19.2 : 수렴 영역

The z-transform converges only for certain values of z. This range of values is known as the Region of Convergence (ROC), which is essential for determining the behavior and stability of the system or signal.

It specifies the region in the complex plane where the z-transform converges.

The ROC can take different forms, like within a circle, outside a circle, or within an annulus.

Consider the exponential discrete-time signal x[n]. The Z-transform is a geometric series, and the ROC corresponds to the region outside a circle with the radius a, centered at the origin.

Depending on the value of the radius, if the ROC includes the unit circle, the system is stable; if it is outside the unit circle, it is unstable.

When the ROC coincides with the unit circle, the system is marginally stable.

The DTFT exists only if the ROC includes the unit circle.

The ROC does not include poles since the z-transform does not converge at poles.

It also affects the inverse z-transform, which retrieves the original time-domain signal.