EE1411EE1411EE141-S04EE141EE141--Spring 2004Spring 2004Digital Integrated Digital Integrated CircuitsCircuitsLecture 20Lecture 20MultipliersMultipliersPowerPowerEE1412EE141-S04Administrative StuffAdministrative Stuff Hw 6 due today Midterm 2 next weekMaterial:–Wires– Complex logic– Arithmetic Review session on WeEE1412EE1413EE141-S04Class MaterialClass Material Last lecture Adders Today’s lecture Multipliers and other arithmetic Intro to powerEE1414EE141-S04MultipliersMultipliersEE1413EE1415EE141-S04The Binary MultiplicationThe Binary MultiplicationZX··Y×Zk2kk0=MN1–+∑==Xi2ii0=M1–∑⎝⎠⎜⎟⎜⎟⎜⎟⎛⎞Yj2jj0=N1–∑⎝⎠⎜⎟⎜⎟⎜⎟⎛⎞=XiYj2ij+j0=N1–∑⎝⎠⎜⎟⎜⎟⎜⎟⎛⎞i0=M1–∑=XXi2ii0=M1–∑=YYj2jj0=N1–∑=withEE1416EE141-S04The Binary MultiplicationThe Binary Multiplicationx+Partial productsMultiplicandMultiplierResult1 0 1 0 1 01 0 1 0 1 01 0 1 0 1 01 1 1 0 0 1 1 1 00 0 0 0 0 01 0 1 0 1 01 0 1 1EE1414EE1417EE141-S04The Array MultiplierThe Array MultiplierY0Y1X3X2X1X0X3HAX2FAX1FAX0HAY2X3FAX2FAX1FAX0HAZ1Z3Z6Z7Z5Z4Y3X3FAX2FAX1FAX0HAZ2Z0EE1418EE141-S04The The MxNMxNArray MultiplierArray Multiplier——Critical PathCritical PathHA FA FA HAHAFAFAFAFAFA FA HACritical Path 1Critical Path 2Critical Path 1 & 2EE1415EE1419EE141-S04CarryCarry--Save MultiplierSave MultiplierHA HA HA HAFAFAFAHAFAHA FA FAFAHA FA HAVector Merging AdderEE14110EE141-S04Multiplier Multiplier FloorplanFloorplanSCSCSCSCSCSCSCSCSCSCSCSCSCSCSCSCZ0Z1Z2Z3Z4Z5Z6Z7X0X1X2X3Y1Y2Y3Y0Vector Merging CellHA Multiplier CellFA Multiplier CellX and Y signals are broadcastedthrough the complete array.( )EE1416EE14111EE141-S04WallaceWallace--Tree MultiplierTree Multiplier6543210 6543210Partial products First stageBit position6543210 6543210Second stage Final adderFA HA(a) (b)(c) (d)EE14112EE141-S04WallaceWallace--Tree MultiplierTree MultiplierPartial productsFirst stageSecond stageFinal adderFA FA FAHA HAFAx3y3z7z6z5z4z3z2z1z0x3y2x2y3x1y1x3y0x2y0x0y1x0y2x2y2x1y3x1y2x3y1x0y3x1y0x0y0x2y1EE1417EE14113EE141-S04WallaceWallace--Tree MultiplierTree MultiplierFAFAFAFAy0y1y2y3y4y5SCi-1Ci-1Ci-1CiCiCiFAy0y1y2FAy3y4y5FAFACCSCi-1Ci-1Ci-1CiCiCiEE14114EE141-S04Multipliers Multipliers ——SummarySummary• Optimization Goals Different Vs Binary Adder• Once Again: Identify Critical Path• Other possible techniques- Data encoding (Booth)- PipeliningFIRST GLIMPSE AT SYSTEM LEVEL OPTIMIZATION- Logarithmic versus Linear (Wallace Tree Mult)EE1418EE14115EE141-S04The Binary ShifterThe Binary ShifterAiAi-1BiBi-1RightLeftnopBit-Slice i...EE14116EE141-S04The Barrel ShifterThe Barrel ShifterSh3Sh2Sh1Sh0Sh3Sh2Sh1A3A2A1A0B3B2B1B0: Control Wire: Data WireArea Dominated by WiringEE1419EE14117EE141-S044x4 barrel shifter4x4 barrel shifterBufferSh3Sh2Sh1Sh0A3A2A1A0Widthbarrel~ 2 pmMEE14118EE141-S04Logarithmic ShifterLogarithmic ShifterSh1 Sh1 Sh2 Sh2 Sh4 Sh4A3A2A1A0B1B0B2B3EE14110EE14119EE141-S04A3A2A1A0Out3Out2Out1Out000--7 bit Logarithmic Shifter7 bit Logarithmic ShifterEE14120EE141-S04PowerPowerEE14111EE14121EE141-S04The Power ChallengeThe Power Challenge 400 million computers in the world 0.16 PW (PetaWatt = 1015 W) of power dissipation Equivalent to 26 nuclear plants! Data centers represent the absolute challenge 1 single server rack is between 5 and 20 kW 100’s of those racks in a single room!EE14122EE141-S04PowerPower--andand--Energy ChallengesEnergy ChallengesCourtesy of IBMPower and energy management and minimizationhave emerged as some of the most dominant roadblocks. The best opportunity lies in a very aggressive scaling and adaptation of supply and threshold values in concert with a careful orchestration of the system activity. 138 W/cm2EE14112EE14123EE141-S04Portability:Portability:Battery storage the limiting factorBattery storage the limiting factor Little change in basic technology store energy using a chemical reaction Battery capacity doubles every 10 years Energy density/size, safe handling are limiting factorEnergy densityof materialKWH/kgGasoline 14Lead-Acid 0.04Li polymer 0.15EE14124EE141-S04Battery ProgressBattery Progress0204060801001201401601940 1950 1960 1970 1980 1990 2000 2010First Commercial UseEnergy Density(Wh/kg) Trend LineNiCdSLANiMHLi-IonReusableAlkalineLi-PolymerFacture 4 over the last 10 years!EE14113EE14125EE141-S04Power Dissipation in CMOSPower Dissipation in CMOS Dynamic power Charging capacitances Dominant today Leakage power Leaky transistors Concern in low-activity, portable devices Short circuit power Static power E.g. pseudo-NMOSEE14126EE141-S04Dynamic Power ConsumptionDynamic Power Consumption() ()∫∫∫====→DDVDDLoutLDDTTDDDDDDVCdvCVdttiVdttPE020010() ()∫∫∫====DDVDDLoutoutLTTLoutCCVCdvvCdttivdttPE020021VddVoutiLCLPMOSNETWORKNMOSA1ANNETWORK210 DDLVCE =→EE14114EE14127EE141-S04Dynamic Power ConsumptionDynamic Power Consumption One half of the power from the supply is consumed in the pull-up network and one half is stored on CL Charge from CLis dumped during the 1→0 transitionVddVoutiLCLPMOSNETWORKNMOSA1ANNETWORK210 DDLVCE =→221DDLRVCE =221DDLCVCE =EE14128EE141-S04Dynamic Power ConsumptionDynamic Power ConsumptionPower = Energy/transition • Transition rate= CLVDD2• f0→1= CLVDD2• f • P0→1= CswitchedVDD2• f Power dissipation is data dependent – depends on the switching probability Switched capacitance Cswitched= CL• P0→1EE14115EE14129EE141-S04Transition Activity and PowerTransition Activity and Power Energy consumed in N cycles, EN:EN= CL• VDD2• n0→1n0→1 – number of 0→1 transitions in N cyclesfVCNnfNEPDDLNNNavg⋅⋅⋅⎟⎠⎞⎜⎝⎛=⋅=→∞→∞→210limlimfNnN⋅=→∞→→1010limαfVCPDDLavg⋅⋅⋅=→210αEE14130EE141-S04“Dynamic” or timing dependent component ➟Type of Logic Function (NOR vs. XOR)“Static” component (does not account for timing)➟Circuit Topology➟Type of Logic Style (Static vs. Dynamic)➟Signal Statistics➟Inter-signal Correlations➟Signal Statistics and CorrelationsFactors Affecting Transition ActivityFactors Affecting Transition ActivityEE14116EE14131EE141-S04Type of Logic Function: NOR vs. XORType of Logic Function: NOR vs. XOR011001010100OutBAExample: Static 2-input NOR GateAssume signal probabilitiespA=1 = 1/2pB=1 = 1/2Then transition probabilityp0→1 = pOut=0 x pOut=1= 3/4 x 1/4 =
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