Continuous-time low-pass filter Butterworth filter

Transcription

1 Continuous-tie ow-pass fiter Butterworth fiter th -order Butterworth fiter with utoff frequeny : ( H Very : (- x 5 x x x Magnitude is onotoniay dereasing in Passband error is worse near H.77 H (s ( s si i s i : poes of H (s e that are in Re s < ; < Odd : { } e that are in Re s < ; < Even : { } ROC: region to the right of a poes Fat: H( H( s H( s s

2 H (s ( s si i H( s H( s ( s si ( s si ( ssi ( ssi i i i i Fro this equation, H ( s H ( s Want H (s to be stabe have poes in Re{s i } < has poe s i s and -s i s the eft-haf-pane poes is for H (s; ( the right-haf-pane poes is for H (-s Fro H ( H( s H( s H s s H s H H ( s H ( s H ( s s s s Fro this equation, H ( s H ( s s has poe at s s i i If is odd, poes: s i ( s i ( th -roots of unity e ; < If is even, poes: s i e s ype I s i e Chebyshev fiter th -order Chebyshev fiter with utoff type I

3 V H ε V (x the th Chebyshev poynoia in x V (x xv (x V - (x H V ε ε Euirippe in passband error is distributed unifory o ahieve a given passband ax. error, require ower-order ( than butterworth Monotoni in stopband heb x,.5 heb x,.7 heb 5 x, x ype II th -order Chebyshev fiter with utoff type II V H ε

4 heb, x x Monotoni in passband Equirippe in stopband th -order Eipti fiter with utoff : H ( ε U Eipti fiter U (x th Jaobian eipti funtion of x Equirippe both in passband and stopband Require saer than Chebyhev to ahieve ax error in passband or stopband Digita fiter ign Od-fashioned DSP paradig Proessing ontinuous-tie signas with the aid of disrete-tie systes wn [ ] yn [ ] yr ( t w t C H D D C yt H ( Obetive: Given H ired ( H (, find and ( Y ( H ( W ( Restrit to band-iited input w (t and < H H, Soution: use Ĥ so that

5 ote: here, we are not saping h (t to get h[n]. So, an t find the reationship using the deonstrution. h[n] is soe signa that, when used, wi ake the whoe disrete syste C, H, D D C at ike H (. Proof Given Ĥ (. W W for, assuing no aiasing W W k k If no aiasing, W or W W p W ; for - W W k k W ( for, assuing no aiasing Y H W, R Fro Y H W, and R Y R H W p. W p W : Substitute Y Y p, Y H W H W ( ( ( ( R H H,, or hus, want ( H ( H at east for C Don t need to worry about. > sine W ew-fashioned DSP paradig there. Design an ipeentabe disrete-tie syste with Ĥ ( Ĥ (

6 IIR fiter ign IIR infinite-duration ipuse response Aways assue h[n] is rea-vaued IIR fiter ign using ipuse invariane Design a disrete-tie ow-pass fiter Ĥ (fro avaiabe ( Ĥ with utoff need to eet ign spes Pik d > ; d ign Via w W d, transate Ĥ ( ; aong with ign spes d H, Design h (t that eets the ontinuous-tie ign spes and stabe & ausa rationa H (s Set hn [ ] h ( n. See whether this work. d d H ( d H k k d > doesn t atter d d d d k > sa aiasing if ( d H d k d Ĥ outside soe bound ( Aiasing ight ause Ĥ ( not to eet origina ign spes (espeiay if Ĥ ( barey does the ob in ontinuous tie If this ours, then shoud over-ign [ ] [ ] st n s d h t ke u t hn d k z un ; z e k kz H ( s ;Re{ s} > axre { s} H( z d ; z > ax z s s z z h (t is stabe, ausa, and has rationa H (s h[n] is stabe, ausa, and has rationa H(z h (t is not band-iited there is aiasing when onvert to h[n] poe H s poe s d e st h ( t ke u( t L H ( s ; Re{ s} > axre{ s } s s s poes of H (s k

7 Stabe iff a poes of rationa H(s ies in Re{s} <. So need Re{s } <, sn n n d s d h[ n] h d ( nd dke u[ n] dk( e un [ ] dk( z u[ n] z poes of H(z s d e kz kz ; n Z z z ze h[ n] d k( z un [ ] H( z d d sd s d z > ax z ax e Re{s } < z s d e < stabe Reverse proess is not uniquey deterined IIR fiter ign using biinear transforation Want a disrete-tie ow-pass fiter need to eet ign spes Pik any d > Ĥ with utoff Via tan, transate Ĥ H, ( (equa height; aong with ign d spes Design h (t / H (s that eets the ontinuous-tie ign spes and stabe & ausa rationa H (s H(z H s z z d work. See whether H ( H ( z e Idea rapezoida approxiation y( t w( τ dτ y( nd y ( n d w nd w n d d yn [ ] yn [ ] ( wn [ ] wn [ ] d ( z Y( z ( z W ( z Y z d z H( z W z z Iaginary axis in s-spae ( s aps onto d ( ( (

8 unit ire in z-spae ( z e e e d e e d e d e e e e e ote: Continuous-tie integrator H I (s s d sin tan os d A of -spae, ie, - < < aps onto - in -spae (and -periodi o aiasing on-inear apping between and pieewise-onstant ot a big probe if ( H, H, s phase harateristis get dangerousy twisted s is a poe of H (s z If Re{S } <, then z < d d s s is a poe of H(z H(z is rationa and stabe, if H (s is rationa, ausa, and stabe Let M z H z H s β M z, ( Re H (s is rationa & stabe { } hen H(z is rationa H(z is stabe ( z < if β and M have sae sign Proof z s s sz z z s <, β is rea, and M is a non-zero integer. β s M M M M β M β β z β s

9 Equaization ( β Re{ s} ( I{ s} ( β Re{ s} ( I{ s} { } I{ } { } I{ } M β s β Re s s z M z β s β Re s s For M > ; want z < z < ( β Re{ s } ( { } I s β Re{ s } β M ( ( I{ s} < β Re{ s} Re { s} < β β Re{ s} Re { s} β Re{ s} <β Re{ s} β Re{ s} < β > ;Re{ s } < For M < ; want z < z > ( β Re{ s } ( { } I s β Re{ s } β Design Ĥ ( to undo effet of Ĝ ( Can set H(z G( z Ex. work if G(z M ( ( I{ s} > β Re{ s} Re { s} > β β Re{ s} Re { s} β Re{ s} >β Re{ s} β Re{ s} > β < ;Re{ s } < z d a poynoia (z degree d H(z k k z - and h[n] k δ[n]k δ[n-] ot aways get ausa/stabe answer Ex. G(z z d a < poynoia (z degree d Get z in H(z and h[n] is not ausa soution: ign so that H(zG(z z a-d and then the resut is sipy a deay

10 Phase φ( In genera, we have H ( H ( e φ( is -periodi, not uniquey deterined due to -utipe abiguity Causa rea-vaued h[n] annot have zero phase H ( H ( β ( H ( e H uness h[n] K δ[n] * h[n] rea H ( H ( ; a n, a (positive integer α Case when H ( H ( e If wn [ ] H y [ n], then [ ] n [ ] wn H H e y n n sipe tie-deay Proof Y W H hn [ ] h [ n n ] nor onstant phase n n DF [ ] Y W H e Y e y n n by tie-shift rue. DF [ ] h n H H DF [ ] n DF [ ] hn H e h n n If h [n] is FIR, then hn [ ] h [ n n ] n n n n e e e e e wi be ausa for n arge enough n n ; rea a α Case when H ( H ( e DF [ ] ( ( α ( n sin n α α n H e hn e e d α for integer n [ ] α, [ ] δ [ ] hn n n [ ] w( t y( t w( t D α α wn e C C D yn y n Proof

11 W W( R > W α e W( α Y W( e > α Y Y k e W ; k d n n e e α ( e α e n α e t tα nα n α n e e e D α e C C D e e ( α β Generaized inear phase: Ĥ ( A( e Rea α, β, A( Can be expressed with β or for rea h[n] Proof * h[n] is rea H ( H ( ( α β ( α β A( e A( e β A( A( e rea So If If β e is rea trivia,, or ( β e, e β ± absorb e β into A( and β β e -, e β ± absorb sign into A( and e β β ruy inear phase when β and A( α Let A( Ĥ (, then H ( H ( e FIR fiter ign FIR fiter with generaized inear phase (g..p. ( α β Every generaized- inear-phase FIR fiter ( A( e types: Ĥ is of one of these 4

12 ype I II III IV hm [ ] hi [ M ] hii [ M ] hiii [ M ] hiv M idpoint M h[n] duration odd even odd even M M, M M M, M h[n] around idpoint even even odd odd #unknown h[n] η A( about even even odd odd A( about even odd odd even α M M M M [ ] β hink about A( as os( os sin( sin fiter H,L L H After shifted by in or (- n I IV I II in n { } A h[ M] hm [ ] os ( I > A hm os > II [ ] { } A hm [ ] sin( III > A hm sin > IV [ ] ype I h [ n ] I Odd duration Even-syetri about idpoint h[m]: h [ M ] h [ M ] α M, β I, > I

13 { } A h[ M] hm [ ] os ( ype II h[n] I Proof H I > [ ] n hne n { } M ( M ( M [ ] [ ] [ ] h M e hm e hm e > { } [ ] [ ] [ ] h M e e hm e hm e M M > hm [ ] { hm [ ] os( } e > Even about : A( A( Proof os ( os( Even about : A( A( Proof os os ( ( ( ( e e M ( e e e e os( ( ( ( ( e e Periodi with period : A( A Proof Even duration ( e e e e os( ( ( ( os os os Even-syetri about idpoint hm [ ] hm [ ] II [ ] [ ] h M h M, > α M, β II [ ] II A hm os > :

14 Proof H II [ ] n ( M M ( { h[ M e ] hm [ e ] } > hne n e h[ M e ] h[ M e ] > M hm [ ] os e > Even about : A( A( os Proof os Odd about : A( A( M Proof os os Re e e e e Re ( e e sin os os Re e e e e Re ( e e sin sin Periodi with period 4: A( 4 A

15 ype III os 4 os 4 os Proof h[n] Odd duration h[m] Odd-syetri about idpoint h[m] : h [ M ] h [ M ] α M, β { } A hm [ ] sin( III ype IV Proof H > III [ ] n hne III, > { } M ( M ( M [ ] [ ] [ ] > { [ ] [ ] } > n h M e hm e hm e e hm e hm e M { ( hm [ ] sin( } e > { hm [ ] sin ( } e > Odd about : A( A( Odd about : A( A( Periodi with period : A( A h[n] Even duration M M Odd-syetri about idpoint h [ ] h [ ] h [ M ] h [ M ], > IV IV : III

16 α M, β [ ] sin IV A hm > Proof [ ] [ ] [ ] { } [ ] [ ] [ ] [ ] > > > > sin sin M M M M M n n IV e h M e h M e h M e M h e e h M e h M e h n H Odd about : A A Even about : A A Periodi with period 4: 4 A A α id-oation of the duration interva β when having odd-syetri h[n] about idpoint(s o see this, odd negative sign in the idde sin - A( (III and IV bok DC bad ow-pass fiter A( (II and III bad high-pass fiter ype I isn t the best sine pass, Casading g..p FIR fiters ats as a g..p FIR i i i i i i i i H H A e α β Adding ( g..p FIR fiters ay not ats as a g..p FIR o see this, onsider the syetry of resuted h[n] Fiter Design tehnique Given Ĥ

17 argeting FIR g..p. fiters Frequeny-saping Design Find rea h[n] FIR g..p. fiter with duration interva n < and H k H k k < (aways exist Step Given Pik fiter type aording to the agnitude of know α, β 3 Set rea A ~ ( whih has required syetry for the targeted fiter type A A H 4 Get equations fro: ote that we now ook at A ~ ( Ak A k for < < Ĥ around, 5 Do one of the foowing: 5. Sove for h[n]-vaues fro the above equations, using inear agebra. 5. Set [ ] Gk A k e [ ] Gk [ ] hn L k αk β nk ψ ; n < Miniize H ; k < ie-doain east-squares ign for genera set of -points:, <L ; L an > H Math (approxiatey at a genera set of -points:, < L not neessariy unifory spaed an prioritize athing regions L Find h[n], n <, suh that A A Let α A ~ (, [ first ] ~ is iniized h α h, α h[ first η ], α k

18 Γ L η atrix with entries fro oeffiients on h s Miniize Γh α ( h Γ Γ Γ α Weighted tie doain east squares fiter ign ε Given weights ε >. LHW L Choose h to iniize ε [ Γh] α or ( h α E ( h α h Γ EΓ Γ Eα Miniize H ( H ( ε L Windowing d by Γ Γ Windowing h [n] with retanguar windowing funtion [ n] [ ] h n n < L hn [ ] L[ nh ] [ n] trunated version of h [n] n L Proof Use Parseva Identity: H H d hn h n [ ] [ ] n L o iniize, set h[n] h [n] ; -L < n < L h[n] isn t ausa shift it by (L- n < L- ( L [ ] [ ] n < L L n L [ ] hn h n h n n L n L n H ( h[ n] e a partia su of H (, whih is h [ ] ( n ( L sin L sin sin sin Proof n ne n

19 ( L L DF k L ( L k δ [ ] δ n k e [ ] DF n k e L L sin sin L sin e sin sinl ( iff L, sin, L 4 4 Centra obe s width derease as inrease L L Area under one side of first side-obe L L roughy the as inreases L ; tie-shift rue 4 sin L sin d d sin sin 4 x x.9 sin x sin d x h n L nh n H H µ µ dµ DF [ ] [ ] [ ] ( H H µ µ dµ ( Start with and inreasing it.

20 Assue that the side obes beyond the first one don t ontribute too uh. First, entra and two first-side- obe of ( µ ( µ Ĥ, so LQWHJUDO is in the passband of area under area under ain obe first-side-obe Ĥ µ s passband so the ext, the right first-side-obe is going out of integra inrease unti a of the right first-side-obe is gone out of ( µ passband. hen, integra area under area under ain obe ( eft first-side-obe Ĥ s so, height of the overshoot is proportiona to the area of the first-side obe hen, Centra obe start going out of Ĥ ( µ s passband so the integra start to derease again (big derease. his is where the transition region ours and it is proportiona to the width of the ain obe. Finay, a of the ain obe is gone out of Ĥ ( µ s passband. he eft firstside ob starts to get out, so the integra inreases again beause ess negative part is inuded. Gibbs Phenoenon (9% overshoot at up Bigger narrower entra obe narrower transition region Sae first-side-obe area sae peak overshoot (Gibbs; L [n] for L < n < L ( for n L PDLQ-OREH ZLGWK VW VLG -OREH KHLJKW avlgh - utoff 5HFWDQJXODU 7ULDQJXODU %DUWOH W - L n DQQ n os DQQLQJ Haing n.54.46os OREHDUHD 4 - G 8 - G 8 - G 8 - G LQLPD[LOWHUGHVLJQ p ast < (in passband where Ĥ ( s first > (in stopband where Ĥ ( s - p ~ transition region width δ passband rippe ax δ VWRSEDQGULSSOHPD[

21 3URWRW\SLFDOSUREOHP DHVLJQDGXUDWLRQ -,5ILOWHU transition region width s - p < a speifi aount iniizes the axiu of δ and δ 3URSHUWLHVRIVROXWLRQ Equirippe in both passband and stopband δ δ nuber of rippes between and is ELJJHU PDOOHU δ δ UHULSSOHV Signa fow graph Any signa fow graph ribing fiter is a reaization of the fiter z Minia/anonia reaization fewest possibe deay branh #deay branhes aount of eory required via the given signa fow graph In genera, H( z p z p( z q z, a proper rationa funtion q( z FIR Order of the fiter degree of q(z (assue fration is in owest ters Ex. Fiter is FIR q(z z n Minia reaization have # deay branhes Order of the fiter yn [ ] ayn [ ] ayn [ ] bwn [ ] bwn [ ] bwn [ ] H ( z b z b z b z az a Diret For II: Controabe anonia reaization Y ( z W( z ( b bz bz Q ( z IIR b bz bz W z az az az az W a z ( z a z Q( z az Q( z az Q( z W( z qn [ ] aqn [ ] aqn [ ] wn [ ] qn [ ] wn [ ] aqn [ ] aqn [ ] Qz

22 w[n] ( Y z b bz bz Q z [ ] [ ] [ ] [ ] Y z bq z bz Q z bz Q z yn bqn bqn bqn q[n] b z - y[n] -a b z - -a b b z b z a H ( z w[n] b b z a z b y[n] z - -a b b z z a H ( z w[n] b a z b y[n] z - -a a z H ( z a z

23 w[n] y[n] z - -a z - -a z - -a - z - ransposed Diret For II: Observabe anonia reaization yn [ ] bwn [ ] ( bwn [ ] ayn [ ] ( bwn [ ] ayn [ ] w[n] b y[n] z - b -a z - b -a Guarantees that it aso reaizes H(z o get fro diret for II, ust reverse a the arrows and swith roes of w[n] and y[n] 'LUHFW RUP, QRQ-PLQLPDOUHDOL]DWLRQ gn [ ] bwn [ ] bwn [ ] bwn [ ] yn [ ] ayn [ ] ayn [ ]

24 [ ] [ ] [ ] [ ] yn gn ayn ayn w[n] b y[n] z - z - b -a z - z - b -a Casade Reaization : a hain of diret for reaizations of the individua st - and nd - order fators 3DUDOOHO IRUP H z ([SDQG XVJSDUWLDO IUDFWLRQV z 5HDOL]HHDFKSLHFHGLUHFWO\ HRRXSLQSDUH OHO Casade of an FIR syste with an IIR syste H ( z p q ( z ( z (Causa FIR Fiters k p( z q ( z FIR IIR k k k h[n] h[ k] δ [ n k] ; H(z h [ k] z ; y[n] h [ n] w[ n k] h[] h[] h[] h[3] h[-] h[-] w[n] z - z - z - z - z - FIR fiters with generaized inear phase RURGG,,,,

25 w[n] z - z - z - ± ± ± z - h[] h[] z - h[] z - h[m] y[n] use for odd syetri h[n] (III RUHYHQ,,,9 w[n] ± z - ± z - ± z - h[] h[] z - h[m] z - y[n] use for odd syetri h[n] (IV