Z - transform

1. Z - transform • Defined as power series • Examples: 1

2. Z - transform • And since • We get 2

3.  +ve and > 1  +ve and = 1  +ve and < 1 Z - transform • Define 3

4. Z - transform • We have • ie • Note that has a pole at on the z-plane. 4

5. Z - transform Note: • (i) If magnitude of pole is > 1 then increases without bound • (ii) If magnitude of pole is < 1 then has a bounded variation • i.e. the contour on the z-plane is of crucial significance. • It is called theUnit circle 5

6. 1 1 The unit circle 6

7. Z – transform properties (i)Linearity • The z-transform operation is linear • Z • Where Z , i = 1, 2 (ii) Shift Theorem Z 7

8. .... Z - transform • Let • Z • But for negative i. 8

9. Z - transform Examples: • (i) Consider generation of new discrete time signal from via • Recall linearity and shift • (ii) Z write 9

10. Z - transform • From • With from earlier result • We obtain Z Z 10

11. Inverse Z - transform • Given F(z) to determine . • Basic relationship is • may be obtained by power series expansion. It suffers from cumulative errors 11

12. Inverse Z - transform • Alternatively • Use form = -1 • otherwise • where closed contour  encloses origin 12

13. Inverse Z - transform • Integrate to yield Examples • (i) write 13

14. Inverse Z - transform • And hence Pole at of Residue (ii) Let where and • To determine 14

15. Inverse Z - transform • From inversion formula • But 15

16. Inverse Z - transform • Hence • Thus 16

17. Inverse Z - transform Note: •  (i) For causal signals for negative i. Thus upper convolution summation limit is in this case equal to k. • (ii) Frequency representation of a discrete-time signal is obtained from its z-transform by replacing where T is the sampling period of interest. (Justification will be given later.) 17