EE1411

Arithmetic Circuits

Chapter 11 (I)Chapter 11 (I)

Arithmetic Circuits:Arithmetic Circuits:Adder DesignsAdder Designs

Rev. 1.0 05/12/2003Rev. 2.0 06/05/2003Rev. 2.2 12/31/2003Rev. 3.0 1/7/2004

EE1412

Arithmetic Circuits

A Generic Digital ProcessorA Generic Digital Processor

MEMORY

DATAPATH

CONTROL

INPU

T-O

UT

PUT

EE1413

Arithmetic Circuits

Building Blocks for Digital ArchitecturesBuilding Blocks for Digital Architectures

Arithmetic and Unit

- Bit-sliced datapath (adder, multiplier, shifter, comparator, etc.)

Memory

- RAM, ROM, Buffers, Shift registers

Control

- Finite state machine (PLA, random logic.)

- Counters

Interconnect

- Switches

- Arbiters

- Bus

EE1414

Arithmetic Circuits

Intel MicroprocessorIntel Microprocessor

9-1

Mux

9-1

Mux

5-1

Mux

2-1

Mux

ck1

CARRYGEN

SUMGEN+ LU

1000um

b

s0

s1

g64

sum sumb

LU : LogicalUnit

SUM

SEL

a

to Cache

node1

RE

G

Itanium has 6 integer execution units like this

EE1415

Arithmetic Circuits

BitBit--Sliced DesignSliced Design

Bit 3

Bit 2

Bit 1

Bit 0

Reg

iste

r

Add

er

Shif

ter

Mul

tipl

exer

ControlD

ata-

In

Dat

a-O

ut

Tile identical processing elements

Word Data Word Data

EE1416

Arithmetic Circuits

ItaniumItanium Integer DatapathInteger Datapath

Fetzer, Orton, ISSCC’02

EE1417

Arithmetic Circuits

AddersAdders

EE1418

Arithmetic Circuits

Several Implementations of AddersSeveral Implementations of Addersq One-bit Full Adder (Cell)q Carry-Ripple Adderq Bit-Serial Adderq Mirror Adderq Transmission-Gate Adderq Manchester Adderq Carry-bypass (Carry-skip) Adderq Carry-Select Adderq Carry lookahead Adderq Conditional-sum Adder (supplement material) q Carry-save Adder ((supplement material)

EE1419

Arithmetic Circuits

FullFull--Adder (FA)Adder (FA)A B

Cout

Sum

Cin Fulladder

Generate (G) = AB

Propagate (P) = A ⊕ B

Delete = A B

EE14110

Arithmetic Circuits

Boolean Function of Binary FullBoolean Function of Binary Full--Adder Adder

iiii

i

ABCCBACBACBA

CBAS

+++=

⊕⊕=

ACBCABC iiO ++=A B

Cout

Sum

Cin Fulladder

)( iii CBACABCS +++=

)( BACABC iO ++=For CMOS Implementation

EE14111

Arithmetic Circuits

Express Sum and Carry as a function of P, G, DExpress Sum and Carry as a function of P, G, D

Define 3 new variable which ONLY depend on A, B

Generate (G) = AB

Propagate (P) = A ⊕ B

Delete = A B

Can also derive expressions for S and Co based on D and P

Propagate (P) = A + BNote that we will be sometimes using an alternate definition for

EE14112

Arithmetic Circuits

CarryCarry--Ripple AdderRipple Adder

Worst-case delay is linear with the number of bits

FA FA FA FA

A0 B0

S0

A1 B1

S1

A2 B2

S2

A3 B3

S3

Ci,0 Co,0

(= Ci,1)

Co,1 Co,2 Co,3

td = O(N)tadder = (N-1)tcarry + tsum

CriticalPath

• Propagation delay (or critical path) is the worst-case delay over all possible input patterns

• A= 0001, B=1111, trigger the worst-case delay• A: 0 à 1, and B= 1111 fixed to set up the worst-

case delay transition.

EE14113

Arithmetic Circuits

Complimentary Static CMOS Full AdderComplimentary Static CMOS Full Adder

•Logic effort of Ci is reduced to 2 (c.f., A and B signals)•Ci is late arrival signal à near the output signal•Co needs to be inverted à Slow down the ripple propagate

A B

B

A

Ci

Ci A

X

VDD

VDD

A B

Ci BA

B VDD

A

B

Ci

Ci

A

B

A CiB

Co

VDD

SOC=

28 Transistors

EE14114

Arithmetic Circuits

Inversion PropertyInversion Property

A B

S

CoCi FA

A B

S

CoCi FA

EE14115

Arithmetic Circuits

Minimize Critical Path by Reducing Inverting StagesMinimize Critical Path by Reducing Inverting Stages

•Exploit Inversion Property•Reduce One inverter delay in each Full-adder (FA) unit

A3

FA FA FA

Even cell Odd cell

FA

A0 B0

S0

A1 B1

S1

A2 B2

S2

B3

S3

Ci,0 Co,0 Co,1 Co,3Co,2

EE14116

Arithmetic Circuits

SubtractorSubtractor[ ] 11)'1(

)'2(

++=++=

+=−

BABofsA

BofsABA

EE14117

Arithmetic Circuits

BitBit--Serial AdderSerial Adder

012345 aaaaaaA =

012345 bbbbbbB =

012345 ssssssS =

• Bit-serial design trades time for area efficiency• Circuit operates at bit rate (N times faster than

word rate for an N-bit data)• Need a controller (counter) for counting the serial data bits

EE14118

Arithmetic Circuits

A Better Structure: The Mirror AdderA Better Structure: The Mirror Adder

Exploring the “Self-Duality”of the Sum and Carry functions

VDD

Ci

A

BBA

B

A

A BKill

Generate"1"-Propagate

"0"-Propagate

VDD

Ci

A B Ci

Ci

B

A

Ci

A

BBA

VDD

SCo

24 transistors!

)( iii CBACABCS +++=

)( BACABC iO ++= )( BACABC iO ++=

)( iii CBACABCS +++=

EE14119

Arithmetic Circuits

Mirror Adder: Stick DiagramMirror Adder: Stick Diagram

CiA B

VDD

GND

B

Co

A Ci Co Ci A B

S

EE14120

Arithmetic Circuits

Mirror Adder DesignMirror Adder Design•The NMOS and PMOS chains are completely symmetrical

•A maximum of two series transistors can be observed in the carry-generation circuitry à for good speed.

•The transistors connected to Ci are placed closest to the output.

•When laying out the cell, the most critical issue is the minimization of the capacitance at node Co.

•The capacitance at node Co is composed of four diffusion capacitances, two internal gate capacitances, and six input gate capacitances, in the connecting adder cell .

EE14121

Arithmetic Circuits

Sizing for Mirror AdderSizing for Mirror Adder• Increase the size of the Carry Stage to about 3 or 4

times the size of the Sum Stage (PMOS/NMOS ratio of 2 is assumed)

EE14122

Arithmetic Circuits

Review of TGReview of TG--based 6T XOR Gatebased 6T XOR Gate

011101110000

FBA

B

B

Truth Table

A=0: Pass B Signal

A=1: Inverting B Signal

B

B

EE14123

Arithmetic Circuits

Review of FullReview of Full--Adder (FA)Adder (FA)A B

Cout

Sum

Cin Fulladder

Generate (G) = AB

Propagate (P) = A ⊕ B

Delete = A B

EE14124

Arithmetic Circuits

TransmissionTransmission--Gate Full Adder (24T)Gate Full Adder (24T)

•Same delay for Sum and Carryà Good for Multiplier design

BAP ⊕=

ii

i

PCCP

CPS

+=

⊕=

APCPC iO ⋅+⋅=

EE14125

Arithmetic Circuits

Manchester CarryManchester Carry--Chain AdderChain Adder

CoCi

Gi

Di

Pi

Pi

VDD

CoCi

Gi

Pi

VDD

φ

φ

Static CircuitsDynamic Circuits

iiiO DGCPC ⋅+⋅= )(

+⋅==

==

,,1

,1,0

iiiO

O

GCPC

C

φ

φ

EE14126

Arithmetic Circuits

Manchester Carry ChainManchester Carry Chain

G2

φ

C3

G3

Ci,0

P0

G1

VDD

φ

G0

P1 P2 P3

C3C2C1C0

Ci = 4 diffusion cap + 2 gate cap

EE14127

Arithmetic Circuits

Manchester CarryManchester Carry--Chain AdderChain Adder

Pi + 1 Gi + 1 φ

Ci

Inverter/Sum Row

Propagate/Generate Row

Pi Gi φ

Ci - 1Ci + 1

VDD

GND CCRR

RCNN

RCt

ij

i

jj

N

iiP

==

+=

⋅= ∑∑

==

, where2

)1(69.0

69.011

EE14128

Arithmetic Circuits

Manchester Adder Circuits (Cont.)Manchester Adder Circuits (Cont.)

q Dynamic stage§ When CLK is low, the output node is pre-charged by the p

pull-up transistor.§ When CLK goes high, the pull-down transistor turns on.§ If carry generate G=AB is true → the output node

discharges.§ If carry propagate P=A+B is true → a previous carry may

be coupled to the output node, conditionally discharging it.

q Static stage§ This requires P to be generated as A⊕B§ The Manchester adder stage improves on the carry-

lookahead implementation.

EE14129

Arithmetic Circuits

CarryCarry--Bypass (CarryBypass (Carry--skip) Adderskip) Adder

Idea: If (P0 ^ P1 ^ P2 ^ P3 = 1), then Co,3= C0

else �k”kill” or “generate”

FA FA FA FA

P0 G1 P0 G1 P2 G2 P3 G3

Co,3Co,2Co,1Co,0Ci,0

FA FA FA FA

P0 G1 P0 G1 P2 G2 P3 G3

Co,2Co,1Co,0Ci,0

Co,3

Mul

tipl

exer

BP=PoP1P2P3

EE14130

Arithmetic Circuits

Manchester Bypass Adder Circuits Manchester Bypass Adder Circuits

Fig6. Manchester adder with carry bypass: (a) simple (b) conflict free

q The control signals T1,T2,and T3 shown in Fig6(b) are generated by:§ T1 = -(P0P1P2)P3

§ T2 = -P3

§ T3 = P0P1P2P3

Wired-OR node

EE14131

Arithmetic Circuits

Delay DefinitionDelay Definitionq tsetup: the fixed overhead time to create the

“generate”and “propagate”signals

q tcarry: the propagation delay through a single bit. The worst case carry-propagation delay through a single stage of M bits is approximately M times larger

q tbypass: the propagation delay through the bypass multiplexer of a single stage

q tsum: the time to generate the sum of the final stage

EE14132

Arithmetic Circuits

CarryCarry--Bypass Adder (cont.)Bypass Adder (cont.)

Carrypropagation

Setup

Bit 0–3

Sum

M bits

tsetup

tsum

Carrypropagation

Setup

Bit 4–7

Sum

tbypass

Carrypropagation

Setup

Bit 8–11

Sum

Carrypropagation

Setup

Bit 12–15

Sum

tadder = tsetup + Mtcarry + (N/M-1)tbypass + (M-1)tcarry + tsum

M bits form a Section à (N/M) Bypass Stages

EE14133

Arithmetic Circuits

Carry Ripple versus Carry BypassCarry Ripple versus Carry Bypass

N

tp

ripple adder

bypass adder

4..8

Wordlength (N) > 4~8 is better for Bypass Adder

EE14134

Arithmetic Circuits

CarryCarry--Select Adder (4Select Adder (4--bit stage) bit stage)

Setup

"0" Carry Propagation

"1" Carry Propagation

Four 2-to-1 Multiplexer

Sum Generation

Co,k-1 Co,k+3

"0"

"1"

P,G

Carry Vector

EE14135

Arithmetic Circuits

CarryCarry--Select AdderSelect Adder

Fig7. Carry-select adder:(a) basic architecture (b) 32-bit carry-select adder example

EE14136

Arithmetic Circuits

Carry Select Adder: Critical PathCarry Select Adder: Critical Path

0

1

Sum Generation

Multiplexer

1-Carry

0-Carry

Setup

Ci,0 Co,3 Co,7 Co,11 Co,15

S0–3

Bit 0–3 Bit 4–7 Bit 8–11 Bit 12–15

0

1

Sum Generation

Multiplexer

1-Carry

0-Carry

Setup

S4–7

0

1

Sum Generation

Multiplexer

1-Carry

0-Carry 0-Carry

Setup

S8–11

0

1

Sum Generation

Multiplexer

1-Carry

Setup

S12–15

summuxcarrysetupadd ttMtMN

tt +⋅+⋅

+=

EE14137

Arithmetic Circuits

Linear Carry Select Linear Carry Select

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Bit 0-3 Bit 4-7 Bit 8-11 Bit 12-15

S0-3 S4-7 S8-11 S12-15

Ci,0

(1)

(1)

(5)(6) (7) (8)

(9)

(10)

(5) (5) (5)(5)

EE14138

Arithmetic Circuits

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Bit 0-1 Bit 2-4 Bit 5-8 Bit 9-13

S0-1 S2-4 S5-8 S9-13

Ci,0

(4) (5) (6) (7)

(1)

(1)

(3) (4) (5) (6)

Mux

Sum

S14-19

(7)

(8)

Bit 14-19

(9)

(3)

Square Root Carry SelectSquare Root Carry Select

N-bit adder with P stages, 1st stage adds M bits

NPP

NMPPPP

MP

PMMMMN

22

)21

(22

)1(

)1()2()1(22

≈⇒≈⇒−+=−

+=

−+++++++= L

summuxcarrysetupadd ttNMttt +++= )2(

EE14139

Arithmetic Circuits

Adder Delays Adder Delays -- Comparison Comparison

Square root select

Linear select

Ripple adder

20 40N

t p(in

uni

t del

ays)

600

10

0

20

30

40

50

EE14140

Arithmetic Circuits

CarryCarry--LookaheadLookahead AddersAdders

q The linear growth of adder carry-delay with the size of the input word for n-bit adder maybe improved by calculation the carries to each stage in parallel.

EE14141

Arithmetic Circuits

LookLook--ahead Adder ahead Adder -- Basic IdeaBasic IdeaAN-1, BN-1A1, B1

P1

S1

•••

••• SN-1

PN-1Ci, N-1

S0

P0Ci,0 Ci,1

A0, B0

)))(((

)(

),,(

0,00111,

2,11,

1,1,,

ikkkkkO

kOkkkkkO

kOkkkOkkkO

CPGPPGPGC

CPGPGC

CPGCBAfC

++⋅+⋅+=

⋅+⋅+=

⋅+==

−−

−−−

−−

…

EE14142

Arithmetic Circuits

•Carry of the I-th stage ---

•Expanding

•For four stages, the appropriate term :

Carry-Lookahead Adders (Cont.)

Generic carry-lookahead adder

iiii CCCPGfC , eDecorrelat),,( 303 +⇒=

signal) Propagate(

signal) Generate(1

iii

iii

iiii

BAP

BAG

CPGC

⊕=

=

⋅+= −

0121211

211

2111

)(CPPPPGPPGPG

CPGPGC

CPGC

iiiiiiii

iiiiii

iiii

…L −−−−

−−−

−−−−

+++⋅+=+⋅+=

⋅+=

in

in

in

in

CPPPPGPPPGPPGPGC

CPPPGPPGPGCCPPGPGC

CPGC

012301231232333

0120122222

010111

000

++++=

+++=++=

⋅+=

EE14143

Arithmetic Circuits

§§Example of LookExample of Look--ahead Addersahead Adders

No carry prorogation delay

⊕=⋅+=+

iii

iiii

cpscpgc 1

1112 cpgc ⋅+=11 cc =

111 cps ⊕=

222 cps ⊕=)( 1112 cpgp ⋅+⊕=

2223 cpgc ⋅+=)( 11122 cpgpg ⋅+⋅+=

112122 cppgpg ⋅⋅+⋅+= Generate S2 without C2

C2

EE14144

Arithmetic Circuits

EE14145

Arithmetic Circuits

Static Static LookaheadLookahead CircuitCircuit

Co k,Gk Pk Gk 1– Pk 1– Co k 2–,

+( )+=

Co k,Gk Pk Gk 1– P k 1– … P1 G0 P0 Ci 0,

+( )+( )+( )+=

Expanding Lookahead equations:

All the way:

Co,3

C i,0

VDD

P0

P1

P2

P3

G0

G1

G2

G3

EE14146

Arithmetic Circuits

Dynamic Dynamic Lookahead Lookahead CircuitCircuit

• The worst-case delay path in this circuit has 6 NMOS transistors in series

• P, G Pattern = ????????

EE14147

Arithmetic Circuits

Summary of Summary of Lookahead Lookahead AdderAdder

• The size and fan-in of the gates needed to implement

this carry-lookahead scheme can clearly get out of

hand as bit number increases.

• The number of stages of lookahead is usually limited

to about 4.

• The circuit and layout are quite irregular compared

with Ripple-carry Adder

EE14148

Arithmetic Circuits

ConditionalConditional--Sum AddersSum Addersq The switch-like structuresq Using pass-transistor logic to achieve an addition time

significantly faster than a ripple-carry adder.q This operation cannot be completed until ci-1, the carry-

out of the (i-1)-th position, is available.q Instance of waiting for the arrival of the carry value,

conditional sum and conditional carry are generated by considering both possible values of the carry-in bit.

q All conditional sums and conditional carries can be generated in parallel

1

011

00

++=

++=

iiii

iiii

baSC

baSC

EE14149

Arithmetic Circuits

Example 1Example 1q A=(1101101)2 and B=(0110110)2

q Sum=(10100011)2 , Cout=1

01

10

1

0

Bit1

01

11

01

01

11

01

Si1

Ci1

10

01

10

10

01

10

Si0

Ci0

010110B

111011A

Bit0Bit2Bit3Bit4Bit5Bit6

C-1=0

EE14150

Arithmetic Circuits

Example 2Example 2q A=(1101101)2 and B=(0110110)2

q Sum=(10100011)2 , Cout=1

0

11

0

Bit1

01

11

001

101

Si1

Ci1

10

01

110

001

Si0

Ci0

010110B

111011A

Bit0Bit2Bit3Bit4Bit5Bit6

C-1=0

EE14151

Arithmetic Circuits

MultiMulti--levels levels q The number of bits grouped together is equal

to 2l at level l.

10

101

11001

Si0(2)

Ci0(2)

01

011

00101

Si1(2)

Ci1(2)

10

101

110

001

Si0(1)

Ci0(1)

01

011

001

101

Si1(1)

Ci1(1)

01

01

11

01

01

11

01

Si1(0)

Ci1(0)

10

10

01

10

10

01

10

Si0(0)

Ci0(0) C-1=0

EE14152

Arithmetic Circuits

TwoTwo--level 3level 3--bit Conditionalbit Conditional--Sum AdderSum Adder

ConditionalCell

ConditionalCell

ConditionalCell

MUX MUX MUX MUX MUX MUX

MUX MUX MUX

a2 b2 a1 b1 a0 b0

C21 C2

0 S21 S2

0 C11 C1

0 S11 S1

0 C01 C0

0 S01 S0

0

c-1

c2 s2 s1 s0

EE14153

Arithmetic Circuits

CarryCarry--save Adder (basics)save Adder (basics)

FullAdder

A B

Ci Co

Sum

iio

iiii

i

ACBCABC

CBACBACBACBA

CBAS

++=

+++=

⊕⊕=

FullAdder

A B Ci

Co S

1 bit

io CBASC ++=+2

Full Adder as a 3:2 CompressorFull Adder as a 3:2 Compressor

EE14154

Arithmetic Circuits

CarryCarry--Save AdderSave Adder

)(

)(

)(

01111

01111

01111

CCCCCC

BBBBBB

AAAAAA

nnn

nnn

nnn

………

−−−

−−−

−−−

=

=

=

)()(2

2

01111

01111

10

1

SSSSSRRRRR

SRNSum

nnn

nnn

ii

……

−−−

−−−

=

+×=

+== ∑

10543

,21

,,

,

NNNNC

NBNA

L=

==

EE14155

Arithmetic Circuits

CarryCarry--Save and CarrySave and Carry--Propagate Adder Propagate Adder (CSA/CPA)(CSA/CPA)

EE14156

Arithmetic Circuits

MultiMulti--input Carryinput Carry--Save AddersSave Adders

§ 4-input CSA tree § 5-input CSA tree § 6-input CSA tree

pipeling

EE14157

Arithmetic Circuits

Summary Summary

q Datapath designs are fundamentals for high-speed DSP, Multimedia, Communication digital VLSI designs.

q Most adders, multipliers, division circuits are now available in Synopsys Designware under different area/speed constraint.

q For details, check “Advanced VLSI Design”notes, or “Computer Arithmetic” textbooks