Lectures 9 IIR Systems: First Order System
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1 EE3054 Sigals ad Systms Lcturs 9 IIR Systms: First Ordr Systm Yao Wag Polytchic Uivrsity Som slids icludd ar xtractd from lctur prstatios prpard by McCllla ad Schafr Lics Ifo for SPFirst Slids This work rlasd udr a Crativ Commos Lics with th followig trms: Attributio Th licsor prmits othrs to copy, distribut, display, ad prform th work. I rtur, licss must giv th origial authors crdit. No-Commrcial Th licsor prmits othrs to copy, distribut, display, ad prform th work. I rtur, licss may ot us th work for commrcial purposs ulss thy gt th licsor's prmissio. Shar Alik Th licsor prmits othrs to distribut drivativ works oly udr a lics idtical to th o that govrs th licsor's work. Full Txt of th Lics This (hidd pag should b kpt with th prstatio /9/ , JH McCllla & RW Schafr
2 FIR systm: Rviw Dscribd by a fdforward diffrc quatio Impuls rspos is fiit duratio (fiit impuls rspos or FIR Charactrid by impuls rspos h, systm fuctio H( (Z-trasform of h ad frqucy rspos H(^w M k k x k h h x y b h 0 * M k k k x b y 0 ( ( H H h IIR Systm: Gral M k k N k k k x b k y a y 0 Wightd avrag of iput sampls Wightd avrag of past output sampls (fdback Still a liar tim-ivariat systm Impuls rspos is ifiitly log grally Calld Ifiit Impuls Rspos (IIR systm
3 Roadmap First discuss first ordr systm Tim domai: output for giv iput, impuls rspos Z-domai: trasfr fuctio, charactriatio by pols, how to comput output usig Z-domai Frqucy rspos Nxt discuss scod ordr systm y a y ay bkx k k 0 Fially to gral IIR systm M y a M y bkx k k 0 ONE FEEDBACK TERM (First Ordr Systm ADD PREVIOUS OUTPUTS y a y b 0 x b x PREVIOUS FEEDBACK FIR PART of th FILTER FEED-FORWARD CAUSALITY NOT USING FUTURE OUTPUTS or INPUTS /9/ , JH McCllla & RW Schafr 6 3
4 FILTER COEFFICIENTS ADD PREVIOUS OUTPUTS y 0.8y 3x x FEEDBACK COEFFICIENT SIGN CHANGE MATLAB yy filtr(3,-,,-0.8,xx /9/ , JH McCllla & RW Schafr 7 COMPUTE OUTPUT /9/ , JH McCllla & RW Schafr 8 4
5 COMPUTE y FEEDBACK DIFFERENCE EQUATION: y 0.8y 5x NEED y- to gt startd y0 0.8y 5x0 /9/ , JH McCllla & RW Schafr 9 AT REST CONDITION y 0, for <0 BECAUSE x 0, for <0 /9/ , JH McCllla & RW Schafr 0 5
6 COMPUTE y0 THIS STARTS THE RECURSION: SAME with MORE FEEDBACK TERMS y a y a y k 0 b k x k /9/ , JH McCllla & RW Schafr COMPUTE MORE y CONTINUE THE RECURSION: /9/ , JH McCllla & RW Schafr 6
7 PLOT y y has ifiit duratio! /9/ , JH McCllla & RW Schafr 3 Is IIR systm LTI? If x0, y0 for <0, ys! Proof 7
8 Proprtis of LTI systm: Rviw Ay LTI systm ca b charactrid by its impuls rspos ht(δ Output to ay iput is rlatd by yx*h IMPULSE RESPONSE h y a y h b 0 δ x u, for 0 h b0 ( a u h has ifiit duratio! /9/ , JH McCllla & RW Schafr 6 8
9 PLOT IMPULSE RESPONSE h b 0 (a u 3(0.8 u /9/ , JH McCllla & RW Schafr 7 Show that for th xampl systm y0.8 y- 5 x y x* h yilds sam rsult as th dirct computatio usig rcursio h5 * 0.8^ u xδ-3δ- δ-3 By liarity ad tim ivariac: y h 3 h- h-3 9
10 Wh x ad h ar both ifiit duratio, umrical computatio of covolutio is grally ifasibl. But w ca still us th rcursio basd o th diffrc quatio, although this is tdious. Z-trasform coms to rscu! Y( X( H( Dtrmi H(, X(, Y( From Y(, dtrmi y (ivrs Z-trasform CONVOLUTION PROPERTY MULTIPLICATION of -TRANSFORMS X( Y( H(X( H( CONVOLUTION i TIME-DOMAIN IMPULSE RESPONSE x y h x h /9/ , JH McCllla & RW Schafr 0 0
11 /9/ , JH McCllla & RW Schafr Systm Fuctio of First Ordr Systm Impuls rspos: Ifiit duratio! Z-trasform (Systm Fuctio H( h ( ( a b u a b H ( 0 u a b h /9/ , JH McCllla & RW Schafr Drivatio of H( Rcall Sum of Gomtric Squc: Yilds a COMPACT FORM r 0 r if ( ( a a b a b a b H > If r <
12 Rcap: FIRST-ORDER IIR FILTER: y a y b 0 x h b0 ( a u Trasform pair H( b 0 a /9/ , JH McCllla & RW Schafr 3 Aothr first ordr systm h y a y b 0 x b x H( b0 ( a u b ( a u b 0 a is a shift b a b b 0 a /9/ , JH McCllla & RW Schafr 4
13 Ca w dtrmi H( mor asily Ca w dtrmi H( w/o dtrmiig h first? YES: by apply Z-trasform to th diffrc quatio! DELAY PROPERTY of X( DELAY i TIME<-->Multiply X( by - Proof: x X( x X( x xl l xl l X( l (l /9/ , JH McCllla & RW Schafr 6 3
14 Z-Trasform of IIR Filtr DERIVE th SYSTEM FUNCTION H( Us DELAY PROPERTY Apply trasform o both sids y a y b 0 x b x Y( a Y( b 0 X( b X( /9/ , JH McCllla & RW Schafr 7 Y( a Y( b 0 X( b X( Y( a Y( b 0 X( b X( ( a Y( (b 0 b X( H( Y( X( b 0 b a B( A( /9/ , JH McCllla & RW Schafr 8 4
15 Exampl DIFFERENCE EQUATION: y 0.8y 3x x READ th FILTER COEFFS: H( 3 Y ( 0.8 X ( /9/ , JH McCllla & RW Schafr 9 POLES & ZEROS ROOTS of Numrator & Domiator H( b 0 b a H( b 0 b a b 0 b 0 b b 0 ZERO: H(0 a 0 a POLE: H( if /9/ , JH McCllla & RW Schafr 30 5
16 EXAMPLE: Pols & Zros VALUE of H( at POLES is INFINITE H ( 0.8 ( H ( 0.8( ( 4 5 H ( 0.8( ZERO at - POLE at 0.8 /9/ , JH McCllla & RW Schafr POLE-ZERO PLOT ZERO at POLE at 0.8 /9/ , JH McCllla & RW Schafr 3 6
17 Stability of th Systm FIRST-ORDER IIR FILTER: y a y b 0 x h b0 ( a u H( b 0 a Pol at a_ Wh a_ < h b 0 (a u 3(0.8 u 7
18 Wh a_ > Show h Systm produc uboudd output for fiit iput! Ustabl! BIBO stability BIBO: boudd iput boudd output Stability from Pol Locatio A causal LTI systm with iitial rst coditios is stabl if all of its pols li strictly isid th uit circl! Our xampl is for st ordr systm with pol oly. Abov statmt is tru for systms of ay ordr, which ca b dcomposd ito sum of first ordr systms. Zro locatios do ot affct systm stability FIR systms ar always stabl (pols at ros oly 8
19 FREQUENCY RESPONSE SYSTEM FUNCTION: H( H( has DENOMINATOR FREQUENCY RESPONSE of IIR W hav H( H(? H(? THREE-DOMAIN APPROACH h H( H(? /9/ , JH McCllla & RW Schafr 37 FREQUENCY RESPONSE EVALUATE o th UNIT CIRCLE H(? H(? /9/ , JH McCllla & RW Schafr 38 9
20 0 /9/ , JH McCllla & RW Schafr 39 FREQ. RESPONSE FORMULA 0.8 ( 0.8 ( H H ( H cos.64 8cos 400, ( π H /9/ , JH McCllla & RW Schafr 40 Frqucy Rspos Plot 0.8 ( H frq(b,a b, a, -0.8
21 UNIT CIRCLE MAPPING BETWEEN ad??? 0? ±π ±? ± π /9/ , JH McCllla & RW Schafr 4 SINUSOIDAL RESPONSE x SINUSOID > y is SINUSOID Gt MAGNITUDE & PHASE from H( if x th y whr H ( H ( H ( /9/ , JH McCllla & RW Schafr 4
22 POP QUIZ Giv: H( 0.8 Fid th Impuls Rspos, h Fid th output, y Wh x cos(0.5π /9/ , JH McCllla & RW Schafr 43 Evaluat FREQ. RESPONSE is at? 0.5π? 0.5π ro at π π /9/ , JH McCllla & RW Schafr 44
23 POP QUIZ: Eval Frq. Rsp. Giv: Fid output, y, wh Evaluat at H ( H( 0.8 ( 0.5π 0.5π 0.8 x cos(0.5π y 5.8cos(0.5π 0.47π /9/ , JH McCllla & RW Schafr 45 THREE DOMAINS Us H( to gt Frq. Rspos? /9/ , JH McCllla & RW Schafr 46 3
24 READING ASSIGNMENTS This lctur focuss o First Ordr Systm Chaptr 8, Scts. 8-, 8-, 8-3, 8-4, Block diagram structur: study by yourslf 8-5: Frqucy rspos of first ordr systm: study by yourslf (Slids
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