离散时间信号处理（英文版）chap7-第2讲.pptVIP

Designing of AF Butterworth lowpass filter Designing of AF 7.1.1 Filter design by Impulse Invariance Transformation principle The parameter Td cannot be used to control aliasing. In practice, to compensate for the aliasing, the AF may be somewhat overdesigned, i.e., designed to exceed the specifications, particularly in the stopband. Transformation principle Transformation from Hc(s) to H(z) Transformation from Hc(s) to H(z) 若模拟滤波器是稳定的，即：所有极点位于s平面左平面，则变化后的数字滤波器的所有极点都位于单位圆内，因此数字滤波器也是稳定的。 冲激响应不变法保证s平面极点与z平面极点有这种代数对应关系，但不等于整个s平面与z平面有这种对应关系。特别是，数字滤波器与模拟滤波器的零点之间就没有这种对应关系，而是随sk和Ak两者而变化。 Example 7.2 Design a DF by Butterworth Discussion: Effect of Td in the Impulse Invariance Discussion: Effect of Td in the Impulse Invariance Summary of Impulse Invariance Steps of designing IIR DF from AF Transform the specifications for the desired DF to those on AF Design the AF, i.e., obtain the system function Hc(s) or impulse response hc(t) Transform Hc(s) or hc(t) to H(z) or h[n] to obtain the desired DF Impulse Invariance Bilinear Transformation —— monotonic in both passband and stopband where —— determine N and Type I Chebyshev filter: —— equiripple in passband and monotonic in stopband Type II Chebyshev filter: —— equiripple in stopband and monotonic in passband Elliptic filter: —— equiripple in both passband and stopband then If —— no aliasing So, can be realized without distortion. —— linear relationship Transformation principle： Frequency mapping: So, the obtained has some distortion. Any practical AF cannot be exactly bandlimited, so aliasing is unavoidable. If approaches zero at high frequencies, the aliasing may be negligibly small, and a useful DF can result from an AF by applying the impulse invariance. 0 s平面虚轴对应z平面单位圆上 s左半平面对应于z平面单位圆内部 s右半平面对应于z平面单位圆外部 Hc(s)先沿s平面虚轴作周期延拓，然后再经过映射关系 映射到z平面，故冲激响应不变法并不是从s平面到z平面的简单代数映射关系，而是多值映射，即：s平面每一条宽为2?/Td的横条都重叠地映射到整个z平面上。 ZT sampling Suppose: s平面上的单极点