Sensing Error Minimization for Cognitive Radio in Dynamic Environment using Death Penalty Differential Evolution based Threshold Adaptation

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1 Sensing Error Minimization for Cognitive Radio in Dynamic Environment using Death Penalty Differential Evolution based Threshold Adaptation Soumyadip Das 1, Sumitra Mukhopadhyay 2 1,2 Institute of Radio Physics and Electronics University of Calcutta Kolkata, India 1 soumya[dot]eltsc09[at]gmail[dot]com, 2 sumitra[dot]mu[at]gmail[dot]com Abstract In cognitive radio technology, constant changes in the environment like change in background noise, movements of the users or transmitters, interferences, etc. require the spectrum sensing parameters like detection threshold to be changed. Else, errors in spectrum sensing arises which either causes interference between transmission from primary and secondary users, or the secondary user misses an opportunity of transmission causing under usage of available spectrum bands. Thus an optimal threshold value must be determined for minimum error. Here, initially, a Death Penalty based Differential Evolution (DPDE) algorithm is proposed for constrained optimization and based on the proposed DPDE, a threshold adaptation algorithm is designed for dynamic sensing error minimization in changing environment. The performance of the proposed algorithm is compared with previously proposed gradient descend based threshold adaptation algorithm and was found to be faster and more accurate. Keywords-Cognitive radio; death penalty differential evolution; dynamic threshold adaptation; energy detection; energy sensing; sensing error minimization. I. INTRODUCTION Scarcity of available spectrum bands is a major issue nowadays for modern applications which require high speed communication. Fixed spectrum allocation scheme leads to underutilization of spectrum [1], [2] allotted to the licensed users called primary users (PU). Cognitive Radio (CR) technology is used for dynamic spectrum allocation such that, secondary users (SU) are able to use the spectrum when it is not in use by the PU, thus avoiding any harmful interference to the PU transmission. The SU senses the available spectrum holes (vacant spectrum bands) and transmits until the PU again starts to retransmit. So spectrum sensing [1]-[6] is a key issue in CR technology. Different spectrum sensing algorithms include energy detection [1], cyclo-stationary feature detection [1, 3], matched filtering [3], poly phase filter banks [6], etc. According to IEEE standard [7], any spectrum sensing technique must detect the presence of the licensed user within two seconds. In practical situation, the factors like background noise, interference, etc lead to constant changes in the environment which requires the sensing parameters to be dynamically changed or adapted for more accurate spectrum sensing. In this paper, adaptation of sensing threshold for energy based spectrum sensing in dynamic environment is studied. In [5-6], Gradient descent optimization based sensing Threshold Adaptation (GTA) algorithm was proposed for dynamic environment. The convergence speed and accuracy of the gradient descent algorithm is highly dependent on the initial guess value (ε ) of threshold and the step size (μ). Repeated trial experimentations are required to find the correct values of ε and μ for efficient functioning of the algorithm. On the other hand, in evolutionary algorithms (EA) [8], initially a number of trial solutions are randomly predicted over the range of possible solutions. Thus repeated trial experimentations are not required to find a correct initial guess solution. Thus, bioinspired evolutionary algorithms are more robust and have been used successfully for solving a number of complex engineering problems. In this paper, the problem of sensing threshold adaptation due to change in environment is formulated as a constrained optimization problem and a differential evolution (DE) [9] based algorithm is proposed for solving it. DE is a robust bio-inspired stochastic search method first proposed by Storn and Price [9]. It has a number of advantages over other EAs [10]. Coding and implementation of DE is much simpler compared to other EAs. Yet it exhibits better performance than most other EAs in solving a wide variety of problems [10]. This facilitates the use of DE to solve real world optimization problems having strict time and resource constraints. Compared to other EAs, DE has fewer control parameters namely amplification factor ( F ) and crossover probability (CR) and their effect on the performance is well studied [10]. In the proposed DE based threshold adaptation method, the DE strategy used is denoted as DE/rand/1/bin strategy [9]. Evolutionary algorithms are inherently designed for unconstrained optimization and require constraint handling techniques for constrained optimization problems. Death penalty [11] is a simple and elementary method for handling constraints. Death penalty constraint handling technique is integrated with DE and Death Penalty Differential Evolution (DPDE) is proposed to solve the above constrained optimization problem of sensing threshold adaptation due to change in environment. The work is undertaken as part of Media Lab Asia Project entitled Mobile Broaand Service Support over Cognitive Radio Networks /16/$ IEEE

2 Figure 1. Death Penalty Adaptation Based Energy detector Figure 1 shows the basic block diagram showing death penalty based adaptation of detection threshold. The remaining sections are arranged as follows: In Section II, spectrum sensing by energy detection is described. In Section III dynamic threshold adaptation for sensing error minimization is modeled as a constrained optimization problem. In Section IV, the basic Differential Evolution (DE) algorithm and Death Penalty constrained handling technique are explained. Then DE is modified to introduce Death Penalty Differential Evolution (DPDE) algorithm as a new technique for constrained optimization. In Section V Death Penalty Differential Evolution based Threshold Adaptation (DPDETA) algorithm is proposed. In Section VI, the experiments are described that evaluate the performance of DPDETA and compared with the Gradient based threshold adaptation (GTA) algorithm [5]. Concluding remarks are given in Section VII. II. SPECTRUM SENSING BY ENERGY DETECTION Spectrum sensing is done to determine whether the primary user (PU) is using the spectrum or not at a particular time instant. In case when the PU is absent, a secondary user is allowed to opportunistically use the spectrum and thus enhance the spectrum utilization. Spectrum sensing is thus a binary hypothesis problem where the null hypothesis (H ) and the alternative hypothesis (H ) are given as follows: H :PU is absent H : PU is in operation Let at the n th time instant, x(n) be the signal received by the SU system; u(n)be an independent identically distributed (i.i.d) Additive White Gaussian Noise (AWGN) with mean zero and variance σ ; s(n) be the PU signal which is also an i.i.d random signal with mean zero and variance σ. The signal received at the secondary receiver for H is given as x(n) = u(n) (1) The signal received at the secondary receiver for H is given as x(n) = s(n) + u(n) (2) In energy based spectrum sensing, the decision rule can be stated as decide for H, if T < ε (3) H, if T ε where, ε is the sensing threshold and Tis the test statistic of the energy detector given as follows T(x) = x(n) (4) There are two types of sensing error probabilities, namely probability of false alarm (P ) and probability of missed detection (P ). The probability that the presence of primary signal is detected when PU is actually absent is called probability of false alarm (P ) and the probability that the primary signal is not detected when PU is actually present is called probability of missed detection ( P ). In terms of complementary error function, erfc( ) the above probabilities can be modeled as [5] P [ε(n)] = Pr[(T(x) > ε(n) H )] = 1 2 erfc ε(n) Nσ 2σ N P = (5) Nσ erfc ε(n) (6) 2σ N where σ is the variance of the received signal x(n) and is given by σ = σ + σ (7) The noise variance σ is assumed to be known beforehand [7] or, may be determined experimentally [14], or can be estimated by modeling uncontaminated PU signal as p-th order AR model along with Yule-Walker equation [5][13]. The PU user power can be calculated by subtracting noise power from the received signal power or the same can be known by IEEE standard [7] and thus SNR can be calculated which is given by γ = σ σ. III. DYNAMIC THRESHOLD ADAPTATION FOR SENSING ERROR MINIMIZATION AS A CONSTRAINED OPTIMIZATION PROBLEM Dynamic environment arises due to background noise and interference, movements of the transmitters and users, etc. False alarm and miss-detection are functions of the detection threshold ε(n). So, with changes in the environment, it is required to adjust ε(n) to minimize the sensing error probabilities. For a high value of P, the secondary user misses an opportunity to use the vacant spectrum. Again, from equation (5), it is noticed that higher the value of ε(n), lesser is the P. So, high value of ε(n) is ideal for the interest of the SU. But, from equation (6), if the ε(n) is too high, P increases, that is, for high sensing threshold, signal with low SNR cannot be detected. Thus, optimal value of threshold ε(n) must be chosen with two goals namely 1. Increasing ε(n) to minimize the probability of false alarm and 2. Decreasing ε(n) to minimize the probability of missed detection The total sensing error (e) is contributed by P and P and is given as follows

3 e[ε(n)] = (1 δ)p + δp (8) where, δ is defined as a weighting parameter, which determines the relative contribution of P and P to the spectrum sensing error e[ε(n)]. Thus, the problem of spectrum sensing error minimization by dynamic threshold adaptation can be formulated as a constrained optimization problem, formally defined as follows min () e[ε(n)] (9a) subject to P [ε(n)] α (9b) and P [ε(n)] β (9c) where, α and β are maximum allowable values of P and P respectively. Gradient descent based threshold adaptation (GTA) algorithm [5] is available to find the optimal threshold. But using the proposed bio-inspired differential evolution (DE) based algorithm, the optimal threshold can be solved much quickly and more accurately as would be explained and proved by experiments in the subsequent sections. IV. CONSTRAINED OPTIMIZATION USING DE: DEATH PENALTY DIFFERENTIAL EVOLUTION (DPDE) DE is a comparatively newer bio-inspired optimization algorithm [9]. It constitutes the steps like mutation, crossover and selection [9]. Here, trial vectors are generated using mutation and crossover operators and selection operator is used to determine the vectors that form the next generation. Say there are D independent variables or optimization parameters. These are encoded to form a D-dimensional solution called an individual. The search process starts with formation of the initial population consisting of individuals chosen randomly with uniform distribution in the search space denoted byx, i = 1,2,3,.., N, where N is the population size and G denotes the generation number. Mutation operator is used to generate a mutant vector for each individual as follows [9] m = x + F x x (10) where x, x and x are different, F [0,2] is the amplification factor. In crossover step, the trial vector is generated as follows [15] t = x, rand(j) CR or j = randn(i) m, rand(j) > CR and j randn(i) (11) where j = 1, 2, D, randn(i) {1,2, D} is a random integer, rand(j) [0,1] is a random number and CR [0,1] is the crossover probability. Finally, the selection operator compares the fitness values of the target vector and the trial vector and selects one of them into the next generation as follows (for a minimization problem) [9] x = t, f(t ) < f(x ) x, f(t ) f(x ) (12) Among the N solutions of the new generation, the one having the best fitness value is the output of that iteration/generation. This process repeats iteratively in search of a better solution considering this new generation as the parent generation on which the mutation, crossover and selection operators act again in the same way. In order to solve constrained optimization problem using the above DE algorithm, it is integrated with the death penalty constraint handling technique [11], [12] and thus Death Penalty Differential Evolution (DPDE) is proposed as described in the next section. A. Death Penatly DE for constrained handling Death penalty is a simple constraint handling method [11]. This method is integrated with DE for constraint handling and we propose Death Penalty DE (DPDE). Here, an evaluation function is defined as eval(x) = f(x) + p (13) where, f(x) is the fitness function, x is the individual and p is the penalty value. A large penalty value like (+ ) is added to the infeasible solutions so that they are rejected during the selection. If F denotes the feasible region, then the penalty value p can be modeled as, p = 0 + if xεf otherwise (14) To integrate the above death penalty technique into DE, the selection operator of equation (12) is modified as x = t, eval(t ) < eval(x ) x (15), eval(t ) eval(x ) Thus the selection operator rejects the infeasible solutions and handles the constraints. V. DEATH PENALTY DE BASED THRESHOLD ADAPTATION (DPDETA) FOR SENSING ERROR MINIMIZATION In this section, Death Penalty DE based threshold adaptation (DPDETA) for sensing error minimization in dynamic environment, is described. The basic scheme is to solve the constrained optimization problem (9) by DPDE. Here, each individual in a population is actually a predicted detection threshold (ε) called as detection threshold individual (DTI). In the initialization step, N initial DTIs are randomly predicted with uniform probability in the range of the possible values of operating threshold. These constitute the initial population. Here e[ε(n)] is the fitness function and overall fitness is evaluated using equation (13) by replacing f(x) with e[ε(n)] to form where ε(n) = x. eval(x ) = e[x ] + p (16) The overall DPDETA algorithm is given in Algorithm 1. Algorithm 1: Death Penalty DE based Threshold Adaptation (DPDETA) algorithm 1. Input parameter: α, β, δ, N, max_gen; 2. Estimate the noise variance (σ ) [13] 3. Measure the variance (σ ) of the received signal x(n) 4. Calculate primary user signal variance σ using (7) 5. Generate N number of random solutions DTI in the search domain determined from error profile 6. Calculate P and P using (5), (6) 7. if P α or P β

4 p = + ; else p = 0; end if 8. Evaluate initial parent generation using equations (8) and (16) 9. while (number of iteration max_gen) do a) Generate mutant vector using (10) b) Generate trial vector using (11) c) Calculate P and P using (5), (6) using trial vector d) if P α or P β p = + ; else p = 0; end if e) Evaluate trial vector using (9) and (16) f) Select between trial vector or target vector using (15) to form new Generation g) Output the DTI having the minimum (best) fitness as the optimal threshold ε = ε(n) 10. end while 11. Repeat from step 2. VI. SIMULATION AND NUMERICAL RESULTS A. Experiments The proposed DPDETA algorithm was tested and its performance was compared with the Gradient based threshold adaptation (GTA) algorithm proposed in [5]. The algorithms were implemented in MATLAB and simulated under changing environmental scenarios which was implemented by changing the SNR of the received signal as -3 db, -2 db, 0 db and 5 db at regular time intervals. Experiments were performed for δ = 0.1, 0.5 and 0.9 keeping α = 0.1and β = 0.2. Steps of the experimentation are as follows: TABLE I. THE LOWEST ATTAINABLE SENSING ERROR VALUE AND THE CORRESPONDING THRESHOLD VALUE(ε) FOR THE DYNAMIC SCENARIOS δ δ=0.1 δ=0.5 δ=0.9 SNR Min. Min. ε ε ε (db) error error Min. Error e e e-7 (i) Using equation (8), the sensing error is calculated for γ = 3 db, 2 db, 0 db and 5 db at δ = 0.1, 0.5 and 0.9 for a range of threshold values. The lowest errors obtained and the corresponding threshold values for each case are shown in Table I. The plots of error against threshold for γ = 3 db, 2 db, 0 db and 5 db at δ = 0.5 are shown in Figure 2 and those for δ = 0.1, 0.5 and 0.9 at γ = 0 db are shown in Figure 2. These plots are called the error profiles and the range of the threshold values, for which the error is the minimum, can be found from them. Thus the search domain of DPDETA is determined. The lowest point in each curve gives the lowest achievable sensing error and the corresponding threshold at that particular scenario. Also, the lowest sensing error achieved by DPDETA and GTA are indicated which will be discussed subsequently. (ii) The GTA algorithm [5] was tested for gradient step size μ = 0.2, 0.5 and 0.9 at δ=0.5. Tolerance was fixed at ε = 10 and SNR was changed at every 1,000,000 iterations as γ = 3 db, 0 db, 3 db and5 db. The values of threshold and sensing error obtained at every iteration are plotted and shown in Figures 3 and 3 respectively to demonstrate the dynamic adjustment of threshold as the SNR changes. The algorithm fails if the initial starting value is not chosen carefully which requires some trial runs. For different values ofμ, the algorithm converges to different threshold values. The difference is larger for higher SNR. Then similar simulations were repeated for μ = 0.5 and δ=0.1, 0.5 and 0.9 respectively. These simulation results are plotted in Figures 4 and 4. Figure 2. Sensing error profile for γ = 3dB, 2dB, 0dB and 5dB at δ = 0.5; Sensing error profile for δ = 0.1, 0.5 and 0.9 at γ = 0dB Figure 3. Values of threshold and sensing error with iterations obtained using GTA [5] for step size μ=0.2,0.5 and 0.9 at δ=0.5

5 (iii) For the proposed DPDETA algorithm, population size is kept as N = 10, and maximum number of generations (max_gen) was fixed to 200. Simulation was done for δ = 0.1, 0.5 and 0.9 with changing values of the SNR as γ = 3 db, 0 db, 3 db and 5 db after every 200 iterations or generations. DPDETA dynamically adjusts the sensing threshold to attain the minimum sensing error when the SNR changes. Simulation results are shown in Figure 5. In Figure 5 the best threshold value obtained in each generation is plotted and in Figure 5 the corresponding minimum error value is plotted. B. Comparison between DPDETA and GTA[5] DPDETA and GTA [5] are compared in terms of speed and accuracy. The results regarding the minimum sensing error, corresponding threshold value (η ), the threshold adaptation time (τ), deviation (η) of the minimum error value from the minimum attainable sensing error and P and P obtained by DPDETA and GTA [5] are indicated in Table II. for each generation of DPDETA is ms and that of each iteration of GTA is ms. Then, the threshold adaptation time (τ) is calculated by multiplying the number of generations or iterations (ξ ) required for threshold adaptation with the execution time for each generation or iteration. These are shown in Table II. In figure 6, the time required by the algorithms to adjust the sensing threshold in each environmental scenario is compared. In almost all scenarios DPDETA was found to converge faster than GTA [5]. Figure 4. Values of the threshold and Dynamic values of the sensing error with iterations obtained using GTA [5] for step size μ=0.5 The time required for threshold adaptation by the algorithms is calculated as follows. First DPDETA and GTA both were run for 50 times each and the system runtimes were recorded using MATLAB. In each run, maximum iteration of GTA was fixed at 100,000 and maximum generation (max_gen) of DPDETA was fixed at Then, average execution times for one run of GTA and DPDETA were calculated from the 50 run times. Finally, these were respectively divided by the number of iterations in each run (i.e. 100,000) in case of GTA and by the number of generations in each run (that is 1000) in case of DPDETA to find the average execution time required for each iteration of GTA and for each generation of DPDETA. The average execution time Figure 5. Best threshold value obtained to minimise the sensing error is plotted against the generation number; Minimum error obtained in each generation, using DPDETA Accuracy of each algorithm is defined in terms of deviation (η ) of the obtained minimum sensing error value from the minimum attainable sensing error (listed in Table I) in a particular environmental scenario. η of DPDETA and GTA [5] are shown in Table II and compared in the bar graph of Figure 7 for each environmental scenario. In most of the cases, η = η = 0, except for SNR = 5 db where DPDETA was much more accurate than GTA and for SNR=-3dB, δ = 0.9, deviation η = η. The minimum errors and the threshold values obtained using DPDETA and GTA [5] are plotted in the error profiles for the different scenarios in Figure 2. Most of these points coincide with the lowest point of the error profile except for the plot of GTA [5] at δ = 0.9 and SNR=5dB which can be confirmed from the bar graph of Figure 7. Also, Figure 3 proves that GTA is not robust for all the scenarios and depends on value of μ selected.thus it can be concluded that DPDETA is much faster and more accurate than GTA [5] in most of the scenarios.

6 TABLE II. THE MINIMUM SENSING ERROR OBTAINED, THE CORRESPONDING THRESHOLD VALUE (ε), THE TIME REQUIRED FOR THRESHOLD ADAPTATION(τ), DEVIATION OF OBTAINED MINIMUM ERROR VALUE FROM THE MINIMUM ATTAINABLE SENSING ERROR (η), P AND P OBTAINED BY DPDETA AND GTA [5] S N R δ Gradient based Threshold adaptation (GTA) algorithm DPDE based Threshold Adaptation (DPDETA) algorithm ε error η DPDETA P fa P m ξ τ(msec) ε error η GTA P fa P m ξ τ(msec) e e e e-5 9.4e-5 9.5e-4 4.9e e-7 4.3e e e e-5 8.3e-5 1.7e-4 9.1e e-7 1e e e e-5 8.0e-5 8.9e-5 1.1e e-7 2.5e e e ACKNOWLEDGMENT The work is undertaken as part of Media Lab Asia Project entitled Mobile Broaand Service Support over Cognitive Radio Networks. Figure 6. Comparison of convergence time taken by GTA and DPDETA algorithm in different environtal scenarios. Figure 7. Comparison of deviation from minimum sensing error at different environtal scenarios by GTA and DPDETA. VII. CONCLUSION A Death Penalty Differential Evolution (DPDE) based threshold adaptation algorithm (DPDETA) for dynamic sensing error minimization in Cognitive Radio is proposed. Whenever any change occurs in the environment, the DPDETA algorithm dynamically changes the sensing threshold with an aim to minimize the sensing error in that changed environment. DPDETA is more robust and responds to a change in SNR much faster than the previously proposed deterministic gradient descend based algorithm [5] for threshold adaptation. Further tests may be done on real signals. Also, dynamic threshold adaptation problem can be solved using other bioinspired algorithms to test their performance. REFERENCES [1] K. B. Letaief and W. Zhang, Cooperative communications for cognitive radio networks, Proc. IEEE, vol. 97, no. 5, pp , May, [2] Y. C. Liang et al., Sensing-throughput tradeoff for cognitive radio networks, IEEE Trans. Wireless Commun., vol. 7, no. 4, pp , Apr, [3] W. Zhang et al., Cooperative spectrum sensing optimization in cognitive radio networks, in 2008 IEEE International Conf. on Commun., Beijing, 2008, pp [4] Q. Zhang et al., Cooperative relay to improve diversity in cognitive radio networks, IEEE Commun. Mag., vol. 47, no. 2, pp , [5] D. R. Joshi et al., Gradient-Based Threshold Adaptation for Energy Detector in Cognitive Radio Systems, IEEE Commun. Lett., vol. 15, no. 1, pp , Jan, [6] D. R. Joshi et al., Dynamic threshold adaptation for spectrum sensing in cognitive radio systems, in 2010 IEEE Radio and Wireless Symposium (RWS), 2010, pp [7] Shellhammer, Stephen J. "Spectrum sensing in IEEE " IAPR Wksp. Cognitive Info. Processing, pp. 9-10, [8] T. Back, Evolutionary algorithms in theory and practice: evolution strategies, evolutionary programming, genetic algorithms, Oxford university press, [9] K. Price et al., Differential evolution: a practical approach to global optimization, Springer Science & Business Media, [10] S. Das and P. N. Suganthan, "Differential evolution: a survey of the state-of-the-art," IEEE Trans. Evol. Comput., vol. 15, no. 1 pp. 4-31, Feb [11] Özgür Yeniay, Penalty function methods for constrained optimization with genetic algorithms. Mathematical and Computational Applications, vol. 10, no. 1, pp , Apr [12] Z. Michalewicz, and M. Schoenauer, "Evolutionary algorithms for constrained parameter optimization problems," Evolutionary computation, vol. 4, no. 1, pp. 1-32, [13] K. K. Paliwal, "Estimation of noise variance from the noisy AR signal and its application in speech enhancement," IEEE Trans. Acoust., Speech, Signal Process, vol. 36, no. 2, pp , Feb [14] M. Wellens and P. Mähönen, "Lessons learned from an extensive spectrum occupancy measurement campaign and a stochastic duty cycle model," Mobile networks and applications, vol. 15, no. 3, pp , 2010.