Perfect Reconstruction FIR Filter Banks Research Articles

A state-space description for perfect-reconstruction wavelet FIR filter banks with special orthonormal basis functions

This paper presents a state space description for wavelet FIR filter banks with perfect reconstruction using special orthonormal basis functions. The FIR structure guarantees the BIBO stability, robustness and improves the filter divergence problem while orthonormal basis functions have characteristics that make them attractive in the modeling of dynamic systems. The state space description presented in this paper has all of those advantages and is minimal.

Design and Factorization of Two-Channel Perfect Reconstruction Filter Banks with Causal-Stable IIR Filters

In this paper, new design and factorization methods of two-channel perfect reconstruction (PR) filter banks (FBs) with casual-stable IIR filters are introduced. The polyphase components of the analysis filters are assumed to have an identical denominator in order to simplify the PR condition. A modified model reduction is employed to derive a nearly PR causal-stable IIR FB as the initial guess to obtain a PR IIR FB from a PR FIR FB. To obtain high quality PR FIR FBs for carrying out model reduction, cosine-rolloff FIR filters are used as the initial guess to a nonlinear optimization software for solving to the PR solution. A factorization based on the lifting scheme is proposed to convert the IIR FB so obtained to a structurally PR system. The arithmetic complexity of this FB, after factorization, can be reduced asymptotically by a factor of two. Multiplier-less IIR FB can be obtained by replacing the lifting coefficients with the canonical signal digitals (CSD) or sum of powers of two (SOPOT) coefficients.

Orthogonal and Biorthogonal $\sqrt 3$-Refinement Wavelets for Hexagonal Data Processing

The hexagonal lattice was proposed as an alternative method for image sampling. The hexagonal sampling has certain advantages over the conventionally used square sampling. Hence, the hexagonal lattice has been used in many areas. A hexagonal lattice allows radic3, dyadic and radic7 refinements, which makes it possible to use the multiresolution (multiscale) analysis method to process hexagonally sampled data. The radic3-refinement is the most appealing refinement for multiresolution data processing due to the fact that it has the slowest progression through scale, and hence, it provides more resolution levels from which one can choose. This fact is the main motivation for the study of radic3-refinement surface subdivision, and it is also the main reason for the recommendation to use the radic3-refinement for discrete global grid systems. However, there is little work on compactly supported radic3 -refinement wavelets. In this paper, we study the construction of compactly supported orthogonal and biorthogonal radic3-refinement wavelets. In particular, we present a block structure of orthogonal FIR filter banks with twofold symmetry and construct the associated orthogonal radic3-refinement wavelets. We study the sixfold axial symmetry of perfect reconstruction (biorthogonal) FIR filter banks. In addition, we obtain a block structure of sixfold symmetric radic3-refinement filter banks and construct the associated biorthogonal wavelets.

Eigenfilter Approach to the Design of One-Dimensional and Multidimensional Two-Channel Linear-Phase FIR Perfect Reconstruction Filter Banks

We present an eigenfilter-based approach for the design of two-channel linear-phase FIR perfect-reconstruction (PR) filter banks. This approach can be used to design 1-D two-channel filter banks, as well as multidimensional nonseparable two-channel filter banks. Our method consists of first designing the low-pass analysis filter. Given the low-pass analysis filter, the PR conditions can be expressed as a set of linear constraints on the complementary-synthesis low-pass filter. We design the complementary-synthesis filter by using the eigenfilter design method with linear constraints. We show that, by an appropriate choice of the length of the filters, we can ensure the existence of a solution to the constrained eigenfilter design problem for the complementary-synthesis filter. Thus, our approach gives an eigenfilter-based method of designing the complementary filter, given a ldquopredesignedrdquo analysis filter, with the filter lengths satisfying certain conditions. We present several design examples to demonstrate the effectiveness of the method.

Multi-Plet Two-Channel Perfect Reconstruction FIR and Causal Stable IIR Filter Banks

This paper proposes a new class of two-channel structural perfect reconstruction (PR) finite-impulse response / infinite-impulse response (FIR/IIR) filter banks (FBs) called the multi-plet FBs. It generalizes the structural PR FBs proposed by Phoong and the conventional triplet FB in that it employs multiple lifting steps similar to the conventional lifting structure. Apart from the important structural PR property, the multi-plet FB can be systematically designed to meet a given specification on the passband/stopband ripples and transition bandwidth: a low-order prototype PR FB with a much wider transition band is first designed using nonlinear optimization in order to obtain a symmetric frequency response and prescribed passband and stopband ripples. As only a few variables are required, its performances can easily be controlled. A subfilter is then designed using second-order cone programming (SOCP) so that the prototype FB can be warped by means of frequency transformation to meet the desired transition bandwidth, while preserving the PR condition, passband/stopband ripples and lifting structure of the prototype FB. The design procedure is very general and it applies to both linear-phase/low-delay FIR and IIR multi-plet FBs. By formulating the K-regularity conditions as a set of linear equality constraints on the subfilter coefficients, multi-plet-based wavelet bases can easily be designed under the SOCP framework. Design examples show that the proposed approach offers more flexibility in controlling the frequency characteristics of the PR FBs and lower design complexity than conventional methods

Erratum to “A Direct Design of Oversampled Perfect Reconstruction FIR Filter Banks”

Provides an overview of the technical articles and features presented in this issue.

Optimal design of synthesis filters in multidimensional perfect reconstruction FIR filter banks using Grobner bases

For a given low-pass analysis filter in an n-dimensional (n-D) perfect reconstruction (PR) FIR filter bank, there exists certain degree of freedom in designing a synthesis filter to make the overall filter bank have PR property, and Quillen-Suslin Theorem can be used to precisely determine the degree of freedom. Designing a PR synthesis filter with optimal frequency response amounts to minimizing the deviation between the frequency responses of the synthesis filter and an ideal filter under the PR constraint. In this paper, a method based on Grobner bases computation is developed for optimally designing critically sampled n-D PR FIR filter banks when one analysis filter is known. This design method can be easily combined with various existing design methods including minimax optimization and weighted least squares optimization, and can incorporate additional design goals including linear phase and bandpass characteristic.

Rate-distortion-optimal subband coding without perfect-reconstruction constraints

We investigate the design of subband coders without the traditional perfect-reconstruction constraint on the filters. The coder uses scalar quantizers, and its filters and bit allocation are designed to optimize a rate-distortion criterion. Using convexity analysis, we show that optimality can be achieved using filterbanks that are the cascade of a (paraunitary) principal component filterbank for the input spectral process and a set of pre and postfilters surrounding each quantizer. Analytical expressions for the pre and postfilters are then derived. An algorithm for computing the globally optimal filters and bit allocation is given. We also develop closed-form solutions for the special case of two-channel coders under an exponential rate-distortion model. Finally, we investigate a constrained-length version of the filter design problem, which is applicable to practical coding scenarios. While the optimal filterbanks are nearly perfect-reconstruction at high rates, we demonstrate an apparently surprising advantage of optimal FIR filterbanks; they significantly outperform optimal perfect-reconstruction FIR filterbanks at all bit rates.

Two-dimensional perfect reconstruction FIR filter banks with triangular supports

We present a theory and design of two-dimensional (2-D) perfect reconstruction (PR) filter banks (FBs) (PRFBs) in which the supports of the analysis and synthesis filters consist of two triangulars. The two-triangular FB can be realized by designing an appropriate 2-D complex prototype whose passband support is a triangle that is a half of a parallelepiped-shaped passband support defined by the sampling matrix. Then a complex prototype filter is modulated by the DFT, and each analysis filter is derived by taking the real part of the modulated output. We show that the two-triangular FB satisfies the condition of permissibility. A necessary and sufficient condition for 2-D PRFBs is derived. Moreover, we present a design method of the 2-D PRFB that minimizes the cost function consisting of the frequency constraint and PR condition. Finally, a design example is presented to confirm the validity of the proposed method.

Design of selective M-channel perfect reconstruction FIR filter banks

A new method is presented for the optimal design of M-channel perfect reconstruction FIR quadrature mirror filter (QMF) banks which is based on the orthogonal matrix concept. The parameters of the analysis filters are adjusted so that they have a linear phase and minimum stopband energy. Comparison with a recently published method reveals that the resulting design procedure gives better stopband characteristics. A numerical design example is included to validate the theory and to demonstrate the superiority of the designed filters from selectivity, reconstruction and distortion points of view.

Minimax design of two-channel low-delay perfect-reconstruction FIR filter banks

The authors deal with the design problem of low-delay perfect-reconstruction filter banks for which the FIR analysis and synthesis filters have equiripple magnitude response. Based on the minimax error criterion, the design problem is formulated in such a manner that the coefficients for the FIR analysis filters can be found by minimising the weighted peak error of the designed analysis filters, subject to the perfect-reconstruction constraints. A design technique based on a modified dual-affine scaling variant of Karmarkar's algorithm, in conjunction with approximation schemes, is then developed for solving the resulting nonlinear optimisation problem. The effectiveness of the proposed design technique is demonstrated by several simulation examples.

Mirror-image symmetric perfect-reconstruction FIR filter banks: Parametrization and design

The problem of designing N– K filters of an N-band multi-rate analysis FIR filter bank, given the rest K filters, so that perfect reconstruction (PR) with an FIR synthesis filter bank is achieved, was recently studied [Kofidis et al., 1996]. A solution procedure was proposed for computing a (particular) solution for the unknown filters and a complete parametrization of the solution set was provided. In this paper, the above problem (referred to as the ( N, K)-problem) is treated in the context of pairwise mirror-image symmetric filter banks, and it is shown to be equivalent to two independent, unconstrained ( N/2, K/2)-problems. The synthesis bank, which is shown to have the mirror-image symmetry too, is easily obtained as a by-product of the solution process. The specific parametrization of the resulting filters is given, and an optimization procedure, leading to practically useful filters, is discussed. The case of linear phase (LP) is also treated, and a complete parametrization of the LP analysis/synthesis polyphase matrices is derived. This extends earlier work to include a much larger class of LP filter banks. The proposed approach leads naturally to a ladder-type realization of the filter bank, with the PR, LP and mirror-image symmetry properties being structurally enforced under both coefficient quantization and round-off errors. A design example of an LP paraunitary FB is presented, which demonstrates the superior numerical performance of the structures developed here over the lattice-based realizations, achieved at practically no extra computational cost.

Analysis, design, and implementation of two-channel linear-phase filter banks: a new approach

The problem of designing two-channel perfect-reconstruction FIR filter banks with linear-phase analysis and synthesis filters is revisited. Based on a new algebraic formulation, all the possible factorized forms for this two-band filter bank are derived. We thus obtain complete and canonical solutions for the filter banks, composed of odd-order symmetric and antisymmetric filters (type-A systems) and for those built with symmetric even order filters (type-B systems). A strong characteristic of these new cascade structures, which, until now, had not been identified, is related to a defectivity property. Taking this into account is the key issue to cover all the FIR solutions and to design cascade structures being robust to the quantization of their parameters. Design examples are provided that illustrate our method.

A new approach to the design of critically sampled M-channel uniform-band perfect-reconstruction linear-phase FIR filter banks

A new approach is presented for the design of uniform-band M-channel perfect-reconstruction (PR) FIR filter banks employing linear-phase analysis and synthesis filters. The technique designs on the impulse responses of the analysis filters directly. The design problem is formulated as an optimization program. The filter bank's PR feature can either be implicitly enforced through a set of mathematical relationships among the analysis filters' coefficients or through a set of constraints in the optimization program. The former approach results in a filter bank whose PR feature's dependency on hardware and software is eliminated or, at least, minimized. The synthesis filters are then obtained by a set of relationships that describe each synthesis filter as a function of the analysis filters. The criterion for optimality is least-squares, where the square of the difference between the ideal and actual frequency responses is integrated over the appropriate frequency bands for all M analysis filters and minimized.

Multidimensional Two-channel Linear Phase FIR Filter Banksand Wavelet Bases with Vanishing Moments

In the paper new design methods for multidimensional two-channel perfect reconstruction (PR) FIR filter banks are described. The filters have linear phase, achieve an arbitrarily high number of van...

Design of oversampled perfect reconstruction FIR filter banks

There are some applications that require oversampled filter banks, for example, adaptive signal processing using multirate filter banks. Little study was done, however, for such filter banks compared with maximally decimated ones. In this paper, we propose a design method for oversampled filter banks which satisfy the perfect reconstruction and linear phase properties. Moreover, we consider the design method for both orthogonal and biorthogonal ones. When the decimation ratio is not a divisor of the number of channels, we find it impossible for analysis and synthesis filters to have the same frequency responses by considering the mechanism of aliasing cancellation. In other words, biorthogonal filter banks can be designed under an arbitrary decimation ratio, while it is restricted to a divisor of the number of channels in the orthogonal case. We first derive the conditions for perfect reconstruction and linear phase for both biorthogonal and orthogonal ones. Then, we present the design method for both cases. Finally, some design examples are shown. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(10): 78–86, 1997

A novel design technique for two-channel linear-phase perfect-reconstruction FIR filter banks

This paper considers the design of two-channel linear-phase perfect-reconstruction FIR filter banks optimal in the least-squares sense. The design problem is formulated to minimize the error energies of the analysis filters subject to perfect-reconstruction constraints. Although it is a quadratic programming problem with nonlinear constraints in the nature, a linearization technique is presented to linearize the design problem such that the core work in finding the optimal coefficients of the analysis filters requires only to solve a sequence of linearized subproblems, which can be solved analytically by using the Lagrange multiplier method. The paper concludes with the design examples which demonstrate the superiority of the proposed method over the existing methods.

A method of designing cosine-modulated 2D perfect reconstruction FIR filter banks with linear phase

This paper proposes a new method for designing cosine-modulated 2D perfect reconstruction FIR filter banks with a linear phase. In conventional cosine-modulated perfect reconstruction filter banks, although the prototype filters have linear phase, it is impossible to obtain linear phases in analysis and synthesis filters. In this paper, the analysis and synthesis filters with linear phases are realized by using a new decomposition scheme. In this scheme, the input signal is decomposed into D+1 subbands in the frequency plane, while the system has 2D channels. These D+1 filters are obtained by cosine modulation of the prototype filter. The remaining D-1 filters are obtained by sine modulation and support the same bandwidth as the cosine-modulated bandpass filters. In this case, the filter bank can satisfy the perfect reconstruction condition by applying certain condition to the prototype filter. The effectiveness of the present method is verified through a design example. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(4): 1–10, 1997

Design of perfect reconstruction FIR filter banks using DFT modulation

Design of perfect reconstruction FIR filter banks using DFT modulation

Design of perfect reconstruction FIR filter banks using DFT modulation

Design of perfect reconstruction FIR filter banks using DFT modulation