[공돌이] ROC and Pole/Zero in Z-transform

The ROC(Region of Convergence) is always bounded by a circle since the convergence condition is on the magnitude |Z|

• The ROC for right-sided sequences (x(n)|ncircle of radius R_x- (if n_o>=0 → x(n) is a causal sequence.)

• The ROC for left-sided sequences (x(n)|n>n_o = 0) is always inside of a
circle of radius R_x+ (if n_o<= 0 → x(n) is a anticausal sequence)

• The ROC for two-sided sequences is always an open ring R_x- < |Z| < R_x+
if it exists.

• The ROC for finite-duration sequences(x(n)|n_2(if n_1<0 → z = ∞ is not in the ROC. if n_2>0 → z = 0 is not in the ROC)

• The ROC cannot include a pole since X(z) converges uniformly in there

• There is at least one pole on the boundary of a of a rational X(z)

• The ROC is one contiguous region, the ROC does not come in pieces.


Pole & Zero in common Z-Transform

Let assume that there is a result of Z - transform of x(n) :

X(z) = (z-b)/(z-a)

1) if z=a, then X(z) goes to infinity, so it can't be exist : so the pole of this transform is z=a

2) if z=b, then X(z) goes to zero, so the zero of this transform is z=b


Propertie랑 기타 잡다한건 나중에