Page 164 - Fundamentals of Radar Signal Processing

(2.122)

This is the reflectivity variation in range, taking into account the azimuth
and elevation averaging at each range due to the nonideal antenna power pattern.
Note that in the limit as the antenna power pattern tends to the ideal E (θ, ϕ) →
2
Gδ (θ, ϕ), then , that is, the “angle-averaged” reflectivity
D
exactly equals the effective reflectivity along the antenna look direction, as
expected.
Applying Eq. (2.122) to Eq. (2.117) leaves (Munson and Visentin, 1989)

(2.123)

or an equivalent equation, using time units instead of range units

(2.124)

Equation (2.123) or (2.124) shows that the complex voltage at the output of a
coherent receiver versus time for a given antenna look direction is the
convolution in the range dimension of the angle-averaged effective reflectivity
function in that look direction, , with the waveform modulation function
x(t).

2.7.4 Noncoherent Scattering
Equation (2.114) and its approximate form (2.117) assume coherent addition of
individual differential scatterer echoes; that is, the complex amplitude
(magnitude and phase) of the total response is the complex sum of the
differential complex echoes. For distributed area or volume clutter contributing
very large numbers of scatterers with essentially random phases such as rain or
natural ground clutter (grass, trees, water, etc.), it is more useful to model the

scatterer reflectivity as having a random phase with either a random or
nonrandom magnitude. The total received signal is then also a random variable,