Frequency Response & Digital Filters
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- Amos Foster
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1 Frquy Rspos & Digital Filtrs S Wogsa Dpt. of Cotrol Systms ad Istrumtatio Egirig, KUTT Today s goals Frquy rspos aalysis of digital filtrs LTI Digital Filtrs Digital filtr rprstatios ad struturs Idal filtrs IIR & FIR filtrs
2 Frquy Rspos of Disrt-Tim Systms Rlatiosip tw ZT & DTFT I goig from t DTFT to t ZT w rpla y. Rplaig wit, ZT will om DTFT. Tis valuatio is quivalt to valuatig t -trasform o t uit irl i t omplx pla. 3 Frquy Rspos of Disrt-Tim Systms Frquy Rspos Aalysis Cosidr a DT trasfr futio, t disrt frquy rspos futio FRFis wr is t disrt frquy i rad/sampl. is tmagitud orgai of t FRF. is t pas of t FRF. 4
3 Frquy Rspos of Disrt-Tim Systms A x si si A y A x os os A y 5 W, t systm rspos is x x y os os Proof: Frquy Rspos of Disrt-Tim Systms Sttig i tis rlatiosip yilds ± 6 Addig ad yilds { } R os Wit, w gt os os
4 Frquy Rspos of Disrt-Tim Systms Systm Rspos to Sampld Siusoids If a DT systm is stal wit trasfr futio, t i stady-stat x A y A x Aos x Asi y A os y A si 7 Frquy Rspos of Disrt-Tim Systms EXAPLE.5.5 os si os.5 si 8
5 Frquy Rspos of Disrt-Tim Systms EXAPLE If a siusoidal iput π y.547si 3 3 os si os.5 si π x si 3 π / 3.547, o is applid π / 3 3 o tf, -.5,-; mag,pasod,pi/3 mag.547 pas Frquy Rspos of Disrt-Tim Systms Bod Plot A Bod plot i t disrt tim is a grap of ad plottd as a futio of, wr is usually ragig from to π..5 os si os.5 si um ; % umrator d -.5; % domiator frqum,d; % DT frquy rspos agitud db ormalid Frquy π rad/sampl Pas dgrs ormalid Frquy π rad/sampl
6 Frquy Rspos from Pol-Zro Loatio Trasfr futio: d a Frquy rspos: d d T magitud: d d to dista from to dista from ad pas: d Liar pas trm Sum of t agls from t ros/pols to uit irl Frquy Rspos of LTI A Grapial Viw Trasfr futio: d W ar goig aroud t irl wit d Frquy rspos: Adoptd from Ela Pusaya, Basis of Digital Filtrs.
7 Frquy Rspos of LTI A Grapial Viw d dista from dista from d to to T magitudof t frquy rspos is giv y tims t produt of t distas from t ros to dividd y t produt of t distas from t pols to d T pasrspos is giv y t sum of t agls from t ros to mius t sum of t agls from t pols to plus a liar pas trm Adoptd from Ela Pusaya, Basis of Digital Filtrs. 3 Frquy Rspos of LTI A Grapial Viw w is los to a pol, t magitud of t rspos riss rsoa. w is los to a ro, t magitud of t rspos falls a ull. Adoptd from Ela Pusaya, Basis of Digital Filtrs. 4
8 Frquy Rspos of LTI Exampl Sour: Aso Amadar, Digital Sigal Prossig: A odr Itrodutio. 5 Exampl Filtrs ad Pol-Zro Plots 6
9 Today s goals Frquy rspos aalysis of digital filtrs LTI Digital Filtrs Digital filtr rprstatios ad struturs Idal filtrs IIR & FIR filtrs 7 Wat is Digital Filtr? Digital filtr is a systm tat prforms matmatial opratios o a disrt-tim sigal ad trasforms it ito aotr squ tat as som mor dsiral proprtis,.g. x Digital filtr y I tis ours, w limit ourslvs to LTI digital filtrs oly. 8
10 Exampl Appliatios of Digital Filtrs ois Rmoval Eltroardiogram ECG ttps://youtu./v3-yzmqu8?t45 9 Exampl Appliatios of Digital Filtrs ois Rmoval Eltroardiogram ECG Low-pass filtrd ECG ttps://youtu./v3-yzmqu8?t45
11 Exampl Appliatios of Digital Filtrs Audio Prossig Digital Filtr Rprstatios x y a x - x x y a y a y a y T liar tim-ivariat digital filtr a dsrid y t liar diffr quatio T ordrof t filtr is t largr of or T ordrof t filtr is t largr of or a X Y Trasfr futio of t filtr is
12 Digital Filtr Struturs: Commo Elmts Addrs: ultiplirs: Dlays: 3 Digital Filtr Struturs EXAPLE: Eo Gratio Eos ar dlayd sigals ad a gratd y t followig diffr quatio: y x αx D, α < wr D is t dlay i sampls. Rfltd soud Dirt soud 4
13 Idal Filtrs agitud Rspos Idal filtrs lt frquy ompots ovr t passad pass troug udistortd gai,wilompotsattstopadarompltlyutoffgai. 5 Idal Filtrs Pas Rspos Idal filtr: admits a liar pas rspos θ wr θ Liar pas oliar pas Output is mrly a dlayd vrsio of iput. Y X Fourir trasform of x 6
14 Liar Pas Rspos Liar-pas filtrsdlay all frquis y t sam amout, try maximally prsrvig wavsap. θ 5 5 oliar pas: θ 3 π Adaptd from Ela Pusaya, Basis of Digital Filtrs. 7 Idal Filtrs Liar Pas Rspos Group Dlay: A masur of liarity of t pas is otaid from t group dlay futio, wi is dfid as d θ τ d T group dlay is ostatw t pas is liar. 8
15 Exampl A simpl idal lowpass filtr d, if, if < π T impuls rspos is giv y d π si o-ausal ad ifiit i duratio. Clarly aot implmtd i ral-tim!. 9 FIR & IIR Filtrs Y X a Fiit Impuls Rspos FIR Filtrs:, o fda. T FIR filtr as o pols, oly ros, i.. y 3 x x x Ifiit Impuls Rspos IIR Filtrs: a y.6y x 3
16 FIR & IIR Filtrs FIR Filrs : t impuls rspos of a FIR filtr lasts oly a fiit tim / 3,,, lswr y 3 x x x IIR Filtrs: t impuls rspos futio of a IIR filtr is o-ro ovr a ifiit lgt of tim y.6y x 3 FIR & IIR Filtrs Staility: FIR filtrs ar always stal. IIR filtrs a ustal aus t filtr av pols i tir trasfr futios. if ot dsigd proprly. Ordr: IIR filtrs ar omputatioally mor ffiit ta FIR filtrs as ty rquir fwr offiits du to t fat tat ty us fda or pols. Pas: FIR filtrs a guaratd to av liar pas. IIR filtrs i gral do ot av liar pas. 3
17 Symmtry oditios for liar pas rspos Cosidr a FIR filtr of ordr. By dfiitio is t -trasform of... Suffiit oditios for t pas liarity of a FIR filtr: If is itr symmtri or atisymmtri aout its tr poit, t filtr pas rspos is a liar futio of Symmtrial impuls rspos: Atisymmtrial impuls rspos:.g. 4.g Symmtry oditios for liar pas rspos Symmtrial impuls rspos: EXAPLE: for 4, T frquy rspos is If is symmtriaout its tr poit, t 4 ad 3:, t pas rspos is giv y os os 34
18 Symmtry oditios for liar pas rspos Atisymmtrial impuls rspos: EXAPLE: for 4, T frquy rspos is si si If is atisymmtriaout its tr poit, t, -4 ad -3:, t pas rspos is giv y π
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