ETE 204 – Digital Electronics

ETE 204 – Digital Electronics

Karnaugh Maps

[Lecture: 6]

Instructor: Sajib RoyLecturer, ETE, ULAB

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Simplification of Logic Functions Logic functions can generally be simplified

using Boolean algebra. However, two problems arise:

– It is difficult to apply to Boolean algebra laws and theorems in a systematic way.

– It is difficult to determine when a minimum solution has been achieved.

Using a Karnaugh map is generally faster and easier than using Boolean algebra.

Summer 2012 ETE 204 - Digital Electronics

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Simplification using Boolean AlgebraGiven: F(A,B,C) = m(0, 1, 2, 5, 6, 7)

Find: minimum SOP expression

Combining terms in one way:

Combining terms in a different way:


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Karnaugh Maps

Like a truth table, a Karnaugh map specifies the value of a function for all combinations of the

input variables.


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Two-variable K-map

0

1

0 1

m 0 m 2

m 3 m 1

BA

row # A B minterm0 0 0 m0

1 0 1 m1

2 1 0 m2

3 1 1 m3


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Two-variable K-map: Example

0 2

1 3

Minterm expansion: F(A,B) = m(0, 1) = A'B' + A'BMaxterm expansion: F(A,B) = (2, 3) = (A'+B).(A'+B')

numeric algebraic

row # A B F0 0 0 11 0 1 12 1 0 03 1 1 0


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Three-variable K-map

row # A B C minterm0 0 0 0 m0

1 0 0 1 m1

2 0 1 0 m2

3 0 1 1 m3

4 1 0 0 m4

5 1 0 1 m5

6 1 1 0 m6

7 1 1 1 m7

m 0 m 4

m 5 m 1

BC

A

m 3 m 7

m 6 m 2

0 0

0 1

1 1

1 0

0 1

Gray Code


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Three-variable K-map: Example

3 7

2 6

0 4

1 5

Minterm expansion: F(A,B,C) = m(2, 3, 4, 6)Maxterm expansion: F(A,B,C) = (0, 1, 5, 7)

row # A B C F0 0 0 0 01 0 0 1 02 0 1 0 13 0 1 1 14 1 0 0 15 1 0 1 06 1 1 0 17 1 1 1 0


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Minimization using K-maps K-maps can be used to derive the

Minimum Sum of Products (SOP) expression Minimum Product of Sums (POS) expression

Procedure: Enter functional values in the K-map Identify adjacent cells with same logical value

Adjacent cells differ in only one bit Use adjacency to minimize logic function

Horizontal and Vertical adjacency K-map wraps from top to bottom and left to right


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Minimization using K-maps Logical Adjacency is used to

Reduce the number number of literals in a term Reduce the number of terms in a Boolean

expression. The adjacent cells

Form a rectangle Must be a power of 2 (e.g. 1, 2, 4, 8, …)

The greater the number of adjacent cells that can be grouped together (i.e. the larger the rectangle), the more the function can be reduced.


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K-maps – Logical Adjacency

Gray code


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Minimization: Example #1

Minimize the following logic function using a Karnaugh map:

F(A,B,C) = m(2, 6, 7)

Specify the equivalent maxterm expansion.


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Minimize the following logic function using a Karnaugh map:

F(A,B,C) = M(1, 3, 5, 6, 7)

Specify the equivalent minterm expansion.


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Use a Karnaugh map to determine the

1. minimum SOP expression2. minimum POS expression

For the following logic function:

F(A,B,C) = m(0, 1, 5, 7)

Specify the equivalent maxterm expansion.


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Use a Karnaugh map to determine the


For the following logic function:

F(A,B,C) = M(0, 1, 5, 7)

Specify the equivalent minterm expansion.


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For the following truth table:

# A B C F0 0 0 0 01 0 0 1 12 0 1 0 03 0 1 1 14 1 0 0 15 1 0 1 06 1 1 0 07 1 1 1 1


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Example #5

Specify the:

1. minterm expansion2. maxterm expansion

Use a K-map to determine the:



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For the following truth table:

# A B C F0 0 0 0 01 0 0 1 12 0 1 0 13 0 1 1 14 1 0 0 05 1 0 1 16 1 1 0 07 1 1 1 0


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Example #6

Specify the:

1. minterm expansion2. maxterm expansion

Use a K-map to determine the:



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Minimal Forms

Can a logic function have more than one minimum SOP expression?

Can a logic function have more than one minimum POS expression?


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K-maps – Two minimal formsF(A,B,C) = m(0,1,2,5,6,7) = M(3,4)


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Questions?


ETE 204 – Digital Electronics

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