Fourier Series of Periodic Functions

Tutorials on Fourier series are presented. In the first part an example is used to show how Fourier coefficients are calculated and in a second part you may use an app to further explore Fourier series of the same function.

Fourier Series and Coefficients

Fourier series may be used to represent periodic functions as a linear combination of sine and cosine functions. If f(t) is a periodic function of period T, then under certain conditions, its Fourier series is given by:

Fourier series of a periodic function f(t).

where n = 1 , 2 , 3 , ... and T is the period of function f(t). an and bn are called Fourier coefficients and are given by

Formula for a0.

Formula for coefficients an.

Formula for coefficients bn.

Example 1
Find the Fourier series of the periodic function f(t) defined by

Formula for function f(t).

graph of the periodic function f(t) in the example

Solution to Example 1
Coefficient a0 is given by

Calculation of a0.

Coefficients an is given by
Calculation of an.

And coefficients bn is given by
Calculation of bn.

A computation of the above coefficients gives
\( a_0 = 0 \), \( a_n = 0 \) and \( b_n = \dfrac{2}{n\pi} (1 - \cos (n \pi)) \)
Note that \( \cos (n \pi) \) may be written as
\( cos (n \pi) = (-1)^n \)
and that bn = 0 whenever n is even.
The given function f(t) has the following Fourier series
Fourier series of f(t).

Interactive Tutorial on Fourier Series

For numerical calculations purposes we cannot include an infinite number of terms in the series above, we therefore define function \( f_N(t) \) with a limited number of terms \( N \) as follows \[ f_N(t) = \sum_{n=1}^{N} \dfrac{2}{n\pi} (1-(-1)^n) \sin(\dfrac{2 n \pi t}{T}) \] The app below may be used to explore the Fourier series of f(t) solved in example 1 above including a limited number of terms \( N \) in the series and see how the graph of function \( f_N(t) \) defined above becomes close to the graph of function f(t) as \( N \) increases.

The default values are \( N = 5 \) and \( T = 4 \) and the series have one term which is a sinusoidal function of period T. Incease \( N \) and compare the graph of the function obtained (in blue) to that of \( f(t) \) (in red) defined in example.

\( N \) = \( T \) =
Change \( N \) and \( T \) and click this button

Hover the mousse cursor on the graph or plotted point to read the coordinates.