Download ECE310 - Lecture 21

ECE310 – Lecture 22
Random Signal Analysis
04/25/01
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Random Signals

The only way to analyze a random
signal is through its


Autocorrelation, and
Power spectral density
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PSD


ESD: Describes how the
signal energy is distributed
in frequency
ESD: the FT of the
autocorrelation for energy
signal
 y f  H f  x f 
2
x f  X  f 
2


PSD: Describes how the
signal power is distributed in
frequency
PSD: the FT of the
autocorrelation for power
signal
G y  f   H  f  Gx  f 
2
1
2
Gx  f   lim X T  f 
T  T
XT  f  
T /2
 j 2ft


x
t
e
dt

T / 2
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The Concept of Randomness



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Random – unpredictable
No cause and effect relationship
Examples
Random signal analysis needs
knowledge from two areas


Probability
Statistics
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Probability Basics



The study of probability is the study of how to
quantitatively estimate the likelihood that an event
will occur, under certain circumstances
Developed in 18th and 19th century for estimating the
probability of winning at casino games
Difficulties in the analysis of random signals in
engineered systems


No game rules
Experimental approach: acquire and analyze the random
signal over a long period of time
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Probability of Event A

nA: the number of A events
N: the total number of events

Probability of event A

nA
Pr  A  lim
N  N
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Disjoint Events

Mutually exclusive
n A  nB
Pr  A  B   lim
 Pr  A  Pr B 
N 
N

Example: 15.2.4 on page 15-10

What’s the probability of tossing a 7 with
two dice on a single throw?
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Independent Events

The probability that both events occur
in independent trials is the product of
their probabilities.
Pr A  B  Pr A PrB

Example: 15.2.1

What is the probability of tossing 3
successive heads with a fair coin?
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Statistics


The study of description and interpretation of data
A set of data is a sequence of numerical values



Discrete random variables
Statistics is to use a few well-chosen descriptors to
characterize the random variable
Descriptors



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Mean
Variance and standard deviation
Covariance
Histogram
Probability density function
Power spectral density
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Mean

Sample mean
1
x
N

N
x
i 1
i
Expected value/Population mean
1
E x    x  lim
N  N



N
x
i 1
i
Sample mean is an estimation of population mean
Example (brighter/darker)
MATLAB: mean()
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Variance and STD


Mean indicates the center of gravity
Standard deviation is the square root of
variance, indicating how far away is each
value from the center of gravity



1 N
 x  lim  xi   x 2  E X  E  X 2
N  N
i 1
MATLAB: std(), var()
mean(x1) = -5.8703e-005
mean(x2) = 8.5495e-006
std(x1) = 0.0324
std(x2) = 0.0289
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Covariance (*)

A measure of how much two random
variables vary together
 XY  EX  E X EY  EY   E XY   E X EY 
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Histogram


A graph indicating what percentage of the
time a random variable spends in various
ranges of values
x=[2 3 4 5 4 3 2 1 6 7 4 5 3 2 3 4]
Example:
hist(x)
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Probability Density Function
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
Raw histogram
1st normalization
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Divide each frequency with total number of
occurrence – relative frequency
2nd normalization

The width of the bin is approaching to zero
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