A FINITE-MEMORY DISCRETE-TIME CONVOLUTION APPROACH FOR THE NON-LINEAR DYNAMIC MODELLING OF S/H-ADC DEVICES

Transcription

1 FINITE-MEMORY DISCRETE-TIME CONVOLUTION PPROCH FOR THE NON-LINER DYNMIC MODELLING OF S/H-DC DEVICES D. Mii, G. Pasini, P.. Taveso 2, F. Filicoi 2, G. Iclano 3 Depaen of Elecical Engineeing, Univesiy of ologna, Viale Risogieno 2, 436 ologna, Ialy Phone Fax e-ail: doenico.ii@ail.ing.nibo.i 2 Depaen of Eleconics, Cope Science and Syses, Univesiy of ologna 3 Depaen of Eleconics, Univesiy of Fienze, Via S. Maa 2, 525 Fienze, Ialy bsac - non-linea dynaic black-box odel fo Saple-Hold and nalog o Digial Convesion devices S/H- DC is poposed in his pape. I deives fo he expansion of a Volea-like seies peviosly inodced by ahos and i is based on a discee convolion in he ie doain which descibes he non-linea dynaics of he syse. The odel allows a fncional descipion of he S/H- DC non-linea behavio as he s of wo blocks which ake sepaaely ino accon he saic and he pelydynaic conibions. The odel chaaceizaion can be caied o by eans of a siple and eliable easeen pocede and he odel analyical epesenaion allows an easy ipleenaion in he faewok of coecial available CD ools fo cici analysis and design. Peliinay expeienal esls which validae he poposed appoach ae pesened in he pape, ogehe wih soe exaples which show odel capabiliies in pedicing device behavio nde non-linea opeaing condiions. Keywods - S/H-DC device, non-linea dynaic odel, Volea seies.. INTRODUCTION Chaaceizaion and odelling of S/H-DC devices ae of basic ipoance in he pefoance analysis of easeen insens based on digial signal pocessing. In he ai of idenify siable pocedes fo he coecion of he device non-idealiies, an high accacy in he pedicion of device behavio nde lage-signal opeaing condiions is needed: his can be obained only if he odel akes ino accon no only he non-lineaiies associaed wih he S/H- DC saic chaaceisic, b also hose which deal wih device dynaics. In ode o chaaceize he non-linea dynaic effecs i has been peviosly shown ha a S/H-DC device wih inp signal s I can be descibed hogh he black-box fncional odel of Fig. [,2]: he acal device is odelled as an ideal insen sapling a signal which is he esl of he inp/op elaionship of a non-linea dynaic syse non-linea syse wih eoy. The non-linea block is hen descibed by he cascade of a pely-linea dynaic syse wih a vially non-finie eoy and of a non-linea one associaed wih finie, elaively sho eoy effecs Fig.2. The fis block of sch a cascade is siply chaaceized by eans of is plse esponse in he ie doain, while he lae can be convenienly odelled by he ncaion o he fis ode inegal of a odified Volea seies [3], peviosly poposed fo he chaaceizaion of non-linea dynaic syses nde he hypohesis of sho daion of nonlinea dynaic effecs. Moe pecisely, he op signal of he non-linea syse in Fig.2 can be fhe descibed as he s of he ops of a eoyless non-linea block and a pely-dynaic non-linea one, if he daion of is eoy can be consideed sho wih espec o he ypical ini peiod of he inp signal s. This hypohesis epesens he only ipoan consain eqied on he behavio of a non-linea dynaic syse in ode o sccessflly apply o i he appoach poposed in [3], and i can be consideed saisfied in he case of a S/H-DC device. In fac, he ideal behavio fo his faily of syses is chaaceized by pely-linea dynaic effecs which ae aken ino accon by he fis block of he cascade in Fig.2, while he nonlinea dynaics epesen an eo soce de o he pesence, in he syse, of acive devices, which ae sally descibed by fas dynaics and can be associaed wih sho eoy. s I NON-LINER MEMORY S/H [ k] -bi DC q [ k] Fig. - Fncional odel fo he chaaceizaion of non-linea dynaic effecs in he cascade of a S/H cici and an -bi DC S/H-DC device.

2 s I LINER MEMORY s NON-LINER SHORT MEMORY S/H-DC DEVICE q [] k Fig.2 - Fncional odel of he S/H-DC device poining o he sepaaion of he non-linea syse wih eoy of Fig. ino he cascade of wo blocks, aking ino accon especively he linea and non-linea dynaics. PURELY-DYNMIC NON-LINER SYSTEM WITH SHORT MEMORY s s I LINER SYSTEM WITH MEMORY S/H-DC DEVICE q [] k NON-LINER MEMORYLESS SYSTEM S Fig.3 - Final fncional odel fo he S/H-DC device, showing he pely-linea block wih non-finie eoy, he eoyless and pely-dynaic wih sho eoy non-linea blocks. The final S/H-DC epesenaion is heefoe shown in Fig.3. non-linea dynaic odel, siable fo he S/H- DC chaaceizaion in he feqency doain, has been peviosly poposed by ahos, saing fo his fncional descipion [2]. In his pape, an accae S/H-DC ie doain-oiened odel, easily idenifiable by eans of convenional easeen pocedes and eliable neical algoihs, will be pesened. 2. THE S/H-DC DISCRETE-TIME CONVOLUTION MODEL The pely-linea dynaic block a he inp of he fncional sce in Fig.2 can be siply chaaceized by eans of he convenional convolion inegal of is plse esponse h wih espec o he signal s I, exended ove a vially non-finie eoy ineval: h s s I d ccoding o he odified Volea seies appoach poposed in [3] fo he odelling of non-linea dynaic syses wih sho eoy, he op of he cascaded non-linea block in Fig. 2 can be descibed as he s of wo conibions, epesened especively by he op S D of a nonlinea eoyless block and he op of a pelydynaic non-linea block Fig. 3: S + 2 In 2, he fis conibion i.e. he op of he non-linea saic block can be wien as a powe seies in he signal s : S [ ] + a z s y s 3! D s fa as he conibion in 2 a he op of he non-linea block is concened, i can be descibed by he following expession: + T D w[ s ] e, T, d 4 whee he dynaic deviaion fncion e, s s 5 epesens he diffeence beween pas vales s of he inp signal of he non-linea syse wih sho eoy shown in Fig.2 wih espec o he insananeos vale s. Soe ipoan consideaions can be ade abo he analyical epesenaion of 3 and 4. Recalling he classical Volea descipion fo a geneic non-linea dynaic syse:

3 h,, s i d i 6! i i is possible o deonsae ha he coefficiens a,2,... of he powe seies 3 can be expessed by eans of he Volea kenels: d i,, a h i 7 Moeove, also he weigh fncion w in 4, which epesens he fis ode kenel of he odified seies [3], can be deived fo kenels h : w b s 8! [ ] s, + h +, 2,, + d s s 2 b 9 Eqaion 4 epesens a fis ode convolion inegal wih espec o he dynaic deviaion fncion e,, nonlinealy conolled by he signal s, and can adeqaely chaaceize he S/H-DC non-linea dynaic effecs if he eoy ineval [ T, ] T daion T M of he non-linea block in Fig.2 is sho wih espec o he ypical ini peiod of s i.e. he peiod of he S/H-DC inp signal s I. In sch condiions, in fac, he dynaic deviaion asses vales ha ae sall even in he pesence of lage flcaions of he signal s. Ths sing expession 2, which is he lineaizaion i.e. he ncaion o he fis ode inegal of he odified Volea seies wih espec o he dynaic deviaion e,, does no inodce elevan eos in he descipion of he non-linea block behavio, since all ohe inegal conibions in he seies ae highe ode infiniesial of e, [3]. y disceising now he inegaion doain of 4 i.e he non-linea eoy ie ino he saion of P inevals of eqal widh, he sae eqaion can be ewien as: w[ s, p ] e, p wp[ s ] e, p p P p p P p whee P T and P T. Eqaion epesens he discee-ie, non-linealy conolled convolion which akes ino accon he non-linea dynaics of he S/H-DC device. 3. MODEL IDENTIFICTION The feasibiliy of he exacion of behavioal odels, sch as hose based on he classical Volea seies, fo nonlinea dynaic syses is songly liied by he difficlies ha ae elaed o he idenificaion of all he kenels which ae needed in ode o achieve a good pedicion accacy. In any cases only a pa of hese kenels can be pacically chaaceized, and he odel folaion sffes fo a song appoxiaion ha edces is pefoances. The poposed appoach ovecoes sch a liiaion, being is fll idenificaion based on a siple and eliable expeienal pocede. I is ipoan o noice ha, being possible o ipose, wiho loss of genealiy, H j o he ansfe fncion of he linea dynaic syse, he fncion z [] coincides wih he saic chaaceisic of he enie device. The saic block of Fig.3 is hs easily idenifiable by eans of convenional DC easeens ove a siable ineval of inp signal vales. Moeove, i has been shown [2] ha in zeo bias, sall-signal opeaing condiions, he non-linea block in Fig.2 can be consideed eoyless. Theefoe, he linea dynaic block a he inp of he syse can be chaaceized hogh sall-signal, zeo bias easeens a diffeen feqencies ove he inp signal s I ypical bandwidh. Finally, by applying o he device a se of R lage-aplide sinsoidal inp signals s I fo diffeen vales of feqency and bias poin, and by easing, fo each es inp a he op, a lin- signal, a siable se of M saples ea syse of RM eqaions in P N + β pn can be obained fo : p P p z [ s ] e N nknowns n, p β pns, l, R, l, M n whee each fncion z [] has been epesened as a polynoial seies ncaed o he N-h ode wih coefficiens β pn n,..., N. If a lage nbe of es fncions is chosen a inp of he S/H-DC device RM >> P N +, he odel idenificaion can be copleed, by solving he ovedeeined linea syse hogh well-known leas sqae ehods. This neical pocede leads o a eliable solion, wiho he need fo non-linea opiizaion algoihs which sally sffe fo convegence pobles, sch as he dependence on he saing poin de o he pesence of liple local inia. 4. EXPERIMENTL RESULTS The poposed ie-doain discee convolion odel has been exaced fo a S/H-DC device peviosly ipleened in he faewok of SPICE cici analysis CD ool. The device has been ealized hogh a sapling swich based on a Schoky diode MD7 bidge cici [4], sepaaed fo he inp and op by wo Op ps OP64. The odel has been hen flly ipleened in he gilen-ds package envionen, which allows o pefo a lage se of diffeenind of cici analyses, boh in ie and feqency doain. The siple, ie-doain oiened aheaical folaion allowed an easy ipleenaion of he odel by eans of convenional ools available a he

4 se-ineface level of he silao, wih no need fo senio faciliies, code pogaing o addiional appoxiaions. In ode o validae he poposed appoach, seveal es have been pefoed, nde diffeen opeaing condiions a he inp of he device. In paicla, seveal lage-aplide sinsoidal signals have been applied, fo diffeen vales of feqency and bias poin. Meased and pediced wavefos a he device op have been hen copaed by coping he ean sqae eo, fo each inp exciaion, ove a veco of ie doain saples. Tab. shows he good ageeen beween eases and pedicions, a odeaely low feqencies as well as a high feqencies, whee he non-linea dynaic effecs in he S/H-DC behavio becoe ipoan. Tab. Mean sqae eo, coped ove a veco of ie saples, beween eased and pediced wavefos a he S/H-DC op. xis indicae he bias poin and feqency vales fo each sinsoidal signal inp. 2 ε.5 MHz 3 MHz 6 MHz 2.5 MHz 25 MHz 5 MHz 2. V V V V V V V V V V V nalyses pefoed by eans of he device odel allows also o qanify he conibion of non-linea dynaic effecs on he S/H-DC behavio. In Fig.4 he ansfe fncion pediced by he odel fo he linea dynaic block of Fig.3 is epesened, fo khz p o 5 MHz. Since he H j2πf c-off feqency of he device can be placed a he vale of MHz, i is ineesing o invesigae he non-linea behavio of he device fo inp exciaions aond his poin of he spec, whee he convee is nea o he lii of is egion of se. H j2πf ad Hz.E+3.E+5.E+7.E+9 f.e+3.e+5.e+7.e+9 Fig. 4 - Magnide and phase of he linea dynaic block ansfe fncion, pediced by he odel fo khz p o 5 MHz. -5 f Hz

5 Fig.5 Magnide of pediced specal coponens d nis a he op of he S/H-DC device, as esl of a sinsoidal inp signal f 2 MHz, aplide:.5v, bias:.5v n haonic balance analysis has been hen pefoed wih a.5v bias,.5v aplide sinsoidal inp signal a feqency f 2 MHz. Fig.5 shows he spec of he wavefo a he op of he device. Fig.6 shows he copaison beween he eased and silaed device DC chaaceisic U V S I V Fig. 6 Copaison beween eased and pediced S/H-DC device saic chaaceisic ove he ange [-2V,2V]. esides showing he high accacy of he odel in pedicing also he saic esponse, i poins o how he eoyless non-linea phenoena ae ild in he device behavio. Ths, all he non-linea effecs ha case he pesence of haonics in he op spec of Fig.5 can be consideed elaed only o he device dynaics. The poposed odel akes ino accon hese non-idealiies, no qanifiable by eans of convenional odels, which sally aibe he non-linea effecs of he device only o is non-ideal DC chaaceisic. Fig.7 epesens he pediced wavefo Fig.7 Meased and pediced --- saples a he op of he S/H-DC device, as esl of a sinsoidal inp signal f 2 MHz, aplide:.5v, bias:.5v See Fig.5 a he S/H-DC op whose spec is shown in Fig.5, copaed o he eased saples. 4. CONCLUSIONS new finie-eoy discee-ie convolion appoach has been poposed fo he non-linea dynaic odelling of S/H-DC devices. The odel akes sepaaely ino accon he saic and pely-dynaic non-lineaiies and can be idenified saing fo expeienal daa, obained hogh convenional easeen pocedes, by applying eliable neical algoihs. Validaion esls have been povided, which confi he odel capabiliy o pedic accaely he syse behavio, nde diffeen opeaing condiions, fo DC p o he feqencies in he egion ha epesens he lii of he device esponse. REFERENCES [] D. Mii, G. Pasini, F. Filicoi, G. Iclano, G. Vannini, R. Rossini, Expeienal evalaion of dynaic non-lineaiies in a S/H-DC device, in Poc. of IMEKO TC-4 Syposi "Elecical Insen in Indsy, Glasgow, Sep. 997, pp [2] D. Mii, G. Pasini, F. Filicoi, G. Iclano, G. Pellegini, Finie eoy non-linea odel of a S/H-DC device, in Poc. of IMEKO TC-4 Syposi on "Developen in Digial Measing Insenaion, Naple, Sep. 998, pp [3] D. Mii, G. Iclano, F. Filicoi, G. Vannini, G. Pasini, G. Pellegini, odified Volea seies appoach fo he chaaceizaion of nonlinea dynaic syses, in Poc. of IEEE Insenaion and Measeen Technology Confeence IMTC/96, ssels, Jn. 996, pp [4] C.F. Coobs, Eleconic Insen ook, McGaw Hill ook Copany, N.Y., 995.