Data Analysis

FOURIER SERIES

Standard formulae

The function $y (x)$ is a periodic function, the period is $2 l$

$y (x) = \frac{a_{0}}{2} + \sum_{i = 1}^{\infty} a_{n} \cos (\frac{n 2 π}{2 l} x) + \sum_{i = 1}^{\infty} b_{n} \sin (\frac{n 2 π}{2 l} x),$		(1.1)
$a_{n} = \frac{1}{l} \int_{- l}^{l} y (x) \cos (\frac{n π}{l} x) d x,$	$b_{n} = \frac{1}{l} \int_{- l}^{l} y (x) \sin (\frac{n π}{l} x) d x .$

The complex form of the Fourier series is

y (x) = \sum_{n = - \infty}^{\infty} c_{n} \exp (i \frac{n π}{l} x),

c_{n} = \int_{- l}^{l} y (x) \exp (- i \frac{n π}{l} x) d x .

(1.2)

The Fourier integral theorem is defined by the relation

y (x) = \lim_{A \to \infty} \frac{1}{2 π} \int_{- A}^{A} \int_{- \infty}^{\infty} y (t) e^{i a (x - t)} d t d a .

(1.3)

The Fourier transform and the inverse transform are

$g (u) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} y (x) e^{- i u x} d x,$	(1.4)
$y (x) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} g (u) e^{i u x} d u .$

Numerical examples

Example 1
The script file pwopro03 gives examples of Fourier series expansions of periodic functions. The first example is a discontinuous skew symmetric function, the second a continuous function with a discontinuous derivative.

Download
Scilab: pwopro03.sci
Octave/Matlab: pwopro03.m

Example 2
The script file pwoprs03 calculates the Fourier series coefficients for damped free vibrations looks at the influence of the number of points. The values of coefficients change but the amplitudes are similar.

Download
Scilab: pwoprs03.sci
Octave/Matlab: pwoprs03.m