10.3 Autocorrelation Function of Ergodic RP 10.4 Power Spectral Density of Ergodic RP 10.5 Normal RP (Gaussian RP)

Transcription

1 ENGG450 Probabiliy ad Saisics for Egieers Iroducio 3 Probabiliy 4 Probabiliy disribuios 5 Probabiliy Desiies Orgaizaio ad descripio of daa 6 Samplig disribuios 7 Ifereces cocerig a mea 8 Comparig wo reames 9 Ifereces cocerig variaces A Radom Processes

2 Iroducio 3 Probabiliy 4 Probabiliy bili disribuios ib i 0 Radom Processes 0. Defiiio of RP 0. Saioary RP & Ergodic RP 0.3 Auocorrelaio Fucio of Ergodic RP 0.4 Power Specral Desiy of Ergodic RP 0.5 Normal RP Gaussia RP 5 Probabiliy desiies Orgaizaio & descripio 6 Samplig disribuios 7 Ifereces.. mea 8 Comparig reames 9 Ifereces.. variaces A Radom processes

3 0.3 Auocorrelaio Fucio of Ergodic RP 3 Df Def: he auocorrelaio fucio R, of a radom process is he esemble average of he produc &, i.e. R, = E [ ] p, ;, d d Suppose he radom process is saioary which implies p, ;, = p, ; where = -. or or R, = = R p, ; d d d if is also ergodic.

4 Def: he auocovariace fucio of a saioary radom process of mea is = E[ ] Def: he auocorrelaio fucio of a saioary radom process is R = E[ + ] If he radom process is ergodic, he = d Def. he covariace of radom variables &Y is = E[ - Y-Y ] Def: he ormalized auocovariace fucio of a saioary radom process wih variace is = [, ]. Def. he ormalized covariace of r.v.s & Y 0 0 E[ Y Y ] E[ ]E[ Y Y ]

5 Auocorrelaio & Auocovariace 5 Fid: he relaioship bewee he auocorrelaio fucio R & auocovariace fucio. Soluio: For simpliciy of oaio, le U= F, V= F+, As E[U V] ] = R U V = E[ U- V- ] = E[UV] E[U ] - E[ V] + E[] E[U ] = E[ V] = E[] = R = E[ F F+ ] = E[ F- F+ - ] Noe: so = R - Def: wo radom processes ad Y are ucorrelaed if E[ Y ] = 0, for ay ad. Def:, E[ covariace E [ ] ] Def. wo radom variables ad are ucorrelaed if = E[ - ] = 0.

6 Properies of Auocorrelaio Fucio R 6 Le, Y & Z be ergodic saioary i radom processes ad heir auocorrelaio fucios be R, R y ad R z respecively. R0 0 R = R- i.e. R is symmeric w.r.. =0. 3 R is a maimum a he origi, i.e. R0 R d d 4 Z = R τ z d R d R z 5 Z = d R v R u du dv

7 Properies of Auocorrelaio Fucio R 7 Noe:, Y & Z are ergodic ad saioary. 6 Z = + Y, where & Y are ucorrelaed processes wih zero mea. R z = R R y Def: wo radom processes & Y are ucorrelaed if E[ Y ]=0 0, for ay ad. 7 Z = Y, where & Y are saisically idepede processes. R z = R R y Def: wo radom processes & Y are saisically idepede if & Y are idepede for ay ad.

8 Properies of Auocorrelaio Fucio R 8 Prove: R0 0 Pf: R 0 Noe: R d d R0 is he mea square value of a radom process. If represes volage or curre io a resisor, he R0 is he average power dissipaed o he resisor. radom sigal geeraor resisor

9 Properies of Auocorrelaio Fucio R 9 Prove: R = R- Proof: R = + d - d = R-

10 3 Prove R0 R Properies of Auocorrelaio Fucio R 0 Proof: Schwarz s Iequaliy f where f & g are arbirary real fucios & he iegrals are over arbirary is. Cosider he case where f =, g = + & he iegral is from - o. d g d f g d d d We have akig he i, he iequaliy implies R0 R0 R R0 R d

11 Properies of Auocorrelaio Fucio R d d 4 Prove: Z = R z d R Proof: R z = + Noe: R = E[ + ] By he defiiio of derivaio ' 0 R d he auocorrelaio fucio R z is { E - } { - 0 }

12 Properies of Auocorrelaio Fucio R d d 4 co Prove : Z = R z d R d R z = E[ ] R R R R R R - R + R 0 dr - dr d d

13 Properies of Auocorrelaio Fucio R 3 d d 4 Prove co: Z = R z Proof co: d R d R z 00 dr - dr d d d 0 R ' R ' 0 d R d d R d

14 Properies of Auocorrelaio Fucio R 4 5 Prove: Z = z d R R u dudv v Proof: d Rz d Z R d R d so R R u z v du dv

15 Properies of Auocorrelaio Fucio R 5 6 Prove: Z = + Y, where & Y are ucorrelaed R z = R + R y Proof: R z = Z Z+ Def: wo radom processes ad y are ucorrelaed if E[ y ] = 0, for ay ad. = [ + Y] [+ + Y+ ] = + + Y Y+ + Y+ + Y + = R + R y

16 Properies of Auocorrelaio Fucio R 6 7 Prove: Z = Y, where & Y are saisically idepede processes R z = R R y Proof: R z = Z Z+ = [ Y] [+ Y+ ] Def: wo radom processes ad Y are saisically idepede if ad Y are idepede, for ay ad. = [ + ] [Y Y+ ] = R R y

17 0.4 Power Specral Desiy of Ergodic RP 7 sigal Noise r = + I may applicaios, we eed o compare he sigal sregh o he oise sregh. Sregh ca be measured as eergy or power depedig o wheher he sigal is aperiodic, periodic or radom. Aperiodic sigal which is assumed o have fiie eergy deermiisic, o-zero oly i a fiie duraio has is sregh measured usig eergy. Periodic sigal deermiisic, o-zero over ifiie duraio ad radom sigal o-zero over ifiie duraio have ifiie eergy have heir sregh measured usig power. Sregh ca also be measured i ime domai or frequecy domai.

18 sigal Sigal Power ad Parseval s heorem 8 r = + Noise Sregh ca also be measured i ime domai or frequecy domai. Paraeval s heorem saes ha sregh i he wo domais are he same. Parseval s heorem for aperiodic sigal: oal eergy compued i ime domai d Χ f df oal eergy compued i frequecy domai Fourier rasform has may versios. I all versios, he raio of he eergy level i frequecy domai ad he eergy i ime domai is cosa. I some versios, he cosas are o equal o. Hece, Parseval s heorem has may versios depedig o he Fourier rasforms used. Here, we assume ha he cosa is.

19 oal eergy E Periodic fucio Aperiodic fucio Ergodic saioary radom process d d is fiiei Isaaeous power P Average power P Parseval s hm Auocorrelaio fucio R d 0 average oal d power eergy 0 c i i d 0 d d power d l im d average Χ f df G f df d d d Fourier Aalysis Fourier Series Fourier rasform Fourier rasform i j j f c i e Χ f e df j f R G f e df i / i j ci e d j f Χ f e d j f G f R e d / G f Χ f

20 Aperiodic -fucio oal eergy y where d Y Power Specral Desiy Power Desiy Specrum 0 f f area = f Y f F { y } df y e jf d Gf power specral desiy [wa/hz] G f df = average power of fhe radom sigal i he bad f ad f [wa] Ergodic radom process Average power d f G f df where G f Χ f df G f 0 f f 0 = average power of he compoe f 0 i he radom sigal = 0 wa Noe: Gf 0 f Gf = G-f

21 Wieer-Khichie heorem h m If radom process is of power specral desiy Gf & auocorrelaio fucio R, he Gf & R cosiue a Fourier rasform pair; i.e., & j f jf G f R τ e d R τ G f e df Noe: R τ τ d. Here, we assume is Pf. - F { R τ } F { } F { - } If F { } Χ f, he F { } If is real, he Χ f F { R τ} Χ f Χ f Χ f G f Χ f. Χ f. ergodic &saioary ad defie G f Χ f I some books, Gf F{R}. h τ d τ h τ 3. R is real ad eve Gf is real ad eve.

22 Whie Noise Def. Whie oise is ay radom sigal whose power specral desiy is equal o a cosa ad so idepede of frequecy. Gf / Noe: By defiiio, Gf is a eve fucio which implies Gf has a egaive frequecy par eacly he same as ha of he posiive frequecy par. η j f η he auocorrelaio is herefore R τ e df δ τ. / Ideal whie oise coais a ifiie amou of power ad is o physically realizable. However, may radom processes are referred o as whie i he sese ha over he frequecy bad of ieres, is Gf is cosa. If Gf is o fla, he oise is said o be colored. f

23 0.5 Gaussia Radom Processes 3 If r.v.s ad Y are Gaussia disribued wih zero mea ad saisically idepede, heir joi p.d.f. p,y is he produc of he margial p.d.f.s p, y y ep { y y } oe : p ep If he r.v.s are o idepede, d he he above equaio is modified d by iroducig a cross-produc erm i he epoe. p, y ep y y y y y As y, so we have y p, y ep y y y y 0 y 0 Y Y covariace Y Y Y Y

24 Def: Radom variable is Gaussia disribued if p ep. Def: Radom variables & Y are joily Gaussia disribued if y y p, y ep wih 0. Y y y h m : If r.v.s & Y are joily Gaussia, he & Y are margially Gaussia disribued. h m : R.v.s & Y are joily Gaussia iff a + b Y is Gaussia disribued for all a & b. h m : If r.v.s & Y are joily Gaussia & ucorrelaed, he & Y are idepede. oe: If r.v.s & Y are idepede of ay disribuio, he & Y are ucorrelaed.

25 -D D Gaussia Disribuio 5 Def If,,, are r.v.s he he colum vecor = [,,, ] is called a radom vecor. Def he correlaio mari [R] ad covariace mari [C] of a -D vecor are [ R ] E[ [ C ] E [ ] - η - η ] where η E [ ] [,,..., ]. E[ ] E[ ] E[ 3 ] E[ ] E[ ] E[ ] E[ 3 ] E[ ] Hece, [R] = E[ 3 ] E[ 3 ] E[ 3 3 ] E[ 3 ] : : : : E[ ] E[ ] E[ 3 ] E[ ]

26 Def,, are zero-mea radom variables wih covariace i [C] h j i l G i if h D j i df i mari [C]. hey are joily Guassia if he -D joi p.d.f. is - ep,...,, - } ] [ { C p ad ] [ of is he deermia where C 3. E ] [ ] [ C h m he r.v.s,,, are joily Gaussia iff he sum 3 j y a + a + + a is a Gaussia r.v. for all a, a,, a..

27 Def A radom process is said o be Gaussia or ormal, if he r.v.s.,,, are joily Gaussia for ay,,,,. he joi p.d.f. p,,;, for a saioary zero-mea ui-variace Gaussia radom process is - τ - ρ τ - ρ ep τ where =, =, E[ ]=0, = ad auocovariace fucio is E[ - - ]/ = E[ + ], Noe: he 3 d -order momes are of = =; 0 of = = & =E[ +]=. he saisics i i of a Gaussia radom process are compleely l deermied by is s -order ad d -order momes.

28 Gaussia Radom Processes 8 Def. wo real radom processes ad are said o be joily Gaussia, or joily ormal, if he r.v.s,,,,,,,m,,,,,,, are joily Gaussia for ay m,,,,,,,,m,,,,,,, ; ha is he joi disribuio of he radom variables is Gaussia. Noe: hree or more radom processes are similarly defied o be joily Gaussia.