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NASA TN D-1078

TECHNICAL NOTE

D-1078

FOURIER SERIES OPERATING PACKAGE

Milton L. Charnow

Goddard Space Fright Center

Greenbelt, Maryland

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

WASHINGTON December 1961

https://ntrs.nasa.gov/search.jsp?R=19980227213 2020-07-09T10:31:59+00:00Z

H

FOURIER SERIES OPERATING PACKAGE

by

Milton L. Charnow

Goddard Space Flight Center

SUMMARY

This report presents a computer program for multiplying,

adding, differentiating, i nt e g r a t i n g, "barring" and scalarly

multiplying "literal" Fourier series as such, and for extracting

the coefficients of specified terms.

CONTENTS

oO

0

Summary .................................

INTRODUCTION ............................

SPECIAL REPRESENTATION OF A SINE,

COSINE, OR CONSTANT ....................

THE FOURIER OPERATING PACKAGE .............

Multiplication .............................

Addition and Subtraction ......................

Differentiation ............................

Integration ...............................

Bar Operation ............................

Scalar Multiplication ........................

Coefficient Extraction .......................

Series Evaluation ..........................

CONCLUDING REMARKS ......................

ACKNOWLEDGMENTS ........................

Appendix A - Nine Term Series in Standard

and Special Form .........................

Appendix B - Flow Charts ......................

Appendix C - Listing of Instructions ...............

Appendix D - Mystic Code ......................

2

3

4

4

5

5

5

6

6

II

25

47

.°.III

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FOURIER SERIES OPERATING PACKAGE

by

Milton L. Charnow

Goddard Space Flight Center

INTRODUCTION

The Hansen Satellite Theory as modified by Musen 1 involves various manipulations

of "literal" Fourier series, as such, before final numerical evaluation. To program this

formulation it is necessary to represent cosines, sines, and constants in a special man-

ner and thus manipulate the various Fourier series.

SPECIAL REPRESENTATION OF A SINE, COSINE,

OR CONSTANT

It was decided to represent any term, including the constant term, of the Fourier

series we are concerned with by two consecutive eight-place floating-point words. The

terms of this series have the general form

A n cos (iF+jE +kW+ 1U)

or

A n sin (iF+jE +kW+ 1U)

and the values of F, E, W, and U are not used until final numerical evaluation. The first

eight-place floating-point word represents the coefficient A n in an entirely standard man-

ner. The second eight-place floating-point word represents the sine or cosine term of up

to four arguments in an artificial manner.

1Musen, P., "A Modified Hansen's Theory as Applied to the Motion of Artificial Satel-lites," NASA Technical Note D-492, November 1960.

Considera term suchasAncos (iF + jE + kW+ IU). Ultil this term is evaluatedweare concernedonlywith thevaluesof An, i, j, k, and1. The ::oefficient A n is represented

by the first of the two eight-digit floating-point words. The four coefficients of the argu-

ments (i, j, k, and 1) are each represented by two digits of tl-e second eight-place floating-

point word. The first two digits represent i and are normalized to 00; the next three pairs

of digits represent j, k, and 1, and are normalized to 50. Thas, the value of i may vary

from 0 to 99, while j, k, and 1 may vary from -49 to +49. A cosine is denoted by a plus

sign; a sine by a minus sign, that is, 1 cos (OF + 0E + 0W + ()U) or cos (0) becomes

+10000000 + 01, +00505050 + 08. Any constant term can be represented as A n cos (0). A

few additional examples will be helpful (see also Appendix A):

Conventional

1/4 cos (IF + 2E - 3W - 2U)

1/'4sin (OF + 1E - 0W + 2U)

1/4

0

Special

25000000+ 00, +01524748 + 08

25000000 + 00, -00515052 + 08

25000000 + 00, +00505050 + 08

00000000 + 00, +00505050 + 08

The convention of having the first non-zero coefficient of th,_ argument word positive was

adopted. Cos(-x) becomes cos(x) and sin(-x) becomes -sin(x). Examples are:

(L2) cos (OF - 3E + lW + 0U) becomes (1/2) co.4 (0F + 3E - 1W + 0U)

and the special representation is 50000000 + (_0, + 00534950 + 08

(1/2) sin (OF - 3E + lW + 0U) becomes -(1/2) sin (OF + 3E - lW + 0U)

and the special representation is -50000000 + 00 - 00534950 + 08.

The first location address of a series contains the num)er of terms of the series.

A series of n terms would be represented by 2n + 1 words the first of which would be

the number n.

THE FOURIER OPERATING PACKAGE

The following series operations are performed by the t'ourier Operating Package:

• Multiplication

• Addition and Subtraction

• Differentiation

• Integration

• Bar (Special operation used in the Hansen sa=ellite theory)

• Scalar Multiplication

• Coefficient Extraction

• Series Evaluation

oOt'--0,-4

Multiplication

Multiplication of two series, where the terms are of the general form described

earlier and the values of F, E, W, and U are not used until the final numerical evalua-

tion, is according to the conventional trigonometric identities:

A cos X'B cos Y = (AB/2) cos (X+Y) + (AB/2) cos (X-Y)

A cos X-B sin Y = (AB/2) sin (X+Y) - (AB/2) sin (X-Y)

A sin X.B cos Y = (AB/2) sin (X+Y) + (AB/2) sin (X-Y)

A sin X'B sin Y = (AB/2) cos (X+Y) + (AB/2) cos (X-Y).

The Fourier Multiplication routine is composed of three major sections: the multiplier,

the collapser, and the arranger.

The Multiplier

Two Fourier series such as

AiA1 + A2A2 + A3A3 + . . . + AnA n (Series A)

and , , , ,BIBI + B2B2 + B3B 3 + . . . + BmB m (Series B)

which are to be multiplied are arranged in descending order of the absolute values of the

coefficients, that is,

IAI > A21> A 3 >...> An and [BI > I B2 > B31>...> Bm]

To facilitate further discussion, we shall denote any term in the A series AxA_,

any term in the B series as ByBy, and any term. in.the resultant, series by CzCz, where

Ax, By and C z are the coefficient words and Ax, By, and C z are the argument words.

The multiplication of the A series by the B series proceeds as follows: The first

term in the A series is multiplied by each term in the B series, then the second term in

the A series is multiplied by each term in the B series, and so on until each term in the

A series has been multiplied by each term in the B series. For exampl% AxBy is compared

AxBy * •with some numerical criterion e. If > e, then CzC z and C(z + 1) C(z + 1) are gen-

erated according to the trigonometric formulas already stated. If I AxBy _- e, then the

value of y is examined. If y > 1 (i.e., By is any term other than the first term), A x is

replaced by A(x + 1) and A(x + 1) is multiplied by B 1. If y = 1, the multiplication of the

two series is terminated since any further A(x + 1) By will also be less than e. The mul-

tiplication process continues until AnA* has been multiplied by BmB* unless the

numerical criterion or space limitations intervene.

The Co llaps er

Every multiplication generates two terms of _wo words each. The purpose of the

collapser is to combine like argument terms. Each _'gument term Cz is compared with,each other argument term previously generated and s_ored. If C z equals any other

argument term, the corresponding coefficient terms are added. Thus, there is no dupli-

cation of terms.

The Arranger

The final step in the multiplication is the arranging of the terms of the series.

ICll is comparedwithlC21 , IC31,etc. If ICzl)ICll, then C 1is replaced by C z and

C_ is replaced by C z. The process continues until the terms are arranged, in descending

order, according to the absolute value of the coefficients.

Addition and Subtraction

Addition or subtraction of two Fourier serie_ is primarily a process of com-

paring argument terms and adding the coefficients of like terms. A_ is compared suc-

cessively with B_ through Bm, A_ with B_ through Brl , etc., until A n has been compared

with B m.* If A x*= By,* the sum of the coefficients (A x _ By) and the argument term A* are* *

stored, and By and By are replaced by zeros, If A_ d)es not equal any By, both A x and,

A x are stored. After all terms in the A series have been compared with all terms in theB series, the remaining B series terms are stored.

Subtraction is accomplished in like manner after changing the signs of each coeffi-

cient term in the B series. The resultant series in each case is processed through the

arranger.

Differentiation

Differentiation, in this application, is with respect to the F variable.

_ A sin (iF +jE +kW+ 1U) - iA cos (iF +jE +kW+ 1U).

Example:

3w,

Thus,

O3

C_

and

{_-_) [I0000000 + 01, - 03524751 + 08 becomes +30000000 + 01, +03524751 + 08.

On completion of the differentiation, the resultant series is processed thru the

arranger.

Integration

Integration, in this application, is with respect to E.

tion of E. Thus,

cos (iF+jE +kW+ 1U)dE- el j +c2k sin (iF +jE +kW+ IU).

Example:

Withc 1 = 1 andc 2 = 1,

f[30000000 + 01, + + + 00, +01525250 08]dE becomes 75OOOOOO _01525250 O8.

The resultant integrated series is also processed through the arranger.

However, W is also a func-

Bar Operation

The bar operation is a special function in the Hansen Satellite Theory. It con-

sists of adding the coefficient of the F argument to the coefficient of the E argument and

substituting zero for the F coefficient. Thus,

Acos (iF+ jE +kW+ 1U) after barring becomes Acos (0F+ (i+ j) E+ kW+ 1U).

Example:

50000000 + 00, + 02534850 + 08 after barring becomes 50000000 + 00, + 00554850 + 08.

Scalar Multiplication

Scalar multiplication is the multiplication of the coefficient A n of each term by

a constant.

Coefficient Extraction

In the Hansen Satellite Theory it is sometimes ne _.essary to use the coefficient

of some term of a Fourier series such as a sine 1F term, cosine 2E term, or the con-

stant term of a series. Let us assume it is necessary to use the constant term. If there

is a constant term in that series, it will be the multiplier of the cos (0), or in special

form, that A n word which multiplies 00505050 + 08. We successively compare each

argument term in the series with cos (0) and extract that A n which multiplies the argu-

ment term cos (0). If no argument term of the series is c(_s (0), a normalized zero

(10000000 + 00 + 00000000 + 00 + 00505050 + 08), is printe(i.

O

Series Evaluation

The numerical values F, E, W, and U are only employed in the Series Evalua-

tion Routine.

To evaluate a Fourier series, the numerical valu_.s of i, j, k, and 1 are multi-

plied by the numerical values of F, E, W, and U, and the s_Lm iF + jE + kW + IU is de-

termined. The sine or cosine of iF + jE + hW + 1U is multiplied by the coefficient A and

the terms are added.

CONCLUDING REMARKS

Appendix A is the special representation of a nine ter_n series. Appendix B presents

flow charts and Appendix C a listing of instructions for the program packages for multi-

plication (including the collapser and arranger), addition t_r subtraction, differentiation,

integration, bar operation, scalar multiplication, coefficie_t extraction, and the evalua-

tion of the final series. Because this program was writter in Mystic Code for the IBM

709, an explanation of Mystic Code is given in Appendix D.

The Fourier operating package can be used with any t]mory that involves representa-

tions of functions by Fourier series. It can also be modified to operate with polynomials

of the form xaybzcu d.

o0

D-O

ACKNOWLEDGMENTS

The author is indebted to Messrs. R. G. Kelly and T. P. Gorman for their aid in the

construction of the package and for the translation into Mystic Code and to Aileen Marlo"

for preparing the flow charts.

AppendixA

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O

Nine Term Series in Standard and Special Form

The following is a nine term series presented in standard and in special form. Note

that in the special form, the first location (address) contains the number of terms in the

series.

Series in standard representation

.29467121 cos (0)

+.00010496334 cos (OF + 0E + 2W + 0U)

+.00005252596 sin (OF + 1E - lW + 0U)

+.000019845618 cos (0F+ 1E - 2W+0U)

+.0000066329604 cos (OF + 1E + 2W + 0U)

+.0000020107054 cos (OF + 2E - 2W + 0U)

-.00000036004597 sin (OF + 0E + lW + 0U)

-.000000055052357 sin (OF + 0E + 3W + 0U)

-.000000031090653 sin (OF + 1E + lW + 0U)

Series in special representation

90000000 + 01

+29467127+00,

+10496334-03,

+52525962-04,

+19845618-04,

+66329604-05,

-20107054-05,

-36004597-06,

-55052357-07,

-31090653-07,

+00505050+08

+00505250+08

-00514950+08

+00514850+08

+00515250+08

+00524850+06

-00505150+08

-00505350+08

-00515150+08

11

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0

Appendix B

Flow Charts

The following are the flow charts for the op-

erating packages for multiplication (including

the collapser and the arranger), addition or

subtraction, bar operation, differentiation, in-

tegration, and series evaluation.

2H

cO

0

?Coq)_t e I

criterion e

Initiate

N l , N 2

Flow Chart for Multiplication

Initiate

A series

counter a

=+2

Initiate

B series

13

A x By > O}

45

I y I

I I_t ermi..e I

for k x l

i k+50[

j+so 1+so IGet By I

Is it + or -?|

sin I c ,s //

M

14

Flow Chart for Multiplication (<'ontinued)

Determine

for By

i' k' + SO

j' +SO l' +50

Determine

i +i' i - i'

J+i' J-J'k +k' k -k'

1 + 1' [ - 1'

i+i' = +,07

\ + I o I

lj+j ' = +,0,-?_

_101-/l

+ I ° =+,O,-P_

_1° I-/

I Change

indicator

to zero

Ib

Change signs

of j+j', k+k °

1+1'; indicate

reversal

fC]lange

indicator

to zero

Change

signs and

indicate

reversal

t

t

0-..j

Gc

Flow Chart for Multiplication (Continued)

15

?Determine I

j+j'+SO j-j'+50k+k'+SO k-k' +50

1+1'+50 1-1'+50

i +i' > 99?

Arranger

Arranger

Form

i+i', j÷j', etc.

i-i', j-j*, etc.

in normalized

form

Was A_ sin

or cos_

16

Flow Chart for Multiplication (Continued)

B ,ino.co. 8 ' norco"sin [ Cos? sin I cos?

Ui

V=_O0

sin x sin y

sin x cos y

Fol'm

- _A_ey cos (X ÷ y)

+lA_y cos (x-y_

C C_ and c,. xC_, x

Form

_ AxBy sin (x+y)

1

+ _ AxBy sin (x-y)

C,C_ and C,. tC_ +x

cos x sin y _J cos x cos y

!Form

1

+ _AxSy :;in (x+y)

1

- _ AxBy _.in (x-y)

CzC _ _ ('i+1C_÷i

Check that _

no sin (0) ]

propagated

Form

I

+ _A_Sy cos (x*y)

1+ W A_By cos (x- y)

CzC _ and Cz+lC_, 1

v

Collal )ser

o_

O"

Flow Chart for Multiplication

Collapser

first

term

lAdd toe f f.

terms (new

coe f f. term=

(Continued)

17

1st term?

B series?

A series?

Arranger

18

Flow Chart for Multiplication (Continued)

Arranger

I

_o

_+2

Ax, l

Is A z < O?

Ax, 1 <

I Store A x Inow larRest

½

-- \,.. I _o/

cO

p--O

Flow Chart for Addition or Subtraction

Add _ubt r _t

_/2, and

counters

/ _+2 . \

Get By rand By

I

19

I InitiAte •counter for

r e_olning B's

Store

By snd B_

20

Flow Chart for Bar Operation

Initiate

Initiste

c_t ^ ^: IIndicate sinl

Determine [i,j,k,I

+Addi+j = j

Replsce i by

is j' > O?

Change signs

j', k, land

indicate change

was A_ sin or cos?

I + 50

Normalize

into A_

I

",4

j O0

Flow Chart for Differentiation

21

oo

o "Initialize 2NInitialize

_+2

Get Ax, A_

Is A_ positive?

22

Flow Chart for Integration

Initiate 2N

InitiRte

a+2

Get A x snd A'_

Indicate sine

or cosine

+i,j,k,I

tJi

Determine

jc kg

jc+kg =

Determine

A z : A_/kg

¢os_

Indicate

printout

Store A s

and A z

23

Flow Chart for Series Evaluation

oor-c_

?Initialize

series total

accumulator,

_, and series

counter a

_+2

Get A_,

+G_t IA_I

Determine

arglJments

i, j,k,l

and add

+Get

absolute

value of

angle

Determine

value of cos

Multiply by A x

AccumulAte

totnls

I

Determine

value of sin

Multlp[y by A x

Accumulate

totals

I

25

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Appendix C

Listing of Instructions

The following is a listing of instructions for

multiplication (including the coUapser and the

arranger), addition or subtraction, differentia-

tion, integration, bar operation, scalar multi-

plication, and coefficient extraction.

The square root instructions are for stand-

ard number representations and therefore have

not been discussed in the report.

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I OOOCO O O O C OOO +1 OOCO

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O OO(OC

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47

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Appendix D

Mystic Code

The Fourier Operating Package described in

this report was written for the IBM 709 in Mys-

tic Code. The foUowing is an explanation of the

Mystic Code.

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