Graphics Programs Reference
In-Depth Information
The sequences of
ÐdotÑ
assignments for which the corresponding ambiguity
function approaches an ideal or a
ÐthumbtackÑ
response are called Costas
codes. A near thumbtack response was obtained by Costas
1
using the following
logic: there is only one frequency per time slot (row) and per frequency slot
(column). Therefore, for an
NN
×
matrix the number of possible Costas codes
is drastically less than
N
!
. For example, there are
N
c
=
4
possible Costas
codes for
N
=
3
, and
N
c
=
40
possible codes for
N
=
5
. It can be shown
that the code density, defined as the ratio
N
c
⁄
N
!
, gets significantly smaller as
N
becomes larger.
There are numerous analytical ways to generate Costas codes. In this section
we will describe two of these methods. First, let
q
be an odd prime number,
and choose the number of subpulses as
Nq
1
=
(4.40)
Define
γ
as the primitive root of
q
. A primitive root of
q
(an odd prime num-
γγ
2
γ
3
…γ
q
1
ber) is defined as
γ
such that the powers
,
,
,
,
modulo
q
generate
every integer from
1
to
q
1
.
In the first method, for an
NN
×
matrix, label the rows and columns, respec-
tively, as
i
=
012…
q
,,, ,
(
2
)
(4.41)
j
=
123…
q
,,, ,
(
1
)
Place a dot in the location
,()
ij
corresponding to
f
i
if and only if
()
j
i
=
(
modulo q
)
(4.42)
In the next method, Costas code is first obtained from the logic described
above; then by deleting the first row and first column from the matrix a new
code is generated. This method produces a Costas code of length
Nq
2
=
.
Define the normalized complex envelope of the Costas signal as
N
∑
1
1
N
τ
1
s
()
=
-------------
s
l
(
t
τ
1
)
(4.43)
l
=
0
exp
(
j
2π
f
l
t
)
≤≤
elsewhere
0
t
τ
1
s
l
()
=
(4.44)
0
1. Costas, J. P., A Study of a Class of Detection Waveforms Having Nearly Ideal
Range-Doppler Ambiguity Properties,
Proc. IEEE
72
, 1984, pp. 996-1009.
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