Cooperative Spectrum Sensing in Cognitive Radio Networks over Correlated Log–Normal Shadowing Marco Di Renzo
Fabio Graziosi, Fortunato Santucci
Telecommunications Technological Center of Catalonia Access Technologies Area Mediterranean Technological Park, Av. Canal Olimpic s/n 08860 Castelldefels, Barcelona, Spain E–Mail:
[email protected]
Dept. of Electrical and Information Engineering Center of Excellence in Research DEWS University of L’Aquila Poggio di Roio, 67040, L’Aquila, Italy E–Mail: {graziosi, santucci}@ing.univaq.it
Abstract— A fundamental component to enable Dynamic Spectrum Access (DSA) capabilities in the future generation of wireless communication systems is the design of robust and efficient spectrum sensing methods to detect licensee users transmitting in a given frequency band. In this context, collaborative methods have been proposed to improve spectrum sensing capabilities over wireless channels, by counteracting shadow–fading phenomena via distributed diversity. However, recent results have shown that the expected benefits of cooperation, which are obtained at the expenses of increased traffic overhead and the need for a control channel, may be significantly hampered when the signals sensed by the cooperative users experience correlated shadow– fading. Moreover, it has been argued that spatial correlation of shadow–fading yields fundamental limits to the performance of cooperative spectrum sensing methods, and practical implications in terms of protocol design and network deployment. Motivated by the above considerations, in this paper we propose an analytical framework for the analysis and design of cooperative spectrum sensing methods over correlated shadow– fading environments, when each cooperative user is equipped with a simple energy–based detector. We will show that the framework requires efficient and accurate methods for modeling the power–sum of correlated Log–Normal Random Variables (RVs), which well describe shadowing phenomena. In particular, three main technical contributions can be found in the present paper: i) we generalize some recently proposed methods to approximate the power–sum of independent Log–Normal RVs to correlated scenarios, ii) we analyze the accuracy of several approximation techniques to compute detection probability over correlated Log–Normal shadowing, which is an unexplored field so far, and iii) we rely on the proposed framework to study the detrimental effect of correlation on the performance of cooperative spectrum sensing methods. Furthermore, numerical results will be shown to validate the proposed framework.
I. I NTRODUCTION Cognitive Radio (CR) is commonly considered a key enabling technology to provide high bandwidth to mobile users via heterogeneous wireless architectures and Dynamic Spectrum Access (DSA) capabilities (see, e.g., [1]–[5]). Moreover, technology forecasts predict that CR will be a critical part of many future wireless systems and networks (see, e.g., [6]–[8]). Broadly speaking, a CR can be defined as “an intelligent wireless communication device that exploits side information about its environment to improve spectrum utilization” [9], and is likely to consist of several components, but mainly of a sensing, decision, and execution unit. Several definitions of CRs exist in the literature [3], [4], [9]. However, a promising solution, which is based on the idea of opportunistic communications, is the so–called interweave paradigm, according to which a CR is defined as “an intelligent wireless communication system that periodically monitors the radio
spectrum, intelligently detects occupancy in the different parts of the spectrum and then opportunistically communicates over spectrum holes with minimal (i.e., no harmful) interference to the active users” [3], [9]. In the light of the above definition of CR, there is a common understanding in the research community that a fundamental element for the successful exploitation of interweave CRs is the design of robust spectrum sensing methods to detect licensee users transmitting over a given frequency band. Accordingly, several spectrum sensing methods have been proposed to enable CR functionalities, and studied via analytical frameworks and experimental activities, see, e.g., [10]–[12] and references therein for a survey, and [13]–[23] for analysis and design of specific spectrum sensing methods. Among the various proposals, cooperative spectrum sensing methods using energy–based detectors in each cooperative user are often considered a good candidate to enable CR functionalities, as they provide a good trade–off for keeping the complexity of every cooperative node at a moderate level, as well as counteracting the limitations of energy–based detection in the low Signal–to–Noise Ratio (SNR) regime via distributed diversity [19], [24]. Accordingly, several studies have been conducted to analyze the performance of such a kind of spectrum sensing methods over a variety of fading channels (see, e.g., [13], [15], [16], [18], [19], [21], [22]). In these papers, the authors have recognized the importance of including the characteristics of wireless propagation for system analysis and design. In particular, they have pointed out the importance of accurately modeling correlated Log– Normal shadow–fading phenomena to properly analyze the impact of distributed cooperation. It has been shown that shadowing correlation can significantly reduce the performance of cooperation, which results in an optimal number of cooperative users yielding the highest cooperative gain and lowest traffic overhead due to cooperation. Moreover, asymptotic analysis based on a different kind of detector has explicit shown the performance limits set by correlated shadowing [14]. In the light of the above results, there is a common understanding about the importance of developing accurate and simple frameworks for the analysis and design of cooperative spectrum sensing methods over correlated Log–Normal shadow–fading environments. Moreover, the importance of accurately modeling correlation has also been reinforced by some recent experiments, which have proposed specific statistical models to well describe Log–Normal shadow–fading correlation for cooperative networks [25]. However, despite there is a significant number of studies for the analysis of
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energy–based cooperative spectrum sensing methods (see, e.g., [10], [11], [13], [15], [16], [19]), as far as correlated Log– Normal shadow–fading environments are considered, performance metrics are typically obtained via extensive numerical simulations, which do not yield, in general, a basis for a systematic system analysis and optimization. One of the main reasons for the absence of sound analytical frameworks to analyze the above mentioned scenario is due to the inherent analytical complexity of handling correlated Log–Normal Random Variables (RVs) if compared to other fading distributions. In order to fully understand this point and set the required background, let us consider, as an illustrative example, a typical cooperative detection scheme where the signals sensed by several cooperative users are sent, via an error-free reporting channel, to a fusion center that combines them by using a Square–Law Combining (SLC) mechanism [10], [11], [13]. In order to analyze the spectrum sensing capabilities of such a system (e.g., for computing the average Detection Probability [15, pp. 75, Eq. (13)]), we would need to know, in closed–form, the Probability Density Function (PDF) of the power–sum of correlated Log–Normal RVs. However, no closed–form results exist to date to handle this problem (see, e.g., [26], [27] and references therein). So, several techniques have been proposed to approximate the power–sum of correlated Log–Normal RVs, each one having its own advantages and disadvantages in terms of accuracy and complexity (see, e.g., [28] and references therein). Although, as the large body of references in, e.g., [27] suggests, modeling the power–sum of Log–Normal RVs is a long– lasting open problem with several contributions on the matter, two fundamental questions arise when these approximation methods are used in the frame of CR networks: Q1) What is theirs accuracy for spectrum sensing analysis?, and Q2) Can they be used in correlated scenarios?. The motivation of these two questions is as follows. Most approximation methods proposed in the literature have been developed for the analysis and design of outage events due to co–channel interference in cellular scenarios. So, most of them have been designed to accurately estimate the Outage Probability (Pout ) only. However, recent results have shown (see, e.g., [28] and references therein) that some of these approximation methods may fail to provide accurate performance predictions when different performance metrics are considered (e.g., to compute the Average Bit Error Probability, ABEP). In order to overcome the limitations of these approximation methods, which typically assume that the power–sum of Log–Normal RVs can be approximated with another Log–Normal RV, some improved non–Log–Normal approximation methods have been proposed, which yield, in general, more accurate estimates at the expenses of an additional computational complexity. However, the main limitation of these techniques is that either they can be used for independent Log–Normal RVs only or their usage and accuracy for correlated RVs have never been investigated so far. Furthermore, even for independent Log– Normal RVs the accuracy provided by all the approximation methods so far proposed in the literature cannot be predicted a priori in CR scenarios, as the computation of the Detection Probability in [15, pp. 75, Eq. (13)]) involves a different weight function with respect to ABEP and Pout , and recent results in [28] show that the shape of that weight function significantly affects the accuracy of the approximation.
So, motivated by the above considerations, the present research work is intended to provide fundamental theoretical advances along two main directions: i) to analyze the accuracy of Log–Normal power–sum approximations for cooperative spectrum sensing, and extend non–Log–Normal approximation methods to correlated scenarios, and ii) to use the developed framework to assess the performance of cooperative CR spectrum sensing in realistic propagation scenarios. More specifically, the following contributions and results are claimed in the present paper: i) we will show that well–known methods (e.g., Fenton–Wilkinson (FW) and Schwartz–Yeh (SY)) for Log– Normal power–sum approximation may fail to provide accurate estimates when used to compute Detection Probability for some system setups, ii) we will extend a recently proposed non–Log–Normal approximation method based on the Pearson system of distributions [27] to the correlated scenario, and will analyze its accuracy, and iii) we will quantify the impact of correlation on the performance degradation of energy–based detection cooperative spectrum sensing methods. The remainder of the manuscript is organized as follows. In Section II, the cooperative spectrum sensing problem will be introduced and formulated. In Section III, some Log–Normal approximation methods will be revised, and the approximation based on Pearson Type IV distribution will be extended to the correlated scenario. Furthermore, the SY approximation method will be revised and extended, which will result in an Improved SY (I–SY) approximation technique. In Section IV, numerical and simulation results will be compared to assess the accuracy of Log–Normal power–sum approximations to compute Detection Probability in CR scenarios, and the impact of correlated shadowing on system performance will be investigated. Finally, Section V will conclude the paper. II. P ROBLEM S TATEMENT Let us consider a typical CR network that performs spectrum sensing operations in a distributed and cooperative fashion (see, e.g., [20, pp. 20, Fig. 3]). In general, cooperative spectrum sensing is composed by four main and subsequent steps: i) every CR performs spectrum sensing locally and independently from each other, ii) every measurement is sent to a common band manager via an error–free reporting channel, iii) based on the collected measurements, the band manager makes a decision about the status of the sensed frequency band, and iv) the band manager broadcasts back the final decision to the cognitive users, thus enabling or not the transmission of one CR over that frequency band. By assuming, similar to [13], that every CR is equipped with an energy–based detector for spectrum sensing, and that the band manager (i.e, fusion center) uses a SLC mechanism to combine the measured energies, the performance of the cooperative spectrum sensing mechanism can be characterized by two performance measures, i.e., False Alarm Probability (Pfa ) and Detection Probability (Pd ), which can be computed as follows [13]: ⎧ λ Γ( LN ⎪ 2) 2 , ⎪ ⎨ Pfa = Γ( LN2σ) 2 (1) +∞ aξ ⎪ P = Q LN λ ⎪ fγt (ξ) dξ ⎩ d σ2 , σ2 0
2
where, the following definitions have been used: i) Γ (·, ·) denotes the incomplete Gamma function [29, pp. 260, Eq.
(6.5.3)], ii) Qm (·, ·) is the generalized Marcum Q–function [15, pp. 73], iii) L is the number of cooperative CRs, iv) N is the number of degrees of freedom of the system [13], v) λ is the detection/decision threshold used by the fusion center in the binary hypothesis testing problem to discriminate between presence and absence of a licensee user, and vi) σ 2 = 1, a = 2. Moreover, fγt (·) is the PDF of the SNR, γt , at the fusion center, which is defined as follows: γt =
L
i=1
γi =
L
Es 2 α N0 i i=1
(2)
where i) Es is the (licensee’s user) signal energy, ii) N0 is the one–sided Power Spectral Density (PSD) of the Additive White 2 L Gaussian Noise (AWGN) at the receiver, and iii) αi i=1 represent the set of channel power gains, which are Log–Normal distributed due to shadowing propagation. In the context of opportunistic spectrum access, the optimization of Pfa and Pd (or, equivalently, the Miss Probability Pm = 1 − Pd ) plays an important role to fully exploit CR capabilities. As a matter of fact, Pfa represents the probability of falsely detect a licensee user where it is not actually transmitting, and so a high Pfa results in a low spectrum utilization. On the other hand, Pm denotes the probability of not detecting a licensee user where it is actually transmitting, and so a high Pm results in a high interference inflicted to the licensee user. As a consequence, a good system design and optimization foresee small values of both Pfa and Pm . Since Pfa in (1) is independent from channel statistics, in the present contribution we are mainly interested in developing a simple but effective framework to compute Pd in (1), which requires a closed–form expression for the PDF of the power– sum of correlated Log–Normal RVs, i.e., fγt (·). In Section III, we will propose and compare several methods to approximate fγt (·), as no closed–form results exist for it to date [27]. III. M ETHODS FOR L OG –N ORMAL P OWER –S UM A PPROXIMATION According to the comments in Section II, the problem of computing Pd (or Pm ) boils down to have general and flexible methods for approximating the power sum of correlated Log– Normal RVs. The problem can be formulated, in general terms, as follows1 . Given a set of L correlated Log–Normal RVs, L {Xi }i=1 , with mean vector (in dB) μ and covariance matrix (in dB) Σ , each one having a PDF as follows2 : 2 (10 log10 (ξ) − μ (i)) 10/ln (10) (3) exp − fXi (ξ) = Σ (i, i) 2Σ Σ (i, i)ξ 2πΣ L what is the PDF, fY (·), of RV Y = i=1 Xi ? Three techniques will be analyzed and compared in what follows to approximate such a PDF. Moreover, we will denote the approximating PDF of Y with f˜Y (·). A. Pearson Type IV Approximation In [27], we have recently proposed a novel accurate method for approximating the power–sum of Log–Normal RVs, which is based on the Pearson system of distributions. However, 2 L 1 For example, {X }L i i=1 = αi i=1 and Y = γt for the spectrum sensing problem described in Section II. 2 We denote with v (i) the i–th element of vector v, and with M (i, j) the element in the i-th row and j–th column of matrix M.
therein the framework has been analyzed for independent Log–Normal RVs, while no results exist to date about the applicability and accuracy of Pearson Type IV approximation for correlated summands. So, motivated by the importance of handling spectrum sensing problems over correlated shadowing environments, a major objective of this contribution is to extend the framework in [27], and, more importantly, analyze its accuracy for computing Pd in (1). According to the Pearson Type IV approximation method, the PDF of Y , fY (·), is approximated as follows: ∼ f˜Y (ξ) fY (ξ) = 2 −m ˜ h = ξ 1 + (10 log10d2(ξ)+u) exp −ν tan−1 10 log10d (ξ)+u (4) ˜ = 10h/ln (10), h is a normalization factor [27], and where h u, m, d, ν are the parameters that define the Pearson Type IV distribution. These latter parameters can be readily computed from the non–central moments of RV YdB = 10 log10 (Y ) via simple algebraic manipulations, as described in [27]. Due to space constraints, these formulas are not reported here. In [27], we have provided a simple formula to compute L the non–central moments of YdB when the RVs {Xi }i=1 are independent. In this paper, we provide a similar formula when the RVs may be generically correlated. This formula can be obtained by plugging into [27, pp. 1001, Eq. (6)] the Moment Generating Function (MGF) of the power–sum of correlated Log–Normal RVs, which has been recently computed in [30, pp. 2694, Eq. (17)] by using Gauss Quadrature Rule (GQR) methods. By avoiding the details of the analytical derivation due to space constraints, we report only the final result in (5), which yields the n–th non–central moment of RV YdB : n
Np Np Np
10 (n) n mYdB = ··· Π (p) {ln [Ω (p)]} ln (10) p1 =1 p2 =1 pL =1 (5) L Np and where p is a vector with elements {pj }j=1 , {xp }p=1 Np {Hp }p=1 are the zeros and the weights of the Np –order Hermite polynomial [29, Table 25.10, pp. 924], and the following functions have been defined: ⎧ L ⎪ ⎪ Π (p) = H √pi ⎪ ⎨ π i=1 L L √ ⎪ sq ⎪ Ω (p) = exp ln(10) ⎪ 2 Σ (i, j) x + μ (i) pj ⎩ 10 i=1
j=1
(6) where Σ sq = UV1/2 , and U and V are the matrices containing the eigenvectors and eigenvalues of Σ , respectively. Note that, although the procedure used to compute (5) is similar to the one used in [27], the accuracy of Pearson Type IV approximation to estimate the power–sum of correlated Log–Normal RVs is an open issue, as well as the accuracy of every Log–Normal approximation method when used for spectrum sensing analysis (as described in Section I). B. Improved Schwartz–Yeh (SY) Approximation The SY approximation method was proposed in 1982 in [31]. The main idea is to approximate the Log–Normal power– sum Y with another Log–Normal RV, as follows: 2 (10 log 10/ln (10) (ξ) − μ ) SY 10 ∼ f˜Y (ξ) = exp − fY (ξ) = 2 2 ξ 2σSY 2πσSY (7)
YdB
YdB
Of course, (8) holds for independent and correlated Log– Normal RVs. The accuracy of the I–SY approximation will be analyzed in Section IV. C. Fenton–Wilkinson (FW) Approximation The FY approximation method was proposed in 1960 in [32]. Similar to the SY method, the main idea is to approximate the Log–Normal power–sum Y with another Log– Normal RV with approximating parameters μF W and σF W 3 . The only difference with SY and I–SY methods is that the approximating parameters are obtained via moment–matching in the linear domain, instead of in the logarithmic domain. In this contribution, we use the FW approximation method described in [33, pp. 1124, Eq. (12)] and called MMA12. This approximation method can be used for both independent and correlated Log–Normal RVs. The analytical formulas are not reported in this contribution due to space constraints, but can be found in [33]. IV. N UMERICAL AND S IMULATION R ESULTS The aim of this section is to analyze the accuracy of the proposed approximations to compute Pm , as a function of the number of cooperative CRs (L), and shadowing correlation among them. In particular, Pm will be obtained, via straightforward numerical integration techniques, from (1) by approximating the PDF of the SNR γt by using the methods described in Section III, i.e., fγt (·) ∼ = f˜γt (·). Analysis will be compared with Monte Carlo simulations to assess its accuracy. The following system setup is considered for performance analysis: i) the detection/decision threshold λ is computed approximating PDF can be obtained from (7) by replacing μSY and with μF W and σF W , respectively.
3 The
σSY
0
10
Monte Carlo Simulation PIV Approximation I−SY Approximation FW Approximation −1
10
Pm
where μSY and σSY are the parameters of the approximating PDF, which are obtained via a moment matching in the logarithmic domain between Y and the approximating Log– Normal RV [26]. According to [31], μSY and σSY are, in general, estimated using an iterative procedure that foresees a Log–Normal approximation in every iterative step. The limitation of this approach is twofold: i) an iterative method is required, which may not be managed in a simple way, and (more importantly), ii) the approximation tends to degrade in accuracy when the number of summands increases, as the approximation error in every iterative step tends to accumulate. In this paper, we propose an alternative method for computing the parameters μSY and σSY in (7), which avoids iterative methods and error accumulation. We call this novel method “Improved– SY (I–SY) approximation”, where the term improved is used to emphasize that μSY and σSY are obtained via log–moment matching without error accumulation. The I–SY approximation can be obtained as a byproduct of the general method proposed for Pearson Type IV approximation. In fact, (5) provides a GQR–closed–form expression to compute the log–moments of Y , i.e., the non–central moments of YdB , without requiring any iterative procedure and, as a consequence, avoiding possible error accumulations. In other words, μSY and σSY in (7) can be computed as follows: ⎧ (1) ⎨ μSY = mYdB 2 (8) ⎩ σSY = m(2) − m(1)
−2
10
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10
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Fig. 1.
Pm vs. number (L) of cooperative CRs (ρ = 0).
according to a Constant False Alarm (CFA) criterion [15] by using the formula for Pfa in (1) with Pfa = 10−3 ; ii) without loss of generality, the Log–Normal RVs are assumed to be μ (i)}L identically distributed with parameters {μ i=1 = 15 dB, L Σ (i, i)}i=1 = 9 dB, and with equal correlation coefficient {Σ
L Σ (i, i) Σ (j, j) ; iii) ρ = ρXi Xj i=j=1 = Σ (i, j) N = 10, iv) Np = 7, and v) Es /N0 = 0 dB. In Figs. 1–4, we have reported the accuracy provided by the proposed approximation methods in the above reference scenario for different values of ρ. We note that these latter values are in agreement with those measured in [25]. By comparing simulation and approximations, the following conclusions can be drawn: i) both FW and I–SY approximation methods provide, in general, less accurate estimates of Pm than Pearson Type IV, which is due to a non–uniform and inaccurate approximation of the PDF of γt over the whole range of values it can take, ii) when low correlation values are considered, FW method shows an error floor as the number of cooperative users increases, which yields an upper bound in estimating Pm and does not allow to well reproduce the performance improvement due to cooperation, iii) Pearson Type IV approximation provides very reliable and accurate estimates of Pm , which work well regardless of the correlation coefficient ρ and the number of cooperative CRs L, and iv) for high correlation values, all approximation methods provide, as expected, the same results. In conclusions, the performed analysis confirms that Pearson Type IV approximation can be well used for the analysis of correlated RVs, and that it provides reliable estimates when used for the analysis and design of dynamic spectrum access via CRs. On the other hand, FW and I–SY methods yield less accurate estimates of Pm . For low correlation values, FW method can significantly over–estimate Pm . V. C ONCLUSIONS In this paper, we have provided analytical frameworks for the analysis of cooperative spectrum sensing techniques over correlated Log–Normal shadowing environments. Novel approximation methods have been introduced to handle correlated scenarios, and their accuracy has been validated via Monte Carlo simulations. R EFERENCES [1] J. Mitola III and G. Q. Maguire Jr., “Cognitive radio: making software radios more personal”, IEEE Personal Commun., pp. 13–18, Aug. 1999.
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10 Monte Carlo Simulation PIV Approximation I−SY Approximation FW Approximation
Monte Carlo Simulation PIV Approximation I−SY Approximation FW Approximation
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Fig. 2.
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Pm vs. number (L) of cooperative CRs (ρ = 0.1)
Fig. 4.
Pm vs. number (L) of cooperative CRs (ρ = 0.5).
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Pm vs. number (L) of cooperative CRs (ρ = 0.3).
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