Properties of Region of Convergence (ROC) of the Z-Transform

Z-Transform

The Z-transform (ZT) is a mathematical tool which is used to convert the difference equations in time domain into the algebraic equations in z-domain.

Mathematically, if $\mathrm{x(n)}$ is a discrete-time signal or sequence, then its bilateral or two-sided Z-transform is defined as −

$$\mathrm{Z \left[x \left(n\right)\right]\:=\:X\left(z\right)\:=\:\sum_{n=-\infty}^{\infty }x\left(n\right)z^{-n}}$$

Where, z is a complex variable.

Region of Convergence (ROC) of Z-Transform

The set of points in z-plane for which the Z-transform of a discrete-time sequence $\mathrm{x \left(n\right)}$, i.e., $\mathrm{X \left(z\right)}$ converges is called the region of convergence (ROC) of $\mathrm{X \left(z\right)}$.

Properties of ROC of Z-Transform

The region of convergence (ROC) of Z-transform has the following properties −

• The ROC of the Z-transform is a ring or disc in the z-plane centred at the origin.
• The ROC of the Z-transform cannot contain any poles.
• The ROC of Z-transform of an LTI stable system contains the unit circle.
• The ROC of Z-transform must be connected region. When the Ztransform $\mathrm{X \left(z\right)}$ is a rational, then its ROC is bounded by poles or extends up to infinity.
• For $\mathrm{x \left(n\right) \:=\: \delta\left(n\right )}$, i.e., impulse sequence is the only sequence whose ROC of Z-transform is the entire z-plane.
• If $\mathrm{x\left(n\right)}$ is an infinite duration causal sequence, then its ROC is $\mathrm{\left|z\right|\:\gt\:a}$, i.e., it is the exterior of a circle of the radius equal to a.
• If $\mathrm{x\left(n\right)}$ is an infinite duration anti-causal sequence, then its ROC is $\mathrm{|z|\:\lt\:b}$, i.e., it is the interior of a circle of the radius equal to b.
• If $\mathrm{x(n)}$ is an infinite duration two-sided sequence, then its ROC is $\mathrm{a\:\lt\:|z|\:\lt\:b}$, i.e., it consists of a ring in the z-plane, which is bounded on the interior and exterior by a pole and does not contain any poles.
• If $\mathrm{x\left(n\right)}$ is a finite duration causal sequence (i.e., right-sided sequence), then its ROC is the entire z-plane except at z = 0.
• If $\mathrm{x\left(n\right)}$ is a finite duration anti-causal sequence (i.e., left sided sequence), then its ROC is the entire z-plane except at $\mathrm{z\:=\: \infty}$.
• If $\mathrm{x \left(n\right)}$ is a finite duration two-sided sequence, then its ROC is the entire z-plane except at z = 0 and $\mathrm{z\:= \: \infty}$.
• The ROC of the sum of two or more sequences is equal to the intersection of the ROCs of these sequences.