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Signals and Systems – 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 $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete-time signal or sequence, then its bilateral or two-sided Z-transform is defined as −
$$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty }}^{\infty }\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z}^{-\mathit{n}}}$$
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 $\mathit{x}\mathrm{\left(\mathit{n}\right)}$, i.e., $\mathit{X}\mathrm{\left(\mathit{z}\right)}$ converges is called the region of convergence (ROC) of $\mathit{X}\mathrm{\left(\mathit{z}\right)}$.
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 $\mathit{X}\mathrm{\left(\mathit{z}\right)}$ is a rational, then its ROC is bounded by poles or extends up to infinity.
For $\mathit{x}\mathrm{\left(\mathit{n}\right)} \:\mathrm{=}\: \mathit{\delta}\mathrm{\left(\mathit{n}\right )}$, i.e., impulse sequence is the only sequence whose ROC of Z-transform is the entire z-plane.
If $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is an infinite duration causal sequence, then its ROC is $\left|\mathit{z} \right|>\mathit{a}$, i.e., it is the exterior of a circle of the radius equal to a.
If $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is an infinite duration anti-causal sequence, then its ROC is $\left|\mathit{z} \right|<\mathit{b}$, i.e., it is the interior of a circle of the radius equal to b.
If $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is an infinite duration two-sided sequence, then its ROC is $\mathit{a}<\left|\mathit{z} \right|<\mathit{b}$, 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 $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a finite duration causal sequence (i.e., right-sided sequence), then its ROC is the entire z-plane except at z = 0.
If $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a finite duration anti-causal sequence (i.e., left sided sequence), then its ROC is the entire z-plane except at $z\:\mathrm{=}\:\mathit{\infty}$.
If $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a finite duration two-sided sequence, then its ROC is the entire z-plane except at z = 0 and $z\:\mathrm{=}\:\mathit{\infty}$.
The ROC of the sum of two or more sequences is equal to the intersection of the ROCs of these sequences.
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