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Transcript of ECE 331 – Digital System Design Karnaugh Maps and Determining a Minimal Cover (Lecture #7) The...
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.
Fall 2010 ECE 331 - Digital System Design 3
Minimization using K-maps
Example:
Minimize the following function using a K-map:
F = m(1, 3, 4, 5, 10, 12, 13)
Fall 2010 ECE 331 - Digital System Design 5
Minimization using K-maps
Example:
Minimize the following function using a K-map:
F = m(0, 2, 3, 5, 6, 7, 8, 10, 11, 14, 15)
Fall 2010 ECE 331 - Digital System Design 7
Minimization using K-maps
Example:
Minimize the following function using a K-map:
F(A,B,C,D) = M(1, 3, 9, 12)
Fall 2010 ECE 331 - Digital System Design 9
Minimization using K-maps
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)
Fall 2010 ECE 331 - Digital System Design 10
Minimization using K-maps
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)
Fall 2010 ECE 331 - Digital System Design 11
Minimization using K-maps
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.
Fall 2010 ECE 331 - Digital System Design 12
Minimization using K-maps
Example:
Using a K-map, minimize the following incompletely specified function:
F = m(1, 3, 5, 7, 9) + d(6, 12, 13)
Fall 2010 ECE 331 - Digital System Design 14
Minimization using K-maps
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)
Fall 2010 ECE 331 - Digital System Design 15
Minimization using K-maps
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)
Fall 2010 ECE 331 - Digital System Design 17
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
Fall 2010 ECE 331 - Digital System Design 18
Implicant
Prime Implicant
Prime Implicant
Implicant
Implicant
Prime Implicant
Implicants and Prime Implicants
Fall 2010 ECE 331 - Digital System Design 21
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
Fall 2010 ECE 331 - Digital System Design 22
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
Fall 2010 ECE 331 - Digital System Design 23
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
Fall 2010 ECE 331 - Digital System Design 24
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.
Determining a Minimal Cover
Fall 2010 ECE 331 - Digital System Design 25
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.
Fall 2010 ECE 331 - Digital System Design 27
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.
Fall 2010 ECE 331 - Digital System Design 28
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
Fall 2010 ECE 331 - Digital System Design 32
Exercise:
Design a 7-Segment Decoder.
7-Segment Decoder
Fall 2010 ECE 331 - Digital System Design 34
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
Fall 2010 ECE 331 - Digital System Design 35
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