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# sis_full_adder_example_lec05 - Lecture 5 Multilevel Logic...

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ECE C03 Lecture 5 1 Lecture 5 Multilevel Logic Synthesis Hai Zhou ECE 303 Advanced Digital Logic Design Spring 2002

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ECE C03 Lecture 5 2 Outline • Mapping 2-level logic to NAND logic, NOR logic • CAD Tools for Multilevel Logic: MIS Algorithm • Factored forms • Flattening • READING: Katz 3.1, 3.2
ECE C03 Lecture 5 3 Multi-Level Logic: Advantages Reduced sum of products form: x = A D F + A E F + B D F + B E F + C D F + C E F + G 6 x 3-input AND gates + 1 x 7-input OR gate (may not exist!) 25 wires (19 literals plus 6 internal wires) 1 2 3 4 5 6 7 1 2 3 4 A A A B B B C C C D D D D E E E E F F F F F F F G G x x Factored form: x = (A + B + C) (D + E) F + G 1 x 3-input OR gate, 2 x 2-input OR gates, 1 x 3-input AND gate 10 wires (7 literals plus 3 internal wires)

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ECE C03 Lecture 5 4 Multi-Level Logic: Conversion of Forms NAND-NAND and NOR-NOR Networks DeMorgan's Law: (A + B)' = A' • B'; (A • B)' = A' + B' Written differently: A + B = (A' • B')'; (A • B) = (A' + B')' In other words, OR is the same as NAND with complemented inputs AND is the same as NOR with complemented inputs NAND is the same as OR with complemented inputs NOR is the same as AND with complemented inputs OR/NAND Equivalence A A B B A 0 0 1 1 A 1 1 0 0 B 0 1 0 1 B 1 0 1 0 A + B 0 1 1 1 A • B 0 1 1 1 A A B B A + B 1 1 1 0 A • B 1 1 1 0 OR OR Nand Nand
ECE C03 Lecture 5 5 Multi-Level Logic: Conversion Between Forms AND/NOR Equivalence It is possible to convert from networks with ANDs and ORs to networks with NANDs and NORs by introducing the appropriate inversions ("bubbles") To preserve logic levels, each introduced "bubble" must be matched with a corresponding "bubble" A A B B A 0 0 1 1 A 1 1 0 0 B 0 1 0 1 B 1 0 1 0 A • B 0 0 0 1 A + B 0 0 0 1 A A B B A • B 1 0 0 0 A + B 1 0 0 0 AND AND NOR NOR

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ECE C03 Lecture 5 6 A B C D A B C D Map AND/OR network to NAND/NAND network NAND NAND NAND
ECE C03 Lecture 5 7 Map AND/OR Network to NOR/NOR Network Step 1 Step 2 NOR NOR Conserve "Bubbles" NOR NOR NOR Conserve "Bubbles" A \A B \B C \C D \D Z Z

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ECE C03 Lecture 5 8 Map OR/AND Network to NAND/NAND Nand Nand Nand Conserve Bubbles! Conserve Bubbles!
ECE C03 Lecture 5 9 Multi-Level Logic: More than Two Levels ƒ = A (B + C D) + B C' Original AND-OR Network Introduction and Conservation of Bubbles Redrawn in terms of conventional NAND Gates C Level 1 Level 2 Level 3 Level 4 G1 D B A F G5 G4 G3 G2 \ C F \B F

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