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# 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 in the general form

• is the numerator polynomial of order with roots ,
• is the denominator polynomial of order with roots .
In general, we assume the order of the numerator polynomial is always lower than that of the denominator polynomial, i.e., . If this is not the case, we can always expand into multiple terms so that is true for each of terms.

Example 1:

Two zeros: , ;
(Three poles: , and .)

Example 2:

As the order of the numerator is higher than that of the denominator , we expand it into the following terms

and get

Equating the coefficients for terms on both sides, we get

Solving this equation system, we get coefficients

and

Alternatively, the same result can be obtained more easily by a long division .

The zeros and poles of a rational are defined as

• Zero: Each of the roots of the numerator polynomial for which is a zero of ;

If the order of exceeds that of (i.e., ), then , i.e., there is a zero at infinity:

• Pole: Each of the roots of the denominator polynomial for which is a pole of ;

If the order of exceeds that of (i.e., ), then , i.e, there is a pole at infinity:

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 on the s-plane.

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