A New SLM Algorithm with Low Complexity for PAPR Reduction in OFDM

D. W. Lim, S. J. Heo, and J. S. No

School of EECS, Seoul National University

Abstract

Thepeak to avemgepwer ratio (PAPR) dishihtion of the @odic orihogonnlfi.quency division multiplexing (OFDM) dain symbol sequence is ewluated and it is shown t h a the OFDM data symbol sequence with short period is wpcted to have high PAPR. Using this result, we identr& the optimal phase sequence set and pmpse mws of a cyclic Hndamard mahu. comtrucfed by m- sequence as phase sequences for selected mapping (SLM) OFDM The cornpuiationnl complen'ry of conventiono! SLM OFDM is i m m e d extensively in pmprtion fo the number of phase sequences. In order to ovmcome this disadvantage, we pmpose new SLM OFDM scheme. Fmm the simulation m u l b fir the standard of IEEE 802.16 of mobile wimless mefmpolitan area nefwork (WMN) , it is shown ihai new SLM OFDM with 2048 camers reduces the compurational complai* up to 51% while it has almost thesame capabiliq ofPAPR duction compared to that of conventionnl SLM OFDM Since the compuintionnl reduction gain i ncmes as the number of caniers incmases, the pmposed scheme is appmpriate for the high data mte OFDMsysfems.

K q w r d s - Orthogonal fkqency division multiplexing (OFDW, peak to average power ratio (PAPR), selected

mapping (SLW

I. Introduction

oahogonal fquency division multiplexing (OFDM) system is one of the strong candidates f a the standard of the next generation mobile radio wmmunication systems and it has been accepted as a standard for the wireless local area network and mobile wireless

metropolitan area network 0. One of the major drawbacks of OFDM system is that it

has a high PAPR The high PAPR brings on signal &tortion at a high power amplifier (IIPA) because of the non-linearity of the HPA. Then, the signal distoItion induces the degradation of hit enor rate (BER). A lot of techniques ([I]-[S]) have been pmposed to reduce PAPR problem of OFDM systems. The &ods can be

classified into two categories. Firsf there is a determinktic method that limits PAPR of OFDM below a ceaain level. w i n g and block coding belong to this category. Clipping is widely used from the siqlicity of its implementation but it inhcduces both in-band and out- of-band radidion Block wdmg [5] reduces PAPR by encoding the input data to codeword that has low PAPR but it increases the redundant symbols seriously.

The second category is based on probabilistic approach. This methcd @roves the characteristic of PAPR statistically without signal distortion. SLM and parfial transmit sequence (FTS) [ I ] are classified as this category. In S W statistically independent alternative OFDM data symbol sequences are generated by multiplying pre- detemined Uphase sequences with an input OFDM data symbal sequence. The symbol sequence with the lowest PAPR is selected f a hammission after IF'FTs. In PTS, the input data symbal sequence is partitioned into a number of disjoint sub-blocks. lFFT is applied to each group and t : ~ signal of each gmup is multiplied by a rotating factor. Then, the PAPR is computed for each set of phase rotating wmbinatiom and wmpared It is hown that SLM is more advantageous than F% if the amount of redundancy is limited but the computational wmplexity is larger than PTS.

In this paper, new SLM OFDM is proposed to reduce PAPRwith low computational complexity and it is shown

that new SLM OFDM reduces the computational complexity while it has almost same capability of PAPR reduction compared to that of conventional SLM OFDM.

II. PAPR Distribution

A OFDMSystem Model The discrete time domain OFDM signal of Ncaniers

inthetimeintervalof O<t SN-Icanbewrittenas

where A[n] is the h u t data symbol of n-th canier and discrete bime t is integer value. The PAPR of OFDM signal, defined as the ratio of the mximum power

divided by the average power of the s~gnal, is expressed as

PAPR 0

The crest factor 5 is defined as

The PAPR of continuous time domain signal can be estimated h m the PAPR of discrete thne do& signal, which is over-sampled with the over-sampling factor mt less than 8.

B PAPR Drsmbution ofperiodic InpuiSymbol Seguence

Let X [ n ] be an input data symbol sequence of length N, which has nonzero complex value only within the interval O < n S N / M - 1 .

nonzero, 0 < n < N / M - 1 X [ n ] =

0, otherwise.

If sequence A [ n ] of length N is g m t e d by repeabing

the non-zero value of X [ n ] M times, A [ n ] is expressed as

Let a [ t ] and x [ t ] denote discrete fime domain OFDM signals of A [ n ] and X [ n ] respectively.

According to t l ~ shit't property of Fourier transformation, a [ f ] can be e.qressed as

M x [ f ] , f = 0 mod M (2)

0, otherwise.

In [I], the PAPR dishbution of a f t ] uith M = l - -

tas been evatuated. In the following, we will derive the PAPR dishbution for M 2 1. We first calculate t l ~

disbibution of instantaneous power of x [ f ] . As X [ n ] has n o m value only withm the interval 0 < n < N / M -1, OFDM signal x [ f ] is givenas

As X [ n ] can be thought as independent zeremean d m variable with variance ui, the average power 0: of OFDM signal x [ f ] is calculated using Parseval's theorem

Since x [ f ] is the sum of N / M nonzero values of ~ [ n ] , becomes a zero-mean wmplex near Gaussian disI5iuted random variable with variance cr: due to the central limit theorem for large N / M . r = 1x1 has Payleigh &bution and is vaitten as

From (2) and (S), nonzero u=lal has Rayleigh

&s t t i ion as follows:

From (4) and (6), the probability that the magnitude of one OFDM signal a [ f ] value does not exceed Q is

given by

In (21, a [ ! ] is zem for t#OmodM. Assuming

a [ t ] to be statistically independent, the probability that 2 0 2 0 at least one magnitude value of OR)M signal a [ t ] /a [ t ] l -T < l b [ f ] l ~ l a [ t ] l + ~ . (14) exceeds a, canbe approximated as

In case of bimy sequence, a [ t ] and b [ t ] have the average power 1. Then, tbe crest factor of b [ t ] can be

(8) written as

Using ParsevalS t h e 0 4 the average power ui of The difference of the crest factor of a [ i ] and b [ [ ]

OFDM signal a It] is calculated as becomes very small for large number of carriers N and . .

(9) relatively small distance D.

D: a2 E {la [t]l] = D;.

Plugging (9) into (a), the probability that the crest factor exceeds some threshold value $", = a,/ua can be writtenas

From (lo), the probability that OFDM signal exceeds the value PAPR, = 1: becomes large as Mincreases.

C PAPR Bound of Two Sequences Let A[n] and B[n] be binary input data symbol

sequences of length N. When the distance of two sequences is D, B [n] can be expressed by

III. New SLM OFDM

A. ConventionalSLMOFDM In conventional SLM OFDM [I], U statistically

indepdent alternative OFDM q u t data symbol sequences are generated by multlplymg predetermined V n c e with an input data symbol sequence, where the predetermined sequence is called aphase sequence.

Atter U IFFTs for alternative OFDM input data symbol sequences, the signal with the lowest PAPR is selected for transmission Assume that the alternative input data symbol sequences are statistically independent.

Using (10) with M = 1, the probability that the crest factor exceeds tbe threshold value canbe written as

d=O U

prsLM,mM {C > 601 =(pro,,, (6 > 601) Fmm (11),1FFTof B[n]-A[n] canbecalculateas N 0

= ( I - ( - ) ) ) I F F T { B [ ~ I -A["] }

The computatiom~ complexity of conventional SLM D-I 12~3 (12) OFDM is increased in proportion to the number of phase

= - - z A [ n d ] . e . fi ,=, sequences.

Let a l l ] and b [ f ] denote OFDM signals of A[n] B. NewSMOFDM and B[n] mpectively, lb[ t ]-a[ t] l can be writtenas New SLM OFDM transform3 input data symbols lnto

2 0'1 j2n%/ a set of alternative time domain signals. Then, the time ib[fl-a[tllsTz A [ n d l e d m i n signal that has the lowest PAPR is selected for d=O

2 0 (I3) transmission. In this scheme, the N = Zn points IFFT is

=- fi '

partitioned into two parts. The first part is IFET of the fkst k stage and the second part is IFFT of the remining (n-k)

From (13), lb[t]l is boundedas stage. Alternative OFDM si& are generated by multiplying different phase sequences with the inkmediate result at k-th stage of IFFT.

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Select the

OFDM signal - with

minimum PAPR

Fig. 1. Block diagram of new SLM OFDM system.

As the intadiate result is used comnly , the only (n- k) stages after the k-th stage of EFT are to be calculated U tvneS for U phase sequences. In order to recover input data symbal sequence from the received signal, the receiver needs to have information on which phase sequence is used m tbe hansmitter. A s i q l e metbod for iransmibing the information is to add the idex of phase sequence mto input data symbol sequence. Ihe bits for hansmittiq the side information are called redundancy bits. As redundancy bits are very imymnf it should be coded for enor detection and emr c o d o n . IfM-QAM constellation and Uphase sequences are used, the number

of redundancy bits is [log, U / r l where r is coding rate. [log, U / r l carriers are set to zero to reserve side information TO hqlement this without furtha increase of computational complexity, the IFFT si& a!& of the coded Index (1-4 are stored m tbe memo~y. Ihq the side information is added after the IFFT of input dan symbol sequence. T k new SLM OFDM shorn m Fig. 1 can be written as

a'" = EFT:+, {P(") (A,,)} +IFFT; (Aidex ) = IFFT;+, {P(u). a:,] +

where IFFT,! T/cates IFIT from i stage to j stage and the size of ElTis N-2".

where U is the number of phase sequences. Wben the phase s q m c e s are multiplied at tbe k-th stage, the number of complex computation of new SLM OFDM is given by

Table 1 gves complexity reduction gam of new SLM OFDM against wnventional SLM OFDM with miom numbers of sapnces andmultiplication stages.

TaMe 1. Computational wmplex redudon gain.

N; Phase Sequences

C. Compfafioml Complexity A. P h e Sequw of Conventioml SLM OFDM

The number of wtqlex computation of conventional In SLM OFDM, the two imprtant problem on phase

SLM OFDM when the IIWLXZ of is N = T, is sequences are how to design phase sequ- to m t e

given by U statistically mdependent OFDM data symbol and how many phase sequences to use. It is h o r n from simnlation d t s that the random squences show the

best performance [3], but there has been no clear way %

iden* the optimal p h e sequence set. It has been used without any mathemtical proof that

the phase sequences are to be orthogod for generating statistically indepadent alternative OFDM data symbol sequences h m an input data symbol sequence.

Fig. 2 shows the complementary d a t i v e distribution function (CCDF) of the PAPR of SLM ODFM system with 64 caniers for s o m phase sequences

where the number U of phase sequences is 4. The performance of bimy random sequence, quaternary random sequence, cyclic Hadamard sequences, Z4 family- A sequences is almost the same. But the performance of Sylvester Hadamard sequences is not so good as that of random or pseudo-mdom sequences though Sylvester Hadamard sequences are orthogonal.

10' 1 5 B 8 B 1 0 1 1 I Z

PAPR,(dB)

Flg. 2. CCDF of the PAPR of SLM OFDM for some phase

seq-

Fmm this result, we can think that the orthogonal property between phase sequences is not a sdicient condition for the optimal phase sequmes in SLM

OFDM. In section lI, we have shown that if an input data symbol sequence of OFDM system is periodic or similar

to periodic sequexe, OFDM signal has the large PAPR with high probability. If a random phase sequence is

d t ip l ied with a periodic input dab symbol sequence, the o b M sequence may get +odic pmperty and has the low PAPR

A cyclic Hadamad matrix is a Hadamard matrix with a property that in the standard form, remobiug the top row and the left-mst column, the rows are cyclic shifts of each 0 t h The mws of cyclic Hadamard matrix are d o g o n a l and have good &c aubcmclation

property (large nmit hctor) except the first row. Thus, the rows of cyclic Hadamard matrix are suited to the phase sequences of conventional SLM OFDM.

B. Phase Sequences ofNew SLM OFDM In new SLM OFDM, the U phase sequences are

multiplied with the intermediate results of IFFT while phase sequences are multiplied with the input data symbol

sequences in SLM OFDM. The computational complexity of new SLM OFDM is reduced as the intermediate dfiplication stage k approaches to the last stage n. But the performance of PAPR reduction hnrases. As there is trade-off between the computational complexity and the performance of PAPR reductioq it is very +rtant to compmmise them by finding an optimal stage h m s ~ t i o n ~ e s u I t s .

For any N = 2", Npoint IFET can be computed from two N/2 point IFET, the N/2 point IFET h m two N/4 point FPT and so on. As the b f o m of k-th stage consists of 2"-' blocks that are NI2"-' point IFFT according to the successive doubllng method, we let the ekmenb of phase sequence in one block have the same value. Met generating sequence R("' of the length Yk, phase sequence P(") of the length Nis made by -6ng e a c h e l e m of R("', NIT- ' h.

In new SLM OFDM, mws of a cyclic Hadamard mtrix areusedas sequences R(").

V. Simulation Results

Sinidations are performed for the stanstandard of IEEE 802.16 [q for mobile WMAN. The OFDM system specified in IEEE 802.16 has 2048 carriers, which are modulated with QPSK, lQAM, and 64QAM s i p 4 constellation according to the banmission rate. The number of used carriers is 1702 and 346 d e r s are set to

zem to shape the power spectrum density of the hansmit signal. The 100,OM) input data symbols are generated randomly to provide the statistics of PAPR

A. Varying the Shge ofMultiplication Fig. 3 shows the ~ a t i o n m u l t s as the d@lication

The PAPR distribution of the periodic OFDh4 data

-NlbLUOFOY<III IS symbols has been evaluated and it has been shown that

ro' the OFDM data symbol sequence with large periodicity is

$ expected to have higbPAPR. The& we have identified the optimal phase sequence set and pmpose rows of a cyclic

. . . . . . . If~&nwrd mami as a plwse scqucnze for Sl.hl 0t1)\1 . . s?stcm. Tlle ne\\ S1.M OFD\I has Lxcn proposed and 11s

Fig. 3. CCDF of PAPR of new and convenbnal SLM

OFDM for some multiplication stage of phase sequences when 16-QAM is used.

stage is varied for ( n -k) = 4,5, and 6. ?he performance of new SLM OFDM is almost the same with

that of conventional SLM OFDM when the multiplication stage is not less than 5.

B. Peformance Comparison between Conventional SLM OFDMand New SLM Fig. 4 shows a comp; performance between

conventional SLM OFDM and new SLM OFDM when h e multiplication stage (n-k) is 5. The new SLM OFDM reduces the computational complexity up to 41% - 51% as the number of sequences Uincmses from 4 to 16.

PAPRJdB)

Fig. 4. CCDF of PAPR of conventional SLM OFDM and new SLM OFDM with 16QAM when the multiplication stage (wk) )s 5.

performance analyzed in reference to the standard of EEE 802.16. As results of simulations, the new SLhl OFDM with 2048 caniers reduces the computational complexity up to 51% while it has almost the same capability of PAPR reduction compared to that of the conventional SLM OFDM Since the computational reduction gain inneases as the number of camem increase, the proposed scheme is appropriate for the high data rate OFDM systems.

References [I] S. H. Miiller, R W. Bad, Rebat F. H Fiscber, and J o h l e s B. H k , 'OFDM with reduced peak-to- average power ratio by multiple signal representation," In Annals of Teecommun., vol. 52, no. 1-2, pp. 58-67, Feb. 1997. [Z] P. V Eetvellt, G. Wade and M. Tomlinson, "Peak to average power reducing for OFDM schemes by selective s d l i n & " Elecfmn. Len., vol. 32, no. 21,pp. 1963- 1964, Oct. 1996. [3] N. Ohkubo and T. Ohtsnki, "Apeak to average power ratio reduction of multicanier CDMA using selected mapping", ininpmc. IEEE Yeh. Techno1 Conf, 2002. [4] M. Breiling, S. H. Miiller and Johannes B. Huba, "SLM peak power reduction without explicit side hfomtion," IEEE Commun. Len., vol. 5 , no. 6, pp. 239- 241, June 2001. [5] J. A. Davis and 1. Jedwab, "Peak-to-man power control in OFDM, Golay complementary sequences, and Reed-MnUer codes," IEEE Trans. Infirm. Theory, vol. 45, no. 7, Nov. 1999. [6] IEEE standard for local and metmplitan area network IEEE Std 802.16am-2003.

VI. Conclusion