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Adaptive secure data transmission method for OSI level 1

by Lallo, Pauli, PhD


Page 111
111
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
eixt )
(
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

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112
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 /
2π

= .
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 /
2π
using Euler’s rule is
)
/
2
sin(
)
/
2
cos( N
nk
j
N
nk π
π

Page 113

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

X (k) =

1


0

N
n=

x(n)[cos(2π
nk / N) jsin(2πnk / N)]

and the inverse DFT (IDFT) as

1
x(n) = N

1


0

N
n=

X (k)[cos(2π
nk / N) + jsin(2πnk / 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.

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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


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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.

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