(Eq.1) |
(Eq.2) |
(Eq.3) |
Time domain | Z-domain | Proof | ROC | |
---|---|---|---|---|
Notation | ||||
Linearity | Contains ROC1 ∩ ROC2 | |||
Time expansion | with | |||
Decimation | ohio-state.edu or ee.ic.ac.uk | |||
Time delay | with and | ROC, except z = 0 if k > 0 and z = ∞ if k < 0 | ||
Time advance[10] | with | |||
First difference backward | with x[n]=0 for n<0 | Contains the intersection of ROC of X1(z) and z ≠ 0 | ||
First difference forward | ||||
Time reversal | ||||
Scaling in the z-domain | ||||
Complex conjugation | ||||
Real part | ||||
Imaginary part | ||||
Differentiation | ROC, if is rational; ROC possibly excluding the boundary, if is irrational[11] | |||
Convolution | Contains ROC1 ∩ ROC2 | |||
Cross-correlation | Contains the intersection of ROC of and | |||
Accumulation | ||||
Multiplication | - |
Signal, | Z-transform, | ROC | |
---|---|---|---|
1 | 1 | all z | |
2 | |||
3 | |||
4 | |||
5 | |||
6 | |||
7 | |||
8 | |||
9 | |||
10 | |||
11 | |||
12 | |||
13 | |||
14 | |||
15 | |||
16 | |||
17 | , for positive integer [11] | ||
18 | , for positive integer [11] | ||
19 | |||
20 | |||
21 | |||
22 |
(Eq.4) |
(Eq.5) |
(Eq.6) |