Comparing the clusters with the different sampling rate following Shannon-Nyquist theorem

 

This Python programming tests how important to set the sampling rate denoted by M.

Shannon-Nyquist requirements claims that the sampling rate M has to be exact: either too small or too big. 

# importing necessary tools

import matplotlib.pyplot as plt

import matplotlib.gridspec as gridspec

import numpy as np

import pandas as pd

import random

import math

import cmath

# Defining the imaginary number

i=complex(0,1)

###### Drawing the full complex field circle with a different step size ######

# Definining the sampling rate

M=10

# Generating the list of the imaginary numbers

ImagList=[]

for j in range(1,M+1):

    ImagList.append((j/M)*i)

# Generating the exponentials of the imaginary numbers in ImagList

ExpList=[cmath.exp(2*math.pi*ImagList[j]) for j in range(len(ImagList))]

# Separating the exponentials into the real part and the imaginary part

ExpListR=[ExpList[j].real for j in range(len(ExpList))]

ExpListI=[ExpList[j].imag for j in range(len(ExpList))]

# # # Uncomment the following lines to show the circle graph

# # Plotting graph for a complex field circle

# plt.rcParams["figure.figsize"] = (9,9)

# plt.plot(ExpListR,ExpListI,'ro')

# plt.xlabel('Real field')

# plt.ylabel('Imaginary field')

# plt.show

###### ###### Comparison of the sampling rates ###### ######

###### ###### to test Shannon-Nyquist theorem  ###### ######

FreqSamples=[3*i,4*i,13*i]

SampleRateMs=[10,100,1000]

ColorList=['b','c','m']

Freq=[]

for k in range(len(SampleRateMs)):

    TempFreq=[FreqSamples[j]/SampleRateMs[k] for j in range(len(FreqSamples))]

    Freq.append(TempFreq)

ExpFreq=[]

for k in range(len(Freq)):

    TempExpFreq=[cmath.exp(2*math.pi*Freq[k][j]) for j in range(len(Freq[k]))]

    ExpFreq.append(TempExpFreq)

ExpFreqR=[]

ExpFreqI=[]

for k in range(len(ExpFreq)):

    BenchR=[1,0,-1,0]

    TempExpFreqR0=[ExpFreq[k][j].real for j in range(len(ExpFreq[k]))]

    TempExpFreqR1=BenchR+TempExpFreqR0

    ExpFreqR.append(TempExpFreqR1)

    BenchI=[0,1,0,-1]

    TempExpFreqI0=[ExpFreq[k][j].imag for j in range(len(ExpFreq[k]))]

    TempExpFreqI1=BenchI+TempExpFreqI0

    ExpFreqI.append(TempExpFreqI1)

panda_df = pd.DataFrame(data = ExpFreq)

display(panda_df)

# Compare plots with the different sampling rates

# Ref. https://stackoverflow.com/questions/37360568/python-organisation-of-3-subplots-with-matplotlib

plt.rcParams["figure.figsize"] = (20,20)

gs = gridspec.GridSpec(3, 3)

for k in range(len(ExpFreq)):

    ax = plt.subplot(gs[0, k]) # row 0, col 1

    plt.plot(ExpFreqR[k],ExpFreqI[k],'ro', color=ColorList[k])

    plt.title(f'Sampling Rate M= {SampleRateMs[k]}')

    plt.xlabel('Real field')

    plt.ylabel('Imaginary field')