Transcript AAE 520 Uncertainity..

AAE 520 Experimental Aerodynamics
Measurement Uncertainty
References
Coleman,H.W and Steele,W.G. Experimentation and Uncertainty Analysis for Engineers, John
Wiley & Sons, 1989
Matlab Signal Processing Toolbox Manual
LabView Manual
Software
Matlab
LabView
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AAE 520 Experimental Aerodynamics
Generic Transducer System
Physical
Property
Transducer
Input
Circuit
Signal
Conditioner
Transmission
Processing
Areas of Concern
•.Accuracy
•.Static Sensitivity
•.Frequency Response
•.Loading (Impedance Matching)
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Display
AAE 520 Experimental Aerodynamics
“degree of goodness”
Anyone comparing results of a mathematical model with experimental data (and perhaps also with
the results of other mathematical models) should certainly consider the "degree of goodness" of
the data when drawing conclusions based on the comparisons. In Figure a the results of two
different mathematical models are compared with each other and with a set of experimental data.
In Figure b the same information is presented, but a range representing the likely amount of error
in the experimental data has been plotted for each data point.It should be immediately clear that
once the "degree of goodness" of the data is taken into consideration, it is fruitless to argue for the
validity of one model over another based only on how well the results match the data.
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AAE 520 Experimental Aerodynamics
Experimental Approach - Questions
1. What question are we trying to answer? (What is the problem?)
2. How accurately do we need to know the answer? (How is the answer to be used?)
3. What physical principles are involved? (What physical laws govern the situation?)
4. What experiment or set of experiments might provide the answer?
5. What variables must be controlled? How well?
6. What quantities must be measured? How accurately?
7. What instrumentation is to be used?
8. How are the data to be acquired, conditioned, and stored?
9. How many data points must be taken? In what order?
10. Can the requirements be satisfied within the budget and time constraints?
11. What techniques of data analysis should be used?
12. What is the most effective and revealing way to present the data?
13. What unanticipated questions are raised by the data?
14. In what manner should the data and results be reported?
(Ref. Coleman and Steel)
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AAE 520 Experimental Aerodynamics
Need for Uncertainty Analysis
Is uncertainty analysis always necessary ?
Four types of Experiments
(a) quick-sort
Flow visualization is used to establish the global nature of a flowfield. The visual results are then used to
determine where to place probes.
In development work, one needs to know whether something will work. A fast low cost yes or no answer
is needed.
In such a cases uncertainty analysis is not essential.
(b) report of field test, development test, or acceptance test
(c) report of research
(d) calibration test
No alternative to Uncertainty Analysis in (b),(c) and (d)
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AAE 520 Experimental Aerodynamics
Replication Levels
Three different replication levels are
Zeroth Order is described by the following conditions time is frozen; the display of
each instrument is considering to be invariant A single sample is taken.
The values of uncertainty at this level are often assigned "one-half the smallest
scale division" or some similar rule of thumb.
First Order: At this order, time is the only variable; with the experiment running, the
display for each instrument is assumed to vary stochastically about a stationary
mean. The first order uncertainty interval includes the timewise variation of the
display and its interpolation uncertainty. Valid estimates of the mean and standard
deviation are obtained.
N'th Order: At this order, time and the instrument identities are considered to be
variables. For each conceptual replication, each instrument is considered to have
been replaced by another of the same type.
An example is the wind tunnel measurements of aircraft model drag in various
facilities around the world using the same model.
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AAE 520 Experimental Aerodynamics
Measurement Error
True Value
NBS Standard
Bias Error
Systematic Error
Remains Constant During Test
Estimated Based On Calibration
or judgement
True Average m
xk Measured Value
Bias Error b
Precision ( Random Error )
Index - Estimate of Standard
Deviation
Total Error
dk
dk = b + ek
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Random Error
ek
AAE 520 Experimental Aerodynamics
True Value
Unbiased, Precise, Accurate
True Value
Biased, Precise, Inaccurate
Unbiased, Imprecise, Accurate if N>>1 Biased, Imprecise, Inaccurate
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AAE 520 Experimental Aerodynamics
• Accuracy
Errors
– Measure of how close the result of the experiment comes to
the “true” value
• Precision
– Measure of how exactly the result is determined without
reference to the “true” value
• Bias Error
– Reproducible inaccuracy introduced by calibration or
technique. Sometimes this error is correctable
• Random Error
– Indefiniteness of result due to finite precision of experiment.
Measure of fluctuation in result after repeated
experimentation
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AAE 520 Experimental Aerodynamics
Normal Distribution ( Gaussian or Bell Curve )
The normal distribution is a two parameter family of curves. The first parameter, m , is the mean.
The second, s, is the standard deviation.
The usual justification for using the normal distribution for modeling is the Central Limit Theorem
which states (roughly) that the sum of independent samples from any distribution with finite mean and
variance converges to the normal distribution as the sample size goes to infinity.
The normal pdf ( probability density function) is:
( xm )2
4.5
2s 2
4.0
3.5
3.0
2.5
sigma, s
Y
1
y
e
s 2

Normal Distribution
2.0
normalized so that the area under the curve = 1.0
1.5
Mean m
1.0
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X
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0.8
0.9
1.0
AAE 520 Experimental Aerodynamics
Parameter Estimation.
A desirable criterion in a statistical estimator is unbiasedness. A statistic is unbiased if the expected
value of the statistic is equal to the parameter being estimated.
Unbiased estimators of the parameters, m, the mean, and s, the standard deviation are:
N
x
x
i
1
N
estimate of the mean, m
[ mean(data) ]
N
S
2
(
x

x
)
 i
1
N 1
estimate of the standard deviation, s
[ std(data) ]
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AAE 520 Experimental Aerodynamics
Data Sample
Signal from Hot Wire in a Turbulent Boundary Layer
Output from an A/D Converter (in counts) at Equal Time Intervals
Long Time Record
Short Time Record
1000
980
960
Amplitude
940
920
900
880
860
840
820
800
0
20
40
60
Time
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80
100
AAE 520 Experimental Aerodynamics
Estimate of the Probability Density Function
[ hist(data,# of bins) ]
800
700
600
500
400
300
200
100
0
850
900
950
1000
1050
1100
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AAE 520 Experimental Aerodynamics
Errors in the Estimate of the Mean
• For a normal Distribution
m  x  t95
s
N
m - true mean
x - estimate of the mean
s- estimate of the standard deviation
N - number of samples
t95 - 95% confidence interval from Students t distribution
t95 = ~2 for N>20
The notation
m  12.5  .5.95
Means we are 95% confident that the mean lies
between 12.0 and 13.0
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AAE 520 Experimental Aerodynamics
Mean
95% Confidence Intervals
0.20
0.18
Std. Dev/Mean
0.16
1.0
0.14
0.12
Error
s/ x
m  x (1  t95
)
N
m  x (1  error )
0.10
0.08
.5
0.06
.3
0.04
.2
0.02
.1
0.00
.05
100
1000
Number of Samples
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10000
AAE 520 Experimental Aerodynamics
Standard Deviation
95% Confidence Interval
2
nS 2
nS
s2  2
2
 p1 (n  1)
 p2 (n  1)
N
S
 (x  x)
i
1
N 1
2
 2 - Chi Square Distributi on
For 95% Confidence Interval
1
p1  (1  Confidence interval)  .025
2
1
p2  (1 + Confidence interval)  .975
2
For n  100
2
1
 2  ( p + 2(n  1)  1)
2


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AAE 520 Experimental Aerodynamics
Standard Deviation
95% Confidence Interval
1.01
p1= .025
True Value/Estimated Value
1.00
0.99
0.98
0.97
p2= .975
0.96
0.95
0.94
0.93
100
1000
Number of Samples
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10000
AAE 520 Experimental Aerodynamics
Propagation of Errors
General Uncertainty Analysis
Suppose the quantity of interest Q is to be calculated from the measured
xquantities
1 , x2 ,........
by the equation
Q  Q( x1 , x2 ,.......)
+ i
If there are variations in the readings, at any instant xiti is
andx inot
that is measured. This consequently causes a variation in Q. Ifi 'the
s
are small then by the Taylor's series expansion the calculated Q is given as
Q*  Q( x1 , x2 ,.......) +
or
dQ
dQ
1 +
 2 + .........
dx1
dx2
Q  Q*  Q( x1 , x2 ,.......) 
dQ
dQ
1 +
 2 + ......
dx1
dx2
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AAE 520 Experimental Aerodynamics
Qthat
Consider that many measurements have been made and
n
is the deviation for each reading set. It's standard deviation over
many readings is given as
1
S Q2 
N 1
N

(Qn ) 2
1
Substituting for Q
1
S Q2 
N 1
N

1
 dQ

dQ
1 +
 2 + ...)n 
(
dx2
 dx1

1  dQ 

S Q2 

N  1  dx1 
 dQ 

+ 
dx
 2
2
N

21
1
2 N

1
22
2
dQ dQ
+ 2( )(
)
dx1 dx2
dQ dQ
+ 2(
)(
)
dx2 dx3
N
 
1
2
+ ..........
1
N
 
2
3
+ ........
1
+ ............................. 

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AAE 520 Experimental Aerodynamics
If the measurements are independent , then there errors
will be uncorrelated
N
and the cross term will be zero. That is in the sums  (1 2 ) n , etc., any
1
term is as likely to be positive as negative (assuming independent
variations)
then these sums tend to zero for large N. Hence
2
 dQ  1
2

S Q  
 dx1  N  1
But
.
1
S1 
N 1
N

2
1
N

1
2
1
2  dQ 

1 +
 dx2  N  1
N

22
1

+......................... 

x1
is the standard deviation in the measurement
1
Q
Therefore, the standard deviation in the calculated result
2
SQ 
is
2
 dQ  2  dQ  2

 S1 + 
 S 2 ......... 

 dx1 
 dx2 

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AAE 520 Experimental Aerodynamics
COMMON SENSE ERROR ANALYSIS
.Examine the data for consistent. No matter how hard one tries, there will always be some data
points that appear to be grossly in error. The data should follow commonsense consistency, and
points that do not appear "proper" should be eliminated. If very many data points fa11 in the
category of "inconsistent," perhaps the entire experimental procedure should be investigated for
gross mistakes or miscalculations.
.Perform a statistical analysis of data where appropriate. A statistical analysis is only appropriate
when measurements are repeated several times. If this is the case, make estimates of such
parameters as standard deviation, etc.
.Estimate the uncertainties in the results. We have discussed uncertainties at length. Hopefully,
these calculations will have been performed in advance, and the investigator will already know the
influence of different variables by the time the final results are obtained.
.Anticipate the results from theory. Before trying to obtain correlations of the experimental data,
the investigator should carefully review the theory appropriate to the subject and try to glean some
information that wi11 indicate the trends the results may take. Important dimensionless groups,
pertinent functional relations, and other information may lead to a fruitful interpretation of the data.
.Correlate the data. The word "correlate" is subject to misinterpretation. In the context here we
mean that the experimental investigator shou1d make sense of the data in terms of physical
theories or on the basis of previous experimental work in the field. Certainly, the results of the
experiments should be analyzed to show how they conform to or differ from previous
(Ref. Holman,
J. P., ”Experimental
Methods for Engineers")
investigations or standards that may be employed
for such
measurements.
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AAE 520 Experimental Aerodynamics
Design of experiments
Using Uncertainty Analysis
• The choice of test and data reduction procedures can have an
important impact on the accuracy of the results
• It is important to specify the level of replication - 0th, 1st or Nth
order
• Reliable means for cross-checking and/or externally validating
the results are necessary
• In experiments where data are reduced by computer,
uncertainty analysis can be done by sequential perturbation
using the data reduction program
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AAE 520 Experimental Aerodynamics
Other Issues
•
•
•
•
Threshold
Resolution
Linearity
Hystersis
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AAE 520 Experimental Aerodynamics
Applications of Curve Fitting
•Generate a Calibration Curve
• Removal of measurement noise.
• Filling in missing data points (for example, if one or more measurements
were missed or improperly recorded).
• Interpolation (estimation of data between data points; for example, if the
time between measurements is not small enough).
• Extrapolation (estimation of data beyond data points; for example, if you
are looking for data values before or after the measurements were taken).
• Differentiation of digital data. (For example, if you need to find the
derivative of the data points. The discrete data can be modeled by a
polynomial, and the resulting polynomial equation can be differentiated.)
• Integration of digital data (for example, to find the area under a curve when
you have only the discrete points of the curve).
• To obtain the trajectory of an object based on discrete measurements of its
velocity (first derivative) or acceleration (second derivative).
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AAE 520 Experimental Aerodynamics
Method of Least Squares
Straight Line Fit (Regression Analysis) to
y  mx + b
Set of Data Points (N points)xi , yi
Problem - Find m and b
di  yi  y (can be )
Vertical Deviation
Minimize the Sum of the Squares of the Deviation
N
N
N
 d   ( yi  y)   yi  (mxi + b)  minimum
2
i
1
2
1
  N 2
 d i   0
m  1

2
1
  N 2
 d i   0
b  1

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AAE 520 Experimental Aerodynamics
xi , yi
d i  yi  y
y
b
y  mx + b
Uncertainty
N
sy 
x
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2
d
 i
1
n2
AAE 520 Experimental Aerodynamics
General Curve Fit
Curve fitting analysis is a technique for extracting a set of curve parameters
or coefficients from the data set to obtain a functional description of the
data set. The algorithm that fits a curve to a particular data set is known as
the Least Squares Method. The error is defined as
e ( a )   f ( x, a )  y ( x ) 
2
where e(a) is the error, y(x) is the observed data set, f(x,a) is the functional
description of the data set, and a is the set of curve coefficients which best
describes the curve.
To solve this system, you set up and solve the Jacobian system generated
by expanding equation. After you solve the system for a, you can
obtain an estimate of the observed data set for any value of x using the
functional description f(x, a).
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AAE 520 Experimental Aerodynamics
5
4.5
Polynomial Fits
to Hot Wire Data
Hot Wire Output (volts)
4
3.5
3
2.5
4th Order
2
1.5
1
20
40
60
80
100
120
Velocity (ft/sec)
140
160
5.5
5
5
4.5
3.5
3
Linear
2.5
2
3.5
3
2.5
2nd Order
2
1.5
1.5
1
20
200
4
4
Hot Wire Output (volts)
Hot Wire Output (volts)
4.5
180
40
60
80
100
120
Velocity (ft/sec)
140
160
180
200
1
20
40
60
80
100
120
Velocity (ft/sec)
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140
160
180
200
AAE 520 Experimental Aerodynamics
Oscilloscope
The oscilloscope is basically a graph-displaying device -- it draws a graph of
an electrical signal. In most applications the graph shows how signals
change over time: the vertical (Y) axis represents voltage and the horizontal
(X) axis represents time. The intensity or brightness of the display is
sometimes called the Z axis.
URL for a description of oscilloscope operation
http://www.tek.com/Measurement/App_Notes/XYZs/intro.html
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AAE 520 Experimental Aerodynamics
Vertical Plates
Horizontal Plates
Electron
Beam
Analog Oscilloscope
CRT
Screen
Vertical Amplifier
Volts/Div
Input Signal
(Waveform to be
Observed)
Delay
Line
Trigger
Time Base
Amplifier
Sweep Generator
Sawtooth Waveform
Time/Div
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AAE 520 Experimental Aerodynamics
Digital Oscilloscope
Input
A/D
CPU
Buss
Amp
Memory
Input
(Trigger)
A/D
D/A
Computer
Interface
Amp
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CRT
AAE 520 Experimental Aerodynamics
Data Acquisition (DAQ) Fundamentals
•The personal computer
•Transducers
•Signal conditioning
•DAQ hardware
•Software
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AAE 520 Experimental Aerodynamics
CPU
Central Processing
Unit
Computer
Address
Data
Control
Input
Device
Keyboard
Disk
A/D Converter
Buss
Memory
Output
Device
CRT
Printer
Disk
D/A Converter
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AAE 520 Experimental Aerodynamics
101011
Sampled Analog
Input Signal
A/D
Converter
Digital Output
B bits/Sample
111
110
Digital
Output
101
100
011
010
001
000
0
1/2 LSB
Full
Scale
Analog Input
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AAE 520 Experimental Aerodynamics
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AAE 520 Experimental Aerodynamics
A/D PERFORMANCE SPECS
Resolution describes the smallest standard incremental change in output voltage of a DAC or the
amount of input voltage change required to increment the output of an ADC between one code change
and the next adjacent code change. A converter with n switches can resolve 1 part in 2 n . The least
significant increment is then 2 bn , or one least significant bit (LSB). In contrast, the most significant bit
(MSB) carries a weight of 2 b1 . Resolution applies to DACs and ADCs, and may be expressed in
percent of full scale or in binary bits. For example, an ADC with 12-bit resolution could resolve 1 part in
2 12 (1 part in 4096) or 0.0244% of full scale. A converter with 10V full scale could resolve a 2.44mV
input change.
Accuracy. An accuracy specification describes the worst case deviation of the
DAC output voltage from a straight line drawn between zero and full scale; it includes all errors.
Quantizing Error ,Gain Error ,Scale Error (full scale error), Hysteresis Error ,Offset Error (zero error)
and Linearity,
Conversion Rate is the speed at which an ADC or DAC can make repetitive data conversions.
Input Impedance of an ADC describes the load placed on the analog source.
Number of Channels
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AAE 520 Experimental Aerodynamics
Analog to Digital Converter
12 Bit A/D - Resolution = one part in
Binary Word - 101100111101
MS
B
Unipolar Mode (0-10 Volts)
Integer
Output
Voltage
4095
.
.
.
.
.
1
0
9.9975
.
.
.
.
.
.0024
0
212  4096
LSB
Volts=10/4096*Icount
Scale
+Full Scale - 1 LSB
.
.
.
.
.
Bipolar Mode
1 LSB
-5.0 to 5.0
0
V=5/2048*(Icount-2048)
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AAE 520 Experimental Aerodynamics
Dynamic Response of Measurement Systems
Zero Order System
y  Kx(t )
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AAE 520 Experimental Aerodynamics
First Order System

y  Kx0 (1  et /  )
dy
+ y  Kx(t )
dt
y  Kx0 e t / 
Impulse response –
First Order System
Step response
- First Order System
1
0.9
 1
0.8
0.9
0.8
 2
0.7
0.7
Amplitude
Amplitude
1
  .5
0.6
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
2
t /
4
6
8
10
0
 1
 2
  .5
0
2
4
t /
Purdue University - School of Aeronautics and Astronautics t / 
6
8
10
AAE 520 Experimental Aerodynamics
First Order System
Sinusoidal Response
y
Kx
sin( t + )
1+  
  tan 1 ( )
amplitude
Sinusoidal Response - First Order System
1
0.8Input
0.6
0.4
0.2
Output
0
-0.2
-0.4
-0.6
-0.8
-10
0.5
1
1.5
2
2.5
time
0
Gain dB
Amplitude Decrease and Phase Shift
-10
-20
-30 -1
10
Phase deg
2 2

10
0
10

10
0
10
1
0
-30
-60
-90
-1
10
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1
AAE 520 Experimental Aerodynamics
Second Order System
d2y
dy
2
2
+
2

+

y

K

x(t )
n
n
n
2
dt
dt
 n - natural frequency
Step response - Second Order System
2
1.8
1.6
 - damping factor
5% overshoot
System comes to5% of static value
Amplitude
  1 - critical damping - no oscillatio ns
  .7 - for fastest response
1.4
1.2
1
0.8
0.6
in half the time for critically damped 0.4
systems
0.2
0
0
5
 nt
10
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15
20
AAE 520 Experimental Aerodynamics
Second Order System - Sinusoidal Response
x  X sin(  t )
y
KX
2 2
2





1     +  2   
   n     n  



sin(  t +  )

 

2

n 
1
  tan  
2 
  

 1     
 n 

Amplitude
(dB)
Phase
(deg)
 / n
Purdue University - School of Aeronautics and Astronautics
AAE 520 Experimental Aerodynamics
Second Order System - Impulse
Response
y  Kx0 e  t sin( 1   2  nt +  )
n
Impulse response - Second Order System
1
Logarithmic Decrement
0.8
0.4
x1
e  nt
d  ln( )  ln  n (t + )   n
x2
e
0.2

Amplitude
0.6
0
d
-0.2
-0.4
2
n 1  2
2
1
2
 2
-0.6
-0.8
0
20
40
60
80
100
Purdue University - School of Aeronautics and Astronautics
1.2
1
1
0.8
0.8
Amplitude Ratio
Amplitude Ratio
AAE 520 Experimental Aerodynamics
1.2
0.6
0.4
0.2
0
0
10
20
30
Frequency (Hz)
40
0
50
1.2
1.2
1
1
0.8
0.8
0.6
0.4
0.2
0
10
20
30
Frequency (Hz)
40
50
High Pass Filter
Removes DC and Low Frequency Noise
(Such as 60, 120 Hz)
Amplitude Ratio
Amplitude Ratio
0.4
0.2
Low Pass Filter
Removes High Frequency Noise
0
0.6
0.6
0.4
0.2
0
10
20
30
Frequency (Hz)
Band Pass
40
50
0
0
10
20
30
Frequency (Hz)
Stop Band
Purdue University - School of Aeronautics and Astronautics
40
50
AAE 520 Experimental Aerodynamics
Filters and Transfer Functions
In general, the z-transform Y(z) of a digital filter’s output y(n) is related to the
z-transform X(z) of the input by:
b(1) + b(2) z 1 + ... + b(nb + 1) z  nb
Y ( z)  H ( z) X ( z) 
X ( z)
1
 na
1 + a(2) z + ... + a(na + 1) z
Filtering with the filter Function
It is simple to work back to a difference equation from the z-transform relation
shown earlier. Assume that a(1) = 1. Move the denominator to the left-hand
side and take the inverse z-transform. In terms of current and past inputs, and
past outputs, y(n) is:
y (n)  b(1) x(n) + b(2) x(n  1) + .. + b(nb + 1) x(n  nb)  a(2) y (n  1) + ... + a(nb + 1) y (n  na)
This is the standard time-domain representation of a digital filter, computed
starting with y(1) and assuming zero initial conditions. This representation’s
progression is
y (1)  b(1) x(1)
y (2)  b(1) x(2) + b(2) x(1)  a (2) y (1)
y (3)  b(1) x(3) + b(2) x(2) + b(3) x(1)  a (2) y (2)  a (3) y (3)
Purdue University - School of Aeronautics and Astronautics
AAE 520 Experimental Aerodynamics
1.2
1
Amplitude Ratio
10
 Rp / 20
Elliptic Filter
Pass Band Ripple
0.8
[b,a]=ellip(5,1,20,.5)
0.6
0.4
0.2
0
10 Rs / 20 Stop Band Ripple
0
10
20
30
Frequency (Hz)
40
50
0
10
20
30
Frequency (Hz)
40
50
200
150
Phase (degrees)
100
50
0
-50
-100
-150
-200
Purdue University - School of Aeronautics and Astronautics
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.2
0.4
0.2
0
10
Frequency
10
0
-1
10
1
Elliptic Filter
0
10
Frequency
10
1
0
-1
10
Bessel Filter
1
0.8
0.8
0.6
0.6
0.4
0.2
0
10
Frequency
Butterworth
1
0
-1
10
0.4
0.2
Magnitude
0
-1
10
Magnitude
1
Magnitude
1
Magnitude
Magnitude
AAE 520 Experimental Aerodynamics
0.4
0.2
0
10
Frequency
10
Chebyshev I Filter
1
0
-1
10
0
10
Frequency
Chebyshev II
Purdue University - School of Aeronautics and Astronautics
10
1
10
1
AAE 520 Experimental Aerodynamics
Effect of Filter Order
Chebyshev II
1
N= 5
Magnitude
0.8
N= 10
N= 2
0.6
0.4
0.2
0
-1
10
0
10
Frequency
10
1
Purdue University - School of Aeronautics and Astronautics
AAE 520 Experimental Aerodynamics
Example Signal
Fs = 100;
t = 0:1/Fs:1;
x =.5+ sin(2*pi*t*5)+.25*sin(2*pi*t*40);
% DC plus 5 Hz signal and 40 Hz signal sampled at 100 Hz for 1 sec
2
Total Signal
1.5
Amplitude (volts)
DC Level
Low Frequency
Signal
1
0.5
High Frequency
0
-0.5
-1
0
0.2
0.4
0.6
Time (sec)
0.8
1
Purdue University - School of Aeronautics and Astronautics
AAE 520 Experimental Aerodynamics
2
1.2
0.6
0.4
Recovers
DC + 3Hz
Amplitude (volts)
Amplitude Ratio
Cheby2
Low Pass
0.8
0.5
Filter
0
-1
0
10
20
30
Frequency (Hz)
40
50
Filtfilt
0
0.2
0.4
0.6
Time (sec)
0.8
1
0
0.2
0.4
0.6
Time (sec)
0.8
1
0
0.2
0.4
0.6
Time (sec)
0.8
1
0
0.2
0.4
0.6
Time (sec)
0.8
1
2
1.2
1.5
Recovers
40 Hz
Cheby2
High Pass
0.8
0.6
Amplitude (volts)
1
Amplitude Ratio
1
-0.5
0.2
0
Original
Signal
1.5
1
0.4
1
0.5
0
-0.5
0.2
-1
0
0
10
20
30
Frequency (Hz)
40
50
2
1.2
1.5
Cheby2
Band Pass
0.8
0.6
Recovers
3Hz
0.4
1
Amplitude (volts)
Amplitude Ratio
1
-1
-1.5
0
10
20
30
Frequency (Hz)
40
50
1.2
2
1
1.5
Recovers
DC + 40Hz
0.8
Cheby2
Stop Band
0.6
0.4
1
0.5
0
-0.5
0.2
0
Amplitude (volts)
Amplitude Ratio
0
-0.5
0.2
0
0.5
-1
0
10
20
30
Frequency (Hz)
40
50
Purdue University - School of Aeronautics and Astronautics
AAE 520 Experimental Aerodynamics
10 Point Averaging Filter
b= 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1
a= 1.0
1
2
0.9
1.5
0.8
Amplitude (volts)
Amplitude Ratio
0.7
0.6
0.5
0.4
0.3
0.2
1
0.5
0
-0.5
0.1
0
0
10
20
30
Frequency (Hz)
40
50
-1
0
0.2
0.4
0.6
Time (sec)
Purdue University - School of Aeronautics and Astronautics
0.8
1