Rate-distortion-optimal subband coding withoutperfect-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     

A general-purpose optimization approach for designing two-channel FIR filterbanks

An efficient general-purpose optimization approach is proposed for designing two-channel finite impulse response (FIR) filterbanks. This technique can be used for optimizing two-channel FIR filterbanks in all alias-free cases proposed in the literature. The generalized problem is to minimize the maximum of the stopband energies of the two analysis filters subject to the given passband and transition band constraints and the given allowable reconstruction error. Therefore, in addition to the perfect-reconstruction filterbanks, nearly perfect-reconstruction banks can be optimized in a controlled manner. The optimization is carried out in two steps. In the first step, for the selected type of the filterbank, a good starting-point filterbank for further optimization is generated using an existing design scheme. The second step involves optimizing the filterbank with the aid of a modified Dutta-Vidyasagar (1977) algorithm. Several examples are included, illustrating the efficiency and the flexibility of the proposed approach.     

A new approach to the design of critically sampled M-channeluniform-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     

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     

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     

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     

Lagrange multiplier approaches to the design of two-channelperfect-reconstruction linear-phase FIR filter banks

The authors present two approaches to the design of two-channel perfect-reconstruction linear-phase finite-impulse-response (FIR) filter banks. Both approaches analyze and design the impulse responses of the analysis filter bank directly. The synthesis filter bank is then obtained by simply changing the signs of odd-order coefficients in the analysis filter bank. The approach deals with unequal-length filter banks. By designing the lower length filters first, one can take advantage of the fact that the number of variables for designing the higher length filters is more than the number of perfect-reconstruction constraint equations. The second approach generalizes the first, and covers the design for all parts of linear phase perfect reconstruction constraint equations     

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.     

Invertible deinterlacing with sampling-density preservation: theory and design

A novel class of deinterlacing for intra-frame/field-based motion-picture coding, such as Motion JPEG2000, as well as an inter-frame-based coding without the support of interlaced scanning video, such as MPEG-1, is developed. This technique has two features: sampling-density preservation and invertibility. These features mean that the amount of deinterlaced pictures is not increased, and the original pictures can be perfectly reconstructed. This deinterlacing technique is a kind of sampling-lattice alteration and is regarded as a generalization of conventional field interleaving and field separation. With the help of multidimensional (M-D) multirate theory, it is shown that the design problem of such a system, that is, invertible deinterlacer with sampling-density preservation, can be replaced to finding a 2/spl times/2 multivariable polynomial matrix with a monomial determinant. This problem resembles the design of two-channel M-D maximally decimated perfect-reconstruction finite-impulse-response (FIR) filterbanks. The inverse system, which is referred to as a reinterlacer , is given as FIR when the dual deinterlacing system is FIR. A practical design procedure is provided by suggesting three constraints considered to be preferable: normalization, regularity, and vertical symmetry. The significance of the procedure is verified by showing some design examples of deinterlacing and reinterlacing filters. Simulation results show that the developed method causes fewer comb-shaped artifacts than conventional field interleaving.     

Iterative reweighted least-squares design of FIR filters

Develops a new iterative reweighted least squares algorithm for the design of optimal L_p approximation FIR filters. The algorithm combines a variable p technique with a Newton's method to give excellent robust initial convergence and quadratic final convergence. Details of the convergence properties when applied to the L_p optimization problem are given. The primary purpose of L_p approximation for filter design is to allow design with different error criteria in pass and stopband and to design constrained L₂ approximation filters. The new method can also be applied to the complex Chebyshev approximation problem and to the design of 2D FIR filters     

一种新型伪镜像滤波器组的实现方法

本文提出了一种新型２Ｍ个通道伪镜像滤波器组的结构和设计方法。本设计方法中，低通原型滤波器采用两级级联线性相位ＦＩＲ的结构，并使得整个分析／综合系统性能近于全恢复，无群时延失真。应用此方法设计伪镜像滤波器组，分析和综合滤波器组的结构和设计方法都比较简单，算法效率高，实现比较容易，这类滤波器组可用于语音的子带编码等。本文最后给出了一个实例。     

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     

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.      

可规划相位FIR滤波器的神经网络设计方法

本文针对可规划相频响应的实系数FIR滤波器的逼近问题,采用一个三层复激活函数前馈神经网络来实现。该网络隐层各神经元的激活函数为复指数函数,将滤波器系数作为隐层各神经元到输出层的连接权值,通过对误差函数的最小化来调整权值,并根据网络特性与所要设计的滤波器的特点,提出了一些实际设计中训练样本集选取与误差加权值设置的规则。依托所采用的神经网络,根据上述规则,进行了两例可规划相频特性的实系数FIR滤波器的设计,结果表明所设计滤波器的相频响应较好地满足了设计要求。     

Theory of rate-distortion-optimal, constrainedfilterbanks-application to IIR and FIR biorthogonal designs

We design filterbanks that are best matched to input signal statistics in M-channel subband coders, using a rate-distortion criterion. Previous research has shown that unconstrained-length, paraunitary filterbanks optimized under various energy compaction criteria are principal-component filterbanks that satisfy two fundamental properties: total decorrelation and spectral majorization. In this paper, we first demonstrate that the two properties above are not specific to the paraunitary case but are satisfied for a much broader class of design constraints. Our results apply to a broad class of rate-distortion criteria, including the conventional coding gain criterion as a special case. A consequence of these properties is that optimal perfect-reconstruction (PR) filterbanks take the form of the cascade of principal-component filterbanks and a bank of pre- and post-conditioning filters. The proof uses variational techniques and is applicable to a variety of constrained design problems. In the second part of this paper, we apply the theory above to practical filterbank design problems. We give analytical expressions for optimal IIR biorthogonal filterbanks; our analysis validates a conjecture by several researchers. We then derive the asymptotic limit of optimal FIR biorthogonal filterbanks as filter length tends to infinity. The performance loss due to FIR constraints is quantified theoretically and experimentally. The optimal filters are quite different from traditional filters. Finally, a sensitivity analysis is presented     

Symmetric Orthogonal Complex-Valued Filter Bank Design by Semidefinite Programming

A new design method for complex-valued two-channel finite impulse response (FIR) filter banks with both orthogonality and symmetry properties is developed. Based on a novel linear matrix inequality (LMI) characterization of trigonometric curves, the optimal design of perfect-reconstruction filter banks is cast into a semidefinite programming (SDP) problem. The dimension of the resulting SDP problem is further reduced by exploiting convex duality. Consequently, the globally optimal solution can be found for any practical filter length and desired regularity order.     

Design of 2-D FIR Filters by a Feedback Neural Network

A Hopfield-type neural network for the design of 2-D FIR filters is proposed. The network is contrived to have an energy function that coincides with the sum-squared error of the approximation problem at hand and by ensuring that the energy is a monotonic decreasing function of time, the approximation problem can be solved. Two solutions are obtained. In the first the 2-D FIR filter is designed on the basis of a specified amplitude response and in the second a filter that has specified maximum passband and stopband errors is designed. The network has been simulated with HSPICE and design examples are included to show that this is an efficient way of solving the approximation problem for 2-D FIR filters. The neural network has high potential for implementation in analog VLSI and can, as a consequence, be used in real-time applications.     

Time-domain design of FIR multirate filter banks

A new time-domain methodology for designing FIR multirate filter banks is proposed. The conditions for perfect reconstruction systems can only be met by a limited number of systems, and consequently one of the major problems is to design analysis and synthesis filters which reduce the reconstruction error to a minimum. A recursive technique is proposed which uses the synthesis filters from one iteration to update the analysis filters for the next. The Letter shows that this is computationally simpler than previously proposed time-domain methods and produces filter banks in which the reconstruction error is reduced to practically acceptable levels.<>     

Three different criteria for the design of two-dimensional zerophase FIR digital filters

An error criterion for the design of FIR filters is proposed. Filters with relatively many free filter coefficients are designed using the Chebyshev, the weighted-least-squares (WLS), and a new partitioned minimax error criterion, and the performance of the filters is compared. A general and fast technique for the WLS design is also presented     

Lowpass minimum phase filter design using IFIR filters

A design procedure for minimum-phase FIR filters using an interpolated FIR (IFIR) filter is proposed. The IFIR technique allows two linear-phase filters of much lower order to be designed thereby making it easier to apply mipizing than is possible in high order FIR prototype filters. In this way, the problem of finding the roots of high order polynomials is overcome