IEEE TRANSACTIONS ON BROADCASTING, VOL. 54, NO. 2, JUNE 2008 257

An Overview: Peak-to-Average Power RatioReduction Techniques for OFDM Signals

Tao Jiang, Member, IEEE, and Yiyan Wu, Fellow, IEEE

Abstract—One of the challenging issues for OrthogonalFrequency Division Multiplexing (OFDM) system is its highPeak-to-Average Power Ratio (PAPR). In this paper, we reviewand analysis different OFDM PAPR reduction techniques, basedon computational complexity, bandwidth expansion, spectralspillage and performance. We also discuss some methods of PAPRreduction for multiuser OFDM broadband communication sys-tems.

Index Terms—Complementary cumulative distribution function(CCDF), high power amplifier (HPA), multiuser OFDM, OFDM,peak-to-average power ratio (PAPR).

I. INTRODUCTION

AS AN attractive technology for wireless communications,Orthogonal Frequency Division Multiplexing (OFDM),

which is one of multi-carrier modulation (MCM) techniques,offers a considerable high spectral efficiency, multipath delayspread tolerance, immunity to the frequency selective fadingchannels and power efficiency [1], [2]. As a result, OFDM hasbeen chosen for high data rate communications and has beenwidely deployed in many wireless communication standardssuch as Digital Video Broadcasting (DVB) and based mobileworldwide interoperability for microwave access (mobileWiMAX) based on OFDM access technology [3].

However, still some challenging issues remain unresolved inthe design of the OFDM systems. One of the major problemsis high Peak-to-Average Power Ratio (PAPR) of transmittedOFDM signals. Therefore, the OFDM receiver’s detection effi-ciency is very sensitive to the nonlinear devices used in its signalprocessing loop, such as Digital-to-Analog Converter (DAC)and High Power Amplifier (HPA), which may severely impairsystem performance due to induced spectral regrowth and de-tection efficiency degradation. For example, most radio systemsemploy the HPA in the transmitter to obtain sufficient transmitspower and the HPA is usually operated at or near the satura-tion region to achieve the maximum output power efficiency,and thus the memory-less nonlinear distortion due to high PAPRof the input signals will be introduced into the communicationchannels. If the HPA is not operated in linear region with largepower back-off, it is impossible to keep the out-of-band power

Manuscript received August 13, 2007; revised December 3, 2007.T. Jiang is with the Department of Electronic and Information Engineering,

Huazhong University of Science and Technology, Wuhan 430074, P. R. China(e-mail: Tao.Jiang@ieee.org).

Y. Wu is with the Communications Research Center, Ottawa, ON K2H 8S2,Canada (e-mail: yiyan.wu@crc.ca).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TBC.2008.915770

below the specified limits. This situation leads to very inefficientamplification and expensive transmitters.

Therefore, it is important and necessary to research on thecharacteristics of the PAPR including its distribution and reduc-tion in OFDM systems, in order to utilize the technical featuresof the OFDM.

As one of characteristics of the PAPR, the distributionof PAPR, which bears stochastic characteristics in OFDMsystems, often can be expressed in terms of ComplementaryCumulative Distribution Function (CCDF). Recently, someresearchers have reported on determination of the PAPR distri-bution based on different theoretics and hypotheses [4]–[10].Moreover, various approaches also have been proposed toreduce the PAPR including clipping [11]–[14], coding schemes[15]–[21], phase optimization [22], [23], nonlinear compandingtransforms [24]–[29], Tone Reservation (TR) and Tone Injec-tion (TI) [30], [31], constellation shaping [32]–[34], PartialTransmission Sequence (PTS) and Selective Mapping (SLM)[35]–[51] and other techniques such as pre-scrambles proposedin [52]. These schemes can mainly be categorized into signalscrambling techniques, such as block codes and PTS etc., andsignal distortion techniques such as clipping.

Although some techniques of PAPR reduction have beensummarized in [53], it is still indeed needed to give a compre-hensive review including some motivations of PAPR reductions,such as power saving, and to compare some typical methodsof PAPR reduction through theoretical analysis and simulationresults directly. An effective PAPR reduction technique shouldbe given the best tradeoff between the capacity of PAPR re-duction and transmission power, data rate loss, implementationcomplexity and Bit-Error-Ratio (BER) performance etc.

In this paper, firstly we investigate the distribution of PAPRbased on the characteristics of the OFDM signals. Then, we an-alyze five typical techniques of PAPR reduction and propose thecriteria of PAPR reduction in OFDM systems in details. Finally,we briefly discuss the issue of PAPR in some broadband com-munication systems correlative with OFDM technology, such asmultiuser OFDM systems.

II. CHARACTERISTICS OF OFDM SIGNALS

Let a block of symbolsis formed with each symbol modulating one of a set of subcar-riers , where is the number of sub-carriers. The subcarriers are chosen to be orthogonal, thatis, , where and is the originalsymbol period. Therefore, the complex envelope of the trans-mitted OFDM signals can be written as

(1)

0018-9316/$25.00 © 2008 IEEE

258 IEEE TRANSACTIONS ON BROADCASTING, VOL. 54, NO. 2, JUNE 2008

Fig. 1. Distribution of PAPR of OFDM signal samples oversampled by different L.

where .Suppose that the input data streams is statistically inde-

pendent and identically distributed (i.i.d.), i.e. the real partand imaginary part are uncorrelated and

orthogonal. Therefore, based on the central limit theorem, whenis considerably large, the distribution of both and

approaches Gaussian distribution with zero meanand variance , where

is the expected value of [27]. In other words, OFDMsignals with large become Gaussian distributed with Proba-bility Density Function (PDF) as [25]

(2)

where is the variance of .Moreover, the Rayleigh nature of original OFDM signals’

amplitude can be gotten and its PDF can be expressed as [30]

(3)

where is the amplitude of OFDM signals.

III. DEFINITION OF PAPR

A. Baseband PAPR

1) Continuous-time PAPRIn general, the PAPR of OFDM signals is defined asthe ratio between the maximum instantaneous power andits average power

(4)

where is the average power of and it can be com-puted in the frequency domain because Inverse Fast FourierTransform (IFFT) is a (scaled) unitary transformation.

2) Discrete-time PAPRThe PAPR of the discrete time sequences typically deter-mines the complexity of the digital circuitry in terms ofthe number of bits necessary to achieve a desired signalto quantization noise for both the digital operation andthe DAC. However, we are often more concerned with re-ducing the PAPR of the continuous-time signals in prac-tice, since the cost and power dissipation of the analogcomponents often dominate.To better approximate the PAPR of continuous-timeOFDM signals, the OFDM signals samples are obtainedby times oversampling. -times oversampled time-do-main samples are -point IFFT of the data block with

zero-padding. Therefore, the oversampled IFFToutput can be expressed as

(5)

Fig. 1 shows the distribution of the PAPR of the OFDMsignals with and . As shown, thelargest PAPR increase happens from to .However, the PAPR does not increase significantly after

. It has shown that is sufficient to get accuratePAPR results [30]. The PAPR computed from the -timesoversampled time domain OFDM signal samples can bedefined as

(6)

where denotes the expectation operator.

JIANG AND WU: OVERVIEW OF PEAK-TO-AVERAGE POWER RATIO REDUCTION TECHNIQUES FOR OFDM SIGNALS 259

B. Passband PAPR

Note that, if is large, an OFDM system usually does notemploy pulse shaping, since the power spectral density of theband-limited OFDM signal is approximately rectangular. Thus,the amplitude of OFDM RF signals can be expressed as

(7)

where is the carrier frequency and . Therefore, thepeak of RF signals is equivalent to that of the complex basebandsignals.

Moreover, the average power of the passband signal is

(8)

Therefore, the passband PAPR is approximately twice thebaseband PAPR, i.e.

(9)

In this paper, we only consider the PAPR of the basebandOFDM signals.

IV. MOTIVATION OF PAPR REDUCTION

A. Nonlinear Characteristics of HPA and ADC

Most radio systems employ the HPA in the transmitterto obtain sufficient transmission power. For the proposed ofachieving the maximum output power efficiency, the HPA isusually operated at or near the saturation region. Moreover,the nonlinear characteristic of the HPA is very sensitive to thevariation in signal amplitudes.

However, the variation of OFDM signal amplitudes isvery wide with high PAPR. Therefore, HPA will introduceinter-modulation between the different subcarriers and in-troduce additional interference into the systems due to highPAPR of OFDM signals. This additional interference leadsto an increase in BER. In order to lessen the signal distortionand keep a low BER, it requires a linear work in its linearamplifier region with a large dynamic range. However, thislinear amplifier has poor efficiency and is so expensive. Powerefficiency is very necessary in wireless communication as itprovides adequate area coverage, saves power consumptionand allows small size terminals etc. It is therefore important toaim at a power efficient operation of the non-linear HPA withlow back-off values and try to provide possible solutions to theinterference problem brought about. Hence, a better solution isto try to prevent the occurrence of such interference by reducingthe PAPR of the transmitted signal with some manipulations ofthe OFDM signal itself.

Large PAPR also demands the DAC with enough dynamicrange to accommodate the large peaks of the OFDM signals.

Although, a high precision DAC supports high PAPR with areasonable amount of quantization noise, but it might be veryexpensive for a given sampling rate of the system. Whereas,a low-precision DAC would be cheaper, but its quantizationnoise will be significant, and as a result it reduces the signalSignal-to-Noise Ratio (SNR) when the dynamic range of DACis increased to support high PAPR. Furthermore, OFDM sig-nals show Gaussian distribution for large number of subcarriers,which means the peak signal quite rarely occur and uniformquantization by the ADCs is not desirable. If clipped, it will in-troduce in band distortion and out-of-band radiation (adjacentchannel interference) into the communication systems.

Therefore, the best solution is to reduce the PAPR beforeOFDM signals are transmitted into nonlinear HPA and DAC.

B. Power Saving

When a HPA have a high dynamic range, it exhibits poorpower efficiency. It has been shown that PAPR reduction cansignificantly save the power, in which the net power saving isdirectly proportional to the desired average output power and itis highly dependent upon the clipping probability level [54].

Suppose that an ideal linear model for HPA, where linear am-plification is achieved up to the saturation point, and thus weobtain

(10)

where is the HPA efficiency and it is defined as, where is the average of the output

power and is the constant amount of power regardless ofthe input power.

To illustrate the power inefficiency of a HPA in terms of thePAPR, we give an example of OFDM signals with 256 subcar-riers and its CCDF has been shown in Fig. 1. In order to guar-antee that probability of the clipped OFDM frames is less than0.01%, we need to apply an input backoff (IBO) equivalent tothe PAPR at the probability level, i.e.( 25.235), referring to Fig. 1, and thus the efficiency of HPA be-comes . Therefore, so low efficiencyis a strong motivation to reduce the PAPR in OFDM systems.

V. DISTRIBUTION OF THE PAPR IN OFDM SYSTEMS

It is known that the CCDF of PAPR can be used to estimatethe bounds for the minimum number of redundancy bits re-quired to identify the PAPR sequences and evaluate the perfor-mance of any PAPR reduction schemes. We can also determinea proper output back-off of HPA to minimize the total degrada-tion according to CCDF. Moreover, we can directly apply dis-tribution of PAPR to calculate the BER and estimate achievableinformation rates. In practice, we usually adjust these designparameters jointly according to simulation results. Therefore, ifwe can use an analytical expression to accurately calculate thePAPR distribution for OFDM systems, it can greatly simplifythe system design process. Therefore, it is of great importanceto accurately identify PAPR distribution in OFDM systems.

Recently, some upper and lower bounds of the PAPR, whichis based on the Rayleigh distribution and Nyquist sampling rate,have been derived. In the OFDM system with M-Phase-Shift-Keying (MPSK) modulation, signal constellation has the same

260 IEEE TRANSACTIONS ON BROADCASTING, VOL. 54, NO. 2, JUNE 2008

amplitude level, and thus the power of each subcarrier is con-stant. Therefore, the PAPR of an MPSK-OFDM signal can beexpressed as [30]

(11)

However, for the OFDM system with square M-QuadratureAmplitude Modulation (MQAM), signal constellation hasvarying signal power levels over different constellation points.When all the subcarriers have the same phase, the maximum ofPAPR occurs. Therefore, according to the conclusion of [55],the upper bound of PAPR in MQAM-OFDM systems can bederived out

(12)

For a relatively large , the lower and upper bounds of thedistribution of the PAPR have been proposed in [7], whichwere developed based on the previous works in conjunctionwith some approximations and parameters obtained throughsimulations. In [8], some bounds analysis has also been de-veloped for both independent and dependent subcarriers inOFDM systems. For independent subcarriers, a generic path forbounding practical constellations was used and discussed. Fordependent subcarriers, some theoretical bounds of distributionsof the PAPR have been obtained in terms of the Euclidiandistance distributions, in which the focus was mainly on binarycodes, such as Bose-Chaudhuri-Hocquenghem (BCH) codes.However, the lower and upper bounds can offer little help incharacterizing the distribution of the PAPR in practical OFDMsystems. In fact, the accurate statistical distribution of the PAPRfor generic OFDM system is what we want.

When the number of the subcarriers is relatively small, theCCDF expression of the PAPR of OFDM signals can be writtenas [4]

(13)

However, (13) does not fit well in OFDM systems with a verylarge [4]. In [5], an empirical approximation expression of theCCDF of the PAPR in OFDM systems has been given as

(14)

It should be noted that (14) lacks theoretical justification andalso yields some discrepancies with the simulation results forlarge , which has been proven in [6].

In [6], an analytical PAPR CCDF expression has been de-veloped, which is based on the level-crossing rate approxima-tion of the peak distribution along with the exact distribution,since the envelope of an OFDM signal can always be consid-ered as an asymptotically Gaussian process in a band-limitedOFDM system. In fact, the theoretical results obtained in [6]were based on the conditional probability of the peak distribu-tion of the OFDM signals when the reference level is given.When the constraint provides a lower bound of , theeffect on the accuracy of the PAPR distribution can be numer-ically evaluated. Indeed, for high , the conditional probabilitythat the peak of the OFDM signals exceeds may be very small.

In this case, the expression of the PAPR CCDF, as shown in [6],can be simplified as

(15)

The approximation of (15) can be made relatively accuratefor a relatively large number of subcarriers by appropriatelyadjusting the reference level of the PAPR. If the range of thePAPR of interest is great, the distribution can be further simpli-fied without loss of the accuracy. In [6], it also has been shownthat the statistical distribution of the PAPR of the OFDM signalsis not so sensitive to the increase of the number of subcarriers.

In coded OFDM systems, it has been proven that the com-plex envelope of the coded OFDM signals can converge weaklyto a Gaussian random process if the number of subcarriers goesto infinity [9]. In [9], a simple approximation of the CCDFof PAPR has been developed by employing the extreme valuetheory, and the expression can be written as

(16)

However, all the above mentioned the expressions of theCCDF have not been considered power distribution strategy inOFDM systems. Similarly, with the help of the extreme valuetheory for Chi-squared-2 process, a more accurate analyticalexpression of the CCDF of PAPR for adaptive OFDM systemswith unequal power allocation to subcarriers has been derivedin [10]

(17)where if the subcarrier at DC is inactive, oth-erwise if the subcarrier at DC is active,in which denotes the number of active subcarriers (car-rying information) and denotes the number of inactivesubcarriers (idle). If the subcarrier at DC is nonzero, it is activesubcarrier; otherwise it is inactive subcarrier. Thus, the numberof the subcarriers should be equal to the sum of and

. denotes the transmission power allocated to the-th subcarrier.

VI. PAPR REDUCTION TECHNIQUES IN OFDM SYSTEMS

In this section, we mainly discuss five typical techniques forPAPR reduction in OFDM systems.

A. Clipping and Filtering

The simplest and most widely used technique of PAPR reduc-tion is to basically clip the parts of the signals that are outsidethe allowed region [11]. For example, using HPA with saturationlevel below the signal span will automatically cause the signalto be clipped. For amplitude clipping, that is

(18)

where is preset clipping level and it is a positive real number.

JIANG AND WU: OVERVIEW OF PEAK-TO-AVERAGE POWER RATIO REDUCTION TECHNIQUES FOR OFDM SIGNALS 261

Generally, clipping is performed at the transmitter. However,the receiver need to estimate the clipping that has occurred andto compensate the received OFDM symbol accordingly. Typi-cally, at most one clipping occurs per OFDM symbol, and thusthe receiver has to estimate two parameters: location and size ofthe clip. However, it is difficult to get these information. There-fore, clipping method introduces both in band distortion andout of band radiation into OFDM signals, which degrades thesystem performance including BER and spectral efficiency.

Filtering can reduce out of band radiation after clipping al-though it can not reduce in-band distortion. However, clippingmay cause some peak regrowth so that the signal after clippingand filtering will exceed the clipping level at some points. To re-duce peak regrowth, a repeated clipping-and-filtering operationcan be used to obtain a desirable PAPR at a cost of computa-tional complexity increase.

As improved clipping methods, peak windowing schemes at-tempt to minimize the out of band radiation by using narrow-band windows such as Gaussian window to attenuate peak sig-nals.

B. Coding Schemes

When signals are added with the same phase, they producea peak power, which is times the average power. Of course,not all code words result in a bad PAPR. Therefore, the goodPAPR reduction can be obtain when some measures are takento reduce the occurrence probability of the same phase of thesignals, which is the key idea of the coding schemes.

A simple block coding scheme was introduced by Jones etal.[15], and its basic idea is that mapping 3 bits data into 4 bitscodeword by adding a Simple Odd Parity Code (SOBC) at thelast bit across the channels. The main disadvantage of SOBCmethod is that it can reduce PAPR for a 4-bit codeword. Later,Wulich applied the Cyclic Coding (CC) to reduce the PAPR[56]. In 1998, Fragiacomo proposed an efficient Simple BlockCode (SBC) to reduce the PAPR of OFDM signals [57]. How-ever, it is concluded that SBC is not effective when the framesize is large. Subsequently, Complement Block Coding (CBC)and Modified Complement Block Coding (MCBC) schemeswere proposed to reduce the PAPR without the restriction offrame size [20], [58]. CBC and MCBC are more attractive dueto their flexibility on choosing the coding rate, frame size andlow implementation complexity. CBC and MCBC utilize thecomplementary bits that are added to the original informationbits to reduce the probability of the peak signals occurrence.

To make comparisons, some results of the PAPR reductionobtained with different coding schemes have been shown inTable I, in which the number of subblock is 2 and the codingrate for MCBC. About 3-dB PAPR reduction can beobtained when coding rate by using CBCwith long frame size. It is also shown that the PAPR reductionsobtained with CBC when coding rate arealmost the same as that when . In addition,when coding rate is 3/4, more than 3-dB more PAPR reductioncan be obtained using MCBC than the other schemes with anyframe size. The flexibility in coding rate choice and low com-plexity makes the proposed CBC and MCBC schemes attractivefor OFDM systems with large frame sizes and high coding rates.

TABLE IPAPR REDUCTION COMPARISON WITH DIFFERENT CODING SCHEMES

In [59], [60], [62], authors used the Golay complementarysequences to achieve the PAPR reduction, in which more than3-dB PAPR reduction had been obtained. Codes with error cor-recting capabilities has been proposed in [61] to achieve morelower PAPR for OFDM signals by determining the relationshipof the cosets of Reed-Muller codes to Golay complementary se-quences. While these block codes reduce PAPR, they also re-duce the transmission rate, significantly for OFDM systems withlarge number of subcarriers.

In fact, let be a code defined over an equal energy con-stellation, denotes the rate and denotes the length of the

, respectively, then has possible codewords. There-fore, it is possible to compute the codewords with large PAPRby trying all the codewords of and computing the peaks ofthe corresponding signals at some selected time points. How-ever, it is little hope for computing the PAPR of an arbitrarycode when is large. Even if it is possible, the complexity isstill too high. Based on this motivates, authors of [22] proposeda novel method of computation and reduction of the PAPR andit mainly introduced a specific phase shift to each coordinate ofall possible codewords where phase shifts are independent of thecodewords and known both to transceiver, then it can be freelyobtained more 4.5-dB PAPR reduction by using the optimizedphase shifts. From this viewpoint, we also consider the codingscheme of PAPR reduction as a special phase optimization.

In summarization, the inherent error control capabilityand simplicity of implementation make coding method morepromising for practical OFDM systems design. However, themain disadvantage of this method is the good performance ofthe PAPR reduction at the cost of coding rate loss.

C. PTS and SLM

In a typical OFDM system with PTS approach to reduce thePAPR, the input data block in is partitioned into disjointsubblocks, which are represented by the vectors

[35] as shown in Fig. 2. Therefore, we can get

(19)

where with or 0. In general, for PTS scheme, the known

subblock partitioning methods can be classified into three cate-gories [35]: adjacent partition, interleaved partition and pseudo-

262 IEEE TRANSACTIONS ON BROADCASTING, VOL. 54, NO. 2, JUNE 2008

Fig. 2. Block diagram of PTS technique.

Fig. 3. Block diagram of SLM technique.

random partition. Then, the subblocks are transformedinto time-domain partial transmit sequences

(20)

These partial sequences are independently rotated by phasefactors . The objectiveis to optimally combine the subblocks to obtain the time-domain OFDM signals with the lowest PAPR

(21)

Therefore, there are two important issues should be solved inPTS: high computational complexity for searching the optimalphase factors and the overhead of the optimal phase factors asside information needed to be transmitted to receiver for thecorrect decoding of the transmitted bit sequence.

Suppose that there are phase angles to be allowed, thuscan has the possibility of different values. Therefore, thereare alternative representations for an OFDM symbol. Toreduce the searching complexity and avoid/reduce the usage ofside information, many extensions of PTS have been developedrecently [63]–[67]. In [66], authors proposed a novel scheme,which is based on a nonlinear optimization approach named assimulated annealing, to search the optimal combination ofphase factors with low complexity. In general, PTS needs

IFFT operations for each data block, and the number of the re-quired side information bits is , where denotesthe smallest integer that does not exceed .

Similarly, in SLM, the input data sequences are multipliedby each of the phase sequences to generate alternative inputsymbol sequences. Each of these alternative input data se-quences is made the IFFT operation, and then the one with thelowest PAPR is selected for transmission [51]. A block diagramof the SLM technique is depicted in Fig. 3. Each data blockis multiplied by different phase factors, each of length ,

, resultingin different data blocks. Thus, the vth phase sequence aftermultiplied is

. Therefore, OFDM signals becomes as

(22)

where , .Among the data blocks , only one

with the lowest PAPR is selected for transmission and the corre-sponding selected phase factors also should be transmittedto receiver as side information. For implementation of SLMOFDM systems, the SLM technique needs IFFT operationand the number of required bits as side information isfor each data block. Therefore, the ability of PAPR reduction inSLM depends on the number of phase factors and the designof the phase factors. Some extension of SLM also have beenproposed to reduce the computational complexity and numberof the bits for side information transmission [36]. For example,

JIANG AND WU: OVERVIEW OF PEAK-TO-AVERAGE POWER RATIO REDUCTION TECHNIQUES FOR OFDM SIGNALS 263

Fig. 4. The spectrums of original OFDM signals and companded signals.

an SLM scheme without explicit side information was proposedin [68].

Although PTS and SLM are important probabilistic schemesfor PAPR reduction, it was already known that SLM can producemultiple time domain OFDM signals that are asymptoticallyindependent, whereas the alternative OFDM signals generatedby PTS are interdependent. PTS divides the frequency vectorinto some subblocks before applying the phase transformation.Therefore, some of the complexity of the serval full IFFT op-erations can be avoided in PTS, so that it is more advantageousthan SLM if the amount of computational complexity is limited[69]. Also it is demonstrated that the PAPR reduction in PTSperforms better than that of SLM. However, the required bits ofthe side information in PTS is larger than that of SLM.

D. Nonlinear Companding Transforms

One of the most attractive schemes is nonlinear compandingtransform due to its good system performance including PAPRreduction and BER, low implementation complexity and nobandwidth expansion.

The first nonlinear companding transform is the -law com-panding, which is based on the speech processing algorithm

-law, and it has shown better performance than that of clip-ping method [24]. -law mainly focuses on enlarging signalswith small amplitude and keeping peak signals unchanged, andthus it increase the average power of the transmitted signals andpossibly results in exceeding the saturation region of HPA tomake the system performance worse.

In fact, the nonlinear companding transform is also an espe-cial clipping scheme. The differences between the clipping andnonlinear companding transform can be summarized as: 1) Clip-ping method deliberately clips large signals when the amplitudeof the original OFDM signals is larger than the given threshold,and thus the clipped signals can not be recovered at the re-ceiver. However, nonlinear companding transforms compandoriginal OFDM signals using the strict monotone increasing

function. Therefore, the companded signals at the transmittercan be recovered correctly through the corresponding inversionof the nonlinear transform function at the receiver; 2) Nonlinearcompanding transforms enlarge the small signals while com-pressing the large signals to increase the immunity of smallsignals from noise, whereas clipping method does not changethe small signals. Therefore, clipping method suffers from threemajor problems: in-band distortion, out-of-band radiation andpeak regrowth after digital analog conversion. As a result, thesystem performance degradation due to the clipping may notbe optimistic. However, nonlinear companding transforms canoperate well with good BER performance while keeping goodPAPR reduction [70].

The design criteria of nonlinear companding transform hasalso been given in [70]. Since the distribution of the originalOFDM signals has been known, such as Rayleigh distributionof the OFDM amplitudes written in (3), we can obtain the non-linear companding transform function through theoretical anal-ysis and derivation according to the desirable distribution of thecompanded OFDM signals. For example, we transform the am-plitude of the original OFDM signals into the desirable distribu-tion with its PDF , . Therefore,the nonlinear transform function can be derived as

(23)

Obviously, this nonlinear companding transform of (23) be-longs to the exponential companding scheme. Based on thisdesign criteria, two types of nonlinear companding transform,which are based on error function and exponential function, re-spectively, have been proposed in [25], [27].

It is well-known that original OFDM signals have a verysharp, rectangular-like power spectrum as shown in Fig. 4. Thisgood property will be affected by the PAPR reduction schemes,e.g. slower spectrum roll-off, more spectrum side-lobes, andhigher adjacent channel interference. Many PAPR reduction

264 IEEE TRANSACTIONS ON BROADCASTING, VOL. 54, NO. 2, JUNE 2008

Fig. 5. Block diagram of TR/TI approaches for PAPR reduction.

schemes cause spectrum side-lobes generation, but the non-linear companding transforms cause less spectrum side-lobes.As seen in Fig. 4, error and exponential companding trans-forms have much less impact on the original power spectrumcomparing to the -law companding scheme. It is the majorreason that the error and exponential companding schemes notonly enlarge the small amplitude signals but also compress thelarge amplitude signals, while maintain the average power un-changed by properly choosing parameters, which can increasethe immunity of small amplitude signals from noise. However,the -law companding transform increases the average powerlevel and therefore requires a larger linear operation region inHPA.

Nonlinear companding transform is a type of nonlinearprocess that may lead to significant distortion and performanceloss by companding noise. Companding noise can be definedthat the noises are caused by the peak regrowth after DAC togenerate in-band distortion and out-band noise, by the excessivechannel noises magnified after inverse nonlinear compandingtransform etc. For out-of-band noise, it needs to be filtered andoversampled. For in-band distortion and channel noises magni-fied, they need to iterative estimation. Unlike Additive WhiteGaussian Noise (AWGN), companding noise is generated bya process known and that can be recreated at the receiver, andsubsequently be removed. In [28], the framework of an iterativereceiver has been proposed to eliminate commanding noise forcompanded and filtered OFDM system.

E. TR and TI

TR and TI are two efficient techniques to reduce the PAPR ofOFDM signals [30]. Fig. 5 describes the block diagram of TRand TI, in which the key idea is that both transmitter and receiverreserve a subset of tones for generating PAPR reduction signals. Note that these tones are not used for data transmission.

In TR, the objective is to find the time domain signal to beadded to the original time domain signal to reduce the PAPR.Let denote complex symbols fortone reservation at reserved tones. Thus, the data vector changesto after tone reservation processing, and this results in anew modulated OFDM signals as

(24)

where . Therefore, the main aim of the TR isto find out the proper to make the vector with low PAPR.To find the value of , we must solve a convex optimiza-tion problem that can easily be cast as a linear programmingproblem.

Similarly, TI also uses an additive correction to optimize in(24). The basic idea of TI is to extend the constellation and thus

the same data point corresponds to multiple possible constella-tion points. One option is to replicate the original shaded con-stellation into serval alternative ones. Therefore, is a transla-tion vector such that . Note that TI needs notrequire the extra side information and the receiver only needsto know how to map the redundant constellations on the orig-inal one. An alternative strategy is to move the constellationpoints by applying an FFT on the clipped time signals, and thesame operations are repeated till all the constellation points arewithin specified boundaries and the PAPR specification of thetime signal is satisfied [71]. Some modifications of TI have beenproposed to obtain good performance including PAPR reductionand low complexity [72].

The TI technique is more problematic than the TR techniquesince the injected signal occupies the frequency band as the in-formation bearing signals. Moreover, the alternative constella-tion points in TI technique have an increased energy and theimplementation complexity increases for the computation theoptimal translation vector.

VII. CRITERIA OF THE PAPR REDUCTION IN OFDM SYSTEMS

As above analyzed, we find most of existing solutions stillhave some drawbacks and the obvious one is the trade-off be-tween PAPR reduction and some factors such as bandwidth. Thecriteria of the PAPR reduction is to find the approach that it canreduce PAPR largely and at the same time it can keep the goodperformance in terms of the following factors as possible.

1) High capability of PAPR reduction: It is primary factor tobe considered in selecting the PAPR reduction techniquewith as few harmful side effects such as in-band distortionand out-of-band radiation.

2) Low average power: Although it also can reduce PAPRthrough average power of the original signals increase, itrequires a larger linear operation region in HPA and thusresulting in the degradation of BER performance.

3) Low implementation complexity: Generally, complexitytechniques exhibit better ability of PAPR reduction. How-ever, in practice, both time and hardware requirements forthe PAPR reduction should be minimal.

4) No bandwidth expansion: The bandwidth is a rare resourcein systems. The bandwidth expansion directly results inthe data code rate loss due to side information (such asthe phase factors in PTS and complementary bits in CBC).Moreover, when the side information are received in errorunless some ways of protection such as channel coding em-ployed. Therefore, when channel coding is used, the lossin data rate is increased further due to side information.Therefore, the loss in bandwidth due to side informationshould be avoided or at least be kept minimal.

5) No BER performance degradation: The aim of PAPR re-duction is to obtain better system performance includingBER than that of the original OFDM system. Therefore,all the methods, which have an increase in BER at the re-ceiver, should be paid more attention in practice. Moreover,if the side information is received in error at the receiver,which may also result in whole erroneous data frame andthus the BER performance is reduced.

6) Without additional power needed: The design of a wirelesssystem should always take into consideration the efficiency

JIANG AND WU: OVERVIEW OF PEAK-TO-AVERAGE POWER RATIO REDUCTION TECHNIQUES FOR OFDM SIGNALS 265

Fig. 6. Comparisons of CCDF based on different PAPR reductions.

of power. If an operation of the technique which reducesthe PAPR need more additional power, it degrades the BERperformance when the transmitted signals are normalizedback to the original power signal.

7) No spectral spillage: Any PAPR reduction techniques cannot destroy OFDM attractive technical features such asimmunity to the multipath fading. Therefore, the spectralspillage should be avoided in the PAPR reduction.

8) Other factors: It also should be paid more attention on theeffect of the nonlinear devices used in signal processingloop in the transmitter such as DACs, mixers and HPAssince the PAPR reduction mainly avoid nonlinear distor-tion due to these memory-less devices introducing into thecommunication channels. At the same time, the cost ofthese nonlinear devices is also the important factor to de-sign the PAPR reduction scheme.

We consider a typical OFDM system with 256 subcarriers(namely ) and 16-QAM constellation in which over-sampled OFDM sequences with the oversampling rate of 4 areused to analyze PAPR reduction and BER performance based ondifferent schemes as shown in the following figures. The presetclipping level has been selected to 80% of the maximum of theoriginal OFDM symbols in the clipping scheme and the numberof the reserved tone is 20 in TR scheme. For PTS scheme, therotation vectors belong to the set and the number ofthe subblocks is 16. Therefore, the searches is for each op-timal PTS. Note that, for PTS and TR schemes, all the side in-formation have not been submitted to the receiver. For nonlinearcompanding transform, the error companding is that proposedin [25], and the exponential companding is based on (23).

As shown in Fig. 6, different curves of the CCDF have beengiven for random original OFDM symbols generated anddifferent PAPR reduction schemes. From Fig. 6, it is very clearthat all schemes can reduce the PAPR largely in OFDM system.However, their performances of the PAPR reduction are dif-

ferent. For example, when , the PAPRs are2.6 dB, 4.5 dB, 6.6 dB, 6.8 dB, 6.9 dB and 11.7 dB for theexponential companding, error companding, PTS, TR, clippingscheme and original OFDM signals, respectively. Obviously,the signals companded by the nonlinear companding transformwith exponential function can reduce the PAPR largest and thePAPR reduction of the clipping scheme is the smallest amongthese typical methods. Although clipping scheme can improveits performance of the PAPR reduction through reducing itspreset clipping level A. However, the performance of the BERwill be degraded largely when its preset clipping level is re-duced [73].

Fig. 7 depicts the performance of BER versus SNR of actualOFDM signals with PAPR reduction based on different schemesover the AWGN channel, in which the typical HPA of the SolidState Power Amplifier (SSPA) has been considered. Note thatSSPA produces no phase distortion and only the AM/AM con-version [25]. In Fig. 7, the performance bounds are obtainedby ignoring the effect of SSPA and directly transmitting theoriginal OFDM signals through the AWGN channels. Gener-ally speaking, the performances of the BER with different PAPRreduction schemes have some degradation from Fig. 7. Specifi-cally, to achieve a BER of , the minimum required SNR is13.8 dB (performance bound). The required SNRs under the ex-ponential companding, PTS, TR, error companding and clippingschemes are 14.9 dB, 15.7 dB, 16.6 dB, 17.5 dB and 25.6 dB,respectively.

Therefore, an efficient PAPR reduction should be the lowestpossible value of PAPR while keeping a minimal level BER. InTable II, we summarize the five typical PAPR reduction tech-niques based on the theoretical analysis and simulation results.

VIII. PAPR REDUCTION FOR MULTIUSER OFDM SYSTEMS

Recently, multiuser OFDM also has received much attentiondue to its applicability to high speed wireless multiple access

266 IEEE TRANSACTIONS ON BROADCASTING, VOL. 54, NO. 2, JUNE 2008

Fig. 7. Comparisons of BER based on different PAPR reductions.

TABLE IICOMPARISON OF DIFFERENT PAPR REDUCTION TECHNOLOGIES

communication systems. In multiuser OFDM system, datastreams from multiple users are orthogonally multiplexed ontothe downlink and uplink subchannels. In a multiuser OFDMsystem, a group of carriers is assigned for each user withadaptive modulation, bit and power allocation. Obviously, thecharacteristics including distribution of the PAPR for each userin uplink multiuser OFDM is the same as that of the PAPRin single user OFDM system since the data of each user willbe transmitted to channels independently in uplink multiuserOFDM system. Therefore, the PAPR can be reduced accordingto these schemes mentioned above in the uplink multiuserOFDM systems. However, the characteristics of the PAPR indownlink multiuser OFDM is different from that of the PAPRin single user OFDM system since the data composed fromdifferent users will be transmitted to channels successivelyin downlink multiuser OFDM system. Therefore, the PAPRreduction is more complicated in an downlink than that inOFDM uplink in multiuser OFDM systems. If downlink PAPRreduction is achieved by some approaches which have beendesigned for OFDM, each user has to process the whole dataframe and then demodulate the assigned subcarriers to extracttheir own information. Thus, it introduces additional processingfor each user at the receiver. Therefore, we mainly describesome modifications of PAPR reduction techniques for thedownlink multiuser OFDM systems.

1) PTS/SLM for PAPR reduction in multiuser OFDM sys-tems: PTS and SLM techniques can easily be modified forPAPR reduction in downlink of multiuser OFDM systems.For PTS, subcarriers assigned to one user are grouped intoone or more subblocks, and then PTS can be applied to sub-blocks for all users. As side information, the selected phasefactor for each subblock can be embedded into the pre-re-served subcarrier in each subblock. Note that, the pre-re-served subcarrier does not undergo the phase rotation ineach subblock. Similarly, some of the subcarriers can beused to transmit side information when the modified SLMis applied to reduce the PAPR for multiuser OFDM sys-tems. All users use the information carried by these subcar-riers to obtain the phase sequence is used at the transmitter,and thus the data for each user can be recovered correctly.

2) TR for PAPR reduction in multiuser OFDM systems: In theTR technique for multiuser OFDM systems, the symbolsin peak reduction subcarriers are optimized for the wholedata frame in both amplitude and phase. At the same time,some peak reduction subcarriers are assigned to each userin the TR for PAPR reduction.

IX. CONCLUSIONS

OFDM is a very attractive technique for wireless communi-cations due to its spectrum efficiency and channel robustness.

JIANG AND WU: OVERVIEW OF PEAK-TO-AVERAGE POWER RATIO REDUCTION TECHNIQUES FOR OFDM SIGNALS 267

One of the serious drawbacks of in OFDM systems is thatthe composite transmit signal can exhibit a very high PAPRwhen the input sequences are highly correlated. In this paper,we described several important aspects, as well as provide amathematical analysis, including the distribution of the PAPR,in OFDM systems. Five typical techniques to reduce PAPRhave been analyzed, all of which have the potential to providesubstantial reduction in PAPR at the cost of loss in data rate,transmit signal power increase, BER performance degradation,computational complexity increase, and so on. We also showedthat it is possible to reduce the PAPR of for multiuser OFDMsystems.

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Tao Jiang (M) received the B.S. and M.S. degreesin applied geophysics from China University ofGeosciences, Wuhan, P. R. China, in 1997 and 2000,respectively, and the Ph.D. degree in informationand communication engineering from HuazhongUniversity of Science and Technology, Wuhan,P. R. China, in April 2004. Then, he joined theBrunel University, UK. In Oct. 2006, he movedto the University of Michigan, USA. He is nowwith the Department of Electronics and InformationEngineering, Huazhong University of Science and

Technology, Wuhan, 430074, P. R. China. He has authored or co-authored over40 technical papers in major journals and conferences and five books/chaptersin the areas of communications. His current research interests include the areasof wireless communications and corresponding signal processing, especiallyfor OFDM, UWB and MIMO systems, cooperative networks, cognitive radioand wireless sensor networks. He serves on the editorial boards of someinternational journals, such as Wiley of Wireless Communications and MobileComputing (WCMC), and as a Technical Program Committee Member forsome major international conferences, including IEEE Globecom, ICC, VTCand Chinacom. Dr. Jiang is a Member of IEEE Communication Society andIEEE Broadcasting Society.

Yiyan Wu (Fellow) received the B.S. degree fromBeijing University of Posts and Telecommunications,and M.S. and Ph.D. degrees in electrical engineeringfrom Carleton University, Ottawa, Canada, in 1986and 1990, respectively. After graduation, he workedat Telesat Canada as a senior satellite communicationsystems Engineer. In 1992, He joined Communica-tions Research Center Canada (CRC) and now is aPrinciple Research Scientist. His research interestsinclude broadband multimedia communications,digital broadcasting and communication systems

engineering. He is an IEEE Fellow, an adjunct professor of Carleton University,Ottawa, Canada; Shanghai Jiaotung University; and Beijing University ofPosts and Telecommunications, China. He is a member of IEEE BroadcastTechnology Society Administrative Committee, and a member of the ATSCBoard of Directors, representing IEEE. He is the Editor-in-Chief of the IEEETRANSACTIONS ON BROADCASTING. He has more than 200 publications andreceived many technical awards for his contribution to the research and devel-opment of digital broadcasting and broadband multimedia communications.