next up previous
Next: Properties of Laplace Transform Up: Laplace_Transform Previous: Zeros and Poles of

Properties of ROC

The existence of Laplace transform $X(s)$ of a given $x(t)$ depends on whether the transform integral converges

\begin{displaymath}X(s)=\int_{-\infty}^\infty x(t)e^{-st} dt
=\int_{-\infty}^\infty x(t)e^{-\sigma t} e^{-j\omega t} dt < \infty \end{displaymath}

which in turn depends on the duration and magnitude of $x(t)$ as well as the real part of $s$ $Re[s]=\sigma$ (the imaginary part of $s$ $Im[s]=j\omega$ determines the frequency of a sinusoid which is bounded and has no effect on the convergence of the integral).

Right sided signals: $x(t)=x(t)u(t-t_0)$ may have infinite duration for $t>0$, and a positive $\sigma>0$ tends to attenuate $x(t)e^{-\sigma t}$ as $t \rightarrow \infty$.

Left sided signals: $x(t)=x(t)u(t_0-t)$ may have infinite duration for $t<0$, and a negative $\sigma<0$ tends to attenuate $x(t)e^{-\sigma t}$ as $t \rightarrow -\infty$.

Based on these observations, we can get the following properties for the ROC:

Example 1: Consider the Laplace transform of a two-sided signal $x(t)=e^{-b\vert t\vert}$:

\begin{displaymath}X(s)={\cal L}[x(t)]={\cal L}[e^{-b\vert t\vert}]
={\cal L}[e^{-bt}u(t)]+{\cal L}[e^{bt}u(-t)] \end{displaymath}

The Laplace transform of the two components can be obtained from the two examples discussed above. From example 1, we get

\begin{displaymath}{\cal L}[e^{-bt}u(t)]=\frac{1}{s+b},\;\;\;\;\;Re[s]>-b \end{displaymath}

and let $b=-a$ in example 2, we have

\begin{displaymath}{\cal L}[e^{bt}u(-t)]={\cal L}[e^{-at}u(-t)]=-\frac{1}{s+a}=-\frac{1}{s-b},
\;\;\;\;\;Re[s]<-a=b \end{displaymath}

Combining the two components, we have

\begin{displaymath}{\cal L}[e^{-b\vert t\vert}]=\frac{1}{s+b}-\frac{1}{s-b}=\frac{-2b}{s^2-b^2},
\;\;\;\;\;-b<Re[s]<b \end{displaymath}

Whether $X(s)$ exists or not depends on $b$. If $b>0$, i.e., $x(t)$ decays exponentially as $\vert t\vert \rightarrow \infty$, then the ROC is the strip between $-b$ and $b$ and $X(s)$ exists. But if $b<0$, i.e., $x(t)$ grows exponentially as $\vert t\vert \rightarrow \infty$, then the ROC is an empty set and $X(s)$ does not exist.

Example 2: Given the following Laplace transform, find the corresponding signal:

\begin{displaymath}X(s)=\frac{1}{(s+1)(s+2)}=\frac{1}{s+1}-\frac{1}{s+2} \end{displaymath}

There are three possible ROCs determined by the two poles $s_{p_1}=-1$ and $s_{p_2}=-2$: In particular, note that only the first ROC includes the $j\omega$-axis and the corresponding time function has a Fourier transform. Fourier transform does not exist in the other two cases.


next up previous
Next: Properties of Laplace Transform Up: Laplace_Transform Previous: Zeros and Poles of
Ruye Wang 2012-01-28