LECTURE DEFINITION

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1 LECTURE 8 Radom Processes 8. DEFINITION A radom process (or sochasic process) is a ifiie idexed collecio of radom variables {X() : T}, defied over a commo probabiliy space. The idex parameer is ypically ime, bu ca also be a spaial dimesio. Radom processes are used o model radom experimes ha evolve i space/ime: Ȃ Received sequece/waveform a he oupu of a commuicaio chael Ȃ Packe arrival imes a a ode i a commuicaio ework Ȃ Thermal oise i a resisor Ȃ Scores of a NBA eam i cosecuive games Ȃ Dailypriceofasock Ȃ Wiigs or losses of a gambler Ȃ Coes i memory cells We are ieresed i several quesios ivolvig radom processes. Ȃ Depedecies of he radom variables of he process. How do fuure received values deped o pas received values? How do fuure prices of a sock deped o is pas values? Ȃ Log-erm averages. Wha is he proporio of ime a queue is empy? Wha is he average oisepoweraheoupuofacircui? Ȃ Exreme or boudary eves. Wha is he probabiliy ha a lik i a commuicaio ework is cogesed? Wha is he probabiliy ha he maximum power i a power disribuio lie is exceeded? Wha is he probabiliy ha a gambler will lose all his capial? Ȃ Esimaio/deecio. How bes ca oe recover a sigal from a oisy waveform? How caweesimaeadnasequecefromoisyreads?

2 2 Radom Processes AradomprocesscabeviewedasafucioX(,ω)ofwovariables,heime T adheoucomeω,where ishespaceofheuderlyigradomexperime.there arehuswowaysoviewi. Firs,forfixed, X(,ω)isaradomvariableover. Ihis view, he radom process is a idex collecio of radom variables. Secod, for fixed ω, X(, ω) is a deermiisic fucio of, called a sample fucio, as illusraed i Figure.. Ihisview,heradomprocessisaradomlydrawfucioi. X(, ) X(, 2 ) X(, 3 ) 2 X(, ) X( 2, ) Figure.. Sample fucios of a radom process 8.2 DISCRETE-TIME RANDOM PROCESSES A radom process is said o be discree-ime if T is a couably ifiie se, e.g., Ă = {0,,2,...}adĂ = {..., 2,,0,+,+2,...}. Ihiscase,heprocessisdeoedbyX, for N,acouablyifiiese,adissimplyaifiiesequeceofradomvariables. A sample fucio for a discree-ime process is called a sample sequece or sample pah. A discree-ime process ca comprise discree, coiuous, or mixed radom variables.

3 8.2 Discree-Time Radom Processes 3 Example.. Le Z Uif[0,]addefiehediscreeimeprocess X = Z, =,2,... The sample pahs are illusraed i Figure.. The firs-order pdf of heprocess, ha is, hesequeceofpdfsof X is f X (x) = x ( )/ = x, x [0,], whichcabeeasilyfoudbydiffereiaig P{X x} = P{Z x / }w.r..x. x 2 Z = x Z = x Z = Figure.. Sample pahs of he radom process i Example.. Iheaboveexample,wespecifiedheradomprocessbydescribigheseofsample pahs ad explicily providig a probabiliy measure over he se of eves (subses of sample pahs). This way of specifyig a radom process has very limied applicabiliy, ad is suied oly for very simple processes. A discree-ime radom process is i geeral specified(direcly or idirecly) by specifyig all is k-h order cdfs(pdfs, pmfs), i.e., he joicdf(pdf,pmf)ofhesamples X,X 2,...,X

4 4 Radom Processes forevery order k adforeveryseof k pois < 2 < < k N. The Kolmogorov exesio heorem, he proof of which is beyod his course, guaraees ha he eire process ca be defied his way. I he followig, we discuss several classes of discree-ime radom processes IID Processes We say ha {X : N} is a IID process if he radom variables X,X 2,... are idepede ad ideically disribued(i.i.d.). Example8.2(Beroulliprocess). X,X 2,...i.i.d.Ber(p). Example8.3(Discree-imewhieGaussiaoise). X,X 2,...i.i.d.N(0,N). Herewe specifiedhe-horder pmfs(pdfs) ofheprocessesbyspecifyig hefirsorder pmf(pdf) ad saig ha he radom variables are idepede. I would be quie difficul o provide he specificaios for a IID process by specifyig he probabiliy measureoverhesubsesofhesample Radom Walk Le Z,Z 2,...,Z,...bei.i.d.,where The(symmeric) radom walk is defied by X 0 = 0, X = i= + w.p. /, Z = w.p. /. Z i = X + Z, =,2,... Agai his process is specified by(idirecly) specifyig all -h order pmfs. The sample pahforaradomwalkisasequeceofiegersasillusraedifigure.. Thefirs-order pmfis P{X = k}asafucioof. Noeha k {, ( 2),..., 2,0,+2,...,+( 2),+} foreve, k {, ( 2),...,,+,+3,...,+( 2),+} forodd. Nowifweleabeheumberof+ sisepsad abeheumberof s,he orequivalely,a = +k/2. Thus k = a ( a) = 2a, P{X = k} = P{(+k)/2headsiidepedecoiosses} = 2 if k iseve. +k 2 Forexample, P{X 5 = 3} = 5/32ad P{X 0 = 6} = 45/024.

5 8.2 Discree-Time Radom Processes 5 x z : Figure.. Asamplepahofheradomwalk Markov Processes Aradom process{x }issaid obemarkov ifhe fuuread hepasare codiioally idepede give he prese. Mahemaically, his ca be rephrased i several ways. For example, if heradom variables X,X 2,... are discree, heheprocess is Markov if p X + X,X 2,...,X (x + x,x 2,...,x ) = p X + X (x + x ) foreveryadevery(x,x 2,...,x + ). Discree-imediscreeMarkovprocessesareofe called Markov chais. Example 8.4. IID processes are Markov. Example 8.5. The radom walk process is Markov. To see his, cosider P{X + = x + X = x,...,x = x } = P{X + Z + = x + X = x,...,x = x } = P{Z + = x + x X = x,...,x = x } (a) = P{Z + = x + x } (b) = P{Z + = x + x X = x } = P{X + Z + = x + X = x } = P{X + = x + X = x }, whereheequaliies i(a)ad(b)followsice Z + isidepedeof(x,...,x ).

6 6 Radom Processes Idepede Icreme Processes Aradomprocess{X }issaidobeidepedeicreme ifheicremes X, X 2 X,..., X X areidepedeforall k adall < 2 < < k. Example.. The radom walk is a idepede icreme process sice X = Z i, X 2 X = X X =. i= 2 Z i, i= + Z i i= + are idepede(as fucios of idepede radom vecors). The idepede icremeproperymakesieasyofidhe-horderpmfsofheradomwalkprocess. For example, P{X 5 = 3, X 0 = 6, X 20 = 0} = P{X 5 = 3, X 0 X 5 = 3, X 20 X 0 = 4} = P{X 5 = 3} P{X 0 X 5 = 3} P{X 20 X 0 = 4} = P{X 5 = 3} P{X 5 = 3} P{X 0 = 4} = = I geeral if a process is idepede icreme, he i is also Markov. To see his, le{x }beaidepedeicremeprocess. The P{X + = x + X = x,...,x = x } = P{X + X = x + x X = x,x 2 X = x 2 x,...,x X = x x } (a) = P{X + X = x + x } (b) = P{X + X = x + x X = x } = P{X + = x + X = x }, where he equaliies i(a) ad(b) follows by he idepede icreme propery of he process. The coverse is o ecessarily rue, e.g., IID processes are Markov bu o idepede icreme.

7 8.3 Coiuous-Time Radom Processes Gauss Markov Process LeZ,Z 2,...bei.i.d. N(0,N),i.e.,{Z }beawhiegaussiaoise(wgn)process.the Gauss Markov process is a firs-order auoregressive process defied by X = Z, X = αx + Z, = 2,3,..., whereαisaparameersuchha α <.ThisprocesscabegeeraedbypassigaWGN process hrough a discree-ime liear ime ivaria sysem, as we will see i Lecure# i more deail. The Gauss Markov process is Markov. I is o, however, idepede icreme. 8.3 CONTINUOUS-TIME RANDOM PROCESSES AradomprocessiscoiuousimeifT isacoiuousse,e.g.,ă = (, )oră + = [0, ). Example. (Siusoidal sigal wih radom phase). Le X() = αcos(ω +Θ), 0, where Θ Uif[0,2π] ad α ad ω are cosas. The sample fucios are illusraed i Figure.. The firs-order pdf of he process is he pdf of X() = αcos(ω +Θ). I Problem.,wefoudiobe f X() (x) = απ (x/α) 2, α < x < +α. Noehahepdfisidepedeof. Acoiuous-ime radomprocess{x(): 0}issaidobeidepedeicreme if X( + s) X() is idepede of {X(u): 0 u } for every s, 0. I paricular, X( ), X( 2 ) X( ),...,X( k ) X( k )areidepedeforeverykad < 2 < < k. A coiuous-ime radom process {X(): 0} is said o be Markov if he fuure {X( + s): s > 0} is codiioally idepede of he pas {X(u): 0 u } give he prese X(). I paricular, X( k+ ) is codiioally idepedeof (X( ),...,X( k )) give X( k )foreverykad < 2 < < k+. Asihediscree-imecase,aidepede icreme process is Markov, bu he coverse does o hold. Ulike discree-ime radom processes, a couiuous-ime radom process cao be fully specified by k-h order disribuios aloe. For example, kowig k-h order disribuiosforeverykdoesoaswerwheherheprocessiscoiuousoro. Thus,for coiuous-ime radom processes, we ofe eed addiioal properies (such as coiuiy). I he followig, we discuss a few famous coiuous-ime radom processes.

8 8 Radom Processes x() α Θ = 0 π 2ω π ω 3π 2ω 2π ω x() Θ = π 4 x() Θ = π 2 Figure.. Sample fucios of he siusoidal sigal wih radom phase Browia Moio Aradomprocess{W(): 0}issaidobeaBrowiamoio(orWieerprocess) if Ȃ W(0) = 0, Ȃ {W()}isidepedeicremewihW() W(s) N(0, s)forevery > s,ad Ȃ W()iscoiuousfor 0almossurely. The k-h order pdf ca be easily compued from he idepede icreme propery. For example, f W( ),W( 2 ),W( 3 )(, 2, 3 ) = f W( )( )f W(2 ) W( )( 2 )f W(3 ) W( 2 )( 3 2 ) = f W( )( )f W(2 )( 2 )f W(3 2 )( 3 2 ) Poisso Process Aradomprocess{N(): 0}issaidobePoissowihrae λ if Ȃ N(0) = 0ad

9 8.4 Mea ad Auocorrelaio Fucios 9 Ȃ {N()}isidepedeicremewihN() N(s) Poisso(λ( s))forevery > s. AsamplepahisshowiFigure.. Here, 2,...arehearrivalimesorhewaiimes ofheeves. Thediffereces, 2,...arecalledheierarrivalimesofheeves. () Figure.. AsamplepahofaPoissoprocess. RecalligfromExample.,hefirsarrivalimeT cabespecifiedbyherelaioship {T > } = {N() = 0}, whichimplieshat isaexp(λ)radomvariable. Moregeerally,heierarrivalimes T,T 2 T,T 3 T 2,...arei.i.d.Exp(λ). 8.4 MEAN AND AUTOCORRELATION FUNCTIONS Foraradomprocess X()hefirsadsecodordermomesare Ȃ meafucio: X() = E[X()]for T. Ȃ auocorrelaiofucio: R X (, 2 ) = E[X( )X( 2 )]for, 2 T. The auocovariace fucio of a radom process is defied as C X (, 2 ) = E[(X( ) E[X( )])(X( 2 ) E[X( 2 )]] = R X (, 2 ) X( ) X( 2 ). Example8.8. ForaIIDprocess X X() = E[X ], R X (, 2 ) = E[X X 2 ] = E[X2 ] = 2, (E[X ]) 2 = 2.

10 0 Radom Processes Example 8.9. For he radom phase sigal process i Example., 2π α X() = E[αcos(ω +Θ)] = cos(ω +θ)dθ = 0, 0 2π R X (, 2 ) = E[X( )X( 2 )] 2π α = 2 2π cos(ω +θ)cos(ω 2 +θ)dθ = 0 2π 0 α 2 4π cos(ω( + 2 )+2θ)+cos(ω( 2 )) dθ = α2 2 cos(ω( 2 )) Example 8.0. For he radom walk, X() = E i= Z i = i= 0 = 0. Tocompueheauocorrelaio fucio,firsassumeha 2 adcosider R X (, 2 ) = E[X X 2 ] = E[X (X 2 X + X )] = E[X 2 ] =. I geeral, R X (, 2 ) = mi{, 2 }. Example 8.. For he Gauss Markov process, X() = E[X ] = E[αX + Z ] = α E[X ]+E[Z ] = α E[X ] = α E[Z ] = 0. Tofidheauocorrelaio fucio,assumefirsha < 2. The, 2 X 2 = α 2 X + i=0 α i Z 2 i. Thus, R X (, 2 ) = E[X X 2 ] = α 2 E[X 2 ]+0, sice X ad Z 2 i are idepede, zero mea for 0 i 2. Nex, o fid

11 8.5 Gaussia Radom Processes E[X 2 ],cosider E[X 2 ] = N, E[X 2 ] = E[(αX + Z ) 2 ] = α 2 E[X 2 ]+N Therefore, i geeral, = α2 α 2 N. R X (, 2 ) = α 2 α 2mi{, 2 } α 2 N. 8.5 GAUSSIAN RANDOM PROCESSES AradomprocessissaidobeGaussiaif [X( ), X( 2 ),..., X( k )] T isagaussiaradomvecorforevery k ad < 2 < < k. Example 8.2. The discree ime WGN process is Gaussia. Example8.3. TheGauss MarkovprocessisGaussia. Ideed,sice X = Z ad X k = αx k + Z k wih Z,Z 2,...i.i.d.N(0,N),wehave X Z X 2 α 0 0 Z X 3 = Z.. 3,. α 2 α 3 0. X α α 2 α Z which is a liear rasformaio of a Gaussia radom vecor ad is herefore Gaussia iself. Example 8.4. The Browia moio is Gaussia(why?). PROBLEMS.. Symmericradomwalk.Le X bearadomwalkdefiedby X 0 = 0, X = i= wherez,z 2,...arei.i.d. wih P{Z = } = P{Z = } = 2. Z i,

12 2 Radom Processes (a) Fid P{X 0 = 0}. (b)approximae P{ 0 X 00 0}usighecerallimiheorem. (c) Fid P{X = k}... Absolue-valueradomwalk.CosiderhesymmericradomwalkX iheprevious problem. Defieheabsolue valueradomprocessy = X. (a) Fid P{Y = k}. (b)fid P{max i<20 Y i = 0 Y 20 = 0}... Discree-ime Wieer process. Le Z, 0, be a discree ime whie Gaussia oise process, i.e., Z,Z 2,... are i.i.d N(0,). Defie heprocess X,, such ha X 0 = 0,ad X = X + Z,for. (a) Is X aidepedeicremeprocess? Jusifyyouraswer. (b)is X agaussiaprocess? Jusifyyouraswer. (c) Fidhemeaadauocorrelaio fuciosof X. (d)specifyhefirs-orderpdfof X. (e) Specifyhejoipdfof X 3,X 5,ad X 8. (f) Fid E(X 20 X,X 2,...,X 0 ). (g) Give X = 4, X 2 = 2,ad0 X 3 4,fidhemiimumMSEesimaeof X 4... A radom process. Le X = Z + Z for, where Z 0,Z,Z 2,... are i.i.d. N(0,). (a) Fidhemeaadauocorrelaio fuciosof{x }. (b)is{x }Gaussia? Jusifyyouraswer. (c) Fid E(X 3 X,X 2 ). (d)fid E(X 3 X 2 ). (e) Is{X }Markov? Jusifyyouraswer. (f) Is{X }idepedeicreme? Jusifyyouraswer... Movig average process. Le X = 2 Z + Z for, where Z 0,Z,Z 2,... are i.i.d. N(0,). Fidhemeaadauocorrelaio fucioof X... Auoregressiveprocess. LeX 0 = 0adX = 2 X +Z for,wherez,z 2,... arei.i.d. N(0,). Fidhemeaadauocorrelaio fucioof X... Radom biary waveform. I a digial commuicaio chael he symbol is represeed by he fixed duraio recagular pulse д() = for0 < 0 oherwise,

13 Problems 3 ad he symbol is represeed by д(). The daa rasmied over he chael is represeed by he radom process X() = A k д( k), for 0, k=0 wherea 0,A,...arei.i.dradomvariables wih (a) Fidisfirsadsecodorderpmfs. A i = + w.p. 2 w.p. 2. (b) Fid he mea ad he auocorrelaio fucio of he process X()... Arrow of ime. Le X 0 be a Gaussia radom variable wih zero mea ad ui variace, ad X = αx + Z for,whereα isafixedcosawih α < ad Z,Z 2,...arei.i.d. N(0, α 2 ),idepedeof X 0. (a) Isheprocess{X }Gaussia? (b)is{x }Markov? (c) FidR X (,m). (d)fidhe(oliear)mmseesimaeof X 00 give(x,x 2,...,X 99 ). (e) FidheMMSEesimaeof X 00 give(x 0,X 02,...,X 99 ). (f) FidheMMSEesimaeof X 00 give(x,...,x 99,X 0,...,X 99 ).