Chapter 3 The Karnough Map 3 - 1

3 - 2

Minimum Forms of Switching Functions 1.Combine terms by using ( Adjacency) Do this repeatedly to eliminates as many literals as possible. A given term may be used more than once because 2. Eliminate redundant terms by using the consensus theorems

Minimum Forms of Switching Functions Example: Find a minimum sum-of-products 3 - 4

Minimum Forms of Switching Functions Example: Find a minimum product-of-sums Eliminate by consensus 3 - 5

Map Representation Boolean n-cube Each n-cube represents a minterm of an n-variable Boolean function. There is 1-to-1 correspondence between each row in the truth table and each vertex in the n-cube. Boolean m-subcube Set of 2 m vertices in which n-m of the variables have the same value at every vertex, while the remaining m variables will take on the 2 m possible combinations of the values 0 and 1. In a standard sum-of-product form, each term will be a subcube of 1- minterms and each 1-minterm will be in at least one subcube

(a) Boolean n-cubes for n = 1, 2, and 3. (b) One of 2-subcubes of a 3-cube

Prime implicant(PI) A subcube that is not contained within any other subcube. Essential prime implicant(EPI) A subcube that includes a 1-minterm that is not included in any other subcube. Karnaugh map(K-map, or map) Map is a 2-dimensional form of Boolean n-cube. An array of squares arranged in rows and columns. Each square corresponds to one vertex of the cube - that is, one minterm of the Boolean function. 1 inside its square when the minterm is 1-minterm. 0 inside its square when the minterm is 0-minterm

(a) Boolean cubes and corresponding Karnaugh-maps for n = 1, 2, 3, and 4. m1m1 m0m0 m0m0 m1m1 m2m2 m3m3 m0m0 m1m1 m3m3 m2m2 m4m4 m5m5 m7m7 m6m6 m0m0 m1m1 m3m3 m2m2 m4m4 m5m5 m7m7 m6m6 m 12 m 13 m 15 m 14 m8m8 m9m9 m 11 m

Two-variable map (a) Example subcubes of size 2 Subcube x’ Subcube y

Chapter 3 The Karnaugh Map 3.1 Introduction to the Karnaugh Map(K-map)  Layout An array of squares arranged in rows and columns. Each square corresponds to one vertex of the cube - that is, one minterm of the Boolean function. A’B’A B’ A’BA B m0m0 m2m2 m1m1 m 03 B A P.114 Map

Map representation of three two-variable Boolean functions. (a) Truth table(b) AND: xy (d) XOR: x’y + xy’(c) OR: x + y

Chapter 3 The Karnaugh Map 3.1 Introduction to the Karnaugh Map(K-map)  Three-variable Maps Three-variable maps have eight squares, arranged in a rectangle P Map 3.3 Another Example Map 3.4 is incorrect

Chapter 3 The Karnaugh Map 3.2 Minimum Sum of Product Expressions using the K-map  Adjacencies on three- and four-variable maps P.123 Map

Three-variable map (b) Example subcubes of size 4(c) Example subcubes of size 2 (a) Map organization z x x’y’ yz xz’

Map representation of carry and sum functions of FA (a) Truth table of FA (b) Carry function: (c) Sum function:

Copyright © 2005 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 3.9 x yz + x yz + xy z + xy z + xyz. P

Chapter 3 The Karnaugh Map 3.1 Introduction to the Karnaugh Map(K-map)  Four-variable Maps A’B’C’D’A’B C’D’A B C’D’A B’C’D’ A’B’C’DA’B C’DA B C’DA B’C’D A’B’C DA’B C DA B C DA B’C D A’B’C D’A’B C D’A B C D’A B’C D’ P.117 Map

Four-variable map (a) Map organization (b) Example of 2-subcubes (c) Example of 3-subcubes y’w x’y xz w’ x’ x’y’z’w’x’y’z’w x’y’zw x’y’zw’ x’yz’w’x’yz’wx’yzwx’yzw’ xyz’w’xyz’wxyzwxyzw’ xy’z’w’xy’z’wxy’zwxy’zw’

Chapter 3 The Karnaugh Map 3.1 Introduction to the Karnaugh Map(K-map)  An Implicant of a Function Implicants of F Minterms A’B’C’D’ A’B’CD A’BCD ABC’D’ ABC’D ABCD AB’CD Groups of 2 A’CD BCD ACD B’CD ABC’ ABD Groups of 4 CD P.121 Map

Chapter 3 The Karnaugh Map 3.1 Introduction to the Karnaugh Map(K-map)  Groups of eight 1’s A’D’ P.119 Map

Representation of greater-than and less-than functions in maps. (a) Truth table (b) Greater-than function: (c) Less-than function: 3 -23

EXAMPLE Map method Using the map method, simplify the Boolean function (a) Prime implicants in the map (c) Two functional expressions (b) PI, EPI, and cover lists w’z’ wz yzw’y PIs w’y’z’wzxyzw’y

Copyright © 2008 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. g=xz+wz+wyz+wxy g=xz+wz+wyz+xyz g=xz+wz+xyz+wxy Example 3.10 g(w, x, y, z) = Σm(2, 5, 6, 7, 9, 10, 11, 13, 15) P

not usedminimum G = ABC + ACD + ABC + ACD Copyright © 2008 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 3.10 P

Chapter 3 The Karnaugh Map 3.5 Five-Variable Maps  A Five Variable Map A five-variable map consists of 2 5 = 32 squares. P.141 Map

Chapter 3 The Karnaugh Map 3.5 Five-Variable Maps  A Five Variable Map Example(1) F(A, B, C, D, E) = ∑ m(2, 5, 7, 11, 18, 21, 23, 27) m 2 +m 18 = A’B’C’DE’ + AB’C’DE’ = B’C’DE’ m 11 +m 27 = A’BC’DE + ABC’DE = BC’DE m 5 +m 7 +m 21 +m 23 = B’CE P.142 Example

Six-variable map (a) Map organization

Six-variable map(cont’d) (b) Example subcubes of size 16 z’w’ xz x’v

Chapter 4 Designing Combinational Systems 4.8 LARGER EXAMPLES A Driver for a Seven-Segment Display The seven-segment display : commonly used for decimal digits. Four inputs : W, X, Y, Z Seven outputs : a, b, c, d, e, f, g P.225 Display Driver abcdefgabcdefg W X Y Z a b c d e f g Figure

Chapter 2 Combinational Systems 2.1 The Design Process for Combinational Systems  The Development of Truth Tables A truth table for the seven-segment display driver. DigitWXYZabcdefg X X X XXXXXXX -1011XXXXXXX -1100XXXXXXX -1101XXXXXXX -1110XXXXXXX -1111XXXXXXX P.42 Table

Chapter 4 Designing Combinational Systems 4.8 LARGER EXAMPLES P.226 Map

Chapter 4 Designing Combinational Systems 4.8 LARGER EXAMPLES Several prime implicants that can be shared. a = W + Y + XZ + X’Z’ b = X’ + YZ + Y’Z’ c = X + Y’ + Z d = X’Z’ + YZ’ + X’Y + XY’Z e = X’Z’ + YZ’ f = W + X + Y’Z’ g = W + X’Y + XY’ + {XZ’ or YZ’} The shared terms are shown in colorful. TypeNumber Number of modulesChip 2-in in in P

Copyright © 2008 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Figure 2.12 NAND gates. Figure 2.13 Alternate symbol for NAND. Figure 2.14 Symbols for NOR gate. P

Conversion and Optimization Rules Replace AND and OR gates with NAND(NOR) gates by using Rules 1 and 2(3 and 4), and eliminate double inverters whenever possible

Translation of Sum of Products and Product of Sums to NAND and NOR Schematics

Example design with static-1 hazard (a) Map representation (b) Logic schematic(c) Timing diagram

Example of hazard-free design (a) Map representation (b) Logic schematic(c) Timing diagram Redundant cover