Nikola Stojanovic, Negovan Stamenkovic

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A case study related to the design the the analog lowpass filter using a set of orthogonal Jacobi polynomials, having four parameters to vary, is considered. The Jacobi polynomial has been modified in order to be used as a filter approximating function. The obtained magnitude response is more general than the response of the classical ultraspherical filter, due to one additional parameter available in orthogonal Jacobi polynomials. This additional parameter may be used to obtain a magnitude response having either smaller passband ripple, smaller group delay variation or sharper cutoff slope. Two methods for transfer function approximation are investigated: the first method is based on the known shifted Jacobi polynomial, and the second method is based on the proposed modification of Jacobi polynomials. The shifted Jacobi polynomials are suitable only for odd degree transfer function. However, the proposed modified Jacobi polynomial filter function is more general. It includes the Chebyshev filter of the first kind, the Chebyshev filter of the second kind, the Legendre filter, Gegenbauer (ultraspherical) filter and many other filters, as its special cases.


Filters, analog circuits, approximation, filter characteristic functions, Jacobi polynomial, orhogonal polznomials

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W. V. Assche and E. Coussement, “Some classical multiple orthogonal polynomials,” Journal of Computational and Applied Mathematics, vol. 127, no. 12, pp. 317 – 347, Jan. 2001,

numerical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials. [Online]. Available:


L. Storch, “Synthesis of constant-time-delay ladder networks using Bessel polynomials,” Proceedings

of the IRE, vol. 42, no. 11, pp. 1666–1675, Nov. 1954.

B. D. Rakovich and V. S. Stojanovich, “On the design of equal ripple delay filters with Chebyshev

stopband attenuation,” Radio and Electronic Engineer, vol. 43, no. 4, pp. 257–265, 1973.

S. Butterworth, “On the theory filter amplifier,” Experimental Wireless and the Radio Engineer, vol. 7,

pp. 536–541, Oct. 1930.

S. C. D. Roy, “Modified chebyshev lowpass filters,” International Journal of Circuit Theory and

Applications, vol. 38, no. 5, pp. 543–549, 2010. [Online]. Available: http://dx.doi.org/10.1002/cta.585

S. Prasad, L. G. Stolarczyk, J. R. Jackson, and E. W. Kang, “Filter synthesis using Legendre polynomials,”

Proceedings of the IEE, vol. 114, no. 8, pp. 1063–1064, Aug. 1967.

M. T. Chryssomallis and J. N. Sahalos, “Filter synthesis using products of Legendre polynomials,”

Electrical Engineering, vol. 81, no. 6, pp. 419–424, 1999.

D. ˇZivaljevi´c, N. Stamenkovi´c, and V. Stojanovi´c, “Nearly monotonic passband low-pass

filter design by using sum-of-squared Legendre polynomials,” International Journal of Circuit

Theory and Applications, vol. 44, no. 1, pp. 147–161, Jan. 2016. [Online]. Available:


Y. H. Ku and M. Drubin, “Network synthesis using Legendre and Hermite polynomials,” J. Franklin

Inst., vol. 273, no. 2, pp. 138–157, Feb. 1962.

I. M. Filanovsky, “Bessel-Butterworth transitional filters,” in 2014 IEEE International Symposium on

Circuits and Systems (ISCAS), June 2014, pp. 2105–2108.

A. Dey, S. Sadhu, and T. K. Ghoshal, “Adaptive Gauss Hermite filter for parameter varying nonlinear

systems,” in 2014 International Conference on Signal Processing and Communications (SPCOM), July

, pp. 1–5.

B. D. Rakovich and V. B. Litovski, “Least-squares monotonic lowpass filters with sharp cutoff,” Electronics

Letters, vol. 9, no. 4, pp. 75–76, Feb. 1973.

A. Budak and P. Aronhime, “Transitional Butterworth-Chebyshev filters,” Circuit Theory, IEEE Transactions

on, vol. 18, no. 3, pp. 413–415, May 1971.

Y. Peless and Murakami, “Analysis and synthesis of tranzitional Butterworth-Thomson filters and

bandpass amplifier,” RCA Rev., vol. 18, no. 3, pp. 60–94, Mar. 1957.

A. Papoulis, “Optimum filters with monotonic response,” Proceedings of the IRE, vol. 46, no. 3, pp.

–609, Mar. 1958.

——, “On monotonic response filters,” Proceedings of the IRE, vol. 47, no. 2, pp. 332–333, Feb. 1959.

M. Fukada, “Optimum filters of even orders with monotonic response,” IRE Transactions on Circuit

Theory, vol. 6, no. 3, pp. 277–281, Sept. 1959.

P. Halpern, “Optimum monotonic low-pass filters,” Circuit Theory, IEEE Transactions on, vol. 16,

no. 2, pp. 240–242, May 1969.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and

Mathematical Tables, 9th ed. New York, Dover: National Bureau of Standards Applied Mathematics

Series 55, 1972.

A. H. Bhrawy, E. H. Doha, S. S. Ezz-Eldien, and R. A. Van Gorder, “A new Jacobi spectral collocation

method for solving 1+1 fractional Schr¨odinger equations and fractional coupled Schr¨odinger systems,”

The European Physical Journal Plus, vol. 129, no. 12, pp. 1–21, 2014. [Online]. Available:


C. Beccari, “The use of the shifted Jacob1 polynomials in the synthesis of lowpass filters,” International

Journal of Circuit Theory and Applications, vol. 7, no. 2, pp. 289–295, 1979.

B. D. Rakovich, “Designing monotonic low-pass filterscomparison of some methods and criteria,”

International Journal of Circuit Theory and Applications, vol. 2, no. 3, pp. 215–221, Sept. 1974.

[Online]. Available: http://dx.doi.org/10.1002/cta.4490020302

D. Topisirovi´c, V. Litovski, and M. Andrejevi´c Stoˇsovi´c, “Unified theory and state-variable implementation

of critical-monotonic all-pole filters,” International Journal of Circuit Theory and Applications,

vol. 43, no. 4, pp. 502–515, Apr. 2015. [Online]. Available: http://dx.doi.org/10.1002/cta.1956

T. V. Hoang and S. Tabbone, “Errata and comments on ”Generic orthogonal moments: Jacobi-Fourier

moments for invariant image description”,” Pattern Recognition, vol. 46, no. 11, pp. 3148 – 3155, Nov.

[Online]. Available: http://www.sciencedirect.com/science/article/pii/S0031320313001817

D. Johnson and J. Johnson, “Low-pass filters using ultraspherical polynomials,” IEEE Transactions on

Circuit Theory, vol. 13, no. 4, pp. 364–369, Dec. 1966.


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