IEEE - Aerospace and Electronic Systems - June 2021 - 27

Maffei et al.
assumed negligible. For signals propagating through free
space vacuum, Maxwell's equations can be elaborated
considering the absence of volume charge density or current
density. Introducing the signal wavenumber k ¼
2p= (where is the electromagnetic carrier wavelength),
the dispersion relation for free space vacuum can be formulated
as
v2 ¼ c2k2
(1)
with respect to the radian frequency v ¼ 2pf where f is
the electromagnetic carrier frequency and c ¼f is both
the phase and group speed. On the contrary, for signal
propagation through slabs ofplasma, a significant physical
complexity arises. Indeed, plasma supports a plethora of
oscillation modes (either as electrostatic or induced electric
fields) depending on many boundary conditions and
variables, including temperature and the occurrence of a
magnetic field. In simple terms, the dispersion relation in
a plasma region can be characterized by the frequency of
plasma media vp; which lies completely within the HF
band taking into account the electron density profiles
shown in [20]. More specifically, for < vp , the dispersion
relation entails absorption on signal propagation,
whereas for v vp; the dispersion relation for free space
vacuum in (1) becomes a reasonable approximation also
in a plasma region. In other words, radar signals (either
GBR or SBR based) may certainly propagate above the
HF band not only in free space but also in plasma media.
Clearly, GBRs must also take into account possible absorption
in specific windows of the electromagnetic spectrum
due to signals inevitably crossing neutral atmospheres
while SBRs can be deployed on orbits above the troposphere.
Yet, it is also well known that the 26.5-40 GHz
range of the Ka-band fitting the WR-28 waveguide is a
suitable spectral window with minor attenuation due to
water vapor and oxygen.
Once the condition of existence of propagation has
been faced, the treatment on channel phenomenology
through free space vacuum and plasma slabs hinges on the
stochastic Helmholtz equation as well as on possible gyrotropics
effects. With regards to the Helmholtz equation, it
can be shown [24] that a weak or moderate plasma turbulence
allows simplifying the propagation model through a
plasma slab predominantly as a forward scattering mechanism
around a small cone with negligible attenuation and
backscattering within the cone itself (thus excluding multiple
scattering effects in the plasma slab). This results in
a Rytov solution (see [24, eqs. (3.24) and (3.25)]) based
on the superposition of contributions for both amplitude
and phase of the emerging field out of a plasma slab and
represents the keystone for modeling weak scintillations
on radar signals as it allows for the applicability of the
central limit theorem (CLT). With regards to gyrotropics
effects [23] and considering magnetized plasma media, it
JUNE 2021
is well known that the original polarization axes of a general
propagating transverse electromagnetic mode (TEM)
are rotated by the so-called Faraday rotation angle.
CHANNEL MODELS
By modeling plasma as a fluid slab made of two components
(i.e., electrons e and ions i) [25], the spatial and temporal
behavior of the electron content can be analyzed by
the electron density ne at a specific spatial location
ðx; y; zÞ at a given instant ðtÞ as per small perturbations
(indicated by the " 1 " subscript) around fixed values (indicated
by the " 0 " subscript), i.e.,
ne x; y; z; tðÞ¼ n0 þ ne1 x; y; z; tðÞ:
(2)
This, in turn, allows introducing a normalized electron
density as per the following r.v. 2 Rþ:
x; y; z; tðÞ¼
ne1 x; y; z; tðÞ
n0
(3)
to describe the medium via spectral and statistical properties
[24]. For example, assuming wide sense stationarity
(WSS), the Wiener-Khinchin theorem represents the
Fourier-based relation between the correlation function of
the normalized electron density Gðx; y; z; tÞ and its spectrum
SðKx;Ky;Kz; vÞ, i.e.,
Gðx; y; z; tÞ
¼ E½ðx0 þ x; y0 þ y; z0 þ z; t0 þ tÞðx0;y0;z0;tÞ
S Kx;Ky;Kz; v
1
ðÞ2p 4
x Z¼þ1
x¼1
¼
y Z¼þ1
y¼1
(4)
z Z¼þ1
z¼1
t Z¼þ1
t¼1
G x; y; z; tðÞ (5)
expjKxx þKyy þKzz vt
dxdydzdt
where E½ indicates the statistical expectation operator. In
principle, from the knowledge of the spectral structure of
the irregularities of plasma patches in (5), it is possible to
characterize the state of the medium and subsequently
analyze detrimental effects on propagating signals. As a
simpler alternative paradigm, it is possible to consider
experimental scintillation parameters [22] such as the
scintillation index
s4ðÞ ¼
v
u
u
t
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E I
hi
ðÞ2
E I
½2
E I
ðÞ
½2
ðÞ
(6)
as a function of the intensity I of the electric field ~E at a
specific wavelength in a given location and temporal
window (i.e., IðÞ¼ j~EðÞj2 where jj is the modulus
operator). In addition to s4 (the wavelength dependence
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