Graphics Programs Reference
In-Depth Information
t 2
---
τ i
(
t µ i
,
) τ t µ µ i t
+
+
τ tt
(12.39)
where the over-bar indicates evaluation a t the state
(
00
,
)
, and the subscripts
denote partial derivatives. For example,
τ t µ
means
2
τ t µ
=
τ i
(
t µ i
,
)
(12.40)
t µ
(
t µ
,
)
=
(
00
,
)
The Taylor series coefficients are
-----  1
2 h
c
-------------
τ
=
(12.41)
cos
β i
2 v
c
-----  β i
τ t µ
=
sin
(12.42)
2 v 2
hc
 β i
------- 
τ tt
=
cos
(12.43)
Note that other Taylor series coefficients are either zeros or very small. Hence,
they are neglected. Finally, we can rewrite the returned radar signal as
ó i t µ i
s i
(
t µ i
,
)
=
A i
cos
[
(
,
) 0
–
]
(12.44)
) t τ tt t 2
ó i t µ i
---
(
,
)
=
f 0
(
1
–
τ t µ µ i
–
–
Observation of Eq. (12.44) indicates that the instantaneous frequency for the
scattere r varies as a linear function of time due to the second order phase
term (this confirms the result we concluded about a scatterer
Doppler history). Furthermore, since this phase term is range-bin dependent
and not scatterer dependent, all scatterers within the same range bin produce
this exact second order phase term. It follows that scatterers within a range bin
have identical Doppler histories. These Doppler histories are separated by the
time delay required to fly between them, as illustrated in Fig. 12.10.
ith
τ tt t 2
f 0
(
2
)
Suppose that there are scatterers within the range bin. In this case, the
combined returns for this cell are the sum of the individual returns due to each
scatterer as defined by Eq. (12.44). In other words, superposition holds, and the
overall echo signal is
I
kth
I
s r
()
=
s i
(
t µ i
,
)
(12.45)
i
=
1
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