8/6/2019 0A9ADAE0d01

1/5

An Algorithm for Reducing the Peak to Average Power Ratio in aMulticarrier Communications SystemVahid Tarokh

AT& T Labs-Research180 Park Avenue

Florham Park, New Jersey 07932tarokh@research.att com

Abstract - One o f t he m a i n i m p l em en t a t i ondisadvantages of a mul t icarr ier communicat ionsys tem i s th e poss ib ly h igh peak to averagepower ra t io of the t ran smi t ted s ignals .

One proposed so lu t ion is given by Jones a ndWilkinson El]. It is based on apply ing a specificphas e s h if t (no t dependen t on t he codeword) toeach d imens ion of the t ransmi t ted codewords .Thes e phas e s h i f t s a r e known b o t h to t h e t r ans-m i t t e r an d t he r ece iver . Th us t he phase s h i f t scan be com pens a t ed for in the receiver wi th-ou t changi ng t he d i s t ance p roper t i e s of t h e code.Therefore , nei ther the ra te nor the code perfor-mance i s sacr i fi ced . Th e phases need only oncebe comp uted off- line a nd t he complexi ty of thesys tem does not increase.

Our cont r ibut ion i s to provide an a lgor i thmto com pu t e t h e phas es t ha t m i n im i ze t h e m ax-i m um peak to average power ra t io t aken overal l poss ib le t ransm i t ted s ignals. Th e new algo-r i t hm enab l es us to apply the Jones-Wi lk insonm e t h o d to var i ous m ed i um l eng t h as well as longcodes of pract ical in teres t , a computat ional t askfor which n o so lu t ion was known before.

W e app l y ou r algor i thm to s om e o f t he codesadapted for use in t he phys ical l ayer of fu t urewireless local area networks by t he ETSI BRANHiper lan-I1 s tandardizat ion c ommit tee . We re -po r t t he com pu t ed r educ t i ons i n peak t o ave ragepower rat io.

I . INTRODUCTIONMulticarrier communications (OFDM) has employed

for wire-line communications and has also been adaptedfor use in the future wireless local area networks. In sucha system (Figure I) , the input bits are encoded into con-stellation symbols using a code C. Then N constellationsymbols are provided to the input of a discrete Fouriertransform (DF T) by a serial to parallel block. The out -put of the DFT is multiplexed and transmitted over thechannel (after modulation). The receiver performs theinverse operations. First the received signal is demulti-plexed and fed into the inverse DFT ( IDF T) block. Aparallel to serial block converts the outpu t of IDFT toa serial sequence and the decoder provides the outputbit stream.

Ham id JafarkhaniAT&T Labs - Research

100 Schulz DriveRed Bank, N J 07701

hamid@research.att comOne of the main implementation disadvantages ofOFDM is the high peak to average power ratio (PAPR)

of the transmitted signals. This forces the use of poweramplifiers with large linear range which translates intohigher cost. In this light, any reduction of the peak toaverage power ratio of the transm itted signals can be ofimportance in engineering OFDM systems.

In this paper, we use the Jones-Wilkinson phase shiftmethod [l] for a given code C. The block diagram of thesystem is given in Figure 2 . However, unlike [I],we pro-vide a polynomial time algorithm for determining therequired optimal phase shifts. The new algorithm en-ables us to apply the Jones-Wilkinson method to variousmedium length as well as long codes of practical inte rest ,a computational task for which no solution was knownbefore.

We apply our algorithm to some of the codes adaptedfor use in the physical layer of futu re wireless local areanetworks by the ETSI BRAN Hiperlan-I1 standardiza-tion committee. We report the computed reductions inpeak to average power ratio.

The organization of the paper is as follows. In Section11, we review the communication model, formulate theproblem and establish the notation. Section 111proposesour proposed algorithm. Some simulation results areprovided in Section V.

11. THESYSTEM ODELThe model of a multicarrier communication system is

given next. At each time T = O,T, 7 , . . . blocks BT ofk bits arrive at the encoder. These k bits a re encoded asa sequence of N constellation symbols from a constella-tion Q with 2= elements. The admissible sequences arecalled codewords, and the ensemble of all possible code-words is a code C of rate R = k / N . For the simplicity ofpresentation, we will assume without loss of generalitythat the elements of Q have equal energy norm alized tobe one.

Upon processing the k input bits BT, he outputof the encoder is a sequence of N constellation sym-bols C O , c1 , . ,C N - 1 . This codeword goes through a se-rial to parallel converter and a DFT (discrete Fouriertransform) device producing a sequence of symbolsCO c1 . C N - 1 .

0-7803-5565-2/99/$10.000 999 lEEE680

8/6/2019 0A9ADAE0d01

2/5

Then I11 The Method of Jones andWilkinson- 1C1 = ciexp[-2dij/N],

i = O

for 1 = 0 , 1 , 2 , . . ,N - 1,where j = 2/7T.This sequence is the input to the RF chain which

produces the transmitted signal. The transmi tted signalat time t is then given by

for 0 5 t 5 $, where fo is the carrier frequency, fs isthe bandwidth of each tone, and R(z) denotes the realpart of z . The relation between the quantities fs and7 epends on whether a guard time is assigned, or acyclic prefix is used and these details have no bearingupon the method presented in this paper. However, wenote that fs = 1/T is commonly assumed in an idealsituation We write cz = d, + e, j , where d, and e , arethe real and imaginary parts of the symbol c , , thus

N-1~ ( t ) [d2c0s(2n(fo+ f s ) t ) + e2szn(2.rr(fo+ zf s) t ) l .

2=0

The receiver receives the signal S ( t ) perturbed bynoise. The R F chain at the receiver down-converts, pro-cesses the received data and obtains estim ates of th eparameters Ct,Z= 0 , 1 , .. ,N - 1. The receiver thenapplies an IDFT on these estimates and generates es-timates of C O , c l , . . . ,CN-1 from which it extracts theblock BT of input bits.

However, there are input bits b, , z = 0 ,+. N - 1 forwhich the terms in (2) add constructively. For instancefor the BPSK constellation, the all zero input bit stream000' . O is mapped to the sequence of all ones producing~ E ~ ' c o s ( 2 5 ~ ( f ozfs)t) at the output of the transmit-ter R F chain. This signal has its peak power N 2 at timet = 0. Thus the peak to average signal power ratio ofthe transmitted signals can be as large as 2N for theBPSK constellation.For any sequence c = c o c l . . .C N - 1 , we will denoteby PAPR( c) the peak t o average power ratio of the cor-responding signal S ( t ) n the range 0 5 t < k. For acode C the peak to average power ratio of C is defined tobe the maximum peak to average power ratio amongstthe codewords of C. That is

PAPR(C) = max[PAPR(c)]. ( 3 )C CThe peak to average power ratio problem can now beformulated.Statement of The Problem: For a given transmis-sion rate and a certain desired error protection level,design an encoder C such that PAPR(C) is minimized.

The choice of a code to provide a certain level of errorprotection at a given ra te is the topic of coding tech-niques and is outside the scope of this paper. We as-sume that a coding expert has produced various can-didate codes that have the desired transmission rate aswell as a certain desired error protection level. For eachcandidate code C the method we propose reduces thePAPR(C) without reducing the rate or error protec-tion capability of C and without increasing its decodingcomplexity. T he method is applied to each possible codeand the one which minimizes the peak to average powerratio is selected.

Consider a code C with codewords of length N .For a codeword c = cocl . . . c N - 1 , we will refer toCO ,c l , . . . CN-1 as respectively the zero-th, the first, . .,and the ( N - 1)-th coordinates of C. To decrease thepeak to average power ratio, Jones and Wilkinson [l ]proposed that the i-th coordinate of all codewords of Cbe shifted by a phase 4; nown both to the transmit-ter and t,he receiver. Since the i -th coordinate of everycodewords is phase-shifted by the same fixed angle I$;,these phase shifts can be compensated for in the receiverfollowing the application of the IDFT. Thus these phaseshifts do not change the error correction capability, thera te of the encoder nor the decoding complexity of thereceiver. Following these initial phase shifts, any arbi-trary codeword c = cocl . . .CN-1 is mapped into

CO exp(j4o) c1 exp(j4i) . . . C N - I exp(j4iv-1).We let C(&, 4 1 , ' . . c$N-~) denote the set of phaseshifted codewords.

The transmit,ted signal s ( t , 40,41, . ,~ N - I ) s

for 0 5 t 5 l / f s .Assuming that < = f o / f s >> 1, it is well-known that

the average power of s ( t , 40,41 , . . ~ N - I ) is k$ =Thus the peak to average power ratio of

N-

the signal s ( t ,40,411,. . , ~ N - I )s given byPAPR(c(40,41,..-,4~-1),6)2 maxOStSl

R (E:; c, exp[-(aT(fo + zfs)t + 4 , ~ ) ~N

which only depends on the paramete rs of the O F D Msystem, the codeword cocl . . .C N - ~ and the phases407 1 1 ' , 4 N - 1 .The important insight of Jones and Wilkinson [l ] istha t following an appropria te choice of the phase shifts,the code C(40, 41 , . . . N - I ) an have a lower peak to

68 1

8/6/2019 0A9ADAE0d01

3/5

average power ratio than C. In fact, for any code C, weshould choose $0 ,$1 , . . $ ~ - 1 such that the peak toaverage power ratio of C($O, 1 , . . . , ~ - 1 ) is minimized.In oth er words,

argrninPAPR(C($o,#l,-..,$N-l),)must be computed.

The minimizing values $0,41,. 4 ~ - 1 eed only becomputed once and they can be tabulated for variouscodes of interest. Because these values are computed off-line, this computation does not add to the complexityof the tr ansm itter or th e receiver. It remains to designan algorithm for computing these minimizing values.

IV The Proposed AlgorithmA The Instantaneous Peak to Aver-age Power RatioWe observe that

PAPR(c)= 2 max g ( t ,C , c ) ,O < t < lwhere

Let

for 0 5 t 5 1. We refer to g ( t ,6 , c ) as the ins tan taneouspeak t o average power ratio of the codeword c and tog ( t ,C ) as the ins tan taneou s peak to average power ratioof the code C. We focus on algorithmic computation ofthese functions.

The next Theorem provides a geometric interpreta-tion of the value of instantaneous peak t o average powerratio at time t and will turn out to be very importantfor computation. The theorem has an analog for QA Mconstellations [3].Theorem I V . l Let C denote a code whose codewordshave equal energy . Le t g ( t , J) be as defined by Equation(4). For each t , t h e v a lu e o f g ( t , C ) i s a tt a in e d b y t h ecodeword c which i s the c loses t to or the farthest code-word f rom the vec tor w ( t , 0, then c maximizesW ( c .w ( t , C ) ) amongst all codewords. This means th at/ I C - w(t,C)1I2 = 2N - 28(c . w ( t , C ) ) is minimized.Alternatively, if W(c . w ( t , C ) ) < 0, then c minimizesW(c .w ( t , C ) ) amongst all codewords. This means tha tIIc - w ( t , C) l lz = 2 N - 2 8 ( c .w ( t ,6 )) is maximized. 0Definition I V . l For an y code C , we define

-c = {- c , c E C} .W e r e fe r t o -c as the negative of c an d -C as the neg-atave code of C . A code C i s s y m m e t r i c if -c E C when-ever c E C .Corollary I V . l Let C denote a code whose codewordshave equal energy. Le t g ( t ,C ) be as defined by E q u a t i o n

e For each t , the va lue of g ( t ,5) i s a t ta ined e i therby the codeword c E C c l o se s t t o w ( t ,6) o r b y t h enegative of the codeword c E C c l o se s t t o w ( t ,Cl.Equiva len t ly , the va lue o fg( t ,6) is at tained ei therby the codeword c E C closest to w ( t , C ) o r b y t h ecodeword c E C c l o se s t t o - w ( t ,C ) . If C i s s y m -m e t r i c , the v al ue o f g ( t , 6 ) is at tained by the code-word c closest to w ( t , C ) .The complex i ty o f f in d in g the codeword of a codeC with th e largest peak to average power ratio ate a c h t i m e i s a t m o s t t w i ce a s m u c h as the decodingcornplexzty of C . I f C i s symmetr ic the complex i tyof f indeng th e codeword o f a code C with the largestpeak to average power ra t io a t each t ime i s a t mos tas much as the decodeng complexity of C.

(4)-

Proof: We refer the reader to [3]. 0It follows from the above tha t if the signal constella-tion points have equal energy and if the code C supportsa soft decision decoding algorithm, then the value g ( t , C )

can be easily computed. Linear block and convolutionalcodes, and more generally trellis codes all have well de-fined trellises, and computation of the closest and far-thest codewords from w ( t , C) can be accomplished usinga straightforward application of the Viterbi algorithm.B Computation of Peak to AveragePower RatioIf the function PAPR(C(&, 1 , . . . , N - I ) , C ) can becomputed accurately for any 90, 1,. .. G N - ~ ,hen wecan apply stan dard minimization techniques to computethe values 40, 1, . , ~ - 1 , that minimize the peak topower average of C($O,41, . N - I ) .To this end, werecall that

PAPR(C($O,$~,...,~N--~),C)2 max g ( t , C ) .o j t gThe following lemma will be very useful in computing

PAPR(C($o, 1 , . . . , N-I) , C ) -

682

8/6/2019 0A9ADAE0d01

4/5

Lemma IV.1 For a n y (, t h e fu n c t i o n g ( t , ) i s con t in -u o u s o n t E [0 , ].

Proof: The proof is given in [3]. 0We conclude from the above lemma that by evalu-ating g ( t , C ) at sufficiently many uniformly spaced sam-

ples, PAPR(C(40,4 1 , . N - I ) ~) can be computed toany degree of accuracy. For each sample point the de-coder of C must be used once to com pute g( t , ) and thenumber of these uniformly spaced samples depends on( which is very large in practice. For instance, for thecarrier frequency of 2 GHZ, 1MHZ available bandwidthand 512 tones, ( s of the order of l o 6 and small changesin t (say of the order causes 2x(t to change sig-nificantly. For most practical systems the complexity ofthe computation of P A P R ( C ( ~ ~ , ~ ~ , . . . , ~ N - ~ ) , C )s-ing just the above uniform sampling is too great.

The application of any minimization method for com-puting the minimizing values of 40,41, . N - I e-quires that PAPR(C(&,# l 1 . . ,~N-I), ) be computedfor many different values of 4 o 141, . .,div-1 and thiscan be infeasible if the number of samples is too large.

This leads us to consider alternative methods forcomputing PAPR(C(40, 1 , . . . ~ N - I ) , ) . A fruitfulapproach is to study the change in the value of g ( t , ()for any fixed t as < changes [3].

Ou r result in [3] is summarized in the following the-orem.Theore m IV.2 Let a code C and phase sh i f t s40,. , ~ N - I be given. Suppose tha t the in terva l [O, 13is divided into L equal subintervals given b y point s t o =0, l = 1 / L , 2 = 2 / L , . . , L = 1. L et

Suppose tha t ( 3 4 N Z . Let 51 = 4N2 an dL = 16N3. T h e n t h e e s t i m a t i o n e r r o r in replacingPAPR(C(40, .. , ~ N - I ) , ) with G,-,((1) is le ss t h a nN T

d B , f o r N 2 2.

Proof: The proof is given in [ 3 ] . 0In view of the above re sults , for large N the com-

plexity of computing G L ( ( ~ )s at most 8N 3 times thecomplexity of soft decoding of C. For small N , we couldincrease both (1 and L to increase the accuracy of re-placing P A P R ( C ( ~ O , . . . , ~ N - ~ ) ,) with G ~ ( c 1 ) .

We can now use any applicable minimization methodto find the minimizing values of 40,...,4~-1orG,-,((l ). Once the answer is computed, we know forthese values of 40, . . , 4 ~ - 1he peak to average powerratio of the phase shifted code is within dB of thecomputed value (for large N) r even less for small N.N Z

V . SIMULATIONES ULTS

40= 0.00 41 = 0.75 42 = 0.75 43 = 0.504 4 = 0.25 4 5 = 0.25 46 = 0.50 47 = 0.504 8 = 0.00 4 9 = 0.75 410 = 0.75 411 = 0.25412 = 0.50 413 = 0.50 414 = 0.75 4 1 5 = 0.00416 = 0.50 417 = 0.00 418 = 0.75 419 = 0.2542 0 0.50 4 21 0.25 4522 = 0.25 4 2 3 = 0.504 24 = 0.50 42 5 = 0.50 426 = 0.75 4 2 7 = 0.754 2 8 = 0.50 4 2 9 = 0.25 $30 = 0.00 4 3 1 = 0.004 3 2 = 0.00 4 3 3 = 0.00 4 3 4 = 0.75 4 3 5 = 0.004 3 6 = 0.75 437 = 0.25 4 3 8 = 0.50 439 = 0.754 4 0 = 0.25 4 4 1 = 0.75 4 4 2 = 0.50 4 4 3 = 0.254 4 4 = 0.00 4 4 5 = 0.00 4 4 6 = 0.75 44 7 = 0.25

Table 1: Phase Shifts of QPSK type for the rate1 /2 BPSK convolutional code. Each number mustbe multiplied by 36 0 to give the phase shift in de-grees.

In this section, we apply the algorithm described inthe previous section to various codes adapted for usein the physical layer of future wireless local area net-works by the ETSI BRAN Hiperlan-II standardizationcommittee. The method we use for finding t,he mini-mizing phases for the function G,-,(C1) is the gradientmethod. Once the phase shifts were computed , theywere rounded to QPSK , 8-PSK and 16-PSK phase shifts.The peak to average power ratio was then recomputedfor the rounded phase shifts.

The 5 GHZ OFDM system of ETSI BRAN Hiperlan-11 standardization committee has 48 subcarriers, withcoding rates of 11 2 and 314. There are 48 subcarri-ers with O FD M symbol duration of 3 microseconds and600 nanoseconds guard interval. The subcarrier spac-ing is 416.666 kHz. The BPSK, QPSK and QAM signalconstellations can be employed corresponding to respec-tively 8, 16 and 32 Mbps for the ra te half code. Inter-leaving is also employed t,o break the possible burstynat ure of errors. To avoid difficulties in A / D and D /Aconverter offsets and carrier feed-through in the R F sys-tem, the subcarrier falling at D C is not used [ a ] . Thecoder is the standard rate 11 2 convolutional code [4].For more details see [ a ] .

For the BPSK case, we have computed the followingphase shifts. In each case the numbers must be mul-tiplied by 360 degrees to give the corresponding phaseshift in degrees. The first set of shifts are shifts of QPSKtype and are given in Table V. The reduction in PAPRis 4.09 dB.

By allowing shifLs of 8-PSK and 16-PSK type, thereduction in PA PR increases respectively to 4.22 dB and4.46 dB. The phase shifts of 8-PSK type are given inTable V.

We note that the Gradient method only convergesto a local minima. In this light, there may be other

683

8/6/2019 0A9ADAE0d01

5/5

$0 = 0.000$4 = 0.250#12 = 0.625416 = 0.375420 = 0.500424 = 0.50042s = 0.375432 = 0.125436 = 0.875440 =0.250444 = 0.875

48 = 0.00041 = 0.6254 5 = 0.2504 9 = 0.875413 = 0.625417 = 0.875421 = 0.250425 = 0.5004 2 9 = 0.250433 = 0.00043 7 0.250$641 = 0.750445 = 0.875

4 2 = 0.625410= 0.875414 = 0.62541 s= 0.750422 = 0.250426 = 0.625430 = 0.125434 = 0.7504313 = 0.6254 4 2 = 0.375446 = 0.750

46 = 0.3754 3 = 0.50047 = 0.625411 = 0.375415 = 0.875419 = 0.1254 2 3 = 0.375427 = 0.750431 = 0.000435 = 0.875439 = 0.750443 = 0.125447 = 0.250

Table 2: Phase Shifts of 8-PSK type for the ratel / 2 BPSK convolutional code. Each number mustbe multiplied by 360 to give the phase shift in de-grees.

/ChannelI

IDFT b e - M u GI

Figure 1: OFDM Block Diagram.Figure 2: The Phase Shifted OFDM Block Diagram.

phase shift values that give a greater reduction in peakto average power ratio for the above code. The useof more powerful minimization methods instead of thegradient method may yield even better results.

REFERENCESA. E. Jones and T. A. Wilkinson, Combined coding er-ror control and increased robustness to system nonlinear-ities in O F D M , Proc. IEEE 46th Vehicular TechnologyConference, pp. 904-908, Atlanta, 1996.R.J. Kopmeiners and R. van Nee, Multirate O F D M Pro-posal, WG3 Tempovary Document 3LTN0920, E T S IE P B R A N 9 , June 1998.V. Tarokh and H. Jafarkhani, On Reducing the Peak toAverage Power Ratio in Multicarrier Communications,IEEE Trans. Comm., submitted.A. J . Viterbi, J. K . Wolf, E. Zehavi, R. Padovani,A Pragmatic Approach to Trellis-Coded Modulation,IEEE Communicat ions Magazine, pp. 11-19, July 1989.

684