1 / 43

Gate-level Minimization

Gate-level Minimization. Although truth tables representation of a function is unique, it can be expressed algebraically in different forms The procedure of simplifying Boolean expressions (in 2-4) is

felcia
Download Presentation

Gate-level Minimization

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Gate-level Minimization • Although truth tables representation of a function is unique, it can be expressed algebraically in different forms • The procedure of simplifying Boolean expressions (in 2-4) is difficult since it lacks specific rules to predict the successive steps in the simplification process. • Alternative: Karnaugh Map (K-map) Method. • Straight forward procedure for minimizing Boolean Function • Fact: Any function can be expressed as sum of minterms • K-map method can be seen as a pictorial form of the truth table. y y x m0 m1 m2 m3 x Two-variable map

  2. Two-variable K-MAP y y x x y y y y x x x x

  3. Two-variable K-MAP y y y y x x x x The three squares can be determined from the intersection of variable x in the second row and variable y in the second column.

  4. Three-Variable K-Map • Any two adjacent squares differ by only one variable. • M5 is row 1 column 01.  101= xy’z=m5 • Since adjacent squares differ by one variable (1 primed, 1 unprimed) • From the postulates of Boolean algebra, the sum of two minterms in adjacent squares can be simplified to a simple AND • For example m5+m7=xy’z+xyz=xz(y’+y)=xz

  5. Three-Variable K-Map Example 1

  6. Three-Variable K-Map Example 2 Simplify:

  7. Three-Variable K-Map Example 3 Simplify:

  8. Three-Variable K-Map Example 3 Simplify:

  9. Three-Variable K-Map Example 4 Given: (a) Express F in sum of minterms. (b) Find the minimal sum of products using K-Map (a)

  10. Three-Variable K-Map Example 4 (continued)

  11. Three-variable K-Map: Observations • One square represents one minterm  a term of 3 literals • Two adjacent squares  a term of 2 literals • Four adjacent squares  a term of 1 literal • Eight adjacent squares  the function equals to 1

  12. Four-Variable K-Map

  13. Four-Variable K-Map Simplify F(w,x,y,z) = S(0,1,2,4,5,6,8,9,12,13,14) Example 5 1

  14. Four-Variable K-Map Example 6 Simplify F(A,B,C,D) = Represented by 0001 or 0000

  15. Prime Implicants • Need to ensure that all Minterms of function are covered • But avoid any redundant terms whose minterms are already covered • Prime Implicant is product Term obtained by combining maximum possible number of adjacent squares • If a minterm in a square is covered by only prime implicant then ESSENTIAL PRIME IMPLICANT Non Essential prime implicant CD, B’C, AD and AB’ Essential prime implicant BD and B’D’

  16. Four-variable K-Map: Observations • One square represents one minterm  a term of 4 literals • Two adjacent squares  a term of 3 literals • Four adjacent squares  a term of 2 literal • Eight adjacent squares  a term of 1 literal • sixteen adjacent squares  the function equals to 1

  17. SUM of PRODUCT and PRODUCT OF SUM Simplify the following Boolean function in: (a) sum of products (b) product of sums Combining the one’s: (a) Combining the zero’s: Taking the the complement: (b)

  18. SOP and POS gate implementation PRODUCT OF SUM (POS) SUM OF PRODUCT (SOP)

  19. Implementation of Boolean Functions • Draw the logic diagram for the following function: F = (a.b)+(b.c) a b F c

  20. Implement a circuit • 2 Level • More than two level • SOP • POS • Implement a circuit using OR and Inverter Gates only • Implement a circuit using AND and Inverter Gates only • Implement a circuit using NAND Gates only • Implement a circuit using NOR Gates only

  21. NAND IMPLEMENTATION

  22. F=AB+CD F=(A’B’)’+(C’D’)’ F=[(AB)’.(CD)’]’=AB+CD TWO LEVEL IMPLEMENT-ATION

  23. F(X,Y,Z)=(1,2,3,4,5,7) SUM OF PRODUCT

  24. CHAPTER 4 COVERT AND TO NAND WITH AND INVER. CONVERT OR TO NAND WITH INVERT OR. SINGLE BUBBLE WITH INVERTER

  25. SIMPLIFICATION WITH TABULATION METHOD DO IT ON BOARD

More Related