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Hamid R. Rabiee
Stochastic Processes
Elements of Stochastic Processes
Lecture II
Fall 2014
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.
Outline
Basic Definitions
Statistics of Stochastic Processes
Stationary/Ergodic Processes
Stochastic Analysis of Systems
Power Spectrum
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.
,...,...,,, 21 nXXXtX
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
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?)
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
Basic Definitions (cont’d)
Example I
Brownian Motion
Motion of all particles (ensemble)
Motion of a specific particle (sample path)
Basic Definitions (cont’d)
Example II
Voltage of a generator with fixed frequency
Amplitude is a random variable
tAtV cos.,
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
Statistics of Stochastic Processes
First-Order CDF of a random process
First-Order PDF of a random process
xtXtxFX Pr,
txFx
txf XX ,,
Statistics of Stochastic Processes (cont’d)
Second-Order CDF of a random process
Second-Order PDF of a random process
22112121 Pr,;, xtXandxtXttxxFX
212121
2
2121 ,;,.
,;, ttxxFxx
ttxxf XX
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 ,;,,
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
Statistics of Stochastic Processes (cont’d)
Autocovariance of a random process
Correlation Coefficient
Example
2211
212121
.
.,,
ttXttXE
ttttRttC
2221112
21 ,,2, ttRttRttRtXtXE
2211
2121
,.,
,,
ttCttC
ttCttr
End of Section 1
Any Question?
Outline
Basic Definitions (cont’d)
Stationary/Ergodic Processes
Stochastic Analysis of Systems
Power Spectrum
Basic Definitions (cont’d)
Example
Poisson Process
Mean
Autocorrelation
Autocovariance
kt
tk
etK .
!
.
tt .
21212
1
21212
221
...
...,
ttttt
tttttttR
2121 ,min., ttttC
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
tYitXtZ .
tX tY
Basic Definitions (cont’d)
Cross-Correlation
Orthogonal Processes
Cross-Covariance
Uncorrelated Processes
2121 ., tYtXEttRXY
212121 .,, ttttRttC YXXYXY
0, 21 ttRXY
0, 21 ttCXY
Basic Definitions (cont’d)
a-dependent processes
White Noise
Variance of Stochastic Process
0, 2121 ttCatt
0, 2121 ttCtt
tttC X2,
Basic Definitions (cont’d)
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
Outline
Basic Definitions
Stationary/Ergodic Processes
Stochastic Analysis of Systems
Power Spectrum
Stationary/Ergodic 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
…
xftxf XX ,
;,,;, 21212112 xxfttxxftt XX
Stationary/Ergodic 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 .
Stationary/Ergodic Processes (cont’d)
Autocovariance of a WSS process
Correlation Coefficient
2 RC
0C
Cr
Stationary/Ergodic Processes (cont’d)
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”
.qC
aC 0
Stationary/Ergodic Processes (cont’d)
Example
SSS
Suppose a and b are normal random variables
with zero mean.
WSS
Suppose “ ” has a uniform distribution in the
interval
tbtatX sin.cos.
tatX cos.
,
Stationary/Ergodic Processes (cont’d)
Example
Suppose for a WSS process
X(8) and X(5) are random variables
eAR .
3200
8.525858222
RRR
XXEXEXEXXE
Stationary/Ergodic Processes (cont’d)
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
Stationary/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
Outline
Basic Definitions
Stationary/Ergodic Processes
Stochastic Analysis of Systems
Power Spectrum
Stochastic Analysis of Systems
Linear Systems
Time-Invariant Systems
Linear Time-Invariant Systems
Where h(t) is called impulse response of the system
babtxaSystembtyatxSystemty ,..
txSystemtytxSystemty
txthtytxSystemty *
Stochastic Analysis of Systems (cont’d)
Memoryless Systems
Causal Systems
Only causal systems can be realized. (Why?)
00 tttxSystemty
00 tttxSystemty
Stochastic Analysis of Systems (cont’d)
Linear time-invariant systems
Mean
Autocorrelation
txEthtxthEtyE **
2*
21121 *,*, thttRthttR xxyy
Stochastic Analysis of Systems (cont’d)
Example I
System:
Impulse response:
Output Mean:
Output Autocovariance:
t
dxty
0
).(
tUdtttht
0
).(
t
dttxEtyE
0
.
0 0
2121 ..,, ddttRttR xxyy
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
Outline
Basic Definitions
Stationary/Ergodic Processes
Stochastic Analysis of Systems
Power Spectrum
Power Spectrum
Definition
WSS process
Autocorrelation
Fourier Transform of autocorrelation
deRS j.
tX
R
Power Spectrum (cont’d)
Inverse trnasform
For real processes
deSR j.2
1
0
0
.cos.1
.cos.2
dSR
dSS
Power Spectrum (cont’d)
For a linear time invariant system
Fact (Why?)
xx
xxyy
SH
HSHS
.
..
2
*
dHStyVar xx
2.
2
1
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
Power Spectrum (cont’d)
Example II
System
Impulse Response
Power Spectrum
txdt
dty
.iH
xxyy SS .2
Acknowledgement
Thank to A. Jalali and M. H. Rohban
Next Lecture
Markov Processes
&
Markov Chains
