Logic Design CS221 1 st Term Logic-Circuit Implementation Cairo University Faculty of Computers and Information

24/10/2009 cs221 – sherif khattab 2 Administrivia lab 2 is divided into two parts subject must include the word CS221 homework 1 due today project ideas due next Saturday by

24/10/2009 cs221 – sherif khattab 3 important concepts clock timing diagram counter datasheet

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

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)‏

24/10/2009 cs221 – sherif khattab 6 don't-care conditions (contd.)‏ ? ? ? ? ? ? ? ? ? ? ? ? X X X X X X X X X X X X

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 = ?

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 =?

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

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

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

24/10/2009 cs221 – sherif khattab 12 NAND implementation (contd.)‏ AND, OR, and NOT gates can represented using NAND gates

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.

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.

24/10/2009 cs221 – sherif khattab 15 how to get a NAND implementation (contd.)‏

24/10/2009 cs221 – sherif khattab 16 how to get a NAND implementation (contd.)‏

24/10/2009 cs221 – sherif khattab 17 how to get a NAND implementation (contd.)‏

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? ? ?

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)‏

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

24/10/2009 cs221 – sherif khattab 21 NOR implementation (contd.)‏ AND, OR, and NOT gates can represented using NOR gates

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.

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.

24/10/2009 cs221 – sherif khattab 24 how to get a NOR implementation (contd.)‏

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 ??

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?

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)‏

24/10/2009 cs221 – sherif khattab 28 NOR implementation NOR ??

24/10/2009 cs221 – sherif khattab 29 NOR implementation

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

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

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

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?

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

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

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

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

24/10/2009 cs221 – sherif khattab 38 parity xor gates are often used in parity generation and checking truth table for even parity generation

24/10/2009 cs221 – sherif khattab 39 parity xor gates are often used in parity generation and checking truth table for even parity checking