Consider following function of t f ( t ) = 1 / a [ u ( t - T0 ) - u ( t - T0 - a

Question

Consider following function of t f ( t ) = 1 / a [ u ( t - T0 ) - u ( t - T0 - a ) ] Here both T0 and a are positive numbers If you pick some values for T0 and a you discover that you have a rectangular pulse starting at T0 having width a and height 1 / a Obtain the Laplace Transform of this pulse by direct integration. If you let a rightarrow 0 you obtain a unit Impulse Let a rightarrow 0 in the expression you obtained for the Laplace Transform of the pulse to obtain Laplace Transform of the shited unit impulse.

Explanation / Answer

int(e^(-st) * f(t), t = 0..infinity) = 1/a * int(e^(-st) * 1, t = Ta .. Ta + a) = 1/a * [-1/s * e^(-st)] t = Ta .. Ta + a = 1/(as) * [e^(-sTa) - e^(-s(Ta + a))] = e^(-sTa)/(as) * (1 - e^(-as))

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