Download Part I: The normal distribution - Department of Electronic Systems

Instrumentation and data acquisition
Spring 2010
Lecture 5: Noise, precision and accuracy
Zheng-Hua Tan
Department of Electronic Systems
Aalborg University, Denmark
[email protected]
Thanks to Christian Fischer Pedersen for providing some
of the slides.
Instrumentation and data acquisition,V, 2010
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Part I: The normal distribution
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The normal distribution
Digital noise generation
Precision and accuracy of data acquisition
Data acquisition system
Instrumentation and data acquisition,V, 2010
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Signal acquisition
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When acquiring a physical signal, the particular measurement could
be affected by many factors: electronic (random) noise, interfering
signals and inaccuracies.
Noise
Generating
process

Acquired
signal
+
Statistics and probability for characterizing signal & noise, reducing
the noise, etc.
 The normal distribution
 Precision and accuracy
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The normal distribution

Normal distribution occur frequently in nature
Signals from random processes usually have a bell-shaped
PDF, called a normal (Gauss) distribution, or a Gaussian.
Basic curve of a normal distribution. No normalization
1
y(x)=exp(-x2)
0.9
0.8
0.7
0.6
y(x)
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0.5
0.4
0.3
0.2
0.1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
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The normal distribution
Raw un-normalized curve: y ( x)  e  x
To get the complete normal distribution PDF
2
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Add adjustable mean
mean, μ
Add adjustable standard deviation, σ
Normalize the curve – area under curve equals to 1
2
2
1
e ( x   ) / 2
2 
Normal distribution, PDF: P( x) 
Normal distribution PDF
0.4
0.14
P(X)
Mean: =0
Std. dev: =1
0.35
Normal distribution PDF
P(X)
Mean: =20
Std. dev: =3
0.12
0.3
0.1
0.25
P(x)
P(x)
0.08
0.2
0.06
0.15
0.04
0.1
0.02
0.05
0
-5
-4
-3
-2
-1
0
x
1
2
3
0
4
5
5
10
15
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Cumulative distribution function
 Integral of PDF gives probability of
samples being within a certain range
Integral
g of the PDF is called cumulative
distribution function, CDF
High resolution numerical integration of
the Gaussian PDF, e.g. -10σ≤x≤10σ
Store results in table for looking up
probabilities (see next slide)
Φ(x) (phi) is the probability that the
value of a normally distributed signal at a
randomly chosen time will be less than x
(expressed in standard deviations
referenced to the mean).
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Cumulative distribution function
For example, Φ(-1)
has a value of
0.1587, indicating
that there is a
15.87% probability
that the value of
the signal will be
between -∞ and
one standard
deviation below
the mean, at any
randomly chosen
time.
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Gaussian Mixture Model
(C. K. Raut, 2004)
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Gaussian Mixture Model
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Mixture weight πj has a form of ’a priori’ probability:
shows relative importance of each component
Component Gaussians can have full, diagonal or
spherical covariance matrix
(C. K. Raut, 2004)
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Part II: Digital noise generation
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The normal distribution
Digital noise generation
Precision and accuracy of data acquisition
Data acquisition system
Instrumentation and data acquisition,V, 2010
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5
Random noise
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An important topic in both electronics and
DSP.
It limits how small of signals an instrument
will measure.
Digital noise generation is required to test the
performance of algorithms that must work in
the presence of noise.

It relies on random number generator
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From a uniform distribution to a Gaussian
(1)
(2)
Adding more and more iid random
variables yields a normal distribution.
To generate a normally distributed noise
signal with arbitrary mean and σ, for
each sample in the signal:
1) Add 12 random numbers
2) Subtract 6 to make the mean zero
3) Multiply by desired σ
4) Add your desired mean
(3)
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Central limit theorem
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Central limit theorem is the mathematical basis for
the aforementioned algorithm (procedure) that
y distributed noise.
creates a normally
Central limit theorem: A sum of random numbers
becomes normally distributed as more and more of
the random numbers are added together.
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This also explains why normally distributed signals are
seen so widely in nature.
So when many different random forces are interacting,
So,
interacting the
resulting PDF is Gaussian/Normal; and this happens often
in real physical systems.
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Part III: Precision and accuracy
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The normal distribution
Digital noise generation
Precision and accuracy of data acquisition
Data acquisition system
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Precision and accuracy
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Precision and accuracy are terms used to describe
systems and methods that measure, estimate or predict
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Objective: Get TRUE value of parameter
Reality: Get MEASURED value of parameter
Discrepancy between true and measured value can be
quantified by accuracy and precision.
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Accuracy: The mean of the measurements are shifted from the
true value. The amount of shift is called accuracy of
measurement.
Precision: The standard deviation of the measurements indicates
the precision, i.e. a slender or broad distribution indicates good or
bad precision respectively.
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Precision and accuracy – cont.
From: S. W. Smith, “The Scientist and Engineer's
Guide to Digital
Signal
Processing”,2010
California Technical Publishing,16
1997
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Precision and accuracy – cont.
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Situation: Good accuracy & poor precision
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Cause of poor precision
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Shift in mean from true value is small
The histogram
g
of measurements is broad
Measurements as group: Good
Measurements as individuals: Poor
Poor repeatability: Measurements do not agree well
Improve precision: Average several measurements
R d
Random
errors
Poor precision indicates

Presence of random noise
From: S. W. Smith, “The Scientist and Engineer's
Guide to Digital
Signal
Processing”,2010
California Technical Publishing,17
1997
Instrumentation
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Precision and accuracy – cont.
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Situation: Poor accuracy & good precision
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Shift in mean from true value is large
The histogram
g
of measurements is slender
Successive readings
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Average several measurements
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Does not improve accuracy
Cause of poor accuracy
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Close in value
All have a large error
Systematic errors
Poor accuracy indicates
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Poor calibration
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Precision and accuracy – cont.
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Identification of problem
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1) Avg. successive readings yield better
measurement?
t?
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Yes: The error is due to precision
No: The error is due to accuracy
2) Calibration corrects the error?
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Yes: The error is due to accuracy
No: The error is due to precision
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Part IV: Data acquisition system
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The normal distribution
Digital noise generation
Precision and accuracy of data acquisition
Data acquisition system
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Anatomy of a data acquisition experiment
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System setup
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Calibration of hardware
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Install the hardware and software
Attach sensors
Provide a known input to the system and record the output.
The rest is to be done by software
Trials
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Acquire data
To achieve a precise, accurate measurement, you need to
perform several data acquisition trials using different
hardware and software configurations.
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The data acquisition system
From MATLAB Data Acquisition Toolbox User's Guide
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Data acquisition hardware
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Multifunction boards
From MATLAB Data Acquisition Toolbox User's Guide
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Making quality measurements
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Matching the sensor range and A/D converter
range
Sampling rate
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Making quality measurements
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Matching the sensor range and A/D converter
range
Sampling rate
Accuracy and precision
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Noise removal
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Summary
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The normal distribution
Digital noise generation
Precision and accuracy of data acquisition
Data acquisition system
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