The z-transform defines a function in the entire complex plane much as the analog Laplace transform is a function in the complex plane:
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The ROC is the set of all values of z for which:
Notation for ROC: r- < |z| <r+
The usual form is:
where N(z) and D(z) are polynomials is the usual form.
Zeros of N(z) are "zeros" of X(z); zeros
of D(z) are "poles" of X(z);
EXAMPLES:
RIGHT SIDED SEQUENCE
Proof:
so: ROC is outside a circle with radius a with a
zero at 0 and a pole at a.
LEFT SIDED SEQUENCE
Proof:
which converges as long as: |a-1 z| < 1 or |z| < |a| to
with a zero at 0 and a pole at a.
i.e., ROC is inside a circle with radius a
COMBINED EXPONENTIAL SEQUENCE
NOTATION NOTE: 
FINITE SEQUENCE
Proof:
Which nas N-1 poles at the origin and N-1 zeros with radius a:
EXAMPLE GIVEN FOR N=12
TWO-SIDED SEQUENCE
if |z| >1/3 and |z| < 1/2
NEED TO MULTIPLY OUT TO SEE THE ZEROS
ARBITRARY FINITE SEQUENCES:
x(n) = K d (n - r)---------> X(z) = K z -r
x(n) = forward difference = d (n +1) - d (n )---> X(z) = z 1 - z 0 = z 1 - 1
x(n) = backward difference = d (n) - d (n-1 )---> X(z) = z 0 - z -1 = 1-z -1
x(n) = d (n)
- 3 d (n-1)
+ 2 d(n-2)------->X(z)
= 1 - 3 z -1 + 2 z -2
SUMMARY OF PROPERTIES:
| 1. | The ROC is a ring or disk in the z-plane centered at the origin: i.e. 0 <= rR <= |z| <= rL |
| 2. | The ROC cannot contain any poles. |
| 3. | If x(n) is a finite-duration sequence, i.e, a seqence that is zero except in a finite interval: -inf < N1 <= n <= N2 < inf then the ROC is the entire z-plane except possibly at z = 0 or z=inf.
Example: x(n) = 2d(n+1) + d(n) + 5d(n-1) - 3d(n-2)
which converges everywhere except at z = 0 or z=inf |
| 4. | If x(n) is a right-sided sequence ( zero for n < N1 < inf), the ROC extend outward from the outermost (largest magnitude) finite pole of X(z) to (and possibly including if 0 <= N1) z = inf. |
| 5. | If x(n) is a left-sided sequence (zero for n > N2 >- inf) the ROC extend inward from the innermost (smallest magnitude) nonzero pole of X(z) to (and possibly including if 0 >= N2) z = 0. |
| 6. | A two-sided sequence is an infinite-duration sequence that is neither right-sided nor left-sided. If x(n) is a two-sided sequence, the ROC will consist of a ring in the z-plane, bounded on the interior and exterior by a pole, and, consistent with property 3, not containing any poles. |
| 7. | The ROC must be a connected region. |
MATLAB DISCUSSION: poly, prony, filter, etc.
