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
This form sometimes used in logic circuit, example:
Objectives: 1. Deriving of logical expression form truth tables. 2. Logical expression simplification methods: a. Algebraic manipulation. b. Karnaugh map (k-map). 1. Deriving of logical expression from
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