CprE 281: Digital Logic
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1 CprE 28: Digital Logic Instructor: Alexander Stoytchev
2 Minimization 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 Example: K-Map for the 2- Multiplexer
5 2- Multiplexer (Definition) Has two inputs: x and Also has another input line s If s=, then the output is equal to x If s=, then the output is equal to
6 Circuit for 2- Multiplexer x s s f x f (b) Circuit (c) Graphical symbol [ Figure 2.33b-c from the textbook ]
7 Truth Table for a 2- Multiplexer [ Figure 2.33a from the textbook ]
8 [ Figure 2.33a from the textbook ] Let s Draw the K-map
9 Let s Draw the K-map
10 Let s Draw the K-map
11 Let s Draw the K-map
12 Let s Draw the K-map
13 Let s Draw the K-map f (s, x, ) = x + s x
14 Let s Draw the K-map f (s, x, ) = x + s x Something is wrong!
15 Compare this with the SOP derivation
16 Let s Derive the SOP form
17 Let s Derive the SOP form
18 Let s Derive the SOP form Where should we put the negation signs? s x s x s x s x
19 Let s Derive the SOP form s x s x s x s x
20 Let s Derive the SOP form s x s x s x s x f (s, x, ) = s x + s x + s x + s x
21 Let s simplify this expression f (s, x, ) = s x + s x + s x + s x
22 Let s simplify this expression f (s, x, ) = s x + s x + s x + s x f (s, x, ) = s x ( + ) + s (x +x )
23 Let s simplify this expression f (s, x, ) = s x + s x + s x + s x f (s, x, ) = s x ( + ) + s (x +x ) f (s, x, ) = s x + s
24 Let s Draw the K-map again [ Figure 2.33a from the textbook ]
25 Let s Draw the K-map again
26 Let s Draw the K-map again
27 Let s Draw the K-map again
28 Let s Draw the K-map again s x The order of the labeling matters.
29 Let s Draw the K-map again s x
30 Let s Draw the K-map again s x
31 Let s Draw the K-map again s x
32 Let s Draw the K-map again s x f (s, x, ) = s x + s This is correct!
33 Two Different Ways to Draw the K-map x x 3 m x x 3 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 x 3 x (a) Truth table m m m 3 m 2 m 4 m 5 m 7 m 6
34 Another Way to Draw 3-variable K-map x x 3 m m m 2 m 3 m 4 m 5 m 6 m 7 x x 3 m m m 3 m 2 m 6 m 7 (b) Karnaugh map x x 3 m 4 m 5 (a) Truth table m m 4 m m 5 m 3 m 7 m 2 m 6
35 Gray Code Sequence of binary codes Consecutive lines vary by only bit
36 Gray Code & K-map s x
37 Gray Code & K-map s x
38 Gray Code & K-map s x These two neighbors differ only in the LAST bit
39 Gray Code & K-map s x These two neighbors differ only in the LAST bit
40 Gray Code & K-map s x These two neighbors differ only in the FIRST bit
41 Gray Code & K-map s x These two neighbors differ only in the FIRST bit
42 Adjacency Rules s x adjacent columns
43 Gray Code & K-map s x These four neighbors differ in the FIRST and LAST bit They are similar in their MIDDLE bit
44 A four-variable Karnaugh map x x x 3 x 4 m m 4 m 2 m 8 m m 5 m 3 m 9 x 3 m 3 m 2 m 6 m 7 m 5 m 4 m m x 4 [ Figure 2.53 from the textbook ]
45 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 3 x 4 x 3 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
46 Adjacency Rules adjacent rows adjacent columns adjacent columns
47 Gray Code & K-map x x2 x3 x4 m m m2 m3 m4 m5 m6 m7 m8 m9 m m m2 m3 m4 m5 x x 3 x 4 x 3 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
48 Gray Code & K-map x x2 x3 x4 m m m2 m3 m4 m5 m6 m7 m8 m9 m m m2 m3 m4 m5 x x 3 x 4 x 3 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
49 Example of a four-variable Karnaugh map [ Figure 2.54 from the textbook ]
50 Example of a four-variable Karnaugh map [ Figure 2.54 from the textbook ]
51 Strategy For Minimization
52 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).
53 Terminology Literal: a variable, complemented or uncomplemented Some Examples: _ X X 2
54 Terminology Implicant: product term that indicates the input combinations for which function output is Example _ x - indicates that x and x yield output of _ x
55 Terminology Prime Implicant Implicant that cannot be combined into another implicant with fewer literals Some Examples x x x 3 x 3 Not prime Prime
56 Terminology Essential Prime Implicant Prime implicant that includes a minterm not covered by any other prime implicant Some Examples x 3 x
57 Terminology Cover Collection of implicants that account for all possible input valuations where output is Ex. Ex. x x 3 + x x 3 + x x 3 x x 3 + x x 3 x x 3
58 Give the Number of Implicants? Prime Implicants? Example Essential Prime Implicants? x x 3
59 Why concerned with minimization? Simplified function Reduce the cost of the circuit Cost: Gates + Inputs Transistors
60 Three-variable function f (x,, x 3 ) = Σ m(,, 2, 3, 7) x 3 x x x 3 [ Figure 2.56 from the textbook ]
61 Example x x 3 x 4
62 Example x x 3 x 4
63 Example x x 3 x 4 x x 3 x 4 x 3 x 4 x x 3 x 4 x 3 x 4
64 Example: Another Solution x x 3 x 4 x x 3 x 4 x 3 x 4 x x 3 x 4 x 3 x 4 x x 4 x x 4 x x 3 x x 3 [ Figure 2.59 from the textbook ]
65 f ( x,, x 4 ) = Σ m(2, 3, 5, 6, 7,,, 3, 4) x x 3 x 4 x x 4 x 3 x 4 x 3 x 4 x x 3 x 3 [ Figure 2.57 from the textbook ]
66 x f ( x,, x 4 ) = Σ m(, 4, 8,,, 2, 3, 5) x 3 x 4 x 3 x 4 x x 3 x x 4 x x 3 x 4 x x 3 x x 4 [ Figure 2.58 from the textbook ]
67 Minimization of Product-of-Sums Forms
68 Do You Still Remember This Boolean Algebra Theorem?
69 Let s prove 4.b
70 Let s prove 4.b
71 Let s prove 4.b
72 Let s prove 4.b
73 Let s prove 4.b
74 Let s prove 4.b They are equal.
75 Grouping Example x x 2 x M M 2
76 Grouping Example x x 2 x x * = M * M 2 = M * M 2
77 Grouping Example x x 2 x x * = M * M 2 = M * M 2
78 Grouping Example x x 2 x x * = M * M 2 = M * M 2
79 Grouping Example x x 2 x x * = M * M 2 = M * M 2 _ ( x + ) * ( x + ) = Property 4b (Combining)
80 POS minimization of f (x,, x 3 ) = Π M(4, 5, 6) x 3 x ( x + x 3 ) ( x + ) [ Figure 2.6 from the textbook ]
81 POS minimization of f ( x,, x 4 ) = Π M(,, 4, 8, 9, 2, 5) x x 3 x 4 ( x 3 + x 4 ) ( + x 3 ) ( x + + x 3 + x 4 ) [ Figure 2.6 from the textbook ]
82 Questions?
83 THE END
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