1. Signal Processing in the Discrete Time Domain Microprocessor Applications (MEE4033) Sogang University Department of Mechanical Engineering

2. Definition of the z -Transform

3. Overview on Transforms The Laplace transform of a function f(t) : The z -transform of a function x(k) : The Fourier-series of a function x(k) :

4. Example 1: a right sided sequence 12345678910-2-3-4-5-6-7-8 k x(k)x(k)... for, is For a signal

5. Example 2: a lowpass filter Suppose a lowpass filter law is where 1/3

6. Example 2: a lowpass filter 2/3 Rearranging the equation above, Signals Transfer function

7. Example 2: a lowpass filter 3/3 Signals Transfer function The block-diagram representation:

8. Example 3: a highpass filter A highpass filter follows: where 1/2 Transfer function

9. z -Transform Pairs Discrete-time domain signal z-domain signal 1/2

10. z -Transform Pairs 2/2 Discrete-time domain signal z-domain signal

11. Example 4: a decaying signal Suppose a signal is for. Find. for z -transform Inverse z -transform

12. Example 5: a signal in z -domain Suppose a signal is given in the z-domain: for z -transform Inverse z -transform From the z-transform table, The signal is equivalent to

13. Properties of the z-Transform

14. Linearity of z-Transform where a and b are any scalars.

15. Example 6: a signal in z-domain Suppose a signal is given in the z-domain: for z -transform Inverse z -transform Since the z-transform is a linear map, Arranging the right hand side,

16. Shift

17. Example 7: arbitrary signals z -transform Inverse z -transform Any signals can be represented in the z-domain: 12345678910-2-3-4 k y(k)y(k) 5 z -transform Inverse z -transform 12345678910-2-3-4 k y(k)y(k) 3 2 1

18. Discrete-Time Approximation Backward approximation Forward approximation Trapezoid approximation

19. Multiplication by an Exponential Sequence

20. Differentiation of X(z)

21. Complex Conjugation

22. Reversal

23. Initial Value Theorem

24. Convolution of Sequences 1/2

25. 2/2 Convolution of Sequences Proof:

26. z-Transform of Linear Systems

27. Linear Time-Invariant System

28. N th -Order Difference Equation z-Transform

29. Stable and Causal Systems Re Im 1 The system G(z) is stable if all the roots (i.e., d i ) of the denominator are in the unit circle of the complex plane.

30. Stable and Causal Systems Re Im 1 The system G(z) is causal if the number of poles is greater than that of zeros (i.e., M N ).

31. Example 8: a non-causal filter Suppose a transfer function is given By applying the inverse z-Transform Therefore, the system is causal if

32. Example 9: open-loop controller Suppose the dynamic equation of a system is Approximating the dynamic equation by The transfer function from U(z) to Y(z) is 1/2

33. Example 9: open-loop controller A promising control algorithm is 2/2 However, the control algorithm is non-causal.

34. Frequency Response of H(z) The z-transform of a function x(k) : The Fourier-transform of a function x(k) : (Recall: Similarity of the z-Transform and Fourier Transform) The frequency response is obtained by setting where T is the sampling period.

35. Example 10: frequency response of a low pass filter Suppose a lowpass filter 1/2 By substituting for z, The magnitude is

36. 2/2 Since, Example 10: frequency response of a low pass filter

37. IIR Filters and FIR Filters An IIR (Infinite Impulse Response) filter is A FIR (Finite Impulse Response) filter is