Property of ROC
Property 1
The ROC of the Z Transformation consists of a ring in the Z-plane centered about the origin.
Property 2
The ROC of the Z Transformation does not contain any poles
Deduction: If a specific value $ Z_0 $ is on ROC, all values of Z on the same circle $ (|Z|=|Z_0|) $ are part of ROC.
Property 3
If x[n] is of "Finite duration", then the ROC is the entire Z-plane, exclude possibly Z=0 and/or z=$ \infty $.
Property 4
If x[n] is "right-sided", and a circle $ (|Z|=r_0) $ is part of ROC, then all finite value $ |Z|>r_0 $ is also part of ROC.
Property 5
If x[n] is "left-sided", and a circle $ (|Z|=r_0) $ is part of ROC, then all value $ 0<|Z|<r_0 $ is also part of ROC.
Property 6
If x(t) is "two sided", and a circle $ (|Z|=r_0) $ is part of ROC, then the ROC contains a ring which includes the circle $ (|Z|=r_0) $.
Property 7
If X(Z) is rational, i.e. $ X(Z)=\frac {P(Z)}{Q(Z)} $ where P(s),Q(s) are polynomial,
Then the ROC is bounded by the poles (i.e. values that make Q(Z)=0) or extendes to infinity.
Property 8
If X(Z) is rational, i.e. $ X(Z)=\frac {P(Z)}{Q(Z)} $ where P(s),Q(s) are polynomial, and x[n] is right-sided
Then the ROC is the area out of the circle containing the outmost pole (i.e. values that make Q(Z)=0 with maximun absolute value).
Property 9
If X(Z) is rational, i.e. $ X(Z)=\frac {P(Z)}{Q(Z)} $ where P(s),Q(s) are polynomial, and x[n] is left-sided
Then the ROC is the area inside the circle containing the innermost non-zero pole (i.e. values that make Q(Z)=0 with minimum absolute value).
