Generalized Chebyshev Function of Arbitrary Order With Real-Frequency Zero Pairs in an Explicit Rational Polynomial Expression

Generalized Chebyshev functions of an arbitrary order with up to three real-frequency zero pairs are derived in an explicit rational polynomial expression. Given in-band ripple level and positions of the zero pairs, numerator of the rational function is a sum of Chebyshev polynomials of the first kind weighted by constants specified by the zeros, and the denominator is simply a successive product of $\left(a_{i}^{2}-\omega^{2}\right)$, given that $\pm a_{i}$ are the designated real-frequency zero pair. Based on these analytical generalized Chebyshev filtering expressions, effort required by computer in synthesis of a lowpass prototype filter is at most a root-searching routine for polynomials. The whole synthesis process is straight-forward and needs neither iteration nor optimization.