Signals and Systems · Laplace and z transforms

Use the bilateral Laplace transform with absolute-convergence ROC

Problem

Use the bilateral Laplace transform with absolute-convergence ROC. Let $u(t)=1$ for $t>0$ and $u(t)=0$ for $t<0$, with $t$ in seconds and $s$ in $\mathrm{s}^{-1}$. Let \[ x(t)=3\,e^{-5t}\,u(t), \] and let $X(s)$ denote its bilateral Laplace transform. (a) Determine $X(s)$ and its ROC. (b) Let $x_{\mathrm{r}}(t)=x(-t)$. Using the defining integral (or a time-reversal theorem whose ROC transformation you state), determine the bilateral Laplace transform of $x_{\mathrm{r}}$ and its ROC. Locate the pole of the reversed-signal transform. (c) Let $a=2.00$ (dimensionless) and define $x_{\mathrm{c}}(t)=x(at)$. Determine the bilateral Laplace transform of $x_{\mathrm{c}}$ and its ROC. Then repeat for $a=-2.00$. In each case report the pole location.

Hint

Time reversal replaces $s$ by $-s$ and reflects the ROC through the origin. Time scaling $t\mapsto at$ replaces $s$ by $s/a$ and multiplies by $1/|a|$.

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