DESIGN OF AN INTEGRATED PROGRAMMABLE FRACTIONAL-ORDER GENERALIZED FILTER. M. Sc. Thesis BERTSIAS PANAGIOTIS R.N. 485

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1 UNIVERSITY OF PATRAS DEPARTMENT OF PHYSICS ELECTRONICS LABORATORY M.Sc. ELECTRONICS & COMMUNICATIONS DESIGN OF AN INTEGRATED PROGRAMMABLE FRACTIONAL-ORDER GENERALIZED FILTER M. Sc. Thei BERTSIAS PANAGIOTIS R.N. 485 Grdute of Deprtment of Phyic Supervior: Cot Pychlino, Profeor Ptr, September 06

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3 DESIGN OF AN INTEGRATED PROGRAMMABLE FRACTIONAL-ORDER GENERALIZED FILTER M. Sc. Thei BERTSIAS PANAGIOTIS R.N. 485 Exmintion Committee: C.Pychlino G.Economou S.Vli Approved by the three-member exmintion committee on 8/09/06. C.Pychlino Profeor G.Economou Profeor S.Vli Aocite Profeor Ptr, September 06 iii

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5 Abtrct Subject of thi M. Sc.Thei i the deign of generlized frctionl () order filter. Thi tructure offer the four tndrd type of filter function, i.e. lowp (LP), highp (HP), bndp (BP) nd bndtop (BS). The election of the type of the filter function i chieved through digitl logic, expreed by n pproprite combintion of bit. Thee bit offer the dvntge of the ttrctive deign flexibility nd preciion to the whole ytem, by progrmming the order well the cutoff frequency of the filter. In ddition, ey undertnding nd ue merge through the pproprite combintion of bit for ech ce. The contribution mde in thi Thei i tht cpcitorle reliztion of frctionlorder filter, uing current-mirror ctive element, re introduced for firt time in the literture. In ddition, thee filter offer electronic djutment of their frequency chrcteritic through digitl progrmming of the correponding bi current nd the relized cling fctor. Initilly, the procedure for pproximting thi type of filter by n pproprite integerorder multifeedbck topology i preented nd the repective deign eqution re ummrized. The bic opertion required for implementing the generlized filter re integrtion nd ummtion. Thu, the topology i relized through fully differentil cpcitorle loy nd lole integrtor topologie, which employ current-mirror ctive element, offering the dvntge of electronic djutment of their time-contnt. A two-integrtor loop filter i relized uing thee integrtor, confirming the bove. Furthermore, the progrmmbility of cpcitorle current-mirror integrtor i preented through Current-Diviion Network (CDN) nd pproprite witching cheme, offering flexibility nd providing comptibility, preciion, ey control well modulrity to the whole ytem. Extending thi concept into generl topology, it i feible to deign generlized frctionl-order filter, which would be controlled through n pproprite digitl logic correponded to pecific number of bit. The ytem implementtion i done uing MOS trnitor operting in the trong inverion region. The deign of the circuit h been performed through the Virtuoo Schemtic Editor of the Cdence oftwre the imultion reult hve been derived through Virtuoo Anlog Deign Environment provided by the AMS CMOS 0.5μm technology. Index term: Generlized frctionl-order filter, Differentil cpcitorle integrtor, Current-mirror, Current-Diviion Network (CDN), Digitl progrmming v

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7 Acknowledgement Thi Thei h been relized during the cdemic yer within the M. Sc. Electronic nd Communiction of the Deprtment of Phyic t Univerity of Ptr. The period until the completion of my grdute tudy provided unique experience for me nd I would like to thnk the people who contributed to thi coure. Initilly, I would like to thnk my upervior, Profeor Cot Pychlino, for guiding nd upporting me nd for the confidence he howed in me. Our collbortion w in high-level nd I owe to him gret grtitude for the opportunity given me to work together nd for the vluble knowledge he imprted me. Furthermore, I would like to thnk Profeor of Phyic Deprtment S.Vli nd G.Economou, who re lo member of the committee, for their guidnce through thi proce. Moreover, I would like to thnk my dvior nd PhD cndidte, Georgi Tirimokou, for her dicuion, ide nd feedbck tht hve been bolutely invluble. Alo, I would epecilly like to thnk my mzing fmily, my prent nd my iter, for the love, upport nd contnt encourgement I hve gotten over the yer. Lt but not let, I would like to thnk ll my friend nd ll the people nerby who were tnding contntly by me in peronl nd friendly level. vii

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9 Tble of Content Abtrct... Acknowledgement... Tble of content... v vi ix Chpter -Introduction.... Frctionl clculu.... Literture overview.... Thei objective & overview... Chpter Frctionl-Order () Generlized Filter Introduction Frctionl-order lowp filter Frctionl-order highp filter Frctionl-order bndp filter Frctionl-order bndtop filter Approximtion of the frctionl-order generlized filter Deign eqution for relizing generlized filter....8 Concluion... 5 Chpter Cpcitorle Integrtor Uing Current-Mirror Active Element Introduction Cpcitorle integrtor uing current-mirror A two-integrtor loop filter uing cpcitorle integrtor....4 Simultion reult....5 Concluion Chpter 4 Progrmmbility of Integrtor with Current-Mirror Introduction Progrmmbility of the bi current Progrmmbility of the gin Simultion reult Concluion ix

10 Chpter 5 Progrmmble Generlized Frctionl-Order Filter Introduction Progrmmbility of the propoed ytem Propoed topology Integrtor with progrmmble brnche Digitl logic Simultion reult Concluion... 0 Chpter 6 Concluion nd Future Work Concluion Future work Reference Appedix.... x

11 Introduction. Frctionl clculu Frctionl clculu h been introduced long time go nd it i generliztion of the trditionl clculu tht led to new concept nd tool with wider generlity nd pplicbility. The bic ide behind thi type of clculu i tht it conider derivtive nd integrl of rbitrry order. Everything trted in 695, when Mrqui de L Hopitl nd Wilhelm Leibniz hd oberved n pprent prdox for the nthderivtive of the liner function f ( x) = x,, if n=/, wht would the reult be, n d x n dx n d y nd generlly for ny function y with derivtive. Following thi inquiition, n dx frctionl clculu w primrily tudy reerved for the bet mind in mthemtic. Fourier, Euler, Lplce re mong the mny tht dbbled with frctionl clculu nd the mthemticl conequence. Mny found, uing their own nottion nd methodology, definition tht fit the concept of non-integer order integrl or derivtive, while the mot fmou but, lo, mot complex of them own to Riemnn- Liouville nd Grunwld-Letnikov. The firt definition i: d dt α α f α ( t) = D f ( t) = Γ ( α ) d dt t ( ) α t τ f ( τ ) 0 dτ (..) where 0 < < nd Γ( ) i the gmm function. The econd definition, which i widely ued in the time domin, i:

12 CHAPTER -Introduction D α f m α j α ( t) = ( t) ( ) n f (( m j) t) j= 0 j (..b) α j where Δt i the integrtion tep nd n = ( ) ( Γ( j α ))/( Γ( α ) Γ( j ) ) j Thu, it i very fortunte tht pplying the Lplce trnform to frctionl derivtive i lo vlid nd it i widely ued to the frequency domin. Thi i the bic force driving force for further ppliction in mny field. Auming zero initil condition nd pplying the Lplce trnform, yield: where α ( ) α = j. ( t) = F( ) α α L0 d t f (..c) Severl method hve been developed to pproximte thi opertor vi integerorder trnfer function nd everl circuit with pecific ctive element through pproprite trnfer function hve been implemented for thi reliztion. The development of frctionl-order circuit i promiing reerch re nd thi i originted from their interdiciplinry nture, they could find ppliction in biochemitry, medicine, electricl engineering nd o on. For exmple, the modeling of vicoelticity well of biologicl cell nd tiue h been performed through the utiliztion of the frctionl-order clculu. Alo, biologicl ignl uch electrocrdiogrm (ECG) nd electroencephlogrph (EEG) hve pectr tht obey to the frctionl-order rule... Literture overview In the literture, mny circuit with everl ctive element hve been ued to pproximte the pproprite trnfer function bed on the forementioned concept. Recently, the deign of frctionl-order filter h gined growing reerch interet, the derived frequency repone exhibit topbnd ttenution equl to 6 ( n α ) db/octve, where n nd (0<<) re the integer nd frctionl prt of the order of the filter. Compred with integer-order filter, which chieve 6 n db/octve ttenution, frctionl filter offer more precie control of the ttenution grdient. Alo, the implementtion of frctionl-order ocilltor i enbling the boot of frequency independent of the cpcitnce vlue. Frctionl-order cpcitor, lo known contnt phe element (CPE), nd frctionl-order inductor (FI) re the mot importnt component for relizing frctionl-order circuit tht offer the benefit of circuit implicity. The min drwbck i tht thee element re not yet commercilly vilble device. Until their commerciliztion, frctionl-order circuit re relized through uitble integer-order pproximtion. Thu, poible olution i the ubtitution of CPE by pproprite

13 CHAPTER -Introduction RC network, uch infinite RC cble or ppropritely configured integer-order RC ldder network, which pproximte their behvior. But it hould be mentioned tht thi procedure i not ey, from the prcticl implementtion point of view. In ddition, on-line djutment of the order nd impednce of the employed frctionlorder element i not poible. In imilr wy, FI could be pproximted through the utiliztion of Generlized Impednce Converter (GIC) with pive element nd CPE; thi olution uffer lo from the prcticl problem relted to the pproximtion of CPE. Furthermore, nother olution i the pproximtion of frctionl-order filter function by pproprite integer-order function. For thi purpoe, dicrete component ctive filter hve been lredy introduced in the literture. The employed ctive element were opertionl mplifier (op-mp), econd-genertion current conveyor (CCII), nd current feedbck opertionl mplifier (CFOA). Due to the employment of pive reitor, the relized time-contnt hve the form: τ=rc nd, thu, the obtined frequency repone uffer from limited ccurcy, which could be minimized uing dditionl utomtic tuning circuitry. In ddition, the derived filter tructure do not hve the cpbility for electronic tuning of their frequency chrcteritic. In order to overcome thi drwbck, compnding frctionl-order filter hve been introduced in the literture. Thee tructure offer the benefit of reitorle reliztion with electronic-djutment of their frequency chrcteritic; in ddition, they hve cpbility for ultr-low voltge opertion. On the other hnd, compnding filter uffer from the increed circuit complexity compred with their liner counterprt. An lterntive wy for relizing reitorle frctionl-order circuit with electronic tuning cpbility of their chrcteritic i uing current-mirror ctive element. Current-mirror re ttrctive cell for performing current-mode nlog ignl proceing. Thi i originted from their imple tructure which provide cpbility for implementing filter with reduced power diiption. Thi i very importnt feture, epecilly in biomedicl ppliction where the bttery life of implntble device hould be mximized. Thu, the reliztion of the required timecontnt nd the employment of the current cling fctor re performed through the utiliztion of the mll ignl trnconductnce prmeter (g m ) of the diode-connected input trnitor.. Thei objective & overview The contribution mde in thi Thei i tht cpcitorle reliztion of frctionl-order filter, uing current-mirror ctive element, re introduced for firt time in the literture. In ddition, thee filter offer electronic djutment of their frequency chrcteritic through digitl progrmming of the correponding bi current nd the relized cling fctor. More pecificlly, generlized frctionl () order filter tructure i propoed which offer the four tndrd type of filter function, i.e. lowp (LP), highp

14 CHAPTER -Introduction (HP), bndp (BP) nd bndtop (BS). The election of the type of the filter function i chieved through digitl logic, expreed by n pproprite combintion of bit. In ddition, the order well the cutoff frequency of the filter re digitlly progrmmed offering n ttrctive deign flexibility. The Thei i orgnized follow: In Chpter, the () order lowp, highp, bndp, nd bndtop filter re preented in term of trnfer function nd the mot importnt frequency chrcteritic. Alo, the procedure for pproximting ech type of filter by n pproprite integer-order multifeedbck topology i given. The concept for relizing loy nd lole integrtor, uing current-mirror ctive element nd without externl cpcitor i preented n Chpter.A deign exmple, two-integrtor loop filter uing the forementioned integrtor i deigned nd evluted. Thi i performed uing the Anlog Deign Environment of Cdence oftwre nd the deign kit provided by the AMS 0.5μm CMOS proce. In Chpter 4, the progrmmbility of cpcitorle current-mirror integrtor i preented. A Current-Diviion Network (CDN) nd pproprite witching cheme re preented for thi purpoe. The vlidity of the derived fully progrmmble current-mirror i verified through imultion reult. A fully digitlly progrmmble, cpcitorle, frctionl-order generlized filter i introduced in Chpter 5. The choice of the type, order, nd cutoff/reonnce frequency of the relized filter function i performed through n pproprite digitl logic ytem which i derived ccording to the concept preented in Chpter 4. In order to get view of the relitic behvior of the ytem, lyout deign of the nlog core of the generlized filter i performed. The derived imultion reult confirm the functionlity of the propoed ytem. The concluion derived through the frmework of thi Thei nd, lo, propol for future reerch work re given in Chpter 6. 4

15 Frctionl-Order () Generlized Filter. Introduction The trnfer function relized by frctionl-order (α) generlized filter re preented in thi Chpter. Thi type of filter i cpble of implementing the four tndrd filter function, i.e. lowp (LP), highp (HP), bndp (BP), nd bndtop (BS). The procedure for pproximting thi type of filter by n pproprite integer-order multifeedbck topology i preented nd, lo, the repective deign eqution re ummrized. 5

16 CHAPTER -Frctionl-Order () Generlized Filter 6. Frctionl-order lowp filter The trnfer function of frctionl-order lowp filter (FLPF) i: ( ) H LP = (..) The mgnitude repone of thi filter i: ( ) ( ) co in j H = απ απ α α (..b) nd it phe repone i: ( ) { } { } = co in in co tn rg rg j H απ απ απ απ α (..c) Tking the eqution in (..b) nd etting the condition ( ) 0 = = p j H d d, the pek frequency ( p ) cn be eily clculted from the eqution below: ( ) ( ) 0 co in = απ α α απ α α α α p p p (..d) Alo, there i nother frequency, which i defined the frequency where -db drop of the pbnd gin pper, nd it i clled hlf-power (-db) frequency ( h ). It i clculted from the next eqution: ( ) 0 co in = h h h h απ απ α α α α (..e)

17 CHAPTER -Frctionl-Order () Generlized Filter 7. Frctionl-order highp filter The trnfer function of frctionl-order highp filter (FHPF) i: ( ) H HP = (..) The mgnitude repone of thi filter i: ( ) ( ) co in j H = απ απ α α (..b) nd it phe repone i: ( ) { } { } ( ) = co in in co tn rg rg j H απ απ απ απ π α α (..c) Tking the eqution in (..b) nd etting the condition ( ) 0 = = p j H d d, following the correponding procedure like the forementioned lowp frctionlorder filter, then the pek frequency ( p ) cn be eily clculted from the eqution below: ( ) ( ) ( ) 0 co in = p p p α απ α απ α α α α (..d) Furthermore, the clcultion of the hlf-power (-db) frequency ( h ) i derived from the following eqution: ( ) 0 co in = h h h h απ απ α α α α (..e)

18 CHAPTER -Frctionl-Order () Generlized Filter 8.4 Frctionl-order bndp filter The trnfer function of frctionl-order bndp filter (FBPF) i: ( ) H BP = (.4.) The mgnitude repone of thi filter i: ( ) ( ) co in j H = απ απ α α (.4.b) nd it phe repone i: ( ) { } { } = co in in co tn rg rg j H απ απ απ απ απ α (.4.c) The pek frequency ( p ) cn be eily clculted from the eqution below, which i derived from the eqution in (.4.b), etting the condition ( ) 0 = = p j H d d : ( ) ( ) 0 co in = p p p α απ α απ α α α α (.4.d) The hlf-power (-db) frequencie ( hl : lower frequency, hu : upper frequency) re clculted from the condition: ( ) ( ) p h j H j H = = = (.4.e)

19 CHAPTER -Frctionl-Order () Generlized Filter 9 The qulity fctor of the filter i clculted from the next eqution: hl hu p Q = (.4.f).5 Frctionl-order bndtop filter The trnfer function of frctionl-order bndtop filter (FBSF) i: ( ) H BS = (.5.) The mgnitude repone of thi filter i: ( ) ( ) ( ) co in in j H = απ απ απ α α α α (.5.b) nd it phe repone i: ( ) { } { } = co in in co tn in co tn rg rg j H απ απ απ απ απ απ α α α (.5.c) Tking the eqution in (.4.b) nd etting the condition ( ) 0 = = p j H d d, the pek frequency ( p ) cn be eily clculted like the forementioned ce. The hlf-power (-db) frequencie ( hl : lower frequency, hu : upper frequency), in the ce of the bndp filter, re clculted from the condition:

20 CHAPTER -Frctionl-Order () Generlized Filter 0 ( ) ( ) p h j H j H = = = (.5.d) Alo, the qulity fctor of the filter i clculted from the next eqution: hl hu p Q = (.5.e).6 Approximtion of the frctionl-order generlized filter A wy for implementing frctionl-order generlized filter i through the employment of frctionl-order cpcitor. However, the min drwbck i tht frctionl-order cpcitor re not currently commercilly vilble. Until their commerciliztion, frctionl-order trnfer function re relized through pproximtion uing uitble integer-order function. Thu, n efficient pproximtion, with regrd to ccurcy nd circuit complexity, i the econd-order expreion, which i derived ccording to the Continued Frction Expnion (CFE), given by (.6.): ( ) ( ) ( ) ( ) ( ) ( ) b b b (.6.) FLPF Subtituting (.6.) into (..), the derived trnfer function i the following: ( ) ( ) D b b b H LP 0 = (.6.b) where the expreion of D() i given by the following eqution: ( ) b b b D = (.6.c)

21 CHAPTER -Frctionl-Order () Generlized Filter FHPF If (.6.) i ubtituted into (..), the derived trnfer function i the following: H HP ( ) = ( ) D 0 (.6.d) Compring (.6.b) nd (.6.d), it i obviou tht the numertor in (.6.d) i derived through the ubtitution of b j of the numertor in (.6.b) by j (j=0,, ) nd multipliction by the fctor. FBPF Subtituting (.6.) into (.4.), the derived trnfer function i the following: ( ) = (.6.e) D 0 H BP ( ) Compring (.6.b) nd (.6.e), it i obviou tht the numertor in (.6.e) i derived through the ubtitution of b j of the numertor in (.6.b) by j (j=0,, ). FBSF The following trnfer function i derived through the ubtitution of (.6.) into (.5.). ( ) ( b ) ( 0 b ) D( ) b = (.6.f) 0 H BS Compring ll the forementioned pproximted eqution, it i eily derived tht ll of the derived trnfer function hve the me denomintor D(), while their numertor cn be derived by the me generl form. Conequently, they could be relized by the me topology jut by chnging the coefficient vlue of both numertor nd denomintor.

22 CHAPTER -Frctionl-Order () Generlized Filter.7 Deign eqution for relizing generlized filter A uitble olution, n outcome of the forementioned concluion, i the Functionl Block Digrm (FBD) of the multifeedbck topology depicted in Figure.. i out τ G i o i o i o i o G i o i o G o i o i o G i in i in τ τ τ i in i in Figure. Functionl Block Digrm for relizing ll filter The relized trnfer function, which i derived from Figure., i given by (.7.): ( ) 0 τ τ τ τ τ τ τ τ τ τ τ τ = G G G G H (.7.) FLPF Compring (.7.) with (.6.b), the reulted deign eqution for the lowp filter re the following: b = τ 0 b b = τ b b = τ = 0 G (.7.b) b b G =

23 CHAPTER -Frctionl-Order () Generlized Filter 0 b b G = b b G = FHPF Compring (.7.) with (.6.d), the deign eqution tht re derived for the highp filter, re the following: b = τ 0 b b = τ b b = τ G = (.7.c) b G = 0 0 b G = 0 = 0 G FBPF The deign eqution for the bndp filter re derived through the comprion of (.7.) with (.6.e): b = τ 0 b b = τ

24 CHAPTER -Frctionl-Order () Generlized Filter b b = τ = 0 G (.7.d) b G = 0 b G = b G = FBSF Compring (.7.) with (.6.f), the reulted deign eqution for the bndtop filter re the following: b = τ 0 b b = τ b b = τ G = (.7.e) b b G = 0 0 b b G = b b G =

25 CHAPTER -Frctionl-Order () Generlized Filter.8 Concluion A frctionl-order () generlized filter h jut been decribed through the conidertion of the four tndrd filter function tht could be implemented. Thee function re by the me topology, jut by chnging the coefficient vlue of both numertor nd denomintor. The bic opertion required for implementing the generlized filter re integrtion nd ummtion. Therefore, current- mode building block re preferble in comprion with their voltge-mode counterprt. Thu, current-mirror could be n ttrctive olution for relizing generlized frctionlorder filter. 5

26 CHAPTER -Frctionl-Order () Generlized Filter 6

27 Cpcitorle Integrtor Uing Current- Mirror Active Element. Introduction Integrtor topologie, which employ current-mirror ctive element, re preented in thi Chpter. According to the nowdy trend for deigning reitorle filter with electronic djutment cpbility of their time-contnt, the employment of current-mirror eem to be n pproprite choice for thi purpoe. The topologie tht will be preented re differentil loy nd lole integrtor well twointegrtor loop filter uing thee integrtor. The deign of thee filter h been performed through the Virtuoo Schemtic Editor of the Cdence oftwre. Furthermore, the imultion reult re hown they hve been derived through Virtuoo Anlog Deign Environment. 7

28 CHAPTER -Cpcitorle Integrtor Uing Current-Mirror. Cpcitorle integrtor uing current-mirror Loy nd lole integrtor cheme uing current-mirror nd cpcitor re demontrted in Figure. () nd (b) repectively. V DD V DD i in I o I o i out i in I o I o I o I o I o i out Mn Mn Mn Mn Mn Mn4 Mn5 C C () (b) Figure. Integrtor uing current mirror () loy nd (b) lole. The correponding trnfer function re given by (..) nd (..b), repectively H ( ) H = τ ( ) = τ (..) (..b) The relized time-contnt τ i given by the formul: τ = C g m, where g m i the mll-ignl trnconductnce of the trnitor M n nd C i the cpcitor connected between the gte nd ource of the trnitor M n. Conidering tht MOS trnitor operte in the trong inverion region, the g m of the trnitor i given by (..c). g m W = n I o (..c) L Thu, the previou expreion could be lterntively written : τ = C / W L n I o, where I o i the bi current, n i the trnconductnce prmeter, which i equl to µ C, where μ n i the mobility of electron nd C ox i n ox the gte oxide cpcitnce, W nd L re the width nd the length of the trnitor M n repectively. Obviouly, it i redily obtined tht the relized time-contnt could be 8

29 CHAPTER -Cpcitorle Integrtor Uing Current-Mirror electroniclly djuted through the bi current I o nd thi would be n importnt point for chieving reitorle reliztion with electronic djutment of their frequency chrcteritic. Conidering only the internl cpcitnce of MOS trnitor, topology in Figure. () become cpcitorle loy integrtor nd thi i demontrted in Figure., where the idel current ource re implemented uing multiple output current-mirror cheme. Mp V DD Mp Mp I o i in i out Mn Mn Figure. Cpcitorle loy integrtor eq The time-contnt τ i, now, given by the formul: g g τ = C g eq m, where C = C C, where C g nd C g re the cpcitie between the gte nd ource of trnitor M n nd M n, repectively. The cpcitnce between the gte nd ource of trnitor i given by (..d), where L ov i the gte overlp length. C g = W L C ox W L ov C ox (..d) In the AMS CMOS 0.5μm technology, µ A L ov 0.µ m, n = 70, nd V C ff ox µ m Inpecting Figure. (b), it i derived tht loy integrtor nd n inverter hve been combined in order to relize lole integrtor. Thu, the requirement for n inverter impoe retriction for the trnpoition of thi circuit into cpcitorle integrtor cheme. More pecificlly, the pect rtio of the trnitor of the inverter, M n4 nd M n5, hve to be very mll in comprion with tht of the trnitor tht form the loy integrtor [i.e. ( W L) inv << ( W L) ] nd thi i not lwy relizble, epecilly in high-frequency ppliction. 9

30 CHAPTER -Cpcitorle Integrtor Uing Current-Mirror In order to overcome thi difficulty, fully differentil topology will be utilized nd the derived cheme i demontrted in Figure.. V DD i in I o I o I o i out- Mn Mn Mn V DD I o I o I o i in- i out Mn4 Mn5 Mn6 () V DD Mp Mp Mp Mp4 i in i out- I o Mn Mn Mn V DD Mp5 Mp6 Mp7 i in- i out Mn4 Mn5 Mn6 (b) Figure. Differentil cpcitorle lole integrtor with () idel current ource (b) trnitor current ource 0

31 CHAPTER -Cpcitorle Integrtor Uing Current-Mirror The time-contnt τ i given, gin, by the formul: trnconductnce g m of the trnitor M n or M n4 would be: g m τ = C g eq m, where the W = n I o nd L the equivlent cpcitnce C eq of the integrtor would be: C eq = Cg Cg Cg, where C g, C g nd C g re the cpcitnce between gte nd ource of trnitor M n, M n, nd M n, or M n4, M n5, nd M n6, repectively.. A two-integrtor loop filter uing cpcitorle integrtor In order to evlute the behvior of the cpcitorle integrtor, two-integrtor loop (biqud) filter will be relized. The Functionl Block Digrm (FBD) of thi filter i preented in Figure.4, where i BP nd i LP re the bndp nd the lowp output current repectively. i in τ τ i BP i LP Figure.4 Functionl Block Digrm of two-integrtor loop filter The trnfer function of the lowp filter i given by the formul (..), while the trnfer function of the bndp filter by the formul (..b). H LP ( ) i i = LP = o (..) in o Q o H BP ( ) i i BP = = (..b) in o Q o o Q

32 CHAPTER -Cpcitorle Integrtor Uing Current-Mirror where the chrcteritic (reonnce) frequency i = τ τ Q =. o τ τ, nd the qulity fctor The implementtion of thi topology in cpcitorle fully differentil form i demontrted in Figure.5, where A, B, C nd D re lie which declre node for voiding too mny wire nd fcilitting the connection. The loy integrtor conit of trnitor M n, M n nd M n of the firt topology nd M n8, M n9 nd M n0 of the econd topology. The lole integrtor conit of trnitor M n4, M n5, M n6 nd M n7 of the firt topology nd M n, M n, M n nd M n4 of the econd topology. Thu, the firt time-contnt τ i given by the formul: Ceq Cg Cg Cg Cg, τ = = = nd the econd time-contnt τ i given by the g g g m formul: τ = m, m, Ceq Cg4 Cg5 Cg6 Cg7 4Cg, g m = g m = g m, where the cpcitnce C g, nd C g, re given by the formul (..d) nd the trnconductnce g m nd g m of the trnitor M n or M n4 would be: trnitor re identicl. g = g = W I L m, m, n o, uming tht the V DD Mp Mp Mp Mp4 Mp5 Mp6 Mp7 Mp8 i in i BP- i LP I o A Mn Mn Mn B Mn4 C Mn5 D Mn6 Mn7 V DD i in- Mp9 Mp0 Mp Mp Mp Mp4 i BP Mp5 i LP- D C B A Mn8 Mn9 Mn0 Mn Mn Mn Mn4 Figure.5 Fully differentil two-integrtor loop cpcitorle filter uing currentmirror ctive elelemnt

33 CHAPTER -Cpcitorle Integrtor Uing Current-Mirror.4 Simultion reult The pecifiction of the deigned filter were: f o = 50MHz nd Q = For thi purpoe, the width W n nd the length L n of the nmos trnitor of the integrtor hve been elected to be 0μm nd μm repectively. The width W p of the pmos trnitor h been elected to be 90μm while their length L p w μm. The vlue of the bi current I o h been et 50μA. The power upply voltge of the filter were V DD =. 5V nd V SS =. 5V. Figure.6 Frequency repone of the two-integrtor loop filter The derived frequency repone re demontrted in Figure.6.The cut-off frequency of the lowp output w f o, LP = 4MHz. Τhe pek frequency of the bndp output w I 0 = 50µA. f, BP = MHz nd the bndwidth i BW = 85MHz t pek 50 The electronic tuning cpbility of the topology i originted from the following expreion: n m o = = = = L I o (.4.) τ C C g Cg Cg Cg g, C τ,,,, g, g m g m g W

34 CHAPTER -Cpcitorle Integrtor Uing Current-Mirror where it i obviou the dependence of the reonnce frequency o from the bi current; more pecificlly the reonnce frequency f o, i proportionl to the qure root of the bi current I o. Thu, concerning the lowp output, for I o = 5µ A, which i the hlf of the initil vlue of bi current, it i expected tht the new cut-off frequency would be / time of the initil cut-off frequency. For I o = 00µ A, which i double to initil bi current, the new cut-off frequency hould be time of the initil frequency. With repective procedure, for I o = 00µ A, the cut-off frequency hould be double to the initil one. Thi i demontrted in Figure.7, where the lowp filter frequency repone h been derived t I o equl to 5μA, 00μA, nd 00μA, repectively. The imultion reult compred with the correponding theoreticl reult re ummrized in Tble. Figure.7 Electronic tuning cpbility of the lowp filter function I o (μa) f o (MHz) imultion theory Tble Reult of electronic tuning of the lowp filter function 4

35 CHAPTER -Cpcitorle Integrtor Uing Current-Mirror Concerning the bndp output, imilr reult re derived, becue of the dependence of frequency o with the bi current I o. Thu, the frequency repone of the repective chnge of bi current, which re demontrted in Figure.8, would be bed on the qure root dependence, like the lowp output nd the repective reult re demontrted in Tble compred with the repective theoreticl one. Figure.8 Electronic tuning of the bndp filter function I o (μa) f pek (MHz) imultion theory Tble Reult of electronic tuning cpbility of the bndp filter A Monte Crlo nlyi h been performed nd it reult, bout the bndwidth of the lowp function, re demontrted in Figure.9. The tndrd devition of thi prmeter w.76mhz nd the men vlue w 4MHz. The reult of thi nlyi, concerning the gin of the lowp function, re demontrted in Figure.0, where the men vlue h been 0.98 nd the tndrd devition h been

36 CHAPTER -Cpcitorle Integrtor Uing Current-Mirror Figure.9 Sttiticl plot bout the bndwidth of the lowp filter Figure.0 Sttiticl plot bout the gin of the lowp filter function Repective reult hve been derived, lo, for the bndp filter function. Sttiticl plot bout the bndwidth of thi function i illutrted in Figure., where the men vlue h been 85MHz nd the tndrd devition h been.5mhz. In Figure., the ttiticl reult for gin re hown, where the men vlue h been nd the tndrd devition Furthermore, ttiticl plot bout the centrl frequency of bndp filter function h been executed nd it reult re 6

37 CHAPTER -Cpcitorle Integrtor Uing Current-Mirror demontrted in Figure., where the men vlue h been 50.54MHz nd the tndrd devition.56mhz. Figure. Sttiticl plot bout the bndwidth of the bndp filter Figure. Sttiticl plot bout the gin of the bndp filter f 7

38 CHAPTER -Cpcitorle Integrtor Uing Current-Mirror Figure. Sttiticl plot bout the centrl frequency of the bndp filter The linerity of the two-integrtor loop filter h been exmined through Periodic Stedy Stte (PSS) nlyi for the lowp nd the bndp output, repectively. The derived reult re demontrted in Figure.4 for the lowp filter function, where for full cle input ignl mplitude (i.e.50μa), the Totl Hrmonic Ditortion (THD) i equl to 0.5%. In Figure.5, the reult for the bndp filter function re illutrted, where for tone f =MHz nd f =.MHz, the Third-Order Input Intercept Point w IIP = 64. 4dBm. Figure.4 Liner performnce of the lowp filter 8

39 CHAPTER -Cpcitorle Integrtor Uing Current-Mirror Figure.5 rd -order inter-modultion ditortion plot of the bndp filter The rm vlue of totl input referred noie of the lowp filter function w 76.6nA, while for the bndp filter function w 0nA. Thu, the dynmic rnge (DR) of the lowp filter, clculted from the formul in (.4.b), will be 59.dB. DR = 0log υ ( in,mx) rm ( input _ referred _ noie) rm (.4.b) ( IIP ) ( input _ referred noie) ) DR = _ dbm rm, dbm (.4.c) The DR of the bndp filter, clculted from the formul (.4.c), will be 4.8dB. 9

40 CHAPTER -Cpcitorle Integrtor Uing Current-Mirror.5 Concluion Fully differentil cpcitorle loy nd lole integrtor topologie, uing current-mirror ctive element, hve been preented. The ttrctive chrcteritic offered by thee topologie re the following: ) bence of ny retriction concerning the pect rtio of trnitor for relizing lole integrtor, b) bence of pive reitor, c) electronic tuning of the frequency chrcteritic of the derived filter. The provided two-integrtor loop filter deign confirmed the bove. Therefore, thee tructure re ttrctive cndidte for relizing frctionl-order filter nd thi will be further dicued in the next Chpter. 40

41 4 Progrmmbility of Integrtor with Current-Mirror 4. Introduction Digitl progrmming i modern wy for deigning circuit nd controlling their chrcteritic, it offer flexibility nd provide modulrity to the whole ytem. Digitl progrmmble logic of the chrcteritic of the integrtor with currentmirror i preented in thi Chpter. A Current-Diviion Network i employed for thi purpoe, in order to chieve the diviion of the min bi current into lower bi current, which would be ued for biing the integrtor. 4

42 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror 4. Progrmmbility of the bi current A Current-Diviion Network (CDN) i ued for digitl progrmming of the chrcteritic of integrtor, through the ue of current-mirror, it demontrted in Figure 4.. V DD Mpo Mpo Mp Mp Mp Mp I REF Mn Mn n o Mo Mo n M M n MM I REF I REF / I REF /4 Mpio Mpi Mpi I OUT Mnio Mni Mni () V DD I REF n o n n CDN I OUT (b) Figure 4. A typicl CDN () circuitry (b) ymbol 4

43 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror The trnitor tht contruct the inverter re M nij nd M pij where j=0,,, The witching trnitor of thee inverter re M j nd M j, where, gin, j=0,,, The input bi current I REF i replicted through the current-mirror formed by trnitor M po -M po. The vlue of the control bit n o of the witching trnitor M o -M o determine the preence of the current I REF t the output of the CDN. Becue of the current-mirror formed by the trnitor M po -M p, current equl to I REF / could be vilble t the output of the CDN. The next tge i fed by current I REF /, through the current-mirror formed by M po -M p. The vlue of the control bit n of the witching trnitor M -M determine the preence of the current I REF / t the output of the CDN. Thi procedure i continuing, depending on the current, which re deirble to be t the output, controlled by the repective j-th digitl bit n j. Thu, the expreion of the totl output current i tht given by (4.). n j I OUT = I REF n j (4.) j= 0 4. Progrmmbility of the gin The bic ide for demontrting the progrmmbility of the gin of currentmirror i hown in Figure 4.. The concept i tht pecific gin could be chieved with the repective output brnch. In the imultion which would be followed, four gin re ued, i.e. G, G, G nd G 4, they re hown in Figure 4.. Switche pi, pi nd ni, ni (i=,,, 4) re ued in order to chooe which pth would be followed for the pproprite gin. If gin i deirble, then the witche pi pecific now) would be on nd the other witche off. nd ni (where i i 4

44 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror VDD Io iin Mnb Mnb Mn p Mpo Mp Mp Mp Mp4 Mp5 p n Giin Mn p p n Giin Mn p p n Giin Mn4 p4 p4 n4 G4iin Mn5 n n n n4 VSS G G G G4 iout Figure 4. Concept of gin progrmmbility of loy integrtor uing current-mirror Thu, ytem, which would operte elector, i necery in order to elect the pproprite brnch of gin nd dible the other brnche. Thi concept i preented in Figure

45 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror VDD Io Mnb Mnb Mpo Mp Mp Mp Mp4 Mp5 iin Mn G G Selector (p) co Giin c c c co c Giin c c Selector (n) A A A A4 Mn Selector (p) Selector (n) Mn Selector (p) Selector (n) Giin Mn4 Selector 4 (p) B B B B4 Selector 4 (n) G4iin Mn5 VSS G G G G4 iout Figure 4. Loy integrtor with progrmmble gin The block decribed elector (n) nd (p) re demontrted in Figure 4.4 nd they hve been implemented by nmos nd pmos trnitor repectively. Thee 45

46 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror cell re being controlled by bit, in order to enble the correponding witch between G, B i nd G, A i repectively. G Mn Bi V SS Mn V SS Mni Mpi c i V DD V DD c i Mpi Mni V DD Mp V SS G Mp Ai () (b) Figure 4.4 Reliztion of elector in Figure 4. () Selector (n), (b) Selector (p) For elector (n), depending on the vlue of the control bit, the input voltge would go through the output voltge or not nd the output voltge would reet, getting the vlue of V (or it would get very low vlue in prctice nerly to zero). More pecificlly, if c = 0, the voltge t node B i would be zero nd the repective brnch i would be dibled. If c i =, the voltge t node B i would be equl to the voltge t node G nd the repective brnch would be ctivted. Repectively, for elector (p), depending on the vlue of the control bit, the input voltge would go through the output voltge or not nd the output voltge would get the vlue of V DD. More pecificlly, if c = 0, the voltge t node A i would be equl to V DD nd the repective i brnch would be dibled. If c i =, the voltge t node A i would be equl to the voltge t node G nd the repective brnch would be ctivted. Thu, the forementioned deign i plying the role of witch, which i on or off, depending on the vlue of the control bit. V DD nd V SS re the power upplie of the circuit. Introducing the ymbol of the CDN in the forementioned generl topology of the current-mirror, n electronic djutment of the bi current nd o the frequency in prllel with n electronic djutment of the gin of the circuitry re chieved. Thi topology i demontrted in Figure

47 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror no n IREF CDN bit VDD Io Mnb Mnb Mpo Mp Mp Mp Mp4 Mp5 iin Mn G G Selector (p) co Giin c c c co c Giin c c Selector (n) A A A A4 Mn Selector (p) Selector (n) Mn Selector (p) Selector (n) Giin Mn4 Selector 4 (p) B B B B4 Selector 4 (n) G4iin Mn5 VSS G G G G4 iout Figure 4.5 Current-mirror with bi nd gin progrmmbility 47

48 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror A more generl topology of the forementioned current-mirror, which i, finlly, the concept of the following imultion, i demontrted in Figure 4.6, which pplie, generlly, for gin G i (i:the number of the deirble gin), being controlled by the repective bit c i. c o c c c V DD I REF Mpo Mp Mp n o n n n CDN bit I o i in i out Mnb Mnb Mn Mn V SS G i Figure 4.6 Abtrct nottion of current-mirror with bi nd gin progrmmbility Bit n i (i=0,,, ) re the control bit of the CDN which djut the bi current tht feed the circuitry. Bit c j (j=0,,, ) re the control bit of the witche. All thee bit could be controlled by only 4 bit, i.e. b o, b, b nd b, the combintion of which would define pecific tte of the current-mirror concerning the deirble frequency nd gin. 4.4 Simultion Reult The imultion of fully progrmmble current-mirror (CM) filter i performed in thi prgrph. The deirble gin re G =0.5, G =, G = nd G 4 =. Alo, CDN of bit i ued nd the input current I REF i ma. The CDN i reponible for the bi current nd o for the frequency djutment of the circuit nd in thi ppliction the deirble frequencie tht would be preented re 5MHz, 50MHz nd 70MHz. An importnt iue, for extrcting the deirble tte of frequency nd gin, i the pect rtio of the trnitor of the forementioned topologie. Strting from the topology in Figure 4.5, the pect rtio of the trnitor M nb, M nb, M n, M po nd M p i 50μm/4μm. The pect rtio of nmos nd pmos trnitor tht relize the current-mirror of the CDN in Figure 4. re 00μm/6μm nd 00μm/6μm repectively. The witching trnitor of CDN hve pect rtio equl to0μm/μm, while the inverter hve been relized uing nmos nd pmos trnitor with pect rtio μm/0.5μm nd μm/0.5μm repectively. The pect rtio of the nmos 48

49 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror trnitor of the witche in Figure 4.4 () i 0μ/0.4μm nd the pect rtio of the pmos trnitor of the witche in Figure 4.4 (b) i 0μm/0.4μm. Alo, the width of the output trnitor in Figure 4.5 i depended on the deirble gin. Thu, the width of the trnitor M n nd M p, which correpond to gin equl to 0.5, i = 5µm. The width of the trnitor M n nd M p, which correpond to gin equl to, i 50 = 50µ m, the width of the trnitor M n4 nd M p4, which correpond to gin equl to, i 50 = 00µ m nd the width of the trnitor M n6 nd M p6, which correpond to gin equl to, i 50 = 50µ m. The length of ll thee trnitor i μm. The power upplie of the topology re V DD =. V ndv SS = 0V Only 4 bit, i.e. b o, b, b nd b, could mnge the control bit of the CDN well the control bit of the witche, thu thee 4 bit control the whole ytem, without chnging the topology. Depending of the combintion of thee bit, bit trem of n i (i=0,,, ) i formed for the pproprite bi current of the CDN, which control the frequency, nd bit trem of c j (j=0,,, ) i formed for the ctivtion of the pproprite brnche of the circuitry. More pecificlly, the firt bit, b o nd b, control the frequency nd the lt bit, b nd b, control the gin nd thi fct i demontrted in Tble. Frequency (MHz) Tble Bit pttern for controlling the CM filter Gin CM Control bit (b o b b b ) (n o n ) Bi current (CDN) Control bit Current Vlue (μa) Χ Χ 00 X X Χ Χ 0 X X Χ Χ 0 X X Χ Χ X X 49

50 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror The lt line with X refer to tte tht would not be ued, o the vlue of bit jut exit with no influence. The firt bit b o nd b control the frequency, thu the firt 4 line refer to frequency equl to 70MHz, the next 4 line refer to 50MHz nd the lt 4 line refer to 5MHz. The lt bit b nd b control the gin, thu the t, 5 th nd 9 th line correpond to gin equl to 0.5, the nd, 6 th, nd 0 th line correpond to gin, the rd, 7 th nd th line refer to gin nd the 4 th,8 th, nd th line correpond to gin equl to. The digitl vlue 0 nd correpond to voltge 0V nd.v repectively. The column with the control bit of the bi current conit of bit (n o n ) tht control the CDN nd produce the pproprite bi current. Ech bit of thi combintion rie through pproprite digitl gte which ue input the forementioned bit b i (i=0,,, ) in order to chnge the vlue of thee bit directly nd utomticlly nd not mnully. The pproprite digitl gte re elected through the following rnugh mp, which ue input the initil bit b i nd output ech bit of the bi current of the CDN. Thu, expreion rie from rnugh mp tht need to be pplied nd correpond to everl combintion of digitl gte. It i reminded tht on rnugh mp of 4 input vrible, the lt line re for the firt vrible, o the firt bit b o, the intermedite line re for the econd vrible, o the econd bit b, the lt column re for the third vrible, o the third bit b nd the intermedite column re for the fourth vrible, o the fourth bit b. For the firt bit of the bi current n o, the following rnugh mp i derived nd ' ' it expreion i given by: no = bobb b, following by the repective topology of digitl gte. The me procedure i following for ll the bit of the bi current nd the repective mp with their expreion nd the pproprite digitl gte re demontrted in Tble. Tble rnugh mp nd digitl gte for the controlling bit of CDN ' ' n o = b o bb b X X X X b o b b b AND n o 50

51 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror ' ' n = bb b bobb X X X X b b b AND OR n b o AND n = 0 ' ' ' ' n = bo bb bb b bobb b X X X X b b b o AND b AND OR n AND 5

52 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror n 4 = b b ' ' ' ' ' ' o bb b bb b bobb b X X X X 0 0 b o b AND b AND b n 4 OR AND AND ' ' n 5 = b b b ' ' ' ' ' o bb b bobb bb b X X X X 0 0 b b b ' o 5

53 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror b o b b AND b AND AND OR n 5 AND AND ' ' n 6 = b b b ' ' ' o bb bobb b b b b ' o ' ' X X X X b b AND b o AND b AND OR n 6 AND 5

54 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror ' ' n 7 = bb bo b bob X X X X 0 b b AND b o AND OR n 7 AND ' ' ' ' ' n 8 = bo b bobb bobb X X X X 0 0 b o b AND b AND OR n 8 b AND 54

55 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror n 9 = ' b o b b ' ' X X X X b o b AND n 9 b n 0 = b ' b X X X X b AND b n 0 n = 0 n = b b ' b X X X X

56 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror b b AND b n In order to control the elector demontrted in Figure 4.4 () nd (b), 4 expreion of bit re required inted of b o tht exit in thi Figure, for electing the pproprite gin, becue of the exitence of 4 elector (n) nd (p) in Figure 4.5. A b nd b re reponible for chnging the gin of the topology, n expreion with only thee bit could be pplied for controlling the forementioned witche. Thu, the necery expreion re following: c o = c = ' ' b b ' bb c = ' bb c = bb It i eily undertndble tht the pproprite gte tht need to be ued re AND gte of two input, deigned the forementioned Figure, which follow the rnugh mp. Conequently, thee 4 bit, b i (i=0,,, ), re reponible to control the opertion of the whole propoed ytem. The vlidity of thi concept i proved through the next Figure, in which the derived frequency repone i demontrted for ll the poible tte. Conequently, in Figure 4.7, the derived frequency repone, with cut-off frequency t 5MHz, i demontrted for ll the deirble gin. Thee vlue extrcted from the forementioned experimentl meurement compred with the repective theoreticl one re demontrted in Tble. 56

57 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror Figure 4.7 Frequency repone (f o =5MHz) for gin 0.5,, nd Tble Reult of the cutoff frequency nd the gin (f o =5MHz) REAL IDEAL Cutoff Cutoff frequency Gin frequency Gin (MHz) (MHz) In Figure 4.8, the me procedure i repeted for cut-off frequency t 50MHz nd the derived reult re hown in Tble 4. 57

58 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror Figure 4.8 Frequency repone (f o =50MHz) for gin 0.5,, nd Tble 4 Reult of the cutoff frequency nd the gin (f o =50MHz) REAL IDEAL Cutoff Cutoff frequency Gin frequency Gin (MHz) (MHz) In Figure 4.9 nd Tble 5, the frequency repone for 70MHz nd the repective reult for the cutoff frequency nd the gin re demontrted repectively. 58

59 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror Figure 4.9 Frequency repone (f o =70MHz) for gin 0.5,, nd Tble 5 Reult of the cutoff frequency nd the gin (f o =70MHz) REAL IDEAL Cutoff Cutoff frequency Gin frequency Gin (MHz) (MHz) In the following Figure , n electronic tuning of the cutoff frequency i demontrted for gin 0.5,, nd repectively. 59

60 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror Figure 4.0 Electronic tuning of the cutoff frequency (gin=0.5) Figure 4. Electronic tuning of the cutoff frequency (gin=) 60

61 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror Figure 4. Electronic tuning of the cutoff frequency (gin=) Figure 4. Electronic tuning of the cutoff frequency (gin=) A Monte Crlo nlyi h been performed (f o =5MHz, gin=0.5) nd it reult re demontrted in Figure 4.4 nd 4.5 repectively. The tndrd devition of the bndwidth h been.0mhz nd the men vlue h been 6.8MHz. The tndrd devition of the gin h been nd the men vlue h been

62 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror Figure 4.4 Sttiticl plot bout bndwidth (f o =5MHz, gin=0.5) Figure 4.5 Sttiticl plot bout gin (f o =5MHz, gin=0.5) The reult for f o =5MHz nd gin= re demontrted in Figure 4.6 nd 4.7 repectively. The tndrd devition of the bndwidth h been.0mhz nd the men vlue h been 5.98MHz. The tndrd devition of the gin h been 0.00 nd the men vlue h been

63 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror Figure 4.6 Sttiticl plot bout bndwidth (f o =5MHz, gin=) Figure 4.7 Sttiticl plot bout gin (f o =5MHz, gin=) Furthermore, the reult for f o =5MHz nd gin= re demontrted in Figure 4.8 nd 4.9 repectively. The tndrd devition of the bndwidth h been.mhz nd the men vlue h been 6.66MHz. The tndrd devition of the gin h been nd the men vlue h been

64 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror Figure 4.8 Sttiticl plot bout bndwidth (f o =5MHz, gin=) Figure 4.9 Sttiticl plot bout gin (f o =5MHz, gin=) Finlly, the reult for f o =5MHz nd gin= re demontrted in Figure 4.0 nd 4. repectively. The tndrd devition of the bndwidth h been.9mhz nd the men vlue h been 6.85MHz. The tndrd devition of the gin h been nd the men vlue h been

65 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror Figure 4.0 Sttiticl plot bout bndwidth (f o =5MHz, gin=) Figure 4. Sttiticl plot bout gin (f o =5MHz, gin=) 65

66 CHAPTER 4-Progrmmbility of Integrtor with Current-Mirror 4.5 Concluion The concept of digitl progrmmbility pplied on integrtor with current mirror h been preented. It ignificnce i lrge becue of the flexibility nd the convenience tht re offered to the deigner, which ue only number of bit tht re combined nd fcilitte the opertion of the circuit. CDN nd elector re prt of the forementioned digitl logic, offering comptibility, preciion nd ey control of the whole propoed deign. Thi concept i going to be ued in more generl topology, which would be preented in the next Chpter. 66

67 5 Progrmmble Generlized Frctionl-Order Filter 5. Introduction In previou Chpter, frctionl-order () generlized filter h been preented through it eqution well the differentil technique of deign with current-mirror. Alo, the progrmmbility of integrtor with current-mirror h been nlyzed in combintion with the principl digitl logic. Combining thee prt into generl propoed topology, it i feible to deign generlized frctionl-order filter, which would be controlled through n pproprite digitl logic correponded to pecific number of bit. The vlidity of thi concept would be confirmed through imultion reult for ll the type of the generlized filter. 67

68 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter 5. Progrmmbility of the propoed ytem The bic purpoe of thi Chpter i to preent generlized frctionl-order filter, which contitute the nlog core, digitlly progrmmble by pecific number of bit, it i demontrted in Figure 5.. b ο b b b V DD I REF i IN Digitl Logic Anlog Core i OUT V SS Figure 5. Functionl block digrm of the propoed ytem Bit b i (i=0,,, ) re reponible for the control of the whole ytem. The digitl logic, which conit of vriou digitl gte, CDN nd elector in Chpter 4, ue thee bit input in order to djut the type of the filter, i.e. lowp, highp, bndp, bndtop, well the order () of the filter. All the poible tte tht re being chieved with thoe 4 bit re hown in Tble. 68

69 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter Tble Bit pttern for controlling the whole ytem Type of filter L o w p H i g h p B n d p B n d t o p X Χ Χ X Order () of filter Sytem Control bit b o b b b X 0 0 X 0 X 0 X The lt line with X refer to tte tht would not be ued, o the vlue of bit jut exit with no influence. 69

70 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter 5. Propoed topology The Functionl Block Digrm (FBD) of the propoed topology h been demontrted in Figure. nd it relized trnfer function h been preented in (.7.). The implementtion of thi topology through current-mirror i hown in the following Figure 5.. Τhe input nd the output of the topology re referred i in nd i out repectively. It i importnt to mention tht ource, i in nd i in-, with hlf of the mplitude re ued input, with the econd one hving phe difference 80 o, becue of the differentil technique tht i propoed. Alo, the current output i the ubtrction of the current i out nd i out- of the bic topology. 70

71 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter Giin G Giin- H VDD Io GIo GIo Io Io Io GIo GIo Io Io Io Io Io GoIo GIo iin Giout Giout C D Giout Giout B E F A Goiout Giout A G H H G G H Mn Mn Mn Mn4 Mn5 Mn6 Mn7 Mn8 Mn9 Mn0 Mn Mn Mn Mn4 Mn5 VSS G G G G Go G G VDD H Io GIo GIo Io Io Io GIo GIo Io Io Io Io Io GoIo GIo iin- Giout Giout D C Giout Giout A F E B Goiout Giout B H G G H H G Mn6 Mn7 Mn8 Mn9 Mn0 Mn Mn Mn Mn4 Mn5 Mn6 Mn7 Mn8 Mn9 Mn0 G G G G Go G VSS iout iout- Figure 5. Propoed topology of the nlog core 7

72 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter It i hown tht different bi current hve been ued for ech integrtor in order to obtin more precie control of the ytem. The implementtion h been performed uing current-mirror with pproprite nmos nd pmos trnitor rel ource, like in Chpter, which feed ech integrtor with the pproprite current. Thi concept would be hown in following more generl Figure. Attention hould be given to the brnche tht provide gin. The propoed topology conit of integrtor nd it i bed on the differentil technique, which h been nlyzed, lo, in Chpter. Thu, the output would be ubtrcted, finlly, nd only one output would be tudied. The firt integrtor conit of 4 trnitor nd it time contnt, bed on the concept nd formul preented in Chpter, i given by the formul: m ( G G ) Ceq Cg τ = = (5..) g g, where C g nd g m re the cpcitnce nd the mll-ignl trnconductnce of the trnitor M n repectively nd they re given by: Cg = ( W L) Cox W Lov Cox W nd g m = n I o. L The econd integrtor conit of 6 trnitor nd, with the me concept, it time contnt i given by the formul: m m ( 4 G G ) Ceq Cg τ = = (5..b) g g, where C g nd g m re the cpcitnce nd the mll-ignl trnconductnce of the trnitor M n5 repectively nd they re given by: W Cg = ( W L) Cox W Lov Cox nd g m = n I o. L The third trnitor conit of 5 trnitor nd it time contnt i given by the formul: m m ( G G ) Ceq o Cg τ = = (5..c) g g, where C g nd g m re the cpcitnce nd the mll-ignl trnconductnce of the trnitor M n repectively nd they re given by: W Cg = ( W L) Cox W Lov Cox nd g m = n I o. L m 7

73 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter 5. Integrtor with progrmmble brnche The forementioned topology preent generlized frctionl-order filter, thu it hould be cpble of relizing lowp, highp, bndp nd bndtop filter, without chnging the topology. In prllel, tudying the prgrph.7 with the pproprite deign eqution, it i derived tht gin G i (i=0,,, ) do not be ued ll together for ech type of filter, ome of them re being zero, depending the type of filter, nd chnge their vlue. Thi obtcle would be overcome through the progrmmble logic tht h been preented in Chpter 4. More pecificlly, gin G nd G re lwy in ue, but with different vlue for ech type nd order of the generlized filter. Gin G i ued only for the highp nd the bndtop tte of the filter where it vlue i nd gin G o i ued in ll the tte except for the highp tte. A mentioned, the generlized filter conit of integrtor, or more preciely 6 integrtor uing the differentil technique. Ech integrtor contin brnche with gin G i (i=0,,, ) nd unit gin. More pecificlly, the firt integrtor contin brnch of G, brnch of G, if it i ued, nd unit brnch. The econd integrtor contin brnch of G, brnch of G, if it i ued, nd three unit brnche. The third integrtor contin brnch of G o, if it i ued, brnch of G, lo if it i ued, nd two unit brnche nd the me concept i followed for the differentil couple with the other integrtor. Thu, ll integrtor hve imilr tructure nd they could be decribed by one topology or ymbol tht i demontrted in Figure 5., by jut dding or ubtrcting output. In thi Figure, G i, G i nd G i (i=0,, ) re the deirble gin of the firt, econd nd third integrtor repectively, n oj, n j,, n j (j=,, ) re the control bit of the CDN of ech integrtor, c o, c,, c re the control bit of G i (i=0,, ) nd w o, w re the control bit of G o nd G. 7

74 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter no n n n IREF CDN bit Io Mpo Mnb Mnb iin Mp p p Mp p p Mn n Mn n n n Gi VSS VDD Mp Mn Gi p p n n Mp4 Mn4 Gi p4 p4 Mp5 Giiout G iout n4 Mn5 n4 G Mp6 iout Mn6 () 74

75 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter CDN V DD n j n οj n j n j I REF i in Integrtor G i i out G i out i out c o c c c w o w G i G o, G (b) Figure 5. Integrtor with progrmmble brnche () circuitry (b) ymbol Thu, the propoed generl topology in Figure 5. through the integrtor ymbol in Figure 5. (b) i demontrted in Figure 5.4, where the multiple output of ech integrtor nd the pproprite node re hown. 75

76 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter Giin G Giin- H iin iin VDD nο n n n IREF A Integrtor co c c c wo w VDD nο n n n IREF B Integrtor co c c c wo w Giout Giout Giout Giout VDD nο n n n VDD nο n n n IREF IREF Giout Integrtor H G Giout G H Integrtor iout C iout D iout iout E B co c c c wo w co c c c wo w VDD nο n n n VDD nο n n n IREF IREF Giout Integrtor G H Giout H G Integrtor iout D iout C iout iout F A co c c c wo w co c c c wo w Goiout Giout iout iout G H Goiout Giout iout iout G H F A iout iout- H G E B Figure 5.4 Propoed topology with block of integrtor with progrmmble brnche 76

77 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter 5.4 Digitl logic Ech integrtor h different bi current which rie from the different control bit of ech CDN n ij (i=0,,,, j=,, ). Thee bit re controlled by bit b i (i=0,,, ), which mnge the control of the whole ytem. Depending of the combintion of thee bit, bit trem of n ij i formed for the pproprite bi current of ech CDN, mnging the frequency of the filter, bit trem of c j (j=0,,,) i formed for the ctivtion of the pproprite brnche of the circuitry concerning the gin G i (i=0,,) nd bit trem of w z (z=0,) i formed for the ctivtion of the pproprite brnche concerning the gin G o nd G. The control bit b i in combintion with the control bit of ech CDN, continution of Tble, re demontrted in Tble -4. Tble Bit pttern for controlling the bi current I o of the firt integrtor Sytem Control bit Bi current I o (CDN) Control bit (b o b b b ) (n o n ) Current Vlue (μa) X X 0 X X 0 X X X X 77

78 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter Tble Bit pttern for controlling the bi current I o of the econd integrtor Sytem Control bit Bi current I o (CDN) Control bit (b o b b b ) (n o n ) Current Vlue (μa) X X 0 X X 0 X X X X Tble 4 Bit pttern for controlling the bi current I o of the third integrtor Sytem Control bit Bi current I o (CDN) Control bit (b o b b b ) (n o n ) Current Vlue (μa) X X 0 X X 0 X X X X 78

79 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter The column with the control bit of the bi current conit of bit (n oj n j ) tht control the CDN nd produce the pproprite bi current. Ech bit of thi combintion rie through pproprite digitl gte which ue input the forementioned bit b i (i=0,,, ) in order to chnge the vlue of thee bit directly nd utomticlly nd not mnully. Thu, expreion rie from rnugh mp for ech CDN tht need to be pplied nd correpond to everl combintion of digitl gte. The concept nd the logic of thee mp hve been lredy mentioned in Chpter 4, nd, in Tble 5, the rnugh mp, with the pproprite digitl gte, for current I o re demontrted. Tble 5 rnugh mp nd digitl gte for the controlling bit of CDN of I o ' ' n o = b o bb b X X X X b o b b b AND n o ' ' ' ' n = bo bb bobb X X X X

80 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter b o b AND b OR n AND ' ' n = bo bb bob X X X X 0 0 b o b AND b OR n AND ' n = b b b o bb b b X X X X 0 0 ' o 80

81 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter b o b b AND b AND OR n AND ' n 4 = b b o b b b b ' o 0 0 X X X X ' b o b AND b OR n 4 AND b 8

82 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter ' n 5 = b o b b ' b X X X X 0 b o b AND OR n 5 b b AND n 6 = b b ' ' ' bb bo bb bobb X X X X b b AND b AND OR n 6 b o AND AND 8

83 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter ' ' ' ' ' ' n 7 = b bb b bb bb b o o X X X X b o b AND b b AND OR n 7 AND ' ' ' n 8 = bb bo bb bobb X X X X

84 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter b b AND b o AND OR n 8 b AND ' ' ' n 9 = b b b b ' o b b ' o X X X X 0 0 b o b AND b b OR n 9 AND n 0 = 0 n = 0 n = 0 Repective rnugh mp for the ret of the forementioned current re derived with the me logic ccording to the previou Tble. Thee mp with the 84

85 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter pproprite expreion of the controlling bit re hown in the end of thi thei in Appendix, in Tble nd. Another importnt iue i the control of G i (i=0,, ). Thee gin chnge their vlue depending on the type nd the order of the filter. In Figure 4.5, digitl witching ytem with 4 deirble gin h been preented through elector tht hve been lredy decribed. In thi Chpter, it i demontrted in Tble, poible tte could be implemented, thu control bit c i (i=0,,, ) nd elector hould be ued in order to mnge thee tte. Thi fct h, lo, been decribed in the preenttion of progrmmble integrtor in Figure 5. (b). The Figure of the elector i imply n expnion of Figure 4.5 with elector now. The expreion of thee bit re the me for ll the integrtor nd they control imultneouly the gin G i (i=0,, ) in prllel with the deirble tte. Uing the initil bic control bit b i (i=0,,, ) input nd the control bit c i (i=0,,, ) output, the following Tble i derived. The pproprite expreion for the control of thee bit re derived through repective rnugh mp hown in Appendix in Tble nd the contruction of the repective digitl gte i implemented with imilr wy to the forementioned propoed one. Tble 6 Bit pttern for controlling the gin G i (i=0,, ) Sytem Control bit Gin G i (i=0,, ) Control bit (b o b b b ) (c o c ) X 0 X 0 X X 85

86 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter Furthermore, gin G o nd G re not ued in ll the type of the generlized filter, fct tht i demontrted in Figure 5. (), dditionl bit w o nd w i hould be ued in order to control the ue of thee gin. G o i 0 in highp filter nd it h different vlue in the other type of the filter, depending on the type nd the order. G i 0 in lowp nd bndp filter nd it h pecific vlue equl to in the other type of the filter. Thu, uing the initil bic control bit b i (i=0,,, ) input nd the control bit w o nd w i output, Tble 7 i derived. The pproprite expreion re derived through rnugh mp, which re demontrted in Appendix in Tble 4, where the pproprite digitl gte re contructed with the forementioned wy. Tble 7 Bit pttern for controlling the gin G o nd G Sytem Control bit Gin G o nd G Control bit (b o b b b ) (w o w ) X 0 X 0 X X Thu, generl topology of ech integrtor, tht h been demontrted in Figure 5., through the forementioned concept with the control bit, i hown in Figure 5.5, which pplie, generlly, for gin G ij (i=0,, nd j=the number of the integrtor, i.e.,, ), being controlled by the repective bit. All thee bit in Figure 5.5 re being controlled by only 4 bit b i (i=0,,, ). 86

87 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter noj nj nj nj IREF CDN bit VDD Io Mpo Mnb Mnb VSS iin co c c Mp Mp Mn Mn Gij Gijiout wo w Mp Mn Go,G Go,iout Mp4 Mn4 iout Figure 5.5 Abtrct nottion of n integrtor with progrmmble brnche 87

88 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter 5.5 Simultion Reult The next nd finl tep i teting the topology in Figure 5.4 with the pproprite digitl gte for controlling the CDN nd the pproprite gin tht hve been nlyzed. The imultion tht would be implemented would include order: = 0., = 0.5 nd = 0. 8 for the lowp nd the highp filter nd = 0. 5, = 0. 7 nd = 0.9 for the bndp nd the bndtop filter, it h been demontrted in Tble. The pecifiction of thi generl filter i hving cut-off frequency t 50MHz for the lowp nd the highp filter nd hving pek frequency t 50MHz for the bndp nd the bndtop filter. In Tble 8, the different vlue of G i (i=0,,, ) for ech type of filter re demontrted. 88

89 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter filter Tble 8 Vlue of G i (i=0,,, ) depending on the type nd the order of Type of filter L o w p H i g h p B n d p B n d t o p Order () of filter Sytem Control bit Gin G i (i=0,,, ) b o b b b G G G G o X X 0 0 Χ Χ Χ Χ X X 0 Χ Χ Χ Χ X X 0 Χ Χ Χ Χ X X Χ Χ Χ Χ In order to derive the pproprite reult, n importnt iue i the dimenion of the trnitor tht hould be determined for ech topology. In Figure 5. (), the pect rtio of the trnitor M nb nd M nb for the firt integrtor i 5μm/0.4μm, for the econd integrtor i μm/0.4μm nd for the third integrtor i 0.5μm/0.4μm. In the me Figure, for ech integrtor, the pect rtio of nmos trnitor M n, M n5 nd M n6 nd pmos trnitor M po, M p, M p5 nd M p6 i 0μm/0.4μm. The pect rtio of 89

90 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter the nmos trnitor M n, M n nd M n4 nd pmos trnitor M p, M p nd M p4 depend on the repective gin of the brnch nd o the deirble tte of the generlized filter. The length of ll thee trnitor i 4μm. Trnitor M n nd M p re ued for the firt integrtor, thu they belong to the brnch with gin G. So, their width i G *0μm depending on the vlue of G from Tble 8. Following thi concept, Tble 9 i derived. Tble 9 Width of trnitor M n, M p of the brnch with G (length=4μm) Type of filter L o w p H i g h p B n d p B n d t o p X X X X Order () of filter Sytem Control bit b o b b b Gin G Width of trnitor M n, M p (μm) X 0 0 Χ X X 0 Χ X X 0 Χ X X Χ X 90

91 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter Trnitor M n nd M p re ued for the econd integrtor, thu they belong to the brnch with gin G. So, their width i G *0μm depending on the vlue of G from Tble 8. Following thi concept, Tble 0 i derived. Tble 0 Width of trnitor M n, M p of the brnch with G (length=4μm) Type of filter L o w p H i g h p B n d p B n d t o p X X X X Order () of filter Sytem Control bit b o b b b Gin G Width of trnitor M n, M p (μm) X 0 0 Χ X X 0 Χ X X 0 Χ X X Χ X 9

92 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter Following the me procedure for the brnch with gin G o, trnitor M n4 nd M p4 re ued for the third integrtor. Thu, their width i G o *0μm depending on the vlue of G o from Tble 8 nd o Tble i derived. Tble Width of trnitor M n 4, M p 4 of the brnch with G o (length=4μm) Type of filter L o w p H i g h p B n d p B n d t o p X X X X Order () of filter Sytem Control bit b o b b b Gin G o Width of trnitor M n4, M p4 (μm) X 0 0 Χ X X 0 Χ X X 0 Χ X X Χ X 9

93 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter The pect rtio nd the tructure of the trnitor of the elector nd the CDN re the me with the repective one in Chpter 4. The power upplie of the topology re V DD =. V ndv SS = 0V nd the digitl vlue 0 nd correpond to voltge 0V nd.v repectively. The bi current I REF i ma. The lyout deign of the nlog core of the generlized filter i hown in Figure 5.6, where the dimenion re 860μm x 05μm. Figure 5.6 Lyout deign of the generlized filter 9

94 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter The derived frequency repone for the lowp filter i demontrted in the following Figure 5.7. Figure 5.7 Frequency repone of lowp filter for =0., =0.5 nd =0.8 The repective reult for the bndwidth nd the lope for ech order re hown in Tble. Thee reult (rel) re compred in prllel with the theoreticl reult (idel). It i reminded tht the lope of lowp frctionl filter i clculting from the formul: -6*n db/octve, where n i the order of the filter (n=α). Tble Bndwidth nd lope for lowp filter Type of filter L o w p Order () of filter 0. Bndwidth (MHz) Slope (db/oct) rel idel rel idel Concerning the highp filter, in Figure 5.8, the derived frequency repone i demontrted. Alo, in the next Tble, the reult for the bndwidth nd the lope 94

95 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter re preented compred with the repective theoreticl one. It i reminded tht the lope of highp frctionl filter i clculting from the formul: 6*n db/octve, where n i the order of the filter (n=α). Figure 5.8 Frequency repone of highp filter for =0., =0.5 nd =0.8 Tble Bndwidth nd lope for highp filter Type of filter H i g h p Order () of filter Bndwidth (MHz) Slope (db/oct) rel idel rel idel In Figure 5.9, the derived frequency repone for the bndp filter i demontrted. The repective reult for the pek frequency, it gin nd the lope for ech order compred with the theoreticl re hown in Tble 4. It i reminded tht the lope t low frequencie of bndp frctionl filter i clculting from the formul: 6*α db/octve, nd the lope t high frequencie i tble t -6 db/octve. 95

96 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter Figure 5.9 Frequency repone of bndp filter for =0.5, =0.7 nd =0.9 Tble 4 Pek frequency, gin nd lope for bndp filter Type of filter B n d p Order () of filter Freq_pek (MHz) freq_pek (db) Slope_low (db/oct) Slope_high (db/oct) rel idel rel idel rel idel rel idel Furthermore, the lt type of filter tht would be teted i the bndtop filter. In Figure 5.0, the derived frequency repone for ll the exmined order i demontrted. The derived reult for the pek frequency nd it gin compred with the repective theoreticl reult re hown in the next Tble 5. 96

97 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter Figure 5.0 Frequency repone of bndtop filter for =0.5, =0.7 nd =0.9 Tble 5 Pek frequency nd gin for bndtop filter Type of filter B n d t o p Order () of filter Freq_pek (MHz) freq_pek (db) rel idel rel idel A common order of ll thoe forementioned type of filter i = Thu, it would be ueful to tke ll the frequency repone together for thi order in order to hve n overll exmintion of the generl filter. Thi fct i demontrted in the following Figure

98 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter Figure 5. Frequency repone of lowp, highp, bndp nd bndtop filter for =0.5 A Monte Crlo nlyi h been performed for the lowp filter (f o =50MHz, α=0.5) nd it reult re demontrted in Figure The tndrd devition of the bndwidth h been.45mhz nd the men vlue h been 44.MHz. Alo, the tndrd devition of the gin h been 0.06 nd the men vlue h been.0. Finlly, the tndrd devition of the lope h been 0.8dB/oct nd the men vlue h been -9.dB/oct. Figure 5. Sttiticl plot bout bndwidth for the lowp filter (=0.5) 98

99 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter Figure 5. Sttiticl plot bout gin for the lowp filter (=0.5) Figure 5.4 Sttiticl plot bout lope for the lowp filter (=0.5) The me nlyi h been, lo, performed for the bndp filter (f pek =50MHz, α=0.7). The reult re hown in Figure Τhe tndrd devition of the bndwidth h been.mhz nd the men vlue h been 5.7MHz. The tndrd devition of the gin h been 0.8dB nd the men vlue h been -.05dB. 99

100 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter Figure 5.5 Sttiticl plot bout bndwidth for the bndp filter (=0.7) Figure 5.6 Sttiticl plot bout gin for the bndp filter (=0.7) Furthermore, the tndrd devition of the lope t low frequencie h been 0.44dB/oct nd t high frequencie it h been 0.4dB/oct. Concerning the men vlue, the firt one h been 4.6dB/oct nd the econd one h been -4.9dB/oct. 00

101 CHAPTER 5-Progrmmble Generlized Frctionl-Order Filter Figure 5.7 Sttiticl plot bout lope t low frequencie for the bndp filter (=0.7) Figure 5.8 Sttiticl plot bout lope t high frequencie for the bndp filter (=0.7) The lt ttiticl reult for thi filter re bout the pek frequency. The tndrd devition h been.5mhz nd the men vlue h been 4.MHz. 0