Region of Convergence for the Z-transform

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 x n z n The ROC for a given x n , is defined as the range of z for which the z-transform converges. Since the z-transform is a power series, it converges when x n z n is absolutely summable. Stated differently, n x n z n 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 . The ROC cannot contain any poles. By definition a pole is a where X z is infinite. Since X z must be finite for all z for convergence, there cannot be a pole in the ROC. If x n is a finite-duration sequence, then the ROC is the entire z-plane, except possibly z 0 or z . A finite-duration sequence is a sequence that is nonzero in a finite interval n 1 n n 2 . As long as each value of x n is finite then the sequence will be absolutely summable. When n 2 0 there will be a z term and thus the ROC will not include z 0 . When n 1 0 then the sum will be infinite and thus the ROC will not include z . On the other hand, when n 2 0 then the ROC will include z 0 , and when n 1 0 the ROC will include z . With these constraints, the only signal, then, whose ROC is the entire z-plane is x n c δ n .

The next properties apply to infinite duration sequences. As noted above, the z-transform converges when X z . So we can write X z n x n z n n x n z n n x n z n We can then split the infinite sum into positive-time and negative-time portions. So X z N z P z where N z n -1 x n z n and P z n 0 x n z n In order for X z to be finite, x n must be bounded. Let us then set x n C 1 r 1 n for n 0 and x n C 2 r 2 n for n 0 From this some further properties can be derived: If x n is a right-sided sequence, then the ROC extends outward from the outermost pole in X z . A right-sided sequence is a sequence where x n 0 for 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 Thus in order for this sum to converge, z r 2 , and therefore the ROC of a right-sided sequence is of the form z r 2 .

If x n is a left-sided sequence, then the ROC extends inward from the innermost pole in X z . A right-sided sequence is a sequence where x n 0 for 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 Thus in order for this sum to converge, z r 1 , and therefore the ROC of a left-sided sequence is of the form z r 1 .

If 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 converges, then both the positive-time and negative-time portions converge and thus X z converges as well. Therefore the ROC of a two-sided sequence is of the form -r 2 z r 2 .

Examples Lets take x 1 n 1 2 n u n 1 4 n u n The z-transform of 1 2 n u n is z z 1 2 with an ROC at z 1 2 .

The z-transform of -1 4 n u n is z z 1 4 with an ROC at z -1 4 .

Due to linearity, X 1 z z z 1 2 z z 1 4 2 z z 1 8 z 1 2 z 1 4 By observation it is clear that there are two zeros, at 0 and 1 8 , and two poles, at 1 2 , and -1 4 . Following the obove properties, the ROC is z 1 2 .

Now take x 2 n -1 4 n u n 1 2 n u n 1 The z-transform and ROC of -1 4 n u n was shown in the example above. The z-transorm of 1 2 n u n 1 is z z 1 2 with an ROC at z 1 2 .

Once again, by linearity, X 2 z z z 1 4 z z 1 2 z 2 z 1 8 z 1 4 z 1 2 By observation it is again clear that there are two zeros, at 0 and 1 16 , and two poles, at 1 2 , and -1 4 . in ths case though, the ROC is z 1 2 .