A Direct Design of Oversampled Perfect Reconstruction FIR Filter Banks

We address a problem to find optimal synthesis filters of oversampled uniform finite-impulse-response (FIR) filter banks (FBs) yielding perfect reconstruction (PR), when we are given an analysis FB, in the case where all the filters have the same length that is twice a factor of downsampling. We show that in this class of FBs, a synthesis FB that achieves PR can be found in closed form with elementary matrix operations, unlike conventional design methods with numerical optimization. This framework allows filter coefficients to be complex as well as real. Due to the extra degrees of freedom in a synthesis FB provided by oversampling, we can determine optimal coefficients of synthesis filters that meet certain criteria. We introduce in this paper two criteria: variance of additive noise and stopband attenuation. We show theoretical results of optimal synthesis filters that minimize these criteria and design examples of oversampled linear-phase FIR FBs and DFT-modulated FBs. Moreover, we discuss applications to signal reconstruction from incomplete channel data in transmission and inverse transform of windowed discrete Fourier transform with 50% overlapping.     

A Generalized Oversampled Structure for Cosine-Modulated Transmultiplexers and Filter Banks

In this paper, a new structure, called the channel split-and-add method, for designing oversampled transmultiplexers and filter banks is presented. The proposed method is based on an initial design with an additional number of bands. The band number is then reduced to the desired value by the proper combination of adjacent and/or nonadjacent bands (subchannels). With the proposed approach it is always possible to perform the filtering tasks at the lowest data rate of the system. An example illustrates the design flexibility achieved with the proposed structure.     

Computation of the Para-Pseudoinverse for Oversampled Filter Banks: Forward and Backward Greville Formulas

Frames and oversampled filter banks have been extensively studied over the past few years due to their increased design freedom and improved error resilience. In frame expansions, the least square signal reconstruction operator is called the dual frame, which can be obtained by choosing the synthesis filter bank as the para-pseudoinverse of the analysis bank. In this paper, we study the computation of the dual frame by exploiting the Greville formula, which was originally derived in 1960 to compute the pseudoinverse of a matrix when a new row is appended. Here, we first develop the backward Greville formula to handle the case of row deletion. Based on the forward Greville formula, we then study the computation of para-pseudoinverse for extended filter banks and Laplacian pyramids. Through the backward Greville formula, we investigate the frame-based error resilient transmission over erasure channels. The necessary and sufficient condition for an oversampled filter bank to be robust to one erasure channel is derived. A postfiltering structure is also presented to implement the para-pseudoinverse when the transform coefficients in one subband are completely lost.      

Design of Oversampled Interleaved DFT Modulated Filter Bank Using 2Block Gauss-Seidel Method

In this paper, we present several new properties of the recently introduced interleaved DFT modulated filter bank and an efficient algorithm for designing the filter bank. The periodicity and symmetry properties of the overall transfer function and aliasing transfer functions are stated. Then the design of the filter bank is formulated into a constrained optimization problem that jointly minimizes the overall distortion and aliasing distortion subject to fixed bounds on the stopband energy, transition-band energy, and passband flatness of the prototype filters. The constrained optimization problem is solved by the 2block Gauss-Seidel method, which alternatively optimizes the analysis PF pair and the synthesis PF pair. Since the overall distortion and aliasing distortion are jointly minimized, the proposed algorithm can lead to filter banks with small reconstruction error, even when the filter banks behave with a low redundancy ratio and short PFs. The convergence of the proposed algorithm is proved. Numerical examples and comparisons with the existing method are included to demonstrate the performance of the proposed algorithm.     

精确重建滤波器组的快速设计方法

介绍了一种线性相位精确重建余弦调制滤波器组的设计方法,并总结了实验结果,提出了一种设计高通道数滤波器组的快速算法。     

Orthonormal FBs With Rational Sampling Factors and Oversampled DFT-Modulated FBs: A Connection and Filter Design

Methods widely used to design filters for uniformly sampled filter banks (FBs) are not applicable for FBs with rational sampling factors and oversampled discrete Fourier transform (DFT)-modulated FBs. In this paper, we show that the filter design problem (with regularity factors/vanishing moments) for these two types of FBs is the same. Following this, we propose two finite-impulse-response (FIR) filter design methods for these FBs. The first method describes a parameterization of FBs with a single regularity factor/vanishing moment. The second method, which can be used to design FBs with an arbitrary number of regularity factors/vanishing moments, uses results from frame theory. We also describe how to modify this method so as to obtain linear phase filters. Finally, we discuss and provide a motivation for iterated DFT-modulated FBs.     

Performance Analysis and Prototype Filter Design for Perfect-Reconstruction Cosine-Modulated Filter Banks With Fixed-Point Implementation

Cosine modulated filter banks have gained popularity for their ability to provide perfect reconstruction (PR) while maintaining an efficient design and implementation. However, this effectiveness is hindered if the filter bank is implemented in the fixed-point domain where quantization, rounding, and overflow occur, and result in reconstruction errors. In this article we demonstrate how to maintain PR of the filter bank when implementing it in fixed-point number format with constant wordlength. We explore how the frequency selectivity of the analysis and synthesis filters changes from the floating point ones due to fixed-point errors and present new design criteria for filter banks that will be implemented in fixed-point number format.     

Design and Performance Analysis of Linear Phase Para-UnitaryMBand Filter Banks for Image Coding

This paper presents the optimal design and performance analysis of the linear phase paraunitary (LPPU)Mband filter banks in the frame of vector quantized image coding. First, by maximizing the unified coding gain, which is a function of intra-band correlation, as well as inter-band energy compaction, the LPPUMband filter banks for the general factorization form are designed. Then, the image coding performances of the LPPUMband filter banks, such as the inter-band energy compaction, the intra-band correlation, and the average entropy are discussed. It is shown asymptotically that, as the filter length increases, the unified coding gain for the LPPUMband filter bank improves and the unified coding gain of the LPPU 4 band filter bank approaches very closely that of the LPPU 8 band filter bank. This observation is also verified by extensive computer simulation on the real images. In addition, the benefit of the directMband decomposition, based on the LPPUMband filter bank, over the tree structure decomposition is discussed, by showing the comparable coding performance and the reduced computational complexity.     

一种可靠的波导滤波器分析设计方法

针对波导滤波器的可生产性和一致性的工程实际需求,提出一种可靠的波导滤波器分析设计方法。利用高频结构仿真软件(HFSS)建模仿真,分析了公差和温度对波导滤波器频率特性的影响,从而优化了设计规范,最终结合散射参数法设计了一款C波段H面的波导滤波器。仿真及实测结果表明,相比于常规的设计方法,该方法更准确、可靠,提高了波导滤波器的可生产性和一致性,满足了工程设计需求。     

Optimization Design of Two-Channel Biorthogonal Graph Filter Banks

Narang and Ortega have constructed a two-channel biorthogonal graph filter bank with compact support. The design method does not consider the spectral response of the kernels. In this letter, we employ optimization approach to design the spectral kernels. The analysis and synthesis kernels are, respectively, optimized with constrained optimization problems, in which the reconstruction error and spectral selectivity are controlled simultaneously. The optimization problems are semidefinite programming (SDP), which can be solved effectively. Numerical examples and comparison are included to show that the proposed approach is more flexible in making trade-off between the spectral selectivity and reconstruction error over the existing method.     

最小延迟M带余弦调制小波滤波器组设计

本文推导了最小延迟任意长度M带余弦调制小波滤波器组的完全重构条件。选择低通原型滤波器最大阻带衰减为优化的目标函数,通常的优化目标函数选用最小平方逼近的方法,本文提出了使用最佳一致逼近的方法。最后用黄金分割和牛顿迭代方法解决非线性约束优化极值问题,得到满足几乎完全重构和小波正则性条件的低通原型滤波器。     

一种基于滤波器组的信号提取方法

在如今复杂的电磁环境中,不可避免地会遇到多个信号同时到达的情况,基于多相滤波组的信道化结构能较好地解决多个信号同时到达的问题,但是当信号带宽过宽时会出现信号跨道问题。跨道后的信号会失去部分重要信息,即使经过信道融合,仍然不能保证信号的完整性。在多个信号同时到达的前提下,为了保持信号的完整性,提出了一种利用全景监视与滤波器组相结合的方法来完成有效信号的完整提取。通过Matlab仿真验证了该方法的正确性、有效性。     

Permutation Based Design of Orthogonal Block Transforms and Filter Banks

In this work, a new efficient design techniquefor orthogonal block transforms, lapped orthogonal transformsand 4-channel perfect reconstruction subband filter banks isdeveloped. The technique consists of permutation and sign changeoperations on a reference vector. This approach can be thoughtof as a generalization of the Hadamard transform in the sensethat the reference vector h₀ (which will be a prototype low-pass filteralso forming one of the basis functions of the transform) willin general have components that are not identically 1's. Thedesign technique, a constructive method based on Hadamard arrays,provides a convenient means to explore new transforms. The meritof our method is that the number of unknowns and equality constraintsare both reduced significantly which render the design proceduremuch more feasible while guaranteeing at the same time linearphase.     

Efficient Design of Cosine-Modulated Filter Banks via Convex Optimization

This paper presents efficient approaches for designing cosine-modulated filter banks with linear phase prototype filter. First, we show that the design problem of the prototype filter being a spectral factor of $2M$th-band filter is a nonconvex optimization problem with low degree of nonconvexity. As a result, the nonconvex optimization problem can be cast into a semi-definite programming (SDP) problem by a convex relaxation technique. Then the reconstruction error is further minimized by an efficient iterative algorithm in which the closed-form expression is given in each iteration. Several examples are given to illustrate the effectiveness of the proposed method over the existing ones.      

用GCM法对发夹型带通滤波器进行分析和优化设计

本文利用Ｃｅｎｅｒａｌｉｚｅｄ Ｃｏｕｐｌｉｎｇ Ｍｏｄｅｌ（ＧＣＭ）法考虑所有线间耦合，对发夹型带通滤波器进行分析，计算结果与实验结果较为一致，然后在分析的基础上进行优化设计，对于给定的发夹型滤波器结构的任意初值，经过优化可得到符合指标的电路尺寸。本文以一只三阶发夹型带通滤波器为例给出了优化设计结果。     

A New Design Method of 9-7 Biorthogonal Filter Banks Based on Odd Harmonic Function

The floating-point implementation of a CDF-9-7 wavelet transform with irrational coefficients on a resource limited hardware platform is a challenging task. This paper presents a new design method of 9-7 biorthogonal wavelet filter bank (FB) based on classical Fourier theory, the so-called odd harmonic function (OHF) analysis. Three types of binary rational 9-7 biorthogonal wavelet FBs have been derived, considering vanishing moments in addition to the rationality of filter coefficients. The extensive experiments for the implementation of the new design on the SPIHT (Set Partitioning In Hierarchical Trees) platform have been conducted and the results show that the performance of the proposed new biorthogonal FBs is equal to, or in several cases outperforms the, CDF-9-7 FB.     

基于遗传算法的两通道完全重构滤波器组的设计

介绍了两通道滤波器组的完全重构条件,利用Euclidean分解算法,将两通道滤波器组的设计问题简化为寻找给定特性的低通滤波器的最佳Euclidean互补滤波器的单变量非线性优化问题,并探讨了采用遗传算法设计此类高度非线性优化问题.最后通过设计例子说明将遗传算法应用到滤波器组的设计中是可行的.     

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。     

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.