Unrolling Carry Recurrence

1
Unrolling Carry Recurrence
2
Carry-Lookahead Equations
3
4-Bit CLA
4
Circuit Structure
5
CLA Complexity
6
Managing CLA Complexity
7
Multilevel CLA Example
8
Radix-16 Addition

• Two Binary Numbers Grouped into Hex Digits
• Block Generate and Propagate Signals in Each
  radix-16 Digit
• Replace c4 Position of CLA Network with Block
  Signals gi,i3 and pi,i3
• Results in 4-bit Lookahead Carry Generator

9
CLA Design
10
Lookahead Carry Generator
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Block Generate and Propagate

• Assume i0 lt i1 lt i2

• Example g0,3 is Generate Signal of Block
  for bits 0-3
• Relationships Allow for Merging of Blocks
• Can Allow Merged Block to Overlap

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Example Merged LAG
x15-12
y15-12
x11-8
y11-8
x7-4
y7-4
x3-0
y3-0
CLA 3
CLA 2
CLA 1
CLA 0
c12
c8
c4
c0
g15-12
p15-12
g11-8
p11-8
g7-4
p7-4
g3-0
p3-0
s15-12
s11-8
s7-4
s3-0
Lookahead Carry Generator
g15-0
p15-0
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CLA Latency
14
CLA Architecture
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Overlapped LAGs

• Overlap Blocks i1,j1 and i0,j0

• Relationships Become

• Useful for Building Trees of Different Shapes

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CLA With LAG
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CLA Latency
Example 64-bit CLA in 13 gate levels since 43
64 Generates final carry out for Fig. 6.5
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Ling Adders
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Ling Adders Wired OR
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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?? g?p ?? p
  p? p??
• 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?? g? p? p p? p??
• 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?? g?p? ?
  g? g?? p??

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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