Convolution between two functions $f(t)$ and $g(t)$ is defined as follows:
\[
f(t) * g(t)=\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau
\]
Let $u(t)$ be the unit-step function, i.e., $u(t)=1$ for $t \geq 0$ and $u(t)=0$ for $t<0$.
What is $f(t) * g(t)$ if $f(t)=\exp (-t) u(t)$ and $g(t)=\sin (t) u(t)$ ?
- $\frac{1}{2}(\exp (-t)-\sin (t)+\cos (t)) u(t)$
- $\frac{1}{2}(\exp (-t)+\sin (t)-\cos (t)) u(t)$
- $\frac{1}{2}(\exp (-t)+2 \sin (t)-\cos (t)) u(t)$
- $\frac{1}{2}(\exp (-t)+\sin (t)-2 \cos (t)) u(t)$
- $\frac{1}{2}(\exp (-t)-\sin (t)+2 \cos (t)) u(t)$