Fourier Series: Example 1 - Square Pulse

 

Trigonometric Form Solution

Harmonic Form Solution

Complex Exponential Form Solution

 

 

Figure 1: Periodic Square Pulse

 

Trigonometric Form Solution

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

1)

2)

3)

For this example we will set: w=0.5 and T=1. The coefficients for the Fourier Series become:

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

Figure 2: Harmonics Addition for 50% Square Pulse

 

Harmonic Form Solution

The coefficients are:

So the Fourier Series for the function are:

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

 

Complex Exponential Form Solution

The coefficients are:

The amplitude and the phase of  the coefficients are:

Figure 3: Amplitude and Phase for Complex Exponential Coefficients

The magnitude becomes zero and the phase changes at the points:

The Fourier Series for the function f(t), setting for example w=0.5 and T=1: