---
tags: 名詞解釋
---

# Butterworth filter (巴特沃斯濾波器)
## 前情提要 - 濾波器的種類
>濾波器常用於訊號處理，例如擷取感測元件輸出的類比訊號時，很容易因外在因素產生雜訊導致失真，故需使用濾波器(filter)除去不必要的雜訊，將特定頻率的訊號顯示出來。<br>
濾波器分為被動(passive filters)與主動濾波器(active filters)，頻率在 1MHz 以上大都會用被動濾波器, 此濾波器由電阻、電容、電感所組成，無功率增益，頻率較難調整,功能僅只過濾雜訊，頻率在1MHz 以下大都會用主動濾波器(active filters) 此濾波器由電阻、電容、主動元件所組成，有功率增益、頻率較容易調整，主動元件包括電晶體、運算放大器等，主動濾波器的優點如下:<br>
1. 利用運算放大器提出適當的訊號增益，使訊號通過濾波器不會衰減。<br>
2. 不使用電感，使濾波器可製成體積小的積體電路。<br>
3. 利用運算放大器的高輸入阻抗與低輸入阻抗，使阻抗匹配容易，不易造成阻抗被配不好而產生負載效應。<br>
4. 利用不同特性的組合，可設計成所想要的頻率響應。<br>
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- Low Pass Filter
==指低於截止頻率的信號都可通過==

- High Pass Filter
==指高於截止頻率的信號可通過==

- Band Pass Filter
==只允許某個頻帶寬度的信號通過==

- Band Reject Filter
==不允許某個頻帶寬度的信號通過==

## Low-Pass Filter Code

import numpy as np
from scipy.signal import butter, lfilter, freqz
import matplotlib.pyplot as plt

def butter_lowpass(cutoff, fs, order=5):
    # calculate the Nyquist frequency
    nyq = 0.5 * fs
    
    # design filter
    normal_cutoff = cutoff / nyq
    b, a = butter(order, normal_cutoff, btype='low', analog=False)
    
    return b, a

def butter_lowpass_filter(data, cutoff, fs, order=5):
    b, a = butter_lowpass(cutoff, fs, order=order)
    y = lfilter(b, a, data)
    
    return y

# Setting standard filter requirements.
#
# cutoff (float) : the cutoff frequency of the filter
# fs     (float) : the sampling rate (取樣頻率)
# order  (int)   : order of the filter, by default defined to 5. 
#
order = 6
fs = 30.0       
cutoff = 3.667  

b, a = butter_lowpass(cutoff, fs, order)

# Plotting the frequency response.
w, h = freqz(b, a, worN=8000)
plt.subplot(2, 1, 1)
plt.plot(0.5*fs*w/np.pi, np.abs(h), 'b')
plt.plot(cutoff, 0.5*np.sqrt(2), 'ko')
plt.axvline(cutoff, color='k')
plt.xlim(0, 0.5*fs)
plt.title("Lowpass Filter Frequency Response")
plt.xlabel('Frequency [Hz]')
plt.grid()


# Creating the data for filteration
T = 5.0         # value taken in seconds
n = int(T * fs) # indicates total samples
t = np.linspace(0, T, n, endpoint=False)

data = np.sin(1.2*2*np.pi*t) + 1.5*np.cos(9*2*np.pi*t) + 0.5*np.sin(12.0*2*np.pi*t)

# Filtering and plotting
y = butter_lowpass_filter(data, cutoff, fs, order)

plt.subplot(2, 1, 2)
plt.plot(t, data, 'b-', label='data')
plt.plot(t, y, 'g-', linewidth=2, label='filtered data')
plt.xlabel('Time [sec]')
plt.grid()
plt.legend()

plt.subplots_adjust(hspace=0.35)
plt.show()