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OCR Text

PROGRAM

DECUS NO.

TITLE

AUTHOR

COMPANY

DATE

SOURCE LANGUAGE

LIBRARY

FOCAL8-226

FREQUENCY TRANSFORMATION PROGRAM

Klaus LIckteig

Institut Fuer Kerntechnik
Technische Universitat Berlin
Berlin, Germany
A p r i l 1972

FOCAL 1969; PAL I I I

ATTENTION
This is a USER program. Other than requiring that i t conform to submittal and review standards,
no quality control has been imposed upon this program by DECUS.
The DECUS Program Library is a clearing house only; i t does not generate or test programs. N o
warranty, express or i m p l i e d , is mode by the contributor. Digital Equipment Computer Users
Society or Digital Equipment Corporation as t o the accuracy or functioning of the program or
related material, and no responsibility is assumed by these parties i n connection therewith.

FREQUENCY TRANSFORMATION PROGRAM
DECUS Program Library Write-up
1.

DECUS N O . FOCAL8-226

ABSTRACT:
V a r i o u s F o u r i e r t r a n s f o r m a t i o n m e t h o d s can be a p p l i e d w h e n u s i n g
the F r e q u e n c y T r a n s f o r m a t i o n P r o g r a m d e s c r i b e d b e l o w .

This

p r o g r a m s h o u l d e x a m i n e in p a r t i c u l a r the a c c u r a c y of the F a s t
F o u r i e r T r a n s f o r m a t i o n F O C A L p r o g r a m d e v e l o p e d by R O T H M A N / 2 /
in c o m p a r i s o n w i t h n o r m a l F o u r i e r

transformations.

T h e r e s u l t is that the F a s t F o u r i e r T r a n s f o r m a t i o n s h o u l d be u s e d ,

NU
if the n u m b e r of d i s c r e t e p o i n t s is

N = 2

(NU

=1,2,3,...).

If not s o , the t r a n s f o r m a t ion m e t h o d w i t h t r a p e z o i d a l i n t e g r a t i o n and
l a g w i n d o w " h a n n i n g " s h o u I d be u s e d .

REQUIREMENTS:
2. 1 H a r d w a r e :

A n 8 - k P D P - S / 1 o r 8 / e c o m p u t e r w i t h an
A S R - 3 3 t e l e t y p e is the m i n i m u m h a r d w a r e .

2. 2 S o f t w a r e :

1 ) F O C A L 1969,

DEC-OB-AJAE-PB

initial dialogue: NO - Y E S
2) U t i l i t y o v e r l a y s f o r F O C A L 1 969 ( 8 - k )
DEC-08-AJ1E-PB
3) if a v a i l a b l e : M O D V - C h o i c e ,
D E C U S No. F O C A L

8-135

4) F N E W - F u n c t ion f o r the F a s t

Fourier

T r a n s f o r m a t ion

LOADING PROCEDURE:
1) L o a d F O C A L 1 969 w i t h the B i N - L o a d e r i n t o f i e l d 0 a n d
s t a r t F O C A L at l o c a t i o n 0 2 0 0 .
2) A n s w e r the i n i t i a l d i a l o g u e w i t h

NO - Y E S

3) S t o p the c o m p u t e r .

L o a d the 8 - k o v e r l a y w i t h the

B I N - L o a d e r i n t o f i e l d 1.
4) If the p r o g r a m M O D V - C h o i c e is not a v a i l a b l e , l e a v e out
this point.
L o a d the D E C U S p r o g r a m M O D V - C h o i c e w i t h the B I N L o a d e r i n t o f i e l d 0. R e s t a r t F O C A L at l o c a t i o n 0 2 0 0
a n d a n s w e r the q u e s t i o n w i t h

N . S t o p the c o m p u t e r .

5) L o a d the F N E W - F u n c t ion w i t h the B I N - L o a d e r i n t o f i e l d 0.
6) R e s t a r t F O C A L at l o c a t i o n 0 2 0 0 .
7) L o a d the F O C A L F r e q u e n c y T r a n s f o r m a t i o n P r o g r a m .
8) S t a r t t h e F O C A L p r o g r a m w i t h the

command.

T h e t e l e t y p e w i l l g i v e a m e s s a g e a n d the p r o g r a m w i l l
e r a s e the g r o u p 1 c o m m a n d s .
9) Y o u h a v e t o w r i t e the l i n e s 2 . 14 a n d 2 . 16 s p e c i a l l y f o r
y o u r p r o b l e m ( f o r d e t a i l s see c h a p t e r 8: c o m m e n t of the
F r e q u e n c y T r a n s f o r m a t i o n P r o g r a m in the I i s t i n g s , g r o u p I
of F O C A L p r o g r a m ) .
1 0) W i t h a

c o m m a n d you w i l I s t a r t the t r a n s f o r m a t i o n .

THEORY
4. 1 Integration Methods
A certain

integral
b
a

can be e v a l u a t e d n u m e r i c a l ly o n l y by a p p r o x i m a t i o n .

the d i f f e r e n t e x i s t i n g i n t e g r a t •on m e t h o d s the f o r m a l ism
i n c r e a s e s t o some e x t e n t f o r m o r e a c c u r a c y .

4 . 1. 1 T r a p e z

integration

W h e n e v a l u a t i n g an i n t e g r a l of a c u r v e w i t h o n l y t w o
ordinates

( t ^ , y ^ ) ; (t^ , y ^ )
2

, you get the g r e a t e s t

error.

T h i s linear interpolation (or trapezoidal

integration)

can be e a s i l y d e v e l o p e d .

4 . 1. 2 S i m p s o n

Integration

In the S i m p s o n

integration

an i n t e g r a l is e v a l u a t e d w i t h t h r e e o r d i n a t e s
(t

J y ) ; ( t , , y , ) ; ( t „ , y „ ) . if the i n t e g r a l has t o be d e t e r o
1 1
2
2

m i n e d f o r a l o n g e r i n t e r v a l , the r e s p e c t i v e f o r m u l a s can be
e x p r e s s e d as f o l l o w s ( e q u a t i o n 1):

H e r e the i n t e r v a l is d i v i d e d i n t o a l i n e a r n u m b e r
of s e g m e n t s of e q u a l w i d t h

2 Fourier

M .

Transformation

W i t h the F o u r i e r t r a n s f o r m a t i o n a f u n c t i o n of the t i m e d o m a i n
is t r a n s f o r m e d i n t o one of the f r e q u e n c y d o m a i n ,

if t h i s f u n c -

t i o n is o n l y g i v e n in d i s c r e t e o r d i n a t e s , t h e f r e q u e n c y
a f t e r the t r a n s f o r m a t i o n is l i m i t e d .

The frequency interval

is:

-i/ca-^i-Y/Y
3

domain

Here

is t h e t i m e i n t e r v a l a n d

the n u m b e r

of o r d i n a t e s . ^ o the l i m i t e d f r e q u e n c y d o m a i n a f t e r the
transformation

= k •

= 0,

Af,

for

k = 0,

1 , . . . , N

or
f

2 • Af,

4.21 F o u r i e r Integral

3 - A f , . . . , N-

A f = 1 / (2-

At)

Transformation

The equation

P(fJ

}-t

s h o w s the F o u r i e r t r a n s f o r m of a f u n c t i o n

y(t) .

If the f u n c t i o n is n o n - e x i s t e n t f o r t i m e s
t F 0 ,

P ( f ) i s as f o l l o w s ( e q u a t i o n 2 ) :

d-/y/iJe'-'^'^'^c/i

P(/; o

o r it is d i v i d e d into r e a l a n d i m a g i n a r y p a r t s ( e q u a t i o n 3 ) :

P(/J - ^-fyiP
o

cosCuAi)

PffJ--^

sib

fyW

UiJ

4 . 2. 2 D e r i v a t i o n f r o m a F o u r i e r

Series

P e r i o d i c v a r i a b l e s c a n be r e p r e s e n t e d in t h e t i m e d o m a i n
b y a F o u r i e r s e r i e s of the f o r m

w h e r e the c o e f f i c i e n t s

and

a r e defined by

(equation 4 ) :
A'-.ttf
a^,= J
o

yl-f) cosl^tc-t)

oli

Y'-^^ S-h(u>^-IJ

4 =

N •

u) , the
K

is the p e r i o d i c of the v a r i a b l e , and

k - t h h a r m o n i c of the f u n d a m e n t a l f r e q u e n c y

, is g i v e n

by ( e q u a t i o n 5 ) :

Z-Tk

If the v a r i a b l e
points

y(t)

is s a m p l e d at

seconds

equally spaced

a p a r t , the F o u r i e r t r a n s f o r m is / 3 /

(equation 6):
Pip .

^[y(0j.E.2^yaja>S

P'bydA.^]

p(/y--aX 2x^473/^ ^''^

H e r e the c o e f f i c i e n t s

and
k

trapezoidal

a r e e v a l u a t e d by
k

integration.

4. 2. 3 F a s t F o u r i e r

Transformation

T h e F a s t F o u r i e r T r a n s f o r m a t i o n is e x t r e m e l y u s e f u l
in the c o n v o l u t i o n of t i m e s e r i e s . T h e a l g o r i t h m has
been w e l l d e s c r i b e d by B R I G H A M / l / a n d R O T H M A N / 2 / .
S i n c e the F a s t F o u r i e r T r a n s f o r m a t i o n F O C A L

program

d e v e l o p e d by R O T H M A N / 2 / is u s e d in t h i s F r e q u e n c y

T r a n s f o r m a t i o n P r o g r a m t o o , t h e r e w i l I be n o d e t a i l e d
d e s c r ipt ion of the F a s t F o u r i e r T r a n s f o r m a t ion h e r e .

4. 3 L a g Windows
W h e n u s i n g the F o u r i e r t r a n s f o r m a t ion f r o m a n u m b e r of
d i s c r e t e o r d i n a t e s , t h e r e a r e u s u a l l y side lobes apart f r o m
the m a i n l o b e .

In o r d e r t o c o n c e n t r a t e the ma in I obe and k e e p

the s i d e lobe as low as p o s s i b i e , the o r d i n a t e s of the f u n c t i o n
c a n be m u l t i p l i e d by a l a g w i n d o w .

T h e l a g w i n d o w ( e q u a t i o n 7)

(t) = 1

for

It j ^

= 0

for

|t I >• t

max
max

is p r a c t i c a i i y of n o c o n s e q u e n c e . A s i m p l e and c o n v e n i e n t
c o m p r o m i s e is r e p r e s e n t e d by the l a g w i n d o w c a l l e d
" b a n n i n g " (equation 6):
D.

it)

= 0.5 (1 f

COS

)

_ Q

lil

It I

'
tn-tajc

A n a l t e r n a t i v e c o m p r o m i s e is r e p r e s e n t e d by the l a g w i n d o w
" h a m m i n g " (equation 9):

T-i
J),

li)

O.St

+ a.tSaxs—-

far

It I

t^^,

1^'

""^

9m*jr

Frequency Transformation

Program

T h e p r e s e n t F O C A L p r o g r a m t r a n s f o r m s a n u m b e r of o r d i n a t e s
( t i m e d o m a i n ) i n t o the f r e q u e n c y d o m a i n . H e r e d i f f e r e n t

integration

and t r a n s f o r m a t i o n m e t h o d s a r e u s e d .

1) S i m p s o n i n t e g r a t i o n :
The F o u r i e r t r a n s f o r m a t i o n takes place a c c o r d i n g to equation 3.
H e r e the i n t e g r a t i o n m e t h o d is t h e S i m p s o n

integration

( e q u a t i o n 1).

2) S i m p s o n i n t e g r a t i o n w i t h " b a n n i n g " w i n d o w :
In a d d i t i o n t o t h e a b o v e m e t h o d 1 t h e l a g w i n d o w

D (t)

"hanning"

( e q u a t i o n 6) i s c o n s i d e r e d t o s m o o t h the c u r v e .

3) S i m p s o n i n t e g r a t i o n w i t h " h a m m i n g " w i n d o w :
in a d d i t i o n t o m e t h o d 1 t h e l a g w i n d o w

(t)

"hamming"

3
( e q u a t i o n 9) is c o n s i d e r e d .

4) T r a p e z

integration:

The F o u r i e r t r a n s f o r m a t i o n takes place a c c o r d i n g to equation
H e r e the i n t e g r a t i o n m e t h o d is t h e t r a p e z o i d a l

integration.

5) T r a p e z i n t e g r a t i o n w i t h " h a n n i n g " w i n d o w :
D _ ( t ) " h a n n i n g " is c o n s l i d e r e d in e q u a t i o n 6 , it
2
is r e d u c e d t o ( e q u a t i o n 1 0 ) :
If t h e l a g w i n d o w

df
1^

p(f) = - i

rck
(^^'^^
k = C ' 7 ^ y

- V

6) F a s t F o u r i e r

Transformation:

The F a s t F o u r i e r T r a n s f o r m a t i o n a c c o r d i n g to R O T H M A N / 2 /
is m a d e .

C o m p a r i s o n of the O i f f e r e n t

Methods

T h e q u a l i t y of a F o u r i e r t r a n s f o r m a t i o n m e t h o d w a s s e e n in the
e r r o r s that o c c u r w h e n a t h e o r e t i c a l p e a k ( t h e a m p l i t u d e of the p e a k
t h e a m p l i t u d e of t h e h a r m o n i c s ; w i d t h of the p e a k a p p r o x i m a t e l y
is e v a l u a t e d b y d i f f e r e n t

zero)

methods.

W i t h the F r e q u e n c y T r a n s f o r m a t i o n P r o g r a m s e v e r a l

operations

w e r e made to f i n d out the q u a l i t y of the i n d i v i d u a l m e t h o d s ,
l a r l y t h a t of the F a s t F o u r i e r T r a n s f o r m a t i o n .

particu-

T h e r e s u l t w a s as

f o l l o w s ( t h e p e r c e n t a g e s b e l o w r e f e r t o the e x a m p l e d e s c r i b e d in
chapter 8):

1) T h e S i m p s o n i n t e g r a t i o n h a s b e t t e r r e s u l t s (as c o u l d h a v e b e e n
e x p e c t e d f r o m the t h e o r y ) t h a n the t r a p e z o i d a l

integration

(without u s i n g a lag w i n d o w ) .

T h e S i m p s o n i n t e g r a t i o n has a s h a r p p e a k at

OJ, = -4

(FSEC)

h o w e v e r the r a t i o of the a m p l i t u d e s (the a m p l i t u d e s of the h i g h e r
h a r m o n i c s t o the a m p l i t u d e of the m a i n l o b e ) is a p p r .
T h e r e i s a n a d d i t i o n a l p e a k at

6,5 •

8 per cent.

( p a r t i c u l a r l y in t h i s

example).

T h e t r a p e z o i d a l i n t e g r a t i o n h a s a s o m e w h a t NA'ider (and t h u s l e s s
c o r r e c t ) m a i n l o b e ; the r a t i o of the a m p l i t u d e s is a p p r .
c e n t ; h o w e v e r t h e r e is n o a d d i t i o n a l p e a k .

10 p e r

2) W h e n u s i n g lag w i n d o w s in the F o u r i e r

transformation,

the c u r v e is s m o o t h e d , i. e. the a m p l i t u d e s of the h i g h e r
harmonics are reduced, whereas

the m a i n lobe b e c o m e s w i d e r .

In the S i m p s o n i n t e g r a t i o n the l a g w i n d o w s " h a n n i n g " a n d
" h a m m i n g " a r e u s e d . T h e " h a n n i n g " w i n d o w (at the r a t i o of the
a m p l i t u d e s of a p p r .

z e r o ) w i d e n s the m a i n lobe a b i t m o r e t h a n the

" h a m m i n g " w i n d o w (at the r a t i o of the a m p l i t u d e s of a p p r .
c e n t ) . T h e a d d i t i o n a l p e a k at

6 , 5 • <Vj

1,5 p e r

was again t h e r e .

3) If the l a g w i n d o w " h a n n i n g " is u s e d in the t r a p e z o i d a l

integration,

the m a i n lobe is a l s o w i d e r , h o w e v e r the r a t i o of the a m p l i t u d e s
is o n l y about 0 . 2 p e r c e n t . T h e r e is n o a d d i t i o n a l p e a k . A p a r t f r o m
t h a t , the i m a g i n a r y p a r t ( t h e o r e t i c a l l y e q u a l z e r o in the e x a m p l e )
is much s m a l l e r t h a n in the S i m p s o n i n t e g r a t i o n .

4) T h e F a s t F o u r i e r T r a n s f o r m a t i o n s h o w s a s h a r p m a i n l o b e ;
the a m p l i t u d e s of the h i g h e r h a r m o n i c s a r e r e d u c e d v e r y q u i c k l y
w h e r e a s the f r e q u e n c y

increases. The imaginary part

K
e v a l u a t e d b e t t e r t h a n w i t h the a b o v e m e n t i o n e d m e t h o d s .

T h e c o m p a r i s o n of the v a r i o u s m e t h o d s s h o w s t h a t the F a s t

Fourier

T r a n s f o r m a t i o n F O C A L p r o g r a m d e v e l o p e d by R O T H M A N / 2 / has
not o n l y a much h i g h e r o p e r a t i n g s p e e d , but that it is a l s o m o r e
a c c u r a t e than the o t h e r d e s c r i b e d m e t h o d s . S o it s h o u W be u s e d , if
the F o u r i e r t r a n s f o r m is t o be e v a l u a t e d out of a t i m e s e r i e s of
NU
N = 2

d i s c r e t e o r d i n a t e s . H o w e v e r it s h o u l d be k e p t in m i n d ,

that the f r e q u e n c y s t e p is
Af

= 1 / (N - At)

in the F a s t F o u r i e r T r a n s f o r m a t i o n F O C A L p r o g r a m a n d t h a t the
r e s u l t s f o r the f r e q u e n c i e s

f > 1 / (2 - A t )

c a n n o t be r e g a r d e d as e x a c t .

However,

if t h e r e is a n u m b e r of

i n t e g r a t i o n w i t h lag window D

(t)

N ^0= 2'^'"'

o r d i n a t e s , the t r a p e z

" h a n n i n g " , ( e q u a t i o n 1 0 ) , is u s e -

f u l , s i n c e t h i s m e t h o d i s c o m p a r a t i v e l y e x a c t . H e r e the f r e q u e n c y s t e p
is
Af = 1 / ( 2 - A t

and the maximum f r e q u e n c y :

max

=1/(2-

At)

LITERATURE
/ 1 /

B R I G H A M , E . O . ; M O R R O W , R. E . :
The fast F o u r i e r

transform

I E E E E p e c t r u m , D e c . 1 9 6 7 , p. 63 - 70

JlJ

R O T H M A N , J. E. :
The F a s t F o u r i e r T r a n s f o r m and its
Implementation
D E C U S C O P E 1 9 6 8 , V o l . 7 , N o . 3 , p. 3 - 1

/3/

U H R I G , R. E . :

R a n d o m N o i s e T e c h n i q u e s in N u c l e a r
Reactor Systems
Ranold P r e s s Comp. , New Y o r k

LISTINGS OF PROGRAMS
T h e F a s t F o u r i e r T r a n s f o r m a t i o n F O C A L p r o g r a m w r i t t e n by
R O T H M A N / 2 / w i l l be r e p e a t e d o n c e m o r e b e l o w , b e c a u s e t h e r e
a r e d i f f e r e n t , s o m e t i m e s e v e n i n c o r r e c t v e r s i o n s in the v a r i o u s
existing publications.

The following listings are attached:
1) L i s t i n g of the F N E W - F u n c t ion f o r the F a s t

Fourier

Transformation
2) L i s t i n g of the F O C A L F r e q u e n c y T r a n s f o r m a t i o n P r o g r a m
3) T h e t e l e t y p e o u t p u t w h e n s t a r t i n g t h e F r e q u e n c y
P r o g r a m a n d an e x a m p l e .

Transformation

/
/

Pwr)!jKAM:

M r H r t O U T l M F F l M t N !• U K F O C A L
19 6 9 ,
I'4ITIAL
DIALuGOc,:
N()-":'cS
f-Nt"/( •<,'')

i-iJUAL

nUK

FAST

UhHSION

n ' O l j H I n.H

A. J An,

i K A N S n O K N A T I UN

s P L U I n I L A T I O'-J

hn ' j ! ; I = 1 J 6
t v AJ.= 1 o 1 J
hLAU=^9 9
>ib

fL-AbAD

I '-I T h, i V,H= 6
iJj I s {. ( = Z4 -,74 -s
SPyJi| K = 9 D O " i

* 3P
ui/i 3 b

b 1b 2

i/W 1 w

3 1 33

P 1 3 3

995 3

bloA

1 09

1 i

oO T T U N ,

< t i\)pP- 1

* 9 1 ''1
/

< r 'V L " '

hi) G A L

P.pRSIiHO

A.I An.

+ 513 3
< !• Pl 5

T \ r r..3 L;<

.L4 S

nIP

rLOATlxio

7 A! )

i- L A I . + 1

T A:< h.

1- ( A H T

Li-ITh

P 3

7 0 9 i

L T A

3 i 3 t>

3 37 7

PLK

•4 UP t i p A

13?
i n i/ i

9 3 3 yi

SP/iO H

i41) 7 L

9 59 5

-I

b'i'L

0 t.r

bio 1

9 59

PIJ 5:4.1

pPALbA'ip

3 !oK

1O 1 3

1 o 3

! OA

99 5 3
3 09 3

p 7 AL
• I'-'ib I

n r<
PUT

3
3

IJi'r_3p:A

IJLA

fLAL+ 1

5PA L P S

LO'-ii'iA

NPKT

ARoUi-ipNT

n L . p. AL
I T I N r LAL+ 1

iJLA

n L AL+ P

hiilLU

TAIJ

KLAL+ 1

ThAMbPOhp

110

LLL

KAK

b 17 (4
5 17 i

309 5

iJLA

l'LAL+ 1

109 5

TAP

i-LAL+P

HIT

b 1 7 K

7 0 09
309 6

HAL

I N SLHT

L)LA

t-LAG+P

K37 7

T 57,

GNTR

nUh

b 17 5

b3bb

.JNP

LOOP

3 17 6

bb3b

• JNP

[

R p4' L! Rpl

T 0

4 A s '4

5 17 7

0 0 01'!

7 P0 0

": "

rip

A5K

7 P0 0

" 1 = " bY

3 163

309 o

b i O 5

1 09

blo7

5 17 3
3 17 9

LOOP,

OrJ T K ,

Ll-Ui.jai

biTS

A A 5 1•

PAST

POPvlf

A ^ i Ij-"! A 7 f

b p S O L 4'

R O T A T To 3
/

ANO

'4 3

JTLAG+P

I Tb

Lpi-T
I'-l

L U O OHUCR

9 o
i4 i j

HLrs
P K O 3 ^ AO

•« 1 P 17
13 17

7 K0 0

OMOP

7 20 A

L O N N A JiJ

* D0MP

LPNTB

Kt3li'

TPHNbPOSp

INTO

ALL

AHOUT

TYPh,

GUMNANu

L I N <

BI T

C-.8Y

MOUU

0 1.01

c
c
c
c

l/i

7, P

. 0 3

9-307 0
K H t i O U E N G",' T H A N S F O K M A T I ON

. 09

. 05

. 00

8-K

MOUV-CKOIGE

. 07
0 . 08
0
0

. 09

. 10

0 .

FOCAL

19 5 9 ,

FOCAL

I N I T I A L

AND

IJT A L O G I J E

CNOT

NO-YES

NEGESSARY)

CUECUSiMO.

8 - 13b)

C
C

c
. 1 1 c

L A N G U A G E :

PKOG R A ' i

FUNCTION:

F"NEN(K,Y)

FOURIER

T R A 4 S I - 0 K N A T I ON

FHEOUENG'7

THE

. 18

1. )

S I M P S O N - I N T E G R A T I UN

. 9 0

c
c
c
c

2. )

S i M P S O N - I N T E i l R A T i ON

AN u

H A N N I NG - w I N DO W

3. )

SIMPSON-INTEGRATION

A<7 u

H AMM I NG - - i l N DO W"

9. )

T R A P E Y - I i . J T E G R A T I ON

5. )

T R A P E Y - I N T E G R A T I 0:\

0 . )

FAST

0 ) . 29

. 2o

0
0

. 28
. 29

I'i

. 30

. 31

0 . 32
0 . 39
0 . 30

. 38

'/'}

. 39

. 9 0

DOMAIN)

A R E T R AN S i - 0 RM E D

SIX

DIFFERENT

CALCULATIONS

DATA

INPU'T:

OUTPUT

1.)

I N P U T

2.)

;JA F A - A R R A Y

O F DATA:

'YOU

. 9 9

T I E T I M E

INTER9/)L

'/i .

. 9 0

O A TA

CN(MAX)-128J

. 97

THEN

0 . /i8

0 2.19

G
f•

(/ 2 .

y^;

0 . 31
. 52

I N P U T

.lA.HE

To
N

WRITE:

i i j

T I=

WRITE

NEWL I N E S
I S

01-

FROM

DAT A ,

I MAG I N A R E P A R T

3 0

TRANSi-ORi-1

M D H AiMN I N i3 - WI N DO

T H E NUMBER

T H E TIME,

0 ] . 9 2

t • 1.

INTO

WILL

T R A 7 S F 0 Hi 1 A T I OX

F 0 IJ rcl E R

9 1

] .

(TIME

l/i

DATA

DOMAIN.

DATA:

FAST

. 19

0] .

NUMBER

FOR T H E

DAT A - A R R A Y S

FIGH

REoUbN G Y ,
OF

NU.

TIE

AM.V

SEGONDS

I S PUNCHED

HEADER

r'OURl ERKUEF F l CI h J F 5

2 . 19

O.06bb

SPEED

REAL A N D

7II.

A N D T H E N DM E E H

O N PA»->ER

TAPE,

0. 0 6 b b

A i7 ; *

I'l! . D o

. 59

! ! ! ! ,

" P L E A S E

0 1. 3 6

"NO.

02.

01 . 38

" P R O B L E M

" T

;: " t

01 . b 1

0 1. OX

(•'• 1 . n 3

. 69

19",

N R I I E
!, "

! " , ! ! ,
T I M E

T H E NhN
N O .

' Y O U HAVE

I N T E R V A L " ,

L I N E S " ,
02.

T O

! , "N

10",

Dur

!, " S P E C I A L L Y

I M E

T H E

NUMBER

F O R

Y O U R

V A R I A B L E S : " , !
O F

D A T A " , ! !

(•)9.

' 1

G h R t (.lijhN G"'

''/it-.

N 2

Mb. MR

! ! ! ! !

02.10

"NUMBER

UAT A-ARRAYS

02.

! ! ! ! ! ! ! ! ! ! ,

" U A T A - A R R A Y :

" ,

TRANSFORMATION

02.

"PLEASE

rlEW

02.

"0 2. lo

? ? ? " , ! !;

L I -MES

b2.

TM=(N- 1)*T 1

02.

MM S

!JF= l / T M / p :

" N UM H E R

DATA

! ! ! ",

Pb.OGHA-i

KU;
%2.

[

00,

! ! , " 0 2 . l9

P 1= 3 . I ' i 1 5 9 3 ;

02.>'i

"OEL T A - T

= " , 4,7. 0 b , T 1 , "

C SEC3 " ,

09.

"TAOCt AX)

= " , %7 . 0 3 ,

CSEG3

02.26

"DELTA-n'

= " , %7 . 0 6 , U r , "

CBY3 ",

02.

X ' ,

02.3 0

02.39

"DATA-ARRAY",

! !,

02.

T = 0,

0 3 , I *T 1, "

! ! ! !

0b. b b
02. 5 0

t :z M, M - 1 ;

i- 1;

= " , 0 3 . 00, N,
I'M, "

%1.

I I I

TSEGI

U A T A " ,

" S I M PSON - I N T LG H AT I O N " ;

i.Ei = - i ;

02.

1)2=1. ;

.0 3 = 1 . ;

02.

" S I M R b i j N - I i J T E G H A T i Ob)

02,

u 1 = 0;

09. 0 9

,99=0.5;

" b l " ! P S ( ) N - I N T E i l H A T l ON

02.00

1)9=0.59;

1)5=0.9b;

02.

/''

" i ' R A ^ E Y - I N T E G R A r i ON "

02.7

02.

7 o

" T H A D R Y - T : \ I T h G R A T l OA)

09.

/•<

8 ;

•{ A ' i i - N
I)

02.

MANNING";

jjb=i').n;

09.09

H A H A I I N G "

>b . ^

02,

02.

S -Or-o;

09.

NiJ='40+l;

02.

2. .3 o
[ NU3

"n A R T

n!)OHTER
T

THAAlShORMATl O N "

[ 1 - 1 2 8 1
I

10,2.80;

C C 9T-,,| u ) - b j ]

2.99,2.99;

•lb. •' 0

09.

S KU=KU- 1 ;

"•3.

I N T E G R A T I O N

S I N A S C H

C-.KUl

9=123

2.88, 2.90;

N = 2 T N U;
2 . 12;

NU = ' 4 u - ] ;

! ! ! ! ! ! ! ! !

G
b

8 =

03.12

0"1 = K * 9 V ;

0b.

CU 1 ]

• "'.-i,

? !

" , X7 . 06, AC I ) , !

09. b'l

03.

28318b

AC I )

02.52
no

'???", !

P2-b.

2 . 9 9 , 2 . 9 9 , 2 . 1

! ! ! ! !

02. 2 2

C i< U ]

Ku,

b. 18;

T=T 1;
D

S K = 0!'1 * P 2 x T

03.18

R P = AC 0 ) + 9 * U 2 * A C

03.20

I P = 9 * 9 2 * AC

l)*FSIf\lCK)

03.

1 = 2, 2, N-2;

T=T+Ti;

03.2b

K P = 94<T i * H P / 3 ;

03.21

S 8 = 8 + 1;

CK-N]

1)*FG0SCH)

I P = - 2 * T 1*1
3.12,3.12;

9/3;
T

!!!!!.;

12
R

2.90

m . 0 1

G T R A P E 7, - I A] T E G R A T I O ' 4

lYA. ME

l/Pi. 1 0

09.12

K=0
13

09.

1 = 1 , 4 - p;

09 . 1 8

R P = C T 1 / 2 ) « C AC 0 ) + 2 * R P + AC N - 1 ) * C C - 1 ) T 8 ) 3

0 9 . -A'l

I P= - C T 1 ) * I p ;

09 . 2 2

8 = '8+ i ;

05. Mi

T R A P b Y - I G T E G R A T I Oi4

0D . 0 2

0n.

1 '4

C C-IV]

12
9.

1 2 , 9 . 125

05.

1=1,9-2;

05.18

R P = C T l / 2 ) * [ AC 0 ) + R P l

03.20

I P= - C T 1 / 2 ) * I P ;

05.22

S 8 =8+1;

MM . M l

G 0 0 1 f'U T

08 . 0 2

04.10

! ! , "

08.

CK-N]

CHY3

3 . 1 2 , 5 . 1 2 ;

! ! ! ! ! ;

G 9 A M IM I N G
G

I".

I'V

U2=99+U5*EG0SC

P1*T/Ti'i3

10.12

93=99+95*KG0Sr

t - 1 * c T'+T 1 ) / T M 3

11.01

r Rr,:'.iuEi4 C'-:'

T R A ' N S I - O R M A T I 0:4

11.00

SI 4 MSiji..!

TEilHATI'l-J

I NAGI NA R E " ,

HEAL
PART

10. 02

H AN N I .M G

10.01

1 J.

! ! ! ! ! ;

S 8= 0

' 0 5 . 12

PART",

-lAMMING

G
S R l = O.0*9q;,teC T + T 1 )

11.10

H= 0 M * P 2 * T ;

11.19

1 9 14

11.1b

RP= H P + 2:< 9 2 * AC I ) * i- UU SC T ) + 9 * 9 3 * AC I + 1 ) * F G O SC 4 1 )

11.18

T P = I P+ 2 * 9 2 * A C I ) * n S I (4 C 8 ) + 9 * 9 3 * P C I + 1 ) * hASPN C 8 1 )

1 1. 20

T = T + T 1+T1

1 1. l b ;

12.'0 1

G OUTPUT

12. 0 2

12.12

%'f, 0 9 , O N , "
PARAMETER

13.01

13.02

13.
lb.

S
s

10
12

Rb=,M;
S
0 " l = C* i h

TELETYPE
", I P, !

" , %, H P , "'

IP=f0

1 9 . 0 1 G r Rr.'RJEM G Y TH AM S E O R 4A T I O N
1 9 . 0 2 G T R A P E 7. - I iN T EG R A T I (.) M
l9 . 0 3 G
14.

H=P1*I * 8 / M

19.19

P P = K P + A C T ) * J - G O SC K )

19.15

I P = T F^+AC T ) * E h I i \ l C H )

15.01

FRE0liE:4G8

15.02

T R A P E Y - I M T E G R A T I OM

TRAIMSFORMATIOIM

15. 0 3

] 0

H = P 1 * I * K / N ;

15.1 9

R P = P P + AC I ) *!-( 1 * F G O SC T )

13.16

I P=I P+ACI )*K

15.

H A N W I Ntl

H 1= 1 + h U O S C P 1 * I / M )

l*r

SI'MCK)

! !

lo.M!
lb. 02
lo.l'/l
lo.lP
16.14
I D . 16
l o . 18
1O.20
16. 2 2
16.24
lo. 26
lo.28

U
(J
S
K
S

16.30

0 = P - S ;

lo. 52
l b . 34
16.36
l b . 38
l b . 'i'8
16. 4 2
1(3. 6 0

S
S
S
I
S
U
s

K C (-1) = X C 0 ) - G I ;

I
s
S
S
S
b

/ AST j-OURIEK

T=P2/N;
S S=N/2;
S L= l;
S (0=5-1;
S {=1-NU
l =0,!\i-i;
S X ( T ) = 0.
S H = AC fO+S) + A C 0 ) ;
S AC 0 + S ) = AC Q ) - A C (0+S) ; S AC (0) = SR
C (0 3
1 6 . 1 8 , 1 6 . 1 8 ; 6 P= (0- 1 ;
G l o . 1-1
CL-NUl
16.2 0 , l b .4 2 , 16. 2 U
L=L+i;
s s=s/2;
s 8=8+1;
-S p = i v j - i ;
s Y = i / [ 2 t c - i)]
G= 1
U= n I T R C P * Y ) ;
S 'C = T * niN E N C M CJ , U )
Gu=f'GO SCK) ;
b SN=F5Ii\jCK)
G R = G O * AC P ) + S ' \ I * < C P ) ;
S ' M = C O * < C P ) - S ' \ l * yR P )
S

S H = G R + A C (-'-),•

S I = G I + K C

O F 1-T T

8 7 . 8 4 ,I*L>F, "
";
S •<=l-NEi'7CNu, I )
b H = 2 * AC 8 ) / N ;
S SI= 2*8C 8 ) /N
8, SR, "
" , SI , !

: Gu

H..E/1SE

WRITE

NO.
N't.

SPr.GIAH/Y
Ojii
il
N

T I E

NEW

L T N E S

0 2 . 14
0 2. 16
F O R 'YOUR

PROBLE"!

WAVE TO U b F I N E T H E V A H I A P L E
T I M E rMTERV/Al.
N U M B E R On wyVFA
14

* . 1 6
*i.>0

S T 1 = 0.
-* ;

Q) 5

S AC P ) = SR; S K C P ) = S I
P = P - i ;
I C G - b ] l b . 3 o , l o . 3 8 , 1 6 .36
G= G+ 1; G 1 6 . 2 4
[0-8+13
16.40, l b . 18, 1 6 . 4 0
;3=0- b; G 1 6 . 2 2
8
u;"= 1 / c T i * c N - 1 ) ) ;
i- 1 = 0 , 4 - 1 ;
u i ?

17. 0 ! 0 O b T P U T
ii .02
G
17.1")
I
1'7 . 1 2 s
17 . 14 T

T H A N b n ' O R M A T I ON

0o66o7

/ • ') • *

P C O )= ACIO)-GR

vlU'lrfcR

O F DATA-ARRAYS

D A T A - ARRAY :

•Tii'i;iER

Of'

iJA.TA

D E L T A - T

A D O 6 67

C SECI

T A b ( 1.1 A X )
IJELTA-F

=
=

1 . O0M
7.46Q6 98

r SEC]
CHZl

DATA- ARRAY

c SEC]
i>i. O-vp/i

0. 0n7
0 . 13 3

0 • 2 l/i " I
0 . 2to7
M.

3 33

0. 4 0 0
0. 4 6 7
o. 5 33
0. 6 M 0
0 . 6 67
0.733
A. 8 00
0. 8 6 7
/). 5 3 3
1 . 000

DATA
1. 00 0 O0 0
0.91354 5
0 . 6to9 1 27
0. 309 0 1 1
0 . 1 0 4 537
0 . 50 0 0 1 0
0. 8 09 0 2 5
0.978152
0 . 9 7 8 1 4 to
O . 4 09 0 08
0.49 998 4
0 . 104 5 0 6
0 . 3 0 9 0 39
o 6 69 1to0
0 . 9 13557
1.000000

',/),

S I M P.SUiM- I N T E G R A T I

(A.

(AiA9,iA

ii.

Sf/H/)")

HEAL

I M AG I i\ yjRE

PART

5 E - 0 1

0 . 0 0 ('i 0 0 0 E + 0 0

•0.889 030t-0 1

M . 4 2 9 1 1 0E+ 0 0

0 . 8 8 8 9 7

1 . 0 \A 0 0
1. 5 0 0 0

• 0 . 8 8 8 7 Q 7 E -

0 1

• i'i. 7 4 9 7 7 4E+ 0 0

2 . 0 0 00

0 . 8 8 8 8 4 5 E -

0 1

•0 . 208 o7 P E - 0 1

• 0 . 8 8 8 8 b2E- 0 1

• 0 . 27 9 222E+ 0 0

i'i'.

2 . 9 0 0 10

1 088 9 0E+ 01

• 0 . 9 8 7 6 28 £ - 0 2

0E- 0 1

• 0 . 3 4to72 9 t , - 0 1

3.50 0 0

•0. 88888 0E- 01

• 0 . 1 6 3 9 i. v 9 E+ 0 0

4.000 0

0. 88888 ME-01

•0. b4o33bE-01

4.5 0 0 0

• 0.888884E-0l

•0 .

0. X

0(000

P . 0 0

5.5 000

•

10 37 2 D E + 0 0

0. 8 8 8 8 8 1 E - 0 1

• 0 . 9 307 1 o E - 0 1

0 . 8 8 8 8 8 4 L - 0 1

• 0 . 6 26;')99 E- 0 1

8E-01

• 0 . 2 4 9 9 17 E+ 0 0

4222 27 E+ 00

• 0 . 2 9 r,4 b 3 E - 0 1

8 8 8 9

1 I E - 0 1

0. 1 4 304 4E+ 0 0

f).

000 0

f>.

P M 0 0

• 0 .

7 . Hl>i0(0

0 .

7 . 5 i'l 0 0

•0 .8889 2 o E - 0 1

• 0 . 4 9 09 1 o E - 0 5

4 . 0 0 0 0

0. 8 8 8 9

• 0 . 14 3 0 3ME+ 0 0

0 . 8 8 8 8 7

4 P E - 0 1

S r i P b O W - I N T E G R A T I O'M
f

C(Yl

i l A i M N I NG
I N/\

HEAL

H . I ' i 0 i/i 0

\ ARE

PAhT

P A H T

2 8 8fa5 4 E - 0 P

0('ii4 0t'5iE+ 0 0

•••. r> 01/1 I'l

0 ,

2 4 9 9 9 8 E+ p i )

(':. 2 1 2 ' - i 4 b E + 0 0

1 .

0. 50 000 2E+ 00

• 0 . 8 5 104 I E - 0 1

0. 2b000faE+ 0 0

• 0. 38 2 o 7 2 E + 0 0

0 0 0 0

0 .

179 3 1 1 E - 0 O

• 0 . 2 b 7 o 8 3E+ 0 0

2 . D 0 0 M

- 0 .

1 2 4 8 38 E - 0O

•0 .

3 . 0 0 0 0

- 0 . 4 f a 4 9 fa 1 E - 0 o

. " 0 0 0

1.5000

piViMH

'4.

1 3 16 0 8 E - 0

4 . 0 0 0 0

- 0 . 4 5 6 18 6 E - 0 o

4.5000

0 . 4 39 b 2 9 E - 0 o

p.0000
-.3000

- 0 .

1 5 34 / i E+ 0 0

• 0 . 1 2 8 07 0 E + 0 0
• 0 . 1 042D6E+ 00
' i/i. 9 4 2 2 3 P E " 1 .

0 1

88 749ME- 0 1

P37 4 3 9 L- 0 b

• 0 . 8 8 1 19 3 E - 0 1

2E-0o

• 0 . 117 0 p 2 L+ 0 0

0. 18999

3 . 0 0 0 0

- 0. 8 33349 E-01

'0.

o.5000

- 0 .

• 0. 4 1 D P 1 PE- 01

7" .

- 0. 8 3334 3E- 01

0 0 0 0

7 . D'/MM
'< . ' 1 i'i 0 I ' l

0 .

lDfafa68E+00
/ 2b 5 18L- 0 b

- >/.. 8 3 3 3 2 4 E - 0 1

148 0 3 3E+ 0 0

0.04M99 2 E - 0 1
0 . 18 l b b 4 E - 0 b
- 0 . 5 4 09 9 1 E - 0 1

1 -SON - I N TEG H A T I O N

HAMMI NG

REAL

I NAG I R A R E
PART

PART

0 . 0 0 0 0
0.

5 0 0 0

1 . A (A (A (A
1 . b00 0
2 . 0 000
2.5000
3 . A (A A'A

0.7109 1 IE-02
0 . 2 2 2 8 8 6 E+ 0 0
0 . 54 7 1 14E+ 0 0
0. 2 2 2 8 9 5E+0 0
0. 7 1 1 09 3 E - 0 2
- 0 . 7 11 1 0 0 E - 0 2
0. 7 1 1 0 5 2 E - 0 2

3000
7.0000
7.5000

- 0 . 7 1 1 09 2r,- 0 2
0.71106EE-0 2
- !A. 7 1 1 0 0 O E - 0 2
M. 7 1 1 0 5 4 E - 0 2
- <A. 7 1 1 0 8 9 E - 0 2
- 0 . 09 3 57 l E - 0 1
- 0 . 187 1 1 2E+ 0 0
69 5 5 0 3 E - 0 1
- 0 . 7 1 1 07 5 E - 02

-'< . 0(/)l/)(/!

- 0 . 09 5 5 4 2 E ~ 0 1

5 0 0 0

4.000 0
4 . 5 000
3.000 0
5.

5 0 0 0

6.00 00
O.

0. 000yi00E+ 00
0 . 2 2 9 4 4 8 E+ 0 0
•0 . 79 08 D 0 E - 0 1
• 0 . 4 1 19 4 8 E + 0 0
• 0. 24 79 3 8 E + 0 0

0. l o 3 5 32E+ 00
•0 . 12059 0E+00
• 0 . 109 0 2 9 E + 0 0
•0.91058 0E-01
• 0 . 8 9 9 8 39 E - 0 1
0 . 8 8 5 1 57 L - 0 1
• 0 . 1 1 2 o 9 7 E+ 0 0
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