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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 $z$ in the general form

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

where $N(z)$ is the numerator polynomial of order $M$ with roots $z_{z_k}, (k=1,2,
\cdots, M)$, and $D(z)$ is the denominator polynomial of order $N$ with roots $z_{p_k}, (k=1,2, \cdots, N)$. In general, we assume the order of the numerator polynomial is lower than that of the denominator polynomial, i.e., $M < N$. If this is not the case, we can always expand $X(z)$ into multiple terms so that $M < N$ is true for each of terms.

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

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


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