LECTURE NOTES ON SWITCHING THEORY AND LOGIC DESIGN 2016 - 2017 II B.Tech II semester (JNTUH-R15) ELECTRONICS AND COMMUNICATION ENGINEERING
UNIT - 1 NUMBER SYSTEMS & CODES NUMBER SYSTEMS & CODES Philosophy of number systems Complement representation of negative numbers Binary arithmetic Binary codes Error detecting & error correcting codes Hamming codes HISTORY OF THE NUMERAL SYSTEMS: A numeral system (or system of numeration) is a linguistic system and mathematical notation for representing numbers of a given set by symbols in a consistent manner. For example, It allows the numeral "11" to be interpreted as the binary numeral for three, the decimal numeral for eleven, or other numbers in different bases. Ideally, a numeral system will: Represent a useful set of numbers (e.g. all whole numbers, integers, or real numbers) Give every number represented a unique representation (or at least a standard representation) Reflect the algebraic and arithmetic structure of the numbers. For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits, with the operations of arithmetic (addition, subtraction, multiplication and division) being present as the standard algorithms of arithmetic. However, when decimal representation is used for the rational or real numbers, the representation is no longer unique: many rational numbers have two numerals, a standard one that terminates, such as 2.31, and another that recurs, such as 2.309999999... . Numerals which terminate have no non-zero digits after a given position. For example,numerals like 2.31 and 2.310 are taken to be the same, except in the experimental sciences, where greater precision is denoted by the trailing zero.
The most commonly used system of numerals is known as Hindu-Arabic numerals.Great Indian mathematicians Aryabhatta of Kusumapura (5th Century) developed the place value notation. Brahmagupta (6th Century) introduced the symbol zero. BINARY The ancient Indian writer Pingala developed advanced mathematical concepts for describing prosody, and in doing so presented the first known description of a binary numeral system.A full set of 8 trigrams and 64 hexagrams, analogous to the 3-bit and 6-bit binary numerals, were known to the ancient Chinese in the classic text I Ching. An arrangement of the hexagrams of the I Ching, ordered according to the values of the corresponding binary numbers (from 0 to 63), and a method for generating thesame, was developed by the Chinese scholar and philosopher Shao Yong in the 11th century. In 1854, British mathematician George Boole published a landmark paper detailing an algebraic system of logic that would become known as Boolean algebra. His logical calculus was to become instrumental in the design of digital electronic circuitry. In 1937, Claude Shannon produced his master's thesis at MIT that implemented Boolean algebra and binary arithmetic using electronic relays and switches for the first time in history. Entitled A Symbolic Analysis of Relay and Switching Circuits, Shannon's thesis essentially founded practical digital circuit design. Binary codes Binary codes are codes which are represented in binary system with modification from the original ones. Weighted Binary codes Non Weighted Codes Weighted binary codes are those which obey the positional weighting principles, each position of the number represents a specific weight. The binary counting sequence is an example.
Reflective Code A code is said to be reflective when code for 9 is complement for the code for 0, and so is for 8 and 1 codes, 7 and 2, 6 and 3, 5 and 4. Codes 2421, 5211, and excess-3 are reflective, whereas the 8421 code is not. Sequential Codes A code is said to be sequential when two subsequent codes, seen as numbers in binary representation, differ by one. This greatly aids mathematical manipulation of data. The 8421 and Excess-3 codes are sequential, whereas the 2421 and 5211 codes are not. Non weighted codes Non weighted codes are codes that are not positionally weighted. That is, each position within the binary number is not assigned a fixed value. Ex: Excess-3 code Excess-3 Code Excess-3 is a non weighted code used to express decimal numbers. The code derives its name from the fact that each binary code is the corresponding 8421 code plus 0011(3). Gray Code The gray code belongs to a class of codes called minimum change codes, in which only one bit in the code changes when moving from one code to the next. The Gray code is non-weighted code, as the position of bit does not contain any weight. The gray code is a reflective digital code which has the special property that any two subsequent numbers codes differ by only one bit. This is also called a unitdistance code. In digital Gray code has got a special place.