Equalization of a coaxial cable for digital transmission : computer-optimized location of poles and zeros of a constant-resistance network to equalize a coaxial cable 1.2/4.4 for high-speed digital transmission (140 Mb/s)

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computer-optimized location of poles and zeros of a

constant-resistance network to equalize a coaxial cable 1.2/4.4 for

high-speed digital transmission (140 Mb/s)

Citation for published version (APA):

Bergmans, T. (1978). Equalization of a coaxial cable for digital transmission : computer-optimized location of poles and zeros of a constant-resistance network to equalize a coaxial cable 1.2/4.4 for high-speed digital transmission (140 Mb/s). (EUT report. E, Fac. of Electrical Engineering; Vol. 78-E-80). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1978

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Computer-optimized location of poles and zeros of a constant-resistance network to equalize a coaxial e:lble L 2/4. 4 for high-speed digital transmission (140 Mb/ s)

by

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AFDELING DER ELEKTROTECHNIEK VAKGROEP TELECOMMUNICATIE

DEPARTMENT OF ELECTRICAL ENGINEERING GROUP TELECOMMUNICATIONS

Equal ization of a coaxial cable for digital transmission.

Computer-optimized location of poles and zeros of a constant-resistance network to equalize a coaxial cable 1.2/4.4 for high-speed digital transmission (140 Mb/s)

by

T. Bergmans

This work was done as part of the research project ECA 6.

TH-Report 78-E-80 ISBN 90/6144/080/7

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C(jw) e(t) E( jw)

"7

f

f{t}

fn F(jw) F (jw) eq H(jw)

ICI

Im

ISI

j k L m MSE n N PED PT(t) P(jw) R Re

Input symbol at the instant

nT

argument

desired equal izer transfer function for the idealized F(jw) approximation of C(jw)

sequence of pulses vol tage source

mean square error between C(jw) and C(jw) frequency

output voltage before the sampler f(t + nT)

s

Fourier transform of f(t) equivalent Nyquist channel cable transfer function

interchannel interference

imaginary part of a complex expression intersymbol interference

imaginary unit: j2

=

-1.

integer index

cab I e I eng th number of levels mean square error

integer index

N

2 is the number of calculated samples in the discrete Fourier transform

peak error distortion

rectangular pulse with duration

T

and ampl itude 1

T

Fourier transform of PT(t -

2)

constant resistance (R) network

real part of a complex expression driving point impedance of the source

terminating resistance time

samp ling ins tant siqnal element time

(5)

{x .} 1-Z Z a Zb y w @3 set of variables in C(jw}

characteristic impedance of the cable impedance in a constant

impedance in a cons tant

progagation constant cable constant

angular frequency

103

resistance network

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SUMMARY

The computer-optimized location of poles and zeros of an equal izer transfer function for high-speed digital transmission is presented.

It is assumed that the equalizer is realized by means of constant resistant networks.

The objective is an equalizer design procedure for 140 Mb/s over the coaxial cable 1.2/4.4 mm with a length of 2 km. To achieve this result the location of poles and zeros is first determined for a cable length of 1 km and a low bitrate, followed by optimizations for higher bitrates and a cable length of 2 km. Starting with a cosine roll-off ampl itude and I inear phase type as a special case of Nyquist's first criterion. an equalizer transfer function C(jw)

is derived. This function is approximated by a transfer function C(jw) described by poles and zeros. The difference between C(jw) and C(jw) is minimized, resulting in a location of poles and zeros The intersymbol interference is further minimized with the mean square error (MSE) as a criterion. Finally the peak error distortion

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Chapter I.

iNTRODUCTiON

in the assumed digital line transmission system (Fig. I.) a sequence of pulses e(t) is transmitted over a cable. Every T seconds the source produces a rectangular pulse.

The symbol an is one out of m equiprobable levels and PT(t)

describes a rectangular pulsewith unit height and duration T. startin9

T T

at the instant t =-]" and ending at t =+2. Therefore,

e(t)

=

L an. PT(t - nT) (1)

n=-oo

r---,e ( t )

IL..--

sou rC---Je

I

·Ir--

cab I e - _detectorthreshold

Fig. I. The digital I ine transmission system.

in general the received signal is disturbed by add i t ive noi se

intersymbol interference

(lSI)

i nterchanne I interference

(leI).

in the receiver the following disturbances can appear: timing errors in the sampler

decision level offset in the threshold detector tolerances in the component values of the equalizer.

in the following only

lSI

is taken into account. The dispersion in the cable causes an overlap of many neighbouring pulses

(interferers). The number of interferers can be rather large. This is illustrated in

of two km coaxial

Fig. 2., which shows the calculated response

T

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time, due to the physical length of the cable has been neglected. [4] Fig. 2. shows the necessity to apply an equalizer for the ISI.

1V

n··

,input

-.JL

16 12 10 8

6

2

o

t

o

T _ _ _ t = 7.18 ns

=+...

-/response

Fig. 2. Response of 2 km coaxial cable

T 1.2/4.4 to the pulse PT(t - 2)'

In this report the equalizer transfer function is characterized by poles and zeros, and is assumed to be real ised by means of constant resistance networks. The location of the poles and zeros is optimized by a computer optimization procedure

[7].

_ t

T

In Chapter 2 the required theory for the optimization is described and in Chapter 3 the developed computer programs are discussed. Chapter 4 shows the results obtained. The Appendix gives the transfer function of the cable in cascade with the equal izer network.

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Chapter 2

THEORY

In the Appendix it is shown that the overall transfer function of the cable in cascade with the equalizer is independent of the fact whether the equal ization network is inserted at the sending end or at the receiving end of the cable. We consider the system as given

in Fig. 3. P (jw) - -... .-11 cab I e H (jw)

I

equa-

I

F (jw) !----4_00j I i ze ~ f-.--... c(jw)

C

(jw)

Fig. 3. The cable in cascade with the equalizer.

The system response to the pulse PT(tJ at the instant ts + nT is given by

f

~

f(t + nTJ

n s

The following definitions are used to characterize the disturbance due to ISI in the binary case (a • {O,l}J.

n D. 00

Ifni

FED = 1:'

'T'

n=- IJ£I 00

f

MSE!;;, 1:' n 2 n=-oo fO

eye opening!;;, 1 - FED

where the primed summation excludes the term with n

=

O.

(2)

(3)

(4)

(5)

(10)

For m-level transmission the PED has to be multiplied by m - 1.

The equal izer design procedure comprises three steps:

1/ Choice of a function F(jw) satisfying Nyquist's first criterion.

A spectrum F(jw) with a cosine roll-off amplitude and a linear phase characteristic has been chosen [1,3].

2/ C(jw) as prescribed by the choice of 1/ is approximated by C(jw) in terms of poles and zeros and a linear multiplier.

C(jw) =x 1

The set of variables {x.} in C(jw) is optimized by means of a

1-computer optimization program.

3/ In this step the special form of F(jw) as assumed in 1/ is dropped. The Nyquist cirterion is approximated by minimizing

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and without any special constraint on the spectrum F(jw). This further optimization is again performed with the aid of a computer program and the results of step 2/ are used as starting values for the optimization procedure.

A detailed description of these steps is given below.

2.1. Special case of Nyquist's first criterion.

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T

The transmitted Signal is a pulse PT(t -

2)

with Fourier transform p(jw)

=

T . wT sm ₂ wT 2 wT -j

2

• e

The cable transfer

functi:~j~:)~~~;ximated

H(jw)

=

e 2

by [4]:

,

where T1 = 0.44.10 -6 for 2 km length coaxial cable 1.2/4.4 mm.

F(jw) is chosen as described in the toregoing.

F(jw) _ _ (

fo~T

(1 + aos

"'J ) .

-jwt e s (7)

(8)

(9)

(11)

Hence, it follows that the desired equalizer e(jw) = . e

o

-jwt

s

fWT1

wT)

/j+1)l

T

2 wT 2 . wT S'ln -2 .wT J -2 e transfer function is for

Iwl

~

211

T

I

211

for

wi>

T (10)

e(jw) is represented by a polar graph. For certain blttime T and cable length L it only depends on the sampling instant t

s' when fO

=

1 is supposed. This sampling instant is chosen in such a way that the polar graph of e(jw) finds itself mainly in some two or three quadrants of the complex plane and turns clockwise with increasing w. This polar graph is approximated by e(jw). The linear multiplier xl

is introduced to s~tisfy the supposition fO

=

1.

The approximation e(jw) is fitted in an optimal way to e(jw). As a criterion for the optimization we choose the minimum of the mean square

error 211/T

le(jw) -

C(jw)

12

dw + ~ ~

2

E2 = _f _f

_le(jw)

I

_dw 0 211/T ~

2

It easy to see that

Ie - el

can b: expressed as a function of modulus and argument of e(jw) and e(jw) in the fo II owi ng way

~2

2

~2 ~ ~

Ie - el

=

lei

+

lei - 21el lei

cos (arg

e -

arg

e)

2.2. General case of Nyquist's first criterion.

In general P(jw) satisfies Nyquist's first cri,terion if

with t

=

0 [2].

s ~ E n=-CJ:) ~ E n=-tXJ Re P(jw +

Im

P(jw +

21T)

n -T = f T 0

211)

n -T

=

0 >

At the discrete instants nT the system response is

(11 )

the

(12)

00 jwnT dJJJ f{nT) _- _2TI1 f F(jw) e , -00 which is rewritten as TI 1

-T

_F _{(jw) e}jwnT dJJJ f(nT)

=

f 2TI _-TI eq

T

wi th 00 1: F(jw + np) 2TI _Iwl ~ TI _T F (jw)

₌

n=-oo eq Iwl

>!!.

0 T

According to the Pa rseva 1 relationship [3]:

TI 00

T

1 f IF (jwll 2 dJJJ

=

T 1:

~.

2TI _eq _{n=-oo n} -TI

T

In the following use is made of the fact that fit)

Then F(jw) is Hermitian. Substituting (4) in (17) yields TI 1

T

MSE

= - -

f

TITt;

0

Substituting F (jw)

=

f

T

in (18) yields

MSE

=

O.

eq 0 In the equation A F(jw)

=

P(jw) H(jw) C(jw), ( 1 4 ) ; ( 15) ( 16) ( 17)

is a real funct ion.

(18 )

(19 )

P and

H

are fixed, so that

MSE

can be minimised as a function of the poles and zeros in C(jw). The sampling instant ts is considered as a parameter. N.B. the linear multipl ier disappears in (18) by division.

Further _TI

T

f

=

2

f Re

F

(jw) dJJJ •

o TI 0 eq

Approximating C(jw) by C(jw) the cable transfer function thusfar was approximated by (8). Now a more accurate approximation is used:

(13)

H(jw)

1

---yL

e ₍₂₁₎

where y and Z are resp. the propagation constant and the characteristic impedance of the cable.

Equation (21) follows from (39) in the Appendix.

R

=

?5n, y and Z are considered as functions of

w.

[4,5]

Chapter 3.

COMPUTER PROGRAMS

The variables in the programs are - the (angular) frequency (w)

- the linear multipl ier, poles and zeros given by

{xi}.

The parameters are - the cable length L

- the sampl ing instant t

B ~

- the number of poles and the number of zeros of C(jw) - the number of levels m.

The information rate of 139.264 Mb/s is converted into multilevel trans-mission at correspondinR_lowertranstrans-mission rates according to

1 139.264 Mb d

T=

2. au

tog m

(22)

Th i s Y i e Ids for

m=

2 139.264 Mbaud

m=

4 69.632 Mbaud

m=

8 46.421 Mbaud

m=

16 34.816 Mbaud.

3.1. Polar graph of C(jw)

The I isting of this program is on page 24.

With this program modulus and argument (converted into degrees) of

expression (10) are calculated as a function of the frequency at various sampl ing instants.

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3.2. Optimum location of poles and zeros (special case)

The 1 isting of this program is on pp. 25 and 26.

With this program the value of (11) is minimized as a function of the variables

{x.}

in C(jw).

'1-The first integrand of (11) is calculated by means of (12) in the procedure "ideen". The second integral has been neglected. Minimizing takes place

in the procedure "minifun" [7]. Required input data:

- cable length L

- transmission rate

liT

- sampl ing instant ts in multiples of T

1 inear amplifier, poles and zeros, given by

{x.}.

'1-3.3. Optimum location of poles and zeros (general case)

The 1 isting of this program is on pp. 27 - 31.

With this program the value of (18) is minimized as a function of the variables

{x.}

in C(jw).

'1-In the procedure "bepzgamma" auxil iary variables for the evaluation of Z and y as needed in (21) are calculated for specified values of w.

The system transfer function (19) is calculated in the procedure "fjomega" (F(jw).

In the procedure "integrobfu" (integrand, object, function)

IF

(jw)

12

eq

is calculated by means of (16). The MSE is calculated in the procedure "objektfunktie" (object-function) according to (18) and (20). Minimizing takes place in the procedure "minifun".

Required inputdata: - cable length L

- transmission rate

liT

- sampl ing instant t in multiples of

T

s

- poles and zeros

{x.}.

'1-3.4. Discrete Fourier transform

A discrete Fourier transform, using a fast Fourier transform algorithm, yields the interfering samples for calculation of the PED to characterize

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the resulting equal isation. The I isting of this program is on pp. 32 - 35.

To calculate the transfer function of the cable there is the procedure "bepgamma". After this F(J'w) is calculated for specified values of w.

Transformation takes place in "FFTlH" followed up by "rebitreverse" [9]. Use is made of the fact that the transformation result consists of real numbers. The frequency I imit is taken at twenty times the Nyquist

frequency (101T Hz). There are 213 discrete values calculated, so a time interval of 409.6 T is considered [10]. In each time T there are 20

discrete values, so the sampl ing instant can be varied in steps of 0.05 T. Required input data:

- cable length L

- transmission rate 11T

- number of discrete values 2N

- number of discrete values in each time T (delta)

- the sampl ing instant is described by an integer, which points to the place in the FFT-array, where the value of

f

is found

o - poles and zeros, linear multipl ier given by

{x.}.

'Z-With this program the

MSE

is also calculated. This value is in accordance with the

MSE

of the optimization program.

Chapter 4.

RESULTS

During the research the following appeared:

1) The first calculations took place with the numerical integration procedure "integral trapex" [8]. Due to

~. d

v'Z-m

aw

w+O u

~w!.;

e 2

the variable

w

must be changed into;'; •

=

00 (23)

At higher bitrates and/or a cable length of 2 km the method described above reaches its maximum number of iterations and the results became

inaccurate. Afterwards the method "simpsom" [8] has been used (the variable;'; is still in the optimization programs).

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variables were used in "minifun", but the derivation of these

derivatives became too compl icated in connection with the many possible pole and zero configurations. In addition to inaccuracy of "integral trapex" arised here. It is decided that the derivatives are calculated

in "minifun": "information" = 0 and "order" = 1 [7].

Incidentally "minifun" produced useless results which was remediable with "order" = O.

3. Seal ing the variables {xi} for "minifun" was necessary. Therefore, time has been scaled from [8] into [~8] and frequency from [Hz] into

[MHz]. At 140 Mbaud and a cable length of 2 km the seal ing [sJ into [ns] and [Hz] into [GHz] was necessary.

Seal ing in such a way that the values of the variables are about

unity serves for guidance. The resulting values of the poles and zeros have to be multipl ied afterwards by 106 and 109 respectively.

4. To equal ize for low transmission rates over 1 km cable, one zero and two complex poles can properly be used as a starting point. Extension takes place by adding one pole and one zero each time and optimizing again.

Exampl e:

(24 )

5. To equal ize for high transmission rates over 2 km cable, a number of identical sections, each having two zeros and two poles, can properly be used as a starting point. Extension takes place by spl itting up the poles and zeros in a following optimization.

Example: 2 + jwxz _}2-+ { -w + x2 2 -w + jwx3 + x4 2 ₊_jwx z + x2 2 + jwxS -w -w + x6 2 + jwx 3 + x4 2 + jwx 7 + -w -w _x8 (25)

(17)

6. Combining the poles and zeros in pairs into second order polynomials yields the advantage that the answer at the question weather these pairs have to be real or complex conjugated is left to the optimi-zation process.

7. To find the most favourable sampling instant (minimum PED) , steps of 0.05 T for the parameter ts is properly usable for the investigated systems.

8. Equal izing 2 km cable at low transmission rates for the ideal ized F(jw) and the approximation (8) for H(jw) aPED < 1 is hardly

possible. At higher transmission rates it appeared to be impossible. A decreasing

~

was combined with an increasing PED! A coarse

approximation (PED small enough) can properly be used to minimise the MSE. Further investigation of this phenomenon yielded the following explanation: for 0;/ <

f

< 0:/ Hz, lC(jw)

I

is much greater than for the other frequencies. Hence, the contribution to the error

(11)

in that region is relatively large. Minimizing (11) by means of "minifun" mainly takes place for these frequencies at the cost of the approxi-mations for the other frequencies. Just the high and low frequency approximations are important for a small PED (MSE).

9. In all examples the resulting PED minimizing

(18)

was smaller than the resulting PED minimizing (11).

In Fig. 7 up to Fig.

14

the polar graphes of C(jw) are given for the chosen sampling instants. Fig.

15

shows the polar graphes of C(jw) and C(,jw) for two different pole-zero configurations, resulting from the minimization of (11) for

L

= 1

km

and

liT

= 34.816

Mbaud.

In Fig. 16

PED (t ) is shown for L

=

1

km.

Most results are found by minimizing

(11);

s

some results are found by minimizing

(18).

The latter procedure takes a lot of computer time! The most favourable sampl ing instants of Fig. 16

are gathered in Fig. 17.

In Fig.

18

the polar graph of C(jw) is given together with that of C(jw) for the specified pole-zero configuration, resulting from the minimi-zation of (11) for

L

=

2

km

and

liT

=

139.264

Mbaud.

This result is the starting point for minimization of

(18)

with t as a parameter.

(18)

See PED (t ) in Fig. 19 and the location of the poles and zeros in Fig.

s

20 for minimum PED (t

=

3.25T). The corresponding time response is

s

shown in Fig. 21.

4.1.Resulting poles and zeros for minimum PED

All PED's are calculated for binary transmission.

L

=

1 kIn. jw + x3 jw + x₄ C(jw) = (jw + xl + jx 2J (jw + xl - jx2J . jw + x5 liT 34.816 Mbaud PED 0.36 t 1.30 T s xl

5.447999@7

x 2

1.364410@8

x3

9.90484666

x 4

1.51477698

x5

3.830472617

L

=

2 kIn. C(jw) liT

=

34.816 Mbaud PED

=

0.56 46.421 Mbaud 69.632 Mbaud 0.40 0.48 1.35 T 1.45 T

7.406336@7

1.138479@8

1.829171@8

2.822890@8

1 .078964@7

9.838389@6

1.410157@8

1 .208746@8

5.542677@7

8.2532711!'7

t

=

1.9 T (only one sampl ing 'instant has been considered).

s 139.264 Mbaud 0.76 1.90 T

5.843068@7

5.853325@8

1.

241175@7

7.253391@9

6.902467@8

(19)

Xl = J.460J19@? x 2

=

1. ?58510@8 x J

=

4.404496@6 x 4

=

1.7J2081@8 x5 = 1.7J2034@8 X 6

=

5. OJ9494@? x?

=

9.66928?@? L

=

2 kIn. C(jw) -w

=

-w 2 + x9 + jwx₁₀ -w 2 + x ll + jwx12 -w liT

=

46.421

Mbaud

FED

=

0.60 2 2 + xl + jwx₂ -w 2 ₊_x5+ jwx 6 2 + x J + jwx4 -w + x? + jwx8 2 + xl;) + _jwx14 jw -w _{- xl?} 2 + x 15 + jwx16 jw -w ₊_xl?

t

=

2.1 T (only one sampl ing instant has been considered).

s xl

=

2.594021&15 x 2

=

?040936@8 x3

=

3.093922@16 x 4

=

2.2918?0@8 x = 8.111945@16 5 x

=

6.483066@8 6 x

=

6.123260@16 ? x8 = 2.648380@8 x9

=

8.090232@16 x 10= 6. ?21882@8 x₁₁

=

5.44?488@16 x 12= 2.4?8860@8 x₁₃

=

8.114935@16 x 14= 6.653J08li8 x 15= 5.J61069@16 x 16= 2.209435@8 x17= 3.21245?a9

(20)

L == 2 km. 2 _{- jwx} 2 ₊ jwx 6 -w 2 ₊_jwx 10 jw -w + xl -W + x5 + x9 C(jw) ₌₌ 2 2 2 2 -W + x3 + jwx₄ -w + x 7 + jwx8 -w + xl l + jwx12 jw

liT

== 69.632

Mbaud

PED

=

0.55

t

=

2.5 T (only one sampling instant has been considered).

$ Xl

=

1.239304~6 x 2

=

3.711036@9 x

=

1.659129@17 3 x

=

3.635496@8 4 x5

=

2.388364@17 x6

=

2.867953@9 x 7 = 1.149789@17 x8 == 4.969062@8 x9

=

1.761166@17 x 10= 2.200041@9 x 11= 1.201828@17 x 12= 4.280197@8 x13= 5. 406587@8 L

=

2 km. C(jw) 2 + x9 -w 2 -w + x 11 2 -w + xl

=

₂ -w + x3 + jwx 10 + jwx 12

liT

=

139.264

Mbaud

PED

=

0.46 t = 3.25 T $ + jwx 2 2 -w + x5 + jwx 6 + jwx 4 2 . -w + x₇+ Jwx8 2 • -w + x13 + Jwx₁₄ jw - x₁₇ 2 -w + x 15 + jwx16 jw + x17 - x 13 + x 13

(21)

Xl = 4.267848t114 x?, = 1.0.96795@8 x.3 = 4.000553@17 x₄= 4.156386@8 x5 = 2.577319@17 x6 = 3.980139@9 x 7 = 6.119170@17 X _{8 - 1.575047@9} -X

=

9 4.016042@17 x₁₀

=

2.021609@9 x 11

=

6.665803@17 x₁₂

=

1.721698@9 x₁₃

=

3.594530@17 x 14

=

1.760119@9 x 15

=

7.313199@17 x₁₆

=

1.515921@9 x₁₇

=

3.274685@8 Chapter 5.

CONCLUSIONS AND FINAL REMARKS

1. A method has been found to minimize the ISI with the result PED < 1 for binary transmission. The procedure starts with a few variables

(poles and zeros). Then after optimization the number of variables is extended and optimized again.

2. For multilevel transmission (m> 2) no open eye diagram is found. This means that the number of poles and zeros is too small or the sampl ing instant is not the most favourable.

3. The significance of minimizing (11) is to produce good starting values for the minimization of (18).

4. To prevent an increasing PED with decreasing E2, in cases that the maximum value of IC(jw)

I

is high, a modification of (11) is necessary

(22)

E

5. From Fig. 17 it appears that binary transmission is prefered to multi-level transmission.

6. The calculated equal izer transfer functions C(jw) represent real isable networks. Specific equal izer configurations, such as constant resistance networks, introduce extra constraints to the location of poles and

zeros.

7. The influence of tolerances in the component values of the equal izers may be a subject for further research.

APPENDIX

The transfer function of a cable in cascade with constant resistance networks is considered. The transfer function of the cable is given [4].

E

Fig. 4. The source and terminating resistance connected to the cable. F Z E

=

2. Rl + Z • R2 + Z • _Rl 1 - R + 1 -yL e

z

_-2yL + Z • e

Constant resistance network inserted at the sending end of the cable:

R

Z

a

R

y, Z

Fig. 5. The constant resistance network inserted at the sending end of the cable.

(23)

E Suppos i ng 2

1

Za • Zb = R R = R 1 (27)

and the applying Thevenin's theorem, the source and the constant resistance network are replaced by an

R emf = E. R + Z

a

with a driving point impedance

R. = R

'Z-Hence, the transfer function is

F R Z E = 2. ""R--'+"'--;;Z- • R + Z • a Z • 1 _ R - Z R + Z -yL e R2 - Z -2yL · R + Z · e 2 (2R) (29 ) (30)

Constant resistance network inserted at the receiving end of the cable:

R

y, Z

Z

a

R

Fig. 6. The constant resistance network inserted at the receiving end of the cable.

Supposing

(24)

F

=

F' R

R + Z

a Now the transfer function is

F R Z R -yL 2 e

Fi-

_{R + Z} _Rl₊ _Z _{R +}_Z

.

_{Rl - Z} a ₁

_-

R - Z -2yL Z e R j + Z R + (33) Conclusion If (34)

then the transfer function

F R Z R --yL 2. e = + Z

.

R + Z • -2yL E R + Z R 1 - (R - Z)2 a R + Z e ( 35)

does not depend on the fact whether the constant resistance network .is inserted at the sending end or at the receiving end of the cable. Applying several constant resistance networks in cascade, the transfer function follows in the same way. Using n networks it is

with F Z R

Fi-

2 · R · + Z · R + Z · 1 -e--yL -2yL . e Z _{ak' bk -}Z - R2 k 1 2 J

=

J J " • • J n n

n

R (36) k=l

The realization of the equal izer transfer function C(jw) with constant resistance networks follows with (37) and

A • n R

C(Jw)

=

n

R + Z ( ' )

k=l ak JW

With C(jw) the system transfer function equals

--yL e -2yL e (38) (39)

(25)

For the coaxial cable 1.2/4.4 it is found [4]

lim Z(jw)

=

75 Q w--><'"

In the computer programe use is made of

(40)

R =R =R=75Q (41)

1 2

~Iith the approximations for high frequencies [4]

Z

=

75 Q (42)

the cable transfer function becames

-(J' +

H(jw)

=

e

'~T

1)

-2

2

So, the system transfer function (41) can be rewritten as

ACKNOWLEDGEMENT

.,-w-r:;-+ 1)

~-f

(43)

(44 ) ( 45)

This paper is the result of the graduation work, carried out in the group Telecommunications ECA. The author wishes to acknowledge the coaches

J. van der Plaats, W.C. van Etten and A.P. Verlijsdonk, members of this group.

REFERENCES

rl] A.M. Giacometti and T.F.S. Hargreaves,

"A 11,0 Mb/s digital transmission system for coaxial cables", Phil ips Telecommunications Review, Vol.

33,

nr.

4,

December

1975,

(26)

[2] A.P. Verlijsdonk,

IITelecommunication systems",

Lecture notes THE, nr. 5.512, page 4.5 (in Dutch). [3] R.W. Luckey, J. Salz and E.J. Weldon jr.,

"Principles of data communication", New York: Mc Graw-Hill, 1968, pp. 75-79. [4] J. van der Plaats,

"Transmission of Information I", Lecture notes THE, 1975, (in Dutch). [5] J. van der Plaats,

"Calculations of the parameters and the transfer function of an arbitrary terminated cable connection in the frequency domain", THE, June 1971, (in Dutch).

[6] M. Abramowitz and I.A. Stegun,

"Handbook of mathematical functions with formulas, graphs and mathematical tables",

Washington:

u.s.

Government Printing Office, December 1965, pp. 379-385. [7] "BEATHE" procedure MINIFUN, for solving non-l inear optimization

problems",

THE computing centre, RC-information nr. 57, March 1974. [8] "Integration",

THE computing centre, PP-3.1.1, October 1974.

[9] "Algol procedures to calculate discrete Fourier transforms", THE computing centre, RC-information nr. 24, July 1970. [10] W.C. van Etten,

"The discrete Fourier transform", THE, November 1970, (in Dutch).

(27)

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(28)

llURROUuHS b7700 HEATHE COMPILER. VER~IuN 2.~.OOO. TUESDAY. JN141

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117 *T7 +T8 *T8 I/IN7 *N7 +N8 *N~ JI; ARGC:=W*IAU-wr*IS +A~G;

ARI,CC:=-c*I\f'CfAN IWI/X[ 18]);

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ARGCC:=A~GCC+ARGH;

ARGH:=ANCIA~( 14 IT3 )1 'IF'ARGh<O'TH[N'ARGH:=ARGH+PI; ARGCC:=ANVC(+ARGH;

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ARuH:=~RLIA~1 N4 IN) J; 'IF'ARGh<O'THEN'ARGH:=ARuH+PI: AR(,CC:=ARGCC-ARGH;

ARuH:=MRLTANI N6 INS II 'IF'ARGh<O'THEN'ARGH:=ARGH+PI: AR'~CC:=ARGCC-ARGH;

ARuH:=ARLTA~1 N8 IN7 I; 'IF'ARG~<O'THEN'ARGH:=ARGH+pI;

ARGCC:=ARuCC-ARGH;

IDEEN:=W*IK*K+KK*KK-2*K*KK*COSIAHGC-ARGCCII; 'END'IOEI:N:

'l'

(29)

'INTEGER' II 'R~AL'G; 'REAL' 'ARRAY'X[*]I 'flI:.GIr-.' 'II,TEGEo'I' J;

G:=SIMPSOM (IDEEN(W.X) .W.O~k.8GR.lb);

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TIME = 39.353 SECONDS ELAPSED I 6.245 SECONDS PROCESSING; 3.215 S~COND~ I/O.

====================================================== =================~======

(30)

BURROUGHS 81100 BEATHE COMPILER, VERSION 2.8.000, fRIDAY, 12/111

E C A I A f STU 0 ERE NIT B ERG MAN 5 I K ABE L K 0 R R E

=

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'BEGIN"fIL~'IN,OUTI 'REAL'L,T,TS,PI,TAUNUL,BGR,MOOO,MOD,w,REO,IMD,SHIfT, BERX,BEIX,BERXAC,BEIXAC,KERX,KEIx,KERXAC,KEIXACI 'REAL"ARRAY'XIl:111! 'INTEGER'N,M,I,CONVERGEU,LOOPI 'INTEGER"ARRAY'XTYPEll:17J,GTYPEll:1JI 'BOOLEAN'CONVI 'BOOLEAN"ARRAY'LINI1:IJ; ~ %%%%~%%%%%%%%%%%%%%%%%%%f%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %

% KABELKORREKTIE MINIMALISATIE VAN DE MSE f

% EQUIVALENT NYQUISTCHANNEL ~

% %

%%%%%%%%%%%%%%%%%~%%%%~%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

'REAL"PROCEDURE' BEPZGAMMAIXl,OKE)!

'VALUE' XloOKEI 'REAL' XII 'BOOLEAN' OKEI 'BEGIN' 'REAL' YI

'If' Xl'LEQ'B'THEN'

'BEGIN"REAL'PI4,LNX21 y:=x1*Xl*x1*Xl/4096; LNX2:=LNIXl/2)1

PI~:=O.1853981633914481

BERX:=1+Y*I-64+Y*1113.11111114+Y*I-32.36345652+Y*12.64191j~7+

Y*I-O.08349609+Y*10.00122552+Y*I-O.00000901)))))))1

BEIX :=X-l+XlI64+H6+Y<>-I-l13. 117111-14+Y* (12.81111142+Y* 1-10.5676 5719+Y*IO.52i856l5+Y*I-O.Ol103667+Y*IO.0001l346))))))) ; BERXAC:=Xl*Xl*XTI'64*(~4+V*114.22222222+V*I-6.068148l0+Y*IO.660 41849+Y*I-O.02609253+Y*IO.00045951+Y*'-O.00000394)))))I); BEIXAC:=X1*IO.5+V*I-IO.66666666+Y*lll~3111~112+V*I-2.31161514+ Y*(O.14611204+V*I-O.00319386+Y*IO.000004609))))))); , If ' OKE 'THEN' 'BEGIN'KERX:=-LNX2*BERX+PI4*BEIX-O.51121566+V*I-59.05819744+Y* 1171.36272133+V*I-60.6097745l+V*15.65539121+Y*I-O.19636341 +Y*IO.00309699+Y*I-O.00002458)))))))1 KEIX:=-LNX2*BEIX-P14*BERX+X1*XI/64*(6.16454936+V*I-142.9182 7687+Y*1124.23569650+Y*(-21.30060904+Y*(1.17509064+ Y*I-0.02695875+Y*IO.00029532)))))))1 KERXAC:=-LNX2*BERXAC-BERX/Xl+PI4*BEIXAC+Xl*xl*Xl/64*1-3.6911 3734+Y*i2l.420340l7+Y*(-11.36433272+Y*ll.41384780+Y*I-0.0613 6358+Y*(0.OOl16l37+v*I-0.0000l075))))))); KEIXAC:=-LNXZ*BEIXAC-BEIX/XI-PI4*BERXAC+Xl*(0.21139217+Y*I-l3.39858846+Y*(19.41l82758+Y*(-4.6~950823+Y*IO.33049424+ Y*(-0.00826707+V*(O.000l1997)))~)))1 'END' 'END' 'ELSE' 'BEGIN"REAL'WPI2,TETARMIN,TETAIMIN,TETARPLU,TETAIPLU,fIRMIN, fIIMIN,fIRPLU.fIIPLU.W2,w2PI,PI,XW2,SQRTX,SINXMIN,COSXMIN, SINXPLU,COSXPLU,EXPMIN,EXPPLU,KONSTl,KONST21 wPI2:=1.2533l4l3733;w2:=1.41421356237IW2PI:=O.39H942280402; PI:=3.14159265361

(31)

Y:=-8/Xl;XW2:=Xl/w2;SQRTX:=SQRT(Xl) I TETARMINi=yo(O.0110_86+YO(0.0+YO(-O.0000906+YO(-0.00002S2+ YO(-O.000003_+YO(O.0000006))))))1 TETAIMIN:=-O.3926991+yo(-O.011048S+yo(-0.000976S+YO(-0.0000901+ YO(O.O+YO(O.OOOOOSl+YO(0.0000019))))))1 FIRMIN:=O.7071068+YO(-O.062S001+YO(-O.0013813+YO(O.OOOOOOS+yo (0.00003_6+YO(O.OOOOl17+YO(O.0000016111)11; FIIMIN:=O.707106H+yO(-O.OOOOOOl+YO(O.OD13811+YO(O.0002_S2+V* (0.0000338+yo(-O.000002_+yo(-O.0000032)1111); COSXMIN:=COS(-XW2+TETAIMIN); SINXMIN:=SIN(-XW2+TETAIMIN); EXPMIN:~EXP(-xw2+TETARMINII YI=8/X11 TETARPLU:=Y*(O.Ol10486+Y*(O.O+Y*(-0.0000906+Y*(-0.OOOO2S2+v* (-0.0000034+Y*(O.00000061111)1; TETAIPLU:=-O.3926991+Y*(-O.OllO_8S+Y*(-0.000976S+Y*(-O.0000901+ Y*(O.O+Y*(0.OOOOOSl+Y*(O.0000019)1)1)); fIRPLU:=O.707106H+Y*(-O.0625001+Y*(-O.0013813+Y*10.OOOOOOS+ Y*(O.0000346+Y*(O.OOOOl17+Y*(O.0000016)1)1)11 FIIPLU:=O.7071068+Y*I-O.OOOOOOl+Y*IO.0013811+Y*IO.0002_52+ Y*IO.0000338+y*(-O.000002_+Y*(-0.000003211)1111 COSXPLU:=COSIXW2+TETAIPLUII SINXPLU:=SIN(XW2+TETAIPLUI; EXPPLU:=EXP(XW2+TETARPLUII KONST1:=WPI20_{EXPMIN/SQRTX;}

KONST2:=W2PI*EXPPLU/SQRTXI KERX:=KONST1*COSXMINI KEIX:=KONSTl*SINXMIN; BE~X:=KONST2*COSXPLU-KEIX/PI; BEIX:=KONST2°SINXPLU+KERX/PII KERXAC:=KEIxofIIMIN-KERxofIRMINI KEIxAC:=-(KEIx*fIRMIN+KERx*fIIMIN); BERXAC:=KONST2*(COSXPLU*fIRPLU-SINXPLU*fIIPLU)-KEIXACIPI; 8EIXAC:=KONST2*(COSXPLU*fIIPLU+SINXPLU*FIRPLUI-KERXACIPI; 'END' 'END'BEPZGAMMAI % % 'REAL"PROCEDURE'fJOMEGA(W,XII

, VALUE' W; 'REAL' WI' REAL' , ARRAY' X I * II

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fI21~ARCTAN(DELTAGCAPII M002:=SQRT(1+DELTAGCAP*DELTAGCAP)I ul:=5.12236096378*W*WDUII

'IF'Ul<95'THEN'

'BEGIN' BEPZGAMMA(Ul,'FALSE')1

(32)

Kl:=3.9044495579~-2*WOWDUII Rl/=KloIBERXOSEIXAC-dEIX*BERXAC)/NOEMERI Ll:=KIO(BERXOBERXAC+BElxoBEIXAC)/NOEMERI 'END' 'ELSE' 'BEGIN' Ul/=3.622oS617316owoWUUII Rl:=UloUl*RlO/12°Ul-l)1 Ll:=Wlo 50 0112 0 Ul-l)/UI/UI+8°IUI-I)0(UI-ltO(UI-I)/UI/UII UI/UlIUlIUl) I 'END' I U2:=2.173212owowDUI; 'IF' U2<6 'THEN'

'BEGIN' R2:=7.S29625706B~-3*WO(SINH(U2)+SIN(U2))OWDUII (COSHIU2)-COSIU2)1-2.6739677746~-41 L21=7.52962S7068~3*W*(SINHIU21-SIN(U2))/(COSH(U2)-COS(U2)) *WDUII 'END' 'ELSE' 'BEGIN' R2:=7.S2962S7068~-3*wowDUI-2.6739677746~-4; L2:=7.S296257068~-3owoWDUII 'END' I RR:=Rl+R21 LR:=Ll+WloLE+L21 FIl:=ARCTAN(RR/LRII MODl:=SQRT(l+RR*RR/LR/LRII

ZMOD:=SQRT(LR/(wl OCAPloMODl/MOD21; ZARG:=(FI2-FIl)/2;

GAMMAMOD:=SQRT(L~*WloCAPoMODl*MOD211

GAMMAARG:=(PI-FII-FI21/21

ALfALI=GAMMAMODOCOS(GAMMAARGloLI BETALI=GAMMAMOD*SIN(GAMMAARGloLI

RGZ:=7S/ZMODI ZGR:=ZMODI7SI Cl:=(RGZ+ZGRI*COS(ZARGII C2:=(ZGR-RGZloSIN(ZARGI; EXPALFAL:=EXPI-ALFALII

EXP2ALFAL:=EXPALFAL*EXPALFAL; COS2BETAL:=COS(2*BETALI; SIN2BETAL:=SIN(20BETALII C3:=EXP2ALFAL*COS2BETALI

C4:=EXP2ALFAL*SIN2BETALI

NR:=2+Cl-(-2+ClloC3-C2 0C41 NII=C2 o (1-C31+(-2+ClloC41 FMOD:=2*SIN(ARGll*EXPALFAL/(SQRT(NR*NR+NI*NlloARGIII

'IF'NR<O'THEN'FARG:=-(Pl+ARGl+BETAL+ARCTAN(NI/NRII+OMTAUO 'ELS£'FARG::-(ARGl+BETAL+ARCTANINI/NRII+OMTAUOI MOD:=FMOD 0SQRT«TI 0Tl +T2 0T2 I/(NI 0Nl +N2 oN2 )0

,tT3 "T3 +T4 0T4 )/CN3 0N3 +N4 0N4 10 ITS °T5 +T6 0T6 I/INS 0NS +N6 0N6 10 IT7 0T7 +T8 0T8 )/IN7 0N7 +N8 ONe III ARG :=FARG-2 0ARCTAN(wl/XI17))+wl oTSI

ARGH:=ARCTAN( T2 ITI )1 'IF'ARGH<Q'THEN'ARGH:=ARGH+PII ARG :=ARG +ARGH;

ARGH:=ARCTAN( T4 IT3 I; 'IF'ARGH<O'THEN'ARGH/=ARGH+PI; ARG :=ARG +ARGH;

ARGH:=ARt:TAN( T6 ITS II 'IF'ARGH<O'THEN'ARGH:=ARGH+PI; ARG :=ARG +ARGH;

ARGH:=ARCTAN( T8 IT7 II 'IF'ARGH<O'THEN'ARGH::ARGH+PI; ARG :=ARG +ARGH;

ARGH: =ARCTAN ( ·N2 IN 1 I; 'IF' ARGH<O' THENl ARGH: =ARGH+P I; ARG :=ARG -ARGH;

ARGH:=ARCTAN( N4 IN] ); 'IF'ARGH<O'THEN'ARGH:=ARGH+PI; ARG :=ARG -ARGH;

ARGH:=ARCTAN( N6 INS ); 'IF'ARGH<O'THEN'ARGH/=ARGH+PI; ARG :=ARG -ARGH;

ARGH:=ARCTAN( N8 IN7 II 'IF'ARGH<O'THEN'ARGH:=ARGh+PI; ARG :=ARG -ARGHI

(33)

'END'

'ENO'FJOMEGAI %

%

'REAL"PROCEOURE'INTEGROBFUIW,X,N,FNULII

'VALUE'W,N,FNULI 'REAL'''; 'INTEGER'N; 'REAL"ARRAY'X(°)1 'HOOLEAN'FNULI

'BEGIN"~EAL'F2EQR,F2EQI,NSHIFT,WPLu,WMIN,wll

'INTEGER'JI Wll=woWI 'IF'W=O'THEN'

'8EGIN' FJOMEGAIO,XI; F2EQRI:REO; 'FOR'J:=l'STEP'l'UNTIL'N'OO'

'BEGIN'NSHIFT:=JoSHIFT; WPLU:=SQRTINSHIFTII FJOMEGAIWPLU,X)I F2EQR:=F2EQR+2 0REO

'EII/O' ;

'1~'FNUL'THEN'INTEGROBFU:=woF2EQR'ELSE'

INTEGROBFUI=woF2EQRoF2EQR 'ENO' 'ELSE'

'BEGIN' FJOMEGAIW,X)I F2EQRI=REOI F2EQI:=IMOI 'FOR'JI=l'STEP'l'UNTIL'N'OO'

'8EGIN'NSHIFTI=SHIFToJI WPLU:=SQRTINSHIFT+wlll WMIN:=SQRTINSHIFT-Wlll

FJOMEGAIWPLU,X); F2£QR:=F2£QR+REO; F2EQI:=F2EQI+IMO;

FJOMEGAIWMIN,XI; F2EQR:.F2EQR+REO; F2EQI:.F2EQI-IMO;

'END' 'END' ; 'IF'FNUL'THEN'INTEGROBFU:=woF2EQR'ELSE' INTEGROBFUI=woIF2EQROF2EQR+F2EQI"F2EQI) 'ENO'INH.GROBFU; % % 'PROCEOURE'OBJEKTFUNKTIEIX,G,111

'INTEGER'II 'REAL'G; 'REAL' 'ARRAY'X(O)I 'I:lEGIN"REAL'FOI GI=SIMPSOMIINTEGROBFUIW,X,N.'FALSE'I,w,0,BGR,J211 FO:=SIMPSOMIINTEGRO~FUIW,X,N,'TRUE'I,W,O,BGR,J2); G:=GoPI/FO/FO/T/2-11 'ENO'OI:lJEKTFUNKTIE; % % 'PROCEOURE'P;; 'PROCEOURE'Q;; $'INCLUOE"'MINIFUN" % % READIIN,I,L,T.TS,X(1),X(2),XIJ),X(4],X(S],X(6].X(7],X(8],X(9].X(lO]. X( 11 j,X( 121,X(13j,X( 14ltX( 15ltX( 161,X( 17]); TI=l/T; TSI-TS"T; WRITEIOUT,c/,"LENGTE_VAN_OE_KABEL_IN __ M.",E17.10,1,''BITTIJO_IN_NSEC. __ I,E.17.10,1,ISAMPLEMOMENT_IN_NSEC. __ ",E17.10>,L,T,TS);

PII=4oARCTANIll; GTYPE(ll:=ll LIN(ll:='FALSE';

'FOR'I:=1'STEP'4'UNTIL'13'OO'XTYPE(11:=XTYPE(I+ll:=11 'FOR'l:=3'STEP'4'UNTIL'lS'OO'XTYPE(ll:=XTYPE(I+l]:=21 XTYPE( 171 :=2;

TAUNUL:=LolO "SQRTIl.121/J; %%% LENGTE L IN METERS %%~

SHIFT:=2~PI/TI BGR:=SQRTIPI/TI;

FJOMEGAIO,XI; MODO:=REOI N:=O;

(34)

'UNTIL'A~S(MOOO/MOOI'GEQ'~1'OR'N=11; NI=N-11

wR ITE (OUT, <h "AANT AL_EKSTRA_NYQU I STINTERIIALL£N". 14. h "LAA TSTE_ VERHOU DINGI,E20.10,1,IMOOULUS_VOOR_OMEGA=O ___ ",E20.10>,N,MODOIMOD,MODOII MINIFUN(X,OBJEKTFUNKTIE.XTYPE.GTYPE,LIN,0.P,Q,1,11,1,~-5,~-5.0,1, CONVERGEO.2. 3,OUT~1 'END'.

==============================================================================

RS DETECTED

=

O.

ENTS = 61. TOTAL SEGMENT SIZE = 4036 wORDS. CORE ESTIMATE

=

6112 wORDS. STI 934 CARDS. 9722 SYNTACTIC ITEMS. 113 DISK SEGMENTS.

AME: (U1643S433)ECA/AFSTUOEREN/TBERGMANS/KABELKORREKTIE. 81100 CODE GENERAT£l ME

=

32.333 SECONDS ELAPSEO; 1.292 SECONDS PROCESSING; 6.091 SECONDS 1/0.

(35)

~URROUGHS b7700 BEATHE COMPILER. VERSION 2.8.000. TUESOAY. ll::ll4

E C A I A F STU D ERE NIT B ERG MAN S I K A ~ ELK U R R E = =

=

= =

= = =

=

= =

=

= =

=

= = =

=

= =

=

= -

=

= 'BEGIN"FILE'IN,O~T; 'REAL'PI,ARG1.ARG2.MOD,W.Wl,T,TAU.L.PERIODE.Tl,Nl,T2.N2,13.N3,T4.N4. T5.N~.T6.N6,T7,N7.T8.N8. Xl,HERX.BEIX,KERX.KEIX.BERXAC.8EIXAC,KERXAC.KtIXAC.wOMEGA. TWEEPI.TAUNUL.Ul.U2.NOEME~.Kl.Rl,Ll,NR.NI.R2.L2.RI0.RR.LR.Lt. FIl.FI2.DELTAGCAP.MODl.MOD2,ZMOD.CAP.lARG.GAMMAMOU.GAMMAARu. OMTAUO.ALFAL.BETAL.Cl.C2.C3,C4.RGZ.ZGR.EXPALFAL.EXPiALFAL. COS2BETAL.SIN2BETAL.NORM.GETAL.PEU,MSE,FMOD.FARG.ARGH; 'INTEGER' J.N.MACHN.DELTA.MACHNl.TS; '800LEAN'FORWARO.OKE; 'REAL"ARRAY'Q(U: 8200]. X(l:18]; %%%%%~%%%~%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%~ i ~

% DISKRETE FOURIERTRANSFORMATIE (INGANGSSIGNAAL. ~

% KABtL EN KORREKTIENETWERKI ~

% BEPALI~G VAN PED EN M5E ~

% ~

%%%%%*%%%~%%%%%%%%%%%%%~%%%%%~%%%%%%%%%%%%%%%%%%%%~

'REAL' 'PROCEDURE' BEPZGAMMA(Xl.OKEI;

, VALUE' Xl, OKE; 'RE AL' Xli' BOOLEAN' OKE' 'BEGIN"REAL' Y; 'IF' Xl'LEQ'b'THEN' 'BEGIN"REAL'PI4,LNX2; Y:=Xl*Xl*Xl*Xl/40~6; LNX2:=LN(Xl/21; PI4:=O.785398163397448; BERX:=1+Y*(-64+Y*(113.77777774+Y*(-32.36345652+Y*(i.641~13~7+ Y*(-0.08349609+V*(0.00122552+V*(-0.0000090111)1111; BEIX:=Xl*Xl/64*(16+V*(-]13.77777774+V*(72.81777742+V*(-IO.~676 5779+Y*(0.52185615+V*(-0.01103667+Y*(O.000l13461111111; BERXAC:=Xl*Xl*Xl/64*(-4+V*(14.22222222+V*(-6.06814810+V*(0.660 47849+V*(-0.02609253+v+(0.00045957+V*C-0.0000039411111II; BEIXAC:=Xl+(0.5+V+(-10.66666666+Y+(11.37777772+V*C-2.31167~14+ Y*(0.14617204+V*(-0.00379386+V*(O.0000046091111111; 'IF'OKE'THEN' '8EGIN'KERX:=-LNX2*BERX+PI4*BEIX-O.57721566+V*(-5~.05B1974~+VD (171.3b272133+V*(-60.60977451+V*(5.6553~121+V*C-O.19636347 +V*(O.00309699+V*(-0.00002458111))II; KEIX:=-LNX2*HEIX-PI4*HERX+Xl*Xl/64*C6.764~493b+f*(-142.~lb2 76b7+'*(124.23569650+Y*(-21.30060904+V*(1.17~0~U64+ V*C-O.02695875+V*(0.000295J21)1)111; KERXAC:=-LNX2*BERXAC-BERX/Xl+PI4*BtIXAC+Xl*Xl*Xl/64*(-3.6911 3734+'*(21.42034017+Y*(-11.36433272+V*(1.413847bO+V*(-0.0613 63~8+Y*(0.OOl16137+V*(-0.OOOOl0751111111; KEIXAC:=-LNX2*BEIXAC-8EIX/Xl-PI4*SERXAC+Xl*(0.iI139217·Y*(-13.3ge58846+Y*(19.41182758+V*(-4.6~950823+Y*CO.j3049424+ Y*(-O.00826707+Y*(O.OOOl19~71111111; 'END' 'END' 'ELSE' 'BEGIN' 'REAL'WPI2.TETARMIN.TETAIMIN.TETARPLU.TETAIPLU.FIRMIN. FIIMIN.FIRPLU.FIIPLU.W2.W2PI.PI.XW2.SQRTX.SINXMIN.C05XMIN. 5INXPLU.CO~XPLU.EXPMIN.EXPPLu.KONST1,KONST2;

(36)

WPI2:=1.2~331413733;W2:=1.41421356237;W2PI:=O.3989~228O~02; PI:=3.1415~26536;

Y:=-8/Xl;AW2:=Xl/W2;SQRTX:=SQ~TeXl);

TETARMIN:=y*eO.Oll0486+Y*(0.0+Y*(-0.0000906+Y*(-0.OOOO252+ Y" (-O.0000034+Y* eO.0000006»»»;

TETAIMIN:=-0.3926991+y*e-0.01I0485+y*e-0.0009165+y*e-0.0000401+ y*eo.0+y*eo.0000051+y"eO.0000019»»»; FIRMIN:=O.7071068+Y*e-0.06250ul+y*e-O.U013~IJ+Y"eO.ooo0005+,· eO.0000346+Y"eO.0000117+Y"CO.U000016»»»; FIIMIN:=0.7071068+y*e-O.000000l+y*eO.0013811+Y*CO.0002452+Y* eO.000033~+y*c-0.0000024+Y*C-U.0000032»»»; COSXMIN:=COSC-Xw2+TETAIMIN); 5INXMIN:=SINC-AW2+TtlAIMIN); EXPMIN:=EXPC-XW2+TETARMIN); Y:=8/Xl; TtTARPLU:=Y*(0.Ol10486+Y*CO.0+Y*(-0.OU00906+Y*(-O.UUOO252+Y* C-0.0000034+y*eO.0000006»»»; TETAIPLU:=-0.3926991+Y*(-0.0110485+Y*(-0.0009165+Y*C-O.OOOO~OI+ Y*CO.0+Y*(0.0000051+Y*CO.0000019»»»; FIRPLU:=0.7071068+Y*(-0.0625001+Y*(-0.0013813+Y*(O.000OOO~+ y*eO.U000346+Y*(0.0000117+y*eO.0000016»»»; FIIPLU:=0.7071068+Y*(-0.000000l+Y*(0.0013811+Y*(0.U002452+ Y*CO.U0003J8+Y*(-0.0000024+Y*(-0.OOOOOJ2»»»; COSXPLU:=COS(XW2+TETAIPLU); SINXPLU:=SINeXw2+TETAIPLU); EXPPLU:=EXpeXW2+TETARPLU); KONSTl:=wPI2*EXPMIN/SQ~TX; KON~T2:=w2PI*EXPPLU/SQRTX; KERX:=KON~TI*COSXMIN; KEIX:=KONSTl*SINXMIN; nERX:=KO~ST2"COSXPLU-KEJX/PI; REIX:=KONST2*SINXPLU+KERX/PI; KERXAC:=KtIX*FIIMIN-KERX*FIR~IN; KEIXAC:=-CKEIX*FIRMIN+KfRX*fIIMIN); 8ERX~C:=KONST2*(COSXPLU*fIRPLU-SINXPLU*FIIPLU)-KE1X~CIPI; 8EIXAC:=KONST2*CCOSXPLU*FIIPLU+SINXPLU*fIRPLU)-KE~XACIPI 'END' 'END'BEPZGAMMA; % %

'REAL' 'PROCEDURE' ABC fORMULE (A.B.C); 'VALUE' A.8.C; 'REAL' A.B.C;

'BEGIN"REAL' DETRM. TWEEA;

DETRM:=S"S-4*A"C; TWEEA:=2*A; 'If'DETRM<O'THEN'

'BEGIN'OETRM:=SQRT(-DETRM);WRITECOUT.<I,"DE_WORTELS_VhN_DIT_TwEEDE

GRAADSPOLYNOOM_ZIJN_TOEGEVOEGU_KOMPLEKS_VOO~".I'''A=''.El 5.6,1.

"B=",t15.6,1,"C=",E15.6.11 ."REELE_DEEL=".E15.6 " _____ _ IMAGINAII,E_ DEEL=".E IS .6. 1 I>. A. B.C.-B/TwEEA. UE. HlM/l wEEA) 'END'

'ELSE'

'dEGIN'DETR~:=SQRT( DETRM);WRITE(OUT.<I"'DE_WORTElS_VAN_DIT_TW~E0E GRAAD~POLYNOOM_ZIJN_BEIDE_REEEL _VQOR",I."A=",ElS.6,1. "B=".t15.6.1."C=".E15.6.11 ,"l-E_OPl.. L=",E.l':>.6 " _____ _

2-E_O~L.=".E15.6.11>.A.B,C,(-U+UETRM)/TwEEA,C-8-0tlRM)ITWEE~)

'END'

'END' ABC FORMLlE;

% %

HEAUCIN.I'L,T.N,UELTA.TS); T:=I/T;

~EAUCIN.I,X[11.A[21.XI31,XI41.XI51.X[61,XI7J.XIUI,XI91,A(iOI.xIIll,

X(121.XI131.XI141.XI151.XI161.XI171,X(181);

(37)

loR I 1 f (OUT. <I. "LENG T E_ VAN_DE_K A8EL _ IN_MEl ERS". E 1 O. j .1." 1 1 JUSDUUR_ V I,N_ CE_INPUIPULSE_IN_SEK.".EIO.3.I"PERIUDETIJD_VAN_DE_U.F.T._IN_SEK.".

f. 12.5';. "AAN T AL_SAMPLEPUNTEN_ VAN_DE_D. F • T .". 1 ~ tI. "SAMPLI:MUMENT". 1',.1 1>,L.l.PERIODE.MACHN.TS);

'FOR'J:=1'SlEP'l'UNTIL'18'DO'WRITE(OUT.<I."X __ ".I3.t:.20.t>.I>,J.XIJ.):

lAUNUL:=L~~-a~SQRT(1.12)/J; %%% LENGTE L IN METERS ~%%

LE:=2.598S65~6825~-7; CAP:=4.7889661438j~-11; RIO:=1.52~47263497~-2:

TWEEPI:=6.2b318~3072; PI:=3.141S926~36; DELTAGCAP:=O;

FI2:=ARCTAN(UELTAGCAP); MOD2:=SQRT(I+DELTAGCAP~DELTAGCAP);

'FOR'J:=1'STEP'l'UNTIL'MACHN1'DO'

'BEGIN'W:=T~EEFI*J/PERIODE; WOMEGA:=SQRT(W); Wl:=w*w;

Ul:=S.122360~6378~-3~wOMEGA;

'IF' Ul<~S 'IHEN'

'BEGIN' BEPZGAMMA(Ul.'FALSE'); NOEMER:=BERXAC~8ERXAC+BfIXAC~dEIXAC; Kl:=3.9u4449S579~-S~WOMEGA; Rl:=Kl*(BERX*BEIXAC-BEIX*BERX~C)/NOEMER; Ll:=Kl~(BlRX~BERXAC+BEIX~BEIXAC)/NOEMtRI 'END' 'lLSE' 'BEGIN' Ul:=3.6220S617316~-3~WOMtGAI Rl:=Ul~Ul*RlO/(2*Ul-1)1 Ll:=W*O.5*~-7~((2*UI-I)/Ul/UI+8*(Ul-1)~(Ul-1)*(UI-I)/UI/UII UI/U1IU1IU1) 1 'END'I U2:=21.7J212~-4*WOMEGAI

'IF' U2<b 'THEN'

'BEGIN' R2:=7.S2962S7068@-6*WOMEGA~(SINH(U2)+SlN(U2)l/ (COSH(U2)-COS(U2»-2.673Q677746~-41 L2:=7.S2962S7068~-6~WOMEGA~(SINH(U2)-SIN(U2»1 (COSH(U2)-LOS(U2»1 'END' 'ELSE' 'BEGIN' ~2:=7.S296257068~-6~WOMEGA-2.673~677746~-4; L2:=7.S2962S7068~-6~WOMEGA; 'END'I RR:=Rl+R21 LR:=Ll+W*LE+L21 FII:=ARCTAN(RR/LR) 1 MODI:=SQRT(l+RR~RR/LR/LR)1 ZMOD:=SQRT(LR/(W~C4P)~MODI/MOD2); ZARG:=(FI2-FIll/21 GAMMAMOO:=SGRT(LR*W*CAP~MODl~MODc); GAMMAARG:=(PI-FII-FI2)/2; ARGl:=w~T/2; OMTAUO:=W*TAUNUL; ALFAL:=GAMMAMOD*COS(GAMMAARG)*L; BETAL:=GAMMAMUD*SIN(GAMMAARG)*L;

RGZ:=15/lMOO; ZGR:=ZMODI1S; CI:=(KGZ+ZGR)~COS(LARb);

C2:=(ZGR-RGZ)*SIN(ZARG); EXPALFAL:=EXP(-ALFAL); E~P2ALFAL:=EXPALFAL*E~PALFAL; Cos2BETAL:=COS(2*BtlAL); SIN2BEIAL:=~IN(2*BETAL) I C3:=EXP2ALFAL~COS2BETAL; C4:=ExP2ALFAL*SIN2KETALI NR:=2+CI-(-Z+CI)*C3-C2*C4; NI:=C2*(1-C3)+(-2+C1)*C4; FMOD:=2*SIN(ARG1)OEXPALFAL/(SQRT(NR*NR+NI~NI)*AR01); 'IF'NR<O'THEN'FARG:=-(PI+ARGI+BETAL+ARCTAN(NI/NR»+OMTAUO 'ELSE'FARG:=-(ARG1+BETAL+ARCTAN(NI/NR»+DMTAUO; W:=W/j9; WI:=W*W;

Tl :=X(2 I-loll T2 :=W*X(3 ): Nl :=XI4 I-wI; N2 :="~Xl5 I; T3 :=XI6 I-loll T4 :=W"XI7 II N3 :=X(1i I-lNlI N4 := .. *~l9 I; TS :=XlIO)-~l; T6 :=W*X(11); N5 :=X(12)-wl; N6 :=w~Xl13);

T7 :=X(14)-.1; T8 :=w"X£lS); N7 :=X(16)-wl; N8 :=w*X(17); MOU:=FMOO*X(ll*SQRT((TI "Tl +T2 *T2 )/(NI *Nl +N~ *N~ l*

(38)

(13 *T3 +T4 *T4 (15 "T5 +T6 "T6 CT7 "T7 +HI "f8 ARG2:=FAR6-Z"ARCTANCW/X[18JII I/CN3 I/CN5 ) I CN 7 "N3 "NS "N7 +N'+ "N4 )" +N6 "Nb ) .. +N8 "No I);

ARGH:=ARCTA~C T2 ITI I; 'IF'ARGH<O'THEN'ARGH:=ARGH+P11 ARG2:=AR62+Af.<GH;

ARGH:=ARCTAN( T4 IT3 II 'IF'ARGH<O'THEN'ARGH:=ARGH+PI; ARG2:=ARG2+ARGH;

ARGH:=ARCTA~( T6 ITS I; 'IF'ARGH<O'THEN'ARGH:=ARGH+PI: AkG2:=ARG2+ARGH;

ARGH:=ARCTA~C T8 IT7 II 'IF'ARGH<O'THEN'ARGH:=AR6H+PI; ARG2:=ARG2+ARGH;

ARGH:=ARCTANC N2 INI I; 'IF'ARGH<O'THEN'ARGH:=ARGH+PI; ARG2:=ARG2-AkGH;

ARGH:=ARCTAN( N4 IN3 I; 'IF'ARGH<O'THEN'ARGH:=ARGH+Pl; ARG2:=ARG2-ARGHI

ARGH:=ARCTA~( N6 INS I; 'IF'ARGH<U'THEN'Af.<GH:=ARGH+PI: ARG2:=ARG2-AkGH;

ARGH:=ARCTA~( N8 IN7 I; 'IF'ARGH<O'TH~N'ARGH:=ARGH+P1;

ARG2:=ARG2-ARGH;

QIJJ:=MOO"CUSCARG21; QIJ+MACHNll:=MUO"SIN(AkGZII 'END' ;

GIOJ:=XIIJ"XIZI/X[4J"X[6J/X[8J"XII0J/X[12J"X(141/X(161;

G(MACHNll:=OI GlOI:=Q[OI "7~/CO.021~01812092"L+l~O);

FFTIH(N,Q,'lRUE'11 RE8ITREVERSECN,Q); PEO:=O; MSE:=O; ~ORM:=Q(TSI; 'FOR'J:=TS+20'STEP'20'uNTIL'81~Z'OO' 'BEGIN'Q[JJ:=Q[~I/NORM;PEO:=PED+ABS(Q[JII;MSt:=MSE+Q[JI*W(JII IIRITECOUT,<tl2.3>.Q(J) 'END'; 'FOR'J:=TS-20'STEP'-20'lJNTIL'1'DO'

'BEGIN'GETAL:=AtiS(Q[JI/NORMI; PED:=P~D+GETAL; MSE:=MSE+GtTAL"GETALI WRITE(OUT,<II,E12.3>,GETALI

'END' ;

INR ITE (oUT. <11111, "PEO". E20.1 0, "_VOOR_", 14, II INTERFERERS">, PEO,408 I : IoRITECOUT,<IIIII."MSE".E20.10,"_VOOR_",14."INTERFERERSII>,MSt,,+081: wRITE(OUT.<IIIII>I; 'FOR'J:=O'STEP'1'UNTIL'1000'DO'WRITECOUT,<EI2.3>.Q(JIINOf.<MI; % % .RITE(OUT,<III."N_U_L_P_U_N_T_E_N",I." ... ""'1»; 'FOR'J:=2'STEP'4'UNTIL'14'OO' A8C FURMULE(1,XIJ+ll"~9,XlJI"_18)1

.RITECOUT,<t16.7>.X(18J"~91;

.R1TE(OUT.<III"'P_0_L_E_N",I." ... ">11

'FOR'J:=4'STEP''+'UNTIL'16'DO' ABC FORMULECl,X(J+lJ"p9,X(JI"~1811

.RITECOUT,<E16.7>.X(181"_91; 'ENC'.

==============================================================================

'ORS DETECTED = O.

·MENTS

=

70. TOTAL SEGMENT SIZE = 1197 WO~OS. CO~E ESTIMATE

=

3749 WO~OS. ST = 215 CARDS, 2734 SYNTACTIC ITEMS, 50 DISK SEGMEN1S.

NAME: (U1643S433ItCA/AFSTUDEREN/TBER6MANS/KA8ELKORREKTlt. 87700 CODE GENERATE IME = 11.352 SECONDS ELAPSED; 2.207 SECONDS PROCtSSING; 3.973 SECONDS 1/0.

(39)

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(40)

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