A comprehensive study of two-variable Hurwitz polynomials

As a result of the varied applications of Hurwitz polynomials in the analysis and design of two-dimensional circuits and systems, several different classes of two-variable Hurwitz polynomials have been defined. The authors provide a comprehensive presentation of the definitions of different classes, the interrelationships among them, and their applications. The place and role of the newly defined principal Hurwitz polynomials among the various Hurwitz polynomials are also brought to light. Using a continuity property of the zeros of two-variable polynomials, new proofs for the test procedures of the different Hurwitz polynomials are presented. Applications of the Hurwitz polynomials in the study of the stability of two-dimensional systems and the properties of two-variable passive networks are summarized     

Computer-aided design of stable recursive digital filters from magnitude specifications

This paper presents a method of designing one- and two-dimensional recursive digital filters making use of the properties of bilinear transformation of strictly Hurwitz polynomials. The result is a stable digital filter in both one- and two-dimensional cases, which requires no further testing for stability. The well known unconstrained minimization technique of Fletcher—Powell is used, making use of a transformation of optimization parameters to satisfy the mild constraints of stability. This method is considered to be more efficient on an overall basis than the existing l_p design technique of Maria and Fahmy. Nor does it rely on Deczky's theorem to ensure a stable filter which may sometimes lead to unstable solutions in the two-dimensional case. An example is provided illustrating the method.     

Synthesis of some non-reciprocal filters employing the Jacobi polynomials

In this paper, we establish that some of the generalized linear phase polynomials defined in Reference 1 are Hurwitz. The demonstration consists in showing that the ratio of two linear phase polynomials is a positive real function. The corresponding network interpretation with closed-form element values is then given, which yields at the same time the reflection-filter synthesis of the non-minimum phase, non-reciprocal transfer functions introduced in Reference 1.     

Direct synthesis techniques for general Chebyshev filters: lowpass,highpass, and bandstop cases

Conventional synthesis techniques for general Chebyshev lowpass, highpass, and bandstop filters are usually based on general Chebyshev filtering function, from which lowpass prototype is set up. By applying the lowpass to lowpass, highpass, or bandstop frequency transformation on realization networks of lowpass prototypes, final realization networks of general Chebyshev lowpass, highpass, and bandstop filters are obtained. In this paper, direct synthesis techniques for general Chebyshev lowpass, highpass, and bandstop filters are discussed. Transmission zeros can be placed wherever they are desired to control the performance of the filters. Unlike conventional synthesis techniques, they derive filtering polynomials constituting network parameters directly in the lowpass, highpass, or bandstop domain, which might find applications in analogue and digital filter design. Several examples are presented for demonstration. Copyright © 2015 John Wiley & Sons, Ltd.     

Pseudopassive two-dimensional recursive digital filters for image processing

This paper describes the theory of two-dimensional digital filters that are pseudopassive with respect to the l_p-norm of the state vector. As the classical pseudopassive digital filters are a subclass of these filters, the respective theorems referred to stability are also generalized. It is shown that this theory is useful for the two-dimensional filters that answer with non-negative-valued responses to non-negative-valued excitations. Such systems are especially suitable for image processing. the synthesis of the l₁-pseudolossless systems is proposed as a tool to guarantee stability of such filters. A technique to obtain local state-space models for such two-dimensional l₁-pseudolossless recursive filters with prescribed spatial responses is given. A ‘Gaussian filter’ design illustrates the technique and shows that the proposed two-dimensional l₁-pseudolossless filters are able to match useful spatial responses.     

A method of design for a class of 2-D filters

This paper discusses a method of designing quadrantally symmetric cascaded two-dimensional (2-D) digital recursive filters by subjecting a one variable approximating function to successive transformations. the needed approximation is done in the one variable domain rather than in the 2-D domain, hence leading to a large reduction of computational labour. Using cepstral techniques each denominator polynomial is spectrally factorized into recursible non-symmetric half plane components. A significant feature of the method is in decoupling the problems of approximation and stability. Consequently spectral factorization needs to be performed only once for each denominator polynomial. Effects of truncation on filter stability are minimized by ensuring rapid convergence of cepstra. the choice of an adequate DFT size in cepstral computations is shown to be an important consideration for many problems associated with spectral decomposition. Attempts are also made to stabilize the unstable transfer function using an existing 2-D discrete Hilbert transform method. Considerable distortion in magnitude characteristics is shown to result on stabilization. Finally the method is illustrated by two examples.     

On the design of 2D IIR digital filters using penalty optimization techniques

In this paper an efficient technique is developed for the design of two-dimensional recursive digital filters in the frequency domain. the formulation of the design problem is based upon a rational approximation method in parallel with a suitable interior penalty technique. the structure of the proposed approach permits one to include in the design optimization problem additional constraints related to the stability, linear phase and symmetries properties of the resultant filter. In contrast with previous design methods, the coefficients of the filters are determined directly and it is not required to apply any ‘inverse’ or stabilization technique. Analytical examples illustrate the features and effectiveness of the proposed approach.     

Realization of a general all‐pole current transfer function by using CBTA

In this article, a general all‐pole current transfer function synthesis procedure using current backward transconductance amplifiers (CBTAs) is proposed. The proposed configuration uses n current backward transconductance amplifiers and n grounded capacitors as the only type of passive elements. The circuit is eligible to realize any all‐pole transfer characteristics with a given strictly Hurwitz (stable) denominator polynomial. Further, it is straightforward to find the values of the passive elements from the coefficients of this polynomial by using the Routh–Hurwitz algorithm as in the realization of a two‐element kind passive network synthesis. In this sense and as far as the author's knowledge, it is the only active structure that can be synthesized like a passive two‐element kind Cauer circuit. The simulations that are performed using PSPICE exhibit satisfactory results coherent with the theory. Copyright © 2011 John Wiley & Sons, Ltd.     

The modified Pascal polynomial approximation and filter design method

Polynomial approximations are extensively used in analog and IIR digital filter design. In this paper, a comprehensive filter design and an optimization procedure are presented explicitly using a filter‐appropriate modified Pascal polynomial. The so‐designed all‐pole Pascal filters exhibit non‐equiripple passband and monotonic transition and stopband responses. The order of the new Pascal filters is calculated from the order inequality which, although it cannot be analytically solved, leads to a nomograph that has been created and is presented here. Inevitably, the mathematical complexity introduced by the nature of the Pascal polynomials makes the analytical expression of the poles of the transfer function unfeasible and for that reason poles are given by means of appropriate tables. The design method is demonstrated in several detailed examples and Pascal filters are compared with their all‐pole counterparts, Butterworth and Chebyshev, over which they reveal certain advantages and disadvantages. Copyright © 2010 John Wiley & Sons, Ltd.     

Robust simultaneous amplitude and phase approximations oriented to bandpass wave digital lattice filters

A straightforward design method is delivered for bandpass wave digital lattice filters satisfying arbitrary amplitude and linear phase specifications. The optimality and efficiency of this method are ensured by two sources. On the one hand, the approximation is carried out directly without the need to apply frequency transformation techniques. On the other, the amplitude and phase specifications are approximated simultaneously. The approximation process is relying on the pre-construction of one of the two branch all-pass functions to exhibit exact linear phase. The other all-pass function is determined such that it controls the amplitude and/or phase responses in the passband and the two stopbands. Accordingly, the approximation problem is reduced to constructing a strictly Hurwitz polynomial specified by its phase within the Nyquist range. The approximation problem is solved by applying interpolation techniques combined with the iterative Remez-exchange algorithm. The realization of the resulting filter is carried out according to explicit formulae. The delivered method is applied through two examples illustrating its efficiency and reliability. © 1998 John Wiley & Sons, Ltd.     

Design of digital filters with maximally flat passband and equiripple stopband magnitude

This paper introduces a general class of digital filters with maximally flat passband magnitude, equiripple stopband magnitude, and different order numerator and denominator. the classical Chebyschev type II (inverse Chebyschev) filters having equal numerator and denominator orders are considered as special members of this filter class. the filter types to be considered are lowpass, highpass and bandpass. the proposed filter class consists both of filters having all the zeros on the unit circle and of filters having some zeros off the unit circle to contribute to the passband shaping. An efficient iterative algorithm for the design of filters of the given type is presented. It is based on shaping the passband and stopband responses alternately until the difference between successive solutions is within given tolerance limits. Several comparisons show that in wideband applications filters with numerator order higher than denominator order present considerable advantages over equivalent Chebyschev type II designs, in terms of multiplication rate and/or frequency selectivity. In narrowband applications, in turn, filters with higher denominator order are shown to be the most effective.     

Generalized linear phase polynomials—application in filter synthesis

In this paper, we give explicit expressions for the study of attenuation and phase characteristics of generalized linear phase polynomials (i.e. Jacobi and generalized Bessel polynomials for transmission line and lumped filters respectively). We present an exact method to find the digital transfer function which exhibits [n/2] to (n?1) simultaneous conditions on amplitude and delay.     

On discrete scattering Hurwitz polynomials

Discrete scattering Hurwitz polynomials play for discrete systems a similar role as scattering Hurwitz polynomials do for classical systems. A number of properties of discrete scattering Hurwitz polynomials are presented. These properties are obtained by deriving them from the corresponding properties of scattering Hurwitz polynomials. This is achieved by considering polynomials associated in a proper way to given polynomials and by examining the nature of the corresponding transformations.     

On the problem of filters with arbitrary amplitude and phase constraints

A simplified method—based on the concept of reflection filters—is presented for designing linear phase filters with arbitrary amplitude to phase constraints. This method starts with transfer functions with 2: 1 amplitude to phase constraints, and through varying the recurrence relation of the polynomials involved, transfer functions with more selective amplitude are obtained.     

Application of direct synthesis techniques to customize filters with complex frequency response

Recently, direct synthesis techniques (DSTs) have been presented for filter synthesis. Unlike conventional synthesis techniques, DSTs derive the filtering polynomials of the filters to be synthesized directly in their own frequency domain. These filtering polynomials are real coefficient so that they might find applications in various fields. Furthermore, DSTs might be used to customize filters with a more complex frequency response, such as asymmetric frequency response or multi‐band frequency response. In this paper, DSTs are compared with some well‐known filter synthesis techniques. Then, the application of DSTs in the design of lumped‐element LC filters, distributed‐element filters, active RC filters, and infinite impulse response digital filters with complex frequency response is discussed. Some examples are presented for demonstration. Copyright © 2015 John Wiley & Sons, Ltd.     

Direct synthesis technique (DST) for complex general Chebyshev filters

Recently, we discussed the concept of direct synthesis technique (DST), in which real‐coefficient filtering polynomials containing all information of the filters to be synthesized are derived directly for realization, and they could find applications in the design of lumped‐element LC filters, active RC filters, and infinite impulse response digital filters. In this paper, another DST for complex general Chebyshev bandpass filters is discussed, which is based on a complex mapping relation and featured by complex‐coefficient filtering polynomials. It is called as complex DST in this paper. Compared with real‐coefficient filtering polynomials whose polarities are determined by the number of their zeros at zero frequency, the polarities of complex‐coefficient filtering polynomials can be easily changed by multiplying imaginary unit j . Such advantage might make their realization more flexible. The analysis shows that conventional coupling matrix could be considered as narrow‐band approximation of network matrix derived by complex DST in the normalized frequency domain. In order to demonstrate the validity of complex DST in this paper, it is applied in the design of classic parallel‐coupled microstrip bandpass filters. Compared with conventional synthesis techniques, complex DST could find out better dimensions and provide more choices for realization and synthesize both even‐order and odd‐order parallel‐coupled microstrip bandpass filters. Copyright © 2017 John Wiley & Sons, Ltd.     

Polynomial Root Determination by an Equivalent Transfer Function Simulation on an Iterative Analog Computer

The methods for determining the roots of polynomials by analog computer techniques are varied, but stability analysis is usually of chief concern. The computer approach used in this paper always produces stable operation, and introduces an equivalent transfer function representation of the polynomial, a linear transformation, and a generalized iterative analog computer program. The accuracy obtainable is not comparable to that of digital programs; however, the procedure is basic to engineering analysis and should not be overlooked from the academic viewpoint.     

IIR低通滤波器在谐波监测中的应用

以IIR数字滤波器的基本理论为依据，结合滤波器的传递函数分子、分母系数固定这一事实，通过选用高密度可编程逻辑器件确定了IIR数字滤波器的硬件实现方案，并按照层次化、模块化、参数化的设计思路，采用VHDL硬件描述语言和原理图两种设计技术进行了IIR滤波器的硬件设计；对设计的低通滤波器进行了系数量化并对其影响进行了分柝最后进行了实际滤波效果测试，验证了设计的正确性。     

Frequency domain identification for undergraduates

A method to synthesize a transfer function from experimentally obtained gain and phase data is presented. The least squares technique discussed minimizes the error between the inverse transfer function and the inverse of the experimental frequency response data. The relevant formulas are derived in a straightforward manner so that undergraduate students can follow the development. The inverse formulation appears to give a better fit to the data than the previous approaches. This is most likely due to the fact that the technique biases the error with the numerator rather than the denominator of the derived transfer function. There is, however, no guarantee that a minimum phase transfer function will result from this technique. The user of this technique must preselect the numerator and denominator orders. The formulation assumes that there are no zeros at the origin. A modification to the basic scheme is presented for this case     

On a generalization of the wave digital filter concept

The design of digital filter structures imitating the behaviour of classical analogue networks has received considerable interest in the literature. Of particular interest has been the Wave Digital Filter first considered by Fettweis. We examine here a generalization in which the relationships between the voltages, currents and wave variables are considered as a linear transformation on the ABCD matrix of the analogue two-port network. The linear transformation is examined in some detail and conditions are derived which impose constraints on the elements of the matrices involved in the transformation. Finally a table giving thirteen transformations known to yield realizable digital filter structures is presented.