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ECE 301 – Digital Electronics
Karnaugh Mapsand
Determining a Minimal Cover
(Lecture #8)
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
Four-variable K-maprow # A B C D minterm
0 0 0 0 0 m0
1 0 0 0 1 m1
2 0 0 1 0 m2
3 0 0 1 1 m3
4 0 1 0 0 m4
5 0 1 0 1 m5
… …
11 1 0 1 1 m11
12 1 1 0 0 m12
13 1 1 0 1 m13
14 1 1 1 0 m14
15 1 1 1 1 m15
Spring 2011 ECE 301 - Digital Electronics 3
Four-variable K-map
A B0 01 01 10 1
m0
m4
m12
m8
m1
m5
m13
m9
Gray code
C D
m
3
m
7
m
1
4
m1
1
0 0
m
2
m
6
m
1
5
m1
0
0 1
1 1
1 0
Gray code
Spring 2011 ECE 301 - Digital Electronics 4
Minimization: Example #7
Use a Karnaugh map to determine the minimum POS expression
For the following logic function:
F(A,B,C,D) = m(0,1,3,4,5,7,8,11,14)
Specify the equivalent maxterm expansion.
Spring 2011 ECE 301 - Digital Electronics 5
Minimization: Example #8
Use a Karnaugh map to determine the minimum SOP expression
For the following logic function:
F(A,B,C,D) = M(0,2,5,7,8,11,13,15)
Specify the equivalent minterm expansion.
Spring 2011 ECE 301 - Digital Electronics 6
Minimization: Example #9
Use a Karnaugh map to determine the
1. minimum SOP expression2. minimum POS expression
For the following logic function:
F(A,B,C,D) = M(0,1,2,3,6,11,14)
What is the cost of each logic circuit?
Spring 2011 ECE 301 - Digital Electronics 7
Karnaugh Maps
Karnaugh maps can also be used to minimize incompletely specified functions.
Spring 2011 ECE 301 - Digital Electronics 8
Minimization: Example #10
Use a Karnaugh map to determine the
1. minimum SOP expression2. minimum POS expression
For the following logic function:
F(A,B,C) = m(4,7) + d(1,3)
Spring 2011 ECE 301 - Digital Electronics 9
Minimization: Example #11
Use a Karnaugh map to determine theminimum SOP expression
For the following logic function:
F(A,B,C,D) = M(0,2,5,6,8,13,15) . D(3,4,10)
Spring 2011 ECE 301 - Digital Electronics 10
Minimization: Example #12
Use a Karnaugh map to determine theminimum POS expression
For the following logic function:
F(A,B,C,D) = m(0,1,2,4,6,8,9,10) + d(3,7,11,13,14)
Spring 2011 ECE 301 - Digital Electronics 11
Determining a Minimal Cover
Spring 2011 ECE 301 - Digital Electronics 12
Literals and Implicants Literal
Each occurrence of a variable or its complement in an expression
Implicant (SOP) ← represents a product term
A single 1 in the K-map A group of adjacent 1's in the K-map
Implicant (POS) ← represents a sum term
A single 0 in the K-map A group of adjacent 0's in the K-map
Spring 2011 ECE 301 - Digital Electronics 13
Prime Implicants
Prime Implicant (SOP) A product term implicant that cannot be
combined with another product term implicant to eliminate a literal.
Prime Implicant (POS) A sum term implicant that cannot be combined
with another sum term implicant to eliminate a literal.
Spring 2011 ECE 301 - Digital Electronics 14
Implicant
Prime Implicant
Prime Implicant
Implicant
Implicant
Prime Implicant
Implicants and Prime Implicants
Additional Prime Implicants?
Spring 2011 ECE 301 - Digital Electronics 15
Identifying Prime Implicants
Spring 2011 ECE 301 - Digital Electronics 16
Identifying Required Terms
Is this term required?
Spring 2011 ECE 301 - Digital Electronics 17
If a minterm is covered by only one prime implicant, that prime implicant is said to be essential, and must be included in the minimum sum of products (SOP).
Essential Prime Implicants
Prime Implicants
Implicants
Essential Prime Implicants
Spring 2011 ECE 301 - Digital Electronics 18
Note: 1’s shaded in blue are covered by only one prime implicant. All other 1’s are covered by at least two prime implicants.
Identifying Essential Prime Implicants
Spring 2011 ECE 301 - Digital Electronics 19
Determining a Minimal Cover• Identify all prime implicants
• Select all essential prime implicants
• Select prime implicant(s) to cover remaining terms by considering all possibilities
Sometimes selection is obvious Sometimes “guess” next prime implicant
Continue, perhaps recursively Try all possible “guesses”
• Determine the Boolean expression May not be unique
Spring 2011 ECE 301 - Digital Electronics 20
Shaded 1’s are covered by only one prime implicant.
Essential prime implicants:
A′B, AB′D′
Then AC′D covers the remaining 1’s.
Determining a Minimal Cover
Spring 2011 ECE 301 - Digital Electronics 21
A Minimal Cover
Thus …
A minimal cover is an expression that consists of the fewest product terms (for a SOP expression) or sum terms (for a POS
expression) and the fewest literals in each term.
Spring 2011 ECE 301 - Digital Electronics 22
Questions?