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Obtain the F.S expression for the waveform shown in the figure,

Obtain the F.S expression for the waveform shown in the figure,

enter image description here

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Solution:

$ C_n=\frac{1}{T_0} \int_t^{t+T_0} x(t) \mathrm{e}^{-j n \omega_0 t} \mathrm{~d} t\\ $

$ =\frac{1}{2 \pi} \int_0^{2 \pi} x(t) \mathrm{e}^{-j n t} \mathrm{~d} t\\ $

$ \begin{aligned} & =\frac{1}{2 \pi}\left(\int_0^\pi \mathrm{e}^{-j n t} \mathrm{~d} t-\int_\pi^{2 \pi} 2 \mathrm{e}^{-j n t} \mathrm{~d} t\right) \\\\ & =\frac{1}{-2 \pi j n}\left(\left(\mathrm{e}^{-j n \pi}-1\right)-2\left(\mathrm{e}^{-j n 2 \pi}-\mathrm{e}^{-j n \pi}\right)\right) \\\\ & =\frac{1}{-2 \pi j n}\left[(-1)^n-1-2\left(1-(-1)^n\right)\right] \\\\ & =\frac{3\left[1-(-1)^n\right]}{2 \pi j n}\\ \end{aligned} $

$ \mathrm{x}(t)=\sum_{n=-\infty}^{\infty} C_n \mathrm{e}^{j n t}\\ $

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