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

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introduction to digital systems
Introduction to Digital Systems

Saint-Petersburg State University

Faculty of Applied Mathematics – Control Processes

prof. Evgeny I. Veremey

Lections 13 ─ 16

Part 3. Analog System Discretization

slide2

Analog Systems Discretization

1

1. The signals sampling

Sampling frequency:

Sampling Period:

Example:

slide3

Analog Systems Discretization

2

Uncertainties of the analog signal

restoration

Zero order hold

and first order hold

restoration

slide4

Analog Systems Discretization

3

The sampling of harmonic continues-time functions

slide5

Analog Systems Discretization

4

The sampling of harmonic continues-time functions

Frequencies are not comparable:

Frequencies are comparable:

slide6

Analog Systems Discretization

5

The aliasing effect

slide7

Analog Systems Discretization

6

2. The base conceptions of analog systems discretization

Analog system:

Approximate calculation of the definite integral

Direct Euler method:

Inverse Euler method:

Trapezium method:

slide8

Analog Systems Discretization

7

Direct Euler method:

Inverse Euler method:

3. Discretization methods for the analog LTI systems

Cauchy formula:

slide9

Analog Systems Discretization

8

Cauchy formula:

Matrix exponent:

Discrete

Analog

Approximate

ADM

Precise

PDM

slide10

Analog Systems Discretization

9

Analog-digital matrices correspondence:

Nonsingular matrix A:

slide11

Analog Systems Discretization

10

Two ways for approximate discretization:

1. Taylor

approximation

2. Pade

approximation

Direct Euler method:

slide12

Analog Systems Discretization

11

Inverse Euler method

(Pade m=0, l=1):

Trapezium method

(Pade m=1, l=1):

slide13

Analog Systems Discretization

12

4. Тhe precise discretization taking into account input signal waveform

LTI discretization with preliminary ZOH:

Case 1. δ-functions sequence:

slide14

Analog Systems Discretization

13

4. Тhe precise discretization taking into account input signal waveform

LTI discretization with preliminary ZOH:

Case 1: δ-functions sequence:

slide15

Analog Systems Discretization

14

Case 2: piecewise constant input signal:

Nonsingular matrix A:

MATLAB implementation:

slide16

Analog Systems Discretization

15

Case 3: piecewise exponential input signal:

Case 4: sequence of rectangular impulses:

slide17

Analog Systems Discretization

16

5. Тhe precise discretization of LTI systems with tf-models

LTI discretization with preliminary ZOH: