Logic Design CS221 1 st Term 2009-2010 K-Map Cairo University Faculty of Computers and Information.
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Transcript of Logic Design CS221 1 st Term 2009-2010 K-Map Cairo University Faculty of Computers and Information.
Logic DesignCS221
1st Term 2009-2010
K-MapK-Map
Cairo University
Faculty of Computers and Information
13/10/2009 cs221 – sherif khattab 2
Administrivia imagine cup (imaginecup.com): tutorial session,
Wednesday @3pm in seminar room bonus winners (correct answer + explanation)
Mostafa Ahmed Sayed - 20080311 Mostafa Mohamed Mostafa - 20070434 Mido Talaat, Aya Elkady - 20080088 Mohamed Mahmoud Sayed - 20080272 Ahmed Emad Samy Yossef - 20080020
lab 2 is significantly longer than lab 1 email subject must include the word CS221 project groups: GA and GB
http://www.fci.cu.edu.eg/~skhattab/cs221/project.php
13/10/2009 cs221 – sherif khattab 3
boolean functions input -> output binary variables (and constants) and logic
operations defined by a truth table for two variables, only 16 boolean functions
possible. why? how many possible truth tables?
for n variables, 22n boolean functions
13/10/2009 cs221 – sherif khattab 4
gate-level minimization Boolean function representation
truth table: one way algebraic form (and logic circuit): many ways simplest and cheapest circuit
minimum # terms -> min. # gates minimum # literals per term -> min. # inputs to each gate not unique
gate-level minimization automatic (logic synthesis software) manual
algebraic: awkward map method: simple, straightforward (Karnaugh map or
K-map)
13/10/2009 cs221 – sherif khattab 5
map diagram of squares each square represents one minterm (one row
in truth table) For 2 variables:
4 minterms x'y', x'y, xy', xy
columns in which x = 1
13/10/2009 cs221 – sherif khattab 7
two-variable map Example: F = x + y = x (y + y') + y (x + x') = xy + xy' + x'y
13/10/2009 cs221 – sherif khattab 8
Gray code only one bit changes
from number to next: 7 -> 0100 8 -> 1100
2-bit Gray code 00, 01, 11, 10
13/10/2009 cs221 – sherif khattab 9
three-variable map 8 minterms column sequence in Gray code row 1 and column 01 -> 1 01 = 5 -> m
5
row x and column y'z -> x y'z
13/10/2009 cs221 – sherif khattab 10
2 adjacent cells each two adjacent cells differ in how many
variables (bits)? sum (OR) of two adjacent cells -> one AND
e.g., m5 + m
7 = xy'z + xyz = xz (y + y') = xz
e.g., m1 + m
5 = x'y'z + xy'z = y'z (x + x') = y'z
13/10/2009 cs221 – sherif khattab 11
2 adjacent cells m
0, m
1 adjacent?
m0, m
5 adjacent?
m0, m
2 adjacent?
m4, m
6 adjacent?
13/10/2009 cs221 – sherif khattab 12
4 adjacent cells sum (OR) of four adjacent cells -> one literal
e.g., m1 + m
3 + m
5 + m
7
001 011 101 111 = x'y'z + x'yz + xy'z + xyz = x'z (y + y') + xz (y + y') = x'z + xz = z (x + x') = z
13/10/2009 cs221 – sherif khattab 14
3-variable map simplification 1 cell -> three literals (e.g., xyz) 2 adjacent cells -> two literals (e.g., xy) 4 adjacent cells -> one literal (e.g., z) 8 adjacent cells -> ??
13/10/2009 cs221 – sherif khattab 19
four-variable map ?? minterms column sequence in Gray code row 01 and column 01 -> 01 01 = 5 -> m
5
row w'x and column y'z -> w'x y'z
13/10/2009 cs221 – sherif khattab 20
4-variable map simplification 1 cell -> four literals (e.g., wxyz) 2 adjacent cells -> three literals (e.g., w'xy) 4 adjacent cells -> two literals (e.g., wz) 8 adjacent cells -> one literal (e.g., x) 16 adjacent cells -> 1 adjacent?
13/10/2009 cs221 – sherif khattab 23
5-variable map ?? minterms the two 4-variable maps on top of each other