ECE 331 – Digital System Design Karnaugh Maps and Determining a Minimal Cover (Lecture #7) The slides included herein were taken from the materials accompanying.

ECE 331 – Digital System Design

Karnaugh Mapsand

Determining a Minimal Cover

(Lecture #7)

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.

Fall 2010 ECE 331 - Digital System Design 2

Four-variable K-map

Each minterm is located adjacent to the four terms with which it can combine.

The 16 cells in the K-mapcorrespond to the 16 rowsin a 4-variable truth table.


Minimization using K-maps

Example:

Minimize the following function using a K-map:

F = m(1, 3, 4, 5, 10, 12, 13)



Example:



Example:


F = m(0, 2, 3, 5, 6, 7, 8, 10, 11, 14, 15)



Example:



Example:


F(A,B,C,D) = M(1, 3, 9, 12)



Example:



Exercise:

Using a K-map derive the minimum sum-of-products (SOP) for the following Boolean expression:

F(A,B,C,D) = m(1, 5, 6, 8, 9, 12, 13, 14)



Exercise:

Using a K-map derive the minimum product-of-sums (POS) for the following Boolean expression:

F(A,B,C,D) = m(1, 5, 6, 8, 9, 12, 13, 14)



Exercise:

Using a K-map derive the minimum Boolean expression for the following function:

F(A,B,C) = M(0, 2, 3, 7, 9, 10, 11, 14)

Note: the minimum Boolean expression may be in either SOP or POS form.



Example:

Using a K-map, minimize the following incompletely specified function:

F = m(1, 3, 5, 7, 9) + d(6, 12, 13)



Example:



Exercise:

Using a K-map derive the minimum sum-of-products (SOP) expression for the following incompletely

specified function:

F(A,B,C,D) = m(1, 5, 9, 13, 14) + d(4, 7, 8, 15)



Exercise:

Using a K-map derive the minimum product-of-sums (POS) expression for the following incompletely

specified function:

F(A,B,C,D) = M(1, 3, 4, 9, 10, 12) . D(2, 6, 11, 14)




Implicants and Prime Implicants Literal

Each appearance of a variable or its complement in an expression.

Implicant (SOP) Any single 1 or any group of 1’s which can be

combined together on a K-map of the function F Represents a product term

Prime Implicant (SOP) A product term implicant that cannot be

combined with another term to eliminate a literal


Implicant

Prime Implicant

Prime Implicant

Implicant

Implicant

Prime Implicant

Implicants and Prime Implicants


Identifying Prime Implicants


Identifying Required Terms

Is this term required?


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


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


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


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.



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.


Introduction to the 7-Segment Decoder


Binary Coded Decimal

Assign a 4-bit code to each decimal digit. A 4-bit code can represent 16 values. There are only 10 digits in the decimal number

system. Unassigned codes are not used.

How do we interpret these unused codes? Hint: think about K-maps.


BCD Digits

Decimal Digit BCD Code

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001


7-Segment Display


7-Segment Display


7-Segment Decoder


Exercise:

Design a 7-Segment Decoder.

7-Segment Decoder


Describing a Function


Describing a Function (SOP)# A B C F

0 0 0 0 0

1 0 0 1 1

2 0 1 0 1

3 0 1 1 0

4 1 0 0 0

5 1 0 1 1

6 1 1 0 0

7 1 1 1 1

F = A'B'C + A'BC' + AB'C + ABC

Minterm Expansion

F = (m1, m

2, m

5, m

7)

Shorthand Notation

F = m(1, 2, 5, 7)

Shorter-hand Notation

corresponds to the row #s


Describing a Function (POS)# A B C F

0 0 0 0 0

1 0 0 1 1

2 0 1 0 1

3 0 1 1 0

4 1 0 0 0

5 1 0 1 1

6 1 1 0 0

7 1 1 1 1

F = (A+B+C)(A+B'+C')(A'+B+C)(A'+B'+C)

Maxterm Expansion

F = (M0, M

3, M

4, M

6)

Shorthand Notation

F = M(0, 3, 4, 6)

Shorter-hand Notation

corresponds to the row #s


Questions?

ECE 331 – Digital System Design Karnaugh Maps and Determining a Minimal Cover (Lecture #7) The slides included herein were taken from the materials accompanying.

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