Topic: Signals and Systems
The Laplace Transform of $x(t)$ formed by convolution operator is $x(t)=\{t\cdot u(t)\}\otimes\{cos~2\pi t\cdot u(t)\}$
- A. $1/[s(s^{2}+4\pi^{2})]$
- B. $2\pi/[s(s^{2}+4\pi^{2})]$
- C. $1/[s^{2}(s^{2}+4\pi^{2})]$
- D. $s^{3}/(s^{2}+4\pi^{2})$
Answer: A. $1/[s(s^{2}+4\pi^{2})]$
Explanation: $x(t)=\{t\cdot u(t)\}\otimes\{cos~2\pi t~u(t)\}$. In the Laplace domain, convolution becomes multiplication: $X(s)=\frac{1}{s^{2}}\cdot\frac{s}{s^{2}+(2\pi)^{2}}=\frac{1}{s(s^{2}+4\pi^{2})}$.