Python中的数字高通巴特沃斯滤波器

在本文中，我们将讨论如何使用Python设计数字高通巴特沃斯滤波器。巴特沃斯滤波器是一种信号处理滤波器，旨在使通带中的频率响应尽可能平坦。让我们按照以下规格来设计滤波器并观察数字巴特沃斯滤波器的幅度、相位和脉冲响应。

什么是高通滤波器？

高通滤波器是一种电子滤波器，它使频率高于某一截止频率的信号通过，而使频率低于截止频率的信号衰减。每个频率的衰减取决于滤波器设计。

数字高通滤波器和数字低通滤波器之间的区别：

最显着的区别在于滤波器的幅度响应，我们可以清楚地观察到，在高通滤波器的情况下，滤波器通过频率高于某个截止频率的信号并衰减频率低于截止频率的信号，而在情况下低通滤波器滤波器通过频率低于特定截止频率的信号，并衰减所有频率高于指定截止值的信号。

规格如下：

• 采样率为 3.5 kHz
• 通带边缘频率 1050 Hz
• 阻带边缘频率 600Hz
• 1 dB 通带纹波
• 最小阻带衰减 50 dB

我们将绘制滤波器的幅度、相位和脉冲响应。

循序渐进的方法：

第 1 步：导入所有必要的库。

Python3

# Import required modules
import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
import math

第 2 步：使用给定的过滤器规格定义变量。

蟒蛇3

# Specifications of Filter
   
 # sampling frequency
f_sample = 3500
   
# pass band frequency
f_pass = 1050
   
# stop band frequency
f_stop = 600
   
# pass band ripple
fs = 0.5
   
# pass band freq in radian
wp = f_pass/(f_sample/2)  
   
# stop band freq in radian
ws = f_stop/(f_sample/2) 
   
# Sampling Time
Td = 1 
   
 # pass band ripple
g_pass = 1
   
# stop band attenuation
g_stop = 50

步骤 3：使用signal.buttord()方法构建过滤器。

蟒蛇3

# Conversion to prewrapped analog frequency
omega_p = (2/Td)*np.tan(wp/2)
omega_s = (2/Td)*np.tan(ws/2)
   
   
# Design of Filter using signal.buttord function
N, Wn = signal.buttord(omega_p, omega_s, g_pass, g_stop, analog=True)
   
   
# Printing the values of order & cut-off frequency!
print("Order of the Filter=", N)  # N is the order
# Wn is the cut-off freq of the filter
print("Cut-off frequency= {:.3f} rad/s ".format(Wn))
   
   
# Conversion in Z-domain
   
# b is the numerator of the filter & a is the denominator
b, a = signal.butter(N, Wn, 'high', True)
z, p = signal.bilinear(b, a, fs)
# w is the freq in z-domain & h is the magnitude in z-domain
w, h = signal.freqz(z, p, 512)

输出：

第 4 步：绘制幅度响应。

蟒蛇3

# Magnitude Response
plt.semilogx(w, 20*np.log10(abs(h)))
plt.xscale('log')
 
plt.title('Butterworth filter frequency response')
plt.xlabel('Frequency [Hz]')
plt.ylabel('Amplitude [dB]')
plt.margins(0, 0.1)
 
plt.grid(which='both', axis='both')
plt.axvline(100, color='green')
plt.show()

输出：

第 5 步：绘制脉冲响应。

蟒蛇3

# Impulse response
imp = signal.unit_impulse(40)
c, d = signal.butter(N, 0.5)
response = signal.lfilter(c, d, imp)
 
# Illustrating impulse response
plt.stem(np.arange(0, 40), imp, markerfmt='D', use_line_collection=True)
plt.stem(np.arange(0, 40), response, use_line_collection=True)
plt.margins(0, 0.1)
 
plt.xlabel('Time [samples]')
plt.ylabel('Amplitude')
plt.grid(True)
plt.show()

输出：

步骤 6：绘制相位响应。

蟒蛇3

# Phase response
fig, ax1 = plt.subplots()
 
ax1.set_title('Digital filter frequency response')
ax1.set_ylabel('Angle(radians)', color='g')
ax1.set_xlabel('Frequency [Hz]')
 
angles = np.unwrap(np.angle(h))
ax1.plot(w/2*np.pi, angles, 'g')
ax1.grid()
ax1.axis('tight')
 
plt.show()

输出：

以下是基于上述方法的完整程序：

蟒蛇3

# import required modules
import numpy as np
import matplotlib.pyplot as plt
from scipy import signal
import math
 
 
# Specifications of Filter
   
 # sampling frequency
f_sample = 3500
   
# pass band frequency
f_pass = 1050
   
# stop band frequency
f_stop = 600
   
# pass band ripple
fs = 0.5
   
# pass band freq in radian
wp = f_pass/(f_sample/2)  
   
# stop band freq in radian
ws = f_stop/(f_sample/2) 
   
# Sampling Time
Td = 1 
   
 # pass band ripple
g_pass = 1
   
# stop band attenuation
g_stop = 50
 
# Conversion to prewrapped analog frequency
omega_p = (2/Td)*np.tan(wp/2)
omega_s = (2/Td)*np.tan(ws/2)
   
   
# Design of Filter using signal.buttord function
N, Wn = signal.buttord(omega_p, omega_s, g_pass, g_stop, analog=True)
   
   
# Printing the values of order & cut-off frequency!
print("Order of the Filter=", N)  # N is the order
# Wn is the cut-off freq of the filter
print("Cut-off frequency= {:.3f} rad/s ".format(Wn))
   
   
# Conversion in Z-domain
   
# b is the numerator of the filter & a is the denominator
b, a = signal.butter(N, Wn, 'high', True)
z, p = signal.bilinear(b, a, fs)
 
# w is the freq in z-domain & h is the magnitude in z-domain
w, h = signal.freqz(z, p, 512)
 
 
# Magnitude Response
plt.semilogx(w, 20*np.log10(abs(h)))
plt.xscale('log')
plt.title('Butterworth filter frequency response')
plt.xlabel('Frequency [Hz]')
plt.ylabel('Amplitude [dB]')
plt.margins(0, 0.1)
plt.grid(which='both', axis='both')
plt.axvline(100, color='green')
plt.show()
 
 
# Impulse Response
imp = signal.unit_impulse(40)
c, d = signal.butter(N, 0.5)
response = signal.lfilter(c, d, imp)
plt.stem(np.arange(0, 40),imp,markerfmt='D',use_line_collection=True)
plt.stem(np.arange(0,40), response,use_line_collection=True)
plt.margins(0, 0.1)
plt.xlabel('Time [samples]')
plt.ylabel('Amplitude')
plt.grid(True)
plt.show()
 
 
# Phase Response
fig, ax1 = plt.subplots()
ax1.set_title('Digital filter frequency response')
ax1.set_ylabel('Angle(radians)', color='g')
ax1.set_xlabel('Frequency [Hz]')
angles = np.unwrap(np.angle(h))
ax1.plot(w/2*np.pi, angles, 'g')
ax1.grid()
ax1.axis('tight')
plt.show()

输出：