Stochastic Processesce.sharif.edu/courses/90-91/1/ce695-1/resources/root...S. Karlin and H. M....
Transcript of Stochastic Processesce.sharif.edu/courses/90-91/1/ce695-1/resources/root...S. Karlin and H. M....
Hamid R. Rabiee
Stochastic Processes
Elements of Stochastic Processes
Lecture II
with thanks to Ali Jalali
1
Overview
Reading Assignment
Chapter 9 of textbook
Further Resources
MIT Open Course Ware S. Karlin and H. M. Taylor, A First Course in Stochastic
Processes, 2nd ed., Academic Press, New York, 1975.
2 Stochastic Processes
Outline
Basic Definitions
Statistics of Stochastic Processes
Stationary Processes
Stochastic Analysis of Systems
Power Spectrum
Ergodic Processes
3 Stochastic Processes
Basic Definitions
Suppose a set of random variables indexed by a parameter
Tracking these variables with respect to the parameter constructs a process that is called Stochastic Process.
i.e. The mapping of outcomes to the real (complex) numbers changes with respect to index.
X (t , β )= (X 1(β), X 2(β), .. . , X n(β ), . ..)
4 Stochastic Processes
Basic Definitions (cont’d)
A stochastic process x(t) is a rule for assigning
to every a function x(t, ).
ensemble of functions: Family of all functions a
random process generates
5 Stochastic Processes
Basic Definitions (cont’d)
With fixed (zeta), we will have a
“time” function called sample path.
Are sample paths sufficient for
estimating stochastic properties of a
random process?
Sometimes stochastic properties of a
random process can be extracted just
from a single sample path. (When?)
6 Stochastic Processes
Basic Definitions (cont’d)
With fixed “t”, we will have a random variable.
With fixed “t” and “zeta”, we will have a real (complex)
number
7 Stochastic Processes
Basic Definitions (cont’d)
Example I
Brownian Motion
Motion of all particles (ensemble)
Motion of a specific particle (sample path)
8 Stochastic Processes
Basic Definitions (cont’d)
Example II
Voltage of a generator with fixed frequency
Amplitude is a random variable
tAtV cos.,
9 Stochastic Processes
Basic Definitions (cont’d)
Equality
Ensembles should be equal for each “beta” and “t”
Equality (Mean Square Sense)
If the following equality holds
Sufficient in many applications
,, tYtX
0,,2 tYtXE
10 Stochastic Processes
Statistics of Stochastic Processes
First-Order CDF of a random process
First-Order PDF of a random process
txFx
txf XX ,,
xtXtxFX Pr,
11 Stochastic Processes
Statistics of Stochastic Processes (cont’d)
Second-Order CDF of a random process
Second-Order PDF of a random process
212121
2
2121 ,;,.
,;, ttxxFxx
ttxxf XX
22112121 Pr,;, xtXandxtXttxxFX
12 Stochastic Processes
Statistics of Stochastic Processes (cont’d)
nth order can be defined. (How?)
Relation between first-order and second-
order can be presented as
Relation between different orders can be
obtained easily. (How?)
ttxftxf XX ,;,,
13 Stochastic Processes
Statistics of Stochastic Processes (cont’d)
Mean of a random process
Autocorrelation of a random process
Fact: (Why?)
dxtxfxtXEt X ,.
212121212121 ,;,..., dxdxttxxfxxtXtXEttR X
2)(
2)(, tXtXttR
14 Stochastic Processes
Statistics of Stochastic Processes (cont’d)
Autocovariance of a random process
Correlation Coefficient
Example
2211
212121
.
.,,
ttXttXE
ttttRttC
2211
2121
,.,
,,
ttCttC
ttCttr
2221112
21 ,,2, ttRttRttRtXtXE
15 Stochastic Processes
End of Section 1
Any Question?
16 Stochastic Processes
Outline
Basic Definitions (cont’d)
Stationary Processes
Stochastic Analysis of Systems
Power Spectrum
Ergodic Processes
17 Stochastic Processes
Basic Definitions (cont’d)
Example
Poisson Process
Mean
Autocorrelation
Autocovariance
21212
1
21212
221
...
...,
ttttt
tttttttR
2121 ,min., ttttC
tt .
kt
tk
etK .
!
.
18 Stochastic Processes
Basic Definitions (cont’d)
Complex process
Definition
Specified in terms of the joint statistics of
two real processes and
Vector Process
A family of some stochastic processes
tX tY
tYitXtZ .
19 Stochastic Processes
Basic Definitions (cont’d)
Cross-Correlation
Orthogonal Processes
Cross-Covariance
Uncorrelated Processes
2121 ., tYtXEttRXY
0, 21 ttRXY
212121 .,, ttttRttC YXXYXY
0, 21 ttCXY
20 Stochastic Processes
Basic Definitions (cont’d)
a-dependent processes
White Noise
Variance of Stochastic Process
0, 2121 ttCatt
0, 2121 ttCtt
tttC X2,
21 Stochastic Processes
22
Basic Definitions (cont’d)
Normal Process
A Process is called Normal, if the random
variables are jointly normal for any
and .
Existence Theorem
For an arbitrary mean function
For an arbitrary covariance function
There exist a normal random process that its
mean is and its covariance is
t
t
21, ttC
21, ttC
tx
)(,..., 21 txtx n
21,...,tt
Stochastic Processes
Outline
Basic Definitions
Stationary Processes
Stochastic Analysis of Systems
Power Spectrum
Ergodic Processes
23 Stochastic Processes
Stationary Processes
Strict Sense Stationary (SSS)
Statistical properties are invariant to shift of time
origin
First order properties should be independent of “t”
or
Second order properties should depends only on
difference of times or
…
;,,;, 21212112 xxfttxxftt XX
xftxf XX ,
24 Stochastic Processes
Stationary Processes (cont’d)
Wide Sense Stationary (WSS)
Mean is constant
Autocorrelation depends on the difference of times
First and Second order statistics are usually
enough in applications.
tXE
RtXtXE .
25 Stochastic Processes
Autocovariance of a WSS process
Correlation Coefficient
Stationary Processes (cont’d)
2 RC
0C
Cr
26 Stochastic Processes
White Noise
If white noise is an stationary process, why do
we call it “noise”? (maybe it is not stationary !?)
a-dependent Process
a is called “Correlation Time”
Stationary Processes (cont’d)
.qC
aC 0
27 Stochastic Processes
Example
SSS
Suppose a and b are normal random variables
with zero mean.
Why is it SSS?
WSS
Suppose “ ” has a uniform distribution in the
interval
Why is it WSS?
Stationary Processes (cont’d)
tbtatX sin.cos.
tatX cos.
,
28 Stochastic Processes
Example
Suppose for a WSS process
X(8) and X(5) are random variables
Stationary Processes (cont’d)
3200
8.525858222
RRR
XXEXEXEXXE
eAR .
29 Stochastic Processes
End of Section 2
Any Question?
30 Stochastic Processes
Outline
Basic Definitions
Stationary Processes
Stochastic Analysis of Systems
Power Spectrum
Ergodic Processes
31 Stochastic Processes
Stochastic Analysis of Systems
Linear Systems
Time-Invariant Systems
Linear Time-Invariant Systems
Where h(t) is called impulse response of the system
babtxagbtyatxgty ,..
txgtytxgty
txthtytxgty *
32 Stochastic Processes
Stochastic Analysis of Systems (cont’d)
Memoryless Systems
Causal Systems
Only causal systems can be realized. (Why?)
00 txgty
00 tttxgty
33 Stochastic Processes
Stochastic Analysis of Systems (cont’d)
Linear time-invariant systems
Mean
Autocorrelation
2
*
21121 *,*, thttRthttR xxyy
txEthtxthEtyE **
34 Stochastic Processes
Stochastic Analysis of Systems (cont’d)
Example I
System:
Impulse response:
Output Mean:
Output Autocovariance:
tUdtttht
0
).(
t
dxty
0
).(
t
dttxEtyE
0
.
0 0
2121 ..,, ddttRttR xxyy
35 Stochastic Processes
Stochastic Analysis of Systems (cont’d)
Example II
System:
Impulse response:
Output Mean:
Output Autocovariance:
txdt
dty
tdt
dth
txEdt
dtyE
2121
2
21 ,.
, ttRtt
ttR xxyy
36 Stochastic Processes
Outline
Basic Definitions
Stationary Processes
Stochastic Analysis of Systems
Power Spectrum
Ergodic Processes
37 Stochastic Processes
Power Spectrum
Definition
WSS process
Autocorrelation
Fourier Transform of autocorrelation
R
tX
deRS j.
38 Stochastic Processes
Power Spectrum (cont’d)
Inverse trnasform
For real processes
deSR j.2
1
0
0
.cos.1
.cos.2
dSR
dSS
39 Stochastic Processes
Power Spectrum (cont’d)
For a linear time invariant system
Fact (Why?)
xx
xxyy
SH
HSHS
.
..
2
*
dHStyVar xx
2.
2
1
40 Stochastic Processes
Power Spectrum (cont’d)
Example I (Moving Average)
System
Impulse Response
Power Spectrum
Autocorrelation
Tt
Tt
dxT
ty .2
1
.
.sin
T
TH
22
2
.
.sin
T
TSS xxyy
T
T
xxyy dRTT
R2
2
..2
12
1
41 Stochastic Processes
Power Spectrum (cont’d)
Example II
System
Impulse Response
Power Spectrum
txdt
dty
.iH
xxyy SS .2
42 Stochastic Processes
Outline
Basic Definitions
Stationary Processes
Stochastic Analysis of Systems
Power Spectrum
Ergodic Processes
43 Stochastic Processes
Ergodic Processes
Ergodic Process
Equality of time properties and statistic properties.
First-Order Time average
Defined as
Mean Ergodic Process
Mean Ergodic Process in Mean Square Sense
T
T
Tt dttXT
tXtXE2
1lim
tXEtXE
02
1.1
lim02
0
2
T
XTt dT
CT
tXE
44 Stochastic Processes
Ergodic Processes (cont’d)
Slutsky’s Theorem
A process X(t) is mean-ergodic iff
Sufficient Conditions
a)
b)
01
lim
0
T
T dCT
0
dC
0lim CT
45 Stochastic Processes