§ 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


(2) Ideal measurement system

1ie oe

45o

ie

oe

2. System structure

(1) Fundamental structure

MeasurementSystems

Calibration SignalGenerator

Disturbance

MeasurandRead out

Standard unit

Controlledenvironment


(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


(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


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


(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


(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


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.


High Precision High Accuracy High Accuracy and Precision

★★★★★

★★★

★★

★

★

★

★

★★ ★★★

★★★★

★

3. Static and Dynamic characteristics

Linearity

Sensitivity(slope)

Input

Output

Threshold Span

(Minimum)Resolution


(1) Static

Frequency range

Span

A

u

1

2A

Delay

Steady state

t


(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)


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


Mathematical analysis


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-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(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



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)


(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


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


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


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


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) 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


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


(1) Linear trend

= +

t t t

x x x


(2) Polynomial trend

Use nonlinear curve fitting for trend removing.

= +

t t t

x x x


(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


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 :


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


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


3. Finite bits representation


(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. 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


Tolerableerror bound

Remove data

Replacing data

t

Note: Other issues include signal clipping, temporary dropouts, etc.

Use predictor-corrector algorithm for replacing wild data

§ 4.1 instrumentation and measurement systems § 4.2 dynamic measurement and calibration § 4.3...

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