ece 301 – digital electronics karnaugh maps (lecture #7) the slides included herein were taken...
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![Page 1: ECE 301 – Digital Electronics Karnaugh Maps (Lecture #7) The slides included herein were taken from the materials accompanying Fundamentals of Logic Design,](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0ef27/html5/thumbnails/1.jpg)
ECE 301 – Digital Electronics
Karnaugh Maps
(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.
![Page 2: ECE 301 – Digital Electronics Karnaugh Maps (Lecture #7) The slides included herein were taken from the materials accompanying Fundamentals of Logic Design,](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0ef27/html5/thumbnails/2.jpg)
Spring 2011 ECE 301 - Digital Electronics 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.
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Spring 2011 ECE 301 - Digital Electronics 3
Simplification using Boolean Algebra
Given: 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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Spring 2011 ECE 301 - Digital Electronics 4
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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Spring 2011 ECE 301 - Digital Electronics 5
Two-variable K-map
0
1
0 1
m 0 m 2
m 3 m 1
B
Arow # A B minterm
0 0 0 m0
1 0 1 m1
2 1 0 m2
3 1 1 m3
![Page 6: ECE 301 – Digital Electronics Karnaugh Maps (Lecture #7) The slides included herein were taken from the materials accompanying Fundamentals of Logic Design,](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0ef27/html5/thumbnails/6.jpg)
Spring 2011 ECE 301 - Digital Electronics 6
Two-variable K-map: Example
0 2
1 3
Minterm expansion: F(A,B) = m(0, 1) = A'B' + A'B
Maxterm expansion: F(A,B) = (2, 3) = (A'+B).(A'+B')
numeric algebraic
row # A B F
0 0 0 1
1 0 1 1
2 1 0 0
3 1 1 0
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Spring 2011 ECE 301 - Digital Electronics 7
Three-variable K-map
row # A B C minterm
0 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
![Page 8: ECE 301 – Digital Electronics Karnaugh Maps (Lecture #7) The slides included herein were taken from the materials accompanying Fundamentals of Logic Design,](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0ef27/html5/thumbnails/8.jpg)
Spring 2011 ECE 301 - Digital Electronics 8
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 F
0 0 0 0 0
1 0 0 1 0
2 0 1 0 1
3 0 1 1 1
4 1 0 0 1
5 1 0 1 0
6 1 1 0 1
7 1 1 1 0
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Spring 2011 ECE 301 - Digital Electronics 9
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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Spring 2011 ECE 301 - Digital Electronics 10
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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Spring 2011 ECE 301 - Digital Electronics 11
K-maps – Logical Adjacency
Gray code
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Spring 2011 ECE 301 - Digital Electronics 12
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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Spring 2011 ECE 301 - Digital Electronics 13
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.
![Page 14: ECE 301 – Digital Electronics Karnaugh Maps (Lecture #7) The slides included herein were taken from the materials accompanying Fundamentals of Logic Design,](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0ef27/html5/thumbnails/14.jpg)
Spring 2011 ECE 301 - Digital Electronics 14
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.
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Spring 2011 ECE 301 - Digital Electronics 15
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.
![Page 16: ECE 301 – Digital Electronics Karnaugh Maps (Lecture #7) The slides included herein were taken from the materials accompanying Fundamentals of Logic Design,](https://reader030.vdocuments.net/reader030/viewer/2022032800/56649d365503460f94a0ef27/html5/thumbnails/16.jpg)
Spring 2011 ECE 301 - Digital Electronics 16
Minimization: Example #5
For the following truth table:
# A B C F
0 0 0 0 0
1 0 0 1 1
2 0 1 0 0
3 0 1 1 1
4 1 0 0 1
5 1 0 1 0
6 1 1 0 0
7 1 1 1 1
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Spring 2011 ECE 301 - Digital Electronics 17
Example #5
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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Spring 2011 ECE 301 - Digital Electronics 18
Minimization: Example #6
For the following truth table:
# A B C F
0 0 0 0 0
1 0 0 1 1
2 0 1 0 1
3 0 1 1 1
4 1 0 0 0
5 1 0 1 1
6 1 1 0 0
7 1 1 1 0
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Spring 2011 ECE 301 - Digital Electronics 19
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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Spring 2011 ECE 301 - Digital Electronics 20
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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Spring 2011 ECE 301 - Digital Electronics 21
K-maps – Two minimal forms
F(A,B,C) = m(0,1,2,5,6,7) = M(3,4)
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Spring 2011 ECE 301 - Digital Electronics 22
Questions?