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Region of Convergence for the Z-transform

Module by: Benjamin Fite, Dan Calderon. E-mail the authors

Summary: (Blank Abstract)

Note: You are viewing an old version of this document. The latest version is available here.

Introduction

With the z-transform, the s-plane represents a set of signals (complex exponentials). For any given LTI system, some of these signals may cause the output of the system to converge, while others cause the output to diverge ("blow up"). The set of signals that cause the system's output to converge lie in the region of convergence (ROC). This module will discuss how to find this region of convergence for any discrete-time, LTI system.

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 =xnzn 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 xnzn x n z n is absolutely summable. Stated differently,
n =|xnzn|< 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 .

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 |Xz|< X z . So we can write

|Xz|=| n =xnzn| n =|xnzn|= 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 n0 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 .

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

  • 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 .

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

  • 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 .

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

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 zz12 z z 1 2 with an ROC at |z|>12 z 1 2 .

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

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 .

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

Due to linearity,

X 1 z=zz12+zz+14=2z(z18)(z12)(z+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 .

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

Example 2

Now take

x 2 n=-14nun12nu(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 (12n)u(n)1 1 2 n u n 1 is zz12 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+zz12=z(2z18)(z+14)(z12) 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=-14nun12nu(n)1 x 2 n -1 4 n u n 1 2 n u n 1 .
Figure 12 (ex2roc2.png)

Graphical Understanding of ROC

Using the demonstration, learn about the region of convergence for the Laplace Transform.

Conclusion

Clearly, in order to craft a system that is actually useful by virtue of being causal and BIBO stable, we must ensure that it is within the Region of Convergence, which can be ascertained by looking at the pole zero plot. The Region of Convergence is the area in the pole/zero plot of the transfer function in which the function exists. For purposes of useful filter design, we prefer to work with rational functions, which can be described by two polynomials, one each for determining the poles and the zeros, respectively.

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