Ling adder

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

4-bit Ling adder, Sklansky architecture:

gm1=0
pm1=0
p0 = a0 OR  b0    
g0 = a0 AND b0    
d0 = a0 XOR b0    
p1 = a1 OR  b1
g1 = a1 AND b1
d1 = a1 XOR b1
p2 = a2 OR  b2
g2 = a2 AND b2
d2 = a2 XOR b2
p3 = a3 OR  b3
g3 = a3 AND b3
d3 = a3 XOR b3
'Level1----------------Distance=2^0=1
'(G,P) = (g,p) o (g,p)=(g,p)
'GLi = gi OR gi-1
'PLi = pi AND pi-1    'Distance=1
GLm1 = 0              'for k<0
PLm2 = 0              'for k<0
'(GL0,PLm1)
GL0 = g0 OR  gm1
PLm1=0                'for k<0
'(GL1,PL0)
GL1 = g1 OR  g0
PL0 = p0 AND pm1      'Distance=1
'(GL3,PL2)
GL3 = g3 OR  g2
PL2 = p2 AND p1       'Distance=1
'Level2---------------------------Distance=2^1=2
'(G,P) = (g,p) o (g',p') = (g OR (p AND g'),p AND p')
'G=g OR (p AND p')
'P=      p AND p'
'(GL2,PL1) o (GL1,PLm1)          '
GL21 =  g2 OR p1 AND GL1         '
'(GL3,PL2) o (GL1,PL0)           '
GL31 = GL3 OR PL2 AND GL1        '
'Ling PsevdoCarry (H)-------------------
Hm1 = GLm1                   'Ling PsevdoCarry
H0  = g0                     'Ling PsevdoCarry
H1  = GL1                    'Ling PsevdoCarry
H2  = GL21                   'Ling PsevdoCarry
H3  = GL31                   'Ling PsevdoCarry
'SUM-------------------------------------------------
'si=(/Hi-1 AND di) OR (Hi-1 AND (di XOR pi-1))
s0 = (1-Hm1) AND d0 OR Hm1 AND (d0 XOR pm1)  's0=d0
s1 = (1-H0 ) AND d1 OR H0  AND (d1 XOR p0 )
s2 = (1-H1 ) AND d2 OR H1  AND (d2 XOR p1 )
s3 = (1-H2 ) AND d3 OR H2  AND (d3 XOR p2 )
s4 = p3 AND H3  'Cout=s4

References
  1. Naffziger, S. (8–10 February 1996). "A Sub-Nanosecond 0.5um 64b Adder Design" (PDF). Digest of Technical Papers, 1996 IEEE International Solid-State Circuits Conference. San Francisco. pp. 362–363. Archived from the original (PDF) on 10 April 2006.
  1. H. Ling, "High Speed Binary Parallel Adder", IEEE Transactions on Electronic Computers, EC-15, p. 799-809, October, 1966.
  2. H. Ling, "High-Speed Binary Adder", IBM J. Res. Dev., vol.25, p. 156-66, 1981.
  3. R. W. Doran, "Variants on an Improved Carry Look-Ahead Adder", IEEE Transactions on Computers, Vol.37, No.9, September 1988.
  4. N. T. Quach, M. J. Flynn, "High-Speed Addition in CMOS", IEEE Transactions on Computers, Vol.41, No.12, December, 1992.
  5. S. Naffziger, "High Speed Addition Using Ling's Equations and Dynamic CMOS Logic", U.S. Patent No. 5,719,803, Issued: February 17, 1998.
  6. G. Dimitrakopoulos, D. Nikolos, "High-Speed Parallel-Prefix VLSI Ling Adders", IEEE Transaction on Computers, Vol.54, No.2, February, 2005.


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