Homework ECE301Fall2008mboutin - Rhea

Property 1

The ROC of the Laplace Transformation consists of vertical strips in the complex plane. It could be empty or the entire plane.

Why? The ROC of X(s) consists of those $s=a+j\omega$ for which the Fourier transform of $x(t)e^{-at}$ converges. This condition only depends of a.

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.

For example, x(t)=blah(u(t+7)-u(t-7)), then the ROC is the entire plane or empty

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.

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.

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)

If X(s) is rational, i.e. $X(s)=\frac {P(s)}{Q(s)}$ with P(s),Q(s) polynomial,

Then the ROC does not contain any zero of Q(s), i.e. the poles of X(s).

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.

If $X(s)=\frac {P(s)}{Q(s)}$ ,

ROC is either bounded by poles or extends to infinity or -infinity.