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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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 expression2. 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 expression2. 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 theminimum 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 theminimum 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) ← 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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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 Cover1. 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

– 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?