Aerospace and Electronic Systems Magazine March 2018 - 35
Wei et al.
where Δi is the incident grazing angle, Δs is the reflection angle,
and Lsea is the sea wave length. When m = 1, the sea wave echo
corresponds to the first-order sea clutter. The reflection angle Δs is
zero, considering that the echo from the target adheres to the sea
surface. The relation of velocity and wavelength of gravity wave
is v p = gLsea 2π .
We can derive the formula of the first-order sea clutter Doppler
frequency under this geometric model:
fB = ±
g
cos ( β 2 )
πλ
cos Δ i + 1
2
(6)
The influence of the ionosphere on the first-order sea clutter Doppler frequency shift is mainly caused by the change of signal phase
path. The variation of the signal phase path is caused by the variation of the ionospheric reflection height, which is determined by
the ionospheric plasma density. The Doppler shift caused by the
ionosphere is expressed as
f ion
4 fh dh
c 2 dt
g ( cos Δ i +1)
2πλ
cos ( β 2 ) + f ion
(10)
where ω0 is the translational frequency and Am and ωm are the amplitude and the frequency of the periodic item, respectively.
Therefore, Bragg frequency broadening caused by ionospheric
phase contamination within a coherent processing interval (CPI)
can be expressed as follows:
1 d t
max
f
ion
2
dt
min
1 d t
fion _ max f ion _ min
2
dt
(11)
From the preceding analysis, the spreading model of the first-order
sea clutter spectrum should be as follows:
g
g
f g max f β max + f ion _ min , −
f g min f β min + f ion _ max
−
πλ
πλ
g
g
f g min f β min + f ion _ min , +
f g max f β max + f ion _ max
+
πλ
πλ
where
cos ( β 2 ) =
φ ( t ) = ω0t + Am sin (ωmt )
(7)
where h is the ionospheric virtual height, c is the speed of light, and
τ is the group time delay.
Taking the impact of the ionosphere into account, the total
Bragg frequency expression can be written as
f Bragg = ±
Ionosphere is a time-varying and nonstationary medium. The nonstationary characteristics of ionosphere induce nonlinear change of
signal phase path, which leads to sea clutter spectrum frequency
broadening. According to the Weierstrass approximation theorem,
we can use a finite p-order polynomial phase model to describe
ionospheric phase contamination. For simplicity, ionospheric
phase contamination can be expressed as a sum of a linear term
and a periodic term, which can be described [17]
1
Ra − L cosθt
+
2 2 Ra 2 + L2 − 2 Ra L cos θt
(8)
(12)
The meaning of each item is shown in Table 1.
According to the analysis, ionospheric contamination is the
dominant broadening factor under a long CPI, which is usually the
case under the ship detection application [14].
From Equation (8), we can see that the characteristic of frequency shift for the first-order sea clutter in the hybrid propagation
mode is a function of the frequency, the azimuth angle, the
grazing angle, the distance, and the ionospheric state.
Table 1.
Meaning of Items in the Spreading Spectrum Model
FIRST-ORDER SEA CLUTTER BROADENING
As shown in Figure 1, because of the wide receiving beamwidth, the resolution cell in a footprint cannot be equivalent to a point but rather is equivalent to a large area. This
phenomenon leads to a corresponding bistatic angle in a
resolution cell that is not a single value β but rather a range
of [βmin, βmax]. Therefore, the measured Doppler frequency
of the first-order sea clutter is not a single value but instead
is within a range of f b ∈ fb min , f b max . This gives rise to
first-order sea clutter broadening. The theoretical broadening value Δfb because of the bistatic angle is calculated as
follows:
Δf b = f b max − f b min
MARCH 2018
(9)
Name
Representation
Interpretation
fgmax
cos i min 1
Maximum quantity
caused by grazing angle
2
cos i max 1
fgmin
2
Minimum quantity
caused by grazing angle
fβmax
Ra L cos t max
1
2
2 2 Ra L2 2RaL cos t max
Maximum quantity
caused by bistatic angle
fβmin
Ra L cos t min
1
2 2 Ra 2 L2 2RaL cos t min
Minimum quantity
caused by bistatic angle
IEEE A&E SYSTEMS MAGAZINE
35
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