Digital electronics is a field of
electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
involving the study of
digital signals and the engineering of devices that use or produce them. This is in contrast to
analog electronics
Analogue electronics ( en-US, analog electronics) are electronic systems with a continuously variable signal, in contrast to digital electronics where signals usually take only two levels. The term "analogue" describes the proportional relat ...
and
analog signals.
Digital
electronic circuits are usually made from large assemblies of
logic gates, often packaged in
integrated circuits. Complex devices may have simple electronic representations of
Boolean logic functions.
History
The
binary number system
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one).
The base-2 numeral system is a positional notatio ...
was refined by
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of math ...
(published in 1705) and he also established that by using the binary system, the principles of arithmetic and logic could be joined. Digital logic as we know it was the brain-child of
George Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in ...
in the mid 19th century. In an 1886 letter,
Charles Sanders Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism".
Educated as a chemist and employed as a scientist for t ...
described how logical operations could be carried out by electrical switching circuits.
[Peirce, C. S., "Letter, Peirce to A. Marquand", dated 1886, '' Writings of Charles S. Peirce'', v. 5, 1993, pp. 541–3. Googl]
Preview
See Burks, Arthur W., "Review: Charles S. Peirce, ''The new elements of mathematics''", ''Bulletin of the American Mathematical Society'' v. 84, n. 5 (1978), pp. 913–18, see 917
PDF Eprint
Eventually,
vacuum tube
A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric potential difference has been applied.
The type known as ...
s replaced relays for logic operations.
Lee De Forest's modification of the
Fleming valve
The Fleming valve, also called the Fleming oscillation valve, was a thermionic valve or vacuum tube invented in 1904 by English physicist John Ambrose Fleming as a detector for early radio receivers used in electromagnetic wireless telegraphy. ...
in 1907 could be used as an
AND gate
The AND gate is a basic digital logic gate that implements logical conjunction (∧) from mathematical logic AND gate behaves according to the truth table. A HIGH output (1) results only if all the inputs to the AND gate are HIGH (1). If not al ...
.
Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is con ...
introduced a version of the 16-row
truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
as proposition 5.101 of ''
Tractatus Logico-Philosophicus
The ''Tractatus Logico-Philosophicus'' (widely abbreviated and cited as TLP) is a book-length philosophical work by the Austrian philosopher Ludwig Wittgenstein which deals with the relationship between language and reality and aims to define th ...
'' (1921).
Walther Bothe
Walther Wilhelm Georg Bothe (; 8 January 1891 – 8 February 1957) was a German nuclear physicist, who shared the Nobel Prize in Physics in 1954 with Max Born.
In 1913, he joined the newly created Laboratory for Radioactivity at the Reich Physi ...
, inventor of the
coincidence circuit In physics and electrical engineering, a coincidence circuit or coincidence gate is an electronic device with one output and two (or more) inputs. The output activates only when the circuit receives signals within a time window accepted as ''at the ...
, shared the 1954
Nobel Prize
The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfr ...
in physics, for creating the first modern electronic AND gate in 1924.
Mechanical
Mechanical may refer to:
Machine
* Machine (mechanical), a system of mechanisms that shape the actuator input to achieve a specific application of output forces and movement
* Mechanical calculator, a device used to perform the basic operations of ...
analog computer
An analog computer or analogue computer is a type of computer that uses the continuous variation aspect of physical phenomena such as electrical, mechanical, or hydraulic quantities (''analog signals'') to model the problem being solved. In ...
s started appearing in the first century and were later used in the medieval era for astronomical calculations. In
World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposing ...
, mechanical analog computers were used for specialized military applications such as calculating torpedo aiming. During this time the first electronic
digital computers were developed, with the term ''digital'' being proposed by
George Stibitz in 1942. Originally they were the size of a large room, consuming as much power as several hundred modern
PCs.
The
Z3 was an
electromechanical computer
A mechanical computer is a computer built from mechanical components such as levers and gears rather than electronic components. The most common examples are adding machines and mechanical counters, which use the turning of gears to increment out ...
designed by
Konrad Zuse
Konrad Ernst Otto Zuse (; 22 June 1910 – 18 December 1995) was a German civil engineer, pioneering computer scientist, inventor and businessman. His greatest achievement was the world's first programmable computer; the functional program ...
. Finished in 1941, it was the world's first working
programmable, fully automatic digital computer. Its operation was facilitated by the invention of the vacuum tube in 1904 by
John Ambrose Fleming
Sir John Ambrose Fleming FRS (29 November 1849 – 18 April 1945) was an English electrical engineer and physicist who invented the first thermionic valve or vacuum tube, designed the radio transmitter with which the first transatlantic ra ...
.
At the same time that digital calculation replaced analog, purely
electronic circuit elements soon replaced their mechanical and electromechanical equivalents.
John Bardeen and
Walter Brattain
Walter Houser Brattain (; February 10, 1902 – October 13, 1987) was an American physicist at Bell Labs who, along with fellow scientists John Bardeen and William Shockley, invented the point-contact transistor in December 1947. They shared the ...
invented the
point-contact transistor at
Bell Labs
Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984),
then AT&T Bell Laboratories (1984–1996)
and Bell Labs Innovations (1996–2007),
is an American industrial Research and development, research and scientific developm ...
in 1947, followed by
William Shockley
William Bradford Shockley Jr. (February 13, 1910 – August 12, 1989) was an American physicist and inventor. He was the manager of a research group at Bell Labs that included John Bardeen and Walter Brattain. The three scientists were jointl ...
inventing the
bipolar junction transistor
A bipolar junction transistor (BJT) is a type of transistor that uses both electrons and electron holes as charge carriers. In contrast, a unipolar transistor, such as a field-effect transistor, uses only one kind of charge carrier. A bipola ...
at Bell Labs in 1948.
At the
University of Manchester
, mottoeng = Knowledge, Wisdom, Humanity
, established = 2004 – University of Manchester Predecessor institutions: 1956 – UMIST (as university college; university 1994) 1904 – Victoria University of Manchester 1880 – Victoria Univ ...
, a team under the leadership of
Tom Kilburn
Tom Kilburn (11 August 1921 – 17 January 2001) was an English mathematician and computer scientist. Over the course of a productive 30-year career, he was involved in the development of five computers of great historical significance. With ...
designed and built a machine using the newly developed
transistor
upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink).
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch ...
s instead of vacuum tubes. Their "
transistorised computer", and the first in the world, was
operational by 1953, and a second version was completed there in April 1955. From 1955 and onwards, transistors replaced vacuum tubes in computer designs, giving rise to the "second generation" of computers. Compared to vacuum tubes, transistors were smaller, more reliable, had indefinite lifespans, and required less power than vacuum tubes - thereby giving off less heat, and allowing much denser concentrations of circuits, up to tens of thousands in a relatively compact space.
While working at
Texas Instruments
Texas Instruments Incorporated (TI) is an American technology company headquartered in Dallas, Texas, that designs and manufactures semiconductors and various integrated circuits, which it sells to electronics designers and manufacturers globa ...
in July 1958,
Jack Kilby
Jack St. Clair Kilby (November 8, 1923 – June 20, 2005) was an American electrical engineer who took part (along with Robert Noyce of Fairchild) in the realization of the first integrated circuit while working at Texas Instruments (TI) in 1 ...
recorded his initial ideas concerning the
integrated circuit (IC), then successfully demonstrated the first working integrated circuit on 12 September 1958.
Kilby's chip was made of
germanium. The following year,
Robert Noyce at
Fairchild Semiconductor invented the
silicon
Silicon is a chemical element with the symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic luster, and is a tetravalent metalloid and semiconductor. It is a member of group 14 in the periodic ta ...
integrated circuit. The basis for Noyce's silicon IC was the
planar process, developed in early 1959 by
Jean Hoerni
Jean Amédée Hoerni (September 26, 1924 – January 12, 1997) was a Swiss-American engineer. He was a silicon transistor pioneer, and a member of the " traitorous eight". He developed the planar process, an important technology for reliably fab ...
, who was in turn building on
Mohamed Atalla's
silicon surface passivation method developed in 1957. This new technique, the integrated circuit, allowed for quick, low-cost fabrication of complex circuits by having a set of
electronic circuits on one small plate ("chip") of
semiconductor material
A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. ...
, normally silicon.
The
metal–oxide–semiconductor field-effect transistor
The metal–oxide–semiconductor field-effect transistor (MOSFET, MOS-FET, or MOS FET) is a type of field-effect transistor (FET), most commonly fabricated by the controlled oxidation of silicon. It has an insulated gate, the voltage of which d ...
(MOSFET), also known as the MOS transistor, was invented by
Mohamed Atalla and
Dawon Kahng
Dawon Kahng ( ko, 강대원; May 4, 1931 – May 13, 1992) was a Korean-American electrical engineer and inventor, known for his work in solid-state electronics. He is best known for inventing the MOSFET (metal–oxide–semiconductor field-effe ...
at Bell Labs in 1959.
The MOSFET's advantages include
high scalability, affordability,
low power consumption, and high
transistor density. Its rapid on–off
electronic switch
In electronics, an electronic switch is an electronic component or device that can switch an electrical circuit, interrupting the current or diverting it from one conductor to another. Electronic switches are considered binary devices because the ...
ing speed also makes it ideal for generating
pulse train
A pulse wave or pulse train is a type of non-sinusoidal waveform that includes square waves (duty cycle of 50%) and similarly periodic but asymmetrical waves (duty cycles other than 50%). It is a term used in synthesizer programming, and is ...
s,
the basis for electronic
digital signals, in contrast to BJTs which, more slowly, generate
analog signals
An analog signal or analogue signal (see spelling differences) is any continuous signal representing some other quantity, i.e., ''analogous'' to another quantity. For example, in an analog audio signal, the instantaneous signal voltage varies ...
resembling
sine waves
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
.
Along with MOS
large-scale integration
An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchip) is a set of electronic circuits on one small flat piece (or "chip") of semiconductor material, usually silicon. Large numbers of tiny ...
(LSI), these factors make the MOSFET an important switching device for
digital circuits
Digital electronics is a field of electronics involving the study of digital signals and the engineering of devices that use or produce them. This is in contrast to analog electronics and analog signals.
Digital electronic circuits are usually ...
. The MOSFET revolutionized the
electronics industry,
and is the most common
semiconductor device.
In the early days of
integrated circuits
An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchip) is a set of electronic circuits on one small flat piece (or "chip") of semiconductor material, usually silicon. Large numbers of tiny ...
, each chip was limited to only a few transistors, and the low degree of integration meant the design process was relatively simple. Manufacturing yields were also quite low by today's standards. The wide adoption of the MOSFET transistor by the early 1970s led to the first
large-scale integration
An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchip) is a set of electronic circuits on one small flat piece (or "chip") of semiconductor material, usually silicon. Large numbers of tiny ...
(LSI) chips with more than 10,000 transistors on a single chip. Following the wide adoption of
CMOS, a type of MOSFET logic, by the 1980s, millions and then billions of MOSFETs could be placed on one chip as the technology progressed, and good designs required thorough planning, giving rise to
new design methods. The
transistor count
The transistor count is the number of transistors in an electronic device (typically on a single substrate or "chip"). It is the most common measure of integrated circuit complexity (although the majority of transistors in modern microprocessors ...
of devices and total production rose to unprecedented heights. The total amount of transistors produced until 2018 has been estimated to be (13
sextillion
Two naming scales for large numbers have been used in English and other European languages since the early modern era: the long and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-Eng ...
).
The
wireless revolution
Wireless communication (or just wireless, when the context allows) is the transfer of information between two or more points without the use of an electrical conductor, optical fiber or other continuous guided medium for the transfer. The most ...
(the introduction and proliferation of
wireless networks
A wireless network is a computer network that uses wireless data connections between network nodes.
Wireless networking is a method by which homes, telecommunications networks and business installations avoid the costly process of introducing c ...
) began in the 1990s and was enabled by the wide adoption of MOSFET-based
RF power amplifier
A radio-frequency power amplifier (RF power amplifier) is a type of electronic amplifier that converts a low-power radio-frequency signal into a higher-power signal. Typically, RF power amplifiers drive the antenna of a transmitter. Design goal ...
s (
power MOSFET and
LDMOS
LDMOS (laterally-diffused metal-oxide semiconductor) is a planar double-diffused MOSFET (metal–oxide–semiconductor field-effect transistor) used in amplifiers, including microwave power amplifiers, RF power amplifiers and audio power amplif ...
) and
RF circuit
Radio-frequency (RF) engineering is a subset of electronic engineering involving the application of transmission line, waveguide, antenna and electromagnetic field principles to the design and application of devices that produce or use signals w ...
s (
RF CMOS
RF CMOS is a metal–oxide–semiconductor (MOS) integrated circuit (IC) technology that integrates radio-frequency (RF), analog and digital electronics on a mixed-signal CMOS (complementary MOS) RF circuit chip. It is widely used in modern wire ...
). Wireless networks allowed for public digital transmission without the need for cables, leading to
digital television
Digital television (DTV) is the transmission of television signals using digital encoding, in contrast to the earlier analog television technology which used analog signals. At the time of its development it was considered an innovative adva ...
,
GPS
The Global Positioning System (GPS), originally Navstar GPS, is a satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of the global navigation satellite sy ...
,
satellite radio,
wireless Internet
Wireless communication (or just wireless, when the context allows) is the transfer of information between two or more points without the use of an electrical conductor, optical fiber or other continuous guided medium for the transfer. The most ...
and
mobile phones through the 1990s2000s.
Properties
An advantage of digital circuits when compared to analog circuits is that signals represented digitally can be transmitted without degradation caused by
noise
Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference aris ...
. For example, a continuous audio signal transmitted as a sequence of 1s and 0s, can be reconstructed without error, provided the noise picked up in transmission is not enough to prevent identification of the 1s and 0s.
In a digital system, a more precise representation of a signal can be obtained by using more binary digits to represent it. While this requires more digital circuits to process the signals, each digit is handled by the same kind of hardware, resulting in an easily
scalable system. In an analog system, additional resolution requires fundamental improvements in the linearity and noise characteristics of each step of the
signal chain
Signal chain, or signal-processing chain is a term used in signal processing and mixed-signal system design to describe a series of signal-conditioning electronic components that receive input (data acquired from sampling either real-time phenom ...
.
With computer-controlled digital systems, new functions can be added through software revision and no hardware changes are needed. Often this can be done outside of the factory by updating the product's software. This way, the product's design errors can be corrected even after the product is in a customer's hands.
Information storage can be easier in digital systems than in analog ones. The noise immunity of digital systems permits data to be stored and retrieved without degradation. In an analog system, noise from aging and wear degrade the information stored. In a digital system, as long as the total noise is below a certain level, the information can be recovered perfectly. Even when more significant noise is present, the use of
redundancy permits the recovery of the original data provided too many errors do not occur.
In some cases, digital circuits use more energy than analog circuits to accomplish the same tasks, thus producing more heat which increases the complexity of the circuits such as the inclusion of heat sinks. In portable or battery-powered systems this can limit the use of digital systems. For example, battery-powered
cellular phone
A mobile phone, cellular phone, cell phone, cellphone, handphone, hand phone or pocket phone, sometimes shortened to simply mobile, cell, or just phone, is a portable telephone that can make and receive calls over a radio frequency link while ...
s often use a low-power analog front-end to
amplify and
tune the radio signals from the base station. However, a base station has grid power and can use power-hungry, but very flexible
software radios. Such base stations can easily be reprogrammed to process the signals used in new cellular standards.
Many useful digital systems must translate from continuous analog signals to discrete digital signals. This causes
quantization errors. Quantization error can be reduced if the system stores enough digital data to represent the signal to the desired degree of
fidelity
Fidelity is the quality of faithfulness or loyalty. Its original meaning regarded duty in a broader sense than the related concept of ''fealty''. Both derive from the Latin word ''fidēlis'', meaning "faithful or loyal". In the City of London fin ...
. The
Nyquist–Shannon sampling theorem
The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that per ...
provides an important guideline as to how much digital data is needed to accurately portray a given analog signal.
In some systems, if a single piece of digital data is lost or misinterpreted, the meaning of large blocks of related data can completely change. For example, a single-bit error in audio data stored directly as
linear pulse-code modulation
Pulse-code modulation (PCM) is a method used to digitally represent sampled analog signals. It is the standard form of digital audio in computers, compact discs, digital telephony and other digital audio applications. In a PCM stream, the am ...
causes, at worst, a single click. Nevertheless, many people use
audio compression to save storage space and download time, even though a single bit error may cause a large disruption.
Because of the
cliff effect
In telecommunications, the (digital) cliff effect or brickwall effect is a sudden loss of digital signal reception. Unlike analog signals, which gradually fade when signal strength decreases or electromagnetic interference or multipath increases, ...
, it can be difficult for users to tell if a particular system is right on the edge of failure, or if it can tolerate much more noise before failing. Digital fragility can be reduced by designing a digital system for robustness. For example, a
parity bit or other error management method can be inserted into the signal path. These schemes help the system detect errors, and then either
correct the errors, or request retransmission of the data.
Construction
![relogio binario](https://upload.wikimedia.org/wikipedia/commons/6/69/Relogio_binario.JPG)
A digital circuit is typically constructed from small electronic circuits called
logic gates that can be used to create
combinational logic
In automata theory, combinational logic (also referred to as time-independent logic or combinatorial logic) is a type of digital logic which is implemented by Boolean circuits, where the output is a pure function of the present input only. This ...
. Each logic gate is designed to perform a function of
boolean logic when acting on logic signals. A logic gate is generally created from one or more electrically controlled switches, usually
transistors
upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink).
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch ...
but
thermionic valves
A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric potential difference has been applied.
The type known as a ...
have seen historic use. The output of a logic gate can, in turn, control or feed into more logic gates.
Another form of digital circuit is constructed from lookup tables, (many sold as "
programmable logic device
A programmable logic device (PLD) is an electronic component used to build reconfigurable digital circuits. Unlike digital logic constructed using discrete logic gates with fixed functions, a PLD has an undefined function at the time of manu ...
s", though other kinds of PLDs exist). Lookup tables can perform the same functions as machines based on logic gates, but can be easily reprogrammed without changing the wiring. This means that a designer can often repair design errors without changing the arrangement of wires. Therefore, in small volume products, programmable logic devices are often the preferred solution. They are usually designed by engineers using
electronic design automation software.
Integrated circuits consist of multiple transistors on one silicon chip, and are the least expensive way to make large number of interconnected logic gates. Integrated circuits are usually interconnected on a
printed circuit board which is a board which holds electrical components, and connects them together with copper traces.
Design
Engineers use many methods to minimize
logic redundancy
Logic redundancy occurs in a digital gate network containing circuitry that does not affect the static logic function. There are several reasons why logic redundancy may exist. One reason is that it may have been added deliberately to suppress tra ...
in order to reduce the circuit complexity. Reduced complexity reduces component count and potential errors and therefore typically reduces cost. Logic redundancy can be removed by several well-known techniques, such as
binary decision diagrams,
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
,
Karnaugh map
The Karnaugh map (KM or K-map) is a method of simplifying Boolean algebra expressions. Maurice Karnaugh introduced it in 1953 as a refinement of Edward W. Veitch's 1952 Veitch chart, which was a rediscovery of Allan Marquand's 1881 ''logi ...
s, the
Quine–McCluskey algorithm
The Quine–McCluskey algorithm (QMC), also known as the method of prime implicants, is a method used for minimization of Boolean functions that was developed by Willard V. Quine in 1952 and extended by Edward J. McCluskey in 1956. As a gener ...
, and the
heuristic computer method. These operations are typically performed within a
computer-aided design system.
Embedded system
An embedded system is a computer system—a combination of a computer processor, computer memory, and input/output peripheral devices—that has a dedicated function within a larger mechanical or electronic system. It is ''embedded'' ...
s with
microcontrollers and
programmable logic controllers are often used to implement digital logic for complex systems that don't require optimal performance. These systems are usually programmed by
software engineers
Software engineering is a systematic engineering approach to software development.
A software engineer is a person who applies the principles of software engineering to design, develop, maintain, test, and evaluate computer software. The term '' ...
or by electricians, using
ladder logic.
Representation
Representations are crucial to an engineer's design of digital circuits. To choose representations, engineers consider different types of digital systems.
The classical way to represent a digital circuit is with an equivalent set of
logic gates
A logic gate is an idealized or physical device implementing a Boolean function, a logical operation performed on one or more binary inputs that produces a single binary output. Depending on the context, the term may refer to an ideal logic gate ...
. Each logic symbol is represented by a different shape. The actual set of shapes was introduced in 1984 under IEEE/ANSI standard 91-1984 and is now in common use by integrated circuit manufacturers. Another way is to construct an equivalent system of electronic switches (usually
transistor
upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink).
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch ...
s). This can be represented as a
truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
.
Most digital systems divide into
combinational and
sequential systems. A combinational system always presents the same output when given the same inputs. A sequential system is a combinational system with some of the outputs fed back as inputs. This makes the digital machine perform a ''sequence'' of operations. The simplest sequential system is probably a
flip flop, a mechanism that represents a
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that ta ...
digit or "
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
". Sequential systems are often designed as
state machines. In this way, engineers can design a system's gross behavior, and even test it in a simulation, without considering all the details of the logic functions.
Sequential systems divide into two further subcategories.
"Synchronous" sequential systems change state all at once when a
clock signal changes state.
"Asynchronous" sequential systems propagate changes whenever inputs change. Synchronous sequential systems are made of well-characterized asynchronous circuits such as flip-flops, that change only when the clock changes, and which have carefully designed timing margins.
For
logic simulation
Logic simulation is the use of simulation software to predict the behavior of digital circuits and hardware description languages. Simulation can be performed at varying degrees of physical abstraction, such as at the transistor level, gate le ...
, digital circuit representations have digital file formats that can be processed by computer programs.
Synchronous systems
![JohnsonCounter2](https://upload.wikimedia.org/wikipedia/commons/e/e8/JohnsonCounter2.png)
The usual way to implement a synchronous sequential state machine is to divide it into a piece of combinational logic and a set of flip flops called a ''state register''. The state register represents the state as a binary number. The combinational logic produces the binary representation for the next state. On each clock cycle, the state register captures the feedback generated from the previous state of the combinational logic and feeds it back as an unchanging input to the combinational part of the state machine. The clock rate is limited by the most time-consuming logic calculation in the combinational logic.
Asynchronous systems
Most digital logic is synchronous because it is easier to create and verify a synchronous design. However, asynchronous logic has the advantage of its speed not being constrained by an arbitrary clock; instead, it runs at the maximum speed of its logic gates. Building an asynchronous system using faster parts makes the circuit faster.
Nevertheless, most systems need to accept external unsynchronized signals into their synchronous logic circuits. This interface is inherently asynchronous and must be analyzed as such. Examples of widely used asynchronous circuits include synchronizer flip-flops, switch
debounce
In electrical engineering, a switch is an electrical component that can disconnect or connect the conducting path in an electrical circuit, interrupting the electric current or diverting it from one conductor to another. The most common type o ...
rs and
arbiters.
Asynchronous logic components can be hard to design because all possible states, in all possible timings must be considered. The usual method is to construct a table of the minimum and maximum time that each such state can exist and then adjust the circuit to minimize the number of such states. The designer must force the circuit to periodically wait for all of its parts to enter a compatible state (this is called "self-resynchronization"). Without careful design, it is easy to accidentally produce asynchronous logic that is unstable—that is—real electronics will have unpredictable results because of the cumulative delays caused by small variations in the values of the electronic components.
Register transfer systems
![Register transfer level - example toggler](https://upload.wikimedia.org/wikipedia/commons/0/01/Register_transfer_level_-_example_toggler.svg)
Many digital systems are
data flow machines. These are usually designed using synchronous
register transfer logic and written with
hardware description language
In computer engineering, a hardware description language (HDL) is a specialized computer language used to describe the structure and behavior of electronic circuits, and most commonly, digital logic circuits.
A hardware description language en ...
s such as
VHDL
The VHSIC Hardware Description Language (VHDL) is a hardware description language (HDL) that can model the behavior and structure of digital systems at multiple levels of abstraction, ranging from the system level down to that of logic gate ...
or
Verilog
Verilog, standardized as IEEE 1364, is a hardware description language (HDL) used to model electronic systems. It is most commonly used in the design and verification of digital circuits at the register-transfer level of abstraction. It is als ...
.
In register transfer logic, binary numbers are stored in groups of flip flops called
registers. A sequential state machine controls when each register accepts new data from its input. The outputs of each register are a bundle of wires called a ''
bus
A bus (contracted from omnibus, with variants multibus, motorbus, autobus, etc.) is a road vehicle that carries significantly more passengers than an average car or van. It is most commonly used in public transport, but is also in use for cha ...
'' that carries that number to other calculations. A calculation is simply a piece of combinational logic. Each calculation also has an output bus, and these may be connected to the inputs of several registers. Sometimes a register will have a
multiplexer
In electronics, a multiplexer (or mux; spelled sometimes as multiplexor), also known as a data selector, is a device that selects between several analog or digital input signals and forwards the selected input to a single output line. The sel ...
on its input so that it can store a number from any one of several buses.
Asynchronous register-transfer systems (such as computers) have a general solution. In the 1980s, some researchers discovered that almost all synchronous register-transfer machines could be converted to asynchronous designs by using first-in-first-out synchronization logic. In this scheme, the digital machine is characterized as a set of data flows. In each step of the flow, a synchronization circuit determines when the outputs of that step are valid and instructs the next stage when to use these outputs.
Computer design
![80486dx2-large](https://upload.wikimedia.org/wikipedia/commons/0/02/80486dx2-large.jpg)
The most general-purpose register-transfer logic machine is a
computer. This is basically an
automatic binary
abacus
The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hi ...
. The
control unit
The control unit (CU) is a component of a computer's central processing unit (CPU) that directs the operation of the processor. A CU typically uses a binary decoder to convert coded instructions into timing and control signals that direct the op ...
of a computer is usually designed as a
microprogram
In processor design, microcode (μcode) is a technique that interposes a layer of computer organization between the central processing unit (CPU) hardware and the programmer-visible instruction set architecture of a computer. Microcode is a lay ...
run by a
microsequencer In computer architecture and engineering, a sequencer or microsequencer generates the addresses used to step through the microprogram of a control store. It is used as a part of the control unit of a CPU or as a stand-alone generator for address ...
. A microprogram is much like a player-piano roll. Each table entry of the microprogram commands the state of every bit that controls the computer. The sequencer then counts, and the count addresses the memory or combinational logic machine that contains the microprogram. The bits from the microprogram control the
arithmetic logic unit,
memory
Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembered ...
and other parts of the computer, including the microsequencer itself. In this way, the complex task of designing the controls of a computer is reduced to a simpler task of programming a collection of much simpler logic machines.
Almost all computers are synchronous. However,
asynchronous computer
Asynchronous circuit (clockless or self-timed circuit) is a sequential digital logic circuit that does not use a global clock circuit or signal generator to synchronize its components. Instead, the components are driven by a handshaking circui ...
s have also been built. One example is the
ASPIDA DLX
The DLX (pronounced "Deluxe") is a RISC processor architecture designed by John L. Hennessy and David A. Patterson, the principal designers of the Stanford MIPS and the Berkeley RISC designs (respectively), the two benchmark examples of RISC de ...
core.
Another was offered by
ARM Holdings.
[ They don't, however, have any speed advantages because modern computer designs already run at the speed of their slowest component, usually memory. They do use somewhat less power because a clock distribution network is not needed. An unexpected advantage is that asynchronous computers do not produce spectrally-pure radio noise. They are used in some radio-sensitive mobile-phone base-station controllers. They may be more secure in cryptographic applications because their electrical and radio emissions can be more difficult to decode.]
Computer architecture
Computer architecture
In computer engineering, computer architecture is a description of the structure of a computer system made from component parts. It can sometimes be a high-level description that ignores details of the implementation. At a more detailed level, the ...
is a specialized engineering activity that tries to arrange the registers, calculation logic, buses and other parts of the computer in the best way possible for a specific purpose. Computer architects have put a lot of work into reducing the cost and increasing the speed of computers in addition to boosting their immunity to programming errors. An increasingly common goal of computer architects is to reduce the power used in battery-powered computer systems, such as smartphone
A smartphone is a portable computer device that combines mobile telephone and computing functions into one unit. They are distinguished from feature phones by their stronger hardware capabilities and extensive mobile operating systems, whic ...
s.
Design issues in digital circuits
Digital circuits are made from analog components. The design must assure that the analog nature of the components doesn't dominate the desired digital behavior. Digital systems must manage noise and timing margins, parasitic inductances and capacitances.
Bad designs have intermittent problems such as glitch
A glitch is a short-lived fault in a system, such as a transient fault that corrects itself, making it difficult to troubleshoot. The term is particularly common in the computing and electronics industries, in circuit bending, as well as among ...
es, vanishingly fast pulses that may trigger some logic but not others, runt pulse In digital circuits, a runt pulse is a narrow pulse that,
due to non-zero rise and fall times of the signal, does not reach a valid
high or low level. A runt pulse may occur when switching between
asynchronous
Asynchrony is the state of not ...
s that do not reach valid threshold voltage
The threshold voltage, commonly abbreviated as Vth or VGS(th), of a field-effect transistor (FET) is the minimum gate-to-source voltage (VGS) that is needed to create a conducting path between the source and drain terminals. It is an important s ...
s.
Additionally, where clocked digital systems interface to analog systems or systems that are driven from a different clock, the digital system can be subject to metastability
In chemistry and physics, metastability denotes an intermediate energetic state within a dynamical system other than the system's state of least energy.
A ball resting in a hollow on a slope is a simple example of metastability. If the ball i ...
where a change to the input violates the setup time
In electronics, a flip-flop or latch is a circuit that has two stable states and can be used to store state information – a bistable multivibrator. The circuit can be made to change state by signals applied to one or more control inputs and ...
for a digital input latch.
Since digital circuits are made from analog components, digital circuits calculate more slowly than low-precision analog circuits that use a similar amount of space and power. However, the digital circuit will calculate more repeatably, because of its high noise immunity.
Automated design tools
Much of the effort of designing large logic machines has been automated through the application of electronic design automation (EDA).
Simple truth table-style descriptions of logic are often optimized with EDA that automatically produce reduced systems of logic gates or smaller lookup tables that still produce the desired outputs. The most common example of this kind of software is the Espresso heuristic logic minimizer
The ESPRESSO logic minimizer is a computer program using heuristic and specific algorithms for efficiently reducing the complexity of digital logic gate circuits. ESPRESSO-I was originally developed at IBM by Robert K. Brayton et al. in 1982. ...
. Optimizing large logic systems may be done using the Quine–McCluskey algorithm
The Quine–McCluskey algorithm (QMC), also known as the method of prime implicants, is a method used for minimization of Boolean functions that was developed by Willard V. Quine in 1952 and extended by Edward J. McCluskey in 1956. As a gener ...
or binary decision diagram
In computer science, a binary decision diagram (BDD) or branching program is a data structure that is used to represent a Boolean function. On a more abstract level, BDDs can be considered as a compressed representation of sets or relations. ...
s. There are promising experiments with genetic algorithms and annealing optimizations.
To automate costly engineering processes, some EDA can take state table
State may refer to:
Arts, entertainment, and media Literature
* ''State Magazine'', a monthly magazine published by the U.S. Department of State
* ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States
* ''Our S ...
s that describe state machines and automatically produce a truth table or a function table
In computer science, a dispatch table is a table of pointers or memory addresses to functions or methods. Use of such a table is a common technique when implementing late binding in object-oriented programming.
Perl implementation
The followin ...
for the combinational logic
In automata theory, combinational logic (also referred to as time-independent logic or combinatorial logic) is a type of digital logic which is implemented by Boolean circuits, where the output is a pure function of the present input only. This ...
of a state machine. The state table is a piece of text that lists each state, together with the conditions controlling the transitions between them and their associated output signals.
Often, real logic systems are designed as a series of sub-projects, which are combined using a ''tool flow''. The tool flow is usually controlled with the help of a scripting language
A scripting language or script language is a programming language that is used to manipulate, customize, and automate the facilities of an existing system. Scripting languages are usually interpreted at runtime rather than compiled.
A scripting ...
, a simplified computer language that can invoke the software design tools in the right order. Tool flows for large logic systems such as microprocessor
A microprocessor is a computer processor where the data processing logic and control is included on a single integrated circuit, or a small number of integrated circuits. The microprocessor contains the arithmetic, logic, and control circ ...
s can be thousands of commands long, and combine the work of hundreds of engineers. Writing and debugging tool flows is an established engineering specialty in companies that produce digital designs. The tool flow usually terminates in a detailed computer file or set of files that describe how to physically construct the logic. Often it consists of instructions on how to draw the transistors
upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink).
A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch ...
and wires on an integrated circuit or a printed circuit board.
Parts of tool flows are debugged by verifying the outputs of simulated logic against expected inputs. The test tools take computer files with sets of inputs and outputs and highlight discrepancies between the simulated behavior and the expected behavior. Once the input data is believed to be correct, the design itself must still be verified for correctness. Some tool flows verify designs by first producing a design, then scanning the design to produce compatible input data for the tool flow. If the scanned data matches the input data, then the tool flow has probably not introduced errors.
The functional verification
Verify or verification may refer to:
General
* Verification and validation, in engineering or quality management systems, is the act of reviewing, inspecting or testing, in order to establish and document that a product, service or system meets ...
data are usually called ''test vectors''. The functional test vectors may be preserved and used in the factory to test whether newly constructed logic works correctly. However, functional test patterns don't discover all fabrication faults. Production tests are often designed by automatic test pattern generation ATPG (acronym for both Automatic Test Pattern Generation and Automatic Test Pattern Generator) is an electronic design automation method/technology used to find an input (or test) sequence that, when applied to a digital circuit, enables automatic t ...
software tools. These generate test vectors by examining the structure of the logic and systematically generating tests targeting particular potential faults. This way the fault coverage {{Unreferenced, date=November 2022
Fault coverage refers to the percentage of some type of fault that can be detected during the test of any engineered system. High fault coverage is particularly valuable during manufacturing test, and techniques ...
can closely approach 100%, provided the design is properly made testable (see next section).
Once a design exists, and is verified and testable, it often needs to be processed to be manufacturable as well. Modern integrated circuits have features smaller than the wavelength of the light used to expose the photoresist. Software that are designed for manufacturability add interference patterns to the exposure masks to eliminate open-circuits, and enhance the masks' contrast.
Design for testability
There are several reasons for testing a logic circuit. When the circuit is first developed, it is necessary to verify that the design circuit meets the required functional, and timing specifications. When multiple copies of a correctly designed circuit are being manufactured, it is essential to test each copy to ensure that the manufacturing process has not introduced any flaws.
A large logic machine (say, with more than a hundred logical variables) can have an astronomical number of possible states. Obviously, factory testing every state of such a machine is unfeasible, for even if testing each state only took a microsecond, there are more possible states than there are microseconds since the universe began!
Large logic machines are almost always designed as assemblies of smaller logic machines. To save time, the smaller sub-machines are isolated by permanently installed ''design for test'' circuitry, and are tested independently. One common testing scheme provides a test mode that forces some part of the logic machine to enter a ''test cycle''. The test cycle usually exercises large independent parts of the machine.
Boundary scan is a common test scheme that uses serial communication with external test equipment through one or more shift register
A shift register is a type of digital circuit using a cascade of flip-flops where the output of one flip-flop is connected to the input of the next. They share a single clock signal, which causes the data stored in the system to shift from one loc ...
s known as ''scan chains''. Serial scans have only one or two wires to carry the data, and minimize the physical size and expense of the infrequently used test logic. After all the test data bits are in place, the design is reconfigured to be in ''normal mode'' and one or more clock pulses are applied, to test for faults (e.g. stuck-at low or stuck-at high) and capture the test result into flip-flops or latches in the scan shift register(s). Finally, the result of the test is shifted out to the block boundary and compared against the predicted ''good machine'' result.
In a board-test environment, serial to parallel testing has been formalized as the JTAG standard.
Trade-offs
Cost
Since a digital system may use many logic gates, the overall cost of building a computer correlates strongly with the cost of a logic gate. In the 1930s, the earliest digital logic systems were constructed from telephone relays because these were inexpensive and relatively reliable.
The earliest integrated circuits were constructed to save weight and permit the Apollo Guidance Computer to control an inertial guidance system for a spacecraft. The first integrated circuit logic gates cost nearly US$50, which in would be equivalent to $. Mass-produced gates on integrated circuits became the least-expensive method to construct digital logic.
With the rise of integrated circuits
An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchip) is a set of electronic circuits on one small flat piece (or "chip") of semiconductor material, usually silicon. Large numbers of tiny ...
, reducing the absolute number of chips used represented another way to save costs. The goal of a designer is not just to make the simplest circuit, but to keep the component count down. Sometimes this results in more complicated designs with respect to the underlying digital logic but nevertheless reduces the number of components, board size, and even power consumption.
Reliability
Another major motive for reducing component count on printed circuit boards is to reduce the manufacturing defect rate due to failed soldered connections and increase reliability. Defect and failure rates tend to increase along with the total number of component pins.
The failure of a single logic gate may cause a digital machine to fail. Where additional reliability is required, redundant logic can be provided. Redundancy adds cost and power consumption over a non-redundant system.
The ''reliability'' of a logic gate can be described by its mean time between failure
Mean time between failures (MTBF) is the predicted elapsed time between inherent failures of a mechanical or electronic system during normal system operation. MTBF can be calculated as the arithmetic mean (average) time between failures of a system ...
(MTBF). Digital machines first became useful when the MTBF for a switch increased above a few hundred hours. Even so, many of these machines had complex, well-rehearsed repair procedures, and would be nonfunctional for hours because a tube burned-out, or a moth got stuck in a relay. Modern transistorized integrated circuit logic gates have MTBFs greater than 82 billion hours (). This level of reliability is required because integrated circuits have so many logic gates.
Fan-out
Fan-out
In digital electronics, the fan-out is the number of gate inputs driven by the output of another single logic gate.
In most designs, logic gates are connected to form more complex circuits. While no logic gate input can be fed by more than one ...
describes how many logic inputs can be controlled by a single logic output without exceeding the electrical current ratings of the gate outputs. The minimum practical fan-out is about five. Modern electronic logic gates using CMOS transistors for switches have higher fan-outs.
Speed
The ''switching speed'' describes how long it takes a logic output to change from true to false or vise versa. Faster logic can accomplish more operations in less time. Modern electronic digital logic routinely switches at , and some laboratory systems switch at more than ..
Logic families
Digital design started with relay logic
Relay logic is a method of implementing combinational logic in electrical control circuits by using several electrical relays wired in a particular configuration.
Ladder logic
The schematic diagrams for relay logic circuits are often calle ...
which is relatively inexpensive and reliable, but slow. Occasionally a mechanical failure would occur. Fan-outs were typically about 10, limited by the resistance of the coils and arcing on the contacts from high voltages.
Later, vacuum tube
A vacuum tube, electron tube, valve (British usage), or tube (North America), is a device that controls electric current flow in a high vacuum between electrodes to which an electric potential difference has been applied.
The type known as ...
s were used. These were very fast, but generated heat, and were unreliable because the filaments would burn out. Fan-outs were typically 5 to 7, limited by the heating from the tubes' current. In the 1950s, special computer tubes were developed with filaments that omitted volatile elements like silicon. These ran for hundreds of thousands of hours.
The first semiconductor
A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. ...
logic family was resistor–transistor logic Resistor–transistor logic (RTL) (sometimes also transistor–resistor logic (TRL)) is a class of digital circuits built using resistors as the input network and bipolar junction transistors (BJTs) as switching devices. RTL is the earliest class ...
. This was a thousand times more reliable than tubes, ran cooler, and used less power, but had a very low fan-out of 3. Diode–transistor logic
Diode–transistor logic (DTL) is a class of digital circuits that is the direct ancestor of transistor–transistor logic. It is called so because the logic gating function (e.g., AND) is performed by a diode network and the amplifying function ...
improved the fan-out up to about 7, and reduced the power. Some DTL designs used two power-supplies with alternating layers of NPN and PNP transistors to increase the fan-out.
Transistor–transistor logic Transistor–transistor logic (TTL) is a logic family built from bipolar junction transistors. Its name signifies that transistors perform both the logic function (the first "transistor") and the amplifying function (the second "transistor"), as o ...
(TTL) was a great improvement over these. In early devices, fan-out improved to 10, and later variations reliably achieved 20. TTL was also fast, with some variations achieving switching times as low as 20 ns. TTL is still used in some designs.
Emitter coupled logic
In electronics, emitter-coupled logic (ECL) is a high-speed integrated circuit bipolar transistor logic family. ECL uses an overdriven bipolar junction transistor (BJT) differential amplifier with single-ended input and limited emitter current to ...
is very fast but uses a lot of power. It was extensively used for high-performance computers, such as the Illiac IV
The ILLIAC IV was the first massively parallel computer. The system was originally designed to have 256 64-bit floating point units (FPUs) and four central processing units (CPUs) able to process 1 billion operations per second. Due to budget cons ...
, made up of many medium-scale components.
By far, the most common digital integrated circuits built today use CMOS logic
Complementary metal–oxide–semiconductor (CMOS, pronounced "sea-moss", ) is a type of metal–oxide–semiconductor field-effect transistor (MOSFET) fabrication process that uses complementary and symmetrical pairs of p-type and n-type MOSFE ...
, which is fast, offers high circuit density and low power per gate. This is used even in large, fast computers, such as the IBM System z
IBM Z is a family name used by IBM for all of its z/Architecture mainframe computers.
In July 2017, with another generation of products, the official family was changed to IBM Z from IBM z Systems; the IBM Z family now includes the newest mod ...
.
Recent developments
In 2009, researchers discovered that memristor
A memristor (; a portmanteau of ''memory resistor'') is a non-linear two-terminal electrical component relating electric charge and magnetic flux linkage. It was described and named in 1971 by Leon Chua, completing a theoretical quartet of fu ...
s can implement a boolean state storage and provides a complete logic family with very small amounts of space and power, using familiar CMOS semiconductor processes.
The discovery of superconductivity has enabled the development of rapid single flux quantum
In electronics, rapid single flux quantum (RSFQ) is a digital electronic device that uses superconducting devices, namely Josephson junctions, to process digital signals. In RSFQ logic, information is stored in the form of magnetic flux quanta ...
(RSFQ) circuit technology, which uses Josephson junction
In physics, the Josephson effect is a phenomenon that occurs when two superconductors are placed in proximity, with some barrier or restriction between them. It is an example of a macroscopic quantum phenomenon, where the effects of quantum mec ...
s instead of transistors. Most recently, attempts are being made to construct purely optical computing
Optical computing or photonic computing uses light waves produced by lasers or incoherent sources for data processing, data storage or data communication for computing. For decades, photons have shown promise to enable a higher bandwidth than the ...
systems capable of processing digital information using nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
optical elements.
See also
* De Morgan's laws
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British math ...
* Logical effort The method of logical effort, a term coined by Ivan Sutherland and Bob Sproull in 1991, is a straightforward technique used to estimate delay in a CMOS circuit. Used properly, it can aid in selection of gates for a given function (including the nu ...
* Logic optimization
Logic optimization is a process of finding an equivalent representation of the specified logic circuit under one or more specified constraints. This process is a part of a logic synthesis applied in digital electronics and integrated circuit d ...
* Microelectronics
Microelectronics is a subfield of electronics. As the name suggests, microelectronics relates to the study and manufacture (or microfabrication) of very small electronic designs and components. Usually, but not always, this means micrometre-s ...
* Unconventional computing
Unconventional computing is computing by any of a wide range of new or unusual methods. It is also known as alternative computing.
The term ''unconventional computation'' was coined by Cristian S. Calude and John Casti and used at the First In ...
Notes
References
Further reading
* Douglas Lewin, ''Logical Design of Switching Circuits'', Nelson,1974.
* R. H. Katz, ''Contemporary Logic Design'', The Benjamin/Cummings Publishing Company, 1994.
* P. K. Lala, ''Practical Digital Logic Design and Testing'', Prentice Hall, 1996.
* Y. K. Chan and S. Y. Lim, Progress In Electromagnetics Research B, Vol. 1, 269–290, 2008, "Synthetic Aperture Radar (SAR) Signal Generation, Faculty of Engineering & Technology, Multimedia University, Jalan Ayer Keroh Lama, Bukit Beruang, Melaka 75450, Malaysia.
External links
Digital Circuit Projects: An Overview of Digital Circuits Through Implementing Integrated Circuits
(2014)
*
MIT OpenCourseWare introduction to digital design class materials ("6.004: Computation Structures")
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Electronic design
Electronic design automation