Page 242 - Fundamentals of Radar Signal Processing

(4.44)

The first integral on the right-hand side of Eq. (4.44) is just the energy E of the
pulse measured in the time domain; the second is, by Parseval’s theorem, also
the energy. Thus

(4.45)

The symmetry property can be proved by substituting –t and –F for t and
D
F , respectively, in the definition in Eq. (4.30)
D

(4.46)

Now make the change of variables s′ = s + t to get

(4.47)

Since A(t, F ) ≡|Â(t, F )|, Eq. (4.35) follows immediately.
D
D
It is reasonable to ask what would be an ideal ambiguity function. The
answer varies depending on the intent of the system design, but a commonly
cited goal is the “thumbtack” ambiguity function of Fig. 4.7, which features a
single central peak with the remaining energy spread uniformly throughout the
delay-Doppler plane. The narrow central peak implies good resolution in both
range and Doppler. The lack of any secondary peak implies that there will be no

range or Doppler ambiguities. The uniform plateau suggests low and uniform
sidelobes, minimizing target masking effects. All of these features are beneficial
for a system designed to make fine-resolution measurements of targets in range
and Doppler or to perform radar imaging. On the other hand, a waveform
intended to be used for target search might be preferred to be more tolerant of
Doppler mismatch so that the Doppler shift of targets whose velocity is not yet
known does not prevent their detection due to a weak response at the matched

filter output. Thus, what is “ideal” in the way of an ambiguity function depends
on the use to which the waveform will be put.