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

Wavelet Packet for Wireless Communications Antony Jamin Petri Mah¨ onen¨ Advanced Communications Chair of Wireless Networks Networks (ACN) SA Aachen University Neuchatel, Switzerland Germany [email protected] [email protected]

Abstract— As proven by the success of OFDM, OFDM signal. Little interest has been given to the multicarrier modulation has been recognized as an evaluation of those system level characteristics in efficient solution for wireless communications. Wave- the case of WPM. form bases other than sine functions could similarly be used for multicarrier systems in order to provide Moreover, the major advantage of WPM is its an alternative to OFDM. flexibility. This feature makes it eminently suitable In this article, we study the performance of wavelet for future generation of communication systems. packet transform modulation (WPM) for transmis- With the ever-increasing need for enhanced per- sion over wireless channels. This scheme is shown formance, communication systems can no longer to be overall quite similar to OFDM, but with some interesting additional features and improved be designed for average performance while as- characteristics. A detailed analysis of the system’s suming channel conditions. Instead, new genera- implementation complexity as well as an evaluation of tion systems have to be designed to dynamically the influence of implementation-related impairments take advantage of the instantaneous propagation are also reported. conditions. This situation has led to the study Index Terms— Wavelet packet modulation, multi- of flexible and reconfigurable systems capable of carrier modulation, orthogonal waveform bases optimizing performance according to the current channel response [3]. A tremendous amount of work has been done recently to fulfill this re- I.INTRODUCTION quirement at the physical layer of communication THOUGH the principle of multicarrier mod- systems: complex equalization schemes, dynamic A ulation is not recent, its actual use in com- bit-loading and power control that can be used to mercial systems had been delayed until the tech- dynamically improve system performance. While nology required to implement it became available WPM can take advantage of all those advanced at reasonable cost [1]. Similarly, the idea of using functionalities designed for multicarrier systems, more advanced transform than Fourier’s as the core we show that it benefits also from an inherent of a multicarrier system has been introduced more flexibility. This feature together with a modular than a decade ago [2]. However, such alternative implementation complexity make WPM a potential methods have not been foreseen as of major interest candidate for building highly flexible modulation and therefore have received little attention. With the schemes. current demand for high performance in wireless Wavelet theory has been foreseen by several communication systems, we are entitled to wonder authors as a good platform on which to build mul- about the possible improvement that wavelet-based ticarrier waveform bases [4], [5], [6]. The dyadic modulation could exhibit compared to OFDM sys- division of the bandwidth, though being the key tems. We address this question in this article by point for compression techniques, is not well suited taking theoretical as well as implementation aspects for multicarrier communication [7]. Wavelet packet into account. bases therefore appear to be a more logical choice Several objectives motivate the current research for building orthogonal waveform sets usable in on WPM. First of all, the characteristics of a mul- communication. In their review on the use of or- ticarrier modulated signal are directly dependent thogonal transmultiplexers in communications [2], on the set of waveforms of which it makes use. Akansu et al. emphasize the relation between filter Hence, the sensitivity to multipath channel distor- banks and transmultiplexer theory and predict that tion, synchronization error or non-linear amplifiers WPM has a role to play in future communication might present better values than a corresponding systems. PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 2

Lindsey and Dill were among the first to pro- conclude by underlining the communications area pose wavelet packet modulation. The theoretical where WPM appears to be a promising technology fondation of this orthogonal multicarrier modu- for future generation transceivers. lation technique and its interesting possibility of leading to an arbitrary time-frequency plane tiling II.COMMUNICATIONSYSTEMARCHITECTURE are underlined [8]. Lindsey has further completed the work in this area by presenting additional The simplified block diagram of the multicarrier results [9]. In particular, it is shown in this last communication system studied in this article is paper that power spectral density and bandwidth shown in Figure 1. The transmitted signal in the efficiency are equal for WPM and standard QAM discrete domain, x[k], is composed of successive modulation. Moreover, WPM is placed into a mul- modulated symbols, each of which is constructed titone communication framework including alterna- as the sum of M waveforms ϕm[k] individually tive orthogonal bases such as M-band wavelet mod- amplitude modulated. It can be expressed in the ulation (MWM) and multiscale modulation (MSM). discrete domain as: Additional research work on more realistic mod- M−1 X X   els of WPM-based transceivers has also been car- x[k] = as,m ϕm k − s M (1) ried out. Maximum likelihood decoding for wavelet s m=0 packet modulation has been addressed by Suzuki where a is a constellation encoded s-th data et al. [10]. Simulations for both flat fading and s,m symbol modulating the m-th waveform. Denoting multipath channels have been performed for a re- T the sampling period, the interval [0, LT − 1] ceiver using a channel impulse response estimator, is the only period where ϕ [k] is non-null for an MLSE and Viterbi decoder. It is shown that the m any m ∈ {0..M − 1}. In an AWGN channel, the use of wavelet packet specific characteristics leads lowest probability of erroneous symbol decision to improved results. The study of an equalization is achieved if the waveforms ϕ [k] are mutually scheme suited for WPM has also been done by m orthogonal, i.e. Gracias and Reddy [11]. In addition, the multi- resolution structure of the wavelet packet wave- hϕm[k], ϕn[k]i = δ[m − n], (2) forms have shown to offer opportunity for improved synchronization algorithms [12], [13]. This has where h·, ·i represents a convolution operation and been exploited for instance in the case of WPM by δ[i] = 1 if i = 0, and 0 otherwise. Luise et al. [14]. All together, the research work In OFDM, the discrete functions ϕm[k] are provides insight on the theoretical performance the well-known M complex basis functions m  of an actually implemented WPM-based , w[t]exp j2π M kT limited in the time domain thus contributing to complete the research work by the window function w[t]. The corresponding required to lead to a fully implementable WPM- sine-shaped waveforms are equally spaced in the based communication system. frequency domain, each having a bandwidth of 2π We focus in this article on reviewing the ad- M and are usually grouped in pairs of similar vantages of wavelet packet modulation in wireless central frequency and modulated by a complex communications. Theoretical background on this QAM encoded symbol. In WPM, the subcarrier modulation scheme is first recalled. Specific issues waveforms are obtained through the WPT. Exactly of using such sets of waveforms for multicarrier as for ODFM, the inverse transform is used to communication systems are underlined, and an ex- build the transmitted symbol while the forward haustive comparison with OFDM is made. Special one allows retrieving the data symbol transmitted. emphasis on the flexibility of this scheme is given. Since wavelet theory has part of its origin in filter Section III reports the performance results of WPM bank theory [15], the processing of a signal through in several typical wireless channel models. The case WPT is usually referred as decomposition (i.e. of multipath channels is detailed, and the perfor- into wavelet packet coefficients), while the reverse mance versus complexity of channel equalization operation is called reconstruction (i.e. from wavelet for such channels is studied. Results obtained are packet coefficients) or synthesis. compared with those obtained with a classical In this paper, we limit our analysis to WPT OFDM scheme, since it is currently the reference that can be defined through a set of FIR filters. in multicarrier systems. Section IV focuses on the Though it would be possible to use other wavelets effect of interference and implementation impair- as well, those cannot be implemented by Mallat’s ments. Then, the implementation complexity of fast algorithm [16] and hence their high complexity a WPM system is also reported, since this is a make them ill-suited for mobile communication. key issue in wireless communication systems. We The synthesis discrete wavelet packet transform PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 3

IWPT Channel WPT

serial

parallel

encoder decoder

Serial to

Parallel to

Constellation Constellation n(t)

Fig. 1. Wavelet packet modulation functional block diagram constructs a signal as the sum of M = 2J wave- previous one(s) ends. The waveforms being M- forms. Those waveforms can be built by J succes- shift orthogonal, the inter-symbol orthogonality is sive iterations each consisting of filtering and up- maintained despite this overlap of consecutive sym- sampling operations. Noting h·, ·i the convolution bols. This allows taking advantage of increased operation, the algorithm can be written as: frequency domain localization provided by longer waveforms while avoiding system capacity loss that  D rec E  ϕj,2m[k] = hlo [k], ϕj−1,m[k/2] normally results from time domain spreading. The waveforms length can be derived from a deteilled D rec E  ϕj,2m+1[k] = hhi [k], ϕj−1,m[k/2] analysis of the tree algorithm [16, Section 8.1.1].  1 for k = 1 Explicitly, the wavelet filter of length L0 generates with ϕ [k] = ∀m 0,m 0 otherwise M waveforms of length

where j is the iteration index, 1 ≤ j ≤ J L = (M − 1) (L0 − 1) + 1. (3) and m the waveform index 0 ≤ m ≤ M − 1. Using usual notation in discrete signal processing, ϕj,m[k/2] denotes the upsampled-by-two version In Daubechie’s wavelet family [17] for instance, the of ϕj,m[k]. For the decomposition, the reverse length Lo is equal to twice the wavelet vanishing operations are performed, leading to the comple- order N. For the order 2 Daubechie wavelet, L is mentary set of elementary blocks constituting the equal to 4, and thus a 32 subcarrier WPT is com- wavelet packet transform depicted in Figure 2. posed of waveforms of length 94. This is therefore In orthogonal wavelet systems, the scaling filter about three times longer than the corresponding rec rec OFDM symbol, assuming no cyclic prefix is used. hlo and dilatation filter hhi form a quadrature mirror filter pair. Hence knowledge of the scaling The construction of a wavelet packet basis is filter and wavelet tree depth is sufficient to design entirely defined by the wavelet scaling filter, hence the wavelet transform [15]. It is also interesting its selection is critical. This filter solely determines to notice that for orthogonal WPT, the inverse the specific characteristics of the transform. In transform (analysis) makes use of waveforms that multicarrier systems, the primary characteristic of are time-reversed versions of the forward ones. In the waveform composing the multiplex signal is communication theory, this is equivalent to using out-of-band energy. Though in an AWGN channel a matched filter to detect the original transmitted this level of out-of-band energy has no effect on waveform. the system performance thanks to the orthogonality A particularity of the waveforms constructed condition, this is the most important source of through the WPT is that they are longer than interference when propagation through the channel the transform size. Hence, WPM belongs to the causes the orthogonality of the transmitted signal family of overlapped transforms, the beginning to be lost. A waveform with higher frequency of a new symbol being transmitted before the domain localization can be obtained with longer time support. On the other hand, it is interesting to use waveforms of short duration to ensure that dec rec the symbol duration is far shorter than the channel h 2 2 h hi hi coherence time. Similarly, short waveforms require less memory, limit the modulation- dec rec delay and require less computation. Those two h 2 2 h lo lo requirements, corresponding to good localization both in time and frequency domain, cannot be Fig. 2. Wavelet packet elementary block decomposition and chosen independently. In fact, it has been shown reconstruction that in the case of wavelets, the bandwidth-duration PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 4

Full name Abbreviated Vanishing Length use pulse for each subcarrier. name order L0 Haar haar 1 2 It is nevertheless possible to translate the M real Daubechie [17] dbN N 2N waveform directly in the complex domain. The Symlets symN N 2N resulting complex WPT is then composed of 2M Coiflet coifN N 6N Discrete Meyr dmey − 62 waveforms forming an orthogonal set. While this is mathematically trivial, this simple fact has not been TABLE I clearly emphasized in the liter- SUMMARY OF WAVELET FAMILY CHARACTERISTICS. ature. This is of particular interest in systems where a baseband signal is desirable. The Asymmetrical Digital Subsciber Line (ADSL) standard falls for product is constant1 [18]. This is usually referred instance in this category [21]. Currently, a real to as the uncertainty principle. baseband signal is constructed by using a 2M-DFT We limit our performance analysis to WPT based fed with M conjugate pair modulation symbols on widely used wavelets such as those given in [22]. An WPT of size M would therefore provide [17]. While there are numerous alternative wavelet an equivalent modulation scheme at roughly half families that could be used as well, a comparative complexity. study would deserve a separate publication by itself. As previously mentioned, we are essentially A. Interesting features for wireless communication interested in wavelets leading to fast transforms In addition to the basic features already men- through the tree algorithm. The primary wavelet tioned, the WPT presents interesting advantages family we have been using in our research is the from a system architecture point-of-view. For in- one from Daubechie [17, p. 115], since it presents stance, the dependence of the WPT on a generating wavelets with the shortest duration. Furthermore, wavelet is a major asset in communication appli- their localization in the frequency domain can be cations. An improved transmission integrity may adequately adjusted by selecting their vanishing be achieved with the aid of diversity [23, Section order, as it is illustrated by two of the curves plotted 14.4]. Space-, frequency-, and time-diversity are in Figure 4. The label dbN is used in the rest of the most common physical diversities exploited. In this article to refer to WPT based on Daubechie addition, WPM provides a signal diversity which is wavelet of vanishing order N. similar to systems to some extent. Further work carried out after her initial re- Such diversity is in fact making joint use of time search led to a near symmetric extension of the and frequency space. Practically, using two differ- dbN family, since it was a demanded feature in ent generating wavelets allows us to produce two applications such as image compression [17, pp. modulated signals that can transmitted on the same 254-257]. Following the notation in [19], we will frequency band and suffer from reduced interfer- refer to this family as Symlets, denoted as symN. ence only. The amount of cross-correlation between The last wavelet we will make reference to has the signals is directly dependent on the generat- been constructed at R. Coiffman demand for image ingwavelets chosen. The best method to be used processing applications [17, pp. 258-259] and will to select suitable wavelets is a topic to be studied be denoted coifN. This wavelet is near symmet- further. This particular feature could be exploited ric, has 2N moments equal to 0 and has length for instance in a cellular communication system, 6N − 1. Table I summarizes the characteristics of where different wavelets are used in adjacent cells the wavelets and families mentioned above. Order in order to minimize inter-cell interference. 4 members of those two last families are as well Another interesting feature of WPM is directly plotted in Figure 4. related to the iterative nature of the wavelet packet Finally, a minor difference between OFDM and transform. The number of subcarriers in OFDM WPM remains to be emphasized. In the former, the systems is usually fixed at design time, and it is set of waveforms is by nature defined in the com- especially difficult to implement an efficient FFT plex domain. WPM, on the other hand, is generally transform having a programmable size. This lim- defined in the real domain but can be also defined itation does not exist in WPT since the transform in the complex domain, solely depending of the size is determined by the number of iteration of the scaling and dilatation filter coefficients [20]. Since algorithm. From an implementation point of view, the most commonly encountered WPT are defined this can be made configurable without significantly in the real domain, it has naturally led the authors to increasing the overall complexity. This permits on- the-fly change of the transform size, which can be 1Both measurement in the frequency and time domain are taken as the domain in which most of the energy of the signal required for different reasons, for instance to recon- is localized. figure a transceiver according to a given communi- PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 5

Frequency Dt(w4,1)

w4,0 w 4,1 Df(w4,1)

w3,1

w1,1 w4,4

w5,10 w5,11

w3,3

Time

Fig. 3. Time-Frequency plane division with semi-arbitrary wavelet packet tree pruning cation protocol2 or, in the event of the appearence carriers could be used for synchronization purpose of cognitive radio [24], the transform size could be in order to take advantage of longer symbols that selected according to the channel impulse response require a small amount of bandwidth. Alternatively, characteristics, computational complexity and link the authors have carried out some research work quality. on the selecting the transform tree according to The last property of the WPT is the semi- the channel impulse response in the frequency arbitrary division of the signal space. Wavelet domain. Preliminary results have not shown signif- packet transform still leads to a set of orthogonal icant improvement in transform complexity versus functions, even if the construction iterations are not link BER, but further work on this issue is still repeated for all sub-branches. From a multicarrier ongoing [26]. communication system perspective, this maps into All together, WPM presents a much higher level having subcarriers of different bandwidth. More- of flexibility than current multicarrier modulation over, since each subcarrier has the same time- schemes. This makes WPM a candidate of choice frequency (TF) plane area, the increase of band- for reconfigurable and adaptive systems such are width is bounded to a decrease of subcarrier symbol the ones likely to compose the next generation of length. This is illustrated in Figure 3, where a non- wireless communication devices. regular decomposition tree is shown together with the corresponding time-frequency plane division. III.PERFORMANCEINVARIOUSCHANNEL This characteristic of WPM has been referred to in MODELS the literature as a multirate system [25], although the term might be misleading in this case. While the We analyze in this section the performance of transmission over the channel is effectively done at WPM in diverse propagation channels. The results different symbol rates, the corresponding subcar- presented have been obtained by simulations only, rier throughput remains identical due to the con- due to the fact that no analytical expressions are stant subcarrier bandwidth-duration product. From available. a communication perspective, such a feature is The derivation of a WPM link BER over an usable for systems that must support multiple data AWGN channel is a trivial issue since, identically streams with different transport delay requirements. to OFDM, orthogonality between subcarriers en- A logical channel requiring lower transport delay sures the link performance is solely dependent on could make use of a wider subcarrier, while some the constellation used on each subcarrier. The case signalling information could be carried within a of a Ricean channel can be derived similarly, as- narrower bandwidth. Especially, those narrow sub- suming that the channel response is time-invariant for the duration of a whole WPM symbol. This 2OFDM-based standards make use of transform sizes ranging is however a less conservative assumption than between 32 and 8192 subcarriers. for OFDM since WPM symbols are several times PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 6

longer than their OFDM counterparts. The case of since inter-symbol interference is removed3. When transmission through the multipath channel hence the path arrival delay τ exceeds the cyclic prefix remains to be studied more in depth. length, the overall error rate increases dramatically, becoming even higher than with the other schemes A. WPM without equalization at a path delay higher than 20. Here we present on comparative results between Overall, the raw BER performance of WPM WPM and OFDM schemes without channel equal- is identical to OFDM with the channel model ization. While the absence of equalization in a real assumed here. Additional results not reported here system would yield a poor performance in most have shown equivalence for different frequency channels, it is nevertheless of interest to consider selective channels when no equalization is used. this case here. The main motivation is to gain Hence the potential of improved performance com- insight on the distortion caused by a given channel pared to OFDM is bound to the design of an to the different modulated signals. Moreover, since efficient equalization scheme for either modulation. optimal equalization schemes for both are differ- ent, the definition of equivalence between systems B. With equalization would require the choice of appropriate comparison The initial interest in multicarrier modulation criteria. schemes was the low complexity of the equalization With the same concern of comparing equivalent they required. In multicarrier systems, equaliza- systems, the OFDM reference system used in this tion is usually divided into pre- and post-detection section has no cyclic prefix. The addition of a equalizers4, according to their position respective to prefix is indeed a technique aiming at rendering the the core transform. In OFDM, a pre-equalizer is not multipath distortion easier to cancel and as such required if the channel delay spread is shorter than can be considered as a form of equalization [27, the cyclic prefix. For longer delay spread, the pre- Section 18.6]. Since no similar artifact is available detection equalizer aims at shortening the apparent for WPM, the comparison is more fair if no cyclic channel impulse response to a value lower or equal prefix is used. In addition, the introduction of the to the cyclic prefix duration. In the case it succeeds, cyclic prefix leads to a bandwidth efficiency loss the symbol at the input of the DFT is free from and thus this would add to the difference between inter-symbol interference. A post-detection equal- modulation schemes. izer composed of a single tap per subcarrier is then We choose to assume here the simple case of a sufficient to compensate for the channel distortion time invariant 2-path channel model as introduced in the frequency domain. Though this low com- by Rummler [23, section 14-1-2]. The first path plexity structure is a major advantage of OFDM, it has unit power and the second path power is 3 dB is limited by the fact that for a channel with long lower. The relative delay of the second path τ is a impulse response, the length of the prefix required simulation parameter. The BER versus the second leads to a significant loss in capacity and transmit path delay τ curves of WPM(db2), WPM(db6), and power. For WPM however, the use of a cyclic prefix WPM(db10) schemes with 32 QPSK modulated is impossible due to the overlapping of consecu- subcarriers is shown in Figure 5. All curves are tive symbols. Hence both inter-symbol interference quite similar and thus the increase of the wavelet (ISI) and inter–symbols inter-carriers interference order has little impact on the performance. The (ISCI) have to be cancelled by the equalization main differences appear to be in the area of where scheme. While no research work addresses this τ is lower than 8 samples. For these values, the issue specifically for WPM, a comparison of the WPM scheme based on the higher order wavelet is efficiency and complexity of pre-detection, post- less sensitive to actual delay. detection, and combined equalization techniques The performance for two different OFDM for DWMT has been studied by Hawryluck et al. schemes appears in Figure 5. For the reason dis- [29]. Research results show clearly that the best cussed, the first has no cyclic prefix (K = 0). The results are obtained by using both pre- and post- second one has a prefix of OFDM symbol duration detection equalization. Due to the similarity with 1 normalized length 4 which corresponds here to this modulation scheme, a similar approach is taken 8 samples. For τ lower or equal to 3 samples, here and we separately analyze the performance of the OFDM scheme without prefix performs slightly both types of equalization. better than the WPM ones. For longer path delays, OFDM performs similarly to the WPM schemes, 3The channel distortion is not cancelled since no equalization with a close resemblance to WPM(db2). OFDM is assumed. 4As in [28], the terms pre- and post-detection equalization are with a cyclic prefix shows much better performance prefered here to time- and frequency-domain equalizers, though for path delays lower than the prefix duration they refer to the same functions PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 7

1) Pre-detection equalization: The objective of work is needed in order to verify the validity of the pre-detection equalizer is slightly different be- this assumption for a particular equalizer training tween an overlapping multicarrier and an OFDM methods with a finite length training sequence. system making use of a cyclic prefix. While in 2) Post-detection equalization: As underlined, the latter case, the equalizer attempts to shorten post detection equalization is used in OFDM sys- the perceived channel impulse response to a value tems in the form of a single tap filter. These taps lower than the cyclic duration, the equalization of aim at inverting the channel transfer function in WPM aims at minimizing the mean square error at the frequency domain. For WPM, the structure its output. of the post-detection equalizer is more complex We compare here the sensitivity of WPM and since it has to remove both ISI and ISCI. Hence, OFDM to imperfect equalization. To avoid the a combined time-frequency equalization structure system’s related issues concerning equalizer train- is required. Denoting η and ν as the number ing, we will further assume that the equalizer taps of equalizer taps in time and frequency domains have reached their near optimal values. Denoting respectively, we can express the estimated symbol ˜l the transfer function of the pre-detection equalizer θk at the output of the post-detection equalizer as Q(z) and the equivalent discrete channel response ν −1 η −1 C(z), the overall link from modulator to demodu- X2 X2 θ˜l = q(∆k, ∆l) θˆl+∆l . (6) lator has an equivalent transfer function of k k+∆k ∆k=− ν +1 η 2 ∆l=− 2 +1 H(z) = C(z) Q(z). (4) The total number of operations per subcarrier Considering the case where the equalization symbol is η×ν, and the complexity grows rapidly if method has converged to the value Q˜(z), we choose a complex structure is used. This is a major concern to approximate H(z) as since with L0 − 1 overlapping symbols, perfect equalization cannot be reached for η ≤ (L0 − 1). Leq −1 X −l Such equalization complexity is impractical in a H(z) = 1 + q(l) z . (5) real system, hence we limit our analysis to smaller l=1 equalizer sizes that give a BER sufficiently low to In the case of perfect equalization, all the coeffi- permit an outer error correction code to reach a cients q(l) are null. To take into account imperfect good performance. equalization, we assume that the set q(l) is nor- We choose to compare the performance of the 2 mally distributed. We denoted σeq the noise power equalization schemes for a typical channel with an in the Leq equalizer taps indexed from 1 to Leq −1. exponentially decaying profile. The decay factor τ0 The sensitivity of WPM and OFDM can thus be is taken equal to 1 and the total number of paths 2 compared as a function of the noise power σeq in is limited to 4. For this channel, the equalizer taps the equivalent transmission filter H(z). Figure 6 have been calculated through the steepest descent displays the results of a WPM link for an average LMS algorithm [23, Section 11-1-2]. The iteration of 100 randomly generated channels as a function step ∆ has been taken equal to 10−3 and the of the length Leq of the resulting filter. The noise number of training symbols has be chosen to ensure 2 power σeq has be set to -5 dBc in order to lead to a the algorithm has converged. significant number of errors. All the schemes used The performance of the equalizer as a function in this simulation assume 16 subcarriers, 16QAM, of its number of taps η in time domain is given and an AWGN channel with 20 dB SNR. De- in Figure 7. The modulation schemes selected here spite the symbols overlapping, WPM performs very are WPM(coif1) and WPM(coif4), in order to allow similarly to OFDM in such scenarios. Moreover, comparison of the effect of better frequency domain no significant difference is perceived between the localization. Considering first the case where the WPM schemes using different wavelet orders. equalizer does not take into account the symbols 2 Alternatively, the effect of the noise power σeq on adjacent subcarriers (i.e. ν = 1), the increase for the same modulation schemes for a fixed value of η brings only very limited improvement, though of Leq has been studied. While some differences the difference is slightly more significant for the exist for some given channel, the results confirm WPM(coif4) scheme. When the two adjacent sub- that all schemes asymptotically reach the same per- carrier symbols are used in the equalizer, increasing 2 formance for any value of Leq and σeq. Thus, WPM η from 1 to 3 taps leads to a larger interference and OFDM exhibit identical sensitivity to pre- reduction of about of 1 dB. A further increase equalization errors. These results have been reached of both ν and η bring no improvement in the under the assumption that the equalizer taps have WPM(coif1) scheme, and only limited interfer- converged identically for both schemes. Further ence reduction in the case of the more frequency PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 8

localized WPM(coif4) scheme. Furthermore, it is devices, since the energy consumption and cost of interesting to notice that, while all the modulation radio-frequency amplifiers increases with the need schemes have shown identical performance when for higher linear dynamic range. No work on this no equalization is used, the addition of a single tap issue has been reported in the literature specifically equalizer per subcarrier reveals the extra robust- for WPM, though the use of a wavelet packet de- ness of the WPM(coif4) in comparison with the rived pulse over non-linear satellite communication WPM(coif1) scheme. channel has been addresses by Dovis et al. [32]. Overall, the post-equalization of a WPM signal Thus, we evaluate the actual sensitivity of different requires at least a 3 × 3-tap structure to remove WPM schemes to non-linear distortion. the largest part of the interference caused by mul- A model of the non-linear amplifier has to be tipath propagation. Less frequency localized WPM chosen. Different approaches have been used in schemes require an even more complex structure the literature [33], [34]. For most complex models, due to an increased inter-subcarrier interference the numerous parameters have to be derived from level. This of course leads to a complexity penalty measurement on an actual device. To keep the anal- when compared with OFDM schemes where a ysis here more generic, we will assume a simple cyclic prefix can be used and only one tap per non-linear, memoryless power amplifier model. For subcarrier is needed. Hence, the complexity gain most applications using medium and lower transmit of WPM emphasized later in this article is reduced power, phase distortion can be assumed to be null. when equalization is considered. Hence, only the amplitude distortion is assumed to 3) Discussion on equalization: The results ob- be significant. Such a simple model is characterized tained emphasise that equalization of WPM in by a transfer function of the form [30, section multipath channels is more complex than that of 6.3.1]:

OFDM. This is essentially due to the use of a cyclic 1 " #− 2p prefix that gives an edge to OFDM, when compared x(t)2p yx(t) = x(t) 1 + (7) to overlapping multicarrier schemes such as WPM. γ Thus, the raw link performance of WPM and OFDM is equivalent only when the cyclic prefix with y(t) the output signal, x(t) the input signal, is not a practical solution. This corresponds to the and γ and p are the backoff margin and the linearity case where the channel impulse response is too long order of the model respectively. A higher value of p to allow reasonable performance with a low number leads to an output versus input amplitude response, of subcarriers. In the latter situation, the system where the output amplitude is smoothly reaching its complexity of the WPM and OFDM schemes is saturation value as the input power increases. For roughly equivalent since OFDM requires more than an average quality radio-frequency power amplifier, a single tap post-equalizer to remove multipath the best fit value for p is found to be between 2 interference. and 3 [30, section 6.3.1]. Note that γ is referred to as the backoff margin and is defined as the IV. ROBUSTNESSTOINTERFERERS, difference between the average and peak power of 5 NON-LINEARDISTORTIONANDSAMPLING the amplifier . OFFSETS Simulation results are displayed in Figure 8, where the link BER for WPM(db2), WPM(sym2), In addition to the performance of a modula- WPM(coif2), WPM(dmey), and OFDM schemes tion scheme in the given propagation condition, is plotted as a function of the backoff margin its sensitivity to implementation impairments is a γ. An AWGN channel with 20 dB SNR has characteristic that makes it a candidate for an actual been assumed. WPM schemes require a backoff communication system. Hence we report in this value 1.5 dB higher than OFDM to achieve equal section the performance of WPM in the presence performance at 10−4 BER. Among the WPM of some of the implementation impairments that are schemes, WPM(dmey) performs best, requiring critical in OFDM systems: the sensitivity to non- about 0.4 dB less power than WPM(db2) and linear distortion and sampling instant error. WPM(sym2) schemes at 10−4 BER. In order to reach insignificant BER degradation due to non- A. Non-linear signal distortion effects linearity, the OFDM signal requires a 5 dB backoff OFDM modulated signals are known to exhibit a value. WPM schemes need a slightly higher value high peak to average power ratio (PAPR) [30], [31]. of 6 dB. This 1 dB difference is rather low and This implies that power amplifiers must have a high 5This differs from the definition commonly used where the dynamic range in order to avoid causing distortion. backoff margin is the difference between the amplifier average This is of concern in power-limited communication power operating point and −1 dB compression point. PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 9

thus should not lead to any significant complexity sensitivity of WPM to synchronization errors. No variation in the radio-frequency section of a modem implementation of the algorithm has been reported using either modulation scheme. and therefore its performance in the presence of imperfect estimation remains to be studied. B. Sampling phase offset Multicarrier modulation signals are by nature C. In presence of a narrow band interferer much more sensitive to synchronization errors than With today’s growing use of wireless systems, single carrier ones [35]. We focus here on the effect the radio-frequency spectrum is becoming more of a non-ideal sampling instant. and more congested by communication signals. In The interference caused by a sampling phase such a condition, future modulation schemes have error ∆τ is of three kinds. There is a gain loss in the to cope not only with channel distortion, but also recovery of the symbol of interest, an inter-carrier with interference originating from other sources interference term, and an inter-symbol inter-carrier as well. Moreover, multicarrier modulation is very interference contribution. This last term originates likely to encounter in-band interfering signal since from the symbol overlapping and thus does not ex- it is usually best suited for wideband communica- ist in OFDM. Hence, the sensitivity of WPM is ex- tion systems. pected to be higher than for OFDM. In addition, the We assume the case of a WPM link communi- BER degradation depends on the auto-correlation cationg over a AWGN channel and exposed to an of each subcarrier waveform, which differs between in-band, un-modulated signal superimposed on the subcarriers. Overall, the BER of the multicarrier signal of interest at the receiver. We denote Pdist signal can be obtained as the average of the BER as the power of the disturbing signal and fdist as over each individual subcarrier. Since no analytical its frequency. The amount of interference endowed closed form solution is available, the sensitivity by each subcarrier k can thus be approximated as of each modulation scheme to sampling instant [28] error has been obtained by simulation. Figure 9 M−1 2 X reports link BER as a function of the sampling PI = Pdist Φk(fdist) (8) phase offset normalized to the sampling period. For k=0 this particular simulation, the channel is modelled where Φk in the Fourier transform of ϕk. In the as AWGN with 20 dB SNR. As it was expected case of waveforms with null out-of-band energy, due to the overlapping of symbols, WPM is more the disturbance will be limited to the subcarrier sensitive than OFDM to an imperfect sampling whose band includes the frequency fdist. With an instant. A BER of 10−4 is achieved for OFDM actual system, additional disturbance is caused by at a normalized sampling error of 27%, while the side lobes of the adjacent subcarriers. The side WPM(db2) requires less than 21%. For the two lobe energy level decreasing with the order of the other WPM schemes, the error tolerated is slightly wavelet, the sensitivity of WPM to a single tone lower, with a maximum of about 18%. disturber can thus be reduced by increasing the Additional results on the sensitivity of WPM order of the generating wavelet. to sampling instant error as a function of the Results obtained through simulation are shown wavelet order is given in Figure 10. The differences in Figure 11, where the BER performance of between the different order of the coif wavelet WPM(db2), WPM(db8), and OFDM links are plot- are limited, except for the lower order one which ted as a function of the disturber normalized fre- tolerates a 2% higher phase error than the oth- quency Fdist. A 16-subcarrier scheme with 16QAM ers. There is no direct relation between the order modulated symbols has been chosen to point out and sensitivity level. Simulation results for sym the effect of the disturber. The disturber has power wavelets which are not shown here have led to equal to the signal of interest, i.e. Pdist = 0 dBc. identical conclusions. The curves obtained for all modulation schemes Overall, the higher sensitivity of WPM implies show clearly that the level of interferance is highly that a more robust sampling instant synchronization dependent on the actual disturber frequency. Hence, scheme is required in comparison with OFDM. On the curve for OFDM shows clearly the higher the other hand, the multi-resolution structure of BER when the disturber frequency corresponds to the wavelet packet waveforms offer the opportu- the center frequency of one subcarrier. The WPM nity for improved synchronization algorithms [13], schemes show a similar effect but with a smoother [12]. This has been exploited for instance in the curve. Considering the average over the whole fre- case of WPM by Luise et al. [14], where the quency band, WPM(coif5) and WPM(dmey) out- proposed synchronization scheme has shown en- perform OFDM. The WPM(coif1) scheme however hanced performance that could compensate for the shows more degradation than OFDM. Overall, the PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 10

WPM schemes seem to be able to perform better by one elementary block is: than OFDM when a wavelet with sufficiently low  l m 4 L0 − 2 adds out-of-band energy is used. For a given wavelet, direct  2 there appear to be a gain in robustness with the PWPT = l m . (9)  4 L0 mults increase in generating wavelet order. 2 Additional results are presented in Figure 12 where d·e denotes the smallest integer higher than where the link BER of the same schemes as pre- its argument. viously are given as a function of the disturber Overall, a transform of size M = 2j is composed relative power Pdist. Its normalized frequency has of N (j) elementary blocks per stage j. Denoting been arbitrary chosen to be 0.1666, which can be R(j) as the input rate of the block of stage j, the verified from the previous figure to corresponds to number of operations for the stage j: the case where WPM leads to an average gain in direct comparison with OFDM. The WPM(coif1) scheme CWPT (j) = N (j) R(j) PWPT (10) ( performs overall quite similarly to OFDM, but N (j) = 2j−1 with , with a robustness to disturber about 2 dB lower. R(j) = 2J−j The WPM(coif5) and WPM(dmey) schemes, on the other hand, provide a much higher robustness than where the transform input signal rate R(J) is OFDM for a disturber power of up to -15 dB. assumed to be equal to unity. The overall number Past this threshold, it is noticeable that OFDM of operations for the WPT is then has instead a lower sensitivity, being undisturbed J for Pdist lower than −20dB while the two WPM X J−1 direct CWPT (J) = 2 PWPT (11) schemes require about 2 dB less. j=1 In general, the degradation in terms of BER of   l m  (2J − 1) 4 L0 − 2 adds the WPM signal due to a single tone disturber is  2 CWPT (J) =  l m . highly dependent on both its frequency and power.  (2J − 1) 4 L0 mults The results obtained have nevertheless shown that 2 WPM schemes are capable of high immunity to dis- This computational complexity can be further turbance when higher order wavelets are selected. compared to that required by the DFT used in OFDM systems. We assume here the implemen- tation of the DFT considered in [36], leading to a V. IMPLEMENTATION COMPLEXITY ESTIMATES complexity of The structure of a WPM-based transceiver is  2J (2J − 1) adds C (J) = . (12) very similar to that of OFDM. This section reports DFT J 2J mults on the implementation complexity of the WPT since it is the building block in which the systems Figure 13 displays the complexity of the WPT differ the most6. Three alternative architectures are relative to the DFT in terms of additions and considered that differ in the implementation of the multiplications. The number of additions is lower elementary blocks. for the WPT for transform sizes higher than 8 with We first consider the direct implementation of the L0 = 4, and for a transform size greater then 32 in elementary blocks as they are shown in Figure 2. In the case where a longer filter with L0 = 16 is used. the case of the forward transform, the elementary The number of multiplications on the other hand blocks consist simply of two up-samplers and two is generally higher for the WPT. With the shorter filter (L = 4), the number of multiplications is FIR filters of length L0. For the inverse transform, 0 an additional adder in required to combine the equal for both 256-point transforms. In the case output of the two filters, so the overall number of of an implementation with a generic processor, operations is slightly lower. We consider further the the cost of both operations is identical. Hence, forward transform only, since the case of the re- Figure 14 shows the total number of operations verse one can be easily derived from it. With filters required by the WPT, again relative to the DFT for different lengths L of the wavelet filter. The WPT of length L0, the number of operations required by 0 each filter is actually equivalent to the complexity complexity is higher for small size transforms, but decreases as the size of the transform increases. required by a L0/2-tap long filter thanks to the zero values samples inserted by the upsampler. Hence, Hence, for a transform size over 64, the number of the number of operations per input sample required operations for the WFT is lower than for the DFT, even in the case of L0 = 16. 6Optimal systems might differ as well in equalization and Overall, the WPT implementation complexity is synchronization schemes. on the same order of the one required by the PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 11

DFT, and might even be lower for medium size communications. This generic modulation has the transforms and wavelet filters of moderate length. potential of becoming a unique multicarrier com- The repartition between the addition/multiplication munication scheme used by devices with various operations leads to the conclusion that the com- constraints. A single scheme could thus be used to plexity is in favor of the WPT in the case of an communicate with small, resource-limited devices implementation in a general purpose signal proces- as well as high capacity, multimedia capable nodes. sor. In the case of a hardware implementation, the While considering such a scenario, it appears that higher complexity of a multiplier in comparison the full capability of WPM can only be exploited with an adder is likely to reduce the difference by a system having the capacity to determine the between the two schemes. The iterative structure best suited configuration. Hence, this fully supports of the WPT is nevertheless very well suited to the fact that WPM is the modulation of choice hardware implementation [37]. For each stage of for smart, environment and resource aware wireless the transform, the product of the number of elemen- systems. tary block by their processing rate is constant and Again, the authors believe that the most inter- 2J−1 J equal to . A complete stage transform can be esting feature of WPM is its inherent flexibility. performed by a single occurrence of the elementary While such flexibility cannot be fully exploited J 2J−1 block running at a clock speed times the with current systems and technologies, it is fore- 2J−1 J symbol rate or by instantiations running at seen that future generation systems will be able times the symbol rate. This provides an appreciable to provide the degree of intelligence required to flexibility in trading-off speed versus silicon area, achieve optimal performance in a given environ- a feature that is suitable for the future generation ment, at a given quality, and for a limited energy of reconfigurable systems. consumption. WPM is then likely to become a strong competitor to multicarrier systems currently VI.CONCLUSIONSANDFURTHERRESEARCH in use. TOPICS Some issues still remain in the way of the wide- We reviewed in this article the advantages of us- spread use of WPM. Although its principle has ing WPM for multicarrier communication systems. been proposed for more than a decade, WPM has An in-depth comparison between this new scheme not been studied in depth yet. Its similarity to both and OFDM has been reported. The performance of OFDM and overlapping multicarrier systems such WPM has been shown to be identical to the latter as CMFB allows the use of the work done for in several reference wireless channels, though at a the latter as a basis for the research developments. higher cost of equalization in multipath channels. This has been the approach taken in this paper Additional results taking into consideration effects in order to evaluate the potential of this new of some non-ideal elements of the system have modulation scheme. The first area where a large shown that WPM is slightly more sensitive than amount of work remains to be done is equalization. OFDM to these commonly encountered types of The overlapping of symbols causes a significant distortion. A comparison of the core transforms has amount of interference that requires an dedicated shown the low complexity potential of WPM based equalization scheme to be studied. A study on systems. Overall, the performance results of WPM lead us synchronization of the WPM signals in both time to conclude that this new modulation scheme is a and frequency domains would most probably lead viable alternative to OFDM to be considered for to efficient algorithms, due to the multi-resolution today’s communication systems. OFDM remains nature of the multiplexed signal. nevertheless a strong competitor thanks to its ca- Finally, it is important to underline that wavelet pability to cope with multipath effects efficiently. theory is still developing. Since the use of wavelet With the demand for ever higher bandwidth, the use packets in has been mainly of cyclic prefix will likely become restricted, and studied by communications engineers, an important hence OFDM and WPM will present equalization potential for improvements is possible if some of complexity of the same order. the specific issues are addressed from a mathemat- The major interest of WPM nevertheless resides ical point-of-view. Wavelet based communication in its ability to fulfill the wide range of require- systems have already shown a number of advan- ments of tomorrow’s ubiquitous wireless commu- tages over conventional systems. It is expected that nications. Hence, WPM has the strong advantage more is still to be pointed out as the knowledge of of being a generic modulation scheme whose ac- this recently proposed scheme gains more interest tual characteristics can be widely customized to within both the wireless communication industry fulfill the various requirements of advanced mobile and research community. PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 12

ACKNOWLEDGMENTS [20] X. Q. Gao, T. Q. Nguyen, and G. Strang, “On two-channel orthogonal and symmetric complex-valued FIR filter banks This work has been supported in part by the and their corresponding wavelets,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Academy of Finland (Grants for projects 50624 and Processing, 2002, pp. 1237–240. 50618). Authors wish to thanks Pr. Hanzo and Dr. [21] W. Y. Chen, DSL: Simulation techniques and standards de- Daly whose comments and suggestions have largely velopment for digital subscriber line systems. Macmillan Technical Publishing, 1998. contributed to improve this paper. [22] ANSI Committee, “ANSI T1.413-1995: Network and Cus- tomer Installation Interfaces - Asymmetric Digital Sub- scriber Line (ADSL) Metallic Interface,” ANSI Standard REFERENCES Committee T1E1.4, 1995. [23] J. G. Proakis, Digital communications, 3rd ed. Mc Graw Hill international editions, 1995. [1] J. A. C. 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Daneshgaran, “Performance nally multiplexed communication,” IEEE Transaction on of wavelet waveforms over linear and non-linear channels,” Signal Processing, vol. 45, no. 5, pp. 1336–1339, May in Proceedings of Wireless Communications and Network- 1997. ing Conference (WCNC), 1999, pp. 1148–1152. [10] N. Suzuki, M. Fujimoto, T. Shibata, N. Itoh, and [33] M. Tummla, M. T. Donovan, B. E. Watkins, and R. North, K. Nishikawa, “Maximum likelihood decoding for wavelet “Voltera series based modeling and compensation of non- packet modulation,” in IEEE Vehicular Technology Con- linearities in high power amplifers,” in Proceedings of ference (VTC), 1999, pp. 2895–2898. International Conference on Acoustics, Speech and Signal [11] S. Gracias and V. U. Reddy, “An equalization algorithm Processing, vol. 3, 1997, pp. 2417–2420. for wavelet packet based modulation schemes,” IEEE [34] G. Chrisikos, C. J. Clark, A. A. Moulthrop, M. S. Muha, Transactions on Signal Processing, vol. 46, no. 11, pp. and C. P. Silva, “A nonlinear ARMA model for sim- 3082–3087, November 1998. ulating power amplifers,” in IEEE MTT-S International [12] F. Daneshgaran and M. Mondin, “Clock synchronisation Microwave Symposium Digest, vol. 2, 1998, pp. 733–736. without self-noise using wavelets,” Electronics Letters, [35] M. Speth, S. Fechtel, G. Fock, and H. Meyr, “Broadband vol. 31, no. 10, pp. 775–776, May 1995. transmission using OFDM: system performance and re- [13] ——, “Symbol synchronisation using wavelets,” in IEEE ceiver complexity,” in International Zurich Seminar on Military Communications Conference, vol. 2, 1995, pp. Broadband Communications, February 1998, pp. 99–104. 891–895. [36] G. Bachman, L. Narici, and E. Beckenstein, Fourier and [14] M. Luise, M. Marselli, and R. Reggiannini, “Clock syn- wavelet analysis. Springer, 2000. chronisation for wavelet-based multirate transmissions,” [37] A. Jamin and P. Mah¨ onen,¨ “FPGA implementation of the IEEE Transactions on Communications, vol. 48, no. 6, pp. wavelet packet transform for high speed communications,” 1047–1054, June 2000. in International Conference on Field Programmable Logic [15] G. Strang and T. Q. Nguyen, Wavelet and filter banks. (FPL), Montpellier, France, September 2002. 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, CBMS Series, 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. PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 13

Wavelet modulation lowpass filter Amplitude versus frequency response |H(f)| for different wavelets 10 'db2', M = 8, 75 taps 'sym4', M = 8, 135 taps 0 'coif4', M = 8, 375 taps 'db6', M = 8, 195 taps −10

−20

−30

−40

−50

Amplitude |H(f)| in dB −60

−70

−80

−90

−100 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Normalized frequency Ω/(2* π) in Hz Fig. 4. Frequency domain localization of different wavelets

10 −1

10 −2

10 −3 BER

10 −4

10 −5 WPM(db2) WPM(db6) WPM(db10) OFDM (K=0) 10 −6 OFDM (K=8) 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Delay between paths Fig. 5. Performance of WPM versus OFDM in a 2-path time-invariant channel. BER is plotted as a function of the delay of arrival of the second path. The delayed path relative power of −3 dBc and the SNR is 20 dB PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 14

−1 10 WPM complex(db2, M=16) WPM complex(db6, M=16) WPM complex(db10, M=16) OFDM (M=16, K=0)

2 −2 σ = −5 dB 10 eq SNR = 20 dB BER

−3 10

−4 10 5 10 15 20 25 L eq

Fig. 6. BER performance of WPM and OFDM schemes with noisy equivalent impulse response, as a function of the length Leq 2 of imperfectly equalized channel response, at σeq = −5dBc.

WPM(coif1) and WPM(coif4) post−detection equalizer performance (All schemes 32 subcarriers, 4−path exp. decay delay profile channel, = 1, SNR = 20 dB) τ0 −5 WPM(coif1), ν = 1 WPM(coif1), ν = 3 −5.5 WPM(coif1), ν = 5 WPM(coif4), ν = 1 −6 WPM(coif4), ν = 3 WPM(coif4), ν = 5 −6.5

−7

−7.5

−8 Interference power [dBc] −8.5

−9

−9.5

−10 1 2 3 4 5 6 7 Number of taps in time domain η Fig. 7. Comparative performance of WPM(coif1) and WPM(coif4) schemes with post-detection equalization, as a function of the number of equalizer taps η in the time domain and with the number of taps ν in the frequency domain as a parameter. PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 15

WPM link BER degradation as a function of PA backoff margin γ (All schemes 64 subcarriers, 16QAM and AWGN channel with 20 dB SNR) 10 −1 WPM(db2) WPM(sym2) WPM(coif2) WPM(dmey) 10 −2 OFDM (K=0)

10 −3 BER

10 −4

10 −5

10 −6 2 3 4 5 6 7 8 9 10 Amplifier backoff margin [dB] Fig. 8. BER versus power amplifier backoff margin γ for different WPM schemes and OFDM.

WPM modulation sensitivity to sampling phase error (All modulation 16 subcarriers, QPSK over AWGN channel with 40 dB SNR) 10 −1

10 −2

10 −3 BER

10 −4

10 −5 WPM (db2) WPM (coif2) WPM (sym2) −6 OFDM (M=16, K=0) 10 0.15 0.175 0.2 0.225 0.25 0.275 0.3 0.325 0.35 0.375 0.4 Sampling phase error normalized to sampling period Fig. 9. Sensitivity of different WPM schemes versus OFDM schemes to sampling phase error, expressed as the link BER versus the normalized sampling phase error. PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 16

WPM(coifN) sensitivity to sampling phase error for different wavelet order (All schemes 32 subcarriers, QPSK and AWGN channel with 20 dB SNR) 10 −1

10 −2

10 −3 BER

10 −4

10 −5 WPM(coif1) WPM(coif2) WPM(coif3) WPM(coif4) −6 WPM(coif5) 10 0.16 0.18 0.2 0.22 0.24 0.26 0.28 Sampling phase error normalized to sampling period Fig. 10. Sensitivity of WPM(coifN) schemes to sampling phase error as a function of the wavelet order, expressed as the link BER versus the normalized sampling phase error.

WPM robustness to single tone disturber, as a function of the disturber frequency = 0 dBc) (P dist (All schemes 16 subcarriers, 16QAM) 0 10 OFDM (K=0) WPM complex(coif1) WPM complex(coif5) WPM complex(dmey)

−1 10 Bit Error Rate (BER)

−2 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Disturber relative frequency

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. PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 17

Sensitivity of WPM and OFDM schemes to a single tone disturber 0 (All schemes 16 subcarriers, 16QAM) 10

−1 10

Bit Error Rate (BER) −2 10

OFDM (K=0) WPM(coif1) WPM(coif5) WPM(dmey) −3 10 −25 −20 −15 −10 −5 0 5 Disturber relative power [dBc]

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.

1 10

0 10

−1 10

−2 10 Lo = 4

Relative amount of ADD Lo = 10 −3 Lo = 16 10 8 16 32 64 128 256 512 1024 Transform size (M)

1 10

0 10

Lo = 4 Lo = 10 Relative amount of MULT −1 Lo = 16 10 8 16 32 64 128 256 512 1024 Transform size (M) Fig. 13. Number of additions and multiplications required by the WPT relatively to the DFT as a function of the transform size M, for different wavelet generating filter lengths L0. PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 18

WPT complexity relative to DFT 1 10

Lo = 4 Lo = 10 Lo = 16

0 10

−1 10 Total amount of operations for WPT relatively to DFT

−2 10 3 4 5 6 7 8 9 10

Number of stages (J)

Fig. 14. WPT complexity relative to that of the DFT as a function of the transform size M = 2J , for different wavelet generating filter lengths L0.