Graphics Programs Reference
In-Depth Information
Waveform Resolution and Ambiguity
173
where is the difference in delay between the two returns. One can assume
that the reference time is , and thus without any loss of generality one may
set . It follows that the two targets are distinguishable by how large or
small the delay
τ
τ
0
τ
0
=
0
can be.
τ
In order to measure the difference in range between the two targets consider
the integral square error between
ε
2
and
. Denoting this error as
,
ψ ()
ψ
t
(
τ
)
it follows that
∞
∫
ε
2
2
=
ψ () ψ
t
(
τ
)
d
(3.123)
∞
Eq. (3.123) can be written as
∞
∫
∞
∫
ε
2
ψ ()
2
)
2
=
d
+
ψ
t
(
τ
d
(3.124)
∞
∞
∞
∫
ψ ()ψ
∗
t
)
∗
()ψ
t
{
(
(
τ
+
(
τ
)
)
d
}
∞
Using Eq. (3.118) into Eq. (3.124) yields
∞
∫
∞
∫
ε
2
u
()
2
ψ
∗
()ψ
t
=
2
d
2
Re
(
τ
)
d
=
(3.125)
∞
∞
∞
∫
∞
∫
j
ω
0
τ
u
()
2
u
∗
()
ut
τ
2
d
2
Re e
(
)
d
The first term in the right hand side of Eq. (3.125) represents the signal energy,
and is assumed to be constant. The second term is a varying function of with
its fluctuation tied to the carrier frequency. The integral inside the right-most
side of this equation is defined as the Ðrange ambiguity function,Ñ
τ
∞
∫
u
∗
()
ut
τ
χ
R
()
=
(
)
d
(3.126)
∞
The maximum value of is at . Target resolvability in range is
measured by the squared magnitude . It follows that if
for some nonzero value of , then the two targets are indistin-
guishable. Alternatively, if for some nonzero value of , then
the two targets may be distinguishable (resolvable). As a consequence, the
most desirable shape for
χ
R
()
τ
=
0
()
2
χ
R
χ
R
() χ
R
=
()
τ
χ
R
() χ
R
≠
()
τ
is a very sharp peak (thumb tack shape) cen-
χ
R
()
tered at
and falling very quickly away from the peak.
τ
=
0
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