FACTA UNIVERSITATIS Series: Electronics and Energetics Vol. 30, No 3, September 2017, pp. 351 - 362 DOI: 10.2298/FUEE1703351S Nikola Stojanović1, Negovan Stamenković2 Received June 14, 2016; received in revised form November 18, 2016 Corresponding author: Nikola Stojanović University of Niš, Faculty of Electronic Engineering, Aleksandra Medvedeva 14, 18000 Niš, Serbia (E-mail: nikola.stojanovic@elfak.ni.ac.rs) FACTA UNIVERSITATIS Series: Electronics and Energetics Vol. 28, No 4, December 2015, pp. 507 - 525 DOI: 10.2298/FUEE1504507S hoRIZoNTAL CURRENT BIpoLAR TRANSISToR (hCBT) – A Low-CoST, hIGh-pERFoRmANCE FLExIBLE BICmoS TEChNoLoGy FoR RF CommUNICATIoN AppLICATIoNS Tomislav Suligoj1, marko Koričić1, Josip Žilak1, hidenori mochizuki2, So-ichi morita2, Katsumi Shinomura2, hisaya Imai2 1University of Zagreb, Faculty of Electrical Engineering and Computing, Department of Electronics, Micro- and Nano-electronics Laboratory, Croatia 2Asahi Kasei Microdevices Co. 5-4960, Nobeoka, Miyazaki, 882-0031, Japan Abstract. In an overview of Horizontal Current Bipolar Transistor (HCBT) technology, the state-of-the-art integrated silicon bipolar transistors are described which exhibit fT and fmax of 51 GHz and 61 GHz and fTBVCEO product of 173 GHzV that are among the highest-performance implanted-base, silicon bipolar transistors. HBCT is integrated with CMOS in a considerably lower-cost fabrication sequence as compared to standard vertical-current bipolar transistors with only 2 or 3 additional masks and fewer process steps. Due to its specific structure, the charge sharing effect can be employed to increase BVCEO without sacrificing fT and fmax. Moreover, the electric field can be engineered just by manipulating the lithography masks achieving the high-voltage HCBTs with breakdowns up to 36 V integrated in the same process flow with high-speed devices, i.e. at zero additional costs. Double-balanced active mixer circuit is designed and fabricated in HCBT technology. The maximum IIP3 of 17.7 dBm at mixer current of 9.2 mA and conversion gain of -5 dB are achieved. Key words: BiCMOS technology, Bipolar transistors, Horizontal Current Bipolar Transistor, Radio frequency integrated circuits, Mixer, High-voltage bipolar transistors. 1. INTRODUCTION In the highly competitive wireless communication markets, the RF circuits and systems are fabricated in the technologies that are very cost-sensitive. In order to minimize the fabrication costs, the sub-10 GHz applications can be processed by using the high-volume silicon technologies. It has been identified that the optimum solution might Received March 9, 2015 Corresponding author: Tomislav Suligoj University of Zagreb, Faculty of Electrical Engineering and Computing, Department of Electronics, Micro- and Nano-electronics Laboratory, Croatia (e-mail: tom@zemris.fer.hr) LowpASS FILTERS AppRoxImATIoN BASEd oN ThE JACoBI poLyNomIALS 1University of Niš, Faculty of Electronic Engineering, Serbia 2University of Priština, Faculty of Natural Science and Mathematics, Serbia Abstract. A case study related to the design the the analog lowpass filter using a set of or- thogonal 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 ultra- spherical filter, due to one additional parameter available in orthogonal Jacobi polyno- mials. This additional parameter may be used to obtain a magnitude response having ei- ther 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 but not orthogonal. It is transformed into orthogonal polynomial when orders are equal and then includes the Chebyshev filter of the first kind, the Cheby- shev filter of the second kind, the Legendre filter, Gegenbauer (ultraspherical) filter and many other filters, as its special cases. Key words: Filters, analog circuits, approximation, filter characteristic function, Jacobi polynomial, orthogonal polynomials. FACTA UNIVERSITATIS (NIŠ) SER.: ELEC. ENERG. vol. 30, no. 1, February 2017, xx-xx Lowpass Filters Approximation Based on the Jacobi polynomials Nikola Stojanović1 and Negovan Stamenković2 1University of Niš, Faculty of electronic engineering, Serbia 2University of Priština, Faculty of Natural Science and mathematics, Serbia Abstract: 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 mag- nitude 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 but not orthogonal. It is transformed into orthogonal polynomial when orders are equal and then 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. Keywords: Filters; analog circuits; approximation; filter characteristic function; Jacobi polynomial; orthogonal polynomials. 1 Introduction The very classical orthogonal polynomials Jacobi, Laguerre and Hermite [1] and their special cases i.e Gegenbauer, Chebyshev and Legendre are widely used in communication theory and particu- larly in the synthesis transfer function of electric filters. the coefficients of the Bessel-thomson filters, which provide maximally flatness of the group delay response in the passband without any ripple, are related to the Bessel polynomials [2]. However, the Bessel type polynomials are not orthogonal on an interval of the x-axis, but in certain cases are orthogonal on a unit circle. Manuscript received on June 9, 2016. Corresponding author: Nikola Stojanović, University of Niš, Faculty of electronic Engineering, A. Medvedeva 14, 1800 Niš, Serbia (e-mail: nikola.stojanovic@elfak.ni.ac.rs). 1 352 N. StojaNoviĆ, N. StameNkoviĆ Lowpass Filters approximation Based on the jacobi Polynomials 353 2 N. Stojanović and N. Stamenković: Apart from Chebyshev polynomials, which are of utmost importance in the synthesis of filters exhibiting a sharp increase in attenuation as the frequency increases above corner frequency, other classes of above mentioned orthogonal polynomials have found many useful applications in the synthesis of electrical filters. in particular, the approximation problem in the synthesis of electrical filters consists of finding a physical realizable function of frequency that shall meet a prescribed set of specifications with regard to its magnitude and/or group delay characteristics. it is known that, for a given filter degree, there is always a trade-off between the magnitude and group delay characteristics. By considering the whole frequency band, the better group delay characteristic is generally associated with the better time domain characteristic [3]. The better time domain characteristic leads to smaller time delay or smaller values of the overshoot in the step response. There are approximations that have a very good magnitude characteristic in detriment of their group delay characteristic, as for example, Butterworth [4], Chebyshev [5], [6], Bernstein [7], Legendre [8] [9] [10] and their derivatives by Ku and Drubin [11]. Converse case occurs with other approximations, as for example, Bessel [12], Gauss [13], Hermite [11] and least-squares monotonic [14] [15], all those filters present optimized characteristics in specific points. transitional filters are alternative filter solutions that perform a trade-off between the magnitude and group delay characteristics. transitional Butterworth-Chebyshev [16] filters are considered with magnitude characteristics that vary gradually from those of the Butterworth filter to those of the Chebyshev filter as a number of pass-band ripples (or the degree of flatness at the origin) is varied. Three degrees of freedom are available for transitional Butterworth-Chebyshev filters: the degree n, the ripple factor ε and the degree of flatness at the origin. the smooth transition is accomplished using the method proposed of Peless and Murakami [17] by finding each pole of the transitional Butterworth-thompson filter as an interpolation between a pole of the Butterworth filter and a corresponding pole of the thompson filter. a special class of filter functions of odd order providing monotonic magnitude characteristic of the resulting filter has first been investigated by Papoulis [18] by means of Legendre polynomials. Subsequently these results have been extended so as to include filters of even degree [19], [20], and also some other functions leading to the same class of filtering networks whose magnitude response is bounded to be monotonic have been derived using a different approach based on the applications of Jacobi polynomials [21]. In this paper, the concept of magnitude response synthesis techniques is extended for orthogo- nal jacoby lowpass filters. Simple modification of orthogonal Jacobi polynomial, suitable for the continuous-time lowpass filter design, is proposed in this paper. if the degree of the filter is given, both indexes (order) of the Jacobi polynomial can be used for smoothly adjusting the filter perfor- mance. The magnitude response obtained is more general than the continuous-time response of the Chebyshev filter because of two additional parameters available with the modified jacobi polyno- mials. It should be noted, the proposed Jacobi approximation covers many of the above-mentioned all-pole filter functions. 352 N. StojaNoviĆ, N. StameNkoviĆ Lowpass Filters approximation Based on the jacobi Polynomials 353 Lowpass filters approximatin based on the jacobi polynomials 3 2 Filter magnitude function in lowpass filter design, assuming all the zeros of the system function are at infinity, the squared magnitude function (insertion loss) can be written as |Hn( jω)|2 = 1 1 + ε 2φ 2n (ω) (1) where ω is frequency variable, ε is a parameter that controls the passband attenuation tolerance, n denotes the degree of the filter and the polynomial φn(ω) is the characteristic (or approximating) function of the filter which is to be selected to give desired magnitude characteristic. The character- istic function is normalized to unity at the pass-band edge frequency ωp, which is also normalized to ωp = 1, then can be written as φn(1) = 1. this conventional procedure for filter design using the insertion loss method includes the de- sign of a lumped element LC ladder lowpass filter known as the lowpass prototype. A more modern procedure uses this network synthesis technique to design filters with a completely specified fre- quency response. the design is simplified by beginning with low-pass filter prototypes that are normalized in terms of impedance and frequency. Transformations are then applied to convert the prototype designs to the desired frequency range and impedance level. in filter design, the characteristic frequency use for frequency normalization is the cutoff fre- quency known as the filter passband corner frequency, and therefore normalized cutoff frequency is equal to 1. For this application, the function φ 2n (x) is required to be an even polynomial ψn(ω 2) = φ 2n (x). If φn(x) is even or odd, then φ 2 n (x) is always even, as is required. Polynomi- als φn(x), which are neither even nor odd, may be also be used in magnitude functions if φn(x) is replaced by φn(x2). Therefore it is necessary that no terms of the form x2k+1 appear in the characteristic function. The Jacobi polynomials P(α,β )n (x) have n distinct zeros for α �= β but they are neither even nor odd. Such type of polynomials are not suitable to be a filter characteristic function. However, Jacobi orthogonal polynomials can be adapted for use in the low-pass filter magnitude functions, as will be shown in the next section. 3 Jacobi polynomial The Jacobi polynomials [22], denoted by P(α,β )n (x) of the degree n, are orthogonal on the interval [−1,1] with respect to the Jacobi weight function w(α,β ) = (1 − x)α (1 + x)β when α,β ≥ −1. We shall refer to α and β as the orders of the Jacobi polynomial. Namely, ∫ 1 −1 P(α,β )m (x)P (α,β ) n (x)w(α,β )(x)dx = h (α,β ) n δn.m, (2)4 N. Stojanović and N. Stamenković: where h(α,β )n = 2α+β +1 2n + α + β + 1 Γ(n + α + 1)Γ(n + β + 1) Γ(n + 1)Γ(n + α + β + 1) , (3) δn.m is Kronecker delta symbol and Γ(·) is well known Gamma function. The Jacobi polynomials are generated by the three-term recurrence relation: P(α,β )0 (x) = 1, P(α,β )1 (x) = 1 2 (α + β + 2)x + 1 2 (α − β ), P(α,β )n+1 (x) = (a (α,β ) n x − b (α,β ) n )P (α,β ) n (x)− c (α,β ) n P (α,β ) n−1 (x), n ≥ 1 (4) where a(α,β )n = (2n + α + β + 1)(2n + α + β + 2) 2(n + 1)(n + α + β + 1) b(α,β )n = (β 2 − α 2)(2n + α + β + 1) 2(n + 1)(n + α + β + 1)1)(2n + α + β ) c(α,β )n = (n + α)(n + β )(2n + α + β + 2) (n + 1)(n + α + β + 1)1)(2n + α + β ) Matlab is an inexpensive an easi-to-use software package and widely available comercial prod- uct that is in widespread in both academia and industry [23]. A Matlab script for evaluating Jacobi polynomials using the above procedure is given in JacobiPoly.m. In addition to Jacobi poly- nomial, proposed Matlab program also evaluates Gegenbauer and Legendre polynomials. JacobyPoly.m function P=JacobiPoly(n,a,b) % Coefficients P of the Jacobi polynomial % They are stored in decending order of powers if nargin == 1, a=0; b=0; elseif nargin == 2, b=a; end p0 = 1; p1 = [(a+b)/2+1,(a-b)/2]; if n == 0, P=p0; elseif n == 1, P=p1; else for k=2:n, d=2*k*(k+a+b)*(2*k-2+a+b); A=(2*k+a+b-1)*(2*k+a+b-2)*(2*k+a+b)/d; B=(2*k+a+b-1)*(aˆ2-bˆ2)/d; 354 N. StojaNoviĆ, N. StameNkoviĆ Lowpass Filters approximation Based on the jacobi Polynomials 355 4 N. Stojanović and N. Stamenković: where h(α,β )n = 2α+β +1 2n + α + β + 1 Γ(n + α + 1)Γ(n + β + 1) Γ(n + 1)Γ(n + α + β + 1) , (3) δn.m is Kronecker delta symbol and Γ(·) is well known Gamma function. The Jacobi polynomials are generated by the three-term recurrence relation: P(α,β )0 (x) = 1, P(α,β )1 (x) = 1 2 (α + β + 2)x + 1 2 (α − β ), P(α,β )n+1 (x) = (a (α,β ) n x − b (α,β ) n )P (α,β ) n (x)− c (α,β ) n P (α,β ) n−1 (x), n ≥ 1 (4) where a(α,β )n = (2n + α + β + 1)(2n + α + β + 2) 2(n + 1)(n + α + β + 1) b(α,β )n = (β 2 − α 2)(2n + α + β + 1) 2(n + 1)(n + α + β + 1)1)(2n + α + β ) c(α,β )n = (n + α)(n + β )(2n + α + β + 2) (n + 1)(n + α + β + 1)1)(2n + α + β ) Matlab is an inexpensive an easi-to-use software package and widely available comercial prod- uct that is in widespread in both academia and industry [23]. A Matlab script for evaluating Jacobi polynomials using the above procedure is given in JacobiPoly.m. In addition to Jacobi poly- nomial, proposed Matlab program also evaluates Gegenbauer and Legendre polynomials. JacobyPoly.m function P=JacobiPoly(n,a,b) % Coefficients P of the Jacobi polynomial % They are stored in decending order of powers if nargin == 1, a=0; b=0; elseif nargin == 2, b=a; end p0 = 1; p1 = [(a+b)/2+1,(a-b)/2]; if n == 0, P=p0; elseif n == 1, P=p1; else for k=2:n, d=2*k*(k+a+b)*(2*k-2+a+b); A=(2*k+a+b-1)*(2*k+a+b-2)*(2*k+a+b)/d; B=(2*k+a+b-1)*(aˆ2-bˆ2)/d; Lowpass filters approximatin based on the jacobi polynomials 5 C=2*(k-1+a)*(k-1+b)*(2*k+a+b)/d; P=conv([A B],p1)-C*[0,0,p0]; p0 = p1; p1 = P; end end end Some properties of the Jacobi polynomials, which are needed here, are as follows P(α,β )n (1) = Γ(n + α + 1) Γ(n + 1)Γ(α + 1) (5) and P(α,β )n (−1) = (−1)nΓ(n + β + 1) Γ(n + 1)Γ(β + 1) (6) Jacobi polynomials have symmetry P(α,β )n (x) = (−1)nP (β ,α) n (x) (7) The following important derivative relation is d dx P(α,β )n (x) = 1 2 (n + α + β + 1)P(α+1,β +1)n−1 (x) (8) 3.1 Shifted Jacobi polynomials In order to use Jacobi polynomials on the interval x ∈ [0,1] we define the so-called shifted jacobi polynomials by introducing the change of variable x �→ 2x − 1. Let the shifted Jacobi polynomials P(α,β )n (2x−1) be denoted by J (α,β ) n (x). The shifted Jacobi polynomials are orthogonal with respect to the weight function w(α,β )s = (1 − x)α xβ in the interval [0,1] with the orthogonality property: ∫ 1 0 w(α,β )s J (α,β ) m (x)J (α,β ) n (x)dx = 1 2n + α + β + 1 Γ(n + α + 1)Γ(n + β + 1) Γ(n + 1)Γ(n + α + β + 1) δn,m (9) The shifted Jacobi polynomials are generated from the three-term recurrence relations [24]: J (α,β ) 0 (x) = 1, J (α,β ) 1 (x) = (α + β + 2)y −(β + 1), J (α,β ) n+1 (x) = (a (α,β ) n x − b (α,β ) n )J (α,β ) n (x)− c (α,β ) n J (α,β ) n−1 (x), n ≥ 1 (10) 354 N. StojaNoviĆ, N. StameNkoviĆ Lowpass Filters approximation Based on the jacobi Polynomials 355 Lowpass filters approximatin based on the jacobi polynomials 5 C=2*(k-1+a)*(k-1+b)*(2*k+a+b)/d; P=conv([A B],p1)-C*[0,0,p0]; p0 = p1; p1 = P; end end end Some properties of the Jacobi polynomials, which are needed here, are as follows P(α,β )n (1) = Γ(n + α + 1) Γ(n + 1)Γ(α + 1) (5) and P(α,β )n (−1) = (−1)nΓ(n + β + 1) Γ(n + 1)Γ(β + 1) (6) Jacobi polynomials have symmetry P(α,β )n (x) = (−1)nP (β ,α) n (x) (7) The following important derivative relation is d dx P(α,β )n (x) = 1 2 (n + α + β + 1)P(α+1,β +1)n−1 (x) (8) 3.1 Shifted Jacobi polynomials In order to use Jacobi polynomials on the interval x ∈ [0,1] we define the so-called shifted jacobi polynomials by introducing the change of variable x �→ 2x − 1. Let the shifted Jacobi polynomials P(α,β )n (2x−1) be denoted by J (α,β ) n (x). The shifted Jacobi polynomials are orthogonal with respect to the weight function w(α,β )s = (1 − x)α xβ in the interval [0,1] with the orthogonality property: ∫ 1 0 w(α,β )s J (α,β ) m (x)J (α,β ) n (x)dx = 1 2n + α + β + 1 Γ(n + α + 1)Γ(n + β + 1) Γ(n + 1)Γ(n + α + β + 1) δn,m (9) The shifted Jacobi polynomials are generated from the three-term recurrence relations [24]: J (α,β ) 0 (x) = 1, J (α,β ) 1 (x) = (α + β + 2)y −(β + 1), J (α,β ) n+1 (x) = (a (α,β ) n x − b (α,β ) n )J (α,β ) n (x)− c (α,β ) n J (α,β ) n−1 (x), n ≥ 1 (10) 6 N. Stojanović and N. Stamenković: where the recursion coefficients are a(α,β )n = (2n + α + β + 1)(2n + α + β + 2) (n + 1)(n + α + β + 1) b(α,β )n = (2n + α + β + 1)(2n2 +(1 + β )(α + β )+ 2n(α + β + 1)) (n + 1)(n + α + β + 1)(2n + α + β ) c(α,β )n = (2n + α + β + 2)(n + α)(n + β ) (n + 1)(n + α + β + 1)(2n + α + β ) (11) The shifted Jacobi polynomial J (α,β )n (x) can be obtained in the polynomial standard form as J (α,β ) n (x) = n ∑ i=0 (−1)n−i Γ(n + α + β + i + 1) Γ(i + 1)Γ(n + α + β + 1) Γ(n + β + 1) Γ(n − i + 1)Γ(β + i + 1) xi (12) Suppose the Jacobi polynomials should be normalized soo’ that φn(1) = 1. According to the polynomial (12), the normalization constant is k(α,β )n = ∑ni=0 a (n) i , where a (n) i are corresponding polynomial coefficients. As an example, Fig. 1 shows the characteristic functions based on the shifted Jacobi polyno- mials for n = 1,2,...,5 in the form φn(x) = xν J (α,β ) m (x2)/k (α,β ) n , where n = ⌊m/2⌋+ ν , the floor function ⌊m/2⌋ rounds the value of m/2 to the nearest integers towards zero, ν = 0 and ν = 1 for n even and odd, respectively. −1.5 −1 −0.5 0 0.5 1 −5 −4 −3 −2 −1 0 1 2 3 4 5 J4 (α,β) J 3 (α,β) J 2 (α,β) J 5 (α,β) J 1 (α,β) Characteristic function shifted Jacobi α=−0.5, β=0.5 x φ n (x )= xν J m(α ,β ) (x 2 ) /k n(α ,β ) Fig. 1. The normalized shifted Jacobi polynomials φn(x) = xν J (α,β ) m (x2) for ν = 0 and ν = 1 for n even and odd, respectively, used in place characteristic function, α = −0.5 and β = 0.5, n = 2m + ν , m = 0,1 and 2. 356 N. StojaNoviĆ, N. StameNkoviĆ Lowpass Filters approximation Based on the jacobi Polynomials 357 6 N. Stojanović and N. Stamenković: where the recursion coefficients are a(α,β )n = (2n + α + β + 1)(2n + α + β + 2) (n + 1)(n + α + β + 1) b(α,β )n = (2n + α + β + 1)(2n2 +(1 + β )(α + β )+ 2n(α + β + 1)) (n + 1)(n + α + β + 1)(2n + α + β ) c(α,β )n = (2n + α + β + 2)(n + α)(n + β ) (n + 1)(n + α + β + 1)(2n + α + β ) (11) The shifted Jacobi polynomial J (α,β )n (x) can be obtained in the polynomial standard form as J (α,β ) n (x) = n ∑ i=0 (−1)n−i Γ(n + α + β + i + 1) Γ(i + 1)Γ(n + α + β + 1) Γ(n + β + 1) Γ(n − i + 1)Γ(β + i + 1) xi (12) Suppose the Jacobi polynomials should be normalized soo’ that φn(1) = 1. According to the polynomial (12), the normalization constant is k(α,β )n = ∑ni=0 a (n) i , where a (n) i are corresponding polynomial coefficients. As an example, Fig. 1 shows the characteristic functions based on the shifted Jacobi polyno- mials for n = 1,2,...,5 in the form φn(x) = xν J (α,β ) m (x2)/k (α,β ) n , where n = ⌊m/2⌋+ ν , the floor function ⌊m/2⌋ rounds the value of m/2 to the nearest integers towards zero, ν = 0 and ν = 1 for n even and odd, respectively. −1.5 −1 −0.5 0 0.5 1 −5 −4 −3 −2 −1 0 1 2 3 4 5 J4 (α,β) J 3 (α,β) J 2 (α,β) J 5 (α,β) J 1 (α,β) Characteristic function shifted Jacobi α=−0.5, β=0.5 x φ n (x )= xν J m(α ,β ) (x 2 ) /k n(α ,β ) Fig. 1. The normalized shifted Jacobi polynomials φn(x) = xν J (α,β ) m (x2) for ν = 0 and ν = 1 for n even and odd, respectively, used in place characteristic function, α = −0.5 and β = 0.5, n = 2m + ν , m = 0,1 and 2.Lowpass filters approximatin based on the jacobi polynomials 7 As shown in Fig. 1, the hump at x = 0 occurs when the filter degree is even. Using (6) size of the hump can be obtained as φ (α,β )m (0) = 1 k(α,β )n P(α,β )m (−1) = 1 k(α,β )n (−1)mΓ(m + β + 1) Γ(m + 1)Γ(β + 1) (13) because J (α,β )n (0) = P (α,β ) n (−1). One can easily show that the size of the hump increases when the degree of the filter increases. For example, for n = 4, (m = 2 and ν = 0) from (13) follow P(−0.5,0.5)2 (−1) = 1.875 and from (12) is k (−0.5,0.5) 2 = 0.3750 then value for hump is φ2(0) = 5. For n = 6 (m = 3 and ν = 0) follow P(−0.5,0.5)3 (−1) = −2.1875, k (−0.5,0.5) 3 = 0.3125 then φ3(0) = −7. Thus, the even degree of the shifted Jacobi polynomial is not suitable as the filter characteristic function. other definitions of the monic shifted jacobi polynomials are given in [22, Chapter 22], Gn(p,q,x), which are also orthogonal in the interval [0,1] with respect to weight function w(x) = (1−x)p−qxq−1 (with q > 0 and p > q − 1), are used for the construction magnitude of the filter’s transfer func- tion [25] [26] [27]. Shifted Jacobi polynomials [22] are related to the Jacobi Polynomials P(α,β )n (x) as [28] Gn(p,q,x) = Γ(n + 1)Γ(n + p) Γ(2n + p) P(p−q,q−1)n (2x − 1) (14) It can be concluded, the shifted Jacobi polynomials J (α,β )n (x) have n distinct positive real zeros in the interval (0,1) but they are neither even nor odd then it can not be used as a characteristic function in the equation (1). However, [xJ (α,β )n (x2)]2 or [xG(p,q,x2)]2 could be used in (1) in place of squared characteristic function φ 2n (ω). 3.2 Modified Jacobi polynomials We propose the following modified jacobi polynomials, based on the summation of two Jacobi orthogonal polynomials which have the same degree n, as J (α,β ) n (x) = P (α,β ) n (x)+ P (β ,α) n (x) (15) where P(α,β )n (x) is above mentioned classical Jacoby orthogonal polynomial in x. One can easily show that modified jacobi polynomial (15) is not orthogonal polynomial except in the case when α = β is. Since Jacobi polynomials P(β ,α)n (x) = (−1)nP (α,β ) n (−x) are not orthogonal polynomials with the respect to the weight function w(α,β )(x) over the interval [−1,1], then the modified orthog- onal Jacobi polynomials (15) are not orthogonal polynomials as the shifted Jacobi polynomials are. However, the resulting degree of modified jacobi polynomial is n, which is pure odd or pure even polynomial in x, and hence the realization of the lowpass filter is possible for all specifications if they are used as characteristic function. 356 N. StojaNoviĆ, N. StameNkoviĆ Lowpass Filters approximation Based on the jacobi Polynomials 357 Lowpass filters approximatin based on the jacobi polynomials 7 As shown in Fig. 1, the hump at x = 0 occurs when the filter degree is even. Using (6) size of the hump can be obtained as φ (α,β )m (0) = 1 k(α,β )n P(α,β )m (−1) = 1 k(α,β )n (−1)mΓ(m + β + 1) Γ(m + 1)Γ(β + 1) (13) because J (α,β )n (0) = P (α,β ) n (−1). One can easily show that the size of the hump increases when the degree of the filter increases. For example, for n = 4, (m = 2 and ν = 0) from (13) follow P(−0.5,0.5)2 (−1) = 1.875 and from (12) is k (−0.5,0.5) 2 = 0.3750 then value for hump is φ2(0) = 5. For n = 6 (m = 3 and ν = 0) follow P(−0.5,0.5)3 (−1) = −2.1875, k (−0.5,0.5) 3 = 0.3125 then φ3(0) = −7. Thus, the even degree of the shifted Jacobi polynomial is not suitable as the filter characteristic function. other definitions of the monic shifted jacobi polynomials are given in [22, Chapter 22], Gn(p,q,x), which are also orthogonal in the interval [0,1] with respect to weight function w(x) = (1−x)p−qxq−1 (with q > 0 and p > q − 1), are used for the construction magnitude of the filter’s transfer func- tion [25] [26] [27]. Shifted Jacobi polynomials [22] are related to the Jacobi Polynomials P(α,β )n (x) as [28] Gn(p,q,x) = Γ(n + 1)Γ(n + p) Γ(2n + p) P(p−q,q−1)n (2x − 1) (14) It can be concluded, the shifted Jacobi polynomials J (α,β )n (x) have n distinct positive real zeros in the interval (0,1) but they are neither even nor odd then it can not be used as a characteristic function in the equation (1). However, [xJ (α,β )n (x2)]2 or [xG(p,q,x2)]2 could be used in (1) in place of squared characteristic function φ 2n (ω). 3.2 Modified Jacobi polynomials We propose the following modified jacobi polynomials, based on the summation of two Jacobi orthogonal polynomials which have the same degree n, as J (α,β ) n (x) = P (α,β ) n (x)+ P (β ,α) n (x) (15) where P(α,β )n (x) is above mentioned classical Jacoby orthogonal polynomial in x. One can easily show that modified jacobi polynomial (15) is not orthogonal polynomial except in the case when α = β is. Since Jacobi polynomials P(β ,α)n (x) = (−1)nP (α,β ) n (−x) are not orthogonal polynomials with the respect to the weight function w(α,β )(x) over the interval [−1,1], then the modified orthog- onal Jacobi polynomials (15) are not orthogonal polynomials as the shifted Jacobi polynomials are. However, the resulting degree of modified jacobi polynomial is n, which is pure odd or pure even polynomial in x, and hence the realization of the lowpass filter is possible for all specifications if they are used as characteristic function.8 N. Stojanović and N. Stamenković: Many of the aforementioned polynomials are special cases of modified jacobi polynomials. For α = β , one can obtain the ultraspherical polynomials (symmetric Jacobi polynomials) [29]. For α = β = ∓1/2, the Chebyshev polynomials of first and second kinds. For α = β = 0, one can obtain the Legendre polynomials. For the two important special cases α = −β ± 1/2, the Chebyshev polynomials of third and fourth kinds are also obtained. Finally, the constants C(α,β )n = J (α,β ) n (1) have to be chosen in such a way that normalization criterion φn(1) = 1 is satisfied, i.e. φn(ω) = J (α,β ) n (ω) C(α,β )n , (16) where C(α,β )n = 1 Γ(n + 1) [Γ(n + α + 1) Γ(α + 1) + Γ(n + β + 1) Γ(β + 1) ] . (17) modified jacobi polynomials are symmetrical in relation to the orders α and β , i.e. J(α,β )n (ω) = J (β ,α) n (ω). table 1 contains the modified jacobi polynomials for α = −0.5 and β = 0.5 up to the ninth degree. table 1. the modified orthogonal jacobi polynomials J(α,β )n (x), α = −0.5, β = 0.5, and n = 0,1,...,10. n J(−0.5,0.5)n (x) 1 2 x 2 3 x2 − 3 4 3 5 x3 − 5 2 x 4 35 4 x4 − 105 16 x2 + 35 64 5 63 4 x5 − 63 4 x3 + 189 64 x 6 231 8 x6 − 1155 32 x4 + 693 64 x2 − 231 512 7 429 8 x7 − 1287 16 x5 + 2145 64 x3 − 429 128 x 8 6435 64 x8 − 45045 256 x6 + 96525 1024 x4 − 32175 2048 x2 + 6435 16384 9 12155 64 x9 − 12155 32 x7 + 255255 1024 x5 − 60775 1024 x3 + 60775 16384 x 10 46189 128 x10 − 415701 512 x8 + 323323 512 x6 − 1616615 8192 x4 + 692835 32768 x2 − 46189 131072 it is important to know where the roots of the modified jacobi polynomials are located. The fastest way to calculate the zeros of the modified jacobi polynomials is by using mathematical programs such as Matlab, Mathematica and Maple. It can be concluded that the modified jacobi 358 N. StojaNoviĆ, N. StameNkoviĆ Lowpass Filters approximation Based on the jacobi Polynomials 359 8 N. Stojanović and N. Stamenković: Many of the aforementioned polynomials are special cases of modified jacobi polynomials. For α = β , one can obtain the ultraspherical polynomials (symmetric Jacobi polynomials) [29]. For α = β = ∓1/2, the Chebyshev polynomials of first and second kinds. For α = β = 0, one can obtain the Legendre polynomials. For the two important special cases α = −β ± 1/2, the Chebyshev polynomials of third and fourth kinds are also obtained. Finally, the constants C(α,β )n = J (α,β ) n (1) have to be chosen in such a way that normalization criterion φn(1) = 1 is satisfied, i.e. φn(ω) = J (α,β ) n (ω) C(α,β )n , (16) where C(α,β )n = 1 Γ(n + 1) [Γ(n + α + 1) Γ(α + 1) + Γ(n + β + 1) Γ(β + 1) ] . (17) modified jacobi polynomials are symmetrical in relation to the orders α and β , i.e. J(α,β )n (ω) = J (β ,α) n (ω). table 1 contains the modified jacobi polynomials for α = −0.5 and β = 0.5 up to the ninth degree. table 1. the modified orthogonal jacobi polynomials J(α,β )n (x), α = −0.5, β = 0.5, and n = 0,1,...,10. n J(−0.5,0.5)n (x) 1 2 x 2 3 x2 − 3 4 3 5 x3 − 5 2 x 4 35 4 x4 − 105 16 x2 + 35 64 5 63 4 x5 − 63 4 x3 + 189 64 x 6 231 8 x6 − 1155 32 x4 + 693 64 x2 − 231 512 7 429 8 x7 − 1287 16 x5 + 2145 64 x3 − 429 128 x 8 6435 64 x8 − 45045 256 x6 + 96525 1024 x4 − 32175 2048 x2 + 6435 16384 9 12155 64 x9 − 12155 32 x7 + 255255 1024 x5 − 60775 1024 x3 + 60775 16384 x 10 46189 128 x10 − 415701 512 x8 + 323323 512 x6 − 1616615 8192 x4 + 692835 32768 x2 − 46189 131072 it is important to know where the roots of the modified jacobi polynomials are located. The fastest way to calculate the zeros of the modified jacobi polynomials is by using mathematical programs such as Matlab, Mathematica and Maple. It can be concluded that the modified jacobi Lowpass filters approximatin based on the jacobi polynomials 9 polynomials, J(α,β )n (x), have n simple real zeros in the closed interval [−1,1]. For example, the zeros of the modified jacobi polynomial of degree 8 with α = −0.5 and β = 0.5 are: {−0.9396926,−0.7660444,−0.5000000,−0.1736482, 0.1736482, 0.5000000, 0.7660444, 0.9396926}. The zeros of J(α,β )n (x) are located symmetrically about x = 0 in the interval −1 < x < 1. Note that modified jacobi polynomials are the only non orthogonal polynomials which are suitable for the synthesis of the filter function given in a closed form. The characteristic functions φn(x) based on the modified jacobi polynomials J α,β ) n (x) are illus- trated in Figure 1 for x in [−1,1] and n = 1,2,...,5. They satisfy the following relationships: for |x| < 1, the characteristic polynomial oscillates around zero and they ripples are bounded by ±1 for α,β ≥ −0.5, φn(0) �= 0 for n even and φn(0) = 0 for n odd. For |x| > 1, the polynomials magnitude increase (decrease) monotonically. x -1 -0.5 0 0.5 1 φ n( x) =[ P n(α ,β ) ( x) +P n(β ,α ) ( x) ]/C n(α ,β ) -3 -2 -1 0 1 2 3 4 5 4 3 2 n=1 α=-0.5, β=0 Modified Jacobi polynomials Fig. 2. the normalized modified orthogonal jacobi polynomials J(α,β )n (ω)/C (α,β ) n used in place characteristic function φn(x), α = −0.5 and β = 0.5, n = 1,...,5. An example is given in Figure 3, which shows the ninth-order modified jacobi lowpass filter and its three partial filters with their individual orders α and β . As mentioned earlier, Jacobi orthogonal polynomial corresponds to the Chebyshev polynomial if α = β = −0.5 which have 3dB ripples in the pass-band. in general, passband ripples are being undesirable, but a value less than 0.5 dB is acceptable in many applications. if α = −0.5 and order β increases, the ripples in the passband decrease smoothly to be unequal and smaller in magnitude. For β > 1.5 the passband response is nearly flat, but the cutoff slope is much steeper than a Butterworth filter cutoff slope. On the other hand, for −1 < β < −0.5 the passband ripples are unequal, but in magnitude are Lowpass filters approximatin based on the jacobi polynomials 9 polynomials, J(α,β )n (x), have n simple real zeros in the closed interval [−1,1]. For example, the zeros of the modified jacobi polynomial of degree 8 with α = −0.5 and β = 0.5 are: {−0.9396926,−0.7660444,−0.5000000,−0.1736482, 0.1736482, 0.5000000, 0.7660444, 0.9396926}. The zeros of J(α,β )n (x) are located symmetrically about x = 0 in the interval −1 < x < 1. Note that modified jacobi polynomials are the only non orthogonal polynomials which are suitable for the synthesis of the filter function given in a closed form. The characteristic functions φn(x) based on the modified jacobi polynomials J α,β ) n (x) are illus- trated in Figure 1 for x in [−1,1] and n = 1,2,...,5. They satisfy the following relationships: for |x| < 1, the characteristic polynomial oscillates around zero and they ripples are bounded by ±1 for α,β ≥ −0.5, φn(0) �= 0 for n even and φn(0) = 0 for n odd. For |x| > 1, the polynomials magnitude increase (decrease) monotonically. x -1 -0.5 0 0.5 1 φ n( x) =[ P n(α ,β ) ( x) +P n(β ,α ) ( x) ]/C n(α ,β ) -3 -2 -1 0 1 2 3 4 5 4 3 2 n=1 α=-0.5, β=0 Modified Jacobi polynomials Fig. 2. the normalized modified orthogonal jacobi polynomials J(α,β )n (ω)/C (α,β ) n used in place characteristic function φn(x), α = −0.5 and β = 0.5, n = 1,...,5. An example is given in Figure 3, which shows the ninth-order modified jacobi lowpass filter and its three partial filters with their individual orders α and β . As mentioned earlier, Jacobi orthogonal polynomial corresponds to the Chebyshev polynomial if α = β = −0.5 which have 3dB ripples in the pass-band. in general, passband ripples are being undesirable, but a value less than 0.5 dB is acceptable in many applications. if α = −0.5 and order β increases, the ripples in the passband decrease smoothly to be unequal and smaller in magnitude. For β > 1.5 the passband response is nearly flat, but the cutoff slope is much steeper than a Butterworth filter cutoff slope. On the other hand, for −1 < β < −0.5 the passband ripples are unequal, but in magnitude are Lowpass filters approximatin based on the jacobi polynomials 9 polynomials, J(α,β )n (x), have n simple real zeros in the closed interval [−1,1]. For example, the zeros of the modified jacobi polynomial of degree 8 with α = −0.5 and β = 0.5 are: {−0.9396926,−0.7660444,−0.5000000,−0.1736482, 0.1736482, 0.5000000, 0.7660444, 0.9396926}. The zeros of J(α,β )n (x) are located symmetrically about x = 0 in the interval −1 < x < 1. Note that modified jacobi polynomials are the only non orthogonal polynomials which are suitable for the synthesis of the filter function given in a closed form. The characteristic functions φn(x) based on the modified jacobi polynomials J α,β ) n (x) are illus- trated in Figure 1 for x in [−1,1] and n = 1,2,...,5. They satisfy the following relationships: for |x| < 1, the characteristic polynomial oscillates around zero and they ripples are bounded by ±1 for α,β ≥ −0.5, φn(0) �= 0 for n even and φn(0) = 0 for n odd. For |x| > 1, the polynomials magnitude increase (decrease) monotonically. x -1 -0.5 0 0.5 1 φ n( x) =[ P n(α ,β ) ( x) +P n(β ,α ) ( x) ]/C n(α ,β ) -3 -2 -1 0 1 2 3 4 5 4 3 2 n=1 α=-0.5, β=0 Modified Jacobi polynomials Fig. 2. the normalized modified orthogonal jacobi polynomials J(α,β )n (ω)/C (α,β ) n used in place characteristic function φn(x), α = −0.5 and β = 0.5, n = 1,...,5. An example is given in Figure 3, which shows the ninth-order modified jacobi lowpass filter and its three partial filters with their individual orders α and β . As mentioned earlier, Jacobi orthogonal polynomial corresponds to the Chebyshev polynomial if α = β = −0.5 which have 3dB ripples in the pass-band. in general, passband ripples are being undesirable, but a value less than 0.5 dB is acceptable in many applications. if α = −0.5 and order β increases, the ripples in the passband decrease smoothly to be unequal and smaller in magnitude. For β > 1.5 the passband response is nearly flat, but the cutoff slope is much steeper than a Butterworth filter cutoff slope. On the other hand, for −1 < β < −0.5 the passband ripples are unequal, but in magnitude are 8 N. Stojanović and N. Stamenković: Many of the aforementioned polynomials are special cases of modified jacobi polynomials. For α = β , one can obtain the ultraspherical polynomials (symmetric Jacobi polynomials) [29]. For α = β = ∓1/2, the Chebyshev polynomials of first and second kinds. For α = β = 0, one can obtain the Legendre polynomials. For the two important special cases α = −β ± 1/2, the Chebyshev polynomials of third and fourth kinds are also obtained. Finally, the constants C(α,β )n = J (α,β ) n (1) have to be chosen in such a way that normalization criterion φn(1) = 1 is satisfied, i.e. φn(ω) = J (α,β ) n (ω) C(α,β )n , (16) where C(α,β )n = 1 Γ(n + 1) [Γ(n + α + 1) Γ(α + 1) + Γ(n + β + 1) Γ(β + 1) ] . (17) modified jacobi polynomials are symmetrical in relation to the orders α and β , i.e. J(α,β )n (ω) = J (β ,α) n (ω). table 1 contains the modified jacobi polynomials for α = −0.5 and β = 0.5 up to the ninth degree. table 1. the modified orthogonal jacobi polynomials J(α,β )n (x), α = −0.5, β = 0.5, and n = 0,1,...,10. n J(−0.5,0.5)n (x) 1 2 x 2 3 x2 − 3 4 3 5 x3 − 5 2 x 4 35 4 x4 − 105 16 x2 + 35 64 5 63 4 x5 − 63 4 x3 + 189 64 x 6 231 8 x6 − 1155 32 x4 + 693 64 x2 − 231 512 7 429 8 x7 − 1287 16 x5 + 2145 64 x3 − 429 128 x 8 6435 64 x8 − 45045 256 x6 + 96525 1024 x4 − 32175 2048 x2 + 6435 16384 9 12155 64 x9 − 12155 32 x7 + 255255 1024 x5 − 60775 1024 x3 + 60775 16384 x 10 46189 128 x10 − 415701 512 x8 + 323323 512 x6 − 1616615 8192 x4 + 692835 32768 x2 − 46189 131072 it is important to know where the roots of the modified jacobi polynomials are located. The fastest way to calculate the zeros of the modified jacobi polynomials is by using mathematical programs such as Matlab, Mathematica and Maple. It can be concluded that the modified jacobi 358 N. StojaNoviĆ, N. StameNkoviĆ Lowpass Filters approximation Based on the jacobi Polynomials 359 Lowpass filters approximatin based on the jacobi polynomials 9 polynomials, J(α,β )n (x), have n simple real zeros in the closed interval [−1,1]. For example, the zeros of the modified jacobi polynomial of degree 8 with α = −0.5 and β = 0.5 are: {−0.9396926,−0.7660444,−0.5000000,−0.1736482, 0.1736482, 0.5000000, 0.7660444, 0.9396926}. The zeros of J(α,β )n (x) are located symmetrically about x = 0 in the interval −1 < x < 1. Note that modified jacobi polynomials are the only non orthogonal polynomials which are suitable for the synthesis of the filter function given in a closed form. The characteristic functions φn(x) based on the modified jacobi polynomials J α,β ) n (x) are illus- trated in Figure 1 for x in [−1,1] and n = 1,2,...,5. They satisfy the following relationships: for |x| < 1, the characteristic polynomial oscillates around zero and they ripples are bounded by ±1 for α,β ≥ −0.5, φn(0) �= 0 for n even and φn(0) = 0 for n odd. For |x| > 1, the polynomials magnitude increase (decrease) monotonically. x -1 -0.5 0 0.5 1 φ n( x) =[ P n(α ,β ) ( x) +P n(β ,α ) ( x) ]/C n(α ,β ) -3 -2 -1 0 1 2 3 4 5 4 3 2 n=1 α=-0.5, β=0 Modified Jacobi polynomials Fig. 2. the normalized modified orthogonal jacobi polynomials J(α,β )n (ω)/C (α,β ) n used in place characteristic function φn(x), α = −0.5 and β = 0.5, n = 1,...,5. An example is given in Figure 3, which shows the ninth-order modified jacobi lowpass filter and its three partial filters with their individual orders α and β . As mentioned earlier, Jacobi orthogonal polynomial corresponds to the Chebyshev polynomial if α = β = −0.5 which have 3dB ripples in the pass-band. in general, passband ripples are being undesirable, but a value less than 0.5 dB is acceptable in many applications. if α = −0.5 and order β increases, the ripples in the passband decrease smoothly to be unequal and smaller in magnitude. For β > 1.5 the passband response is nearly flat, but the cutoff slope is much steeper than a Butterworth filter cutoff slope. On the other hand, for −1 < β < −0.5 the passband ripples are unequal, but in magnitude are 10 N. Stojanović and N. Stamenković: larger than 1, but these values of β (also for α ) have no practical significance. it is shown that the passband ripple can be adjusted to improve the linearity of the group delay response near the ω = 0. Normalized frequency, ω 10-1 100 S to pb an d at te nu at io n, d B 0 10 20 30 40 50 60 70 P as sb an d at te nu at io n, d B 0 1 2 3 5 10 15 20 G ro up d el ay , s Modified Jacobi, N=9 α=-0.5, β=0 α=-0.5, β=0.5 α=-0.5, β=1.5 Butterworth Fig. 3. the frequency responses of the 9th degree modified jacobi filters. Generally, for microwave applications modified orthogonal jacobi as filter function may be also used. the most widely used filters in microwave applications are a band-pass filters [30]. Using lowpass to bandpass frequency transformation of lumped element lowpass filter, the series inductor converts to the series resonator and parallel capacitor converts to the parallel resonator. Richards transformation can be used to emulate the inductive and capacitive behaviour of the lumped circuit elements into distributive element consist the transmission line sections, and Kuroda’s identities can be used to facilitate the conversion between the various transmission line realizations. in the application where approximation of the filter magnitude function based on the Christofel- Darboux formula for classical orthonormal Jacobi polynomials gives excellent results [31] [32], this method cannot be applied to the modified jacobi filters, because it is non orthogonal. In this case, it should either generate the sum of the product modified jacobi polynomial, or Christoffel-Darboux formula be applied separately to the both orthonormal Jacobi polynomials as: A2n(ω 2) =[p (α,β ) 0 (ω)] 2 +[p(α,β )1 (ω)] 2 + ···+[p(α,β )n (ω)]2 +[p(β ,α)0 (ω)] 2 +[p(β ,α)1 (ω)] 2 + ···+[p(β ,α)n (ω)]2 (18) where p(α,β )i (ω), i = 1,2,...,n are orthonormal Jacobi polynomials with respect to the weight function w(α,β )(ω) = (1 − ω)α (1 + ω)β and p(β ,α)i (ω), i = 1,2,...,n are also orthonormal Jacobi polynomials but with respect to the other weight function w(β ,α)(ω) = (1 − ω)β (1 + ω)α . The 360 N. StojaNoviĆ, N. StameNkoviĆ Lowpass Filters approximation Based on the jacobi Polynomials 361 10 N. Stojanović and N. Stamenković: larger than 1, but these values of β (also for α ) have no practical significance. it is shown that the passband ripple can be adjusted to improve the linearity of the group delay response near the ω = 0. Normalized frequency, ω 10-1 100 S to pb an d at te nu at io n, d B 0 10 20 30 40 50 60 70 P as sb an d at te nu at io n, d B 0 1 2 3 5 10 15 20 G ro up d el ay , s Modified Jacobi, N=9 α=-0.5, β=0 α=-0.5, β=0.5 α=-0.5, β=1.5 Butterworth Fig. 3. the frequency responses of the 9th degree modified jacobi filters. Generally, for microwave applications modified orthogonal jacobi as filter function may be also used. the most widely used filters in microwave applications are a band-pass filters [30]. Using lowpass to bandpass frequency transformation of lumped element lowpass filter, the series inductor converts to the series resonator and parallel capacitor converts to the parallel resonator. Richards transformation can be used to emulate the inductive and capacitive behaviour of the lumped circuit elements into distributive element consist the transmission line sections, and Kuroda’s identities can be used to facilitate the conversion between the various transmission line realizations. in the application where approximation of the filter magnitude function based on the Christofel- Darboux formula for classical orthonormal Jacobi polynomials gives excellent results [31] [32], this method cannot be applied to the modified jacobi filters, because it is non orthogonal. In this case, it should either generate the sum of the product modified jacobi polynomial, or Christoffel-Darboux formula be applied separately to the both orthonormal Jacobi polynomials as: A2n(ω 2) =[p (α,β ) 0 (ω)] 2 +[p(α,β )1 (ω)] 2 + ···+[p(α,β )n (ω)]2 +[p(β ,α)0 (ω)] 2 +[p(β ,α)1 (ω)] 2 + ···+[p(β ,α)n (ω)]2 (18) where p(α,β )i (ω), i = 1,2,...,n are orthonormal Jacobi polynomials with respect to the weight function w(α,β )(ω) = (1 − ω)α (1 + ω)β and p(β ,α)i (ω), i = 1,2,...,n are also orthonormal Jacobi polynomials but with respect to the other weight function w(β ,α)(ω) = (1 − ω)β (1 + ω)α . The Lowpass filters approximatin based on the jacobi polynomials 11 orthonormal Jacobi plynomials are: p(α,β )n (ω) = √ 2n + α + β + 1 2α+β +1 Γ(n + 1)Γ(n + α + β + 1) Γ(n + α + 1)Γ(n + β + 1) P(α,β )n (ω) (19) and p(β ,α)n (ω) = √ 2n + α + β + 1 2α+β +1 Γ(n + 1)Γ(n + α + β + 1) Γ(n + α + 1)Γ(n + β + 1) P(β ,α)n (ω) (20) where P(β ,α)n (ω) and P (β ,α) n (ω) are the orthogonal Jacobi polynomials which can be evaluated by the proposed Matlab program. By using Christoffel-Darboux formula equation (18) is reduced to: A2n(ω 2) = k(α,β )n k(α,β )n+1 [dp(α,β )n+1 dω p(α,β )n − dp(α,β )n dω p(α,β )n+1 ] + k(β ,α)n k(β ,α)n+1 [dp(β ,α)n+1 dω p(β ,α)n − dp(β ,α)n dω p(β ,α)n+1 ] (21) where k(α,β )n and k (β ,α) n are leading coefficients of the orthonormal jacobi polynomials p (α,β ) n (ω) and p(β ,α)n (ω), respectively. the following filter approximating function for n = 5, α = −0.5 and β = 0.5 is given as an example: A10(ω 2) = 325.9493ω 10 − 488.9240ω 8 + 244.4620ω 6 − 40.7437ω 4 + 3.8197ω 2 + 2.5 according to the definition, the characteristic function should be normalized so that is unit, A10(1) = 1, at the cutoff frequency, ωp = 1. 4 Conclusion In this paper, we intended to illuminate the usage of Jacobi orthogonal polynomials in the design of time-continuous low-pass filter transfer function. Since jacobi polynomial cannot be directly used as filter characteristic function, we suggested shifted jacobi polynomials and proposed a simple modification of jacobi polynomials to use as a filter characteristic function. the modified jacobi polynomials are not orthogonal, but they are suitable for the filter transfer function approximation. Filter degree, maximum passband attenuation and two indexes of Jacobi polynomials are four parameters that adjust the performance of the filter. the new modified jacobi polynomials are implemented to approximate the lowpass filter transfer function in such a way that they are used directly as filter characteristic function (as standard orthogonal polynomials: 360 N. StojaNoviĆ, N. StameNkoviĆ Lowpass Filters approximation Based on the jacobi Polynomials 361 Lowpass filters approximatin based on the jacobi polynomials 11 orthonormal Jacobi plynomials are: p(α,β )n (ω) = √ 2n + α + β + 1 2α+β +1 Γ(n + 1)Γ(n + α + β + 1) Γ(n + α + 1)Γ(n + β + 1) P(α,β )n (ω) (19) and p(β ,α)n (ω) = √ 2n + α + β + 1 2α+β +1 Γ(n + 1)Γ(n + α + β + 1) Γ(n + α + 1)Γ(n + β + 1) P(β ,α)n (ω) (20) where P(β ,α)n (ω) and P (β ,α) n (ω) are the orthogonal Jacobi polynomials which can be evaluated by the proposed Matlab program. By using Christoffel-Darboux formula equation (18) is reduced to: A2n(ω 2) = k(α,β )n k(α,β )n+1 [dp(α,β )n+1 dω p(α,β )n − dp(α,β )n dω p(α,β )n+1 ] + k(β ,α)n k(β ,α)n+1 [dp(β ,α)n+1 dω p(β ,α)n − dp(β ,α)n dω p(β ,α)n+1 ] (21) where k(α,β )n and k (β ,α) n are leading coefficients of the orthonormal jacobi polynomials p (α,β ) n (ω) and p(β ,α)n (ω), respectively. the following filter approximating function for n = 5, α = −0.5 and β = 0.5 is given as an example: A10(ω 2) = 325.9493ω 10 − 488.9240ω 8 + 244.4620ω 6 − 40.7437ω 4 + 3.8197ω 2 + 2.5 according to the definition, the characteristic function should be normalized so that is unit, A10(1) = 1, at the cutoff frequency, ωp = 1. 4 Conclusion In this paper, we intended to illuminate the usage of Jacobi orthogonal polynomials in the design of time-continuous low-pass filter transfer function. Since jacobi polynomial cannot be directly used as filter characteristic function, we suggested shifted jacobi polynomials and proposed a simple modification of jacobi polynomials to use as a filter characteristic function. the modified jacobi polynomials are not orthogonal, but they are suitable for the filter transfer function approximation. Filter degree, maximum passband attenuation and two indexes of Jacobi polynomials are four parameters that adjust the performance of the filter. the new modified jacobi polynomials are implemented to approximate the lowpass filter transfer function in such a way that they are used directly as filter characteristic function (as standard orthogonal polynomials: 12 N. Stojanović and N. Stamenković: Chebyshev or Legendre). These methods of approximation can be used to provide filters with adjustment of the passband ripple, group delay deviation or cutoff slope. Acknowledgment This work is supported by Serbian Ministry of Education, Science and Technological Development, Project No. 32009TR. References [1] 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, nu- merical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0377042700005033 [2] 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. [3] 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, Apr. 1973. [4] S. Butterworth, “on the theory filter amplifier,” Experimental Wireless and the Radio Engineer, vol. 7, pp. 536–541, Oct. 1930. [5] H. G. Dimopoulos, “Optimal use of some classical approximations in filter design,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 54, no. 9, pp. 780–784, Sep. 2007. [6] S. C. D. Roy, “modified chebyshev lowpass filters,” International Journal of Circuit Theory and Ap- plications, vol. 38, no. 5, pp. 543–549, 2010.[Online]. Available: http://dx.doi.org/10.1002/cta.585 [7] R. Ramiz and H. Sedef, “General method for designing and simulating of resistively terminated LC ladder filters,” FACTA UNIVERSITATIS, Series: Electronics and Energetics, vol. 12, no. 3, pp. 79–94, 1999. [8] S. Prasad, L. G. Stolarczyk, J. R. Jackson, and E. W. Kang, “Filter synthesis using Legendre polyno- mials,” Proceedings of the IEE, vol. 114, no. 8, pp. 1063–1064, Aug. 1967. [9] M. T. Chryssomallis and J. N. Sahalos, “Filter synthesis using products of Legendre polynomials,” Electrical Engineering, vol. 81, no. 6, pp. 419–424, 1999. [10] D. Živaljević, N. Stamenković, and V. Stojanović, “Nearly monotonic passband low-pass filter design by using sum-of-squared Legendre polynomials,” International Journal of Circuit Theory and Appli- cations, vol. 44, no. 1, pp. 147–161, Jan. 2016. [Online]. Available: http://dx.doi.org/10.1002/cta.2068 [11] 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. [12] i. m. Filanovsky, “Bessel-Butterworth transitional filters,” in 2014 IEEE International Symposium on Circuits and Systems (ISCAS), Jun. 2014, pp. 2105–2108. [13] 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), Jul. 2014, pp. 1–5. 12 N. Stojanović and N. Stamenković: Chebyshev or Legendre). These methods of approximation can be used to provide filters with adjustment of the passband ripple, group delay deviation or cutoff slope. Acknowledgment This work is supported by Serbian Ministry of Education, Science and Technological Development, Project No. 32009TR. References [1] 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, nu- merical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0377042700005033 [2] 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. [3] 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, Apr. 1973. [4] S. Butterworth, “on the theory filter amplifier,” Experimental Wireless and the Radio Engineer, vol. 7, pp. 536–541, Oct. 1930. [5] H. G. Dimopoulos, “Optimal use of some classical approximations in filter design,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 54, no. 9, pp. 780–784, Sep. 2007. [6] S. C. D. Roy, “modified chebyshev lowpass filters,” International Journal of Circuit Theory and Ap- plications, vol. 38, no. 5, pp. 543–549, 2010.[Online]. Available: http://dx.doi.org/10.1002/cta.585 [7] R. Ramiz and H. Sedef, “General method for designing and simulating of resistively terminated LC ladder filters,” FACTA UNIVERSITATIS, Series: Electronics and Energetics, vol. 12, no. 3, pp. 79–94, 1999. [8] S. Prasad, L. G. Stolarczyk, J. R. Jackson, and E. W. Kang, “Filter synthesis using Legendre polyno- mials,” Proceedings of the IEE, vol. 114, no. 8, pp. 1063–1064, Aug. 1967. [9] M. T. Chryssomallis and J. N. Sahalos, “Filter synthesis using products of Legendre polynomials,” Electrical Engineering, vol. 81, no. 6, pp. 419–424, 1999. [10] D. Živaljević, N. Stamenković, and V. Stojanović, “Nearly monotonic passband low-pass filter design by using sum-of-squared Legendre polynomials,” International Journal of Circuit Theory and Appli- cations, vol. 44, no. 1, pp. 147–161, Jan. 2016. [Online]. Available: http://dx.doi.org/10.1002/cta.2068 [11] 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. [12] i. m. Filanovsky, “Bessel-Butterworth transitional filters,” in 2014 IEEE International Symposium on Circuits and Systems (ISCAS), Jun. 2014, pp. 2105–2108. [13] 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), Jul. 2014, pp. 1–5. 12 N. Stojanović and N. Stamenković: Chebyshev or Legendre). These methods of approximation can be used to provide filters with adjustment of the passband ripple, group delay deviation or cutoff slope. Acknowledgment This work is supported by Serbian Ministry of Education, Science and Technological Development, Project No. 32009TR. References [1] 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, nu- merical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0377042700005033 [2] 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. [3] 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, Apr. 1973. [4] S. Butterworth, “on the theory filter amplifier,” Experimental Wireless and the Radio Engineer, vol. 7, pp. 536–541, Oct. 1930. [5] H. G. Dimopoulos, “Optimal use of some classical approximations in filter design,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 54, no. 9, pp. 780–784, Sep. 2007. [6] S. C. D. Roy, “modified chebyshev lowpass filters,” International Journal of Circuit Theory and Ap- plications, vol. 38, no. 5, pp. 543–549, 2010.[Online]. Available: http://dx.doi.org/10.1002/cta.585 [7] R. Ramiz and H. Sedef, “General method for designing and simulating of resistively terminated LC ladder filters,” FACTA UNIVERSITATIS, Series: Electronics and Energetics, vol. 12, no. 3, pp. 79–94, 1999. [8] S. Prasad, L. G. Stolarczyk, J. R. Jackson, and E. W. Kang, “Filter synthesis using Legendre polyno- mials,” Proceedings of the IEE, vol. 114, no. 8, pp. 1063–1064, Aug. 1967. [9] M. T. Chryssomallis and J. N. Sahalos, “Filter synthesis using products of Legendre polynomials,” Electrical Engineering, vol. 81, no. 6, pp. 419–424, 1999. [10] D. Živaljević, N. Stamenković, and V. Stojanović, “Nearly monotonic passband low-pass filter design by using sum-of-squared Legendre polynomials,” International Journal of Circuit Theory and Appli- cations, vol. 44, no. 1, pp. 147–161, Jan. 2016. [Online]. Available: http://dx.doi.org/10.1002/cta.2068 [11] 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. [12] i. m. Filanovsky, “Bessel-Butterworth transitional filters,” in 2014 IEEE International Symposium on Circuits and Systems (ISCAS), Jun. 2014, pp. 2105–2108. [13] 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), Jul. 2014, pp. 1–5. 12 N. Stojanović and N. Stamenković: Chebyshev or Legendre). These methods of approximation can be used to provide filters with adjustment of the passband ripple, group delay deviation or cutoff slope. Acknowledgment This work is supported by Serbian Ministry of Education, Science and Technological Development, Project No. 32009TR. References [1] 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, nu- merical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0377042700005033 [2] 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. [3] 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, Apr. 1973. [4] S. Butterworth, “on the theory filter amplifier,” Experimental Wireless and the Radio Engineer, vol. 7, pp. 536–541, Oct. 1930. [5] H. G. Dimopoulos, “Optimal use of some classical approximations in filter design,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 54, no. 9, pp. 780–784, Sep. 2007. [6] S. C. D. Roy, “modified chebyshev lowpass filters,” International Journal of Circuit Theory and Ap- plications, vol. 38, no. 5, pp. 543–549, 2010.[Online]. Available: http://dx.doi.org/10.1002/cta.585 [7] R. Ramiz and H. Sedef, “General method for designing and simulating of resistively terminated LC ladder filters,” FACTA UNIVERSITATIS, Series: Electronics and Energetics, vol. 12, no. 3, pp. 79–94, 1999. [8] S. Prasad, L. G. Stolarczyk, J. R. Jackson, and E. W. Kang, “Filter synthesis using Legendre polyno- mials,” Proceedings of the IEE, vol. 114, no. 8, pp. 1063–1064, Aug. 1967. [9] M. T. Chryssomallis and J. N. Sahalos, “Filter synthesis using products of Legendre polynomials,” Electrical Engineering, vol. 81, no. 6, pp. 419–424, 1999. [10] D. Živaljević, N. Stamenković, and V. Stojanović, “Nearly monotonic passband low-pass filter design by using sum-of-squared Legendre polynomials,” International Journal of Circuit Theory and Appli- cations, vol. 44, no. 1, pp. 147–161, Jan. 2016. [Online]. Available: http://dx.doi.org/10.1002/cta.2068 [11] 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. [12] i. m. Filanovsky, “Bessel-Butterworth transitional filters,” in 2014 IEEE International Symposium on Circuits and Systems (ISCAS), Jun. 2014, pp. 2105–2108. [13] 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), Jul. 2014, pp. 1–5. 12 N. Stojanović and N. Stamenković: Chebyshev or Legendre). These methods of approximation can be used to provide filters with adjustment of the passband ripple, group delay deviation or cutoff slope. Acknowledgment This work is supported by Serbian Ministry of Education, Science and Technological Development, Project No. 32009TR. References [1] 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, nu- merical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0377042700005033 [2] 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. [3] 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, Apr. 1973. [4] S. Butterworth, “on the theory filter amplifier,” Experimental Wireless and the Radio Engineer, vol. 7, pp. 536–541, Oct. 1930. [5] H. G. Dimopoulos, “Optimal use of some classical approximations in filter design,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 54, no. 9, pp. 780–784, Sep. 2007. [6] S. C. D. Roy, “modified chebyshev lowpass filters,” International Journal of Circuit Theory and Ap- plications, vol. 38, no. 5, pp. 543–549, 2010.[Online]. Available: http://dx.doi.org/10.1002/cta.585 [7] R. Ramiz and H. Sedef, “General method for designing and simulating of resistively terminated LC ladder filters,” FACTA UNIVERSITATIS, Series: Electronics and Energetics, vol. 12, no. 3, pp. 79–94, 1999. [8] S. Prasad, L. G. Stolarczyk, J. R. Jackson, and E. W. Kang, “Filter synthesis using Legendre polyno- mials,” Proceedings of the IEE, vol. 114, no. 8, pp. 1063–1064, Aug. 1967. [9] M. T. Chryssomallis and J. N. Sahalos, “Filter synthesis using products of Legendre polynomials,” Electrical Engineering, vol. 81, no. 6, pp. 419–424, 1999. [10] D. Živaljević, N. Stamenković, and V. Stojanović, “Nearly monotonic passband low-pass filter design by using sum-of-squared Legendre polynomials,” International Journal of Circuit Theory and Appli- cations, vol. 44, no. 1, pp. 147–161, Jan. 2016. [Online]. Available: http://dx.doi.org/10.1002/cta.2068 [11] 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. [12] i. m. Filanovsky, “Bessel-Butterworth transitional filters,” in 2014 IEEE International Symposium on Circuits and Systems (ISCAS), Jun. 2014, pp. 2105–2108. [13] 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), Jul. 2014, pp. 1–5. 12 N. Stojanović and N. Stamenković: Chebyshev or Legendre). These methods of approximation can be used to provide filters with adjustment of the passband ripple, group delay deviation or cutoff slope. Acknowledgment This work is supported by Serbian Ministry of Education, Science and Technological Development, Project No. 32009TR. References [1] 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, nu- merical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0377042700005033 [2] 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. [3] 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, Apr. 1973. [4] S. Butterworth, “on the theory filter amplifier,” Experimental Wireless and the Radio Engineer, vol. 7, pp. 536–541, Oct. 1930. [5] H. G. Dimopoulos, “Optimal use of some classical approximations in filter design,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 54, no. 9, pp. 780–784, Sep. 2007. [6] S. C. D. Roy, “modified chebyshev lowpass filters,” International Journal of Circuit Theory and Ap- plications, vol. 38, no. 5, pp. 543–549, 2010.[Online]. Available: http://dx.doi.org/10.1002/cta.585 [7] R. Ramiz and H. Sedef, “General method for designing and simulating of resistively terminated LC ladder filters,” FACTA UNIVERSITATIS, Series: Electronics and Energetics, vol. 12, no. 3, pp. 79–94, 1999. [8] S. Prasad, L. G. Stolarczyk, J. R. Jackson, and E. W. Kang, “Filter synthesis using Legendre polyno- mials,” Proceedings of the IEE, vol. 114, no. 8, pp. 1063–1064, Aug. 1967. [9] M. T. Chryssomallis and J. N. Sahalos, “Filter synthesis using products of Legendre polynomials,” Electrical Engineering, vol. 81, no. 6, pp. 419–424, 1999. [10] D. Živaljević, N. Stamenković, and V. Stojanović, “Nearly monotonic passband low-pass filter design by using sum-of-squared Legendre polynomials,” International Journal of Circuit Theory and Appli- cations, vol. 44, no. 1, pp. 147–161, Jan. 2016. [Online]. Available: http://dx.doi.org/10.1002/cta.2068 [11] 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. [12] i. m. Filanovsky, “Bessel-Butterworth transitional filters,” in 2014 IEEE International Symposium on Circuits and Systems (ISCAS), Jun. 2014, pp. 2105–2108. [13] 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), Jul. 2014, pp. 1–5. 362 N. StojaNoviĆ, N. StameNkoviĆ Lowpass Filters approximation Based on the jacobi Polynomials PB 12 N. Stojanović and N. Stamenković: Chebyshev or Legendre). These methods of approximation can be used to provide filters with adjustment of the passband ripple, group delay deviation or cutoff slope. Acknowledgment This work is supported by Serbian Ministry of Education, Science and Technological Development, Project No. 32009TR. References [1] 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, nu- merical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0377042700005033 [2] 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. [3] 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, Apr. 1973. [4] S. Butterworth, “on the theory filter amplifier,” Experimental Wireless and the Radio Engineer, vol. 7, pp. 536–541, Oct. 1930. [5] H. G. Dimopoulos, “Optimal use of some classical approximations in filter design,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 54, no. 9, pp. 780–784, Sep. 2007. [6] S. C. D. Roy, “modified chebyshev lowpass filters,” International Journal of Circuit Theory and Ap- plications, vol. 38, no. 5, pp. 543–549, 2010.[Online]. Available: http://dx.doi.org/10.1002/cta.585 [7] R. Ramiz and H. Sedef, “General method for designing and simulating of resistively terminated LC ladder filters,” FACTA UNIVERSITATIS, Series: Electronics and Energetics, vol. 12, no. 3, pp. 79–94, 1999. [8] S. Prasad, L. G. Stolarczyk, J. R. Jackson, and E. W. Kang, “Filter synthesis using Legendre polyno- mials,” Proceedings of the IEE, vol. 114, no. 8, pp. 1063–1064, Aug. 1967. [9] M. T. Chryssomallis and J. N. Sahalos, “Filter synthesis using products of Legendre polynomials,” Electrical Engineering, vol. 81, no. 6, pp. 419–424, 1999. [10] D. Živaljević, N. Stamenković, and V. Stojanović, “Nearly monotonic passband low-pass filter design by using sum-of-squared Legendre polynomials,” International Journal of Circuit Theory and Appli- cations, vol. 44, no. 1, pp. 147–161, Jan. 2016. [Online]. Available: http://dx.doi.org/10.1002/cta.2068 [11] 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. [12] i. m. Filanovsky, “Bessel-Butterworth transitional filters,” in 2014 IEEE International Symposium on Circuits and Systems (ISCAS), Jun. 2014, pp. 2105–2108. [13] 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), Jul. 2014, pp. 1–5. Lowpass filters approximatin based on the jacobi polynomials 13 [14] B. D. Rakovich and v. B. Litovski, “Least-squares monotonic lowpass filters with sharp cutoff,” Elec- tronics Letters, vol. 9, no. 4, pp. 75–76, Feb. 1973. [15] D. mirković, m. a. Stošović, P. Petković, and v. Litovski, “Design of iiR digital filters with crit- ical monotonic passband amplitude characteristic - A case study,” FACTA UNIVERSITATIS, Series: Electronics and Energetics, vol. 29, no. 2, pp. 269–283, 2016. [16] a. Budak and P. aronhime, “transitional Butterworth-Chebyshev filters,” Circuit Theory, IEEE Trans- actions on, vol. 18, no. 3, pp. 413–415, May 1971. [17] 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. [18] a. Papoulis, “optimum filters with monotonic response,” Proceedings of the IRE, vol. 46, no. 3, pp. 906–609, Mar. 1958. [19] ——, “on monotonic response filters,” Proceedings of the IRE, vol. 47, no. 2, pp. 332–333, Feb. 1959. [20] m. Fukada, “optimum filters of even orders with monotonic response,” IRE Transactions on Circuit Theory, vol. 6, no. 3, pp. 277–281, Sep. 1959. [21] P. Halpern, “optimum monotonic low-pass filters,” Circuit Theory, IEEE Transactions on, vol. 16, no. 2, pp. 240–242, May 1969. [22] 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. [23] M. Lutovac and D. T. sić, “Symbolic signal processing and system analysis,” FACTA UNIVERSITATIS, Series: Electronics and Energetics, vol. 16, no. 3, pp. 423–431, 2003. [24] a. H. Bhrawy, e. H. Doha, S. S. ezz-eldien, and R. a. van Gorder, “A new Jacobi spec- tral collocation method for solving 1+1 fractional Schrödinger equations and fractional coupled Schrödinger systems,” The European Physical Journal Plus, vol. 129, no. 12, pp. 1–21, 2014. [Online]. Available:http://dx.doi.org/10.1140/epjp/i2014-14260-6 [25] C. Beccari, “the use of the shifted jacob1 polynomials in the synthesis of lowpass filters,” Interna- tional Journal of Circuit Theory and Applications, vol. 7, no. 2, pp. 289–295, 1979. [26] B. D. Rakovich, “Designing monotonic low-pass filterscomparison of some methods and criteria,” In- ternational Journal of Circuit Theory and Applications, vol. 2, no. 3, pp. 215–221, Sep. 1974. [Online]. Available: http://dx.doi.org/10.1002/cta.4490020302 [27] D. topisirović, v. Litovski, and m. andrejević Stošović, “Unified theory and state-variable implemen- tation 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 [28] 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. 2013. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0031320313001817 [29] 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. [30] Z. D. Milosavljević and M. V. Gmitrović, “Realizable band-pass filter structures with optimal redun- dancy parameters,” FACTA UNIVERSITATIS, Series: Electronics and Energetics, vol. 13, no. 1, pp. 131–141, 2000. 14 N. Stojanović and N. Stamenković: [31] V. D. Pavlović and Ć. B. Dolićanin, “mathematical foundation for the Christoffel-Darboux formula for classical orthonormal jacobi polynomials applied in filters,” Scientific Publications Of The State Univ. Of Novi Pazar, Series A: Appl. Math. Inform. And Mech.,, vol. 3, no. 2, pp. 139–151, 2011. [32] V. D. Pavlović et al., “New class of filter functions generated most directly by the Christoffel-Darboux formula for classical orthonormal Jacobi polynomials,” Scientific Publications Of The State Univ. Of Novi Pazar, Series A: Appl. Math. Inform. And Mech.,, vol. 5, no. 1, pp. 23–33, 2013.