Chapter 2 Sampling and Construction. New Words. Contents. 2.1 Types of Signals in CCS 2.2 Sampling Process and Its Mathematical Description 2.3 Sampling Theorem and Some Problems 2.4 Reconstruction 2.5 Selection of Sampling Rate 2.6 Summarization. Clock. S/H &A/D. Computer. D/A.
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Chapter 2
Sampling and Construction
2.1 Types of Signals in CCS
2.2 Sampling Process and Its Mathematical Description
2.3 Sampling Theorem and Some Problems
2.4 Reconstruction
2.5 Selection of Sampling Rate
2.6 Summarization
Clock
S/H &A/D
Computer
D/A
Holder
Process
Sensor
Figure 2.1: Schematic diagram of a CCS
continuous time signal
discrete time signal
analog signal
discrete signal
digital signal
Digital
signal
Analog
signal
S/H
Quantization
Coding
D
A
B
C
Sampleandhold circuit
AD converter Quantization
Coding
Figure 2.2 AD converter
Analog
signal
Digital
signal
Decoding
Holder
H
G
F
Decoding
DA converter
Hold circuit
Fig. 2.3 Computercontrolled system
Figure 2.4 Simplification of Figure 2.3
p
p
t
T
T
t
T
t
2.2.1 Description of the sampling process
Sampling period: T (s); Sampling frequency: f=1/T Hz
Sampling angle frequency: rad/s
Sampling process is unavoidable in computercontrolled systems.
1
t
0
1
t0
t
2.2.2 Time domain description for ideal sampler and
ideal sampled signal
(1) Definition of Dirac function
(2) Dirac function at t=t0
(3) An important property of Dirac function
1
2T
T
0
T
2T
t
(4) Description of the ideal sampler
(5) Time domain description of the ideal sampled signal
(2.1)
1
where T(t) is the periodic function. And Eq. (2.1) is the
time domain description of the ideal sampled signal and
its diagram is shown in Fig. 2.7.
2.2.3 Frequency domain description for ideal sampled signal
(1) Fourier series
(2) Fourier transform
As T approaches infinity, we define
Frequency domain description of the ideal sampled signal
Example:
2.2.4 Laplace Transform of sampled signal
1
0 p
T T+p
T T+p
2.2.5 Time domain description of real sampled signal
The Laplace transformations of the both sides of the above equation
2.2.6 Frequency domain description of real sampled signal
2.2.6 Frequency domain description of real sampled signal
2.3.1 Frequency folding phenomena
2.3.2 Sampling theorem
s 2max
2.3.3 Some problems
1. Sample signal with disturbance
2. Aliasing or Frequency Folding
(2.4)
2.4.1 Ideal reconstruction
Conditions:
Drawback:
2.4.2 Nonideal reconstruction
where
Figure 2.15 (a) Magnitude and (b) phase of
zero

order hold
Input and Output of ZOH
2.4.3 Postsampling Filters
Cause:
2.5.1 The sampling theorem’s limit
2.5.2 According to risetime of the system
where Tr is the rise time. For firstorder systems, the
rise time is equal to the time constant. For a second
order system with damping and natural frequency 0,
rise time is given by
where = cos.
2.5.3 For a digital PIDcontroller
2.5.4 For most common process variables :
Type of variable Sampling time, s
Flow 13
Level 510
Pressure 15
Temperature 1020
2.5.5 For A closedloop control system
DA converter
4. Let the input of the ideal sampler be x(t), please give time domain description and frequency domain description of the ideal sampled signal.
5. Let the continuous signal x(t)=et (L[x(t)]=1/(s+1)), please select the sampling angle frequency to guarantee x*(t) to be reconstructed to x(t).
Note: L[x(t)]=1/(s+1)