random signals and systems chapter 10 - ntut.edu.twhtwu/courses/fall2009rp/rp... · 2009-09-09 ·...
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RP Chapter 10 Stochastic Processes 1
Random Signals and Systems
Chapter 10
RP Chapter 10 Stochastic Processes 2
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RP Chapter 10 Stochastic Processes 3
RP Chapter 10 Stochastic Processes 4
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RP Chapter 10 Stochastic Processes 5
Note: Two kinds of Averages
(1) Ensemble average: E[X(t0)]
(2) Time average: time average of x(t,s0) over a period of time
RP Chapter 10 Stochastic Processes 6
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RP Chapter 10 Stochastic Processes 7
RP Chapter 10 Stochastic Processes 8
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RP Chapter 10 Stochastic Processes 9
RP Chapter 10 Stochastic Processes 10
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RP Chapter 10 Stochastic Processes 11
RP Chapter 10 Stochastic Processes 12
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RP Chapter 10 Stochastic Processes 13
RP Chapter 10 Stochastic Processes 14
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RP Chapter 10 Stochastic Processes 15
RP Chapter 10 Stochastic Processes 16
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RP Chapter 10 Stochastic Processes 18
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RP Chapter 10 Stochastic Processes 19
RP Chapter 10 Stochastic Processes 20
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RP Chapter 10 Stochastic Processes 21
RP Chapter 10 Stochastic Processes 22
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RP Chapter 10 Stochastic Processes 23
RP Chapter 10 Stochastic Processes 24
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RP Chapter 10 Stochastic Processes 25
RP Chapter 10 Stochastic Processes 26
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RP Chapter 10 Stochastic Processes 27
RP Chapter 10 Stochastic Processes 28
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RP Chapter 10 Stochastic Processes 29
,...2,1,0!
)]([)( )(01 01 =−
= −− mem
ttmP ttm
Mλλ
Note: The Poisson random variable, M=N(t1)-N(t0) has the PMF
RP Chapter 10 Stochastic Processes 30
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RP Chapter 10 Stochastic Processes 31
Note: the Poisson Process is memoryless. That is, the probability of an arrival
duringany instant is independent of the past history of the process.
RP Chapter 10 Stochastic Processes 32
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RP Chapter 10 Stochastic Processes 33
RP Chapter 10 Stochastic Processes 34
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RP Chapter 10 Stochastic Processes 35
RP Chapter 10 Stochastic Processes 36
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RP Chapter 10 Stochastic Processes 37
RP Chapter 10 Stochastic Processes 38
Note: The memoryless property of a Poisson process with rate can be viewed
from the exponential interarrival time as follows:λ
xnn etXxtXp λ−=>+> ]|[
Meaning: If the arrival has not occurred by time t, the additional time until
the arrival, Xn – t, has the distribution as Xn.
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RP Chapter 10 Stochastic Processes 39
RP Chapter 10 Stochastic Processes 40
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RP Chapter 10 Stochastic Processes 41
Note; Any arrival of N(t) is independently labeled either type 1 with probability p
or type 2 with probability 1-p. Such decomposition of N(t) is called Bernoulli
decomposition.
RP Chapter 10 Stochastic Processes 42
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RP Chapter 10 Stochastic Processes 43
RP Chapter 10 Stochastic Processes 44
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RP Chapter 10 Stochastic Processes 45
RP Chapter 10 Stochastic Processes 46
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RP Chapter 10 Stochastic Processes 47
Note: (1) The independent increments property of the Brownian motion process:
W(t+r) = W(t)+ [ W(t+r) - W(t) ]
Where W(t+r) - W(t) is independent of W(t)
(2) The Poisson process also holds the independent increments property.
RP Chapter 10 Stochastic Processes 48
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RP Chapter 10 Stochastic Processes 49
RP Chapter 10 Stochastic Processes 50
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RP Chapter 10 Stochastic Processes 51
RP Chapter 10 Stochastic Processes 52
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RP Chapter 10 Stochastic Processes 53
RP Chapter 10 Stochastic Processes 54
Note: The Cov [X(t1),X(t2)] reveals how much the process is likely to change over
the time period between t1 and t2.. A high covariance indicates that the
sample function is unlikely to change. A covariance near zero suggests the
rapid change.
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RP Chapter 10 Stochastic Processes 55
RP Chapter 10 Stochastic Processes 56
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RP Chapter 10 Stochastic Processes 58
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RP Chapter 10 Stochastic Processes 59
RP Chapter 10 Stochastic Processes 60
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RP Chapter 10 Stochastic Processes 61
RP Chapter 10 Stochastic Processes 62
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RP Chapter 10 Stochastic Processes 64
)()()( )()( 11xfxfxf XtXtX == +τNote: for a stationary process x(t),
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RP Chapter 10 Stochastic Processes 65
RP Chapter 10 Stochastic Processes 66
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RP Chapter 10 Stochastic Processes 67
RP Chapter 10 Stochastic Processes 68
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RP Chapter 10 Stochastic Processes 69
RP Chapter 10 Stochastic Processes 70
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RP Chapter 10 Stochastic Processes 72
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RP Chapter 10 Stochastic Processes 74
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RP Chapter 10 Stochastic Processes 76
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RP Chapter 10 Stochastic Processes 78
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RP Chapter 10 Stochastic Processes 80
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RP Chapter 10 Stochastic Processes 82
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RP Chapter 10 Stochastic Processes 83
RP Chapter 10 Stochastic Processes 84
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RP Chapter 10 Stochastic Processes 85
RP Chapter 10 Stochastic Processes 86
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RP Chapter 10 Stochastic Processes 87
RP Chapter 10 Stochastic Processes 88
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RP Chapter 10 Stochastic Processes 89
RP Chapter 10 Stochastic Processes 90
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RP Chapter 10 Stochastic Processes 91
RP Chapter 10 Stochastic Processes 92
End of Chapter 10