ECE 331 – Digital System Design
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Transcript of ECE 331 – Digital System Design
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ECE 331 – Digital System Design
Minimizing Boolean Expressions using K-maps,The Minimal Cover,
andIncompletely Specified Boolean Functions
(Lecture #6)
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ECE 331 - Digital System Design 2
Minimizing SOP Expressions
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ECE 331 - Digital System Design 3
Remember …
1. A Canonical SOP expression can be derived from a Truth table.
2. A shorthand notation can be used to describe a SOP expression.
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ECE 331 - Digital System Design 4
Minimizing SOP Expressions# 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
Canonical Sum-of-Products
F = (m1, m
2, m
5, m
7)
Shorthand Notation
F = m(1, 2, 5, 7)
Shorter-hand Notation
corresponds to the row #s
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ECE 331 - Digital System Design 5
Minimizing SOP Expressions
Exercise:
Given the following Canonical SOP expression:
F(A,B,C) = m(1, 2, 5, 7)
1. Write out the expression in terms of the minterms.2. Minimize the SOP expression using a K-Map
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ECE 331 - Digital System Design 6
Minimizing SOP Expressions
Exercise:
Given the following Canonical SOP expression:
F(A,B,C) = m(0, 2, 3, 6)
1. Write out the expression in terms of the minterms.2. Minimize the SOP expression using a K-Map
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ECE 331 - Digital System Design 7
Minimizing SOP Expressions
Exercise:
Given the following Canonical SOP expression:
F(A,B,C,D) = m(0, 4, 8, 10, 11, 12, 13, 15)
1. Write out the expression in terms of the minterms.2. Minimize the SOP expression using a K-Map
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ECE 331 - Digital System Design 8
Minimizing POS Expressions
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ECE 331 - Digital System Design 9
Minimizing POS Expressions# 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)
Canonical Product-of-Sums
F = (M0, M
3, M
4, M
6)
Shorthand Notation
F = M(0, 3, 4, 6)
Shorter-hand Notation
corresponds to the row #s
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ECE 331 - Digital System Design 10
Minimizing POS Expressions
Exercise:
Given the following Canonical POS expression:
F(A,B,C) = M(4, 5, 6)
1. Write out the expression in terms of the minterms.2. Minimize the POS expression using a K-Map
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ECE 331 - Digital System Design 11
Minimizing POS Expressions
Exercise:
Given the following Canonical POS expression:
F(A,B,C) = M(1, 2, 3, 5)
1. Write out the expression in terms of the minterms.2. Minimize the POS expression using a K-Map
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ECE 331 - Digital System Design 12
Minimizing POS Expressions
Exercise:
Given the following Canonical POS expression:
F(A,B,C,D) = M(0, 1, 4, 8, 9, 12, 15)
1. Write out the expression in terms of the minterms.2. Minimize the POS expression using a K-Map
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ECE 331 - Digital System Design 13
Selecting a Minimal Cover
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ECE 331 - Digital System Design 14
Definitions Literal: each appearance of a variable, either
uncomplemented or complemented Implicant: a product term that implies F=1 Prime Implicant: an implicant that cannot be
combined into another implicant that has fewer literals
Cannot be further minimized Essential Prime Implicant: a prime implicant
that covers a minterm uniquely.
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Definitions
F A B
C 0 0 0 1 1 1 1 0
0
1
Which are the implicants, prime implicants, and essential prime implicants?
1 1 0 0
1 1 1 0
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ECE 331 - Digital System Design 16
Definition: Implicants
F A B
C 0 0 0 1 1 1 1 0
0
1
Implicants: A'B'C', A'B'C, A'BC', A'BC, ABC (minterms)
1 1 0 0
1 1 1 0
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Definition: Implicants
F A B
C 0 0 0 1 1 1 1 0
0
1
Implicants: A'B'C', A'B'C, A'BC', A'BC, ABC (minterms)A'C', A'C, A'B', A'B, BC (pairs of minterms)
1 1 0 0
1 1 1 0
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Definition: Implicants
F A B
C 0 0 0 1 1 1 1 0
0
1
Implicants: A'B'C', A'B'C, A'BC', A'BC, ABC (minterms)A'C', A'C, A'B', A'B, BC (pairs of minterms) A' (quartet of minterms)
1 1 0 0
1 1 1 0
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ECE 331 - Digital System Design 19
Definition: Prime Implicants
F A B
C 0 0 0 1 1 1 1 0
0
1
Prime Implicants: BC,A'
1 1 0 0
1 1 1 0
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ECE 331 - Digital System Design 20
Definition: Essential Prime Implicants
F A B
C 0 0 0 1 1 1 1 0
0
1
Essential Prime Implicants: BC, A'
1 1 0 0
1 1 1 0
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Definitions
Minimal Cover (SOP): the sum (ORing) of prime implicants
Solution may not be unique For SOP, must cover each “1” at least once A minimal solution is one with the fewest
product terms in the SOP expression, and the fewest literals in each product term.
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Selecting a Minimal Cover (SOP) 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”
Write minimum Boolean expression May not be unique
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ECE 331 - Digital System Design 23
Example:
Determine the minimal cover for the following K-Map:
Selecting a Minimal Cover
a b c d
0
00 01 11 10
1 1 0
1 1 1 0
1 0 1 1
0 0 1 1
00
01
11
10
F
1. Identify Prime Implicants2. Identify Essential Prime Implicants3. Determine Minimal Cover
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Example #1
Prime Implicants: a'b'd, bc', ac, a'c'd, ab, b'cd
Essential Prime Implicants: bc', ac
Minimal Cover (SOP): F = a'b'd + bc' + ac
a b c d
0
00 01 11 10
1 1 0
1 1 1 0
1 0 1 1
0 0 1 1
00
01
11
10
F
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ECE 331 - Digital System Design 25
Example:
Determine the minimal cover for the following K-Map:
Selecting a Minimal Cover
1. Identify Prime Implicants2. Identify Essential Prime Implicants3. Determine Minimal Cover
y z w x
0
00 01 11 10
0 0 0
1 1 1 0
1 0 1 1
0 0 0 0
00
01
11
10
F
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ECE 331 - Digital System Design 26
Example #2
Prime Implicants: xy'z', w'xy', w'xz, xyz, wxy, wxz'
Essential Prime Implicants: none
Minimal Cover: F = xy'z' + w'xz + wxy
F = w'xy' + xyz + wxz'
y z w x
0
00 01 11 10
0 0 0
1 1 1 0
1 0 1 1
0 0 0 0
00
01
11
10
F
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ECE 331 - Digital System Design 27
Incompletely Specified Functions
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Incompletely Specified Functions
Some minterms should (or will) never occur. For example, Binary Coded Decimal (BCD)
These are considered “don't care” outputs. When minimizing a SOP expression using a K-
Map, treat a “don't care” as a “1” whenever it is beneficial.
When minimizing a POS expression using a K-Map, treat a “don't care” as a “0” whenever it is beneficial.
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ECE 331 - Digital System Design 29
Incompletely Specified Functions
Exercise:
Derive the minimum SOP expression for the following incompletely specified logic function:
F(A,B,C,D) = m(2, 4, 5, 6, 10) + D(12, 13, 14, 15)
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ECE 331 - Digital System Design 30
Incompletely Specified Functions
Exercise:
Derive the minimum POS expression for the following incompletely specified logic function:
F(A,B,C,D) = m(2, 4, 5, 6, 10) + D(12, 13, 14, 15)