Optimal Wavelet Design for Multicarrier Modulation with Time Synchronization Error D. Karamehmedović, M. K. Lakshmanan, H. Nikookar International Research Center for Telecommunications and Radar (IRCTR) Department of Electrical Engineering, Mathematics and Computer Science Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands m.k.lakshmanan@tudelft.nl, h.nikookar@tudelft.nl Abstract— Wavelet Packet based Multi-Carrier Modulation (WPMCM) is an efficient transmission technique which has the advantage of being a generic scheme whose characteristics can be customized to fulfill a design specification. However, WPMCM is sensitive and vulnerable to time synchronization errors because its symbols overlap. In this paper, we design new wavelets to alleviate WPMCM’s vulnerability to timing errors. First, a filter design framework that facilitates the development of new wavelet bases is built. Then the expressions for errors due to time offset in WPMCM transmission are derived and stated as a convex optimization problem. Finally, an optimal filter that best handles these deleterious effects is designed by means of Semi Definite Programming (SDP). Through computer simulations the performance advantages of the newly designed filter over standard wavelet filters are proven. I. INTRODUCTION Wavelet Packet based Multi Carrier Modulation is a multiplexing method that uses orthogonal waveforms derived from a wavelet packet transform to combine a collection of parallel signals into a single composite signal. WPMCM is similar to the traditionally popular Fast Fourier Transform (FFT) based Orthogonal Frequency Division Multiplexing (OFDM) in the sense that both use orthogonal waveforms as subcarriers and achieve high spectral efficiency by allowing subcarriers’ spectra to overlap. The adjacent subcarriers do not interfere with each other as long as the orthogonality between the subcarriers is preserved. The difference between OFDM and WPMCM is in the shape of the subcarriers and the way they are generated. OFDM makes use of Fourier bases while WPMCM uses wavelet packet bases which are generated from a class of FIR filters called paraunitary filters. WPMCM was first proposed by Lindsey who laid out the theoretical foundations and propounded its use as an alternative to OFDM [1]. His idea has since been carried forward by many researchers. Maximum likelihood decoding for wavelet packet modulation has been addressed by Suzuki [2]. The study of an equalization scheme suited for WPMCM has been conducted by Gracias [3]. In [4]–[5] an investigation on the performance of WPMCM systems in the presence of time offset is performed. The greatest motivation for pursuing WPMCM systems is in the flexibility and adaptability that they offer. Unlike OFDM where the carriers are static sine/cosine bases, WPMCM uses wavelets whose features can be tailored to satisfy an engineering demand. Different wavelets result in different subcarriers leading to different transmission characteristics. By careful selection of proper wavelets it is possible to optimize WPMCM performance in terms of bandwidth efficiency, frequency selectivity of subcarriers, sensitivity to synchronization errors, PAPR, etc. The WPMCM is a developmental system and a lot of key research questions still remain to be addressed before it can become practically viable. One of them is their high sensitivity and vulnerability to time synchronization errors because of the overlap in WPMCM symbols. The goals of this paper are two-fold. First to utilize wavelet theory to establish a framework that facilitates the design of new wavelet bases. Then to focus on the reduction of time- synchronization errors in WPMCM by developing a family of new wavelets that better cope with infarctions caused by time offset. To this end the expressions for Inter Carrier Interference (ICI) and Inter Symbol Interference (ISI) in WPMCM transmission caused by time offset are first derived and stated as a convex optimization problem. Then an optimal filter that best handles these deleterious effects is derived using optimization algorithms. Through computer simulations the performance advantages of the newly designed filter over standard wavelet filters is demonstrated. The rest of the paper is organized as follows. Section II outlines the basics of WPMCM transceiver while section III discusses the criteria for two-channel filter bank. Time offset in WPMCM is discussed in Section IV. In Section V the optimization problem is formulated and finally numerical results are presented in Section VI followed by conclusions which appear in Section VII. II. WAVELET PACKET BASED MULTI-CARRIER MODULATION The wavelet packet theory can be viewed as an extension of Fourier analysis. The basic idea of both transformations is the same: projecting an unknown signal on a set of known basis functions to obtain insights on the nature of the signal. Any function S(n) in 2 ( ) L ℜ can be expressed as the sum of weighted wavelet packets. In communication systems, this means that a signal can be seen as the sum of modulated wavelet packets leading to the idea of WPMCM. The WPMCM signal is composed of symbols obtained from a sum of modulated and weighted wavelet packet waveforms ξ . In the discrete time domain this signal S(n) can be expressed as: log ( ) 2 1 , 0 () ( ) N N k uk u k Sn a n uN ξ - = = - ∑∑ (1) In (1) N denotes the number of subcarriers while u and k are the symbol and subcarrier indices, respectively. The constellation symbol modulating k th subcarrier in u th symbol is represented as a u,k . The sub-index log 2 (N) denotes the levels of decomposition required to generate N subcarriers. It is well known from the theory of wavelets that compactly supported orthonormal wavelet bases can be obtained from two-band paraunitary filter banks [6]–[8]. Time and frequency limited wavelet packet bases ξ(t) can be derived by iterating discrete half-band high g[n] and low-pass h[n] filters, recursively defined by: 2 1 2 1 1 () 2 [] ( 2 ) () 2 [] ( 2 ) p p l l l n p p l l l n t hn t n t gn t n ξ ξ ξ ξ + + + = - = - ∑ ∑ (2) In (2) the subscript l denotes the level in the tree structure and superscript p indicates the tree depth. The number of channels This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings. 978-1-4244-4148-8/09/$25.00 ©2009