A digital comparator or magnitude comparator is a hardware electronic device that takes two numbers as input in binary form and determines whether one number is greater than, less than or equal to the other number. Comparators are used in central processing units (CPUs) and microcontrollers (MCUs). Examples of digital comparator include the CMOS 4063 and 4585 and the TTL 7485 and 74682.
For the equality of A and B, all variables (for i=0,1,2,3) must be 1.
So the equality condition of A and B can be implemented using the AND operation as
The binary variable (A=B) is 1 only if all pairs of digits of the two numbers are equal.
Inequality
In order to manually determine the greater of two binary numbers, we inspect the relative magnitudes of pairs of significant digits, starting from the most significant bit, gradually proceeding towards lower significant bits until an inequality is found. When an inequality is found, if the corresponding bit of A is 1 and that of B is 0 then we conclude that A>B.
This sequential comparison can be expressed logically as:
(A>B) and (A < B) are output binary variables, which are equal to 1 when A>B or A<B respectively.
alternative comparator without using XNOR (using NOR gate)
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be represented as a base-ten floating-point number:
In electronics, a multiplexer, also known as a data selector, is a device that selects between several analog or digital input signals and forwards the selected input to a single output line. The selection is directed by a separate set of digital inputs known as select lines. A multiplexer of inputs has select lines, which are used to select which input line to send to the output.
In electronics, an analog-to-digital converter is a system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a digital signal. An ADC may also provide an isolated measurement such as an electronic device that converts an analog input voltage or current to a digital number representing the magnitude of the voltage or current. Typically the digital output is a two's complement binary number that is proportional to the input, but there are other possibilities.
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one).
Two's complement is a mathematical operation to reversibly convert a positive binary number into a negative binary number with equivalent value, using the binary digit with the greatest place value to indicate whether the binary number is positive or negative. It is used in computer science as the most common method of representing signed integers on computers, and more generally, fixed point binary values. When the most significant bit is a one, the number is signed as negative. (see Converting from two's complement representation, below).
In mathematics and computing, the method of complements is a technique to encode a symmetric range of positive and negative integers in a way that they can use the same algorithm (hardware) for addition throughout the whole range. For a given number of places half of the possible representations of numbers encode the positive numbers, the other half represents their respective additive inverses. The pairs of mutually additive inverse numbers are called complements. Thus subtraction of any number is implemented by adding its complement. Changing the sign of any number is encoded by generating its complement, which can be done by a very simple and efficient algorithm. This method was commonly used in mechanical calculators and is still used in modern computers. The generalized concept of the radix complement is also valuable in number theory, such as in Midy's theorem.
An adder, or summer, is a digital circuit that performs addition of numbers. In many computers and other kinds of processors adders are used in the arithmetic logic units (ALUs). They are also used in other parts of the processor, where they are used to calculate addresses, table indices, increment and decrement operators and similar operations.
A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are due to inexactness in the representation of real numbers and the arithmetic operations done with them. This is a form of quantization error. When using approximation equations or algorithms, especially when using finitely many digits to represent real numbers, one of the goals of numerical analysis is to estimate computation errors. Computation errors, also called numerical errors, include both truncation errors and roundoff errors.
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set. Alternative names are switching function, used especially in older computer science literature, and truth function, used in logic. Boolean functions are the subject of Boolean algebra and switching theory.
In Boolean algebra, any Boolean function can be expressed in the canonical disjunctive normal form (CDNF) or minterm canonical form and its dual canonical conjunctive normal form (CCNF) or maxterm canonical form. Other canonical forms include the complete sum of prime implicants or Blake canonical form, and the algebraic normal form.
Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic needed to express the languages in them. For example, PH, the union of all complexity classes in the polynomial hierarchy, is precisely the class of languages expressible by statements of second-order logic. This connection between complexity and the logic of finite structures allows results to be transferred easily from one area to the other, facilitating new proof methods and providing additional evidence that the main complexity classes are somehow "natural" and not tied to the specific abstract machines used to define them.
In digital electronics, a NAND gate (NOT-AND) is a logic gate which produces an output which is false only if all its inputs are true; thus its output is complement to that of an AND gate. A LOW (0) output results only if all the inputs to the gate are HIGH (1); if any input is LOW (0), a HIGH (1) output results. A NAND gate is made using transistors and junction diodes. By De Morgan's laws, a two-input NAND gate's logic may be expressed as A • B=A+B, making a NAND gate equivalent to inverters followed by an OR gate.
A carry-lookahead adder (CLA) or fast adder is a type of electronics adder used in digital logic. A carry-lookahead adder improves speed by reducing the amount of time required to determine carry bits. It can be contrasted with the simpler, but usually slower, ripple-carry adder (RCA), for which the carry bit is calculated alongside the sum bit, and each stage must wait until the previous carry bit has been calculated to begin calculating its own sum bit and carry bit. The carry-lookahead adder calculates one or more carry bits before the sum, which reduces the wait time to calculate the result of the larger-value bits of the adder.
XOR gate is a digital logic gate that gives a true output when the number of true inputs is odd. An XOR gate implements an exclusive or from mathematical logic; that is, a true output results if one, and only one, of the inputs to the gate is true. If both inputs are false (0/LOW) or both are true, a false output results. XOR represents the inequality function, i.e., the output is true if the inputs are not alike otherwise the output is false. A way to remember XOR is "must have one or the other but not both".
The XNOR gate is a digital logic gate whose function is the logical complement of the Exclusive OR (XOR) gate. It is equivalent to the logical connective from mathematical logic, also known as the material biconditional. The two-input version implements logical equality, behaving according to the truth table to the right, and hence the gate is sometimes called an "equivalence gate". A high output (1) results if both of the inputs to the gate are the same. If one but not both inputs are high (1), a low output (0) results.
A successive-approximation ADC is a type of analog-to-digital converter that converts a continuous analog waveform into a discrete digital representation using a binary search through all possible quantization levels before finally converging upon a digital output for each conversion.
In electronics, a subtractor – a digital circuit that performs subtraction of numbers – can be designed using the same approach as that of an adder. The binary subtraction process is summarized below. As with an adder, in the general case of calculations on multi-bit numbers, three bits are involved in performing the subtraction for each bit of the difference: the minuend, subtrahend, and a borrow in from the previous bit order position. The outputs are the difference bit and borrow bit . The subtractor is best understood by considering that the subtrahend and both borrow bits have negative weights, whereas the X and D bits are positive. The operation performed by the subtractor is to rewrite as the sum .
A negative base may be used to construct a non-standard positional numeral system. Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative—that is to say, the base b is equal to −r for some natural number r.
The Karnaugh map is a method of simplifying Boolean algebra expressions. Maurice Karnaugh introduced it in 1953 as a refinement of Edward W. Veitch's 1952 Veitch chart, which was a rediscovery of Allan Marquand's 1881 logical diagram aka Marquand diagram but with a focus now set on its utility for switching circuits. Veitch charts are also known as Marquand–Veitch diagrams or, rarely, as Svoboda charts, and Karnaugh maps as Karnaugh–Veitch maps.
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid.
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