ACTIVE RC FILTERS ECEN 622 TAMU

Transcription

1 EEN 6 TAMU ATIE FILTES. Baic Building Blck Firt-Order Filter Secnd-Order Filter, uing multiple S Secnd-Order Filter, uing ne S ( Op Amp) State-ariable Biquad. Nn-Ideal Active Filter Uing S ( Op Amp) v. S ( trancnductance Amp) Secnd-Order Nn-idealitie Fully Differential erin Fully Balanced, Fully Symmetric Balance ircuit 3. Intrductin t Matlab and Simulink fr filter Deign and filter apprximatin technique By Edgar Sánchez-Sinenci

2 6 (ESS) A ATIE - FILTES The baic building blck i illutrated belw A H i A Let u aume that A, then H Next we cnider particular cae i H Integratr i H Differentiatr By Edgar Sánchez-Sinenci

3 r F = an be: EXAMPLE: Let Auming ideal p amp A H 0,. F F F F Then uing () F ( K F F ) n p z (4) H F z p p z F F p p z z By Edgar Sánchez-Sinenci 3

4 Particular cae are eaily derived frm (3) and (4) Integratr: 0, H F F F F Differentiatr ;, F 0 H F F Lw-Pa: 0 F H High-Pa: H F F F F F F By Edgar Sánchez-Sinenci 4

5 One ple and ne zer i F F i F F F F F F (5) What are the key difference between Eq. (4) and (5)? Exercie. Obtain the tranfer functin f the fllwing circuit. i F F F A By Edgar Sánchez-Sinenci 5

6 Secnd-Order Filter Baed n a Tw-Integratr Lp. We can deign a ecnd-rder filter by cacading tw inverter. i.e. F F F F i v i F F F F F F F F F F F F F F F F (6) What are the lcatin f the ple? p, F F F F F F F F F F F F 4 F F F F By Edgar Sánchez-Sinenci 6

7 T have cmplex ple it require that 0? F F F F F F F F Which it i impible t atify. Therefre, cacading tw firt-rder filter yield a ecnd-rder filter with nly real ple. The general frm f the ecnd rder tw-integratr lp ha the fllwing tplgy. 3 F F i - H 3 F F F F By Edgar Sánchez-Sinenci (7a) 7

8 Nte the imilarity f Eq. (7a) with (). Al berve that A - need t be inerted befre r after the ecnd inverter t yield a negative feedback lp. Let u cnider the fllwing filter where 3 3,,, F F, F F F F Thu Eq. (7a) yield: H F F 3 F F F F F F H F F F 3 F F F Q By Edgar Sánchez-Sinenci 8

9 By injecting in different current umming nde a general biquad filter can be btained. F r F LP = 3 3 r F BP = r/k F /K 3 - BP = 3 F 3 F F F F K F F F F F F K 3 F F F F 3 By Edgar Sánchez-Sinenci F 3 F F F F F K F 3 K 3 9

10 Exercie. Obtain the exprein f and. Mre general biquad exprein and tplgie can be btained by adding a ummer. in -K 0 S T K 0 in K 0 K 30 -K 0 -K 30 Exercie 3. Draw an active- tplgy f the blck diagram hw abve. Exercie 4 a) Fr nly 0 btain and when intead f the reitr F /K 3 a capacitr K 4 F i ued. b) Fr nly 3 0 btain when the reitr 3 i replaced by a capacitr K HP F. By Edgar Sánchez-Sinenci 0

11 By uing al the pitive input f the p amp ther ueful filter can be btained. A Example. Phae hifter = =, =, = and A with By Edgar Sánchez-Sinenci

12 Sallen and Key Bandpa Filter K i 3 K i a nn-inverting amplifier Uing Ndal Analyi (3) () 0 - () 3 3 K i K K H i By Edgar Sánchez-Sinenci

13 A particular cae i fr = = 3 =, = = Then r fr a given ; and Q and Q 4 K K 4 Exercie 5. Prve the tranfer functin i a BP filter f the fllwing circuit Q -K i H By Edgar Sánchez-Sinenci i K K K K 3

14 In the pat befre I fabricatin, active filter implementatin preferred ne p amp tructure. One very ppular type i the Sallen and Key unity gain implementatin. A4 i LP Sallen - Key One al ppular tplgy i the auch Filter H LP 4 Q Q Q i 3 H 3 3 LP auch Filer By Edgar Sánchez-Sinenci 4

15 Anther technique fr analyi and deign baed n tate-variable ue building blck. IUIT EPESENTATION S / Nte: and can take any value including. -K F -K S K 3 4 K K 5 5 By Edgar Sánchez-Sinenci 5 5

16 Let u apply t a tw-integratr lp plu Man ule. -K - -K Q -K K S / / in 3 Fr Secnd-tplgy By Edgar Sánchez-Sinenci Kin KQ K K K in KQ KK K K K Q K K Q in in K K K HP BP 6

17 Next we hw that we can g frm an Active- repreentatin int a blck diagram r vice vera HP BP LP KHN Biquad Filter 5 i 3 HP -5/4 BP S / / 6 7 LP 5 KQ 3 4 By Edgar Sánchez-Sinenci 7

18 Q Q 3 5 i HP K K By Edgar Sánchez-Sinenci

19 EEN 6 TAMU ATIE FILTES. Baic Building Blck Firt-Order Filter Secnd-Order Filter, uing multiple S Secnd-Order Filter, uing ne S ( Op Amp) State-ariable Biquad. Nn-Ideal Active Filter Uing S ( Op Amp) v. S ( trancnductance Amp) Secnd-Order Nn-idealitie Fully Differential erin Fully Balanced, Fully Symmetric Balance ircuit 3. Intrductin t Matlab and Simulink fr filter Deign and filter apprximatin technique By Edgar Sánchez-Sinenci 9

20 ELEN 6 (ESS) Nn-Ideal Active- Integratr Op Amp Nn-Idealitie i Integratr ae H H 0 A GB A ; GB GB ; H A where GB GB A ; GB j ; H j tan GB j 90 A A By Edgar Sánchez-Sinenci 0

21 tan GB tan A j ; i.e. GB H j M GB j GB M 0 GB GB i.e. GB M 0. ~ 5% errr By Edgar Sánchez-Sinenci

22 It fllw that the ideal -6 db/ctave rll-ff expected frm an ideal integratr change t - db/ctave at the frequency f the paraitic ple given by p t db which may be apprximated by, A -6 db/ctave t fr t 0 /A / t - db/ctave By Edgar Sánchez-Sinenci

23 In general Q L t GB T j jx then we define the integratr Q-factr by Q Q I I X t A j By Edgar Sánchez-Sinenci 3

24 Making an analgy f Q L f an inductr Q L L L L L Ly Part Fr an integratr ne can btain Q I GB GB GB A j v i GB S v Miller Integratr By Edgar Sánchez-Sinenci 4

25 Hw can we cmpenate thi degradatin f perfrmance? a) v i A GB v If we make GB Ideally we btain i A Thi integratr yield a pitive Q I A j v i A v By Edgar Sánchez-Sinenci 5

26 6 (ESS) i ATIE INTEGATO: Ple Shift and Preditrtin A() + - i i A A A() (a) (b) A Let A() Then (b) becme A i A 3dB 3dB 3dB 3dB 3dB ; where A i the D gain and 3dB the dminant ple in pen lp. 3dB A 3dB A3dB 3dB () Let GB=A 3dB G. Daryanani, Principle f Active Netwrk Synthei and Deign, Jhn Wiley and Sn, 976. By Edgar Sánchez-Sinenci 6

27 The rt f the denminatr are GB GB 4 P 3dB, GB / (3a) Uing the apprximatin X) / X/ fr X<<, then GB GB P 3dB, GB (3b) Thu the rt yield P A P GB A GB By Edgar Sánchez-Sinenci 7

28 The Bde Plt Lk Like By Edgar Sánchez-Sinenci 8

29 PEDISTOTION; FEQUENY OMPENSATION In rder t relax the bandwidth p amp requirement ne can ue a r n the Miller Integratr. That i i A() Equivalent i i i Ue GB r GB B. Wu and Y. hiu, A 40nm MOS Derivative-Free IF Active- BPF with Prgrammable Bandwidth and enter Frequency Achieving Over 30dBm IIP3, IEEE JSS, By l. Edgar 50, N. Sánchez-Sinenci 8, pp , Augut 05. 9

30 30 i i F A L L G m Uing S v. IS in Active- Filter The mtivatin i t ue OTA (IS) intead f mre pwer hungry Op Amp (S) i L m m m L m i F i F F i g g g g then, A If A 0 Fr By Edgar Sánchez-Sinenci

31 Uing S Signal flw graph: x i x + x F = 0 i F + F x A = A x x = i F + F + + F β + F Uing Man rule: i = F A + F = + A + F F / + A + F Thu a A the gain becme F / By Edgar Sánchez-Sinenci 3

32 Uing S x Uing Man rule: i x x + x = 0 + L = G m x x = i = G m + / L x β Signal flw graph: i + x G m + / L + i = G m + + / L G = m + / L + / + G m + + L What cnditin d we need t impe n G m fr prper peratin? By Edgar Sánchez-Sinenci 3

33 Uing S: Acceptable G m ange Signal flw graph: x G m i + x + / L i = / + G m + + L + Nte frm the ignal flw graph that having a negative feedback lp require G m > Fr the gain t apprach the ideal gain f /, we need G m + + L G m L Thu, guaranteeing thi ecnd cnditin autmatically guarantee the negative feedback cnditin By Edgar Sánchez-Sinenci 33

34 Uing S: Practical nideratin x G m L In practice, L = Lad where i the OTA utput reitance and Lad i the external lad impedance One huld nte that G m and are nt independent ince increaing current t increae G m will reduce. T a firt rder, ne can cnider A = G m t be cntant. Finally, in cacaded filter deign, the lad f ne tage i the input reitr f the next ne. We can thu aume Lad = a a realitic cnditin ( Lad = place a ler cntraint n G m ) Subtituting fr L a decribed abve yield the fllwing cntraint n G m : G m + / + / A + / By Edgar Sánchez-Sinenci 34

35 Numerical Example Ideal S and S cmpnent frm adence were ued t imulate the abve circuit. Tw cnfiguratin were teted:. Unity gain inverting amplifier ( = = ). Ly integratr with crner frequency MHz ( = ) T have a fair cmparin, the value f A wa fixed t 30 fr bth the S and S implementatin (thu the S had = 30/G m ). Thi i a typical value fr the vltage gain f a ingle tage amplifier. In all tet, = 00kΩ, the utput reitance f the S i et t kω and a lad capacitance i added t the utput f the amplifier t give an utput ple at 0 MHz. With thee number, the cntraint n G m i G m 54 μs The value f G m wa wept frm 30 μs t 600 μs and the imulatin reult are hwn in the fllwing lide. By Edgar Sánchez-Sinenci 35

36 Simulatin eult: Inverting Amplifier S epne Increaing G m S epne Increaing G m By Edgar Sánchez-Sinenci 36

37 Simulatin eult: Ly Integratr Increaing G m Increaing G m S epne S epne By Edgar Sánchez-Sinenci 37

38 ncluin It i pible t ue S (OTA) intead f S (Opamp) in active- filter in rder t avid uing ctly buffer tage. Prper perfrmance requirement place a lwer limit n the trancnductance f the OTA ued. Uing a trancnductance f 0-5x the minimum requirement yield a cmparable perfrmance t a deign emplying an Opamp implementatin. By Edgar Sánchez-Sinenci 38

39 It can be hwn that fr equal deviatin i /k Q A A A 3 BP GB A, the Tw-Thma filter ha the fllwing Qa Q 4Q GB L r r - L r 4Q GB and ; 4Q k GB GB Imprved verin by replacing nninverting integratr: r r BP - L Q a Q k GB kq GB GB Imprved integratr verin with pitive Q I By Edgar Sánchez-Sinenci 39

40 Hw t generate Fully-Differential Filter baed n Single-Ended erin? i - + i Single Ended i i i i Fully-Differential erin By Edgar Sánchez-Sinenci 40

41 Particular ae. Aume n v i - i Available. i i - + i i - i State -ariable Filter 03 i Q X HP - + BP LP Symmetric cnditin 03 3 Q X ead fully balanced - fully ymmetric circuit frm 607. By Edgar Sánchez-Sinenci 4

42 i K 3 + HP K 0 K Q K 03 BP K 0 LP K K KQ 3 Q,, K K KHN State ariable Tw-Integratr Filter Ue Man ule: LP i K3K0K K0KQ K 0 0 K 0 K 03 K K 0 K 3 Q K 0 K K 0 0 K 0 K 03 Next we cnider the fully-differential verin f the KHN filter. By Edgar Sánchez-Sinenci 4

43 X 03 3 Q i i Q LP LP X 03 KHN Fully-Differential erin By Edgar Sánchez-Sinenci 43

44 Hw can we take advantage f imprved cmbinatin f ± Q I in fully differential verin? Q in in /k A+ - /k A+ - Q Q SAME! in in /k A+ - /k A- + Q By Edgar Sánchez-Sinenci 44

45 6 (ESS) Effect f Nn-Ideal Op Amp n the Tw-Thma Biquad Q i /K ( ) A = ( ) ( ) A = ( ) r r ( ) A 3 = ( ) BP LP - LP When A i (i,,3) are finite, the denminatr becme f the tranfer functin yield: D Q Q 3Q A A AA Q A A A A A 3 A A A A QA A A QA A By Edgar Sánchez-Sinenci 45

46 GB Let A i i, i,,3 GBi Furthermre aume the range f interet and, Q. A GB i i Then D() becme: D() GB GB GB 3 3 GB GB Q GB D Q a a a GB GB GB 3 Thu a r fr a and Q a, then GB GB GB GB GB 3 GB By Edgar Sánchez-Sinenci 46

47 Qa Q Q GB GB GB 3 Fr equal GB GB GB 3 Q a Q 4Q GB Q a Q 4Q GB, fr 4Q GB Nte that fr a table filter 4Q r GB Q 4 GB By Edgar Sánchez-Sinenci 47

48 EEN 6 (ESS) TAMU KEY FILTE PAAMETES IN ATIE- FILTES Dynamic ange Signal-T-Nie ati Ttal Output Nie Nie Pwer Spectral Denity Ttal Area eitr and apacitr can be expreed a: r where r, and c c are the nrmalized filter value. The reitr pwer diipatin fr a inuidal input yield P f i i H i H i r f Where H f i the tranfer functin frm the input t the terminal f reitr i eference. L. th et all, General eult fr eitive Nie in Active and MOSFET- Filter, IEEE Tran n ircuit and Sytem II, l. 4, N., pp , December 995. By Edgar Sánchez-Sinenci 48

49 Fcuing n the nie reitr, the pwer pectral denity i given by S f 4kT H f 4kT r H f The definitin f H f i pictrially hwn belw: () + i - H + - f Thu, the ttal utput nie (mean quared value) due t the reitr becme N S f df In practice the upper limit f the integratin i limited t a ueful practical value. By Edgar Sánchez-Sinenci 49

50 The ignal-t-rati fr a given i and frequency f i given by SN and max H BP BP f D H P where P f f,max i i max H f N P N,max i the maximum pecified pwer diipatin in the reitr. max f H f Let u cnider a ecnd - rder BP filer example c Q c Q Uing the fllwing ntatin the abve H c Q fcjf jf Q f c jf f c BP yield Then (3) (4) By Edgar Sánchez-Sinenci 50

51 P i f a a Q c Fr the biquad hwn belw i Q/a Q/a /a - OBP and N kt Q / a a a N a OBP prprtinal t (a). limited by linearity and by reitr pwer diipatin which i By Edgar Sánchez-Sinenci 5

52 Fully Differential Fully Balanced ircuit What i the prblem with ingle-input / ingle-utput? i n A F n n F A 0 A Fr i F id ( id icm Hw t lve thi prblem? icm ) N eliminatin f cmmn-mde ignal. F F ( ) Fr ( ) i id icm F ( ) N cmmn-mde utput. By Edgar Sánchez-Sinenci 5

53 Hw t btain a fully differential circuit? We will dicu tw ptential apprache Apprach Apprach F F P P n n F F F F F ( ) ( ) F ( ) F ( ) F ( ) ; D cnditin n P n P emark: enitive t M ignal n P F P n F P emark: Mre rbut t reject cmmn-mde ignal P By Edgar Sánchez-Sinenci 53

54 Firt-Order FB Lw Pa with Op Amp *.ubckt pamp nn inv ut rin nn inv 00K egain 0 (nn, inv) 00K rpen K cpen u eut 3 0 (, 0) rut 3 ut 50.end *vin 3 3 ac.0 vin 3 0 ac.0 x 4 pamp x 4 pamp 3 K 3 4 K B 3 4 K BB 3 K F K FB K F 4 0 K FB 4 0 K u B u A u B u rdummy 3 3.ac dec 0 0Hz 0KHz.prbe.end i i B BB F F A B FB FB B By Edgar Sánchez-Sinenci 54

55 Fully Balanced T-T Active- Implementatin Q i K Q K Q i K K Q By Edgar Sánchez-Sinenci 55

56 6 Active Filter By Edgar Sánchez-Sinenci Texa A&M Univerity Intrductin t Matlab and Simulink Fr Filter Deign By Edgar Sánchez-Sinenci 56

57 Example : Ideal Integratr = K = 0.59mF By Edgar Sánchez-Sinenci 57

58 Bde Plt: Ideal Integratr (Matlab) =tf( ); =e3; =0.59e-3; h=/(**); figure() bde(h) grid minr %eitr alue %apacitr alue %h= ()/i() %reate Bde Plt %Add grid t plt H= gcr; %change X-axi unit h.axegrid.xunit = Hz ; %Set unit t Hz ple(h); zer(h); %calculate h ple %calculate h zer Phae (deg) Magnitude (db) Bde Diagram Frequency (Hz) By Edgar Sánchez-Sinenci 58

59 Bde Plt: Ideal Integratr (Simulink) ) reate Mdel uing Gain, Integratr, and In/Out blck ) G t: Tl => ntrl Deign=> Linear Analyi 3) Then pre: Linearize mdel By Edgar Sánchez-Sinenci 59

60 Tw-Thma Biquad (Simulink) By Edgar Sánchez-Sinenci 60

61 Output Wavefrm (Scpe) By Edgar Sánchez-Sinenci 6

62 Integratr Nn-ideal amplifier clear clc =tf(''); =; %eitr alue =0.59e-3; %apacitr alue h=-/(**); %h= ()/i() figure() bdemag(h) hld n f=e3; fr i=:5; GBW=*pi*f; A=GBW/; Beta=/(+/(*)); h=-/(**)*/(+/(a*beta)); hld n bdemag(h,{*pi*,*pi*e5}) f=0*f; end Magnitude (db) grid minr %Add grid t plt h= gcr; %change X-axi unit h.axegrid.xunit = 'Hz'; %Set unit t Hz legend('ideal', 'GBW=kHz','GBW=0kHz', 'GBW=00kHz','GBW=MHz', 'GBW=0MHz',) By Edgar Sánchez-Sinenci Bde Diagram Frequency (Hz) ideal GBW=kHz GBW=0kHz GBW=00kHz GBW=MHz GBW=0MHz 6

63 Filter Apprximatin: Lw-Pa Butterwrth The quared magnitude f a lw-pa butterwrth filter i given by: By Edgar Sánchez-Sinenci 63

64 Ple-zer plt Ple-er Map Imaginary Axi eal Axi By Edgar Sánchez-Sinenci 64

65 Bde Plt 0 Bde Diagram -0 Magnitude (db) Phae (deg) Frequency (rad/ec) By Edgar Sánchez-Sinenci 65

66 Lw-pa hebyhev Filter Ue the Matlab chebap functin t deign a ecnd rder Type I hebyhev lw-pa filter with 3dB ripple in the pa band w=0:0.05:400; % Define range t plt [z,p,k]=chebap(,3); [b,a]=zptf(z,p,k); % nvert zer and ple f G() t plynmial frm bde(b,a) grid minr; By Edgar Sánchez-Sinenci 66

67 Lw-pa hebyhev Filter 0 Bde Diagram Magnitude (db) Phae (deg) Frequency (rad/ec) By Edgar Sánchez-Sinenci 67

68 Lw-pa hebyhev Filter % Anther way t write the cde! w=0:0.0:0; [z,p,k]=chebap(,3); [b,a]=zptf(z,p,k); G=freq(b,a,w); xlabel('frequency in rad/'); ylabel('magnitude f G()'); emilgx(w,ab(g)); title('type hebyhev Lw-Pa Filter'); Grid; By Edgar Sánchez-Sinenci 68

69 Lw-pa hebyhev Filter Type hebyhev Lw-Pa Filter By Edgar Sánchez-Sinenci 69

70 Invere hebyhev Uing the Matlab chebap functin, deign a third rder Type II hebyhev analg filter with 3dB ripple in the tp band. w=0:0.0:000; [z,p,k]=chebap(3,3); [b,a]=zptf(z,p,k); G=freq(b,a,w); emilgx(w,ab(g)); xlabel('frequency in rad/ec'); ylabel('magnitude f G()'); title('type hebyhev Lw-Pa Filter, k=3, 3 db ripple in tp band'); grid By Edgar Sánchez-Sinenci 70

71 Invere hebyhev Type hebyhev Lw-Pa Filter, k=3, 3 db ripple in tp band Magnitude f G() Frequency in rad/ec By Edgar Sánchez-Sinenci 7

72 Elliptic Lw-Pa Filter Ue Matlab t deign a fur ple elliptic analg lw-pa filter with 0.5dB maximum ripple in the pa-band and 0dB minimum attenuatin in the tp-band with cutff frequency at 00 rad/. w=0: 0.05: 500; [z,p,k]=ellip(4, 0.5, 0, 00, ''); [b,a]=zptf(z,p,k); G=freq(b,a,w); plt(w,ab(g)) title('4-ple Elliptic Lw Pa Filter'); grid By Edgar Sánchez-Sinenci 7

73 Elliptic Lw-Pa Filter 4-ple Elliptic Lw Pa Filter By Edgar Sánchez-Sinenci 73

74 Tranfrmatin Methd Tranfrmatin methd have been develped where a lw pa filter can be cnverted t anther type f filter by imply tranfrming the cmplex variable. Matlab lplp, lphp, lpbp, and lpb functin can be ued t tranfrm a lw pa filter with nrmalized cutff frequency, t anther lw-pa filter with any ther pecified frequency, r t a high pa filter, r t a band-pa filter, r t a band eliminatin filter, repectively. By Edgar Sánchez-Sinenci 74

75 LPF with nrmalized cutff frequency, t anther LPF with any ther pecified frequency Ue the MATLAB buttap and lplp functin t find the tranfer functin f a third-rder Butterwrth lw-pa filter with cutff frequency fc=khz. % Deign 3 ple Butterwrth lw-pa filter (wcn= rad/) [z,p,k]=buttap(3); [b,a]=zptf(z,p,k); % mpute num, den cefficient f thi filter (wcn=rad/) f=000:500/50:0000; % Define frequency range t plt w=*pi*f; % nvert t rad/ec fc=000; % Define actual cutff frequency at KHz wc=*pi*fc; % nvert deired cutff frequency t rad/ec [bn,an]=lplp(b,a,wc); % mpute num, den f filter with fc = khz Gn=freq(bn,an,w); % mpute tranfer functin f filter with fc = khz emilgx(w,ab(gn)); grid; xlabel('adian Frequency w (rad/ec)') ylabel('magnitude f Tranfer Functin') title('3-ple Butterwrth lw-pa filter with fc= khz r wc =.57 kr/') By Edgar Sánchez-Sinenci 75

76 LPF with nrmalized cutff frequency, t anther LPF with any ther pecified frequency 3-ple Butterwrth lw-pa filter with fc= khz r wc =.57 kr/ Magnitude f Tranfer Functin adian Frequency w (rad/ec) By Edgar Sánchez-Sinenci 76

77 High-Pa Filter Ue the MATLAB cmmand chebap and lphp t find the tranfer functin f a 3-ple hebyhev high-pa analg filter with cutff frequency fc = 5KHz. % Deign 3 ple Type hebyhev lw-pa filter, wcn= rad/ [z,p,k]=chebap(3,3); [b,a]=zptf(z,p,k); % mpute num, den cef. with wcn= rad/ f=000:00:00000; % Define frequency range t plt fc=5000; % Define actual cutff frequency at 5 KHz wc=*pi*fc; % nvert deired cutff frequency t rad/ec [bn,an]=lphp(b,a,wc); % mpute num, den f high-pa filter with fc =5KHz Gn=freq(bn,an,*pi*f); % mpute and plt tranfer functin f filter with fc = 5 KHz emilgx(f,ab(gn)); grid; xlabel('frequency (Hz)'); ylabel('magnitude f Tranfer Functin') title('3-ple Type hebyhev high-pa filter with fc=5 KHz ') By Edgar Sánchez-Sinenci 77

78 High-Pa Filter 3-ple hebyhev high-pa filter with fc=5 KHz Magnitude f Tranfer Functin Frequency (Hz) By Edgar Sánchez-Sinenci 78

79 Band-Pa Filter Ue the MATLAB functin buttap and lpbp t find the tranfer functin f a 3-ple Butterwrth analg band-pa filter with the pa band frequency centered at f = 4kHz, and bandwidth BW =KHz. [z,p,k]=buttap(3); % Deign 3 ple Butterwrth lw-pa filter with wcn= rad/ [b,a]=zptf(z,p,k); % mpute numeratr and denminatr cefficient fr wcn= rad/ f=00:00:00000; % Define frequency range t plt f0=4000; % Define centered frequency at 4 KHz W0=*pi*f0; % nvert deired centered frequency t rad/ fbw=000; % Define bandwidth Bw=*pi*fbw; % nvert deired bandwidth t rad/ [bn,an]=lpbp(b,a,w0,bw); % mpute num, den f band-pa filter % mpute and plt the magnitude f the tranfer functin f the band-pa filter Gn=freq(bn,an,*pi*f); emilgx(f,ab(gn)); grid; xlabel('frequency f (Hz)'); ylabel('magnitude f Tranfer Functin'); title('3-ple Butterwrth band-pa filter with f0 = 4 KHz, BW = KHz') By Edgar Sánchez-Sinenci 79

80 Band-Pa Filter.4 3-ple Butterwrth band-pa filter with f0 = 4 KHz, BW = KHz. Magnitude f Tranfer Functin Frequency f (Hz) By Edgar Sánchez-Sinenci 80

81 Band-Eliminatin (band-tp) Filter Ue the MATLAB functin buttap and lpb t find the tranfer functin f a 3-ple Butterwrth band-eliminatin (band-tp) filter with the tp band frequency centered at f = 5 khz, and bandwidth BW = khz. [z,p,k]=buttap(3); % Deign 3-ple Butterwrth lw-pa filter, wcn = r/ [b,a]=zptf(z,p,k); % mpute num, den cefficient f thi filter, wcn= r/ f=00:00:00000; % Define frequency range t plt f0=5000; % Define centered frequency at 5 khz W0=*pi*f0; % nvert centered frequency t r/ fbw=000; % Define bandwidth Bw=*pi*fbw; % nvert bandwidth t r/ % mpute numeratr and denminatr cefficient f deired band tp filter [bn,an]=lpb(b,a,w0,bw); % mpute and plt magnitude f the tranfer functin f the band tp filter Gn=freq(bn,an,*pi*f); emilgx(f,ab(gn)); grid; xlabel('frequency in Hz'); ylabel('magnitude f Tranfer Functin'); title('3-ple Butterwrth band-eliminatin filter with f0=5 KHz, BW = KHz') By Edgar Sánchez-Sinenci 8

82 Band-Eliminatin (band-tp) Filter 3-ple Butterwrth band-eliminatin filter with f0=5 KHz, BW = KHz Magnitude f Tranfer Functin Frequency in Hz By Edgar Sánchez-Sinenci 8

83 Hw t find the minimum rder t meet the filter pecificatin? The fllwing functin in Matlab can help yu t find the minimum rder required t meet the filter pecificatin: Buttrd fr butterwrth hebrd fr chebyhev Elliprd fr elliptic hebrd fr invere chebyhev By Edgar Sánchez-Sinenci 83

84 alculating the rder and cutff frequency f a invere chebyhev filter Deign a 4MHz Invere hebyhev apprximatin with Ap gain at paband crner. The tp band i 5.75MHz with -50dB gain at tp band. clear all; Fp = 4e6; Wp=*pi*Fp; F=.4375*Fp; W=*pi*F; Fplt = 0*F; f = e6:fplt/e3:fplt ; w = *pi*f; Ap = ; A = 50; % hebrd help yu find the rder and wn (n and Wn) that %yu can pa t cheby cmmand. [n, Wn] = chebrd(wp, W, Ap, A, ''); [z, p, k] = cheby(n, A, Wn, 'lw', ''); [num, den] = cheby(n, A, Wn, 'lw', ''); bde(num, den) By Edgar Sánchez-Sinenci 84

85 Bde Plt 0 Bde Diagram Magnitude (db) Phae (deg) Frequency (rad/ec) By Edgar Sánchez-Sinenci 85

86 eference [] S. T. Karri, Signal and Sytem with Matlab mputing and Simulink Mdeling, Fifth Editin. Orchard Publicatin [] Matlab Help File By Edgar Sánchez-Sinenci 86