Simulaig adom Volage or Curre Sources I SPICE epor NDT1-07-007 3 July 007
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1. Iroducio I his repor we cosider how o simulae radom volage or curre sources i SPICE. The geeral approach described below ivolves ierpolaig a simulaed whie oise sequece. The word ierpolaio here is used i he sese familiar o umerical aalyss Zarowski [1]; see Chaper 6. The simulaed whie oise sequece is obaied ouside of SPICE by aoher sofware ool. I priciple, his could be quie lierally ay oher geeral use programmig laguage or sofware ool capable of oupuig simulaed radom sequeces. I his repor we employ GNU Ocave.1.73 as he ool for creaig simulaed radom sequeces. We also work wih he SPICE implemeaio kow as SwicherCAD III a.k.a. LTspice by Liear Techology Corporaio. The simulaio approach provided here is quie differe from he approaches suggesed i []. I is suggesed i [] ha radom sources be simulaed i oe of wo mai ways: 1 Use he oise models for elecroic devices e.g., diodes. Use arbirary behavioral volage/curre sources i combiaio wih he rad or radom fucios provided by SwicherCAD III. Boh approaches require creaig a suiable circui or subcircui ha SwicherCAD III mus simulae o produce he desired resul. These will be collecively referred o as circui-based approaches o radom source simulaio. The aleraive ierpolaio approach cosidered here will be described wihi he framework of he heory of cyclosaioary CS radom processes Garder ad Fraks [3], Garder [4], Garder, Napoliao, Paura [5]. This is ecessary o properly accou for he geerally periodic srucure ihere i he simulaed radom process. For reasos eplaied i laer secios we will ierpolae based o usig secod-order cardial B- splies Chui [6]. I will be see ha ierpolaig a whie oise sequece does o lead o a aalog whie oise process. I pracice, he ierpolaio sraegy oly approimaes badlimied aalog whie oise Ziemer ad Traer [7]. However, his will be see o be adequae for he simulaio of oise i elecroic sysems. There is a relaioship bewee he ierpolaio approach ad he uiform samplig heorem for lowpass sigals [7] 1. This will be remarked upo a he appropriae jucure. Some possible advaages of he ierpolaio approach are: 1 A leas i priciple here is ulimied corol over simulaed process saisical characerisics afforded o he user. For eample, he approach bypasses he pseudoradom umber geeraor PNG ha is buil i o SwicherCAD III. I is herefore possible for a user o work wih wha migh be a saisically beer PNG. 1 This samplig heorem is also kow as Shao s samplig heorem, or also as he Whiaker-Shao- Koel ikov WSK samplig heorem A. I. Zayed, Advaces i Shao s Samplig Theory, CC Press, 1993. Noe ha o claim is beig made ha he PNG i SwicherCAD III acually eeds o be improved. 3
There is o eed o creae a circui or subcircui wihi SwicherCAD III o simulae he desired radom process. I his way he simulaor is uburdeed. 3 Some of he radom process simulaio mehods proposed i [] eed fudge facors o ge higs o come ou jus righ. This is error-proe ad o coveie. Some possible disadvaages of he ierpolaio approach are: 1 The user mus work wih a sofware ool ouside of SwicherCAD III creaig sysem iegraio issues. Eedig he approach preseed here o accommodae oher problems implies adequae mahemaical sophisicaio o he par of he user. 3 There may be a ruime pealy i havig SwicherCAD III read i possibly log simulaed radom process descripios from e files. I is o a all clear a his wriig which of he ierpolaio approach or he circui-based approaches are more ruime efficie ad uder wha codiios. These liss may o be ehausive. Fially, he aalyical resuls preseed here are ceraily well kow. The oly ecepio migh be he aalyical i.e., closed-form epressios for he cyclic auocorrelaio fucios i Secio 4. This repor emphasizes he issue of whieess ad oher maers e.g., radom process descripios i erms of probabiliy desiy fucios are omied.. Usig GNU Ocave wih SPICE A ypical SPICE implemeaio will provide he abiliy o simulae various ypes of ideal ad idepede volage or curre sources. However, he simples cosrucios ivolve workig wih such eiies as siusoids, epoeials, recagular pulses periodic or moo-cycle, ad piecewise-liear PWL cosrucios. This repor emphasizes he simulaio of aalog radom processes usig PWL cosrucios as hese are so easy o work wih. A specific ad simple eample of how o impleme his approach usig GNU Ocave.1.73 ad SwicherCAD III appears i he Appedi. 3. Whie Noise I his repor we emphasize he problem of approimaig whie oise. The reaso for his is simply ha his is a basic buildig block for o-whie colored oise processes. Whie oise may be filered o arrive a radom processes wih highly diverse specral characerisics. 4
Whie Noise Sequece As oed i he Iroducio we may simulae a radom aalog sigal by ierpolaig a simulaed radom sequece here deoed by { }. We do o make ay assumpios abou he joi probabiliy disribuios for he sequece of radom variables ha model he eperimeal realizaio { }. I paricular, he Gaussia assumpio is o esseial here. We oly assume ha he sequece is a leas wide-sese saioary WSS defied i [7]; see Secio 5., ad also ha i is a zero mea ad whie oise WN process: E[ ] 0, E[ m ] m for all iegers, ad m 3.1 E [A] deoes he epeced value of radom variable A. As well, { } is he Kroecker dela sequece a.k.a. ui sample sequece. This sequece is equal o oe for = 0 bu is oherwise zero-valued. Noe ha he oise variace is he same hig as he double-sided power specral desiy N0 / discussed i [7] see Eamples 5.5, ad 5.6 ad he subsecio o oise-equivale badwidh. Whie Aalog Sigal A aalog whie radom process has he followig properies which are i close aalogy o he previously discussed whie oise sequece. Suppose his process is deoed. I is WSS. We will also assume ha i is zeromea ad so E[ ] 0 for all ime real-valued. The auocorrelaio fucio is E[ ] 3. Here is he Dirac impulse coceraed a he origi 0. The power specral desiy PSD of aalog whie oise is he Fourier rasform of 3.. We employ he followig commoplace defiiios of he Fourier rasform FT ad he iverse Fourier rasform IFT, respecively: F F{ f } j f e d, 1 f F { F } 1 F e j d 3.3 As well f so if f is i Hz he ω will be i radias/secod. Therefore, he PSD of aalog whie oise is S F{ } 3.4 5
Typical uis are V / Hz vols-squared per Herz. I is appare ha 3.4 is a cosa for all frequecies. Moreover, o real-world radom process ca be ruly whie sice 3.4 implies ha such a process has ifiie power. I oher words, whie oise is a mahemaical ficio, albei a very coveie oe. A more pracical aleraive o whie oise is he cocep of badlimied whie oise. I is defied i erms of PSD accordig o S 0, B, oherwise 3.5 Thus, his PSD is some o-zero cosa value oly over some fiie bad of frequecies. The uderlyig radom process ow has fiie power 3. We may defie B via B B. The auocorrelaio fucio correspodig o 3.5 is via he IFT B j e d si B B B sic B 3.6 where sic = si /. This is he sic-pulse. This defiiio 4 is implemeed i GNU Ocave.1.73 as a buil-i fucio. Fially, we remark ha 3.1 ad 3. ivolves a abuse of oaio ha is commo i he sigal processig commuiy. Tha is, ormally a radom variable is deoed i upper case. Thus, we should properly say ha X is he h radom variable i some bi-ifiie sequece of radom variables whereas is a paricular realizaio of his radom variable i.e., he paricular oucome of a radom eperime. This i ur implies ha, for eample, we should have wrie E [ X ] raher ha E [ ]. However, coe may be used o ascribe he correc meaigs i pracice, hece he populariy of his abuse. If he reader is o comforable wih his sor of hikig ad hese kids of disicios he we refer him/her o basic es o applied probabiliy for egieers or scieiss of which here are may o choose from. Oe eample of such a e is Leo-Garcia [8]. A iroducio o applied probabiliy heory also appears i [7]. 4. Piecewise-liear Ierpolaio I his secio we defie a cerai PWL aalog radom process. We will also sudy cerai of is saisical characerisics. This is where he heory of cyclosaioary radom processes will come io play. Ulimaely we show how he PWL process may be used o adequaely approimae badlimied whie oise. 3 However, process is sill o physically realizable because i is o causal. To syhesize i requires passig whie oise hrough a ideal lowpass filer which is a o-causal sysem. 4 A aleraive defiiio is sic = si/. 6
Firs of all, defie he ui sep fucio 1 u 0, 0, 0 4.1 We will be workig wih his fucio quie a lo. Now suppose ha we obai { } by uiformly samplig he WSS aalog process. Tha is,, where 0 is he samplig period. If sigal is zero mea ad whie Secio 3 he he sample sequece { } will be zero mea ad whie accordig o 3.1. Ne we form a ew radom aalog sigal by liear ierpolaio which is formally described by he series 1 u u 1 4. where are he samplig imes. Sudy of 4. reveals ha i liearly ierpolaes he pois, ad 1, 1. The radom process 4. is oly a approimaio o. We shall ivesigae he qualiy of his approimaio laer o. A Secod-order Cardial B-splie epreseaio The represeaio for our PWL approimaio i 4. is very cumbersome o work wih for aalyical purposes. There is a much more cocise represeaio based o workig wih cardial B-splies [6] which we cosider here. These are deoed by N m i [6] which represes he mh order cardial B-splie. The firs order B-splie is N1 u u 1 4.3 A skech reveals ha his is simply a ui heigh square pulse defied suppored o he ui ierval [0,1]. The higher-order splies are obaiable i varied ways such as repeaed iegraio via 1 Nm Nm1 d 0 4.4 Acually, his is a process of repeaed liear covoluio ad so we also have 7
N m N N N 1 * * * 1 1 m facors 4.5 The aserisk * deoes liear covoluio accordig o he familiar defiiio which is f f * g g d 4.6 Cosideraio of 4.5 ad 4.3 ells us ha N m oly has suppor o he ierval [0,m]. Ye aoher formula for he geeral B-splie is Equaio 4.1.1 i [6] m 1 k m m1 Nm 1 k u k 4.7 m 1! k k 0 I is appare from 4.7 ha splies are piecewise-polyomial cosrucios. paricular he secod-order B-splie is piecewise-liear PWL: I N u 1 u 1 u 4.8 A plo appears i Fig. 1 below. p N. For coveiece we will from ow o wrie 1 Secod-order Cardial B-splie p 0.8 0.6 p 0.4 0. 0 0 0.5 1 1.5 Figure 1: The secod-order cardial B-splie N i Chui [6]. 8
Wih o loss of geeraliy we will assume ha he legh duraio of a lie segme is 1ui secods if you prefer. This appare resricio will laer be removed by he elemeary applicaio of he scalig propery for he Fourier rasform. As a resul he messy epressio 4. is ow replaced by he ice ad compac epressio p 1 4.9 For his o be a rue ierpolaig series Chaper 6 of [1] i is ecessary ha 4.9 recover he give sequece whe evaluaed o he iegers. This holds rue here sice k p k 1 k 4.10 To see his i helps o view he plo i Fig. 1, bu oly lookig a he cases for ieger. esuls o he Process Saisics We mus ow sudy he mea ad he auocorrelaio of he radom process 4.9. We shall discover ha eve hough { } is a whie oise process, is o eve WSS much less whie. We shall prove ha i is cyclosaioary CS, ad he use he mahemaical framework afforded by his heory o desig so ha i adequaely approimaes a badlimied whie oise process. The mea value of he radom process i 4.9 is m E[ ] E p 1 E[ ] p 1 0 4.11 for all. The auocorrelaio fucio is, E[ ] p m p m m 4.1 We observe ha boh he ime origi which is defied wih respec o ad he lag τ deermie his fucio. This implies ha he radom process is o-saioary i spie of he fac ha { } is a whie oise sequece described by 3.1. However, i is easy o cofirm ha 1,, 4.13 9
10 Tha is, he auocorrelaio fucio is periodic wih respec o ad he period is oe ui. This meas ha is a cyclosaioary CS radom process. As suggesed i [3], [4] or [5] we may herefore epress 4.1 as a comple Fourier series j e, 4.14 where he Fourier series coefficies are give by 1 0, d e j 4.15 The fucio is called a cyclic auocorrelaio fucio, ad he iegers i his coe are called he cycle frequecies. More geerally, if he process had period T he he cycle frequecies would be /T. From 4.1 Equaio 4.15 ca be rewrie as d e p p d e p p j m m m j 1 4.16 The correspodig PSD } { F S is called he cyclic specrum a cycle frequecy. We may ry o simplify 4.1 by averagig over oe period o elimiae depedece wih respec o. The resul is d p p d p p d p p d m m m, ~ 1 1 0 1 0 4.17 This amous o preedig ha is a saioary process. This is a oversimplificaio. However, i is impora o observe ha
~ 0 4.18 I fac, as oed i [3] or [5] he Fourier series epasio i 4.14 will be ha of a WSS radom process provided ha 0 for all 0. For a whie process we also 0 require ha have he form i 3.. Ye i is clear from p give i 4.8 ha he iegral a he ed of 4.17 is o a Dirac pulse. I is o eve a good approimaio o oe. Ad from 4.16 i should also be clear ha 0 for all τ whe 0. Eac epressios for he cyclic auocorrelaio fucios are cosidered below. Now we eed o ivesigae if here are ay codiios uder which ca usefully approimae a whie oise process. This ivolves a more deailed sudy of he srucure of ad especially of he cyclic specra S. I fac, we see ha S j j p p e e p e j p e d P P j d p e p e d d j j d d via 4.19 This is a simple ad iformaive epressio. Because we are workig wih splie ierpolas i is easy o ge closed-form epressios for our paricular problem. ecall from 4.5 ha p N1 N1 4.0 while from he direc applicaio of he defiiio of he Fourier rasform j / si / F{ N1 } e 4.1 / Hece by applicaio of he covoluio heorem for he Fourier rasform we have P j si / [ F{ N1 }] e / 4. 11
Therefore, he las equaio i 4.19 ow becomes S si si 4.3 The special case = 0 may be sigled ou ad is 4 0 si / S / 4.4 Now observe ha F 4 j si / { N1 * N1 * N1 * N1 } e / 4.5 The shifig heorem for he Fourier rasform saes ha F { f } e j F 4.6 Neglecig he real-valued scalig by i 4.4 we ow see ha 4.4 ad 4.5 represe he same pulse ecep for a relaive ime delay of wo uis. eferrig o 4.5 0 we ow see ha is really jus a fourh-order cardial B-splie. Oce agai, his is ohig like a Dirac pulse. I is easy o see ha he cyclic specra are real-valued ad always o-egaive. There is also he symmery codiio S S 4.7 Therefore, we really eed oly cosider 0. Figure below displays he firs few cyclic specra assumig ha 1. I 4.14 he erms for 0 may be regarded as ierferece erms. We see from Fig. ha hese erms seem o quickly weake wih icreasig. Also, ad imporaly, he ierferece is weak i he viciiy of 0 ha is, close o DC. If we hik i erms of ryig o approimae badlimied whie oise 0 isead of whie oise he S approimaes 3.5 provided ha is small eough. We will defie his more precisely a lile laer o. 1
Figure : The firs hree cyclic specra = 0,1, i 4.3 ad 1 for PWL ierpolaio syhesis of a radom process. The closed-form epressio 4.3 was easily obaied. We also have a simple epressio 0 for if we choose o make use of 4.7. Uforuaely, he epressios for whe 0 are o so ice hough hey are obaiable. We will provide hem maily for compleeess hough hey are o esseial o our mai purpose. The process of derivig closed-form epressios for higher-order cyclic auocorrelaio fucios is simplified by a careful sudy of he regios of suppor of he fucios ivolved, ad by workig wih p as give by 4.8. I paricular, Fig. 1 idicaes ha p is o-zero oly for [0,]. The iegrad of 4.16 las equaio has he produc p p. Ploig his o he -ais for various τ we coclude ha 0 for [,] 4.8 Also from a sudy of 4.16 we fid he symmery codiio j e 4.9 13
This meas ha we really oly eed o look a eiher [,0], or [0,]. Moreover, he iegraio limis i 4.16 ca be resriced o [0,] isead of, he eire real umber lie. Addiioal sudy of he srucure of he iegrad i 4.16 reveals ha we eed he iegral j f a, b, c, d a b c e d d 4.30 I is udersood ha f a, b, c, d 0 for all d. Noe ha 1 1 e d e, e d e, e d e 3 4.31 Of course, for our problem 4.30 becomes j, ad i fac. From 4.31 we fid ha a f a, b, c, d e I ca be show ha Q, b a c b 3 j e a d 4.3 4.33 where [0,], ad 1,,3,. The Big Q i 4.33 is give by Q, f 1,,0, f, 1,0, 1 f, 1,, d f 4, 4,4 1, 1 4.34 The d i he secod erm o he righ-had side of 4.34 is evaluaed accordig o d 1, 1, 1 4.35 Tha is, here are wo cases o cosider. We coclude his subsecio by observig ha * [ ] 4.36 This implies ha he doubly-ifiie summaio i 4.14 ca be rewrie as 14
0 j, e[ e ] 1 4.37 The operaio e[g] meas ake he real par of g. Therefore, we eed oly sudy he cyclic auocorrelaio fucios for 0,1,,3,. A Simple Desig Approach As has bee oed above i is o possible o syhesize whie oise by ay meas if for o oher reaso ha ifiie power sigals do o eis. This poses o real problem for elecroic circui simulaio, however. I pracice a elecroic circui is badlimied. Suppose is badwidh is BW. A ipu sigal wih compoes havig frequecies less ha BW will pass, bu ay compoes wih frequecies above BW will be removed. Therefore, i pracice if is badlimied whie oise of badwidh B > BW he elecroic circui will respod o i as hough he oise were ruly whie accordig o 3. or 3.4. Now, we have aalyzed he specral characerisics of he PWL radom process i 4.9 ad discovered ha he specrum is reasoably fla for small ω. However, we eed pracical desig crieria o eploi his idea. A lile laer o we will make use of he scalig propery for he Fourier rasform which is F{ a} X / a / a a 0 4.38 This will allow us o work wih 0 1. We ow specify a simple approach for doig so. Power levels are ormally epressed i decibels db: db S 10log10 S 4.39 0 Clearly, S 0 which is he peak value of S. This PSD rolls off moooically from is peak value uil he firs zero which occurs a. ecallig ha f, ad defiig as before B B we will cosider he PSD S o be fla for f [ 0, B] such ha 0 0 10log 10 S 0 0 S B 0 A db 4.40 where A db is some accepable amou of aeuaio over [0,B] e.g., A db { 3,,1,0.5,0.1}. From Fig. he choice of 3 db or db aeuaio is probably raher crude for wo reasos: 15
0 1 The PSD S is o really very fla over [0,B] i his case. The erms i 4.37 for 0 will be associaed wih i-bad periodic ierferece. ecall ha B for 3 db of aeuaio is he half-power poi. I pracice we shall likely prefer AdB 1dB. From A db 10log 10 A i is o hard o use 4.40 o obai he followig oliear equaio for B: si B 10 AdB / 40 B 0 4.41 Simple umerical mehods o solve his are preseed i [1]. GNU Ocave has a oliear programmig oolbo wih rouies o solve 4.41 e.g., fmibd 5. AdB B assumes 1 3 0.31839 0.6150 1 0.18599 0.5 0.13190 0.1 0.05913 0 Table 1: Cuoff frequecies for which S is accepably fla [ 0, B ], B B. Table 1 above liss various cuoff frequecies B for cerai aeuaios. We may assume 1secod ad so B i Table 1 has he uis of Hz. Clearly, B is o very big ad so we ca oly simulae whie oise ipus o very arrowbad elecroic sysems. To accommodae real-world badwidhs we eed o reduce. For a desired oise badwidh B B ad upo applyig 4.38 he desired segme legh is give by B secods 4.4 B This follows because we use / i place of. From 4.38 we have B B ad so 4.4 immediaely arises. Now we choose B BW wih he choice based o a accepable flaess; B i 4.4 is appropriaely chose from Table 1. As we epec, he flaer he PSD of Fig. case = 0 i he desired bad, he smaller we eed o make B ad he shorer will become. 5 Fucio fmibd implemes he golde secio search mehod for locaig he miimum of a scalarvalued fucio wih a scalar argume. The problem of solvig 4.41 ca be covered o a simple fucio miimizaio problem. 16
For a es sigal from 4.9 of desired duraio T secods we eed o geerae N T BT B 4.43 lie segmes. oudig o a ieger value is assumed i evaluaig 4.43. We prefer o use as small a N as possible ad so should be as large as possible o miimize he umber of required segmes. Usig fewer lie segmes places less of a burde o SPICE. However, a good qualiy es sigal meas ha B mus be small ad so N will be large. So, here is a ieviable radeoff ivolved here. As a simple eample suppose ha we wa a badlimied whie oise process wih badwidh B 1 MHz. Suppose ha A db 1 db is a adequae amou of flaess for our purpose. Therefore, from 4.4 we mus employ a PWL syhesis wih segme leghs of 0.18599 / B 0.18599 sec. If we ow wish o creae a es sigal of legh T 100 sec he from 4.43 we eed N 100 / 0.18599 538 lie segmes. If he sequece { } is Gaussia he he GNU Ocave rad fucio may be used o obai he sequece elemes, assumig a Gaussia process is desired. The e secio pois ou ha he ideal mehod of ierpolaig he sequece { } ivolves usig sic-pulses. Bu if sic-pulses are he ideal choice he why do we use hem isead of PWL ierpolaio? ecall ha si sic 4.44 This sigal is o fiiely suppored as i is defied over he eire real umber lie. I is much messier o impleme sources i SPICE usig 4.44 ha PWL cosrucios. Moreover, ryig o accuraely approimae a sic-pulse wih simpler fucios e.g., PWL approimaio eeds a complicaed represeaio due o he slow decay of he sic-pulse wih respec o ime. ecall as well ha fucio 4.44 is he impulse respose of a ideal lowpass aalog filer. The heory of filer desig ceers aroud creaig useful causal approimaios o 4.44. The plehora of mehods for doig his really uderscores he difficuly of obaiig ice approimaios o 4.44. 5. Sic-pulse Ierpolaio Here we show ha he ideal ierpolaig fucio is really he sic-pulse. I was briefly argued a he ed of Secio 4 ha his choice is o so easy o impleme hece our pracical preferece for he PWL cosrucio i 4.9. Neverheless, cosiderig he ideal case gives us a frame of referece o judge he qualiy of he approimaio 4.9. I gives us a useful perspecive o he desig equaios 4.4, ad 4.43. 17
I his secio p is redefied o be 4.44. I his case 1 P u u 0,, oherwise 5.1 The ierpolaig series 4.9 is ow replaced wih si c 5. Wih his we are coiuig o assume ha he spacig bewee cosecuive elemes of { } is 1secod. We may cofirm ha 5. does ideed ierpolae he ipu sequece because k si c k k k 5.3 Similarly o 4.1 we ow have, si c si c 5.4 ad 4.13 also holds. However, we will see ha i 5. is acually WSS! This is mos easily show by viewig epressios for he cyclic specra. Of course j si c si c e d 5.5 which are he cyclic auocorrelaio fucios. The geeral epressio a he ed of 4.19 coiues o hold i our ew siuaio. Thus, S F{ } P P 5.6 This is evaluaed usig 5.1. Immediaely we discover ha P S 0, 0, 0 5.7 We coclude ha defied by he series 5. is WSS ad also badlimied whie oise wih badwidh B B radias/secod. Thus, B 0. 5 Hz. 18
This implies ha if our elecroic circui agai has badwidh BW ad ha B is he desired oise badwidh such ha B BW 0. 5we mus space he sequece samples accordig o samplig ierval 0.5 B secods 5.8 Comparig his epressio wih 4.4 while keepig i mid he eries of Table 1 we see ha usig sic-pulse ierpolaio requires fewer pulses o simulae a badlimied whie oise process of some pre-deermied duraio T compared o he umber of lie segmes eeded i he PWL syhesis of Secio 4. Sic-pulse ierpolaio is more efficie i his way ha PWL ierpolaio. Bu agai he sic-pulse is harder o work wih i pracice. The coecio bewee 5.8 ad Shao s samplig heorem should be clear. ecall ha a aalog sigal badlimied o B Hz mus be sampled a a miimum rae of B Hz i order for he acquired samples o permi a perfec recosrucio of he eire origial aalog sigal via a sic-pulse ierpolaig series such as ha illusraed by 5.. Appedi The followig GNU Ocave scrip wries a SPICE direcive specifyig a ideal idepede volage source o he file myv1.: ------------------------------------------------------------------------------------------------------------ % % Make_myV1.m % fucio Make_myV1 oufile = fope"myv1.","w"; fprifoufile,"%s","v1 V1 0 PWL"; = [ 0.1.9 1.0]; = [ 0 1 1 0 ]; for k = 1:legh fprifoufile,"%8.4f%8.4f",k,k; edfor; fprifoufile,"%s"," ser=0 Cpar =0"; fcloseoufile; 19
Whe he above scrip is ru he followig e will be wrie o file myv1.. I is a SPICE direcive creaig he desired volage source via piecewise-liear PWL cosrucio: V1 V1 0 PWL 0.0000 0.0000 0.1000 1.0000 0.9000 1.0000 1.0000 0.0000 ser=0 Cpar =0 SwicherCAD III favors schemaic-based descripios of circuis. The simples eample of how o make use of oupu from Make_myV1.m is illusraed as follows which is a schemaic coaied i a file called ArbiaryVolageSource.asc: We see he SPICE direcive.ic myv1.. Whe he rasie simulaio is ru he file myv1. will be read ad ierpreed as a direcive o impleme he desired source. Here he direcive specifies a rapezoidal pulse shape. Cosequely, ypical circui sigal races are as follows: I is o hard o edi he previous GNU Ocave scrip o syhesize ay desired PWL es sigal icludig simulaed radom processes desiged accordig o he heory i he mai body of he prese repor see especially he las subsecio of Secio 4. Limied eperimeaio shows ha SwicherCAD III will accep source descripios coaiig a leas housads of lie segmes. 0
Ackowledgme Various sofware ools were employed i he producio of his repor. GNU Ocave.1.73 ad SwicherCAD III a SPICE implemeaio by Liear Techology Corporaio boh ruig uder MS Widows XP were all employed i he umber cruchig ad he producio of illusraive figures. PDF Olie, a free olie file ype coversio service was used o cover he MS Word versio of his repor io pdf. Ipu from he LTspice echical group paricularly from H. Seewald is much appreciaed. efereces 1. C. J. Zarowski, A Iroducio o Numerical Aalysis for Elecrical ad Compuer Egieers, Joh Wiley ad Sos, 004.. hp://ech.groups.yahoo.com/group/ltspice/ 3. W. A. Garder, L. E. Fraks, Characerizaio of Cyclosaioary adom Sigal Processes, IEEE Tras. o Iformaio Theory, vol. IT-1, o. 1, Ja. 1975, pp. 4-14. 4. W. A. Garder, Eploiaio of Specral edudacy i Cyclosaioary Sigals, IEEE Sigal Processig Magazie, April 1991, pp. 14-36. 5. W. A. Garder, A. Napoliao, L. Paura, Cyclosaioariy: Half a Ceury of esearch, Sigal Processig, v. 86, 006, pp. 639-697. 6. C. K. Chui, A Iroducio o Waveles, Academic Press, 199. 7.. E. Ziemer, W. H. Traer, Priciples of Commuicaios: Sysems, Modulaio, ad Noise, Hougho Miffli, 1976. 8. A. Leo-Garcia, Probabiliy ad adom Processes for Elecrical Egieerig d ediio, Addiso-Wesley, 1994. 1