# 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:                