A c o m p u l a t i o n p r o g r a m f o r n u m e r i c a l f i l t e r i n g 

M . A T T O L I N I ( * ) - L . D A L L ' O L I O ( * * ) - V . G U E R R I E R I ( * * ) - P . R A N DI ( * ) 

Ricevuto il 9 Aprile 1968 

RIASSUNTO. — Viene descritto u n p r o g r a m m a , scritto in linguaggio 
F O R T R A N IV, per il filtraggio numerico di serie t e m p o r a l i . Tale pro-
g r a m m a è s t a t o p r o v a t o con u n calcolatore del tipo IBM 7094/7040 DCS. 

L ' o p e r a z i o n e di filtraggio numerico p e r m e t t e di e s t r a r r e o sopprimere, 
in u n a serie t e m p o r a l e , onde di f r e q u e n z a prefissate i n d i p e n d e n t e m e n t e da 
variazioni di fase ed ampiezza con cui esse si possono presentare. 

SUMMARY. — A c o m p u t a t i o n p r o g r a m , w r i t t e n in F O R T R A N IV 
language, for n u m e r i c a l filtering of time series is described. I t was tested 
with an I B M 7094/7040 DCS c o m p u t e r . 

The numerical filtering operation allows to e x t r a c t or suppress in a 
time series prefixed f r e q u e n c y waves i n d i p e n d e n t l y by t h e change of phases 
and wideness w i t h which t h e y can a p p e a r . 

M E T H O D . 

I t h a s b e e n d e s c r i b e d , i n a p r e v i o u s p a p e r ('), a filtering o p e r a t i o n 
of t i m e series o n e q u i s p a c e d d a t a , c o r r e s p o n d i n g t o a p r e f i x e d r e s p o n s e . 

I n t h e f o l l o w i n g p a p e r is d e s c r i b e d a c o m p u t a t i o n p r o g r a m b a s e d 
o n t h e m e t h o d d e v e l o p e d i n ('). 

A n u m e r i c a l filtering o p e r a t i o n o n a n e q u i s p a c e d d a t a series 
ni(i = 1,2, ) is o b t a i n e d b y c o m p u t i n g a n e w s e r i e s : 

+s 
n'k = Sj aj nk+i , k^r + 1 , [1] 

— r 

(*) I s t i t u t o di Fisica Università di Bologna. Gruppo I t a l i a n o di Fisica 
Cosmica. Consiglio Nazionale delle Ricerche. 

(**) Centro di Calcolo di Bologna. Comitato Nazionale per l ' E n e r g i a 
Nucleare. 



2 0 K M . A T T O L I N I - L . D A L I . ' O L I O - V . G U E R R I E R T - P . R A N D I 

w h e r e t h e (r + s + 1) q u a n t i t i e s a j a r e called coefficients or w e i g h t s 
of t h e n u m e r i c a l filter, as d e s c r i b e d in ('). T h e series n ' k c o n t a i n s t h e 
a l l o w e d f r e q u e n c i e s c o m p u t e d b y m e a n s of t h e used filter. 

F o r o b t a i n i n g a new series n " * w h i c h does n o t c o n t a i n t h e s a m e 
f r e q u e n c y b a n d s , one m u s t r e v e r s e t h e coefficients sign of t h e filter, 
e x c e p t t h e c e n t r a l one w h i c h is r e p l a c e d w i t h t h e d i f f e r e n c e b e t w e e n 
n u m b e r 1 a n d t h e c e n t r a l coefficient, a n d t h e so o b t a i n e d filter 
m u s t b e u s e d on t h e original d a t a series. 

T h e t w o o p e r a t i o n s , n u m e r i c a l f i l t e r i n g a n d t h e d i f f e r e n c e f r o m 
t h e original series, a r e e q u i v a l e n t t o t h e r e s u l t i n g series n " k — v ' k . 

T h e choice of t h e coefficients of t h e filter is s t r i c l y c o n n e c t e d 
t o t h e k n o w l e d g e of t h e p h y s i c a l p r o b l e m a n d i t d e p e n d s on t h e 
r e s t r i c t i o n s i m p o s e d 011 t h e f r e q u e n c y b a n d t o s e a r c h a n d also on t h e 
i n d e p e n d e n c e of t h e r e s u l t s . 

D E S C R I P T I O N O F T H E P R O G R A M . 

T h e p r o g r a m h a s b e e n c a r r i e d o u t a t " C e n t r o di Calcolo del 
ON E N " in B o l o g n a u s i n g t h e F O R T R A N IV l a n g u a g e , v e r s i o n 13, 
u n d e r t h e I B S Y S 7094/7040 DCOS m o n i t o r c o n t r o l . 

T h e d a t a set t h a t c a n b e e l a b o r a t e d is p r a c t i c a l l y u n l i m i t e d ; 
t h e m a x i m u m allowed set of coefficients of t h e filter is 99. T h i s 
d a t a set is e x t r e m e l y s u f f i c i e n t f o r m a n y p r o b l e m s b u t i t c a n also b e 
i n c r e a s e d b y m o d i f y n g t h e a p p r o p r i a t e s t a t e m e n t s . 

S o m e t i m e s i t m a y b e n e c e s s a r y t o c o m p u t e series of filtering 
o p e r a t i o n s on t h e s a m e d a t a , a n d , i n t h i s case, i t will b e s u f f i c i e n t 
t o follows t h e s u g g e s t i o n s d e s c r i b e d a f t e r f o r o b t a i n i n g series of com-
p u t a t i o n s a t a l o w e s t t i m e . 

I t h a s also f o r e s e e n t h e e l a b o r a t i o n of s e v e r a l d a t a series w i t h 
t h e s a m e coefficients or w i t h d i f f e r e n t coefficients f o r g a i n i n g t h e set-
u p t i m e , i n t h e c o m p u t e r , of t h e r u n n i n g p r o g r a m . 

B e s i d e s , i t h a s b e e n c o n s i d e r e d t h e n e c e s s i t y of g r o u p p i n g t h e 
i n i t i a l of filtered series d a t a b e f o r e a n u m e r i c a l filtering: i t allows t o 
r e d u c e t h a t r a n d o m noise t h a t s h o u l d give u o i n f o r m a t i o n a b o u t 
t h e t e s t i n g series. 

T h e p r o b l e m t o t r e a t will s u g g e s t , t i m e a f t e r t i m e , t h e r i g h t choice 
f o r g r o u p p i n g . 

T h e p r o b l e m c o n s i s t s i n k e e p i n g t h e t i m e p o s i t i o n of t h e o r i g i n a l 
d a t a , n a t u r a l l y p r o v i d i n g t h a t t h e n u m b e r of t h e coefficients (r -f s + 1) 



A C O M P U T A T I O N P R O G R A M F O R N U M E R I C A L F I L T E R I N G 21 5 

will b e o d d . T h e c o m p u t e d coefficient n'k is set in t h e m i d d l e of t h e 
i n t e r v a l only if t h e c o n d i t i o n r = s is satisfied. T h e c o n d i t i o n s r = s 
m u s t b e satisfied in o r d e r t o use t h i s p r o g r a m c o r r e c t l y . 

A f u r t h e r e x p e d i e n t , t o a v o i d over-excess of figures f o r s u b s e q u e n t 
e l a b o r a t i o n s a n d w i t h no d a m a g e t o t h e clearness of t h e p r o b l e m a n d 
to t h e a m o u n t of t h e g a i n a b l e i n f o r m a t i o n , consists i n s e l e c t i n g t h e 
s i g n i f i c a n t figures, in t h e r e s u l t s of e v e r y o p e r a t i o n s , f o r t h e considered 
p r o b l e m . 

B e s i d e s , t h i s e x p e d i e n t allows t o rescale t h e r e s u l t s as desired so 
as t o b e seen i n a n a p p r o p r i a t e scale. 

F u r t h e r e x p e d i e n t s h a v e b e e n described below. 

O R D E R O F D A T A C A R D S D E C K S E T - X J P . 

All c a r d s m u s t b e i n c l u d e d in t h e o r d e r s h o w n b e l o w . 

[A] = t i m e series d a t a p a r a m e t e r c a r d . 

[B] = v a r i a b l e f o r m a t c a r d f o r t i m e series i n p u t d a t a . 

[C] = t i m e series d a t a . 

[D] = E N D p u n c h e d in col. 1-3. 

[E] = c o m m e n t c a r d . 

[F] = p a r a m e t e r c a r d f o r a filter o p e r a t i o n . 

[G] = v a r i a b l e f o r m a t c a r d f o r o u t p u t d a t a t o b e p u t on logical 
t a p e N T P O . 

[II] = v a r i a b l e f o r m a t c a r d f o r o u t p u t d a t a t o b e p r i n t e d . 

[I] = filter coefficients. 

R e p e a t [E] t h r o u g h [I] as desired. 

[J] = b l a n k c a r d . 

[K] = b l a n k c a r d . 

R e p e a t [A] t h r o u g h [K] as desired. 

[L] = b l a n k c a r d (to n o t i f y t h e p r o g r a m t h a t t h e e n t i r e j o b s is 
c o m p l e t e d a n d c o n t r o l m u s t b e r e t u r n e d t o t h e M o n i t o r ) . 



20K 
M. ATTOLINI - L. DALI.'OLIO - V. G U E R R I E R T - P . R A N D I 

E X A M P L E O F D A T A D E C K S E T - U P . 

1. R e p e a t e d as desired (see page 3) 
2. Repeated as desired (see page 3) 

C A R D P R E P A R A T I O N . 

Card 
columns 

Corre-
s p o n d i n g 
variable 

[A] card 

1 - 2 N D S 

3 - 4 

5 - 6 

7 - 8 

N T P I 

K E 

I T 

D E S C R I P T I O N 

F O R M A T ( 4 1 2 ) 

— n u m b e r of t i m e series d a t a c o n t a i n e d i n e a c h 
i n p u t c a r d . 
—- l o g i c a l t a p e n u m b e r of i n p u t (5 if i n p u t is f r o m 
p u n c h e d c a r d s ) . 
— n u m b e r of d e c i m a l figures if c o n v e r s i o n is u s e d (*) 
— = 0 if i n p u t c o n v e r s i o n is r e q u i r e d (*) 

^ » » » » n o t r e q u i r e d (*). 

(*) One m u s t use conversion wlien t i m e series i n p u t d a t a are p u n c h e d 
as integer c o n s t a n t s h a v i n g t h e sign (hole zone) over t h e less significant 
figure of t h e n u m b e r . I n this case each field of w i d t h w, c o n t a i n i n g t h e 
n u m b e r , is considered like it were f o r m e d of two sequential fields: t h e 1st of 
i n t e g e r - t y p e of wliidth w — I, t h e 2nd of a l p h a m e r i c - t y p e of w i d t h 1. 
i.e.: if n u m b e r 12.125 is p u n c h e d according to t h e f o r m a t specification 
P 7 . 3 no conversion is required, being a F o r t r a n n u m b e r (in this case 
IT = 0). The c h a r a c t e r string 12121^, (where N means holes 5 a n d 11, 
m i n u s sign overpunched) columns of t h e card p u n c h e d in a field of w i d t h 
w = 5 for having t h e same value as t h e above one, m u s t be read according 
t o t h e f o r m a t specification (14, A l ) and must be KE = 3 (decimal figures 
of t h e number) a n d IT ^ 0. 



A C O M P U T A T I O N P R O G R A M F O R N U M E R I C A L F I L T E R I N G 21 5 

C a r d 
' c o l u m n s 

C o r r e -
s p o n d i n g D E S C R I P T I O N 
v a r i a b l e 

[B] card FORMAT (13A6,A2) 

1-80 F R 1 — f o r m a t of t i m e series i n p u t d a t a m u s t b e like 
(A3, s p e c i f i c a t i o n s f o r i n p u t d a t a ) . C o l u m n s 1-3 
c a n n o t b e u s e d . T h i s r e s t r i c t i o n is u s e d t o allow 
a n i n d e f i n i t e n u m b e r of t i m e series i n p u t d a t a c a r d s , 
t h e e n d of w h i c h is r e a c h e d w h e n t h e p r o g r a m 
r e a d s t h e [D] c a r d . 

[G] card F R 1 f o r m a t (see [B] c a r d ) 
4-n E L 
(n < 80) ( K A , — t i m e series i n p u t d a t a (those d a t a will b e t h e 

K B ) 1 s t file of a t a p e w h e r e logical n u m b e r is 3. T h i s 
file is a d d r e s s e d b y m e a n s t h e v a l u e 1 a s s i g n e d t o t h e 
N F E L v a r i a b l e ) . T h e i n i t i a l t i m e series d a t a a n d 
successive filtered ones a r e r e c o r d e d on logical t a p e 
3 t o f o r m a s e q u e n c e of files. E a c h file c o n t a i n s a 
s e q u e n c e of r e c o r d e d d a t a . A s d e s i r e d , t o filter 
d a t a r e c o r d e d i n t o o n e of t h o s e files, one m u s t 
assign t o t h e N F E L p a r a m e t e r t h e v a l u e of t h e file 
a d d r e s s (its p h y s i c a l p o s i t i o n on t h e reel) a s s u m e d 
p u r i n g t h e file r e c o r d i n g . 

[/)] card FORMAT ( A 3 ) 

1 - 3 K E N T ) — t h e w o r d E N D must b e p u n c h e d in t h e cor-
r e s p o n d i n g c o l u m n s . 

T h i s c a r d n o t i f y t h e p r o g r a m t h a t all t h e c a r d s 
c o n t a i n i n g t i m e series i n p u t d a t a h a v e been r e a d . 

[£] card FORMAT (13A6,A2) 
1-80 F C O M — t h i s is a c o m m e n t c a r d (i.e. i t m a y c o n t a i n s a 

c o m m e n t a b o u t t h e n u m e r i c a l filter o p e r a t i o n ) . 

[i1] card FORMAT ( 9 1 2 , E l l . 6 ) 
1-2 N F E L — specifies t h e a d d r e s s of t h e file of logical t a p e 3 

c o n t a i n i n g t h e i n p u t d a t a f o r t h e filter o p e r a t i o n 
( N F E L = 1 a d d r e s s e s i n i t i a l i n p u t d a t a ) (see 
[C] c a r d ) . 

3-4 N — n u m b e r of filter coefficients. 
5-0 N D P — n u m b e r of d a t a p e r r e c o r d t o b e p u t on logicel 

t a p e N T P O . 
(Must a l w a y s b e N D P > 0). 



20K M. A T T O L I N I - L . D A L I . ' O L I O - V . G U E R R I E R T - P . R A N D I 

C a r d 
c o l u m n s 

[jP] car 
7 - 8 

9 - 1 0 

1 1 - 1 2 

1 3 - 1 4 

1 5 - 1 6 

1 7 - 1 8 

1 9 - 3 2 

C o r r e -
s p o n d i n g 
v a r i a b l e 

d (cont.) 
N E 

N T P O 

M S T A 

M P E E 

I C 

K A 2 
A l 

D E S C R I P T I O N 

— n u m b e r of i n p u t d a t a t o b e g r o u p p e d b e f o r e 
e x e c u t i n g filter o p e r a t i o n . 
— logical t a p e n u m b e r on w h i c h (if M P E E ^ 0) 
o u t p u t d a t a a r e p u t (i.e. f o r p u n c h i n g off-line). 
— 1 t o p r i n t o u t p u t 

0 n o t t o p r i n t o u t p u t 
— 1 t o p u t o u t p u t on logical t a p e N T P O 

0 n o t t o p u t o u t p u t on logical t a p e N T P O 
— 1 if o u t p u t c o n v e r s i o n f o r r e c o r d i n g on logical 
t a p e N T P O is r e q u i r e d 

0 if o u t p u t c o n v e r s i o n is n o t r e q u i r e d . 
— n u m b e r of figures f o r o u t p u t d a t a . 
— rescale f a c t o r . 

T h e rescale f a c t o r ( A l ) allows t h e p r o g r a m t h e 
rescaling of t h e o b t a i n e d r e s u l t s . K A 2 allows t h e 
user t o fix t h e n u m b e r of o u t p u t figures of t h e 
rescaled r e s u l t s . 
i.e.: if t h e filter o p e r a t i o n h a s given t h e n u m b e r 
23456 a n d u s e r desires t o o u t p u t o n l y t h e n u m b e r 34 
(2nd a n d 3 r d figures of t h a t r e s u l t ) h e m u s t fix 
K A 2 = 2 a n d A l = 0.01. 

[G] card F O R M A T ( 1 3 A 6 , A 2 ) 

1 - 8 0 F E 2 — f o r m a t of o u t p u t d a t a t o b e p u t on logical t a p e 
N T P O (see [ I f ] c a r d ) if t h e s e o u t p u t a r e n o t desired 
( M P E E = 0 in [ F ] c a r d ) t h i s c a r d c a n also b e a 
b l a n k c a r d . 

if M P E E ^ 0 t h e f o r m a t specified must b e l i k e 
(13, s p e c i f i c a t i o n s f o r o u t p u t d a t a ) . C o l u m n s 1 - 3 
of o u t p u t r e c o r d s t o b e p u t on logical t a p e N T P O 
c o n t a i n a c a r d s e q u e n c e n u m b e r . « s p e c i f i c a t i o n f o r 
o u t p u t d a t a » m u s t b e like , I w , AL, 
w h e r e w = KA2-1 if I C = 1, or like I w 
w h e r e w = K A 2 if I C = 0 f o r each o u t p u t d a t a . 

[H] card FORMAT (13A6,A2) 

— f o r m a t of o u t p u t d a t a t o b e p r i n t e d . If t h i s 
o u t p u t is n o t desired (MSTA = 0 in [H] c a r d ) t h i s 
c a r d c a n also b e a b l a n k c a r d . 



A COMPUTATION PROGRAM FOR NUMERICAL F I L T E R I N G 21 5 

D E S C R I P T I O N 

If M S T A # 0 t h e f o r m a t specified must b e like 
(15, specifications f o r o u t p u t d a t a ) . 

T h e 1st 5 c h a r a c t e r s of e a c h p r i n t i n g line a r e t h e 
line s e q u e n c e n u m b e r . " s p e c i f i c a t i o n s f o r o u t p u t 
d a t a " m u s t b e like I w , w h e r e w = K A 2 
f o r each o u t p u t d a t a . 

F O R M A T ( 8 X , 1 2 F 6 . 5 ) 

— filter coefficients. (These m u s t b e p u n c h e d i n 
t h e following o r d e r : c e n t r a l coefficient a n d t h o s e 
a t i t s r i g h t ) . 

[J], [ K ] a n d [L] m u s t b e b l a n k c a r d s . 

R e p e a t [E] t h r o u g h [I] c a r d s t o do o t h e r n u m e r i c a l filter o p e r a t i o n 
or using i n i t i a l d a t a ( N F E L = 1) or u s i n g o u t p u t d a t a o b t a i n e d f r o m 
t h e initial ones (see [C] c a r d ) . 

C O M M E N T . 

I n t h e f o l l o w i n g p a g e s it lias b e e n r e p r o d u c e d t h e list of t h e 
d e s c r i b e d p r o g r a m . A s a n e x a m p l e of u s a g e of t h e r e s u l t s i t h a s 
b e e n also r e p r o d u c e s a p l o t t e r (see F i g . 1) o b t a i n e d h a n d l i n g t h e 
7091/7010 D C S o u t p u t s b y a n a d d i t i o n a l p r o g r a m r u n n i n g on a n 
I B M 1620 c o m p u t e r . 

T h e a u t h o r s h a v e m a i n l y c o n t r i b u t e d each in t h e i r a c t i v i t y field. 

A C K N O W L E D G M E N T S . 

P r o f . M. Galli g a v e t h e o p p o r t u n i t y of v e r y i n t e r e s t i n g discussion. 
P r o f . E . C l e m e n t e l , D r . A. Chiarini, D r . C. Lolli m a d e e a s y t h e I B M 
7094/7010 D C S c o m p u t e r s u t i l i s a t i o n a t t h e " C e n t r o di Calcolo del 
C N E N " of B o l o g n a ( S H A R E I n s t a l l a t i o n B I ) . 

Corre-
Card 

s p o n d i n g 
colums . . . v a r i a b l e 

[E] card (cont.) 

[I] card 

9-80 CO E F 

R E Q U I R E D R O U T I N E S : R F 5 K 0 1 S D A n . 3 0 6 1 

a n d R F 5 K 0 3 S D A n . 3057 



J A N U A R Y 1 9 6 6 

2 3 2 5 2 7 2 9 U.T. 

F i g u r e 1 - C a p t i o n : A) Original D a t a ; B) F i l t e r e d D a t a ; C) F i l t e r e d D a t a b y r e v e r s e d l>) filter coefficients. 



A C O M P U T A T I O N P R O G R A M F O R N U M E R I C A L F I L T E R I N G 2 1 5 

DIMENSION FCOMl 1U1.FR1 ( 1 U ) . F R 2 ( 1 U I , F R 3 I 1 U ) . E L I 5 0 3 0 ) . F I L I 5 0 0 0 ) , 
1 K F IL ( 5 0 0 0 ) , N R E C I 1 0 0 ) , C O E F l 1 0 0 ) 

DIMENSION K A ( U 0 ) , K 3 i U 0 ) 
ECUIVALENCE(FIL,<FIL) 
DATA KEND/3HEND/.ICIFT/10737U182U/ 

530 READ!5,203 ) NDS,NTPI,KE,IT 
IFINTPI.EQ.O.OR.NDS.EQ.O) GO TO 1530 

2 0 3 F 0 R M A T ( 9 I 2 , E 1 4 . 6 ) 
KE =1 0*»KE 
REWIND 3 
RE AD( 5 , 2 0 2 ) (FR1 ( < ) . K. = 1. 11+) 

2 0 2 FORMAT(13A6,A2) 
C 
c 

END FILE 3 
NFIIE=1 
*JTDT = 0 
I F t I T . L E . 0 1 G 0 TO 5 0 0 

50 3 R E A D I N T P I , F R 1 ) K F I N , ( K A ( J ) , K B ( J ) , J = 1 ,NDS) 
IFIKF1N.EC.KEND1G0 TO 501 
DO 5 0 2 J = 1« NDS 
KB!1)= MOD(KB 1 J ) / K I FT , 1 6 ) 

50 2 E L l J ) = I S I G N t K A ( J ) » 1 0 + I A B S I M O D ( K B ( l ) , 1 0 ) ) , K B l l ) ) / K E 
NT OT =NT 0T + 1 
WRITEl31NT0T,I ELI J ) , J = 1 ,NDS) 
GO TO 503 

C 
50 0 R E A D ! N T P I . F R 1 ) K F I N , ( E L ( J ) , J = 1 , N D S ) 

IFIKFIN.EQ.KENDJGO TO 501 
NTOT=NTOT+1 
W R I T E I 3 J N T 0 T , ( E L ( J ) * J - 1 , N D S ) 
GO TO 500 

C 
501 END FILE 3 

NREC(NFILE)=NTOT»NDS 
536 READ! 5 , 2 0 0 ) (FCOMlK) t K=l, 1 U) 
200 FORMAT!13A6,A2) 

WRITE« 6 , 2 0 1 JI FCOMlK),K = 1,11») 
20 1 FORMAT 1 1 H 1 , 6 1 / ) , 2 8 X , 1 3 A 6 , A 2 / / 2 8 X , 8 0 < 1 H - ) ) 

C 
C 

506 READ« 5 , 2 0 3 ) NFEL,1,NDP.NR,NTPO,MSTA,MPER,IC,KA2,A1 
IFINFEL.EQ.O.OR.NFEL.GT.NFILE.OR.N.EO.O. OR. NREC(NFEL). EQ. 0)GOT0530 
READ! 5 , 2 0 2 ) ( FR2(K),K = 1 , 1 l » ) f ( F R 3 ( K ) , K * l , 1 l » > 
REWIND l* 
MFIL=NFILE-NFEL+2 
CALL BAKFILt3.MFIL) 

C 
C 

M1 =N/2 
M2=M1•1 
READ! 5 , 3 3 7 ) ( C O E F l I ) , I = M2,N) 

337 FORMAT» 8 X , 1 2 F 6 . 5 ) 
DO 50 1=1,Ml 
M3=N-I•1 
EL( I ) = 0 . 

50 COEFlI)=COEF(M3) 
KA2=10»»KA2 



2 1 (i M . A T T O L I N I - L . D A L L ' O L I O - V . G U E R R I E R I - P . R A N D I 

NN=NTOT-NVR+1 
DO 8 J = 11 NVR 
EL(J)=EL(NN) 

8 NN=NN+1 
I2=NVR 

C 

NSP=K3/NDP-
GO TO 513 

17 NSP» 1 
IF(K3« EQ.NDP) GO TO 513 
K4=K3+1 
DO 514 MM=K4,NDP 

514 F IL{MM)=0» 
513 DO 15 J 1 = 1 » K3, NDP 

J2=J1+NDP-1 
WRITE141 ( F I L I J ) , J = J 1 , J 2 ) 

505 IFI(MSTA+MPER1.LE.0) GO TO 301 
DO 16 H J = J 1 , J 2 
I FIFILIMJ l . N E . O . ) GO TO 416 
KFIL(MJ)=0 
GO TO 16 

416 KFIL(MJ ) =MOD( I F I X ( F I L « M J ) » M » S I G N ( . 5 , F I L Í H J ) J ) , KA2 ) 
16 CONTINUE 

I F ( M S T A . G T . 0 ) W R I T E ( 6 , F R 3 ) J N P , ( K F I L l J ) , J = J 1 , J 2 ) -
IFIMPER.LT.1) GO TO 301 
I F( I C . E Q . l ) GO TO 613 
WRITE(8.FR2) J N P . I K F I K J ) , J = J 1 , J 2 J 
GO TO 301 

613 DO 300 J = J 1 , J 2 
J J = J - J 1 + 1 
K A ( J J ) = I A B S ( K F I L ( J ) / 1 0 ) 

300 K B ( J J ) = M O D ( K F I L ( J ) , 1 0 ) » K I F T 
WRITEINTP0.FR2) J N P , ( K A ( J ) , K B < J ) , J = l . J J ) 

301 JNP=JNP+1 
15 CONTINUE 

GO TOI 511> 1101,K<K 
511 NRE = NRE-NTOT«-IG-l 

IG=NVR*1 
IFINRE.LE.O I GO TO 104 
GO TO 11 

104 IF(NVR.LT.NR»N!G3 TO 110 
KKK=2 
GO TO 105 

1 10 HFIL«MFIL-1 
CALL SKPFIL(3»HFIL) 
NF ILE=NFIL E +1 
REWIND 4 

C 
C 

JNP=JNP-1 
I NP = 0 
IF{JNP.LT.NDS)G0 TO 507 
K 1 = J NP/NDS 
DO 508 MM=1 « <1 
J l = 1 
J 2=NDP 
DO 509 MJ = 1 «NDS 



A C O M P U T A T I O N P R O G R A M F O R N U M E R I C A L F I L T E R I N G 21 5 

JNP = 1 
NVR=0 
Ml = 1 
KKK = 1 
IFINR.GT.1)G0 TO 13 
NR= 1 
GO TO 111 

13 M 1=2 
M 1 = M 1*NR 

DO 600 I=M2,M1 
600 E L I I ) = 0 . 

M2=M1+ 1 
111 IG=M2 

C 
c 

K 1=4800 
NRE=NREC(NFEL) 

11 IF(K1.GT.NRE)M=NRE 
K1=K1+1 G— 1 
DO 1 11 = 1 6 , K 1 , NDS 
12=11+NDS-l 

1 R E A D t 3 ) N U M , ( E L ( J ) , J = I 1 , 1 2 ) 
105 NT0T=I 2 

W R I T E ( 6 , 7 6 7 ) I E L ( J ) , J = 1 , N T 0 T ) 
T67 FORMAT t 2 4 F 5 . 0 ) 

NRAG=NTOT/NR 
K3 = NRAG-N<-1 
IF(KKK.EQ.2)GO TO 18 
KRES=M0D(K3,NDP) 
K3=K3-KRES 

18 GO TO < 7 0 1 , 7 0 2 ) , N 1 
C 
C 

702 NR 1 = 1 
NR2=NR 
NRAG=K3+N-1 
FRAG=NR 
DO 20 KRA=1,NRAG 
EL(KRA)=EL(NR 1 ) 
NR1=NR1+1 
DO 21 IRA=NR1,NR2 

21 EL(KRA)=EL(KRA)+EL( IRA) 
EL1KRA)=EL(KRA)/FRAG 
NR1=NR2+1 

20 NR2=NR2+NR 
C 
C 

701 DO 12 KK=1,K3 
F IL(KK)=EL(KK)»COEF(1 ) 
KZ = KK 
DO 14 MK = 2 « N 
KZ = KZ + 1 

14 F1L(KK)=FIL(KK)+EL(KZ)»C0EF(MK) 
12 CONTINUE 

C 
C 

IFIKKK.EQ.2) GO TO 17 
NVR=NT0T-K 3»NR 



2 IK M. A T T O L I N I - L. D A L L ' O L I O - V. G U E R R I E R I - P . R A N D I 

READ I 4 ) ( F I L I J ) « J = J1> J 2) 
J 1 =J2 + 1 

509 J2=J2+NDP 
J l = l 
J 2 = NDS 
DO 510 MJ=1,SDP 
INP=INP +1 
WRIT E ( 3 ) I N P » ( F I L I J ) » J = J 1 , J 21 
J 1 =J 2+1 

510 J2=J2+NDS 
508 CONTINUE 
507 KM=MODlJNP.NDS) 

IFIKM.EQ.OÍGO TO 561 
J 1 = 1 
J2=NDP 
DO 5 1 2 HJ=1 « KM 
READ!U)IFILI J ) , J * J 1 , J 2 ) 
J 1=J 2+1 

512 J2=J2+NDP 
IF{KKK.NE.21K3=N0P 
NTRA=KM»NDP-NDP+K3 
KM=NDS-MODINTRA,NDS) 
IFIKM.EQ.OJGO TO 5 1 5 
DO 516 MJ=1,KM 
MM=NTRA+MJ 

516 FILI MM)=0. 
515 KM=NTRA/NDS+1 

IF(MODtNTRA.NDS).EQ.O) KM=KM-1 
J 1 = 1 
J 2 = NDS 
DO 517 MJ=1,(M 
INP=INP+1 
W R I T E ( 3 ) I N P . ( F I L I J ) . J = J 1 . J 2 ) 
J 1 =J 2+1 

517 J2=J2+NDS 
561 NT0T=INP 

GO TO 501 
C 
c 

1530 STOP 
END 

T I M I N G : 

1) t o t e s t 960 i n p u t d a t a of a t i m e s e r i e s w i t h 2 n u m e r i c a l 
filtering e a c h of 69 c o e f f i c i e n t s g r o u p p i n g i n i t i a l d a t a 2 b y 2, i t t a -
k e s 1 ' 1 2 " ( i n c l u d i n g t h e m o n i t o r t i m e a n d t h e p r o g r a m c o m p i l a t i o n , 
t h e r e f o r e t h e t i m e of c o m p u t a t i o n is o n l y 2 0 " ) . 

2) t o t e s t 3 4 5 6 i n p u t d a t a of a t i m e s e r i e s w i t h 3 n u m e r i c a l 
filtering of 9, 69, 3 5 c o e f f i c i e n t s r e s p e c t i v e l y g r o u p p i n g 2 b y 2 o n c r o s s i n g 
f r o m I o t o 2° i t t a k e s 1 ' 4 3 " ( i h c l u d i n g t h e m o n i t o r t i m e a n d t h e 
p r o g r a m c o m p i l a t i o n : t h e r e f o r e t h e t i m e of c o m p u t a t i o n is o n l y 3 2 " ) . 

R E F E R E N C E S 
(*) GALLI M., RANDI P . , On the Design of the Optimum Numerical Filter with 

a Prefixed Response. " A n n a l i di Geofisica " , X X , 4, (1967).