Gate Level Minimization. Lecture 06,07,08 By : Ali Mustafa

Transcription

1 Gate Level Minimization Lecture 6,7,8 By : Ali Mustafa

2 Outline Introduction The Karnaugh Map Method Two-Variable Map Three-Variable Map Four-Variable Map Five-Variable Map Product of Sums Simplification Don t-care Conditions

3 Introduction The Map Method K map provides a pictorial method of grouping together expressions with common factors and therefore eliminating unwanted variables. The K map can also be described as a special arrangement of a truth table.

4 Choice of Blocks We can simplify function by using larger blocks Do we really need all blocks? Can we leave some out to further simplify expression? Function needs to contain special type of blocks They are called Essential Prime Implicants Need to define new terms Implicant Prime implicant Essential prime implicant

5 Terminology

6 Procedure for Simplifying Boolean Functions.Generate all PIs of the function. 2.Include all essential PIs. 3.For remaining minterms not included in the essential PIs, select a set of other PIs to cover them, with minimal overlap in the set. 4.The resulting simplified function is the logical OR of the product terms selected above.

7 Examples to illustrate terms

8 2 variable map example F = m3 F = m + m2 + m3

9 Two Variable Map Consider the following map. The function plotted is: Z = f(a,b) = A + AB

10 3-variable K Map

11 Example : 3-variable K Map (Cont..)

12 Example : 3-variable K Map

13 Another Example : 3-variable K Map F = yz + xz

14 3-variable K Map

15 Self Tasks Solve the expression using K MAP

16 Solution of Self Tasks

17 4-variable Karnaugh Map

18 4-variable Karnaugh Map K map can be extended to 4 variables: Top cells are adjacent to bottom cells. Leftedge cells are adjacent to right-edge cells. Note variable ordering (WXYZ).

19 Four-variable Map Simplification One square represents a minterm of 4 literals. A rectangle of 2 adjacent squares represents a product term of 3 literals. A rectangle of 4 squares represents a product term of 2 literals. A rectangle of 8 squares represents a product term of literal. A rectangle of 6 squares produces a function that is equal to logic.

20 Example : 4-variable K Map

21 Example Simplify the following Boolean function g(a,b,c,d) = Σm(,,2,4,5,7,8,9,,2,3). First put the function g( ) into the map, and then group as many s as possible.

22 Example: 4-variable Karnaugh Map Example: F(w,x,y,z)=Σ(,,2,4,5,6,8,9,2,3,4)

23 Simplifying logic Function using 4-variable K-Map

24 5-variable Karnaugh Map

25 Example: 5-variable Karnaugh Map F(A,B,C,D,E) = Σ(,2,4,6,9,3,2,23,25,29,3)

26 Product of Sums Minimization How to generate a product of sums from a Karnaugh map? Use duality of Boolean algebra (DeMorgan law) Look at s in map instead of s Generate blocks around s Gives inverse of function Use duality to generate product of sums Example: F = Σ(,,2,5,8,9,) F = AB+ CD + BD F = (A +B )(C +D )(B +D)

27 You can easily do product of sum too

28 You can easily do product of sum too

29 Example: POS minimization

30 Don't Care Conditions There may be a combination of input values which will never occur if they do occur, the output is of no concern. The function value for such combinations is called a don't care. They are usually denoted with x. Each x may be arbitrarily assigned the value or in an implementation. Don t cares can be used to further simplify a function

31 Minimization using Don t Cares Treat don't cares as if they are s to generate PIs. Delete PI's that cover only don't care minterms. Treat the covering of remaining don't care minterms as optional in the selection process (i.e.they may be, but need not be, covered).

32 Minimization example F(w,x,y,z) = Σ(,3,7,,5) and d(w,x,y,z) = Σ(,2,5) What are possible solutions?

33 Another Example

34 Self Tasks Self reading chapter 3 (Revision) 3. to 3.5 Chapter 3 exercise Self reading chapter 2 (Revision) 2. to 2.28 Chapter 2 exercise Except 2.,2.5,2.6,

35 Recap

36 2 Variable K-Map A B B B A Group A A Group 2 B F= A + B

37 A BC 3 Variable K-Map B C B C BC BC Group A m m m3 m2 A B Group 2 A m4 m5 m7 m6 AC IGNORE F= A B + AC

38 A BC 3 Variable K-Map B C B C BC BC Group A A m m m3 m2 Group 2 A m4 m5 m7 m6 C F= A + C

39 A BC 3 Variable K-Map B C B C BC BC Group A m m m3 m2 A M4 m5 m7 m6 F=

40 3 Variable K-Map A A BC B C B C BC BC A BC m m m3 m2 A m4 m5 m7 m6 AB C A B C ABC F= A B C + A BC + AB C + ABC

41 CD AB A B A B AB AB 4 Variable K-Maps C D C D CD CD m m m3 m2 m4 m5 m7 m6 m2 m3 m5 m4 m8 m9 m m Group AB F= AB

42 CD AB A B A B AB AB 4 Variable K-Maps C D C D CD CD m m m3 m2 m4 m5 m7 m6 m2 m3 m5 m4 m8 m9 m m Group BC F= BC

43 CD AB A B A B AB AB 4 Variable K-Maps C D C D CD CD m m m3 m2 m4 m5 m7 m6 m2 m3 m5 m4 m8 m9 m m Group B D F= B D

44 CD AB A B A B AB AB 4 Variable K-Maps C D C D CD CD m m m3 m2 m4 m5 m7 m6 m2 m3 m5 m4 m8 m9 m m Group BD F= BD

45 CD AB A B A B AB AB 4 Variable K-Maps C D C D CD CD m m m3 m2 m4 m5 m7 m6 m2 m3 m5 m4 m8 m9 m m Group BD Group 2 B D F= BD + B D

46 5 Variable K-Maps m m m3 m2 m4 m5 m7 m6 m2 m3 m5 m4 m8 m9 m m m6 m7 m9 m8 m2 m2 m23 m22 m28 m29 m3 m3 m24 m25 m27 m26 A A Adjacent Rows BC DE BC DE

47 5 Variable K-Maps m m m3 m2 m4 m5 m7 m6 m2 m3 m5 m4 m8 m9 m m m6 m7 m9 m8 m2 m2 m23 m22 m28 m29 m3 m3 m24 m25 m27 m26 A A Adjacent Columns BC DE BC DE

48 Simplify 5 Variable K-Maps F = m(,2,4,6,9,,3,5,7,2,25,27,29,3) BC DE B C A A D E D E DE DE D E D E DE DE m m m3 m2 BC B C DE m6 m7 m9 m8 B C m4 m5 m7 m6 B C m2 m2 m23 m22 BC m2 m3 m5 m4 BC m28 m29 m3 m3 BC m8 m9 m m BC m24 m25 m27 m26 F = A B E + AD E + BE

49 Simplify 5 Variable K-Maps F = m(,2,5,7,3,5,8,2,2,23,28,29,3) BC DE B C A A D E D E DE DE D E D E DE DE m m m3 m2 BC B C DE m6 m7 m9 m8 B C m4 m5 m7 m6 B C m2 m2 m23 m22 BC m2 m3 m5 m4 BC m28 m29 m3 m3 BC m8 m9 M m BC m24 m25 m27 m26 F = A B C E + ACD + CE + F = A B C E + B C DE + ACD + CE F = B C E (A + D) + ACD + CE B C DE

50 6 Variable K-Maps CD EF A B CD EF A B m m5 m3 CD EF AB CD EF AB m47 m63

51 Simplify 4 Variable K-MAP Find the groups and write the desired equation m m m3 M2 m m m3 m2 M m m3 m2 m4 m5 m7 M6 m4 m5 m7 m6 m4 m5 m7 m6 m2 m3 m5 m4 m2 m3 m5 m4 m2 m3 m5 m4 m8 m9 m m m8 m9 m m m8 m9 m m

52 Self Task Minimize the following expression using K- Map and realize it using number of gates Y = m(,5,7,9,,3,5) Y = m(,3,5,9,,3) Y = m(4,5,8,9,,2,3,5)

53 Product of Sum K-MAP Two ways to express Y = (,,2,5,8,9,) Y = (,2,3,5,7)

54 CD AB A B A B AB POS K-MAP Y = (,,2,5,8,9,) Make POS expression C D C D CD CD m m4 m m5 m3 m7 m2 m6 F = CD + AB + BD Applying De Morgan Law F = (C +D )(A +B )(B +D) AB m2 m8 m3 m9 m5 m m4 m

55 A POS K-MAP Simplify Y = (,2,3,5,7) BC B C B C BC BC A m m m3 m2 B +C A+C A m4 m5 m7 m6 A +C But in POS we consider as a positive logic SO F= (B +C )(A+C)(A +C )

56 Self Tasks. Simplify the following expression with POS A. Y = (,2,3,7) B. Y = (,2,8,) C. Y = (,,2,5,8,,3) 2. Simplify the following expression with SOP & POS A. Y = X Z + Y Z + YZ + XY 3. Exercise questions 3. to 3.4