Explicit expression for the characteristic function of generalized legendre filters

Rakovich, B.D.; Popović, M.

doi:10.1002/cta.4490060406

Cited by 6 publications

(7 citation statements)

References 12 publications

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Section: 'Ylmentioning

confidence: 99%

“…the solution of the optimum problem is determined in terms of the shifted Jacobi polynomials almost by inspection; for example, for the general lowpass filter we have so that the polynomial v k ( y ) must be the kth degree shifted Jacobi polynomial with parameters p and q such that p -q = a + l and q -l = v + u -$ and, taking into account the normalization condition (3), we have the solution in the form (13) Similarly in the case of monotonic lowpass filters if it is required to minimize the area under the derivative The solution is given by (14) with Y = 0 and a = 1, and this new solution differs from that introduced by Rak~viC. '~ in the sense that the error integral is evaluated along w 2 instead of along w ; hereafter we will refer to this new solution as the modified least squares filter.…”

Section: 'Ylmentioning

confidence: 99%

“…'~ in the sense that the error integral is evaluated along w 2 instead of along w ; hereafter we will refer to this new solution as the modified least squares filter. These general formulae (13) and (14) depend on a number of parameters, of which: u (non-negative integer) must be viewed as a constraint and is related to the multiplicity of the zero at the z.' controls the weight function near the origin and may assume negative values, so that it can contrast the a controls the weight function near cut-off and may assume negative values.…”

Section: 'Ylmentioning

confidence: 99%

“…13) with u = 0 or 1 and v = 0 yields all polynomials that somehow have been used in filter synthesis, from Chebyshev (a = -1.5) to Butterworth (a + OO), passing through the Gegenbauer polynomials,' Legendre2 (a = -l), associated LegendreZv4 ( a = 0) and so on. ( a = O), Halpern'.''…”

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confidence: 99%

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The use of the shifted jacobi polynomials in the synthesis of lowpass filters

Beccari

1979

Circuit Theory & Apps

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Section: 'Ylmentioning

confidence: 99%

Section: 'Ylmentioning

confidence: 99%

Section: 'Ylmentioning

confidence: 99%

mentioning

confidence: 99%

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The use of the shifted jacobi polynomials in the synthesis of lowpass filters

Beccari

1979

Circuit Theory & Apps

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“…Some methods for introducing optimisation of the transfer properties in the approximation problem of amplitude, phase and delay are recommended in Rakovich (1974Rakovich ( , 1977, Beccari (1977Beccari ( , 1979, Rakovich and Popovich (1978) and Beccari and Molinaro (1981).…”

Section: Introductionmentioning

confidence: 99%

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

Ilić

Pavlović

2011

International Journal of Electronics

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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 [71, þ1] 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.

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Monotonic, critical monotonic, and nearly monotonic low‐pass filters designed by using the parity relation for Jacobi polynomials

Stojanović

Stamenković²,

Živaljević

2017

Circuit Theory & Apps

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A new class of continuous-time low-pass filter using a set of Jacobi polynomials, with all transmission zeros at infinity, is described. The Jacobi polynomial has been adapted by using the parity relation for Jacobi polynomials in order to be used as a filter approximating function. The resulting class of polynomials is referred to as a pseudo Jacobi polynomials, because they are not orthogonal. The obtained magnitude response of these filters is more general than the magnitude response of the classical ultraspherical filter, because of one additional degree of freedom available in pseudo Jacobi polynomials. This additional parameter may be used to obtain a magnitude response having either smaller passband ripples or sharper cutoff slope. Monotonic, critical monotonic, or nearly monotonic passband filter approximating functions can be also generated. It is shown that proposed pseudo Jacobi polynomial filter approximation also includes the Chebyshev filter of the first kind, the Chebyshev filter of the second kind, the Legendre filter, and many transitional filter approximations, as its special cases. Several examples are presented, and detailed formulas including the practical suggestions for their efficient implementation are also provided. The proposed nearly monotonic filter is compared with the least-square-monotonic filters, designed as critical monotonic, in details. The advantages of the new filters are discussed.

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Explicit expression for the characteristic function of generalized legendre filters

Cited by 6 publications

References 12 publications

The use of the shifted jacobi polynomials in the synthesis of lowpass filters

The use of the shifted jacobi polynomials in the synthesis of lowpass filters

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

Monotonic, critical monotonic, and nearly monotonic low‐pass filters designed by using the parity relation for Jacobi polynomials

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