# Adaptive secure data transmission method for OSI level 1

Appendix 1: Mathematical background of

some transforms

A1 Definition of the Fourier Transform

Definition of the class

^{p}

_{L }is as follows:

Suppose

_{∞}

<

≤

_{p}

_{1 }. The function f on

_{)}

,

(

_{∞}

_{−∞ }is said to be of class

^{p}

L

^{(written }

^{p}

L

_{f }

_{∈ }) if

∫

^{∞}

∞

−

^{∞}

<

dx

x

f

^{p}

)

(

For each

^{1}

L

_{f }

_{∈ }, the integral

∫

^{∞}

∞

−

^{dt}

t

f

e

^{ixt })

(

exists for all real x.

The Fourier transform F of

^{1}

L

_{f }

_{∈ }formula is defined by

∫

^{∞}

∞

−

=

_{dt}

t

f

e

x

F

^{ixt })

(

)

(

^{, }

^{∞}

<

<

∞

−

_{x}

F(x) is continuous at x, which is shown using the Lebesque convergence theorem

[Gol70].

Definition of the Fourier transform on

^{2}

1

_{L}

_{L }

_{∩ }is also shown in [Gol70]:

It turns out that if

^{2}

L

_{f }

_{∈ }then the Fourier transform F of f is also in

^{2}

L

^{and}

2

½

2

^{)}

2

( f

F

_{π}

= , where

_{p}

_{f }is defined to be

∫

^{∞}

∞

−

p

p

_{dx}

x

f

^{/}

1

)

)

(

(

The symbol

_{p}

_{f }is read as the

^{p}

_{L }norm of f.

A2 Inverse Fourier Transform

If we know that a function

_{F }is the Fourier transform of some

^{1}

L

_{f }∈ we can determine

the function

_{f }from the values

_{F(x) }of

_{F}. The inversion

_{f(t) }is as follows [Gol70]:

If

_{∫}

^{∞}

∞

−

=

_{dt}

t

f

e

x

F

^{ixt }

^{)}

(

)

(

Then

_{∫}

^{∞}

∞

−

−

=

_{dx}

x

F

e

t

f

^{itx }

^{)}

(

2

1

)

(

_{π}

These equations may be written symmetrically by replacing f(t) with

_{)}

(

2

1

_{t}

f

π

^{as}

∫

^{∞}

∞

−

=

_{dt}

t

f

e

x

F

^{ixt })

(

2

1

)

(

_{π}

∫

^{∞}

∞

−

−

=

_{dx}

x

F

e

t

f

^{itx })

(

2

1

)

(

_{π}

The functions F and f are called a pair of Fourier transforms i.e., F is the Fourier

transform of f and vice versa. Such pairs are of great importance in the analysis of

electrical impulses etc. The Fourier transform is valid for both periodic and nonperiodic

f(t). All signals encountered in the real world easily meet the requirements.

[Mar62, Gol70]

A3 Discrete Fourier Transform (DFT)

The DFT is defined in references an operation on an N-point vector [x(0),x(1),…,x(N -

1)] as

∑

−

=

=

1

0

)

(

)

(

N

n

nk

N

W

n

x

k

_{X }, for k = 0, 1, 2, …, N -1

where

^{N}

j

N

^{e}

W

^{/}

2

_{π}

−

= .

The operation is a transformation from the N-point vector in time domain to another N-

point vector X(k) in frequency domain. The definition is interpreted as a frequency

sampling of the discrete-time Fourier transform.

A4 Inverse DFT (IDFT)

The inverse DFT (IDFT) can be computed using a forward DFT algorithm. The formula

for the IDFT is nearly identical to that for the forward DFt, except for a minus sign in

the exponent and a factor 1/N as

∑

−

=

−

=

1

0

)

(

1

)

(

N

n

nk

N

W

k

X

N

n

_{x }, for k = 0, 1, 2, …, N -1

A simplification for the part

^{N}

nk

j

e

^{/}

2

_{π}

^{− }using Euler’s rule is

)

/

2

sin(

)

/

2

cos( N

nk

j

N

nk

_{π}

π

^{−}

The Euler’s rule lets us state a more familiar form of DFT as

X ^{(}k^{) }^{=}

1

∑

0

−

N

n_{=}

x^{(}n^{)[cos(2}^{π}

nk ^{/ }N^{) }^{− }j^{sin(2}^{π}nk ^{/ }N^{)]}

and the inverse DFT (IDFT) as

1

x^{(}n^{) }^{= }_{N}

1

∑

0

−

N

n_{=}

X ^{(}k^{)[cos(2}_{π}

nk ^{/ }N^{) }^{+ }j^{sin(2}_{π}nk ^{/ }N^{)]}

Now we can calculate amplitude and phase from the complex value of _{X(k) }as

X ^{(}k^{) }^{= }a ^{+ }jb

X ^{(}k^{) }^{=}

φ_{(}_{t}_{) }^{= }_{tan}

a

−^{1}

2

+ _{b}

b

a

2

To compute the Fourier Transform digitally we do perform a numerical integration. The

result (DFT) is an approximation to a true Fourier Transform. A limitation is the finite

time record of input signal (finite-length vector). The calculations are made at discrete

points on the frequency and time domain. The frequency spacing of the result is the

reciprocal of the time record length.

A5 Fast Fourier Transform (FFT)

The Fast Fourier Transform (FFT) is an algorithm for computing the Discrete Fourier

Transform first described in [Coo65]. FFT is a fast algorithm for computing the DFT.

FFT is used and described in references MATLAB [Bur94] and SPW [Com90].

References

[Bur94] Burrus, C. S., et. al., _{Computer-Based Exercises for Signal Processing Using}_{MATLAB}^{• }_{, }Prentice-Hall International, Inc., Englewood Cliffs, NJ, 1994.

[Com90] Comdisco Systems, Inc., _{Signal Processing Worksystem}, January 31, 1990.

[Coo65] Cooley, J. W., Tukey, J. W., An algorithm for the machine calculation of complex

Fourier series, _{Mathematics of Computation}, 19, 90, pp. 297-301, 1965.

[Gol70] Goldberg R. R., _{Fourier Transforms}, The Syndics of the Cambridge University

Press, Cambridge, 1970.

[Mar62] Margenau, H., _{The Mathematics of Physics and Chemistry}, D. Van Nostrand

Company, INC. Princeton, NJ, USA, March 1962.

113

Appendix 2: A high-level block diagram of a

dmt/ofdm system

Reference

[Ram02] Ramírez-Mireles, F., et. al., The Benefits of Discrete Multi-Tone (DMT)

Modulation for VDSL Systems, Ikanos Communications, framirez@ikanos.com, Version

1.4, October 11, 2002.

114

Appendix 3: xDSL Capacity versus Distance

Reference

[Mar03] De Marchis, G., Tutorial on Technologies, TelCon srl, ITU-T workshop Outside

plant for the Access Network, Hanoi 24 November 2003.

115