Exclusive XOR Gate

XOR gate (sometimes called EOR, EXOR, and pronounced as Exclusive OR) is a digital logic gate that results in true (either 1 or HIGH) output when the number of true inputs is an odd count. An XOR gate implements an exclusive OR, i.e., a true output result if one, and only one, of the gate inputs, is true. If both the inputs are false (either 0 or LOW) or if both are true, there results in a false output. The XOR represents the inequality function, that is, the output is true if the inputs are not identical, and then the output is false. An easy way to remember XOR is "must have one or the other, but not both".

XOR can also be viewed as an additional modulo 2. Resultantly, XOR gates are used to implement binary addition in computers. A half adder has an XOR gate and an AND gate. Other uses include comparators, subtractors, and controlled inverters. XOR gate is a hybrid logic gate that has 2 inputs that perform the Exclusive Disjunction operation.

The XOR gate operation is similar to the OR gate’s; few inputs vary. The output of the XOR gate is also called 'an odd function' because it gives '1' when an odd number of ones are present at the inputs. Also, an XOR gate is a combinational logic circuit that generates the parity bit in the transmitter. The XOR gate is different from the inclusive OR gate because, in inclusive OR, it allows both the possibilities (if A or B are true, or if both are true, the output value would be 'true'). Still, in XOR, it only allows one possibility (if any one of A or B is true, only then, the output will be 'true').

XOR gate comes in the category of arithmetic gates. Arithmetic gates are sets of gates that perform arithmetic and bitwise operations on binary numbers. Therefore, XOR also contributes to the functioning of ALU.

Symbol of XOR Gate

There exist three schematic symbols for XOR gates. They can be given the traditional ANSI, DIN, and IEC symbols. In a few cases, the DIN symbol is used with '⊕' instead of '#.'

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The logic symbols (⊕, Jpq, and v) can be used to represent an XOR operation in algebraic expressions.

The C-like languages use the caret symbol (^) to represent a bitwise XOR. (It is to remember that the caret does not denote the logical conjunction (AND) in these languages, despite the similarity of the symbol).

XOR Truth Tables

There are two kinds of truth XOR Table, which are with 2 inputs and 3 inputs. Let us have a look at both.

Truth Table of 2 input XOR Gate

The Boolean expression representing the 2

input XOR gate is written as, Y = (A⊕B) = A.B + A. B

XOR Gate with inputs A and B

Here, we can observe that the XOR gate gives output as true only if only one of the inputs is 'true.’ Otherwise, the output delivered will be 'false' or '0'.

(The above algebraic expression represents the XOR gate with inputs A and B.)

The XOR Gate Truth Table for 3 Input XOR Gate

If an XOR gate were to accept three or more inputs and form and if those inputs were true, then it would, in effect, be a one-hot detector (indeed, this case is only for two inputs). However, it is rarely implemented this way in practice.

The Boolean expression for the three-input XOR logic gate is represented by,

Y = A⊕B⊕C = ABC + ABC + ĀBC + ABC

(The above algebraic expression represents the XOR gate with inputs A, B, and C.)

XOR Gate with Inputs A, B, and C

Note:

1. A⊕A = 0

2. А ⊕ A'  = 1

3. A ⊕ 0 = A

4. A⊕1 = A

here, A' is read as 'not A'

XOR Gate Equivalent Circuit

We can produce XOR gates using other logic gates like OR, AND, NOT, NOR, NAND gates. These can be done by connecting the following gates in suitable forms. Also, an XOR gate can be made by combining 2 NOT gates, 2 AND gates, and 1 OR gate.

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The figure represented above used three different logic gates to form an XOR, which is one of the significant disadvantages. Besides, one of the easier ways to form the XOR gate is only to use the combination of the NOR gate and the NAND gate.

Construction of XOR Gate Using NAND Gate

By connecting 4 NAND gates, we can make an XOR gate.

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Construction of XOR Gate Using NOR Gate

We can make an XOR gate by connecting a minimum of 5 NOR gates. The desired output after connecting NOR gates in the following way would be Y = (AB' + A'B)

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Applications of XOR Gate

There are numerous counts of various applications of the XOR gate. Let us have a look at a few of them.

Uses in Addition

This is one of the applications of the XOR gate, and it can be used as a one-bit adder that adds any of the two bits together to produce one bit as output. For instance, if we add 1 plus 1 in binary, we can expect a two-bit answer, as 10 (it means 2 in decimal). Exclusive OR gate also has wide use in sequential and combinational circuits. 

Because the trailing sum bit in this output has been achieved with the XOR gate, the preceding carry bit is calculated using an AND gate. This is the primary principle in Half Adders. A slightly larger Full Adder circuit can be chained together to add longer binary numbers.

The example of a half adder circuit diagram is shown below.

Pseudo-Random Number Generator

The PRN (Pseudo-Random Number)

Generators, specifically called Linear feedback shift registers, are described as exclusive-or operation. Thus, a suitable setup of XOR gates can model a linear feedback shift register to generate random numbers.

Correlation and Sequence Detection

The XOR gates form a 0 output when both inputs are matched. When searching for a PRN sequence or a specific bit pattern in a very long data sequence, an XOR gates series can be used to compare a string of bits from the available data sequence against the target sequence in parallel.

Then, the number of 0 outputs can be counted to determine how the data sequence matches the target sequence well. Correlators are used in various communications devices such as the CDMA receivers and decoders for channel codes and error correction. In a CDMA receiver, the correlators are used to extract a particular PRN sequence from a combined collection of the PRN sequences.

An example of a full adder circuit diagram can be represented as below.

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Binary to Gray / Gray to Binary Conversion

As gray code is unweighted and difficult to use in arithmetic operations. So, XOR is used in binary to gray code / gray to binary code conversion.

In Dealing with a Basic Logic Comparator

As an XOR gate can generate a logic '1' output (when its 2 input bites are unequal). In this case, it can be used in dealing with a basic logic comparator.

Combinational Logic Circuit Minimization

XOR gates can be used to process the given two or more inputs such as to generate at least one output signal based on the logic function of each logic gate.

Some More Applications of XOR Gate are

When its two input bits are unequal, an XOR gate is typically found in applications dealing with a basic logic comparator that generates a logic "1" output (that represents 'high' or 'true')

• XOR logic acts as a simple system for encrypting and decrypting data, i.e., performing an XOR of a digitized message with a binary key produces encrypted ciphertext. Hence, XOR gates are a fundamental building block of cryptographic circuits.

• XOR gate is mainly used in the operation of half and full adders.

• The concept of XOR gate is also used in the implementation of microprocessors and circuits for digital signal processing.