The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function

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1 A gnraliation of th frquncy rsons function Th convolution sum scrition of an LTI iscrt-tim systm with an imuls rsons h[n] is givn by h y [ n] [ ] x[ n ] Taing th -transforms of both sis w gt n n h n n Y y n x n ( [ ] [ ] [ ] h [ ] x[ n ] n ( l+ h[ ] x[ l] l n Or, Thrfor, Thus, Y( l h x [ ] [l] l Y( H(X( X ( Y( h[ ] X ( H( Hnc, H ( Y( / X ( Th function H(, which is th -transform of th imuls rsons h[n] of th LTI systm, is call th transfr function or th systm function Th invrs -transform of th transfr function H( yils th imuls rsons h[n] 3 5 Consir an LTI iscrt-tim systm charactri by a iffrnc quation x[ n ] y[ n ] Its transfr function is obtain by taing th -transform of both sis of th abov quation Thus H( Or, quivalntly as ( H( An altrnat form of th transfr function is givn by H( ( ξ ( λ

2 7 Or, quivalntly as H ( ( ξ ( ( ξ, ξ,..., ξ ar th finit ros, an λ λ, λ ar th finit ols of H(,,... If >, thr ar aitional ( ros at If <, thr ar aitional ( ols at 8 For a causal IIR igital filtr, th imuls rsons is a causal squnc Th ROC of th causal transfr function H ( ( ξ ( ( is thus xtrior to a circl going through th ol furthst from th origin Thus th ROC is givn by > max λ 9 Examl-Consir th -oint movingavrag FIR filtr with an imuls rsons /, n h[n], othrwis Its transfr function is thn givn by ( n H ( [ n ( ] Th transfr function has ros on th unit circl at j π /, Thr ar ols at an a singl ol at 8 Th ol at.5 xactly cancls th 7 ro at -.5 Th ROC is th ntir - -lan xct Imaginary Part Ral Part Examl-A causal LTI IIR igital filtr is scrib by a constant cofficint iffrnc quation givn by y[ n] x[ n ]. x[ n ] + x[ n 3] +.3 y[ n ]. y[ n ] +. y[ n 3] Its transfr function is thrfor givn by H ( Altrnat forms:. + H ( (. + j.8(. j.8 (.3(.5 + j.7(.5 j.7 ot:pols farthst from hav a magnitu.7 ROC: >.7 Imaginary Part Ral Part

3 3 Frquncy Rsons from Transfr Function If th ROC of th transfr function H( inclus th unit circl, thn th frquncy rsons H ( of th LTI igital filtr can b obtain simly as follows: H ( j H ( ω For a ral cofficint transfr function H( it can b shown that H ( H ( H *( H ( H ( H ( H ( Frquncy Rsons from Transfr Function For a stabl rational transfr function in th form H ( ( ξ ( ( th factor form of th frquncy rsons is givn by ω j H ( ( ( ( 5 Frquncy Rsons from Transfr Function It is convnint to visuali th contributions of th ro factor ( an th ol factor ( from th factor form of th frquncy rsons Th magnitu function is givn by H ( ( Frquncy Rsons from Transfr Function which rucs to H ( Th has rsons for a rational transfr function is of th form arg H ( j ω arg( / + ω( + ω arg( j arg( 7 Frquncy Rsons from Transfr Function Th magnitu-squar function of a ralcofficint transfr function can b comut using H ( ( ( ( ( * * 8 Th factor form of th frquncy rsons ω j H ( ( ( ( is convnint to vlo a gomtric intrrtation of th frquncy rsons comutation from th ol-ro lot as ω varis from to π on th unit circl 3

4 9 Th gomtric intrrtation can b us to obtain a stch of th rsons as a function of th frquncy A tyical factor in th factor form of th frquncy rsons is givn by ( ρ j φ whr ρ jφ is a ro if it is ro factor or is a ol if it is a ol factor As shown blow in th -lan th factor ( ρ j φ rrsnts a vctor starting at th oint ρ jφ an ning on th unit circl at As ω is vari from to π, th ti of th vctor movs countrclocis from th oint tracing th unit circl an bac to th oint As inicat by H ( th magnitu rsons H ( at a scific valu of ω is givn by th rouct of th magnitus of all ro vctors ivi by th rouct of th magnitus of all ol vctors 3 Liwis, from arg H ( j ω arg( / + ω( + arg( arg( w obsrv that th has rsons at a scific valu of ω is obtain by aing th has of th trm / an th linar-has trm ω( to th sum of th angls of th ro vctors minus th angls of th ol vctors Thus, an aroximat lot of th magnitu an has rsonss of th transfr function of an LTI igital filtr can b vlo by xamining th ol an ro locations ow, a ro (ol vctor has th smallst magnitu whn ω φ

5 5 To highly attnuat signal comonnts in a scifi frquncy rang, w n to lac ros vry clos to or on th unit circl in this rang Liwis, to highly mhasi signal comonnts in a scifi frquncy rang, w n to lac ols vry clos to or on th unit circl in this rang A causal LTI igital filtr is BIBO stabl if an only if its imuls rsons h[n] is absolutly summabl, i.., S h [ n] < n W now vlo a stability conition in trms of th ol locations of th transfr function H( Th ROC of th -transform H( of th imuls rsons squnc h[n] is fin n by valus of r for which h[ n] r is absolutly summabl Thus, if th ROC inclus th unit circl, thn th igital filtr is stabl, an vic vrsa In aition, for a stabl an causal igital filtr for which h[n] is a right-si squnc, th ROC will inclu th unit circl an ntir -lan incluing th oint An FIR igital filtr with boun imuls rsons is always stabl 7 8 On th othr han, an IIR filtr may b unstabl if not sign rorly In aition, an originally stabl IIR filtr charactri by infinit rcision cofficints may bcom unstabl whn cofficints gt quanti u to imlmntation Examl-Consir th causal IIR transfr function H ( Th lot of th imuls rsons cofficints is shown on th nxt sli 9 3 5

6 3 Amlitu h[n] Tim inx n As can b sn from th abov lot, th imuls rsons cofficint h[n] cays raily to ro valu as n incrass 3 Th absolut summability conition of h[n] is satisfi Hnc, H( is a stabl transfr function ow, consir th cas whn th transfr function cofficints ar roun to valus with igits aftr th cimal oint: ^ H ( A lot of th imuls rsons of h^[n] is shown blow Amlitu h^[n] In this cas, th imuls rsons cofficint h^[n] incrass raily to a constant valu as n incrass Hnc, th absolut summability conition of is violat Thus, H^ ( is an unstabl transfr function Tim inx n 3 35 Th stability tsting of a IIR transfr function is thrfor an imortant roblm In most cass it is ifficult to comut th infinit sum S n h [ n] < For a causal IIR transfr function, th sum S can b comut aroximatly as K h [ n] S K n 3 Th artial sum is comut for incrasing valus of K until th iffrnc btwn a sris of conscutiv valus of S K is smallr than som arbitrarily chosn small numbr, which is tyically For a transfr function of vry high orr this aroach may not b satisfactory An altrnat, asy-to-tst, stability conition is vlo nxt

7 37 Consir th causal IIR igital filtr with a rational transfr function H( givn by H ( Its imuls rsons {h[n]} is a right-si squnc Th ROC of H( is xtrior to a circl going through th ol furthst from 38 But stability rquirs that {h[n]} b absolutly summabl This in turn imlis that th DTFT of {h[n]} xists ow, if th ROC of th -transform H( inclus th unit circl, thn H ( j H ( ω H ( 39 Conclusion: All ols of a causal stabl transfr function H( must b strictly insi th unit circl Th stability rgion (shown sha in th -lan is shown blow unit circl j Im j j stability rgion R Examl- Th factor form of is H ( H ( (.9 (.93 which has a ral ol at.9 an a ral ol at.93 Sinc both ols ar insi th unit circl, H( is BIBO stabl Examl- Th factor form of is H^ ( H^ ( ( (.85 which has a ral ol on th unit circl at an th othr ol insi th unit circl Sinc both ols ar not insi th unit circl, H( is unstabl 7

the output is Thus, the output lags in phase by θ( ωo) radians Rewriting the above equation we get

the output is Thus, the output lags in phase by θ( ωo) radians Rewriting the above equation we get Th output y[ of a frquncy-sctiv LTI iscrt-tim systm with a frquncy rspons H ( xhibits som ay rativ to th input caus by th nonro phas rspons θ( ω arg{ H ( } of th systm For an input A cos( ωo n + φ, < n

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