International Symposium on Information Theory and its Applications, ISITA2008Auckland, New Zealand, 7-10, December, 2008

New SLM Scheme for PAPR Reduction in OFDM Signals

Kyu-Hong Kim†, Hyun-Bae Jeon†, Jong-Seon No†, and Dong-Joon Shin‡

† Dept. of Electrical Eng. and Computer ScienceSeoul National Univ. Seoul 151-744, Korea

E-mail: {adageneral,lucidream}@ccl.snu.ac.krjsno@snu.ac.kr

‡ Dept. of Electronics and Computer Eng.Hanyang Univ. 133- 791 Seoul, Korea

E-mail: djshin@hanyang.ac.kr

Abstract

In this paper, we propose a new selected mapping(SLM) scheme for reducing peak to average power ratio(PAPR) of orthogonal frequency division multiplexing(OFDM) signals, called bit SLM scheme. In the con-ventional SLM, the alternative symbol sequences aregenerated by rotating the phases of the M-ary symbolsin the frequency domain. However, in the proposedscheme, the alternative symbol sequences are generatedby multiplying the data in the binary expression bythe binary phase sequences before mapping to quadra-ture amplitude modulation (QAM) symbols. Simula-tion results show that the bit SLM scheme has betterPAPR reduction performance compared to the conven-tional SLM scheme, especially for the QAM modulatedOFDM signal for the small size of inverse fast Fouriertransform (IFFT).

1. INTRODUCTION

Orthogonal frequency division multiplexing(OFDM) can support high data rate and have highreliability to provide video, voice, and data com-munication. But due to the high peak to averagepower ratio (PAPR) feature, an OFDM signal maysuffer from significant inter-modulation and undesiredout-of-band radiation [1], [2] when it is passed througha nonlinear device such as a high power amplifier(HPA) [3].

Several techniques have been proposed to mitigatethe PAPR problem in the OFDM signals such as clip-ping [4], block coding [5], [6], selected mapping (SLM)[7], partial transmit sequence (PTS) [8]. The key ideaof SLM scheme is to create different signals by mul-tiplying mutually independent phase sequences beforeinverse fast Fourier transform (IFFT) and select the

one with the smallest PAPR.In this paper, we propose a new SLM scheme, called

bit SLM. The difference between the conventional SLMand the bit SLM is the way of generating alternativesymbol sequences. In the conventional SLM, alterna-tive symbol sequences are generated by rotating thephases of the M-ary symbols in the frequency domainbut in the proposed scheme, the alternative symbol se-quences are generated by multiplying the data in thebinary expression by the binary phase sequences beforemapping to quadrature amplitude modulation (QAM)symbols.

2. CONVENTIONAL SLM SCHEME

The OFDM signal sequence a = [a0, a1, · · ·, aN−1]using N = 2n subcarriers can be expressed as

at =1√N

N−1∑

k=0

Akej2π kN t, 0 ≤ t < N − 1 (1)

where A = [A0, A1, · · ·, AN−1] is an input symbol se-quence usually modulated by using phase shift keying(PSK) or QAM and t stands for a discrete time index.

The PAPR of the transmitted OFDM signal can bedefined as

PAPR(a) .=max0≤t<N

|at|2

E [|a|2] (2)

where E[·] denotes the expectation operator.To generate U alternative symbol sequences, an

input symbol sequence is multiplied by U differ-ent phase sequences, each of length N , Pu =[Pu,0 Pu,1· · ·Pu,N−1], 1 ≤ u ≤ U . The firstphase sequence P1 is usually the all-1 sequence.Then, the alternative symbol sequences Au =[Au,0 Au,1· · ·Au,N−1], 1 ≤ u ≤ U , are generated. We

Binary

Source

M1

QAM

modulator

B1 A1

IFFTa1

M2

QAM

modulator

B2A2

IFFT

a2

MU

QAM

modulatorIFFT

Select

a

signal

with

the

lowest

PAPR

~~au

AUaU

B

BU

Figure 1: Block diagram of the bit SLM scheme.

use the expression Au = A ⊗ Pu to represent thecomponent-wise multiplication, Au,n = AnPu,n, 0 ≤n ≤ N − 1. After U different alternative symbol se-quences are transformed with IFFT, the OFDM signalsequence au = IFFT(Au) with the lowest PAPR is se-lected for transmission.

In [9], if we assume that alternative symbol se-quences are mutually independent, the complementarycumulative distribution function (CCDF) expressionfor the SLM OFDM symbol can be given as

Pr(PAPR(au) > PAPR0) = (1− (1− e−PAPR0)N )U .(3)

But, in the case of small N , at is not approximatelycomplex Gaussian distributed and each alternativesymbol sequence is mutually dependent [9]. Thus, theCCDF expression (3) cannot apply to the SLM OFDMsymbol in general.

3. BIT SLM SCHEME

3.1. Bit SLM Scheme

The block diagram of bit SLM is shown in Fig. 1. Inthe bit SLM scheme, the alternative symbol sequencesare generated by multiplying data in the binary expres-sion of input symbol sequence by the binary phase se-quences before mapping to QAM symbols. We choosebinary m-sequences as phase sequences. The length ofm-sequence is the same as the length of input symbolsequence in the binary expression, i.e, N(log2 M) forM-QAM.

The alternative bit sequences Bu in the bit SLMscheme are generated by multiplying the input bit se-quence B by the binary phase sequence Mu. We usethe expression Bu = B⊗Mu to represent the bit-wise

multiplication as

Bu = [Bu,0, Bu,1, . . ., Bu,N(log2 M)−1], 1 ≤ u ≤ U

= B⊗Mu

= [B0Mu,0, B1Mu,1, . . ., BN(log2 M)−1Mu,N(log2 M)−1)](4)

where Bi ∈ {±1}, M is the modulation orderof M-QAM and Mu,i ∈ {±1} is the binary m-sequence with an extra +1 padded. It is very sim-ple and easy to generate the alternative symbol se-quences. For example, for the input bit sequenceB = [1, 1,−1, 1,−1,−1, 1, 1] and the phase sequenceMu = [1,−1, 1, 1,−1,−1,−1, 1], the alternative sym-bol sequence is given as Bu = [1,−1,−1, 1, 1, 1,−1, 1].Then Bu can be mapped to QAM symbols to generatethe alternative symbol sequence Au. After alternativesymbol sequences Au’s are IFFTed individually, theOFDM signal au = IFFT(Au) with the lowest PAPRis selected for transmission, where u is expressed as

u = arg min1≤u≤U

PAPR(au). (5)

In the bit SLM scheme, alternative symbol se-quences are generated before mapped to QAM sym-bols and thus we can change not only the phase butalso the magnitude of the symbols in the input symbolsequence with QAM modulation. From this, we cangenerate more mutually independent alternative sym-bol sequences and have higher probability of selectingthe one with the lower PAPR compared with the con-ventional SLM scheme. The bit SLM scheme does notincrease the computational complexity compared withthe conventional SLM scheme because multiplicationsof the input bit sequence by binary phase sequences arenot burdensome compared to IFFT.

3.2. Cyclic Hadamard Matrix

A Hadamard matrix HN is an N×N square matrixwith elements +1 and −1 satisfying HNHT

N = NIN ,where IN is the N × N identity matrix. A cyclicHadamard matrix is a Hadamard matrix with an addi-tional property that by removing the top row and theleft-most column of HN , the rows become cyclic shiftsof each other. The rows of the cyclic Hadamard ma-trix are orthogonal and each rows is aperiodic [8]. Thebinary sequence with ideal autocorrelation can be usedas the row of the cyclic Hadamard matrix, for exam-ple, m-sequence. Thus, the rows of cyclic Hadamardmatrix could be good phase sequences for the bit SLMscheme.

(a) 16-QAM.

(b) 64-QAM.

Figure 2: Comparison of PAPR reduction performance ofthe conventional and bit SLM schemes with N = 64.

4. SIMULATION RESULTS

The PAPR reduction performances of the conven-tional SLM and the bit SLM schemes are investigatedby numerical analysis. The OFDM signals used for thesimulation have N = 64 and 128, where 16-QAM and64-QAM are considered. Their performance is evalu-ated by the CCDF computed by using the four timesoversampling.

Figs 2 and 3 compare the PAPR reduction perfor-mances of the bit SLM and conventional SLM schemesfor N = 64 and 128, respectively, when 16-QAM and64-QAM are used. They show that the PAPR reduc-tion performance of the bit SLM scheme with M-QAMis better than that of the conventional SLM, especiallywhen N is small and U is large.

(a) 16-QAM.

(b) 64-QAM.

Figure 3: Comparison of PAPR reduction performance ofthe conventional and bit SLM schemes with N = 128.

The reason why the proposed bit SLM scheme hasbetter PAPR reduction performance can be explainedas follows. The power |at|2 of an OFDM signal hasa central chi-square distribution with two degrees offreedom and the average power of an OFDM signalsequence is defined as

Pav =1N

N−1∑t=0

|at|2, 0 ≤ t < N − 1. (6)

Therefore, Pav also follows chi-square distribution andthe probability that the minimum average power Pmin

among U alternative OFDM signals is less than thethreshold Po is given by

Pr(Pmin < Po) = 1−Pr(Pav > Po)U = 1−Qχ22N

(NPo)U

(7)

1 2 3 4 5 6 7 8 9 10

x 105

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Mo

vin

g A

vera

ge o

f P

ow

er(

dB

)

Symbol Number

0 1 2 3 4 5 6 7 8 9 10

x 105

3

4

5

6

7

8

1 2 3 4 5 6 7 8 9 10

x 105

3

4

5

6

7

8

So

rted

PA

PR

(dB

)

PAPR(Conv. SLM)

PAPR(bit SLM)

Power(Conv. SLM)

Power(bit SLM)

Figure 4: Relation of PAPR and moving average power ofthe conventional SLM and the bit SLM with N = 64.

where Qχ22N

is the right-tail probability for chi-squaredistribution with 2N degrees of freedom. It means thatthe number U of alternative symbol sequences increasesthe probability that the minimum average power Pmin

among U alternative OFDM signals is less than Po.Since the right-tail probability for chi-square distribu-tion decreases exponentially as N increases, we needmore alternative symbol sequences to have the smallerminimum average power for larger N .

To show what influence this result has on the PAPRreduction, we sorted IFFT output signals in the in-creasing order of PAPR values and analyzed theirpower values using moving average filter of 1000 taps.Fig. 4 shows that the OFDM signal with higher PAPRhas larger average power. That is, a moving averagepower of the selected OFDM signal by SLM scheme isproportional to PAPR statistically. Since the averagepowers of U alternative OFDM signals have differentvalues in the bit SLM scheme, the PAPR reduction ofthe bit SLM scheme is better than the conventionalone which generates U alternative OFDM signals hav-ing the same average power. Fig. 4 also shows that themoving average power of the selected signal from U al-ternative symbol sequences in the bit SLM is smallerthan that of the conventional SLM scheme and the dif-ference is bigger at higher PAPR. Average power dif-ference between two PAPR schemes is about 0.2dB andthis does not affect the BER performance.

5. CONCLUSIONS

We proposed a new SLM scheme for PAPR reduc-tion in the OFDM systems. In the bit SLM scheme, the

alternative symbol sequences are generated by multi-plying the input symbol sequence in binary expressionby the binary phase sequences before mapping to QAMsymbols. Simulation results show that the proposedscheme has better PAPR reduction performance thanthe conventional one. The performance improvementof PAPR reduction for the bit SLM scheme becomeslarger as N decreases.

References

[1] R. O’neal and L. N. Lopes, “Envelope variationand spectral splatter in clipped multicarrier sig-nals,” in Proc. PIMRC’95, Sep. 1995, pp. 71–75.

[2] J. Tellado, Multicarrier Modulation with Low PAR: Applications to DSL and Wireless, Norwell, MA:Kluwer, 2000.

[3] H. Ochiai, “Performance analysis of peak powerand band-limited OFDM system with linear scal-ing,” IEEE Trans. on Wireless Communications,vol. 2, no. 5, pp. 1055–1065, Sep. 2003.

[4] D. Wulich and L. Goldfeld, “Reduction of peakfactor in orthogonal multicarrier modulation byamplitude limiting and coding,” IEEE Trans.Commun., vol. 47, no. 1, pp.18–21, Jan. 1999.

[5] A. Jones, T. Wilkinson, and S. Barton, “Blockcoding scheme for reduction of peak to mean en-velope power ratio of multicarrier transmission se-hcmes,” Electron. Lett., vol. 30, no. 25, pp. 2098-2099, Dec. 1994.

[6] J. A. Davis and J. Jedwab, “Peak-to-meanpower control in OFDM, golay complementary se-quences, and Reed-Muller codes,” IEEE Trans.Inf. Theory, vol. 45, no. 7, pp. 2397–2417,Nov. 1999.

[7] D.-W. Lim, S.-J. Heo, and J.-S. No, “On the phasesequence set of SLM OFDM scheme for a crest fac-tor reduction,” IEEE Trans. Signal Process., vol.12, no. 2, pp. 93–96, Feb. 2005.

[8] D.-W. Lim, C.-W. Lim, J.-S. No, and H. Chung,“A new PTS OFDM scheme with low complexityfor PAPR reduction,” IEEE Trans. Broad., vol.54, no. 1, pp. 77–82, Mar. 2006.

[9] G. T. Zhou and L. Peng, “Optimality condition forselected mapping in OFDM,” IEEE Trans. SignalProcess., vol. 54, no. 8, pp. 3159–3165, Aug. 2006.