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172 MATLAB Simulations for Radar Systems Design
where we used the notation
to indicate that the output is a function of
s o
(
tf d
;
)
both time and Doppler frequency.
3.10. Waveform Resolution and Ambiguity
As indicated by Eq. (3.93) the radar sensitivity (in the case of white additive
noise) depends only on the total energy of the received signal and is indepen-
dent of the shape of the specific waveform. This leads us to ask the following
question: If the radar sensitivity is independent of the waveform, then what is
the best choice for a transmitted waveform? The answer depends on many fac-
tors; however, the most important consideration lies in the waveformÓs range
and Doppler resolution characteristics.
As discussed in Chapter 1 , range resolution implies separation between dis-
tinct targets in range. Alternatively, Doppler resolution implies separation
between distinct targets in frequency. Thus, ambiguity and accuracy of this
separation are closely associated terms.
3.10.1. Range Resolution
Consider radar returns from two stationary targets (zero Doppler) separated
in range by distance . What is the smallest value of so that the returned
signal is interpreted by the radar as two distinct targets? In order to answer this
question, assume that the radar transmitted pulse is denoted by
R
R
,
s ()
s () A ()
=
cos
(
f 0 t
+
φ ()
)
(3.117)
where
is the carrier frequency,
is the amplitude modulation, and
is
f 0
A ()
φ ()
the phase modulation. The signal
can then be expressed as the real part of
s ()
the complex signal
, where
ψ ()
ψ () A () e j ω 0 t
(
–
φ ()
)
u () e j ω 0 t
=
=
(3.118)
and
–
j φ ()
u () A () e
=
(3.119)
It follows that
s () Re ψ ()
=
{
}
(3.120)
The returns from both targets are respectively given by
s r 1
() ψ t
=
(
–
τ 0
)
(3.121)
s r 2
() ψ t
=
(
–
τ 0
–
τ
)
(3.122)
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