Digital electronics
Digital electronics represent signals by discrete bands of analog levels, rather than by a continuous range. All levels within a band represent the same signal state. Relatively small changes to the analog signal levels due to manufacturing tolerance, signal attenuation or parasitic noise do not leave the discrete envelope, and as a result are ignored by signal state sensing circuitry.
In most cases the number of these states is two, and they are represented by two voltage bands: one near zero volts and a higher level near the supply voltage, corresponding to the "false" ("0") and "true" ("1") values of the boolean domain respectively.
Digital techniques are useful because it is easier to get an electronic device to switch into one of a number of known states than to accurately reproduce a continuous range of values.
Digital electronic circuits are usually made from large assemblies of logic gates, simple electronic representations of Boolean logic functions.
Advantages
One advantage of digital circuits when compared to analog circuits is [2] that signals represented digitally 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 on a compact disc as about 6 billion 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, each digit is handled by the same kind of hardware. 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 of 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 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.
Disadvantages
In some cases, digital circuits use more energy than analog circuits to accomplish the same tasks, thus producing more heat. In portable or battery-powered systems this can limit use of digital systems.
For example, battery-powered cellular telephones often use a low-power analog front-end to amplify and tune in 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 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 stores enough digital data to represent the signal 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 logic can be designed to catch unused states and trigger a reset sequence or other error recovery routine.
Digital memory and transmission systems can use techniques such as error detection and correction to use additional data to correct any errors in transmission and storage.
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 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.
Digital logic gates, which are also known as combinational logic gates or simply 'logic gates', are digital IC's whose output at any time is determined by the states of its inputs at that time. Since logic gates are digital IC's, their input and output signals can only be in one of two possible digital states, i.e., logic '0' or logic '1'. Thus, the logic state in which the output of a logic gate will be put in depends on the logic states of each of its individual inputs.
The primary application of logic gates is to implement 'logic' in the flow of digital signals in a digital circuit. Logic in its ordinary sense is defined as a branch of philosophy that deals with what is true and false, based on what other things are true and false. This essentially is the function of logic gates in digital circuits - to determine which outputs will be true or false, given a set of inputs that can either be true (logic '1') or false (logic '0').
The response output (usually denoted by Q) of a logic gate to any combination of inputs may be tabulated into what is known as a truth table. A truth table shows each possible combination of inputs to a logic gate and the combination's corresponding output. Table 1, which describes the various types of logic gates, provides a truth table for each of them as well.
Interestingly, the operation of logic gates in relation to one another may be represented and analyzed using a branch of mathematics called Boolean Algebra which, like the common algebra, deals with manipulation of expressions to solve or simplify equations. Expressions used in Boolean Algebra are called, well, Boolean expressions.
Table 1. Logic Gates and their Properties
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Gate
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Description
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Truth Table
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AND Gate
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The AND gate is a logic gate that gives an output of '1' only when all of its inputs are '1'. Thus, its output is '0' whenever at least one of its inputs is '0'. Mathematically, Q = A · B.
|
|
A
|
B
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Output Q
|
|
0
|
0
|
0
|
|
0
|
1
|
0
|
|
1
|
0
|
0
|
|
1
|
1
|
1
|
|
|
OR Gate
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The OR gate is a logic gate that gives an output of '0' only when all of its inputs are '0'. Thus, its output is '1' whenever at least one of its inputs is '1'. Mathematically, Q = A + B.
|
|
A
|
B
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Output Q
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|
0
|
0
|
0
|
|
0
|
1
|
1
|
|
1
|
0
|
1
|
|
1
|
1
|
1
|
|
|
NOT Gate
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The NOT gate is a logic gate that gives an output that is opposite the state of its input. Mathematically, Q = A.
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|
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NAND Gate
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The NAND gate is an AND gate with a NOT gate at its end. Thus, for the same combination of inputs, the output of a NAND gate will be opposite that of an AND gate. Mathematically, Q = A · B.
|
|
A
|
B
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Output Q
|
|
0
|
0
|
1
|
|
0
|
1
|
1
|
|
1
|
0
|
1
|
|
1
|
1
|
0
|
|
|
NOR Gate
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The NOR gate is an OR gate with a NOT gate at its end. Thus, for the same combination of inputs, the output of a NOR gate will be opposite that of an OR gate. Mathematically, Q = A + B.
|
|
A
|
B
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Output Q
|
|
0
|
0
|
1
|
|
0
|
1
|
0
|
|
1
|
0
|
0
|
|
1
|
1
|
0
|
|
|
EXOR Gate
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The EXOR gate (for 'EXclusive OR' gate) is a logic gate that gives an output of '1' when only one of its inputs is '1'.
|
|
A
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B
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Output Q
|
|
0
|
0
|
0
|
|
0
|
1
|
1
|
|
1
|
0
|
1
|
|
1
|
1
|
0
|
|
Logic gates may be thought of as a combination of switches. For instance, the AND gate, whose output can only be '1' if all its inputs are '1', may be represented by switches connected in series, with each switch representing an input. All the switches need to be activated and conducting (equivalent to all the inputs of the AND gate being at logic '1'), for current to flow through the circuit load (equivalent to the output of the AND gate being at logic '1').
An OR gate, on the other hand, may be represented by switches connected in parallel, since only one of these parallel switches need to turn on in order to energize the circuit load.
In Boolean Algebra, the AND operation is represented by multiplication, since the only way that the result of multiplication of a combination of 1's and 0's will be equal to '1' is if all its inputs are equal to '1'. A single '0' among the multipliers will result in a product that's equal to '0'. The Boolean expression for 'A AND B' is similar to the expression commonly used for multiplication, i.e., A·B.
The OR operation, on the other hand, is represented by addition in Booelean Algebra. This is because the only way to make the result of the addition operation equal to '0' is to make all the inputs equal to '0', which basically describes an 'OR' operation. The Boolean expression for 'A OR B' is therefore A+B.
The NOT operation is usually denoted by a line above the symbol or expression that is being negated: A = NOT(A). The NAND operation is simply an AND operation followed by a NOT operation. The NOR operation is simply an OR operation followed by a NOT operation. The symbols used for logic gates in electronic circuit diagrams are shown in Figure 1.
Figure 1. Logic Gate Symbols
One of the most useful theorems used in Boolean Algebra is De Morgan's Theorem, which states how an AND operation can be converted into an OR operation, as long as a NOT operation is available. De Morgan's Theorem is usually expressed in two equations as follows:
(A·B) = A + B; and
(A+B) = A · B.
De Morgan's Theorem has a practical implication in digital electronics - a designer may eliminate the need to add more IC's to the design unnecessarily, simply by substituting gates with the equivalent combination of other gates whenever possible. Since NAND and NOR gates can be used as NOT gates, de Morgan's Theorem basically implies that any Boolean operation may be simulated with nothing but NAND or NOR gates. This is why NAND and NOR gates are also called universal gates.
Boolean Algebra, also known as the 'algebra of logic', is a branch of mathematics that is similar in form to algebra, but dealing with logical instead of numerical relationships. It was invented by George Boole, after whom this system was named. Thus, instead of variables that represent numerical quantities as in conventional algebra, Boolean algebra handles variables that represent two types of logic propositions: 'true' and 'false'.
Boolean algebra has become the main cornerstone of digital electronics, since the latter also operates with two logic states, '1' and '0', represented by two distinct voltage levels. Boolean algebra's formal interpretation of logical operators AND, OR, and NOT has allowed the systematic development of complex digital systems from simple logic gates, that now not only include circuits that perform mathematical operations, but intricate data processing as well. Tables 1 to 4 summarize the definitions of logical operators and their basic mathematical properties as represented in Boolean algebra.
Table 1. Elementary Logic Gate Actions
|
OR
0+0=0
0+1=1
1+0=1
1+1=1
|
AND
0·0=0
0·1=0
1·0=0
1·1=1
|
NOT
0=1
1=0
|
NOR
0+0=1
0+1=0
1+0=0
1+1=0
|
NAND
0·0=1
0·1=1
1·0=1
1·1=0
|
Table 2. Single-Variable Logic Gate Actions
|
OR
A+0=A
A+1=1
A+A=A
A+A=1
|
AND
A·0=0
A·1=A
A·A=A
A·A=0
|
NOT
A=NOT(A)
NOT(A)=A
|
NOR
A+0=A
A+1=0
A+A=A
A NOR A=0
|
NAND
A·0=1
A·1=A
A·A=A
A NAND A=1
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Table 3. Multi-Variable Boolean Equalities
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A+B=B+A
(A+B)+C=A+(B+C)
A·B=B·A
(A·B)·C=A·(B·C)
A·(B+C)=A·B+A·C
|
Table 4. De Morgan's Theorem
