Summary

DescriptionMinimum and maximum phase responses.gif	English: Shows the phase responses of a minimum and maximum phase responses when ${\displaystyle H(z)}$ is a monomial with ${\displaystyle a=0.8}$. Both filters have the same gain response. Top : Minimum phase filter. Bottom : Maximum phase filter. Left : Nyquist diagram. Right : Phase responses
Date	21 September 2016
Source	Own work
Author	fdeloche

Licensing

I, the copyright holder of this work, hereby publish it under the following license:

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/4.0CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 truetrue

Generation code

Minimumphase.py

# coding: utf-8
'''Generate an animation showing the phase response for a minimum and maximum phase system'''
__author__      = "fdeloche"

# In[1]:

get_ipython().magic(u'matplotlib inline')
import sys
import numpy as np
import matplotlib.pyplot as pl
from matplotlib.animation import FuncAnimation

# In[2]:

createGif=True
pl.rc('xtick', labelsize=20)
pl.rc('ytick', labelsize=20)
pl.rc('font', weight='bold')

# In[3]:

fig, ((ax1, ax2), (ax3, ax4)) = pl.subplots(2, 2, figsize=(15, 15))
#fig.set_tight_layout(True)
a_x = 0.8
a_y=0.

m=1000

A = a_x + 1j*a_y
a_mod = np.abs(A)
Ainv = 1./A
a_xbis = np.real(Ainv)
a_ybis = -np.imag(Ainv)
# r^2 =  

x_lim_a = -0.3
x_lim_b = 1.9
y_lim = 1.1

t = np.linspace(0, 1, num=m)

ax1.scatter(0, 0, linewidth=6, color='blue')
ax1.scatter(1, 0, linewidth=4, color='blue')
ax1.set_xlim([x_lim_a, x_lim_b])
ax1.set_ylim([-y_lim, y_lim])

ax1.set_title('$1-az^{-1}$', fontsize=35)

ax1.plot(t, np.zeros(m), color='blue', linewidth=4)
ax1.plot(1+a_mod*np.cos(2*np.pi*t), a_mod*np.sin(2*np.pi*t), color='black')
ax1.axis('off')
ax1.text(-0.2, 0.1, "$(0, 0)$", fontsize=30, color='blue')
ax1.text(1-0.1, 0.1, "$(1, 0)$", fontsize=30, color='blue')



ax3.set_title('$\overline{a}(1-\overline{a}^{\ -1}z^{-1})$', fontsize=35)
ax3.scatter(0, 0, linewidth=6, color='blue')
ax3.scatter(np.abs(A), 0, linewidth=4, color='blue')
ax3.set_xlim([x_lim_a, x_lim_b])
ax3.set_ylim([-y_lim, y_lim])

ax3.plot(t*np.abs(A), np.zeros(m), color='blue', linewidth=4)
ax3.plot(np.abs(A)+np.cos(2*np.pi*t), np.sin(2*np.pi*t), color='black')
ax3.axis('off')
ax3.text(-0.1, 0.1, "$(0, 0)$", fontsize=30, color='blue')
ax3.text(-0.1+np.abs(A), 0.1, "$(\overline{a}, 0)$", fontsize=30, color='blue')

Z = np.cos(2*np.pi*t) - 1j*np.sin(2*np.pi*t)
G = np.angle(1-A*Z)

ax2.set_title('Phase response', fontsize=25)
ax2.plot(2*np.pi*t, G, color='blue', linewidth=2)
ax2.plot(2*np.pi*t, 0*t, color='black')

G2 = np.angle(1-np.conj(Ainv)*Z)

#ax4.set_title('Phase response', fontsize=25)
ax4.plot(2*np.pi*t, G2, color='blue', linewidth=2)
ax4.plot(2*np.pi*t, 0*t, color='black')


ax2.set_ylim([-np.pi, np.pi])

ax4.set_ylim([-np.pi, np.pi])

ax2.set_xlim([0, 6.283])

ax4.set_xlim([0, 6.283])

'''
ax2.spines["top"].set_visible(False)
ax2.spines["right"].set_visible(False)

ax4.spines["top"].set_visible(False)
ax4.spines["right"].set_visible(False)
'''

# In[4]:

line1, = ax1.plot(1-np.abs(A)*t*1, t*0, color='blue', linewidth=4)
line2, = ax3.plot(np.abs(A)-t*1, t*0, color='blue', linewidth=4)
line3, = ax1.plot((1-np.abs(A))*t*1, t*0, color='red', linewidth=4)
line4, = ax3.plot((np.abs(A)-1)*t, t*0, color='red', linewidth=4)

point1 = ax1.scatter(1-np.abs(A), 0, linewidth=5, color='red')
point2 = ax3.scatter(np.abs(A)-1, 0, linewidth=5, color='red')

line5, = ax2.plot(0*t, G[0]*t, color='red', linewidth=4)
line6, = ax4.plot(0*t, G2[0]*t, color='red', linewidth=4)

point3 = ax2.scatter(0, G[0], color='red', linewidth=5)
point4 = ax4.scatter(0, G2[0], color='red', linewidth=5)

# In[5]:

n_frames = 55
def update(i):
    t0 = i*1./n_frames
    B = [a_mod*np.cos(2*np.pi*t0), -a_mod*np.sin(2*np.pi*t0)]
    line1.set_xdata(1-t*B[0])
    line1.set_ydata(-t*B[1])
    C = [np.cos(2*np.pi*t0), -np.sin(2*np.pi*t0)]
    line2.set_xdata(np.abs(A)-t*C[0])
    line2.set_ydata(-t*C[1])
    
    line3.set_xdata((1-B[0])*t)
    line3.set_ydata(-t*B[1])
    line4.set_xdata((np.abs(A)-C[0])*t)
    line4.set_ydata(-t*C[1])
    
    point1.set_offsets((1-B[0], -B[1]))
    point2.set_offsets((np.abs(A)-C[0], -C[1]))
    
    line5.set_xdata(2*np.pi*t0+0*t)
    line6.set_xdata(2*np.pi*t0+0*t)

    Z0 = np.cos(2*np.pi*t0) - 1j*np.sin(2*np.pi*t0)
    G0 = np.angle(1-A*Z0)
    G20 = np.angle(1-np.conj(Ainv)*Z0)
    
    line5.set_ydata(G0*t)
    line6.set_ydata(G20*t)
    
    point3.set_offsets((2*np.pi*t0, G0))
    
    point4.set_offsets((2*np.pi*t0, G20))
    return line1, line2, line3, line4, point1, point2, line5, line6, point3, point4

# In[ ]:

anim = FuncAnimation(fig, update, frames=np.arange(0,n_frames), interval=50, blit=True)
if(createGif):
    anim.save('result.gif', dpi=30, writer='imagemagick')
else:
    pl.show()

# In[ ]: