Polyphase Conditions and Structures for 2-D Quincunx FIR Filter Banks Having Quadrantal or Diagonal Symmetries

In this brief, we derive conditions on the polyphase matrix of 2-D finite-impulse response (FIR) quincunx filter banks, for the filters in the filter bank to have quadrantal or diagonal symmetry. These conditions provide a framework for synthesizing polyphase structures which structurally enforce the symmetry. This is demonstrated by constructing examples of small parameterized matrix structures which satisfy the above conditions, thus giving perfect reconstruction FIR quincunx filter banks with quadrantal or diagonally symmetric short-kernel (i.e., short-support) filters. It is also shown that cascades of the above constructed small structures can be used to construct filters of higher order.     

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 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.     

A new class of two-channel biorthogonal filter banks and waveletbases

Proposes a novel framework for a new class of two-channel biorthogonal filter banks. The framework covers two useful subclasses: i) causal stable IIR filter banks. ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for such a class. Filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity. The properties of such a class are discussed in detail. The design of the analysis/synthesis systems reduces to the design of a single transfer function. Very simple design methods are given both for FIR and IIR cases. Zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity property. In the IIR case, two new classes of IIR maximally flat filters different from Butterworth filters are introduced. The filter coefficients are given in closed form. The wavelet bases corresponding to the biorthogonal systems are generated. the authors also provide a novel mapping of the proposed 1-D framework into 2-D. The mapping preserves the following: i) perfect reconstruction; ii) stability in the IIR case; iii) linear phase in the FIR case; iv) zeros at aliasing frequency; v) frequency characteristic of the filters     

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.<>     

一种基于迭代短卷积算法的低复杂度并行FIR滤波器结构

该文基于快速卷积算法,提出一种适用于线性相位FIR滤波器的并行结构。该结构采用快速卷积算法减少子滤波器个数,同时让尽可能多的子滤波器具有对称系数,然后利用系数对称的特性减少子滤波器模块中的乘法器数量。对于具有对称系数的FIR滤波器,提出的并行结构能够比已有的并行FIR结构节省大量的硬件资源,尤其当滤波器的抽头数较大时效果更明显。具体地,对一个4并行144抽头的FIR滤波器,提出的结构比改进的快速FIR算法(Fast FIR Algorithm, FFA)结构节省36个乘法器(14.3%),23个加法器(6.6%)和35个延时单元(11.0%)。     

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     

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     

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     

A Mapping-Based Design for Nonsubsampled Hourglass Filter Banks in Arbitrary Dimensions

Multidimensional hourglass filter banks decompose the frequency spectrum of input signals into hourglass-shaped directional subbands, each aligned with one of the frequency axes. The directionality of the spectral partitioning makes these filter banks useful in separating the directional information in multidimensional signals. Despite the existence of various design techniques proposed for the 2-D case, to our best knowledge, the design of hourglass filter banks in 3-D and higher dimensions with finite impulse response (FIR) filters and perfect reconstruction has not been previously reported. In this paper, we propose a novel mapping-based design for the hourglass filter banks in arbitrary dimensions, featuring perfect reconstruction, FIR filters, efficient implementation using lifting/ladder structures, and a near-tight frame construction. The effectiveness of the proposed mapping- based design depends on the study of a set of conditions on the frequency supports of the mapping kernels. These conditions ensure that we can still get good frequency responses when the component filters used are nonideal. Among all feasible choices, we then propose an optimal specification for the mapping kernels, which leads to the simplest passband shapes and involves the fewest number of frequency variables. Finally, we illustrate the proposed techniques by a design example in 3-D, and an application in video denoising.     

Complex, linear-phase filters for efficient image coding

With the exception of the Haar basis, real-valued orthogonal wavelet filter banks with compact support lack symmetry and therefore do not possess linear phase. This has led to the use of biorthogonal filters for coding of images and other multidimensional data. There are, however, complex solutions permitting the construction of compactly supported, orthogonal linear phase QMF filter banks. By explicitly seeking solutions in which the imaginary part of the filter coefficients is small enough to be approximated to zero, real symmetric filters can be obtained that achieve excellent compression performance     

Multirate filter banks with block sampling

Multirate filter banks with block sampling were recently studied by Khansari and Leon-Garcia (1993). In this paper, we want to systematically study multirate filter banks with block sampling by studying general vector filter banks where the input signals and transfer functions in conventional multirate filter banks are replaced by vector signals and transfer matrices, respectively. We show that multirate filter banks with block sampling studied by Khansari and Leon-Garcia are special vector filter banks where the transfer matrices are pseudocirculant. We present some fundamental properties for the basic building blocks, such as Noble identities, interchangeability of down/up sampling, polyphase representations of M-channel vector filter banks, and multirate filter banks with block sampling. We then present necessary and sufficient conditions for the alias-free property, finite impulse response (FIR) systems with FIR inverses, paraunitariness, and lattice structures for paraunitary vector filter banks. We also present a necessary and sufficient condition for paraunitary multirate filter banks with block sampling. As an application of this theory, we present all possible perfect reconstruction delay chain systems with block sampling. We also show some examples that are not paraunitary for conventional multirate filter banks but are paraunitary for multirate filter banks with proper block sampling. In this paper, we also present a connection between vector filter banks and vector transforms studied by Li. Vector filter banks also play important roles in multiwavelet transforms and vector subband coding     

Reduction of Symmetric Complex Filters

Due to their linear-phase property, symmetric filters are an interesting class of finite-impulse-response (FIR) filters. Moreover, symmetric FIR filters allow an efficient implementation. In this paper we extend the classical definition of Hermitian symmetry to a more general symmetry that is also applicable to complex filters. This symmetry is called generalized-Hermitian symmetry. We show the usefulness of this definition as it allows for a unified treatment of even and odd-length filters. Central in this paper is a theorem on the reduction of generalized-Hermitian-symmetric filters to Hermitian-symmetric filters, both with finite precision coefficients. A constructive proof of this theorem is presented and an associated procedure for reducing generalized-Hermitian-symmetric filters is derived. Two of the examples show the application of the reduction procedure and the achieved savings on arithmetic costs. Finally, all three examples show that a special instance of the generalized-Hermitian-symmetric filters with finite precision coefficients, may have lower arithmetic costs than the Hermitian-symmetric filter from which it is derived.     

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     

Novel Area-Efficient FPGA Architectures for FIR Filtering With Symmetric Signal Extension

This paper presents four novel area-efficient field-programmable gate-array (FPGA) bit-parallel architectures of finite impulse response (FIR) filters that smartly support the technique of symmetric signal extension while processing finite length signals at their boundaries. The key to this is a clever use of variable-depth shift registers which are efficiently implemented in Xilinx FPGAs in the form of shift register logic (SRL) components. Comparisons with the conventional architecture of FIR filter with symmetric boundary processing show considerable area saving especially with long-tap filters. For instance, our architecture implementation of the 8-tap low Daubechies-8 FIR filter achieves ~ 30% reduction in the area requirement (in terms of slices) compared to the conventional architecture while maintaining the same throughput. Two of the above-cited novel architectures are dedicated to the special case of symmetric FIR filters. The first architecture is highly area-efficient but requires a clock frequency doubler. While this reduces the overall processing speed (to a maximum of 2), it does maintain a high throughput. Moreover, this speed penalty is cancelled in bi-phase filters which are widely used in multirate architectures (e.g., wavelets). Our second symmetric FIR filter architecture saves less logic than the first architecture (e.g., 10% with the 9-tap low Biorthogonal 9&7 symmetric filter instead of 37% with the first architecture) but overcomes its speed penalty as it matches the throughput of the conventional architecture.     

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.      

一种新的调制型混合滤波器组设计算法

针对经典的准确重构混合滤波器组设计问题,提出一种调制型混合滤波器组(MHFB)的设计算法,推导了分析滤波器组系数矩阵的行列式与原型滤波器系数的解析关系,给出了具有普适性的综合滤波器组解的一般形式,并讨论了因果、稳定系统的设计方法。针对多通道HFB设计复杂的问题,提出一种FIR形式综合滤波器组的优化设计算法,适用于多通道HFB的设计。仿真结果验证了算法的有效性。     

Linear phase cosine modulated maximally decimated filter banks withperfect reconstruction

We propose a novel way to design maximally decimated FIR cosine modulated filter banks, in which each analysis and synthesis filter has a linear phase. The system can be designed to have either the approximate reconstruction property (pseudo-QMF system) or perfect reconstruction property (PR system). In the PR case, the system is a paraunitary filter bank. As in earlier work on cosine modulated systems, all the analysis filters come from an FIR prototype filter. However, unlike in any of the previous designs, all but two of the analysis filters have a total bandwidth of 2π/M rather than π/M (where 2M is the number of channels in our notation). A simple interpretation is possible in terms of the complex (hypothetical) analytic signal corresponding to each bandpass subband. The coding gain of the new system is comparable with that of a traditional M-channel system (rather than a 2M-channel system). This is primarily because there are typically two bandpass filters with the same passband support. Correspondingly, the cost of the system (in terms of complexity of implementation) is also comparable with that of an M-channel system. We also demonstrate that very good attenuation characteristics can be obtained with the new system     

Optimal design of nonuniform multirate filter banks

The design of general nonuniform filter banks is studied. Contrary to uniform filter banks, in nonuniform filter banks, it may not be possible to achieve perfect reconstruction, but in some cases by using optimization techniques, we can design acceptable filter banks. Here, the initial finite impulse response (FIR) analysis filters are designed according to the characteristics of the input. By the design procedure, the FIR synthesis filters are found so that theH-norm of an error system is minimized over all synthesis filters that have a prespecified order. Then, the synthesis filters obtained in the previous step are fixed, and the analysis filters are found similarly. By iteration, theH-norm of the error system decreases until it converges to its final value. At each iteration, the coefficients of the analysis or synthesis filters are obtained by finding the least squares solution of a system of linear equations. If necessary, the frequency characteristics of the filters can be altered by adding penalty terms to the objective function.This research was supported by the Natural Sciences and Engineering Research Council of Canada.