introduction to digital logic · 2 algorithmic simplification...
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Introduction to Digital Logic
K-Maps
2
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Put in sum of products form
Put in sum of minterms form
Find the maximal p-terms
Find the minterms that only have one maximal p-term
Those are the start of the expression
Remove minterms that have that p-term
Find the maximal p-term that occurs the most times. Pick one if tie.Remove minterms that contain that maximal p-term Pick another maximal p-term, add to the expression, remove minterm
3
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Find all of the minterms
ABCD ABC'D
A'BCD A'BCD' A'BC'D A'BC'D'
ABC'D' A'BC'D'
AB'CD A'B'CD
AB'CD' A'B'CD'
4
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
ABCD ABC'D
A'BCD A'BCD' A'BC'D A'BC'D'
ABC'D'
AB'CD A'B'CD
AB'CD' A'B'CD'
5
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Find the maximal p-terms
ABCD (BD, CD) ABC'D (BC', BD)
A'BCD (BD, A'B, CD, A'C) A'BCD' (A'C, A'B) A'BC'D (BC', BD, A'B)
A'BC'D' (A'B, BC')
ABC'D' (BC')
AB'CD (CD, B'C) A'B'CD (CD, B'C, A'C)
AB'CD' (B'C) A'B'CD' (B'C, A'C)
6
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Find minterms that only have one maximal p-term
ABCD (BD, CD) ABC'D (BC', BD)
A'BCD (BD, A'B, CD, A'C) A'BCD' (A'C, A'B) A'BC'D (BC', BD, A'B)
A'BC'D' (A'B, BC')
ABC'D' (BC')
AB'CD (CD, B'C) A'B'CD (CD, B'C, A'C)
AB'CD' (B'C) A'B'CD' (B'C, A'C)
7
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Write those as the start to the expression
ABCD (BD, CD) ABC'D (BC', BD)
A'BCD (BD, A'B, CD, A'C) A'BCD' (A'C, A'B) A'BC'D (BC', BD, A'B)
A'BC'D' (A'B, BC')
ABC'D' (BC')
AB'CD (CD, B'C) A'B'CD (CD, B'C, A'C)
AB'CD' (B'C) A'B'CD' (B'C, A'C)
BC' + B'C
8
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Remove any minterms that have those maximal p-terms
ABCD (BD, CD)
A'BCD (BD, A'B, CD, A'C) A'BCD' (A'C, A'B)
BC' + B'C
9
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Find the maximal p-term that occurs the most times. Pick one if tie.
ABCD (BD, CD)
A'BCD (BD, A'B, CD, A'C) A'BCD' (A'C, A'B)
BC' + B'C
10
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Remove minterms that contain that maximal p-term Pick another maximal p-term, add to the expression, remove minterm
ABCD (BD, CD)
BC' + B'C + A'B
11
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
BC' + B'C + A'B + CD
12
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
Boxes represent the minterms
Values grouped together
13
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
Find all the minterms and put a 1 in those boxes
14
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1 1 1
1 1
15
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1 1 1
1 1 1
16
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1
1 1 1 1
1 1 1
1
17
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1
1 1 1 1
1 1 1
1
18
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
All the minterms are represented
19
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
Group into 8, 4, 2, 1. The bigger the group, the better.
20
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
Represented by maximal p-term BC’
21
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
22
K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
23
K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
24
K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
25
K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
All maximal p-terms are represented
26
K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
Find mniterms represented by only one p-terms. Find 1s only covered by one box.
27
K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
BC’ + B’C
However, all 1s are not covered
28
K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
BC’ + B’C + AB + CD
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