Vlastimir D. Pavlović

New class of filter functions generated most directly by the Christoffel–Darboux formula for classical orthonormal Jacobi polynomials

The new originally capital general solution of determining the prototype filter function as the response that satisfies the specifications of all pole low-pass continual time filter functions of odd and even order is presented in this article. In this article, two new classes of filter functions are proposed using orthogonal and orthonormal Jacobi polynomials. The approximation problem of filter function was solved mathematically, most directly applying the summed Christoffel–Darboux formula for the orthogonal polynomials. The starting point in solving the approximation problem is a direct application of the Christoffel–Darboux formula for the initial set of continual Jacobi orthogonal polynomials in the finite interval in full respect to the weighting function with two free real parameters. General solution of the filter functions is obtained in a compact explicit form, which is shown to enable generation the Jacobi filter functions in a simple way by choosing the numerical values of the free real parameters. For particular specifications of free parameters, the proposed solution is used with the same criterion of approximation to generate the appropriate particular filter functions as are: the Gegenbauer, Legendre and Chebyshev filter functions of the first and second kind as well. The examples of proposed filter functions of even and odd order are illustrated and compared with classical solutions.

New class of filter functions generated most directly by Christoffel-Darboux formula for Gegenbauer orthogonal polynomials

A new original formulation of all pole low-pass filter functions is proposed in this article. The starting point in solving the approximation problem is a direct application of the Christoffel-Darboux formula for the set of orthogonal polynomials, including Gegenbauer orthogonal polynomials in the finite interval [−1, +1] with the application of a weighting function with a single free parameter. A general solution for the filter functions is obtained in a compact explicit form, which is shown to enable generation of the Gegenbauer filter functions in a simple way by choosing the value of the free parameter. Moreover, the proposed solution with the same criterion of approximation could be used to generate Legendre and Chebyshev filter functions of the first and second kind as well. The examples of proposed filter functions of even (10th) and odd (11th) order are illustrated. The approximation is shown to yield a good compromise solution with respect to the filter frequency characteristics (magnitude as well as phase characteristics). The influence of tolerance of the filter critical component (inductor) on the proposed magnitude and group delay characteristics of a resistively terminated LC lossless ladder filter is analysed as well. The proposed filter functions are superior in terms of the excellent magnitude characteristic, which approximates an ideal filter almost perfectly over the entire pass-band range and exhibits the summed sensitivity function better than that of a Butterworth filter. In the article, we present the filter function solution that exhibits optimum amplitude as well as optimum group delay characteristics that are of crucial importance for implementation of digital processing as well as RF analogue parts of communication networks. Derivation of the other band range filter functions, which could be realised either by continuous or digital filters, is also generally possible with the procedure proposed in this article.

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.

A New Class of Low Complexity Low-Pass Multiplierless Linear-Phase Special CIC FIR Filters

This letter presents a new class of selective low-pass multiplierless linear-phase special CIC FIR filter functions of low complexity given in an explicit cascaded compact form. Several examples of the proposed filters are illustrated. They are compared in a fair way with conventional CIC FIR filters for the equal values of group delay and the same number of cascade sections. An example of the new class of the proposed seventh order filter with nine cascades is presented, where its stopband attenuation has high value of 143.82 dB, while corresponding conventional CIC FIR filter with the same filter order and the number of cascades has only 115.18 dB.

An explicit form of all-pole filter function with decreasing envelope of the summed sensitivity function

In this paper, the all-pole lowpass filter function with the decreasing envelope of the summed sensitivity in the pass-band is considered. The filter transfer function with maximal number of the ripples of the summed sensitivity in the pass-band is obtained in the explicit form, by the application of the Chebyshev polynomials of the first kind. The slope of the decrease of the summed sensitivity envelope can be controlled by a free-parameter α. We derived a new approximation function in order to achieve small summed sensitivity in the filter pass-band. Consequently, the sensitivity analysis was carried out and a comparison of the summed sensitivity and the group delay with respect to the classic all-pole filters was given. The approach presented in this paper is based on that the minimization of the summed sensitivity is important for the reduction of the deviation of the magnitude response caused by temperature changes of the continuous-time active filters implemented into the analog front end or as programmable chips. Copyright

An Analytical Method for the Multiplierless 2-D FIR Filter Functions and Hilbert Transform in

A novel analytical method for the new class of linear-phase multiplierless 2-D finite-impulse response (FIR) digital filter functions generated by applying the Christoffel-Darboux formula for classical Chebyshev polynomials of the first and the second kind, in a compact explicit representative form, is proposed in this brief. Correct transformation from a continuous 2-D domain into the z domains without residuum and without errors is described. The proposed solution with high selectivity is a filter function in the z1 domain and the Hilbert transformer in the z2 domain. A corresponding example of the new class of linear-phase highly selective 2-D FIR digital filters of low complexity, which is obtained by the proposed approximation technique, is shown in this brief.

z2

In this paper we consider the lowpass (all-pole) filters which have non-monotonical amplitude characteristic in the passband. The mean-square approximation with Chebyshev weights, with the application of two Lagranges multiplicators for the values of the transfer function at the cut-off frequency and at ω→0. gives a new class of filters which is convenient considering selectivity and the amplitude characteristic slope in the passband. Choosing the value of the parameter α we define the value of the transfer function at ω→0, while the parameter β defines the ratio between the dominating attenuation maximum in the passband and the attenuation at ω→0.

Domain

ABSTRACT A new CIC filter architecture, i.e. CIC (cascaded-integrator-comb) FIR (finite impulse response) filter functions, designed by introducing spreading of the delays in the comb stages is presented here. The design forms of modified CIC filters that approximate simultaneously both magnitude and group delay specifications are given. The benefit of the novel filter class is demonstrated by a few example filter functions and their comparisons with existing classical CIC structures and one technique given in the literature. The effect of the compensation filter on the frequency response (reduction of the passband droop) is also shown. The novel cascaded-filter architecture has valuable benefits: wider passband range, higher insertion loss in stopband, as well as smaller values of integer coefficients of the impulse response.

Least-square low-pass filters using Chebyshev polynomials

In this article, the approximation problem of a continuous analogue filter function of even and odd order is solved mathematically most directly applying the proposed generalised Christoffel–Darboux formula for two continuous orthogonal polynomials (Chebyshev first and second kind) on the equal finite segment of orthogonality in a compact explicit representative form. A set of analytic expressions of the proposed formula for the representative examples of odd-orders is given. Additionally, these expressions are applied in generating new excellent class all pole low-pass prototype analogue filter functions of odd orders (7, 9, 11, and 13th order). Based on the generated functions, the fully symmetric doubly resistively terminated LC ladder filter networks are realised by Darlington realisation. The generated filters are analysed, and their characteristics are presented. The effects of finite tolerance of a critical reactance of these ladder filters on the filter characteristics are considered. The filters generated by the proposed formula show excellent properties and possess important advantages in comparison to the corresponding filters generated by other methods.

New CIC Filter Architecture: Design, Parametric Analysis and Some Comparisons

In this article, the all-pole low-pass filter function with mini-max for the summed sensitivity function in the pass-band is considered. With the application of Chebyshev polynomials of the first kind, the proposed filter function is obtained in an explicit form with a maximum number of oscillations of the summed sensitivity function in the pass-band. The calculation of the filter function is derived by using the summed sensitivity function as a starting point. New original approximation function is derived in order to achieve a mini-max summed sensitivity function in the filter pass-band. Sensitivity analysis is carried out and a comparison of the summed sensitivity and the group delay of the proposed and classical all-pole filters is given. Minimisation of the summed sensitivity function is important for reduction of the deviation of the magnitude response caused by temperature changes of the continuous-time active filters implemented into the analogue front end or as programmable chips.