Adder (electronics)

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An adder 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 or ALU. 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.

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Although adders can be constructed for many number representations, such as binary-coded decimal or excess-3, the most common adders operate on binary numbers. In cases where two's complement or ones' complement is being used to represent negative numbers, it is trivial to modify an adder into an adder–subtractor. Other signed number representations require more logic around the basic adder.

Binary adders

Half adder

Half adder logic diagram Half Adder.svg
Half adder logic diagram
Half adder in action Halfadder.gif
Half adder in action

The half adder adds two single binary digits A and B. It has two outputs, sum (S) and carry (C). The carry signal represents an overflow into the next digit of a multi-digit addition. The value of the sum is 2C + S. The simplest half-adder design, pictured on the right, incorporates an XOR gate for S and an AND gate for C. The Boolean logic for the sum (in this case S) will be A′B + AB′ whereas for the carry (C) will be AB. With the addition of an OR gate to combine their carry outputs, two half adders can be combined to make a full adder. [1] The half adder adds two input bits and generates a carry and sum, which are the two outputs of a half adder. The input variables of a half adder are called the augend and addend bits. The output variables are the sum and carry. The truth table for the half adder is:

InputsOutputs
ABCS
0000
1001
0101
1110
Half adder using NAND gates only. Half adder using NAND gates only.jpg
Half adder using NAND gates only.

Full adder

Logic diagram for a full adder. Full-adder logic diagram.svg
Logic diagram for a full adder.
Full adder in action. A full adder gives the number of 1s in the input in binary representation. Fulladder.gif
Full adder in action. A full adder gives the number of 1s in the input in binary representation.
Schematic symbol for a 1-bit full adder with Cin and Cout drawn on sides of block to emphasize their use in a multi-bit adder 1-bit full-adder.svg
Schematic symbol for a 1-bit full adder with Cin and Cout drawn on sides of block to emphasize their use in a multi-bit adder

A full adder adds binary numbers and accounts for values carried in as well as out. A one-bit full-adder adds three one-bit numbers, often written as A, B, and Cin; A and B are the operands, and Cin is a bit carried in from the previous less-significant stage. [2] The full adder is usually a component in a cascade of adders, which add 8, 16, 32, etc. bit binary numbers. The circuit produces a two-bit output. Output carry and sum typically represented by the signals Cout and S, where the sum equals 2Cout + S.

A full adder can be implemented in many different ways such as with a custom transistor-level circuit or composed of other gates. One example implementation is with S = ABCin and Cout = (AB) + (Cin ⋅ (AB)).

In this implementation, the final OR gate before the carry-out output may be replaced by an XOR gate without altering the resulting logic. Using only two types of gates is convenient if the circuit is being implemented using simple integrated circuit chips which contain only one gate type per chip.

NOR Full adder NorAdder.svg
NOR Full adder

A full adder can also be constructed from two half adders by connecting A and B to the input of one half adder, then taking its sum-output S as one of the inputs to the second half adder and Cin as its other input, and finally the carry outputs from the two half-adders are connected to an OR gate. The sum-output from the second half adder is the final sum output (S) of the full adder and the output from the OR gate is the final carry output (Cout). The critical path of a full adder runs through both XOR gates and ends at the sum bit s. Assumed that an XOR gate takes 1 delays to complete, the delay imposed by the critical path of a full adder is equal to

The critical path of a carry runs through one XOR gate in adder and through 2 gates (AND and OR) in carry-block and therefore, if AND or OR gates take 1 delay to complete, has a delay of

The truth table for the full adder is:

InputsOutputs
ABCinCoutS
00000
00101
01001
01110
10001
10110
11010
11111

Adders supporting multiple bits

Ripple-carry adder

4-bit adder with logical block diagram shown 4-bit ripple carry adder.svg
4-bit adder with logical block diagram shown
Decimal 4-digit ripple carry adder. FA = full adder, HA = half adder. RippleCarry2.gif
Decimal 4-digit ripple carry adder. FA = full adder, HA = half adder.

It is possible to create a logical circuit using multiple full adders to add N-bit numbers. Each full adder inputs a Cin, which is the Cout of the previous adder. This kind of adder is called a ripple-carry adder (RCA), since each carry bit "ripples" to the next full adder. Note that the first (and only the first) full adder may be replaced by a half adder (under the assumption that Cin = 0).

The layout of a ripple-carry adder is simple, which allows fast design time; however, the ripple-carry adder is relatively slow, since each full adder must wait for the carry bit to be calculated from the previous full adder. The gate delay can easily be calculated by inspection of the full adder circuit. Each full adder requires three levels of logic. In a 32-bit ripple-carry adder, there are 32 full adders, so the critical path (worst case) delay is 3 (from input to carry in first adder) + 31 × 2 (for carry propagation in latter adders) = 65 gate delays. [3] The general equation for the worst-case delay for a n-bit carry-ripple adder, accounting for both the sum and carry bits, is

A design with alternating carry polarities and optimized AND-OR-Invert gates can be about twice as fast. [4]

4-bit adder with carry lookahead 4-bit carry lookahead adder.svg
4-bit adder with carry lookahead

Carry-lookahead adder

To reduce the computation time, engineers devised faster ways to add two binary numbers by using carry-lookahead adders (CLA). They work by creating two signals (P and G) for each bit position, based on whether a carry is propagated through from a less significant bit position (at least one input is a 1), generated in that bit position (both inputs are 1), or killed in that bit position (both inputs are 0). In most cases, P is simply the sum output of a half adder and G is the carry output of the same adder. After P and G are generated, the carries for every bit position are created. Some advanced carry-lookahead architectures are the Manchester carry chain, Brent–Kung adder (BKA), [5] and the Kogge–Stone adder (KSA). [6] [7]

Some other multi-bit adder architectures break the adder into blocks. It is possible to vary the length of these blocks based on the propagation delay of the circuits to optimize computation time. These block based adders include the carry-skip (or carry-bypass) adder which will determine P and G values for each block rather than each bit, and the carry-select adder which pre-generates the sum and carry values for either possible carry input (0 or 1) to the block, using multiplexers to select the appropriate result when the carry bit is known.

A 64-bit adder 64-bit lookahead carry unit.svg
A 64-bit adder

By combining multiple carry-lookahead adders, even larger adders can be created. This can be used at multiple levels to make even larger adders. For example, the following adder is a 64-bit adder that uses four 16-bit CLAs with two levels of LCUs.

Other adder designs include the carry-select adder, conditional sum adder, carry-skip adder, and carry-complete adder.

Carry-save adders

If an adding circuit is to compute the sum of three or more numbers, it can be advantageous to not propagate the carry result. Instead, three-input adders are used, generating two results: a sum and a carry. The sum and the carry may be fed into two inputs of the subsequent 3-number adder without having to wait for propagation of a carry signal. After all stages of addition, however, a conventional adder (such as the ripple-carry or the lookahead) must be used to combine the final sum and carry results.

3:2 compressors

A full adder can be viewed as a 3:2 lossy compressor: it sums three one-bit inputs and returns the result as a single two-bit number; that is, it maps 8 input values to 4 output values. Thus, for example, a binary input of 101 results in an output of 1 + 0 + 1 = 10 (decimal number 2). The carry-out represents bit one of the result, while the sum represents bit zero. Likewise, a half adder can be used as a 2:2 lossy compressor, compressing four possible inputs into three possible outputs.

Such compressors can be used to speed up the summation of three or more addends. If the addends are exactly three, the layout is known as the carry-save adder. If the addends are four or more, more than one layer of compressors is necessary, and there are various possible designs for the circuit: the most common are Dadda and Wallace trees. This kind of circuit is most notably used in multipliers, which is why these circuits are also known as Dadda and Wallace multipliers.

See also

Related Research Articles

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Adder–subtractor

In digital circuits, an adder–subtractor is a circuit that is capable of adding or subtracting numbers. Below is a circuit that does adding or subtracting depending on a control signal. It is also possible to construct a circuit that performs both addition and subtraction at the same time.

In CPU design, the use of a sum-addressed decoder (SAD) or sum-addressed memory (SAM) decoder is a method of reducing the latency of the CPU cache access and address calculation. This is achieved by fusing the address generation sum operation with the decode operation in the cache SRAM.

In Boolean algebra, any Boolean function can be put into 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.

Fredkin gate Universal reversible logic gate, applied in quantum computing

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Carry-lookahead adder

A carry-lookahead adder (CLA) or fast adder is a type of adder electronics adder used in digital logic. A carry-look ahead 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. The Kogge–Stone adder (KSA) and Brent–Kung adder (BKA) are examples of this type of adder.

Carry-skip adder

A carry-skip adder is an adder implementation that improves on the delay of a ripple-carry adder with little effort compared to other adders. The improvement of the worst-case delay is achieved by using several carry-skip adders to form a block-carry-skip adder.

Carry-select adder

In electronics, a carry-select adder is a particular way to implement an adder, which is a logic element that computes the -bit sum of two -bit numbers. The carry-select adder is simple but rather fast, having a gate level depth of .

Brent–Kung adder

The Brent–Kung adder, proposed in 1982, is an advanced binary adder design, having a gate level depth of .

Early completion is a property of some classes of asynchronous circuit. It means that the output of a circuit may be available as soon as sufficient inputs have arrived to allow it to be determined. For example, if all of the inputs to a mux have arrived, and all are the same, but the select line has not yet arrived, the circuit can still produce an output. Since all the inputs are identical, the select line is irrelevant.

XOR gate logic gate

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; 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".

Carry-save adder type of digital adder

A carry-save adder is a type of digital adder, used in computer microarchitecture to compute the sum of three or more n-bit numbers in binary. It differs from other digital adders in that it outputs two numbers of the same dimensions as the inputs, one which is a sequence of partial sum bits and another which is a sequence of carry bits.

Kogge–Stone adder

In computing, the Kogge–Stone adder is a parallel prefix form carry look-ahead adder. Other parallel prefix adders (PPA) include the Brent–Kung adder (BKA), the Han–Carlson adder (HCA), and the fastest known variation, the Lynch–Swartzlander spanning tree adder (STA).

Subtractor

In electronics, a subtractor 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 .

The serial binary adder or bit-serial adder is a digital circuit that performs binary addition bit by bit. The serial full adder has three single-bit inputs for the numbers to be added and the carry in. There are two single-bit outputs for the sum and carry out. The carry-in signal is the previously calculated carry-out signal. The addition is performed by adding each bit, lowest to highest, one per clock cycle.

Negative base

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.

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Arithmetic logic unit digital circuits

An arithmetic logic unit (ALU) is a combinational digital electronic circuit that performs arithmetic and bitwise operations on integer binary numbers. This is in contrast to a floating-point unit (FPU), which operates on floating point numbers. An ALU is a fundamental building block of many types of computing circuits, including the central processing unit (CPU) of computers, FPUs, and graphics processing units (GPUs). A single CPU, FPU or GPU may contain multiple ALUs.

Garbled circuit is a cryptographic protocol that enables two-party secure computation in which two mistrusting parties can jointly evaluate a function over their private inputs without the presence of a trusted third party. In the garbled circuit protocol, the function has to be described as a Boolean circuit.

Open Quantum Assembly Language is an intermediate representation for quantum instructions. The language was first described in a paper published in July 2017, and source code was released as part of IBM's Quantum Information Software Kit (Qiskit) for use with their IBM Q Experience cloud quantum computing platform. The language has similar qualities to traditional hardware description languages such as Verilog.

References

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  3. Satpathy, Pinaki (2016). Design and Implementation of Carry Select Adder Using T-Spice. Anchor Academic Publishing. p. 22. ISBN   978-3-96067058-2.
  4. Burgess, Neil (2011). Fast Ripple-Carry Adders in Standard-Cell CMOS VLSI. 20th IEEE Symposium on Computer Arithmetic. pp. 103–111.
  5. Brent, Richard Peirce; Kung, Hsiang Te (March 1982). "A Regular Layout for Parallel Adders". IEEE Transactions on Computers . C-31 (3): 260–264. doi:10.1109/TC.1982.1675982. ISSN   0018-9340.
  6. Kogge, Peter Michael; Stone, Harold S. (August 1973). "A Parallel Algorithm for the Efficient Solution of a General Class of Recurrence Equations". IEEE Transactions on Computers . C-22 (8): 786–793. doi:10.1109/TC.1973.5009159.
  7. Reynders, Nele; Dehaene, Wim (2015). Written at Heverlee, Belgium. Ultra-Low-Voltage Design of Energy-Efficient Digital Circuits. Analog Circuits and Signal Processing Series. Analog Circuits And Signal Processing (ACSP) (1 ed.). Cham, Switzerland: Springer International Publishing AG Switzerland. doi:10.1007/978-3-319-16136-5. ISBN   978-3-319-16135-8. ISSN   1872-082X. LCCN   2015935431.

Further reading