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Logic Design CS221 1 st Term 2009-2010 Logic-Circuit Implementation Cairo University Faculty of Computers and Information
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24/10/2009 cs221 – sherif khattab 2 Administrivia lab 2 is divided into two parts email subject must include the word CS221 homework 1 due today project ideas due next Saturday by email
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24/10/2009 cs221 – sherif khattab 3 important concepts clock timing diagram counter datasheet
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24/10/2009 cs221 – sherif khattab 4 K-map simplification a Boolean function is represented by a truth table function value (0 or 1) at each combination of inputs truth table => K-map K-map simplification combining adjacent squares
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24/10/2009 cs221 – sherif khattab 5 don't-care conditions in some Boolean functions, we do not care whether the output is 0 or 1 for some input combinations these don't-care conditions allow for simpler expressions Example: conversion from BCD-code to excess-3 code how many inputs? how many outputs? truth table (next slide)
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24/10/2009 cs221 – sherif khattab 6 don't-care conditions (contd.) 101 0 ? ? ? ? 101 1 ? ? ? ? 110 0 ? ? ? ? 101 0 X X X X 101 1 X X X X 110 0 X X X X
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24/10/2009 cs221 – sherif khattab 7 don't-care conditions (contd.) K-map for z don't-care conditions can be treated as 1 or 0, whichever gives a simpler expression m 12, m 14, m 10 : 1's m 13, m 15, m 11 : 0's z = ?
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24/10/2009 cs221 – sherif khattab 8 don't-care conditions (contd.) Example 2: F = ∑(1,3,7,11,15) with don't-care minterms d = ∑(0,2,5) F = w'x' + yzF = w'z' + yz =?
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24/10/2009 cs221 – sherif khattab 9 Boolean function implementation convert Boolean function expression into logic circuit Example: F = B'D' + B'C' + A'C'D
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24/10/2009 cs221 – sherif khattab 10 AND-OR implementation any Boolean function can be represented using only AND, OR, and NOT gates why? sum-of-minterms and sum-of-product forms can be directly converted into two-level AND-OR implementation
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24/10/2009 cs221 – sherif khattab 11 NAND implementation NAND gates are easy to fabricate any Boolean function can be represented using only NAND gates why? answer in the next 2 slides
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24/10/2009 cs221 – sherif khattab 12 NAND implementation (contd.) AND, OR, and NOT gates can represented using NAND gates
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24/10/2009 cs221 – sherif khattab 13 NAND implementation (contd.) any Boolean function can be represented using only AND, OR, and NOT gates AND, OR, and NOT gates can represented using NAND gates Then? NAND is a universal gate.
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24/10/2009 cs221 – sherif khattab 14 how to get a NAND implementation first, note that NAND gate can be: start with AND-OR implementation insert pairs of bubbles (NOT gates) works with alternating levels of AND and OR gates: AND-OR, AND-OR-AND-OR, etc.
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24/10/2009 cs221 – sherif khattab 15 how to get a NAND implementation (contd.)
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24/10/2009 cs221 – sherif khattab 16 how to get a NAND implementation (contd.)
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24/10/2009 cs221 – sherif khattab 17 how to get a NAND implementation (contd.)
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24/10/2009 cs221 – sherif khattab 18 how to get a NAND implementation (contd.) combining 1's in a K-map gives: sum-of-products? product-of-sums? ? ?
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24/10/2009 cs221 – sherif khattab 19 how to get a NAND implementation (contd.) Boolean function in truth table, minterms, or algebraic form represent the function in a K-map simplify the K-map convert the resulting SoP into AND-OR logic circuit convert the AND-OR circuit into NAND circuit as described earlier (by inserting pairs of bubbles)
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24/10/2009 cs221 – sherif khattab 20 NOR implementation NOR gates are easy to fabricate any Boolean function can be represented using only NOR gates why? answer in the next 2 slides
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24/10/2009 cs221 – sherif khattab 21 NOR implementation (contd.) AND, OR, and NOT gates can represented using NOR gates
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24/10/2009 cs221 – sherif khattab 22 NOR implementation (contd.) any Boolean function can be represented using only AND, OR, and NOT gates AND, OR, and NOT gates can represented using NOR gates Then? NOR is a universal gate.
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24/10/2009 cs221 – sherif khattab 23 how to get a NOR implementation first, note that NOR gate can be: start with OR-AND implementation insert pairs of bubbles (NOT gates) works with alternating levels of AND and OR gates: AND-OR, AND-OR-AND-OR, etc.
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24/10/2009 cs221 – sherif khattab 24 how to get a NOR implementation (contd.)
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24/10/2009 cs221 – sherif khattab 25 how to get a NAND implementation (contd.) combining 1's in a K-map gives: sum-of-products? product-of-sums? NAND NOR ??
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24/10/2009 cs221 – sherif khattab 26 product-of-sums from K-map simplify the function's complement (F') where are the squares of F'? combine adjacent 0's get an algebraic form for F' get complement of F': (F')' = F how?
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24/10/2009 cs221 – sherif khattab 27 product-of-sums from K-map Example: simplify the following K-map in product- of-sums form F' = AB + CD + BD' F = (A' + B')(C' + D')(B' + D)
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24/10/2009 cs221 – sherif khattab 28 NOR implementation NOR ??
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24/10/2009 cs221 – sherif khattab 29 NOR implementation
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24/10/2009 cs221 – sherif khattab 30 wired-and wiring of two NAND gates => AND no physical AND gate -> one-level => less delay F = (AB)'.(CD)' = (AB + CD)' AND-OR-INVERT
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24/10/2009 cs221 – sherif khattab 31 wired-or wiring of two NOR gates => OR no physical OR gate -> one-level => less delay F = (A+B)' + (C+D) = ((A+B) (C+D))' OR-AND-INVERT
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24/10/2009 cs221 – sherif khattab 32 other two-level implementations how many two-level implementations from the four gates: AND, OR, NAND, NOR? AND-AND AND-ORAND-NANDAND-NOR OR-AND OR-OR OR-NAND OR-NOR NAND-AND NAND-OR NAND-NAND NAND-NOR NOR-AND NOR-OR NOR-NAND NOR-NOR
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24/10/2009 cs221 – sherif khattab 33 other two-level implementations 8 degenerate (AND-AND) : one operation 8 nondegenerate AND-AND AND-ORAND-NANDAND-NOR OR-AND OR-OR OR-NAND OR-NOR NAND-AND NAND-OR NAND-NAND NAND-NOR NOR-AND NOR-OR NOR-NAND NOR-NOR which ones are duals? which ones are equivalent?
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24/10/2009 cs221 – sherif khattab 34 AND-NOR, NAND-AND how to get AND-NOR of a function? F' in sum-of-products (AND-OR) F in AND-NOR
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24/10/2009 cs221 – sherif khattab 35 OR-NAND, NOR-OR how to get OR-NAND of a function? F' in product-of-sums (OR-AND) F in OR-NAND
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24/10/2009 cs221 – sherif khattab 36 xor implementation can we simplify this K-map? note that the 1's are in the odd rows => XOR
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24/10/2009 cs221 – sherif khattab 37 xor implementation can we simplify this K-map? note that the 1's are in the odd rows => XOR
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24/10/2009 cs221 – sherif khattab 38 parity xor gates are often used in parity generation and checking truth table for even parity generation
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24/10/2009 cs221 – sherif khattab 39 parity xor gates are often used in parity generation and checking truth table for even parity checking
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