Electronic – What are values of fundamental frequency and the harmonics for the below signal g(t)

This stumped me for a while as well but it is actually quite simple.

The general Fourier series:

\$ g(x) = a_0 + \displaystyle \sum_{n=1}^{\infty}[a_n sin(n x) + b_n cos(n x) ] \$

I am not going to do all the math but if you use the orthogonal signal space \$ \left\{{1, cos(nx),sin(nx)}\right\} n\in N \$ you can derive the fourier series terms as:

\$ a_n = \dfrac{1}{\pi} \displaystyle \int_{-\pi}^{\pi} g(x)cos(n x)\,\mathrm{d}x\$

\$ b_n = \dfrac{1}{\pi} \displaystyle \int_{-\pi}^{\pi} g(x)sin(n x)\,\mathrm{d}x\$

And:

\$ a_0 = \dfrac{1}{2\pi} \displaystyle \int_{-\pi}^{\pi} g(x)\,\mathrm{d}x\$

The "regular" Fourier series

\$ g(x) = \dfrac{a_0}{2} + \displaystyle \sum_{n=1}^{\infty}[a_n sin(n x) + b_n cos(n x)]\$

For the sake of symmetry the \$a_0\$ term was redefined as:

\$ a_0 = \dfrac{1}{\pi} \displaystyle \int_{-\pi}^{\pi} g(x)\,\mathrm{d}x\$

to be similar to \$a_n\$ and \$b_n\$. Therefore the \$\dfrac{1}{2}\$ term was added to the expression of the Fourier series.

Reference:

Generalized Fourier Series
Fourier Series

If you calculate some more of them, you will come to the conclusion that almost every odd coefficient \$a_n\$ is a real positive number, except for \$a_3, a_9, a_{15}...\$, \$a_{3+6k}\$, where \$k\$ is a non-negative integer. These are all negative. (And every even coefficient is zero.) For those, which are positive numbers, the phase is 0 degrees/radians; for those, which are negative, the phase is -180°/\$-\pi\$ radians, because negating a periodic signal is equal to shifting its phase by -180°. Be careful using arctan, as it has a value range of \$\pm 90°\$ but both 0° and -180° has the same tangent value, zero.

Here I plotted the values and the phase spectrum from \$n=0\$ to \$n=21\$.

Edit: The first image is obviously not the amplitude (as it can't be negative), just the values of \$a_n\$ from 0 to 21. My bad.