ECE 555 Digital Circuits & Components

ECE555ECE555Lecture 10Lecture 10

Nam Sung KimUniversity of Wisconsin – Madison

Dept. of Electrical & Computer Engineering

1

ECE 555 Digital Circuits & Components

OutlineOutline Multiplier

• Array Multiplier• CSA/Wallace Tree Multiplier

Shifter

Datapath Bitslice Organization

2

ECE 555 Digital Circuits & Components

MultiplicationMultiplication Example

3

1100 : 1210 0101 : 510

ECE 555 Digital Circuits & Components

MultiplicationMultiplication Example

4

1100 : 1210 0101 : 510 1100

ECE 555 Digital Circuits & Components

MultiplicationMultiplication Example

5

1100 : 1210 0101 : 510 1100 0000

ECE 555 Digital Circuits & Components

MultiplicationMultiplication Example

6

1100 : 1210 0101 : 510 1100 0000 1100

ECE 555 Digital Circuits & Components

MultiplicationMultiplication Example

7

1100 : 1210 0101 : 510 1100 0000 1100 0000

ECE 555 Digital Circuits & Components

MultiplicationMultiplication Example

8

1100 : 1210 0101 : 510 1100 0000 1100 000000111100 : 6010

ECE 555 Digital Circuits & Components

MultiplicationMultiplication Example

M x N-bit multiplication• Produce N M-bit partial products• Sum these to produce M+N-bit product

9

1100 : 1210 0101 : 510 1100 0000 1100 000000111100 : 6010

multiplier

multiplicand

partialproducts

product

ECE 555 Digital Circuits & Components

General Multiplication FormGeneral Multiplication Form Multiplicand: Y = (yM-1, yM-2, …, y1, y0)

Multiplier: X = (xN-1, xN-2, …, x1, x0)

Product:

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1 1 1 1

0 0 0 0

2 2 2M N N M

j i i jj i i j

j i i j

P y x x y

x0y5 x0y4 x0y3 x0y2 x0y1 x0y0

y5 y4 y3 y2 y1 y0

x5 x4 x3 x2 x1 x0

x1y5 x1y4 x1y3 x1y2 x1y1 x1y0

x2y5 x2y4 x2y3 x2y2 x2y1 x2y0

x3y5 x3y4 x3y3 x3y2 x3y1 x3y0

x4y5 x4y4 x4y3 x4y2 x4y1 x4y0

x5y5 x5y4 x5y3 x5y2 x5y1 x5y0

p0p1p2p3p4p5p6p7p8p9p10p11

multiplier

multiplicand

partialproducts

product

ECE 555 Digital Circuits & Components

Dot DiagramDot Diagram Each dot represents a bit (or partial product)

11

partial products

multiplier x

x0

x15

ECE 555 Digital Circuits & Components

Array MultiplierArray Multiplier

12

y0y1y2y3

x0

x1

x2

x3

p0p1p2p3p4p5p6p7

B

ASin Cin

SoutCout

BA

CinCout

Sout

Sin

=

CSAArray

CPA

critical path BA

Sout

Cout CinCout

Sout

=Cin

BA

ECE 555 Digital Circuits & Components

Rectangular ArrayRectangular Array Squash array to fit rectangular floorplan

13

y0y1y2y3

x0

x1

x2

x3

p0

p1

p2

p3

p4p5p6p7

ECE 555 Digital Circuits & Components

Wallace Tree MultiplierWallace Tree Multiplier Using 3:2 compressor (FA!)

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0123456

0123456

0123456

0123456

x0

x3

ECE 555 Digital Circuits & Components

Wallace Tree MultiplierWallace Tree Multiplier Using 3:2 compressor (FA!)

x0y0x0y1x0y2x0y3

x1y0x1y1x1y2x1y3

x2y0x2y1x2y2x2y3

x3y0x3y1x3y2x3y3

+ +

+

ECE 555 Digital Circuits & Components

Wallace Tree MultiplierWallace Tree Multiplier Using 3:2 compressor (FA!)

x0y0x0y1x0y2

x0y3

x1y0

x1y1

x1y2x1y3 x2y0

x2y1x2y2

x3y0x3y1x3y3

+ +

+++x2y3

x3y2

+

+

ECE 555 Digital Circuits & Components

Wallace Tree MultiplierWallace Tree Multiplier

17

Partial products

First stage

Second stage

Final adder

FA FA FA

HA HA

FA

x3y3

z7 z6 z5 z4 z3 z2 z1 z0

x3y2x2y3

x1y1x3y0 x2y0 x0y1x0y2

x2y2x1y3

x1y2x3y1x0y3 x1y0 x0y

ECE 555 Digital Circuits & Components

Parallel Programmable ShiftersParallel Programmable Shifters

Dat

a InControl =

Dat

a O

ut

Shift amountShift directionShift type (logical, arith, circular)

Shifters used in multipliers, floating point units

Consume lots of area if done in random logic gates

ECE 555 Digital Circuits & Components

A Programmable Binary ShifterA Programmable Binary Shifter

rgt nop left

Ai

Ai-1 Bi-1

Bi

Ai Ai-1rgt nop left Bi Bi-1

A1 A00 1 0 A1 A0

A1 A01 0 0 0 A1

A1 A00 0 1 A0

0

ECE 555 Digital Circuits & Components

4-bit Barrel Shifter4-bit Barrel Shifter

A0

A1

A2

A3

B0

B1

B2

B3

Sh1

Sh2

Sh3

Sh0 Sh1 Sh2 Sh3

Example: Sh0 = 1 B3B2B1B0 = A3A2A1A0

Sh1 = 1 B3B2B1B0 = A3A3A2A1

Sh2 = 1 B3B2B1B0 = A3A3A3A2

Sh3 = 1 B3B2B1B0 = A3A3A3A3

Area dominated by wiring

ECE 555 Digital Circuits & Components

4-bit Barrel Shifter Layout4-bit Barrel Shifter Layout

BufferSh3Sh2Sh1Sh0

A3

A2

A1

A0

Widthbarrel ~ 2 pm N N = max shift distance, pm = metal pitchDelay ~ 1 fet + N diff caps

Widthbarrel

Only one Sh#active at a timel

ECE 555 Digital Circuits & Components

8-bit Logarithmic Shifter8-bit Logarithmic Shifter

A3

A2

A1

A0

!Sh1Sh1 !Sh2Sh2 !Sh3Sh3

B0

B1

B2

B3

ECE 555 Digital Circuits & Components

8-bit Logarithmic Shifter Layout8-bit Logarithmic Shifter Layout

Widthlog ~ pm(2K+(1+2+…+2K-1)) = pm(2K+2K-1) K = log2 NDelay ~ K fets + 2 diff caps

A0

B3

B2

B1

B0

A1

A2

A3

1 2 4

ECE 555 Digital Circuits & Components

Datapath Bit-Sliced OrganizationDatapath Bit-Sliced Organization

27

Control Flow

Bit 0

Bit 1

Bit 2

Bit 3

Tile identical bit-slice elements

Re

gis

ter

File

Pip

elin

e R

egis

ter

Ad

der

Sh

ifter

Pip

elin

e R

egis

ter

Mu

ltip

lexe

r

Mu

ltip

lexe

r

Data Flow

Pip

elin

e R

egis

ter

From I$

Pip

elin

e R

egis

ter

To/From D$

decoder