Design of two-dimensional digital filters using singular-valuedecomposition and balanced approximation method

It is shown that the singular-value decomposition (SVD) of the sampled amplitude response of a two-dimensional (2-D) digital filter possesses a special structure: every singular vector is either mirror-image symmetric or antisymmetric with respect to its midpoint. Consequently, the SVD can be applied along with 1-D finite impulse response (FIR) techniques for the design of linear-phase 2-D filters with arbitrary prescribed amplitude responses which are symmetrical with respect to the origin of the (ωΨω₂) plane. The balanced approximation method is applied to linear-phase 2-D FIR filters of the type that may be obtained by using the SVD method. The method leads to economical and computationally efficient filters, usually infinite impulse response filters, which have prescribed amplitude responses and whose phase responses are approximately linear     

Generating 2-D FIR filter functions by Christoffel–Darboux formula for Chebyshev polynomials of the second kind

Global Christoffel–Darboux formula for different polynomials has already been used for the filter design. Here, this formula for orthonormal Chebyshev polynomials of the second kind and for two independent variables is applied in generating novel class linear-phase two-dimensional (2-D) finite impulse response (FIR) digital filter functions. In this way, 2-D filters with some specific features including economy, phase linearity, symmetry and selectivity are designed. Representative examples of the 2-D FIR digital filters of a new class obtained by the proposed approximation technique are given. A filter generated by the proposed approach is compared with the corresponding one generated by the procedure from literature.     

1D and 2D economical FIR filters generated by Chebyshev polynomials of the first kind

Christoffel–Darboux formula for Chebyshev continual orthogonal polynomials of the first kind is proposed to find a mathematical solution of approximation problem of a one-dimensional (1D) filter function in the z domain. Such an approach allows for the generation of a linear phase selective 1D low-pass digital finite impulse response (FIR) filter function in compact explicit form by using an analytical method. A new difference equation and structure of corresponding linear phase 1D low-pass digital FIR filter are given here. As an example, one extremely economic 1D FIR filter (with four adders and without multipliers) is designed by the proposed technique and its characteristics are presented. Global Christoffel–Darboux formula for orthonormal Chebyshev polynomials of the first kind and for two independent variables for generating linear phase symmetric two-dimensional (2D) FIR digital filter functions in a compact explicit representative form, by using an analytical method, is proposed in this paper. The formula can be most directly applied for mathematically solving the approximation problem of a filter function of even and odd order. Examples of a new class of extremely economic linear phase symmetric selective 2D FIR digital filters obtained by the proposed approximation technique are presented.     

Design of 3-D Noncausal Filters with Small Roundoff Noise and No Overflow Oscillations

The contribution of this paper consists of two individual parts. First, an invertible mapping technique is presented for 3-D digital system design, and it is applied to approximate 3-D noncausal filters in the spatial domain. Secondly, an algorithm is proposed for obtaining a structure for 3-D IIR filters with small roundoff noise and no overflow oscillations. The design of noncausal filters can be carried out by three steps: 1), a given noncausal impulse response is transformed into the first octant using the proposed 3-D invertible mapping technique; 2), the transformed impulse response in the first octant is approximated by balanced model reduction of 3-D separable denominator systems;3), the resultant 3-D IIR filter is transformed back to the original coordinates.     

任意型一维FIR数字滤波器设计新方法

本文介绍了一种设计任意型（宽带、窄带、低阶、高阶、低通、高通、带通、带阻）一维FIR数字滤波器新方法．给出设计不同类型滤波器的统一方法。各种类型滤波器脉冲响应的设计公式简单、计算方便。该方法消除了Gibbs现象．不仅克服了优化设计中收敛速度、设计精度及初值选取等问题，而且减少设计高阶滤波器的时间。最后，本文给出了各种类型滤波器设计的仿真结果。     

Realization of 2-D linear-phase FIR filters by using thesingular-value decomposition

The singular-value decomposition (SVD) technique is investigated for the realization of a general two-dimensional (2-D) linear-phase FIR filter with an arbitrary magnitude response. A parallel realization structure consisting of a number of one-dimensional (1-D) FIR subfilters is obtained by applying the SVD to the impulse response of a 2-D filter. It is shown that by using the symmetry property of the 2-D impulse response and by developing an appropriate unitary transformation, an SVD yielding linear-phase constituent 1-D filters can always be obtained so that the efficient structures of the 1-D linear-phase filters can be exploited for 2-D realization. It is shown that when the 2-D filter to be realized has some specified symmetry in its magnitude response, the proposed SVD realization would yield a magnitude characteristic with the same symmetry. An analysis is carried out to obtain tight upper bounds for the errors in the impulse response as well as in the frequency response of the realized filter. It is shown that the number of parallel sections can be reduced significantly without introducing large errors, even in the case of 2-D filters with nonsymmetric magnitude response     

Design of two-channel linear-phase QMF banks based on real IIR all-pass filters

The design of two-channel linear-phase quadrature mirror filter (QMF) banks constructed by real infinite impulse response (IIR) digital all-pass filters is considered. The design problem is appropriately formulated to result in a simple optimisation problem. Using a variant of Karmarkar's algorithm, the optimisation problem can be efficiently solved through a frequency sampling and iterative approximation method to find the real coefficients for the IIR digital all-pass filters. The resulting two-channel QMF banks possess an approximately linear phase response without magnitude distortion. The effectiveness of the proposed technique is achieved by forming an appropriate Chebyshev approximation of the desired phase response and then finding its solution from a linear subspace in a few iterations. Finally, several simulation examples are presented for illustration and comparison     

Digital filter design techniques in the frequency domain

Digital filtering is the process of spectrum shaping using digital components as the basic elements. Increasing speed and decreasing size and cost of digital components make it likely that digital filtering, already used extensively in the computer simulation of analog filters, will perform, in real-time devices, the functions which are now performed almost exclusively by analog components. In this paper, using the z-transform calculus, several digital filter design techniques are reviewed, and new ones are presented. One technique can be used to design a digital filter whose impulse response is like that of a given analog filter; other techniques are suitable for the design of a digital filter meeting frequency response criteria. Another technique yields digital filters with linear phase, specified frequency response, and controlled impulse response duration. The effect of digital arithmetic on the behavior of digital filters is also considered.     

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.     

Image data compression using subband coding

A general reversive subband coding system with 2-D infinite impulse response filters is proposed. The system considered guarantees perfect image reconstruction (free of phase distortions). Application of wave digital filters is considered. A new technique of high-frequency source encoding is proposed. The experiments with real images prove high efficiency of the technique proposed.     

Optimization of two-dimensional IIR filters with nonseparable and separable denominator

We present algorithms for the optimization of two-dimensional (2-D) infinite impulse response (IIR) filters with separable or nonseparable denominator, for least squares or Chebyshev criteria. The algorithms are iterative, and each iteration consists of solving a semidefinite programming problem. For least squares designs, we adapt the Gauss-Newton idea, which outcomes to a convex approximation of the optimization criterion. For Chebyshev designs, we adapt the iterative reweighted least squares (IRLS) algorithm; in each iteration, a least squares Gauss-Newton step is performed, while the weights are changed as in the basic IRLS algorithm. The stability of the 2-D IIR filters is ensured by keeping the denominator inside convex stability domains, which are defined by linear matrix inequalities. For the 2-D (nonseparable) case, this is a new contribution, based on the parameterization of 2-D polynomials that are positive on the unit bicircle. In the experimental section, 2-D IIR filters with separable and nonseparable denominators are designed and compared. We show that each type may be better than the other, depending on the design specification. We also give an example of filter that is clearly better than a recent very good design.     

A new closed form method for design of variable bandwidth linear phase FIR filter using Bernstein multiwavelets

In this paper, a new method for the design of variable bandwidth linear-phase finite impulse response filters using Bernstein polynomial Multiwavelets is proposed. In this method, approximation has been achieved by linearly combining the fixed coefficient linear phase filters with Bernstein multiwavelets, which are used to tune bandwidth of the filter. Optimisation has been achieved by minimising the mean square error between the desired and actual filter response which leads to a system of linear equations. The matrix elements can be expressed in form of Toeplitz-plus-Hankel matrix, which reduces the computational complexity. The simulation results illustrate significant improvement in errors in passband (e_p), and stopband (e_s) as compared to earlier published work.     

A computational method for the LU decomposition of rectangularmatrices and its implications to the realization of 2-D digital filters

The realization of 2-D digital filters based on the lower-upper triangular decomposition of the coefficient matrix is investigated. A numerical method based on the QA decomposition, which has some important characteristics, is proposed for reaching the LU structure. The coefficients in the final LU structure have values favorable to fixed-point arithmetic implementation. Furthermore, the QR structure can be used for the realization and possesses good numerical characteristics in terms of the approximate decomposition scheme. The symmetry in the impulse response coefficient matrix of an octagonally symmetric 2-D FIR filter is utilized to reduce the computational effort spent in the decomposition and the total number of multipliers in the final realization structure     

Non-fragile <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> filter designs for polytopic two-dimensional systems in Roesser model

This paper is concerned with the problem of non-fragile H ₂ and H _∞ filter designs for two-dimensional (2-D) discrete systems in Roesser model with polytopic uncertainties. The filters to be designed are assumed to be with additive norm-bounded coefficient variations which reflect the imprecision in filter implementation. The complicated filter design problem is successfully tackled by using the slack variable technique and imposing a structural restriction on the slack matrix. Explicit expressions of the non-fragile H ₂ and H _∞ filters are given in terms of solutions to a set of linear matrix inequalities (LMIs). An illustrative example is provided to demonstrate the feasibility and effectiveness of the proposed method.     

Implementation of Digital Unbiased FIR Filters with Polynomial Impulse Responses

This paper addresses the design and implementation of digital unbiased finite impulse response (FIR) filters with polynomial impulse response functions. The transfer function, its fundamental properties, and a general block-diagram are discussed for the impulse response represented with the l-degree Taylor series expansion. As a particular results, we show a fundamental identity uniquely featured to such filters in the transform domain. For low-degree impulse responses, the transfer functions are found in simple closed forms and represented in compact block-diagrams. The magnitude and phase responses are also analyzed along with the group delays. A comparison with predictive FIR filters is given. As examples of applications, filtering of time errors of local clocks is discussed along with the low-pass filter design employing a cascade of the unbiased FIR filters.     

Design of two-dimensional half-plane recursive digital filters with octagonal symmetry

A design method for two-dimensional (2-D) half-plane recursive digital filters is proposed. Constraints in the parameters of the designed transfer function are imposed to ensure octagonal symmetry in the magnitude response. Optimization algorithms are used to minimize thel _p norm. The stability of these filters is checked easily by evaluating a small number of predetermined inequalities. Twelve designs are presented to show the characteristics and efficiency of the method proposed.The research reported in this paper was suppored by the Natural Sciences and Engineering Research Council of Canada under Grant A7397.     

A novel nonnegative decomposition method and its application to 2-D digital filter design

In designing two-dimensional (2-D) digital filters in the frequency domain, an efficient technique is to first decompose the given 2-D frequency domain design specifications into one-dimensional (1-D) ones, and then approximate the resulting 1-D magnitude specifications using the well-developed 1-D filter design techniques. Finally, by interconnecting the designed 1-D filters one can obtain a 2-D digital filter. However, since the magnitude responses of digital filters must be nonnegative, it is required that the decomposition of 2-D magnitude specifications result in nonnegative 1-D magnitude specifications. We call such a decomposition the nonnegative decomposition. This paper proposes a nonnegative decomposition method for decomposing the given 2-D magnitude specifications into 1-D ones, and then transforms the problem of designing a 2-D digital filter into that of designing 1-D filters. Consequently, the original problem of designing a 2-D filter is significantly simplified.     

A weighted least-squares method for the design of stable 1-D and2-D IIR digital filters

We present a new approach to the least-squares design of stable infinite impulse response (IIR) digital filters. The design is accomplished by using an iterative scheme in which the denominator polynomial obtained from the preceding iteration is treated as a part of the weighting function, and each iteration is carried out by solving a standard quadratic programming problem that yields a stable rational function. When the iteration converges, a stable and truly least-squares solution is obtained. The method is then extended to address the least-squares design of stable IIR two-dimensional (2-D) filters. Examples are included to illustrate the proposed design techniques     

具有任意频响的二维FIR数字滤波器设计的闭式最小二乘解

本文研究具有任意频响特性的二维ＦＩＲ数字滤波器的最小二乘设计问题。     

Design of High-Order Extrapolated Impulse Response FIR Filters with Signed Powers-of-Two Coefficients

Expensive multiplication operations can be replaced by simpler additions and hardwired shifters so as to reduce power consumption and area size, if the coefficients of a digital filter are signed power-of-two (SPT). As a consequence, FIR digital filters with SPT coefficients have been widely studied in the last three decades. However, most approaches for the design of FIR filters with SPT coefficients focus on filters with length less than 100. These approaches are not suitable for the design of high-order filters because they require excessive computation time. In this paper, an approach for the design of high-order filters with SPT coefficients is proposed. It is a two-step approach. Firstly, the design of an extrapolated impulse response (EIR) filter is formulated as a standard second-order cone programming (SOCP) problem with an additional coefficient sensitivity constraint for optimizing its finite word-length effect. Secondly, the obtained continuous coefficients are quantized into SPT coefficients by recasting the filter-design problem into a weighted least squares (WLS) sequential quadratic programming relaxation (SQPR) problem. To further reduce implementation complexity, a graph-based common subexpression elimination (CSE) algorithm is utilized to extract common subexpressions between SPT coefficients. Simulation results show that the proposed method can effectively and efficiently design high-order SPT filters, including Hilbert transformers and half-band filters with SPT coefficients. Experiment results indicate that 0.81N∼0.29N adders are required for 18-bit N-order FIR filters (N=335∼3261) to meet the given magnitude response specifications.