SNR. However, the matched filtering method requires a priori knowledge of several primary signal’s information such as pilot and frame structure. For the feature detection method which relies on cyclostationarity, the sufficient signal information must be given as well. The energy detection differs from both method because it can work without any primary’s signals parameters. In this method, the test statistic is energy of received signal samples makes it simple to be implemented. Unfortunately, the energy detection’s performance is among the lowest at low SNR due to SNR wall [4]. Several methods which exploit the distribution were proposed such in [6] and in our previous work at [5]. In this paper, we proposed another similiar method based on the normality test using Jarque-Bera (JB) test statistic. The skewness and the kurtosis of received signal are used to calculate the JBtest statistic of the samples. The values are compared with a predefined threshold to distinguish between occupied spectrum and white space. The threshold is calculated from empirical estimation of system’s noise. Evaluation by simulation shows that the proposed method performs better than similar previous proposed method in [6]. The rest of this paper describes problem formulation of spectrum sensing problem, explanation of proposed method, result of performance evaluation and comparison by simulation and ended with conclusion.
c 978-1-4799-7447-4/14/$31.00 2014 IEEE
II. S YSTEM M ODEL A cognitive radio finds the unused frequency spectrum by continuous sensing the primary’s transmission activity on a certain band. When the channel is being used, spectrum sensing detects the signals from the primary user which naturally will contain also noise. Spectrum sensing will decide a channel is vacant if it does not detect the presence of the primary signal or only noise is detected. There is a chance that primary signal can not be detected due to its level is much lower compare to the noise. Suppose r(n) is the discrete time received signal, s(n) is the primary signal, and w(n) is additive white Gaussian noise (AWGN). When the channel is being used by primary user, spectrum sensing module receives signals together with the noise: r(n) = s(n) + w(n). If the channel is vacant, then only noise will be detected: r(n) = w(n). Suppose N is the number of received samples, spectrum sensing has to decide among two possible conditions: H0 : r(n) = w(n)
n = 0, 1, , N − 1
(1)
H1 : r(n) = s(n) + w(n)
n = 0, 1, , N − 1
(2)
where H0 denotes as vacant channel condition while H1 as a condition of the channel is being used. The main aim of spectrum sensing is to make a correct decision based on its received samples even at low signal level compare to noise or low SNR. The false detection will happen when the detector decide channel is occupied (H1 ) but the true is vacant (H0 ). The probability of this decision is called probability of false alarm: Pf = P (H1 |H0 ). The main objective of spectrum sensing is to maximize probability of detection: Pd = P (H1 |H1 ). The detector has to achieve high Pd even at low signal level (SNR). The other constrain is in the limited knowledge of primary signal parameter. We have to consider a method that does not need any primary signal’s parameter or blind spectrum sensing method. III. P ROPOSED S PECTRUM S ENSING M ETHOD We propose spectrum sensing method based on the distribution difference. When the channel is idle, H0 : r(n) = w(n). w(n) is AWGN, Gaussian sample sequences. w(n) ∼ 2 N (µw , σw ) i.e. Gaussian random variable with the probability density function (pdf) of: 1 1 2 p exp − 2 (w − µw ) (3) p(w) = 2 2σw 2πσw When the channel is occupied, H1 : r(n) = s(n)+w(n). s(n) is received signal propagating from primary user which already includes the wireless channel effects. The probability of s(n) will have several possibilities depends on the multipath fading channel model, such as rician, rayleigh, etc. Rayleigh can be considered as general condition in wireless environments. The rayleigh random variable has the probability density function (pdf) of: s s2 (4) p(s) = 2 exp − 2 σs 2σs
where σs2 is the signal’s variance/power. As s and w are independent, the probability density function of received signal then will be p(r) = p(s) ⊗ p(w) [7]. The pdf of r(n) can be calculated as: Z ∞ p(s)p(r − s)ds p(r) = 0 Z ∞ s2 1 (r − s)2 s exp − 2 exp − =√ ds 2 2σs 2σw 2πσw σs2 0 (5) which will be: p(r) =
σs r
e 2 + σ 2 )3/2 (σw s Φ
−
r2 2 +σ 2 ) 2(σw s
r σs p 2 + σ2 σw σw s
! +√
2 − r σw e 2σw2 2 + σ2 ) 2π (σw s
(6)
Where Φ(x) is a cumulative distribution function of standard normal random variable. Based on the distribution difference, the conditions of hypothesis testing equation becomes: 2 • H0 : distribution of r(n) = w(n) will be N (µw , σw ) 2 • H1 : distribution of r(n) = s(n) + w(n) 6= N (µw , σw ) Based on the distribution of received samples: r = r0 r1 ... rN −1 we have to draw a decision of detection. Our goal is to design an efficient spectrum sensing method which works based on this distribution difference. There are several statistical methods to analyze the distribution from measured samples. In the proposed method we use JB test. The JB (Jarque-Berra) test is a goodnessof-fit test which measures the departure of samples from Gaussian/normal distribution [8]. In this method, we don’t need to know the statistical parameters such as the mean and the variance. The JB test statistic is defined as: ! 2 ns (K − 3) 2 JB = S + (7) 6 4 where ns is the number of samples; S is the sample skewness and K is the sample kurtosis. Given sample of vector r with the length of ns , with unknown mean and variance, the JB test performs testing of the null hypothesis that the sample comes from a normal distribution. The skewness (S) is defined as: Pns 3 1 µ3 µ3 i=1 (ri − r) ns S= 3 = = (8) P 3/2 3/2 σ 2 ns (σ 2 ) 1 i=1 (ri − r) ns The skewness is usually used to measure the symmetry/asymmetry of the data from the mean. Negative skewness shows the data are more spread out to the left side of the mean. If the skewness is positive, the distribution of data are spread out more to the right. The skewness of any perfectly symmetric distribution, such as Gaussian distribution, is zero. The kurtosis (K) is defined as: Pns 4 1 µ4 µ4 i=1 (ri − r) ns = (9) K= 4 = 2 2 Pns σ 2 (σ 2 ) 1 (r − r) i i=1 ns
The kurtosis of the normal distribution is 3. A distribution with the kurtosis less than 3 tends to have flatter density than the Gaussian one. On the other hand, if the kurtosis is more than 3, its density has sharper peak and longer tails than the Gaussian. Following the skewness and the kurtosis of Gaussian distribution, a vector of samples should have JB equal to zero if its density is Gaussian. Due to limited number of available samples, the JB will vary even for AWGN. The JB values have to be compared to a threshold to make decision on hypothesis testing. Cognitive radio performs signal detection on samples of r(n) after several pre-processing steps including downconverting, filtering and Fast Fourier transform (FFT) [6]. We have to arrange a N-points FFT on r(n) to get R(k): R(k) =
NFX F T −1
x(n) exp −j2π (k − 1)
n=1
n−1 NF F T
1 ≤ k ≤ NF F T − 1
(10)
Vector of R(k) are complex valued, R = Re(R) + j Im(R). JB test can be applied to Re(R), Im(R), or kRk. When the condition is H0 , the distribution of Re(R) and Im(R) should be Gaussian. In order to maximize the information, for the test statistic (T ) of our proposed method, we used the both parts: skewness(Re(R)) + skewness(Im(R)) 2 kurtosis(Re(R)) + kurtosis(Im(R)) Kt = 2 St =
(11)
Since we perform spectrum sensing using N available samples, our test statistic is: ! 2 N (Kt − 3) 2 T = St + (12) 6 4 For the large number of T , based on the central limit theorem, its distribution can be approximated as Gaussian with the mean µT and variance σT2 . Probability of T higher than a certain value for example t will follow: Z ∞ − 12 (T −µT )2 1 2σ T P (T > t) = p e dT 2πσT2 t t − µT =Q (13) σT where Q() is marqum-q function. The threshold of our proposed method then will be: λt = µT + σT Q−1 (Pf )
(14)
To get the threshold, we set the probability of false alarm to be 0.1 (P (t > λ) = Pf = 0.1). The mean µT and varianceσT of the test statistic are measured in the condition of H0 . For set of certain number of FFT points and also number of samples (N ), we can find the specific threshold. Once we get the threshold, we can use it to perform detection for spectrum sensing.
IV. P ERFORMANCE E VALUATION The proposed method was evaluated by Monte Carlo simulations. The objective was to find the performance especially the probability of detection. The test statistic as described above was calculated and compared with the threshold to get the detection rate. As the common main interest for detectors is to find its sensitivity, i.e. its performance at low level s(n), we performed the simulation for several signal’s levels or several SNR. To ensure the experiments were statistically correct, we repeated the simulation for about 1000 times. First step is to find the threshold. The threshold is taken empirically from the noise samples. The additive white Gaussian noise (AWGN) samples were generated. JB test to complex valued samples was performed to the FFT output according to equation 11 and equation 12. As mentioned before, the mean and variance are taken to calculate the threshold. We calculate threshold using equation 14 for N-FFT of 2048 and 8196. Number of samples are 30000, 60000, and 120000. For the threshold calculation, as common requirement, we set the probability of detection to 10 %. After we got the threshold, we evaluated the proposed method for the DTV signals [9] as primary signal s(n). The signal’s parameters are: • frequency: 545 MHz, • signal bandwidth’s: 6 MHz, • sampling rate: 50 MHz • oversampling: 8/7 bandwidth The result of digital sample at ADC output with speed will equal to 6857 samples per 1 ms. If we have to perform sensing and have to make decision in 5 ms, it will be equal to about N = 30, 000 samples. The noise (w(n)) was added and the signal (s(n)) was scaled to achieve certain signal to noise ratio (SNR). After to be scaled thhe input of the simulation was: r(n) = scaling(s(n)) + w(n)
(15)
As comparison, we also simulated the signals by method proposed in [6]. Compare to that previous method, our method differs in the test statistic and the threshold. The previous method used the amplitude of FFT’s output to be the input of the test statistic. As for the threshold, they proposed a threshold formula . First experiment used N-FFT = 2048. The number of sample N for detection is 30000. Detection rate was counted for several SNR. The figure 1 shows that the performance of proposed method gives better detection rate than the previous method mainly due to our method can adapt to the noise when the number of sample (N) changes as well as the FFT points (N-FFT). Figure 2 shows that performance of the proposed detector at several number of samples: N = 30000, N = 60000 and N = 120000. The probability function or histogram built based on empiric samples will converge to its precise pdf when the number of samples is higher. This result in higher detection rate as confirmed on this figure. Figure 3 describes the role of FFT to the performance of the detector. Here again we took the detection rate to measure it. We conducted the simulation for 2 different FFT points:
N-FFT = 2048 N-FFT = 8192 The two N-FFTs above also parameters includes in the digital TV standard. The figure shows that roles of the number FFT to the performance of the detector. The point number of FFT shows the resolution of the received signals. FFT with 8196 points reveals more information than N-FFT = 2048. From the figure 2 and 3 we can conclude that increase the size of FFT gives more impact on the detection rate compare to the use of more samples per-frame (N). •
•
Fig. 3: Detection performance for N-FFT=2048 & NFFT=8192
Fig. 1: Detection performance at N-FFT=2048, N=30000
not need any information on the parameters of the primary signal. The target of analysis is the real and imaginary part of FFT output for every frame of N samples. The test statistic for each frame is compared to the predefined threshold to make a decision. The threshold is set based on empiric analysis of the noise. This makes the method can adapt to the noise’s characteristic. According to our monte carlo simulation results, for the digital TV test signal, our proposed method performs better than previous JB test sensing method. Detection performance increases as the FFT size as well as the number of samples increase. R EFERENCES
Fig. 2: Detection performance for N=30000, N=60000 & N=120000 V. C ONCLUSION In this paper, we propose a blind spectrum sensing method based on JB (Jarque-Bera) test. The proposed method does
[1] J.Mitola and G.Maguire, “Cognitive radio: Making software radios more personal,” IEEE Personal Communications, vol. 6, no. 3, p. 1, 1999. [2] H. Harada, “White space communication systems: An overview of regulation, standardization and trial,” IEICE Transactions on Communications, vol. E97-B, no. 2, pp. 261–274, February 2014. [3] T.Yucek and H.Arslan, “A survey of spectrum sensing algorithm for cognitive radio applications,” IEEE Communications Surveys and Tutorials, vol. 11, no. 1, pp. 116–130, 2009. [4] R.Tandra and A.Sahai, “SNR walls for signal detection,” IEEE Journal of Selected Topics in Signal Processing, vol. 2, no. 1, pp. 4–17, February 2008. [5] A. Subekti, Sugihartono, N. R. Syambas, and A. B. Suksmono, “A HOS based spectrum sensing for cognitive radio in noise of uncertain power,” in Information and Communication Technology (ICoICT), 2014 2nd International Conference on, May 2014, pp. 511–514. [6] L.Lu, H.-C. Wu, and S. Iyengar, “A novel robust detection algorithm for spectrum sensing,” IEEE Journal on Selected Areas in Communications, vol. 29, no. 2, pp. 305–315, 2011. [7] H.Taub and D.L.Schilling, Principles of Communication Systems. McGraw-Hill, 1989. [8] C. M. Jarque and A. K. Bera, “A test for normality of observations and regression residuals,” International Statistical Review, vol. 55, no. 2, pp. 163–172, August 1987. [9] I.Garrison, “DTV channel characterization,” in Conference on Information Sciences and Systems (CISS) 2001, 2001.
