V L S I IM P L E M E N T A T IO N O F D IG IT A L F IL T E R S
by S R E E N IV A S A C H A R S U N D E R B .E ., 1985 a n d M .E ., 1987 A n n a U niv ersity , M a d ra s, In d ia A D IS S E R T A T IO N S U B M IT T E D IN P A R T IA L F U L F IL M E N T O F T H E R E Q U IR E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H IL O S O P H YA
( J C LP
T L \.) in th e D e p a rtm e n t ofA C U L fV t ),r <d</)IDUA !'L S T U D T S E le c tric a l a n d C o m p u te r E n g in e e rin g
______________________ W e a c c e p t th is d is s e rta tio n as co n fo rm in g
Dt an to the required standard DATE.
D r. A. A n to n .o u , S u p e rv iso r, D e p t, o f E le c t. &; C o m p . E ng.
D r. F . Sfl-G uibaly, C o -S u p erv iso r, D e p t, of E le c t. & C o m p . E n g .
Dr. N. J . D im o p o a lo s, D e p a r tm e n t M e m b e r, D e p t, o f E le c t, h C o m p . E ng.
D r. R . V a h ld iec k , G r a d u a te A d v iso r, D e p t, o f E le c t. & C o m p . E n g .
D r. I). M . M ille r, O u ts id e M e m b e r, D e p t, o f C o m p u te r S cien ce
D r. M . A. S id -A h m e d , E x te r n a l E x a m in e r, U n iv e rs ity o f W in d s o r
© S R E E N IV A S A C H A R S U N D E R , 1992 U N IV E R S IT Y O F V IC T O R IA
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S u p erv iso rs: D r. A . A n to n io u a n d D r. F. E l-G u ib a ly
ABSTRACT
In th is th e sis we d e sc rib e a m e th o d of m a p p in g o n e -d im e n sio n a l a n d m u ltid im e n sional filte r a lg o rith m s o n to sy sto lic a r c h ite c tu re s u sin g th e z - d o m a in a p p ro a c h . In th is a p p ro a c h th e filte r a lg o rith m is first tra n s f o rm e d in to its c o rre sp o n d in g d o m ain e q u iv a le n t a n d re c u rsiv e ex p re ssio n s s im ila r to single a s s ig n m e n t codes a re d e riv e d using H o rn e r’s ru le o r o th e r p o ly n o m ia l e v a lu a tio n te c h n iq u e s . B y o b ta in in g d iffe re n t re c u rsiv e ex p re ssio n s, d iffe re n t sy sto lic s tr u c tu r e s ca n b e d e riv e d . T h e c h a ra c te ris tic s of th e s e s tr u c tu r e s ca n e a sily b e d e d u c e d fro m th e re c u rsiv e ex p re ssio n s. T h e m u ltid im e n s io n a l filters d e riv e d a re m o d u la r a n d h ie ra rc h ic a l, i.e ., th e th re e -d im e n s io n a l s tr u c tu r e s a re o b ta in e d fro m th e tw o -d im e n s io n a l ones w hich a r e in t u r n o b ta in e d from o n e -d im e n sio n a l s tr u c tu r e s .
In c o n sid e rin g t h e d esig n of a n y a rra y p ro c e sso r, it is i m p o r ta n t to co n sid e r th e d esig n o f th e p ro c e ssin g e le m e n ts involved. T h e m o s t i m p o r ta n t a n d d e m a n d in g o p e r a tio n in th e s e e le m e n ts is th e m u ltip lic a tio n . F o u r d iffe re n t m u ltip lie rs a re d e sig n e d in w h ich th e n u m b e r o f o p e ra tio n s re q u ire d to p ro d u c e t h e d e sire d re s u lt is re d u c e d . T h e re d u c e d n u m b e r of o p e r a tio n s a lo n g w ith t h e a d v a n ta g e s o f v ery -la rg e -sc a le in te g ra tio n tech n o lo g y in te r m s of in c re a se d d e v ic e d e n s ity a n d fa s te r sw itc h in g m a k e th e s e m u ltip lie rs p o te n tia l c a n d id a te s in h ig h -s p e e d sig n al p ro c e ssin g a p p lic a tio n s . T h e first m u ltip lie r is an a re a -c rfic ie n t m u ltip lie r t h a t uses a p p r o x im a te ly 50% o f th e a r e a of a fu ll p a ra lle l m u ltip lie r. In th is m u lti p lie r o n ly th e u n iis y ie ld in g th e m o s t sig n ific a n t p a r t o f t h e p r o d u c t a re u s e d . In a d d itio n , a c o rre c tio n un:c is in c o rp o r a te d to m in im iz e t h e e r ro r re s u ltin g fro m c irc u m v e n tin g th e u se of u n its y ie ld in g th e le a s t sig n ific a n t p a r t o f th e p r o d u c t. T h e seco n d m u ltip lie r is b a se d on th e m o d ifie d o c ta l B o o th a lg o rith m in w hich fo u r-b it se g m e n ts o f th e m u ltip lie r a re sc a n n e d a n d c o rre s p o n d in g o p e r a tio n s ef fected o n th e m u ltip lic a n d . T h e th i r d m u ltip lie r is a d im in ish e d -1 m u ltip lie r t h a t finds a p p lic a tio n in th e F e rm a t n u m b e r- th e o re tic tra n s f o rm . In th is m u ltip lie r th e
u se of a tr a n s la t o r is c irc u m v e n te d a n d a novel te c h n iq u e for tra n s la tio n is in c o r p o ra te d in th e m u ltip lie r s tr u c tu r e . T h e fo u rth m u ltip lie r is one t h a t p e rfo rm s an in n e r- p ro d u c t o p e r a tio n w ith o u t th e u se of an a c c u m u la to r th e re b y re s u ltin g in in c re a se d sp e e d a n d re d u c e d area.
F in a lly , we d iscu ss th e V L S I im p le m e n ta tio n s of th r e e of th e m u ltip lie rs m e n tio n e d a b o v e , a s e c o n d -o rd e r d ig ita l filte r, a n d a sin g le p ro c e ssin g e le m e n t th a t ca n b e u s e d as a b a sic u n it in d e sig n in g o n e -d im e n sio n a l a n d m u ltid im e n sio n a l d ig ita l filte rs . S o m e a ss o c ia te d p ro b le m s in d ig ita l-filte r s tr u c tu r e s , viz., th e q u a n tiz a tio n a n d overflow lim it-c y c le o sc illa tio n s, h a v e been ta k e n in to co n sid e ra tio n a n d w ays h av e b e e n su g g e ste d for th e ir e lim in a tio n .
E x a m in e rs : D r. A. A n to n io u , S u p e rv iso r, D e p t, o f E lec t. V, C o m p . E ng. D r. F . E l - G u i b j y , C o -S u p e rv iso r, D e p t, of E le c t, fe C o m p . E ng. D r. N . J . D im o p o u ld s, D e p a r tm e n t M e m b e r, D e p t, o f E le c t. &r C o m p . E ng. D r. R . V ahldie.ck, G r a d u a te A d v iso r, D e p t, of E le c t. L C o m p . E ng. D r. D. M . M ille r, O u ts id e M e m b e r, D e p t, of C o m p u te r Science D r. M . A . S id -A h m e d , E x te rn a l E x a m in e r, U n iv e rsity o f W in d so r
T a b le o f C o n te n ts
T it le P a g e i A b s tr a c t ii T ab le o f C o n te n ts iv L ist o f T a b les ix L ist o f F ig u res x A b b r e v ia tio n s x v A c k n o w le d g e m e n ts x v ii D e d ic a tio n x v iii 1 I n tr o d u c tio n 1 1.1 V L S I a rra y p ro c esso rs ... 2 1.2 S y sto lic a r r a y s ... 3 1.3 R ev iew of p re v io u s w o r k ... 4 1.3.1 M a p p in g m e th o d o lo g ie s ... 4 1.3.2 M u ltip lie r d e s i g n ... 8 1.3.3 D ig ita l-filte r i m p l e m e n t a t i o n ... 8 1.4 O u tlin e of t h e s i s ... 102 M a p p in g m e th o d o lo g y for o n e -d im e n sio n a l d ig ita l filte r s 13 2.1 I n t r o d u c t i o n ... 13
T A B L E O F C O N T E N T S
2.2 S ig n al flow g ra p h a p p r o a c h ... 2.3 O n e -d im e n sio n a l n o n re c u rsiv e d ig ita l f i l t e r s ... 2.4 ^ -d o m a in a p p r o a c h ... 2.5 O n e -d irn e n sio n a l re c u rsiv e d ig ita l f i l t e r s ... 2 6 E x a m p l e s ... 2.6.1 S e c o n d -o rd e r re c u rsiv e f i l t e r s ... 2.6 .2 D e c im a to rs a n d I n t e r p o l a t o r : ! ... 2.7 C o n c l u s i o n s ...
3 M a p p in g m e th o d o lo g y for m u ltid im e n s io n r 1 d ig ita l filters
3.1 I n t r o d u c t i o n ... 3.2 T w o -d im e n sio n a l re c u rs 5 vc- filters ... 3.3 T h re e -d im e n s io n a l re c u rsiv e f i l t e r s ... 3.4 C o n c l u s i o n s ...
4 D e s ig n o f efficien t m u ltip lie r s
4.1 I n t r o d u c t i o n ... 4.2 A re a-efficien t p a ra lle l m u l t i p l i e r ... 4.2.1 E rr o r a n a l y s i s ... 4 .2 .2 In c re a se of sp ee d in th e tr u n c a te d m u l t i p l i e r ... 4 .2 .3 R e d u c tio n o f a r e a in th e tr u n c a te d m u l ti p l i e r ... 4.2 .4 T w o ’s c o m p le m e n t m u l t i p l i c a t i o n ... 4 .2 .5 P ip e lin e d tr u n c a te d m u l t i p l i e r ... 4 .2 .6 A p p lic a tio n of th e t r u n c a te d m u ltip lie r in d ig ita l filters . . 4 2.7 Q u a si-se ria l tr u n c a te d m u l ti p l i e r ... 4.3 M u ltip lie r based on th e m o d ified o c ta l B o o th a lg o rith m . . . .
4 .3.1 T h e Q M B m u l t i p l i e r ... 4 .3 .2 T h e O M B m u l t i p l i e r ... 4 .3 .3 E x te n s io n to tw o ’s c o m p le m e n t m u l t i p l i c a t i o n ... 4.4 D im in is h e d -1 m u ltip lie r for F e rm a t n u m b e r t r a n s f o r m ... 4.4.1 D im in ish e d -1 m u ltip lie r u sin g th e p a ra lle l m u ltip lie r . . . .
v 13 Hi 25 30 36 36 41 15 46 16 17 51 58 59 50 60 63 69 71 71 73 76 80 82 82 87 91 92 93
4.4.2 R e sid u e r e d u c t i o n ... 96 4.4.3 C o m p a r i s o n ... 99 4.4.3.1 A r e a ... 99 4 .4.3.2 S p e e d ... 100 4.5 A c c m u la to r-m u ltip lie r ... 100 4.5.1 C o m p a riso n s ... 102 4.5.1.1 A r e a ... 102 4 .5 .1 .2 S p e e d ... 103 4.6 C o n c l u s i o n s ... . . . . 103
5 V L S I im p le m e n ta tio n o f m u ltip lie r s 105 5.1 I n t r o d u c t i o n ... 105
5.2 Im p le m e n ta tio n of th e tr u n c a te d m u l t i p l i e r ... 106
5.2.1 F e a t u r e s ... 106
5.2.2 Icon a n d blo ck d ia g ra m of th e tr u n c a te d m u ltip lie r . . . . 106
5.2.3 F u n c tio n a l d e s c rip tio n o f th e b lo ck d ia g ra m ... 107
5.2.-1 T im in g d i a g r a m ... 108 5.2.5 P h y sic a l c h a r a c t e r i s t i c s ... 109 5.2.6 L a y o u t of th e tr u n c a te d m u l ti p l i e r ... 109 5.3 Im p le m e n ta tio n of t h e O M B m u l t i p l i e r ... I l l 5.3.1 F e a t u r e s ... I l l 5.3.2 Icon a n d blo ck d ia g ra m of th e O M B m u l t i p l i e r ... I l l 5.3.3 F u n c tio n a l d e s c rip tio n o f th e b lo ck d ia g ra m ... 112
5.3.4 T im in g d i a g r a m ... 112
5.3.5 P h y sic a l c h a r a c t e r i s t i c s ... 112
5.3.6 L ay o u t of th e O M B m u l t i p l i e r ... 115
5.4 Im p le m e n ta tio n of a d im in is h e d -1 m u l t i p l i e r ... 115
5.4.1 F e a t u r e s ... 115
5.4.2 Icon a n d b lock d ia g ra m o f th e d im in ish ed -1 m u ltip lie r . . . 116
T A B L E O F ( O N T E N T S vii 5.4.4 T im in g d i a g r a m ... 117 5.4.5 P h y sic a l c h a r a c te r i s t i c s ... , 117 5.4.6 T e s tin g s t r a t e g y ... 119 5.4.7 L ayout of th e d im in is h e d- 1 m u l t i p l i e r ... 120 5.5 C o n c l u s i o n s ... 129 6 V L S I im p le m e n ta tio n o f filters 122 6.1 I n t r o d u c t i o n ... 122 6.*2 B a c k g r o u n d ... 122 6.3 I m p le m e n ta tio n of th e se c o n d -o rd e r d ig ita l f i l l e r ... 125 6.3.1 F e a t u r e s ... 125 6.3.2 Icon a n d block d ia g ra m o f th e f i l t e r ... 126
6.3.3 F u n c tio n a l d e s c rip tio n o f tiie b u ild in g b l o c k s ... 12N 6.3.4 T im in g d i a g r a m ... 135
6.3.5 P h y sic a l c h a r a c t e r i s t i c s ... 13X 6.3.6 T e stin g s t r a t e g y ... 13X 6.3.7 L ay o u t o f th e s e c o n d -o rd e r d ig ita l f i l t e r ... 131)
6.4 I m p le m e n ta tio n o f a sy sto lic P I C ... I ll 6.4.1 F e a t u r e s ... PH 6.4.2 Icon a n d block d ia g ra m o f th e systolic PIC . . . 141
6.4.3 F u n c tio n a l d e s c rip tio n o f th e block d i a g r a m ... 113
6.4.4 T im in g d i a g r a m ... ] (3
6.4.5 P h y sic a l c h a r a c t e r i s t i c s ... 144
6.4.6 T e stin g s t r a t e g y ... M5 6.4.7 L ay o u t of th e sy sto lic PIC ... 145
6.5 C o m p a r i s o n s ... 145
6 . 6 C o n c l u s i o n s ... j.pj 7 C o n c lu sio n s 1 4 7 7.1 C o n t r i b u t i o n ... 147
I N
L ist o f T a b les
4.1 T h e tr u t h ta b le of th e e n c o d e r alo n g w ith th e m a th e m a tic a l o p e r a tio n s effec te d by th e vario u s th re e -b it se q u e n c e s of th e m u ltip lie r. SI 4.2 T h e t r u th ta b le of th e e n c o d e r alo n g w ith th e m a th e m a tic a l o p e r
a tio n s effec te d by th e vario u s four-bit se q u e n c e s o f t h e m u ltip lie r. SS
5.1 C h ip s ta tis tic s of th e tr u n c a te d a n d p a ra lle l m u ltip lie r s ... 10!) 5.2 G a te -le v e l s ta tis tic s o f th e tr u n c a te d a n d p a ra lle l m u ltip lie rs . . . . 1 ]() 5.3 C h ip s ta tis tic s of th e O M B a n d Q M B m u ltip lie rs ... I l l 5.4 G a te -le v e l s ta tis tic s o f th e O M B a n d Q M B m u ltip lie r s ... I l l 5.5 C h ip s ta tis tic s of th e d im in is h e d -1 m u ltip lie r ... 1 IS 5.6 G a te -le v e l s ta tis tic s o f th e d im in ish ed -1 m u ltip lie r ... 11!)
6.1 C o n tro l sig n a ls for th e se c o n d -o rd e r d ig ita l filte r... 136 6.2 C h ip s ta tis tic s of th e se c o n d -o rd e r d ig ita l filte r... 13.S 6.3 G a te -le v e l s ta tis tic s of th e sec o n d -o rd e r d ig ita l filte r... 13!) 6.4 C h ip s ta tis tic s of th e sy sto lic P E ... I l l 6.5 G a te -le v e l s ta tis tic s of th e sy sto lic P E ... I l l
L ist o f F ig u res
2.1 D e p e n d e n c e g ra p h of (2 .3 )... 17 2.2 S ignal flow g ra p h of (2 .4 )... 18 2.4 M ap p in g of ih e signal flow g ra p h of (2 .3 ) o n to a s y sto lic array . (a)
T h e sy sto lic s tr u c tu r e , (b ) D e ta ils o f P E in v o lv e d ... 19 2.4 S ignal flow g ra p h of (2.7) in w inch filte r in p u ts a re b ro a d c a s t a n d
o u tp u ts a re p r o p a g a te d ... 20 2.7) M a p p in g of th e signal flow g ra p h of (2.4) o n to a s y sto lic a rra y (a)
T h e sy sto lic s tr u c tu r e , (b ) D e ta ils o f P E in v o lv e d ... 21 2.(5 S ignal flow g ra p h of (2.9) in w hich filte r in p u ts a n d o u tp u ts a r e
p r o p a g a te d ... 22 2.7 M a p p in g of (2.9) o n to a sy sto lic a r c h ite c tu re , (a ) T h e sy sto lic a rra y .
(b ) D e ta ils o f th e P E in v o lv e d ... 23 2.8 S ignal flow g ra p h of th e m o d ified fo rm of F ig . 2.6 sh o w in g re d i
re c tio n o f th e filte r in p u ts a n d th e fo rm a tio n o f p a r tia l p ro d u c ts a t th e h y p e r p la n e s ... 24 2.9 S y sto lic re a liz a tio n of th e sig n al flow g ra p h o f F ig. 2 . 8 ... 25 2.10 S y sto lic re a liz a tio n of th e sig n al flow g ra p h of F ig . 2 .8 ... 26 2.11 M ap p in g o f (2.21) o n to a sy sto lic a r c h ite c tu r e , (a ) T h e sy sto lic
a rray , (b ) D e ta ils of P E in v o lv e d ... 32 2.12 M ap p in g o f (2.24) o n to a sy sto lic a r c h ite c tu r e , (a) T h e sy sto lic
L I S T O F F I G U R E S
2.13 M a p p in g of (2.26) a n d (2.27) o n to sy sto lic a r c h ite c tu re s , (a) T h e sy sto lic array , (b ) D e ta ils of ? E invo lv ed in m a p p in g (2.26). (c) D e ta ils of P E in v o lv ed in m a p p in g (2 .2 7 )... 2.14 M a p p in g of (2.30) o n to a sy sto lic a i- a y ... 2.15 M a p p in g of (2.31) onco a sy sto lic a rra y ... 2.16 (a ) M a p p in g of (2.32) o n to a sy sto lic a rra y , (b ) M a p p in g of (2.33)
o n to a sy sto lic a r ra y ... 2.17 M a p p in g o f (2.35) o n to a sy sto lic a rra y ... 2.18 S y sto lic s tr u c tu r e for a d e c im a to r. (a) T h e sy sto lic a rra y , (b) D e
ta ils of P E in v o lv e d
2.19 A n a lte r n a tiv e sy sto lic s tr u c tu r e for a d e c im a to r. (a) T h e sy sto lic a rray , (b ) D e ta ils of P E in v o lv e d ...
3.1 S y sto lic a r ra y for a 2-D re c u rsiv e filter u sin g S ch em e 1 for a w indow o f size 3 x 3 ... 3.2 S y sto lic a r ra y for a 2-D re c u rsiv e filte r u sin g S ch em e 2 for a w in d o w
o f size 3 x 3 ... 3.3 S y sto lic a r ra y tb ” a 2-D re c u rsiv e filte r u sin g S ch em e 3 for a w indow
o f size 3 x 3 ... 3.4 S y sto lic a r r a y for a 3-D re c u rsiv e filter u sin g sch e m e sim ila r to t h a t
o f S c h e m e 1 for a w indow of size 3 x 3 x 3 . (a ) T h e sy sto lic a rra y , (b ) D e ta ils of th e P E in v o lv e d ... ...
4.1 A n 8 x 8 m u ltip lic a tio n usin g p a ra lle l m u ltip lie r w h e re A , IIA a n d FA a re t h e A N D , h a lf-a d d e r a n d fu ll-a d d e r cells, re sp ectiv ely , (a ) M u ltip lie r blo ck d ia g ra m , (b ) D e ta ils of A H A cell, (c) D e ta ils of A FA c e ll... 4.2 R e p re s e n ta tio n of A , B , a n d P in te rm s of th e ir m o s t a n d le a s t
4.3 D e ta ils o f g e n e ra tio n of P . T h e p a r tia l re s u lts are p la c e d h o riz o n
ta lly a c c o rd in g to th e ir b in a ry w e ig h ts... 64
4.4 S ectio n s in th e p a ra lle l m u ltip lie r g e n e ra tin g th e fo u r te rm s « f P . T h e sh a d e d i >n re p re s e n ts th e cells t h a t g e n e ra te d is c a rd e d re su lts d u e to tr u n c a t io n ... 65
4.5 A tr u n c a te d m u ltip lie r for an 8 x 8 b it m u ltip lic a tio n ... 66
4.6 V a ria tio n o f th e e x p e c te d value of t h e e rro r w ith N ... 69
4.7 V a ria tio n of th e s ta n d a r d d e v ia tio n o f th e e r r o r w ith N ... 70
4.8 N M M te c h n iq u e to in c re a se th e s p e e d of th e tr u n c a te d m u ltip lie r. 70 4.9 V a ria tio n o f th e m tio of th e a re a o f a tr u n c a te d m u ltip lie r to t h a t o f a full m u ltip lie r w ith N ... 72
4.10 A n 8 x 8 tw o ’s co m p le m e n t m u ltip lic a tio n u s in g p a ra lle l m u ltip lie r w h ere N I) is a NaN D g a te cell, (a) M u ltip lie r b lock d ia g ra m , (b ) D e ta ils o f N FA c e ll... 74
4.11 P ip e lin in g te c h n iq u e for t h e la s t row of fu ll-a d d e r cells. R: 1 -b it re g iste r, R IIA : h a lf a d d e r followed b y a 1-bit re g iste r, X : E x clu siv e-O R g a te , R X : E x clu siv e-e-O R g a te follow ed b y a 1-bit re g is te r. . . . 75
4.12 P a r tia l p ro d u c ts for a 4 X 4 m u ltip lic a tio n ... 76
4.13 A n 8 x 8 fu lly p ip e lin e d tr u n c a te d m u ltip lie r. S P 7 b it sh ift re g is te r , A S R : A se t of A N D g a te s follow ed by a r e g is te r ... 77
4.14 A low pass G IC w ave d ig ita l filte r... 78
1.15 V a ria tio n o f th e o u t p u t noise P S D o f a low p ass G IC w ave d ig ita l filte r u sin g s ta n d a r d a n d tr u n c a te d m u ltip lie r s ... 79
1.16 Q u a si-se ria l tr u n c a te d m u ltip lie r b lo ck d ia g r a m ... 81
1.17 A n 8 x 8 c o n v e n tio n a l Q M B m u ltip lie r. C S A : C a rry -sa v e a d d e r. . 84
4.18 O p e ra tio n o f an 8 x 8 c o n v e n tio n a l Q M B m u ltip lie r ... 85
4.19 A n 8 x 8 p a ra lle l Q M B m u ltip lie r... 86
4.20 A n 8 x 8 O M P m u ltip lie r ... 89
L I S T O F F I G U R E S xiii
4.22 A fa st 8 x 8 O M B m u ltip lie r ... 91
4.23 A m o d ifie d 4 x 4 p a ra lle l m u ltip lie r t h a t y ield s a 9 -b it p ro d u c t , (a) M u ltip lie r b lo c k d ia g ra m , (b) D e ta ils of A F A l cell, (o ' D e ta ils of A FA 2 cell, (d ) D e ta ils o f A H A c e ll... 95
4.24 O u tp u t c o n tro lle r ... 96
4.25 N e g a to r... 97
4.26 B lock d ia g ra m o f a m o d ifie d 4 x 4 d im in is h e d -1 o ip e lin e d m u ltip lie r. A ll th e R a re 4 -b it re g is te rs e x c e p t th e o n es b efo re a n d a fte r th e n e g a to r w h ich a r e 5 -b it re g iste rs. D is a 1 -b it flip-flop... 98
4.27 A n 8 x 8 a c c u m u la te -m u ltip lie r ... 101
5.1 Icon for th e tr u n c a te d m u ltip lie r ... 106
5.2 B lo ck d ia g ra m fo r th e tr u n c a t e d m u ltip lie r ... 107
5.3 S c h e m a tic d ia g ra m of th e c o rre c tio n u n i t ... 108
5.4 T im in g d ia g ra m for th e tr u n c a te d m u ltip lie r ... 109
5.5 P h o to m ic r o g ra p h of th e 16 x 16 tr u n c a te d m u ltip lie r ... 110
5.6 Icon fo r th e O M B m u ltip lie r ... I l l 5.7 S c h e m a tic d ia g ra m for th e o c ta l e n c o d e r ... 113
5.8 T im in g d ia g ra m for th e O M B m u ltip lie r ... 114
5.9 P h o to m ic r o g ra p h of th e O M B m u ltip lie r ... 115
5.10 Icon for th e d im in ish e d -1 m u ltip lie r ... 116
5.11 T im in g d ia g ra m of th e d im in is h e d -1 m u ltip lie r ... 118
5.12 F lip -flo p u se d in th e s c a n - p a th ... 119
5.13 P h o to m ic ro g rc ip h of th e d im in ish ed -1 m u ltip lie r ... 120
6.1 S a tu r a tio n c h a r a c te r is tic s ... 124
6.2 Ico n fo r th e s e c o n d -o rd e r re c u rsiv e d ig ita l f ilte r ... 126
6.3 B lo ck d ia g ra m o f th e s e c o n d -o rd e r re c u rsiv e d ig ita l f ilte r... 127
6.4 B lock d i .g r a m o f X coeffreg u n i t ... 128
6.6 B lock d ia g ra m of X sig reg u n i t ... 130
6.7 B lock d ia g ra m of Y sigreg u n i t ... 130
6.8 / n 8 X 8 ite ra tiv e m u ltip lie r t h a t in c lu d e s th e overflow d e te c tio n u n it, (a ) T h e m u ltip lie r u n it, (b ) O verflow d e te c tio n u n i t ...132
6.9 S c h e m a tic d ia g ra m of th e m a g n itu d e t r u n c a t o r ... 133
6.10 B lock d ia g ra m of t h e m a g n itu d e d e c o d e r... 134
6.11 B lock d ia g ra m of th e s a tu r a tio n - a r ith m e tic u n i t ... 135
6.12 B lock d ia g ra m of th e c o n tro lle r... 137
6.13 T im ir g d ia g ra m fo r th e se c o n d -o rd e r f ilte r ... 138
6.14 P h o to m ic ro g ra p h o f th e s e c o n d -o rd e r d ’ ita l f ilte r ... 140
6.15 Icon for th e sy sto lic P E ... 141
6.16 B lock d ia g ra m of th e sy sto lic P E ... . 142
6.17 T im in g d ia g ra m fo r th e sy sto lic P E ... 143
A b b r e v ia tio n s
XV
A C M A c c u m u la to r-m u ltip lie r
A F A A N D g a te follow ed b y a full a d d e r A H A A N D g a te follow ed b y a h a lf a d d e r A S R A N D g a te follow ed b y a sh ift re g iste r C A D C o m p u te r a id e d d esign
C M C C a n a d ia n M ic ro e le c tro n ic s C o rp o ra tio n C M O S C o m p le m e n ta ry m e ta l-o x id e se m ic o n d u c to r
C M O S 3 D L M N o rth e rn T ele co m E le c tio n ic s 3 -m icro n sin g le-p o ly silic o n , d o u b le-le v el m e ta l, p-w ell C M O S p rocess
C M O S 4 S N o rth e rn T ele co m E le c tro n ic s 1.2-m icron d o u b le -p o ly silic o n , d o u b le-le v el m e ta l, tw in -w ell C M O S process
C S A C a rry -sa v e a d d e r C S F C a sc a d e d s e c o n d -o rd e r filte r D F T D is c re te F o u rie r tra n s f o rm D G D e p e n d e n c e g ra p h D S M D esig n scale m ic ro n F A F ull a d d e r F I F O F h rst-in -first-o u t
F N T F e rm a t n u m b e r- th e o re tic tra n sfo rm G IC G e n e ra liz e d b n m itta n c e co n v e rto r IC I n te g r a te d c iiv m t
L S B L e a st sig n ifican t b it M A C M u ltip lie r-a c c u m u la to r M -D M u ltid im e n sio n a l M S B M o st sig n ific a n t b it
N F A N A N D g a te follow ed b y a full a d d e r N H A N A N D g a te follow ed b y a h a lf a d d e r N M M N o n -a d d itiv e m u ltip lic a tiv e m o d u le N T E N o rth e rn T ele co m E le c tro n ic s N T T N u m b e r th e o r e tic tra n s f o rm 0 1 IB O c ta l m o d ified B o o th a lg o rith m P E P ro c e ssin g e le m e n t P G A P in g rid a rra y P L A P r o g ra m m a b le logic a r r a y P S D P o w er s p e c tr a l d e n sity Q M B Q u a r te r n a r y m o d ified B o o th a lg o rith m R H A A h a lf a d d e r follow ed by a 1-bit re g iste r
R IA R e g u la r ite r a tiv e a lg o rith m RX An E X C L IJS IV E -O R g a te follow ed b y a 1 -b it re g is te r S FG S ignal flow g ra p h SR S hift re g is te r SSF S ingle se c o n d -o rd e r filter S1JF S p e e d -u p fa c to r SVD S in g u la r value d e c o m p o sitio n T M S T est m o d e selec t
V LSI V e ry -larg e -sca le in te g ra tio n W I)F W ave d ig ita l filte r
A ck n o w led g m en ts
XVll
I w ish to e x p re ss m y g r a titu d e to m y s u p e rv iso rs, D r. A. A n to n io u a n d D r. F . E l-G u ib a ly of th e D e p a rtm e n t of E le c tric a l a n d C o m p u te r E n g in e e rin g , fo r th e ir e n c o u ra g e m e n t, g u id a n c e , a n d a d v ic e d u rin g th e co u rse of th is research a n d for t h e ir h elp in th e p r e p a ra tio n of th is th esis.
T h e a s s is ta n c e re n d e re d by th e C a n a d ia n M ic ro e le c tro n ic s C o rp o ra tio n in th e f a b ric a tio n o f V L S I chip s for th is p ro je c t is fully ack n o w led g ed . I am g ra te fu l to D r. D. M . M ille r o f th e D e p a r tm e n t o f C o m p u te r S cience in th is re g a rd .
F in a n c ia l a s s is ta n c e re c e iv e d fro m D r. A. A n to n io u a n d I5r. F . E l-C u ib a ly (th ro u g h t h e N a tu r a l S cien ces a n d E n g in e e rin g R esea rch (lounc.il of C a n a d a a n d th e M ic ro n e t, N a tio n a l C e n tre s o f E x ce lle n ce C ro g ra m ) is g ra te fu lly ac know ledged.
I a m g ra te fu l t o m y p a r e n ts fo r m a k in g it p o ssib le for m e to b ec o m e w h a t I a m a n d to g et as fa r as I h av e. M y w ife, R a m a , has c o n tr ib u te d to th is th esis in m a n y in ta n g ib le w ay s for w h ich I w ish to re c o rd m y v e ry sin c e re g ra titu d e .
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I n tr o d u c tio n
T h e in c re a sin g d e m a n d for p ro c essin g sp ee d a n d overall sy ste m p e rfo rm a n c e in m o d e rn sig n al a n d im a g e p ro c e ssin g a p p lic a tio n s n e c e s s ita te s a sp ec ia liz ed c o m p u tin g tech n o lo g y . T h e a v a ila b ility of lo w -co st, h ig h -d e n sity , h ig h -sp ee d very- la rg e -sc a le in te g ra tio n (V L S I) d ev ice s a n d e m e rg in g c o m p u te r-a id e d design (C A D ) fa c ilitie s p re sa g e a m a jo r b re a k th ro u g h in th e design a n d a p p lic a tio n of m assiv ely p a r a lle l p ro c esso rs. In p a r tic u la r , V L S I m ic ro e le c tro n ic s tech n o lo g y h a s in sp ired m a n y in n o v a tiv e d esig n s in a r ra y p ro c esso r a r c h ite c tu re s . 'I'h is tr e n d has now b e c o m e a m a jo r focus o f a tt e n t io n in g o v e rn m e n ts, in d u s trie s , a n d u n iv e rsitie s. In t h e la s t d e c a d e , th e r e h as b ee n a d r a m a tic w o rld w id e g ro w th in re searc h a n d d e v e lo p m e n t on th e s y s te m a tic m a p p in g of various sig n al a n d image* p ro c essin g a p p lic a tio n s o n to V L S I a r c h ite c tu re s .
M o d e rn sig n al a n d im a g e p ro c essin g te c h n o lo g y d e p e n d s c ritic a lly o n th e d e vice a n d a r c h ite c tu r a l in n o v a tio n s of th e c o m p u tin g h a rd w a re . S e q u e n tia l sy ste m s a re in a d e q u a te for re a l-tim e p ro c essin g sy ste m s; th e a d d itio n a l c o m p u ta tio n a l c a p a b ility a v a ila b le th ro u g h V L S I c o n c u rre n t a rra y p ro c e sso rs will b e c o m e a n e cessity . In m o s t re a l-tim e d ig ita l signal p ro c e ssin g a p p lic a tio n s , g e n e ra l-p u rp o se p a ra lle l c o m p u te r s c a n n o t offer s a tis fa c to ry p ro c e ssin g sp ee d due* to sev e re d e m a n d s im p o s e d by sy ste m o v e rh e a d . T h e re fo re , s p e c ia l-p u rp o s e a rra y p ro cesso rs w ill b e c o m e th e o n ly a p p e a lin g a lte r n a tiv e . L e t us c o n sid e r a re a l-tim e a p p lic a tio n in o r d e r to s u b s t a n ti a te th e ab o v e claim . In d ig ita l v id eo p ro c essin g , it is usual to
A ssu m in g t h a t th e r e a re 24 fram es a rriv in g p e r seco n d a n d t h a t 10 o p e ra tio n s p e r p ix e l a re c a rrie d o u t, th e n u m b e r o f o p e r a tio n s p e r sec o n d to b e c a rrie d o u t w ould b e a p p r o x im a te ly 107. W ith t h e p re s e n t te c h n o lo g y for th e g e n e ra l-p u rp o s e p ro c e sso rs, it is d ifficu lt to ac h ie v e th is sp e e d d u e to th e classic m e m o ry access b o ttle n e c k p ro b le m s a n d sy s te m o v e rh e a d .
C u r r e n t p a ra lle l c o m p u te rs c a n b e p u t in to th r e e s tr u c tu r a l classes: v e c to r p ro c e sso rs, sh a re d m e m o ry s y ste m s, a n d a r r a y p ro c esso rs [ 1 ]-[2]. T h e first tw o classes b elo n g to th e g e n e ra l-p u rp o se c o m p u te r d o m a in . T h e d e v e lo p m e n t of th e s e s y s te m s re q u ire s a c o m p lic a te d d esig n of c o n tro l u n its a n d o p tim iz e d sch em es for a llo c a tio n of m a c h in e re so u rces. T h e th ir d class, h ow ever, b e lo n g s to t h e d o m a in of sp e c ia l-p u rp o se c o m p u te rs. T h e d esig n of su c h s y s te m s re q u ire s a b ro a d kno w led g e of th e re la tio n s h ip b e tw e e n p a ra lle l-c o m p u tin g h a r d w a re a n d so ftw a re s tr u c tu r e s . It is th is class of a rra y s t h a t offers a p ro m is in g s o lu tio n to m e e t re a l-tim e p ro c essin g re q u ire m e n ts .
1.1
V L S I a rra y p r o c e s s o r s
A s o lu tio n to m e e t re a l-tim e sig n al p ro c e ssin g re q u ire m e n ts is to u se sp ec ia l- p u rp o s e a rra y p ro c esso rs a n d to m a x im iz e th e p ro c e ssin g c o n c u rre n c y b y e ith e r p ip e lin e o r p a ra lle l p ro c e ssin g or b o th . A n efficient s y s te m c a n b e a c h ie v e d if t h e a rra y e n ta ils a b a la n c e d d is tr ib u tio n o f p ro c e sso r w ork lo ad w h ile o b se rv in g t h e re q u ire m e n t of d a t a lo cality , i.e ., s h o rt c o m m u n ic a tio n p a th s . T h e s e p ro p e rtie s o f load d is tr ib u tio n a n d in fo rm a tio n flow se rv e as g u id e lin e s to th e d e sig n e r o f V L S I a rra y s . O n e such sp e c ia l-p u rp o se V L S I a r ra y is th e sy sto lic a r ra y w h ich e n ta ils a m a ssiv e a m o u n t of co n c u rre n cy . In th e follow ing se c tio n w e s h a ll d iscu ss th e c o n c e p t of sy sto lic a rra y s a n d th e ir s u ita b ility as s p e c ia l-p u rp o s e c o m p u te r s for d ig ita l signal p ro c essin g a p p lic a tio n s .
1.2
S y s to lic a rra y s
A s y sto lic sy s te m c o n sists o f a set of re g u la rly in te rc o n n e c te d p ro c esso rs, each c a p a b le o f p e rfo rm in g a se t of o p e ra tio n s [3]. In a sy sto lic array , d at a flows b etw e en p ro c e sso rs in a rh y th m ic fa sh io n , p assin g th ro u g h m a n y p ro c essin g e le m e n ts ( P F s) befo re i t re tu r n s to m em o ry , m u c h as b lo o d c irc u la te s from th e h e a r t th ro u g h th e v a sc u la r sy ste m s a n d b ack to th e h e a r t. In th is fashion d a t a e x tra c te d from m e m o ry is u sed b y m a n y p ro c esso rs in a p a ra lle l a n d / o r p ip e lin e d fash io n th e re b y im p ro v in g memo* ' u tiliz a tio n . T h e m a jo r fa c to rs for a d o p tin g sy sto lic a rra y s for s p e c ia l-p u rp o s e p ro c e sso rs are:
1. S im p le a n d re g u la r d esig n
2. C o n c u rre n c y a n d local c o m m u n ic a tio n 3. S u ita b ility for c o m p u te - b o u n d a p p lic a tio n s
In in te g r a te d - c ir c u it tech n o lo g y , t h e co st of c o m p o n e n ts is d ro p p in g d r a m a t i cally w h e re a s th e c o st of d esig n grow s w ith th e c o m p le x ity of th e s y ste m . S pecial- p u rp o s e s y ste m s a r e seld o m p ro d u c e d in la rg e q u a n titie s a n d in su ch cases p a rt costs a r e less im p o r t a n t th a n d esig n c o sts. A s a co n se q u e n c e , th e d esig n c o st of s p e c ia l-p u rp o s e s y s te m s m u s t b e re la tiv e ly sm a ll for th e m to be m o re a ttra c tiv e ' th a n g e n e ra l-p u rp o s e s y ste m s. M oreo v er, if a sp ec ia l-p u rp o se' s y s te m ele*sign is c o m p o se d of a few ty p e s of sim p le P E s t h a t a re u sed re p e titiv e ly w ith sim ple' in te rfa c e s , g re a t sa v in g s in te r m s of c o st c a n b e ac h ie v ed . F u rth e rm o re ', sim ple' a n d r e g u la r s y s te m s a r e lik ely to be m o d u la r a n d th e re fo re a d a p ta b le to vario u s p e rfo rm a n c e goals.
A n im p o r ta n t fa c to r in t h e sp ee d o f a c o m p u tin g sy ste m is th e use- o f c o n c u rre n c y . F or s p e c ia l-p u rp o s e s y ste m s, c o n c u rre n c y d e p e n d s on th e unelerlying a lg o rith m s em p lo y e d . M assiv e p a ra lle lism c a n b e achiever! if th e a lg o rith m is fo rm u la te d su ch t h a t h ig h d e g re e s of p ip e lin in g a n d m u ltip ro c e s s in g can be in tr o d u ce d .
S y sto lic a rra y s a r c d e sig n e d for c o m p u te -b o u n d p ro b le m s t h a t a re b a se d on re g u la r re c u rre n c e e q u a tio n s [4]-[10j. C o n se q u e n tly , th e y h a v e b e e n u sed in th e a re a s of d ig ita l filte rin g , im a g e a n d sp ee ch p ro c essin g , a n d m a tr ix a lg e b r a to n a m e a few a p p lic a tio n s [4]-[14]. S ev eral ty p e s o f sy sto lic a rra y s h av e b e e n p ro p o se d d e p e n d in g on th e ty p e of s tr u c tu r e em p lo y e d , e.g ., lin e a r, tr ia n g u la r o r h e x a g o n a l.
In an effo rt to o b ta in sy sto lic s tr u c tu r e s for c o m p u te -b o u n d p ro b le m s , sev e ra l m a p p in g m e th o d o lo g ie s h a v e b e e n p ro p o se d to m a p a lg o rith m s d ire c tly o n to sy sto lic a rra y s to o b ta in m a x im u m c o n c u rre n c y b y u sin g p ip e lin in g a n d p a ra lle l p ro c essin g . In th e follow ing se c tio n we sh all rev iew so m e o f th e to p ic s t h a t a re re le v a n t to th is th esis.
1 .3
R e v ie w o f p r e v io u s w o rk
1.3.1 M a p p in g m e th o d o lo g ie s
M an y signal p ro c e ssin g a lg o rith m s ca n b e e x p re sse d as a s e t of ite r a tiv e s t a t e m e n ts a n d such a lg o rith m s a r e c a lle d re g u la r ite r a tiv e a lg o rith m s (R IA ) [5]. T h e co m m o n c h a ra c te ris tic of m a n y of th e p ro p o se d m e th o d o lo g ie s for m a p p in g R IA s o n to ite r a tiv e a rra y s is th e u se o f a tra n s f o rm a tio n a l a p p ro a c h t h a t in v o lv es t r a n s fo rm in g th e a lg o rith m d e s c rip tio n s to ite r a tiv e s ta te m e n ts t h a t a re a m e n a b le to V L S I im p le m e n ta tio n . D is tin c t tra n s f o rm a tio n a l s y s te m s fo r sy sto lic d esig n c a n be c h a ra c te riz e d by th e m a n n e r in w hich th e a lg o rith m s a r e d e s c rib e d , th e ty p e of fo rm a l m o d e ls u se d , a n d th e ty p e of tr a n s f o rm a tio n s used .
In th e m e th o d o lo g y p ro p o se d by L am a n d M o sto w [15], a n a lg o rith m o b ta in e d by so ftw a re tra n s f o rm a tio n s fro m a h ig h -lev el sp e c ific a tio n , w h ich re s u lts in seg m e n ts o f c o d e e x e c u te d re p e a te d ly w ith a re g u la r p a tt e r n o f d a t a ac cesses, is m a p p e d o n to a sy sto lic d esign d e sc rib e d b y a s tr u c t u r e a n d a d riv e r. T h e s t r u c tu r e d e sc rib e s th e h a rd w a re P E s (w hich a re fu n c tio n a lly e q u iv a le n t to th e co d e s e g m e n ts ), in te rc o n n e c tio n s , a n d in p u t- o u tp u t p o r ts . T h e d riv e r defines d a t a s tre a m s in te rm s o f th e o rig in a l v a ria b le s in th e a lg o rith m .
In [16], ail a lg e b ra ic r e p re s e n ta tio n is d e riv e d from th e m a th e m a tic a l re p re s e n ta tio n of th e a lg o rith m . T h e ca n o n ic al a lg e b ra ic r e p re s e n ta tio n c o n sists of two ex p re ssio n s of th e ty p e s (a ) v = A v + b .r, a n d (b ) // = c 7'v , w h e re x re p re se n ts th e in p u t, y re p re s e n ts th e o u tp u t , an d v re p re s e n ts in te r m e d ia te v a ria b le s. T h e m a tr ix A an d th e c o lu m n v e c to rs b a n d c re p re se n t th e d e la y s b e tw e e n th e in te r m e d ia te v a ria b le s a n d each e n tr y is e ith e r 0 o r z ~k, w h e re ' is th e co m p lex v a ria b le in th e 0 tra n s f o rm d o m a in , w hich in th e tim e d o m a in re p re s e n ts a u n it d elay a n d k c o rre sp o n d s to th e n u m b e r of delay s. A lg eb raic tra n s f o rm a tio n s a re th e n a p p lie d to th is re p re s e n ta tio n . T h e re a re tw o m a jo r ty p e s o f tra n s fo rm a tio n s , n a m e ly , re tim in g a n d A’-sIow ing [16], t h a t d e te r m in e th e d is trib u t ion o f d e la y s a n d th e la te n c y p e rio d s of th e sy sto lic array . In th is m e th o d , v e c to r v is tra n s fo rm e d to a v e c to r u = D v , w h e re th e m a tr ix D is alw ay s a d iag o n al m a tr ix w hose d ia g o n al e le m e n ts a r e th e d elay s. B eca u se o f th is, th e n u m b e r of p o ssib le st ru c tu re s t h a t c a n b e o b ta in e d is lim ite d .
In t h e m e th o d p ro p o se d by M oldovan [6]-[7], an a lg e b ra ic m odel of i,iie alg o r ith m is d eriv e d fro m a se t o f re c u rre n c e re la tio n s , s im ila r to th o s e used in so ftw a re c o m p ile rs. T h is m o d e l c o n sists o f a s tr u c tu r e d s e t of in d e x e d c o m p u ta tio n a l sp ac e w h ere e a c h n o d e re p re s e n ts a se t of c o m p u ta tio n s . T h e a lg e b ra ic re p re s e n ta tio n of th e a lg o rith m is th e n tra n s f o rm e d b y local a n d g lo b al tra n s f o rm a tio n s , b o cal tra n s f o rm a tio n s are u se d to re w rite c o m p u ta tio n s t h a t a re m a p p e d in to th e f u n c tio n a l an d s tr u c t u r a l sp e c ific a tio n s of th e P E s o f th e sy sto lic a r c h ite c tu re . G lo b al tr a n s f o rm a tio n s , c o m p o se d o f tim e a n d sp ac e tra n s f o rm a tio n s , a r e used to r e s tr u c tu r e t h e a lg o rith m . T h e y a re chosen in su ch a way t h a t the* n ew alg o r ith m h a s a set o f d e p e n d e n c ie s t h a t a re a m e n a b le to V L S I im p le m e n ta tio n . T im e tra n s f o rm a tio n s d e te r m in e th e e x e c u tio n tim e o f th e a lg o rith m a n d th e tim in g for d a ta c o m m u n ic a tio n s . S p a c e tra n s f o rm a tio n s d e te r m in e th e in te rc o n n e c tio n s an d th e d ire c tio n s o f d a t a flow.
A n e x te n s io n t o th e w o rk by M oldovan was c a rrie d o u t by M ira n k e r a n d W in k ler [8]. In th is m e th o d a n a lg o rith m is re p re s e n te d as e ith e r a m a th e m a tic a l
ex p re ssio n or a cy c lic -lo o p p ro g ra m . T h e m a th e m a tic a l ex p re ssio n s a r e re w rit te n u s in g th e p ro p e rtie s o f th e o p e r a to rs in an ad h o c m a n n e r . T h e o r e tic a lly th is m e th o d c a n b e a p p lie d to a n y a lg o rith m a lth o u g h a s y s te m a tic d e sig n seem s p o ssib le o n ly for th o se a lg o rith m s d e sc rib e d b y p ro g ra m s w ith loops.
In th e m e th o d d e s c rib e d by C a p e llo a n d S te ig litz [9], s t a r ti n g fro m a set of re c u rre n c e e q u a tio n s d e sc rib in g t h e a lg o rith m , a c a n o n ic a l re p re s e n ta tio n is o b ta in e d by a d jo in in g an in d e x re p re s e n tin g tim e to th e d e fin itio n of re c u rre n c e . E ach in d e x is a s s o c ia te d w ith a d im e n sio n of a g e o m e tric sp a c e , w h e re e a c h p o in t c o rre sp o n d s to a tu p le of indices on w hich a se t re c u rre n c e s is defin ed . To e a c h such p o in t, a s e t o f c o m p u ta tio n s is a sso c ia te d , a n d its im p le m e n ta tio n is left u n sp ec ifie d . T h o se c o m p u ta tio n s a re m a p p e d d ire c tly in to fu n c tio n a l sp e c ific a tio n s o f th e P E s in t h e sy sto lic a r c h ite c tu re . F ro m th e g e o m e tric r e p r e s e n ta tio n in co n ju n c tio n w it! a n o rd e rin g ru le, th e topology, th e size of th e a r c h ite c tu r e , an d th e tim in g a r e d e riv e d s y s te m a tic a lly . B y s e le c tin g d iffe re n t g e o m e tric tra n s f o rm a tio n s, d is tin c t re p re s e n ta tio n s an d th e ir c o rre sp o n d in g a r c h ite c tu r e s a r e deriv ed .
In [4], a sig n a l flow g ra p h (S F G ) re p re s e n tin g an a lg o rith m is first d eriv e d . T h e n o d e s of th e SFG c o rre sp o n d to th e fu n c tio n a l d e s c rip tio n o f th e P E s of t h e a r c h ite c tu re . L o c a liz a tio n ru les a re th e n a p p lie d to d e riv e a re g u la r a n d te m p o ra lly lo calize d S F G . T h e lo c a liz a tio n p ro c e d u re c o n sists of se le c tin g c u t-s e ts of t h e S F G a n d re a llo c a tin g sc a le d delay s to edges le a v in g a n d e n te r in g each c u t-s e t in su ch a w ay th a t a t le a st o n e u n it o f tim e is allow ed fo r c o m m u n ic a tin g a sig n al b e tw e e n tw o n o d e s. D e la y s are c o m b in e d w ith o p e ra tio n a l m o d u le s t o o b ta in a full d e s c rip tio n o f th e o p e r a tio n o f a b asic sy sto lic m o d u le . T h e re s u ltin g S F G can b e m a p p e d d ire c tly o n to a sy sto lic a r ra y b y m a p p in g b a sic m o d u le s in to P E s a n d ed g e s in to in te rc o n n e c tio n s . T im in g a n d d a t a m o v e m e n ts a re d e riv e d fro m th e b a sic m o d u le s d u e to t h e lo calize d s p a tia l a n d te m p o ra l c h a r a c te ris tic s of t h e S F G .
Q u in to n [10] p ro p o se d a m e th o d b ase d on ex p re ssin g a p ro b le m a s a set o f u n ifo rm re c u rre n c e e q u a tio n s over a d o m a in c o n sistin g o f a s e t o f in d e x p o in ts.
In th is m e th o d , g iven a s y ste m o f n u n ifo rm re c u rre n c e e q u a tio n s defin ed o ver som e d o m a in D € Z M a n d w ith so m e c h a ra c te ris tic d e p e n d e n c y v ec to rs, a tim in g fu n c tio n t h a t m a p s p o in ts o f D o n to tim e is fo u n d . T h is re q u ire s th e id e n tific a tio n of a c o n v e x sp ac e o f fe asib le so lu tio n s fro m w h ic h o n e c a n b e ch osen h e u rist ically. S uch a sp a c e c a n b e fo u n d fro m th e k n o w led g e o f th e d e p e n d e n c y v ec to rs a n d I) ( D c a n b e th o u g h t of as th e in d ex s e t of th e re c u rre n c e s ). N e x t, an allo c a tio n fu n c tio n is ch o sen , w h ich p ro je c ts D alo n g so m e ch o se n d ire c tio n su ch that, tw o p o in ts in D w ith t h e sa m e im a g e u n d e r th e tim in g fu n c tio n d o n o t m a p o n to th e sam e p o in t in sp ac e. O n ce th e tim in g a n d a llo c a tio n fu n c tio n s a re k now n, th e sy sto lic a r ra y c a n b e s y s te m a tic a lly g e n e ra te d .
In t h e m e th o d a d v a n c e d by C o h en [17], s t a r ti n g fro m a m a th e m a tic a l e x p re s sion in v o lv in g s u b s c rip te d v aria b les, a new e x p re ssio n , w h e re a w ell-defined sh ift o p e r a to r is u se d to m o d el d is p la c e m e n ts in tim e o r sh ifts in sp ac e, is d eriv e d . S y m b o lic m a n ip u la tio n is u se d to tra n s f o rm th e d e riv e d m a th e m a tic a l ex p ressio n in to e q u iv a le n t on es by u sin g th e p ro p e rtie s of th e sh ift a n d fu n c tio n a l o p e ra to rs in th e ex p re ssio n .
In m a n y of th e a p p ro a c h e s u sin g m a tr ix tra n s f o r m a tio n s , th e n u m b e r of s tr u c tu re s p o ssib le is lim ite d b e c a u se of th e re s tr ic tio n s in th e n u m b e r o f fe asib le tr a n s fo rm a tio n s . M o reo v er, e sp e c ia lly in d ig ita l filte rin g , th e c o m p le x ity o f th e tr a n s fo r m a tio n a l a p p ro a c h in cre ases as th e d im e n sio n o f th e filte r in c re a se s. T h e in d ex sp ac e fo r a o n e -d im e n sio n a l (1-D ) filte r is tw o -d im e n sio n a l (2 -D ) a n d t h a t of a 2-D filte r is fo u r-d im e n sio n a l (4-D ) a n d so o n . A s th e d im e n sio n of th e filter in creases, firstly , it m a y be d ifficu lt to o b ta in a m a tr ix r e p re s e n ta tio n of th e p ro b le m a n d , sec o n d ly , th e sc h e d u lin g a n d p ro je c tio n v e c to rs u sin g th e S F G m e th o d , for in s ta n c e , b e c o m e c o m p lic a te d .
In t h i s th e sis, w e d e sc rib e a m e th o d for m a p p in g 1-D a n d m u ltid im e n sio n a l (M -D ) d ig ita l filte r a lg o rith m s o n to sy sto lic a r c h ite c tu r e s u sin g th e z -d o m a in a p p ro a ch . T h is m e th o d is m o re g e n e ra l th a n th e o n e m e n tio n e d in [17] a n d e a sie r th a n m a n y of th e m e th o d s h a t u se t h e tra n s f o rm a tio n a l a p p ro a c h m e n tio n e d
above. A ny filte r a lg o rith m is first tra n s fo rm e d in to its c o rre sp o n d in g 2-d o m a in e q u iv a le n t. D ifferen t s tr u c tu r e s a re o b ta in e d by re o rd e rin g th e s u m m a tio n s a n d delay s involved in th e filte i a lg o rith m th e re b y c irc u m v e n tin g th e u se of m a tr ix tra n s fo rm a tio n . T h e m e th o d m e n tio n e d in th is th e s is c a n b e a p p lie d to o b ta in m a n y a d v a n ta g e o u s s tr u c tu r e s t h a t can sa tisfy a se t of d e s ira b le o r p re s e t c r ite r ia such as laten c y , lo cality , a n d m o d u la rity .
1.3.2
M u ltip lie r d esig n
In c o n sid e rin g th e d esign of an y a rra y p ro c e sso r, it is i m p o r ta n t to co n sid e r th e design o f th e P E s invo lv ed . T h e m o st im p o r ta n t o p e r a tio n in a n y P E is m u ltip li c a tio n . C u rre n tly , th e m u ltip lie r a r e a a n d tim e a r e s till th e d o m in a n t fa c to rs in d e te rm in in g th e size an d sp e e d of o p e ra tio n of th e sy s te m . In th e d esig n of m u lti pliers d iscu ssed in th e l i t e r a t u r e [18]-[25], a lo t of effo rt h as b e e n d ire c te d to w a rd s in c re a sin g th e sp e e d of o p e r a tio n a n d d e c re a sin g th e a r e a b y u sin g th e a d v a n ta g e s of V L S I te c h n o lo g y in te r m s of in c re a se d d ev ice d e n s ity a n d f a s te r sw itch in g . H ow ever, ii m u ltip lic a tio n a lg o rith m s a re d esig n ed su ch t h a t th e n u m b e r of o p e r a tio n s re q u ire d t o p ro d u c e th e d esire d re s u lt is le d u c e d th e n , to g e th e r w ;th th e ad v an t ages of V L S I tech n o lo g y a g re a t re d u c tio n in a r e a a n d in c re a se in sp ee d o f o p e ra t ion can b e a c h ie v ed sim u lta n e o u sly . In th is th e s is w e d e sc rib e , in a d d itio n , m u ltip lie r sch e m e s th a t a r e s u ita b le n o t o n ly in t h e a r c h ite c tu r e s p ro p o se d h e re b u t a lso in m a n y o th e r a r c h ite c tu re s u se d in sig n a l p ro c e ssin g a p p lic a tio n s .
1.3.3
D ig ita l-filte r im p le m e n ta tio n
As w as m e n tio n e d e a rlie r, th e re s id f of a d v a n c e s in V L S I fa b ric a tio n te c h n o lo g y has b ro u g h t a b o u t a d r a m a tic re d u c tio n in th e c o st of in fo rm a tio n p ro c e ssin g . O ne a re a in w h ich th is effect is m o st p ro n o u n c e d is th e field o f re a l-tim e sig n al p ro cessin g . In th is a re a th e c o n tin u o u s flow o f d a t a in c o n ju n c tio n w ith th e c o m p le x ity of m a n y o f th e a lg o rith m s im p o ses sev e re c o m p u ta tio n a l d e m a n d s t h a t often c a n n o t b e satisfied by g e n e ra l-p u rp o se m a c h in e s o r c o m p o n e n ts . S a m p le
9
ra te s d e p e n d on th e a p p lic a tio n , ra n g in g fro m a r o u n d 8 kH z for speech sy ste m s to te n s a n d h u n d re d s of M H z for re a l-tim e ra d a r p ro cesso rs.
T h e s e d e m a n d s can b e m e t in p rin c ip le by new sy ste m a r c h ite c tu re s w hich ex p lo it so m e of th e p o te n tia l c o n c u rre n c y t h a t is in h e re n t in th e u n d e rly in g a l g o rith m s . V L S I te c h n o lo g y offers th e p o te n tia l to im p le m e n t su ch a rc h ite c tu re s . T h ro u g h th is te c h n o lo g y w e e x p e c t to see th e im p le m e n ta tio n of pow erful re a l-tim e signal p ro c e ssin g a lg o rith m s t h a t p re v io u sly h av e b e e n only of th e o re tic a l in te re st. H ow ever, th e v e ry a d v a n c e s in d e v ic e tech n o lo g y t h a t cau sed th is re v o lu tio n also b rin g a n ew c h a lle n g e to p ro d u c t d e v e lo p m e n t of V L S I sy s te m s . W ith o u t a d vances in design m e th o d o lo g y a n d to o ls, m a n u fa c tu rin g c a p a b ility an d a lg o rith m d e v e lo p m e n t w ill fa r e x c e e d o u r c a p a c ity fo r sy ste m design [26].
D e sig n of h ig h sa m p le r a te n o n re c u rsiv e filters h a s re ceiv ed c o n sid e ra b le in te r est in t h e la s t tw o d e c a d e s, b o th in th e c o n te x t of b it-p a ra lle l [27]-[28] as well as b it-s e ria l im p le m e n ta tio n s [29]-[31]. T h ro u g h b it-p a ra lle l d esig n s, im p le m e n ta tio n of h ig h sa n p le r a te n o n re c u rsiv e filte r ch ip s ru n n in g u p to 300 M H z has b ec o m e po ssib le. O n t h e o th e r h a n d , h ig h s a m p le r a te re c u rsiv e filters h av e n o t received m u ch a tt e n ti o n d u e to in te r n a l re c u rsio n o r lo o p in g t h a t n e g a te s th e p o ssib il ity of p ip e lin in g . P a s t effo rts on h ig h -sp e e d re c u rsiv e filte r s tr u c tu r e s h av e been b a se d o n blo ck filte r s tr u c tu r e s , w h e re a b lo ck of in p u ts is p ro c e sse d to g e n e ra te a b lo ck o f o u tp u ts , a n d th e signals a re p ro c e sse d in n o n -o v e rla p p in g b locks [32]. A lth o u g h m a n y b lo c k re c u rsiv e filte r s tr u c tu r e s e x is te d for a lo n g tim e , th e y w ere q u ite c o m p le x to im p le m e n t. W ave d ig ita l filters (W D F s ) , a class of re cu rsiv e d ig ita l filte rs t h a t a re closely re la te d to classica l filte r n e tw o rk s, have received c o n sid e ra b le in te r e s t sin ce th e s e s tr u c tu r e s e x h ib it a d e s ira b le p ro p e rty t h a t th e fre q u e n c y re sp o n se of th e s e filters is less se n sitiv e to coefficient v a ria tio n [33] an d c o n se q u e n tly v a rio u s a p p ro a c h e s to d ire c t V L S I im p le m e n ta tio n of W D F s have b ee n le p o r te d [34]-[36]. T h o u g h th e W D F s a re v e ry ea sily a m e n a b le to V LSI im p le m e n ta tio n , t h e y c a n n o t b e d ec o m p o se d in to m o d u la r P F s t h a t ca n b e used for a n y ty p e o f filte rin g , v iz ., low p ass, h ig h p a ss, e tc . In o th e r w ords a iow pass
W I)F s tr u c tu r e c a n n o t b e used fo r o th e r ty p e s of filte rin g u n le ss sp e c ia l k in d s of a d a p to r s are u sed .
In th e p a s t, th e effect of lim it-c y c le o sc illa tio n s d u e to q u a n tiz a tio n h a s n o t been ta k e n in to c o n sid e ra tio n in th e im p le m e n ta tio n of re c u rsiv e , d ire c t-fo rm , d ig ita l-filte r s tr u c tu r e s . A s a co n se q u en ce , t h e a p p lic a tio n o f su c h re c u rs iv e filte rs has b e e n lim ite d . In th is th e sis w e also d iscu ss th e im p le m e n ta tio n o f sy sto lic d ig ita l-filte r s tr u c tu r e s o b ta in e d in d ire c t-fo rm t h a t is efficient in a r e a a n d in w h ic h c irc u its to c irc u m v e n t q u a n tiz a tio n a n d overflow lim it cycles a r e in c o rp o ra te d .
1.4
O u tlin e o f t h e s is
T h is th e sis is o rg a n iz e d in th re e p a r ts . T h e first p a r t, c o m p risin g C h a p te r s 2 and 3, d eals w ith th e m a p p in g o f 1-D a n d M -D d ig ita l-filte r a lg o rith m s o n to sy sto lic a rra y s. T h e seco n d p a r t, C h a p te r 4, d eals w ith fo u r d iffe re n t m u ltip lie r s tr u c tu r e s t h a t c a n b e u s e d in t h e sy sto lic a rra y s p ro p o s e d h e re , in p a r tic u la r , an d in a r c h ite c tu re s u se d in d ig ita l sig n al p ro c e ssin g , in g e n e ra l. T h e la s t p a r t , c o m p risin g C h a p te r s 5 a n d 6, d e a ls w ith t h e V L S I im p le m e n ta tio n o f so m e o f th e m u ltip lie rs d iscu ssed in C h a p te r 4 a n d so m e o f th e s e c o n d -o rd e r d ig ita l filte r s tr u c tu r e s d isc u sse d in C h a p te r 2.
In C h a p te r 2, tw o a p p ro a c h e s fo r th e m a p p in g o f d ig ita l filte r a lg o rith m s o n to h a rd w a re are d iscu ssed . O n e is b a s e d on th e S F G a n d th e o th e r on th e 2-d o m a in c h a ra c te riz a tio n o f th e filte r a lg o rith m . In th e 2-d o m a in a p p ro a c h , a n y filte r a lg o rith m is first tra n s fo rm e d in to its c o rre s p o n d in g 2-d o m a in e q u iv a le n t a n d by re o rd e rin g th e s u m m a tio n a n d d elay s in th e tra n s f o rm e d e q u a tio n , se v e ra l s tr u c tu r e s a re o b ta in e d t h a t s a tisfy th e d e sig n c r ite r ia , su c h as la te n c y , lo c a lity , an d m o d u la rity . T h e in c o n v e n ie n c e o f usin g th e S F G to o b ta in 1-D re c u rs iv e filte r s tr u c tu r e s a n d th e efficacy of u sin g th e 2-d o m a in m e th o d a r e also d iscu ssed . W e la te r u se only th e 2-d o m a in a p p ro a c h to d e riv e s y sto lic s tr u c tu r e s fo r d ig ita l filte rs th a t a r e m o d u la r w ith lo cal d a t a c o m m u n ic a tio n s .
11
s tr u c tu r e s for 2-D , 3-D , a n d M -D d ig ita l filters. All th e filter s tr u c tu r e s o b ta in e d a re m o d u la r a n d h ie ra rc h ic a l. T e c h n iq u e s to c irc u m v e n t so m e in h e re n t p ro b lem s in ra s te r-s c a n n e d im ag es, like line a n d fra m e w ra p -a ro u n d p ro b le m s, are also co n sid e re d . T h e c h a p te r co n c lu d e s w ith a c o m p a riso n of th e v ario u s s tr u c tu r e s .
In C h a p te r 4, fo u r d iffe re n t m u ltip lie rs a re d e sc rib e d . T h e first m u ltip lie r is an area-effic ie n t m u ltip lie r t h a t uses o n ly a b o u t 50% o f th e a r e a of a c o n v e n tio n a l full p a r a lle l m u ltip lie r. In m o s t sig n al p ro c e ssin g a p p lic a tio n s , o o tli th e in p u t a n d th e o u t p u t w ord le n g th s of a s y s te m a r e th e sam e. A n N x N m u ltip lie r p ro d u c es a p r o d u c t of 2 N b its , o f w h ich o n ly N b its a re used . T h e m u ltip lie r d esig n ed h ere avoids t h e use o f all th e r e d u n d a n t cells w h ich y ield th e N b its t h a t a re tr u n c a te d . A c o rre c tio n u n it is in c o r p o r a te d t h a t re d u c e s th e c o n c o m ita n t erro r.
T h e seco n d m u ltip lie r is b a s e d on t h e m o d ified o c ta l B o o th a lg o rith m . hi th is a lg o rith m , four b it-s e g m e n ts o f th e m u ltip lie r a re sc a n n e d a n d th e c o rre sp o n d in g o p e r a tio n s effected o n th e m u ltip lic a n d . In th is m e th o d , h o w ever, a n o il-triv ia l m u ltip lic a tio n o f a n u m b e r b y th r e e is p re s e n t w hich is effected as an a d d itio n of th e n u m b e r in q u e s tio n w ith a le ft-s h ifte d v ersio n of th e n u m b e r. T h is involves an e x tr a d e la y as a r e s u lt o f th is a d d itio n . In o rd e r to im p ro v e th e speed o f o p e ra tio n we a d v a n c e a m u ltip lie r b a se d o n th e o c ta l m o d ifie d B o o th a lg o rith m in w hich th e r e s u lts of t h e n o n -triv ia l o p e r a tio n a r e p r e c o m p u te d u sin g an e x te r n a l c a rry lo o k -a h e a d a d d e r th u s av o id in g th e e x t r a delay.
T h e t h ir d m u ltip lie r finds a p p lic a tio n in th e F e rm a t n u m b e r- th e o re tic tr a n s form . H e re , th e n u m b e rs , w h ic h a re in te g e rs, a re re p re s e n te d in d im in is h e d -1 re p re s e n ta tio n . H ith e r to d im in ish e d -1 m u ltip lie rs h a v e used tr a n s la to r s to co n v e rt n u m b e rs fro m th e ir d im in is h e d- 1 re p re s e n ta tio n to th e c o rre s p o n d in g b in a ry value. In th e m u ltip lie r p ro p o s e d , th e u se of a tr a n s la to r is c irc u m v e n te d a n d a novel te c h n iq u e to in c o r p o r a te th is o p e r a tio n o f tr a n s la tio n in th e m u ltip lie r s t r u c t u r e is d e s c rib e d . A s a c o n se q u e n c e , th e a r e a is re d u c e d a n d t h e sp eed o f o p e r a tio n of th e m u ltip lie r is in c re a se d .
coil-ju n c tio n w ith an a c c u m u la to r w h e re ea ch m u ltip lic a tio n is follow ed b y an a d d itio n of th e n u m b e r s to re d in th e a c c u m u la to r. T h e o u t p u t is o b ta in e d w h en t h e final p a ir o f n u m b e rs a re m u ltip lie d a n d a d d e d to th e re s u lt s to re d in th e a c c u m u la to r. In C h a p te r 4, w e p ro p o se a s t r u c tu r e t h a t p e rfo rm s a n in n e r p ro d u c t t h a t c irc u m v e n ts th e use o f a n a c c u m u la to r th e re b y re s u ltin g in in c re a s e d s p e e d a n d re d u c e d area.
In C h a p te r 5, we d e sc rib e th e VLSI im p le m e n ta tio n o f th re e o f th e m u ltip lie r s tr u c tu r e s d isc u sse d in C h a p te r 4. T h e se ch ip s h a v e b e e n s im u la te d u sin g S IL O S a n d im p le m e n te d in 1.2/i C M 0 S 4 S technology.
In C h a p te r 6, we d e a l w ith th e V L S I im p le m e n ta tio n of a s e c o n d -o rd e r re cu rsiv e filte r a n d a sin g le P E p ro p o se d in C h a p te r 2. In th e im p le m e n ta tio n of th e s e c o n d -o rd e r d ig ita l filte r, an ite r a tiv e m u ltip lie r s tr u c tu r e h a s b e e n in c o rp o ra te d w hich sig n ific a n tly re d u ces th e silicon a r e a of t h e ch ip . In b o th d esig n s, viz., th e se c o n d -o rd e r d ig ita l filte r a n d th e sy sto lic P E , u n its to e lim in a te b o th q u a n tiz a tio n a n d overflow lim it cycles h av e b e e n in c o rp o r a te d . A c o m p a riso n of th e se c o n d -o rd e r filte r b u ilt as a single u n it a n d t h e s e c o n d -o rd e r filte r b u ilt u sin g a ca sc a d e o f th e P E s h a s also b ee n c a rrie d o u t in te rm s o f ro u n d o ff noise a n d a r e a
x tim e co m p lex ity .
C h a p te r 2
M a p p in g m e th o d o lo g y for
o n e -d im e n sio n a l d ig ita l filte r s
2 .1
I n tr o d u c t io n
A s h a s b e e n m e n tio n e d e a rlie r, th e re a r e sev eral m a p p in g m e th o d o lo g ie s for th e m a p p in g of d ig ita l- filte r a lg o rith m s o n to h a rd w a re . H ow ever, we h av e chosen th e sig n a l flow g ra p h (S F G ) a p p ro a c h to c o m p a re o u r a p p ro a c h w ith sin ce it in v o lv es m a tr ix tra n s f o rm a tio n s t h a t a r e ty p ic a l of m o st o f th e o th e r a p p ro a c h e s. In a d d itio n , t h e S F G serv es as a to o l for d a t a flow a n a ly sis of th e u n d e rly in g a lg o rith m .
In th is c h a p te r , se v e ra l sy sto lic a rc h ite c tu re s for 1-D n o n re c u rsiv e an d recursive* d ig ita l filte rs u s in g th e S F G a n d 2-d o m a in a p p ro a c h e s a re d e riv e d . It is show n t h a t th e S F G a p p ro a c h is n o t effectiv e in th e sense t h a t it lack s th e v e r s a tility o f th e 2- d o m a in a p p ro a c h . As a p a r tic u la r a p p lic a tio n o f th e 2-d o m a in a p p ro a c h , sy sto lic s tr u c tu r e s s u ita b le for s e c o n d -o rd e r d ig ita l filte rs, d e c im a to rs , a n d in te rp o la to rs a r e d e riv e d .
2 .2
S ig n a l flo w g r a p h a p p r o a c h
P a ra lle l im p le m e n ta tio n s o f a n a lg o rith m ca n b e o b ta in e d u sin g tw o a p p ro a c h e s v iz ., v e c to riz in g a se q u e n tia l a lg o rith m an d u sin g re c u rsiv e e q u a tio n s an d s in
gle a s sig n m e n t co d es. V e c to riz in g c o m p ilers p ro c ess a so u rc e c o d e w r itte n as a s e q u e n tia l code to g e n e ra te m a c h in e in s tru c tio n s t h a t c a n b e e x e c u te d in p a r a l lel. H ow ever, a v e c to riz in g c o m p ile r do es n o t re w rite t h e so u rc e co d e to u tiliz e th e in h e re n t c o n c u rre n t p a ra lle lism . T h e r e a re la n g u a g e s like O C C A M [4] d ev e l o p ed fo r p a ra lle l m a c h in e s t h a t d e sc rib e a c o n c u rre n t c o m p u tin g s y s te m as a se t o f in d e p e n d e n t pro cesses t h a t u se lo cally d efin e d v a ria b le s a n d c o m m u n ic a te v ia p re d efin ed c h a n n e ls. H ow ever, to o ls a re n e e d e d t o d efine th e c o n c u rre n c y a n d p a r allelism w ith in a n a lg o rith m b efo re co d in g it u sin g la n g u a g e s d e sig n e d fo r p a ra lle l m ach in e s.
D e p e n d e n c e g ra p h s (D C s ) a n d S F G s a r e to o ls t h a t d e s c rib e t h e d a t a flow in a n a lg o rith m w hich allow th e h a rd w a re d e sig n e r to s tu d y a n y u n d e rly in g p a ra lle lis m . A D C e x h ib its th e p a ra lle lism in a n a lg o rith m in th e fo rm of a re g u la rly r e p e a tin g p a tte r n of d a t a flow. T h ro u g h clev e r m a n ip u la tio n o f th e d a ta -flo w d ire c tio n s , an S F G is d e riv e d th a t c a n b e u sed to o b ta in a p a r a lle l h a r d w a re s t r u c t u r e to im p le m e n t th e a lg o rith m .
B efo re we d iscu ss D G s a n d S F G s, le t us e x a m in e t h e c o n c e p t o f sin g le a ssig n m e n t co d e [4]. C o n sid e r th e F O R T R A N co d e fo r a m a tr ix - v e c to r m u ltip lic a tio n
c = A b g iven by D O 10 N = 1,4 C (N ) = 0.0 D O 10 K = 1,4 C (N ) = C (N ) + A (.N ,K ) * B(I<) 10 C O N T IN U E
I t c a n b e seen t h a t C (N ) is o v e rw ritte n m a n y tim e s to sa v e s to ra g e sp ac e. M oreover, C (N ) is e v a lu a te d a f te r C ( N - l) is e v a lu a te d . If su ch a co d e w e re to b e im p le m e n te d in h a rd w a re , it w ould re s u lt in a d esig n t h a t is in effic ien t in sp ee d . A lg o rith m s c a n b e d e sc rib e d in su ch a w ay t h a t e a c h v a ria b le is a ssig n e d o n ly o n e v alu e d u rin g th e e x e c u tio n . T h is d e s c rip tio n is said to b e in sin g le a s s ig n m e n t code fo rm . T h e ab o v e F O R T R A N co d e ca n b e w r itte n in th e sin g le a s s ig n m e n t code fo rm as
