Obtain Fourier series of the following full wave rectified sine wave shown in the figure,

Solution:

$ \boldsymbol{a}_0=\frac{2}{T} \int_0^{\frac{T}{2}} x(t) d t=\frac{2}{1} \int_0^{\frac{1}{2}} x(t) d t=\left[2 \int_0^{\frac{1}{2}} \sin \pi t d t\right]=2\left[-\frac{\cos \pi t}{\pi}\right]_0^{\frac{1}{2}}=-\frac{2}{\pi}\left[\cos \frac{\pi}{2}-\cos 0\right]=\frac{\mathbf{2}}{\boldsymbol{\pi}}\\ $

$ \boldsymbol{a}_{\boldsymbol{n}}=\frac{4}{T} \int_0^{\frac{T}{2}} x(t) \cos n \Omega_0 t d t=\frac{4}{1} \\\int_0^{\frac{1}{2}} \sin \pi t \cos n 2 \pi t d t=2 \int_0^{\frac{1}{2}}[\sin ((1+2 n) \pi t)+\sin ((1-2 n) …