Fourier Series: Example 3 - Parabolic Function

 

Trigonometric Form Solution

Harmonic Form Solution

Complex Exponential Form Solution

 

Figure 1: Periodic Parabolic function

 

Trigonometric Form Solution

The coefficients for the Trigonometric Form of the Fourier Series are:

1)

2)

3)

Using the above coefficients, we can calculate the Fourier Series of f(t):

Figure 2: Harmonics Addition for Parabolic function

 

Harmonic Form Solution

The coefficients are:

So the Fourier Series for the function:

This is the same like the Trigonometric Form, as expected, since there are no sine factors.

 

Complex Exponential Form Solution

The coefficients are:

Now, because:

It is:

Also for n=0 it is:

Figure 3: Amplitude and Phase for Complex Exponential Coefficients

The Fourier Series for the function f(t):

Now because:

It is: