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Int. J. Electron. Commun. (AEÜ) 66 (2012) 1006– 1010

Contents lists available at SciVerse ScienceDirect

International Journal of Electronics andCommunications (AEÜ)

j our na l ho mepage: www.elsev ier .de /a eue

ow-complexity PAPR reduction technique for OFDM systems using modifiedidely linear SLM scheme

. Yanga,b,∗, Y.M. Siub, K.K. Soob, S.W. Leungb, S.Q. Lia

National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, Sichuan 610054, ChinaDepartment of Electronic Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong Special Administrative Region

r t i c l e i n f o

rticle history:eceived 17 October 2011ccepted 14 May 2012

eywords:rthogonal frequency division

a b s t r a c t

In this paper, a modified widely linear selective mapping (MWL-SLM) scheme is proposed to reduce thepeak-to-average power ratio (PAPR) of the orthogonal frequency-division multiplexing (OFDM) systems.In the proposed MWL-SLM scheme, through partition one complex signals into two real signals andcombining the linear properties of the Fourier Transform, at most 4M2 candidate signals can be obtainedbut only require M inverse fast Fourier transform (IFFT) operations. As a result, the proposed SLM scheme

ultiplexing (OFDM)eak-to-average power ratio (PAPR)elected mapping (SLM)omputational complexity

has the ability to generate more candidates when compared with conventional SLM (C-SLM) and widelylinear SLM (WL-SLM). Therefore, MWL-SLM outperforms C-SLM and WL-SLM for the same computationalcost of IFFT operations. Alternatively, for a given number of candidates, MWL-SLM has slightly inferiorPAPR reduction performance to C-SLM and WL-SLM but requires less IFFT operations to be implemented,thus resulting in a lower computational complexity. Simulation results demonstrate the effectiveness ofthe proposed scheme.

. Introduction

Due to its tremendous advantages such as high spectral effi-iency, immunity to inter-symbol interference (ISI) and robustnessgainst channel fading [1], orthogonal frequency division mul-iplexing (OFDM) has been adopted as a standard technique in

any high-rate data transmission systems, such as European Digi-al Audio Broadcasting (DAB) [2], Digital Video Broadcasting (DVB)3], WLAN standards (802.11), WiMax (802.16) [4] and 3GPP Longerm Evolution (LTE).

However, some drawbacks are still unresolved in the designf OFDM system. One of the major obstacles to the practicalmplementation of an OFDM system is the relatively high peak-to-verage power ratio (PAPR) of the transmitted signal [5], which mayccasionally reach the saturation region of power amplifier, andesult in significant in-band distortion and undesirable out-of-bandadiation. To deal with this problem, a number of techniques haveeen proposed over the past decade, including clipping [6], block

oding [7], active constellation extension (ACE) [8], tone reserva-ion (TR) [9,10], tone injection (TI) [11], companding transformchemes [12,13] and multiple signal representation techniques

∗ Corresponding author at: National Key Laboratory of Science and Technology onommunications, University of Electronic Science and Technology of China, Sichuan10054, China.

E-mail address: eelyang@uestc.edu.cn (L. Yang).

434-8411/$ – see front matter © 2012 Elsevier GmbH. All rights reserved.ttp://dx.doi.org/10.1016/j.aeue.2012.05.003

© 2012 Elsevier GmbH. All rights reserved.

such as partial transmit sequence (PTS) [14] and selective mapping(SLM) [15].

Among them, multiple signal representation is one of the mostpromising techniques because it is simple to implement, no dis-tortion in the transmitted signal, high bandwidth efficiency, highpower efficiency and significant improvement of the statistics ofthe PAPR. However, the conventional SLM (C-SLM) and PTS schemesuffer from higher computational complexity due to a bank ofIFFT operations required to produce candidate signals, which mayhinders its practical application in the systems. To overcome theinherent high computational complexity, several SLM and PTSschemes have been proposed to improve the PAPR reduction per-formance and reduce the computational complexity [16–22], butthese schemes still have substantial requirement on the computa-tional complexity.

The focus of this paper is to further reduce the computationalcomplexity of the SLM scheme. In the proposed scheme, named asmodified widely linear SLM (MWL-SLM) scheme, one complex sig-nal vector has been first partitioned into two real signal vectors, andthen transforms them into time domain with real IFFT operations.

After using the properties of Fourier Transform and reconstruct-ing two real signal vectors to obtain new candidates, at most4M2 alternative candidates can be generated while only M com-

plex IFFT operations is required. Therefore, the proposed schemecan remarkably reduce the computational complexity when com-pared with the C-SLM scheme and the widely linear SLM (WL-SLM)scheme.

mmun. (AEÜ) 66 (2012) 1006– 1010 1007

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T2[xn, k] = x(n−k)N⇔ T2[Xn, k] = Xn · e−j2�kn/N (7)

L. Yang et al. / Int. J. Electron. Co

The rest of this paper is organized as follows. After this intro-uction, in Section 2, the PAPR problem of a typical OFDM system

s formulated and then the principle of the C-SLM scheme and WL-LM scheme is explained. The main idea of our proposed MWL-SLMcheme, the analysis of the overall candidates as well as the com-arison of computational complexity for the proposed scheme with-SLM and WL-SLM are presented in Section 3. Simulation resultsre presented and discussed in Section 4. Finally, Section 5 givesoncluding remarks.

. Preliminaries

.1. PAPR in OFDM

Consider an OFDM system with N subcarriers, the mathematicalepresentation of the OFDM signal is given by

(t) = 1√N

N−1∑l=0

X(l)ej2�l �ft, 0 ≤ t ≤ T (1)

here X(l) is the data symbol carried by the lth subcarrier, �f is thedjacent subcarrier separation, and T is the OFDM signal duration.o ensure that all the subcarriers are orthogonal each other, theFDM symbol duration should be T = 1/�f.

The PAPR of the transmitted OFDM signal in (1) is defined as

APR = 10log10

Max0≤t≤T

|x(t)|2

Pav(2)

here Pav is the average power of x(t).In general, to approximate the true PAPR of continuous time

ignal x(t), a discrete time signal samples x[n] with L oversample ofyquist rate is required. The oversampling technique can be real-

zed by inserting (L − 1)N zeros in the middle of x and the samples[n] can be represented as

[n] = 1√LN

LN−1∑l=0

X̄(l)ej2� ln /N, 0 ≤ n ≤ LN − 1 (3)

here X̄ = [X(0), . . . , X(N/2 − 1), 0, . . . , 0︸ ︷︷ ︸(L−1)N

, X(N/2), . . . , X(N − 1)]T .

t was shown in [23] that an oversampling factor of four is sufficiento approximate the exact PAPR.

.2. Conventional SLM (C-SLM)

In C-SLM scheme, M statistically independent sequences Xm(1 ≤ ≤ M) are generated from the original sequence X with M indi-

idual phase sequences Pm = [Pm(0), . . . , Pm(N − 1)], which can beritten as

Xm = [Xm(0), . . . , Xm(N − 1)]

= X � Pm

= [X(0)Pm(0), . . . , X(N − 1)Pm(N − 1)], 1 ≤ m ≤ M

(4)

here � is denoted as the component-wise multiplication of thewo vectors. In general, each symbol of the phase sequences Pm

hould have unit magnitude to preserve the power, and the firsthase sequence P1 is usually the all one sequence 1N. Then a set ofandidates xm can be generated as

m = IFFT(Xm) = IFFT(X � Pm), 1 ≤ m ≤ M (5)

The one xm∗with the lowest PAPR among all the candidates

m(1 ≤ m ≤ M) is selected and transmitted, the block diagram ofhe C-SLM scheme is given in Fig. 1.

Fig. 1. Block diagram of the conventional SLM scheme.

2.3. Widely linear SLM (WL-SLM)

In [19], a simple and low complexity SLM, named as widely lin-ear SLM (WL-SLM), was proposed. In WL-SLM, real and imaginaryparts of candidate signal are separated and individually treated.Therefore, more candidates can be obtained through linear super-position of the several real and imaginary parts of candidates signal.

Denote XmI

def= Re(Xm) and XmQ

def= Im(Xm) be the real and the imag-inary part of the Xm(1 ≤ m ≤ M), respectively. The time-domainsignal xm

I and xmQ could be obtained when transforming Xm

I andXm

Q with M ‘real-to-complex’ IFFTs, respectively. It is worthwhile tonote that each element Pm(k) in Eq. (4) is in {±1}. In this paper, wecall xm

I and xmQ as the real and imaginary candidates.

Last, select one real xiI and one imaginary xq

Q candidate from the

set of M, which could generate total M2 candidate time-domain sig-

nals. From these, the candidate x(m∗I,m∗

Q) = x

m∗I

I + xm∗

QQ with lowest

PAPR is selected. Fig. 2 showed the structure of the WL-SLM scheme[19].

3. The proposed SLM scheme with low complexity

3.1. Modified widely linear SLM (MWL-SLM)

In order to further reduce the complexity of the SLM scheme,in the paper, we try to utilize the real xi

I and imaginary xqQ candi-

date in WL-SLM scheme and the properties of Fourier Transformto reconstruct the new candidates. The proposed method providessimple PAPR reduction procedure by processing the OFDM symbolat the time domain.

Given N dimension frequency domain signal vector X and thecorresponding time domain signal vector x. First, let us recall someproperties of the fast Fourier transform (FFT), which will be used inthe proposed scheme.

(a) The time domain signal reversal

T1[xn] = x(N−n)N=

{xn, n = 0

xN−n, 1 ≤ n ≤ N − 1⇔ T1[Xn] = X(N−n)N

(6)

Fig. 2. The structure of the WL-SLM scheme.

1008 L. Yang et al. / Int. J. Electron. Commun. (AEÜ) 66 (2012) 1006– 1010

MWL

c

T

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miw

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T

wot

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Fig. 3. Proposed

where k is the circular shift number and (n)N denoted as themodulus of n after division N.

When using property (b), we select k = N/2, which imply that theorresponding frequency domain signal:

2

[Xn,

N

2

]= Xn · e−j�n =

{−1 · Xn, n is odd

+1 · Xn, n is even(8)

The selected parameter assures that the phase rotation factorsn Xn are in {±1}. In the rest of this paper, we use T2[Xn] to standor T2[Xn,N/2].

The advantage of the above Fourier transformations is that wean make use of the FFT properties (the transform pairs) to obtainp[xk], 1 ≤ p ≤ 2 directly from xk in the time domain instead ofrom IFFT[Tp[xk]] in order to avoid additional IFFTs. It is worthwhileo notice that these transformations can be used individually orointly on different part (real or imaginary) to generate the new can-idates. We use the notation T12[ ] = T1[T2[ ]] for the combinationsf the about two transformations.

Without loss of generality, one may choose to perform transfor-ations only on imaginary part xq

Q . For example, no transformations made on the real part candidate. Then the new candidates can

ritten as

x1 = xI + j · xQ

x2 = xI + j · T1[xQ ]

x3 = xI + j · T2[xQ ]

x4 = xI + j · T12[xQ ]

(9)

Lemma: For (9), transformation of Tp[ ], 1 ≤ p ≤ 2 on xI willbtain the same PAPR as the transformation of Tp[ ], 1 ≤ p ≤ 2 onQ.

Prove:

1[x2] = T1[xI] + j · T1[T1[xQ ]] = T1[xI] + j · xQ

here T1[x2] can be seen as the transformation of T1[ ] on xI. Obvi-usly, T1[x2] has the same PAPR as the x2 in (9) because it is onlyhe signal reversal of the x2.

2[x3] = T2[xI] + j · T2[T2[xQ ]] = T2[xI] + j · xQ

here T2[x3] is the transformation of T2[ ] on xI, which has the sameAPR as the x3 in (9).

In the following, we employ the FFT properties to combine withhe WL-SLM scheme to search for the optimal sequence of MWL-LM scheme. The procedure of the proposed MWL-SLM scheme cane described as follows.

-SLM structure.

Step (1) Partition the original complex frequency domain vector Xinto two real vectors, named as XI and XQ, respectively.

Step (2) Same as the conventional SLM scheme, generate M timedomain vectors xm

I and xmQ (1 ≤ m ≤ M) by (4) and (5),

respectively.Step (3) For xm

Q (1 ≤ m ≤ M), by the use of the FFT properties (6), we

can generate 2M candidates xm′Q (1 ≤ m′ ≤ 2M). Then using

(7), total 4M candidates xm′′Q (1 ≤ m′′ ≤ 4M) are obtained.

Step (4) Select one from xiI , (1 ≤ i ≤ M) as the real part of the candi-

date and one from xqQ , (1 ≤ q ≤ 4M) as the imaginary part

of the candidate. Then, combine them into one new candi-date x(i,q) = xi

I + j · xqQ , 1 ≤ i ≤ M, 1 ≤ q ≤ 4M. Through

this step, total 4M2 candidates could be generated. Fromthem, the candidate x(i∗,q∗) = xi∗

I + j · xq∗Q with the lowest

PAPR is selected for transmission.

The transmitter structure of the proposed MWL-SLM scheme isshown in Fig. 3.

3.2. Analysis of computational complexity

In this section, the computational complexity for C-SLM [15],WL-SLM [19] and MWL-SLM are analyzed. Assume that total 4M2

candidates are generated for each SLM scheme for the sake of fair-ness of comparison.

It is well known from the literature that a LN-point IFFT requiresLN/2 log2LN numbers of complex multiplication and LN log2LNnumbers of complex addition. For C-SLM scheme with 4M2 indi-vidual phase sequences requires 4M2 IFFT operations. Therefore,the total numbers of complex multiplication and complex addi-tion for the C-SLM scheme are 2M2LN log2LN and 4M2LN log2LN,respectively. While for the WL-SLM scheme, 2M IFFT operations arerequired to generate 2M real part and imaginary part of candidates,the numbers of complex multiplication and complex addition arereduced to MLN log2LN and 2MLN log2LN, respectively. After com-bining the 2M real and imaginary part of candidates, total 4M2

candidates can be obtained. This step requires extra 4M2LN num-bers of complex addition. As for our proposed MWL-SLM scheme,only M IFFT operations are required to generate M real and imag-inary part of candidates. The numbers of complex multiplicationand complex addition are further reduced to MLN/2 log2LN andMLN log2LN, respectively. The combination of the real and imagi-nary part requires extra 4M2LN numbers of complex addition.

The computational complexity reduction ratio (CCRR) [24] of the

proposed MWL-SLM scheme over the C-SLM and WL-SLM schemeis defined as

CCRR =(

1 − Complexity of the MWL-SLMComplexity of the C-SLM or WL-SLM

)× 100% (10)

L. Yang et al. / Int. J. Electron. Commun. (AEÜ) 66 (2012) 1006– 1010 1009

Table 1Computational complexity of the different SLM scheme when N = 256 (where Mu is denoted as multiplication and Ad is denoted as addition).

C-SLM WL-SLM MWL-SLM CCRRC (%) CCRRW (%)

M = 2, total candidates are 16Mu 81,920 20,480 10,240 12.50 50Ad 163,840 57,344 36,864 22.50 64.29

15,360 8.33 5067,584 18.33 68.75

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6 7 8 9 10 11 1210

-4

10-3

10-2

10-1

100

PAPR (dB)

Pr(

PA

PR

>P

AP

R0)

original OFDM

C-SLM, 4M2=16

C-SLM, 4M2=36

WL-SLM,2M=4

WL-SLM,2M=6

MWL-SLM,M=2

MWL-SLM,M=3

M = 3, total candidates are 36Mu 184,320 30,720

Ad 368,640 98,304

The CCRR of the MWL-SLM scheme over the C-SLM and WL-SLMcheme for typical values of N and M are given in Table 1, whereCRRC denoted as the CCRR of the MWL-SLM scheme over the C-LM scheme and CCRRW denoted as the CCRR of the MWL-SLMcheme over the WL-SLM scheme. Here, we assume that the threechemes generate the same numbers of candidates. It is shownn Table 1 that the computational complexity of the MWL-SLMcheme is reduced rapidly with the increase of M.

. Simulation results

In this section, we describe the computer simulations are used toerify the PAPR reduction performance of the proposed MWL-SLMcheme. To obtain the complementary cumulative density functionCCDF = Pr[PAPR > PRPR0]), where PAPRo is a certain thresholdalue that is usually given in decibels relative to the root meanquare (RMS) value. 106 independent OFDM blocks were randomlyenerated, where the numbers of subcarriers are set to be N = 256nd data symbols are modulated using the 16-QAM constellation.he oversampling factor with L = 4 are examined here.

The CCDFs of the C-SLM, WL-SLM and MWL-SLM scheme with IFFT operations are plotted in Fig. 4. From the above analysis,

he C-SLM, WL-SLM and MWL-SLM scheme could generate M, M2

nd 4M2 candidates, respectively, after running M IFFT operations.learly, the WL-SLM and MWL-SLM scheme outperforms the C-SLMcheme in PAPR reduction. Especially for the MWL-SLM scheme, theAPR at CCDF = 0.1% is improved about 1.7 dB when compared withhe C-SLM scheme with the M = 2 and M = 3, respectively.

Fig. 5 shows the CCDFs of the C-SLM, WL-SLM and MWL-SLMcheme with 4M2 candidates. The results in Fig. 4 show that theAPR performance of the MWL-SLM scheme with the 16 (when

= 2) and 36 (when M = 3) candidates is only slightly inferior tohe C-SLM scheme by about 0.25 dB and 0.35 dB, respectively. How-ver, the MWL-SLM scheme has a much lower computational costecause it requires M IFFT operations only compared with 4M2 IFFT

ig. 4. CCDFs of the C-SLM, WL-SLM and MWL-SLM scheme with M IFFT operations.

0

Fig. 5. CCDFs of the C-SLM and the MWL-SLM scheme with 4M2 candidates.

operations of the C-SLM scheme when both schemes have 4M2

candidates.

5. Conclusion

Compared with the conventional SLM and WL-SLM scheme, thispaper has presented a modified widely linear SLM scheme to fur-ther reduce the PAPR of an OFDM signal. In the proposed scheme,new candidates are generated using the properties of the FourierTransform and real/imaginary parts of the candidate signal. Simu-lation result has shown that the proposed scheme performs betterthan the conventional SLM and WL-SLM scheme under the samenumber of IFFT operations. On the other hand, for the same num-ber of candidates, the proposed scheme has similar performance asthe conventional SLM scheme and WL-SLM scheme but the com-putational complexity is significantly reduced because of fewernumbers of IFFT operations.

Acknowledgements

This work was supported in part by the National Natural ScienceFoundation of China under Grant 61001088 and Grant 61032002,Specialized Research Fund for the Doctoral Program of HigherEducation under Grant 20100185120007 and Important NationalScience and Technique Supporting Project of China under Grant2012ZX03001030-003 and 2012ZX03004002-002.

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