Linear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let

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1 It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr function with ral impuls rspons h[ Lt h[ If is to hav a linar-phas, its frquncy rspons must b of th form j j c+β whr c and β ar constants, and H, calld th amplitud rspons, also calld th ro-phas rspons, is a ral function of For a ral impuls rspons, th magnitud rspons is an vn function of, i.., j H Sinc j, th amplitud rspons is thn ithr an vn function or an odd function of, i.. ± Th frquncy rspons satisfis th rlation j H H* or, quivalntly, th rlation j c+β j c+β If H is an vn function, thn th abov rlation lads to j β jβ implying that ithr β or β π From j j c+β w hav j c+β H Substituting th valu of β in th abov w gt jc n ± ± h[ 6 Rplacing with in th prvious quation w gt l ± h[ l] l Making a chang of variabl l n, w rwrit th abov quation as ± h[

2 As, w hav n h[ Th abov lads to th condition h[, n n with c / Thus, th FIR filtr with an vn amplitud rspons will hav a linar phas if it has a symmtric impuls rspons 8 If H is an odd function of, thn from j c+β j c+β jβ jβ w gt as Th abov is satisfid if β π/ or β π / Thn j j c+β rducs to j jc j 9 Th last quation can b rwrittn as jc n j j h[ As, from th abov w gt l j h[ l] l Making a chang of variabl l n w rwrit th last quation as l j h[ l] l Equating th abov with j h[ n w arriv at th condition for linar phas as h[, n with c / Thrfor, a FIR filtr with an odd amplitud rspons will hav linar-phas rspons if it has an antisymmtric impuls rspons Sinc th lngth of th impuls rspons can b ithr vn or odd, w can dfin four typs of linar-phas FIR transfr functions For an antisymmtric FIR filtr of odd lngth, i.., vn h[/] W xamin nxt th ach of th 4 cass

3 Typ : 8 Typ : Typ : Symmtric Impuls Rspons with Odd Lngth In this cas, th dgr is vn Assum 8 for simplicity Th transfr function is givn by H ] ] ] 3] + h [ 4] 5] 6] ] 8] 8 3 Typ 3: 8 Typ 4: 4 Bcaus of symmtry, w hav h[] 8], h[] ], h[] 6],andh[3] 5] Thus, w can writ 8 ] + ] + + h [ ] + 3] + 4] 4 { h[ ] + ] h [ ] + 3] + 4]} Th corrsponding frquncy rspons is thn givn by j4 H {h[]cos4 ]cos3 + h []cos 3]cos 4]} Th quantity insid th bracs is a ral function of, and can assum positiv or ngativ valus in th rang π 5 6 Th phas function hr is givn by θ + β whr β is ithr or π, and hnc, it is a linar function of Th group dlay is givn by dθ τ 4 d indicating a constant group dlay of 4 sampls In th gnral cas for Typ FIR filtrs, th frquncy rspons is of th form j / whr th amplitud rspons H, also calld th ro-phas rspons, is of th form H / ] + h[ cos n 8 3

4 9 Exampl-Considr [ H which is sn to b a slightly modifid vrsion of a lngth- moving-avrag FIR filtr Th abov transfr function has a symmtric impuls rspons and thrfor a linar phas rspons 6 ] A plot of th magnitud rspons of H along with that of th -point movingavrag filtr is shown blow Magnitud modifid filtr moving-avrag /π ot th improvd magnitud rspons obtaind by simply changing th first and th last impuls rspons cofficints of a moving-avrag MA filtr It can b shown that w an xprss H which is sn to b a cascad of a -point MA filtr with a 6-point MA filtr Thus, H has a doubl ro at, i.., π Typ : Symmtric Impuls Rspons with Evn Lngth In this cas, th dgr is odd Assum for simplicity Th transfr function is of th form H ] ] ] 3] + h [ 4] 5] 6] ] 3 Making us of th symmtry of th impuls rspons cofficints, th transfr function can b writtn as ] + ] + ] + 3] + / / / 5/ / { h[ ] + ] + 3/ / / / ] + 3] + } Th corrsponding frquncy rspons is givn by j/ H {h[]cos ]cos ]cos h[3]cos } As bfor, th quantity insid th bracs is a ral function of, and can assum positiv or ngativ valus in th rang π 4 4

5 5 Hr th phas function is givn by θ + β whr again β is ithr or π As a rsult, th phas is also a linar function of Th corrsponding group dlay is τ indicating a group dlay of sampls 6 Th xprssion for th frquncy rspons in th gnral cas for Typ FIR filtrs is of th form j / whr th amplitud rspons is givn by + / + H h[ cos n Typ 3: Antiymmtric Impuls Rspons with Odd Lngth In this cas, th dgr is vn Assum 8 for simplicity Applying th symmtry condition w gt 4 3 { h[ ] ] ] 3] } 8 Th corrsponding frquncy rspons is givn by j4 jπ / {h[]sin4 ]sin3 ]sin 3]sin } It also xhibits a linar phas rspons givn by θ + π + β whr β is ithr or π 9 Th group dlay hr is τ 4 indicating a constant group dlay of 4 sampls In th gnral cas j / j whr th amplitud rspons is of th form / H h[ sin n 3 Typ 4: Antiymmtric Impuls Rspons with Evn Lngth In this cas, th dgr is vn Assum for simplicity Applying th symmtry condition w gt / / / 5/ / { h[ ] ] 3/ / / / ] 3] } 5

6 3 Th corrsponding frquncy rspons is givn by j / jπ / {h[]sin ]sin ]sin h[3]sin } It again xhibits a linar phas rspons givn by θ + π + β whr β is ithr or π 3 Th group dlay is constant and is givn by τ In th gnral cas w hav j whr now th amplitud rspons is of th form j / + / + H h[ sin n 33 Gnral Form of Frquncy Rspons In ach of th four typs of linar-phas FIR filtrs, th frquncy rspons is of th form j / jβ Th amplitud rspons H for ach of th four typs of linar-phas FIR filtrs can bcom ngativ ovr crtain frquncy rangs, typically in th stopband 34 Exampl Considr th causal Typ FIR transfr function H Its amplitud and phas rsponss ar givn by H 6 6cos + 4cos cos3 θ xt, considr th causal Typ FIR transfr function Its amplitud and phas rsponss ar givn by H H θ + π 35 xt, considr th causal Typ FIR transfr function Its amplitud and phas rsponss ar givn by H θ + π j ot: H H 36 6

7 3 Hnc, H and H hav idntical magnitud rsponss but phas rsponss diffring by π as shown blow Amplitud 5 5 Amplitud rspons of H /π Phas, radians 5-5 Phas rsponss of H and H H H /π 38 Exampl Considr th causal Typ FIR transfr function H Its amplitud and phas rsponss ar givn by H sin + 4sin + sin3 θ 3 π 3 + xt, considr th causal Typ FIR transfr function H Its amplitud and phas rsponss ar givn by H4 H3 π θ ot: 4 3 j H3 H Hnc, H 3 and H 4 hav idntical magnitud rsponss but phas rsponss diffring by π as shown blow Amplitud Amplitud rspons of H /π Phas, radians Phas rsponss of H 3 and H 4 H 3 H /π Th magnitud and phas rsponss of th linar-phas FIR ar givn by j θ + β, for + β π, for < Thgroup dlay in ach cas is τ ot that, vn though th group dlay is constant, sinc in gnral H is not a constant, th output wavform is not a rplica of th input wavform 4 4

8 43 ot that, vn though th group dlay is constant, sinc in gnral is not a constant, th output wavform is not a rplica of th input wavform An FIR filtr with a frquncy rspons that is a ral function of is oftn calld a rophas filtr Such a filtr must hav a noncausal impuls rspons 44 Phas FIR Transfr Considr first an FIR filtr with a symmtric impuls rspons: h[ Its transfr function can b writtn as h[ By making a chang of variabl m n, w can writ h[ + m m h[ h[ m] h[ m] m m 45 Phas FIR Transfr But, m m m] Hnc for an FIR filtr with a symmtric impuls rspons of lngth + w hav H H A ral-cofficint polynomial satisfying th abov condition is calld a mirror-imag polynomial MIP 46 Phas FIR Transfr ow considr first an FIR filtr with an antisymmtric impuls rspons: h[ h[ Its transfr function can b writtn as h[ h[ By making a chang of variabl m n, w gt + m h[ h[ m] H m Phas FIR Transfr Hnc, th transfr function of an FIR filtr with an antisymmtric impuls rspons satisfis th condition A ral-cofficint polynomial satisfying th abov condition is calld a antimirror-imag polynomial AIP Phas FIR Transfr It follows from th rlation ± that if ξ o is a ro of, so is / ξ o Morovr, for an FIR filtr with a ral impuls rspons, th ros of occur in complx conjugat pairs Hnc, a ro at ξ o is associatd with a ro at ξ * o

9 49 Phas FIR Transfr Thus, a complx ro that is not on th unit circl is associatd with a st of 4 ros givn by ± jφ r, ± φ j r A ro on th unit circl appar as a pair ± jφ as its rciprocal is also its complx conjugat 5 Phas FIR Transfr Sinc a ro at ±is its own rciprocal, it can appar only singly ow a Typ FIR filtr satisfis with dgr odd Hnc implying H, i.., must hav a ro at 5 Phas FIR Transfr Likwis, a Typ 3 or 4 FIR filtr satisfis H H Thus implying that must hav a ro at On th othr hand, only th Typ 3 FIR filtr is rstrictd to hav a ro at sinc hr th dgr is vn and hnc, 5 Phas FIR Transfr Typical ro locations shown blow Typ Typ 3 Typ Typ 4 53 Phas FIR Transfr Summariing Typ FIR filtr: Eithr an vn numbr or no ros at and Typ FIR filtr: Eithr an vn numbr or no ros at, and an odd numbr of ros at 3 Typ 3 FIR filtr: An odd numbr of ros at and 54 Phas FIR Transfr 4 Typ 4 FIR filtr: An odd numbr of ros at, and ithr an vn numbr or no ros at Th prsnc of ros at ± lads to th following limitations on th us of ths linar-phas transfr functions for dsigning frquncy-slctiv filtrs 9

10 Phas FIR Transfr A Typ FIR filtr cannot b usd to dsign a highpass filtr sinc it always has a ro A Typ 3 FIR filtr has ros at both and, and hnc cannot b usd to dsign ithr a lowpass or a highpass or a bandstop filtr Phas FIR Transfr A Typ 4 FIR filtr is not appropriat to dsign lowpass and bandstop filtrs du to th prsnc of a ro at Typ FIR filtr has no such rstrictions and can b usd to dsign almost any typ of filtr 55 56