RESEARCH Open Access PAPR reduction of OFDM using … · 2.3 Tone reservation scheme to reduce the...

Hung et al. EURASIP Journal on Wireless Communications and Networking 2013, 2013:244http://jwcn.eurasipjournals.com/content/2013/1/244

RESEARCH Open Access

PAPR reduction of OFDM using invasive weedoptimization-based optimal peak reduction toneset selectionHo-Lung Hung1, Chung-Hsen Cheng2 and Yung-Fa Huang3*

Abstract

Tone reservation (TR) is one of the attractive techniques to reduce peak-to-average power ratio (PAPR) inorthogonal frequency division multiplexing system. As conventional TR technique requires exhaustive searchingover all the combinations of the given peak reduction tone (PRT) sets, it results in computational complexity thatincreases exponentially with the number of the subcarriers. In this paper, we aim to obtain a desirable PAPRreduction with low computational complexity. Since the process of searching the optimal PRT set can becategorized as combinatorial optimization with some variables and constraints, we propose a novel scheme, whichis based on a nonlinear optimization approach named as invasive weed optimization method, to search theoptimal combination of PRT set with low complexity. To validate the analytical results, extensive simulations havebeen conducted, showing that the proposed schemes can achieve significant reduction in computationalcomplexity while keeping good PAPR reduction. As results of simulations, the proposed scheme shows almost thesame PAPR reduction performance as compared with the genetic algorithm-based TR method which has beenknown to have the best performance and obtains near-optimal PRT sets.

Keywords: Orthogonal frequency division multiplexing; Peak-to-average power ratio reduction; Tone reservation;Invasive weed optimization

1 IntroductionOrthogonal frequency division multiplexing (OFDM) hasbeen attracting substantial attention due to its excellentperformance under severe channel condition [1-4]. OFDMhas been standardized in many wireless applications withhigh-speed data transmission such as terrestrial digitalaudio broadcasting and digital video broadcasting [2] andalso has been implemented in wireless local area networksand wireless metropolitan area networks due to its robust-ness to multipath fading and bandwidth efficiency andother advantages. However, some challenging issues stillremain unresolved in design of OFDM systems. One of themajor problems is its sensitivity to peak-to-average powerratio (PAPR) of transmitted signals, and some schemeshave been proposed to reduce the PAPR in OFDM systems

* Correspondence: [email protected] of Information and Communication Engineering, ChaoyangUniversity of Technology, Taichung 413, TaiwanFull list of author information is available at the end of the article

© 2013 Hung et al.; licensee Springer. This is anAttribution License (http://creativecommons.orin any medium, provided the original work is p

[3,4]. High PAPR results in a way that an OFDM receiver'sdetection efficiency becomes very sensitive to nonlineardevices used in a signal processing loop, such as digital-to-analog converter (DAC) and high-power amplifier,which may severely impair performance due to inducedspectral regrowth and detection efficiency degradation[5,6]. Therefore, it is very important to accurately identifyPAPR distribution in OFDM systems to work out someeffective 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 reductiontechniques in OFDM systems is summarized in [3,4]. Itis known that clipping [4] is the simplest method, butit degrades the bit error rate (BER) of the system andresults in out-of-band noise and in-band distortion.Among all existing techniques of PAPR reduction,selective mapping (SLM) [7] and partial transmit se-quence (PTS) [8-10] are very attractive due to theirgood PAPR reduction without the restriction on the

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number of subcarriers. However, in SLM technique, therequirement of multiple IFFT operations increases theimplementation complexity. The PTS technique uses aniterative routine similar to the trial-and-error methodin finding optimum phase factors that leads to lowerPAPR. However, the PTS technique requires an ex-haustive search over all combinations of allowed phasefactors, whose complexity increases exponentially withthe number of subblocks. Hence, it achieves consider-able PAPR reduction without distortion, but the highcomputational complexity of multiple Fourier trans-forms is a problem in practical systems [3,4,8-10]. As aresult, for all these search methods, either computa-tional complexity is still high or PAPR reduction per-formance is not good enough.To alleviate this problem, many PAPR reduction

techniques have been proposed in the literature [3] foran overview. One of the classical and most populartechniques is known as tone reservation (TR) since nodata is transmitted over the dedicated subcarriers[11-17]. In [18], a tone reservation algorithm has beendeveloped where several subcarriers are set aside forPAPR reduction. Since the subcarriers are orthogonal,the additive signal on unused subcarriers causes no dis-tortion to the data-bearing subcarriers. The TR techniqueattracted much attention for reducing PAPR for currentand future OFDM standard systems because TR providesgood PAPR reduction performance without BER perform-ance degradation and signal distortion. In addition, the TRtechnique is simple and effective, and it causes no interfer-ence to the data signal. TR does not require the exchangeof side information between transmitter and receiver.However, one of the disadvantages of TR is the in-crease in mean power of the transmitted signal becauseof corrective signal addition. Also, the computationalcomplexity of the optimization algorithm is to calcu-late the optimized corrective tones which reduce theoriginal signal's PAPR [11-15]. Therefore, practically, itis not realizable for a large number of peak reductiontone (PRT) set. Moreover, for these schemes, either theperformance in PAPR reduction is suboptimal or thecomputational complexity is still high.To tackle the complexity issue of TR, we formulate

the sequence search of TR as a particular combinatorialoptimization problem. To reduce the complexity for op-timal peak reduction tone (OPRT) set, some stochasticsearch techniques [12-17] have recently been proposedbecause they could obtain the desirable PAPR reductionwith a low computational complexity. Hence, the com-putational complexity is not significantly reduced. As aconsequence, the key question is how to decrease thecomplexity while maintaining a PAPR reduction close tothat of OPRT. In this paper, we take a fresh look at TRfor PAPR reduction and propose solutions for both

above-mentioned problems. In order to reduce compu-tational complexity, some approaches have been pro-posed recently. Different evolutionary algorithms suchas 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 colonizationof invasive weeds in nature which is based on the biologyand ecology of weeds. In this paper, we propose a newlysuboptimal PRT set selection scheme based on a modi-fied IWO algorithm, which can efficiently reduce thePAPR of the OFDM signals. In the TR scheme, a smallnumber of unused subcarriers called peak reductioncarriers (PRCs) are reserved to reduce the PAPR, andthe goal of the TR scheme is to find the optimal valuesof the PRCs that minimize the PAPR of the transmittedOFDM system. The proposed scheme can search thebetter combination of the initial PRT set to reducedPAPR. Simulation results show that the IWO-PRToptimization scheme can achieve superior PAPR re-duction performance and at the same time requires farless computational complexity than the previousOPRT techniques. The rest of this paper is organizedas follows. In Section 2, a typical OFDM system isgiven, the PAPR problem is formulated, and then PRTis explained. Then, IWO is proposed to search the op-timal combination of PRT set for reduced PAPR inSection 3. Sections 4 and 5 discuss the simulation re-sults and conclusions, respectively.

2 OFDM system model and PAPR definition2.1 OFDM system modelIn an OFDM system, a high-rate data stream is split intoN low-rate streams that are transmitted simultaneouslyby subcarriers, where N is the number of subcarriers.Each of the subcarriers is independently modulatedusing phase-shift keying (PSK) or quadrature amplitudemodulation (QAM). Inverse discrete Fourier transform(IDFT) generates the ready-to-transmit OFDM signal.For an input OFDM block X = [X0, X1,…, XN − 1]

T, eachsymbol in X modulates one subcarrier of { f0, f1,…, fN − 1}.The N subcarriers are orthogonal, i.e., fn = nΔf, where Δ

f ¼ 1NT and T is the symbol period. The complex enve-

lope of the discrete-time transmitted OFDM signal inone symbol period is given by [4]

xn ¼ 1ffiffiffiffiN

pXN−1

i¼0

Xiej2πni=LN ; n ¼ 0; 1; 2;…; LN−1; ð1Þ

L is the oversampling factor, where L = 4, which isenough to provide an accurate approximation of thePAPR [4] and xn is the nth signal component in OFDMoutput symbol. However, OFDM output symbols typically


have a large dynamic envelope range due to the superpos-ition process performed at the IFFT stage in the transmit-ter. PAPR is widely used to evaluate this variation of theoutput envelope. PAPR is an important factor in thedesign of both high-power amplifier (PA) and DACand for generating error-free (or with minimum errors)transmitted OFDM symbols and also preventing thePA to work in nonlinearity region. It is shown in [3,4] thatchoosing L = 4 is sufficient to approximate the peak valueof the continuous time OFDM signals.

2.2 Peak-to-average power ratioThe PAPR of the discrete-time baseband OFDM signal isdefined as the ratio of the maximum peak power dividedby the average power of the OFDM signal [3,4,11], whichis referred to as the PAPR, and is defined as

PAPR xð Þ ¼ 10 log10

max0≤n≤LN−1 xnj j2E xnj j2� � ; ð2Þ

where max |xn|2 is the maximum value of the OFDM sig-

nal power, and E[·] denotes the expected value operation.In principle, PAPR reduction techniques are concerned inthe reduction of max |xn|

2. By applying the central limittheorem, assuming that the number of subchannels is suf-ficiently large, the time domain symbol is approximately azero-mean complex that is Gaussian distributed and thepower distribution becomes a central chi-square distribu-tion with two degrees of freedom.

2.3 Tone reservation scheme to reduce the PAPRIn this paper, we consider the selection of the OPRT setfor tone reservation [11-18,29-31] scheme to reducethe PAPR of an OFDM signal. TR scheme requires asacrifice in data transmission efficiency because somesubcarriers in an OFDM symbol should be reserved aspeak reduction tones which are used only to reducePAPR without carrying data. The size of PRT plays acritical role in TR scheme. To achieve lower PAPR,more subcarriers should be reserved as PRTs whichreduces data transmission efficiency. TR technique ispart in the reduction of the PAPR of an OFDM signalby reserving a few tones within the transmitted bandwidthand by assigning them the appropriate values [18,29,30].In this paper, we formulate the optimal PRT set selectionproblem as a constrained combinatorial optimizationand propose the application of the IWO method [19] tosolve the problem.In order to reduce the PAPR of OFDM signal using

tone reservation scheme, some subcarriers are reservedas peak reduction tones set which are used to generatepeak-canceling signal. In the TR-based OFDM scheme,peak reduction tones are reserved to generate PAPRreduction signals. These reserved tones do not carry

any data information, and they are only used for re-ducing PAPR. Specifically, the peak-canceling signalC = [C0,C1,…,CN − 1]

T generated by reserved PRT is addedto the original time domain signal X = [X0, X1,…, XN − 1]

T

to reduce its PAPR. The PAPR-reduced signal can beexpressed as [11-13]

Y ¼ xþ c ¼ Ω Xþ Cð Þ ð3Þ

where c =ΩC is the peak-canceling signal in the timedomain and C = [C0,C1,…,CN − 1]

T is the avoid signaldistortion; the data vector X and the peak reduction vectorC lie in disjoint frequency domains, and X and C are notallowed to be non-zero on the same subcarriers, that is

Xn þ Cn ¼ Xn; n ∈RC ;Cn; n ∈R;

�ð4Þ

where R = {i0, i1,…, iM − 1} denotes the index set of thedata-bearing tones (subcarriers), RC denotes the index setof the complementary set of R in N = {0, 1,…,N − 1}, andM < N is the size of peak reduction tone set. With theTR scheme, the PAPR of the peak-reduced OFDM signalY = [ y0, y1,…, yN − 1]

T is then redefined as [11-13]

PAPR Yð Þ ¼max xn þ cnj j2

0≤n≤N−1

Ε xnj j2� � : ð5Þ

The PAPR reduction performance of the TR schememainly depends on the size of the PRT set, the max-imum number of iteration, and the selection of PRTs.Thus c must be chosen to minimize the maximum ofthe peak-reduced OFDM signal Y, i.e.,

c optð Þ ¼ argminmax xn þ cnj j2C 0≤n<N

: ð6Þ

To compensate for this effect, we may simply follow anapproach similar to [12], to obtain the PRT sets. In thispaper, to obtain the optimum c(opt), Equation 7 can bereformulated as the following optimization problem:

minC

E;

subject to E ≥ 0;xn þ cnj j2≤E;

ð7Þ

which is a quadratically constrained quadratic program(QCQP) problem [11] and E is an optimization param-eter. In order to find the optimal values of the PRCs,the TR optimization problem can be formulated as aQCQP problem. However, computing the exact solu-tion to this QCQP problem is generally challenging.Therefore, the focus of the work is on the second ap-proach. To reduce the complexity of QCQP, Tellado in[11] proposed a simple gradient algorithm to provide


an approximate solution with reduced complexity. Thisiterative algorithm is used in this paper because of itslow complexity. Although the optimum of a QCQP ex-ists, the solution requires a high computational cost ofO(NsN

2L), where Ns is a complex number space. However,several low-complexity methods have been proposed[11-13,18] and have achieved sufficient accurate subopti-mal solutions. Although the optimum of a QCQP exists,the solution requires a high computational cost. To reducethe complexity of the QCQP, Tellado in [11,18,30] pro-posed a simple gradient algorithm to iteratively approachc and to update the vector as follows:

c iþ1ð Þ ¼ c ið Þ−αip k−kið Þð Þn� � ð8Þ

where αi is a scaling factor relying on the maximum peakfound at the ith iteration, p = [p0, p1,…, pN − 1]

T is calledthe time domain kernel, and p = [((k − ki))N] denotes acircular shift of p to the right by a value ki, where ki iscalculated by

ki ¼ arg max xk þ c ið Þk

�� k

ð9Þ

For ease of presentation, the time domain kernel isexpressed as

P ¼ p0; p1;…; pN−1½ �T ¼ ΩP ð10Þwhere P = [P0, P1,…, PN − 1]

T, Pn ∈ {0, 1} is called the fre-quency domain kernel whose elements are defined by

Pn ¼ 0; n ∈ RC

1; n ∈ R

�ð11Þ

Then, the optimal frequency domain kernel P corre-sponds 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 onthe time domain kernel P, and the best performance canbe achieved when the time domain kernel p is a discreteimpulse because the maximum peak can be canceledwithout affecting other signal samples at each iteration.However, in order for the time domain kernel to be adiscrete impulse, all the tones should be allocated to thePRT set. As the number of reserved tones becomes larger,the PAPR reduction performance is improved, but thedata transmission rate decreases.After I iterations of this algorithm, the peak-reduced

OFDM signal is obtained:

Y ¼ xþ c Jð Þ ¼ x−XI

i¼1

aip k−kið Þð ÞN� � ð12Þ

From Equations 8 to 12, it can be found that the PAPRreduction performance of the TR-based OFDM system

depends on the selection of the time domain kernel,which is only a function of the PRT set. When there isa single discrete pulse, the best PAPR reduction per-formance can be obtained because the maximal peakat location can be canceled without distorting othersignal samplings. However, it is impractical because asingle discrete pulse will result in all tones beingassigned to the PRT set. So we should select the timedomain kernel such that it not only reduces the peakat location but also suppresses the other big values atlocation. To find the optimal PRT set, in mathematicalform, we require solving the following combinatorialoptimization problem:

R� ¼ arg minR

p1; p2;…; pN−1½ �T�� ∞ ð13Þ

where ‖·‖j denotes the j-norm and ∞ − norm refers tothe maximum values. It is known that this problem isNP-hard because the time domain kernel p must beoptimized over all possible discrete sets R [30]. whichrequires an exhaustive search of all combination ofpossible PRT set, i.e. possible combination numbers ofPRT set are searched, where denotes the binomialcoefficient. It is a non-deterministic polynomial time(NP)-hard problem and cannot be solved for the num-ber 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 proposedas the candidates of PRT set. Although the consecutivePRT set and the equally spaced PRT set are the sim-plest selections of PRT set, their PAPR reduction per-formance are inferior to that of the random PRT setoptimization. However, the random PRT set optimizationrequires a larger PRT set sampling, and the associatedcomplexity limits the application of such a technique.A variance minimization method in [31] is developedto solve the NP-hard problem, and it is just a modifiedversion of the random PRT set optimization, which alsohas the drawback of high computational cost. In [15], across entropy method was proposed to solve the problem.It obtains better results than the existing selectionmethods, but it requires a larger population or samplingsize. These limitations of the existing methods motivate usto find an efficient method to obtain a nearly optimal PRTset. As mentioned before, we propose an IWO-based PRTset selection method for the purpose of having a very lowcomputational complexity.

2.4 PRT position search based on IWO algorithmA detailed description of the IWO used for searchingthe nearly optimal PRT set positions is described as fol-lows. An initial population of plants (weed) is randomlygenerated. Each plant sequence is a vector of lengthN, and each element of the vector is a binary zero or


one depending on the existence of a PRT at that position(one denotes existence and zero denotes non-existence).The number of the PRT in each binary vector is M < N.The S plants are denoted as {A1,A2,⋯,As}. Each Au is abinary vector of length. For each plant Au, the PRT set Ruis the collection of the locations whose elements are one.Then, the frequency domain kernel corresponding to thePRT set is obtained using Equation 11, and the timedomain kernel Pu ¼ pu0 ; p

u1 ;⋯; puN−1

� �is obtained using

Equation 10. The merit (secondary peak) of the sequenceis defined as [12]

m Auð Þ ¼ pu1 ; pu2 ;…; pN−1

� �T�� ∞: ð14Þ

The T sequences (called elite sequences) with the lowestmerits are maintained for the next population generation.The best merit of the sequences is defined as

m̂ ¼ min1≤u≤N

m Auð Þ: ð15Þ

3 Minimizing PAPR using modified invasive weedoptimization-based PRT set3.1 IWO algorithmIn recent years, Mehrabian and Lucas proposed a novelpopulation-based stochastic search method called IWOmechanism to achieve global optimization [19]. IWO isa numerical stochastic optimization technique inspiredby the colonization of weeds. As a result of investiga-tion, weeds have shown to be very robust and adaptiveto changes in the environment, where capturing theirproperties leads to a very powerful optimization tech-nique [19-25]. Most importantly, the IWO method hasshown its robustness in practice. Hence, by applying theIWO algorithm into the PRT scheme, it could provide away to reduce the PAPR of OFDM transmitted signal.The performance evaluation of the proposed scheme forPAPR reduction and computation complexity is given inthis paper. We show that not only the PAPR is reducedbut also the complexity and processing time is de-creased. However, to our knowledge, IWO [19-24] hasnot yet been used for the same purpose till now. In thiswork, we extend the classical IWO algorithm for hand-ling PAPR problems. We propose a hybrid schedule todecrease the variance of the seed populations in IWOand also use the concept of particle swarm optimization(PSO) [25-28] for choosing the best maximum numberof population members that will survive to the nextgeneration.IWO, first designed and developed by Mehrabian and

Lucas [19], is a relatively novel numerical stochasticoptimization algorithm inspired from colonization ofinvasive weeds. A weed is any plant growing where it isnot wanted; any tree, vine, shrub, or herb may qualify

as a weed in any specified geographical area, dependingon the situation. Weeds have shown a very robust andadaptive nature that renders them undesirable plantsin agriculture. In a D-dimensional search space, a weedwhich represents a potential solution of the objectivefunction is represented as p = (p1, p2,…, pN − 1). Firstly,p weeds, called a population of plants, are initializedwith random growth position, and then each weed pro-duces seeds depending on its fitness and the colony'slowest fitness and highest fitness to simulate the naturalsurvival of the fittest process. The number of seeds eachplant produces increases linearly from minimum pos-sible seed production to its maximum; the generatedseeds are being distribution randomly in the search areaby normal distribution with mean the equal to zero anda variance parameter decreasing over the number of it-eration. By setting the mean parameter equal to zero,the seeds are distributed randomly such that they locatenear the parent plant, and by decreasing the variance overtime, the fitter plants are grouped together and in-appropriate plants are eliminated over time. The generalscheme for the IWO algorithm is shown in Algorithm 1,which consists of four main procedures: initialization,reproduction, spatial dispersal, and competitive exclusionoperator, respectively. [20-24]:

Algorithm 1 Invasive weed optimization algorithm 1

The produced seeds in this step are being dispreadover the search space by normally distributed randomnumbers with the mean equal to the location of theproducing plants and varying standard deviations. Thestandard deviation (SD) at the present time step can beexpressed by

piter ¼iterMAX‐iterð ÞniterMAXð Þn

� pinitial−pfinalð Þ þ pfinal; ð16Þ

to generate a new feasible population by randomly addingor removing PRTs, where iterMAX is the maximum num-ber of iterations, witer is the SD at the present time step,and n is the nonlinear modulation index. This alter-ation ensures that the probability of dropping a seed ina distant area decreases non-linearly at each time step,


which results in the grouping of fitter plants and theelimination of inappropriate plants. Figure 1 shows thestandard deviation over the course of a run with 200 it-erations and different modulation indexes.

3.2 Particle swarm optimizationKennedy [26] invented PSO as one of the most powerfulmembers of the class of stochastic search techniques in2001. They are originally defined to solve NP-completeproblems. However, two optimization techniques arecompared in this section. One advantage of PSO [27]over the GA is its algorithmic simplicity. A GA comprisesparameters of its major operators which are crossover,mutation, and elitism. The parameters are population size,probability of mutation, probability of crossover, and selec-tion. However, PSO has one simple operator, velocity calcu-lation. The benefit of having a small number of operators isthe reduction of computation and the elimination of theneed to select the best operator for a given optimization.Another difference between the PSO and GA is the abilityto control convergence. Mutation and crossover rates cansubtly affect the convergence of the GA, but not as effect-ively as the inertial weight. Fogle [27] indicated that thedecrease of inertial weight significantly increases theswarm's convergence. This type of control allows its useto determine the rate of convergence, and the final level ofstagnation is ultimately achieved. Stagnation occurs in theGA when eventually all the individuals possess primarilythe same genetic code. In that case, the gene pool is sohomogeneous that crossover has little or no effect, andeach 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 socialbehavior of a swarm of bees to search the location withthe most flowers in a field, the optimization procedureof PSO is based on a population of particles which fly inthe solution space with velocity dynamically adjusted

Figure 1 Standard deviation over the course of the run.

according to its own flying experience and the flyingexperience of the best among the swarm. During thePSO process, each potential solution is represented asa particle with a position vector and a moving velocity rep-resented as x and v, respectively. Thus for a K-dimensionaloptimization, the position and velocity of the jth par-ticle can be represented as xj = (xj,1, xj,2,…, xj,K) andvj = (vj,1, vj,2,…, vj,K). Like a GA, the PSO also beginsby generating a population of particles at random. Ateach time step, an associated value for each particle isevaluated in accordance with a function called the fitnessfunction which is critically defined and configured from aconsideration of the search objective. The value normallycalled the fitness indicates the goodness of the solution.The position of the individual best fitness which theith particle has been achieved so far; that of the highestfitness which has been obtained among all the particlesin the population so far are known as the personal best(denoted as xbestj ) and the global best (denoted as xbest),

respectively, and both are stored to generate the newvelocity of jth particle. During the process, each particleadjusts its velocity according to its own experience, andthe position of the best of all particles moves toward thebest solution.In the meantime, a condition is also set during the fol-

lowing step which controls the algorithm when it stops byeither setting it to obtain an acceptable target solution orto run for a set maximum number of search iterations G.If the algorithm does not stop, after a time step, Δt, thenew velocity vi(t +Δt) for particle i is updated by

vi t þ Δtð Þ ¼ w � vi tð Þi þ c1 � randðÞ � xbesti tð Þ−xi tð Þ �

þc2 � randðÞ � xbest tð Þ−xi tð Þ �;

ð17Þ

where vi(t) is the old velocity of particle i at time t.Significantly, in the real implementation of a PSO, timet and t + Δt generally represent the current and nextiterations, respectively, so Δt in Equations 17 and 18 isalways omitted in most literature. This equation clearlyindicates that the new velocity is related to the old vel-ocity weighted by w(t) and is also associated with theposition of the particle itself and of the global best oneby factors c1 and c2, respectively. The parameters c1 and c2which are set to constant value are therefore known asthe cognitive and social rates, respectively, since theyrepresent the weighting of the acceleration terms thatpull the individual particle toward the personal best andglobal best positions. The inertia weigh w in Equation 17is employed to manipulate the impact of the previoushistory of velocities on the current velocity. Conversely,in the later stages, convergence toward the global optimashould be enhanced to find the optimum solution


efficiently. Therefore, w resolves the trade-off betweenthe global and local exploration ability of the swarm.Ratnaweera et al. [28] introduced time-varying acceler-ation coefficient, which reduces the cognitive componentand increases the social component of acceleration co-efficient. For the purpose of intending to simulate theslight unpredictable component of the natural swarmbehavior, two random functions rand() are applied toindependently provide uniformly distributed numbersin the range [0,1] to stochastically vary the relative pullof the personal and global best particles. Based on theupdated velocities, new position for particle j is computedaccording the following equation:

xj t þ 1ð Þ ¼ xj tð Þ þ vj t þ 1ð Þ: ð18Þ

The populations of particles are then moved according tothe new velocities and locations calculated by Equations 17and 18 and tend to cluster together from different di-rections. Thus, the evaluation of each associate fitnessvalues of the new population of particles begins again.The algorithm runs through these processes iterativelyuntil it stops. In Equation 17, the inertial weight factorprovides the necessary diversity to the swarm by changingthe momentum of particles and hence avoids the stagna-tion of particles at the local optima. Usually, it needs todefine a maximum velocity for each modulus of velocityvector, which is often set as the upper limit of each modu-lus of position vector. This helps to control the unnecessaryexcessive roaming of particles outside the predefined searchspace. The role of w is considered critical for the conver-gence behavior of PSO. Given a user-specified maximumweight wmax and a minimum weight wmin, the inertialweight w is updated as the following monotonicallydecreasing function of the iteration g:

wgþ1 ¼ wg þ wmin−wmaxð ÞGmax

; ð19Þ

where Gmax is the predefined maximum number of it-erations and g is the iteration number. It has beendemonstrated that 0.9 for wmax and 0.4 for wmin cangreatly improve the performance of PSO.

3.3 The hybrid IWO/PSO algorithmIWO and PSO algorithms are both derived from biologicbehavior. Although these algorithms go through similarevolution processes, including selection, reproduction,recombination, and mutation, they are two approacheswith different propagation methods. In the hybrid algo-rithm, the IWO algorithm plays the role of guiding theevolution and the PSO algorithm works as an assistant.The interaction between the dispersion method of theIWO algorithm and the velocity of the PSO algorithmcontrols the balance between local exploitation and global

exploration in the problem space. The process of thehybrid algorithm is formulated in detail as follows:

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

Step 2. Seeds grow into plants. Each seed is assessedaccording to its fitness value, which is obtained fromthe fitness function defined to represent thegoodness of the solution. After the fitness value isassigned to the corresponding seed, the seed growsinto 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 fitnessvalues. The ith plant denotes the ith initial position,which is pbest. The most satisfactory fitness value isthe best position of the colony gbest.

Step 4. The velocities and positions of all plants aremodified. The next velocities and positions of allplants are renewed based on Equations 17 and 18:

vi t þ Δtð Þ ¼ w� vi tð Þi þ c1 � randðÞ � xbesti tð Þ−xi tð Þ �

þ c2 � randðÞ � xbest tð Þ−xi tð Þ �

;

and xi t þ 1ð Þ ¼ xi tð Þ þ vi t þ 1ð Þ:where t is the iterative time, vi and xi are the velocityand position of the ith plant, respectively, w is theinertia weight, and c1 and c2 are the learning factors.

Step 5. The flowering plants produce new seeds. Thenumber of seeds produced by each plant depends onits fitness value ranking and decreases from themaximum possible seed production Θmax to theminimum Θmin.

Step 6. The seeds are dispread over the solution space bynormally distributing random numbers with meanequal to the locations of the producing seeds andvarying standard deviations. The SD at the presenttime step can be expressed as Equations 16 and 20:

piter ¼iterMAX‐iterð ÞniterMAXð Þn

� pinitial−pfinalð Þ þ pfinal

xi t þ 1ð Þ ¼ xi t þ 1ð Þ þ rand� piterð20Þ

where itermax is the maximum number of iterations,pinitial and pfinal are the initial and final standarddeviations, respectively, and n is the non-linearmodulation index.

Step 7. The new seeds are ranked with their parentsbased on their fitness values and find their positionsin the colony. The seeds with a higher ranking growinto flowering plants, and those with a lower


ranking in the colony are eliminated to reach themaximum number of plants in the colony.

Step 8. The surviving seeds can produce new seedsbased on their ranking in the colony. Theprocess is repeated until either the maximum

Algorithm 2 Invasive weed opt

number of iterations is reached or the fitnesscriterion is met.

Therefore the proposed IWO-PSO-based PRT positionsearch algorithm can be summarized in Algorithm 2:

imization algorithm 2


4 Results and discussionsLet us now proceed to the application of the IWO-basedtone reservation method in OFDM PAPR reduction.In the simulations, we consider an OFDM system withN = 64,128 and 256 subcarriers; data symbols aremodulated using the QPSK and16-QAM constellation,and the number of reserved PRT set is at W = 8 and 16,respectively. In order to generate the complementary cu-mulative distribution function (CCDF) = (PAPR > PAPR0)of the PAPR, 10−4 OFDM blocks are generated ran-domly, where the transmitted signal is oversampled bya factor of L = 4.Figure 2 compares the average PAPR reduction per-

formance of the IWO-TR, CE-TR, and GA-TR with thePRT set for the same iteration numbers. Figure 2 showsthat the average PAPR reduction performance of theIWO-TR algorithm is better than those of CE-TR andGA-TR with different modulations. When the iterationnumber equals 120, the IWO-TR algorithm convergesto 5.8 dB PAPR with 16-QAM. Although the GA-TRalgorithm is almost simple, its convergence speed isthe slowest among the three methods. When the iter-ation number is 100, its average PAPR is approximately6.3 dB, which is 0.52 dB larger than the IWO-TR algo-rithm in 120 iterations. The CE-TR algorithm con-verges to 6 dB PAPR in 120 iterations, however, whichis approximately 0.2 dB larger than IWO-TR algorithmin the same iterations. Note that the PAPR perform-ance of the IWO algorithms and the CE method isclose to optimal. This shows that adding flexibility tothe choice of the number of iterations and modulationscan improve the convergence speed of the algorithm.Finally, this paper shows the trade-off between the num-ber of iterations and modulation for the PAPR reduction.The distribution of the PAPR bears stochastic charac-

teristics in a practical OFDM system, which often can be

Figure 2 Relationship of PAPR reduction with the number ofiterations and modulation for different techniques.

expressed in terms of CCDF. The CCDF itself can beused to estimate the bounds for the minimum numberof redundancy bits required to identify the PAPRsequences and evaluate the performance of any PAPRreduction schemes. Therefore, we mainly discuss theshapes of CCDF obtained by different searching methodsfor the combination of phase factors. In this section, thesystem performance with applied proposed IWO-basedPRT scheme is evaluated based on the PAPR CCDF [4]by computer simulation. The cumulative distributionfunction (CDF) of the amplitude of a sampling signal iscomputed as CDF = 1 − exp(PAPR0), and the CCDF [2,3]could be defined as

CCDF ¼ Pr PAPR > PAPR0ð Þ¼ 1−Pr PAPR≤PAPR0ð Þ ¼ 1− 1− exp −PAPR0ð Þð ÞN :

ð21Þ

In Figure 3, some results of the CCDF of the PAPR aresimulated for the OFDM system with 64 subcarriers and16-QAM modulation, the number of the reserved tonesW = 8 employing random partition and the numbers ofiterations are used for PAPR reduction. In Figure 3, weselect the GA mentioned, in which the trial of PRT setcombination is randomly generated to compare theperformance of PAPR reduction with that of the IWOmethod. It is evident that IWO can provide consistentimprovements of PAPR reduction than GA with somesignificant gains in the low PAPR range. In Figure 3,the CCDFs for OFDM without PAPR reduction and foroptimal PRT with high enumeration are included.When CCDF = (PAPR > PAPR0) = 10− 4, the PAPR0 ofthe original OFDM, GA-, CE- and IWO-TR are 12, 7.2,6.5, and 6.30 dB with iterations 50, respectively. However,when Pr(PAPR > PAPR0) = 10− 4, the PAPR performance ofour proposed IWO-based PRT scheme with 30 iterationnot only is almost the same as that of the GA-based PTSscheme with 50 iteration but is also having a much lowercomputational complexity. In general, in order to obtainan optimal PAPR, a search for the number of iterationsmust be accomplished.As shown in Figure 4, as the maximum number of it-

erations is increased, the CCDF of the PAPR has a bet-ter performance. The comparison is carried out underthe same conditions of initial population of candidatesolutions and number of iterations. Figure 4 comparesthe PAPR reduction performance of the proposedIWO algorithm based PRT sets, GA, and CE algorithmfor the same iterations. The same maximum iterationnumber is set to 50 for GA-TR, CE-TR, and IWO-TRschemes. When Pr(PAPR > PAPR0) = 10− 4, the PAPR ofthe original OFDM is 12.2 dB. Using the GA-TR algo-rithm, IWO-TR algorithm, and CE-TR algorithm with128 subcarriers, the PAPRs are approximately reduced

PAPRO(dB)

2 4 6 8 10 12 14 16 18 20

P[P

AP

R>

PA

PR

O]

10-4

10-3

10-2

10-1

100

Original OFDMGA-TR iteration=50IWO-TR iteration=50CE-TR iteration=30CE-TR iteration=10IWO-TR iteration=30CE-TR iteration=50GA-TR iteration=10 IWO-TR iteration=10GA-TR iteration=30

Figure 3 PAPR CCDF comparison of different maximum numbers of iterations of the IWO method for N = 64, W = 8.


to 7.23, 7.1, and 6.56 dB, respectively. From Figure 3, itcan be seen that apart from the PAPR by OPTS in [11],the PAPR reduction performance of the IWO-PTS is thebest among all of the methods for the same or almost thesame search complexity.Figure 5 shows a comparison of the CCDF of PAPR

for the conventional OFDM, the GA-PRT, the CE-PRT,and the proposed IWO-PRT set in an OFDM systemwith N = 256 subcarriers. Figure 5 illustrates that theproposed IWO scheme with iteration = 30 outperforms

PA

4 6 8 10

P[P

AP

R>

PA

PR

O]

10-4

10-3

10-2

10-1

100

Figure 4 PAPR CCDF comparison of different maximum numbers of i

the GA method with iteration = 50 and can achieve al-most the same PAPR reduction performance as boththe conventional PTS and SD-PTS schemes. Since thecomputation complexity reduction ratio increases asthe number of subcarriers, the proposed scheme becomesmore suitable for the high data rate OFDM system.Next, a fair comparison of the performance-complexitytrade-offs for the different stochastic PTS searchingstrategies with N and W, respectively, is provided inFigures 3, 4, and 5, where the average PAPR reduction

PRO(dB)

12 14 16 18 20

CE-TR iteration=10IWO-TR iteration=30IWO-TR iteration=50IWO-TR iteration=30GA-TR iteration =30 GA-TR iteration=50IWO-TR iteration=50IWO-TR iteration=10GA-TR iteration=10Original OFDM

terations of the IWO method for N = 128 and W = 16.

PAPRO(dB)

2 4 6 8 10 12 14 16 18

P[P

AP

R>

PA

PR

O]

10-4

10-3

10-2

10-1

100

OFDMCE-PRT iteration=50IWO-PRT iteration=50GA-PRT iteration=50CE-PRT iteration=30GA-PRT iteration=10CE-PRT iteration=10IWO-PRT iteration=10GA-PRT iteration=30PSO M=4,W=4IWO-PRT iteration=30

Figure 5 PAPR CCDF comparison of different maximum numbers of iterations of the IWO method for N = 256 and W = 32.


is the function of the number of sample. It can be seenthat (1) it is beneficial to select more samples so thatthe average PAPR reduction performance can be improved,and (2) the application of the IWO method to solve theoptimum PRT set problem yields an enhanced trade-offin the low average PAPR range of approximately 6.72and 8.76 dB for W= and W = 16, respectively. Therefore,in this case, we proposed an improved TR scheme, whichis based on the TR algorithm, to reduce the PAPR ofOFDM system. In summary, based on the proposedIWO searching for the optimal combination of PRTset, we conclude that (1) the proposed IWO methodcan reduce the computational complexity as shown inFigures 3, 4, and 5, and (2) the proposed IWO method

Figure 6 Average PAPR reduction comparison of the proposed schem

can obtain almost the same PAPR reduction as that ofoptimal PRT sequences illustrated in Figure 2.In Figure 6, we compare the PAPR reduction perform-

ance of the IWO-PRT with different modulations, withthe same size of population P = 30, and with the samemaximum iteration number I = 30 for W = 16. Whensampling = 3,000, the average PAPR of the GA-PRTwith QPSK and 16-QAM are 8.12 and 7 dB, respect-ively. The average PAPR by the IWO-PRT with QPSKand 16-QAM are 7.2 and 4.85 dB, respectively. FromFigure 6, it can be discovered that the difference of thePAPR between QPSK and 16-QAM is important. Littleperformance improvement can be obtained by differentmodulation schemes.

e with different modulation QPSK and QAM.


5 ConclusionsThis paper presents an IWO-based method used to searcha nearly optimal PRT set in the TR method for the im-provement of peak-to-average power ratio performanceof the OFDM system. The aim of this paper is to reviewthe conventional PAPR reduction schemes and the variousdifferent evolutionary algorithms based on conventionalPAPR reduction schemes to achieve a low computationalcomplexity. The simulations have been conducted andproven that IWO is an effective method to yield an en-hanced trade-off between PAPR reduction and com-plexity. Since the computational complexity reductionratio increases as the number of subcarriers increases,the proposed scheme becomes more suitable for thehigh data rate OFDM systems.

Competing interestsThe authors declare that they have no competing interests.

Authors' informationHLH received his M.S. degree in electrical engineering from the University ofDetroit Mercy, Michigan, USA in 1994 and his Ph.D. degree in electricalengineering from National Chung Cheng University, Chia-Yi, Taiwan in 2007.From1995 to 2006, he was a lecturer with the Department of ElectricalEngineering, Chienkuo Technology University, Taiwan. Since 2007, he was anassociate professor of the Department of Electrical Engineering, ChienkuoTechnology University, Taiwan. His current research interests are in wirelesscommunications, detection of spread-spectrum signal, wireless sensornetworks, evolutionary computation, and intelligent systems. HLH serves asan associate editor for the Telecommunication Systems. YFH is a professor atthe Department of Information and Communication Engineering, inChaoyang University of Technology.

AcknowledgementsThe works of H-L Hung and Y-F Huang were supported in part by theNational Science Council (NSC) of Taiwan under grant NSC 101-2221-E-324-024.

Author details1Department of Electrical Engineering, Chienkuo Technology University,Changhua City 500, Taiwan. 2Metal Industries Research & DevelopmentCentre, Taichung 40768, Taiwan. 3Department of Information andCommunication Engineering, Chaoyang University of Technology, Taichung413, Taiwan.

Received: 18 May 2013 Accepted: 26 September 2013Published: 17 October 2013

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doi:10.1186/1687-1499-2013-244Cite this article as: Hung et al.: PAPR reduction of OFDM using invasiveweed optimization-based optimal peak reduction tone set selection.EURASIP Journal on Wireless Communications and Networking2013 2013:244.

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