Page 324 - Fundamentals of Radar Signal Processing

signal is timed to overlap exactly with the echo from a target at range R .
0
What will be the duration at the mixer output of the beat frequency tone
between the echo from a scatterer at the leading edge of the range window
(200 km – 150 m) and the LFM reference? Assuming a rectangular window
(i.e., no Hamming window), what will be the range resolution at the
leading edge of the window?

15. It was stated that range skew at the output of a stretch processor could be
corrected with a filter having the frequency response H(Ω) =
2
exp(–jΩ τ/2β ) where β = 2πβ is in radian frequency units. Show that the
Ω
Ω
group delay function d (Ω) of this filter meets the stated requirement,
g
namely d (–β · δt /τ) = –δt seconds. Group delay in seconds is defined as
b
g
Ω
b
dg(Ω) = – dΦ(Ω)/dΩ, where Φ(Ω) = arg [H(Ω)].
16. Assuming a sampling rate of F samples per second at the stretch mixer
s
output, convert the analog frequency response H(Ω) of the previous
problem to an equivalent discrete-time frequency response H(ω). Also
give the expression for H(ω) in the particular case when F is chosen to
s
match the stretch mixer output bandwidth of Eq. (4.111).

17. Explicitly compute the loss in processing gain LPG and the processing loss
PL as a function of K for a triangular window of odd length K + 1 (so K is
even) defined according to

Numerically evaluate the result for K = 4 and K = 20 and give the answers
in dB. What are the asymptotic values in dB for LPG and PL as K →∞?
The following facts may be useful (be careful about the limits):

(Hint: sum just the first half of the triangle, then use symmetry to get the
sum of the whole function. Be careful not to double-count any samples.)

18. Consider the array steering factor E(θ ) of Eq. (4.123) and use the weights
0
given in Eq. (4.124) with |a | = 1 for all n. Assume the phases of the
n
weights are computed for a wavelength λ and steering angle θ , but the
0
0
waveform bandwidth is approximately 10 percent of the nominal frequency
so that the effective wavelength varies over the range of (1 ± 0.05)λ .
0
Derive an equation that gives the new angle θ at which E(θ) will be
maximum in terms of λ , θ , and the actual wavelength λ. When the actual
0
0
wavelength is 5 percent larger than λ and the design steering angle is θ =
0
0