Content and Objective

Show how to get frequency response out of poles and zeros

Importing and Plotting Options



In [1]:

    
import numpy as np

from scipy import signal

import matplotlib
import matplotlib.pyplot as plt

%matplotlib inline



In [2]:

    
# plotting options 
font = {'size'   : 20}
plt.rc('font', **font)
plt.rc('text', usetex=True)

matplotlib.rc('figure', figsize=(18, 6) )

Show Example

defining frequency regime



In [3]:

    
delta_Omega = .01
Omega = np.arange( 0, 2 * np.pi + delta_Omega, delta_Omega )

defining poles and zeros



In [4]:

    
# constant pre-factor of transfer function
const = -2

# switch for choosing different pol-zero constellations
switch = 3

if switch == 1:
    zeros = np.array( [ -1, 1] )
    poles = np.array( [ .95j, -.95j ] )

elif switch == 2:
    zeros = np.array( [] )
    poles = np.array( [2] )
    
else:
    const = 0.5
    zeros = np.array( [ 0,0,0] ) 
    poles = np.array( [ .5, .25 + 1j * .8, .25 - 1j * .8] )

plotting zeros and poles as well as unit circle



In [5]:

    
plt.subplot(121)

plt.plot( np.real( zeros ), np.imag( zeros ) , 'o')
plt.plot( np.real( poles ), np.imag( poles ) , 'x')

phi = np.linspace(0, 2*np.pi, 256)
unit_circle = np.exp(1j*phi)
plt.plot( np.real( unit_circle ) , np.imag( unit_circle ) )

plt.grid( True )
plt.xlabel( 'real' )
plt.xlabel( 'imag' )









    Out[5]:





Text(0.5, 0, 'imag')

get frequency response



In [6]:

    
# construct numerator and denominator by multiplying according distances
numerator = np.prod( [ ( np.exp( 1j * Omega ) - z ) for z in zeros ], axis=0 )
denominator = np.prod( [ ( np.exp( 1j * Omega ) - p ) for p in poles ], axis=0 )

# get frequency response
H = const * numerator / denominator

plotting frequency response



In [7]:

    
plt.subplot(121)
plt.plot( Omega, np.abs( H ), label='$|H(\\Omega)|$' )

plt.grid( True )
plt.xlabel('$\\Omega$')
plt.ylabel('$|H(\\Omega)|$')
plt.legend(loc='upper right')

plt.subplot(122)
plt.plot( Omega, np.angle( H ), label='$\\angle H(\\Omega)$' )

plt.grid( True )
plt.xlabel('$\\Omega$')
plt.ylabel('$\\angle \\Omega$')
plt.legend(loc='upper right')









    Out[7]:





<matplotlib.legend.Legend at 0x7f82504ec5c0>

get impulse response



In [8]:

    
t, h = signal.dimpulse( ( zeros, poles, const, 1 ) )



In [9]:

    
plt.stem( t, h[0] )

plt.xlim( (-1, 40) )
plt.grid( True )
plt.xlabel('$n$')
plt.ylabel('$h[n]$')









    Out[9]:





Text(0, 0.5, '$h[n]$')



In [ ]: