Question

Find the Laplace transform of the signal x(t)=te^-2|t|.

(a) \(-\frac{1}{(s-2)^2} + \frac{1}{(s+2)^2}\)

(b) \(\frac{1}{(s-2)^2} + \frac{1}{(s+2)^2}\)

(c) \(\frac{1}{(s-2)^2} – \frac{1}{(s+2)^2}\)

(d) \(-\frac{1}{(s-2)^2} – \frac{1}{(s+2)^2}\)

I have been asked this question during an interview.

The question is from The Laplace Transform topic in portion Laplace Transform and System Design of Signals and Systems

Answer 1

Right choice is (a) \(-\frac{1}{(s-2)^2} + \frac{1}{(s+2)^2}\)

The explanation is: Given  x(t)=te^-2|t|

L{x(t)} = X(s) = \(\int_{-∞}^∞ x(t) e^{-st} \,dt = \int_{-∞}^∞ te^{-2|t|} e^{st} \,dt \)

=\(\int_{-∞}^0 te^{2t} e^{-st} \,dt + \int_0^∞ te^{-2t} e^{-st} \,dt = -\frac{1}{(s-2)^2} + \frac{1}{(s+2)^2}\).