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Digital Signal Processing 17 (2007) 272–279

www.elsevier.com/locate/dsp

Novel study on PAPRs reduction in wavelet-based multicarriermodulation systems

Haixia Zhang a,∗, Dongfeng Yuan a, Matthias Pätzold b

a School of Information Science and Engineering, Shandong University, PR Chinab Faculty of Engineering and Science, Agder University College, Norway

Available online 5 September 2006

Abstract

Utilizing the advantage of concentrating energy to certain subspaces of the discrete wavelet transform (DWT), a novel peak-to-average power ratio (PAPR) reduction method for wavelet-based multicarrier modulation (WMCM) systems is proposed andpresented in this paper. Both theoretical analyses and simulations are carried out to prove the feasibility of the proposed method.PAPRs are investigated by employing different thresholds. The distortions of the wavelet transform caused by the threshold aremeasured in terms of the mean square error (MSE). This paper also investigates the bit error rate (BER) performance of thresholdcontrolled WMCM systems. All results given in this paper are based on a limited class of wavelets. However, this method can beadapted to WMCM system with other kinds of wavelets as well, since almost all kinds of wavelets have this similar characteristicused in the threshold control method. Therefore, we conclude that the proposed threshold method is an efficient technique forWMCM systems.© 2006 Elsevier Inc. All rights reserved.

Keywords: PAPRs; Threshold; Wavelet; Multicarrier modulation

1. Introduction

The next generation of wireless systems will require higher data quality than current cellular mobile radio systemsand should provide higher bit rate services. In other words, the next generation of wireless systems are supposedto have a better quality and coverage, be more powerful and bandwidth efficient, and be deployed in diverse en-vironments [1]. The fundamental phenomenon which makes reliable wireless transmission difficult is time-varyingmultipath fading. Theoretically, the most effective technique to mitigate multipath fading in a wireless channel istransmitter power control [2]. If the channel conditions, as experienced by the receiver on one side of the link, areknown at the transmitter side, then the transmitter can predistort the signal in order to diminish the impact of thechannel. Multicarrier modulation (MCM) has been proposed to be an ideal way to combat the impact from the chan-nel states. Orthogonal frequency division multiplexing (OFDM) can be considered as a special case of MCM. It hasbeen applied to systems such as DVB, DAB, HiperLAN/2 [3] and digital subscriber line standard ADSL [3,4] for itspowerful capability of handling high bit rate transmissions and for combating multipath fading. One of the key tech-

* Corresponding author.E-mail address: [email protected] (H. Zhang).

1051-2004/$ – see front matter © 2006 Elsevier Inc. All rights reserved.doi:10.1016/j.dsp.2006.08.002

H. Zhang et al. / Digital Signal Processing 17 (2007) 272–279 273

nical points of OFDM is the use of guard intervals (GIs) or cyclic prefixes (CPs), which convert the linear convolutioninto a cyclic one. This kind of conversion can be exploited to simplify the equalization in the receiver by substitutingfrequency domain equalization for time domain equalization [3,5]. The other advantage of using GIs or CPs lies in thefact that intersymbol interferences caused by sidebands of pulse forming filter can be diminished. As the GI or CPsmust be discarded in the receiver, system throughput is greatly reduced. Therefore, to increase the information datarate, some other orthogonal bases with higher spectrum containment must be found.

In recent years, wavelet bases have been introduced into the communication field as an alternative approach toMCM [6–12]. In [6], Lindsey discusses the possibility of applying multidimensional signals to orthogonally mul-tiplexed communications. The inter-symbol interference/inter-carrier interference (ISI/ICI) of MCM schemes withwavelet base and Fourier base is compared in [9]. Reference [12] presents a wavelet-based OFDM (DWT-OFDM)that can support much higher spectrum efficiency than Fourier-based OFDM (DFT-OFDM). Reference [13] investi-gates the bit error rate (BER) performance of MCM systems with different orthogonal bases and shows that they haveadvantages under specified channel conditions.

A significant problem in MCM systems is the possibility of high peaks in the transmitted signals, which emphasizesthe necessity to increase the dynamic range of the corresponding linear amplifier of the communication system or,otherwise, to clip the signals. The latter yields an undesirable intercarrier and out-of-band radiation [10,12,14] andresults in degraded system performance. Thus, one wishes to avoid transmission schemes using orthogonal base withhigh peaks. We have investigated PAPR reduction methods in MCM systems with different orthogonal bases. Ourresults show that Fourier based MCM (OFDM) schemes outperform those based on wavelets with respect to PAPRreductions regardless of which wavelet is chosen [15]. To the best of the authors’ knowledge, there is no report aboutPAPR reduction in WMCM systems in the open literature. The motivation of this paper is to investigate wavelet basedMCM techniques thoroughly and make such schemes much more useful. Here, we propose a novel threshold controlmethod to reduce the PAPR of WMCM systems using a specific property of the wavelet transform.

The remaining parts of this paper are organized as follows. In Section 2, we first consider the contracture of waveletbase for WMCM system and give an example of two level wavelet transform. Then in Section 3, we depict the PAPRreduction method. With theoretical analysis, we show that the proposed method is an efficient method to reduce PAPRin WMCM system. The distortion of the wavelet transform caused by the threshold is measured in terms of the MSEin Section 4. Numerical simulation results of BER distortion caused by PAPR reduction method in WMCM systemare also shown in Section 4.

2. Construction of wavelet bases

The wavelet transform is a kind of technique derived from the Fourier transform. The most important differencebetween these two transformations is that individual wavelet functions are localized in space, while Fourier sine andcosine functions are not. This localization feature of wavelet, along with wavelets’ localization of frequency, provideslots of special characteristics. This makes wavelet transforms different from Fourier transforms. It can provide mainsidelobes of much lower magnitude than those of Fourier transforms [16]. This is also one of the reasons why we haveused wavelet bases to modulate symbols in MCM systems.

In the following, let Z+ denote the set of nonnegative integers, i.e., Z+ = {0,1,2, . . .}. The wavelet packets aredefined recursively by a sequence of functions using pairs of quadrature mirror filters (QMFs) h(k) and g(k) oflength L:

w2n(t) = √2

∑k∈Z+

h(k)wn(2t − k), (1)

w2n+1(t) = √2

∑k∈Z+

g(k)wn(2t − k), n ∈ Z+. (2)

The sequences h(k) and g(k) correspond to the discrete impulse response of a QMF bank with perfect reconstruction,where the relationship g(k) = (−1)kh(1 − k) holds. The function w0(t) is the unique fixed point of the first two-scaleequation obtained from (1) when n = 0. It is exactly the scaling function from a multiresolution analysis (MRA).Similarly, w1(t) is the corresponding wavelet function from (2). The elements of the set {wn(t)}n∈N are called thewavelet packet functions, which have two useful properties. Denote 〈x, y〉 the integral operator, then, we have

274 H. Zhang et al. / Digital Signal Processing 17 (2007) 272–279

Fig. 1. The decomposition and reconstruction of wavelet functions.

⟨wn(x − j),wn(x − k)

⟩ = δj,k, j, k ∈ Z+, (3)⟨w2n(x − j),w2n+1(x − k)

⟩ = 0, j, k ∈ Z+. (4)

Equation (3) states that each individual wavelet packet function is orthogonal with all its nonzero translations. Thisproperty will be utilized to eliminate ISI. Equation (4) states that every pair of packet functions from the same parentpacket are orthogonal at all translations. Therefore, {wn(t)}n∈N is a set of orthogonal functions. In WMCM, thebaseband sequence {ck} is obtained from the wavelet reconstruction algorithm according to

c(j)k = 1√

2

∑l∈Z+

[g(n − 2l)c

(j−1)l + h(n − 2l)d

(j−1)l

], (5)

where {c(j)k } and {d(j)

k } are the kth symbols of the j th subband. The transmitted baseband signals are again transformedinto subband signals using the wavelet decomposition algorithm. Thus,

c(j−1)k = 1√

2

∑l

r(2k − l)c(j)l , (6)

d(j−1)k = 1√

2

∑l

η(2k − l)c(j)l , (7)

where {rk} and {ηk} are the decomposition sequences of the wavelet. By choosing different {rk} and {ηk}, we can getdifferent kind of wavelets.

Figure 1 gives the decomposition and reconstruction process of two-level wavelet functions. The symbol “↓” meansdecomposition and “↑” means reconstruction.

3. A novel PAPR reduction method

As mentioned in the first part of this paper, the BER performance of MCM systems with a wavelet orthogonal baseis better than that employing a Fourier base in certain environments. However, their PAPR is higher than that of MCMsystem employing Fourier base. As it is known, if the transmitted peak power is limited, no matter by regularity orapplication constraint, the average power allowed by MCM will be reduced. This will in turn reduce the transmissionrange of MCM systems. Thus, to maintain spectral efficiency, a linear amplifier with a large dynamic range is needed.This would degrade the power efficiency greatly, which should be avoided. Therefore, to increase the efficiency ofwavelet based MCM systems, methods are required to reduce the high PAPR to economize the power consumption.

The discrete wavelet transform (DWT) is a type of batch processing, which analyses a finite length time domainsignal by breaking up the initial domain in two parts: the detail and approximation information [17]. The approxima-tion domain is successively decomposed into detail and approximation domains. We use the properties of the discretewavelet transform that the DWT is scattered. This means only few coefficients of DWT dominates the representation.This property is widely used in image processing, such as wavelet de-noising [18–20]. Using this properly in WMCMsystems, we can reduce the PAPR with little reconstruction loss. The theoretical analysis is as follows.

The transmitter of a WMCM system is shown in Fig. 2. Let x(n) be the signal obtained after orthogonal modulation.Then, the PAPR can be defined as

PAPR(dB) = 10 log10maxn{|x(n)|2}

E{|x(n)|2} = 10 log10max{|x(n)|2, n = 0,1, . . . ,N − 1}

1 ∑N−1 |x(n)|2 . (8)
N n=0


Fig. 2. Transmitter of a wavelet based WMCM system.

Fig. 3. Receiver of a wavelet based MCM system.

Since wavelet transforms always concentrate energy on some given number of bases, we can introduce{xT (n) = 0, if |x(n)|2 < T,

xT (n) = x(n), if |x(n)|2 � T ,(9)

a threshold T and compare it with the energy of each orthogonal base. Then, we define a sequence xT (n).Assume that there are M basis functions whose energies are smaller than the threshold T . Then, let us define a new

sequence x1(n):

x1(i) = xT (n), xT (n) = 0, i = 0,1, . . . ,N − M − 1; n = 0,1, . . . ,N − 1. (10)

Reconsidering (8), the PAPR can now be written as

PAPRN = 10 log10 g(n), g(n) = max{|x1(n)|2, for n = 0,1, . . . ,N − M − 1}1

N−M

∑N−M−1n=0 |x1(n)|2 . (11)

Let the threshold T fulfil the inequality T < 1N

∑N−1n=0 |x(n)|2.

Then, there exist:

1

N

N−1∑n=0

|x(n)|2 <1

N − M

N−M−1∑n=0

|x1(n)|2. (12)

From (12), (11), and (8), we obtain

PAPR > PAPRN. (13)

Equation (12) is reasonable, since the denominator is reduced more than the numerator. As we mentioned, the wavelettransform can concentrate the energy on certain basis functions, and the remaining basis functions only carries verylittle energy, so that the condition imposed on the threshold T can always be satisfied. Therefore, we can exploitthis properly of the wavelet transform to reduce the PAPR. Equations (8)–(13) show that this method is an effectiveone in reducing the PAPR. The only difficulty with this method is choosing proper threshold T . From (9) and (8),it is easy to realize that the higher the value of T is, the larger M becomes, which results in a lower PAPR value.On the other hand, the higher the value of T is, the higher are the distortions of transmitted signals, which resultsin higher information loss and in BER performance distortions. Given a certain value of T , different wavelet basisfunctions perform different in reducing PAPR due to their own characteristics. In this paper, we choose Haar waveletsas orthogonal base and we investigate the BER performance of MCM system for different values of T .

The proposed PAPR reduction method can easily be implemented in the receiver. We only need an additionalforward transfer channel which carries the label information of subspaces whose energy is set to zero. Then, we padzeros in the subspaces which are set to zero at the transmitter. After the serial to parallel transformation, DWT can beperformed to get de-orthogonal signals (see Fig. 3).


4. Mean square error caused by threshold

To test the proposed threshold method in practice, we simulated a threshold controlled WMCM system with theDaubechies (dB), Biorthogonal (bio), and Haar wavelets. Let MSE E2 be a measure of the performance loss ofWMCM systems with threshold control according to

E2 =∑N

n=1(yT(n) − y(n))2

N, (14)

where yT(n) denotes the signal after the discrete wavelet transform in systems with threshold control, and y(n) denotesthe corresponding signal in systems without threshold control. In systems without channel coding, what should bedone after obtaining y(n) is that to demodulate the received signal according to the modulation scheme. AssumingBPSK in our WMCM system, we found that the decision results are always correct if E2 < 1 is always satisfied, sincethe decision region is 1 in case of BPSK. The generalization to M-ary modulation schemes is trivial. The MSE E2 ofdifferent wavelet bases introduced by the threshold T is shown in Fig. 4. As we can see from the presented results,Haar wavelet is the best one for our proposed threshold control method. For the same value of T , Haar wavelets resultin the lowest MSE.

In Fig. 4, we show the curves of the MSE vs the threshold T . As a result of threshold control, the mean andmaximum reduction of the PAPR value is shown in Fig. 5, from which it can be seen that the proposed thresholdmethod can result in significant improvements with respect to PAPR reduction. The range of PAPR reduction enlargeswith increasing the threshold T , which can easily be explained by (11) and (12). Increasing T results in a largerdenominator, and thus in a lower PAPR. The only sufficient condition that guarantees correctness of the results isT < 1

N

∑N−1n=0 |x(n)|2. So, our system can obtain a lower PAPR, since we have premeditated this inequality in our

system. This explains the simulation results shown in Fig. 5. Taking Biorthogonal 5.5 as an example, it can be seenfrom Figs. 4 and 5 that a maximum reduction of PAPR over 8 dB can be obtained when using a threshold T of 0.25.In this case, the MSE caused by using the threshold method is only about 0.0125, which is neglectable comparing thedecision region of BPSK. Another important point which we can obtain from our simulated results is that differentkinds of wavelet perform differently under the same conditions. Biorthogonal wavelets can bring much more PAPRdecrease than the Daubechies wavelets, at the same time, they induce lower MSE. The Daubechies wavelet 10 cangive more PAPR reduction but brings higher MSE compared to the Daubechies wavelet 4. We can also find that theperformance is also different when using the same wavelet but with different parameters.

Fig. 4. Mean square error E2 in terms of the threshold T .


Fig. 5. The minimum and maximum reduction of PAPR as a function of the threshold T .

Table 1Mean and maximum PAPR reduction range with variant T

T 0.1 0.2 0.3 0.4

PAPR reduction (dB) Mean 0.4577 0.8050 1.1440 1.5059Max 1.0189 1.6974 2.1440 2.7476

5. Results analysis

In wavelet based MCM systems with threshold control, the Haar wavelets are chosen as an example to demonstratethe impact of the threshold T on the BER performance. The input serial bit stream is transferred to 512 parallelsub-serial bit streams. In other words, the subcarrier number is N = 512. Assuming that the transmission channel isperfectly estimated, and that the signal is perfectly recovered from the channel distortions, we study the performanceof the proposed method with respect to the PAPR reduction range and the BER distortion. In Table 1, the PAPRreduction is measured in terms of maximum value and mean value. These results are obtained from WMCM systemwith Haar wavelet.

The curves depicted in Fig. 6 show the relationship between the BER and the SNR for various values of thethreshold T . Through comparison of the BER of MCM systems with and without the threshold method, we can findthat only a small BER performance loss is obtained when employing a sufficiently small threshold T . For example,the BER performance decreases only from 0.75 × 10−4 to 0.55 × 10−4 if the SNR equals 12 dB, T = 0.2. ComparingTable 1 with Fig. 6, it can easily be seen that to get the same BER performance in systems with and without thethreshold scheme, only 0.5 dB additional transmission power should be added to system with threshold method if theSNR equals 12 dB, T = 0.3. At the same time, a mean value 1.1440 dB and a maximum value 2.1440 dB can besaved. So, as a whole, the proposed method can economize transmission power efficiently.

6. Conclusions

A novel PAPR reduced method, threshold control method, in a WMCM system is proposed and presented. Themethod makes use of the special property of discrete wavelets transmission that only a few numbers of large coeffi-cients dominates the representation. The main goals are to decrease PAPR of a WMCM system and make the WMCM


Fig. 6. BER performance with variant T .

a much more feasible MCM system. In addition, we introduce a variable E2 to measure the mean square error causedby the proposed method. From both theoretical and simulation aspects, we prove that the proposed method is aneffective way in PAPR reduction for wavelet based MCM systems. All the results shown in this paper are based onlimited kinds of wavelets. However, this threshold control PAPR reduction method can be adapted to WMCM systemsusing other wavelets as an orthogonal base as well, since wavelets have the similar property of concentrating energyin certain subspaces. So, we conclude that the threshold control PAPR reduction method can be an efficient way toreduce the PAPR of WMCM systems. It should be mentioned that we haven’t considered the multipath interferencein this paper.

Acknowledgments

The authors would like to say thanks to the following foundations for their financial support: National ScientificFoundation of China (No. 60372030); National Mobile Communications Research Laboratory, Southeast University;State Key Laboratory of Integrated Services Network, Xidian University.

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Haixia Zhang received her B.E. degree from the Department of Communication and Information Engineering, Guilin Univer-sity of Electronic Technology, China, in 2001, got her M.Eng degree in communication and information systems in the School ofInformation Science and Engineering, Shandong University, China, in June 2004. Now she is a Ph.D. candidate joint in Schoolof Information Science and Engineering, Shandong University, China, and Munich University of Technology, Germany. Till now,she has published more than thirty papers in journals and conference proceedings. Her current research interests include channelcoding, multicarrier modulation, MIMO techniques, and cross layer design.

Dongfeng Yuan received his M.S. degree from the Department of Electrical Engineering, Shandong University, China, in 1988,and got his Ph.D. degree from the Department of Electrical Engineering, Tsinghua University, China, in January 2000. Currentlyhe is a full professor and deputy dean in the School of Information Science and Engineering, Shandong University, China. He is asenior member of IEEE and a senior member of China Institute of Communications and China Institute of Electronics.

From 1993 to 1994, he was a visiting professor in the Electrical and Computer Department at the University of Calgary, Alberta,Canada, from 1998 to 1999 a visiting professor in the Department of Electrical Engineering in the University of Erlangen, Germany,from 2001 to 2002 a visiting professor in the Department of Electrical Engineering and Computer Science in the University ofMichigan, Ann Arbor, USA. He has published over 200 papers in technical journals and at international conferences in his researchfield recently. His research interests include multilevel coding and multistage decoding, space-time coded modulation, turbo codes,LDPC codes, OFDM techniques for high speed transmission in 4G, and unequal error protection characteristics in multimediatransmission in fading channels.

Matthias Pätzold received the Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering from Ruhr-University Bochum,Bochum, Germany, in 1985 and 1989, respectively, and the Habil. degree in communications engineering from the TechnicalUniversity of Hamburg-Harburg, Hamburg, Germany, in 1998. From 1990 to 1992, he was with ANT Nachrichtentechnik GmbH,Backnang, Germany, where he was engaged in digital satellite communications. From 1992 to 2001, he was with the Departmentof Digital Networks at the Technical University of Hamburg-Harburg. Since 2001, he has been a full professor of mobile com-munications with Agder University College, Grimstad, Norway. He is author of the books “Mobile Radio Channels—Modelling,Analysis, and Simulation” (in German) (Vieweg, Wiesbaden, Germany, 1999) and “Mobile Fading Channels” (Wiley, Chichester,UK, 2002). His current research interests include mobile radio communications, especially multipath fading channel modelling,multi-input–multi-output (MIMO) systems, channel parameter estimation, and coded-modulation techniques for fading channels.

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