Understanding Z-Transform: Geometric Evaluation and Properties

1. The inverse z-transform Any path in the ROC to make the integral converge Example ROC |z|>1/3

2. The inverse z-transform Example ROC 1/4 <|z|<1/3 Example ROC |z|<1/4

3. The inverse z-transform Example ROC 0 <|z|< Example |az-1|<1

4. Geometric evaluation of the Fourier Transform from the pole-zero plot First order ROC Unit circle x 1

5. Geometric evaluation of the Fourier Transform from the pole-zero plot ROC Unit circle 1 x

6. Geometric evaluation of the Fourier Transform from the pole-zero plot 2nd order ROC x Unit circle 1 x

7. Properties of z-transform (1) Linearity with ROC R1 with ROC R2 with ROC containing

8. (2) Time shifting with ROC R1 with ROC R1 with addition or deletion of poles at z=0 or infinite (3) Scaling in the z-domain with ROC |z0|R1 with ROC R1 with addition or deletion of poles at z=0 or infinite

9. (3) Scaling in the z-domain with ROC |z0|R1 with ROC R1 with addition or deletion of poles at z=0 or infinite x x

10. (4) Time reversal with ROC R1 with ROC 1/R1

11. (5) Time expansion if n is a multiple of k otherwise with ROC (R)1/k (6) Conjugation with ROC R with ROC R real

12. (7) Convolution with ROC R1 with ROC R2 Example

13. Differentiation in z-domain with ROC R1 Example

14. Initial value theorem for n<0 Example, check the correctness of a z-transform