Design of orthonormal symmetric wavelet filters using real allpass filters

Abstract

In this paper, a class of real-valued orthonormal symmetric wavelet filters is constructed by using allpass filters, and a new method for designing the allpass-based wavelet filters with the given degrees of flatness is proposed. The proposed method is based on the formulation of a generalized eigenvalue problem by using the Remez exchange algorithm and considering the flatness condition. Therefore, a set of filter coefficients can be easily computed by solving the eigenvalue problem, and the optimal solution in the minimax sense is obtained through a few iterations. Furthermore, the design of the maximally flat allpass-based wavelet filters is also included as a specific case, but it has a closed-form solution that is the same as in Selesnick (IEEE Trans. Signal Process. 46 (4) (April 1998) 1138–1141) so that the iteration procedure is not needed. Finally, some examples are designed to investigate the filter characteristics, and it is shown that the number of delay elements strongly influences the filter magnitude responses.

Zusammenfassung

In dieser Arbeit wird die Konstruktion einer Klasse reellwertiger, orthonormaler und symmetrischer Waveletfilter als Allpaßfilter betrachtet. Es wird eine neue Methode für den Entwurf der Allpaß-Waveletfilter mit gewähltem Grad an Glattheit vorgeschlagen. Die vorgeschlagene Methode beruht auf der Formulierung eines verallgemeinerien Eigenwert-problems bei Berücksichtigung der Glattheitsbedingung und der Anwendung des Remezschen Austausch-Algorithms! Deshalb können die Filterkoeffizienten nach Lösung des Eigenwertproblems leicht berechnet werden und die optimale Lösung wird nach wenigen Iterationen erreicht. Der Entwurf des maximal glatten Allpaß-Waveletfilters ist als Spezialfall enthalten. Da seine Lösung in geschlossener Form angegeben werden kann und diese dieselbe ist wie die in Selesnick (IEEE Trans. Signal Process. 46 (4) (April 1998) 1138–1141) vorgeschlagene, ist in diesem Fall die Iterationsprozedur verzichtbar. Es werden einige Beispiel-Wavelets entworfen, um deren Filter-Charakteristik zu untersuchen. Dabei wird gezeigt, daß die Zahl der Verzögerungselemente den Amplitudengang der Filterantwort beeinflußt.

Résumé

Nous construisons dans cet article une classe de filtres d'ondelettes symétriques, orthonormaux, et à valeurs réelles à l'aides de filtres passe-tout, et présentons une méthode nouvelle de conception des filtres d'ondelettes basés sur les filtres passe-tout et ayant un degré de platitude donné. La méthode proposéee est basée sur la formulation d'un problème de valeurs propres généralisées et utilise l'algorithme d’échange de Remez pour une platitude donnée. De ce fait, un ensemble de coefficients de filtre peut être aisément calculé en résolvant le problème de valeurs propres, et la solution optimale au sens minimax est obtenue en quelques itérations. De plus, la conception des filtres d'ondelettes maximalement plats est également incluse comme cas particulier, mais elle a une solution analytique identique à celle de Selesnick (IEEE Trans. Signal Process. 46 (4) (April 1998) 1138–1141) et ne requiert done pas de procédure itérative. Enfin, quelques exemples sont conçus pour étudier les caractéristiques des filtres, et il est montré que le nombre d’éléments de retard influence considérablement les résponses en amplitude du filtre.

Introduction

The discrete wavelet transform (DWT), which is implemented by a two-channel perfect reconstruction filter bank (PRFB), has been applied extensively to digital signal and image processing [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. In many applications such as digital image coding, wavelets are required to be real since the signal is real-valued in general. In this paper, we restrict ourselves to real-valued wavelet filters. In addition to orthonormality, one desirable property for wavelets is symmetry, which corresponds to the phase linearity of the wavelet filters. It is known [10] that FIR filters (corresponding to the compactly supported wavelets) can easily realize the linear phase. However, it had been proven in [4] that there does not exist any real-valued compactly supported orthonormal symmetric wavelets except the Haar wavelet. To obtain symmetric wavelets, at least one of the above properties has to be given up. One possible solution to this dilemma is to construct compactly supported biorthogonal symmetric wavelets [4], [11], [14]. In [2], [14], biorthogonal symmetric wavelets have been used in image coding application and are required close to orthonormal. Another possible solution is to construct orthonormal symmetric wavelets by using IIR filters [5], [7]. In [5], a class of orthonormal symmetric wavelet filters has been constructed by using real allpass filters, but the design method for these allpass-based wavelet filters is not discussed. A closed-form solution for the maximally flat allpass-based wavelet filters is given in [7], but only the case of K=1 and even N is described, where K is the number of delay elements and N is the order of allpass filter, as is explained in Section 2. It is known [6] that frequency selectivity is a useful property for many applications such as signal processing, but the maximally flat filter has a poor frequency selectivity in general. For this reason, the wavelet filters are required to have the best-possible frequency selectivity for the given degrees of flatness, i.e., the given number of vanishing moments (indication of regularity).

In this paper, we consider the design of the orthonormal symmetric wavelet filters proposed in [5], and propose a new method for designing the allpass-based wavelet filters with the given degrees of flatness. The proposed method is based on the formulation of a generalized eigenvalue problem by using the Remez exchange algorithm and considering the flatness condition. Therefore, a set of filter coefficients can be easily obtained by solving the eigenvalue problem [15], [16], and the optimal solution in the minimax sense is attained through a few iterations. The proposed design algorithm is computationally efficient because it retains the speed inherent in the Remez exchange algorithm. Furthermore, the design of the maximally flat allpass-based wavelet filters that have the maximal degrees of flatness is also included in the proposed method as a specific case, but it has a closed-form solution that is the same as in [7] so that the iteration procedure is not needed. Finally, we design some examples to investigate the filter characteristics, and show the effects of the number of delay elements on the filter magnitude responses.

This paper is organized as follows. Section 2 describes a class of real-valued orthonormal symmetric wavelet filters composed of allpass filters. Section 3 presents a design method for the allpass-based wavelet filters with the given degrees of flatness based on the formulation of a generalized eigenvalue problem by using the Remez exchange algorithm. Section 4 shows some design examples to demonstrate the effectiveness of the proposed method, and investigate the filter characteristics. Conclusions are given in Section 5.