Delay and Power Reduction in Deep Submicron Buses

Transcription

1 Georga Sae Uversy Georga Sae Uversy Compuer Scece Theses Deparme of Compuer Scece 5--5 Delay ad Poer Reduco Deep Submcro Buses Sharareh Babvey Follo hs ad addoal orks a: hps://scholarorks.gsu.edu/cs_heses Par of he Compuer Sceces Commos Recommeded Cao Babvey, Sharareh, "Delay ad Poer Reduco Deep Submcro Buses." Thess, Georga Sae Uversy, 5. hps://scholarorks.gsu.edu/cs_heses/5 Ths Thess s brough o you for free ad ope access by he Deparme of Compuer Scece a Georga Sae Uversy. I has bee acceped for cluso Compuer Scece Theses by a auhorzed admsraor of Georga Sae Uversy. For more formao, please coac scholarorks@gsu.edu.

2 COPYRIGT BY Sharareh Babvey 5

3 Delay ad Poer Reduco Deep Submcro Buses By: Sharareh Babvey Major Professor: Commee: A. P. Preehy Saed Belkasm Ale Zelkovsky Elecroc Verso Approved Offce of Graduae Sudes College of Ars ad Sceces Georga Sae Uversy May 5

4 Delay ad Poer Reduco Deep Submcro Buses By: Sharareh Babvey Uder he Drco of: A. P. Preehy ABSTRACT As echology scales do, couplg beee odes of he crcus creases ad becomes a mpora facor ercoeco aalyss. I may cases lke he deep submcro echology DSM, he couplg beee les er-re capacace s srog ad he poer cosumed by parasc capacace s o-eglgble [-6]. I hs ork, e employ he dffereal lo-egh ecodg [] o reduce eergy ad delay rasmsso cos o DSM buses. We propose a eumerao mehod ha reduces he ecoder able-sze from O ords o O ords, for a -b DSM bus, ad so reduces he memory compley sgfcaly ad faclaes eergy ad delay reduco due o addressg ad fechg daa from large lookup ables. We modfy he eergy ad delay equaos for DSM buses ad develop e represeaos erms of umber of he same ad oppose dreco rasos o he bus ad use hem our ercoec aalyss. We also use hese equaos o develop formulas for compug he

5 mea rasmsso cos per b o DSM buses for boh dffereal lo-egh ecodg ad ucoded schemes. Usg he ercoec aalyss, e compue a help codeord from he se of useleced codeords o he fly ad assg o each seleced codeord. Ths lo compley sep cosss of smple operaos ad eables us o ga more cos reduco hou creasg he able sze or umber of he bus les. The smulao resuls for 6-b, 3-b ad 64-b buses a mamum rae oly oe era le added sho ha he proposed ecodg scheme acheves more ha % cos reduco, ad performs more ha.5% beer ha o he orgal dffereal lo-egh scheme, he ors case. IDEX WORDS: Deep Sub-Mcro bus, Poer reduco, Delay reduco

6 ACKOWLEDGEMETS I ould lke o hak my advsor Dr. A. P. Preehy for gvg me he prvlege o ork uder her supervso. I ll alays remember her kdess, help ad suppor. I ould lke o hak Dr. Seve W. McLaughl for all hs corbuos o hs ork ad hs valuable gudeles ad commes. I ould also lke o hak he commee members Dr. Saed Belkasm ad Dr. Ale Zelkovsky for reveg my hess ad her valuable commes. I especally a o hak Dr. Suderrama for all hs help, advce ad suppor durg my graduae sudes a GSU ad for all hs hard ork ad devoo o he Compuer Scece Deparme. v

7 TABLE OF COTETS ACKOWLEDGEMETS.. v LIST OF FIGURES.. v LIST OF TABLES v ITRODUCTIO TERMS AD DEFIITIOS.. 4. TRASMISSIO RATE. 4. -ARY VS. BIARY AMMIG WEIGT 4.4 PRASE REPRESETATIO. 5.5 LEXICOGRAPIC ORDERIG. 5 3 RELATED WORKS 6 3. DEEP SUBMICRO BUSES 6 3. EERGY COSUMPTIO I DEEP SUBMICRO BUSES DELAY I DEEP SUBMICRO BUSES 3.4 LOW TRASITIO-ACTIVITY ECODIG FOR DEEP DSM BUSES. 3.5 DIFFERETIAL LOW-WEIGT ECODIG APPROAC ECODIG SCEME DECODIG SCEME EERGY REDUCTIO BY DIFFERETIAL LOW WEIGT ECODIG 5 4 EERGY AD DELAY AALYSIS FOR DSM BUSES 6 v

8 4. COST FUCTIO FOR EERGY AD DELAY I DSM BUSES TRASMISSIO COST AALYSIS. 4.. COST COMPARISO FOR TRASITIO CODEWORDS C DISCOT AD C COT 4.. MEA TRASITIO COST FOR CODEWORDS TRASMISSIO COST FOR DIFFERETIAL LOW-WEIGT ECODIG TRASMISSIO COST FOR TE UCODED SCEME TE PROPOSED ECODIG SCEME LOW-WEIGT ORDER OPERATIOS FOR TE LOW-WEIGT ORDERED UMBERS LOW-WEIGT COUT LOW-WEIGT DIVISIO LOW-WEIGT ADDITIO LOW-WEIGT SUBTRACT TE PROPOSED ECODIG SCEME TE PROPOSED EUMERATIO SCEME USIG ELP CODEWORDS SIMULATIO RESULTS 49 6 COCLUSIOS. 5 BIBLIOGRAPY.. 54 APPEDIX 56 v

9 LIST OF FIGURES Fgure 3. Older slco echology..7 Fgure 3. Deep Sub-Mcro echology.. 7 Fgure 3.3 Appromae DSM bus model.9 Fgure 3.4 DSM bus model for a bus h o parallel les Fgure 3.5 a Trasm ucoded daa, b Trasm coded daa, c Sgalg..3 Fgure 3.6 The dffereal lo-egh ecodg ad decodg..4 Fgure 3.7 Eergy reduco by lo-egh ecodg ad decodg 5 Fgure 5. The Lo-Wegh cou algorhm 38 Fgure 5. The Lo-Wegh dvso algorhm 39 Fgure 5.3 The Lo-Wegh addo algorhm 43 Fgure 5.4 The Lo-Wegh subrac algorhm 44 v

10 LIST OF TABLES Table 4. Traso cos for C co ad dffere al bus saes. Table 5. Lo-Wegh orderg for eres of se P k, k Table 5. Mappg 4-b daa sequeces o 5-b codeords 46 Table 5.3 Comparso of dffere rasmsso mehods for 6-b bus..49 Table 5.4 Comparso of dffere rasmsso mehods for 3-b bus..5 Table 5.5 Comparso of dffere rasmsso mehods for 64-b bus..5 Table 5.6 Mappg 4-b daa sequeces o 5-b codeords 5 v

11 Chaper Iroduco Curre red of echology requres scalg do he sze of devces ad creasg he clock frequeces. Such scalg leads o epoeal crease leakage curre ad decrease ose mmuy for hgh speed crcus. Shrkg feaure szes resuls more ercoec levels packed closer ogeher. Reducg he re areas decreases he oal re capacace. oever, as echology scales do, o may les are solaed or shelded aymore. Iducve effecs ad couplg beee odes of he crcu crease ad become a mpora facor ercoeco aalyss. As frequecy of operao creases, addoal meal paers or groud plaes may be requred for ducve sheldg. As supply volage s scaled or reduced, crossalk has become a ssue for all clock ad sgal rg levels. I may cases, lke he deep submcro echology DSM, he poer dsspao, delay ad crossalk deped o he cross-acves beee he odes, as ell. For DSM buses, he couplg beee les er-re capacace s much sroger ha he couplg beee dvdual les ad groud ad he eergy cosumpo caused by parasc

12 capacace s o-eglgble. Thus, boh he rsc capacace ad parasc capacace should be ake o accou [, 8]. The ear erm soluo adoped by he dusry s he use of her meallzao, reduced aspec raos ad less aggressve scalg of delecrc cosa o decrease he capacace. There are some lmaos for fulfllg hs goal, as some requred characerscs of cera echologes lke area, curre per area, ec. may be volaed. A se soluo s o employ ecodg o reduce he effec of he capacace boh for he esg echologes, ad fuure echologes h smaller capacace. Ecodg corols he raso acvy, ad so he cross-couplg effec ad schg acves o he DSM bus by corollg he daa sequeces rasmed o he bus [, 7-5]. I [, 7], P. Sorads, A. Chadrakasa ad V. Tarokh dscussed o fudameal quesos. Wha s he mmum eergy requred per formao b for commucag hrough DSM buses? Ad, s he mmum eergy achevable usg codg? They roduced equaos for he mmum eergy requremes ad shoed ha he mmum eergy s asympocally achevable usg codg. They also proposed a smple dffereal codg scheme he dffereal lo-egh codg ha acheved mos of he possble eergy reduco. Ths effce scheme employs dffereal codg ad selecs raso codeords of smaller ammg eghs for rasmsso o he bus. Thus, s based o reducg he raso acvy o he bus. oever, hs codg scheme requres ables of sze O for a -b bus. Also, does o ake o accou some characerscs of he DSM buses ha ca be eploed o reduce he rasmsso cos furher. For eample, o a DSM bus f o adjace les chage boh from o or vce versa same dreco rasos, less eergy s cosumed compared o o rasos

13 o o ocosecuve les. Coversely, f o adjace les chage oppose drecos he more eergy s cosumed, compared o o rasos o o ocosecuve les. The same dreco ad oppose dreco bus rasos ca be eploed o acheve more eergy ad delay reduco o he DSM bus. I hs ork, e propose a eumerao mehod ha reduces he requred able-sze of he dffereal lo-egh ecoder from O ords o O ords, for a -b bus. Thus, he eumerao reduces he memory compley sgfcaly, especally he s large. Furhermore, faclaes less poer dsspao ad delay, as elmaes he eed for addressg ad fechg daa from very large ables memory. We modfy he eergy ad delay equaos for DSM buses furher ad develop a e represeao erms of umber of he same ad oppose dreco rasos o he bus. These equaos are very helpful for cos aalyss he DSM buses. We also develop equaos for compug he mea rasmsso cos per b o DSM buses for boh dffereal lo-egh ecodg ad ucoded schemes. We use smple operaos lke compleme ad roae, o compue help codeords from he se of useleced codeords o he fly, ad assg hem o he seleced codeords. oe ha he help codeords eable us o acheve more cos reduco, hou creasg he memory compley or umber of he bus les. The smulao resuls sho ha he proposed ecodg scheme acheves more cos poer ad eergy reduco, compared o he orgal dffereal lo-egh scheme. 3

14 Chaper Terms ad Defos I hs chaper, e prese some defos ad erms ha are used hroughou hs ork.. Trasmsso Rae The rasmsso rae R s compued as R m/, he m-b daa sequeces D {, } m are rasmed o a bus h > m les. Se {, } m cludes all bary sequeces of legh m ad so has m dffere elemes. I fac, rasmsso rae s a measure of ho may useful bs are rasmed o he bus durg each use of bus.. -ary vs. Bary A bary sequece of legh s composed of elemes he se {, }. For eample, s a bary sequece of legh 6. Smlarly, a -ary sequece of legh s composed of elemes he se {,, -}..3 ammg Wegh ammg egh of a gve bary sequece s he umber of oes ha sequece. For eample D has hammg egh equal o 4 ad e sho by W D 4. 4

15 .4 Phrase Represeao We may break each daa sequece of legh l o phrases, here each phrase cosss of d zeros ad ermaes h sgle ozero symbol for bary sequeces. The phrase represeao [6] uses he legh of each phrase o represe he daa sequece. Usg hs represeao, each l b bary daa sequece D s mapped o a uque phrase represeao p from he alphabe {, }, hose legh l s equal o he ammg egh of D [6]. We deoe he mappg fuco by f phrase ad e have f phrase D p. For eample, he phrases represeg he 8-b daa D are:. The phrase represeao for D s he sequece3 of legh 4. Thus, f phrase 3. For a gve l he fuco f phrase D s oe o oe ad has a verse. The verse fuco epads a phrase represeao o a uque bary sequece of legh l. For eample, for l 8, f - phrase 3..5 Lecographc Orderg Le {,, 3 } k deoe all he -ary permuaos of k symbols chose from he se {,, 3, } ad le S be a subse of hs se. By a lecographc orderg for S e mea he usual dcoary orderg h he assumpo ha < < 3 < < - [6]. Specfcally, for, k S ad y y, y y k S e have < y f here ess k such ha < y ad j y j for all j <. Le S, u deoe he umber of elemes S for hch he frs u coordaes are, u. The, he lecographc de or rak of s gve by [7]: S k j j S,,, j, 5

16 Chaper 3 Relaed Works I hs chaper, e roduce he Deep Submcro Buses DSM ad he mpora facors of eergy cosumpo ad delay o hese buses. We also epla ho o measure eergy ad delay DSM buses. We prese he cocep of lo raso-acvy ecodg [, 7-5] for reducg poer cosumpo ad delay o DSM buses. Fally, e epla deal, he dffereal lo-egh ecodg [] ha s a smple ad effce lo raso-acvy scheme upo hch e base our proposed ecoder. 3. Deep Submcro Buses Curre red of echology requres scalg do he sze of devces ad crease clock frequeces. Such scalg leads o epoeal crease leakage curre ad decrease ose mmuy for hgh speed crcus. Shrkg feaure szes resuls more ercoec levels packed closer ogeher. Reducg he re areas decreases he oal re capacace. oever, as echology scales do, o may les are solaed 6

17 or shelded aymore. Couplg beee odes of he crcu creases ad becomes a mpora facor ercoeco aalyss. Fgures 3. ad 3. compare he old slco echology ad he deep submcro echology DSM [, 8, ad ]. Fgure 3. The older slco echology [8]. Fgure 3. The Deep Sub-Mcro echology [8]. 7

18 3. Eergy Cosumpo Deep Submcro Buses Cosder a bus le carryg a bary sequece,,. The eergy cosumpo a me k for chargg ad dschargg he parasc capacace beee he le ad groud depeds oly o he o values k- u ad k u. I oher ords, he raso cos depeds o he al ad fal values of he bus. Equao 3. geeralzes hs propery o bus les. I hs equao, u [ u u ] s he al sae of he bus, ad u u u u ] s he fal sae. [ u T a u u 3. The raso acvy T a of a crcu ode or le s a useful poer measure he he ode le s decoupled from oher acve odes he crcu []. For such odes he poer cosumpo ca be smply compued by E T a C L V dd /. Equao 3. compues he eergy cosumpo for a bus h les. I hs equao C L s he capacace beee he ode ad groud. E z T C V a L dd / u u C V L dd / 3. Couplg beee odes mples ha poer dsspao depeds also o her crossacves ad herefore, Equao 3. s o vald aymore. I may cases lke he deep Submcro Buses, couplg beee les s eve sroger ha he couplg beee a le ad he groud. 8

19 9 Fgure 3.3 shos a appromae model for a DSM echology bus [-3]. Ths model cosders he boudary capacors due o frgg effecs, capacors due o couplg beee a le ad he groud, ad couplg beee o adjace les. Fgure 3.3 A DSM bus model [3]. The eergy cosumpo durg raso from he al bus sae u o he fal bus sae u s gve by [3, 6]: / dd T T V u u C u u u u E 3.3 Equao 3.4 shos he mar C, here L I c c. c I s he er-re capacace per u legh beee adjace les ad c L s he parasc capacace per u legh beee each le ad groud. The cosa s a real o-egave parameer ha depeds o physcal parameers of les, such as geomery, sze, dsace beee he les ad he maufacurg echology. For moder deep sub-mcromeer echologes, ca be as hgh as 8 for eample.3- m echologes. c L C. : : : : : : 3.4

20 For he obsolee o-submcro SM echologes s praccally zero ad C reduces o a scalar mar. Thus, he case of DSM echologes >, ulke he case of SM, he cos fuco has erms correspodg o eracos beee he values rasmed o dffere les. Fgure 3.4 shos more deals of he appromae model of a DSM bus h o parallel les [4]. There s a drver for each le Each le s coeced o V dd. A drver ca be modeled by a volage source h a seres ressace r d. Fgure 3.4 DSM bus model for a bus h o parallel les [4]. 3.3 Delay Deep Submcro Buses A sgfca ose source curre processes s he couplg beee les. Ths couplg s eve sroger ha he couplg beee a le ad he groud. Cosder a bus h les, each of legh L. Le u [ u u ] be he al codo of he bus, u k Vk L uk here,, {, } u s he volage a he sar of he le k of he bus. Also,

21 ] [ u u u u s he fal codo of he bus, here } {,,, k k k u L V u s he volage a he ed of le k of he bus Fgure 3.4. The delay of each le k s ofed h D k. The sum of he delays he bus s compued by Equao 3.5, as follos [4, 5]. I hs equao, r s he ressace of le per u legh. Also, r T s a cosa depedg o ressace of each le ad he le drver per u legh, ad he le legh. T T k k r L u u C u u u u D, 3.5 r L r r d T Equao 3.6 shos he mar C. I hs equao, L I c c, c I s he er-re capacace per u legh beee adjace les ad c L s he parasc capacace per u legh beee each le ad groud. c L C. : : : : : : Lo Traso-Acvy Ecodg for Deep DSM Buses Equaos 3.3 ad 3.5 sho ha for DSM buses, eergy ad delay deped o he sequece of daa rasmed o he bus. Thus, e may decrease eergy ad delay o he bus by corollg he sequece of daa rasmed o he bus.

22 Ecodg s a effce ay for corollg he daa sequece ha s rasmed o he bus. Usg ecodg e may map he hgh cos codeords o some oher codeords h loer coss. Oe approach s o add redudacy he form of era bus les. To be effecve, he ecoder ad decoder sysems should have small eergy cosumpo ad delay values [, 7-5]. Defo : Code C s a mappg from elemes se Q m {, } m o he elemes se Q {, }, here > m. Each sequece d Q m s mapped o oe ad oly oe eleme d Q ha sasfes a propery of eres. As a resul, some elemes of Q are o used a all. Cosder a bus h les ad le Q {, } be he se of all bary vecors of legh. I a ucoded scheme, ay daa sequece d Q ca be rasmed o he bus. Fgure 3.5 depcs boh ucoded ad coded rasmsso o he bus [9]. Addg era les o he bus, hle he daa sream s uchaged creases he capacy of he rasmsso chael he bus. Cosder a -b bus, hch s supposed o rasm daa sequeces d Q. The bus has he poeal o rasm dffere sequeces, hle here are oly m < dffere daa sequeces. Thus, some bus saes are o used. Oe may choose he ucoded bus saes o be he epesve oes.

23 Fgure 3.5 a Trasm ucoded daa, b Trasm coded daa, c Sgalg. [9] 3.5 Dffereal Lo-Wegh Ecodg Approach 3.5. Ecodg Scheme The Bus sae a me s he sequece hch s o he bus a me. The ecodg scheme [] cludes he follog seps:. Eed he bus from m les o > m les.. Choose m codeords ou of all possble codeords of legh > m such ha he ammg egh for he seleced codeords are smaller ha a gve hreshold. choose codeords of lo ammg eghs. 3

24 3. Map each bus pu daa d of legh m o a vald codeord c of legh, usg fuco F. Fuco F uses a able o complee he mappg. 4. XOR s, he bus sae a he curre me h he codeord correspodg o he bus pu daa. So, pu s c s o he bus. As he code ords have small egh, here ll be small umber of rasos o he bus, ad so he rasmsso cos decreases. Fgure 3.6 depcs he ecodg ad decodg schemes he dffereal lo-egh ecodg approach Decodg Scheme Each receved codeord s s decoded [] o oba he correspodg pu daa, usg he follog seps:. Use delay o access he prevous bus sae s ad compue codeord c s s.. Use he fuco F o fd he correspodg daa pu d for codeord c. Fgure 3.6 depcs he ecodg ad decodg schemes he dffereal lo-egh ecodg approach. Fgure 3.6 The dffereal lo-egh ecodg ad decodg []. 4

25 3.6 Eergy Reduco by Dffereal Lo Wegh Ecodg Fgure 3.7 shos smulao resuls for he dffereal lo egh scheme. I hs fgure m/ s he code rae umber of useful bs rasmed per bus use. E b Y s eergy per b for he ecoded scheme ad E b X s eergy per b for ucoded daa. oe ha E b Y E b X he o bus les or oo may bus les are added. For eample, for 4-b daa he opmal eergy reducos happes a rae m/.5, here 4 ou of see 4-b sequeces are chose as codeords. Fgure 3.7 Eergy reduco by lo-egh ecodg ad decodg []. 5

26 Chaper 4 Eergy ad Delay Aalyss for DSM Buses I hs chaper, e aalyze he rasmsso eergy ad delay equaos ad sho ha he eergy cosumpo ad delay for a gve DSM bus are dffere oly by a scalar erm. We roduce a commo cos fuco hch ca corol boh. Reducg he cos fuco by ecodg or ay oher approach resuls boh eergy ad delay reduco. We modfy he eergy ad delay equaos for DSM buses ad develop e represeaos erms of umber of he same ad oppose dreco rasos o he bus ad use hem our ercoec aalyss. The, e drve closed form formulas for compug he mea cos per b over all possble al bus saes, for boh dffereal lo-egh ecodg [] ad ucoded approaches. 4. Cos Fuco for Eergy ad Delay DSM Buses Usg he equaos roduced Chaper 3, he eergy cos for raso T from he al bus sae, o he fal bus sae z z, z z s compued by Equao 4., here z, {, }, T z-, he cosa K / s a scalar depedg o he drver volage ad C s he capacy mar. V dd 6

27 7 K C T f z E, T C T C T f T, 4. C L c. : : : : : : Smlarly, he delay s compued by Equao 4., here T z- ad T r L k ' s a scalar depedg o he legh ad ressace per legh for each bus le. oe ha he capacace mar C ad he scalars K ad K deped o physcal characerscs of he bus. ', K C T f z D k k 4. T C T C T f T, C L c. : : : : : : Equaos 4. ad 4. dcae ha for a gve DSM bus, he eergy ad delay are dffere oly by a scalar erm ad deped o he fuco f T, C. Ths fuco s a measure of he rasmsso eergy ad delay. We call he cos fuco ad e focus o decreasg he rasmsso eergy ad delay cos o he DSM bus by decreasg hs

28 8 cos fuco. Cos fuco f T, C depeds o he sequece of he daa rasmed o he bus. Thus, e may decrease f T, C ad cosequely, he rasmsso eergy ad delay o he DSM buses by corollg he sequece of daa rasmed o he bus. For a DSM bus h les, he raso vecor s deoed by T,, here shos he raso acvy for le. Therefore, f T, C ca be compued as follos: T C T C T f T, [ ] L c :. : : : : : : Dvde boh sdes of Equao 4.3 by cosa c L resuls Equao 4.4. T L C C T f :. : /, Ad, e complee he mar mulplcao ad smplfy furher o ge Equao 4.5.

29 /, L C C T f L C C T f /, 4.5 Kog ha T, z-, each for could have he follog values: o from raso z ad f o from raso z ad f o raso z f 4.6 Usg Equaos 4.5 ad 4.6, e ca see ha he o-raso case leaves C L C T f /, uchaged, ad so mposes o era eergy or delay cos o he sysem. oever, each raso o a le adds as much as o he cos fuco. Moreover, each same dreco raso o adjace les boh les go from o or vce versa decreases he cos by, hle a oppose dreco raso o o adjace les oe goes from o ad he oher from o creases he cos by. I geeral, f codeord c resuls T rasos o he DSM bus from hch rasos are he same dreco ad - he oppose dreco, he he follog formula gves he oal raso cos o DSM buses for codeord c. T C C T f L /, 4.7

30 Our goal s o employ ecodg o map hgh cos daa sequeces h large cos fuco values o codeords h loer coss. Thus, e are eresed codeords ha resul mmum umber of rasos less raso acvy. Meahle, e lke he same dreco rasos o happe o adjace les ad he oppose dreco rasos o o-adjace les, as far as possble. O he oher had, e cao be very selecve o he codeords, as decreases he umber of codeords seleced for rasmsso o a DSM bus of a gve legh ad so decreases he rasmsso rae. 4. Trasmsso Cos Aalyss I hs seco, e compare he mea cos for dffere raso codeords of ammg egh o vesgae ho he locao of oes he codeord affecs he rasmsso cos. Le deoe a ru of > zeros. C co deoes a raso codeord h a couous ru of s ad C dsco deoes a raso codeord for hch here s a leas oe zero beee ay o cosecuve s. Recall ha a raso vecor h adjace s may lead o same dreco or oppose dreco adjace rasos based o he al bus sae. The queso s ho does he raso codeords C co ad C dsco compare, erms of rasmsso cos. We sar h compug he mea cos for a gve codeord of ammg egh over dffere al bus saes. For a raso codeord h ammg egh, here are eacly ozero rasos. Suppose ha he umber of he same dreco ad he oppose dreco rasos are ad -, respecvely. Thus, usg Equao 4.7, he cos fuco ca be compued as follos.

31 f T, C / C L 4.8 I he follog secos, e frs gve closed form equaos for he mea cos per b of raso sequeces lke C co ad C dsco, ad he e geeralze hese equaos for codeords of ay gve form. 4.. Cos Comparso for Traso Codeords C dsco ad C co For C dsco, he raso cos s smply depede of he al bus sae S o. For C co hoever, S o s a mpora facor. Ay zero s equvale o o raso ad so o cos. For adjace s, e should defy ho may rasos are he same dreco ad ho may are o. oe ha for a codeord of egh, here may be a mos - pars of cosecuve rasos, some of hch are he same dreco ad some he oppose dreco. The umber of all possble combaos of selecg ou of objecs s compued by!!!. Smlarly, gves he umber of all possble combaos of - oppose dreco rasos ou of - pars of cosecuve rasos. Mulplcao by a facor of s requred, as he rasos could be from o or from o. Table 4. summarzes dffere values of, as ell as he cos per raso codeord for dffere values of - ad -. For eample, cosder he raso sequece C co. There are possble al bus saes, amely S ad S ha resul - ad

32 cosequely. There are four possble values ha resul - ad so, hch are S, S, S, S. Fally, S ad S resul - ad. - for - oppose dreco #of possble combaos rasos Traso Cos per codeord compued by Equao Table 4. Traso cos for C co ad dffere al bus saes. Recall he follog properes of he combao operao. 4.9, 4. For odd values of, e have: k / / 4.

33 3 k / / 3 4. Combe Equaos o oba he follog equao. k / 4.3 Smlarly, for eve values of, e have: k / / / 4.4 k / / / / / / Ad fally, combe Equaos 4.9, 4. ad 4.4 o oba he follog equao. k / / 4.6

34 4 Usg Table 4., he oal cos over all possble al bus saes, deoed by COST s compued as follos: 5 3 COST 4.7 Use he assocave propery of summao o regroup he erms parehess of Equao 4.7 ad oba o groups as facors of ad. COST 4.8 Usg Equao 4.9 e oba: COST 4.9

35 5 o, e smplfy he summao erm Equao 4.9 o smplfy COST furher. For odd values of -, replace h - Equao 4.. / / 3 4. Ad use Equao 4. o ge: / / 3

36 6 So, e fally oba: k 4. Smlarly, for eve values of - e oba Equao 4. as follos. / / / / / / 3 k 4. Equaos 4. ad 4. dcae ha: 4.3

37 Combg Equaos 4.9 ad 4.3, resuls : COST 4.4 There are dffere al bus saes S. Therefore, he mea cos for C co over dffere al bus saes, deoed by CosCco s compued as follos. Cos Cco COST / 4.5 Equao 4.5 shos ha he mea raso cos over all possble orgal bus saes S o s smply, for boh C dsco, ad C co. 4.. Mea Traso Cos for Codeords We ca geeralze he above propery o ay codeord of hammg legh, ad of he geeral from C l, here ad deoe a ru of > zeros ad oes, respecvely. The eger l shos he umber of rus of oes umber of s he codeord. Theorem : The mea cos per b, over all possble al bus saes s he same for all codeords of equal ammg egh, ad s compued by. 7

38 Proof: We use mahemacal ferece o prove ha he cos per b for C geeral s equal o. a. I Seco 4.. e shoed ha he mea raso cos over all possble orgal bus saes S o for C C co s. Thus, he al ferece codo s sasfed. b. We sho ha f he mea raso cos over all possble orgal bus saes for C k s he s so, for C k C k, as ell. The proof s as follos. Table 4. dcaes ha he rasmsso cos may be oe of he values c, c 3 c - - based o he umber of he same ad oppose dreco rasos. Suppose s he umber of al bus saes ha resul cos c ad he ammg egh for C k deoed by C k s equal o. For raso sequece C k, e have: COST Accordg o he ferece assumpo, COST 4.7 8

39 Ad so, he rgh had sdes of Equaos 4.6 ad 4.7 are equal ad e have he follog: Suppose he legh for he k h ru of oes C k s. Thus, he ammg egh for C k s ad from Equao 4.5 he follog Equao holds: Cos Ck COST / Ck 4.9 Usg Equao 4.6 h, for oly he las ru of oes C k, e have: To compue he oal cos for C k C k, e should cosder C k, as ell as he las ru of oes. We compue he cos for each of hese separaely, ad he add up he dvdual coss o oba he oal cos. Equao 4.3 gves he oal cos for C k. 9

40 COST C 4.3 Usg assocave propery of he summao, e place he frs erms he ere parehess oe group, ad he secod erms aoher oe o oba: COST Ck

41 3 The, e smplfy he erms sde he brackes usg Equaos 4.9 ad 4.3 ad facor hese erms ou. [ ] 3 C k COST oe ha ad e ca smplfy he erms sde he brackes, usg Equao 4.8. COST C k 4.3 Ad fally from Equao 4.9, / Cos k Ck C k C COST 4.33 Equao 4.33 proves sep b of he ferece ad so e ca coclude ha he mea raso cos per b, over all possble al bus saes S o s smply, for ay gve codeord C l.

42 3 4.3 Trasmsso Cos for Dffereal Lo-Wegh Ecodg I hs scheme, oly he codeords h ammg egh smaller ha a hreshold ma are seleced. As eplaed he prevous seco, he mea cos over al bus saes s he same for all he codeords of he same ammg egh ad ca be compued by. Thus, mea eergy per b for a dffereal lo-egh code of legh s as follos: ma ma ma ma b E Ad so, ma ma b E 4.34

43 Trasmsso Cos for Ucoded Scheme I he ucoded scheme, every sequece he se S {, } may be rasmed o he bus. Thus, he raso vecors are all elemes of se S ad he mea eergy per b s compued as follos:.5 ma b E 5. b E 4.35 As Equao 4.35 shos he mea cos per b s a cosa depedg o he o ecodg s used. oe ha rasmsso rae hs case s equal o. Thus, he rae s deal, hle he cos s hgh compared o he ecoded case.

44 Chaper 5 The Proposed Ecodg Scheme As eplaed he prevous chapers, he dffereal lo egh ecodg s a effce ad lo compley scheme ha acheves mos of he rasmsso cos reduco possble, o DSM buses []. I hs chaper, e propose a eumerao mehod o be used h he dffereal loegh codg. Ths eumerao mehod reduces he mappg able sze from O ords o O ords, for a -b bus. Thus, reduces he memory compley sgfcaly, especally he s large. Furhermore, faclaes less poer dsspao ad delay, as elmaes he eed for addressg ad fechg daa from very large ables memory. 34

45 Furhermore, e eplo he same dreco ad oppose dreco rasos o acheve more cos reduco. For each seleced lo egh codeord, e compue a help codeord from hose hgh egh codeords hch are o seleced o he fly. The help codeords resul loer cos for some al bus saes. They are compued by smple operaos lke compleme ad roae ad so reduce he cos hou ay era memory or bus les. Ths chaper s orgazed as follos. Frs e propose a e se for represeg he codeords ad he Lo-Wegh order for hs se. We also defe a umber of operaos o hs se. The, e epla he proposed ecodg scheme ad applcao of he Lo- Wegh order he proposed eumerao process, deal. A he ed, e prese he elp codeords ad smulao resuls. For he erms ad defos used he follog secos, refer o Chaper. 5. Lo-Wegh Order Le se P k {,, 3 } k deoe all he -ary permuaos of k symbols o ecessarly dsc chose from he alphabe {,, 3 } such ha he sum of he dgs of each permued sequece s smaller ha or equal o. Thus, permuao p P k s of legh k ad may be deoed as d, d d k here d {,, 3 }. For smplcy, e deoe p h d d d k he < o comma beee symbols. We defe he se P k, k as he uo of ses P, P ad P. Usg he phrase represeao roduced Chaper.4, each permuao p P k may be epaded o oba a bary sequece c. oe ha he cosra o he sum of he symbols of p P k guaraees ha he legh of c s smaller ha or equal o ad rasmg o a DSM bus requres o more ha les. I oher ords, he legh of he bary sequece correspodg o 35

46 each permuao s lmed o. We requre hs cosra, as e have a lmed umber of bus les for daa rasmsso. Recall he lecographc orderg roduced Chaper.5. Ths orderg scheme cosders he rghmos dg as he leas sgfca dg LSD ad he lefmos oe as he mos sgfca dg MSD, compug he lecographc de rak. Aleravely, e defe he lo-egh order for hch he rghmos symbol s he MSD ad he rak for, k P k, k s compued by he Equao 5.. I hs equao, k S, u deoes he umber of elemes P k, k for hch he frs u coordaes are, u. Also, P k deoes he cardaly of se P k. S k h h k j k S j P,,,, 5. j oe ha eed he Lo-Wegh order eeds o dffere values of k. Specfcally, for, k P k ad y y, y y k P k e have < y f k < k. So, f - phrase has a loer ammg egh compared o f - phrasey. We eplo hs orderg o decrease he rasmsso cos by choosg he loer egh codeords before ad sooer ha he hgher egh oes. For eample, Table 5. shos he elemes of se P k h k 5 ad from alphabe {,, 3, 4, 5}, sored Lo-Wegh order. 36

47 de p Ide p Table 5. Lo-Wegh order for eres of se P k, k Operaos for Lo-Wegh Ordered umbers 5.. Lo-Wegh Cou Lo-Wegh cou eables us o fd he -b codeord C ha follos appears codeord afer a gve -b codeord C, Lo-Wegh order. The algorhm s depced Fgure 5.. I hs algorhm C deoes he f - phrase of a eleme p se P k {,, 3 } k, k. 37

48 f C ermaes /*case */ Shf C o rgh by o oba C. else Fd he frs phrase p C. f here ess a afer phrase p C /*case */ C Shf oly phrase p o lef by else /*case 3*/ C Iser a a he begg of he sequece ad remove a p C Move he zeros of he frs phrase C o he ed of he sequece. Fgure 5. The Lo-Wegh cou algorhm. Eamples: Case. I Table 5., he h ery s C f - phrase ad he h ery s C f - phrase 3. Case. I Table 5., he 9 h eleme s C f - phrase 4, p bs o 4 of C, C ad C f - phrase. oe ha s he h ery Table 5.. Case 3. I Table 5., he 5 h eleme s C f - phrase 3, p bs 3 o 5 of C, C ad C f - phrase. oe ha s he 6 h ery Table

49 5.. Lo-Wegh Dvso Fgure 5. shos he dvso algorhm. The Lo-Wegh dvso of a eger by a ls of egers y, y y k s deoed by f dv, y, y y k. Smlar o ormal eger dvso, hs operao s mplemeed by successve subracos. The oly dfferece s ha sead of subracg a fed eger dvded all he me, each me oe of he elemes he dvded ls s subraced from he dvder. The operao f dv, y, y y k resuls he quoe q ad remader r f ad oly f y y y q r, here r < y q. /*Lo-Wegh Dvde algorhm*/, q Whle y - y q q ; r ; Fgure 5. The Lo-Wegh dvso algorhm Lo-Wegh Addo Cosder he frs vecor of egh Lo-Wegh order for dffere values of. For eample, for 4, hese vecors are,,, ad. Recall ha he correspodg bary codeords are,,,, respecvely. We deoe such codeords h o dcae ha he umber of oes s. 39

50 Lo-Wegh addo eables us o fd he -b codeord C ha appears a gve eger base codeords afer C, he frs codeord of egh Lo- Wegh order. Ths operao s requred for he eumerao process ad e epla, deal. oe ha he umber of ralg zeros for f s -. We defe y -. I fac, y - s he umber of ralg zeros for f. Cosder ad s phrase phrase phrase C represeao p d, d d, here d for. The follog rules are used o add p h a a eger base he Lo-Wegh addo. Rule : f a {, y-} he add eger a o he lefmos phrase represeao p. Rule : f a y y for y, he he d lefmos dg he phrase represeao p s creased by. Defe f y y y y, y Rule 3: f X f for y, he he 3rd lefmos dg he phrase represeao p s creased by y-. Defe y y f f, 4

51 y Rule 4: f X f for y, he he 4 h lefmos dg he phrase represeao p s creased by y-. Defe f y f, k, k y k y The fal Rule: f X f for y, he he h las lefmos dg he k phrase represeao p s creased by y-. oe ha f y f y f y for, k. For eample, cosder p d d d 3 d 4, he phrase represeao of c for 9 ad Ls a he ed of hs Chaper hch shos all codeords h 4 ad 9 Lo-egh order. Recall ha y - 6. If X he d creases by. If X he d creases by. If X y - 5 he d creases by 5. If X y 6 he d creases by. If X y y 6 5 he d creases by. If X y y he d creases by 5. 4

52 If X f y 6.7/ he d 3 creases by. If X f y f y 5 36 he d 3 creases by. If X f y f y f he d 3 creases by 5. If X f y he d 4 creases by. If X f y f y he d 4 creases by. If X f y f y f he d 4 creases by 5. For ordary egers of ay base, ay gve dg creases over equal dsaces deermed by base. For eample, le be a aural umber. The eger appears egers afer base. Smlarly, he eger 3 appears egers afer. oever, usg he Lo-Wegh order, he dgs of phrase represeao p of a gve bary sequece c, do o crease o equal ervals. Ths fac s dcaed by he rules of he Lo-Wegh addo roduced above. For eample, cosder Ls a he ed of hs chaper. There are 5 egers beee ad, hle egers appear beee ad 3. Alhough a fed erval over hch a gve dg creases does o es, s possble o predc ho he erval chages for each dg of p as he above rules mply. To crease phrase p of legh by X base, e compue he amou of crease for he rghmos dg p he mos sgfca dg Lo-Wegh order sequeces 4

53 usg he Lo-Wegh dvso operao. The, e updae X, ad repea he same procedure for he follog dgs ul he lefmos dg he leas sgfca dg Lo-Wegh order sequeces s processed or X s zero. The follog algorhm complees Lo-Wegh addo O me ad O space. /*Lo-Wegh crease by X algorhm*/ /*X - */ /* p d d d, d for */ /*Fd y, he umber of ralg zeros of he */ /* sequece represeed by p*/ y - for each dg d, Y y, y-, y-, for couer o - Y y j Y j Evaluae f dv X, Y o ge q ad r /* Updae X */ Xr /* Updae phrase p*/ d d q /* Updae y */ y y q d d r Fgure 5.3 The Lo-Wegh addo algorhm. 43

54 5..4 The Lo-Wegh Subrac Cosder he phrase represeao p as he frs vecor of egh Lo- Wegh order. Also, cosder p as a gve codeord of same ammg egh as p. Suppose ha codeord p appears c codeords afer p, Lo-Wegh order. The Lo-Wegh subrac operao akes p ad p ad compues c, he dsace beee p ad p. /*Lo-Wegh subrac algorhm*/ /* p d d d, d for */ y - for each dg d, c Y y, y-, y-, for couer o - Y y j Y j c c d j Y j d Fgure 5.4 The Lo-Wegh subrac algorhm. 5.3 The Proposed Ecodg Scheme For a rasmg m-b daa e requre a leas m bus les. We crease umber of bus les o > m les, such ha m/ s greaer ha or equal o he desred rasmsso rae. The ecodg dea s o fd a fuco ha maps a m-b daa sequece o a lo ammg egh -b codeord. 44

55 Le L be he ls of all bary sequeces {, } m, ormal ascedg order. For eample, usg a deg sysem sarg a zero, he de for he all zeros sequece s zero ad he de for he all oes sequece s m -. Also, ls L s he ls of all -ary sequeces P k, k, Lo-Wegh order. We epad usg f - phrase ad map he h eleme of L as he codeord for he h eleme of L, for m -. Table 5. shos he 5-b codeords assged o 4-b daa sequeces. oe ha he codeords are 5-b bary sequeces h lo, ad ammg eghs. Ths ecodg scheme ca be compleed o he fly, usg eumerao. 5.4 Eumerao Scheme The queso s ho o map a gve m-b sequece o a -b codeord o he fly hou sorg he daa, codeord pars a able. Recall ha he able sze gros epoeally h daa legh ad elmag he able s so useful erms of memory compley. Le L be a ls hose k h eleme s he umber of -b bary sequeces h eacly k oes ammg egh k. We deoe he k h eleme of L by, ad form aoher k ls L cumulave by cumulave summao of ls L. Thus, he k h eleme of ls L cumulave s k k ad shos he umber of codeords h k. Ths ls gros learly h ad does o eed a large memory space. Furhermore, based o Equao 5., e may keep oly half of he ls ad compue he res o he fly ad save more space. k k, k k 5. 45

56 Ide L Bary Daa L Codeord Phrase Codeord Represeao Table 5. Mappg 4-b daa sequeces o 5-b codeords. The proposed ecodg scheme allos replacg he able of daa, codeord pars h L cumulave ls ad eables eumerave codg. The eumerao scheme cludes he follog seps for mappg daa D o codeord C.. Use L cumulave ls o fd he ammg egh of codeord C by Equao 5.3. k D < k k k C k 5.3. Use L cumulave ls o fd he umber of codeords h ammg eghs smaller ha he ammg egh of codeord C deoed by C. L cumulave C Form he frs codeord C h ammg egh Lo-Wegh orderg. 46

57 C A srg of oes folloed by - zeros h he geeral form:,,, 4. Fd he codeord ha appears c D - codeords afer C Lo-Wegh order. For eample, cosder fdg he 5-b codeord correspodg o he 4-b daa D hch s decmal. L {, 5,,, 5, } L cumulave {, 6, 6, 6, 3, 3} I hs eample, 6 D < 6, D ad L cumulave, C, ad D - 9. Thus, o fd he correspodg codeord for D e should fd he codeord C ha appears 9 codeords afer C Lo-Wegh orderg. Thus, C, as Table 5. shos. Smlarly, a he decoder sde, e ca use eumerao o fd he daa D for a receved codeord C, o he fly. The dfferece s ha e have o use C ad compue s Lo- Wegh rak hch s he correspodg daa. Recall ha a he ecoder e used he daa rak o compue C. The follog seps complee he eumerao for a he decoder:. Cou he umber of oes of codeord C o fd s ammg egh C.. Use L cumulave ls o fd he umber of codeords h ammg eghs smaller ha C. L cumulave C Form C as before, a he ecoder. 4. Codeord C appears c codeords afer C. Use he Lo-Wegh Subrac o fd c. 47

58 5. Compue D c. 5.5 Usg help Codeords We shoed ha he codeords of loer ammg eghs have less mea rasmsso cos per b, compared o larger egh oes. O he oher had, for some al bus saes, a hgh egh raso codeord may resul smaller cos. For eample, f he al bus sae S he he raso codeord c s more cosly ha c, hle has a loer ammg egh. Therefore, afer selecg he lo egh codeords, e assg a help codeord from he useleced hgher egh codes o each seleced lo egh codeord. Before rasmsso o he bus, e compue he rasmsso cos for boh he codeord ad he help codeord. The codeord hch resuls smaller cos s placed ad rasmed o he bus. Ths approach helps o ga more cos reduco, compared o he orgal dffereal lo-egh codg scheme. We add aoher sep o hs o reduce he cos more. Cosder he phrase represeao p p p p k of he era codeord assged o a gve lo egh codeord. Phrase roao of p o he rgh o oba p p k p p p k- s helpful o reduce he cos, as ell. For eample, cosder codeord c he compleme s c ad p, hch cao help reduce he rasmsso cos of c for ay al bus saes. Phrase roao of p hoever, resuls p, or equvalely, hch has a smaller rasmsso cos for al bus saes lke cccc, here here are o resrcos o each ad ay ru of cosecuve bs depced by c mus be all or all. 48

59 5.6 Smulao Resuls We employed he proposed ecodg mehod o rasm daa o 6-b, 3-b ad 64- b buses a he mamum possble rae, here oly oe era le bus s added. Tables compare he ucoded ad he dffereal lo-egh mehods h he proposed mehod. As he ables sho, all cases he proposed mehod resuls beer cos poer ad delay reducos. We ca acheve more poer reduco by addg more bus les. The ables sho ha he proposed mehod resuls poer reduco, for he mos lmed case ors case, here oly oe era bus le s alloed. Moreover, he proposed mehod requres O ables, here s he mamum ammg egh of he seleced codeords. Recall ha he lo-egh ecodg scheme requres O ables for -b buses. Thus, here s a cosderable reduco sze of ables ad he sysem memory requremes. Mehod k E/codeord E/b E ormalzed Ucoded Df. lo-egh Proposed Table 5.3 Comparso of dffere rasmsso mehods for 6-b bus. Accordg o Table 5.3, he proposed ecodg acheves 8.5% compared o ucoded ad / % mproveme compared o he dffereal lo-egh ecodg. Ths able shos he smulao resuls for e6 radomly chose pu samples. 49

60 Mehod k E/codeord E/b E ormalzed Ucoded Df. lo-egh Proposed Table 5.4 Comparso of dffere rasmsso mehods for 3-b bus. Table 5.4 shos ha he proposed ecodg acheves 4.5% compared o ucoded ad -5.6/ % mproveme compared o he dffereal lo-egh mehod. Ths able shos he smulao resuls for 8e6 radomly chose pu samples. Mehod k E/codeord E/b E ormalzed Ucoded Df. Lo-egh e Proposed e Table 5.5 Comparso of dffere rasmsso mehods for 64-b bus. Table 5.5 shos ha he proposed ecodg reduces he rasmsso cos.87% oally ad -3.95e/3.373e.6% more ha he dffereal lo-egh 5

61 mehod for a 64-b DSM bus. Ths able shos he smulao resuls for e6 radomly chose pu samples. We also compared o mehods for fdg he helpg codeords: compleme ad phrase roae ad compleme oly. Table 5.6 shos he resuls for a 8-b bus. As he able dcaes he compleme ad phrase roae mehod resuls beer poer reduco compared o compleme oly. Era Codeord Seleco Mehod Comp. & Phrase Rae E/ codeord E/b Improveme vs. Df. Lo-egh 8/ / % Roae Compleme 8/ / % Table 5.6 Mappg 4-b daa sequeces o 5-b codeords. oe ha our smulaos are for he hghes possble rae ere oly oe era le s added. We ca acheve more reduco by addg more les. oever, e cao add oo may les as decreases he rae he umber of useful rasmed bs per bus use. For opmal umber of bus les refer o Seco

62 Chaper 6 Coclusos I he deep submcro echology DSM, he couplg beee les er-re capacace s much sroger ha he couplg beee dvdual les ad groud ad he eergy caused by parasc capacace s o-eglgble [-6]. Equaos of he DSM buses dcae ha he eergy ad delay deped o he sequece of daa rasmed o he bus. Ecodg s a effce ay for corollg he rasmed sequece o he bus ad may ecodg schemes have bee used o reduce eergy ad delay o DSM buses, he leraure [7-5]. oe ha o be effecve, he ecoder ad decoder sysems should have small eergy ad delay values. The dffereal lo-egh codg [] s a smple dffereal scheme ha acheves mos of he possble eergy reduco, from a formao heory po of ve. Ths effce scheme employs dffereal codg ad selecs raso codeords of smaller ammg eghs for rasmsso o he bus. Thus, s based o reducg he raso acvy o he bus. oever, hs codg scheme requres ables of sze O for a - le bus, hch s a sgfca overhead, especally he he umber of bus les s 5

63 large. Also, does o ake o accou some characerscs of he DSM buses ha ca be eploed o reduce he rasmsso cos furher. We proposed a eumerao mehod ha reduces he requred able-sze of he dffereal lo-egh ecoder from O ords o O ords, for a -le bus ad acheves cosderable ga erms of memory compley. Furhermore, faclaes less poer dsspao ad delay, as elmaes he eed for addressg ad fechg daa from very large ables memory. We also defed a cos fuco hch could corol boh he eergy ad delay o DSM buses. We modfed he eergy ad delay equaos of DSM buses ad developed a e represeao erms of umber of he same ad oppose dreco rasos o he bus. We used hese equaos our ercoec aalyss o reduce he rasmsso cos furher ad o develop closed form formulas for compug he mea rasmsso cos per b o DSM buses for boh dffereal lo-egh ecodg ad ucoded schemes. We employed smple operaos lke compleme ad roae, o compue help codeords from he se of useleced codeords o he fly, ad assg hem o he seleced codeords. oe ha he help codeords eable us o acheve more cos reduco, hou creasg he memory compley or umber of he bus les. The smulao resuls for 6-b, 3-b ad 64-b buses a mamum rae oly oe era le added sho ha he proposed ecodg scheme acheves more ha % cos reduco, ad performs more ha.5% beer ha o he orgal dffereal loegh scheme, he ors case. 53

64 Bblography. P. Sorads, V. Tarokh, A. Chadrakasa, Mamum achevable eergy reduco usg codg h applcaos o deep sub-mcro buses, IEEE Ieraoal Symposum o Crcus ad Sysems, Vol., pp. I-85 - I-88, May.. D. Sylveser, Wu Chemg, Aalycal modelg ad characerzao of deepsubmcromeer ercoec, Proceedgs of he IEEE, Vol. 89, o. 5, pp , May. 3. P.P. Sorads, A. Chadrakasa, A bus eergy model for deep submcro echology, IEEE Trasacos o Very Large Scale Iegrao VLSI Sysems, Vol., o. 3, pp , Jue. 4. P.P. Sorads, A. Chadrakasa, Reducg bus delay submcro echology usg codg, Proceedgs of he Desg Auomao Coferece, pp. 9 4, Ja.. 5. T. Sakura, Closed-form epressos for ercoeco delay, couplg, ad crossalk VLSI, IEEE Trasacos o Elecro Devces, Vol. 4, o., pp. 8 4, Ja P.P. Sorads, A. Chadrakasa, Poer Esmao ad Poer Opmal Commucao Deep Sub-Mcro Buses: Aalycal Models ad Sascal Measures, Joural of Crcus, Sysems ad Compuers, Vol., o. 6, pp ,. 7. P. Sorads, V. Tarokh, A. Chadrakasa, Eergy reduco VLSI compuao modules: a formao-heorec approach, IEEE Trasacos o Iformao Theory, Vol. 49, o. 4, pp , Aprl M.R. Sa, W.P. Burleso, Bus-ver codg for lo-poer I/O, IEEE Trasacos o Very Large Scale Iegrao Sysems, Vol. 3, o., pp , March P.P. Sorads, A. Chadrakasa, Lo poer bus codg echques cosderg er-re capacaces, Proceedgs of he IEEE Cusom Iegraed Crcus Coferece, pp. 57 5, May.. D. Ross, V.E.S. va Djk, R.P. Klehors, A.. eulad, C. Mera, Codg scheme for lo eergy cosumpo faul-olera bus, Proceedgs of he Eghh IEEE Ieraoal Workshop o O-Le Tesg, pp. 8, July. 54

65 . V. Sudararaja, K. K. Parh, Daa rasmsso over a bus h peak-lmed raso acvy, Proceedgs of he ASP Desg Auomao Coferece, pp. 4, Ja... S. Ramprasad,. R. Shabhag, I.. ajj, A codg frameork for lo-poer address ad daa busses, IEEE Trasacos o Very Large Scale Iegrao Sysems, Vol. 7, o., pp., Jue S. Ramprasad,. R. Shabhag, I.. ajj, Sgal codg for lo poer: fudameal lms ad praccal realzaos, IEEE Trasacos o Crcus ad Sysems II: Aalog ad Dgal Sgal Processg, Vol. 46, o. 7, 93 99, July D. Marculescu, R. Marculescu, M. Pedram, Iformao heorec measures for poer aalyss, IEEE Trasacos o Compuer-Aded Desg of Iegraed Crcus ad Sysems, Vol. 5, o. 6, pp , Jue R. Shabhag, A mahemacal bass for poer-reduco dgal VLSI sysems, IEEE Trasacos o Crcus ad Sysems II: Aalog ad Dgal Sgal Processg, Vol. 44, o., pp , ov S. Daa, S. W. Mclaughl, Opmal Block Codes for M-ary Rulegh- Cosraed Chaels, IEEE Tras. o Iformao Theory, Vol. 47, o. 5, July. 7. T. M. Cover, Eumerave source codg, IEEE Tras. Iform. Theory, Vol. IT-9, pp , Ja M.A. Elgamel, M.A. Bayoum, Iercoec ose aalyss ad opmzao deep submcro echology, IEEE Crcus ad Sysems Magaze, Vol. 3, o. 4, pp. 6-7, M.R. Sa, W. P. Burleso, Lo-poer ecodgs for global commucao CMOS VLSI, IEEE Trasacos o Very Large Scale Iegrao Sysems, Vol. 5, o. 4, pp , Dec hp://.syopsys.com/producs/lr/fleroue_p.pdf. 55

66 Apped Ls Ls of he codeords h ammg egh equal o 4 for 9, Lo-egh order. X p c

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