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#Quaternary half adder truth table full
For example, a four-bit binary adder would require four full adders of the type shown in Fig. A cascade arrangement of these adders can be used to construct adders capable of adding binary numbers with a larger number of bits.
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However, a single full adder circuit can be used to add one-bit binary numbers only. The full adder of the type described above forms the basic building block of binary adders. 3.7(a) redrawn with the portion of the circuit representing a half-adder replaced with a block. Figure 3.7(a) shows logic implementation of Equations above. The whole idea of writing the Boolean expressions in this modified form was to demonstrate the use of a half-adder circuit in building a full adder. The other is also the output of an AND gate whose inputs are Cin and the output of an EX-OR operation on A and B. One of them is the AND output of A and B. Similarly, Boolean expression above can be implemented by ORing two minterms. Karnaugh Map for the sum and carry out of a full adderīoolean expression above can be implemented with a two-input EX-OR gate provided that one of the inputs is Cin and the other input is the output of another two-input EX-OR gate with A and B as its inputs. Similarly, the expression for CARRY output can be rewritten as follows: The expressions for SUM and CARRY outputs can be rewritten as follows: A full adder can also be seen to comprise two half-adders and an OR gate. As is clear from the two maps, the expression for the SUM (S) output cannot be simplified any further, whereas the simplified Boolean expression for Cout is given by the equationįigure shows the logic circuit diagram of the full adder. Karnaugh maps for the two expressions are given in Fig. We will do so with the help of the Karnaugh mapping technique. The next step is to simplify the two expressions. The Boolean expressions for the two output variables are given in Equation below for the SUM output (S) and in above Equation for the CARRY output (Cout): These expressions are then simplified by using any of the simplification techniques described in the previous chapter. In order to arrive at the logic circuit for hardware implementation of a full adder, we will firstly write the Boolean expressions for the two output variables, that is, the SUM and CARRY outputs, in terms of input variables. A half-adder can be used for addition of LSBs only.įigure shows the truth table of a full adder circuit showing all possible input combinations and corresponding outputs. A full adder is therefore essential for the hardware implementation of an adder circuit capable of adding larger binary numbers. We have a similar situation for the other higher column bits. As a result, when we add the next adjacent higher column bits, we would be required to add three bits if there were a carry from the previous addition. We record the sum under the LSB column and take the carry, if any, forward to the next higher column bits. We begin with the addition of LSBs of the two numbers. Let us recall the procedure for adding larger binary numbers. The full adder circuit overcomes the limitation of the half-adder, which can be used to add two bits only. Such a building block becomes a necessity when it comes to adding binary numbers with a large number of bits. Full AdderĪ full adder circuit is an arithmetic circuit block that can be used to add three bits to produce a SUM and a CARRY output. 3.3, it could also be implemented by using an appropriate arrangement of either NAND or NOR gates. However, these two expressions can certainly be represented in different forms using various laws and theorems of Boolean algebra to illustrate the flexibility that the designer has in hardware-implementing as simple a combinational function as that of a half-adder.Īlthough the simplest way to hardware-implement a half-adder would be to use a two-input EX-OR gate for the SUM output and a two-input AND gate for the CARRY output, as shown in Fig. While the first one representing the SUM output is that of an EX-OR gate, the second one representing the CARRY output is that of an AND gate. The Boolean expressions for the SUM and CARRY outputs are given by the equations belowĪn examination of the two expressions tells that there is no scope for further simplification. Figure 3.2 shows the truth table of a half-adder, showing all possible input combinations and the corresponding outputs. Such a circuit thus has two inputs that represent the two bits to be added and two outputs, with one producing the SUM output and the other producing the CARRY. A half-adder is an arithmetic circuit block that can be used to add two bits.