Fractal Scaling or Scale-invariant Radar

Fractal Scaling or Scale-invariant Radar
Universal Journal of Physics and Application 11(1): 13-32, 2017
DOI: 10.13189/ujpa.2017.110103
Fractal Scaling or Scale-invariant
Radar: A Breakthrough into the Future
A.A. Potapov1,2,3
Kotel'nikov Institute of Radio-Engineering and Electronics (IREE RAS), Russian Academy of Sciences, Russia
College of Information Science and Technology / College of Cyber Security, Jinan University (JNU), China
Cooperative Chinese-Russian Laboratory of Informational Technologies and Signals Fractal, China
Copyright©2017 by authors, all rights reserved. Authors agree that this article remains permanently open access under the
terms of the Creative Commons Attribution License 4.0 International License
Abstract Results of application of theory of fractal and
chaos, scaling effects and fractional operators in the
fundamental issues of the radio location and radio physic are
presented in this paper. The key point is detection and
processing of super weak signals against the background of
non-Gaussian intensive noises. The main ideas and strategic
directions in synthesis of fundamentally new topological
radar detectors of low-contrast targets / objects have been
considered. The author has been investigating these issues
for exactly 35 years and has obtained results of the big
scientific and practical worth. The reader is invited to look at
the fundamental problems with the synergetic point of view
of non-Markovian micro- and macro- systems. The results of
big practical and scientific importance obtained by the author
were published in four summary reports of the Presidium of
Russian academy of science (2008, 2010, 2012, and 2013)
and in the report for the Government of Russian Federation
Fractal, Scaling, Texture, Chaos, Radar,
1. Introduction
Intensive development of modern radar technology
establishes new demands to the radiolocation theory [1,2].
Some of these demands do not touch the theory basis and
reduce to the precision increase, improvement and
development of new calculation methods. Other ones are
fundamental and related to the basis of the radiolocation
theory. The last demands are the most important both in the
theory and in practice.
Radar detection of unobtrusive and small objects near the
ground and sea surface and also in meteorological
precipitations is an extremely hard problem [1]. One should
take into account that the noise from the sea surface and
vegetation has nonstationary and multi-scale behavior
especially at high incidence angles of the sensing wave.
The entire current radio engineering is based on the
classical theory of an integer measure and an integer
calculation. Thus an extensive area of mathematical analysis
which name is the fractional calculation and which deals
with derivatives and integrals of a random (real or complex)
order as well as the fractal theory has been historically turned
out “outboard” (!) [2-14]. At the moment the integer
measures (integrals and derivatives with integer order),
Gaussian statistics, Markov processes etc. are mainly and
habitually used everywhere in the radio physics, radio
electronics and processing of multidimensional signals. It is
worth noting that the Markov processes theory has already
reached its satiation and researches are conducted at the level
of abrupt complication of synthesized algorithms. Radar
systems should be considered with relation to open
dynamical systems. Improvement of classical radar detectors
of signals and its mathematical support basically reached its
saturation and limit. It forces to look for fundamentally new
ways of solving of problem of increasing of sensitivity or
range of coverage for various radio systems.
In the same time I'd like to point out that it often occurs in
science that the mathematical apparatus play a part of
“Procrus-tean bed” for an idea. The complicated
mathematical symbolism and its meanings may conceal an
absolutely simple idea. In particular the author put forward
one of such ideas for the first time in the world in the end of
seventies of XX century. To be exact he suggested
introducing fractals, scaling and fractional calculation into
the wide practice of radio physics, radio engineering and
radio location [2,6-24]. Now after long intellectual battles
my idea has shown its advantages and has been positively
perceived by the majority of the thoughtful scientific
community. For the moment the list of the author's and
pupils works counts more than 800 papers including 23
monographs on the given fundamental direction. Nowadays
it is absolutely clear that the application of ideas of scale
invariance - "scaling" along with the set theory, fractional
Fractal Scaling or Scale-invariant Radar: A Breakthrough into the Future
measure theory, general topology, measure geometrical
theory and dynamical systems theory reveals big
opportunities and new prospects in processing of
multi-dimensional signals in related scientific and
engineering fields. In other words a full description of
processes of modern signal and fields processing is
impossible basing on formulas of the classical mathematics
In this paper presented alternative solutions of actual
problems of modern radiolocation are based on the ideas and
methods of new scientific fundamental direction “Fractal
Radio Physics and Fractal Radio Electronics: Designing of
Fractal Radio Systems”. This direction was initiated by
Professor A. Potapov since 1980 at the IREE RAS
(Kotel’nikov Institute of Radio Engineering and Electronics
of the Russian Academy of Sciences, Moscow, Russia) and
currently it is widely developed in his works and
acknowledged in the Scientific World.
2. Theory of Fractional Measure and
Nonintegral Dimension
The main property of fractals is a nonintegral value of its
dimension. Development of the dimension theory began with
works of Poincare, Lebesgue, Brauer, Urysohn and Menger.
Sets which are negligibly small and indistinguishable in
terms of Lebesgue measure in one meaning or another
appear in various fields of mathematics. To distinguish such
sets with pathologically complex topological structure it is
necessary to use non-traditional characteristics of smallness,
for example capacity, potential, Hausdorff measures and
dimension and so on. Application of Hausdorff fractional
dimension which is tightly related to conceptions of entropy,
fractals and strange attractors in the dynamic systems theory
turned out to be the most fruitful [2,3,5-24].
Conception of Hausdorff measure and dimension is one of
those required conceptions which must be mastered
organically before every researcher can become the fractal
specialist and deterministic chaos specialist. This fractional
dimension is determined by p - dimensional measure with an
arbitrary real positive number p which was introduced by
Hausdorff in 1919. In general the measure conception is
related neither with metrics nor with topology. However the
Hausdorff measure can be built in a random metric space
basing on its metrics and the Hausdorff dimension itself is
related with the topological dimension.
Conceptions introduced by Hausdorff are based on the
Caratheodory theory (1914). Let us assume that ( M , ρ ) is a
metric space, F is a family of subsets of set M and f is such a
function on F that 0 ≤ f (G ) ≤ ∞ when C ∈ F and f(Ø) = 0.
Let us build auxiliary measures mεf and then the main
measure Λ f in the following way. When E ⊂ M and
ε > 0 the value of mεf is defined as the infimum of set of
mεf = inf
∑ f (G )
over every possible countable ε -coverings of {G } , G ∈ F .
It results from inequation mεf 1 ( E ) ≥ mεf 2 ( E ) for ε 2 > ε1
that the limit exists
Λ ( E ) = lim mεf ( E ) = sup mεf ( E ) .
ε →0+
It is clear that mεf and Λ (E ) are exterior measures on M.
Let ρ (a, B) > ε > 0 . Let us consider a random ε -covering of
{Gi } set A  B which consists of a certain number of sets.
Then families {A  Gi } and {B  Gi } do not intersect and
cover sets A and B respectively and so
mεf ( A  B) ≥ mεf ( A) + mεf ( B)
Λ f ( A  B) = Λ f ( A) + Λ f ( B) .
Class of Λ f – measurable sets of space M forms a σ –
ring which the exterior measure Λ f is regular on. They
also call measure Λ f as the result of application of
Caratheodory theory to function f and exterior measure mεf
as the approximating measure with order ε . Measure Λ f
represents properties of function f and family F quite fine
although it is not an extension of f usually.
We point out two simple statements which describe
behaviour of approximating measures at a decreasing
sequence C1 ⊃ C2 ⊃ ... of compact subsets of space M. If
elements of family F are open subsets of M then
lim mεf (Gi ) = mεf (  Ci ) .
i →∞
i =1
If 0 < ε 0 < ε and f ( S ) = inf { f (T )} : T ∈ F , S ⊂ IntT ,
d (T ) ≤ ε for every such S ∈ F that
d ( S ) ≤ ε 0 lim mεf (Gi ) ≤ mεf 0 (  Ci ) ,
i →∞
i =1
where d is the diameter of the sets, Int is the set of all the
internal points of set T.
Let us assume that X is a limited compact metric space, F
is the family of all the nonempty compact sets from X,
function f: F → [0,+∞] continuous in regard to Hausdorff
metric and f (C ) > 0 for all such C ∈ F that d (C ) > 0 . If
A1 ⊂ A2 ⊂ A3 ⊂ ... form an increasing sequence of subsets
of space X then
lim mεf ( Ak ) = mεf (  Ak )
k →∞
k =1
Let us define h – Hausdorff measure. Let h(r ) be a
continuous monotonic increasing function of r (r ≥ 0)
Universal Journal of Physics and Application 11(1): 13-32, 2017
which h(0) = 0 for. We indicate the class of such functions
as H0. By applying the Caratheodory construction to function
f ( E ) = h[d ( E )] where E ≠ Ø and f(Ø) = 0 (here d (E ) is
the diameter of set E) we get Λ h - the Caratheodory
measure which is called as the Hausdorff h-measure. If at
that h(r ) = γ (α )r α where α is a fixed positive number
which is not necessarily an integer and γ (α ) is a positive
constant which depends only on α then the Hausdorff
h-measure is called as the α -dimensional measure or the
Hausdorff α -measure H α which is a Borel regular measure.
One can imagine the construction of Hausdorff h-measure
in the following way. Let us cover α with a random sequence
of circles Cv with radius rv ≤ ε (ε > 0; v = 1, 2, ...) and
mark the infimum of respective sums
∑ h( r )
v =1
mhε (α , h) ≥ 0 . This number increases with decrease of ε . By
Λ h ( E ) = lim mhε (α , h)
ε →0
given number α 0 is the Hausdorff-Besicovitch dimension.
If they use balls of the same size for covering during
determination of the Hausdorff H α -measure then such a
measure is called as entropic. Then dimension (10) is called
as entropic or a Kolmogorov dimension. For sets of positive
k-dimensional Lebesgue measure both dimensions coincide
and equal K. The Hausdorff-Besicovitch dimension
describes the exterior property of a set. Therefore it is
appropriate to introduce a conception of the
Hausdorff-Besicovitch dimension at a point which would
describe its internal structure.
In this case number
α E ( x0 ) = lim α 0 ( E  On ( x0 ))
n →∞
linear measure ( h(r ) = 2πr ), planar measure ( h(r ) = πr 2 )
and logarithmic measure ( h(r ) = 1 / ln r ).
E1 ⊂ E2
Λ h ( E1 ) ≤ Λ h ( E2 ) that is the Hausdorff h-measure is a
monotonically increasing set function. With using an
h-measure the dimension of set is defined in the following
way. If 0 < Λ h ( А) < ∞ then h is called as the metric
by definition or there is the point of “jumping” α 0 such that
Hα (E ) = ∞ for α < α 0 and Hα ( E ) = 0 for α > α 0 . And
contracting neighborhoods of point x0 ∈ M .
Each limited closed set E of m-dimensional Euclidean
space contains point x0 ∈ E such that
space M. By choosing various functions as h(r ) we get:
E ⊂ M or Hα ( E ) = 0 for every α > 0 , then α 0 ( E ) = 0
is called a local Hausdorff-Besicovitch dimension of set E at
point x0 . Here {On ( x0 )} is a random sequence of
Limit (8) is the Hausdorff exterior h-measure which is a
Borel regular measure at σ - ring of Λ h - measureable sets of
Hα1 ( E ) = ∞ for every positive α 1< α 2 . Hence for set
0 ≤ Λ h ( E ) ≤ +∞ .
dimension (the Hausdorff dimension) of set A. If h(r ) = cr α
and 0 = Λ h ( А) < ∞ then the dimension of set A is indicated
as α , here с is a constant. Sets with a certain dimension
have an h-measure equal to 0 for each exterior dimension and
an h-measure equal to ∞ for each lowest dimension.
Next generalization of the dimension conception is a
Hausdorff-Besicovitch dimension, which is introduced using
nonnegative numbers α 0 = α 0 ( E ) in the form of equation
α 0 ( E ) = sup{α : Hα ( E ) ≠ 0} = inf {α : Hα ( E ) = 0} (10)
for set E. The Hausdorff-Besicovitch dimension of a set is
defined by behaviour of Hα (E ) not as function of E but as
function of α .
Correctness of definition (10) confirms the following
property of H α -measure. If Hα (E ) < ∞ then Hα ( E ) = 0
for every α 2 > α1 . If measure H α 2 ( E ) is a non-zero then
α E ( x0 ) = α 0 ( E ) .
Function α E (x) is called as a function of the local
Hausdorff-Besicovitch dimension if
0 ≤ α E ( x) ≤ α 0 ( E ) for every x ∈ M ,
α E ( x) = 0 , if set E is closed and x ∉ E ,
α E ( x) = 0 for all the isolated points of set E.
The Hausdorff-Besicovitch dimension is a metric
conception but there is its fundamental association with the
topological dimension dim E which was determined L.S.
Pontryagin, L.G. Shnirelman. They introduced a conception
of the metric order in 1932: the infimum of the
Hausdorff-Besicovitch dimension for every metric of
compact E equals its topological dimension dim E ≤ α ( E ) .
One of widely used methods for evaluating the sets
Hausdorff dimension which is known as the principle of
masses allocation was proposed by Frostman in 1935.
Sets which have the fractional Hausdorff-Besicovitch
dimension are called fractals sets or fractals. More strictly,
set E is fractal (a fractal) in a general sense (in the B.
Mandelbrot sense) if its topological dimension does not
coincide with the Hausdorff-Besicovitch dimension, to be
exact α 0 ( E ) > dim E . For example the set E of all the surd
points [0; 1] is fractal in a general sense since α 0 ( E ) = 1 ,
dim E = 0 . Set E is called fractal (a fractal) in the narrow
sense if α 0 ( E ) is not an integer. A set which is fractal in the
narrow sense is also fractal in a general sense.
Fractal Scaling or Scale-invariant Radar: A Breakthrough into the Future
As it was shown by A.S. Besicovitch for the first time in
1929 there were deep discrepancies between Lebesgue sets
and fractals. First of all, these features concern densities.
Geometric properties of fractal set E are determined by
behaviour of function
D ( x, ε ) =
H α ( E  O( x, ε ))
for small ε , wherex is a random point of set E.The higher α
which is the density of set E at point x is
Dα ( E , x) = limD( x, ε ) ,
ε ≤0
and the lower α which is the density of set E at point x is
Dα ( E , x) = lim D( x, ε ) .
ε ≤0
When Dα ( E , x) = Dα ( E , x) then their generalized value
is called as an α -density of set E at point x and it is
identified as Dα ( E , x) . If ε → 0 + then Dα ( E , x) and
Dα ( E , x) are called as right-side, if ε → 0 − then they are
called as left-side, if ε → 0 they are called two-side the
upper and lower α -density respectively.
question by calculating the number of resolution elements
covering the object. Surface area S in this case would be
equal to:
S ≡ S (λ ) = N (λ )δ (λ )
where δ (λ ) - the square of a resolution element of the radar;
N (λ ) - number of resolution elements required to cover the
object, λ - the wavelength of the radar, as it was already
noted for a simple object (Figure 1, d) value S (λ ) = const .
For the RI on Figure 1,a and 1,b one can build dependence
S (λ ) = f (λ ) and assuming that δ (λ ) = K (λ ) , where K is a
known function then one can build dependence
S (λ ) = f (δ ) . It happens that the measured square S is
described well by formula
S (λ ) = kλ − D
Then just taking the logarithm we can calculate parameter
D. Dependence which determines a fractal signature D(t, f,
r ) of a RI by itself is shown on Figure 1,e. This dependence
describes a space fractal cepstrum of an image (this
conception was introduced by the author in nineties of XX
3. To the “Fractal” Conception in Radio
In general terms a radar image (RI) can always be
presented as a set of elements Xk, whose values are
proportional to the scattering cross-section (SCS) of a k-th
element of resolution of the radar [6-10]. In Figure 1, a the
RI of the terrain which was obtained at wavelength λ = 8.6
mm from a helicopter is shown. In Figure 1, b the RI of the
same terrain region which was obtained by a radar at
wavelength λ ≈ 30 cm is shown. Both images are
two-dimensional with gray level proportional to SCS.
Let us suppose that for every RI a surface (Figure 1, c)
with height h which is proportional to the gray level is built.
Let us suppose that we need to measure the square of the
resulting surface. On RI which corresponds λ ≈ 30 cm the
square will be less than for RI on λ ≈ 8.6 mm since the
smaller wavelength the more terrain details can be
A probing electromagnetic wave is some kind of a "ruler"
in this case. At that an increasingly finer structure of
time-spatial signals or wave fields begins to have an effect.
If we have a RI which was obtained at even shorter waves
then its square will be bigger and so on. By decreasing the
wavelength λ we will get increasing values of the squares.
Then the question arises: and what is the square of the
surface which the RI was obtained from in reality? If the
surface is covered with simple objects, for example a
rectangular eminence (Figure 1, d), and sizes of this
eminence are much higher than the wave length then the
squares of objects on the RI will be approximately equal for
short and long waves. Then we would answer the mentioned
Figure 1. Examples explaining the matter of fractal processing (a - d) and
a fractal space signature (e)
The fractional parameter D is called the
Hausdorff-Besicovitch dimension or the fractal dimension
[3,5,7,8].For RIs of objects with simple geometric form
(rectangles, circles, smooth curves) this dimension coincides
with the topological one that is it equals 2 for
two-dimensional RIs and it is determined by the slope of
straight lines (18) in binary logarithmic coordinates.
However the value of D for majority of images of real
coverings and meteorological formations turns out to be
higher than the topological dimension D0 = 2 that
emphasizes its complexity and random nature.
4. Textural and Fractal Measures in
Radio Physics and Radar
A radar along with observation objects and radio waves
propagation medium forms a space-time radar-location
Universal Journal of Physics and Application 11(1): 13-32, 2017
probing channel. During the process of radio location the
useful signal from target is a part of the general wave field
which is created by all reflecting elements of observed
fragments of the target surrounding background, that is why
in practice signals from these elements form the interfering
component. It is worthwhile to use the texture conceptions to
create radio systems for the landscape real inhomogeneous
images automatic detecting [6-8]. A texture describes spatial
properties of earth covering images regions with locally
homogenous statistical characteristics. Target detecting and
identification occurs in the case when the target shades the
background region at those changing integral parameters of
the texture.
Many natural objects such as a soil, flora, clouds and so on
reveal fractal properties in certain scales [5-10]. Today
analysis of natural textures is undergone by significant
changes due to use of metrics taken from the fractal
geometry. After a texture they introduced the conception of
fractals that is signs based on the fractional measure theory
for fundamentally different approach of solving modern
radio location problems. The fractal dimension D or its
signature in different regions of the surface image is a
measure of texture i.e. properties of spatial correlation of
radio waves scattering from the corresponding surface
regions. At already far first steps the author initiated a
detailed research of the texture conception during the process
of radio location of the earth coverings and objects against its
background. Further on a particular attention was paid to
development of textural methods of objects detecting against
the earth coverings background with low ratios of
signal/background (see for example [6-10,14,20,24] and
5. Textural Measures and Textural
Regions of background reflections which are united in a
general texture conception are always presented around a
detectable target. It allows proposing new approaches to
detecting extensive low-contrast targets against the
background of earth coverings in obtained radar images (RI)
or multidimensional signals. Analysis of experimentally
obtained extensive data bases in aggregate with visual
research of degree of complexity of profiles of isolines of
scattered radiation which was fixed on optical and radio
images brought the author to ideas of synergetic
developments of ensembles of fractal signs based on
synthesis of scaling invariants with fractional measure
properties in eighties of XX century [6-8].
Unlike tone and colour which relate to image separate
fragments a texture relates to more than one fragment. We
think that the texture is a matrix or a fragment of space
properties of images regions with homogenous statistical
characteristics. Textural signs are based on statistical
characteristics of levels of intensity of image elements and
relate to probabilistic signs whose random values are
distributed over all classes of natural objects. A decision on
texture belonging to one or another class can be made only
basing on specific values of signs of the given texture. In this
case it is usual to say about a texture signature. Classic radar
signatures include time, spectrum and polarized features of
the reflected signal. In our view the texture signature is a
distribution of general totality of dimensions for the given
texture in scenes of the same kind as the given one.
When it is possible to decompose a texture two main
factors are revealed. The first one correlates a texture with
non-derivative elements which form the entire image and the
second one serves for describing a spatial dependence
between them. Tone non-derivative elements by itself
represent image fields which are characterized by certain
values of brightness proportional to the intensity of the
reflected signal which in turn depends on values of the
normalized effective cross-section σ ∗ of the earth surface.
Since a conception of normalized effective cross-section is
meaningful only for a spatially homogenous object then
consequently the texture of an image of the real earth surface
is determined by space changes of σ ∗ .
Everything pointed above allows setting mutual
relationships between conceptions of normalized effective
cross-sections of underlying surface and its texture. When a
small part of the image is characterized by a minor change of
typical non-derivative elements then the dominant property
of this part is the value of the normalized effective
cross-section. At a visible change of the brightness of these
elements the dominant property is put in the texture. In other
words when decreasing the number of distinguishable typical
non-derivative elements in an image the part of energy signs
(in particular σ ∗ ) increases. In fact for one resolution
element the energy signs are the only signs. If the number of
distinguishable typical non-derivative elements increases
then textural signs begin dominating.
It turned out that use of textural signs is extremely useful
during detecting of low-contrast targets on images of any
nature. Application of optical and radar images of the earth
surface allows supplementing conventional signs with new
quite significant ones which allow decreasing the signatures
overlapping. The space organization of a texture can be
structural, functional and probabilistic [6,25]. Texture signs
describe representative properties general for the given class
of textures.
During the process of the statistical analysis of textures
they use statistics of the first or second order. When using
statistics of the second order the textural signs are directly
extracted using matrixes of distribution of probability of
space dependence of brightness gradation P which is also
called as a matrix of gradients distribution. This method was
proposed in [25]. It was experimentally shown in [6,25] that
signs based on parameters of correlation functions do not
estimate an image texture so good as the signs determined
over the gradient matrix P do.
Let us briefly consider the classical approach to obtaining
textural signs [25]. Also let us assume that the image under
consideration is rectangular and has Nx resolution elements
Fractal Scaling or Scale-invariant Radar: A Breakthrough into the Future
horizontally and Ny elements vertically. At that G = {1,2,...,N}
is a set of N quantized brightness values. Then image I is
described by a function of brightness values from set G that
is I: Lx×Ly →G, где Lx = {1,2,..., Nx} and Ly = {1,2,..., Ny} are
horizontal and vertical space zones respectively. The
collection of Nx and Ny is a collection of resolution elements
in a scan pattern. Matrix of gradients distribution P contains
relative frequencies pij of presence of image neighbor
elements which are placed at distance d from each other with
brightness i,j G. Usually they distinguish horizontal (α=
0o), vertical (α = 90o) and transversally diagonal (α = 45o
and α = 135o) elements pairs.
Let us formulate conceptions of adjacent or neighbor
elements [25]. Consider Figure 2 and the central pixel on it
which painted as a dark small circle with eight neighbor
pixels around it. Resolution elements 1 and 5 are the nearest
neighbor elements and the angle between them equals zero.
Resolution elements 2 and 6 are the nearest neighbor
elements which angle α = 135о. Consequently elements 3
and 7 are the nearest neighbor elements which angle 90о
and elements 4 and 8 are the nearest neighbor which angle
45о with regard to the central pixel.
Figure 2. Diagram of formation of gradients distribution matrix Р
It should be noted that this information is purely spatial
and does not relate to brightness levels. Then we assume that
the information about textural signs is properly determined
by matrix P of relative frequencies which two neighbor
elements separated with distance d appear on the image with.
At that one element has brightness i and other elements has
brightness j.
In case of need the respective normalization of frequencies
for matrixes of gradients distribution can be easily done. For
d=1, α=0 we have 2Ny (Nx -1) pairs of adjoining horizontally
to each other resolutions elements.For d=1, α=45о we get
only 2(Ny-1)(Nx-1) pairs of adjoining diagonally to each
other resolutions elements. After getting M pairs of adjoining
to each other resolution elements matrix of gradients
distribution P is normalized by dividing every element by M.
Number of arithmetic operations which are needed for
processing images using this method is directly proportional
to NxNy. Frequently used linear integral Fourier and Adamar
transforms require NxNylog(NxNy) operations. Besides saving
of time during processing big data arrays we need to keep
just two strings of data about the image in the operational
computer memory during calculation matrixes P.
The first calculation of the full ensemble of 28 textural
signs and a detailed synchronous analysis of textural
signatures for real (optical and radar in the range of
millimeter waves (MW) at wave 8.6 mm) and synthesized
textures as well was performed in IREE RAS in 1985 and
fully presented in [6]. The full-sized experiments were
carried out in co-operation with Central Design Bureau
"Almaz". At that the task of calculation of textural signs
taking into account the signatures drift at the season change
was formulated and solved. We also note that in [25]
questions of informativity of all 28 textural signs were not
considered and there is no estimation of windows size impact
to accuracy of determination of textural signs. Choice of
window sizes is caused by the fact that a texture is
determined by the neighborhood of the image point.
It turned out that for windows with size 3×3 or 5×5 pixels
statistical textural measures act more as detectors of
brightness drops than as texture meters though at that the
calculation time is reduced [6]. Too big windows sizes may
distort the results due to impact of structures margins and
images edges. However the big window allows reaching a
high statistical confidence. Windows 20×20 pixels are the
most effective for textural processing of aerospace photos of
farming lands, pastures, woodlands and other similar objects.
When changing the window sizes from 80×80 to 20×20
pixels the numeric values of textural signs changed by
5...10 %. Further change of windows size resulted in
considerable distortion of textural signs.
Compactness of areas of textural signs existence for RI
textures gives us a possibility to guess that classification of
earth coverings and targets detection sometimes is carried
out more precisely using RI. However, interconnecting of
optical and radio engineering systems mutually
complements their main advantages and increases general
informativity. The scale invariance and the rotation
invariance is reached by selecting a particular step of
discretization while digitization of texture (usually it is about
an autocorrelation interval) and operation of averaging signs
values on four scanning directions during computer
Earlier the author proposed for the first time and
implemented with his colleagues the following
nontraditional effective methods of signals detection at small
ratios signal - background𝑞𝑞02 : the dispersion method on the
basis of f-statistics [6], method of detection using the linearly
simulated standards [6] and the method of direct use of
ensemble of textural signs or textural signatures [6]. The
most complete description of performance potential of
textural methods of processing of optical and radar images
was presented in [6,10] where for the first time the prospects
of using textural signatures when detecting of weak radar
Universal Journal of Physics and Application 11(1): 13-32, 2017
signals while the ratio signal/background 𝑞𝑞02 is about unity or
less was proved.
As a result of theoretical and experimental researches it
was also shown that determination of textural signs reduces
the effect of passive interferences from the earth surface and
improves extraction and detection of weak signals.
Moreover the important advantage of textural methods of
processing is a capability of neutralization of speckles on
coherent images of the earth surface which were obtained by
synthetic-aperture radar.
6. Methods of Determination of Fractal
Dimension D and Fractal Signatures
When using the fractal approach it is natural to focus
attention on description and also processing of radio physical
signals and fields exceptionally in the fractional measure
space with application of hypothesis of the physical scaling
and distributions with heavy tails or stable distributions.
Fractal and scaling methods of processing of signals, wave
fields and images are in the wide sense based on that part of
information which was irretrievably lost when using the
classical processing methods. In other words the classical
methods of signals processing basically select only that
information component which is related to the integer-valued
Fractal methods can function at all signal levels:
amplitude, frequency, phase and polarized. Nothing of the
kind exists in the world literature before the author's
researches and works.
The absolute worth of Hausdorff-Besicovitch dimension
is the possibility of its experimental determining [6-8]. Some
set can be measured with d-dimensional (d is an integer)
samples with side l1 . Then number of samples N1 covering
the set will be: N 1 = A / l1d . Value d must be based on
preliminary information about the set's dimension.
Theoretically, if d is less than the topological dimension then
N1 → ∞ , and if d > R
where R
is the Euclidean space
then N1 → 0 . The sample with size l 2 will give estimation
N 2 = A / l2d , then the similarity dimension will be:
D = − log l2 / l1 N 2 / N 1 .
Let us define the Hausdorff dimension in the following
way. Let's consider some set of points N0 in a d-dimensional
space. If there are N( ε ) - dimensional sample bodies (cube,
sphere) with typical size ε needed to cover that set, at that
N (ε ) ≈ 1 / ε D , when ε → 0
is determined by the similarity law.
The practical implementation of the method described
above faces the difficulties related to the big volume of
calculations. It is due to the fact that one must measure not
just the ratio but the upper bound of that ratio to calculate the
Hausdorff-Besicovitch dimension. Indeed, by choosing a
finite scale which is larger than two discretes of the temporal
series or one image element we make it possible to "miss"
some peculiarities of the fractal.
Building of the fractal signature D(t, f, r ) [6-8,26] or
dependence of estimates of kind (19) and (20) on the
observation scale often helps to solve this problem Figure 1,e.
Also the fractal signature describes the spatial fractal
cepstrum of the image. In V.A. Kotelnikov IREE RAS
besides the classical correlation dimension we developed
various original methods of measuring the fractal dimension
including methods: dispersing, singularities accounting, on
functionals, triad, basing on the Hausdorff metric, samplings
subtraction, basing on the operation "Exclusive OR" and so
on [7,8,10,11]. During the process of adjustment and
algorithms mathematical modeling our own data were used:
air photography (AP) and radar images (RI) at long
millimeter waves [6]. Enduring season measurements of
scattering characteristics of the earth coverings were already
naturally conducted at wavelength 8.6 mm by the author
from board of a flying laboratory located in helicopter in the
eighties of XX.
A significant advantage of dispersing dimension is its
implementation simplicity, operation speed and calculations
efficiency. In 1998 we proposed to calculate the fractal
dimension using the locally dispersing method (see for
example [2,7-11,15,17,22,26-28]). Parameters of the
algorithms which measure fractal signatures D affect
measurements errors strongly enough. In the developed
algorithms they use two typical windows: a scale one and a
measuring one. The unbiassed measurements can be carried
out when using the scale windows which exceed sizes of the
measuring window. One selects the necessary measurements
scale using the scale window. This window defines the
minimum and maximum values of scales which the scaling is
observed in. That is why the scale window serves for
selection of the object to be recognized and its following
description in the framework of fractal theory. An image
brightness local variance or image intensity is determined by
the measuring window using common statistical methods.
The locally dispersing method of measurements of the fractal
dimension D is based on measuring a variance of the image
fragments intensity/brightness at two spatial scales:
ln σ 22 − ln σ 12
ln δ 2 − ln δ1
In formula (21) σ 1 , σ 2 are root-mean-squares at the first
δ 1 and second δ 2 scales of image fragment, respectively.
Accuracy characteristics of the locally dispersing method
were investigated in [15,17,26-28]. Determination of
one-dimensional fractal signatures over the area of images
under investigation in different directions gives the new
technique of measuring the anisotropy of surface images. It
should be noted that the proposed locally dispersing method
of measuring the fractal signatures allows direct obtaining of
empirical distributions of fractal dimensions D.
It is proved in [15,26,28] that in the Gaussian case the
dispersing dimension of a random sequence converges to the
Fractal Scaling or Scale-invariant Radar: A Breakthrough into the Future
Hausdorff dimension of a corresponding stochastic process.
The essential problem is that any numerical method includes
a discretization (or a discrete approximation) of the process
or object under analysis and the discretization destroys
fractal features. Development of a special theory based on
the methods of fractal interpolation and approximation is
needed to fix this contradiction. Various topological and
dimensional effects during the process of fractal and scaling
detecting and processing of multidimensional signals were
studied by the author in [2,7-11,14-24,26-28].
7. Fractal Processing of Signals and
Images against the Background of
High-intensity Interferences and
The author was the first who shows that the fractal
processing excellently does for solving modern of
processing the low-contrast images and detecting superweak
signals in high-intensity noise when the modern radars do
not practically function [2,6-11,14,16]. When using the
fractal approach, as it was pointed out above, it is natural to
focus attention on description and also on processing of radio
physical signals (fields) exceptionally in the fractional
measure space with use the hypothesis of the scaling and
universal distributions with “heavy tails” or stable
distributions [7,8,29].
The author's developed fractal classification was
personally approved by B. Mandelbrot [9,10] in USA in
2005. It is presented on Figure 3 where the fractal properties
are described on the assumption that D0 is the topological
dimension of the space of embeddings.
The textural and fractal digital methods under author's and
his pupils development (Figure 4) allow to partially
overcome a prior uncertainty in radar problems using the
geometry or the sampling topology (one- or
multidimensional) [6,16]. At that topological peculiarities of
the sampling get very important as opposed to the average
realizations which have different behavior.
It turns out that the concepts of fractal signatures and
fractal spectra are very helpful for measurements. For
example, these concepts are effectively applied to solve
problems of detection of low-contrast targets and weak
signals in the presence of intense non-Gaussian (!)
interference. The methods of fractal processing should take
into account the scaling effect of real radio signals and
electromagnetic fields. The introduction of a fractional
measure and scaling invariants necessitates the predominant
use of power-series probabilistic distributions. These
distributions result from feedback that amplifies events. Note
that, for distributions with heavy tails, sample means are
unstable and carry little information because the law of large
numbers cannot be applied in this situation.
Figure 3. The author's classification of fractal sets and signatures
Figure 4. Textural and fractal methods of processing of low-contrast
images and superweak signals in high-intensity noise
Figure 5 shows the general view of distributions with
fractal dimension D. At fractal processing of realizations of
signals in noise it is shown, that at the relations a SNR
+10 dB we precisely measure statistics of a signal. With
reduction of value aside negative values (for example,
-3 dB) there is a displacement of a maximum final fractal
distributions aside values fractal dimensions of noise or a
Universal Journal of Physics and Application 11(1): 13-32, 2017
handicap. Thus always in a vicinity of value fractal
dimensions of a useful component there is “a heavy tail”
fractal distribution, reaching stable size, about 20 %. The
given tendency is kept and at much smaller values, equal
SNR -10 dB and -20 dB, as shown in Figure 5.
Figure 5. Empirical fractal distributions with heavy tails for images
observed in the presence of an intense Gaussian noise: (1 and 3) scene A, (2
and 4) scene B, (1 and 2)
q02 = −10
dB, and (3 and 4)
q02 = −20 dB
Thus, the algorithms of fractal pattern recognition based
on the paradigm «topology of targets is their fractal
dimension» [7,8]. The methodological basis of the fractal
pattern recognition algorithms is the rejection of topological
constants and a description of the types of targets using
features of fractal dimensions D or fractal target signature.
High sensitivity of estimation of functionals of
non-integral dimension to the presence of a continuous
contour in images suggests a large potential of fractal
filtering of the contours of objects in strong interference
(Figure 6). The observation was made using ground-based
telescope, the distance between it and objects was about 800
km. These data are presented in the book [11]. None of the
modern methods of digital processing can provide
comparable objects resolution!
Figures 7-9 show selected results of fractal nonparametric
filtering of low-contrast objects. Aircraft images were
masked by an additive Gaussian noise. In this case, the SNR
ratio was -3 dB. It is seen in the figures that all desired
Figure 6. The initial image of space complex at the time of joining
«Shuttle» – «Peace» (a) and the results of (b – d) the fractal processing
(targets clustering) for various values of the threshold D of topological
fractal nonparametric detector
Figure 7. Source image
in the noise. The optimum mode of
information is hidden
filtering of necessary contours or objects is chosen by the
operator using the spatial distribution of fractal dimensions
D of a scene. This distribution is determined automatically
and is shown in the right panel of the computer display
This concept can be widely applied to solve modern
problems of radar, correlation-extremely navigation,
artificial intelligence, and dynamic systems. The algorithms
developed by the authors for calculating fractal signatures
are efficient over an extremely wide range of physical sizes
of characteristic image details and provide detection
estimation for scaling effects, including even those masked
by noise.
Figure 8. Source image and noise SNR ratio was -3 dB
Fractal Scaling or Scale-invariant Radar: A Breakthrough into the Future
Figure 9. Results of fractal filtration image on Figure 7
Many other examples of fractal scaling processing of
low-contrast images and real weak signals from the practice
of radar, telecommunications, astronomy, materials science,
medicine and many other disciplines are given in
labeled all of this briefly and expressively – “The Fractal
Paradigm” [18,22,23,30-37].
The fractal geometry is a huge and of genius merit of
mathematician B. Mandelbrot. But its radio physical/radio
engineering implementation is a merit of the Russian (now it
is international) scientific school of fractal methods and
fractional operators under the supervision of Professor A.A.
Potapov (V.A. Kotel’nikov IREE RAS, see also the author's
web page
In modern situation all the attempts to belittle their
meaning and to reckoning only on the classical knowledge is
endures the intellectual fiasco. Union of specified problem
triad in common “Fractal Analysis and Synthesis” therefore
creates base of “Fractal Radio Systems” (2005) – Figure 10,
“Global Fractal–Scaling Method” (2006), and “Fractal
Paradigm” (2011) [7-14,18,19,24,30,34-37].
The work obtained in Kotel'nikov IREE of RAS by author
and his apprentices is based on the theoretical and
experimental results in scheduled introduction of the fractals,
fractional integration-differentiation and the scaling effects
in radio physics, radio engineering, and some contiguous
scientific directions (Figure 11). We have published a
sufficiently large number of works for each direction from
the data from Figure 11 in Russia and abroad.
8. Designed Breakthrough Technologies
and Fractal Radio Systems
A critical distinction between the author's proposed fractal
and scaling methods and classical ones is due to
fundamentally different (fractional) approach to the main
components of a physical signal. It allowed us to come to the
new level of informational structure of the real non-Markov
signals and fields. Thus this is the fundamentally new radio
For 35 years of scientific researches the global fractal and
scaling method designed by author has justified itself in
many applications. This is a challenge to time in a way. I
Figure 10. Author conception of fractal radio systems and devices
Universal Journal of Physics and Application 11(1): 13-32, 2017
Figure 11.
Sketch of author's development of new information technologies based on fractals, fractional operators and scaling effects
9. Conception of Fractal Radio Elements
(Fractal Capacitor), Fractal Antennas
and Fractal Radio Systems
As it follows from above, significantly positive results in
area of justification and development of different methods of
digital fractal filtering of weak multidimensional stochastic
signals are obtained. The third stage of the work on creation
and development of breakthrough informational
technologies for solving modern problems of radio physics
and radio electronics, which was begun in IREE RAS in
2005, is characterized by transformation to design of fractal
element base of fractal radio systems on the whole. Creation
of the first reference dictionary of fractal signs of targets
classes and permanent improvement of algorithmic supply
were the main points during the development and
prototyping of the fractal nonparametric detector of radar
signals (FNDRS) in the form of a back-end processor.
Basing on the obtained results we can speak about design of
not only fractal blocks (devices) but also about design of a
fractal radio system itself [7-14,18,19,24,31-37]. Such
fractal radio systems (Figure 10) which structurally include
(beginning with the input) fractal antennas and digital fractal
detectors are based on the fractal methods of information
processing and they can use fractal methods of modulation
and demodulation of radio signals in the long view
Fractal antennas are extremely effective during
development of two-frequency or multi-frequency radio
location and telecommunication systems. The structures
form of such antennas is invariant to certain scale
transformations that is an electrodynamics similarity is
observed. As it is known, spiral and log periodic antennas are
the most obvious examples of frequency-independent
antennas. Fractal antennas were the next step in building of
new ultra broadband and multiband antennas. The scaling of
fractal structures gives them multiband properties in an
electromagnetic sense [7-9,38-42]. Multi-frequency radio
measurements along with fractal processing of the obtained
information are a serious alternative to existing methods of
enhancing the signal-to-noise ratio. Since every target has its
own typical scales then one can directly determine a new
signs class (except for the pointed above) in the form of
fractal-and-frequency signatures by selecting the search
frequency grid [11,28,31].
Unlike the classical methods when smooth antenna
diagrams (AD) are synthesized an idea of realization of
radiation characteristics with a repetitive structure at
Fractal Scaling or Scale-invariant Radar: A Breakthrough into the Future
arbitrary scales initially underlies the fractal synthesis theory.
It gives a possibility to design new regimes in the fractal
radio dynamics, to obtain fundamentally new properties and
fractal radio elements as well (for example a fractal
capacitor) [39].
Application of a recursive process theoretically allows to
create a self-similar hierarchical structure up to separate
conductive tracks in a microchip and in nanostructures. In
practice the sum of random values converges not to Gaussian
distributions but to Levi stable distributions with heavy tails
(i.e. fractal distributions - paretians) quite frequently.
Simulation of Levi distributed random values can lead to
processes of anomalous diffusion which is described with
fractional derivatives on space and/or time variables. In
substance, equations with fractional derivatives describe
non-Markov processes with memory.
Physical simulation of fractional integral and differential
operators allows creating radio elements with passive
elements, simulating nonlinear impedances Z (ω ) with
frequency scaling
generalization of Cantor set at physical level allows to
proceed to so called Cantor blocks in the planar technology
of molecular nanostructures.
Application of fractal structures also allows to create
media which show complex reflecting and transmitting
properties in a wide frequency range and able to simulate
three-dimensional photon and magnon crystals which are the
new media of information transfer (for more details see
[7,8,10]). One can select a configuration and sizes of fractal
structures and check such unusual properties for a frequency
range on the scheme on Figure 13. The pickup antenna (is not
shown) was placed closely to the fractal plates.
where 0 ≤ η ≤ 1 , A − const , ω - angular frequency basing on
the modern nanotechnologies. For that purpose the model of
impedance Z (ω ) was created in the form of an unlimited
chain (continuous) fraction. In case of a finite stage of
building the equivalent circuit for RC chains with using the
n-th matching fraction for the given continuous fraction one
can adjust frequency ranges which the necessary power law
of impedance of the form ω −η will be observed in. In this
particular case we will for the first time realize a "non-linear"
fractal capacitor [8,10,39].
Thus, independently of the [4], our model of the
impedance Z (ω) in the form of an endless chain (continuous)
fraction was created. In the case of the final stage of
construction of the equivalent electrical circuit for RC chains
when the corresponding n-th fraction of the considered
continued fraction is used, we can adjust the frequency bands
in which there will be a power-law dependence of the
(Figure 12). In this case, we first implement
in practice nonlinear «fractal capacitor» or the fractal
Figure 12.
An example of the implementation of the fractional operator
d −1/ 2 dt −1/ 2 or fractal capacitor
Basing on nanophase materials one can also create planar
and volume nanostructures which simulate the considered
above “fractal” radio elements and radio devices of
microelectronics i.e. the question is about building an
element base of new generation. In particular, an elementary
Figure 13. Scheme of "fractal" experiments
On the right on Figure 13 pictures of a secondary
electromagnetic field for fractal and copper plates are shown.
One can see that the "superwave" fractal structure slows
down the directional radiation while a metallic plate does not
reveal this function. Such "superwave" properties mean that
a fractal plate can act as a compact reflector.
Thus fractal structures always have a self-similar series of
resonances which lead to logarithmic periodicity of working
zones. The related topologic fractal structure allows to
modulate the electromagnetic waves transmission coefficient.
The lowest frequency of weakening corresponds to wave
lengths which can significantly enhance the outer sizes of the
fractal plate and makes such fractal structures be the
superwave reflectors. The obtained results allow to extend
the applied above calculation method on the basis of
algorithms of a numerical solution of hyper-singular integral
equations to a wide class of electrodynamic problems which
appear during researches of fractal magnon crystals, fractal
resonators, fractal screens, fractal radar barriers and also
other fractal frequency-selective surfaces and volumes
which are required for realizing the fractal radio systems.
The fractal radio systems proposed by the author reveal
new opportunities in the modern radio electronics and can
have the widest outlooks of practical application.
Promising elements of fractal radio electronics include
also functional elements fractal impedances which are
implemented based on the fractal geometry of the conductors
on the surface (fractal nano-structures) and in space (the
fractal antenna), the fractal geometry of micro-relief surface
of substrates or fractal structure of polymer composites, etc.
Universal Journal of Physics and Application 11(1): 13-32, 2017
10. Principles of Fractal-scaling or
Scale-invariant Radio Location
At the moment world investigations on the fractal radio
location are conducted exceptionally in V.A. Kotelnikov
IREE RAS [2,6-24,26-28,30-37,39-52].
In accordance with requirements to the promising radars
let us consider a generalized functional scheme of the
classical system - Figure 14. On the one hand it is quite
simple and on the other hand it contains all the basically
necessary elements.
Also the case in point here can be both single-channel
radar station (RS) and a multi-channel RS. A synchronizing
device provides work coordination for every element of an
RS scheme.
Electromagnetic energy is generated and radiated by
means of a transmitting device which consists of a modulator,
a high-frequency generator and a transmitting antenna.
Reflected signals arrive to a receiving antenna. A receiving
device performs all the necessary transformations of arriving
signals related to their separation, amplification, extraction
from noise.
From the information of Figure 14 one can directly
proceed to fractal radar. On Figure 15 there are almost all
points of application of hypothetical or now projectable
fractal algorithms, elements, nodes and processes which can
be introduced into the scheme on Figure 14. Ideology of a
fractal radar [7-9,24,30,36,37,43,46,48,51,52] is based on
conception of fractal radio systems - Figure 10.
radiolocation and it aims to ascertain what's done and things
to do in this field on the basis of the fractal theory.
Investigations carried out showed the correctness of the path
chosen by the author (since 1980) to improve the
radiolocation technique.
Figure 14.
The generalized functional scheme of classical radar
11. Postulates of Fractal Radar
Fractal radar defined in [7-9,24,30,36,46,48,52] is based
on four main postulates:
1) intelligent signal processing based on the theory of
fractional measure, scaling effects and fractional
operator’s theory;
2) Hausdorff dimension or fractal dimension D of a signal
or a radar image (RI) is directly connected with the
topological dimension;
3) robust non-Gaussian probability distributions of the
fractal dimension of the processed signal;
4) “Maximum topology with a minimum of energy” for
the received signal. It allows to take advantages of
fractal scaling information processing more effectively.
The key point of fractal approach is to focus on describing
and processing of radar signal (fields) exclusively in the
space of fractional measure with the use of the scaling
hypothesis and distributions with heavy-tailed or stable
distributions (non-Gaussian). Fractal-scaling processing
methods of signals, wave fields and images are in a broad
sense based on the pieces of information, which isn’t usually
taken into account and irretrievably lost if classical methods
of processing are applied.
This work is concerned with the main radio physical area –
Figure 15. The points of application of fractals, scaling and fractional
operators in a classical radar for proceeding to a fractal radar
It is necessary to think about the processing of the input
signals with a low threshold at high levels of false alarm and
then a transition to a low level of false alarms. Moreover, the
false alarm probability is never measured in real time. In
principle [7-9,24,36,37,52], we need a new metric, and the
new parameters of radar detection.
12. Strategic Directions in Synthesis of
New Topological Radar Detectors of
Low-Contrast Targets
Intensive development of modern radar technology
establishes new demands to the radiolocation theory. Some
of these demands do not touch the theory basis and reduce to
the precision increase, improvement and development of
new calculation methods. Other ones are fundamental and
Fractal Scaling or Scale-invariant Radar: A Breakthrough into the Future
related to the basis of the radiolocation theory. The last
demands are the most important both in the theory and in
Radar detection of unobtrusive and small objects near the
ground and sea surface and also in meteorological
precipitations is an extremely hard problem [1,2,6-9,48,52].
One should take into account that the noise from the sea
surface and vegetation has nonstationary and multi-scale
behavior especially at high incidence angles of the sensing
Often, variety of subjacent coverings, conditions of radar
observation and maintenance of the objects mentioned above
leads to the fact that almost always signal-to-noise ratio 𝑞𝑞02
for these tasks fills in the area of negative (in decibels) values,
that is 𝑞𝑞02 < 0 dB. It makes the classical radar methods and
algorithms of detection non-applicable in most cases that are
use of energy detectors (when likelihood ratio is exclusively
defined by the energy of an input signal) is impossible.
Detection of low-contrast objects against the background
of natural high-intensity noise mentioned above inevitably
requires us to be able to propose and then to calculate some
fundamentally new property which differs from the
functionals related to the noise and signal energy.
We think that the initial information comes from different
radio systems in the form of a one-dimensional signal and a
radar image - Figure 16 [9,36,37,51,52]. The system of initial
radio systems and consideration of a radar image and a
one-dimensional signal in the millimeter waves (MMW)
range was already presented by the author in [6]. Now a
fractal radar, a MIMO - radar and unmanned aerial vehicles
(UAV) are included into the pattern of Figure 16.
The fractal radar conception is presented in [9, 48] in
detail, the MIMO-radar conception is considered in
[9,37,48,52]. The main idea of fractal MIMO-radars is use of
fractal antennas and fractal detectors [7,8,15,31,48]. An
ability of fractal antennas of simultaneous operating at
several frequencies or radiating a wideband sensing signal
drastically increases the number of degrees of freedom. It
determines many important advantages of such a kind of
radio location and sufficiently broadens opportunities of
All the currently existing methods and signs of detection
of unobtrusive objects against the background of
high-intensity reflections from the sea, ground and
meteorological formations are presented on the Figure 17
As compared with usual detection methods, the
fractal-scaling or scale-invariant methods proposed by
Professor A. Potapov, can effectively improve the
signal/interference relation and considerably increase the
probability of target detection. Methods under consideration
are suitable both for usual radars and for SAR, and also for
MIMO systems for multi-positioning radiolocation.
Figure 16. Radio systems of the initial information
Figure 17. Signs and methods of detection of low-contrast objects against
the background of high-intensity ground noise
Universal Journal of Physics and Application 11(1): 13-32, 2017
Also in terms of Weierstrass function for one-dimensional
fractal scattering surface we obtained scattering field
absolute value dependences on incident angle and surface
fractal dimension D. In subsequent computer calculations,
we used the above expression for the coherence function (CF)
Ψk - Figure 17:
Ψk = Es (k1 ) Es (k 2 ) ,
of the fields Es(k) scattered by the fractal surface [6-9,47].
We can show that the tail intensity of signals reflected by a
fractal surface is described by power functions:
I (t ) ~ 1 /(t ' ) 3− D .
Result (24) is very important because, for standard cases,
the intensity of a reflected quasi-monochromatic signal
decreases exponentially. Thus, the shape of a signal scattered
by a fractal statistically rough surface substantially differs
from the shape of a scattered signal obtained with allowance
for classical effects of diffraction by smoothed surfaces
Fractal (Hausdorff) dimension D or its signature at
different surface-mapping parts are simultaneously and
texture measure [6-8] i. e. properties of spatial correlation of
radio scattering by corresponding surface patches.
Fractal signatures including spectra of fractal dimensions
and fractal cepstrum represent the attribute vector uniquely
determining wide class of targets and objects than the use of
fractal dimension values. Thus, we can specify the propose
structure of the fractal radiolocation detector of target classes
consisting of edge detector and fractal signature calculator.
Obtained signatures are compared with the signature
database and the decision concerning the presence or
absence of the object is made in accordance with some
The general conception of the textural or fractal detector is
presented on Figure 18. The set of textural or fractal signs is
determined basing on the received radio signal or image.
Then a decision of signal presence H1 or its absence H0 is
done in the threshold device at threshold value П and certain
level of probability of a false alarm F.
Figure 18. Conception of textural or fractal detector
Values of fractal dimension D, Hurst exponents H for
multi-scale surfaces, Holder exponents, values of lacunarity
and so on can be also used as signs. Hurst exponent
H = 3 – D
H = 2 – D
for a radar image and
for a one-dimensional signal.
Some original variants of generalized structures of radar
fractal detectors are presented in Figure 19. One can
synthesize all kinds of fractal detectors from these schemes.
The structure aggregated scheme of the fractal detector of
radar signals is presented in Figure 19, a. It includes the
contour filter and the fractal cepstrum calculator. After
comparison with the database of standard fractal cepstrum
one makes a decision at the compare facility. Further
concretization of the FNDRS structure scheme is presented
in Figure 19, b. Input signal (radar image, 1-D sampling)
comes into the input transducer. It is intended for preliminary
preparation of analyzed sampling. This preparation includes
either compulsory noise (in the case when radar
low-resolution analog-digital converter is used) or, for
example, contrast compression (in the case of sampling with
high dynamic range).
Figure 19.
Initial (a) and detailed (b) structures of fractal signal detectors
Prepared input sampling comes into the edge detector.
Operation of this facility is based on the measurement of the
local fractal dimension over all sampling elements. It is
important that one can execute the preliminary detection in
accordance with the empiric distribution of local fractal
dimensions obtained at the edge detector output over all
sampling elements [15,28,31].
After the edge detector the input sampling actually
represents a binary array. “Units” in this array mean the
belonging of corresponding sampling element to the contour
of some object. If several objects are present in the sampling,
the question about their division arises. In our facility the
cluster selector realizes this task.
Obtained subset of initial sampling containing the
contours of one object comes to the input of the signature
calculator. This facility creates several “smoothed”
samplings in accordance with the expected observation
scales and calculates the “length” and “area”. The detection
process is realized by means of the comparison of the fractal
sampling signature with the database fractal signatures (or
data bank).
A detector on the basis of the Hurst exponent (Figure 20)
Fractal Scaling or Scale-invariant Radar: A Breakthrough into the Future
works with using one or several search frequencies of radar.
The Hurst exponent H reflects irregularity of a fractal object
– (25) and (26). The less exponent H the more irregular a
fractal object. So, the Hurst exponent gets higher when an
object appears.
Figure 20.
Calculation of the correlation integral C (r ) was conducted
using the F. Takens theorem on a sampling out of 50 000
counts which corresponds to the angle of incidence of an
electromagnetic wave θ =500. The following values were
obtained: D = 1 + 1,84 ≈ 2,8 ; m = 7; λ1 ≥ 0.6 bit/s; τ max ≈
1700ms when the reflected signal intensity correlation time
is τ ≈ 210 ms and the wind velocity is 3 m/s (Figure 22).
Fractal detector basing on the Hurst exponent
On Figure 21 there is the scheme of fractal detector with
autoregressive estimation of the power spectrum of the
interference from the Earth surface.
Figure 22. A kind of the screen of a computer with dependences D for
radar-tracking signals
Figure 21. Fractal detector with autoregressive estimation of the spectrum
of the interference and the Hurst exponent
The autoregressive model represents a linear model of
prediction which estimates the power spectrum of the
interference from the surface and forms its autocorrelation
matrix. The autoregressive equation describes relation
between current and preceding counts of a sampled
stochastic process. Earlier, in the eighties of XX we were
resolving the problem of autoregression on the basis of
canonical system of Yule-Walker equations with transform
of brightness histograms [6]. Thus in the detector on Figure
16 real fractal properties of the power spectrum on the basis
of autoregressive spectral estimation which are applied for
detection of low-contrast objects are used. We used much the
same detectors during the textural processing of APG and RI
as early as the eighties of XX.
I should note that the correlation dimension which
requires a big size sampling cannot be considered as
detection statistics (see Figure 17) and this is impossible in
Hence, if the current conditions are measured within the
accuracy of 1 bit then the whole predictive power in time will
be lost for about 1.7 c. At that the interval of prediction of
radar signal intensity is about 8 times the correlation time.
The obtained results show that a correct description of the
process of radio waves scattering requires not more than 5
independent variables. The correlation integral C (r ) can
also be used as a mean of separation of modes of the
deterministic chaos and white noise – Figure 23. Calculation
of the classical Henon attractor (Figure 24) was conducted
with the purpose of verification of adequacy of the created
Figure 23. A kind of the screen of a computer with dependences D for
Gaussian noise
13. Strange Attractors in the Phase
Space of Reflected Radar Signals in
Millimeter Wave Range
A deterministic chaos mode was discovered during radio
location of plant covering at wave length 2.2mm [11].
Estimations of fractal dimension D, nest dimension m,
maximum Lyapunov exponent λ1 and prediction time τ max
were used to measure and reconstruct the strange attractor.
Figure 24. Cross section: Henon's attractor
Universal Journal of Physics and Application 11(1): 13-32, 2017
Dependences of fractal dimensions D and correlation
integrals of radar processes under examination of millimeter
waves scattering by a birch (1) and spruce (2 and 3) forests
with D ≈ 2.6 are given on Figure 25.
Fractal-scaling or scale-invariant radar.
The multi-radar.
MIMO radar.
Cognitive radar.
This list of studied questions, of course, is supposed to be
expanded and refined in the future. The author has been
dealing with it for nearly 40 years of his scientific career.
15. Officially Admitted Results of the
Fractal Researches
Figure 25. Scattering of millimeter waves by a birch (1) and spruce (2 and
3) forests
The obtained results along with a family of fractal
distributions underlie the new dynamical model of signals
scattered by plant coverings. The proposed model of
electromagnetic waves scattering by earth coverings has a
fundamental difference from existing classical models [6-8].
14. A New Direction in the Theory of
Statistical Solutions
Fast development of the fractal theory in radar and radio
physics led to establishing of the new theoretical direction in
modern radar. It can be described as «Statistical theory of
fractal radar». This direction includes (at least at the initial
stage) the following fundamental questions:
1. The theory of the integer and fractional measure.
2. Caratheodory construction in the measure theory.
3. Hausdorff measure and Hausdorff-Besicovitch
4. The theory of topological spaces.
5. The dimension theory.
6. The line from the point of view of mathematician.
7. Non-differentiable functions and sets.
8. Fundamentals of the theory of probability.
9. Stable probability distributions.
10. The theory of fractional calculus.
11. The classical Brownian motion.
12. Generalized Brownian motion.
13. Fractal sets.
14. Anomalous diffusion.
15. The main criteria for statistical decision theory in radar.
16. Wave propagation in fractal random media.
17. Wave scattering generalized Brownian surface.
18. Wave scattering surface on the basis of
non-differentiable functions.
19. Difraсtals.
20. Cluster analysis.
21. Theory and circuitry of fractal detectors.
Results of our scientific activity on fractal-and-scaling
processing of information in the presence of high intensity
noise and fractal radio systems and radio elements as well in
V.A. Kotelnikov IREE RAS are published in four summary
reports of The Presidium of the Russian Academy of Science
(Scientific achievements of RAS. - M: Nauka, 2008, 2010,
2012, 2013) and in The Report For the Russian Federation
Government. About results of realization of the 2008 - 2012
Program of fundamental scientific researches of the state
academies of sciences in 2011. In three volumes, - M.:
Nauka, 2012. We'll briefly cite the text of these officially
admitted achievements.
In book “The summary report of the Presidium of the
Russian Academy of Sciences. Scientific achievements
of the Russian Academy of Science in 2007” (M.:
Nauka, 2008, pp. 204) in subsection “Location systems”
computational systems” (p. 41) there is the following
text: “A reference dictionary of fractal signs of optical
and radio images which is necessary for realization of
fundamentally new fractal methods of processing of
radar information and synthesis of high-informative
devices of detection and recognition of weak signals
against the background of high intensity non-Gaussian
noise was created. It was determined that for effective
solving of radar problems and designing of fractal
detectors of multidimensional radio signals, the
fractional dimension, fractal signatures, fractal
cepstrum and also textural signatures of the place noise
are essential (IREE RAS)” - 2007, published in 2008.
In book “The summary report of the Presidium of the
Russian Academy of Sciences. Scientific achievements
of the Russian Academy of Science in 2009” (M.:
Nauka, 2010, pp. 616) in subsection “Location systems.
Geo-information technologies and systems” of section
“Nanotechnologies and informational technologies” (p.
24) there is the following text: “The principles of
designing of new, fractal adaptive radio systems and
fractal radio elements for modern problems of radio
engineering and radio location are proposed and shown
by experiments for the first time in the world. The
operating principle of such systems and elements is
based on introduction of fractional transformations of
radiated and received signals in a fractional dimension
Fractal Scaling or Scale-invariant Radar: A Breakthrough into the Future
space with taking into account its scaling effects and
non-Gaussian statistics. It allows to get the new level of
informational structure of the real non-Markov signals
and fields (IREE RAS)” - 2009, published in 2010.
In book “The summary report of the Presidium of the
Russian Academy of Sciences. Scientific achievements
of the Russian Academy of Science in 2011” (M.:
Nauka, 2012, pp. 620) in subsection “Location systems.
Geo-information technologies and systems” of section
“Informatics and informational technologies” (p. 199 200) and in book “The Report for the Russian
Federation Government. On the results of realization of
the fundamental scientific researches 2008-2012
Program of national academies of sciences in 2011. In
three volumes” (M.: Nauka, 2012, pp. 1016) on p. 242
there is the following text: “A systematic research of
electrodynamic properties of fractal antennas was
carried out basing on the fractal analysis. Broadband
and multiband properties of fractal antennas and
dependence of resonances number on the number of
fractal iteration are confirmed. It is shown that an
effective realization of frequency-selective media and
protective screens which deform a radar portrait of the
target is possible using miniature fractal antennas.
Fractal frequency-selective 3D-media or fractal
“sandwiches” (engineering radio electronic micro- and
Nano constructions) are studied. (IREE RAS)” - 2011,
published in 2012.
In book “The summary report of the Presidium of the
Russian Academy of Sciences. Scientific achievements
of the Russian Academy of Science in 2012” (M.:
Nauka, 2013, pp. 616) in subsection “Elemental base of
microelectronics, Nano electronics and quantum
computers. Materials for micro- and Nano electronics.
Nano- and microsystem engineering. Solid-state
electronics” of section “Nanotechnologies and
informational technologies” (p. 195) there is the
following text: “It is determined that the integer-valued
quantum Hall effect is a physical base of memristor
functioning. The relations between the current and the
voltage for a random memristor type have been
obtained. The results are addressed to practical
realization of memristors as new elements of electronic
circuits. (Research Institute Of Applied Mathematics
And Automation Of Kabardino-Balkaria Research
Centre RAS, IREE RAS)” - 2012, published in 2013.
16. Conclusions
This work is concerned with the main radio physical area –
radiolocation and it aims to ascertain what's done and things
to do in this field on the basis of the fractal theory.
Investigations carried out showed the correctness of the path
chosen by the author (since 1980) to improve the
radiolocation technique.
In particular, over period of thirty-five years this resulted
in invention, creation and development of the new kind and
method of radiolocation, namely, fractal-scaling or
scale-invariant radiolocation. This implies radical changes in
the structure of theoretical radiolocation itself and in its
mathematical apparatus also.
Earlier fractals made up the thin amalgam on the strong
science frame of the XX century ending. In the modern
situation attempts to humiliate their significance and rely
only on the classical knowledge suffered an intellectual
The detailed analysis of all works published by the
author’s is not an aim of this chapter. Nevertheless, the
acquaintance with the author’s investigations in this area
should substantially help to large group of experts and more
accurately determine the practical application ways of the
fractal theory to solve the radio physical and radiolocation
problems. I consider that the “sampling topology” problem
[6-10,16] is one of the most important in radio electronics,
and I am also convinced that without fractals and scaling all
signal-detection theory loses its causal meaning for the
signal and noise conceptions.
The functional principle “Topology maximum at energy
minimum” for receivable signal permitting effective
application of advantages of the fractal-scaling information
processing was introduced by the author. This refers to the
adaptive target signal processing. Application of the fractal
principle results in the soul-searching in the detection field of
movable and immovable objects at the intensive disturbance
and noise background.
In this chapter, the author touched upon only the most
important problems connected with the application of the
fractals and scaling effects in statistical radio physics and
radiolocation. In the development of fractal directions many
important periods have already passed including the
establishment period of this field of science. However we
will have to solve many problems. It is the solution technique
(approach) is the most valuable but not results and
implementations. This method was created by Professor A.
Potapov [2,6-24,26-28,30-37,39-52]. Scientific results
obtained in recent years are the initial material for further
development and substantiation of practical application of
fractal methods in modern fields of radio physics, radio
engineering, radiolocation, electronics and information
controlling systems.
This work was supported in part by the project of
International Science and Technology Center No. 0847.2
(2000 – 2005, USA), Russian Foundation for Basic Research
(projects №№ 05-07-90349, 07-07-07005, 07-07-12054,
07-08-00637, 11-07-00203), and also was supported in part
by the project “Leading Talents of Guangdong Province”, №
00201502 (2016–2020) in the JiNan University (Guangzhou,
Universal Journal of Physics and Application 11(1): 13-32, 2017
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Proc. of CHAOS 2015 Int. Conf., Paris, Henri Poincaré
Institute, 559–573, 2015.
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Nonholonomic, Fractal and Linked Structures in Relativistic
Continuous Medium, Electrodynamics, Quantum Mechanics
and Cosmology: In three volumes, A.A. Potapov Ed.
LENAND, Moscow, 2015.
[26] A. A. Potapov. New Information Technology in Radar
Detection of Low-Contrast Targets Based on Probabilistic
Texture and Fractal Features, J. Communications Technology
and Electronics, vol. 48, No 9, 1012–1029, 2003.
[13] A. A. Potapov, Wu Hao, J. Foukzon, S. A. Podosenov, E. R.
Men’kova. Fractals in Radar, Fields, Control and
Low-Contrast Target Detection, Ed. by Dr. Phys.-Math. Sci.,
Professor A. A. Potapov, China, 2016, in press.
[14] A. A. Potapov, The Textures, Fractal, Scaling Effects and
Fractional Operators as a Basis of New Methods of
Information Processing and Fractal Radio Systems Designing,
Proc. SPIE, Vol. 7374, 73740E-1–3740E-14, 2009.
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with Fractal Signatures, Pattern Recognition and Image
Analysis, Vol. 8, No 2, 226–229, 1998.
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(Moscow), Vol. 2, No 1, 4–13. 2004.
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Selection and Fractal Antennas, Proc. 1st Int. Workshop on
Mathematical Modeling of Physical Processes in
Inhomogeneous Media, Mexico, Guanajuato, 44–46, 2001.
[27] A. A. Potapov and V. A. German. Processing of Optic and
Radar Images of the Earth Surface by Fractal Methods, J.
Communications Technology and Electronics, Vol. 45, No 8,
853–860, 2000.
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fractal dimension and fractal signatures of a multidimensional
stochastic signal, J. Communications Technology and
Electronics, Vol. 49, No 12, 1370–1391, 2004.
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for Sums of Independent Random Variables, Addison–Wesley,
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Paradigm in Fundamental Radar Problems, Book of abstracts
int. conf. Dynamics, Bifurcations and Strange Attractors,
dedicated to the memory of L. P. Shil’nikov (1934–2011),
Nizhniy Novgorod, 2013.
[31] Yu. V. Gulyaev, S. A. Nikitov, A. A. Potapov, and V. A.
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Fractal Scaling or Scale-invariant Radar: A Breakthrough into the Future
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No.8,909–916, 2006.
Detectors, Zbornik radova Konferencije MIT - Matematicke i
Prirodno-matematicki fakultet Ulverziteta u Pristini (Serbia),
539-552, 2014.
[32] A. A. Potapov, E. M. Il’yin, E. P. Chigin, V. A. German.
Development and Structure of the First Etalon Dictionary of
Fractal Properties of Target Classes, Electromagnetic
Phenomena. Vol.5, No.2, 15, 107-142, 2005.
[44] A. A. Potapov, A. E. Rassadin. Feynman Integrals, Fractal
Paradigm and New Point of View on Hydroacoustics, Eurasian
Physical Technical Journal, Vol.12, No.1 (23), 3-13, 2015.
[33] A. A. Potapov. On application of the Hurst exponent H in
adaptive fractal processing of information and synthesis of
new classes of fractal H– signals, Review of the Applied and
Industrial Mathematics, Vol. 15, No 6, 1121-1123, 2008.
[45] A. A. Potapov. Oscillator with Fractional Differential Positive
Feedback as Model of Fractal Dynamics, J. Computational
Intelligence and Electronic Systems, Vol.3, No.3, 236–237,
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of the Applied and Industrial Mathematics, Vol.14, No.4,
742-744, 2007.
[46] A. A. Potapov. Oscillations, waves, structures and systems
illustrated by examples of the global fractal-scaling method
(measure zero sets, singularity, scaling, sample topology,
sprites, jets, elves, memristors, oscillators, fractal mazes,
robust antenna arrays and fractal detectors), Non-Linear
World (Moscow), Vol.12, No.4, 3–38, 2014.
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in Fundamentally New Dynamic Fractal Signal Detectors,
Proc. the Eighth Int. Kharkov Symposium on Physics and
Waves-MSMW’13, Ukraine, Kharkov, IRE NASU, 644–647,
[36] A. A. Potapov. The Fractal-Scaling Radiolocation: Formation
History 1980–2015, Chaotic Modeling and Simulation
(CMSIM), Vol. 3, 317–331, 2016.
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Fractal Antennas and Fractal Detectors in the MIMO Systems,
Book of abstracts 9th int. conf. (CHAOS’ 2016) on Chaotic
Modeling, Simulation and Applications, UK, London, Senate
House, University of London, 85, 2016.
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Capacitor, Fractional Operators and Fractal Impedances,
Non-Linear World (Moscow), Vol. 4, No 4-5, 172–187, 2006.
[40] Yu. V. Gulyaev, S. A. Nikitov, A. A. Potapov, and A. G.
Davydov. Design of Fractal Radio Systems: Numerical
Analysis of Electromagnetic Properties of the Sierpinski
Fractal Antenna, J. Communications Technology and
Electronics, Vol.50, No.9, 988–993, 2005.
[41] A. A. Potapov, and E. N. Matveev. Fractal Electrodynamics.
Scaling of the Fractal Antennas Based on Ring Structures and
Multiscale Frequency Selective 3D Media and Fractal
Sandwiches: Transition to Fractal Nanostructures, J.
Communications Technology and Electronics, vol.55, No.10,
1083–1101, 2010.
[42] S. Sh. Rekhviashvili, and A. A. Potapov. Memristor and the
Integral Quantum Hall Effect, J. Communications Technology
and Electronics, Vol.57, No.2, 189–191, 2012.
[43] A. A. Potapov. The Global Fractal Method, Fractal Paradigm
and the Fractional Derivatives Method in Fundamental Radar
Problems and Designing of Revolutionary Radio Signals
[47] A. A. Potapov, A. V. Laktyun’kin. Frequency Coherence
Function of a Space–Time Radar Channel Forming Images of
an Anisotropic Fractal Surface and Fractal Objects, J.
Communications Technology and Electronics, Vol.60,
No.9,962-969, 2015.
[48] A. A. Potapov. Fractal scaling or scale-invariant radiolocation
principles and their application in radars with synthetic
aperture, III All-Russian Sci. Tech. Conf. “Development
prospects of long-range detection radar, integrated systems,
VKO infoware complexes and control and information
processing complexes: “RTI VKO systems - 2015”,
Radiotekhnicheskii Inst. im. A. L. Mintsa, Moscow, MGTU
im. Baumana, 573–590, 2015.
[49] A. A. Potapov, A. E. Rassadin. Feynman integrals and
radiation of waves in two-and three-dimensional spaces, Int.
Conf.-School «Dynamics, Bifurcations and Chaos 2016 (DBC
III)», Nizhniy Novgorod, Lobachevsky State University of
Nizhniy Novgorod, 54, 2016.
[50] A. A. Potapov, V. A. German, A. A. Pahomov. Processing of
Images Obtained From Unmanned Aerial Vehicles in the
Regime of Flight over Inhomogeneous Terrain with
Fractal-Scaling and Integral Method, Proc. CIE Int. Conf. on
Radar «Radar 2016», ed. Wu Shunjun, China, Chinese
Institute of Electronics (CIE), 585–587, 2016.
[51] A. A. Potapov A.A. Strategic Directions in Synthesis of New
Topological Radar Detectors of Low-Contrast Targets against
the Background of High-Intensity Noise from the Ground, Sea
and Precipitations, Proc. CIE Int. Conf. on Radar «Radar
2016», ed. Wu Shunjun, China, Chinese Institute of
Electronics (CIE), 692–696, 2016.
[52] A. A. Potapov. Fractality and scaling problems in radio
location and radio physics with new methods of detection of
low-contrast targets against a background of high intensity
noise, XV Int. Academic Congress "Fundamental and Applied
Studies in the Modern World", United Kingdom, Oxford
University Press, Oxford, 314-322, 2016.
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