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Chapter 2 - Part 1 - PPT - Mano & Kime - 2nd Ed

Slide 1 Lecture 7 Karnaugh Maps

Lecture 7 Karnaugh Maps

Slide 2 Outline

Outline Circuit Optimization Literal cost Gate input cost Two-Variable Karnaugh Maps Three-Variable Karnaugh Maps

Slide 3 Circuit Optimization

Circuit Optimization Goal: To obtain the simplest implementation for a given function Optimization is a more formal approach to simplification that is performed using a specific procedure or algorithm Optimization requires a cost criterion to measure the simplicity of a circuit Two distinct cost criteria we will use: Literal cost (L) Gate input cost (G) Gate input cost with NOTs (GN)

Slide 4 Literal Cost

D Literal a variable or its complement Literal cost the number of literal appearances in a Boolean expression corresponding to the logic circuit diagram Examples: F = BD + A C + A F = BD + A C + A + AB F = (A + B)(A + D)(B + C + )( + + D) Which solution is best? Literal Cost D B C B B D C B C L=8 L=11 L=10

Slide notes

2nd Literal Cost = 11 3rd Literal Cost = 10 The first solution is best

Slide 5 Gate Input Cost

Gate Input Cost Gate input costs - the number of inputs to the gates in the implementation corresponding exactly to the given equation or equations. (G - inverters not counted, GN - inverters counted) For SOP and POS equations, it can be found from the equation(s) by finding the sum of: all literal appearances the number of terms excluding terms consisting only of a single literal,(G) and optionally, the number of distinct complemented single literals (GN). Example: F = BD + A C + A F = BD + A C + A + AB F = (A + )(A + D)(B + C + )( + + D) Which solution is best? D B C B B D C B D B C G=11,GN=14 G=15,GN=18 G=14,GN=17

Slide notes

G = 15, GN = 18 (second value includes inverter inputs) G = 14, GN = 17 1st solution is best

Slide 6 Cost Criteria (continued)

Example 1: F = A + B C + Cost Criteria (continued) A B C F B C L = 5 L (literal count) counts the AND inputs and the single literal OR input. G = L + 2 = 7 G (gate input count) adds the remaining OR gate inputs GN = G + 2 = 9 GN(gate input count with NOTs) adds the inverter inputs

Slide 7 Cost Criteria (continued)

Example 2: F = A B C + L = 6 G = 8 GN = 11 F = (A + )( + C)( + B) L = 6 G = 9 GN = 12 Same function and same literal cost But first circuit has better gate input count and better gate input count with NOTs Select it! Cost Criteria (continued) B C A A B C F C B F A B C A

Slide 8 Why Use Gate Input Counts?

Why Use Gate Input Counts? CMOS logic gates: Each input adds: P-type transistor to pull-up network N-type transistor to pull-down network Adobe Systems

Slide 9 Boolean Function Optimization

Boolean Function Optimization Minimizing the gate inputs reduces circuit cost. Some important questions: When do we stop trying to reduce the cost? Do we know when we have a minimum cost? Two-level SOP & POS optimum or near-optimum functions Karnaugh maps (K-maps) Graphical technique useful for up to 5 inputs

Slide 10 Two Variable K-Maps

Two Variable K-Maps A 2-variable Karnaugh Map: Similar to Gray Code Adjacent minterms differ by one variable y = 0 y = 1 x = 0 m 0 = m 1 = x = 1 m 2 = m 3 = y x y x y x y x

Slide 11 K-Map and Truth Tables

K-Map and Truth Tables The K-Map is just a different form of the truth table. Example Two variable function: We choose a,b,c and d from the set {0,1} to implement a particular function, F(x,y). Function Table K-Map Input Values (x,y) Function Value F(x,y) 0 0 a 0 1 b 1 0 c 1 1 d y = 0 y = 1 x = 0 a b x = 1 c d

Slide 12 Karnaugh Maps (K-map)

Karnaugh Maps (K-map) A K-map is a collection of squares Each square represents a minterm The collection of squares is a graphical representation of a Boolean function Adjacent squares differ in the value of one variable Alternative algebraic expressions for the same function are derived by recognizing patterns of squares The K-map can be viewed as A reorganized version of the truth table

Slide 13 Some Uses of K-Maps

Some Uses of K-Maps Finding optimum or near optimum SOP and POS standard forms, and two-level AND/OR and OR/AND circuit implementations for functions with small numbers of variables Demonstrate concepts used by computer-aided design programs to simplify large circuits

Slide 14 K-Map Function Representation

K-Map Function Representation Example: F(x,y) = x For function F(x,y), the two adjacent cells containing 1 s can be combined using the Minimization Theorem: F = x y = 0 y = 1 x = 0 0 0 x = 1 1 1 x y x y x ) y , x ( F = + =

Slide 15 K-Map Function Representation

K-Map Function Representation Example: G(x,y) = x + y For G(x,y), two pairs of adjacent cells containing 1 s can be combined using the Minimization Theorem: G = x+y y = 0 y = 1 x = 0 0 1 x = 1 1 1 ( ) ( ) y x y x xy y x y x ) y , x ( G + = + + + = Duplicate x y

Slide 16 Three Variable Maps

Three Variable Maps A three-variable K-map: Where each minterm corresponds to the product terms: Note that if the binary value for an index differs in one bit position, the minterms are adjacent on the K-Map yz=00 yz=01 yz=11 yz=10 x=0 m0 m1 m3 m2 x=1 m4 m5 m7 m6 yz=00 yz=01 yz=11 yz=10 x=0 x=1 z y x z y x z y x z y x z y x z y x z y x z y x

Slide 17 Alternative Map Labeling

Alternative Map Labeling Map use largely involves: Entering values into the map, and Reading off product terms from the map. Alternate labelings are useful: y z x 1 0 2 4 3 5 6 7 x y z z y y z z 1 0 2 4 3 5 6 7 x 0 1 00 01 11 10 x

Slide 18 Example Functions

Example Functions By convention, we represent the minterms of F by a "1" in the map and leave the minterms of blank Example: Example: Learn the locations of the 8 indices based on the variable order shown (x, most significant and z, least significant) on the map boundaries y x 1 0 2 4 3 5 6 7 1 1 1 1 z x y 1 0 2 4 3 5 6 7 1 1 1 1 z F

Slide 19 Combining Squares

Combining Squares By combining squares, we reduce number of literals in a product term, reducing the literal cost, thereby reducing the other two cost criteria On a 3-variable K-Map: One square represents a minterm with three variables Two adjacent squares represent a product term with two variables Four adjacent terms represent a product term with one variable Eight adjacent terms is the function of all ones (no variables) = 1.

Slide 20 Example: Combining Squares

Example: Combining Squares Example: Let Applying the Minimization Theorem three times: Thus the four terms that form a 2 × 2 square correspond to the term "y". y = z y yz + = z y x z y x z y x z y x ) z , y , x ( F + + + = x y 1 0 2 4 3 5 6 7 1 1 1 1 z

Slide 21 Three-Variable Maps

Three-Variable Maps Reduced literal product terms for SOP standard forms correspond to rectangles on K-maps containing cell counts that are powers of 2. Rectangles of 2 cells represent 2 adjacent minterms; of 4 cells represent 4 minterms that form a pairwise adjacent ring. Rectangles can contain non-adjacent cells due to wrap-around at edges

Slide 22 Three-Variable Maps

Three-Variable Maps Topological warp of 3-variable K-maps that shows all adjacencies:

Slide notes

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Slide 23 Three-Variable Maps

Three-Variable Maps Example Shapes of 2-cell Rectangles: y 0 1 3 2 5 6 4 7 x z X Y YZ X Z

Slide 24 Three-Variable Maps

Three-Variable Maps Example Shapes of 4-cell Rectangles: Read off the product terms for the rectangles shown y 0 1 3 2 5 6 4 7 x z Y Z Z

Slide 25 Three Variable Maps

Three Variable Maps z) y, F(x, = y 1 1 x z 1 1 1 z z y x + y x K-Maps can be used to simplify Boolean functions by systematic methods. Terms are selected to cover the 1s in the map. Example: Simplify

Slide 26 Summary

Summary Circuit Optimization Literal cost Gate input cost Two-Variable Karnaugh Maps Three-Variable Karnaugh Maps

End of slides

Table of Contents

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