Scholarship at UWindsor
Scholarship at UWindsor
Electronic Theses and Dissertations Theses, Dissertations, and Major Papers
1-1-1984
Image processing using a two-dimensional digital convolution
Image processing using a two-dimensional digital convolution
filter.
filter.
Rajendra P. Rathi University of Windsor
Follow this and additional works at: https://scholar.uwindsor.ca/etd
Recommended Citation Recommended Citation
Rathi, Rajendra P., "Image processing using a two-dimensional digital convolution filter." (1984). Electronic Theses and Dissertations. 6788.
https://scholar.uwindsor.ca/etd/6788
NOTE TO USERS
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C O N V O L U T I O N F I L T E R
by
B d j e n d r a P. R a t n i
A t h e s i s
p r e s e n t e d to the U n i v e r s i t y of W i n d s o r in p a r t i a l f u l f i l l m e n t of t h e
r e q u i r e m e n t s t o e t h e d e q r e e of M a s t e r of A p p l i e d S c i e n c e
in
D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g
W i n d s o r , O n t a r i o , 1984
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A t w o - d i m e n s i o n a l d i q i t a l c o n v o l u t i o n f i l t e r (re
f e r r e d a s C o n v o l v e r ) e m p l o y i n q the F a s t n u m b e r t h e o r e t i c
t r a n s f o r m (FNTT) a l q o r i t h m was b u i l t in t he D e p a r t m e n t as an
e x t e r n a l p e r i p h e r a l to the 3BL tuini-computer f o r I m a q e P r o
c e s s i n g . The p u r p o s e of t h is r e s e a r c h is to a n a l y s e the d e
-siqn, s u q q e s t i m p r o v e m e n t s , p r o v i d e a w o r k i n q s y s t e m and i l
l u s t r a t e the use of the c o n v o l v e r t hr o uq h v a r i o u s e x a m p l e s .
The t h e s i s d e s c r i b e s the t h e o r e t i c a l bacicqround a n d
the n a rd va re i m p l e m e n t a t i o n or the c o n v o l v e r . A d e t a i l e d
e x p l a i n a t i o n of the d e s i g n c o n s i d e r a t i o n s has been d e v e l o p e d
to pr ov id e an e a s y and c o m p l e t e r e f e r e n c e f or the user.
S e v e r a l c o m p a r i s o n s have been p r e s en te d, as part of a n a l y
sis, to e s t a b l i s h t he e f f i c i e n c y of the t e c h n i q u e s used i n
the d e s i q n or tne c o n v o l v e r . T i m i n q d i a q r a m s have b e e n p r e
p a r e d to f a c i l i t a t e the u n d e r s t a n d i n g of the p r o c e s s i n g of
s i q n a l s t hrouqh t he filter. T h r o u q h - p u t r a t e c a l c u l a t i o n s
are i n c l u d e d to i n d i c a t e the s p e e d of p r o c e s s i n g .
A s y s t e m a t i c way to wr it e the i n t e r f a c i n g s o f t w a r e
has teen e x p l a i n e d . A d i r e c t o r y of the a v a i l a b l e s o f t w a re ,
and a table of the m a i n I n t e g r a t e d C i r c u i t s used in the c o n
v o l v e r is i nc luded. S o f t w a r e has been w r i t t e n to make the
Carl and Viola Glos
I am deeply Indebted to Dr. W.C. Miller and Dr. G.A. Jullien for
their constant help, support and guidance throughout the course of ray
degree. I express my sincere gratitude to Dr. M. Shridhar for his valuable
advices and encouragements. I express ray sincere thanks to Dr. M. Ahmadi,
Dr. M. Sid-Ahraed, and Dr. J. J. Soltis for their constructive comments
and criticisms. I am also thankful to Mrs. S.A. Ouellette and Mrs. L.J.
Kennedy for their invaluable assistance.
I sincerely acknowledge the support provided by the American-Nepal
Education Foundation, Oregon, U.S.A. In particular I am grateful to
Dr. Hugh B. Wood, Executive Director of the Foundation, for his personal
interest in me.
I am especially grateful to Mr. Carl M. Glos and Mrs. Viola Glos
whose guidance has been a perennial source of inspiration for me in my
personal life throughout my degree at Windsor. I also thank Mr. Edwin D.
Reimer and Mrs. Elfrieda Reimer for being my wonderful friends at Windsor.
I express my sincerest thanks and gratitude to my parents for their help,
love and patience. I also wish to mention my gratefulness to Dr. K.K.
Aggarwal, my professor at Regional Engineering College, Kurukshetra, India,
A B S T B A C T ... i i
A C K H O U L E D G E N E N T S ... i v
T A B L E OF C O N T E N T S ... . . . v i
T A B L E O F F I G U B E S ... T i i
L I S T OF T A B L E S ... ix
L I S T O F A B B R E V I A T I O N S ... x i
C h a p t e r . page
I. I N T R O D U C T I O N ... 1
O b j e c t i v e and o u t l i n e of the R e s e a r c h work . . . 1
T h e s i s O r g a n i z a t i o n . . . 8
II. D I G I T A L F I L T E R I N G U S I N G F A S T N U M B E R T H E O R E T I C
T R A N S F O R M T E C H N I Q U E S ... 10
I n t r o d u c t i o n ... . . . 1 0 D e f i n i t i o n of N u m b e r T n e o r e t i c T r a n s f o r m . . . . 12 M o d u l a r A r i t h m e t i c ... . . . 1 J
An E x a m p l e of c o n v o l u t i o n u s i n g NTT when M is a prime . . . 2 1 P r a c t i c a l C o n s i d e r a t i o n s in c h o o s i n q &,N a n d M
for a n N T T ... 23 D e s i g n c o n s i d e r a t i o n tor NTT u s ed in C o n v o l v e r . 2h
c n o o s i n q M ... . 2 5
c h o o s i n g N a n d 8 . . . . . . 2 7
O R D E R E D - I N P U T - O S D E B E D - O U T P U T - N I T a l q o r i t h m . . . 28 R e s i d u e to B i n a r y C o n v e r s i o n . . . . ... 29
Use of MHC in o b t a i n i n g the f i n a l r e s u l t . . 30
c o n c l u s i o n ... . . . . . . 3 1
III. I M P L E M E N T A T I O N OF B U T T E R F L Y AND A N A L Y S I S O N T H E
C O N V O L V E R ... 3 3
I n t r o d u c t i o n . ... - ... . . . 3 3
m o d u l a r a p p r o a c h ... 36 c a l c u l a t i n g the e n t r i e s in the L o o k - u p t a b l e s 38 m e m o r y s a v i n g by the use o£ s u b - m o d u l a r
a p p r o a c h . . . ...40
f u r t n e r r e d u c t i o n in m e mo ry r e g u i r e m e n t s . . 43
t i m in g c o n s i d e r a t i o n for the b u t t e r f l y . - . . . 44
T h r o u g h p u t R a t e C o n s i d e r a t i o n s (0100-al go ri thm) 46
s e r i a l s e g u e n t i a l p r o c e s s i n g . . . 4 6
S p e e d c o n s i d e r a t i o n in tne C o n v o l v e r . . . . 49
C a s c a d e P r o c e s s i n g . . . . . . 5 1
t h r e e - m e m o r y s t r u c t u r e f or f a s t e r p r o c e s s i n g . . 52
usrng a c o m p l e t e b u t t e r f l y for f a s t e r s p e e d . . 54
A bri e f C o m p a r i s o n of F F T and F N T T b u t t e r f l y
i m p l e m e n t a t i o n s . . . 5 6
C o m p a r i s o n of 1 - D - r a d i x - 2 and 2 - D - r a d i x - 2
b u t t e r f l i e s . . . . . . 5 8
C o m p a r i s o n of R/B c o n v e r s i o n m e t h o d s . . . 60 C o n c l u s i o n . . . 6 2
IV. H A R D W A R E AMD F U N C T I O N A L D E T A I L S O F T H E C O N V O L V E R . 69
i n t r o d u c t i o n . . . . . . . . 6 9
O v e r v i e w of the H a r d w a r e ... . . . 7 0
S y s t e m C l o c x s a n d R e s e t L o g i c ... . . . . 7 5
H S D / c o n v o l v e r i n t e r f a c e . . . 7 6
Line D r i v e r s / R e c e i v e r s . ... . . . . 7 8
NTT s t a g e / b u t t e r f l y c o u n t e r . . . 7 8
I n t e r f a c e c o n t r o l logi c . . . 7 9
L D I M G o p e r a t i o n ... . 8 1 L D C O F o p e r a t i o n ... 81 C L 8 M C N T o p e r a t i o n ... . . . 8 2 S D F I L D T o p e r a t i o n . . . ... . . . . 8 2
S DC A HOT o p e r a t i o n ... . . . 8 2
MEMORY B U F F E R S ... 84
M e m o r y W r i t e A d d r e s s G e n e r a t o r . . . . . . . 8 4
M e m o ry R e a d A d d r e s s G e n e r a t o r . . . 8 5 T wi d d l e F a c t o r / T r a n s , of F i l t e r C o ef r. Audr.
G e n r a t o r ... . . . 8 5
M e m o r y A d d r e s s M u l t i p l e x e r . . . 8 6
M e m o r y B u t t e r MEM 1 a n d M E M 2 ... 87 M e m or y B u f f e r T C O F M E M ... 87 T w i d d l e F a c t o r E P R G M s ... . . . 8 8 The B u t t e r f l y U n i t ... . . 8 8 P i p e l i n e T i m i n g . . . 9 2
R e s i d u e to B i n a r y C o n v e r t e r . . . 9 2
using the c o n v o l v e r . . . ... 95
V. F I L T E R I N G A P P L I C A T I O N S A N D F I L T E R I N G OF L A R G E
M A T R I C E S ... 98
t r a n s m i s s i o n s [ 1 6 ] ... ... 100 Low Pass F i l t e r i n q ... 10 1
Hiqh Pass F i l t e r i n q ... ... 104
H o m o - m o r p h i c F i l t e r i n q . . . 106
E x a m p l e s of the I m a q e F i l t e r i n q . . . 107
F i l t e r in q or I m a q e s of l ar qe r d i m e n s i o n s . . . 112
B l o c k - m o d e f i l t e r i n q ... 113
B l o c k - M o d e F i l t e r i n q A l q o r i t h m . . . 115
W r a p - a r o u n d Error C o n s i d e r a t i o n for Fi rs t
3 1 o c k ... 119
N u m be r or B l o c k s to be P r o c e s s e d . . . 120
T i m i n q C o n s i d e r a t i o n in P r o c e s s i n q L a r q e r
I m a q e s ... . . . 120
c o n c l u s i o n s ... . . . 123
VI. C O N C L U S I O N S ... 124
A p pe n di x p a o e
A. D E V E L O P M E N T OF 0 I 0 0 - N T T A L G O R I T H M ... 128
B. P R O C E D U R E TO U S B AN HSD D E V I C E ON S E L M INI
C O M P U T E R ... 132
C. D I R E C T O R Y O F A V A I L A B L E S O F T W A R E R E L A T E D T O T H E
C O N V O L V E R ...138
R E F E R E N C E S ... 139
F i gu re
3.1 A 2-D r a d i x - 2 B u t t e r f l y in O I O O
a l q o r i t h m
3.2 S u b - m o d u l a r I n d e x T a b l e s
3 . 3 A m o d u l o 19 ROM a r r a y m u l t i p l i e r
(sub -moduli 7 a n d d)
3.4 The B u t t e r f l y c o m p u t a t i o n u n it
3.5 T i m i n g D i a g r a m and tleqister C o n t e n t s
m the C o n v o l v e r b u t t e r f l y
3.6 G r a p h i c a l r e p r e s e n t a t i o n of 0 1 0 0 - a l q o r i t h m
for the c a s e of N=d (1-0-raaix-2)
3 .7 A T hr ee M e m o r y B u f f e r S t r u c t u r e for
I m p r o v e d s p e e d of p r o c e s s i n g
3.8 T i m i n g C o n s i d e r a t i o n in the use of M e m o r y
C o n f i g u r a t i o n s n o w n in F i q . (3.7)
3 . 9 I n t e r c o n n e c t i o n s in the use of a
J .11 I m p l e m e n t a t i o n ◦£ the n i x e d Radix
C o n v e r s i o n M e t h o d
4.1 O v e r v i e w of the C o n v o l v e r h a r d w a r e
4.2 T h e F i l t e r I n t e r f a c e and I/O C o n t r o l
Unit
4.3 The M e m or y B u f f e r O r g a n i z a t i o n
4.4 The B u t t e r f l y C o m p u t a t i o n U n i t
4.5 The p i p e l i n e t i m in g d ia g r a m
4.6 The R e s i d u e to B i n a r y C o n v e s i o n Unit
5. 1 T he r a d i a l c r o s s - s e c t i o n of Low P a s s F i l t e r s
5.2 The r a d i a l c r o s s - s e c t i o n of High P a s s F i l t e r s
5. 3 A c r o s s - s e c t i o n of a Homo-tnorphic F i l t e r
5.4 to 5.20 E x a m p l e s of the f i l t e r i n q
t h r o u g h the use of the c o n v o l v e r
T a b l e
3.1 Memo ry R e q u i r e m e n t s by D i r e c t and
S u b m o d u l a r a p p r o a c h of M u l t i p l i c a t i o n
a n d the M e m o r y S a v i n q R at io (MSH)
3.2 C o m p a r i s o n of T i m e r e q ui re d for
c o n v o l u t i o n by d i f f e r e n t m e t h o d s of
P r o c e s s i n q
3.3 C o m p a r i s o n of b u t t e r f l y c o m p u t a t i o n a l
r e q u i r e m e n t (1-D -r ad if-2 a n d
2 -D - radix -2 )
3.4
C o m p a r i s o n of C RT andMac
i m p l e m e n t a t i o n4.1 Main f e a t u r e s of the c o n v o l v e r
4 . 2 Main I C ' s a n d th ei r usa qe in tae C o n v o l v e r
5.1 A p p r o x i m a t e C o m p u t a t i o n t i me us in q
d i f f e r e n t m e t h o d s or c o n v o l u t i o n
5.3 P r o c e s s i n q T i m e by the u se of d i f f e r e n t s i ze
of B a s i c b l o c k for l a r q e r i m a q e s
A / 0 A u a l o q to D i q i t a l
B / R B i n a r y to R e s i d u e C o n v e r s i o n
C R T C h i n e s e R e m a i n d e r T h e o r e m
F I R F i n i t e I m p u l s e R e s p o n s e
F NT F e r m a t Numbe r T r a n s f o r m
U S D H i q n S p e e d D e vi ce
MEM M e m o r y B u f fe r
MRC M i x e d R a d i x C o n v e r s i o n M e t h o d
NTT N u mb er T h e o r e t i c T r a n s f o r m
O I O O O r d e r e d I n p u t o r d e r e d o u t p u t
R / B R e s i d u e to B i na ry C o n v e r s i o n
R NS R e s i d u e N u m b e r S y s t e m
RI R a de r T r a n s f o r m
d G e n e r a t o r used in d e f i n i t i o n of NTT
PL i ch p r i m e m o d ul i
ri i th r e s x d u e
|. | m o p e r a t i o n m o d u l o H
* Si qn to r e p r e s e n t m u l t i p l i c a t i o n
I N X B O D O C T I O M
1-1 O B J E C T I VE A N D O U T L I N E OF T H J R E S E A R C H W O R K
High s pe e d d i g i t a l f i l t e r i n q of t w o - d i m e n s i o n a l s i g n a l s
is an e s s e n t i a l e l e m e n t of c o n t e m p o r a r y r e s e a r c h o n s i q n a l
p r o c e s si nq . Th e a p p l i c a t i o n a r e a s i n c l u d e i m a q e p ro c e s s i n q ,
p a t t e r n r e c o g n i t i o n , d i g i t a l c o m m u n i c a t i o n and r o b o t i c v i
sion. T h e p r e - p r o c e s s i n q of s i q n a l s is a l s o i m p o r t a n t for
i m a g e s o b t a i n e d from s p a c e e x p l o r a t i o n p h o t o q r a p h s , r a d i o
-qraphs , n uc le ar m e d i c a l imaqes, and g e o p h y s i c a l data. T h e
f i ltering , or the c o n v o l u t i o n of i m a q e s w i t h a f i l t e r k e r
nel, c an be a c h i e v e d e i t n e r by d i r e c t c o m p u t a t i o n or i n d i
rectly , by the use of a t r a n s r o r m h avi nq the c y c l i c c o n v o l u
t i o n p ro perty.
One of the i n d i r e c t t e c n n i q u e s for tne c o n v o l u t i o n is
use of the D i s c r e t e F o u r i e r T r a n s f o r m (DF T) . T h e use of D FT
ror c o n v o l u t i o n b e c a m e p o p u l a r when C o o l e y a n d T u k e y [23 1
i n t r o d u c e d t he e f f i c i e n t Fast F o u r i e r T r a n s f o r m (FFT) a l g o r
i t h m to c o m p u t e D F T r e s u l t i n g in a s i g n i f i c a n t s a v i n q i n
c o m p u t a t i o n and p e r f o r m a n c e i m p r o v e m e n t o v e r the d i r e c t
method. The FFT u s e s the c y c l i c p ro p e r t y of the c o m p l e x e x
p o n e n t i a l f u n c t i o n to r e d u c e the n u m b e r of m u l t i p l i c a t i o n s .
-ing s p e e d s t i l l r e m a i n s p r o p o r t i o n a l to the (complex) m u l t i
p l i c a t i o n time. A t t e m p t s were m a d e to r e d u c e t h e m u l t i p l i
c a t i o n time, for e x a m p l e , Liu and Peled f 25) p r o p o s e d to us e
a b i t - s l x c i n q a l q o r i t h m a nd t a b l e l o o k - u p s c h e m e to r e p l a c e
the c o n v e n t i o n a l m u l t i p l i e r . Also, tn er e xs a d e f i n i t e
d r a w b a c k in the use of DP T for m a c h i n e i m p l e m e n t a t i o n , that
th er e xs f i n i t e p r e c x s i o n r e p r e s e n t a t i o n or the t r a n s c e n d e n
tal m u l t i p l i e r f u n c t x o n s . T h e s e a p p r o x i m a t i o n s c o n t r i b u t e to
the e r r o r no is e in t h e output.
An a l t e r n a t i v e t r a n s f o r m d o m a i n t e c h n i q u e w h i c h has a t
trac ted c o n s i d e r a o l e i n t e r e s t in last few y e a r s i s the u s e
or the N u m b e r T h e o r e t i c T r a n s f o r m (NTT). F e r m a t N u mb er
T r a n s f o r m s (FNT) a n d R ad er T r a n s f o r m s ( R T) , w hi ch a r e s p e
-c x f i -c N T T ' s hav e b ee n i m p l e m e n t e d [ 7 , 1U). A v e r y a t t r a c t i v e
m e t h o d of i m p l e m e n t a t i o n or N T T ' s for c o n v o l u t i o n is, h o w e v
er, o ver the r i a q s that are i s o m o r p h x c to d i r e c t sums of
G a l o x s F i e l d s £ 1 3 1. T h x s i m p l i e s the use of the R e s i d u e N u m
ber S y s t e m (RNS) , w h i c h i t s e l f is of p a r t i c u l a r i n t e r e s t in
l i q i t a l s i g n a l p r o c e s s i n q £ 3 1 1 o e c a u s e of the p a r a l l e l n a
ture or its a r i t h m e t i c . T h e RNS was e x t e n s i v e l y x n v e s t x q a t e d
by S z a b o and T a n a k a [ 3 ] in 1967 for use in the d e s i q n of a
q e n e r a l p u r p os e c o m p u t e r . R e s x d u e t e c h n i q u e s , h o w e ve r, d id
not r e c e i v e w i d e - s p r e a d a t t e n t i o n b e c a u s e tne x e r r i t e c o r e
raeraorxes u s e d at t h at t i m e w e re too e x p e n s i v e and bulk y to
With the c u r r e n t a d v a n c e s in s e m i - c o n d u c t o r m e m o r y
t e ch n ol oq y, the i m p l e m e n t a t i o n of NTT's for s i q n a l p r o c e s s
inq is a v i a b l e a l t e r n a t i v e to c o n v e n t i o n a l m e t h o d s I" 7
1-T h o u q h the t r a n s f o r m d o m a i n r e p r e s e n t a t i o n of NTT s e q u e n c e s
nas no known p r a c t i c a l i n t e r p r e t a t i o n , its i m p l e m e n t a t i o n
for c o n v o l u t i o n is m e a n i n q f u l . T h e only r e s t r i c t i o n to be
o b s e r v e d is that the d a t a p oi nt s are s m a l l e n o u q h (scaled)
so that the f i n a l r e s u l t d o e s not p r o d u c e a d a t a p o i n t
q r e a t e r than the r t n q modulus. Also, s i n c e the t r a n s f o r m is
d e f i n e d over a f i n i t e rinq, the r e s u l t s a r e exa ct . F ur th er,
in RNS a r i t h m e t i c i m p l e m e n t a t i o n , a m u l t i p l i c a t i o n can be
r e p l a c e d by a tao le l q o k u p o p e r a t i o n , a n d thus the t h r o u q h
-put r a t e can ue e x p e c t e d to be hiqh w i th r e l a t i v e l y low
n a r d w a r e cost. T n e n u m b e r of b i t s used f o r d at a r e p r e s e n t a
tion, nowever, s h o u l d b e sm al l so t h a t m e m o r i e s r e q u i r e d f or
l o o k - u p tables are c o m m e r c i a l l y a v a i l a b l e . J u l l i e n f 51 s u q
-q e s t e d a m e th o d to i m p l e m e n t m u l t i p l i c a t i o n s w h i c h r e s u l t s
m t r e m e n d o u s m e m o r y s a v i n q a nd r ed u c e s m e m o r y r e q u i r e m e n t
to a v i a b l e size, s t i l l us in q l o o k - u p tables.
T h e t r a n s f o r m d o m a i n t e c h n i q u e s a r e only a t t r a c t i v e
when o n e or t he f a s t a l q o r i t h m s are e m p l o y e d in t h e i r c o m p u
tation. F a s t F o u r i e r T r a n s f o r m type a l q o r i t n m s c a n be a p
p l i e d to c o m p u t e the NTT. T h e heart of s u c n f a s t a l q o r i t h m
is a c o m p u t a t i o n a l u n i t c a l l e d a " B u t t e r f l y " . One o r m ul t i
d i m e n s i o n a l b u t t e r r l i e s h a v e oeeu u se d to c o m p u t e the t r a n s
-i th m £ 4] -is or p a r t i c u l a r i n t e r e s t s i n c e it e l i m i n a t e s the
d a t a p r e - s h u f f l i n q . T h i s f u r t h e r r e q u i r e s unity t wi dd l e
f a c t o r s at the la st s t a q e of the b u t t e r r f l y c o a p u t a t i o n .
Hence, t he t r a n s f o r m of the c o e f f i c i e n t s can be m u l t i p l i e d
by the t r a n s f o r m of the d a t a p o i n t s at the l a s t s t a q e of
b u t t e r f l y c o m p u t a t i o n i n s t e a d o f the m u l t i p l i c a t i o n by tne
t w i d d l e factors. T h i s r e s u l t s in s a v i n q in the t o t a l n u m b e r
of c o m p u t a t i o n s .
for its p r a c t i c a l use, a n y RNS a r i t h m e t i c s t r u c t u r e r e
q u i r e s B i na ry to R e s i d u e (B/R) a nd R e s i d u e t o B i n a r y (R/B)
c o n v e r t e r units. S e v e r a l h a r d w a r e t e c h n i q u e s [ 3 ] a re a v a i l a
ble ror a B/R c o n v e r t e r . The s e p a r a t e n e e d of s u c h a B/R c a n
be a v o i d e d if the A /D c o n v e r t e r u s e d q i v e s the b i n a r y o u t p u t
w h i c h a l so is a residue. T h i s can be a c a s e w h e n the rinq
m o d u l u s is l a rq e r t h a n the p o s s i b l e m a x i m u m v a l u e of a n y
d a t a point. T h e t i n a l o u t p u t s o b t a i n e d f r o m a two or m o r e
moduli, however, h av e to ne c o m b i n e d to o b t a i n the result.
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 (CRT) is o n e of tne m e t h o d s
[3], but it s u f f e r s f r o m the d i s a d v a n t a q e that it needs a
mod M adder, w he re M is tne d y n a m i c ranqe. T h e o t h e r met ho d
is via the use of a M i x e d R a d i x C o n v e r s i o n (MRC) t e c h ni qu e .
The m u l t i p l i c a t i o n n e e d e d in t h is m e t h o d c an be i m p l e m e n t e d
usi nq l o o k - u p tables. T h i s m e t h o d has c o m p u t a t i o n a l a d v a n
-ta qe s o v e r the C R T w n e n f e w e r m o d u l i a r e used.
H a r d w a r e r e a l i z a t i o n s are n o r m a l l y fi xe d for a s p e c i f i c
-t e c h n i q u e s c a n be u s ed ia p r o c e s s i n q of i m a q e s of l a r q e r d i
m e n s i o n s £ 18 1 th an that of the b a s i c b l o c k size. S u c h a l q o r
-i th m s are a t t r a c t -i v e wh en w o r k i n q in a l i m i t e d m e m o r y s y s
tem, s u c h as t nat of a m i ni -c o m p u t e r .
A s p e c i a l p u r p o s e d i q i t a l s i q n a l p r o c e s s o r is a d e d i
c a t e d piece of h a r d w a r e whose f u n c t i o n i s t o p e r f o r m a s p e
c i f i c set of p r o c e s s i n q a l q o r i t h m s (in r e a l time) a s a s e l f
c o n t a i n e d s u b s y s t e m . O b v i o u s l y a l l the s i q n a l p r o c e s i n q a l
q o r i t h m s c an be i m p l e m e n t e d on a q e n e r a l p u r p o s e c o m p u t e r ,
h o w e v e r the speed of such i m p l e m e n t a t i o n s on q e n e r a l p u r p o s e
c o m p u t e r s a r e not p a r t i c u l a r l y a t t r a c t i v e . M a n y i n d u s t r i a l
n e e d s h a ve only o ne a p p l i c a t i o n in mind, f o r e x a m p l e , f a u l
ty pact d e t e c t i o n in an a s s e m b l y line. S e c o n d l y , m o s t q e n
eral p u r p o s e c o m p u t e r a r c h i t e c t u r e s c an n o t n o r m a l l y handl e
s i m u l t a n e o u s c o m p u t a t i o n s . A d e d i c a t e d pie ce of hardwa re ,
h ow ever, is d e s i q n e d to h a n d l e a la rq e n u m b e r of c o m p u t a
tions, a n d e m p l o y s a p a r a l l e l p r o c e s s i n q and p i p e l i n i n q to
a c h i e v e s p e e d s s e v e r a l o r d e c s of m a q n i t u d e f a st er t h a n q e n
e r a l p u r p o s e c o m p u t e r s .
T h i s r e s e a r c h is an e x t e n s i v e i n v e s t i q a t i o n
into the p r o c e s s o r a r c h i t e c t u r e of a F a s t 2 - d i a e n s i o n a l D i
S i g n a l s and S y s t e m s qrou p, Dept. of E l e c t r i c a l E n g i n e e r i n g
at the U n i v e r s i t y of W i n d s o r [ 29 1. A l t h o u g h d i f f e r e n t h a r d
ware s t r u c t u r e s for the r e a l i z a t i o n of F a s t F o u r i e r T r a n s
f o r m s have b e e n p r o p o s e d , p r o c e s s i n q i m a g e s a nd o t h e r i n h e
r e n t l y t w o - d i m e n s i o n a l s i q n a l s u s i n g a F a s t N u m b e r T h e o r e t i c
T r a n s f o r m (FNTT) with a t w o - d i m e n s i o n a l b u t t e r f l y s t r u c t u r e
is a r e l a t i v e l y r e c e n t method. T h e v a r i o u s c o m p o n e n t s wh ic h
m a k e up the c o m p l e t e p r o c e s s o r a r e e x a m i n e d in t h is thesis.
In t h e r e a l i z a t i o n of d i g i t a l s y s t e m s usi nq s p e c i a l
p u r p o s e h ar dware, the c o n c e p t s of p a r a l l e l i s m , m u l t i p l e x i n g ,
and p i p e l i n g are ox g r e a t i m p o rt a nc e in a c h i e v i n g a m a x i mu m
v a l u e of p e r f o r m a n c e - c o s t ra ti o for the p a r t i c u l a r a p p l i c a
t i o n b ei n g c o n s i d e r e d . T h e t h e o r e t i c a l c o n s i d e r a t i o n s u s e
ful with r e s p e c t to 'speed and c os t t r a d e - of fs ' a r e r e v i e w e d
in this work.
The m e m or y a r c h i t e c t u r e n e e d e d for i m p l e m e n t a t i o n of
t w o - d i m e n s i o n a l O r d e r e d - I n p u t - O r d e r e d - O u t p u t NTT a l g o r i t n m
is i n v e s t i g a t e d in l i g n t of the s p e e d / c o s t t r a d e - o f f . T h e
i m p l e m e n t a t i o n of s u c h a b u t t e r r l y is d e s c r i D e d in detail.
T h e u s e of t a b l e l o o k - u p lor m a t h e m a t i c a l o p e r a t i o n s , in
p a r t i c u l a r m u l t i p l i c a t i o n , by a s u b - m o d u l a r a p p r o a c h , is
in-ves tiqated.
S e v e r a l d e s i q u a s p e c t s u s e d in i m p l e m e n t a t i o n of the
p r o c e s s o r a re c o m p a r e d to e s t a b l i s h the e f f i c i e n c y of the
m e n s i o n a l b u t t e r f l y . S e ve r a l o t h e r c o m p a r i s o n s are made to
s u p p o r t the a r c h i t e c t u r e used in tne c o n v o l v e r . F o r i n s
tance, the use of the M i x e d R a d i x C o n v e r s i o n m e t h o d is j u st
i f i e d c o m p a r e d to the C h i n e s e R e m a i n d e r T h e o r e m in the i m
p l e m e n t a t i o n of R e s i d u e to B i n a r y c o n v e r t e r . T h e
i m p l e m e n t a t i o n of m u l t i p l i c a t i o n ny s u b - m o d u l a r l o o k - u p t a
ble and by the use or d i r e c t R O M m u l t i p l i e r s i s c o m p a r e d .
T i m i n q c o n s i d e r a t i o n s a re m a d e for s e r i a l s e q u e n t i a l
p r o c e s i n q (used in the c onvolver) and c a s c a d e p r o c e s s i n q ,
and s p e e d / c o s t (efficiency) c o n s i d e r a t i o n for t h e s e m e t h o d s
are i n v e s t i q a t e d £oj; v i d e o - r a t e p r o c e s s i n q spe ed. T w o
s t r u c t u r e s , namely, a t h r e e - m e m o r y b u f f e r s t r u c t u r e and u s e
of a 'complete' b u t t e r f l y s t r u c t u r e , h a v e been p r o p o s e d to
i m p r o v e the p r o c e s s i n q speed. T h e t i m i n q d i a q r a m s w i t h r e
s p e c t to r e q i s t e r c o n t e n t s in the b u t t e r f l y of the c o n v o l v e r
are presented.
S e v e r a l e x a m p l e s or imaqe f i l t e r i n q a r e p r e s e n t e d to
i l l u s t r a t e tne a p p l i c a t i o n of the proc es so r . T h e e x a m p l e s
are t a k e n from w e ll d e f i n e d imaqes. k s i m p l e and a p p r o x i m a t e
m e t h o d to o b t a i n t he c o e f f i c i e n t s of a t w o — a i m e s n s i o n a l f i
n i t e i m p u l s e r e s p o n s e f i l t e r i s d e s c r i b e d . S e v e r a l s t a n d a r d
f i l t e r s are used tor I m a q e S m o o t h e m n q , I m a q e E n h a n c e m e n t
and o t h e r f e a t u r e e x t r a c t i o n on imaqes. T he r e s u l t s o b
t a i n e d from three d i f f e r e n t m e t h o d s in s o f t w a r e , n a m e l y d i
T h e use of b l o c k - m o d e f i l t e r i n q is i n v e s t i q a t e d in f i l
t e r i n q of v e r y larqe s e q u e n c e s , in a l i m i t e d m a i n - m e m o r y
system. T h e c h o i c e of the b a s i c - b l o c k s i z e is a t r a d e - o f f
w i th speed. T h e o r e t i c a l c o m p a r i s o n s f o r t h i s t r a d e - o f f ar e
presented.
1.2 X H E S I S o r g a n i z a t i o n
C h a p t e r - 2 p r o v i d e s the t h e o r e t i c a l b a c k q r o u n d on the
a p p l i c a t i o n of F a s t N u mb e r T h e o r e t i c T r a n s f o r m t e c h n i q u e s in
d i q i t a l f i l t e r i n q of tvo d i m e n s i o n a l s e q u e n c e s . I t d e t a i l s
the m o d u l a r a r i t h m e t i c , a l q e b r a i c c o n s t r a i n t s to be o b s e r v e d
in the use of the NTT a nd the r e s t r i c t i o n s i m p o s e d f r o m
p r a c t i c a l poi nt of view. F u r t h e r it d e s c r i b e s the c o n c e p t s
of the 2-diraensional O I O O - N T T a l qo r i t h m , a n d the m e th od of
Mir ed Radix c o n v e r s i o n used in the r e s i d u e to b i n a r y c o n v e r
sion. T n e d e s i q n c o n s i d e r a t i o n s used in the c o n v o l v e r a r e
d e t a i l e d in t h is c h a p te r .
The i m p l e m e n t a t i o n of the t r a n s f o r m c o m p u t a t i o n a l e l e
ment, the n u t te rf ly , a nd the m u l t i p l i c a t i o n in the b u t t e r f l y
u si nq the s u b - m o d u l a r a p p r o ac h, are d e s c r i b e d in C h a p t e r - 3 .
A n u m b e r of c o m p a r i s o n s vita r e s p e c t to speed, c o s t and m e
m o r y s t o r a q e are i n c l u d e d in t hi s part to d e s c r i b e t h e p e r
f o r m a n c e of tne p r o c e s s o r . V a r i o u s t i m i n q d i a q r a m s are a l s o
i n c l u d e d in tnis part.
C h a p t e r - 4 d e a l s w i th the h a r d w a r e a n d the f u n c t i o n a l
The Hiqh S p e e d D e v i c e (HSD) i n t e r f a c e of the P i l t e r with the
m i n i - c o m p u t e r SEL and t h e c o n t r o l loqic are e x a m i n e d . T h e
d i s c u s s i o n on the s o f t w a r e of the HSD is i n c l u d e d in the a p
pendix. The s te p s for the use of the c o n v o l v e r are d e s c r i b e d
in this c ha pter.
In t he next part, C h a p t e r - 5 , we d e t a i l the r e s u l t s of
f i l t e r i n q by the use of s e v e r a l s t a n d a r d f i l t e r s on test
ia-aqes. T h e a p p l i c a t i o n s in mind were I m a q e S m o o t h e n i n q and
E dqe E n h a n c e m e n t . T h i s final p art c o n s i d e r s the p r o c e s s i n q
of la rq e a r r a y s (larqer than c a n be p r o c e s s e d in o n e block)
uy b l o c k - m o d e f i l t e r i n q . The time of p r o c e s s i n q , w h i c h d e
pends on bl oc k s i z e has been compared. C h a p t e r - 6 p r e s e n t s
D I G I T A L F I L T E R I N G USING F A S T N U M B E R T H E O R E T I C T R A N S F O R M T E C H N I Q U E S
2,1 I N T R O D U C T I O N
In d i g i t a l i m a q e p r o c e s s i n g , as well as in o t h e r areas,
it is d e s i r a b l e to f i l t e r a t w o - d i m e n s i o n a l d i s c r e t e s i g n a l
x(i#1) by c o n v o l v i n g that s i g n a l with the t w o - d i m e n s i o n a l
d i g i t a l pulse r e s p o n s e of a p p l i e d f i l t er h(i,j) p r o d u c i n g an
o u t pu t s i g n a l y ( i , j ) . T h e t w o - d i m e n s i o n a l c o n v o l u t i o n is d e
fined as
y (i,i) = x * n
= £ X x(k,l) .h(i-k,j-l) (2-1)
K90
i# 1= 0 , 1 , ... M-1
where t he s e q u e n c e s x, h a nd y a re a s s u m e d to have
s q u a r e s h a p e of d i m e n s i o n (NxN) , (LxL) a n d (iixa) r e s p e c t i v e
ly, M ^ N+L-1.
P r o c e s s i n g s i g n a l s w ith a d i g i t a l c o m p u t e r or with s p e
c ial purp os e d i g i t a l h a r d w a r e i n v o l v e s the i m p l e m e n t a t i o n of
c o m p u t a t i o n a l s c h e m e s on s e q u e n c e s of n umbers. For exam pl e,
Eqn.{2.1) c a n be i m p l e m e n t e d hy a c t u a l l y takinq the sum of
p r o d u c t s as d e f i n e d or by i n d i r e c t methods. T h e i n d i r e c t
t r a n s f o r m of the product. T he i n d i r e c t m e t h o d s a r e a t t r a c
tive, b e c a u s e v i t h v i a b l e r e s t r i c t i o n s on the l e n q t h of s e
quences, c o m p u t a t i o n a l l y e f f i c i e n t a l g o r i t h m s c an be d e v e l
o p e d which h a ve a d v a n t a q e s o v e r d i r ec t m e t h o d s in t er m s of
s p e e d and thus the c o s t of filtering. T h e m o s t c o m m o n t e c h
n i q u e to r e d u c e t he c o m p u t a t i o n a l cost o f c o n v o l u t i o n is by
the use of the D i s c r e t e F o u r i e r T r a n s f o r m c o m p u t e d via u se
of Fast F o u r i e r T r a n s f o r m (FFT) al qo r it hm .
It is i n t e r e s t i n g to note that the FFT has b ee n used to
c o m p u t e c o n v o l u t i o n s and many h a r d w a r e s t r u c t u r e s h av e been
i m p l e m e n t e d [ 4 , 2 4 ] wifh s l i q h t v a r i a t i o n s to the b a s i c a l
q o r i t h m s u g g e s t e d by C o o l e y and T u c k e y [ 23 1 . Each s t r u c t u r e
l o o k s at the h a r d w a r e / s p e e d t r a d e - o f f a s s o c i a t e d with both
the c o m p u t a t i o n a l e l e m e n t s a n d the s u p p o r t i n g s t r u c t u r e .
However, this p r o c e d u r e is t i m e - c o n s u m i n g on m i n i - c o m p u t e r s
even with m u l t i p l i c a t i o n n a r d w a r e i n s t a l l e d , due to the
l a r q e numoer of c o m p l e x m u l t i p l i c a t i o n s r e q u i re d. F ur th er
there is c o n s i d e r a b l e b u i l d - u p of r o u n d - o f f e r r o r b e c a u s e of
the fin ite p r e c i s i o n in r e p r e s e n t i n q r e al n u m b e r s on d xq i t a l
co mp u te rs . F i l t e r d e s i g n s usinq ROM o r i e n t e d HNS a r i t h m e t i c
u n i t s [ 1 1 ] a n d i m p l e m e n t a t i o n of the FFT vitn the use of R e
s i d u e N u mb er S y s t e m [ 3 1 1 have been s u q g e s t e d for i m p r o v e d
e ff i c i e n c y , s i n c e f o r c o n v o l u t i o n we a re o n l y i n t e r e s t e d in
the C y c l i c C o n v o l u t i o n P r o p e r t y (CCP) of the t r a n s f o r m , it
v o l ve d i n FFT t w i d d l e f a c t o r s h a ve been i n v e s t i g a t e d . H u m b e r
T h e o r e t i c T r a n s f o r m s ( N T T ) , w h i c h a r e d e f i n e d as part of
G e n e r a l i z e d D i s c r e t e F o u r i e r T r a n s f o r m s (GDFT) , use i n t e g e r
t wi d d l e f a c t o r s a nd h a v e g a i n e d c o n s i d e r a b l e i n t e r e s t for
s e v e r a l y e a r s a s a c l a s s of s i g n a l p r o c e s s i n g a l g o r i t h m s .
The h a r d w a r e o f t h e I m a g e c o n v o l v e r u s e s H u m b e r T h e o r e t i c
T r a n s f o r m f o r e m p l o y i n g the i n d i r e c t m e t h o d of f i l t e r i n g .
2.2
DEFINITION OF MIJHBBB TBBQBBTIC IBABSFQBH
N u m b e r T h e o r e t i c T r a n s f o r m s are d e f i n e d as p a r t of a
c l a s s of G e n e r a l i z e d D i s c e r e t e F o u r i e r T r a n s f o r m s and a r e
c o m p u t e d o v e r f i n i t e f i e l d s [13],
*k
r nk ) x e
n
M
-nk
(
2.
2)
M
w h e r e N is the s e q u e n c e l e n gt h and M r e p r e s e n t s the m o d u l u s
of the f i e l d a r i t h m e t i c ; the g e n e r a t o r 5 is a n Nth ro ot of
uni ty (S**N= 1; S**N1j^1 m o d M f or 1 £ N K N ) a n d N e xi st s. It
has b een s u g g e s t e d t h a t N T T * s be i m p l e m e n t e d in r i n g s w h i c h
w h er e the pi are p r i s e s and r r e p r e s e n t s the d e q r e e of t he
e x t e n s i o n fields. T h e r e s u l t s of the o p e r a t i o n c a n be r e c o v
e re d by either u si n q the C h i n e s e R e m a i n d e r T h e o r e m or a m i x
ed radix c o n v e r s i o n [3 1 a l q o r i t n m . T h i s a m o u n t s to impl
e-me nt in q t he NTT u s i n q the R e s i d u e Number S y s t e m (RNS) and
the i nh er en t p a r a l l e l i s m of RNS i m p l e m e n t a t i o n can be m ade
to a d v a n t a q e to o b t a i n f a s t e r speed of p r o c e s s i n q . We
rir st d i s c u s s s om e of the b a s i c c o n c e p t s of RNS from n u mb er
t h e or y r e l e v a n t to the NTT in the next s ec ti o n.
2.3 M O D U L A R A R I T H M E T I C
D E F I N I T I O N - 2.1: T w o i n t e q e r s a a nd b a r e s a i d to be c o n q r
-uent mod a if
a = b + k.H (2.4)
w h e r e k is s o m e i n t e q e r an d H is the modulus. T he b is r e s i
due of a mod M when
0 < b < a
and is w r i t t e n as
a = b (mod i1)
D E F I N I T I O N - 2 . 2 : All i n t e q e r s are c o n q r u e n t mod M to s om e
i n t e q e r in the f i n i t e se t ( 0, 1 , 2 , ... #M-1) and let the s e t
of e l e m e n t s be c o m b i n e d by two d i f f e r e n t o p e r a t i o n s '+ 1 a n d
both mod a . T h e n tnis s et is c a l l e d the rinq of i n t e q
D E F I N I T I O H - 2 . 3 : If in d cinq of i n t e q e r s m u l t i p l i c a t i v e
i n v e r s e s exi st for a ll n o n z e r o i nt e qe rs , t h is rinq is k n o w n
as a Fi eld . It c an he sh ow n that Zm is a fie ld if and onl y
if a is a prime. The se t of a ll i n v e r t i b l e e l e m e n t s of a
rinq is a qroup with respect to the o p e r a t i o n of m u l t i p l i c a
tion and is c a l l e d a " m u l t i p l i c a t i v e qroup".
The f o l l o w i n q 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 are d ef ined
in m o d u l a r a r i t h m e t i c .
1. Addit io n: Ex am pl e, 7 + 1 2 = 2 (mod 17)
2. N eq ation : Example, - 7 = 1 0 (mod 17)
3. S u b t r a c t i o n : Exa mp le , 7 - 1 2 = 7 + ( - 1 2 ) = 7 + 5=12 (mod 17)
4. M u l t i p l i c a t i o n ; Example, 7 x 12 = 1 6 (mod 17)
5. M u l t i p l i c a t i v e I n v er se : M u l t i p l i c a t i v e I n v e r s e of an
i n te q e r b rn Zm e x i s t s if a nd only if b a n d W are r e
l a t i v e l y prime. In tnat c a s e b is an i n t e q e r such
t h at bxb' -1 (mod M ) . It oay be ho wever n o t e d that
w h e n M is a n o n - p r i m e inteqer, no t a i l m e m b e r s of the
set nave m u l t i p l i c a t i v e i nverses.
-i
Example : 7 =5 (mod 17)
for 7xb-1 (mod 17)
3 1 =5 (mod 14) as 3x5 = 15=1 (mod 14)
b u t 2 1 (mod 14) d o e s n o t exist.
6. D ivison: x/y e x i s t s if and only if y has an i n v e r s e
a nd x/y is c o n t a i n e d in the rinq. In that c a s e x/y
= x. y~*.
D E F I N I T I O N - 2.4: It pi is a prime, the e le m e n t s
fO, 1, 2,.. - - pi-1) f o r m a field with a d d i t i o n a nd m u l t i p l i c a
tion m o d u lo pi. In any fin it e field the n u m b e r of e l e m e n t s
must be a power of a p ri me ( p i * * r i ) , where ri is a p o s i t i v e
i n te q e r and an e l e m e n t (primitive root) m us t exist, p o w e r s
of whi ch c a n q e n e r a t e all the n o n - z e r o e l e m e n t s of t h e
field. Suc h a n e l d is c o m m o n l y d e n o t e d by the s y m b o l
GF(pi**ri) a nd is c a l l e d a G a l o i s Field r 9 ].
D E F I N I T I O N - 2.5: T he R e s i d u e r e p r e s e n t a t i o n of an i n t e q e r in
the BNS ta kes the r o rm of an L - t u p l e
X = (x 1 , x 2 , ... . xl)
of the le ast p o s i t i v e r e s i d u e s with r e s p e c t to the set of
m odu li
( m 1 , m 2 , - . . . .ml)
Tne r a n q e of n u m b e r s w h i c h can be u n i q u e l y c o d e d in RNS are
L
0 s< X < rr mi = N
A s i q n e d i n t e q e r s y s t e m can be d e v e l o p e d a t t a c h i n q a p o s i
tive siqn to n u m b e r s in the r a n q e ( 0 , 1 , N/2-1) for H
e v e n or {0,1,.... (N- 1)/2} for d odd, and a n e q a t i v e siqn to
the n u mb er in the ranqe ( d / 2 , M / 2 + 1 ,... N-1} or
{ ( M + 1 ) / 2 , ... M - 1} r e s p e c t i v e l y . The o p e r a t i o n s in RNS c a n
be c a r r i e d i n d e p e n d e n t l y for e a c u of the moduli. T he c o r r e c t
a n s w e r s would be o b t a i n e d r e q a r d l e s s of i n t e r m e d i a t e o v e r
flows of a n a r i t h m e t i c c o m p u t a t i o n if t n e r esu lt is w i t n i n
As m e n t i o n e d in i n t r o d u c t i o n , f o r the e x i s t e n c e of
t r a n s f o r m s w it h the DPT s t r u c t u r e g i v e n in Sqn. (2-2) and
h a v i n g the C y c l i c C o n v o l u t i o n P r o p e r t y ( C C P ) , it is n e c e s
s ar y that an i n t e g e r e x i s t that is an Nth roo t of unity. He
will c o n s i d e r t hi s p r o b l e m usinq m o du l a r a r i t h m e t i c .
F i r s t E u l e r ' s f u n c t i o n a) {M) is d e f i n e d as the n u m be r of
i n t e q e r s in 2m t a a t a r e r e l a t i v e l y pr im e to ti. O b v i o u s l y
then for M a prime n u m b e r <D(H)=M-1. If M is a c o m p o s i t e n u m
ber and its prime f a c t o r e d form is d e n o t e d by
r1 r2 rl
tl=(p1) • (P2) (pi)
. i
K
then the q e n e r a l e x p r e s s i o n for i is [9 1
u) (M) =d (1-1/p D . ( 1 - 1 / p 2 ) ----(1- 1/pl) .
= fl (pi-1) (2.5)
tM
I H E O a E N - 2 . 1: E uler's t h eo r e m s t a t e s that for e v e r y S prime
to M
(M)
6 =1 (mod M)
For M p ri me this r e d u c e s to F e r m a t ' s theorem.
T H E O Q E M - 2 . 2: F e r m a t ' s t he orem s t a t e s that for M a p ri m e
number,
( M - U
w hi c h h o l d s for al l n o n z e r o e l e m e n t s of Zm s i n c e they a re
all r e l a t i v e l y prime to H if M i s a prime.
T h e r e a re c e r t a i n r o o t s of unity t h at a re of p a r t i c u l a r
i n t e r e s t . I f N is t he l e a s t p o s i t i v e i n t e q e r such that
N N 1
& =1 (mod N) ; 3 £1 (mod d); 1 ^ N 1 < N (2.7)
then o is s a i d to oe a root of unity of or de r N, or & is a
p r i m i t i v e Nth r o o t of unity.
It t h e o r d e r of & is e q u a l to u»(M) , t h e n & is c a l l e d a
p r i m i t i v e root. If M is a prime and 8 i s a p r i m i t i v e root,
tne set of i n t e q e r s
k
X = {& (mod a ) , k = 0 , 1 , 2 , . . . , a - 2 ] ( 2 . 8 )
is the t ot a l set or n o n z e r o e l e m e n t s in Zm, and all n o n z e ro
e l e m e n t s in Zm c an i»e q e n e r a t e d by p o w e r s of t h e p r i m i t i v e
r o o t. Tn is , t h u s c n a r a c t e r i z e s the e n t i r e field. T he n o n z e r o
c l a s s e s of Zm f or m a c y c l i c m u l t i p l i c a t i v e q r o u p o f o rd e r
M-1 {1,2, a-1) , with m u l t i p l i c a t i o n m o d u l o M, i s o m o r p h i c
to the a u d i t i o n q r o u p {0,1,... H-2) wrth a d d x t i o n modu lo
a -
1
.E ul er 's t h e o r e m i m p l i e s that if & is of o rd e r N then N
mu st d x v i d e tf(M), d e n o t e d by N|u)(M). If a is a pr im e it c a n
be sho wn that r o o t s of o r d e r N exist if a nd o nly if N| (M-1)
and the ro ot s are q i v e n ny
w h e r e So d e n o t e s a p r i m i t i v e root, M o r e g e n e r a l l y if & is a
r oot of o r d e r N then
& ** k is o f or d er N/k if k|H
&***. is of ord er N if N and k are
r e l a t i v e l y prime.
T h is i m p l i e s that the n u m b e r of r o o t s of o r d e r N is
qi ve n by d)(N) and, ther ef or e , the n u m b e r of p r i m i t i v e ro ot s
is d>(d(M)). T h e s e r e l a t i o n s a ll o w one to c a l c u l a t e all of
the coots of all p o s s i b l e o r d e rs f rom one p r i m i t i v e root.
Example:
L e t d=7, Z m = { 0 , 1 ,2 , 3, 4, 5,6:• +
j>(U =1 i) (2) = 1 <h (3) =2
0)(4)=2 a) (5) =4 cfi(6)=2
1) (7) =6
C o n s i d e r r a i s i n g e a c n e l e m e n t of Z7 to p o w e r s f r om 1 to 6
(mod 7) , T ab- (2. 1) .
T a b l e - (2.1)
0 1 2 3 4 6 6
1 I 1 1 1 1 1 1 1
2 I 1 2 4 1 2 4 1
3 1 1 3 2 6 4 5 1
4 I 1 4 2 1 4 2 1
5 I 1 5 4 6 2 3 1
6 I 1 6 1 6 1 6 1
T a b l e - (2.2) B o o t s of o r d e r N
N r o o t s of order N
1 1
2 6
3 2,4
6 3,5
O n l y those N t ha t d i v i d e <i (tf) - S (7) =6 h a v e ro ot s that b elo nq
to them. Tne nurnoer of roots is qiv en by a) (N) and the n u m
ber of p r i m i t i v e r o o t s is 3(td{tt))=2 a n d t h e y a r e 3 and 5.
Note that both of the p r i m i t i v e r o o t s q e n e r a t e all the n o n
z e r o elements.
o> (H) - 1
For a n o n p r i m e M, 6 nas an i n v e r s e qiven by & if 5 and
H a re r e l a t i v e l y prime. It c a n be noted t na t for N a c o m p o
s i t e r a t h e r t han a prim e number, Zm is not a field s i n c e a ll
e l e m e n t s w i l l not h a v e inverses. There is no p r i m i t i v e root
that will q e n e r a t e the e n t i r e rinq, o n l y s u b s e t s with i(i!)
elements. Let a n a v e tne f o l l o w i n q u n i q ue pr im e f a c t o r i z a
tion.
r 1 r2 rl
d = IP 1) - (p2) (pi)
Hhen the a r i t h m e t i c has to be p e r f o r m e d mod M, it can
be p e r f o r m e d m o d u l o e a c n prime p o w e r (pi)**ri s e p a r a t e l y £9 1
a nd the final r e s u l t mod a can be o b t a i n e d u s i n q the C h i n e s e
B e m a i n d e r t h e o r e m £ 3 1. When the a r i t h m e t i c mod (pi**ri) is
Now we r e t u r n to the d i s c u s s i o n on the d e s i q n of the
NTT processor. He n o t i c e t he f o l l o v i n q r e q u i r e m e n t s for t h e
CCP to e x i s t and t h e NTT to be d e f i n e d o v er the finite
field.
T H E O R E M - 2 . 3 : A l e n q t h N t r a n s f o r m h a v i n g the DFT s t r u c t u r e
will i m p l e me n t c y c l i c c o n v o l u t i o n if a n d o n l y if th er e e x
i sts an inverse of N and an e l e m e n t S, a root of unity of
o r d e r N , i . e . , N is t he l e a s t p o s i t i v e i n t e q e r such that
&**N =1
T h i s is a very q e n e c a l r e s u l t a p p l y i n q to both rin qs
a nd rields th at are f i n i t e or i n f i n i t e a nd it h as been d e
v e l o p e d from a v a r i e t y of p o i n t s of view [25 1. For M a c o m
p o s it e n u mb er as r e p r e s e n t e d in Eqn. (2-5) , we can o b t a i n t he
r e s u l t s of o p e a t i o n mod rt by c o m b i n i n q the r e s u l t s o b t a i n e d
from the o p e r a t i o n m c d u l o e a ch ( p i * * r i ) .
T her ef or e, the l e n qt h N n u m b e r t h e o r e t i c t r a n s f o r m h a v
ing the CCP in Zm must a l s o have the C C P in 2fpi**ri} for
i= 1,2,....l. T n i s r e q u i r e s that & (mod pi**ri) be an i nt eq e r
of order N a n d must e x i s t in Z ( pi* *r i) , i . e . , N is the l e a s t
p o s i t i v e i n t e q e r s u c h t h a t
& * * N =1 (mod p i * * r i ) , i = 1 , 2 , . . . . l .
F u r t h e r m o r e , s i n c e the i n v e r s e t r a n s f o r m r e q u i r e s N ; the
i n v e r s e of N s h o u l d e x i s t in Z { p i * * r i } , or, N s h o u l d be r e
l a t i v e l y pri me to M. Now we fiad that by E u l e r * s t he orem
N| a) (pi**ri) , i = 1 , 2 , ... 1.
S in c e N is r e l a t i v e l y pr im e to a (or its factors)
N| (pi-1) i = 1 , 2 , ... 1.
N|q c d ( p 1 - 1 , p 2 — 1, . . . . pl-1)
We d e f i n e 0(M) as the g r e a t e s t c o m m o n d i v i s o r (qcd) of the
( P i - D
0 (H) =qcd ( p1-1,p2- 1, .. ..pl-1)
ther efore, N|0(M)
T h i s q i v e s us
T H E O R E M -2.4: A l e n q t h N t r a n s f o r m h a vi ng the DPT s t r u c t u r e
will i m p l e m e n t c y c l i c c o n v o l u t i o n mod M if a nd o n l y if
NIO (M)
and this e s t a b l i s n e s the ma xi mu m t r a n s f o r m l e n q t h in Zm as
N m a x = 0 (M)
T h i s is a v e r y i m p o r t a n t t h e o r e m that s t a t e s e x a c t l y what
the pos si bl e t r a n s f o r m l e n q t h s for a q iv e n m o d u l u s are.
2.3.1 Afl E x a m p l e of c o n v o l u t i o n using NTT w h e n M is a
p r i m e
C o n s i d e r two s e q u e n c e s
x= (2,-2, 1,0)
h — (1,2,0,0)
wh os e c o n v o l u t i o n is desired. F r o m o v e r f l o w c o n s i d e r a t i o n ,
it is s u f f i c i e n t ir we d e f i n e the t r a n s f o r m s o v er GF(17)
tt= 1 7 N=4
Now si nc e M=17, the i n t e q e r 2 is of o r d e r 8
or,
1 1 1 1
1 4 4* *2 4**3
T = 1 4**2 4**4 4**6
1 4**3 4**6 4**9
~1 1 1 1
1 4 16 13
T = 1 1b 1 1b (mod 17)
1 13 16 4
=-4 (mod 17) = 13 (mod 17) and
T r a n s f o r m a t i o n H a t r i x is
1 1 1 1
1 13 16 4
= 13 1 16 1 16
1 . 4 16 13
The T r a n s f o r m s of x and h are q iv e n by
1 1 1 1 2
1 4 16 13 15
= 1 16 1 16 1
_1 13 16 4 _ 0
= t 19 10, 3, 9 1 (mod 17]
s i m i l a r l y H = [ 3 , 9 , 1 6 , 1 0 1
and thus Y = X.H
= T 3 , 5 , 1 2 , 5 1 (mod 17)
T a k i n q the i n ve r s e t r a n s f o r m of y,
y = (2,2,14,2) (mod 17)
A c c ording to our assumption, i nt eg e rs are s u p p o s e d to lie
b e tw e en -8 a nd 8- T he re fo re
y = (2,2,-1,2)
which is the c o r r e c t answer.
2.4 P R A C T I CAL C O N S I D E R A T I ONS IN C H O O SING & , N AND M FOB AN
NTT
Al th o ug h the cla ss of all p o s s i b l e number theoretic
tra ns forms s ee m s very l ar ge at fi rs t c o n s i d e r a t i o n (in fact,
i n f i n i t e ) , c l o s e r e x a m i n a t i o n sh o ws that very few s ee m to be
a t t r a c t i v e for use in s i gn al processing. Agarwa l and Burrus
f 6 ] s um me r is e the c r i t e r i a which would make a p ar ti cu la r NTT
to be a t tr a c t i v e in c o m p a r i s o n to other i m p l e m e n t a t i o n s of
convolution. T h ey list that for NTT to ne c o m p u t a t i o n a l l y
e ff icient three r e g u i r e m e n t s are:
1. (a) N s h o u l d be highly c o m p o s i t e (preferably a power
of 2) for a fast tF T-type a l g o r i t h m to exist and
(b) N s n o u l d be large e n ou gh for pract ic al seg ue n ce
lenths
2. since c om p l e x m u l t i p l i c a t i o n s take most of the c o m p u
t a ti on time in c a l c u l a t i n g the FFT, it is i mp or ta nt
that m u l t i p l i c a t i o n by powers of S be a s i m pl e o p e r a
tion. For m a c h i n e imp le me ntation, this is p o s si bl e
if the p owers of & have b i n ar y r e p r e s e n t a t i o n s with
very few bits; p r e fe r ab ly also a power of two, where
i m p l e m e n t a t i o n s , H should have a b inary r e p r e s e n t a
tion with a very few bits and s h o u l d be la rge e n o uq h
to prevent ove rflow.
U n f o r t u n a t e l y the c o n d i t i o n s gi ve n by above t h e o r e m s in
s e c - (2.3) do not q iv e a s y s t e m a t i c way of d e t e r m i n i n g the
"best" cnoices. U s u a l l y M nas to be s e l e c t e d f ir st and N
and & are d e te r m i n e d s ui tably. When the m o d u l u s fl is cho sen
to be a Ferm at Num be r as
t
2
M = Ft =2 + 1 (2. 10)
b
= 2 + 1 , b = 2 * * t
then a p r om is i nq c l a s s of NTT's can be o b t a i n e d £8]. Such
t r a n s f o r m s a r e c a l l e d F e rm at N u m b e r T r a n s f o r m (FNT)- A s p e
cial c l a s s of such t r a n s f o r m is when the value of S is c h o
sen t>=SQHI (2.) . T h e s e t r a n s f o r m s are d e s c r i b e d by Ba de r and
are known as Rader T r a n s f o r m s £9],
2.5 DESIGN C O N S I D E R A T I O N FOR NTJ U S E D IB C O N V O L V E R
As we m e n t i o n e d in section-(2-h) , i t is u s u a l l y but not
a l w a y s the c a se that a value of M is c h o s e n first and s u i t a
ble t r a n s fo rm l e n q t h N a n d the g e n e r a t o r & is d e te rm i ne d. We
