EE1411EE1411EECS141EE141EE141--Fall 2006Fall 2006Digital Integrated Digital Integrated CircuitsCircuitsLecture 28Lecture 28Adders, MultipliersAdders, MultipliersROMROMEE1412EECS141AnnouncementsAnnouncements Homework 10 due on ThursdayEE1412EE1413EECS141Class MaterialClass Material Last lecture Adders Today’s lecture Finish adders Multipliers ROM Reading Chapters 11, 12EE1414EECS141CarryCarry--LookaheadLookaheadAddersAddersEE1413EE1415EECS141LookaheadLookahead--Basic IdeaBasic IdeaCok,fAkBkCok,1–,,()GkPkCok 1–,+==AN-1, BN-1A1, B1P1S1••••••SN-1PN-1Ci, N-1S0P0Ci,0Ci,1A0, B0EE1416EECS141LookaheadLookahead: Topology: TopologyCok,GkPkGk1–Pk1–Cok 2–,+()+=Cok,GkPkGk1–Pk1–…P1G0P0Ci0,+()+()+()+=Expanding Lookahead equations:All the way:Co,3Ci,0VDDP0P1P2P3G0G1G2G3EE1414EE1417EECS141Logarithmic LookLogarithmic Look--Ahead AdderAhead AdderA7FA6A5A4A3A2A1A0A0A1A2A3A4A5A6A7Ftp∼ log2(N)tp∼ NEE1418EECS141Carry Carry LookaheadLookaheadTreesTreesCo0,G0P0Ci0,+=Co1,G1P1G0P1P0Ci0,++=Co2,G2P2G1P2P1G0P+2P1P0Ci0,++=G2P2G1+()=P2P1()G0P0Ci0,+()+G2:1P2:1Co0,+=Can continue building the tree hierarchically.EE1415EE1419EECS141Tree AddersTree Adders16-bit radix-2 Kogge-Stone tree(A0, B0)(A1, B1)(A2, B2)(A3, B3)(A4, B4)(A5, B5)(A6, B6)(A7, B7)(A8, B8)(A9, B9)(A10, B10)(A11, B11)(A12, B12)(A13, B13)(A14, B14)(A15, B15)S0S1S2S3S4S5S6S7S8S9S10S11S12S13S14S15EE14110EECS141EE1416EE14111EECS141EE14112EECS141Tree AddersTree Adders(a0, b0)(a1, b1)(a2, b2)(a3, b3)(a4, b4)(a5, b5)(a6, b6)(a7, b7)(a8, b8)(a9, b9)(a10, b10)(a11, b11)(a12, b12)(a13, b13)(a14, b14)(a15, b15)S0S1S2S3S4S5S6S7S8S9S10S11S12S13S14S1516-bit radix-4 Kogge-Stone TreeEE1417EE14113EECS141Sparse TreesSparse Trees(a0, b0)(a1, b1)(a2, b2)(a3, b3)(a4, b4)(a5, b5)(a6, b6)(a7, b7)(a8, b8)(a9, b9)(a10, b10)(a11, b11)(a12, b12)(a13, b13)(a14, b14)(a15, b15)S1S3S5S7S9S11S13S15S0S2S4S6S8S10S12S1416-bit radix-2 sparse tree with sparseness of 2EE14114EECS141EE1418EE14115EECS141Tree AddersTree Adders(A0, B0)(A1, B1)(A2, B2)(A3, B3)(A4, B4)(A5, B5)(A6, B6)(A7, B7)(A8, B8)(A9, B9)(A10, B10)(A11, B11)(A12, B12)(A13, B13)(A14, B14)(A15, B15)S0S1S2S3S4S5S6S7S8S9S10S11S12S13S14S15Brent-Kung TreeEE14116EECS141Example: Domino AdderExample: Domino AdderVDDClkPi= ai + biClkaibiVDDClkGi = aibiClkaibiPropagate GenerateEE1419EE14117EECS141Example: Domino AdderExample: Domino AdderVDDClkkPi:i-k+1Pi-k:i-2k+1Pi:i-2k+1VDDClkkGi:i-k+1Pi:i-k+1Gi-k:i-2k+1Gi:i-2k+1Propagate GenerateEE14118EECS141Example: Domino SumExample: Domino SumVDDClkGi:0ClkSumVDDClkdClkGi:0ClkSi1ClkdSi0KeeperEE14110EE14119EECS141ZX··Y×Zk2kk0=MN1–+∑==Xi2ii0=M1–∑⎝⎠⎜⎟⎜⎟⎜⎟⎛⎞Yj2jj0=N1–∑⎝⎠⎜⎟⎜⎟⎜⎟⎛⎞=XiYj2ij+j0=N1–∑⎝⎠⎜⎟⎜⎟⎜⎟⎛⎞i0=M1–∑=XXi2ii0=M1–∑=YYj2jj0=N1–∑=withThe Binary MultiplicationThe Binary MultiplicationEE14120EECS141x+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 1The Binary MultiplicationThe Binary MultiplicationEE14111EE14121EECS141Y0Y1X3X2X1X0X3HAX2FAX1FAX0HAY2X3FAX2FAX1FAX0HAZ1Z3Z6Z7Z5Z4Y3X3FAX2FAX1FAX0HAZ2Z0The Array MultiplierThe Array MultiplierEE14122EECS141HA FA FA HAHAFAFAFAFAFA FA HACritical Path 1Critical Path 2Critical Path 1 & 2()()[]()()andsumcarrymulttNtNtNMt⋅−+⋅−+⋅−+−≈ 1121The MThe M--byby--N Array Multiplier: N Array Multiplier: Critical PathCritical PathEE14112EE14123EECS141ABPCiVDDAAAVDDCiAPABVDDVDDCiCiCoSCiPPPPPSum GenerationCarry GenerationSetupTransmissionTransmission--Gate Full AdderGate Full AdderBalanced tsumand tcarryEE14124EECS141CarryCarry--Save MultiplierSave MultiplierHA HA HA HAFAFAFAHAFAHA FA FAFAHA FA HAVector Merging Adder()()mergeandcarrymultttNtNt+⋅−+⋅−= 11EE14113EE14125EECS141Multiplier Multiplier FloorplanFloorplanSCSCSCSCSCSCSCSCSCSCSCSCSCSCSCSCZ0Z1Z2Z3Z4Z5Z6Z7X0X1X2X3Y1Y2Y3Y0Vector Merging CellHA Multiplier CellFA Multiplier CellX and Y signals are broadcastedthrough the complete array.( )EE14126EECS141WallaceWallace--Tree MultiplierTree Multiplier6543210 6543210Partial products First stageBit position6543210 6543210Second stage Final adderFA HA(a) (b)(c) (d)EE14114EE14127EECS141WallaceWallace--Tree MultiplierTree MultiplierPartial productsFirst stageSecond stageFinal adderFA FA FAHA HAFAx3y3z7z6z5z4z3z2z1z0x3y2x2y3x1y1x3y0x2y0x0y1x0y2x2y2x1y3x1y2x3y1x0y3x1y0x0y0x2y1EE14128EECS141WallaceWallace--Tree MultiplierTree MultiplierFAFAFAFAy0y1y2y3y4y5SCi-1Ci-1Ci-1CiCiCiFAy0y1y2FAy3y4y5FAFACCSCi-1Ci-1Ci-1CiCiCiEE14115EE14129EECS141Multipliers Multipliers ––SummarySummary Optimization goals different than in binary adder Once again: Identify critical path Other possible techniques Logarithmic versus linear (Wallace Tree Mult) Data encoding (Booth) PipeliningFirst glimpse at system level
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