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EE 315 Class 32On Region of Convergence (ROC)

• Poles are never part of the ROC.

• For a z -transform X [z ] of a signal of nite-duration, the region of convergence consists of almost the entire complex plane with possiblythe exception of the origin and/or z = ∞ .

• For a z -transform X [z ] of a signal of innite-duration:

• A right sided signal x [n ] has a region of convergence |z | > R 1 whereR 1 is the largest radius of the poles of X [z ]. That is, the region of convergence is the outside of a circle of radius R 1 .

• An left sided signal x [n ] has as region of convergence the inside of the circle dened by the smallest radius R 2 of the poles of X [z ] or|z | < R 2 .

• A two-sided signal x [n ] has as region of convergence R 1 < |z | < R 2 ,or the inside of a torus of inside radius R 1 and outside radius R 2corresponding to the maximum and minimum radii of the poles of X r [z ] and X l [z ] which are the z -transforms of the right and leftcomponents of x [n ].