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Karnaugh Maps [Lecture: 6] Instructor: Sajib Roy Lecturer, ETE, ULAB. ETE 204 – Digital Electronics. Simplification of Logic Functions. Logic functions can generally be simplified using Boolean algebra . However, two problems arise: - PowerPoint PPT Presentation
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ETE 204 – Digital Electronics
Karnaugh Maps
[Lecture: 6]
Instructor: Sajib RoyLecturer, ETE, ULAB
2
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
3
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:
Summer 2012 ETE 204 - Digital Electronics
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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.
Summer 2012 ETE 204 - Digital Electronics
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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
Summer 2012 ETE 204 - Digital Electronics
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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
Summer 2012 ETE 204 - Digital Electronics
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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
Summer 2012 ETE 204 - Digital Electronics
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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
Summer 2012 ETE 204 - Digital Electronics
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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
Summer 2012 ETE 204 - Digital Electronics
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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.
Summer 2012 ETE 204 - Digital Electronics
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K-maps – Logical Adjacency
Gray code
Summer 2012 ETE 204 - Digital Electronics
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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.
Summer 2012 ETE 204 - Digital Electronics
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Minimization: Example #2
Minimize the following logic function using a Karnaugh map:
F(A,B,C) = M(1, 3, 5, 6, 7)
Specify the equivalent minterm expansion.
Summer 2012 ETE 204 - Digital Electronics
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Minimization: Example #3
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.
Summer 2012 ETE 204 - Digital Electronics
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Minimization: Example #4
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 minterm expansion.
Summer 2012 ETE 204 - Digital Electronics
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Minimization: Example #5
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
Summer 2012 ETE 204 - Digital Electronics
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Example #5
Specify the:
1. minterm expansion2. maxterm expansion
Use a K-map to determine the:
1. minimum SOP expression2. minimum POS expression
Summer 2012 ETE 204 - Digital Electronics
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Minimization: Example #6
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
Summer 2012 ETE 204 - Digital Electronics
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Example #6
Specify the:
1. minterm expansion2. maxterm expansion
Use a K-map to determine the:
1. minimum SOP expression2. minimum POS expression
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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?
Summer 2012 ETE 204 - Digital Electronics