6、模拟滤波器的频率响应、特性及综合设计

模拟滤波器的频率响应、特性及综合设计

1. 频率响应的向量表示法

频率响应的表达式为：
[H(j\omega) = G\frac{(j\omega - s_{z1})(j\omega - s_{z2})(j\omega - s_{z3})\cdots(j\omega - s_{zM})}{(j\omega - s_{p1})(j\omega - s_{p2})(j\omega - s_{p3})\cdots(j\omega - s_{pN})}]
其中，每个因子 (j\omega - a_i - jb_i = -a_i + j(\omega - b_i) = r_ie^{j\varPhi_i})，这里 (a_i + jb_i) 对应于极点或零点，且：
[r_i = \sqrt{a_i^2 + (\omega - b_i)^2}]
[\varPhi_i = \arctan(\frac{\omega - b_i}{-a_i})]
将其代入频率响应表达式可得：
[H(j\omega) = G\frac{(r_{z1}e^{j\varPhi_{z1}})(r_{z2}e^{j\varPhi_{z2}})(r_{z3}e^{j\varPhi_{z3}})\cdots(r_{zM}e^{j\varPhi_{zM}})}{(r_{p1}e^{j\varPhi_{p1}})(r_{p2}e^{j\varPhi_{p2}})(r_{p3}e^{j\varPhi_{p3}})\cdots(r_{pN}e^{j\varPhi_{pN}})} = G\frac{r_{z1}r_{z2}r_{z3}\cdots r_{zM}e^{j(\varPhi_{z1}+\varPhi_{z2}+