Digital Logic Design. Malik Najmus Siraj

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1 Digital Logic Design Malik Najmus Siraj

2 LECTURE 4

3 Today s Agenda Recap 2 s complement Binary Logic Boolean algebra

4 Recap Computer Arithmetic Signed numbers Radix and diminished radix complement

5 2 s complement representation

6 2 s complement calculation

7 Complement Summary

8 Signed number with complement

9 Arithmetic with Radix complement

10 Binary Logic

11 Binary Logic

12 Evaluation of Logic function

13 Multiple input

14 Algebra

15 Boolean Algebra

16 Boolean algebra addition and multiplication li

17 Boolean addition

18 Boolean Multiplication

19 Theorems of Boolean algebra

20 Commutative Law

21 Duality principle Every Algebraic expression deducible from the postulates of Boolean algebra remains valid if the operator of identity elements are change.

22 Associative Law

23 Associative Law Associative Law for Multiplication A.(B.C) = (A.B).C

24 Distributive Law A.(B + C) = A.B + A.C

25 Rules of Boolean Algebra 1. A + 0 = A 7. AA= A.A A 2. A + 1 = 1 3. A.0 = 0 4. A.1 = A 8. A. = 0 A 9. A = A 10.A + A.B = A 5. A + A = A 11.A + A.B= A + B 6. A + A = 1 12.(A+B).(A+C) (A+C) = A+B.C

26 Examples x+x = x 1. = (x+x).1 2. =(x+x). (x+x ) 3. =x+xx 4. =x+0 5. =x

27 Continue.. x.x = x 1. x.x x.x + x.x 3. x(x+x ) 4. x.1 5. X There are different theorems and their proofs given on page 40 and 41.

28 Demorgan s Theorems First Theorem A.B A + B Second Theorem A + B A. B A + B = A. B

29 Demorgan s Theorems Any number of variables X. Y. Z = X + Y + Z X + Y + Z = X. Y. Z Combination of variables ( A + B. C).( A. C + B) = ( A + B. C) + ( A. C + B) = A. ( B. C ) + ( A. C ). B = A. B + A. C + A. B + B. C = Digital A Logic. B Design@CASE + A. C + by Najmus B. C Siraj

30 Operator Precedence 1. Parentheses 2. NOT 3. AND 4. OR Examples (x+y) ; x y

31 Boolean Analysis of Logic Circuits Boolean Algebra provides concise way to represent operation of a logic circuit Complete function of a logic circuit can be determined by evaluating the Boolean expression using different input combinations F = (AB+C )D

32 Boolean Analysis of Logic Circuits C AB + C ( AB + C) D From the expression, the output is a 1 if variable D = 1 ( AB + C) =1 if AB=1 or C=0 Rows in the truth table is 2 n n is the number of variables in the function

33 Boolean Analysis of Logic Circuits Inputs Output Inputs A B C D F A B C D F Output

34 Simplification using Boolean Algebra AB + A(B+C) + B(B+C) = AB + AB + AC + BB +BC = AB + AC + B + BC = AB + AC + B = B + AC

35 Simplified Circuit

36 Lecture 06

37 Recap

38 Today s agenda Duality yprincipalp Complement of a function Minterms and Maxterms Sum of Product and Product of Sum

39 Duality Principle Every algebraic expression deducible from the postulates of Boolean algebra remains valid if the operator and the identity elements are interchanged 1. x+0=x x.1=x 2. (x+y) = x y xy (x.y) y) = x +y x+y

40 Example 2.1 x(x +y) = xx +xy = 0+xy = xy x+x y = xx+xy+x x+x y = x(x+y)+x (x+y) = x+y (x+y)(x+y ) = xx+xy +xy+yy = x(1+y +y) = x xy+x z+yz = xy+x z+yz(x+x ) = xy+x z+yzx+yzx = xy(1+z) + x z(1+y) = xy+x z (x+y)(x +z)(y+z) = (x+y)(x +z)

41 Complement of a function The generalized form of DeMorgan s theorem states that the complement of a function is obtained by interchanging AND and OR operators and complementing each literal Example 2.22

42 Example solution

43 Easy way to fined complement A simpler way to find complement of a function is to take the dual of the function and complement each literal. Examples

44 Minterms and Maxterms Minterms for two variable are x y, x y, xy, xy. There are eight minterms for 3 variables function. 2 n minterms for n variable. Maxterms for two variable are x+y, x+y,x +y, x +y

45

46 X Y Z f

47 Example 2.4 (SOP)

48

49 Example 2.5(POS) Each term missing one variable So

50 Conversion between Canonical forms

51

52

53

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