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CananInforma*onTheoristBeHappyinaCenterforInforma*on
Storage?
JackKeilWolf
CMRR,UCSD
PadovaniLecture
2010SchoolofInforma*onTheory
USC
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This talk is dedicated to David Slepian
who taught me all that I know about
informa*ontheoryandalotmore.
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Introduc*on
• Averyshortpictorialhistoryofmyrela*onshipwith
RobertoPadovaniandhowIendedupteachingatUCSD.
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TheUniv.ofPennsylvania
1952-1956
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PrincetonUniversity
1956-1959
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USAF
1960-1963
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NYU(Uptown)
1963-1965
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BrooklynPoly
1965-1973
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UMASS
1973-1984
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RobertoPadovaniandMe
• RobertoPadovaniwasoneofmygraduatestudentsatthe
UniversityofMassachuse\s.
• HisM.S.thesiswasontheperformanceoferrordetec*ng
codesandhisPh.D.thesiswasonthedesignand
performanceoftrelliscodes.
• HejoinedLinkabitCorpora*onupongradua*onfromUMass.
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RobertoPadovaniandMe
• HejoinedQualcommshortlya_eritwasfounded.
• OneofthefirstQualcommproductswasaTCMchipbaseduponapragma*ccodingschemeheco-developed.
• HewastheprinciplearchitectofQualcomm’shighspeedcellulardatasystem.
• HeispresentlyCTOofQualcomm.
• Heisagreatfriendandaterrificboss.
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Adver*sementforTCMChip
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RobertoPriortoGivinganInvited
LectureatUMassin2008
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RobertoPresen*ngtheLecture
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RobertoAnsweringQues*onsAbout
theFutureofCommumica*ons
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UCSD
• In1983anewinterdisciplinaryresearchcenterwasbeingformedatUCSD.
• ItwascalledtheCenterforMagne*cRecordingResearch(CMRR)
andwasconcernedwitheduca*ngstudentsandpursuingresearchinmagne*crecording.
• Itsoundedinteres*ngtomebecause:
– Ourkidshadallle_Amherstandwewerelookingforsomethingnew.
– IhadworkedwithGoriedUngerboeckatIBMZurichoncodingforapar*alresponsechannelwhichIlearnedwasamodelforthemagne*crecordingchannel.
– UCSDwasinSanDiego.
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Loca*on,Loca*on,Loca*on
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AMinorProblem
• Iknewnothingaboutmagne*crecording.
• NotonlydidInotknowhowtospellcoercivitybutthefirst
*meImen*oneditinatalkImispronouncedit.
• ButUCSDreluctantlymademeanofferasthefirstfaculty
memberinCMRR.
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AdvicefromOtherstoMe
• Berlekamphadwri\en: – "Communica*onlinkstransmitinforma*onfromheretothere.
Computermemoriestransmitinforma*onfromnowtothen.“
– Thatsoundedverygoodtome.
• Butmanyofmyverysmartfriendssaid: – Magne*crecordingisboring.
– Notonlyisitboringbutitisadeadend!Alltheadvanceshave
beenmade.Thefutureliesin…• Op*calrecording
• Holographicrecording
• Etc.,etc.,etc.
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1956:IBMRAMACFirstMagne*cHard
Drive
TotalCapacity=5Mbyte
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GigabyteDriveCirca1983
IBM3380
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A20102TerabyteDrive
$119.00Amazon.com
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Arealdensityhasbeenincreasedmorethan250million:meswithrespecttothe
firstRAMACin1956from0.002Mbit/in2to500Gbit/in2intoday
Weexpectmuchhigherarealdensityinthefuture,i.e.,1Tbit/in2and10Tbit/in2
HistoricalArealDensityIncrease
ofHardDiskDrives
25
JackWolfArrivesatCMRR
*
CAGR=Cumula*veAnnualGrowthRate
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1980’s:IBM3380Drive
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CoatofArmsofInforma*onTheorist
WorkinginDigitalRecordingError
Correction
Encoder
Modulation
Encoder
Write
Equalization
Equalization
and Detection
Modulation
Decoder
Error
Correction
Decoder
Channel
Channel Encoder
Channel Decoder
Notethatthechanneliscontrollable(butbyphysicists)
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SomeItemsofInteresttoan
Informa*onTheorist• Thechannel(whichismadeupofthemagne*cmediaand
thewriteandreadheads)keepschanging.
– Bigimprovementsinrecordingdensityhavebeenachievedhere!!
• Theerrorcorrec*ngcodeusedforthelast25yearsisaReed
Solomoncodeinconjunc*onwithahardinputdecoder.
– ButLDPCcodesanditera*vedecodingareontheway!!
• Thepurposeofthemodula*oncodeistopreventcertain
badsequencesfrombeingwri\en.
– Toaninforma*ontheorist,thisiscodingforthenoiselesschannel.
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Informa*onTheoristsLikeSimple
Models• Thewritesignalisplainvanilla+1/-1basebandbinarydata.
(NoQAM,M-PSK,etc.)
• AnAWGNchanneliso_enusedasafirstorder
approxima*onforthechannelmodel.Buttheactual
channelisreallymuchmorecomplex.
• Atlowrecordingdensi*esthereisessen*allynoISIso
matchedfilter(bitbybit)detec*onisop*mal(foranAWGN
channel).
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SomeItemsofInteresttoan
Informa*onTheorist• However,athigherrecordingdensi*es,theISIcannotbe
ignored.
• Toachievehigherrecordingdensi*es,in1984theindustryabandonedbitbybitdetec*onandadoptedpar*alresponsesignalingwithViterbisequencedetec*ontocombatISI.
• IBMcalleditPRML(sincetheywantedtoavoidtheuseofViterbi’sname).
• EverydiskdrivetodayusessomeformofViterbidetec*on.
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WhySuchAmazingProgress?
• Itdependsuponwhoyoutalkto.
– Physicistscreditadvancedmaterialsforheadsanddisks.
– Mechanicalengineerscreditadvancedmechanics.
– Informa*ontheoristscreditapplica*onsofShannontheory.
• Onees*mateisthatabout20%ofthe“progress”wasduetoadvancesinsignalprocessing.
• However,advancesinallfieldswererequiredtomakethesystemwork.
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ProgressasSeenByaPhysicist
2007NobelPrize
inPhysicswasawardedtotheinventorsofthe
GMRhead
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TheTrueSourcesofProgress
• Manydifferenttechnologicaladvancesledtothisamazingprogress.
• Newinven*onsweretheenablingtechnology.
• However,theconstantprogressbetweentheintroduc*onofthesenewinven*ons,wastheresultof scaling(i.e.,shrinkingthedimensionsofeverything).
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Longitudinal magnetic recording (LMR)
technology
Perpendicular magnetic recording (PMR)
technology
Limit was around 150 Gbit/in2
It was achieved by 500 Gbit/in2 today
PMRtechnology- Highanisotropymaterial
- Ver:calalignmentofmagne:za:on- Muchsmallerbitispossible
41
Longitudinalvs.Perpendicular
Recording
Writing is due to flux leakingfrom the write head to the disk.
Reading is due from flux leakingfrom the disk to the read head.
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ShingledWriteProcess
disk
100 nm
Gapis100nmbutbitsare25nm.Howcanthisbe??
100 nm
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Modula*onCodes
• Thepurposeofmodula*oncodesisto prohibittheoccurrenceofcertaintroublesomesequencessuchassequenceswhichcauseexcessiveISIorwhichmake*mingrecoverydifficult.
• Themostwellknownexampleofamodula*oncodeistheso-called(d,k )code,wherenorunof0’slongerthank orlessthand ispermi\ed.d andk arenonnega*veintegersforwhichk>d .
• InearlyGbytedrives(circa1980),(2,7)and(1,7)codeswereused.Today,varia*onson(0,k )codesareused.
• Shannondiscussedsuchcodesintheverybeginningofhis1948paperinasec*oncalleden*tled“TheDiscreteNoiselessChannel”.
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ShannonStatueatCMRR
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ClaudeShannon
• Inhisclassic1948paper,Shannonshowedthatforlargen,thenumberoflengthnconstrainedsequences,N(n),isapproximately2Cn.Thequan*tyC iscalledthecapacityoftheconstrainedsystem.
• Saidinanotherway
• Therateofacode,R,isthe(average)ra*oofthenumberofunconstraineddigitstoconstraineddigits.ShannonshowedthatthereexistscodesatrateR,ifandonlyif
R<C .
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Compu*ngtheCapacity
• Shannon(1948)gavetwoequivalentmethodsforcompu*ng
thecapacitywhichareapplicableto(d,k)codes.
Firstmethod:
• Forfinitek,N(n)sa*sfiesthelineardifferenceequa*on:
N(n)=N(n-(d+1))+N(n-(d+2))+…+N(n-(k+1)).
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Compu*ngtheCapacity
• Bystandardmethodsofsolvinglineardifferenceequa*onsShannonshowedthatCisequaltothebase2logarithmofthelargestrealrootoftheequa*on:
xk+2-xk+1-xk-d+1+1=0.
SecondMethod:
• Shannonshowedthatthecapacityisequaltothebase2
logarithmofthelargesteigenvalueoftheadjacencymatrixofagraphwhichgeneratesthecodesymbols.
• Weillustratethesetwomethodsfora(1,2)code(i.e.,d=1andk=2).
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Compu*ngtheShannonCapacityof
Binary(1,2)Codes Firstmethod
•
Ifd=1andk=2,theequa*on xk+2-xk+1-xk-d+1+1=0
becomes
x4-x3-x2+1=0.
• Thelargestrealrootofthisequa*onis1.3247andit’sbase2
logarithmis0.4057.
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Compu*ngtheShannonCapacityof
Binary(1,2)Codes SecondMethod
• Aconstraintgraphthatgeneratescodewordsina(1,2)codeis:
• Theadjacencymatrixofthisgraphis:
010
101
100.
0 0
11
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Example:ARate1/2(2,7)Code
• UsingeitherofShannon’smethods,thecapacity,C ,ofa(2,7)codeisfoundtobe0.5174.However,Shannondidnottellushowtoconstructcodesatratesnearoratcapacity.
• Avariablelength,fixedrate,R=½,(2,7)code:
informa*onphrases codewords
10 0100
11 1000
000 000100
010 100100 011 001000
0010 00100100
0011 0000100000100011
000
1011
010011
01.
Thecodewords
formaprefix-freecodesocanbe
decoded.
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Example:(2,7)Codes
ThiscodewasusedtocombatISIinsystemsusingbitbybitdetec*on:
Nocoding:channelbitspacing=T
1011000
Withrate½(2,7)coding:channelbitspacing=T /2
01001000000100
Minimumsepara*on
between1’s=T
minimumsepara*on
between1’s=3T /2
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ConstrainedCodes(2-Dimensions)
• Codingtheoristsarealsointerestedin2-dimensional
constrainedbinarycodes:i.e.,constrainedbinaryarrayswherethebinarydigitsarearrangedinanarrayofrowsand
columns.
• Suchcodesmighthaveapplica*onin2-dimensionalstorage.
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ConstrainedCodes(2-Dimensions)
• Wewilluseasanexample,a2-dimensionalarraywhere
everyrowandeverycolumnsa*sfiesa1-dimensional
(d,k)constraint.
• Otherinteres*ngconstraintsexist.
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A2-Dimensional(1,2)Array
0100101001001... 1001010010010...
0010100100101...
0101001001010...
1010010010101... ...
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ConstrainedCodes(2-Dimensions)
• Letthearrayhavemrowsandncolumns.
• N(m,n):thenumberofarraysthatsa*sfythe2-
dimensionalconstraint.
• Thenthe2-dimensionalcapacity,C2,isdefinedas:
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Capacityof(d,k)ConstrainedArrays
• For2-dimensional(d,k)constraints,C 2existsbutShannon
didn’ttellushowtocomputeit.
• Tothisday,forrectangularconstraints,theexactvalueofthe
capacityisunknownexceptforthetrivialcaseswhereC 2=0
orC2=1.
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ECCCodes
• ReedSolomoncodesareusedintoday’sharddiskdrives.
• Weareonthevergeofseeingtheintroduc*onofLDPC
codeswithitera*vedecodinginHDD.
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PressReleaseAugust4,2010
• …announcesitslow-densityparitycheck(LDPC)-based…deviceiscurrentlyshippinginmainstream2.5-inchmobileharddiskdriveproducts.…
• Today’sHDDdatarecoveryarchitecturesaremostlybasedonconcatenatedcodingschemeswhichuseReedSolomonerrorcorrec*oncodes,inventedalmost50yearsago.…
• Now,byusing…LDPC-basedsolu*ons,HDDvendorscancon*nuetodoublethestoragecapacityoftheirdrivesevery18months.…
• currentLDPC-baseddevicereducesthenumberoferrors
readfromadiskfrom1in100to1in100Millionbitsofdata,rela*vetothepreviously-usedconcatenatedcodingschemes.
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TheFutureofHDD’s
• Itispossiblethatthearealdensitywillsaturateverysoonusingthepresenttechnology.
• Asthesizeofthestoredbitshrinks,thepresentmagne*cmaterialwillnotholdit’smagne*za*on.
• Thisiscalledthesuperparamagne*ceffect.
• Itisbelievedthataradicallynewsystemwillberequiredtoovercomethiseffect.
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TheFutureofDiskDrives
• Twosolu*onsarebeingpursuedtoovercomethesuperparamagne*ceffect.
– Onesolu*onistouseamagne*cmaterialwithamuchhighercoercivity.Theproblemwiththissolu*onisthatyoucannotwriteonthematerialatroomtemperaturesoyouneedtoheatthemediatowrite.Thisisdonewithalaser
– Thesecondapproachiscalledpa\ernedmediawherebitsarestoredonphysicallyseparatedmagne*c“islands”separatedbyaseaofnon-magne*cmaterial.
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FutureTechnology?
HAMR-HeatAssisted
Magne*cRecording
Pa\ernedMedia
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Wri*ngonPa\ernedMedia
OrdinaryMediaPa\ernedMedia
Inordinarymedia,onecanwriteabitanywhereonthemagne*csurface.
Inpa\ernedmediaonemustwriteeachbitonamagne*cisland.Thisisa
difficulttasksinceonecannotreadandwritesimultaneously.
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ShingledRecordingforPa\erned
MediaTime Data Islands / Recorded Bits
0 __ x x x x x x x x x
1 0 0 0 0 0 x x x x x
2 1 0 1 1 1 1 x x x x
3 1 0 1 1 1 1 1 x x x
4 0 (wrtten late) 0 1 1 1 0 0 0 0 x
5 0 (written late) 0 1 1 1 0 0 0 0 0
Note that if the data bit written late is the same as the previous bit, thereis no error in the recorded bit!!!
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Mathema*calModelAt*mei=1,2,3,... X i databit{0,1}
Y i recordedbit {0,1}
Zistateofchannel {0,1}
Z i =0ifdatabitiswri\enoncorrectisland Z i=1ifdatabitiswri\enlate
Then: Y i = X i if Z i =0
Y i = X i-1if Z i=1.
Thus: Y i = X i ⊕( X i ⊕ X i-1) Z i
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PreviousExample
X 0 1 1 0 0 1 0 ...
Z 0 0 0 1 1 0 1 ...
Y 0 1 1 1 0 1 1 ...
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ASimpleRate½CodeandChannel
Capacity• Considerthetrivialbinaryrate½codewhereeachdatabitMi is
recordedtwice.
• Thatis,assumethat X 2i-1= X 2i =Mi ε{0,1}.Thensince
Y 2i = X 2i ⊕( X 2i ⊕ X 2i-1) Z 2i
Y 2i =Mi independentofthevalueof Z 2i .
• Adecodercandecodethisrate½codewithzeroerrorprobability
justbyobservingthevaluesofY withevenindicesandthusthezeroerrorcapacityofthischannelisatleast½.
• Asaresult,alowerboundtothecapacityofthechannelis½independentofthesta*s*calmodelassumedforthe Z process.
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TwoDifferentModelsfor Z
• Random Z
– { Z i }isBernouliwithparameter p:B( p)
– Thatis,{ Z i }arei.i.d.,and p=Pr [ Z i =1]
• 2-stateGilbertmodel Z :G( p0,1 ,p1,0 )
Z i =0 Z i =1
p0,1
p1,0
1-p0,1
1-p1,0
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ChannelCapacity
• Foranymodelforthe Z process,thechannelcapacityis
definedintheusualmanner.
• WecallthecapacityoftheBernoullistatemodelwith
parameter p,C B( p),andwecallthecapacityoftheGilbertstatemodelwithparameters( p0,1 ,p1,0 ),C G( p0,1 ,p1,0 ).
• FortheBernoulistatemodelonecanprovethat:
C B( p)=C B(1-p)andforGilbertstatemodel,onecanprovethat:
C G( p0,1 ,p1,0 )=C G( p1,0 ,p0,1).
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ProofthatC B( p)=C B( p)=C B(1- p)
Details are given in the paper “Write Channel Model for Bit-Patterned Media Recording” which will appear in theIEEE Transactions of Magnetics.
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BernoulliModel
• Theparameterspacecanthereforebereducedtothe
interval pε[0,1/2].
•
Furthermorethesamesymmetryargumentholdsfornot justtherate-maximizinginputdistribu*on,butforallinput
distribu*ons.
• ThecapacityoftheBernoullimodelisupperboundedby
theachievablerateforagenie-aideddecoder,i.e.,onewiththe{ Z }processrealiza*onknown.
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UpperBoundonCapacityforthe
BernouliModel• Giventherealiza*onofthe Z process,whenever Z i−1=1and Z i =0 ,
thevalueof X i-1cannotbedeterminedfromtheY process.
• ThustheBernoullistatechannelisequivalenttoacorrelated
symmetricerasurechannelwithaverageerasureratePr { Z i−1=1, Z i =0 }= p(1-p).
• Theresul*ngerasurechanneliscorrelatedsince,erasuresbeing
dependenton1to0transi*onsinthe Z process,twoconsecu*ve
bitscannotbeerased.
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UpperBoundonCapacityforthe
BernouliModel
• Thecapacityofacorrelatedsymmetricerasurechannelisthesameasthatofamemorylesssymmetricerasure
channelwiththesameerasureprobability.
• Therefore,C B( p)<[1−p(1-p)].
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Example
Z 0 0 0 1 1 0 ...
Y 0 1 1 1 0 1 ...
X 0
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Example
Z 0 0 0 1 1 0 ...
Y 0 1 1 1 0 1 ...
X 0 1
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Example
Z 0 0 0 1 1 0 ...
Y 0 1 1 1 0 1 ...
X 0 1 1
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Example
Z 0 0 0 1 1 0 ...
Y 0 1 1 1 0 1 ...
X 0 1 1 0
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Example
Z 0 0 0 1 1 0 ...
Y 0 1 1 1 0 1 ...
X 0 1 1 0 ? 1
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BernouliModel• Theinputdistribu*onthatmaximizesthemutual
informa*onisunknownsonoclosedformexpressionhas
beenfoundforthecapacityoftheBernoulimodel.
• Howeverlowerboundsonthecapacitycanbefoundby
assumingpar*cularformsfortheinputdistribu*on(e.g.,an
i.i.d.inputprocess).
• Very*ghtupperandlowerboundshavebeenfoundforthe
symmetricinforma*onratewhentheinputisuniformandi.i.d.Anaccuratees*mateofthesymmetricinforma*on
ratehasbeenobtainedusingtheBCJRalgorithm.
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BernouliModel
SymmetricInforma*onRate
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BernouliModel
SymmetricInforma*onRate
• Notethatonthepreviousslide,forsomevaluesof p,an
upperboundtothesymmetricinforma*onrateisstrictly
lessthan½.
• Butweknowthatthetruecapacityisgreaterthanorequal
to½forallvaluesof p.
• Thisshowsthatforthesevaluesof p,thecapacityachieving
inputprocessisnoti.u.d.
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BernouliModel
• Toexplorethelossinachievablerateduetoani.i.d.input
weconsideredasymmetricfirstorderbinaryMarkovinput
wherePr { X i =1| X i-1=0 }=Pr { X i =0 | X i-1=1}=β.
• Upperandlowerboundswerefoundforthemutual
informa*onforaMarkovianinputasafunc*onofβand p.
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BernouliModelwithMarkovianInput
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ComparisonoftheSymmetric
Informa*onRateandtheInforma*onRateforaMarkovianSource
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SummaryforBernouliStateModel
• ItwasfoundthatfortheBernoulimodelfor Z ,considerablegainsinthereliabletransferratearepossiblebyusingan
inputwithmemory.
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GilbertModel
• Weknowlessaboutcompu*ngthecapacityforthismodel
thanfortheBernoulimodel
• ByusingagenietoinformthedecoderoftheZprocesswe
canobtainanupperboundtothecapacity.Againtheresult
isacorrelatederasurechannelwithaverageerasure
probabilityPr { Z i-1=1,Z i =0 }= p1,0 p0,1/( p1,0 +p0,1)resul*nginthe
upperboundforthecapacity:
C G( p0,1 ,p1,0 )<1–[ p1,0 p0,1/( p1,0 +p0,1)].
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Inser*onsandDele*ons
• Foranymodelfor Z onecaninterprettheeffectsofthe
channelintermsofinser*onsanddele*ons.
• If Z i-1
=0and Z i =1,thenY
i-1=Y
i =X
i-1sothereisaninser*onof
X i-1intheY sequence.Ifontheotherhand Z j-1=1and Z j =0,
thenY j-1=X j-2andY j =X jsothereisadele*onof X j-1intheY
sequence.
• Notethatinthismodel,inser*onsanddele*onsalternateinoccurrenceandinser*onsarearepeatofthepreviousdata
digit.
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Inser*onsandDele*ons
• Example
X 0 1 1 0 0 1 0 ...
Z 0 1 1 1 1 0 0
Y 0 0 1 1 0 1 0 ...
inser*ondele*on
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SomeFinalRemarksONHDD’s
• Peoplehavebeenpredic*ngthedeathofmagne*charddisk
drivesformanyyears.
• Lackinga“prognos*scope”,itisdifficulttopredicthowlong
theHDDwillremainthestoragedeviceofchoice.
• However,magne*charddiskdrivesseemstobea“catwith
ninelives”havingbeatoutallcompe*torsinthepast.
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91
CodingforFlashMemories
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92
FlashMemory
• Flashisanon-vola*lememorywhichisfast,powerefficient
andhasnomovingparts.
• Electricallyprogrammedanderased.
• Usedin:
– Digitalcameras
– LowcapacityIPODS
– Mobilephones – Laptopcomputers
– Hybriddrives
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FlashMemoriesStructure
Arrayofcellsmadefromfloa*nggatetransistors.
Cellsaresubdividedintoblocksandthenintopages.
Thecellsareprogrammedbypulsingelectronsviahot-electron
injec*on.
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94
FlashMemoriesStructure
Arrayofcellsmadefromfloa*nggatetransistors.
Cellsaresubdividedintoblocksandthenintopages.
Thecellsareprogrammedbypulsingelectronsviahot-electron
injec*on.
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95
FlashMemoriesStructure
Eachcellcanhaveqlevels,
representedbydifferent
amountsofelectrons.
Intoday’sproducts,q=2,4,8or16.
Inordertoreduceacelllevel,
allthecellsinthatblockmustbe
resettolevel0beforerewri*ng.
–AVERYEXPENSIVEOPERATION
Arrayofcellsmadefromfloa*nggatetransistors.
Cellsaresubdividedintoblocksandthenintopages.
Thecellsareprogrammedbypulsingelectronsviahot-electron
injec*on.
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96
FlashMemoryStructure
• Thememoryconsistsofblocks
– Thesizeofeachblockis128(or256) KB.
– Eachblockconsistsof64 (or128)pages.
– Thesizeofeachpageis2KB.
• Wri*ng–Writesequen*allytothenextavailable
page.
• Erasing–Canonlyeraseanen*reblock!
Page 1
Page 2
Page 3
!
Page 63
Page 64
! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !
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97
FlashMemoryConstraints
• Theenduranceofflashmemoriesisrelatedtotothenumber
of*mestheblocksareerased.
• Insinglelevelflashwithq=2,ablockcantolerate~104-105
erasuresbeforeitstartsproducingexcessiveerrors.
– SLC:SingleLevelCell
• Thelargerthevalueofq,thelesstheendurance.
– MLC:Mul*LevelCell
• TheGoal:Represen*ngthedataefficientlysuchthatblock
erasuresarepostponedasmuchaspossible.
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98
ExperimentDescrip*on
• Foreachblockthefollowingstepswererepeated:
• Theblockwaserased.
• Pseudo-randomdatawaswri\entotheblock.
• Thedatawasreadandcomparedtofinderrors.
• Remark:
• Theexperimentwasdoneunderlabcondi*ons.Otherfactorssuchas
temperaturechange,intervalsbetweenerasuresandmul*plereadings
beforeerasureswerenotconsidered.
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99
RawBERforSLCblock
!106
!10-4
Guaranteedlife*mebythemanufacturer
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100
RawBERforMLCblock
!105
!10-3
Guaranteedlife*mebythemanufacturer
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AnIntroduc*ontoWOM-codes
• WOM-codesallowustowriteseveral*mestothesameblockofmemorywithouterasing.
• Example:In1982,RivestandShamirfoundawaytowrite2
bitsofinforma*ontwiceusingonly3cells.
• WedenoteaWOMcodethatwritesk*mestoncellsasa<V 1,V 2,…,V k >/ncodewhereV i isthenumberofmessageswri\enontheithwrite.
• ThustheRivestShamircodeisa<4,4>/3codewithk =2.
101
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TheRivest-ShamirCode
102
DataBits FirstWrite SecondWrite
00 000 111
01 001 110
10 010 101
11 100 011
Example1:FirstWrite:Wantto
storedata01:
Write001tomemory.
SecondWrite:Wanttostoredata
10:
Write101tomemory.
Ifwewanttowritethesamedataonthe
secondwrite,wedonotchangewhatiswri\enonthefirstwrite.
Example2:FirstWrite:Wantto
storedata01
Write001tomemory.
SecondWrite:Wanttostoredata
01:Leave001inmemory.
Notethatwhengoingfromfirstwrite
tosecondwrite,no1’sareerased.
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RateofaWOM-code
• Therateoftheithwriteis:
Bitsofinforma*on
Totalnumberofbits
log2(V i)
n
• ThetotalrateofaWOM-codeisR=∑(Ri).
• TheRivestShamircodehasR1=R2=2/3andR=4/3.
Ri=
Ri=
103
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WOMCapacityRegion
ThecapacityregionofabinaryWOMcodewithtwowritesis
C WOM={(R1,R2)|∃ p∈[0,0.5],R1≤h( p),R2≤1– p}
whereh( p)=- plog2( p)–(1- p)log2(1- p).
R=R1+R2<h( p)+1–p.
Therighthandsideismaximizedfor p=1/3yielding
Rmax=log2(3)=1.58…104
Our Construc*on for a 2 Write WOM
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OurConstruc*onfora2-WriteWOM
Code• Choosealinearcode(n,k )withparitycheckmatrixH.Letr=n-k
sothatH isanr -rowbyn-columnmatrixofrankr.
• Foravectorv ϵ{0,1}n,letH (v )bethematrixH with0’sreplacing
thecolumnsthatcorrespondtotheposi*onsofthe1’sinv .
• Onfirstwrite,writeonlythosevectorsv suchthatrank(H (v ))=r .
LetV 1={v ϵ{0,1}n|rank(H (v ))=r }.
• ThenR1=log2|V 1| /n.
Our Construc*on for a 2 Write WOM
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OurConstruc*onfora2-WriteWOM
Code• Assumethate1isvectorwri\enonthefirstwrite.
• Secondwrite: – Consideradatavectors2ofr bits.
–
Finde2suchthatH (e1)⋅e2=s1⊕s2,wheres1=H
⋅e1. – Asolu*one2existssincerank(H (e1))=r .
– Writee1⊕e2tomemory.
• Decodingonthesecondwrite: – Mul*plythestoredvectore1⊕e2byH :
H ⋅(e1⊕e2)=He1⊕He2=s1⊕(s1⊕s2)=s2
• Thus,R2=(n-k)/nandR=R1+R2=[log2|V 1|+(n-k)]/n.
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WOM-CodeConstruc*on:AnExample
• LetH betheparitycheckmatrixofa(7,4)Hammingcode.
– For a vectorv ϵ{0,1}n
,letH (v )bethematrixH with0’sinthecolumnsthatcorrespondtotheposi*onsofthe1’sinv .
• Onthefirstwrite,weprogramonlyvectorsv suchthatrank(H (v ))=3,
V 1={v ϵ{0,1}n|rank(H (v ))=3}.
• ForHasshownabove,|V 1
|=1+7+21+35+(35-7)=92.Thus,wecanwriteoneof92messagesatthefirstwrite.
• Encodinganddecodingofthefirstwritearedonewithalookuptable.
• Saywewritee1=0101100.
1110100
1011010
1101001
H =n=#ofcolumns=7
r =#ofrows=3
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WOM-CodeConstruc*on:AnExample
e1=0101100• Notethats1=H ⋅e1=010.
• Insert0’sinthecolumnsofH thatcorrespondto1’sinthefirstwrite.ThisnewmatrixisH (e1):
H =
• Wecannowwriteamessageoflengthr =3bits.Saywewanttostores2=011onsecondwrite.
• Wanttofindavectore2suchthatH (e1)·e2=s1⊕s2.
• s1⊕s2=001
• Choosee2=0000001.
• Thenwewritee1⊕e2=0101101
1110100
1011010
1101001
1010000
1010010
1000001
1010000
10100101000001
=H(e1 )
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WOM-CodeConstruc*on:AnExample
• Notethatwecandecodebymul*plyingbyH :
– Wecanwrite92messagesatthefirstwritesoV 1=92and
R1=log2(92)/7=0.932,R2=3/7=0.429andR=1.361whichisbe\er
thantheRivest-Shamirconstruc*on.
– However,R1=R2fortheRivest-Shamirconstruc*on.
1110100
10110101101001
0
1
0
11
0
1
. =
0
11
= s2
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SomeMoreResults
• ThebestvalueofRpreviouslyachievedfortwowriteswasbyWuwhoobtainedR=1.371.Weobtainedmanycodesthatbe\eredthisresult.
• FortheGolay(24,12)code,weobtainedR=1.4547.
• FortheGolay(23,11)code,weobtainedR=1.4632.
• ChoosingthecodeasthedualoftheReed-Muller(4,2)code,weobtainedR=1.4566.
• IfR1>R2,wecanlimitthenumberofmessagesusedonthefirstwritesothatR1=R2
andR=2R1.DoingthisforthedualoftheReed-Muller(4,2)code,weobtainedR=1.375.
110
Computer Search for Good 2-Write
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ComputerSearchforGood2-Write
WOMCodes
• Constructarandommatrixofsizen×r andrankr.
• CyclethroughallvectorsoflengthnandHammingweightat
mostn-r .
• Foreachvectorv ,zerooutcolumnsofthematrixwhere1’s
existinv.
• Computetherankofthematrix.Ifitisthesameasthe
originalrank,addoneto|V 1|.
• Onceweknow|V1|,wecancomputetheratewith
R=(1/n)(log2|V 1|+r )
111
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TimeSharing
• Ifweknowtwocodeswithrates(R1 ,R2)and(R3 ,R4),wecan
achieveanyratepair
(t*R1+(1-t )*R3 ,t*R2+(1-t )*R4)
fortara*onalnumberbetween0and1.
112
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SomeAchievableRatePairsandCapacity
forWOMWithTwoWrites
113
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114
MoreAchievableRatePairsandCapacityfor
WOMWithTwoWrites
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CodesforMorethan2Writes
t -#ofWrites LowerBoundOur
Construc*onUpperBound
3 1.58 1.66 log4=2
4 1.75 1.95 log5=2.32
5 1.75 1.99 log6=2.58
6 1.75 2.14 log7=2.8
7 1.82 2.15 log8=3
8 1.88 2.23 log9=3.17
9 1.95 2.23 log10=3.32
10 2.01 2.27 log11=3.46
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Ifyouhaven’tguessedalready,theanswerto
theques*on:
Can an Informa*on Theorist Be Happy in a
CenterforInforma*onStorage?
isaresoundingyes.
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MyCollaboratorsonthisTalk
PaulSiegel AravindIyengar EitanYaakobi
withSco\Kayser
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SomeoftheRestoftheCastatCMRR
AndaSpecialThankstoAllofMy
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Ph.D.Students
Altekar,ShirishA.
Armstrong,AlanJ.
Aviran,SharonBaggen,ConstantP.M.J.
Barndt,RichardD.
Bender,Paul
Bernal,RobertW.
Bridwell,JohnD.Bunin,BarryJ.
Caroselli,JosephP.
Chiang,Chung-Yaw
Demirkan,Ismail
Dorfman,VladimirEddy,ThomasW.
Ergul,FarukR.
Fitzpatrick,James
Fredrickson,L.J.French,CatherineAnnFriedmann,ArnonA.
Goldberg,JasonS
Gupta,DevVart
Hartman,PaulD.
Ho,KelvinK.Y.Karakulak,Seyhan
Kerdock,AnthonyM.
Kim,ByungGuk
Klein,TheodoreJ.
Knudson,KellyJ.Kurkoski,BrianM.
Lee,Patrick
Levi,Karl
Liff,AllanI.
Lin,YinyiMa,HowardH.
Ma,JoongS.
MacDonald,CharlesE.
Mangano,DennisT.Marrow,MarcusMasnick,Burt
McEwen,PeterA.
Miller,John
Milstein,LaurenceB.
Padovani,RobertoPanwar,ShivendraS.
Pasternack,Gerald
Paerson,JohnD.
Philips,ThomasK.
Prohazka,CraigG.
Raghavan,SreenivasaA.
Ritz,Mordechai
Rodriguez,ManoelA.
Schiff,Leonard
Souvignier,ThomasV.Trismen,Robert
Wainberg,Stanley
Walvick,Edward
Weathers,AnthonyD.Zehavi,EphraimZhang,Wenlong
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Thankyouforyourkinda\en*on.