Optimal Linear Soft Fusion Schemes for Cooperative Sensing in ...
Optimal Linear Soft Fusion Schemes for Cooperative Sensing in Cognitive Radio Networks Bin Shen, Kyungsup Kwak
Zhiquan Bai
Graduate School of Information Tech. & Telecom. Inha University, Incheon, 402-751, Korea Email: [email protected], [email protected]
School of Information Science and Engineering Shandong University, Jinan, 250-100, China Email: [email protected]
Abstract—In this paper, three optimal linear soft fusion schemes for cooperative sensing are proposed on the basis of corresponding optimality criteria, namely Neyman-Pearson (N-P), deflection coefficient maximization (DCM), and linear quadratic optimization (LQO). Multiple cooperative secondary users (SU) in the cognitive radio (CR) network simply serve as relays to provide space diversity for the fusion center (FC) to obtain the global test statistic. After the ideal optimal fusion weights are acquired, iterative weight setting strategies are utilized to implement them in practice. Analysis and simulation results illustrate that the proposed N-P and DCM schemes yield significant improvements on the sensing performance and the iterative weight setting algorithm can effectively approach the ideal performance of these two schemes. As for the LQO scheme which operates on the received signal covariance matrices merely, it is capable of providing satisfactory performance with sufficient cooperative SUs. Index Terms—Energy detection, optimal fusion, cooperative spectrum sensing, cognitive radio.
I. I NTRODUCTION In order to solve the spectrum scarcity problem, cognitive radio (CR) technique has emerged in recent years as a promising paradigm to exploit the spectrum opportunities, which are restricted by the current rigid spectrum allocation scheme [1] [2]. Since CRs are inherently lower priority or secondary users (SU), the fundamental requirement for them is to avoid interference to the potential primary users (PU) in the vicinity. To detect the PU signal with unknown location, structure and strength, energy detection serves as the optimal spectrum sensing scheme when the detector only knows the power of the received signal. Moreover, energy detection is the most commonly used strategy in spectrum sensing due to its implementation simplicity. However, there are several factors that prevent energy detectors from operating in a reliable manner, e.g. multipath fading/shadowing [3] and noise power fluctuating [4] [5]. These factors imply the necessity of user cooperation in the CR networks [6]- [10]. Ref [6], [7] and [8] mainly investigate hard fusion schemes for cooperative spectrum sensing, e.g. OR, AND, and Majority fusion rules. Apparently and straightforwardly, soft fusion is superior to hard fusion since it demands more communication This research was supported by the MKE (The Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute for Information Technology Advancement)(IITA-2009-C1090-0902-0019).
bandwidth of the control channel for conveying the sensing information between the cooperative SUs and the fusion center (FC). Compared to hard fusion schemes, the problem of requesting too much control channel bandwidth in soft fusion can be effectively solved by quantization of local observations at the expense of additional noise and a signalto-noise ratio (SNR) loss at the receiver [11]. In [9], under the assumption that SNR information of each cooperative SU is readily available, a non-linear optimization enigma is used to formulate the cooperative spectrum sensing problem, which might be somehow difficult to implement in practice. In [10], an optimal soft combination scheme is proposed, based on some approximation in the target optimality function and the assumption that cooperative SUs in the network experience independently and identically distributed (IID) fading effects. It is thereby proved to be identical to a maximal ratio combination (MRC) strategy. In this paper, we propose three optimal linear soft fusion schemes for cooperative spectrum sensing in low SNR scenarios, based on corresponding optimality criteria, namely Neyman-Pearson (N-P), deflection coefficient maximization (DCM), and linear quadratic optimization (LQO). The derived optimal soft fusion weight vectors require partial a priori information of the channel or the covariance matrix of the received signal, which are nuisance parameters extremely challenging to acquire in practice. To implement the proposed optimal soft fusion schemes, iterative weight setting algorithms are hence also proposed. Analysis and simulations verify the superior performance of the proposed optimal fusion schemes and the effectiveness of the iterative weight setting algorithms. The rest of this paper is organized as follows. Section II describes the system model. Section III develops the proposed three optimal linear soft fusion schemes and the iterative weight setting algorithms are presented in section IV. Simulations and discussions are given in section V. Section VI concludes the paper. II. S YSTEM M ODEL A. Spectrum Sensing Observation Relaying We consider K cooperative SUs are dispersed over a certain geographical area via some upper layer algorithms and they
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simply serve as relays to provide space diversity. In the first phase, the signal received at the i-th relay Ri is H0 , 0 ≤ t ≤ Ts n (t), xi (t) = √ i (1) EP U hi s(t) + ni (t), H1 , i = 1, 2, · · · , K √ where s(t) is the transmitted PU signal with amplitude EP U (hereinafter we assume E[s2 (t)] = 1, and E[.] denotes the expectation operation), and hi is the channel gain between PU and Ri , which accommodates the effects of channel shadowing, channel loss and fading, etc. ni (t) is the complex additive white Gaussian noise with zero mean and variance σi2 , and we assume ni (t) and s(t) are mutually independent. H0 and H1 are the hypotheses of the PU signal being absent and present, respectively. Ts is the spectrum sensing interval. Upon receiving signal in the first phase, each relay will simply acts in an amplify-and-forward (AAF) manner, and the signal received by the FC from Ri in the second phase is ¯ i xi (t) + nFC (t) zi (t) = Ei h √ ¯ n (t) + nFC (t), (2) H0 Eh = √ i i i ¯ EP U Ei hi hi s(t) + n ˜ FC,i (t), H1 ¯ i the channel gain where Ei is the transmit power of Ri , h between the FC and Ri , and nFC (t) the noise at the FC with 2 . Again, we assume that 1) nFC (t) zero mean and variance σFC is independent with both ni (t) and s(t); 2) nFC (t) is the same for each of the relays; 3) the individual sensing observations are relayed to the FC in an orthogonal manner that the FC can easily discern the K observations captured at different SUs. We can now write the received signals at the FC in a more compact form, H0 Π0 × n(t) + nFC (t)1, (3) Z(t) = Π1 × s(t)1 + n ˜ FC (t), H1 where signals Z(t) = [z1 (t), z2 (t), · · · , zK (t)]T are received by the FC, n(t) = [n1 (t),n2 (t),· · · , nK (t)]T are the noise components at the K relays, n ˜ F C (t) = T [˜ nFC,1 (t), n ˜ FC,2 (t), · · · , n ˜ FC,K (t)] are the combined K noise components at the FC, and 1 is the column vecare diagonal matrices tor of all ones. Π0√ and Π1 √ √ ¯ ¯ ¯ T = [ E , E , · · · , h h with π 0 2 2 √1 1 √EK hK ] and πT 1 = √ ¯ 1 , EP U E2 h 2 h ¯ 2 , · · · , EP U EK h K h ¯K ] on [ EP U E1 h 1 h their diagonals, respectively. B. Energy Measuring and Fusing at the Fusion Center The soft fusion is carried out at the FC. It first measures the received signal energies from the K relays, Ts H0 Z0 , 2 |Z(t)| dt = (4) Z= H1 Z , 1 0 where test statistics Z0 = [Z0,1 , Z0,2 , · · · , Z0,K ]T and Z1 = [Z1,1 , Z1,2 , · · · , Z1,K ]T are captured within the sensing interval Ts and the frequency bandwidth W . The captured PU signal energies in test statistics Z1 can be
represented by the sum of 2Ts W virtual samples [12] 1 2Ts W 2 θi = |z1,i,j − n ˜ FC,i,j | j=1 2W = γi N0,i W Ts ¯ i |2 Ts , = EP U Ei |hi |2 |h
(5)
where {θi }K i=1 make up the PU signal energy vector T θ = [θ1 , θ2 , · · · , θK ] , z1,i,j and n ˜ FC,i,j are the sample ˜ FC,i (t) at time instant of the relayed signal zi (t) and n j under2 hypothesis H1 , respectively. Additionally, N0,i = ¯ i | σ 2 + σ 2 /W is the equivalent one-sided noise power Ei |h i FC spectral density corresponding to the i-th relayed signal, and γi is the SNR of the i-th relayed signal. When Ts W is asymptotically large (e.g., larger than 100) [13], we can well approximate the test statistics Z as normal distributed variables, according to the central limit theorem (CLT), with means and variances μ0,i = E[Z0,i ] = N0,i Ts W, H0 (6) 2 2 = V ar[Z0,i ] = N0,i Ts W. δ0,i μ1,i = E[Z1,i ] = N0,i Ts W + θi , H1 (7) 2 2 = V ar[Z1,i ] = N0,i Ts W + 2N0,i θi . δ1,i Based on Z and by allocating different weight coefficients to them, the FC fuses the K observations into a global statistic, Zc =
K
ωi Zi = ω T Z,
(8)
i=1 T
where ω = [ω1 , ω2 , · · · , ωK ] is the vector of weighting coefficients. The combining weight for the signal from a particular SU represents its contribution to the global decision. Consequently, Zc has its means and variances given by T H0 ω μ0 , ¯ Zc = E[Zc ] = (9) H1 ω T μ1 , ⎧K 2 2 ⎪ ⎪ H0 ωi = ω T Σ0 ω, ⎨ δ0,i (10) V ar[Zc ] = i=1 K ⎪ 2 2 T ⎪ ⎩ δ1,i ωi = ω Σ1 ω, H1 i=1
where mean vectors are μ0 = [μ0,1 , μ0,2 , · · · , μ0,K ]T and μ1 = [μ1,1 , μ1,2 , · · · , μ1,K ]T (note that μ1 −μ0 = θ); Σ0 and 2 2 2 , δ0,2 , · · · , δ0,K ]T Σ1 are diagonal matrices with δ 20 = [δ0,1 2 2 2 and δ 21 = [δ1,1 , δ1,2 , · · · , δ1,K ]T on the diagonals, respectively. It is worth noting that the statistics Z do not have to be conditionally independent though hereby we utilize the independent case for the illustration purpose, i.e., with Σ0 and Σ1 diagonal. If the elements of Z are spatially correlated with each other, then the covariance matrix Σ1 is non-diagonal and the subsequent analysis will continue to hold [9]. Given a global threshold λ at the FC, the corresponding probabilities of false alarm and detection are
λ − μT0 ω λ − μT1 ω , PD = Q , (11) PFA = Q ω T Σ0 ω ω T Σ1 ω √ +∞ where Q(x) = x exp(−t2 /2)dt/ 2π.
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III. O PTIMAL L INEAR S OFT F USION S CHEMES A. Neyman-Pearson Criterion Based Fusion Scheme For a cooperative spectrum sensing algorithm, a common standard for setting the global threshold is the NeymanPearson criterion, which guarantees the probability of detection is maximized with a constant false-alarm probability. Setting the threshold λ for a desired probability of false alarm PFA,DES , we obtain the probability of detection in the NeymanPearson framework,
η ω T Σ0 ω + μT0 ω − μT1 ω , (12) PD = Q ω T Σ1 ω where η is Q−1 (PFA,DES ) and Q−1 (.) is the inverse function of Q(.). Observing that Q(.) is a strictly monotonically decreasing function, we obtain the optimal weight vector ω opt,NP as ω opt,NP = argmin (ω) ω
η ω T Σ0 ω + μT0 ω − μT1 ω . = argmin ω ω T Σ1 ω
(13)
To the best knowledge of the authors, currently there is no closed-form solution in the literature for acquiring the optimal weights in (13). By setting ∂(ω)/∂ω = 0 and solving this equation, we reach the un-normalized optimal weight vector ω T Σ0 ω −1 ∗ −1 θ Σ0 (IK + αNP Σ−1 ω opt,NP = 0 Σ1 ) (14) η ω T Σ1 ω −1 −1 = βNP Σ−1 θ, 0 (IK + αNP Σ0 Σ1 )
The detection performance in Neyman-Pearson framework is then given by ⎛ ⎞ T −1 −1 η θ T A−2 Σ−1 0 θ − θ A Σ0 θ (NP) ⎠ . (16) PD = Q ⎝ T −2 −2 θ A Σ0 Σ1 θ The detection performance PD in (16) is actually a probability conditioned on the PU signal energy vector θ, which is a composite random variable vector. The statistically averaged PD is hence, ⎛ ⎞ T −1 −1 η ξ T A−2 Σ−1 0 ξ − ξ A Σ0 ξ (NP) ⎠pθ (ξ)dξ, P¯D = Q⎝ T −2 −2 Θ1 ⊂RK + ξ A Σ0 Σ1 ξ (17) where pθ (.) is the joint probability distribution function (PDF) of the multi-variable vector θ, Θ1 is a subset of RK + which contains all θ vectors leading to the H1 decision, and RK + is the positive K-dimensional vector space. Even if Σ1 (NP) is not a diagonal matrix, P¯D in (17) still holds true, since the non-diagonal elements of Σ1 are still actually related to θ. B. DCM Criterion Based Fusion Scheme From (9) and (10) it is clear that the weight vector ω plays an important role in determining the PDFs of the global test statistic Zc under both hypotheses. To measure the effect of the PDF on the detection performance, we introduce a deflection coefficient (DC) [14] 2
[ω T (μ1 − μ0 )]2 (E[Zc |H1 ] − E[Zc |H0 ]) = . V ar[Zc |H0 ] ω T Σ0 ω (18) The deflection coefficient d2DC (ω) provides a good measure of the detection performance, since it characterizes the variance-normalized distance between the centers of two conditional PDFs of Zc . Therefore, the optimal weight vector ω opt,DC is defined as the one that maximizes the distance d2DC (ω) (19) ω opt,DC = argmax d2DC (ω). d2DC (ω) =
where IK is the identity matrix of order K, βNP is a scalar imposing no effect on (ω) which can be set to 1, and αNP is a non-negative loading term, which can be optimized by αopt,NP = argmin (ω opt,NP (αNP )) αNP ⎛ ⎞ T −1 −1 (15) θ − θ A Σ θ η θ T A−2 Σ−1 0 0 ⎠, = argmin ⎝ αNP θ T A−2 Σ−2 0 Σ1 θ where A = IK + αNP Σ−1 0 Σ1 . The validity of the second equation in (15) is based on the assumption that Σ0 and Σ1 are symmetric matrices, which is inherently satisfied in practice. It is apparent that the derived weight vector ω opt,NP = ω ∗opt,NP /ω ∗opt,NP ( . 2 denotes the Euclidean norm) in (14) is not a closed-form solution and finding the optimal value of αNP involves numerical evaluation. To simplify the problem, we proposed another scheme in the following subsection which comes out with a closed-form optimal weight vector. For a given PFA,DES , minimization of the function (ω) through ω opt,NP yields a maximum probability of detection PD . The desired constant false alarm probability is guaranteed when the statistical properties of noise components under hypothesis H0 are readily available for threshold setting. This is due to the fact that threshold λ is dynamically determined by η ω Topt,NP Σ0 ω opt,NP + ω Topt,NP μ0 .
ω
According to the Schwarz inequality, we obtain 1/2
−1/2
[ω T (μ1 − μ0 )]2 = [ω T Σ0 Σ0
(ω T (μ1 − μ0 ))]2
≤ (ω T Σ0 ω)(θ T Σ−1 0 θ),
(20)
where the equation is satisfied only when the optimal weights is −1 (21) ω ∗opt,DCM = cΣ−1 0 (μ1 − μ0 ) = cΣ0 θ, where c is a constant imposing no effect on d2DC (ω) and thus can be set to 1. After normalization, the optimal soft fusion weights are ω opt,DCM = ω ∗opt,DCM /ω ∗opt,DCM 2 . With a desired probability of false alarm PFA,DES , the detection performance in the DCM framework is given by ⎛ ⎞ T −1 T −1 θ Σ θ − θ Σ θ η 0 0 (DCM) ⎠, = Q⎝ (22) PD θ T Σ−2 Σ θ 1 0
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(DCM)
where PD is maximized in the sense that the distance between the two PDFs of Zc under hypotheses H0 and H1 is maximized to the most by ω opt,DC . Similarly to the previous case, the statistically averaged PD is ⎛ ⎞ T −1 T −1 ξ Σ ξ − ξ Σ ξ η 0 0 (DCM) ⎠pθ (ξ)dξ, = Q⎝ P¯D T −2 Θ2 ⊂RK + ξ Σ0 Σ1 ξ (23) where Θ2 is a subset of RK + which contains all θ vectors leading to the H1 decision. C. Linear Quadratic Optimization Based Fusion Scheme
IV. I MPLEMENTATION OF P ROPOSED F USION S CHEMES
In this subsection, we derive the optimal weight vectors in terms of linear quadratic optimization. The sensing problem is formulated as a Rayleigh quotient and the corresponding optimal weight vector is defined as the one maximizing the Rayleigh quotient. The generalized Rayleigh quotient of the global test statistic Zc is defined by Γ(ω) R(Σ1 , Σ0 ; ω) =
ω T Σ1 ω , ω T Σ0 ω
(24)
where Γ(ω) bears resemblance to the SNR of Zc . Its value is 1 under H0 whereas under H1 , the real symmetric matrix Σ1 contains the PU signal component and the associated Γ(ω) is proportional to the real SNR of the PU signal. The optimal weight vector ω opt,Ray is defined as the one that maximizes Γ(ω) ω opt,Ray = argmax Γ(ω).
(25)
ω
The generalized Rayleigh quotient R(Σ1 , Σ0 ; ω) in (24) can be reduced to the Rayleigh quotient R(D, CTω) through the transformation D = C−1 Σ1 C−T [15], where C is the Cholesky decomposition of matrix Σ0 . It is well known that the Cholesky decomposition of Σ0 is a decomposition of the symmetric, positive-definite matrix Σ0 into a lower triangular matrix and the transpose of the lower triangular matrix, Σ0 = CCT .
(26)
The proposed optimal soft fusion schemes require partial a priori knowledge of the channel, which has to be estimated for setting the derived weight vectors. A. Estimation of PU Signal Energy Vector The optimal weighting vectors in (14) and (21) are mainly determined by the signal energy quantities EP U |hi |2 , under ¯ i , the relay power Ei , the assumption that the channel gains h the noise variances σi2 and σF2 C are readily available through estimation at the FC before the sensing operation begins. We utilize two matrices to keep records of PU’s behaviors, in which the current sensing data Zl is categorized and stored in either a Presence or an Absence matrix for future reference, according to the current global decision. In other words, if it is decided that the current data Zl contain the PU signal energies, it is stored in a K-by-L Presence matrix Z(P ) in a first-in-first-out (FIFO) manner; otherwise, it is stored in a K-by-L Absence matrix Z(A) and meanwhile a zero column vector is stacked into Z(P ) since the fusion center has decided that no PU signal energy is contained in Zl . The estimates of K {θi }K i=1 and {N0,i }i=1 for the current statistic are calculated via simple arithmetic operations: ⎧ l−1 ⎪ (P ) (A) ⎪ ⎪ θi,l = ζ1 ρl |Zi,j − Zi,j |, ⎨ j=l−L (30) l−1 ⎪ (A) ⎪ 0,i,l = 1 ⎪ ρ Z , N j i,j ⎩ Ts W j=l−L
Now we obtain the Rayleigh quotient as Γ(ω) = R(Σ1 , Σ0 ; ω) = R(D, CTω) =
vector opt,Ray as the unit eigenvector corresponding to the largest eigenvalue of D, i.e. eigmax (D). It is now derived that the optimal combining weights in the LQO framework, opt,Ray , is only corresponding to the largest eigenvalue of D. Furthermore, it is worth noting that as a blind cooperative sensing scheme, the LQO scheme offers robustness against the dynamically changing environment because it merely operates on the received signal covariance matrices. The threshold-setting should be practically considered in advance, since for a desired false alarm probability, a calibration of the global threshold is indispensable.
D , T T
(27)
where = CT ω. Notice that R(A, x ) = R(A, x), for x = cx, ∀c = 0, therefore we will solve for with a unit norm ||||2 = 1, max T D, (28) s.t. T = 1. By using the scalar Lagrange multiplier λLag ∈ R, we obtain L() = T D + λLag ( T − 1),
(29)
where taking the derivative of L() with respect to , i.e. ∂L()/∂ = 0, we can easily find the optimal weight
where ζ is the common coefficient of {θi }K i=1 which can be practically set to 1, l is the time index of the current sensing data, ρj is the forgetting factor which is usually a scalar, and L is the reference matrix depth, which should be carefully tuned on the basis of estimation or prediction of the channel gain varying velocity. Additionally, to be more compact for implementation, the proposed estimations in (30) can be implemented as: (A) 1 (P ) θi,l ≈ L−1 L θi,l−2 + L Zi,l−1 − Zi,l−1 , (31) 0,i,l−2 + 1 Z (A) , 0,i,l ≈ L−1 N N L
where forgetting factors 1/L independent of l.
L
l−1 {ρj }j=l−L
i,l−1
are all set to a constant
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B. Estimation of Received Signal Covariance Matrix The LQO based optimal fusion scheme needs the maximum eigenvectors of the covariance matrices Σ1 and Σ0 . We now develop a method to estimate the covariance matrices using the received signal only. Considering the current estimate of (l) , the covariance matrix can be estimated Σ1 is denoted by Σ 1 on-line as (l) = Σ 1
l−2 j=l−L
(j) + (1 − ρj Σ 1
l−1
probability. Although the target optimality function of the DCM scheme is only part of the N-P’s, it has a nearly undiscernible performance difference compared to the N-P scheme, because in the low SNR regime, the difference between their optimality functions is too small. As for the LQO scheme, it is an eigenvector based blind sensing solution, which requires less a priori knowledge than the N-P and DCM schemes, and therefore has inferior performance.
ρj )Z(l−1) ZT(l−1) ,
j=l−L
L − 1 (l−2) 1 + Z(l−1) ZT(l−1) (with ρj = 1/L), ≈ Σ1 L L (32) where Z(l) are the sensing statistics at time index l, which comes from the sensing observations stored in the PU signal reference matrix Z(P ) . (l) can be obtained in the similar way The estimate Σ 0 described above in (32). With the estimated signal covariance matrices, we can attain the needed eigenvectors. Additionally, the method of categorizing and storing the sensing data Z(l) based on the global decisions, is somewhat different to the one described in the previous subsection. If the current decision is drawn as H0 , the current sensing observation Z(l) will be inserted into both of Z(P ) and Z(A) ; otherwise, the current observation will be only stored in Z(P ) . In this way, (l) would actually be if there is no PU signal, the estimate Σ 1 (l) , which does not corrupt close or identical to the estimate Σ 0 the detection since the contribution of its eigenvectors to the test statistic would be low.
Fig. 1.
ROC performance of the proposed optimal fusion schemes.
V. S IMULATIONS AND D ISCUSSIONS In this section, the proposed soft fusion schemes are evaluated via simulations. The basic parameters are fixed and set as Ts W = 400, L = 12 and K = 8. Each simulation consisted of 105 iterations. The channel gains between each SU and the target PU are experiencing IID Rayleigh fading. For simplicity, we assume that the PU signal power and the channel gains have constant values for each sensing interval Ts . We further assume the channel gains vary from (observation) period to period while their PDFs are determined by the fading characteristic of the channels. Hereafter, we suppose that the noise variances 2 K }i=1 in (6) are distributed around an average level δ02 {δ0,i with a deviation d. The noise variance deviation d is normally distributed as N (0, ρd δ02 ). ρd indicates the location difference factor in the network area, which is jointly determined by the ¯ i , and σ 2 (ρd = 0.2 in simulations). locations of the SUs, Ei , h i Fig. 1 gives the receiver operating characteristic (ROC) curves of the prosed optimal fusion schemes in IID rayleigh fading environment. In order to demonstrate the superior performance of the proposed optimal fusion schemes, we provide the performance of conventional MRC scheme as well. The N-P scheme’s superiority lies in that its target optimality function is the complete one taken from the mathematical expression of the PU signal detection
Fig. 2. ROC Performance of the LQO scheme with different number of cooperative SUs (SNR = -15dB).
In order to investigate the impact of user number on the collaboration gain in sensing, we give the ROC curves in Fig. 2, which demonstrate that the detection performance of the LQO scheme is enhanced with increase in the number of users. The certifies that the performance deterioration in the absence of a priori knowledge of the channels is effectively alleviated.
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VI. C ONCLUSIONS In this paper, optimal linear soft fusion schemes for cooperative spectrum sensing in CR networks are investigated and analyzed. Based on Neyman-Pearson criterion, DCM criterion, and linear quadratic optimization, three optimal fusion schemes are proposed to combine the sensing observations collected from the cooperative SUs. Analysis and simulations illustrate that the proposed optimal soft fusion schemes yield significant improvements of the spectrum sensing performance in the CR network.
∂dDC (ω)/∂ω = 0, where from
(38)
∂ ω T θ/ ω T Σ0 ω
∂dDC (ω) = ∂ω ∂ω θ ω T Σ0 ω − (ω T θ)(Σ0 ω)/ ω T Σ0 ω (39) = ω T Σ0 ω = 0, we reach the equation
A PPENDIX
θω T Σ0 ω − (ω T θ)(Σ0 ω) = 0.
A. N-P Optimal Weights The optimal fusion weights for the N-P scheme are obtained by solving the equation ∂(ω)/∂ω = ∂[1 (ω)/2 (ω)]/∂ω ∂1 (ω) ∂2 (ω) (ω) − (ω) 2 1 ∂ω ∂ω (33) = 2 2 (ω) = 0, where functions 1 (ω) = η ω T Σ0 ω − ω T θ and 2 (ω) = ω T Σ1 ω. Upon substituting the derivatives in (33) with Σ0 ω ∂1 (ω) = η − θ, ∂ω ω T Σ0 ω Σ1 ω ∂2 (ω) = , ∂ω ω T Σ1 ω
(34)
we obtain ηΣ0 ω ω T Σ1 ω −θ ω T Σ0 ω Σ1 ω + (ω T θ − η ω T Σ0 ω) = 0. ω T Σ1 ω
(35)
After some mathematical manipulations, the equation in (35) is again given as (36) (ηα12 Σ0 − ηα02 Σ1 )ω + α0 (ω T θ)Σ1 ω = α0 α12 θ, where α0 = ω T Σ0 ω, α1 = ω T Σ1 ω, and the solution to the above equation is α0 −1 −1 Σ (IK + αNP Σ−1 θ 0 Σ1 ) ηα1 0 −1 −1 θ = βNP Σ−1 0 (IK + αNP Σ0 Σ1 )
ω ∗opt,NP =
B. DCM Optimal Weights The optimal fusion weights for the DCM scheme are obtained according to the Schwarz’s inequality in (20) or by solving the equation
(37)
where αNP = (α0 ω T θ − ηα02 )/(ηα12 ), (αNP > 0) and it is still a function of ω, which requires numerical search for its optimal value leading to the minimum (ω) in (15).
(40)
Since ω T Σ0 ω and ω T θ are scalars, we can obtain ω ∗opt,DCM =
ω T Σ0 ω −1 Σ0 θ = βDC Σ−1 0 θ. ωT θ
(41)
R EFERENCES [1] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 23, no. 2, pp. 201-220, Feb. 2005. [2] Q. Zhao and B. M. Sadler, “A survey of dynamic spectrum access: signal processing, networking, and regulatory policy,” IEEE Signal Processing Magazine, pp. 79-89, May 2007. [3] D. Cabric, S. M. Mishra, and R. W. Brodersen, “Implementation issues in spectrum sensing for cognitive radios,” Proc. of 38th Asilomar Conf. Signals Systems, Computers, pp. 772-776, 2004. [4] R. Tandra and A. Sahai, “Fundamental limits on detection in low SNR under noise uncertainty,” Proc. of the Wireless Communications Symposium on Signal Processing 2005, June 2005. [5] A. Sonnenschein and P. M. Fishman, “Radiometric detection of spreadspectrum signals in noise of uncertain power,” IEEE Trans. on Aerosp. Electron. Syst., vol. 28, no. 3, pp. 654-660, July 1992. [6] A. Ghasemi and E. S. Sousa, “Impact of user collaboration on the performance of sensing-based opportunistic spectrum access,” Proc. of IEEE VTC 2006 Fall, Sept. 2006. [7] S. M. Mishra, A. Sahai and R. W. Brodersen, “Cooperative sensing among cognitive radios,” Proc. of IEEE ICC 2006, June 2006. [8] W. Zhang, R. K. Mallik, and K. B. Letaief, “Cooperative spectrum sensing optimization in cognitive radio networks,” Proc. of IEEE International Conference on Communications (ICC’08), pp. 3411-3415, May 2008. [9] Z. Quan, S. Cui, and A. H. Sayed, “Optimal linear cooperation for spectrum sensing in cognitive radio networks,” IEEE Journal of Selected Topics in Signal Processing, vol. 2, no. 1, pp. 28-40, Feb. 2008. [10] J. Ma and Y. Li, “Soft Combination and Detection for Cooperative Spectrum Sensing in Cognitive Radio Networks,” Proc. of IEEE GLOBECOM 2007, pp. 3139-3143, Dec. 2007. [11] C. W. Helstronm, “Improved multilevel quantization for detection of narrowband signals,” IEEE Trans. Aerosp. Electron. Syst., vol. 24, no. 2, pp. 142-147, March 1988. [12] H. Urkowitz, “Energy detection of unknown deterministic signals,” Proceedings of IEEE, vol. 55, April 1967, pp. 523-231. [13] R.F. Mills and G.E. Prescon, “A comparison of various radiometer detection models,” IEEE Transactions on Aerospace and Electronic Systems, vol. 32, no. 1, pp. 467-473, Jan. 1996. [14] H. V. Poor, An Introduction to Signal Detection and Estimation, 2nd ed. Springer-Verlag, New York, 1994. [15] R. E. Prieto, “A general solution to the maximization of the multidimensional generalized rayleigh quotient used in linear discriminant analysis for signal classification,” Proc. IEEE ICASSP 2003, no. VI, pp. 157-160, 2003.
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Zhiquan Bai
Graduate School of Information Tech. & Telecom. Inha University, Incheon, 402-751, Korea Email: [email protected], [email protected]
School of Information Science and Engineering Shandong University, Jinan, 250-100, China Email: [email protected]
Abstract—In this paper, three optimal linear soft fusion schemes for cooperative sensing are proposed on the basis of corresponding optimality criteria, namely Neyman-Pearson (N-P), deflection coefficient maximization (DCM), and linear quadratic optimization (LQO). Multiple cooperative secondary users (SU) in the cognitive radio (CR) network simply serve as relays to provide space diversity for the fusion center (FC) to obtain the global test statistic. After the ideal optimal fusion weights are acquired, iterative weight setting strategies are utilized to implement them in practice. Analysis and simulation results illustrate that the proposed N-P and DCM schemes yield significant improvements on the sensing performance and the iterative weight setting algorithm can effectively approach the ideal performance of these two schemes. As for the LQO scheme which operates on the received signal covariance matrices merely, it is capable of providing satisfactory performance with sufficient cooperative SUs. Index Terms—Energy detection, optimal fusion, cooperative spectrum sensing, cognitive radio.
I. I NTRODUCTION In order to solve the spectrum scarcity problem, cognitive radio (CR) technique has emerged in recent years as a promising paradigm to exploit the spectrum opportunities, which are restricted by the current rigid spectrum allocation scheme [1] [2]. Since CRs are inherently lower priority or secondary users (SU), the fundamental requirement for them is to avoid interference to the potential primary users (PU) in the vicinity. To detect the PU signal with unknown location, structure and strength, energy detection serves as the optimal spectrum sensing scheme when the detector only knows the power of the received signal. Moreover, energy detection is the most commonly used strategy in spectrum sensing due to its implementation simplicity. However, there are several factors that prevent energy detectors from operating in a reliable manner, e.g. multipath fading/shadowing [3] and noise power fluctuating [4] [5]. These factors imply the necessity of user cooperation in the CR networks [6]- [10]. Ref [6], [7] and [8] mainly investigate hard fusion schemes for cooperative spectrum sensing, e.g. OR, AND, and Majority fusion rules. Apparently and straightforwardly, soft fusion is superior to hard fusion since it demands more communication This research was supported by the MKE (The Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute for Information Technology Advancement)(IITA-2009-C1090-0902-0019).
bandwidth of the control channel for conveying the sensing information between the cooperative SUs and the fusion center (FC). Compared to hard fusion schemes, the problem of requesting too much control channel bandwidth in soft fusion can be effectively solved by quantization of local observations at the expense of additional noise and a signalto-noise ratio (SNR) loss at the receiver [11]. In [9], under the assumption that SNR information of each cooperative SU is readily available, a non-linear optimization enigma is used to formulate the cooperative spectrum sensing problem, which might be somehow difficult to implement in practice. In [10], an optimal soft combination scheme is proposed, based on some approximation in the target optimality function and the assumption that cooperative SUs in the network experience independently and identically distributed (IID) fading effects. It is thereby proved to be identical to a maximal ratio combination (MRC) strategy. In this paper, we propose three optimal linear soft fusion schemes for cooperative spectrum sensing in low SNR scenarios, based on corresponding optimality criteria, namely Neyman-Pearson (N-P), deflection coefficient maximization (DCM), and linear quadratic optimization (LQO). The derived optimal soft fusion weight vectors require partial a priori information of the channel or the covariance matrix of the received signal, which are nuisance parameters extremely challenging to acquire in practice. To implement the proposed optimal soft fusion schemes, iterative weight setting algorithms are hence also proposed. Analysis and simulations verify the superior performance of the proposed optimal fusion schemes and the effectiveness of the iterative weight setting algorithms. The rest of this paper is organized as follows. Section II describes the system model. Section III develops the proposed three optimal linear soft fusion schemes and the iterative weight setting algorithms are presented in section IV. Simulations and discussions are given in section V. Section VI concludes the paper. II. S YSTEM M ODEL A. Spectrum Sensing Observation Relaying We consider K cooperative SUs are dispersed over a certain geographical area via some upper layer algorithms and they
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simply serve as relays to provide space diversity. In the first phase, the signal received at the i-th relay Ri is H0 , 0 ≤ t ≤ Ts n (t), xi (t) = √ i (1) EP U hi s(t) + ni (t), H1 , i = 1, 2, · · · , K √ where s(t) is the transmitted PU signal with amplitude EP U (hereinafter we assume E[s2 (t)] = 1, and E[.] denotes the expectation operation), and hi is the channel gain between PU and Ri , which accommodates the effects of channel shadowing, channel loss and fading, etc. ni (t) is the complex additive white Gaussian noise with zero mean and variance σi2 , and we assume ni (t) and s(t) are mutually independent. H0 and H1 are the hypotheses of the PU signal being absent and present, respectively. Ts is the spectrum sensing interval. Upon receiving signal in the first phase, each relay will simply acts in an amplify-and-forward (AAF) manner, and the signal received by the FC from Ri in the second phase is ¯ i xi (t) + nFC (t) zi (t) = Ei h √ ¯ n (t) + nFC (t), (2) H0 Eh = √ i i i ¯ EP U Ei hi hi s(t) + n ˜ FC,i (t), H1 ¯ i the channel gain where Ei is the transmit power of Ri , h between the FC and Ri , and nFC (t) the noise at the FC with 2 . Again, we assume that 1) nFC (t) zero mean and variance σFC is independent with both ni (t) and s(t); 2) nFC (t) is the same for each of the relays; 3) the individual sensing observations are relayed to the FC in an orthogonal manner that the FC can easily discern the K observations captured at different SUs. We can now write the received signals at the FC in a more compact form, H0 Π0 × n(t) + nFC (t)1, (3) Z(t) = Π1 × s(t)1 + n ˜ FC (t), H1 where signals Z(t) = [z1 (t), z2 (t), · · · , zK (t)]T are received by the FC, n(t) = [n1 (t),n2 (t),· · · , nK (t)]T are the noise components at the K relays, n ˜ F C (t) = T [˜ nFC,1 (t), n ˜ FC,2 (t), · · · , n ˜ FC,K (t)] are the combined K noise components at the FC, and 1 is the column vecare diagonal matrices tor of all ones. Π0√ and Π1 √ √ ¯ ¯ ¯ T = [ E , E , · · · , h h with π 0 2 2 √1 1 √EK hK ] and πT 1 = √ ¯ 1 , EP U E2 h 2 h ¯ 2 , · · · , EP U EK h K h ¯K ] on [ EP U E1 h 1 h their diagonals, respectively. B. Energy Measuring and Fusing at the Fusion Center The soft fusion is carried out at the FC. It first measures the received signal energies from the K relays, Ts H0 Z0 , 2 |Z(t)| dt = (4) Z= H1 Z , 1 0 where test statistics Z0 = [Z0,1 , Z0,2 , · · · , Z0,K ]T and Z1 = [Z1,1 , Z1,2 , · · · , Z1,K ]T are captured within the sensing interval Ts and the frequency bandwidth W . The captured PU signal energies in test statistics Z1 can be
represented by the sum of 2Ts W virtual samples [12] 1 2Ts W 2 θi = |z1,i,j − n ˜ FC,i,j | j=1 2W = γi N0,i W Ts ¯ i |2 Ts , = EP U Ei |hi |2 |h
(5)
where {θi }K i=1 make up the PU signal energy vector T θ = [θ1 , θ2 , · · · , θK ] , z1,i,j and n ˜ FC,i,j are the sample ˜ FC,i (t) at time instant of the relayed signal zi (t) and n j under2 hypothesis H1 , respectively. Additionally, N0,i = ¯ i | σ 2 + σ 2 /W is the equivalent one-sided noise power Ei |h i FC spectral density corresponding to the i-th relayed signal, and γi is the SNR of the i-th relayed signal. When Ts W is asymptotically large (e.g., larger than 100) [13], we can well approximate the test statistics Z as normal distributed variables, according to the central limit theorem (CLT), with means and variances μ0,i = E[Z0,i ] = N0,i Ts W, H0 (6) 2 2 = V ar[Z0,i ] = N0,i Ts W. δ0,i μ1,i = E[Z1,i ] = N0,i Ts W + θi , H1 (7) 2 2 = V ar[Z1,i ] = N0,i Ts W + 2N0,i θi . δ1,i Based on Z and by allocating different weight coefficients to them, the FC fuses the K observations into a global statistic, Zc =
K
ωi Zi = ω T Z,
(8)
i=1 T
where ω = [ω1 , ω2 , · · · , ωK ] is the vector of weighting coefficients. The combining weight for the signal from a particular SU represents its contribution to the global decision. Consequently, Zc has its means and variances given by T H0 ω μ0 , ¯ Zc = E[Zc ] = (9) H1 ω T μ1 , ⎧K 2 2 ⎪ ⎪ H0 ωi = ω T Σ0 ω, ⎨ δ0,i (10) V ar[Zc ] = i=1 K ⎪ 2 2 T ⎪ ⎩ δ1,i ωi = ω Σ1 ω, H1 i=1
where mean vectors are μ0 = [μ0,1 , μ0,2 , · · · , μ0,K ]T and μ1 = [μ1,1 , μ1,2 , · · · , μ1,K ]T (note that μ1 −μ0 = θ); Σ0 and 2 2 2 , δ0,2 , · · · , δ0,K ]T Σ1 are diagonal matrices with δ 20 = [δ0,1 2 2 2 and δ 21 = [δ1,1 , δ1,2 , · · · , δ1,K ]T on the diagonals, respectively. It is worth noting that the statistics Z do not have to be conditionally independent though hereby we utilize the independent case for the illustration purpose, i.e., with Σ0 and Σ1 diagonal. If the elements of Z are spatially correlated with each other, then the covariance matrix Σ1 is non-diagonal and the subsequent analysis will continue to hold [9]. Given a global threshold λ at the FC, the corresponding probabilities of false alarm and detection are
λ − μT0 ω λ − μT1 ω , PD = Q , (11) PFA = Q ω T Σ0 ω ω T Σ1 ω √ +∞ where Q(x) = x exp(−t2 /2)dt/ 2π.
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III. O PTIMAL L INEAR S OFT F USION S CHEMES A. Neyman-Pearson Criterion Based Fusion Scheme For a cooperative spectrum sensing algorithm, a common standard for setting the global threshold is the NeymanPearson criterion, which guarantees the probability of detection is maximized with a constant false-alarm probability. Setting the threshold λ for a desired probability of false alarm PFA,DES , we obtain the probability of detection in the NeymanPearson framework,
η ω T Σ0 ω + μT0 ω − μT1 ω , (12) PD = Q ω T Σ1 ω where η is Q−1 (PFA,DES ) and Q−1 (.) is the inverse function of Q(.). Observing that Q(.) is a strictly monotonically decreasing function, we obtain the optimal weight vector ω opt,NP as ω opt,NP = argmin (ω) ω
η ω T Σ0 ω + μT0 ω − μT1 ω . = argmin ω ω T Σ1 ω
(13)
To the best knowledge of the authors, currently there is no closed-form solution in the literature for acquiring the optimal weights in (13). By setting ∂(ω)/∂ω = 0 and solving this equation, we reach the un-normalized optimal weight vector ω T Σ0 ω −1 ∗ −1 θ Σ0 (IK + αNP Σ−1 ω opt,NP = 0 Σ1 ) (14) η ω T Σ1 ω −1 −1 = βNP Σ−1 θ, 0 (IK + αNP Σ0 Σ1 )
The detection performance in Neyman-Pearson framework is then given by ⎛ ⎞ T −1 −1 η θ T A−2 Σ−1 0 θ − θ A Σ0 θ (NP) ⎠ . (16) PD = Q ⎝ T −2 −2 θ A Σ0 Σ1 θ The detection performance PD in (16) is actually a probability conditioned on the PU signal energy vector θ, which is a composite random variable vector. The statistically averaged PD is hence, ⎛ ⎞ T −1 −1 η ξ T A−2 Σ−1 0 ξ − ξ A Σ0 ξ (NP) ⎠pθ (ξ)dξ, P¯D = Q⎝ T −2 −2 Θ1 ⊂RK + ξ A Σ0 Σ1 ξ (17) where pθ (.) is the joint probability distribution function (PDF) of the multi-variable vector θ, Θ1 is a subset of RK + which contains all θ vectors leading to the H1 decision, and RK + is the positive K-dimensional vector space. Even if Σ1 (NP) is not a diagonal matrix, P¯D in (17) still holds true, since the non-diagonal elements of Σ1 are still actually related to θ. B. DCM Criterion Based Fusion Scheme From (9) and (10) it is clear that the weight vector ω plays an important role in determining the PDFs of the global test statistic Zc under both hypotheses. To measure the effect of the PDF on the detection performance, we introduce a deflection coefficient (DC) [14] 2
[ω T (μ1 − μ0 )]2 (E[Zc |H1 ] − E[Zc |H0 ]) = . V ar[Zc |H0 ] ω T Σ0 ω (18) The deflection coefficient d2DC (ω) provides a good measure of the detection performance, since it characterizes the variance-normalized distance between the centers of two conditional PDFs of Zc . Therefore, the optimal weight vector ω opt,DC is defined as the one that maximizes the distance d2DC (ω) (19) ω opt,DC = argmax d2DC (ω). d2DC (ω) =
where IK is the identity matrix of order K, βNP is a scalar imposing no effect on (ω) which can be set to 1, and αNP is a non-negative loading term, which can be optimized by αopt,NP = argmin (ω opt,NP (αNP )) αNP ⎛ ⎞ T −1 −1 (15) θ − θ A Σ θ η θ T A−2 Σ−1 0 0 ⎠, = argmin ⎝ αNP θ T A−2 Σ−2 0 Σ1 θ where A = IK + αNP Σ−1 0 Σ1 . The validity of the second equation in (15) is based on the assumption that Σ0 and Σ1 are symmetric matrices, which is inherently satisfied in practice. It is apparent that the derived weight vector ω opt,NP = ω ∗opt,NP /ω ∗opt,NP ( . 2 denotes the Euclidean norm) in (14) is not a closed-form solution and finding the optimal value of αNP involves numerical evaluation. To simplify the problem, we proposed another scheme in the following subsection which comes out with a closed-form optimal weight vector. For a given PFA,DES , minimization of the function (ω) through ω opt,NP yields a maximum probability of detection PD . The desired constant false alarm probability is guaranteed when the statistical properties of noise components under hypothesis H0 are readily available for threshold setting. This is due to the fact that threshold λ is dynamically determined by η ω Topt,NP Σ0 ω opt,NP + ω Topt,NP μ0 .
ω
According to the Schwarz inequality, we obtain 1/2
−1/2
[ω T (μ1 − μ0 )]2 = [ω T Σ0 Σ0
(ω T (μ1 − μ0 ))]2
≤ (ω T Σ0 ω)(θ T Σ−1 0 θ),
(20)
where the equation is satisfied only when the optimal weights is −1 (21) ω ∗opt,DCM = cΣ−1 0 (μ1 − μ0 ) = cΣ0 θ, where c is a constant imposing no effect on d2DC (ω) and thus can be set to 1. After normalization, the optimal soft fusion weights are ω opt,DCM = ω ∗opt,DCM /ω ∗opt,DCM 2 . With a desired probability of false alarm PFA,DES , the detection performance in the DCM framework is given by ⎛ ⎞ T −1 T −1 θ Σ θ − θ Σ θ η 0 0 (DCM) ⎠, = Q⎝ (22) PD θ T Σ−2 Σ θ 1 0
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(DCM)
where PD is maximized in the sense that the distance between the two PDFs of Zc under hypotheses H0 and H1 is maximized to the most by ω opt,DC . Similarly to the previous case, the statistically averaged PD is ⎛ ⎞ T −1 T −1 ξ Σ ξ − ξ Σ ξ η 0 0 (DCM) ⎠pθ (ξ)dξ, = Q⎝ P¯D T −2 Θ2 ⊂RK + ξ Σ0 Σ1 ξ (23) where Θ2 is a subset of RK + which contains all θ vectors leading to the H1 decision. C. Linear Quadratic Optimization Based Fusion Scheme
IV. I MPLEMENTATION OF P ROPOSED F USION S CHEMES
In this subsection, we derive the optimal weight vectors in terms of linear quadratic optimization. The sensing problem is formulated as a Rayleigh quotient and the corresponding optimal weight vector is defined as the one maximizing the Rayleigh quotient. The generalized Rayleigh quotient of the global test statistic Zc is defined by Γ(ω) R(Σ1 , Σ0 ; ω) =
ω T Σ1 ω , ω T Σ0 ω
(24)
where Γ(ω) bears resemblance to the SNR of Zc . Its value is 1 under H0 whereas under H1 , the real symmetric matrix Σ1 contains the PU signal component and the associated Γ(ω) is proportional to the real SNR of the PU signal. The optimal weight vector ω opt,Ray is defined as the one that maximizes Γ(ω) ω opt,Ray = argmax Γ(ω).
(25)
ω
The generalized Rayleigh quotient R(Σ1 , Σ0 ; ω) in (24) can be reduced to the Rayleigh quotient R(D, CTω) through the transformation D = C−1 Σ1 C−T [15], where C is the Cholesky decomposition of matrix Σ0 . It is well known that the Cholesky decomposition of Σ0 is a decomposition of the symmetric, positive-definite matrix Σ0 into a lower triangular matrix and the transpose of the lower triangular matrix, Σ0 = CCT .
(26)
The proposed optimal soft fusion schemes require partial a priori knowledge of the channel, which has to be estimated for setting the derived weight vectors. A. Estimation of PU Signal Energy Vector The optimal weighting vectors in (14) and (21) are mainly determined by the signal energy quantities EP U |hi |2 , under ¯ i , the relay power Ei , the assumption that the channel gains h the noise variances σi2 and σF2 C are readily available through estimation at the FC before the sensing operation begins. We utilize two matrices to keep records of PU’s behaviors, in which the current sensing data Zl is categorized and stored in either a Presence or an Absence matrix for future reference, according to the current global decision. In other words, if it is decided that the current data Zl contain the PU signal energies, it is stored in a K-by-L Presence matrix Z(P ) in a first-in-first-out (FIFO) manner; otherwise, it is stored in a K-by-L Absence matrix Z(A) and meanwhile a zero column vector is stacked into Z(P ) since the fusion center has decided that no PU signal energy is contained in Zl . The estimates of K {θi }K i=1 and {N0,i }i=1 for the current statistic are calculated via simple arithmetic operations: ⎧ l−1 ⎪ (P ) (A) ⎪ ⎪ θi,l = ζ1 ρl |Zi,j − Zi,j |, ⎨ j=l−L (30) l−1 ⎪ (A) ⎪ 0,i,l = 1 ⎪ ρ Z , N j i,j ⎩ Ts W j=l−L
Now we obtain the Rayleigh quotient as Γ(ω) = R(Σ1 , Σ0 ; ω) = R(D, CTω) =
vector opt,Ray as the unit eigenvector corresponding to the largest eigenvalue of D, i.e. eigmax (D). It is now derived that the optimal combining weights in the LQO framework, opt,Ray , is only corresponding to the largest eigenvalue of D. Furthermore, it is worth noting that as a blind cooperative sensing scheme, the LQO scheme offers robustness against the dynamically changing environment because it merely operates on the received signal covariance matrices. The threshold-setting should be practically considered in advance, since for a desired false alarm probability, a calibration of the global threshold is indispensable.
D , T T
(27)
where = CT ω. Notice that R(A, x ) = R(A, x), for x = cx, ∀c = 0, therefore we will solve for with a unit norm ||||2 = 1, max T D, (28) s.t. T = 1. By using the scalar Lagrange multiplier λLag ∈ R, we obtain L() = T D + λLag ( T − 1),
(29)
where taking the derivative of L() with respect to , i.e. ∂L()/∂ = 0, we can easily find the optimal weight
where ζ is the common coefficient of {θi }K i=1 which can be practically set to 1, l is the time index of the current sensing data, ρj is the forgetting factor which is usually a scalar, and L is the reference matrix depth, which should be carefully tuned on the basis of estimation or prediction of the channel gain varying velocity. Additionally, to be more compact for implementation, the proposed estimations in (30) can be implemented as: (A) 1 (P ) θi,l ≈ L−1 L θi,l−2 + L Zi,l−1 − Zi,l−1 , (31) 0,i,l−2 + 1 Z (A) , 0,i,l ≈ L−1 N N L
where forgetting factors 1/L independent of l.
L
l−1 {ρj }j=l−L
i,l−1
are all set to a constant
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B. Estimation of Received Signal Covariance Matrix The LQO based optimal fusion scheme needs the maximum eigenvectors of the covariance matrices Σ1 and Σ0 . We now develop a method to estimate the covariance matrices using the received signal only. Considering the current estimate of (l) , the covariance matrix can be estimated Σ1 is denoted by Σ 1 on-line as (l) = Σ 1
l−2 j=l−L
(j) + (1 − ρj Σ 1
l−1
probability. Although the target optimality function of the DCM scheme is only part of the N-P’s, it has a nearly undiscernible performance difference compared to the N-P scheme, because in the low SNR regime, the difference between their optimality functions is too small. As for the LQO scheme, it is an eigenvector based blind sensing solution, which requires less a priori knowledge than the N-P and DCM schemes, and therefore has inferior performance.
ρj )Z(l−1) ZT(l−1) ,
j=l−L
L − 1 (l−2) 1 + Z(l−1) ZT(l−1) (with ρj = 1/L), ≈ Σ1 L L (32) where Z(l) are the sensing statistics at time index l, which comes from the sensing observations stored in the PU signal reference matrix Z(P ) . (l) can be obtained in the similar way The estimate Σ 0 described above in (32). With the estimated signal covariance matrices, we can attain the needed eigenvectors. Additionally, the method of categorizing and storing the sensing data Z(l) based on the global decisions, is somewhat different to the one described in the previous subsection. If the current decision is drawn as H0 , the current sensing observation Z(l) will be inserted into both of Z(P ) and Z(A) ; otherwise, the current observation will be only stored in Z(P ) . In this way, (l) would actually be if there is no PU signal, the estimate Σ 1 (l) , which does not corrupt close or identical to the estimate Σ 0 the detection since the contribution of its eigenvectors to the test statistic would be low.
Fig. 1.
ROC performance of the proposed optimal fusion schemes.
V. S IMULATIONS AND D ISCUSSIONS In this section, the proposed soft fusion schemes are evaluated via simulations. The basic parameters are fixed and set as Ts W = 400, L = 12 and K = 8. Each simulation consisted of 105 iterations. The channel gains between each SU and the target PU are experiencing IID Rayleigh fading. For simplicity, we assume that the PU signal power and the channel gains have constant values for each sensing interval Ts . We further assume the channel gains vary from (observation) period to period while their PDFs are determined by the fading characteristic of the channels. Hereafter, we suppose that the noise variances 2 K }i=1 in (6) are distributed around an average level δ02 {δ0,i with a deviation d. The noise variance deviation d is normally distributed as N (0, ρd δ02 ). ρd indicates the location difference factor in the network area, which is jointly determined by the ¯ i , and σ 2 (ρd = 0.2 in simulations). locations of the SUs, Ei , h i Fig. 1 gives the receiver operating characteristic (ROC) curves of the prosed optimal fusion schemes in IID rayleigh fading environment. In order to demonstrate the superior performance of the proposed optimal fusion schemes, we provide the performance of conventional MRC scheme as well. The N-P scheme’s superiority lies in that its target optimality function is the complete one taken from the mathematical expression of the PU signal detection
Fig. 2. ROC Performance of the LQO scheme with different number of cooperative SUs (SNR = -15dB).
In order to investigate the impact of user number on the collaboration gain in sensing, we give the ROC curves in Fig. 2, which demonstrate that the detection performance of the LQO scheme is enhanced with increase in the number of users. The certifies that the performance deterioration in the absence of a priori knowledge of the channels is effectively alleviated.
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VI. C ONCLUSIONS In this paper, optimal linear soft fusion schemes for cooperative spectrum sensing in CR networks are investigated and analyzed. Based on Neyman-Pearson criterion, DCM criterion, and linear quadratic optimization, three optimal fusion schemes are proposed to combine the sensing observations collected from the cooperative SUs. Analysis and simulations illustrate that the proposed optimal soft fusion schemes yield significant improvements of the spectrum sensing performance in the CR network.
∂dDC (ω)/∂ω = 0, where from
(38)
∂ ω T θ/ ω T Σ0 ω
∂dDC (ω) = ∂ω ∂ω θ ω T Σ0 ω − (ω T θ)(Σ0 ω)/ ω T Σ0 ω (39) = ω T Σ0 ω = 0, we reach the equation
A PPENDIX
θω T Σ0 ω − (ω T θ)(Σ0 ω) = 0.
A. N-P Optimal Weights The optimal fusion weights for the N-P scheme are obtained by solving the equation ∂(ω)/∂ω = ∂[1 (ω)/2 (ω)]/∂ω ∂1 (ω) ∂2 (ω) (ω) − (ω) 2 1 ∂ω ∂ω (33) = 2 2 (ω) = 0, where functions 1 (ω) = η ω T Σ0 ω − ω T θ and 2 (ω) = ω T Σ1 ω. Upon substituting the derivatives in (33) with Σ0 ω ∂1 (ω) = η − θ, ∂ω ω T Σ0 ω Σ1 ω ∂2 (ω) = , ∂ω ω T Σ1 ω
(34)
we obtain ηΣ0 ω ω T Σ1 ω −θ ω T Σ0 ω Σ1 ω + (ω T θ − η ω T Σ0 ω) = 0. ω T Σ1 ω
(35)
After some mathematical manipulations, the equation in (35) is again given as (36) (ηα12 Σ0 − ηα02 Σ1 )ω + α0 (ω T θ)Σ1 ω = α0 α12 θ, where α0 = ω T Σ0 ω, α1 = ω T Σ1 ω, and the solution to the above equation is α0 −1 −1 Σ (IK + αNP Σ−1 θ 0 Σ1 ) ηα1 0 −1 −1 θ = βNP Σ−1 0 (IK + αNP Σ0 Σ1 )
ω ∗opt,NP =
B. DCM Optimal Weights The optimal fusion weights for the DCM scheme are obtained according to the Schwarz’s inequality in (20) or by solving the equation
(37)
where αNP = (α0 ω T θ − ηα02 )/(ηα12 ), (αNP > 0) and it is still a function of ω, which requires numerical search for its optimal value leading to the minimum (ω) in (15).
(40)
Since ω T Σ0 ω and ω T θ are scalars, we can obtain ω ∗opt,DCM =
ω T Σ0 ω −1 Σ0 θ = βDC Σ−1 0 θ. ωT θ
(41)
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