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1 . Sigals ad Sysems Sigal () Fucio of ime, space, ec () Coais iformaio (3) f is a whole sigal ad f() is he value of sigal a ime. (4) Domai of sigal: 's for which i is defied Coiuous-ime sigal Domai of sigal is R. Discree-ime sigal Domai of sigal is I. Ui ad dimesio of sigal easure of sigal Properies of sigal Power ad eergy Coiuous-ime Discree-ime Isaaeous power p () x () p [ ] x [ ] Average power P x () d P x [ ] T P lim x ( ) d P T T lim x T [ ] Eergy E x () d E x [ ] T T T E lim x ( ) d () Sigal wih fiie eergy sigal: E ad P () Sigal wih fiie average power: E ad P E lim x [ ], fid a example., fid a example. hp://ewoo.com Eug Je Woo

2 Trasformaios of he idepede variable Coiuous-ime Discree-ime Time shif x ( ) x [ ] Time reversal x( ) x[ ] Time scalig x( ) x[ ] Combiaio x( ) x[ ] Characerisics of sigal Coiuous-ime Discree-ime Periodic x() x( T) (fudameal) period, T x[ ] x[ ] (fudameal) period, Aperiodic o periodic o periodic Eve x() x( ) x[ ] x[ ] Odd x() x( ) x[ ] x[ ] Eve par Evlx () q lx () x( ) q Ev x [ ] x [ ] x[ ] Odd par Odlx() q lx() x( ) q Od x[ ] x[ ] x[ ] l q l q l q l q Coiuous-ime expoeial ad siusoidal sigal () Real expoeial sigal: x () Ce, CRad ar a a () Complex expoeial sigal: x () Ce, C Ce Cad a( r ) C r ( ) r I his geeral case, x () Ce e Ce lcos( ) si( ) q. (3) Periodic complex expoeial: x () e x ( T) e e e We eed e T. Therefore, (a) If, x () T. (b) If, T Tmi ( T) T = fudameal period. Why T mi? (4) Siusoidal sigal: x () Acos( ), T, f (a) ad T : secod, sec, or s (b) : radias per secod or rad/s, fudameal frequecy (c) f : cycles per secod or H (her), fudameal frequecy hp://ewoo.com Eug Je Woo

3 (d) : radias (wha is oe radia?) (e) Wha are power, eergy, average ad rms value of x()? (f) Plo ad describe he volage sigal, v () cos( ) 4 (g) Calculae power, eergy, average ad rms value of v() i (f). [V]. (5) Euler's formula: e cos si A A (a) Acos( ) e e e e A R e o e A (b) A e e A si( ) e e A Imo e ( ) ( ) (6) Harmoically relaed complex expoeials: () e, I oe ha f T. Ca you fid a iverse relaioship? Discree-ime expoeial ad siusoidal sigal () Real expoeial sigal: x [ ]Ce C, e, C R ad, R () Complex expoeial sigal: x [ ] C, C Ce Cad e C I his geeral case, x [ ] C lcos( ) si( ) q. ( ) (3) Periodic complex expoeial: x [ ] e x ( ) e e e We eed e. Therefore, (a) Whe, x [ ]. (b) Whe, m = fudameal period if he ieger, m has o commo facor wih he ieger,. Why? ( ) (c) Sice e e e e, x[] is periodic i wih period. (4) Siusoidal sigal: x [ ] Acos( ) (a) : dimesioless (b) Wha are power, eergy, average ad rms value of x()? hp://ewoo.com 3 Eug Je Woo

4 (c) Plo ad describe he volage sigal, v [ ] cos( ) [V]. (d) Calculae power, eergy, average ad rms value of v[] i (c). (5) Euler's formula: e cos si A A (a) Acos( ) e e e e A R e o e A (b) A e e A si( ) e e A Imo e ( ) ( ) (6) Harmoically relaed complex expoeials: [ ] e, I oe ha ad [ ] e ( ). Ca you fid a iverse relaioship? (7) Revisi he periodiciy of e w.r.. ( ) (a) e e e e (b) is he highes frequecy sice e ( ). (c), are he lowes frequecies sice e e (d) Plo cos for,,,,,,,,. Fid a rule (8) Revisi he periodiciy of e w.r.. ( ) (a) Sice x [ ] e x ( ) e e e, we eed e. (b) Therefore, m for some mi or m. (c) I oher words, e is periodic if is a raioal umber. (9) Revisi harmoics of e wih a commo period of. ( ) ( )( ) ( ) (a) [ ] e [ ] e e e (b) There are oly disic periodic expoeials ad hey are hp://ewoo.com 4 Eug Je Woo

5 l[ ], [ ],, [ ] q or, e, e,, e o. 4 ( ) o or e Coiuous-ime ui impulse as ui sep fucio R S, () Ui sep fucio is u () T. Plo u(). Is u() coiuous? Is u(), differeiable? du() () Ui impulse fucio is (). Ad, u () ( ) d ( ) d. d, (3) Derive () ad u() from () T, oherwise. (4) Samplig propery: x ( ) ( ) x ( ) ( ) R S Discree-ime ui impulse as ui sep fucio, () Ui sep is u [ ] T. Plo u[]., R S R S, () Ui impulse is [ ] T. Plo [ ],. [ ] u[ ] u[ ]. Ad, u [ ] [ m] [ m]. m m (3) Samplig propery: x[ ] [ ] x[ ] [ ] Sysem () Sysem rasforms ipu sigals io oupu sigals () Sysem is a fucio mappig ipu sigals io oupu sigals Ipu-oupu represeaio Coiuous-ime sysem Sysem rasforms ipu io oupu: x() y() hp://ewoo.com 5 Eug Je Woo

6 Discree-ime sysem Sysem rasforms ipu io oupu: x[ ] y[ ] Sysem iercoecios () Series or cascade iercoecio () Parallel iercoecio (3) Feedbac iercoecio (4) Combiaio Basic sysem properies () emoryless vs. wih memory () Iveribiliy ad iverse sysem (3) Causaliy (4) Sabiliy (boudedess) (5) Time ivariace (6) Lieariy hp://ewoo.com 6 Eug Je Woo

7 . Samplig Samplig heorem () Impulse rai samplig (a) Samplig fucio is a impulse rai, p () ( T) P ( ) ( s ) T (b) T is samplig period ad samplig frequecy is s T or f s s T (c) xp() x () p () xt ( ) ( T) () Samplig heorem: b g Xp( ) X( ) P( ) X ( s) T Le x( ) be a bad-limied sigal wih X ( ) = for. The x () is uiquely deermied by is samples x( T),,,, if where s T. Give hese samples, we ca recosruc x( ) by geeraig a periodic impulse rai i which successive impulses have magiudes ha are successive sample values. This impulse rai is he processed hrough a ideal lowpass filer wih gai T ad cuoff frequecy greaer ha ad less ha s. The resulig oupu sigal will exacly equal x(). s Sigal recosrucio from samples (ierpolaio) () Recosruced sigal usig a filer h() is x () x () h() x( T) h( T) ctsi( c) () For a ideal lowpass filer h(), h () r c p ad hp://ewoo.com 7 Eug Je Woo

8 ct si c( T) xr () x( T) ( T) b c g Aliasig: effec of udersamplig C/D ad D/C coversio: discree-ime processig of coiuous-ime sigal () Sigal coversio (a) Coiuous-o-discree-ime coversio: aalog-o-digial coverer (A/D coverer or ADC) (b) Discree-o-coiuous-ime coversio: digial-o-aalog coverer (D/A coverer or DAC) () Sigal represeaio: x [ ] x ( T) ad y [ ] y ( T) (3) oe ha d c T xp() xc() p() xc( T) ( T) Xp( ) xc( T) e ad d c x [ ] x ( T) X ( e ) x [ ] e x ( T) e d c d d c. Therefore, X ( e ) X ( T) ad T. Furhermore, d X d p ( e ) Xc ( ) T T b g hp://ewoo.com 8 Eug Je Woo

9 3. Liear Time-Ivaria (LTI) Sysems Discree-ime LTI sysems: covoluio sum () Samplig propery: x [ ] x [ ] [ ] () Assume [ ] h [ ], he y [ ] xh [ ] [ ] from lieariy. (3) y [ ] xh [ ] [ ] xh [ ] [ ] xh [ ] [ ] x [ ] h [ ] from imeivariace. Coiuous-ime LTI sysems: covoluio iegral () Samplig propery: x () x( ) ( ) d () Assume ( ) h ( ), he y () x( ) h () d from lieariy. (3) y( ) x( ) h ( ) d x( ) h ( ) d x( ) h( ) d x( ) h( ) from imeivariace. Properies of LTI sysems () Commuaive propery: x [ ] h [ ] h [ ] x [ ] ad x () h () h () x (). b g ad () Disribuive propery: x [ ] h[ ] h[ ] x [ ] h[ ] x [ ] h[ ] l q. Is his parallel iercoecio? x () h() h() x () h() x () h() b g b g ad (3) Associaive propery: x [ ] h[ ] h[ ] x [ ] h[ ] h[ ] l q l q. Is his series iercoecio? x() h () h () x() h () h () emoryless LTI sysem h [ ] [ ] or h () () hp://ewoo.com 9 Eug Je Woo

10 Iverible LTI sysem () Discree-ime LTI sysem is iverible if h [ ] h[ ] h [ ] [ ]. () Coiuous-ime LTI sysem is iverible if h () h() h () (). Causal LTI sysem () Discree-ime LTI sysem is causal if h [ ] for <. The, y [ ] xh [ ] [ ]. () Coiuous-ime LTI sysem is causal if h () for <. The, y () x( ) h ( ) d. Sable LTI sysem () Discree-ime LTI sysem is causal if h [ ], i.e. absoluely summable. () Coiuous-ime LTI sysem is causal if h( ) d, i.e. absoluely iegrable. Ui sep respose of LTI sysem () For discree-ime LTI sysem, s [ ] u [ ] h [ ] h [ ], h [ ] s [ ] s [ ]. () For coiuous-ime LTI sysem, s () u () h () h( ) d, h ds() (). d Liear cosa-coefficie differeial equaio a d y () b d x x y d () () () d Causal LTI hp://ewoo.com Eug Je Woo

11 Liear cosa-coefficie differece equaio a y[ ] b x[ ] x[ ] y[ ] Causal LTI hp://ewoo.com Eug Je Woo

12 4. Fourier Series Respose of LTI sysem o complex expoeials () Eigefucios ad eigevalues s s s (a) Coiuous-ime ( e H( s) e ): if x( ) e, s ( ) s s y ( ) h( ) x ( ) d h( ) e d e h( ) e d s Defie he sysem fucio, Hs () h() e d. The, y( ) H( s) e s. (b) Discree-ime ( H( ) ): if x[ ], y [ ] hx [ ] [ ] h [ ] h [ ] Defie he sysem fucio, H ( ) h[ ]. The, y[ ] H( ). () Superposiio priciple s s (a) Coiuous-ime: if x( ) a e, y( ) a H( s ) e. (a) Discree-ime: if x[ ] a, y[ ] a H( ). Fourier series represeaio of coiuous-ime periodic sigal () Represeaio x() x( T) for all T x () ae ae a ae a e If sigal is real, x * () x (). The, a * ad a or * wih a Ae x () a ae ae a Acos wih a B C x( ) a B cos C si () Coefficie deermiaio xe () ae e. hp://ewoo.com Eug Je Woo

13 T T T ( ) () x e d ae e d a e d T T T ( ), if T e d cos( ) d si( ) d, if T a x() e d x() e d T T T (3) Fourier series pair of periodic coiuous-ime sigal x () ae ae T T T a x ( T ) () e d x () e d T T where a are Fourier series coefficies or specral coefficies of x(). oe ha a x( ) d average value over oe period T. T (4) Covergece of he Fourier series (a) Over ay period, x() mus be absoluely iegrable; ha is, x () d so ha T a. (b) I ay fiie ierval of ime, x() is of bouded variaio; ha is, here are o more ha a fiie umber of maxima ad miima durig ay sigle period of he sigal. (c) I ay fiie ierval of ime, here are oly a fiie umber of discoiuiies. Furhermore, each of hese discoiuiies is fiie. Properies of coiuous-ime Fourier series - Boh x() ad y() are periodic wih T ad T. Their Fourier coefficies are a ad b, respecively. Sigal Fourier coefficie Lieariy Ax() By () Aa Bb Time shifig x( ) ae Frequecy shifig e x() a Cougaio x * () Time reversal x( ) * a a Time scalig x( ),, period of T a Periodic ( ) ( ) T covoluio x y d Tab hp://ewoo.com 3 Eug Je Woo

14 uliplicaio x() y() Differeiaio dx() d Iegraio x() d, fiie wih a Cougae symmery Real x() l ab l a l a * a a a a ad a a Rea Re a ad Im a Im a Real ad eve Real ad eve x() a real ad eve Real ad odd Real ad odd x() a purely imagiary ad odd Eve x decomposiio e() x() x( ), real Rea Odd x decomposiio o() x() x( ), real Ima Parseval's relaio x() d a T T Fourier series represeaio of discree-ime periodic sigal () Represeaio x[ ] x[ ] for all x a a e a e [ ] [ ] ( ) () Coefficie deermiaio ( ) a x[ ] e (3) Fourier series pair of periodic coiuous-ime sigal x a e a e [ ] ( ) a x[ ] e x[ ] e ( ) where a are specral coefficies of x[]. hp://ewoo.com 4 Eug Je Woo

15 Properies of discree-ime Fourier series - Boh x[] ad y[] are periodic wih ad. Their Fourier series coefficies are a ad b, respecively, ad are periodic wih. Sigal Fourier coefficie Lieariy Ax[ ] By [ ] Aa Bb Time shifig ( ) x[ ] ae Frequecy shifig ( ) e x[ ] a Cougaio x * [ ] Time reversal x[ ] * a a Time scalig x[ m], if lm x( m) [ ], oherwise where l is a ieger Periodic x[][ ryr] covoluio r uliplicaio x[ y ] [ ] Firs differece x [ ] x [ ] Ruig sum Cougae symmery m ab a l ab l l ( ) x[ ], fiie ad periodic wih a Real x[] e e a a ( ) * a a a a a a Re Re ad ad a a Ima Ima Real ad eve Real ad eve x[] a real ad eve Real ad odd Real ad odd x[] a purely imagiary ad odd Eve x decomposiio e[ ] x[ ] x[ ], real Rea Odd x decomposiio o[ ] x[ ] x[ ], real Ima Parseval's relaio x[ ] a Fourier series ad LTI sysems () Frequecy respose (a) Coiuous-ime: s H ( ) H ( s ) h ( ) e d hp://ewoo.com 5 Eug Je Woo

16 (b) Discree-ime: () Superposiio priciple e He ( ) H ( ) he [ ] (a) Coiuous-ime: if x () ae, y () ahe ( ) e. (a) Discree-ime: if x [ ] ae, y ah e e. [ ] ( ) Filerig () Frequecy-shapig filers () Frequecy-selecive filers hp://ewoo.com 6 Eug Je Woo

17 5. Coiuous-Time Fourier Trasform Fourier rasform pair () Fourier rasform or Fourier iegral x () X( ) e d () Iverse Fourier rasform where X ( ) X( ) x( ) e d is called he specrum of x(). (3) For a periodic sigal ~ x() wih period T, a T T ~() xe d xe () T T a T X ( ) sice d wih x () R S T x~ (), T T., oherwise (4) Covergece of he Fourier rasform (a) x() is absoluely iegrable; ha is, x () d. (b) x() has a fiie umber of maxima ad miima wihi ay fiie ierval. (c) x() has a fiie umber of discoiuiies wihi ay fiie ierval. Furhermore, each of hese discoiuiies mus be fiie. Fourier rasform for periodic sigals () X( ) ( ) x () ( ) e d e () I geeral, X( ) a( ) x () ae hp://ewoo.com 7 Eug Je Woo

18 Properies of coiuous-ime Fourier rasform - x() X ( ) ad y() Y( ) Sigal Fourier rasform Lieariy ax() by() ax ( ) by( ) Time shifig x ( ) e X( ) Frequecy shifig e x() X( ( )) Cougaio x * () X * ( ) Time reversal x( ) X ( ) Time/frequecy x( ) F scalig a X I HG a K J Covoluio x() y() X ( ) Y( ) uliplicaio x() y() X( ) Y( ) Differeiaio dx() X ( ) d Iegraio xd () X( ) X( ) ( ) Cougae Real x() * X( ) X ( ) symmery X( ) X( ) X( ) X( ) Re lx( ) q Re lx( ) q Im X( ) Im X( ) l q l q Real ad eve Real ad eve x() X ( ) real ad eve Real ad odd Real ad odd x() X ( ) purely imagiary ad odd Eve x x x decomposiio e( ) () ( ) q, real Re lx( ) q Odd x x x decomposiio o () () ( ) q, real Im lx( ) q Parseval's relaio x () d X( ) d hp://ewoo.com 8 Eug Je Woo

19 Sysem characeried by liear cosa-coefficie differeial equaio () A class of coiuous-ime LTI sysem wih a d y () b d x x y d () () () d Causal LTI () Frequecy respose b Y( ) H( ) X( ) a ( ) ( ) hp://ewoo.com 9 Eug Je Woo

20 6. Laplace Trasform (Bilaeral) Laplace rasform () Defiiio wih s s X() s x() e d x () L X() s () Fourier rasform Flx () q X( ) X() s s (3) ROC (regio of covergece) (4) Calculus problem algebraic problem Raioal Laplace rasform () s () X() s Ds () () Poles (3) Zeros ROC () The ROC of X(s) cosiss of srips parallel o he -axis i he s-plae. () For raioal Laplace rasforms, he ROC does o coai ay poles. (3) If x() is of fiie duraio ad is absoluely iegrable, he he ROC is he eire s- plae. (4) If x() is righ-sided, ad if he lie Relq s is i he ROC, he all values of s for which Relq s will also be i he ROC. (5) If x() is lef-sided, ad if he lie Relq s is i he ROC, he all values of s for which Relq s will also be i he ROC. (6) If x() is wo-sided, ad if he lie Relq s is i he ROC, he he ROC will cosis of a srip i he s-plae ha icludes he lie Relq s. (7) If he Laplace rasform X(s) of x() is raioal, he is ROC is bouded by poles or exeds o ifiiy. I addiio, o poles of X(s) are coaied i he ROC. (8) If he Laplace rasform X(s) of x() is raioal, he if x() is righ-sided, he ROC is he regio i he s-plae o he righ of he righmos pole. If x() is lef-sided, he ROC is he regio i he s-plae o he lef of he lefmos pole. hp://ewoo.com Eug Je Woo

21 Iverse Laplace rasform () Defiiio wih s s x () X() s e ds x () X() s () I pracice, for raioal Laplace rasforms, use he parial fraio expasio. L Geomeric evaluaio of frequecy respose from pole-ero plo Properies of Laplace rasform Sigal Laplace rasform ROC x() x() x() X () s X() s X() s R R R Lieariy ax() bx() ax() s bx() s A leas R R Time shifig x ( ) s e X() s R Shifig i he s- s e x() X( s s ) Shifed versio of R, i.e. domai sroc if ( ss) R Time scalig x( a) F a X s Scaled ROC, i.e., a sroc if s a R Cougaio x () * * X ( s ) R Covoluio x() x() X() s X() s A leas R R Differeiaio i d he ime domai d x sx () s A leas R Differeiaio i x() d he s-domai ds X() s R Iegraio i he ime domai x( ) d Iiial- ad fialvalue problem x() for < ad x() coais o impulses or higher order sigulariies a = HG I K J mrelq s X() s A leas R s x( ) lim sx( s) s lim x ( ) lim sx( s) s r LTI sysem ad Laplace rasform () Y() s H() s X () s where H( s) is he rasfer fucio or sysem fucio () Causiliy (a) The ROC of H( s) for causal LTI sysem is a righ-half plae hp://ewoo.com Eug Je Woo

22 (b) For a raioal H(), s he ROC of H( s) for causal LTI sysem is he righ-half plae o he righ of he righmos pole (3) Sabiliy (a) A LTI sysem is sable iff he ROC of is H( s) icludes he eire -axis (b) A causal LTI sysem wih raioal H( s) is sable iff all poles of H( s) lie i he lef-half plae, i.e., all poles have egaive real pars Sysem fucio algebra ad bloc diagram represeaios () Parallel iercoecio: h () h() h () Hs () H() s H () s () Series or cascade iercoecio: h () h() h () Hs () H () sh () s Ys () H() s (3) Feedbac iercoecio: Hs () X() s H () s H () s Uilaeral Laplace rasform s (a) Defiiio: X () s x() e d (b) ROC is always a righ-half plae UL x () X() s UL x () l q hp://ewoo.com Eug Je Woo

23 7. Discree-Time Fourier Trasform Discree-ime Fourier rasform pair () Discree-ime Fourier rasform x X e [ ] ( ) e d () Iverse Fourier rasform X( e ) x[ ] e where X( e ) is called he specrum of x[]. (3) Covergece of he discree-ime Fourier rasform: x[] is absoluely summable; ha is, x ( ) or x ( ). Discree-ime Fourier rasform for periodic sigals () x [ ] e X( e ) ( l) l ( ) () I geeral, x [ ] ae X( e ) a( ) Properies of discree-ime Fourier rasform - x [ ] X( e ) ad y [ ] Ye ( ) Sigal Fourier rasform Lieariy ax[ ] by[ ] ax ( e ) by( e ) Time shifig x [ ] e X( e ) Frequecy shifig e x[ ] Xe ( ) Cougaio x * [ ] X * ( e ) Time reversal x[ ] e X( e ) hp://ewoo.com 3 Eug Je Woo

24 R S Xe e Time expasio x m x [ ] [ ], if T, oherwise where m is a ieger Covoluio x[ ] y[ ] X( e ) Y( e ) uliplicaio x[ ] y[ ] X e ( ) ( ) Y( e ) d Differecig i x[ ] x[ ] ( e ) X( e ) ime Accumulaio X( e ) x [ ] e Differeiaio i frequecy Cougae symmery x[ ] X( e ) ( ) dx ( e ) d Real x[] * X( e ) X ( e ) X( e ) X( e ) o o X( e ) X( e ) Re X( e ) Re X( e ) o o Im X( e ) Im X( e ) Real ad eve Real ad eve x[] X( e ) real ad eve Real ad odd Real ad odd x[] X( e ) purely imagiary ad odd Eve x x x decomposiio e [ ] [ ] [ ] Re ox( e ) Odd x x x decomposiio o [ ] [ ] [ ] Im X( e o ) Parseval's relaio x [ ] X( e ) d Sysem characeried by liear cosa-coefficie differece equaio () A class of discree-ime LTI sysem wih () Frequecy respose a y[ ] b x[ ] x[ ] y[ ] Causal LTI hp://ewoo.com 4 Eug Je Woo

25 He ( ) Ye ( ) X( e ) be ae hp://ewoo.com 5 Eug Je Woo

26 8. Z-Trasform (Bilaeral) -rasform () 정의 ( re ) Z lx [ ] q X( ) x [ ] () Fourier rasform 과의관계 Flx [ ] q X( e ) X( ) xe [ ] e Z x [ ] X( ) Z Z x [ ] X( e ) e d Im plae e si Re cos 3 b g r re r cos si e ad cos si ad e cos si ( ) X( e ) X( e ) 4 T (3) ROC (regio of covergece) (a) 수렴조건 : x [ ] X( ) hp://ewoo.com 6 Eug Je Woo

27 (b) ROC : ROC rig 의모양을가진다. m r : ROC 는원점을중심으로하는 m r X e (c) : ROC ( ) 가수렴, X e ( ) 가 aalyic 한경우 ; X( e ) ( ) ( ) 인경우는? Raioal form () () X() D () (a) x[ ] 이 real 또는 complex expoeials 의합으로표현될때 (b) x[ ] 의길이가유한할때 () Pole 과 ero (a) X( ) 을 ero 라함 (b) X( ) 을 pole 이라함 (3) 예제 : x [ ] a u [ ] X() a a e 이고, X () 가수렴하기위해서는 a 즉, a 이어야함. 따라서, X () a a 이고 ROC 는 a 이다. Im plae a Re a 만약, a = 이라면, X() ( ) 이다. hp://ewoo.com 7 Eug Je Woo

28 (4) 예제 : x [ ] a u[ ] e e 이 a X() a u[ ] a a a a 고, X () 가수렴하기위해서는 a 즉, a 이어야함. 따라서, X () a 이고 ROC는 a 이다. Im plae a Re a 만약, a = 이라면, X() ( ) 이다. (5) 예제 : x [ ] R S T a,, oherwise a a X() a a e e 이고, = 에 ( a a ) 개의 pole 이있고, ( ) 개의 ero 가 ae 있다. 따라서, ROC 는 인전체공간이된다.,,,, 에 hp://ewoo.com 8 Eug Je Woo

29 Im plae a Re 선형성 (Lieariy) l q () Z ax [ ] bx [ ] ax ( ) bx ( ) K () x [ ] a u [ ] i i Z X() K K a i i i a i RST, : max a i i UVW (3) x [ ] bu[ ] K i i Z X() K K b i i i b i RST, : mi b i i UVW (4) 예제 : x [ ] u [ ] u [ ] 3 F H G I K J F HG I K J Z? F (5) 예제 : x [ ] u [ ] u[ ] H G I K J 4 F H G I K J Z? ROC 의성질 () The ROC of X() cosiss of a rig i he -plae ceered abou he origi. () The ROC does o coai ay poles. (3) If x[] is of fiie duraio, he he ROC is he eire -plae excep possible = ad/or =. (4) If x[] is a righ-sided sequece, ad if he circle r is i he ROC, he all fiie values of for which r will also be i he ROC. (5) If x[] is a lef-sided sequece, ad if he circle r is i he ROC, he all fiie hp://ewoo.com 9 Eug Je Woo

30 values of for which r will also be i he ROC. (6) If x[] is wo-sided, ad if he circle r is i he ROC, he he ROC will cosis of a rig i he -plae ha icludes he circle r. (7) If he -rasform X() of x[] is raioal, he is ROC is bouded by poles or exeds o ifiiy. (8) If he -rasform X() of x[] is raioal, he if x[] is righ-sided, he ROC is he regio i he -plae ouside he ouermos pole. If x[] is causal, he ROC also icludes =. Iverse -rasform () Defiiio wih re ( 실제로는거의사용하지않음 ) Z x [ ] X( ) d x [ ] Z X( ) C () I pracice, for raioal -rasforms, use ispecio or parial fraio expasio. (a) Ispecio au Z [ ], a a Z au [ ], a (b) Parial fracio expasio X() b a - > poles a = - < eros a = - eros a ad poles a b a a hp://ewoo.com 3 Eug Je Woo

31 X() C m b b X() a e e c d r A Cm Br m d r, i m d e i if sm! d A d sigle poles d A d X sm gb ig s muliple pole of order m e () R sm d s S U dw X w smb i g T e V dw W r B r Br[ r] R S b g T b g A d u[ ], d A d u[ ], d 9 8 예제 : X() 3 (c) Power series expasio: X() x[] 예제 : X() 예제 : X() loge a bg a, w d i 및 Z x [ ] [ ] [ ] [ ] [ ] a Z x [ ] R S b g T a,, Geomeric evaluaio of frequecy respose from pole-ero plo hp://ewoo.com 3 Eug Je Woo

32 Properies of -rasform Sigal Laplace rasform ROC x[ ] x[ ] x[ ] X () X() X() R R R Lieariy ax[ ] bx[ ] ax() bx() A leas R R Time shifig x [ ] X() R, excep possible addiio or deleio of he origi Scalig i he - domai e x[ ] x[ ] a x[ ] Time reversal x[ ] Time expasio R S xr [], r x( )[ ] T, r X( e ) X( ) X( a ) X( ) X( ) R R R Scaled versio of R Ivered R Cougaio x [ ] * * X ( ) R Covoluio x[ ] x[ ] X() X() A leas R R Firs differece x[ ] x[ ] ( ) X( ) A leas R Accumulaio Differeiaio i he -domai Iiial-value problem x [ ] X() x[ ] dx () R d x[ ] for < x[ ] lim X( ) b g 예제 : x r u re [ ] cos [ ] u [ ] re u [ ] Z u [ ], re Z u [ ] F H G re I K J re m A leas R m r r hp://ewoo.com 3 Eug Je Woo

33 re u [ ] Z F H G re I K J re rcos X() rcos r, r Complex covoluio Z () w [ ] x [ x ] F [ ] W () X v X v v dv C H G I K J b g F () w [ ] x [ x ] [ ] W () X e X e e e, periodic covoluio Parseval s relaio () x [ ] x [ ] * * C X v X v v dv b g e e e * * () x[ ] x[ ] X e X e d Auocorrelaio () 정의 : C [ ] x[ ] x[ ] xx () -rasform: C () xx [][ ] x [] x [ ] xx x [ ] X( ) X X bg e LTI sysem ad rasform () Y () H() X () where H() is he rasfer fucio or sysem fucio () Causiliy hp://ewoo.com 33 Eug Je Woo

34 (a) A discree-ime LTI sysem is causal iff he ROC of H() is he exerior of a circle, icludig ifiiy (b) A discree-ime LTI sysem wih raioal H() is causal iff - he ROC is he exerior of a circle ouside he ouermos pole, ad - he order of he umeraor of H() cao be greaer ha he order of he deomiaor (3) Sabiliy (a) A LTI sysem is sable iff he ROC of is H() s icludes he ui circle (b) A causal LTI sysem wih raioal H() s is sable iff all poles of H() s lie iside he ui circle, i.e., all poles have magiudes smaller ha oe Sysem fucio algebra ad bloc diagram represeaios () Parallel iercoecio: h [ ] h[ ] h [ ] H () H() H () () Series or cascade iercoecio: h [ ] h[ ] h [ ] H () H () H () Y () H() (3) Feedbac iercoecio: H () X() H () H () Uilaeral -rasform () Defiiio: X () x[] () ROC is always he exerior of a circle UZ x [ ] X( ) UZ x [ ] l q hp://ewoo.com 34 Eug Je Woo

35 9. Time ad Frequecy Characeriaio of Sigals ad Sysems agiude-phase represeaio of Fourier rasform () X X e ( ) ( ) X ( ) X( e ) or X( e ) X( e ) e () agiude (a) X( ) : eergy desiy specrum (b) X( ) (3) Phase d : eergy i he frequecy bad, d (c) X( ) : relaive magiudes of complex expoeials ha mae up x() (a) X( ) : relaive phase of complex expoeials ha mae up x() ad grealy affecs he sigal (b) Impora i some case (image) ad o impora i some oher case (soud) LTI sysem () Y( ) H( ) X ( ) or Ye ( ) He ( ) Xe ( ) () Y( ) H( ) X( ) where H( ) is he gai or magiude disorio (3) Y( ) H( ) X( ) where H( ) is he phase shif or phase disorio (4) Liear ad oliear phase (a) Liear phase: delay wihou disorio, H( ) e y( ) x( ) delay of He ( ) e y [ ] x [ ] delay of (b) oliear phase: delay wih disorio (5) All-pass sysem: H( ) or He ( ) d d (6) Group delay: ( ) lh( ) q or ( ) ohe ( ) d d hp://ewoo.com 35 Eug Je Woo

36 (a) Pricipal phase (b) Uwrapped phase (c) Dispersio: differe frequecy compoes delayed by differe amous Bode plo () Coiuous-ime (a) agiude: log H( ) versus log f (b) Phase: H( ) versus log f () Discree-ime (a) agiude: log He ( ) versus (b) Phase: H( e ) versus Ideal ad oideal filers () Frequecy domai specificaios (a) Passbad (b) Trasiio bad (c) Sop bad (d) Passbad ripple (e) Sopbad ripple () Time domai specificaios (a) Rise ime (b) Overshoo (c) Rigig frequecy (d) Selig ime Firs-order coiuous-ime sysems () Differeial equaio: dy () y () x () d (a) is he ime cosa () Frequecy respose: H( ) (3) Impulse respose: h () e u () hp://ewoo.com 36 Eug Je Woo

37 (4) Sep respose: s () h () u () e u () (5) Bode plo (a) log H( ) log ( ) (b) H( ) a ( ) Secod-order coiuous-ime sysems () Differeial equaio: d y () dy() y () x () d d (a) is he dampig raio (b) is he udamped aural frequecy () Frequecy respose: H( ) ( ) ( ) b cgb cg c c where c where (3) Impulse respose c c (a) If, h () e e u () (a) If, h () e u () (4) Sep respose (a) If, s h u e e () () () u () c c (a) If, s () e e u () (5) Bode plo R S T RL S T L (a) log H( ) log 4 P c F H G I K J O P Q c OU Q PV W F H G I KJ U V W hp://ewoo.com 37 Eug Je Woo

38 (b) H( ) a (6) oe ha (a) H( ) (b) If F H G b b g I g K J. 77, max arg mi H( ) (c) If. 77, H( ) decreases moooically (d) Qualiy facor, Q defies a measure of he sharpess of he pea Firs-order discree-ime sysems () Differece equaio: y [ ] ay [ ] x[ ], a (a) a deermies he rae of he sysem respose () Frequecy respose: He ( ) ae (3) Impulse respose: h [ ] a u [ ] a (4) Sep respose: s [ ] h [ ] u [ ] u [ ] a (5) Bode plo (a) He ( ) (b) He ( ) a e a acos L a si a cos Secod-order discree-ime sysems O QP () Differece equaio: y [ ] rcos y [ ] r y [ ] x [ ], r ad (a) r corolls he rae of decay of h[] hp://ewoo.com 38 Eug Je Woo

39 (b) deermies he frequecy of oscillaio () Frequecy respose He ( ) rcos e r e re e re e e e (a) If ad, He ( ) A B + where re e re e e e e e A ad B si si (b) If, He ( ) (c) If, He ( ) (3) Impulse respose e re e re (a) If ad, h [ ] Are Bre u [ ] r b g (a) If, h [ ] r u [ ] (c) If, h [ ] r u [ ] (4) Sep respose b gb g L L F G H L b g b g e e O QP si e I F J G e K H r O r u r QP b si (a) If ad, s [ ] re re A B u G re J G re JP [ ] r (a) If, s [ ] r ( ) [ ] r r IO JP KQP g u [ ] hp://ewoo.com 39 Eug Je Woo

40 L b g b g r r (c) If, s [ ] r ( )( r) u[ ] r r r (5) If A ad B are real for he case where ad, He ( ) ede ede O QP wih d ad d I correspods o he differece equaio, y [ ] ( d d) y [ ] dd y [ ] x [ ]. The, he frequecy respose is A B He ( ) de de d where A d d d ad B d d. Impulse respose ad sep respose are h[ ] Ad Bd u[ ] ad L F HG I d d s [ ] A B u [ ] d KJ. d F HG IO KJ QP hp://ewoo.com 4 Eug Je Woo

41 . Dualiies i Fourier Trasform Coiuous ime Discree ime Time domai Frequecy domai Time domai Frequecy domai Fourier series coiuous ime periodic i ime x () ae discree frequecy aperiodic i frequecy a xe () d T T discree ime periodic i ime x [ ] a e ( ) discree frequecy periodic i frequecy a ( ) xe [ ] Fourier rasform coiuous ime aperiodic i ime x () X( ) e d coiuous frequecy aperiodic i frequecy X( ) x( ) e d discree ime aperiodic i ime x X e [ ] ( ) e d coiuous frequecy periodic i frequecy X( e ) x[ ] e hp://ewoo.com 4 Eug Je Woo

42 . 표본화와표본화주파수의변환 (Samplig ad Samplig Frequecy Coversio) 표본화 (Samplig) 연속-시간신호 xc () 를 T 시간간격으로표본화하는경우를생각해보자. 이때, xc () 는 bad-limied 신호로서최대주파수가 f 이라가정한다. 편의상연속-시간신호의각주파수는 로나타내고이산-시간신호의각주파수는 로나타낸다. xc () 의연속-시간 Fourier 변환쌍은다음과같다. xc() Xc( ) e d Xc( ) xc( ) e d 임펄스열을 xc () 에곱하는것으로표본화과정을모델링하고, 표본화후의신호를 x () 라하자. s s () ( T) S ( ) s s T T F I HG K J b x () x ()() s x () ( T) x ( T) ( T) s c c 이때, s f s 는표본화주파수이다. 그러면, x s( ) 의연속-시간 T Fourier 변환은다음과같이두가지로구할수있다. b g s c T xc( T) ( T) e d xc( T) e Xs( ) Xc( ) S( ) Xc s T X ( ) x ( ) e d x ( T) ( T) e d s c g 연속-시간신호 x () 에서시간축을 T 로나누어정규화 (ormalie) 하고, s hp://ewoo.com 4 Eug Je Woo

43 임펄스열을이산-시간수열 (sequece) 로변환하여얻어지는이산-시간신호를 x[ ] 이라하면, x [ ] x( T) 이고, x[ ] 의이산-시간 Fourier 변환쌍은다음과같다. x [ ] X( e ) e d X( e ) x[ ] e 그런데, X ( ) x ( T) e x[ ] e s c T T c 이므로 X ( ) X( e ) X( e ) s T 이다. 따라서, 두각주파수사이에는 T 의관계가성립하고, 이산-시간신호 x[ ] 의 Fourier 변환은다음의두가지로 나타낼수있다. T T X( e ) Xc s T b g X( e ) Xc T T T F HG I K J Coiuous Time Divide Time Axis by T Discree or ormalied Time [ ] xc ( T ) uliply Time Axis by T x Coiuous Frequecy T uliply Frequecy Axis by T Divide Frequecy Axis by T Discree or ormalied Frequecy T hp://ewoo.com 43 Eug Je Woo

44 Xc ( ) Xs ( ) s T s X( e ) s b s s T g s s T T m b m T m g b s T g 4 T s yquis 표본화정리 (yquis samplig heorem) 표본화주파수와신호의최대주파수사이에다음의관계가성립하면, 이득이 T 이고차단주파수가 인이상적인저역통과필터를이용하여표본화하여얻어진이산-시간신호 x[ ] 으로부터원래의연속-시간신호 xc () 를오차없이완벽하게복원할수있다. 또는 s s 또는 m 이때, 을 yquis rae 이라하고 s 를 yquis 주파수라고한다. 연속 - 시간신호의복원 (Recosrucio of coiuous-ime sigal) x[ ] 을이득이 T 이고차단주파수가 인이상적인저역통과필터에통과시키면 ierpolaio 에의해원래의연속-시간신호 x () 가복원된다. 표본 c hp://ewoo.com 44 Eug Je Woo

45 화후의신호 x ( ) x ( T) ( T) x[ ] ( T) 를이득이 T 이고 차단주파수가 T 보자. s s c 인이상적인저역통과필터에통과시켰다고가정해 si T h () T 이므로, 저역통과필터의출력은 H( ) R S T T, T, oherwise ~ si ( T) T x () xs () h () x [ ] ( T) T 가되며, x[ ] 들이 sic 함수에의해 ierpolaio 되어서 ~ x () x() 가된다. c 표본화주파수변환 (Samplig frequecy coversio) () 정수배의 dowsamplig 표본화주기 T 로표본화한신호 x[ ] 으로부터, 표본화주기 T T 로 표본화한신호 x 이고, 이므로, 이고, 이다. [ ] 을구하는경우를생각해보자. X x [ ] x[ ] x ( T) x ( T ) c c X( e ) Xc T T T r ( e ) X T T T F HG c r 이제 r i으로하면, 이고 i ( ) 이고, X R S T i ( e ) X T T T T c i F HG F HG I K J I K J IU K JV W hp://ewoo.com 45 Eug Je Woo

46 이다. 위의 X( e ) Xc T T T 이므로, Xe F HG ( i) e I K J 에서 i Xc T T T X ( e Xe ) i i F HG ( ) 가된다. Aliasig 이발생하지않기위해서는 m T 또는 T m 또는 m 또는 s 이어야한다. e I K J hp://ewoo.com 46 Eug Je Woo

47 Xc ( ) Xs ( ) T Xs ( ) s s s s T T X( e ) T s s s T m X m ( e ) 4 T s T m m 4 T s 최초의표본화주파수가충분히크지않아서 ( 즉, s ), - dowsamplig 에의해 aliasig 이발생할수밖에없는경우에는, 차단주파수가 인이상적인디지털저역통과필터를사용한후, dowsamplig 한다. 이러한과정을 decimaio 이라한다. hp://ewoo.com 47 Eug Je Woo

48 Xc ( ) X( e ) T m X ( e ) 3 m T 3 4 T s He ( ) c 3 4 T s ~ X( e ) H( e ) X( e ) T c 3 4 T s 4 T s ~ X ( e ) T 4 T s hp://ewoo.com 48 Eug Je Woo

49 () 정수배의 upsamplig 표본화주기 T 로표본화한신호 x[ ] 으로부터, 표본화주기 T T L로 표본화한신호 xl [ ] 을구하는경우를생각해보자. 이 L 의배수인경우에는 xl[ ] x[ L] xc( T L) xc( T) 이고, 이 L 의배수가아닌경우에는 x [ ] 으로한다. 즉, L 또는, xl[ ] R S T 이다. 이산 - 시간 Fourier 변환을하면 따라서, X F H x [ L],, L, L,, oherwise xl[ ] x[ ] [ L] L L XL( e ) x[ ] [ L] e x[ ] e X( e ) G J L ( e ) I K 은 X( e ) 의 축을 L 로나누어줌으로써구해진다 ( 즉, L ). 이것은 축을 L 배만큼압축함을의미한다. 연속-시간각주파수와의관계는 T T L 이다. 따라서, upsamplig 은그자체에의한 aliasig 이발생하지않는다. 원래의신호 x[ ] 이충분히큰표본화주파수에의해표본화된신호여서 aliasig 이발생하지않았다고가정하자. 이득이 L 이고차단주파수가 L 인이상적인디지털저역통과필터에 xl [ ] 을통과시키면, ierpolaio 에의해그출력 ~ x L [ ] 은원래의연속-시간신호 xc () 를 T T L 의표본화주기로빠르게표본화한결과와동일하게된다. hp://ewoo.com 49 Eug Je Woo

50 Xc ( ) X( e ) T X T L m ( e ) L 4 T s m He ( ) L c L 4 T L sl ~ X ( e ) H( e ) X ( e ) L LTT L 4 T L s L m 4 T L s L (3) 유리수배의표본화주파수변환 L-ierpolaor 의출력을 -decimaor 의입력에연결하는 cascade 결합에의해 L 배의표본화주파수를얻을수있다. hp://ewoo.com 5 Eug Je Woo

51 . LTI 시스템의주파수영역해석 (Aalysis of LTI Sysem i Frequecy Domai) LTI (Liear Time Ivaria) sysem LTI 시스템의입출력관계는다음과같다. y [ ] h [ ] x [ ] xh [ ] [ ] Z Y () H() X () 주파수응답 (frequecy respose) 은 에서, Ye ( ) He ( ) Xe ( ) e 이고 Ye ( ) 따라서 He ( ) 이다. 이때, X( e ) () He ( ) Ye ( ) : magiude respose 또는 gai X( e ) () arg He ( ) He ( ) Ye ( ) Xe ( o ) : phase respose 또는 phase shif o o : group delay d (3) ( ) grd He ( ) He ( ) d 여기에서, group delay 는 umber of samples 을그단위로가지며, 위상의선형성을나타내는척도이다. 만약 group delay 가상수가아니라면위상지연이비선형적임을의미한다. 이상적인 LPF 및 HPF 이상적인 LPF 는다음과같은주파수응답을가진다. 즉, H lp R S, c ( e ) T, c hp://ewoo.com 5 Eug Je Woo

52 H lp ( e ) c c 따라서, c hlp[ ] Hlp( e ) e d e d c R c, S e c si c, T si c 이고, 이것은 ocausal sysem 이다. 이상적인 HPF는다음과같은주파수응답을가진다. 즉, c H ( e ) H ( e ) hp lp H hp ( e ) c c si c 따라서, hhp[ ] [ ] hlp[ ] [ ] 이고, 역시 ocausal sysem 이다. 이상적인지연 (delay) 입력신호를 d 샘플만큼지연시키기만하는시스템을고려해보자. hd[ ] [ d] 이고, Hd ( e ) [ d ] e e d 이다. 따라서, H d ( e ) 이고 H ( e ), d 이다. 위상지연이주파수의선형함수임을주목하라. 이를 liear phase 라고함. d hp://ewoo.com 5 Eug Je Woo

53 Liear phase 특성을가지는이상적인 LPF 앞의이상적인 LPF 가 d 샘플만큼의이상적인지연을가진다면, H lp ( e ) R S T d e, 이다. 이역시 ocausal sysem 이다., c 그리고 h c lp si c( d) [ ] ( ) d Liear cosa coefficie differece equaio 으로표현되는 LTI 시스템 이에속하는 LTI 시스템은다음의식으로표현된다. a y[ ] b x[ ] 따라서, 전달함수는 raioal fucio 이되고 b Y () b H () X() a a Z a Y () b X () 이며, 우리는다음의사항들을알수있다. () 분자의 () 분모의 c e 은 ero a c d e 은 pole a d e e c d 와 pole a = 를의미한다. 와 ero a = 를의미한다. (3) h [ ] for < 즉, 시스템이 causal 하다면, ROC 는가장밖에위치한 pole 의바깥쪽이된다. (4) 시스템이 sable 하기위해서는 ROC 가반드시단위원 (ui circle) 즉 을포함하여야한다. (5) 시스템이 causal 하고또 sable 하기위해서는모든 pole 들은단위원 내부에위치하여야한다. Iverse 시스템전달함수 H() 를가지는 LTI 시스템의 iverse 시스템을고려해보자. 즉, G () HH () i () 또는 g [ ] h [ ] hi [ ] [ ] 이어야하므로, iverse 시스템의전달함수는 hp://ewoo.com 53 Eug Je Woo

54 Hi () H () 또는 H e i ( ) He ( ) 이다. 만약, Hi () 가존재한다면, 이는 H() 에의한효과를완벽하게상쇄할 수있음을의미한다. 그러나, 어떤 에서 He ( ) = 이라면, iverse 시스템은 존재하지않게되고, 이는한번 이되면되살릴수있는방법이없음을의미 한다. 이때, H() 의 ero 의위치와 He ( ) = 의관계는? 만약, iverse 시스템, H I () 이존재하고 H () 와 H () 가모두 causal 및 sable 하다면, 이는 H() 의모든 pole 들과모든 ero 들이단위원내부에위치하여야함을의미한다. 이와같은시스템을 miimum phase sysem 이라고한다. i Impulse respose: IIR ad FIR 전달함수 H() 의 pole 들은모두 sigle pole 이라면, 다음과같이전개할 수있다. 즉, 따라서, impulse respose 는 X() r A Br d r, i if sigle poles h [ ] B[ r] Ad u [ ] r r 이다. 이때, 모든 A 들이다 이면 h [ ] 의길이가유한하고, 그렇지않으면 h [ ] 은무한한길이를가진다. 따라서, () FIR (fiie impulse respose): A,, o pole, all ero 예 : h [ ] R S T a,, oherwise : FIR () IIR (ifiie impulse respose): A, for some, a leas oe pole 예 : y [ ] ay [ ] x[ ] h [ ] a u [ ] : IIR Frequecy respose of raioal sysem fucio () Gai i db = log He ( ) hp://ewoo.com 54 Eug Je Woo

55 - Aeuaio i db = log He ( ) = (Gai i db) () argohe ( ) He ( ) Ye ( ) Xe ( ) - Pricipal value 로계산되며, 따라서 ( ) 의값을가진다. He - 따라서, 의 discoiuiy 가발생할수있다. - 이러한 discoiuiy 를모두제거하여연속함수로만드는것은 phase uwrappig 이라한다. d (3) ( ) grdohe ( ) ohe ( ) d (4) Pole-ero plo 으로부터 frequecy respose 를추론하는것이가능하다. 즉, 단위원을돌면서 - Pole 이가까워지면 gai이증가하고 - Zero 가가까워지면 gai이감소한다. (5) 예 : a - H (), a re : r, r, r 및,, - H () - H () a b g, a re : r, r, r 및,, e e e. 683 e H 와 H 사이의관계 Liear cosa coefficie differece equaio 으로표현이가능한 LTI 시스템 의전달함수 H() 는 raioal fucio 이다. 이때, H H( e ) 와 H H( e ) () H ad # of poles ad eros 사이에는다음의관계들이성립한다. U V W () H wihi a scale facor fiie umber of choices for H S R T H ad # of poles ad eros hp://ewoo.com 55 Eug Je Woo

56 (3) If H() is miimum phase, he H H ad H H wih uow. (4) Pole 과 ero 들의관계 * * * He ( ) He ( ) H( e ) HH ( ) ( ) C ( ) b H () a re, e e * c d e re, r e, r e * b C () HH () ( ) * a F H G I K J * b ad H ( ) * a * e e * c d * ec ec * ed ed e 따라서, C() 의 pole 과 ero 들은각각 real 이거나아니면 cougae * reciprocal pair 이어야한다. 예를들면, 다음의두시스템에서 H H 이나 H H 이다. 즉, H () e e 5., H 4 4 () e8. e e8. e (5) Uceraiy: 만약, H H () () a a * 이라면 * * * * C() H() H ( ) H () H ( ) e e 4 4 e8. e e8. e 이므로불확실성이존재한다. 따라서, all-pass sysem 을분석해보자. All-pass sysem 다음과같은전달함수를가지는시스템을 all-pass 시스템이라고한다. * a Hap( ), a, a a hp://ewoo.com 56 Eug Je Woo

57 * * e a ae Hap( e ) e H ap( e ) ae ae 일반적인 all-pass 시스템의전달함수는 r c * d e e e e Hap() d * e e e e 이고, d 및 e 일때 sable 하고 causal 하게된다. Pole 과 ero 는각 b c rg개씩이고, 다음과같은특성을가진다. 각 () H ( e ) ap () H ( e ) ap (3) ( ) (4) iimum phase sysem 따라서, phase 또는 group delay compesaio 에이용된다. iimum phase sysem iimum phase sysem 은다음과같은특징을가진다. () 모든 pole 과 ero 는단위원내부에존재한다. () Iverse sysem 이존재한다. (3) iimum phase 이다. (4) H H 그리고 H H wih uow 전달함수가 raioal fucio 인모든 LTI 시스템의전달함수는 miimum phase 전달함수와 all-pass 전달함수의곱으로표현할수있다. 즉, H () H () H () mp 만약, H() 가단위원밖에하나의 ero 를가지고나머지의모든 pole 과 ero 는단위원내부에있는경우를가정해보자. 그러면, c 에대하여 * * c H() H() c H() c Hmp() Hap() c 이다. 위와같은경우에, ap e e * H() 의 sable 하고 causal 한 iverse sysem 은 hp://ewoo.com 57 Eug Je Woo

58 H () i H 가된다. 이때, () mp G () HH () () H () 이고, Ge ( ) 이며 Ge ( ) H ( e ) 이다. ap i 예제 : 다음시스템의 iverse sysem 은? Iverse filerig 한후의결과는? e e e e H (). 9e. 9e 5. e 5. e ap iimum phase sysem 의주요특성 () iimum phase lag: H Hmp Hap 및 H ap H 단, He ( ) 를위해서는 He ( ) h [ ] 의조건이필요하다. () iimum group delay: grdlq H grdhmps grdhaps 및 grdh ap s grdlq H grdh mp s (3) iimum eergy delay: H mp H H mp h h mp [ ] [ ] h [ ] He ( ) d Hmp( e ) d hmp[ ] 다음과같이 parial eergy 를정의하면 이고 m E [ ] hm [ ] hm [ ] h [ m] m m 따라서, 대부분의에너지가 = 근처에집중되어있음을알수있다. mp 참고로 maximum phase sysem 은모든 ero 가단위원외부에존재하고, hp://ewoo.com 58 Eug Je Woo

59 maximum eergy delay 의특성을가진다. hp://ewoo.com 59 Eug Je Woo