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1Engineering Measurements
Indrawanto
Mechanical Engineering Department
Faculty of Mechanical and Aerospace Engineering
ITB
2013
Textbook
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Textbook
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Purpose of Measurement
The purpose of a measurement system is to present an observer with a numerical value corresponding to the variable being measured
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
2General structure of measurement
system
The elements of measurement system can be defined as
Sensing element: device which detects and responds to measurand
Signal conditioning element: amplify, filter, integrate, differentiate, convert freq. to voltage, etc.
Signal processing element
Data presentation element
Measurements Direct/Indirect comparison (rulers, balance scale, interferometer) Calibrated system (odometer, spring scale, pressure gage)
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Example of Measurement system
Tire Pressure Gage Bourdon Type Pressure Gage
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Computers Readout SystemsSequential Sampling Data Acquisition
Simultaneous Sampling Data Acquisition
MUX = multiplexer (switch)S/H = sample & hold (hold voltage while ADC reads)ADC = analog to digital converter
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
An Example of a Measurement system
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
3Block diagram symbols
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Static characteristics of
measurement system elements
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Systematic characteristics
Accuracy
Precision
Ideal straight line
Non-linearity
Sensitivity
Environmental effects
Hysteresis
Resolution
Wear and ageing
Error bands
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Static characteristics of
Measurements Accuracy: difference between measured and true values
(error); typically specified by a maximum value.
error = = xmeas - xexact
Precision: difference between measured values during
repeated measurements of the same quantity.
Types of error
Accuracy &Precision
Precision(can be calibrated)
Accuracy(can be averaged)
Neither
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
4Ideal straight line
( )MINMINMAX
MAXMINMIN IIII
OOOO
=
aIKO +=IDEAL
MINMIN
MINMAX
MINMAX
KIOaIIOOK
==
==
intercept line-straight ideal
slope line-straight ideal
Ideal straight line
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Definition of Non-Linearity
( ) ( ) ( ) ( ) ( )
( ) =
=
=++++++=
=
++=+=
mq
q
m
m
qq IaIaIaIaIaaIO
OON
INaKIIOaKIIOIN
0
2210
MINMAX
%100
f.s.d of procentage as NLMax
or
LL
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Sensitivity
Rate of change of O with respect to I
INK
IO
dd
dd
+=
Sensitivity: change of an instruments output per unit change in the measured quantity
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Environmental effects
( ) IIMM IKIIKINaKIO ++++=Modifying Interfering
Thermocouple
sensitivity.
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
5Hysteresis
( ) ( ) ( )%100
..
=
=
MINMAX
II
OOHdsfprocashysteresisMax
IOIOIHHysteresis
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Example: Histerisys, Backlash in Gears
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Resolution
Resolution is defined as the largest change in I that
occurs without any corresponding change in O.
Resolution expressed as a percentage of f.s.d.
%100
MINMAX
R
III
Resolution and potentiometer example.
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Wear and Ageing
Measurement elements change slowly but
systematically throughout their life.
One example is the stiffness of a spring k(t)
decreasing slowly with time due to wear, i.e.
k(t) = k0 bt
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
6Error bands
Nonlinearity, hysteresis and resolution effects in many sensors and transducers are small that is difficult and not worthwhile to exactly quantify each individual effect. In these cases the manufacturer defines the performance of the element in terms of error band.
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Error bands (con.)
Error bands and rectangular probability density function.
Probability density function.Indrawanto, FTMD-ITB,
Engineering Measurements, 2013
Generalized model of a system
element
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Generalized model of a system element
(strain gauge)
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
7Generalized model of a system element
(thermocouple)
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Generalized model of a system element
(accelerometer)
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Statistical characteristics
Probability & Statistics provides a way to deal with uncertainty
Bias error: systematic errors that can be removed by calibrationPrecision error: random error that isnot directly controllable (by definition), could be minimized by careful design and statistical analysis
Bias:calibrationconsistent human errorloading error
Precision:disturbancesnoisevariable conditions
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Statistical characteristics (con.)
+
+
=2
22
22
2
III
MM
IO IO
IO
IO
II
MM
IIOI
IOI
IOO
+
+
=
Small deviation from the mean value
Standard deviation of output for a single element
Normal probabilitydensity function
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
8Statistical characteristics (con.)
( ) IIMM IKIIKINaIKO ++++=
( ) ( )
= 2
2
2exp
21
pi
OOOpo
Mean value of output for a single element
the corresponding probability density function
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Statistical characteristics (con.)
Statistical variations amongst a batch of similar
elements tolerance
( ) ( )
= 2
200
000
2exp
21
RR
RRRppi
The Gaussian probability density function
= mean value of distribution0R
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Statistical characteristics (con.)
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Statistical characteristics (con.)
( ) IIMM IKIIKINaIKO ++++=
+
+
+
+
+
= L22
22
22
22
22
aIII
IM
Ia
OKO
IO
IO
IOO
KIM
Mean value of output for a batch of elements
Standard deviation of output for a batch of elements
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
9Model for chromelalumel thermocouple.
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Model for millivolt to current temperature transmitter.
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Calibration
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Calibration
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
10
Calibration
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Measurement error of a system of
ideal elements
Ideal output
O = On = K1 K2 K3 Ki KnI
E = (K1K2K3Kn -1)
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Example
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
General calculation of system
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
11
General calculation of system
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Example
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
12
Model for recorder
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Summary of calculation
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Error band
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Error Reduction Technique
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
13
Error Reduction Technique
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Error Reduction Technique
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Closed-loop
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Dynamic characteristics of
measurement systems
Dynamic behavior is described by the heat balance equation:
Rate of heat inflow rate of heat outflow = rate of change sensor heat content
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
14
( )( )[ ]
( )
F
F
F
TTdt
TdUAMC
dtTdMCTTAU
TTdtdMC
TTAUW
=+
=
=
=
i.e.
0 content heat sensor of increase of rate
watts
First order linear differential equationIndrawanto, FTMD-ITB,
Engineering Measurements, 2013
==
secs
WJ
mCmWCkgJkg
21-02-
-10-1
The quantity MC/UA has the dimensions of time and is refered to as the time constant tttt
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
( ) ( ) transformLaplace of Definition 0
dttfesf st
=
( ) ( )[ ] ( ) ( )sTsTTsTs F=+ 0( ) ( ) ( )
( ) ( ) ( )sTsT
sTsTsTs
F
F
=+
=+
1si.e.
Using Laplace table
At initial conditions equal zero
eq. diff.order first Linear FTTdtTd =+
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Laplace transform
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
15
Transfer function
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Analogous First Order Elements
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Analogous First Order Elements
RCVVt
V
VVt
VRC
t
VCt
qiCVq
iRVV
EE
in
==+
=+
===
=
,dd
i.e.dd
dd
dd
current , Charge
IN
IN
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Analogous First Order Elements
kFF
t
F
FFt
Fk
kkF
x
t
xFF
MM
==+
=+
==
==
,
dddd
stiffness springNm,nt displaceme
constant dampingNsmdd
IN
IN
1
1IN
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
16
Analogous First Order Elements
kCRCR
k
CCCRRCg
ACRRCRgRA
MCCUA
RCRUAMC
MMMM
gggg
FFFFFF
FFF
ThThThThTh
1,; Mechanical
, Electrical
,; Fluidic
,
1; Thermal
M ====
===
====
====
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Mass-spring-damper model for elastic
sensor
FkxxxmxmxkxF
=++
=
=
&&&
&&&
onaccelerati mass forceresultant
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
sec/rad m
kn = km2
=
Fk
xt
x
t
x
nn
=++ 1d
d2d
d12
2
2
Undamped natural frequency
Damping ratio
( ) ( )xxxx
xxxFFF&&&& ==
==,
0,0
Fk
xt
x
ktx
km =++ 1
dd
dd
2
2
Linear second order differential equation
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( )sFk
sxxsxsxxssxsnn
=++ 102001 22
&
( ) ( )sFk
sxssnn
=
++
1121 22
( )( ) ( )sGksFsx 1
=
( )121
12
2 ++=
ss
sG
nn
Transfer function for a second-order element
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
17
Series R-L-C circuit
= +
+
where =
( = charge on the capacitance)
thus 2
2+
+
1
=
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Identification of the dynamics of an element
Step response of first order elements
( ) ( ) ( ) ( )sssfsGsf i +== 11
0
Expressing in partial fractions
( ) ( ) ( ) sB
s
Ass
sf ++
=
+=
111
0
B=1, A=-t
( ) ( ) ( ) /111
11
0+
=
+=
sssssf
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Using Laplaces table( )
=
tsf exp10
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Second order elements
( )121
12
2 ++=
ss
sG
nn
( )sss
sf
nn
++
=
1211
22
0
( )s
C
ss
BAssf
nn
+
++
+=
121 220
Output signal due to a unit step input
Expressing inPartial functions
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
18
( ) ( )
( )( ) ( )
( )( ) ( ) ( ) ( )222222
222
220
111
121
221
++
++
+=
++
+=
++
+=
nn
n
nn
n
nn
n
nn
n
ss
s
s
s
s
s
ss
s
ssf
( ) ( )2011
n
n
n ssssf
+
+=
( ) ( )tetf ntn += 110
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
( ) ( ) ( ) ( )
+= ttetf nntn 222
0 1sin1
1cos1
( ) ( ) ( ) ( )
+= ttetf nntn 1sinh1
1cosh1 22
20
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Sinusoidal response of first order elements
( ) ( ) ( )220 11
++=
s
Is
sf
( ) ( ) 220 1 ++
++
=
s
cBss
Asf
( ) ( ) ( )2222222
1,
1,
1
+=
+
=
+=
ICIBIA
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
( ) ( ) ( ) ( )
( ) ( ) ( )( ) ( )
( ) ( ) ( )
+
+
++
++=
+
+
++
++
++=
+
+
++
++=
222222
2
22
2222
2222
2
222222
2
0
sincos11
11
111
111
1
111
1
s
sIs
I
s
sI
s
I
s
sIs
Isf
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
19
( ) ( )2222 1sin,11
cos
+
=
+=
( ) ( ) ( )4444 34444 21
44 344 21
termSinusoidal
22
termTransient
/220 sin11
++
++
= t
Ie
Isf t
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
-40
-30
-20
-10
0
M
a
g
n
i
t
u
d
e
(
d
B
)
10-2
10-1
100
101
102
-90
-45
0
P
h
a
s
e
(
d
e
g
)
Bode Diagram
Frequency (rad/sec)
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
-40
-30
-20
-10
0
10
20
M
a
g
n
i
t
u
d
e
(
d
B
)
10-1 100-180
-135
-90
-45
0
P
h
a
s
e
(
d
e
g
)
Bode Diagram
Frequency (rad/sec)
( )121
12
2 ++=
ss
sG
nn
( )( ) ( ) 121
12
2 ++=
jj
jG
nn
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
( )
=
2
22
2
2
2
41
1
nn
jG
( )
=
221
/1/2
tanargn
njG
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
20
Dynamic errors in measurement
systems
( )sOi( ) ( )sIsO 21 =
Complete measurement system with dynamics
K2G2(s) KiGi(s) KnGn(s)K1G1(s)
1 2 3 4Input i.e. true signal
Output i.e. measured signal
( ) ( )sIsI 1= ( )sI i( )sO2 ( )sIn ( )sO
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
( ) ( ) ( )tItOtE =
( )( ) ( ) ( ) ( ) ( ) ( )sGsGsGsGsGsI
sOni LL21==
Transfer function for complete measurement system
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Simple measurement system with
dynamics
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
( ) ( ) ( )}2001{20 200101.0 4 tEeBeAetutT tttM ++=
( ) ( ) ( ) 2420011
11011
1011120
+
++=
sss
s
( ) ( ) ( )
+
+
++= 24 200101.0
120s
DCss
Bs
As
( ) ( )tTtTtE TM = )(( )}2001{20 200101.0 4 tEeBeAe ttt ++=
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
21
( ) ( ) ( ) ( ) +== tIjwGtOtItI sinsin
( ) ( )( ) ( )( )2524 105.2101101101 1 jjjjjG ++++=( )
( )( ) ( ) 10.0]}10105.21[1011001{1
42581
+++=
=jG
( ) ( ) ( ) =
8510tan10tan10tan0)(arg 21411jG
( ) ( ){ }tttE sin85sin1.020 =Indrawanto, FTMD-ITB,
Engineering Measurements, 2013
Periodic Functions
A function f(t) is said to be periodic it is defined for all real t and if there is some
positive number p such that
( ) ( )tfptf =+For any integer n
( ) ( )tfnptf =+ for all tIf f(t) and g(t) have period p, then
( ) ( ) ( )tbgtafth += (a,b constant)
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Periodic Signals
A periodic signal may have:
( )L
L
+++
+++=
tbtbtataatf
0201
02010
2sinsin2coscos
This may be written as
( )
=
=
++=1 2
000 sincosn n
nn tnbtnaatf
where
22
Periodic Signals Contd
( )
dttmtnb
dttmtna
dttmadttmtf
T nn
T nn
T T
01
0
01
0
000
cossin
coscos
coscos
0
0
0 0
+
+
=
=
=
Multiplying both sides by cos mw0t and integrating over a period of f(t),
we obtain
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Periodic Signals Contd
The first term integrates to zero. We multiply each term of the seriesin parentheses by cos w0t and integrate term by term to obtain
( )
=
=
+
=
100
01
00
0
0 0
cossin
coscoscos
n Tn
T n Tn
dttmtnb
dttmtnadttmtf
Three possible cases occur
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
The resulting integrals are
nmdttntmI
nmTnm
dttntmI
nmTnm
dttntmI
T
T
T
, all,0cossin
0,2/,0
coscos
0,2/,0
sinsin
0
0
0
003
0002
0001
==
=
==
=
==
and
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Periodic Signals Contd
( )
= 2cos
00
0
Tadttmtf m
T
( ) =0
0,cos2 00 T
m mdttmtfTa
( )=0
00
sin2
Tm dttmtfTb
For n=m, the integral gives T0/2, which yields
Multiplying both sides by T0/2
In similar manner it can be shown that
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
23
Example
A square wave signal f(t) is defined as
And periodically extended outside this interval. The average value is zero, so a0=0, therefore ams are
( )
24
Periodic Signals
( )( )( ) ( ) Ttfttf
Ta
ttntfT
b
ttntfT
a
T
T
T
Tn
T
Tn
over of valueaveraged1
dsin2
dcos2
2/
2/0
2/
2/ 1
2/
2/ 1
==
=
=
+
+
+
( ) =
=
=
=
++=n
n
n
n
nn tnbtnaatf1 1
110 sincos Fourier series for periodic signal
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Periodic Signals Contd
( )
( ) ( ) ( )
( ) ( ) ( ){ }
=
=
=
=
=
=
+=
+=
=
n
n
nn
n
n
nn
n
n
n
tntnjnGItE
tnjnGItO
tnItI
1111
111
11
sinsin
sin
sin
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Damped harmonic oscillator
Excitation
Response
0 = 02 =
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Response of damped harmonic oscillator to
sawtooth excitation
Excitation
Response
02 = 03 =0 =
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
25
Response of damped harmonic oscillator to
sawtooth excitation
20 =
0 =2
3 0 =
Excitation
Response
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
( ) ( ) ( ) ( ) 014.07,020.05,033.03,100.0 jGjGjGjG
( ) ( ) ( )[( ) ( )]00
00
937sin002.0925sin004.0
903sin011.085sin100.080
++
+=
tt
tttTM pi
( )
++++= LtttttTT 7sin7
15sin513sin
31
sin80pi
( ) ( )( ) ( ) 00
00
937arg,925arg903arg,85arg
jGjGjGjG
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Solution of differential equation using Laplace Transform
Partial-fraction Expansion: Distinct Real Roots
( ) ( )( )( )( )3142
++
++=
sss
sssY
( )31
321
++
++=
s
Cs
Cs
CsY
( )( )( )( )( )( )
( )( )( )
( ) 61
142
23
342
38
3142
33
12
01
=
+
++=
=
+
++=
=
++
++=
=
=
=
s
s
s
ss
ssC
ss
ssC
ss
ssC
( ) ( ) ( ) ( )tetetty tt 1611
231
38 3
=
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Solution of differential equation using Laplace
Transform
Partial-fraction Expansion: Distinct Complex Roots
( ) ( )11
2 ++=
ssssF
( )12321
++
++=
ss
CsCs
CsF
( ) 101 == =sssFC( ) ( ) 11 322 =++++ sCsCss
( ) ( ) 4322121
211
++
++=
s
s
ssF
( ) ( )ttetetf tt 1sin3
1cos1 43
2/432/
=
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
26
Solution of differential equation using Laplace
Transform
Partial-fraction Expansion: Distinct Repeated Roots
( ) ( )( )2213++
+=
ss
ssF
( ) ( )2321
221 ++
++
+=
s
Cs
Cs
CsF
( ) ( ) ( )( ) ( )
( ) ( ) 1132
22
2231
22
23
2
22
1211
=
+
+=+=
=+=
=
+
+=+=
=
=
=
=
=
ss
s
s
s
s
ssFsC
sFsdsdC
s
ssFsC
( ) ( ) ( )tteeetf ttt 122 22 =Indrawanto, FTMD-ITB,
Engineering Measurements, 2013
Example 2
A pressure sensor has the following transfer function
Input signal Pin to the sensor can be represented as follow
Show the sensor output due to the second harmonic of the input signal
( ) ( )( ) 121+
==
ssPsP
sGin
out
0 1-1
10
T second
P barin
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Signals
Deterministic signals: a deterministic signal is
one whose value at any future time can be
exactly predicted
Random signals: a random signal is one whose
value at any future time can not be exactly
predicted
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Deterministic and Random Signals
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
27
Statistical Representation of Random Signals
Sampling of a random signal
T0 is sufficiently long, i.e. N is sufficiently largeThe signal is stationary, i.e. long term statistical quantities do not change with time
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Statistical Representation of Random Signals
Contd
( ) ttyT
yT
d1 00
0=
=
=
=
Ni
iiyN
y1
1
( )[ ] tytyT
T
d10
0
2
0
2 = ( )
=
=
=
Ni
ii yyN 1
22 1
Mean for continuous signal
Mean for sampled signal
Standard deviation for continuous signal
Standard deviation for sampled signal
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Statistical Representation of Random Signals
Contd
0=y
=0
02
0rms d
1 Tty
Ty
=
=
=
Ni
iiyT
y1
2
0rms
1
For special case
The r.m.s. value is
or
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Probability Density Function p(y)
Probability Density Function is function of signal value y and is a measure of the probability that the signal will have a certain range of values
Sampling of a random signalIndrawanto, FTMD-ITB,
Engineering Measurements, 2013
28
Probability Density Function p(y)
mjNn
jP
j
j
,,1,
samples ofnumber totalsectionth in the occurs sample timesofnumber
K==
=The probability
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
( )jjj
nnnN
PPPC
+++=
+++=
L
L
21
21
1
( )( )NnnnN
N
nnnN
C
m
mm
=+++==
+++=
L
L
21
21
since11
1
The cumulative probability
The final value of Cj when j=m is:
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
( ) jy CyP 0lim=Cumulative probability distribution function
( )yPyp
dd
=
Probability density function
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
( ) yypPP yyy ==+,( )= 2
121
d,
y
yyyyypP
( ) ( )
= 2
2
2exp
21
pi
yyyp
Probability for signal that lie between y and y + y is
Probability for signal that lie between y1 and y2 is
The Gaussian probability density function is:
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
29
Power Spectral Density
Signal power is a stationary quantity which can be used to quantify random signals.
( ) =
=
=
=
++=n
n
n
n
nn tnbtnaaty1 1
110 sincos
The Fourier series for a signal with period T0with yaT mean,/2 001 == pi Indrawanto, FTMD-ITB,
Engineering Measurements, 2013
( )
( )
=
=
=
=
=
=
=
=
==
==
Ni
ii
Ni
iin
Ni
ii
Ni
iin
Ni
nyN
TinyN
b
Ni
nyN
TinyN
a
111
111
2sin2sin2
2cos2cos2
pi
pi
Fourier series coefficients for sampled signal
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Power Spectrum
( )2221 nnn baw += nn wwwwW ++++= L210Power due to nth harmonic Cumulative power
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Power Spectrum Contd
( ) nWW 01lim= ( ) ddW
w =
Power spectral density in watts sec rad-1 or watts/Hz
Cumulative power function
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
30
Autocorrelation function
( ) ( ) ( ) =
0
0 00
d1limT
Tyyttyty
TR
Autocorrelation function Ryy() of the signal is the relation between Ryyand time delay .
A Simple Correlator
Autocorrelation function of a continuous signal
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Autocorrelation function Contd
( ) ( )[ ]
12
0 10
2
00
1
2
cos
d22cos1lim2
d1limcos2
0
0
0
0
b
ttT
bt
TbR
T
T
T
Tyy
=
+=
( ) ( ) += tbty 1sin If( ) ( ) ( )[ ]
( )[ ]{ } ttT
b
tttT
bR
T
T
T
Tyy
d22coscos1lim
dsinsin1lim
0
0
0
0
0 1121
0
2
0 110
2
+=
++=
Thus the autocorrelation function of a sinusoidal signal is cosine function of the same frequency, but the phase information in the sine wave is lost
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Autocorrelation function Contd
K,2,1,0, == mTm( )
=
=
=
Ni
imiiNyy
yyN
TmR1
1lim
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Autocorrelation function Contd
( ) tbtbtbty 131211 3sin2sinsin ++=A periodic signal which is the sum of three harmonics
The autocorrelation function
( ) 133
22
21 3cos
22cos
2cos
2bbbRyy ++=
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
31
Autocorrelation function Contd
( ) ( )( ) ( )
pi
dcos2
dcos
0
0
=
=
yy
yy
R
R
( )
C
yy
A
AARC
C
sin
sindcos0
0
=
==
The power spectrum can be obtained from the autocorrelation function by Fourier analysis. Similarly the autocorrelation function can be obtained from the power spectrum by adding the harmonics together. For random signals, Ryy() and () are related by the Fourier transform:
For a signal with () constant up to C and zero for higher frequencies then
Indrawanto, FTMD-ITB, Engineering Measurements, 2013
Summary
In order to specify a random signal we need to know:
deviation standard and meanfunction density yprobabilit
or
either
function ncorrelatio autodensity spectralpower
or
either
To specify amplitude behavior
To specify frequency/time behavior
or
( ) ( ) dd1lim000
2
0
0
0
==
T
Tyyty
TR
( ) TOT2RMS0 WyRyy ==The autocorrelation coefficient at zero time delay
Indrawanto, FTMD-ITB, Engineering Measurements, 2013