Lecture 5. Karnaugh-Map

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1 Lecture 5 - Lecture 5 Karnaugh-Map

2 Lecture 5-2 Karnaugh-Map Set Logic Venn Diagram K-map

3 Lecture 5-3 K-Map for 2 Variables

4 Lecture 5-4 K-Map for 3 Variables C C C

5 Lecture 5-5 Logic Expression, Truth Table, K-Map, and Logic Diagram F = + Logic Expression F Truth Table K-Map Logic Diagram (= Digital Circuit Represenation) F

6 Lecture 5-6 Non-Unique Circuits for the Same Function F = + ecause F = + = ( + )+ = + + F F

7 Lecture 5-7 Key Terms Literal - a variable or the complement of a variable. ex) X, Y, X Product Term (=Cube) : a single literal or a product of two or more literals. ex) Z,, WYZ, W Y Z. product term can be represented by a rectangle in a K-map and we will see why later. Sum-of-product (=SOP) : a sum of product terms. ex) W+Y, +Z+ Sum Term. Product-of-Sums (POS). Normal Term - a product or sum term in which no var. appears more than once. ex) Z, Y+W n-variable minterm - a normal product term with n literals. ex) W, W (3-variable minterm)

8 Lecture 5-8 True Table gain Let s derive the logical expression from a truth table. X Y Z F F = X Y Z + X Y Z (= XZ) truth table provides a Sum-of-Product (SOP) form naturally. : each product is a minterm in this case. K-map also provides a Sum-of-Product (SOP) form naturally. : each product does not have to be a minterm as we will see.

9 Lecture 5-9 K-Map Indexing Methods X Z Y Z Our Choice

10 Lecture 5 - Property of K-Map One cell in K-map represents a minterm. We can get an ND-OR (Sum-of-Product) style circuits easily from K-map. Two adjecent cells in K-map contain the same variable in positive and negative forms. Z X Y Z X Y Z

11 Lecture 5 - djacency in K-Map () cell is adjacent to the cells if they share a line.. Z WX YZ

12 Lecture 5-2 djacency in K-Map (2) Note that the cell on the boundary are adjacent to the cell on the other side. (You need some imagination power!!) Z WX YZ

13 Lecture 5-3 Motivation of K-Map Simplification If we express K-map using logic expression, only the cells with show up in Sum-of-Product form. Z X Y Z F = X Y Z + X Y Z X Y Z Two product terms that differ only in a variable can be combined into one. Ex) X Y Z + X Y Z = XZ(Y+Y) = XZ We like simpler form. How can we get simler expression from K-map?

14 Lecture 5-4 Introduction to K-Map Simplification We use a ractangle to specify that the product term in that ractangle can be combined to one product. Z X Z How to read the expression for a ractangle? - Find out the variable whose polarity (positive or negative) is consistent in the ractangle.

15 Lecture 5-5 Excercise on a Ractangle Reading Z WX YZ

16 Lecture 5-6 Minterm Numbers in Truth Table and K-Map X Y Z minterm no Z F =X Y Z + X Y Z can be described by F = (, 7)

17 Lecture 5-7 K-Map Simplification Examples F = (,,2,3) C F C F = C + C + C + C = C + C C F = C + C =

18 Lecture 5-8 Z Z

19 Lecture 5-9 Z F = X Z + X Y Z Z F = X Z + Y Z (etter) Lesson - Combine cells using as large ractangle as possible.

20 Lecture 5-2 Z Impossible cover (Only cells in a rectangle can be circled. Why?) Z Redundant cover Lesson - Use as few ractangless possible.

21 Lecture 5-2 djacent Covers CD D D The two covers differ in only one variables if the two covers are adjecent. Then the two covers can be combined into one cover.

22 Lecture 5-22 Partially Overlapped Covers CD D D The two covers differ in more than one variables. The two covers are partially overlapped. Then the two covers cannot be combined.

23 Lecture 5-23 djacent Covers with Different Sizes CD D D The two covers have different size. The two covers differ in more than one variables. Then the two covers cannot be combined.

24 Lecture 5-24 djacency in K-Map (3) cube is adjacent to another cube if they share a segment along one side and their sizes are the same. Z WX YZ

25 Lecture 5-25 The Covers That Can e Combined They must be adjecent. Correct covers can have 2 n cells in it = 4, = 8, = 6, so on...

26 Lecture 5-26 Practice Z Z CD CD

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