**Topological Processing of Spatio-Temporal Electromagnetic Field Shapes**

**Abstract.** This paper is on
the spatio-temporal signals with the topologically modulated electromagnetic
fields. The carrier of the digital information is the topological scheme
composed of the separatrices-manifolds and equilibrium positions of the field.
The signals and developed hardware for their processing in the space-time
domain are considered.

Key-words:
Electromagnetic field, dynamical system, 3-manifolds, topological computing,
artificial spatial intelligence

**1.
Introduction**

During
last several months the increased attention has been paid to topology. Topology
was established by H. Poincare, and it has been passed more then 100 years of
the fruitful developments. It studies the global characteristics of objects
ignoring their exact geometrical features. It is possible to find many
applications of this science, but one of them is very attractive in the
modeling of the intelligence. The human brain, mostly, operates with the
qualitative defined information, and, the visually based processing is
prevailing towards the logical handling of digital-like information. It allows
defining that one of the types of human brain activity as the “spatial
intelligence.” The development of the logical theory and the hardware modeling
is one of the most difficult tasks. The major part of the results is on the
image recognizing by pixel-to-pixel methods or holographic-based optical
recognition systems.

The
proposed paper is on the electronic-based hardware to handle the
electromagnetic, topologically modulated impulses that carry the information by
their field maps and impulse magnitudes. It allows developing the electronic
gates processing the images on the gate level and simulating the brain spatial
activity by electronics.

** **

**2.
Field-Force Lines Pictures and their Topological Schemes**

Since
the earlier years, it has been known that the fields surrounding the electrical
charges and magnets have certain spatial shapes. M. Faraday used the idea of
the field-force line pictures to visualize the fields and explain the found
electromagnetic phenomena. Later, J.C. Maxwell used the field-force lines to
create his system of equations for the electromagnetic field.

Mathematically,
the field-force lines are computed by the following system of differential
equations if the transient fields are pre-simulated:

_{}

_{}

where
_{} are
the radii-vectors of the field-force lines of the electric _{} and magnetic _{} fields,
accordingly, _{} are the parametrical variables
normalized by the coefficients _{}, and *t* is the time. In
general case, the equations are the
non-autonomous dynamical systems, and they can
be transformed to the autonomous ones by introducing a new variable_{}:

_{}

_{}

where _{} and _{} are the
radii-vectors and _{} and _{} are the field
vectors in the extended phase-space. Then, the dimension of the phase spaces
of the systems
and is
four. Besides, the vector fields _{}and _{} are governed by the
Maxwell equations, boundary and initial conditions.

Qualitatively,
the systems
and
are described by the ordered sets of separatrices of their extended 4-D phase
spaces and the equilibrium positions where the fields are zero. These elements
compose the topological scheme of a field similarly to the well-studied case on
a plane [1,2]. Taking into account that the dimension of the systems
and is
four in general case, the separatrices of the phase space can be the 1-D, 2-D
and 3-D manifolds. Last of them have not been studied well [3-5]. For example,
only recently the Poincarè conjecture was proofed in [6,7].

In
electromagnetics, the topological models of the dynamical systems
and
are used not only to analyze the pre-computed fields but to derive the
qualitative solutions of the boundary electromagnetic problems [8,9]. These
topological solutions of electromagnetic problem consist of analytical or
semi-analytical composing of topological schemes according to the given
boundary condition. It gives a rough model of the field used for qualitative
estimations or as an initial approximation for further problem treatment [10].

The
developed theory and derived solutions for the electromagnetic problems in the
space-frequency domain allowed proposing the idea on the topological modulation
of the field [8,11]. Logical variables are assigned to different topological
schemes, and the signal is a series of field impulses with their discretely
modulated spatio-temporal forms. In general case, these digital shapes are
composed from the manifolds of different dimensions, including the 3-D ones,
and the developed hardware not only detects these topologically different field
shapes but processes them according to the Boolean, predicate and constraint
logics [11-19].

**3.
Topologically Modulated Signals and Transmission Lines**

The
idea of logical handling of topological field shapes was proposed during a
research on the qualitative theory of the electromagnetic boundary problems. It
was found that the field topological schemes are changed discretely, and for
the electric and magnetic fields, it is written each own topological scheme _{},
correspondingly. Taking into account the time- dependence of the fields, the
considered schemes are the objects that have a certain shape in the 4-D phase
space of the equations
and .

The
schemes of the *i-th* and *j-th* fields can be non-homeomorphic to
each other _{}, and then a natural number *i*
or *j *can be assigned to each scheme. For example, the modes of a
transmission line can have non-homeomorphic topological schemes of their fields,
and they are numbered by the modal numbers. Additionally, the non-homeomorphic
schemes can be composed from arbitrary combinations of modes, as well.
Generally say, the number of modes and their combinations is infinite for a
given transmission line, but for a certain frequency band only a few of them
are the propagating modes. A set of two non-homeomorphic topological schemes of
the propagating modes can correspond to the binary system_{}. Manipulation of the
topological schemes according to a digital modulation signal is the topological
field modulation. The digital operations with these signals are the topological
computing.

To
support such signals, it was considered a number of waveguides. Among them are
the rectangular and corrugated waveguides, coupled strip-slot line, coupled
microstrip line and coupled strip line [12]. The first ones are the dispersive
waveguides, and the modal impulses excited in such waveguides have
non-separable spatial and temporal field part, and they are described with the
extended autonomous systems
and .

A
much more studied case is the signaling along the transversal electromagnetic
(TEM) waveguides like the strip transmission line. In this case, the dynamical
system describing the discretely modulated signal is still non-autonomous, but
each impulse of the field can be expressed by the separable spatial and
temporal parts, i.e. the systems
and
are the locally autonomous ones. It allowed avoiding some difficulties to
define and describe the signals and hardware for them.

An
example of signal transmitted along the coupled strip line is shown in Fig. 1.
The last waveguide consists of two conducting strips placed at the dielectric
substrate covered by the grounded conducting plates. This line supports the
even and odd modes. The first mode has the magnetic plane symmetry of the
field at the central line axis. The odd mode field is symmetrical regard to the
electric central plane. The modal field topological schemes of these modes are
not homeomorphic to each other, and they are assigned to the logical “1” and
“0”, correspondingly. Then, the topologically modulated signal is a series of modal impulses as shown in Fig. 1, and
the signal “topology” is changed from impulse to impulse.

Besides
the field topology, the carrier of information is the impulse magnitude, and
the signals introduced in [11] are the two-placed objects. The predicate logic
theory for them was considered in [14].

Fig. 1. Electric
field of topologically modulated impulses in coupled strip transmission line at
the fixed moment of time. The grounding shields are not shown.

** **

** **

** **

** **

** **

**4.
Hardware Theory for Topologically Modulated Signals**

In
general, the proposed topologically modulated signals are more complicated
objects then the 3-D manifolds [3-7]. The signal topological schemes consist of
an ordered set of vector separatrices-manifolds, including the 3-D ones, in
general case, and the field equilibrium positions.

The
developed theory and circuitry allows not only detects the different
spatio-temporal field shapes but compare them according to the Boolean,
predicate and constrained logic. In [12-14], the reconfigurable “in-the-fly”
hardware is shown. It was proposed to build such hardware using analogies with
the matched filters when the outputs of a gate are tuned to a certain modal
type or a spatio-temporal signal topology [12-14,15].

The
theory of the components was created using the conservation law written for the
power flows _{} in each *n-th* arm
cross-section and the energy *W* stored in the gate volume and expressed
through the field geometry from
and [13-16]:

_{}

This
expression reminds the Ricci flow equation used by topologists to study the
manifolds [3-7]. It shows the evolution of the geometry of the excited fields
to a steady state if a transient happens with the incident field. The geometry
of the field force line maps inside the transformer and in the output arms
depends nonlinearly on the input fields and the relationships of their
amplitudes. The maps can be changed discretely by a smooth variation of the
incident field parameters. Particularly, it allows controlling the field
distribution of the output fields. If the output field is close to a
propagating mode of the *n-th* output, then the matching conditions allow
transmitting the maximal power of the incident signal to this output. Other
terminals can be isolated from the input if this signal excites the evanescent
modes in them. Then, the switching of the input signal from one output to
another can be realized choosing the input signal parameters. Additionally, the
semiconductor components can realize the time signal processing.

**5. Logical Circuitry**

The first hardware for
topologically modulated signals was proposed in 1992 [11]. It deals with the
signals in the waveguides with the strong dispersion to whom a complete theory
and
should be applied. As well, the low-dispersion microstrip and strip lines were
considered, and the logic circuitry was patented.

The
first experimental results are shown in [17] where the microwave impulses are
switched by a passive circuit into different circuit arms. The developed
switches allowed proposing the AND, NOT and OR gates handling the information
written to the field topology.

The
components for the predicate logic are considered in [14,18,19]. They allow
handling the information contained in the signal amplitudes and topological
schemes, and they are the two-place signals.

Additionally
to the predicate circuitry, the topologically modulated impulses allow
designing the reconfigurable gates [13,14]. For this purpose, one of the
logical levels can be assigned to control the logical operation of the gate.
Depending on the signal magnitude level, the NOT gate, for example, can be
transformed into a follower for the information contained in the spatial maps
of the electromagnetic signals. Next, the amplitude relationship can define
the type of logical operations in a universal gate OR/AND that handling the
topological charts. Then, this gate can be incorporated into a flip-flop of
reconfigurable logic, and even a reconfigurable processor can be designed.

The
proposed and studied two-place signals are pertinent for reversible
computations. In [11], several passive circuits are studied that are reversible
physically and logically. They consist of completely passive components based
on the interference of the electromagnetic waves or combinations of passive and
semiconductor components. Later, similar way was studied in [20].

**6. Conclusions and
Future Research**

It
has been considered the electromagnetic signals carrying the digital
information by their spatio-temporal field structures and magnitudes. The
spatio-temporal carrier is the topological scheme of the extended 4-D phase
space of differential equations for field-force lines composed of the
manifolds-separatrices and equilibrium field positions. The performed research
touches the detecting and logical processing such topological signals. These
signals are processed by the logical circuitry designed on the spatio-time
filtration.

The
research relates to the developments of the principles and hardware for
modeling the spatial artificial intelligence [21]. Last one is defined as “a
skill” to handle the spatio-temporal information. It is supposed that the
information process is based on the parallel-like effects and structures, and
the human mind and body handle the global spatial information, at the first, on
the qualitative level [14,22,23]. It means that the topological approaches to
the signal processing and computing are a key component to understand and model
such a type of intellectual skills [18].

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G.
A. Kouzaev

Department
of Electronics and Telecommunications

Norwegian
University of Science and Technology-NTNU

Trondheim
N-7491, Norway, E-mail: guennadi.kouzaev@iet.ntnu.no