Ling adder explained

In electronics, a Ling adder is a particularly fast binary adder designed using H. Ling's equations and generally implemented in BiCMOS. Samuel Naffziger of Hewlett-Packard presented an innovative 64 bit adder in 0.5 μm CMOS based on Ling's equations at ISSCC 1996. The Naffziger adder's delay was less than 1 nanosecond, or 7 FO4.^[1]

Equations

Ling adder, architecture Skllansky, radix-2, 4-bit

In Borland Turbo Basic 1.1: '--- Step 0 ------------ Warning --------------------------------------- P00 = A0 OR B0 '1dt, Initial only CLA & Ling Propagate (not in PPA) G00 = A0 AND B0 '1dt, Initial CLA & Ling & PPA Generate D00 = A0 XOR B0 '1dt, Only Ling Initial half bit generate (P0 in PPA) P10 = A1 OR B1 '1dt G10 = A1 AND B1 '1dt D10 = A1 XOR B1 '1dt P20 = A2 OR B2 '1dt G20 = A2 AND B2 '1dt D20 = A2 XOR B2 '1dt P30 = A3 OR B3 '1dt G30 = A3 AND B3 '1dt D30 = A3 XOR B3 '1dt '--- Step 1, Ling Propagate and Generate ------ LG01 = G00 '1dt LG11 = G10 OR G00 '2dt LP11 = P10 '1dt, Sklansky architecture LG21 = G20 '1dt, Sklansky architecture LP21 = P20 AND P10 '2dt LG31 = G30 OR G20 '2dt '--- Step 2, Ling PseudoCarry (H) --------------------------- H0 = LG01 '1dt H1 = LG11 '2dt H2 = LG21 OR (LP11 AND LG11) '4dt TTL, Sklansky architecture ' 1dt 1dt 2dt H3 = LG31 OR (LP21 AND LG11) '4dt TTL ' 2dt 2dt 2dt '--- Sum ----------------------------------------- S0 = (D00) '1dt S1 = (D10 AND 1-H0) OR ((D10 XOR P00) AND H0) '4dt TTL S2 = (D20 AND 1-H1) OR ((D20 XOR P10) AND H1) '5dt TTL S3 = (D30 AND 1-H2) OR ((D30 XOR P20) AND H2) '7dt TTL S4 = ((P30) AND H3) '5dt TTL, S4=C4=Cout ^[2]

Ling adder, architecture Kogge-Stone, radix-2, 4-bit

'--- Step 0 ------------ Warning --------------------------------------- P00 = A0 OR B0 '1dt, Initial only CLA & Ling Propagate (not in PPA) G00 = A0 AND B0 '1dt, Initial CLA & Ling & PPA Generate D00 = A0 XOR B0 '1dt, Only Ling Initial half bit generate (P0 in PPA) P10 = A1 OR B1 '1dt G10 = A1 AND B1 '1dt D10 = A1 XOR B1 '1dt P20 = A2 OR B2 '1dt G20 = A2 AND B2 '1dt D20 = A2 XOR B2 '1dt P30 = A3 OR B3 '1dt G30 = A3 AND B3 '1dt D30 = A3 XOR B3 '1dt '--- Step 1 ---------------------------- LG01 = G00 '1dt, Ling Generate LP11 = P10 AND P00 '2dt, Ling Propagate, Kogge-Stone architecture LG11 = G10 OR G00 '2dt LP21 = P20 AND P10 '2dt LG21 = G20 OR G10 '2dt, Kogge-Stone architecture LG31 = G30 OR G20 '2dt '--- Step 2, Ling PsevdoCarry ---- H0 = LG01 '1dt H1 = LG11 '2dt H2 = LG21 OR (LP11 AND LG01) '4dt TTL, Kogge-Stone architecture ' 2dt 2dt 1dt H3 = LG31 OR (LP21 AND LG11) '4dt TTL ' 2dt 2dt 2dt '--- Sum ----------------------------------------- S0 = (D00) '1dt S1 = (D10 AND 1-H0) OR ((D10 XOR P00) AND H0) '4dt TTL S2 = (D20 AND 1-H1) OR ((D20 XOR P10) AND H1) '5dt TTL S3 = (D30 AND 1-H2) OR ((D30 XOR P20) AND H2) '7dt TTL S4 = ((P30) AND H3) '5dt TTL, S4=C4=Cout ^[3]

External links

H. Ling, "High Speed Binary Parallel Adder", IEEE Transactions on Electronic Computers, EC-15, p. 799-809, October, 1966.
H. Ling, "High-Speed Binary Adder", IBM J. Res. Dev., vol.25, p. 156-66, 1981.
R. W. Doran, "Variants on an Improved Carry Look-Ahead Adder", IEEE Transactions on Computers, Vol.37, No.9, September 1988.
N. T. Quach, M. J. Flynn, "High-Speed Addition in CMOS", IEEE Transactions on Computers, Vol.41, No.12, December, 1992.
S. Naffziger, "High Speed Addition Using Ling's Equations and Dynamic CMOS Logic", U.S. Patent No. 5,719,803, Issued: February 17, 1998.
G. Dimitrakopoulos, D. Nikolos, "High-Speed Parallel-Prefix VLSI Ling Adders", IEEE Transaction on Computers, Vol.54, No.2, February, 2005.

Notes and References

S. . Naffziger . A Sub-Nanosecond 0.5um 64b Adder Design . Digest of Technical Papers, 1996 IEEE International Solid-State Circuits Conference . San Francisco . 8–10 February 1996 . 362–363 . https://web.archive.org/web/20060410222424/http://lap.epfl.ch/courses/comparith/Papers/9-Nafziger-Ling-Adder.pdf . 10 April 2006 . dead.
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