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Wavelet Packet Modulation for Wireless Communications PUBLISHED IN WIRELESS COMMUNICATIONS & MOBILE COMPUTING JOURNAL, MARCH 2005, VOL. 5, ISSUE 2 1 Wavelet Packet Modulation 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. COMMUNICATION SYSTEM ARCHITECTURE 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 modem, 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.
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