Cognitive radio has been proposed as a promising technol- ... II. NEYMAN-PEARSON THEOREM AND ENERGY. DETECTION. We use hypothesis H0 and H1 to denote âsignal .... question arises: if we only have partial information or rough.
2014 IEEE 25th International Symposium on Personal, Indoor and Mobile Radio Communications
Robust Energy Detection for Cognitive Radio Yonghong Zeng and Meng Wah Chia Institute for Infocomm Research, A∗ STAR, 1 Fusionopolis Way, Singapore 138632 Emails: {yhzeng, mwchia}@i2r.a-star.edu.sg
Abstract—Spectrum sensing is the fundamental technology in cognitive radio to learn the radio environment. Energy detection is one of the simplest method for spectrum sensing and has the best performance in theory for signals without special features. However, it is well-known that energy detection is vulnerable to noise uncertainty. There is still no good approach to solve the problem. On the other hand, robust hypothesis testing has been known for a long time, which is a general paradigm to deal with uncertain signal and noise. In this paper, we use this paradigm to tackle the noise uncertainty problem. It is shown that the noise uncertainty can be formed as an ǫ-contamination model. Then it is proved that the robust hypothesis testing turns to robust energy detection for independent signal samples with Gaussian distribution. Methods are found for calculating the parameters in the testing. Simulations show that the robust energy detection achieves better average performance than the energy detection on uncertain noise environment. I. I NTRODUCTION Cognitive radio has been proposed as a promising technology to optimize the spectrum usage efficiency. It has been used in some applications and is gauged as a major technology for the future. The key principle of cognitive radio is spectrum sharing or spectral reusing. It allows secondary users/networks to communicate over the spectrum licensed/allocated to primary users, as long as the interference caused by the secondary users to the primary users is acceptable. To control its interference, the secondary users are required to frequently detect the presence of the primary users. Spectrum sensing is a fundamental technology for a cognitive radio to detect the presence of primary users. In recent years, it has reborn as a very active research area despite its long history. There have been quite a few sensing methods for different situations and with different requirements [1– 4]. Among them, energy detection is one of the simplest method and has the best performance in theory for signals without special features. However, it is well-known that energy detection is vulnerable to noise uncertainty [3, 5, 6]. The noise uncertainty causes the energy detection to have a signalto-noise (SNR) wall [3, 5, 6]. So far, there is still no good approach to solve the problem. To deal with signal and noise distribution uncertainty, robust hypothesis testing has been proposed for a long time [7, 8]. In this paper, we use the robust hypothesis testing as a general paradigm to tackle the noise uncertainty problem. It is proved that the robust hypothesis testing turns to robust energy detection for independent signal samples with Gaussian distribution, if the signal and noise distributions satisfy an ǫ-contamination model. Furthermore, it is shown that the noise uncertainty can be formed as an ǫ-contamination model.
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Thus we obtain a robust energy detection to tackle the noise uncertainty problem. Methods are found for calculating the parameters in the testing. Simulations show that the robust energy detection achieves better average performance than the energy detection on uncertain noise environment. II. N EYMAN -P EARSON T HEOREM AND E NERGY DETECTION
We use hypothesis H0 and H1 to denote “signal absent” and “signal present”, respectively. The system model of the received signal is given as H0 : x(n) = η(n),
(1)
H1 : x(n) = s(n) + η(n), n = 0, 1, · · · , N − 1,
(2)
where η(n) is the noise plus possible interference, s(n) is the received source signal and N is the number of samples. Note that s(n) is the primary signal after going through a propagation channel. Without loss of generality, we assume that the signal and noise are real numbers. For complex signals, we can extract the real and imagine parts so that the real signal model can be used with the sample size being doubled. The results in the following can be extended to complex signals as well. The purpose of spectrum sensing is to determine which hypothesis is correct based on the received signal. If H1 is chosen, the detector may further give information on the signal waveform and modulation schemes in some applications. The probability of detection, Pd , and probability of false alarm, Pf a , are defined as follows: Pd = P (H1 |H1 ) , Pf a = P (H1 |H0 ) .
(3)
In general a sensing algorithm is said to be “optimal” if it achieves the highest Pd for a given Pf a , with a fixed number of samples, though there could be other criteria to evaluate the performance. For notation simplicity, we stack the signal and noise samples into vectors: x = [x(0) x(1) · · · x(N − 1)]T , T
s = [s(0) s(1) · · · s(N − 1)] , η = [η(0) η(1) · · · η(N − 1)]T .
(4) (5) (6)
Based on the vector form, the hypothesis testing problem can be reformulated as
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H0 : x = η, H1 : x = s + η.
(7) (8)
A. Neyman-Pearson Theorem One of the fundamental theorem in detection is the NeymanPearson (NP) theorem [9–11], which is the basis for many detection methods. NP theorem says that the likelihood ratio test (LRT) maximizes the probability of detection for a given probability of false alarm. The test statistic of LRT is defined as p(x|H1 ) TLRT = , (9) p(x|H0 ) where p(·) denotes the probability density function (PDF). The detection procedure is: if TLRT > γ, the decision is H1 ; otherwise, the decision is H0 , where γ is a threshold. Although LRT gives the best detector, it is hard to implement it in practical applications, as LRT requires exact distributions (PDFs) given in (9) to make a decision. In fact, the PDF of random vector x under H1 is determined by the noise distribution, the wireless channels, and the source signal distribution, while the PDF of x under H0 is related to the noise distribution. Hence, we have to obtain the information of propagation channel as well as the signal and noise distributions before using the LRT. However, in most applications, especially in cognitive radio, it is almost impossible to obtain such information. If the signal samples x(n) are independent over time n, we can decouple the PDFs in LRT to p(x|H0 ) p(x|H1 )
= =
N −1 Y
n=0 N −1 Y
p(x(n)|H0 ),
(10)
p(x(n)|H1 ).
(11)
power may change over time, which yields the so-called noise uncertainty problem [5, 6, 13]. Let σ ˆη2 = αση2 be the expected noise power, where ση2 is the actual noise power and α is called the noise uncertainty factor. The upper bound of α (in dB scale), that is, B = sup{10 log10 α},
is called the noise uncertainty bound. Usually we assume that α in dB scale, i.e., 10 log10 α, is uniformly distributed in the interval [−B, B] [5] (This assumption may not always be correct. We use this assumption only for analysis purpose). In practice, the noise uncertainty bound of a receiving device is normally below 2 dB [5, 17], while the environment/interference noise uncertainty can be much larger [5]. Similarly, the signal s(n) may also have uncertainty as the propagation channel is unpredictable.
D. Robustness of energy detection It has been shown that the energy detection can be very unreliable when there is noise uncertainty [5, 6, 13, 14, 18]. Here we just give the major results. Let γ σ ˆη2 be the threshold for ED, where γ is a constant. As described above, the actual noise power is ση2 = σ ˆη2 /α. It has been proved in [14] that, for given threshold γ and noise uncertainty bound B, the expected (average) probability of detection and probability of false alarm are respectively: r ! Z 10B/10 5 γt − (µt + 1) N ¯ dt, (14) Pd = Q µt + 1 2 log(10)Bt 10−B/10 P¯f a =
n=0
B. Energy detection
TED =
N −1 1 X |x(n)|2 . N n=0
Z
10B/10
10−B/10
Furthermore, if the noise and signal samples are independent and identically distributed (iid), and have Gaussian distributions, i.e., η(n) ∼ N (0, ση2 I) and s(n) ∼ N (0, σs2 I), the LRT reduces to the well-known energy detector (ED) [12– 14], where the test statistic is given as follows (by discarding irrelevant constant terms): (12)
Even if the signal is not Gaussian distributed, the LRT still approaches to ED at low SNR [15, 16]. The test statistic is compared with a threshold to make a decision. Obviously the threshold should be related to the noise power. Hence energy detection needs a priori information of the noise variance (power). It has been shown that energy detection is very sensitive to the inaccurate estimation of the noise power [3, 5, 6].
C. Noise uncertainty Many detection methods rely on the noise power to set the threshold. Accuracy of the noise power knowledge thus becomes crucial to the success of such methods. This is especially true for energy detection. Unfortunately, the noise
(13)
where
r
Q (γt − 1)
1 Q(t) = √ 2π
Z
N 2
+∞
!
e−v
5 dt, log(10)Bt 2
/2
dv,
(15)
(16)
t
and µ = σs2 /ˆ ση2 is the average SNR. It has been proved that an ED has a SNR wall if there is noise uncertainty [6, 14]. In fact, based on the Pd and Pf a expressions above, it is shown in [14] that there exists a positive number µ0 such that ED cannot achieve reliable sensing for any µ < µ0 and can find a threshold to achieve reliable sensing for any µ > µ0 . Furthermore, methods have been found to calculate the SNR wall µ0 at given noise uncertainty bound [6, 14]. III. ROBUST
HYPOTHESIS TESTING
The searching for robust detection methods has been of great interest in the field of signal processing and other areas as well. Max-min approach is one of the useful paradigms to design robust detectors, which maximizes the worst case detection performance. The robust hypothesis testing [7, 8] is a famous technique based on this approach. We denote the PDF of a received signal sample by f0 at hypothesis H0 and f1 at hypothesis H1 . If these two functions are known, we know that the LRT described above is
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optimal. Unfortunately, in practice it is very hard, if possible, to obtain these two functions exactly due to noise uncertainty, interference, and channel impairment. Thus an interesting question arises: if we only have partial information or rough estimation of the PDFs, how we design the detector? It is difficult to answer this question as we first need to quantify how partial or how rough the information is. One special case is that we only know f1 and f0 fall into certain classes. For example, they belong to the so-called ǫ-contamination class given by H0 : f0 ∈ F0 , F0 = {(1 − ǫ0 )f00 + ǫ0 g0 }, H1 : f1 ∈ F1 , F1 = {(1 − ǫ1 )f10 + ǫ1 g1 },
(17)
where fj0 (j = 0, 1) is the nominal PDF under hypothesis Hj , gj is an arbitrary density function, and ǫj in [0, 1] is the maximum degree of contamination. Suppose that we do not have f0 and f1 exactly, but we know fj0 and ǫj (an upper bound for contamination), j = 0, 1. The problem becomes designing a detection scheme to minimize the worst-case error probability (e.g., probability of false alarm ˆ plus probability of mis-detection), that is, to find a detector Ψ such that ˆ = arg min Ψ
max
Ψ (f0 ,f1 )∈F0 ×F1
(Pf a (f0 , f1 , Ψ)+1−Pd (f0 , f1 , Ψ)).
(18) It is proved in [7] that the optimal test statistic is a “censored” version of the LRT given by ˆ = TCLRT = Ψ
N −1 Y
r(x(n)),
(at H1 ) are respectively
r0 (t) =
r(t) =
f1 (t) , f00 (t)
c , 1
c1 < f10 (t) f00 (t)
< c2
TCLRT =
(23)
N −1 Y
rq (x(n)),
(24)
n=0
where rq (t) =
c , 2
r0 (t) ≥ c2
t2
2 √ 1 e 2(1+1/δ)ςη 1+δ
c1 ,
, c1 < r0 (t) < c2 r0 (t) ≤ c1
(25)
Now the major issue is the determination of the constants c1 and c2 . As proved in [7, 19], c1 and c2 must be chosen such that q0 (t) and q1 (t) are density functions: Z ∞ Z ∞ q0 (t)dt = 1, q1 (t)dt = 1, (26) −∞
−∞
where
q0 (t) = q1 (t) =
(1 − ǫ0 )f00 (t), r0 (t) < c2 1−ǫ0 0 r0 (t) ≥ c2 c2 f1 (t), (1 − ǫ1 )f10 (t), c1 (1 − ǫ1 )f00 (t),
r0 (t) > c1 r0 (t) ≤ c1
2(1 − ǫ0 ) 2(1 − ǫ0 )(0.5 − Q(d¯2 )) + Q(dˆ2 ) = 1, c2 2(1 − ǫ1 )c1 (0.5 − Q(d¯1 )) + 2(1 − ǫ1 )Q(dˆ1 ) = 1,
(20)
≤ c1
and c1 , c2 are nonnegative numbers related to ǫ0 , ǫ1 , f00 , and f10 [7, 19]. Note that if choosing c1 = 0 and c2 = +∞, the test is the conventional LRT with respect to nominal PDFs, f00 and f10 . IV. ROBUST
f10 (t) , δ = ςs2 /ςη2 . f00 (t)
(27) (28)
After some mathematical derivations, we can prove that c1 and c2 must satisfy the following equations:
≥ c2 f10 (t) f00 (t)
(22)
At this case, the robust hypothesis testing turns to
(19)
where f10 (t) f00 (t)
(21)
For notation simplicity, we denote
n=0
c , 20
2
− t2 1 e 2ςη , 2πςη 2 − 2t 2 1 f10 (t) = √ q e 2(ςη +ςs ) . 2π ςη2 + ςs2
f00 (t) = √
(30)
where
ENERGY DETECTION
One particularly interesting case is the iid Gaussian distributed signals with noise and signal uncertainty. We wonder how the robust hypothesis testing can be used at this case. We know that the major problem of the energy detection is the noise and signal uncertainty. Thus it is interesting to use the robust hypothesis testing paradigm to deal with the noise and signal uncertainty. We assume that the nominal distributions of noise and signal samples are Gaussian, and the nominal noise variance and signal variance are ςη2 and ςs2 , respectively. Thus the nominal PDFs of noise samples η(n) and received signal samples x(n)
(29)
1/2 √ d¯2 = 2(1 + 1/δ) log(c2 1 + δ) , 1/2 √ 2 , log(c2 1 + δ) dˆ2 = δ 1/2 √ d¯1 = 2(1 + 1/δ) log(c1 1 + δ) , 1/2 √ 2 dˆ1 = log(c1 1 + δ) . δ
(31) (32) (33) (34)
For given ǫ0 , ǫ1 and δ, we can use equation (29) and (30) to find c2 and c1 , respectively, although closed-form expression is not available. We can further simplify the test statistic. Taking logarithm of the test statistic, we can derive that
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log(TCLRT ) =
N −1 X n=0
log(rq (x(n))),
(35)
where log(rq (x(n))) log(c |x(n)|2 ≥ d2 2 ), 2 |x(n)| 2 √1 = 2(1+1/δ)ςη2 + log 1+δ , d1 < |x(n)| < d2 log(c1 ), |x(n)|2 ≤ d1
and
d1 = d¯21 ςη2 , d2 = d¯22 ςη2 .
Similarly, the signal uncertainty can also be expressed as an ǫ-contamination model: f1 (t) = (1 − ǫ1 )f10 (t) + ǫ1 g1 (t),
(36) where
2
− t2 1 e 2ˆσl , ǫ1 = 1 − σ ˆl /ˆ σx , f10 (t) = √ 2πˆ σl 2 2 t − t2 1 − g1 (t) = √ (e 2ˆσx2 − e 2ˆσl ), 2πˆ σx ǫ1
(37)
We can simplify the expression by normalizing the term. In fact, let LTn = 2(1 + 1/δ)ςη2 log(rq (x(n)) + (1 + 1/δ)ςη2 log(1 + δ). (38) Then we can verify that |x(n)|2 ≥ d2 d2 , 2 |x(n)| , d1 < |x(n)|2 < d2 LTn = (39) d1 , |x(n)|2 ≤ d1
Thus the “robust hypothesis testing” at this case is actually a robust energy detection (RED) that has a test statistic as follows: N −1 1 X TRED = LTn . (40) N n=0 It is possible to extend this result to signal and noise with non-Gaussian distributions, like the derivation in [15, 16]. The result can be extended for complex signal and noise as well. To save space, we will not discuss them in detail here. V. R ELATIONSHIP BETWEEN NOISE UNCERTAINTY ǫ- CONTAMINATION MODEL
AND
In the last section, we have derived the robust energy detection assuming that the noise and signal distributions satisfy the ǫ-contamination model. We know that a particular noise/signal model is the noise uncertainty model discussed in Section II.C. It is interesting to exam the relationship between the two models. Let σl2 be the lower bound of the noise power and ση2 be the actual noise power. Assuming Gaussian distribution, we can write the PDF of the noise samples as 2
− t2 1 e 2ση . f0 (t) = √ 2πση
(41)
It is easy to prove that the PDF can be turned to f0 (t) = (1 − ǫ0 )f00 (t) + ǫ0 g0 (t),
(42)
where 2
− t2 1 ǫ0 = 1 − σl /ση , f00 (t) = √ e 2σl , 2πσl 2 t2 − 2σ − t2 1 2 g0 (t) = √ (e η − e 2σl ). 2πση ǫ0
(45)
(43) (44)
It can be verified that ǫ0 ∈ [0, 1], and f00 (t) and g0 (t) are density functions. Thus, (42) is indeed an ǫ-contamination model.
(46) (47)
where σ ˆl2 is the lower bound of the signal power and σ ˆx2 is the actual signal power. However, here the parameters ǫ0 and ǫ1 are related to the actual noise power and signal power. In practice, it is hard to obtain the exact values of them. If we know the noise and signal uncertainty bound (in dB scale) to be B0 and B1 , the nominal values for the parameters can be chosen as ǫ0 = 1 − 10−B0 /20 , ǫ1 = 1 − 10−B1 /20 .
(48)
VI. S IMULATIONS It is assumed that the nominal noise power and the noise uncertainty bound are known. The actual noise power at each Monte-Carlo test, which is unknown to the receiver, is generated by using the noise uncertainty model. As discussed in Section IV and V, the parameters d¯1 , d¯2 and ǫ0 , ǫ1 are related to SNR and noise uncertainty bound, which means different parameters at different SNRs and noise uncertainty bounds. To reduce complexity, in the simulations we fix the parameters at all cases, which are calculated based on the noise uncertainty bound B0 = 1 dB and SNR=0 dB. The threshold is set based on the nominal noise power and noise uncertainty bound at given probability of false alarm Pf a = 0.01 by using simulations. We first consider two types of real signals: BPSK signal and 4PAM (pulse-amplitude modulation) with real Gaussian distributed noise. The number of samples is N = 2000. The average Pd for the ED and RED at different noise uncertainties are shown in figure 1 and figure 2 for BPSK signal and 4PAM signal, respectively, where ED-rdB and RED-rdB denote ED and RED, respectively, with noise uncertainty bound rdB. A simulation result for complex Gaussian distributed noise and 16QAM signal is given in 3. Simulations show that, if there is noise uncertainty, RED does improve the overall (average) performance over the ED. However, the improvement is not very significant. If there is no noise uncertainty, RED is slightly worse than ED (no wonder as ED is optimal at this case). In summary, although RED improves the performance over ED, it is still far from satisfaction. RED alone may still not be reliable in practice. We can combine it with other approaches like cooperative sensing to increase the reliability, which needs further investigations in the future.
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1
0.9
0.9
Probability of detection
Probability of detection
1
0.8
0.7 ED−2dB ED−1.5dB ED−1dB ED−0.5dB ED−0dB RED−2dB RED−1.5dB RED−1dB RED−0.5dB RED−0dB
0.6
0.5
0.4
0.3 −4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
0.8
0.7
0.6
0.5
0.4
1
−4
SNR (dB)
Fig. 1.
Average performance of RED with noise uncertainties (BPSK)
Fig. 3.
Probability of detection
0.9
0.8 ED−2dB ED−1.5dB ED−1dB ED−0.5dB ED−0dB RED−2dB RED−1.5dB RED−1dB RED−0.5dB RED−0dB
0.7
0.6
0.5
0.4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
SNR (dB)
Fig. 2.
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
SNR (dB)
1
0.3 −4
ED−2dB ED−1.5dB ED−1dB ED−0.5dB ED−0dB RED−2dB RED−1.5dB RED−1dB RED−0.5dB RED−0dB
Average performance of RED with noise uncertainties (4PAM)
VII. C ONCLUSIONS We have derived the RED and methods for calculating the associated parameters based on the robust hypothesis testing. We have also proved that the noise uncertainty can be formed as an ǫ-contamination model. Simulations have shown that RED achieves better average performance than ED on uncertain noise environment. However, the improvement of RED over ED is limited. How to combine RED with other approaches remains to be investigated. R EFERENCES [1] T. Yucek and H. Arslan, “A survey of spectrum sensing algorithms for cognitive radio applications,” IEEE Communications Surveys & Tutorials, vol. 11, no. 1, pp. 116–130, 2009. [2] “Sensing techniques for cognitive radio-state of the art and trends,” in IEEE SCC41-P1900.6 White Paper, 2009.
Average performance of RED with noise uncertainties (16QAM)
[3] Y. H. Zeng, Y.-C. Liang, A. T. Hoang, and R. Zhang, “A review on spectrum sensing for cognitive radio: challenges and solutions,” EURASIP Journal on Advances in Signal Processing, vol. 2010, no. Article ID 381465, pp. 1–15, 2010. [4] B. Wang and K. J. R. Liu, “Advances in cognitive radio networks: a survey,” IEEE Journal of Selected Topics in Signal Processing, vol. 5, pp. 5–23, Feb. 2011. [5] A. Sahai and D. Cabric, “Spectrum sensing: fundamental limits and practical challenges,” in Proc. IEEE International Symposium on New Frontiers in Dynamic Spectrum Access Networks (DySPAN), (Baltimore, MD), Nov. 2005. [6] R. Tandra and A. Sahai, “SNR walls for signal detection,” IEEE Journal of Selected Topics in Signal Processing, vol. 2, pp. 4–17, Feb. 2008. [7] P. J. Huber, “A robust version of the probability ratio test,” Ann. Math. Stat., vol. 36, pp. 1753–1758, 1965. [8] P. J. Huber, “Robust estimation of a location parameter,” Ann. Math. Stat., vol. 35, pp. 73–104, 1964. [9] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory, vol. 2. Prentice Hall, 1998. [10] H. V. Poor, An introduction to signal detection and estimation. SpringerVerlag, 1988. [11] H. L. Van-Trees, Detection, estimation and modulation theory. John Wiley & Sons, 2001. [12] H. Urkowitz, “Energy detection of unkown deterministic signals,” Proceedings of the IEEE, vol. 55, no. 4, pp. 523–531, 1967. [13] A. Sonnenschein and P. M. Fishman, “Radiometric detection of spreadspectrum signals in noise of uncertainty power,” IEEE Trans. on Aerospace and Electronic Systems, vol. 28, no. 3, pp. 654–660, 1992. [14] Y. H. Zeng, Y.-C. Liang, A. Hoang, and E. C. Y. Peh, “Reliability of spectrum sensing under noise and interference uncertainty,” in IEEE ICC Workshop, (Dresden, Germany), June 2009. [15] A. Sahai, N. Hoven, and R. Tandra, “Some fundamental limits on cognitive radio,” in Allerton Conference on Communication, Control, and Computing, Oct. 2004. [16] S. K. Zheng, P. Y. Kam, Y.-C. Liang, and Y. H. Zeng, “Spectrum sensing for digital primary signals in cognitive radio: A bayesian approach for maximizing spectrum utilization,” IEEE Trans. Wireless Communications, vol. 12, no. 4, pp. 1774–1782, 2013. [17] S. Shellhammer and R. Tandra, Performance of the Power Detector with Noise Uncertainty. doc. IEEE 802.22-06/0134r0, July 2006. [18] Y. H. Zeng and Y.-C. Liang, “Robust spectrum sensing in cognitive radio (invited paper),” in IEEE PIMRC Workshop, (Istanbul, Turkey), Sept. 2010. [19] S. A. Kassam and H. V. Poor, “Robust techniques for signal processing: a survey,” Proceedings of IEEE, vol. 73, no. 3, pp. 433–482, 1985.
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