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ECE 301 – Digital Electronics
Karnaugh Maps
(Lecture #7)
The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6th Edition, by Roth and Kinney,
and were used with permission from Cengage Learning.
Spring 2011 ECE 301 - Digital Electronics 2
Simplification of Logic Functions
Logic functions can generally be simplified using Boolean algebra.
However, two problems arise:– It is difficult to apply to Boolean algebra laws
and theorems in a systematic way.– It is difficult to determine when a minimum
solution has been achieved. Using a Karnaugh map is generally faster and
easier than using Boolean algebra.
Spring 2011 ECE 301 - Digital Electronics 3
Simplification using Boolean Algebra
Given: F(A,B,C) = m(0, 1, 2, 5, 6, 7)Find: minimum SOP expression
Combining terms in one way:
Combining terms in a different way:
Spring 2011 ECE 301 - Digital Electronics 4
Karnaugh Maps
Like a truth table, a Karnaugh map specifies the value of a function for all combinations of the
input variables.
Spring 2011 ECE 301 - Digital Electronics 5
Two-variable K-map
0
1
0 1
m 0 m 2
m 3 m 1
B
Arow # A B minterm
0 0 0 m0
1 0 1 m1
2 1 0 m2
3 1 1 m3
Spring 2011 ECE 301 - Digital Electronics 6
Two-variable K-map: Example
0 2
1 3
Minterm expansion: F(A,B) = m(0, 1) = A'B' + A'B
Maxterm expansion: F(A,B) = (2, 3) = (A'+B).(A'+B')
numeric algebraic
row # A B F
0 0 0 1
1 0 1 1
2 1 0 0
3 1 1 0
Spring 2011 ECE 301 - Digital Electronics 7
Three-variable K-map
row # A B C minterm
0 0 0 0 m0
1 0 0 1 m1
2 0 1 0 m2
3 0 1 1 m3
4 1 0 0 m4
5 1 0 1 m5
6 1 1 0 m6
7 1 1 1 m7
m 0 m 4
m 5 m 1
BC
A
m 3 m 7
m 6 m 2
0 0
0 1
1 1
1 0
0 1
Gray Code
Spring 2011 ECE 301 - Digital Electronics 8
Three-variable K-map: Example
3 7
2 6
0 4
1 5
Minterm expansion: F(A,B,C) = m(2, 3, 4, 6)
Maxterm expansion: F(A,B,C) = (0, 1, 5, 7)
row # A B C F
0 0 0 0 0
1 0 0 1 0
2 0 1 0 1
3 0 1 1 1
4 1 0 0 1
5 1 0 1 0
6 1 1 0 1
7 1 1 1 0
Spring 2011 ECE 301 - Digital Electronics 9
Minimization using K-maps K-maps can be used to derive the
Minimum Sum of Products (SOP) expression Minimum Product of Sums (POS) expression
Procedure: Enter functional values in the K-map Identify adjacent cells with same logical value
Adjacent cells differ in only one bit Use adjacency to minimize logic function
Horizontal and Vertical adjacency K-map wraps from top to bottom and left to right
Spring 2011 ECE 301 - Digital Electronics 10
Minimization using K-maps Logical Adjacency is used to
Reduce the number number of literals in a term Reduce the number of terms in a Boolean
expression.
The adjacent cells
Form a rectangle Must be a power of 2 (e.g. 1, 2, 4, 8, …)
The greater the number of adjacent cells that can be grouped together (i.e. the larger the rectangle), the more the function can be reduced.
Spring 2011 ECE 301 - Digital Electronics 11
K-maps – Logical Adjacency
Gray code
Spring 2011 ECE 301 - Digital Electronics 12
Minimization: Example #1
Minimize the following logic function using a Karnaugh map:
F(A,B,C) = m(2, 6, 7)
Specify the equivalent maxterm expansion.
Spring 2011 ECE 301 - Digital Electronics 13
Minimization: Example #2
Minimize the following logic function using a Karnaugh map:
F(A,B,C) = M(1, 3, 5, 6, 7)
Specify the equivalent minterm expansion.
Spring 2011 ECE 301 - Digital Electronics 14
Minimization: Example #3
Use a Karnaugh map to determine the
1. minimum SOP expression2. minimum POS expression
For the following logic function:
F(A,B,C) = m(0, 1, 5, 7)
Specify the equivalent maxterm expansion.
Spring 2011 ECE 301 - Digital Electronics 15
Minimization: Example #4
Use a Karnaugh map to determine the
1. minimum SOP expression2. minimum POS expression
For the following logic function:
F(A,B,C) = M(0, 1, 5, 7)
Specify the equivalent minterm expansion.
Spring 2011 ECE 301 - Digital Electronics 16
Minimization: Example #5
For the following truth table:
# A B C F
0 0 0 0 0
1 0 0 1 1
2 0 1 0 0
3 0 1 1 1
4 1 0 0 1
5 1 0 1 0
6 1 1 0 0
7 1 1 1 1
Spring 2011 ECE 301 - Digital Electronics 17
Example #5
Specify the:
1. minterm expansion2. maxterm expansion
Use a K-map to determine the:
1. minimum SOP expression2. minimum POS expression
Spring 2011 ECE 301 - Digital Electronics 18
Minimization: Example #6
For the following truth table:
# A B C F
0 0 0 0 0
1 0 0 1 1
2 0 1 0 1
3 0 1 1 1
4 1 0 0 0
5 1 0 1 1
6 1 1 0 0
7 1 1 1 0
Spring 2011 ECE 301 - Digital Electronics 19
Example #6
Specify the:
1. minterm expansion2. maxterm expansion
Use a K-map to determine the:
1. minimum SOP expression2. minimum POS expression
Spring 2011 ECE 301 - Digital Electronics 20
Minimal Forms
Can a logic function have more than one minimum SOP expression?
Can a logic function have more than one minimum POS expression?
Spring 2011 ECE 301 - Digital Electronics 21
K-maps – Two minimal forms
F(A,B,C) = m(0,1,2,5,6,7) = M(3,4)
Spring 2011 ECE 301 - Digital Electronics 22
Questions?
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