Digital electronics and digital (electronic) circuits are electronics that handle digital signals (discrete bands of analog levels). All levels within a band of values represent the same information state . In most cases, the two voltage bands are one and the same voltage. These correspond to the false and true values of the Boolean domain . Because of this discretization , relatively small changes to the analog signal levels of the discrete envelope,
In contrast, analog circuits operate one analog signals Whose performance is more subject to manufacturing tolerance , signal attenuation and noise . Digital techniques are useful because it is easy to get an electronic device to switch into one of a number of known states.
Digital electronic circuits are usually made from wide assemblies of logic gates (Often printed on integrated circuits ), simple electronic representations of Boolean logic functions . 
The binary number system was refined by Gottfried Wilhelm Leibniz (published in 1705) and he also established that by using the binary system, the principles of arithmetic and logic could be combined. Digital logic as we know it was the brain-child of George Boole , in the mid 19th century. Boole died young, but his ideas lived on. In an 1886 letter, Charles Sanders Peirce described how logical operations could be carried out by electrical switching circuits.  Eventually, vacuum tubes replaced relays for logic operations. Lee De Forest ‘s modification, in 1907, of the Fleming valve can be used as an AND logic gate. Ludwig Wittgenstein introduced a version of the 16-row truth table as proposition 5.101 of Tractatus Logico-Philosophicus (1921). Walther Bothe , inventor of the coincidence circuit , got part of the 1954 Nobel Prize in physics, for the first modern electronic and gate in 1924.
Mechanical analog computers started appearing in the first century and were later used in the medieval era for astronomical calculations. In World War II , mechanical analog applications were used for specialized military applications such as calculating torpedo aiming. During this time the first electronic digital computers were developed. Originally they were the size of a large room, consuming as much power as several hundred modern personal computers (PCs). 
The Z3 was an electromechanical computer designed by Konrad Zuse , finished in 1941. It was the world’s first programmable , fully automatic digital computer.  Its operation was facilitated by the invention of the vacuum tube in 1904 by John Ambrose Fleming .
Purely electronic circuit elements and their analog and digital counterparts. The bipolar junction transistor was invented in 1947. From 1955 onwards transistors replaced vacuum tubes in computer designs, giving rise to the “second generation” of computers.
Compared to vacuum tubes, transistors have many advantages: they are smaller, and require less power than vacuum tubes, so give off less heat. Silicon junction transistors were much more reliable than vacuum tubes and had longer, indefinite, service life. Transistorized computers could contain a large amount of space.
At the University of Manchester , a team under the leadership of Tom Kilburn designed and built a machine using the newly developed transistors instead of valves.  Their first transistorised computer and the first in the world, was operational by 1953 , and a second version was completed there in April 1955.
While working at Texas Instruments , Jack Kilby Recorded His initial ideas concernant the integrated system in July 1958 successfully integrated Demonstrating the first working example is 12 September 1958.  This new technology allowed for quick, low-cost production of complex circuits by Having a set of electronic circuits on one small plate (“chip”) of semiconductor material , normally silicon .
In the early days of simple integrated circuits, the technology was large scale limited each chip to a few transistors, and the low degree of integration meant the design process was relatively simple. Manufacturing yields were also low by today’s standards. As the technology progressed, millions, then trillions  of transistors could be placed on one chip, and good designs required thorough planning, giving rise to new design methods .
An advantage of digital circuits when compared to analog circuits is that it can be transmitted without degradation due to noise .  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. An hour of music Can Be Stored were compact disc using about 6 trillion binary digits.
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 contents, each digit is handled by the Sami kind of hardware, resulting and in year 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 .
Computer-controlled digital systems can be controlled by software, allowing new functions to be added without changing hardware. Often this can be done outside the factory by updating the product’s software. So, the product’s design errors can be corrected 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 degrades 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 as well as the inclusion of heat sinks. In portable or battery-powered systems.
For example, battery-powered cellular phones often use a low-power analog-to- amplify and tune in the radio signals from the base station. However, a very flexible radio software . Such base stations can be easily reprogrammed to process the signals used in new cellular standards.
Digital circuits are sometimes more expensive, especially in small quantities.
Most 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 shifts to the desired degree of fidelity . The Nyquist-Shannon sampling theorem 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. Because of the cliff effect , 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 at least ask for a new copy of the data. In a state-machine, the state transition can be designed to catch unused states and trigger a reset sequence or other error recovery routine.
Digital transmission and transmission systems.
On the other hand, some techniques used in digital systems make those systems more vulnerable to single-bit errors. These techniques are acceptable when the underlying bits are reliable enough that such errors are highly unlikely.
A single-bit error in audio data stored directly as a linear pulse code modulation (such as on a CD-ROM ) causes, at worst, a single click. Instead, many people use audio compression to save storage space and download time, even though a single-bit error may corrupt the entire song.
A digital circuit is typically constructed from small electronic circuits called logical gates that can be used to create combinational logic . 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 purpose thermionic valves -have seen historic use. The output of a logic gate can, in turn, control or feed into more logic gates.
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 designed by engineers using electronic design automation software (see below for more information) to perform some type of function.
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.
Each logic symbol is represented by a different shape. The actual set of shapes was introduced in 1984 under IEEE / ANSI standard 91-1984. “The logic symbol given under this standard is being increasingly used now and have even started appearing in the literature published by the manufacturers of digital integrated circuits.” 
Another form of digital circuit is constructed from lookup tables, (many sold as ” programmable logic devices “, though other kinds of PLDs exist). Lookup tables can perform the same functions as machines based on logic gates, but can easily be 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 devices are often the preferred solution. They are usually designed by engineers using electronic design automation software.
When the volumes are medium to broad, and the logic can be slow, gold Involves complex algorithms or sequences, Often a small microcontroller is programmed to make an embedded system . These are usually programmed by software engineers .
When it comes to controlling, the conventional solution is a programmable logic controller , or PLC. These are usually programmed by electricians, using ladder logic .
Structure of digital systems
Engineers use many methods to minimize the complexity. When the complexity is less, the circuit also has fewer errors and less electronics, and is therefore less expensive.
The most widely used simplification is a minimization algorithm like the Espresso heuristic logic minimizer [ needs update ] within a CAD system, although historically, binary decision diagrams , an automated Quine-McCluskey algorithm , truth tables , Karnaugh maps , and Boolean algebra have been used .
Representations are crucial to an engineer’s design of digital circuits. Some analysis methods only work with particular representations.
The classical way to represent a digital circuit with an equivalent set of logic gates . Another way, often with the least electronics, is to construct an equivalent system of electronic switches (usually transistors ). One of the easiest ways is to simply have a memory containing a truth table. The inputs are fed into the memory address and the outputs of the memory become the outputs.
For automated analysis, these representations have digital file formats that can be processed by computer programs. Most digital engineers are very careful to select computer programs (“tools”) with compatible file formats.
Combinational vs. Sequential
To choose representations, engineers consider types of digital systems. Most digital systems divide into ” combinational systems ” and ” sequential systems .” A combinational system always presents the same output when given the same inputs. It is basically a representation of a set of logic functions, as already discussed.
A sequential system is a combinational system with some 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 digit or ” bit “.
Sequential systems are often designed as state machines . In this way, engineers can design a system of 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 at once, when a “clock” signal changes state. “Asynchronous” sequential systems . Synchronous sequential systems are made of well-asynchronous circuits such as flip-flops, which change only when the clock changes, and which have carefully designed timing margins.
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.” Each time a ticks signal, the state register captures the feedback from the previous state of the combinational logic, and feeds it back to an unchanging input to the combinational part of the state machine. The fastest rate of the clock is set by the most time-consuming logic calculation in the combinational logic.
The state register is just a representation of a binary number. If the states in the state machine are numbered (easy to arrange), the logic function is somewhat similar to that of the next state.
As of 2014, most digital logic is synchronous because it is easier to create and verify a synchronous design. However, asynchronous logic is thought to be superior because its speed is not 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.
Nevertherless, most systems require synchronous logic circuits. These are inherently asynchronous in their design and must be analyzed as such. Examples of widely used asynchronous circuits include synchronizer flip-flops, switch debouncers 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. Then 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 such 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
Many digital systems are data flow machines. These are usually designed using synchronous register transfer logic , using hardware description languages such as VHDL or Verilog .
In register transfer logic, binary numbers are Stored in groups of flip flops called Expired registers . The outputs of each register are a bundle of wires called a ” bus ” 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 on its input, so it can store a number from any one of several buses. Alternatively, the outputs of several items may be connected to a bus through buffers that can turn off the output of all the devices except one. A sequential state machine controls when each register accepts new data from its input.
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 a set of data flows. In each step of the flow, an asynchronous “synchronization circuit” determines when the outputs of that step are valid, and present a signal that says, “grab the data” It turns out that just a few relatively simple synchronization circuits are needed.
The most general-purpose register-transfer logic machine is a computer . This is basically an automatic binary abacus . The control unit of a computer is usually designed as a microprogram run by a microsequencer . A microprogram is much like a player-piano roll. Each table entry or “word” 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 and other parts of the computer, including the microsequencer itself. AT ”
In this way, the complex task of designing the controls of a computer is reduced to a simpler task.
Almost all computers are synchronous. However, true asynchronous computers have also been designed. One example is the Aspida DLX core.  Another was offered by ARM Holdings . Speed advantages have not materialized, because modern computer designs already run at the speed of their slowest component, usually memory. These do use somewhat less power because a clock distribution network is not needed. An anxpected advantage is that asynchronous computers do not produce spectrally-pure radio noise, so they are used in some 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 is a specialized engineering activity that tries to arrange the registers, calculation logic, buses and other parts of the computer in the best way for some purpose. Computer architects have applied large quantities of computer-generated computer systems to reduce the cost and increase the speed and immunity to programming errors of computers. An increasingly common goal is to reduce the power used in a battery-powered computer system, such as a cell phone. Many computer architects serve an extended apprenticeship as microprogrammers.
Design issues in digital circuits
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Digital circuits are made from analog components. The design must ensure that the analog nature of the components does not dominate the desired digital behavior. Parasitic inductors and capacitances, and filter power connections.
Badges have intermittent problems such as “glitches”, vanishingly fast pulses that may trigger some logic but not others, ” runt pulses ” that do not reach valid “threshold” voltages, or unexpected (“undecoded”) combinations of logic states.
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 Where a change to the input Violates the set-up time for a digital input latch. This situation will be self-resolve, but it will take a random time, and while it persists can result in invalid signals being propagated within the digital system for a short time.
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. On the other hand, in the high-precision domain (for example, where 14 or more bits of precision are needed), analog circuits require much more power and area than digital equivalents.
Automated design tools
To save costly engineering effort, much of the effort of designing large logic machines has been automated. The computer programs are called ” electronic design automation tools” or just “EDA.”
Simple truth table-style descriptions of what are the outputs of a product. The most common example of this kind of software is the Espresso heuristic logic minimizer .
Most practical algorithms for Optimizing broad logic systems use algebraic manipulations or binary decision diagrams , and there are promising experiments with genetic algorithms and annealing optimizations .
To automate costly engineering processes, some EDA can take state tables that describe state machines and automatically produce a table or a function table for the combinational logic 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 the belonging output signals.
It is common for the function tables of such computer-generated state-machines to be optimized with minimal software such as Minilog .
Often, real logic systems are designed as a series of sub-projects, which are combined using a “tool flow.” The tool flow is usually a “script,” a simplified computer language that can invoke the software design tools in the right order.
Tool flows for large logic systems, such as microprocessors, can combine the work of hundreds of engineers.
Writing and debugging tools are an integral part of digital designs. The tool for the termination of a computer in a computer system. Often it consists of instructions to draw the transistors and wires on an integrated circuit or a printed circuit board .
Parts of tool flows are “debugged” by verifying the outputs of simulated logic. 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 correct, the design itself must be verified for correctness. Some tool flows verify designs by first producing a design, and then scanning the design to produce compatible input data for the tool flow. If the scanned data matches the input data, then the tool is probably not introduced errors.
The functional verification data are usually called “test vectors”. The functional test vectors may be preserved and used in the factory to test that newly constructed logic works correctly. However, functional test patterns do not discover common manufacturing faults. Production tests are often designed by software tools called ” test pattern generators “. These generate test vectors by examining the structure of the logic and systematically generating tests for particular faults. This way the fault coverage can approach 100% étroitement, Provided the design is testable Properly made (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 the ability to expose the photoresist. Manufacturability software adds interference patterns to the 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 properly designed circuit are manufactured, it is essential to test each manufacturing process. 
A large logic machine (say, with more than a hundred logical variables) can have an astronomical number of possible states. Obviously, in the factory, testing every state is impractical if testing each state takes a microsecond, and there are more states than the number of microseconds since the universe began. Unfortunately, this ridiculous-sounding case is typical.
Fortunately, large logic machines are almost always designed as assemblies of smaller logic machines. To save time, the smaller sub-machines are permanently installed and are tested independently.
One common test scheme known as “scan design” moves test bits serially (one after another) from external test equipment through one or more serial shift registers known as “scan chains”. Serial scans have the data and the 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 and capture the test Result in flip-flops and / or latches in the scan shift register (s). Finally, the result of the test is shifted to the block boundary and against the predicted “good machine” result.
In this paper, we present the results of the JTAG study .
Another common test scheme provides a test mode that forces some of the logic machine to enter a “test cycle.” The test cycle usually exercises large independent parts of the machine.
Several numbers determine the practicality of a system of digital logic: cost, reliability, fanout and speed. Engineers explored numerous electronic devices to get an ideal combination of these features.
The cost of a logic gate is crucial, because it is necessary to build a computer or other advanced system. Since the majority of a digital computer is simply an interconnected network of logic gates, the overall cost of building a computer correlates strongly with the price per logic gate. In the 1930s, the earliest digital logic systems were constructed from telephone relays because these were inexpensive and relatively reliable. After that, engineers always used the cheapest available electronic switches that could still fulfill the requirements.
The earliest integrated circuits were a happy accident. They Were constructed not to save money, order 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 $ 50 (in 1960 dollars, when an engineer earned $ 10,000 / year). To everyone’s surprise, by the time the circuits were mass-produced, they had become the least-expensive method of constructing digital logic. Improvements in this technology have driven all subsequent improvements in cost.
With the rise of integrated circuits , the absolute number of chips used represents 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. A major motive for reducing component is to reduce the manufacturing defect rate and increase reliability, as every soldered connection is a potentially bad one, so the defect and failure rates tends to increase along with the total number of component pins.
For example, in some logic families, NAND gates are the simplest digital gate to build. All other logical operations can be implemented by NAND gates. NAND gates, then the remaining gates could be used to implement other logical operations like logical and . This could eliminate the need for a separate chip containing those different types of gates.
The “reliability” of a logic gate describes its mean time between failure (MTBF). Digital machines often have millions of logic gates. Also, most digital machines are “optimized” to reduce their cost. The result is that the failure of a single machine will cause a digital machine to stop working. It is possible to design machines to be more reliable by using redundant logic which will not malfunction as a result of the failure of any single gate, Raises the financial cost and also increases the power of the machine and may increase the power it consumes.
Digital machines first become useful when the MTBF for a switch has 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 (8.2 · 10 10 hours),  and need them because they have so many logic gates.
Fanout 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 fanout is about five. Modern electronic logic gates using CMOS transistors for switches have fanouts near fifty, and can sometimes go much higher.
The “switching speed” describes how many times per second an inverter (an electronic representation of a “logical” function) can change from true to false and back. Faster logic can accomplish more operations in less time. Digital logic first became useful when switching speeds got above 50 Hz , because that was faster than a team of humans operating mechanical calculators. Modern electronic digital logic switches Routinely at 5 GHz (5 × 10 9 Hz), and Some laboratory systems switch at more than 1 THz (1 × 10 12 Hz).
Design started with relays . Relay logic was relatively inexpensive and reliable, but slow. Occasionally a mechanical failure would occur. Fanouts were typically about 10, limited by the resistance of the coils and arcing on the contacts from high voltages.
Later, vacuum tubes were used. These were very fast, but generated heat, and were unreliable because the filaments would burn out. Fanouts were typically 5 … 7, limited by the heating from the tubes. In the 1950s, special “computer tubes” were developed with filaments that omitted volatile elements like silicon. These are:
The first semiconductor logic family was resistor-transistor logic . This was a thousand times more reliable than tubes, ran cooler, and used less power, purpose Had a very low fan-in of 3. diode-transistor logic Improved the fanout up to about 7, and the Reduced power. Some DTL designs use two power-supplies with alternating layers of NPN and PNP transistors to increase the fanout.
Transistor-logic transistor (TTL) was a great improvement over these. In early devices, fanout 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 is very fast but uses a lot of power. It was extensively used for high-performance computers made up of many medium-scale components (such as the Illiac IV ).
By CMOS logic , which is fast, the high current density and low-power per gate. This is used Even in broad, fast computers, Such As the IBM System z .
In 2009, Researchers Discovered That memristors can Implement a boolean state storage (similar to a flip flop , involvement and logical inversion ) supplier providing a full logic family with very small water equivalent of space and power, using familiar CMOS semiconductor processes. 
Josephson junctions instead of transistors. The discovery of superconductivity has enabled the development of rapid single flux quantum (RSFQ) . Most recently, attempts are made to construct purely optical computing systems capable of processing digital information using nonlinear optical elements.
- Boolean algebra
- Combinational logic
- De Morgan’s laws
- Formal verification
- Hardware description language
- Integrated circuit
- Logical effort
- Logic family
- Logic gate
- Logic minimization
- Logic simulation
- Sequential logic
- Claude E. Shannon
- Transparent latch
- Unconventional computing
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