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Properties of ROC

Whether the z-transform $X(z)$ of a function $x[n]$ exists depends on whether or not the transform summation converges

\begin{displaymath}X(z)=\sum_{n=-\infty}^\infty x[n]z^{-n}
=\sum_{n=-\infty}^\i...
...-\infty}^\infty x[n] \vert z\vert^{-n} e^{-j\omega n} < \infty \end{displaymath}

which in turn depends on the duration and magnitude of $x[n]$ as well as the magnitude $\vert z\vert=r=e^\sigma$ (the phase of $z$ $\angle{z}=\omega$ determines the frequency of a sinusoid which is bounded and has no effect on the convergence of the integral).

Right sided signals: $x[n]=x[n]u[n-n_0]$ may have infinite duration for $n>0$, and a $z$ value with $\vert z\vert=e^{\sigma}>1$ tends to attenuate $x[n]\vert z\vert^{-n}$ as $n \rightarrow \infty$.

Left sided signals: $x[n]u[n_0-n]$ may have infinite duration for $n<0$, and a $z$ value with $\vert z\vert=e^{\sigma}<1$ tends to attenuate $x[n]\vert z\vert^{-n}$ as $n \rightarrow -\infty$.

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

Example 1:

\begin{displaymath}X(z)=\sum_{n=-3}^5 x[n]z^{-n} \end{displaymath}

When $z=0$, $z^{-n}=\infty$ for $n>0$, when $z=\infty$, $z^{-n}=\infty$ for $n<0$. Therefore neither $z=0$ nor $\vert z\vert=\infty$ are included in the ROC.

Example 2:

\begin{displaymath}x[n]=a^{\vert n\vert}=a^n u[n]+a^{-n} u[-n-1] \end{displaymath}

The Z-transform is linear, and $X(z)$ is the sum of the transforms for the two terms:

\begin{displaymath}{\cal Z}[a^n u[n]]=\frac{1}{1-az^{-1}},\;\;\;\;\;(\vert z\ver...
...ac{-1}{1-a^{-1}z^{-1}},\;\;\;\;\;(\vert z\vert<1/\vert a\vert) \end{displaymath}

If $\vert a\vert<1$, i.e., $x[n]$ decays when $\vert n\vert\rightarrow\infty$, the intersection of the two ROCs is $\vert a\vert<\vert z\vert<1/\vert a\vert$, and we have:

\begin{displaymath}{\cal Z}[x[n]]=\frac{1}{1-az^{-1}}-\frac{1}{1-a^{-1}z^{-1}}
=\frac{a^2-1}{a}\frac{z}{(z-a)(z-1/a)} \end{displaymath}

However, if $\vert a\vert>1$, i.e., $x[n]$ grows without a bound when $\vert n\vert\rightarrow\infty$, the intersection of the two ROCs is a empty set, the Z-transform does not exist.

Example 3: Given the following z-transform, find the corresponding signal:

\begin{displaymath}X(z)=\frac{1}{(1-\frac{1}{3}z^{-1})(1-2z^{-1})}
=-\frac{1/5}{1-\frac{1}{3}z^{-1}}+\frac{6/5}{1-2z^{-1}} \end{displaymath}

The two poles are $z_{p_1}=1/3$ and $z_{p_2}=2$, respectively. The $X(z)$ has three possible ROCs associated with three different time signals $x[n]$: In particular, note that only the last ROC includes the circle $\vert z\vert=1$ and the corresponding time function $x[n]$ has a discrete Fourier transform. Fourier transform of the other two functions do not exist.


next up previous
Next: Properties of Z-Transform Up: Z_Transform Previous: Zeros and Poles of
Ruye Wang 2014-10-28