CprE 281: Digital Logic
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1 CprE 28: Digital Logic Instructor: Alexander Stoytchev
2 Karnaugh Maps CprE 28: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev
3 Administrative Stuff HW4 is out It is due on Monday Sep 4 pm It is posted on the class web page I also sent you an with the link.
4 Administrative Stuff Homework Solutions are posted on BlackBoard
5 Quick Review
6 Do You Still Remember This Boolean Algebra Theorem?
7 Let s prove 4.a
8 Let s prove 4.a
9 Let s prove 4.a
10 Let s prove 4.a
11 Let s prove 4.a
12 Let s prove 4.a They are equal.
13
14 A B C D F Motivation An approach for simplifying logic expressions How do we guarantee we have reached minimum SOP/POS representation?
15 Two-Variable K-Map
16 Karnaugh Map (K-map) View the function in a visual form Same information as truth table Easier to group minterms x m x m m 2 m 3 m m 2 m m 3 (a) Truth table (b) Karnaugh map [ Figure 2.49 from the textbook ]
17 Minterms x m x m m m m 2 3 m m 2 m 3
18 Minterm Example x x m m m m 2 3 m + m 3
19 Minterm Example x x m m m m 2 3 m + m 3 _ x + x =
20 Grouping Example x x 2 x m m
21 Grouping Example x x 2 x x + = m + m = m + m
22 Grouping Example x x 2 x x + = m + m = m + m
23 Grouping Example x x 2 x x + = m + m = m + m + x = x x Property 4a (Combining)
24 Grouping Rules Group s with rectangles Both sides a power of 2: x, x2, 2x, 2x2, x4, 4x, 2x4, 4x2, 4x4 Can use the same minterm more than once Can wrap around the edges of the map Some rules in selecting groups: Try to use as few groups as possible to cover all s. For each group, try to make it as large as you can (i.e., if you can use a 2x2, don t use a 2x even if that is enough).
25 Two-Variable K-map x m x m m 2 m 3 m m 2 m m 3 (a) Truth table (b) Karnaugh map [ Figure 2.49 from the textbook ]
26 Step-By-Step Example x
27 . Draw The Map x x
28 2. Fill The Map m x x m m 2 m 3 m m 2 m m 3
29 2. Fill The Map m x x m m 2 m 3
30 3. Group m x x m m 2 m 3
31 3. Group m x x m m 2 m 3
32 3. Group m x x m m 2 m 3
33 4. Write The Expression x x
34 4. Write The Expression x x _ x +
35 Writing The Expression Find which variable is constant x _ x is constant
36 Writing The Expression Find which variable is constant x x is constant
37
38 These are all valid groupings
39 These are also valid But try to use larger rectangles if possible.
40 This one is valid too In this case the result is the constant function.
41 Why are these two not valid?
42 Let s Find Out x x 2 x m m 3
43 Let s Find Out x x 2 x x + = m + m 3 = m + m 3
44 Let s Find Out x x 2 x x + = m + m 3 = m + m 3
45 Let s Find Out x x 2 x x + = m + m 3 = m + m 3 x + x = x + x We can t use Property 4a here. This can t be simplified.
46 Three-Variable K-Map
47 Location of three-variable minterms x m x m m 2 m m 2 m 6 m 4 m 3 m 4 m m 3 m 7 m 5 m 5 (b) Karnaugh map m 6 m 7 (a) Truth table [ Figure 2.5 from the textbook ]
48 Location of three-variable minterms x (a) Truth table m m m 2 m 3 m 4 m 5 m 6 m 7 x m m m 3 m 2 m 6 m 7 (b) Karnaugh map m 4 m 5 Notice the placement of Variables Binary pair values Minterms
49 Gray Code Sequence of binary codes Two neighboring lines vary by only bit
50 Gray Code & K-map s x
51 Gray Code & K-map s x
52 Adjacency Rules adjacent columns
53 Adjacency Rules adjacent columns As if the K-map were drawn on a cylinder
54 Adjacency Rules m 4 m m 2 m 5 m m 3 adjacent columns As if the K-map were drawn on a cylinder
55 These are valid groupings
56 These are valid groupings
57 These are valid groupings
58 These are valid groupings
59 These are valid groupings
60 This is a valid grouping
61 Some invalid groupings
62 Examples of three-variable Karnaugh maps [ Figure 2.52 from the textbook ]
63 Four-Variable K-Map
64 A four-variable Karnaugh map x x x 4 m m 4 m 2 m 8 m m 5 m 3 m 9 m 3 m 2 m 6 m 7 m 5 m 4 m m x 4 [ Figure 2.53 from the textbook ]
65 A four-variable Karnaugh map x x2 x3 x4 m m m2 m3 m4 m5 m6 m7 m8 m9 m m m2 m3 m4 m5 x x 4 m m m 5 m 3 m 2 m 6 m 4 m 2 m 3 m 7 m 5 m 4 x m 8 m 9 m m x 4
66 Adjacency Rules adjacent rows adjacent columns adjacent columns
67 Adjacency Rules adjacent rows As if the K-map were drawn on a torus adjacent columns
68 Adjacency Rules adjacent rows m 2 m 8 m m 6 m 4 m 4 As if the K-map were drawn on a torus adjacent columns
69 x x 4 Some Valid Groupings x x x x 4 m m 4 m 2 m 8 m m 4 m 2 m 8 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 x x x 4 x x x 4 m m 4 m 2 m 8 m m 4 m 2 m 8 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4
70 x x 4 Some Valid Groupings x x x x 4 m m 4 m 2 m 8 m m 4 m 2 m 8 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 x x x 4 x x x 4 m m 4 m 2 m 8 m m 4 m 2 m 8 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4
71 x x 4 Some Valid Groupings x x x x 4 m m 4 m 2 m 8 m m 4 m 2 m 8 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 x x x 4 x x x 4 m m 4 m 2 m 8 m m 4 m 2 m 8 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4
72 x x 4 Some Valid Groupings x x x x 4 m m 4 m 2 m 8 m m 4 m 2 m 8 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 x x x 4 x x x 4 m m 4 m 2 m 8 m m 4 m 2 m 8 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4
73 x x 4 Some Invalid Groupings x x x x 4 m m 4 m 2 m 8 m m 4 m 2 m 8 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 x x x 4 All sides must be powers of 2. x x x 4 m m 4 m 2 m 8 m m 4 m 2 m 8 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4
74 x x 4 Some valid Groupings x x x x 4 m m 4 m 2 m 8 m m 4 m 2 m 8 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 x x x 4 All sides must be powers of 2. x x x 4 m m 4 m 2 m 8 m m 4 m 2 m 8 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4 m m 5 m 3 m 2 m 6 m 3 m 7 m 5 m 4 m 9 m m x 4
75 Example of a four-variable Karnaugh map [ Figure 2.54 from the textbook ]
76 Example of a four-variable Karnaugh map [ Figure 2.54 from the textbook ]
77 Example of a four-variable Karnaugh map [ Figure 2.54 from the textbook ]
78 Example of a four-variable Karnaugh map [ Figure 2.54 from the textbook ]
79 Five-Variable K-Map
80 A five-variable Karnaugh map x x 4 x x 4 x 5 = x 5 = f = x + x x 4 + x x 5 [ Figure 2.55 from the textbook ]
81 Questions?
82 THE END
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