Basic Measurement Techniques
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0 x(t) Basic Statistical properties
Marco Tarabini
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Introduction
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If a physical phenomenon of interest is random, then each time history measurement x(t) of that phenomenon represents a unique set of circumstances which is not Iikely to be repeated in other independent measurements of that same phenomenon. Hence, to completely define all properties of the phenomenon, it is necessary to conceptually think in terms of all the time history measurements {x(t)} that might have been made.
For the usual case of engineering interest where the phenomenon produces continuous time history data, an infinite number of such conceptual measurements is required to fully describe the phenomenon; that is. N → ∞. lt follows that the instantaneous amplitude of the phenomenon at a specific time t1 in the future or from a different experiment cannot ho determined from an exact equation, but instead must ho defined in probabilistic terms.
©Marco Tarabini
Analysis of random data
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Basic statistical properties to describe single random records:
• Mean and mean square values
• Probability density functions
• Autocorrelation functions
• Autospectral density functions
Joint statistical properties for pairs of random processes:
• Joint probability density functions
• Cross correlation functions
• Cross spectral density functions
• Frequency response functions
• Coherence function
→
Let us deal with basic statistical properties…
©Marco Tarabini
Analysis of random data
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Common applications of probability density and distribution functions, beyond a basic probabilistic description:
• Evaluation of normality
• Indication of non linear effects
• Analysis of extreme values
Primary application of correlation:
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Detection of periodic behavior
Extraction of signals from noise
Measurement of time delays
0.06
p(x)
0.05
0.04
0.03
0.02
0.01
0
-5
-4
-3
-2
-1
0 x(t) Basic Statistical properties
Marco Tarabini
1
2
3
4
5
Introduction
2
If a physical phenomenon of interest is random, then each time history measurement x(t) of that phenomenon represents a unique set of circumstances which is not Iikely to be repeated in other independent measurements of that same phenomenon. Hence, to completely define all properties of the phenomenon, it is necessary to conceptually think in terms of all the time history measurements {x(t)} that might have been made.
For the usual case of engineering interest where the phenomenon produces continuous time history data, an infinite number of such conceptual measurements is required to fully describe the phenomenon; that is. N → ∞. lt follows that the instantaneous amplitude of the phenomenon at a specific time t1 in the future or from a different experiment cannot ho determined from an exact equation, but instead must ho defined in probabilistic terms.
©Marco Tarabini
Analysis of random data
3
Basic statistical properties to describe single random records:
• Mean and mean square values
• Probability density functions
• Autocorrelation functions
• Autospectral density functions
Joint statistical properties for pairs of random processes:
• Joint probability density functions
• Cross correlation functions
• Cross spectral density functions
• Frequency response functions
• Coherence function
→
Let us deal with basic statistical properties…
©Marco Tarabini
Analysis of random data
4
Common applications of probability density and distribution functions, beyond a basic probabilistic description:
• Evaluation of normality
• Indication of non linear effects
• Analysis of extreme values
Primary application of correlation:
•
•
•
•
•
Detection of periodic behavior
Extraction of signals from noise
Measurement of time delays
References: 35 Bendat Piersol : RANDOM DATA ch3 The Scientist and Engineer’s Guide to Digital Signal Processing ch2 (on the Internet) Bendat Piersol ENGINEERING APPLICATION OF CORRELATION AND SPECTRAL ANALYSIS ch2 www.misure.mecc.polimi.it Corsi Moschioni ©Marco Tarabini