Cooperative Spectrum Sensing with Per-User Power ... - IEEE Xplore
2012 IEEE 23rd International Symposium on Personal, Indoor and Mobile Radio Communications - (PIMRC)
Cooperative Spectrum Sensing with Per-User Power Constraints Vahid Jamali† , Bijan Golkar‡ , Soheil Salari† , Mahmoud Ahmadian†, and Elvino S. Sousa‡ Department of Electrical and Computer Engineering K. N. Toosi University of Technology, Tehran, Iran ‡ University of Toronto, Toronto, Canada v jamali [email protected], {bijan,sousa}@comm.utoronto.ca, {salari,mahmoud}@eetd.kntu.ac.ir †
Abstract—In collaborative spectrum sensing, the presence of the primary user is detected at a central entity, known as the fusion center. This center collects the information from the secondary users and decides on the occupancy of the desired frequency band. In the conventional strategy, the secondary users transmit their initial observations toward the fusion center with their maximum transmit powers. In this paper, however, we consider the problem of beamforming among the secondary users with individual power constraints. Correlated shadow fading has been considered in the channel gains between the primary transmitter and the secondary users as well as the channel gains between the secondary users and the fusion center. We consider the problem of maximizing the probability of detection for a required probability of false alarm. Most previous works have considered the total power constraint while in practical scenarios, each secondary user has a limited battery lifetime. An algorithm is developed which efficiently solves the problem via second order cone programming (SOCP) in an iterative manner. An approximation of the original problem is studied which reduces the computational complexity of the iterative procedure. The Monte Carlo simulations confirm the effectiveness of the spectrum sensing framework compared to the conventional strategy.
I. I NTRODUCTION Cognitive radio has gained considerable research interest due to its high potential to enhance the spectral efficiency [1]. It is well known that spectrum sensing is a fundamental problem for realization of cognitive radio networks especially under shadow fading. Among different spectrum sensing techniques, energy detection is the most popular method addressed in the literature due to its low computational complexity and not requiring a priori knowledge of the primary signal [2], [3]. The detection performance of spectrum sensing techniques with a single secondary user significantly degrades with poor channel conditions between the target-under-detection and the secondary user. In order to improve the reliability of spectrum sensing, multiple secondary users can collaborate to conduct spectrum sensing and take advantage of spatial diversity. It has been shown that cooperative spectrum sensing techniques can alleviate the problem of corrupted detection, especially when the collaborating secondary users experience independent channel conditions from the target [3], [4]. Generally, each cooperative spectrum sensing process consists of two successive phases: 1) The sensing phase in which all secondary users attempt to observe the primary
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band; 2) The reporting phase during which they transmit their observations to a fusion center, where a final decision about the existence of the primary user is made. Several studies have shown that the achievable performance of cooperative spectrum sensing methods depends on the assumed fading channel model [3], [5]–[7]. Specifically in [6], it is argued that shadow-fading is the major aspect to be considered in the sensing channel (channel between primary transmitter and secondary user) when detecting digital TV signals, while multipath fading is expected to have a less important impact. The channel impairment on the reporting channel (channel between a secondary user and the fusion center) also affects the overall performance of the sensing procedure; however, most previous works have neglected this issue by assuming ideal or just noisy reporting channels. To study the imperfect effect of the reporting phase, Zhang et al. in [8], have proposed a relay diversity technique with an algebraic coding approach to improve the performance of cooperative spectrum sensing when some secondary users cannot report their decisions to the fusion center due to heavy shadowing. In addition, a clustering technique is utilized in [9] where in each cluster, all the secondary users forward there observations to one of the nodes in the cluster called cluster head which acquires the best reporting channel. Then the cluster head transmits the observation to the fusion center. Recently, Xiong et al. in [10] have proposed a beamforming technique among secondary users who communicate the locally measured signal-to-noise ratios (SNRs) to the fusion center by the assumption of total power limitation. The problem of beamforming among the different nodes has been investigated deeply in the literature [11], [12]; however, only [10] utilized it in the cooperative sensing framework. In this paper, we propose a novel cooperative spectrum sensing framework assuming correlated shadow fading in both the sensing and the reporting channels. The secondary terminals are viewed as a distributed multi-antenna transmitter, which reports the local SNR levels to a fusion center. Unlike [10], where a total power constraint for the secondary users was assumed, we consider individual power constraints. An efficient algorithm with the guaranteed convergence to an -optimal solution is proposed which solves the problem via second order cone programming (SOCP) in an iterative
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manner. To further reduce the computational complexity of the iterative procedure, we approximate the optimization problem with a single SOCP problem which yields a closed form solution for the beamforming coefficients. The remainder of the paper is organized as follows. In section II, we describe the system model and problem formulation. The first part of section III is devoted to the global probability of detection maximization under individual secondary user power constraints. In the second part, we present the approximation method. Simulation results are given in section IV. Finally the main results are summarized in section V. Notation: Throughout the paper, we use uppercase boldface letters to represent matrices and lowercase bold letters to denote vectors. In addition, (.)T and (.)H stand for transpose and Hermitian transpose of a vector, respectively. λmax (A) and υmax (A) represent principal eigenvalue and eigenvector of matrix A, respectively. Also E{.} stands for statistical expectation and diag(a) denotes a diagonal matrix with diagonal elements given by vector a. we use . to show Euclidean norm of a vector and (.), (.) and |.| to denote the real part, imaginary part and amplitude of a complex number, respectively and sign(.) stands for the sign of a real number. N (μ, σ 2 ) represents a normal distribution with mean μ and variance σ 2 . Finally, 0 and 1 are column vectors with elements all equal to zero and one, respectively, and I is an identity matrix. II. S YSTEM M ODEL Let us consider a cognitive radio network with M secondary users. Fig. 1 illustrates the deployment scenario. The users are randomly deployed in a circle with radius Rs at a distance R from the primary transmitter. The secondary users located within the primary coverage range (Rp ) or in the guard range (Rg ) potentially interfere with the primary transmission [13]. Therefore, the secondary users are allowed to access the primary band only when the distance R is greater than the interference range of the primary network defined as Ri = Rp + Rg . The size of the secondary network is assumed to be small compared to R, i.e. Rs R, such that all secondary users experience the same pathloss from the primary transmitter. A two-phase detection framework is proposed: During the sensing phase, the secondary users observe the primary band and estimate their local SNRs. In the reporting phase the secondary users forward the estimated SNRs to a central entity, referred to as the fusion center, where the final decision about the primary band is obtained. The fusion center can either be one of the secondary users or a separate control node in the network. Let us assume that the primary transmitter broadcasts Pp s, where s is the normalized information symbol, i.e. E{|s|2 } = 1, and Pp denotes the power of the primary transmitter. The received signal at the i-th secondary user can be written as y = Pp αβi hi s + ni (1)
SU 2
PT Rg
Rs
SU M
Rp
SU 1
R l FC
PR
PT: Primary Transmitter, PR: Primary Receiver, SUi : i’th Secondary User, FC: Fusion Center R p : Primary Range, R s : Secondary Range, Rg : Guard Range
Fig. 1.
The deployment scenario
where α, βi and hi denote the pathloss, shadowing and multipath fading components of the channel between the primary transmitter and the i-th secondary user (i = 1, 2, · · · , M ), respectively. The pathloss is proportional to the inverse power of the distance from the primary transmitter, i.e. α ∝ 1/Rκ where κ is the pathloss exponent. The shadowing is modeled statistically with a correlated multivariate lognormal distribution. Assuming βi = 100.1ωi , we have ω ∼ N (0, Σ) where ω = [ω1 , ω2 , . . . , ωM ]T and Σ is the covariance matrix of ω. Without loss of generality, the correlation properties of ω can be modeled as a decreasing exponential function [13], i.e. Σi,j = σ 2 ρdij ,
i, j = 1, . . . , M
where Σi,j is the (i, j)-th entry of the covariance matrix, σ 2 denotes the variance of {ωi }M i=1 and dij is the distance between the i-th and the j-th secondary users. Parameter ρ is the correlation coefficient between the secondary users separated by unit distance and related to the correlation distance Dc by the relation ρ = e−1/Dc [14]. Moreover, the multipath fading is modeled as i.i.d zero-mean and unit variance complex Gaussian random variable, i.e. hi ∼ CN (0, 1). Finally, ni is the zero-mean additive white Gaussian noise (AWGN) at the i-th secondary user with variance σn2 . Let xi denote the average SNR level of the primary transmission at the i-th secondary user on a logarithmic scale. Then, xi can be written as xi = ωi − 10κlog10 (R) + K
(2)
P 10log10 ( σ2p ). n
where K = In the assumed network model, the secondary users are viewed as a distributed transmitter with M antenna elements. This multi-antenna structure forwards the estimated SNR levels in the logarithmic scale multiplied by the beamforming vector f = [f1 , f2 , . . . , fM ]T to the fusion center. The received signal at the fusion center can be written as
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y=
M α βi hi fi xi + n = f H Hx + n i=1
(3)
h ]T ), where H = diag([ α β1 h1 , α β2 h2 , . . . , α βM M T and x = [x1 , x2 , . . . , xM ] . The pathloss (α ), shadowing (βi ) and multipath fading (hi ) components of the reporting channels are modeled similar to that of the sensing channels. A quasi-static fading condition is assumed for the reporting channel gains such that the channel realizations stay fixed for the duration of each data transmission to the fusion center. Moreover, n denotes the noise component at the fusion center and is modeled as a zero-mean additive white Gaussian noise with variance σn2 . A hypothesis testing problem is formulated by choosing between the white space hypothesis H0 which by definition occurs when R > Ri and the occupied space hypothesis H1 when R ≤ Ri . By modifying the definition of hypothesis H1 to the worst case scenario, namely R = Ri , a simple binary Neyman-Pearson hypothesis testing problem is formed [3] ¯ f H HΣHH f + σ 2 ) y ∼ N (f H Hμ(R), n
(4)
¯ + K, with ¯ = μ(R)1 ¯ and μ(R) ¯ = −10κlog (R) where μ(R) 10 ¯ = Ri + l , l > 0 (white space) H0 : R (5) ¯ = Ri (occupied space) H1 : R
III. P ROBABILITY OF D ETECTION M AXIMIZATION UNDER I NDIVIDUAL P OWER C ONSTRAINT Our aim is to maximize the probability of detection, Pd , while the power of each secondary user is limited to its power budget. Mathematically, we want to solve the following optimization problem max
H1
(6)
H0
The performance of the spectrum sensing procedure can be evaluated by the probability of detection (reliability) and the probability of false alarm (efficiency) formulated as ⎞ ⎛ H τ − f Hμ(R|H1 ) ⎠ (7) Pd = P (y > τ |H1 ) = Q ⎝ f H HΣHH f + σn2 ⎞ ⎛ H τ − f Hμ(R|H ) 0 ⎠ (8) Pf = P (y > τ |H0 ) = Q ⎝ H H 2 f HΣH f + σn where Q(.) is the complementary cumulative distribution function, which calculates the tail probability of a zero mean unit variance Gaussian variable. The sensing performance of the secondary network depends largely on the beamforming vector, f , and the test threshold, τ . Assuming a target probability of false alarm Pf = ψ, the test threshold can be calculated as ¯ 0 ) + Q−1 (ψ) f H HΣHH f + σn2 (9) τ = fH Hμ(R|H Substituting (9) in (7), we get ⎛
where Pimax is the maximum allowable transmit power of the i-th secondary user and Pi can be written as Pi = E{|fi xi |2 } = ξ|fi |2 = fH Ci f, i = 1, 2, . . . , M (12) where Ci = ξdiag(ii ) and ii is the ith column of the identity 2 2 matrix. ξ = E{x2i } = e2c σ (π0 e2cμ0 + π1 e2cμ1 ) and π1 = P (H1 ), π0 = P (H0 ) denote the probability that secondary users are located within or beyond the interference range of the primary network, and c = (ln10)/10 [10]. Considering that Q(.) is a monotonically decreasing function, the optimization problem in (11) is equivalent to fH Hδδ H HH f f f HΣHH f + σn2 subject to fH Ci f ≤ Pimax , i = 1, 2, . . . , M max
¯ 1 ) − μ(R|H ¯ 0 ). where δ = μ(R|H
(13)
A. Iterative Approach We begin by rewriting (13) into the following equivalent form (14) max t f
subject to
fH Hδδ H HH f ≥t fH HΣHH f + σn2 fH Ci f ≤ Pimax , i = 1, 2, . . . , M
Since Σ is a symmetric and positive definite matrix, using Cholesky decomposition Σ = LLH , leads to fH HΣHH f = LH HH f2 . Then, by applying the following definitions
HL 0 Hδ f ˜ A= ,b= ,f= 0 1 0T σn
⎞
max ˜ f
(10)
H
It can be seen that the optimization problem in (13) is a nonconvex problem. Therefore, in the sequel of this section, we have proposed two different approaches to find the beamforming vector. In what follows, we first formulate the problem as a standard SOCP and propose an iterative algorithm to find the solution. Subsequently, an approximate solution is presented with considerably lower complexity.
we obtain
f H Hδ ⎠ Pd = Q ⎝Q−1 (ψ) − H H 2 f HΣH f + σn
(11)
subject to Pi ≤ Pimax , i = 1, 2, . . . , M
By considering the optimal likelihood ratio test with a test threshold τ as the decision rule, we have y ≷ τ
Pd
f
t
H H subject to t˜f A2 ≤ |˜f b|2 H ˜f C˜i 2 ≤ Pimax , i = 1, 2, . . . , M √ ˜ i = ξdiag(ii ). where C
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(15)
Note that for any value of t, the above problem is convex in ˜f and can be solved using interior-point methods. However, the joint optimization over t and ˜f is a quasi-convex optimization problem which can be solved using the bisection algorithm [16]. We next show how for a specific value of t, (15) can be expressed as a SOCP. An interesting observation here is that if ˜f satisfies the constraints in (15), so does ejθ ˜f for any arbitrary phase shift θ, and the value of the objective function is maintained. Thus, H without loss of generality we can assume that ˜f b is a real number [17]. Using this assumption, (15) can be rewritten as the following feasibility problem ˜f (16) √ H H ˜ ˜ subject to tf A ≤ |(f b)| H (˜f b) = 0 H ˜f C˜i ≤ Pimax , i = 1, 2, . . . , M
Algorithm 1 The Proposed Iterative Approach 1: Initialize: 2: i ← 0 max T 3: fl ← √1ξ P1max , P2max , . . . , PM , then set
4:
i=1
maximum Pd achieved by solving individual power constraints {Pimax }M i=1 . This is due to the fact that the total power constraint results in a larger feasibility area than the individual fH Hδδ H HH f power constraints. Therefore, we choose u(0) = fHuHΣHH f +σu2 u u n , where fu is the optimal solution obtained in [10] with the aforementioned total power constraint. Then we can initialize (0) (0) and follow the bisection the value of t to its midpoint u +l 2 procedure outlined in Algorithm 1 to find the beamforming coefficients. B. Approximation Approach As explained in the previous subsection, the -optimal solution requires an iterative procedure in which the feasibility of the problem should be checked in each iteration. To avoid such a computational complexity, we perform an approximation and then maximize the resulted function instead of the original cost function. The optimization problem in (11) can be reformulated as max f
fH Hδδ H HH f fH (HΣHH +
2 σn f2 I)f
subject to fH Ci f ≤ Pimax , i = 1, 2, . . . , M
(17)
Solve problem (6) in [10] with PTmax ← set fu ← fopt , and u(0) ←
5:
Set t(0) ←
M i=1
Pimax , then
H H fH u Hδδ H fu H 2 fH u HΣH fu +σn
u(0) +l(0) 2 (i)
T and ˜f ← fTl , 1
while u(i) − l > do if If (16) is feasible for t(i) then ˜f(i) ← f obtained by the feasibility check in the 8: previous step 9: Set l(i+1) ← t(i) , u(i+1) ← u(i) and t(i+1) ← 6: 7:
find
Now, for any specific value of t, (16) is in the form of a standard SOCP [17]. Let us assume that the optimal value of t lies in the interval [l(0) , u(0) ]. We expect that the optimal beamforming results in a higher performance that the conventional strategy in which each secondary user transmits with its maximum power. Therefore, by satisfying the individual power con2 straints with equality ξ|fi | = Pimax , i = 1, 2, . . . , M , the ¯ HH ¯ H fl fH Hδδ , lower bound can be initialized as l(0) = fHl HΣ 2 ¯ H ¯ H fl +σn max max max lT 1 where fl = √ξ P1 , P2 , . . . , PM . Moreover, the maximum achievable Pd with the total power constraint M PTmax = Pimax is larger than or at least equal to the
¯ H ¯H fH l Hδδ H fl ¯ H ¯ H fl +σ2 HΣ fH n l
l(0) ←
u(i+1) +l(i+1) 2
10: 11: 12: 13: 14: 15:
else (i) Set ˜f ← f(i−1) , l(i+1) ← l(i) , u(i+1) ← t(i) and (i+1) +l(i+1) t(i+1) ← u 2 end if i←i+1 end while √ (i) Set ˜fopt ← ˜f and Pdmax ← Q(Q−1 (ψ) − t(i) )
The maximum value of f occurs when all the inequality constraints are satisfied with equality,then we have: f = max PTmax max = √1 P1 + P2max + · · · + PM ξ . Therefore, by ξ max PT substituting f = in the denominator, we get ξ max f
fH Hδδ H HH f fH (HΣHH +
2ξ σn PTmax I)f
(18)
subject to fH Ci f ≤ Pimax , i = 1, 2, . . . , M It is worth mentioning that the difference between the upper bound and the original cost function tends to zero as P max → ∞. As discussed before, the above objective function is upperbounded by the Raleigh Ritz inequality and is maximized when fapp = ζz (19) σ2 ξ
n I)−1 Hδδ H HH }. Now ζ where z = υmax {(HΣHH + P max T should be selected such that none of the inequality constraints are violated or i = 1, 2, . . . , M (20) ζ 2 ξ|zi |2 ≤ Pimax ,
where zi is the ith elements of the vector z. To satisfy all the above inequality constraints, we have Pjmax Pimax , j = arg min (21) ζ= 2 2 i ξ|zj | |zi | In the conventional strategy, we assume that all secondary users transmit with their maximum power. Therefore, the
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0
0
10
10
Probability of Miss−detection
Probability of Miss−detection
−1
−1
10
−2
10
10
−2
10
Iterative Method, η= 0.5 Approximation Method, η= 0.5 Conventional Method, η= 0.5 Iterative Method, η= 0.4 Approximation Method, η= 0.4 Conventional Method, η= 0.4 Iterative Method, η= 0.3 Approximation Method, η= 0.3 Conventional Method, η= 0.3
−3
10 Optimal Beamforming,Total Power Constraint Beamforming with Iterative Approach, Individual Power Constraint Approximation Method, Individual Power Constraint Conventinal Method
−4
−3
10
10 0
5 10 15 20 25 30 35 40 45 Sum of Individual Powers Cunsumed by the Secondary Users(dBm)
50
min ) versus Fig. 2. Minimum achievable probability of miss-detection (Pmd the sum of individual transmit power consumed by the secondary users for M = 5 and η = 0.5.
0
5 10 15 20 25 30 35 40 45 Sum of Individual Powers Consumed by the Secondary Users(dBm)
50
min ) versus Fig. 3. Minimum achievable probability of miss-detection (Pmd the sum of individual transmit power consumed by the secondary users for M = 5 and different values of η.
0
10
(22)
−1
Probability of Miss−detection
amplitude of their coefficients are Pimax , i = 1, 2, . . . , M |fi | = ξ
Then under individual power constraints, the probability of detection for the conventional strategy is given by ⎞ ⎛ H H HH f Hδδ ¯ ¯ f ⎠ Pdcov = Q ⎝Q−1 (ψ) − H (23) ¯ H ¯ H f + σn2 f HΣ
10
−2
10
Iterative Method, M = 5 Approximation Method, M = 5 Conventional Method, M = 5 Iterative Method, M = 10 Approximation Method, M = 10 Conventional Method, M = 10 Iterative Method, M = 15 Approximation Method, M = 15 Conventional Method, M = 15
−3
10
IV. S IMULATION R ESULTS
−4
10
The simulation results are based on 1000 independent realizations of the secondary users locations. For each realization, 10000 independent channel realizations are generated. We consider the bandwidth of the primary system to be 6M Hz (Typical TV band). Rp is assumed to be 30km and the secondary users report their observations to the fusion center over the reporting channel with a bandwidth of 240kHz. In addition, the following system parameters are assumed: π0 = π1 = 0.5, σ = 2.3, Rg = Rs = 5km, l = 15km, K = 180, κ = 4, ψ = 0.01 and a background noise power spectral density of −170dBm/Hz (The simulation parameters are adopted from [3], [20] and [21]). In Fig. 2, the minimum achievable global probability of min miss-detection (Pmd = 1 − Pdmax ) versus sum of individual transmit power consumed by the secondary users is illustrated when the sum of allowable individual power, i.e. M ( m=1 Pimax ), changes from 0 to 50dBm. We consider five secondary users with the following power constraints: P1 /3 = P2 /2 = P3 /2 = P4 = P5 denoted by [32211]. It is seen that as the overall individual power constraints of the secondary users increase, a better performance is achievable. However, the curves saturate as the power constraints get weaker. The proposed -optimal beamforming strategy
0
5 10 15 20 25 30 35 40 45 Sum of Individual Power Consumed by the Secondary Users(dBm)
50
min ) versus Fig. 4. Minimum achievable probability of miss-detection (Pmd the sum of individual transmit power consumed by the secondary users for η = 0.5 and different number of secondary users.
outperforms the case where the secondary users utilize all their allowable power. This is due the fact that in conventional strategy, there is no difference in fusing the observations while in our proposed method we differentiate between the observations by allocating the secondary users powers. As expected, the performance is indeed slightly worse than that of the same problem introduced in [10] with total power constraint but as a cost of this little degradation, we have considered a more practical scenario and we could assure that there is no violation in even the individual power constraints. In addition, the lower complexity of the approximation method, when compared to the -optimal beamforming strategy, has compromised the performance. c Fig. 3 is depicted for different values of η = D Rs with M = 5. As expected, a lower value of η results in a better performance due to the higher independence of the observations
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R EFERENCES
Sum of Individual Powers Consumed by the Secondary Users(dBm)
60 Iterative Method, M = 5 Approximation Method, M = 5 Conventional Method, M = 5 Iterative Method, M = 10 Approximation Method, M = 10 Conventional Method, M = 10 Iterative Method, M = 15 Approximation Method, M = 15 Conventional Method, M = 15
50
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−10
0
5
10
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M
Sum of Individual Power Constraints(dBm)
40
45
50
Pimax
m=1
Fig. 5. Total transmit power consumed by the secondary users versus the sum max ) of allowable individual transmit power of secondary users ( M m=1 Pi for η = 0.5 and different values of M .
at the secondary users. The same figure is plotted for η = 0.5 and different number of secondary users in Fig. 4. The power constraints M = 5 : [32211], M = 10 : [3332221111] and M = 15 : [333332222211111] are considered. It should be noted that a lower probability of miss-detection is achieved as the number of secondary users increases. In Fig. 5, the sum of the consumed power by the secondary users versus the sum of the allowable transmit power is depicted. In this figure, we have shown that our proposed beamforming strategy saves total transmit power of the secondary user compared to the case where secondary users utilize all their allowable power to forward their observations, thus we can conclude that power constraints do not hold at equality. In addition, the consumed power of the secondary users decreases as M increases. V. C ONCLUSION In this paper, we present a cooperative spectrum sensing strategy with per-user power constraints. A distributed beamforming method is proposed based on which the local SNR estimates of the secondary users are transmitted to a central entity, known as the fusion center. A hypothesis testing problem is solved at the fusion center which decides on the occupancy of the primary frequency band. The problem is formulated as an optimization problem. An iterative procedure is presented which achieves an -optimal solution. Subsequently, a suboptimal solution with lower complexity is proposed. Correlated shadow fading is considered in the channel gains between the primary transmitter and the secondary users as well as the channel gains between the secondary users and the fusion center. The numerical results show that the proposed beamforming strategy outperforms the conventional strategy. In addition, the approximation method performs well given the lower computational complexity of the method.
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Cooperative Spectrum Sensing with Per-User Power Constraints Vahid Jamali† , Bijan Golkar‡ , Soheil Salari† , Mahmoud Ahmadian†, and Elvino S. Sousa‡ Department of Electrical and Computer Engineering K. N. Toosi University of Technology, Tehran, Iran ‡ University of Toronto, Toronto, Canada v jamali [email protected], {bijan,sousa}@comm.utoronto.ca, {salari,mahmoud}@eetd.kntu.ac.ir †
Abstract—In collaborative spectrum sensing, the presence of the primary user is detected at a central entity, known as the fusion center. This center collects the information from the secondary users and decides on the occupancy of the desired frequency band. In the conventional strategy, the secondary users transmit their initial observations toward the fusion center with their maximum transmit powers. In this paper, however, we consider the problem of beamforming among the secondary users with individual power constraints. Correlated shadow fading has been considered in the channel gains between the primary transmitter and the secondary users as well as the channel gains between the secondary users and the fusion center. We consider the problem of maximizing the probability of detection for a required probability of false alarm. Most previous works have considered the total power constraint while in practical scenarios, each secondary user has a limited battery lifetime. An algorithm is developed which efficiently solves the problem via second order cone programming (SOCP) in an iterative manner. An approximation of the original problem is studied which reduces the computational complexity of the iterative procedure. The Monte Carlo simulations confirm the effectiveness of the spectrum sensing framework compared to the conventional strategy.
I. I NTRODUCTION Cognitive radio has gained considerable research interest due to its high potential to enhance the spectral efficiency [1]. It is well known that spectrum sensing is a fundamental problem for realization of cognitive radio networks especially under shadow fading. Among different spectrum sensing techniques, energy detection is the most popular method addressed in the literature due to its low computational complexity and not requiring a priori knowledge of the primary signal [2], [3]. The detection performance of spectrum sensing techniques with a single secondary user significantly degrades with poor channel conditions between the target-under-detection and the secondary user. In order to improve the reliability of spectrum sensing, multiple secondary users can collaborate to conduct spectrum sensing and take advantage of spatial diversity. It has been shown that cooperative spectrum sensing techniques can alleviate the problem of corrupted detection, especially when the collaborating secondary users experience independent channel conditions from the target [3], [4]. Generally, each cooperative spectrum sensing process consists of two successive phases: 1) The sensing phase in which all secondary users attempt to observe the primary
978-1-4673-2569-1/12/$31.00 ©2012 IEEE
band; 2) The reporting phase during which they transmit their observations to a fusion center, where a final decision about the existence of the primary user is made. Several studies have shown that the achievable performance of cooperative spectrum sensing methods depends on the assumed fading channel model [3], [5]–[7]. Specifically in [6], it is argued that shadow-fading is the major aspect to be considered in the sensing channel (channel between primary transmitter and secondary user) when detecting digital TV signals, while multipath fading is expected to have a less important impact. The channel impairment on the reporting channel (channel between a secondary user and the fusion center) also affects the overall performance of the sensing procedure; however, most previous works have neglected this issue by assuming ideal or just noisy reporting channels. To study the imperfect effect of the reporting phase, Zhang et al. in [8], have proposed a relay diversity technique with an algebraic coding approach to improve the performance of cooperative spectrum sensing when some secondary users cannot report their decisions to the fusion center due to heavy shadowing. In addition, a clustering technique is utilized in [9] where in each cluster, all the secondary users forward there observations to one of the nodes in the cluster called cluster head which acquires the best reporting channel. Then the cluster head transmits the observation to the fusion center. Recently, Xiong et al. in [10] have proposed a beamforming technique among secondary users who communicate the locally measured signal-to-noise ratios (SNRs) to the fusion center by the assumption of total power limitation. The problem of beamforming among the different nodes has been investigated deeply in the literature [11], [12]; however, only [10] utilized it in the cooperative sensing framework. In this paper, we propose a novel cooperative spectrum sensing framework assuming correlated shadow fading in both the sensing and the reporting channels. The secondary terminals are viewed as a distributed multi-antenna transmitter, which reports the local SNR levels to a fusion center. Unlike [10], where a total power constraint for the secondary users was assumed, we consider individual power constraints. An efficient algorithm with the guaranteed convergence to an -optimal solution is proposed which solves the problem via second order cone programming (SOCP) in an iterative
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manner. To further reduce the computational complexity of the iterative procedure, we approximate the optimization problem with a single SOCP problem which yields a closed form solution for the beamforming coefficients. The remainder of the paper is organized as follows. In section II, we describe the system model and problem formulation. The first part of section III is devoted to the global probability of detection maximization under individual secondary user power constraints. In the second part, we present the approximation method. Simulation results are given in section IV. Finally the main results are summarized in section V. Notation: Throughout the paper, we use uppercase boldface letters to represent matrices and lowercase bold letters to denote vectors. In addition, (.)T and (.)H stand for transpose and Hermitian transpose of a vector, respectively. λmax (A) and υmax (A) represent principal eigenvalue and eigenvector of matrix A, respectively. Also E{.} stands for statistical expectation and diag(a) denotes a diagonal matrix with diagonal elements given by vector a. we use . to show Euclidean norm of a vector and (.), (.) and |.| to denote the real part, imaginary part and amplitude of a complex number, respectively and sign(.) stands for the sign of a real number. N (μ, σ 2 ) represents a normal distribution with mean μ and variance σ 2 . Finally, 0 and 1 are column vectors with elements all equal to zero and one, respectively, and I is an identity matrix. II. S YSTEM M ODEL Let us consider a cognitive radio network with M secondary users. Fig. 1 illustrates the deployment scenario. The users are randomly deployed in a circle with radius Rs at a distance R from the primary transmitter. The secondary users located within the primary coverage range (Rp ) or in the guard range (Rg ) potentially interfere with the primary transmission [13]. Therefore, the secondary users are allowed to access the primary band only when the distance R is greater than the interference range of the primary network defined as Ri = Rp + Rg . The size of the secondary network is assumed to be small compared to R, i.e. Rs R, such that all secondary users experience the same pathloss from the primary transmitter. A two-phase detection framework is proposed: During the sensing phase, the secondary users observe the primary band and estimate their local SNRs. In the reporting phase the secondary users forward the estimated SNRs to a central entity, referred to as the fusion center, where the final decision about the primary band is obtained. The fusion center can either be one of the secondary users or a separate control node in the network. Let us assume that the primary transmitter broadcasts Pp s, where s is the normalized information symbol, i.e. E{|s|2 } = 1, and Pp denotes the power of the primary transmitter. The received signal at the i-th secondary user can be written as y = Pp αβi hi s + ni (1)
SU 2
PT Rg
Rs
SU M
Rp
SU 1
R l FC
PR
PT: Primary Transmitter, PR: Primary Receiver, SUi : i’th Secondary User, FC: Fusion Center R p : Primary Range, R s : Secondary Range, Rg : Guard Range
Fig. 1.
The deployment scenario
where α, βi and hi denote the pathloss, shadowing and multipath fading components of the channel between the primary transmitter and the i-th secondary user (i = 1, 2, · · · , M ), respectively. The pathloss is proportional to the inverse power of the distance from the primary transmitter, i.e. α ∝ 1/Rκ where κ is the pathloss exponent. The shadowing is modeled statistically with a correlated multivariate lognormal distribution. Assuming βi = 100.1ωi , we have ω ∼ N (0, Σ) where ω = [ω1 , ω2 , . . . , ωM ]T and Σ is the covariance matrix of ω. Without loss of generality, the correlation properties of ω can be modeled as a decreasing exponential function [13], i.e. Σi,j = σ 2 ρdij ,
i, j = 1, . . . , M
where Σi,j is the (i, j)-th entry of the covariance matrix, σ 2 denotes the variance of {ωi }M i=1 and dij is the distance between the i-th and the j-th secondary users. Parameter ρ is the correlation coefficient between the secondary users separated by unit distance and related to the correlation distance Dc by the relation ρ = e−1/Dc [14]. Moreover, the multipath fading is modeled as i.i.d zero-mean and unit variance complex Gaussian random variable, i.e. hi ∼ CN (0, 1). Finally, ni is the zero-mean additive white Gaussian noise (AWGN) at the i-th secondary user with variance σn2 . Let xi denote the average SNR level of the primary transmission at the i-th secondary user on a logarithmic scale. Then, xi can be written as xi = ωi − 10κlog10 (R) + K
(2)
P 10log10 ( σ2p ). n
where K = In the assumed network model, the secondary users are viewed as a distributed transmitter with M antenna elements. This multi-antenna structure forwards the estimated SNR levels in the logarithmic scale multiplied by the beamforming vector f = [f1 , f2 , . . . , fM ]T to the fusion center. The received signal at the fusion center can be written as
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y=
M α βi hi fi xi + n = f H Hx + n i=1
(3)
h ]T ), where H = diag([ α β1 h1 , α β2 h2 , . . . , α βM M T and x = [x1 , x2 , . . . , xM ] . The pathloss (α ), shadowing (βi ) and multipath fading (hi ) components of the reporting channels are modeled similar to that of the sensing channels. A quasi-static fading condition is assumed for the reporting channel gains such that the channel realizations stay fixed for the duration of each data transmission to the fusion center. Moreover, n denotes the noise component at the fusion center and is modeled as a zero-mean additive white Gaussian noise with variance σn2 . A hypothesis testing problem is formulated by choosing between the white space hypothesis H0 which by definition occurs when R > Ri and the occupied space hypothesis H1 when R ≤ Ri . By modifying the definition of hypothesis H1 to the worst case scenario, namely R = Ri , a simple binary Neyman-Pearson hypothesis testing problem is formed [3] ¯ f H HΣHH f + σ 2 ) y ∼ N (f H Hμ(R), n
(4)
¯ + K, with ¯ = μ(R)1 ¯ and μ(R) ¯ = −10κlog (R) where μ(R) 10 ¯ = Ri + l , l > 0 (white space) H0 : R (5) ¯ = Ri (occupied space) H1 : R
III. P ROBABILITY OF D ETECTION M AXIMIZATION UNDER I NDIVIDUAL P OWER C ONSTRAINT Our aim is to maximize the probability of detection, Pd , while the power of each secondary user is limited to its power budget. Mathematically, we want to solve the following optimization problem max
H1
(6)
H0
The performance of the spectrum sensing procedure can be evaluated by the probability of detection (reliability) and the probability of false alarm (efficiency) formulated as ⎞ ⎛ H τ − f Hμ(R|H1 ) ⎠ (7) Pd = P (y > τ |H1 ) = Q ⎝ f H HΣHH f + σn2 ⎞ ⎛ H τ − f Hμ(R|H ) 0 ⎠ (8) Pf = P (y > τ |H0 ) = Q ⎝ H H 2 f HΣH f + σn where Q(.) is the complementary cumulative distribution function, which calculates the tail probability of a zero mean unit variance Gaussian variable. The sensing performance of the secondary network depends largely on the beamforming vector, f , and the test threshold, τ . Assuming a target probability of false alarm Pf = ψ, the test threshold can be calculated as ¯ 0 ) + Q−1 (ψ) f H HΣHH f + σn2 (9) τ = fH Hμ(R|H Substituting (9) in (7), we get ⎛
where Pimax is the maximum allowable transmit power of the i-th secondary user and Pi can be written as Pi = E{|fi xi |2 } = ξ|fi |2 = fH Ci f, i = 1, 2, . . . , M (12) where Ci = ξdiag(ii ) and ii is the ith column of the identity 2 2 matrix. ξ = E{x2i } = e2c σ (π0 e2cμ0 + π1 e2cμ1 ) and π1 = P (H1 ), π0 = P (H0 ) denote the probability that secondary users are located within or beyond the interference range of the primary network, and c = (ln10)/10 [10]. Considering that Q(.) is a monotonically decreasing function, the optimization problem in (11) is equivalent to fH Hδδ H HH f f f HΣHH f + σn2 subject to fH Ci f ≤ Pimax , i = 1, 2, . . . , M max
¯ 1 ) − μ(R|H ¯ 0 ). where δ = μ(R|H
(13)
A. Iterative Approach We begin by rewriting (13) into the following equivalent form (14) max t f
subject to
fH Hδδ H HH f ≥t fH HΣHH f + σn2 fH Ci f ≤ Pimax , i = 1, 2, . . . , M
Since Σ is a symmetric and positive definite matrix, using Cholesky decomposition Σ = LLH , leads to fH HΣHH f = LH HH f2 . Then, by applying the following definitions
HL 0 Hδ f ˜ A= ,b= ,f= 0 1 0T σn
⎞
max ˜ f
(10)
H
It can be seen that the optimization problem in (13) is a nonconvex problem. Therefore, in the sequel of this section, we have proposed two different approaches to find the beamforming vector. In what follows, we first formulate the problem as a standard SOCP and propose an iterative algorithm to find the solution. Subsequently, an approximate solution is presented with considerably lower complexity.
we obtain
f H Hδ ⎠ Pd = Q ⎝Q−1 (ψ) − H H 2 f HΣH f + σn
(11)
subject to Pi ≤ Pimax , i = 1, 2, . . . , M
By considering the optimal likelihood ratio test with a test threshold τ as the decision rule, we have y ≷ τ
Pd
f
t
H H subject to t˜f A2 ≤ |˜f b|2 H ˜f C˜i 2 ≤ Pimax , i = 1, 2, . . . , M √ ˜ i = ξdiag(ii ). where C
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(15)
Note that for any value of t, the above problem is convex in ˜f and can be solved using interior-point methods. However, the joint optimization over t and ˜f is a quasi-convex optimization problem which can be solved using the bisection algorithm [16]. We next show how for a specific value of t, (15) can be expressed as a SOCP. An interesting observation here is that if ˜f satisfies the constraints in (15), so does ejθ ˜f for any arbitrary phase shift θ, and the value of the objective function is maintained. Thus, H without loss of generality we can assume that ˜f b is a real number [17]. Using this assumption, (15) can be rewritten as the following feasibility problem ˜f (16) √ H H ˜ ˜ subject to tf A ≤ |(f b)| H (˜f b) = 0 H ˜f C˜i ≤ Pimax , i = 1, 2, . . . , M
Algorithm 1 The Proposed Iterative Approach 1: Initialize: 2: i ← 0 max T 3: fl ← √1ξ P1max , P2max , . . . , PM , then set
4:
i=1
maximum Pd achieved by solving individual power constraints {Pimax }M i=1 . This is due to the fact that the total power constraint results in a larger feasibility area than the individual fH Hδδ H HH f power constraints. Therefore, we choose u(0) = fHuHΣHH f +σu2 u u n , where fu is the optimal solution obtained in [10] with the aforementioned total power constraint. Then we can initialize (0) (0) and follow the bisection the value of t to its midpoint u +l 2 procedure outlined in Algorithm 1 to find the beamforming coefficients. B. Approximation Approach As explained in the previous subsection, the -optimal solution requires an iterative procedure in which the feasibility of the problem should be checked in each iteration. To avoid such a computational complexity, we perform an approximation and then maximize the resulted function instead of the original cost function. The optimization problem in (11) can be reformulated as max f
fH Hδδ H HH f fH (HΣHH +
2 σn f2 I)f
subject to fH Ci f ≤ Pimax , i = 1, 2, . . . , M
(17)
Solve problem (6) in [10] with PTmax ← set fu ← fopt , and u(0) ←
5:
Set t(0) ←
M i=1
Pimax , then
H H fH u Hδδ H fu H 2 fH u HΣH fu +σn
u(0) +l(0) 2 (i)
T and ˜f ← fTl , 1
while u(i) − l > do if If (16) is feasible for t(i) then ˜f(i) ← f obtained by the feasibility check in the 8: previous step 9: Set l(i+1) ← t(i) , u(i+1) ← u(i) and t(i+1) ← 6: 7:
find
Now, for any specific value of t, (16) is in the form of a standard SOCP [17]. Let us assume that the optimal value of t lies in the interval [l(0) , u(0) ]. We expect that the optimal beamforming results in a higher performance that the conventional strategy in which each secondary user transmits with its maximum power. Therefore, by satisfying the individual power con2 straints with equality ξ|fi | = Pimax , i = 1, 2, . . . , M , the ¯ HH ¯ H fl fH Hδδ , lower bound can be initialized as l(0) = fHl HΣ 2 ¯ H ¯ H fl +σn max max max lT 1 where fl = √ξ P1 , P2 , . . . , PM . Moreover, the maximum achievable Pd with the total power constraint M PTmax = Pimax is larger than or at least equal to the
¯ H ¯H fH l Hδδ H fl ¯ H ¯ H fl +σ2 HΣ fH n l
l(0) ←
u(i+1) +l(i+1) 2
10: 11: 12: 13: 14: 15:
else (i) Set ˜f ← f(i−1) , l(i+1) ← l(i) , u(i+1) ← t(i) and (i+1) +l(i+1) t(i+1) ← u 2 end if i←i+1 end while √ (i) Set ˜fopt ← ˜f and Pdmax ← Q(Q−1 (ψ) − t(i) )
The maximum value of f occurs when all the inequality constraints are satisfied with equality,then we have: f = max PTmax max = √1 P1 + P2max + · · · + PM ξ . Therefore, by ξ max PT substituting f = in the denominator, we get ξ max f
fH Hδδ H HH f fH (HΣHH +
2ξ σn PTmax I)f
(18)
subject to fH Ci f ≤ Pimax , i = 1, 2, . . . , M It is worth mentioning that the difference between the upper bound and the original cost function tends to zero as P max → ∞. As discussed before, the above objective function is upperbounded by the Raleigh Ritz inequality and is maximized when fapp = ζz (19) σ2 ξ
n I)−1 Hδδ H HH }. Now ζ where z = υmax {(HΣHH + P max T should be selected such that none of the inequality constraints are violated or i = 1, 2, . . . , M (20) ζ 2 ξ|zi |2 ≤ Pimax ,
where zi is the ith elements of the vector z. To satisfy all the above inequality constraints, we have Pjmax Pimax , j = arg min (21) ζ= 2 2 i ξ|zj | |zi | In the conventional strategy, we assume that all secondary users transmit with their maximum power. Therefore, the
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0
0
10
10
Probability of Miss−detection
Probability of Miss−detection
−1
−1
10
−2
10
10
−2
10
Iterative Method, η= 0.5 Approximation Method, η= 0.5 Conventional Method, η= 0.5 Iterative Method, η= 0.4 Approximation Method, η= 0.4 Conventional Method, η= 0.4 Iterative Method, η= 0.3 Approximation Method, η= 0.3 Conventional Method, η= 0.3
−3
10 Optimal Beamforming,Total Power Constraint Beamforming with Iterative Approach, Individual Power Constraint Approximation Method, Individual Power Constraint Conventinal Method
−4
−3
10
10 0
5 10 15 20 25 30 35 40 45 Sum of Individual Powers Cunsumed by the Secondary Users(dBm)
50
min ) versus Fig. 2. Minimum achievable probability of miss-detection (Pmd the sum of individual transmit power consumed by the secondary users for M = 5 and η = 0.5.
0
5 10 15 20 25 30 35 40 45 Sum of Individual Powers Consumed by the Secondary Users(dBm)
50
min ) versus Fig. 3. Minimum achievable probability of miss-detection (Pmd the sum of individual transmit power consumed by the secondary users for M = 5 and different values of η.
0
10
(22)
−1
Probability of Miss−detection
amplitude of their coefficients are Pimax , i = 1, 2, . . . , M |fi | = ξ
Then under individual power constraints, the probability of detection for the conventional strategy is given by ⎞ ⎛ H H HH f Hδδ ¯ ¯ f ⎠ Pdcov = Q ⎝Q−1 (ψ) − H (23) ¯ H ¯ H f + σn2 f HΣ
10
−2
10
Iterative Method, M = 5 Approximation Method, M = 5 Conventional Method, M = 5 Iterative Method, M = 10 Approximation Method, M = 10 Conventional Method, M = 10 Iterative Method, M = 15 Approximation Method, M = 15 Conventional Method, M = 15
−3
10
IV. S IMULATION R ESULTS
−4
10
The simulation results are based on 1000 independent realizations of the secondary users locations. For each realization, 10000 independent channel realizations are generated. We consider the bandwidth of the primary system to be 6M Hz (Typical TV band). Rp is assumed to be 30km and the secondary users report their observations to the fusion center over the reporting channel with a bandwidth of 240kHz. In addition, the following system parameters are assumed: π0 = π1 = 0.5, σ = 2.3, Rg = Rs = 5km, l = 15km, K = 180, κ = 4, ψ = 0.01 and a background noise power spectral density of −170dBm/Hz (The simulation parameters are adopted from [3], [20] and [21]). In Fig. 2, the minimum achievable global probability of min miss-detection (Pmd = 1 − Pdmax ) versus sum of individual transmit power consumed by the secondary users is illustrated when the sum of allowable individual power, i.e. M ( m=1 Pimax ), changes from 0 to 50dBm. We consider five secondary users with the following power constraints: P1 /3 = P2 /2 = P3 /2 = P4 = P5 denoted by [32211]. It is seen that as the overall individual power constraints of the secondary users increase, a better performance is achievable. However, the curves saturate as the power constraints get weaker. The proposed -optimal beamforming strategy
0
5 10 15 20 25 30 35 40 45 Sum of Individual Power Consumed by the Secondary Users(dBm)
50
min ) versus Fig. 4. Minimum achievable probability of miss-detection (Pmd the sum of individual transmit power consumed by the secondary users for η = 0.5 and different number of secondary users.
outperforms the case where the secondary users utilize all their allowable power. This is due the fact that in conventional strategy, there is no difference in fusing the observations while in our proposed method we differentiate between the observations by allocating the secondary users powers. As expected, the performance is indeed slightly worse than that of the same problem introduced in [10] with total power constraint but as a cost of this little degradation, we have considered a more practical scenario and we could assure that there is no violation in even the individual power constraints. In addition, the lower complexity of the approximation method, when compared to the -optimal beamforming strategy, has compromised the performance. c Fig. 3 is depicted for different values of η = D Rs with M = 5. As expected, a lower value of η results in a better performance due to the higher independence of the observations
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R EFERENCES
Sum of Individual Powers Consumed by the Secondary Users(dBm)
60 Iterative Method, M = 5 Approximation Method, M = 5 Conventional Method, M = 5 Iterative Method, M = 10 Approximation Method, M = 10 Conventional Method, M = 10 Iterative Method, M = 15 Approximation Method, M = 15 Conventional Method, M = 15
50
40
30
20
10
0
−10
0
5
10
15
20
25
30
35
M
Sum of Individual Power Constraints(dBm)
40
45
50
Pimax
m=1
Fig. 5. Total transmit power consumed by the secondary users versus the sum max ) of allowable individual transmit power of secondary users ( M m=1 Pi for η = 0.5 and different values of M .
at the secondary users. The same figure is plotted for η = 0.5 and different number of secondary users in Fig. 4. The power constraints M = 5 : [32211], M = 10 : [3332221111] and M = 15 : [333332222211111] are considered. It should be noted that a lower probability of miss-detection is achieved as the number of secondary users increases. In Fig. 5, the sum of the consumed power by the secondary users versus the sum of the allowable transmit power is depicted. In this figure, we have shown that our proposed beamforming strategy saves total transmit power of the secondary user compared to the case where secondary users utilize all their allowable power to forward their observations, thus we can conclude that power constraints do not hold at equality. In addition, the consumed power of the secondary users decreases as M increases. V. C ONCLUSION In this paper, we present a cooperative spectrum sensing strategy with per-user power constraints. A distributed beamforming method is proposed based on which the local SNR estimates of the secondary users are transmitted to a central entity, known as the fusion center. A hypothesis testing problem is solved at the fusion center which decides on the occupancy of the primary frequency band. The problem is formulated as an optimization problem. An iterative procedure is presented which achieves an -optimal solution. Subsequently, a suboptimal solution with lower complexity is proposed. Correlated shadow fading is considered in the channel gains between the primary transmitter and the secondary users as well as the channel gains between the secondary users and the fusion center. The numerical results show that the proposed beamforming strategy outperforms the conventional strategy. In addition, the approximation method performs well given the lower computational complexity of the method.
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