is monotonically decreasing function of Ω, it is also called maximally flat at the

Transcription

1 Le.8 Aalog Filte Desig 8. Itodtio: Let s eview aalog filte desig sig lowpass pototype tasfomatio. This method ovets the aalog lowpass filte with a toff feqey of adia pe seod, alled the lowpass pototype, ito patial aalog lowpass, highpass, badpass, ad badstop filtes with thei feqey speifiatios. 8. Bttewoth Filtes 8.. Bttewoth low-pass filte LPF A typial feqey espose fo a Bttewoth low-pass filte of ode is show i Fig. 8.. j + / Popeties: 8. j 0 fo all j fo all fiite j db Fig.8. Bttewoth LPF /s j is mootoially deeasig ftio of, it is also alled maximally flat at the oigi sie all deivatives exist ad ae zeo. As, we get ideal espose. The omalized LP Bttewoth is obtaied whe: ad / se. Sbstittig S j i eq. 8., ad eaage to get the LP Bttewoth poles, the: S [ + / ] Fo odd, S K k π /, k 0,,,..., 8.a Fo eve, k π / + π /, k 0,,,..., 8..b S K Fo stable ad asal filte: S S S B S 8.3 LP poles k B S : Bttewoth polyomial of ode see Table. 54

2 LP: Left half plae. Example: Fid the tasfe ftio S fo the omalized Bttewoth filte of ode oe. Soltio: applyig eq.8.a, whee, k 0, S 0 0 S S π. Usig eq. 8.3 ad takig LP poles S : S S S S + I m S * * R e S S -S 8.. Aalog- to aalog tasfomatio To obtai Bttewoth filtes with toff feqeies othe tha ad /se. It is oveiet to se ad /se. Bttewoth filtes as pototypes ad apply aalog-to-aalog tasfomatio see Table. The tasfomatioal method is ot limited i its appliatio to Bttewoth filtes. 55

3 8..3 Desig Eqatios of Bttewoth Filtes: A Bttewoth LPF Filte of ode is give by the followig eqatio: log } 0. k 0. k 0 { 0 / log0 / ee, / /, see Table. Whee, k, k,, ad ae the pass-bad gai ad stop-bad atteatio with thei elative feqeies espetivelysee Table. To satisfy o eqiemet at exatly, the: / 0 0. k / 8.5a To satisfy o eqiemet at / 0 0. k exatly, the: / 8.5b is the toff feqey at 3dB Example : desig a aalog Bttewoth LPF that has a db btte toff feqey of 0 ad/se. ad at least 0 db of atteatio at 30 ad/se. Soltio: Applyig eq. 8.4, whee k - db, k -0 db, ad/se log { 0 log 0 / 0 0 / 30 } 0. 0 To satisfy o eqiemet at exatly, the: 0 ad/se., ad / 8 0 / ad / se Fom Table of omalized Bttewoth LPF ad/ se with 4 : S S S + S S + Usig Table ad applyig LP LP tasfomatio, S S /.3836, ad eaagig: S S S S S Fo Bttewoth PF: - Pt / / i eqatio 8.4, ad fid its ode.see Table 56

4 - Use Table to fid the omalized Bttewoth LPF eqatio with ode. 3- Apply LP P tasfomatio, S / S, ad eaage the eqatio obtaied i step. Fo Bttewoth BPF: - Callate mi { A, B } sig eqatios give i Table. Fid the filte ode sig eq Use Table to fid the omalized Bttewoth LPF eqatio with ode. S + l 3- Apply LP BP tasfomatio, S, ad eaage the eqatio S obtaied i step Fo Bttewoth BSF: Refe to Table to see the vaiables. l Fig. 8. Bttewoth BPF Example 3: Desig a aalog Bttewoth BPF with the followig /s: A db ppe ad lowe toff feqeies of 50 z ad 0 Kz. A stop-bad atteatio of at least 0 db at 0 z ad 45 kz. Soltio: π ad / se. π 45 x x0 5 ad / se. π 0 x x 0 5 ad / se. l π ad / se Callate mi { A, B } mi.5053, by sig eqatios give i Table. Apply eq. 8.4 to fid:.89 3 Fom Table of omalized Bttewoth LPF ad/ se with 3: 3 S 3 S + S + S + Apply LP BP tasfomatio by sbstittig S S S + l l 7 S S , i the above eqatio ad eaage it to obtai BPF as.w 57

5 8.3 Chebyshev Filtes: Thee ae two types of Chebyshev Filtes: - Oe otaiig a ipple i the pass-bad type. - Oe otaiig a ipple i the stop-bad type. j + ε T 8.6 T is the th ode Chebyshev polyomial whee T 0 x, ad T x x as listed i Table 3. ε is a paamete hose to povide the pope pass-bad ipple. Fig. 8.3 shows omalized Chebyshev Filtes of both types. odd eve Fig. 8.3 Nomalized Chebyshev filtes of type fo odd, ad eve 8.3. Desig Eqatios of Chebyshev Filtes: log 0 [ g + g ] 8.7 log0 [ + ] 0log / [/ A ] stopbad atteatio 8.8a 0 db / g [ A / ε ] 8.8b S K V 0 b K V K S S 0 / LPF poles 0 + ε K K V S odd eve 8.9 Table 4 gives V S fo to 0 ad ε oespodig to 0.5,,, ad 3 db ipples. Table 5 gives the zeos {poles of S } fo the same ad ε. 58

6 8.3. Desig steps of Chebeshev LPF, PF, BPF, ad BSF : - Use the bakwad desig eqatios fom Table to obtai omalized LPF eqiemets. -Callate A sig eq. 8.8a 3-Callate g fom eq. 8.8b, the apply eq.8.7 to fid the ode. 4- Use Table 4 ad Table 5 to fid the Chebeshev Filte eqatio with ode. 5- Apply LP LP o P o BP o BS tasfomatio Table ad eaage the eqatio obtaied i step 4. Example 4: Desig a Chebshev filte to satisfy the followig speifiatios: -Aeptable pass-bad ipple of db -Ctoff feqey of 40 ad/se. 3- stop-bad atteatio of 0 db o moe at 5 ad/se. Soltio: Fom Table / 5/ 40.3 ad/se. 0log / [/ A ] 0, A 0, sig ε db see Table 4 ad Table 5 0 Applyig eq. 8.8b, the g 3.0 log [ ] , odd log0 [.3.3 ] + Fom Table 4 with 5 ad ε db S S S S S S Usig poles fom Table 5: S S S S S S Usig Table ad applyig LP LP tasfomatio, S S / 40, ad eaagig the above eqatio: S S S S S LPF S Notes:. Bttewoth o maximally flat amplitde; as the ode is ieased the espose beomes flatte i the pass-bad ad the atteatio is geate i the stop-bad. 59

7 . Chebshev Filte has a shape toff; i.e., a aowe tasitio bad best amplitde espose tha a Bttewoth filte of the same ode 3. Chebshev Filte povides pooest phase espose most oliea. The Bttewoth filte ompomise betwee amplitde ad phase this is oe of the easos fo its widespead poplaity. 8.4 Ellipti Filtes: A LP ellipti filte povides a smalle tasitio width ad is optimm i the sese that o othe filte of the same ode has a aowe tasitio width fo a give pass-bad ipple ad stop-bad atteatio. Fig. 8.4a omalized ellipti LPF Fig. 8.4b ellipti LP filtes types Fig. 8.4 a shows a omalized ellipti LPF ad Fig. 8.4 b shows ellipti LP filtes of type odd, ad type eve Desig steps of Ellipti LPF, PF, BPF, ad BSF : sig Table 6. Loate k aeptable pass-bad ipple db, ad k stop-bad atteatio db.. Callate sig Table, pp At olm, take a vale less tha. 4. The filte ode is the fa left of that ow, ad the oeffiiets fo the filte ae fod i all ows oespodig to that. 5. Aodig to, the omalized ellipti LPF eqatios ae: / S + A o 0 i S, odd 8.0 a S + S S + B S + B o i i o i 60

8 S + A S 0 S, eve 8.0 b B / 0 i i S + Bi S + 6- Apply LP LP o P o BP o BS tasfomatio Table ad eaage the eqatio obtaied i step 5. o i Notes: Fo omalized ellipti filte, geometi mea, ad /,the 0.5, ad 0.5 Fo ot omalized ellipti filte, The /. /, whee / 0 ad / ellipti hebeshev Bttewoth Example 5 : Fid the tasfe ftio fo a ellipti LPF with db toff vale at 0000 ad/se., ad a stop-bad atteatio of 40 db fo all past 4400 ad/se. Soltio: { } / /000 5/6 ad / 0 / 4400/000 6/5 /.44, k db, ad k 40 db. Fom Table 6, 4 Applyig eq. 8.0 b, Whee: 0 0.0, A , B , ad B , i A , B , ad B i S S 0.0 S S S S S Apply LP LP tasfomatio Table, whee 0 geometi mea 000. Sbstittig S S / 000 i the above eqatio: S S S S S S S LPF 6