Reading lo l lo 4 1 Randomprocessesy To this point we have considered only finite collections of RVs i e fado vectors The remainder of the course will be s on the case where we have an infinite collection of RVs which we call a rado proc.es RP A discrete RP is a countable collection of RVs Xn EIR nes where S is a countably infinite set Typically we take S N EI Assa a sequence of bits is transmitted over a noisy channel ad bits are flipped independently with probability p so that Xa new is a B Ipl RP A cnt fe RP is an uncountable collection of RVs Xt c IR tec where T is an uncountable subset of IR Typically I To T or T CR EI County process NE wle Ne counts the number of occurrences of so event up to five t
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Reading lo l lo 4
1Randomprocessesy
To this point we have considered only finite collections ofRVs i e fado vectors The remainder of the course will be son the case where we have an infinite collection of RVs which
we call a rado proc.es RP
A discrete RP is a countable collection of RVs
XnEIR nes
where S is a countably infinite set Typically we take S N
EI Assa a sequence of bits is transmitted over a noisy channel
ad bits are flippedindependently with probability p so that
Xa new is a B Ipl RP
A cnt fe RP is an uncountable collection of RVs
Xt cIR tec
where T is an uncountable subset of IR Typically I To T or T CR
EI Countyprocess NE wle Ne counts thenumber of occurrences
of so eventup to five t
Relaf.vn SapleSpRecall that a RU is a faction from the sample space to IR
For a RP we really here a set X Iw and we can view
a RP in two ways
1 Fix n then view each Xncw as a RV
2 Fix w whichgives the Sequence X w telco
This sequence is called a realization or a sanpkp.at
The second way ismore common and aligns better with
the
notion of a realization of a RV
EI R F P Te 2e Bl6,2 3 Unifko zig
BorelSignaalgebra we may cover
this later if the allowsuiew
Xe w cos za f Et w
RV Since this is a faction of w
viewTela
c First drawof w
t
Three Second drawof w
Characterization of RPS
For RVs and RVecs we spent a great deal of the explicity writing
down distributions from which we can desire parameters such as
the mean variance and covariance We will discuss the analog for
RPs next week For now we will focus on the first near and
second order variance covariace statistics of RPS
For a RP the me f ct.in is
my t EL te
For a RP 3 tin correlating between two RVsadXs is
Rx It a ELIE Xa sort.us called auto
Ext Xt cos 2T ft 0 where 0 unit E a a
E Xt El coskafttosa
cos tuft 01 doTi
propertyofcosine
R t s EExeXs E cos zaftig cos Zafra
E ces 2Tifftts t 20 t cos felt s
cos zaf t s
Properties of Correlation
1 symmetry Rx It s Rx la t
2 positive semidefiniteness for any function t
Half Rx It a a dtdss.co
this is thefunctional version of Lt RaIo
3 Candy Schwarz
11214111 1EllieXa EVEEXi E VRxlt.tl3xfa.sl
More second orderquantities of interest
A RP with ETH Rx It t are is called a sqq process
for a RP Xe the covariacefactibetween Xf and Xs is
tis ELIA EE 3 Xs EEXs
For t e 12ps Xe ad Ye the crosscorrelationfact is
Ralt a Ellie's
For t e 12ps Xe ad Yt the crosscovariacefact is
cult a E Itt Ethel Ys EETs
Rx ft s mxltlm.is
Station_it captures thenotionthat the statistics of a RP d not
change over time We typically consider twodefinitions of stationarity
strict ad wide and the latter appears muchmore frequently in
practice
A RP is instigating if for any finite collection
of a fires t tu all jointprobabilities donot depend on the
tie shift 1st i e
P Xt s n Xe c B P Xt at Xtrae
A RP B called strictystation ifit is nth order strictly
stationary for all finite positiven
EdXe Z htt is a strictlystationary RP
In general strict statienarily3 toostrong of an assumption to use and
it is very difficultto prove for interesting RPs
We instead
focus vainly on wide sense stationaryWss RPs
A RP Exe is wishing luss if
Li E Xe Xs Ks t the men does not change over t.me
Ii EEXets is a factiononly of t s only
for less RPs we often write Rx It A as a univariate function
of the difference 7 i e we write Rx T
EJ Xt cos Zaft G is a loss RP since we showed
Rx Itis cos zaf t s
UssthroughtIn signal processing
we often consider what happens when we pass
a NSS process a signal througha LTI system leg a filter
Xt Ye
As with deterministic signalsit is more convenient to analyze
these systems in theFourier domain
The powerspectraldensity PSD of a USS process is the Fourier
transform of the correlation faction
Sx f R e e j Stdx
We can also invert the PSD to find the autocorrelation
Rx K IS f d Adf
TI Any real symmetric PSD 12 12satisfies
c Rx lo 3 12 1 I2 Sx1ft is real symmetric
3 Self 20
Proof Maybe on homework
Back to LTI Syste s analyay in the tire domain we Lace
x
Yt htt Xt ne dix
Let's look at the rear and correlation f etions
myItt EEE htt EExe c Idt
x
on Itt Heldtre
where mitt ax does not depend on it since Xe a boss
Now let's look at the correlation function
Ry It s EEE YsE 11h14Xt eh lol Xs o dodt
If htt h lo E Xt ets a do IT
Ihle 1h10 Rx ft si te d do IT i
which depends only ont t if lying
Let 3 ad Ye3 be two WSS Rps If the cross correlated
Rx It s depends only on t s we say Xtal Ye are
jo.ntywss.me
For Ye 1h10 Xt DO we have
E Exits EIXt.LT 0 Xs odO
fhk4E XtXs o dO
IhlO Rxlt stO dO
So we can let t t s ud write
Rxy It h lol Rx exo do
Substituting into c above we see that
Ryle hfdkx.ir Adp
i.e Ry is the convolution ofh ad Rxy This cnet.ua esexa in
these objects in the frequencydomain Let
Hlf fth T ed fide
be the transfer fretion of htt First a definition
The crosspowospectly Cassis of two jointly WSS Rps
is the Fourier transform of the cross correlation faction
Gwen the cross correlation factionabove we can evaluate the
CPSD of 3 ad 3 whichalsogives us thePSD of Yt
SMH Hlf 5 18complex conjugateand
Sy f Hlf S f
H f 12518
ProbleyLet Xt be a boss RP with correlation fate
12 1 c Ae Kl
Find the second moment of the RV
Y Xs Xz
Solution
EET Ellis xD
Xs't 2XsXr
212 10 212 13
2A l ek
Problemy
Let Ye A cos wt of where w and O are constants and
A is a RV Is Xt WSS
Solution
For Xe to be boss we need its near to be independent of time
and its correlation tobe a Letter only of the fine difference
vi It E Ext
EIA cos lotto
Cos cotto ETA
So Milt is only independent of time if EAT o
Rx It a EL xc.isI A cos cotton cos usted
Et 12 cos cotton cos usted
E 1 2 cosCultist 20 cos lult ss
which depends on ad A so Xe is not Wess
RobleyLet Xe be Wss with 12 1 4 e and let h have
transfer function Hlf jh af.FI 12 1 1 ad Rylewhere Ye is the output of the system
Solutionfirst find S y ft then take the inverse
FT Please feel free
to use a FT table of yourchoice for these
5 f H f Sdf
Using a table we see that
S f zI ef 42
Note that j2f is the FT of a differentiator so faking