Page 159 - Fundamentals of Radar Signal Processing

a three-dimensional convolution of the effective reflectivity with a convolution

kernel comprising the antenna two-way voltage pattern in the angle coordinates
and the pulse modulation function in the range coordinate. Specifically,

(2.116)

where the symbols ∗, ∗ , and ∗ denote convolution over the indicated
ϕ
θ
t
coordinate. Now assume the antenna pattern is symmetric in the two angular
coordinates, as is often the case; rescale the time variable to units of range; and
replace θ and ϕ with general angular variables θ and ϕ. These substitutions
0
0
finally give

(2.117)

Equations (2.116) and (2.117) are stated as approximate convolutions
because of the finite integration limits in the angular variables, which arise due
to the periodicity in angle of the antenna pattern and scene reflectivity. A full
discussion of a spherical convolution-like equation similar to Eq. (2.114),

including development of the Fourier transform relations, is given in Baddour
(2010). Nonetheless, like a linear convolution, Eq. (2.117) computes the output
at a given point in space as a local average of the reflectivity distribution,
weighted by the antenna pattern and waveform. For most antennas and pulses,
these patterns concentrate most of their energy in a relatively small finite region

defined by the mainlobe for the antenna pattern and the pulse duration for the
waveform. Consequently, the output signal can be expected to behave like a true
linear convolution.
The convolutional model of Eq. (2.117) is an important result. Its
significance is that it allows interpretation of the measured data as the result of a
linear filtering process, so that Fourier transform relations between y(θ, ϕ, R),
2
ρ′(R, θ, ϕ), E (θ, ϕ), and x(t) can be established and applied to model signal
properties, determine sampling rates, and so forth. For example, the range
resolution of the measured reflectivity function is seen to be limited by the pulse
duration. (In Chap. 4 it will be seen that the introduction of matched filtering
will significantly change this statement.) Similarly, for a conventional scanning
radar, the angular resolution will be determined by the antenna beamwidth. (In

Chap. 8 it will be seen that the introduction of synthetic aperture techniques also
significantly changes this statement.) It also follows from the filtering action of
2
x(t) and E (θ, ϕ) that the bandwidth of the measured reflectivity function in
range and angle is limited by the bandwidth of the waveform modulation
function and antenna power pattern. This observation will be used in Chap. 3 to