Signals in Natural Domain
Chapter 7 :  The Z-transform
 
Properties of the ROC
  1. The ROC of $ X(z)$ consists of an annular region in the z-plane, centered about the origin. This property follows from equation (7.3), where we see that convergence depends on $ r$ only.
  2. The ROC does not certain any poles. Since at poles $ X(z)$ does not converge.
  3. The ROC is a connected region in z-plane. This property is proved in complex analysis.
  4. If $ \{x[n]\}$ is a right sided sequence, i.e. $ x[n]=0$, for  $ n<n_{0}$, and if the circle $ \vert Z\vert =r_{0}$ is in the ROC, then all finite values of $ z$, for which $ \vert z\vert > r_{0}$  will also be in the ROC.
    For a right sided sequence
    $\displaystyle X(z)=\sum\limits^{\infty}_{n=n_{0}} x[n]z^{-n}$

    If $ n_{0}$ is negative then we can write
                                                                  $\displaystyle X(z)=\sum\limits^{0}_{n=n_0}x[n]z^{-n}+\sum\limits^{\infty}_{n=1}x[n]z^{-n}$

    Let $ Z=re^{jw}$, with $ r>r_{0}$, then, $ X(z)$ exists if
    $ \sum\limits^{-1}_{n=n_{0}}\vert x[n]\vert r^{-n}+\sum\limits^{\infty}_{n=0}\vert x[n]\vert r^{-n}$ is finite.
    The first summation is finite as it consists of a finite number of terms. In the second summation note that each term is less than $ \vert x[n]\vert r^{-n}_{0}$ as. Since $ \sum\limits^{\infty}_{n=1} \vert x[n]\vert r^{-n}_{0}$ is finite by our assumption that circle with radius $ r_{0}$ lies in ROC, the second sum is also finite. Hence values of z such that $ \vert z\vert > r_{0}$ lies in ROC, except when. At $ z=\infty$, the first summation will became infinite. So if $ n_{0}\geq 0$, i.e. the sequence $ x[n]$ is causal, the value $ z=\infty$ will lie in the ROC.
  5. If $ \{x[n]\}$ is left sided sequence, i.e. $ x[n]=0,\,n>n_0$ and $ \vert z\vert =r_{0}$ lies in the ROC, the values of $ z$ function $ 0<\vert z\vert<r_{0}$ also lie in the ROC.
    The proof is similar to the property 4. The point $ z=0$, will lie in the ROC if the sequence is purely
    anticausal $ (x[n]=0,\,n>0)$
  6. If $ \{x[n]\}$ is non zero for, $ n_{1}\leq n \leq n_{2}$, then ROC is entire z-plane except possibly $ z=0$, and/or. In this case the $ X(z)$ consists of finite number of terms and therefore it converges if each term infinite which is the case when $ z$ is different from 0 or. $ z=0$ lies in ROC, if $ n_{2}\leq{0}$, and $ z=\infty$ lies in the ROC if.
  7. If $ \{x[n]\}$ is two-sided sequence and if circle $ \vert z\vert =r_{0}$ is in ROC, then ROC will consist of annular region in z-plane, which includes. We can express a two sided sequence as sum of a right sided sequence and a left sided sequence. Then using property 4 and 5 we get this property. Using
    property 2 and 3 we see what ROC will be banded by circles passing through the poles.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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