Peak to Average Ratio Reduction for Multicarrier Transmission:a ReviewDavid DECLERCQ and Georgios B. GIANNAKIS1 IntroductionOrthogonal Frequency Division Multiplexing (OFDM) and Digital Multi Tone (DMT) aremulticarrier modulation techniques that have been recently proposed in several communica-tion areas such as transmission over Asymmetric Digital Subscriber Line (ADSL) or wirelesstransmission in Digital Audio/Video Broadcasting (DAB/DVB) [Sari95, Salt98]. Both of thesetechniques split the available transmission bandwith into several narrowband carriers whichcarry di�erent signals simultaneously. The carriers are called frequencies or tones. This kindof modulation has been �rst proposed a long time ago [Wang66] and has the great property ofbeing able to reconstruct the signal without distortion despite the overlapping carrier spectra.Moreover, a very attractive aspect of OFDM/DMT modulation is that the demultiplexing(at the transmission) and the multiplexing (at the reception) can be e�ciently implementedwith fast Fourier transforms. The main drawback of this modulation is the nearly Gaussianbehaviour of the data stream that has to be sent through the channel. Unlike other types ofdata modulation which have \bounded" constellations, the OFDM/DMT stream exhibits largeexcursions from its average power, forcing the ampli�ers at the transmitter to operate in theirnonlinear region. This causes severe signal distorsion, which must be cancelled or reduced tomaintain good detection performances at the receiver. Because the important advantages ofOFDM/DMT modulation in a lot of practical areas, a growing attention has been given tosolve the problem of amplifying OFDM/DMT symbols. In this report, we focus on methodswhich try to reduce the impulsiveness of the OFDM/DMT process by adding redundancy orsignal transformation at the transmitter. In another report, we will study methods that takeinto account a model for the nonlinear ampli�er, in order to compensate for its distorsions[Decl99b]. We won't study either the techniques involving coded versions of OFDM, in whichthe goal is to �nd codes that exhibit good peaks properties. Note that those methods thatuse codes can be jointly implemented with the peak reduction schemes presented here. Theoutline of this report is the following. In the �rst part, we remind what the OFDM/DMTscheme is, and fully describe its main properties. Then, a complete study of the Peak toAverage Ratio (PAR) is performed. Finally, we present three methods found in the literaturethat aim to reduce the PAR of the OFDM transmit data, and compare their performanceswith extensive simulations in terms of PAR reduction at the transmitter.1

2 Review of OFDM/DMT modulation2.1 Description of the OFDM/DMT TransmitterA typical OFDM/DMT system at the transmitter is depicted in �gure 1.x nX k / /

S

P

P

S

Order N

IDFTLPF HPA

CyclicPrefix

X N-1

X 0 x 0

x N-1Figure 1. Description of the OFDM/DMT modulatorFirst, the high rate data stream is converted to N parallel streams which are IDFT-transformed in order to send the streams over the N orthogonal subcarriers.xn = 1pN N�1Xk=0 Xk exp�2j�knN � 8n 2 f0 : : : N � 1g (1)remarkThe normalization factor 1=pN is prefered in (1) to have the same mean power at the inputand the output of the IDFT transformation. Of course, we must use the same normalizationfactor for the dual DFT at the receiver.Then, a cyclic pre�x of length Np is added to what we will call in the following theOFDM symbol in order to compensate for the dispersion of the transmission channel. Thispre�x has the same role as the classical guard time inserted in block transmission, but inthe OFDM/DMT scheme, a cyclic repetition of the symbols is prefered for reasons that wewill discuss later. The data is then serialized and feed a FIR Linear Pulse Filter (LPF) anda nonlinear High Power Ampli�er (HPA) that boosts the transmitted average power. Notethat this kind of modulation can clearly be interpreted as a convolutive coding scheme withcomplex exponentials.remarkIn ADSL communication, the channel model is baseband and carries only real valued data.In the single carrier case, we would add another block that would transmit the in-phase andquadrature components. In the case of OFDM/DMT modulation, because of the properties ofthe Fourier Tranform, N=2 complex information symbols are taken, symmetrized to obtain ablock of size N , which gives a real valued OFDM symbol through the IDFT transform. Notethat in ADSL, we transfer N=2 information symbols over N carriers.The input stream Xk can be drawn from any constellation (M-QAM, M-PSK, etc). InOFDM, the power allocated to di�erent subcarriers is the same and the sub-streams come2

from the same constellation. In DMT however, the transmitter uses information about thechannel dispersion to allocate di�erent powers over the tones. Typically, if a tone frequencyfalls in a deep null of the multipath channel, one allocates little power because the informationcarried by this tone will be highly distorted at the reception. Therefore, in DMT modulation,the tones which have larger power usually carry symbols drawn from larger constellations.remarkWe don't take into account the techniques that use codes coupled with multicarrier modulation(Coded-OFDM). In these techniques, the codes are used to stenghten the tones which have alow SNR, corresponding to deep channel nulls.One way to optimally allocate power among the sub-carriers at the transmitter is to usethe well known water pouring theorem [Bing90]. However, this requires the knowledge ofthe channel transfer function at the transmitter, which can be done only in two-way channelapplication (this is the case for example, for ADSL communications). Moreover, when theSNR is su�ciently high, performance doesn't degrade signi�cantly by allocating the samepower among the carriers. We will assume, in the following of this report, that it is the caseand we will deal with OFDM modulation, carrying information drawn from a unique type ofconstellation.The input symbols Xk, when converted into the subchannels, will be refered as frequencysymbols because they are carried by the N tones/frequencies of the OFDM modulation. Thedata xn at the output of the IDFT will then be refered as time domain samples, and thevector built with N time domain values x = x0!N�1 obtained from the IDFT of the Ncarriers X = X0!N�1 is called an OFDM symbol.2.2 Equalization in OFDM/DMTLet us denote the impulse response of the multipath channel that the data go through byh(t). The receiver does, as in most communication systems, the dual operations of those atthe transmitter (cf. �gure 2). First a matched �lter recovers the digital time domain stream.Assuming perfect synchronisation, the received digital samples areyn = L�1Xl=0 h(lTs)xn�l + wn n = �Np : : : N � 1 (2)Ts being the symbol duration, LTs the channel spread length and wn the Additive WhiteGaussian Noise (AWGN).The cyclic pre�x is then removed and the ow is converted to N parallel sub-streams. Thena DFT is applied to switch back the data to the frequency domain, the channel equalizationis performed, and a �nal hard-decision is made to obtain symbol estimates X̂k.Thanks to the cyclic pre�x introduced at the transmitter - rather than a guard interval -the DFT operation performed at the receiver turns the time domain convolution in (2) intoa frequency domain multiplication without any side e�ect. This comes from the fact thatthe OFDM symbol is observed by the channel as a periodic concatenation of itself. We havethereforeYk = 1pN N�1Xn=0 yn exp��2j�knN � 3

N-1y

0y

/S

P

ny/

P

S

N-1Y

Y 0

RemovePrefix

MatchedFilter

nr

Order

DFT

N Equalization

Detection

N-1

X k

X 0

XFigure 2. Description of the OFDM/DMT receiver= DFT (h � xn + wn) = H � kNTs� :Xk +Wk 8k = 0 : : : N � 1 (3)Note that due to the DFT properties, Wk is still a complex AWGN with the same varianceas wn.As we can see in eq. (3), thanks to the modulation scheme, the inter-symbol interferenceusually expressed in convolutive terms in the time domain, is transformed to a simple blockmultiplicative interference, rendering the equalization much easier.remarkWhen the channel spread is greater than the cyclic pre�x length, some additionnal time-domainequalization is required prior to the DFT.The equalization of the channel is usually performed in two steps: (i) by means of atransmitted train of pulses, the channel impulse response is estimated at the receiver byaveraging the received blocks 1. The spacing between two pulses must therefore be at leastthe length of the channel spread, (ii) from the estimated channel taps, the equalization is donein the frequency domain with a simple Automatic Gain Control block [Rein94].The key feature of OFDM/DMT transmission is this equalization technique, which isperformed in the frequency domain. This renders this modulation very robust to frequencyselective fading. Actually, it transforms the frequency selective transfer function of the channelinto N at subchannels, each being easily equalized. This is con�rmed by a recent comparisonof a Coded-OFDM scheme with its corresponding single carrier system, but for which theequalization is carried out in the frequency domain [Aue98]. The authors show that the twosystems perform similarly for low and medium code rates.2.3 Ampli�cation of OFDM symbolsA major problem of OFDM/DMT modulation is the ampli�cation of OFDM symbols.Whereas in single carrier transmission the symbol constellations are bounded, the peak ofthe complex enveloppe of the OFDM symbol may be N times higher than the maximum ab-solute value of the input constellation. Those high peaks usually lie in the nonlinear range ofthe ampli�er, resulting in out of band radiations [Paul98] and performance degradation at thereceiver.1this technique is known in the literature as sounding4

The two kind of nonlinear ampli�ers reported in the literature that are used in digitalcommunication are the Travelling Wave Tube Ampli�er (TWTA) and the Solid State PowerAmpli�er (SSPA).A lot of methods have been proposed to cancel the e�ects due to the nonlinear distortions inthose ampli�ers. Most of them use approximate inverse formulas of the AM/PM and PM/PMcharacteristics of the nonlinear transfer function. Several implementation methods have beenproposed, the most relevant relying on polynomial approximations. All these methods convertthe nonlinear behaviours of the ampli�ers into a clipper (or soft limiter). The nonlinear e�ectsare then cancelled beyond a threshold, represented by the saturation point of the ampli�ers(the maximum power that can be allocated by the ampli�er). This is summerized in �gure 3for the TWTA.A satA sat

AM/AMAM/PM

AM/AMAM/PM

Input AmplitudeO

utpu

t Am

plitu

de/P

hase

Out

put A

mpl

itude

/Pha

se

Input Amplitude

PredistorsionNonlinear

Figure 3. Cancellation of the nonlinear behaviour of a TWTA ampli�erA discussion of these techniques together with a detailled presentation of the nonlinearampli�ers and their e�ects will be made in [Decl99b].In this report, we will then assume that the ampli�er transfer function has been turnedinto a soft limiter and therefore, the only distorsion that remains is a clipping e�ect. Tosimplify the description of the e�ects of the nonlinear ampli�ers to OFDM modulation, wewill not take into account the cyclic pre�x because it doesn't bring additionnal information,and therefore doesn't a�ect the performance study under consideration.remarkDespite the fact that the power ampli�er operate in continuous time, we will restrict ourstudy to the discrete time domain. The results cannot be directly extended from discrete tocontinuous time without some assumptions [Tell98b]. A way to quantify the modi�cationsbrought by the continuous time conversion is to oversample the discrete stream and study thee�ects of the nonlinear ampli�er on the oversampled data. It has been pointed out in [Tell98]that an oversampling by a factor of 4 is su�cient to simulate continuous time, for the presentproblem. We will discuss these points based on simulations in section 5.5

3 Peak to Average Ratio in OFDM symbols3.1 De�nitionThe Peak to Average Ratio (PAR) of an OFDM symbol is de�ned asPAR(x) = maxn20:::N�1 jxnj2IE hjxnj2i (4)This quantity is a good measure of the impulsiveness of a block symbol: if the underlyingprocess exhibits sample values that are very far from its average power, the PAR will be large.For example:� for an i.i.d. binary process taking its values in f0; 1g, we have PAR = 2,� for a i.i.d. M-PSK mapping, PAR = 1, which is the lower bound of the PAR,� for a 16-QAM constellation, with the same probabilities for the symbols, PAR = 1:8.When passing through the IDFT block, the impulsiveness of the data increases a lot,yielding a PAR which can be at most N times larger that the input one. Of course, this upperbound is not tight at all and the PAR must be interpreted as a random variable (dependingon the block realization), and studied with probabilistic means.3.2 Statistical propertiesAssuming that the OFDM symbol size is large and that the input data are uncorrelated,the central limit theorem applies and the time domain transmitted ow is approximatelydistributed as complex Gaussian with zero mean and variance �2x = �2X (with the normalizationfactor 1=pN for the IDFT operation). When the input stream is no longer white, but isstrongly mixing - which is the case when correlation is introduced before the IDFT, as inCoded-OFDM schemes - the output is still a nearly Gaussian complex process, but the variance�2x depends on the input correlation. Moreover, the stronger the correlation present at theinput of the IDFT is, the larger the OFDM symbol length N should be, for the Gaussianapproximation to be accurate.Let us assume that the output of the IDFT is e�ectively Gaussian distributed with variance�2x. Hence, its modulus un = jxnj is Rayleigh distributed with probability density function:p(u) = 2u�2x exp �u2�2x! (5)The undesirable e�ect of the OFDM symbol lies in the tails of that Rayleigh distribution,corresponding to the presence of high peaks, far greater than the average power. It is thenuseful to study the occurence of such peaks. That is the probability that the amplitude of asample within the OFDM symbol exceeds a treshold x0 > 0,Prob(jxj � x0) = Z +1x0 p(u)du = exp �x20�2x! (6)6

If the samples of the OFDM symbol are moreover assumed to be independent - which isobviously not the case for �nite block sizes, but the assumption becomes more and more validas N tends to in�nity - we get:Prob� maxn20:::N�1 jxnj � x0� = 1� N�1Yn=0 Prob (jxnj � x0)= 1� 1� exp �x20�2x!!N (7)and then, we obtain the corresponding probability for the Peak to Average Ratio:Prob PAR(x) � PAR0 = x20�2x! = 1� (1� exp(�PAR0))N (8)This probability, corresponding to the Complementary Cumulative Distribution Func-tion (CCDF) of the PAR, is then a function of the number N of sub-carriers used in theOFDM/DMT modulation and of the average power of the input constellation through �2x. Inorder to verify the accuracy of this approximate probability, we have drawn on �gure 4 thisoutage probability (8) as a function of the threshold PAR0, for di�erent block sizes and a16 � QAM constellation. The corresponding Monte Carlo simulations are shown in dashedlines. We can see a good agreement between the theoretical curves and the simulations, evenfor small block sizes. This shows that that assumptions that have been made can be consideredas valid, at least for N � 64.Those curves are useful to quantify the improvements of methods for reducing the PAR.Indeed, it represents the probability that an OFDM symbol has peaks that exceed a giventhreshold, being �xed by power or reliability requirements.3.3 Relation to SER4 PAR Reduction methodsOne of the possible solutions to avoid the clipping of too many samples of the OFDM symbolbt the HPA is to transform it so that the Peak to Average Ratio becomes lower, implyingless impulsive behaviour. Such methods have been studied extensively, and can be classi�edin two major schemes. The �rst possibility is to add some redundancy to the original OFDMsymbol, so that the complete symbol (data+code) has a lower PAR. Two groups of researchershave independently developped this kind of technique [Gath97, Tell98a], and we present one ofthem in section 4.1. The other possibility, is to transform the input symbols from the originalconstellation (M-QAM, etc) prior to the IDFT, in order to provide a time domain OFDMsymbol after the IDFT with lower peaks. Many di�erent ideas based on the transformation ofthe input constellation have been proposed in the literature [Mull97, Ksch98, Tell98a], and wepresent two of the most relevant in section 4.2 in detail. A comparative study of performancesis then derived in section 5.7

4 5 6 7 8 9 10 11 1210

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10−3

10−2

10−1

100

PAR0

Pro

b(P

AR

(x)>

PA

R 0)

N=64

N=256

N=1024

Figure 4. CCDF of the Peak to Average Ratio for di�erents block sizes. The approximation basedon the nearly Gaussian distribution of the IDFT are drawn in solid line, the dashed lines representestimations based on 10000 Monte Carlo runs.

8

4.1 Adding Redundancy at the Transmitter: Tone ReservationBy adding a structured time domain code c to x, or equivalently a frequency domain code Cto X, one changes the statistical distribution of the OFDM symbol, and therewith its Peak toAverage Ratio. The goal is then to �nd convenient codes that reduce the PAR, with a smallincreased complexity at the transmitter and the receiver. The additional side informationrequired to recover the original data - which represents the added redundancy - should not betoo large either.The method proposed in [Tell98a], assumes that some frequencies are reserved for thiscode and therefore carry no data information. This method is called Tone Reservation. Thespeci�cation of the L chosen tone locations is of major importance and we will discuss thatpoint at the end of this section. Let us assume that the L tones fk1 : : : kLg have been �xed atthe beginning of the transmission and that they won't change until the transmission is overor some new information about the channel is fed back to the transmitter.Given those tone locations, one has to optimize the values of the codes that �ll the unusedfrequencies in order to minimize the PAR of the new OFDM symbol. The frequency domaincode vector C has then only L nonzero values and let us denote ~C = [Ck1 : : : CkL ]T the vectorcollecting those values. We recall that the reserved locations don't carry data informationXik = 0 8k = 1 : : : L. The criterion that leads to the estimated codes is:~̂C = Argmin~C � maxn=0:::N�1 jxn + cnj� (9)where cn is naturally a function of ~C because it is the IDFT of the complete vector C takenat instant n.This minimization problem is convex in ~C and can be solved as a Linear Programmingproblem of complexity O(LN2). In [Tell98a], it is shown that by optimizing the signal toclipping noise ratio instead of PAR and by using a gradient algorithm, one can obtain similarperformance with a complexity O(N). The optimization is in this case done on the timedomain code c. We refer to that paper for a complete description of the method. After somecalculations, one arrives at the following iterative algorithm:c(k+1) = c(k) � ��kp[((n� nk))N ] nk = Argmaxn ���xn + c(k)n ��� (10)where �k is a scale factor depending on the maximum peak found at iteration k, p iscalled by the authors Peak Reduction Kernel (PRK) and is only a function of the tone lo-cations fi1 : : : iLg. Therefore, one needs to calculate the PRK only at the beginning of thetransmission. The notation p[((n � nk))N ] means that the kernel has been circulary shiftedin time by a value of nk. This kernel has its maximum - in the time domain - at n = 0 andits aim is to decrease the high peak found at nk, without increasing the other values of theOFDM symbol at n 6= nk too much. Then, a pertinent choice for p - and therefore for thereserved tones - is obtained by minimizing its secondary peak.A good choice for the tone locations is to select them pseudo-randomly, more structuredchoices like adjacent tones or equi-spaced tones yield higher secondary peaks. The authorspropose an intensive random search algorithm to �nd good locations, arguing that the opti-mization of this kernel needs to be done only once at the beginning of the transmission.remark 9

A problem may arise if we want to apply this method to DMT modulation. In that case, one canobviously not choose the locations randomly, because di�erent powers are allocated to di�erentfrequencies. Indeed, the reserved tones should transmit little power, and this constraint mightnot be compatible with the reduction of the secondary peak.As a �rst conclusion regarding this method, we wish to discuss its implications in termsof performance loss of the overall communication system. We will use those points in ourcomparison of PAR reduction methods in section 5.1. redundancy/side informationThis is the drawback of the method: reserving more tones for the codes achieves morePAR reduction, but the transmission is at a lower rate. If the same power is allocatedamong the carriers, and then the same number of bits is transmitted in each tone, thepercentage of overall redundancy is LN 100%, where L tones are reserved for the codes.2. transmit power increaseThe power increase is hard to estimate because it depends on the values of the codes andtherefore of the OFDM symbol realization. We will estimate the average power increasefor a given redundancy in section 5.3. complexityThis is clearly the main advantage of this method: the complexity at the transmitter isO(N) and the receiver needs only to set the L tones corresponding to the code to zero.4.2 Playing with the constellationsIn this section, we will tackle methods based on the transformation of the input constellation.As a matter of fact, these method depend on the type of constellation we are dealing with,and generalizing their use to any type of constellations might not be immediate. We will thenrestrict our study of these methods to M �QAM constellations.4.2.1 Selected MappingA �rst, though very general, idea proposed in [Bauml96] is to create a \generalized alphabet"of OFDM symbols xu, each representing the same data information. Once the alphabet U ofsize U is created, the task is to choose among the possible representations, the OFDM symbolwith the lowest PAR. Mathematically, we wish to transmit:~x = Arg minxu 2U (PAR(xu)) (11)A possible solution to generate the alphabet U is to rotate the input symbols in the QAMconstellation, prior to the IDFT. We must therefore select U phase vectors, each of size N :'u = [ej'u;1 : : : ej'u;N ]T 8 0 � u � U � 1and build the candidate OFDM symbols by multiplying component by component the originalblock-symbol X with these phase vectors and take their IDFT:xu = IDFT (X:diag('u)) 8 0 � u � U � 1 (12)10

This method is called selected mapping (SLM). As pointed out in [Bauml96] and [Mull97],the optimization problem may not be computationally feasible if the size of the OFDM blocksare large, and more importantly, if the size U of the OFDM alphabet U is large, which isrequired in order to achieve substantial PAR reduction. Moreover, thsi method requires thestorage of all the phase vectors f'ug0�u�U�1, both at the transmitter and at the receiver.Because of these substential drawbacks, we have chosen not to fully describe this method,focusing on the two methods described in the following sections.4.2.2 Partial Transmit SequencesMotivated by the method presented above, M�uller and Huber have proposed a related scheme,based on the rotation of sub-blocks of the original constellation [Mull97]. The description ofthe method is summarized on the diagram in �gure 5. The frequency-domain OFDM symbol(before IDFT) is split in U disjoint sub-blocks. For each sub-blockXu, the tones being used bythe other sub-blocks are set to zero. Each block is then rotated with an angle 'u - multipliedby a complex exponential, and the sub-blocks are �nally added to feed the IDFT operator.We show the example of a 3-block partition, where the rectangular pulses represent symbolictone locations and not the information carried by these tones. We remark that this methodcan handle M-PSK input constellations also.remarkthe Xu are not really sub-blocks, because each has size N , but they represent a non-overlappingdecomposition of the original OFDM symbol: X =PuXu.The new OFDM symbol obtained after this transformation is:~x = IDFT UXu=1Xu ej'u! = UXu=1xu ej'u (13)ϕje 2

ϕje 3

ϕj 1e

X| |kx n

N-1

N-1

N-1

N-10

0

0

0

IDFT

Figure 5. Partial Transmit Sequence method for PAR reduction.The optimization of the angles is a complicated task because the general problem leads to11

a time consuming nonlinear optimization.'̂ = Arg min'1:::'u maxn=0:::N�1 ����� UXu=1xu;n ej'u�����! (14)We must then limit the number of sub-blocs in the partition and restrict the choice of candidateangles to a discrete alphabet. The authors in [Mull97] chose the alphabet 'u 2 f0; �2 ; �; 3�2 g,which is pertinent from a computational point of view because no multiplication is required tocompute (13). The actual angles are then obtained by exhaustive search over all the possiblecombinations. Therefore, besides the implementation simpli�cations, the optimization remainsa NP -complete problem, with exponential complexity in the number of sub-blocks and thenumber of possible values for the angles. Computational complexity at the transmitter is thena severe limitation of this method.remarkThe authors propose in other papers additionnal simpli�cations of their method, for instanceby introducing di�erential encoding at the transmitter, but none of these modi�cations removethe exponential complexity of the angles optimization.Lets turn our attention to the choice of the sub-blocks. As explained in [Mull97b], theaperiodic autocorrelation function between the sub-blocs plays an important role. Actually,it appears in the expression of the upper bound of PAR obtained in [Tell97]. The choice ofpseudo-random partition should reduce this autocorrelation, resulting in a better PAR reduc-tion. This implies additionnal memory storage and computation time both at the transmitterand at the receiver, where the knowledge of the partition is needed. But since the deter-mination of the sub-blocks needs to be done only at the beginning of the transmission, thisrepresent a little loss. However, we must be careful and quantify the relative additional PARreduction brought by the pseudo-random choice, compared to the simplest case of adjacentsub-blocks. Indeed, adjacent sub-blocks would require no extra memory, with the number ofsub-blocks as the only side information.1. redundancy/side informationThe partition is chosen once for the complete transmission and therefore no on-lineredundancy is needed to transmit that information. The values of the angles need to beknown at the receiver to recover the transformed symbols. If the values of the angles aretaken among a known alphabet, we need only to transmit the U indices that correspondto the U sub-blocks. Moreover, if we decide to leave the �rst block unchanged and rotatethe U � 1 remaining, the redundancy for an OFDM symbol is then (U � 1) log2(W ),where W is the number of allowed discrete phases.2. transmit power increaseThere is no power increase in this method.3. complexityThe compexity is exponential with the number of sub-blocks and the log-size of theangles' alphabet, even when the simpli�cations proposed are taken into account. Fur-thermore, the receiver needs to performN rotations in the complex plane, which requireshigh computation at the receiver. If we choose the values proposed above for the angles,12

simple symetries in the complex plane replace the multiplications and the complexity atthe receiver becomes acceptable.4.2.3 Tone InjectionThe Tone Injection is another method that makes use of transformed input constellationsto choose an OFDM symbol which carry identical information, but with a lower PAR. Theidea is to generalize the QAM input constellation so that a QAM symbol has several identicalrepresentations in the complex plane. Each candidate point is obtained by shifting the originallocation of the symbol by a value of D, either for the real part, the imaginary part, or both:this is depicted in �gure 6.This method has been independently proposed by Motorola ISG [Ksch98] and in [Tell98a].In [Ksch98], the authors use a lattice expansion of the M-QAM constellation, which corre-sponds to a value of D = dpM , where d is the minimum distance between 2 symbols in thecomplex plane andM is the number of constellation points. In [Tell98a], the authors allow anypossible value for the shift range D, advicing that it should not reduce the minimum distanceof the constellation, in order to keep the same symbol error rate at the receiver: D � dpM .D

Re

Im

A8

AA A3

A4

A5A6A7

1

A0

2

d

Figure 6. Generalized constellation for QAM-16 mapping. Each symbol has 9 equivalent representa-tions.The important advantage of that method is that it requires no side information at all.Indeed, the receiver need only to know the value ofD and by performing a modulo-D operation,every symbol in the generalized constellation will fall into the original M-QAM constellation.Note moreover that the complexity at the receiver is negligible because only 2 modulo-Doperations are required - one for the real part, one for the imaginary part.13

Beside its promising structure and properties, this method has an important drawback: avery large complexity at the transmitter. Actually, if we make L shifts to the OFDM symboland consider K possible candidates for each QAM-M symbol (8 in the case of �gure 6), theoverall complexity at the transmitter would be:O��NL�KL�which is far too much to be practically implemented. One then has to introduce heuristiccriteria to reduce this complexity. The authors in [Tell98a] propose an iterative D-shiftingmethod based on the following remarks:� Because we consider symbols driven from a larger constellation, the average transmitpower will increase signi�cantly. We naturally wish this increased power to be as low aspossible. If we take as an example the constellation in �gure 6, we should only considerA6 as a candidate point for A0. Moreover, it is obvious that shifting symbols which arecloser to the origin of the original constellation would result in a greater power increase.One should then allow only the symbols located on the outer ring to be shifted.� Furthermore, the shifting of the symbol at the frequency location k0 has to reduce thePAR, by reducing the highest peak in the time domain at the location n0. Going backto the �gure 6, if Xk0 = A0 is transformed into A6, the new value of the time-domainOFDM symbol at location n0 would be~xn0 = xn0 � jDpN exp�2j�k0n0N � (15)Knowing that n0 is �xed (location of the peak), k0 must be chosen to reduce the PARby having j~xn0 j � jxn0 j. In practice this yield a large number of required tests and highcomplexity algorithms.remarkIn their contribution [Tell98a], the authors consider only the case of PAR reduction in ADSLtransmission, where the time domain OFDM symbol is real valued. In that case, the addition-nal quantity in (15) turns to a sine function, reducing the complexity of the iterative scheme.The strategy is the following: (i) �nd the location of maximum value of OFDM symboln0 (ii) restrict the possible candidates to shifting the symbols belonging to the outer ring ofthe constellation and consider only shifts that don't increase too much the transmit power,(iii) choose among the remaining possibilities the shift that reduces the absolute value of theOFDM symbol at location n0 the most, (iv) iterate the process until su�cient PAR reductionis achieved, or maximum power allocation is reached.1. redundancy/side informationNo redundancy is required to recover the data at the receiver.2. transmit power increaseThis is the �rst drawback of the method. We use a larger constellation to represent14

the same information, which requires a greater average transmit power. Of course thepower increase depends on the symbols that are shifted. If we restrict our discussion tothe constraints introduced above and a 16-QAM constellation, the percentage of powerincrease for L shifts would be on the average:16L10N 100% (16)For example, with N = 256 carrier, L = 10 shifts would lead to 6:25% power increase.3. complexityThe complexity of the full optimization at the transmitter is O��NL�KL�, whichcannot be implemented for practical situations. Even with the heuristical restrictionstaken into account, the complexity remains very large and di�cult to estimate becauseit is dependent of the OFDM symbol realisation. Moreover, the choice of the shiftsbecome highly sub-optimal. The advantages of this method should invite researchers to�nd more e�cient optimization methods.The complexity at the receiver, for a single OFDM symbol, is of 2N comparisons andL substractions/additions (if we restrict the shift distance to D). This complexity isnegligible compared to the overall complexity of the receiver.5 ResultsWe wish to make a fair comparison of the above presented methods, in terms of the loss ofe�ciency they bring to the communication system. We will then compare the PAR reductionperformances of these three methods for (i) the same amount of added redundancy, (ii) thesame complexity.We have decided to make the comparison of these methods in terms of their ability to reducethe Peak to Average Ratio at the transmitter. Because none of them rely on the knowledgeof the channel or the model of the nonlinear HPA (an ideal clipper is assumed), the achievedperformances at the transmitter - in terms of PAR reduction - should not di�er signi�cantlyfrom the corresponding symbol error rate at the receiver. Of course, this assumption is validonly if the additional processing at the receiver is done perfectly and the side informationneeded is transmitted without error. We will then only compare the method through CCDFof the PAR, which represents the probability that an OFDM symbol has a PAR greater thanthe one �xed by the transmission requirements.The input constellation used for the simulations has been chosen to be 16-QAM, but nosubstantial di�erence in the results has been found with 64-QAM or 16-PSK constellations.All the results are presented for OFDM transmission over N = 256 carriers. We will alsoaddress the problem that the methods are performed in the discrete domain although thee�ective PAR should be measured in the continuous domain.remarkWe must keep in mind that transforming the OFDM symbol may change its average power in away that cannot be found in closed form. That is the case for the TR method in which we adddeterministic values to the OFDM symbol, the optimization of which are symbol-dependent. In15

our simulations, we must then estimate IE hjxnj2i by its time average value over the OFDMsamples. The PAR that we wish to calculate is then:dPAR(x) = maxn20:::N�1 jxnj21N PN�1n=0 jxnj2We will moreover indicate, when necessary, the amount of additional power that has been usedto send the transformed OFDM symbols. Denoting the modi�ed OFDM block by ~x, we take asde�nitions:�P = IE hj~xnj2 � jxnj2iIE hjxnj2i 100% �PdB = 10 log100@IE hj~xnj2iIE hjxnj2i1AFor the TI method, knowing the heuristical constraints that we include in the optimization stepand the size of the original constellation, the average increased power per OFDM symbol canbe calculated using the formula (16), as for the PTS method, there is no power increase.5.1 Comparison in terms of redundancyBecause of the exponential complexity required by the PTS method, we have decided to�x an amount of redundancy based on a reasonable time computation for that method. It isunderstud that we mean reasonable complexity for our comparative study and not for practicalimplementation.� Partial Transmit SequencesAs pointed out before, the redundancy needed to recover the data is (U � 1) log2(W )where U is the number of sub-blocks andW the size of the angle alphabet. Regardless ofthe complexity, we may choose U = 6 andW = 6 ('u = 2k�6 ). We take the ratio betweenthe number of bits added over the total number of bits that represents information as ameasure of the percentage of redundancy:R = nb: bits addednb: bits info: � 100% = 12:924N = 1:26% (17)if we transmit QAM � 16 symbols over N carriers.remarkNote that in this case, the values of the angles are crucial to recover the data at the re-ceiver, and this information should be sent safely, that is with a powerful error correctingcode. This would raise the redundancy just derived.� Tone ReservationUsing the same measure of redundancy, if we use L = 3 carriers to carry codes, theredundancy is: R = nb: bits addednb: bits info: � 100% = LN � L = 1:18% (18)16

This represents the redundancy if we assume that the mean power of the codes is thesame as the 16-QAM mean power of the constellation. This is of course not true and theactual power per sub-channel is generally (from the simulations) greater for the codesthan for the data. We have decided not to take this increased power into account in ourdetermination of the redundancy.� Tone InjectionAs pointed out before, there is no redundancy needed in this method. For comparisonpurposes, we can introduce an equivalent redundancy corresponding to the power increasebrought by the shifted symbols. Considering the computational restrictions pointed outin section 4.2.3, for a QAM-16, only 16 candidate symbols are added to the originalconstellation points. Lets assume that instead of shifting symbols before the IDFT,we wish to transmit information mapped in this expanded constellation. That wouldrequire only 1 additionnal bit to code a shifted candidate symbol. If we consider L = 13shifts, we arrive at R = nb: bits addednb: bits info: � 100% = L4N = 1:27% (19)The results are shown on �gure 7. The increased power for the TR method has been found�P = 14% (�PdB = 0:55 dB), as for the TI method it is �P = 8:13% (�PdB = 0:34 dB). Itappears that the best method for reducing the PAR allowing a certain amount of redundancyis the Tone Injection. This was expected because no redundancy is actually required for thismethod and the power increase is reasonable, being even less than for the Tone Reservation.As mentionned earlier, the two main drawbacks of this latter method are the large amountof redundancy needed to achieve good PAR reduction, and the power increase that resultsfrom the codes - based on the fact that they don't come from a �nite alphabet. Those pointsare veri�ed by the simulations. The odd behaviour of the TI results at probabilities below10�2 comes most probably from the sub-optimality of the considered method. In fact, whendeciding the shifts, we �x too many constraints in order to reduce the complexity.5.2 Comparison in terms of complexityWe will in this section take into account the complexity at the transmitter. It is useless tocompare the complexities of the optimum versions of the methods, because they cannot beimplemented. The derivation of the actual complexity of each method is then rather di�cultbecause of the large number of constraints introduced. Hence, we have decided to comparethe methods in terms of their computation time during the simulation. We are aware thatthis is an heuristic point of view because it depends on the programs we have written, whichhave not really been optimized for computation time. This study however gives indicationsabout the PAR reduction that can be achieved with a given complexity.We have indicated in table 1 the time in seconds that had been necessary to compute10000 PAR reduction in each method (the programs have been written in MATLAB). Theresults in terms of CCDF are reported on �gure 8. The parameters of each method have beenchosen as:� Partial Transmit SequencesU = 5 sub-blocks and W = 4 angles ('u 2 f0; �2 ; �; 3�2 g).17

3 4 5 6 7 8 9 10 11 1210

−4

10−3

10−2

10−1

100

PAR0

Pro

b(P

AR

(x)>

PA

R 0)

originalTR : L=3PTS : U=6 W=6TI : L=13

Figure 7. Comparison of the 3 presented PAR reduction schemes with the same amount of redundancy.� Tone ReservationNote that with the iterative algorithm proposed in section 4.1, the complexity doesnot depend on the number of reserved tones, which is not the case for the optimumestimation done with the Linear Programming. For a fair comparison, we have thendecided to compare the sub-optimum implementations of the two other methods withthe optimum implementation of Tone Reservation. The number of reserved tones hasbeen �xed to L = 13.� Tone InjectionBecause of the high complexity of this method, only L = 3 shifts have been allowedper OFDM symbol. Note that if we had dealt with real valued OFDM symbols (ADSLcase), we would have consider almost 4 times more shifts.PTS TR TITimecomputation 3341 3423 4304Table 1. Time computation necessary for each PAR reduction method. The time is indicated inseconds.The additional mean power for the TR method has been estimated to �P = 11:5%(�PdB = 0:47 dB), and for the TI method, the formula (16) gives �P = 1:87% (�PdB =0:08 dB). Clearly, on can see that the Tone Reservation method outperforms the two others,18

which was expected because of the very small complexity of this method. The PAR reduc-tion achievement appears really good, but this result should be mitigated by the amount ofredundancy and power added. The performance loss of the TI method - compared to �gure7 - is simply amazing: this emphasizes the fact that one should �nd more relevant methodto choose the shifts in the expanded constellation, both less computationaly intensive andless restrictive. The method of Partial Transmit Sequences is the one which exhibits the lesschanges between �gure 7 and �gure 8.

3 4 5 6 7 8 9 10 11 12

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10−2

10−1

100

PAR0

Pro

b(P

AR

(x)>

PA

R 0)

originalTR : L=13PTS : U=5 W=4TI : L=3

Figure 8. Comparison of the 3 PAR reduction schemes with the same complexity at the transmitter.5.3 Continuous PAR reduction versus discrete PAR reductionAs already discussed in this report and in [Tell98, Tell98b], the Peak to Average Ratio valueis important at the input of the ampli�er, which lies in the analog part of the transmitter. Infact, the value of the PAR in the analog domain (Continuous PAR: C-PAR) is larger than itscorresponding value in the discrete domain (Discrete PAR: D-PAR). More importantly, themethods described here rely on criteria that have been considered for the D-PAR, and theiractual ability to reduce the C-PAR might be discutable. In [Tell98], the author introduces acompetitive optimization criterion for the PTS method, based on the minimization of aperiodiccorrelation between sub-blocks. It is shown that the use of this criterion achieves more C-PAR reduction despite the fact that less D-PAR reduction has been found. However, thecomputational complexity of this new criterion is even larger than the one proposed by M�uller& al., and its practical implementation remains a challenge. In [Tell98b], the authors adviceusing their methods at a higher rate (2 times the Nyquist rate) to increase the performance19

on the C-PAR while the achievable D-PAR reduction remains unchanged. This is naturallyat the cost of increased complexity at the transmitter.The optimization of each method regarding the C-PAR is not the purpose of this report,and we have decided to simply quantify the loss of C-PAR reduction when D-PAR criteria areused. We have drawn, in dotted lines, on �gure 9 the results obtained by the three methodsin the con�guration of �gure 7, and their corresponding Continuous PAR reduction in solidlines. The continuous PAR has been simulated with an oversampling by a factor of 4 at theoutput of the parallel to serial converter.

4 5 6 7 8 9 10 11 12 1310

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D−PAR: originalC−PAR: originalTR: L=3PTS: U=6 W=6TI: L=13

Figure 9. Comparison of PAR reduction for the 3 methods. The dotted lines represent digital measureof the PAR and the solid lines represent analog (over-sampled) measure of the PAR.We can see that whereas the PAR of an OFDM symbol without coding increases only byless than 0:5dB when measuring the continuous PAR instead of the discrete PAR, the e�ecton the proposed techniques is much more disastrous. Each method performs worse on C-PARreduction by more than 3dB compared to the D-PAR reduction. This shows the necessity ofconsidering techniques that use the C-PAR as a criterion rather than the D-PAR. A lot ofwork is still to be done, either to adapt the existing techniques, or to �nd other criteria andmethods based on the analog OFDM signal.remarkThe shape of the linear pulse �lter should also be considered since this �lter operates beforethe nonlinear ampli�er. We have not considered its e�ect in our simulations.20

6 ConclusionIn this repotr, we have compared 3 di�erent methods for reducing the PAR of OFDM/DMTmodulated transmissions. Reducing the Peak to Average Ratio at the transmitter allow thenonlinear ampli�ers to operate at a higher input backo� and/or reduce the distorsions thatcould introduce inter-carrier interference and thereby increase the symbol error rate at thereceiver. Other techniques based on a model for the nonlinearity describing the ampli�ers willbe reported in a future contribution [Decl99b].After recalling the main features of OFDM/DMT modulator and demodulator, we focusedon the study of the Peak to Average Ratio, and showed the e�ect of the clipping noise generatedby the high power ampli�er to the symbol error rate at the receiver by giving the closed formof the SNR in presence of this clipping noise.Three methods aiming to reduce the PAR in OFDM transmissions have then been pre-sented: the Tone Reservation (TR), the Partial Transmit Sequences (PTS) and the ToneInjection (TI).They have been compared through thorough simulations. We have compared the 3 methodswhen the same amount of (i) redundancy and (ii) complexity was allowed. The best method interms of complexity is TR whereas the best in terms of redundancy is TI - which corroboratethe previous remarks. As a �nal conclusion we will try to give our opinion on the di�erentcapabilities of each method.� The TR method has the only advantage of a very low complexity at the transmitter.The extra redundancy and average power required for large PAR reduction is too majora drawback for that method to be advised. One can limit those increases by selectingonly a small number of code-tones and adding a power constraint to the optimizationcriterion, but this would result in a lower PAR reduction.� Regarding the PTS method, it has been shown, in the cited papers, that we nearlyachieve the maximum PAR reduction by selecting only a few blocks and taking only afew allowed values for the rotation angles. Hence, if we take a look at the results, thismethod seems to be limited compared to the 2 others. However, the restriction to a�nite alphabet for the possible values of the angles could be a too restrictive assumptionto optimize the angle values.� The TI method is very promising, mainly because it requires no side information, areasonable increased average power and a simple decoding technique. The only problemis the complexity at the transmitter. This complexity can become acceptable when smallinput constellations are considered, and when transmitting along the ADSL channel,the real valued property of the OFDM ow simplifying the choice of the shifts. Thegood results found should encourage further investigation to make this scheme easier toimplement for larger constellations and bandpass applications.References[Aue98] V. Aue and G.P. Fettweis, A comparison of the Performance of Linearly Equal-ized Single Carrier and Coded OFDM over Frequency Selective Fading Channelsusing the Random Coding Technique, In ICC-98, Atlanta, GA, June 1998.21

[Bauml96] R. Fisher R. B�auml and J. Huber, Reducing the Peak-to-Average PowerRatio of Multicarrier Modulation by Selected Mapping, Elec. Letters, 32(22)pp.2056-2057, Oct. 1996.[Bing90] J.A. Bingham, Multicarrier Modulation for Data Tranmission: an Idea whose Timehas come, IEEE comm. mag., pages 5-22, May 1990.[Decl99b] D. Declercq and G. Giannakis, Mitigation of Nonlinear Distorsions due toHigh Power Ampli�ers, Technical report, SPINCOM, univ. Minnesota, USA., 1999.[Gath97] A. Gatherer and M. Polley, Controlling Clipping Probability in DMT Trans-mission, In ASILOMAR-97, Paci�c Grove, CA, 1997.[Ksch98] A. Narula F. Kschischang and V. Eyuboglu, A New Approach to PAR Con-trol in DMT Systems, ITU, Q4/15(NF-083) pp.1-5, May 1998.[Mull97] S.H. M�uller and J.B. Huber, A Comparison of Peak Reduction Schemes forOFDM, In GLOBECOM-97, Phoenix, AZ, 1997.[Mull97b] S.H. M�uller and J.B. Huber, A novel Peak Power Reduction Scheme forOFDM, In PIMRC-97, pages 1090-1094, Helsinki, Finland, 1997.[Paul98] M. Pauli and H.P. Kuchenbecker, On the Reduction of Out-of-Band Radiationof OFDM Signals, In ICC-98, pages 1304-1308, Atlanta, GA, June 1998.[Rein94] C. Reiners and H. Rohling, Multicarrier Transmission Technique in CellularMobile Communication Systems, In IEEE Vech. Tech. Conf., pages 1645-1649, 1994.[Salt98] B.R. Saltzberg, Comparison of Single-Carrier and Multitone Digital Modulationfor ADSL Applications, IEEE comm. mag., pages 114-121, Nov 1998.[Sari95] G. Karam H. Sari and I. Jeanclaude, Transmission Techniques for DigitalTerrestrial TV Broadcasting, IEEE comm. mag., 33(2) pp.100-109, Feb 1995.[Tell97] C. Tellambura, Upper Bound on the Peak Factor of N-Multiple Carriers, Elec.Letters, 33 pp.1608-1609, Sept. 1997.[Tell98b] J. Tellado and J.M. Cioffi, Further Results on Peak-to-Average Ratio Reduc-tion, Technical report, ISL, Stanford univ., 1998.[Tell98a] J. Tellado and J.M. Cioffi, Peak Power Reduction for Multicarrier Transmis-sion, In GLOBECOM-98, Sydney, Autralia, Nov 1998.[Tell98] C. Tellambura, A coding Technique for Reducing Peak-to-Average Power Ratioin OFDM, In GLOBECOM-98, pages 2783-2786, Sydney, Autralia, Nov 1998.[Wang66] R.W. Wang, High Speed Multichannel Data Transmission with Bandlimited Or-thogonal Signals, Bell Syst. Tech Journal, 45 pp.1775-1796, Dec 1966.22