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Arange demo with Fourier Series

Fourier Series

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
sns.set()

Basis functions (\(\sin(\theta), \sin(2\theta) ... )\)

fs = 100
thetas = np.arange(0,2*np.pi,1/fs)

plt.figure(figsize = [20,6])

for i in range(1,6):
    amp = 1*1/float(i)

    '''Y= A sin(n.theta)'''
    ys = amp*np.sin(i*thetas)

    lb="sin("+str(i)+ r'$\theta$)'
    plt.plot(thetas,ys, label= lb)

plt.xlabel("Angle ("+r'$\theta$)') 
plt.ylabel(r'$\sin(n\theta)$')
plt.legend()
plt.show()

png

Basis function Periodicity
//(sin(\theta)//) 2$\pi$
$sin(2\theta)$ $\pi$
$sin(3\theta)$ 2$\pi$/3
$sin(4\theta)$ $\pi$/4

Basis functions (\(\cos(\theta), \cos(2\theta) ...)\)

fs = 100
thetas = np.arange(0,2*np.pi,1/fs)

plt.figure(figsize = [20,6])

for i in range(1,6):
    amp = 1*1/float(i)

    '''Y= A sin(n.theta)'''
    ys = amp*np.cos(i*thetas)

    lb = "cos("+str(i)+ r'$\theta$)'
    plt.plot(thetas,ys, label=lb)

plt.xlabel("Angle ("+r'$\theta$)') 
plt.ylabel(r'$\cos(n\theta)$')
plt.legend()
plt.show()

png

Basis function Periodicity
$cos(\theta)$ 2$\pi$
$cos(2\theta)$ $\pi$
$cos(3\theta)$ 2$\pi$/3
$cos(4\theta)$ $\pi$/4

Basis functions (\(\sin(\theta), \sin(\theta/2) ...)\)

fs = 100
thetas = np.arange(0,20*np.pi,1/fs)

plt.figure(figsize = [20,6])

for i in range(1,10):
    amp = 1*float(i)

    '''Y= A sin(n.theta)'''
    ys = amp*np.sin(thetas/float(i))

    lb = "sin("+r'$\theta$/'+str(i)+")"
    plt.plot(thetas,ys, label= lb)

plt.xlabel("Angle ("+r'$\theta$)') 
plt.ylabel(r'$\sin(\theta/n)$')
plt.legend()
plt.show()

png

Assignment: What are the periodicites?

Basis functions (\(\cos(\theta), \cos(\theta/2) ...)\)

fs = 100
thetas = np.arange(0,20*np.pi,2*np.pi/fs)

plt.figure(figsize = [20,6])

for i in range(1,9):
    amp = 1*float(i)

    '''Y= A sin(n.theta)'''
    ys = amp*np.cos(thetas/float(i))

    lb="cos("+r'$\theta$/'+str(i)+")"
    plt.plot(thetas,ys, label= lb)

plt.xlabel("Angle ("+r'$\theta$)') 
plt.ylabel(r'$\cos(\theta/n)$')
plt.legend()
plt.show()

png

Assignment: What are the periodicites?

A : Fourier Series: Square Wave

img

1. Individual basis function terms:

Here we want to plot the individual terms in the above series to compere the periodicity and amplitudes.

L = 1
ns =100
thetas = np.arange(0,2,1/ns)

plt.figure(figsize = [15,4])

for n in [i for i in range(10) if i%2 !=0]:

    '''creating individual terms in the series'''
    fs = (4/np.pi)*(1/n)*np.sin(n*np.pi*thetas/float(L))

    plt.plot(thetas,fs)

plt.xlabel("Angle ("+r'$\theta$)') 
plt.ylabel("Basis terms")
plt.show()

png

2.Subsequent development by adding terms

Here we want to observe the plote of terms (e.g., first term, sum of first two terms, sum of first three terms) to realize how final series is developed by contibution of individual terms.

L = 1
ns =100
thetas = np.arange(0,2,1/ns)
plt.figure(figsize = [15,4])

k = 1
for n in [i for i in range(10) if i%2 != 0]:
    if n == 1:
        '''taking care of first term'''
        fi = (4/np.pi)*(1/n)*np.sin(n*np.pi*thetas/float(L))
        plt.plot(thetas,fi, label = "first term")
        ff = fi

    else:
        '''terms following first term'''
        fi = ff + (4/np.pi)*(1/n)*np.sin(n*np.pi*thetas/float(L))
        plt.plot(thetas,fi, label = "sum of "+str(k)+" terms")

        ff = fi

    k = k+1

plt.legend()
plt.xlabel("Angle ("+r'$\theta$)') 
plt.ylabel('sum of n - terms')
plt.show()

png

3. Sum of all n-terms

L = 1
ns =100
thetas = np.arange(0,10,1/ns)
plt.figure(figsize = [15,4])

'''including n = 1,3,5,...19th terms in basis expansison'''
fs = sum([(4/np.pi*(1/n)*np.sin(n*np.pi*thetas/float(L)))\
                for n in [i for i in range(20) if i%2 !=0]])

plt.plot(thetas,fs,label ="sawtooth wave")
plt.legend()
plt.xlabel("Angle ("+r'$\theta$)') 
plt.ylabel('Square wave')
plt.show()

png

Assignment

Develope the Fourier series of Triangular wave following aboves method: img

B: Fourier Transform

Experiment 1: Signal without noise

fs = 100.0
ts= np.arange(0,10,1/fs)
freq_cos = [2,4,6,8,10,12]
freq_sin = [1,3,5,7,9]

'''prepare candidate signals'''
sigs = [np.cos(2*np.pi*f1*ts) + np.sin(2*np.pi*f2*ts)\
                    for f1,f2 in zip(freq_cos,freq_sin)]

'''resultant signal'''
rsig = sum(sigs)

fft_rsig = np.fft.fft(rsig)
fft_freq = np.fft.fftfreq(len(fft_rsig),1/fs)

plt.figure(figsize = [18,10])

plt.subplot(311)
plt.plot(ts,rsig)
plt.title("Original signal in Time Domain")

plt.subplot(312)
plt.stem(fft_freq, abs(fft_rsig.real),\
                    label="Real part")
plt.title("FFT in Frequency Domain")
plt.ylim(0,1000)
plt.xlim(0,15)
plt.legend(loc=1)

plt.subplot(313)
plt.stem(fft_freq, abs(fft_rsig.imag),\
                label="Imaginary part")
plt.xlabel("frequency (Hz)")
plt.legend(loc=1)
plt.xlim(0,15)
plt.ylim(0,1000)
plt.show()

png

Experiment 2: Signal with noise

fs = 100.0
ts= np.arange(0,10,1/fs)
freq_cos = [2,4,6,8,10,12]
freq_sin = [1,3,5,7,9]

'''prepare candidate signals'''
sigs = [np.cos(2*np.pi*f1*ts) + np.sin(2*np.pi*f2*ts) \
        for f1,f2 in zip(freq_cos,freq_sin)]

'''resultant signal + noise'''
noise = 10*np.random.rand(len(rsig))
rsig_noise = sum(sigs) + noise

fft_rsig_noise = np.fft.fft(rsig_noise)
fft_freq_noise = np.fft.fftfreq(len(fft_rsig_noise),1/fs)

plt.figure(figsize = [18,10])

plt.subplot(311)
plt.plot(ts,rsig_noise)
plt.title("Original signal with noise in Time Domain")

plt.subplot(312)
plt.stem(fft_freq_noise, abs(fft_rsig_noise.real),\
                        label="Real part")
plt.title("FFT in Frequency Domain")
plt.ylim(0,1000)
plt.xlim(0,15)
plt.legend(loc=1)

plt.subplot(313)
plt.stem(fft_freq_noise, abs(fft_rsig_noise.imag),\
                        label="Imaginary part")
plt.xlabel("frequency (Hz)")
plt.legend(loc=1)
plt.xlim(0,15)
plt.ylim(0,1000)
plt.show()

png