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High-speed Addition with Bipolar Digital Circuits. Matthew W. Ernest Rensselaer Polytechnic Institute. Carry types: Carry Select. Compute possible results in parallel Select when actual carry-in available Requires internal carry for blocks, e.g. ripple Delay: O(f(n/b) +b)

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high speed addition with bipolar digital circuits

High-speed Addition with Bipolar Digital Circuits

Matthew W. Ernest

Rensselaer Polytechnic Institute

carry types carry select
Carry types: Carry Select
  • Compute possible results in parallel
  • Select when actual carry-in available
  • Requires internal carry for blocks, e.g. ripple
  • Delay: O(f(n/b) +b)
  • Area: O(f(n/b)·b+b)
  • Affected by block sizing

1

1

0

0

carry select delay path
Carry Select Delay Path
  • t=0..4: Each block operates in parallel
  • t=5: Carry-out of first block selected by carry-in, no activity in second block
  • t=6: Carry-out of second block selected by carry-out of first

1

b7…b4

0

1

b7…b4

0

6

5

4

3

2

1

0

t

lengthening non critical paths
Lengthening non-critical paths
  • t=0..4: Each block operates in parallel
  • t=5: Carry-out of first block selected by carry-in, additional bit handled during delay
  • t=6: Carry-out of lengthened second block selected by carry-out of first

1

b8…b4

0

1

b7…b4

0

6

5

4

3

2

1

0

t

carry select delay
Carry Select Delay
  • td: delay of circuit
  • tg: delay of gate
  • tm: delay of mux
  • N: # of stages
  • ci: bits in stage i

Given: td = tg· ci + N · tm

If: tg· ci+1£ tg· ci + tm

Define: s = ci+1 - ci£ tm / tg

minimizing delay via stage size
Minimizing delay via stage size

td / tg = éÖ2 B s + s/2ù

N =étd / tg - s/2 ± Ö(td / tg - s/2)2 - 2 B sù

s

c1 =td / tg- Ns

carry types block carry look ahead
Carry Types: Block carry look-ahead
  • A block propagates a carry if all bits in the block propagate a carry
  • A block generates a carry if a bit generates a carry and all succeeding bits propagate
  • Delay: O(log n)
  • Area: O(n log n)
carry vs pseudocarry
Carry vs. Pseudocarry

Cout=Gn+ Pn• Gn-1 +…+Pn• Pn-1• ... P0• Cin

If G=A•B

and P=A+B

then

G=G•P

Cout= Pn•Gn+ Pn• Gn-1 +…+Pn• Pn-1• ... P0• Cin

Cout= Pn(Gn+ Gn-1 +…+Pn-1• ... P0• Cin)

Cout= Pn•Hn

Hn =Gn+ Gn-1 +…+Pn-1• ... P0• Cin

deriving block pseudocarry from carry lookahead terms
Deriving Block Pseudocarry from Carry Lookahead Terms

Block Generate:

Gi•j0= Gij + PijGij-1i + … + PijPij-1iPij-2i•••Gi0

If G=A•B

and P=A+B

then

G=G•P

Gi•j0= PijGij + PijGij-1i + … + PijPij-1iPij-2i•••Gi0

Gi•j0= Pij(Gij + Gij-1i + … + Pij-1iPij-2i•••Gi0)

Hi•j0= Gij + Gij-1i + … + Pij-1iPij-2i•••Gi0

generalized pseudocarry equations
Generalized Pseudocarry Equations

H2s= G1s+1 + G1s

Hi+js= Hjs+i + Ijs+i-1•His

Hi+j+ks= Hks+I+j + Iks+I+j-1•Hjs+i + Iks+I+j-1• Ijs+i-1•His

Ip+qt= Iqt+p•Ipt

Ip+q+rt= Irt+q+p•Iqt+p•Ipt

generating sums using pseudocarry
Generating Sums Using Pseudocarry

Sn=AnÅBnÅCn-1

If

Tn=AnÅBn

Cm= Pm•Hm

then

Sn=TnÅPn-1Hn-1

pseudocarry blocks
Pseudocarry Blocks

H2s

H2s

H2s

H2s

H2s

H2s

H2s

H2s

H2s

H2s

H2s

H2s

H2s

H2s

H2s

H2s

H6s

H6s

H6s

H6s

H6s

H18s

H14s

H32s

single ended vs double ended
Any function of inputs
  • Fan-in limited by supply voltage
  • Limited to simple functions
  • Large fan-in
Single-ended vs. Double-ended
look ahead gate w fully differential logic
Look-ahead gate w/ fully differential logic

Hn-2

Hn-2

Hn-1

Hn-1

In-1

In-1

In

In

Hn-1

Hn-1

Hn

Hn

In

In

Hn

Hn

mixed input look ahead gates
Hn

Hn-1

Vr

Hn

Vr

In

In

Mixed input look-ahead gates
  • In(Hn+ Hn-1) + In•Hn
  • Hn+ In•Hn-1
  • Two series-gated levels for three inputs
mixed input look ahead gates1
Hn

Hn-1

Hn

Hn-2

Hn-1

Hn

In-1

In-1

In

In

Mixed input look-ahead gates
  • In In-1(Hn+ Hn-1 + Hn-2) + In In-1(Hn+ Hn-1) + In• In-1• Hn
  • Hn+ In•Hn-1 + In• In-1• Hn-2
  • Three series-gated levels for five inputs
adder comparision
Adder comparision

CSel

PCLA

Ripple

CLA

Bits

C

B

A

32

32

12

12

9

6

5

64

64

20

16

12

7

6

pseudocarry tree oscillator
Pseudocarry Tree Oscillator

Select

0

1

31

32

1

B

A

Cin

Cout

minimize balance wire length
Minimize/Balance Wire Length

Isometric Alignment

cascode output stage
Cascode Output Stage
  • Eliminates capacitive coupling between input and output
  • Shortens rise time, but increases delay
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