AbstractâA normal factor graph (NFG) based approach for cooperative spectrum sensing in cognitive radio over time varying and frequency non-selective ...
Factor Graph based Cooperative Spectrum Sensing in Cognitive Radio Over Time-varying Channels Debasish Bera, Indrajit Chakrabarti, Priyadip Ray, and S. S. Pathak G. S. Sanyal School of Telecommunications, IIT Kharagpur, West Bengal, India.
Abstract—A normal factor graph (NFG) based approach for cooperative spectrum sensing in cognitive radio over time varying and frequency non-selective fading channels is presented in this paper. An NFG based representation of a distributed cognitive radio system is first presented and then a Sum-ProductAlgorithm (SPA) based analysis is developed for inference. The spectrum sensing problem is modelled as a distributed binary hypothesis testing problem. A Neyman-Pearson (NP) based likelihood ratio test statistic is derived for optimal sensing. As exact theoretical analysis of the system level probability of detection and probability of false alarm is very difficult, we provide an approximation which performs satisfactorily in the moderate to high signal-to-noise ratio (SNR) regime. The proposed NFG based spectrum sensing approach is computationally scalable to large networks and performs well under time varying channel conditions. Extensive simulation results are provided to validate our proposed approximation.
I. I NTRODUCTION AND R ELATED W ORK Next generation communication networks based on Dynamic Spectrum Access (DSA) [1] is evolving to meet the spectrum scarcity problem. Cognitive radio (CR) [2] is the key enabling technology of DSA which dynamically and intelligently use the unused spectrum. Spectrum sensing or primary user (PU) detection is the first critical step for dynamic spectrum management for utilization of the unused spectrum by unlicensed or secondary users (SUs or CRs). In fixed spectrum allocation process, a particular frequency band at a particular place is restricted to the license holders of the band at that place. In many countries, all frequency bands have been completely allocated to specific uses. However, it has been observed that most licensed spectrum is often under-utilized (sparse in frequency). Even all the frequency bands are not used simultaneously (sparse in time) in all geographical locations (sparse in space). All these cases may be exploited in spectrum crisis situations with the aid of CR networks [3]. Sensing techniques can be classified as: (a)Local Sensing: It is sensitive to fading, shadowing, and model uncertainty but simple in terms of implementation and computation. (b)Cooperative Sensing: Information from multiple SUs are jointly used for PU detection. It combats many random factors like noise, shadowing and enhances the accuracy, reliability of detection at the cost of complexity. The cooperative sensing strategy can further be categorized as Centralized Cooperative Spectrum Sensing (CCSS) and Decentralized Cooperative Spectrum Sensing (DCSS). In CCSS a central unit, called fusion centre (FC), collects local sensing information (hard decisions) from CRs, identifies the available spectrum and broadcast this information to other
CRs or directly control CR traffic. Spectrum sensing is basically a detection problem. Therefore, CCSS can be viewed as centralized (with FC) distributed detection (with multiple detectors). Well known fusion rules are AND, OR, voting (ideal SU-FC channel), equal gain combiner (EGC), maximum ratio combining (MRC), likelihood ratio testing (LRT) (nonideal SU-FC channel) based rule. Among these, AND, OR, voting, MRC, EGC based rules are suboptimal even for large number of sensors and LRT is the optimal solution [4]. LRT is implemented using either Neyman-Pearson criterion (maximization of prob. of detection subject to a constraint on prob. of false alarm) or Bayes criterion (minimization of prob. of error) [5]. CCSS provides globally optimal performance with higher communication cost between SUs and FC. It is simple to implement but not immune to node failure. Probabilistic modelling and inference of large-scale wireless communication systems, such as cooperative spectrum sensing, is often computationally very expensive due to the presence of complex dependencies between the system’s random variables. Graphical models [6] with good message-passing algorithms like Belief-Propagation (BP) algorithm [7] or SumProduct-Algorithm (SPA) [8] are efficient, faster inference technique with superior convergence. It is well suited for representing probabilistic relationships among random variables and draw approximate inferences where exact solution is intractable (NP-hard). It exploits the factorization property of joint probability functions captured by the logical structures of the graphs. There are several well-known graphical models: Bayesian Network (BN), Markov Random Field (MRF), Tanner graph (TG), and Factor graph (FG). FG is a generalization of TG, suitable for model based design, and efficient for computation of marginal distributions of each observed variables using probability propagation with SPA. Besides, factor graphs are more general (any BN, MRF or TG can be encoded as FG with no increase in its representation size) [9] and it expresses the structure of factorization of a mathematical relation between variables and local functions. Therefore we have considered FG, precisely normal (Forney-style) factor graph (NFG) [8], as our graphical model and solved the cooperative spectrum sensing problem using SPA as message passing. In this paper, we develop normal factor graph based mathematical models for CCSS scheme over time varying, frequency non-selective fading channel and proposed a new distribution for probability of detection with interval of confidence and shown how the distribution varies for different SNRs. We draw
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non-selective and time varying fading channel, where hk is Rayleigh distributed r.v. with parameter σk i.e. hk ∼ R(σk ), σk is the s.d. of original complex Gaussian r.v. from which the Rayleigh variable (hk ) is generated. Also, let, PUs states are independent i.e. PUs are memoryless. Therefore, the k-th SU observes a complex baseband equivalent signal (xk ) as: xk = hk s + w k
Fig. 1. Block diagram for system model of Centralized Cooperative Spectrum Sensing (CCSS) technique with K SUs (sensors), one FC and one PU. Final decision is made at the FC.
the inferences using SPA over normal factor graph. Therefore, we try to connect two active research area, namely approximate inference methods in graphical models and decentralized (team) Neyman-Pearson detection methods in CR scenario. This paper is structured as follows: Section II explains the basic system model and problem formulation for CCSS in different channel conditions. The system analysis for CCSS over time varying, frequency flat channel condition is presented in Section III. Section IV deals with analytical solutions for CCSS scheme and properties of the new distribution. The simulation results are explained in Section V. Finally, section VI concludes the paper with future research path directions. II. S YSTEM M ODEL FOR CCSS S CHEME A. Brief Description Cooperative spectrum sensing plays a crucial role for successful implementation of the CR technology. Imperfect channel knowledge affects the performance of cooperative schemes. Therefore proper system modelling is very important for its practical realizations. The block diagram of CCSS system is shown in Fig.1. This model has also been used in [10]. The system consists of one primary user, K secondary users, and one fusion centre at one place. All SUs are sensing the state (active or not active) of PU simultaneously. It is assumed that the sensing duration should be much smaller than the PU’s average busy-to-idle and idle-to-busy state transition periods. Otherwise, sensing outcomes will not be meaningful for the corresponding utilization period. We assume that SUs are transmitting at lesser power than PU and energy detectors are used at each SU for local decisions. Through out this paper, v1K denotes the set {v1 , ..., vK }, vk denotes vector {vk1 , ..., vkN } over N sampling interval, P (x) denotes probability density function (pdf), F (x) denotes cumulative distribution function (CDF), and no distinction is considered between random variables and their values. B. Problem Formulation According to the CCSS model given in Fig.1, all SUs are monitoring the same frequency band at which PU is transmitting. The baseband-equivalent signal s ∈ {0, 1}, is transmitted by the PU and propagated to the k-th SU over a frequency
(1)
2 where wk ∼ CN (0, σw ) i.e. w k has circularly symmetric k complex Gaussian (CSCG) distribution. At the k-th user, the signal is mapped onto another signal uk = γk (x2k ) which is then transmitted to FC where the final decision is made. The channel between each SU and FC is assumed to be noisy (binary symmetric channels (BSC) or additive white Gaussian noise (AWGN)) and characterized by P (yk |uk ), where yk denotes the received signal at the FC from k-th SU. Therefore, for BSC with crossover probability αk , the received signal at FC from k-th SU is:
yk = uk with probability 1 − αk =∼ uk with probability αk
(2)
Otherwise, for AWGN channel with additive noise nk ∼ CN (0, σn2 k ), the signal received at FC from k-th SU: yk = uk + nk ; where, yk , uk ∈ {0, 1}. The spectrum sensing can be modelled as binary hypothesis testing problem with null and alternative hypotheses: H0 : Primary user is idle or not active i.e. s = 0 H1 : Primary user is busy or active i.e. s = 1 Alternatively, it becomes a binary hypothesis testing problem to decide whether or not the mean received power at SUs is higher than the expected power (threshold). It is assumed that SUs use the spectrum whenever they detect a spectral hole (white space). The constraint in our system is the probability of interference with PU’s transmission i.e. the prob. of making an erroneous decision about the presence of PU under H0 (Pf = probability of false alarm). Therefore, for efficient utilization of spectrum, the system design needs to minimize the probability of missed detection (Pm ) or maximize the probability of detection (Pd ) subject to the constraint that Pf ≤ predetermined threshold (τ ). It becomes simple, onesided, Neyman-Pearson (NP) hypothesis testing problem [5]. III. S YSTEM A NALYSIS FOR CCSS S CHEME It is well recognized that proper analysis of a system depends on its exact probabilistic model. The CCSS model defined in section-II largely depends on different channel conditions considered between PU-SU and SU-FC. Analytical discussions for different channel models have been presented in this article. Factor graph and SPA based approach to cooperative spectrum sensing is already addressed in [10]. The probability of detection (Pd ) and likelihood ratio test statistic are reformulated in [11] by considering non-central chi-square distribution under H1 . All previous works have considered only time invariant, frequency flat fading channels i.e. hk is ∼ CN (0, σh2 k ) or constant for N samples in time respectively. But in practical CR environment the channel
is time varying (slow) over N sampling interval. Therefore, consideration of time varying channel is very relevant to the context of spectrum sensing in CR network. A. Channel Model:PU-SU as frequency-flat, time-varying (slow) fading with AWGN and SU-FC as BSC or AWGN and PU-SU-FC channels are independent The spectrum sensing can be viewed as a system-on-graph problem. We are trying to solve distributed detection (with FC) problem by passing messages (probability values) through SPA over NFG. The complex envelop of the received signal at the input of demodulators of SUs are corrupted by multiplicative k ). Rayleigh fading (hk ) and AWGN (w 1) Probability Models: Now, we have to find likelihood functions, P (y1K |s = 0) and P (y1K |s = 1) i.e. a detection problem is mapped to Bayesian inference problem by finding the likelihoods. The joint probability distribution, K K K P (s, hK 1 , x1 , u1 , y1 ), represents the CCSS model of Fig.1. The likelihood function, P (y1K |s), can be computed as: K K K K K K P (s, hK 1 , x1 , u1 , y1 )du1 dx1 dh1
= P (s, y1K )
K K K K K K or, P (y1K |s) = P (hK 1 , x1 , u1 , y1 |s)du1 dx1 dh1
(3)
K K K The joint distribution of interest, P (hK 1 , x1 , u1 , y1 |s), can be factorized further and for hard local decisions it becomes: K K K K K K K P (hK 1 , x1 , u1 , y1 |s) = P (y1 , h1 , x1 , u1 |s) K = P (yk |uk )I(uk = γk (tk ))P (tk |s, hk )P (hk )
Fig. 2. Normal factor graph for joint distribution of interest of eq.(4), while using hard decisions at SU’s. The graph is shown for two PU-SU-FC channels.
(1)MP (hk )→hk = P (hk ) and (2)Myk →P (yk |uk ) = 1
(3)MP (yk |uk )→uk= P (yk |uk ) and (4)Muk →I(uk )=P (yk |uk ) (5)MI(uk )→tk = P (yk |0)I(tk < τk ) + P (yk |1)I(tk > τk ) (6)Mtk →P (tk |s,hk )=P (yk |0)I(tk < τk ) + P (yk |1)I(tk > τk ) (7)Mhk →P (tk |s,hk ) = P (hk ) (8)MP (tk |s,hk )→s = P (tk |s, hk )dhk dtk × (6) × (7)[Cont.] = P (hk )[P (yk |uk = 0)I(tk < τk )+ tk hk
P (yk |uk = 1)I(tk > τk )]P (tk |s, hk )dhk dtk = P (yk |0) P (tk |s)dtk + P (yk |1) P (tk |s)dtk tk τk
The final message of interest is: MP (tk |s,hk )→s
= P (yk |0) P (tk |s)dtk +P (yk |1) P (tk |s)dtk tk τk
where, τk = local threshold, and hk P (tk |s, hk )P (hk )dhk = P (tk |s) is obtained by marginalizing it over hk . The final message, from node P (tk |s, hk ) to edge s, is computed in terms of likelihood probability and CDF of tk |s. In the following section we will find CDF of tk under hypotheses H0 and H1 . IV. LRT S TATISTICS FOR THE CCSS S CHEME This section presents the detail analytical explanations in different channel conditions for CCSS model. Given, xk = 2 2 hk s + w k , tk = |xk | where, wk ∼ CN (0, σw ) and, k hk ∼ R(σk ) i.e. hk is a Rayleigh distributed with variance k as s = 0 and σh2 k = 0.4292 × σk2 . Now, under H0 : xk = w 2 therefore, xk ∼ CN (0, σw ). Thus x is complex Gaussian k k under H0 . Consider N number of samples as sensing interval. We can conclude tk is chi-square distributed with 2N degrees of freedom and CDF of tk under H0 is, F (tk |H0 ) = P (tk |H0 )dtk = tk τk tk τk
P (tk |s)dtk
(13)
tk
