DLDA

Number Systems

Decimal Numbers

Decimal number system has base 10 as it uses 10 digits from 0 to 9. In decimal number system, the successive positions to the left of the decimal point represents units, tens, hundreds, thousands and so on.

Binary Numbers

The binary system is a number system where numbers are represented using the digits 0 and 1. It is used in the computers, since they work with two voltage levels, and thus their natural number system is the binary system (1 - ON, 0 - OFF).

Number System

Octal Numbers

The base-8 number system is named octal and makes use of digits 0 to 7. In computing, sometimes the octal number system is used instead of hexadecimal. Its main advantage is that it does not require using symbols different to the numbers 0 to 7. However, to work with bytes of sets of bytes, assuming that a byte is an 8-bit word, the hexadecimal system is more comfortable, every byte thus defined is completely represented by two hexadecimal digits.

Hexadecimal Numbers

The hexadecimal number system or hexadecimal system (sometimes shortened as Hex) is a number system that uses 16 symbols: from 0 to 9 and A, B, C, D, E and F. Its current use is highly linked to Computer Science, since the computers use the byte or octet as a basic memory unit. Since a byte has 8 bits, they can be represented with two hexadecimal digits.

Codes

ASCII

Character data is represented in a computer by using standardized numeric codes which have been developed. The most widely accepted code is called the American Standard Code for Information Interchange ( ASCII). The ASCII code associates an integer value for each symbol in the character set, such as letters, digits, punctuation marks, special characters, and control characters.

Gray Code

The reflected binary code (RB C), also known just as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). The reflected binary code was originally designed to prevent spurious output from electromechanical switches.

Codes

BCD Code

Binary-coded decimal (BCD) is a class of binary encoding of decimal numbers where each decimal digit is represented by a fixed number of bits, usually four or eight.

Excess-3

Excess-3 is a self-complementary binary-coded decimal (BCD) code and numeral system. It is a biased representation. Excess-3 code was used on some older computers as well as in cash registers and hand-held portable electronic calculators of the 1970s, among other uses.

Boolean Algebra

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the prime operations are addition and multiplication, the main operations of Boolean algebra are the conjunction and denoted as ∧, the disjunction or denoted as ∨, and the negation not denoted as ¬. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes numeric relations.

Boolean Algebra

The basic operations of Boolean algebra are as follows:

AND (conjunction), denoted x∧y (sometimes x AND y or Kxy), satisfies x∧y = 1 if x = y = 1 and x∧y = 0 otherwise.

OR (disjunction), denoted x∨y (sometimes x OR y or Axy), satisfies x∨y = 0 if x = y = 0 and x∨y = 1 otherwise.

NOT (negation), denoted ¬x (sometimes NOT x, Nx or !x), satisfies ¬x = 0 if x = 1 and ¬x = 1 if x = 0.

Logic Gates

Logic gate is an idealized or physical device implementing a Boolean function; that is, it performs a logical operation on one or more binary inputs and produces a single binary output

The AND gate is so named because, if 0 is called "false" and 1 is called "true," the gate acts in the same way as the logical "and" operator

The NAND gate operates as an AND gate followed by a NOT gate. It acts in the manner of the logical operation "and" followed by negation. The output is "false" if both inputs are "true."

The NOR gate is a combination OR gate followed by an inverter. Its output

"true" if both inputs are "false.

The OR gate gets its name from the fact that it behaves after the fashion of the logical inclusive "or." The output is "true" if either or both of the inputs are "true."

Logic Gates

A logical inverter , sometimes called a NOT gate to differentiate it from other types of electronic inverter devices, has only one input. It reverses the logic state.

The XNOR (exclusive-NOR) gate is a combination XOR gate followed by an inverter. Its output is "true" if the inputs are the same, and"false" if the inputs are different

The XOR ( exclusive-OR ) gate acts in the same way as the logical "either/or." The output is "true" if either, but not both, of the inputs are "true." The output is "false" if both inputs are "false" or if both inputs are "true."

An adder is a digital circuit that performs addition of numbers. The half adder adds two binary digits called as augend and addend and produces two outputs as sum and carry; XOR is applied to both inputs to produce sum and AND gate is applied to both inputs to produce carry. The full adder adds 3 one bit numbers, where two can be referred to as operands and one can be referred to as bit carried in. And produces 2-bit output, and these can be referred to as output carry and sum.

A ripple carry adder is a logic circuit in which the carry-out of each full adder is the carry in of the succeeding next most significant full adder. It is called a ripple carry adder because each carry bit gets rippled into the next stage.

Half Subtractor is used for subtracting one single bit binary digit from another single bit binary digit.A logic Circuit Which is used for Subtracting Three Single bit Binary digit is known as Full Subtractor.

A BCD adder adds two BCD digits and produces output as a BCD digit. A BCD or Binary Coded Decimal digit cannot be greater than 9.For implementing a BCD adder using a binary adder circuit IC 7483, additional combinational circuit will be required, where the Sum output S3−S0 is checked for invalid values from 10 to 15

Part 1

Multiplexer is a device that has multiple inputs and a single line output. The select lines determine which input is connected to the output, and also to increase the amount of data that can be sent over a network within certain time. It is also called a data selector.

De-multiplexer is also a device with one input and multiple output lines. It is used to send a signal to one of the many devices. The main difference between a multiplexer and a de-multiplexer is that a multiplexer takes two or more signals and encodes them on a wire, whereas a de-multiplexer does reverse to what the multiplexer does.

An encoder is a device, circuit, transducer, software program, algorithm or person that converts information from one format or code to another. The purpose of encoder is standardization, speed, secrecy, security, or saving space by shrinking size. Encoders are combinational logic circuits and they are exactly opposite of decoders. They accept one or more inputs and generate a multibit output code.

In digital electronics, a decoder can take the form of a multiple-input, multiple-output logic circuit that converts coded inputs into coded outputs, where the input and output codes are different e.g. n-to-2n , binary-coded decimal decoders. Decoding is necessary in applications such as data multiplexing, 7 segment display and memory address decoding.

The 74181 is a bit slice arithmetic logic unit (ALU), implemented as a 7400 series TTL integrated circuit. The first complete ALU on a single chip,[1] it was used as the arithmetic/logic core in the CPUs of many historically significant minicomputers and other devices.

Part 2

The purpose of a Digital Comparator is to compare a set of variables or unknown numbers, for example A (A1, A2, A3, …. An, etc) against that of a constant or unknown value such as B (B1, B2, B3, …. Bn, etc) and produce an output condition or flag depending upon the result of the comparison. For example, a magnitude comparator of two 1-bits, (A and B) inputs would produce the following three output conditions when compared to each other.

A>B,A=B,A>B

The J,JK,T,D flip-flop is the modification of the SR flip flop

SR latch

A latch (also called a flip-flop) is a fundamental component of data storage. A single latch can hold 1-bit of data, increase that number by many orders of magnitude and you can create kilo-, mega-, giga-, even tera-bytes of memory. Of course, like most digital circuits, latches are made out of digital logic gates!

In electronics, a flip-flop or latch is a circuit that has two stable states and can be used to store state information. A flip-flop is a bistable multivibrator. The circuit can be made to change state by signals applied to one or more control inputs and will have one or two outputs. It is the basic storage element in sequential logic. Flip-flops and latches are fundamental building blocks of digital electronics systems used in computers, communications, and many other types of systems.

Part 1

A Mealy Machine is an FSM whose output depends on the present state as well as the present input.It can be described by a 6 tuple (Q, ∑, O, δ, X, q0) where Q is a finite set of states.

∑ is a finite set of symbols called the input alphabet.

O is a finite set of symbols called the output alphabet.

δ is the input transition function where δ: Q × ∑ → Q

X is the output transition function where X: Q × ∑ → O

q0 is the initial state from where any input is processed (q0 ∈ Q).

Moore machine is an FSM whose outputs depend on only the present state.A Moore machine can be described by a 6 tuple (Q, ∑, O, δ, X, q0) where

Q is a finite set of states.

∑ is a finite set of symbols called the input alphabet.

O is a finite set of symbols called the output alphabet.

δ is the input transition function where δ: Q × ∑ → Q

X is the output transition function where X: Q → O

q0 is the initial state from where any input is processed (q0 ∈ Q).

Synchronous Counters are so called because the clock input of all the individual flip-flops within the counter are all clocked together at the same time by the same clock signal

A Johnson counter is a modified ring counter, where the inverted output from the last flip flop is connected to the input to the first. The register cycles through a sequence of bit-patterns. The MOD of the Johnson counter is 2n if n flip-flops are used. The main advantage of the Johnson counter counter is that it only needs half the number of flip-flops compared to the standard ring counter for the same MOD.

Asynchronous Counters use flip-flops that are serially connected together so that the input clock pulse ripples through the counter giving them the name of ripple counter

Part 2

Sequence generator A digital logic circuit whose purpose is to produce a prescribed sequence of outputs. Each output will be one of a number of symbols or of binary or q-ary logic levels. The sequence may be of indefinite length or of predetermined fixed length. A binary counter is a special type of sequence generator. Sequence generators are useful in a wide variety of coding and control applications.

In digital circuits, a shift register is a cascade of flip flops, sharing the same clock, in which the output of each flip-flop is connected to the 'data' input of the next flip-flop in the chain, resulting in a circuit that shifts by one position the 'bit array' stored in it, 'shifting in' the data present at its input and 'shifting out' the last bit in the array, at each transition of the clock input.