PAPR reduction of OFDM using invasive weed optimizationbased optimal peak reduction tone set selection
 HoLung Hung^{1},
 ChungHsen Cheng^{2} and
 YungFa Huang^{3}Email author
https://doi.org/10.1186/168714992013244
© Hung et al.; licensee Springer. 2013
Received: 18 May 2013
Accepted: 26 September 2013
Published: 17 October 2013
Abstract
Tone reservation (TR) is one of the attractive techniques to reduce peaktoaverage power ratio (PAPR) in orthogonal frequency division multiplexing system. As conventional TR technique requires exhaustive searching over all the combinations of the given peak reduction tone (PRT) sets, it results in computational complexity that increases exponentially with the number of the subcarriers. In this paper, we aim to obtain a desirable PAPR reduction with low computational complexity. Since the process of searching the optimal PRT set can be categorized as combinatorial optimization with some variables and constraints, we propose a novel scheme, which is based on a nonlinear optimization approach named as invasive weed optimization method, to search the optimal combination of PRT set with low complexity. To validate the analytical results, extensive simulations have been conducted, showing that the proposed schemes can achieve significant reduction in computational complexity while keeping good PAPR reduction. As results of simulations, the proposed scheme shows almost the same PAPR reduction performance as compared with the genetic algorithmbased TR method which has been known to have the best performance and obtains nearoptimal PRT sets.
Keywords
1 Introduction
Orthogonal frequency division multiplexing (OFDM) has been attracting substantial attention due to its excellent performance under severe channel condition[1–4]. OFDM has been standardized in many wireless applications with highspeed data transmission such as terrestrial digital audio broadcasting and digital video broadcasting[2] and also has been implemented in wireless local area networks and wireless metropolitan area networks due to its robustness to multipath fading and bandwidth efficiency and other advantages. However, some challenging issues still remain unresolved in design of OFDM systems. One of the major problems is its sensitivity to peaktoaverage power ratio (PAPR) of transmitted signals, and some schemes have been proposed to reduce the PAPR in OFDM systems[3, 4]. High PAPR results in a way that an OFDM receiver's detection efficiency becomes very sensitive to nonlinear devices used in a signal processing loop, such as digitaltoanalog converter (DAC) and highpower amplifier, which may severely impair performance due to induced spectral regrowth and detection efficiency degradation[5, 6]. Therefore, it is very important to accurately identify PAPR distribution in OFDM systems to work out some effective measures to curb PAPR.
To reduce the PAPR of OFDM signals, numerous techniques have been proposed in the literature[3–17]. A comprehensive tutorial review of PAPR reduction techniques in OFDM systems is summarized in[3, 4]. It is known that clipping[4] is the simplest method, but it degrades the bit error rate (BER) of the system and results in outofband noise and inband distortion. Among all existing techniques of PAPR reduction, selective mapping (SLM)[7] and partial transmit sequence (PTS)[8–10] are very attractive due to their good PAPR reduction without the restriction on the number of subcarriers. However, in SLM technique, the requirement of multiple IFFT operations increases the implementation complexity. The PTS technique uses an iterative routine similar to the trialanderror method in finding optimum phase factors that leads to lower PAPR. However, the PTS technique requires an exhaustive search over all combinations of allowed phase factors, whose complexity increases exponentially with the number of subblocks. Hence, it achieves considerable PAPR reduction without distortion, but the high computational complexity of multiple Fourier transforms is a problem in practical systems[3, 4, 8–10]. As a result, for all these search methods, either computational complexity is still high or PAPR reduction performance is not good enough.
To alleviate this problem, many PAPR reduction techniques have been proposed in the literature[3] for an overview. One of the classical and most popular techniques is known as tone reservation (TR) since no data is transmitted over the dedicated subcarriers[11–17]. In[18], a tone reservation algorithm has been developed where several subcarriers are set aside for PAPR reduction. Since the subcarriers are orthogonal, the additive signal on unused subcarriers causes no distortion to the databearing subcarriers. The TR technique attracted much attention for reducing PAPR for current and future OFDM standard systems because TR provides good PAPR reduction performance without BER performance degradation and signal distortion. In addition, the TR technique is simple and effective, and it causes no interference to the data signal. TR does not require the exchange of side information between transmitter and receiver. However, one of the disadvantages of TR is the increase in mean power of the transmitted signal because of corrective signal addition. Also, the computational complexity of the optimization algorithm is to calculate the optimized corrective tones which reduce the original signal's PAPR[11–15]. Therefore, practically, it is not realizable for a large number of peak reduction tone (PRT) set. Moreover, for these schemes, either the performance in PAPR reduction is suboptimal or the computational complexity is still high.
To tackle the complexity issue of TR, we formulate the sequence search of TR as a particular combinatorial optimization problem. To reduce the complexity for optimal peak reduction tone (OPRT) set, some stochastic search techniques[12–17] have recently been proposed because they could obtain the desirable PAPR reduction with a low computational complexity. Hence, the computational complexity is not significantly reduced. As a consequence, the key question is how to decrease the complexity while maintaining a PAPR reduction close to that of OPRT. In this paper, we take a fresh look at TR for PAPR reduction and propose solutions for both abovementioned problems. In order to reduce computational complexity, some approaches have been proposed recently. Different evolutionary algorithms such as genetic algorithm (GA)[12] and cross entropy (CE) method[13] have been used for PAPR applications. Contrast to GA and CE, invasive weed optimization (IWO) algorithm[19–24] and particle swarm optimization[25–28] are inspired from the phenomena of colonization of invasive weeds in nature which is based on the biology and ecology of weeds. In this paper, we propose a newly suboptimal PRT set selection scheme based on a modified IWO algorithm, which can efficiently reduce the PAPR of the OFDM signals. In the TR scheme, a small number of unused subcarriers called peak reduction carriers (PRCs) are reserved to reduce the PAPR, and the goal of the TR scheme is to find the optimal values of the PRCs that minimize the PAPR of the transmitted OFDM system. The proposed scheme can search the better combination of the initial PRT set to reduced PAPR. Simulation results show that the IWOPRT optimization scheme can achieve superior PAPR reduction performance and at the same time requires far less computational complexity than the previous OPRT techniques. The rest of this paper is organized as follows. In Section 2, a typical OFDM system is given, the PAPR problem is formulated, and then PRT is explained. Then, IWO is proposed to search the optimal combination of PRT set for reduced PAPR in Section 3. Sections 4 and 5 discuss the simulation results and conclusions, respectively.
2 OFDM system model and PAPR definition
2.1 OFDM system model
L is the oversampling factor, where L = 4, which is enough to provide an accurate approximation of the PAPR[4] and x_{ n } is the n th signal component in OFDM output symbol. However, OFDM output symbols typically have a large dynamic envelope range due to the superposition process performed at the IFFT stage in the transmitter. PAPR is widely used to evaluate this variation of the output envelope. PAPR is an important factor in the design of both highpower amplifier (PA) and DAC and for generating errorfree (or with minimum errors) transmitted OFDM symbols and also preventing the PA to work in nonlinearity region. It is shown in[3, 4] that choosing L = 4 is sufficient to approximate the peak value of the continuous time OFDM signals.
2.2 Peaktoaverage power ratio
where max x_{ n }^{2} is the maximum value of the OFDM signal power, and E[·] denotes the expected value operation. In principle, PAPR reduction techniques are concerned in the reduction of max x_{ n }^{2}. By applying the central limit theorem, assuming that the number of subchannels is sufficiently large, the time domain symbol is approximately a zeromean complex that is Gaussian distributed and the power distribution becomes a central chisquare distribution with two degrees of freedom.
2.3 Tone reservation scheme to reduce the PAPR
In this paper, we consider the selection of the OPRT set for tone reservation[11–18, 29–31] scheme to reduce the PAPR of an OFDM signal. TR scheme requires a sacrifice in data transmission efficiency because some subcarriers in an OFDM symbol should be reserved as peak reduction tones which are used only to reduce PAPR without carrying data. The size of PRT plays a critical role in TR scheme. To achieve lower PAPR, more subcarriers should be reserved as PRTs which reduces data transmission efficiency. TR technique is part in the reduction of the PAPR of an OFDM signal by reserving a few tones within the transmitted bandwidth and by assigning them the appropriate values[18, 29, 30]. In this paper, we formulate the optimal PRT set selection problem as a constrained combinatorial optimization and propose the application of the IWO method[19] to solve the problem.
Then, the optimal frequency domain kernel P corresponds to the characteristic sequence of the PRT set R, and the maximum peak is always p because it is a {0, 1} sequence. The PAPR reduction performance depends on the time domain kernel P, and the best performance can be achieved when the time domain kernel p is a discrete impulse because the maximum peak can be canceled without affecting other signal samples at each iteration. However, in order for the time domain kernel to be a discrete impulse, all the tones should be allocated to the PRT set. As the number of reserved tones becomes larger, the PAPR reduction performance is improved, but the data transmission rate decreases.
where ∥·∥_{ j } denotes the jnorm and ∞  norm refers to the maximum values. It is known that this problem is NPhard because the time domain kernel p must be optimized over all possible discrete sets R[30]. which requires an exhaustive search of all combination of possible PRT set, i.e. possible combination numbers of PRT set are searched, where denotes the binomial coefficient. It is a nondeterministic polynomial time (NP)hard problem and cannot be solved for the number of tones envisaged in practical systems. In[18, 30], the consecutive PRT set, the equally spaced PRT set, and the random PRT set optimization were proposed as the candidates of PRT set. Although the consecutive PRT set and the equally spaced PRT set are the simplest selections of PRT set, their PAPR reduction performance are inferior to that of the random PRT set optimization. However, the random PRT set optimization requires a larger PRT set sampling, and the associated complexity limits the application of such a technique. A variance minimization method in[31] is developed to solve the NPhard problem, and it is just a modified version of the random PRT set optimization, which also has the drawback of high computational cost. In[15], a cross entropy method was proposed to solve the problem. It obtains better results than the existing selection methods, but it requires a larger population or sampling size. These limitations of the existing methods motivate us to find an efficient method to obtain a nearly optimal PRT set. As mentioned before, we propose an IWObased PRT set selection method for the purpose of having a very low computational complexity.
2.4 PRT position search based on IWO algorithm
3 Minimizing PAPR using modified invasive weed optimizationbased PRT set
3.1 IWO algorithm
In recent years, Mehrabian and Lucas proposed a novel populationbased stochastic search method called IWO mechanism to achieve global optimization[19]. IWO is a numerical stochastic optimization technique inspired by the colonization of weeds. As a result of investigation, weeds have shown to be very robust and adaptive to changes in the environment, where capturing their properties leads to a very powerful optimization technique[19–25]. Most importantly, the IWO method has shown its robustness in practice. Hence, by applying the IWO algorithm into the PRT scheme, it could provide a way to reduce the PAPR of OFDM transmitted signal. The performance evaluation of the proposed scheme for PAPR reduction and computation complexity is given in this paper. We show that not only the PAPR is reduced but also the complexity and processing time is decreased. However, to our knowledge, IWO[19–24] has not yet been used for the same purpose till now. In this work, we extend the classical IWO algorithm for handling PAPR problems. We propose a hybrid schedule to decrease the variance of the seed populations in IWO and also use the concept of particle swarm optimization (PSO)[25–28] for choosing the best maximum number of population members that will survive to the next generation.
IWO, first designed and developed by Mehrabian and Lucas[19], is a relatively novel numerical stochastic optimization algorithm inspired from colonization of invasive weeds. A weed is any plant growing where it is not wanted; any tree, vine, shrub, or herb may qualify as a weed in any specified geographical area, depending on the situation. Weeds have shown a very robust and adaptive nature that renders them undesirable plants in agriculture. In a Ddimensional search space, a weed which represents a potential solution of the objective function is represented as p = (p_{1}, p_{2}, …, p_{N  1}). Firstly, p weeds, called a population of plants, are initialized with random growth position, and then each weed produces seeds depending on its fitness and the colony's lowest fitness and highest fitness to simulate the natural survival of the fittest process. The number of seeds each plant produces increases linearly from minimum possible seed production to its maximum; the generated seeds are being distribution randomly in the search area by normal distribution with mean the equal to zero and a variance parameter decreasing over the number of iteration. By setting the mean parameter equal to zero, the seeds are distributed randomly such that they locate near the parent plant, and by decreasing the variance over time, the fitter plants are grouped together and inappropriate plants are eliminated over time. The general scheme for the IWO algorithm is shown in Algorithm 1, which consists of four main procedures: initialization, reproduction, spatial dispersal, and competitive exclusion operator, respectively.[20–24]:
3.2 Particle swarm optimization
Kennedy[26] invented PSO as one of the most powerful members of the class of stochastic search techniques in 2001. They are originally defined to solve NPcomplete problems. However, two optimization techniques are compared in this section. One advantage of PSO[27] over the GA is its algorithmic simplicity. A GA comprises parameters of its major operators which are crossover, mutation, and elitism. The parameters are population size, probability of mutation, probability of crossover, and selection. However, PSO has one simple operator, velocity calculation. The benefit of having a small number of operators is the reduction of computation and the elimination of the need to select the best operator for a given optimization. Another difference between the PSO and GA is the ability to control convergence. Mutation and crossover rates can subtly affect the convergence of the GA, but not as effectively as the inertial weight. Fogle[27] indicated that the decrease of inertial weight significantly increases the swarm's convergence. This type of control allows its use to determine the rate of convergence, and the final level of stagnation is ultimately achieved. Stagnation occurs in the GA when eventually all the individuals possess primarily the same genetic code. In that case, the gene pool is so homogeneous that crossover has little or no effect, and each successive generation is the same as the first.
In this context, the population is called a swarm and the individuals are called particles. Resembling the social behavior of a swarm of bees to search the location with the most flowers in a field, the optimization procedure of PSO is based on a population of particles which fly in the solution space with velocity dynamically adjusted according to its own flying experience and the flying experience of the best among the swarm. During the PSO process, each potential solution is represented as a particle with a position vector and a moving velocity represented as x and v, respectively. Thus for a K  dimensional optimization, the position and velocity of the j th particle can be represented as x_{ j } = (x_{j,1}, x_{j,2}, …, x_{j,K}) and v_{ j } = (v_{j,1}, v_{j,2}, …, v_{j,K}). Like a GA, the PSO also begins by generating a population of particles at random. At each time step, an associated value for each particle is evaluated in accordance with a function called the fitness function which is critically defined and configured from a consideration of the search objective. The value normally called the fitness indicates the goodness of the solution. The position of the individual best fitness which the i th particle has been achieved so far; that of the highest fitness which has been obtained among all the particles in the population so far are known as the personal best (denoted as${\mathbf{x}}_{j}^{\text{best}}$) and the global best (denoted as x^{best}), respectively, and both are stored to generate the new velocity of j th particle. During the process, each particle adjusts its velocity according to its own experience, and the position of the best of all particles moves toward the best solution.
where G_{max} is the predefined maximum number of iterations and g is the iteration number. It has been demonstrated that 0.9 for w_{max} and 0.4 for w_{min} can greatly improve the performance of PSO.
3.3 The hybrid IWO/PSO algorithm

Step 1. Seeds are produced. First, the variables that need to be optimized should be determined. Each variable is initiated in its solution space. The solution of each variable is a particle, and a set of particles for the variables form a seed; i.e., each seed is an initial solution of the problem. A number of seeds constitute a colony.

Step 2. Seeds grow into plants. Each seed is assessed according to its fitness value, which is obtained from the fitness function defined to represent the goodness of the solution. After the fitness value is assigned to the corresponding seed, the seed grows into a flowering plant, i.e., a weed.

Step 3. Each plant finds its position in the colony. A group of plants are ranked based on their fitness values. The i th plant denotes the i th initial position, which is p_{best}. The most satisfactory fitness value is the best position of the colony g_{best}.

Step 4. The velocities and positions of all plants are modified. The next velocities and positions of all plants are renewed based on Equations 17 and 18:$\begin{array}{ll}\phantom{\rule{6.5pt}{0ex}}{\mathbf{v}}_{i}\left(t+\mathrm{\Delta}\mathit{t}\right)& =w\times {\mathbf{v}}_{i}{\left(t\right)}^{i}+{c}_{1}\times \text{rand}()\times \left({x}_{i}^{\text{best}}\left(t\right){x}_{i}\left(t\right)\right)\\ \phantom{\rule{1em}{0ex}}+{c}_{2}\times \text{rand}()\times \left({x}^{\text{best}}\left(t\right){x}_{i}\left(t\right)\right),\\ \phantom{\rule{1.5em}{0ex}}\mathrm{and}\phantom{\rule{0.5em}{0ex}}{x}_{i}\left(t+1\right)={x}_{i}\left(t\right)+{v}_{i}\left(t+1\right).\end{array}$
where t is the iterative time, vi and xi are the velocity and position of the i th plant, respectively, w is the inertia weight, and c1 and c2 are the learning factors.

Step 5. The flowering plants produce new seeds. The number of seeds produced by each plant depends on its fitness value ranking and decreases from the maximum possible seed production Θ_{max} to the minimum Θ_{min}.

Step 6. The seeds are dispread over the solution space by normally distributing random numbers with mean equal to the locations of the producing seeds and varying standard deviations. The SD at the present time step can be expressed as Equations 16 and 20:$\begin{array}{l}{p}_{\text{iter}}=\left[\frac{{\left({\text{iter}}_{\text{MAX}}\u2012\text{iter}\right)}^{n}}{{\left({\text{iter}}_{\text{MAX}}\right)}^{n}}\right]\left({p}_{\text{initial}}{p}_{\text{final}}\right)+{p}_{\text{final}}\\ {x}_{i}\left(t+1\right)={x}_{i}(t+1)+\text{rand}\times {p}_{\text{iter}}\end{array}$(20)
where iter_{max} is the maximum number of iterations, p_{initial} and p_{final} are the initial and final standard deviations, respectively, and n is the nonlinear modulation index.

Step 7. The new seeds are ranked with their parents based on their fitness values and find their positions in the colony. The seeds with a higher ranking grow into flowering plants, and those with a lower ranking in the colony are eliminated to reach the maximum number of plants in the colony.

Step 8. The surviving seeds can produce new seeds based on their ranking in the colony. The process is repeated until either the maximum number of iterations is reached or the fitness criterion is met.
Therefore the proposed IWOPSObased PRT position search algorithm can be summarized in Algorithm 2:
4 Results and discussions
Let us now proceed to the application of the IWObased tone reservation method in OFDM PAPR reduction. In the simulations, we consider an OFDM system with N = 64,128 and 256 subcarriers; data symbols are modulated using the QPSK and16QAM constellation, and the number of reserved PRT set is at W = 8 and 16, respectively. In order to generate the complementary cumulative distribution function (CCDF) = (PAPR > PAPR_{0}) of the PAPR, 10^{4} OFDM blocks are generated randomly, where the transmitted signal is oversampled by a factor of L = 4.
5 Conclusions
This paper presents an IWObased method used to search a nearly optimal PRT set in the TR method for the improvement of peaktoaverage power ratio performance of the OFDM system. The aim of this paper is to review the conventional PAPR reduction schemes and the various different evolutionary algorithms based on conventional PAPR reduction schemes to achieve a low computational complexity. The simulations have been conducted and proven that IWO is an effective method to yield an enhanced tradeoff between PAPR reduction and complexity. Since the computational complexity reduction ratio increases as the number of subcarriers increases, the proposed scheme becomes more suitable for the high data rate OFDM systems.
Authors' information
HLH received his M.S. degree in electrical engineering from the University of Detroit Mercy, Michigan, USA in 1994 and his Ph.D. degree in electrical engineering from National Chung Cheng University, ChiaYi, Taiwan in 2007. From1995 to 2006, he was a lecturer with the Department of Electrical Engineering, Chienkuo Technology University, Taiwan. Since 2007, he was an associate professor of the Department of Electrical Engineering, Chienkuo Technology University, Taiwan. His current research interests are in wireless communications, detection of spreadspectrum signal, wireless sensor networks, evolutionary computation, and intelligent systems. HLH serves as an associate editor for the Telecommunication Systems. YFH is a professor at the Department of Information and Communication Engineering, in Chaoyang University of Technology.
Declarations
Acknowledgements
The works of HL Hung and YF Huang were supported in part by the National Science Council (NSC) of Taiwan under grant NSC 1012221E324024.
Authors’ Affiliations
References
 van Nee R, Prasad R: OFDM for Wireless Multimedia Communications. Boston: Artech House; 2000.Google Scholar
 Digital Video Broadcasting (DVB): Framing Structure, Channel Coding and Modulation for Digital Terrestrial Television. SophiaAntipolis: ETSI; 2009. ETSI EN 300 744 V1.6.1Google Scholar
 Han SH, Lee JH: An overview of peaktoaverage power ration reduction techniques for multicarrier transmission. IEEE Wireless Commun. 2005, 12(2):5665. 10.1109/MWC.2005.1421929View ArticleGoogle Scholar
 Jiang T, Yiyan W: An overview: peaktoaverage power ratio reduction techniques for OFDMsignals. IEEE Trans. on Broadcasting 2008, 54(2):257268.View ArticleGoogle Scholar
 Litsyn S, Wunder G: Generalized bounds on the crestfactor distribution of OFDM signals with applications to code design. IEEE Trans. Inform. Theory 2006, 52: 9921006.MathSciNetView ArticleMATHGoogle Scholar
 Jiang T, Yao W, Guo P, Song Y, Qu D: Two novel nonlinear companding schemes with iterative receiver to reduce PAPR in multicarrier modulation systems. IEEE Trans. Broadcast. 2006, 52(2):268273. 10.1109/TBC.2006.872992View ArticleGoogle Scholar
 Baml RW, Fisher RFH, Huber JB: Reducing the peaktoaverage power ratio of multicarrier modulation by selected mapping. Electron. Lett. 1996, 32(22):20562057. 10.1049/el:19961384View ArticleGoogle Scholar
 Wang Y, Chen W, Tellambura C: A PAPR reduction method based on artificial bee colony algorithm for OFDM Signals. IEEE Trans. on Wireless Commun. 2010, 9(10):29942999.View ArticleGoogle Scholar
 Hung HL, Huang YF: PAPR reduction in OFDM system using differential evolutionbased partial transmit sequences scheme. IET Commun. 2012, 6(11):14831488. 10.1049/ietcom.2010.0997MathSciNetView ArticleGoogle Scholar
 Taspinar N, Kalinli A, Yildrim M: Partial transmit sequences for PAPR reduction using parallel tabu search algorithm in OFDM systems. IEEE Communications Letter 2011, 15(9):974976.View ArticleGoogle Scholar
 Tellado J: Peak to Average Power Reduction for Multicarrier Modulation. Ph.D. dissertation: Stanford University; 2000.Google Scholar
 Wang Y, Chen W, Tellambura C: Genetic algorithm based nearly optimal peak reduction tone set selection for adaptive amplitude clipping PAPR reduction. IEEE Trans. Broadcast. 2012, 58(3):462471.View ArticleGoogle Scholar
 Chen JC, Chiu MH, Yang YS, Li CP: A suboptimal tone reservation algorithm based on crossentropy method for PAPR reduction in OFDM systems. IEEE Trans. Broadcast. 2011, 57(3):752757.View ArticleGoogle Scholar
 Li H, Jiang T, Zhou Y: An improved tone reservation scheme with fast convergence for PAPR reduction in OFDM systems. IEEE Trans. Broadcast. 2011, 57(4):902906.View ArticleGoogle Scholar
 Chen JC, Li CP: Tone reservation using nearoptimal peak reduction tone set selection algorithm for PAPR reduction in OFDM systems. IEEE Signal Processing Letters 2010, 17(11):933936.View ArticleGoogle Scholar
 Guel D, Palicot J, Louet Y: Tone reservation technique based on geometric method for orthogonal frequency division multiplexing peaktoaverage power ratio reduction. IET Commun. 2010, 4(17):20652073. 10.1049/ietcom.2009.0808View ArticleGoogle Scholar
 Gazor S, AliHemmati R: Tone reservation for OFDM Systems by maximizing signaltodistortion ratio. IEEE Trans. Commun. 2012, 11(2):762770.Google Scholar
 Tellado J, Cioffi JM: PAR Reduction in Multicarrier Transmission System (ANSI Document, T1E1.4 Technical subcommittee, contribution number 97–367. Washington DC: ANSI; 1997.Google Scholar
 Mehrabian AR, Lucas C: A novel numerical optimization algorithm inspired from weed colonization. Ecological Informatics 2006, 1: 355366. 10.1016/j.ecoinf.2006.07.003View ArticleGoogle Scholar
 Roy GG, Das S, Chakraborty P, Suganthan PN: Design of nonuniform circular antenna arrays using a modified invasive weed optimization algorithm. IEEE Trans. Antennas Propag. 2011, 59(1):110118.View ArticleGoogle Scholar
 Karimkashi S, Kishk AA: Invasive Weed Optimization and its Features in Electromagnetics. IEEE Trans. Antennas Propag. 2010, 58(4):12691278.View ArticleGoogle Scholar
 Bai YY, Xiao S, Liu C, Wang BZ: A hybrid IWO/PSO algorithm for pattern synthesis of conformal phased arrays. IEEE Trans. Antennas Propag. 2013, 6(4):23282332.View ArticleGoogle Scholar
 Karimkashi S, Kishk AA, Zhang G: Modelling of aperiodic array antennas using infinitesimal dipoles. IET Microwaves Antennas & Propagation 2012, 6(7):761767. 10.1049/ietmap.2011.0402View ArticleGoogle Scholar
 Foudazi A, Mallahzadeh AR: Pattern synthesis for multifeed reflector antennas using invasive weed optimisation. IET Microwaves, Antennas & Propagation 2012, 6(14):15831589. 10.1049/ietmap.2012.0045View ArticleGoogle Scholar
 Kennedy J, Eberhart RC: Swarm Intelligence. San Mateo: Morgan Kaufmann; 2001.Google Scholar
 Hung HL: Interference cancellation for HNNPSO multiuser detection of UWB systems over multipath fading channel. Springer Telecommunication Systems 2011, 12: 113.Google Scholar
 Fogle DB: Evolutionary Computation: Toward a New Philosophy of Machine Intelligence. 2nd edition. IEEE Press: Piscataway; 2000.Google Scholar
 Ratnaweera AJ, Halgamuge SK, Watsom HC: Selforganizing hierarchical particle swarm optimizer with timevarying acceleration coefficients. IEEE Trans. Evolutionary Computation 2004, 8(3):240255. 10.1109/TEVC.2004.826071View ArticleGoogle Scholar
 Tellado , Cioffi JM: Peak Power Reduction for Multicarrier Transmission: Information Systems Laboratory. Stanford: Stanford University; 1999.Google Scholar
 Tellado J: Multicarrier Modulation with Low PAPR. Massachusetts: Kluwer; 2000.Google Scholar
 Lim DW, Noh HS, No JS, Shin DJ: Near optimal PRT set selection algorithm for tone reservation in OFDM systems. IEEE Trans. Broadcast. 2008, 54(3):454460.View ArticleGoogle Scholar
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