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Question 21.39: A waveform, f(t), is defined as follows: f(t) = {2   0<t......

A waveform, f(t), is defined as follows:

f(t)= \begin{cases}2 & 0<t \leqslant 1.25 \\ 0 & 1.25<t \leqslant 1.5\end{cases}

and f(t) is periodic with period of 1.5. Find the Laplace transform of the waveform.

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\begin{aligned} \mathcal{L}\{f(t)\} & =\frac{\int_0^{1.5} \mathrm{e}^{-s t} f(t) \mathrm{d} t}{1-\mathrm{e}^{-1.5 s}} \\ & =\frac{\int_0^{1.25} 2 \mathrm{e}^{-s t} \mathrm{~d} t+\int_{1.25}^{1.5} 0 \mathrm{e}^{-s t} \mathrm{~d} t}{1-\mathrm{e}^{-1.5 s}} \\ & =\frac{2\left(1-\mathrm{e}^{-1.25 s}\right)}{s\left(1-\mathrm{e}^{-1.5 s}\right)} \end{aligned}

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