# Region of Convergence Properties

## Region of Convergence Properties

ROC Region of Convergence -

Definition: ROC is the set of all values of z for which X(Z) will have finite value.

Before going to ROC properties, first it is necessary to study ROC for finite and infinite duration sequences.

### ZT and ROC for finite duration sequence

a) Right handed / Positive /causal sequence-

e.g. x[n] = [1 (origin), 2, 5, 7, 0, 1]

Solution: X(z) = 1 + 2z-1+ 5z-2 + 7z-3 + z-5,

ROC: entire z plane except z = 0

b) Left handed / Negative /Anti-causal sequence-

c) Both sided sequence -

### ZT and ROC for infinite duration sequence

a) Right handed / Positive /causal sequence-

b) Left handed / Negative /Anti-causal sequence-

c) Both sided sequence -

### Properties of ROC

ROC is a ring or disk centered at the origin.

ROC must be a connected region.

## References

- WikiNote Foundation