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REDUCING THE PEAK-TO-AVERAGE POW ER RATIO IN OFDM RADIOTRANSMISSION SYSTEMSThomas May and Hermann Rohling

The authors are with the Institute of Telecomm unicationsTechnical University of Braunschweig , Germanymay @ifn.ing.tu-bs.de, ohling@ifn.ing.tu-bs.de

Abstruct- An important difficulty which has to be solvedin OFDM transmission systems is the large peak-to-averagepower ratio of the OFDM signal. Without any measures, thesignal is limited by the power amplifier in the transmitterwhich causes interference both of the signal itself and in ad-jacent frequency bands. In this paper a method is proposedwhich considerably reduces the peak-to-average power ra-tio of the OFDM signal by means of signal processing.

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I . I N T R O D U C T I O NOFDM systems allow the transmission of high d ata rates overbroadband radio channels w ith frequency-selective fading with-out having to provide a powerful channel equalizatio n. If differ-ential modulation is applied, no channel estimation is requiredat all. For this reason, the complexity of OF DM systems can be

much lower compared with a single carrier transmission sy stem.On the other hand, a difficulty about OFDM signals is the

fact that they have a very large peak-to-average power ratio.In the transmitter, the maximal output power of the amplifierwill limit the peak amplitude of the signal and this effect willproduce interference both within the OF DM band and in adja-cent frequency bands. Furthermore, in OFDM systems with aguard interval, a rectangular pulse must be used for mod ulation.The corresponding pulse spectrum is a sinc function which alsocauses out-of-band interference. These effects, particularly theproblem of amplitude limitation of OFDM signals, have beendiscussed in several publicatio ns.

In this paper, we propose a manipulation of the OFD M signallike in [6 ] in order to remove p eaks of the signal which exceed agiven amplitude threshold. The proposed m ethod does not pro-duce out-of-band interference and with this con dition it causesminimal interference within the O FDM band.

11. PROPOSALS I N TH E L I T E R A T U R EIn most of the publications about amplitude limitation ofOFDM signals it is assumed that it can be achieved by pre-

distortion of the signal that the amplifier behaves like an ideallimiter. This means that the sign al is amplified linearly up toa maximal input amplitude A0 and larger amplitudes are lim-ited to Ao, see Fig. . Based on this assumption, we also modelthe amplifier as an ideal limiter with amplitude threshold A0 inthis paper. The input power of the amplifier as compared to thethreshold is described by the input backoff IB O = 1 0 log 2where Ps s the average OFDM signal power.

00 0 5 1 1 5 2normalized input amplitude

Fig 1 Ideal limiter with normalized input and output amplitude. inaxiinalinput amplitude A0 = 1

In literature, two kinds of approaches are investigated whichassure that the transmitted OFD M signal s ( t )does not exceedthe amplitude A0 if a given inpu t backoff is used. The firstmethod makes use of redundancy in such a way that any datasequence leads to an OFDM signal with Is( t) l 5 -4 0 or thatat least the probability of higher amplitude peaks is greatly re-duced. This approach does not result in interference of theOFDM signal .

In the second kind of approach, the OFD M signal is manip-ulated by a correcting function which eliminates the amplitudepeaks. The out-of-ban d interference caused by the correctingfunction is zero or negligib le. How ever, interference of theOF DM sign al itself is tolerated to a certain extent. This ap-proach will be investigated and optimized in the following.A. inserting Redundancy

One way to avoid high amplitude peaks is to select from themultitude of all possible OF DM blocks those which fulfill thecondition Is(t) 5 A0 at a given input backoff. These suitableOFDM blocks could be assigned to different data bit sequenceslike a code. If e.g. only lo- of all possible blocks are se-lected, the system can transmit approximately 30 bits less perOFDM block, which could be acceptable. H owever, it seems tobe necessary to find a way of constructing complying codes. Ifthese codes cannot be constructed, the effort for this approachin terms of memory is gene rally too large. Even for a simplesystem with 16 subcarriers and QPSK, billions of assignmentswould have to be stored. In [2] short block cod es are employedin order to enable a lower input backoff in OFDM systems withfour or eight subcarriers. These codes can simultaneously be

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mailto:ifn.ing.tu-bs.demailto:rohling@ifn.ing.tu-bs.demailto:rohling@ifn.ing.tu-bs.demailto:rohling@ifn.ing.tu-bs.demailto:ifn.ing.tu-bs.de

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2. . . , .. . , . .

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0 0 20 40 60 80 100 120t

continuous signal-samples X

Fig. 2. The continuous-time OFDM signals exceeds the amplitude thresholdA D n spite of clipping.

used for error correction.In [4] procedures are proposed in which the same data se-quence can be represented by several different OFDM blocks.The transmitter generates all possible signals corresponding to

a data sequence and chooses the most suitable one for transmis-sion. The receiver must ad ditionally be told which of the sign alshas been chosen. Th is can be achieved with little redundancy. Ifdifferential mod ulation is applied between adjacent sub carriers,the receiver does not even need any sid e information. H owever,in this case on several subcarriers reference sym bols are trans-mitted for the differential demodulation. This scheme allowse.g. to decrease the input backoff from 12 dB to 9 dB at thesame level of out-of-band interferen ce.A proposal which realizes an OFD M transmission w ith a con-stant envelope using 50% redundancy is proposed in [3]. In this

scheme, instead of one OF DM block two blocks are transmit-ted which are calculated from s ( t ) .However, this calculation isnon-linear and causes itself out-of-band interference. The ob-jective of this approach is not to avoid out-of-band interferencebut to avoid interference of the OFDM signal.B. Correcting the OFDM - Signal

The second approach in which the OF DM signal is correctedwith a suitable function avoids out-of-band interference but ittolerates interference of the OFDM signal itself. In the simplestcase, the sampled signal is limited to the am plitude A0 [ I ] . Thismethod is termed clipping. Clipping does not cause out-of-bandinterference if s ( t ) s not oversampled. H owever, without over-sampling the analog signal after the D/A conversion w ill exceedthe amplitude threshold, see Fig. 2

This effect has to be considered. Furthermore, the OFDMsignal must be filtered because of the rectangular pulse. Forboth reasons, oversampling of the signal is necessary. In [5] th eproposal is made to apply clipping to the oversampled signalwhich causes out-of-band interference. This is taken care of bya FIR filter which also removes the side lobes of the modula-tion pulse. How ever, the filter leads to new am plitude peaks i nthe signal, but, after all, the peak-to-average power ratio of thesignal is reduced by this meth od.

In [ 6 ] , he OFDM signal is corrected by multiplying i t with

a correcting function k ( t ) . If the signal exceeds the amplitudethreshold A0 at the times tn , hen the corrected signal c ( t ) sc ( t ) = s ( t ) l i ( t ) where

n

Thus, the signal is attenuated by a Gaussian fun ction at all po -sitions where it has high am plitude peaks. T he spectrum C ( )of the corrected signal is

where B, are constants and U; is the variance of the Gaussianfunc tiong ( t) in the f requency domain. So , the correction broad-ens the signal spectrum by the width of a Gaussian function withthe variance U; = 1/2ru2The G aussian functio n which is usedfor the correction should be narrow in the time domain so thatonly a small interval of the signal is attenuated. It should alsobe narrow in the frequency domain so that the signal spectrumis broadened as little as possible. If for an OFDM system withthe bandwidth B a Gaussian function with U = 5 / B is chosen,then uf = B/lOn, which results in an acceptable broadeningof the signal spectrum.Obviously, with this method it can be achieved to limit theamplitude of the oversampled signal without causing out-of-band interference, except in a narrow frequency band adjacentto the OFDM band. However, if many amplitude pe,aks haveto be corrected, the entire signal is attenuated and the peak-to-average power ratio of the signal cannot be improved beyond acertain figure.This scheme of correcting the OFD M signal can be realizedfor any number of subcarriers and it does not need any redun-dancy. It causes interferenc e of the OFDM signal, but this is ofsecondary importance in a fading environment in which OFDMis typically applied. The important task is to avoid out-of-bandinterference.

111. A D D I T I V EORRECTING FUNCTIONSEach manipulation of a signal can be seen as an addiitive cor-rection. The corrected signal can be written as

c ( t ) = s ( t )+ k ( t ) wheren

The correcting function k ( t ) composed with the auxiliaryfunction g (t ) which must be normalized so that g ( 0 ) =: 1. This

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correction limits the signal s ( t ) o A0 at the positio ns t,of am-plitude peaks. O f course, the correction could c ause the signalto exceed the amp litude threshold at a different position. Weigno re this effect now and w e will se e that it is negligible later.In the following, we determine an auxiliary function g ( t ) whichproduces no out-of-band interference and causes interference ofthe OFDM signal with minimal power.For the sake of simplicity, only a single OFDM modulationinterval with the extent TN = l /Af is considered which isrepeated periodically. Th e auxiliary functio n is also periodical,then. In this case, the spectra of both the signal and th e auxiliaryfunction are discrete. Thus, an auxiliary function with a spec-trum limited to the OFDM band can b e entirely described in thefrequency do main by its samples at the subcarrier frequencies.

I0.3

O . j J t.2 I-

0.15-0.1

0.05n 0 100 200 300 400 500 600t

Fig. 3. Example of a correcting function k ( t ) fo r an OFDM signal with 12 8subcarriers, oversam plingN-1g ( t ) = Gk e j z n k A f t

k=O

k=OThe interference power is minimized, if we minimize the powerpg O f d t ) .

If an auxiliary function g ( t ) is chosen such that the phases ofthe coefficients Gk are not all equal, then a different functionwith lower power could be found in the following way: Thepower of the function g ( t ) with the coefficients G(, = lGklhas the same power as g ( t ) an d g ( 0 ) > g ( 0 ) = 1. If g ( t )is normalized according to Eqn. (2), the result is an auxiliaryfunction with lower power than g ( t ) . Therefore, the auxiliaryfunction with minimal power has coefficients Gk which are allequal. We obtain

N-1.Pg = Gi

k = O

Taking Eqn. (2) into account, Py is minimal, if we choseG k = l / N for all coefficients. Then , Eqn. (1) yields

k=ON (3)

n=-mM sinc(7rBt)eJnBt (5)

This approximation is valid for t E [ - T N / ~ ,N / ~ ] .hesinc function is repeated periodically because only a singlemodulation interval is considered which is repeated periodi-cally, too, leading to a discrete spectrum. If the entire OFDMsignal is considered and not only a single modulation interval,

then the auxiliary function is not periodical and a single sincfunction with a c ontin uous spectrum is obtained as it is given inEqn. (5 ) .

This function can correct an amplitude peak in the OFDMsignals with minim al interference of the signal and without anyout-o f-band interference. For practical application however, theextent of the sinc function in time must be limited by window-ing.

If the OFDM signal is not oversampled, then the sampledauxiliary function g(nAt) s zero for all n # 0. The correctionschem e is identical with clipp ing in this case.

The spectrum of the corrected signal is

= S ( f )+ AnG(f)e-lZTtmf

Each correction of an amplitude peak causes interference oneach subcarrier and th e power of the correcting function is dis-tributed evenly to all subcarriers.

This correcting scheme has been applied to an OFDM trans-mission system with 128 subcarriers i n a simulation The signals ( t ) has been oversampled by a factor of four and normalized sothat the signal power is one. Then the signal has been correctedwith k ( t ) . For the correction, the amplitude threshold A,) hasbeen set according to the inpu t backoff which has been chosen.After the correction, the signal has been limited to the ampli-tude Ao i n order to take into account the limitation of amplitudepeaks which may have remained.Fig. 3shows an example of a correcting function. In Fig.4, the power density spectrum of the interference caused by thecorrection and the limitation of the signal is shown for an inputbackoff I BO = 4 dB. It can be seen that the interference poweris concentrated within the OF DM bandwidth Despite the cor-rection of the OFDM signa l, the signal still happens to exceedthe amplitude threshold Ao. For th is reason, the limiter causes

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105 3 A8 7 6 5 4 3 2 1 0IBO [dB]Fig. 4. Power density spectrum of the interference of the received signal which Fig. 6. Signal-to-interference ratio S I R adjacent to the OFDM band in theis caused by correction and limitation at I BO = 4 B case of correction with sinc functions and without any correction

40sinc functions*5 - -no correction+

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--

UIp" 20 -. 15 -10 -

5 -10 8 6 4 2 0IBO [dB]

Fig. 5 . Signal-to-interference ratio S I R n the OFDM band in the case of cor-rection with Gaussian functions (multiplicative), with sinc functions (addi-tive) and without any correction

out-of-band interference the pow er of which is more than 60 dBbelow the signal power.The signal-to-interference power ratio S I R can be deter-mined both within the OFDM band and in the adjacent fre-quency bands. The S I R in the OFDM band is the quotientof the power densities of the signal and of th e interference. As

the SIR in the adjacent frequency band we de note the quotientof the signal power density in the OFD M band and the averagepower density of the interference in a frequency band with thebandwidth B/10, directly adjacent to the OFDM band.

In Fig. 5 the resulting S I R in the OFDM band is depicted forcorrection with Gaussian fu nctio ns (multiplicative), with sincfunctions (additive) and without any correction. In each case,the signal has additionally been limited according to the IBO.Without any correction we obtain the least interference power,because the signal is modified only w here the amplitude thresh-old of the limiter is exceeded. The correcting functions ad di-tionally modify the signal within a certain area around the am-plitude peaks.

Th e SIR in the adjacent frequency bands is shown i n Fig. 6.The correction of the signal has the effect of an attenuation.

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-" 8 7 6 5 4 3 2 1 0IBO [dB]Fig. 7 . Attenuation of the signal due to the correction with Gaussian functions.with sinc functions and with limitation of the signal

This is o bvious in the case of the multiplicative correction withGaussian functions, but the additive correction with sinc func-tions and the limitation of the signal also reduce thle signalpower. The effective attenuation of the signal due to the cor-rection and the limitation is displayed in Fig. 7

Becau se of this attenua tion o f the signal, the real inpiut powerof the amplifier is lower than the IB O suggests which the cor-rection of the signal was designed for. Note that IBO refers tothe OFDM signal before the correction. The correction withsinc functions causes a lower attenuation than the correctionwith Gaussian functions so that the same IBO leads to a higherinput power of th e amplifier in this case.

For the AWGN channel an d DQPSK modulation, Fig. 8shows the bit error rate as a function of the input backciff 1RO.In the corresponding simu lation , the noise power S has beenfixed so that i l i / N = 18 dB. Thus, the signal-to-noise ratioS N R depends on the IBO. Still, the input backoffrefers to theOF DM signal before Correction.Fo r a WSSUS radio channel with multipath propagation andconsequently with frequency-selective fading, Fig. 9 showssimulation results with *4Z/AT = 30 dB. In this case, the in-terference due to the correction is less grave, because it is atten-uated by the radio channel as well, whereas the noise is added

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0 1

0.01am

0 001

0 000110 0 6 4 2 0IBO [dB]Fig. 8. Bit error rate B E R as a function of the input backoff I B O of the limiter(amplitude threshold Ao) for the AWGN channel with Ai /N 18dB

no correction --t-sinc functions --3tGaussian functions+0.01

[rLum

Jf I , i

0.00110 0 6 4 2 0IBO [de]Fig. 9. Bit error rate BERas a function of the input backoff IB O of the limiter(amplitude threshold Ao ) for a channel with frequency-selective Rayleighfading and A: /N 30 dB

to the attenuated signal. Accordingly, a lower input backoff isoptimum than in the case of the AWGN ch annel.

IV . CONCLUSIONIn this paper the problem of out-of-band interference inOFDM transmission sys tems has been discussed which resultfrom limit ing the signal amplitu de. The approach of correctingth e OFDM signal with a suitab le func tion has been analyzed. Inthis approach the signal is modified in such a way that a given

amplitude threshold of the signal is not exceeded after the cor-rection.With the restriction that the correcting function does notcause any out-of-band interference, the correcting function hasbeen given which has minimal power and thus causes minimalinterference power within the OFDM band.

The interference power which is produced by the correctingfunction has been determined for the proposed method and for amultiplicative correction with Gaussian functions. Both for theAWGN channel and for a channel with frequency-selective fad-ing with a fixed noise power the resultin g bit error rate has beenevaluated as a function of the input backoff of the power am-plifier. It has been shown that for DQPSK modulation an inputbackoff of only 4 dB is reasonable with the proposed scheme.

REFERENCESR. ONeill, L. B. Lopes: Performance of Amplitude Limited MultitoneSignals, Proc. IEEE VTC 94 , pp. 1675-1679, June 1994A. E. Jones, T. A. Wilkinson: Combined Coding for Error Control andIncreased Robustness to System Nonlinearities in OFDM, Proc. IEEEVTC 96, Atlanta, 1996R. Dinis, A. G u m CEPB-OFDM: A New Technique for Multicar-rier Transmission with saturated Power Amplifiers, Proc. IEEE ICCS96, Singapore, Nov. 1996S . H. Muller, R. W. Bauml, R. F.H. Fischer, J. B. Huber: OFDM with Re-duced Peak-to-Average Power Ratio by Multiple Signal Representation,Annals of Telecommunicat ions, Vol. 52 . pp. 58-67, Feb. 1997X. Li, L. J . Cimini: Effects of Clipping and Filtering on the Performanceof OFDM , Proc. IEEE VTC 97, Phoenix, May 1997M. Pauli, H.-P. Kuchenbecker: Reduzierung der durch Nichtlinearitatenhervorgerufenen Auaerbandstrahlung bei einein Mehrtragerverfahren.(in german), ITG Fachberichte 135 Mobile Koinmunikation. 1995

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