Mulvre me Sere Anl
Le { : } be Mulvre me ere. Denon: () = men vlue uncon o { : } = E[ ] or. (,) = Lgged covrnce mr o { : } = E{[ - ()][ - ()]'} or,
Denon: e me ere { : } onr e jon drbuon o,,, e me e jon drbuon o,,, or ll ne ube,,..., o nd ll coce o.
In ce en μ( ) nd ( ) E μ (,) = E{[ - ][ - ]'} = E{[ + - ][ + - ]'} = E{[ - - ][ 0 - ]'} = ( - ) or,. or.
Denon: e me ere { : } wel onr : or. nd or,. μ() μ (,) = ( - )
In ce () = E{[ + - ][ - ]'} = Cov( +, ) clled e Lgged covrnce mr o e proce { : }
e Cro Correlon Funcon nd e Cro Specrum
Noe: j () = (,j) elemen o (), cov, j nd clled e cro covrnce uncon o j nd. j j 0 jj clled e cro correlon uncon o j nd. 0
) Denon: j e clled e cro pecrum o j j nd. Noe: nce j () j (-) en j () comple. ) I j () = c j () - q j () en c j () clled e Copecrum (Concden pecrl den) nd q j () clled e qudrure pecrum
) I j () = A j () ep{ j ()} en A j () clled e Cro Amplude Specrum nd j () clled e Pe Specrum.
Denon: clled e Specrl Mr pp p p p p j p p F
e Mulvre Wener-Kncn Relon (p-vre) nd p p p p e Σ F d e p p p p F Σ
Lemm: Aume en F() : j ) Pove emdene: *F() 0 * 0, were n comple vecor. ) Hermn:F() = F*() = e Adjon o F() = e comple conjuge rnpoe o F()..e. j () =.
Corrollr: e c F() pove emdene lo men ll qure ubmrce long e dgonl ve pove deermnn Hence j j jj 0 nd or j jj j * j j j jj
Denon: K j j jj = Squred Coerenc uncon Noe: K j
Denon: j j rner uncon oced w j nd.
Applcon nd Emple o Mulvre Specrl Anl
Emple I - Lner Fler
denoe bvre me ere w zero men. Le =..., -, -, 0,,,... : Suppoe e me ere { : } conruced ollow:
e me ere { : } d o be conruced rom { : } b men o Lner Fler. E ' ' ' E E ' ' ' E ' ' '
connung ' ' ' d e ' ' ' d e ' ' ' d e e e ' ' ' d e e e ' ' '
connung e e d e A d u e pecrl den o e me ere { : } : e A
Commen A: A e clled e rner uncon o e lner ler. A clled e Gn o e ler wle rg A clled e Pe S o e ler.
Alo E E E
connung d e d e d A e
u cro pecrum o e bvre me ere : : A e
Commen B: K = Squred Coerenc uncon. A A
Emple II - Lner Fler w ddve noe e oupu
denoe bvre me ere w zero men. Le =..., -, -, 0,,,... : Suppoe e me ere { : } conruced ollow: v e noe {v : } ndependen o e ere { : } (m be we)
E v v E v E E ' ' ' v v E v E ' ' ' vv ' ' ' d e d e e vv
connung e A were A e vv d u e pecrl den o e me ere { : } : A vv
Alo E v E v E E
connung d e d e d A e
u cro pecrum o e bvre me ere : : A e
u K = Squred Coerenc uncon. A A A vv vv Noe o Sgnl Ro
Mulvre me Sere Anl
e Cro Correlon Funcon nd e Cro Specrum
Noe: j () = (,j) elemen o (), cov, j nd clled e cro covrnce uncon o j nd. j j 0 jj clled e cro correlon uncon o j nd. 0
) Denon: j e clled e cro pecrum o j j nd. Noe: nce j () j (-) en j () comple. ) I j () = c j () - q j () en c j () clled e Copecrum (Concden pecrl den) nd q j () clled e qudrure pecrum
) I j () = A j () ep{ j ()} en A j () clled e Cro Amplude Specrum nd j () clled e Pe Specrum. Noe: nd j j e j j e
now j j cov, cov, j cov, j j nd j j j e e j e * j
Denon: K j j jj = Squred Coerenc uncon Noe: K j
Denon: j j rner uncon oced w j nd.
Emple I - Lner Fler
denoe bvre me ere w zero men. Le =..., -, -, 0,,,... : Suppoe e me ere { : } conruced ollow:
u e pecrl den o e me ere { : } : e A Commen : A e clled e rner uncon o e lner ler. A clled e Gn o e ler wle rg A clled e Pe S o e ler.
e cro pecrum o e bvre me ere : : A e
Commen B: K = Squred Coerenc uncon. A A
Emple II - Lner Fler w ddve noe e oupu
denoe bvre me ere w zero men. Le =..., -, -, 0,,,... : Suppoe e me ere { : } conruced ollow: v e noe {v : } ndependen o e ere { : } (m be we)
e e pecrl den o e me ere { : } : A e cro pecrum o e bvre me ere : e A vv
Alo nd or = 0,,,..., m. ) co( ) n( I X X X b ) co( ) n( I Y Y Y b
Fnll b b Y X I b b b b b b X Y I b b b b I comple conjuge o
Noe: nd ) ep( I ) ep( C ) ep( I ) ep( C
Alo nd ) ep( I ) ep( C ) ep( I ) ep( C
Alo nd ) ep( I ) ep( C ) ep( I ) ep( C
e mple cro-pecrum, copecrum & qudrure pecrum
Recll e perodogrm ) co( ) n( I mpoc epecon 4 (). I Smlrl e mpoc epecon o 4 (). An mpoc unbed emor o () cn be obned b dvdng b 4. I
e mple cro pecrum ) ep( ˆ C I
e mple copecrum cˆ ˆ Re C co( )
e mple qudrure pecrum qˆ C Im ˆ n( )
e mple Cro mplude pecrum, Pe pecrum & Squred Coerenc
A Recll Cro Amplude Specrum c q Pe Specrum n q c K Squred Coerenc uncon c q
u er mple couner pr cn be dened n mlr mnner. Nmel ˆ A mple Cro Amplude Specrum ˆ mple Pe Specrum n cˆ q ˆ qˆ cˆ \ Kˆ ˆ ˆ ˆ cˆ ˆ qˆ ˆ mple Squred Coerenc uncon
Conen Emon o e Cro-pecrum ()
Dnell Emor
= e Dnell Emor o e Copecrum d d r r d c d c ˆ ˆ, = e Dnell Emor o e qudrure pecrum d d r r d q d q ˆ ˆ,
Weged Covrnce Emor
ˆ,, c q were w m wm C co ˆ,, w m wm C n : 0,,, w m weg uc w 0 ) 0 w ) w m m m w m re equence o ) w m 0 or m.
Agn once e Copecrum nd Qudrure Specrum ve been emed, e Cro pecrum, Amplude Specrum, Pe Specrum nd Coerenc cn be emed generll ollow ung eer e ) Dnell Emor or b) e weged covrnce emor o c () nd q ():
Nmel Aˆ cˆ qˆ ˆ n ˆ Kˆ cˆ qˆ qˆ cˆ ˆ cˆ ˆ ˆ ˆ qˆ ˆ