I would like to do some examples so my classmates can see a general way of solving problems from this chapter. All problems come from the textbook.
Question 1
Lets start with a region of convergence problem. Chapter 9 Problem 7
$ \frac{(s-1)}{(s+2)(s+3)(s^2+s+1)} $
Given the equation we know that there are 4 poles.
s0 = -2
s1 = -3
s3 = $ \frac{-1}{2}+\frac{3^{.5}}{2}j $
s3 = $ \frac{-1}{2}-\frac{3^{.5}}{2}j $
Given these poles, the regions of convergence are as follows...
Re{s} > $ \frac{-1}{2} $
-2 < Re{s} < $ \frac{-1}{2} $
-3 < Re{s} < -2
Re{s} < -3
Question 2
Find the Laplace Transform of the following equation...
$ X(s) = \frac{2(s+2)}{s^{2}+7s+12} $, (Re{s} > -3)
Using partial fraction expansion, we get...
$ X(s) = \frac{4}{s+4} - \frac{2}{s+3} $
Using the inverse Laplace transform we conclude that...
$ x(t) = 4(e^{-4t})u(t) - 2(e^{-3t})u(t) $
