Examples

Fourier coefficients for a square wave

Fourier coefficients for a square wave

Problem: Find the Fourier series for the square wave given by $f(x)=1$ for $x\in [0,\pi ]$ and extended to an odd periodic function.

Solution: As an odd function, this signal will be represented by the sine series alone. So $a_n=0$ for all $n$, while

$$\begin{array}{rl}{b}_n& =\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\sin (nx)dx\\ & \\ & =\frac{1}{\pi }{\int }_{-\pi }^0-1\cdot \sin (nx)dx+\frac{1}{\pi }{\int }_0^{\pi }+1\cdot \sin (nx)dx\\ & =\frac{2}{\pi }{\int }_0^{\pi }\sin (nx)dx\\ & ={\frac{2}{\pi }\frac{-1}{n}\cos (nx)|}_0^{\pi }=-\frac{2}{n\pi }(\cos (n\pi )-1)\\ & =\{\begin{array}{ll}\frac{4}{n\pi }& n\text{ odd}\\ 0& n\text{ even}.\end{array}\end{array}$$

The final Fourier series for the square wave is then:

$$f(x)=\frac{4}{\pi }\sin (x)+\frac{4}{3\pi }\sin (3x)+\frac{4}{5\pi }\sin (5x)+\frac{4}{7\pi }\sin (7x)+\cdots$$