使用 Scipy – Python设计 IIR 带通 Chebyshev Type-1 滤波器

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 切比雪夫滤波器？

IIR Chebyshev 是一种与巴特沃斯一样的线性时间不变滤波器，但是，与巴特沃斯滤波器相比，它具有更陡峭的滚降。切比雪夫滤波器根据通带纹波和截止纹波等参数进一步分为切比雪夫I型和切比雪夫II型。

切比雪夫滤波器与巴特沃斯滤波器有何不同？

与巴特沃斯滤波器相比，切比雪夫滤波器具有更陡峭的滚降。

什么是切比雪夫 I 型滤波器？

Chebyshev Type-I 通过在通带中加入相等的纹波，最大限度地减少了整个通带上理想和实际频率响应之间的绝对差异。

规格如下：

• 通带频率：1400-2100Hz
• 阻带频率：1050-24500 Hz
• 通带纹波：0.4dB
• 阻带衰减：50 dB
• 采样频率：7 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 numpy as np
import scipy.signal as signal
import matplotlib.pyplot as plt

第 2 步：定义用户定义的函数，例如mfreqz()和impz()。

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)
      
    # 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 = 7000  # Sampling frequency in Hz
fp = np.array([1400, 2100])  # Pass band frequency in Hz
fs = np.array([1050, 2450])  # Stop band frequency in Hz
Ap = 0.4  # Pass band ripple in dB
As = 50  # stop band attenuation in dB

步骤 4：计算截止频率

蟒蛇3

# Compute pass band and stop band edge frequencies
wp = fp/(Fs/2)  # Normalized passband edge frequencies w.r.t. Nyquist rate
ws = fs/(Fs/2)  # Normalized stopband edge frequencies

第 5 步：计算截止频率和阶数

蟒蛇3

# Compute order of the  Chebyshev type-1 filter using signal.cheb1ord
N, wc = signal.cheb1ord(wp, ws, Ap, As)
  
# Print the order of the filter and cutoff frequencies
print('Order of the filter=', N)
print('Cut-off frequency=', wc)

输出：

步骤 6：计算滤波器系数

Python

# Design digital Chebyshev type-1 filter
# using signal.cheby1 function
z, p = signal.cheby1(N, Ap, wc, 'bandpass')
  
# Print numerator and denomerator coefficients
# of the filter
print('Numerator Coefficients:', z)
print('Denominator Coefficients:', p)

输出：

步骤 7：绘制幅度和相位响应

蟒蛇3

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

输出：

步骤 8：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：

完整代码：

蟒蛇3

# import required library
import numpy as np
import scipy.signal as signal
import matplotlib.pyplot as plt
  
  
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)
      
    # 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 = 7000  # Sampling frequency in Hz
fp = np.array([1400, 2100])  # Pass band frequency in Hz
fs = np.array([1050, 2450])  # Stop band frequency in Hz
Ap = 0.4  # Pass band ripple in dB
As = 50  # stop band attenuation in dB
  
  
# Compute pass band and stop band edge frequencies
wp = fp/(Fs/2)  # Normalized passband edge frequencies w.r.t. Nyquist rate
ws = fs/(Fs/2)  # Normalized stopband edge frequencies
  
# Compute order of the  Chebyshev type-1
# filter using signal.cheb1ord
N, wc = signal.cheb1ord(wp, ws, Ap, As)
  
# Print the order of the filter and cutoff frequencies
print('Order of the filter=', N)
print('Cut-off frequency=', wc)
  
# Design digital Chebyshev type-1 filter using 
# signal.cheby1 function
z, p = signal.cheby1(N, Ap, wc, 'bandpass')
  
  
# Print numerator and denomerator coefficients of the filter
print('Numerator Coefficients:', z)
print('Denominator Coefficients:', p)
  
# Call mfreqz 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：