12. Stochastic Processes - California Institute of Technologyee162.caltech.edu/notes/lect12.pdf ·...
Transcript of 12. Stochastic Processes - California Institute of Technologyee162.caltech.edu/notes/lect12.pdf ·...
1
Let denote the random outcome of an experiment. To every such outcome suppose a waveform
is assigned.The collection of such waveforms form a stochastic process. The set of and the time index t can be continuousor discrete (countably infinite or finite) as well.For fixed (the set of all experimental outcomes), is a specific time function.For fixed t,
is a random variable. The ensemble of all such realizationsover time represents the stochastic
ξ
),( ξtX
}{ kξ
Si ∈ξ
),( 11 itXX ξ=
),( ξtX
12. Stochastic Processes
t1
t2
t
),(n
tX ξ
),(k
tX ξ
),(2
ξtX
),(1
ξtX
Fig. 12.1
),( ξtX
0
),( ξtX
Introduction
2
process X(t). (see Fig 12.1). For example
where is a uniformly distributed random variable in represents a stochastic process. Stochastic processes are everywhere:Brownian motion, stock market fluctuations, various queuing systemsall represent stochastic phenomena.
If X(t) is a stochastic process, then for fixed t, X(t) representsa random variable. Its distribution function is given by
Notice that depends on t, since for a different t, we obtaina different random variable. Further
represents the first-order probability density function of the process X(t).
),cos()( 0 ϕω += tatX
ϕ
})({),( xtXPtxFX
≤=
),( txFX
(12-1)
(12-2)
(0,2 ),π
dx
txdFtxf X
X
),(),( =∆
3
For t = t1 and t = t2, X(t) represents two different random variablesX1 = X(t1) and X2 = X(t2) respectively. Their joint distribution is given by
and
represents the second-order density function of the process X(t).Similarly represents the nth order densityfunction of the process X(t). Complete specification of the stochasticprocess X(t) requires the knowledge of for all and for all n. (an almost impossible taskin reality).
})(,)({),,,( 22112121 xtXxtXPttxxFX
≤≤= (12-3)
(12-4)
),, ,,,( 2121 nn tttxxxfX
),, ,,,( 2121 nn tttxxxfX
niti , ,2 ,1 , =
21 2 1 2
1 2 1 21 2
( , , , )( , , , )
X
X
F x x t tf x x t t
x x
∂=
∂ ∂∆
4
Mean of a Stochastic Process:
represents the mean value of a process X(t). In general, the mean of a process can depend on the time index t.
Autocorrelation function of a process X(t) is defined as
and it represents the interrelationship between the random variablesX1 = X(t1) and X2 = X(t2) generated from the process X(t).
Properties:
1.
2.
(12-5)
(12-6)
*1
*212
*21 )}]()({[),(),( tXtXEttRttR
XXXX== (12-7)
.0}|)({|),( 2 >= tXEttRXX
(Average instantaneous power)
( ) { ( )} ( , )
Xt E X t x f x t dxµ +∞
−∞= = ∫∆
* *1 2 1 2 1 2 1 2 1 2 1 2( , ) { ( ) ( )} ( , , , )
XX XR t t E X t X t x x f x x t t dx dx= = ∫ ∫
∆
5
3. represents a nonnegative definite function, i.e., for anyset of constants
Eq. (12-8) follows by noticing that The function
represents the autocovariance function of the process X(t).Example 12.1Let
Then
.)(for 0}|{|1
2 ∑=
=≥n
iii tXaYYE
)()(),(),( 2*
12121 ttttRttCXXXXXX
µµ−= (12-9)
.)(
∫−=
T
TdttXz
∫ ∫
∫ ∫
− −
− −
=
=T
T
T
T
T
T
T
T
dtdtttR
dtdttXtXEzE
XX
2121
212*
12
),(
)}()({]|[|
(12-10)
niia 1}{ =
),( 21 ttRXX
∑∑= =
≥n
i
n
jjiji ttRaa
XX
1 1
* .0),( (12-8)
6
Similarly
,0}{sinsin}{coscos
)}{cos()}({)(
0 0
0
=−=+==
ϕωϕωϕωµ
EtaEta
taEtXEtX
).(cos2
)}2)(cos()({cos2
)}cos(){cos(),(
210
2
210210
2
20102
21
tta
ttttEa
ttEattRXX
−=
+++−=
++=
ω
ϕωω
ϕωϕω
(12-12)
(12-13)
Example 12.2
).2,0(~ ),cos()( 0 πϕϕω UtatX += (12-11)
This gives
∫ ===π ϕϕϕϕ π
2
0 }.{sin0cos}{cos since 2
1 EdE
7
Stationary Stochastic ProcessesStationary processes exhibit statistical properties that are
invariant to shift in the time index. Thus, for example, second-orderstationarity implies that the statistical properties of the pairs {X(t1) , X(t2) } and {X(t1+c) , X(t2+c)} are the same for any c. Similarly first-order stationarity implies that the statistical properties of X(ti) and X(ti+c) are the same for any c.
In strict terms, the statistical properties are governed by thejoint probability density function. Hence a process is nth-orderStrict-Sense Stationary (S.S.S) if
for any c, where the left side represents the joint density function of the random variables andthe right side corresponds to the joint density function of the randomvariables A process X(t) is said to be strict-sense stationary if (12-12) is true for all
),, ,,,(),, ,,,( 21212121 ctctctxxxftttxxxf nnnn XX+++≡
(12-12)
)( , ),( ),( 2211 nn tXXtXXtXX ===
).( , ),( ),( 2211 ctXXctXXctXX nn +=′+=′+=′
. and ,2 ,1 , , ,2 ,1 , canynniti ==
8
For a first-order strict sense stationary process,from (12-12) we have
for any c. In particular c = – t gives
i.e., the first-order density of X(t) is independent of t. In that case
Similarly, for a second-order strict-sense stationary processwe have from (12-12)
for any c. For c = – t2 we get
),(),( ctxftxfXX
+≡
(12-16)
(12-15)
(12-17)
)(),( xftxfXX
=
[ ( )] ( ) , E X t x f x dx a constant.µ+∞
−∞= =∫
), ,,(), ,,( 21212121 ctctxxfttxxfXX
++≡
) ,,(), ,,( 21212121 ttxxfttxxfXX
−≡ (12-18)
9
i.e., the second order density function of a strict sense stationary process depends only on the difference of the time indices In that case the autocorrelation function is given by
i.e., the autocorrelation function of a second order strict-sensestationary process depends only on the difference of the time indices Notice that (12-17) and (12-19) are consequences of the stochastic process being first and second-order strict sense stationary. On the other hand, the basic conditions for the first and second order stationarity – Eqs. (12-16) and (12-18) – are usually difficult to verify.In that case, we often resort to a looser definition of stationarity,known as Wide-Sense Stationarity (W.S.S), by making use of
.21 τ=− tt
.21 tt −=τ
(12-19)
*1 2 1 2
*1 2 1 2 1 2 1 2
*1 2
( , ) { ( ) ( )}
( , , )
( ) ( ) ( ),
XX
X
XX XX XX
R t t E X t X t
x x f x x t t dx dx
R t t R R
τ
τ τ
=
= = −
= − = = −∫ ∫
∆
∆
10
(12-17) and (12-19) as the necessary conditions. Thus, a process X(t)is said to be Wide-Sense Stationary if(i)and(ii)
i.e., for wide-sense stationary processes, the mean is a constant and the autocorrelation function depends only on the difference between the time indices. Notice that (12-20)-(12-21) does not say anything about the nature of the probability density functions, and instead deal with the average behavior of the process. Since (12-20)-(12-21) follow from (12-16) and (12-18), strict-sense stationarity always implies wide-sense stationarity. However, the converse is not true in general, the only exception being the Gaussian process.This follows, since if X(t) is a Gaussian process, then by definition
are jointly Gaussian randomvariables for any whose joint characteristic function is given by
µ=)}({ tXE
(12-21)
(12-20)
),()}()({ 212*
1 ttRtXtXEXX
−=
)( , ),( ),( 2211 nn tXXtXXtXX ===nttt ,, 21
11
where is as defined on (12-9). If X(t) is wide-sense stationary, then using (12-20)-(12-21) in (12-22) we get
and hence if the set of time indices are shifted by a constant c to generate a new set of jointly Gaussian random variables
then their joint characteristic function is identical to (12-23). Thus the set of random variables and have the same joint probability distribution for all n and all c, establishing the strict sense stationarity of Gaussian processes from its wide-sense stationarity.
To summarize if X(t) is a Gaussian process, thenwide-sense stationarity (w.s.s) strict-sense stationarity (s.s.s).
Notice that since the joint p.d.f of Gaussian random variables dependsonly on their second order statistics, which is also the basis
),( ki ttCXX
1 ,
( ) ( , ) / 2
1 2( , , , )XX
n n
k k i k i kk l k
X
j t C t t
n eµ ω ω ω
φ ω ω ω =
−∑ ∑∑= (12-22)
12
1 1 1 1
( )
1 2( , , , )XX
n n n
k i k i kk k
X
j C t t
n eµω ω ω
φ ω ω ω = = =
− −∑ ∑∑= (12-23)
niiX 1}{ =
niiX 1}{ =′
⇒
),( 11 ctXX +=′)(,),( 22 ctXXctXX nn +=′+=′
12
for wide sense stationarity, we obtain strict sense stationarity as well.From (12-12)-(12-13), (refer to Example 12.2), the process
in (12-11) is wide-sense stationary, butnot strict-sense stationary.
Similarly if X(t) is a zero mean wide sense stationary process in Example 12.1, then in (12-10) reduces to
As t1, t2 varies from –T to +T, variesfrom –2T to + 2T. Moreover is a constantover the shaded region in Fig 12.2, whose area is given by
and hence the above integral reduces to
),cos()( 0 ϕω += tatX
2zσ
.)(}|{|
212122 ∫ ∫− −
−==T
T
T
Tz dtdtttRzEXX
σ
21 tt −=τ)(τ
XXR
)0( >ττττττ dTdTT )2()2(
21
)2(21 22 −=−−−−
.)1)((|)|2)((2
2 2||
21
2
2
2 ∫∫ −−−=−=
T
t TT
T
tz dRdTRXXXX
τττττσ τ
(12-24)
T− T
T−τ
τ
τ−T2
2t
1t
Fig. 12.2
21tt −=τ
13
Systems with Stochastic InputsA deterministic system1 transforms each input waveform intoan output waveform by operating only on the time variable t. Thus a set of realizations at the input corresponding to a process X(t) generates a new set of realizations at the output associated with a new process Y(t).
),( itX ξ)],([),( ii tXTtY ξξ =
)},({ ξtY
Our goal is to study the output process statistics in terms of the inputprocess statistics and the system function.
1A stochastic system on the other hand operates on both the variables t and .ξ
][⋅T⎯⎯ →⎯ )(tX ⎯⎯→⎯ )(tY
t t
),(i
tX ξ),(
itY ξ
Fig. 12.3
14
Deterministic Systems
Systems with Memory
Time-Invariantsystems
Linear systems
Linear-Time Invariant(LTI) systems
Memoryless Systems
)]([)( tXgtY =
)]([)( tXLtY =Time-varying
systemsFig. 12.3
.)()(
)()()(
∫
∫∞+
∞−
∞+
∞−
−=
−=
τττ
τττ
dtXh
dXthtY( )h t( )X t
LTI system
15
Memoryless Systems:The output Y(t) in this case depends only on the present value of the input X(t). i.e.,
(12-25))}({)( tXgtY =
Memorylesssystem
Memorylesssystem
Memorylesssystem
Strict-sense stationary input
Wide-sense stationary input
X(t) stationary Gaussian with
)(τXX
R
Strict-sense stationary output.
Need not bestationary in any sense.
Y(t) stationary,butnot Gaussian with
(see (12-26)).).()( τητ
XXXYRR =
(see (9-76), Text for a proof.)
Fig. 12.4
16
Theorem: If X(t) is a zero mean stationary Gaussian process, andY(t) = g[X(t)], where represents a nonlinear memoryless device, then
Proof:
where are jointly Gaussian random variables, and hence
)(⋅g
)}.({ ),()( XgERRXXXY
′== ητητ (12-26)
212121 ),()(
)}]({)([)}()({)(
21dxdxxxfxgx
tXgtXEtYtXER
XX
XY
∫ ∫=
−=−= τττ
(12-27)
)( ),( 21 τ−== tXXtXX
* 1
1 2
/ 21 2
1 2 1 2
* *
12 | |
(0) ( )
( ) (0)
( , )
( , ) , ( , )
{ } XX XX
XX XX
X X
x A x
T T
A
R R
R R
f x x e
X X X x x x
A E X X LL
π
ττ
−−=
= =
⎛ ⎞= = =⎜ ⎟⎝ ⎠
∆
17
where L is an upper triangular factor matrix with positive diagonal entries. i.e.,
Consider the transformation
so that
and hence Z1, Z2 are zero mean independent Gaussian random variables. Also
and hence
The Jacobaian of the transformation is given by
. 0
22
1211 ⎟⎠
⎞⎜⎝
⎛=l
llL
IALLLXXELZZE ===−− −− 11 *1**1* }{}{
* * *1 * 1 2 21 2 .x A x z L A Lz z z z z− −= = = +
22222121111 , zlxzlzlxzLx =+=⇒=
1 1 1 2 1 2( , ) , ( , )T TZ L X Z Z z L x z z− −= = = =∆ ∆
18
Hence substituting these into (12-27), we obtain
where This gives
.|||||| 2/11 −− == ALJ
2 21 2
1/ 211 1 12 2 22 2
11 1 22 2 1 21 2
12 2 22 2 1 21 2
/ 2 / 21 1| | 2 | |
1 2
1 2
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
XY J A
z z
z z
z zR l z l z g l z e e
l z g l z f z f z dz dz
l z g l z f z f z dz dz
πτ +∞ +∞ − −
−∞ −∞
+∞ +∞
−∞ −∞+∞ +∞
−∞ −∞
= + ⋅
=
+
=
∫ ∫
∫ ∫
∫ ∫
22
212 22
22
11 1 1 22 2 21 2
12 2 22 2 22
/ 2
2
2
1 2
2
12
/ 212
( ) ( ) ( )
( ) ( )
( ) ,
z z
z
z
u lll
l z f z dz g l z f z dz
l z g l z f z dz
e
ug u e du
π
π
−
+∞ +∞
−∞ −∞+∞
−∞
+∞ −−∞
+
=
∫ ∫
∫
∫
0
22 2 .u l z=
19
222
222 22
2
2
( )
/ 2112 22 2
( )( )
( ) ( )
( ) ( ) ( ) ,
u
XY
uu
XX
f u
u
df uf u
du
u
l
lulR l l g u e du
R g u f u du
πτ
τ
+∞ −−∞
′− =−
+∞
−∞
=
′= −
∫
∫
Hence).( gives since 2212* τ
XXRllLLA ==
the desired result, where Thus if the input to a memoryless device is stationary Gaussian, the cross correlation function between the input and the output is proportional to theinput autocorrelation function.
),()}({)(
})()(|)()(){()(
τητ
ττ
XXXX
XXXY
RXgER
duufugufugRR uu
=′=
′+−= ∫∞+
∞−∞+∞−
0
)].([ XgE ′=η
20
Linear Systems: represents a linear system if
Let
represent the output of a linear system.Time-Invariant System: represents a time-invariant system if
i.e., shift in the input results in the same shift in the output also.If satisfies both (12-28) and (12-30), then it corresponds to a linear time-invariant (LTI) system.LTI systems can be uniquely represented in terms of their output to a delta function
][⋅L
)}({)( tXLtY =
)}.({)}({)}()({ 22112211 tXLatXLatXatXaL +=+ (12-28)
][⋅L
)()}({)}({)( 00 ttYttXLtXLtY −=−⇒=
(12-29)
(12-30)
][⋅L
LTI)(tδ )(th
Impulse
Impulseresponse ofthe system
t
)(th
Impulseresponse
Fig. 12.5
21
Eq. (12-31) follows by expressing X(t) as
and applying (12-28) and (12-30) to Thus)}.({)( tXLtY =∫
∞+
∞−−=
)()()( ττδτ dtXtX
(12-31)
(12-32)
(12-33).)()()()(
)}({)(
})()({
})()({)}({)(
∫∫
∫
∫
∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
−=−=
−=
−=
−==
ττττττ
ττδτ
ττδτ
ττδτ
dtXhdthX
dtLX
dtXL
dtXLtXLtY
By Linearity
By Time-invariance
then
LTI
∫
∫∞+
∞−
∞+
∞−
−=
−=
)()(
)()()(
τττ
τττ
dtXh
dXthtYarbitrary
input
t
)(tX
t
)(tY
Fig. 12.6
)(tX )(tY
22
Output Statistics: Using (12-33), the mean of the output processis given by
Similarly the cross-correlation function between the input and outputprocesses is given by
Finally the output autocorrelation function is given by
).()()()(
})()({)}({)(
thtdth
dthXEtYEt
XX
Y
∗=−=
−==
∫
∫∞+
∞−
∞+
∞−
µτττµ
τττµ
(12-34)
).(),(
)(),(
)()}()({
})()()({
)}()({),(
2*
21
*21
*21
*21
2*
121
thttR
dhttR
dhtXtXE
dhtXtXE
tYtXEttR
XX
XX
XY
∗=
−=
−=
−=
=
∫
∫
∫
∞+
∞−
∞+
∞−
∞+
∞−
ααα
ααα
ααα*
*
(12-35)
23
or
),(),(
)(),(
)()}()({
})( )()({
)}()({),(
121
21
21
2*
1
2*
121
thttR
dhttR
dhtYtXE
tYdhtXE
tYtYEttR
XY
XY
YY
∗=
−=
−=
−=
=
∫
∫
∫
∞+
∞−
∞+
∞−
∞+
∞−
βββ
βββ
βββ*
).()(),(),( 12*
2121 ththttRttRXXYY
∗∗=
(12-36)
(12-37)
h(t))(tX
µ )(tY
µ
h*(t2) h(t1)⎯⎯⎯ →⎯ ),( 21 ttRXY⎯→⎯ ⎯→⎯ ),( 21 ttRYY
),( 21 ttRXX
(a)
(b)
Fig. 12.7
24
In particular if X(t) is wide-sense stationary, then we haveso that from (12-34)
Also so that (12-35) reduces to
Thus X(t) and Y(t) are jointly w.s.s. Further, from (12-36), the output autocorrelation simplifies to
From (12-37), we obtain
XXt µµ =)(
constant.a cdhtXXY
,)()(
µττµµ == ∫
∞+
∞−(12-38)
)(),( 2121 ttRttRXXXX
−=
(12-39)
).()()(
,)()(),( 21
2121
τττ
τβββ
YYXY
XYYY
RhR
ttdhttRttR
=∗=
−=−−= ∫∞+
∞−
(12-40)
).()()()( * ττττ hhRRXXYY
∗−∗= (12-41)
. ),()()(
)()(),(
21*
*2121
ttRhR
dhttRttR
XYXX
XXXY
−==−∗=
+−= ∫∞+
∞−
ττττ
ααα∆
25
From (12-38)-(12-40), the output process is also wide-sense stationary.This gives rise to the following representation
LTI systemh(t)
Linear system
wide-sense stationary process
strict-sense stationary process
Gaussianprocess (alsostationary)
wide-sense stationary process.
strict-sensestationary process(see Text for proof )
Gaussian process(also stationary)
)(tX )(tY
LTI systemh(t)
)(tX
)(tX
)(tY
)(tY
(a)
(b)
(c)
Fig. 12.8
26
White Noise Process:W(t) is said to be a white noise process if
i.e., E[W(t1) W*(t2)] = 0 unless t1 = t2.W(t) is said to be wide-sense stationary (w.s.s) white noise if E[W(t)] = constant, and
If W(t) is also a Gaussian process (white Gaussian process), then all of its samples are independent random variables (why?).
For w.s.s. white noise input W(t), we have
),()(),( 21121 tttqttRWW
−= δ (12-42)
).()(),( 2121 τδδ qttqttRWW
=−= (12-43)
White noiseW(t)
LTIh(t)
Colored noise
( ) ( ) ( )N t h t W t= ∗
Fig. 12.9
27
and
where
Thus the output of a white noise process through an LTI system represents a (colored) noise process.Note: White noise need not be Gaussian.
“White” and “Gaussian” are two different concepts!
)()()(
)()()()(*
*
τρτττττδτ
qhqh
hhqRnn
=∗−=
∗−∗=(12-45)
.)()()()()(
** ∫∞+
∞−+=−∗= αταατττρ dhhhh (12-46)
(12-44)
[ ( )] ( ) ,
WE N t h dµ τ τ+∞
−∞= ∫ a constant