Processamento Digital de Sinal

Transcription

1 Deparaeo de Elecróica e Telecouicações da Uiversidade de Aveiro Processaeo Digial de ial Processos Esocásicos uar ado Processes aioar ad ergodic Correlaio auo ad cross Fucio Covariace Fucio Esiaes of he fucio: usig saples Correlaio ad Covariace Marices

2 ado vecors 3 Eaple: Joi Gaussia ado Vecor E VA COV,... COV, E COV, VA K COV, VA E f T ep.5 - K - / / K- covariace ari K- deeria of K

3 Y Eaple I: D- vecor >>=.5*rad,; >>=.*rad,; >> plo,,'*' >> ais- - Mea Vecor = C Covariace ari K = Eaple II: D-vecor >>=rad, >>plo,:,,:,'*' = K = Wha is he differece bewee EAMPLE I ad EAMPLE II? 3

4 fdp: probabili desi fucio E pdf: probabili desi fucio E

5 ado Process ealizaio Geeralizaio of he cocep of rado vecors: each er is he value of a rado variable i ie isa ado Processes: pes epo segudos - 3 aosras aosras - 3 eposegudos Tie coiuous or discree, rado variable coious or discree 5

6 Eaple I: Beroulli ado variable 5 realizaios of rado process Each saple alog realizaios is a rado variable Beroulli Eaple II: siusoid wih rado phase 3 ealizaios cos Phase : uiforl disribued rado variable,

7 Eaple III:siusoid wih rado apliude ealizaios of rado process. Acos Apliude: uiforl disribued rado variable Epeced Value ad Variace E E Variável aleaória VA E VA E 7

8 Coiuos ie: Auocorrelaio e Auocovariace Fucios ecod order Descripors:, C,, E,, Discree ie: Auocorrelaio ad Auocovariace ecod order descripors, C,, E,,

9 9 aioar Processes,..., i i p p i Idepede sae as a idepede rado vecor p p aioar Idepede ad ideicall disribued IID is saioar. Eaples: firs ad secod order descripio,,.5 se se A VA C se se A E se Ase iusoid : apliude u i f o r e l d i s r i b u e d,.,, cos cos,, C se iusoid : phase uiforel d i s r i b u e d,. Eaple : Epeced value is cosa o depedig o Auocovariace/Auocorrelaio: fucios of ie differeces.

10 W- Wide ese aioar The epeced value ea is cosa ce coiuous ou ce discree The auocorrelaio/ auocovariace: ou k g coious ou k g k discree Fucio of he differece bewee ie isas. Ergodic Process All sasisics ca be esiaed usig oe realizaio of he process The operaor epecaio is subsiued b averages alog he ie. For isace, r li r li

11 Ergodic process: ea Mea alog ie average of he saples of oe realizaio ea alog realizaios group ea T T T i d Esiaes alog ie /3 Auocorrelaio fucio Give saples he esiae is - scalig facor : oe, biased, ubiased ad oralized aiu of he fucio equal o

12 Esiaes alog ie /3 Auocovariace Fucio C C - scalig facor: oe, biased, ubiased ad oralized aiu equal o Esiiavas o epo 3/3 Cross- Correlaio ad Cross- Covariace fucios Give saples of a discree ie rado process / Y C Y Y Y Y C Value. a =

13 Eaple: siusoid phase rado variable - 8 The four realizaios ado process where K is he rado variable cos 8 k {,,,3}, values of rado variable wih equal probabili K Eaple: auocorrelaio fucio =:5 =cos*pi/8* subplo5 se,,grid o r,d=corr; subplo5 sed,r,'k', grid o r,d=corr,'biased'; subplo53 sed, r,'k',grid o r3,d=corr,'ubiased'; subplo5 sed,r3,'k',grid o r,d=corr,'coeff' subplo55 sed,r,'k',grid o

14 Esiaes versus heoreical.8. biased aalical ubiased Aalical resul d.5cos 8 d Eaple: Gaussia oise 5 whie oise ubiased biased d >>=rad,;.79 >> sd.885 ea

15 Eaple: Cross- correlaio fucio >> =rad,; >> subplo3 >>se,grid o >> =filer.5.5,,; >> subplo3, >>se,grid o >> r,d=corr,, biased ; >> subplo33, >>sed,r,grid o Auocorrelaio ari ad rado processes L elees of he auocorrelaio fucio orgaized io L L ari L L L oe: he auocorrelaio fucio is a eve fucio 5

16 Covoluio wih ari aipulaios Assuig wo sequeces L ad,,,,, L h The covoluio is h L h h h L L Lab. Assigee?????? h T T The ipulse respose ca also be approiaed b I h c T T if,

17 Quesios I- how ha T Esiaio of he auocorrelaio ari, wih he diagoal eries calculaed wih, -, -. aples. II- how ha T Y Y Y L Appedi: Leas quares oluio Give a sse of equaios A b If he colus of A are liearl idepede, he sse has a uique leas squares soluio give b A A T A T b The oal squared error TE T A b e b b b b 7

18 loger Appedi: eaple Bibliograph Paolo Pradoi, Mari Vierli, igal Processig for Couicaio Chaper 8 secios, ad 3 Available olie: hp:// Joh Proakis, Digial igal Processig, Preice Hall, 7. 8