Property of ROC
Property 1
The ROC of the Laplace Transformation consists of vertical strips in the complex plane. It could be empty or the entire plane.
Property 2
If x(t) is of "Finite duration", i.e. there exists a $ t_m $ such that x(t)=0 when $ |t|>t_m $,
and if $ \int_{-\infty}^\infty|x(t)|^2dt $ is finite for all values of s,
Then the ROC is the entire complex plane.
Property 3
If x(t) is "left sided", i.e. there exists a $ t_m $ such that x(t)=0 when $ t>t_m $, then whenever a vertical line is in the ROC, the half plane left of that line is also in the ROC.
Property 4
If x(t) is "right sided", i.e. there exists a $ t_M $ such that x(t)=0 when $ t<t_M $, then whenever a vertical line is in the ROC, the half plane right of that line is also in the ROC.
Property 5
If x(t) is "two sided", i.e. there exists no $ t_m $ such that x(t)=0 for $ t>t_m $ and no $ t_M $ such that x(t)=0 for $ t<t_M $, then the ROC is either empty of it is a strip in the complex plane. (only one strip)
Property 6
If X(s) is rational, i.e. $ X(s)=\frac {P(s)}{Q(s)} $ where P(s),Q(s) are polynomial,
Then the ROC does not contain any zero of Q(s), i.e. the poles of X(s).
Property 7
If $ X(s)=\frac {P(s)}{Q(s)} $ and x(t) is right sided,
Then the ROC is the half plane starting from the vertical line through the pole with the largest real part and extending to infinity.
If $ X(s)=\frac {P(s)}{Q(s)} $ and x(t) is left sided,
Then the ROC is the half plane starting from the vertical line through the pole with the smallest real part and extending to -infinity.
Property 8
If $ X(s)=\frac {P(s)}{Q(s)} $,
ROC is either bounded by poles or extends to infinity or -infinity.
