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.     

Minimax design of two-channel IIR QMF banks with arbitrary groupdelay

The paper deals with the minimax design of two-channel infinite impulse response (IIR) QMF banks with arbitrary group delay, for which the IIR analysis filters and the resulting filter bank possess the frequency response optimal in the minimax (L_∞) sense. Utilising a lattice structure for the denominators of the IIR analysis filters, a design technique is presented based on an approximation scheme and a weighted least-squares (WLS) algorithm, previously developed by one of the authors for solving the resulting design problem that is basically a nonlinear optimisation problem. During the design process, this technique finds the tap coefficients for the numerator and the reflection coefficients for the denominator of the prototype IIR analysis filter simultaneously. The stability of the designed prototype IIR analysis filter is ensured by incorporating an efficient stabilisation procedure to make all of the reflection coefficient values fall between -1 and +1. Computer simulations show the effectiveness of the proposed design technique     

Design of two-channel linear-phase QMF banks based on real IIR all-pass filters

The design of two-channel linear-phase quadrature mirror filter (QMF) banks constructed by real infinite impulse response (IIR) digital all-pass filters is considered. The design problem is appropriately formulated to result in a simple optimisation problem. Using a variant of Karmarkar's algorithm, the optimisation problem can be efficiently solved through a frequency sampling and iterative approximation method to find the real coefficients for the IIR digital all-pass filters. The resulting two-channel QMF banks possess an approximately linear phase response without magnitude distortion. The effectiveness of the proposed technique is achieved by forming an appropriate Chebyshev approximation of the desired phase response and then finding its solution from a linear subspace in a few iterations. Finally, several simulation examples are presented for illustration and comparison     

A new class of two-channel biorthogonal filter banks and waveletbases

Proposes a novel framework for a new class of two-channel biorthogonal filter banks. The framework covers two useful subclasses: i) causal stable IIR filter banks. ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for such a class. Filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity. The properties of such a class are discussed in detail. The design of the analysis/synthesis systems reduces to the design of a single transfer function. Very simple design methods are given both for FIR and IIR cases. Zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity property. In the IIR case, two new classes of IIR maximally flat filters different from Butterworth filters are introduced. The filter coefficients are given in closed form. The wavelet bases corresponding to the biorthogonal systems are generated. the authors also provide a novel mapping of the proposed 1-D framework into 2-D. The mapping preserves the following: i) perfect reconstruction; ii) stability in the IIR case; iii) linear phase in the FIR case; iv) zeros at aliasing frequency; v) frequency characteristic of the filters     

Minimax design of two-channel nonuniform-division filterbanks using IIR allpass filters

The design of two-channel linear-phase nonuniform-division filter (NDF) banks constructed by infinite impulse response (IIR) digital allpass filters (DAFs) in the sense of L/sub /spl infin// error criteria is considered. First, the theory of two-channel NDF bank structures using two IIR DAFs is developed. Then, the design problem is appropriately formulated to result in a simple optimization problem. Utilizing a variant of Karmarkar's algorithm, we can efficiently solve the optimization problem through a frequency sampling and iterative approximation method to find the coefficients for the IIR DAFs. The resulting two-channel NDF banks can possess approximately linear-phase response without magnitude distortion. The effectiveness of the proposed technique is achieved by forming an appropriate Chebyshev approximation of a desired phase response and then to find its solution from a linear subspace in a few iterations. Several simulation examples are presented for illustration and comparison.     

Two-channel IIR QMF banks with approximately linear-phase analysisand synthesis filters

Perfect linear-phase two-channel QMF banks require the use of finite impulse response (FIR) analysis and synthesis filters. Although they are less expensive and yield superior stopband characteristics, perfect linear phase cannot be achieved with stable infinite impulse response (IIR) filters. Thus, IIR designs usually incorporate a postprocessing equalizer that is optimized to reduce the phase distortion of the entire filter bank. However, the analysis and synthesis filters of such an IIR filter bank are not linear phase. In this paper, a computationally simple method to obtain IIR analysis and synthesis filters that possess negligible phase distortion is presented. The method is based on first applying the balanced reduction procedure to obtain nearly allpass IIR polyphase components and then approximating these with perfect allpass IIR polyphase components. The resulting IIR designs already have only negligible phase distortion. However, if required, further improvement may be achieved through optimization of the filter parameters. For this purpose, a suitable objective function is presented. Bounds for the magnitude and phase errors of the designs are also derived. Design examples indicate that the derived IIR filter banks are more efficient in terms of computational complexity than the FIR prototypes and perfect reconstruction FIR filter banks. Although the PR FIR filter banks when implemented with the one-multiplier lattice structure and IIR filter banks are comparable in terms of computational complexity, the former is very sensitive to coefficient quantization effects     

Generalized Chebyshev Filters for the Design of IIR Filters and Filter Banks

This paper introduces the generalized IIR Chebyshev filters. The proposed filters are obtained by applying bilinear transformation to the corresponding analog filters. The novelty of the method is the introduction of a new rational Chebyshev function, which includes Chebyshev Type I and Chebyshev Type II IIR filters as special cases. The application of the proposed digital filters to design perfect reconstruction two-channel filter banks is described. The proposed filters can be applied in orthogonal discrete wavelet transform.     

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 “predesigned” analysis filter, with the filter lengths satisfying certain conditions. We present several design examples to demonstrate the effectiveness of the method.      

近似完全重构IIR余弦调制滤波器组的优化设计

提出一种新的近似完全重构因果稳定的IIR余弦调制滤波器组的设计方法。基于预先给定的极点值,IIR原型滤波器的设计问题可以简化成一个凸极大值极小化的优化问题,从而采用二阶锥规划法求解。所得余弦调制滤波器组具有良好的频率特性和合理的完全重构误差。所设计的原型滤波器是因果稳定的,并且其多相因子分母相同,简化了完全重构条件,可以用来进一步优化得到的完全重构系统。     

Perfect-reconstruction filter banks having linear-phase FIR filterswith equiripple response

The design of equiripple linear-phase analysis and synthesis FIR filters of two-channel perfect-reconstruction (PR) filter banks is formulated as the minimization of a weighted peak-error under both linear inequality (arising from the desired responses of the analysis filters) and nonlinear equality (PR) constraints. The effectiveness of a proposed method to solve the design problem (a modified dual-affine scaling variant of Karmarkar's (1989) algorithm and an approximation scheme) is illustrated through several design examples     

Design of two-channel low-delay IIR nonuniform-division filter banks using L/sub 1/ error criteria

The design of a two-channel nonuniform-division filter (NDF) bank with infinite impulse response (IIR) analysis/synthesis filters and low group delay in the sense of L/sub 1/ error criteria is considered. The problem formulation results in a nonlinear optimisation problem. Based on a variant of Karmarkar's algorithm, the optimisation problem is solved through a frequency sampling and iterative approximation technique to find the tap coefficients and the reflection coefficients for the numerator and the denominator of the IIR analysis filters. An efficient stabilisation procedure ensures that the reflection coefficients lie in (-1, 1). Simulation results are provided for illustration and comparison.     

Design of face-centred orthorhombic filter banks

In this paper, a method to design the two-channel FIR linear-phase (LP) face-centred orthorhombic (FCO) filter banks with equiripple magnitude responses and perfect-reconstruction (PR) is presented. The necessary conditions of lengths of LP FCO filter banks satisfying the PR constraint are derived. An interior-point algorithm is utilized to optimize the peak ripples of the analysis filters and a first-order approximation skill is introduced to satisfy the PR constraint. The simulation example is presented to illustrate the effectiveness of this proposed design technique.     

Wavelet Packets Analysis Based on IIR Two-channel Bi-orthogonal Filter Banks

1IntroductionAccordingtoitSrelationwithwaveletanalysis,tWo-channelfilterbankscanbeusednotonlyintheimplementationofwavelettransformandinversetransform,butalsoinwavelet'sconsmichon.AndIIRfiltersarewellknowntohaveshadertransihonregions,lowcomplexity,lowreconstrUctionermrandaPPro~lylinearphase.Weconstr'UctIIRhi-orthogonalPerfectreconstructionfilterbanksusinganallpassfunchonA(z)whoseamplitUdecharacterishcsisallpassandwhosephasecharacterishcsisanapproximatelylinearPhaseinthepass-band.Andweacc…     

Adaptive McClellan transformations for quincunx filter banks

A technique is developed for the design of 2-D nonseparable two-channel filter banks for a quincunx sampling lattice, where the isopotentials of the frequency response can be optimized and adapted to the input signal's statistics. By employing known odd-length symmetric linear phase filter banks as the l-D prototype filters for 2-D filters parameterized by the McClellan transformation, conditions are derived such that the resulting 2-D two-channel filter bank retains the perfect-reconstruction or aliasing-free properties of the 1-D prototype two-channel filter bank. A particular two-parameter transformation function is developed that has sufficient flexibility to adapt its orientation in any direction and whose optimization involves a simple constrained least-squares problem in which the feasible set lies within a circle. The results have practical applications in many areas of image and video processing where multirate filter banks are used     

Pade table, continued fraction expansion, and perfectreconstruction filter banks

We investigate the relationships among the Pade table, continued fraction expansions and perfect reconstruction (PR) filter banks. We show how the Pade table can be utilized to develop a new lattice structure for general two-channel bi-orthogonal perfect reconstruction (PR) filter banks. This is achieved through characterization of all two-channel biorthogonal PR filter banks. The parameterization found using this method is unique for each filter bank. Similar to any other lattice structure, the PR property is achieved structurally and the quantization of the parameters of the lattice does not effect this property. Furthermore, we demonstrate that for a given filter, the set of all complementary filters can be uniquely specified by two parameters, namely, the end-to-end delay of the system and a scalar quantity. Finally, we investigate the convergence of the successive filters found through the proposed lattice structure and develop a sufficient condition for this convergence     

Analysis, design, and implementation of two-channel linear-phasefilter 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 general formulation of modulated filter banks

This paper presents a general framework for maximally decimated modulated filter banks. The theory covers the known classes of cosine modulation and relates them to complex-modulated filter banks. The prototype filters have arbitrary lengths, and the overall delay of the filter bank is arbitrary, within fundamental limits. Necessary and sufficient conditions for perfect reconstruction (PR) are derived using the polyphase representation. It is shown that these PR conditions are identical for all types of modulation-modulation based on the discrete cosine transform (DCT), both DCT-III/DCT-IV and DCT-I/DCT-II, and modulation based on the modified discrete Fourier transform (MDFT). A quadratic-constrained design method for prototype filters yielding PR with arbitrary length and system delay is derived, and design examples are presented to illustrate the tradeoff between overall system delay and stopband attenuation (subchannelization)     

On Bounds of Shift Variance in Two-Channel Multirate Filter Banks

Critically sampled multirate FIR filter banks exhibit periodically shift variant behavior caused by nonideal antialiasing filtering in the decimation stage. We assess their shift variance quantitatively by analysing changes in the output signal when the filter bank operator and shift operator are interchanged. We express these changes by a so-called commutator. We then derive a sharp upper bound for shift variance via the operator norm of the commutator, which is independent of the input signal. Its core is an eigensystem analysis carried out within a frequency domain formulation of the commutator, leading to a matrix norm which depends on frequency. This bound can be regarded as a worst case instance holding for all input signals. For two channel FIR filter banks with perfect reconstruction (PR), we show that the bound is predominantly determined by the structure of the filter bank rather than by the type of filters used. Moreover, the framework allows to identify the signals for which the upper bound is almost reached as so-called near maximizers of the frequency-dependent matrix norm. For unitary PR filter banks, these near maximizers are shown to be narrow-band signals. To complement this worst-case bound, we derive an additional bound on shift variance for input signals with given amplitude spectra, where we use wide-band model spectra instead of narrow-band signals. Like the operator norm, this additional bound is based on the above frequency-dependent matrix norm. We provide results for various critically sampled two-channel filter banks, such as quadrature mirror filters, PR conjugated quadrature filters, wavelets, and biorthogonal filters banks.     

Design of IIR linear-phase QMF banks based on complex allpasssections

A new method for the design of two-channel, perfect reconstruction, analysis/synthesis QMF banks is presented. The filters of the banks are IIR, power complementary, linear phase, and are represented by means of complex allpass functions. Design procedures based both on numerical approximation and on a flatness constraint imposed on the frequency responses of the filters are given     

Digital filter bank design quadratic-constrained formulation

Formulate the filter bank design problem as an quadratic-constrained least-squares minimization problem. The solution of the minimization problem converges very quickly since the cost function as well as the constraints are quadratic functions with respect to the unknown parameters. The formulations of the perfect-reconstruction cosine-modulated filter bank, of the near-perfect-reconstruction pseudo-QMF bank, and of the two-channel biorthogonal linear-phase filter bank are derived using the proposed approach. Compared with other design methods, the proposed technique yields PR filter banks with much higher stopband attenuation. The proposed technique can also be extended to design multidimensional filter banks