A complete factorization of paraunitary matrices with pairwisemirror-image symmetry in the frequency domain

The problem of designing orthonormal (paraunitary) filter banks has been addressed in the past. Several structures have been reported for implementing such systems. One of the structures reported imposes a pairwise mirror-image symmetry constraint on the frequency responses of the analysis (and synthesis) filters around π/2. This structure requires fewer multipliers, and the design time is correspondingly less than most other structures. The filters designed also have much better attenuation. We characterize the polyphase matrix of the above filters in terms of a matrix equation. We then prove that the structure reported in a paper by Nguyen and Vaidyanathan (1988), with minor modifications, is complete. This means that every polyphase matrix whose filters satisfy the mirror-image property can be factorized in terms of the proposed structure     

基于提升方法的滤波器组因子分解

讨论了FIR滤波器组的分解.2通道完全重构FIR 子波变换分解可为有限步的提升步骤,使用Laurent多项式的辗转相除法给出了这种分解的一个代数方法的证明;证明了二通道子波变换的分解定理不能平行推广到2M通道滤波器组.提出使用M-通道滤波器组构造2M-通道滤波器组,它由多相矩阵的分块化和提升方法实现,这种方法易于构造非线性滤波器组,如整数变换.     

Design techniques for silicon compiler implementations ofhigh-speed FIR digital filters

Architecture design techniques for implementing both single-rate and multirate high throughput finite impulse response (FIR) digital filters are explored, with an emphasis on those which are applicable to automated integrated circuit layout techniques. Various parallel architectures are examined based on the criteria of achievable throughput versus hardware complexity. Well-known techniques for reduced complexity and computation time are briefly summarized, followed by the introduction of several new techniques which offer further gains in both throughput and circuitry reduction. An architecture for mirror-symmetric polyphase filter banks is derived which exploits the coefficient symmetry between multiple filters to reduce hardware. Finally, the evolution of a silicon compiler which utilizes all of these techniques is presented, and results are given for compiled filters along with comparisons to other compiled and custom FIR filter chips     

General formulation of shift and delta operator based 2-D VLSI filter structures without global broadcast and incorporation of the symmetry

Having local data communication (without global broadcast of signals) among the elements is important in very large scale integration (VLSI) designs. Recently, 2-D systolic digital filter architectures were presented which eliminated the global broadcast of the input and output signals. In this paper a generalized formulation is presented that allows the derivation of various new 2-D VLSI filter structures, without global broadcast, using different 1-D filter sub-blocks and different interconnecting frameworks. The 1-D sub-blocks in z-domain are represented by general digital two-pair networks which consist of direct-form or lattice-type FIR filters in one of the frequency variables. Then, by applying the sub-blocks in various frameworks, 2-D structures realizing different transfer functions are easily obtained. As delta discrete-time operator based 1-D and 2-D digital filters (in \(\gamma \) -domain) were shown to offer better numerical accuracy and lower coefficient sensitivity in narrow-band filter designs when compared to the traditional shift-operator formulation we have covered both the conventional z-domain filters as well as delta discrete-time operator based filters. Structures realizing general 2-D IIR (both z- and \(\gamma \) -domains) and FIR transfer functions (z-domain only) are presented. As symmetry in the frequency response reduces the complexity of the design, IIR transfer functions with separable denominators, and transfer functions with quadrantal magnitude symmetry are also presented. The separable denominator frameworks are needed for quadrantal symmetry structures to guarantee BIBO stability and thus presented for both the operators. Some limitations of having exact symmetry with separable 1-D denominator factors are also discussed.     

Theory and design of signal-adapted FIR paraunitary filter banks

We study the design of signal-adapted FIR paraunitary filter banks, using energy compaction as the adaptation criterion. We present some important properties that globally optimal solutions to this optimization problem satisfy. In particular, we show that the optimal filters in the first channel of the filter bank are spectral factors of the solution to a linear semi-infinite programming (SIP) problem. The remaining filters are related to the first through a matrix eigenvector decomposition. We discuss uniqueness and sensitivity issues. The SIP problem is solved using a discretization method and a standard simplex algorithm. We also show how regularity constraints may be incorporated into the design problem to obtain globally optimal (in the energy compaction sense) filter banks with specified regularity. We also consider a problem in which the polyphase matrix implementation of the filter bank is constrained to be DCT based. Such constraints may also be incorporated into our optimization algorithm; therefore, we are able to obtain globally optimal filter banks subject to regularity and/or computational complexity constraints. Numerous experiments are presented to illustrate the main features that distinguish adapted and nonadapted filters, as well as the effects of the various constraints. The conjecture that energy compaction and coding gain optimization are equivalent design criteria is shown not to hold for FIR filter banks     

FIR filter banks for hexagonal data processing

Images are conventionally sampled on a rectangular lattice. Thus, traditional image processing is carried out on the rectangular lattice. The hexagonal lattice was proposed more than four decades ago as an alternative method for sampling. Compared with the rectangular lattice, the hexagonal lattice has certain advantages which include that it needs less sampling points; it has better consistent connectivity and higher symmetry; the hexagonal structure is also pertinent to the vision process. In this paper, we investigate the construction of symmetric FIR hexagonal filter banks for multiresolution hexagonal image processing. We obtain block structures of FIR hexagonal filter banks with 3-fold rotational symmetry and 3-fold axial symmetry. These block structures yield families of orthogonal and biorthogonal FIR hexagonal filter banks with 3-fold rotational symmetry and 3-fold axial symmetry. In this paper, we also discuss the construction of orthogonal and biorthogonal FIR filter banks with scaling functions and wavelets having optimal smoothness. In addition, we present a few of such orthogonal and biorthogonal FIR filters banks.     

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     

Compactly Supported Orthogonal and Biorthogonal $sqrt 5$-Refinement Wavelets With 4-Fold Symmetry

Recently, radic5 -refinement hierarchical sampling has been studied and radic5-refinement has been used for surface subdivision. Compared with other refinements, such as the dyadic or quincunx refinement, radic5-refinement has a special property that the nodes in a refined lattice form groups of five nodes with these five nodes having different x and y coordinates. This special property has been shown to be very useful to represent adaptively and render complex and procedural geometry. When radic5-refinement is used for multiresolution data processing, radic5-refinement filter banks and wavelets are required. While the construction of 2-D nonseparable (bi)orthogonal wavelets with the dyadic or quincunx refinement has been studied by many researchers, the construction of (bi)orthogonal wavelets with radic5-refinement has not been investigated. The main goal of this paper is to construct compactly supported orthogonal and biorthogonal wavelets with radic5 -refinement. In this paper, we obtain block structures of orthogonal and biorthogonal radic5-refinement FIR filter banks with 4-fold rotational symmetry. We construct compactly supported orthogonal and biorthogonal wavelets based on these block structures.     

Compactly supported orthogonal and biorthogonal square root 5-refinement wavelets with 4-fold symmetry

Recently, square root 5 -refinement hierarchical sampling has been studied and square root 5-refinement has been used for surface subdivision. Compared with other refinements, such as the dyadic or quincunx refinement, square root 5-refinement has a special property that the nodes in a refined lattice form groups of five nodes with these five nodes having different x and y coordinates. This special property has been shown to be very useful to represent adaptively and render complex and procedural geometry. When square root 5-refinement is used for multiresolution data processing, square root 5-refinement filter banks and wavelets are required. While the construction of 2-D nonseparable (bi)orthogonal wavelets with the dyadic or quincunx refinement has been studied by many researchers, the construction of (bi)orthogonal wavelets with square root 5-refinement has not been investigated. The main goal of this paper is to construct compactly supported orthogonal and biorthogonal wavelets with square root 5 -refinement. In this paper, we obtain block structures of orthogonal and biorthogonal square root 5-refinement FIR filter banks with 4-fold rotational symmetry. We construct compactly supported orthogonal and biorthogonal wavelets based on these block structures.     

Two-Channel FIR Filter Banks Utilizing the FRM Approach

The frequency-response masking (FRM) approach has been introduced as a means of generating narrow transition band linear-phase finite impulse response (FIR) filters with a low arithmetic complexity. This paper proposes an approach for synthesizing two-channel maximally decimated FIR filter banks utilizing the FRM technique. For this purpose, a new class of FRM filters is introduced. Filters belonging to this class are used for synthesizing nonlinear-phase analysis and synthesis filters for two types of two-channel filter banks. For the first type, there exist no phase distortion and aliasing errors, but this type suffers from a small amplitude distortion as for the well-known quadrature mirror filter (QMF) banks. Compared to conventional QMF filter banks, the proposed banks lower significantly the overall arithmetic complexity at the expense of a somewhat increased overall filter bank delay in applications demanding narrow transition bands. For the second type, there are also small aliasing errors, allowing one to reduce the arithmetic complexity even further. Efficient structures are introduced for implementing the proposed filter banks, and algorithms are described for maximizing the stopband attenuations of the analysis and synthesis filters in the minimax sense subject to the given allowable amplitude and/or aliasing errors. Examples are included illustrating the benefits provided by the proposed filter banks.     

Orthogonal and Biorthogonal FIR Hexagonal Filter Banks With Sixfold Symmetry

Recently, hexagonal image processing has attracted attention. The hexagonal lattice has several advantages in comparison with the rectangular lattice, the conventionally used lattice for image sampling and processing. For example, a hexagonal lattice needs less sampling points; it has better consistent connectivity; it has higher symmetry; and its structure is plausible to human vision systems. The multiresolution analysis method has been used for hexagonal image processing. Since the hexagonal lattice has high degree of symmetry, it is desirable that the hexagonal filter banks designed for multiresolution hexagonal image processing also have high order of symmetry, which is pertinent to the symmetry structure of the hexagonal lattice. The orthogonal or prefect reconstruction (PR) hexagonal filter banks that are available in the literature have only threefold symmetry. In this paper, we investigate the construction of orthogonal and PR finite impulse response (FIR) hexagonal filter banks with sixfold symmetry. We obtain block structures of 7-size refinement (seven-channel two-dimensional) orthogonal and PR FIR hexagonal filter banks with sixfold rotational symmetry. $sqrt{7}$-refinement orthogonal and biorthogonal wavelets based on these block structures are constructed. In this paper, we also consider FIR hexagonal filter banks with axial (line) symmetry, and we present a block structure of FIR hexagonal filter banks with pseudo-sixfold axial symmetry.      

Integer Linear Programming-Based Bit-Level Optimization for High-Speed FIR Decimation Filter Architectures

Analog-to-digital converters based on sigma-delta modulation have shown promising performance, with steadily increasing bandwidth. However, associated with the increasing bandwidth is an increasing modulator sampling rate, which becomes costly to decimate in the digital domain. Several architectures exist for the digital decimation filter, and among the more common and efficient are polyphase decomposed finite-length impulse response (FIR) filter structures. In this paper, we consider such filters implemented with partial product generation for the multiplications, and carry-save adders to merge the partial products. The focus is on the efficient pipelined reduction of the partial products, which is done using a bit-level optimization algorithm for the tree design. However, the method is not limited only to filter design, but may also be used in other applications where high-speed reduction of partial products is required. The presentation of the reduction method is carried out through a comparison between the main architectural choices for FIR filters: the direct-form and transposed direct-form structures. For the direct-form structure, usage of symmetry adders for linear-phase filters is investigated, and a new scheme utilizing partial symmetry adders is introduced. The optimization results are complemented with energy dissipation and cell area estimations for a 90 nm CMOS process.     

Restoring Coefficient Symmetry in Polyphase Implementation of Linear-Phase FIR Filters

Polyphase implementation of FIR filters effectively reduces the multiplication rate and data storage in a multirate system. However, the coefficients of the polyphase components are no longer symmetric even though the overall filter has a symmetric (or anti-symmetric) impulse response. In this paper, we introduce a new technique that recasts pairs of the original polyphase components as sums or differences of auxiliary pairs of symmetric and anti-symmetric impulse response filters. The coefficient symmetry of these auxiliary polyphase components can be fully exploited to reduce arithmetic complexity without undue complications. Our new technique makes use of the fact that the impulse responses of the non-symmetric polyphase components exist in time-reversed pairs which can be synthesized from pairs of symmetric and anti-symmetric impulse response filters. This results in a factor-of-two reduction in the number of multipliers required to implement the polyphase components.     

AD6635中FIR滤波器的多相实现

通过分析多通道数字下变频器件AD6635的结构特点,提出了一种多相有限冲击响应(FIR)滤波器的设计方法,并给出了相应的参数.这种方法通过并联多个数据通路处理同一信号来构成多相滤波器.仿真证明,采用这种方法构成的多相滤波器,与非多相滤波器相比,在过渡带宽度、通带波纹、阻带衰减等主要滤波性能指标上有显著提高.     

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.     

Role of anticausal inverses in multirate filter-banks .II. The FIRcase, factorizations, and biorthogonal lapped transforms

For pt. I see ibid., vol.43, no.5, p.1090, 1990. In part I we studied the system-theoretic properties of discrete time transfer matrices in the context of inversion, and classified them according to the types of inverses they had. In particular, we outlined the role of causal FIR matrices with anticausal FIR inverses (abbreviated cafacafi) in the characterization of FIR perfect reconstruction (PR) filter banks. Essentially all FIR PR filter banks can be characterized by causal FIR polyphase matrices having anticausal FIR inverses. In this paper, we introduce the most general degree-one cafacafi building block, and consider the problem of factorizing cafacafi systems into these building blocks. Factorizability conditions are developed. A special class of cafacafi systems called the biorthogonal lapped transform (BOLT) is developed, and shown to be factorizable. This is a generalization of the well-known lapped orthogonal transform (LOT). Examples of unfactorizable cafacafi systems are also demonstrated. Finally it is shown that any causal FIR matrix with FIR inverse can be written as a product of a factorizable cafacafi system and a unimodular matrix     

Frame-theoretic analysis of oversampled filter banks

We provide a frame-theoretic analysis of oversampled finite impulse response (FIR) and infinite impulse response (FIR) uniform filter banks (FBs). Our analysis is based on a new relationship between the FBs polyphase matrices and the frame operator corresponding to an FB. For a given oversampled analysis FB, we present a parameterization of all synthesis FBs providing perfect reconstruction. We find necessary and sufficient conditions for an oversampled FB to provide a frame expansion. A new frame-theoretic procedure for the design of paraunitary FBs from given nonparaunitary FBs is formulated. We show that the frame bounds of an FB can be obtained by an eigen-analysis of the polyphase matrices. The relevance of the frame bounds as a characterization of important numerical properties of an FB is assessed by means of a stochastic sensitivity analysis. We consider special cases in which the calculation of the frame bounds and synthesis filters is simplified. Finally, simulation results are presented     

Multidimensional orthogonal filter bank characterization and design using the Cayley transform.

We present a complete characterization and design of orthogonal infinite impulse response (IIR) and finite impulse response (FIR) filter banks in any dimension using the Cayley transform (CT). Traditional design methods for one-dimensional orthogonal filter banks cannot be extended to higher dimensions directly due to the lack of a multidimensional (MD) spectral factorization theorem. In the polyphase domain, orthogonal filter banks are equivalent to paraunitary matrices and lead to solving a set of nonlinear equations. The CT establishes a one-to-one mapping between paraunitary matrices and para-skew-Hermitian matrices. In contrast to the paraunitary condition, the para-skew-Hermitian condition amounts to linear constraints on the matrix entries which are much easier to solve. Based on this characterization, we propose efficient methods to design MD orthogonal filter banks and present new design results for both IIR and FIR cases.     

On the perfect reconstruction problem in N-band multirate maximallydecimated FIR filter banks

The problem of finding N-K filters of an N-band maximally decimated FIR analysis filter bank, given K filters, so that FIR perfect reconstruction can be achieved, is considered. The perfect reconstruction condition is expressed as a requirement of unimodularity of the polyphase analysis filter matrix. Based on the theory of Euclidean division for matrix polynomials, the conditions the given transfer functions must satisfy are given, and a complete parameterization of the solution is obtained. This approach provides an interesting alternative to the method of the complementary filter in the case of N>2,K相似文献