COMPLEX FILTERS AS CASCADE OF BUFFERED GINGELL STRUCTURES: DESIGN FROM BAND-PASS CONSTRAINTS

Transcription

1 COMPLEX FILTERS AS CASCADE OF BUFFERED GINGELL STRUCTURES: DESIGN FROM BAND-PASS CONSTRAINTS A Thesis presented t the Faculty f Califrnia Plytechnic State University, San Luis Obisp In Partial Fulfillment f the Requirements fr the Degree Master f Science in Electrical Engineering by Nicle Marie Hay June 207

2 207 Nicle Marie Hay ALL RIGHTS RESERVED ii

3 COMMITTEE MEMBERSHIP TITLE: Cmplex Filters as Cascade f Buffered Gingell Structures: Design frm Band-Pass Cnstraints AUTHOR: Nicle Marie Hay DATE SUBMITTED: June 207 COMMITTEE CHAIR: Vladimir Prdanv, Ph.D. Assciate Prfessr f Electrical Engineering COMMITTEE MEMBER: David Braun, Ph.D. Prfessr f Electrical Engineering COMMITTEE MEMBER: Wayne Pilkingtn, Ph.D. Assciate Prfessr f Electrical Engineering iii

4 ABSTRACT Cmplex Filters as Cascade f Buffered Gingell Structures: Design frm Band-Pass Cnstraints Nicle Marie Hay Cmplex filters are multi-input, multi-utput netwrks designed t discriminate based upn the relative phase difference between input signals. Cmplex filters find applicatin in mdern wireless systems fr single sideband transmissin and image-reject receptin. This thesis presents ne active cmplex filter implementatin using tw peratinal amplifiers per stage, termed type-ii tplgy. The type-ii riginates frm the passive RC-CR plyphase tplgy presented by Gingell in his 973 paper, Single sideband mdulatin using sequence asymmetric plyphase netwrks. This new tplgy gains several advantages ver existing cmplex filter implementatins, namely cascadability (multiple sectins placed in series t create a higher-rder respnse) withut altering the characteristics f each individual stage. In additin t describing the derivatin f the tplgy and its perfrmance relative t existing tplgies, this thesis investigates the passband characteristics f a general higher-rder filter and prvides a passband-centric design apprach thrugh derivatins f clsed frm expressins fr passband gain and bandwidth. The thesis includes a five-stage design example using this apprach in additin t an implementatin, its characterizatin, and its cmparisn t the derived expressins and simulatins. Keywrds: Cmplex filter, psitive frequency, negative frequency, quadrature, cascade iv

5 TABLE OF CONTENTS Page LIST OF TABLES... vi LIST OF FIGURES... vii CHAPTER. INTRODUCTION.... Purpse f Study....2 Psitive and Negative Frequency Differential Circuits (Multi-input Netwrks) Cmplex Netwrks Frequency Behavir f Gingell-type st Order Cmplex Filter LITERATURE REVIEW Image Rejectin Applicatin Phase-Splitting Applicatin The Hilbert Transfrm Passive Implementatins Active Implementatins TYPE-II TOPOLOGY Derivatin Crss-cupling Design Example MODELING OF BAND-PASS REGION Passband Gain as a Functin f N and k Even and Odd Orders Passband 3dB Bandwidth as a Functin f N and k Calculatin DESIGN WITH BANDWIDTH CONSTRAINT APPLICATIONS BIBLIOGRAPHY APPENDIX v

6 LIST OF TABLES Table Page. X = cs θ, Y = sin θ, X + jy = e ϕ X = sin θ, Y = cs θ, X + jy = e ϕ Cmparisn f Gingell s and Type II transfer functins Ntch frequency values Ideal values fr each stage f inverting tplgy E96 values fr inverting tplgy E96 values fr inverting tplgy, R2 as a parallel cmbinatin f tw R Cmpnent values fr design example Cmparisn f calculatin, simulatin, and measured values Matlab Values and LTspice Simulatin Cmparisn f measured and calculated bandwidth Cmpnent values fr N=4 design Cmpnent values fr N= Cmpnent values fr N= Cmparisn f bandwidth measurements vi

7 LIST OF FIGURES Figure Page. Cmpnent definitin and rtatin directin definitin, a. X and Y cmpnents f a rtating vectr, b. Cunterclckwise (psitive) rtatin, c. Clckwise (negative) rtatin Characterizing a fully differential amplifier, cmmn mde (left), differential mde (right) Characterizing cmplex netwrks, psitive sequence (left), negative sequence (right) Fur-input, fur-utput cmplex netwrk, psitive sequence respnse Single stage RLC ntch filter (a) symmetric ple-zer plt, RLC ntch filter, (b) single real ple and imaginary zer, asymmetrical respnse, (c) single real ple and negative imaginary zer, asymmetrical respnse Gingell s RC ladder netwrk Directly buffered Gingell tplgy Simplified, Buffered, Gingell tplgy (2-input) Active highpass/lwpass structure (nn-inverting) input tplgy realizes inverting LP TF with respect t v and nn-inverting HP with respect t v "Inverting" tplgy Single-stage cmplex filter Inputs: COS, -SIN prduces a ntch at f C = 3.62kHz Inputs: COS, SIN, prduces a bandpass at f C = 3.62kHz Inputs: COS, COS prduces an all-pass circuit Frequency dmain analysis f circuit in Figure 4: prduces a 90 phase difference, V in a 3.62kHz signal, V in2 grunded Crss-cupling all stages, schematic Crss-cupling all stages, simulatin respnse S-plane plt f three-stage filter, crss-cupled S-plane plt f three-stage filter, secnd stage nt-crss cupled N crss-cupling between first and secnd stage, schematic N crss-cupling between first and secnd stage, simulatin respnse S-plane plt f three-stage filter, all stages nt-crss cupled Three-stage filter, all stages nt crss-cupled, schematic Three-stage filter, all stages nt crss-cupled, simulatin respnse Five-stage filter all stages nt crss-cupled, simulatin respnse Inverting tplgy, 5 stages cascaded, ideal cmpnent values Stp-band perfrmance f 5 th rder cascade, ideal cmpnent values Passband perfrmance f 5 th rder cascade, ideal cmpnent values Stp-band perfrmance f 5 th rder cascade, E96 cmpnents Passband perfrmance f 5 th rder cascade, E96 cmpnents Stp-band perfrmance f 5 th rder cascade, E96 cmpnents, R2 as parallel cmbinatin f tw R vii

8 34. Passband perfrmance f 5 th rder cascade, E96 cmpnents, R2 as a parallel cmbinatin f tw R Fine-stage active cmplex filter prttype implemented using five dual MCP6282 p-amp chips n perfbard stage design perfrms 90 phase-splitting fr five different frequencies, (tp t bttm, left t right 47Hz, 32Hz, 703Hz,.508kHz, and 3.34kHz) Filter magnitude fr image-reject peratin, passband Filter magnitude fr image-reject peratin, stpband Plt f theretical transfer functin and measured data fr 5-stage circuit Simulatin: driving with imperfect quadrature, 90 (blue), 89 (red), 88 (green) Simulatin: driving with imperfect quadrature, 90 (blue), 89 (red), 9 (green) Simulatin: driving with imperfect quadrature, 2% vltage variatin (red), ideal (blue) Transfer functin plt fr multiple stage, effective bandpass, filter Determining system central frequency using ple-zer plts Pass-band plt fr N=3, 5, 7 (nrmalized frequency, k = ) Pass-band plt fr N=2, 4, 6 (nrmalized frequency, k = ) Passband respnse fr single-stage, peak magnitude f 3dB, magnitude greater than 0dB fr all psitive frequency range Ple-zer plt fr k = dB Nrmalized Bandwidth vs. N fr k =, k= means identical stages Nrmalized 3dB bandwidth fr tw stage tplgy vs. k Nrmalized 3dB bandwidth fr fur stage tplgy vs. k Nrmalized 3dB bandwidth fr six stage tplgy vs. k Nrmalized d3b bandwidth fr eight stage tplgy vs. k Nrmalized d3b bandwidth fr three stage tplgy vs. k Nrmalized 3dB bandwidth fr five stage tplgy vs. k Nrmalized 3dB bandwidth fr seven stage tplgy vs. k Human Ear Sensitivity [27] Stpband fr N=4 design Passband fr N=4 design Stpband fr N= Passband fr N= Stpband fr N= Passband fr N= viii

9 . INTRODUCTION. Purpse f Study Cmplex filters cnsist f tw-input, tw-utput netwrks designed t accept a pair f quadrature signals and discriminate based n the relative phase between thse tw signals. They find use in mdern wireless systems fr single sideband transmissin r fr image-reject receptin [-6]. Bth passive and active implementatins exist; in the categry f active implementatins, p-amps [2] r peratinal transcnductance amplifiers [3] are the mst cmmn active cmpnents. The passive Gingell tplgy, extensively studied fr its prperties and lw sensitivity t cmpnent variatin, stands ut as ne f the mst well-knwn structures [5] [6]. This thesis intrduces ne pssible implementatin f a tw p-amp buffered Gingell type structure and discusses the design methdlgy f higher-rder filters based n bandpass cnstraints. Use f minimal active cmpnents fr buffering purpses in a multi-stage Gingell netwrks presents the pprtunity t develp unique design strategies and examine the difficulties f implementing ver-cnstraint designs. The fllwing prvides an verview f the cncept f psitive and negative frequencies, differential and cmplex netwrks, and a well-knwn existing implementatin f a cmplex filter. This leads t the intrductin f a new tplgy, termed type-ii, which requires the fewest number f active cmpnents. This tplgy shares similarities in design with anther tplgy, called type-i, discussed by Jhnstn in, Cmplex filters as a cascade f buffered Gingell

10 structures: design frm stp-band cnstraints [7]. Because f the brad tpic f cmplex filters and extensive characterizatin f the new tplgies, the study resulted in tw theses. Cmplex filters as a cascade f buffered Gingell structures: design frm stp-band cnstraints fcuses n the type-i tplgy, investigatin f stp-band characteristics, and Mnte Carl Analysis. This thesis reprt fcuses n the type-ii tplgy, investigatin f passband characteristics, and a passband-centric design apprach. This includes examinatin f the type-ii tplgy fr mathematical cnfirmatin, implementatin results, and general design guidelines and issues. The analysis makes use f a mathematical mdeling tl and circuit simulatin prgram in additin t mathematical derivatins and necessary apprximatins t explain the tplgy derivatin and perfrmance..2 Psitive and Negative Frequency Figure. Cmpnent definitin and rtatin directin definitin, a. X and Y cmpnents f a rtating vectr, b. Cunterclckwise (psitive) rtatin, c. Clckwise (negative) rtatin The cmpnents f a rtating vectr expressed in cmplex number ntatin, i.e. x(t) + jy(t), can als be expressed as cmplex expnentials, Ae *+(,) (Figure a). Cmplex number ntatin crrespnds t the Cartesian crdinate system, while cmplex expnential ntatin crrespnds t the plar crdinate cnventin. A 2

11 vectr with fixed magnitude and cunterclckwise directin f rtatin (Figure b) expressed as a cmplex expnential: A 2 cs t + j A 2 sin t = Aejt ( ) Based n the sign f the end cmplex expnential expnent, we define this vectr s directin f rtatin as psitive. Similarly, a clckwise rtating vectr (Figure c) expressed mathematically: A 2 cs t j A 2 sin t = Ae jt ( 2) Based n the sign f the expnent, we define this rtatin as negative. Interchanging the x and y cmpnents in the cunterclckwise case () results in: A 2 sin t + j A 2 cs t = A 2 e jt ( 3) Observe that expressin (3) equates t the clckwise case (2). T further illustrate that interchanging the x and y cmpnents f a rtating vectr results in a change in the directin f the vectr s rtatin, Table and Table 2 prvide the psitin f the vectr in the S-plane fr bth the clckwise and cunterclckwise cases. Table. X = cs θ, Y = sin θ, X + jy = e ϕ Psitin (Fig. b) θ( ) φ( )

12 Table 2. X = sin θ, Y = cs θ, X + jy = e ϕ Psitin (Fig. c) θ ( ) φ( ) Differential Circuits (Multi-input Netwrks) As mentined previusly, a cmplex filter is a tw-input, tw-utput netwrk. T better understand its prperties, first examine anther tw-input, tw-utput netwrk, the differential amplifier. Tw characteristics describe a differential amplifier: cmmn-mde characteristics, and differential-mde characteristics. Driving ne amplifier input with a small, likely sinusid, signal and driving the ther input with a signal 80 ut f phase with the first signal determines differential-mde characteristics (Figure 2, right). Cnnecting the tw inputs t each ther and driving them with the same small signal determines cmmn-mde characteristics (Figure 2, left). These tw measurements cmbined represent a perfrmance specificatin, cmmn mde rejectin rati (CMRR); CMRR measures hw well the amplifier rejects cmmn-mde (alike) input signals cmpared t differential-mde input gain [8]. A 89 = differential mde vltage gain A C9 = cmmn mde vltage gain CMRR = A 89 A C9 4

13 Figure 2. Characterizing a fully differential amplifier, cmmn mde (left), differential mde (right) In summary, pairs f sinusids with well-defined phase difference prduce the characterizing respnses f a differential amplifier. Tw sinusids in phase determine cmmn-mde characteristics. A pair f sinusids with an 80 phase shift between the tw determines the differential-mde characteristics..4 Cmplex Netwrks Figure 3. Characterizing cmplex netwrks, psitive sequence (left), negative sequence (right) A cmplex netwrk simultaneusly receives tw input signals and prvides tw utput signals, this pair f signals equates t a vectr (r cmplex) quantity. Cmplex filters prvide tw different respnses based n the relative phase difference between tw applied input signals [9]. These respnses are referred t as psitive frequency respnse (Figure 3, left) and negative frequency respnse (Figure 3, right). The psitive frequency respnse requires a phase difference between the input signals f 90 : input = J K cs t J K input 2 = J K sin t j J K ( 4) ( 5) 5

14 When cnsidered as a cmplex quantity, these tw respnses appear as a cunterclckwise rtating vectr (). The negative frequency respnse requires a phase difference f negative 90 : input = J K cs t J K input 2 = J K sin t j J K ( 6) ( 7) When cnsidered as a cmplex quantity, these tw respnses appear as a clckwise rtating vectr (2). The tw respnses f a cmplex filter shuld, fr the purpse f discriminating between psitive and negative phase difference inputs, exhibit drastically different respnses. One respnse shuld pass the signal, analgus t a bandpass filter, while the ther respnse shuld stp the signal, similar t a ntch r bandstp filter. Cmplex filters als perate as 90 phase splitters. Because f the asymmetric respnse fr psitive versus negative frequency signals, a cmplex filter prduces quadrature signals n the utput. Observe that a real sinusid equates t a cmbinatin f a psitively rtating vectr and a negatively rtating vectr: A cs t j0 = A 2 ejt + A 2 e jt ( 8) This last expressin (8) is a frm f Euler s identity [0]. The fur-input, fur-utput cmplex netwrk functins as tw cmplex netwrks in parallel: 6

15 Figure 4. Fur-input, fur-utput cmplex netwrk, psitive sequence respnse A fur-input, fur-utput cmplex netwrk accepts fur signals, each 90 ut f phase with each ther. Figure 4 depicts a psitive sequence where the first input shifts 90 ahead f the secnd input, the secnd input shifts 90 ahead f the third input, and s n. This cmplex filter respnds t a psitive sequence by prducing the signals with the same relative phase but larger magnitude n the utput. The filter respnds t a negative sequence by prducing n utput signal (nt shwn). The Gingell netwrk, discussed in the next sectin, is an example f a fur-input, fur-utput cmplex netwrk..5 Frequency Behavir f Gingell-type st Order Cmplex Filter The fllwing discussin intrduces the st rder cmplex filter respnse as a partial respnse f a cmmn 2 nd rder RLC ntch filter (Figure 5) respnse. 7

16 Figure 5. Single stage RLC ntch filter Equatins 9-3 derive the transfer functin f the circuit in Figure 5: H s = P QR P QR ST Z VW = sl + Y = Z[ WVSY ZW ZW ( 9) ( 0) H s = \ [ RQ]^ \R \ [ RQ]^ \R ST ( ) H s = Z [ WVSY Z [ WVSZWTSY H s = Z [ S ^ QR Z [ S \_ Q S ^ QR ( 2) ( 3) This secnd-rder transfer functin cntains tw zers lcated n the j-axis at ±j Y, which create a ntch respnse. Lking at the denminatr f (3), and with VW the knwledge that entirely real ples create the desired respnse, the ples f the transfer functin are fund at ay. This crrespnds t a critically damped system. VW Nte that this symmetric arrangement f zers (Figure 6a) prduces a symmetric respnse, meaning that the filter prvides an identical respnse fr bth psitive and negative frequencies; in ther wrds, this exhibits nn-cmplex filter prperties. 8

17 Figure 6. (a) symmetric ple-zer plt, RLC ntch filter, (b) single real ple and imaginary zer, asymmetrical respnse, (c) single real ple and negative imaginary zer, asymmetrical respnse In his 973 Paper, Single Sideband Mdulatin using Sequence Asymmetric Plyphase Netwrks, Gingell intrduces a filter tplgy (Figure 7) which implements a single real ple and single negative imaginary zer (Figure 6c, Figure 6b) [4]. Figure 7. Gingell s RC ladder netwrk The equatins belw describe the relatinships between the inputs and utputs: V cd,y = Y ZWTSY V efy + V cd,k = ZWT ZWTSY V efy + ZWT ZWTSY V efk ( 4) Y ZWTSY V efk ( 5) When cmbined, equatins (4) and (5) prduce a single transfer functin in matrix frm: 9

18 V V ut ut2 + j / = j / + j / j / + j / V V + j / in in2 ( 6) This transfer functin implements a single real ple and single imaginary zer when a pair f quadrature signals drive the circuit. Because f the asymmetry alng the jaxis, the filter displays an asymmetric respnse as well. In ther wrds, this filter exhibits a different respnse based n the sign r phase difference f the input signals, i.e. a cmplex filter respnse. Gingell s RC ladder netwrk accepts fur inputs, hwever tw f these equate t 80 -shifted versins f the ther tw. In additin, V in2 is the same as V in shifted by 90 ; in ther wrds, V in and V in2 represent a cmplex pair f signals. Because f these prperties, this netwrk characterizes as bth differential and cmplex. Figure 8. Directly buffered Gingell tplgy As discussed in the literature review, the cmplex respnse f this filter finds uses in a variety f fields, hwever, in sme cases, the desired frequency range f cmplex respnse prves wider than the netwrk can prvide. In this case, cascading prvides the desired respnse [7]. Fr passive netwrks (Figure 7), cascading shifts 0

19 the ple lcatin. Buffering the utput reslves this issue; unfrtunately buffering the circuit shwn in Figure 7 requires fur buffers, ne fr each utput, fr each individual netwrk (referred t as a stage ) cascaded. With the gal f reducing the ttal number f active cmpnents, because tw f the inputs are inverted versins f the ther tw, the netwrk simplifies t nly tw inputs by adding an inverting stage internal t the netwrk (Figure 9). Figure 9. Simplified, Buffered, Gingell tplgy (2-input) Further reductin f the number f active cmpnents fllws in Sectin 3 Type- II Tplgy.

20 2. LITERATURE REVIEW 2. Image Rejectin Applicatin The cmplex filter finds use in image rejectin signal prcessing []. Larrwe s paper n band-pass quadrature filters discusses the use f quadrature filters fr the use f image rejectin [2]. 2.2 Phase-Splitting Applicatin The cmplex filter als perates as a phase-splitter r quadrature-generatr. Darlingtn discussed this applicatin in 950 [3]. Saraga presents a general tplgy fr using tw phase-shift netwrks t create a single-sideband mdulatr [4]. Hward examines the subject f single-sideband methds in amateur radi applicatins [5]. Gingell references the ppularity f cmplex netwrks fr singlesideband generatin and discusses the idea f the alternative f quadrature mdulatin [4]. Anther existing tplgy intrduces the use f a single cmplex lwpass filter as a substitute fr the tw phase-shift netwrk mentined previusly [6]. 2.3 The Hilbert Transfrm The Hilbert transfrm is the equivalent peratin f a phase-splitter. Papers by Feldman, Jhanssn, and Liu discuss the prperties and applicatins f the Hilbert transfrm [7] [8] [9]. Hsu cnsiders the applicatin f the Hilbert transfrm fr finding instantaneus frequency and intrduces a new methd, the sculating circle methd [20]. The Hilbert transfrm als finds applicatins in the medical area f prcessing electrencephalgraphy (ECG) readings [2]. 2

21 2.4 Passive Implementatins Existing passive implementatins f cmplex filters are sme f the lder incarnatins f phase-splitting circuitry. In his 955 Realizatin f cnstant phase difference, Bell Labs Darlingtn discusses a passive all-pass structure that creates a cnstant phase shift [3]. Saraga discusses alternatives t this all-pass structure in his Wide-band phase splitting netwrks [4]. Bedrsian builds n the LC netwrks presented in Darlingtn and Saraga s wrk t prvide nrmalized design curves fr the LC all-pass netwrk [22]. Bth Darlingtn and Saraga s tplgies rely n the use f inductrs; Hward presents a RC netwrk [5]. Gingell presents a fur-input, furutput passive plyphase netwrk (the main basis fr this thesis) [4]. 2.5 Active Implementatins Active implementatins f cmplex filters appear in bth in lder and newer applicatins. Larrwe presents an active implementatin f a cmplex bandpass filter [2]. Wiebach presents design analysis f several existing active and passive implementatins in additin t an investigatin n the effects f cmpnent tlerances [23]. Hutchins discusses three active all-pass phase splitting netwrks: Llyd s All-Pass, Budak s All-Pass, and the State Variable All-Pass [24]. Stikvrt als discusses the implementatin f a plyphase filter with p-amps [25]. 3

22 3. TYPE-II TOPOLOGY 3. Derivatin This sectin presents the cascadable st rder cmplex filter structure, type-ii tplgy. The tplgy cnsists f a pair f dual-input structures fr which deriving the transfer functin requires the applicatin f superpsitin. Figure 0 shws a twinput tplgy that realizes a highpass respnse with respect t V in and a lwpass respnse with respect t V in2. Figure 0. Active highpass/lwpass structure (nn-inverting) Superpsitin reveals the relatinship between the inputs and utputs: H = V ut V in = H 2 = V ut V in2 = scr scr+ scr+ ( 7) ( 8) V cd,y = ZWT ZWTSY V efy + Y ZWTSY V efk ( 9) Altering the abve structure t utilize bth f the peratinal amplifier s inputs (Figure ) maintains a high pass respnse n the utput with respect t the psitive terminal (V 2 ) and builds an inverting lw pass respnse with respect t the negative terminal (V ). 4

23 Figure. 2-input tplgy realizes inverting LP TF with respect t v and nn-inverting HP with respect t v2 The transfer characteristics belw derived again by use f superpsitin: H Y = g hij[ g kl^ = ZWT ZWTSY H K = g hij[ g kl[ = ay YSZWT ( 20) ( 2) V cd,k = ZWT YSZWT V efy s + ay YSZWT V efk s ( 22) The circuit shwn in Figure is nt cmplex n its wn, but it can replace the inverter and buffer (and adjacent circuitry) preceding V ut2 in Figure 9; this results in the tplgy in Figure 2. Figure 2. "Inverting" tplgy 5

24 This cmplex filter has the same transfer functin magnitude as Gingell s RC netwrk (Figure 7, equatins (4) and (5)) if C 2 = C and R 2 = R/2. LTspice testing cnfirmed the filter s perfrmance (Figure 3). The filter cmpnents set a center frequency f 3.62kHz. Figure 3. Single-stage cmplex filter Setting V t a csine functin and V 2 t a negative sine functin prduces a ntch respnse n bth utputs (Figure 4). Nte that the V ut respnses (in red) lies exactly underneath the V ut2 respnses (in blue) in bth Figure 4 and Figure 5. 6

25 Figure 4. Inputs: COS, -SIN prduces a ntch at f C = 3.62kHz Setting V t a csine functin and V 2 t a sine functin prduces a bandpass respnse n bth utputs (Figure 5); nte that nce again, the tw respnses lie n tp f each ther. Figure 5. Inputs: COS, SIN, prduces a bandpass at f C = 3.62kHz 7

26 Setting V and V 2 t the same csine functin prduces neither the ntch respnse nr the bandpass respnse; this prduces an all-pass respnse (Figure 6). Observe that the nearly cnstant magnitude respnse, hwever, the phase respnse f the secnd utput varies. Figure 6. Inputs: COS, COS prduces an all-pass circuit Setting V 2 t grund and V t a 3.62kHz sine functin prduces a sinusid respnse n bth utputs (Figure 7). Nte that the utputs shift apprximately 90 ut f phase with each ther and V ut magnitude equals V in magnitude. 8

27 Figure 7. Frequency dmain analysis f circuit in Figure 4: prduces a 90 phase difference, V in a 3.62kHz signal, V in2 grunded The abve cnfirms the tplgy s functinality as bth a psitive and negative frequency discriminatr and as a quadrature signal generatr. Nte that as a phase splitter, the magnitude f the quadrature utputs is nly the same at the designed center frequency. Generating quadrature ver a wider range requires higher-rder netwrks. 3.2 Crss-cupling This sectin returns t the riginal intent f designing a cascadable tplgy fr the purpse f creating bradband frequency respnses. T cascade multiple stages, first ntice the sign change in V ut2 with respect t V in and V in2 (Table ). 9

28 Original Gingell Table 3. Cmparisn f Gingell s and Type II transfer functins Type II Inverting V npy = scr + V scr tuy + scr + V tuk V cd,y = scr + V scr efy + scr + V efk V npk = scr scr + V tuy + scr + V scr tuk V cd,k = scr + V efy + scr V efk Because f this sign change, it is necessary t crss cuple by cnnecting V ut f the stage t V in2 f the next stage and V ut2 f the stage t V in f the next stage. This intrduces an additinal 80 phase shift, which equates t a negative sign; this crrects the sign change in verall transfer functin f the cascaded filter. See the appendix fr a full explanatin n Gingell respnse t psitive and negative sequences. The figures belw shw the results f crss-cupling and nt crss-cupling in different frms, prving the necessity f crss-cupling t maintain the desired system respnse. Figure 9 and Figure 20 demnstrate the results f crss-cupling, while Figure 23, Figure 26, and Figure 27 shw the undesired respnse that ccurs withut crss-cupling. Figure 9 shws the result f crss-cupling the stages t each ther. The respnse after each stage shws the results f the cascaded preceding stages building upn each ther. Frm Figure 20, nte that crss-cupling keeps the zers (crrespnding t the ntches in the frequency respnse) n the negative prtin f the frequency axis. 20

29 Figure 8. Crss-cupling all stages, schematic Figure 9. Crss-cupling all stages, simulatin respnse Figure 20. S-plane plt f three-stage filter, crss-cupled 2

30 Nt crss-cupling a stage (in this case, just the cnnectin between the first and secnd stages) results in a sign change f the lcatin f the zers. Figure 2 shws the s-plane plt, demnstrating hw the zers f the stages after the nn-crss cupled stage change sign. Figure 23 shws the simulated respnse. Figure 2. S-plane plt f three-stage filter, secnd stage nt-crss cupled Figure 22. N crss-cupling between first and secnd stage, schematic 22

31 Figure 23. N crss-cupling between first and secnd stage, simulatin respnse Nt crss-cupling all the stages cause the sign f the zers t alternate with respect t adjacent frequencies (Figure 24). Figure 24. S-plane plt f three-stage filter, all stages nt-crss cupled 23

32 The figures belw shw the simulatin results f nt crss-cupling any stages. Figure 26 shws the respnse f a three-stage filter, and Figure 27 shws the respnse f a five-stage filter. Figure 25. Three-stage filter, all stages nt crss-cupled, schematic Figure 26. Three-stage filter, all stages nt crss-cupled, simulatin respnse 24

33 Figure 27. Five-stage filter all stages nt crss-cupled, simulatin respnse 3.3 Design Example The fllwing design example demnstrates the perfrmance f a 5-stage filter and the effects f cmpnent matching and tlerances in additin t the cnstructin f a prttype filter. The methd used fr determining the ntch frequency fr each stage invlves placing each ntch equidistantly in a lg scale alng the frequency range f interest. Such placement results in near equi-ripple stpband characteristics. This example uses a ntch spacing rati f k = 2.8. Table 4. Ntch frequency values Frequency (Hz) f 47 f f f f Figure 28 is a schematic f the five filter stages. 25

34 Figure 28. Inverting tplgy, 5 stages cascaded, ideal cmpnent values d 26

35 Using the equatin fr the RC time cnstant and chsing reasnable values fr the capacitr (C stage = C stage2 = 0nF, C stage3 = 3.3nF, C stage4 = C stage5 = nf), Table 5 shws the calculated resistr values fr each stage (crrespnding t each stage frequency). Figure 29 shws the results f the simulatin with ideal cmpnent values. Table 5. Ideal values fr each stage f inverting tplgy Resistr Stage Stage 2 Stage 3 Stage 4 Stage 5 R k 49.9k k k k R2 = Y R k K 24.7k k 5.876k k Figure 29. Stp-band perfrmance f 5 th rder cascade, ideal cmpnent values 27

36 Figure 30. Passband perfrmance f 5 th rder cascade, ideal cmpnent values Runding t standard % resistr values, als knwn as EIA E96 values, results in Table 6 [26]. Figure 3 shws the results f simulating with standard % values. Table 6. E96 values fr inverting tplgy Resistr Stage Stage 2 Stage 3 Stage 4 Stage 5 R 08k 49.9k 68.k 05k 47.5k R2 53.6k 24.9k 34k 52.3k 23.7k 28

37 Figure 3. Stp-band perfrmance f 5 th rder cascade, E96 cmpnents Figure 32. Passband perfrmance f 5 th rder cascade, E96 cmpnents T investigate the effects f cmpnent variatin, Table 7 shws the values f R2 when calculated as a parallel cmbinatin as R. Figure 33 shws the results f simulating with the values shwn in Table 7. 29

38 Table 7. E96 values fr inverting tplgy, R2 as a parallel cmbinatin f tw R Resistr Stage Stage 2 Stage 3 Stage 4 Stage 5 R 08k 49.9k 68.k 05k 47.5k R2 54k 24.95k Figure 33. Stp-band perfrmance f 5 th rder cascade, E96 cmpnents, R2 as parallel cmbinatin f tw R 30

39 Figure 34. Passband perfrmance f 5 th rder cascade, E96 cmpnents, R2 as a parallel cmbinatin f tw R While the filter cmpsed f standard % resistrs fr R and a parallel cmbinatin f R fr R2 perfrmed better in simulatin, using the values in Table 6 perfrm mre practically fr prttyping. Table 8 lists the prttype cmpnent values. The prttype utilizes the MCP μA, 5MHz Rail-t-Rail Op Amp with a 5V supply. Figure 35 shws the prttype after building. Stage Table 8. Cmpnent values fr design example Ntch Frequency (Hz) E96 Capacitr (nf) Cmpnent Values E96 Resistr (kω) 3

40 Figure 35. Fine-stage active cmplex filter prttype implemented using five dual MCP6282 p-amp chips n perfbard Because f the sensitivity t cmpnent tlerances demnstrated in simulatin, the prttype is cnstructed n perfbard t avid the stray capacitance cmmn with slderless breadbards. Perfbard (DOT PCB) cnsists f cpper pads in the same standard spacing as a breadbard n a hard printed circuit bard-type material. T verify the prttype s perfrmance as a brad-band phase-splitter, drive the filter with ne input grunded and the ther receiving a sinusid signal. Measure the tw filter utputs with the input signal set t the individual stage center frequencies (Figure 36). The phase difference between the tw signals in each capture is apprximately 90. The magnitudes f the tw signals remain nearly the same ver the large span f test frequencies. 32

41 Figure stage design perfrms 90 phase-splitting fr five different frequencies, (tp t bttm, left t right 47Hz, 32Hz, 703Hz,.508kHz, and 3.34kHz) Using a pair f quadrature signals n the input stimulates the circuit s asymmetric frequency respnse (Figure 37 and Figure 38). 5 Inverting utput under 90/0.3Vpp quadrature drive Output (db) Frequency (Hz) Figure 37. Filter magnitude fr image-reject peratin, passband 33

42 Inverting utput under 90/80,.3Vpp quadrature drive Output (db) Frequency (Hz) Figure 38. Filter magnitude fr image-reject peratin, stpband Finally, in rder t judge the perfrmance f the prttype, key characteristics are measured and cmpared t simulatin results and mathematical mdeling (Table 9, Figure 39). Table 9. Cmparisn f calculatin, simulatin, and measured values Simulatin Measurement Stpband Attenuatin 29.9dB 9.5dB Peak Gain 0.8dB.0dB 3dB Cutff Passband Frequencies 38Hz, 3.54kHz 30Hz, 3.70kHz Stpband End Ntch Frequencies 45Hz, 3.33kHz 47Hz, 3.42kHz Belw find figure and definitins f parameters in Table 9. 34

43 Figure 39. Plt f theretical transfer functin and measured data fr 5-stage circuit Equatin 26 mathematically represents the system transfer functin; Figure 39 shws the transfer functin graphed alng with the data pints shwn in Figure 37 and Figure 38 fr cmparisn. Fr further explanatin f the rigin f this functin, see chapter 4. H x = 20 lg Yv YS YS w [ y w ([ y) YS w y [ YS w (y) [ YSz YS z [ YS{z YS {z [ YSK {z YS K {z [ ( 23) T quantify the perfrmance f a filter, first define the measurement parameters and definitins. Fur main quantities describe the filter: stpband bandwidth, stpband attenuatin, passband gain, and passband bandwidth. Stpband bandwidth, BW stp, defines the range f frequencies in-between the filter s lwest and highest frequency ntch; nte that this is nt a -3dB bandwidth, find the reasning in [7]. 35

44 Stpband attenuatin, A m, defines the distance frm unity gain (0dB) t the highest peak within the stpband regin. Passband gain, G, cnstitutes the maximum amunt f gain in the passband respnse. Passband bandwidth, BW pass, is a -3dB bandwidth; it defines the range f frequencies fr which the magnitude respnse measures less than 3dB away frm the passband gain, G. Figure 39 shws rughly 2dB less attenuatin in the stpband, A m, than expected accrding t the mathematical mdel, similarly, Table 9 shws apprximately 0dB less attenuatin when cmpared t simulatin results. One surce f this difference cmes frm cmpnent tlerances. Hwever, because the lbes in the measured respnse seem t fllw the lbes f the expected respnse, imperfect quadrature drive used t test the filter culd als cntribute t this difference. Figure 40 belw shws the effects f driving the input with signals with a 89 phase difference (in red) r a 88 phase difference (in green) instead f 90 (in blue). In additin, Figure 4 shws a cmparisn f having less than 90 phase difference (89 in red) and having mre than 90 phase difference (9 in green). 36

45 Figure 40. Simulatin: driving with imperfect quadrature, 90 (blue), 89 (red), 88 (green) Figure 4. Simulatin: driving with imperfect quadrature, 90 (blue), 89 (red), 9 (green) The abve figures demnstrate the effects f driving with imperfect phase n the quadrature inputs. The surce culd als have variatins n the vltage. The same 37

46 circuit is simulated with input surces varying 2% in magnitude (Figure 42). This shws the significant effect that driving surce accuracy has n the circuit respnse. Figure 42. Simulatin: driving with imperfect quadrature, 2% vltage variatin (red), ideal (blue) 38

47 4. MODELING OF BAND-PASS REGION As described in the previus sectin, fur design cnstraints exist: passband bandwidth, passband gain, stpband bandwidth, and stpband gain. Only tw parameters adjust directly in design: number f stages, N, and frequency spacing, k. Because f this mismatch, the design is inherently ver cnstrained. This sectin explres the tw passband characteristics as a functin f N and k in an effrt t prvide general clsed-frm equatins fr the purpse f filter design (see Chapter 5 design example). 4. Passband Gain as a Functin f N and k This sectin prvides mathematical basis fr the bserved gain in the filter s passband respnse. This analysis makes use f a free prgram called Graph which allws pltting f functins and data pints. Figure 43 shws the ttal respnse f an 8-stage filter (equatin 24) as well as the individual stage respnses. Pltting the respnse ver nrmalized frequency, x, defined as the riginal frequency range divided by the center frequency, F c, f the cascaded respnse re-centers the respnse arund an x-axis value f fr the purpse f generalizing the investigatin. This figure als defines the referencing cnventin fr the center, r peak, frequency (f n ) crrespnding t each individual stage and fr the areas f interest referred t later as the mid-band regin and the edge regin. 39

48 H x = 20 lg Yv YS w y y YS w y y YS w y [ y [ YS w y [ y YS w y y [ YS w y y YS w y YSz { YSz{ { [ YS w [ YS z { [ YS z{ { [ y YSz{ [ { YSz{ { ( 24) YS z{ [ { [ YS z{ { [ Figure 43. Transfer functin plt fr multiple stage, effective bandpass, filter Each individual stage in the cascade prvides a single peak at f f, where n crrespnds t the stage number. The numerical values f f f are chsen equidistantly frm each ther n a lgarithmic scale; this spacing prduces nearly equi-ripple stpband characteristics (see Figure 39-Figure 42). Mathematically, this means: } [ }^ = } } [ = } l } l~^ = k ( 25) where k crrespnds t a number termed frequency spacing, the rati between the equidistant peak frequencies. Realize that the center frequency, F c, fr a ttal respnse differs depending n even r dd filter rder. Fr dd system filter rder, the center frequency cincides with the central peak; fr example, fr an N = 3 filter with peaks at f Y, f K, and f the 40

49 center frequency is f K. Fr even system filter rder, the center frequency falls between the tw mst central frequencies; fr example an N = 8 filter, like the ne shwn in Figure 38, has an verall center frequency between f and f. Visualize this in a ple-zer plt (Figure 44) where the dashed arc represents the verall center frequency. Naming frequencies with respect t the verall center frequency prves useful in future design relatinship derivatins. Instead f f f where n crrespnds t stage number, the frequencies becme f S9 r f a9 where m the relative psitin cmpared t F c. Fr an N = 3 filter, f Y becmes f ay, f K becmes F c, and f becmes f SY. The frequencies use the same k relatinship determined befre. With respect t the center frequency, f S9 and f a9 can relate by: } ~^ ˆ = Y {, } ]^ ˆ = k, } ][ ˆ = k K, } ± ˆ = k ±9 ( 26) Hwever, fr an even rder filter, because the center frequency des nt match up with an actual stage peak, F c, the individual peak frequencies relate t the center differently. Fr an N = 2 filter, f Y becmes f ay, f K becmes f SY, and F C centers between the tw: } ~^ ˆ = Y {, } ]^ ˆ = k, } ][ ˆ = k /K, } ± ˆ = k ±9/K ( 27) 4

50 Figure 44. Determining system central frequency using ple-zer plts Nte that the numbering cnventin described abve and used fr the fllwing discussins arbitrarily assigns cnsecutive frequency peak numbers t crrespnding stage number, i.e. f K crrespnds t the center frequency fr the secnd stage. Assume this fr referencing purpses; see [7] fr a discussin n the tpic f stage arrangement. 4.. Even and Odd Orders T determine a general equatin fr maximum mid-band gain, first examine the example nrmalized magnitude transfer functins, H N where N represents system rder, fr different rder filters (equatins 27-32). First bserve that dd-rder filters (equatins 27-29) express differently frm even-rder filters (equatins 30-32) and therefre a different generalized equatin becmes necessary fr each case. H x = 20 lg Yv YS w y YS w y [ YSz YS z [ YS{z YS {z [ ( 28) H x = 20 lg Yv YS w y [ YS w [ y [ YS w y YS w y [ YSz YS z [ YS{z YS {z [ YS{[ z YS { [ z [ ( 29) 42

51 H Πx = 20 lg Yv YS w y YS w [ y YS w y [ YS w [ y [ YS w y YS w y [ YSz YS z [ YS{z YS {z [ YS{[ z YS { [ z [ YS{ z YS { z [ ( 30) Figure 45. Pass-band plt fr N=3, 5, 7 (nrmalized frequency, k = ) H K x = 20 lg Yv YS w y YSz { YS w [ y YS z { [ ( 3) H x = 20 lg Yv YS w y y YS w y y YS w y YSz { YSz{ { ( 32) [ YS w [ YS z { [ YS z{ { [ y H x = 20 lg Yv YS w y [ y YS w y [ y YS w y y [ YS w y y YS w y YSz { YSz{ { YSz{[ { ( 33) [ YS w [ YS z { [ YS z{ { [ YS z{ [ { [ y 43

52 Figure 46. Pass-band plt fr N=2, 4, 6 (nrmalized frequency, k = ) T find a clsed-frm expressin fr peak gain, examine the N = 3 case (Equatin 29, shwn again belw). H x = 20 lg Yv + x k + x k K + x + x + kx K + kx K Nte that the first term within the lg functin crrespnds t the lwest frequency, f, which crrespnds als t f -. The secnd term crrespnds t the center frequency, F c, r f 2, and the third term crrespnds the highest frequency, f 3 r f +. The peak gain ccurs at the center frequency, F c ; in the nrmalized expressin, this means x shuld equal. 44

53 H x = 20 lg Yv YS^ y YS ^ y [ YSY YS Y [ YS{ Y ^ YS y = 20 lg YS { Y [ Yv 20 lg Yv ^ y {SY ^ y { [ SY YS ^ y [ K K YS{ YS { [ = 2 YS{ YS { [ ( 34) Factring ut Y frm the numeratr and denminatr f first term makes the first and third terms identical (the Y factr cancels ut) and the secnd term becmes a cnstant, independent f k. The first and last terms have the same cntributin t peak gain. Because f the lg functin, multiplicatin becmes additin and the functin simplifies: H x = 20 lg Yv lg Yv {SY { [ SY + 20 lg Yv 3dB + 40 lg Yv YS{ YS{ YS{ = 3dB lg YS { [ Yv YS { [ = YS { [ ( 35) Realizing that this generalizes fr any dd N because the cntributin frm additinal stages branches ut symmetrically abut the central frequency: ~^ [ { SY Odd Peak Gain = 20 lg Yv lg š Y Yv ( 36) { [ SY Where N crrespnds t the number f stages and k crrespnds t frequency spacing. The case fr an even value f N is different again because f the individual peak frequencies relatinship t the system center frequency, F c. Fr the N = 2 case (Equatin 32): 45

54 H K x = 20 lg Yv + x k + x k + x k K + x k K Again, setting the nrmalized frequency, x, t t find the peak gain (see Figure 45): H K x = 20 lg Yv YS ^ y YS ^ y 20 lg Yv ^ y [ YSY { ^ y YS Y { [ {SY YS { { [ SY = 20 lg Yv YS ^ y YS ^ y [ YS { YS { [ = YS { [ ( 37) Similar t the N = 3 case, this time factring ut a value f Y reveals that the first and secnd term equate. Again using the prperties f lg: H K x = 20 lg Yv {SY { [ SY + 20 lg Yv YS { YS { [ = 2 20 lg Yv YS { YS { [ = 40 lg Yv YS { YS { [ ( 38) Frm this, the equatin can be expanded fr a general even N case: ^ [ Even Peak Gain = 40 lg { ~ [SY š Y Yv ( 39) { [ ~^SY Where N crrespnds t the number f stages and k crrespnds t frequency spacing. Table 0 validates the derived expressins. It cntains the results f evaluating equatins 37 and 40 with Matlab and simulating with ideal cmpnent values in LTspice. 46

55 Table 0. Matlab Values and LTspice Simulatin Stage Mid-Band Gain (db) Matlab LTspice Passband 3dB Bandwidth as a Functin f N and k This sectin derives a mathematical expressin fr the 3dB bandwidth f the filter s passband respnse. While equatins 37 and 40 prvide clsed-frm expressins fr peak gain as a functin f the number f stages and the frequency spacing, there is n cmparably simple way f deriving such an expressin fr the 3dB bandwidth. The fllwing calculatins depend n a methd f curve fitting perfrmed in Excel. Remember that because a cmplex filter s primary use requires its asymmetric frequency respnse, the passband bandwidth must cmpare t the stpband bandwidth fr prper functinality. In ther wrds, the passband shuld measure wide enugh t pass the desired signal while the stp-band shuld crrespndingly size t reject any image signals. Cnsider als that certain cmbinatins f filter rder and frequency spacing (k) can result in bandwidth shrinkage. This prblem cmmnly arises when a higher rder filter cmbines with very clse frequency spacing in an attempt t achieve higher stpband attenuatin. 47

56 4.2. Calculatin First cnsider the unique case f a single stage (N=) filter. Figure 47 shws the passband respnse f a single stage filter. Under the assumptin that -3dB frequencies dwn frm the maximum passband gain define the passband bandwidth, the single stage filter has theretically infinite bandwidth. Figure 47. Passband respnse fr single-stage, peak magnitude f 3dB, magnitude greater than 0dB fr all psitive frequency range Next, cnsider the case f N larger than and k =. With k set t, the individual stage ntch frequencies are the same, visually, this lk like Figure

57 Figure 48. Ple-zer plt fr k = Fr N f 2 r larger, 3dB bandwidth is finite, unlike the N = case. Figure 49 shws the measurements fr nrmalized 3dB bandwidth versus N fr k = and N ranging frm 2 t 5. System Order Vs. Nrmalized Bandwidth Nrmalized Bandwidth (Hz) System Order - N Figure 49. 3dB Nrmalized Bandwidth vs. N fr k =, k= means identical stages Frm this figure, curve fitting determines equatin 40. BW žzza{ Y = (v. ) ( a Y) ( 40) 49

58 The next examinatin is f the relatinship between frequency spacing, k, and the nrmalized system bandwidth. The figures belw (Figure 50-Figure 56) graph this relatinship ver reasnable values f k [7] fr N = 2, 4, 6, 8, 3, 5, 7. Fr each value f N, and expnential curve f the frm belw fits the pints: y = A exp [α x] where A and α depend n N and x crrespnds t k. 7 Nrmalized bandwidth Vs. K fr N = 2 Bandwidth (nrmalized) y = 3.528e 0.958x k Value Figure 50. Nrmalized 3dB bandwidth fr tw stage tplgy vs. k 50

59 Figure 5. Nrmalized 3dB bandwidth fr fur stage tplgy vs. k Figure 52. Nrmalized 3dB bandwidth fr six stage tplgy vs. k 5

60 Figure 53. Nrmalized d3b bandwidth fr eight stage tplgy vs. k Figure 54. Nrmalized d3b bandwidth fr three stage tplgy vs. k 52

61 Nrmalized bandwidth vs. k value fr N = 5 Nrmalized bandwidth y = e x k value Figure 55. Nrmalized 3dB bandwidth fr five stage tplgy vs. k Nrmalized bandwidth vs. k fr N = y = e.524x Nrmalized bandwidth k value Figure 56. Nrmalized 3dB bandwidth fr seven stage tplgy vs. k Frm the figures abve, all dependencies are fitted reasnably well using an expnential curve. 53

62 Building n equatin (40) and using systematic trial and errr determines the fllwing equatin fr nrmalized bandwidth: BW žzz = (v. ) e(y~^)( ~^. ) ( ay) ( 4) Table shws the accuracy f this equatin. The largest difference between measured bandwidth and bandwidth calculated using equatin (4) calculates t less than %. This equatin als applies t the k = case. Fr k =, the expnential term becmes unity and equatin (4) equates t equatin (40). Table. Cmparisn f measured and calculated bandwidth N k Measured Calculated % Difference

63 5. DESIGN WITH BANDWIDTH CONSTRAINT Designing a filter fcuses n chsing number f stages (N), and the ntch frequency spacing (k), t meet given filter specificatins. As discussed in Sectin 4, nly tw f the fur quantities f interest may be enfrced; the design is vercnstraint. T explre the effects f filter rder n bandwidth, the fllwing three designs demnstrate the prcess and perfrmance f a 4-stage, 5-stage, and 6-stage filter. Fr these design examples, the desired bandwidth crrespnds t the apprximate mst sensitive range f human hearing (Figure 57), 400Hz t 5kHz. This frequency range encmpasses the bandwidth used fr telephny cmmunicatins, 400Hz t 3.5kHz. The crner frequencies determine the central frequency f the whle filter (Equatin 4). f C = f Y f K = = Hz ( 42) Thus, the filter has a central frequency f 2.449kHz. The nrmalized bandwidth, fund by dividing the bandwidth by the central frequency, becmes

64 Figure 57. Human Ear Sensitivity [27] With bandwidth determined, rearranging the equatin fr passband bandwidth (equatin 4) yields the frequency spacing parameter k. k = ln žzz ay v.v + ( 43) ay. Each f the design examples belw uses equatin (43) t determine crrespnding k fr the three values f N. 56

65 N=4 Design: Chsing a 4-stage parameter, k = ay. ln. ay v. + = ( 44) EdgeAtten = ay. = db ( 45) ^ Pass band Gain even = 40 lg { ~ [ [SY = 40 [ lg Yv { [ ~^SY Yv = db ( 46) š Y š Y K. Œ [ ~^SY K. Œ ~^ [SY Nte that because this is an even rder filter, n individual center frequency equates t the verall filter central frequency. The individual frequencies derive frm the relatinship expressed in (27). The cmpnents chsen in Table 2 cme frm the general equatin fr the RC time cnstant. Stage Table 2. Cmpnent values fr N=4 design Stage Center Frequency (Hz) Cmpnent Values Standard % Capacitr (nf) Standard % Resistr (kω) 3.934k 40.2/80.6/20k 2 0.5k 5.8/3.6k/ k /63.4/ /53.4/3.3 57

66 Stp-band Bandwidth: 9.45kHz Stp-band Attenuatin: 23.5dB Figure 58. Stpband fr N=4 design Nte that stp-band bandwidth measures frm the first stp-band ntch t the last stp-band ntch (refer t [7] fr reasning), as such, this is a cnservative estimate f stp-band bandwidth. 58

67 Pass-band Bandwidth: 4.0 khz Passband Gain: 8.83dB Figure 59. Passband fr N=4 design 59

68 N=5 Design: k = ay. ln. ay v. + = ( 47) ~^ { SY Pass band gain dd = 20 lg Yv lg = 20 lg K. Yv { [ SY Yv lg SY [ [ Yv = db ( 48) š Y š Y K. [ SY ~^ Stage Stage Center Frequency (Hz) Table 3. Cmpnent values fr N=5 Cmpnent Values Standard % Capacitr (nf) Standard % Resistr (kω) 2.448k 64.9/30/ k 28/56.2/ k.8/23.7/ k 0 5/30./ /7.5/7.8 Attenuatin bandwidth: 3.0 khz Attenuatin flr : 26. db Figure 60. Stpband fr N=5 60

69 Band-pass bandwidth: 4.75 khz Band-pass gain: 0.36 db Figure 6. Passband fr N=5 6

70 N=6 Design: k = ay. ln. ay v. + = 2.7 ( 49) EdgeAtten = ay. = db ( 50) ^ Pass band Gain even = 40 lg { ~ [ [SY = 40 [ lg Yv { [ ~^SY š Y Yv =.786 db ( 5) š Y K.YŒ ~^ [SY K.YŒ [ ~^SY Stage Stage Center Frequency (Hz) Table 4. Cmpnent values fr N=6 Cmpnent Values Standard % Capacitr (nf) Standard % Resistr (kω) 3.609k 44.2/88.7/ /4.2/ k 9.3/8.7/ k /59/ /27/ /90.9/22.6 Figure 62. Stpband fr N=6 62

71 Stp-band bandwidth: 6.7kHz Stp-band attenuatin: 28.5 db Pass-band bandwidth: 4.8 khz Pass-band gain:.75 db Figure 63. Passband fr N=6 Table 5. Cmparisn f bandwidth measurements N Stp-band Bandwidth Pass-Band Bandwidth kHz 4.0 khz khz 4.75 khz 6 6.7kHz 4.8 khz Fr a 4-stage implementatin, the cnservative stp-band bandwidth becmes smaller than the pass-band bandwidth, suggesting that the filter is unable t stp all the unwanted frequencies. In the 6-stage implementatin, the stp-band bandwidth becmes larger than the pass-band bandwidth. The clsest match between stp-band and pass-band bandwidths ccurs with the 5-stage filter. 63