DISCRETE-TIME RANDOM PROCESSES

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Crystal Terry
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1 DISCRT-TIM RNDOM PROCSSS Rado Pocsss Dfiitio; Ma ad vaiac; autocoatio ad autocovaiac; Ratiosip btw ado vaiabs i a sig ado pocss; Coss-covaiac ad coss-coatio of two ado pocsss; Statioa Rado Pocsss Statioait; Joit wid ss statioait of two ado pocsss; Poptis of t autocoatio of a WSS pocss: godicit of Rado Pocsss Ma godicit; utocoatio godicit utocoatio ad utocovaiac Matics Wit Nois T Pow Spctu Siga oc Sst Uppsaa Uivsitt - DISCRT-TIM RNDOM PROCSSS disct-ti ado pocss siga is a coctio sb of disct-ti sigas is a itg disct-ti ado pocss is a idd squc of RVs at a ctai fid ti istat g Not: disct-ti ado siga is atd to sb of disct-ti siga disct-ti siga a sig aizatio o obsvatio Siga oc Sst Uppsaa Uivsitt - ap ado pocss RP siusoid cos as a ado apitud tat assus a itg ub btw o ad si ac wit qua pobabiit P/6 fo 6 How a disct-ti sigas is RP associatd wit? cosists of a sb of si disct-ti sigas : cos cos 6 6cos ac siga sows up wit qua pobabiit P[ ]/6 T st sb of disct-ti sigas is fiit Qustio? Giv a ado pocss cos w t apitud is a ado vaiab at istat tat assus a itg ub btw o ad si ac wit qua pobabiit ow a qua pobab disct-ti sigas a t i a sb? Siga oc Sst Uppsaa Uivsitt -3 Siga oc Sst Uppsaa Uivsitt -4

2 ado pocss s fo difft viwpoits Fo a sap spac poit of viw a disct-ti siga i cospods to i i Ω RP at a fid ti is a RV wit F P{ } ad f df d Fo a difft is a difft RV Tus a disctti ado pocss is a idd squc of RVs Siga oc Sst Uppsaa Uivsitt -5 Ma ad Vaiac of a Rado Pocss T a of is dfid as { } f d T vaiac of is dfid as σ { } ad σ a o dtiistic squcs wit t sa id as i dtiistic is t cosquc of sb avag {} wic ids { } ad { σ } σ o t fist-od statistics If is a fuctio of aot RP ζ wit f ζ i g[ ζ ] t is [ ] g ζ g f ζ d Siga oc Sst Uppsaa Uivsitt -6 utocoatio ad utocovaiac T autocoatio btw ad : { } T autocovaiac: c { [ ][ ]} o c Fo c σ Ras: Bot ad c av two idics ad T povid ifoatio about t dg of ia dpdc btw two RVs ad i t sa pocss T pa a ipotat o i siga odig spctu stiatio ad Wi fitig Siga oc Sst Uppsaa Uivsitt -7 Haoic pocss wit ado pas i a-vaud aoic ado pocss is a ado pocss wit a fo of a siusoid si w is a fid costat Cosid t cas w t apitud is a fid costat but t pas is a ado vaiab T pas ado vaiab is uifo distibutd ov t itva [ i ; < f 5 ; otwis Fid ad Siga oc Sst Uppsaa Uivsitt -8

3 Siga oc Sst Uppsaa Uivsitt -9 Haoic pocss wit ado pas Soutio si si si d d f [ ] [ ] [ ] cos cos cos si si b otig: [ ] cos cos / si si B B B ad [ ] [ ] cos cos d Siga oc Sst Uppsaa Uivsitt - Haoic pocss wit ado pas 3 ii cop-vaud aoic ado pocss is of t fo p w ad a fid costats ad t pas is aso a uifodistibut ado vaiab i itva [ ] Fid ad Soutio si cos p j [ ] [ ] p p j j [ ] [ ] p p j j T a- ad cop-vaud aoic pocsss av a zo a ad a autocoatio tat o dpds o - Siga oc Sst Uppsaa Uivsitt - Coss-covaiac ad Coss-coatio T coss-coatio of two RPs ad is T coss-covaiac of two RPs ad is [ ][ ] c o c Ras: T autocoatio ad autocovaiac a t spcia cass of t coss-coatio ad coss-covaiac spctiv fo Coss-coatio is v usfu i t appicatios i siga dtctio Siga oc Sst Uppsaa Uivsitt - ap 3 Coss-coatio Cosid two RPs ad w is ow wit ad ad is t covoutio of wit a dtiistic squc i Fid i ad ii

4 Ratiosip btw ado pocsss Two ado pocsss ad a said to b ucoatd if c o quivat fo a ad Two ado pocsss ad a said to b otogoa if If ad a ucoatd ad o of t o bot as a zo a t t autocoatio of t su of z is z Siga oc Sst Uppsaa Uivsitt -3 Ma ad autocoatio of a soa siga soud poducd b a subai is odd as s si ad w it is civd b a soa it is a ois sius wit ado pas si v ssuig tat is costat ad v is t abit ois wit v ad v σ vδ & ucoatd wit Fid ad { } { si } { v } si v / cos σ vδ is costat ad o dpds o o cos σ vδ o dpds o Siga oc Sst Uppsaa Uivsitt -4 Statioa Rado Pocsss Statioait i ts of dsit fuctios: RP is said to b st-od statioa if t st od dsit fuctio is idpdt of ti f fo a f T a ad vaiac a costat ad σ σ RP is said to b d-od statioa if t d-od joit dsit fuctio f dpds o but ot o ad spctiv i f f d-od statioa pocss as d-od ti-sift-ivaiat statistics g wic dpds o o sic If a RP is d-od statioa t it wi b st-od statioa Siga oc Sst Uppsaa Uivsitt -5 Statioa Rado Pocsss Stict-ss statioait: ado pocss is said to b statioa of od L if t ado pocss ad av t sa Lt-od joit dsit fuctios ado pocss is said to b statioa i t stict ss o stictss statioa if it is statioa fo a ods L Siga oc Sst Uppsaa Uivsitt -6

5 Statioa Rado Pocsss Statioait i ts of sb avag & wid ss statioait: Wid Ss Statioait ado pocss is said to b wid-ss statioa WSS if t foowig t coditios a satisfid: T a of t pocss is a costat T autocoatio dpds o o t diffc i 3 T vaiac of t pocss is fiit c < T aoic pocss wit ado pas s is WSS sic dpdig o c / c 4 Wid ss statioait of a aoic pocss wit uifo distibutd ado apitud Cosid a a-vaud aoic ado pocss si w t fquc ad t pas a fid costats but t apitud is a ado vaiab tat is uifo distibutd ov t itva [b c] wit c>b Dti t statioait of t ado pocss Soutio T a of is { } { si } b c si si wic dpds o Tfo is ot WSS Siga oc Sst Uppsaa Uivsitt -7 Siga oc Sst Uppsaa Uivsitt -8 Poptis of t autocoatio squc of a WSS pocss: Popt St Fo a WSS RP Fo a a RP It foows fo { } { } Popt Ma-squa vau { } Popt 3 Maiu vau Popt 4 Piodicit If of a WSS RP as fo so t is piodic wit piod Fo ap 5 cos is piodic wit a piod of Qustios : Wic os of t foowig autocoatios is vaid fo WSS ado pocsss? i ii iii iv v δ δ δ vi δ δ δ sws: i No; ii Ys; iii No; iv No; v No; vi Ys Siga oc Sst Uppsaa Uivsitt -9 Siga oc Sst Uppsaa Uivsitt -

6 Joit wid ss statioait of two ado pocsss: utocoatio ad utocovaiac Matics Two RPs ad a said to b joit wid-ss statioa if ad a WSS ad if dpds o o t diffc i H R { } p p p p p p p p p p p p w [ p ] T H T is t Hitia taspos ad { } is t autocoatio Not tat t { } a awas a! Siga oc Sst Uppsaa Uivsitt - Siga oc Sst Uppsaa Uivsitt - utocoatio ad utocovaiac Matics If a ado pocss is WSS t R bcos p p H R { } p p p T autocoatio ati R is a p p squa ati Poptis of autocoatio ati: Popt T autocoatio ati of a WSS ado pocss is a Hitia Topitz ati R Top{ p } Popt T autocoatio ati of a WSS ado pocss is ogativ dfiit R > Popt 3 T igvaus λ of t autocoatio ati of a WSS ado pocss a a-vaud ad ogativ Siga oc Sst Uppsaa Uivsitt -3 Siga oc Sst Uppsaa Uivsitt -4

7 ap 5 Dti wt o ot t foowig atics a vaid autocoatio atics: 3 4 j 4 j j i R 5 ii R 5 iii R 3 j 4 j 3 3 j j j 4 i R is ot a vaid autocoatio ati sic it is a-vaud ad ot stic ii R is ot a vaid autocoatio ati it sic t tis aog t diagoa is ot a-vaud iii R 3 is a vaid autocoatio ati sic it is a Hitia Topitz ati R Top{ 4 j j} ad ogativ dfiit 3 W stud t godicit of a ado pocss? T a & autocoatio of RP { } ad { } a dtid fo t sb avags of a possib disct-ti sigas i t sb W do w stud t godicit of a RP? If o o sig aizatio of is avaiab is it possib to dti ad fo? If possib ow to fid ad fo? d wat coditios av to b satisfid? Siga oc Sst Uppsaa Uivsitt -5 Siga oc Sst Uppsaa Uivsitt -6 godicit of a Rado Pocss Ma godicit N Dfiitio If t sap a ˆ N of a WSS pocss N covgs to i t a-squa ss i { ˆ N } N t t pocss is said to b godic i t a ad w wit i ˆ N N I od fo i { ˆ N } N it is cssa ad sufficit tat ˆ N b asptotica ubiasd i { ˆ N } ad i Va{ ˆ N } N N godic Tos Ma godic To Lt to b a WSS ado pocss wit autocovaiac squc c cssa ad sufficit coditio fo to b godic i t a is N i c N N Ma godic To Lt to b a WSS ado pocss wit autocovaiac squc c Sufficit coditios fo to b godic i t a a tat c < i c Siga oc Sst Uppsaa Uivsitt -7 Siga oc Sst Uppsaa Uivsitt -8

8 6 Statioait ad godicit of a ado pocss Cosid a ado pocss w is a ado vaiab tat is uifo distibutd ov t itva [b c] wit c>b Dti t statioait ad t a godicit of t ado pocss Siga oc Sst Uppsaa Uivsitt -9 6 Statioait ad godicit of a ado pocss Soutio b c { } wic is a costat { } { } 3 3 c c c b c cb b f d d b b c b 3 c b 3 wic is a costat it c { } Va c b is a costat fo a ad c < Tus t pocss is WSS N Sic i c c b is ot zo du to b c it foows N N fo to tat is ot godic i t a Siga oc Sst Uppsaa Uivsitt -3 utocoatio godicit Dfiitio N If t sap autocoatio ˆ N of a widss statioa pocss covgs to N i t a-squa ss i ˆ N t t pocss is said to b N autocoatio godic ad w wit i ˆ N N Siga oc Sst Uppsaa Uivsitt -3 Wit Nois Wit ois v is a WSS pocss wit autocovaiac c σ δ v v It is sip a squc of ucoatd ado vaiabs ac avig a vaiac of σ v i v ad v a ucoatd fo sic c v o v vv fo Kowdg of o dos ot p i t stiatio of t ot usig a ia stiato Not wit ois as v so tat c v v vv v t is a ifiit vait of wit ois ado pocsss W? bcaus t ucoatd ado vaiabs ca b a ifiit vait of difft distibutio ad dsit fuctios g wit Gaussia ois ad wit Boui ois Siga oc Sst Uppsaa Uivsitt -3

9 T Pow Spctu Pow spctu o pow spcta dsit of a RP is dfid as P j F{ } j Pow spctu ca b obtaid usig t z-tasfo of P z z a b dtid fo t ivs tasfo j j j F { P } P d o { P z } T Pow Spctu Sic a RP is a sb of disct-ti sigas w ca coput t Foui o z- tasfo of ac disct-ti siga i F{ } o Z{ } i t pocss; but w ca ot coput t Foui o z- tasfo of t RP itsf i w soud ot do F{} o Z{} istad w cacuat t pow spctu of a ado pocss j P wic is t Foui tasfo of a dtiistic squc Siga oc Sst Uppsaa Uivsitt -33 Siga oc Sst Uppsaa Uivsitt -34 Poptis of t pow spctu of a WSS ado pocss j j Popt St Fo a WSS RP P P wic as a-vaud o P z P / z j j If is a t t pow is v P P wic ipis tat P z P z j Popt Positivit P j P d Popt 3 Tota pow Qustios Wic os of t foowig pow spctu is a vaid fo WSS ado pocsss? Wic os of t foowig pow spctu is a vaid fo WSS ado pocsss? z i P z 5 z z ii P z z j iii P z iv P 3 z 3 z cos j v P ad vi 8cos P j 8si sws: i Ys; ii No; iii Ys; iv No; v Ys; vi No Siga oc Sst Uppsaa Uivsitt -35 Siga oc Sst Uppsaa Uivsitt -36

10 ap 7 T pow spctu i T pow spctu of t aoic pocss wit ado pas i ap Soutio Fo ap it foows tat / cos ad t pow spctu is j j j P cos j j j [ δ δ ] j j w t DTFT atio X δ is usd j P is a v ad ogativ Siga oc Sst Uppsaa Uivsitt -37 ap 7 T pow spctu ii T pow spctu of t ado pocss tat as a autocoatio squc w < Soutio Fo t dfiitio it foows tat P j j j j j j j j cos j j wic is a & ogativ ad is v sic P P O { P z Z Z } z z Siga oc Sst Uppsaa Uivsitt -38 MTLB fuctios fo ado pocsss >> ad % cats uifo distibutd ado pocsss >> ad % cats Gaussia oa ado pocsss >> topitz R % poducs a Topitz ati >> co % auto- ad coss-coatio fuctio stiats >> cov % auto- ad coss-covaiac fuctio stiats T MTLB fuctios fo studig ado vaiabs a usfu fo ado pocsss Us p to oo at t dtaid dsciptios of t fuctios Siga oc Sst Uppsaa Uivsitt -39

DISCRETE-TIME RANDOM PROCESS (2)

DISCRETE-TIME RANDOM PROCESS (2) DISCRT-TIM RANDOM PROCSS RANDOM PROCSSS Rando Pocesses Discete-tie ando pocesses: Mean and vaiance: Autocoeation and autocovaiance: Reationsip between ando vaiabes in a singe ando pocess: Coss-covaiance