Signals and Systems

6.21 POLE-ZERO PLOT

If X(z) represents the rational z-transform, it can be expressed as

 

 

The poles and zeros are the values of z for which the value of X(z) are ∞ and 0. In ROC X(z) has the finite value whereas X(z) has the value ∞ at poles. Therefore, the poles does not lie in the ROC of X(z). Again, X(z) has the finite value at zeros. Hence zeros lie in the ROC of X(z). We can write the Eq. (6.77) in the short hand form as

 

 

From Eq. (6.77), we have

 

 

From Eq. (6.79), we conclude that

 

X(z) = 0 for z = z₁,z₂,z₃,∙∙∙∙∙∙∙∙∙, z_M i.e., ...