The compliment of the function expressed in terms of sum of min terms can be obtained by taking sum of missing min terms in the original functions e.g. f= ∑(4, 5, 7) = m7 + m5 + m4 Its compliment is f’ = ∑(0, 1, 2, 3, 6) = m0 + m1 + m2 + m3 + m6 The compliment […]
Digital Electronics
Boolean Equations: Different forms
Standard form: In a standard form we don’t have to compulsorily write all the literals in all the terms of an expression. e.g. f = xyz + y + x f = x’y’z’ + x’yz f= xy + x’y’ Canonical form: In a canonical form we have to compulsorily write all the literals in all the terms […]
Consensus Theorem
Consensus theorem: Given a pair of terms for which a variable appears in one term and its compliment in the other term then consensus term is formed by ANDing the original terms together leaving out the selected variable and its compliment. e.g. Find consensus term out of the two terms X.Y & X’.Z […]
Boolean algebra laws
There are following laws in Boolean algebra: Associative Law: This law states that if we have 3 variables x, y, z then X*(Y*Z) = (X*Y)*Z Commutative Law: This law states that X*Y = Y*X Identity element: If e is the identity then we have the relation with the Boolean algebra e*x = x*e=x Hence 0 is identity for […]
Logic Gates
Following are the basic logical operations which we can operate on binary variables: AND: This operation is represented by dot (.) If we two binary variables as x and y then we can represent AND operation by z=x.y and resultant of the operations is also a binary variable. Following table represents result of AND of every […]
Binary Logic
Binary logic deals with variables which have two discrete values and those two values can be true or false, on or off, yes or no etc. But we think in terms of 1 & 0. Those variables are called binary variables. In digital circuits we represent the higher value by 3V to 5V and lower […]
Example 2: Hamming code and Parity bit
Eg. Now form a hamming code for 5-bit information bits 10110 with odd parity m=5 and we have to follow 2p >= m + p + 1 The value of p as 4 to satisfy 24 (16) >= 5 + 4 + 1 but p=3 doesn’t satisfy as 23 (8) >= 5 + 3 + 1 So […]
Example 1: Hamming code & parity bit
Eg. So let’s form hamming code using 4-bit message bits 1101 with parity bits as even parity bit and check how it is able to detect and correct error. As we have already decided parity bit positions and their corresponding message bits for a 4-bit message For the moment we have hamming code as P1 P2 1 P3 1 […]
Parity bit relation with message bits
E.g. Consider the parity bit P1 and we have to find the position of message bits which we’ll cover with this parity bit. Firstly write the binary equivalents of positions of message bit Bit1 bit2 bit3 bit4 bit5 bit6 bit7 Parity parity parity P1 P2 M1 P3 M2 M3 M4 001 010 011 […]
Hamming code
This code is used for single error correction i.e. using this code we can detect only single error. In parity bit method we used only single extra bit but in this method number of extra bits (which also are parity bits) vary with the number of bits of the message. Suppose we have the number […]