Number_system

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Unformatted text preview: osition. Two outputs are necessary because the sum of three binary digits ranges in value from 0 to 3, and binary 2 and 3 need two digits. These two outputs are designated by the symbols S (for sum) and C (for carry). The truth-table of Figure 6.38 exactly defines the function of full-adder. The l's and O's for the output variables are determined from the arithmetic sum of the three input variables. When all input variables are 0, the output is 0 for both C and S. The S output is equal to 1 when only one input is equal to 1 or when all three inputs are equal to 1. The C output is 1 if two or three inputs are equal to 1. Inputs Outputs A B D C s 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 Figure 6.38. Truth table for a full-adder. The sum-of-products expressions for the two outputs can be directly obtained from the truth table and is given below: Although the expression fcjr S cannot be simplified, it is possible to simplify the expression for C as follows: The S output from the second half-adder is the exclusive-OR of D and the output of the first half-adder giving: S=(A-B+A-B]-D+(A-B+A-B)-D = AB AB D + ABD+ABD = A + B • A + B D+ABD+ABD =(A+B)-(A+B)-D+A-B-D+A-B-D = (AA+AB+AB + BB)-D+AB D+AB D =(AB+ABJD+ABD+AB D = ABD+A§;D+AB D+AB D = ABD+A-BD+ABD+ABD And we have the carry output C = lA-B + AB -D + A-B = ABD+ABD+A-B(D+D) This can be simplified as before to C = AB + AD + BD A Parallel Binary Adder Parallel binary adders are used to add two binary numbers. For example, if we want to add two four-bit numbers, we need to construct a parallel four-bit binary adder as shown in Figure 6.41. Such an adder requires one half-adder (denoted by HA) and three fulladders (denoted by FA). The binary numbers being added are A 4 A3 A2 A! and B4 B3 B2 Bi and the answer is: A4 A3 A2 A1, + B4 B3 B2B1, S5 S4 S3 S2 S1 The first column requires only a half-adder. For any column above the first, there may be a carry from the preceding column. Therefore, we must use a full adder for ea...
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This document was uploaded on 04/07/2014.

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