digggital integrated circuits -...

Digital Integrated Digital Integrated g gg gCircuitsCircuits

Arithmetic CircuitsArithmetic Circuits

EE141© Digital Integrated Circuits2nd Arithmetic Circuits1

A Generic Digital ProcessorA Generic Digital ProcessorA Generic Digital ProcessorA Generic Digital Processor

MEMORYU

T

CONTROL

UT

-OU

TPU

DATAPATH

INPU


Building Blocks for Digital ArchitecturesBuilding Blocks for Digital ArchitecturesBuilding Blocks for Digital ArchitecturesBuilding Blocks for Digital Architectures

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

Memory- RAM, ROM, Buffers, Shift registers

Control- Finite state machine (PLA, random logic.)

Counters- Counters

Interconnect- SwitchesSwitches- Arbiters- Bus


An Intel MicroprocessorAn Intel MicroprocessorAn Intel MicroprocessorAn Intel Microprocessor

-1 M

ux

-1 M

ux

CARRYGENg64a

9- 5-

ck1sum sumb

SUM

SEL

to Cachenode1

REG

9-1

Mux

2-1

Mux SUMGEN

+ LUb

s0

s1

LU : Logical

1000um

gUnit

Itanium has 6 integer execution units like this


BitBit Sliced DesignSliced DesignBitBit--Sliced DesignSliced DesignControlControl

Bit 3

Bit 2ern utBit 2

Bit 1R

egist

er

Add

er

Shift

er

Mul

tiple

xe

Dat

a-In

Dat

a-O

u

Bit 0M

Tile identical processing elements


BitBit--Sliced DatapathSliced DatapathBitBit--Sliced DatapathSliced DatapathFrom register files / Cache / Bypass

Shifter

Multiplexers

Adder stage 1

WiringLo Lo Lo

Adder stage 2

Wiring

oopback Bus

opback Bus

oopback Bus

Adder stage 3

Bit slice

Bit slice

Bit slice

Bit slice e 0

e 2e 163

Sum Select


To register files / Cache

Itanium Integer DatapathItanium Integer DatapathItanium Integer DatapathItanium Integer Datapath

EE141© Digital Integrated Circuits2nd Arithmetic Circuits7Fetzer, Orton, ISSCC’02

AddersAddersAddersAdders


FullFull--AdderAdderFullFull AdderAdderA B

CoutCin Fulladder

Sum


The Binary AdderThe Binary AdderThe Binary AdderThe Binary AdderA B

CoutCin Fulladder

Sum

adder

S A B C⊕ ⊕S A B Ci⊕ ⊕=

A= BCi ABCi ABCi ABCi+ + +i i i iCo AB BCi ACi+ +=


Express Sum and Carry as a function of P G DExpress Sum and Carry as a function of P G DExpress 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, BGenerate (G) = ABPropagate (P) = A ⊕ BDelete = A B

Can also derive expressions for S and Co based on D and PNote that we will be sometimes using an alternate definition for


Propagate (P) = A + B

The RippleThe Ripple--Carry AdderCarry AdderThe RippleThe Ripple--Carry AdderCarry Adder


Complimentary Static CMOS Full AdderComplimentary Static CMOS Full AdderComplimentary Static CMOS Full AdderComplimentary Static CMOS Full Adder


Inversion PropertyInversion PropertyInversion PropertyInversion Property

A B A B

CoCi FA CoCi FA

S S


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


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

V

VDD VDD A

VDD

B

A

A BKill

"0"-Propagate

Ci

Ci

BBA

Ci

AGenerate"1"-Propagate

Ci

CiSCo

BBA A B Ci

B

A

24 transistors


Mirror AdderMirror AdderMirror AdderMirror AdderStick Diagram

VDD

CA BB A C C C A BCiA BB A Ci Co Ci A B

Co

GND

S


GND

The Mirror AdderThe Mirror AdderThe Mirror AdderThe Mirror Adder•The NMOS and PMOS chains are completely symmetrical. p y yA maximum of two series transistors can be observed in the carry-generation circuitry.Wh l i t th ll th t iti l i i th i i i ti•When laying out the cell, the most critical issue is the minimization of the capacitance at node Co. The reduction of the diffusion capacitances is particularly important.

•The capacitance at node Co is composed of four diffusion capacitances, two internal gate capacitances, and six gate

it i th ti dd llcapacitances in the connecting adder cell .•The transistors connected to Ci are placed closest to the output.O l th t i t i th t h t b ti i d f•Only the transistors in the carry stage have to be optimized for optimal speed. All transistors in the sum stage can be minimal size.


Transmission Gate Full AdderTransmission Gate Full AdderTransmission Gate Full AdderTransmission Gate Full Adder

P

AVDD

VDDCi

S

P

PPA A Ci

S

P

P Sum Generation

BVDD

AP

AB VDD

C

P

P Carry GenerationCi

ACi

CoCi

P

P Carry Generation

S t PSetup


Manchester Carry ChainManchester Carry ChainManchester Carry ChainManchester Carry ChainV

PVDD

Pi

VDD

φ

CGi

Pi CoCi

φ

CoCi

Di

Gi

i

Pi φ


Manchester Carry ChainManchester Carry ChainManchester Carry ChainManchester Carry ChainVDD

φ

C

P0

DD

P1 P2 P3

G2

C3

G3Ci,0 G1G0

φ

C3C2C1C0


Manchester Carry ChainManchester Carry ChainManchester Carry ChainManchester Carry ChainStick Diagram

Propagate/Generate Row

g

Pi + 1 Gi + 1 φPi Gi φ

VDD

CiCi - 1Ci + 1

GND

Inverter/Sum Row


CarryCarry--Bypass AdderBypass AdderCarryCarry Bypass AdderBypass AdderP0 G1 P0 G1 P2 G2 P3 G3 Also called

Carry-SkipFA FA FA FA

Co,3Co,2Co,1Co,0Ci,0Carry-Skip

P0 G1 P0 G1 P2 G2 P3 G3BP=PoP1P2P3

FA FA FA FACo,2Co,1Co,0Ci,0

Co,3

ultip

lexe

rM

u

Idea: If (P0 and P1 and P2 and P3 = 1)Idea: If (P0 and P1 and P2 and P3 = 1)then Co3 = C0, else “kill” or “generate”.


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


Carry Ripple versus Carry BypassCarry Ripple versus Carry BypassCarry Ripple versus Carry BypassCarry Ripple versus Carry Bypass

tp

ripple adder

bypass adder

N4..8


N

CarryCarry--Select AdderSelect AdderCarryCarry Select AdderSelect AdderSetup

"0" Carry Propagation"0"

P,G

y p g

"1" Carry Propagation

0

"1" 1 Carry Propagation

MultiplexerC k 1 C

1

Multiplexer

S G i

Co,k-1 Co,k+3

Carry Vector

Sum Generation


Carry Select Adder: Critical Path Carry Select Adder: Critical Path Carry Select Adder: Critical Path Carry Select Adder: Critical Path


Linear Carry Select Linear Carry Select Linear Carry Select Linear Carry Select Bit 0-3 Bit 4-7 Bit 8-11 Bit 12-15

Setup

"0" Carry

Setup

"0" Carry

Setup

"0" Carry

Setup

"0" Carry

(1)

y

"1" Carry

"0"

"1"

y

"1" Carry

"0"

"1"

y

"1" Carry

"0"

"1"

y

"1" Carry

"0"

"1"

(1)

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

Multiplexer Multiplexer Multiplexer MultiplexerCi,0

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

(9)

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

Sum Generation Sum Generation Sum Generation Sum Generation

S0-3 S4-7 S8-11 S12-15 (10)


Square Root Carry Select Square Root Carry Select Square Root Carry Select Square Root Carry Select S

Bit 0-1 Bit 2-4 Bit 5-8 Bit 9-13 Bit 14-19

Setup

"0" Carry "0"

Setup

"0" Carry "0"

Setup

"0" Carry "0"

Setup

"0" Carry "0"

(1)

"1" Carry "1"

"1" Carry "1"

"1" Carry "1"

"1" Carry "1"

(1)

(3) (4) (5) (6) (7)(3)

Multiplexer

Sum Generation

Multiplexer

Sum Generation

Multiplexer

Sum Generation

Multiplexer

Sum Generation

Ci,0

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

S

(8)

Sum Generation Sum Generation Sum Generation Sum Generation

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

Sum

S14-19 (9)


Adder Delays Adder Delays -- Comparison Comparison Adder Delays Adder Delays -- Comparison Comparison


LookAhead LookAhead -- Basic IdeaBasic IdeaLookAhead LookAhead -- Basic IdeaBasic Idea


LookLook--Ahead: TopologyAhead: TopologyLookLook--Ahead: TopologyAhead: Topology


Logarithmic LookLogarithmic Look--Ahead AdderAhead AdderLogarithmic LookLogarithmic Look Ahead AdderAhead AdderFA0

A7A6A5A4A3A2A1

0

A0A1

tp∼ N

A2A3

A4

FA4A5

A6

Atp∼ log2(N)

A7


Carry Lookahead TreesCarry Lookahead TreesCarry Lookahead TreesCarry Lookahead Trees

C G CCo 0, G0 P0Ci 0,+=

Co 1, G1 P1 G0 P1P0 Ci 0,+ +=

Co 2, G2 P2G1 P2 P1G0 P+ 2 P1P0Ci 0,+ +=

G2 P2G1+( )= P2P1( ) G0 P0Ci 0+( )+ G 2:1 P2:1Co 0+=G2 2G1( ) 2 1( ) G0 0Ci 0,( ) G 2:1 2:1Co 0,

Can continue building the tree hierarchically.


Tree AddersTree AddersTree AddersTree AddersS 0 S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S 10 S 11 S 12 S 13 S 14 S 15S S S S S S S S S S S S S S S S

(A0,

B0)

(A1,

B1)

(A2,

B2)

(A3,

B3)

(A4,

B4)

(A5,

B5)

(A6,

B6)

(A7,

B7)

(A8,

B8)

(A9,

B9)

10, B

10)

11, B

11)

12, B

12)

13, B

13)

14, B

14)

15, B

15)

16-bit radix-2 Kogge-Stone tree

( ( ( ( ( ( ( ( ( (

(A1

(A1

(A1

(A1

(A1

(A1


Tree AddersTree AddersTree AddersTree AddersS

0

S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

S11

S12

S13

S14

S15

(a0,

b0)

(a1,

b1)

(a2,

b2)

(a3,

b3)

(a4,

b4)

(a5,

b5)

(a6,

b6)

(a7,

b7)

(a8,

b8)

(a9,

b9)

(a10

, b10

)

(a11

, b11

)

(a12

, b12

)

(a13

, b13

)

(a14

, b14

)

(a15

, b15

)

16-bit radix-4 Kogge-Stone Tree


Sparse TreesSparse TreesSparse TreesSparse Trees

S1

S3

S5

S7

S9

S11

S13

S15

S0

S2

S4

S6

S8

S10

S12

S14

a 0, b 0)

a 1, b 1)

a 2, b 2)

a 3, b 3)

a 4, b 4)

a 5, b 5)

a 6, b 6)

a 7, b 7)

a 8, b 8)

a 9, b 9)

0, b 10

)

1, b 11

)

2, b 12

)

3, b 13

)

4, b 14

)

5, b 15

)

(a (a (a (a (a (a (a (a (a (a

(a10

(a11

(a12

(a13

(a14

(a15

16 bit radix 2 sparse tree with sparseness of 2


16-bit radix-2 sparse tree with sparseness of 2

Tree AddersTree AddersTree AddersTree Adders

S 0 S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S 10 S 11 S 12 S 13 S 14 S 15

B0)

B1)

B2)

B3)

B4)

B5)

B6)

B7)

B8)

B9)

B10

)

B11

)

B12

)

B13

)

B14

)

B15

)

(A0,

(A1,

(A2,

(A3,

(A4,

(A5,

(A6,

(A7,

(A8,

(A9,

(A10

, B

(A11

, B

(A12

, B

(A13

, B

(A14

, B

(A15

, B

B t K T


Brent-Kung Tree

Example: Domino AdderExample: Domino AdderExample: Domino AdderExample: Domino AdderVDD

VDD Clk Gi = aibi

Clk Pi= ai + bi

a b

ai

Clk

ai bi

Clk

bi

Clk

Propagate Generate


Example: Domino AdderExample: Domino AdderExample: Domino AdderExample: Domino Adder

VDDVDD

ClkClkk

P

Pi:i-2k+1

Clkk

P

Gi:i-2k+1

Pi:i-k+1

Pi-k:i-2k+1

Gi:i-k+1

Pi:i-k+1

Gi-k:i-2k+1

P t G tPropagate Generate


Example: Domino SumExample: Domino SumExample: Domino SumExample: Domino SumVDD

Clk

VDD

Clkd

Keeper

Clk

Gi:0

Sum

Clkd

Clk Si0

Clk

Clkd

Gi:0

S 1

Clk

Si1


MultipliersMultipliersMultipliersMultipliers


The Binary MultiplicationThe Binary MultiplicationThe Binary MultiplicationThe Binary MultiplicationM N 1

Z X·· Y× Zk2k

k 0=

M N 1–+

∑= =

M 1⎛ ⎞ N 1⎛ ⎞

Xi2i

i 0=

M 1–

∑⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

Yj2j

j 0=

N 1–

∑⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

=

N 1⎛ ⎞M 1XiYj2

i j+

j 0=

N 1–

∑⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

i 0=

M 1–

∑=

iM 1–

with

X Xi2i

i 0=∑=

Y Y 2j

N 1–


Y Yj2j

j 0=∑=

The Binary MultiplicationThe Binary MultiplicationThe Binary MultiplicationThe Binary Multiplication


The Array MultiplierThe Array MultiplierThe Array MultiplierThe Array Multiplier


The MxN Array MultiplierThe MxN Array MultiplierC iti l P thC iti l P th—— Critical PathCritical Path

HA FA FA HA

HAFAFAFA Critical Path 1

Critical Path 2

C i i l P h 1 & 2FAFA FA HA Critical Path 1 & 2


CarryCarry--Save MultiplierSave MultiplierCarryCarry Save MultiplierSave MultiplierHA HA HA HA

FAFAFAHA

FAHA FA FA

FAHA FA HA

Vector Merging Adder


Multiplier FloorplanMultiplier FloorplanMultiplier FloorplanMultiplier FloorplanX0X1X2X3

SCSCSCSC Z0

Y1

Y0

HA Multiplier Cell

SCSCSCSC

0

Y2

i C

FA Multiplier Cell

SCSCSCSC

Z1

Y3

Vector Merging Cell

X and Y signals are broadcastedSCSCSCSC

CCCC

Z2

X and Y signals are broadcastedthrough the complete array.( )

SSSS

Z3Z4Z5Z6Z7


WallaceWallace--Tree MultiplierTree MultiplierWallaceWallace--Tree MultiplierTree Multiplier


WallaceWallace--Tree MultiplierTree MultiplierWallaceWallace--Tree MultiplierTree Multipliery0 y1 y2

FACi-1

y0 y1 y2 y3 y4 y5

FA

y3

Ci-1Ci

FA FACi-1

Ci-1

Ci

Ci

y4

Ci-1Ci

FACi-1Ci

FA

y5

C FA

i 1i

FA

S

Ci FA

CC S


SC

Multipliers Multipliers ——SummarySummaryMultipliers Multipliers ——SummarySummary

• Optimization Goals Different Vs Binary Adder

• Once Again: Identify Critical Path

Oth ibl t h i• Other possible techniques- Logarithmic versus Linear (Wallace Tree Mult)- Data encoding (Booth)- Pipelining

FIRST GLIMPSE AT SYSTEM LEVEL OPTIMIZATION


ShiftersShiftersShiftersShifters


The Binary ShifterThe Binary ShifterThe Binary ShifterThe Binary ShifterRight Leftnopg p

Ai Bi

Ai-1 Bi-1

Bit-Slice i

...


The Barrel ShifterThe Barrel ShifterThe Barrel ShifterThe Barrel ShifterA3

B3

Sh1A2

B3

B

Sh2A1

B2

: Data Wire

Sh3

1

A

B1: Control Wire

A0

B0

Sh3Sh2Sh1Sh0

Area Dominated by Wiring


y g

4x4 barrel shifter4x4 barrel shifter4x4 barrel shifter4x4 barrel shifterA3

A2A2

A 1

A 0

B ffSh3S h2Sh 1Sh0

BufferWidthbarrel ~ 2 pm M


Logarithmic ShifterLogarithmic ShifterLogarithmic ShifterLogarithmic ShifterSh1 Sh1 Sh2 Sh2 Sh4 Sh4

A3 B3

A2 B22

A1 B1A1

A

B1

A0 B0


00--7 bit Logarithmic Shifter7 bit Logarithmic Shifter00--7 bit Logarithmic Shifter7 bit Logarithmic Shifter

A3Out3

A 2

Out3

2

A1

Out2

1

A

Out1

A0 Out0


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