| Input | Output | ||
| A | B | A XOR B | |
XOR gate (sometimes EOR, or EXOR and pronounced as Exclusive OR) is a digital logic gate that gives a true (1 or HIGH) output when the number of true inputs is odd. An XOR gate implements an exclusive or (
\nleftrightarrow
An XOR gate may serve as a "programmable inverter" in which one input determines whether to invert the other input, or to simply pass it along with no change. Hence it functions as a inverter (a NOT gate) which may be activated or deactivated by a switch.[1] [2]
XOR can also be viewed as addition modulo 2. As a result, XOR gates are used to implement binary addition in computers. A half adder consists of an XOR gate and an AND gate. The gate is also used in subtractors and comparators.[3]
A ⋅ \overline{B}+\overline{A} ⋅ B
(A+B) ⋅ (\overlineA+\overlineB)
(A+B) ⋅ \overline{(A ⋅ B)}
A ⊕ B
There are three schematic symbols for XOR gates: the traditional ANSI and DIN symbols and the IEC symbol. In some cases, the DIN symbol is used with ⊕ instead of ≢. For more information see Logic Gate Symbols.
The "=1" on the IEC symbol indicates that the output is activated by only one active input.
The logic symbols ⊕, Jpq, and ⊻ can be used to denote an XOR operation in algebraic expressions.
C-like languages use the caret symbol ^ to denote bitwise XOR. (Note that the caret does not denote logical conjunction (AND) in these languages, despite the similarity of symbol.)
The XOR gate is most commonly implemented using MOSFETs circuits. Some of those implementations include:
XOR gates can be implemented using AOI logic. [4]
The metal–oxide–semiconductor (CMOS) implementations of the XOR gate corresponding to the AOI logic aboveare shown below.
On the left, the nMOS and pMOS transistors are arranged so that the input pairs
A ⋅ \overline{B}
\overline{A} ⋅ B
A ⋅ B
\overline{A} ⋅ \overline{B}
If inverted inputs (for example from a flip-flop) are available, this gate can be used directly. Otherwise, two additional inverters with two transistors each are needed to generate
\overline{A}
\overline{B}
The AOI implementation without inverted input has been used, for example, in the Intel 386 CPU.[6]
The XOR gate can also be implemented by the use of Transmission gates with pass transistor logic.
This implementation uses two Transmission gates and two inverters not shown in the diagram to generate
\overline{A}
\overline{B}
\overline{A}
The trade-off with the previous implementation is that since transmission gates are not ideal switches, there is resistance associated with them, so depending on the signal strength of the input, cascading them may degrade the output levels.
The previous transmission gate implementation can be further optimized from eight to six transistors by implementing the functionality of the inverter that generates
\overline{A}
B
\overline{B}
The two leftmost transistors mentioned above, perform an optimized conditional inversion of A when B is at a logic high using pass transistor logic to reduce the transistor count and when B is at a logic low, their output is at a high impedance state. The two in the middle are a transmission gate that drives the output to the value of A when B is at a logic low and the two rightmost transistors form an inverter needed to generate
\overline{B}
As with the previous implementation, the direct connection of the inputs to the outputs through the pass gate transistors or through the two leftmost transistors, should be taken into account, especially when cascading them.
If a specific type of gate is not available, a circuit that implements the same function can be constructed from other available gates. A circuit implementing an XOR function can be trivially constructed from an XNOR gate followed by a NOT gate. If we consider the expression
(A ⋅ \overline{B})+(\overline{A} ⋅ B)
As alternative, if different gates are available we can apply Boolean algebra to transform
(A ⋅ \overline{B})+(\overline{A} ⋅ B)\equiv(A+B) ⋅ (\overlineA+\overlineB)
(A+B) ⋅ \overline{(A ⋅ B)}
An XOR gate circuit can be made from four NAND gates. In fact, both NAND and NOR gates are so-called "universal gates" and any logical function can be constructed from either NAND logic or NOR logic alone. If the four NAND gates are replaced by NOR gates, this results in an XNOR gate, which can be converted to an XOR gate by inverting the output or one of the inputs (e.g. with a fifth NOR gate).
An alternative arrangement is of five NOR gates in a topology that emphasizes the construction of the function from
(A+B) ⋅ (\overlineA+\overlineB)
(A ⋅ \overline{B})+(\overline{A} ⋅ B)
For the NAND constructions, the upper arrangement requires fewer gates. For the NOR constructions, the lower arrangement offers the advantage of a shorter propagation delay (the time delay between an input changing and the output changing).
XOR chips are readily available. The most common standard chip codes are:
Literal interpretation of the name "exclusive or", or observation of the IEC rectangular symbol, raises the question of correct behaviour with additional inputs.[12] If a logic gate were to accept three or more inputs and produce a true output if exactly one of those inputs were true, then it would in effect be a one-hot detector (and indeed this is the case for only two inputs). However, it is rarely implemented this way in practice.
It is most common to regard subsequent inputs as being applied through a cascade of binary exclusive-or operations: the first two signals are fed into an XOR gate, then the output of that gate is fed into a second XOR gate together with the third signal, and so on for any remaining signals. The result is a circuit that outputs a 1 when the number of 1s at its inputs is odd, and a 0 when the number of incoming 1s is even. This makes it practically useful as a parity generator or a modulo-2 adder.
For example, the 74LVC1G386 microchip is advertised as a three-input logic gate, and implements a parity generator.[13]
XOR gates and AND gates are the two most-used structures in VLSI applications.[14]
The XOR logic gate can be used as a one-bit adder that adds any two bits together to output one bit. For example, if we add 1 plus 1 in binary, we expect a two-bit answer, 10 (i.e. 2 in decimal). Since the trailing sum bit in this output is achieved with XOR, the preceding carry bit is calculated with an AND gate. This is the main principle in Half Adders. A slightly larger Full Adder circuit may be chained together in order to add longer binary numbers.
In certain situations, the inputs to an OR gate (for example, in a full-adder) or to an XOR gate can never be both 1's. As this is the only combination for which the OR and XOR gate outputs differ, an OR gate may be replaced by an XOR gate (or vice versa) without altering the resulting logic. This is convenient if the circuit is being implemented using simple integrated circuit chips which contain only one gate type per chip.
Pseudo-random number (PRN) generators, specifically linear-feedback shift registers (LFSR), are defined in terms of the exclusive-or operation. Hence, a suitable setup of XOR gates can model a linear-feedback shift register, in order to generate random numbers.
XOR gates may be used in simplest phase detectors.[15]
An XOR gate may be used to easily change between buffering or inverting a signal. For example, XOR gates can be added to the ouput of a seven-segment display decoder circuit to allow a user to choose between active-low or active-high output.
XOR gates produce a 0 when both inputs match. When searching for a specific bit pattern or PRN sequence in a very long data sequence, a series of XOR gates can be used to compare a string of bits from the data sequence against the target sequence in parallel. The number of 0 outputs can then be counted to determine how well the data sequence matches the target sequence. Correlators are used in many communications devices such as CDMA receivers and decoders for error correction and channel codes. In a CDMA receiver, correlators are used to extract the polarity of a specific PRN sequence out of a combined collection of PRN sequences.
A correlator looking for 11010 in the data sequence 1110100101 would compare the incoming data bits against the target sequence at every possible offset while counting the number of matches (zeros):
1110100101 (data) 11010 (target) 00111 (XOR) 2 zero bits 1110100101 11010 00000 5 zero bits 1110100101 11010 01110 2 zero bits 1110100101 11010 10011 2 zero bits 1110100101 11010 01000 4 zero bits 1110100101 11010 11111 0 zero bits Matches by offset: . : : : : : : : ----------- 0 1 2 3 4 5
In this example, the best match occurs when the target sequence is offset by 1 bit and all five bits match. When offset by 5 bits, the sequence exactly matches its inverse. By looking at the difference between the number of ones and zeros that come out of the bank of XOR gates, it is easy to see where the sequence occurs and whether or not it is inverted. Longer sequences are easier to detect than short sequences.
f(a,b)=a+b-2ab
f(0,0)=0+0-2 ⋅ 0 ⋅ 0=0
f(0,1)=0+1-2 ⋅ 0 ⋅ 1=1
f(1,0)=1+0-2 ⋅ 1 ⋅ 0=1
f(1,1)=1+1-2 ⋅ 1 ⋅ 1=0
f(a,b)=|a-b|