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8.4 THE CARRY LOOK-AHEAD ADDER                                      345


                  sum of two positive numbers only if C n ^ C n-\. Thus, a sign-bit overflow error detector
                  can be implemented by

                                            SError Det = Cn ®C n-\,                  (8.4)

                  requiring that the sign-bit carry-in C n-\ be accessible, which it may not be for 1C chips.
                  Another approach permits a detector to be designed that depends only on the external
                  inputs to and sum bit from the sign-bit stage. A further inspection of truth table for an FA
                  in Fig. 8.2c indicates that a sign-bit overflow error can occur only if A = B when C out / S
                  for the (n — l)th stage. Shown in Fig. 8.1 la is the truth table for the sign-bit overflow error
                  conditions based on this fact. From this truth table there results the expression


                                  ^ErrorDet = Sn-\(A n-\B n-\) + £„_! (A n _i B n-\ ),  (8.5)

                  which permits the use of the 2-to-l MUX implementation shown in Fig. 8.1 Ib. For purposes
                  of comparison, the implementation of Eq. (8.4) is given in Fig. 8. lie.



                  8.4 THE CARRY LOOK-AHEAD ADDER
                  The ripple-carry (R-C) adder discussed in Section 8.3 is satisfactory for most applications
                  up to 16 bits at moderate speeds. Where larger numbers of bits must be added together at
                  high speeds, fast adder configurations must be used. One clever, if not also intuitive, design
                  makes use of the modular approach while reducing the propagation time of the R-C effect.
                  The approach has become known as the carry look-ahead (CLA) adder. In effect, the CLA
                  adder "anticipates" the need for a carry and then generates and propagates it more directly
                  than does a standard R-C adder.
                    The design of the CLA adder begins with a generalization of Eqs. (8.2). For the zth stage
                  of the ripple-carry adder, the sum and carry-out expressions are


                                          C. — A • ffi /?• £F> C-
                                          Oj — r\i d7 Ui \X? \*si
                                            = Sum of the i th stage
                                                                                      (8.6)
                                            = Carry-out of the zth stage

                  From the expression for C i+\ it is concluded that C i+\ = 1 is assured if (A, 0 5/) = 1 and
                  Ci = l.orif AiB{ = 1.
                    Next, it is desirable to expand Eqs. (8.6) for each of n stages, beginning with the
                  1st (first) stage. To accomplish this, it is convenient to define two quantities for the zth
                  stage,

                                     (  G,^ = A,• • B t: = Carry Generate I  ,     (8.7)
                                        Pi = Aj 0 Bj = Carry Propagate]
                                                        y


                  which are shown in Fig. 8.3 to be the intermediate functions P(H) and G(L) in the full
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