Shift Invariant Biorthogonal Discrete Wavelet Transform for EEG Signal Analysis

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Since the discovery of the compactly supported conjugate quadrature filter (CQF) based discrete wavelet transform (DWT) (Smith & Barnwell, 1986; Daubechies, 1988), a variety of data and image processing tools have been developed. It is well known that real-valued CQFs have nonlinear phase, which may cause image blurring or spatial dislocations in multi-resolution analysis. In many applications the CQFs have been replaced by the biorthogonal discrete wavelet transform (BDWT), where the low-pass scaling and high-pass wavelet filters are symmetric and linear phase. In VLSI hardware the BDWT is usually realized via the ladder network-type filter (Sweldens, 1988). Efficient lifting wavelet transform algorithms implemented by integer arithmetic using only register shifts and summations have been developed for VLSI applications (Olkkonen et al. 2005). In multi-scale analysis the drawback of the BDWT is the sensitivity of the transform coefficients to a small fractional shift [0,1] τ ∈ in the signal, which disturbs the statistical comparison across different scales. There exist many approaches to construct the shift invariant wavelet filter bank. Kingsbury (2001) proposed the use of two parallel filter banks having even and odd number of coefficients. Selesnick (2002) has described the nearly shift invariant CQF bank, where the two parallel filters are a half sample time shifted versions of each other. Gopinath (2003) generalized the idea by introducing the M parallel CQFs, which have a fractional phase shift with each other. Both Selesnick and Gopinath have constructed the parallel CQF bank with the aid of the all-pass Thiran filters, which suffers from nonlinear phase distortion effects (Fernandes, 2003). In this book chapter we introduce a linear phase and shift invariant BDWT bank consisting of M fractionally delayed wavelets. The idea is based on the B-spline interpolation and decimation procedure, which is used to construct the fractional delay (FD) filters (Olkkonen & Olkkonen, 2007). The FD B-spline filter produces delays τ =N/M (N, M∈N , N= 0,...,M1). We consider the implementation of the shift invariant FD wavelets, especially for the VLSI environment. The usefulness of the method was tested in wavelet analysis of the EEG signal waveforms.