1987 new 2n dct algorithms suitable for vlsi implementation

8/20/2019 1987 New 2n Dct Algorithms Suitable for Vlsi Implementation

1/4

NEW 2 DCT ALGORITHMS SUITABLE

FOR

VLSI IMPLEMENTATION

PierreD U H A M E LandH ed iH ' M ID A

C N E T / P A B / R P E ,

38-40 ,

r u e d u G & n & r a l L e c l e r c , 9 2 1 3 1 I s s y - l e s - M o u l i n e a u x

( F R A N C E )

ABSTRACT

S m a l le n g t hi sc r e t eo s i n er a n s f o r m sD C T ' S )

a r e s e do rm a g e a t ao m p r e s s i o n . In t h a t case,

l e n g t h

8 or 16

D C T ' sr e e e d e d to b e e r f o r m e d

at v i d e o r a t e .

We

p r o p o s e w o e wm p l e m e n t a t i o n

of

D C T' sw hich

h a v ee v e r a ln t e r e s t i n ge a t u r e s ,

as

f a r

as

VLSI

i m p l e m e n t a t i o n

is

c o n c e r n e d .

A f i r s tne ,s i n godu lo -a r i t hm e t i c ,e e d sn l y

o n em u l t i p l i c a t i o ne rn p u to i n t ,

so

t h a t

a

s i n g l e

m ul t i p l i e r

is

needed on -ch ip .

A e c o n d n e , a s e d n a d e c o m p o s i t i o n of t h eD C T

i n t oo l y n o m i a lr o d u c t s ,n dv a l u a t i o n of t h e s e

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

i n

a

v e r ym a l lh i p , i t h

a

g r e a te g u l a r i t yn d

t e s t a b i l i t y .u r t h e r m o r e ,h ea m et r u c t u r ea n

b es e do r FFT c o m p u t a t i o n yh a n g i n g n l yh e

R O M - p a r t of t h e c h i p .

B o th ew r ch i t e c tu r e s , ar e m a in ly a sed n a n e w

f o r m u l a t i o n

of a

l e n g t h - 2D C T

as a

c y c l i cc o n v o l u t i o n ,

w h ich is e x p l a i n e d nh e i r s t e c t i o n of t h e a p e r .

I

INTRODUCTION

In t h e e c e n t e a r s ,m a n y fast D C T l g o r i th m sw e r e

p r o p o s e d , m o n gw h i c h h r e e r e

of

m a j o r n t e r e s t

t h eH E N - F R A L I C K

[ I ]

alg or i th m, B .G. LEE

[31,

and V ETTER LI -N U SSB A U M ER [41 a lgo r i t hm .

T h ei r s tne ,e i n gr o p o s e do r a l o n gi m e ,a s

b e e no n s i d e r e do r VL SI i m p l e m e n t a t i o ne v e r a l

t i m e s , l t h o u g h t o e s o tm e e t h em i n i m u m r i t h -

m e t i c c o m p l e x i t y

Dl.

T h e t h e r n e sm e e th em i n i m u m n o w n u m b e r o f

b g t h m u l t i p l i c a t i o n s a n d a d d i t i o n s to i m p l e m e n t

a

l e n g t h

2C Tl g o r i t h m .u r t h e r m o r e ,ta se e nh o w n

t h a t , if t h e s ea l g o r i t h m sc o u l db e m p r o v e d , h e a m e

a p p r o a c h w o u l d a l s o i m p r o v e

a

w h o l e c l a s s of a l g o r i t h m

( L e . Dan d 2-D FFT 's , DST

---) [ 6 ]

F r o m

a

p r a c t i c a l

po in t of v i ew ,h e l g o r i t h m b y L E E a s r e a t e r

r e g u l a r i ty h a n h e V E T T E R L I - N U S S B A U M E R a l g o r i t h m ,

b u ta so o ro u n d o f fo i s ee r f o r m a n c e s ,u eo

t h e l /cos coe f f i c i en t s .B o t h of t h e mh a v eb e e n m p l e -

m e n t e d n h a r d w a r e ( o r s il i c o n )

[51.

W i t hh o s e o n s i d e r a t i o n s i n m i n d , n e a n see t h a t

t h e r e is s t i l l o m e e e d o rD C T l g o r i th m sm e e t i n g

t h e f o l l o w i n g t h r e e c h a r a c t e r i s t i c s a l t o g e t h e r

-

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

- g r e a te g u l a r i t y of t h er a p ht h ev a i l a b i l i t y o f

a l e n g t hN / 2D C T ns ide o f

a

l e n g t h N D C T is o f t e n

requ i r ed ) ,

-

g o o d n o i s e p e r f o r m a n c e s .

Whi le i t is p o s s i b l eob t a i nc l a s s i ca l l go r i t hm s

m e e t i n gh e s eh r e eo i n t st h ea p e re s c r i b i n g

t h e m is u n d e rh er o c e s s of b e i n gr i t t en ) ,e

p r o p o s e n h i s a p e r w o o m p l e t e l y e w p p r o a c h e s

t h a ta v ee v e r a ln t e r e s t i n ge a t u r e s ,

as

f a r

as

V LS I i m p l e m e n t a t i o n

is

c o n c e r n e d .

F u r t h e r m o r e ,h e s ep p r o a c h e sr ee s c r i b e dr o m

c o n s i d e r a t i o n sh a ta v eh e o r x t i c a lm p o r t a n c e ,

s i n c e o rh e i r s ti m e ,e n g t hD C T ' s r e t a t e d

i n e r m s

of

c y c l i cc o n v o l u t i o n ,w h i c ha l l o w s oo b t a i n

q u i c k l y its m u l t i p l i c a t i v e o m p l e x i t yw eh u s b t a i n

a

s e c o n d e r i v a t i o n o f a result by M.T. HEIDEMAN).

We w i l li ve i n t h i s a p e rn l yke t ch es o f p roo f s

f o r h e e r i v a t i o n s o f t h e l g o r i t h m s , i n c e u r i m

is to s h o w h a t h eu n d e r s t a n d i n g of t h e m a t h e m a t i c a l

u n d e r l y i n gt r u c t u r e

of

t h e C Ta ne a d to n e w

e f f i c i e n t a l g o r i t h m s .

11.

THE LENGTH

2

DCT AS POLYNOMIAL PRODUCTS

T h e D C T

is

d e f i n e d

as

f o l l o w s

1 2n

(1)

x k = i

cos 2 i + l )

k

4N

i O

T h eq u i v a l e n c ee t w e e nh eb o v e C Tor N=2

a n d

a

c y c l i co n v o l u t i o n is o b t a i n e dh r o u g hw o

p e r m u t a t i o n s

of

t h en p u t a r i a b l e s .

T h e i r s t n e , l r e a d y i v e n in4] is used to c h a n g e

t h ee r m s2 i + l ) i n

( I )

i n t o4 i t l ) L e t

us d e f i n e

eq . (1) c a n h e n b e w r i t t e n as

N - 1

2.rr

( * )

k

= x

I cos 4 i + l )

k

4Ni

O

T h ee c o n dn e

will

a l l ow

t o

c h a n g e a p r o d u c t of

i n d i c e s

4 i + l )

( 4 k + l ) n t o a

s u m of i nd i ces u - + v

k

T h i sesu l t is o b t a i n e dh r o u g hh es e of a o n e

to o n eo r r e s p o n d a n c ee t w e e nh e set

of

i n t e g e r s

of t h eo r m4 i + l ) , i-0, --

p o w e r s

of

m o d u l o 2 .

+ 2 .

w r i t e

2 -1 an dh eu c c e s s i v e

i t is a lw aysoss ib l e to


2/4

U .

(3) 4 i + l

=

< >2n+2

i =o

, _ _ _ _ ~ - 1

T h e p e r m u t a t i o n ( 4 ) is t h e n a l w a y s f e a s i b l e

x '

<

5 i

> - 1

4N

4) X' l i =

4

a n d e q .

(3)

b e c o m e s as fo l l ow s

L e t u s n o wo n s i d e re p a r a t e l yh ev e n X Z k

a n do d d e r m s X 2 k + l of t h e D CT .

I t

is

w e l ln o w n ,n da i r l yb v i o u sr o mq . (1)

t h a t X 2 k is t h e u t p u t o f

a

D C T of l eng thN /2 .

H e n c e , h e o l l o w i n gd e c o m p o s i t i o n ,o n h eo d d e r m s

w i l l a p p l y e c u r s i v e l yo n h e D C T ' s o f r e d u c e d e n g t h s .

W h e no n s i d e r i n gn l yh ed de r m sX 2 k + l

I

e q .I ) is n o w y m m e t r i c a l i n

i

a n d k, a n d h e w o

p e r m u t a t i o n s e s c r i b e d b o v e r e o w e a s i b l en k.

( W i t hh e n l y i f f e r e n c eh a th e r e r eN / Z + t e r m s

X Z k + l ,a n d N t e r m s x Zic l , t h u s e s u l t i n g n h e

-

t e r m

of eq. 7. ( s e e [XI f o rm o r e e t a i l s ) .H e n c e ,w e a v e

as

a

r e s u l t

w h e r e

L e t

us

n o w d e f i n e h e f o ll o w i n g p o l y n o m i a l s

N/2-1

N 1

( 1 0 )

V z)

=

LA

\

i

O

k

k

x . z

i

2

C O S

<

si

>4N

z

i

eq.

6 )

c a n n o w b e r e f o r m u l a t e d as

(11) Y(z)

=

X(z)

. V(z)

m o d

zN I

L e . t h e d de r m s of t h eD C T a n e t a te d as

a

p o l y n o m i a l p r o d u c t

of

l e n g t h N / 2 .

T h i sa nep p l i e de c u r s i v e l y to t h ee n g t h N / 2

D C T r i s i n g r o m h e o m p u t a t i o n

of

t h ee v e n e r m s ,

a n d so on ,h u sesu l t i ngn

a

c o m p l e t eo r m u l a t i o n

o f t h e D C T a s p o l y n o m i a l p r o d u c t s .

W h e no n s i d e r i n gh e s eo l y n o m i a lr o d u c t s ,t

is

e a s i l ye c o g n i z e dh a th eo l y n o m i a l sn v o l v i n g

t h en p u t

of

t h e C Tr el le d u c t i o n s of X(z)

m o d u l o h e c y c l o t o m i c f a c t o r s of

xN-1

(N=Zn).Knowing

t h i s , n e a n

see

t h a th ew h o l e

set

o fo lynom ia l

p r o d u c t s is e q u i v a l e n t t o a cyc l i convo lu t i onLe .

a p o l y n o m i a lr o d u c to d u l o x -1) f o l l o w e d

r e d u c t i o no d u l oh ey c l o t o m i ca c t o r s of x -1.

T h ee q u e n c e to b ey c l i c a l l yo n v o l v e d i t hh e

i n p u t a t a e m a i n s

to

b eound .B u t , i ncew e n o w ,

b yu c c e s s i v ep p l i c a t i o n s

of

eq .

IO)

t o t h eD C T ' s

o f d e c r e a s i n ge n g t h

N,

N / 2 , N / 4

----

t h e x p r e s s i o n

o f t h e u n k n ow no l y n o m i a lo d u l oh ey c l o t o m i c

f a c t o r s ,t is e a s y

to

r e c o n s t r u c th en i t i a ln e ,

g i v e n n eq .

(12)

N

f i a

We

h a v eo ws t a b l i s h e dh a th e C T

of

l e n g t h

N =Z n c a n b e o b t a i n e d as show n n i g . ( I ) .

111

THEORETICAL CONSEQUENCES

As

a

s i d e e s u l t , t

is

n o wv e r y a s y to g e ta nu p p e r

b o u n d n h em u l t i p l i c a t i v e o m p l e x i t y of t h e e n g t h

2 DCT

I t h a sb e e n h o w nb yW I N O G R A D [91, t h a t h em u l t i -

p l i c a t i v e o m p l e x i t y o f

a

cyc l i c onvo lu t i on of l eng th

2 is g iven by

(13)

p(conv. 2 )

=

2 -n -1

F u r t h e r m o r e ,n e of t h eu l t i p l i c a t i o n sn v o l v e d

i nh e C To m p u t e d as a convo lu t i on ,

as

s h o w n

inig. ( I )

is

t r i v i a l ( V(z) mod. x-1 = 1). We t h e n

o b t a i n , as a n u p p e r b o u n d

(14)

p(DCT 2 )

=

2 -n - 2

I t

is

p o s s i b l e b u tm o r en t r i c a t e ) t o s h o w , y s m g

s o m et h e re s u l t s

of

W I N O G R A D t h a th i sp p e r -

b o u n d s l s o h e o w e rb o u n d .T h i s e s u l tw a s l r e a d y

ob t a in ed by M .T . H EID EM A N [ I

11.

C o n s e q u e n c e s

of

p r a c t i c a l m p o r t a n c ec a nb eo b t a i n e d

b y b s e r v a t i o n

of

t a b l e 1, c o n t a i n i n gh e o m p a r i s o n

b e t w e e n h i s o w e rb o u n da n d h ep r a c t i c a la l g o r i t h m s

f o r s h o r t - l e n g t h s

In fact, o b s e r v a t io n of t a b l e 1 t e l l s

us

t h a t h e c o m p u -

t a t i o n of a D C T of l e n g t h 2 n e e d sm o r eh a n n e

m u l t i p l i c a t i o n e r o i n tw h a t e v e rh el g o r i t h mw i l l

be. As

a

c o n s e q u e n c e , i f

a

D C Th i p is n e e d e d to

w o r k n e a l i m e at v i d e oa t e s w h i c h is t h e case

i na n yC Tp p l i c a t i o n s ) ,m p l e m e n t a t i o n of

a

D C Til leedoreh a nn eu l t i p l i e rn - ch ip .

42.2.2

18 6


3/4

N

CHEN

EE

ETTCRLI

ower bound

4

8 16

6

6 26 32 32

44

T a b l e

1

c o m p a r i s o n

of

t h eo w e ro u n dn dh e

p r a c t i c a l a l g o r i t h m s

IV. THE DCT COMPUTED BY NTT

N e v e r t h e l e s s ,h e r e is s t i l l

a

way of o b t a i n i n g

a

D C Tl g o r i t h mi thn eu l t i p l i ca t i one ro i n t

( h e n c e o n e m u l t i p l ie r o n - c h i p )

S i n c ew e a v e o w s t a b l i s h e dh eD C T

as a

c y c l i c

c o n v o l u t i o n ,w e a nu s eN u m b e rT h e o r e t i cT r a n s f o r m s

(NTT) 1121 t o c o m p u t e h e co n v o l u t i o n , a n d h e sc h e m e

of fig. ( I ) n o w b e c o m e s as shown n ig . (2).

F u r t h e r m o r e ,i n c eh eo m p u t a t i o n of t h ee s u l t

m o d u l oh ey c l o t o m i ca c t o r s of x N - I

is

o b t a i n e d

as i n t e r m e d i a t ea r i a b l e sn s i d eh en v e r s eT T ,

t h eu t t e r f l i e sh o w n i n

fig. (2)

c a nei m p l i f i e d

w i t hh ea s t p e r a t i o n sn v o l v e dnh e o m p u t a t i o n

of t h e N TT -' o x. T h i s e s u l t s n h e i a g r a m h o w n

in ig .

3

f o r N = 8 .

I t s h o u l db en o t e d h a t h i sc o r r e s p o n d s

to a

f a v o r a b l e

case f o r N T T ' s to be used

S i n c eN T T ' s r e e n e r a l l y

p e r f o r m e d n h o r t - l e n g t h

s e q u e n c e s N = 1 6 e e m s to

b e a m a x i m u m ) ,w e v o i d

t h es u a lr o b l e mr i s i n g

inN T Tha t ,u e to t h e

r e l a t i o n s h i pe t w e e n a t h e N roo t o f un i t y , N ,

t h ee n g t h of t h er a n s f o r m ,n d M t h eo d u l u s

(

a N

1 mod M),

it

is o f t e nm p o s s i b l e

to

use 2

as

a

r o o t of u n i t yt h u sv o i d i n gu l t i p l i ca t i ons

i n h e N T T )

for

e v e n m o d e r a t e e n g t h s .

- What is n e e d e d to c o m p u t eh eD C T is r ea l l y a

c y c l i c c o n v o l u t i o n , a n d h e r e

is

n o n e e d

of

t h e o v e r l a p -

a d drv e r l a p - s a v el g o r i t h m s to o b t a i n

a

l i nea r

c o n v o l u t i o n , as

is

n e e d e d n

F I R

f i l t e r i ng .

-

T h em o d u l oa r i t h m e t i c

is

n o t u c h

a

prob l em , i nce ,

w i t h h eg i v e nc o n s t r a i n t s ,w ec a nw o r km o d u l o a F e r -

m a t u m b e r , r

a

p s e u d oF e r m a t u m b e r [131 w hich

g i v e s o n e of t h e C i m p l e s t k n o w n m o d u l o - a r i t h m e t ic [141.

In t h i s case, In flg. (3) r e p r e s e n t sn l y a s h i f t ,

a n d a n em p l e m e n t e d b y

a

r o t a t i o n o f t h en p u t

w o r d at

a

b i t eve l .

- F u r t h e r m o r e , s i n c e

a

g r e a t p r e c i s i o n o n h e

X k

is of-

t e nn e e d e d u s e

of

D C T na d a p t a t i v e e e d b a c k o o p s ) ,

t h e e e d

of

grea t e r o rd l eng ths hens ing TT ' s ,

c o m p a r e d to t h es u a l case,

is

n o tu c h a w as t e .

V DCT 3Y DISTRIBUTED ARITHMETIC

A nothe ros s ib i l i t y

is

to

use t h ee c o m p o s i t i o n

of

t h e C Tn t oo l y n o m i a lr o d u c t s , as exp l a ined i n

s e c t i o n 11 a n dh e noo m p u t eh e s eo l y n o m i a l

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

L e t

us

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

u si ng t h e d i s t r i b u t e d a r i t h m e t i c

t h

b e h e n n e r p r o d u c t to b e c o m p u t e d , a n d

t h e e x p r e s s i o n

of

x i i n t e r m s ot i t s b i n a r y r e p r e s e n t a t i o n

(2 s c o m p l e m e n t ) .e t us k n o write xi a t a b i t

l e v e l n

(15)

a n d r e v e r s e h e w o r e s u l t i n g s u m s . We g e t

L-1 6 - 1

L-1

(17) = - a . x i o .t (E i x i j

2 - j

j =

i s 0

In t h i s q u a t i o n , h ed o u b l e u m is a s u c c e s s i v e h i f t

a n dd d of e l e m e n t a r ye r m sb e t w e e nr a c k e t s ) ,

e a c he r me i n gnn n e rr o d u c te t w e e n

a i

a n d a v e c t o r of b i t s ( x . . , i = O , ---N-I).

L e t f b e h i s u n c t i o n . f depends n N b i n a r y a r i a -

b l e s ,e n c ea na k e

ZN

d i f f e r en ta lues . If t he se

v a l u e s r e t o r e dn

a

ROM

at

t h e d d r e s s corres-

ponding to t h ei n a r yo n f i g u r a t i o n

of

t h en p u t

b i t s ,nm p l e m e n t a t i o n o f t h en n e rr o d u c t b y

d i s t r i b u t e d a r i t h m e t i c

is

as

show n n ig. 4.

When used n a D C T l g o r i t h m , h ed i s t r i b u t e d r i t h -

m e t i cm p l e m e n t a t i o n of t h eo l y n o m i a lr o d u c t

w i l l e q u i r eo n e n n e rp r o d u c tc o m p u t a t i o np e rc o e f f i -

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

tod e c o m p o s e h e n i t i a lD C T n t op o l y n o m i a lp r o d u c t s

(see

fig. 5).

A n u m b e r of r e m a r k s a r e of i n t e r e s t

- S i n c eh e ROM is a d d r e s s e d b y t h e i t s

of

s a m e

w e i g h t o f t h eo u t p u t s

of

t h eb u t t e r f l i e s , h e s eb u t t e r -

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

- T h e p e e d of a c i r c u i t m p l e m e n t i n g h i s r c h i t e c -

t u r e w i ll b e i m i t e d n l y by t h e u t p u t c c u m u l a t o r .

If t h e e q u i r e d p e e d is l o w e r , t is poss ib l e to r e d u c e

t h ei z e o f t h ei r c u i t by us ing t hee l a t i o n s h i p s

b e t w e e nh ei f f e r e n tn n e rr o d u c t sn v o l v e d 181,

in

a

m a n n e r e r yim i l a r to t h a t x p l a i n e dn [ I 51

f o r t h e c o m p u t a t i o n of convo lu t i on .

All t h e o m b i n a t i o n s o f t h e n p u t a t a r e e r f o r -

m e dn e r i a l r i t h m e t i c .H e n c e ,h ee s u l t i n g r c h i -

t e c t u r e

is

v e r ye g u l a rn da s i l ym p l e m e n t e d .

- S i n c eh et r u c t u r e of t h ee c o m p o s i t i o n o f t h e

D C Tn t oo l y n o m i a lr o d u c t s is t h ea m e

as

f o r

o t h e rr a n s f o r m s ,h ea m et r u c t u r ea nl s oe

u s e do rh eo m p u t a t i o n s of F o u r i e rr a n s f o r m s

by chang ing on ly he ROM p a r t of t h e c h i p .

I

VI.

CONCLUSIQN

We

h a v ei r s tx p l a i n e dh eq u i v a l e n c ee t w e e n

D C T a n d c y c l i c c o n v o l u t i o n .

T h u s ,w e s e dh i se l a t i o n s h i po b t a i n e wD C T

a l g o r i t h m si t ho m eh a r a c t e r i s t i c su i t a b l eo r

V LS I i m p l e m e n t a t i o n .

O t h e rl g o r i t h m sa n

also

b eb t a i n e di t hu c h

a n a p p r o a c h .F u r t h e rw o r kw i l l b e e p o r t e d .

422 3

1807


4/4

REFERENCES

s11

[21

[31

[41

151

[61

W.H. CHEN,

C.

H A R R ISO N , S. C. FR A L IC K

A fast c o m p u t a t i o n a lA l g o r i t h mo rhe Dis-

c r e t e C o s i n e T r a n s f o r m .

I E E Erans .nomm. , Vol. COM-2509,

S e p t e m b e r1977, pp. 1004-1011.

A. JALAL I, K .R . RAO, A High -speedD C T

P r o c e s s o ro re a l - t i m eroc es s i ng of N TSC

col or TV s ignal .E E Erans .nlec .om p .

Vol. EMC.4 ,02, May 198 2, pp. 278-2 86.

B Y E O N G E E ,Ae wl g o r i t h moo m p u t e

t h eD i s c r e t eC o s i n eT r a n s f o r m . E E ET r a n s .o n

ASSP. Vol ASSP-32,06 ,ecem ber984 ,

M. VETTERLI, H . NUSSBAUMER, A s i m p l e

F F T n dD C T l g o r i t h m sw i t h ed u c e d u m b e r

of ope ra t i on S igna l r oces s ing , V ol. 6 ,04,

M.

V ETTER LI ,

A.

L I G T E N B E R G, A d i s c r e t e

c o s i n er a n s f o r m h i p , E E E o u r n a l of

Select.

Areas .n om. , Vol. SA C -4 , O 1 . anua ry 986 ,

P.D U H A M EL Im pl em e n ta t i o n o f sp l i t -r ad ix

FFT

a l g o r i t h m so ro m p l e x ,e a l ,n de a l -

sym m et r i cda t a . EEETrans .onAS SP , Vol. 34,

n02,pp.285-295,April1986.

pp.1243-1245.

A ugus t1984, pp.267-278.

pp.49-65.

16

16

-.13

Fig.

1 T h e o m p u t a t l o n of a l e n g t h

2 DCT

b a se d

on a c y c l i c c o n v o l u t l o n

P.U H A M E L D i s p o s i t i fer a n s f o r m e e

e n o s i n u sd ' u n i g n a ln u m 6 r i q u e6 c h a n t i l l o n n i .

F r e n c ha t e n t ,9 6 0 1 6 2 9 ,eb rua ry 1986 .

P. DUHAM EL Disposi t i f ded 6 t e r m i n a t i o nd e

l ar a n s f o r m k e u m k r i q u e ' u ni g n a l . r e n c h

pa t en tn 8612431 ,Sep t em ber 1986 .

S.

W I N O G R A D S o m ei l i nea ro r m s h o s e

m u l t i p l i c a t i v eo m p l e x i t ye p e n d snh e

f i e ld o f cons t an t s .M a th .ys t .heo ry ,977 ,

L. AUSLANDER, S. WINOGRAD Themul t ipl i -

c a t i v ec o m p l e x i t y of c e r t a i ns e m i - l i n e a rs y s t e m s

d e f i n e d b ypolynom ials . Adv. in App liedM athe -

m a t i c s . Vol. 1 , n03, pp.257-299,1980.

M . T . H E I D E M A N ,r i v a t eo m m u n i c a t i o n .

H.J.NUSSBAUMER,

Fast

F o u r i e r T r a n s f o r m a n d

C onvo lu t i onlgo r i t hm s . p r i nge r -V er l ag ,981 .

R.C.AGARWAL,

C.S.

B U R R U S Fast convo lu -

t i o n ss i n ge r m a tu m b e rr a n s f o r m si t h

a p p l i c a t i o n o i g i t a l i lt e r i n g . E E ET r a n s . n

L.M. LEIBOW'ITZ A sim pl ifi ed b i n a r ya r i t h m e -

t i co rh e e r m a tN u m b e r r a n s f o r m .E E E

Trans .n ASSP, Vol. 24,p .56-359,976.

S. C H U , C.S . B U R RU S A p r i m e f a c t o r F F T a l -

g o r i t h m u s i n gd i s t r i b u t e da r i t h m e t i c . E E E T r a n s .

on A SSP, Vol. 30, n02, p . 17-226,Apr i l 982.

Vole10,pp.169-180.

ASSP,

VOI.

22, pp. 87-97,1974.

t

4

Fig 4

Imp1ementa t lon

of

a nn n e rro d u c t b y

d l s t r l b u t e d a r l t h rn e t l c

Fig. 2 G e n e r a l s c h e m e

of

t h e

DCT

c o rn p u l e d b y

N T l

8

ig.

5 T h eD C T

of

length 8

by

distributed a r l t h r n e t l c

1808

1987 new 2n dct algorithms suitable for vlsi implementation

Documents