Optimum Chebyshev lowpass filter with a pair of imaginary axis zeros

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.