Combinational Logic Part 2: Karnaugh maps (quick)

Sum of Minterms Implementation OR all of the minterms of truth table for which the function value is 1 F = m 0 + m 2 + m 5 + m 7 F = X’Y’Z’ + X’YZ’+ XY’Z + XYZ 2

Sum of Products Implementation Simplifying sum-of-minterms can yield a sum of products Difference is that each term need not have all variables Resulting gates  ANDs and one OR F = Y’ + X’YZ’ + XY 3

Two-Level Implementation Sum of products has 2 levels of gates Fig 2-6 4

More Levels of Gates? What’s best? ♦ Hard to answer ♦ More gate delays (more on this later) ♦ But maybe we only have 2-input gates 5

Product of Maxterms Implementation Can express F as AND of Maxterms for all rows that should evaluate to 0 or This makes one Maxterm fail each time F should be 0 6

Product of Sums Implementation ORs followed by AND 7

Karnaugh Map Graphical depiction of truth table A box for each minterm ♦ So 2 variables, 4 boxes ♦ 3 variable, 8 boxes ♦ And so on Useful for simplification ♦ by inspection ♦ Algebraic manipulation harder 8

K-Map from Truth Table Examples There are implied 0s in empty boxes 9

Function from K-Map Can generate function from K-map Simplifies to X + Y (in a moment) 10

In Practice: Karnaugh maps were mildly useful when people did simplification Computers now do it! We’ll cover Karnaugh maps as a way for you to gain insight, ♦ not as real tool 11

Three-Variable Map Eight minterms Look at encoding of columns and rows 12

Simplification Adjacent squares (horizontally or vertically) are minterms that vary by single variable Draw rectangles on map to simplify function Illustration next 13

Example instead of 14

Adjacency is cylindrical Note that wraps from left edge to right edge. 15

Covering 4 Squares is 16

Another Example Help me solve this one 17

In General One box -> 3 literals Rectangle of 2 boxes -> 2 literals Rectangle of 4 boxes -> 1 literal Rectangle of 8 boxes -> Logic 1 (on 3-variable map) ♦ Covers all minterms 18

Slight Variation Overlap is OK. No need to use full m 5 -- waste of input 19

4-variable map At limit of K-map 20

Also Wraps (toroidal topology) 21

Systematic Simplification  A Prime Implicant is a product term obtained by combining the maximum possible number of adjacent squares in the map into a rectangle with the number of squares a power of 2.  A prime implicant is called an Essential Prime Implicant if it is the only prime implicant that covers (includes) one or more minterms.  Prime Implicants and Essential Prime Implicants can be determined by inspection of a K-Map.  A set of prime implicants "covers all minterms" if, for each minterm of the function, at least one prime implicant in the set of prime implicants includes the minterm. Chapter 2 - Part 2 22

DB CB B D A Example of Prime Implicants  Find ALL Prime Implicants B’D’ and BD are ESSENTIAL Prime Implicants C BD CD BD Minterms covered by single prime implicant DB B C D A AD BA Chapter 2 - Part 2 23

Prime Implicant Practice  Find all prime implicants for: Chapter 2 - Part 2 24

Prime Implicant Practice  Find all prime implicants for: B D A 11 1 C 1 1 A CB DB Chapter 2 - Part 2 25

Algorithm to Find An Optimal Expression for A Function  Find all prime implicants.  Include all essential prime implicants in the solution  Select a minimum cost set of non-essential prime implicants to cover all minterms not yet covered.  The solution consists of all essential prime and the selected minimum cost set of non-essential prime implicants minimum cost selected minimum cost Chapter 2 - Part 2 26

The Selection Rule  Obtaining a good simplified solution: Use the Selection Rule Chapter 2 - Part 2 27

Prime Implicant Selection Rule  Minimize the overlap among prime implicants as much as possible.  In the solution, make sure that each prime implicant selected includes at least one minterm not included in any other prime implicant selected. Chapter 2 - Part 2 28

Selection Rule Example  Simplify F(A, B, C, D) given on the K- map B D A C B D A C 1 1 Essential Minterms covered by essential prime implicants Selected Chapter 2 - Part 2 29

Don’t Care So far have dealt with functions that were always either 0 or 1 Sometimes we have some conditions where we don’t care what result is Example: dealing with BCD ♦ Only care about first 10 30

Mark With an X In a K-map, mark don’t care with X Simpler implementations Can select an X either as 1 or 0 31

Example or What would we have if Xs were 0? 32

Selection Rule Example with Don't Cares  Simplify F(A, B, C, D) given on the K-map. Selected Minterms covered by essential prime implicants 1 1 x x x x x 1 B D A C x x x x x 1 B D A C 1 1 Essential Chapter 2 - Part 2 33

Product of Sums Example  Find the optimum POS solution: Hint: Use and complement it to get the result. F 34