Demonstration of Fourier Series using Python Code

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FOURIER SERIES:

In mathematics, a Fourier series is a way to represent a wave-like function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a set of simple oscillating functions, namely sine and cosine with the harmonics of periods. So, Fourier series are used in the analysis of periodic functions.

Fourier Series:
$begin{displaymath}A_0 + sum_{n = 1}^{infty} (A_ncos(nx) + B_nsin(nx)).end{displaymath}$
where,

$begin{displaymath}left{begin{array}{lclr} a_0 &=& displaystyle frac{1}{2p... ..._{-pi}^{pi} f(x) sin(nx)dx,& 1 leq n. end{array}right.end{displaymath}$

Here i used python programming tool instead of manual calculation to represent the Fourier

Series with some examples

PYTHON CODE:
import numpy as np
import matplotlib.pyplot as plt
resolution = 0.0001
x = np.arange(-np.pi,np.pi,resolution)
square = np.array(x) ……
See more

OUTPUT WAVEFORM:

Python Output:
Output:
>>> a0
0.99998642294279794 (~1)
>>> b1
-0.63661977194539721 (-2/ $π$ )
>>> b3
-0.21220658952264121 (-2/ 3 $π$ )

1 comment

Sanjeev says:

March 24, 2015 at 3:50 am

1) Naming consistency between A_n and a_n, B_n and b_n
2) Add comments to the python code
3) Waveforms needs to make more sense. Instead of calling it first harmonic can you say sin(1*x)*f(x).
4) Help decode the output of the python code
DC, first, third
Plot of
DC+a_1*sin(x)+a_3*sin(3x)

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