Auto-correlation. Window Selection: Hamming. Hamming Filtered Power Spectrum. White Noise Auto-Covariance vs. Hamming Filtered Noise

Transcription

1 Correltio d Spectrl Alsis Applictio 4 Review of covrice idepedece cov cov with vrice : ew rdom vrile forms. d For idepedet rdom vriles - Autocorreltio Autocovrice cptures covrice where I geerl. for oise process tht Recll R oise Power the DC vlue theorem For white oise ] [ π i S R DFT e S R Zero-Me Gussi oise Power Spectrum E{P }. R

2 Auto-correltio Widow Selectio: mmig filtermmig; R R. >> for :56 R sum.*circshift'-'; ed mmig Filtered Power Spectrum White oise Auto-Covrice vs. mmig Filtered oise Filtered oiseimge imoisei gussi ; _utocov corroiseimge; figure;imgesc_utocov/8*8;colormpgr;is'imge' Imge oise Field Autocovrice Ufiltered figure;imgescfftshiftsfft_utocov/8*8;colormpgr;is'imge' Imge oise Field Power Spectrum

3 Filtered wc.6; order ; mmig Widow _utocov corroiseimge_filtered; figure;imgesc_utocov/8*8;colormpgr;is'imge' Filtered wc.6; order ; mmig Widow _utocov corroiseimge_filtered; figure;imgesc_utocov/8*8;colormpgr;is'imge' Imge oise Field Autocovrice Imge oise Field Power Spectrum fmri Simultio Widowig vs. Filterig Widow pplied i temporl or sptil domi to reduce spectrl lege d rigig rtifct Widows fll ito specilied set of fuctios geerll used for spectrl lsis Filter pplied to reduce oise i.e. oise mtchig or to degrde or improve sptil resolutio Some cross-over: oe method of filter desig is the widow method which uses widow fuctios for freuec spce modultig fuctios. Widowig vs. Filterig Mthemticll g f p "Filter" g t f t w t "Widow" I G ξ F ξ P ξ G f F f W f Filterig MP 574

4 Outlie Review of FIR/IIR Filters Z-trsform Differece Eutio Filter Desig Widowig Power Spectr Correltio d Covolutio Emple from Prof. olde s otes Widowig d Spectrl Estimtio Weier/Adptive Filters Decovolutio -Trsform s Alsis Tool Smpled versio discrete versio of the Lplce trsform: e st where T is the smplig period. DFT d -trsform re relted: e iωt where s e iωt T i e h ω Lplce to -Trsform iω s h dt t f e s F st iω Cotiuous FT Discrete FT Im Re ω s o-cusl sigls uit circle -Trsform d Lier Sstems Stted more geerll: o o D h h The iverse - trsform T{f} f g h Differece Eutio Implemettio Shift theorem of -trsform: M M X Y X Differece Eutio Implemettio Shift theorem of -trsform: M M X Y X FIR

5 FIR Coefficiets d Impulse Respose FIR filter: h for FIR filters FIR vs. IIR filters Fiite impulse respose FIR implies lier sstem tht is lws stle There re o poles Ifiite impulse respose IIR is ol stle if poles re iside the uit circle or pole coicides with ero. IIR Sstem IIR Stilit Zeros o t: - Poles t:.5±.5.75 o Im uit circle o Re fvtoolba fvtoolba B [ - -]; A[ ]

6 fvtoolba Ustle B [ - -]; A[ ] Ustle Fiite impulse respose FIR B [ - -]; A[ ] B [ - -]; A[] Defiitio of Stilit h < FIR filter Desig Widowig Simpl tructe IIR filter Rectgulr Widow: hd h Otherwise h h w e ω d d e ω W

7 Mtl: fdtool

8 filter i Mtl FILTER Oe-dimesiol digitl filter. Y FILTERBAX filters the dt i vector X with the filter descried vectors A d B to crete the filtered dt Y. The filter is "Direct Form II Trsposed" implemettio of the stdrd differece eutio: * * *-... *- - * *- Eportig Filter Coefficiets