International Conference on Intelligent Control and Information Processing August 13-15, 2010 - Dalian, China
Joint Energy and Autocorrelation Based Spectrum Sensing Algorithm for Cognitive Radios Lei Yang, Zhe Chen, and Fuliang Yin Abstract— In cognitive radio networks, spectrum sensing is a key technology to detect the presence of the primary user. In this paper, a joint energy and autocorrelation based spectrum sensing algorithm is proposed. The proposed detection algorithm is based on the assumption that the primary user’s signal is not white. We compare the performance of the proposed detector and the conventional energy detector in both additive white Gaussian noise channel and multipath fading environments, simulation results show that the sensing performance is improved when the proposed detector is adopted.
I. I NTRODUCTION
C
OGNITIVE radio (CR) is an intelligent wireless communication system, it can sense and get information from the surrounding environment. The CR technology has recently been researched due to its ability to adapt the wireless environment by changing the operating parameters [1], [2]. A standard for wireless regional area networks (WRANs) [3] is proposed by the IEEE 802.22 Working Group, which adopts the CR technology. Spectrum sensing is a key technology of the cognitive radio, which can obtain the knowledge about the spectrum usage and the presence of the primary user [4], [5]. If the spectrum band is not occupied by the primary users, it will be known as the ”spectrum hole” and the secondary users can use the spectrum hole temporarily [1]. The secondary users need to periodically detect the presence of the primary user, if the primary user appears, the secondary users should change to another free channel immediately in order to minimize the interference to the primary user. The CR system divides working period into sensing period (quiet time) and transmission period. The secondary users stop transmitting in the sensing period. A challenge for cognitive radio design is minimizing the quiet time to improve the efficiency of the spectrum. The frequency bands used by WRANs are licensed for the analog and digital TV broadcasting and wireless microphones. The draft of the standard demands the sensing period is less than 2s, the probability of detection is 0.9 and the probability of false alarm is 0.1. The signals of the analog TV, digital TV, and wireless microphone should be detected when the SNRs are larger than 1dB, −21dB and −12dB, respectively. A number of different methods for spectrum sensing are proposed for identifying the presence of the primary users’ signal. We can classify these methods as energy detection Lei Yang, Zhe Chen, and Fuliang Yin are with the School of Electronic and Information Engineering, Dalian University of Technology, Dalian 116023, China. (email: yanglei
[email protected];
[email protected]; fl
[email protected]).
c 978-1-4244-7050-1/10/$26.00 2010 IEEE
418
based sensing, matched filtering based sensing, cyclostationarity based sensing and autocorrelation based sensing. The energy detection methods do not need any prior knowledge of the primary user’s signal [6]. The computation load is low and it is robust to unknown multipath fading. The matched filtering based sensing is known as the optimum method for detection of primary users when the transmitted signal is known [7]. The main advantage of matched filtering is the short time to achieve a certain probability of false alarm or probability of miss detection [4]. Cyclostationarity feature detection is a method for detecting primary user transmissions by exploiting the cyclostationarity features of the received signals [8], [9]. The cyclostationarity based detection algorithms can differentiate noise from primary users’ signals. This is a result of the fact that noise is wide-sense stationary with no correlation while modulated signals are cyclostationarity. The Matched filter detection and the cyclostationarity detection methods require lots of prior knowledge of the primary users, which are hard to be obtained in the real environment. Recently, the autocorrelation based sensing methods are proposed in [10], [11], [12]. The variance of the noise does not need to be estimated because of the “self-normalizing” feature of the narrow band signals. These detectors can detect the primary user’s signal based on the property that most of the narrow band signals’ autocorrelation matrices are not diagonal. In this paper, a new spectrum sensing algorithm is proposed based on the assumption that the primary user’s signal is not white. The reason why the received signal is correlated is discussed in [11]. We noticed that the correlation feature of the received signal does not considered by the conventional energy detector. Thus when the variance of noise is already detected by the sensors, if we consider the power and the autocorrelation of the signal jointly, the sensing performance will be improved. The rest of this paper is organized as follows: In Section II, we introduce the system model of the CR system. In Section III-A, we describe the basic algorithm of the conventional energy detector. The proposed joint energy and autocorrelation based detector is presented in Section IIIB. Some simulation results are presented in Section IV to show the performance of the proposed detector. Finally, the conclusions of this paper are given in Section V. Some notations are used throughout the paper. Bold uppercase symbols, e.g., A, are used to denote matrices. Bold lowercase symbols, e.g., a, are used to denote vectors. E{·} denotes the expectation. · denotes the integer-floor function.
II. S YSTEM M ODEL Let fs be the sampling rate, Ts = 1/fs be the sampling period. s(t) is the primary user’s signal and w(t) is the white noise. The sensing period is [0, T ]. Assume the noise w(t) 2 2 is stationary with zero mean and variance σw . σw is already detected by the sensors. The primary user’s signal suffers a multipath fading and is uncorrelated with the noise. The L length channel vector h is defined as h = [h(0), · · · , h(L − 1)]T . Then the received signal is given by x(t) = η
L−1 l=0
h(l)s(t − l) + w(t),
0≤t≤T
(1)
where η ∈ {0, 1} denotes the presence or absence of the primary user. If η = 1, the primary user appears, otherwise, there is only noise in the frequency spectrum. The samples number is N = T /Ts . We define x(n) = x(nTs ), s(n) = s(nTs ) and w(n) = w(nTs ) for simplifying the notations. L−1 Let u(n) = l=0 h(l)s(n − l). The variance of u(n) is σu2 . Then the received signals have two hypotheses H0 : x(n) = w(n) (2) H1 : x(n) = u(n) + w(n)
where H0 means there exists a primary user in the frequency band, while H1 means the primary user is absence.
When σu2 is known by the detector, the probability of detection with a given threshold λ is determined as Pd,ED = Q(
In this paper, we assume the signal samples are correlated when the primary user is presence. This assumption is reasonable because lots of the original signal is correlated, the primary user’s signal suffers a multipath fading, and since most of the signal is narrow band, when the received signal is over sampled, it will be highly correlated [11]. So we can get the conclusion that the autocorrelation matrix of the primary user’s signals is not diagonal. Define R(τ )= E{x(n)x(n + τ )} N −τ −1 = lim x(n)x(n + τ ) N →∞
N −1 1 2 x (n) N n=0
ˆ )= R(τ
(3)
Under hypothesis H0 , there is only noise in the received signal, thus z is chi-square distributed with N degrees of freedom. When N is large, z can be considered as a Gaussian 2 distributed random variable with mean σw and variance 4 2σw /N . The probability of false alarm can be derived as ∞ (z−σw2 ) 1 Pf,ED = e 2σw4 /N dz (4) 2 2σw π/N γ
Hence, for a given probability of false alarm Pf,ED , the threshold λ of an energy detector can be derived as 2 λ = σw (1 + 2/N Q−1 (Pf,ED )) (5) √ ∞ −t2 /2 where Q(x) = (1/ 2π) x e dt is the normal Qfunction. The test statistic of the energy detector is as follows TED =
N −1
2
x (n)
(6)
n=0
419
(8)
n=0
is the autocorrelation function of x(n). Since the samples number is finite in the real environment, the true value ˆ ) as the of R(τ ) can not be obtained. Thus, we use R(τ estimation value of R(τ ), which is defined as follows
A. Energy Detector
z=
(7)
B. Joint Energy and Autocorrelation Based Spectrum Sensing Algorithm
III. D ESIGN OF S PECTRUM D ETECTOR The energy detector is a basic spectrum sensing method, it does not need any information of the signal to be detected and is robust to unknown multipath fading. It has low computational and implementation complexities. The power of the received signal is
2 λ/σw ) 2 +1 σu2 /σw
1 N −τ
N −τ −1
x(n)x(n + τ )
(9)
n=0
We have some theorems for the estimation value of the autocorrelation function. ˆ ) is a Gaussian Theorem 1: Under hypothesis H0 , R(τ 4 distributed variable with zero mean and variance σw /(N −τ ) when τ = 0. Proof: Let x and y are two independent normally distributed variables with zero means and variances σx2 and σy2 . v is the product of x and y. It is easy to observe that the probability density function of v is given by ∞ ∞ −x2 /(2σ2 ) −y2 /(2σy2 ) x e e √ √ pxy (v)= δ(xy − v)dxdy σy 2π −∞ −∞ σx 2π =
) K0 ( σ|v| x σy πσx σy
(10)
where δ(·) is a delta function and Kn (·) is a modified Bessel function of the second kind. The mean of v is 0, and the variance of v is σx2 σy2 . Let a, b, c be the independent normally distributed variables with zero means and variances σ 2 , then the variables ab and bc are zero means and the variances are σ 4 . The covariance of ab and bc is cov{(ab)(bc)}= E{(ab − E{ab})(bc − E{bc})} = E{(ab)(bc)} = E{ab2 c} = E{a}E{b2 }E{c} = 0 thus, ab is uncorrelated with bc.
(11)
1.0
According to the discussions above and Central Limit ˆ ) is Theorem, the distribution of R(τ (12)
ˆ ) is a Gaussian distributed variable, the probaThus, R(τ bility density function is p(x) =
2 σw
1 2π/(N − τ )
e
(N −τ )x2 − 2σ4 w
(13)
ˆ Theorem 2: Under hypothesis H0 , R(0) is independent ˆ with R(τ ) when τ = 0. Proof: Since there is only noise in the frequency band, ˆ and w(n) is white. If we have enough samples, R(0) is equal ˆ to the variance of the noise, and R(τ ) is always zero, thus ˆ ) is independent with R(0). ˆ R(τ Under hypothesis H0 , let ⎡ 2 ⎤ ⎤ ⎡ ˆ Rw (0) σw ⎢ 0 ⎥ ⎢ R ˆ w (τ1 ) ⎥ ⎢ ⎥ ⎥ ⎢ r=⎢ ⎥ m = ⎢ .. ⎥ .. ⎣ ⎦ ⎣ ⎦ . . ˆ w (τk ) R
⎛
⎜ ⎜ ⎜ C=⎜ ⎜ ⎝
4 2σw N
0
0 2 σw N −τ1
0 .. . 0
··· ..
···
0 .. .
.
0
0 2 σw N −τk
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Under hypothesis H1 , we have ⎡ ˆ ⎡ 2 ˆ u (0) ⎤ Rw (0) + R σw + σu2 ⎢ R ⎢ Ru (τ1 ) ˆ w (τ1 ) + R ˆ u (τ1 ) ⎥ ⎢ ⎥ ⎢ r=⎢ ⎥ m=⎢ .. .. ⎣ ⎦ ⎣ . . ˆ ˆ R (τ u k) Rw (τk ) + Ru (τk )
1 (2π)(k+1)
det(C)
e(r−m)
T
C−1 (r−m)
0.8 0.7 0.6 0.5 0.4 0.3
JECD,N=2000, =0.0045 JECD,N=1000, =0.0022 ED,N=2000 ED,N=1000
0.2 0.1 0.0 -14
-13
-12
-11
-10
-9
-8
-7
-6
SNR(db) Fig. 1. Pd comparison of the proposed detector and the energy detector for the AWGN channel.
ˆ w (τ ). The test statistic can be derived similarly when and R k > 1. When k = 1 and under hypothesis H0 , the test statistic can be simplifies as TJECD =
2 2 ˆ w (0) − σw ˆ w (τ )2 2(R (N − τ )R ) + 4 4 σw N σw
(16)
Finally, the test statistic inequations can be derived as H0 : TJECD < λ H1 : TJECD > λ
⎤
(17) (18)
IV. S IMULATION R ESULTS AND D ISCUSSIONS
⎥ ⎥ ⎥ ⎦
where Ru (τ ) is the autocorrelation function of the signal u(n), and C is not a diagonal matrix. Since the autocorrelation function of the primary user’s signal is hard to be obtained in the real environment, similar with the energy detector, we can obtain the threshold by the giving probability of false alarm under hypothesis H0 . According to Theorem 2, C is a diagonal matrix when the primary user does not exist. Since ˆ w (0), R ˆ w (τ1 ), . . . , R ˆ w (τk ) are all Gaussian distributed, and R independent with each other, the joint probability density ˆ w (0), R ˆ w (τ1 ), . . . , R ˆ w (τk ) is function of R p(r; H0 ) =
0.9
probability of detection
ˆ ) ∼ N (0, (N − (N − τ )R(τ
4 ) τ )2 σ w
(14)
Based on the discussions above, we introduce the following test statistic for the proposed spectrum sensing algorithm (r − m)T C−1 (r − m) (15) N In order to simplify the derivation, let k = 1, which ˆ w (0) means that the vector r contain only two elements, R TJECD =
420
In this section, the Monte Carlo simulation is used to evalute the performance of the proposed detector. We compare the proposed detector (denoted by JECD) and the conventional energy detector (denoted by ED). Assume the primary user is a narrow band signal, the modulation is 16QAM, signal bandwidth fc = 2MHz, and the carrier frequency fb = 200kHz. The sampling frequency fs = 5MHz. We only consider k = 1 to reduce the computations. The correlation coefficient of the signal u(n) is defined as ρ=
Ru (τ ) σu2
(19)
Fig. 1 shows the comparison of the proposed detector and the energy detector for additive white Gaussian noise (AWGN) channel. Pf = 0.1, ρ = 0.7. We can observe that the probability of detection of the proposed detector increases when the SNR of the primary user is higher. When the samples number is larger, the performance gets better. However, the total sensing time increases, and the detector’s computational is higher . Fig. 2 illustrates the receiver operating characteristic (ROC) curves of the proposed detector and the energy detector when the samples number is different. The SNR of the primary user is −10dB, ρ = 0.7. We can observe that
1.0
0.9
0.9
0.8
0.8
0.7
0.7
probability of detection
probability of detection
1.0
0.6 0.5 0.4 0.3
JECD,N=2000 JECD,N=1000 ED,N=2000 ED,N=1000
0.2 0.1 0.0 0.0
0.1
0.2
0.3
0.4
0.5
SNR(db)
0.6
0.7
0.8
0.9
0.5 0.4 0.3
0.1 0.0 0.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
probability of detection
1.0
0.6 0.5 0.4 0.3
JECD,N=2000, =0.0045 JECD,N=1000, =0.0022 ED,N=2000 ED,N=1000
0.1 0.0 -14
-13
-12
-11
-10
-9
-8
-7
0.1
0.2
0.3
0.4
0.5
SNR(db)
0.6
0.7
0.8
0.9
1.0
Fig. 4. ROC curves of the proposed detector and the energy detector with different samples numbers for the multipath fading channel.
1.0
0.2
JECD,N=2000 JECD,N=1000 ED,N=2000 ED,N=1000
0.2
Fig. 2. ROC curves of the proposed detector and the energy detector with different samples numbers for the AWGN channel.
probability of detection
0.6
0.6 0.5 0.4 0.3 0.2
JECD,N=2000, =0.0045 JECD,N=1000, =0.0022
0.1 0.0 -0.9
-6
SNR(db)
-0.7
-0.5
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
correlation coefficient
Fig. 3. Pd comparison of the proposed detector and the energy detector for the multipath fading channel.
Fig. 5. Pd versus the correlation coefficient ρ with different samples numbers.
the performance of the proposed detector is improved when the samples number is larger. In order to test the performance of the proposed detector in the multipath fading channel, we further show the following experimental results in Fig. 3 and Fig. 4. The correlation coefficients are 0.7, The channel vectors h = [0.8, −0.5, 0.3, 0.4]T in both of the simulations. In Fig. 3, Pf = 0.1. In Fig. 4, SNR = −10dB. We can observe that when the correlation coefficients are same, the performance of the proposed detector in the multipath environment is nearly the same with the AWGN environment. Next, we observe the sensitivity of the proposed detector to the correlation coefficient. In Fig. 5, the correlation coefficient ρ is change from −0.9 to 0.9. The simulation results shows when |ρ| is larger, the detection probability of
the proposed algorithm is better.
421
V. C ONCLUSIONS In this paper, we propose a joint energy and autocorrelation based spectrum sensing algorithm. The proposed detection algorithm is based on the assumption that the autocorrelation matrix of the primary user’s signal is not diagonal. We compare the performance of the proposed detector and the conventional energy detector, simulation results show that when we use the proposed spectrum sensing algorithm, the detection performance is improved. In our future work, we will derive the closed-form expressions for the false alarm and detection probabilities of the proposed algorithm in the AWGN and multipath fading channel.
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