LTI Systems with Feedback, IIR Filter Design

Transcription

1 Lectue 27 Outine: LTI Systems with Feedbac, IIR Fite Design Announcements: Last HW wi be posted, due Thu June 7, no ate HWs Pactice fina wi be posted today Fina exam announcements on next side Feedbac in discete LTI Systems IIR fite design

2 Fina Exam Detais Time/Location: Monday June, 3:36:3 in this oom. Food aftewads. Open boo and notes you can bing any witten mateia you wish to the exam. Cacuatos not aowed. PDF bowsing devices aowed. Coves a cass mateia; emphasis on postmt mateia (ectues 4 on) See ectue ppt sides fo mateia in the eade that you ae esponsibe fo Pactice fina wi be posted today, 25 EC pts fo taing it (not gaded). Can be tuned in any time up unti exam, Sons given when you tun in you answes if done by pm June, so we have time to emai them to you Fina exam fom 26 did not incude IIR fite design My fina eview on W 6/6 in cass. Geogia Th 6/7 46pm (eview + OHs) OHs befoe fina: Me: today afte cass, 6/6, Wed, 2:3 (hee); 6/8, Fi, by appt. (equest by Thu 5pm) Regua Tuesday section and TA OHs this wee pus Th 6/7 ~56pm (Geogia) Sun, 6/, 22pm (John), Mon 6/, 22pm (Maavia)

3 Review of Last Lectue DT systems impemented with deay/ampifie eements A system with output feedbac geneay has infinite impuse esponse Can compute impuse esponse by hand ; had to do LTI Systems descibed by diffeentia equations Exampe: 2 nd ode feedbac system [ ] [ ] M N n x b n y a ( ) ( ) X b Y a M N ø ö ç ç è æ ø ö ç ç è æ ( ) ( ) ( ) A B a b H N M ( ) ( )( ) e e H j j > +, cos q q q ( ) W W W cos 2 j j j e e e H q ( ) ( ) 2 H [ ] ( ) [ ] [ ] u n n h n n + q:

4 Review Continued Diect Fom IIR Fite Impementation This ectue wi discuss IIR fite design Appoximates continuous time fites with owe compexity than FIR designs

5 Feedbac in discete LTI Systems + e[n] g[n] x [n] S y[n] G() ± G( ) x ( t) y( t ( ) ( ) )! G H h[n] H () Motivation fo Feedbac Can mae an unstabe system stabe; mae a system ess sensitive to distubances; mae it cose to idea Can have negative effect: mae a stabe system unstabe Tansfe Function T() of Feedbac System: Y()G()E(); E()X()±Y()H() Exampe: Popuation Gowth: y[n]2y[n]+x[n][n] Poe at 22b, Stabe fo.5<b<.5 Equivaent System ( ) ( ) Y G T ( ) X! G T ( ) ( ) H ( ) 2( b ) ( )

6 IIR Fite Design x(t) H c (s) y c (t) Discussed FIR fite design of h w [n] fo desied IIR noncausa fite H c (jw) befoe MT ADC Sampe time T x[n] y H() DAC y d [n] d(t) Convet H c (jw) to DT IIR fite as H c (jw)þh d (e jw ) fo WwT Now conside design of IIR H() to appoximate H c (s) Have moe toos at ou disposa fo this design (tansfoms) Low compexity impementation of IIR fites vs FIR (via feedbac) Two methods: tempoa and fequency esponse matching Tempoa (not on HW/exam): Design H() so that fo a given x(t) (impuse, step), y d [n]y c (nt); intoduces aiasing if T>p/W

7 Fequency Response (FR) Matching FR Matching: Biatea Tansfom (BLT) Noninea invetibe mapping between spane and pane Can use moe genea mapping fo any ea C, <C Discete time fite design of H() unde BLT: Ensues that if H c (s) stabe, H() aso stabe with no aiasing Thee is waping due to noninea mapping, esp. at high fequencies Mapping between s and pane: s yieds jw axis in spane Þ unit cice in pane eft/ight haf of spane Þ inside/outside unit cice Lefthaf poes of spane map inside cice Stabiity peseved in the mapping to H()

8 s/ pane mapping Phase: fo w <<p/t <w< p<w<p So entie jw axis of spane mapped to one evoution of the pane aong unit cice No aiasing of the fequency esponse, but waps

9 Exampe: 2 nd ode Buttewoth BLT: IR

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11 Main Points Feedbac stabiies unstabe systems; obtains desied tansfe function; can mae stabe system unstabe Easy to anaye systems with feedbac using tansfoms IIR fite design using biinea tansfom entais a noninea mapping fom s to pane Fo desied CT fite tabe, ensue DT fite is aso stabe Avoids aiasing Cosey matches CT fite fo ow fequencies Waps high fequencies