ELCE5180 Digital Signal Processing

Transcription

1 ELCE580 Digital Sigal Prossig Assigmt : Disrt Fourir Trasform DFT am Class&Stut ID Aim. To stuy t Disrt Fourir Trasform.. Us DFT to aalyz t DTFT. 3. Us t FFT to imlmt t fast ovolutios. Itroutio Fast Fourir Trasform FFT: FFT is a algoritm to ru t amout of omutatios ivolv i t DFT. MATLAB rovis a futio all fft to omut t FFT. DFT is t saml vrsio of DTFT j X X 0 3 Fast ovolutio: T ov futio omut t ovolutio btw two fiit-uratio squ a is vry ffiit for smallr valus of <50. For larg valus of it is ossibl to s u t ovolutio usig t FFT tis algoritm is all fast ovolutio. x FFT x x * x IFFT FFT Rquirmts. Lt x b a 4-oit squ: x { } j a. Plot t magitu a as of isrt-tim Fourir trasform X. b. Comutr t 4-oit DFT of x us stm to lot its magitu a as. Vrify tat t abov DFT is t saml vrsio of DTFT. It may b lful to ombi t abov two lots i o gra usig t ol futio.. Lt x b a 64-oit squ by aig 60 zros usig t zros futio t j omutr t 64-oit DFT. Ca w obtai otr samls of t DTFT X?

2 . Lt a fiit-lgt squ b giv by x j Plot t DTFT X of t abov squ usig DFT as a omutatio tool. Coos t lgt of DFT so tat tis lot aars to b a smoot gra. 3. Cosir t squ x os0.48 os0.5 W wat to trmi its strum bas o t fiit umbr of samls. a. Dtrmi a lot t DTFT of x 0 0. b. Dtrmi a lot t 00-oit DFT of t abov x by aig 89 zros.. Dtrmi a lot t DTFT of x To illustrat t iffr btw t ig-sity strum a t ig-rsolutio strum. 4. Lt x b a L-oit uiformly istribut raom umbr btw [0] a lt b a L-Poit Gaussia raom squ wit ma 0 a varia Usig ra a ra futio. Dtrmi t avrag xutio tims of liar ovolutio a fast ovolutio for L= i wi t avrag is omut ovr t 00 tims ralizatios for a L.

3 ELCE580 Digital Sigal Prossig am Assigmt : IIR Filtr Dsig Class&Stut ID Aim. Dsig aalog low-ass filtrs.. Stuy a aly filtr trasformatios to obtai igital low-ass filtrs. Itroutio T basi tiqu of IIR filtr sig is to trasform wll-ow aalog filtrs to igital filtrs usig omlx-valu maig. Tr ar two wily us trasformatios: amly imuls ivaria trasformatio a biliar trasformatio. A tyial sifiatio of a low-ass igital filtr is sow i Fig. - i wi ba [ 0 ] is all assba a is t tolra or ril tat w ar willig to at i t ial assba rsos Ba [ st ] is all stoba a is t orrsoig tolra or ril ba ] is all trasitio ba [ st R is t assba ril i B a A s is t stoba attuatio i B. A w av R 0log A 0log s A tyial sifiatio of a low-ass aalog filtr is sow i Fig. - i wi a j is all magitu-squar rsos is a assba ril aramtr is t assba utoff frquy i ra/s A is a stoba attuatio a st is t stoba utoff frquy i ra/s. A w av

4 R s 0log A 0log 0 0 A A 0 0 R 0 As 0 j st a j 0 / R A s A 0 st B Fig. - Digital low-ass filtr sifiatios Fig. - Aalog low-ass filtr sifiatios 4 Buttrwort lo-ass filtrs: T magitu-squar rsos of a -orr Buttrwort aalog loass filtr is giv by a j Wr is t orr of t filtr a is t 3B utoff frquy i ra/s. T ss of t sig i t as Buttrwort filtr is to obtai t orr a t utoff frquy from t aramtrs R st a A s. A w av t sig quatios as follow log 0 R 0 0 log0 R 0 0 st As 0 0 As 0 st 0 Wr t oratio x mas oos t smallst itgr lagr ta x. A to satisfy t

5 sifiatios xatly at or st w obtai from t orrsoig quatio abov. MATLAB rovis a futio all buttr to sig Buttrwort aalog filtr a t futio is ivo by [ba] = buttrw's' r is t orr of t filtr W is t 3B utoff frquy a t strig s mas to sig a aalog filtr. 5 Cbysv low-ass filtrs: T magitu-squar rsos of a Cbysv-I filtr is a j Wr x is t -orr Cbysv olyomail. W av t sig quatios as follow aros A aros st MATLAB rovis a futio all by to sig Cbysv-I aalog filtr a t futio is ivo by [ba] = byrw's' T magitu-squar rsos of a Cbysv-II filtr is a j T sig quatios for Cbysv-II rototy ar similar to tos of Cbysv-I xt tat st MATLAB rovis a futio all by to sig Cbysv-II aalog filtr a t futio is ivo by [ba] = byasw's' 6 Aalog -to-igital filtr trasformatios: MATLAB rovis a futio all imivar to aly imuls ivaria trasformatio a a futio all biliar to aly biliar trasformatio. 8 T frqs futio rturs t frquy rsos of aalog filtrs a t frqz futio

6 rturs t frquy rsos of igital filtrs. 9 It is a fat tat t buttr by a by futios a b us to sig igital filtr irtly. A tr is also aotr st of filtr sig futios amly t buttor bor a bor futios wi a rovi filtr orr a utoff frquy W. Ts futio us biliar trasformatio baus of its sirabl avatags. owvr you ar ot suos to us ts futios to sig igital filtr irtly i tis xris. Rquirmts. Dsig a low-ass igital filtr to satisfy 0. R B st 0. 3 A s 5B Us a Buttrwort rototy a imuls ivaria trasformatio. Us a Cbysv-I rototy a imuls ivaria trasformatio. 3 Us a Cbysv-II rototy a imuls ivaria trasformatio. 4 Us a Buttrwort rototy a biliar trasformatio. 5 Us a Cbysv-I rototy a biliar trasformatio. 6 Us a Cbysv-II rototy a biliar trasformatio. Commt o t rsults. Is tis a satisfatory sig? Wy?

7 ELCE580 Digital Sigal Prossig am Assigmt 3: FIR Filtr Dsig Class&Stut ID Aim To stuy two sig tiqus amly t wiow sig a t frquy samlig sig for liar-as FIR filtrs. Itroutio Liar-as FIR filtrs. T imuls rsos must b symmtri tat is W t ass of symmtry a atisymmtry ar ombi wit o a v w obtai four tys of liar-as FIR filtrs. Frquy rsos futios for a of ts tys av som j uliar xrssios a sas. To stuy ts rsoss w writ as j j j j wr is a amlitu rsos futio a ot a magitu rsos futio. T amlitu rsos is a ral futio but uli t magitu rsos wi is always ositiv T amlitu rsos may b bot ositiv a gativ. A is a as futio wi is a otiuous liar futio. W t imuls rsos is symmtry os 3- W t imuls rsos is atisymmtry si 3- Ty- Symmtrial imuls rsos o

8 0 os a 0 a a Ty- Symmtrial imuls rsos v os b b Ty-3 Atisymmtrial imuls rsos o 0 si Ty-4 Atisymmtrial imuls rsos vv si MATLAB rovis a futio all zroas to alulat t amlitu rsos futio. A t as futio a b alulat by 3- or 3-. Wiow sig tiqus. T basi ia bi t wiow sig is to oos a ror ial frquy-sltiv filtrwi always as a oausalifit-uratio imuls rsosa t to truator wiowits imuls rsos to obtai a liar-as a ausal FIR filtr.trfor t masis i tis mto is o sltig a aroriat wiowig futio a a aroriat ial filtr.w will ot a ial frquy-sltiv filtr by j wi as a uity magitu gai a liar-as aratristis ovr its assbaa zro rsos ovr its stoba.a ial low ass filtr of bawit is giv by

9 j 0 j Tt imuls rsos of tis filtr is giv by j si F ot tat is symmtri wit rst to a fat usful for liar-as FIR filtrs. To obtai a FIR filtr from o as to truat o bot sis.tat is b tougt of as big form by t rout of follows: w a a wiow futio a w as Tr ar svral is of wiow futios.i Tabl 3. w rovi a summary of fix wiow futio aratristis i trms of tir trasitio witsas a futio of a tir miimum stoba attuatios i B.Bot t aroximat as wll as t xat trasitio bawits ar giv. Wiow am Tabl 3. Summary of ommoly us wiow futio aratristis Wiow futio frquy Si Lob MagituB Rtagular -3 aratristis Mai Lob Wit 4 Trasitio Wit Sifiatio Mi.Stoba AttuatioB.8 - aig -3 ammig -4 Blama MATLAB rovis svral futios to imlmt wiow futios abov Tabl 3. MATLAB futios for imlmtig wiow futio MATLAB Wiow am MATLAB Wiow am futio futio boxar Rtagular blama Blama aig aig aisr Kaisr ammig ammig

10 3Frquy samlig sig tiqus.giv t ial lowass filtr j oos t filtr lgt a t saml j at quisa frquis btw 0 a j 0 T w av IDFT 0 j / si / si j w writ as j r j For a liar-as FIR filtr w av a * W is a ral squw av r r 0 Ty- a Ty- 0 Ty-3a Ty-4 4 T frqz futio rturs t frquy rsos of igital filtrs. 5 It is a fat tat t fir a fir futios a b us to sig FIR igital filtr irtly. owvr you ar ot suos to us ts futios to sig igital filtr irtly i tis xris. Rquirmts. W av to lot t amlitu rsos futio as

11 futio amlitu rsos a as rsos.. Dsig a low-ass igital filtr to satisfy 0. R 0. 5B st 0. 3 A s 50B Coos Rtagular wiow futio aig wiow futio ammig wiow futio a Blama wiow futio to sig t low-ass filtr rstivly.commt o t rsults. Is tis a satisfatory sig? Wy? 3. Dsig a ba-ass igital filtr wit t followig sifiatios 0. st A s 60B R B R B 0. st 8 A s 60B Coos a aroriat wiow futio to sig t ssary ba-ass filtr. 4. Dsig a low-ass igital filtr usig frquy samlig aroa to satisfy Coos =0 0. R 0. 5B st 0. 3 A s 50B Coos =40 so tat w av o saml i t trasitio ba ot ts samls by T= Coos =60 so tat w av two saml i t trasitio ba ot ts samls by T=0.595 T= Commt o t rsults. Is tis a satisfatory sig?