Fourier Transform - FFT

 so what is FFT or the fourier transform.

what is the use of the transform

ok an example would be a couple of data points in a digital world

and you want to  transform the data.

so why we need to transform the data 

consider a couple of digital data for sound wave and you want to do some manipulation on the data ,sine it resembles a sine wave 

we plot the data points as a sin wave 

 

your x axis would be time 

y axis would be your data 

The theory here is sound wave has a mix of signals so sin wave of 

different frequencies ,since we have collected data over time

we have to transform time data to freq data

 

in the freq domain our

x axis would be angular frequency 

y axis would be amplitude.

 

 you can find the link for the FFT and inverse FFT for  a sin wave 

 

 

https://github.com/naveez-alagarsamy/matplotlib/blob/main/fourier.ipynb

 

if u cant acess the link use the below code to run locally

 

import matplotlib.pyplot as plt
import numpy as np
from numpy.fft import fft, ifft

sample_rate = 8  # 8 samples in 1 second
sample_interval = 2*np.pi/sample_rate

x = np.arange(0, 2*np.pi, sample_interval) 
y = np.sin(x)

#adding another sin wave with freq =4
freq = 4
y += np.sin(freq*2*np.pi*x)

y_hat = np.arange(len(y))

print(y_hat)
#y = np.sin(2x)

plt.plot(x, y)
plt.show()


print(fft(y))
print(np.abs(fft(y)))

#fourier transform 
X = fft(y)

#angular freq which is = (2 * pi * n/sample_interval * sample rate )
w  = (2* np.pi* y_hat)/(sample_interval * sample_rate )
print(np.abs(w))

plt.figure(figsize = (12, 6))
plt.subplot(121)

plt.stem(np.abs(w), np.abs(X), 'b', \
         markerfmt=" ", basefmt="-b")
plt.xlabel('Freq (Hz)')
plt.ylabel('Amplitude ')
plt.xlim(0, 10)

plt.subplot(122)
plt.plot(x, ifft(X), 'r')
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.tight_layout()
plt.show()