4x4 signed multiplication 2’s complement: x= –x n–1 2 n–1 +unsigned y*x=z

6-1

• 4x4 Signed Multiplication2’s Complement: X= –xn–12n–1+Unsigned

Y*X=Z

Multiplications

6-2

ADD Y if xi =1( i=0 to n–2); Shift

SUB Y if xn-1 =1; Shift

EX:Y= – 5, X=3

2’s C Sequential MUL 1-1

1 0 1 1 = -5 0 0 1 1 = 3+= 1 0 1 1Shift 1 1 0 1 1+= 1 0 0 0 1Shift 1 1 0 0 0 1Shift 1 1 1 0 0 0 1Shift 1 1 1 1 0 0 0 1

6-3

ADD Y if xi =1( i=0 to n–2); Shift

SUB Y if xn-1 =1; Shift

2’s C Sequential MUL 1-2

1 0 1 1 = -5 1 1 0 1 = -3

1 0 1 1Shift 1 1 0 1 1Shift 1 1 1 0 1 1+= 1 0 0 1 1 1Shift 1 1 0 0 1 1 1 = 0 0 0 1 1 1 1Shift 0 0 0 0 1 1 1 1

6-4

1’s C: X= –xn–1(2n–1–ulp) +Unsigned

A(0)= Y(ulp) if xn–1 =1 else A(0)=0

ADD Y if xi =1( i=0 to n–2); Shift

SUB Y if xn-1 =1; Shift

EX1: Y= 5, X= – 3

1’s C Sequential MUL 1-1

6-5

5* – 3 = – 22 ???

1’s C Sequential MUL 1-2

0 1 0 1 = 5 1 1 0 0 = -3A(0) 0 1 0 1Shift 0 0 1 0 1Shift 0 0 0 1 0 1+= 0 1 1 0 0 1Shift 0 0 1 1 0 0 1 0 1 0 1= 1 1 0 1 0 0 1Shift 1 1 1 0 1 0 0 1

6-6

5* – 3 = – 15

1’s C Sequential MUL 1-3

0 1 0 1 0 0 0 =5 1 1 0 0 =-3A(0) 0 1 0 1Shift 0 0 1 0 1Shift 0 0 0 1 0 1+= 0 1 1 0 0 1Shift 0 0 1 1 0 0 1 0 1 0 1 0 0 0= 1 1 1 0 0 0 0Shift 1 1 1 1 0 0 0 0

6-7

– 5* – 3 = 10 !!! What’s wrong???

1’s C Sequential MUL 1-4

1 0 1 0 = -5 1 1 0 0 = -3A(0) 1 0 1 0Shift 1 1 0 1 0Shift 1 1 1 0 1 0+= 1 1 0 0 0 1 0Shift 1 1 0 0 0 1 0 =1 0 0 0 1 0 1 0Shift 0 0 0 0 1 0 1 0

6-8

End around carry: – 5* – 3 = 12!!!???

1’s C Sequential MUL 1-5

1 0 1 0 = -5 1 1 0 0 = -3A(0) 1 0 1 0Shift 1 1 0 1 0Shift 1 1 1 0 1 0+= 1 1 0 0 0 1 0Shift 1 1 0 0 0 1 1 =1 0 0 0 1 0 1 1Shift 0 0 0 0 1 1 0 0

6-9

Add 1s instead of 0’s: – 5* – 3 = 15 O.K.

1’s C Sequential MUL 1-6

1 0 1 0 1 1 = -5 1 1 0 0 = -3A(0) 1 0 1 0Shift 1 1 0 1 0Shift 1 1 1 0 1 0+= 1 1 0 0 1 0 1Shift 1 1 0 0 1 1 0 =1 0 0 0 1 1 1 0Shift 0 0 0 0 1 1 1 1

6-10

2’C : n-bit adder & 2n-bit Shift Reg.1’C : 2n-bit adder & 2n-bit Shift Reg.

Compare 2’sC & 1’sC Seq. MUL

6-11

• Two SUB; (n-2) ADD X*Y=Z=(z1, z2, …zn)

Multiplications(2)

6-12

• 4x4 Signed MultiplicationSign extensionOne SUB; (n-2) ADD

Multiplications(3)

6-13

• 4x4 Signed Multiplicationall ADD (2n)

Multiplications(4)

6-14

• 4x4 Signed Multiplicationall ADD (2n–1)

Multiplications(5)

6-15

Serial Multiplications (1)

Rin

LastBit

Rout

CLR

XY

CLRY

CLRPin

CinCLR CLR

X

0

1 Pout

Cout

(a)

MUX

FA0

1

0

1FA

CLR

FA0

1

Control path

0

XYXYPinCinRinLastBit

PoutCoutRout

#1

01

FirstBit

LastBit

(b)

X

Y

P

CLR CLR

(d)(c)

0

XYXYPinCinRinLastBit

PoutCoutRout

#2 XYXYPinCinRinLastBit

PoutCoutRout

#(N-1)

6-16

• Serial Multiplication Scheme

Serial Multiplications(2)

0 1 2 3

X iY i

X0Y0

6-17

• Serial Multiplication Scheme

Serial Multiplications(3)

0 1 2 3

X iY iZ0

X1Y1

W10 +W01

W11

6-18

• Serial Multiplication Scheme

Serial Multiplications(4)

0 1 2 3

X iY iZ1

X2Y2

W22V2 +W20 +W02

W21 +W12

6-19

• Serial Multiplication Scheme

Serial Multiplications(5)

0 1 2 3

X iY iZ2

X3Y3

W33V3 +W30 +W03

V4 +W31 +W13

W32 +W23

6-20

• Serial Multiplication Scheme

Serial Multiplications(6)

0 1 2 3

X iY iZ3

V4 +W30 +W03

V5 +W31 +W13

V6 +W32 +W23

W33+W33 +W33

6-21

• Serial Multiplication Scheme

Serial Multiplications(7)

0 1 2 3

X iY iZ4

V5 +W30 +W03

V6 +W31 +W13

+W33 +W33

V7

+W32 +W23

6-22

• Serial Multiplication Scheme

Serial Multiplications(7)

0 1 2 3

X iY iZ5

V6 +W30 +W03

V7 +W31 +W13

+W33 +W33

+W32 +W23

6-23

• Serial Multiplication Scheme

Serial Multiplications(7)

0 1 2 3

X iY iZ6

V7 +W30 +W03

6-24

Serial/Parallel Multiplier(1-1)

C4

C3

C9 C8 C7 C6 C5 C4 C3

W00W10W20W30

W01W11W21W31

W02W12W22W32

W0'3W1'3W2'3W3'3

b

P0P1P2P3P4P5P6P7

4

C1

C

C2

C3

C4

C2

MSB

3

W30

X 31

X 32

C2

6-25

Serial/Parallel Multiplier(1-2)

FA FFR

a3

FFR

FA FFR

a2

FFR

FA FFR

a1

FFR

FA FFR

a0

FFR

STR

MSB

bi& & & &

&

FFR

FA FFR

FFR

6-26

Serial/Parallel Multiplier(2)

C4 C3 C2

W00W10W20W'30

W01W11W21W'31

W02W12W22W'32

W'03W'13W'23W33

11

P0P1P2P3P4P5P6P7

C1

C2

C3

C4inverter

C1

MSB

FFR

&

a3

FA FFR

&

a2

FFR

FA FFR

&

a1

FFR

FA

&

a0

FFRSTR

MSB

bi

FAFA FA

MSB

FFR

FFR

FFR

FFR

FFR

6-27

Serial/Parallel Multiplier(3-1)

C2A

C2B

C4A C4B C3A C3B C2A C2B C1A

W00W10W20W30

W01W11W21W31

W02W12W22W32

W0'3W1'3W2'3W3'3

b

P0P1P2P3P4P5P6P7

2A

C1B

C

C1A

C2B

C2A

C1B

MSB

3

W30

X 31

X 32

C1A

C1B

C1A

C2B

C2A

6-28

Serial/Parallel Multiplier(3-2)

FA

a3

FA

a2

FA

a1

FA

a0MSB

InA& & & &

&

FA

InB& & & &

a3 a2 a1 a0

FA FA FA FA

FF

FAFF

Rising edgetriggered

falling edgetriggered 0

6-29

Serial/Parallel Multiplier(4-1)

C8

C7 C6 C5

C4 C3 C2

W00W10W20W'30

W01W11W21W'31

W02W12W22W'32

W'03W'13W'23W33

1

11

P0P1P2P3P4P5P6P7

STR

C1

C1

C2

C3C4

C5

C1

1

C6

1

C7

1

C8

Discard by

MB

Next STR

6-30

Serial/Parallel Multiplier(4-2)

FA3 SF3R

&

a3

CF3P

FA2 SF2R

&

a2

CF2R

FA1 SF1R

&

a1

CF1R

FA0 SF0R

&

a0

CF0R

STR

MB

bi

D

1

MUX

FFR

D

0

MUX

orD

0

MUX

T2/T4

S@0/S@1

CLK

T5/T7

T1/T1

T2/T4

T3/T2

T1/T2

T4/T6 T3/T5

T4 /T5T6/T5 T5/T4 T4/T3 T3/T2

T7/T5 T4/T7 T3/T6 T2/T5

T6/T5 T6/T4 T4/T6 T4/T3 T3/ T2

T2/T1T2/

T3T3/T4

T4 /T7

T7 /T4

T6 /T3

T4 /T2

T4 / T3

T5 /T4

T7/T5

T6 /T7

P

CF1P

6-31

Serial/Parallel Multiplier(4-3)

Table 1 Test patterns

STR MB Bi An-1 an-2...a0 SFs CFs

T1 1 0 1 0 1 1111...11 1000...00

T2 0 1 0 1 1 1000...00 0111...11

T3 0 0 1 1 0 0011...11 0100...0

T41 0 1 1 0 1 0101...11 0000...0

repeat vector T4 (n-1) Times shift out 0000....0

6-32

Serial/Parallel Multiplier(5)

Table 2 Area and delay of the logic elements and multipliers

NAND2 Inverter XOR2 FF FAArea 1 0.5 1.5 3.5 7.5Delay 1 0.5 2.3 3.8 4.6

[2] n n (n-1) 2(n-1) 2(n-1)[1] n+1 n+1 n 2n+1 n+1[4] n 1 n 2n n

Table 3 Comparisons

clock last clock Area Delay AT Testable I/O pins[2] 12.2 4.6n+8.1 25n-23.5 16.8n-4.1 420n2-497.3n+98.35 ? 2n+4

[1] 12.2 12.2 17.5n+12.5 24.4n+12.2 427n2+518n+152.5 ? n+4

[4] 11.7 11.7 17n-0.5 23.4n 397.8n2-11.7n Yes n+4

6-33

• Reduce Partial Product terms

• Accelerate Addition

• 3 Types of MultiplicationParallel MULHS Seq. MULArray MUL

High-Speed Multiplications

6-34

• Booth AlgorithmConcept: A* 0011...110= A*0100…0(–1)0 Ex. Old PP#=4, New PP# =2

Reduce Partial Product terms

6-35

• Booth AlgorithmA*X A*YConversion Table (right)Start from LSB (add a 0) Overlap 1 bitEX: A*01110011 A*0

11 1 0 0 1 1 (0) A* 100(–1)0 10(–1)

Booth Algorithm

xi xi 1 yi Commnet

0 0 0 0 sting

0 1 1 End of 1s

1 0 ( 1) Start of 1s

1 1 0 1 string

6-36

• 0100*0111 0100*100(–1)

Booth Algorithm(2)

xi xi 1 yi Commnet

0 0 0 0 sting

0 1 1 End of 1s

1 0 ( 1) Start of 1s

1 1 0 1 string

0 1 0 0 = 4 1 0 0 1 = 7

1 1 0 0Shift 1 1 1 0 0Shift 1 1 1 1 0 0Shift 1 1 1 1 1 0 0+= 0 0 1 1 1 0 0Shift 0 0 0 1 1 1 0 0

6-37

• 1011*1101 1011*0(–1)1 (–1)

Booth Algorithm(3)

xi xi 1 yi Commnet

0 0 0 0 sting

0 1 1 End of 1s

1 0 ( 1) Start of 1s

1 1 0 1 string

1 0 1 1 = 5 0 1 1 1 = 3

0 1 0 1Shift 0 0 1 0 1+= 1 1 0 1 1Shift 1 1 1 0 1 1 = 0 0 1 1 1 1Shift2 0 0 0 0 1 1 1 1

6-38

• DrawbacksADD/SUB Variable Inefficient for isolate 1s

• Modified Booth Alg.Scan 3-bit at a timeOverlap 1-bit If n= Even, it can handl

e 2’s C #

Modified Booth Algorithm

xi xi 1 xi 2 yi yi 1

0 0 0 0 00 0 1 0 10 1 0 0 10 1 1 1 01 0 0 ( 1) 01 0 1 0 ( 1)1 1 0 0 ( 1)1 1 1 0 0

6-39

• Original 1011*1101 1011*0(–1)1 (–1)

• Now 1011*110 1(0) 1011*0(–1)01

Modified Booth Algorithm(2)

xi xi 1 xi 2 yi yi 1

0 0 0 0 00 0 1 0 10 1 0 0 10 1 1 1 01 0 0 ( 1) 01 0 1 0 ( 1)1 1 0 0 ( 1)1 1 1 0 0

6-40

• Find min. +/– for MUL: Find min. SD representation Find min. |yi|

• Z=(z1, z2, …zn) is min. if zizi+1 =0

(111) is min. but zizi+1 0

• Canonical RecodingFind such ZUsing sequence step

Canonical Recoding

xi+1 xi ci zi ci+1

0 0 0 0 00 0 1 1 00 1 0 1 00 1 1 0 11 0 0 0 01 0 1 ( 1) 11 1 0 ( 1) 11 1 1 0 1

6-41

• EX: Assume C0=0

X=011001 z0=1, c1=0

X=01100 z1=0, c2=0

X=0110 z2=0, c3=0

X=011 z3= –1, c4=1

X=01 z4=0, c5=1

X=(0)0 z5=1, c6=0

Z= 10(–1)001

Canonical Recoding(2)

xi+1 xi ci zi ci+1

0 0 0 0 00 0 1 1 00 1 0 1 00 1 1 0 11 0 0 0 01 0 1 ( 1) 11 1 0 ( 1) 11 1 1 0 1

6-42

• Conventional CSM

Array Multipliers

x4 x3 x2 x1 x0

y4 y3 y2 y1 y0

1 W’40 W30 W20 W10 W00

W’41 W31 W21 W11 W01

W’42 W32 W22 W12 W02

W’43 W33 W23 W13 W03

W44 W’34 W’24 W’14 W’04

Z9 Z8 Z7 Z6 Z5 Z4 Z3 Z2 Z1 Z0

6-43

• Conventional CSM

Array Multipliers(2)

Z1

Z2

Z3

Z4

Z5

0Z

Z9

Z8

Z7

Z6

FA

FA

FA FA

FA

FAFA

FAFA

FA

FAFA

FAFA

FA

i jW

ij= AND( X , Y )

W00

1

0

W'

W'

W

W

W'

W

W

W W W WW

W

W

W

W

W'

W'

W'

W 1001

12 02

20112130

03

4031

22

13

0414

23

32

41

42

W33

W'24W'34

43

FA

FACin

In1 In2

CoutSum

W44

FAFAFA FA

0 0 0

i jW'

ij= NAND( X , Y )

6-44

• Pezaris Array Multiplier– 1 = – 2•1+10 = – 2•0+ 0

Array Multipliers (3-0)

x4 x3 x2 x1 x0

y4 y3 y2 y1 y0W40 W30 W20 W10 W00

W41 W31 W21 W11 W01W42 W32 W22 W12 W02

W43 W33 W23 W13 W03

W44 W34 W24 W14 W04Z9 Z8 Z7 Z6 Z5 Z4 Z3 Z2 Z1 Z0

6-45

• Pezaris Array Multiplier

Array Multipliers(3)

Z1

Z2

Z3

Z4Z

5

0Z

Z9

Z8

Z7

Z6

FA

FA

FA FA

FA

FAFA

FAFA

FA

FAFA

FAFA

FA

i jW

ij= AND( X , Y )

W000

W

W

W

W

W

W

W

W W W WW

W

W

W

W

W

W

W

W 1001

12 02

20112130

03

4031

22

13

0414

23

32

41

42

W33

W24W34

43

FA

FAC

A B

CoS

W44

FAFAFA FA

0 0 0FA- 2Co+S=A-B-C

FA

- 2Co-S=-A-B-C

-2Co+S=A+B+C

6-46

• Modified Pezaris Array Multiplier

Array Multipliers(4)

Z1

Z2

Z3

Z4Z

5

0Z

Z9

Z8

Z7

Z6

FA

FA

FA FA

FA

FAFA

FAFA

FA

FAFA

FAFA

FA

i jW

ij= AND( X , Y )

W000

W

W

W

W

W

W

W

W W W WW

W

W

W

W

W

W

W

W 1001

12 02

20112130

03

4031

22

13

0414

23

32

41

42

W33

W24W34

43

FA

FAC

A B

CoS

W44

FAFAFA FA

0 0 0FA- 2Co+S=A-B-C

FA

- 2Co-S=-A-B-C

-2Co+S=A+B+C

0

FA2Co-S=B+C-A

6-47

• Baugh-Wooley Array Multiplier–24x4 •yi2i (i=0 to 3)= –(0,0, x4y3,…, x4y0) 24

ADD (1, 1, y’3,…, y’0) 24 ADD 24 if x4=1

or ADD 0 if x4=0

[(1, x’4, x4y’3,…, x4y’0) ADD (0,1,0…, x4)]24

So is –24y4 •xi2i (i=0 to 3)

{(1, x’4 +y’4, [x4y’3+x’3y4],…, [ x4y’0 + y4x’0 + x4+y4]} 24

Array Multipliers (5-01)

6-48

• Baugh-Wooley Array Multiplier

Array Multipliers (5-02)

x4 x3 x2 x1 x0

y4 y3 y2 y1 y0

W40’ W30 W20 W10 W00

W41’ W31 W21 W11 W01

W42’ W32 W22 W12 W02

W43’ W33 W23 W13 W03

W44 W3’4 W2’4 W1’4 W0’4

1 x’4 x4

y’4 y4

Z9 Z8 Z7 Z6 Z5 Z4 Z3 Z2 Z1 Z0

6-49

• Baugh-Wooley Array Multiplier

Array Multipliers(5)

Z1

Z2

Z3

Z4Z

5

0Z

Z9

Z8

Z7

Z6

FA

FA

FA FA

FA

FAFA

FAFA

FA

FAFA

FAFA

FA

i jW

ij= AND( X , Y )

W000

W

W

W

W

W

W

W

W W W WW

W

W

W

W

W

W

W

W 1001

12 02

20112130

03

40'31

22

13

0'41'4

23

32

41'

42'

W33

W2'4W3'4

43'

FA

FACin

In1 In2

CoutSum

W44

FAFAFA FA

0 0 0

i jW

i'j= AND( X' , Y )

FA

X4

Y4

FA

FA

X'4 Y'

4

1

6-50

• The On-the-fly CSM

Array Multipliers(6)

Z1

Z2

Z3

Z4Z

5

0Z

Z9 Z

8Z7

Z6

HA

HA

FA FA

FA

HAFA

HA

FA

FA

FA

FA

FA

i jW

ij= AND( X , Y )

W001

W'

W'

W

W

W'

W

W

W W W WW

W

W

W

W

W'W'W 10

01

12 02

20112130

03

4031

22

13

0414

23

32

41W'42

W33

W'24W'34

43

FACin

In1 In2

CoutSum

W44

i jW'ij

= NAND( X , Y )BB

FAHA

X4

Y4

FA

T

FA

FA

FA

HA

T

M

HA

13

4

23

4

4.55 3.5

2.5

1.5

0.5

5.5

44.5

5.5

6.56.50.5

6-51

• The On-the-fly

Array Multipliers(7)

(d)

MUX0 1

M Cell

VDD

(b)

Binary AdditionOn-the-fly

T Cell

MUX0 1

+

MUX0 1

+

R Cell

+

+ +MUX0 1

+

+

+

MUX0 1

+

MUX0 1

MUX0 1

T Cell

T Cell

HA Cell

M Cell

M Cell

B Cell

Carry-out

T

T

T

RHA

M

MM

B B B

(c)

MUX0 1

+

M* Cell

B* Cell

MUX0 1

(e)

6-52

• The Gray code– Binary to Gray:(Parallel)

• g n-1=bn-1

• gi=bi+1bi

– Gray to Binary:(Sequence)• b n-1=gn-1

• bi=bi+1gi

On-the-fly Application(1)

Binary Gray

000 000

001 001

010 011

011 010

100 110

101 111

110 101

111 100

6-53

• The Gray code adder

On-the-fly Application(2)

+

ga gb

+ Gray code to binary conversion

n-bit Binary adder

+ Binary to Gray code conversion

(n-1)TXOR

TADDn

TXOR

TotalDelay: n T

XORT

ADDn+

G/B G/B

B/G

log n T2 XOR

Chain:

Tree:

G/B Delay

(1+log n) 2

TADDn

+

for chain structure

for tree structure

TXOR

6-54

• The Gray code adder

On-the-fly Application(3)

Binary Addition

+

+

MUX0 1

Z

+

ZZZ

++

MUX0 1

MUX0 1

+

+

gan-1

+

gbn-1

+ga

n-2gb

n-2

++

ga1

gb1

++

ga0gb

0

MUX0 1

MUX0 1

+ + +

n-1 n-2 1 0

G/B

B/G

TXOR

TXOR

TXOR

TXORTXOR

TXOR

2

(n-1)

n

(n+1)

(n+2)

Delay:

On-the-fly

6-55

• Delay: Max{TCi,Tdi, Taibj}+kTFA

• P5-0=A3-0*B1-0 + C3-0 + D1-0

Additive Multiply Module (AMM)

FA FA FA FA

FA FA FA FA

0 5 C3 0 5 C2 0 4 C1 0 3 C0

d0

a3b0 a2b0 a1b0 a0b0

d1

a3b0 a2b0 a1b0 a0b0

AMMA3-0 B3-0

C3-0 D3-0

P5-0

1 4

5 9 5 9 4 8 3 7 2 6

6-56

AMM (2)

3-07-4

5-07-2

9-411-6

9-4

13-815-10

11-6

6-57

AMM (3)

AMM 5-0

7-2

9-4

11-6

9-4

13-8

15-10

11-6

0 0 03-0 1-0

5-2 1-0554321

10 10 9 8 7 610 10 9 8 6 4

15 15 14 13 12 1115 15 14 13 11 9

20 20 19 18 17 1620 20 19 18 16 14

25 25 24 23 22 21

6-58

• Same Column Same a4k, a4k+1, a4k+2, a4k+3

• Same Diagonal Same b2i, b2i+1

• 4m*4m unsigned multiplicationuses m*(2m)=2m2 AMMsDelay= TAND+Tdia+Tcol

Tdia= 5mTFA

Tcol= 5(3m–1) TFA

AMM (4)

m

2m

. . .

. . .