## 1. Introduction

The discrete wavelet transform (DWT) algorithms have a firm position in multi-scale processing of biomedical signals, such as EMG and EEG. The DWT algorithms were initially based on the compactly supported conjugate quadrature filters (CQFs) (Smith & Barnwell, 1986; Daubechies, 1988). However, a drawback in CQFs is due to the nonlinear phase effects such as spatial dislocations in multi-scale analysis. This is avoided in biorthogonal discrete wavelet transform (BDWT) algorithms, where the scaling and wavelet filters are symmetric and linear phase. The biorthogonal filters are usually constructed by a ladder-type network called lifting scheme (Sweldens, 1988; ITU-T, 2000). Efficient lifting BDWT structures have been developed for microprocessor and VLSI environment (Olkkonen et al. 2005; Olkkonen & Olkkonen, 2008). Only integer register shifts and summations are needed for implementation of the analysis and synthesis filters.

A severe obstacle in multi-scale DWT analysis is the dependence of the total energy of the wavelet coefficients in different scales on the fractional shifts of the analysed signal. If we have a discrete-time signal

In this book chapter we review the shift invariant DWT algorithms for multi-scale analysis of biomedical signals. We describe a dual-tree DWT, where two parallel CQF wavelet sequences form a Hilbert pair, which warrants the shift invariance. Next we review the construction of the shift invariant BDWT, which is based on the novel design of the Hilbert transform filter. Finally, we describe the FFT based computation of the analytic signal and the implementation of the shift invariant quadrature mirror filter (QMF) bank.

## 2. Shift invariant CQF bank

In the following we describe a shift invariant DWT algorithm based on two parallel real- valued CQF banks. The conventional CQF DWT bank consists of the

where

Let us denote the frequency response of the z-transform filter as

Then we obtain the relations

where * denotes complex conjugation. The tree structured implementation of the two parallel real-valued CQF filter banks is described in Fig. 2. In M-stage CQF tree the frequency response of the wavelet sequence is

Next we construct a phase shifted parallel CQF filter bank consisting of the scaling filter

where

We have

We may note that the phase shifted CQF bank (6,8) obeys the PR condition (2). The frequency response of the M-stage CQF wavelet sequence is

(9) |

where the phase function

By selecting the phase function

the scaling filters (6) are half-sample delayed versions of each other. By inserting (11) in (10) we have

The wavelet sequences (5,9) yielded by the CQF bank (1) and the phase shifted CQF bank (6,8) can be interpreted as real and imaginary parts of the complex wavelet sequence

The requirement for the shift-invariance comes from

where

The result (12) indicates that if the scaling filters are the half-sample delayed versions of each other, the resulting wavelet sequences are not precisely Hilbert transform pairs. There occurs a phase error term

The two parallel BDWT trees can be considered to form a complex wavelet sequence by defining the Hilbert transform operator

By filtering the real-valued signal

whose magnitude response is zero at negative side of the frequency spectrum

Let us consider the complex wavelet sequence at the first stage (Fig. 6).The wavelet sequence is obtained by decimation of the high-pass filtered analytic signal

The frequency spectrum of the undecimated wavelet sequence

A key feature of the dual-tree wavelet transform is the shift invariance of the decimated analytic wavelet coefficients. The frequency spectrum of the decimated wavelet sequence of the fractionally delayed signal

## 2. Shift invariant BDWT filter bank

The two-channel BDWT filter bank is of the general form

where the scaling filter

An essential result is related to the modification of the BDWT bank (Olkkonen & Olkkonen, 2007a).

*Lemma 1:* If the scaling filter

where

In the following we apply *Lemma 1* for constructing the shift invariant BDWT filter bank. We describe a novel method for constructing the Hilbert transform filter

where the

The Hilbert transform filter is then obtained as

The Hilbert transform filter is inserted in the BDWT bank using the result of *Lemma 1* (22). The modified prototype BDWT filter bank is

A highly simplified BDWT filter bank can be obtained by noting that in (25)

The modified BDWT filter bank (27) can be realized by the Hilbert transform filter

An integer-valued Hilbert transform filter can be constructed by the B-spline transform (see details Olkkonen & Olkkonen, 2011b). The frequency response of the Hilbert transform filter shows a maximally flat magnitude spectrum. The phase spectrum corresponds to an ideal Hilbert transformer (15).

The Hilbert transform filter in Fig. 4 can be replaced by the Hilbert transform operator (16), which yields an analytic signal. This avoids the need for two parallel filter banks. In the following we describe a FFT based method for computation of the analytic signal and the implementation of the shift invariant quadrature mirror filter (QMF) bank.

## 3. FFT based computation of analytic signal

The fast Fourier transform of the signal

where

The analytic signal is then computed using the inverse FFT transform

The weighting sequence in (29) can be eliminated by writing

Now, for even n we have

and for odd n

For zero mean signal

The odd points of the analytic signal are then computed from (33). We call this as the reconstruction property of the zero mean analytic signal. In the following we present a novel shift invariant QMF bank, which utilizes the reconstruction property of the analytic signal.

## 4. Shift invariant QMF bank

In QMF bank the scaling and wavelet filters obey the relation

The shift invariant tree structured QMF DWT is described in Fig. 5. The FFT based Hilbert transform operator

where

The reconstruction consists of the summation of the decimated signals. We obtain

i.e. the summation of the decimated signals produces the even points

## 5. Conclusion

The dual-tree DWT algorithms have appeared to outperform the real-valued DWTs in several applications such as denoising, texture analysis, speech recognition, processing of seismic signals and multiscale-analysis of neuroelectric signal analysis (Olkkonen et al. 2006; Olkkonen et al. 2007b, Olkkonen & Olkkonen, 2010, Olkkonen & Olkkonen 2011a).

Selesnick (2002) noted that a half-sample time-shift between the scaling filters in parallel CQF banks yields a nearly shift invariant DWT, where the wavelet bases form a Hilbert transform pair. However, the multi-scale analyses of neuroelectric signals have revealed that the first stages of wavelet sequences are quite poorly shift invariant. We reanalysed the condition and observed a phase-error term

In this book chapter we described a novel shift invariant dual-tree BDWT (27) based on *Lemma 1* (22) and the Hilbert transform filter (25). In many respects the shift invariant BDWT bank outperforms the previous nearly shift invariant DWT approaches. The Hilbert transform filter assisted BDWT yields precisely shift invariant wavelet sequences, which permits the statistical analyses between scales in multi-scale analyses of biomedical signals such as EMG and EEG.

The Hilbert transform filter in Fig. 4 can be replaced by the Hilbert transform operator (16), which yields an analytic signal. This avoids the need for two parallel filter banks. In this work we described a FFT based method for computation of the analytic signal and the implementation of the shift invariant QMF bank. As a clear advantage of the half-band QMF structure is that the frequency responses of the scaling and wavelet filters are mirror symmetric with respect to