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Region of Convergence for the Z-transform

Module by: Benjamin Fite

Summary: (Blank Abstract)

The Region of Convergence

The region of convergence, known as the ROC, is important to understand because it defines the region where the z-transform exists. The z-transform of a sequence is defined as
Xz=∑n=-∞∞xnz-n Xz n x n z n (1)
The ROC for a given xn x n , is defined as the range of z z for which the z-transform converges. Since the z-transform is a power series, it converges when xnz-n x n z n is absolutely summable. Stated differently,
∑n=-∞∞|xnz-n|<∞ n x n z n (2)
must be satisfied for convergence.

Properties of the Region of Convergencec

The Region of Convergence has a number of properties that are dependent on the characteristics of the signal, xn x n .
  • The ROC cannot contain any poles. By definition a pole is a where Xz X z is infinite. Since Xz X z must be finite for all zz for convergence, there cannot be a pole in the ROC.
  • If xn x n is a finite-duration sequence, then the ROC is the entire z-plane, except possibly z=0 z 0 or |z|=∞ z . A finite-duration sequence is a sequence that is nonzero in a finite interval n 1 ≤n≤ n 2 n 1 n n 2 . As long as each value of xn x n is finite then the sequence will be absolutely summable. When n 2 >0 n 2 0 there will be a z-1 z term and thus the ROC will not include z=0 z 0 . When n 1 <0 n 1 0 then the sum will be infinite and thus the ROC will not include |z|=∞ z . On the other hand, when n 2 ≤0 n 2 0 then the ROC will include z=0 z 0 , and when n 1 ≥0 n 1 0 the ROC will include |z|=∞ z . With these constraints, the only signal, then, whose ROC is the entire z-plane is xn=cδn x n c δ n .
finite.png
Figure 1: An example of a finite duration sequence.
The next properties apply to infinite duration sequences. As noted above, the z-transform converges when |Xz|<∞ X z . So we can write
|Xz|=|∑n=-∞∞xnz-n|≤∑n=-∞∞|xnz-n|=∑n=-∞∞|xn||z|-n X z n x n z n n x n z n n x n z n (3)
We can then split the infinite sum into positive-time and negative-time portions. So
|Xz|≤Nz+Pz X z N z P z (4)
where
Nz=∑n=-∞-1|xn||z|-n N z n -1 x n z n (5)
and
Pz=∑n=0∞|xn||z|-n P z n 0 x n z n (6)
In order for |Xz| X z to be finite, |xn| x n must be bounded. Let us then set
|xn|≤ C 1 r 1 n x n C 1 r 1 n (7)
for n<0 n 0 and
|xn|≤ C 2 r 2 n x n C 2 r 2 n (8)
for n≥0 n 0 From this some further properties can be derived:
  • If xn x n is a right-sided sequence, then the ROC extends outward from the outermost pole in Xz X z . A right-sided sequence is a sequence where xn=0 x n 0 for n< n 1 <∞ n n 1 . Looking at the positive-time portion from the above derivation, it follows that
    Pz≤ C 2 ∑n=0∞ r 2 n|z|-n= C 2 ∑n=0∞ r 2 |z|n P z C 2 n 0 r 2 n z n C 2 n 0 r 2 z n (9)
    Thus in order for this sum to converge, |z|> r 2 z r 2 , and therefore the ROC of a right-sided sequence is of the form |z|> r 2 z r 2 .
rtsided1.png
Figure 2: A right-sided sequence.
rtsided2.png
Figure 3: The ROC of a right-sided sequence.
  • If xn x n is a left-sided sequence, then the ROC extends inward from the innermost pole in Xz X z . A right-sided sequence is a sequence where xn=0 x n 0 for n> n 2 >-∞ n n 2 . Looking at the negative-time portion from the above derivation, it follows that
    Nz≤ C 1 ∑n=-∞-1 r 1 n|z|-n= C 1 ∑n=-∞-1 r 1 |z|n= C 1 ∑k=1∞|z| r 1 k N z C 1 n -1 r 1 n z n C 1 n -1 r 1 z n C 1 k 1 z r 1 k (10)
    Thus in order for this sum to converge, |z|< r 1 z r 1 , and therefore the ROC of a left-sided sequence is of the form |z|< r 1 z r 1 .
lefsided1.png
Figure 4: A left-sided sequence.
lefsided2.png
Figure 5: The ROC of a left-sided sequence.
  • If xn x n is a two-sided sequence, the ROC will be a ring in the z-plane that is bounded on the interior and exterior by a pole. A two-sided sequence is an sequence with infinite duration in the positive and negative directions. From the derivation of the above two properties, it follows that if r 2 <|z|< r 2 r 2 z r 2 converges, then both the positive-time and negative-time portions converge and thus Xz X z converges as well. Therefore the ROC of a two-sided sequence is of the form r 2 <|z|< r 2 r 2 z r 2 .
twosided1.png
Figure 6: A two-sided sequence.
twosided2.png
Figure 7: The ROC of a two-sided sequence.

Examples

To gain further insight it is good to look at a couple of examples.
Example 1 
Lets take
x 1 n=12nun+14nun x 1 n 1 2 n u n 1 4 n u n (11)
The z-transform of 12nun 1 2 n u n is zz-12 z z 1 2 with an ROC at |z|>12 z 1 2 .
ex1roc1a.png
Figure 8: The ROC of 12nun 1 2 n u n
The z-transform of -14nun -1 4 n u n is zz+14 z z 1 4 with an ROC at |z|>-14 z -1 4 .
ex1roc1b.png
Figure 9: The ROC of -14nun -1 4 n u n
Due to linearity,
X 1 z=zz-12+zz+14=2zz-18z-12z+14 X 1 z z z 1 2 z z 1 4 2 z z 1 8 z 1 2 z 1 4 (12)
By observation it is clear that there are two zeros, at 0 0 and 18 1 8 , and two poles, at 12 1 2 , and -14 -1 4 . Following the obove properties, the ROC is |z|>12 z 1 2 .
ex1roc2.png
Figure 10: The ROC of x 1 n=12nun+-14nun x 1 n 1 2 n u n -1 4 n u n
Example 2 
Now take
x 2 n=-14nun-12nu-n-1 x 2 n -1 4 n u n 1 2 n u n 1 (13)
The z-transform and ROC of -14nun -1 4 n u n was shown in the example above. The z-transorm of -12nu-n-1 1 2 n u n 1 is zz-12 z z 1 2 with an ROC at |z|>12 z 1 2 .
ex2roc1.png
Figure 11: The ROC of -12nu-n-1 1 2 n u n 1
Once again, by linearity,
X 2 z=zz+14+zz-12=z2z-18z+14z-12 X 2 z z z 1 4 z z 1 2 z 2 z 1 8 z 1 4 z 1 2 (14)
By observation it is again clear that there are two zeros, at 0 0 and 116 1 16 , and two poles, at 12 1 2 , and -14 -1 4 . in ths case though, the ROC is |z|<12 z 1 2 .
ex2roc2.png
Figure 12: The ROC of x 2 n=-14nun-12nu-n-1 x 2 n -1 4 n u n 1 2 n u n 1 .

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