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Region of Convergence for the Z-transform

Module by: Benjamin Fite. E-mail the author

Summary: (Blank Abstract)

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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

X⁢z=∑ n =−∞∞x⁢n⁢z−n Xz n x n z n
(1)
The ROC for a given x⁢n 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 x⁢n⁢z−n x n z n is absolutely summable. Stated differently,
∑ n =−∞∞|x⁢n⁢z−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, x⁢n x n .

  • The ROC cannot contain any poles. By definition a pole is a where X⁢z X z is infinite. Since X⁢z X z must be finite for all zz for convergence, there cannot be a pole in the ROC.
  • If x⁢n 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 x⁢n 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 x⁢n=c⁢δ⁢n x n c δ n .

Figure 1: An example of a finite duration sequence.
Figure 1 (finite.png)

The next properties apply to infinite duration sequences. As noted above, the z-transform converges when |X⁢z|<∞ X z . So we can write

|X⁢z|=|∑ n =−∞∞x⁢n⁢z−n|≤∑ n =−∞∞|x⁢n⁢z−n|=∑ n =−∞∞|x⁢n|⁢|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
|X⁢z|≤N⁢z+P⁢z X z N z P z
(4)
where
N⁢z=∑ n =−∞-1|x⁢n|⁢|z|−n N z n -1 x n z n
(5)
and
P⁢z=∑ n =0∞|x⁢n|⁢|z|−n P z n 0 x n z n
(6)
In order for |X⁢z| X z to be finite, |x⁢n| x n must be bounded. Let us then set
|x⁢n|≤ C 1 ⁢ r 1 n x n C 1 r 1 n
(7)
for n<0 n 0 and
|x⁢n|≤ 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 x⁢n x n is a right-sided sequence, then the ROC extends outward from the outermost pole in X⁢z X z . A right-sided sequence is a sequence where x⁢n=0 x n 0 for n< n 1 <∞ n n 1 . Looking at the positive-time portion from the above derivation, it follows that
    P⁢z≤ 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 .

Figure 2: A right-sided sequence.
Figure 2 (rtsided1.png)
Figure 3: The ROC of a right-sided sequence.
Figure 3 (rtsided2.png)

  • If x⁢n x n is a left-sided sequence, then the ROC extends inward from the innermost pole in X⁢z X z . A right-sided sequence is a sequence where x⁢n=0 x n 0 for n> n 2 >−∞ n n 2 . Looking at the negative-time portion from the above derivation, it follows that
    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 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 .

Figure 4: A left-sided sequence.
Figure 4 (lefsided1.png)
Figure 5: The ROC of a left-sided sequence.
Figure 5 (lefsided2.png)

  • If x⁢n 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 X⁢z 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 .

Figure 6: A two-sided sequence.
Figure 6 (twosided1.png)
Figure 7: The ROC of a two-sided sequence.
Figure 7 (twosided2.png)

Examples

To gain further insight it is good to look at a couple of examples.

Example 1

Lets take

x 1 ⁢n=12n⁢u⁢n+14n⁢u⁢n x 1 n 1 2 n u n 1 4 n u n
(11)
The z-transform of 12n⁢u⁢n 1 2 n u n is zz−12 z z 1 2 with an ROC at |z|>12 z 1 2 .

Figure 8: The ROC of 12n⁢u⁢n 1 2 n u n
Figure 8 (ex1roc1a.png)

The z-transform of -14n⁢u⁢n -1 4 n u n is zz+14 z z 1 4 with an ROC at |z|>-14 z -1 4 .

Figure 9: The ROC of -14n⁢u⁢n -1 4 n u n
Figure 9 (ex1roc1b.png)

Due to linearity,

X 1 ⁢z=zz−12+zz+14=z⁢(2⁢z−18)(z−12)⁢(z+14) X 1 z z z 1 2 z z 1 4 z 2 z 1 8 z 1 2 z 1 4
(12)
By observation it is clear that there are two zeros, at 0 0 and 116 1 16 , and two poles, at 12 1 2 , and -14 -1 4 . Following the obove properties, the ROC is |z|>12 z 1 2 .

Figure 10: The ROC of x 1 ⁢n=12n⁢u⁢n+-14n⁢u⁢n x 1 n 1 2 n u n -1 4 n u n
Figure 10 (ex1roc2.png)

Example 2

Now take

x 2 ⁢n=-14n⁢u⁢n−12n⁢u⁢(−n)−1 x 2 n -1 4 n u n 1 2 n u n 1
(13)
The z-transform and ROC of -14n⁢u⁢n -1 4 n u n was shown in the example above. The z-transorm of (−12n)⁢u⁢(−n)−1 1 2 n u n 1 is zz−12 z z 1 2 with an ROC at |z|>12 z 1 2 .

Figure 11: The ROC of (−12n)⁢u⁢(−n)−1 1 2 n u n 1
Figure 11 (ex2roc1.png)

Once again, by linearity,

X 2 ⁢z=zz+14+zz−12=z⁢(2⁢z−18)(z+14)⁢(z−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 .

Figure 12: The ROC of x 2 ⁢n=-14n⁢u⁢n−12n⁢u⁢(−n)−1 x 2 n -1 4 n u n 1 2 n u n 1 .
Figure 12 (ex2roc2.png)

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