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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, 6 th Edition, by Roth and Kinney, and were used with permission from Cengage Learning.
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Karnaugh Maps and Determining a Minimal Cover

Feb 03, 2022

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Page 1: Karnaugh Maps and Determining a Minimal Cover

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.

Page 2: Karnaugh Maps and Determining a Minimal Cover

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

Page 3: Karnaugh Maps and Determining a Minimal Cover
Page 4: Karnaugh Maps and Determining a Minimal Cover

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) = Σ m(0,1,3,4,5,7,8,11,14)

Specify the equivalent maxterm expansion.

Page 5: Karnaugh Maps and Determining a Minimal Cover

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) = Π M(0,2,5,7,8,11,13,15)

Specify the equivalent minterm expansion.

Page 6: Karnaugh Maps and Determining a Minimal Cover

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) = Π M(0,1,2,3,6,11,14)

What is the cost of each logic circuit?

Page 7: Karnaugh Maps and Determining a Minimal Cover

Spring 2011 ECE 301 - Digital Electronics 7

Karnaugh Maps

Karnaugh maps can also be used to minimize incompletely specified functions.

Page 8: Karnaugh Maps and Determining a Minimal Cover

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) = Σ m(4,7) + Σ d(1,3)

Page 9: Karnaugh Maps and Determining a Minimal Cover

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) = Π M(0,2,5,6,8,13,15) . Π D(3,4,10)

Page 10: Karnaugh Maps and Determining a Minimal Cover

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) = Σ m(0,1,2,4,6,8,9,10) + Σ d(3,7,11,13,14)

Page 11: Karnaugh Maps and Determining a Minimal Cover

Spring 2011 ECE 301 - Digital Electronics 11

Determining a Minimal Cover

Page 12: Karnaugh Maps and 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

Page 13: Karnaugh Maps and Determining a Minimal Cover

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.

Page 14: Karnaugh Maps and Determining a Minimal Cover

Spring 2011 ECE 301 - Digital Electronics 14

Implicant

Prime

Implicant

Prime

Implicant

Implicant

Implicant

Prime

Implicant

Implicants and Prime Implicants

Additional Prime Implicants?

Page 15: Karnaugh Maps and Determining a Minimal Cover

Spring 2011 ECE 301 - Digital Electronics 15

Identifying Prime Implicants

Page 16: Karnaugh Maps and Determining a Minimal Cover
Page 17: Karnaugh Maps and Determining a Minimal Cover

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

Page 18: Karnaugh Maps and Determining a Minimal Cover

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

Page 19: Karnaugh Maps and Determining a Minimal Cover

Spring 2011 ECE 301 - Digital Electronics 19

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

Page 20: Karnaugh Maps and Determining a Minimal Cover

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

Page 21: Karnaugh Maps and 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.

Page 22: Karnaugh Maps and Determining a Minimal Cover

Spring 2011 ECE 301 - Digital Electronics 22

Questions?