2013 8th Intenational Confeence on Communications and Netwoking in China (CHINACOM) Cyclic Constellation Mapping Method fo PAPR Reduction in OFDM system Yong Cheng, Jianhua Ge, Jun Hou, and Fengkui Gong the State Key Lab. of Integated Sevice Netwoks, Xidian Univesity, Xian, P. R. China Email: Chengyong0626@gmail.com, jhge@xidian.edu.cn, j.hou.xidian@gmail.com, fkgong@xidian.edu.cn Abstact In this pape, a novel cyclic constellation mapping method (CCM) is poposed to educe the peak-to-aveage powe atio (PAPR) of othogonal fequency division multiplexing (OFDM) signals. By popely choosing the move paamete, this method can obtain a consideable PAPR pefomance. Moeove, in ode to demodulate the signal coectly at the eceive side, a pilot-aided estimation method is also intoduced. Simulation esults show that the poposed method achieves a bette PAPR and bit-eo ate (BER) pefomance than the conventional patial tansmit sequences (PTS) while maintaining low computational complexity. Index Tems othogonal fequency division multiplexing, peak-to-aveage powe atio, cyclic constellation mapping, pilotaided estimation. I. INTRODUCTION Othogonal fequency division multiplexing (OFDM) [1] has dawn explosive attention in a numbe of wieless communication standads including the IEEE 802.11 a/g, the IEEE 802.16 and 3GPP LTE, due to the advantages of high spectal efficiency, and obustness to the fading channel. Howeve, it suffes fom high peak-to-aveage powe atio (PAPR) and thus causes in-band distotion and out-of-band adiation. Theefoe, many PAPR eduction techniques [2]-[3] have been pesented, such as distotion scheme clipping and filteing [4], [5], and distotionless scheme selective mapping (SLM) [6], PTS [7]- [11] and coding scheme [12]. Among them, distotionless scheme is a pomising PAPR eduction scheme. Howeve, it equies lage numbe of invese fast Fouie tansfom (IFFT) at the tansmitte, causing high complexity. Theefoe, in this pape, a novel cyclic constellation mapping method is poposed to educe the PAPR with low computational complexity. By moving the oiginal constellation to the popely position, it can significantly impove the PAPR pefomance. In addition, a pilot-aided estimation method is also intoduced to demodulate the signal coectly.simulation esults show that the poposed scheme outpefoms the conventional patial tansmit sequence at the same complexity. The outline of the pape is oganized as follows. Section II descibes the system and powe amplifie model. Section III intoduces the poposed cyclic constellation mapping method fo PAPR eduction. The pefomance of the poposed scheme is evaluated in Section IV and it is followed by conclusions in Section V. II. SYSTEM MODEL In this section, we fist intoduce the high PAPR poblem in OFDM system. Then a widely-used powe amplifie model, solid-state powe amplifie (SSPA), is descibed. A. OFDM Model An OFDM signal nomally consists of N subcaies modulated by M-ay quadatue amplitude modulation (M-QAM) o phase shift keying (PSK). Let X= [X 0,X 1,..., X N 1 ] T donate the fequency domain tansmit signal. Then, the time domain OFDM signal can be given by x k = 1 N N n=1 2π j X n e JN nk, (1) whee J epesents the ovesampling facto. In this pape, the ovesampling opeation is achieved by inseting (J 1)N zeos in the middle of X, i.e., [X 0,...,X N/2 1, 0,..., 0 }{{},X N/2,...,X N 1 ]. (2) (J 1)N zeos The PAPR of the J-times ovesampled tansmitted signal is defined as max x k 2 0 k<jn 1 PAPR(x) =, (3) 2] E [ x k whee E [ ] denotes the expectation opeation. In [13], accuate continuous-time PAPR equies J 4. A widely used function to measue PAPR eduction pefomance is the complementay cumulative distibution function (CCDF), which epesents the pobability of the PAPR of an OFDM signal exceeding a given theshold PAPR 0, CCDF =P(PAPR > PAPR 0 ). (4) On the othe hand, consideing the fading channel in the eal communication system, the pilot-aided OFDM is needed in ode to obtain the channel state infomation. Theefoe, the subcaies in a pilot-aided OFDM signal can be divided into N s data caies and N p pilot caies, which is patitioned as: { D n, n Υ X n = P n, n Υ, (5) 107 978-1-4799-1406-7 2013 IEEE
Fig. 1. m ˆm X n X n xk Y n The block diagam of the OFDM system with CCM. y k applied to the data caies. Theefoe, the new tansmit signal can be given by x m (k) = 1 N N n=1 2π j S m (X n )e JN nk, (8) and the PAPR eduction poblem can be witten as the following intege optimization poblem whee Υ denotes the set of pilot caies indices, and Υ denotes the data one. In addition, D n and P n ae the data sequence and pilot sequence, espectively. B. Powe Amplifie Model Since a powe amplifie has its own linea ange, a high PAPR will cause a nonlinea distotion. To evaluate the effect of a PAPR eduction method, a powe amplifie model is needed. The SSPA model with amplitude modulation (AM)/AM amplitude distotion can be descibed as, x SSPA(x) = [ ) ] 1, (6) 2p 2p 1+ ( x C whee x, p and C ae the input signal, contol paamete and the maximum output amplitude at the satuation point, espectively. III. THE NOVEL CYCLIC CONSTELLATION MAPPING METHOD In this section, a novel cyclic constellation mapping (CCM) method is intoduced to educe the PAPR. Then, a channel estimation method is also designed to estoe the signal with low complexity. The block diagam of the OFDM system with CCM is depicted in Fig. 1, and the algoithm will intoduce in detail below. A. The Cyclic Constellation Mapping Method In a typically OFDM system, the oiginal M-QAM constellation has a fixed mapping method, i.e., the squae 64-QAM shown in Fig. 2a. The main idea of CCM is to cyclically move the mapping constellation, shown in Fig. 2b. Thus, these exta move can be exploited to educe the PAPR. Equation (7) illustates this modification. In this equation, M = log 2 M 2 1, d and S m (X n ) denote the minimum distance between constellation points and the new mapping constellation when the cyclic move numbe equals to m, espectively. Note that this opeation is only m (opt) =min m PAPR(x m), (9) whee m (opt) is the optimal move of the constellation. Accoding to the cyclic opeation mentioned above, the poposed CCM algoithm can be easily summaized as follows. Algoithm 1 : The Cyclic Constellation Mapping Method fo PAPR Reduction 1: Initialization: set up the maximum cyclic move numbe V.Letm (opt) = m =0, PAPR(x (opt) )=INF and x (opt) =0. 2: Obtain a new mapping constellation S m (X n ) by using (7). 3: Calculate PAPR(x m (k)) and x m (k) by using (3) and (8). 4: if PAPR(x m (k)) <PAPR(x (opt) ) then 5: Let PAPR(x (opt) )=PAPR(x m (k)). 6: Stoe x m (k) as the best solution x (opt). 7: end if 8: Let m = m +1. 9: if m>= V then 10: Tansmit x (opt). 11: else 12: Go to step 2. 13: end if B. Channel Estimation In ode to demodulate the signal coectly of CCM method, we need to send the optimal cyclic move numbe m (opt) to the eceive side. Thee have some papes [10], [11] intoducing the method to send the infomation to eceive. Hee, a simple pilot-aided estimation method is poposed. Since the optimal cyclic move numbe m (opt) is a decimal numbe, taking the condition that V equals 64 fo example, m (opt) can be tuned into a 6 bits binay numbe. This infomation can be sent by using the otation of the pilot caies. Accoding to the size of Υ, fo one pilot caie we can send q bits of infomation. So that m (opt) can be divided into v pats, S m 1 (X n )+d S m (X n )= Re{S m 1 (X n )} + j (Im{S m 1 (X n )} d) S m 1 (X n ), Re{S m 1 (X n )} <M d Re{S m 1 (X n )} >M d &Im{S m 1 (X n )} > M d othewise (7) 108
(a) Oiginal Squae 64-QAM Constellation (b) Cyclically Moved Constellation Fig. 2. The Cyclic Constellation Mapping Method. Subset 1 1 Subset 2 1 (1) 1 1 (2) 1 1 (1) 2... 1 (2) 2... 1 (1) v (2) v ˆH p () =Y p ()/P () =H p ()+n p (),/ U, (15) Subset l 1 () l 1 1... () l 2... 1 () l v 1 whee n p () is the eo of estimation. By using diffeent intepolating method, thee has Fig. 3. Allocation of Pilot Sequence. ˆH p ( )=f( ˆH p ()), U, (16) log2 V v = q. (10) The otation facto R(n) of the pilot caie is given by R(n) = { 1 n/ U (l) v n U, (11) whee U is the set of otation caies, v (l) is the otation facto of the infomation of pat v in subset l. Hee, l is used to impove the accuacy of the estimation. (l) 2π v,i = ej 2 q i, (12) whee i means the seial numbe of the q bits to be sent. Theefoe, the tansmitted pilot caies can be descibed as P (n) =R(n)P (n). (13) At the eceive side, afte the FFT opeation, the eceived pilot signal can be witten as Y p (n) =H p (n)r(n)p (n)+n(n), (14) whee H p (n) is the fading channel fequency esponse in pilot caies, n(n) is the additive white Gaussian noise (AWGN). Denoting that ˆHp is the estimation of H p. Accoding to the Least Squaes (LS) estimation [14], whee f can be any intepolation method, such as linea intepolation, quadatic intepolation and spline intepolation, etc. Finally, the estimation of the otation factos can be teated as a Maximum Likelihood (ML) estimation, ˆ v = min Y p ( ) ˆH p ( ) (l) v,i P ( ) 2. (17) 1 i 2 q l By doing this, m (opt) can be ecoveed at the eceive and then ˆH p ( ) can be estimated coectly. C. Complexity Analysis fo CCM The computational complexity of the cyclic constellation mapping method can be sepaated into two pats: cyclically move the mapping constellation and the IFFT opeation. All these calculation can be descibed in the numbe of additions and multiplications. Note that the complexity of the evesion of a eal numbe also can be teated as one addition in compute calculation. Theefoe, the computational complexity of one CCM move is counted as: 1. N additions in cyclically moving the mapping constellation fo one move; 2. An IFFT opation, which equies JNlog 2 JN additions and 1 2 JNlog 2JN multiplications. Fom the analysis above, since the complexity of IFFT is much highe than the cyclic mapping constellation. Thus, the complexity of the CCM method can appoximately be egaded as V IFFTs. 109
Fig. 4 shows the pobability of each optimal m in 10 6 andom OFDM date blocks. As shown in Fig. 4, each totally cyclic move numbe has almost the same pobability to achieve a lowest PAPR. Theefoe, we can decease the maximum cyclical move numbe V, in ode to educe the algoithm complexity of high ode constellation, especially in 128-QAM o 256-QAM. 0.02 Detection Pobability 10 0 10 1 10 2 p=1 l=1 q=1 l=3 q=2 l=3 q=3 l=3 q=2 l=5 q=3 l=5 Pobability(m (opt) =m) 0.015 0.01 0.005 Aveage Pobability Individual Pobability 10 3 10 4 15 10 5 0 5 10 15 20 E /N (db) b 0 0 0 10 20 30 40 50 60 m Fig. 4. The pobability distibution of the optimal m. IV. SIMULATION RESULTS In this section, numeous simulations ae conducted to pesent the pefomance of the poposed pilot estimation and CCM method. In ode to compae the successfully estimating pobability, CCDF and BER pefomance, 10 6 andom 64- QAM OFDM symbols ae geneated with fou times ovesampling facto and 256 subcaies. Hee, the SSPA model is descibed in (6) with C =1.2 and p =3and signals ae nomalized in fequency domain. A. Estimation Pefomance In ode to compae the capacity of ou estimation algoithm, infomation fo one pilot caie q is applied as 1, 2, 3 with subset numbe l= 1, 3 o 5. When q = 1, the otation set is v {±1}; when q { =2and 3, the otation sets ae v {±1, ±j} and v ±1, ±j, ± 2 2 ± 2 2 }. j As shown in Sec. III, using these diffeent contol paametes, the numbes of pilot caies is needed as shown in (18). The detail data is shown in Table I. log2 V Υ =2l +1. (18) q In simulation, the fading channel consists of five multipath components with delays of [0,1,2,3,4] samples with equivalent aveage gain. LS estimation and linea intepolation is involved as descibed in Sec. III. In Fig. 5, the detection capacity in ou algoithm is shown. It can be easily gotten that highe l and lowe q offe bette pefomance. Tabel I summaizes the successfully estimating ate at 99.9% and the cost of pilot caies. Note that when q goes to 3, the estimation capacity declines apidly. And l should not set to 1, since when deep fading channel appeas, the estimation will be in a mess. Fig. 5. Detection pobability in diffeent mode. TABLE I THE COMPLEXITY AND SUCCESSFUL DETECTION PROBABILITY WITH DIFFERENT INFORMATION BIT q AND SUBSET NUMBER l Indices q l Υ E b /N 0 of successfully detection ate at 99.9% 1 1 1 13 18.2 db 2 1 3 37-0.8 db 3 2 3 19 6.1 db 4 3 3 13 11.6 db 5 2 5 31-0.4 db 6 3 5 21 8.5 db Accoding to Fig. 5 and Table I, although the pefomance of the 2nd contol paamete is a little bette than that of the 5th, the cost of pilot is lage. Thus, the 5th contol paametes is ecommended in eal OFDM systems. Because the pilot estimation algoithm can achieve the successful estimating ate to 99.9% in -0.4dB, in the following simulation, we assume that thee is no pediction eo in the eceive. B. CCDF and BER Pefomance To compae CCDF and BER pefomance, PTS algoithm is intoduced in simulation. M and W in PTS ae the numbe of subblocks and the size of the phase facto set, espectively. Table II shows the computational complexity of CCM and PTS with diffeent factos. In Fig. 6, the CCDF fo diffeent maximum cyclic move numbe V is shown. When P(PAPR > PAPR 0 )=10 3, the PAPR of the oiginal OFDM signal is 11.7 db, while that of the CCM with V =64is 7.6 db. The CCM method outpefoms PTS with 0.6 db and 1.1 db in 8 and 64 IFFTs. What s moe, CCM using 16 times of IFFT outpefoms PTS using 64 times of IFFT with 0.4 db, and CCM with 64 IFFTs outpefoms PTS with 128 IFFTs with 0.3 db. Theefoe, it 110
CCDF (P[PAPR>PAPR 0 ]) 10 0 10 1 10 2 10 3 Oiginal PTS M=4 W=2 CCM V=8 PTS M=4 W=4 CCM V=16 PTS M=8 W=2 CCM V=64 10 4 5 6 7 8 9 10 11 12 PAPR (db) 0 BER 10 1 10 2 10 3 10 4 10 5 10 6 Oiginal without SSPA Oiginal +SSPA PTS M=4 W=2 +SSPA CCM V=8 +SSPA PTS M=4 W=4 +SSPA CCM V=16 +SSPA PTS M=8 W=2 +SSPA CCM V=64 +SSPA 10 15 20 25 30 35 E /N (db) b 0 Fig. 6. PAPR CCDF pefomance of CCM and PTS fo a 256 subcaies OFDM signals. TABLE II COMPARISON OF CCM AND PTS IN COMPLEXITY, PAPR AND BER PERFORMANCE Schemes Complexity PAPR at 10 3 Eo floo CCM V =8 8 IFFTs 9.0 3.3E-5 CCM V =16 16 IFFTs 8.3 8.7E-6 CCM V =64 64 IFFTs 7.6 1.2E-6 PTS M=4 W =2 8 IFFTs 9.6 7.9E-5 PTS M=4 W =4 64 IFFTs 8.7 2.4E-5 PTS M=8 W =2 128 IFFTs 7.9 6.3E-6 can be concluded that CCM has a good PAPR pefomance than PTS even with a lowe complexity. Fig. 7 pesents the BER pefomance though a AWGN channel. Consideing BER at 10 4, CCM outpefoms PTS with 3.2 db and 6.0 db at the complexity of 64 and 8 IFFTs, espectively. In addition, when the signal passing a SSPA, thee will appea an eo floo. The eo floo of these two algoithm in diffeent contolling factos is shown in Table II. The eo floo of the oiginal OFDM signal with SSPA model is 4.8E-4, and when using CCM method, the eo floo is educed to 1.2E-6, which is fa bette than the PTS. Theefoe, a bette PAPR pefomance will educe the nonlineaity of the powe amplifie. V. CONCLUSION In this pape, a novel cyclic constellation mapping method is poposed to educe the high PAPR of OFDM signals. A pilotaided method is also epesented in ode to demodulate the signal coectly. Simulation esults show that the poposed C- CM method can achieve a bette BER and PAPR pefomance with lowe computational complexity than the conventional PTS method. Fig. 7. BER pefomance of oiginal, CCM and PTS fo a 256 subcaies OFDM signals. ACKNOWLEDGMENT The wok was suppoted in pat by the National Science and Technology Majo Poject of the Ministy of Science and Technology of China unde Gant 2012ZX03001027, the National High-tech R&D Pogam of China (863 Pogam) unde Gant 2012AA011701, the National Natual Science Foundation of China unde Gant 61001207 and the 111 Poject (B08038). REFERENCES [1] R. van Nee and R. Pasad, OFDM fo Wieless Multimedia Communications. Boston, MA: Atech House, 2000. [2] S. H. Han and J. H. Lee, An oveview of peak-to-aveage powe atio eduction techniques fo multicaie tansmission, IEEE Wieless Commun., vol. 12, no. 2, pp. 56-65, Ap. 2005. [3] J. Tellado, Multicaie Modulation with Low PAR: Applications to DSL and Wieless. Kluwe Academic Publishes, 2000. [4] T. Lee and H. Ochiai, Expeimental analysis of clipping and filteing effects on OFDM systems, in Poc. IEEE ICC 2010, South Afica, Cape Town, May 2010. [5] Y. C. Wang and Z. Q. Luo, Optimized iteative clipping and filteing fo PAPR eduction of OFDM signals, IEEE Tans. Commun., vol. 59, no. 1, pp. 33-37, Jan. 2011. [6] H. Jeon, J. No and D. Shin, A low-complexity SLM scheme using additive mapping sequences fo PAPR eduction of OFDM signals, IEEE Tans. Boadcast., pp. 1-10, Ma. 2011. [7] Y. Wang, W. Chen, and C. Tellambua, A PAPR eduction method based on atificial bee colony algoithm fo OFDM signals, IEEE Tans. Wieless Commun, vol. 9, no. 10, pp. 2994-2999, Oct. 2010. [8] J. Hou, J. H. Ge, and J. Li, Peak-to-aveage powe atio eduction of OFDM signals using PTS scheme with low computational complexity, IEEE Tans. Boadcast., vol. 57, no. 1, pp. 143-148, Ma. 2011. [9] B. Wang, P. H. Ho and C. H. Lin, OFDM PAPR Reduction by Shifting Null Subcaies Among Data Subcaies, IEEE Commun. Lett., vol. 16, no. 9, pp. 1377-1379, Sep. 2012. [10] L. Guan, T. Jiang, D. Qu and Y. Zhou, Joint Channel Estimation and PTS to Reduce Peak-to-Aveage-Powe Radio in OFDM Systems Without Side Infomation, IEEE Signal Pocess. Lett., vol. 17, no. 10, pp. 883-886, Oct. 2010. 111
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