Digital Logic

Summary

Digital logic is electronic technology constructed using the discrete mathematical principles of Boolean algebra, which is based on binary calculation, or the base 2 counting system. The underlying principle is the relationship between two opposite states, represented by the numerals 0 and 1. The various combinations of inputs utilizing these states in integrated circuits permit the construction and operation of many devices, from simple on-off switches to the most advanced computers.

Definition and Basic Principles

Digital logic is built upon the result of combining two signals that can have either the same or opposite states, according to the principles of Boolean algebra. The mathematical logic is based on binary calculation, or the base 2 counting system. The underlying principle is the relationship between two opposite states, represented by the numerals 0 and 1.

The states are defined in modern electronic devices as the presence or absence of an electrical signal, such as a voltage or a current. In modern computers and other devices, digital logic is used to control the flow of electrical current in an assembly of transistor structures called gates. These gates accept the input signals and transform them into an output signal. An inverter transforms the input signal into an output signal of exactly opposite value.

An AND gate transforms two or more input signals to produce a corresponding output signal only when all input signals are present. An OR gate transforms two or more input signals to produce an output signal if any of the input signals are present. Combinations of these three basic gate structures in integrated circuits are used to construct NAND (or not-AND) and NOR (or not-OR) gates, accumulators, flip-flops, and numerous other digital devices that make up the functioning structures of integrated circuits and computer chips.

Background and History

Boolean algebra is named for George Boole (1815–64), a self-taught English scientist. This form of algebra was developed from Boole's desire to express concrete logic in mathematical terms; it is based entirely on the concepts of true and false. The intrinsically opposite nature of these concepts allows the logic to be applied to any pair of conditions that are related as opposites.

The modern idea of computing engines began with the work of Charles Babbage (1791–1871), who envisioned a mechanical “difference engine” that would calculate results from starting values. Babbage did not see his idea materialize, though others using his ideas were able to construct mechanical difference engines.

The development of the semiconductor junction transistor in 1947, attributed to William Shockley, John Bardeen, and Walter Brattain, provided the means to produce extremely small on-off switches that could be used to build complex Boolean circuits. This permitted electrical signals to carry out Boolean algebraic calculations and marked the beginning of what has come to be known as the digital revolution. These circuits helped produce the many modern-day devices that employ digital technology.

How It Works

Boolean Algebra. The principles of Boolean algebra apply to the combination of input signals rather than to the input signals themselves. If one associates one line of a conducting circuit with each digit in a binary number, it becomes easy to see how the presence or absence of a signal in that line can be combined to produce cumulative results. The series of signals in a set of lines provides ever larger numerical representations, according to the number of lines in the series. Because the representation is binary, each additional line in the series doubles the amount of information that can be carried in the series.

Bits and Bytes. Digital logic circuits are controlled by a clock signal that turns on and off at a specific frequency. A computer operating with a CPU (central processing unit) speed of 1 gigahertz (10₉ cycles per second) is using a clock control that turns on and off 1 billion times each second. Each clock cycle transmits a new set of signals to the CPU in accord with the digital logic circuitry. Each individual signal is called a bit (plural byte) of data, and a series of 8 bits is termed one byte of data. CPUs operating on a 16-bit system pass two bytes of data with each cycle, 32-bit systems pass four bytes, 64-bit systems pass eight bytes, and 128-bit systems pass sixteen bytes with each clock cycle.

Because the system is binary, each bit represents two different states (system high or system low). Thus, two bits can represent four (or 2₂) different states, three bits represents eight (or 2₃) different states, four bits represents sixteen (or 2₄) different states, and so on. A 128-bit system can therefore represent 2₁₂₈ or more than 3.40 × 10₃₈ different system states.

Digital Devices. All digital devices are constructed from semiconductor junction transistor circuits. This technology has progressed from individual transistors to the present technology in which millions of transistors can be etched onto a small silicon chip. Digital electronic circuits are produced in “packages” called integrated circuit, or IC, chips. The simplest digital logic device is the inverter, which converts an input signal to an output signal of the opposite value. The AND gate accepts two or more input signals such that the output signal will be high only if all of the input signals are high. The OR gate produces an output high signal if any one or the other of the input signals is high.

All other digital logic devices are constructed from these basic components. They include NAND gates, NOR gates, X-OR gates, flip-flops that produce two simultaneous outputs of opposite value, and counters and shift registers, which are constructed from series of flip-flops. Combinations of these devices are used to assemble accumulators, adders, and other components of digital logic circuits.

One other important set of digital devices is the converters that convert a digital or analog input signal to an analog or digital output signal, respectively. These find extensive use in equipment that relies on analog, electromagnetic, signal processing.

Karnaugh Maps. Karnaugh maps are essential tools in designing and constructing digital logic circuits. A Karnaugh map is a tabular representation of all possible states of the system according to Boolean algebra, given the desired operating characteristics of the system. By using a Karnaugh map to identify the allowed system states, the circuit designer can select the proper combination of logic gates that will then produce those desired output states.

Applications and Products

Digital logic has become the standard structural operating principle of most modern electronic devices, from the cheapest wristwatch to the most advanced supercomputer. Applications can be identified as programmable and nonprogrammable.

Nonprogrammable applications are those in which the device is designed to carry out a specific set of operations as automated processes. Common examples include timepieces, CD and DVD players, cellular telephones, and various household appliances. Programmable applications are those in which an operator can alter existing instruction sets or provide new ones for the particular device to carry out. Typical examples include programmable logic controllers, computers, and other hybrid devices into which some degree of programmability has been incorporated, such as gaming consoles, GPS (global positioning system) devices, and even some modern automobiles.

Digital logic is utilized for several reasons. First, the technology provides precise control over the processes to which it is applied. Digital logic circuits function on a very precise clock frequency and with a rigorously defined data set in which each individual bit of information represents a different system state that can be precisely defined millions of times per second, depending on the clock speed of the system. Second, compared with their analog counterparts, which are constructed of physical switches and relays, digital circuits require a much lower amount of energy to function. A third reason is the reduced costs of materials and components in digital technology. In a typical household appliance, all of the individual switches and additional wiring that would be required of an analog device are replaced by a single small printed circuit containing a small number of IC chips, typically connected to a touchpad and LCD (liquid crystal display) screen.

Programmable Logic Controller. One of the most essential devices associated with modern production methods is the programmable logic controller, or PLC. These devices contain the instruction set for the automated operation of various machines, such as CNC (computer numerical control) lathes and milling machines, and all industrial robotics. In operation, the PLC replaces a human machinist or operator, eliminating the effects of human error and fatigue that result in undesirable output variability. As industrial technology has developed, the precision with which automated machinery can meet demand has far exceeded the ability of human operators.

PLCs, first specified by General Motors Corporation (now Company) in 1968, are small computer systems programmed using a reduced-instruction-set programming language. The languages often use a ladder-like structure in which specific modules of instructions are stacked into memory. Each module consists of the instructions for the performance of a specific machine function. More recent developments of PLC systems utilize the same processors and digital logic peripherals as personal computers, and they can be programmed using advanced computer programming languages.

Digital Communications. Digital signal processing is essential to the function of digital communications. As telecommunication devices work through the use of various wavelengths of the electromagnetic spectrum, the process is analog in nature. Transmission of an analog signal requires the continuous, uninterrupted occupation of the specific carrier frequency by that signal, for which analog radio and television transmission frequencies are strictly regulated.

A digital transmission, however, is not continuous, being transmitted as discrete bits or packets of bits of data rather than as a continuous signal. When the signal is received, the bits are reassembled for audio or visual display. Encryption codes can be included in the data structure so that multiple signals can utilize the same frequency simultaneously without interfering with each other. Data reconstruction occurs at a rate that exceeds human perception so that the displayed signal is perceived as a continuous image or sound. The ability to interleaf signals in this way increases both the amount of data that can be transmitted in a limited frequency range and the efficiency with which the data is transmitted.

A longstanding application of digital logic in telecommunications is the conversion of analog source signals into digital signals for transmission, and then the conversion of the digital signal back into an analog signal. This is the function of digital-to-analog converters (DACs), and analog-to-digital converters (ACDs). A DAC uses sampling to measure the magnitude of the analog signal, perhaps many millions of times per second depending upon the clock speed of the system. The greater the sampling rate, the closer its digital representation will be to the real nature of the analog signal. The ACD accepts the digital representation and uses it to reconstruct the analog signal as its output.

One important problem that exists with this method is called aliasing, in which the DAC analog output correctly matches the digital representation, but at the wrong frequencies. Twenty-first century telecommunications technology has almost completely switched to an all-digital format that does not use analog signals.

Servomechanisms. Automated processes controlled by digital logic require other devices by which the function of the machine can be measured. Typically, a measurement of some output property is automatically fed back into the controlling system and used to adjust functions so that desired output parameters are maintained. The adjustment is carried out through the action of a servomechanism, a mechanical device that performs a specific action in the operation of the machine. Positions and rotational speeds are the principal properties used to gauge machine function.

In both cases, it is common to link the output property to a digital representation such as Gray code, which is then interpreted by the logic controller of the machine. The specific code value read precisely describes either the position or the rotational speed of the component, and is compared by the controller to the parameters specified in its operating program. Any variance can then be corrected and proper operation maintained.

Social Context and Future Prospects

At the heart of every consumer electronic device is an embedded digital logic device. The transistor quickly became the single most important feature of electronic technology, and it has facilitated the rapid development of everything from the transistor radio to space travel. Digital logic, as embedded devices, is becoming an ever more pervasive feature of modern technology; an entire generation has now grown up not knowing anything but digital computers and technology. Even the accoutrements of this generation are rapidly being displaced by newer versions, as tablet computers and smartphones displace more traditional desktop personal computers, laptops, and cell phones. The telecommunications industry in North America made a government-regulated switch-over to digital format in the early twenty-first century, making analog transmissions a relic of the past.

Research to produce new materials for digital logic circuits and practical quantum computers is ongoing. The eventual successful result of these efforts, especially in conjunction with the development of nanotechnology, will represent an unparalleled advance in technology that may usher in an entirely new age for human society.

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