Chapter 14 Sections 4.3, 4.4 and 4 - University of Texas at Dallas
§ 4.1 Instrumentation and Measurement Systems § 4.2 Dynamic Measurement and Calibration § 4.3...
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Transcript of § 4.1 Instrumentation and Measurement Systems § 4.2 Dynamic Measurement and Calibration § 4.3...
§ 4.1 Instrumentation and Measurement Systems
§ 4.2 Dynamic Measurement and Calibration
§ 4.3 Data Preparation and Analysis
§ 4.4 Practical Considerations
Chapter 4 Data Acquisition and PreprocessingChapter 4 Data Acquisition and PreprocessingR. J. Chang
Department of Mechanical EngineeringNCKU
1. Introduction (1) Signal and Noise
§§ 4.1 4.1 Instrumentation and Measurement Systems(1 Instrumentation and Measurement Systems(1))
*: convolution
Signal (S)
Deterministic (D)
Stochastic (S)
Deterministic (D) Stochastic (S)Noise (S)
D¡¯D S¡¯D
D¡¯S S¡¯S
Output dataInput signal (S)
Noise (N)
(3) Measurement quality ─ S/N ratio
10 (dB)
10 (dB)
log
log
Signal average powerSignal/Noise
Noise average power
Signal R.M.S
Noise R.M.S
§§ 4.1 4.1 Instrumentation and Measurement Systems(2 Instrumentation and Measurement Systems(2))
(2) Ideal measurement system
1ie oe
45o
ie
oe
2. System structure
(1) Fundamental structure
MeasurementSystems
Calibration SignalGenerator
Disturbance
MeasurandRead out
Standard unit
Controlledenvironment
§§ 4.1 4.1 Instrumentation and Measurement Systems(3 Instrumentation and Measurement Systems(3))
(2) Three stages structure
Sensor/Transducer
Variable Manipulation
Unit
Data Transmission
Unit
Signal Readout CPU/Processor
Unknown Measurand
Signal Information
Representation
Calibration Signal Source
NoiseNoise Noise
Signal modulationModulated
signalOutput signal
Signal
First stage Second stage Third stage
§§ 4.1 4.1 Instrumentation and Measurement Systems(4 Instrumentation and Measurement Systems(4))
(3) Sensor/Transducer
(a) Passive
(b) Active
Open loop type:
Servo type:
§§ 4.1 4.1 Instrumentation and Measurement Systems Instrumentation and Measurement Systems(5)(5)
Applied effect Output signal
Electrical energy
Applied effect Output signal
Electrical energy
Applied effect Output signal+-
Ex : Accelerometer for vibration data
Stress and strain gauges for deflection data
Microphone for acoustical data
Seismometers for seismic data
STM probe for atom distribution
§§ 4.1 4.1 Instrumentation and Measurement Systems Instrumentation and Measurement Systems(6)(6)
3. Signal and Standard units (1) Signal space
Outputsignal
OO
OE
E
E
M
M
M
C
C
C
Modulatedsignal
Inputsignal
O ─ Optics
E ─ Electricity
M ─ Mechanics
C ─ Chemistry
§§ 4.1 4.1 Instrumentation and Measurement Systems Instrumentation and Measurement Systems(7)(7)
(2) International system of unit (SI)
(a) Base units
Length ─ meter(m)
Mass ─ Kilogram(kg)
Time ─ second(s)
Current ─ ampere(A)
Temperature ─ kelvin(K)
Amount of sunstance ─ mole(mol)
Luminous intensity ─ candela(cd)
(b) Supplement units
Plane angle ─ radian(rad)
Solid angle ─ steradian(sr)
§ § 4.14.1 Instrumentation and Measurement Systems Instrumentation and Measurement Systems(8)(8)
1. Instrument characteristics
MeasurementSystems
StaticCharacteristics
K
Pure dynamicCharacteristics
P(s)
+
-Calibration input/output characteristics G(s)
Disturbance
MeasurandError
distribution
§§ 4.2 4.2 Dynamic Measurement and Calibration Dynamic Measurement and Calibration(1)(1)
2. Uncertainties and Error
(1) Causes
Variations of system parameters
External disturbance
Uncertain operational state
(2) Classification
Absolute and relative
System and random
Static and dynamic
Human and instrument
§§ 4.2 4.2 Dynamic Measurement and Calibration Dynamic Measurement and Calibration(2)(2)
(3) System and Random error
Definition:
Error sources: System ─ calibration, operation, loading
Random─ noise, bits operation
True value
System error
Mean value
Random errordensity distribution
§§ 4.2 4.2 Dynamic Measurement and Calibration Dynamic Measurement and Calibration(3)(3)
Systematic error = ||True value – Estimated value||
Random error = Distribution deviation of estimated value
(4) Accuracy and Precision Physical meaning:
Data interpretation: Accuracy error ─ systematic error
Precision error─ random error
Note: Errors of sample mean and variance need to be further analyzed statistically.
§§ 4.2 4.2 Dynamic Measurement and Calibration Dynamic Measurement and Calibration(4)(4)
High Precision High Accuracy High Accuracy and Precision
★★★★★
★★★
★★
★
★
★
★
★★ ★★★
★★★★
★
3. Static and Dynamic characteristics
Linearity
Sensitivity(slope)
Input
Output
Threshold Span
(Minimum)Resolution
§§ 4.2 4.2 Dynamic Measurement and Calibration Dynamic Measurement and Calibration(5)(5)
(1) Static
Frequency range
Span
A
u
1
2A
Delay
Steady state
t
§§ 4.2 4.2 Dynamic Measurement and Calibration Dynamic Measurement and Calibration(6)(6)
(2) Dynamic
Time domain delay
20 ( )
60 1000 ~ 1
u
l
log dB
dB
Span
Ex : means range
4. Calibration and Measurement
(1) Characteristics calibration
Measurementsystems
*x *y
Given Readout
Slope
*x ix x
*y
iy
y
m̂
3 yS
3 ySb̂
§ § 4.24.2 Dynamic Measurement and Calibration Dynamic Measurement and Calibration(7)(7)
Sy :Sample standard deviation
ˆˆ ˆ( ) 3 yy mx b S
(2) Measurement applications
Measurementsystems
ox oy
Unknown Readout
Calibrationtransferfunction
ˆox
Unknown
ox
oy
3 xS 3 xS
y
x
§ § 4.24.2 Dynamic Measurement and Calibration Dynamic Measurement and Calibration(8)(8)
x y
x
S S
x y
2 22
1
m̂
b̂ ˆ () 3
m̂o o S
1. Digitization
(1) Signal transmission
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(1)(1)
Sampleand Hold
(S/H)
A/DConverter
t k0 1 2
MSB LSB
k=0
k=1...
Digital signalAnalog signal Sampled signal
(2)Signal classification
Discrete, Dt Continuous, Ct
Discrete, Da Da-Dt Da-Ct
Continuous, Ca Ca-Dt Ca-Ct
TimeAmplitude
0 T 2T t
0 T 2T t
D1
D2
D3
0 T 2T t
0 T 2T t
(1) Analog (Ca-Ct) (2) Sample (Da-Ct)
(3) Discrete (Ca-Dt) (4) Digital (Da-Dt)
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(2)(2)
2. Interface card and Operation
RVi
Vc
CVo
HoldSampler
+
-
Sample/Hold control
Input voltage
Output voltage
h t
Vi
Vo
, ViVo
Sample
Sample/Holdcontrol signal
t
HoldHoldHoldHold Hold
ZOHt
t tt
EX: A Capacitance S/H device
(1) Sample and Hold
Ideal sampler with zero-order hold(ZOH), h<< ∆t
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(3)(3)
Mathematical analysis
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(4)(4)
ZOHx t( ) Dx t( )
Discrete signal
t
x t x t t t
N
D
k 0
( ) ( ) ( k )
1 s tx
s
X s( )
x t( )
X s( )
x t( )
X s*( )
x t*( )
t
*
0
*
X ( ) ( )
X ( ) ( ) ts
k ts
k
z
s x k x
s x z
─
─e
Discrete Laplace transform T
Z - transform
(2) A/D and D/A converter
SolidLatches
andSwitches
Rg
Vo
D3
D2
D1
D0
ChipSelection MSB
LSBMSB LSB4-Bits
VrReference
voltage
R/23
R/22
R/21
R/20
+
-
Gain adjustment
Output voltage
On/Off
V oo o
i
s s s s
s
g 3 2 1r 3 2 1
R(V ) ( ) ( 2 2 2 2 )
R0, Off
1, On
EX: 4 bits D/A converter
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(5)(5)
4-BitsD/A
MSB LSB
Storage register
Shift register
+
-
Clock
Vin
Analoginput
Digital output
10 0 1 10
0.67
1.33
2.00
2.67
3.33
4.00
4.675.33
6.00OFF
ON
ONON
TestMSB
TestBit2
TestBit3
TestLSB
Conversion time
t
Voltage (V)D/A output
5V
Analog input
If Vi =10V for rating value
11 1 1
1110
For 5V analog input
4
100.667
2 1
LSB V
EX: Successive approximation 4 bits A/D converter
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(6)(6)
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(7) (7)
(3)Computer sampling
A/DDigital
ComputerD/A
Clock
WriteRead
Analoginput
Analogoutput
Voltage
Output
t
Sample time
Computationaldelay
k-1 k k+1 k+1 t
A/D conversion procedureStart
Get A/D GainA/D Channel
Setup Board (StopBoard and ClearData Register)
Send Read A/D,Set Gain,Channel Low Byte Ready? Y,ReadHigh Byte Ready? Y,Read
Check Status Register
Output Converted Data
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(8)(8)
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(9)(9)
2. Filtering (1) Types of filter
- ( )= ( ) ( )
( ) :
Y( )=H( ) X( )
H( ):
y t h x t d
h
Time - Domain
Impulse response function of linear filter
Frequency - Domain
Transfer function
LinearFilter
x(t) y(t)
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(10)(10)
(a) Analog filter Pass band – High pass, Low pass, Band pass, Band
stop, …
Power supply – Active filter, Passive filter
(b) Digital filter
Pass band
Algorithm – Batch type, Recursive type
Realization – Software filter, Hardware filter
Adaptive / Non-adaptive
(b) Practical filter EX : Butterworth low–pass filter
(2) Analog filter
(a) Ideal filter
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(11)(11)
2
2nc
c
1H( )
1 ( )
:
j
Cut -off frequency
Low Pass Band Pass High Pass
Gain
c
c l u c
(3) Digital filter
(a) Nonrecursive filter
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(12)(12)
i k i k k i kk k
y h x h x
M
M
Req’d : Stable operation
BIBO stability iff kh
IIR filterInfinite impulse response
FIR filterFinite impulse response
(b) Recursive filter
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(13)(13)
1i i k i k
k
y x h y
M
c
Fourier transform
Y( )H( )
X( ) 1 ( ( ))
:
kkh j t
t
c
exp
Sampling time
( )
H( )1 k
k
z j t
zh z
Define exp
c
EX : 2nd – order recursive filter
(a) Low pass (b) Band pass (c)High pass
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(14)(14)
0 00 0
2 20 0 0
2 2
( )( )H( ) c
( )( )
2 cos c
2 cos
p p
j j
j jp p
p p p
z r e z r ez
z r e z r e
z r z r
z r z r
2
2 2p p p
zH(z)
z 2r cos z r
c
2 2p p p
z(z 1)H(z)
z 2r cos z r
c
2
2 2p p p
(z 1)H(z)
z 2r cos z r
c
rp
Unit circle
p
Poles
Zeros
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(15)(15)
(4) Filter design Filter specs → Analog filter → Digital filter → Realization
Filter specs → Digital filter → Realization
(5) Filter realization a. Software digital filter b. Hardware digital filter EX : DSP realization- Use key operations Convolution Correlation Filtering Discrete transformation c. Error – Finite bits operation, Round off error
Discretization
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(16) (16)
Continuous and Discrete transformations
( ) ( 1)n-1 0
( ) ( 1)m-1 0
q q
p p
n n
m m
y y
u u
Differential
Eq.Difference
Eq.
ContinuousTransferFunction
G(s)
DiscreteTransferFunction
H(z)
ContinuousState Eq.
DiscreteState Eq.
1 n
0 1 m
( ) b ( 1) b ( )
a ( ) a ( 1) a ( )
y k y k y k
u k u k u k
n
m
1m-1 0
1n-1 0
p p( )
q q
m m
n n
s sG s
s s
10 1 m
11 n
a a a( )
1 b b
m
n
z zH z
z z
x x u
y x u
A B
C E
( 1) ( ) ( )
( ) ( ) ( )
x k x k u k
y k x k u k
C D
..
..
..
..
..
..
Continuous systems Discrete systems
3. Trend removing
x(t) = r(t) + z(t) r(t): deterministic trend
Effects of trend: (a) Distortion in correlation and spectral estimations (b) Nullify the estimation of low frequency spectral content (c) Finite bits computation problem (d) Possible extraction of deterministic time function for obtaining stationary data
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(17)(17)
Linear least square estimation
2 2
2
2 2
( , ) 1
ˆˆˆ
N ( )( )ˆN ( )
ˆN ( )
i i
i i i i
i i
i i i i i
i i
t x i
x t
x t x t
t t
x t t t x
t t
:
:
:
Data set N
Linear relationship m b
Parameter estimation m
( )( )-( )( ) b
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(18)(18)
(1) Linear trend
= +
t t t
x x x
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(19)(19)
(2) Polynomial trend
Use nonlinear curve fitting for trend removing.
= +
t t t
x x x
§§ 4.3 4.3 Data Preparation and Analysis Data Preparation and Analysis(20)(20)
(3) Periodic trend
Use low-pass filter for trend removing.
+=
t t t
x x x
1. Finite sampling frequency
Sampling frequency ωs is finite. (1) Aliasing issue (a) Line spectrum
§§ 4.4 4.4 Practical Considerations Practical Considerations(1)(1)
Fourierspectrum
Sufficientsamplingfrequency
Insufficientsamplingfrequency
Signal frequency
Mirror image
s
2
Signal frequency
Mirror image
s
2s
2
s
2
Before sampling After sampling
(b) Continuous spectrum
§§ 4.4 4.4 Practical Considerations Practical Considerations(2)(2)
Fourierspectrum
Sufficientsamplingfrequency
Insufficientsamplingfrequency
Signal frequency
Mirror image
s
2
s
2s
2
s
2
Before sampling After sampling
Overlap signal andimage frequency
(2) Solution (a) Nyquist sampling rule Nyquist frequency :
§§ 4.4 4.4 Practical Considerations Practical Considerations(3)(3)
max2s
(b) Practical solution max3 ~ 5s
Cascaded Anti-aliasing filter
2S
N
max : Maximum frequency of signal content
Practical low-pass filterIdeal low-pass filter
2. Finite record length
§§ 4.4 4.4 Practical Considerations Practical Considerations(4)(4)
wy t x t t Y X W
w(t): Window function
t t
w(t) y(t)x(t)
tT
(1) Leakage problem§§ 4.4 4.4 Practical Considerations Practical Considerations(5)(5)
(a) Ideal record length
(b) Non-ideal record length
t
T1=Tp
y(t)
A
A'
t
T2
t
T2
y(t) x(t)
A
AA
x(t)Tp
t
x(t)
t
A
A
(2) Solution
Use zero taping
Data mirror extension
Use improved window functions
§§ 4.4 4.4 Practical Considerations Practical Considerations(6)(6)
3. Finite bits representation
§§ 4.4 4.4 Practical Considerations Practical Considerations(7)(7)
(1) Issues
Finite resolution
Fixed point
Floating point
(2) Solution
Higher bits representation
Proper coding scheme
Least Significant Bit (LSB)
Most Significant Bit (MSB)
§§ 4.4 4.4 Practical Considerations Practical Considerations(8)(8)
4. Wild point
Causes: Data transmission loss Digital bit error(1) Issues (a) Raising overall noise level in estimated spectrum (b) Produce spurious frequencies in estimated power spectrum
t
Possible wild point
(2) Solution
§§ 4.4 4.4 Practical Considerations Practical Considerations(9)(9)
Tolerableerror bound
Remove data
Replacing data
t
Note: Other issues include signal clipping, temporary dropouts, etc.
Use predictor-corrector algorithm for replacing wild data