Sunday, July 19, 2009

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} < \infty \end{displaymath}

which in turn depends on the duration and magnitude of x[n] as well as the magnitude $\vert z\vert=r=e^\sigma$. If x[n] is right sided (i.e., x[n]=0 for n<n0), it may have infinite duration for n>0, and the larger |z| the more quickly x[n]z-n decays as $n\rightarrow \infty$. On the other hand, if x[n] is left sided (i.e., x[n]=0 for n>n0), it may have infinite duration for n<0,>z| the more quickly x[n]z-n decays as $n\rightarrow -\infty$. The angle $\angle{z}=\omega$ determines the frequency of a sinusoid which is bounded and has no effect on the convergence of the integral. Based on these observations, we can get the following properties for the ROC:

  • If x[n] is of finite duration, then the ROC is the entire z-plane (the z-transform summation converges, i.e., X(z) exists, for any z) except possibly z=0 and/or $z=\infty$.

  • The ROC of X(z) consists of a ring centered about the origin in the z-plane. The inner boundary can extend inward to the origin in some cases, and the outer can extend to infinity in other cases.

  • If x[n] is right sided and the circle |z|=r0 is in the ROC, then any finite z for which |z|>r0 is also in the ROC.

  • If x[n] is left sided and the circle |z|=r0 is in the ROC, then any z for which 0<|z|<r0 is also in the ROC.

  • If x[n] is two-sided, then the ROC is the intersection of the two one-sided ROCs corresponding to the two one-sided parts of x[n]. This intersection can be either a ring or an empty set.

  • If X(z) is rational, then its ROC does not contain any poles (by definition $X(z)\vert _{z=z_p}=\infty$ dose not exist). The ROC is bounded by the poles or extends to infinity.

  • If X(z) is a rational z-transform of a right sided function x[n], then the ROC is the region outside the outmost pole. If x[n]=0 for n<0 src="http://fourier.eng.hmc.edu/e102/lectures/Z_Transform/img76.gif" alt="$z=\infty$" align="bottom" border="0" height="17" width="59">.

  • If X(z) is a rational z-transform of a left sided function x[n], then the ROC is inside the innermost pole. If x[n]=0 for $n \ge 0$ (anti-causal), then the ROC includes z=0.

  • Fourier transform $X(e^{j\omega})$ of discrete signal x[n] exists if the ROC of the corresponding z-transform X(z) contains the unit circle |z|=1 or $z=e^{j\omega}$
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