next up previous
Next: Properties of ROC Up: Laplace_Transform Previous: Region of Convergence (ROC)

Zeros and Poles of the Laplace Transform

All Laplace transforms in the above examples are rational, i.e., they can be written as a ratio of polynomials of variable $s$ in the general form

\begin{displaymath}
X(s)=\frac{N(s)}{D(s)}=\frac{\sum_{k=0}^M b_k s^k}{\sum_{k=0...
...
=\frac{\prod_{k=1}^M (s-s_{z_k})}{\prod_{k=1}^N (s-s_{p_k})}
\end{displaymath}

In general, we assume the order of the numerator polynomial is always lower than that of the denominator polynomial, i.e., $M < N$. If this is not the case, we can always expand $X(s)$ into multiple terms so that $M < N$ is true for each of terms.

Example 1:

\begin{displaymath}X(s)=\frac{s^2-2s+2}{s^3+5 s^2+12 s+8}
=\frac{s^2-2s+2}{(s+1)...
...8)}
=\frac{[s-(1+j)][s-(1-j)]}{[s-(-1)][s-(-2+2j)][s-(-2-2j)]} \end{displaymath}

Two zeros: $s_{z1}=1+j$, $s_{z2}=1-j$;
(Three poles: $s_{p1}=-1$, $s_{p2}=-2+2j$ and $s_{p3}=-2-2j$.)

Example 2:

\begin{displaymath}X(s)=\frac{s^2-3}{s+2} \end{displaymath}

As the order of the numerator $M=2$ is higher than that of the denominator $N=1$, we expand it into the following terms

\begin{displaymath}X(s)=\frac{s^2-3}{s+2}=A+Bs+\frac{C}{s+2} \end{displaymath}

and get

\begin{displaymath}s^2-3=(A+Bs)(s+2)+C=Bs^2+(A+2B)s+(2A+C) \end{displaymath}

Equating the coefficients for terms $s^k$ $(k=0, 1, \cdots, M)$ on both sides, we get

\begin{displaymath}B=1,\;\;\;A+2B=0, \;\;\; 2A+C=-3 \end{displaymath}

Solving this equation system, we get coefficients

\begin{displaymath}A=-2; \;\;\; B=1, \;\;\; C=1 \end{displaymath}

and

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

Alternatively, the same result can be obtained more easily by a long division $(s^2-3) \div (s+2)$.

The zeros and poles of a rational $X(s)=N(s)/D(s)$ are defined as

On the s-plane zeros and poles are indicated by o and x respectively. Obviously all poles are outside the ROC. Essential properties of an LTI system can be obtained graphically from the ROC and the zeros and poles of its transfer function $X(s)$ on the s-plane.


next up previous
Next: Properties of ROC Up: Laplace_Transform Previous: Region of Convergence (ROC)
Ruye Wang 2012-01-28