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

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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

Location of disturbing sources

Identification of propagation paths and velocities

©Marco Tarabini

Analysis of random data

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Application of spectral density functions:

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Determination of system properties form input data and output data Prediction of output data from input data and system properties Identification of input data from output data and system properties Specification of dynamic data for test programs

Identification of energy and noise sources

Optimum linear prediction and filtering

©Marco Tarabini

Signals: usually they are represented in time

domain

Peak-to-peak

amplitude (pk.pk)

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Max positive

value

Time

Max negative amplitude

©Marco Tarabini

Time domain analysis

AVERAGE VALUE

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1T

μx lim xt dt

T T 0

2

X(t)

1

0

-1

-2

0

0.1

0.2

Time [s]

©Marco Tarabini

0.3

Time domain analysis

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1T2

MEAN SQUARE VALUE

Ψ lim x t dt

T T 0

1T2

2

RMS VALUE

RMS Ψ lim

0 x t dt

T

T

2

(Root mean square)

2

1

X(t)

Y2 is related to the

average power of a

signal

0

-1

-2

0

©Marco Tarabini

0.05

0.1

Time [s]

0.15

0.2

RMS: root mean square

RMS is a measure of a signal’s average power. Instantaneous power delivered to a resistor is P=[V(t)]2/R.To get average power, integrate and divide by the period.

1 t 0 T

V2

1

2

Vt RMS

Pavg

R T t

R

0

Solving

for Vrms:

VRMS

1 t 0 T

2

Vt

T t

0

• An AC voltage with a given RMS value has the same heating (power) effect as a DC voltage with that same value.

• All the following voltage waveforms have the same RMS value, and should indicate 1.000 VAC on an RMS meter

Waveform

Vpeak

Vrms

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1.733 v

1.414 v

1v

1v

Sine

1.414

1

Triangle

1.733

1

Square

1

1

©Marco Tarabini

DC

1

1

All = 1 WATT

Time domain analysis

VARIANCE

10

1T

2

σ lim xt ...