I have the following question:
Pole-zero plot of x(t) and y(t) are given below:
The signal $g(t)$ and $h(t)$ are defined as $g(t)=x(t)e^{-3t}$ and $h(t)=y(t)*e^{-t}u(t)$. If $g(t)$ and $h(t)$ are both absolutely integrable, determine whether the signals $g(t)$, $h(t)$ are left-sided/right-sided.
My try:
I take the laplace transform of both the signals and get $G(s)=X(s+3)$ and $H(s)=Y(s)\cdot \frac{1}{s+1}$
Also because both $g(t)$ and $h(t)$ are absolutely integrable, their transforms must be stable so both must have $jw$-axis in their respective ROC.
As we can see $X(s+3)$ shifts the pole-zero plot to the left by $3$ units so we have all the poles in the left $s$-plane and ROC would be $Re\{s\}>-1$ hence $g(t)$ is right sided.
Similarly $H(s)$ has all the poles in the left $s$-plane and ROC is again $Re\{s\}>-1$ hence $h(t)$ is right sided.
Is this reasoning correct?