Aerospace and Electronic Systems Magazine December 2017 - 13

Anderson
ize the definition of radar cross section, as first reported in
[10] and elaborated for the HF radar situation in [11], specifically by introducing a power series model for the nonlinear
response of materials and mapping this into a hierarchy of
nonlinear radar cross sections. The concept is most clearly visualized by writing out the first few terms of the resulting scalar radar equation for the received power, with the frequency
dependence explicitly written,
 PR ( f ) =

PT ( f ) GT ( f ) GR ( f ) λ 2

( 4π R )( 4π R ) 4π
2
T

+ 

2
R

σ1 ( f )

PT1 GT1 ( f1 ) PT2 ( f 2 ) GT2 ( f 2 ) GR ( f ) λ 2

( 4π R )( 4π R ) ( 4π R ) 4π
2
T1

2
T2

2
R

(3)

× δ ( f − f1 − f 2 ) df1 df 2
+ 

PT1 GT1 ( f1 ) PT2 ( f 2 ) GT2 ( f 2 ) GT3 ( f 3 ) GR ( f ) λ

( 4π R )( 4π R ) ( 4π R ) 4π
2
T1

2
T2

2

2
R

× δ ( f − f1 − f 2 − f 3 )σ 3 ( f1 , f 2 , f 3 ) df1 df 2 df 3 + 

Figure 11.

Predicted signal-to-noise ratio for a slightly nonlinear vessel in sea clutter under
quiet (rural) noise conditions, observed with an HFSWR [12].

may not be as easily detected as their PEC equivalents, their scat-

where the symbols have their customary meaning. The first
term on the right hand side is immediately recognizable as
the right hand side of the standard radar equation. The second
term integrates the contributions from all pairs of frequencies
whose nonlinear products fall at the designated frequency f,
and so on for higher order terms.
The two most important points to draw from this expansion are (i) the power law decay with range increases its exponent as the order of the nonlinearity increases, and (ii) the
radar cross section is defined in multidimensional frequency
space. To see how this can be useful in principle, consider a
nonlinear target immersed in an environment whose constitutive relations are highly linear. We may choose as an example,
a ship on the sea surface, ships being notoriously nonlinear
owing to oxidation and other physicochemical processes in
the marine environment, as well as their onboard electronic
equipment. A radar operating in the linear mode would have
Figure 12.
to detect the ship against a strong background of sea clutter.
Full-scale radar cross section measurements (joined by blue curve) overlaid on
In contrast, one able to operate in nonlinear mode would repredictions based on (i) metal components only (green curve), and (ii) composite
ceive target echoes at the nonlinear product frequency where
scattering model including dielectric and sea water contributions (red curve).
there would be zero sea clutter. As sea clutter can exceed
target echo strength by 60 dB or more, there is huge reward
tering properties at HF can in principle be determined. If used
if the nonlinear mode can be realized. In practice there are sevwisely, computational electromagnetics codes can be valuable
eral challenging obstacles, explored in [12], but it turns out that,
tools for developing the phenomenological models from which
at least for HF surface wave radar, even a small amount of nonoptimum radar operating strategies can be developed. Of course,
linearity can support detection at OTH ranges. Figure 11 shows
oversimplification of the scattering model can lead to drastic erone model calculation; in this example an HFSWR is considered,
ror. As a real-world example, consider Figure 12, which compares
and the signal-to-noise ratio is plotted as a function of range for
two theoretical predictions of the frequency dependence of the HF
quadratic and cubic nonlinearities. The figure also shows the imradar cross section of a fiberglass boat with experimental data.
provements in detectability achievable by application of higher
The green curve was generated by ignoring the contributions
order statistical analysis. For more details see [12].
of the dielectric materials of the boat construction and laboriously
Many other targets of interest to HF radar cannot be modelled
cataloguing and modelling the full complement of metallic compoas perfectly electrically-conducting (PEC) obstacles, including
nents. The other curve was derived using a model that took proper
small boats of primarily wooden construction, speedboats made
account of the electrical properties of the "transparent" dielectric
from fibreglass reinforced plastics, and even some advanced airshell. This rather obvious outcome is still not universally apprecicraft designed to have reduced radar cross section. While these
DECEMBER 2017

IEEE A&E SYSTEMS MAGAZINE

11



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