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Find the coefficient of Fourier Series for the given signal, $$ \mathrm{x}(t)=\sum_{n=-\infty}^{\infty} \delta\left(t-n T_0\right) $$
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Solution:

$ C_n=\frac{1}{T_0} \int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} x(t) \mathrm{e}^{-j n \omega_0 t} \mathrm{~d} t $

$ =\frac{1}{T_0} \int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \delta(t) \mathrm{e}^{-j n \omega_0 t} \mathrm{~d} t $

$ \text { Since: } \mathrm{x}(\mathrm{t}) \cdot \delta(t)=\mathrm{x}(0) \cdot \delta(t) \text { and area under impulse is unity } $

$ =\frac{1}{T_0} \int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \delta(t) \mathrm{e}^0 \mathrm{~d} t $

$ =\frac{1}{T_0} \int_{-\frac{T_0}{2}}^{\frac{T_0}{2}} \delta(t) \mathrm{d} t=\frac{1}{T_0} $

$ C_n=\frac{1}{T_0} $

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