Wavelet packet division multiplexing and wavelet packet design under timing error effects

Abstract

Wavelet packet division multiplexing (WPDM) is a multiple signal transmission technique in which the message signals are waveform coded onto wavelet packet basis functions for transmission. The overlapping nature of such waveforms in time and frequency provides a capacity improvement over the commonly used frequency division multiplexing (FDM) and time division multiplexing (TDM) schemes while their orthogonality properties ensure that the overlapping message signals can be separated by a simple correlator receiver. The interference caused by timing offset in transmission is examined. A design procedure that exploits the inherent degrees of freedom in the WPDM structure to mitigate the effects of timing error is introduced, and a waveform that minimizes the energy of the timing error interference is designed. An expression for the probability of error due to the presence of Gaussian noise and timing error for the transmission of binary data is derived. The performance advantages of the designed waveform over standard wavelet packet basis functions are demonstrated by both analytical and simulation methods. The capacity improvement of WPDM, its simple implementation, and the possibility of having optimum waveform designs indicate that WPDM holds considerable promise as a multiple signal transmission technique. I. INTRODUCTION W AVEFORM coding [1] is usually employed in a digital communication system to convert the message data into continuous waveforms in order to provide better immunity against noise, fading, or jamming during transmission. To achieve this aim, the various schemes of waveform coding endevor to make the distance between the waveforms in the coded signal as large as possible, i.e., to make the cross-correlation coefficient between any pair of waveforms as small as possible [2]. The smallest possible value of the cross-correlation coefficient is 1 when the waveforms are antipodal, with a distance of between its member waveforms being the waveform energy); however, this may only serve for a single binary signal. On the other hand, an orthogonal set of waveforms has all the crosscorrelation coefficients equal to zero and has a distance equal

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