Margin Adaptive Resource Allocation for Multi user OFDM Systems by Particle Swarm Optimization and Differential Evolution

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1 Margin Adaptive Resource Allocation for Multi user OFDM Systems by Particle Swarm Optimization and Differential Evolution Imran Ahmed, Sonia Sadeque, and Suraiya Pervin Northern University Bangladesh, Dhaka, Bangladesh Simon Fraser University, Burnaby, Canada University of Dhaka, Dhaka, Bangladesh Abstract Like many wireless systems, Orthogonal Frequency Division Multiplexing (OFDM needs proper allocation of limited resources such as total transmit power and available frequency bandwidth among the users to meet their service requirements. In this paper, different versions of two evolutionary approaches, Differential Evolution (DE and Particle Swarm Optimization (PSO have been applied for adaptive sub carrier and bit allocations to minimize the overall transmit power of a multi user OFDM system. Each user will be assigned a number of sub carriers with at least one minimum sub carrier even at the worst case. Then the number of bits and the transmit power level for each user are calculated to obtain the optimum requirements. Simulation results show that both the approaches outperform the conventional static and many other dynamic resource allocation schemes in multi user scenario. The results also reveal that the employed two different schemes of DE show better performances than the original and modified versions of PSO. I. INTRODUCTION Being very efficient in combating Inter Symbol Interference (ISI and in the use of available bandwidth, Orthogonal Frequency Division Multiplexing (OFDM is considered as one of the most promising transmission techniques in wideband wireless systems [1], [2]. Multi user OFDM allows multiple users to share the sub carriers in each OFDM frame [3]. Allocation of sub carriers to the users with the best signal to noise ratio (SNR improves the performance of the system. The system performance can be further enhanced by employing resource allocation techniques including number of bits and sub carrier allocation for each user in response to the channel state information (CSI. Two classes of resource allocation schemes exist in OFDM systems: fixed resource allocation [3] and dynamic resource allocation [4],[5],[6],[7]. Fixed resource allocation schemes, such as time division multiple access (TDMA and frequency division multiple access (FDMA, assign an independent dimension, e.g. time slot or sub channel, to each user. A fixed resource allocation scheme is not optimal since the scheme is fixed regardless of the current channel condition. On the other hand, dynamic resource allocation allocates a dimension adaptively to the users based on their channel gains. Due to the time varying nature of the wireless channel, dynamic resource allocation makes full use of multi user diversity to achieve higher performance. Two classes of optimization techniques have been proposed in the dynamic multi user OFDM literature: margin adaptive (MA [7] and rate adaptive (RA [6]. The objective of MA technique is to achieve the minimum overall transmit power given the constraints on the users data rate and bit error rate (BER. On the other hand, the objective of RA technique is to maximize total throughput, where total transmit power and BER are assumed to be constant. These optimization problems are nonlinear and hence computationally intensive to solve. In this paper, although we confine ourselves within MA resource allocation, it can be easily extended for RA schmes. In [7], Wong et. al. proposed an iterative searching algorithm that applies Lagrangian relaxation for optimum multi user sub carrier, bit and power allocation. The algorithm is close to the lower bound of power requirement with high and complex computation. The algorithm proposed in [8], however, over simplifies the sub carrier allocation but could not fully utilize the multi user diversity. In [5], an iterated water-filling algorithm is proposed; the algorithm can acquire similar performance as Wong s algorithm and avoids the computational complexity. Y. B. Reddy et. al. introduced Genetic Algorithm (GA in resource allocation with significant improvements [9]. [1] applies Particle Swarm Optimization (PSO in allocating multi user OFDM resources with some improvements. Q. Feng et. al. worked on differential evolution (DE algorithm for resource allocation of OFDM system and showed some significant improvements [11]. In this paper, PSO [12], [13], [14] has been modified and applied to minimize the total transmit power to allocate sub carriers and number of bits for multi user OFDM systems. The modifications are done in such a way, so that it overcomes the shortcomings of original PSO. In addition, two versions of DE [15], [16] have been deployed as optimizers, so that they can search over diversified spaces more efficiently and quickly. The functions of the considered two types of DE have been compared with those of the original and modified versions of PSO. The overall performances of modified PSO (MPSO and DE methods are further compared with some of the existing fixed and dynamic sub carrier and bit allocation schemes. We show that, the performance and convergence of the conventional PSO have been improved by introducing dynamic inertia weight and by inserting generation index in position update equation, respectively. Simulation results show that, for higher number of users, two types of DE algorithm outperform the considered existing algorithms and even the MPSO, but take longer time for convergence. The rest of the paper is organized as follows: In Section II, the problem of dynamic resource allocation is formulated in multi user OFDM system and system model is described. The MPSO and DE based description for this problem and the parameters for these evolutionary approaches are discussed in Section III. Section IV gives the numerical results and the discussion of the system performance and finally Section V finishes the paper by giving conclusion.

2 II. SYSTEM MODEL In this paper, we consider a multi user OFDM system having K users with (k = 1, 2,..., K and N sub carriers with (n = 1, 2,..., N. The resource allocator at base station allots a subset of N sub carriers to each user and determines the number of bits per each assigned sub carrier on downlink transmission. b n,k {, 1, 2,..., B M } signifies the number of bits for nth sub carrier and kth user, where B M denotes the maximum number of information bits that can be transmitted associated with each sub carrier for a particular modulation scheme. The allocator is assumed to have perfect instantaneous CSI. The channel is modeled as slow varying Rayleigh faded and its components have independent identically distributed (i.i.d complex values with zero mean and unit variance. Let h n,k represents the magnitude of instantaneous channel gain of nth sub carrier and kth user. Additive white Gaussian noise (AWGN is considered in the system with elements of complex value, zero mean and unit variance. The required transmission power of nth sub carrier and kth user at a specified BER, P b for b n,k bits is given by [4], where p n,k = f(b n,k (1 f(b n,k = N ( ( 2 Q 1 Pb ( 2 b n,k 1. (2 3 4 Here N denotes the noise power spectral density and Q 1 denotes the inverse Q function where Q(x = 1 2π e t2 2 dt. In multi user scenario, not more than one user is considered to share a particular sub carrier. Mathematically it is expressed as { 1, bn,k λ n,k = (3, b n,k = The required total transmission power, P can be written as follows [4] P = N f(b n,k λ n,k. (4 The sub carrier and bit allocation problem for minimizing the total transmit power at a constant P b can be formulated as subject to N arg min b n,k,h n,k N f(b n,k x λ n,k (5 λ n,k = N for b n,k {, 1, 2,..., B M } and R k = N b n,k for k = 1, 2,..., K. R k > needs to satisfy n=1 for practical realization of allocation. Moreover, it should be noted that, while satisfying (5, transmit power of each user, p k can also be allocated along other resources (which is termed as power allocation in the literature. Although we confine ourselves within sub carrier and bit allocations in this paper, it can easily be extended for power allocation by satisfying MA optimization. III. APPLICATION OF MPSO AND DE AS OPTIMIZERS A. MPSO Like other evolutionary computation techniques, PSO is a population based search algorithm and is initialized with a population of random solutions, called particles [12]. Each particle in PSO is also associated with a velocity. Particles fly through the search space with velocities which are dynamically adjusted according to their historical behaviors. Therefore, the particles have a tendency to fly towards the better and better search area over the course of search process [13], [14]. The original PSO algorithm is discovered through simplified social model simulation. The PSO algorithm works on the social behavior of particles in the swarm. Therefore, it finds the global best solution by simply adjusting the trajectory of each individual towards its own best location and towards the best particle of the entire swarm at each time step (generation. Optimization by MPSO requires position and velocity values as initial population (swarm set [12]. For defining the problem of multi user OFDM systems, we need to obtain position and velocity matrices as initial population (swarm sets. To this end, we form a channel matrix, H of K rows and N columns where each of the elements denotes channel gain for a definite user using a definite sub carrier. From the channel gains, we form a bit matrix, B of same size of H according to water filling algorithm [4]. According to H and B, we form velocity and position matrices with M rows and N columns each, where M denotes the size of initial population (swarm for each sub carrier. The position matrix consists of user indices. The original PSO updates the components of position and velocity matrices (x p,q and v p,q, respectively from ith generation to (i + 1th generation according to the following [12] v i+1 p,q = wv i p,q + c 1 r i 1 ( ζ i p,q x i p,q + c2 r i 2 ( ψ i q x i p,q (6 x i+1 p,q = xi p,q + vi p,q. (7 for p {1, 2,, M}, q {1, 2,, N}. Here, w is the inertia weight, c 1 and c 2 are positive constants, r i 1 and ri 2 are two random variables within the range [ : 1]. ζ i p,q and ψ i q are the local and global optimum values, respectively for a specific iteration index, i. For every generation, we verify (5. After a number of generations, we obtain the optimum result for a definite arrangement of user index from which we can obtain sub carrier and bit arrangements. It was observed from [1] that, in comparison to GA, PSO needs more iterations to converge to the optimum value. In (7, we see that the previous position has been added with the newly obtained velocity to get the new position which reveals the mismatch of dimensions. To give an idea of timing information to the update equations, we introduced the generation index in the position update equation, which gives the following expression x i+1 p,q = x i p,q + i v i p,q. (8 where i denotes the generation index normalized by total number of iterations. In conventional PSO, inertia weight was static throughout all the iterations. In this paper, we decrease the inertia weight from a relatively large value to a small value through the course of the PSO run. The reason behind it is that, the large inertia weight facilitates global search while the small inertia weight facilitates local search and this dynamic

3 change of inertia weight helps towards better results [14]. The inertia weight, w in (6, is replaced by the dynamic inertia weight w i, which can be defined as follows w i = w h i w s (9 where w s = (w h w l /w r. Here w h, w l and w r define maximum, minimum and incremental rate of inertia weight. B. DE The DE algorithm is a population based algorithm like GA using the similar operations like crossover, mutation and selection. The main difference in constructing better solutions is that GA relies on crossover while DE relies on mutation operation. This main operation is based on the differences of randomly sampled pairs of solutions in the population and it is defined as mutation. Based on the mutation, we define two different schemes of DE algorithm as DE 1 (DE/rand/1 and DE 2 (DE/rand/1 with per-vector-dither[17]. Both of the schemes require three control parameters: weight factor (F, crossover rate (C R, and population size (N P. The initially generated populations are moved towards the optimum solution by carrying out mutation, crossover and selection operation for each generation. Optimization by DE 1 and DE 2 requires 3 random matrices as initial populations. These matrices consist user indices and each of the matrices are of N P rows and N columns. In every generation, we shuffle the three matrices by using a rotate matrix which is also randomly generated. However, for a definite generation i, we then perform the mutation operation by differential variation and form a mutant matrix, m X i. This operation depends on the type of DE and for DE 1 and DE 2, the operations for each of the elements of m X i are mx i p,q = ri (3 p,q + F (r i r i (1 (1p,q (2p,q and mx i p,q = ri (3 p,q + (r i (1p,q r i (2p,q ((1 Fr + F, (11 respectively for p {1, 2,, N R } and q {1, 2,, N}. r i (1 p,q, r i (2 p,q and r i (3 p,q denote the (p, q elements of 3 random matrices at generation, i. The mutant matrix, m X i of ith generation is then compared with the solution matrix, X i 1 of (i 1th generation according to the H and B by satisfying (5. Crossover operator C R is used to increase the diversity of mutant matrix. This constant represents the probability of trial vector inherits parameter values from the mutant matrix. Mutant individual and target individual are subjected to crossover to generate the trial individual according to C R. After crossover, selection operation is performed to obtain the new solution matrix, X i for ith generation. After a definite number of generations, the convergence is achieved and we get the sub carrier and number of bits from the converged user indices. IV. NUMERICAL RESULTS In this section we discuss on the simulations and their results under different conditions. TABLE I SPECIFICATIONS OF MPSO Parameter Value Initial Swarm Size 25 Generations 1 to 1 c c w h 1.2 w l.1 TABLE II SPECIFICATIONS OF DE Parameter Value Initial Population Size 25 Generations 1 to 1 F.85 C R.9 A. Specifications In the simulations, slow varying Rayleigh fading channel has been used and it has been assumed to be known to the resource allocator at base station. The total transmitted power and bandwidth have been taken as.1w and 1MHz. The overall bit error rate (BER is taken as 1 3. The total bandwidth is divided into 64 sub carriers for different number of users whereas the user locations are assumed to be equally distributed. 2, 4, 6 and 8 users have been considered in different aspects. We take b n,k {, 2, 4, 6} which specify no modulation, 4PSK, 16QAM and 64QAM, respectively. The parameters of MPSO and DE follow the specifications of Table I and II. B. Results Fig. 1 shows minimization of total transmit power vs. number of generations which actually shows the nature of convergence curves of different optimization schemes. First, we see that, the rate of convergence has been improved by modifying the PSO which was the shortcoming of original PSO in comparison to other evolutionary approaches like GA [1]. The introduction of generation index in the position update equation of MPSO helps to achieve the faster convergence. Moreover, the final optimum result is also improved by using MPSO over the original version of it. By using dynamic inertia weight, the searching operation has become more efficient for MPSO. The high initial value of inertia weight helps to achieve global optimum search space first. Then the gradual decrease of inertia weight facilitates the local search. The rates of convergence and final converged values of DE 1 and DE 2 are also comparable to MPSO. It should be noted that, the number of generations does not explicitly indicate the amount of time required by the optimization schemes. For instance, it can be shown that, the time required by DE 1 amd DE 2 for each generation takes more time than that by MPSO. Table III shows 1 trial runs to minimize total transmit power for 4 user OFDM system using different algorithms. It is evident from most of the observations that, we get better results for two branches of DE. The reason behind showing this table is that, each of the evolutionary algorithms provide

4 Minimum transmit power (in dbm MPSO PSO DE 1 DE Generation Index Fig. 1. Convergence curves of different algorithms for allocating sub carrier and bits of 4 user OFDM systems. TABLE III MINIMUM TRANSMIT POWER (IN dbm USING DIFFERENT ALGORITHMS FOR 4 USER OFDM SYSTEM WITH DIFFERENT RUN TIMES. ALL THE SIMULATIONS HAVE BEEN CARRIED OUT ON A PC (PROCESSOR: INTEL(R, CORE(TM 2 CPU, 1.73 GHZ, RAM: 248 MB Run Index MPSO DE 1 DE Mean nearly same results for all the trial runs. This result is more highlighted with respect to different number of users in Fig. 2. Here, we show minimization of total transmit power vs. number of users with different algorithms 1. We see that all the dynamic allocation algorithms outperform fixed TDMA and FDMA schemes. Although Wong s [7] algorithm 2 shows an excellent performance for lower number of users, its performance declines for high number of users. On the contrary, MPSO and DE algorithms do not provide good performance for lower number of users at all. But their performances improve for high number of users. This is mainly due to the fact that, in comparison to other methods, evolutionary 1 The specifications of GA follows [18] 2 The specifications of Wong s algorithm follows [7] Minimum transmit power (in dbm TDMA (fixed FDMA (fixed Algorithm of Wong GA PSO MPSO DE1 DE Number of users Fig. 2. Minimum total transmit power vs. number of users for different algorithms for multi user OFDM systems. TABLE IV EXECUTION TIME (IN SECOND REQUIRED TO REACH A TARGET VALUE OF 3.45 dbm FOR 4 USER OFDM SYSTEM IN DIFFERENT TRIAL RUN WITH DIFFERENT ALGORITHMS. ALL THE SIMULATIONS HAVE BEEN CARRIED OUT ON A PC (PROCESSOR: INTEL(R, CORE(TM 2 CPU, 1.73 GHZ, RAM: 248 MB Run Index PSO MPSO DE-1 DE-2 [7] algorithms can handle efficiently with large number of data in terms of performance and complexity [1]. Moreover, among the considered evolutionary approaches, DE 2 is found to be the best in minimizing the total transmit power. Because, relatively higher value of C R (.9 is chosen for simulations, which means 9 percent of the elements of the trial vector were identical to those of the mutant vector and it implies a high density. It seems that, due to the high crossover constant, the path length is increased without a significant higher speed to approach the minimum. With each generation, the individuals got closer to each other and converged before they reach the minimum. On the other hand, if C R is chosen too small then more generations are likely to be needed to find the minimum or it might even not find the global minimum value [15]. In Table IV, we compare the execution time required to reach a target value of 3.45 dbm by different algorithms for 4 user OFDM system. It is of high importance to calculate the convergence time of the evolutionary algorithms for practical implementation. Several conclusions can be drawn from the Table IV. First we see that, MPSO shows better efficiency in execution time in comparison to its original version. Moreover,

5 execution time of MPSO is also comparable to that of [7] for all the considered trial runs which makes MPSO as practically feasible to implement. Although it is evident from Fig. 2 that both the versions of DE outperform MPSO for higher number of users, their time requirements are relatively higher than MPSO. As greedy selection scheme is used in DE, the trial vector yields a better cost function value compared to the parameter vector. As a result, DE shows the significant improvement in the overall performance, although its execution time is relatively higher than the other ones. However, DE-2 needs less time to converge than DE-1 because of the variation in calculating the differential variation. DE-2 uses some randomly generated value with F to calculate the difference vector which emerges to minimize the total execution time in reaching the optimum value [16]. TABLE V NUMBER OF FUNCTIONS EVALUATED (RUN TIME FUNCTIONS OF THE ALGORITHMS OF DIFFERENT TRIAL RUN FOR MA OPTIMIZATION FOR 4 USER OFDM SYSTEM FOR A TARGET VALUE OF 3.45 dbm. ALL THE SIMULATIONS WERE CARRIED OUT ON A PC (PROCESSOR: INTEL(R, CORE(TM 2 CPU, 2. GHZ, RAM: 4 GB. Run Index MPSO DE-1 DE Table V represents the number of functions evaluated by MPSO, DE-1 and DE-2 to reach a target value of 3.45 dbm. 1 different trials have been made for evaluating the results for each of the algorithms. It should be noted that, for MPSO, we counted the number of position and velocity update equations used for total generations, and for DE algorithms, we counted the number of equations for selection, cross over and mutation operations for total generations. Like previous case, MPSO evaluates less functions than DE-1 and DE-2 to reach a specified target value. However, DE-2 requires less number of functions to be evaluated than DE-1. So, in terms of usage of functions, MPSO is more efficient than DE algorithms although DE algorithms perform relatively better than MPSO. REFERENCES [1] R. Chang. Orthogonal frequency division multiplexing. US. Patent-3, , January 197. [2] Y. Wu and W. Y. Zou. Orthogonal Frequency Division Multiplexing: A Multi- Carrier Modulation Scheme. IEEE Trans. Consumer Electronics, 41: , August [3] E. Lawrey. Multiuser OFDM. Proc. International Symposium on Signal Processing and its Applications, August [4] J. Jang and K. B. Lee. Transmit power adaptation for multiuser OFDM systems. IEEE J. Select. Areas Commun., 21: , February 23. [5] I. Kim, H. L. Lee, B. Kim, and Y. H. Lee. On the use of linear programming for dynamic subchannel and bit allocation in multiuser OFDM. Proc. IEEE Global Communications Conf., 6: , November 21. [6] W. Rhee and J. M. Cioffi. Increasing in capacity of multiuser OFDM system using dynamic subchannel allocation. Proc. IEEE VTC, May 2. [7] C. Wong, R. S. Cheng, K. B. 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CONCLUSION In this paper, we allocated sub carrier and number of bits for multi user OFDM systems by minimizing total transmit power using MPSO and DE algorithms. Both the algorithms outperform other existing considered algorithms for higher number of users. However, DE performs relatively better than MPSO but takes more time to converge. Moreover, the time requirement for the convergence of MPSO is comparable to other real time existing algorithm.