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

the inverse FT

Rate da 12 4742

T e

We can apply the same idea to get Ryle via Sylt

Sy f 1HHH2S lf

f jzaflljzafjs.CH

Ryle day Rie

Rulee 42

l T2 e