M645-POBBILIY & NDOM POCESS UNI-IV-COELION ND SPECL DENSIIES By K.VIJYLKSHMI Dp. of pplid mhmics
COELION ND SPECL DENSIIES Dfiniion: uo Corrlion h uo Corrlion of rndom procss {x}is dfind by xx xx im vrg pproch, E[ ] or, E[ ] xx, Lim d
Propris of CF.If {} is W.S.S procss, hn i xx xx ii i. xx xx funcionof isnvn xx iii E [ ] xx
.i If {} is Sionry procss nd i hs no priodic componns, hn Lim xx whr E[ ] ii If {} is Sionry procss nd i hs no priodic componns, hn xx will hv priodic componn wih h sm priod. iiiif is rgodic,zro mn nd hs no priodic componns, hn Lim xx
POBLEMS.Find h mn nd uocorrlion of h poisson procss. Soluion: h probbiliy lw of poisson procss {} wih prmr is,,,...,! } { n n n P P n n h mn of h poisson procss is! ] [ x x x x x x xp E
x x x! ] [ ] [ x x x p x x x x p x E ] [ x x x xp x p x x
! x x E x x x x x x!
Vr x E[ ] [ E x]
h uo corrlion of h poisson procss is xx, E[ ] E { [ ]} E { [ ] E[ ]} E [ ] E [ ] E [ ]}
Sinc is procss of indpndn incrmns.,., if xx }, min{
.Find h uocorrlion funcion of h priodic im funcion sin Soluion: d Lim, d Lim sin sin, d Lim sin sin, d Lim ] cos [cos,
d Lim d Lim cos cos, d Lim d cos cos, Lim Lim sin cos, cos, cos,
Soluion: W know h xx E[ ] L III ly xx E[ ] P P xx E[ P P ] xx E[ P] E[ P ] xx
4.h uo corrlion funcion of sionry rgodic procss wih no priodic componns is givn by 5 4 6 Find h mn nd vrinc of {}. Soluion: 5 4 6
hn by h propry, Lim xx whr E[ ] Lim [ 5 4 6 ] 5 E [ ] 5 lso by propry E [ ] 54 xx 5 4 6 9
Vrinc Vr x E[ ] [ E ] 9-5 4 Mn5 Vrinc4
5. sionry rndom procss {x} hs n CF givn by 5 6.5 Find h mn nd vrinc. Soluion: 36 4 5 36 6.5 4 Lim xx
4 6.5 36 5 Lim 4 5. 6 36 5 Lim 5. 6 5 4 Mn 9 4 65 36 5 ] [ E
Vr x E[ ] [ E ] 9-45 Hnc Mn nd vrinc5
6. sionry rndom procss {x} hs n CF givn by 9 Find h mn of h rndom vribl Y d nd vrinc of.
Soluion: 9 By h propry of CF Lim xx 9 Lim 9 9 9 E[ ] 3
E[ ] 9 9 Vr x E[ ] [ E ] 3 9
Givn Y d E[ Y ] E[ d] E[ Y ] E[ ] d 3 d 3[ ] 3 6
7.Find h mn squr vlu of h rndom procss whos uo corrlion is cos Soluion: Givn cos Mn squr vlu E [ ] cos
8.If {} is WSS procss wih uo corrlion funcion nd if Y Show h YY
Soluion: L.H.S ] [ Y Y E YY ]} ][ {[ E ] [ E ] [ ] [ ] [ ] [ E E E E [ ] [ ] [ ] [ E E E E
.H.S [ ] [ ] [ ] [ E E E E
Soluion: Y Y sin θ p. d. f f θ, < θ < π π
] [,.. Y Y E F C YY ] sin sin [ 4 θ θ E cos cos 4 θ θ E cos cos cos cos 4 4 θ θ θ θ E
} cos {cos } {cos } {cos 4 4 θ θ θ θ E E E E [ ] θ θ θ π θ θ π π π 4 cos } cos{ 8 cos 4 cos 4 4 4 4 4 4 E d d
8 cos{ 8 4 4 4 4 4 4 4 4 } cos{ 8 4 4 4 [ ] cos 8, 4 YY
COSS COELION FUNCION Dfiniion h cross corrlion funcion of wo rndom procsss nd Y is dfind by or xy xy, E[ Y ], E[ Y ] If nd Y r ls joinly WSS procss, hn xy E [ Y ] nd yx E [ Y ]
Orhogonl procsss Indpndnc of ndom Procsss wo rndom procsss nd Y r sisiclly indpndn if xy, E[ ] E[ Y ],
Propris of cross corrlion funcion If nd Y r WSS rndom procsss, hn xy yx xy xx yy 3 xy [ ] xx yy
Soluion: From h Cuchy-Schwrz inquliy { [ ]} E Y E[ ] E[ Y ]
{ [ ]} E Y E[ ] E[ Y ] [ xy ] xx yy xy xx yy
Soluion: f θ, < θ < π π
h cross corrlion of nd Yis xy, E [ Y ] [ cos θ sin θ ] E E [ sin θ sin ] E[sin θ ] E[sin ]
] [sin ] [sin θ E E θ θ π sin sin d θ π sin cos { } π sin cos cos 4
{ } π sin cos cos 4 { } sin cos cos 4 sin 4 sin xy
Joinly W.S.S procsss
Soluion: h cross corrlion of nd Yis xy, E [ Y ] E [ cos Bsin Bcos sin ] E B E [ cos cos sin sin ] cos sin E B sin cos
E B E cos [ cos cos sin sin ] sin E B sin cos sinc E B E E B E E B σ xy [ sin cos cos ], σ sin xy σ, sin Which implis nd Y r Joinly W.S.S procsss.
POWE SPECL DENSIY S i d i S d π Equion nd r winr-khinchin rlion.
Soluion: d S i σ α d i sin cos cos i d σ α
σ α cos d i x cosbxdx b σ α α
.h uo corrlion funcion of h rndom vribl is givn by for < Find h powr spcrum of h procss. Soluion: S S i d cos i sin d
S cos i sin d S cos d i S cos d
d S cos cos sin cos
cos [ ] cos S 4 sin
Soluion: S π i d b cos i sin d π
b cos i sin d π b i π cos d b cos d π
d b cos π b cos sin π [ ] π b cos
4.h powr spcrl dnsiy of rndom procss is givn by jβ S α, W < W W Drmin h CF of h procss. Soluion: S π i d π W W α jβ W i d
W W i d W j β α π W W i i W j j W j β β α π β β α π i i i i i i W j W W j sin cos sin π β π β π α W W W W
5.Find h CF of h procss {} for which h PSD is givn by Soluion: S, < S π i d cos i sin d π
cos i sin d π cos d π π sin cos sin 3
3 sin cos sin π 3 sin cos sin π [ ] π sin cos sin 3
Soluion: S π i d i d π
π d i π i i π i i i π sin
7.h CF of h poisson procss is givn by ε ε ε ε < >,, nd Find h PSD of h procss. Soluion: d S i
ε ε ε ε ε ε d d d S i i i ε ε ε ε d d S i i ε ε ε d F cos
sin cos F ε ε ε ε cos ε F ε F sin 4 ε ε
From h bl of Fourir rnsform πδ F sin ε πδ S sin 4 ε ε F
8.Show h h PSD of rndom procss is rl nd vrify h Soluion: S S S i d S cos i sin d S cos d i sin d
S cos d i sin d S cos d lso S S isrl cos d cos d S
Soluion: 4 5 9 4 S 4 9 4 5 4
4 5 4 4 5 4 3 5 4 3 5 3 8
8 3 5 3 4 From h bl of Fourir rnsforms α α α 8 5.. 3 3 4 8 6 5 6 5 h mn squr vlu of h procss
COSS SPECL DENSIY S Y nd Y i d S Y Y i d
Soluion: Y S Y π i d Y π W W ib W i d
W W i i W ib i W ib π π d W ib i W W Y π iw iw iw iw W ib i ib i ib π iw iw iw iw iw iw W jb b i
π i b jb W iw iw iw iw iw iw b b π i W sinw cosw sinw Y sin W b cosw b sin W π W πw [ W b sin W bw cos W ]
Soluion: h im vrg of cross corrlion is Lim Y, d B Lim sin d B Lim cos d B sin 4 { sin } Lim
B sin 4 { sin } Lim B sin B sin Considr h Fourir rnsform of h im vrg of cross corrlion is S Y B F sin
S Y B F sin From h Fourir rnsform bl F [ sin ] jπ[ δ δ ] S Y jπb δ δ [ ]