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V L S I S Y S T M S

The Center or Advanced Computer

Studies Unive rsity of Louisiana

at Lafayette L afayette Louisiana

USA

overs the b road spect rum of VLSI ar i thmet ic

us t om m em ory o rgan i za t i on and da t a t r ans fe r t he ro le o f

are descr ip t ion languages c lock schedul ing low-pow er

esign micro e lect ro mechan ical sys tems and n oise analys is

ectr ical engineers in indus t ry gove rnm ent and academ ia.

Over the years the fund am enta l s of the f ie ld have evolved to

f topics and a broad range o f pract ice . To

pass such a wide range of knowledge the sect ion focuses

concep ts m ode l s and equa t i ons t ha t enab le the

des ign engineer to analyze des ign and predic t the behav ior of

large-scale systems. While design formulas and tables are

l i sted emph as i s is p laced on the key concepts and the theor ies

unde r ly ing the processes . In ord er to do so the ma ter ia l is

re inforced wi th f requent examples and i l lus t ra t ions .

The com pi l a t i on o f t h i s s ec ti on wou l d no t have been pos -

s ib le wi th out the dedica t ion and effor t s of the sect ion edi tor

and con t r i bu t i ng au t ho r s . I w i sh t o t ha nk t he m a ll .

W ai -Ka i C hen

Ed i t o r

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Logari thmic and Res idue Number

System s for V L SI ri thm etic

T h a n o s S t o u r ai t is

Department of Electrical and

Computer Engineering

University of P atras

Greece

1 .1 I n t r o d u c t io n . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . 179

1 .2 LNS Bas ics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

1 2 1 L N S a n d L i n e a r R e p r e s e n t a t i o n s • 1 2 2 L N S O p e r a t i o n s 1 2 3 L N S a n d P o w e r

D i s s i p a t i o n

1 .3 T h e R e s id u e N u m b e r S y s t e m . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. 185

1 3 1 R N S B a s ic s • 1 3 2 R N S A r c h i t e c t u r e s • 1 3 3 E r r o r T o l e r a n c e i n R N S S y s t e m s •

1 3 4 R N S a n d P o w e r D i s s i p a t i o n

R e f e re n c e s . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . 190

1 1 I n t r o d u c t i o n

e ry la rge -sca le in tegra te d c i rcu i t (VLSI) a r i thm et ic un i t s a re

e s sen t ia l f o r th e o p e r a t i o n s o f t h e d a t a p a th s a n d /o r t h e a d -

dress ing uni t s o f mic ro proces sors , d ig i ta l s igna l processors

(DSPs) , a s we l l a s da ta -process ing appl ica t ion-spec i f ic in te -

gra ted c i rcu i t s (ASICs) and programmable in tegra ted c i rcu i t s .

h e i r o p t im iz e d r e a l i z at i o n , in t e r ms o f p o w e r o r e n e r g y c o n -

su mp t io n , a r e a , a n d /o r sp e e d , i s imp o r t a n t f o r me e t in g

d e m a n d in g o p e r a t i o n a l sp e c if i c at i o n s o f su c h d e v i ce s .

I n m o d e r n V L S I d e s ig n fl o w s , t h e d e s ig n o f s t a n d a r d a r i t h -

e t ic un i t s i s ava i lab le f rom des ign l ib ra r ie s . These uni t s

mp lo y b in a r y e n c o d in g o f n u mb e r s , su c h a s o n e s o r tw o s

o m p l e m e n t , o r s i g n m a g n i t u d e e n c o d i n g t o p e r f o r m a d -

u l t i p l ic a t i o n s . I f n o n s t a n d a r d o p e r a t i o n s a r e

r e d , o r i f h i g h p e r f o r m a n c e c o m p o n e n t s a re n e e d e d , t h e n

des ign o f specia l a r i thme t ic un i t s i s necessa ry . In th is case,

a r i t h me t i c sy s t e m i s o f u tmo s t im p o r t a n c e .

T h e im p a c t o f a r i t h me t i c i n a d ig i ta l sy s te m i s n o t o n ly

d e f in i t i o n o f t h e a r c h i t e c tu r e o f a r i t h me t i c

Ar i th m et ic a f fec ts seve ral levels o f the des ign abs t rac -

t ma y r e d u c e t h e n u m b e r o f o p e r a t i o n s , t h e

ign a l a ct iv i ty , a n d t h e s t r e n g th o f t h e o p e r a to r s . T h e c h o i c e

a y l e a d t o su b s t a n t i a l p o w e r s av in g s, r e d u c e d

T h i s c h a p t e r d e sc r ib e s tw o a r i t h me t i c sy s t e ms th a t e mp lo y

n o n s t a n d a r d e n c o d i n g o f n u m b e r s . T h e l o g a r it h m i c n u m b e r

sy s t e m ( L N S ) a n d t h e r e s id u e n u mb e r sy s t e m ( R N S ) a r e

s in g l e d o u t b e c a u se t h e y h a v e b e e n sh o w n to o f f e r imp o r t a n t

a d v a n tag e s i n t h e e f f i c ie n c y o f t h e i r o p e r a t i o n a n d ma y b e a t

t h e s a m e t im e m o r e p o w e r - o r e n e r g y - e f fi c i e n t, f as t er , a n d /o r

sma l l e r t h a n o th e r sy s te ms .

A l t h o u g h a d e t a il e d c o m p a r i s o n o f p e r f o r m a n c e o f th e s e

sy s te ms to t h e i r c o u n te r p a r t s i s n o t o f f e r e d h e r e , o n e m u s t

k e e p in m in d t h a t su c h c o m p a r i so n s a r e o n ly me a n in g f u l

w h e n th e sy s te ms u n d e r q u e s t i o n c o v e r t h e s a me d y n a m ic

r a n g e a n d p r e se n t t h e s a me p r e c i s io n o f o p e r a t i o n s . T h i s

necess i ty usua l ly t r ans la te s in ce r ta in da ta word lengths ,

which , in the i r tu rn , a f fec t the o pe ra t in g cha rac te r i s t ic s of the

sys tems.

1 2 L N S B a s i c s

Tradi t iona l ly , LNS has been cons ide red a s an a l te rna t ive

to f l o a t i n g - p o in t r e p r e se n t a t i o n ( K o r e n , 1 9 9 3 ; S to u r -

a it is , 1986) . The orga niza t io n of an LNS wo rd i s show n in

Figure 1.1.

T h e L N S ma p s a l i n e a r r e a l n u mb e r X to a t r i p l e t a s

follows:

© 2005 by AcademicPress

79

reserved

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180 T h a n o s S t o u r a i t i s

I G U R E

1 1

n n 1 . . . 0

Sx .. .

x = lOg b [X l *

The Organization o f an (n + 1)-bit LNS Digital Wo rd

X LN_~S Zx , Sx,

x = log b [X[),

(1.1)

Sx i s the s ign of X , b i s th e b a se o f t h e l o g a r i t h m ic

e p r e se n t a t io n , a n d Z x is a single-bit f lag, which, when asser ted,

eno tes tha t X is ze ro . A ze ro f lag i s r equi r ed b ecause log b X is

o t a f in i te nu m be r for X = O. S imi la rly , s ince the loga r i thm of

a n e g a t i v e n u mb e r i s n o t a r e a l n u mb e r , t h e s i g n i n f o r ma t io n

of X is s tored in f lag

Sx.

L o g a r i t h m x = l o g b x l i s enco ded as a

a n d i t m a y c o m p r i s e a n u m b e r o f k i n t eg e r a n d

I f ractio nal bits .

T h e i n v e r se ma p p in g o f a lo g a r i t h mic t r i p l e

Z x , S x , x )

to a

b e r X is d e f in e d b y:

Z x , 5 x , X ) LN~S- ' X : X = ( 1 - Z x ) - 1 ) S x b x . ( 1 . 2 )

1 2 1 LNS and Linear Rep resentat ions

w o imp o r t a n t i s su e s i n a f i n i t e w o r d l e n g th n u mb e r sy s t e m

are the

r n g e

o f n u m b e r s t h a t c a n b e r e p r e s en t e d a n d t h e

r e c i s i o n of the rep resen ta t ion (Koren , 1993).

Let (k,

1,

b ) - L N S d e n o te a n L N S o f i n t eg e r a n d f r a c t i o n a l

l, re spec t ive ly , and of base b . These three

e t e r m in e t h e p r o p e r t i e s o f t h e L N S a n d c a n b e

o mp u te d so t h a t t h e L N S me e t s c e r t a in sp e c i f i c a t i o n s . F o r

x a mp le , f o r a k , l , b ) - L N S to b e c o n s id e r e d a s e q u iv a l e n t to

n n-b i t l inea r f ixed-poin t sys tem, the fo l lowing two res t r ic -

i o n s m a y b e p o se d :

1 . T h e tw o r e p r e se n t a t i o n s sh o u ld e x h ib i t e q u a l a v e r a g e

r e p r e se n t a t i o n a l e r r o r .

2 . T h e tw o r e p r e se n t a t i o n s sh o u ld c o v e r e q u iv a l e n t d a t a

ranges.

Th e ave rage repres enta t io na l e r ror , ~ave , i s de f ined as:

Xmax

E g r e l ( X )

X Xmin

8 a ve = ( 1 . 3 )

X m a x - X m i n q - 1 '

here min a n d X m~ d e f in e t h e r a n g e o f re p r e se n t ab l e

numbers in each sys tem and where gre l (X) i s the re la t ive

r e p r es e n t a ti o n a l e r r o r o f a n u m b e r X e n c o d e d i n a n u m b e r

sy s t em. T h i s e r r o r i s , i n g e n e ra l , a f u n c t i o n o f t h e v a lu e o f X

and i t i s de f ined as :

I x - 2 1

gre1(X) -- - - , (1 .4 )

X

in w h ic h X i s t h e a c tu a l v a lu e a n d X i s t h e c o r r e sp o n d in g v a lu e

r e p r e sen t a b l e i n t h e sy s t em. N o t i c e t h a t X ¢ X d u e t o t h e

f in it e le n g th o f t h e w o r d s . A ssu min g th a t t h e l o g a r i t h m o f X

i s r e p r e se n t e d a s a tw o ' s c o mp le me n t n u mb e r , t h e r e l a t i v e

rep rese ntat ion al err or ~rel, LNS for a (k, l , b)-L N S is ind ep en d-

e n t o f X a n d , t h e r e f o r e , is e q u a l t o t h e a v e r a g e r e p r e se n t a t io n a l

e r ror . I t i s g iven by [ re fe r to K oren (1993) for the case b = 2] .

g a v e , L N S = g r e l , L N S = b 2 - I -

1.

1 . 5 )

D u e to f o r mu la 1 . 3 , t h e a v e r a g e r e p r e se n t a t i o n a l e r r o r f o r t h e

n-b i t l inea r f ixed-p oin t case is g iven by:

g a v e , F X P - - 2 n - 1 i = 1 ~

(1.6)

w h ic h , b y c o m p u t in g t h e su m o n t h e r i g h t - h a n d s id e , c a n b e

wr i t ten a s :

t~(2 n) + ~/

gave, FXP

- -

2 n - 1 ' (1.7)

w h e r e ~ is th e E u le r g a m ma c o n s t a n t a n d f u n c t i o n ~ i s d e f in e d

t h r o u g h :

t~(x) = d in F( x) , (1.8)

w h e r e F ( x ) i s t h e E u le r g a mma f u n c t i o n .

I n t h e f o l l o w in g , t h e ma x imu m n u mb e r r e p r e se n t a b l e i n

e a c h n u m b e r s y s t e m i s c o m p u t e d a n d u s e d t o c o m p a r e

th e r a n g e s o f th e r e p r e se n t a t i o n s . N o t i c e t h a t d i f f e re n t

f i g u r e s c o u ld a l so h a v e b e e n u se d f o r r a n g e c o mp a r i so n ,

such as the ra t io X m a x X m i n (S toura i t i s , 1986) . The maxi-

mu m n u mb e r r e p r e se n t a b l e b y a n n - b i t l i n e a r i n t e g e r i s

2 - 1 ; t h e r e f o r e t h e u p p e r b o u n d o f t h e f i x e d - p o in t r a n g e i s

g iven by:

F X P 2 n

X m a~ = - - 1 . 1 . 9)

T h e m a x im u m n u m b e r r e p r e se n t a b l e b y a (k , l, b ) -L N S e n c o d -

ing 1.1 is as follows:

L N S b 2 k + l - 2 - I

Xmax = (1.10)

T h e r e f o r e , a c c o r d in g t o t h e e q u iv a l e n c e r e s t r i c t i o n s p o se d

above , to make an LNS equiva len t to an n-b i t l inea r f ixed-

p o in t r e p r e se n t a t io n , t h e f o l l o w in g i n e q u a li t ie s sh o u ld b e s i -

m ul tan eou s ly sat is fied :

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Logar ithmic an d Residue Num ber Sys tems for VL SI Ar i thmet i c

181

LNS FXP

Xma _> Xm~~. (1.11)

E . . . L N S ~ g a v e , F X P (1.12)

, f rom equa t ions 1 .5 and 1 .7 thro ugh 1 .10 the fo l lowing

I * 2~)-~-~7 (1.13)

l = - l o g 2 lo g b ( l + 2 ~ - 1 [

k = [log2 log b (2 ~ - 1) + 2 -1 - 1)7. (1.14)

es of k and l that corresp ond to var ious values of n for

seen in Table 1 .1 , where for each

M t hou gh t he wo rd l eng th s k and I com pu t ed v i a equa t ions

1 .13 and 1 .14 me et the po sed equivalence specif icat ions of

equ ation s 1.11 an d 1.12, LNS is capable of cove ring a signifi-

cant ly larger range than the eq uivalent f ixed-poin t representa-

t ion . Let neq denote the word length of a f ixed-poin t sys tem

t ha t can cover t he r ange o f fe red by an LNS de f i ned t h rough

equ ation s 1.13 an d 1.14. Equivalent ly, let neq be th e sm allest

integer, which satisfies:

2 r~ q - - 1 > b 2 k + 1 - 2 . ( 1 . 1 5 )

From equa t ion 1 .15 , i t fo l lows that :

n e q = [ ( 2 k + 1 - - 2 - t ) l o g 2 b 1 .

(1.16)

I t s hou l d be s t r e s s ed t ha t when r t eq ~ n, the preci s ion of the

par t icu lar f ixed-poin t sys tem i s bet ter than that of the LNS

der ived by equat ions 1 .13 and 1 .14 . Equ at ion 1 .16 reveals that

the par t icu lar LNS, whi le meet ing the preci s ion o f an n-b i t

l inear representat ion , in fact covers the range provided by an

neq-bit l inear system.

O f course, the average (relat ive) erro r is no t the on ly wa y to

compare the accuracy of comput ing sys tems. Especia l ly t rue

for s ignal process ing sys tems, one m ay use the s ignal - to-noise

rat io (SNR), assuming that quant izat ion errors represent

noise , to com pare the preci s ion of two sys tems. In that case ,

by equa t i ng t he S NR s o f t he LNS a nd t he f i xed -po in t s y s tem

that covers the requi red dynamic range, the in teger and f rac-

t i ona l word l eng th s o f t he LNS m ay be com pu t ed .

1 2 2 L N S O p e r a t i o n s

M appin g o f equat ion 1 .1 i s of pract ical in teres t because i t can

s impl i fy cer ta in ar i thm et ic ope rat ions ( i. e. , i t can reduc e the

implementat ion complexi ty , a l so cal led s t rength , of several

operators ) . Fo r example , due to the proper t ies of the logar i thm

func t i on , the m u l t i p li ca t i on o f t wo l i nea r num bers , X : bx

and Y = by , i s reduced t o t he add i t i on o f t he i r l oga r i thm i c

images , x and y .

The bas i c a r i thm et i c ope ra t i ons and t he i r LNS coun t e rpa r t s

are sum ma rized in Table 1 .2, where , for s impl ic i ty and wi tho ut

loss o f general i ty, the zero flag zx is om i t t ed an d i t is a s s um ed

that X > Y. Table 2 reveals that , whi le the co mp lexi ty of mos t

opera t i ons i s r educed , the com pl ex i t y o f LNS add i t i on and

LNS subtract ion is s ignificant . In part icular, for d = Ix - Y l ,

LNS add i t i on r equ ires t he com pu t a t i on o f the non l i nea r func -

t ion:

s~ d) = logb (1 + b - d ,

(1.17)

and s ub t rac t ion r equ i res t he com pu t a t i on o f t he non l i nea r

funct ion:

s, d)

= logb (1 -

b-d).

1.18)

Equat ions 1 .7 and 1 .8 subs tant ia l ly l imi t the d ata wo rd lengths

for which LNS can o ffer ef f ic ien t VLSI implem entat ions . The

T A B L E 1 . 1 C o r r e s p o n d e n c e o f n, k , l , a n d n eq f o r V a r i o u s B a s e s b

n b = 1 .5 b = 2 b = 2 . 5

k 1 rteq k 1 rteq k 1 tleq

5 3 2 6 3 3 9 2 3 7

6 4 3 10 3 4 9 2 4 7

7 4 4 10 3 5 9 3 5 12

8 4 5 10 3 5 9 3 6 12

9 4 5 10 4 6 17 3 7 12

10 5 6 20 4 7 17 3 7 12

11 5 7 20 4 8 17 3 8 12

12 5 8 20 4 9 17 4 9 23

13 5 9 20 4 10 17 4 10 23

14 5 10 20 4 11 17 4 I I 23

15 5 11 20 4 12 17 4 12 23

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182 T h a n o s S t o u r a i t i s

TABLE 1.2 Bas ic Linear Arithmetic Operations and T heir LNS Counterparts

Linear operation Logarithmic operation

Multiply

Divide

Root

Power

Add

Subtract

W = X Y = b ~ b = b ~+r

w = x m = b~)m

W = X + Y = b ~ + b y = b X l + b y ~ )

w = x _ y = ~ _ b Y = b ~ l _ b y x )

w = x + y , S w = S x X O R s y

w = x - y, sw = sxXOR sy

w = ~, rn, integer, sw = sx

w = r ex , m , inte ger , Sw = sx

w = x+ log b 1 + by-x), sw = Sx

w = x + log~ 1 - bY -X) , sw = sx

M u l t i p l y /

d i v i d e / /

A d d /

s u b t r a c t

IOgb(1

+ _ b I x - y l

, S u m / d i f f e r e n c e

P r o d u c t / q u o t i e n t

IG UR E 1.2 The O rganiza t ion o f a Bas ic LNS Processor: the processor comp rises an adder , two mult ip lexers, a sign-invers ion uni t , a look -up

able , and a f ina l adder . I t m ay perfo rm the four op era t ions o f addi t ion, subtrac t ion, mult ip l ica tion, or d ivision.

r g a n i z a t io n o f a n L N S p r o c e s s o r t h a t c a n p e r f o r m t h e f o u r

c o p e r a t i o n s o f a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c at i o n , o r

i v i s i o n i s s h o w n i n F i g u r e 1 . 2 . N o t e t h a t t o i m p l e m e n t L N S

u b t r a c t i o n i.e ., t h e a d d i t i o n o f t w o q u a n t i t ie s o f o p p o s i t e

i gn ) a d i f f e r en t m e m o r y l o o k - u p t a b le L U T ) i s r e q u i re d .

T h e m a i n c o m p l e x i t y o f an L N S p r o c e s s o r is th e i m p l e m e n -

a t i o n o f t h e L U T s f o r s t o r i n g t h e v a l u e s o f th e f u n c t i o n s s a d )

n d s s d ) . A s t r a i g h t f o r w a r d i m p l e m e n t a t i o n i s o n l y f ea s ib l e

o r s m a l l w o r d l e n g t h s . A d i f f e r e n t t e c h n i q u e c a n b e u s e d f o r

r w o r d l e n g t h s b a s e d o n t h e p a r t i t i o n i n g o f a n L U T i n t o

n a s s o r t m e n t o f sm a l l e r L U T s. T h e p a r t i c u l a r p a r t i t i o n i n g

e s p o s si b l e d u e t o t h e n o n l i n e a r b e h a v i o r o f t h e a d d i t i o n

s u b t r a c t i o n f u n c t i o n s , l o g b 1 + b - a ) a n d l o g b 1 - b d ) ,

i v e l y , t h a t a r e d e p i c t e d i n F i g u r e 1 .3 f o r b = 2. B y

f f e r e n t m i n i m a l w o r d l e n g t h r e q u i r e d b y

s o f f u n c t i o n s a m p l e s, t h e o v e r al l si ze o f t h e L U T is

r e s s e d , l e a d i n g t o a L U T o r g a n i z a t i o n o f F i g u r e 1 .4 . I n

t o t h e a b o v e t e c h n i q u e s , r e d u c t i o n o f t h e s iz e o f

e m o r y c a n b e a c h i ev e d b y p r o p e r s e le c ti o n o f th e b a s e o f

I t t u r n s o u t t h a t t h e s a m e b a s e s th a t y i e l d

i m u m p o w e r c o n s u m p t i o n f o r th e L N S ar it h m e ti c u n i t

g t h e b i t a c t i vi ty , a s m e n t i o n e d i n t h e n e x t s e c t io n ,

o r e s u lt i n m i n i m u m L U T s iz es .

T o u s e t h e b e n e f i ts o f L N S , a c o n v e r s i o n o v e r h e a d i s r e -

q u i r e d i n m o s t c a se s t o p e r f o r m t h e f o r w a r d L N S m a p p i n g

d e f i n e d b y e q u a t i o n 1 .1 . It is n o t e d t h a t c o n v e r s i o n s o f e q u a -

t i o n s 1 .1 a n d 1 .2 ar e r e q u i r e d i f a n L N S p r o c e s s o r r e c e iv e s

i n p u t o r t r a n s m i t s o u t p u t l i n e a r d a t a i n d i g it a l f o r m a t . S i n ce

a l l a r i t h m e t i c o p e r a t i o n s c a n b e p e r f o r m e d i n t h e l o g a r i t h m i c

d o m a i n , o n l y a n i n i ti a l c o n v e r s i o n i s im p o s e d ; t h e r e f o r e , a s t h e

a m o u n t o f p r o c e ss i n g i m p l e m e n t e d i n L N S g r o w s, t he c o n t r i -

b u t i o n o f th e c o n v e r s i o n o v e r h ea d t o p o w e r d i s s ip a t io n a n d t o

a r e a - t i m e c o m p l e x i t y b e c o m e s n e g l i g ib l e b e c a u s e i t r e m a i n s

c o n s t a n t .

I n s t a n d - a l o n e D S P s y s t e m s , th e a d o p t i o n o f a d i f f e re n t

s o l u t i o n t o t h e c o n v e r s i o n p r o b l e m i s p o s si b le . I n p a r t ic u l a r ,

t h e L N S f o r w a r d a n d i n v e r s e m a p p i n g o v e r h e a d c a n b e m i t i -

g a t e d b y c o n v e r t i n g t h e a n a l o g d a t a d i r e c t ly i n t o d i g i ta l l o g a -

r i t h m s .

LNS rithmetic Exam ple

L e t X = 2 .75 , Y = 5 .65 , an d b = 2 . P e r f o r m t h e o p e r a t i o n s X . Y ,

X + Y , v / X a n d y 2 u s i n g t h e L N S .

I n i t i a l l y , t h e d a t a a r e t r a n s f e r r e d t o t h e l o g a r i t h m i c d o m a i n

a s i m p l i e d b y e q u a t i o n 1 . h

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1 L o g a r i t h m i c a n d R e s id u e N u m b e r S y s t e m s f o r V L S I A r i t h m e t i c 1 8 3

2 4 6

(a) : S a (O)

0 8

0 6

0 4

0 2

b)

: s s

d)

-3

i

6 8

F I G U R E 1 .3 T h e F u n c t i o n s sa(d) a n d s , ( d ): A p p r o x i m a t i o n s r e q u i r e d f o r L N S a d d i t i o n a n d s u b t r a c t i o n .

I G U R E 1 . 4 T h e P a r t i t i o n i n g o f t h e L U T : T h e p a r t i t i o n i n g s to r e s

h e a d d i t i o n a n d s u b t r a c t i o n f u n c t i o n s i n t o a s e t o f s m a l l e r L U T s,

t o m e m o r y c o m p r e s s i o n .

X L N S Z x , S x , X = l o g I x l

= ( 0 , O , x = l o g 2 2 . 7 5 ) = (0 , O , 1 . 4 5 9 4 ).

( 1 . 1 9 )

y L N S ( Z y , S y , y = l o g 2 ] Y I )

= ( 0 , 0 , y = l o g 2 5 . 6 5 ) = ( 0 , 0 , 2 . 4 9 8 3 ) .

( 1 . 2 0 )

U s i n g t h e L N S i m a g e s f r o m e q u a t i o n s 1 .1 9 a n d 1 .2 0 , t h e

r e q u i r e d a r i t h m e t i c o p e r a t i o n s a r e p e r f o r m e d a s f o l lo w s : T h e

o g a r i t h m i c i m a g e w o f t h e p r o d u c t W = X • Y is g i v e n b y :

W = X + y = 1 . 4 5 9 4 + 2 . 4 9 8 3 = 3 . 9 5 7 7 . ( 1 . 2 1 )

s b o t h o p e r a n d s a r e o f t h e s a m e s i g n ( i. e. , Sx = sy = 0 ) , t h e

s i g n o f t h e p r o d u c t i s s~ = 0 . I n a d d i t i o n , b e c a u s e

Zx

¢ 1 a n d z y ¢ 1 , t h e r e s u l t is n o n - z e r o ( i . e. , z ~ = 0 ) .

T o r e t r i e v e t h e a c t u a l r e s u l t W f r o m e q u a t i o n 1 .2 1 , i n v e r s e

c o n v e r s i o n o f 1 .2 i s u s e d a s f o l l o w s :

W = ( 1 - z w ) ( - 1 ) sw 2 W = 2 3 9 5 7 7 = 1 5 . 5 3 7 7 . ( 1 . 2 2 )

B y d i r e c t l y m u l t i p l y i n g X b y Y , i t i s f o u n d t h a t W = 1 5 . 5 3 75 .

T h e d i f f er e n c e o f 0 .0 0 0 2 i s d u e t o r o u n d - o f f e r r o r d u r i n g t h e

c o n v e r s i o n f r o m l i n e a r to t h e L N S d o m a i n .

T h e c a l c u l a t i o n o f t h e l o g a r i t h m i c i m a g e w o f W = v ~ i s

p e r f o r m e d a s f o l lo w s :

w = - x = - 1 . 4 5 9 4 = 0 . 7 2 9 7 . ( 1 . 2 3 )

2 2

T h e a c t u a l r e s u l t i s r e t r i e v e d a s f o l l o w s :

W = 2 0 . 7 29 7 = 1 . 6 5 8 3 .

( 1 . 2 4 )

T h e c a l c u l a t i o n o f t h e l o g a r i t h m i c i m a g e w o f W = X 2 c a n b e

d o n e a s :

W = 2 - 1 . 4 5 9 4 = 2 . 9 1 8 8 . ( 1 . 2 5 )

A g a i n , t h e a c t u a l r e s u l t i s o b t a i n e d a s :

W = 2 2 9 18 8 = 7 . 5 6 2 2 .

( 1 . 2 6 )

T h e o p e r a t i o n o f lo g a r i t h m i c a d d i t i o n i s r a t h e r a w k w a r d ,

a n d i ts r e a l i z a t io n i s u s u a l l y b a s e d o n a m e m o r y L U T o p e r -

a t i o n . T h e l o g a r i t h m i c i m a g e w o f t h e s u m W = X + Y i s a s

f o l l o w s :

w = m a x ( x , y ) + l o g 2 ( 1 + 2 min(x y)-ma x(x y)) ( 1 . 2 7 )

= 2 . 4 9 8 3 + l o g 2 ( 1 + 2 - 1 0 3 89 )

( 1 . 2 8 )

= 3 . 0 7 0 4 . ( 1 . 2 9 )

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184 T h a n o s S t o u r a i t i s

T h e a c tu a l v alu e o f th e su m W = X + Y i s o b t a in e d a s :

W = 2 3 0 7 0 4 = 8 4 0 0 1

1.30)

1 2 3 L N S a n d P o w e r D i s s i p a t i o n

Pow er d iss ipa t ion m inim iza t io n i s soug ht a t a l l leve ls of des ign

b s t r a c t i o n , r a n g in g f r o m so f tw a r e a n d h a r d w a r e p a r t i t i o n in g

d o w n to t e c h n o lo g y - r e l a t e d i ssue s. T h e a v e ra g e p o w e r d i s sip a -

io n i n a c i r c u i t i s c o mp u te d v i a t h e r e l a ti o n sh ip :

P a v e = ~ l k C L g f f l d 1 . 3 1 )

C L i s the to ta l swi tch ing

capac i tance , Vda i s the supply vol tage , and a i s the ave rage

c t iv i ty in a c lock pe r iod .

L N S i s a p p l i c a b le f o r l o w - p o w e r d e s ig n b e c a u se i t r e d u c e s

p l e x i t y o f c e r ta in a r i t h me t i c o p e r a to r s a n d t h e b i t

c t ivity.

i s s ip a t i o n a n d L N S A r c h i te c tu r e

s p r o p e r t i e s o f th e l o g a r i t h m f u n c t io n t o r e d u c e

gth o f seve ra l a r i thm et ic ope ra t io ns ; thus , i t leads to

p l e x i t y sa ving s. B y r e d u c in g t h e a r e a c o mp le x i ty o f o p e r -

i t c h in g c a p a c i t a n c e CL o f e q u a t io n 1 . 3 1 c a n b e

o r e , r e d u c t io n i n l a t e n c y a ll ow s f o r f u r th e r

in su p p ly v o l t a g e, w h ic h a l so r e d u c e s p o w e r d i s s ip a -

C h a n d r a k a sa n a n d B r o d e r se n , 1 99 5 ). A s tu d y o f t h e

f t h e c h o i c e o f th e n u m b e r s y s te m o n t h e Q R D - R L S

r e v e a le d t h a t L N S o f f e rs a c c u r a c y c o m p a r a b l e t o

f f l o a t i n g - p o in t o p e r a t i o n s b u t o n ly a t a f r a c t i o n o f t h e

a n c e p e r i t e r a t i o n o f t h e a lg o r i t h m S a c h a a n d

1 9 98 ). T h e r e d u c t io n o f a v er a g e sw i t c h e d c a p a c i t an c e o f

th e s im p l i f i c at i o n o f b a s ic a r i t h me t i c

o w n in T a b l e 1 . 2 . I t c a n b e s e e n t h a t n - b i t

c a t i o n a n d d i v i si o n a r e r e d u c e d t o k + / ) - b i t a d d i t io n

su b t r a c t i o n , r e sp ec t iv e ly , w h i l e t h e c o m p u ta t i o n o f

roots

c t iv e ly. F o r t h e c o m m o n c ases o f sq u a re

r o o t

e ra t io n i s r edu ced to le f t o r r igh t sh i f t r e spec t -

ly . For exam ple , a ssume tha t a n-b i t ca r ry- save a r ray mul t i -

, wh ich has a co mp lexi ty of n 2 - n 1-b i t fu ll adde rs FAs),

r ep laced by an n-b i t adde r , a ssum ing k + l = n has a com -

o f n FA s f o r a r i p p l e - c a r r y imp le me n ta t i o n K o r e n ,

1 9 93 ). T h e r e f o r e , mu l t i p l i c a t i o n c o m p le x i ty is r e d u c e d b y a

rcL,

given as:

/ . /2 _ n

rcL

- - - - - n - 1. 1 .32)

n

A d d i t i o n a n d su b t r a c t i o n , h o w e v e r , a r e c o m p l i c a t e d in L N S

T o p e r a t i o n f o r t h e e v a lu a t io n o f

1Ogb 1 ± by - X ) , a l t h o u g h d i f f e r e n t a p p r o a c h e s h a v e b e e n

p r o p o se d i n t h e l i t e ra tu r e O r g in o s

et aL,

1995; Pa l iouras and

Stoura i t i s , 1996). An L UT op era t i on requ i re s a ROM of n × 2 ~

bi ts , a s ize tha t c an inh ib i t use o f LNS for la rge va lues of n .

I n a n a t t e mp t t o so lv e t h i s p r o b l e m , e f f i ci e n t t a b l e r e d u c t io n

t e c h n iq u e s h a v e b e e n p r o p o se d T a y lo r

et al . ,

1988) . As a resu lt

o f t h e a b o v e a na ly si s, a p p l i c a t io n s w i th a c o m p u ta t i o n a l l o a d

d o m i n a t e d b y o p e r a t io n s o f s i m p l e L N S i m p l e m e n t a t i o n c a n

b e e x p e c t e d t o g a in p o w e r d i s s ip a t i o n r e d u c t io n d u e t o t h e

L N S imp a c t o n a r c h i t e c tu r e c o mp le x i ty .

S in c e mu l t i p l i c a t i o n - a d d i t i o n s a r e imp o r t a n t i n D S P a p p l i -

c a t io n s , t h e p o w e r r e q u i r e me n t s o f a n L N S a n d a l in e a r f ix e d -

p o in t a d d e r - mu l t i p l i e r h a v e b e e n c o mp a r e d . I t h a s b e e n

r e p o r t e d t h a t a p p r o x i m a t e l y a tw o t i m e s r e d u c t i o n i n p o w e r

diss ipa t ion i s poss ib le for ope ra t io ns w i th wo rd s izes of 8 to 14

bi ts Pa l iouras and S toura i t i s , 2001) . Given a suf f ic ien t nu m be r

o f c o n se c u t iv e mu l t i p l i c a t i o n - a d d i t i o n s , t h e L N S imp le m e n ta -

t i o n b e c o me s mo r e e f f i c i e n t f r o m th e l o w - p o w e r d i s s ip a t i o n

v i e w p o in t , e v e n w h e n a c o n s t a n t c o n v e r s io n o v e r h e a d is t a k e n

in to c o n s id e r a t i o n .

P o w e r D i s s i p a t io n a n d L N S E n c o d i n g

T h e e n c o d i n g o f d a t a t h r o u g h l o g a r it h m s o f v a ri o u s b a s es

impl ie s va r ia t ions in the b i t ac t iv i ty i .e ., the a f ac tor of

e q u a t io n 3 1 a n d , t h e r ef o r e , t h e p o w e r d i s s ip a ti o n ) P a l i o u r as

and Stourait is , 1996, 2001) .

A ssu min g a u n i f o r m d i s t r i b u t i o n o f l i n e a r n - b i t i n p u t

n u m b e r s , t h e d i s t r i b u t i o n o f b i t a s s e rt i o n s o f t h e c o r r e sp o n d -

in g L N S w o r d s r ev ea ls t h a t L N S c a n b e e x p lo i t e d t o r e d u c e t h e

ave rage ac t iv ity . Le t P0~ l i ) be the b i t a sse r t ion probab i l i t ie s

i .e ., the prob abi l i ty of the / th b i t t r ans i t ion f rom 0 to 1).

A ssu min g th a t d a t a a r e t e mp o r a r i l y i n d e p e n d e n t , i t h o ld s t h a t:

p o ~ i ) = p o i )p ~ i ) = 1 - pl i) )P~ i), 1.33)

w h e r e P0 i ) a n d P l i ) is th e p r o b a b i l i t y o f t h e / t h b i t b e in g 0 o r

1 , r e sp e c ti v ely . D u e t o t h e a s su mp t io n o f u n i f o r m d a t a d i s t r i -

b u t i o n , i t h o ld s t h a t :

1

p o i ) = p l i ) 2 1.34)

which , due to equa t ion 1 .33 , g ives :

1

p o - . l i )

= - . 1 . 3 5)

4

T h e r e f o r e , al l b it s i n t h e l i n e a r f i x e d - p o in t r e p r e se n t a t i o n e x -

h ib i t an equa l P0 ~ l i ) , i = 0 , 1 . . . . n - 1 .

A c t iv i ti e s o f t h e b i ts i n a n L N S - e n c o d e d w o r d a r e q u a n t i f i e d

u n d e r s imi l a r a s su mp t io n s . S in c e t h e r e i s a n o n e - to - o n e c o r -

r e sp o n d e n c e o f li n e a r f i x e d - p o in t v a lu e s t o t h e i r L N S ima g e s

d e f in e d b y e q u a t io n 1 . 1 , t h e L N S v a lu es f o l l o w a p r o b a b i l i t y

func t ion ident ica l to the f ixed-poin t case . In f ac t , the LNS

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L o g a r it h m i c a n d R e s i du e N u m b e r S y s t e m s f o r V L S I A r i t h m e t i c 185

i n g c a n b e c o n s i d e re d as a c o n t i n u o u s t r a n s f o r m a t i o n o f

~' r an do m va r iab le X, which i s a word in th e l inea r

H e n c e , t h e tw o d i s c r et e r a n d o m v a r ia b l es f o l l o w th e

Th e pLNS probabi l i t ie s o f b i t a sse r t ions in LNS words , h ow -

, a r e n o t c o n s t a n t a s P0 -* l ( i ) o f e q u a t i o n 1 . 35 ; t h e y d e p e n d

th e f o l l o w in g e x p e r ime n t i s p e r f o r me d . F o r a l l p o s -

l e v a lu e s o f X in a n - b i t s y s t em, t h e c o r r e s p o n d in g L lo g x j

( i ) fo r each b i t a re co m pute d . Th en , poL~Sl i ) i s co m pu ted a s

a t ion 1 .33 . Th e ac tua l a sse r t ion probabi l i t ie s for the b i t s

LNS word , ~LNS i ) , a re depic te d in F igure 1 .5 . I t can be

1 0-~1

( i ) for the more s ign i f ican t b i t s i s subs tan t ia l ly

e r than P0-~ l ( i ) for the less s ign i f ican t b i ts . M oreove r , i t can

( i ) depen ds o n b . This beh avior , which i s due

i n h e r e n t d a t a c o m p r e s s i o n p r o p e r t y o f t h e l o g a r it h m

d s t o a r e d u c t i o n o f t h e a v e r a g e ac t i v i ty i n t h e

rd . T he ave rage ac t iv i ty sav ings pe rcentage , S . . . i s

P01

0 . 2 5

0 . 2

0 . 1 5

0.1

0 . 0 5

b = 2 . 5

b = 2

b = 1 . 5 - - - -

4

1 2 3 4 5

A) n = 8

6 7 8

POl

0 . 2 5

0 . 2

0 . 1 5

0.1

0 . 0 5

b=2 5

b=2

b=1 5

1 2 3 4 5 7 8 9 1 0 1 1 1 2

B ) n = 1 2

E 1.5 Activities Against Bit Significance i (in an LNS Word

8 and n = 12) and V arious Values of the Base b. The horizon-

of the corresponding n-bit fixed-point

K- ~k+1-1 ~,LNS /

/ - . , i=o t ' °~ l ( i ) 100 ,

Saw = 1

(1.36)

w h e r e p F X P i ) = 1 / 4 f o r i = O , 1 . . . . n - - 1 ; t h e w o r d

l e n g th s k a n d l a r e c o m p u t e d v i a e q u a t i o n s 1 . 1 3 a n d 1 .1 4 ,

a n d n d e n o t e s t h e l e n g th o f t h e f i x e d - p o in t s y s t e m. T h e s a v in g s

pe rcentag e Save i s dem on st ra t ed in F igu re 1 .6(A) fo r va r ious

v a lu e s o f n a n d b, a n d t h e p e r c en t a g e is f o u n d t o b e m o r e t h a n

15 in cer ta in cases.

A s imp l i e d b y t h e d e f i n i t i o n o f n eq i n e q u a t i o n 1 .1 6 , h o w -

ever , the l inea r sys tem tha t p rov ides an equiv a len t r ange to tha t

of a (k, l , b )-LNS, r equi re s neq b i t s . I f the r edu ced prec is ion of a

k, l ,

b) -LNS, com par ed to an nCq-bi t f ixed -poin t sys tem, is

acceptab le for a pa r t icu la r ap pl ica t ion , S'av i s used to desc r ibe

the r e la t ive e f f ic iency of

L N S ,

ins tead of equa t ion 1 .36 , where :

x -~k+l-1 -LNS .~ \

2_,i=0

P o - - . ] t t )~

Savo= 1 - -5G _100

(1.37)

Savings pe rcentage S~v i s demonst ra ted in F igure 1 .6(B) for

va r ious va lues of n and b . Savings a re fou nd to exceed 50 in

som e cases . No t ice tha t F igure 1 .6 r evea ls tha t , fo r a pa r t icu la r

w o r d l e n g th n , t h e p r o p e r s e l e c t io n o f l o g a r i t h m b a s e b c a n

s igni f ican t ly a f fec t the ave rage ac t iv i ty . The re fore , the c hoice o f

b i s imp o r t a n t i n d e s ig n in g a l o w - p o w e r L N S - b a s e d s y st e m.

F in a l l y , i t s h o u ld b e n o t e d t h a t o v e r h e a d i s imp o s e d f o r

l i n e a r - t o - l o g a r i t h mic a n d l o g a r i t h mic - to - l i n e a r c o n v e r s io n .

C o n v e r s io n o v e r h e a d c o n t r i b u t e s a d d i t i o n a l a r e a a n d t ime

c o m p le x i t y as w e ll as p o w e r d i s s i p a ti o n . A s t h e n u m b e r o f

o p e r a t i o n s g r o w s , h o w e v e r , t h e c o n v e r s io n o v e r h e a d r e ma in s

c o n s t a n t ; t h e r e f o r e, t h e o v e r h e a d 's c o n t r i b u t i o n t o t h e o v e r a ll

b u d g e t b e c o m e s n e g l i gib le .

1 3 The Residue Nu m ber System

A d i f f e r e n t c o n c e p t t h a n t h e n o n l i n e a r l o g a r i t h mic t r a n s f o r -

ma t io n i s f o l lo w e d b y ma p p in g o f d a t a t o a p p r o p r i a t e ly s el ec -

t e d f i n i t e fi el ds . T h i s m a y b e a c h i e v e d t h r o u g h t h e u s e o f o n e o f

t h e m a n y a v a i l ab l e v e r s io n s o f t h e r e s id u e n u m b e r s y s t e m

(RNS) (Szabo and Tanaka , 1967) . RNS a r i thmet ic f aces d i f f i -

c u l ti e s w i th s i g n d e t e c t i o n , d iv is i o n , a n d m a g n i tu d e c o m p a r i -

son . These d i f f icu l t ie s may outwe igh the bene f i t s i t p re sents

f o r a d d i t i o n , s u b t r a c t i o n , a n d mu l t i p l i c a t i o n a s f a r a s g e n e ra l

c o mp u t in g i s c o n c e r n e d . I t s u s e i n s p e c i a l i z e d c o mp u ta t i o n s ,

l ike those for s igna l p rocess ing , o f fe r s many advantages . RNS

has been used to of fe r supe r ior f au l t to le rance capabi l i t ie s

as we l l a s h igh-speed , sma l l - a rea , and/or s ign i f ican t power -

d iss ipa t ion sav ings in the des ign o f s igna l p rocess ing a rch i tec -

tures for F IR f i l te r s (F rek ing and Pa rh i , 1997) and o the r c i r -

c u i ts ( C h r e n , 1 9 98 ). R N S m a y e v e n r e d u c e t h e c o m p u ta t i o n a l

l o a d i n c o mp le x - n u mb e r p r o c e s s in g ( T a y lo r et a l . , 1985) , thus

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1 8 6 T h a n o s S t o u r a i t i s

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/ x / \ \ ~ ~ .

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R E 1 . 6 P e rcen t age o f A verage A c t iv i t y R edu c t ion f ro m U se o f L N S . T he pe rcen t ag e i s com par ed t o n -b i t and t o neq-b it l i nea r f ixed-p o in t

o u s b a s e s b o f t h e l o g a r i t h m . T h e d i a g r a m r ev e al s t h a t t h e o p t i m a l s e l e c t io n o f b d e p e n d s o n n , a n d i t c a n l e a d t o s i g n if ic a n t p o w e r

i d i n g s p e e d a n d p o w e r s a v i n g s a t t h e a l g o r i t h m i c l e v e l o f

1 3 1 R N S B a s i c s

p s a n a t u r a l n u m b e r X i n t h e r a n g e [ 0 , M - 1 ],

u N I m i t o a n N - t u p l e o f r e s i d u e s x i :

x R N S { X1 X . . . . X N }

( 1 . 3 8 )

x i = X ) m i , ( )m~ d e n o t e s t h e m o d rni o p e r a t i o n a n d

m i is a m e m b e r o f th e s e t o f th e c o - p r i m e i n te g e rs

= { m l , m 2 . . . . . m N } c a ll ed m o d u l i . C o - p r i m e i n te g e r s

g c d ( m i , m j ) = 1, i ¢ j . T h e s e t

d u l i is ca l le d th e b a s e o f R N S . T h e m o d u l o o p e r -

a t i o n

( X ) m

r e t u r n s t h e i n t e g e r r e m a i n d e r o f t h e i n t e g e r d i -

v i s i o n x d i v m (i .e ., a n i n t e g e r k s u c h t h a t x = m . l + k ) w h e r e

l i s a n i n t e g e r .

R N S i s o f in t e r e s t b e c a u s e b a s i c a r i t h m e t i c o p e r a t i o n s c a n b e

p e r f o r m e d i n a d i g i t -p a r a l l e l c a r r y - fr e e m a n n e r , s u c h a s in :

z i = ( x i o Y i )m ~ , ( 1 . 3 9 )

w h e r e i = 1 , 2 . . . . . N a n d w h e r e t h e s y m b o l o s ta n d s f o r

a d d i t i o n , s u b t r a c t i o n , o r m u l t i p l i c a ti o n . E v e r y i n t e g e r i n t h e

ran ge 0 _< X < I - IN _I m i h a s a u n i q u e R N S r e p r e s e n t a t i o n .

I n v e rs e c o n v e rs i o n m a y b e a c c o m p l i s h e d b y m e a n s o f t h e

C h i n e s e r e m a i n d e r t h e o r e m ( C R T ) o r th e m i x e d - r a d i x c o n v e r -

s i o n ( S o d e r s t r a n d e t a l . , 1 9 8 6 ). T h e C R T r e t r ie v e s a n i n t e g e r

f r o m i t s R N S r e p r e s e n t a t i o n a s :

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L o g a r i th m i c a n d R e s id u e N u m b e r S y s t e m s f o r V L S I A r i t h m e t ic

1 8 7

( 1 . 4 0 )

ro T, = G , M = M H N _ l m i , a n d m 7 1 i s t h e m u l t i p l i c a t i v e

r s e o f ~ m o d u l o m i ( i . e . , a n i n t e g e r s u c h t h a t

~ 1 ) m i = 1) .

U s i n g a n a s s o c i a t e d m i x e d r a d i x s y s t e m , i n v e r s e c o n v e r s i o n

y a l so b e p e r f o r m e d b y t r a n s la t in g t h e r e s i du e r e p r e s en t a -

m i x e d r a d i x r e p r e s e n t a t i o n . B y c h o o s i n g t h e R N S

d u l i t o b e t h e w e i g h t s in th e m i x e d r a d i x r e p r e s e n t a t i o n , t h e

m a p p i n g i s f a c i li t a te d b y a s s o c i a t in g t h e m i x e d r a d i x

e R N S . S p e c i f ic a l ly , a n i n t e g e r 0 < X < M c a n

d b y N m i x e d r a d i x d ig i t s (x ~l . . . . x ~ ) a s :

X = m l ) 4 - . . . 4 -

x 3 ( m 2 m l )

m ( m N - l m N 2 . '

4 - 4 m l 4 - X ll ,

( 1 . 4 1 )

X~{X1 X2

X 3 } = { 1 0 ) 3 , 1 0 ) 5 , 1 0 ) 7 1

= { 1 , 0 , 3 } .

( 1 . 4 4 )

Y ~ { y l , Y 2 ,

Y3} = {(5)3 , (5)5 , ( 5 ) 7 } = {2, O , 51. (1 .4 5)

T h e R N S i m a g e o f t h e s u m Z = X + Y i s o b t a i n e d a s:

z R N S { z I , Z 2 , Z 3 } = { 1 4- 2 3 , 0 4- 055 , 3 4 - 5 ) 7 ) 1 . 4 6 )

= { 0 , 0 , 1 } .

T o r e t r ie v e th e i n t e g e r t h a t c o r r e s p o n d s t o t h e R N S r e p r e s e n t a -

t i o n { 0, 0 , 1 } b y a p p l y i n g t h e C R T o f e q u a t i o n 1 . 40 , th e

f o l l o w i n g q u a n t i t i e s a r e p r e c o m p u t e d : M = 3 - 5 - 7 = 1 0 5 ,

--ml = -5=1°5 3 5 , ~ = - 5 -1 0 5 2 1 ,

~ _ 1 0 5 ~ _ =

15, / 7 / 1 1 - 1 = 2 ,

m ~ 1 = 1 , an d m 33 1 = 1 . T h e v a l u e o f t h e s u m i n i n t e g e r

f o r m i s o b t a i n e d b y a p p l y i n g e q u a t i o n 1 .4 0

e r e 0 _< ~ <

m i ,

i = 1 . . . N , a n d t h e

x i

c a n b e g e n e r a t e d

e q u e n t i a l l y f r o m t h e

x i

u s i n g o n l y r e s i d u e a r i t h m e t i c , s u c h a s

in:

( x )

1 ~ ~ X l

m l

x ~ = ( m ~ - l ( x - - X t l ) ) m 2

( m 2 1 ( m l l ( X X t l ) )

= - - - - 4 ) ) m 3 ,

( 1 . 4 2 )

a n d s o o n , o r a s i n t h e f o l l o w i n g :

X 1 = X 1

4 ( ( x 2 , - 1

= - - X l ) m l m 2 ) m 2

( ( ( X 3 t - 1 - 1

= - - x l ) m 1 m 3 - - x 2 ) m 2 m 3 ) m 3

XN ' ' ) m l l m m - - ~ ) m 2 1

( ( ( ( x u - x l

t - - 1

m , - . . . - x N 1 ) m , _ l m N ) m N .

( 1 . 4 3 )

T h e d i g it s x I c a n b e g e n e r a t e d s e q u e n t i a l ly t h r o u g h r e s i d u e

s u b t r a c t i o n a n d m u l t i p l i c a t i o n b y t h e f i x e d m~ 1 . T h e s e q u e n -

t ia l n a t u r e o f c a l c u l a t io n i n c r ea s e s t h e l a t e n c y o f t h e r e s i d u e s

c o n v e r s i o n t o b i n a r y n u m b e r s .

T h e s e t o f R N S m o d u l i i s o f t e n c h o s e n s o t h a t t h e i m p l e -

m e n t a t i o n o f t h e v a r i o u s R N S o p e r a t i o n s ( e .g ., a d d i t i o n ,

m u l t i p l i c a t i o n , a n d s c a l in g ) b e c o m e s e f fi c ie n t . A c o m m o n

c h o i c e i s t h e s e t o f m o d u l i { 2 n - 1 , 2 , 2 + 1 }, w h i c h m a y

a l so f o r m a s u b s e t o f t h e b a s e o f R N S .

R N S A r i t h m e t i c E x a m p l e

C o n s i d e r t h e b a s e B = { 3 , 5 , 7 } a n d t w o i n t eg e r s X = I 0 a n d

Y = 5 . T h e R N S i m a g e s o f X

a n d

Y a r e a s w r i t t e n h e r e :

Z = X 4- Y = ( 3 5 ( 2 . 0 ) 3 4 - 2 1 ( 1 . 0 ) 5 4 - 1 5 ( 1 - 1 )7 )1 0 5

= (15)105 = 15 . (1 . 47 )

T o v e r i f y t h e r e s u l t o f e q u a t i o n 1 .4 6 , n o t i c e t h a t

X 4 - Y = 1 0 4 - 5 = 1 5 a n d t h at :

R N S

15- - -+ { (15 )3 , (15 )5 , (15 )7 } = {0 , 0 , 1} = {Z l , z2, z 3 } , ( 1 . 4 8 )

w h i c h i s t h e r e s u l t o b t a i n e d i n e q u a t i o n 1 .4 6. T h e s a m e i n t e g e r

m a y b e r e tr i e v e d b y u s i n g a n a s s o c i a t e d m i x e d r a d i x s y s t e m

d e f i n e d b y e q u a t i o n 1 .4 1 as :

Z = 4 - 1 5 + Z ' 3 4 -

Z' ,

w i th 0 < Z tl < 3 , 0 _< z~ < 5 , 0 _< z~ < 7 and t he fo l l o w in g :

Z tl z Z 1 ~ 0

Z ~ = ( 3 I ( Z 2 - - Z t l ) ) 5 = ( 2 " z 2 5 = 0

a n d

z 3 ( 5 - 1 [ 3 - 1 ( z 3 - z , ) - 4 ] ) 7 = - Z l) - 3 . 2 57

= 1 - 0 ) - 3 . 0 ) 7 = 1

( 1 . 4 9 )

s o t h a t Z = 1 . 1 5 + 0 . 3 + 0 = 15.

1 .3 .2 R N S A r c h i t e c t u r e s

T h e b a s i c a r c h i t e c t u r e o f a n R N S p r o c e s s o r i n c o m p a r i s o n t o a

b i n a r y c o u n t e r p a r t i s d e p i c t e d i n F i g u r e 1 .7 . T h i s f i g u r e sh o w s

t h a t t h e w o r d l e n g t h n o f t h e b i n a r y c o u n t e r p a r t i s p a r t i -

t i o n e d i n t o N s u b w o r d s , t h e r e s id u e s , t h a t c a n b e p r o c e s s e d

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188 Thanos Stouraitis

A ) S t r uc tu r e o f a B ina r y A r ch i t ec tu r e

F I G U R E 1 7

O

o

n M ~ ~ _ i n M

B) Cor responding RNS Processor

Basic Architectures

n

n d e p e n d e n t l y a n d a r e o f w o r d l e n g th s i g n i f ic a n t ly sma l l e r

han n . The a rch i tec ture in F igure 1 .7 a ssumes , wi thout loss

f g e n er a li ty , t h a t t h e m o d u l i a r e o f e q u a l w o r d l e n g th . T h e i t h

e si d u e c h a n n e l p e r f o r m s a r i t h m e t i c m o d u l o mi.

M o s t imp le m e n ta t i o n s o f a r i t h me t i c u n i t s f o r R N S c o n s i st o f

n a c c u mu la to r a n d a mu l t i p l i e r a n d a r e b a s e d o n R O Ms o r

et al. (1983) have ana lyzed the e f f ic iency of

I imp le m e n ta t i o n s o f R N S a d d e rs . Mo r e o v e r , im-

t a t i o n s o f a r i t h me t i c u n i t s t h a t o p e r a t e i n a fi n i te i n t e g e r

ing R m) and tha t a re ca l led AUras a re of fe red in the l i te ra ture

(S toura i t i s , 1993). Th ey a re less cos t ly and re qui re le ss a rea and

r d w a r e c o m p l e x i t y a n d p o w e r c o n s u m p t i o n . T h e y a r e

n t i n u o u s ly d e c o m p o s in g t h e r e s id u e b i t s t h a t c o r -

e s p o n d t o p o w e r s o f 2 t h a t a r e l a r g e r t h a n o r e q u a l t o 2 n , u n t i l

h e y a re r e d u c e d t o a s e t o f b i t s th a t c o r r e s p o n d t o a s u m o f

e r s o f 2 tha t i s le ss than 2 n , whe re n = [ log2 m 1 . This

e c o m p o s i t i o n i s imp le m e n te d b y u s in g f u ll a d d e r ( FA ) a rr a y s.

F o r a ll mo d u l i , t h e F A - b a s ed A U ra s a re s h o w n to e x e c u t e m u c h

f a s te r as w e l l a s h a v e mu c h s ma l l e r h a r d w a r e c o mp le x i t y a n d

i m e - c o m p l e x i t y p ro d u c t s t h a n R O M - h a se d g e n er al m u l t i -

p l ie rs . S ince the AUras use fu l l adde r s a s the i r bas ic un i t s , th ey

l e a d t o m o d u la r a n d r e g u l a r de s ig n s , w h ic h a r e i n e x p e n siv e a n d

e a s y t o imp le m e n t i n V L S I .

1 3 3 E r r o r T o l e r a n c e i n R N S S y s t e m s

B e c a u s e t h e r e i s n o i n t e r a c t i o n a mo n g d ig i t s ( c h a n n e l s ) i n

r e s id u e a r i t h me t i c , a n y e r r o r s g e n e r a t e d a t o n e d ig i t c a n n o t

p r o p a g a t e a n d c o n t a min a t e o th e r c h a n n e l s d u r in g s u b s e q u e n t

o p e r a t i o n s , g iv e n t h a t n o c o n v e r s io n h a s o c c u r r e d f r o m th e

R N S to a w e ig h t e d r e p r e s e n t a t io n .

I n a d d i t i o n , b e c a u s e t h e r e i s n o w e ig h t a s s o c i a te d w i th t h e

RNS res idues (d ig i t s ) , i f any d ig i t beco m es cor rupte d , the

assoc ia ted channe l may be eas i ly ident i f ied and dea l t wi th .

B a s ed o n t h e a m o u n t o f r e d u n d a n c y t h a t i s b u i l t i n a n R N S

processor , the f au l ty channe ls m ay be r ep laced or jus t i so la ted ,

w i th t h e r e st o f t h e s y s te m o p e r a t i n g in a s o f t f a i lu r e mo d e ,

b e in g a l l o w e d t o g r a c e fu l l y d e g r a d e i n to a c c u r a te o p e r a t i o n s o f

r e d u c e d d y n a mic r a n g e . P r o v id e d t h a t t h e r e ma in in g d y n a mic

r a n g e c o n t a in s t h e r e s u lt s , th e r e i s n o p r o b l e m w i th t h i s d e g -

r a d a t i o n .

T h e mo r e r e d u n d a n t a n R N S i s , t h e e a s i e r i t i s t o i d e n t i f y

a n d c o r r e c t e r r o r s. A r e d u n d a n t R N S ( R R N S ) u s e s a n u m b e r r

o f m o d u l i i n a d d i t i o n t o t h e N s t a n d a r d m o d u l i t h a t

a re necessa ry for cove r ing the des i r ed dynamic range . Al l

N ÷ r mo d u l i mu s t b e re l a ti v e ly p r ime . I n a n R R N S, a n u m b e r

X i s p r e s e n t e d b y a t o t a l o f N n o n r e d u n d a n t r e s id u e d ig i t s

{X2 . . . . XN} p lus r r ed un dan t r e s idue d ig i t s

{ X N 1 . . . . XN r}.

IIN+rm

i s r e p r e s e n t ed b yf t h e t o t a l n u m b e r o f s ta te s,

MR = i=~ ,

the RRNS. Th e M = I IiN1mi f i r st s ta tes cons t i tu te it s leg i t im-

a t e r a n g e w h i l e a n y n u m b e r t h a t l ie s i n t h e r a n g e M, MR), is

called i l legit imate .

A n y s in g l e e r r o r mo v e s a l e g i t ima t e n u mb e r X in to a n

i l l e g i t ima t e n u mb e r X t. O n c e i t is v e ri f ie d t h a t t h e n u m b e r

be ing te s ted i s il leg i t ima te , i t s d ig i t s a re d isca rded one b y one ,

u n t i l a l e g i t ima t e r e p r e s e n t a t i o n i s f o u n d . T h e d i s c a r d e d d ig i t

w h o s e o mi s s io n r e s u l t s i n t h e l e g i t ima t e r e p r e s e n t a t i o n i s

t h e e r r o n e o u s o n e . A c o r r e c t d ig i t c a n t h e n b e p r o d u c e d

b y e x t e n d in g t h e b a s e o f t h e r e d u c e d R N S th a t p r o d u c e d

th e l e g i t ima t e r e p r e s e n t a t i o n . T h e a b o v e e r r o r - l o c a t i n g - a n d -

c o r r e c t in g p r o c e d u r e c a n b e imp le m e n te d i n a v a r i e ty o f

w a ys . A s s u min g t h a t t h e m ix e d r a d ix r e p r e s e n t a t i o n s o f al l

t h e r e d u c e d R N S r e p r e s e n t a t i o n s c a n b e e f f ic i e n t ly g e n e r a t e d ,

the leg i t ima te one can be eas i ly ident i f ied by checking the

highes t o rde r m ixed rad ix d ig i t aga ins t ze ro . I f i t is ze ro ,

the r epresenta t ion i s leg i t ima te .

Mix e d r a d ix r e p r e s e n t a t i o n s a s s o c i a t e d w i th t h e R N S

n u mb e r s c a n b e u s e d t o d e t e c t o v e r f l o w s a s w e l l a s t o d e t e c t

a n d c o r r e c t e r r o r s i n r e d u n d a n t R N S s y s t e ms . F o r e x a mp le ,

t o d e t e c t o v e r f l o w s , a r e d u n d a n t mo d u lu s mN+l i s a d d e d t o

th e b a s e a n d t h e c o r r e s p o n d in g h ig h e s t o r d e r mix e d r a d ix

dig i t

aN.l

i s f o u n d a n d c o mp a r e d t o z e r o . A s s u min g t h a t t h e

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1 Logarithmic and Residue Nu mb er Systems for VLSI Ari thm etic 189

mb e r b e in g t e s t e d f o r o v e r f l o w i s n o t l a r g e e n o u g h to

th e a u g m e n te d r a n g e o f t h e r e d u n d a n t sy s t e m, o v e r-

M o n te C a r lo r u n s . I t i s o b se r v e d t h a t R N S p e r f o r m s b e t t e r th a n

tw o ' s c o mp le me n t r e p r e se n t a t i o n f o r a n t i c o r r e l a t e d d a t a a n d

s l ig h t ly w o r se t h a n s ig n - m a g n i tu d e a n d tw o ' s c o m p le m e n t

r e p r e se n t a t io n s f o r u n c o r r e l a t e d a n d c o r r e l a t e d se q u e n c es .

1 .3 .4 R N S a n d P o w e r D i s s i p a t i o n

2500

(Freking and Pa rh i , 1997) . By em ploy ing b ina ry- l ike RNS f i l te r

2000

truc tures ( Ibrah im, 1994) , i t has been repor ted tha t RNS

educes the b i t ac t iv i ty up to 38 in (4 × 4) -b i t mul t ip l ie r s . 1500

mic a l l y w i th t h e e q u iv a l e n t b in a r y w o r d l e n g th , R N S c a n

1 0 0 0

o le r a t e a l a r g e r r e d u c t io n i n t h e su p p ly v o l t a g e t h a n t h e

c o r r e sp o n d in g b in a r y a r c h i t e c tu r e w h i l e a c h i e v in g a p a r t i c u - 5 00

a r d e l a y sp e c i f i c a t i o n . T o d e mo n s t r a t e t h e o v e r a l l imp a c t o f

the RNS on the po we r budge t of an F IR fi lter , F rek ing and

Parh i (1997) r epor t tha t a f i l te r un i t wi th 16-b i t coe f f ic ien ts

a n d 3 2 - b i t d y n a m ic r a n g e , o p e r a t i n g a t 5 0 MH z , d i s s ip at e s

2 6 . 2 mW o n a v e r a g e f o r a tw o ' s c o mp le me n t imp le me n ta t i o n ,

w h i l e t h e R N S e q u iv a l e n t a r c h i t e c tu r e d i s sip a te s 3 . 8 mW . 2 5 0 0

H e n c e , p o w e r d i s s ip a t i o n r e d u c t io n b e c o me s mo r e s i g n i f i c a n t

a s t h e n u m b e r o f f i lt e r t a p s i n c re a ses , a n d a 3 - f o ld r e d u c t io n i s 2 0 0 0

poss ib le for fi l ter s wi th m ore than 100 taps .

L o w - p o w e r m a y a l so b e ac h i e v e d v i a a d i f f e r en t R N S imp le - 1 50 0

me n ta t i o n . I t h a s b e e n su g g e s t e d t o o n e - h o t e n c o d e t h e r e s i -

d u e s i n a n R N S - b a se d a r c h i t e c tu r e , t h u s d e f in in g one hot RNS 1000

( O H R ) ( C h r e n , 1 9 98 ). I n s t e a d o f e n c o d in g a r e s id u e v a lu e

xi

in

a c o n v e n t io n a l p o s i t i o n a l n o t a t i o n , a n ( m - 1 ) - b i t w o r d i s 5 00

e mp lo y e d . I n t h i s w o r d , t h e a s se r ti o n o f t h e i t h b i t d e n o t e s

the re s idue va lue xi T h e o n e - h o t a p p r o a c h a l l o w s f o r a f u r th e r

r e d u c t io n i n b i t a c t i v i t y a n d p o w e r - d e l a y p r o d u c t s u s in g r e s i -

d u e a r i t h me t i c . O H R i s f o u n d to r e q u i r e s imp le c i r c u i t s f o r

process ing . The power reduc t ion i s r endered poss ib le s ince a l l

bas ic ope r a t ions ( i. e. , add i t ion , subt rac t ion , and mu l t ip l ica - 25 00

t ion) a s we l l as the RN S-spec i f ic ope ra t ion s o f sca l ing

( i . e . , d iv i s io n b y c o n s t a n t ) , mo d u lu s c o n v e r s io n , a n d i n d e x

2000

c o m p u ta t i o n a r e p e r f o r m e d u s in g t r a n sp o s i t i o n o f h i t l in e s

a n d b a r r e l shi f te r s. T h e p e r f o r ma n c e o f t h e o b t a in e d r e s id u e

1 5 0 0

a r c h i t e c tu r e s is d e mo n s t r a t e d t h r o u g h th e d e s ig n o f a d i r e c t

d ig i ta l f r equ ency synthes ize r tha t exhib i t s a pow er -de lay pro d- 1000

u c t r e d u c t io n o f 8 5 o v e r t h e c o n v e n t io n a l a p p r o a c h ( C h r e n ,

1998). 500

RNS Signa l Activity for G aussian Input

T h e b i t a c t i v i t y i n a n R N S a r c h i t e c tu r e w i th p o s i t i o n a l l y e n -

c o d e d r e s id u e s h as b e e n e x p e r im e n ta l l y s t u d i e d f o r t h e e n c o d -

ing of 8-b i t da ta us in g the b ase {2 , 151} , wh ich prov ides

a l i n e a r f ix e d - p o in t d y n a m ic r a n g e o f a p p r o x im a te ly 8 .2 4 b i t s.

A ssu min g d a t a s a mp le d f r o m a G a u ss i a n p r o c e s s , t h e

bi t a sse r t ion ac t iv i t ie s of the p a r t icu la r RNS, an 8 -b i t s ign-

ma g n i tu d e , a n d a n 8 - b i t tw o ' s - c o mp le me n t sy s t e m a r e me a s -

ured an d co mp ared . T he re su l t s a re depic ted in F igure 1 .8 for 100

T C

2 0 4 0 6 0 8 0 1 0 0

( A ) S t r o n g l y A n t i c o r r e l a t e d G a u s s i a n D a t a

R N S

* ' ' - ' ' ' , ' ' ' ' - T O

~ SM

2 0

4 0 6 0 . 8 0 1 0 0

( B ) U n c o r r e l a t e d G a u s s i a n D a t a

R N S

S M

2 0 4 0 6 0 8 0 1 0 0

( C ) S t r o n g l y C o r r e l a t e d G a u s s i a n D a t a

FIGUR E 1.8 Num ber o f Low-to-High Transitions. (A) This figure

shows strongly anticorrelated (p = -0.99) Gaussian data for two's

complement, R NS , and sign-magnitude num ber syste m s for 100

Monte C arlo runs. (B) Shown here are uncorrelated (p = 0) Gaussian

data; (C) This figure illustrates strongly correlated (p = 0.99) Gauss-

ian data

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1 9 0 T h a n o s S t o u r a i t i s

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