- 57 -

Calculation of f orrnant frequencies

of twin-tube and n-tube approximations

of the vocal tract.

- J.G.Blom, E.O.Kappner -

- 58 -

Contents.

1. Summary

2. Introduction

3. Basic piinci�J�s 3.1. Four-pole th�ory

3. 2. Bi-!;eGtion method of Bolzano

4. Flow-chart nar�ati.ves·:. · ·:: ·

4. l. Twin-tube .pro,gram .

4.2. n-tube program

5. Applications

5.l. Twin-tube program

s. 2. n-tube program

References

Appendix I

p. 59

59

60 64

65 67

70 71

72

{l) Flow-chart twin-tube program 73 (2) Flow-chart calculations twin-tube program 75

Appendix II

(l} Plow-chart n-tube program

(2) Flow-chart calculations n-tube program

Appendix IIl Listing twin-tube prog:ram

Appendix IV

Listing n-tube program

Appendix V Output twin-tube program

Appendix VI Output n-tube program

76 78

80

84

90

92

..

- 59 -

1. Sununa;r,y

In this report two programs in Basic Fortran IV are described, which

calculate the formant frequencies of models of the vocal tract. The

first program calculates the formant frequencies of the so.called

twin-tube model. The second, which can be considered as a further

development of the twin-tube program,takes a series of 20 connected

tubes, each with uniform cross-area, as an approximation of the vocal

tract. In both programs the calculations are based on the theory of

four terminal networks and the bi-section iteration method of Bolzano.

The programs are primarily written for an IBM 1130 system, but can

easily be modified for other systems. The console typewriter is used

for the input of the parameters. The programs are self-instructing.

2. Introduction

The literature on the calculation of formant frequencies mentions

several models of the vocal tract.They all are one-dimensional models.

In most cases the boundary conditions are: u = 0 (u = velocity at

the vocal cords, and p = 0 {p z sound pressure) at the lips. Two

types of models can be distinguished;

(a) models in which the cross area is a continuous function of place,

(b) models in which the vocal tract is approximated by a number of

tubes with uniform cross-areas.

Examples of type (a) are presented in calculations based on Webster's

horn equation, as done, for example, by Ungeheuer (1). A well-known

model of type (b) is the so-called twin-tube, which was presented

first by Dunn (2), and later by Fant (3). Mol' s fo rmula (4), which i<·

built on other parameters, is easier to handle. Since the twin-tube

equation is trancendental, in the general case its solution can only

be found by numerical methods.

In the case of vocal tract models consisting of a greater number of

tubes, it is hardly possible to derive manageable equations. Using a computer, it is possible to avoid the derivation of the equation, as

will be shown later on { 4.2.6.).

In this report two programs are presented. Both are written in Basic

Fortran IV for an IBM 1130 system, but can easily be modified for

other systems. The first program, written by Blom, is a numerical

solution of the twin-tube equation. The second, the n-tube program,

- 60 -

written by Blom and Kappner, uses an algorithm to determine the formant frequencies of· a chain of an arbitrary number of tubes. For practical reasons the number of tubes is limited to 20 in this program. In modifications for other computing systems this limit can be adapted to the precision of the system. Further work in this field is in progress, and will be reported upon later. I n this report the following notations are used: s is the cross-area of the tube, u is the velocity of the air particles, v is the volwne velocity v = S.u ) , c is the velocity of sound propagation, p is the sound pressure, p is the density, f is the frequency, w is the angular frequency ( :,.) =·2nf )

3. Basic Principles

I I k 1.1

I � I I

Po , I P1 I I : Si I I \ • \ VO ' " v1 '

Fig.1

For a tube with uniform cross area, which has a length 1 and a . 1 cross area s1 at both ends

where A 1 , a1

, the relation between pressure and volume velocity Cp0,v0 and p1,v1 respectively) is given by:

_

=

C1 and D1 are the four-pole coefficients:

•'

- 61 -

wl1 .... Ai

= cos B1 -

c

s 1 w11 c1

c i sin I 01 .::::

pc c

I s1 $2 s_ :>

----------

Po VQ P1 P2

P3 v3 v1 v2

F'ig. 2

wl 1 i .f?E. sin

s 1 c

wl1 cos

c

s n-2 s s n-1 n

�" pn-3 v n-3

p v n-2 n-1 vn-2

v n

I

For n connected tubes with uniform cross areas, with lengths and

cross areas respectively 11, 12,13, . . . ,ln

and s1, s2,s3, ... ,sn

' and

pressure and volume velocity at the end of each as given in fig. 2,

we have:

If we define:

then we have

Assuming that the closed ending ( vocal cords ) is at the first tube,

- 62 -

and the open ending (lips) at the n1th, we have as boundary

conditions:

Thus:

v = 0 0

Then the condition

and 0

D = 0 (1) must be fulfilled. This is an equation in w

us the resonance frequencies of the system.

the solutions give

For a small number of tubes, the matrix multiplication can easily

be carried out. In the case n = 2 ( twin-tube ) , the condition (1)

becomes:

or:

i s

1 sin

wl1

pc c

wl1

cos -

c

i

cos

.2..£ sin s2 t�l2 -- -

c

Introducing the parameters:

s1 5

2

l = total length � lJ

+ 12

k = ratio of the cross areas

=

=

excentricity := � 11 -1

relative excentricity =

wl2 + w1,_

cos cos c c

wl1 wl2

sin -- sin --

c c

I

( see fig. 3 )

this equation can be reduced as follows:

wl2 0 =

c

= 0

and

- 63 -

[ wl � cos c + cos

wl - · 1-k 2wll 0 cos 1+k

cos -- = c c

wl 1-k or: cos - -c l+kcos

<.i.161 c

= 0

I r<:: �11

{'j, = tl - 11

::

1 12

, , I

I ' I

• s2 I I I I \ \ \ \ \

I I !E--- 'l 1 fl I � � -)!

�-(11 + 12) Fig. 3

11

cos

( 2)

I ?If

= -2-C 1 ?.- 11)

6 has been introduced as an absolute value. Since it only appears

under the cosine function, its sign is of no consequence.

The equation (2) is the well-known twin-tube eq�ation in the form

given by Mel. The twin-t4be program starts from this equation.

Since the complexity of the expres s ion D increases rapidly with

the number o� tubes, a s lightly different approach must be chosen

for the n-tube program ( see 4.2.S ) .

- 64 -

3.2. Bi-section method of Bolzano. ---------------------------------

Suppose f(x) is a continuous and bounded function on ( O ,�). We want to approximate the values of x, fo� which f (x} = O . ( Let us assume that such valuei:; exist.) Let these values be denoted as x1 , x2 , • • . ,xn , · · · · successively. Suppose that we can find a certain value d, (d > 0) , such that for each pair of successive zeroes holds : xn - xn-i > d , and a value c (c > O) , such that x1> c . Then we construct a series of trials, denoted as x = t t t O' 1' 2· • • • •

t , n in the following way.

2

The numbers O, I, 2, • • • denote the

successive trials t0, t1

, t2, • • o •

x

Fig.4 ;

The first tri.al is : t0 = c. We dc-termine the sign of f ( t0) . The next trial is: t = c + d, and the sign of f {t1) is determined.

• 1 If this sign is the same as the sign of f (t0) , the next trial is: t2 = c + 2d . In this way we continue, every time determining the sign of f(tn) . As long as this sign is the same as that of f(t0), the n�xt trial is obtained by adding d to the preceding. We assume that f(t ) t 0 for each n • (If f (t )=O for a certain n , n n the zero is found, and the process is stopped.) We continue, until we have got a trial tk = c + kd, ·for which the sign of f (tk) is opposite to the sign of f (t0) • Then the zero x1 must be between the last two trials, tk-l and tk. We n"ow "go back­ward" in steps of %d ( that means: tk+l = tk-�d ; -tk+2:::: tk - d) , again determining the sign of each tn' until the first trial tk+i for which the sign of f (tk+i> is opposite to the sign of f(tk) ( Thus: i �

1 or i = 2. ) Then x1 must be between the last two trials, and we "go forward" in steps of �d. This process is con­tinued until x1 is approximat�d sufficiently accurately. To approximate the value of �2 , we repeat this process: we start from the approximated value of x1 and take as the first trial x1 + d , �nd so on. The decision " go forward " or " go backward " must now depend on signs opposite to those, determining this decision when approxima�.ing x1 .

- 65 -

It is clear that the error in the approximated value of x2

is independent of that of x.1. The iteration method here described is known as the bi-section

method of Bolz�no. It is well suited to high speed computers.

4. Flow-chart narratives.

Both programs are self-instructing. The instructions are p£inted on the console printer. The keyboard is used for the input of the parameters, the punching of data decks thus being avoided.

Appendix III is a listing of the twin-tube program. Appendix I con­sists of: (l) a flow-chart of the twin-tube program, (2) a separate flow-chart of the calculations carried out in the

twin-tube program ( 11CALCULATE" in flow ... chart ( l)) . Observing the following remarks, the course of the program can easily be read from these appendices.

4.1.1. c is defined as 35000 cm/sec, a normal value for the warm air in the vocal tract. By means of d.ita en try switch 14 this parameter can be changad (stutements 1�.iO ff. ). In statement 145 a test is built in for extreme values of c.

4.1.2: If data switch 15 is on, the progrnm stops.

4.l.3. Every time when reading par��eter values, the program tests the field. This is realised by reading all the places out of field; if a value + 0 is encountered, an error message is produced.

4.1.4. The counter INT causes the instruction in statement 255 to be printed once only, namely after the first run of calculations.

4.1.5. Three tests for the values of the twin-tube parameters are

built in (statements 195, 205, 215.) . In the case of extreme values, the program gives error messages . The tests are the following: (a) 1 must be between 10 and 25 cm. This is only because of the

practical object of the program� calculating models of the vocal

- 66 -

tract. Technically the program could go beyond these limits. Difficulties only arise at very great values for 1, for example: a model having k = 8 and B = 0 , would produce, at an 1-value of more than l meter, frequencies of the first and second formant so close to each other, thJt they would come within one single approximation step {see 3.2. and 4.1.7.).

,. {b) k must be between 0 .1 and 10. This limitation is for mathematical reasons. Writing the twin-tube equation {3.1.,(2}) as:

wl 1-k wBl cos c:- = �+k cos -c-

the solutions can be found as the intersection points of two cosine functions, one having maximal and minimal values of +1 and -1 ,

1-k 1-k the other of l+k and - l+k . Both for great and small values of 1-k . k, the value of l+k is close to 1.

In this case, in the neighbourhood of those values for w for which both functions are about maximum ( or minimum ) , there are two intersection points close to each other (see fig.5). It now may happen that these points come within one approximation step,which means that two formant frequencies may be missed.

1 ------------ ---1-k .,,,,,,. - - - - -- - - - - - - -.-S.--C:i....._ 1+k

Fig.5

(c) e must be between -1 and +1 . By definition, e has no physical meaning for other values.

4.1.6. Statement 245 + 1 tests fo� the sign of the expression D (3.1.,(1)), which for the twin-tube has the form of 3.1.,(2). Two parameters T1 and T2 are used, each of which is either +1 or -1.

- 67 -

rhe sign of T2 changes every time that the sign of D is changed;

the sign of 'I' 1 changes every time the program starts to approximate

another formant frequency. This is the way in which the decision

"go forward/go backward" is governed.

Since 0.1 s k s 10, (see 4.1.5.,(a)}, it follows that:

3.nd: wl cos c

0 < � < :l , 1+k

l-k cos �is 1 > 0 for small values of w • 1+k c rhe first trial t0 ( s�e 3.2.), which only served to determine the

sign of the expression D for small values of w , can be omitted.

Both T1 and T2 are first defined as - 1 .

4.1.7. The.value of F ( trial frequency ) is first defined as

zero, and DF (increment of F ) as 32 Hz. Therefore the trials t0,t1 ,

t2, • . . • are: 32, 6 4,96, . . . • Hz, until the sign of D changes. In other

#ords, our assumption is that the difference between two successive

formant frequencies is more than 32 llz, and the first formant

frequency is more than 32 Hz.

4.1.8. In statement 235 the step length is devided by 2, and 235 + l tests for the step len<;Jth: if this is·less than� Hz, the process is

3topped. Statement 250 represents a rounding-off procedure: � Hz is

added to the last Lrial, and the figures to the right of the decimal

?Oint are deleted. Tllis deletl.o:n is effected by the FORMAT-statement;

this is not mentioned in the flow-chart.. In this way we round off

to the nearest whole number of Hz. If the fractional part of the

last trial is exactly .S, rounding off occurs to the next greater

�hole number of Hz.

4.2. n-tube program.

�ppendix IV is a listing of the n-tube program. Appendix II contains

the flow-chart, and, just as in the case of the twin-tube program,

a separate flow�chart fqr the calculations. The following remarks

refer to these appendices.

- 68 -

4.2.1. In three places the flew-chart has the words: WRITE INSTRUCTION,

followed by a number in brackets. These numbers refer to the sections

of Appendix VI where the respective instructions are shown.

4.2.2. There are two input modes:

mode 1: all tubes have the same length�

mode 2: there are tubes which have different lengths.

4.2.3. The following table gives a survey of the parameters to be

entered through the keyboard and of the data entry switches by means

of which the parameter values can be changed.

���-2f _E�E��S�! MODE

NMB KTOT

c S{I)

In the case of mode 1:

��en!ns input mode,see 4.2.2 number of tubes ($ 20) number of formant

frequencies to be

calculated (s 4 )

velocity of sound

diameter of the Ith

tube

(I c: 1 I • • • I NMB)

LN length of each tube

In the case of mode 2:

LNGTH(I) length of the Ith tube

( I:::: 1. I • • • • I NMB )

9eEe-�n�EY-��!�£h l 2

3 4

9, to change diameters

of all tubes

8 I to change diameter

of one tube only

5

7, to change lengths

of all tubes

6 I to change length

of one tube only

If data switch 6 or 8 is on, the program first inquires which tube

must be changed.

Data switch 15 causes the program to stop.

The course of the program when testing for the different data switches

can easily be read from the flow-chart.

- 69 -

Contrary to the twin-tube program, no tests are built in for

parameter.values and for the field. The program, therefore, is

supposed to work in an error-free environment. Errors in the

calculated formant frequencies arc data-dependent.

4.2.4. The diameters of the tubes are specified in the input.

Since the calculations are in terms of cross areas, the diameters have to be converted accordingly. Only the ratios of the cross

areas are of importance, so it is sufficient to square the

diameters ( statements 205-210).

4.2.5. Contrary to the twin-tube program we do not have an equation

to start from. We therefore proceed as follows. The trial

frequency is filled in in the four-pole coefficients of the dif­

ferent tubes, so that these become constants. With these constants,

the matrix multiplication is carried out, which gives us the nwn­erical value of the expression D for this trial.Depending on this

value we take the next trial as described in 3.2.

In the program this is realised as follows.

After defining the trial frequency, the unit matrix is loaded in a buffer AMAT (360 ff.).

A matrix BMAT is defined as the fou:s�-pole matrix of the first tube

in which the trial frequency is filled in ( depending on the mode:

statements 365 ff. and 375 ff, or 380 ff.). Subsequently &1-1.AT x BMAT

is calculated (385 ff.), and this product is stored in buffer

AMAT ( 385+4 ) . Now BMAT is defined as the four-pole matrix of the

second tube, and AMA'!' x BMAT is calculated agaj.n, and this product

is stored buffer AMAT, and so on. When this loop is ended, the buf­

fer AMAT contains the product of the four-pole matrices of all the

tubes. Statement 410 + 1 tests for the lower-right element of this

product matrix.

4.2.6. The 2x2-matrices to be multiplied contain real numbers in the

principal diagonal, the other two elements being imaginary.The

product of two of such matrices is of the same form again. Only the

lower-right element of the product matrix is used, which is real.

As a consequence of the imaginary numbers, certain terms in the

product have minus signs.These are stated in the matrix multiplicatior

in the statements 385 ff.; the factors i themselves are omitted.

4.2.7. In the lower-right element of the product matrix, the factors

pc from the four-pole coefficients cancel each other out. This is

- 70 -

why they are left out in the calculations.

4.2.8. F (trial frequency) is first defined as 32 Hz, and DF

(increment) as 64 Hz. The assumption, therefore, is that the first

formant frequency is more than 32 Hz, and the difference between

two successive formant frequencies is more than 64 Hz.

4. 2. 9. Two parameters •r1 and T2 are used in the same way as in the

twin-tube program (see 4.1.6.). For the twin:-tube the sign of the

expression D for small values of w is known beforehand. In the n-tube

program this Sign is determined in the first trial (32 Hz) and

depending on this sign a parameter TT is set to +1 or -1 ( 395 ff.).

5. Applications.

��!�-'.!'i�n:��E�-E!99!�· Appendix V shows an output of the twin-tube program.

5.1.l. In the first section , marked (1)1 the twin-tube parameters

are defined on the basis of a print-plotted twin-tube model and an

instruction is printed, prescribing how to enter the parameters.

When the "program start"-key is pressed, three pairs of brackets

are printed, in which the parameter values have to be filled in.

After pressing the "end of field"-key the calculations are carried

out and the formant frequencies are printed on the next line. By

pressing the "program start"-key once again, new parameter values

can be entered.

5. l . 2. In section ( 2) , the parameters of the model for ( a ] , a tube

with a constant cross area, are filled in. The formant frequencies

are the odd harmonics, produced by an organ pipe closed at one side

and open at the other, and having a length of 17.5 cm ( the average

length of the male vocal tract ) .

5.1.3 In section (3) the parameter values of the model of Dutch [a]

are filled in (see Mol, (4)).

5.1.4. In section (4) the velocity of sound is defined as 34000 cm/s

(data switch 14), and the same twin-tube parameters as those in

section (3) are filled in. As is to be expected, a lower value for

c results in lower formant frequencies.

- 71 -

5.l.5. In sections (5), (6) and (7), the parameter l, k and S, respectively, are out of their specified ranges ( see 4 .1. 5.) •

In each case an error message is produced and new brackets are

printed. In section (8) all the parameters are within the specified

ranges again.

����-�=��e§_Ef99E��· The following remarks refer to the output of the n-tube program,

reproduced in appendix VI.

5.2.l. In section (l) , an instruction is printed and some of the

parameters are defined. After pressing the "program start"-key ,

brackets are printed to specify the mode, the number of tubes, the

desired number of formant frequencies, and the .velocity of sound.

5.2.2. In the case of mode l,the instruction in section (3) is

printed. After this, brackets are printed for the specification of

the dimensions of the tubes. In section (4) the model for [a ] is

calculated. Section (5) shows the output when calculating the

twin-tube model for Dutch [al, first for c = 35000 cm/s and then

for c = 340 0 0 cm/s, just as is appendix V.

Whereas in the twin-tube program the ratio of the cross areas is used as a parameter, in the n-tube program it is the diameters of

the tubes that have to be specified.

5. 2.3. In section (6) LN is changed by means of data switch 5. The

computer calculates again, the other parameters remaining unchanged.

A comparison with section (5) shows that greater lengths result in

lower formant frequencies ( the "law of proportional growth" , see

Ungeheuer (1)).

5.2.4. If data switch 1 is on, the program branches to the

beginning, immediately after the first instruction (section (7)).

The mode is now defined as 2. An instruction is printed which is

slightly different from that printed in the case of mode 1.

5.2.5. As can be seen in section (8) , the specification of the

dimensions of the tubes takes two lines when the number of tubes is

more than 10 .. First ten pairs of brackets are printed on one line,

and after the values for the first ten tubes have been filled in

and the "end of field"-key has been pressed, the remaining pairs of

brackets are printed on the next line.

5.2.6. Section (9) is produced by setting data switch 8 on. Only

the diameter of the 18th tube is changed; the remaining parameter

values are left unchanged.

- 72 -

References.

(1) Ungeheuer G.: Elemente eincr akustischen Theorie der

Vokalartikulation. (Springer 1962} (2) Dunn H.K.: The calculation of vowel resonances, and an

electrical vocal tract. (J.A.S.A.,vol.22,p.740 ff.,1950)

( 3) Fant G.: Acoustic theory of speech production.

(Mouton 1960)

(4) Mol H.: Fundamentals of phonetics II. (Mouton 1970)

- 73 -

Appendix I.

(1) Flow-chart twin-tube program.

130,

START

.INT = 0

c = 35000.

WRITE INSTRUCTION

125 PAUSE

on

on off 135 WRITE:"V( ) " ,__ __________ _,

READ C

14o no

-=>--�ERROR MESSAGE

160 . WRITE:"RESET DATA ENTRY:

SWITCH 14 AND PRESS PROGRAM $TART KEY"

165 WRITE BRACKETS FOR SPECIFYING PARAMETERS

READ K, BE'l'A , L

on

PAUSE

155 no

'·!

- 74 -

no 18 �----1 ERROR MESSAGE

200 no 195 =::=----�ERROR MESSAGE

21 0 no

205 _:;;;-----t ERROR MESSAGE

215 no

yes

CALCULATE

250+1 WRITE FOR(K), K=1,4

INT = !NT + 1

+

WRITE:"PRESS PROGRAM START KEY TO ENTER NEW PARAMETERS. SET

DATA ENTRY SWITCH 15 AND PRESS PROGRAM START KEY TO. STOP."

225

260 WRITE: "END OF TWNTE"

STOP

- 75 -

(2) Flow-chart calculations twin-tube program.

230 p

R

245

DF = DF/2. 235

0 250

= 6.283825 .. LNGTH I c

( = p • BETA ..

= ( 1. .. AK)/(1. + AK)

F = o.

DO 250 K = 1,4 ----- ---- -- - ---,

N = 0

= 32.

T1 = (-1.)**K

N = N + 1 240

T2 = (-i.) .. N

F = F - T2•DF

+

0

I FOR(K) = F + .5 --- - ---- - - --- - - l

- 76 -

Appendix II. (l) Flow-chart n-tube program.

START

WRITE INSTRUC,TION ( 1)

PAUSE

WRITE BRACKETS FOR 145 SPECIFICATION OF

MODE, NMB, KTOT, C

READ MODE, NMB, KTOT, C

1

155 170 WRITE INSTRUCTION (3) WRITE INSTRUCTION (7)

WRITE : "LN ( ) " 160 ----..---.J WRITE BRACKETS

SPECIFICATION LENGTHS AD LN

READ (LNGTH(I),1=1,NMB)

WHITE BRACKETS SPECIFICATION

DIAMETERS

READ (S(I),I=11NMB)

190

SQUARE DIAME'I'ERS (I=1,NMB) 205-210

CALCULATE 315 ff.

WRITE (FOR(K),K=1,KTOT) 415+1

PAUSE 215

on

on

420( STOP) e)

- -, • r o5

WRITE:'. LN=(

READ LN

305 WRITE:"I=(

READ I

WRITE:" S (I)= (

READ S(I)

S(I)=S(I)••2

310

- 77 -

off

) rt

) "

) "

un ?30

�/RITE: 11KTO'l' ( ) II

E AD KTOT

on 2l�O

WRITE:11C= ( ) II

READ C

on

250 WRITE: "NMB ( ) "

1

on 275

VJRI TE� 11 I= ( / ----......

READ I

) II

WRITE:"LNGTH(I)=( ) 11

EAD LNGTH(I)

on

WRITE BRACKETS SPECIFICATION

LENGTHS

READ (LNGTH(I),I=1,NMB)

- 78 -

(2} Flow-chart calculations n-tube program.

315

1 2

320 I.NN ::. f1.283825*LN/C 325 DO 330 1=1 ,tlMBi--------

365

BMAT1 -

BMAT4 =

LNG(I):6.283825•LNGTE(I)/C

33.5 F = 32.

M = 0

DO 415 K=1,KTOT ------- �------------­'

N 0

DF 64.

N = N+1 3/.i.5

T2 = -1.•o.N

350

., I

+

0 F = F - T2•DF 355

AMA'l' = I 360

2

COS(LNN•F)

BMAT1

P = SIN (I .NN • F' ) I I I � I I

CJ

- 79 -

�----------- DO 390 1= 1,NMB 370

375

Rf-A'.AT2 =

BMA'I'3 =

I

1 2

380 P/S(I) p = SIN(F•LNG(I)) P• S (T) BMAT1 = COS(F•LNG(I))

BMAT4 =- BMA'l'1 PMA�'2 = r/S(I)

BMAT3 = p,. s (I)

CMAT = AMA'l'• BM/\T 385

'--------------- AMAT = CMA'I'

400 T'I'

11-10 'I' 1 =

DF = DF/2.

+

+

+

TT·-�••K

0

1. 405

+

4 FOR(K) = F + . 5 15L-----------�--'

I I �·�3 II FC� *L!ST l\LL *Inc�1CA�n,TY�E��JTf�.��YBO�RD,1132p�!\T[�,nAD[RTA�E,crs�.�LOTTfRl

c T 1, \TOO 10 (

c

c r '-

c ( r � c , \.

(

(

�-·'

, ,., - ·�

l c:. . ,,

?r: 2� 7A . ' _, '.2C:.

4� 4'.:

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N T U6 0 Lt 5 0 "\ T U B 0 4 6 0 N T U B C 4 7 C

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00 Vl

1 6 5 \ D r = \ n o + 1 I i:- < � � ::> r) - 1 l 1 7 C t l 7 C' , 1 7 5

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lv '< ! T E ( l , 8 5 >

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1 7 5 W C< ! T £ 1 l t 9 0 )

! F' ( �• N ) l R 0 t l R O , l R 5

1 � 0 'r'l '< I T E <l' t 1 "3 5 l ! l , ! = 1 , N �-B l

't.' Q I T f < 1 , 7 0 l < ( l H A A K ( I I t ! = 1 , 8 J t J = l • �\ ,\;8 ) R E A D ( 6 , 7 5 l I �- N G T H I 1 I , I = 1 t � �" B l G 0 T O 1 9 0

1 ? 5 rl '< l T ( ( l t l 3 5 l < ? t T = l t l C J

W R J T [ l l t 7 0 l : l ! H A A ( ! I l t l = l 1 8 l t J = ! , l O l

R E A D < 6 , 7 c; J < LN G T H < I l , I = l , l � l 1J '< I T E < 1 , l 3 5 l I 1 , T = l 1 ' N '-'B l W R I T f < l , 7 0 l < I I ,_.. A A '< I I l , I = : , � ) , J = l , N ,'� : R � A !°) l 6 t 7 5 > < L � G T � ( l l • I = l l t �� "' e l

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2 1 5 P � U $ E

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G O T O f � 2 0 t 7 2 C J , � S T C P

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,..., ·, c f � 3

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<X> O"I

2 3 0 ,._, � I T E I l t 1 0 0 I R E A C, 1 6 e 6 0 > A K T () T

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G I') T 0 I 2 4 0 t 7. 4 �· l t �· C 2 4 0 W � I T E C l e l 0 5 l

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G � T O l � 5 0 e 2 5 5 l t � � � B 2 5 1) IJ � I T f ( l t 9 � l

R E A D C 6 e 6 0 l A N � B

/IJ V B = A N !'I B � N = Wl.8 - 1 0 G O T 0 < 1 6 0 e l 7 5 l t � � D f

2 5 5 G 0 T 0 C 7 6 0 t ?. 7 0 l t �0 D E

2 6 C' C .4. L L D /I T S �" 1 5 , N � � I

G O T O C 2 6 5 t 3 0 0 l t N L � 2 6 '7 W � P f: < l , 5 5 J

R f-: A G ( 6 , 6 0 l U.; G '.) T O 3 0 0

; 1 0 C A L L D A T S � < 6 t N L l G 0 T C < 2 7 5 e 2 8 0 l t N L

2 7 5 I\' R I T E < 1 ' l l 5 >

R f A D l 6 t 6 0 l A I I = A ! �/ � J T [ ( 1 . 1 2 0 >

R !=: A D l 6 t 1 2 5 l U,I G T H ( T ) 2 e o � A L L C A T S W l 7 t N L >

G 0 T " < 2 8 5 t ?- 0 0 l t N L 2 iJ 5 W R I T E 1 l e 9 0 )

! F f � .•; l 2 9 0 , 2 9 0 , 2 9 5

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� E A D C !> , 7 5 : ( L f\: G T H ( I l • l = l t \!� 6 ) G O T � 3 0 C

2 9 5 W Q I T E < l t l 3 S > t I , I = l t l O >

W Q T T E C J , 7 0 l < I I � AAK < I l t l = l • ? l , J = l t l O l R � A f� c 6 , 7 5 l ( l N G T H < ! ) t I = l , l 0 l W R l ! F l l t l 3 5 l l l t ! = l l , N � B l 1-I <:.; l ' ( C l t 7 0 l ( I i ·�A A � I I l t I = l ' 8 l t J = l t \! N l

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� T UB l 3 8 C

f\: T UB 1 3 9 C f\: T Uo l 4 0 0 f\: T U 8 1 4 1 0 N T U B 1 4 2 0 N T U8 l 4 3 0 N T U9 1 4 4 0 N T U8 1 4 5 0 N T UB 1 4 6 0 N T UB 1 4 7 0 N T U9 1 4 8 0

N T U8 1 4 9 0 N T U8 1 5 0 0 N T UB l 5 1 0

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co ....J

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G � T r ( 3 0 5 t 3 l 0 l t ".' S

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I = A I r\' � l'!' E < l t l 3 0 l

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t"' = O · D ·'.) ° 4 1 5 l( = l t K T () T l\J = 0

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/l. 1� A T 2 .,, O . A V A T 3 :: � . f\ "! A T 4 = l .

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o V A T 3 = P * S < J >

P A G E � 5

t\ T U tH 5 9 0

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f\ T U B l 8 2 0 N T U8 l 8 3 0

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CX> CX>

G f') T 0 3 8 5 3 8 (' F' = S I N < F * L �;r, I l I l

B v A T l = C 8 S I F * L N G 'I I l > B V. A T 4 = B 't< A T 1 B t.' A T ? = P I S ( I l 3 M A T 3 = 0 * 5 1 1 )

:: A 5 ( a,.i A. T l : /dv"A T l * B � A T } - A V f � 2 * B � A T 1

3 9 0

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( >.' /\ T I• A � fl T l 6. � A T i:

= A � /I T 4 = ( •/ fi T 1

= C '•1A T 2

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* B'-' A T 2

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l\ '; lJ £3 2 0 9 0 N T U B 2 1 0 C \ T :J C 2 1 1 0

l\ T U b 2 l 2 0 '- T Ju 2 1 3 0

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P iWGHAf.I T \ i lJ T E , T•.N T E C O l l PUT E S F l , F 2 , r 3 , F 4 u r T l l [ T\. I IH U (; [ f·I O D E L o r- T J : [ V O C A L T i�J\ C T . U E F I IJ I T I O N O F PAHAJ.I E T E RS .

. . . . . . . . . . . . . . . . . . . . .

0 1

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x [) x X 0 . ) L

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T O C l :A:� G E v S ET Df\T,'\ r:: rn ;� y S\ . I T C I : 1 4 . U S [ K E Y B<11\ R D T <J F. I J T E i< r/\ IU\l i [ T [ i{ V A L U l: S .

x

LI S E A D E C l i l/\ L PO l l J T J\l'W P I U :: s s [ I m (I F F I F. L l l l� f:: Y . I I- Y O U S T l( I K F. A � J iW !·H� K !: Y P : � r. s s i: I U \ S F. F l f:: L D K !: Y /\ Im H E E 1H E i< Pi\RAf-1 E T E H S . F I : �sT P i{ E S S P I UJ (; RJ\i·i STAin K E Y .

K LJ E T J\ L ) ) ) V = 3 5 0 0 0 . r;r.:/ S

1 . 1 7 • !j

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PiU : s s P i W G �/\1·1 S T A l n K E Y T O F. 1H F. R : H .: \ . p ,\ IV \l l E T E H S . S ET [)/\ T A r:rn;<Y S\d T C H 1 5 At-:O P H E S S p q o r. ;L\l l S T. \ IH K E Y T O S T O P .

( 1 ) .

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I rt" §. ro "O 11 0 l.Q 11 � .

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l ( )

3 4 0 0 0 . R E S E T DJ\TJ\ EIH H Y S\1 I T C I � 1 4 i\IW P!l E S S P :WC iUd i START K F:: Y .

1: p, E T 1\ L

J( 4)

V = � 4 0 0 0 . C t l / S 8 . 1 7 . 5

F 1 = 7 li 2 • C I S F 2 = 1 1 8 2 • C I s r 3 = 2 7 O 4 • C I S F t1 = 3 1 2 5 . c I S

K 8 E T A L l ( . ) ( ) ( ) V = 3 4 0 0 0 . C M/ S 5 . . 5 3 0 ( 5 ) * * L O U T S I D E H/\ N G E 1 0 2 5 •

J c , , * * ... tff E 1l PA.�Al lET E K S A N O P i l E S S E t-!D O F F I E L D K [ Y . � !� l\ E T /\ L l ( ) ( ) V = 3 4 0 0 0 . C i l / S

1 2 . 1 7 . 5 (6) * * K O U T S l lH: H /\ M G I:: 0 . 1 , 1 0 , J * * E N T [ H PA i�/\l·l [T [ i { S A N [) P i U � S S c rr n ·o r r I [ L [) n: Y .

!.: n F. T /\ L l ) V = 3 4 0 0 0 . C l·i / S 8 . 2 . 1 7 . 5 ( ? )

* * B E T J\ O U T S I O E llM�GE - 1 , + l , J * * [ N T E R r/\:{/\l \ET E ll S /\IW P R [ S S J: P O O F r I E L D K E Y .

1: e E T A L l ( ) C ) C ) V = 3 4 0 0 0 • Cl-1/ S . 1 2 5 . 2 5 1 5 . ( 8 )

F l = 2 5 2 . C / S F 2 = 1 7 9 2 . C / S F 3 = 2 9 G 8 . C / S r 4 = 3 7 0 9 . C / S _J EIJD O F nm T E

P RO<;HAl·I rn u rs [

IHIJ B E COMPUT E S F. l , F 2 , F 3 , F 4 O F A I J rx - r u 1n : : 10f)f: L O F T l l E V O C A L T n l'.r:T .

u s i: I r� P lJT f.1 0 0 [ 1 I F ,\ L L T U G E S r A v r:. r 1 · i:: S /\1 1 E L E !� G T l i . I F H ! I S I S IJOT T l l [ C ,'\S [ u s r: l i� PJ T I HJ U E 2 . M M B = N UH l3 E i! o r T U R E S .

w ; B S H lJ U L O B E Si iAL L [ i� T H � !J o ;� [ 0 U A L T O 2 0 .

K T O T = NU ! i l1 E R o r F U Rf.l/\IHS Y li lJ l . A fH T O l './\ V F: C IJ l- i P U T E n . K T O T S l ! O IJ L O B [ S l 1/\ L L E ! l T l l/\N O r{ E O U;\ l T n 4 . C = V E L O C I T Y U F- S O LliW .

U � E K E y !)()/\ i"W T O on E n T H E PA it/\I i [ T E t1 S . IJ !:> F. D E C l l l;\ l PO l iH S A r m p ;l r: s s [ ! J f) o r F I E L D K r: Y . I F Y O U S T H I k F. A \ 1 1lufJG K F. Y P ii E S S [ .{ ;\ S f: r: I E U1 K :: Y td�f) H [ E :H E H PAHt.f.l [ T E H S .

T u C l l A N G E M U O E , rm B , KTOT U ll c S !. T T l ' E F O L L U \ . t :� r, [)/, T A S\; t T C f l E S ,\ r ; n P R E S S P 1 H 1 r �Al1 S T t. R T K E Y J.lUDE 1 fu.1B 2 K T O T 3 c 1,

I F Y O U C l !/\ N G E l lO D E , W IB KTOT J\I J D C 1-I U S T l: [ ;l [ ;:iH [ :� C O . S ET DAT/\ S \ i l T C l l 1 5 l\tJO P R E S S P IW f. R l\i1I S T /\IH l� E Y T O ST O P .

F I H S T P H E S S P P.O G R /\ 1 1 S T fd H K E Y .

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1 . 1 . 4 . 3 5 0 0 0 .

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Nl·: n = l·IO U T H O P nl I i � G

T 0 C I i A I� G E U J S [ T D A T A Si.; I T C I : 5 •

T (; C t lAll.J G E C l � [ S l i4 G L E S ( l ) S ET l l :� T f\ S 1 i l"f C l l 8 . TU C 1 1 f\ N G F. A L L T I : E S ( I ) S ET D A T ,'\ S \'. l T C I � ._, •

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c = <

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l (3)

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c ) ( ) ( ) c ) ( ) ( ) ( . G • G . G • (i • G 3 •

)

7 8 ) ( ) (

. () . G

1 7 l S ) ( )

3 • 3 •

9 J 0

) ( ) . 6 . 6

l ( 6 )

J

( 7 )

( 8 )

I

'° .c.

S ( I ) 1 2

) ( 1 . 1 . 2

1 1 1 2 ) (

1 . 9 1 . 7

F l = 3 7 7 . l l Z

F 2 = I G 7 7 . l !Z F 3 = 2 7 4 4 . l ! L

F 4 = 3 4 8 2 . l ! L

I = (

1 8 . S ( I ) = (

1 . 1

F l = 3 5 7 . I l l

F 2 = l u l 3 . l l Z

r 3 = 2 7 3 5 . l � Z F 4 : 3 5 1 0 . �' Z

3 4

) ( ) ( ) (

1 . 4 1 . G

1 3 1 4 ) ( ) ( ) (

1 . G 1 . 5

)

) & 7 8 9 1 0 ) ( ) ( ) ( ) ( ) (

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I 1 5 J G 1 7 1 8

) ( ) ( , ) ( ) 1 . 4 1 .l • .J 1 . 2 1 . 3

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