Question

Find the Laplace transform of the periodic function below. f(t) = { 8 if 0 < t < 1 0 if i<t<2 ; f(t + 2) = f(t) f(0) 2 3 -4 -6 7 Q

Answer 1

If a function is periodic with period p , so that $f(p&plus;t)=f(t)$ ,and let $f_{1}(t)$ is one period of the function then

$L\left \{ f(t) \right \}\Leftrightarrow L\left \{ f_{1}(t) \right \}\times \frac{1}{1-e^{-sp}}$

Hence the Laplace Transform of the periodic function f(t) with period p , equals the Laplace Transform of one cycle of the function, divided by $(1-e^{-sp})$

$L\left \{ u(t-c) \right \}\Leftrightarrow \frac{e^{-cs}}{s}$

$f_{1}(t)=8[(u(t)-u(t-1))]$

$L\left \{ f_{1}(t) \right \}\Leftrightarrow \frac{8}{s}-\frac{8e^{-s}}{s}$

$L\left \{ f_{1}(t) \right \}\Leftrightarrow \frac{8(1-e^{-s})}{s}$

Hence the laplace transform of f(t) is

$L\left \{ f(t) \right \}\Leftrightarrow \frac{8(1-e^{-s})}{s(1-e^{-2s})}$$L\left \{ f(t) \right \}\Leftrightarrow \frac{8(1-e^{-s})}{s(1-e^{-2s})}$