Graphics Programs Reference
In-Depth Information
Costas showed that the output of the matched filter is
N
∑
1
N
∑
1
1
----
χτ
f
D
(
,
)
=
exp
(
j
2π
lf
D
τ
)
Φ
ll
τ
f
D
(
,
)
lq
τ
+
(
(
lq
)τ
1
,
f
D
)
(4.45)
l
=
0
q
=
ql
0
≠
sin
α
α
τ
τ
1
-----------
Φ
lq
τ
f
D
(
,
)
=
τ
1
-----
exp
(
j
β
j
2π
f
q
τ
)
,
τ
1
≤
(4.46)
απ
f
l
=
(
f
q
f
D
) τ
1
(
τ
)
(4.47)
βπ
f
l
=
(
f
q
f
D
) τ
1
(
+
τ
)
(4.48)
Three-dimensional plots of the ambiguity function of Costas signals show
the near thumbtack response of the ambiguity function. All sidelobes, except
for few around the origin, have amplitude . Few sidelobes close to the ori-
gin have amplitude , which is typical of Costas codes. The compression
ratio of a Costas code is approximately
1
⁄
N
2
⁄
N
N
.
4.4.2. Binary Phase Codes
Consider the case of binary phase codes in which a relatively long pulse of
width is divided into smaller pulses; each is of width . Then,
the phase of each sub-pulse is randomly chosen as either or radians rela-
tive to some CW reference signal. It is customary to characterize a sub-pulse
that has phase (amplitude of +1 Volt) as either Ð1Ñ or Ð+.Ñ Alternatively, a
sub-pulse with phase equal to (amplitude of -1 Volt) is characterized by
either Ð0Ñ or Ð-.Ñ The compression ratio associated with binary phase codes is
equal to , and the peak value is times larger than that of the long
pulse. The goodness of a compressed binary phase code waveform depends
heavily on the random sequence of the phases of the individual sub-pulses.
τ'
N
∆τ
=
τ'
⁄
N
0
π
0
π
ξ ' ∆τ
=
⁄
N
One family of binary phase codes that produces compressed waveforms with
constant sidelobe levels equal to unity is the Barker code.
Fig. 4.14
illustrates a
Barker code of length seven. A Barker code of length is denoted as .
There are only seven known Barker codes that share this unique property; they
are listed in
Table 4.1
. Note that and have complementary forms that
have the same characteristics. Since there are only seven Barker codes, they
are not used when radar security is an issue.
n
B
n
B
2
B
4
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