Digital Integrated Circuits (83-313) · Booth Recoding 10 Multiplying by ‘0’ is redundant. Can we reduce the number of partial products? Based on the observation that We can turn

Digital Integrated Circuits

(83-313)

Semester B, 2015-16

Lecturer: Adam Teman

TAs: Itamar Levi, Robert Giterman 1

Lecture 9:

Multipliers

Grade School Multiplication

2

Multiplication using serial addition

3

Binary Multiplication

4

multiplicand

multiplier

partial

product

array

double precision product

N

2N

N can be formed in parallel

Serial Shift and Add

5

2

for RCAserial addert O N t O N

Array Multiplier

6

Y0

Y1

X3 X2 X1 X0

X3

HA

X2

FA

X1

FA

X0

HA

Y2X3

FA

X2

FA

X1

FA

X0

HA

Z1

Z3Z6Z7 Z5 Z4

Y3X3

FA

X2

FA

X1

FA

X0

HA

Z2

Z0

Many Critical Paths

7

1 2 1 1mult carry sum addt M N t N t N t

Carry-Save Multiplier

8

HA HA HA HA

FAFAFAHA

FAHA FA FA

FAHA FA HA

Vector Merging Adder

1 1mult carry AND merget N t N t t

Multiplier Floorplan

9

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 broadcasted

through the complete array.

( )

Booth Recoding

10

Multiplying by ‘0’ is redundant.

Can we reduce the number of partial products?

Based on the observation that

We can turn sequences of 1’s into sequences of 0’s

For example: 0111=1000-0001

So we can introduce a ‘-1’ bit and recode the multiplier:

For example, the number 56

1

0

2 2 1n

i n

i

Radix-2 Booth Recoding

11

Parse multiplier from left to right

For each change from 0 to 1, encode a ‘1’

For each change from 1 to 0, encode a ‘-1’

For bit 0, assume bit i=-1 is a 0

Example: 0011 0111 0011

Modified (Radix-4) Booth Recoding

12

Radix-2 Booth Recoding doesn’t work for parallel hardware implementations, since:

A worst case (010101010101010) doesn’t reduce the number of partial products.

Variable length recoders (according to the length of ‘1’ strings) cannot be implemented efficiently.

Instead, just assume a constant length recoder.

First apply standard booth recoding.

Next encode each pair of bits:

This can be summarized in a truth table:

Partial Product Selection Table

Multiplier Bits Recorded Bits

000 0

001 + Multiplicand

010 + Multiplicand

011 +2 × Multiplicand

100 -2 × multiplicand

101 - Multiplicand

110 - Multiplicand

111 0

Modified (Radix-4) Booth Recoding

13

For example, let’s take our previous example:

0011 0111 0011 = 01 0-1 10 0-1 01 0-1

We could have done this by using the table:

0 0 1 1 0 1 1 1 0 0 1 1

To implement this we need pretty simple hardware:

Mi

yj

Xi

yj-1

2Xi

PPij

Booth

Selector

Booth

Encoder

x2i+1

x2i

x2i-1

Tree Multipliers

14

Can we further reduce the multiplier delay

by employing logarithmic (tree) structures?

PP1PP2 PP3PP4PP5

+

CLA

Result

PP6PP7PP8

+ +

+ +

PP0

+

+

Wallace-Tree Multiplier

15

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

16

Wallace-Tree Multiplier

17

6 5 4 3 2 1 0 6 5 4 3 2 1 0

Partial products First stage

Bit position

6 5 4 3 2 1 0 6 5 4 3 2 1 0

Second stage Final adder

FA HA

(a) (b)

(c) (d)

HA

Pipelining Multipliers

18

Pipelining can be applied to most multiplier structures: