Block p and g Generators

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Block p and g Generators
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Carry Determination as Prefix Computations

• Two Contiguous (or Overlapping) Blocks (g, p)
  and (g, p)
• Merged Block (g, p) g g gp p
  pp
• Large Group Generates Carry if
• left group generates carry
• right group generates and left group propagates

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Carry Operator,

• Define Operator Over (g, p) Pairs (g, p) (g,
  p) (g, p)
• g g gp p pp
• is Associative (g, p) (g, p)
  (g, p) (g, p) (g, p) (g,
  p) (g, p) (g, p) (g, p)

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Carry Operator, (cont)

• is NOT Commutative (g, p) (g, p) ?
  (g, p) (g, p)
• This is Easy to See Because g g gp ?
  g gp

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Prefix Adders
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Carry Determination

• Assume Adder with NO cIN ci1 g0,i
• Carry Enters i1 Block iff Generated in Block
  0,i
• Assume Adder with cIN 1
• Viewed as Generated Carry from Stage -1 p-1
  0, g-1 cIN
• Compute g-1,i For All i
• Formulate Carry Determination as

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Prefix Computation
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Prefix Sums Analogy

• Designs for Prefix Sums Can be Converted to
  Carry Computation
• Replace Adder with Operator
• Addition IS Commutative, Order Doesnt Matter
• Can Group (g, p) In Anyway to Combine Into Block
  Signals (as long as order is preserved)
• (g, p) Allow for Overlapping Groups, Prefix
  Sums Does Not (sum would contain some values
  added two or times)

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Prefix Sum Network
(adder levels)
( of adders)
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Another Way for Prefix Sums

• Compute the Following Firstx0x1 x2x3
  x4x5 ... xk-2xk-1
• Yields the Partial Sums, s1, s3, s5, ..., sk-1
• Next, Even Indexed Sums Computed As s2j
  s2j-1 x2j

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Alternative Prefix Sum Network
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Comparison of Prefix Sum Networks

• First Design Faster lg2(k) versus
  2lg2(k)-2 (levels)
• First Design has High Fan-out Requirements
• First Design Requires More Cells (k/2)lg2k
  versus 2k-2-lg2k
• Second Design is Brent-Kung Parallel Prefix Graph
• First Design is Kogge-Stone Parallel Prefix Graph
  (fan-out can be avoided by distributing
  computations)

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Brent-Kung Network
independent,so single delay
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Kogge-Stone Network
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Area/Levels of Prefix Networks
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Hybrid Parallel Prefix Network