확률 변수 및 확률과정의 기초 확률 변수 및 확률과정의 기초 6.5 Stationary Random Processes A discrete-time or continuous-time random process X(t) is stationary if ) , , ( ) , , ( 1 ) ( , ), ( 1 ) ( , ), ( 1 1 k t X t X k t X t X x x F x x F k k … … … … τ τ + + = for all time shifts τ, all k, and all choices of sample times t 1 t k times t 1 ,…,t k.
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확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
6.5 Stationary Random Processes
A discrete-time or continuous-time random process X(t)is stationary ify
),,(),,( 1)(,),(1)(,),( 11 ktXtXktXtX xxFxxFkk
…… …… ττ ++=
for all time shifts τ, all k, and all choices of sample times t1 tktimes t1,…,tk.
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Jointly stationary
1 1
( ) ( )( ), , ( ) ( ), , ( )k j
X t Y tX t X t Y t Y t′ ′… …
For two processes and ,the joint cdf's of and
k j
do not depend on the placement of the time originfor all and and all choices of sampling times
1, , k
jt t…
p g and 1, , jt t′ ′…
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
The first-order cdf of a stationary random process must be independent of time.p→ FX(t)(x) = FX(t+τ)(x) = FX (x) all t, τ.→ mX (t) = E[X(t)] = m for all t.
VAR[X(t)] = E[(X(t)-m)2] = σ2 for all t.
The second-order cdf of a stationary random process can depend only on the time difference between the p ysamples.
∵ The values of the random telegraph at the times t1,…, tk
][][][ 1121 1121 −−− −=−===− kknnnnn yySPyySPySP
kk
g p 1 k
is determined by the number of occurrences of the Poisson process in the time intervals (tj, tj+1).
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Similarly
])(,,)([ 11 =+=+ kk atXatXP ττ …
])()([
])()([])([ ])(,,)([
112211
11
=+=+×
=+=+=+=kk
atXatXP
atXatXPatXP
ττ
τττ
Conditional probability (ex 6.22)
])()([ 11 −− =+=+× kkkk atXatXP ττ
p y ( )
{ }12 ( )1
1 12[ ( ) ( ) ]
j jt tj je a a
P X t X t
α +− −+
⎧ + =⎪⎪⎨
if
{ }1
1 12 ( )
1
2[ ( ) ( ) ]1 12
j j
j j j jt t
j j
P X t a X t ae a aα +
+ +− −
+
⎪= = = ⎨⎪ − ≠⎪⎩
if
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
cf) [ ] ==±=±=∞∞
∑∑22 )()(
])([1)0(1)(
αα jtt
j tt
tNPXtXP integer even
===
−−
=∑∑
2
00
11)!2(
)()!2(
)( αα αα
tttt
j
tt
j jtee
jt
+++
+=+=
−
−−−−
22
2
1111
)1(21)(
21 αααα
αα
tttt eeee
where
cf) Poisson process
−+−=++= 22
!21,
!21 αααα αα eewhere
cf) Poisson process
tk
ektktNP λλ −==!)(])([
k!])([
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
1 1[ ( ) ( ) ]
1j j j jP X t a X t aτ τ+ ++ = + =
⎧ { }{ }
1
1
2 ( )1
2 ( )
1 121 1
j j
j j
t tj j
t t
e a a
e a a
α τ τ
α τ τ
+
+
− + − −+
− + − −
⎧ + =⎪⎪= ⎨⎪ ≠
if
if{ } 112
j jj je a a +
⎪ − ≠⎪⎩
if
The joint probabilities differ only in the first term.→ P[X(t ) ] and P[X(t + ) ]→ P[X(t1) = a1] and P[X(t1+τ) = a1]
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
cf)]1)0([]1)0(1)([]1)([ ===== XPXtXPtXP
{ } { }1111]1)0([]1)0(1)([
22
−=−==+ XPXtXP
tt{ } { }1
121
211
21
21 22 −⋅++⋅= −− ee tt αα
1]1)([
2
=−=
=
tXP
1]1)0([1])([])([
2]1)([
1111 =±===+==∴
==
XPatXPatXP
tXP
withτ2
])([2
])([])([ 1111
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
])([])([21]1)0([ 1111 =+≠=⇒≠±= atXPatXPXPIf τ
,1]1)0([ ==XPIf
[ ] 1]1)0()([)( XXPXP
The process forgets the
[ ]
{ } if 2 111
1]1)0()([)(
ae
XatXPatXP
t
⎪⎪⎧ =+
⋅====
− αThe process forgets the initial conditionand settles down into steady state
{ }
{ } if 2 11212
ae t⎪⎪⎩
⎪⎪⎨
−=−=
− αsteady state→ stationary behavior.
{ }
large becomes as 111 21])([
2
tatXP →==
⎪⎩
2
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Wide-Sense Stationary Random Processes
Cannot determine whether a random process is stationary
Can determine whethermX (t) = m for all t.CX (t1, t2) = CX (t1-t2) for all t1, t2
Function of t1- t2 only
Jointly Wide-Sense Stationary
X(t) is wide-sense stationary (WSS).
Jointly Wide Sense StationaryX(t) and Y(t) are both wide-sense stationary.Cross-Covariance depends only on t1-t2p y 1 2
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
NoteX(t) is Wide-Sense Stationary( ) y→ auto covariance CX (t1, t2) = CX (τ ) and
auto correlation RX (t1, t2) = RX (τ ) where τ = t1-t2
NoteAll stationary random processes are wide-sense stationary.All stationary random processes are wide sense stationary.Some wide-sense stationary processes are not stationary.
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Ex. 6.29Xn : Consist of two interleaved sequences of independent r.v.’s.n q p
1, [ 1]2nn P X = ± =
⎡ ⎤
For even
1 9 1, , [ 3]3 10 10n nn P X P X⎡ ⎤= = = − =⎢ ⎥⎣ ⎦
For odd
Xn is not stationary since its pmf varies with n.
X nm = 0)(
for
for
i
jiX jiXE
jiXEXEjiC
⎪⎩
⎪⎨⎧
==
≠==
1][
0][][),(
2
.Stationary Sense-Wide : nX→
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Properties of Autocorrelation Function of WSS Process
Average power of the process.RX (0) = E[X2(t)] for all t.X ( ) [ ( )]
Even function of τRX (τ ) = E[X(t +τ) X(t)] = E[X(t) X(t +τ)] = RX (-τ )
Measure of the rate of change of a random process.The change in the process from time t to t+τ :
[ ]2
22
)}()0({2]))()([(
]))()([()()(
ττ
ετετ
XX RRtXtXE
tXtXPtXtXP
−=
−+≤
>−+=>−+
22 εε=≤
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
cf) Markov inequalityXEaXP ][][ ≤≥
aaXP ][ ≤≥
Observation:If RX (0)-RX (τ ) is small, the probability of a large change in X(t)in seconds is smallin τ seconds is small.
cf) R (0)-R (τ ) is smallcf) RX (0) RX (τ ) is small → RX (τ ) drops off slowly.
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
RX (τ ) is maximum at τ = 0Proof) )
1E[XY]2 ≤ E[X2]E[Y2] for any two r.v.’s X and Y.- Can be proved using the approach used to prove |ρ|≤1. - HW
0)}()0({2]))()([( 2 dRRXdXE 0)}()0({2]))()([( 2 =−=−+ dRRtXdtXE XX
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Let X(t) = m + N(t), where N(t) is a zero-mean process for which RN (τ ) → 0 as τ → ∞, N ( ) ,then
( ) [( ( ))( ( ))]R E m N t m N tτ τ= + + +2
2 2
( ) [( ( ))( ( ))]
2 [ ( )] ( )
( )
X
N
R E m N t m N t
m mE N t R
m R m
τ τ
τ
τ τ
= + + +
= + +
= + → → ∞as
Note
( ) Nm R mτ τ= + → → ∞as
NoteRX (τ ) approaches the square of the mean of X(t) as τ → ∞.
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Summary: Three type of components
∞→→ ττ as0)(R①
+=
∞→→
ττ
ττ as
22
1
)()(
0)(
dRR
R
XX
X
②
①
∞→→ ττ as 23 )( mRX③
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
WSS Gaussian Random Processes
If a Gaussian random process is wide-sense stationary, then it is also stationary.
Proof) The joint pdf of a Gaussian random process is completelyThe joint pdf of a Gaussian random process is completely determined by the mean mX (t) and autocovariance CX (t1, t2).
X(t) is wide sense stationary → its mean is constant its autocovariance is only the function of the difference of the sampling times t-t → the joint pdf of X(t) depends only onsampling times ti tj → the joint pdf of X(t) depends only on this set of differences → invariant with respect to time shiftsThus the process is also stationary
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Cyclostationary Random Processes
)(
),,,( 21)(,),(),( 21 ktXtXtX
xxxF
xxxFk
……
For all k, m and all choices of sampling times t1,…,tk
),,,( 21)(,),(),( 21 kmTtXmTtXmTtX xxxFk
…… +++=
Wide-Sense Cyclostationary.f h d i f i i i i h:If the mean and autocovariance functions are invariant with
respect to shifts in the time origin by integer multiples of T
)()( tTt),(),(
)()(
2121 ttCmTtmTtCtmmTtm
XX
XX
=++=+
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
NoteIf X(t) is cyclostationary, then X(t) is also wide-sense
l icyclostationary.
X(t) is a cyclostationary process with period TX(t) is a cyclostationary process with period T.→ X(t) is stationarized by observing a randomly phase-shifted version of X(t)shifted version of X(t)XS (t) = X(t + Θ), Θ uniform in [0, T],
where Θ is independent of X(t).where Θ is independent of X(t).→ If X(t) is a cyclostationary, XS (t) is a stationary random process.p
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
If X(t) is a wide-sense cyclostationary random process, then XS (t) is a wide-sense stationary random processS ( ) y p
∫=T
Xs dttmT
tXE0
)(1)]([
∫
∫
+=T
XX dtttRT
R
T
s 0
0
),(1)( ττ ∫Ts 0
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
6.6 Continuity, Derivatives and Integrals of Random Processes
The system having dynamics: described by linear differential eqs.Each sample function of a random process: deterministic signalInput to the system: Sample function of continuous-time random
process O t t f th t A l f ti f th dOutput of the system: A sample function of another random
process Probabilistic methods for addressing the continuityProbabilistic methods for addressing the continuity, differentiability and integrability of random processes
cf) A random process: the ensemble of sample functions ) p p
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Mean Square Continuity
X(t, ζ) : A particular deterministic sample function for each point ζ in S of random processp ζ p
The continuity of the sample function at a point t0 forThe continuity of the sample function at a point t0 for each point ζ :If given any ε > 0 there exists a δ > 0 such that |t-t0| < δIf given any ε > 0 there exists a δ > 0 such that |t t0| < δimplies that |X(t, ζ)-X(t0, ζ)| < ε
)()(lim ζζ tXtX ),(),(lim 00
ζζ tXtXtt
=→
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
All sample functions of the random process are continuous at t0, then the random process is continuous0, pThe continuity of random process in a probabilistic sense is considered.Mean square continuity: The random process X(t) is continuous at the point t0 in
)()(l.i.m. 00
tXtXtt
=→
The random process X(t) is continuous at the point t0 in the mean square sense if
02
0 0]))()([( tttXtXE →→− as
Note: Mean square continuity does not imply that all the sample functions are continuous
00 0]))()([( tttXtXE →→ as
sample functions are continuous
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Considering the mean square difference:),(),(),(),(]))()([( 0000
20 ttRttRttRttRtXtXE XXXX +−−=−
Therefore, if RX (t1 t2) is continuous in both t1 and t2 at
Therefore, if RX (t1, t2) is continuous in both t1 and t2 at the point (t0, t0), then X(t) is mean square continuous at the point t0.p 0
If X(t) is mean square continuous at t0, then the mean (t) s ea squa e co t uous at t0, t e t e eafunction mX (t) must be continuous at t0.
)()(lim 0tmtm XX = )()( 00
XXtt→
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Proof2
02
00 )]()([]))()([()]()([VAR0 tXtXEtXtXEtXtX −−−=−≤
If X(t) is mean square continuous L H S → 0 as t → t then
20
20
20 )]()([)]()([]))()([( tmtmtXtXEtXtXE XX −=−≥−∴
If X(t) is mean square continuous, L.H.S. → 0 as t → t0, then R.H.S. → 0, i.e., mX (t) → mX (t0)
Note: If X(t) is mean square continuous at t0, then we can interchange the order of the limit and the gexpectation
⎥⎦⎤
⎢⎣⎡= )(l.i.m.)]([lim tXEtXE ⎥⎦⎢⎣ →→
)()]([00 tttt
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
For the WSS random process X(t),))()0((2]))()([( 2 ττ RRtXtXE −=−+
: If RX (τ ) is continuous at τ = 0, then the WSS random
))()0((2]))()([( 00 ττ XX RRtXtXE =+
X ( ) ,process X(t) is mean square continuous at every point t0.
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Mean Square Derivatives
The derivative of a deterministic functionζζε ),(),(li tXtX −+
: this limit may exist for some sample functions and it may fail to ε
ζζε
),(),(lim0→
exist for other sample functions
Mean Square DerivativetdXtXtX )()()( ζζ
dttdXtXtXtX )(),(),(l.i.m.)(
0≡
−+≡′
→ εζζε
ε
Provided that the mean square limit exists, that is,
0)(),(),(lim2
=⎥⎤
⎢⎡
⎟⎞
⎜⎛ ′−
−+ tXtXtXE ζζε 0)(lim0 ⎥
⎥⎦⎢
⎢⎣
⎟⎠
⎜⎝→
tXEεε
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Note: The existence of the mean square derivative does not imply the existence of the derivative for all sample p y pfunctions.
The mean square derivative of X(t) at the point t exists if
)(2
ttR∂
exists at the point (t1, t2) = (t, t)
),( 2121
ttRtt X∂∂
e sts at t e po t (t1, t2) (t, t)
Proof) Use the Cauchy criterion) y
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
If the random process X(t) is WSS
)()(22
ttRttR ∂∂ )(),(
2
2121
2121
XX
d
ttRtt
ttRtt
∂⎞⎛∂
−∂∂
=∂∂
)()( 2211
τττ XX RttR
dd
t ∂∂
−=⎟⎠⎞
⎜⎝⎛ −−
∂∂
=
The mean square derivative of a WSS random process X(t) exists if RX(τ ) has derivatives up to order two at τ=0.
exists,if,processrandomGaussianaFor )()( tXtX ′
processrandomGaussianabemust then )(tX ′
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Mean of X’(t))()(li)()(l i)]([ tXtXEtXtXEXE ⎤⎡ −+⎤⎡ −+′ εε
)()()(li
)()(lim)()(l.i.m.)]([00
tdtmtm
tXtXEtXtXEtXE
XX −+
⎥⎦⎤
⎢⎣⎡ +
=⎥⎦⎤
⎢⎣⎡ +
=′→→
εε
εε
εεε
)()()(lim0
tmdt X
XX ==→ εε
The cross-correlation between X(t) and X’(t))()(l i m)()( 22 tXtXtXEttR ⎥⎤
⎢⎡ −+
=ε
)(),(),(lim
l.i.m.)(),(
2121
0121,
ttRttRttR
tXEttR
XX
XX
∂=
−+=
⎥⎦⎢⎣=
→′
εεε
),(lim 212
0ttR
t X∂==
→ εε
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
The autocorrelation of X’(t)
)()( tXtX ⎤⎡ ⎫⎧ + ε
)()(
)()()(l.i.m.),( 211
021
ttRttR
tXtXtXEttRX
+
⎥⎦
⎤⎢⎣
⎡ ′⎭⎬⎫
⎩⎨⎧ −+
=→
′
εε
εε
)(),(lim 2,1,21,
0
ttRttR XXXX
∂
−+= ′′
→ εε
ε
),(
2
21,1
ttRt XX
∂
∂∂
= ′
),( 2121
ttRtt X∂∂
∂=
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
For the WSS random process X(t)
∂∂ )()()( 212
, ττ
τ XXXX RttRt
R
⎫⎧
∂∂
−=−∂∂
=′
)()()( 2
2
2121
ττ
τ XXX RttRtt
R∂∂
−=⎭⎬⎫
⎩⎨⎧
−∂∂
∂∂
=′
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Mean Square Integrals
Imply the integral of a random process in the sense of mean square convergenceq g
The integral of the random process X(t)The integral of the random process X(t)The mean square limit of the sequence In as the width of the subintervals approaches zero:subintervals approaches zero:
∑=
Δ=n
kkkn tXI
1)(
∑∫ Δ=′′=→Δ k
kk
t
ttXtdtXtY
k
)(l.i.m.)()(00
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Conditions that ensure the existence of the mean square integral
0, 0)()(2
→ΔΔ→⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
Δ−Δ∑ ∑ kjj k
kkjj tXtXE as
:The Cauchy criterion
⎥⎦⎢⎣ ⎭⎩ j k
Expanding the square inside the expected value
⎤⎡∑∑∑∑ ΔΔ=⎥
⎦
⎤⎢⎣
⎡ΔΔ
j kkjkjXkj
j kkj ttRtXtXE ),()()(
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
The limit of the right hand side approaches a double integral
∫ ∫∑∑ =ΔΔt t
kk dudvvuRttR )()(lim ∫ ∫∑∑ =ΔΔ→ΔΔ t t X
j kkjkjX dudvvuRttR
kj 0 0
),(),(lim0,
The mean square integral of X(t) exists if the double integral of the autocorrelation function exists
If X(t) is a mean square continuous random process, then its integral exists.then its integral exists.
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
The mean and autocorrelation function of Y(t)
[ ]∫∫ ′′⎤⎡ ′′tt
tdtXEtdtXEt )()()( [ ]
∫
∫∫′′=
′′=⎥⎦⎤
⎢⎣⎡ ′′=
t
X
ttY
tdtm
tdtXEtdtXEtm00
)(
)()()(
∫t X tdtm0
)(
∫∫ ⎥⎤
⎢⎡= 21 )()()(
ttdvvXduuXEttR
∫ ∫
∫∫=
⎥⎦⎢⎣=
1 2
00
),(
)()(),( 21
t
t
t
t X
ttY
dudvvuR
dvvXduuXEttR
∫ ∫0 0t t X
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
6.7 Time Averages of Random Processes and Ergodic Theorems
To estimate the mean mX (t) of a random process X(t, ζ),
∑N
X )(1)(ˆ ξ
where N is the number of repetitions of the experiment
∑=
=i
iX tXN
tm1
),()(ˆ ξ
where N is the number of repetitions of the experiment
I ti ti th t l ti f ti fIn estimating the mean or autocorrelation functions from the time average of a single realization
∫−=
T
TTdttX
TtX ),(
21)( ξ
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
Ergodic theorems : when time averages converge to the ensemble average (expected value)g ( p )cf) Strong law of large numbers :
: if Xn is an iid discrete-time random process with finite mean E[Xn] = m, then
11lim =⎥⎤
⎢⎡
=∑n
i mXP
i ld i l b
1lim1
⎥⎦
⎢⎣
∑=
∞→ iin
mXn
P
∫T
dttXtX )(1)( ξ yields a single number ⇒ consider process for which mX (t) = m
∫−=
TTdttX
TtX ),(
2)( ξ
확률 변수 및 확률과정의 기초확률 변수 및 확률과정의 기초
An ergodic theorem for the time average of wide-sense stationary processesy p