1. Introduction to Digital Systems Saint-Petersburg State University Faculty of Applied Mathematics – Control Processes Lections 13 ─ 16 prof. Evgeny I. Veremey Part 3. Analog System Discretization

2. Analog Systems Discretization 1 1. The signals sampling Sampling Period: Sampling frequency: Example:

3. Analog Systems Discretization 2 Uncertainties of the analog signal restoration Zero order hold and first order hold restoration

4. Analog Systems Discretization 3 The sampling of harmonic continues-time functions

5. Analog Systems Discretization 4 The sampling of harmonic continues-time functions Frequencies are not comparable: Frequencies are comparable:

6. Analog Systems Discretization 5 The aliasing effect

7. Analog Systems Discretization 6 2. The base conceptions of analog systems discretization Analog system: Direct Euler method: Approximate calculation of the definite integral Inverse Euler method: Trapezium method:

8. Analog Systems Discretization 7 Direct Euler method: Inverse Euler method: 3. Discretization methods for the analog LTI systems Cauchy formula:

9. Analog Systems Discretization 8 Cauchy formula: Matrix exponent: Analog Discrete Approximate ADM Precise PDM

10. Analog Systems Discretization 9 Analog-digital matrices correspondence: Nonsingular matrix A:

11. Analog Systems Discretization 10 Two ways for approximate discretization: 1. Taylor approximation 2. Pade approximation Direct Euler method:

12. Analog Systems Discretization 11 Inverse Euler method (Pade m=0, l=1): Trapezium method (Pade m=1, l=1):

13. Analog Systems Discretization 12 4. Тhe precise discretization taking into account input signal waveform LTI discretization with preliminary ZOH: Case 1. δ-functions sequence:

14. Analog Systems Discretization 13 4. Тhe precise discretization taking into account input signal waveform LTI discretization with preliminary ZOH: Case 1: δ-functions sequence:

15. Analog Systems Discretization 14 Case 2: piecewise constant input signal : Nonsingular matrix A: MATLAB implementation:

16. Analog Systems Discretization 15 Case 3: piecewise exponential input signal : Case 4: sequence of rectangular impulses :

17. Analog Systems Discretization 16 5. Тhe precise discretization of LTI systems with tf-models LTI discretization with preliminary ZOH:

18. Analog Systems Discretization 17