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Multirate Filtering: Filter-Design Exercise in MATLAB (Chinese Version)

Module by: Douglas L. Jones, Swaroop Appadwedula, Matthew Berry, Mark Haun, Jake Janovetz, Michael Kramer, Dima Moussa, Daniel Sachs, Brian Wade, Patrick Frantz, Kanglin Wang. E-mail the authors

Based on: Multirate Filtering: Filter-Design Exercise in MATLAB by Douglas L. Jones, Swaroop Appadwedula, Matthew Berry, Mark Haun, Jake Janovetz, Michael Kramer, Dima Moussa, Daniel Sachs, Brian Wade

Summary: You will design a low-pass finite impulse-response filter using the zero-placement method in MATLAB. The filter can be used as an anti-aliasing and anti-imaging filter in a multirate system.

滤波器设计练习

使用置零的方法为Multirate Filtering: Introduction中的多频采样系统设计滤波器。一个长度为 N N 的FIR滤波器的 z z-变换在 z-1 z -1 中是一个多项式，这个多项式可被因式分解为 N−1 N 1 项。

H⁢z= h 0 + h 1 ⁢z-1+ h 2 ⁢z-2+⋯=( z 1 −z-1)⁢( z 2 −z-1)⁢( z 3 −z-1)⁢⋯ H z h 0 h 1 z -1 h 2 z -2 ⋯ z 1 z -1 z 2 z -1 z 3 z -1 ⋯
(1)

利用这一关系在单位圆上的 ±⁢3⁢π8 ± 3 8 , ±⁢π2 ± 2 , ±⁢5⁢π8 ± 5 8 , ±⁢3⁢π4 ± 3 4 , ±⁢7⁢π8 ± 7 8 和 ±⁢π ± 处放置12个复数零来设计一个低通滤波器（用于多频采样系统中的anti-aliasing和anti-imaging滤波器）。你设计的这个滤波器既可被用作 FIR 1，也可被用作 FIR 3。而对于滤波器 FIR 2 则通过在单位圆上的 ±⁢π4 ± 4 and ±⁢3⁢π4 ± 3 4 处放置四个等间距的零来设计一个低通滤波器（用于多频采样系统的 anti-aliasing 和 anti-imaging 滤波器）。调整设计好的滤波器参数以确保在任何频率下增益都不超过1。

在MATLAB中根据给定的零点来计算滤波器参数，并以此来设计滤波器。你可以使用MATLAB的poly函数。在MATLAB中键入help poly来获取使用细节。

在确定了滤波器参数后，使用MATLAB中的freqz函数来画出频率响应曲线。你会发现这些滤波器的频率响应具有很大的增益。通过除以适当数值的方法来调整滤波器参数，以确保增益最大不超过1。所得到的频率响应是否和你根据z-平面上零点位置所预计的频率响应相一致？

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