Windowing in FIR Filter Design. Design Summary and Examples
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1 Lctur 3 Outi: iowig i FIR Fitr Dsig. Dsig Summary a Exams Aoucmts: Mitrm May i cass. i covr through FIR Fitr Dsig. 4 ost, 5% ogr tha usua, 4 xtra ays to comt (u May 8) Mor tais o say Thr wi b o aitioa urig th wk of th MT Extra crit for ractic MT (ost shorty) My Ohs say wi b bfor cass (:3:3 outsi cassroom) Rviw of Last Lctur Th Art a Scic of iowig Summary of FIR Fitr Dsig FIR Raizatio: Dirct Form Exams: LPF, Diffrtiator, ibrt
2 Rviw of Last Lctur Imus a frq. rsos matchig miimizs th tim/frq. omai rror btw sir fitr a its FIR aroximatio IR Match: h [ ] [ ] ha FR Match: ò ( ) a ( ) å By Parsva s, rrors ar th sam; IR matchig yis ìh [ ] M / h [ ] Shar wiowig causs Gibbs homo (wiggs) Causa Dsig Ca mak h a [] causa by aig ay of M/ M ê ë ù [ ] h é ú û Grou ay fi as a í î > M Oft costat ( ay) or icwis costat ( ay r frq. grou) Grou ay that is ot costat ca itrouc istortio / M h a ( ) ( ) Ð ( ) a
3 Art a Scic of iowig iow sig is crat as a atrativ to th shar timwiowig i h a [] Us to mitigat Gibbs homo iow fuctio (w[], >M/) giv by iow ocausa FIR sig: h [ ] w[ ] h [ ] w[ ] h [ ] w a Frqucy rsos smooths Gibbs i a ( ) w ( ) ( q ) ( ( q ) ò ) q [ ]«( ) Dsig oft tras wiggs i mai vs. siobs w
4 Tyica iow Dsigs boxcar(m+), M 8 triag(m+), M 8 haig(m+), M 8 hammig(m+), M 8 w[].5 w[].5 w[].5 w[] M M 6.8 Boxcar Triaguar Boxcar Triaguar ( ).6.4. og ( ) M 6 M 6.8 ammig aig ammig aig ( ).6.4. og ( )
5 Summary of FIR Dsig ar giv a sir rsos h [] which is gray ocausa a IIR Exams ar ia owass, baass, highass fitrs May b riv from a cotiuoustim fitr Choos a fitr uratio M+ for M v Largr M tais mor comxity/ay, ss aroximatio rror Dsig a gth M+ wiow fuctio w[], ra a v, to mitigat Gibbs whi kig goo aroximatio to h [] Cacuat th ocausa FIR aroximatio h a [] Cacuat th ocausa wiow FIR aroximatio h w [] A ay of M/ to h w [] to gt h[]
6 FIR Raizatio: Dirct Form M x[ ] x[ ] x [ ] x [ ] [ ] x[ M ] x M + D D D h [] h [] h [] h [ M ] h[m ] y [ ] åh[ k] x[ k] M k Cosists of M ay mts a M+ mutiirs Ca itrouc iffrt ays at iffrt frq. comots of x[] i iscuss mor wh w covr z trasforms Efficit immtatio usig DiscrtFourir Trasform (DFT) Nxt cass toic
7 Exam: Lowass Fitr Ia LPF with cutoff c : LPF, c ( ) ì í î m othrwis c h æ sic è c c ö [ ] ø LPF, c h LPF, c [ ]
8 Lowass Fitr, c /, M 3 Lowass Fitr, c /, M 3 Imus Rsos.4. Boxcar h[] Imus Rsos.4. ammig h[] Dsir ( ).5 Dsir ( ) Magitu Rsos.5 Boxcar ( ) Magitu Rsos.5 ammig ( ) Dsir < ( ) 4 Dsir < ( ) Phas Rsos (ra) Boxcar <( ) Phas Rsos (ra) ammig <( )
9 Exam: Diffrtiator (t oy, s rar for tais) Ia Diffrtiator: imagiary a o h [] obtai via itgratio by arts ( ) T T c ø ö è æ [ ] ( ) ( ) ( ) [ ] [ ] ( ) ïî ï í ì ¹ ø ö è æ ú û ù ê ë é + ø ö è æ ò si cos T T T T T h
10 Exam: ibrt Trasform (t oy, s rar for tais) Ia ibrt Trasform (cts tim): Trasat to iscrt tim: Imagiary a o i Us IDFT to gt IR Itgrat gativ/ositiv havs ( ) ï î ï í ì < > sg w w w w w c ( ) ø ö è æ sg T c [ ] ( ) ò ò h sg [ ] ( ) ( ) [ ] ( ) ïî ï í ì ø ö è æ ò ò o v cos h ra a o i.
11
12 Mai Poits FIR sig tais choic of wiow fuctio to mitigat Gibbs Goa is to aroximat sir fitr without Gibbs/wiggs Dsig traoffs ivov mai ob vs. siob sizs Tyica wiows: rctag (boxcar), triag, aig, a ammig FIR sig for sir h [] tais ickig a gth M, sttig h a []h [], M/, choosig wiow w[] with h w []h[]w[]to mitigat Gibbs, a sttig h[]h w [M/] to mak sig causa FIR immt ircty usig M ay mts a M+ mutiirs Ca itrouc grou ay Efficity immt with DFT (xt toic) Exam sigs for LPF, Diffrtiator, a ibrt Trasformr ammig smooths out wiggs from rctaguar wiow Itroucs mor istortio at trasitio frqucis tha rctaguar wiow omwork covrs iffrtiator i mor tai, a a high ass fitr
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