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Autocorrelation Function:

Because the Maple and Mathematica scriptsused are too large for this paper, we present the stepswe followed for computing the autocorrelationfunction (R[n]) and the power spectral density (PSD)of the MLT 3 5 code. The Moore state transitiondiagram of MLT 3 5 is symmetrical and contains 40states, so let N=20.

1. First at all, we denote that the state matrix T canbe written using the matrixes T 1 and T 2 :

=12

21

T T

T T

T M

LL

M

=

p

p p

p p

p p

p p

p p p

p p

p p

p p

p

p p

p p

p p

p p p

p

p

p

p

T

10000000000000000000

0100000000000000000

0010000000000000000

0001000000000000000

0000100000000000000

100000000000000000000100000000000000000

0010000000000000000

0001000000000000000

0000100000000000000

0000000000000000000

0000000000000010000

0000000000000001000

0000000000000000100

00000000000000000100000000000000000000

00000000000000010000

00000000000000001000

00000000000000000100

00000000000000000010

1

0 0 0 0

0 0 0 0

1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

1 11 1

0

0 1

0 0 0 0 0 0 0 00 0

00000rds = -1rds = -2 rds = 0 rds = 1 rds = 2 rds = 3

Transmit +1in these states

Transmit 0in these states

Transmit -1in these states

Mealy state transition diagram for MLT3 5 code

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R[0]

R[1]R[2]

R[3]

R[4]

R[5]

R[6]

R[7]

R[8]

R[9]

R[10]

=

0000000000000000000

00000000000000000000

00000000000000000000

00000000000000000000

00000000000000000000

0000000000000000000

00000000000000000000

00000000000000000000

00000000000000000000

00000000000000000000

00000000000000000001

00000000000000000000

00000000000000000000

0000000000000000000000000000000000000000

00000000000000000001

0000000000000000000

0000000000000000000

0000000000000000000

0000000000000000000

2

p

p

p

p

p

p

p

p

T

[ ]000000000011111111111 = A

= 101

101

101

101

101

10101010101010101010101

101

101

101

101

1p p p p p p p p p p p p p p p p p p p p

w

2. The output matrix is:[ ]11 A A A = M

1=3. The vector w contains the stationary probabilities

p1, p2, , p 2N and matrix d= diag(w). We can use

the fact that: [ ]1121

www M=4. The stationary probabilities was found solving the

following equations system:

==+1

)(

1

1211t

N I w

wT T w,

where I N is a line matrix of ones.5. Now we can calculate the autocorrelation

function (AKF), using the formula:t n AT d An R = )(][

And using Maple, we get:

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21

]0[ = R

p R =103

103

]1[

32

54

52

51

101

]2[ p p p R +=

5432

58

29

527

1033

56

103

]4[ p p p p p R +++=

65432

57

65

535

535

3325

21

]5[ p p p p p p R +++=

65432

513

544

566

556

529

59

103

]6[ p p p p p p R +++=

8765432

516

546

554

526

58

1037

1023

107

101

]7[ p p p p p p p p R ++++=

98765432 85

1545

2845

3185

23410241

549

1033

54

101

]8[ p p p p p p p p p R ++++=

1098765432

572

5318

5701

5986

5959

5686

5384

5174

121027

103

]9[ p p p p p p p p p p R +++++=111098765432

5109

5521

51283

104151

52374

52012

51348

101511

5348

5118

521

]10[ p p p p p p p p p p p R +++++=

Power Spectral Density:Because of the big computing volume for

PSD, we have two ways: to use the Bennett formula(1) or to try some methods [Pet] to reduce thisvolume and use the formula (2).

(1)

+= =1 ))1(2cos(][]0[1),( k n k f k R RT p f W

(2)

=++=

nn C zC T T

n f C

T p f W )]Re(2[

1)(

1),( 21

2

02

where we considered: G1 = a vector containing the Fourier transformsof the input wave forms. This means that, if for apulse wave form

=otherwise

T T t t s,0

]2

,2

[,1)( , f

T f f S

= )sin(

)(

or for a Dirac wave form)()( t t s = , 1)( = f S ,

we can write 11 ASG = .In the followings we shall consider S(f)=1.

)( 111 wdiagGG =

.

*1G = hermitic transpose of 1G .

)2sin()2cos(2 T f jT f e z T f j +== == T f f n normalized frequency with respect

to the input bit rate.

= N 1 a vector of N ones,= N I a N x N identity matrix. 110 GwC =

*1111 )( Gw I GC N

t N +=

*1

121112 )1( GT T w I zGC N

t N +=

)1(21

+= .In our case 1= , which denotes 0= . So weget:

00 =C *111 GGC =

*1

12112 )( GT T I zGC N =

.

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Page 5

In case of using Bennetts formula (1), the PSDexpression obtained with Waterloo Maple is given in[Appendix 2] and the results are illustrated below:

We observe that in case we use formula (2), itmight be useful to find a way to simplify thecomputing of matrix 121 )(

T T I z N .

Let us note that: [ ]10101 OSG M=

and =t O

SG

10

*10

*1

L ,

where:[ ]SSSSSSSSSSS =10 and[ ]000000000010 =O .

Let = DDCC

BB AA

T T I z N M

LL

M1

21 )( and

=WW ZZ

YY XX

T T I z N M

LL

M

)( 21 .

Using all this, it denotes:

[ ] =t O

S

DDCC

BB AA

OSC

10

*10

10102L

M

LL

M

M

[ ] [ ] *10102 S AASC = .So, having the matrix )( 21 T T I z N , it may beeasily demonstrated that 11 )( = ZZ WW YY XX AA ,which is all we need and it is much easier to computethan 121 )(

T T I z N .

The PSD formula resulted after Mathematicacomputing is revealed in [Appendix 3] and thegraphical results are shown in the plots below:

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Conclusions:Even if the PSD formula cant tell us too

much about the code, the 2D and 3D plots are moresignificant. It is shown that the MLT 3 5 codeminimize high frequencies spectral components andthe d.c. component tends to zero.

As further work, it might be interesting

simulate and synthesize an MLT 3 m encoder startingfrom VHDL or Verilog code or from state transitiondiagram using tools as ModelSim or FPGA Express.

References:

[Alx] Alexandru, N. D. and Morgenstern, G.,[1998], Digital Line Codes and SpectralShaping, MatrixRom, Bucharest

[Ans] ANSI X3T12, [1994], FDDI Twisted PairPhysical Layer Medium Dependent (TP-PMD), American National Standard

[Car] Cariolaro, G. L. and Tronaca, G. P., [1974],Spectra of block coded digital signals,IEEE Trans., COM-22, (10), pp. 1555-1556

[Cok] Cook, J. W., [1994], Spectra of a class of run-length limited MLT3 codes, Electron.Lett., 1994, 30, (16), pp. 1284-1285

[Col] Coles, A. N., [1995], New pseudoternaryline code for high-speed twisted pair datalinks, Electronics Letters, 9 th Nov. 1995,vol. 31, no. 23, pp. 1976-1977

[Dra] Drajic, D. and Petrovic, G., [1980], PowerSpectra of HDB n signals, Electron. Lett.,1980, vol. 16, no. 8, pp. 289-291

[Mow] Mowbray, M., Coles, A. N. andCunningham, D. G., [1993], New 5B/ 6TCode for Data Tranmission on UnshieldedTwisted Pair Cable, Electronics Letters,vol. 30, pp. 340-343, Feb. 1993

[Oec] [2001], Optimized EngineeringCorporation, http://www.optimized.com

[Pet] Petrovic, G., [1979], Power Spectrum of Balanced Digital Signal Generated byMarkov Source, Electron. Lett., 1979, vol.15, no. 24, pp. 769-770

Appendix 1:C code for MLT 3 m encoding

/ / MLT 3-m Encoder# include # include < stdlib.h># include # include # include < string.h># define MAX 100

char InStr[MAX],UD,InAnt,In;int B,RDS,RDSStr[MAX];int t,i,m,OutAnt,OutStr[MAX],Out;

void ReadIn(void){printf("\ nGive the input binary stream (maximum 100

bits): \ n");t=1;do{t=0;gets(InStr);for(i=0;i

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Out=-1;RDS--;

}else{Out=1;RDS++;UD='U';

}else if (OutAnt= =1)

if (InAnt=='0')if (RDS(-(int)(m/ 2)))

{Out=0;UD='U';

}else {Out=0;}

else / / InAnt== '1'{Out=0;}

else / / In=='0'if (OutAnt==0)

Out=0;else

if (OutAnt==1){

if (RDS(-(int)(m/ 2)))

{Out=-1;RDS--;

}else

{Out=1;RDS++;

UD='U';}

InAnt=In;OutAnt=Out;

}

void WriteOut(void){printf("\ nInput stream:\ n");for(i=0;i

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Appendix 2:PSD Bennett approach

W[fn,p] = -218/ 5*cos(20*Pi*fn)*p^11+(1042/ 5*cos(20*Pi*fn)+ 144/ 5*cos(18*Pi*fn))*p^10+(-16*cos(16*Pi*fn)-636/ 5*cos(18*Pi*fn)-2566/ 5*cos(20*Pi*fn))*p^9+(1402/ 5*cos(18*Pi*fn)+32/ 5*cos(14*Pi*fn)+4151/ 5*cos(20*Pi*fn)+308/ 5*cos(16*Pi*fn))*p^8+ (-568/ 5*cos(16*Pi*fn)-92/ 5*cos(14*Pi*fn)-4748/ 5*cos(20*Pi*fn)-1972/ 5*cos(18*Pi*fn))*p^7+(108/ 5*cos(14*Pi*fn)-26/ 5*cos(12*Pi*fn)+4024/ 5*cos(20*Pi*fn)+636/ 5*cos(16*Pi*fn)+1918/ 5*cos(18*Pi*fn)-14/ 5*cos(10*Pi*fn))*p^6+ (-1372/ 5*cos(18*Pi*fn)+16/ 5*cos(8*Pi*fn)+ 2*cos(10*Pi*fn)+88/ 5*cos(12*Pi*fn)-52/ 5*cos(14*Pi*fn)-2696/ 5*cos(20*Pi*fn)-468/ 5*cos(16*Pi*fn))*p^5+ (-32/ 5*cos(12*Pi*fn)+1511/ 5*cos(20*Pi*fn)+241/ 5*cos(16*Pi*fn)-9*cos(8*Pi*fn)-16/ 5*cos(14*Pi*fn)-

12/ 5*cos(6*Pi*fn)+768/ 5*cos(18*Pi*fn)-106/ 5*cos(10*Pi*fn))*p^4+(23/ 5*cos(6*Pi*fn)+106/ 5*cos(10*Pi*fn)-696/ 5*cos(20*Pi*fn)-348/ 5*cos(18*Pi*fn)+112/ 5*cos(12*Pi*fn)-98/ 5*cos(16*Pi*fn)+54/ 5*cos(8*Pi*fn)+37/ 5*cos(14*Pi*fn)+8/ 5*cos(4*Pi*fn))*p^3+(236/ 5*cos(20*Pi*fn)-66/ 5*cos(10*Pi*fn)-13/ 5*cos(6*Pi*fn)-58/ 5*cos(12*Pi*fn)+24*cos(18*Pi*fn)-4/ 5*cos(4*Pi*fn)+33/ 5*cos(16*Pi*fn)-33/ 5*cos(8*Pi*fn)-23/ 5*cos(14*Pi*fn))*p^2+(3/ 5*cos(6*Pi*fn)-

27/ 5*cos(18*Pi*fn)+7/ 5*cos(14*Pi*fn)+12/ 5*cos(8*Pi*fn)+18/ 5*cos(12*Pi*fn)-3/ 5*cos(2*Pi*fn)-10*cos(20*Pi*fn)-2/ 5*cos(4*Pi*fn)+ 5*cos(10*Pi*fn)-8/ 5*cos(16*Pi*fn))*p+1/ 2-1/ 5*cos(14*Pi*fn)+3/ 5*cos(18*Pi*fn)-cos(10*Pi*fn)+1/ 5*cos(4*Pi*fn)-1/ 5*cos(6*Pi*fn)+ cos(20*Pi*fn)-3/ 5*cos(8*Pi*fn)-3/ 5*cos(12*Pi*fn)+1/ 5*cos(16*Pi*fn)+3/ 5*cos(2*Pi*fn).

Appendix 3:PSD Petrovic approach

W[fn,p]=

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