Wideband Spectrum Sensing in Cognitive Radio Networks

Wideband Spectrum Sensing in Cognitive Radio Networks Zhi Quan†, Shuguang Cui‡ , Ali H. Sayed†, and H. Vincent Poor§ † arXiv:0802.4130v1 [cs.IT] 28 Feb 2008 ‡ Department of Electrical Engineering, University of California, Los Angeles, CA 90095 Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843 § Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 Email: {quan, sayed}@ee.ucla.edu; cui@ece.tamu.edu; poor@princeton.edu Abstract— Spectrum sensing is an essential enabling functionality for cognitive radio networks to detect spectrum holes and opportunistically use the under-utilized frequency bands without causing harmful interference to legacy networks. This paper introduces a novel wideband spectrum sensing technique, called multiband joint detection, which jointly detects the signal energy levels over multiple frequency bands rather than consider one band at a time. The proposed strategy is efficient in improving the dynamic spectrum utilization and reducing interference to the primary users. The spectrum sensing problem is formulated as a class of optimization problems in interference limited cognitive radio networks. By exploiting the hidden convexity in the seemingly non-convex problem formulations, optimal solutions for multiband joint detection are obtained under practical conditions. Simulation results show that the proposed spectrum sensing schemes can considerably improve the system performance. This paper establishes important principles for the design of wideband spectrum sensing algorithms in cognitive radio networks. I. I NTRODUCTION Spectrum sensing is an essential functionality of cognitive radios since the devices need to reliably detect weak primary signals of possibly-unknown types [1]. In general, spectrum sensing techniques can be classified into three categories: energy detection [2], matched filter coherent detection [3], and cyclostationary feature detection [4]. Since non-coherent energy detection is simple and is able to locate spectrumoccupancy information quickly, we will adopt it as a building block for constructing the proposed wideband spectrum sensing scheme. There are previous studies on spectrum sensing in cognitive radio networks with focus on cooperation among multiple cognitive radios [1] [5] [6] via distributed detection approaches [7] [8]. However, they are limited to the detection of signals on a single frequency band. In [9], two decision-combining approaches were studied: hard decision with the AND logic operation and soft decision using the likelihood ratio test [7]. It was shown that the soft decision combination of spectrum sensing results yields gains over hard decision combining. In [10], the authors exploited the fact that summing signals from two secondary users can increase the signal-to-noise ratio (SNR) and detection reliability if the signals are correlated. In [11], a generalized likelihood ratio test for detecting the presence of cyclostationarity over multiple cyclic frequencies was proposed and evaluated through Monte Carlo simulations. Along with these works, we have developed a linear cooperation strategy [12] [13] based on the optimal combination of the local statistics from spatially distributed cognitive radios. Generally speaking, the quality of the detector depends on the level of cooperation and the bandwidth of the control channel. The literature of wideband spectrum sensing for cognitive radio networks is very limited. An early approach is to use a tunable narrowband bandpss filter at the RF front-end to sense one narrow frequency band at a time [14], over which the existing narrowband spectrum sensing techniques can be applied. In order to operate over multiple frequency bands at a time, the RF front-end requires a wideband architecture and the spectrum sensing usually involves the estimation of the power spectral density (PSD) of the wideband signal. In [15] and [16], the wavelet transform was used to estimate the PSD over a wide frequency range given its multi-resolution features. However, none of the previous works considers making joint decisions over multiple frequency bands, which is essential for implementing efficient cognitive radios networks. In this paper, we introduce the multiband joint detection framework for wideband spectrum sensing in individual cognitive radios. Within this framework, we jointly optimize a bank of multiple narrowband detectors in order to improve the opportunistic throughput capacity of cognitive radios and reduce their interference to the primary communication systems. In particular, we formulate wideband spectrum sensing into a class of optimization problems. The objective is to maximize the opportunistic throughput in an interference limited cognitive radio network. By exploiting the hidden convexity of the seemingly non-convex problems, we show that the optimization problems can be reformulated into convex programs under practical conditions. The multiband joint detection strategy allows cognitive radios to efficiently take advantage of the unused frequency bands and limit the resulting interference. The rest of this paper is organized as follows. In Section II, we describe the system model for wideband spectrum sensing. In Section III, we develop the multiband joint detection algorithms, which seek to maximize the opportunistic throughput. The proposed spectrum sensing algorithms are examined by numerical examples in Section IV and conclusions are drawn in Section V. Subbands occupied by primary users Spectrum holes 0 1 Fig. 1. 1 0 1 0 0 0 1 1 A schematic illustration of a multiband channel. II. S YSTEM M ODELS A. Wideband Spectrum Sensing Consider a primary communication system (e.g., a multicarrier modulation based system) over a wideband channel that is divided into K non-overlapping narrowband subchannels. In a particular geographical region and time, some of the K subchannels might not be utilized by the primary users and are available for opportunistic spectrum access. Multiuser orthogonal frequency division multiplexing (OFDM) is an ideal candidate for such a scenario since it makes the subband manipulation easy and flexible. We model the occupancy detection problem on subchannel k as one of choosing between H0,k (“0”), which represents the absence of primary signals, and H1,k (“1”), which represents the presence of primary signals. An illustrative example where only some of the K bands are occupied by primary users is depicted in Fig. 1. The underlying hypothesis vector is a binary representation of the subchannels that are allowed for or prohibited from opportunistic spectrum access. The crucial task of spectrum sensing is to sense the K narrowband subchannels and identify spectral holes for opportunistic use. For simplicity, we assume that the high-layer protocols, e.g., the medium access control (MAC) layer, can guarantee that all cognitive radios keep quiet during the detection interval such that the only spectral power remaining in the air is emitted by the primary users in addition to background noises. In this paper, instead of considering a single subband at a time, we propose to use a multiband detection technique, which jointly takes into account the detection of primary users across multiple frequency bands. We next present the system model. B. Received Signal Consider a multi-path fading environment, where h(l), l = 0, 1, . . . , L − 1, denotes the discrete-time channel impulse response between the primary transmitter and cognitive radio receiver, with L as the number of resolvable paths. The received baseband signal at the CR front-end can be represented as r(n) = L−1 X l=0 h (l) s (n − l) + v(n), n = 0, 1, . . . , N − 1 (1) where s(n) is the primary transmitted signal at time n (after the cyclic prefix has been removed) and v(n) is additive complex white Gaussian noise with zero mean and variance σv2 ,
2 i.e., v(n) ∼ CN 0, σv . In a multi-path fading environment, the wideband channel exhibits frequency-selective features [17] [18] [19] and its discrete frequency response is given by L−1 1 X h(n)e−j2πnk/N , Hk = √ N n=0 k = 0, 1, . . . , K − 1 (2) where L ≤ N . We assume that the channel is slowly varying such that the channel frequency responses {Hk }K−1 k=0 remain constant during a detection interval. In the frequency domain, the received signal at each subchannel can be estimated by first computing its discrete Fourier transform (DFT): N −1 1 X r(n)e−j2πnk/N Rk = √ N n=0 k = 0, 1, . . . , K − 1 = Hk Sk + Vk , (3) where Sk is the primary transmitted signal at subchannel k and L−1 1 X Vk = √ v(n)e−j2πnk/N , N n=0 k = 0, 1, . . . , K − 1 (4) is the received noise in frequency domain. The random variable Vk is independently and normally distributed
with 2 2 zero mean and variance σ , i.e., V ∼ CN 0, σ k v v , since
v(n) ∼ CN 0, σv2 and the DFT is a linear operation. Without loss of generality, we assume that the transmitted signal Sk , the channel gain Hk , and the additive noise Vk are independent of each other. C. Signal Detection in Individual Bands Here, we consider signal detection in a single narrowband subchannel, which will constitute a building block for multiband joint detection. To decide whether the k-th subchannel is occupied or not, we test the following binary hypotheses H0,k : Rk = Vk H1,k : Rk = Hk Sk + Vk , k = 0, 1, . . . , K − 1 (5) where H0,k and H1,k indicate, respectively, the absence and presence of the primary signal in the k-th subchannel. For each subchannel k, we compute the summary statistic as the sum of received signal energy over an interval of M samples, i.e., Yk = M−1 X m=0 |Rk (m)|2 , k = 0, 1, . . . , K − 1 (6) and the decision rule is given by Yk H1,k R γk , H0,k k = 0, 1, . . . , K − 1 where γk is the corresponding decision threshold. (7) RF Front-End R0 (m) Down Conversion R1 (m) Remove Prefix, Var (Yk ) = 2M σv4
2M σv2 + 2|Hk |2 σv2 H0,k H1,k PM1 | · |2 m=0 Y0 0 Y1 1 YK1 K1 H0 /H1 H0 /H1 FFT PM1 m=0 | · |2 H0 /H1 RK1 (m) Joint Detection (9) for k = 0, 1, . . . , K − 1. Thus, we write these statistics compactly as Yk ∼ N (E (Yk ) , Var (Yk )), k = 0, 1, . . . , K −1. Using the decision rule in (7), the probabilities of false alarm and detection at subchannel k can be respectively calculated as   γk − M σv2 (k) √ (10) Pf (γk ) = Pr (Yk > γk |H0,k ) = Q σv2 2M and | · |2 …... h(n) s(n) + w(n) and variance  and Serial-toParallel Convert m=0 …... A/D PM1 …... For simplicity, we assume that the transmitted
signal at each subchannel has unit power, i.e., E |Sk |2 = 1. This assumption holds when primary radios deploy uniform power transmission strategies given no channel knowledge at the transmitter side. According to the central limit theorem [20], Yk is asymptotically in M normally distributed with mean  M σv2 H0,k
E (Yk ) = (8) M σv2 + |Hk |2 H1,k Fig. 2. A schematic representation of multiband joint detection for wideband spectrum sensing in cognitive radio networks. users. For a given threshold vector γ, the probabilities of false alarm and detection can be compactly represented as h iT (0) (1) (K−1) P f (γ) = Pf (γ0 ), Pf (γ1 ), . . . , Pf (γK−1 ) (12) and h iT ! P d (γ) = Pd(0) (γ0 ), Pd(1) (γ1 ), . . . , Pd(K−1) (γK−1 )
(13) γk − M σv2 + |Hk |2 (k) p Pd (γk ) = Pr (Yk > γk |H1,k ) = Q σv 2M (σv2 + 2|Hk |2 ) respectively. Similarly, the probabilities of miss can be written (11) in a vector as h iT where Q(·) denotes the complementary distribution function (0) (1) (K−1) P m (γ) = Pm (γ0 ), Pm (γ1 ), . . . , Pm (γK−1 ) (14) of the standard normal distribution. The choice of the threshold γk leads to a tradeoff between (k) (k) where Pm (γk ) = 1 − Pd (γk ), k = 0, 1, . . . , K − 1, the probability of false alarm and the probability of miss1 , compactly written as P m (γ) = 1− P d (γ), with 1 the all-one Pm = 1 − Pd . Specifically, a higher threshold will result in a vector. smaller probability of false alarm and a larger probability of Consider a cognitive radio sensing the K narrowband submiss, and vice versa. channels in order to opportunistically utilize the unused ones The probabilities of false alarm and miss have unique for transmission. Let rk denote the throughput achievable over implications for cognitive radio networks. Low probabilities the k-th subchannel if used by cognitive radios, and r = of false alarm are necessary in order to maintain possible (k) T high throughput in cognitive radio systems, since a false alarm [r0 , r1 , . . . , rK−1 ] . Since 1−Pf measures the opportunistic spectrum utilization of subchannel k, we define the aggregate would prevent the unused spectral segments from being acopportunistic throughput capacity as cessed by cognitive radios. On the other hand, the probability of miss measures the interference from cognitive radios to R (γ) = rT [1 − P f (γ)] (15) the primary users, which should be limited in opportunistic threshold vector γ. Due to the inspectrum access. These implications are based on a typical which is a function of the (k) (k) herent trade-off between P assumption that if primary signals are detected, the secondary f (γk ) and Pm (γk ), maximizing users should not use the corresponding channel and that if no the sum rate R(γ) will result in large P m (γ), hence causing primary signals are detected, then the corresponding frequency harmful interference to primary users. The interference to primary users should be limited in a band will be occupied by secondary users. cognitive radio network. For a widband primary communicaIII. M ULTIBAND J OINT D ETECTION tion system, the impact of interference induced by cognitive In this section, we present the multiband joint detection devices can be characterized by a relative priorityT vector over framework for wideband spectrum sensing, as illustrated in the K subchannels, i.e., c = [c0 , c1 , . . . , cK−1 ] , where ck Fig. 2. The design objective is to find the optimal threshold indicates the cost incurred if the primary user at subchannel vector γ = [γ0 , γ1 , . . . , γK−1 ]T so that the cognitive radio k is interfered with. Suppose that J primary users share a system can make efficient use of the unoccupied spectral portion of the K subchannels and each primary user occupies a we define the aggregate interference segments without causing harmful interference to the primary subset Sj . Consequently, P (i) to primary user j as i∈Sj ci Pm (γi ). In special cases where 1 The subscript k is omitted whenever we refer to a generic frequency band. each primary user is equally important, we may have c = 1. To summarize, our objective is to find the optimal thresholds {γk }K−1 k=0 of these K subchannels, collectively maximizing the aggregate opportunistic throughput subject to constraints on the aggregate interference for each primary user and individual constraints on the subbands. As such, the optimization problem for a multi-user primary system can be formulated as max s.t. R (γ) X (i) ci Pm (γi ) ≤ εj , j = 0, 1, . . . , J − 1 (P1) with the optimization variables γ = [γ0 , γ1 , . . . , γK−1 ] . The constraint (16) limits the interference on each subchannel with α = [α0 , α1 , . . . , αK−1 ]T , and the last constraint in (17) dictates that each subchannel should achieve at least a minimum opportunistic spectrum utilization that is proportional to 1−βk . For the single-user primary system where all the subchannels are used by one primary user, we have J = 1. Intuitively, we could make some observations on the multiband joint detection. First, the subchannel with a higher opportunistic rate rk should have a higher threshold γk (i.e., a smaller probability of false alarm) so that it can be highly used by cognitive radios. Second, the subchannel that carries a higher priority primary user should have a lower threshold γk (i.e., a smaller probability of miss) in order to prevent harmful interference by secondary users. Third, a little compromise on those subchannels carrying less important primary users might boost the aggregate rate considerably. Thus, in the determination of the optimal threshold vector, it is necessary to strike a balance among the channel condition, the opportunistic throughput, and the relative priority of each subchannel. The objective and constraint functions in (P1) are generally nonconvex, making it difficult to efficiently solve for the global optimum. In most cases, suboptimal solutions or heuristics have to be used. However, we find that this seemingly nonconvex problem can be made convex by reformulating the problem and exploiting the hidden convexity. We observe the fact that the Q-function is monotonically non-increasing allows us to transform the constraints in (16) and (17) into linear constraints. From (16), we have k = 0, 1, . . . , K − 1. k = 0, 1, . . . , K − 1 2 2M σv2 + 2 |Hk | (19)  Q−1 (1 − αk ) . (20) Similarly, the combination of (10) and (17) leads to γk ≥ γmin,k k = 0, 1, . . . , K − 1 (k) rk Pf (γk ) X i∈Sj (P2) (21) (i) ci Pm (γi ) ≤ εj , j = 0, 1, . . . , J − 1 γmin,k ≤ γk ≤ γmax,k , k = 0, 1, . . . , K − 1. (23) (24) Although the constraint (24) is linear, the problem is still nonconvex. However, it can be furthermore transformed into a tractable convex optimization problem in the regime of low probabilities of false alarm and miss. To establish the transformation, we need the following results. (k) Lemma 1: The function Pf (γk ) is convex in γk if (k) Pf (γk ) ≤ 21 . (k) Proof: Taking the second derivative of Pf (γk ) from (10) gives "
2 # (k) d2 Pf (γk ) γk − M σv2 −1 d exp − = √ dγk2 4M σv4 2π dγk "
2 # γk − M σv2 γk − M σv2 √ exp − = . (25) 4M σv4 2M σv2 2π (k) Since Pf (γk ) ≤ 21 , we have γk ≥ M σv2 . Consequently, the (k) second derivative of Pf (γk ) is greater than or equal to zero, (k) which implies that Pf (γk ) is convex in γk . (k) Lemma 2: The function Pm (γk ) is convex in γk if (γk ) ≤ 12 . Proof: This result can be proved using a similar technique to that used to prove Lemma 1. By taking the second (k) derivative of (11), we can show that Pd (γk ) is concave, and (k) (k) hence Pm (γk ) = 1 − Pd (γk ) is a convex function. Recall that the nonnegative weighted sum of a set of convex functions is also convex [21]. The problem (P1) becomes a convex program if we enforce the following conditions: (k) Pm 1 1 and 0 < βk ≤ , 2 2 k = 0, 1, 2, . . . , K − 1. (26) This regime of probabilities of false alarm and miss is that of practical interest in cognitive radio networks. With the conditions in (26), the feasible set of problem (P2) is convex. The optimization problem takes the form of minimizing a convex function subject to a convex constraint, and thus a local maximum is also the global maximum. Efficient numerical search algorithms such as the interior-point method can be used to solve for the optimal solutions [21]. Alternatively, we can formulate the multiband joint detection problem into another optimization problem that minimizes 0 < αk ≤ where σv K−1 X (18) Substituting (11) into (18) gives   ∆ 2 γmax,k = M σv2 + |Hk | + r  min (16) (17) T γk ≤ γmax,k (22) Consequently, the original problem (P1) has the following equivalent form s.t. P m (γ)  α P f (γ)  β (k) h i √ γmin,k = σv2 M + 2M Q−1 (βk ) . k=0 i∈Sj 1 − Pd (γk ) ≤ αk , where the interference from cognitive radios to the primary communication system, subject to some constraints on the aggregate opportunistic throughput, i.e., minimize (P3) |Hk |2 .50 .30 .45 .65 .25 .60 .40 .70 r (kbps) 612 524 623 139 451 409 909 401 c 1.91 8.17 4.23 3.86 7.16 6.05 0.82 1.30 T r [1 − P f (γ)] ≥ δ P m (γ)  α with δ the required minimum aggregated rate and γ the optimization variables. Like problem (P1), this problem can be transformed into a convex optimization problem by enforcing the conditions in (26). The result will be illustrated numerically later in Section IV. IV. S IMULATION R ESULTS In this section, we numerically evaluate the proposed spectrum sensing schemes. Consider a multiband single-user OFDM system in which a wideband channel is equally divided into 8 subchannels. Each subchannel has a channel gain Hk between the primary user and the cognitive radio, a throughput rate rk if used by cognitive radios, and a cost coefficient ck indicating a penalty incurred when the primary signal is interfered with by the cognitive radio. For each subchannel k (0 ≤ k ≤ 7), it is expected that the opportunistic spectrum utilization is at least 50%, i.e., βk = 0.5, and the probability that the primary user is interfered with is at most αk = 0.1. For simplicity, it is assumed that the noise power level is σv2 = 1 and the length of each detection interval is M = 100. This example studies multiband joint detection in a single cognitive radio. The proposed spectrum sensing algorithms are examined by comparing with an approach that searches a uniform threshold to maximize the aggregate opportunistic throughput. We randomly generate the channel condition between the primary user and the cognitive radio, the opportunistic throughput over each subchannel, and the cost of interference of each subchannel. One realization example is given in Table I. We maximize the aggregate opportunistic throughput over the 8 subchannels subject to some constraints on the interference to the primary users, as formulated in (P1). Fig. 3 plots the maximum aggregate opportunistic rates against the aggregate interference to the primary communication system. It can be seen that the multiband joint detection algorithm with optimized thresholds can achieve a much higher opportunistic rate than that achieved by the one with uniform threshold. Note that in the reference algorithm, the uniform threshold is searched to maximize the achievable rate for a fair comparison. That is, the proposed multiband joint detection algorithm makes better use of the wide spectrum by balancing the conflict between improving spectrum utilization and reducing the interference. In addition, it is observed that the aggregate opportunistic rate increases as we relax the constraint on the aggregate interference ε. An alternative example is depicted in Fig. 4, showing the numerical results of minimizing the aggregate interference subject to the constraints on the opportunistic throughput as 3500 Aggregate Opportunistic Throughput (kbps) P f (γ)  β Multiband Joint Detection Uniform Threshold 3000 2500 2000 0.12 0.13 0.14 Aggregate Interference 0.15 0.16 Fig. 3. The aggregate opportunistic throughput capacity vs. the constraint on the aggregate interference to the primary communication system. formulated in (P3). It can be observed that the multiband joint detection strategy outperforms the one using uniform thresholds in terms of the induced interference to the primary users for any given opportunistic throughput. For illustration purposes, the optimized thresholds and the associated probabilities of miss and false alarm are given in Fig. 5 for (P1) and (P3). To summarize, these numerical results show that multiband joint detection can considerably improve the spectrum efficiency by making more efficient use of the spectral diversity. 2.8 2.6 Multiband Joint Detection Uniform Threshold 2.4 Aggregate Interference st. cT P m (γ) TABLE I PARAMETERS U SED IN S IMULATIONS 2.2 2 1.8 1.6 1.4 1.2 1 2400 2500 2600 2700 2800 2900 3000 Aggregate Opportunistic Throughput (kbps) 3100 Fig. 4. The aggregate interference to the primary communication system vs. the constraint on the aggregate opportunistic throughput. Thresholds 130 120 110 100 Prob. Miss Det. 90 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0.05 0 Prob. False Alarm 0 0.1 0.6 P1 P3 0.4 0.2 0 0 1 2 3 4 5 6 7 Fig. 5. The optimized thresholds and the associated probabilities of miss and false alarm: (P1) ε = 1.25 and (P3) δ = 3224 kbps. V. C ONCLUSION In this paper, we have proposed a multiband joint detection approach for wideband spectrum sensing in cognitive radio networks. The basic strategy is to take into account the detection of primary users across a bank of narrowband subchannels jointly rather than to consider only one single band at a time. We have formulated the joint detection problem into a class of optimization problems to improve the spectral efficiency and reduce the interference. By exploiting the hidden convexity in the seemingly nonconvex problems, we have obtained the optimal solution under practical conditions. The proposed spectrum sensing algorithms have been examined numerically and shown to be able to perform well. ACKNOWLEDGMENT This research was supported in part by the National Science Foundation under Grants ANI-03-38807, CNS-06-25637, ECS-0601266, ECS-0725441, CNS-0721935, CCF-0726740, and by the Department of Defense under Grant HDTRA-071-0037. R EFERENCES [1] D. Cabric, S. M. Mishra, and R. Brodersen, “Implementation issues in spectrum sensing for cognitive radios,” in Proc. 38th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, Nov. 2004. [2] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory. Prentice Hall, Upper Saddle River, NJ, 1998. [3] H. V. Poor, An Introduction to Signal Detection and Estimation. Springer-Verlag, New York, 1994. [4] S. Enserink and D. Cochran, “A cyclostationary feature detector,” in Proc. 28th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, Oct. 1994. [5] D. Cabric, A. Tkachenko, and R. W. 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