low complexity slm for papr in ofdm systems

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 12, DECEMBER 2005

Low-Complexity Selected Mapping Schemes forPeak-to-Average Power Ratio Reductionin OFDM SystemsChin-Liang Wang, Senior Member, IEEE, and Yuan Ouyang, Student Member, IEEE

AbstractOrthogonal frequency-division multiplexing (OFDM)is an attractive transmission technique for high-bit-rate communication systems. One major drawback of OFDM is the highpeak-to-average power ratio (PAPR) of the transmitters outputsignal. The selected mapping (SLM) approach provides goodperformance for PAPR reduction, but it requires a bank of inversefast Fourier transforms (IFFTs) to generate a set of candidatetransmission signals, and this requirement usually results in highcomputational complexity. In this paper, we propose a kind oflow-complexity conversions to replace the IFFT blocks in theconventional SLM method. Based on the proposed conversions,we develop two novel SLM schemes with much lower complexitythan the conventional one; the first method uses only one IFFTblock to generate the set of candidate signals, while the secondone uses two IFFT blocks. Computer simulation results show that,as compared to the conventional SLM scheme, the first proposedapproach has slightly worse PAPR reduction performance andthe second proposed one reaches almost the same PAPR reductionperformance.Index TermsInverse fast Fourier transform (IFFT), orthogonalfrequency-division multiplexing (OFDM), peak-to-average powerratio (PAPR) reduction, selected mapping (SLM).

I. INTRODUCTION

O

WING to the high spectral efficiency and the immunity tomultipath channels, orthogonal frequency-division multiplexing (OFDM) is a promising technique for high-rate datatransmission [1]. This transmission technique has been adoptedfor a number of applications, such as the standard for digitalradio audio broadcasting (DAB) [2], the standard for digitalvideo broadcasting (DVB) [3], the standard for asymmetric digital subscriber line (ADSL) service over twisted-pair phone lines[4], and the IEEE 802.11a standard for wireless local area networks [5].One major disadvantage associated with an OFDM systemis the high peak-to-average power ratio (PAPR) of the transmitters output signal, where the range of PAPR is proportionalto the number of subcarriers used in the system. Due to thehigh PAPR feature, an OFDM signal may suffer from significant

Manuscript received July 12, 2004; revised January 23, 2005. This work wassupported by the National Science Council of the Republic of China under GrantNSC 92-2213-E-007-072. This work was presented in part at the 2004 IEEE Vehicular Technology ConferenceFall (VTC2004-Fall), Los Angeles, CA, Sept.2004. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Yuan-Pei Lin.The authors are with the National Tsing Hua University, Institute ofCommunications Engineering, Hsinchu 300, Taiwan, R.O.C. (e-mail:[email protected]; [email protected]).Digital Object Identifier 10.1109/TSP.2005.859327

intermodulation distortion and undesired out-of-band radiation[6], [7] when it is passed through a nonlinear device, such as atransmit power amplifier.The conventional solution to the PAPR problem is to back-offthe operating point of the nonlinear power amplifier; althoughsimple, this approach usually causes a significant power efficiency penalty. There have been a number of methods proposedfor reducing the PAPR in OFDM systems. Of them, the clipping method is to deliberately clip the peak amplitude of theOFDM signal to some desired maximum level [8][11]. Sincethe clipping procedure is a nonlinear process, it may result inin-band distortion (self-interference) and out-of-band radiation.Another method for PAPR reduction is based on the use ofcoding schemes [12][15], where the original data sequence ismapped onto a longer sequence with a lower PAPR in the corresponding OFDM signal. Basically, a coding scheme would involve a large look-up table and is more suitable for those OFDMsystems with a small number of subcarriers.The selected mapping (SLM) [16][18] and partial transmitsequences (PTS) [19][22] approaches have received considerable attention in recent years for providing improved PAPR statistics of an OFDM signal. In the SLM approach, one OFDMsignal with the lowest PAPR is selected for transmission at thetransmitter from a set of sufficiently different candidate signals, which all represent the same data sequence. Each candidate signal is actually the inverse fast Fourier transforms (IFFTs)of the original data sequence multiplied by an individual phaserotation vector. In the PTS approach, the transmitter partitionsthe original data sequence into a number of disjoint subblocksand then optimally combines the IFFTs of all the subblocks togenerate an OFDM signal with low PAPR for transmission. Unlike the clipping method, PTS and SLM do not have adverse effects on the signal spectrum, but they require a bank of IFFTs togenerate a set of candidate signals. To detect the OFDM signalat the receiver, appropriate side information indicating how thetransmitter generates the output signal is embedded in the transmitted signal with error control codes. In general, the SLM andPTS techniques can provide pretty good PAPR reduction performance, but each of them may require a high computationalload due to the need of a bank of IFFTs. If the same number ofIFFT blocks is used, PTS may generate more candidate signalsfor selection and would achieve better performance than SLM,at the cost of a more complicated optimization process for combining IFFTs of all the subblocks.In this paper, the focus is on reducing the computational complexity of the SLM approach. In [23], we proposed two low-

1053-587X/$20.00 2005 IEEE

WANG AND OUYANG: LOW-COMPLEXITY SELECTED MAPPING SCHEMES

complexity conversions (conversions A and B) to replace half ofthe IFFT blocks used in the conventional SLM approach. Theseconversions use one IFFT output signal to produce another IFFToutput signal. In this paper, we extend the conversions A and Bin [23] to form a new kind of low-complexity conversions andthen use them to replace the IFFTs in the conventional SLM approach. Based on these conversions, we develop two new SLMschemes that have much lower complexity than the conventionalone; the first proposed scheme uses only one IFFT block alongwith some conversions to generate a set of candidate signals,while the second one uses two IFFT blocks along with someconversions. Computer simulation results show that, as compared to the conventional SLM scheme, the first proposed approach has slightly worse PAPR reduction performance, and thesecond proposed one reaches almost the same PAPR reductionperformance.The rest of this paper is organized as follows. In Section II, wedescribe the OFDM signal model and the conventional SLM approach. In Section III, we first present the idea of the previouslyproposed conversions in [23], and then extend the rationale todevelop a new kind of conversions. In Section IV, we describetwo new SLM schemes based on the proposed new conversions.The complexity analyses are also shown in this section. Section V presents performance comparisons of the new and theconventional SLM scheme in terms of PAPR reduction and thebit error rate (BER). Finally, brief conclusions are given in Section VI.II. BASIC PRINCIPLES AND THE SLM APPROACHThe OFDM technique divides a high-rate data stream into anumber of low-rate streams. Each low-rate stream is transmittedsimultaneously over a number of orthogonal subcarriers. Thecomplex baseband OFDM signal can be represented as(1)whereis the data symbol carried by the th subcarrier,is the frequency difference between subcarriers, is the OFDMsymbol duration, and is the number of subcarriers. To ensurethat all the subcarriers are orthogonal each other, the OFDM, i.e., the inverse of thesymbol duration should befrequency spacing of subcarriers. With these notations, the sampling period of the time-domain transmitted signal can be ex. For an OFDM system, the transmitterpressed byand receiver can easily be implemented by the IFFT and fastFourier transform (FFT), respectively.The PAPR of the transmitted OFDM signal in (1) can be defined as(2)Usually the OFDM signal is processed by digital signal processors, field-programmable gate arrays, or some specific digital circuits; therefore, we will express it in discrete time. If weby a sampling rate of, we may miss somesamplesignal peaks and get optimistic results for the PAPR. For better

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Fig. 1. Block diagram of the SLM approach.

approximating the true PAPR in the discrete-time case, we usuby a factor of , i.e., the sampling rate isally oversample. It was shown in [7] that an oversampling factor of four issufficient to approximate the true PAPR.statistically independent sequencesIn the SLM approach,are first generated from the same data sequence and then theone with the lowest PAPR is selected for transmission, as shownin Fig. 1. Let the data sequence be expressed as an -dimensional vectorand the th phase rotation vector be denoted by, wherethe th elementwith a random-generated phase. Then we can generate the frequency-domain version (vector ) of the th candidate signal by performing arraymultiplication (carrier-wise multiplication) of the data vectorand th phase rotation vector as follows:

(3)where

..

(4).

is referred to as the phase rotation matrix corresponding to thephase rotation vector . To simplify the array multiplication in(3), we choose the elements,in vectorfrom the set.A set of transmission candidate signals can be generatedby performing the IFFTs of the frequency-domain vectors ,. The candidate signal that has the lowestwherePAPR is selected for transmission. As mentioned before, oversampling each candidate signal is required for having a betterapproximation of the true PAPR. To oversample a candidatezeros in the middle of thesignal , we first insertdata vectorto form an-dimensional data vector given. Then, weby-dimultiply this zero-padded data vector by anmensional phase rotation vector, wherecan be anarbitrary value for. After this, an-point IFFT is performed to generate the oversampledversion of .

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where

Fig. 2. Idea of conversion with

T

is the IFFT matrix given by

...

for IFFT computation.

If the complementary cumulative distribution functionand all(CCDF) of the a candidate signal iscandidate signals are quite different (i.e., independent),thethen the CCDF of the selected signal for transmission will be. Apparently, The SLM method lowers theprobability of the PAPR exceeding some threshold at theadditional IFFTs. Note that, for correctlyexpense ofrecovering the transmitted data, the receiver has to know whichhas been actually used. A straightforphase rotation vectorward way to transmit this information, called side information,is to transmit the number of the used vector . Side information is usually embedded in the transmitted signal and protectedby error control codes so that the erroneous detection of thecan be neglected. In this paper, we will not discussvectorhow the side information is transmitted.III. SOME CONVERSION METHODS FOR IFFT COMPUTATIONA. The Previous WorkIn [23], we proposed two conversions to replace the IFFTin the SLM approach. The idea of these conversions is shownare two IFFT output signals correin Fig. 2. Signals andandsponding to the frequency-domain signals, respectively, of the SLM approach. Here,is a phaserotation matrix corresponding to a phase rotation vector . Thepreviously proposed conversions in [23] use one IFFT outputsignal to generate another IFFT output signal . We can getfrom Fig. 2 that(5)(6)

...

...

..

.

...(7)

with

. From (5), we can easily obtain(8)

wheredenotes the inverse ofThen, we have

(i.e., the FFT matrix).(9)

In other words, we can obtain a conversion matrixputing from as follows:

for com(10)

It was indicated in [23] that, when the vector(corresponding to the matrix) is of the form(formA),therequires onlycomplex additions. Forconversion withexample, in the case of, the conversion matrixis shown in (11) at the bottom of the page, where we can seethat there are four ones in each row and hence the conversionin (11) requires only 3 16 complex additions.with matrixWe called this kind of conversions (corresponding to form A)as conversion A in [23].is considered, the converAs an oversampling factorrequires zero multiplications andcomplexsion withadditions, where the computational complexity is much smaller-point IFFT. It was also indicated in [23] thatthan that of theis of the formif the vector(form B), the conversion with(called conversion B) requireszero multiplications andcomplex additions. Apparently,these conversions (conversions A and B) involve much lower

(11)


TABLE ICOMPUTATIONAL COMPLEXITY COMPARISON OF THE CONVERSION WITHAND THE IFFT WITH/WITHOUT ZERO PADDING CONSIDERED

T

computational complexity than the-point IFFT for computing .It should be noted that, when oversampling is used, we actuzeros in the middle of the data vector .ally insertThe arithmetic operations with these padded zeros can be neglected and removed from the computational complexity of the-point IFFT. Therefore, the numbers of complex multiplica-point IFFT for the oversamplingtions and additions of theand,case are respectivelyand. Table I showsinstead ofthe complexity comparison of the conversion withand theIFFT with/without zero padding considered. We can see that thestill has significantly smaller computationalconversion withcomplexity even when the zero-padding effect is considered.B. Problem Statements for Finding More ConversionMatricesIn this section, we will derive the relationship between theand the conversion matrix. Basedphase rotation vectoron the derived relationship, we can put some constraints on theand the phase rotation vectorto getconversion matrixthe solutions. From (6), we can know that the frequency-dois equal to the carrier-wise multiplication ofmain vectorthe frequency-domain vector and the phase rotation vector. Hence, from the convolution property of digital signal processing, the time-domain vector is a circular convolution ofthe time-domain vector and the IFFT of the phase rotationin (9) is a circulant matrixvector . This implies thatcorresponding to this convolution. If we let, then (10) can be rewritten as(12)whereis a circularly down-shifted version of the columnvector by elements. If an oversampling factor is used, theis ofdimension and can be representedmatrixin (12). It is interesting to notesimply by replacing withthat the first column vector of the matrixis the IFFT of theis given, thevector . Thus once the phase rotation vectorcolumn vector and the corresponding conversion matrixcan be determined.In order to keep the conversion process withto have thecomplex additionssame low computational complexity ofand zero multiplications as that described in Section III-A, wehave to put the following constraints on the column vector .

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EIGHT TYPES OF ^

TABLE IIAND

^t

FOR

WITH PERIOD 4

1) The number of nonzero elements in must be equal to 4.2) Each of the nonzero elements in must belong to the set.In addition to the above constraints, the corresponding phasemust contain no zero elements such that allrotation vectorsubcarriers signals of the OFDM system will not be blockedout in any subcarrier.

C. The General Form of Conversion MatricesThe Proposed Low-Complexity Conversions

:

In this section, we will find the solutions of vectorsandunder the above constraints. We first note that periof period 4 can satisfy the first constraintodic vectorsin Section III-B, since this kind of vectorscan producefour nonzero elements at fixed positions in its correspondingtime-domain vectors ; for example, the phase rotation vectorgiven inSection III-A is one of such solutions and the corresponding.vectorof period 4It should be noted that not all periodic vectorscan satisfy the second constraint mentioned above. Choosingfor each of the four nonzeroarbitrarily a value fromelements in , we can easily list all possible 256 vectors forthat satisfy both of the first and second constraints. Eliminatingthose vectors that have zero elements in their corresponding(ofvectors , we can get eight types of such solutions ofperiod 4) and (with four nonzero elements in fixed positions),andfor simplicity. Letdenoted bybe the 4-dimensional vector formed by the elements of oneperiod ofandbe the corresponding IFFT of . Thenis actually a four-dimensional vectorit can be shown thatconsisting of all the nonzero elements in . All these eightand are given in Table II.types ofSince all the phase rotation vectorsconstructed by thebasic-period vectorsshown in Table II have the same pein the SLMriod of 4, the corresponding candidate signalsscheme will have many similarities, i.e., having high correlations. In other words, if we use these phase rotation vectorsfor the SLM method, the PAPR reduction performance will degrade. To avoid the disadvantage of having the same period forall the phase rotation vectors but still satisfy the constraints de,scribed in Section III-B, we design a new column vector

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matrix


andas follows:

, for the conversion

zeros

zeros

zeros

zeros

zeroswhere zerosis the

(13)is a row vector withzero elements andth element of the vectorin Table II, i.e.,. For example

zeros

zeros

G

Fig. 3. Example of the first proposed SLM method using the conversion.process with

zeroszeros

zeros

forand

and. In (13), we assume that is a power of 2. According to this assumption, it can be checked thatis equal to the column vector obtained by the IFFT of thevector of period 4; for a given and different , the period ofthe corresponding frequency-domain vectorwill be different; the proof is given in the Appendix . This meansthat, for agiven in Table II, we can derivephase rotation vectors that all have different periods and can satisfy the first and second constraints described in Section III-B.contains noIn the appendix , we also show that the vectorzero elements, which means no subcarriers signals are blockedis a good choice for the first column vector ofout. Thus,and we define this new kind of conthe conversion matrixversion matrices as

(14)whereis a circularly down-shifted version of thebyelements. As an example,column vectorforand.It should be reminded that, in the above discussion, we constrain the computational complexity of conversion matrixto becomplex additions. If we change the number of, wecomplex additions required for the conversion matrixand , and hence get anwould get another type of vectorsother kind of conversion matrices.IV. TWO NEW SLM SCHEMES BASED ON THE PROPOSEDCONVERSIONSIn the previous discussion, we know that the phase rota,andtion vectors of the conversion matriceshave different periods. We may use the corresponding conversions to replace the IFFT blocks in theconventional SLM method because the candidate signals produced by these conversions will have little correlation. Fig. 3,shows an example of the proposed SLM scheme for, and. In this figure, we can see that only oneIFFT block is used and the other candidate signals are produced

Fig. 4. Example of the second proposed SLM scheme using the conversion.process with

G

by the conversions based on, and. Theiscandidate signal produced by the conversion matrix,equivalent to the IFFT of the phase-rotated vectoris the phase rotation matrix corresponding to thewherephase rotation vector. In this example, the computationalcomplexity of the proposed SLM scheme is significantly re-point IFFT blocks to 7complexduced from 7additions. For the general case, the computational complexity is-point IFFT blocks toreduced fromcomplex additions.Fig. 4 shows an example of another proposed SLM scheme.As shown in this figure, the upper part of this proposed schemeis the same as the first proposed scheme, but at the lower part,and anwe use a randomly-generated phase rotation vectoradditional IFFT block to create another set of candidate signals, which are also produced by the proposed conversion ma. In this example, the computational complexity oftricesthe second proposed SLM scheme is significantly reduced from-point IFFT blocks to 6complex additions. For6the general case, the computational complexity is reduced from-point IFFT blocks tocomplex additions.Table III shows a comparison of the computational complexity of the conventional SLM scheme, the first proposedSLM scheme, and the second proposed SLM scheme. We cansee that both of our proposed SLM schemes involve muchlower computational complexity than the conventional one. Itshould be noted that, following the same structure of the second


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TABLE IIICOMPUTATIONAL COMPLEXITY COMPARISON OF THE CONVENTIONALSLM SCHEME, THE FIRST PROPOSED SLM SCHEME, AND THE SECONDPROPOSED SLM SCHEME

G

TABLE IVUSED FOR SIMULATIONSCONVERSION MATRICESPROPOSED SLM SCHEME

OF THE

FIRST

Fig. 5. Comparison of PAPR reduction performance of the conventional andthe first proposed SLM scheme.

proposed SLM scheme, we can produce more different sets ofcandidate signals by using additional different random vectorsand additional IFFT blocks.Like the conventional SLM method, both of our proposedSLM schemes need to transmit side information to indicatewhich candidate signal is chosen. Here, we assume that thisside information can be correctly detected at the receiver side., or the effect ofThus, the effect of the conversion withandcarrier-wise multiplication of the phase rotation vectorthe data vector, can be removed at the receiver by using thisside information, just like the conventional SLM method.V. SIMULATION RESULTSPAPR reduction performances of the conventional andproposed SLM schemes were investigated by computersimulations. The OFDM system used for simulations hassubcarriers, 16-QAM modulation, and 20 MHz bandwidth. The performance is evaluated by the CCDFs computed. The phase rotation vecby an oversampling factor ofused in the conventional SLM method are randomlytorsgenerated, i.e., their elements are randomly chosen from the; such a conventional SLM method should havesetbetter PAPR reduction performance than other SLM methodsthat use other kinds of vectors .The first proposed SLM scheme used in our simulations em, andfor,ploys conversion matricesas shown in Fig. 3. For other values of , the conversion maused in simulations are listed in Table IV. Fig. 5tricesshows a comparison of PAPR reduction performance for thefirst proposed SLM scheme and the conventional one. From thisfigure, we can see that, for the same value of , the proposedSLM scheme has slight performance degradation as compared

M

M

M

Fig. 6. PAPR reduction performance of the first proposed SLM method for= 32,= 42, and= 57.

to the conventional one, where the amount of degradation is lessthan those for. Thefor the cases ofreason of this phenomenon is that the conversion matricesstill have some correlation among their corresponding phase ro(the number of cantation vectors. Hence, as the value ofdidate signals) increases, the proposed SLM scheme degradesmore in PAPR reduction performance. Fig. 6 shows the PAPRreduction performance of the first proposed SLM method for,, and(the case using all the proposed conversion matrices). From this figure, we can see theis aperformance is the same for these three cases. So,for the first proposed SLM scheme.limit onFig. 7 shows a performance comparison in PAPR reductionof the second proposed SLM scheme and the conventional one.The second proposed SLM scheme produces two sets of candidate signals in the way shown in Fig. 4, where the conversionused in simulations for different values ofarematriceslisted in Table V. From Fig. 7, we can see that both schemesreach almost the same PAPR reduction performance, which is

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Fig. 7. Comparison of PAPR reduction performance of the conventional andthe second proposed SLM scheme.

G

TABLE VUSED FOR SIMULATIONS OF THE SECONDCONVERSION MATRICESPROPOSED SLM SCHEME

better than that of the first proposed scheme. The reason of thisphenomenon is that the additional randomly generated vectorof the second proposed SLM scheme makes the candidate signals more different (or dissimilar). It should be remindedthat the second proposed scheme uses one more IFFT block ascompared to the first proposed one.Since the elements of most phase rotation vectors used inthe proposed SLM schemes have different magnitudes, modulation symbols at different tones may have different gains and thesignal power of some tones may be attenuated. This may causedegradation in BER performance. Fig. 8 shows a comparisonof BER performance of the conventional and the first proposedSLM scheme for additive white Gaussian noise (AWGN) chan,nels. From this figure, we can see that, to achieve a BER ofthe proposed scheme requires 1.3 dB more signal-to-noise ratio. For the case(SNR) than the conventional scheme for, these two schemes have almost the same BER perofformance; this is due to the fact that the elements of the phaserotation vector used have the same magnitude, and thus modulation symbols at different tones will have equal gain. It shouldbe noted that, although the first proposed SLM scheme has BER, it achieves lower PAPR with much lessdegradation forcomputational complexity than the conventional scheme.Fig. 9 shows a comparison of BER performance of the firstand the second proposed SLM scheme for AWGN channels.From this figure, we can see that the second proposed SLM

Fig. 8. Comparison of BER performance of the conventional and the firstproposed SLM scheme for AWGN channels.

Fig. 9. Comparison of BER performance of the first and the second proposedSLM scheme for AWGN channels.

scheme with candidate signals has almost the same BER percandidate sigformance as the first proposed method withnals. This phenomenon is due to the fact that these two schemesuse the same set of conversion matrices to generate the candidatesignals and the probability of selecting a certain conversion matrix to generate the signal for transmission is the same for bothof them. Note that the BER degradation can be further avoidedby using an extended structure of the second proposed method,where we use more different random vectors to produce moredifferent sets of candidate signals and each set consist of onlytwo candidate signals. With this particular scheme, about half


of the computational complexity, i.e., half of IFFT blocks, willbe reduced as compared to the conventional SLM method.VI. CONCLUSIONIn this paper, we have proposed a kind of low-complexity-point IFFT computation in the SLM apconversions forproach withcandidate signals, where is the oversamplingfactor. These conversions use one IFFT output signal to produce other IFFT output signals, i.e., the candidate signals. Wehave utilized them to replace the IFFT blocks in the conventionalSLM approach and proposed two new SLM schemes. Both ofthe proposed SLM schemes have much lower complexity thanthe conventional one; the first proposed SLM scheme uses onlyone IFFT block and the computational complexity required for-point IFFT blocks is reduced tothe originalcomplex additions; the second proposed scheme usestwo IFFT blocks and the computational complexity required-point IFFT blocks is reduced tofor the originalcomplex additions. Computer simulation resultshave shown that, as compared to the conventional SLM scheme,the first proposed approach has slight performance degradationin PAPR reduction, and the second proposed one reaches almostthe same performance. By using more different randomly generated vectors along with more IFFT blocks for the second proposed SLM scheme, we can generate more candidate signalsfor transmission and thus achieve better PAPR reduction performance. With the low-complexity and good-performance features, the proposed SLM schemes are rather attractive for use inOFDM-based communication systems.APPENDIXIn this appendix, we will show that the phase rotation vector(i.e., the FFT of the vector) contains no zero elements.has no zero elTo prove this, we first show that. Then, according to the first stateements forcontains no zero elements forment, we show that.be the -transform of the vector given in Table II,Leti.e.,(15)into (15), we can check thatSubstituting all possiblehave no zeros on the unit circle of the -plane. Denoteas the discrete-time Fourier transform (DTFT) of the vector .for all and . FromThen we can easily obtaingiven in Section III-C, we havethe definition of

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, and then we can conclude thatcontains no zero.elements forhas no zero elements forTo prove that the vectorand, we note that theis the factor-of-2 up-sampling signal ofvector, i.e.,the vector

otherwise.From (17), the -transform of the vectorexpressed as

(17)can be

(18)Accordingly, the DTFT of the vectorby

is given

(19)This is a two-fold repetition of, indicating that theforDTFT is compressed by a factor of 2. Sinceall and the elements of the FFT of the vectorare the uniform samples of the DTFTon, the vectorthe -axis betweencontains no zero elements for.Note that the vectoris also the factor-of-2. Thus, folup-sampling signal of the vector, we can concludelowing the same procedure untilcontains no zero elements forandthat.is a 2-fold repetition of, the period ofSinceis half that of. Continuing this process until, we can also conclude that the period of the phase rotationvectoris reduced by a factor of 2 as the value of decreasesby one.REFERENCES

zeros(16)This means that the phase rotation vectoris thediscrete Fourier transform of the vector . Therefore, the elecan be obtained by uniformlyments of the vectoron the -axis betweensampling the DTFT

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[6] R. ONeill and L. N. Lopes, Envelope variations and spectral splatterin clipped multicarrier signals, in Proc. 1995 IEEE Int. Symp. Personal,Indoor and Mobile Radio Commun. (PIMRC 95), Sep. 1995, pp. 7175.[7] J. Tellado, Multicarrier Modulation with Low PAR: Applications to DSLand Wireless. Norwell, MA: Kluwer, 2000.[8] D. Wulich and L. Goldfeld, Reduction of peak factor in orthogonalmulticarrier modulation by amplitude limiting and coding, IEEE Trans.Commun., vol. 47, no. 1, pp. 1821, Jan. 1999.[9] X. Li and L. J. Cimini Jr, Effects of clipping and filtering on the performance of OFDM, IEEE Commun. Lett., vol. 2, no. 5, pp. 131133,May 1998.[10] D. J. G. Mestdagh, P. Spruyt, and B. Biran, Analysis of clipping effectin DMT-based ADSL system, in Proc. 1994 IEEE Int. Conf. Commun.(ICC 94), vol. 1, May 1994, pp. 293300.[11] R. Gross and D. Veeneman, Clipping distortion in DMT ADSL systems, Electron. Lett., vol. 29, pp. 20802081, Nov. 1993.[12] T. A. Wilkinson and A. E. Jones, Minimization of the peak-to-meanenvelope power ratio of multicarrier transmission schemes by blockcoding, in Proc. 1995 IEEE Veh. Technol. Conf. (VTC95), Chicago,IL, Jul. 1995, pp. 825829.[13] J. A. Davis and J. Jedwab, Peak-to-mean power control in OFDM,Golay complementary sequences and Reed-Muller codes, IEEE Trans.Inform. Theory, vol. 45, no. 11, pp. 23972417, Nov. 1999.[14] A. E. Jones, T. A. Wilkinson, and S. K. Barton, Block coding schemefor reduction of peak to mean envelope power ratio of multicarrier transmission schemes, Electron. Lett., vol. 30, pp. 20982099, Dec. 1994.[15] K. G. Paterson and V. Tarokh, On the existence and construction ofgood codes with low peak-to-average power ratios, HP, Tech. Rep.HPL-1995-51 990 407.[16] M. Breiling, S. H. Mller, and J. B. Huber, SLM peak-power reduction without explicit side information, IEEE Commun. Lett., vol. 5, pp.239241, Jun. 2001.[17] S. Mller, R. Buml, R. Fischer, and J. Huber, OFDM with reducedpeak-to-average power ratio by multiple signal representation, Ann.Telecommun., vol. 52, pp. 5867, 1997.[18] R. W. Baml, R. F. H. Fischer, and J. B. Huber, Reducing the peak-toaverage power ratio of multicarrier modulation by selected mapping,Electron. Lett., vol. 32, pp. 20562057, Oct. 1996.[19] S. H. Mller and J. B. Huber, A novel peak power reduction schemefor OFDM, in Proc. 1997 IEEE Int. Symp. Personal, Indoor and Mobile Radio Communications (PIMRC 97), vol. 3, Helsinki, Finland, Sep.1997, pp. 10901094., OFDM with reduced peak-to-average power ratio by optimum[20]combination of partial transmit sequences, Electron. Lett., vol. 33, pp.368369, Feb. 1997.[21] S. G. Kang, J. G. Kim, and E. K. Joo, A novel subblock partition schemefor partial transmit sequence OFDM, IEEE Trans. Commun., vol. 45,no. 9, pp. 333338, Sep. 1999.[22] L. J. Cimini Jr. and N. R. Sollenberger, Peak-to-average power rationreduction of an OFDM signal using partial transmit sequences, IEEECommun. Lett., vol. 4, no. 3, pp. 8688, Mar. 2000.[23] C.-L. Wang, M.-Y. Hsu, and Y. Ouyang, A low-complexity peak-to-average power ratio reduction technique for OFDM systems, in Proc.2003 IEEE Global Telecommun. Conf. (GLOBECOM 2003), San Francisco, CA, Dec. 2003, pp. 23752379.

Chin-Liang Wang (S85M87SM04) was bornin Tainan, Taiwan, R.O.C., in 1959. He received theB.S. degree in electronics engineering from NationalChiao Tung University (NCTU), Hsinchu, Taiwan, in1982, the M.S. degree in electrical engineering fromNational Taiwan University, Taipei, Taiwan, in 1984,and the Ph.D. degree in electronics engineering fromNCTU in 1987.He joined the faculty of National Tsing Hua University (NTHU), Hsinchu, Taiwan, in 1987, where heis currently a Professor of the Department of Electrical Engineering and the Institute of Communications Engineering. Duringthe academic year 19961997, he was on leave at the Information Systems Laboratory, Department of Electrical Engineering, Stanford University, Stanford,CA, as a Visiting Scholar. He served as the Director of the Institute of Communications Engineering from 1999 to 2002. Since August 2002, he has beenthe Director of the Universitys Computer and Communications Center. His research interests include wireless spread-spectrum CDMA communication systems, OFDM-based communication systems, equalization, interference cancellation, multiuser detection, synchronization, and VLSI for communications andsignal processing.Dr. Wang was a recipient of the Distinguished Teaching Award sponsored bythe Ministry of Education, R.O.C., in 1992. He received the Acer Dragon ThesisAward in 1987 and the Acer Dragon Thesis Advisor Awards in 1995 and 1996.In the academic years 19931994 and 19941995, he received the OutstandingResearch Awards from the National Science Council, R.O.C. He received theHDTV Academic Achievement Award from the Digital Video Industry Development Program Office, Ministry of Economic Affairs, R.O.C., in 1996. He wasalso the Advisor on several technical works that won various awards in Taiwan,including the Outstanding Award of the 1993 Texas Instruments DSP ProductDesign Challenge in Taiwan, the Outstanding Award of the 1994 Contest onDesign and Implementation of Microprocessor Application Systems sponsoredby the Ministry of Education, R.O.C., the Outstanding Award of the 1995 Student Paper Contest sponsored by the Chinese Institute of Engineers, and the1995 and 2000 Master Thesis Awards of the Chinese Institute of Electrical Engineering. He served as an Associate Editor for the IEEE TRANSACTIONS ONSIGNAL PROCESSING from 1998 to 2000 and has been an Editor for Equalization for the IEEE TRANSACTIONS ON COMMUNICATIONS since 1998.

Yuan Ouyang (S02) was born in Kaoshiung,Taiwan, R.O.C., in 1972. He received the B.S.degree in electrical engineering from National SunYat-Sen University, Kaoshiung, Taiwan, in 1994,and the M.S. degree in electrical engineering fromNational Tsing Hua University, Hsinchu, Taiwan,in 1996. He is currently working toward the Ph.D.degree in the Department of Electrical Engineering,National Tsing Hua University.His research interests include synchronizationand peak-to-average power ratio reduction forOFDM-based communication systems.

low complexity slm for papr in ofdm systems

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