Question

Find the Laplace transform of the signal x(t) = sin⁡(\(\frac{t}{2}\))u(\(\frac{t}{2}\)).

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

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

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

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

I had been asked this question in my homework.

My question is from Properties of the Laplace Transform topic in chapter Laplace Transform and System Design of Signals and Systems

Answer 1

The correct option is (d) \(\frac{2}{(2s)^2+1}\)

The best explanation: We know that sin⁡t u(t) ↔ \(\frac{1}{s^2+1}\)

Scaling property states that f(at) ↔ \(\frac{1}{a} F(\frac{s}{a})\)

\(sin⁡(\frac{t}{2})u(\frac{t}{2}) \leftrightarrow \frac{1}{(\frac{1}{2})} \frac{1}{\Big[(\frac{s}{1/2})^2+1\Big]} \leftrightarrow \frac{2}{(2s)^2+1}\).