论滤波器设计中的双线性变换关系

The Bilinear Transformation Relation in Filter Design

本文首先根据保角映射理论的分式线性变换,推导了s平面到z平面单值对应映射的一般性双线性变换关系,建立了滤波器双线性变换设计方法的数学依据。然后根据数字滤波器与原型模拟滤波器低频特性近似相等的设计思想求定关系式中的映射常数,引出了可供工程实用的常用设计式,形成了滤波器的双线性变换设计方法。比直接导出该式显得有层次,而且便于说明按不同的设计思想和要求确定映射常数,可得不同的工程设计用式。

In this paper, the relation of single value correspondence mapping from s plane to z plane has been derived according to the conformal mapping theory of fractional linear substitution. The mathematical basis of the bilinear transformation filter design procedures is estabeished. Then, the mapping constants of the transformation are chosen. Thereupon, the practical formula of s with the parameters T and z is derived for the filter design. Compared with the nomal way by which the same formula is directly derived, the process mentioned in this paper has the following virtues: (1)The idea of the formula becomes clear. (2)Several different engineering design formulae may be obtained.