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,
Parity Code STLD/Digital Electronics by Ravinder Nath Rajotiya - August 5, 2019May 10, 20210 Error Detecting Codes: When we transmit a computer code over a medium, the possibility of a bit being lost due to transient or any other reason become the point of interest. During data sharing, the flow of information on the channel require high data integrity for the system to properly work on that data. So, a good mechanism of checking the loss of data integrity is needed so that the system is at least able to detect such violation of data integrity or loss of information in transmission. Different Types of error in communication: One bit error Burst error One Bit error: During transmission there is a chance of getting a bit lost or complemented to indicate loss of data integrity. The receiver should
Computer Codes STLD/Digital Electronics by Ravinder Nath Rajotiya - August 4, 2019May 10, 20210 Computer Codes: The second category of codes are those codes which cannot be represented by any analytical relation, hence these codes are also called as non-analytical code.Non-Analytical code. These are the following codes: Non-Weighted Codes: Excess-3 Codes, Gray Codes, BCD greater than 4-bits Self Complementing codes: 2421, 5211, Excess-3 (XS-3) Sequential Codes: 8421 and XS-3 Cyclic Code: Gray Code Reflecting Code: Gray Code Error Detection and Correcting Code: Parity, Hamming Code Gray Code This code also known as a unit distance code. This code is best suited as a shat position encoder. Going from any one position to next position, there is only one bit change in the code. The pattern formed by the gray code is the mirror image, so the code is also called as reflected code. Figure shows the formation of gray
Signed Number Systems STLD/Digital Electronics by Ravinder Nath Rajotiya - July 31, 2019May 10, 20210 Signed Number System In signed number systems, a number is represented with two parts; one the sign bit, and the rest bits denote the magnitude of the number. Signed number are represented using the following representation: Sign and Magnitude number system: 1's complement Number System 2's Complement Number System Sign and Magnitude number system In this system, the MSB is set to 0(zero) to indicate +ve number and is set to 1(one) to indicate the -ve number. Signed numbers are represented as: Complete the following Table +7 Sign Magnitude +7 0 1 1 1 -6 1 1 1 0 +25 -24 +14 -15 -10 +0 -0 Complement method: Complement of a number: Complement of a number is done to represent its -ve / or positive representation. Thus the complement of a +ve number is the -ve number, and complement of -ve number is +ve number. There are different types of