European Journal of Scientific Research ISSN 1450-216X Vol.33 No.3 (2009), pp.411-428 © EuroJournals Publishing, Inc. 2009 http://www.eurojournals.com/ejsr.htm
Multi-Objective Optimization Using the Bees Algorithm in Time-Varying Channel for MIMO MC-CDMA Systems Fares Sayadi Department of Electrical, Electronic & Systems Engineering Faculty of Engineering & Built Environment, Universiti Kebangsaan Malaysia 43600 UKM Bangi Selangor Darul Ehsan, Malaysia E-mail:
[email protected] Tel: +60 (3) 8921 6300 / 6312; Fax: +60 (3) 8921 6146 Mahamod Ismail Department of Electrical, Electronic & Systems Engineering Faculty of Engineering & Built Environment, Institute of Space Science (ANGKASA) Level 2, Faculty of Engineering & Built Environment Building Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor Darul Ehsan, Malaysia E-mail:
[email protected] Tel: +60 (3) 89216853 / 6855; Fax: +60 (3) 89216856 Norbahiah Misran Department of Electrical, Electronic & Systems Engineering Faculty of Engineering & Built Environment, Institute of Space Science (ANGKASA) Level 2, Faculty of Engineering & Built Environment Building Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor Darul Ehsan, Malaysia E-mail:
[email protected] Tel: +60 (3) 89216853 / 6855; Fax: +60 (3) 89216856 Kasmiran Jumari Department of Electrical, Electronic & Systems Engineering Faculty of Engineering & Built Environment, Institute of Space Science (ANGKASA) Level 2, Faculty of Engineering & Built Environment Building Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor Darul Ehsan, Malaysia E-mail:
[email protected] Tel: +60 (3) 89216853 / 6855; Fax: +60 (3) 89216856 Abstract This paper proposes novel application of the Bees algorithm to multi objective optimization problems for the uplink of a multi carrier code division multiple access (MCCDMA) system based on orthogonal frequency division multiplexing (OFDM). The multiple input multiple output (MIMO) channel from users, which moves at vehicular speed, to the base-station (BS) is time variant. For time varying channels the computational complexity of the two linear minimum mean square error (LMMSE) filters due to be computed at every time instant for channel estimation and multiuser detection (MUD), is nevertheless high. Thus, we develop a novel implementation of the multi objective
Multi-Objective Optimization Using the Bees Algorithm in Time-Varying Channel for MIMO MC-CDMA Systems
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optimization problems based on Bees algorithm throughout the iterative multiuser detection for a MIMO MC-CDMA system. This approach achieves intense trade-off between bit error rate (BER) performance and computational complexity with the benefit of utilizing fewer control parameters for time-varying channels as compared to the representative counterparts that is validated by simulations.
Keywords: Bees algorithm, multi objective optimization, multiuser detection, time variant channel, MIMO MC-CDMA
1. Introduction The multiple-input multiple-output (MIMO) communication is one of the most promising technologies to fulfill higher data rates and a better link reliability in order to provide more advanced services. Indeed, MIMO systems are proficient of achieving increased data rates and improved link reliability compared to single-antenna systems without necessitating additional bandwidth or transmits power (Telatar, 1999; Gesbert et al, 2003). Multicarrier code division multiple access (MC-CDMA) based on orthogonal frequency division multiplexing (OFDM) is robust to multipath fading in a broadband channel, multiuser interference (MUI), and multiple access interference (MAI) or other narrowband links co-existent in the same bandwidth (Portier et al, 2006). In such a system, detection algorithms require excessively computational complexity due to the linear minimum mean-square error (LMMSE) filters utilized for multiuser detection and channel estimation (Mecklenbräuker et al, 2006; Zemen et al, 2006). In (Abrao et al, 2008; Oliveira et al, 2008), the authors use the single objective optimization problem which employed unitary Hamming distance local search also offer good performance, whereas it is insufficient in time variant MIMO channels also with low efficiency beneath high order modulation and high or medium-high system loading scenarios (i.e., inefficiently handle MAI and computational complexity). This paper considers a novel implementation of the multi objective optimization problems (i.e., multiuser detection and channel estimation problems) accompanied by swarming intelligence approach (e.g., the Bees algorithm) is utilized to time-varying channels for a MIMO MC-CDMA system. Our model enables recognizing and erects a Pareto optimality set (non-dominated solutions) for adaptation Bee algorithm. Our new algorithm allows intense trade-off between bit error rate (BER) performance and computational complexity. Section 2 is exhibited the system model. The proposed algorithm is introduced in section 3. Section 4 is detailed the computational complexity. The simulation results are presented in 5. Finally, Section 6 concludes the paper. We denote transpose of a matrix and its conjugate transpose by and , respectively. A diagonal matrix is written as . The real and imaginary part of a complex variable is presented by and , respectively. The signum function is given by , and is the set of real numbers. The largest (correspondingly smallest) integer, is represented by ( ). lower (greater) or equal than
2. System Model Let us consider the transmitting simultaneously and synchronously of active users for an uplink MC-CDMA communication system which equipped with single transmit antenna and receive antennas at the base station as well as the channel model. Supposing, appropriate synchronization between all the single antenna terminals is attainable; the single input multiple output system (SIMO) antennas). The identical time and can be seen as a MIMO communication system (i.e., frequency domain spreading method are utilized by the single antenna transmitters. The data sym-
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Fares Sayadi, Mahamod Ismail, Norbahiah Misran and Kasmiran Jumari
bol of the user with symbol period spreading sequence onds, such that Oliveira et al, 2008):
independently spread over all parallel subcarriers using with each of chips having the chip interval sec-
. The transmitted signal of the
user as follows (Abrao et al, 2008; (1)
where Each
is the central frequency of the subcarrier, , and is the symbol energy. symbol is taken independent and identically distributed (i.i.d.) from a
QPSK constellation (i.e., complex alphabet set ) . The bandwidth of signature waveform cannot be arbitrarily broad due to the restricted attainability of the spectrum. Extra, to accommodate a large number of users, the signature waveforms require to be organized meticulously. The spreading sequence assigned to the subcarrier of the user can be expressed as: (2) where
and
are the chip waveform and the so that
chip sequence. Supposing,
for normalization purposes (i.e., the signature
). Note that the total (i.e., the MCwaveforms have normalized energy as CDMA equivalent) processing gain is We practically presume that Rayleigh fading is flat over all receive antennas as well as each subcarrier which approximately time-invariant during two successive transmission periods (i.e., the symbol period is much smaller than the coherence time due to trivial variations of the channel amplitude and phase are to be pursued throughout the coherence period). Hence, the corresponding channels impulse response of the subcarrier, user, and receive antenna is given by: (3) is a Rayleigh distributed random variable and the phase is uniformly where the amplitude . In the following, we will ignore the time index unless necessary. Therefore, distributed in the the received signal on the subcarrier, receive antenna, respecting all users can be expressed as: (4) where
is the additive white Gaussian noise (AWGN) with bilateral power spectral density given
by . Each of the receivers (for all users) performs signal demodulation in each of the subcarriers and passed through a matched filter (i.e., channel phase restitution), producing Subsequent to these two operations, the resulting signal is dispatched to an intelligent search assisted MUD delineated in Section 3. The channel state information (CSI) is obligated to be appraised at the receiver either by training or some blind methods. The equivalent baseband demonstration of the received signal on the subcarrier of the receive antenna (i.e., Eq. (4)), can be expressed in the matrix and vector notations as (Wei et al, 2004): (5)
Multi-Objective Optimization Using the Bees Algorithm in Time-Varying Channel for MIMO MC-CDMA Systems
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where, (6)
sampled AWGN noise vector on the subcarrier of the The denoted by . The matched filter (MF) output corresponding to the receive antenna can also be explicated in vector notation as follows:
(7) (8) (9) (10) receive antenna is subcarrier from (11) (12)
where is the filtered noise vector (i.e., after MF) with correlation matrix is given by:
and the
(13) is the auto and cross correlation of the spreading code. The conventional detector (CD) concerning user' symbol evaluation which linearly combines decision variables overall subcarriers and receive antennas: (14) where is the linearly combined decision variable (LCDV), demonstrated as: where
(15) Consequently, the entire estimation from all users is the vector constitution: (16) At the receiver, to minimize the influences of the MAI the optimum multiuser detection (OMUD) are utilized, which jointly estimates the symbols by selecting the symbol combination (for all users among all possible combinations in the four constellation points (e.g., QPSK)) associated with the minimal residual error (i.e., metric distance) as (Abrao et al, 2008; Oliveira et al, 2008; and Verd´ul, 1998). Due to the received signals on each element is faded independently, a sovereign log-likelihood function (LLF) is employed for each antenna, by supposing the antennas adequately partitioned. The LLF for the subcarrier in the receive antenna, elucidated as a detached optimization problem with only real-valued variables as (Oliveira et al, 2008): (17) ;
; ;
;
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Fares Sayadi, Mahamod Ismail, Norbahiah Misran and Kasmiran Jumari
The expanded real vector of denoted by The represents a real vector produced from an arbitrary candidate symbol vector (i.e., is the decision vector) and the feasible region in the (i.e., is the symbol alphabet dimension and length of the candidatedecision space given by vector denoted by ). The shows an estimate of the channel coefficients matrix, and depicts the estimated amplitude of the user. The maximum likelihood detection (MLD) finds the in the decision space such that maximizes the Eq. (17) as follows:
where
(18) is a single or multi objective function that takes into consideration some combination
method considering , for subcarriers and receiver antennas in Eq. (17). The estimating of time-variant frequency response demonstrates performance of the iterative receiver structure since the effective spreading sequence truthfully depends on the factual channel realization. The maximum variation in time of the wireless channel is upper-bounded by maximizing normalized one-sided Doppler bandwidth (Zemen et al., 2005): (19) where is the maximum (supported) velocity, and are the carrier frequency and the speed of light, respectively. Time-limited snapshots of the band-limited fading process span a subspace with very small dimensionality (Zemen et al., 2005). The same subspace is spanned by discrete prolate spheroidal (DSP) sequences are defined as (Slepian, 1978): (20) The sequences are doubly orthogonal over the infinite set and the finite set , band-limited by and maximally energy concentrated on . The Slepian basis function is the time-limited DSP sequences. The eigenvalue are primped such that the time-variant frequency selective channel for the duration of a is projected onto the subspace spanned by linear superposition of the first single data block, Slepian sequences, and is approximated as: (21) where the matrix
contains all the subcarrier coefficients for and the vector The dimension for practical issues (Zemen et al., 2005; Zemen et al., 2006). Hence, the estimated coefficient on the subcarrier, user, and receive antenna
every subcarrier is order of 3 to 5 complex channel is denoted by
and the normalization factors of the weights are delineated by the maximum ratio combining (MRC) scheme. Consequently, the MUD for time-variant MIMO channel is performed by substituting Eq. (3) with Eq. (21), resulting in Equations (17) and (18). This particular arrangement will be fundamental for the algorithm which we investigate in section 3.
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3. Bees Algorithm-Assisted Multi Objective Optimization Problems for MIMO Mc-CDMA Systems 3.1. Multi Objective Optimisation and Pareto Optimality Solution Utilizing the MLD, the computational complexity augments exponentially with the number of users, according to the combinatorial optimization problem in Eq. (18), which necessitates a comprehensive search in the decision space feasibilities of the decision vector. The primary heuristic approach is the linearly combined log-likelihood functions antennadiversity-aided strategy (LC-LLFs), i.e., each candidate-vector on the subcarrier of the receive antenna is linearity combined considering all the subcarriers and receiver antennas as (Yen et al., 2003): (22) Assuming the independent channel fading related with different receive antennas yield to Hence, resulting the data evaluation corresponding to disparate antennas may not identical due to deep fading condition in some antennas. From an implementation point of view, the single objective optimization problem based on local search MUD has two main drawbacks. Firstly, an irreducible BER floor emerges under high or medium-high system loading situation for all diversity order . This disadvantage are elucidated from two characteristics: a) the linear single-objective function (i.e., LC-LLF) performance in (22) is influenced by the system loading; b) the local search heuristic method (LSHM) bring about impressive imperfection to attain the global optimum beneath high signal to noise ratio and extreme MAI (Oliveira et al, 2008). Secondly, the LSHM is inadequate or low efficiency technique in time variant MIMO channels also under high order modulation. Therefore, we are considering multi objective function optimization problems which involve multiple and often conflicting objectives. The general optimization problem can be expressed mathematically as (Pham et al., 2007): (23)
where the objective functions are denoted by , and is the column vector comprises termed equality and termed inequality constraints are depicted by independent variables. The and respectively. Taken cooperatively, , and are acknowledged as the problem function (Deb, 2001). Desiring to minimize all the objective functions simultaneously is indicated by the word 'minimize'. A multi objective optimization problem solving is investigated which linearly combined the different objective functions, where contribution of each function is associated to a weight (Abrao et al, 2008). The weighting multi objective version (e.g., weighting particle swarm optimization (WOPSO)), which considers as well independent and combined LLF (i.e., each optimized function employing developed methods for single objective function problems) as following: where the first
(24) log-likelihood objective functions are individually applied to the candidate-vector ; fitness values are the LLFs associated to the receive antennas, given by
while the fitness value is the LC-LLFs (i.e., according to Eq. (22). Practically, in the majority cases, due to the
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Fares Sayadi, Mahamod Ismail, Norbahiah Misran and Kasmiran Jumari
independent channel fading on segregated receive antennas, it is impossible to take the advisable optimal consequence of the particle-candidate for Hence, two major drawbacks are emerged by transforming a multi objective optimisation problem into a single objective optimization scenario, and may be interpreted from two aspects: a) the weight apportioned to some objective functions may not be appropriate; b) not all the solutions are prevalently discovered; resulting in the degradation of the overall linear combination significance. The proposed algorithm is established upon the multi objective problem solution method (i.e., the genuine way) which takes into account all objective functions simultaneously. Assuming, the objective functions exist at least partly conflicting also encompass different units to avoid a single optimal solution with regards to every objective function which reaches its optimum. This is achieved by computing the set of all non-dominated solutions (i.e., the Pareto optimal set (Arora, 2004)), in multi objective optimisation tasks. The Pareto optimality (Pareto efficiency) which defines solutions for multi objective optimisation problems as a superior solution concept, by revelation a solution in the feasible solution space if there is no other possible solution in the solution space that decreases at least one objective function without intensifying another one. 3.2. The Bees Algorithm The Bees Algorithm (BA) is a novel populace-based search algorithm. The algorithm imitates the food foraging behaviour of honeybees' colony. In its basic version, the algorithm executes a kind of vicinity search combined with random search and can be used for optimization.The main steps of the BA are depicted in Figure 1 as (Pham et al., 2007; Pham et al., 2005; Pham et al., 2006). The algorithm necessitates a number of parameters to be set as following: • number of scout bees ( ). • number of sites selected for neighborhood search (out of visited sites) ( ). • number of bees recruited for the selected sites ( ). ) (a patch is a region in the search space that includes the vis• the initial size of each patch ( ited site and its neighborhood). • the stopping criterion Figure 1: Pseudo Code of the Bees Algorithm (Pham et al., 2007)
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Steps Initialize population with random solutions. Evaluate fitness of the population. While (stopping criterion not met) //Forming new population. Select sites for neighborhood search. Determine the patch size. Recruit bees for selected sites and evaluate fitness. Select the representative bee from each patch. Amend the Pareto optimal set. Abandon sites without new information. Assign remaining bees to search randomly and evaluate their fitness. End While
The algorithm starts by initializing scout bees randomly distributed in the search space. The sites visited by the scout bees are evaluated using the fitness function (i.e. the performance of the candidate solutions) in step 2. In step 4, the non-dominated sites are appointed as 'selected sites' and preferred for vicinity search. If there are more than non-dominated sites in the population, the first will be picked out as it is impossible to distinguish between them. If there are less than non-
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dominated sites, from the predominated ones which have been dominated only once, the rest will be picked out and this subroutine is continued pending an adequate number of sites have been chosen. In step 5, a great patch dimension is elected initially. For each patch, the preliminary size is maintained unaltered as long as the recruited bees are able to discover superior solutions in the vicinity. The patch size is lessened, where the neighborhood search appears with a bar to progress. This approach aspires impenetrably searching to achieve more exploitative local search for surrounding of the local optimum. This step is called the 'shrinking method' (Pham et al., 2007). In step 6, the algorithm investigates around the chosen sites. Due to impossible to always rank the solution candidates in the multi objective optimization adaptation of the Bees Algorithm, all the selected sites encompass the same number of recruited bees to seek around the vicinity. If original bee is dominated by one of the recruit bees, the representative bee will be the innovative non-dominated bee, in step 7. Hence in step 8, a nondominated solution will be added to the Pareto optimal set or vice versa (i.e., if the solution is dominating the other solutions in the generated Pareto optimal set, the dominated ones will be eliminated from the set) thorough the fundamental BA which allows establishing multi objective optimization problem solutions. In step 9; acquiring no improvement in the solution by employing the shrinking method, it is supposed that the patch is centered upon a local peak of performance in the solution space (i.e., once the vicinity search has discovered a local optimum) resulting in supplementary development is impossible (i.e., the investigation of the patch is discontinued). This step is called 'abandon sites without new information' (ASWNI) (Pham et al., 2007). In step 10, to reconnoiter new potential solutions, the remaining bees in the population are nominated randomly around the search space. At the each terminating iteration, the colony has two roles to its renovated population: delegates from the selected patches, and scout bees allocated to behavior random searches. These steps are reiterated pending a stopping criterion is satisfied. 3.3. Bees Algorithm-Assisted Multi Objective Optimization Problems This subsection describes the adaptation of the Bees Algorithm to search the maximum probility detection with lowest rank computation. Since the optimal MIMO-MUD problem entangles a search procedure across the finite number of potential solutions, the BA is an exemplary candidate to resolve this problem which can be a fascinating and extremely efficient method solving multi objective optimization problems while on duty in discrete and finite search space. The Bees algorithm-assisted multi objective optimization problems (BAMOP) are proposed as detection algorithm. To achieve the converged optimal solution, the fundamental fitness function as Eq. (18), is employed. The ML detection problem can be explicated as the minimization of the squared Euclidean distance (i.e., -dimensional finite discrete search set (Damen et al., 2003): residual error) to a target vector over (25) represents the complex also time-variant channel matrix which is depicted in section 2. It is where important to point out that this approach can be applied to higher order modulations ( -PSK or QAM), just varying the subset in order to accomplish the desired format modulation. Our algorithm simultaneously considers all objective functions and computes all non-dominated solutions, which involve nonlinear objective functions can be formulated as following: (26) where and are the residual error and computational complexity marginal costing. Evaluating the fitness of solution is based on Eq. (26). Not all combinations of parameters which can support acceptable functions. There are limitations which should be considered regarding to the practical properties of the MIMO MC-CDMA communication systems such as etc. In the BAMOP, all the objective functions rely on the cooperation between the participants. The forager bee assesses various characteristics associated to the food source selection such as its
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Fares Sayadi, Mahamod Ismail, Norbahiah Misran and Kasmiran Jumari
nearness to the hive, wealth of the energy, flavor of its nectar, and the liberation (ease) or complexity (difficulty) of extracting this energy. The positions of the food sources represent the feasible solution space and the nectar amounts of the food sources correspond to the quality (fitness) of the related solution which contains several parameters as mentioned above. At the pseudo first step, a randomly distributed initial population of solutions (food source positions) is generated by the BAMOP which is equal to the number of scout bees. Each solution is a -dimensional vector (i.e., is the number of optimization parameters or dimension of the problems), and the population of the positions (solutions) is subordinate to the iterative search process, of the forager bees. At the initialization step, the bees randomly picked out a set of food sources and their nectar are evaluated according to utilizing the energy usage (cost of computational complexity) and nectar quality (the BER performance in terms of the signal to noise ratio). An adaptation on the position (solution) is generated by an artificial bee (in her memory) contingent upon the local information (visual information), which tests the nectar amount (fitness value) of the new source (solution). The food sources are memorized and chosen to the neighborhood search by foragers based on the nectar amount. The forager bees share their information through waggle dancing of long duration with the probability is proportional to the food source profitability (Tereshko et al., 2005; Alok Singh, 2009) and more bees will be allocated to these spaces as for the recruitment is comparative to advantageousness of a food source. If the representative is a non-dominated solution, it will be added to the Pareto optimal set or vice versa. The patch inspection is discontinued by employing the shrinking method, when the vicinity search has discovered a local optimum. A new food source is randomly settled by one of the remaining bees (a scout bee) and substituted for the abandoned one, when the nectar of a food source is abandoned by the bees. Therefore, the best sits is found from each patch and the random detection spaces form the population for the next round of search is determined. These procedures are protracted until either the best fitness value has stabilised (i.e., there is no improvement in the found solution) or the specified maximum number of iterations has been reached. The probability value related to a food source that is selected by a recruit bee , can be expressed as: (27) The fitness value of the solution denoted by which is proportional to the nectar amount of the food source in a relative position. Assuming each solution comprises of optimization parameters be a solution with parameter values To find out a solution in the also neighborhood of (denoted by ),a solution parameter and a new solution are are identical to with the exception of the selected chosen randomly. The parameter values of parameter value, Hence, the can be expressed as: (28) The BAMOP constructs a candidate food position from the old one in memory, depending on visual information concerning scout bees (i.e., the value of the selected parameter in ), as following: (29) where is resolved randomly which has to be dissimilar from also . The Organization of the food production of neighbor food sources around and the comparison of two which is an uniform variate in We note food positions visually by a bee is depicted by that the perturbation on the position follows the difference between the parameters as and resulting in the step length is adaptively decreased according to the shrinking method (i.e., the search achieves the optimum solution in the search space). The parameter is positioned to a satisfactory value, regarding to the resulting value falls outside the adequate range. The BAMOP simulates the substitution abandon site with new position by exhibiting a position randomly and exchanging it with the abandon one. In proposed algorithm, a position is supposed to be forsaken which cannot be
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superior further during a predestined number of iterations. The value of predestined number of iterations is called 'border break' (border break = ) for forsaking which is a significant control parameter of the BAMOP algorithm. Supposing the abandoned source is , the substitution abandon site with new position can be expressed as: (30) is constructed and evaluated according to its performance compared to old Therefore, source by remaining bee after each candidate source position. In the memory, the new food source is substituted with previous source which has identical or better nectar than the previous one (i.e., amend the Pareto optimal set by implementing the greedy selection process). The measurement of how rapidly the honeybee colony discovers and exploits a lately realized food source (i.e., feasible solutions of the multi objective optimization problems) is depicted by the recruitment rate. The struggle for survival by the bee colony is contingent on the resourceful exploitation of the finest food resources rapid access. We note that the prosperous solution of multi objective optimization problems is associated to the comparatively rapid detection of best solutions particularly in real time solution. Hence, for a stout search process, exploration and exploitation processes are obliged to execute jointly. In the search space, the scout bees supervise the exploration process whereas recruit bees accomplish the exploitation process. The pseudo-codes of the proposed algorithm are depicted in Figure 2. Figure 2: Pseudo Code of the BAMOP Algorithm Steps 1.
initialize population with random solutions
2.
evaluate fitness of the population
3. for 4. compute new solutions for the scout bees by Eq. (29) and evaluate their fitness 5. utilize the greedy selection process to determine the patch size 6. compute the probability values for the solutions by Eq. (27) and evaluate their fitness 7. produce the new solutions for the recruit from the solutions picked out contingent upon 8. amend the Pareto optimal set for the representative bees by execution the greedy selection process 9. define the ASWNI (if exists) 10. substitute ASWNI with a latest randomly created solution by Eq. (30) 11. remember the finest solution performed so far 12. end
4. Computational Complexity In order to execute a competent efficiency measure of the proposed algorithm the computational complexity is delineated which varies to attain the convergence for the different strategies. That will be assessed in terms of the requisite operations (Golub et al., 1996). The multiplication, addition, comparison and random number generation are taken into account as operations. The computational complexity can be expressed as a function which contains the number of users ( ), receivers ( ), subcarriers ( ), iterations needed for convergence the time-variant channel parameter ( , and population size ( ). The cost functions computations in Eq. (26) are the most momentous group factors which determine the detector computational complexity. The computational complexity of proposed algorithm is considered as follows: i. The terms and in Eq. (17) can be evaluated outside the iterations loop and adopted constant during the detector search. These terms can be computed with
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Fares Sayadi, Mahamod Ismail, Norbahiah Misran and Kasmiran Jumari operations which are done times (i.e., one for each subcarrier on each antenna per Slepian dimension), proceeding to:
ii. Evaluation through cost function with a. for each candidate-vector:
times in each iteration requires to:
b. for multiplication calculation: c. for comparison implementation: d. for random number generation: Therefore, the total complexity order concerning BAMOP, can be obtained as: (31) Consequently, the computational complexity based on the number of operations is compared with other representative counterpart as the WOPSO in Table 1. Requiring operations for the unitary Hamming distance search-based strategies such as 1-optimum local search (1-LS) and its simplified (1sLS) also simulation annealing (SA) is presented as (Abrao et al, 2008; Oliveira et al, 2008). Table 1: Algorithm OMUD 1-LS 1-sLS
Comparison of Computational Complexity Operations (
SA WOPSO BAMOP
5. Numerical Results and Discussion In this section, all simulations utilize the main system and channel parameters are illustrated in Table 2 as (Abrao et al, 2008). The realizations of the time-variant frequency selective fading channel sampled at the chip rate are created employing an exponentially decaying power delay profile as (Clarke, 1968) and (Correia, 2001): (32) with root mean square delay spread The autocorrelation for every channel tap is defined by Clarke spectrum with resolvable paths ( ). The system activates at carrier frequency users move with velocity , resulting in Doppler bandwidth In Table 2, the suitable is depending upon system configuration and channel conditions (Oliveira et al, 2008). rate for Nonetheless, a mean value can be defined empirically which eliminates a full a priori information of the system and channel varying for terminating criterion. In the following subsections, numerical results are demonstrated for BER performance, in terms of , system loading ( ) and convergence. In addition, for proposed algorithm the computational complexity are evaluated in order to quantify the complexity order.
Multi-Objective Optimization Using the Bees Algorithm in Time-Varying Channel for MIMO MC-CDMA Systems Table 2:
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System, Algorithm and Channel Parameters
Parameter
Adopted Values S/MIMO MC-CDMA System:
# Rx antennas Code sequences Processing gain Subcarriers # mobile users Received SNR Common Parameters: Swarm pop. size # iterations BAMOP Parameter: Problem dim. WOPSO/SA Parameters (Abrao et al, 2008): For WOPSO: Velocity bound factor Velocity weight factor Weight particle factors For SA: # iterations @ Initial temperature Cooling rate 1-LS and 1-sLS Parameters: none Rayleigh Channel: Carrier frequency Delay spread Chip rate Doppler bandwidth # resolvable paths
5.1. BER Performance The performance of proposed algorithm is compared to few optimization methods. In this work, we implement the equivalent population number and the maximum evaluation number for all problems even though it is a matter of fact that the control parameters meaningfully influence the algorithm performance. 5.1.1. Performance for and System Loading ( ) Figure 3 demonstrates the BER performance of the BAMOP versus signal to noise ratio (SNR) (regularly, in dB) for different system loading (labeled 'BD'), receive antenna elements, and populations as compared to the CD, and the 1-sLS-MUD (labeled 'HD') (the 1-sLS-MUD has better performance than the other heuristic MUD, i.e., the1-Ls, SA, and WOPSO algorithms as (Oliveira et al, 2008)). From this figure, one can state that the heuristic algorithms achieve acceptable performance in medium loading scenario, whereas those are inadequate in time-variant MIMO channels also with low efficiency under high order modulation and high or medium-high system loading scenarios. Even though the growing loading irritates a performance reduction, the BAMOP-MUD technique can sustain more users at a preferred BER, particularly when SNR and enhance. However, our algorithm is more robust to channel-variant error for different system loading as well as high
for all diversity
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Fares Sayadi, Mahamod Ismail, Norbahiah Misran and Kasmiran Jumari
order. These advantages can be explained from two aspects: firstly, the multi objective function (Eq. (18)) is uninfluenced by the system loading; secondly, the proposed algorithm is created upon the multi objective problem solution technique which considers all objective functions concurrently, to avoid a single optimal solution with regards to every objective function (i.e., to achieve the global optimum). Figure 3: BER Performance versus SNR for BAMOP-MUD MIMO MC-CDMA. 0
10
-1
BER
10
-2
10
CD Nr=2 CD Nr=4 HD Nr=2
-3
10
HD Nr=4 BD Nr=2 BD Nr=4
-4
10
0
2
4
6
8 10 Eb/N0 (dB)
and
12
14
16
18
with perfect CSI, and QPSK.
0
10
-1
BER
10
-2
10
CD Nr=2 CD Nr=4 HD Nr=2
-3
10
HD Nr=4 BD Nr=2 BD Nr=4
-4
10
0
2
4
6
and
8 10 E b/N0 (dB)
12
14
16
18
with imperfect CSI, and QPSK.
5.1.2. Performance for Convergence Table 3 depicts convergence speed evaluation for different system loading scenarios (e.g., medium, ; medium-high, and full, ). The following expressions can be made from simulation results as compared to the other representative counterparts (Abrao et al, 2008; Oliveira et al, 2008): • for the 1-Ls, 1-sLS and SA algorithms: under perfect CSI, these accomplish a flat BER performance subsequent to convergence while lessen the convergence speed of search into a local minimum under medium-high or high system loading.
Multi-Objective Optimization Using the Bees Algorithm in Time-Varying Channel for MIMO MC-CDMA Systems • •
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for the WOPSO algorithm: the mission essential iteration number for convergence increase with an enhance in the number of receive antennas. for the BAMOP technique: the proposed algorithm enhances the convergence speed by operating the distinction of randomly established components of the parent and a randomly selected solution from the population. Hence, the compulsory iteration numbers for convergence reduces with an increase in the number of receive antennas also system loading, resulting in the flat BER performance adjacent to the OMUD accomplishment.
Table 3:
Convergence Performance for BAMOP-MUD
[5 4 3 3] [4 3 2 2] [3 2 2 1] [5 4 3 3] [24 25 28 29]
[6 5 4 4] [5 4 3 2] [4 3 2 2] [6 5 4 4] [38 41 46 48]
[9 8 7 6] [5 4 3 3] [4 3 3 2] [9 8 7 6] [59 67 76 80]
5.2. The computational complexity 5.2.1. Numerical Complexity for different We compare the computational complexity quantities according to Table 1. The complexity of the optimization algorithms are evaluated in more details taking into account the complexity factor (CF) and complexity reduction (CR) as: ,
(33)
(34) where is the computational complexity for OMUD, OCFC is the complexity to compute one cost function, and the complexity of the algorithm is denoted by . To demonstrate numerically the , which computational cost of different detection schemes, simulations are performed for varied are exhibited in Table 4 and Figure 4. The simulation parameters are selected as: , and the other ones as delineated in Table 2. The following observations can be made from simulation results: • employing the 1-Ls, 1-sLS and SA methods : the CF decreases partly with the number of receives antennas since each antenna has its own LLF. • utilizing the WOPSO algorithm: the CF lessens with , demonstrating that the WOPSO neces-sitates some further fine adjustment on the input parameters. • applying the BAMOP technique ratifies the number of CF varies slightly with the receive antennas and problem dimensions due to randomly initial solution of BAMOP-MUD which operates only over the MAI, resulting in a lesser amount of candidate-vectors are evaluated. Consequently, the computational complexity of the proposed algorithm is approximately independent of and has a significant complexity diminution compared to mentioned optimization methods as confirmed in Table 4 and Fig. 4.
425 Table 4:
Fares Sayadi, Mahamod Ismail, Norbahiah Misran and Kasmiran Jumari Comparison of Complexity Factor (CF): 0
5
10
15
20
7.5369 7.5361 7.5358 7.5357 5.5674 5.5665 5.5662 5.5661
7.7288 7.5355 7.5348 7.5346 5.7018 5.5617 5.5607 5.5652
7.7301 7.5359 7.5342 7.5338 5.7066 5.5601 5.5599 5.5648
7.8072 7.5352 7.5348 7.5340 5.7638 5.5642 5.5625 5.5659
7.8083 7.5350 7.5352 7.5343 5.7642 5.5611 5.5608 5.5650
3.6495 3.3785 3.2881 3.2430
3.7376 3.3780 3.2880 3.2428
3.7409 3.3773 3.2880 3.2425
3.7482 3.3770 3.2878 3.2426
3.7489 3.3782 3.2879 3.2428
7.7903 7.7856 7.7840 7.7832
7.9892 7.7856 7.7795 7.7782
7.9905 7.7856 7.7814 7.7801
8.0746 7.7856 7.7823 7.7799
8.0785 7.7856 7.7810 7.7778
9.5564 9.4197 9.3741 9.3514
9.8006 9.4227 9.3887 9.3674
9.8129 9.4256 9.3872 9.3663
9.9193 9.4230 9.3865 9.3668
9.9291 9.4235 9.3874 9.3678
1-LS
1-slS
BAMOP
SA
WOPSO
Figure 4: Computational Complexity Reduction (CR): considering and QPSK. -7
3
2.5
x 10
1-LS 1-sLS SA WOPSO BAMOP
CR
2
1.5
1
0.5
0
5.2.2. Numerical Complexity for different Loading ( ) which evaluate the In the sequel, simulations are executed for several loading conditions, computational complexity in terms of operations requirement due to growing the users' impact factor as the growing MAI. Figure 5 demonstrates the performance of the BAMOP-MUD for , and other parameters as depicted in Table 2. One can note that, due to lengthen candidate-vector which executes extra number of neighbors within each iteration, a loading augmentation
Multi-Objective Optimization Using the Bees Algorithm in Time-Varying Channel for MIMO MC-CDMA Systems
426
entails more computational operations so that reassure the BAMOP-MUD convergence. Furthermore, linear relation between the numbers of receives antennas and the computational complexities can be achieved. Figure 5: Computational complexity versus system loading (
) (in terms of number of operations) for
BAMOP-MUD Algorithm, considering mediate receive antennas.
,
and
9
10
Computational Complexity (number of operations)
8
10
7
10
WOPSO Nr=1 1-sLS
Nr=1
BAMOP Nr =1 WOPSO Nr=2 1-sLS 6
10
Nr=2
BAMOP Nr =2 WOPSO Nr=3 1-sLS
Nr=3
BAMOP Nr =3 WOPSO Nr=4 1-sLS 5
10 0.2
Nr=4
BAMOP Nr =4
0.3
0.4
0.5 0.6 0.7 System Loading
0.8
0.9
1
6. Conclusion A novel application of the Bees Algorithm multiuser detection for QPSK MC-CDMA system under time-varying MIMO channels scenarios was proposed and considered in real-valued multi objective optimization problems, which can be expanded to higher order modulations. In this work, the performance of the BAMOP algorithm was compared to the 1-Ls, 1-sLS, SA, and WOPSO optimization algorithms. The 1-Ls, 1-sLS, and SA utilize single objective function to construct new or candidate solutions from the current ones while the WOPSO employs developed methods for single objective func-
427
Fares Sayadi, Mahamod Ismail, Norbahiah Misran and Kasmiran Jumari
tion problems by preserving the best solution found in the population (i.e., it can be employed for cultivating innovative velocities). Whereas, the BAMOP algorithm was established upon the multi objective problem solution method which takes into account all objective functions simultaneously also might be substitutes the finest solution exposed with a randomly created solution. The proposed algorithm has a superior BER performance also improved the convergence speed as compared to the algorithms considered in this paper, mostly when the number of receive antennas and SNR enhance under medium-high and high system loading. The computational complexity of the BAMOP was evaluated in terms of operations. The requiring operations to converge enhances and slightly increase, respectively with the system loading and the receive antenna diversity order, except it is independent of the SNR. The simulation results demonstrated that BAMOP-MUD converges quickly with comparatively low computational complexity. The proposed algorithm uses less common control parameters: maximum iteration number and population size (i.e., it is a simple and flexible technique).From the results attained in section 5, it can be concluded that the BAMOP consents intense trade-off between BER performance and computational complexity on multimodal and multivariable problems than other algorithms considered in this paper.
Acknowledgement The authors would like to acknowledge the support of the Malaysian Ministry of Science, Technology and Innovation for funding this research work under IRPA grant: UKM-GUP-NBT-08-29-120.
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