Lecture 17 Adders and Multipliers - bwrcs.eecs.berkeley.edu
Transcript of Lecture 17 Adders and Multipliers - bwrcs.eecs.berkeley.edu
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EE241 - Spring 2000Advanced Digital Integrated Circuits
Lecture 17
Adders and Multipliers
UC Berkeley EE241 B. Nikoli c
Logarithmic Lookahead Adders
A7
F
A6A5A4A3A2A1
A0
A0
A1
A2A3
A4
A5
A6
A7
F
tp∼ log2(N)
tp∼ N
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Tree Adders
lmG ppP ⋅=
lmmG gpgG ⋅+=
m – more significantl – less significant
Start from the input P, G, and continue up the tree2-bit groups, then 4-bit groups, …
( ) ( ) ( )lmlmmllmm ppgpgpgpgpg ⋅⋅+=•= ,,,),(
Kogge, Stone, Trans on Comp,’73
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Tree Adders
( ) ( )jkjkkikijiji PGPGPG :1:1:::: ,,),( −−•=
( ) ( ) ),(,,),( GPPPGPGPGPG =⋅⋅+=•
( ) ( )jvjvkikijiji PGPGPG :::::: ,,),( •=
Associativity:
Idempotency:
Associativity + Idempotency:
jki >≥
jki >≥ kv >
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Examples
( ) ( ) ( ) ( ){ }{ }{ }00:3,0:34:74:78:118:1112:1512:15 ,1,,),( cPGPGPGPG ••••
( ) ( ) ( ) ( )00014141515 ,1,...,, cPGPGPG ••••
16-bit ripple carry
16-bit carry lookahead with 4-bit groups
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Brent-Kung Adder(G0,P0)(G1,P1)
(G2,P2)
(G3,P3)
(G4,P4)(G5,P5)
(G6,P6)
(G7,P7)
Co,0
Co,1Co,2
Co,3
Co,4
Co,5
Co,6
Co,7
tadd ∼ log2(N)Brent, Kung, Trans on Comp, 3/82
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Tree Adders
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Ling Adder
Variation of CLA
Ling, IBM J. Res. Dev, 5/81
1−⋅+= iiii GpgG
1−⊕= iii GpS
iii bap ⊕=
iii bag ⋅=
11 −− ⋅+= iiii HtgH
11 −−+⊕= iiiiii HtgHtS
iii bat +=
iii bag ⋅=
Ling’s equations
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Ling Adder
1−⋅+= iiii GpgG
( ) 1
11
−
−−
⋅++=
⋅++=
iiii
iiiiii
Gpgg
GpGggG
1−⋅+= iiii GtgG 11 −− ⋅+= iiii GtgH
Ling’s equation
Doran, Trans on Comp 9/88
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Ling Adder
01231232333 gtttgttgtgG +++=
0121223
00121122233
gttgtgg
gtttgttgtgH
+++=
+++=
Conventional
Ling
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HP Adder
Naffziger, ISSCC’96
01234 ppppi =
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HP Adder
Carry rippleSum select
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Hybrid Adders
Dobberpuhl, JSSC 11/92 DEC Aplha 21064
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DEC Adder
l Combination:» 8-bit tapered pre-discharged Manchester
carry chains, with Cin = 0 and Cin = 1» 32-bit LSB carry-lookahead» 32-bit MSB conditional sum adder» Carry-select on most significant bits» Latch-based timing
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Z X·· Y× Zk2k
k 0=
M N 1–+
∑= =
Xi2i
i 0=
M 1–
∑
Yj2j
j 0=
N 1–
∑
=
XiYj2i j+
j 0=
N 1–
∑
i 0=
M 1–
∑=
X Xi 2i
i 0=
M 1–
∑=
Y Yj2j
j 0=
N 1–
∑=
with
Binary Multiplication
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1 0 1 1
1 0 1 0 1 0
0 0 0 0 0 0
1 0 1 0 1 0
1 0 1 0 1 0
1 0 1 0 1 0
×
1 1 1 0 0 1 1 1 0
+
Partial Products
AND operation
Binary Multiplication
2N bits in the final sum
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HA FA FA HA
FA FA FA HA
FA FA FA HA
X0X1X2X3 Y1
X0X1X2X3 Y2
X0X1X2X3 Y3
Z1
Z2
Z3Z4Z5Z6
Z0
Z7
Array Multiplier
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Shift-and-Add Multiplier
l Standard adder and shift-in the multiplicand
l Shift the result as well and addl N cyclesl Parallel adders add more hardware
(adders) instead.
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HA FA FA HA
HAFAFAFA
FAFA FA HA
Critical Path 1
Critical Path 2
Critical Path 1 & 2
MxN Array Multiplier— Critical Path
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H A HA HA HA
FAFAFAHA
FAHA FA FA
FAHA FA HA
Vector Merging Adder
Carry-Save Multiplier
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A
B
P
Ci
VDDA
A A
VDD
Ci
A
P
AB
VDD
VDD
Ci
C i
Co
S
Ci
P
P
P
P
P
Identical Delays for Carry and Sum
Adder Cells in Array Multiplier
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SCSCSCSC
SCSCSCSC
SCSCSCSC
SC
SC
SC
SC
Z0
Z1
Z2
Z3Z4Z5Z6Z7
X0X1X2X3
Y1
Y2
Y3
Y0
Vector Merging Cell
HA Multiplier Cell
FA Multiplier Cell
X and Y signals are broadcastedthrough the complete array.( )
Multiplier Floorplan
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FA
FA
FA
FA
y0 y1 y2
y3
y4
y5
S
Ci-1
Ci-1
Ci-1
Ci
Ci
Ci
FA
y0 y1 y2
FA
y3 y4 y5
FA
FA
CC S
Ci-1
Ci-1
Ci-1
Ci
Ci
Ci
Wallace-Tree Multiplier
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Wallace-Tree Multiplier
Wallace,Trans on Comp. 2/64
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Wallace-Tree Multiplier
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Tree Multipliers
l Time is proportional to log Nl Wiring is complicatedl Different wire lengthsl Optional pipelining