Introduction to Karnaugh-Map STLD/Digital Electronics by Ravinder Nath Rajotiya - August 7, 2019May 10, 20210 Karnaugh Map (K-map) K-Map is a pictorial representation of the Boolean function. It is a systematic method of simplifying the Boolean expression. A K-Map is an arrangement of the adjacent cell , each cell representing the minterm or the maxterm of the SOP or the POS equations. The number of adjacent cells in a K-Map depends on the number of input variables in an equation. The number of cell equals 2N where N is the number of input variables. Let us see how the minterm or the maxterms of the SOP or POS equations are represented. Two variable K-Map Table in figure-1 on the left is a truth table for two input variables. The output is shown in column marked ‘F’. m0, m1, m2,
Boolean Function STLD/Digital Electronics by Ravinder Nath Rajotiya - August 6, 2019May 10, 20210 Boolean Functions The following steps are generally followed while designing the logic circuits using Boolean functions. Analysis of the given statement of the problem to find the number of variables Writing the truth table from the given statement Conversion of the truth table into a logic function/ Boolean expression using standard product of sums(SOP) or product of sums (POS) Simplify the Boolean expression Create the logic circuit using logic gates SOP and POS equations Sum of Product (SOP) equation The SOP equations are formed by ORing the product or the min-terms. Formation of Minterm and Maxterm A B C Minterm(mi) Maxterms(Mi) 0 0 0 A’B’C’ =m0 A+B+C =M0 0 0 1 A’B’C =m1 A+B+C’ =M1 0 1 0 A’BC’ =m2 A+B’+C =M2 0 1 1 A’BC =m3 A+B’+C’ =M3 1 0 0 AB’C’ =m4 A’+B+C =M4 1 0 1 AB’C =m5 A’+B+C’ =M5 1 1 0 ABC’ = m6 A’+B’+C =M6 1 1 1 ABC = m7 A’+B’+C’ =M7 Min terms The input combinations for which the output equals ‘1’ is called
Boolean Theorems STLD/Digital Electronics by Ravinder Nath Rajotiya - August 6, 2019May 10, 20210 Duality Principle: Dual of an expression can be obtained by : Changing all + with . Complement 0’s and 1’s i.e. 1’=0 and 0’=1 Keep the variable sam (i.e. do not complement the variable) Example: Find the dual of : A(B+C) = AB + AC Solution: Find the Dual of F(xyz)=(x+y) (x+z)(y+z) Solution: (x+y) (x+z)(y+z) = (xx+xz+xy +yz)(y+z) = (x+xz+xy+yz)(y+z) = (x +xy +yz )(y+z) = (x+yz)(y+z) = xy +xz + yyz + yz = xy + xz + yz Also dual of (x+y) (x+z)(y+z) = xy + xz + yz Therefore, we find that in this case dual of a function is the function itself 3. For n variables expression haw many total dual functions are possible Solution: Which of the following is TRUE S1: The dual of NAND function is NOR S2:The dual of
Axioms and Laws of Boolean Algebra STLD/Digital Electronics by Ravinder Nath Rajotiya - August 5, 2019May 10, 20210 Boolean Algebra Axioms and Laws of Boolean Algebra Boolean algebra devised in 1864 by George Boole, is a system of mathematical logic. It is an algebraic system consisting of a set of element (0,1) associated with a Boolean variable and two binary operators AND and OR and a uniry operator NOT. In mathematics, an identity is a statement true for all possible values of its variable or variables. Axioms AND Operation 0.0=0 0.1=0 1.0=0 1.1=1 2. OR Operation 0+0=0 0+1=1 1+0=1 1+1=1 3. NOT Operation 0’ = 1 1’ = 0 Identities: In mathematics, an identity is a statement true for all possible values of its variable or variables. Identity Boolean Relation Symbol Additive Identity A + 0 = A A + 1 = 1 A + A = A A + A’ = 1 Multiplicative Identity A . 0 = 0 A . 1 =
Hamming Code STLD/Digital Electronics by Ravinder Nath Rajotiya - August 5, 2019May 10, 20210 Hamming Code Hamming code is an improvement over the other parity systems, this approach enable us to detect one bit error that might get introduced during transmission in every byte of information, However, this is done at the cost of increased data size due to introduction of multiple parity bits. Calculating the Hamming Code The Hamming method make use of multiple parity bits at bit positions 1,2,4,8,.......... Whereas VRC/HRC method was able to detect a single bit error in a block of data, the help of Hamming code enable us to detect and correct single bit error in every byte that is being sent or received. The following step helps in creating the Hamming code word: 1st of all mark all bit positions (1,