Utilize the L transforms of the unit step and the delayed 8

Utilize the L - transforms of the unit step and the delayed 8unit step functions to work out the R- transform of the following function f(t) = {a less than or equal to t less than or equal to T b T b are constants.

Solution

Laplace transform formulas for step function and unit step function

L[U(t)]=1/s

L[U(t-a)]=e^-as/s

From Wave form

f(t) =a[U(t)-U(t-T)]+bU(t-T)

Applying Laplace transform

F(s) =a[1/s - e^-Ts/s] +b(e^-Ts/s)

F(s)=(1/s)[1+(b-a)e^-Ts]