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     

Design of two-channel IIR filter banks with arbitrary group delay

The nonlinear optimisation problem that results from considering the design of a two-channel nonuniform division filter bank is solved. This is 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 examples are provided for illustration     

Minimax design of two-channel low-delay perfect-reconstruction FIRfilter 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 (1989) 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     

Minimax design of two-channel nonuniform-division FIR filter banks

The paper deals with the minimax design of two-channel nonuniform-division filter (NDF) banks. Based on a linearisation scheme, the design problem is formulated as an optimisation problem with linear constraints. The authors present a method to design a two-channel NDF bank using a modified dual-affine scaling variant of Karmarkar's (1984) algorithm. This method provides the optimal results that the linear-phase FIR analysis and synthesis filters have equiripple stopband response and the resulting NDF bank also shows equiripple reconstruction error behaviour. The effectiveness of the proposed design technique is demonstrated by several simulation examples     

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     

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     

Convergence results on the design of QMF banks

This correspondence considers the design of quadrature mirror filter (QMF) banks whose analysis and synthesis filters are FIR and have linear phase. The design criterion is of least-squares type. An iterative computation algorithm is proposed, and its convergence is proven that offers new insight to the design of QMF banks and relates it to a more general nonlinear optimization problem     

Efficient iterative design method for cosine-modulated QMF banks

An iterative algorithm for the design of multichannel cosine-modulated quadrature mirror-image filter (QMF) banks with near-perfect reconstruction is proposed. The objective function is formulated as a quadratic function in each step whose minimum point can be obtained using a closed-form solution. This approach has high design efficiency and leads to filter banks with high stopband attenuation and low aliasing and amplitude distortions. The proposed approach is then extended to the design of multichannel cosine-modulated QMF banks with low reconstruction delays, which are often required, especially in real-time applications. Several design examples are included to demonstrate the proposed algorithms, and some comparisons are made with existing designs     

Efficient design method of cosine-modulated QMF banks satisfying PR property

In this paper the design problem of perfect-reconstruction cosine-modulated QMF banks has been formulated as a quadratic-constrained least-squares (QCLS) minimization problem in which all constrained matrices of the QCLS optimization problem are symmetric and positive definite. A cost function which is a convex function of desired prototype filter coefficients is constructed so that this kind of QCLS optimization problems can be efficiently solved. So a global minimizer of this problem can be easily obtained. Results of two design examples are presented to support the derivations and analyses.     

Minimax design of two-dimensional parallelogram filter banks

The authors present a technique for the minimax design of two-dimensional (2-D) parallelogram filter bank (PFB) systems with linear-phase analysis/synthesis filters. To achieve perfect reconstruction, the required analysis filters must have parallelogram-shaped frequency responses. In general, the original design problem is found to be an optimisation problem with nonlinear constraints. The authors present a linearisation approach to reformulate the design problem. As a result, updating the filter coefficient vector at each iteration for the original design problem can be accomplished by searching the gradient of the linearised optimisation problem. They further present an efficient method based on a modified Karmarkar's algorithm for computing the required gradient vector and finding the required step size analytically. Therefore the filter coefficients can easily be computed by solving only linear equations at each iteration during the design process. The effectiveness of the proposed technique is shown by computer simulations     

The design of two channel IIR QMF bank directly from analogue prototype

This article presents a new method for designing two-band quadrature mirror filter (QMF) banks. Analytic solution of the filter coefficients is obtained in the s-domain. The characteristic function of the new filter has one multiple pole in the pass-band and one multiple zero in the stop-band. As a consequence, the transfer function has a single ripple, both in the pass-band and in the stop-band. Lowpass and highpass filter pair is double complementary in the s-plane. The gain responses of the lowpass and highpass filter pair at the crossover frequency are approximately 3?dB below their maximum values. Using the bilinear transformation, the analogue filter transfer function is mapped into the corresponding digital filter transfer function, so that the analogue crossover frequency is mapped into the digital frequency corresponding to ω?=?π/2. The filter characteristic exhibits Butterworth-like behaviour near the origin and Chebyshev-like behaviour (of the third order) near the pass-band edge.     

General analysis of two-band QMF banks

The two-channel perfect-reconstruction quadrature-mirror-filter banks (PR QMF banks) are analyzed in detail by assuming arbitrary analysis and synthesis filters. Solutions where the filters are FIR or IIR correspond to the fact that a certain function is monomial or nonmonomial, respectively. For the monomial case, the design problem is formulated as a nonlinear constrained optimization problem. The formulation is quite robust and is able to design various two-channel filter banks such as orthogonal and biorthogonal, arbitrary delay, linear-phase filter banks, to name a few. Same formulation is used for causal and stable PR IIR filter bank solutions     

A direct approach to the design of QMF banks via frequency domainoptimization

This paper studies the design of quadrature mirror filter (QMF) banks via frequency domain optimization. A direct approach is adopted that gives the necessary and sufficient condition for perfect reconstruction (PR). While analysis filter banks are designed to achieve frequency domain specifications required for subband coding, synthesis filter banks are designed to minimize the reconstruction error in frequency domain. The criterion used to measure the reconstruction error is H_∞ or Chebyshev norm (sup-norm). State-space solutions are derived for the H_∞ optimization, and numerical algorithms are developed to obtain the optimal synthesis filter bank. Moreover, the asymptotic PR property is established for optimal H_∞ solution of the synthesis filter bank     

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 and VLSI implementation of QMF banks

The authors consider the design of multirate filterbanks for applications such as subband coding with IIR QMF (quadrature mirror filter) pairs. These offer reduced complexity and low latency at the expense of the loss of exact linear phase. In particular, consideration is given to the use of all-pass sections where linear phase is approximately achieved by being part of the objective in numerical optimisation experiments. This approach compares favourably with previous IIR based approaches. Finite wordlength design using simulated annealing shows that low coefficient wordlength may be used. This leads to efficient realisations with three-port adaptors. Using pipelining implementation, a flexible VLSI architecture is designed that can be used for a variety of subband decompositions. Layout and simulation of the design have been performed.     

Design of QMF banks based on derivative information

New L₂ objective functions for the design of quadrature mirror filter (QMF) banks are proposed. They are based on the derivative information of the reconstruction error. Simple and explicit matrix-form formulas for the proposed objective functions are derived. Efficient design methods are proposed by incorporating a separability technique into the derived optimality conditions on prototype filters. The proposed design methods need only solve linear equation iteratively without nonlinear optimisation. Design examples demonstrate that good low-delay QMF banks and linear-phase QMF banks can be obtained in only a few iterations. Compared with the conventional approach, the new approach leads to QMF banks with larger stopband attenuation and smaller reconstruction errors     

A closed form design method for the two-channel quadrature mirror filter banks

This paper presents a simple and efficient closed form method for designing two-channel linear phase quadrature mirror filter (QMF) banks with prescribed stopband attenuation and channel overlap. The proposed method is based on optimum passband edge frequency, which is calculated using empirical formulas instead of using optimization algorithm. Different window functions are used to design the prototype filter for QMF banks. When compared to other existing methods, the proposed method reduces computation time (CPU time) and amplitude distortion (e _am ), which results in a simpler and efficient design procedure for the applications where the design must be carried out in real or quasi-real-time. Several design examples are included to illustrate the proposed method and its improved performances over other exiting methods. An application of the proposed method is considered in the area of subband coding of ultrasound image.     

On two-channel filter banks with directional vanishing moments.

The contourlet transform was proposed to address the limited directional resolution of the separable wavelet transform. One way to guarantee good approximation behavior is to let the directional filters in the contourlet filter bank have sharp frequency response. This requires filters with large support size. We seek to isolate the key filter property that ensures good approximation. In this direction, we propose filters with directional vanishing moments (DVM). These filters, we show, annihilate information along a given direction. We study two-channel filter banks with DVM filters. We provide conditions under which the design of DVM filter banks is possible. A complete characterization of the product filter is, thus, obtained. We propose a design framework that avoids 2-D factorization using the mapping technique. The filters designed, when used in the contourlet transform, exhibit nonlinear approximation comparable to the conventional filters while being shorter and, therefore, providing better visual quality with less ringing artifacts. Furthermore, experiments show that the proposed filters outperform the conventional ones in image approximation and denoising.     

Design of low-delay FIR QMF banks using the Lagrange-multiplierapproach

A new approach for the design of two-channel perfect reconstruction FIR filter banks with short reconstruction delays is presented. A low-order filter is first designed and the objective function of the filter bank is formulated as a quadratic programming problem with linear constraints. The Lagrange-multiplier method is then used to design a higher-order filter. The method is simple, efficient, and flexible and leads to a closed-form solution. A design example is included to illustrate the advantages of the method