Guardbands in Random Testing

Transcription

1 Guardbad Rado Te Güer Kez Techche Uerä lauhal Iu für Iforak Erzrae 3868 lauhal-zellerfeld Tel ax kezforak.u-clauhal.de brac The faul coerae of a rado e ca be eaed b faul ulao. If he ulao erfored b aoher rado euece ha hoe ued uder e or a faul ale ued, a rado dfferece beee he ulao reul ad he faul coerae ha o be codered. The ulao reul u be larer ha he faul coerae ha ha o be uaraeed. The dfferece called uardbad. I he aer he drbuo of he faul coerae ad he drbuo of he dfferece ha bee dered b he aheacal odel of deedel deecable faul. ferard correced u exereal daa. The coaro beee heor ad exere uel a feaure of rado e, o hch o aeo ha bee ad he a. The correlao he faul deeco roce ca o be ored deer uardbad. he fal reul relao for uardbad calculao are e. Ke ord e of dal crcu, rado e, faul ulao Iroduco The o ora araeer of a dal e he faul coerae. I he fraco of deecable faul fro a e of aued faul. U rado aer a ul he faul coerae deed al o he uber of e aer ad le o he e aer elf. Hoeer, a rado arable. The aer deal h he follo roble Ho uch faul coerae ca be uaraeed f he faul ulao ha bee erfored h oher radol eleced u aer ha hoe ued uder e? Th ueo ere racce. Rado aer are ofe ued elfe fuco, bu alo lo co e e. ef he e ol b he uber of e aer ha a lo of adaae oer he alerae, cou, or ad roce a lare ua of exacl defed aer [], []. or he e of a crcu uder oerao he u aer are o ko adace. The faul ulao h a arorae ale of aer he ol a o eae he faul coerae. I a alcao o eouh o ko he aerae faul coerae. The alue ha ca be uaraeed reured. ak uao are, f he faul ulao ha bee doe h a ale of faul. Sulao reul ad faul coerae dffer b a rado aou ad a loer boud ha o be uaraeed for he faul coerae. The er uardbad ha bee ake fro aaloue e. Te a araeer, e.. a olae, he eaured alue u be beer ha he alue ha hould be uaraeed b he e [3]. The dfferece, he o called uardbad, ecear o reduce he robabl ha oe ad oher druo dur eaur ll caue ha bad dece ll be clafed a ood oe. The

2 roble h he faul coerae ak. The faul coerae hould o be loer ha a e boud. Ohere, he uber of bad dece clafed a ood oe ll be oo lare. uardbad calculao eed he drbuo of he araeer uder eao. Seco deelo a aheacal odel o calculae he drbuo, he ea alue ad he arace of he faul coerae ou of deeco robable. Bac feaure are dcued. or he arace a uer boud ha bee foud. Seco 3 decrbe he uardbad roble. Seco deal h he uardbad ze, f he faul coerae eaed b a faul ulao h oher rado aer ha hoe ued uder e. The ecear ze of he uardbad for a ulao h a faul ale ll be dcued eco 5. rbuo of he faul coerae The deeco robabl ( ) of a faul he robabl ha he faul ll be deeced b a le radol eleced u aer. ore dealed ad ore eeral exlaao he ereed reader a fd [], [], [], [5]. To calculae he deeco robable () for a euece of u aer, eerall he boal odel ued. faul ll be deeced b u aer f a lea oe of he u aer deec he faul ( ) ( ) () I baed o he auo ha real rado aer are ued. I ea ha aer a occur b chace ulle e he euece. The boal aroach alo a cloe aroxao for a eudo-rado e (o reeo of aer oble) f he e e uch horer ha a exhaue e [6]. Euao () ca be lfed ( ) e h l () 5% % % 5% % % 69,35%,3%,536% 5,9%,%,5% or all deeco robable ( ) e (3) I he coex of uardbad calculao he faul coerae a rado arable. B chace ca ake alue beee zero ad oe. To duh he rado arable faul coerae fro a exacl ko faul coerae, he Greek leer ξ ll be ued. No our e dea ar. We roduce auxlar rado arable, oe for each faul ha hould be oe f he faul deeced ad zero f o deeced. The dea behd h ha he faul coerae he ea alue of hee auxlar arable ξ ( ) ζ ( ) ( - uber of aued faul). The drbuo of each of he auxlar arable ζ ( ) ( ζ ) ζ P P faul udeecable faul deecable () (5)

3 Ther ea alue are eual o he deeco robable E( ζ ( ) ) ( ) The arace ( ζ ( ) ) ( ( ) ) ( ) () The follo reuoe ha he faul he crcu are deeced deedel of each oher. Proerl eak, o rue. a faul hare corol ad oberao codo. Reul fro ha, he ae local alue a lea a a ar of he u are elble for faul deeco. O he oher had, ould o be oble o calculae he drbuo of he faul coerae hou h auo. ddoal robable ould be eeded of he kd robabl ha faul deecable f faul (u)deecable. Thoe daa are o aalable. Therefore, fr he odel of deedel deecable faul ued. Secod, he reul euao are erfed b exere. rbuo Uder he auo of deedel deecable faul all arao of deecable ad udeecable faul hae o be coled. The robabl of each arao ha o be deered. or all arao h a cera uber of deecable faul he robable hae o be added u. Hoeer, becaue of he exoeal roh of he uber of arao h aroach ol ood for a all uber of faul. ure ho a beer alorh. Tak he drbuo of faul ad he deeco robabl of faul + he drbuo of + faul calculaed. The ar drbuo ha of he auxlar rado arable for he fr faul ζ ( ) h he realzao zero ad oe. ro h he drbuo of he fr ad he ecod faul ζ P + ζ h he realzao {,, } calculaed,... The alculao e of h alorh ro ol h he uare of he uber of faul. (6)

4 Œ Œ z z z } G G J P( ( ) ) ( ) ξ P( ( ) ) ( ) O O ξ for o P( ξ ( ) ) P ξ ( ) ( ) P( ξ ( ) ) P ξ ( ) O for o O P ξ ( ) P ξ + P ξ ( ) ( ) ( ) ( ) rbuo for o 3 faul ( ), ( ), ( ) 6 5,6,, 3 3, 6 faul faul 3 faul rbuo for 6, ad 6 faul ( ),5,6,,8 3, 3,,8,9, 3,5 6,8,9, # $ $ " " # $ $ " " $ $% % % % &' ' &' ' ' '( ( ( ( )* * )* * + +,--- - * * / ?????? B B B B B B E E E E E E GHHH H H H I I I JKKK K I I IK I I IK YZ Z X X YZ Z X X Z Z L L NNN N L LN O O PQQQ Q L LN O OQ R R STTT T VW W O OQ R RT U U VW W R RT U U W W [ [ [ [ [ [ \] ] ] \] ] ] ] ] ] _``` ` ` ` ` ` ` a a bccc c a ac d d efff f a ac d df d df h k kl l k kl l l l l l o o o o o o r r r r r r u u u u u u x x x x x x z{{{ { { { }~~~ ~ ~ ~ ƒ 6 faul ˆ ˆ ˆ ŠŠŠ Š ˆ ˆ ˆŠ ˆ ˆ ˆŠ Œ, faul 6 faul ure lorh o calculae he drbuo of he faul coerae h exale ( ξ ( ) - drbuo of he faul coerae for faul ( ) - deeco robabl of faul - uber of deecable faul - uber of aued faul - uber of e aer) The exale ure ho ha he faul coerae coere o a oral drbuo h a ro uber of faul. roof, reuo deedel deecable faul, ca be foud []. ure ho he faul coerae of a larer cobaoal crcu h uck-a faul. I alo earl oral drbued. 3,6 3, uber of deeced faul,6 freuec fuco 3% 3, 8 % 3, 3 %, ,3 3, 3,5 3,6 - uber of e aer - uber of deeced faul

5 ure aul coerae of he bechark crcu c35 [8] (reul of a faul ulao h, dffere rado euece uber of faul 3, 65 reduda faul are reoed fro faul l ad euale faul are ued o oe faul ) ea alue The ea alue of a u of deede rado arable he u of he ea alue of he uad. The faul coerae he ea alue of he auxlar arable ζ ( ). So, he u ha ll o be dded b he uber of aued faul. E( ξ ( ) ) ( ) e Euao (8) allo he follo cocluo for he aerae faul coerae. for he le deeco robable, coere o oe h a ro uber of e aer. The deeco robable a crcu ar b oe order of aude. uual, he aor of faul eal deecable. The ll be foud h a fe hudred rado aer. The faul coerae ro fa a he be of he e, reach abou 8% o 9%. all fraco of he faul hard o deec. To coer he la fe erce of faul a co a creae he e leh b ulle order of aude. I racce, a ere ueo heher he faul coerae or he reured e leh ca be eaed b a uch horer e euece ha hoe ued uder e. I ould allo o ae a hue aou of ulao e. Euao (8) ho ha oble. or hor e euece he roh of he faul coerae deed alo excluel o he deeco robable of ea o deec faul. The effec of he hard o deec faul all coaro o he arao of he ulao reul. No ee cocluo abou he order of aude of he deeco robable for he hard o deec faul ca be dra. Whou h forao oble o redc he reured e leh for a faul coerae hher ha he ulao reul or for he coerae of a uch loer e e. The faul ulao o eae he faul coerae ha o be erfored h he ae uber of e aer a laed for he e. Varace The arace of he u of deede rado arable he u of he arace of he, uad. or he faul coerae he ea alue of he auxlar arable ξ ( ( ) ) ( ) ( ) ( ) e e (9) I coere o zero h a ro uber of faul. If he e e er lo, he aor of faul ha deeco robable cloe o oe ( ). The uber of uad corbu euao (9) o he arace becoe aller h he e leh ad o alo he arace. Whou ko he le deeco robable a uer boud of he arace ca be e. Whe all deeco robable are eual, ha a axu for a e ea alue ad a e uber of faul. The drbuo of he axu a boal drbuo. The roof e he aedx. ζ (8)

6 ( ( ) ) ( ) ξ E ξ E ξ () Th ueuao allo o eaure he effec of he correlao he faul deeco roce. Le u roduce a araeer ε ( ξ( ) ) ( ξ) ξ ε E E ( ) ccord o euao () ε ca o be larer ha oe for ucorrelaed faul. Hoeer, ca exceed he boud f erdeedece he deeco roce ex. The follo ll llurae h. Le u aue ha he uber of faul for a e crcu ha bee doubled b l or cou each faul o e. Th correod o a faul e h ulle ar of euale faul or faul ha ll be deeced ala ulaeoul. The rck ll eher chae he ea alue of he faul coerae or he arace. Ol he uber of aued faul double. d o ε becoe o e a lare a for a e of deede faul. Oboul, b a furher creae of he uber of euale faul, ε a becoe larer ha oe. Uuall, euale faul are reoed fro he faul l before faul ulao. ro each cla of euale faul ol oe ake. Th ha alo be doe h he faul l ued o roduce ure. Bu here are oher deedece. The corol ad oberao ah are lar for a faul. So, faul reure arl he ae u aer ad ll be deeced ofe b he ae radol eleced e aer. Table colu ho he alue of ε for he exere ure. lhouh, eualece hae bee reoed, he uber look a f o aerae u o 5 ad ore faul ould hae bee deeced each rado euece h he ae aer. faul ulao h all 3,66 uck-a faul faul ulao h a ale of, faul () faul ulao h a ale of 3 faul E ξ ( ) ξ ( ) ε E ξ ( ) ξ ( ) ε E ξ ( ) ξ( ) ε 6 3 8,6 3, 6, 88.5% 93.5% 9.6% 99.% 99.% 99.8%.8%.88%.8%.%.8%.5% % 93.% 9.5% 99.% 99.% 99.8%.%.%.63%.8%.%.8% % 9.6% 98.% 99.% 99.9% %.8%.%.6%.36%.% Table The araeer ε for a colee uck-a faul e ad for o faul ale (crcu c35 [8], ea alue ad arace hae bee eaed b faul ulao h, dffere rado euece) Wh a ro e leh ad a ro faul coerae he erdeedece decreae. ll faul h hh deeco robable are deeced alo b each rado euece of he correod e leh. The do o corbue o he arace. Beee he harder o deec faul are robabl alo oe erdeedece lef. Bu he effec of he afel deecable faul oueh he.

7 Ierdeedece he faul deeco roce do o ol creae he arace. The ca alo chae he hae of he drbuo. The reao for a lare rou of faul deecable b he ae or alo he ae e of u aer. Le u aue a crcu h faul, here 8 faul ll be deeced ala ulaeoul. Thu, he uber of deecable faul led o {,,, 8, 9, }. The drbuo of he faul coerae dded o rae. ure 3 ho h effec for a real crcu h uck-a faul. ro he oo ad he dace of he rae ca be cocluded ha he faul rou co of abou 8 faul h a deeco robabl of abou 5.,6 œ ž Ÿ Ÿ Ÿ 96, % %,5,,3, š œ ž Ÿ Ÿ Ÿ œ ž Ÿ Ÿ Ÿ 96, 8% 6% % % % % œ ž Ÿ Ÿ Ÿ,5,6,,,5,6 9,6,3,,5 š œ ž Ÿ Ÿ Ÿ ž Ÿ œ ure 3 rbuo of he faul coerae of he crcu c6 [8]. far a he hae bee foud euale faul hae bee reoed fro he faul l. Noehele, he reul look a f oe cla of euale faul ha bee foroe (685 uck-a faul reul of a faul ulao h, dffere rado euece) 3 Guardbad The faul coerae of a e e a ual araeer, aufacurer uaraee for. I u be a lea a lare a a e loer boud. Hoeer, he faul coerae a rado arable hch ca ake a alue beee zero ad oe b chace. loer boud ca ol be e h a all error robabl (call he fraco of a erce u o oe erce) P ( ξ ) α ()

8 ª P(ξ () «P( ξ () ) b e d P(ξ () «f E( ξ ()) c a b c d e f faul coerae robabl de aerae faul coerae uaraeed faul coerae area rooroal o he robabl ha he faul coerae oo lo area rooroal o he robabl ha he faul coerae hh eouh ure Oe-de eral eao of he uaraeed faul coerae The aerae faul coerae u be larer ha he uaraeed faul coerae b a cera aou called uardbad G. ho he la eco, he faul coerae ofe earl oral drbued. The uardbad for a oal rado arable u be abou o e larer ha he adard deao ( ) G k ξ h α Φ k (3) (α - error robabl Φ( k) - alue of he adardzed oral drbuo). The adard deao of he faul coerae ca be eaed b euao () u he ea alue ad he araeer ε G k ε ( ( ξ) ξ E E The larer he uber of faul he aller ca be he uardbad. The ea alue ad he arace of he faul coerae ξ ha o be eaed. Th reure a faul ulao h faul ad rado aer. The reul of he faul ulao he eaed ea alue of he faul coerae. I dffer fro he ea alue b a rado aou. The faul coerae elf alo dffer fro he ea alue b a rado aou. So, boh rado dfferece hae o be codered for he uardbad (ure 5). a ()

9 faul coerae of he ko rado euece P ξ () ulao reul faul coerae of a uko rado euece P ξ () E ξ () coo ea alue dfferece P ξ () error robabl ξ () uardbad ure 5 Guardbad beee he faul coerae of a ko rado euece ad he alue ha ca be uaraeed for a arbrar rado euece Sulao h aoher rado euece The faul coerae ll be deered b aoher rado euece ha hoe ued uder e. The uber of rado aer hould be he ae. I h cae, he ulao reul ad he faul coerae are o deede rado arable h he ae drbuo. The arace of he u of o deede rado arable he u of he arace. I double. The adard deao a he uare roo of he arace creae b he facor. B he ae aou he uardbad ha o be creaed coaro o euao () here he ea alue ha bee aued o be exacl ko ( ( ξ) ξ G E E k ε The reured uardbad ca be reduced b reea he faul ulao h z dffere rado euece of leh. The eaed ea alue becoe he ea alue of he ulao reul. The ecear ze of he uardbad reduce do o ( ) ( ξ) ξ G + z k E E ε (ulle ulao reul allo alo a ore accurae eao of he arace.) (5) (6)

10 5 Sulao h a faul ale The ulao h a faul ale co le ulao e. I reur, he arace of he ulao reul hher. U he uer boud (), could be aued ha he arace ( ξ ) ro coerel rooroal o he reduco of he faul ale. The adard deao ( ξ ) ro h he roo. The drbuo becoe broader ad flaer (ure 6). 3,65 faul (colee), faul 3 faul 3,6, 3, 3, 3, ² ± ² 9 8 ² ± ³ 8 6 µ ², uber of rado aer 3.%.8%.6%.% ± œ œ œ ² ±.%.5%.% ± µ œ œ œ.%.5% 8% 9% 96% faul coerae 9% 95% % faul coerae ure 6 rbuo of he faul coerae of dffere ale of uck-a faul (crcu c35 [8], ulao h, rado euece er faul e) Eecall for lare faul ale, he uardbad ze deed uo heher he ulao erfored h he ae or a dffere rado euece ha he e. U dffere rado euece, ulao reul ad faul coerae are o deede rado arable. The arace add ( ξ ξ ) ( ξ ) ( ξ ) + () U he uer boud for he arace, euao () becoe ε ε E E (8) ( ξ ξ ) + ( ξ) ( ξ)

11 I he faul ulao erfored b he ae rado aer a he e, he ulao reul alread he exac coerae for of faul. Th fraco doe o corbue o he arace of he dfferece. I coaro o euao (8) he arace of he dfferece aller b he facor ( ξ ξ ) + ( ξ) ( ξ) ε ε E E (9) or a all faul ale he arace of he faul coerae ca be eleced. The (9) alo cloe o oe facor ( ( ξ) ξ E E ( ξ ξ ) ε The ze of he uardbad G k ε ( ( ξ) ξ E E Euao () e he reo ha he uardbad u be creaed coerel rooroal o he roo of he uber of ulaed faul. The real rooro are uch beer. Table ho alo he ea alue ad he arace of he ulao h faul ale. The creae of he arace uch aller ha could hae bee execed accord o euao (). Wh he reduco of he ze of he faul ale he araeer ε alo becoe aller. The araeer ε ha bee roduced o uaf he erdeedece he faul deeco roce. The ore faul auo are drbued a e crcu, he ore erdeedece are o be execed ad ce era. The reao oboul ha he uber of corol ad oberao ah led a crcu. a aued faul u hare corol ad oberao codo. I ea ha lar u aer ll deec he. The arace doe deed le o he oal uber of faul bu ore o he uber of rou of lar deecable faul. The uber of hoe rou led b he crcu rucure. Selec a faul ale reduce al he uber of lar deecable faul h he rou ad o o uch he uber of rou. So, he arace le effeced. Of caue, h exlaao lfed. urher eao are reured o uderad he heoeo of erdeedece he faul deeco roce beer. The araeer ε a eaure for he effcec of a faul ulao. The a of a faul ulao a cloe redco of he faul coerae. The co deed o he uber of aued faul. cocluo fro h aer alo ha a colee faul ulao doe o rea, exce f erfored h he ae aer a he e. The reduco of he adard deao ou of relao o he uber of ulaed faul. faul ulao h a faul ale ore ecooc. Bede, u he ae aer ha he e a co ob. I rode he exac alue of he faul coerae for he aued faul. Hoeer, hee are o he faul o be foud uder e. Beee he faul coerae of odelled faul ad of real faul alo a rado dfferece ha o be codered. Bu h aoher roble ha hould o be oled h aer. () ()

12 ¹ Suar The faul coerae of a rado e a rado arable. I ol, bu o ala oral drbued. The ea alue coere h a ro uber of e aer o %. Bu o oble o eae he ea alue for a lo e euece b a faul ulao h a uch horer e euece. or he arace a uer boud ha bee foud hch deed ol o he ea alue ad he uber of aued faul. Th boud hold f all aued faul are deecable deedel of each oher. Ierdeedece he deeco roce creae he arace far beod h boud. So, erdeedece ca be uafed ad eaured b ulao exere. uardbad calculao ha o ake o accou o rado arable. The reul of he faul ulao ad he faul coerae dffer fro he coo ea alue b a rado aou. So, he arace of boh rado arable hae o be added. The fal reul ha he uardbad u be aroxael G 5 ulao_ reul ( ulao_ reul) uber_ of_ulaed_ faul Eecall, f a faul coerae cloe o % ha o be uaraeed, he reured uardbad lare coaro o he alloed dfferece o %. I, becaue he uardbad reduce ol rooroal o he roo of he er (-ulao_reul). ere cocluo ha he uardbad for faul ale do creae uch le ha coerel rooroal o he roo of he uber of ulaed faul. Reduc he uber of faul, he facor before he roo alo becoe aller. I becaue here are le erdeedece a faul ale ha a colee faul e. So, look o be ore effce o erfor he faul ulao for a rado e ol h a faul ale ad o h he hole faul e. Referece [] Illa, R. J. Self-eed daa flo loc e aroach. I IEEE e ad Te of couer, No., 985, 5-58 [] Lobard,. Sa,. (ed) Te ad dao of VLSI ad ULSI. Kluer cadec Publher, 988 [3] Wlla, R. H. Hak,.. The effec of uardbad o error roduco e. Euroea Te of., Roerda, 993, - [] ad, R. Blache, G. bou rado faul deeco of cobaoal eork. IEEE Traaco o ouer No. 6, 96, [5] r,.. Sch leel rado aer eabl aal. -5, 988, [6] h,. K. cluke, E. J. Te leh for eudo rado e. Ieraoal Te oferece, 985, 9-9 [] Kez, G. Self e of dal crcu ool, de aroach ad ofare uor. eraochrf, rede Uer of Techolo, 99 ( Gera)

13 [8] Brlez,. uara, H. eural el of cobaoral bechark crcu ad a are ralaor ORTRN. I. Sou o rcu ad Se Secal Seo o TPG ad aul Sulao, 985 edx Proof for ueuao () Le u ubue all deeco robable b he u of he ea alue ad a dfferece ( ) E( ξ( ) ) + δ h E ξ ( ) Iered () ( ξ( ) ) ( E( ( ) ) ) E ( ) ( ) ( ) ξ + δ ξ δ E( ξ( ) ) ( E( ξ( ) )) The lfed ueuao δ rue for all δ. ad δ