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 /9/008 003, JH McCllla & RW Schafr
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
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 /9/008 003, JH McCllla & RW Schafr 6 3
FILTER COEFFICIENTS ADD PREVIOUS OUTPUTS y 0.8y 3x x FEEDBACK COEFFICIENT SIGN CHANGE MATLAB yy filtr(3,-,,-0.8,xx /9/008 003, JH McCllla & RW Schafr 7 COMPUTE OUTPUT /9/008 003, JH McCllla & RW Schafr 8 4
COMPUTE y FEEDBACK DIFFERENCE EQUATION: y 0.8y 5x NEED y- to gt startd y0 0.8y 5x0 /9/008 003, JH McCllla & RW Schafr 9 AT REST CONDITION y 0, for <0 BECAUSE x 0, for <0 /9/008 003, JH McCllla & RW Schafr 0 5
COMPUTE y0 THIS STARTS THE RECURSION: SAME with MORE FEEDBACK TERMS y a y a y k 0 b k x k /9/008 003, JH McCllla & RW Schafr COMPUTE MORE y CONTINUE THE RECURSION: /9/008 003, JH McCllla & RW Schafr 6
PLOT y y has ifiit duratio! /9/008 003, JH McCllla & RW Schafr 3 Is IIR systm LTI? If x0, y0 for <0, ys! Proof 7
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! /9/008 003, JH McCllla & RW Schafr 6 8
PLOT IMPULSE RESPONSE h b 0 (a u 3(0.8 u /9/008 003, 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
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 /9/008 003, JH McCllla & RW Schafr 0 0
/9/008 003, JH McCllla & RW Schafr Systm Fuctio of First Ordr Systm Impuls rspos: Ifiit duratio! Z-trasform (Systm Fuctio H( h 0 0 0 ( ( a b u a b H ( 0 u a b h /9/008 003, JH McCllla & RW Schafr Drivatio of H( Rcall Sum of Gomtric Squc: Yilds a COMPACT FORM r 0 r 0 0 0 0 0 if ( ( a a b a b a b H > If r <
Rcap: FIRST-ORDER IIR FILTER: y a y b 0 x h b0 ( a u Trasform pair H( b 0 a /9/008 003, 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 /9/008 003, JH McCllla & RW Schafr 4
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 /9/008 003, JH McCllla & RW Schafr 6 3
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( /9/008 003, 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( /9/008 003, JH McCllla & RW Schafr 8 4
Exampl DIFFERENCE EQUATION: y 0.8y 3x x READ th FILTER COEFFS: H( 3 Y ( 0.8 X ( /9/008 003, 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 /9/008 003, JH McCllla & RW Schafr 30 5
EXAMPLE: Pols & Zros VALUE of H( at POLES is INFINITE H ( 0.8 ( H ( 0.8( ( 4 5 H ( 0.8( 4 5 0 ZERO at - POLE at 0.8 /9/008 003, JH McCllla & RW Schafr 3 9 0 POLE-ZERO PLOT ZERO at - 0.8 POLE at 0.8 /9/008 003, JH McCllla & RW Schafr 3 6
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
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
FREQUENCY RESPONSE SYSTEM FUNCTION: H( H( has DENOMINATOR FREQUENCY RESPONSE of IIR W hav H( H(? H(? THREE-DOMAIN APPROACH h H( H(? /9/008 003, JH McCllla & RW Schafr 37 FREQUENCY RESPONSE EVALUATE o th UNIT CIRCLE H(? H(? /9/008 003, JH McCllla & RW Schafr 38 9
0 /9/008 003, JH McCllla & RW Schafr 39 FREQ. RESPONSE FORMULA 0.8 ( 0.8 ( H H ( H 0.8 0.8 0.8 0.8 0.8 0.64 4 4 4 4.6 cos.64 8cos 8? @ 400, 0.04 8 8 ( 0, @ π H /9/008 003, JH McCllla & RW Schafr 40 Frqucy Rspos Plot 0.8 ( H frq(b,a b, a, -0.8
UNIT CIRCLE MAPPING BETWEEN ad??? 0? ±π ±? ± π /9/008 003, 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 ( /9/008 003, JH McCllla & RW Schafr 4
POP QUIZ Giv: H( 0.8 Fid th Impuls Rspos, h Fid th output, y Wh x cos(0.5π /9/008 003, JH McCllla & RW Schafr 43 Evaluat FREQ. RESPONSE is 0 0.8 at? 0.5π? 0.5π ro at π π /9/008 003, JH McCllla & RW Schafr 44
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 5.8.309 y 5.8cos(0.5π 0.47π /9/008 003, JH McCllla & RW Schafr 45 THREE DOMAINS Us H( to gt Frq. Rspos? /9/008 003, JH McCllla & RW Schafr 46 3
READING ASSIGNMENTS This lctur focuss o First Ordr Systm Chaptr 8, Scts. 8-, 8-, 8-3, 8-4, 8-5 8-3. Block diagram structur: study by yourslf 8-5: Frqucy rspos of first ordr systm: study by yourslf (Slids 36-45 4