Tone Reservation for Peak-to-Average Power Ratio Reduction inOFDM under Different Optimization Constraints

Christian Siegl and Robert F.H. Fischer

Lehrstuhl fur Informationsubertragung, Friedrich–Alexander–Universitat Erlangen–NurnbergCauerstrasse 7/LIT, 91058 Erlangen, Germany, Email: {siegl,fischer}@LNT.de

Abstract— The high peak-to-average power ratio (PAR) ofthe transmit signal is considered to be one major drawback oforthogonal frequency-division multiplexing (OFDM) systems.This paper analyzes the tone reservation (TR) technique forPAR reduction. It is shown that the good performance of TRis paid with an increase in average transmit power. Therefore,an extension of the TR scheme is considered, which limits theaverage power. This increase of power efficiency comes alongwith a decrease of PAR reduction; a trade-off is possible. Inaddition to that, the TR scheme is compared with selectedmapping (SLM), another popular PAR reduction technique.Simulations show that SLM outperforms TR in terms of PARreduction and power efficiency. Finally, a combined approachof TR and SLM is proposed. Although, this scheme does notoffer the PAR reduction capabilities of pure TR it has lowercomplexity.

I. INTRODUCTION

One of the most serious drawbacks of orthogonal frequen-cy-division multiplexing (OFDM) [1] is the high peak-to-average power ratio (PAR) of the time-domain transmitsignal. Non-linear power amplifiers at the transmitterfront-end cause signal clipping, which, in turn, leads tosignal distortion and out-of-band radiation. In order toavoid violation of spectral masks by out-of-band radiation,operation at large input power back-off is required. Todecrease the input power back-off and therefore increasepower efficiency, a transmitter-sided preprocessing to re-duce the transmit signal’s peak-power is indispensable.

Over the last years, several strategies were discussedin literature to overcome this issue. One very populartechnique is tone reservation (TR) [2]. With TR theminimization of the PAR of the transmit signal can beformulated as a convex optimization problem [3], whichcan be solved efficiently.

Main idea of this scheme is to reserve a number ofsubcarriers to produce a redundant, additive signal whichreduces the peak-power. As TR works on reserved sub-carriers (tones) no additional signal processing is requiredat the receiver. It only has to be aware of the positions ofthe reserved subcarriers and ignores these signals. Explicittransmission of side-information is not necessary in thiscase. This is done inherently as the reserved subcarriersare not available for transmitting information. Due to thesereasons, tone reservation is quite popular for practicalimplementations and therefore is specified in 3GPP [4]as one possible PAR reduction technique, for instance.

This work was supported by Deutsche Forschungsgemeinschaft (DFG)within the framework TakeOFDM under grant FI 982/1-1.

This paper gives a short review of the TR techniqueand addresses the problem of an increased transmit power.Moreover, the original TR algorithm is extended in orderto limit the maximum possible increase in transmit power.In addition to that, these techniques are compared withselected mapping (SLM) [5], another popular techniquefor reducing the peak-power. Finally, a combination of TRand SLM is derived, which selects the best suited additivesignal on the reserved tones among several given ones.

The paper is organized as follows: Section II defines thesystem model. The ideas and results of tone reservation arediscussed in Section III, whereby a novel PAR reductiontechnique based on selection among different additivesignals is introduced in Section IV. Finally, Section Vdraws some conclusions.

II. SYSTEM MODEL

Subsequently, an OFDM system with D subcarriers is con-sidered. The frequency-domain OFDM symbol is denotedby the vector1A = [Ad], d = 0, . . . ,D − 1, of lengthD. Each element Ad is i.i.d.ly drawn from an arbitraryM -ary QAM constellation. The frequency-domain OFDMsymbol is transformed via the inverse discrete Fouriertransform (IDFT) into the time-domain OFDM symbola = IDFT{A} with a = [ak], k = 0, . . . ,D − 1 andak = 1/

√D

∑D−1d=0 Ad exp(j2πkd/D). The numerical

results assume a 4-QAM modulation transmitted withOFDM using D = 512 subcarriers.

Due to the superposition of D carriers, the OFDM time-domain signal exhibits a high peak-to-average power ratio

PAR def=‖a‖2

∞‖a‖2

2

. (1)

This PAR definition is based on the, so-called, short-termaverage power, i.e., the average power over the actualconsidered OFDM frame.

III. TONE RESERVATION FOR PAR REDUCTION

In order to reduce the peak-power of the transmit signal,common PAR reduction techniques either introduce (i) dis-tortion of the information carrying signal, (ii) redundancywhich leads to rate-loss, and/or (iii) an increase of averagetransmit power.

1Notation: Throughout this paper all vectors are assumed to be rowvectors and are denoted as bold letters. Vectors in the frequency domainare written as capital letters, whereby vectors in time domain as lowercase letters. The L-norm of a vector is written as ‖ · ‖L; (·)T gives thetranspose.

A. Idea of Tone Reservation

The idea of tone reservation is to define two sets, I+

and I−, of D+ active and D− reserved subcarriers,respectively, with D+ + D− = D. Both index sets arepredetermined and common to transmitter and receiver andmust fulfill the following conditions:

I+ ∪ I− = {1, . . . ,D} and I+ ∩ I− = ∅. (2)

The information carrying signal is only transmitted overthe active subcarriers, i.e., Ad = 0 for d ∈ I−. The signalof the reserved subcarriers C = [Cd] with Cd = 0 ford ∈ I+ can be chosen arbitrarily. The time-domain OFDMframe is now given by

x = a + c = IDFT{A + C}. (3)

In time domain, the additive signal c influences thewhole information carrying signal a. Hence, the aim isto choose the frequency-domain additive signal C suchthat it minimizes the PAR of x.

A block diagram of this PAR reduction scheme isdepicted in Fig. 1.

IDFT

Algorithm

A a x

c

σ2A σ2

a σ2x

σ2c

Fig. 1. Block diagram of PAR reduction through additive signal.

As I+ and I− are disjoint sets, after discrete Fouriertransform at the receiver it is possible to distinguishbetween information carrying signal A and additive signalc. Hence, no signal distortion occurs.

However, as a number D− of subcarriers is not usedfor transmission of information it can be regarded asintroduced redundancy. Noteworthy, the additive signal ccan arbitrarily chosen, i.e., the norm ‖c‖2

2 is not restrictedand, moreover, an increase of average transmit poweroccurs.

In order to compare the impact of these issues ondifferent setups of the PAR reduction system we regardthe long-term power efficiency

ηTRdef=

σ2aD+

σ2xD

=σ2

aD+

σ2aD+ + σ2

cD−=

1

1 + σ2c

σ2a· D−

D+

, (4)

the fraction between (long-term) average power σ2a =

E{|ak|2

}of the information carrying signal a and the

average power σ2x = E

{|xk|2}

of the transmit signal x.The long-term average power of the additive signal isgiven by σ2

c = E{|ck|2

}.

The peak-to-average power ratio of a given OFDMframe after applying the TR technique is defined by

PAR def=‖x‖2

∞‖x‖2

2

. (5)

In this definition the power of the additive signal c is con-sidered. Definition (5) is reasonable, as we are interestedin the PAR of the actual transmitted signal x.

Other definitions in literature dealing with the TRtechnique, do not incorporate the additive signal in orderto take solely the peak-reducing characteristics of TR intoaccount and may rather be characterized as peak-powerreduction schemes.

B. Review of the TR Convex Optimization Problem

Subsequently, we address the problem to find a suitableadditive signal c. According to [2], the PAR reductionproblem can be formulated as a convex optimizationproblem:

min p

subject to (ak + qTkC)(ak + qT

kC)∗ ≤ p ∀k ,(6)

whereby the vector qTk of length D represents the kth row

of the inverse Fourier matrix [6], i.e., qk = [qk,d] withqk,d = 1/

√D exp(j2πkd/D).

Optimization problems of this type are extensively dis-cussed in literature. According to [3] a solution can, e.g.,be found by application of interior-point methods.

Unfortunately, finding an optimum solution of the addi-tive signal c is quite computational exhaustive. Althoughlow complex implementations do exist, e.g., [2], [7], [8],the complexity is one serious drawback of TR. In additionto that, the complexity is not fixed for each OFDM framebut may vary for different information carrying signals a.

As the additive signal C has to be calculated for everyOFDM frame individually, the (short-term) average powerof one transmitted OFDM frame varies strongly. This facthelps to reduce the PAR, but also leads to a loss in powerefficiency (4).

C. TR with Limited Average Transmit Power

In order to increase the power efficiency, the optimiza-tion problem from (6) can be extended by an additionalconstraint. We define a factor ξ by which the (short-term)average transmit power of one OFDM frame might bemaximal exceeded, i.e., ξ = ‖x‖2

2/σ2a − 1. The corre-

sponding optimization problem is now given by

min p

subject to (ak + qkC)(ak + qkC)∗ ≤ p ∀k

CHC ≤ D−σ2a(1 + ξ D

D−) .

(7)

A solution can also be found according to [3].

D. Numerical Results and Discussion

1) Tone Reservation: The top plot of Fig. 2 showsnumerical results of the complementary cumulative dis-tribution function (ccdf) of the tone reservation techniquefor various number D+ of used subcarriers. The respectivenumbers are D+ = 486 (≈ 95%), D+ = 507 (≈ 99%),D+ = 510, and D+ = 511. Evidently, the higher thenumber of reserved tones, the better the PAR reduction.

However, these significant gains, achieved through thetone reservation technique suffer from several drawbacks.As the additive signal c does not carry any information

4 6 8 10 12 1410

−5

10−4

10−3

10−2

10−1

100

10 log10(PARth) −→

Pr{

PA

R>

PA

Rth}−

→originalTRSLM

0

2

4

6

8

10

12

σ2 c/σ

2 a−

1[%

]−→

80

85

90

95

100

ηT

R/

SLM

[%]−→

486 507 510 5110

10

20

30

40

50

60

bits

ofre

dund

ancy

D+ −→original

Fig. 2. Numerical Results of tone reservation (blue) with D+ = 486(�), 507 (�), 510 (×), and 511 (◦) and, for comparison, of selectedmapping (magenta curves) with U = 4.5 · 1015 (�), 1024 (�), 16(×), and 4 (◦); the results of the original (not PAR reduced) signal aredepicted in red. Top: ccdf of PAR; second row: fractional increase oftransmit power; third row: power efficiency η; bottom: required bits ofside information; D = 512, M = 4.

it increases the average transmit power. In addition tothat, the reserved carriers are not available for transmissionof information. With a given modulation alphabet, theselost carriers can be interpreted as equivalent redundancybits. The three bottom plots of Fig. 2 show the long-termfractional power increase σ2

x/σ2a−1 (in %) (top), the long-

term power efficiency ηTR (middle), and the number ofequivalently used redundancy bits given by log2(M) ·D−(bottom) for various numbers D+ of active subcarriers.

Transmitting data over D+ = 511 subcarriers, i.e.,

using D− = 1 subcarrier for PAR reduction does alreadylead to a significant gain in PAR distribution. However,the transmit power increases on average by approx. 4.5%whereby the power efficiency ηTR decreases to approx.95%; the equivalent number of redundancy bits is twobits per OFDM frame. Using even less subcarriers fortransmission (D+ = 510 or D+ = 507) leads to highergains in PAR reduction, but also leads to an increasedaverage transmit power (up to more than 10% or less than90% in power efficiency) and redundancy bits (up to 10bits per OFDM frame).

Interestingly, for a (rather) small number of used sub-carriers D+ = 486 (i.e., D− = 26) where the peak-power reduction performance is extraordinary good, thefractional power increase is even less than for D+ = 507.In this case, the additional signal c offers so many degreesof freedom that less power is sufficient to achieve verygood performance. However, the number of equivalentlyrequired redundancy bits is extremely large (more than 50bits) and not tolerable.

Fig. 3 shows the results of TR if the maximum increaseof transmit power of the OFDM frames is limited to ξ.Here, we consider only D+ = 507 and four values forξ = 0.12, 0.1, 0.5, and 0.01. Evidently, the higher theallowed power increase, the better is the PAR reductionand the results converge to the ones of pure TR.

According to the formulation of the optimization prob-lem, the parameter ξ defines a limit on the short-termaverage power. An optimum solution for c may not exploitthis limit, but stay below. Hence, this limit is a worst-case assumption on the long-term average power σ2

x. Forreference, the limit ξ is also depicted in the two bottomplots in Fig. 3.

Even if the loss in PAR performance is significant withthis scheme, now a trade off between PAR reduction per-formance and power efficiency is present. The equivalentredundancy is not affected here and equal to pure TR andtherefore not depicted.

Finally it can be stated, the PAR reduction performanceis largely governed by an increase in transmit power. Onlyfor a (comparatively) small number of used subcarriers theTR technique is able to reduce the peaks.

2) Comparison with Selected Mapping: In literature,selected mapping2 [5] is one popular technique for PARreduction in OFDM systems. Advantages of SLM com-pared to TR are as follows. The “best suited” transmitsignal is determined by straightforward calculations, i.e.,U calculations of IDFT and decision metric (PAR) arerequired. It is not necessary to design a complex algorithmin order to solve the optimization problem (6). Moreover,

2The idea of SLM is to generate out of the original OFDM frameseveral, say U , different signal representations via U different mappings.Out of these signal candidates, the best one, i.e, the one exhibiting thelowest PAR is chosen for transmission. At the receiver after equalization,the original data has to be reconstructed by inverting the applied mapping.Hence, side information, in terms of an index of the applied mapping,has to be transmitted.

A practical way to transmit the side information has been proposedin [9]. The so-called scrambler variant of SLM distributes the sideinformation inherently over the whole OFDM frame and does not requireits explicit transmission.

4 6 8 10 12 1410

−5

10−4

10−3

10−2

10−1

100

10 log10(PARth) −→

Pr{

PA

R>

PA

Rth}−

→originalξ = 1%ξ = 5%ξ = 10%ξ = 20%pure TR

0

2

4

6

8

10

12

σ2 c/σ

2 a−

1[%

]−→

ξ

20 10 5 180

85

90

95

100

ηT

R/

SLM

[%]−→

ξ [%] −→

ξ

originalTR

Fig. 3. Numerical results of tone reservation with limited transmit power(blue); for reference the results of pure tone reservation are depicted ingreen, of the original signal in red; Top: ccdf of PAR; middle: fractionalincrease of transmit power; bottom: power efficiency η; D = 512, D+ =507, M = 4.

as SLM operates on different signal representations basedon the same modulation scheme no explicit increase of thetransmit power occurs.

In order to decode the signal correctly at the receiver,the mapping applied on the current OFDM frame has to betransmitted. This redundancy is given by �log2(U)� bits.Efficient techniques to transmit the redundancy are known[9]. Thus, the fractional increase of transmit power due tothe redundancy is given by �log2(U)�/(D log2(M)). Thisfact lowers the power efficiency due to the PAR reductiontechnique, which reads for SLM

ηSLM =D log2(M)σ2

a − �log2(U)�σ2a

D log2(M)σ2a

= 1− �log2(U)�D log2(M)

.

(8)In Fig. 2 the respective results for SLM are additionally

depicted. The number of candidates are chosen such thatthe number of required redundancy bits are comparable,i.e., U = 4, U = 16, U = 1024, and U = 4.5 · 1015.The later two values seem to be far from being practicalas the complexity is extraordinary high. However, in [10]

a variant of SLM is proposed, which reduces the perfor-mance of assessing U candidates to complexity of only2√

U . Hence, the performance of SLM with U = 1024 isachievable by having the complexity of U = 64, whichis tolerable. Nevertheless, U = 4.5 · 1015 is a ratherimpractical setup. Due to this reason, the given ccdf curvesare analytic results based on Gaussian signalling [5]. Theresults of Fig. 2 show, that TR is outperformed by SLMwith respect to PAR reduction and power efficiency.

Hence, the peak-power reducing capabilities of SLMare more significant than of tone reservation as its goodperformance is primarily based on increasing the averagetransmit power.

IV. PAR REDUCTION THROUGH SELECTED TONE

RESERVATION

The basic idea of the tone reservation technique is tocalculate the additive time-domain signal c which reducesthe PAR of the actual transmit signal x. This calculationis done in an optimum way by solving the convex op-timization problem (6) or (7), respectively. The additivesignal c has two effects on the transmit signal: on theone hand it decreases the peaks but on the other hand theaverage transmit power is increased. However, both effectsare beneficial on the PAR but an increased average powerlowers the power efficiency (4).

Assuming the transmission system tolerates some losson the power efficiency, it is possible to design a novelPAR reduction technique which can be regarded as acombination of the ideas of selected mapping and tonereservation.

Instead of applying different mappings on the originalOFDM frame it is also possible to create the U multiplesignal representations by adding arbitrary chosen signalswith support restricted to the I− reserved subcarriers tothe initial signal a and select the best one for transmission.The huge benefit of this technique is, that these U additivesignals can a-priori be chosen and transformed into timedomain. Thus only one IDFT has to be calculated perOFDM frame (instead of U calculations required for theSLM technique), whereby the number U of assessed signalcandidates might be much larger than with SLM.

The subsequent approaches use U arbitrarily chosencomplex sequences c.

Approach I: The D− subcarriers (index in I−) aredrawn from i.i.d. complex Gaussian random variables withvariance σ2

aD+/D−. Numerical results showed, that asimple addition of c, i.e., considering candidates x =a+c, does not lead to mentionable gains in PAR reduction.

Approach II: Next, we introduce a weighted additionof c with the complex scaling factor γ. The transmitsignal is hence given by x = a + γc. In order findthe best scaling factor γ for a given sequence c anoptimization over its absolute value |γ| and its phasearg(γ) has to be accomplished. Numerical simulations ofthis scheme were conducted, whereby an search over asufficiently large number of points γ within the complexplane has been performed. The points have been chosenwith equally distributed phase (in the range 0, . . . , 2π) and

absolute value (in the range starting from |γ| = 0 linearlyincreasing to |γ| =

√ξD/D− + 1, whereby ξ represents

again the worst-case power increase per OFDM frame).Noteworthy, the total number of assessed candidates isgiven by the product of U with the number of assessedvalues γ.

Now, the results of numerical simulations showed abetter performance in PAR reduction. However, it appearsthat almost all chosen values of γ exhibited the maximumpossible absolute value whereby the phase is equallydistributed.

Approach III: Hence, a more useful approach is torestrict the weighting factor’s absolute value to |γ| =√

ξD/D− + 1. Here, only an optimization over the phaseis required. Given an additive sequence c, the resultingPAR values are non-monotonously distributed over thephase of γ. Thus, finding the optimum value is quiteexhaustive. We propose, to assess certain given values,here, e.g., {0, . . . , 7} · π/4.

Fig. 4 shows numerical results of this technique com-pared to pure tone reservation. For each value of D+,the absolute value |γ| (or ξ, respectively) is chosen thatthe increase of average power or power efficiency areequal to the ones of pure TR (cf. Fig. 2). Evidently,the performance increases if the number of candidates isincreased.

Unfortunately, this approach is significantly outper-formed by the pure tone reservation scheme, whereby forD+ = 511 and D+ = 510 the loss is only approximately1 dB. However, the benefit of this approach is the reducedcomplexity. With selected TR the complexity is mainlygoverned by the number of PAR evaluations (given bythe product of U with the number of assessed valuesof γ). Hence, the complexity is fixed per OFDM frame.Our simulations of pure TR indicate that it requiresmuch higher complexity than selected TR. However, aquantitative comparison remains to be done. Moreover,another drawback of pure TR is that the complexity perOFDM frame is not fixed but varying.

V. CONCLUSIONS AND FUTURE WORK

This paper analyzes the tone reservation technique for PARreduction in OFDM systems. Although TR offers quitegood performance in PAR reduction, it suffers from severaldrawbacks. The most annoying point is the increase oftransmit power which lowers the power efficiency.

Comparing TR with SLM in terms of equal (equivalent)redundancy, it is always outperformed by SLM, i.e., SLMoffers better PAR reduction performance and better powerefficiency. However, one unattractive point of SLM mightbe that receiver-side signal processing is necessary toinvert the applied mapping. In contrast to that, TR is apure transmitter-sided scheme.

The power inefficiency of TR might be avoided if anadditional constraint, which limits the average power, isintroduced in the optimization problem. As this methodreduces PAR reduction performance a trade-off betweenboth issues can be achieved.

4 6 8 10 12 1410

−5

10−4

10−3

10−2

10−1

100

10 log10(PARth) −→

Pr{

PA

R>

PA

Rth}−

→

originalsel. TR U = 8sel. TR U = 64pure TR

Fig. 4. Numerical results of selected tone reservation (light blue)according to Approach III for U = 8 (dashed) and U = 64 (solid);as for each of the U candidates eight different phase rotations havebeen assessed, the total number of assessed candidates is 64 or 512,respectively; the number of reserved subcarriers is D+ = 486 (�),507 (�), 510 (×), and 511 (◦); for reference, the results of pure tonereservation are depicted blue, of the original signal red. D = 512,M = 4.

Another TR variant, considered in this paper, is selectedtone reservation, which finds the best suited additivesignal among several predetermined ones. This approachdoes not offer the huge PAR reduction performance asTR but is simple to implement. However, this approachis rather heuristic. Further work could be spend on anefficient design of the additive sequences and an suitedoptimization of the scaling factor γ.

Finally it can be stated that the PAR reduction perfor-mance of TR rather originates from the increased averagepower than in a decreased peak power.

REFERENCES

[1] J.A.C. Bingham, “Multicarrier Modulation for Data Transmission:An Idea Whose Time Has Come,” IEEE Communications Maga-zine, pp. 5–14, May 1990.

[2] J. Tellado, Peak to Average Power Reduction for MulticarrierModulation, Ph.D. thesis, Stanford University, 2000.

[3] S. Boyd and L. Vandenberghe, Convex Optimization, CambridgeUniversity Press, 2004.

[4] 3GPP, TR 25.814: Physical layer aspect for evolved UniversalTerrestrial Radio Access (UTRA), Oct. 2006.

[5] R. Bauml, R.F.H. Fischer, and J.B. Huber, “Reducing the Peak-to-Average Power Ratio of Multicarrier Modulation by SelectedMapping,” IEE Electronics Letters, pp. 2056–2057, Nov. 1996.

[6] A.V. Oppenheim and R.W. Schafer, Discrete-Time Signal Process-ing, Prentice-Hall, Upper Saddle River, 1999.

[7] B.S. Krongold and D.L. Jones, “A New Tone Reservation Methodfor Complex-Baseband PAR Reduction in OFDM Systems,” in Pro-ceedings of IEEE International Conference on Acoustics, Speech,and Signal Processing (ICASSP), May 2002.

[8] A. Aggarwal and T.H. Meng, “Minimizing the Peak-to-AveragePower Ratio of OFDM Signals Using Convex Optimization,” IEEETransactions on Signal Processing, pp. 3099–3110, Aug. 2006.

[9] M. Breiling, S. Muller-Weinfurtner, and J.B. Huber, “SLM Peak-Power Reduction without Explicit Side Information,” IEEE Com-munications Letters, pp. 239–241, June 2001.

[10] R.F.H. Fischer, “Widely-Linear Selected Mapping for Peak-to-Average Power Ratio Reduction in OFDM,” IEE ElectronicsLetters, pp. 766–767, July 2007.