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.

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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

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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

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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) = 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) = 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 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?

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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 expression2. minimum POS expression

For the following logic function:

F(A,B,C) = m(4,7) + d(1,3)

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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)

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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)

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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) ← 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

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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) A sum term implicant that cannot be combined

with another sum term implicant to eliminate a literal.

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Implicant

Prime Implicant

Prime Implicant

Implicant

Implicant

Prime Implicant

Implicants and Prime Implicants

Additional Prime Implicants?

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Identifying Prime Implicants

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Identifying Required Terms

Is this term required?

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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

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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

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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

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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

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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?