Design Of 2-D Recursive Digital Filters Using Nonsymmetric Half-Plane Allpass Filters

A novel structure using recursive nonsymmetric half-plane (NSHP) digital allpass filters (DAFs) is presented for designing 2-D recursive digital filters. First, several important properties of 2-D recursive DAFs with NSHP support regions for filter coefficients are investigated. The stability of the 2-D recursive NSHP DAFs is guaranteed by using a spectral factorization-based algorithm. Then, the important characteristics regarding the proposed novel structure are discussed. The design problem of 2-D recursive digital filters using the novel structure is considered. We formulate the problem by forming an objective function consisting of the weighted sum of magnitude, group delay, and stability-related errors. A design technique using a trust-region Newton-conjugate gradient method in conjunction with the analytic derivatives of the objective function is presented to efficiently solve the resulting optimization problem. The novelty of the presented 2-D structure is that it possesses the advantage of better performance in designing a variety of 2-D recursive digital filters over existing 2-D filter structures. Finally, several design examples are provided for conducting illustration and comparison.     

Efficient design of digital filters for 2-D and 3-D depth migration

Two- and three-dimensional (2-D and 3-D) depth migration can be performed using 1-D and 2-D extrapolation digital filters, respectively. The depth extrapolation is done, one frequency at a time, by convolving the seismic wavefield with a complex-valued, frequency- and velocity-dependent, digital filter. This process requires the design of a complete set of extrapolation filters: one filter for each possible frequency-velocity pair. Instead of independently designing the frequency- and velocity-dependent filters, an efficient procedure is introduced for designing a complete set of 1-D and 2-D extrapolation filters using transformations. The problem of designing a desired set of migration filters is thus reduced to the design of a single 1-D filter, which is then mapped to produce all the desired 1-D or 2-D migration filters. The new design procedure has the additional advantage that both the 1-D and 2-D migration filters can be realized efficiently and need not have their coefficients precomputed or tabulated     

On the properties of linear spectral transformations for 2-dimensional digital filters

A set of eight linear spectral transformations which can be used in the design of two-dimensional digital filters is studied from a group-theoretic point of view. Several properties of the transformations, some of them known and some of them new, are deduced and are then applied in the implementation of 2-D digital filters. It is shown that trade-offs exist which can be used to reduce either the amount of memory required for the programming or the amount of data manipulation.     

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.     

Design of zero-phase recursive 2-D variable filters with quadrantal symmetries

The digital filters with adjustable frequency-domain characteristics are called variable filters. Variable filters are useful in the applications where the filter characteristics are needed to be changeable during the course of signal processing. In such cases, if the existing traditional constant filter design techniques are applied to the design of new filters to satisfy the new desired characteristics when necessary, it will take a huge amount of design time. So it is desirable to have an efficient method which can fast obtain the new desired frequency-domain characteristics. Generally speaking, the frequency-domain characteristics of variable filters are determined by a set of spectral parameters such as cutoff frequency, transition bandwidth and passband width. Therefore, the characteristics of variable filters are the multi-dimensional (M-D) functions of such spectral parameters. This paper proposes an efficient technique which simplifies the difficult problem of designing a 2-D variable filter with quadrantally symmetric magnitude characteristics as the simple one that only needs the normal one-dimensional (1-D) constant digital filter designs and 1-D polynomial approximations. In applying such 2-D variable filters, only varying the part of 1-D polynomials can easily obtain new desired frequency-domain characteristics.     

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     

Three-dimensional rotated digital filters: Design,stability, and applications

Three-dimensional (3-D) signal processing offers many advantages over two-dimensional (2-D) processing, because it preserves 3-D correlations. In this paper the design and the stability of 3-D rotated filters are considered. These filters are designed by rotating a one-dimensional (1-D) digital filter in 3-D space. The rotated filters are valuable in the design of various 3-D filters which possess prescribed spectral specifications. An efficient algorithm for the design of 3-D lowpass (LP) digital filters, with approximately spherically symmetric magnitude responses, is introduced. To achieve the desirable spectral characteristics, a number of 3-D rotated filters is cascaded. The stability of the spherically symmetric filters designed is considered, and stable realizations are proposed. The relation between the cut-off isopotential sphere of the 3-D filter and the cut-off frequency of the 1-D filter employed in the design, is derived. Finally, configurations that result in highpass (HP) and bandpass (BP) filters are proposed. Examples of LP, HP, and BP filters, designed on the basis of the method proposed, are presented.This research was supported by the Public Benefit Foundation, Alexander S. Onassis.     

基于巴特沃斯逼近的二维IIR数字滤波器的设计

本文给出了一种基于巴特沃斯逼近的二维ＩＩＲ数字滤波器的设计方法，得到了由基本的全通节级联，并联实现的各种二维滤波器函数，包括，镜象对称互补滤波器，扇形滤波器和具有任意矩形通、阻带的滤波器，结果表明，这种实现结构具有通有灵敏度低、滤波器系数少的优点，并且由于巴特沃斯逼近的最大平坦性，得到的滤波器具有良好的相位特性。     

Design of 2-D Linear Phase Digital Filters Using Schur Decomposition and Symmetries

In this paper we present a new and numerically efficient technique for designing 2-D linear phase octagonally symmetric digital filters using Schur decomposition method (SDM) and the diagonal symmetry of the 2-D impulse response specifications. This technique is based on two steps. First, the 2-D impulse response matrix is decomposed into a parallel realization of k sections, each comprising two cascaded linear phase SISO 1-D FIR digital filters. It is shown that using the symmetry property of the 2-D impulse response matrix and the fact that the left and right eigenspaces obtained by SDM are transpose of each other, the design problem of two 1-D digital filters is reduced to the design problem of only one 1-D digital filter in each section.     

On the use of continued fraction filter structures for the design of 2-D digital filters

In this paper a method for the realization of 2-D recursive digital filters of second order by means of continued fraction filter structures is presented. The proposed method is based on the interconnection of basic filter structures of first order and gives also the set of all 2-D transfer functions that are realizable by these structures. The general rule for interconnecting these basic structures is described. It is proved that, on the set of second order fraction filter structures, some classes of structures and ordering relations can be defined, so that the set of all the classes form a partially ordered set (poset). Those structures which least constrain the coefficients of the transfer function are defined maximum elements of the poset. It is shown that among all possible continued fraction structures of second order only six are suitable for realization of two dimensional filters. Some results following from the use of spectral transformations are discussed.     

Design of 2-D FIR narrow transition band filters by double transformations using GA approach

In this investigation, subfilters are cascaded in the design of a 2-D narrow transition band FIR digital filter with double transformations, a transformation from wide transition band subfilter into 1-D narrow transition band filter and a McClellan transformation from 1-D filter into 2-D filter. The traditional method for designing a 2-D FIR digital filter with a narrow transition band yields very high orders. The difficulty of the design and implementation will increase with orders exponentially. Numerous identical low-order subfilters are cascaded together to simplify the design of a high-order 2-D filter compared to traditional design method. A powerful genetic algorithm (GA) is presented to determine the best coefficients of the McClellan transformation. It can be used to design any contours of arbitrary shape for mapping 1-D to 2-D FIR filters very effectively. A generalized McClellan transformation is presented, and can be used to design 2-D complex FIR filters. Various numerical design examples are presented to demonstrate the usefulness and effectiveness of the presented approach.

Frequency domain design of FIR and IIR Laplacian of Gaussian filters for edge detection

This work addresses the design of LoG filters in the frequency domain within a structure formed by the cascade of quasi-Gaussian and discrete Laplacian filters. The main feature of such a structure is that it requires half the number of convolutions of the classical structure in which the LoG transfer function is expressed as the sum of two separable transfer functions of 1-D Gaussian and LoG type. Such a perspective allows one to rephrase the design of IIR and FIR filters for edge detection as a frequency domain approximation problem solvable by standard digital filter design tools. The zero-phase IIR solutions have a good performance at low orders and approximation errors practically independent of the aperture parameter. The characteristics of the nearly linear-phase IIR filters solving the problem suggest the consideration of linear-phase FIR filters with zeros constrained on the unit circle. The use of such filters leads to remarkable computational savings with respect to the filters designed by impulse response sampling. The agreement between the edge values obtained by the filters designed according to the scheme proposed in this work and those obtained by standard techniques is very good.Work carried out with the financial support of the C.N.R.-Progetto Finalizzato Robotica, contract no. 91.01942.PF67.     

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.     

Closed-form design and efficient implementation of variable digital filters with simultaneously tunable magnitude and fractional delay

This paper proposes a closed-form solution for designing variable one-dimensional (1-D) finite-impulse-response (FIR) digital filters with simultaneously tunable magnitude and tunable fractional phase-delay responses. First, each coefficient of a variable FIR filter is expressed as a two-dimensional (2-D) polynomial of a pair of parameters called spectral parameters; one is for independently tuning the cutoff frequency of the magnitude response, and the other is for independently tuning fractional phase-delay. Then, the closed-form error function between the desired and actual variable frequency responses is derived without discretizing any design parameters such as the frequency and the two spectral parameters. Finally, the optimal solution for the 2-D polynomial coefficients can be easily determined through minimizing the closed-form error function. We also show that the resulting variable FIR filter can be efficiently implemented by generalizing Farrow structure to our two-parameter case. The generalized Farrow structure requires only a small number of multiplications and additions for obtaining any new frequency characteristic, which is particularly suitable for high-speed tuning.     

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     

The synthesis of two-dimensional passive n-ports containing lumped elements

Two-dimensional (2-D) passive networks are of interest e.g. for use as reference filters for two-dimensional wave digital filters. Necessary properties of the impedance matrix and scattering matrix, respectively, of such n-ports have been established, but not yet been shown to be also sufficient for a given two-variable rational matrix to be the impedance matrix or scattering matrix, respectively, of a passive network containing lumped elements. In the design of 2-D passive n-ports it will be however of great interest whether this mentioned feature can be used as a basis for ageneral synthesis procedure.In this paper it is shown that this is the case. The method presented for the synthesis of 2-D multiports is based mainly on a paraunitary bordering of the given scattering matrix of the desired network in order to obtain the scattering matrix of alossless 2-D multiport, which can be realized by using known procedures. The socalled spectral factorization of a two-variable para-Hermitian polynomial matrix which is nonnegative definite forp =j w plays a crucial role in the design approach presented. No restrictions are made concerning the coefficients of the given rational scattering matrix; they may be either real or complex, so as to include even complex networks which are of special interest for multidimensional systems.     

Some difficulties with the double bilinear transformation in 2-D recursive filter design

It is shown that the double bilinear transformation approach for designing a 2-D IIR digital transfer function from a predetermined 2-D analog transfer function, may in certain cases lead to unstable solutions.     

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     

Direct design of transitional Butterworth-Chebyshev recursive digital filters

A simple method for the approximation of an all-pole transfer function directly in the digital domain for the design of recursive digital filters is described. The transfer function of these filters, referred to as transitional Butterworth-Chebyshev filters, has properties that lie between Butterworth and Chebyshev functions.<>