11 Projjal Gupta,
Member, IEEE
Electronics and Communication EngineeringSRM Institute of Science and Technology, KattankulathurEmail : [email protected]
Abstract —Binary addition is one of the most primitive andmost commonly used applications in computer arithmetic. A largevariety of algorithms and implementations have been proposedfor binary addition. Huey Ling proposed a simpler form of CLAequations which rely on adjacent pair bits (ai, bi) and (ai1,bi1). Along with bit generate and bit propagate, we introduceanother prefix bit, the half sum bit. Ling adder increases thespeed of n-bit binary addition, which is an upgrade from theexisting Carry-Look-Ahead adder. Several variants of the carrylook-ahead equations, like Ling carries, have been presented thatsimplify carry computation and can lead to faster structures. Lingadders, make use of Ling carry and propagate bits, in order tocalculate the sum bit. As a result, dependency on the previous bitaddition is reduced; that is, ripple effect is lowered. This paperprovides a comparative study on the implementation of the abovementioned high-speed adders.
Keywords — Ling Adder, High Speed Binary Adder, Binary Addi-tion.
I. I
NTRODUCTION
The family of Ling adders is a particularly fast adder and isdesigned using H. Ling’s equations and generally implementedin BiCMOS. It is an upgrade to the already existing Carry-Look-Ahead Adders and is mathematically faster, as it requireslesser steps for the computation of a sum. The circuit of a Lingadder is particularly more complex, and is less favourable foruse in VLSI systems due to its complexity and it requires farmore extra components than traditional systems. The circuitis divided into 4 parts, which can be denoted by H. Ling’sequations. II. A
NALYSIS OF L ING ’ S E QUATION
A. Initial Generation of Bits
Ling Adders require to form the bit generate and bit pro-pogate that are used in the regular Carry look ahead adders. Itis denoted by the 3 symbols g i , p i and d i .The generate and propagate bits follow CLA so they can bedenoted as, g i = a i · b i p i = a i + b i However, Ling adder requires an extra half bit term whichlater on simplifies the circuit design, while increasing theoverall efficiency of the adder. This half bit generate is denotedby d i and can be mathematically shown by, d i = a i ⊕ b i The above mentioned Generate Bit g i and Propagate Bit p i are used further to derive the Ling Generates, which areterms that will go on to simplify the final equation. This isparticularly important because these generates will form thebase of the Ling adder circuit design. These are denoted by G ∗ i and P ∗ i B. The CLA Basis of Ling’s Equations
The CLA depends upon the carry out term of the previousfor the new carry terms. c i + i = g i + p i · c i which is similar to the Simple Ripple Adder which usesthe carry output of the preceding data bits for forward addition.Similarly based on the above concept, Ling created a newtheoretical carry generate which he denoted by H. This is usedlater on in the Adder to generate the sum S i . The term H isgiven by, H i = c i + c i − where, c i = H i · p i Introduction of ling carry H i is one of the major reasonswhy Ling Adder is a fast yet complex adder. Use of Lingcarry equations decreases the number of boolean terms duringits operations, but increases the design complexity. a r X i v : . [ c s . A R ] A ug C. Ling Generate and Propagate
Ling proposed the use of Ling Propagate and Ling Generateto simplify the operations of the Ling adder. It is veryimportant, as this is the first step where we can see how theterms are generated by using the i th and ( i − th terms. Theseterms can be derived by G ∗ i = g i + g i − and P ∗ i = p i · p i − Ling generate and propagate terms are used to calculate theLing carry term H . Later on in the Adder design, the sumterms are directly influenced by the all the Ling terms. D. Ling Sum Term
The final sum term for the i th pair terms of a and b aredevised by following Ling sum equations, which take in lessernumber of inputs, and hence decrease the lag in the system.If we assume that all input gates have only two inputs, wecan see that calculation of CLA carry C requires 5 logic levels,whereas that for ling carry H requires only four. Although thecomputation of carry is simplified, calculation of the sum bitsusing Ling carries is much more complicated. The sum bit,when calculated by using traditional carry, is given to be, s i = d i ⊕ c i − We note that we require to use both the carry output andhalf-bit term from the first operation block. c i = H i · p i Using the above term in the Ling Sum Equation, s i = d i ⊕ p i − · H i − on break down, s i = H (cid:48) i − ⊕ d i + H i − ( d i ⊕ p i − ) Hence, the output value for a i + b i is given by s i and c i .III. L ING C ARRY E QUATION
A. General expansion and Substitution
Earlier in the CLA basis subsection of Ling Equationanalysis, we came across 2 equations, H i = c i + c i − and, c i = H i · p i Since we know that, c i +1 = g i + p i · c i Thus we can write the Carry Output as, H i = g i + g i − + p i − · g i − + p i − · p i − · g i − + ..... + p i − · p i − · p i − · ..... · p · p · g But we know that, G ∗ i = g i + g i − P ∗ i = p i · p i − Thus we can simplify the H equations to Ling generate-propagate terms. B. Application in 4-Bit System
In a 4-bit adder design, we require the terms H , H , H and H .From the Ling generate-propagate equations and the ex-panded Ling Carry equation in the previous subsection, wecan write the 4 terms as, H = G + P · G H = G + P · G H = G H = G It is noted that the complexity of the system will increasewith increase of Input terms.IV. L
OGIC DESIGN OF IT L ING A DDER
From all the above sections and designs, we can design the4 bit Ling Adder. As per the design, The flowing outputs are passed throughbasic OR and AND gates to satisfy H equations.The carry issafely calculated as c c = H · p Similarly, each block present in the Logic Diagram representsan operation step described in each subsection of the logicanalysis.The use of free gates in the circuit represent the operations usedto calculate H i . The initial g − and p − dont exist during thecase i = 0 and hence they are taken as logical f alse or zero value by grounding them.Effectively, the overall circuit follows the final equation s = a + b and generates a carry term in-case the overall sum exceeds the4-bit output range. V. PCB
AND
CAD D
ESIGN
The system can be designed in real time by the use of actuallogic gate ICs belonging to the 74xx family. These ICs usuallyconsist of 16 (DIL16) or 14 (DIL14) pins, and require lowpower. From the above Logisim design of the Ling Adder,we can start designing the same circuit on any EDA or CADsoftware. Due to its high complexity, the circuit has to bedesigned on the both sides of a pcb and uses multiple vias forthe on-board connections.VI. C
ONCLUSION
Hence, a basic 4-Bit Ling adder circuit was designedaccording to Huey Ling’s equations. In 4-Bit arithmeticsystem, the CLA requires 5 terms, whereas the Ling adderrequires a maximum input of 4 terms, thereby decreasing thetime required for computation. When this adder is cascadedfor higher number of Bit input terms, the CLA will comeacross an increase in operation times. But in a ling adder,this time increase would be much lesser than the other binaryadders. A
CKNOWLEDGMENT