Lecture 19 - Notes

February 26, 2016

Region of Convergence Continued

Properties

  1. The Region of Convergence consist of a ring centered around the origin.
  2. The Discrete-Time Fourier Transform of $x[n]$ is convergent if and only if the Region of Convergence of the Z-Transform of $x[n]$ contains the unit circle.
  3. The Region of Convergence does not contain any poles.
  4. If $x[n]$ is of finite duration, then the Region of Convergence is the entire z-plane, except possibly $z = 0$ or $z = \infty$.
  5. If $x[n]$ is a right-sided sequence the Region of Convergence extends outward from the outermost finite pole (possibly to $z = \infty$).
  6. If $x[n]$ is a left-sided sequence the Region of Convergence extends inward from the innermost finite pole (possibly including $z=0$).
  7. If $x[n]$ is a two-sided sequence then the Region of Convergence is a ring bounded by interior and exterior poles.
  8. The Region of convergence must be a connect region.
  9. If $x[n]$ and $y[n]$ have Regions of Convergence, $R_x$ and $R_y$, then a linear combination, $v[n] = ax[n] + by[n]$, will have the Region of Convergence that include the Regions of Convergence of $x[n]$ and $y[n]$, $R_v \supseteq R_x \cap R_y$.

These properties will be provided on the midterm.

Example

Find the region of convergence of