Page 97 - Fundamentals of Radar Signal Processing

to the integral in Eq. (2.23) over the cross-range variables for the Gaussian case
is (Probert-Jones, 1962)

(2.24)

where θ and ϕ are the 3-dB beamwidths in azimuth and elevation. For first-
3
3
order calculations, the much simpler assumption is frequently made that the
antenna power pattern P(θ, ϕ) is a constant equal to the gain G over the 3-dB
2
beamwidths and zero elsewhere, so that the integral reduces to G θ ϕ , a value
3 3
2.5 dB higher than that of Eq. (2.24). Using this approximation, Eq. (2.23)
reduces to the range equation for volume scatterers:

(2.25)

Unlike the point scatterer case described by Eq. (2.11) or (2.19), the received
2
power in the volume scattering case of Eq. (2.25) decreases only as R instead
4
of R . The reason is that the size of the radar resolution cell, which determines
the extent of the scatterers contributing to the received power at any one instant,
increases as R due to the spreading of the antenna beam in angle at longer
2
ranges.
Finally, the area scattering case will be considered. This model is used
for the RCS of electromagnetic scattering from the ground, forest, ocean, and

other surfaces. The area scattering case must further be divided into two
subcases depending on whether the range extent of the scatterers contributing to
the echo is limited by the antenna elevation beamwidth or by the range
resolution.
1
First assume that the scattering surface is represented by a flat plane and
consider the extent of the mainlobe on the surface. The cross-range extent is
simply R θ , where R is the nominal range to the center of the illuminated area.
0
0 3
To estimate the down-range extent, consider Fig. 2.1 that shows the boresight
vector intersecting the scattering plane at a grazing angle of δ radians. The
extent of the beam “footprint” in the down-range dimension is therefore R ϕ /sin
0 3
δ meters. Now suppose a pulse of range resolution ΔR is transmitted as shown
in Fig. 2.2. Regardless of the antenna footprint, the range extent of scatterers

within the resolution cell, and therefore backscattering energy at any instant, is
ΔR/cosδ meters.