## 1. Introduction

The discrete wavelet transform (DWT) has an established position in processing of signals and images in research and industry. The first DWT structures were based on the compactly supported conjugate quadrature filters (CQFs) (Smith & Barnwell, 1986; Daubechies, 1988). However, a drawback in CQFs is related to the nonlinear phase effects such as image blurring and spatial dislocations in multi-scale analyses. On the contrary, in biorthogonal discrete wavelet transform (BDWT) the scaling and wavelet filters are symmetric and linear phase. The biorthogonal filters (BFs) are usually constructed by a ladder-type network called lifting scheme (Sweldens, 1988). The procedure consists of sequential down and uplifting steps and the reconstruction of the signal is made by running the lifting network in reverse order. Efficient lifting BF structures have been developed for VLSI and microprocessor environment (Olkkonen et al. 2005; Olkkonen & Olkkonen, 2008). The analysis and synthesis filters can be implemented by integer arithmetics using only register shifts and summations. Many BDWT-based data and image processing tools have outperformed the conventional discrete cosine transform (DCT) -based approaches. For example, in JPEG2000 Standard (ITU-T, 2000), the DCT has been replaced by the lifting BFs.

One of the main difficulties in 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 signal

In this book chapter we review the methods for constructing the shift invariant CQF and BF wavelet sequences. We describe a dual-tree wavelet transform, where two parallel CQF wavelet sequences form a Hilbert pair, which warrants the shift invariance. Next we review the construction of the BF wavelets and show the close relationship between the CQF and BF wavelets. Then we introduce a novel Hilbert transform filter for constructing shift invariant dual-tree BF banks.

## 2. Design of the shift invariant CQF

The CQF DWT bank consists of the

where

The tree structured implementation of the real-valued CQF filter bank is described in Fig. 2. Let us denote the frequency response of the z-transform filter as

Correspondingly, we have the relations

where * denotes complex conjugation. 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

and

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

where the phase function

If we select 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

where the sign function is defined as

In this work we apply the Hilbert transform operator in the form

Our result (12) reveals 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

which warrants that the M-stage CQF wavelet sequence and the phase error corrected sequence are a Hilbert transform pair.

## 3. Biorthogonal discrete wavelet transform

The first DWT structures were based on the compactly supported conjugate quadrature filters (CQFs) (Smith & Barnwell, 1986), which have unavoided nonlinear phase effects in multi-resolution analyses. On the contrary, in biorthogonal discrete wavelet transform (BDWT) the scaling and wavelet filters are symmetric and linear phase. The two-channel biorthogonal filter (BF) bank is of the general form

where the scaling filter

## 4. Relationships between CQF and BF wavelet transforms

In the following treatment we use a short notation for the binomial term

which appears both in the CQF and BF banks. Using the binomial term the CQF bank can be written as

For the PR condition of the CQF bank ( ) the following is valid for K odd

On the other hand, the PR condition of the BF bank gives

Both PR conditions are identical if we state

The above relation (25) gives a novel way to design of the biorthogonal wavelet filter bank based on the CQF bank and vice versa. The polynomials

**Lemma 1:** If the scaling filter

where

## 5. Hilbert transform filter for construction of shift invariant BF bank

In BF bank the shift invariance is not an inbuilt property as in CQF bank. In the following we define the Hilbert transform filter

where

where the *c*_{k} coefficients are optimized so that the frequency response follows approximately

In this work we define the half-sample delay filter more generally as

The quadrature mirror filter

The frequency response of the filter

Comparing (27) and using the IIR filter notation (30) we obtain the Hilbert transform filter as

The Hilbert transform filter is inserted in the BF bank using the result of Lemma 1 (26). The modified prototype BF filter bank is

The BF bank (34) can be highly simplified by noting the following equivalents concerning on (33)

By inserting (35) in (34) we obtain a highly simplified FB bank

The modified BF bank (36) can be realized by the Hilbert transform filter

By filtering the real-valued signal

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

The wavelet sequence is obtained by decimation of the high-pass filtered analytic signal

The result (40) means that the decimation does not produce aliasing but the frequency spectrum is dilated by two. The frequency spectrum of the undecimated wavelet sequence

An integer-valued half-delay filter

## 6. Conclusion

It is well documented that the real-valued DWTs are not shift invariant, but small fractional time-shifts may introduce significant differences in the energy of the wavelet coefficients. Kingsbury (2001) showed that the shift invariance is improved by using two parallel filter banks, which are designed so that the wavelet sequences constitute real and imaginary parts of the complex analytic wavelet transform. The dual-tree discrete wavelet transform has been shown to outperform the real-valued DWT in a variety of applications such as denoising, texture analysis, speech recognition, processing of seismic signals and neuroelectric signal analysis (Olkkonen et al. 2006; Olkkonen et al. 2007b).

Selesnick (2002) made an observation that a half-sample time-shift between the scaling filters in parallel CQF banks is enough to produce the shift invariant wavelet transform. In this work we reanalysed the condition and observed a phase-error term

In multi-scale DWT analysis the complex wavelet sequences should be shift invariant. This requirement is satisfied in the Hilbert transform-based approach (Olkkonen et al. 2006, Olkkonen et al. 2007b), where the signal in every scale is Hilbert transformed yielding strictly analytic and shift invariant transform coefficients. The procedure needs FFT-based computation which may be an obstacle in many digital signal processor realizations. To avoid this we conducted the novel shift invariant dual-tree BF bank (36) based on the Hilbert transform filter (33). This highly simplified BF bank is yielded by *Lemma 1* and the equivalence (35) of the Hilbert transform filter (33). In many respects the BF bank (36) outperforms the previous nearly shift invariant DWT approaches.