PAPR analysis in Wavelet Packet ModulationMatthieu Gautier ∗�, Christian Lereau ∗, Marylin Arndt ∗

∗France Telecom R&D28, chemin du vieux chêne38243 Meylan - FRANCE

Joël Lienard ��GIPSA-Lab

961 rue de Houille Blanche38402 St Martin d’Heres - FRANCE

Abstract— In this paper, the Peak-to-Average-Power-Ratio(PAPR) for multicarrier systems with different pulse shapes isstudied. First, the influence of the pulse shaping characteristicson the PAPR is investigated. Then, from the gained insights, aninnovative approach is proposed where the waveform is adaptedto the environment. To this end, a Wavelet Packet Modulation isintroduced and its PAPR is compared with the one of OrthogonalFrequency Domain Multiplexing modulation. Simulation resultsshow that multicarrier systems using the wavelet approachsignificantly reduces the PAPR for a low number of channelsand hence the cost of the transceiver.

I. INTRODUCTION

Orthogonal Frequency Division Multiplexing (OFDM) iscurrently the modulation used for high data rate transmissionin frequency-selective fading channels [1]. A major drawbackof OFDM modulation at the transmitter side is the high Peak-to-Average Power Ratio (PAPR) of the transmitted signal.PAPR reduction techniques [2]-[5] have been proposed toreduce the PAPR problem in OFDM transmitter. Some tech-niques use coding, in which the data sequence is embeddedin a larger sequence and only a subset of all the possiblesequences is used to exclude patterns with high PAPR [2].These coding techniques reduce PAPR, but they also reducethe transmission rate. Recently, multiple signal representationtechniques have been proposed. These include the partial trans-mit sequence technique [3], selected mapping technique [4]and interleaving technique [5]. These techniques require sideinformations to be transmitted from the transmitter to thereceiver to recover the original data block from the receivedsignal.In this paper, we propose a novel PAPR solution based on thepulse shaping characteristics of multicarrier modulation. Weshow that the PAPR decreases with an appropriate choice ofthe pulse shaping.In order to design a multicarrier modulation with a low PAPR,a given solution is to use wavelet theory. Its application tofilter bank and its extension to wavelet packet decompositionallow the construction of orthogonal bases used to modulatethe data as a multicarrier system. A multicarrier modulationbased on wavelet packet transform is called WPM (WaveletPacket Modulation).Wavelet theory applied to multicarrier modulation has beenstudied in previous works [6]-[9]: it has been shown that theWPM is efficient for wired transmission [6]. In a wirelessenvironment [7][8], performances of WPM depend on theequalization techniques and a study [9] shows that the use of

complex wavelets could reduce time and frequency dispersivechannel interferences.Indeed, major improvements of this paper are the study ofthe link between PAPR and pulse shaping for multicarriermodulation transmitter, and the evaluation of PAPR for theWPM transmitter.In the following, a PAPR analysis is performed for multicarriermodulation in order to define a selection criterion of the pulseshaping (Sec. II). In Sec. III, wavelet packet modulation isintroduced. In Sec. IV, the choice of the proposed criterionand a useful asymptotic function is discussed in function ofthe number of channels. Simulation results are given in Sec. Vto show the performance of the system in different situations.Finally, conclusions from simulations are drawn in Sec. VI.

II. PAPR ANALYSIS IN MULTICARRIER MODULATION

Let M be the number of channels in the multicar-rier scheme. The multicarrier modulated signal s(t) resultsfrom a linear combination of the pulse shaping functions{ψm,n(t), n ∈ Z, m = 0, . . . ,M − 1} weighted with thetransmitted symbols xm[n]:

s(t) =+∞∑

n=−∞

M−1∑m=0

xm[n]ψm,n(t). (1)

OFDM modulation [1] uses a rectangular pulse shapingof duration Ts and orthogonality is obtained with a carrier

spacing 1/Ts. By noting ΠTs0 (t) =

{1 if 0 ≤ t < Ts

0 elsethe

rectangular function, ψm,n(t) is then expressed by:

ψm,n(t) = e−j2π mTs

tΠTs0 (t− nTs). (2)

In the linear case (when no power amplifier is used) andfor large values of M , the OFDM spectrum goes to an idealbandlimited rectangular spectrum. This means that the OFDMsignal within each block appears as Gaussian with very highvariations from one sample to the next. Thus, the powerspectral density of the modulated signal will be broadened bythe nonlinear distortions of a High Power Amplifier (HPA).The PAPR is one way to measure such variation of thetransmitted signal. The PAPR is defined as [10]:

PAPR =max |s(t)|2

Es, (3)

with Es = E{|s(t)|2

}where E {.} denotes the statistical

average operator.

In order to concentrate the study on the effect of ψm,n(t)on the PAPR, we only choose multicarrier modulationwith Es = 1 and we limit the study to 4-QAM symbolswith constant envelope. We also consider that the samepulse shaping ψ(t) is used on every subcarrier, i.e.{ψm,n(t) = ψ(t − nTs)e−j2πm t

Ts , m = 0, . . . ,M − 1}. Inthis case, we can write the theorem below:

Theorem 1: For a multicarrier modulation with M channelsthat uses a modulated pulse shaping ψ(t), the PAPR of thetransmitted signal is upper bound by

PAPR ≤M max0≤t≤Ts

|ψ(t)|2 , (4)

and PAPR ≤M when ψ(t) is a rectangular pulse shapingused by OFDM modulation.For the use in multicarrier modulation, we make use ofthe pulse shaping characteristics and we postulate for lowPAPR transmission:

max0≤t≤Ts

|ψ(t)|2 < 1. (5)

Proof : Assuming a 4-QAM multicarrier system with Es = 1,Eq. 3 becomes:

PAPR ≤ PAPRMAX =1M

max0≤t≤Ts

(M−1∑m=0

|ψm,n(t)|

)2

. (6)

If the same pulse shaping is used on every subcarrier, themaximum PAPR becomes:

PAPRMAX =1M

max0≤t≤Ts

(M−1∑m=0

∣∣∣ψ(t− nTs)e−j2πm tTs

∣∣∣)2

,

(7)

=1M

max0≤t≤Ts

(M−1∑m=0

|ψ(t)|

)2

, (8)

=1M

max0≤t≤Ts

(M |ψ(t)|)2 , (9)

= M max0≤t≤Ts

|ψ(t)|2 . (10)

Eq. (10) is the upper bound defined in Eq. 4.

According to theorem 1, the proposed solution is to use pulseshaping that reduces max |ψ(t)|2 without decreasing Es. Inthis paper, we propose to use the WPM to reduce the PAPR.Indeed, this modulation allows a large choice of pulse shapingψ(t) by choosing different wavelets.

III. WPM SYSTEM DESCRIPTION

The idea of wavelet packet modulation [6] is to use waveletpackets as the pulse shaping functions.Wavelet packets: The concept of wavelet transform has beenextended by establishing the theory for libraries of orthonor-mal bases which were obtained by filling out the binary tree tosome uniform depth as shown in Fig. 1. This decomposition

allows a uniform analysis of the spectrum. The obtainedfunctions are wavelet packets, which are recursively definedby:

p2m(t) =√

2∑

n

hnpm(2t− n), (11)

p2m+1(t) =√

2∑

n

gnpm(2t− n). (12)

hn and gn are a quadrature mirror filter (QMF) pair. hn

is a lowpass filter while gn is a highpass filter. They areconnected by the relation gn = (−1)nh1−n.

Fig. 1. Uniform wavelet packet decomposition.

Definition 1 [11]: A wavelet packet base of L2(R) is given byall orthonormal bases chosen among the functions:

{pml,n(t) = 2

l2 pm(2lt− n), (l, n) ∈ (Z,Z), m ∈ N}. (13)

Therefore, any function f(t) of L2(R) can be decomposedon the base {pm

l,n(t), (l, n) ∈ (Z,Z)}:

s(t) =∑m,n

aml,np

ml,n(t). (14)

All these coefficients aml,n constitute the DWPT (Discrete

Wavelet Packet Transform) of s(t) and the inversetransform is called IDWPT (Inverse Discrete WaveletPacket Transform).

Wavelet Packet Modulation: From eq. (14), we can see thatany function s(t) of L2(R) can be expressed as the sum ofweighted wavelet packets. In communication systems, thismeans that a signal can be seen as the sum of modulatedwavelet packets, which gives the idea of wavelet packetmodulation [6]: the transmitter transforms the symbols fromthe wavelet domain to the time domain with an IDWPT andthe receiver transforms the received signal from the timedomain to the wavelet domain with a DWPT.By choosing M = 2l, ψm,n(t) = pm

l,n(t) and ym[n] = aml,n,

the tree shown in Fig. 1 represents an example ofWPM demodulation for a M= 8 subcarriers system. Theorthogonality of wavelet packets gives a perfect reconstructionsystem for an ideal transmission without interferences.

Complexity: In [8], we compute the complexity CWPM ofWPM. Based on [12], we show that the number of MAC

(Multiplication and ACcumulation) needed to compute oneWPM symbol is

CWPM = LM log2M, (15)

where L denotes the number of coefficients of filters h andg. We note that the complexity of WPM increases with theorder of the wavelet but it varies in the same way as OFDMmodulation according to the number of carriers M .

PAPR illustration: WPM allows flexibility in the pulse shapingdesign and the choice depends on the transmission environ-ment. In this study, we use Daubechies wavelets which givesignificant results on PAPR reduction. They are parameterizedby their order. The higher the wavelet’s order is, the higherthe time dispersion but the computation complexity will behigher (filters length L increase with the order). Coefficientshn have been published by I. Daubechies in [13].The idea that WPM gives lower PAPR than OFDM modulationis illustrated by Fig. 2. It shows time representations ofan OFDM signal and a WPM signal using 6th Daubechieswavelet. We can see that the maximum peak of WPM signalis lower than the one of OFDM signal.

Fig. 2. Time vision of wavelet’s peak reduction.

To limit the PAPR, one method is the clipping [14]. A clippinglevel is set on Fig. 2 at |Amplitude| = 1.5. We can seethat the number of peaks above this reference is higher forOFDM modulation than for WPM. For the same clipping level,WPM should be less sensitive to nonlinear HPA than OFDMmodulation.

IV. ASYMPTOTIC DISTRIBUTION OF THE PAPR

In II, PAPR is evaluated based on the maximum peak poweras shown in eq. (4). However, for the hardware implementa-tion, the evaluation of the power distribution is more importantthan only the maximum power evaluation. In this part, wepresent a study of asymptotic PAPR distribution that standsfor both OFDM and WPM modulation.In both OFDM and WPM systems, the signal s(t) goinginto the channel is a sum of random symbols modulatingorthogonal basis functions. From the central limit theorem,it is claimed [15] that for a large value of channels M , themulticarrier signal can be modelled as a zero-mean Gaussian

distributed random variable with variance σ2s . Introducing the

variable y = |s|2, we obtain the central chi-square distribu-tion [16]:

py(y) =1√

2πyσs

e− y

2σ2s (16)

The complementary cumulative distribution function (CCDF)of the PAPR is defined as:

CCDF (PAPR0) = Prob [PAPR > PAPR0] . (17)

Using the result in (16), so, the CCDF of the PAPR can becalculated as:

CCDF (PAPR0) = 1− (1− e−PAPR0)N . (18)

The distributions obtained by the conventional analysis, how-ever, does not fit those of the PAPR of the OFDM signalsobtained by computer simulations, even for very large M .In [15], Van Nee and Prasad gave an empirical approximation:

CCDF (PAPR0) = 1− (1− e−PAPR0)αN , (19)

where α is a parameter determined by computer simulationto be 2.8. It should be noted that this approximation lackstheoretical justification.

Now, we evaluate the values of M where the pulse shapingcriteria defined in II can be used. Fig. 3 shows the PAPR versusdifferent amounts of channels M for OFDM modulation andWPM using the 6th Daubechies wavelet. The theoreticaldistribution function is also plotted. It is shown that for largevalues of M , OFDM and WPM have quite the same PAPRand the theoretical cumulative distribution function is accurate.However, for low values of M , WPM outperforms OFDMmodulation.

Fig. 3. PAPR as a function of the number of channels M .

We conclude that Theorem 1 is right for low values of Mwhereas theoritical function defined by (19) is right for high

values. In the following, to evaluate the influence of WPMon the PAPR, we choose M = 128 subcarriers. In this case,Fig. 4 shows a 0.6 dB difference between OFDM and WPMusing the 6th Daubechies wavelet.

V. SIMULATION RESULTS

We use computer simulations to evaluate the performanceof the proposed PAPR reduction technique. As a performancemeasure for the proposed technique, we use the CCDF ofthe PAPR and Bit Error Rate (BER) with a nonlinear poweramplifier.Performances of the proposed system are compared to OFDMmodulation for a multicarrier systems with 4-QAM symbolsmodulated on M=128 subcarriers. Fig. 4 shows the CCDF of

Fig. 4. CCDF as a function of PAPR.

the proposed technique for different Daubechies wavelets. Itis shown that the OFDM signal has a PAPR which exceeds11.3 dB for less than 0.1% of the OFDM data blocks. The0.1% PAPR of the proposed technique is 11.1 dB when 3th

Daubechies wavelet is used and 10.7 dB when 6th Daubechieswavelet is used, resulting 0.2 dB and 0.6 dB reduction in0.1% PAPR respectively.A 5.1 dB reduction is reached with the 20th Daubechieswavelet; this result is very interesting but, according to (15),the complexity of this system is higher than the complexityof OFDM modulation.

Now, we evaluate the influence on the BER. To approximatethe effect of nonlinear HPA, we adopt Rapp’s model foramplitude conversion [15]. The relation between amplitudeof the normalized input signal s(t) and amplitude of thenormalized output signal g(s) of the nonlinear power amplifieris given by:

g(s) =s

(1 + s2p)1/(2p), (20)

where p is a parameter that represents the nonlinear charac-teristic of the HPA. The power amplifier approaches linear

amplifier as p gets larger. We choose p = 3, which is a goodapproximation of a general power amplifier [15]. The phaseconversion of the power amplifier is neglected in this paper.Fig. 5 shows BER curve for OFDM modulation and WPMusing the 3th and the 20th Daubechies wavelets. They arecompared to the BER curve of a 4-QAM modulation withoutdistortion. This figure shows that WPM is less sensitive to

Fig. 5. BER as a function of signal to noise power ratio Eb/N0.

the nonlinear HPA than OFDM modulation. Using the 3th

Daubechies wavelet increases the SNR of 1 dB at BER =10−3 compared to OFDM modulation. WPM using the 20th

Daubechies wavelet is very close to the ideal case with nodistortion.PAPR reduction capability and the BER of the proposedtechnique depend on the choice of the wavelet. Specifically,the amount of PAPR reduction increases as the wavelet’s ordergets larger. We can achieve low BER with the 20th Daubechieswavelet.

VI. CONCLUSIONS

In this paper, a novel proposition based on the characteristicsof the pulse shaping has been introduced to reduce PAPR inmulticarrier schemes. Applied to wavelets, this proposal leadsto the multicarrier modulation based on wavelet pulse shaping.The proposed modulation is compared to OFDM modulationand achieves a significant reduction in PAPR for a low numberof channels.Furthermore, it is possible to trade-off the amount of PAPRreduction and the complexity by using different versions ofwavelet used in WPM. An important advantage of proposedtechnique is that wavelets allow flexibility in the system’sdesign and their choice depends on the transmission envi-ronment. We achieve an efficient reconfigurable multicarriermodulation.

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