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Interference threshold based on energy setting in cognitive radios is a non-convex optimization problem [4]. The convergence of optimization techniques like Genetic algorithm (GA) takes several iterations to fix this threshold. Here, an attempt made to use Differential evolution (DE) method for optimization after formulating the objective functions. The advantages with this method were three fold over GA. They were, a. A reduced number of iterations, b. Marginal improvement and consistency of throughput and c. Localization of the best solution. The comparative results are presented and discussed.

The increasing demand for mobility and portability in communication equipment has made wireless communication an indispensable field of research. The focus today is to construct wireless networks or nodes that can dynamically reconfigure their transmission and reception parameters. Spectrum sensing and its utilization are thus ubiquitous. This concept is called a cognitive radio and the first main function of such a system is to detect the unused spectrum and transmit in it without interference to primary users.

Newer wireless technologies crop up everyday providing better service to the users. The 3G or third generation of wireless networks are only being rolled out in most countries and research is already on to take communication to the next level with 4G or Long Term Evolution (LTE) [

Simultaneously, newer techniques are also being developed for making better utilization of the wireless resources available. One such technology is cognitive radio [

The organization of the paper is as follows. In section II, we explain in detail the sensing problem for threshold based detection of spectral holes. In section III, we present an analysis of the optimization techniques that can be used for this process and explain why we choose differential evolution. Section IV is dedicated to our proposed method of threshold based spectrum sensing using differential evolution. In Section V, we present the results obtained and compare the performance of differential evolution with that of genetic algorithms. Section VI concludes the paper, where we restate our major observations and future work.

The sensing problem under consideration here is same as the one used in [_{0} and H_{1}.

H_{0} represents a spectral hole and H_{1} represents an occupied channel. V_{k} is the Gaussian noise with power, H_{k} is the channel impulse response in the frequency domain and S_{k} is the frequency spectrum of the signal.

In order to find out whether a channel is occupied or free, we use the following energy detector.

where γ_{k} is the threshold for the k^{th} sub-band and N is the number of measurements made for each sub-band. When N is large enough, the random variables of the energy detector Y_{k} can be considered to follow a normal distribution, that is Y_{k}~N (µ_{0,k},) for the state H_{0,k} and Y_{k}~N (µ_{1,k},) for the state H_{1,k}. The sensing performance can be quantified based on the probability of identifying a spectral hole, as in (2), and the probability of identifying an occupied channel as a hole, as in (3).

In (2) and (3), P_{f} and P_{d} are the probability of false alarm and probability of detection of a transmission respectively. Here represents the probability of correctly detecting a spectral hole as a spectral hole, and not the Bayesian probability of occurrence of one event given another has occurred. This representation holds for similar probability terms encountered in (2), (3), (4) and (5).

The consideration that the energy detector specified in (1) will follow a normal distribution on taking a large number of measurements N allows us to compute the values of P_{f} and P_{d} from the cdf of the normal distribution as shown below in (4) and (5).

The γ_{k} values should be optimized such that the channel interference is minimum and the throughput achieved by the secondary user is maximum. For this purpose, we use the throughput achievable through each of the k channels, and the interference cost that has to be incurred on transmitting through the channels. The objective functions for the optimization can then be formulated, as in (6) and (7), using these values and the probability of identifying a spectral hole and the probability of missed detection.

where P_{d}_{ }(γ) and P_{f}_{ }(γ) are k dimensional vectors with each element representing the probability of detection and probability of false alarm correspondingly in the respective sub-band; r is the vector containing the throughputs attainable through each channel and c is the vector containing interference costs incurred on transmitting through each of the channel.

The non-convex nature of the objective function posed above limits us from using the conventional optimization methods like convex maximization. Hence, we considered optimizing using direct search algorithms such as genetic algorithm [

Existing methods for threshold optimization in spectrum sensing use genetic algorithms [

The working of the differential evolution algorithm is simple.

1) It has a population of candidate solutions that are sometimes called agents.

2) Positions of existing agents are combined using a mathematical formula to move the agents around the search space.

3) If there is an improvement from the old position to the new position of an agent, then it is accepted as a member of the population, else it is discarded.

4) This process is iteratively repeated to arrive at the optimal solution.

It has been shown through extensive experimental testing that the convergence properties of differential evolution are better than that of genetic algorithms [

This has prompted us to utilize differential evolution for optimizing the threshold values involved in energy based spectrum sensing.

We apply the differential evolution optimization technique to the sensing problem under consideration here, with (6) and (7) as the objective functions. A set of N threshold vectors, each of the form

.

with random values, is taken as the initial population set. G denotes the generation number. This set represents the potential candidates for optimization and is the first generation of the differential evolution scheme discussed below [

The best threshold set in each generation is the one which gives optimal values for throughput (6) and interference (7).

The next generation of vectors is generated as follows. For every vector, t_{i}_{,G} (target vector), the following three steps are performed.

1) Mutation: Three mutually distinct vectors t_{r}_{1,G}, t_{r}_{2,G}, t_{r}_{3,G} are taken such that . A mutant vector/donor vector is generated according to

where F [0, 2] is a constant which controls the magnitude of the differential variation.

2) Crossover: The diversity of the vector set is increased by developing a trial vector as

;; U[0, 1]; rndI is a random integer from []; rndI ensures that at least one element from v_{i}_{,}_{ G}_{ +}_{ 1} is incorporated into u_{i}_{,}_{ G}_{ +}_{ 1}_{.}

CR is the crossover constant [0, 1] and its value is chosen by the user. A higher value of CR causes more elements from the mutant vector v_{i}_{,}_{ G}_{ }_{+}_{ 1} to get incorporated in the trial vector u_{i}_{,}_{ G}_{ }_{+}_{ 1}. It has been shown in [

3) Selection: The trial vector u_{i}_{,}_{ G}_{ }_{+}_{ 1} is compared with the target vector v_{i}_{,}_{ G}_{ }_{+}_{ 1} and the one that gives the best values for R and I is passed on to the next generation as t_{i}_{,}_{ G}_{ }_{+}_{ 1}.

The algorithm is continued till the optimum threshold vector is found.

The optimization of the threshold values was done using both GA and DE. In order to compare the performance of the two methods, we use the number of function evaluations required as a parameter. Further, we also plot the interference vs. throughput curves of the final population with interference as the independent variable, for both DE and GA.

Simulations were done using MATLAB and the values of r, c and H are the same as used in [

The number of function evaluations that an optimization algorithm requires to converge to the optimum value is an important parameter for measuring the performance. The lower the number of function evaluations required, lesser the computational load and faster the convergence.

On optimizing the threshold using both GA and DE, we found that the number of function evaluations required by DE is far less than that of GA.

front after optimizing using DE, and the number of function evaluations required. The green dot in _{1}(x) and the maximum value of F_{2}(x) are used to fix an origin and the green dot or the optimum point is the point that has the minimum distance from the origin that was previously fixed.

In Figures 1 and 2,

DE requires less than two times the number of function evaluations as GA and this was found to be consistent over a number of optimizations. This lesser number of function evaluations means that the computational cost of optimizing using DE is far less than GA.

Another parameter for comparing the performance is the throughput achieved for various values of aggregate interference.

We can see from the graph that for most values of interference, the throughput obtained in the case of DE is greater than that of GA. This increased throughput for the same value of interference is favorable to the secondary user.

It can be seen from

population in GA are uniformly scattered and hence visually localizing the best member becomes difficult. However, in the case of DE, we can see that the best member can be easily localized.

From the above comparisons, DE has been found to be more advantageous than GA in the following ways.

1) The lesser number of function evaluations means that the rate of convergence is better in the case of DE than GA.

2) The higher throughput achieved means that DE converges to a better optimum value when compared to GA.

3) The accuracy of convergence is also evident from the scatter plots in Figures 1 and 2, since the best member is localized in the case of DE, whereas the same cannot be said for GA.

A cognitive radio network aims to maximize spectrum utilization by detecting unused spectrum and letting the secondary users utilize it. Setting the threshold for energy based spectrum sensing is a non-convex optimization process and the optimization should be such that the probability of error decreases. In this paper, we have first explained the sensing problem involved, the requirement of optimization and the objective functions. We have then stated the disadvantage of generic algorithms in these circumstances, which is the large number of iterations it takes to converge. We then propose a method for nonconvex threshold optimization using differential evolution and then compare the results obtained from differential evolution and genetic algorithms. Based on three factors, namely the number of function evaluations, the marginal increase in the throughput achieved and the easiness of localizing the best solution, we conclude that differential evolution has certain definite advantages over genetic algorithms in optimizing the threshold for energy based spectrum sensing.