Design and Realization of Bireciprocal Lattice Wave Discrete Wavelet Filter Banks

In this paper, designs and efficient realizations of special types of IIR wavelet filter banks are introduced. These special types are the bireciprocal lattice wave discrete wavelet filter banks (BLW-DWFBs). It is believed that these IIR filter banks possess superior band discriminations and perfect roll-off frequency characteristics with respect to their FIR counterparts. 5 th and 7 th order bireciprocal lattice wave digital filters (BLWDFs) are first derived. They are designed to simulate scaling and wavelet functions of six-level wavelet transform. Such IIR filter banks are then realized as all-pass sections. Computationally-efficient realizations by such sections indicate that the resulting IIR discrete wavelet filter banks can lead to hardware implementations with less-complexity


Introduction
The main advantage of infinite-impulse response (IIR) filter banks is that a good frequency selectivity and orthonormality are not mutually exclusive properties for IIR filter banks, as they are in a finite-impulse response (FIR) case [1].In 2004, S. Damjanovic, et al. [2] presented the design and characteristics of orthonormal two-band QMF banks, with perfect reconstruction (PR) and linear-phase properties.The corresponding wavelet structures were derived with their wavelet and scaling functions.Five iterations of the analysis filter bank in low-pass branch were used for such purpose with no implementations.In 2004 also, R. Yamashita et al. [3] proposed an IIR half-band filter with an arbitrary degree of flatness.The application of such IIR half-band filters in filter banks and wavelets was presented.The stability of such filter was guaranteed.Several design examples were illustrated with no comments on realization.
Many other examples on the designed orhtonormal wavelet transform implemented with IIR filter pairs were considered by S. Damjanovic, et al. [1] in 2005.Then the frequency transformations of such wavelet IIR filter banks were also presented by S. Damjanovic, et al. [4] in 2005.In 2006, low complexity half-band IIR filters were presented and realized by L. Milic, et al. [5] using two path polyphase structures utilizing all-pass filters as components.Such realization was accomplished with no-phase linearity.Recently, many attempts have been reported concerning FPGA implementations and other hardwares of two-band wavelet IIR filter banks [6]- [8].
In this paper, new solutions are presented for the design problem of generating the wavelet transform by iteration of orthonormal two-band power-complementary IIR wavelet filter banks, with PR properties.The structure of such IIR wavelet filter bank is based on the bireciprocal lattice wave digital filters (BLWDF) to simulate a two channel wavelet filter bank, resulting in a bireciprocal lattice wave discrete wavelet filter bank (BLW-DWFB).The designed lattice structures are composed of two parallel real all-pass sections.2 nd order to 1st order all-pass filter section reduction method is applied via down sampling position alteration in the designed structure.Consequently, the resulting BLW-DWFB structures possess efficient computations.Suggesting, 1 st order wave adaptors for the hardware implementations of all 1st order all-pass filter sections in the structure, open the way for efficient FPGA implementations of such BLW-DWFBs.The generations of scaling and wavelet functions concerning these structures are also conducted in this paper.
This paper is divided into five major sections.Besides this introductory section, the next section reviews the design and realization of IIR wavelet filter banks.In section 3, the proposed design methods for 5 th and 7 th order IIR PR wavelet filter banks are described.Software and hardware efficient realizations are presented in Section 4. Section 5 includes the generation of scaling and wavelet functions after six-level analysis filter banks.Section 6 highlights some conclusions.

IIR Wavelet Filter Banks
A very efficient way for representing the QMF bank can be obtained by using polyphase structure [9].QMF banks, composed of two all-pass filters, are known to be one of the best circuits for building up a multi-channel IIR filter banks.They can completely eliminate the aliasing error and amplitude distortion [10].Fig. 1 shows a two channel all-pass filter based IIR QMF banks with polyphase realization [11].
In Fig. 1 the polyphase components are the 2 nd order all-pass filters A 0 (z 2 ) and A 1 (z 2 ) with the following transfer functions: where is the value of the multiplier coefficient in the i th all-pass section.
Fig. 1 Polyphase realization of the IIR wavelet filter bank.
Let H 0 (z) and H 1 (z) denote the transfer functions of the lowpass and highpass filter of the analysis part of the two-channel quadrature-mirror filter (QMF) bank, and let G 0 (z) and G 1 (z) denote, respectively the transfer functions of the lowpass and highpass filter of the synthesis part.By choosing transfer functions to satisfy the following conditions: ) then, the filter bank will possess both perfect reconstruction and orthonormality properties [12].Analysis filters H 0 (z) and H 1 (z) can respectively, be written for the low-pass side as and for the high-pass side as

The Proposed Design for IIR PR Wavelet Filter Banks
In this section, analytic solutions for the design problems of intermediate filters (whose characteristics are between IIR Butterworth and Daubechies filters) are proposed.The half band filter's poles are placed on an imaginary axis of the complex z-plane, where one of them is placed at the origin and the remaining conjugate complex pole pairs are located between and .All zeros of such filters are located on the unit circle, where three of them are placed at z = -1 for 5 th order filter and five of them at z = -1 for 7 th order filter to meet the flatness condition.
In applications of filter banks and wavelets, an important role is played by the regularity of the low-pass prototype filter; a feature which is closely related to the flatness on the magnitude response of the filter at the nyquist frequency ω = π.In constructing orthonormal bases of wavelets from iterated filter banks, a greater number of zeros of the low-pass filter at ω = π results in more regular wavelets [13].On the other hand, the number of vanishing moments of the wavelets can be obtained by the multiplicity of zeros at z = -1.It can be shown that having a maximum number of zeros at z= -1, implies a maximally flat characteristic for the filters involved [14].As an example, the Butterworth filter has a maximally flat magnitude response as it has all zeros at z = -1 and the highest possible regularity order, but it has the worst frequency selectivity.

5 th order intermediate IIR filter design
This proposed filter, as shown in Fig. 2, has five zeros and five poles.The values of these poles and zeros can be found out depending on the desired magnitude response that is shown in Fig. 3.
The transfer function of such IIR filter depends on the positions of its poles and zeros as in Fig. 2. Such transfer function is given by where a, b, α and β are constants less than 1, as shown in Fig. 2, with k as a magnitude scaling factor.Equation ( 5) can be factorized and reordered as Equation ( 6) is found as a function of .In order to be used in terms of , both numerator and enominator of ( 6) can be multiplied by , yielding ( ) Finding the transfer function of such 5 th order filter means an intermediate IIR filter has been concluded.The corresponding IIR filter can be implemented on BLWDF bases as a parallel connection of two all-pass IIR sections of the type shown in Fig. 1.By Gazsi method [15] that uses an alternative pole technique as illustrated in Fig. 4, the transfer functions of the two all-pass sections ( ) and ( ) can then be derived.
In Fig. 4, the poles of the area R 1 can formulate the transfer function of the all-pass section ( ) and the poles of the area R 2 can formulate the transfer function of the all-pass section ( ).The pole at the origin represents the delay element z -1 in (3).So, Substituting ( 8) and ( 9) in (3) to form the low-pass transfer function of BLWDF as The filter coefficients can be calculated by equating (11) to the general filter function (7).Referring to Fig. 2, the following equation can be written ( ) and from Fig. 3, the magnitude response at ω = 0.5π is Also, substituting in equation ( 14), results in

Conclusions
Designs and efficient realizations of bireciprocal lattice wave discrete wavelet filter banks (BLW-DWFBs) have been introduced.5 th and 7 th order bireciprocal lattice wave digital filters (BLWDFs) have been derived and well-designed to suit for wavelet filter bank utilizing all-pass sections with computationally-efficient realizations.The resulting IIR discrete wavelet filter banks can be hardware-implemented with less-complexity.
Moreover, Applying Sum-Power-Two method for the implementations of all multiplier values in different all-pass sections, Promising efficient multiplierless implementations can be obtained for such IIR wavelet filter banks on FPGA.This is to be considered as a future work concerning the FPGA implementations of these IIR wavelet filter banks.