1. Chapter 6: Sampled Data Systems and the z-Transform 1

2. Sampled-data system A sampled-data system operates on discrete-time rather than continuous-time signals. 2

3. 6.1 The sampling process A sampler is basically a switch that closes every T seconds. 3

4. r(t): continuous signal r*(t): sampled signal q: amount of time the switch is closed 4

5. 5

6. If q is neglected, the operation is called “ideal sampling” 6

7. Ideal sampling Ideal sampling of a continuous signal can be considered as a multiplication of the continuous signal, r(t), with a pulse train, P(t). The pulse train: 7

8. Ideal sampling Let, r(t) = 0, for t<0, then, 8

9. The z-transform Taking Laplace transform of the sampled signal r * (t) gives: Let us define z = e sT. Then, This is the definition of the z-transform. 9

10. The z-transform Notice that the z-transform consists of an infinite series in the complex variable z, i.e. the r(nT ) are the coefficients of this power series at different sampling instants. 10

11. The z-transform The z-transform is used in sampled data systems just as the Laplace transform is used in continuous-time systems. We will look at how we can find the z-transforms of some commonly used functions. We first give a closer look at the D/A converter. 11

12. D/A converter as a zero-order hold (ZOH) D/A converter converts the sampled signal r ∗ (t) into a continuous signal y(t). D/A can be approximated by a ZOH circuit. The ZOH remembers the last information until a new sample is obtained, i.e. it takes the value r(nT) and holds it constant for nT ≤ t < (n + 1)T, and the value r (nT) is used during the sampling period. 12

13. The zero-order hold (ZOH) The impulse response of a ZOH: The transfer function of ZOH is 13

14. The zero-order hold (ZOH) 14 A sampler and zero- order hold can accurately follow the input signal if the sampling time T is small compared to the transient changes in the signal.

15. Example 6.1 Figure 6.10 shows an ideal sampler followed by a ZOH. Assuming the input signal r (t) is as shown in the figure, show the waveforms after the sampler and also after the ZOH. Answer 15

16. The z-transform of some common functions 16

17. Unit Step Function 17

18. Useful closed-form series summations 18

19. Unit Ramp Function 19

20. Exponential Function 20

21. General Exponential Function 21

22. Sine Function 22

23. Discrete Impulse Function 23

24. Delayed Discrete Impulse Function 24

25. 25

26. The z-Transform of a Function Expressed as a Laplace Transform Given a function G(s), find G(z) which denotes the z-transform equivalent of G(s). It is important to realize that G(z) is not obtained by simply substituting z for s in G(s)! 26

27. Example 6.2 Given Determine G(z). 27

28. Answer: Using Inverse Laplace transform Partial fraction Inverse Laplace transform Substitute t = nT gives Finally, 28

29. Method 2: Laplace to z-transform table From table in Appendix A So, 29