Adaptive Generation Method of OFDM Signals in SLM Schemes for Low-complexity

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1 Adaptive Generation Method of OFDM Signals in SLM Schemes for Low-compleity Kee-Hoon Kim, Hyn-Seng Joo, Jong-Seon No, and Dong-Joon Shin 1 ariv: v1 [cs.it] 31 Ag 212 Abstract There are many selected mapping (SLM) schemes to redce the peak-to-average power ratio (PAPR) of orthogonal freqency division mltipleing (OFDM) signals. Beginning with the conventional SLM scheme, there have been proposed many low-compleity SLM schemes inclding Lim s, Wang s, and Baely s SLM schemes typically. In this paper, we propose an adaptive generation (AG) method of OFDM signals in SLM schemes. By generating the alternative OFDM signals adaptively, nnecessary comptational compleity of SLM schemes can be removed withot any degradation of their PAPR redction performance. In this paper, we apply the AG method to varios SLM schemes which are the conventional SLM scheme and its low-compleity versions sch as Lim s, Wang s, and Baely s SLM schemes. Of corse, the AG method can be applied to most of eisting SLM schemes easily. The nmerical reslts show that the AG method can redce their comptational compleity sbstantially. Inde Terms Low-compleity, orthogonal freqency division mltipleing (OFDM), peak-to-average power ratio (PAPR), selected mapping (SLM). I. INTRODUCTION Orthogonal freqency division mltipleing (OFDM) is a mlticarrier modlation method tilizing the orthogonality of sbcarriers. OFDM has been adopted as a standard modlation method in several K.-H. Kim, H.-S. Joo, and J.-S. No are with the Department of Electrical Engineering and Compter Science, INMC, Seol National University, Seol, , Korea ( kkh@ccl.sn.ac.kr, joohs@ccl.sn.ac.kr, jsno@sn.ac.kr). D.-J. Shin is with the Department of Electronic Engineering, Hanyang University, Seol, , Korea ( djshin@hanyang.ac.kr).

2 2 wireless commnication systems sch as digital adio broadcasting (DAB), digital video broadcasting (DVB), IEEE wireless local area network (WLAN), and IEEE wireless metropolitan area network (WMAN). Similar to other mlticarrier schemes, OFDM has a high peak-to-average power ratio (PAPR) problem, which makes its straightforward implementation qite costly. Ths, it is highly desirable to redce the PAPR of OFDM signals. Over the last decades, varios techniqes to redce the PAPR of OFDM signals have been proposed sch as clipping [1],[2], coding [3], active constellation etension (ACE) [4], tone reservation (TR) [5], partial transmit seqence (PTS) [6], and selected mapping (SLM) [7]. Among them, SLM and PTS are widely sed becase they show good PAPR redction performance withot bit error rate (BER) degradation. However, they reqire many inverse fast Forier transforms (IFFTs), which case high comptational compleity. It is well known that SLM scheme is more advantageos than PTS scheme if the amont of side information (SI) is limited. However, the comptational compleity of SLM scheme is larger than that of PTS scheme. Therefore, many modified SLM schemes with low-compleity have been proposed [8] [13]. We review the representative low-compleity SLM schemes among them. Firstly, Lim proposed the low-compleity SLM scheme eploiting the signals at an intermediate stage of IFFT in [1]. In [1], the signals at an intermediate stage of IFFT are mltiplied by phase rotation vectors designed to do not destroy the orthogonality between sbcarriers. Secondly, Wang proposed the lowcompleity SLM scheme sing conversion matrices in [11] and [12]. In [12], one IFFT block is reqired to generate the original OFDM signal seqence and it is converted to many alternative OFDM signal seqences by mltiplying conversion matrices. Thirdly, in [13], Baely proposed the low-compleity SLM scheme eploiting the characteristics of high power amplifier (HPA) in OFDM systems. That is, Baely s SLM scheme only tests phase rotation vectors ntil an OFDM signal seqence with PAPR less than the satration point of HPA is fond. In this paper, an adaptive generation (AG) method of alternative OFDM signals in SLM schemes is proposed. This methodology can be applied to almost all eisting SLM schemes inclding the conventional SLM scheme and its low-compleity versions sch as Lim s, Wang s, and Baely s SLM schemes. Aided by the proposed AG method, the SLM schemes can be implemented with lower comptational compleity than their comptational compleity withot the AG method. Nmerical reslts show that the AG method can redce the comptational compleity of the SLM schemes sbstantially althogh the AG method is applied to the low-compleity SLM schemes which are already modified to have low-compleity. It is meaningfl becase the proposed AG method do not degrade any PAPR redction performance of the

3 3 SLM schemes. The rest of this paper is organized as follows. In Section II, the conventional SLM scheme and its representative three low-compleity versions are reviewed. In Section III, we introdce the AG method and apply it to the conventional SLM scheme. And we analyze the comptational benefit of the AG method stochastically. In Section IV, we briefly describe the applications for the low-compleity SLM schemes. The comptational benefit of the AG method is evalated throgh the nmerical reslts in Section V and conclsions are given in Section VI. II. SLM SCHEMES : CONVENTIONAL SLM SCHEME AND LOW-COMPLEITY SLM SCHEMES In this section, we briefly introdce OFDM systems, the conventional SLM scheme, and its representative low-compleity versions sch as Lim s, Wang s, and Baely s SLM schemes. We handle these three low-compleity SLM schemes becase these SLM schemes are freqently cited in many literatres and have good PAPR redction performance with redced comptational compleity. A. Conventional SLM Scheme In this paper, we se the pper case = {(), (1),..., (N 1)} for the inpt symbol seqence and the lower case = {(), (1),..., (N 1)} for the OFDM signal seqence, where N is the nmber of sbcarriers. The relation between the inpt symbol seqence in freqency domain and the OFDM signal seqence in time domain can be epressed by IFFT as or where W = e j 2π N and n N 1. (n) = = IFFT() (1) N 1 k= (k)w kn (2)

4 4 1 P 2 P 3 P N-point IFFT N-point IFFT N-point IFFT Select the one with minimm PAPR Transmit... U P U N-point IFFT U Fig. 1. A block diagram of the conventional SLM scheme. The conventional SLM scheme in [7] is described in Fig. 1, which generates U alternative OFDM signal seqences, 1 U. To generate U alternative OFDM signal seqences, U distinct phase rotation vectors P known to both transmitter and receiver are sed, where P ={P (), P (1),, P (N 1)} with P (k) = e jφ (k), φ (k) [, 2π), 1 U. P 1 is the all-one vector for generating the original OFDM signal seqence and ths 1 =. The inpt symbol seqence is mltiplied by each phase rotation vector P element by element. Then the inpt symbol seqence is represented by U different alternative inpt symbol seqences, where (k) = (k)p (k), 1 U. These U alternative inpt symbol seqences are IFFTed to generate U alternative OFDM signal seqences = IFFT( ) and their components powers and PAPR vales are calclated. Finally, the alternative OFDM signal seqence ũ having the minimm PAPR is selected for transmission as ( ) ũ = arg min ma (n) 2 E[ (n) 2 ]. (3) 1 U Note that the SI on ũ needs to be transmitted in order to properly demodlate the received OFDM signal seqence at the receiver.

5 5 Psedo code 1: the conventional SLM scheme 1: γ. 2: for = 1, 2,, U 3: generate processing one N-point IFFT. 4: if PAPR of < γ 5: γ PAPR of. 6: ũ. 7: end if 8: end for() 9: transmit ũ. For accrate nderstanding of the conventional SLM scheme, psedo code for the conventional SLM scheme is given as Psedo code 1. The vale of γ in Psedo code 1, which is called as intermediate minimm PAPR vale, is the minimm vale among the PAPR vales of the alternative OFDM signal seqences already generated. B. Low-compleity SLM Schemes There have been many low-compleity SLM schemes which are modified versions of the conventional SLM scheme. We briefly review the representative three low-compleity SLM schemes. They have lower comptational compleity than that of the conventional SLM scheme with the same nmber of alternative OFDM signal seqences. 1) Lim s SLM Scheme [1]: It is already known that one N-point IFFT consists of n = log 2 N stages. In Lim s SLM scheme, the N-point IFFT is processed from 1-st stage to (n r)-th stage, not the n-th stage, to the inpt symbol seqence, where r is the nmber of remaining stages. And U phase rotation vectors, which are designed to do not destroy the orthogonality between the sbcarriers, are mltiplied to the otpt from the (n r)-th stage of IFFT. Then, for each mltiplied otpt, the remaining stages (i.e., from (n r + 1)-th stage to n-th stage) are processed and these U otpts become U alternative OFDM signal seqences. Among them, the OFDM signal seqence with minimm PAPR is transmitted. For selection, γ is sed similarly to the conventional SLM scheme.

6 6 2) Wang s SLM Scheme [12]: In Wang s SLM scheme, the inpt symbol seqence is IFFTed and its otpt, the original OFDM signal seqence, is mltiplied by U 1 different N N matrices which are called as conversion matrices to generate alternative OFDM signal seqences. Totally, the U alternative OFDM signal seqences are generated. Among them, the OFDM signal seqence with minimm PAPR is transmitted. For selection, γ is sed similarly to the conventional SLM scheme. 3) Baley s SLM Scheme [13]: The generation procedre of alternative OFDM signal seqences in Baely s SLM scheme is the same as the case of the conventional SLM scheme. Bt, the selection strategy is different. For simplicity, let s assme that the HPA is linear p to the satration point γ. Then, achieving a PAPR vale less than γ does not help to improve the system power efficiency. That is, Baely s SLM scheme only tests phase rotation vectors ntil an OFDM signal seqence with PAPR less than γ is fond. With overwhelming low probability, all U alternative OFDM signal seqences have larger PAPR vales than γ. In this case, Baely s SLM scheme selects the one with minimm PAPR among them althogh the PAPR of that is larger than γ. III. ADAPTIVE GENERATION METHOD OF OFDM SIGNALS IN THE CONVENTIONAL SLM SCHEME In this section, we introdce the AG method and its application to the conventional SLM scheme in [7] which is the most basic SLM scheme. A. Adaptive Generation Method of OFDM Signals In Psedo code 1, when the conventional SLM scheme generates -th alternative OFDM signal seqence, their components powers and PAPR vale are observed after the -th alternative OFDM signal seqence, = { (), (1),..., (N 1)}, is flly generated. However, this methodology is inefficient in the comptational sense. The alternative OFDM signal seqences are sccessively generated from 1-st to U-th as Psedo code 1. Sppose that γ is the intermediate minimm PAPR vale p to the ( 1)-th alternative OFDM signal seqence. And net, while generating the -th alternative OFDM signal seqence, if an OFDM signal having larger power than γe[ (n) 2 ] is generated, we can stop this generation procedre immediately. The reason is that, in this case, this -th alternative OFDM signal seqence is never selected. Of corse, the PAPR redction performance of the conventional SLM scheme is not effected by this stop. Fig. 2 represents this AG method pictorially. In Fig. 2, the -th decimation-in-time (DIT) IFFT block in Fig. 1 is shown when N = 8. The IFFT block generates the -th alternative OFDM signals in

7 7 decimated inde order. And the generating procedre can be stopped immediately after the third OFDM signal, (2), is generated becase (2) has larger power than γe[ (n) 2 ]. Withot the AG method, the remaining OFDM signals (6), (1), (5), (3), (7) wold be flly generated. : OFDM signals generated : OFDM signals which do not need to be generated Power P 8-point IFFT () (4) (2) (6) (1) (5) (3) (7) Generation in decimated order 2 E[ ( n) ] () (4) (2) (6) (1) (5) (3) (7) Generation in decimated order Fig. 2. An eample of the AG method at an IFFT in the conventional SLM scheme when N = 8. The important qestion is: Is it possible that a part of an alternative OFDM signal seqence can be generated by processing a part of one IFFT, that is, by lower comptational compleity than the comptational compleity of one IFFT. If possible, stop in the AG method can remove the nnecessary comptational compleity of the conventional SLM scheme. The answer is possible. And, similarly, the cases of almost all SLM schemes inclding Lim s, Wang s, and Baley s SLM schemes are possible, too. B. Partial Generation of OFDM Signals by Partially Processing an IFFT In the comptational sense, it is known that the N-point IFFT has totally N log 2 N points which have to be compted by some comple additions and/or mltiplications. Generally, the comptational compleity of the IFFT is indced from these points and we call them as c-points in this paper. Fig. 3 shows a 8-point IFFT strctre in DIT and there are 24 c-points marked by dashed circles.

8 8 () (1) (2) (3) (4) (5) (6) (7) W W W 2 W 3 W W 2 W W 2 W W W W () (4) (2) (6) (1) (5) (3) (7) Fig. 3. An 8-point IFFT strctre in DIT and its c-points. In the strctre of the IFFT, some part of the OFDM signals can be generated by processing the corresponding part of the IFFT. For instance, in the Fig. 4, (), (4), (2) can be generated by not the fll IFFT bt the partial IFFT. That is, 11 c-points can be compted instead of 24 c-points to generate these three OFDM signals. () (1) (2) (3) (4) W W 2 W () (4) (2) (5) (6) (7) Fig. 4. The reqired part of the 8-point IFFT to generate the OFDM signals (), (4), (2). In analogos ways, we can also sccessively generate the OFDM signals (), (4), (2) by one and one as follows. Firstly, () is generated by compting the seven c-points. Net, (4) is generated by compting the one c=point additively to the c-points for (). Finally, (2) is generated by compting the three c-points additively to the c-points for (4).

9 9 C. Conventional SLM Scheme with the Proposed AG Method By combining the contents of sbsections III-A and III-B, we describe the conventional SLM scheme with the AG method. Psedo code 2 shows the detailed procedre of the conventional SLM scheme aided by the AG method. At third and forth lines in Psedo code 2, the alternative OFDM signals are generated in decimated inde order. That is, the OFDM signals of the -th OFDM signal seqence are sccessively generated by one and one by compting the reqired part of the IFFT additively as we described in sbsection III-B. As fifth line in Psedo code 2, the generation procedre can be stopped based on the vale of γ. Then, we can remove the nnecessary comptational compleity from flly generating the alternative OFDM signal seqences. Psedo code 2: the conventional SLM scheme aided by the AG method 1: γ. 2: for = 1, 2,, U 3: for n =, N/2, N/4,, N 1 4: generate (n) by processing the reqired part of one N-point IFFT additively. 5: if (n) 2 /E[ (n) 2 ] > γ 6: go to 11. 7: end if 8: end for(n) 9: γ PAPR of. 1: ũ. 11: end for() 12: transmit ũ. Fig. 5 shows a block diagram of the conventional SLM scheme aided by the AG method. Ecept the first IFFT block, at the each IFFT block, generation of the each alternative OFDM signal seqence is adaptively processed based on the vale of γ and can be stopped dring the generation procedre.

10 1 1 P 1 N-point IFFT 1 P 2 2 N-point IFFT (adaptively processed) 2 3 P 3 3 N-point IFFT (adaptively processed) Select the one with minimm PAPR Transmit... U P U N-point IFFT (adaptively processed) U Fig. 5. A block diagram of the conventional SLM scheme aided by the proposed AG method. D. Comptational Compleity Analysis In this sbsection, we analyze the comptational compleity of the conventional SLM scheme with the AG method stochastically. As other literatres, we only consider the comptational compleity to generate alternative OFDM signal seqences. 1) Comptational Compleity for Partially Processing an IFFT: In this sbsbsection, we describe the comptational compleity reqired to process a partial IFFT. We regard the comptational compleity reqired to process flly one N-point IFFT as T. And there are N log 2 N c-points have to be compted in one N-point IFFT. We define roghly the comptational compleity for one c-point as t T N log 2 N. (4) And, considering the N-point IFFT strctre consisting of smaller-point IFFTs, the comptational compleity to generate a, 1 a N, OFDM signals in the way of sbsection III-B is easily obtained by K(a) = { n 1 }t + { a a 1 2 n 1 2n 1 }t (5) where n = log 2 N. Trivially, K(N) = N log 2 Nt = T which means processing flly one N-point IFFT.

11 re la tiv e c o m p ta tio n a l c o m p le ity to g e n e ra te a O F D M s ig n a ls.7.6 K (a )/T a Fig. 6. Relative comptational compleity reqired to generate a OFDM signals at a 128-point IFFT. To have insight in the comptational sense, K(a)/T verss a is plotted as Fig. 6 for a 128-point IFFT. It is remarkable that the plot in Fig. 6 behaves in a near linear fashion. For instance, a = 64 at the -ais corresponds to.5 at the y-ais, which means that one can generate a half of an OFDM signal seqence by a half cost of the comptational compleity to process one 128-point IFFT. 2) Probability Distribtion of the Nmber of Alternative OFDM Signals Generated: In this sbsbsection, we derive the probability distribtion of the nmber of alternative OFDM signals generated at the each IFFT block in the conventional SLM scheme when the AG method is applied. We remark the fndamental isses. In this paper, or analysis is based on the heristic assmption that the baseband OFDM signal is characterized as a comple Gassian, which becomes accrate as the nmber of sbcarriers increases de to the central limit theorem [2]. Now, assming that the OFDM signal is comple Gassian, the amplitde of the OFDM signal is Rayleigh distribted and it can be easily shown that the probability that one OFDM signal (n) is smaller than γe[ (n) 2 ] is denoted as

12 12 Γ and given by ( ) (n) 2 Γ P E[ (n) 2 ] < γ =1 e γ. (6) Moreover, as the assmption in [7], we assme that the components in an OFDM signal seqence are mtally independent from the central limit theorem. And we also assme that U alternative OFDM signal seqences are mtally independent. The above assmptions are valid only when the Nyqist sampling rate is sed in OFDM systems. We denote the nmber of OFDM signals generated at the -th IFFT block as a random variable A for 2 U. The distribtion of A depends on the vale of γ after ( 1)-th alternative OFDM signal seqence is generated. And γ is eqal to the PAPR vale of the conventional SLM scheme with 1 alternative OFDM signal seqences. Therefore, the probability mass fnctions (PMFs) of A can be represented as p A (a ) = 1 p A P AP R SLM( 1) (a γ) f P AP RSLM( 1) (γ)dγ (7) where a is an integer within [1, N], p A P AP R SLM( 1) (a γ) is the conditional PMF of A given the vale of P AP R SLM( 1), and f P AP RSLM( 1) (γ) is the probability density fnction (PDF) of the random variable P AP R SLM( 1). The random variable P AP R SLM( 1) means the PAPR vale from the conventional SLM scheme generating 1 alternative OFDM signal seqences. And the range of integral in (7) comes from the definition of PAPR. and With Γ, we describe the fnctions p A P AP R SLM( 1) (a γ) and f P AP RSLM( 1) (γ) as Γ a 1 (1 Γ), 1 a N 1 p A P AP R SLM( 1) (a γ) = Γ N 1, a = N f P AP RSLM( 1) (γ) = d dγ F P AP R SLM( 1) (γ) (8) = d dγ (1 (1 ΓN ) 1 ), (9) respectively, where F P AP RSLM( 1) (γ) is the cmlative distribtion fnction (CDF) of P AP R SLM( 1). Combining (5) and (7), the epectation of the comptational compleity of the conventional SLM scheme aided by the AG method is given by N T + K(a 2 )p A2 (a 2 ) + + a 2=1 N K(a U )p AU (a U ). (1) a U =1

13 13 Trivially, the comptational compleity of the conventional SLM scheme generating U alternative OFDM signal seqences withot the AG method is UT. 3) Comparison between the Analytical and Nmerical Reslts: In this sbsbsection, we compare the analytical reslts and the nmerical reslts of the comptational benefit of the AG method. The nmber of sbcarriers is N = 64 and the Nyqist sampling rate is sed. And the nmbers of the alternative OFDM signal seqences are U = 2, 3,, R e la tiv e c o m p ta tio n a l c o m p le ity (% ) U a n a ly tic a l re s lts n m e ric a l re s lts Fig. 7. Relative comptational compleity of the conventional SLM scheme with the AG method compared to the case withot the AG method for N = 64 and varios U. Fig. 7 shows the relative comptational compleity reqired to the conventional SLM scheme with the AG method compared to the case withot the AG method (i.e., UT ). The analytical reslts are given by the vale of (1) to UT ratio and the nmerical reslts are given by testing 1 5 randomly generated inpt symbol seqences by compter simlations. For small U, two reslts are similar. However, for large U, there is a little gap between the two reslts. This difference comes from the assmption, the U alternative OFDM signal seqences are mtally independent. In practical sitations, there are correlations between the U alternative OFDM signal seqences.

14 14 IV. ADAPTIVE GENERATION METHOD OF OFDM SIGNALS IN THE LOW-COMPLEITY SLM SCHEMES In this section, we briefly introdce Lim s [1], Wang s [12], and Baley s [13] SLM schemes applied by the AG method. The basic methodology of these applications are similar to the conventional SLM scheme case. The AG method can be applied to other SLM scheme beyond these SLM schemes analogosly. A. Lim s SLM Scheme with the AG Method In Lim s SLM scheme with the AG method, the common IFFT processed from 1-st stage to (n r)-th stage is the same. Bt, the remaining stages of U IFFTs can be adaptively processed based on the vale of γ. Clearly, it is possible that a part of an alternative OFDM signal seqence can be generated by partially processing the remaining stages of the IFFT. B. Wang s SLM Scheme with the AG Method Clearly, some part of an alternative OFDM signal seqence can be generated by mltiplying the corresponding partial colmns of the conversion matri to the original OFDM signal seqence. That is, the U 1 conversion matri-vector mltiplications can be adaptively processed based on the vale of γ. C. Baely s SLM Scheme with the AG Method In Baely s SLM scheme, there is the vale of γ, the satration point of HPA, instead of γ. With the AG method, each IFFT is adaptively processed based on γ. That is, while alternative OFDM signals are being generated, the IFFT process can be stopped if an OFDM signal having PAPR larger than γ E[ (n) 2 ] is generated. Then, the net alternative OFDM signal seqence has to be generated and tested by γ in the same manner. As we described in sbsection II-B, with overwhelming low probability, all the U alternative OFDM signal seqences have larger PAPR vales than γ. In this case, Baely s SLM scheme with the AG method select the one with minimm PAPR among them by finishing the U partially processed IFFT blocks.

15 15 V. NUMERICAL RESULTS In this section, we present some nmerical reslts. The OFDM signal seqences are for times oversampled by inserting zeroes into (alternative) inpt symbol seqences. We obtain the nmerical reslts from the simlations to present the comptational benefit of the proposed AG method eactly. The reslts are given by testing 1 5 randomly generated inpt symbol seqences. As other literatres, the comptational compleity to generate alternative OFDM signal seqences is only considered for the comparison. Clearly, the PAPR redction performance is not degraded by the AG method and ths we compare only the comptational compleity. The redction of the comptational compleity does not depend on the modlation order and 16-qadratre amplitde modlation (16-QAM) is sed for all simlations. A. Nmerical Reslts for the Conventional SLM Scheme For the conventional SLM scheme, we simlate the OFDM system when N = 256 and N = 124. Table I shows the comptational compleity of the conventional SLM scheme with the AG method and that withot the AG method. As we mentioned, the comptational compleity of the conventional SLM scheme withot the AG method is UT. Table I shows that the AG method can redce the comptational compleity of the conventional SLM scheme sbstantially. For instance, when U = 32, the comptational compleity of the conventional SLM scheme aided by the AG method is redced to be almost 33% of its original cost. Table I shows that the nmber of sbcarriers N does not effect the comptational benefit of the AG method. And the AG method has a large benefit as U increases. TABLE I THE COMPUTATIONAL BENEFIT OF THE AG METHOD FOR THE CONVENTIONAL SLM SCHEME. U = 8 U = 16 U = 32 Conventional SLM withot AG (a) 8T 16T 32T N = 256 Conventional SLM with AG (b) 4.21T 6.69T 1.82T (b)/(a) (%) Conventional SLM withot AG (c) 8T 16T 32T N = 124 Conventional SLM with AG (d) 4.22T 6.65T 1.7T (d)/(c) (%)

16 16 B. Nmerical Reslts for the Low-compleity SLM Schemes Beyond the conventional SLM scheme, we also present the nmerical reslts when we apply the AG method to the three low-compleity SLM schemes. The nmber of sbcarriers is fied to N = 256. Table II shows the comptational compleity of Lim s SLM scheme with the AG method and that withot the AG method. In Lim s SLM scheme, the nmber of the remaining stages is r = 5 which is the garanteed vale to have good PAPR redction performance. The AG method can redce the comptational compleity sbstantially of Lim s SLM scheme. TABLE II THE COMPUTATIONAL BENEFIT OF THE AG METHOD FOR LIM S SLM SCHEME [1]. U = 8 U = 16 U = 32 Lim s SLM withot AG (a) 4.5T 8.5T 16.5T Lim s SLM with AG (b) 2.46T 3.48T 5.1T (b)/(a) (%) Table III shows the comptational benefit when the AG method is applied to Wang s SLM scheme. In Table III, we present the comparison of the nmber of comple additions which are reqired for conversion matri-vector mltiplications in Wang s SLM scheme. And Wang s SLM scheme has a constraint on U and the cases of U = 4, 8, 12 are simlated. TABLE III THE COMPUTATIONAL BENEFIT OF THE AG METHOD FOR WANG S SLM SCHEME [12]. U = 4 U = 8 U = 12 Wang s SLM withot AG (a) 9,216 21,54 33,792 Wang s SLM with AG (b) 4,933 9,288 12,82 (b)/(a) (%) Table IV shows the comptational compleity when the AG method is applied to Baely s SLM scheme. In this case, the nmber of alternative OFDM signal seqences is fied to U = 16. In Table IV, the comptational benefit of the AG method depends on the vale of γ, the satration point of HPA. For γ = 8.dB, the proposed AG method can redce the comptational compleity of Baely s SLM scheme to be almost 55% of its original cost.

17 17 TABLE IV THE COMPUTATIONAL BENEFIT OF THE AG METHOD FOR BAELY S SLM SCHEME [13]. γ = 7.5dB γ = 8.dB γ = 8.5dB Baely s SLM scheme withot AG (a) 8.3T 3.24T 1.73T Baely s SLM scheme with AG (b) 5.12T 1.81T 1.28T (b)/(a) (%) It is remarkable that the AG method has a large comptational benefit when we apply the AG method to the low-compleity SLM schemes which are already modified for low-compleity. That is, the AG method can be combined effectively to almost all eisting SLM schemes to have mch lower comptational compleity. VI. CONCLUSIONS There are many SLM schemes inclding the conventional SLM scheme and its low-compleity versions. The SLM schemes generate the alternative OFDM signal seqences. We proposed the AG method of the alternative OFDM signals in the SLM schemes. With the AG method, the comptational compleity of the SLM schemes can be redced sbstantially. It is meaningfl that the application of the AG method does not degrade any PAPR redction performance of the SLM scheme. In this paper, we described the application for the conventional SLM scheme and its stochastic analysis is also given. And, we briefly described the applications for the three representative low-compleity SLM schemes. Nmerical reslts show the comptational benefit of sing the AG method. It is remarkable that the AG method is also effective for the low-compleity SLM scheme which is already modified to have low-compleity. We anticipate that the AG method can be applied to many other SLM schemes beyond the SLM schemes described in this paper. ACKNOWLEDGMENT This work was spported by the National Research Fondation of Korea (NRF) grant fnded by the Korea government (MEST) (No ). REFERENCES [1] R. Oneal and L. N. Lopes, Envelope variation and spectral splatter in clipped mlticarrier signals, in Proc. IEEE PIMRC, Sep. 1995, pp

18 18 [2] H. Ochiai and H. Imai, Performance of the deliberate clipping with adaptive symbol selection for strictly band-limited OFDM systems, IEEE J. Sel. Areas. Commn., vol. 18, no. 11, pp , Nov. 2. [3] Y.-C. Tsai, S.-K. Deng, K.-C. Chen, and M.-C. Lin, Trbo coded OFDM for redcing PAPR and error rates, IEEE Trans. Wireless Commn., vol. 7, no. 1, pp , Jan. 28. [4] B. S. Krongold and D. L. Jones, PAR redction in OFDM via active constellation etension, IEEE Trans. Broadcast., vol. 49, no. 3, pp , Sep. 22. [5] J. Tellado and J. M. Cioffi, Mlticarrier Modlation With Low PAR, Application to DSL and Wireless. Norwell, MA: Klwer Academic Pblisher, 2. [6] S. H. Müller, R. W. Bäml, R. F. H. Fischer, and J. B. Hber, OFDM with redced peak-to-average power ratio by mltiple signal representation, Ann. Telecommn., vol. 52, no.1 2, pp , Feb [7] R. W. Bäml, R. F. H. Fischer, and J. B. Hber, Redcing the peak-to-average power ratio of mlticarrier modlation by selected mapping, Electron. Lett., vol. 32, no. 22, pp , Oct [8] A. Ghassemi and T. A. Glliver, Partial selective mapping OFDM with low compleity IFFTs, IEEE Comm. Lett., vol. 12, no. 1, pp. 4 6, Jan. 28. [9] H.-B. Jeon, J.-S. No, and D.-J. Shin, A low-compleity SLM scheme sing additive mapping seqences for PAPR redction of OFDM signals, accepted for pblication in IEEE Trans. Broadcast., Apr [1] D.-W. Lim, J.-S. No, C.-W. Lim, and H. Chng, A new SLM OFDM scheme with low compleity for PAPR redction, IEEE Signal Process. Lett., vol. 12, no. 2, pp , Feb. 25. [11] C.-L. Wang and Y. Oyang, Low-compleity selected mapping schemes for peak-to-average power ratio redction in OFDM systems, IEEE Trans. Signal Process., vol. 53, no. 12, pp , Dec. 25. [12] C.-L. Wang and S.-J. K, Novel conversion matrices for simplifying the IFFT comptation of an SLM-based PAPR redction scheme for OFDM systems, IEEE Trans. Commn., vol. 57, no. 7, pp , Jl. 29. [13] R. J. Baely and G. T. Zho, MAP metric for blind phase seqence detection in selected mapping, IEEE Trans. Broadcast., vol. 51, no. 4, pp , Dec. 25.