Page 374 - Engineering Digital Design
P. 374
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