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

+1 vote
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) $\frac{2}{(2s)^2+1}$

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

by (42.1k points)
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) ↔ $\frac{1}{a} F(\frac{s}{a})$

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

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