Chapter 2 Probability and Stochastic Processes

Transcription

1 Chapte Pobablty ad Stochastc Pocesses

2 Table of Cotets Pobablty Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes Fuctos of Radom Vaables Statstcal Aveages of Radom Vaables Some Useful Pobablty Dstbutos Uppe Bouds o the Tal Pobablty Sums of Radom Vaables ad the Cetal Lmt Theoem Stochastc Pocesses Statstcal Aveages Powe Desty Spectum Respose of a Lea Tme-Ivaat System to a Radom Iput Sgal Samplg Theoem fo Bad-Lmted Stochastc Pocesses Dscete-Tme Stochastc Sgals ad Systems Cyclostatoay Pocesses

3 Itoducto Pobablty ad stochastc pocess s mpotat : The statstcal modelg of souces that geeate the fomato; The dgtzato of the souce output; The chaactezato of the chael though whch the dgtal fomato s tasmtted; The desg of the eceve that pocesses the fomatobeag sgal fom the chael; The evaluato of the pefomace of the commucato system. 3

4 . Pobablty Sample space o ceta evet of a de epemet: { } S,,3,4,5,6 The s outcomes ae the sample pots of the epemet. A evet s a subset of S, ad may cosst of ay umbe of sample pots. Fo eample: { } A,4 The complemet of the evet A, deoted by A, cossts of all the sample pots S that ae ot A: A {,3,5,6 } 4

5 Two evets ae sad to be mutually eclusve f they have o sample pots commo that s, f the occuece of oe evet ecludes the occuece of the othe. Fo eample: A {,4}; B {,3,6 } A ad A ae mutually eclusve evets. The uo sum of two evets a evet that cossts of all the sample pots the two evets. Fo eample: C,,3 D A. Pobablty { } A S 5 {,,3, } B C 6

6 . Pobablty The tesecto of two evets s a evet that cossts of the pots that ae commo to the two evets. Fo eample: E B C {,3 } Whe the evets ae mutually eclusve, the tesecto s the ull evet, deoted as φ. Fo eample: A A φ 6

7 . Pobablty Assocated wth each evet A cotaed S s ts pobablty PA. Thee postulatos: PA 0. The pobablty of the sample space s PS. Suppose that A,,,, ae a possbly fte umbe of evets the sample space S such that A Aj φ; j,,... The the pobablty of the uo of these mutually eclusve evets satsfes the codto: P A P A 7

8 . Pobablty Jot evets ad jot pobabltes two epemets If oe epemet has the possble outcomes A,,,,, ad the secod epemet has the possble outcomes B j, j,,,m, the the combed epemet has the possble jot outcomes A,B j,,,,, j,,,m. Assocated wth each jot outcome A,B j s the jot pobablty P A,B j whch satsfes the codto: 0 P A, B Assumg that the outcomes B j, j,,,m, ae mutually m eclusve, t follows that: P A, B P A j If all the outcomes of the two epemets ae mutually eclusve, m the: P A, B j j 8 j j

9 . Pobablty Codtoal pobabltes The codtoal pobablty of the evet A gve the occuece of the evet B s defed as: P A, B P A B P B povded PB>0. P A, B P A B P B P B A P A P A, B s tepeted as the pobablty of A That s, P A, B deotes the smultaeous occuece of A ad B. If two evets A ad B ae mutually eclusve, A B φ, the PAB 0. If B s a subset of A, we have A B B ad P A B. B. 9

10 . Pobablty Bayes theoem: If A,,,...,, ae mutually eclusve evets such that ad B s a abtay evet wth ozeo pobablty, the PAB, P B A P A PA B PB PB A PA PA epesets the a po pobabltes ad PA B s the a posteo pobablty of A codtoed o havg obseved the eceved sgal B. A j S j j 0

11 Statstcal depedece Whe the evets A ad B satsfy the elato PA,BPAPB, they ae sad to be statstcally depedet. Thee statstcally depedet evets A, A, ad A 3 must satsfy the followg codtos:. the, of does ot deped o the occuece the occuece of If A P B A P B A, B P A P B P B A P B A P,,,,, A P A P A P A A A P A P A P A A P A P A P A A P A P A P A A P. Pobablty

12 .. Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes Gve a epemet havg a sample space S ad elemets s S, we defe a fucto s whose doma s S ad whose age s a set of umbes o the eal le. The fucto s s called a adom vaable. Eample : If we flp a co, the possble outcomes ae head H ad tal T, so S cotas two pots labeled H ad T. Suppose we defe a fucto s such that: + s H s - s T Thus we have mapped the two possble outcomes of the co-flppg epemet to the two pots +,- o the eal le. Eample : Tossg a de wth possble outcomes S{,,3,4,5,6}. A adom vaable defed o ths sample space may be ss, whch case the outcomes of the epemet ae mapped to the teges,,6, o, pehaps, ss, whch case the possble outcomes ae mapped to the teges {,4,9,6,5,36}.

13 Gve a adom vaable, let us cosde the evet {.. Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes } whee We wte the pobablty of deote t smply by F,.e., s ay eal umbe the teval -,. ths evet as P F P, - < < The fucto F s called the pobablty dstbuto fucto of the adom vaable. It s also called the cumulatve dstbuto fucto CDF. 0 F F 0 ad F. ad 3

14 .. Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes Eamples of the cumulatve dstbuto fuctos of two dscete adom vaables. 4

15 .. Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes A eample of the cumulatve dstbuto fucto of a cotuous adom vaable. 5

16 .. Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes A eample of the cumulatve dstbuto fucto of a adom vaable of a med type. 6

17 .. Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes The devatve of the CDF F, deoted as p, s called the pobablty desty fucto PDF of the adom vaable. p F df, d p u du, < < < < Whe the adom vaable s dscete o of a med type, the PDF cotas mpulses at the pots of dscotuty of F: p P δ 7

18 8 { }. the aea ude the PDF the age s smply the evet The pobablty of. whee,, falls a teval adom vaable Detemg the pobablty that a d p F F P P F F P P P < < < < + < + >.. Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes

19 9 Multple adom vaables, jot pobablty dstbutos, ad jot pobablty destes: two adom vaables. 0,,, : Note that,, PDFs. the vaables ae called of obtaed fom tegatg ove oe ad The PDFs,,,, Jot PDF :,,, Jot CDF : F F F F d d p p p p d p p d p F p du du u u p P F.. Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes magal

20 Multple adom vaables, jot pobablty dstbutos, ad jot pobablty destes: multdmesoal adom vaables Supposethat Jot CDF Jot PDF F F.. Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes F p, p,,...,,,...,,..., d,,, 4,..., F,,,,..., 0.,,,...,, ae adom vaables. 4 P d, 4, p, p u 5 F,, u 4,...,,,..., u,...,.,...,,..., du du... du

21 Codtoal pobablty dstbuto fuctos cosde two adom vaables: Pobablt y of the evet < s the pobablt y that the adom vaable P < F.. Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes u, u du du p u, u u du p u, F, F p F codtoe p d o du p u, u p u du du du < du du.

22 Codtoal pobablty dstbuto fuctos cosde two adom vaables: Dvde both umeato ad deomato by ad let 0 The codtoal CDF of gve the adom vaable s gve by:, / F, / P F F.. Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes p u u p u du dudu, p Obseve that F 0 ad F. p u du *

23 .. Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes Codtoal pobablty dstbuto fuctos cosde two adom vaables: p, p p, p p Dffeetatg Equato * wth espect to, we obta p We may epess the jot PDF, tems of the codtoal PDFs, p o p, as: p p p 3

24 .. Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes Codtoal pobablty dstbuto fuctos cosde multdmesoal vaables: Jot PDF of the adom vaables,,,...,,,,...,,,...,,...,,..., p p p k k+ k+ Jot codtoal CDF:,,...,,..., F k k+... k p + p k+,...,,,..., k, k+,...,,..., k, k+,...,,,...,,..., 0 k k+ u, u,..., u,,..., dudu... du k k k F F F 4

25 .. Radom Vaables, Pobablty Dstbutos, ad Pobablty Destes Statstcally depedet adom vaables: If the epemets esult mutually eclusve outcomes, the pobablty of a outcome s depedet of a outcome ay othe epemet. The jot pobablty of the outcomes factos to a poduct of the pobabltes coespodg to each outcome. Cosequetly, the adom vaables coespodg to the outcomes these epemets ae depedet the sese that the jot PDF factos to a poduct of magal PDFs. The multdmesoal adom vaables ae statstcally depedet f ad oly f: F p,,..., F F... F,,..., p p... p 5

26 .. Fuctos of Radom Vaables Poblem: gve a adom vaable, whch s chaactezed by ts PDF p, deteme the PDF of the adom vaable Yg, whee g s some gve fucto of. Whe the mappg g fom to Y s oe-to-oe, the detemato of py s elatvely staghtfowad. Howeve, whe the mappg s ot oe-to-oe, as the case, fo eample, whe Y, we must be vey caeful ou devato of py. 6

27 .. Fuctos of Radom Vaables Tasfomato of a adom vaable 7

28 8 Eample.-: Ya+b; whee a ad b ae costat ad assume that a>0. + a b y p a y p a b y F d p a b y P y b a P y Y P y F Y Y a y-b -.. Fuctos of Radom Vaables

29 .. Fuctos of Radom Vaables Eample.- cot.: Ya+b; whee a ad b ae costat ad assume that a>0. 9

30 30 Eample.-: Ya 3 +b; a>0. The mappg betwee ad Y s oe-to-oe. [ ] elatoshp betwee the two PDFs yelds the desed Dffeetato wth espect to a b y Y Y p a b y a y p y a b y F a b y P y b a P y Y P y F.. Fuctos of Radom Vaables

31 .. Fuctos of Radom Vaables Eample.-3: Ya +b; a>0. The mappg betwee ad Y s ot oe-to-oe. 3

32 3 Eample.-3 cot.: Ya +b; a>0. Dffeetatg wth espect to y, we obta the PDF of Y tems of the PDF of the fom: + a b y F a b y F y F a b y P y b a P y Y P y F Y Y [ ] [ ] [ ] [ ] Y p y b a p y b a p y a y b a a y b a +.. Fuctos of Radom Vaables

33 .. Fuctos of Radom Vaables Eample.-3 cot.: Yga +b; a>0 has two solutos: y b y b, a a p Y y cossts of two tems coespodg to these two solutos: p [ y b / a] p [ y b / a] py y + g [ y b / a] g [ y b / a I geeal, suppose that,,, ae the eal oots of gy. The PDF of the adom vaable Yg may be epessed as p py y g ] 33

34 .. Fuctos of Radom Vaables Fucto of oe adom vaable dscete case: Suppose s a dscete adom vaable that ca have oe of values,,,. Let g be a scala-valued fucto. Yg s a dscete adom vaable that ca have oe of m, m, values y, y,, y m. If g s a oe-to-oe mappg, the m wll be equal to. If g s may-to-oe, the m wll be smalle tha. The CDF of Y ca be obtaed easly fom the CDF fucto of as: P Y y P whee the sum s ove all values of that map to y. 34

35 Fucto of oe adom vaable cotuous case: If s a cotuous adom vaable, the the pdf of Yg ca be obtaed fom the pdf of. Let y be a patcula value of Y ad let,,, be oots of the equato yg. That s yg g yg. P.. Fuctos of Radom Vaables y < Y y + y fy y y as y 0 P[ { : y < g y + y} ] 35

36 .. Fuctos of Radom Vaables Fucto of oe adom vaable cotuous case: y 36

37 Fucto of oe adom vaable cotuous case: < < + < < + + < < P y Y y y P P Sce the slope f Y y.. Fuctos of Radom Vaables y g y f Y y f g g P < < + f + f + f 3 3 of g f k g y + s y /, we have y g f g y + f g y g y 3 37

38 Fucto of seveal adom vaables cotuous case: Goal: fd the jot dstbuto of adom vaables Y, Y,, Y gve the dstbuto of elated adom vaables,,,,, whee Y g,,...,,,,..., Let s stat wth a mappg of two adom vaables oto two othe adom vaables: Y g, Y g Suppose ad y plae such that y <.. Fuctos of Radom Vaables g,,,,..., k ae the k, oots of y g, g,. We eed to fd the ego the, < y + y ad y < g, < y + y 38,

39 .. Fuctos of Radom Vaables Fucto of seveal adom vaables cotuous case: Thee ae k such egos as show the followg fgue fo the case of k3. 39

40 .. Fuctos of Radom Vaables Fucto of seveal adom vaables cotuous case: Each ego cossts of a paallelogam ad the aea of each paallelogam s equal to y jot pdf of Y J f, ad Y Y, Y y, y as g g y k g g f J, J,,, whee J the Jacoba of the tasfomato s defed as, By summg the cotbuto fom all egos, we obta the s 40

41 .. Fuctos of Radom Vaables Fucto of seveal adom vaables cotuous case: We ca geealze ths esult to the -vaate case as,,..., f y y y Y,,..., whee J deotes the Jacoba of the tasfomato defed by the followg detemat whe k f g g g J evaluatg at the th soluto g, g,..., g of y g [ g,g,..., g ]: J g g g g g g at,,..., g g g 4

42 Eample.-4 Y Y p Y The j A A y, p a y Y j Jacoba of.. Fuctos of Radom Vaables j,...,, whe e y,,..., { b } j A s ths ae a mat. j b j the Y j, elemets 4,,..., of tasfom ato s the J vese det A. b j yj, b j yj,..., b j j j mat j y A j - det. A

43 ..3 Statstcal Aveages of Radom Vaables The mea o epected value of, whch chaactezed by ts PDF p, s defed as: E m p d Ths s the fst momet of adom vaable. The -th momet s defed as: E p d Defe Yg, the epected value of Y s: [ g ] E Y E g p d 43

44 ..3 Statstcal Aveages of Radom Vaables The -th cetal momet of the adom vaable s: [ ] E Y E m m p d Whe, the cetal momet s called the vaace of the adom vaable ad deoted as : E m p d [ ] E E m I the case of two adom vaables, ad, wth jot PDF p,, we defe the jot momet as: k k p, 44 E d d

45 The jot cetal momet s defed as: E..3 Statstcal Aveages of Radom Vaables [ m m ] k m k If k, the jot momet ad jot cetal momet ae called the coelato ad the covaace of the adom vaables ad, espectvely. The coelato betwee ad j s gve by the jot momet: j 45 m p, d j p, j dd j E d

46 The covaace betwee ad j s gve by the jot cetal momet: µ E[ m m ] j..3 Statstcal Aveages of Radom Vaables m m m m + m m j j j j j j p, p, j j j E j mm j The mat wth elemets μ j s called the covaace mat of the adom vaables,,,,,. j 46 j d d d d p, j j j m m m m j d d j j j p, m m j j d d + m m j j

47 ..3 Statstcal Aveages of Radom Vaables Two adom vaables ae sad to be ucoelated f E j E E j m m j. Ucoelated Covaace μ j 0. If ad j ae statstcally depedet, they ae ucoelated. If ad j ae ucoelated, they ae ot ecessay statstcally depedetly. Two adom vaables ae sad to be othogoal f E j 0. Two adom vaables ae othogoal f they ae ucoelated ad ethe oe of both of them have zeo mea. 47

48 ..3 Statstcal Aveages of Radom Vaables Chaactestc fuctos The chaactestc fucto of a adom vaable s defed as the statstcal aveage: jv E e ψ jv Ψjv may be descbed as the Foue tasfom of p. The vese Foue tasfom s: jv p ψ jv e dv π Fst devatve of the above equato wth espect to v: dψ jv jv j e p d dv 48 e jv p d

49 Chaactestc fuctos cot. Fst momet mea ca be obtaed by: dψ jv E m j dv v 0 Sce the dffeetato pocess ca be epeated, -th momet ca be calculated by: d ψ jv E j dv v 0 Suppose the chaactestc fucto ca be epaded a Taylo sees about the pot v0: ψ jv..3 Statstcal Aveages of Radom Vaables d ψ jv v 49 jv ψ jv E 0 dv! 0 0! v

50 Chaactestc fuctos cot. Detemg the PDF of a sum of statstcally depedet adom vaables: jvy Y ψ Y jv E e E ep jv E Sce p If..3 Statstcal Aveages of Radom Vaables, jv e... the adom vaables,..., Y p p [ ψ jv ] jv e p,,..., dd... d ae statstca lly depede t,... p ae d depede t ad detcall y dstbute d ψ jv 50 ψ jv Y ψ jv

51 ..3 Statstcal Aveages of Radom Vaables Chaactestc fuctos cot. The PDF of Y s detemed fom the vese Foue tasfom of Ψ Y jv. Sce the chaactestc fucto of the sum of statstcally depedet adom vaables s equal to the poduct of the chaactestc fuctos of the dvdual adom vaables, t follows that, the tasfom doma, the PDF of Y s the - fold covoluto of the PDFs of the. Usually, the -fold covoluto s moe dffcult to pefom tha the chaactestc fucto method detemg the PDF of Y. 5

52 ..3 Statstcal Aveages of Radom Vaables Chaactestc fuctos fo -dmesoal adom vaables If,,,,, ae adom vaables wth PDF p,,,, the -dmesoal chaactestc fucto s defed as: ψ jv, jv,..., jv E ep j v... ep j v p,,..., dd... d Fo two dmesoal chaactestc fucto: j v + v ψ jv, jv e p, d d E ψ jv, v v 5 jv v v 0

53 53..4 Some Useful Pobablty Dstbutos Bomal dstbuto: Let what s the pobablty dstbuto fucto of Y? whee the,,,..., ae statstcally d, Y p P P 0 : s of PDF!!! 0 0 k y p p k k y k Y P y p Y k k k p p k k Y P k k k k k k δ δ

54 ..4 Some Useful Pobablty Dstbutos Bomal dstbuto: The CDF of Y s: F y P Y y whee [y] deotes the lagest tege m such that m y. [ y] k k p p k 0 k The fst two momets of Y ae: E Y p E Y p p The chaactestc fucto s: p ψ + p + jv j ν p pe p 54

55 55 Ufom Dstbuto The fst two momets of ae: The chaactestc fucto s: 3 b a ab b a Y E b a E b a ab b a E b a E a b jv e e j jva jvb ν ψ..4 Some Useful Pobablty Dstbutos

56 Gaussa Nomal Dstbuto The PDF of a Gaussa o omal dstbuted adom vaable s: p e π whee m s the mea ad s the vaace of the adom vaable. u m du t dt The CDF s: F..4 Some Useful Pobablty Dstbutos + e π m ef m / m um / du π m efc / e t dt 56

57 ..4 Some Useful Pobablty Dstbutos Gaussa Nomal dstbuto: 57

58 Gaussa Nomal Dstbuto ef ad efc deote the eo fucto ad complemetay eo fucto, espectvely, ad ae defed as: ef..4 Some Useful Pobablty Dstbutos e t t dt ad efc e dt ef π 0 ef--ef, efc--efc, ef0efc 0, ad ef efc0. Fo >m, the complemetay eo fuctos s popotoal to the aea ude the tal of the Gaussa PDF. π 58

59 ..4 Some Useful Pobablty Dstbutos Gaussa Nomal Dstbuto The fucto that s fequetly used fo the aea ude the tal of the Gaussa PDF s deoted by Q ad s defed as: t / Q e dt efc 0 π 59

60 ..4 Some Useful Pobablty Dstbutos Gaussa Nomal Dstbuto The chaactestc fucto of a Gaussa adom vaable wth mea m ad vaace s: jv m / jvm ψ / v jv e e d e π The cetal momets of a Gaussa adom vaable ae: k [ ] k 3 k eve k E m µ k 0 odd k The oday momets may be epessed tems of the cetal momets as: [ ] k k k E m k µ 0 60

61 6 Gaussa Nomal Dstbuto The sum of statstcally depedet Gaussa adom vaables s also a Gaussa adom vaable.. ad vaace s Gaussa - dstbuted wth mea Theefoe, ad whee / / y y y y v jvm v jvm Y m Y m m e e jv jv Y y y ψ ψ..4 Some Useful Pobablty Dstbutos

62 Ch-squae dstbuto If Y, whee s a Gaussa adom vaable, Y has a chsquae dstbuto. Y s a tasfomato of. Thee ae two type of ch-squae dstbuto: Cetal ch-squae dstbuto: has zeo mea. No-cetal ch-squae dstbuto: has o-zeo mea. Assumg be Gaussa dstbuted wth zeo mea ad vaace, we ca apply.-47 to obta the PDF of Y wth a ad b0; p Y y..4 Some Useful Pobablty Dstbutos p [ g [ y b y b / a] + / a] p [ g [ y b / a y b / a] ] 6

63 ..4 Some Useful Pobablty Dstbutos Cetal ch-squae dstbuto The PDF of Y s: p y The CDF of Y s: Y e πy y /, y 0 F Y y y y u / u du e du p 0 Y π 0 u The chaactestc fucto of Y s: ψ Y jv jv / 63

64 Ch-squae Gamma dstbuto wth degees of feedom. Y,,,,...,, ae statstcally depedet ad..4 Some Useful Pobablty Dstbutos detcally dstbuted d Gaussa adom vaables wth zeo mea ad vaace. The chaactestc fucto s: ψ Y jv / jv The vese tasfom of ths chaactestc fucto yelds the PDF: / -y/ py y y e, y 0 / Γ 64

65 ..4 Some Useful Pobablty Dstbutos Ch-squae Gamma dstbuto wth degees of feedom cot.. Γ p s the gamma fucto, defed as : Γ p Γ p Γ 0 t p -! π p dt, Γ Whe, the dstbuto yelds the epoetal dstbuto. e t p > 0 p a tege > 0 3 π 65

66 ..4 Some Useful Pobablty Dstbutos Ch-squae Gamma dstbuto wth degees of feedom cot.. The PDF of a ch-squae dstbuted adom vaable fo seveal degees of feedom. 66

67 67 Ch-squae Gamma dstbuto wth degees of feedom cot.. The fst two momets of Y ae: The CDF of Y s: Y E Y E y + Γ y u Y y du e u y F 0 / / / 0,..4 Some Useful Pobablty Dstbutos

68 ..4 Some Useful Pobablty Dstbutos Ch-squae Gamma dstbuto wth degees of feedom cot.. The tegal CDF of Y ca be easly mapulated to the fom of the complete gamma fucto, whch s tabulated by Peaso 965. Whe s eve, the tegal ca be epessed closed fom. Let m/, whee m s a tege, we ca obta: F Y m y / y y e k 0 k! k, y 0 68

69 No-cetal ch-squae dstbuto If s Gaussa wth mea m ad vaace, the adom vaable Y has the PDF: p Y..4 Some Useful Pobablty Dstbutos y+ m / ym y e cosh πy The chaactestc fucto coespodg to ths PDF s:, y 0 ψ Y jv jv / e jm v / jv 69

70 No-cetal ch-squae dstbuto wth degees of feedom Y..4 Some Useful Pobablty Dstbutos,,,,...,, ae statstcally depedet ad detcally dstbuted d Gaussa adom vaables wth mea m,,,...,, ad detcal vaace equal The chaactestc fucto s: to. ψ Y jv jv / jv ep m jv 70

71 ..4 Some Useful Pobablty Dstbutos No-cetal ch-squae dstbuto wth degees of feedom The chaactestc fucto ca be vese Foue tasfomed to yeld the PDF: y /4 s + y/ s py y e I / y, y 0 s whee, s s called the o-cetalty paamete: s m ad Iα s the αth-ode modfed Bessel fucto of the fst kd, whch may be epeseted by the fte sees: α + k / Iα, 0 k! Γ α + k+ k 0 7

72 7 No-cetal ch-squae dstbuto wth degees of feedom The CDF s: The fst two momets of a o-cetal ch-squaedstbuted adom vaable ae: + y u s Y du s u I e s u y F 0 / / 4 / s s s Y E s Y E y Some Useful Pobablty Dstbutos

73 ..4 Some Useful Pobablty Dstbutos No-cetal ch-squae dstbuto wth degees of feedom Whe m/ s a tege, the CDF ca be epessed tems of the geealzed Macum s Q fucto: m + a / Qm a, b b e Im a d a m k a + b / b Q a, b + e Ik ab k 0 a k a + b / b whee Q a, b e Ik ab, b > a > 0 k 0 a u By usg ad let a s, t s easly show: s y FY y Qm, 73

74 ..4 Some Useful Pobablty Dstbutos Raylegh dstbuto Raylegh dstbuto s fequetly used to model the statstcs of sgals tasmtted though ado chaels such as cellula ado. Cosde a cae sgal s at a fequecy ω 0 ad wth a ampltude a: s a ep jω t 0 The eceved sgal s s the sum of waves: s whee a ep ep [ j ω t + θ ] ep[ j ω t + θ ] jθ 0 a ep jθ 0 74

75 Raylegh dstbuto Defe : We have: whee:..4 Some Useful Pobablty Dstbutos ep jθ + a y cosθ a cosθ + ad sθ sθ Because s usually vey lage, the dvdual ampltudes a ae adom, ad 3 the phases θ have a ufom dstbuto, t ca be assumed that fom the cetal lmt theoem ad y ae both Gaussa vaables wth meas equal to zeo ad vaace: cosθ y y y j a a sθ + jy 75

76 76 Raylegh dstbuto Because ad y ae depedet adom vaables, the jot dstbuto p,y s The dstbuto p,θ ca be wtte as a fucto of p,y : + ep, π y y p p y p ep, cos s s cos / / / /,, π θ θ θ θ θ θ θ θ p y y J y p J p..4 Some Useful Pobablty Dstbutos

77 77 Raylegh dstbuto Thus, the Raylegh dstbuto has a PDF gve by: The pobablty that the evelope of the eceved sgal does ot eceed a specfed value R s gve by the coespodg cumulatve dstbuto fucto CDF: othewse 0 0, / 0 e d p p R π θ θ 0, ep / 0 / du e u F u R..4 Some Useful Pobablty Dstbutos

78 Raylegh dstbuto Mea: Vaace:..4 Some Useful Pobablty Dstbutos mea 0.49 Meda value of s foud by solvg:.77 Moets of R ae: π E[ R] p d π E[ R ] E [ R] p d meda E[ R π k 0 k / Γ + ] k meda 0 p d Most lkely value: ma { p R }. 78

79 ..4 Some Useful Pobablty Dstbutos Raylegh dstbuto 79

80 ..4 Some Useful Pobablty Dstbutos Raylegh dstbuto Pobablty That Receved Sgal Does t Eceed A Ceta Level R F p u du R 0 u u ep 0 u ep ep 0 du 80

81 Raylegh dstbuto Some Useful Pobablty Dstbutos Mea value: E[ R] p d mea 0 ep ep d d 0 0 ep ep d π ep d 0 π π π.533 8

82 8 Raylegh dstbuto: Mea squae value: ep ep ep ep ep ] [ + d d d d p R E Some Useful Pobablty Dstbutos

83 83 Raylegh dstbuto Vaace: 0.49 ] [ ] [ π π R E R E..4 Some Useful Pobablty Dstbutos

84 84 Raylegh dstbuto Most lkely value Most Lkely Value happes whe: dp / d ep ep 0 ep ep 4 p d dp..4 Some Useful Pobablty Dstbutos

85 85 Raylegh dstbuto Chaactestc fucto,..., 0,,! ; ; the cofluet hyupegeometc fucto : s ;, whee ;, s cos 0 / 0 / 0 / / 0 + Γ Γ Γ + Γ + + β β α β α β α π ψ k k v jv R k k k F a F e v j v F d v e j d v e d e e jv..4 Some Useful Pobablty Dstbutos

86 Raylegh dstbuto Chaactestc fucto cot. epessed as :..4 Some Useful Pobablty Dstbutos Beauleu 990 has show that F, ; a e a k 0 F, a k k k! ; a may be 86

87 Geealzed Raylegh dstbuto Cosde..4 Some Useful Pobablty Dstbutos R depedet, detcally dstbuted zeo mea Gaussa adom vaables. YR s ch-squae dstbuted wth degees of feedom. PDF s gve by: p R, whee,,,...,, ae statstcally / Γ e /, 0 87

88 88 Geealzed Raylegh dstbuto Whe s a eve umbe,.e., m, the CDF of R ca be epessed the closed fom: Fo ay tege, the k-th momet of R s: 0,! 0 / k e F m k k R 0, / Γ + Γ k k R E k k..4 Some Useful Pobablty Dstbutos

89 Rce dstbuto Whe thee s a domat statoay o-fadg sgal compoet peset, such as a le-of-sght LOS popagato path, the small-scale fadg evelope dstbuto s Rce. s scatteed waves 'ep[ j ω t + θ ] + [ +..4 Some Useful Pobablty Dstbutos A + + A + y + A cosθ y sθ 0 dect waves Aep jω t jy]ep jω t 0 0 ep[ j ω t 0 + θ ] 89

90 Rce dstbuto..4 Some Useful Pobablty Dstbutos By followg smla steps descbed Raylegh dstbuto, we obta: + A A ep I0 fo A 0, 0 p 0 fo < 0 whee π A A cosθ I0 ep dθ π 0 s the modfed zeoth-ode Bessel fucto. I 0 0! 90

91 Rce dstbuto..4 Some Useful Pobablty Dstbutos The Rce dstbuto s ofte descbed tems of a paamete K whch s defed as the ato betwee the detemstc sgal powe ad the vaace of the mult-path. It s gve by KA / o tems of db: A k db 0 log [db] The paamete K s kow as the Rce facto ad completely specfes the Rce dstbuto. As A 0, K - db, ad as the domat path deceases ampltude, the Rce dstbuto degeeates to a Raylegh dstbuto. 9

92 Rce dstbuto..4 Some Useful Pobablty Dstbutos 9

93 93 Rce dstbuto [ ] [ ] [ ] [ ] j a j t j t j jy A t j A t j t j A t j j t j A t j j a t j A t j a s ep 'ep whee ep ep ep 'ep ep ep 'ep ep ep ep ep θ θ θ ω ω ω θ ω ω ω θ ω ω θ ω θ ω..4 Some Useful Pobablty Dstbutos

94 Rce dstbuto Defe : We have: ad..4 Some Useful Pobablty Dstbutos 'ep jθ + A a + cosθ y a + cosθ + ad A sθ sθ Because s usually vey lage, the dvdual ampltudes a ae adom, ad 3 the phases θ have a ufom dstbuto, t ca be assumed that fom the cetal lmt theoem ad y ae both Gaussa vaables wth meas equal to zeo ad vaace: y y j cosθ a a y sθ + jy 94

95 95 Rce dstbuto Because ad y ae depedet adom vaables, the jot dstbuto p,y s The dstbuto p,θ ca be wtte as a fucto of p,y : + ep, π y y p p y p y y J y p J p θ θ θ θ θ θ θ cos s s cos / / / /,,..4 Some Useful Pobablty Dstbutos

96 96 Rce dstbuto cos ep ep cos ep s cos ep ep, θ π θ π θ θ π π θ A A A A A y p..4 Some Useful Pobablty Dstbutos

97 97 Rce dstbuto + othewse 0 0 cos ep ep, gve by : pobablty desty fucto pdf The Rce dstbuto has a 0 0 d A A d p p π π θ θ π θ θ..4 Some Useful Pobablty Dstbutos

98 Rce dstbuto..4 Some Useful Pobablty Dstbutos Just as the Raylegh dstbuto s elated to the cetal chsquae dstbuto, the Rce dstbuto s elated to the ocetal ch-squae dstbuto. To llustate ths elato, let Y +, whee ad ae statstcally depedet Gaussa adom vaables wth meas m,,, ad commo vaace. Y has a o-cetal ch-squae dstbuto wth ocetalty paamete s m + m. The PDF of Y, obtaed fom Equato.-8 fo, s: s + y / py y e I 0 y, y 0 98

99 Rce dstbuto Defe a ew adom vable R fom Equato.-40 by a smple chage of + s / s pr e I, 0 0 Ths PDF chaactezes the statstc of the evelope of a sgal coupted by addtve aowbad Gaussa ose. It s also used to model the sgal statstcs of sgals tasmtted though some ado chaels. The CDF of R s obtaed fom Equato.-4: F R..4 Some Useful Pobablty Dstbutos 99 y. The PDF of R, obtaed vaable,s : s Q,, 0 Q s gve.-3

100 Geealzed Rce dstbuto R, whee,,..., Gaussa adom vaables wth meas m, detcal vaaces equal to. paamete s The adom vaable R dstbuto wth degees of by Equato.-8: p R..4 Some Useful Pobablty Dstbutos gve by Equato.-9. s / / e Y + s 00 ae statstcally depedet,,,,...,, ad has a o - cetal ch -squae / feedom ad o - cetalty I / s Its PDF s gve 0

101 0 Geealzed Rce dstbuto The CDF s: I the specal case whe m/ s a tege, we have: The k-th momet of R s:. gve by Equato.- whee s F F P Y y P R P F Y Y R 0,, s Q F m R 0 ; ;, / / + Γ + Γ k s k F k e R E s k k..4 Some Useful Pobablty Dstbutos

102 Nakagam m-dstbuto Fequetly used to chaacteze the statstcs of sgals tasmtted though mult-path fadg chaels. PDF s gve by Nakagam 960 as: p R..4 Some Useful Pobablty Dstbutos Γ m Ω m m Ω E E[ R the fadg fgue. Ω R Ω, ] The paamete m s defed as m m e m / Ω m the ato of momets, called 0

103 Nakagam m-dstbuto The -th momet of R s: E..4 Some Useful Pobablty Dstbutos Γ m + / R Γ m Ω m / By settg m, the PDF educes to a Raylegh PDF. 03 PDF fo the Nakagam m-dstbuto. Show wth Ω.

104 Logomal dstbuto: Let..4 Some Useful Pobablty Dstbutos l R, whee ad vaace. s omally dstbuted wth mea m The PDF of R s gve by: p l m / e π 0 < 0 0 The logomal dstbuto s sutable fo modelg the effect of shadowg of the sgal due to lage obstuctos, such as tall buldgs, moble ado commucatos. 04

105 Multvaate Gaussa dstbuto Assume that,,,,, ae Gaussa adom vaables wth meas m,,,,; vaaces,,,,; ad covaaces μ j,,j,,,. The jot PDF of the Gaussa adom vaables,,,,, s defed as p..4 Some Useful Pobablty Dstbutos,...,, / / π det M M deotes the covaace mat wth elemets {μ j }; deotes the colum vecto of the adom vaables; m deote the colum vecto of mea values m,,,,. M - deotes the vese of M. deotes the taspose of. ep m M m 05

106 ..4 Some Useful Pobablty Dstbutos Multvaate Gaussa dstbuto cot. Gve v the -dmesoal vecto wth elemets υ,,,,, the chaactestc fucto coespodg to the - dmetoal jot PDF s: jv ψ jv E e ep jm v v Mv 06

107 ..4 Some Useful Pobablty Dstbutos B-vaate o two-dmesoal Gaussa The bvaate Gaussa PDF s gve by: p, π ρ m ρ m m + m ep π ρ m µ m,, µ E m m m M µ µ j ρj, j, 0 ρj j ρ ρ M, M ρ ρ ρ 07

108 ..4 Some Useful Pobablty Dstbutos B-vaate o two-dmesoal Gaussa ρ s a measue of the coelato betwee ad. Whe ρ0, the jot PDF p, factos to the poduct p p, whee p,,, ae the magal PDFs. Whe the Gaussa adom vaables ad ae ucoelated, they ae also statstcally depedet. Ths popety does ot hold geeal fo othe dstbutos. Ths popety ca be eteded to -dmesoal Gaussa adom vaables: f ρ j 0 fo j, the the adom vaables,,,,, ae ucoelated ad, hece, statstcally depedet. 08

109 Lea tasfomato of Gaussa adom vaables Let YA, A - Y, whee A s a osgula mat. The Jacoba of ths tasfomato s J/det A. The jot PDF of Y s: p y / / π det M det A whee m..4 Some Useful Pobablty Dstbutos / π det Q y Am ad / ep Q AMA The lea tasfomato of a set of jotly Gaussa adom vaables esults aothe set of jotly Gaussa adom vaables. 09 ep A y m M A y m y m Q y m y y

110 ..4 Some Useful Pobablty Dstbutos Pefomg a lea tasfomato that esults statstcally depedet Gaussa adom vaables Gaussa adom vaables ae statstcally depedet f they ae pawse-ucoelated,.e., f the covaace mat Q s dagoal: QAMA D, whee D s a dagoal mat. Sce M s the covaace mat, oe soluto s to select A to be a othogoal mat A A - cosstg of colums that ae the egevectos of the covaace mat M. The, D s a dagoal mat wth dagoal elemets equal to the egevalues of M. To fd the egevalues of M, solve the chaactestc equato: detm-λi0. 0

111 ..5 Uppe Bouds o the Tal Pobablty Chebyshev equalty Suppose s a abtay adom vaable wth fte mea m ad fte vaace. Fo ay postve umbe δ: Poof: P δ m δ m δ m p d m p d δ pd δ P m δ m δ

112 ..5 Uppe Bouds o the Tal Pobablty Chebyshev equalty It s appaet that the Chebyshev equalty s smply a uppe boud o the aea ude the tals of the PDF py, whee Ym,.e., the aea of py the tevals -,δ ad δ,. Chebyshev equalty may be epessed as: [ FY δ FY δ] δ o, equvaletly, as [ F m + δ F m δ] δ

113 Chebyshev equalty Aothe way to vew the Chebyshev boud s wokg wth the zeo mea adom vaable Y-m. Defe a fucto gy as: g..5 Uppe Bouds o the Tal Pobablty Y Uppe-boud gy by the quadatc Y/δ,.e. The tal pobablty Y δ Y < δ ad E Y 0 [ g Y ] P δ [ ] Y E Y g Y E g Y Y δ y E δ δ δ δ 3

114 ..5 Uppe Bouds o the Tal Pobablty Chebychev equalty A quadatc uppe boud o gy used obtag the tal pobablty Chebyshev boud Fo may pactcal applcatos, the Chebyshev boud s etemely loose. 4

115 Cheoff boud..5 Uppe Bouds o the Tal Pobablty The Chebyshev boud gve above volves the aea ude the two tals of the PDF. I some applcatos we ae teested oly the aea ude oe tal, ethe the teval δ, o the teval -, δ. I such a case, we ca obta a etemely tght uppe boud by ove-boudg the fucto gy by a epoetal havg a paamete that ca be optmzed to yeld as tght a uppe boud as possble. Cosde the tal pobablty the teval δ,. vy δ Y δ g Y e ad g Y s defed as g Y 0 Y < δ whee v 0 s the paamete to be optmzed. 5

116 Cheoff boud..5 Uppe Bouds o the Tal Pobablty The epected value of gy s E v Y δ [ g y ] P Y δ E e Ths boud s vald fo ay υ 0. 6

117 Cheoff boud..5 Uppe Bouds o the Tal Pobablty The tghtest uppe boud s obtaed by selectg the value of that mmzes Ee υy-δ. A ecessay codto fo a mmum s: d v Y δ E e 0 dv d dv d Ee E e dv E Y δ vy δ vy δ vy δ [ ] e vδ v Yδ vy [ ] 0 e E Ye δ E e E vy vy Ye δe e 0 7

118 Cheoff boud Let vˆ be the soluto, pobablty s : vˆ δ vy ˆ δ e E e P Y the uppe boud o the oe -sded tal A uppe boud o the lowe tal pobablty ca be obtaed a smla mae, wth the esult that vˆ δ vˆ Y δ e E e δ < 0 P Y..5 Uppe Bouds o the Tal Pobablty 8

119 Cheoff boud..5 Uppe Bouds o the Tal Pobablty Eample.-6: Cosde the Laplace PDF pye - y /. The tue tal pobablty s: δ δ y e dy e P Y δ 9

120 Cheoff boud Eample.-6 cot. vy E Ye Sce..5 Uppe Bouds o the Tal Pobablty v + v vy δ vy E e E Ye vˆ δ P Y v fo δ >> : + 0, + δ δ δ 0 vy e vˆ + + δ δ we obta + v v must be postve + δ δ δ δ e fo Chebyshev boud P Y E e v δ + v δ 0

121 Cheoff boud fo adom vaable wth a ozeo mea: Let Y-m, we have: P Y δ P m δ P m + δ P δ m δm g 0 < δm v δm g e Defe: ad uppe-bouded as: v ˆ vˆ δm We ca obta the esult: P δm e E e Fo δ<0 P m δ P m + δ P δ E v δ m e P..5 Uppe Bouds o the Tal Pobablty vˆδ v e E e m ˆ δ m m

122 ..6 Sums of Radom Vaables ad the Cetal Lmt Theoem Sum of adom vaables Suppose that,,,,, ae statstcally depedet ad detcally dstbuted adom vaables, each havg a fte mea m ad a fte vaace. Let Y be defed as the omalzed sum, called the sample mea: The mea of Y s Y y m E E Y m

123 3 Sum of adom vaables The vaace of Y s: A estmate of a paamete ths case the mea m that satsfes the codtos that ts epected value coveges to the tue value of the paamete ad the vaace coveges to zeo as s sad to be a cosstet estmate. [ ] m m m m E E E m E m E Y m E Y j j j j j y y Sums of Radom Vaables ad the Cetal Lmt Theoem

124 4 Weak law of lage umbes The tal pobablty of Y ca be uppe-bouded. The Chebyshev equalty appled to Y s: I the lmt as, the above equato becomes: The pobablty that the estmate of the mea dffes fom the tue mea m by moe tha δδ>0 appoaches zeo as appoaches fty. The uppe boud coveges to zeo elatvely slowly,.e., vesely wth. δ δ δ δ m P m Y P y y 0 lm δ m P..6 Sums of Radom Vaables ad the Cetal Lmt Theoem

125 5 The Cheoff boud [ ]. the s ay oe of Note : ep ep 0. ad whee ep v v v v v m m m m y e E e e E e v E e v E m v E P m P m P Y m m m δ δ δ δ δ δ δ δ δ δ δ > +..6 Sums of Radom Vaables ad the Cetal Lmt Theoem *

126 ..6 Sums of Radom Vaables ad the Cetal Lmt Theoem The Cheoff boud cot. The paamete υ that yelds the tghtest uppe boud s obtaed by dffeetatg Equato * ad settg the devatve equal to zeo. Ths yelds the equato: E v v e δ E e 0 The boud o the uppe tal pobablty s ˆ m [ ] v m v P ˆδ δ m e E e, δ m > m I a smla mae, the lowe tal pobablty s uppebouded as: vˆδ v m ˆ P Y δ e E e, δ < m [ ] m m 6

127 Cetal lmt theoem Cosde the case whch the adom vaables,,,,, beg summed ae statstcally depedet ad detcally dstbuted, each havg a fte mea m ad a fte vaace. Defe a omalzed adom vaable U wth a zeo mea ad ut vaace. m U,,,..., Let..6 Sums of Radom Vaables ad the Cetal Lmt Theoem Y U. Note that Y has zeo mea ad ut vaace. We wsh to deteme the CDF of Y the lmt as. 7

128 ..6 Sums of Radom Vaables ad the Cetal Lmt Theoem Cetal lmt theoem The chaactestc fucto of Y s: ψ Y U jv ψu jv jvy jv E e E ep jv ψ U Whee U deotes ay of the U, whch ae detcally dstbuted. 8

129 Cetal lmt theoem Epad the chaactestc fucto of U a Taylo sees: ψ EU 9 jv 3 v v v 3 U j + j EU EU + EU 3...! 3! Sce EU 0,ad, the above equato smplfes to: v v ψ U j + R v, whee R v, / deotes the emade. We ote that R v, appoaches zeo as. ψ Y..6 Sums of Radom Vaables ad the Cetal Lmt Theoem v R v, v R v, jv + lψ Y jv l +

130 ..6 Sums of Radom Vaables ad the Cetal Lmt Theoem Cetal lmt theoem Fo small value of, l + v R v, v R v, lψ Y jv v / lm lψ Y jv v lmψ Y jv e The above equato s the chaactestc fucto of a Gaussa adom vaable wth zeo mea ad ut vaace. We ca coclude that the sum of statstcally depedet ad detcally dstbuted adom vaables wth fte mea ad vaace appoaches a Gaussa CDF as

131 ..6 Sums of Radom Vaables ad the Cetal Lmt Theoem Cetal lmt theoem Although we assumed that the adom vaables the sum ae detcally dstbuted, the assumpto ca be elaed povded that addtoal estctos ae mposed o the popetes of the adom vaables. If the adom vaables ae..d., sum of 30 adom vaables s adequate fo most applcatos. If the fuctos f ae smooth, values of as low as 5 ca be used. Fo moe fomato, efeed to Came 946 ad Papouls Pobablty, Radom Vaables, ad Stochastc Pocesses, pp. 4-, 3d Edto. 3

132 . Stochastc Pocesses May of adom pheomea that occu atue ae fuctos of tme. I dgtal commucatos, we ecoute stochastc pocesses : The chaactezato ad modelg of sgals geeated by fomato souces; The chaactezato of commucato chaels used to tasmt the fomato; The chaactezato of ose geeated a eceve; The desg of the optmum eceve fo pocessg the eceved adom sgal. 3

133 Itoducto. Stochastc Pocesses At ay gve tme stat, the value of a stochastc pocess s a adom vaable deed by the paamete t. We deote such a pocess by t. I geeal, the paamete t s cotuous, wheeas may be ethe cotuous o dscete, depedg o the chaactestcs of the souce that geeates the stochastc pocess. The ose voltage geeated by a sgle essto o a sgle fomato souce epesets a sgle ealzato of the stochastc pocess. It s called a sample fucto. 33

134 Itoducto cot.. Stochastc Pocesses The set of all possble sample fuctos costtutes a esemble of sample fuctos o, equvaletly, the stochastc pocess t. I geeal, the umbe of sample fuctos the esemble s assumed to be etemely lage; ofte t s fte. Havg defed a stochastc pocess t as a esemble of sample fuctos, we may cosde the values of the pocess at ay set of tme stats t >t >t 3 > >t, whee s ay postve tege. I geeal, the adom vaables t t,,,...,, ae chaactezed statstcally by the jot PDF p,,...,. t t t 34

135 . Stochastc Pocesses Statoay stochastc pocesses Cosde aothe set of adom vaables t + t, t + t,,...,, whee t s a abtay tme shft. These adom t + t t + t t + t vaables ae chaactezed by the jot PDF p,,...,. The may o may ot be detcal. fo all t jot PDFs of p the adom vaables,,..., p,,..., t t ad all, t s sad to be statoay t the Whe they ae detcal,.e., Whe the jot PDFs ae dffeet, the stochastc pocess s o-statoay. t + t t + t t ad t + t t + t,,,...,, stct sese. whe 35

136 .. Stochastc Pocesses Aveages fo a stochastc pocess ae called esemble aveages. The th momet of the adom vaable s defed as : E p t t t d t I geeal, the value of the th momet wll deped o the tme stat t f the PDF of Whe the pocess s statoay, t depeds o t p p t + t t. fo all t. Theefoe, the PDF s depedet of tme, ad, as a cosequece, the th momet s depedet of tme. t 36

137 Two adom vaables:.. Stochastc Pocesses t,,. The coelato s measued by the jot momet:, Sce ths jot momet depeds o the tme stats t ad t, t s deoted by φt, t. φt, t s called the autocoelato fucto of the stochastc pocess. Fo a statoay stochastc pocess, the jot momet s: Aveage powe the pocess t: φ0e t. t E p d d t t t t t t t t t t E φ t, t φ t t φ τ φ τ E E E φ τ t t + τ t + τ t t t τ ' ' 37

138 .. Stochastc Pocesses Wde-sese statoay WSS A wde-sese statoay pocess has the popety that the mea value of the pocess s depedet of tme a costat ad whee the autocoelato fucto satsfes the codto that φt,t φt -t. Wde-sese statoaty s a less stget codto tha stct-sese statoaty. If ot othewse specfed, ay subsequet dscusso whch coelato fuctos ae volved, the less stget codto wde-sese statoaty s mpled. 38

139 .. Stochastc Pocesses Auto-covaace fucto The auto-covaace fucto of a stochastc pocess s defed as: µ t {[ ][ ]}, t E t m t t m t φ t, t m t m t Whe the pocess s statoay, the auto-covaace fucto smplfes to: µ t, t µ t t µ τ φ τ m Fo a Gaussa adom pocess, hghe-ode momets ca be epessed tems of fst ad secod momets. Cosequetly, a Gaussa adom pocess s completely chaactezed by ts fst two momets. 39

140 .. Stochastc Pocesses Aveages fo a Gaussa pocess p Suppose that t s a Gaussa adom pocess. At tme stats tt,,,,, the adom vaables t,,,,, ae jotly Gaussa wth mea values mt,,,,, ad auto-covaaces: µ t [ ], t j E t m t t m t j,, j,,..., If we deote the covaace mat wth elemets μt,t j by M ad the vecto of mea values by m, the jot PDF of the adom vaables t,,,,, s gve by:,,..., ep m M m / / π det M If the Gaussa pocess s wde-sese statoay, t s also stct-sese statoay. 40

141 .. Stochastc Pocesses Aveages fo jot stochastc pocesses Let t ad Yt deote two stochastc pocesses ad let t t,,,,, Y t j Yt j, j,,,m, epeset the adom vaables at tmes t >t >t 3 > >t, ad t >t >t 3 > >t m, espectvely. The two pocesses ae chaactezed statstcally by the jot PDF: p t, ' ' ' t...,, t, y, y,..., y t t t m The coss-coelato fucto of t ad Yt, deoted by φ y t,t, s defed as the jot momet: φy t, t E t Yt The coss-covaace s: t t t t t t y p, y d dy µ t, t φ t, t m t m t y y y 4

142 .. Stochastc Pocesses Aveages fo jot stochastc pocesses Whe the pocess ae jotly ad dvdually statoay, we have φ y t,t φ y t -t, ad μ y t,t μ y t -t : φ τ EY E Y EY φ τ y t t + τ τ τ y ' ' ' ' t t t t The stochastc pocesses t ad Yt ae sad to be statstcally depedet f ad oly f : p,,...,, y ', y ',..., y ' p,,..., p y ', y ',..., y ' t t t t t m m fo all choces of t ad t ad fo all postve teges ad m. The pocesses ae sad to be ucoelated f t φ t, t E E Y t, t 0 y t t t t µ y t t t t 4

143 .. Stochastc Pocesses Comple-valued stochastc pocess A comple-valued stochastc pocess Zt s defed as: Z t t + jy t whee t ad Yt ae stochastc pocesses. The jot PDF of the adom vaables Z t Zt,,,,, s gve by the jot PDF of the compoets t, Y t,,,,. Thus, the PDF that chaactezes Z t,,,,, s: p,,...,, y, y,..., y t t t t t t The autocoelato fucto s defed as: φ zz t, t E Zt Z t E t + jy t t jy t φ t, t + φyy t, t + j φy t, t φy t, t { } 43 **

144 .. Stochastc Pocesses Aveages fo jot stochastc pocesses: Whe the pocesses t ad Yt ae jotly ad dvdually statoay, the autocoelato fucto of Zt becomes: φ zz t, t φzz t t φzz τ φ ZZ τ φ * ZZ-τ because fom **: φ zz τ EZZ t ' ' ' ' t EZ Z EZZ τ φ t zz τ + τ t t t+ τ 44

145 .. Stochastc Pocesses Aveages fo jot stochastc pocesses: Suppose that Ztt+jYt ad WtUt+jVt ae two comple-valued stochastc pocesses. The coss-coelato fuctos of Zt ad Wt s defed as: φ t t E Z W, E t + jy t U t jv t + + zw t t { φ } u t, t φyu t, t j φyu t, t φu t, t Whe t, Yt,Ut ad Vt ae pawse-statoay, the coss-coelato fucto become fuctos of the tme dffeece. φ τ EZ W EZ ' W ' EWZ ' ' zw t t τ φ τ t + τ t t t + τ wz 45

146 .. Powe Desty Spectum A sgal ca be classfed as havg ethe a fte ozeo aveage powe fte eegy o fte eegy. The fequecy cotet of a fte eegy sgal s obtaed as the Foue tasfom of the coespodg tme fucto. If the sgal s peodc, ts eegy s fte ad, cosequetly, ts Foue tasfom does ot est. The mechasm fo dealg wth peodc sgals s to epeset them a Foue sees. 46

147 .. Powe Desty Spectum A statoay stochastc pocess s a fte eegy sgal, ad, hece, ts Foue tasfom does ot est. The spectal chaactestc of a stochastc sgal s obtaed by computg the Foue tasfom of the autocoelato fucto. The dstbuto of powe wth fequecy s gve by the fucto: j τ Φ φ τ π f f e dτ The vese Foue tasfom elatoshp s: φ τ Φ f e j πfτ df 47

148 .. Powe Desty Spectum 0 φ 0 Φ f df E t Sce φ0 epesets the aveage powe of the stochastc sgal, whch s the aea ude Φf, Φf s the dstbuto of powe as a fucto of fequecy. Φf s called the powe desty spectum of the stochastc pocess. If the stochastc pocess s eal, φτ s eal ad eve, ad, hece Φf s eal ad eve..-3 If the stochastc pocess s comple, φτφ*-τ ad Φf s eal because: * * j π fτ * j π fτ' Φ f φ τ e dτ φ τ ' e dτ ' jπ fτ e dτ f φτ Φ 48

149 .. Powe Desty Spectum Coss-powe desty spectum Fo two jotly statoay stochastc pocesses t ad Yt, whch have a coss-coelato fucto φ y τ, the Foue tasfom s: j τ Φ f e π f d y φ τ Φ y f s called the coss-powe desty spectum. Φ * y.-5 * jπfτ * f φ τ e dτ φ τ φ y y If t ad Yt ae eal stochastc pocesses y jπfτ τ e dτ Φ f y y e τ jπfτ j f f π τ φ τ e dτ f Φ y f Φy f Φ Φ * y y y dτ 49

150 ..3 Respose of a Lea Tme-Ivaat System to a Radom Iput Sgal Cosde a lea tme-vaat system flte that s chaactezed by ts mpulse espose ht o equvaletly, by ts fequecy espose Hf, whee ht ad Hf ae a Foue tasfom pa. Let t be the put sgal to the system ad let yt deote the output sgal. y t h τ tτ dτ Suppose that t s a sample fucto of a statoay stochastc pocess t. Sce covoluto s a lea opeato pefomed o the put sgal t, the epected value of the tegal s equal to the tegal of the epected value. my E Y t h τ E tτ dτ m h τ dτ mh 0 The mea value of the output pocess s a costat. 50

151 ..3 Respose of a Lea Tme-Ivaat System to a Radom Iput Sgal The autocoelato fucto of the output s: φ yy * t t E Y Y, t t h h [ ] * * β h α E t β t α * β h α φ t t + α β dαdβ dαdβ If the put pocess s statoay, the output s also statoay: * + φ τ h α h β φ τ α β dαdβ yy 5

152 ..3 Respose of a Lea Tme-Ivaat System to a Radom Iput Sgal The powe desty spectum of the output pocess s: j φ τ τ π fτ Φ yy f yy e d by makg τ 0 τ+α-β h α h β φ τ + α β e dτdαdβ Φ f H f * jπ fτ The powe desty spectum of the output sgal s the poduct of the powe desty spectum of the put multpled by the magtude squaed of the fequecy espose of the system. 5

153 ..3 Respose of a Lea Tme-Ivaat System to a Radom Iput Sgal Whe the autocoelato fucto φ yy τ s desed, t s usually ease to deteme the powe desty spectum Φ yy f ad the to compute the vese tasfom. jπ fτ φyy τ Φyy f e df Φ j π fτ f H f e df The aveage powe the output sgal s: φ 0 yy Φ f H f df Sce φ yy 0E Y t, we have: f H f Φ df 0 53 vald fo ay Hf.

154 ..3 Respose of a Lea Tme-Ivaat System to a Radom Iput Sgal Suppose we let Hf fo ay abtaly small teval f f f, ad H f 0 outsde ths teval. The, we have: f f Φ f Ths s possble f a oly f Φ f 0 fo all f. df 0 Cocluso: Φ f 0 fo all f. 54

155 ..3 Respose of a Lea Tme-Ivaat System to a Radom Iput Sgal Eample.- Suppose that the low-pass flte s ected by a stochastc pocess t havg a powe desty spectum Φ fn 0 / fo all f. A stochastc pocess havg a flat powe desty spectum s called whte ose. Let us deteme the powe desty spectum of the output pocess. The tasfe fucto of the low-pass flte s: H f H R + f R jπ fl + jπfl / + π L/ R f R 55

156 ..3 Respose of a Lea Tme-Ivaat System to a Radom Iput Sgal Eample.- cot. The powe desty spectum of the output pocess s: f Φ yy N0 + π L / R f The vese Foue tasfom yelds the autocoelato fucto: φ yy τ N0 + / RN0 e 4L R/ L π L R τ 56 f e jπ fτ df

157 Coss-coelato fucto betwee yt ad t φ y Fucto of t -t..3 Respose of a Lea Tme-Ivaat System to a Radom Iput Sgal, [ ] * * t t E Y h α E t α t t t h The stochastc pocesses Wth t -t τ, we have: α φ t t α dα φy t t t ad Y t ae jotly statoay. φ y τ h α φ τ α dα I the fequecy doma, we have: Φ y f Φ f H f If the put pocess s whte ose, the coss coelato of the put wth the output of the system yelds the mpulse espose ht to wth a scale facto. 57 dα

158 ..4 Samplg Theoem fo Bad-Lmted Stochastc Pocesses A detemstc sgal st that has a Foue tasfom Sf s called bad-lmted f Sf0 fo f >W, whee W s the hghest fequecy cotaed st. A bad-lmted sgal s uquely epeseted by samples of st take at a ate of f s W samples/s. The mmum ate f N W samples/s s called the Nyqust ate. Samplg below the Nyqust ate esults fequecy alasg. 58

159 ..4 Samplg Theoem fo Bad-Lmted Stochastc Pocesses The bad-lmted sgal sampled at the Nyqust ate ca be ecostucted fom ts samples by use of the tepolato fomula: s t s W s πw t W πw t W whee {s/w} ae the samples of st take at t/w, 0,±, ±,. Equvaletly, st ca be ecostucted by passg the sampled sgal though a deal low-pass flte wth mpulse espose htsπwt/πwt. 59

160 ..4 Samplg Theoem fo Bad-Lmted Stochastc Pocesses The sgal ecostucto pocess based o deal tepolato ca be llustated the followg fgue: 60

161 ..4 Samplg Theoem fo Bad-Lmted Stochastc Pocesses A statoay stochastc pocess t s sad to be badlmted f ts powe desty spectum Φf0 fo f >W. Sce Φf s the Foue tasfom of the autocoelato fucto φτ, t follows that φτ ca be epeseted as: φ τ φ W s πw τ W πw τ W whee {φ/w} ae the samples of φτ take at τ/w, 0,±, ±,. 6

162 ..4 Samplg Theoem fo Bad-Lmted Stochastc Pocesses If t s a bad-lmted statoay stochastc pocess, the t ca be epeseted as: t W s πw t W πw t W whee {/W} ae the samples of t take at t/w, 0,±, ±,. Ths s the samplg epesetato fo a statoay stochastc pocess. A 6

163 ..4 Samplg Theoem fo Bad-Lmted Stochastc Pocesses The samples ae adom vaables that ae descbed statstcally by appopate jot pobablty desty fuctos. The sgal epesetato A s easly establshed by showg that: s πw t E W πw t t 0 Equalty betwee the samplg epesetato ad the stochastc pocess t holds the sese that the mea squae eo s zeo. 63 W W

164 ..5 Dscete-Tme Stochastc Sgals ad Systems Dscete-tme stochastc pocess cosstg of a esemble of sample sequeces {} ae usually obtaed by ufomly samplg a cotuous-tme stochastc pocess. The mth momet of s defed as: [ m] m p The autocoelato sequece s: E d * * φ, k E k k p, The auto-covaace sequeces s: * µ, k φ, k E E k k d d k 64

165 ..5 Dscete-Tme Stochastc Sgals ad Systems Fo a statoay pocess, we have φ,k φ-k, μ,k μ-k, ad µ k φ k whee m E s the mea value. A dscete-tme statoay pocess has fte eegy but a fte aveage powe, whch s gve as: E φ 0 The powe desty spectum fo the dscete-tme pocess s obtaed by computg the Foue tasfom of φ. Φ f φ 65 e m j πf

166 ..5 Dscete-Tme Stochastc Sgals ad Systems The vese tasfom elatoshp s: φ Φ f jπf The powe desty spectum Φf s peodc wth a peod f p. I othe wods, Φf+kΦf fo k0,±,±,. The peodc popety s a chaactestc of the Foue tasfom of ay dscete-tme sequece. e df 66

167 ..5 Dscete-Tme Stochastc Sgals ad Systems Respose of a dscete-tme, lea tme-vaat system to a statoay stochastc put sgal. The system s chaactezed the tme doma by ts ut sample espose h ad the fequecy doma by the fequecy espose Hf. H f h jπf The espose of the system to the statoay stochastc put sgal s gve by the covoluto sum: y e h k k k 67

168 ..5 Dscete-Tme Stochastc Sgals ad Systems Respose of a dscete-tme, lea tme-vaat system to a statoay stochastc put sgal. The mea value of the output of the system s: k my E y h k E k k 0 m h k m H whee H0 s the zeo fequecy [dect cuet DC] ga of the system. 68

169 ..5 Dscete-Tme Stochastc Sgals ad Systems The autocoelato sequece fo the output pocess s: φyy k E y y + k j j h h j E + k j φ h h j k j+ By takg the Foue tasfom of φ yy k ad substtutg the elato Equato.-49, we obta the coespodg fequecy doma elatoshp: Φ yy f Φ f H f Φ yy f, Φ f, ad Hf ae peodc fuctos of fequecy wth peod f p. 69

170 Cyclostatoay Pocesses Fo sgals that cay dgtal fomato, we ecoute stochastc pocesses wth statstcal aveages that ae peodc. Cosde a stochastc pocess of the fom: t a g t T whee {a } s a dscete-tme sequece of adom vaables wth mea m a Ea fo all ad autocoelato sequece φ aa kea* a +k /. The sgal gt s detemstc. The sequece {a } epesets the dgtal fomato sequece that s tasmtted ove the commucato chael ad /T epesets the ate of tasmsso of the fomato symbols. 70

171 The mea value s: Cyclostatoay Pocesses E t E a g t T a m g tt The mea s tme-vayg ad t s peodc wth peod T. The autocoelato fucto of t s: φ t+ τ, t E t+ τ t m m aa m τ E a a g t T g t+ mt m g t T g t τ mt φ + 7

172 Cyclostatoay Pocesses We obseve that φ t+ τ + kt, t+ kt φ t+ τ, t fo k±,±,. Hece, the autocoelato fucto of t s also peodc wth peod T. Such a stochastc pocess s called cyclostatoay o peodcally statoay. Sce the autocoelato fucto depeds o both the vaables t ad τ, ts fequecy doma epesetato eques the use of a two-dmesoal Foue tasfom. The tme-aveage autocoelato fucto ove a sgle peod s defed as: T φ τ φ, T t+ τ t dt T 7

173 Cyclostatoay Pocesses Thus, we elmate the te depedece by dealg wth the aveage autocoelato fucto. The Foue tasfom of φ τ yelds the aveage powe desty spectum of the cyclostatoay stochastc pocess. Ths appoach allows us to smply chaacteze cyclostatoay pocess the fequecy doma tems of the powe spectum. The powe desty spectum s: j π fτ Φ f e d φ τ τ 73