wavelet packet modulation for wireless communications€¦ · published in wireless communications...

PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 1

Wavelet Packet Modulation forWireless CommunicationsAntony Jamin Petri Mahonen

Advanced Communications Chair of Wireless NetworksNetworks (ACN) SA Aachen University

Neuchatel, Switzerland [email protected] [email protected]

Abstract— As proven by the success of OFDM,multicarrier modulation has been recognized as anefficient solution for wireless communications. Wave-form bases other than sine functions could similarlybe used for multicarrier systems in order to providean alternative to OFDM.

In this article, we study the performance of waveletpacket transform modulation (WPM) for transmis-sion over wireless channels. This scheme is shownto be overall quite similar to OFDM, but withsome interesting additional features and improvedcharacteristics. A detailed analysis of the system’simplementation complexity as well as an evaluation ofthe influence of implementation-related impairmentsare also reported.

Index Terms— Wavelet packet modulation, multi-carrier modulation, orthogonal waveform bases

I. I NTRODUCTION

A THOUGH the principle of multicarrier mod-ulation is not recent, its actual use in com-

mercial systems had been delayed until the tech-nology required to implement it became availableat reasonable cost [1]. Similarly, the idea of usingmore advanced transform than Fourier’s as the coreof a multicarrier system has been introduced morethan a decade ago [2]. However, such alternativemethods have not been foreseen as of major interestand therefore have received little attention. With thecurrent demand for high performance in wirelesscommunication systems, we are entitled to wonderabout the possible improvement that wavelet-basedmodulation could exhibit compared to OFDM sys-tems. We address this question in this article bytaking theoretical as well as implementation aspectsinto account.

Several objectives motivate the current researchon WPM. First of all, the characteristics of a mul-ticarrier modulated signal are directly dependenton the set of waveforms of which it makes use.Hence, the sensitivity to multipath channel distor-tion, synchronization error or non-linear amplifiersmight present better values than a corresponding

OFDM signal. Little interest has been given to theevaluation of those system level characteristics inthe case of WPM.

Moreover, the major advantage of WPM is itsflexibility. This feature makes it eminently suitablefor future generation of communication systems.With the ever-increasing need for enhanced per-formance, communication systems can no longerbe designed for average performance while as-suming channel conditions. Instead, new genera-tion systems have to be designed to dynamicallytake advantage of the instantaneous propagationconditions. This situation has led to the studyof flexible and reconfigurable systems capable ofoptimizing performance according to the currentchannel response [3]. A tremendous amount ofwork has been done recently to fulfill this re-quirement at the physical layer of communicationsystems: complex equalization schemes, dynamicbit-loading and power control that can be used todynamically improve system performance. WhileWPM can take advantage of all those advancedfunctionalities designed for multicarrier systems,we show that it benefits also from an inherentflexibility. This feature together with a modularimplementation complexity make WPM a potentialcandidate for building highly flexible modulationschemes.

Wavelet theory has been foreseen by severalauthors as a good platform on which to build mul-ticarrier waveform bases [4], [5], [6]. The dyadicdivision of the bandwidth, though being the keypoint for compression techniques, is not well suitedfor multicarrier communication [7]. Wavelet packetbases therefore appear to be a more logical choicefor building orthogonal waveform sets usable incommunication. In their review on the use of or-thogonal transmultiplexers in communications [2],Akansuet al. emphasize the relation between filterbanks and transmultiplexer theory and predict thatWPM has a role to play in future communicationsystems.


Lindsey and Dill were among the first to pro-pose wavelet packet modulation. The theoreticalfondation of this orthogonal multicarrier modu-lation technique and its interesting possibility ofleading to an arbitrary time-frequency plane tilingare underlined [8]. Lindsey has further completedthe work in this area by presenting additionalresults [9]. In particular, it is shown in this lastpaper that power spectral density and bandwidthefficiency are equal for WPM and standard QAMmodulation. Moreover, WPM is placed into a mul-titone communication framework including alterna-tive orthogonal bases such as M-band wavelet mod-ulation (MWM) and multiscale modulation (MSM).

Additional research work on more realistic mod-els of WPM-based transceivers has also been car-ried out. Maximum likelihood decoding for waveletpacket modulation has been addressed by Suzukiet al. [10]. Simulations for both flat fading andmultipath channels have been performed for a re-ceiver using a channel impulse response estimator,an MLSE and Viterbi decoder. It is shown that theuse of wavelet packet specific characteristics leadsto improved results. The study of an equalizationscheme suited for WPM has also been done byGracias and Reddy [11]. In addition, the multi-resolution structure of the wavelet packet wave-forms have shown to offer opportunity for improvedsynchronization algorithms [12], [13]. This hasbeen exploited for instance in the case of WPM byLuise et al. [14]. All together, the research workprovides insight on the theoretical performanceof an actually implemented WPM-based modem,thus contributing to complete the research workrequired to lead to a fully implementable WPM-based communication system.

We focus in this article on reviewing the ad-vantages of wavelet packet modulation in wirelesscommunications. Theoretical background on thismodulation scheme is first recalled. Specific issuesof using such sets of waveforms for multicarriercommunication systems are underlined, and an ex-haustive comparison with OFDM is made. Specialemphasis on the flexibility of this scheme is given.Section III reports the performance results of WPMin several typical wireless channel models. The caseof multipath channels is detailed, and the perfor-mance versus complexity of channel equalizationfor such channels is studied. Results obtained arecompared with those obtained with a classicalOFDM scheme, since it is currently the referencein multicarrier systems. Section IV focuses on theeffect of interference and implementation impair-ments. Then, the implementation complexity ofa WPM system is also reported, since this is akey issue in wireless communication systems. We

conclude by underlining the communications areawhere WPM appears to be a promising technologyfor future generation transceivers.

II. COMMUNICATION SYSTEM ARCHITECTURE

The simplified block diagram of the multicarriercommunication system studied in this article isshown in Figure 1. The transmitted signal in thediscrete domain,x[k], is composed of successivemodulated symbols, each of which is constructedas the sum ofM waveformsϕm[k] individuallyamplitude modulated. It can be expressed in thediscrete domain as:

x[k] =∑

s

M−1∑m=0

as,m ϕm

[k − sM

](1)

where as,m is a constellation encodeds-th datasymbol modulating them-th waveform. DenotingT the sampling period, the interval[0, LT − 1]is the only period whereϕm[k] is non-null forany m ∈ 0..M − 1. In an AWGN channel, thelowest probability of erroneous symbol decisionis achieved if the waveformsϕm[k] are mutuallyorthogonal, i.e.

〈ϕm[k], ϕn[k]〉 = δ[m− n], (2)

where〈·, ·〉 represents a convolution operation andδ[i] = 1 if i = 0, and0 otherwise.

In OFDM, the discrete functionsϕm[k] arethe well-known M complex basis functionsw[t]exp

(j2π m

M kT)

limited in the time domainby the window functionw[t]. The correspondingsine-shaped waveforms are equally spaced in thefrequency domain, each having a bandwidth of2πM and are usually grouped in pairs of similarcentral frequency and modulated by a complexQAM encoded symbol. In WPM, the subcarrierwaveforms are obtained through the WPT. Exactlyas for ODFM, the inverse transform is used tobuild the transmitted symbol while theforwardone allows retrieving the data symbol transmitted.Since wavelet theory has part of its origin in filterbank theory [15], the processing of a signal throughWPT is usually referred as decomposition (i.e.into wavelet packet coefficients), while the reverseoperation is called reconstruction (i.e. from waveletpacket coefficients) or synthesis.

In this paper, we limit our analysis to WPTthat can be defined through a set of FIR filters.Though it would be possible to use other waveletsas well, those cannot be implemented by Mallat’sfast algorithm [16] and hence their high complexitymake them ill-suited for mobile communication.The synthesis discrete wavelet packet transform


IWPT WPT

Con

stel

latio

n en

code

r

Channel

n(t)

Con

stel

latio

n de

code

r

Ser

ial t

o pa

ralle

l

Par

alle

l to

seria

l

Fig. 1. Wavelet packet modulation functional block diagram

constructs a signal as the sum ofM = 2J wave-forms. Those waveforms can be built byJ succes-sive iterations each consisting of filtering and up-sampling operations. Noting〈·, ·〉 the convolutionoperation, the algorithm can be written as: ϕj,2m[k] =

⟨hrec

lo [k], ϕj−1,m[k/2]⟩

ϕj,2m+1[k] =⟨hrec

hi [k], ϕj−1,m[k/2]⟩

with ϕ0,m[k] =

1 for k = 10 otherwise

∀m

where j is the iteration index,1 ≤ j ≤ Jand m the waveform index0 ≤ m ≤ M − 1.Using usual notation in discrete signal processing,ϕj,m[k/2] denotes the upsampled-by-two versionof ϕj,m[k]. For the decomposition, the reverseoperations are performed, leading to the comple-mentary set of elementary blocks constituting thewavelet packet transform depicted in Figure 2.In orthogonal wavelet systems, the scaling filterhrec

lo and dilatation filterhrechi form a quadrature

mirror filter pair. Hence knowledge of the scalingfilter and wavelet tree depth is sufficient to designthe wavelet transform [15]. It is also interestingto notice that for orthogonal WPT, the inversetransform (analysis) makes use of waveforms thatare time-reversed versions of the forward ones. Incommunication theory, this is equivalent to usinga matched filter to detect the original transmittedwaveform.

A particularity of the waveforms constructedthrough the WPT is that they are longer thanthe transform size. Hence, WPM belongs to thefamily of overlapped transforms, the beginningof a new symbol being transmitted before the

dec lo h 2

2 dec hi h rec

hi h

rec lo h

2

2

Fig. 2. Wavelet packet elementary block decomposition andreconstruction

previous one(s) ends. The waveforms beingM -shift orthogonal, the inter-symbol orthogonality ismaintained despite this overlap of consecutive sym-bols. This allows taking advantage of increasedfrequency domain localization provided by longerwaveforms while avoiding system capacity loss thatnormally results from time domain spreading. Thewaveforms length can be derived from a deteilledanalysis of the tree algorithm [16, Section 8.1.1].Explicitly, the wavelet filter of lengthL0 generatesM waveforms of length

L = (M − 1) (L0 − 1) + 1. (3)

In Daubechie’s wavelet family [17] for instance, thelength Lo is equal to twice the wavelet vanishingorderN . For the order2 Daubechie wavelet, L isequal to4, and thus a32 subcarrier WPT is com-posed of waveforms of length94. This is thereforeabout three times longer than the correspondingOFDM symbol, assuming no cyclic prefix is used.

The construction of a wavelet packet basis isentirely defined by the wavelet scaling filter, henceits selection is critical. This filter solely determinesthe specific characteristics of the transform. Inmulticarrier systems, the primary characteristic ofthe waveform composing the multiplex signal isout-of-band energy. Though in an AWGN channelthis level of out-of-band energy has no effect onthe system performance thanks to the orthogonalitycondition, this is the most important source ofinterference when propagation through the channelcauses the orthogonality of the transmitted signalto be lost. A waveform with higher frequencydomain localization can be obtained with longertime support. On the other hand, it is interestingto use waveforms of short duration to ensure thatthe symbol duration is far shorter than the channelcoherence time. Similarly, short waveforms requireless memory, limit the modulation-demodulationdelay and require less computation. Those tworequirements, corresponding to good localizationboth in time and frequency domain, cannot bechosen independently. In fact, it has been shownthat in the case of wavelets, the bandwidth-duration


Full name Abbreviated Vanishing Lengthname order L0

Haar haar 1 2Daubechie [17] dbN N 2NSymlets symN N 2NCoiflet coifN N 6NDiscrete Meyr dmey − 62

TABLE I

SUMMARY OF WAVELET FAMILY CHARACTERISTICS.

product is constant1 [18]. This is usually referredto as the uncertainty principle.

We limit our performance analysis to WPT basedon widely used wavelets such as those given in[17]. While there are numerous alternative waveletfamilies that could be used as well, a comparativestudy would deserve a separate publication byitself. As previously mentioned, we are essentiallyinterested in wavelets leading to fast transformsthrough the tree algorithm. The primary waveletfamily we have been using in our research is theone from Daubechie [17, p. 115], since it presentswavelets with the shortest duration. Furthermore,their localization in the frequency domain can beadequately adjusted by selecting their vanishingorder, as it is illustrated by two of the curves plottedin Figure 4. The labeldbN is used in the rest ofthis article to refer to WPT based on Daubechiewavelet of vanishing orderN .

Further work carried out after her initial re-search led to a near symmetric extension of thedbN family, since it was a demanded feature inapplications such as image compression [17, pp.254-257]. Following the notation in [19], we willrefer to this family asSymlets, denoted assymN.The last wavelet we will make reference to hasbeen constructed at R. Coiffman demand for imageprocessing applications [17, pp. 258-259] and willbe denotedcoifN. This wavelet is near symmet-ric, has 2N moments equal to0 and has length6N − 1. Table I summarizes the characteristics ofthe wavelets and families mentioned above. Order4 members of those two last families are as wellplotted in Figure 4.

Finally, a minor difference between OFDM andWPM remains to be emphasized. In the former, theset of waveforms is by nature defined in the com-plex domain. WPM, on the other hand, is generallydefined in thereal domain but can be also definedin the complexdomain, solely depending of thescaling and dilatation filter coefficients [20]. Sincethe most commonly encountered WPT are definedin the real domain, it has naturally led the authors to

1Both measurement in the frequency and time domain aretaken as the domain in which most of the energy of the signalis localized.

use pulse amplitude modulation for each subcarrier.It is nevertheless possible to translate theM realwaveform directly in the complex domain. Theresulting complex WPT is then composed of2Mwaveforms forming an orthogonal set. While this ismathematically trivial, this simple fact has not beenclearly emphasized in the telecommunication liter-ature. This is of particular interest in systems wherea baseband signal is desirable. The AsymmetricalDigital Subsciber Line (ADSL) standard falls forinstance in this category [21]. Currently, a realbaseband signal is constructed by using a2M -DFTfed with M conjugate pair modulation symbols[22]. An WPT of sizeM would therefore providean equivalent modulation scheme at roughly halfcomplexity.

A. Interesting features for wireless communication

In addition to the basic features already men-tioned, the WPT presents interesting advantagesfrom a system architecture point-of-view. For in-stance, the dependence of the WPT on a generatingwavelet is a major asset in communication appli-cations. An improved transmission integrity maybe achieved with the aid of diversity [23, Section14.4]. Space-, frequency-, and time-diversity arethe most common physical diversities exploited. Inaddition, WPM provides asignaldiversity which issimilar to spread spectrum systems to some extent.Such diversity is in fact making joint use of timeand frequency space. Practically, using two differ-ent generating wavelets allows us to produce twomodulated signals that can transmitted on the samefrequency band and suffer from reduced interfer-ence only. The amount of cross-correlation betweenthe signals is directly dependent on the generat-ingwavelets chosen. The best method to be usedto select suitable wavelets is a topic to be studiedfurther. This particular feature could be exploitedfor instance in a cellular communication system,where different wavelets are used in adjacent cellsin order to minimize inter-cell interference.

Another interesting feature of WPM is directlyrelated to the iterative nature of the wavelet packettransform. The number of subcarriers in OFDMsystems is usually fixed at design time, and it isespecially difficult to implement an efficient FFTtransform having a programmable size. This lim-itation does not exist in WPT since the transformsize is determined by the number of iteration of thealgorithm. From an implementation point of view,this can be made configurable without significantlyincreasing the overall complexity. This permitson-the-flychange of the transform size, which can berequired for different reasons, for instance to recon-figure a transceiver according to a given communi-


Frequency

Time

4 , 4 w

0 , 4 w

1 , 4 w

3 , 3 w

1 , 3 w

10 , 5 w 11 , 5 w

1 , 1 w

) ( 1 , 4 w t D

) ( 1 , 4 w f D

Fig. 3. Time-Frequency plane division with semi-arbitrary wavelet packet tree pruning

cation protocol2 or, in the event of the appearenceof cognitive radio [24], the transform size could beselected according to the channel impulse responsecharacteristics, computational complexity and linkquality.

The last property of the WPT is the semi-arbitrary division of the signal space. Waveletpacket transform still leads to a set of orthogonalfunctions, even if the construction iterations are notrepeated for all sub-branches. From a multicarriercommunication system perspective, this maps intohaving subcarriers of different bandwidth. More-over, since each subcarrier has the same time-frequency (TF) plane area, the increase of band-width is bounded to a decrease of subcarrier symbollength. This is illustrated in Figure 3, where a non-regular decomposition tree is shown together withthe corresponding time-frequency plane division.This characteristic of WPM has been referred to inthe literature as amultirate system [25], althoughthe term might be misleading in this case. While thetransmission over the channel is effectively done atdifferent symbol rates, the corresponding subcar-rier throughput remains identical due to the con-stant subcarrier bandwidth-duration product. Froma communication perspective, such a feature isusable for systems that must support multiple datastreams with different transport delay requirements.A logical channel requiring lower transport delaycould make use of a wider subcarrier, while somesignalling information could be carried within anarrower bandwidth. Especially, those narrow sub-

2OFDM-based standards make use of transform sizes rangingbetween32 and8192 subcarriers.

carriers could be used for synchronization purposein order to take advantage of longer symbols thatrequire a small amount of bandwidth. Alternatively,the authors have carried out some research workon the selecting the transform tree according tothe channel impulse response in the frequencydomain. Preliminary results have not shown signif-icant improvement in transform complexity versuslink BER, but further work on this issue is stillongoing [26].

All together, WPM presents a much higher levelof flexibility than current multicarrier modulationschemes. This makes WPM a candidate of choicefor reconfigurable and adaptive systems such arethe ones likely to compose the next generation ofwireless communication devices.

III. PERFORMANCE IN VARIOUS CHANNEL

MODELS

We analyze in this section the performance ofWPM in diverse propagation channels. The resultspresented have been obtained by simulations only,due to the fact that no analytical expressions areavailable.

The derivation of a WPM link BER over anAWGN channel is a trivial issue since, identicallyto OFDM, orthogonality between subcarriers en-sures the link performance is solely dependent onthe constellation used on each subcarrier. The caseof a Ricean channel can be derived similarly, as-suming that the channel response is time-invariantfor the duration of a whole WPM symbol. Thisis however a less conservative assumption thanfor OFDM since WPM symbols are several times


longer than their OFDM counterparts. The case oftransmission through the multipath channel henceremains to be studied more in depth.

A. WPM without equalization

Here we present on comparative results betweenWPM and OFDM schemes without channel equal-ization. While the absence of equalization in a realsystem would yield a poor performance in mostchannels, it is nevertheless of interest to considerthis case here. The main motivation is to gaininsight on the distortion caused by a given channelto the different modulated signals. Moreover, sinceoptimal equalization schemes for both are differ-ent, the definition of equivalence between systemswould require the choice of appropriate comparisoncriteria.

With the same concern of comparing equivalentsystems, the OFDM reference system used in thissection has no cyclic prefix. The addition of aprefix is indeed a technique aiming at rendering themultipath distortion easier to cancel and as suchcan be considered as a form of equalization [27,Section 18.6]. Since no similar artifact is availablefor WPM, the comparison is more fair if no cyclicprefix is used. In addition, the introduction of thecyclic prefix leads to a bandwidth efficiency lossand thus this would add to the difference betweenmodulation schemes.

We choose to assume here the simple case of atime invariant 2-path channel model as introducedby Rummler [23, section 14-1-2]. The first pathhas unit power and the second path power is3 dBlower. The relative delay of the second pathτ is asimulation parameter. The BER versus the secondpath delayτ curves of WPM(db2), WPM(db6), andWPM(db10) schemes with 32 QPSK modulatedsubcarriers is shown in Figure 5. All curves arequite similar and thus the increase of the waveletorder has little impact on the performance. Themain differences appear to be in the area of whereτ is lower than 8 samples. For these values, theWPM scheme based on the higher order wavelet isless sensitive to actual delay.

The performance for two different OFDMschemes appears in Figure 5. For the reason dis-cussed, the first has no cyclic prefix(K = 0). Thesecond one has a prefix of OFDM symbol durationnormalized length1

4 which corresponds here to8 samples. Forτ lower or equal to 3 samples,the OFDM scheme without prefix performs slightlybetter than the WPM ones. For longer path delays,OFDM performs similarly to the WPM schemes,with a close resemblance to WPM(db2). OFDMwith a cyclic prefix shows much better performancefor path delays lower than the prefix duration

since inter-symbol interference is removed3. Whenthe path arrival delayτ exceeds the cyclic prefixlength, the overall error rate increases dramatically,becoming even higher than with the other schemesat a path delay higher than 20.

Overall, the raw BER performance of WPMis identical to OFDM with the channel modelassumed here. Additional results not reported herehave shown equivalence for different frequencyselective channels when no equalization is used.Hence the potential of improved performance com-pared to OFDM is bound to the design of anefficient equalization scheme for either modulation.

B. With equalization

The initial interest in multicarrier modulationschemes was the low complexity of the equalizationthey required. In multicarrier systems, equaliza-tion is usually divided into pre- and post-detectionequalizers4, according to their position respective tothe core transform. In OFDM, a pre-equalizer is notrequired if the channel delay spread is shorter thanthe cyclic prefix. For longer delay spread, the pre-detection equalizer aims at shortening the apparentchannel impulse response to a value lower or equalto the cyclic prefix duration. In the case it succeeds,the symbol at the input of the DFT is free frominter-symbol interference. A post-detection equal-izer composed of a single tap per subcarrier is thensufficient to compensate for the channel distortionin the frequency domain. Though this low com-plexity structure is a major advantage of OFDM, itis limited by the fact that for a channel with longimpulse response, the length of the prefix requiredleads to a significant loss in capacity and transmitpower. For WPM however, the use of a cyclic prefixis impossible due to the overlapping of consecu-tive symbols. Hence both inter-symbol interference(ISI) and inter–symbols inter-carriers interference(ISCI) have to be cancelled by the equalizationscheme. While no research work addresses thisissue specifically for WPM, a comparison of theefficiency and complexity of pre-detection, post-detection, and combined equalization techniquesfor DWMT has been studied by Hawrylucket al.[29]. Research results show clearly that the bestresults are obtained by using both pre- and post-detection equalization. Due to the similarity withthis modulation scheme, a similar approach is takenhere and we separately analyze the performance ofboth types of equalization.

3The channel distortion is not cancelled since no equalizationis assumed.

4As in [28], the terms pre- and post-detection equalization areprefered here to time- and frequency-domain equalizers, thoughthey refer to the same functions


1) Pre-detection equalization:The objective ofthe pre-detection equalizer is slightly different be-tween an overlapping multicarrier and an OFDMsystem making use of a cyclic prefix. While inthe latter case, the equalizer attempts to shortenthe perceived channel impulse response to a valuelower than the cyclic duration, the equalization ofWPM aims at minimizing the mean square error atits output.

We compare here the sensitivity of WPM andOFDM to imperfect equalization. To avoid thesystem’s related issues concerning equalizer train-ing, we will further assume that the equalizer tapshave reached their near optimal values. Denotingthe transfer function of the pre-detection equalizerQ(z) and the equivalent discrete channel responseC(z), the overall link from modulator to demodu-lator has an equivalent transfer function of

H(z) = C(z) Q(z). (4)

Considering the case where the equalizationmethod has converged to the valueQ(z), we chooseto approximateH(z) as

H(z) = 1 +Leq−1∑

l=1

q(l) z−l. (5)

In the case of perfect equalization, all the coeffi-cientsq(l) are null. To take into account imperfectequalization, we assume that the set

q(l)

is nor-

mally distributed. We denotedσ2eq the noise power

in theLeq equalizer taps indexed from 1 toLeq−1.The sensitivity of WPM and OFDM can thus becompared as a function of the noise powerσ2

eq inthe equivalent transmission filterH(z). Figure 6displays the results of a WPM link for an averageof 100 randomly generated channels as a functionof the lengthLeq of the resulting filter. The noisepowerσ2

eq has be set to -5 dBc in order to lead to asignificant number of errors. All the schemes usedin this simulation assume 16 subcarriers, 16QAM,and an AWGN channel with 20 dB SNR. De-spite the symbols overlapping, WPM performs verysimilarly to OFDM in such scenarios. Moreover,no significant difference is perceived between theWPM schemes using different wavelet orders.

Alternatively, the effect of the noise powerσ2eq

for the same modulation schemes for a fixed valueof Leq has been studied. While some differencesexist for some given channel, the results confirmthat all schemes asymptotically reach the same per-formance for any value ofLeq andσ2

eq. Thus, WPMand OFDM exhibit identical sensitivity to pre-equalization errors. These results have been reachedunder the assumption that the equalizer taps haveconverged identically for both schemes. Further

work is needed in order to verify the validity ofthis assumption for a particular equalizer trainingmethods with a finite length training sequence.

2) Post-detection equalization:As underlined,post detection equalization is used in OFDM sys-tems in the form of a single tap filter. These tapsaim at inverting the channel transfer function inthe frequency domain. For WPM, the structureof the post-detection equalizer is more complexsince it has to remove both ISI and ISCI. Hence,a combined time-frequency equalization structureis required. Denotingη and ν as the numberof equalizer taps in time and frequency domainsrespectively, we can express the estimated symbolθl

k at the output of the post-detection equalizer as

θlk =

ν2−1∑

∆k=− ν2 +1

η2−1∑

∆l=− η2 +1

q(∆k,∆l) θl+∆lk+∆k. (6)

The total number of operations per subcarriersymbol isη×ν, and the complexity grows rapidly ifa complex structure is used. This is a major concernsince with L0 − 1 overlapping symbols, perfectequalization cannot be reached forη ≤ (L0 − 1).Such equalization complexity is impractical in areal system, hence we limit our analysis to smallerequalizer sizes that give a BER sufficiently low topermit an outer error correction code to reach agood performance.

We choose to compare the performance of theequalization schemes for a typical channel with anexponentially decaying profile. The decay factorτ0

is taken equal to 1 and the total number of pathsis limited to 4. For this channel, the equalizer tapshave been calculated through the steepest descentLMS algorithm [23, Section 11-1-2]. The iterationstep ∆ has been taken equal to10−3 and thenumber of training symbols has be chosen to ensurethe algorithm has converged.

The performance of the equalizer as a functionof its number of tapsη in time domain is givenin Figure 7. The modulation schemes selected hereare WPM(coif1) and WPM(coif4), in order to allowcomparison of the effect of better frequency domainlocalization. Considering first the case where theequalizer does not take into account the symbolson adjacent subcarriers (i.e.ν = 1), the increaseof η brings only very limited improvement, thoughthe difference is slightly more significant for theWPM(coif4) scheme. When the two adjacent sub-carrier symbols are used in the equalizer, increasingη from 1 to 3 taps leads to a larger interferencereduction of about of 1 dB. A further increaseof both ν and η bring no improvement in theWPM(coif1) scheme, and only limited interfer-ence reduction in the case of the more frequency


localized WPM(coif4) scheme. Furthermore, it isinteresting to notice that, while all the modulationschemes have shown identical performance whenno equalization is used, the addition of a single tapequalizer per subcarrier reveals the extra robust-ness of the WPM(coif4) in comparison with theWPM(coif1) scheme.

Overall, the post-equalization of a WPM signalrequires at least a3 × 3-tap structure to removethe largest part of the interference caused by mul-tipath propagation. Less frequency localized WPMschemes require an even more complex structuredue to an increased inter-subcarrier interferencelevel. This of course leads to a complexity penaltywhen compared with OFDM schemes where acyclic prefix can be used and only one tap persubcarrier is needed. Hence, the complexity gainof WPM emphasized later in this article is reducedwhen equalization is considered.

3) Discussion on equalization:The results ob-tained emphasise that equalization of WPM inmultipath channels is more complex than that ofOFDM. This is essentially due to the use of a cyclicprefix that gives an edge to OFDM, when comparedto overlapping multicarrier schemes such as WPM.Thus, the raw link performance of WPM andOFDM is equivalent only when the cyclic prefixis not a practical solution. This corresponds to thecase where the channel impulse response is too longto allow reasonable performance with a low numberof subcarriers. In the latter situation, the systemcomplexity of the WPM and OFDM schemes isroughly equivalent since OFDM requires more thana single tap post-equalizer to remove multipathinterference.

IV. ROBUSTNESS TO INTERFERERS,NON-LINEAR DISTORTION AND SAMPLING

OFFSETS

In addition to the performance of a modula-tion scheme in the given propagation condition,its sensitivity to implementation impairments is acharacteristic that makes it a candidate for an actualcommunication system. Hence we report in thissection the performance of WPM in the presenceof some of the implementation impairments that arecritical in OFDM systems: the sensitivity to non-linear distortion and sampling instant error.

A. Non-linear signal distortion effects

OFDM modulated signals are known to exhibit ahigh peak to average power ratio (PAPR) [30], [31].This implies that power amplifiers must have a highdynamic range in order to avoid causing distortion.This is of concern in power-limited communication

devices, since the energy consumption and cost ofradio-frequency amplifiers increases with the needfor higher linear dynamic range. No work on thisissue has been reported in the literature specificallyfor WPM, though the use of a wavelet packet de-rived pulse over non-linear satellite communicationchannel has been addresses by Doviset al. [32].Thus, we evaluate the actual sensitivity of differentWPM schemes to non-linear distortion.

A model of the non-linear amplifier has to bechosen. Different approaches have been used inthe literature [33], [34]. For most complex models,the numerous parameters have to be derived frommeasurement on an actual device. To keep the anal-ysis here more generic, we will assume a simplenon-linear, memoryless power amplifier model. Formost applications using medium and lower transmitpower, phase distortion can be assumed to be null.Hence, only the amplitude distortion is assumed tobe significant. Such a simple model is characterizedby a transfer function of the form [30, section6.3.1]:

y(x(t)

)= x(t)

[1 +

(x(t)γ

)2p]− 1

2p

(7)

with y(t) the output signal,x(t) the input signal,andγ andp are the backoff margin and the linearityorder of the model respectively. A higher value ofpleads to an output versus input amplitude response,where the output amplitude is smoothly reaching itssaturation value as the input power increases. Foran average quality radio-frequency power amplifier,the best fit value forp is found to be between2and 3 [30, section 6.3.1]. Note thatγ is referredto as the backoff margin and is defined as thedifference between the average and peak power ofthe amplifier5.

Simulation results are displayed in Figure 8,where the link BER for WPM(db2), WPM(sym2),WPM(coif2), WPM(dmey), and OFDM schemesis plotted as a function of the backoff marginγ. An AWGN channel with 20 dB SNR hasbeen assumed. WPM schemes require a backoffvalue 1.5 dB higher than OFDM to achieve equalperformance at10−4 BER. Among the WPMschemes, WPM(dmey) performs best, requiringabout 0.4 dB less power than WPM(db2) andWPM(sym2) schemes at10−4 BER. In order toreach insignificant BER degradation due to non-linearity, the OFDM signal requires a 5 dB backoffvalue. WPM schemes need a slightly higher valueof 6 dB. This 1 dB difference is rather low and

5This differs from the definition commonly used where thebackoff margin is the difference between the amplifier averagepower operating point and−1 dB compression point.


thus should not lead to any significant complexityvariation in the radio-frequency section of a modemusing either modulation scheme.

B. Sampling phase offset

Multicarrier modulation signals are by naturemuch more sensitive to synchronization errors thansingle carrier ones [35]. We focus here on the effectof a non-ideal sampling instant.

The interference caused by a sampling phaseerror∆τ is of three kinds. There is a gain loss in therecovery of the symbol of interest, an inter-carrierinterference term, and an inter-symbol inter-carrierinterference contribution. This last term originatesfrom the symbol overlapping and thus does not ex-ist in OFDM. Hence, the sensitivity of WPM is ex-pected to be higher than for OFDM. In addition, theBER degradation depends on the auto-correlationof each subcarrier waveform, which differs betweensubcarriers. Overall, the BER of the multicarriersignal can be obtained as the average of the BERover each individual subcarrier. Since no analyticalclosed form solution is available, the sensitivityof each modulation scheme to sampling instanterror has been obtained by simulation. Figure 9reports link BER as a function of the samplingphase offset normalized to the sampling period. Forthis particular simulation, the channel is modelledas AWGN with 20 dB SNR. As it was expecteddue to the overlapping of symbols, WPM is moresensitive than OFDM to an imperfect samplinginstant. A BER of10−4 is achieved for OFDMat a normalized sampling error of 27%, whileWPM(db2) requires less than 21%. For the twoother WPM schemes, the error tolerated is slightlylower, with a maximum of about 18%.

Additional results on the sensitivity of WPMto sampling instant error as a function of thewavelet order is given in Figure 10. The differencesbetween the different order of thecoif waveletare limited, except for the lower order one whichtolerates a 2% higher phase error than the oth-ers. There is no direct relation between the orderand sensitivity level. Simulation results forsymwavelets which are not shown here have led toidentical conclusions.

Overall, the higher sensitivity of WPM impliesthat a more robust sampling instant synchronizationscheme is required in comparison with OFDM. Onthe other hand, the multi-resolution structure ofthe wavelet packet waveforms offer the opportu-nity for improved synchronization algorithms [13],[12]. This has been exploited for instance in thecase of WPM by Luiseet al. [14], where theproposed synchronization scheme has shown en-hanced performance that could compensate for the

sensitivity of WPM to synchronization errors. Noimplementation of the algorithm has been reportedand therefore its performance in the presence ofimperfect estimation remains to be studied.

C. In presence of a narrow band interferer

With today’s growing use of wireless systems,the radio-frequency spectrum is becoming moreand more congested by communication signals. Insuch a condition, future modulation schemes haveto cope not only with channel distortion, but alsowith interference originating from other sourcesas well. Moreover, multicarrier modulation is verylikely to encounter in-band interfering signal sinceit is usually best suited for wideband communica-tion systems.

We assume the case of a WPM link communi-cationg over a AWGN channel and exposed to anin-band, un-modulated signal superimposed on thesignal of interest at the receiver. We denotePdist

as the power of the disturbing signal andfdist asits frequency. The amount of interference endowedby each subcarrierk can thus be approximated as[28]

PI = Pdist

M−1∑k=0

∣∣∣Φk(fdist)∣∣∣2 (8)

where Φk in the Fourier transform ofϕk. In thecase of waveforms with null out-of-band energy,the disturbance will be limited to the subcarrierwhose band includes the frequencyfdist. With anactual system, additional disturbance is caused bythe side lobes of the adjacent subcarriers. The sidelobe energy level decreasing with the order of thewavelet, the sensitivity of WPM to a single tonedisturber can thus be reduced by increasing theorder of the generating wavelet.

Results obtained through simulation are shownin Figure 11, where the BER performance ofWPM(db2), WPM(db8), and OFDM links are plot-ted as a function of the disturber normalized fre-quencyFdist. A 16-subcarrier scheme with 16QAMmodulated symbols has been chosen to point outthe effect of the disturber. The disturber has powerequal to the signal of interest, i.e.Pdist = 0 dBc.The curves obtained for all modulation schemesshow clearly that the level of interferance is highlydependent on the actual disturber frequency. Hence,the curve for OFDM shows clearly the higherBER when the disturber frequency corresponds tothe center frequency of one subcarrier. The WPMschemes show a similar effect but with a smoothercurve. Considering the average over the whole fre-quency band, WPM(coif5) and WPM(dmey) out-perform OFDM. The WPM(coif1) scheme howevershows more degradation than OFDM. Overall, the


WPM schemes seem to be able to perform betterthan OFDM when a wavelet with sufficiently lowout-of-band energy is used. For a given wavelet,there appear to be a gain in robustness with theincrease in generating wavelet order.

Additional results are presented in Figure 12where the link BER of the same schemes as pre-viously are given as a function of the disturberrelative powerPdist. Its normalized frequency hasbeen arbitrary chosen to be0.1666, which can beverified from the previous figure to corresponds tothe case where WPM leads to an average gain incomparison with OFDM. The WPM(coif1) schemeperforms overall quite similarly to OFDM, butwith a robustness to disturber about 2 dB lower.The WPM(coif5) and WPM(dmey) schemes, on theother hand, provide a much higher robustness thanOFDM for a disturber power of up to -15 dB.Past this threshold, it is noticeable that OFDMhas instead a lower sensitivity, being undisturbedfor Pdist lower than−20dB while the two WPMschemes require about2 dB less.

In general, the degradation in terms of BER ofthe WPM signal due to a single tone disturber ishighly dependent on both its frequency and power.The results obtained have nevertheless shown thatWPM schemes are capable of high immunity to dis-turbance when higher order wavelets are selected.

V. I MPLEMENTATION COMPLEXITY ESTIMATES

The structure of a WPM-based transceiver isvery similar to that of OFDM. This section reportson the implementation complexity of the WPTsince it is the building block in which the systemsdiffer the most6. Three alternative architectures areconsidered that differ in the implementation of theelementary blocks.

We first consider the direct implementation of theelementary blocks as they are shown in Figure 2. Inthe case of the forward transform, the elementaryblocks consist simply of two up-samplers and twoFIR filters of lengthL0. For the inverse transform,an additional adder in required to combine theoutput of the two filters, so the overall number ofoperations is slightly lower. We consider further theforward transform only, since the case of the re-verse one can be easily derived from it. With filtersof lengthL0, the number of operations required byeach filter is actually equivalent to the complexityrequired by aL0/2-tap long filter thanks to the zerovalues samples inserted by the upsampler. Hence,the number of operations per input sample required

6Optimal systems might differ as well in equalization andsynchronization schemes.

by one elementary block is:

PdirectWPT =

4⌈

L02

⌉− 2 adds

4⌈

L02

⌉mults

. (9)

whered·e denotes the smallest integer higher thanits argument.

Overall, a transform of sizeM = 2j is composedof N (j) elementary blocks per stagej. DenotingR(j) as the input rate of the block of stagej, thenumber of operations for the stagej:

CWPT (j) = N (j)R(j)PdirectWPT (10)

with

N (j) = 2j−1

R(j) = 2J−j,

where the transform input signal rateR(J) isassumed to be equal to unity. The overall numberof operations for the WPT is then

CWPT (J) =J∑

j=1

2J−1 PdirectWPT (11)

CWPT (J) =

(2J − 1)(4

⌈L02

⌉− 2

)adds

(2J − 1)(4

⌈L02

⌉)mults

.

This computational complexity can be furthercompared to that required by the DFT used inOFDM systems. We assume here the implemen-tation of the DFT considered in [36], leading to acomplexity of

CDFT (J) =

2J (2J − 1) addsJ 2J mults

. (12)

Figure 13 displays the complexity of the WPTrelative to the DFT in terms of additions andmultiplications. The number of additions is lowerfor the WPT for transform sizes higher than8 withL0 = 4, and for a transform size greater then32 inthe case where a longer filter withL0 = 16 is used.The number of multiplications on the other handis generally higher for the WPT. With the shorterfilter (L0 = 4), the number of multiplications isequal for both 256-point transforms. In the caseof an implementation with a generic processor,the cost of both operations is identical. Hence,Figure 14 shows the total number of operationsrequired by the WPT, again relative to the DFT fordifferent lengthsL0 of the wavelet filter. The WPTcomplexity is higher for small size transforms, butdecreases as the size of the transform increases.Hence, for a transform size over 64, the number ofoperations for the WFT is lower than for the DFT,even in the case ofL0 = 16.

Overall, the WPT implementation complexity ison the same order of the one required by the


DFT, and might even be lower for medium sizetransforms and wavelet filters of moderate length.The repartition between the addition/multiplicationoperations leads to the conclusion that the com-plexity is in favor of the WPT in the case of animplementation in a general purpose signal proces-sor. In the case of a hardware implementation, thehigher complexity of a multiplier in comparisonwith an adder is likely to reduce the differencebetween the two schemes. The iterative structureof the WPT is nevertheless very well suited tohardware implementation [37]. For each stage ofthe transform, the product of the number of elemen-tary block by their processing rate is constant andequal to2J−1. A completeJ stage transform can beperformed by a single occurrence of the elementaryblock running at a clock speedJ 2J−1 times thesymbol rate or by2J−1 instantiations running atJtimes the symbol rate. This provides an appreciableflexibility in trading-off speed versus silicon area,a feature that is suitable for the future generationof reconfigurable systems.

VI. CONCLUSIONS AND FURTHER RESEARCH

TOPICS

We reviewed in this article the advantages of us-ing WPM for multicarrier communication systems.An in-depth comparison between this new schemeand OFDM has been reported. The performance ofWPM has been shown to be identical to the latterin several reference wireless channels, though at ahigher cost of equalization in multipath channels.Additional results taking into consideration effectsof some non-ideal elements of the system haveshown that WPM is slightly more sensitive thanOFDM to these commonly encountered types ofdistortion. A comparison of the core transforms hasshown the low complexity potential of WPM basedsystems.

Overall, the performance results of WPM lead usto conclude that this new modulation scheme is aviable alternative to OFDM to be considered fortoday’s communication systems. OFDM remainsnevertheless a strong competitor thanks to its ca-pability to cope with multipath effects efficiently.With the demand for ever higher bandwidth, the useof cyclic prefix will likely become restricted, andhence OFDM and WPM will present equalizationcomplexity of the same order.

The major interest of WPM nevertheless residesin its ability to fulfill the wide range of require-ments of tomorrow’s ubiquitous wireless commu-nications. Hence, WPM has the strong advantageof being a generic modulation scheme whose ac-tual characteristics can be widely customized tofulfill the various requirements of advanced mobile

communications. This generic modulation has thepotential of becoming a unique multicarrier com-munication scheme used by devices with variousconstraints. A single scheme could thus be used tocommunicate with small, resource-limited devicesas well as high capacity, multimedia capable nodes.While considering such a scenario, it appears thatthe full capability of WPM can only be exploitedby a system having the capacity to determine thebest suited configuration. Hence, this fully supportsthe fact that WPM is the modulation of choicefor smart, environment and resource aware wirelesssystems.

Again, the authors believe that the most inter-esting feature of WPM is its inherent flexibility.While such flexibility cannot be fully exploitedwith current systems and technologies, it is fore-seen that future generation systems will be ableto provide the degree of intelligence required toachieve optimal performance in a given environ-ment, at a given quality, and for a limited energyconsumption. WPM is then likely to become astrong competitor to multicarrier systems currentlyin use.

Some issues still remain in the way of the wide-spread use of WPM. Although its principle hasbeen proposed for more than a decade, WPM hasnot been studied in depth yet. Its similarity to bothOFDM and overlapping multicarrier systems suchas CMFB allows the use of the work done forthe latter as a basis for the research developments.This has been the approach taken in this paperin order to evaluate the potential of this newmodulation scheme. The first area where a largeamount of work remains to be done is equalization.The overlapping of symbols causes a significantamount of interference that requires an dedicatedequalization scheme to be studied. A study onsynchronization of the WPM signals in both timeand frequency domains would most probably leadto efficient algorithms, due to the multi-resolutionnature of the multiplexed signal.

Finally, it is important to underline that wavelettheory is still developing. Since the use of waveletpackets in telecommunications has been mainlystudied by communications engineers, an importantpotential for improvements is possible if some ofthe specific issues are addressed from a mathemat-ical point-of-view. Wavelet based communicationsystems have already shown a number of advan-tages over conventional systems. It is expected thatmore is still to be pointed out as the knowledge ofthis recently proposed scheme gains more interestwithin both the wireless communication industryand research community.


ACKNOWLEDGMENTS

This work has been supported in part by theAcademy of Finland (Grants for projects 50624 and50618). Authors wish to thanks Pr. Hanzo and Dr.Daly whose comments and suggestions have largelycontributed to improve this paper.

REFERENCES

[1] J. A. C. Bingham, “Multicarrier modulation for data trans-mission: an idea whose time has come,”IEEE Communi-cations Magazine, vol. 28, no. 5, pp. 5–14, May 1990.

[2] A. Akansu, P. Duhamel, X. Lin, and M. de Courville, “Or-thogonal transmultiplexers in communications: a review,”IEEE Transactions on Signal Processing, vol. 46, no. 4,pp. 979–995, April 1998.

[3] R. Lackey and W. Upma, “Speakeasy: the military soft-ware radio,” IEEE Communications Magazine, vol. 33,no. 5, pp. 56–61, May 1995.

[4] B. G. Negash and H. Nikookar, “Wavelet-based multicar-rier transmission over multipath wireless channels,”Elec-tronics Letters, vol. 36, no. 21, pp. 1787–1788, October2000.

[5] C. J. Mtika and R. Nunna, “A wavelet-based multicarriermodulation scheme,” inProceedings of the40th MidwestSymposium on Circuits and Systems, vol. 2, August 1997,pp. 869–872.

[6] G. W. Wornell, “Emerging applications of multirate signalprocessing and wavelets in digital communications,” inProceedings of the IEEE, vol. 84, 1996, pp. 586–603.

[7] N. Erdol, F. Bao, and Z. Chen, “Wavelet modulation: aprototype for digital communication systems,” inIEEESouthcon Conference, 1995, pp. 168–171.

[8] A. R. Lindsey and J. C. Dill, “Wavelet packet modulation:a generalized method for orthogonally multiplexed com-munications,” inIEEE 27th Southeastern Symposium onSystem Theory, 1995, pp. 392–396.

[9] A. R. Lindsey, “Wavelet packet modulation for orthogo-nally multiplexed communication,”IEEE Transaction onSignal Processing, vol. 45, no. 5, pp. 1336–1339, May1997.

[10] N. Suzuki, M. Fujimoto, T. Shibata, N. Itoh, andK. Nishikawa, “Maximum likelihood decoding for waveletpacket modulation,” inIEEE Vehicular Technology Con-ference (VTC), 1999, pp. 2895–2898.

[11] S. Gracias and V. U. Reddy, “An equalization algorithmfor wavelet packet based modulation schemes,”IEEETransactions on Signal Processing, vol. 46, no. 11, pp.3082–3087, November 1998.

[12] F. Daneshgaran and M. Mondin, “Clock synchronisationwithout self-noise using wavelets,”Electronics Letters,vol. 31, no. 10, pp. 775–776, May 1995.

[13] ——, “Symbol synchronisation using wavelets,” inIEEEMilitary Communications Conference, vol. 2, 1995, pp.891–895.

[14] M. Luise, M. Marselli, and R. Reggiannini, “Clock syn-chronisation for wavelet-based multirate transmissions,”IEEE Transactions on Communications, vol. 48, no. 6, pp.1047–1054, June 2000.

[15] G. Strang and T. Q. Nguyen,Wavelet and filter banks.Wellesley-Cambridge Press, 1996.

[16] S. Mallat, A wavelet tour of signal processing, 2nd ed.Academic Press, 1999.

[17] I. Daubechies,Ten lectures on wavelets. SIAM, CBMSSeries, April 1992.

[18] C. K. Chui,An introduction to wavelets. Academic Press,1992, vol. 1.

[19] The MathWorks Inc., “Wavelet toolbox user’s guide, ver-sion 2,” September 2000.

[20] X. Q. Gao, T. Q. Nguyen, and G. Strang, “On two-channelorthogonal and symmetric complex-valued FIR filter banksand their corresponding wavelets,” inProceedings of IEEEInternational Conference on Acoustics, Speech and SignalProcessing, 2002, pp. 1237–240.

[21] W. Y. Chen,DSL: Simulation techniques and standards de-velopment for digital subscriber line systems. MacmillanTechnical Publishing, 1998.

[22] ANSI Committee, “ANSI T1.413-1995: Network and Cus-tomer Installation Interfaces - Asymmetric Digital Sub-scriber Line (ADSL) Metallic Interface,” ANSI StandardCommittee T1E1.4, 1995.

[23] J. G. Proakis,Digital communications, 3rd ed. Mc GrawHill international editions, 1995.

[24] J. Mitola and G. Q. Maguire, “Cognitive radio: Makingsoftware radios more personal,”IEEE Personal Communi-cations, vol. 6, no. 4, pp. 13–18, August 1999.

[25] W. Yang, G. Bi, and T.-S. P. Yum, “Multirate wirelesstransmission system using wavelet packet modulation,” inProceedings of Vehicular Technology Conference (VTC),vol. 1, 1997, pp. 368–372.

[26] D. Daly, “Efficient multi-carrier comunication on the digi-tal subscriber loop,” Ph.D. dissertation, Dept. of ElectricalEngineering, University College Dublin, May 2003.

[27] L. Hanzo, W. Webb, and T. Keller,Single- and multi-carrier quadrature amplitude modulation - Principles andapplications for personnal communications, WLANs andbroadcasting. John Wiley & Sons, 2000.

[28] S. D. Sandberg and M. A. Tzannes, “Overlapped discretemultitone modulation for high speed copper wire commu-nications,” IEEE Journal on Selected Area in Communi-cations, vol. 13, no. 9, pp. 1571–1585, December 1995.

[29] M. Hawryluck, A. Yongacoglu, and M. Kavehrad, “Ef-ficient equalization of discrete wavelet multi-tone overtwisted pair,” in Proceedings of Broadband Communica-tions, Accessing, Transmission and Networking Confer-ence, 1998, pp. 185–191.

[30] R. van Nee and R. Prasad,OFDM for wireless multimediacommunications. Artech house, 2000.

[31] A. R. S. Bahai and B. R. Saltzberg,Multicarrier digitalcommunications. Kluwer Academic / Plenum Publishers,1999.

[32] F. Dovis, M. Mondin, and F. Daneshgaran, “Performanceof wavelet waveforms over linear and non-linear channels,”in Proceedings of Wireless Communications and Network-ing Conference (WCNC), 1999, pp. 1148–1152.

[33] M. Tummla, M. T. Donovan, B. E. Watkins, and R. North,“Voltera series based modeling and compensation of non-linearities in high power amplifers,” inProceedings ofInternational Conference on Acoustics, Speech and SignalProcessing, vol. 3, 1997, pp. 2417–2420.

[34] G. Chrisikos, C. J. Clark, A. A. Moulthrop, M. S. Muha,and C. P. Silva, “A nonlinear ARMA model for sim-ulating power amplifers,” inIEEE MTT-S InternationalMicrowave Symposium Digest, vol. 2, 1998, pp. 733–736.

[35] M. Speth, S. Fechtel, G. Fock, and H. Meyr, “Broadbandtransmission using OFDM: system performance and re-ceiver complexity,” in International Zurich Seminar onBroadband Communications, February 1998, pp. 99–104.

[36] G. Bachman, L. Narici, and E. Beckenstein,Fourier andwavelet analysis. Springer, 2000.

[37] A. Jamin and P. Mahonen, “FPGA implementation of thewavelet packet transform for high speed communications,”in International Conference on Field Programmable Logic(FPL), Montpellier, France, September 2002.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

10

Wavelet modulation lowpass filterAmplitude versus frequency response |H(f)| for different wavelets

Normalized frequency Ω/(2*π) in Hz

Ampl

itude

|H(f)

| in

dB

’db2’, M = 8, 75 taps’sym4’, M = 8, 135 taps’coif4’, M = 8, 375 taps’db6’, M = 8, 195 taps

Fig. 4. Frequency domain localization of different wavelets

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 3210

−6

10−5

10−4

10−3

10−2

10−1

Delay between paths

BER

WPM(db2)WPM(db6)WPM(db10)OFDM (K=0)OFDM (K=8)

Fig. 5. Performance of WPM versus OFDM in a 2-path time-invariant channel. BER is plotted as a function of the delay ofarrival of the second path. The delayed path relative power of−3 dBc and the SNR is20 dB


5 10 15 20 2510

−4

10−3

10−2

10−1

Leq

BE

RWPM complex(db2, M=16)WPM complex(db6, M=16)WPM complex(db10, M=16)OFDM (M=16, K=0)

σeq2 = −5 dB

SNR = 20 dB

Fig. 6. BER performance of WPM and OFDM schemes with noisy equivalent impulse response, as a function of the lengthLeq

of imperfectly equalized channel response, atσ2eq = −5dBc.

1 2 3 4 5 6 7−10

−9.5

−9

−8.5

−8

−7.5

−7

−6.5

−6

−5.5

−5

WPM(coif1) and WPM(coif4) post−detection equalizer performance(All schemes 32 subcarriers, 4−path exp. decay delay profile channel, τ

0 = 1, SNR = 20 dB)

Number of taps in time domain η

Inte

rfere

nce

pow

er [d

Bc]

WPM(coif1), ν = 1WPM(coif1), ν = 3WPM(coif1), ν = 5WPM(coif4), ν = 1WPM(coif4), ν = 3WPM(coif4), ν = 5

Fig. 7. Comparative performance of WPM(coif1) and WPM(coif4) schemes with post-detection equalization, as a function of thenumber of equalizer tapsη in the time domain and with the number of tapsν in the frequency domain as a parameter.


2 3 4 5 6 7 8 9 1010

−6

10−5

10−4

10−3

10−2

10−1

WPM link BER degradation as a function of PA backoff margin γ(All schemes 64 subcarriers, 16QAM and AWGN channel with 20 dB SNR)

Amplifier backoff margin [dB]

BER

WPM(db2)WPM(sym2)WPM(coif2)WPM(dmey)OFDM (K=0)

Fig. 8. BER versus power amplifier backoff marginγ for different WPM schemes and OFDM.

0.15 0.175 0.2 0.225 0.25 0.275 0.3 0.325 0.35 0.375 0.410

−6

10−5

10−4

10−3

10−2

10−1

WPM modulation sensitivity to sampling phase error(All modulation 16 subcarriers, QPSK over AWGN channel with 40 dB SNR)

Sampling phase error normalized to sampling period

BER

WPM (db2)WPM (coif2)WPM (sym2)OFDM (M=16, K=0)

Fig. 9. Sensitivity of different WPM schemes versus OFDM schemes to sampling phase error, expressed as the link BER versusthe normalized sampling phase error.


0.16 0.18 0.2 0.22 0.24 0.26 0.2810

−6

10−5

10−4

10−3

10−2

10−1

WPM(coifN) sensitivity to sampling phase error for different wavelet order(All schemes 32 subcarriers, QPSK and AWGN channel with 20 dB SNR)

Sampling phase error normalized to sampling period

BER

WPM(coif1)WPM(coif2)WPM(coif3)WPM(coif4)WPM(coif5)

Fig. 10. Sensitivity of WPM(coifN ) schemes to sampling phase error as a function of the wavelet order, expressed as the linkBER versus the normalized sampling phase error.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510−2

10−1

100

Bit

Err

or

Ra

te (

BE

R)

Disturber relative frequency

WPM robustness to single tone disturber, as a function of the disturber frequency (Pdist

= 0 dBc)

(All schemes 16 subcarriers, 16QAM)

OFDM (K=0)WPM complex(coif1)WPM complex(coif5)WPM complex(dmey)

Fig. 11. Link BER in the presence of a single tone disturber as a function of the disturber frequency, for WPM(coif1), WPM(coif5),WPM(dmey), and OFDM schemes.


−25 −20 −15 −10 −5 0 510−3

10−2

10−1

100

Sensitivity of WPM and OFDM schemes to a single tone disturber(All schemes 16 subcarriers, 16QAM)

Bit

Err

or R

ate

(BE

R)

Disturber relative power [dBc]

OFDM (K=0)WPM(coif1)WPM(coif5)WPM(dmey)

Fig. 12. Link BER in the presence of a single tone disturber as a function of the disturber power, for WPM(coif1), WPM(coif5),WPM(dmey), and OFDM schemes.

8 16 32 64 128 256 512 102410

−3

10−2

10−1

100

101

Transform size (M)

Rel

ativ

e am

ount

of A

DD

Lo = 4Lo = 10Lo = 16

8 16 32 64 128 256 512 102410

−1

100

101

Transform size (M)

Rel

ativ

e am

ount

of M

ULT

Lo = 4Lo = 10Lo = 16

Fig. 13. Number of additions and multiplications required by the WPT relatively to the DFT as a function of the transform sizeM , for different wavelet generating filter lengthsL0.


3 4 5 6 7 8 9 1010−2

10−1

100

101

Number of stages (J)

Tot

al a

mou

nt o

f ope

ratio

ns fo

r W

PT

rel

ativ

ely

to D

FT

WPT complexity relative to DFT

Lo = 4Lo = 10Lo = 16

Fig. 14. WPT complexity relative to that of the DFT as a function of the transform sizeM = 2J , for different wavelet generatingfilter lengthsL0.

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