Cross-Entropy-Based Sign-Selection Algorithms for Peak-to-Average Power Ratio Reduction of OFDM Systems

Transcription

1 4990 IEEE TRASACTIOS O SIGAL PROCESSIG, VOL. 56, O. 10, OCTOBER 2008 Cross-Etropy-Based Sig-Selectio Algorithms for Peak-to-Average Power Ratio Reductio of OFDM Systems Luqig Wag ad Chitha Tellambura Abstract Sig-selectio uses a set of subcarrier sigs to reduce the peak-to-average power ratio (PAR) of orthogoal-frequecy-divisio multiplexig (OFDM). However, the computatioal complexity (worst-case) is expoetial i, the umber of subcarriers. Suboptimal sig-selectio algorithms, achievig differet tradeoffs betwee the PAR reductio ad complexity, have thus bee developed. For example, the deradomizatio method achieves high PAR reductio of (log ) with relatively high complexity of ( ). O the other had, selective mappig (SLM) ad partial trasmit sequeces (PTS) sacrifice the achievable PAR reductio for lower complexity. I this paper, we develop two ew cross-etropy (CE)-based sig-selectio algorithms. Our algorithms simultaeously updates the probabilities of the sigs of all subcarriers. As show i Sectio II-D, the first algorithm obtais a PAR lower tha the above methods with a complexity level of ( ). However, if the umber of iteratios is fixed, this algorithm obtais the same PAR reductio as deradomizatio, but with ( log ) complexity. Practical PAR reductio algorithms require that the extra cost of PAR reductio must be small. Therefore, we propose the secod algorithm, which adaptively adjusts the probability of elite samples, ad stops wheever a PAR threshold is reached. Our secod algorithm achieves up to 95% complexity savigs over the first (with oly a 0.4-dB PAR reductio loss). The simulatios cofirm the complexity advatages of the proposed algorithms compared to SLM ad deradomizatio. Idex Terms Cross-etropy (CE) method, importace samplig, orthogoal-frequecy-divisio multiplexig (OFDM), peak-to-average power ratio (PAR) reductio. I. ITRODUCTIO Sice orthogoal-frequecy-divisio multiplexig (OFDM) suffers from the high peak-to-average power ratio (PAR) [1], may PAR-reductio techiques have bee developed ([2] ad refereces therei). Of these, probabilistic PAR reductio techiques are especially suitable whe the umber of subcarriers is large ( 64) [3], [4]. The sig-selectio method [5], [6] is a probabilistic techique ad the basic idea is to use a set of subcarrier sigs to reduce the PAR. This gives a maximum of 2 ( is the umber of subcarriers) potetial OFDM symbols for a give iput. Clearly, the complexity of fidig the optimal set of sigs is expoetial i. Existig sig-selectio algorithms ca be divided ito two groups: 1) algorithms with low-complexity, e.g., selected mappig (SLM) ad partial trasmit sequeces (PTS) [3], [4], which, however, oly obtai moderate PAR reductios; ad 2) algorithms with large PAR reductio, e.g., the deradomizatio method [6], which limit the PAR growth to O(log ), but have relatively high complexity of O( 2 ). The cross-etropy (CE) method is a iterative procedure for combiatorial optimizatio [7]. Each iteratio ivolves geeratig a radom sample accordig to a probability distributio ad the updatig the parameters of the probability distributio i order to produce better samples i the ext iteratio. I [8], we developed a CE-based sig-se- Mauscript received Jue 15, 2007; revised March 13, First published Jue 20, 2008; curret versio published September 17, The associate editor coordiatig the review of this paper ad approvig it for publicatio was Dr. Kostas Berberidis. This work has bee supported i part by SERC. The authors are with the Departmet of Electrical ad Computer Egieerig, Uiversity of Alberta, Edmoto, AB T6G 2V4, Caada ( wlq@ece.ualberta.ca; chitha@ece.ualberta.ca). Digital Object Idetifier /TSP lectio algorithm for PAR reductio. I this paper, we further develop two CE based PAR reductio algorithms. Large PAR reductio is obtaied with lower complexity by simultaeously modifyig the probabilities of the sigs of all subcarriers via the CE method. I cotrast, the deradomizatio method modifies the sigs oe by oe. With a fixed umber of samples, our algorithms obtai the same PAR reductio with a O( log ) complexity. Practical PAR reductio algorithms require that the extra cost of PAR reductio must be small. By adaptively adjustig the probability of elite samples, ad stoppig wheever a PAR threshold is reached, our secod algorithm achieves up to 95% complexity savigs over the first (with oly a 0.4-dB PAR reductio loss). The simulatios show that, for the same level of PAR reductio, our algorithms have lower complexity tha the SLM ad deradomizatio methods. To the best of our kowledge, our work is this first applicatio of the CE method to PAR reductio. This correspodece is orgaized as follows. Sectio II describes the geeral sig-selectio method ad develops two CE-based PAR reductio algorithms. The simulatio results are provided i Sectio III comparig the ew algorithms, the SLM ad deradomizatio methods. The correspodece cocludes i Sectio IV. II. CE METHOD FOR PAR REDUCTIO A. PAR Defiitios OFDM maps a block of iputs bits to a set of possibly complex symbols Xk chose from a M -ary sigal costellatio QM, e.g., pulse amplitude modulatio (PAM), phase shift keyig (PSK), or quadrature amplitude modulatio (QAM). The set of symbols is coverted to time-domai samples via a Iverse discrete Fourier trasform (IDFT). For a iput OFDM block X = [X0;...;X01], the discrete-time basebad equivalet sigal x() ca thus be expressed as x() = p 1 01 Xke j2k=j ; =0; 1;...; 0 1 (1) where J represets the oversamplig factor, which must be J 4 for sufficiet accuracy [9], [10]. I this paper, we choose J = 4for our simulatios. The PAR is the ratio max jx()j 2 PAR(X) E jx()j 2 : (2) B. Sig-Selectio Method A sig sequece c = [c0;...; c01] is poitwise multiplied with the iput OFDM block X to give ^X =[X0c0;...;X01c01]: The time-domai OFDM sigal of X is ^x(; c) = p 1 01 ckxke j2k=j : (3) ote that the average power of ^x(; c) is the same as that of x() for all c i.e., the sigs do ot alter the average power. Sice the sigs are to be chose to miimize the PAR, the resultig combiatorial optimizatio problem is mi c L(c) subject to: c 2f1; 01g (4) where L(c) = max j^x(; c)j 2 is the real-valued cost fuctio, ad f1; 01g is the set of -dimesioal biary vectors. ote that depedig o a specific implemetatio, some of the sigs may be fixed, say, oly s variables sigs are used X/$ IEEE Authorized licesed use limited to: UIVERSITY OF ALBERTA. Dowloaded o December 17, 2009 at 15:04 from IEEE Xplore. Restrictios apply.

2 IEEE TRASACTIOS O SIGAL PROCESSIG, VOL. 56, O. 10, OCTOBER Each elemet of d is modeled a idepedet Beroulli radom variable with the probability distributio P (d k =1)=10 P (d k =0)= p k, k =0;...; 0 1. The probability distributio is Fig. 1. The 91-poit hexagoal costellatio. The optimal solutio of (4), say, c 3 may be trasmitted to the receiver as side iformatio for correctly decodig the trasmit symbols. O the other had, several techiques do, i fact, avoid side iformatio [4], [5]. I this paper, we uses the hexagoal costellatio to elimiate side iformatio [11]. Fig. 1 illustrates a example of the 91-poit hexagoal (91-Hex) costellatio. The 64 poits marked by ad are used to carry six iformatio bits. Thus, the 91-Hex costellatio has the same throughput ad the same miimum Euclidia distace as the square 64QAM costellatio. The sigs of 27 outer poits ca be modified (betwee ad 2) to miimize the PAR. Therefore, o side iformatio is required at the receiver. Here, the mea of s is s =27. To reduce the complexity of (4), suboptimal solutios are commoly used. For example, the SLM method radomly geerates K sig sequeces fc1;...; ckg[3]. For a give OFDM block, a sig sequece ci leadig to the lowest PAR is selected from C as a suboptimal solutio. This algorithm is fast whe K is small. However, it does ot reach highly accurate solutios because K is chose to reduce complexity. The deradomizatio method iteratively optimizes each sig by miimizig the probability that the PAR is larger tha a threshold [6]. Whe all sigs of the subcarriers are allow to chage, it limits the PAR to O(log ) with O( 2 ) complexity. ote that, to obtai the same level of PAR reductio, SLM requires prohibitively high complexity. Oe may fid a better approximatio c 3 by usig the CE techique. The basic idea is to estimate a discrete probability distributio that will geerate a solutio close to c 3. C. The CE Sig-Selectio (CESS) Algorithms for PAR Reductio Without loss of geerality, we cosider the case that s =. Let c =102d, i.e., we geerate the sig sequece c 2f+1; 01g from a biary vector d 2f0; 1g. Let L(d) =max j^x(;10 2d)j 2. Each iteratio of the CE method for the optimal solutio c 3 of (4) ivolves two phases [7]: 1) radomly geeratig a set of samples d1;...; d with respect to a probability distributio f (d; p), where p is a probability parameter vector; 2) update p usig L(d1);...; L(d) so that f (d; p) geerates better samples i the ext iteratio. 01 f (d; p) = p d k (1 0 p k) 10d : (5) The CE method optimizes p = [p0;p1;...;p 01], which will geerate a early optimal solutio d 3 that leads to low PAR. However, d 3 occurs with a very small probability. Estimatig this probability usig the Mote Carlo method requires a large umber of samples of d. Istead, the CE method estimates a relatively large probability Pr[L(d) ], where is a relatively large threshold. Fewer samples are required to estimate this probability. The probability vector p is updated so that most samples geerated by f (d; p) satisfy L(d). The likelihood of d 3 amog these samples is icreased. By iteratively lettig! 0, f (d; p) coverges to a optimum pdf f (d; p 3 ) that geerates d 3 with a miimum umber of samples, ad coverges to L(d 3 ). I each iteratio, p 3 ca be aalytically foud by solvig [7] 1 I fl(d )gr l f (di; p) =0 (6) where I fxg =1whe x ad is 0 otherwise, ad di are geerated usig f (d; p). The partial derivative of (5) is give l f (d; p) = d k d k k p k 1 0 p k where d k is the kth elemet of d. Substitutig (7) ito (6), the optimal p k ca be foud as [7] p 3 k = I fl(d )gd ik I fl(d )g for a give, where d ik is the kth elemet of di. Our sig-selectio algorithm ca thus be summarized as follows [8]. Algorithm 1 (CESS): 1) Let p0 1, p k =0:5, k =1;...; 0 1. Let =0:1. Defie s = de, where dae represets the smallest iteger that is greater tha or equal to a. 2) Geerate samples of di with respect to p, ad calculate their PAR (i.e., L(di)). 3) Sort these samples i ascedig order accordig to L(di). Deote the obtaied PAR value sequece as [L0;...L 01]. 4) Fid as = L, ad update p usig (8). 5) If 0 <p k < 1 for some k, go to Step 2 ad repeat this procedure usig the updated p k. Otherwise. 6) Output the optimal sig sequece as 1 c 3 =10 2p. The optimizatio will coverge to p k = 0or 1 for all k. Alteratively, we may also stop the optimizatio after K iteratios, ad select the sample with lowest PAR for trasmissio. ow, we propose a modified CESS algorithm with threshold (MCESST). This algorithm stops the optimizatio whe the PAR of a OFDM sigal is reduced to withi the amplifier s liear rage. Based o the full adaptive CE method (FACE) [7], MCESST uses 1 If all are either 1 or 0, p ca oly geerate oe sample d=p. (8) Authorized licesed use limited to: UIVERSITY OF ALBERTA. Dowloaded o December 17, 2009 at 15:04 from IEEE Xplore. Restrictios apply.

3 4992 IEEE TRASACTIOS O SIGAL PROCESSIG, VOL. 56, O. 10, OCTOBER 2008 elite samples to adaptively adjust. That is, istead of usig, we defie the umber of elite samples, e. After geeratig samples d 1 ;...; d, ad sortig the correspodig L(d i ) i ascedig order (deotig as L 1;...;L ), a threshold ca be foud such that = L. The = P (L(d) ) ^ (9) where ^ = e =. Based o the cetral limit theory, ^ is a Gaussia radom variable with mea. Its variace decreases to 0 whe goes to 1. We ca also rewrite (9) as P (L(d) ^) e =, where ^ is a estimatio of. It is a radom variable with mea, ad its variace decreases to 0 whe goes to 1. The modified CESS algorithm starts with a relatively small umber mi. Let the iput back off (IBO) threshold be A. Our algorithm ca be summarized as follows. Algorithm 2 (Modified CESS With Threshold): 1) Iitialize A, e, mi, max, ic. Let = mi. 2) Let p 0 1, ad iitialize p k =0:5, k =0;...; ) Geerate samples d i with respect to p, ad calculate their PAR (i.e., L(d i )). Wheever a d k leads to PAR A, output c 3 =10 2d k ad stop the algorithm. Otherwise. 4) Sort these samples i ascedig order accordig to L(d i). Deote the obtaied PAR value sequece as [L 0 ;...L 01 ]. 5) Fid as = L, ad update p usig (8). 6) If p 2f0; 1g, output c 3 =10 2p ad stop the algorithm. Otherwise. 7) If < max, let = + ic, the, go to Step 3. We further exploit the idea of Tabu search [12] to lower the complexity. At the last several iteratios where most p k are 1 or 0, the samples cotai may duplicated samples. Therefore, whe a sample is geerated, it may be with previously geerated samples. If it has bee previously geerated, its PAR eeds ot be computed agai. ote that a list cotaiig the samples of d ad their PAR is also required i (8). We utilize this list for comparisos. Simulatio results show that about 15% iverse fast Fourier trasforms (IFFTs) ca be saved. D. Covergecy ad Complexity Discussio 1) Covergecy of Our Algorithms: The CE covergece is proved i [13]. A basic assumptio of the covergece is that the eighborhood of the optimum solutio is also early optimum. Our optimizatio problem satisfies this assumptio. The eighborhood i our case is defied i terms of the Hammig distace. The PAR differece betwee the optimal sig sequece c 3 ad aother sig sequece c 1 withi Hammig distace D from c 3 ca be bouded as max j^x(; c3 )j 2 0 max j^x(; c 1)j 2 8 D jx k j where k i deote the positio that c 1 differs from c 3. Cosequetly, whe is large, a small D always leads to ear-optimal solutios. Let the set C = fc i : c i =10 2d i;l(d i) g which cotais s good sig sequece samples that lead to small PAR. Ituitively, our algorithms cout the umber of 1 s that appear i C to estimate how the sigs will be i eighborhood of the optimal solutio ad use this iformatio to update p. If most good samples have a positive sig at the kth positio, most probably, the eighborhood of the optimal solutio will have a positive sig at this positio, ad p k is the updated to close to 0. I this way, most of the samples geerated i the ext iteratio have a small Hammig distace to C. A local optimum that has 2 a PAR o larger tha those of C is thus esured. O the other had, eve a p k is close to 0, there is still a chace to geerate some samples havig a egative sig at the kth positio. The samples may also have small PAR, which gives the possibility to escape from the local optimum ad coverge towards the global optimum. 2) Complexity of Our Algorithms for Fidig a ear-optimal Solutio: We first aalyze the complexity of our algorithms. ote that, to fid a ear optimal solutio, o threshold should be used i Algorithm 2. i terms of the umber of samples required to fid a ear optimal solutio. I the (i 0 1)th iteratio, our algorithms estimate the probability =Pr[L(d) ] ad optimize p such that most probably L(d) i the ith iteratio. The accuracy of estimatig is determied by e. Therefore, we cosider the complexity of fidig a ear optimal solutio whe e is fixed. We assume that e is large ad is small such that the samples geerated i each iteratio describe p with egligible error. Also assume that the optimizatio leads to egligible error. The, i the ith iteratio, o sample with L(d) >would be geerated. I other words, the size of the search space at the ith iteratio, deoted as S i ad 2 S 1 = 2, is reduced to S i S i01. If a ear optimal solutio is foud i the Kth iteratio, we have S mi = S K = K01 S 1 = K01 2 where S mi 1 is the umber of the ear optimal solutios. The K = log 2 Smi 0 s +1: log 2 The total umber of samples geerated i the optimizatio is K. Simulatio results show that must be proportioal to i order to obtai the same PAR distributio for differet. Therefore, the total samples required to fid a ear optimal solutio is of the order O( 2 ). Exhaustive search may search 2 samples to fid the optimal solutio. 3 SLM ad deradomizatio could ot fid a ear optimal solutio. 3) Complexity Compariso for Fidig Suboptimal Solutios: We may fix the umber of samples or set up a threshold (as i Algorithm 2) to fid suboptimal solutios with low complexity. Each sample is calculated usig FFT. Thus, the complexity of our algorithms for fidig suboptimal solutios is O( log ) multiplicatios, which is of the same order as SLM. However, simulatios show that SLM requires more samples (i.e., higher complexity) to obtai the same PAR as CESS. Deradomizatio iteratively calculates the sigs as c k = 0sig 2J =1 sih k01 r=1 c ra r sih(a k ) 2 r=k+1 cosh(a r ) (10) where sig(x) is the sig of x, is a costat, ad fa k g is the set of the real ad imagiary parts of X k e j2k=j, which are calculated ad stored before calculatig c k. Computig fa k g requires O( 2 ) multiplicatios. The memory requiremet for storig fa k g is 2J 2 float-poit real umbers. Computig cosh(a r ) requires O( 2 ) multiplicatios ad O( 2 ) hyperbolic fuctios. Thus, calculatig r=k+1 cosh(ar) requires O( 3 ) multiplicatios ad O( 3 ) hyperbolic fuctios. The rest of (10) requires O( 2 ) multiplicatios ad O( 2 ) hyperbolic fuctios. Therefore, the complexity of deradomizatio is O( 3 ) multiplicatios ad O( 3 ) hyperbolic 2 The mea of is =(1+ ). 3 It is easy to see that, to fid a ear optimal solutio, the average umber of samples that exhaustive search eeds to search is expoetial i. Authorized licesed use limited to: UIVERSITY OF ALBERTA. Dowloaded o December 17, 2009 at 15:04 from IEEE Xplore. Restrictios apply.

4 IEEE TRASACTIOS O SIGAL PROCESSIG, VOL. 56, O. 10, OCTOBER Fig. 2. PAR reductio compariso of CESS, SLM, ad the deradomizatio method with =128ad the 91-Hex costellatio. fuctios. ote that the complexity of hyperbolic fuctios is much higher tha that of multiplicatios. By usig more memory, we may calculate ad store the results of r=k+1 cosh(ar) for all k before the optimizatio. The, the total memory requiremet is 4J 2 float-poit real umbers, but the complexity is reduced to O( 2 ) multiplicatios ad O( 2 ) hyperbolic fuctios. A greedy algorithm is proposed i [14] based o deradomizatio, with which the sigs are iteratively calculated as c k =arg mi c 2f+1;01g k a ic i where a i = [a i], = 1;...; 2J ad k = 2;...;, are vectors each havig 2J items. It is proved that p (opt:) = log 2J. Calculatig a i requires O( 2 ) multiplicatios. Calculatig c k, k = 2..., requires 4J( 0 1) power-p operatios 4. Therefore, the complexity of this algorithm is O( 2 ). Its memory requiremet is 2J 2 float-poit real umbers (to store all a i ). The Algorithm 3 i [14] also has a complexity of O( 2 ) by otig that the cosh fuctio used i this algorithm is defied o legth-(2j) vectors. I summary, the complexity of our algorithm is smaller tha that of SLM ad deradomizatio 5. I Sectio III, we compare the PAR reductio of these algorithms by simulatio. III. UMERICAL RESULTS A. Performace Compariso of CESS, SLM ad the Deradomizatio Method We ow compare CESS with SLM ad the deradomizatio method [6], where =128ad the 91-Hex costellatio is used. Fig. 2 plots the PAR complemetary cumulative distributio fuctio (CCDF), also called the clip probability, defied as F () = Pr[PAR(X) >]. The CESS algorithm is used with =0:1.Ata10 04 clip probability, CESS with =30obtais a 5.3- db PAR reductio, which is 0.6 db larger tha the deradomizatio method. The average umber of samples geerated by CESS i this case is 176. With the same complexity, the PAR 4 The optimal is usually large. For example, 6 or 7 whe = 128 ad = 4. 5 Deradomizatio ad the greedy algorithm also require a large umber of additios, which may be a large burde i some implemetatios, e.g., usig the multiplier-accumulator (MAC) [15], where the complexity of multiplicatios is comparable to additios. p p ; Fig. 3. PAR reductio compariso of CESS ad SLM for the same complexity with the 91-Hex costellatio. TABLE I SIMULATIO PARAMETERS FOR MODIFIED CESS WITH THRESHOLD reductio of SLM is 0.2 db smaller tha that of CESS at a clip probability, ad is 0.4 db smaller tha that of CESS at a clip probability. By icreasig the complexity, CESS with =60obtais a 5.8-dB PAR reductio, which is about 1.2 db larger tha the deradomizatio method. We ext compare the PAR reductio of CESS ad SLM for the same complexity. We cosider two cases with = 128 ad = 256, respectively. We choose =0:1 for CESS with = 10, 20, 30 ad 40, respectively. For each case of, we cout the average umber of samples geerated by CESS, ad the use it as the umber of cadidates for SLM. Fig. 3 illustrates this compariso. ote that, whe =0:1 ad = 10, CESS is the same as SLM sice i this case oly oe sample leads to L(d i ) i each iteratio of CESS. We see that, with the same complexity, CESS leads to much smaller average PAR tha SLM. B. Performace of the Modified CESS With Threshold ow, we compare the modified CESS with threshold (MCESST) ad SLM for a 128-subcarrier OFDM system OFDM blocks are simulated. Sice the PAR of a typical sigle-carrier sigal usig squareroot raised-cosie pulse shapig with a roll-off factor of 0.35 is about 6 7 db [16], we compare MCESST ad SLM with three thresholds A = 6, 6.5, ad 7 db (relative to the average power). Whe applyig a threshold to SLM, we select a large cadidate set cotaiig 10 4 sig sequece samples, but stop the SLM optimizatio wheever a cadidate that leads to a PAR below the threshold is foud. The parameters for MCESST are selected as i Table I so that the PARs of virtually all iput OFDM blocks are reduced to below A. We also iclude i this table, out of the 10 5 simulated OFDM blocks, the umbers of bad OFDM blocks (deoted as bad blocks ) whose PARs are larger tha A after optimizatio, as well as the worst PAR. With these settigs, MCESST ad SLM obtai the same PAR reductio for each threshold used. We thus compare their complexity i terms of the average umber of IFFTs. Table II shows the complexity Authorized licesed use limited to: UIVERSITY OF ALBERTA. Dowloaded o December 17, 2009 at 15:04 from IEEE Xplore. Restrictios apply.

5 4994 IEEE TRASACTIOS O SIGAL PROCESSIG, VOL. 56, O. 10, OCTOBER 2008 TABLE II AVERAGE UMBER OF SAMPLES REQUIRED BY THE MODIFIED CESS WITH THRESHOLD AD SLM compariso. MCESST has less complexity tha SLM for all threshold settigs. IV. COCLUSIO I this paper, we developed two cross-etropy sig-selectio algorithms for OFDM PAR reductio. Our algorithms simultaeously updates the probabilities of the sigs of all subcarriers. The first algorithm obtais ear optimal solutios with O( 2 ) complexity. If the umber of iteratios is fixed, this algorithm obtais the same PAR reductio as deradomizatio, but with O( log ) complexity. By adaptively adjustig the probability of elite samples, ad by stoppig wheever a PAR threshold is reached, our secod algorithm achieves much lower complexity tha the first. The simulatios cofirm the complexity advatages of the proposed algorithms compared to SLM ad deradomizatio. REFERECES [1] S. Litsy, Peak Power Cotrol i Multicarriervarshe Commuicatios, 1st ed. Cambridge, U.K.: Cambridge Uiv. Press, [2] S. H. Ha ad J. H. Lee, A overview of peak-to-average power ratio reductio techiques for multicarrier trasmissio, IEEE Wireless Commu. Mag., vol. 12, o. 2, pp , [3] S. H. Müller ad J. B. Huber, A ovel peak power reductio scheme for OFDM, i Proc. 8th IEEE It. Symp. Persoal, Idoor Mobile Radio Commu. (PIMRC), Helsiki, Sep. 1 4, 1997, vol. 3, pp [4] A. Jayalath ad C. Tellambura, SLM ad PTS peak-power reductio of OFDM sigals without side iformatio, IEEE Tras. Wireless Commu., vol. 4, o. 5, pp , Sep [5] L. Wag ad C. Tellambura, A ovel PAR reductio techique for OFDM systems usig adaptive mappig, i Proc. 15th It. Cof. Wireless Commu., Calgary, AB, Caada, Jul. 7 9, 2003, pp [6] M. Sharif ad B. Hassibi, Existece of codes with costat PMEPR ad related desig, IEEE Tras. Sigal Process., vol. 52, o. 10, pp , Oct [7] R. Rubistei ad D. Kroese, The Cross-Etropy Method: A Uified Approach to Combiatorial Optimizatio, Mote Carlo Simulatio, ad Machie Learig. ew York: Spriger-Verlag, [8] L. Wag ad C. Tellambura, Peak-to-average power ratio reductio of OFDM systems usig cross etropy method, i Proc. 17th It. Cof. Wireless Commu., Calgary, AB, Caada, Jul , 2005, pp [9] C. Tellambura, Phase optimisatio criterio for reducig peak-to-average power ratio i OFDM, IEE Electro. Lett., vol. 34, o. 2, pp , Ja [10] G. Wuder ad H. Boche, Peak value estimatio of badlimited sigals from their samples, oise ehacemet, ad a local characterizatio i the eighborhood of a extremum, IEEE Tras. Sigal Process., vol. 51, o. 3, pp , Mar [11] A. Pezeshk ad B. H. Khalaj, Exteded hexagoal costellatios as a meas of multicarrier PAPR reductio, i Proc. 1st EurAsia Cof. Iformatio Commuicatio Techology (EurAsia-ICT ), Shiraz, Ira, Oct , 2002, vol. 2510/2002, pp [12] F. Glover, Tabu search Part I, ORSA J. Comput., vol. 1, o. 3, pp , [13] L. Margoli, O the covergece of the cross-etropy method, A. Oper. Res., vol. 134, o. 1, pp , Feb [14] M. Sharif ad B. Hassibi, Towards reducig the gap betwee PMEPR of multicarrier ad sigle carrier sigals, i Proc. 6th IEEE Workshop Sigal Processig Advaces i Wireless Commu., Ju. 5 8, 2005, pp [15] F. Elguibaly, A fast parallel multiplier-accumulator usig the modified booth algorithm, IEEE Tras. Circuits Syst. II, vol. 47, pp , Sep [16] S. Miller ad R. O Dea, Peak power ad badwidth efficiet liear modulatio, IEEE Tras. Commu., vol. 46, pp , Dec Authorized licesed use limited to: UIVERSITY OF ALBERTA. Dowloaded o December 17, 2009 at 15:04 from IEEE Xplore. Restrictios apply.