Lecture 7: Karnaugh Map, Don t Cares

Transcription

1 EE210: Switching Systems Lecture 7: Karnaugh Map, Don t Cares Prof. YingLi Tian Sept. 28, 2016 Department of Electrical Engineering The City College of New York The City University of New York (CUNY) 1

2 The Karnaugh Map (K-map) K-map is a graphical approach to finding minimum SOP expressions (prime implicants) for function simplification. K-map is very useful for small design problems of 3-4 variables (up to 6 variables) Solutions for Problems of more that 6 variables can be found in Chapter 4, we will NOT cover them in this course. 2

3 Implicant An implicant is a rectangle of 1, 2, 4, 8,... (any power of 2) 1 s. That rectangle may not include any 0 s.

4 Implicant -- 2 The implicants of F are Minterms Groups of 2 Groups of 4 A B C D A CD CD A B CD BCD A BCD ACD ABC D B CD ABC D ABC ABCD ABD AB CD

5 Prime Implicant A prime implicant is an implicant that (from the point of view of the map) is not fully contained in any one other implicant. An essential prime implicant is a prime implicant that includes at least one 1 that is not included in any other prime implicant. prime implicant, but not essential prime implicant

6 Basic Rules of Karnaugh maps Anytime you have N variables, you will have 2 N possible combinations, and 2 N places in a truth table or K-Map. In a Karnaugh Map of any size, crossing a vertical or horizontal cell boundary is a change of only one variable -- no matter how many variables there are. Each single cell that contains a 1 represents a minterm in the function, and each minterm can be thought of as a "product" term with N variables. To combine variables, use groups of 2, 4, 8, etc. A group of 2 in an N-variable Karnaugh map will give you a "product" term with N-1 variables. A group of 4 will have N-2 variables, etc. You will never have a group of 3, a group of 5, etc. 6

7 Create K-map from Expressions f (x, y, z)= x yz + x yz + xy z + xy z + xyz

8 Create K-map from Expressions f (x, y, z)= x yz + x yz + xy z + xy z + xyz

9 Create K-map from Expressions f (A, B, C, D)= A B C D + A BC D + AB C D + ABC D + AB C D + A B CD + A BCD + AB CD + A B CD + ABCD + AB CD = m(0, 2, 3, 4,7, 8, 9, 10, 11, 13, 14)

10 Get expression from K-Map Practice 1 Write out the numerical expression of the following K-map 10

11 Get expression from K-Map Practice 2 Write out the numerical expressions of the following K-maps 11

12 Finding Minimum SOP Using K-Map 12

13 K-map to SOP -- Method 1. Find all essential prime implicants. Circle them on the map and mark the minterm(s) that make them essential with an asterisk (*). Do this by examining each 1 on the map that has not already been circled. It is usually quickest to start with the most isolated 1 s, that is, those that have the fewest adjacent squares with 1 s in them. 2. Find enough other prime implicants to cover the function. Do this using two criteria: a. Choose a prime implicant that covers as many new 1 s (that is, those not already covered by a chosen prime implicant). Must be 2, 4, 8, rectangles. b. Avoid leaving isolated uncovered 1 s. 13

14 K-map to SOP Example 1 minimum all prime implicants f = y z + wyz + w xz

15 K-map to SOP Example 2 f = x yz + x yz + xy z + xy z + xyz x y + x y + x z x y + x y + y z

16 K-map to SOP Example 3 not used, too many isolated 1s. minimum G = A BC + A CD + ABC + AC D

17 K-map to SOP Example 4 g = xz + wz + w yz + wx y g = xz + wz + w yz + x yz g = xz + wz + x yz + w xy

18 Multiple Solutions of Minimum SOP If there are multiple solutions, all minimum solutions must have the same number of terms and literals. 18

19 Practice 1: 19

20 Practice 1 solution 1: f = a c d + bc d + acd + b cd 20

21 Practice 1 -- solution 2 : f = a b d + a bc + abd + ab c 21

22 Practice 2: 22

23 Practice 2 -- Solution: F = A C D + AC D + A CD + ACD + B D + AB F = A C D + AC D + A CD + ACD + B D + B C F = A C D + AC D + A CD + ACD + AB + B C 23

24 K-Map with Don t Cares A prime implicant is a rectangle of 1, 2, 4, 8, 1 s or X s not included in any one larger rectangle. Thus, from the point of view of finding prime implicants, X s (don t cares) are treated as 1 s. An essential prime implicant is a prime implicant that covers at least one 1 not covered by any other prime implicant (as always). Don t cares (X s) do not make a prime implicant essential. 24

25 K-Map with Don t Cares minimum other p.i.s F = BD + A C D + AB C

26 Practice 3: 26

27 Practice 3 -- Solution: g 1 = c d + ab + b d + a cd g 2 = c d + ab + b d + a b c g 3 = c d + ab + ad + a b c

28 Finding Minimum POS Using K-Map 1. Map the complement of the function. (If there is already a map for the function, replace all 0 s by 1 s, all 1 s by 0 s and leave X s unchanged.) 2. Find the minimum sum of products expression for the complement of the function (using the techniques of the last two sections). 3. Use DeMorgan s theorem (P11) to complement that expression, producing a product of sums expression. P11a: (a + b) = a b P11aa: (a + b + c ) = a b c

29 Announcement: HW3 due today. HW4 is out today, due on 10/6/2016. Review Chapter Next class (Chapter ): Finding POS Five and Six Variable K-Maps Multiple Output Problems 29