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Zeros and Poles of Z-Transform

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

where is the numerator polynomial of order with roots , and is the denominator polynomial of order with roots . In general, we assume the order of the numerator polynomial is 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.

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:

Most essential behavior properties of an LTI system can be obtained graphically from the ROC and the zeros and poles of its transfer function on the z-plane.

Next: Properties of ROC Up: Z_Transform Previous: Region of Convergence and
Ruye Wang 2014-10-28