Page 157 - Fundamentals of Radar Signal Processing

2.7 Spatial Models

Previous sections have dealt with models of Doppler shift and the received
power (both mean value and statistical fluctuations) of radar echoes from a
single resolution cell. In this section, the variation in received complex voltage
or power as a function of the spatial dimensions of range and angle will be
considered. It will be seen that the observed complex voltage can be viewed as
the output of a linear filter with the weighted variation in reflectivity over range
or angle as its input. A similar result holds for power when the reflectivity field

has a random phase variation. These relationships will lay the groundwork for
an analysis of data sampling requirements and range and angle resolution in
subsequent chapters.

2.7.1 Coherent Scattering
Consider a stationary pulsed radar. At time zero it transmits the equivalent
complex signal

(2.110)

Assume that has unit amplitude so that the transmitted signal amplitude is
represented by the term . This signal echoes off a differential scatterer of
cross section dσ(R, θ, ϕ) at coordinates (R, θ, ϕ). The baseband complex
reflectivity or just reflectivity of the differential scatterer is, from Eq. (2.50),

2
dζ(R, θ, ϕ) exp[jψ(R, θ, ϕ) so that dσ =|dζ| . The term involving ψ accounts for a
possible constant phase shift on reflection at the scatterer surface. The antenna
9
is assumed to be mechanically scanning in either or both angle coordinates with
one-way voltage pattern E(θ,ϕ) so that at the time of transmission it is steered in

the direction (θ ,ϕ ). Then analogously to Eq. (2.16), the differential received
0
0
voltage is

(2.111)

where E(θ,ϕ) is the one-way antenna voltage pattern. Equation (2.110) can be
simplified by separating the reflectivity terms and the terms which depend on
spatial location and collapsing all of the other system-dependent amplitude

terms into a single constant . The term dζ exp(jψ) is termed the