Page 177 - Fundamentals of Radar Signal Processing

and then the instantaneous Doppler shift was shown to be a linear function

of time. What kind of conic section curve describes the range variation if
the approximation to the square root is not made? Derive the formula for
the instantaneous Doppler shift in this case.

29. Consider again the series approximation to range referred to in the previous
problem. Find the maximum absolute value of t such that the magnitude of
the dropped fourth-order term in t in the series approximation [see Eq.
(2.101)] is less than 10 percent of the magnitude of the retained second-
order term. (This condition is a limit on the amount of data that can be

collected while still using the approximation to the range.) Give the
numerical value of the maximum allowable t when v = 100 m/s and R = 10
0
km.

30. Suppose the reflectivity distribution ρ(θ, ϕ, R) consists of a single isolated
point scatterer at coordinates (θ , ϕ, R), i.e., ρ(θ, ϕ, R) = ρ δ (θ – θ )δ (ϕ –
t D
t
t
t
D
t
ϕ)δ (R – R). Determine y(θ , ϕ , R ) of Eq. (2.114). What determines the
0
0
0
t
D
t
shape of this function in the azimuth (θ) dimension for fixed ϕ and t?
Repeat for the elevation and fast time dimensions with the other two
variables fixed.
31. The first zero of the function J (x) occurs at x ≈ 3.8317. What is the ratio of
1
the Rayleigh azimuth beamwidth of a circular aperture of diameter D with
uniform illumination to the azimuth beamwidth of a rectangular antenna of
the same width D, also with uniform illumination? Figure 2.27 can be used
as an approximate check on the result.
32. Consider a scatterer at elevation h above the ground plane, and an incoming
EM plane wave at a grazing angle of δ radians. What is the difference in
path lengths between the “single bounce” direct reflection (path #1 in the

adjoining figure) and the “double bounce” multipath reflection (path #2), as
a function of δ and h? If h = 50 m and the radar slant range resolution is ΔR
= 20 m, will the double-bounce echo appear in the same range bin as the
single-bounce echo?

33. Continuing with the previous problem, assume the range bin spacing is
greater than the path length difference so that both echoes combine in a