5. Random Processes. 5-3 Deterministic and Nondeterministic Random Processes

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1 5. Radom Processes 5- Itroducto 5- Cotuous ad Dscrete Radom Processes 5-3 Determstc ad Nodetermstc Radom Processes 5-4 Statoary ad Nostatoary Radom Processes 5-5 rgodc ad Noergodc Radom Processes 5-6 Measuremet of Process Parameters 5-7 Smoothg Data wth a Movg Wdow Average opcs Cotuous ad Dscrete Radom Processes Classfcato of Radom Processes Determstc ad Nodetermstc Radom Processes Statoary ad Nostatoary Radom Processes Wde Sese Statoary o me Averages ad Statstcal Mea o Autocorrelato early tro rgodc ad No-ergodc Radom Processes rgodcty Measuremet of Process Parameters o Statstcs Smoothg Data wth a Movg Wdow Average Smulatg a Radom Process Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 of 8 C 3800

2 Chapter 5: Radom Processes A radom process s a collecto of tme fuctos ad a assocated probablty descrpto. Whe a cotuous or dscrete or mxed process tme/space ca be descrbe mathematcally as a fucto cotag oe or more radom varables. A susodal waveform wth a radom ampltude. A susodal waveform wth a radom phase. A sequece of dgtal symbols, each takg o a radom value for a defed tme perod (e.g. ampltude, phase, frequecy). A radom walk (-D or 3-D movemet of a partcle) he etre collecto of possble tme fuctos s a esemble, desgated as x t, where oe partcular member of the esemble, desgated as x t, s a sample fucto of the esemble. I geeral oly oe sample fucto of a radom process ca be observed! hk of: where A ad w are kow costats. t Aswt, 0 Note that oce a sample has bee observed x t As wt the fucto s kow for all tme, t. Note that, x t s a secod tme sample of the same radom process ad does ot provde ay ew formato about the value of the radom varable. x t A w s t here are may smlar esembles egeerg, where the sample fucto, oce kow, provdes a cotug soluto. I may cases, a etre system desg approach s based o ether assumg that radomess remas or s removed oce actual measuremets are take! For example, commucatos there s a sgfcat dfferece betwee coheret (phase ad frequecy) demodulato ad o-coheret (.e. ukow startg phase) demodulato. O the other had, aother measuremet a dfferet evromet mght measure x t A swt I ths space the radom varables could take o other values wth the defed rages. hus a etre esemble of possbltes may exst based o the radom varables defed the radom process. Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 of 8 C 3800

3 For example, assume that there s a kow AM sgal trasmtted: s t b At swt at a udetermed dstace the sgal s receved as y t b At swt, 0 he receved sgal s mxed ad low pass fltered x x t h t y t cosw t ht b At swt cosw t,0 t h t y t cosw t h t b At 0.5s wt s,0 If the flter removes the wt term, we have x t h t y t cosw t b A t s, 0 Notce that based o the value of the radom varable, the output ca chage sgfcatly! From producg o output sgal, ( 0, ), to havg the output be postve or egatve ( 0to or to ). P.S. hs s ot how you perform o-coheret AM demodulato. o perform coheret AM demodulato, all I eed to do s measured the value of the radom cos wt, where. varable ad use t to sure that the output s a maxmum (.e. mx wth m m t Note: the phase s a fucto of frequecy, tme, ad dstace from the trasmtter. ermology to lear: Cotuous vs. dscrete Determstc vs. odetermstc Statoary vs. ostatoary rgodc vs. oergodc Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 3 of 8 C 3800

4 Degrees of freedom xercse 5-. a) If t s assumed that ay radom process ca be descrbed by pckg oe descrptor from each of the four pars defed, how may classes of radom processes ca be descrbed. Cotuous vs. dscrete Determstc vs. odetermstc Statoary vs. ostatoary rgodc vs. oergodc 4 thgs that oe or the other s selected. herefore, Descrptor s 4 6 b) If you cosdered mxed processes whch two are combed (from the classes defed above), how may possble class descrpto of the mxed process ca be defed. radom processes each wth 6 classes. herefore Descrptor s 6 56 hgs get messy way too fast Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 4 of 8 C 3800

5 Cotuous ad Dscrete Radom Processes A cotuous radom process s oe whch the radom varables, such as t, t, t, ca assume ay value wth the specfed rage of possble values. A more precse defto for a cotuous radom process also requres that the probablty dstrbuto fucto be cotuous. xamples are: hermal agtato ose a coductor, Shot ose electroc tubes or trasstors, Wd velocty As a excepto, uform dstrbutos are usually thought of as cotuous. A dscrete radom process s oe whch the radom varables, such as t t,, t, ca assume ay certa values (though possbly a fte umber of values). A more precse defto for a dscrete radom process also requres that the probablty dstrbuto fucto cosst of umerous dscotutes or steps. Alterately, the probablty desty fucto s better defed as a probablty mass fucto the pdf s composed of delta fuctos. xamples are: ADC outputs rucated or rouded umbers A mxed radom process cossts of both cotuous ad dscrete compoets. he probablty dstrbuto fucto cossts of both cotuous regos ad steps. he pdf has both cotuous regos ad delta fuctos. xamples are: Half-wave or full-wave rectfers Dscrete tme samples (a sample ad hold devce) Aalog lmters (saturated or lmted Op-Amp outputs) Note that all of the above sgals may be cosdered cotuous tme or may be sampled at ay arbtrary tme. Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 5 of 8 C 3800

6 xercse 5-. A radom tme fucto has a mea value of ad ampltude that has a expoetal dstrbuto. hs fucto s multpled by a susod of ut ampltude ad phase uformly dstrbuted over (0, p). a) Classfy the product as cotuous, dscrete or mxed. cotuous ampltude dstrbuto ad a cotuous tme sgal wth a cotuous (uform) dstrbuto b) After passg the above sgal through a deal hard lmter, s the output cotuous, dscrete or mxed. he sgal has bee dscretzed value; therefore, oe would expect a pmf descrpto of the sgal behavor. c) If the susod s passed through a half-wave rectfer before multplyg the expoetally dstrbuted tme fucto, s the output cotuous, dscrete or mxed. he flat spot (due to half-wave rectfcato) defes a pmf level amog the cotuous outputs possble, makg ths a mxed sgal. Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 6 of 8 C 3800

7 Problem 5-. Classfy each of the followg radom processes as cotuous, dscrete or mxed. a) A radom process whch the radom varable s the umber of cars per mute passg a traffc couter. b) he thermal ose voltage geerated by a resstor. c) he radom process defed problem 5-.. (Dscrete sample addto) d) he radom process that results whe a Gaussa radom process s passed through a deal half-wave rectfer. e) he radom process that results whe a Gaussa radom process s passed through a deal full-wave rectfer. f) A radom process havg sample fuctos of the form: t A cos B t where A s a costat, B s a R.V. expoetally dstrbuted from 0 to f, ad theta s uformly dstrbuted from 0 to p. Dscrete: (a) ad (c) Cotuous: (b), (e), ad (f) Mxed: (d) A strog ht: Is the desty fucto cotuous or dscrete (delta fucto compoets) or both? Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 7 of 8 C 3800

8 Determstc ad Nodetermstc Radom Processes A odetermstc radom process s oe where future values of the esemble caot be predcted from prevously observed values (a sequece of radom varables versus oe radom varable the model equato). May of the radom processes that you wll deal wth are odetermstc, every sample fucto measured must be descrbed based o t s probablty dstrbuto ad desty. xample clude: a process composed of depedet, detcally dstrbuted radom sample Browa moto Radom walk ypcal defto of ose A determstc radom process s oe where oe or more observed samples allow all future values of the sample fucto to be predcted (or pre-determed). For these processes, a sgle radom varable may exst for the etre esemble. Oce t s determed (oe or more measuremets) the sample fucto s kow for all t. xamples clude: Rado stato carrer frequecy ad phase values at a locato Fourer coeffcets of a measured sgal t A cos f t B s f t 0 0 where both A ad B are depedet radom varables that are fxed for ay partcular sample fucto of the possble esemble. A expoetal decay t A exp t, for t 0 where both A ad are depedet radom varables 0 he flght path of a baseball or football Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 8 of 8 C 3800

9 Problem 5-3. State whether each of the radom processes descrbed problem 5-. s determstc or odetermstc. a) A radom process whch the radom varable s the umber of cars per mute passg a gve traffc couter. b) he thermal ose voltage geerated by a resstor. c) he radom process defed problem 5-.. (Dscrete sample addto) d) he radom process that results whe a Gaussa radom process s passed through a deal half-wave rectfer. e) he radom process that results whe a Gaussa radom process s passed through a deal full-wave rectfer. f) A radom process havg sample fuctos of the form: t A cos B t where A s a costat, B s a R.V. expoetally dstrbuted from 0 to f, ad theta s uformly dstrbuted from 0 to p. Determstc: (f) No-determstc: (a), (b), (c), (d), (e) Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 9 of 8 C 3800

10 Statoary ad Nostatoary Radom Processes he probablty desty fuctos for radom varables tme have bee dscussed, but what s the depedece of the desty fucto o the value of tme, t, whe t s take? If all margal ad ot desty fuctos of a process do ot deped upo the choce of the tme org, the process s sad to be statoary (that s t does t chage wth tme). All the mea values ad momets are costats ad ot fuctos of tme! For ostatoary processes, the probablty desty fuctos chage based o the tme org or tme. For these processes, the mea values ad momets are fuctos of tme. I geeral, we always attempt to deal wth statoary processes or approxmate statoary by assumg that the process probablty dstrbuto, meas ad momets do ot chage sgfcatly durg the perod of terest. xamples: Resstor values (ose vares based o the local temperature) Wd velocty (vares sgfcatly from day to day) Humdty (though t ca chage rapdly durg showers) he requremet that all margal ad ot desty fuctos be depedet of the choce of tme org s frequetly more strget (tghter) tha s ecessary for system aalyss. A more relaxed requremet s called statoary the wde sese: where the mea value of ay radom varable s depedet of the choce of tme, t, ad that the correlato of two radom varables depeds oly upo the tme dfferece betwee them. hat s t ad t t t t 0 R 0 for t t You wll typcally deal wth Wde-Sese Statoary Sgals. the autocorrelato fucto s a key. Note: f a process s slowly tmg varyg, measuremets are used to estmate the curret mea ad varace so that we ca assume these are correct over the measuremet perod. hs creates a class of adaptve systems that ca follow slow varatos tme ad allow operato to cotue. Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 0 of 8 C 3800

11 Chapter 6 Iformato early: he Autocorrelato Fucto For a sample fucto defed by samples tme of a radom process, how alke are the dfferet samples? Defe: ad t he autocorrelato s defed as: R t t, t dx dx x x f x, x he above fucto s vald for all processes, statoary ad o-statoary. For WSS processes: For o-wss processes: R R t t t t R, t, t t t R t t, If the process s ergodc, the tme average s equvalet to the probablstc expectato, or lm x t xt dt x t xt ad R As a ote for thgs you ve bee computg, R t t R dx x f x, 0 0 lm x t dt x t Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 of 8 C 3800

12 Problem 5-4. State whether each of the radom processes descrbed problem 5-. may reasoably be cosdered statoary or ostatoary. If you descrbe a process as ostatoary, state the reaso for ths clam. a) A radom process whch the radom varable s the umber of cars per mute passg a gve traffc couter. b) he thermal ose voltage geerated by a resstor. c) he radom process defed problem 5-.. (Dscrete sample addto) d) he radom process that results whe a Gaussa radom process s passed through a deal half-wave rectfer. e) he radom process that results whe a Gaussa radom process s passed through a deal full-wave rectfer. f) A radom process havg sample fuctos of the form: t A cos B t where A s a costat, B s a R.V. expoetally dstrbuted from 0 to f, ad theta s uformly dstrbuted from 0 to p. Statoary: (b), (d), (e), (f) No- Statoary: (a) traffc chages based o tme of day, (c) depeds upo tme from 0-9, ostatoary, the beyod 9 t becomes statoary. Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 of 8 C 3800

13 rgodc ad Noergodc Radom Processes rgodcty deals wth the problem of determg the statstcs of a esemble based o measuremets from a sample fucto of the esemble. For ergodc processes, all the statstcs ca be determed from a sgle fucto of the process. hs may also be stated based o the tme averages. For a ergodc process, the tme averages (expected values) equal the esemble averages (expected values). hat s to say, x f x dx lm t dt or better stated as lm t dt Note that ergodcty caot exst uless the process s statoary! A oergodc process s oe where ay or all of these propertes do ot exst. All ostatoary processes are oergodc. A example of a statoary process that s ot ergodc s. t Y swt, 0 ad Y a r. v. For ths radom process, the esemble cossts of two radom varable, phase ad ampltude. For ths process, Y takes o dfferet values for dfferet sample fuctos, whle all processes take o the same radom phase oce t s kow. Whle t s exceedgly dffcult to prove ergodcty, t s customary to assume ergodcty for most of the problems that we wll deal wth (uless there s a obvous reaso ot to, lke ostatoary). Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 3 of 8 C 3800

14 Problem 5-5. State whether each of the radom processes descrbed problem 5-. may reasoably be cosdered ergodc or o-ergodc.. a) A radom process whch the radom varable s the umber of cars per mute passg a gve traffc couter. b) he thermal ose voltage geerated by a resstor. c) he radom process that results whe a Gaussa radom process s passed through a deal half-wave rectfer. d) A radom process havg sample fuctos of the form: t A cos B t where A s a costat, B s a R.V. expoetally dstrbuted from 0 to f, ad theta s uformly dstrbuted from 0 to p. rgodc: (b), (c) No- rgodc: (a) traffc chages (ot statoary) ad (d) sgfcat dffereces esemble members, therefore, tme based statstcs vary wth A. Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 4 of 8 C 3800

15 Measuremet of Process Parameters For a statoary process, the statstcal parameters of a radom process are derved from measuremets of the radom varables t at multple tme staces, t. As mght be atcpated, the statstcs may be geerated based o oe sample fucto. herefore, t s ot possble to geerate a esemble average. If the process s ergodc, the tme average s equvalet to the esemble average. As mght be expected, ergodcty s typcally assumed. x f x dx lm t dt Further, sce a fte tme average s ot possble, the statstcal values (sample mea ad varace) defed Chapter 4 are used to estmate the approprate momets. For a cotuous tme or dscrete sample fucto, ths becomes (smplfed) 0 t dt or N N k 0 t dt or etc N N k Note: there are otato challeges hear, whle the bar hat otatos worked for the tme based average, t wll ot work for powers so the book ad otes otatos may be a problem! As may be expected, we wll be formg the tme/sample statstcal elemets ad comparg or computg the probablstc elemets to see f they are equvalet (ergodcty). I some cases based statstcs that may be able to be corrected to become equvalet wll also be detfed. Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 5 of 8 C 3800

16 Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 6 of 8 C 3800 As before, for a radom varable dt t dt t 0 0 he dscrete verso of ths s a repeat of the sample mea Sample Mea, For dscrete samples, t s desred ad, fact, assumed that the observed samples are spaced far eough apart tme to be statstcally depedet. he d momet eeded to compute the varace of the estmated sample mea s computed as For depedet for for,

17 Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 7 of 8 C 3800 Ad the varace of the mea estmate s Var As before, as the umber of values goes to fty, the sample mea becomes a better estmate of the actual mea. As may be expected, the ubased estmate of the sample varace ca be defed as ˆ ˆ xercse 5-6. Show that the estmate of the varace s a ubased estmate of the true varace. ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

18 A Process for Determg Statoarty ad rgodcty a) Fd the mea ad the d momet based o the probablty b) Fd the tme sample mea ad tme sample d momet based o tme averagg. c) If the meas or d momets are fuctos of tme o-statoary d) If the tme average mea ad momets are ot equal to the probablstc mea ad momets or f t s ot statoary, the t s o ergodc. xamples: x f x dx ad x f x dx x ˆ lm x t dt ad x lm x t dt xamples: ) x t cost. Determstc, statoary, ergodc ) t A. x, for A a zero mea, ut varace Gaussa radom varable. Determstc, statoary, o-ergodc A. Determstc, statoary the mea but ot the varace(!), o-ergodc 3) x t A. swt, for A a uformly dstrbuted radom varable, t 4) x t B. rect, for B =+/- wth prob. 50%. No-determstc, Zero mea, statoary, ergodc (Bpolar/Bary commucatos)., for A ad/or theta radom varables... Determstc, statoary the mea but ot the varace(!), o-ergodc 5) x t A coswt Notes ad fgures are based o or take from materals the course textbook: Probablstc Methods of Sgal ad System Aalyss (3rd ed.) by George R. Cooper ad Clare D. McGllem; Oxford Press, 999. ISBN: B.J. Bazu, Sprg 05 8 of 8 C 3800