在 Scipy 中使用双线性变换方法设计 IIR 高通巴特沃斯滤波器 – Python

IIR 代表无限脉冲响应，它是许多线性时间不变系统的显着特征之一，其特点是脉冲响应 h(t)/h(n)在某个点后不会变为零，而是无限持续.

什么是 IIR 高通巴特沃思？

它基本上就像一个具有无限脉冲响应的普通数字高通巴特沃斯滤波器。

规格如下：

• 通带频率：2-4 kHz
• 阻带频率：0-500Hz
• 通带纹波：3dB
• 阻带衰减：20 dB
• 采样频率：8 kHz
• 我们将绘制滤波器的幅度、相位、脉冲、阶跃响应。

循序渐进的方法：

第 1 步：导入所有必要的库。

Python3

# import required library
import numpy as np
import scipy.signal as signal
import matplotlib.pyplot as plt

步骤 2：定义用户定义函数mfreqz()和impz() 。 mfreqz 是幅度和相位图的函数，impz 是脉冲和阶跃响应的函数。

蟒蛇3

def mfreqz(b, a, Fs):
    
    # Compute frequency response of the filter 
    # using signal.freqz function
    wz, hz = signal.freqz(b, a)
  
    # Calculate Magnitude from hz in dB
    Mag = 20*np.log10(abs(hz))
  
    # Calculate phase angle in degree from hz
    Phase = np.unwrap(np.arctan2(np.imag(hz), np.real(hz)))*(180/np.pi)
  
    # Calculate frequency in Hz from wz
    Freq = wz*Fs/(2*np.pi)  # START CODE HERE ### (≈ 1 line of code)
  
    # Plot filter magnitude and phase responses using subplot.
    fig = plt.figure(figsize=(10, 6))
  
    # Plot Magnitude response
    sub1 = plt.subplot(2, 1, 1)
    sub1.plot(Freq, Mag, 'r', linewidth=2)
    sub1.axis([1, Fs/2, -100, 5])
    sub1.set_title('Magnitute Response', fontsize=20)
    sub1.set_xlabel('Frequency [Hz]', fontsize=20)
    sub1.set_ylabel('Magnitude [dB]', fontsize=20)
    sub1.grid()
  
    # Plot phase angle
    sub2 = plt.subplot(2, 1, 2)
    sub2.plot(Freq, Phase, 'g', linewidth=2)
    sub2.set_ylabel('Phase (degree)', fontsize=20)
    sub2.set_xlabel(r'Frequency (Hz)', fontsize=20)
    sub2.set_title(r'Phase response', fontsize=20)
    sub2.grid()
  
    plt.subplots_adjust(hspace=0.5)
    fig.tight_layout()
    plt.show()
  
# Define impz(b,a) to calculate impulse response
# and step response of a system input: b= an array
# containing numerator coefficients,a= an array containing 
#denominator coefficients
def impz(b, a):
      
    # Define the impulse sequence of length 60
    impulse = np.repeat(0., 60)
    impulse[0] = 1.
    x = np.arange(0, 60)
  
    # Compute the impulse response
    response = signal.lfilter(b, a, impulse)
  
    # Plot filter impulse and step response:
    fig = plt.figure(figsize=(10, 6))
    plt.subplot(211)
    plt.stem(x, response, 'm', use_line_collection=True)
    plt.ylabel('Amplitude', fontsize=15)
    plt.xlabel(r'n (samples)', fontsize=15)
    plt.title(r'Impulse response', fontsize=15)
  
    plt.subplot(212)
    step = np.cumsum(response)  # Compute step response of the system
    plt.stem(x, step, 'g', use_line_collection=True)
    plt.ylabel('Amplitude', fontsize=15)
    plt.xlabel(r'n (samples)', fontsize=15)
    plt.title(r'Step response', fontsize=15)
    plt.subplots_adjust(hspace=0.5)
  
    fig.tight_layout()
    plt.show()

第 3 步：使用给定的过滤器规格定义变量。

蟒蛇3

# Given specification
Fs = 8000  # Sampling frequency in Hz
fp = 2000  # Pass band frequency in Hz
fs = 500  # Stop Band frequency in Hz
Ap = 3  # Pass band ripple in dB
As = 20  # Stop band attenuation in dB
  
# Compute Sampling parameter
Td = 1/Fs

步骤 4：计算截止频率

蟒蛇3

# Compute cut-off frequency in radian/sec
wp = 2*np.pi*fp  # pass band frequency in radian/sec
ws = 2*np.pi*fs  # stop band frequency in radian/sec

第 5 步：预包裹截止频率

蟒蛇3

# Prewarp the analog frequency
Omega_p = (2/Td)*np.tan(wp*Td/2)  # Prewarped analog passband frequency
Omega_s = (2/Td)*np.tan(ws*Td/2)  # Prewarped analog stopband frequency

步骤 6：计算巴特沃斯滤波器

蟒蛇3

# Compute Butterworth filter order and cutoff frequency
N, wc = signal.buttord(Omega_p, Omega_s, Ap, As, analog=True)
  
# Print the values of order and cut-off frequency
print('Order of the filter=', N)
print('Cut-off frequency=', wc)

输出：

步骤 7：通过signal.butter( )函数使用 N 和 wc 设计模拟巴特沃斯滤波器。

蟒蛇3

# Design analog Butterworth filter using N and
# wc by signal.butter function
b, a = signal.butter(N, wc, 'high', analog=True)
  
# Perform bilinear Transformation
z, p = signal.bilinear(b, a, fs=Fs)
  
# Print numerator and denomerator coefficients 
# of the filter
print('Numerator Coefficients:', z)
print('Denominator Coefficients:', p)

输出：

步骤 8：绘制幅度和相位响应

蟒蛇3

# Call mfreqz function to plot the
# magnitude and phase response
mfreqz(z, p, Fs)

输出：

步骤 9：绘制脉冲和阶跃响应

蟒蛇3

# Call impz function to plot impulse and 
# step response of the filter
impz(z, p)

输出：

下面是实现：

蟒蛇3

# import required library
import numpy as np
import scipy.signal as signal
import matplotlib.pyplot as plt
  
# User defined functions mfreqz for 
# Magnitude & Phase Response
def mfreqz(b, a, Fs):
      
    # Compute frequency response of the filter
    # using signal.freqz function
    wz, hz = signal.freqz(b, a)
  
    # Calculate Magnitude from hz in dB
    Mag = 20*np.log10(abs(hz))
  
    # Calculate phase angle in degree from hz
    Phase = np.unwrap(np.arctan2(np.imag(hz), np.real(hz)))*(180/np.pi)
  
    # Calculate frequency in Hz from wz
    Freq = wz*Fs/(2*np.pi)  # START CODE HERE ### (≈ 1 line of code)
  
    # Plot filter magnitude and phase responses using subplot.
    fig = plt.figure(figsize=(10, 6))
  
    # Plot Magnitude response
    sub1 = plt.subplot(2, 1, 1)
    sub1.plot(Freq, Mag, 'r', linewidth=2)
    sub1.axis([1, Fs/2, -100, 5])
    sub1.set_title('Magnitute Response', fontsize=20)
    sub1.set_xlabel('Frequency [Hz]', fontsize=20)
    sub1.set_ylabel('Magnitude [dB]', fontsize=20)
    sub1.grid()
  
    # Plot phase angle
    sub2 = plt.subplot(2, 1, 2)
    sub2.plot(Freq, Phase, 'g', linewidth=2)
    sub2.set_ylabel('Phase (degree)', fontsize=20)
    sub2.set_xlabel(r'Frequency (Hz)', fontsize=20)
    sub2.set_title(r'Phase response', fontsize=20)
    sub2.grid()
  
    plt.subplots_adjust(hspace=0.5)
    fig.tight_layout()
    plt.show()
  
# Define impz(b,a) to calculate impulse 
# response and step response of a system
# input: b= an array containing numerator 
# coefficients,a= an array containing 
#denominator coefficients
def impz(b, a):
      
    # Define the impulse sequence of length 60
    impulse = np.repeat(0., 60)
    impulse[0] = 1.
    x = np.arange(0, 60)
  
    # Compute the impulse response
    response = signal.lfilter(b, a, impulse)
  
    # Plot filter impulse and step response:
    fig = plt.figure(figsize=(10, 6))
    plt.subplot(211)
    plt.stem(x, response, 'm', use_line_collection=True)
    plt.ylabel('Amplitude', fontsize=15)
    plt.xlabel(r'n (samples)', fontsize=15)
    plt.title(r'Impulse response', fontsize=15)
  
    plt.subplot(212)
    step = np.cumsum(response)  # Compute step response of the system
    plt.stem(x, step, 'g', use_line_collection=True)
    plt.ylabel('Amplitude', fontsize=15)
    plt.xlabel(r'n (samples)', fontsize=15)
    plt.title(r'Step response', fontsize=15)
    plt.subplots_adjust(hspace=0.5)
  
    fig.tight_layout()
    plt.show()
  
  
# Given specification
Fs = 8000  # Sampling frequency in Hz
fp = 2000  # Pass band frequency in Hz
fs = 500  # Stop Band frequency in Hz
Ap = 3  # Pass band ripple in dB
As = 20  # Stop band attenuation in dB
  
# Compute Sampling parameter
Td = 1/Fs
  
# Compute cut-off frequency in radian/sec
wp = 2*np.pi*fp  # pass band frequency in radian/sec
ws = 2*np.pi*fs  # stop band frequency in radian/sec
  
# Prewarp the analog frequency
Omega_p = (2/Td)*np.tan(wp*Td/2)  # Prewarped analog passband frequency
Omega_s = (2/Td)*np.tan(ws*Td/2)  # Prewarped analog stopband frequency
  
# Compute Butterworth filter order and cutoff frequency
N, wc = signal.buttord(Omega_p, Omega_s, Ap, As, analog=True)
  
# Print the values of order and cut-off frequency
print('Order of the filter=', N)
print('Cut-off frequency=', wc)
  
# Design analog Butterworth filter using N and
# wc by signal.butter function
b, a = signal.butter(N, wc, 'high', analog=True)
  
# Perform bilinear Transformation
z, p = signal.bilinear(b, a, fs=Fs)
  
# Print numerator and denomerator coefficients of the filter
print('Numerator Coefficients:', z)
print('Denominator Coefficients:', p)
  
# Call mfreqz function to plot the magnitude
# and phase response
mfreqz(z, p, Fs)
  
# Call impz function to plot impulse and step
# response of the filter
impz(z, p)

输出：