University of Windsor University of Windsor
Scholarship at UWindsor
Scholarship at UWindsor
Electronic Theses and Dissertations Theses, Dissertations, and Major Papers
1-1-1971
A Hardware Fixed-Point Multiplier/Divider.
A Hardware Fixed-Point Multiplier/Divider.
Nicholas Arpad Balatoni University of Windsor
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Balatoni, Nicholas Arpad, "A Hardware Fixed-Point Multiplier/Divider." (1971). Electronic Theses and Dissertations. 6655.
https://scholar.uwindsor.ca/etd/6655
by
NICHOLAS ARPAD BALATONI
A Thesis
Submitted to the F a c u l t y o f Graduate Stud ies through the Department o f E l e c t r i c a l Engineering in P a r t i a l F u l f i l l m e n t o f the Requirement f o r the Degree of
Master of Applied Science a t the U n i v e r s i t y o f Windsor
UMI Number: EC53048
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ABSTRACT
T h i s work des cr ibe s the design and implementation o f a lo w - c o s t
Hardware F i x e d - P o i n t M u l t i p l i e r / D i v i d e r to be used p r i m a r i l y w i t h
the PDP-8/S computer.
I t was deemed advantageous t o have the use o f such a de v ic e
i n s i t u a t i o n s where n e i t h e r an Extended A r i t h m e t i c Element nor the
Data Break F a c i l i t y were a v a i l a b l e and where a t the same time f a s t
M u l t i p l i c a t i o n and D i v i s i o n o p e r a t i o n s were r e q u i r e d - such a p p l i c a
t i o n s may occur i n o n - l i n e c o n t r o l and i n g r a p h i c s . The dev ic e
c on s tr u c t e d has a maximum M u l t i p l i c a t i o n time o f 90 microseconds,
a minimum M u l t i p l i c a t i o n time o f 18 microseconds, and a D i v i s i o n time
o f 155 microseconds. The speed thus obtained i s a t l e a s t 15 times
f a s t e r than the speed obt ain ed using s ub ro ut in e M u l t i p l i c a t i o n and
D i v i s i o n .
The de v ic e c o n s i s ts o f t h r e e 1 2 - b i t r e g i s t e r s which make up the
main working a r e a s , a s e r i a l A d d e r / S u b t r a c t e r , two 4 - b i t counters
and the a p p r o p r i a t e M u l t i p l i e r and D i v i d e r C o n t r o l l e r s a t an ap
proximate c o s t o f $2000.
To ac hi e v e M u l t i p l i c a t i o n the Leech A l g o r i t h m was used i n
which the l a s t t h r e e d i g i t s o f the M u l t i p i i e r a re compared and ac
c o r d i n g l y t he a p p r o p r i a t e o p e r a t i o n i s c a r r i e d o u t . For D i v i s i o n ,
the Booth A l g o r i t h m was adopted. In t h i s method the f i r s t b i t o f the
D i v i s o r and the f i r s t b i t o f the P a r t i a l Remainder a re compared and
f e a t u r e s and the M u l t i p l i e r / D i v i d e r proved to be a v a l u a b l e a d d i t i o n
ACKNOWLEDGEMENTS
The aut hor wishes t o express hi s a p p r e c i a t i o n to
Dr. P. A .V . Thomas f o r his s u p e r v i s i o n and continuous a s s i s ta n c e
durin g the course o f t h i s r e s e a r c h . Fur th erm or e, the a u t h o r ' s thanks
go to the N a t i o n a l Research Council o f Canada f o r i t s generous
ACKNOWLEDGEMENTS « V
TABLE OF CONTENTS Vt
LIST OF ILLUSTRATIONS v i i i
LIST OF TABLES i x
Chapter
I- INTRODUCTION
1.1 P r e l i m i n a r y Co n s id e r a ti o n s 1
1 . 2 General O u t l i n e 5
I I PHILOSOPHY OF DESIGN
2.1 Algo rithm s Used 9
2 . 1 . 1 M u l t i p l i c a t i o n Algo rithm s 9
2 . 1 . 2 D i v i s i o n Al gor it hm s 17
2 . 2 Basic Timing Requirements: M u l t i p l i e r 20
2 . 3 Basic Timing Requirements: D i v i d e r 22
I I I SYSTEM LOGIC
3.1 " M u l t i p l i c a n d / D i v i s o r " R e g i s t e r (AR) 26
3 . 2 "Ac c um ul a to r/ Di v id e nd /M o s t S i g n i f i c a n t
R e s ul t" R e g i s t e r (BR) 28
3 . 3 “M u l t i p l i e r / Q u o t i e n t / L e a s t S i g n i f i c a n t
R e s ul t" R e g i s t e r tDR) 31
3 . 5 Pulse Counter 43
3 . 6 Cycle Counter 48
3 . 7 L e f t / R i g h t S h i f t Pulse and Level C o n t ro l s 53
3 . 8 M u l t i p l i e r C o n t r o l l e r 55
3 . 9 D i v i d e r C o n t r o l l e r 59
3 . 1 0 I / O , C l o c k , and Flag C i r c u i t s 61
3.11 T o t a l Sequence o f O per ati on 64
IV PERFORMANCE
4 . 1 Cost R e a l i z a t i o n 67
4 . 2 Speed R e a l i z a t i o n 68
4 . 3 R e l i a b i l i t y and Accuracy w i t h E r r o r
Discussion 71
V USE AND LIMITATIONS
5.1 I n s t r u c t i o n Set 73
5 . 2 M u l t i p l i e r Programme 76
5 . 3 D i v i d e r Programme 77
5 . 4 Some L i m i t a t i o n s 77
5 . 5 F i n a l Conclusions and Suggestions 80
APPENDIX
A Sample C a l c u l a t i o n s 82
B Sample R es ult s 87
REFERENCES 89
1 - General Arrangement o f R e g i s t e r s , Adder/
S u b t r a c t e r , Counters and C o n t r o l l e r s 7
2 Flow Diagram f o r M u l t i p l i c a t i o n 16
3 Flow Diagram f o r D i v i s i o n 21
4 Timing Pulses f o r M u l t i p l i e r 23
5 Timing Pulses f o r D i v i d e r 25
6 " M u l t i p l i c a n d / D i v i s o r " R e g i s t e r (AR) 27
7 • "A c c um ul a to r/ Di v id e nd /M o s t S i g n i f i c a n t
Re s ul t" R e g i s t e r (BR) 29
8 " M u l t i p l i e r / Q u o t i e n t / L e a s t S i g n i f i c a n t
R e s u l t " R e g i s t e r (DR) 32
9 A d d e r / S u b t r a c t e r 42
10 Pulse Counter 44
11 Cycle Counter 50
12 L e f t / R i g h t S h i f t Pulse and Level Control 54
13 M u l t i p l i e r C o n t r o l l e r 56
14 D i v i d e r C o n t r o l l e r 60
LIST OF TABLES
T a b l e Page
l . a Booth A l g o r i t h m f o r M u l t i p l i c a t i o n 11
l . b Leech A lg o ri t h m f o r M u l t i p l i c a t i o n 13
2 Booth A l g o r i t h m f o r D i v i s i o n 19
3 H a l f - A d d e r f o r ( X+Y ) 36
4 F u l l - A d d e r f o r ( X+Y ) 37
5 H a T f - S u b t r a c t e r f o r ( X-Y ) 38
6 F u l l - S u b t r a c t e r f o r ( X-Y ) 39
1.1 P r e l i m i n a r y C o n s i d e r a t i o n s .
When thought i s given to the design o f a M u l t i p l i e r / D i v i d e r ,
some p r e l i m i n a r y s p e c i f i c a t i o n s must f i r s t be o u t l i n e d . T h i s in c l u d e s
a re v ie w o f a l r e a d y e x i s t i n g require men ts w i t h thought being given to
the environment in which the de v ic e w i l l be used and t o e x i s t i n g f e a s
i b l e equipment and components t h a t a r e a v a i l a b l e to the d e s i g n e r .
The need t h a t gave r i s e t o the i n i t i a l , idea o f a Hardware
F i x e d - P o i n t M u l t i p l i e r / D i v i d e r came about because o f the a v a i l a b i l i t y
o f an i n s t a l l a t i o n t h a t c o n s is ts o f a PDP-8/S D i g i t a l Computer and
i t s a s s o c ia t e d p e r i p h e r a l d e v i c e s . T h is computer is one o f the
e a r l i e r l o w - c o s t mini-computers o f the D i g i t a l Equipment Company.
The PDP-8/S computer i s equipped w i t h a bus-type I / O channel making
i t s u i t a b l e f o r the a d d i t i o n o f 64 p e r i p h e r a l d e v i c e s . These devices
a re e x t e r n a l to the computer proper and a re r e a d i l y programmable
through t h e r e g u l a r I / O channels w i t h the a i d o f the I / O s k i p f a c i l i t y .
The a r i t h m e t i c u n i t provided i n the PDP-8/S i s a o n e - b i t f u l l
s e r i a l a d d e r, and, using i t , s u b t r a c t i o n may be performed by f i r s t
making one number ne g a ti v e and then per forming an a d d i t i o n . With
t he hi g h e r l i n e o f PDP computers t h e r e is an Extended A r i t h m e t i c
E l e m e n t ( E A E ) a v a i l a b l e w h i c h c a n a l s o b e u s e d f o r M u l t i p l i c a t i o n
and f o r D i v i s i o n . The EAE performs p a r a l l e l a r i t h m e t i c op e ra t i o n s
on p o s i t i v e b in a r y numbers o n l y . Because o f i t s p a r a l l e l n a t u r e , i t
2
from 9 . 0 to 2 1 . 0 microseconds and a d i v i s i o n in a tim e o f 3 6 . 5 m i c r o
seconds. However, on these hig he r l i n e s o f PDP computers, the sub
r o u t i n e f o r signed m u l t i p l i c a t i o n r e q u i r e s 4 0 . 5 to 6 6 . 0 microseconds
and the s u b r o u t i n e f o r signed d i v i s i o n r e q u i r e s 6 5 . 0 microseconds.
By comparison, i t i s p o s s i b l e t o perform signed m u l t i p l i c a t i o n and
signed d i v i s i o n on the PDP-8/S using s u b r o u t i n e s , but the m u l t i p l i
c a t i o n s u b r o u t i n e , as observed in an ac tua l t e s t programme, r e q u i r e d
up t o 6 m i l l i s e c o n d s ; t h i s i s o f course v e ry slow. I t must be
considered f u r t h e r t h a t memory and processor i n s t r u c t i o n times on the
hi g h e r l i n e s o f computers range from 1 . 5 to 4 . 0 microseconds,
compared w i t h times o f 2 0 . 0 t o 4 0 . 0 microseconds per i n s t r u c t i o n on
th e PDP-8/S. The EAE f u r t h e r m o r e r e q u i r e s t h e presence o f the Data
Break F a c i l i t y a v a i l a b l e on the PDP-8 but not a v a i l a b l e on the PDP-8/S.
Thus, times o f up to 100 microseconds f o r m u l t i p l i c a t i o n and 160
microseconds f o r d i v i s i o n , ex c lu d in g sto ra g e t i m e s , would be deemed
q u i t e rea s o na b le and advantageous f o r the proposed d e v i c e .
I t can be e a s i l y concluded then t h a t s u b r o u t i n e m u l t i p l i c a t i o n
and d i v i s i o n a r e not f e a s i b l e t o use i n a p p l i c a t i o n s where l o g i c a l
d e c i s i o n s i n v o l v i n g these two o p e r a t i o n s , or where f r e q u e n t use o f
these op e ra ti on s , a r e r e q u i r e d t o t a k e place r a p i d l y . Such a p p l i
c a t i o n s may e a s i l y occur i n o n - l i n e c o n tr o l and in g r a p h i c s .
U n d e n i a b l y , the s p e e d o f the de v ic e and i t s c os t must be o f
prime i n t e r e s t . To gain speed two t h i n g s may be co n si de re d, namely,
the type o f components t h a t w i l l be used and the type o f a r i t h m e t i c
d e v i c e com pa ti bl e w i t h the PDP-8/S, DEC components p r o v i d e the l o g i c a l
types o f modules t o be used. The two types o f n e g a t i v e l o g i c
modules a v a i l a b l e a re the 2MHZ and the 10 MHZ modules. Thus, to gain
a t best f i v e times the speed, the high speed components should be
used which would p r a c t i c a l l y double the cost o f t he proposed d e v i c e .
S i m i l a r i l y , t o gain speed in the a r i t h m e t i c , p a r a l l e l
o p e r a t i o n s should be adopted. The d i f f e r e n c e i n c o s t i s however
p r o h i b i t i v e due t o t he f a c t t h a t i n s t e a d o f having one A d d e r / S u b t r a c t
e r as f o r s e r i a l a r i t h m e t i c , in p a r a l l e l a r i t h m e t i c one Adder/Sub
t r a c t e r i s r e q u i r e d f o r each b i t o f the 12 b i t r e g i s t e r s used.
The answer t o the re q ui re m e n t problem must l i e then i n the
time c r i t e r i a placed on a s i n g l e m u l t i p l i c a t i o n and on a s i n g l e
d i v i s i o n . To gain an idea o f thes e t i m e s , an examination of the
c a p a b i l i t i e s and requirements o f the a l g o r it h m s used i s c a l l e d f o r .
The a l g o r i t h m s , found in S e c t io n 2 . 1 , suggest t h a t s i x c y c le s a r e r e
q u ir e d f o r M u l t i p l i c a t i o n , and tw e lv e cy c le s f o r D i v i s i o n .
Co nsid er ing a p a r a l l e l A d d e r / S u b t r a c t e r and 4 pulse times per c y c l e ,
M u l t i p l i c a t i o n would r e q u i r e 24 pulse times o r le s s .due to the
e x i s t e n c e o f the Fa s t Op er ati o n which can occur under c e r t a i n con
d i t i o n s t o be discussed l a t e r . Considering 3 pulse times per c y c l e
and 11 f u l l c y c le s plus one s h o r t c y c l e , d i v i s i o n would r e q u i r e 34
puls e t i m e s . I f a s e r i a l A d d e r / S u b t r a c t e r i s c o n s i d e r e d w i t h t w e l v e
pulse times per A d d i t i o n or S u b t r a c t i o n , a M u l t i p l i c a t i o n would
r e q u i r e 90 pulse times or l e s s , and d i v i s i o n would r e q u i r e 155 pulse
4
I f the most expensive proposal were adopted ( 10MHZ modules
and p a r a l l e l o p e r a t i o n ) M u l t i p l i c a t i o n and D i v i s i o n times o f 5 and
7 microseconds r e s p e c t i v e l y could be a c hie ve d. Since an I / O
i n s t r u c t i o n time on the PDP-8/S is 38 microsecond^, such speed i s not
j u s t i f i e d f o r the added cost because o f a l l th e overhead, or
unused, t i m e .
I f the s e r i a l A d d e r / S u b t r a c t e r were used and 10MHZ modules,
18 and 30 microseconds may be r e a l i z e d f o r t he M u l t i p l i c a t i o n and
D i v i s i o n o p e ra t i o n s r e s p e c t i v e l y . Th is would indeed c o n f i n e the
d e v i c e o p e r a t i o n time w i t h i n one I / O i n s t r u c t i o n t i m e . However, i f
the p r i c e d i f f e r e n c e i s considered between t he 10MHZ and 2MHZ
modules, and the f a c t too t h a t some i n t e r f a c i n g i s r e q u i r e d between
the two types i f used i n com bi n a tio n, the proposal i s s t i l l an
u n j ' u s t i f i a b l y expensive one.
L e t the proposal which was f i n a l l y accepted be now c o n s id e r e d ,
t h a t i s , using s e r i a l a r i t h m e t i c and 2MHZ components. I t was found
t h a t the maximum speed o b t a i n a b l e , due t o c a r r y propa gat ion in
counters ( 4 stages) and i n s h i f t r e g i s t e r s is 1.3MHZ. However,
c o n s id e r in g 1MHZ o p e r a t i o n , each pulse time ( tim e from the s t a r t
o f a pulse t o the s t a r t o f the n e x t ) i s one microsecond. Th is then
y i e l d s a M u l t i p l y time o f 90 microseconds or l e s s , and a D i v i s i o n time
o f 1 5 5 mic r o s e c o n d s . C o n s i d e r i n g t h e p r e s e n c e o f the F a s t - O p e r a t i o n ,
the minimum time f o r a M u l t i p l i c a t i o n i s 18 microseconds ( 6 cy c le s
o f 3 puls e times e a c h ) . An I / O Skip f a c i l i t y i s a v a i l a b l e a t no
D i v i s i o n o p e r a t i o n should exceed one I / O i n s t r u c t i o n t im e . The
M u l t i p l i c a t i o n O p e r a t i o n , the main purpose o f the d e v i c e , never
exceeds 3 1 / 0 times and in most cases i t can be completed w i t h i n
2 I / O t i m e s , i e . 76 microseconds. Since i t i s expected t h a t
D i v i s i o n w i l l be much les s used than M u l t i p l i c a t i o n , t he performance
times thus ac qu ire d a re q u i t e a c c e p t a b l e . At t h i s t i m e i t should be
f u r t h e r poin te d out t h a t the d e v ic e w i l l be a b l e t o M u l t i p l y and
D i v i d e 1 2 - b i t signed b i n a r y numbers which g iv e s the added advantage
o f no t having t o perform sign conversions by s o f t w a r e . Th is would
i n d i c a t e an advantage even over the EAE, a u n i t which f u r t h e r r e q u i r e s
the Data Break F a c i l i t y . Furth erm ore , since t he M u l t i p l i e r / D i v i d e r
i s an e x t e r n a l d e v i c e , i t i s r e a d i l y usable n o t o n l y w i t h the PDP-8/S
but w i t h o t h e r PDP 1 2 - b i t word computers.
1 . 2 General O u t l i n e .
I t i s p o s s ib l e to d e s c r i b e the M u l t i p l i e r / D i v i d e r by d i s
cussing i n general terms t he major se c tio ns o f i t s block diagram.
Because one may m u l t i p l y two 1 2 - b i t signed b i n a r y numbers, by
the n a t u r e o f M u l t i p l i c a t i o n the r e s u l t w i l l y i e l d i n many cases a
b i n a r y number t h a t i s more than 1 2 - b i t s but never more than 2 4 - b i t s
lon g. Thus, i t i s necessary to prov ide a double le n g th r e g i s t e r from
which the r e s u l t can be e x t r a c t e d in two t w e lv e b i t segments through
the r e g u l a r programmed I / O cha n ne ls . The u n i t then r e q u i r e s two
s i n g l e le n g th r e g i s t e r s which a t the end o f the o p e r a t i o n w i l l
6
r e s u l t .
Fu rth er m or e, i t i s necessary t o load i n - t h r o u g h t he I / O
c ha n n e ls - the two numbers on which the a r i t h m e t i c o p e r a t i o n i s to be
performed. In order t o r e t a i n t he M u l t i p l i c a n d w h i l e the i n t e r m e d i a t e
o p e r a t i o n s , a d d i t i o n and s u b t r a c t i o n , t a k e p l a c e a t h i r d r e g i s t e r i s
necessary. I t i s found t h a t d u r i n g the D i v i s i o n o p e r a t i o n t h i s
t h i r d r e g i s t e r w i l l c o n ta i n the D i v i s o r .
For i d e n t i f i c a t i o n purposes, l e t the r e g i s t e r used t o s t o r e
the M u l t i p l i c a n d / D i v i s o r be c a l l e d the A - R e g i s t e r (AR). Also the
r e g i s t e r t o hold t he Most S i g n i f i c a n t R e s u l t or in D i v i s i o n , the
D i v i d e n d , s h a l l be c a l l e d the B - R e g i s t e r ( B R ), and the t h i r d r e g i s t e r ,
t o hold th e L e a s t S i g n i f i c a n t R e s u l t / M u l t i p l i e r / Q u o t i e n t , s h a l l be
c a l l e d the D - R e g i s t e r ( D R ) .
These t h r e e r e g i s t e r s p r o v i d e the main working areas i n t o
which data i s lo a d e d , i n which a l l s h i f t i n g t ake s p l a c e , in which
the i n t e r m e d i a t e r e s u l t s a r e held and from which the f i n a l answers a re
e x t r a c t e d . F i g u r e 1 gi ve s the Block Diagram form of the M u l t i p l i e r /
D i v i d e r .
Also noted i n F i g u r e 1 , the f o u r t h major bl o ck in the
diagram i s th e A d d e r / S u b t r a c t e r . An i n s p e c t i o n o f th e a l g o r it h m s in
Se c ti on 2. 1 shows t h a t i n some cases i t i s necessary t o perform an
A d d i t i o n b e t w e e n t h e c o n t e n t s o f AR and BR, a n d i n o t h e r c a s e s a
S u b t r a c t i o n o f the conte nts o f AR from BR i s r e q u i r e d . I n 2 ‘ s
complement a r i t h m e t i c , a S u b t r a c t i o n may be performed by complement
From C o n t r o l l e r s
1 i
Add
✓ I From
Clock
Bi t Delay
Control 1er D i v i d e r Counters
Mul t i pi i e r
C o n t r o l ! e r
D-Reg i s t e r B - R e g i s t e r
A-Regi s t e r
Adder/ S u b t r a c t e r
FIGURE 1.
General Arrangement o f
R e g i s t e r s , A d d e r / S u b t r a c t e r ,
8
when the F u l l A d d e r / S u b t r a c t e r was developed from) Boolean Functions
( r e f e r t o Se c ti on 3 . 4 ) , t t was found t h a t a F u l l Adder and a F u l l
S u b t r a c t e r were r e a d i l y combined such t h a t t h e A d d i t i o n or S u bt ra c
t i o n o f two 1 2 - b i t signed numbers y i e l d e d the c o r r e c t l y signed
1 2 - b i t r e s u l t , w i t h o u t the need f o r the complement and increment
o p e r a t i o n .
The blo ck l a b e l l e d "COUNTERS" serves t h e purpose o f
g e n e r a t i n g and d i s t i n g u i s h i n g from each o t h e r the t w e lv e OS pulses
necessary f o r A d d i t i o n or S u b t r a c t i o n , and t h e c o n t r o l pulses OA, OB,
OX and OY used f o r s h i f t purposes. There a r e two counters however,
one being the Pulse Counter and t he o t h e r being the Cycle Counter.
These a r e d e a l t w i t h in d e t a i l in Sect ions 3 . 5 and 3 . 6 r es p e c
t i v e l y .
Not i nc lu ded in the blo ck diagram a re two major s e c tio ns
t h a t c o n t r o l the M u l t i p l i c a t i o n and D i v i s i o n processes which are
d e t a i l e d i n Sections 3 . 8 and 3 . 9 r e s p e c t i v e l y . These u n i t s compare
the a p p r o p r i a t e d i g i t s o f AR, BR and DR, y i e l d i n g d e c i s io n s as to
which pulses a re a p p l i e d to the r e g i s t e r s and whether A d d i t i o n ,
S u b t r a c t i o n or n e i t h e r i s t o be c a r r i e d out i n each c y c l e o f the
p a r t i c u l a r o p e r a t i o n .
For s i m p l i c i t y i n t he block diag ram , t h e c l o c k , f l a g s and
I / O c o n t r o l c i r c u i t s a r e a l s o o m i t t e d , b u t t h e y a r e d i s c u s s e d a t
C h a p t e r II
P H I L O S O P H Y OF DESIGN
2.1 A lgo rit hm s Used.
2 . 1 . 1 M u l t i p l i c a t i o n A l g o r i t h m s .
The most common forms o f auto ma tic m u l t i p l i e r s a r e the s e r i a l
and p a r a l l e l t y p e , as discussed by P . A .V . Thomas T h e i r g r e a t
l i m i t a t i o n i s t h a t they a r e a b l e t o handle p o s i t i v e numbers o n l y ,
such t h a t a c o r r e c t i v e s u b ro ut in e must be int ro du ce d when n e g a t i v e
numbers must a ls o be handled. T h e i r mode o f o p e r a t i o n i s one o f
continued a d d i t i o n , one a d d i t i o n f o r each b i t o f the e n t i r e word
l e n g t h . This occurs i n a s e r i a l m u l t i p l i e r where only one adder is
u t i l i z e d . I n the p a r a l l e l m u l t i p l i e r , one adder i s r e q u i r e d f o r
each b i t i n the word l e n g t h . To the above l i m i t a t i o n must then be
added one o f slow speed or high c o s t r e s p e c t i v e l y in the two cases
mentioned.
Since i t was decided t o use th e s e r i a l mode o f o p e r a t i o n , a
(o\
scheme suggested by Dr s. A.D. Booth and K .H .V . B o o t h v 'was
i n v e s t i g a t e d . This scheme i n v o lv e s a s e r i a l m u l t i p l i e r which y i e l d s
the c o r r e c t l y signed product o f two signed i n p u t numbers. The
method compares the l e a s t s i g n i f i c a n t b i t o f the M u l t i p l i e r w i t h an
10
s u b t r a c t s the M u l t i p l i c a n d to the P a r t i a l Product i n the accumulator.
T h is occurs i f these two d i g i t s t h a t a r e compared d i f f e r from each
o t h e r . I f the two d i g i t s a re the same, the P a r t i a l Product i s l e f t
u n a l t e r e d . The c o n te n t o f the M u l t i p i i e r R e g i s t e r and Accumulator
a r e s h i f t e d one pl ac e t o the r i g h t i n each c y c l e f o r the number of
cyc les as t h e r e are b i t s in a word l e n g t h . On a 1 2 - b i t word on
low speed modules and s e r i a l o p e r a t i o n t h i s would mean a M u l t i p l i c a
t i o n time o f a pp ro x im a t e ly 170 microseconds.
I n quest f o r a f a s t e r method o f M u l t i p l i c a t i o n , the method
proposed by J.W. Leech^^was adopted i n which the l a s t two
s i g n i f i c a n t d i g i t s o f the M u l t i p l i e r R e g i s t e r , along w i t h the e x t r a
M u l t i p l i e r R e g i s t e r b i t , a r e compared and the M u l t i p l i e r and P a r t i a l
Product a r e r i g h t - s h i f t e d t w i c e , i n t h i s manner almost h a lv i n g the
M u l t i p l i c a t i o n time w i t h o n l y a s l i g h t in c r e a s e in c o s t .
The Booth A l g o ri t h m f o r M u l t i p l i c a t i o n may be simply s t a t e d
in t a b u l a r form as shown in Ta b le l . a . In th e t a b l e , the P a r t i a l
Product i s given t he s h o r t form PP, the M u l t i p l i e r , MLR, and the
M u l t i p l i c a n d i s shortened to MND.
Motitig t h a t o n l y two d i g i t s a re compared, the l a s t M u l t i p l i e r
b i t and the e x t r a b i t a s s oc ia te d w i t h the M u l t i p l i e r R e g i s t e r , on ly
f o u r combinations o f these two b i t s can occur. A c c o r d i n g l y , the
TABLE 1 .a
BOOTH ALGORITHM FOR MULTIPLICATION
M u l t i p l i e r D i g i t s
O p e r a ti o n Performed
bn bn+l
—
*
•—
1
o
o
—
«
o
—
*
o S h i f t PP and MLR one p la c e r i g h t
Add MND t o PP and s h i f t PP and MLR one p l a c e to th e r i g h t
S u b t r a c t MND from PP and s h i f t PP and MLR one p l a c e to th e r i g h t
S h i f t PP and MLR one p l a c e t o the r i g h t
r e l a t i v e l y si mp le . Booth and Booth f u r t h e r o f f e r a r e c u r s i v e formula
f o r the above process and t h i s formula is giv e n i n Equation ( 1 )
below.
an-p = O / 2 ) *an- p + i + ( b n- p + l ~ bn - p ^ M; f o r P=1 > ( h - 1 )
In t h i s equ ati o n a'n i s t he v a lu e o f the accumulator a t t he p
stage o f the proces s, and M i s the v a l u e o f t h e M u l t i p l i c a n d .
On the basis o f t h i s r e c u r s i v e f o r m u l a , Booth and Booth g i v e
a c o n s t r u c t i v e pr oof by means o f a s t a g e - w i s e s u b s t i t u t i o n o f the
v al ues o f 1p 1. A f t e r the s u b s t i t u t i o n s , a l l the equations are
r\ 1
m u l t i p l i e d by a power o f 2 , namely 2 p“ , then t he equa tions are
summed and r ea r ra n ge d to form an equation o f the type a-j = b.M and
i n t h i s way the A l g o ri t h m i s proved.
As noted e a r l i e r , the r e c u r s i v e for mula encountered by Booth
and Booth i s a r e l a t i v e l y simple one due t o t h e f a c t t h a t o n l y two
d i g i t s a r e compare and on ly the f o u r p o s s i b l e combinations must be
accounted f o r . When a t t e n t i o n i s d i r e c t e d t o the Leech A l g o r i t h m ,
i t i s found t h a t t h r e e d i g i t s are compared. Th is g iv e s r i s e to
e i g h t d i f f e r e n t p o s s i b l e patterns-. To show t h i s , the Leech
A l g o r i t h m f o r M u l t i p l i c a t i o n i s now s t a t e d i n t a b u l a r form as shown
below in T a b l e l . b in which a l l a b b r e v i a t i o n s o f Ta b le l . a hold and
a l l s h i f t i n g i s t o the r i g h t .
TABLE l . b
LEECH ALGORITHM FOR MULTIPLICATION
M u l t i p i i e r D i g i t s
O per ati on Performed
bn - l bn bn+l
0 0 0 F a s t - O p e r a t i o n , 2 P o s t - S h i f t s o n ly
0 1 0 Add MND to PP, 2 P o s t - S h i f t s
1 0 0 1 P r e - S h i f t , S u b t r a c t MND from PP, 1 P o s t - S h i f t
1 1 0 S u b t r a c t MND from PP, 2 P o s t - S h i f t s
0 0 1 Add MND to PP, 2 P o s t - S h i f t s
0 1 1 1 P r e - S h i f t , Add MND t o PP, 1 P o s t - S h i f t
1 0 1 S u b t r a c t MND from PP, 2 P o s t - S h i f t s
any l i t e r a t u r e , and as a consequence, i t was decided t o c o n s t r u c t a
r e c u r s i v e formula t h a t would com ple te ly d e s c r i b e the A l g o r i t h m . A
more complex for mu la was expected and the equa tion t h a t was f i n a l l y
found i s as shown i n Equation ( 2 ) .
an - 2 p = ( 1 / 8 ) - an - 2 p + 2 ^ 2" ^ 1 / 2 ^ * ^ bn - 2 p + brt-2p+l T 2bn - 2 p - l ^ "
l bn - 2 p " bn - 2 p + l 1 ^ - H + 0 / 2 ) • ( l bn-2 p + 2 + bn-2p+3 "
2bn - 2 p + l ^ * ^ " l bn-2p+2 " bn - 2 p + 3 I ^ + ^ bn - 2 p “ bn - 2 p - l ^
^ " bn - 2 p " bn - 2 p - l ^ * l bn - 2 p “ bn - 2 p + l ^ * M
The n o t a t i o n used f o r Equation ( 1 ) a l s o a p p l i e s f o r Equation ( 2 ) .
The c o e f f i c i e n t , 1 / 2 , i n Equation ( 1 ) appears because a r i g h t s h i f t
i s performed d u ri n g each c y c l e . This r i g h t s h i f t i s the d i g i t a l
e q u i v a l e n t o f m u l t i p l y i n g by 1 / 2 . Thus i t i s not s u r p r i s i n g to f i n d
t he c o e f f i c i e n t 1 / 8 i n Equation ( 2 ) f o r , i f two p o s t - s h i f t s and one
p r e - s h i f t t a k e p la c e b e f o r e the a d d i t i o n or s u b t r a c t i o n , the c o e f
f i c i e n t m u l t i p l y i n g the 1 / 8 i s 1. I f o n l y two p o s t - s h i f t s have
occurred be f o r e the a d d i t i o n or s u b t r a c t i o n , t h i s c o e f f i c i e n t i s 2.
Th i s i s a l s o the case i f a p o s t - s h i f t and a p r e - s h i f t haye taken
p l a c e . I f on ly one p o s t - s h i f t occurred b e f o r e the a d d i t i o n or
s u b t r a c t i o n , the c o e f f i c i e n t i s 4 . The m u l t i p l y i n g c o e f f i c i e n t
b e f o r e the accumulator term thus becomes 1 / 8 , 1 / 4 , 1 / 4 or 1 / 2
depending on whether 3 , 2 or 1 s h i f t s have o c c u rre d.
in a c o n s t r u c t i v e p r o o f , th e equ ati on a t each stage must be
m u l t i p l i e d by the power o f two t h a t has occurred i n t he previous -i, r
s t a g e .a nd to t h a t power must a l s o be added th e number o f p o s t - s h i f t s
and p r e - s h i f t , i f any, t h a t have occurred j u s t b e f o r e the a d d i t i o n
or s u b t r a c t i o n t h a t i s about to t a k e pla ce d u rin g t h i s s t a g e .
Due t o the c o m p l e x it y o f the equations and the d i f f i c u l t i e s
which a r i s e from i t , a c o n s t r u c t i v e pr oo f w i l l not be o f f e r e d here.
However, a sample c a l c u l a t i o n showing the method i s given in the Ap
pe n di x . The mere purpose o f the above d is c u s s io n i s t o p o i n t out
how the c o m p le x it y Of the r e c u r s i v e for mula in c r e a s es as more
d i g i t s o f the M u l t i p l i e r a re compared a t each stage o f the
M u l t i p l i c a t i o n process. In passing i t may be f u r t h e r mentioned t h a t ,
i f an a l g o r i t h m were designed such t h a t f o u r d i g i t s would be
compared a t a t i m e , t h r e e r i g h t s h i f t s would have t o be c a r r i e d out
a t each stage and the whole M u l t i p l i c a t i o n could be c a r r i e d out in
f o u r c y c l e s . The d i f f e r e n c e in the time r e q u i r e d f o r a M u l t i p l i c a t i o n
i n t h i s way would o f course not j u s t i f y th e c o m p l e x i t y t h a t would
a r i s e i n th e c o n t r o l c i r c u i t s o f the process.
Thus, the Leech A l g o r i t h m was adopted f o r M u l t i p l i c a t i o n
because th e time f o r one M u l t i p l i c a t i o n could be halved i n
comparison to the Booth Method and because the d i f f e r e n c e i n costs
was found to be a c c e p t a b l e even when the in c r e a s e in the c os t o f
the more complex c o n t r o l l e r l o g i c was c on s id e re d.
The Flow Diagram f o r M u l t i p l i c a t i o n i s given i n Fi g ur e 2 where
16
FIGURE 2.
Flow Diagram f o r M u l t i p l i c a t i o n
CYCLES PULSES OPERATION
F i r s t
5
Cycles
OA Advance CC; Set Add/Sub F F ; O pt io n a l P r e - S h i f t o f BR and DR.
> DRio=t
(Fa«
J
Not (
)Rn =DRb ; t Op. )
DR-, n- D R - . D R . ) n _ - I * *, .
10i 11 b Perform A d d i t i o n or i S u b t r a c t i o n during a l l
(1 2) Long Oper ations
OX Op ti on a l P o s t - S h i f t o f BR and DR
OY
P o s t - S h i f t o f BR and DR; C l e a r Pulse Counter
L a s t
Cycle
OA Advance CC; Set Add/Sub FF; Op tional P r e - S h i f t o f BR and DR.
>
Fas
>
X
Long 0
t Op.
' ieZL
P* Perform A d d i t i o n or S u b t r a c t i o n during a l l
Long Oper ations (1 2)
OX Opti ona l P o s t - S h i f t o f BR and DR
c y c le s r e q u i r e d and w i t h r e s p e c t to the number o f pulses r e q u i r e d
per c y c l e .
2 . 1 . 2 D i v i s i o n A l g o r it h m s .
To achi ev e D i v i s i o n on a b i n a r y base computer, t h e r e a r e two
a l t e r n a t i v e processes a v a i l a b l e , namely, the standard t r i a l and
e r r o r method and the n o n - r e s t o r i n g method.
In the f i r s t case, the D i v i s o r i s compared w i t h the P a r t i a l
Remainder a t each s u b - o p e r a t i o n , and i f the former q u a n t i t y i s the
l e s s e r , a s u b t r a c t i o n i s performed and a ' 1 ' i s added to the l e a s t
s i g n i f i c a n t b i t o f the Q u o t i e n t R e g i s t e r . I f , however, the D i v i s o r
i s g r e a t e r than the P a r t i a l Remainder, then nothing i s done except
a zero i s added to the l e a s t s i g n i f i c a n t end o f the Q u o ti e n t
R e g i s t e r . The Q u o t ie n t and the P a r t i a l Remainder a re then doubled
( i e . s h i f t e d l e f t one p l a c e ) and the u n i t i s ready f o r the ne x t sub
o p e r a t i o n i n the D i v i s i o n process.
To ac hieve the above process i n hardware, the e a s i e s t method
t o use i s t o s u b t r a c t the Q u o ti e n t from the P a r t i a l Remainder, t e s t
the sign and, i f n e g a t i v e , add the D i v i s o r back t o the P a r t i a l
Remainder. The l i m i t a t i o n s o f t h i s method a re t h a t only p o s i t i v e
numbers can be handled and the method i s slow c o n s id e r in g the number
o f r e - a d d i t i o n s t h a t must take p l a c e .
The s e c o n d m e t h o d , w h i c h was f o u n d t o b e p r e f e r a b l e an d w h i c h
was adopted f o r t h i s work, i s the n o n - r e s t o r i n g method. As (2)
18
p o s i t i v e and n e g a t i v e numbers and r e q u i r e s no r e - a d d i t \ o n s t o t a k e
p l a c e . The time o f a l l D i v i s i o n o p e ra t i o n s i s thus f i x e d a t as niany
word le n g th s as t h e r e a r e b i t s i n one word. I n t h i s case a word i s
t w e l v e b i t s long.
L e t i t be assumed t h a t the D i v i s o r (m) i s s t o r e d i n the
A-R e g i s t e r ( A A-R ) , the Dividend ( a ) i n B - A-R e g i s t e r (BA-R ), and the Q u o ti e n t
(b ) w i l l appear i n the D - R e g i s t e r (DR) a t th e completion o f the
D i v i s i o n process. The Booth A l g o ri t h m as s t a t e d by Booth and Booth
i s given in Ta b le 2.
I t should be noted t h a t i n the D i v i s i o n process i n t e g e r s are
not ob tai n ed i n the Q u o t i e n t . The form o f th e answer i s such t h a t
a b i n a r y p o i n t e x i s t s a f t e r the sign b i t . As an example the number
0 . 1 0 000 000 000 r e p re s e n ts the r e s u l t 1 / 2 . T h i s occurs because by
t h e n a t u r e o f the A l g o ri t h m the Q u o t i e n t , a/m = b , i s generated o n l y
i n cases where m > a . I t i s thus r e q u i r e d t o s c a l e th e v a l u e o f m
i n cases where m < a u n t i l m > a. The D i v i s i o n i s then c a r r i e d out
and the Q u o t i e n t may be r e - s c a l e d to o b t a i n the t r u e v a lu e o f the
r e s u l t .
Booth and Booth g i v e a c o n s t r u c t i v e p r o o f i n d e t a i l based on
t he f o l l o w i n g r e c u r s i v e for mu la :
r k = 2 r k - i + 0 - 2bk )-"'
where r^ i s the remainder a f t e r the k o p e r a t i o n and b. i s the ■f* h
k d i g i t o f the P a r t i a l Remainder ( b ) . A s t a g e -w i s e s u b s t i t u t i o n
TABLE 2.
BOOTH ALGORITHM FOR DIVISION
Sub-Operation R e s u l t o f Comparison O p e r a ti o n Performed
1 to ( n - 1 ) ARo = BR0 a ) Add 2 " ( n_1) t o DR
b) S h i f t BR and DR l e f t
c) S u b t r a c t AR from BR
1 to ( n - 1 ) AR0 f BRq a) Add zero to DR
b) S h i f t BR and DR l e f t
c) Add AR t o BR
20
a p p r o p r i a t e power o f two. The equ ations thus ob tai n ed a re then added
and the r e s u l t i n g equ ati o n i s r ea r ra n ge d u n t i l an expression is
obt aine d i n the form a/m = Q u o t i e n t . The r e s u l t generated i s c o r r e c t
i n a l l r es p e c ts except f o r the s i g n . To c o r r e c t the sign u n i t y i s
added to the r e s u l t . The above scheme ge n e ra te s the t r u e Q u o t i e n t ,
a/m, r e g a r d l e s s o f the signs o f 'a* and 'm*.
I n T a b l e 2. i t i s shown t h a t 2 - ^n_^ i s added to the Q u o ti e n t
i n t he t w e l f t h or l a s t c y c l e . Th is i s o p t i o n a l s in c e i t s purpose i s
t o g i v e t he l e a s t biased r o u n d - o f f e r r o r as w i l l be discussed in
S e c t i o n 4 . 3 . A sample c a l c u l a t i o n i s given in the Appendix.
The Flow Diagram f o r D i v i s i o n i s given i n F i g u r e 3 , in which
a D i v i s i o n i s de s c ri be d w i t h r e s p e c t t o the number o f cyc les
r e q u i r e d and w i t h r e s p e c t t o the number o f pulses r e q u i r e d per c y c l e .
2 . 2 Basic Timing Requirements: M u l t i p l i e r .
According to the M u l t i p l i c a t i o n A l g o r i t h m , s i x basic cy c le s
a re r e q u i r e d to accomplish the M u l t i p l i c a t i o n o p e r a t i o n . The pulse
req uir eme nts i n each c y c l e are no t n e c e s s a r i l y t he same because 15
pulses a re r e q u i r e d in a c y c l e i n v o l v i n g a Long Op er ati o n and on ly
3 pu ls es a r e necessary i n a c y c l e i n v o l v i n g a Fa s t O p e r a ti o n .
I n both types o f c y c l e s , the f i r s t pulse i s the OA pu ls e .
T h is p u l s e i s n e c e s s a r y i n c a r r y i n g o u t a p r e - s h i f t , an i n i t i a l
r i g h t s h i f t o f BR and DR, f o r s e t t i n g a f l i p - f l o p t o block ou t
the o p t i o n a l OX p u l s e , f o r g e n e r a t i n g the Jam Pulse i n t he Fa s t
FIGURE 3.
Flow Diagram f o r D i v i s i o n
CYCLES PULSES OPERATION
Fi r s t
10
Cycles
OA
Advance CC; Set Add/Sub FF; Add ' 1 ' or 'O' t o DR
OB L e f t S h i f t BR and DR
OS
( 1 2 ) Perform A d d i t i o n or S u b t r a c t i o n
L a s t
Cycl e
OA Advance CC; Set Add/Sub F F ; Add ' 1 ‘ or 'O' to DR
OB L e f t S h i f t BR and DR
OS (1 2 )
Perform A d d i t i o n or S u b t r a c t i o n
On 0S-|2 Complement DRQ and Produce
22
de s ir e d s t a t e .
In t he Fast O p e r a t i o n , the Jam Pulse advances the Pulse
Counter by 12. I n the Long O p e r a t i o n , a f t e r t h e OA p u l s e , 12 pulses
a r e produced t o c a r r y out the a d d i t i o n or s u b t r a c t i o n . These tw e lv e
OS pulses a re channeled t o AR and BR, and a l s o t o the A d d e r / S u b t r a c t e r .
A f t e r the OA pulse i n the Fa s t O p e r a t i o n , and a f t e r O S ^ in
the Long O p e r a t i o n , two more pulses a r e produced, OX and OY. Each
o p e r a t i o n r e q u i r e s two r i g h t s h i f t s due to t h e number o f b i t s being
compared. When a p r e - s h i f t has taken pl ac e on pulse OA, the r i g h t
s h i f t o f BR and DR a re blocked out on t he OX p u l s e , and the mand
a t o r y second r i g h t s h i f t occurs on the OY p u l s e . When no p r e - s h i f t
has taken p l a c e , the r e q u i r e d two r i g h t s h i f t s are performed using
the OX and OY p u ls e s .
In F i g u r e 4 a re shown diagrams o f t he t i m i n g pulses f o r the
d i f f e r e n t types o f cy c le s encountered along w i t h the t o t a l sequence
o f pulses f o r a complete M u l t i p l i c a t i o n .
2 . 3 Basic Timing Requirements: D i v i d e r .
No a l g o r i t h m i c s h o r t c u t s have been i n c o r p o r a t e d i n t o the
D i v i s i o n process, making the f i r s t eleven cy c le s i d e n t i c a l as a l l
pulses a re mandatory.
The process begins w i t h g e n e r a t i n g an OA p u l s e . On t h i s
pulse the A d d / S u b t r a c t F l i p - F l o p is s e t t o t he d e s i r e d s t a t e and
a ls o a ' 1 ' or 'O' i s added i n t o the l e a s t s i g n i f i c a n t b i t o f DR,
FIGURE 4 .
M u l t i p l i e r Timing
Long O p e r a t i o n , 2 P o s t - S h i f t s
0A
---I
I
I
I
i
I
I I
OS
OX
OY
Long O pe ra tio n w i t h P r e - S h i f t
OA
OS
OX
OY
F a s t O per ati on
OA
OS
OX
OY
Complete T y p i c a l M u l t i p l i c a t i o n
OA
OS
OX
OY
I I
I
I
I
I
I I
1 I
L
I_____ L
I I I J . . J . J 1 1 1 .... 1 1 1
1 1 1 1 1 1
24
the Cycle Counter by one.
The next pulse t h a t i s generated is t he OB p u l s e , the s o l e
purpose o f which is t o perform a s i n g l e l e f t s h i f t on BR and DR.
T h i s s h i f t could not be performed on the OA pulse s inc e DR-j-j r e q u i r e s
a t l e a s t 400 nanoseconds to s e t t l e when i n i t i a l l y pulsed by OA.
A f t e r OB, the tw e lv e OS pulses are gen e ra te d. These pulses
perform the a d d i t i o n or s u b t r a c t i o n as d i c t a t e d by t he A d d / S u b t ra c t
F l i p - F l o p .
Once the el even i d e n t i c a l cy c le s have been completed, the
* D i v i s i o n i s e s s e n t i a l l y f i n i s h e d . However, as s t a t e d in the
d is c u s s io n o f the a l g o r i t h m , a t the completion o f the D i v i s i o n o p e ra
t i o n the sign b i t i s i n c o r r e c t . Thus a t w e l f t h c y c l e i s necessary to
c o r r e c t t h i s c o n d i t i o n . I t c o n s i s t s o f a s i n g l e pulse which changes
the sign o f the Q u o t i e n t by complementing DRQ. A c t u a l l y i t was
found t h a t since the D - R e g i s t e r i s pulsed o n l y on the OB p u l s e s ,
the 0 S-J2 pulse o f the e l e v e n t h c y c l e can be used t o c o r r e c t the
s i g n .
I n F ig u r e 5 a r e shown an example o f t he e le v e n i d e n t i c a l cycles
and an example o f the t w e l f t h c y c l e pulse t i m i n g , w h i c h occurs in
r e a l i t y on the O S ^ pulse and i s used f o r s h u t - o f f and f o r the
F i r s t Eleven I d e n t i c a l Cycles
OA — I--- :_____ ;______
QB — 1---
---Eleve nth Cycle S h u t - O f f Timing
OA---— ^---
---O B ______ I____________________________
---26
CHAPTER I I I
SYSTEM LOGIC
3.1 M u l t i p l i c a n d / D i v i s o r R e g i s t e r (AR).
T h i s r e g i s t e r , as i t s name i m p l i e s , i s m a i n l y used f o r the
s t or age o f the M u l t i p l i c a n d and the D i v i s o r . By n e c e s s i t y , the
sto ra g e must be n o n - d e s t r u c t i v e s in c e these two numbers a r e necessary
i n each stage o f the M u l t i p l i c a t i o n or D i v i s i o n o p e r a t i o n s . The
c i r c u i t diagram o f AR i s shown i n FIGURE 6.
The f a c i l i t i e s necessary a r e the d i r e c t c l e a r , load i n , and
t he r i g h t s h i f t o p e r a t i o n s , in a d d i t i o n to which i t must be a
c i r c u l a t i n g s h i f t r e g i s t e r in or de r t o r e t a i n th e 1 2 - b i t number
s tor ed in i t . I t was found t h a t the R-205 F l i p - F l o p was v e r s a t i l e
enough t o meet the above r e q u i r e m e n t s . The bussed d i r e c t c l e a r
t e r m i n a l s achi ev e th e c l e a r o p e r a t i o n . The t h i r d DCD gate in each
F l i p - F l o p achieves the l o a d - i n o p e r a t i o n by bussing the pulse i n p u t .
Also the l e v e l i n p u t o f each DCD gate is the corresponding ou tpu t l i n e
from the Accumulator of t he PDP-8/S.
The DCD gates a t the gated Set and gated C l e a r t e r m i n a l s a re
used f o r s h i f t i n g by cro s s -c o n n e c ti n g the o u t p u t o f the previous
F l i p F l o p t o the l e v e l in p u t o f the DCD gates o f the f o l l o w i n g F l i p
-F l o p . T h u s when t h e p u l s e i n p u t s o f t h e s e DCD g a t e s a r e pu l s e d , t h e
s h i f t i n g o f the con tents o f one F l i p - F l o p t o t he n e x t i s a c hi e v e d.
To c r e a t e the c i r c u l a t i n g r e g i s t e r , the ou t p u t o f the l a s t F l i p - F l o p
28
F I i p - F l o p o f AR.
Fur th er m or e, i t i s necessary t o make the conte nts o f A R^
a v a i l a b l e t o the A d d e r / S u b t r a c t e r a t each i n s t a n c e o f s h i f t i n g . AR
thus becomes t he r e q u i r e d c i r c u l a t i n g sto ra g e r e g i s t e r .
3 . 2 Ac c um u la t o r /D iv id e n d /M o s t S i g n i f i c a n t R e s u l t R e g i s t e r ( B R ) .
Beside p r o v i d i n g a f a c i l i t y f o r l o a d in g i n t he D i v i s o r , t h i s
r e g i s t e r i s a l s o used as an accumulator t o hold the P a r t i a l Product
d u ri n g the i n t e r m e d i a t e steps o f M u l t i p l i c a t i o n o p e r a t i o n . The
c i r c u i t diagram o f BR i s shown in FIGURE 7 .
Due t o the f a c t t h a t D i v i s i o n was a l s o include d in the hard
w a re , i t became necessary t o pro v id e not only the r i g h t s h i f t but
a l s o the l e f t s h i f t f a c i l i t y . Fu rth er m or e, t he d i r e c t c l e a r , load
i n and unload f a c i l i t i e s were a l s o found necessary. Thus, t h e DEC
R201 modules, which pro v id e f i v e DCD i n p u t gates were s e l e c t e d . Two
o f these gates a r e used f o r r i g h t s h i f t i n g as n e c e s s i t a t e d i n
M u l t i p l i c a t i o n , two more a r e used f o r l e f t s h i f t i n g as r e q u i r e d in
D i v i s i o n , and the f i f t h DCD gate i s used f o r lo a d i n g in d a t a .
To achi ev e the unload f a c i l i t y , one o u t p u t ga te had to be
provided f o r each F l i p - F l o p in the r e g i s t e r . The R123 NAND-gate
module was s e l e c t e d f o r t h i s purpose so t h a t , when these gates are
p u l s e d , th e conte nts o f
BR
a re t r a n s f e r r e d t o the PDP-8/S accumulatorthrough the I / O c a b l e s . The outputs o f th e f i r s t b i t , i e .
B R q ,
a r ecompared i n D i v i s i o n w i t h the outputs o f ARQ t o det erm ine the
o p e r a t i o n t h a t i s to be performed. Thus the out puts o f
B R
q a r e fedCO.
o
o
4 ■*.
k.
i I*
30
must be made a v a i l a b l e t o the A d d e r / S u b t r a c t e r in M u l t i p l i c a t i o n and
i n D i v i s i o n . Furthermore the outputs o f BR^-j must a l s o be s up pl ie d
t o DRq in M u l t i p l i c a t i o n v i a the r i g h t s h i f t l e v e l c o n t r o l , f o r ,
e f f e c t i v e l y in t he process BR and DR form a double l e n g th r e g i s t e r
f o r the 2 4 - b i t r e s u l t o f M u l t i p l i c a t i o n .
A g a in , r i g h t s h i f t i s achieved by the c ro s s -c o n n e c ti o n o f the
o u t p u t of the lower b i t to the l e v e l i n p u t o f the ne x t hig her b i t .
L e f t s h i f t on the o t h e r hand i s accomplished by the c ro s s -c o n n e c ti o n
o f th e out put o f the hi g h e r b i t to t he l e v e l i n p u t o f the DCD ga te
o f the next lower b i t . I n both types o f s h i f t i n g the r e s p e c t i v e p a i r s
o f DCD gates a re pulsed a t the same time when r e q u i r e d v i a the
pulse in p u t s o f these g a t e s .
During the i n t e r m e d i a t e o p e r a t i o n s o f A d d i t i o n or S u b t r a c t i o n ,
the DCD g a t e l e v e l in p u t s o f BRq (on r i g h t s h i f t s ) a re obtained
from t he o u t p u t o f the A d d e r / S u b t r a c t e r . Fu rt h er m o r e, on the r i g h t
s h i f t s p o s s i b l e on OA, OX and OY d u ri n g M u l t i p l i c a t i o n , the f i r s t
b i t , namely BRQ, must remain unchanged. Thus BRQ i s r i g h t s h i f t e d
o n ly on the t w e l v e OS p u l s e s , whereas BR-j t o BR^ a re r i g h t s h i f t e d
on a l l compulsory r i g h t s h i f t s namely on OA and OY, or on OX and
OY. The r i g h t s h i f t DCD gate pulse i n p u t s a r e thus bussed on BR-j to
BR^ whereas the r i g h t s h i f t pulse in p u t s o f BRq a re obtained d i r e c t l y
f r o m t h e OS p u l s e s o u r c e . I n D i v i s i o n , th e l e f t s h i f t pulse inp ut s a re
obtained f o r a l l BR b i t s from t he OB pulse source (with the a i d o f
3 . 3 M u l t i p l i e r / Q u o t i e n t / L e a s t S i g n i f i c a n t R e s u l t R e g i s t e r (DR).
The o p e r a t i n g c h a r a c t e r i s t i c s o f DR a re indeed s i m i l a r t o
those o f BR, because DR i s e f f e c t i v e l y the lower s e c t i o n o f a double
le n g th r e g i s t e r . The D - R e g i s t e r provides a p l a c e o f sto ra g e f o r the
M u l t i p l i c a n d and f o r the L ea st S i g n i f i c a n t Product in M u l t i p l i c a t i o n
and a l s o f o r the Q u o t i e n t i n D i v i s i o n . The c i r c u i t diagram o f DR i s
give n i n FIGURE 8.
Both r i g h t and l e f t s h i f t c a p a b i l i t i e s are included in DR
f o r M u l t i p l i c a t i o n and D i v i s i o n r e s p e c t i v e l y , as w e l l as the d i r e c t
c l e a r , load i n and unload f a c i l i t i e s . The DEC R201 F l i p - F l o p s were
used and two o f i t s f i v e DCD gates were used f o r the r i g h t s h i f t ,
two f o r the l e f t s h i f t and the remaining DCD g a t e was used f o r
load in g in t he M u l t i p l i c a n d d i r e c t l y from the PDP-8/S Accumulator.
I d e n t i c a l t o BR, the R123 Nand gate module was used f o r
unloading d a t a back to the PDP-8/S Accumulator, thus u t i l i z i n g one
g a t e a t t h e ' I 1 p u tp u t o f each F l i p - F l o p o f DR, DRQ t o DR^-j.
A t t h i s p o i n t however DR's s i m i l a r i t y to BR ends. There
i s a t h i r t e e n t h F l i p - F l o p inclu ded i n DR, namely DR^, making DR a
t h i r t e e n b i t r e g i s t e r . The purpose o f t h i s e x t r a b i t i s t o pro v id e
a f a c i l i t y f o r comparing the l a s t t h r e e b i t s of DR as r e q u i r e d in
th e M u l t i p l i c a t i o n A l g o r i t h m . Thus b i t s DR-j q, DR-j-j and DR^ are
e x a m i n e d d u r i n g e a c h c y c l e , w i t h the understanding t h a t DR^ is zero
a t the f i r s t stage o f the M u l t i p l i c a t i o n process. Because a t o t a l
o f t w e lv e r i g h t s h i f t s o f DR a r e used f o r a complete M u l t i p l i c a t i o n ,
32 to CD :r P CC
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U o' IU <C r/ Ul
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34
stage w i l l be found in DRb a f t e r t he completion o f the s i x t h or
f i n a l c y c l e . S h i f t i n g i n DR i s achieved by t h e same cro s s -c o n n e c ti o n
and pu l s in g method as descr ibed f o r BR i n S e c t i o n 3 . 2 .
During M u l t i p l i c a t i o n the ' 1 ' and ' 0 ' outputs o f DR-j q, DR-j^,
and DR^ a r e routed to the M u l t i p l i e r C o n t r o l l e r f o r comparison pu r
poses, and during the OA, OX, and OY pulse times the contents o f BR^
a r e r out ed to DRq by means o f the r i g h t s h i f t l e v e l c o n t r o l . T h i s
Ri g h t S h i f t Level Control i s de s c ri be d i n S e c t io n 3 . 7 .
L
During t he D i v i s i o n o p e r a t i o n DR^ i s n o t used. C o n s i d e r a ti o n
must n e v e r t h e l e s s be given to i t s conte nts because DR i s l e f t
s h i f t e d i n each c y c l e . I t i s found t h a t DR^ must be s e t to zero a t
a l l times in D i v i s i o n in or der t o have DR ^ i n the zero s t a t e
such t h a t i t w i l l be p o s s i b l e t o add a 1 or 0 t o i t on the OA pulse
w i t h o u t o v e r f l o w . When the l e f t s h i f t occurs on OB, DR-j-| w i l l again
become zero ready to accept a 1 or 0 on the OA pulse o f the next
c y c l e . I t may be mentioned t h a t t he ‘ Add 1 t o DR-^' pulse from the
D i v i d e r C o n t r o l l e r i s a p p l i e d t o the s e t t e r m i n a l o f DR-j-j. Thus when
the pulse i s p r e s e nt a ' 1 ‘ i s added to the D - R e g i s t e r and when the
pulse i n p u t i s absent nothing i s done, t h a t i s , e f f e c t i v e l y a 'O' i s
added to DR-j-j.
Because the D i v i s i o n A lg o ri t h m y i e l d s t h e c o r r e c t magnitude
but th e wrong sign o f the Q u o t i e n t , th e t w e l f t h c y c l e o p e r a t i o n ,
changing the s i g n , must be performed on DRQ. I n s t e a d o f a l l o w i n g
the c l o c k to run any f u r t h e r and having t o f i n d i t necessary to bloc k
