Question 8.10: Laplace transforms of two time-scaled rectangular pulses Fin......

We have already found the Laplace transform of u(t), which is 1/s, σ > 0. Using the linearity and time-shifting properties,

u(t)-u(t-a) \stackrel{\mathcal{L}}{\longleftrightarrow}  \frac{1-e^{-a s}}{s} ,  all σ.

Now, using the time scaling property,

\mathrm{u}(2 t)-\mathrm{u}(2 t-a) \stackrel{\mathcal{L}}{\longleftrightarrow} \frac{1}{2} \frac{1-e^{-a s / 2}}{s / 2}=\frac{1-e^{-a s / 2}}{s},  all σ.

This result is sensible when we consider that u(2t) = u(t) and u(2t − a) = u(2(t − a/2)) = u(t − a/2).