Design of linear phase M-channel perfect reconstruction FIR filterbanks

We propose two approaches to design M channel nonparaunitary filter banks that satisfy perfect reconstruction (PR) and linear phase (LP) properties. In the first approach, the PR condition is imposed on only a high-pass filter. Although this method does not require nonlinear optimization, it has a demerit in that the order of a high-pass filter becomes high. In the second approach, two filters are optimized simultaneously using a Lagrange-Newton method. We can design PR filter banks that have the same length. The PR constraint is also formulated as a linear and nonlinear equation of the analysis filter coefficients. Finally, some design examples are included     

Oversampled cosine-modulated filter banks with arbitrary systemdelay

Design methods for perfect reconstruction (PR) oversampled cosine-modulated filter banks with integer oversampling factors and arbitrary delay are presented. The system delay, which is an important parameter in real-time applications, can be chosen independently of the prototype lengths. Oversampling gives us additional freedom in the filter design process, which can be exploited to find FIR PR prototypes for oversampled filter banks with much higher stopband attenuations than for critically subsampled filter banks. It is shown that for a given analysis prototype, the PR synthesis prototype is not unique. The complete set of solutions is discussed in terms of the nullspace of a matrix operator. For example, oversampling allows the design of PR filter banks having unidentical prototypes (of equal and unequal lengths) for the analysis and synthesis stage. Examples demonstrate the increased design freedom due to oversampling. Finally, it is shown that PR prototypes being designed for the oversampled case can also serve as almost-PR prototypes for critically subsampled cosine-modulated pseudo QMF banks     

Discrete Lagrangian Methods for Designing Multiplierless Two-Channel PR-LP Filter Banks

In this paper, we present a new search method based on the theory of discrete Lagrange multipliers for designing multiplierless PR (perfect reconstruction) LP (linear phase) filter banks. To satisfy the PR constraints, we choose a lattice structure that, under certain conditions, can guarantee the resulting two filters to be a PR pair. Unlike the design of multiplierless QMF filter banks that represents filter coefficients directly using PO2 (powers-of-two) form (also called Canonical Signed Digit or CSD representation), we use PO2 forms to represent the parameters associated with the lattice structure. By representing these parameters as sums or differences of powers of two, multiplications can be carried out as additions, subtractions, and shifts. Using the lattice representation, we decompose the design problem into a sequence of four subproblems. The first two subproblems find a good starting point with continuous parameters using a single-objective, multi-constraint formulation. The last two subproblems first transform the continuous solution found by the second subproblem into a PO2 form, then search for a design in a mixed-integer space. We propose a new search method based on the theory of discrete Lagrange multipliers for finding good designs, and study methods to improve its convergence speed by adjusting dynamically the relative weights between the objective and the Lagrangian part. We show that our method can find good designs using at most four terms in PO2 form in each lattice parameter. Our approach is unique because our results are the first successful designs of multiplierless PR-LP filter banks. It is general because it is applicable to the design of other types of multiplierless filter banks.     

Two-dimensional perfect reconstruction FIR filter banks withtriangular 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     

无约束法设计完全重构M带滤波器组

本文提出了一种新的正交和双正交完全重构M带滤波器组的设计方法.这种方法是建立在多项式分解基础之上,可在无约束条件下设计,所设计的滤波器组具有结构化完全重构特点,对于双正交滤波器组还可以方便设计系统时延.和已有的方法相比,新方法具有低设计复杂度特点.文中还描述了设计过程,最后给出了设计例子和结果.     

两通道完全重构滤波器组的设计方法:因式分解法

本文首先研究了两通道滤波器组设计的完全重构的条件以及Euclidean多项式的性质,提出了基于因式分解的两通道完全重构滤波器组的设计方法.该方法不需要进行强的非线性优化计算而可以实现真正的结构化的完全重构滤波器组.文中描述了这种方法的推导过程,给出了设计步骤,最后通过给出的设计例子,说明该方法是有效的.     

Analytical design of 3-D wavelet filter banks using themultivariate Bernstein polynomial

The design of 3-D multirate filter banks where the downsampling/upsampling is on the FCO (face centred orthorhombic) lattice is addressed. With such a sampling lattice, the ideal 3-D sub-band of the low-pass filter is of the TRO (truncated octahedron) shape. The transformation of variables has been shown previously to be an effective technique for designing M-D (multidimensional) filter banks. A design technique is presented for the transformation function using the multivariate Bernstein polynomial which provides a good approximation to the TRO sub-band shape. The method is analytically based and does not require any optimisation procedure. Closed form expressions are obtained for the filters of any order. Another advantage of this technique is that it yields filters with a flat frequency response at the aliasing frequency (ω₁, ω₂ , ω₃)=(π, π, π). This flatness is important for giving regular discrete wavelet transform systems     

两通道完全重构滤波器组的设计方法:因式分解法

本文首先研究了两通道滤波器组设计的完全重构的条件以及Euclidean多项式的性质 ,提出了基于因式分解的两通道完全重构滤波器组的设计方法 .该方法不需要进行强的非线性优化计算而可以实现真正的结构化的完全重构滤波器组 .文中描述了这种方法的推导过程 ,给出了设计步骤 ,最后通过给出的设计例子 ,说明该方法是有效的     

基于多项式分解理论的低时延完全重构两通道滤波器组的设计

Euclid多项式分解算法可以用于滤波器组的设计,该文首先讨论了Euclid分解算法与低时延两通道完全重构的滤波器组设计理论,推导出可实现分解的条件,并从理论上加以证明,由于Euclid分解算法具有非唯一性,该文提出了一种新的算法以确定唯一的分解,并将这种算法用于具有低时延特性的两通道全重构滤波器组的设计,最后,通过给出的基于分解方法的设计例子,说明该方法是有效的。     

Design of NPR DFT-Modulated Filter Banks via Iterative Updating Algorithm

In this paper, an efficient algorithm is proposed to design nearly-perfect-reconstruction (NPR) DFT-modulated filter banks. First, the perfect-reconstruction (PR) condition of the oversampled DFT-modulated filter banks in the frequency domain is transformed into a set of quadratic equations with respect to the prototype filter (PF) in the time domain. Second, the design problem is formulated as an unconstrained optimization problem that involves PR condition and stopband energy of the PF. With the gradient vector of the objective function, an efficient iterative algorithm is presented to design the PF, which is updated with linear matrix equations at each iteration. The algorithm is identified as a modified Newton’s method, and its convergence is proved. Numerical examples and comparison with many other existing methods are included to demonstrate the effectiveness of the proposed method.     

Design of Biorthogonal Filter Banks Using a Multi-objective Genetic Algorithm for an Image Coding Scheme

In this paper, we present an optimization method based on a multi-objective Genetic Algorithm (GA) for the design of linear phase filter banks for an image coding scheme. To be effective, the filter banks should satisfy a number of desirable criteria related to such a scheme. Instead of imposing the entire PR condition as in conventional designs, we introduce flexibility in the design by relaxing the Perfect Reconstruction (PR) condition and defining a PR violation measure as an objective criterion to maintain near perfect reconstruction (N-PR) filter banks. Particularly in this work, the designed filter banks are near-orthogonal. This has been made possible by minimizing the deviation from the orthogonality in the optimization process. The optimization problem is formulated as a constrained multi-objective, and a modified Nondominated Sorting Genetic Algorithm NSGAII is proposed in this work to find the Pareto optimal solutions that achieve the best compromise between the different objective criteria. The experimental results show that the filter banks designed with the proposed method outperform significantly the 9/7 filter bank of JPEG2000 in most cases. Furthermore, the filter banks are near orthogonal. This is very helpful, especially where embedded coding is required.     

Design of 2D oversampled linear phase DFT modulated filter banks via modified Newton's method

In this paper, the structure of the 2D oversampled DFT modulated filter banks is analyzed and a spatial-domain condition of a filter bank without transfer function distortion is derived. Based upon the spatial-domain condition, a modified Newton's method is presented for fast design of 2D oversampled linear phase (LP) DFT modulated filter banks with nearly perfect reconstruction (NPR). We formulate the design problem into an unconstrained optimization with a fourth-order objective function, which is the weighted sum of the transfer function distortion of the filter bank and the stopband energy of the prototype filter (PF). The optimization is solved by the modified Newton's method, where each of iterations updates the PF by a set of linear equations. It is proved that the iteration process fast converges to a stationary point of the objective function. Compared with the existing methods, the new method is fast in computation and can design 2D filter banks with a large number of subbands.     

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)     

Subspace approach for the design of cosine-modulated filter bankswith linear-phase prototype filter

A method for designing perfect reconstruction (PR) prototypes for paraunitary cosine-modulated filter banks is presented. The design procedure is based on a subspace approach that allows linear combinations of even-length linear-phase PR prototype filters in such a way that the resulting filter is also a linear-phase PR prototype. Within a given subspace, the weights of the optimal linear combination can easily be computed via an eigenanalysis. The filter design is carried out iteratively while the PR property is guaranteed throughout the design process. No nonlinear optimization routine is needed. As a special case, the proposed approach allows the design of discrete-coefficient prototypes, which are of great interest for efficient hardware implementations     

用提升方法设计M带LPPR-FIR滤波器组

 本文研究了如何用提升方法构造M带PR-FIR滤波器组,并且考虑了一种特殊情形的线性相位(Linear Phase,LP)性质,及其优化设计问题,即如何转化为一个二次型优化问题.     

Design of practically perfect-reconstruction cosine-modulated filter banks: a second-order cone programming approach

Designing optimal perfect-reconstruction (PR) and near PR (NPR) cosine-modulated filter banks is essentially a constrained nonlinear minimization problem. This paper proposes two second-order cone-programming based algorithms for designing NPR and practically PR cosine-modulated filter banks with improved performance relative to several established design methods.     

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     

用提升方法设计M带LPPR-R滤波器组

本文研究了如何用提升方法构造M带PR-FIR滤波器组,并且考虑了一种特殊情形的线性相位(LinearPhase,LP)性质,及其优化设计问题,即如何转化为一个二次型优化问题.     

Design of nonseparable 3-D filter banks/wavelet bases usingtransformations of variables

The authors present a technique to design two-channel filter banks in three dimensions where the sampling is on the FCO (face centred orthorhombic) lattice, The ideal 3-D sub-band is of the truncated octahedron shape. The design technique is based transformation of variable method equivalent to the generalised McClellan transformation. The filters are FIR, have linear phase and achieve perfect reconstruction. Although the sub-band shape is quite complicated, the ideal frequency characteristics are well approximated. This is illustrated with an example. The technique provides the flexibility of controlling the frequency characteristics of the filters with ease. The filters can be implemented quite efficiently due to the highly symmetrical nature of the coefficients of the transformation. The authors also modify and extend the basic design technique to impose the zero property (the number of zeros of the filter transfer function at the aliasing frequency) on the sub-band filters. This property is important when the filter bank is used iteratively in a tree-structured manner as a discrete wavelet transform system and the issue of regularity arises. Several design examples are presented to illustrate the design technique     

一类冗余比为2的二维DFT调制滤波器组的设计

本文构建了一类冗余比为2的二维线性相位的双原型离散傅立叶变换（DFT）调制滤波器组。利用原型滤波器的多相位分解,推导出了该滤波器组的完全重构(PR)条件。基于该PR条件,我们将滤波器组的设计归结为一个关于原型滤波器的多相位分量的无约束优化问题。由于原型滤波器是线性相位的,多相位分量之间具有一定的关系,因此我们可以简化该优化问题。仿真结果验证了滤波器组PR条件的正确性。同时,仿真表明了优化算法的有效性,设计所得的滤波器组重构误差很小、频率特性较好,基本满足实际应用的需要。