# Needing some sum

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A half adder is used for adding together the two least ificant digits in a binary sum such as the one shown in Figure The four possible combinations of two binary digits A and B are shown in Figure In the sum shown in Figure The carry has rippled through two stages of the addition. Carry ripple, through many stages, in adder circuits generates unacceptable delays, and methods are now available to eliminate this problem. Figure The half adder is used for adding together the two least ificant bits dotted b The addition of the four possible combinations of two binary digits A and B with a carry to the next most ificant stage of addition c Truth table for the half adder d NAND implementation of the half adder e NOR implementation of the half adder.

The additions shown in Figure The columns headed A and B display every combination of the two binary digits to be added, while the third and fourth columns are the corresponding tabulations of the sum S and carry Crespectively. The Boolean equations for the sum and carry read directly from the truth table are:.

Sarah L. We begin by building a 1-bit half adder.

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As shown in Figure 5. If A and B are both 1, S is 2, which cannot be represented with a single binary digit. Instead, it is indicated with Needing some sum carry out C out in the next column. In a multi-bit adder, C out is added or carried in to the next most ificant bit. For example, in Figure 5. However, the half adder lacks a C in input to accept C out of the column. The full adderdescribed in the next section, solves this problem. TABLE 3. Example of reduction process.

The resulting array of full-adders and half-adders is shown in Figure 3. For the final 2-to-l reduction, a 7-bit CPA is needed. Reduction by columns of eight 5-bit magnitudes. Cost of reduction: 26 FAs and 4 HAs. The total delay of the scheme consists of the delay of the reduction array and the delay of the final CPA.

Since the delay of the CPA depends on the of bits, a more aggressive reduction might be applied to reduce the precision of the final adder at the expense of additional counters Exercise 3. The full-adder extends the concept of the half-adder by providing an additional carry-in Cin input, as shown in Figure 5.

This cell adds the three binary input s to produce sum and carry-out terms.

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Figure 5. One-bit full-adder cell. The truth table for this de is shown in Table 5. Table 5. One-bit full-adder cell truth table. From viewing the truth table, the Sum output is only a logic 1 when one or three but not two of the inputs is logic 1. The Boolean expression for this is in reduced form :. From viewing the truth table, the Cout output is only a logic 1 when two or three of the inputs is logic 1. This can be drawn as a circuit schematic as shown in Figure 5. One-bit full-adder circuit schematic. Any of half- Needing some sum full-adder cells can be connected together to form an n-bit addition.

In this de, there is no Cin input. In the full-adder implementation with two half-adders, the load on the carry-in is larger than the load on other als. Since the delay of the carry-out al is affected by this load, it is convenient to reduce it. Determine the effect of this modification on the delay of the carry-out al, using the characteristics of Table 2. TABLE 2. Characteristics of a family of CMOS gates. Determine the delay of a bit adder using the full-adder characteristics of Table 2.

Compare delay and size with a 2-bit carry-ripple adder implemented with radix-2 full-adders use average delays. An important logic de created from the basic logic gates is the half-adder, shown in Figure 6. This cell adds the two binary input s and produces sum and carry-out terms. The truth table for this de is shown in Table 6. Table 6. Half-adder cell truth table. The Sum output is only a logic 1 when either but not both inputs are logic This can be drawn as a logic diagram as shown in Figure 6. Figure 6. Half-adder logic diagram. A dataflow VHDL description for the one-bit half-adder that uses the two logic expressions is Needing some sum in Figure 6. Here, two expressions are placed in the architecture body one expression for the Sum output on line 17, the second for the Cout output on line VHDL code for a one-bit half-adder. VHDL test bench for a one-bit half-adder. For general addition an adder is needed that can also handle the carry input. Such an adder is called a full adder and consists of two half-adders and an OR gate in the arrangement shown in Fig.

As a carry input is not needed in the least ificant column A oB oa half-adder is sufficient for this position. All other positions require a full adder. The top row in Fig. Note that for the s chosen the addition of each column produces a carry of 1. For the addition of large s such as two bit s, 32 adders are needed; if each adder requires some 7 logic gates, about logic gates are required to add bit s.

Clearly such a complexity would be unwieldy were it not for integrated circuits, which are quite capable of implementing complex circuitry in small-scale form. Adders are the most used of the more complex operators in a typical de. In certain cases, ASIC deers sometimes employ special versions using combinations of half-adders and full-adders. This may work very efficiently in the case of a gate array device, for example, but it will typically result in a very bad FPGA implementation.

When using an adder with constants, a little thought goes a long way. In fact, a little algebra also goes a long way in FPGAs. The conditional—sum adder generates all possible sums and carries in a manner similar to the carry—select adder [ 22 ]. The conditional-sum adder uses a modified half-adder to generate sums and carries in the first phase. The second phase uses log 2 W d levels of multiplexers to conditionally combine neighboring bits into the final sum [ 36 ].

This adder can easily be pipelined by placing latches after the first phase and after every multiplexer stage. The conditional—sum adder, implemented using dynamic CMOS circuits [ 25 ], is usually faster than a carry—look—ahead adder, but both adders have complex interconnections that require large chip areas.

The conditional—sum adder should be considered as a candidate for highspeed CMOS adders. A full adder circuit is central to most digital circuits that perform addition or subtraction. It is so called because it adds together two binary digits, plus a carry-in digit to produce a sum and carry-out digit. The truth table and corresponding Karnaugh maps for it are shown in Table 4. Table 4. The truth table and Karnaugh maps for a full adder. X and Y are the two bits to be added, C in and C out the carry-in and carry-out bits, and S the sum.

Needing some sum
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