CHAPTER 5 KARNAUGH MAPS
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1 CHAPTER 5 1/36 KARNAUGH MAPS This chapter in the book includes: Objectives Study Guide 5.1 Minimum Forms of Switching Functions 5.2 Two- and Three-Variable Karnaugh Maps 5.3 Four-Variable Karnaugh Maps 5.4 Determination of Minimum Expressions 5.5 Five-Variable Karnaugh Maps 5.6 Other Uses of Karnaugh Maps 5.7 Other Forms of Karnaugh Maps Programmed Exercises Problems
2 Objectives 2/36 Topics introduced in this chapter: 1. Given a function (completely or in completely specified) of three to five variable, plot it on a Karnaugh map. The function may be given in minterm, maxterm, or algebraic form. 2. Determine the essential prime implicants of a function from a map. 3. Obtain the minimum sum-of-products or minimum product-of-sums form of a function from the map. 4. Determine all of the prime implicants of a function from a map. 5. Understand the relation between operations performed using the map and the corresponding algebraic operation.
3 5.1 Minimum Forms of Switching Functions 3/36 1. Combine terms by using XY XY X Do this repeatedly to eliminates as many literals as possible. A given term may be used more than once because X X X 2. Eliminate redundant terms by using the consensus theorems.
4 4/36 Fundamentals of Logic Design Chap Minimum Forms of Switching Functions ab bc c b b a abc abc c ab bc a c b a c b a F m c b a F (0,1,2,5,6,7) ),, ( ac bc b a abc abc c ab bc a c b a c b a F Example: Find a minimum sum-of-products
5 5/36 Fundamentals of Logic Design Chap Minimum Forms of Switching Functions ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( D C D B A D C C B A D B A D C B D C B C B A D B A D C B A D C B A D C B A D C B A D C B A D C B A Eliminate by consensus Example: Find a minimum product-of-sums
6 5.2 Two- and Three-Variable Karnaugh Maps 6/36 A 2-variable Karnaugh Map
7 5.2 Two- and Three-Variable Karnaugh Maps 7/36 Truth Table for a function F (a)
8 5.2 Two- and Three-Variable Karnaugh Maps 8/36 Truth Table and Karnaugh Map for Three-Variable Function
9 5.2 Two- and Three-Variable Karnaugh Maps 9/36 Location of Minterms on a Three-Variable Karnaugh Map
10 10/36 Fundamentals of Logic Design Chap Two- and Three-Variable Karnaugh Maps ),, ( M M M M M m m m c b a F Karnaugh Map of F(a, b, c) = m(1, 3, 5) = M(0, 2, 4, 6, 7)
11 5.2 Two- and Three-Variable Karnaugh Maps 11/36 Karnaugh Maps for Product Terms
12 5.2 Two- and Three-Variable Karnaugh Maps 12/36 Given Function f ( a, b, c) abc b c a
13 5.2 Two- and Three-Variable Karnaugh Maps 13/36 Simplification of a Three-Variable Function F T1 T2 a c b c
14 5.2 Two- and Three-Variable Karnaugh Maps 14/36 Complement of Map in Figure 5-6(a) F T 1 T 2 c ab
15 5.2 Two- and Three-Variable Karnaugh Maps 15/36 Karnaugh Maps Which Illustrate the Consensus Theorem Consensus term is redundant
16 5.2 Two- and Three-Variable Karnaugh Maps 16/36 Function with Two Minimal Forms F m(0,1,2,5,6,7 )
17 5.3 Four-Variable Karnaugh Maps 17/36 Location of Minterms on Four-Variable Karnaugh Map
18 5.3 Four-Variable Karnaugh Maps 18/36 Plot of acd + a b + d f ( a, b, c, d) acd a b d
19 5.3 Four-Variable Karnaugh Maps 19/36 Simplification of Four-Variable Functions
20 5.3 Four-Variable Karnaugh Maps 20/36 Simplification of an Incompletely Specified Function Don t care term
21 5.3 Four-Variable Karnaugh Maps 21/36 Figure s of f f x z wyz w y z x y 0s of f f y z wxz w xy f ( y z)( w x z)( w x y) minimum product of sum for f
22 5.4 Determination of Minimum Expressions Using Essential Prime Implicants 22/36 - Implicants of F : Any single 1 or any group of 1 s which can be combined together on a Map ( 관련항 ) - prime Implicants of F : A product term if it can not be combined with other terms to eliminate variable ( 주항 ) It is not Prime implicants since it can be combined with other terms Prime implicants It is not Prime implicants since it can be combined with other terms Prime implicants Fundamentals of Logic Design Chap. 5
23 5.4 Determination of Minimum Expressions Using Essential Prime Implicants 23/36 Determination of All Prime Implicants
24 5.4 Determination of Minimum Expressions Using Essential Prime Implicants 24/36 Because all of the prime implicants of a function are generally not needed in forming the minimum sum of products, selecting prime implicants is needed. - CD is not needed to cover for minimum expression -B C, AC, BD are essential prime implicants - CD is not an essential prime implicants Fundamentals of Logic Design Chap. 5
25 5.4 Determination of Minimum Expressions Using Essential Prime Implicants 25/36 1. First, find essential prime implicants ( 필수주항 ) 2. If minterms are not covered by essential prime implicants only, more prime implicants must be added to form minimum expression. 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. A BD A C A B D ACD or BCD
26 5.4 Determination of Minimum Expressions Using Essential Prime Implicants 26/36 Flowchart for Determining a Minimum Sum of Products Using a Karnaugh Map
27 5.4 Determination of Minimum Expressions Using Essential Prime Implicants 27/36 1. A B covers 1 6 and its adjacent essential PI 2. AB D covers 1 10 and its adjacent essential PI 3. AC D is chosen for minimal cover AC D is not an essential PI
28 5.5 Five-variable Karnaugh Maps 28/36 5 변수카노맵 4 변수카노맵위에하나를더위치시켜 3 차원으로만들수있다. 하층은 m 0 에서 m 15 까지, 상층은 m 16 에서 m 31 까지 4 변수카노맵각각의사각형을대각선으로나눠 2 차원으로만들수도있다. A A 00 DE BC A 1/0 BC DE
29 5.5 Five-Variable Karnaugh Maps 29/36 Five-Variable Karnaugh Map
30 5.5 Five-Variable Karnaugh Maps 30/36 Figure 5-22
31 5.5 Five-Variable Karnaugh Maps 31/36 Figure 5-23 F( A, B, C, D, E) m(0,1,4,5,13,15,20,21,22,23,24,26,28,30,31) F Resulting minimum solution A B D ABE ACD A BCE AB C or P1 P2 P3 P4 B CD
32 5.5 Five-Variable Karnaugh Maps 32/36 Figure 5-24 F( A, B, C, D, E) m(0,1,3,8,9,14,15,16,17,19,25,27,31) Final solution C D E F B C D B C E A C D A BCD ABDE or P 1 P 2 P 3 P 4 P 5 AC E Fundamentals of Logic Design Chap. 5
33 5.6 Other Uses of Karnaugh Maps 33/36 Fig Show Equivalence of two functions using karnaugh map minturm expansion of f is f maxterm expansion of f is f m(0,2,3,4,8,10,11,15) same M (1,5,6,7,9,12,16,14) Figure 5-25 Factoring expression
34 5.6 Other Uses of Karnaugh Maps 34/36 Minimize solution Figure 5-26 F ABCD B CDE A B BCE Using the consensus theorem: F ABCD B CDE A B BCE ACDE minimum solution : F A B BCE ACDE ACDE
35 5.7 Other Forms of Karnaugh Maps 35/36 Figure Veitch Diagrams
36 5.7 Other Forms of Karnaugh Maps 36/36 Figure Other Forms of Five-Variable Karnaugh Maps
37 5.7 Other Forms of Karnaugh Maps 37/36 Figure Other Forms of Five-Variable Karnaugh Maps
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