Python中的数字带阻巴特沃斯滤波器

在本文中，我们将讨论如何使用Python设计数字带阻巴特沃斯滤波器。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。

什么是数字带阻滤波器？

带通滤波器是一种滤波器，它通过一个范围内的频率并拒绝该范围外的频率。

它与高通、低通和带通的区别：

通过观察带通滤波器的幅度响应可以发现主要差异。在带阻滤波器中，指定频率范围之间的所有信号都被滤波器拒绝。

规格如下：

• 采样率为 12 kHz。
• 通带边缘频率为 2100 Hz 和 4500 Hz。
• 阻带边缘频率为 2700 Hz 和 3900 Hz。
• 通带纹波为 0.6 dB。
• 最小阻带衰减为 45 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：

在开始之前，首先，我们将创建一个用户定义的函数来转换边缘频率，我们将其定义为 convert() 方法。

Python3

# explicit function to convert
# edge frequencies
  
  
def convertX(f_sample, f):
    w = []
  
    for i in range(len(f)):
        b = 2*((f[i]/2) / (f_sample/2))
        w.append(b)
  
    omega_mine = []
  
    for i in range(len(w)):
        c = (2/Td)*np.tan(w[i]/2)
        omega_mine.append(c)
  
    return omega_mine

第 1 步：导入所有必要的库。

蟒蛇3

# 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 = 12000
  
# pass band frequency
f_pass = [2100, 4500]
  
# stop band frequency
f_stop = [2700, 3900]
  
# pass band ripple
fs = 0.5
  
# Sampling Time
Td = 1
  
# pass band ripple
g_pass = 0.6
  
# stop band attenuation
g_stop = 45

第 3 步：使用 signal.buttord()函数构建过滤器。

蟒蛇3

# Conversion to prewrapped analog
# frequency
omega_p = convertX(f_sample, f_pass)
omega_s = convertX(f_sample, f_stop)
  
# 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
# N is the order
print("Order of the Filter=", N)
  
# Wn is the cut-off freq of the filter
print("Cut-off frequency= {:} 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, 'bandpass', 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)
  
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

# Frequency 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
  
# explicit function to convert
# edge frequencies
def convertX(f_sample, f):
    w = []
  
    for i in range(len(f)):
        b = 2*((f[i]/2)/(f_sample/2))
        w.append(b)
  
    omega_mine = []
  
    for i in range(len(w)):
        c = (2/Td)*np.tan(w[i]/2)
        omega_mine.append(c)
  
    return omega_mine
  
# Specifications of Filter
# sampling frequency
f_sample = 12000
  
# pass band frequency
f_pass = [2100, 4500]
  
# stop band frequency
f_stop = [2700, 3900]
  
# pass band ripple
fs = 0.5
  
# Sampling Time
Td = 1
  
# pass band ripple
g_pass = 0.6
  
# stop band attenuation
g_stop = 45
  
# Conversion to prewrapped analog
# frequency
omega_p = convertX(f_sample, f_pass)
omega_s = convertX(f_sample, f_stop)
  
# 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
# N is the order
print("Order of the Filter=", N)
  
# Wn is the cut-off freq of the filter
print("Cut-off frequency= {:} 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, 'bandpass', 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()
  
# Frequency 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：