fuee20260103r 13520-69195-1-LE 041_053.indd FACTA UNIVERSITATIS Ser.: Elec. Energ. vol. 27, no. 4, May 2025, pp. 99-111 https://doi.org/ Original scientific paper OPTIMUM CHEBYSHEV LOWPASS FILTER WITH A PAIR OF IMAGINARY AXIS ZEROS Nikola Stojanović1, Ivan Krstić2, Negovan Stamenković3 1University of Nǐs, Faculty of Electronic Engineering 2University of Kragujevac, Faculty of Engineering 3University of Prǐstina, Faculty of Natural Science and Mathematics ORCID iDs: Nikola Stojanović https://orcid org/0000-0003-3723-8840 Ivan Krstić https://orcid org/0000-0001-7583-3152 Negovan Stamenković https://orcid org/0000-0003-4025-5342 Abstract: The paper compares the characteristics of optimum Chebyshev filters with a finite transmission zero pair of arbitrary multiplicity to those of optimum Chebyshev allpole filters. By introducing a transmission zero pair (single or multiple) at a real frequency into the transcendental form of the Chebyshev polynomial, the filter achieves a specified minimum attenuation extreme value in the stopband, thereby improving the cutoff slope. Additionally, the paper presents a new method for deriving a rational polynomial form of the optimum Chebyshev filtering function from its transcendental form, which is essential for determining the poles of the filter’s transfer function. The method is straightforward and does not rely on optimization or recursive formulas. The proposed approach is validated and illustrated using an example. Although this approximation is primarily intended for microwave filter applications, it can also be applied to both analog and digital signal processing. Key words: Optimum Chebyshev filters, finite transmission zeros, multiple zeros, half- power bandwidth. 1. Introduction In recent publications [1–4], the authors proposed an optimum Chebyshev (C) ap- proximation to design all-pole lowpass filters. Optimum C filters are a variant of C fil- ters that offer an optimum solution for both analog and digital signal processing, where Manuscript received on Mart 25, 2025. Corresponding author: N. Stojanović University of Nǐs, Faculty of Electronic Engineering. E-mail: nikola.stojanovic@elfak.ni.ac.rs. 99 FACTA UNIVERSITATIS Series: Electronics and Energetics Vol. 39, No 1, March 2026, pp. 41 - 53 https://doi.org/10.2298/FUEE2601041S Nikola Stojanović1, Ivan Krstić2, Negovan Stamenković3 Received March 4, 2025; accepted April 23, 2025 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 1University of Niš, Faculty of Electronic Engineering, Niš, Serbia 2University of Kragujevac, Faculty of Engineering, Kragujevac, Serbia 3University of Priština, Faculty of Natural Science and Mathematics, Serbia Abstract. The paper compares the characteristics of optimum Chebyshev filters with a finite transmission zero pair of arbitrary multiplicity to those of optimum Chebyshev allpole filters. By introducing a transmission zero pair (single or multiple) at a real frequency into the transcendental form of the Chebyshev polynomial, the filter achieves a specified minimum attenuation extreme value in the stopband, thereby improving the cutoff slope. Additionally, the paper presents a new method for deriving a rational polynomial form of the optimum Chebyshev filtering function from its transcendental form, which is essential for determining the poles of the filter’s transfer function. The method is straightforward and does not rely on optimization or recursive formulas. The proposed approach is validated and illustrated using an example. Although this approximation is primarily intended for microwave filter applications, it can also be applied to both analog and digital signal processing. Key words: Optimum Chebyshev filters, finite transmission zeros, multiple zeros, half- power bandwidth. © 2026 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper OPTIMUM CHEBYSHEV LOWPASS FILTER WITH A PAIR OF IMAGINARY AXIS ZEROS ORCID IDs: Nikola Stojanović https://orcid org/0000-0003-3723-8840 Ivan Krstić https://orcid org/0000-0001-7583-3152 Negovan Stamenković https://orcid org/0000-0003-4025-5342 100 N. Stojanović, I. Krstić, N. Stamenković the magnitude response of the transfer function is of primary interest. Key features of this approximation include minimum return loss within the half-power bandwidth and maximum out-of-band rejection. In other words, the extremal properties of the Cheby- shev approximation are preserved. Since the characteristic function of the lowpass filter1 represents the ratio of reflected power to transmitted power, the ripple factor minimizes the area below the characteristic function within the half-power passband. The value of the optimum ripple factor, εo, is unique for a given filter degree, whether even or odd, and the transfer functions of the optimum C filters can be systematically cataloged [1]. No all-pole transfer function offers a lower (or equal) return loss with a steeper cutoff slope. The small value of εo leads to reduced out-of-band attenuation. For example, the ripple factor of the optimum 7th degree C all-pole filter is only εo = 0.0935 and decreases as the filter degree increases. Therefore, it is necessary to improve out-of- band attenuation while preserving the passband ripple at εo. One or more pairs of poles can be added to its characteristic function to enhance the performance of the stopband. These poles may be distinct or coincident. There are two approaches to incorporating poles into the optimum C characteristic function, depending on whether it is expressed in polynomial or transcendental form. In the first approach [5–8], a pair of transmission zeros of multiplicity m at the real frequency ω = ±jω0 is introduced into the characteristic function of the optimum C filter, modifying it as follows: Ψn(ω) = ε2oC 2 n(ω)(ω 2 0 − 1)2m/(ω2 − ω2 0) 2m. This modifi- cation preserves the filter degree n and the ripple bandwidth ωr = 1. To determine the pole pair ±ω0 of the characteristic function, which also serves as a transmission zero pair, two nonlinear equations must be solved while ensuring the minimum stopband attenuation is satisfied. As a result, the passband ripples are no longer equal but in- crease toward the passband edge. This approach to improving stopband performance comes at the cost of distorting the extremal properties of the Chebyshev polynomial. The second approach involves the approximation of generalized C filters [9–13], which exhibit equiripple characteristics in both the passband and stopband. In this case, the filtering function of degree n takes the form of a generalized C filtering function: Cn(ω) = cosh { ∑n i=1 acosh [(ω ωi − 1)/(ωi − ω)]}, where ωi represents the position of the ith transmission zero [14]. Some transmission zeros may coincide at the same location (ωi = ω0), while others may be at infinity. An optimization process is required to determine the number and positions of the transmission zeros. Then, a recursive technique is employed to generate the generalized C transfer and reflection rational polynomials given to the determined driving point impedance of a doubly terminated LC ladder network. Ultimately, this approach results in a lossless coupling matrix resonator network that functions as a microwave bandpass filter [14]. It can be concluded that generalized Chebyshev filters preserve the extremal properties of C polynomials and are primarily designed for microwave filter applications, particularly in coupling matrix resonator networks. Consequently, this approximation is unsuitable for improving the out-of-band attenuation of optimum C filters, as their application is not limited to microwave filters but extends to all types of filters where the magnitude 1The characteristic function of the Chebyshev filter is Ψn(ω2) = Pr PL = ε3C2 n(ω), where Pr is reflected power and PL is transmitted power. Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 101 response is of primary importance. The primary objective of this paper is to enhance the out-of-band rejection of an op- timum C all-pole filter by introducing a pair of transmission zeros (single or multiple) in its transcendental form. The attenuation level of the out-of-band lobe is determined by the position of the transmission zero, which is computed using the Newton-Raphson method to solve a system of nonlinear equations. This approach ensures that the extremal properties of the Chebyshev filtering function are preserved. Explicit ratio- nal polynomial expressions are derived for the transcendental C function of arbitrary degree, incorporating a pair of transmission zeros with arbitrary multiplicity. The proposed mathematical framework is presented in detail. A multiple transmission zero pair is used because it allows the odd-degree LC ladder network to be both symmetrical and reciprocal. The validity and efficiency of the proposed method are demonstrated by approxi- mating a seventh-degree rational optimum C filter with a single pair, double pair, and triple pair of real transmission zeros. The properties of these filters are tabulated, and a comparative analysis is provided for all optimum seventh-degree C filters, including the all-pole filter. 2. Rational Optimum C Filters Approximation The squared magnitude of the transmission coefficient of the optimum C filter, representing the ratio of the transmitted power, PL, to the power available from the source, Pa, is given by |S21 (jω)|2 = PL Pa = 1 1 + [ εoCn ( ω ωr )]2 , (1) where Cn(ω/ωr) = cosh[n acosh(ω/ωr)] represents the transcendental form of the C filtering function. When multiplied by εo, this function becomes the optimum all-pole C filtering function. Here, εo is the ripple factor derived for the optimum C filters [1], while ωr is the ripple bandwidth, which normalizes the argument of the characteristic function to make it dimensionless. 2.1. Filtering function derivation The magnitude correction of the transmission coefficient of the optimum C filter to en- hance out-of-band rejection begins by introducing a symmetric pole pair of multiplicity m into the transcendental form of the nth degree C filtering function (1), as follows: Cn,m ( ω ωr ) = cosh { (n− 2m) acosh ( ω ωr ) +m acosh [ α ( ω ωr )] +m acosh [ ᾱ ( ω ωr )]} (2) where α ( ω ωr ) = ω ωr χ− 1 χ− ω ωr and ᾱ ( ω ωr ) = ω ωr χ+ 1 χ+ ω ωr 42 N. STOJANOVIĆ, I. KRSTIĆ, N. STAMENKOVIĆ Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 43 100 N. Stojanović, I. Krstić, N. Stamenković the magnitude response of the transfer function is of primary interest. Key features of this approximation include minimum return loss within the half-power bandwidth and maximum out-of-band rejection. In other words, the extremal properties of the Cheby- shev approximation are preserved. Since the characteristic function of the lowpass filter1 represents the ratio of reflected power to transmitted power, the ripple factor minimizes the area below the characteristic function within the half-power passband. The value of the optimum ripple factor, εo, is unique for a given filter degree, whether even or odd, and the transfer functions of the optimum C filters can be systematically cataloged [1]. No all-pole transfer function offers a lower (or equal) return loss with a steeper cutoff slope. The small value of εo leads to reduced out-of-band attenuation. For example, the ripple factor of the optimum 7th degree C all-pole filter is only εo = 0.0935 and decreases as the filter degree increases. Therefore, it is necessary to improve out-of- band attenuation while preserving the passband ripple at εo. One or more pairs of poles can be added to its characteristic function to enhance the performance of the stopband. These poles may be distinct or coincident. There are two approaches to incorporating poles into the optimum C characteristic function, depending on whether it is expressed in polynomial or transcendental form. In the first approach [5–8], a pair of transmission zeros of multiplicity m at the real frequency ω = ±jω0 is introduced into the characteristic function of the optimum C filter, modifying it as follows: Ψn(ω) = ε2oC 2 n(ω)(ω 2 0 − 1)2m/(ω2 − ω2 0) 2m. This modifi- cation preserves the filter degree n and the ripple bandwidth ωr = 1. To determine the pole pair ±ω0 of the characteristic function, which also serves as a transmission zero pair, two nonlinear equations must be solved while ensuring the minimum stopband attenuation is satisfied. As a result, the passband ripples are no longer equal but in- crease toward the passband edge. This approach to improving stopband performance comes at the cost of distorting the extremal properties of the Chebyshev polynomial. The second approach involves the approximation of generalized C filters [9–13], which exhibit equiripple characteristics in both the passband and stopband. In this case, the filtering function of degree n takes the form of a generalized C filtering function: Cn(ω) = cosh { ∑n i=1 acosh [(ω ωi − 1)/(ωi − ω)]}, where ωi represents the position of the ith transmission zero [14]. Some transmission zeros may coincide at the same location (ωi = ω0), while others may be at infinity. An optimization process is required to determine the number and positions of the transmission zeros. Then, a recursive technique is employed to generate the generalized C transfer and reflection rational polynomials given to the determined driving point impedance of a doubly terminated LC ladder network. Ultimately, this approach results in a lossless coupling matrix resonator network that functions as a microwave bandpass filter [14]. It can be concluded that generalized Chebyshev filters preserve the extremal properties of C polynomials and are primarily designed for microwave filter applications, particularly in coupling matrix resonator networks. Consequently, this approximation is unsuitable for improving the out-of-band attenuation of optimum C filters, as their application is not limited to microwave filters but extends to all types of filters where the magnitude 1The characteristic function of the Chebyshev filter is Ψn(ω2) = Pr PL = ε3C2 n(ω), where Pr is reflected power and PL is transmitted power. Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 101 response is of primary importance. The primary objective of this paper is to enhance the out-of-band rejection of an op- timum C all-pole filter by introducing a pair of transmission zeros (single or multiple) in its transcendental form. The attenuation level of the out-of-band lobe is determined by the position of the transmission zero, which is computed using the Newton-Raphson method to solve a system of nonlinear equations. This approach ensures that the extremal properties of the Chebyshev filtering function are preserved. Explicit ratio- nal polynomial expressions are derived for the transcendental C function of arbitrary degree, incorporating a pair of transmission zeros with arbitrary multiplicity. The proposed mathematical framework is presented in detail. A multiple transmission zero pair is used because it allows the odd-degree LC ladder network to be both symmetrical and reciprocal. The validity and efficiency of the proposed method are demonstrated by approxi- mating a seventh-degree rational optimum C filter with a single pair, double pair, and triple pair of real transmission zeros. The properties of these filters are tabulated, and a comparative analysis is provided for all optimum seventh-degree C filters, including the all-pole filter. 2. Rational Optimum C Filters Approximation The squared magnitude of the transmission coefficient of the optimum C filter, representing the ratio of the transmitted power, PL, to the power available from the source, Pa, is given by |S21 (jω)|2 = PL Pa = 1 1 + [ εoCn ( ω ωr )]2 , (1) where Cn(ω/ωr) = cosh[n acosh(ω/ωr)] represents the transcendental form of the C filtering function. When multiplied by εo, this function becomes the optimum all-pole C filtering function. Here, εo is the ripple factor derived for the optimum C filters [1], while ωr is the ripple bandwidth, which normalizes the argument of the characteristic function to make it dimensionless. 2.1. Filtering function derivation The magnitude correction of the transmission coefficient of the optimum C filter to en- hance out-of-band rejection begins by introducing a symmetric pole pair of multiplicity m into the transcendental form of the nth degree C filtering function (1), as follows: Cn,m ( ω ωr ) = cosh { (n− 2m) acosh ( ω ωr ) +m acosh [ α ( ω ωr )] +m acosh [ ᾱ ( ω ωr )]} (2) where α ( ω ωr ) = ω ωr χ− 1 χ− ω ωr and ᾱ ( ω ωr ) = ω ωr χ+ 1 χ+ ω ωr 42 N. STOJANOVIĆ, I. KRSTIĆ, N. STAMENKOVIĆ Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 43 102 N. Stojanović, I. Krstić, N. Stamenković and χ is a dimensionless coefficient. This normalization ensures that the arguments of both hyperbolic functions are also dimensionless. In particular, if χ approaches infinity or if m = 0, then (2) reduces to the filtering function C of an all-pole filter. Since εo is associated with the optimum C filter, the function εoCn,m(ω) represents the filtering function of the rational optimum C filter. Without loss of generality, ωr can be set to one, ensuring that Cn,m(1) = 1. The multiple pole pair of the C filtering function (2) is located at an out-of-band frequency, producing a lobe. The frequency response shows ωm, where the lobe reaches a local minimum attenuation. By setting the first derivative of (2) to zero, the frequency ωm can be determined in closed form as: ωm = ±  χ2 + 2mχ  χ2 − 1 n− 2m , (3) where ωr = 1. The coefficient χ serves as a degree of freedom to adjust the minimum out-of-band insertion loss level (ILs in dB). Its value is obtained by solving the equation εoCn,m(ωm) = Amin, where Amin = √ 10ILs/10 − 1. The process of determining χ such that the filtering function (2) produces Amin/εo at frequency ωm is iterative [15]. Given specified values of n and m, the two unknowns, χ and ωm, may be determined numerically by solving two nonlinear equations. These nonlinear equations can be expressed as: f1(χ, ωm) =χ2 + 2mχ  χ2 − 1 n− 2m − ω2 m = 0 f2(χ, ωm) =Cn,m(ωm) + Amin εo = 0 (4) The Newton-Raphson iterative formula can be applied to solve this system of nonlinear equations. In kth iteration, the solution is given by  χ(k+1) ω (k+1) m  =  χ(k) ω (k) m  −   ∂f1(χ (k), ω (k) m ) ∂χ ∂f1(χ (k), ω (k) m ) ∂ωm ∂f2(χ (k), ω (k) m ) ∂χ ∂f2(χ (k), ω (k) m ) ∂ωm   −1 ×  f1(χ (k), ω (k) m ) f2(χ (k), ω (k) m )  (5) The initial value χ(0) can be any value greater than one, while ω (0) m is determined by substituting χ(0) into f1(χ (0), ωm). Note that the Newton-Raphson method is imple- mented by the Mathematica’s built-in function FindRoot. This procedure can be applied to filtering functions with an arbitrary passband ripple. However, if εo is used to calculate Amin, then Cn,m(ω) is considered the optimum filtering function. 2.2. Rational polynomial derivation Once χ is determined, the rational polynomial must be derived from the transcendental form of the characteristic function (2) to obtain the transfer function (1). The filtering 44 N. STOJANOVIĆ, I. KRSTIĆ, N. STAMENKOVIĆ Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 45 102 N. Stojanović, I. Krstić, N. Stamenković and χ is a dimensionless coefficient. This normalization ensures that the arguments of both hyperbolic functions are also dimensionless. In particular, if χ approaches infinity or if m = 0, then (2) reduces to the filtering function C of an all-pole filter. Since εo is associated with the optimum C filter, the function εoCn,m(ω) represents the filtering function of the rational optimum C filter. Without loss of generality, ωr can be set to one, ensuring that Cn,m(1) = 1. The multiple pole pair of the C filtering function (2) is located at an out-of-band frequency, producing a lobe. The frequency response shows ωm, where the lobe reaches a local minimum attenuation. By setting the first derivative of (2) to zero, the frequency ωm can be determined in closed form as: ωm = ±  χ2 + 2mχ  χ2 − 1 n− 2m , (3) where ωr = 1. The coefficient χ serves as a degree of freedom to adjust the minimum out-of-band insertion loss level (ILs in dB). Its value is obtained by solving the equation εoCn,m(ωm) = Amin, where Amin = √ 10ILs/10 − 1. The process of determining χ such that the filtering function (2) produces Amin/εo at frequency ωm is iterative [15]. Given specified values of n and m, the two unknowns, χ and ωm, may be determined numerically by solving two nonlinear equations. These nonlinear equations can be expressed as: f1(χ, ωm) =χ2 + 2mχ  χ2 − 1 n− 2m − ω2 m = 0 f2(χ, ωm) =Cn,m(ωm) + Amin εo = 0 (4) The Newton-Raphson iterative formula can be applied to solve this system of nonlinear equations. In kth iteration, the solution is given by  χ(k+1) ω (k+1) m  =  χ(k) ω (k) m  −   ∂f1(χ (k), ω (k) m ) ∂χ ∂f1(χ (k), ω (k) m ) ∂ωm ∂f2(χ (k), ω (k) m ) ∂χ ∂f2(χ (k), ω (k) m ) ∂ωm   −1 ×  f1(χ (k), ω (k) m ) f2(χ (k), ω (k) m )  (5) The initial value χ(0) can be any value greater than one, while ω (0) m is determined by substituting χ(0) into f1(χ (0), ωm). Note that the Newton-Raphson method is imple- mented by the Mathematica’s built-in function FindRoot. This procedure can be applied to filtering functions with an arbitrary passband ripple. However, if εo is used to calculate Amin, then Cn,m(ω) is considered the optimum filtering function. 2.2. Rational polynomial derivation Once χ is determined, the rational polynomial must be derived from the transcendental form of the characteristic function (2) to obtain the transfer function (1). The filtering Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 103 function can be simplified using the hyperbolic addition formula: acosh(α)+acosh(ᾱ) = acosh(γ), where α and ᾱ are defined in equation (2), and γ is given by γ = αᾱ+ √ (α2 − 1)(ᾱ2 − 1). (6) After simple manipulation, the following expression for γ (ω) can be derived: γ(ω) = ω2 ( 2χ2 − 1 ) − χ2 χ2 − ω2 , (7) and (2) can be rewritten in a simplified form as Cn,m(ω) = cosh [ (n− 2m) acosh(ω) +m acosh(γ) ] . (8) Using the identity acosh(x) = log ( x+ √ x2 − 1 ) , the following expression is obtained Cn,m(ω) = cosh [ (n− 2m) log ( ω + √ ω2 − 1 ) +m log ( γ + √ γ2 − 1 )] , (9) or in a more compact form Cn,m(ω) = cosh { log [( ω + √ ω2 − 1 )n−2m × ( γ + √ γ2 − 1 )m]} . (10) After log and cosh operations, we obtain the algebraic form: Cn,m(ω) = 1 2 [( ω + √ ω2 − 1 )n−2m ( γ + √ γ2 − 1 )m + ( ω − √ ω2 − 1 )n−2m ( γ − √ γ2 − 1 )m] . (11) Substituting (7) into √ γ2 − 1 yields the following expression: √ γ2 − 1 = 2χω √ χ2 − 1 √ ω2 − 1 χ2 − ω2 . (12) Rewriting γ ± √ γ2 − 1 in the form a ± b √ ω2 − 1 allows (11) to be expressed as the quotient of two algebraic functions: Cn,m(ω) = 1 2 ( ω + √ ω2 − 1 )n−2m( a+ b √ ω2 − 1 )m + ( ω − √ ω2 − 1 )n−2m( a− b √ ω2 − 1 )m (χ2 − ω2)m (13) where a = ω2 ( 2χ2 − 1 ) − χ2 is the numerator of (7), and b = 2χω √ χ2 − 1 is the numerator of (12) excluding the multiplicative factor of √ ω2 − 1. Utilization of the binomial expansion formula gives ( a± b √ ω2 − 1 )m = A2m(ω)± B2m+1(ω)√ ω2 − 1 (14) 44 N. STOJANOVIĆ, I. KRSTIĆ, N. STAMENKOVIĆ Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 45 104 N. Stojanović, I. Krstić, N. Stamenković where A2m(ω) = ⌊m/2⌋∑ k=0 ( m 2k ) am−2k b2k(ω2 − 1)k and B2m+1(ω) = ⌊(m−1)/2⌋∑ k=0 ( m 2k + 1 ) am−2k−1 b2k+1(ω2 − 1)k+1 are polynomials of degrees 2m and 2m+ 1, respectively. For m = 1 and m = 2, these polynomials are A2 (ω) = ω2 ( 2χ2 − 1 ) − χ2, B3 (ω) = 2χω3 √ χ2 − 1− 2χω √ χ2 − 1, (15) and A4 (ω) = A2 2 (ω) + 2χω √ χ2 − 1B3 (ω) , B5 (ω) = 2A2 (ω)B3 (ω) . (16) By substituting A2m(ω) and B2m+1(ω) into (13), it can be rearranged as follows: Cn,m(ω) = 1 2 [ (ω + √ ω2 − 1)n−2m + (ω − √ ω2 − 1)n−2m ] A2m(ω) + [ (ω + √ ω2 − 1)n−2m − (ω − √ ω2 − 1)n−2m ] B2m+1(ω)√ ω2−1 (χ2 − ω2)m (17) In the numerator of (17), two hypergeometric functions can be rewritten regarding two Chebyshev polynomials. The first function is (x+ √ x2 − 1)n+(x− √ x2 − 1)n = 2Tn(x), where Tn(x) is the Chebyshev polynomial of the first kind, while the second function is (x+ √ x2 − 1)n − (x− √ x2 − 1)n = 2Un(x) √ x2 − 1, where Un(x) is the Chebyshev polynomial of the second kind. Using these relationships, (17) is finally expressed in the form of the rational optimum C filtering function: Cn,m(ω) = A2m(ω)Tn−2m(ω) +B2m+1(ω)Un−2m−1(ω)( χ2 − ω2 )m = Nn (ω) D2m (ω) (18) in which the two terms √ ω2 − 1 in the addend cancel each other out. This filtering function is optimum because εo is embedded in the calculation of χ. The polynomial Nn (ω) is a purely odd or purely even function and satisfies the condition given in [14]. It can be noted that Cn,m(1) = 1, i.e. ωr = 1. Polynomial Nn(ω) can be obtained using only two auxiliary polynomials, derived from the binomial expansion formula (14), along with two known Chebyshev polyno- mials of the first and second kinds. The degrees of the purely even polynomial A2m(ω) and the purely odd polynomial B2m+1(ω) depend solely on the multiplicity of the zero pair of the transmission coefficient and not on the filter degree n. If the number of zero pairs increases, the filter degree n stays the same, eliminating the need for optimization 46 N. STOJANOVIĆ, I. KRSTIĆ, N. STAMENKOVIĆ Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 47 104 N. Stojanović, I. Krstić, N. Stamenković where A2m(ω) = ⌊m/2⌋∑ k=0 ( m 2k ) am−2k b2k(ω2 − 1)k and B2m+1(ω) = ⌊(m−1)/2⌋∑ k=0 ( m 2k + 1 ) am−2k−1 b2k+1(ω2 − 1)k+1 are polynomials of degrees 2m and 2m+ 1, respectively. For m = 1 and m = 2, these polynomials are A2 (ω) = ω2 ( 2χ2 − 1 ) − χ2, B3 (ω) = 2χω3 √ χ2 − 1− 2χω √ χ2 − 1, (15) and A4 (ω) = A2 2 (ω) + 2χω √ χ2 − 1B3 (ω) , B5 (ω) = 2A2 (ω)B3 (ω) . (16) By substituting A2m(ω) and B2m+1(ω) into (13), it can be rearranged as follows: Cn,m(ω) = 1 2 [ (ω + √ ω2 − 1)n−2m + (ω − √ ω2 − 1)n−2m ] A2m(ω) + [ (ω + √ ω2 − 1)n−2m − (ω − √ ω2 − 1)n−2m ] B2m+1(ω)√ ω2−1 (χ2 − ω2)m (17) In the numerator of (17), two hypergeometric functions can be rewritten regarding two Chebyshev polynomials. The first function is (x+ √ x2 − 1)n+(x− √ x2 − 1)n = 2Tn(x), where Tn(x) is the Chebyshev polynomial of the first kind, while the second function is (x+ √ x2 − 1)n − (x− √ x2 − 1)n = 2Un(x) √ x2 − 1, where Un(x) is the Chebyshev polynomial of the second kind. Using these relationships, (17) is finally expressed in the form of the rational optimum C filtering function: Cn,m(ω) = A2m(ω)Tn−2m(ω) +B2m+1(ω)Un−2m−1(ω)( χ2 − ω2 )m = Nn (ω) D2m (ω) (18) in which the two terms √ ω2 − 1 in the addend cancel each other out. This filtering function is optimum because εo is embedded in the calculation of χ. The polynomial Nn (ω) is a purely odd or purely even function and satisfies the condition given in [14]. It can be noted that Cn,m(1) = 1, i.e. ωr = 1. Polynomial Nn(ω) can be obtained using only two auxiliary polynomials, derived from the binomial expansion formula (14), along with two known Chebyshev polyno- mials of the first and second kinds. The degrees of the purely even polynomial A2m(ω) and the purely odd polynomial B2m+1(ω) depend solely on the multiplicity of the zero pair of the transmission coefficient and not on the filter degree n. If the number of zero pairs increases, the filter degree n stays the same, eliminating the need for optimization Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 105 or recursion formulas. As a result, the mathematical framework is significantly simpler than the well-known solution presented in [14] for generalized Chebyshev filters. To restore the half-power point ωc as the edge of the passband of the optimum C filter, it is necessary to determine the frequency at which (2) or (18), multiplied by εo, first reaches a value of 1. This requires solving the equation2 εoCn,m(ω/ωr) = 1. The solution is given by ωc/ωr = λ > 1, where λ is a dimensionless parameter. The new value of the edge of the ripple band, ωr = ωc/λ, can be used to renormalize the filtering function (18) to the half-power bandwidth. For convenience and without loss of generality, the half-power point can be set to one (ωc = 1), resulting in a new ripple band edge of ωr = 1/λ < 1. This value is then used to renormalize the optimum rational C filtering function. C(ε) n,m(ω) = εo Nn(ω) D2m(ω) ∣∣∣∣ ω →ωλ = Nn(ω) D2m(ω) (19) ensuring that C(ε) n,m(1) = 1. The ripple factor εo is an embedded parameter in the rational optimum C approximation, arising from the renormalization process related to the half-power bandwidth. This is denoted by the superscript (ε) in (19). 2.3. Examples and Comparison The frequency responses of the filtering functions of the optimum 7th-degree C filter for m = 1, 2 and 3, expressed in the rational polynomial form (18) and scaled by εo, are shown in Fig. 1. To illustrate that the ripples are equal, the scale within the ripple band −1 ≤ ω ≤ 1 is magnified 1000 times. The ripples reach values of ±εo at the edges of the ripple band ωr = 1. -4 -3 -2 -1 0 1 2 3 4 Normalized frequency, ω -600 -400 -200 0 200 400 600 ǫ o C n ,m (ω ) E n la rg ed sc al e in p as sb an d 0.2 -0.2 m = 1 m = 2 m = 3 Fig. 1. The 7th-degree optimum C filtering functions, ILs = 50 dB. The calculation of the rational optimum C filtering function (18), which features a pair of triple poles whose properties are summarized in Table 1, is presented. The 2This calculation explicitly states that ωr ̸= 1 can be used for frequency renormalization. 46 N. STOJANOVIĆ, I. KRSTIĆ, N. STAMENKOVIĆ Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 47 106 N. Stojanović, I. Krstić, N. Stamenković optimum ripple factor is εo = 0.0935, while the coefficient χ = 1.611042, is determined to achieve a minimum stopband insertion loss of ILs = 50 dB (or 316.1867 times). The two auxiliary polynomials are given by: A6(ω) = 281.86ω6 − 473.981ω4 + 213.666ω2 − 17.484, B7(ω) = 281.858ω7 − 614.885ω5 + 415.276ω3 − 82.248ω These polynomials, along with the first-degree Chebyshev polynomial of the first kind and the zero-degree Chebyshev polynomial of the second kind, are used to construct the rational polynomial form of the optimum C filtering function (18), multiplied by εo: C7,3(ω) = 0.0935 −52.712ω7 + 101.817ω5 − 58.8106ω3 + 9.32565ω (ω2 − 2.59551)3 (20) ensuring that C7,3(1) = 0.00935. To achieve a normalized half-power bandwidth of one, the parameter λ = 1.060296 is calculated for the final renormalization of equation (20) multiplied with the optimum ripple factor. This yields the final rational optimum C filtering function (19): C(ε) 7,3(ω) = Nn(ω) D2m(ω) = −79.408ω7 + 136.43ω5 − 70.098ω3 + 9.8872ω (1.1242ω2 − 2.59546)3 ensuring that C(ε) 7,3(1) = 1. For comparison, the properties of other optimum C filtering functions of the 7th-degree are also listed in Table 1. In all approximations, the optimum value of the ripple factor, εo = 0.0935, is used to calculate χ, ensuring an insertion loss of ILs = 50 dB at the frequency ωm. Table 1. Properties of the 7th-degree optimum C filtering functions Properties Characteristic function m = 0 m = 1 m = 2 m = 3 λ 1.097126 1.077877 1.062136 1.060296 χ ∞ 1.500606 1.455552 1.611042 ωm none 1.709891 2.042391 3.847784 ωr 0.911472 0.927748 0.941498 0.943132 Area 0.028670 0.023813 0.019922 0.019531 Table 1 provides a two-part comparison: the first part contrasts the all-pole op- timum C characteristic function with its rational counterparts, while the second part compares the rational optimum C functions among themselves. The area below the characteristic function within half power passband3 (Area) is essential for compari- son. The first comparison reveals that the all-pole optimum C filtering function has the highest Area, i.e. the reflected power, while its ripple band is the smallest. The second comparison shows that as m increases, Area decreases, whereas the ripple band expands. Consequently, the area between the ripple band edge frequency ωr and the half-power frequency ωc = 1 shrinks. 3The area is computed using the integral: Area = ∫ 1 0 [C(ε) n,m(ω)]2dω. 48 N. STOJANOVIĆ, I. KRSTIĆ, N. STAMENKOVIĆ Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 49 106 N. Stojanović, I. Krstić, N. Stamenković optimum ripple factor is εo = 0.0935, while the coefficient χ = 1.611042, is determined to achieve a minimum stopband insertion loss of ILs = 50 dB (or 316.1867 times). The two auxiliary polynomials are given by: A6(ω) = 281.86ω6 − 473.981ω4 + 213.666ω2 − 17.484, B7(ω) = 281.858ω7 − 614.885ω5 + 415.276ω3 − 82.248ω These polynomials, along with the first-degree Chebyshev polynomial of the first kind and the zero-degree Chebyshev polynomial of the second kind, are used to construct the rational polynomial form of the optimum C filtering function (18), multiplied by εo: C7,3(ω) = 0.0935 −52.712ω7 + 101.817ω5 − 58.8106ω3 + 9.32565ω (ω2 − 2.59551)3 (20) ensuring that C7,3(1) = 0.00935. To achieve a normalized half-power bandwidth of one, the parameter λ = 1.060296 is calculated for the final renormalization of equation (20) multiplied with the optimum ripple factor. This yields the final rational optimum C filtering function (19): C(ε) 7,3(ω) = Nn(ω) D2m(ω) = −79.408ω7 + 136.43ω5 − 70.098ω3 + 9.8872ω (1.1242ω2 − 2.59546)3 ensuring that C(ε) 7,3(1) = 1. For comparison, the properties of other optimum C filtering functions of the 7th-degree are also listed in Table 1. In all approximations, the optimum value of the ripple factor, εo = 0.0935, is used to calculate χ, ensuring an insertion loss of ILs = 50 dB at the frequency ωm. Table 1. Properties of the 7th-degree optimum C filtering functions Properties Characteristic function m = 0 m = 1 m = 2 m = 3 λ 1.097126 1.077877 1.062136 1.060296 χ ∞ 1.500606 1.455552 1.611042 ωm none 1.709891 2.042391 3.847784 ωr 0.911472 0.927748 0.941498 0.943132 Area 0.028670 0.023813 0.019922 0.019531 Table 1 provides a two-part comparison: the first part contrasts the all-pole op- timum C characteristic function with its rational counterparts, while the second part compares the rational optimum C functions among themselves. The area below the characteristic function within half power passband3 (Area) is essential for compari- son. The first comparison reveals that the all-pole optimum C filtering function has the highest Area, i.e. the reflected power, while its ripple band is the smallest. The second comparison shows that as m increases, Area decreases, whereas the ripple band expands. Consequently, the area between the ripple band edge frequency ωr and the half-power frequency ωc = 1 shrinks. 3The area is computed using the integral: Area = ∫ 1 0 [C(ε) n,m(ω)]2dω. Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 107 3. Transmission Coefficient The rational squared magnitude function of the transmission coefficient of the op- timal C filter is obtained by substituting the all-pole characteristic function with the rational characteristic function—given by equation (19) squared—into equation (1). The transmission coefficient S21(s) can then be determined from (1) using the stan- dard procedure of analytic continuation over the entire s-plane, which corresponds to replacing ω with −js in the given |S21(jω)|2. The pole locations of the transmission coefficient can be found by solving for the roots of the polynomial in its denominator4, given by: D2 2m(−js) +N 2 n(−js) = 0. (21) From these roots, the poles pi = σi ± jωi, i = 1, 2, . . . , n, that lie in the left half-plane of s-plane are selected. The pole-zero positions of the 7th-degree transfer functions that correspond to the optimum all-pole Chebyshev and the rational optimum C char- acteristic functions for m = 1, 2, and 3 are shown in Fig. 2. -1.5 -1 -0.5 0 Real Part -1.5 -1 -0.5 0 0.5 1 1.5 Im a g in a ry P a rt m = 3 m = 2 m = 1 m = 0 Fig. 2. Pole-zero positions of 7th-degree optimum C filters. The zero locations for m = 1, 2 and 3 are represented by ⋄, □ and ◦, respectively. The differences in pole positions of all transfer functions are minor. However, the calculation of the critical quality factor, defined as Qc = −0.5|p1|/σ1, where p1 = σ1 ± jω1 is the pole pair that is closest to the imaginary axis, is of practical interest. This factor is commonly used to compare the sensitivity of networks and its values are listed in Table 2. The value of Qc increases with increasing m. However, the difference in Qc between m = 2 and m = 3 is minimal, indicating that their passband sensitivities are almost the same. Fig. 3 compares the steady-state responses, including the insertion loss in dB, given by IL = 20 log10 (|S21(jω)|), and the group delay responses of the filters, whose 4Since |S21(jω)|2 = 1 1+ N2 n(ω) D2 2m(ω) ∣∣∣∣∣ ω=−js = D2 2m(−js) D2 2m(−js)+N2 n(−js) , follows (21). 48 N. STOJANOVIĆ, I. KRSTIĆ, N. STAMENKOVIĆ Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 49 108 N. Stojanović, I. Krstić, N. Stamenković pole-zero positions are shown in Fig. 2. All filters share the same half-power band- width (normalized to 1) and exhibit nearly identical passband characteristics. Table 2 provides the ripple band, ωr = 1/λ, and transmission zero, ω0 = χ/λ, for each filter. The main difference lies in their stopband behavior. 10−1 100 Normalized frequency, ω 0 10 20 30 40 50 60 70 S to p b a n d in se rt io n lo ss , d B ωs 0 0.1 0.2 P a ss b a n d R ip p le , d B 0 10 20 30 40 G ro u p d el a y, s m = 3 m = 2 m = 1 m = 0 Fig. 3. The steady-state responses of the 7th-degree optimum C lowpass filters for m = 0, 1, 2, and 3. The stopband edge frequencies, ωs, are defined as the frequencies where the out- of-band attenuation first reaches the specified minimum value of ILs = 50 dB. These values are listed in Table 2. In Fig. 3, the stopband edge frequency is explicitly assigned only for m = 0, as it is easily identifiable for the other three filters. It may be seen that all rational optimum C filters have one attenuation lobe, each with a level of 50 dB, at frequencies ω′ m, which are listed in Table 2. The cutoff slope, commonly defined as the first derivative of the transmission coef- ficient (1) at the half-power point, of rational optimum C filter can be calculated using the first derivative of its characteristic function as CS = d dω |S21(jω)| ∣∣∣∣ ω=1 = − 1 2 √ 2 d dω C(ε) n,m(ω) ∣∣∣∣ ω=1 (22) since C(ε) n,m(1) = 1. The cutoff slopes of all considered 7th-degree filters are provided in Table 2. Increasing the multiplicity of the transmission zero results in a lower stopband edge frequency and steeper cutoff slope. However, increasing the transmission zero multiplicity beyond two is unnecessary, as the improvement becomes negligible. Notably, the magnitude response of the filter with a triple zero pair exhibits a broad stopband region with extremely high attenuation. All group delay responses increase monotonically in the passband. The differences become evident only in the peak value near the passband edge, which increases as m 50 N. STOJANOVIĆ, I. KRSTIĆ, N. STAMENKOVIĆ Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 51 108 N. Stojanović, I. Krstić, N. Stamenković pole-zero positions are shown in Fig. 2. All filters share the same half-power band- width (normalized to 1) and exhibit nearly identical passband characteristics. Table 2 provides the ripple band, ωr = 1/λ, and transmission zero, ω0 = χ/λ, for each filter. The main difference lies in their stopband behavior. 10−1 100 Normalized frequency, ω 0 10 20 30 40 50 60 70 S to p b a n d in se rt io n lo ss , d B ωs 0 0.1 0.2 P a ss b a n d R ip p le , d B 0 10 20 30 40 G ro u p d el a y, s m = 3 m = 2 m = 1 m = 0 Fig. 3. The steady-state responses of the 7th-degree optimum C lowpass filters for m = 0, 1, 2, and 3. The stopband edge frequencies, ωs, are defined as the frequencies where the out- of-band attenuation first reaches the specified minimum value of ILs = 50 dB. These values are listed in Table 2. In Fig. 3, the stopband edge frequency is explicitly assigned only for m = 0, as it is easily identifiable for the other three filters. It may be seen that all rational optimum C filters have one attenuation lobe, each with a level of 50 dB, at frequencies ω′ m, which are listed in Table 2. The cutoff slope, commonly defined as the first derivative of the transmission coef- ficient (1) at the half-power point, of rational optimum C filter can be calculated using the first derivative of its characteristic function as CS = d dω |S21(jω)| ∣∣∣∣ ω=1 = − 1 2 √ 2 d dω C(ε) n,m(ω) ∣∣∣∣ ω=1 (22) since C(ε) n,m(1) = 1. The cutoff slopes of all considered 7th-degree filters are provided in Table 2. Increasing the multiplicity of the transmission zero results in a lower stopband edge frequency and steeper cutoff slope. However, increasing the transmission zero multiplicity beyond two is unnecessary, as the improvement becomes negligible. Notably, the magnitude response of the filter with a triple zero pair exhibits a broad stopband region with extremely high attenuation. All group delay responses increase monotonically in the passband. The differences become evident only in the peak value near the passband edge, which increases as m Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 109 increases. For m = 2 and m = 3, the group delays are nearly identical. The peak group delay values, τm, are calculated and listed in Table 2. Another important parameter to consider is the reflection coefficient |S11(jω)|2 = [ C(ε) n,m(ω) ]2 1 + [ C(ε) n,m(ω) ]2 (23) The return loss frequency responses in decibels, RL = 20 log10 (|S11(jω)|), of the 7th- degree optimum C filters with for m = 0, 1, 2 and 3 are shown in Fig. 4. The return loss levels of all filters are determined with εo and exhibit the equiripple behavior. The differences in the spacing of reflection zeros among 7th-degree optimum C filters are sufficient for functional tuning. However, the optimum all-pole C filter has the widest separation, while for the rational optimum C filters, this separation slightly decreases as m increases. Additionally, as the filter degree increases, the zero separation decreases. 10−1 100 Normalized frequency, ω -40 -35 -30 -25 -20 -15 -10 -5 0 R et u rn lo ss , d B m = 3 m = 2 m = 1 m = 0 Fig. 4. The return loss responses of the 7th-degree optimum C lowpass filters for m = 0, 1, 2, and 3. Table 2. Properties of 7th-degree optimum C filters Properties Filters m = 0 m = 1 m = 2 m = 3 ω0 = χ/λ ∞ 1.392187 1.370401 1.519427 ω′ m = ωm/λ none 1.586262 1.922885 3.628922 ωr = 1/λ 0.911472 0.927748 0.941498 0.943132 Qc 5.3421 6.6824 8.3906 8.6179 ωs 1.7358 1.3457 1.2674 1.2892 CS 16.94209 21.55732 27.22896 27.85117 τm 14.9337 18.1713 22.4434 23.1394 50 N. STOJANOVIĆ, I. KRSTIĆ, N. STAMENKOVIĆ Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 51 110 N. Stojanović, I. Krstić, N. Stamenković The magnitude correction of the transmission coefficient can be applied to both odd- and even-degree filters. Introducing a single transmission zero significantly en- hances the out-of-band rejection of the all-pole filter by steepening the cutoff slope and lowering the stopband edge frequency. While increasing the multiplicity of the trans- mission zero provides a slight improvement in the cutoff slope, extending it beyond two is unnecessary, as it leads to higher sensitivity and increased network complexity. However, a triple-zero pair results in a magnitude response with a wide stopband region and extremely high attenuation in the frequency band around the transmission zero. 4. Conclusion This paper presents an efficient and straightforward procedure for correcting the magni- tude function in optimal C filters, specifically designed for low-pass filters that prioritize magnitude specifications. The Chebyshev filtering function is modified by introduc- ing a symmetric pair of poles of arbitrary multiplicity into its transcendental form, preserving the extremal properties of the passband’s equal-ripple behavior. A new and efficient method is proposed for deriving a rational polynomial from the transcendental form of the optimum C filtering function that exhibits a transmission zero pair, single or multiple, at a finite frequency. As an example, the derivation of a 7th-degree rational polynomial from the transcendental optimum Chebyshev filtering function is demonstrated for cases of single, double, and triple pole pairs. The proposed approach enhances the cutoff slope and slightly reduces reflection power at the filter’s input terminals compared to the optimum C all-pole filter. How- ever, a single or double transmission pole pair is generally preferred for filter design, as increasing the multiplicity provides minimal improvements in the cutoff slope and return loss while increasing hardware complexity, sensitivity, and tuning time. Acknowledgment: The authors would like to express their gratitude to Professor V. S. Stojanović from the University of Nǐs, Serbia for his invaluable comments and suggestions. References [1] S. Nikolić, N. Stojanović, N. Stamenković, and I. Krstić, “Optimum allpole filters with Chebyshev passband magnitude response,” AEU - International Journal of Electronics and Communica- tions, vol. 135, no. 6, p. 153740, 2021, doi:10.1016/j.aeue.2021.153740. [2] N. Stojanović, N. Stamenković, and I. Krstić, “An improved design method for even-degree op- timum allpole filters with equiripple passband responses,” AEU - International Journal of Elec- tronics and Communications, vol. 159, no. 2, p. 154469, 2023, doi:10.1016/j.aeue.2022.154469. [3] N. Stojanović, I. Krstić, and N. Stamenković, “Recursive digital filters with optimum equiripple passband magnitude characteristic,” AEU - International Journal of Electronics and Commu- nications, vol. 170, no. 10, p. 154851, 2023, doi:10.1016/j.aeue.2023.154851. [4] N. Stojanović, I. Krstić, and N. Stamenković, “Performance analysis of optimum Chebyshev filters at microwave frequencies,” AEU - International Journal of Electronics and Communica- tions, vol. 187, no. 12, p. 155502, 2024, doi:10.1016/j.aeue.2024.155502. [5] M. Agarwal and A. Sedra, “On designing sharp cutoff low-pass filters,” IEEE Trans- actions on Audio and Electroacoustics, vol. 20, no. 2, pp. 138–141, Jun. 1972, doi:10.1109/TAU.1972.1162359. 52 N. STOJANOVIĆ, I. KRSTIĆ, N. STAMENKOVIĆ Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 53 110 N. Stojanović, I. Krstić, N. Stamenković The magnitude correction of the transmission coefficient can be applied to both odd- and even-degree filters. Introducing a single transmission zero significantly en- hances the out-of-band rejection of the all-pole filter by steepening the cutoff slope and lowering the stopband edge frequency. While increasing the multiplicity of the trans- mission zero provides a slight improvement in the cutoff slope, extending it beyond two is unnecessary, as it leads to higher sensitivity and increased network complexity. However, a triple-zero pair results in a magnitude response with a wide stopband region and extremely high attenuation in the frequency band around the transmission zero. 4. Conclusion This paper presents an efficient and straightforward procedure for correcting the magni- tude function in optimal C filters, specifically designed for low-pass filters that prioritize magnitude specifications. The Chebyshev filtering function is modified by introduc- ing a symmetric pair of poles of arbitrary multiplicity into its transcendental form, preserving the extremal properties of the passband’s equal-ripple behavior. A new and efficient method is proposed for deriving a rational polynomial from the transcendental form of the optimum C filtering function that exhibits a transmission zero pair, single or multiple, at a finite frequency. As an example, the derivation of a 7th-degree rational polynomial from the transcendental optimum Chebyshev filtering function is demonstrated for cases of single, double, and triple pole pairs. The proposed approach enhances the cutoff slope and slightly reduces reflection power at the filter’s input terminals compared to the optimum C all-pole filter. How- ever, a single or double transmission pole pair is generally preferred for filter design, as increasing the multiplicity provides minimal improvements in the cutoff slope and return loss while increasing hardware complexity, sensitivity, and tuning time. Acknowledgment: The authors would like to express their gratitude to Professor V. S. 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STAMENKOVIĆ Optimum Chebyshev Lowpass Filter with a Pair of Imaginary-Axis Zeros 53