Page 160 - Fundamentals of Radar Signal Processing

determine the range and angle sampling requirements.

2.7.2 Variation with Angle
Now consider the variation in reflectivity with angle for a fixed range, say R .
0
Define the range-averaged effective reflectivity

(2.118)

This is the reflectivity variation in angle, taking into account the range averaging
at each angle due to the finite pulse length. Note that in the limit of very fine
range resolution, i.e., if the pulse modulation x(2R/c) → δ (R – R ), then
0
D
, that is, the “range-averaged” reflectivity would exactly
equal the effective reflectivity evaluated at the range of interest R .
0
Applying Eq. (2.118) to Eq. (2.117) gives

(2.119)

where again symmetry of the antenna pattern has been assumed in the second

line. Equation (2.119) is a special case of Eq. (2.117) showing that the complex
voltage at the output of a coherent receiver for a fixed range and a scanning
antenna is approximately the convolution in the angle dimensions of the range-
averaged effective reflectivity function evaluated at the range R , with
0
2
the antenna two-way voltage pattern E (θ, ϕ).
As mentioned earlier, the interpretation of Eq. (2.119) as a linear
convolution is an approximation. Suppose that the elevation angle ϕ is fixed, and
consider only the variation in azimuth angle θ. Because the integration is over a
full 2π radians and the integrand is periodic in θ with period 2π, the integration
over azimuth is a circular convolution of periodic functions.
This would not appear to be the case if instead θ is fixed and ϕ varies
because the integrand is over a range of only π radians. However, one could

equally well replace Eq. (2.119) as