第一类切比雪夫数字滤波器分析与设计

Analysis and Design of Chebyshev Kind Digital Filters

本文根据变系数微分方程(1-x^2)(dy^2/dx^2)-x(dy/dx)+ny=0导出了第一类切比雪夫多项式 C_n(x)及第二类切比雪夫多项式 D_n(x)。用二项式展开式得出了 C_n(x) 的前5个多项式,找出 C_n(x)的零点分布的特点,得到 C_n(x)的波形图,接着合成了第一类切比雪夫滤波波形,得到了第一类切比雪夫模平方函数 A(Ω~2),推导了 A(Ω~2)的极点公式和第一类切比雪夫模拟滤波器的传递函数 H(s),直至用 Z 平面的极点表示出第一类切比雪夫数字滤波器的传递函数 H(Z),进而以实例说明了第一类切比雪夫数字低通、高通、带通、带阻滤波器的设计方法.

In this paper the Chebyshev polynomial kind 1 C_n(x)and Chebyshev polyomial kind 2 D_n(x) are derived from the varying-coefficient differential equation(1-x^2)(d^2y/dx^2)-x(dy/dx)+n^2y=0。The first six polynomials of C_n(x) are obtained with binomial expansion.Characteristics of distribution for zeros are found out.Waveform polts of C_n(x)are obtained.The Chebyshev kind 1 filter waveforms are synthesized.Squared-magnitude function A(Ω~2)of Chebyshev kind 1 filter is obtained.Pole formula of A(Ω~2)and analogf ilter's transfer function H(s)of Chebyshev kind 1 are derived,and the transfer fu- nction H(z)of Chebyshev kind 1 digital filter is expressed in terms of poles of z-plane.Some examples are given to illustrate the design opproach of Chebyshev kind 1 digital lowpass,highpss,baodpass and bandstop filters. Finally,the program and expe rimental results are provided.