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Unit-1 Introduction
- Digital Signals and Waveforms: What is signal? ▲ A signal is a time varying physical phenomenon which is intended to convey information. ▲ A signal is a function of time. For example, voice signal, video signal, signals telephone lines etc.
Continuous Time and Discrete Time Signals ▲ A signal is said to be continuous when it is defined for all instants of time.
▲ A signal is said to be discrete when it is defined at only discrete instants of time.
Analog and Digital Signals: ▲ A signal is said to be analog if it has infinite number of possible amplitude values. It can be continuous or discrete time signals.
▲ A signal is said to be digital if it has finite number of possible amplitude values.
Analog System : Analog systems process analog signals which can take any value within a range. For example, the output from a speaker or a microphone. Digital System : Digital systems process digital signals which can take only a limited number of values, usually just two values. The general purpose digital computer is a best known example of digital system.
- Digital Logic and operation ▲ Digital logic is the representation of signals and sequences of digital circuit through numbers. ▲ It is the basis for digital computing and provides a fundamental understanding on how circuits and hardware communicate within a computer. ▲ Digital logic is typically embedded into most electronic devices including calculators, computers, video games and watches. ▲ Digital logic is of two types: positive logic and negative logic. ▲ The binary signals at the inputs and outputs of any logic gate have one of two values. One signal value represents logic-1(e.g. +5V) and the other logic-0(e.g. 0V). ▲ The higher signal level is designated by H and the lower signal level by L.
▲ Choosing the high-level H to represent logic-1 defines a positive logic system. Choosing the low-level L to represent logic-1 defines a negative logic system. Digital Operation : ▲ The operations performed in digital electronics are called digital operations. Some common digital operations are counting, arithmetic operation and logic operations. ▲ The counting operation is performed by “Counters”. ▲ The arithmetic operations are performed by arithmetic and logic unit and are addition, subtraction, multiplication and division and are accomplished with other digital circuits. ▲ The logic operations too are performed by ALU and they include inversion (NOT), AND and OR. ▲ Selecting a single output of multiple inputs (MULTIPLEXING) or giving out many outputs with single inputs (DEMULTIPLEXING) can also be treated as digital operation. ▲ Similarly, the process of encoding and decoding are also digital operations. These operations are performed by different data processing circuits like Multiplexes, de-multiplexes, encoder, decoder etc.
- Digital Computer and Integrated Circuits (IC) ▲ (^) A digital computer is a machine that stores data in a numerical format and performs operations on that data using mathematical manipulation. This type of computer typically includes some sort of device to store information, some method for input and output of data and components that allows mathematical operations to be performed on stored data. ▲ In short, digital computer is a computer that performs calculations and logic operations with quantities represented as digit usually is binary number system.
Integrated Circuits: An Integrated circuit is an associa�on (or connec�on) of various electronic devices such as resistors, capacitors and transistors fabricated to a semiconductor material such as silicon or germanium. It is also called as a chip or microchip. An IC can func�on as an amplifier, rec�fier, oscillator, counter, �mer and memory. Some�me ICs are connected to various other systems to perform complex func�ons. Types of ICs: ICs can be categorized into two types:
- Analog or Linear ICs
- Digital or logic ICs ▲ Further there are certain ICs which can perform as a combina�on of both analog and digital func�ons. Analog or Linear ICs: They produce con�nuous output depending on the input signal. From the name of the IC we can deduce that the output is a linear func�on of the input signal. Op-amp (opera�onal amplifier) is one of the types of linear ICs which are used in amplifiers, �mers and counters, oscillators etc. Digital or Logic ICs : Unlike Analog ICs, Digital ICs never give a continuous output signal. Instead it operates only during defined states. Digital ICs are used mostly in microprocessor and various memory applications. Logic gates are the building blocks of Digital ICs which operate either at 0 or 1. Classifica�on of Integrated Circuits : Integrated circuits are categorized according to the number of logic gates or the complexity of the circuits within a single chip with the general classifica�on for the number of individual gates given as: ■ Small Scale Integra�on or (SSI) – Contain up to 10 transistors or a few gates within a single package such as AND, OR, NOT gates. ■ Medium Scale Integra�on or (MSI) – between 10 and 100 transistors or tens of gates within a single package and perform digital opera�ons such as adders, decoders, counters, flip-flops and mul�plexers. ■ Large Scale Integra�on or (LSI) – between 100 and 1,000 transistors or hundreds of gates and perform specific digital opera�ons such as I/O chips, memory, arithme�c and logic units. ■ Very-Large Scale Integra�on or (VLSI) – between 1,000 and 10,000 transistors or thousands of gates and perform computa�onal opera�ons such as processors, large memory arrays and programmable logic devices. ■ (^) Super-Large Scale Integra�on or (SLSI) – between 10,000 and 100,000 transistors within a single package and perform computa�onal opera�ons such as microprocessor chips, micro-controllers, basic PICs and calculators.
▲ The sign-magnitude approach represents a signed number in a natural manner. With 4 bits we can only represent numbers in the range − 7 ≤ x ≤ +7. In general, if there are n bits, then we can cover all numbers in the range ±(2n-1^ - 1). Note that with n - 1 bits, any value from 0 to 2 n-1^ - 1 can be represented. However, this approach leads to a confusion because there are two representations for the number zero (0000 means +0; 1000 means −0). ▲ In complement approach, positive numbers have the same representation as they do in the sign- magnitude representation. However, in this technique negative numbers are represented in a different manner. ▲ In the ones complement approach, a negative number, − x , is the complement of its positive representation. For example let us find the ones complement representation of 0100 2 (+4 (^) 10). The complement of 0100 is 1011, and this denotes the negative number -410. Table below summarizes 4-bit integers and their interpretations using ones complement numbers.
▲ The ones complement approach does not handle negative numbers naturally. In other words, if the number is negative (when the sign bit is 1), its magnitude is not obvious from its ones complement. To determine its magnitude, one needs to take its ones complement. For example, consider the number
- The most significant bit indicates that this is a negative number. Because the number is
negative, its magnitude cannot be obtained by directly looking at 110110. Instead, one needs to take the ones complement of 110110 to obtain 001001. The value of 001001 as a sign-magnitude number is +9. On the other hand, 110110 represent −9 in ones complement form. Like the sign-magnitude representation, the ones complement approach does not increase the range of numbers covered by a fixed number of bit patterns. For example, 4 bits cover the range −7 to +7. The same range is obtained with sign-magnitude representation. Note that the confusion of two distinct representations for zero exists in the ones complement approach. ▲ In twos complement method; positive integers are represented in the same manner as they are in the sign-magnitude method. In other words, if the sign bit is zero, the number is positive and its magnitude can be directly obtained by looking at the remaining n - 1 bits. However, a negative number -x can be represented in twos complement form as follows:
- Represent + x in sign magnitude form and call this result y
- Take the ones complement of y to get y −^ (or y ´)
- y−^ + 1 is the twos complement representation of -x. ▲ Table below lists 4-bit integers along with their twos complement forms.
▲ From the above table, it can be concluded that:
- Twos complement form does not provide two representations for zero.
- Twos complement form covers up to −8 in the negative side, and this is more than can be achieved with the other two methods. In general, with n bits, and using twos complement approach, one can cover all the numbers in the range -(2 n -1^ ) to +(2 n -1^ - 1). Floating-point representation : They include fractional numbers such as 12.25, -7.50, 0.0 etc. ▲ The usual method used by computers to represent real numbers is floating-point notation. ▲ There are many varieties of floating-point notation and each has individual characteristics. ▲ The key concept is that a real number is represented by a number, called mantissa, times a base raised to an integer power, called an exponent. ▲ (^) The base is usually fixed, and the mantissa and exponent vary to represent different real numbers. ▲ For example, if the base is fixed at 10, the number 387.53 could be represented as 38753 ×10-2^. The mantissa is 38753 and the exponent is -2. ▲ Let us assume a floating-point notation represented by a 32-bit string consisting of a 24 bit mantissa followed by an 8-bit exponent and the base is fixed at 10.
▲ On the other hand, in an odd parity scheme, the parity bit is added in such a way that the number of 1's in the message and the parity bit is an odd number.
▲ For example, suppose a message to be transmitted is 0110. If even parity is used by the transmitting computer, the transmitted data along with the parity will be 00110. On the other hand, if odd parity is used, the data to be transmitted will be 10110.
▲ The parity computation can be implemented in hardware by using exclusive-OR gates. Usually for a given message, the parity bit is generated using either an even or odd parity scheme by the transmitting computer. The message is then transmitted along with the parity bit. At the receiving end, the parity is checked by the receiving computer. If there is a discrepancy, the data received will obviously be incorrect. ▲ With a single parity bit, an error due to a single bit change can be detected. Errors due to 2-bit changes during transmission will go undetected. In such situations, multiple parity bits are used. Unit-3 Combina�onal Logic Design 3.1 Basic Logic Gates: NOT, OR and AND ▲ Logic gates are electronic circuits that operate on one or more input signals to produce an output signal. ▲ Electrical signals such as voltages or currents exist throughout a digital system in either one of two recognizable values (bi-state 0 or 1). Voltage-operated circuits respond to two separate voltage ranges that represent a binary variable equal to logic 1 or logic 0. ▲ The graphics symbols used to designate the three types of gates AND, OR, and NOT are shown in Figure below:
Fig: Graphic Symbols
▲ These circuits, called gates, are blocks of hardware that produce a logic-1 or logic-0 output signal if input logic requirements are sa�sfied. ▲ Note that four different names have been used for the same type of circuits: digital circuits, switching circuits, logic circuits, and gates. ▲ AND and OR gates may have more than two inputs. NOT gate is single input circuit, it simply inverts the input.
Fig: Timing Diagram ▲ The two input signals X and Y to the AND and OR gates take on one of four possible combina�ons: 00, 01, 10, or
- These input signals are shown as �ming diagrams, together with the �ming diagrams for the corresponding output signal for each type of gate. The horizontal axis of a �ming diagram represents �me, and the ver�cal axis shows a signal as it changes between the two possible voltage levels. The low level represents logic 0 and the high level represents logic 1. 3.2 Universal Logic Gates: NOR and NAND ▲ A universal gate is a gate which can implement any Boolean func�on without need to use any other gate type. The NAND and NOR gates are universal gates. ▲ In prac�ce, this is advantageous since NAND and NOR gates are economical and easier to fabricate and are the basic gates used in all IC digital logic families. NAND Gate is a Universal Gate: ▲ To prove that any Boolean func�on can be implemented using only NAND gates, we will show that the AND, OR, and NOT opera�ons can be performed using only these gates. ■ Implemen�ng an Inverter Using only NAND Gate: All NAND input pins connected to the input signal A gives an output A’.
■ Implemen�ng AND Using only NAND Gates: The AND is replaced by a NAND gate with its output complemented by a NAND gate inverter.
■ Implemen�ng OR Using only NAND Gates: The OR gate is replaced by a NAND gate with all its inputs complemented by NAND gate inverters.
▲ Thus, the NAND gate is a universal gate since it can implement the AND, OR and NOT func�ons. NOR Gate is a Universal Gate: ▲ To prove that any Boolean func�on can be implemented using only NOR gates, we will show that the AND, OR, and NOT opera�ons can be performed using only these gates. ■ Implemen�ng an Inverter Using only NOR Gate: All NOR input pins connect to the input signal A gives an output A’.
■ Implemen�ng OR Using only NOR Gates: The OR is replaced by a NOR gate with its output complemented by a NOR gate inverter.
■ Implemen�ng AND Using only NOR Gates: The AND gate is replaced by a NOR gate with all its inputs complemented by NOR gate inverters.
▲ Thus, the NOR gate is a universal gate since it can implement the AND, OR and NOT func�ons.
