1063-7397/01/3002- $25.00 © 2001 MAIK “Nauka /Interperiodica”
Russian Microelectronics, Vol. 30, No. 2, 2001, pp. 118–124. Translated from Mikroelektronika, Vol. 30, No. 2, 2001, pp. 59–66.
Original Russian Text Copyright © 2001by Arutyunov.
To date, many methods of fractal analysis in various
ﬁelds of science and technology have been suggested.
Before extensive activities in this area, mathematicians
used the concept of Hausdorff–Bezikovich (HB)
dimension. In its very essence, this dimension is related
not to topology but to
metric, i.e., to a way of construct-
ing a set considered.
HB dimension can take any val-
ues, which causes some scientists to speak of frac-
tional-dimension space. In fact, there exist objects with
dimension intermediate between point and line, line
and surface, and even surface and volume. These
objects are not topological manifolds, and Mandelbrojt
suggested to call them fractals. The best known fractals
are Cantor set, Brownian curve on a plane, Sierpinsky
carpet, Koch curve, and others. They exemplify inter-
mediate (e.g., uncountable) sets, which are convention-
ally called fractals. For them, the HB dimension
exceeds their topological dimension. It is generally
accepted today that fractals are frequently encountered
in many physical processes and phenomena, especially
in microelectronics, nanoelectronics, and materials sci-
ence. The concept of fractals can be said to have arrived
in time as a response to demands of science.
The subsequent evolution of science has shown that
the fractal theory is rich in application in various phys-
ical and chemical sciences. This is especially true for
scanning probe (tunnel and force) microscopy and frac-
tal materials science as applied to micro- and nanoelec-
tronics [1–3, 7–10].
Earlier, different deﬁnitions of fractal dimension
were studied at length for
out that the value of dimension is deﬁnition-indepen-
dent. Later, Mandelbrojt introduced a new term:
dates back to Euler.
Mandelbrojt cites Snapper and Troyer: “Roughly
speaking, self-afﬁne fractal is afﬁne geometry devoid
of any chance to measure lengths, areas, angles, etc.
Afﬁne geometry can be thought of as a very depleted
object. On the contrary, it is very plentiful.” Mandel-
brojt has shown that self-afﬁne fractals offer consider-
able scope for contemporary science.
While similarity transformation scales a geometri-
cal ﬁgure up or down, afﬁne (linear) transformation
acts differently in various directions. Moreover, a ran-
of Brownian motion remains (statisti-
cally) unchanged at “diagonal afﬁne” transformation.
Mandelbrojt has shown that generally it is necessary to
use several various fractal dimensions even in the case
of strictly self-similar sets, for which the dimension is
unique. Among these dimensions, the most important
are those obtained when a mass inside a sphere is found
and when a fractal is covered by identical cells. It
turned out that, irrespective of deﬁnition, only the fact
of interpolation or extrapolation is essential. In these
cases, we come to radically different values of dimen-
sion: local (which is valid for below-critical scales) and
global (valid for above-critical scales). Both do not
depend on the way we used to determine the dimension
(mass ﬁnding or cell coverage). For example, in the suf-
ﬁciently general case of recurrent construction of frac-
tals, the single radix
is replaced by two radixes
; then, instead of the classical definition for fractal
, we arrive at dimensions of different
degrees of generality. The deﬁnition
for local fractal dimension, while
global dimension. An intermediate, hole-related
dimension is often considered as well. It has the single
value for all scales,
Digital Simulation of Fractal Measurements
P. A. Arutyunov
Moscow State Institute of Electronics and Mathematics, Moscow, Russia
Received August 25, 2000
—The principles of digital signal processing are extended for the existing methods of fractal measure-
ments in terms of Hausdorff–Bezikovich and Mandelbrojt dimensions. Fractal dimension as a rational number
or the ratio of residues
of two equations of measurement (instrument-response equations) is rep-
resented in the form of an equivalent digital model of the ratio
of two sampling rates in some linear digital
system. It is shown that such a system is described by generalized convolution, which, in turn, is represented as
a cascade connection of interpolators and decimators with integer-valued conversion coefﬁcients. For error-free
processing of fractal dimension, it is necessary to convert a rational number to an integer and use Farey fractions
in terms of unimodular and multimodular arithmetic. The Farey fractions can then be used as references points
in metrological scales.