凹陷濾波器與梳形濾波器的設計與驗證

Abstract

摘要

本次實驗的目的是要設計出可降低某一特定頻率干擾的濾波器，也就是凹陷濾波器，但通常干擾會形成諧波，所以干擾在基頻的奇數倍數的頻率也會有能量，且會持續遞減，此時就要設計可同時降低某一特定頻率與它的奇數倍數頻率的濾波器，也就是梳形濾波器。 簡單設計出來的濾波器除了會降低目標頻率的能量，也會影響到附近頻率的能量，可能會將重要的信號也一併濾掉，所以還會利用Shank’s recursive的觀念去改良，將濾波器的影響範圍縮小，就會變成IIR notch/comb filter。 最後再去比對凹陷濾波器與梳形濾波器改良前後的濾波效果。

本次實驗是根據課本第147~150頁與第163頁的觀念來設計四個濾波器，先計算要過濾的頻率所對應的角度，再假設轉移函數H(z)並將算式推導成只含有實數的部分，再將H(z)正規化，最後將函式Z轉換就可得到濾波器的使用參數。 本次實驗是使用Python撰寫，將計算出的參數代入signal.freqz()就可得到濾波器的幅度響應與相位響應，將計算出的參數與未濾波的信號代入signal.lfilter()就可得到濾波後的信號。

觀察結果可發現只過濾目標頻率的效果會比有同時過濾奇數倍頻率的效果還要差，而有調整成IIR filter的濾波也會比較好。

Abstract

The purpose of this experiment is to design a filter that can reduce the interference of a certain frequency, that is, the notch filter, but usually the interference will form harmonics, so the interference will also have energy at odd multiples of the fundamental frequency, and It will continue to decrease. At this time, it is necessary to design a filter that can simultaneously reduce a certain frequency and its odd multiples, that is, a comb filter. The simply designed filter will not only reduce the energy of the target frequency, but also affect the energy of nearby frequencies. It may also filter out important signals. Therefore, the concept of Shank's recursive will be used to improve the filter. When the scope of influence is reduced, it will become an IIR notch/comb filter. Finally, compare the filtering effects of the notch filter and the comb filter before and after the improvement.

In this experiment, four filters are designed according to the concepts on pages 147 to 150 and 163 of the textbook. First, calculate the angle corresponding to the frequency to be filtered, and then assume the transfer function H(z) and derive the formula as only For the part containing real numbers, normalize H(z), and finally perform Z-transformation of the function to obtain the filter parameters. This experiment is written in Python. Substitute the calculated parameters into signal.freqz() to get the magnitude response and phase response of the filter, and substitute the calculated parameters and unfiltered signal into signal.lfilter() to get The filtered signal.

After observing the results, it can be found that the effect of filtering only the target frequency will be worse than the effect of filtering odd multiples at the same time, and the filtering adjusted to the IIR filter will be better.