Question

Find the Laplace transform of the signal x(t) = e^-2t cos⁡(200πt)u(t).

(a) \(\frac{s}{s^2+(200π)^2}\)

(b) \(\frac{s}{s^2-(200π)^2}\)

(c) \(\frac{s-2}{(s-2)^2+(200π)^2}\)

(d) \(\frac{s+2}{(s+2)^2+(200π)^2}\)

The question was posed to me in a job interview.

Asked question is from Properties of the Laplace Transform topic in division Laplace Transform and System Design of Signals and Systems

Answer 1

Right choice is (d) \(\frac{s+2}{(s+2)^2+(200π)^2}\)

The explanation: Given x(t) = e^-2t  cos⁡(200πt)u(t)

We know that L{cos⁡ωt u(t)} = \(\frac{s}{s^2+ω^2}\)

∴L{cos⁡(200πt)u(t)} = \(\frac{s}{s^2+(200π)^2}\)

Frequency shifting property states that L{e^-at x(t)} = X(s+a)

L{e^-2t cos⁡(200πt)u(t)} = L{cos⁡(200πt)u(t)}|s=s+2 = \(\Big[\frac{s}{s^2+(200π)^2}\Big]_{s=s+2} = \frac{s+2}{(s+2)^2+(200π)^2}\).