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Signal Processing in the Discrete Time Domain

Microprocessor Applications (MEE4033). Signal Processing in the Discrete Time Domain. Sogang University Department of Mechanical Engineering. Definition of the z -Transform. Overview on Transforms. The Laplace transform of a function f ( t ) :.

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Signal Processing in the Discrete Time Domain

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  1. Microprocessor Applications (MEE4033) Signal Processing in the Discrete Time Domain Sogang UniversityDepartment 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 x(k) . . . k -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 For a signal for , is

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

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

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

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

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

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

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

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

  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: Arranging the right hand side, Since the z-transform is a linear map, z-transform for Inversez-transform

  16. Shift

  17. Example 7: arbitrary signals Any signals can be represented in the z-domain: y(k) 5 z-transform Inversez-transform k -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 y(k) 3 z-transform 2 1 Inversez-transform k -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

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

  19. Multiplication by an Exponential Sequence

  20. Initial Value Theorem

  21. 1/2 Convolution of Sequences

  22. 2/2 Convolution of Sequences Proof:

  23. z-Transform of Linear Systems

  24. Linear Time-Invariant System

  25. Nth-Order Difference Equation z-Transform

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

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

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

  29. 1/2 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

  30. 2/2 Example 9: open-loop controller A promising control algorithm is However, the control algorithm is non-causal.

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

  32. 1/2 Example 10: frequency response of a low pass filter Suppose a lowpass filter By substituting for z, The magnitude is

  33. 2/2 Example 10: frequency response of a low pass filter Since ,

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

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