# 切比雪夫滤波器

## 特性

### I型切比雪夫滤波器

I型切比雪夫滤波器最为常见。

n阶第一类切比雪夫滤波器的幅度与频率的关系可用下列公式表示[1]

${\displaystyle G_{n}(\omega )=\left|H_{n}(j\omega )\right|={\frac {1}{\sqrt {1+\epsilon ^{2}T_{n}^{2}\left({\frac {\omega }{\omega _{0}}}\right)}}}}$

• ${\displaystyle |\epsilon |<1}$
• ${\displaystyle |H(\omega _{0})|={\frac {1}{\sqrt {1+\epsilon ^{2}}}}}$ 是滤波器在截止频率${\displaystyle \omega _{0}}$的放大率 (注意: 常用的以幅度下降3分贝的频率点作为截止频率的定义不适用于切比雪夫滤波器!)
• ${\displaystyle T_{n}\left({\frac {\omega }{\omega _{0}}}\right)}$${\displaystyle n}$切比雪夫多项式[2]

#### 切比雪夫多项式

${\displaystyle T_{n}(\Omega )=\cos(n\cdot \arccos \ \Omega );0\leq \Omega \leq 1}$
${\displaystyle T_{n}(\Omega )=\cosh(n\cdot \operatorname {(} arccosh\Omega );\Omega >1}$

${\displaystyle T_{n}\left({\frac {\omega }{\omega _{0}}}\right)=a_{0}+a_{1}{\frac {\omega }{\omega _{0}}}+a_{2}\left({\frac {\omega }{\omega _{0}}}\right)^{2}+\,\cdots \,+a_{n}\left({\frac {\omega }{\omega _{0}}}\right)^{n};0\leq \omega \leq \omega _{0}}$
${\displaystyle T_{n}\left({\frac {\omega }{\omega _{0}}}\right)={\frac {\left({\frac {\omega }{\omega _{0}}}{\sqrt {\left({\frac {\omega }{\omega _{0}}}\right)^{2}-1}}\right)^{n}+\left({\frac {\omega }{\omega _{0}}}{\sqrt {\left({\frac {\omega }{\omega _{0}}}\right)^{2}-1}}\right)^{-n}}{2}};\omega >\omega _{0}}$
n 切比雪夫多项式
0 1
1 ${\displaystyle \Omega }$
2 ${\displaystyle -1+2*\Omega ^{2}}$
3 ${\displaystyle 4\Omega ^{3}-3\Omega }$
4 ${\displaystyle 1+8\Omega ^{4}-8\Omega ^{2}}$
5 ${\displaystyle 16\Omega ^{5}-20\Omega ^{3}+5\Omega }$
6 ${\displaystyle -1+32\Omega ^{6}-48\Omega ^{4}+18*\Omega ^{2}}$
7 ${\displaystyle 64\Omega ^{7}-112\Omega ^{5}+56\Omega ^{3}-7\Omega }$
8 ${\displaystyle 1+128\Omega ^{8}-256\Omega ^{6}+160\Omega ^{4}-32\Omega ^{2}}$
9 ${\displaystyle 256\Omega ^{9}-576\Omega ^{7}+432\Omega ^{5}-120\Omega ^{3}+9\Omega }$
10 ${\displaystyle -1+512\Omega ^{10}-1280\Omega ^{8}+1120\Omega ^{6}-400\Omega ^{4}+50\Omega ^{2}}$

${\displaystyle \epsilon =1}$,切比雪夫滤波器的幅度波动= 3分贝。

### II型切比雪夫滤波器

II型切比雪夫滤波器的转移函数为：

${\displaystyle \left|H(\omega )\right|^{2}={\frac {1}{1+{\frac {1}{\epsilon ^{2}T_{n}^{2}\left(\omega _{0}/\omega \right)}}}}}$

${\displaystyle \epsilon ={\frac {1}{\sqrt {10^{0.1\gamma }-1}}}}$ 分贝。

5分贝衰减度相当于ε = 0.6801; 10分贝衰减度相当于 ε = 0.3333。

-3分贝频率fH截止频率 fC 有如下关系：

${\displaystyle f_{H}=f_{C}\cosh \left({\frac {1}{n}}\cosh ^{-1}{\frac {1}{\epsilon }}\right)}$

## 使用范围

• 如果需要快速衰减而允许通频带存在少许幅度波动，可用第一类切比雪夫滤波器；如果需要快速衰减而不允许通频带存在幅度波动，可用第二类切比雪夫滤波器。

## 参考文献

1. ^ Rolf Schaumann et al, p295
2. ^ Rolf Schaumann p295-298
• Rolf Schaumann,Haiqiao Xiao, Mac E.van Valkenburg, Analog Filter Design, 2nd Indian Edition, Oxford University Press, 2013
• Adel S. Sedra, Peter O. Brackett, Filter Theory and Design:Active and Passive, Matri Publishers Inc,1978