ECE 301 – Digital Electronics
Karnaugh Maps
and
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.
Four-variable K-map
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row #
0
A
0
B
0
C
0
D
0
minterm
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
…
11
0
1
0
1
m5
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
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Minimization: Example #7
Use a Karnaugh map to determine the
minimum POS expression
For the following logic function:
F(A,B,C,D) = Σ m(0,1,3,4,5,7,8,11,14)
Specify the equivalent maxterm expansion.
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Minimization: Example #8
Use a Karnaugh map to determine the
minimum SOP expression
For the following logic function:
F(A,B,C,D) = Π M(0,2,5,7,8,11,13,15)
Specify the equivalent minterm expansion.
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Minimization: Example #9
Use a Karnaugh map to determine the
1. minimum SOP expression
2. minimum POS expression
For the following logic function:
F(A,B,C,D) = Π M(0,1,2,3,6,11,14)
What is the cost of each logic circuit?
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Karnaugh Maps
Karnaugh maps can also be used to minimize
incompletely specified functions.
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Minimization: Example #10
Use a Karnaugh map to determine the
1. minimum SOP expression
2. minimum POS expression
For the following logic function:
F(A,B,C) = Σ m(4,7) + Σ d(1,3)
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Minimization: Example #11
Use a Karnaugh map to determine the
minimum SOP expression
For the following logic function:
F(A,B,C,D) = Π M(0,2,5,6,8,13,15) . Π D(3,4,10)
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Minimization: Example #12
Use a Karnaugh map to determine the
minimum POS expression
For the following logic function:
F(A,B,C,D) = Σ m(0,1,2,4,6,8,9,10) + Σ d(3,7,11,13,14)
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Determining a Minimal Cover
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Literals and Implicants
●
Literal
–
●
●
Each occurrence of a variable or its complement in
an expression
Implicant (SOP)
–
A single 1 in the K-map
–
A group of adjacent 1's in the K-map
Implicant (POS)
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← represents a product term
← represents a sum term
–
A single 0 in the K-map
–
A group of adjacent 0's in the K-map
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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)
–
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A sum term implicant that cannot be combined
with another sum term implicant to eliminate a
literal.
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Implicants and Prime Implicants
Prime
Implicant
Implicant
Implicant
Implicant
Prime
Implicant
Prime
Implicant
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ECE 301 - Digital Electronics
Additional
Prime Implicants?
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Identifying Prime Implicants
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Essential Prime Implicants
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
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Identifying Essential Prime Implicants
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.
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Determining a Minimal Cover
1. Identify all prime implicants
2. Select all essential prime implicants
3. 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”
4. Determine the Boolean expression
–
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May not be unique
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Determining a Minimal Cover
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.
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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.
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Questions?
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