Multi-Level Logic: Conversion of FormsMulti-Level Logic: Conversion of FormsMulti-Level Logic: Mapping Between FormsMulti-Level Logic: CAD Tools for SimplificationMulti-Level Logic: CAD Tools for SimplificationMulti-Level Logic: CAD Tools for SimplificationNumber SystemsNumber SystemsNumber RepresentationsNumber SystemsEE 231 Digital ElectronicsFall 01Week 4-1Multi-Level Logic: Conversion of FormsNAND-NAND and NOR-NOR NetworksDeMorgan's Law: A + B = A • B; A • B = A + BWritten differently: A + B = A • B; A • B = A + BA B A B A B A B A B In other words, OR = NAND with complemented inputsAND = NOR with complemented inputsNAND = OR with complemented inputsNOR = AND with complemented inputs≡A B ≡≡A B A B ≡EE 231 Digital ElectronicsFall 01Week 4-2Multi-Level Logic: Conversion of FormsExample: Map AND/OR network to NAND/NAND networkNANDVerify equivalenceof the two formsNANDNANDAABBCCDDZZZABCD==+AB CDCDAB +=EE 231 Digital ElectronicsFall 01Week 4-3Multi-Level Logic: Mapping Between FormsExample: Map AND/OR network to NOR/NOR networkNORNORAStep 1Step 2NORDConserve"Bubbles"NORNORConserve"Bubbles"BCA’B’ZC’D’ZEE 231 Digital ElectronicsFall 01Week 4-4Multi-Level Logic: CAD Tools for SimplificationDecomposition:Take a single Boolean expression and replace with collection of newexpressions:F = A B C + A B D + A' C' D' + B' C' D'(12 literals)F rewritten as:F = X Y + X' Y'X = A BY = C + D(4 literals)ABBefore DecompositionAfter DecompositionFFAAACXBBDCCBDDYCDEE 231 Digital ElectronicsFall 01Week 4-5Multi-Level Logic: CAD Tools for SimplificationExtraction: common intermediate subfunctions are factored outF = (A + B) C D + EG = (A + B) E'H = C D Ecan be re-written as:F = X Y + EG = X E'H = Y EX = A + BY = C D(11 literals)(7 literals)Before ExtractionAfter ExtractionYXE AA BBF FEDCCDG GECDAHHBEE 231 Digital ElectronicsFall 01Week 4-6Multi-Level Logic: CAD Tools for SimplificationFactoring: expression in two level form re-expressed in multi-level formF = A C + A D + B C + B D + Ecan be rewritten as:F = (A + B) (C + D) + E(9 literals)(5 literals)Before Factoring After FactoringAABBCCDDCDEABFFEEE 231 Digital ElectronicsFall 01Week 4-7Number SystemsSign and Magnitude Representation0000011100111011111111101101110010101001100001100101010000100001+0+1+2+3+4+5+6+7-0-1-2-3-4-5-6-70 100 = + 4 1 100 = - 4+-High order bit is sign: 0 = positive (or zero), 1 = negativeThree low order bits is the magnitude: 0 (000) to 7 (111)Two representations for 0 are: 0000 and 1000EE 231 Digital ElectronicsFall 01Week 4-8Number SystemsOnes Complement0000011100111011111111101101110010101001100001100101010000100001+0+1+2+3+4+5+6+7-7-6-5-4-3-2-1-00 100 = + 4 1 011 = - 4+-To negate a number simply flip all the bits.Still two representations of 0!EE 231 Digital ElectronicsFall 01Week 4-9Number RepresentationsTwos Complement0000011100111011111111101101110010101001100001100101010000100001+0+1+2+3+4+5+6+7-8-7-6-5-4-3-2-10 100 = + 4 1 100 = - 4+-To negate: Twos complement = Ones complement + 1Only one representation for 0Easier to implement addition and subtractionEE 231 Digital ElectronicsFall 01Week 4-10Number SystemsOverflow ConditionsAdd two positive numbers to get a negative numberor two negative numbers to get a positive number-1-10000000100100011100001010110010010011010101111001101011111101111+1+2+3+4+5+6-8-7-6-5-4-3-2+00000000100100011100001010110010010011010101111001101011111101111+1+2+3+4+5+6-8-7-6-5-4-3-2+0+7+7-7 - 2 = +75 + 3 =
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