notes-326-set5

# notes-326-set5 - Karnaugh Maps Objectives This section...

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1 1 Elec 326 Karnaugh Maps Karnaugh Maps ± Objectives This section presents a technique for simplifying logical expressions. It will: ² Define Karnaugh and establish the correspondence between Karnaugh maps and truth tables and logical expressions. ² Show how to use Karnaugh maps to derive minimal sum- of-products and product-of-sums expressions. ² Introduce the concept of "don't care" entries and show how to extend Karnaugh map techniques to include maps with don't care entries. ± Reading Assignment ² Sections 2.6 and 2.7 from the text 2 Elec 326 Karnaugh Maps Karnaugh Map Definitions ± A Karnaugh map is a two-dimensional truth-table. Unlike ordinary ( i.e. , one-dimensional) truth tables, however, certain logical network simplifications can be easily recognized from a Karnaugh map. Two-Variable Maps Truth Table 0 0 1 1 0 1 0 1 0 1 1 0 AB Z A B Z Type 1 Map 0 1 1 0 B Z Type 2 Map A 01 0 1 0 1 1 0 Three-Variable Maps Type 2 Map 1 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 Z B,C A C Type 1 Map A B Z 0 1 1 1 1 0 0 0 Truth Table 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 ABC 0 1 1 0 1 0 0 1 Z

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2 3 Elec 326 Karnaugh Maps Four-Variable Maps Truth Table 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ABC D 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 Z Type 2 Map Z C,D 00 01 10 11 00 01 11 10 A,B 1 0 0 1 1 1 1 1 1 1 1 1 1 0 A B C D Z 1 0 0 0 1 1 1 1 1 1 1 1 1 1 0 Type 1 Map 0 0 0 4 Elec 326 Karnaugh Maps ± The interpretation of a type 1 map is that the rows or columns labeled with a variable correspond to region of the map where that variable has value 1. ± Numbering of Karnaugh Map Squares. A = 1 region C = 1 region D = 1 region B = 1 region A B C D A B C D A B C D A B C D Z C,D 00 01 10 11 00 01 10 11 A,B 0 1 3 2 4 5 7 12 13 15 14 8 9 10 11 6 01 00 01 10 11 Z B,C A 0 1 3 2 4 7 5 6 B Z A 0 1 0 1 3 2
3 5 Elec 326 Karnaugh Maps ± Exercise: Plot the following expression on a Karnaugh map. Z = (A•B) (C+D) A C B D Z = A•B•(C+D)' + (A•B)'•(C+D) = A•B•C'•D' + (A'+B')•(C+D) = A•B•C'•D' + A'•C + A'•D + B'•C + B'•D 1 1 1 1 1 1 1 1 1 1 00 0 0 0 0 6 Elec 326 Karnaugh Maps Minimal Sum-Of-Products Expressions ± Ordering of Squares ² The important feature of the ordering of squares is that the squares are numbered so that the binary representations for the numbers of two adjacent squares differ in exactly one position. ³ This is due to the use of a Gray code (one in which adjacent numbers differ in only one position) to label the edges of a type 2 map. ³ The labels for the type 1 map must be chosen to guarantee this property. ² Note that squares at opposite ends of the same row or column also have this property (i.e., their associated numbers differ in exactly one position).

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4 7 Elec 326 Karnaugh Maps ± Merging Adjacent Product Terms Z= m 5 + m 13 = A'•B•C'•D + A•B•C'•D = (A'+A)•B•C'•D = 1•B•C'•D = B•C'•D ² Example A B C D Z 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 A' B C' D A B C' D Z B C' D Z B A C D Z 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 A•B•C' A' • C • D 8 Elec 326 Karnaugh Maps ± For k-variable maps, this reduction technique can also be applied to groupings of 4,8,16,. ..,2k rectangles all of whose binary numbers agree in (k-2),(k-3),(k- 4),. ..,0 positions, respectively.
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notes-326-set5 - Karnaugh Maps Objectives This section...

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