Graphics Programs Reference
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176
MATLAB Simulations for Radar Systems Design
Finally, one can define the corresponding velocity resolution as
c
∆
f
d
2
f
0
c
2
f
0
τ'
∆
v
=
-----------
=
-----------
(3.140)
Again observation of Eqs. (3.138) and (3.116) indicate that the output of the
matched filter and the ambiguity function (when ) are similar to each
other. Consequently, one concludes that the matched filter preserves the wave-
form Doppler resolution properties as well.
τ
=
0
3.10.3. Combined Range and Doppler Resolution
In this general case, one needs to use a two-dimensional function in the pair
of variables ( ). For this purpose, assume that the complex envelope of the
transmitted waveform is
τ
f
d
,
ψ ()
u
()
e
j
2π
f
0
t
=
(3.141)
Then the delayed and Doppler-shifted signal is
)
e
j
2π
f
0
(
f
d
)
t
(
τ
)
ψ'
t
(
τ
)
=
ut
τ
(
(3.142)
Computing the integral square error between Eqs. (3.142) and (3.141) yields
∞
∫
∞
∫
∞
∫
ε
2
2
ψ ()
2
ψ
∗
() ψ'
t
=
ψ () ψ'
t
(
τ
)
d
=
2
d
2
Re
(
τ
)
d
(3.143)
∞
∞
∞
which can be written as
∞
∫
∞
∫
2
Re e
j
2π
f
0
(
f
d
)τ
)
e
j
2π
f
d
t
ε
2
u
()
2
u
()
u
∗
t
=
2
d
(
τ
d
(3.144)
∞
∞
Again, in order to maximize this squared error for
τ
≠
0
one must minimize the
last term of Eq. (3.144).
Define the combined range and Doppler correlation function as
∞
∫
)
e
j
2π
f
d
t
u
()
u
∗
t
χτ
f
d
(
,
)
=
(
τ
d
(3.145)
∞
In order to achieve the most range and Doppler resolution, the modulus square
of this function must be minimized for
and
. Note that the output of
τ
≠
0
f
d
≠
0
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