Global Positioning System Reference
In-Depth Information
|
F
(ω)
|
T
ω
arg
F
(ω)
(
0
,π)
...
ω
...
(
0
,
−
π)
FIGURE 1.2. Top: Magnitude spectrum
|
F
(ω)
|
of rectangular pulse. Notice that
F
(ω)
has
zeros at
±
T
, .... Bottom: Phase spectrum arg
(
F
(ω))
of rectangular pulse. Notice
that arg
(
F
(ω))
has jumps equal to
π
at
±
2
T
4
,
±
2
T
4
T
,
±
, ....
metrically around
t
=
0. The equation for the pulse is
1
,
|
t
|≤
T
/
2,
(
)
=
f
t
(1.7)
0
,
otherwise.
Let the frequency be
f
in Hz [cycle/s] and
ω
=
2
π
f
[radian/s]. Then the
Fourier transform of
f
(
t
)
is
T
sinc
ω
T
sin
ω
2
ω
T
2
F
(ω)
=
=
.
(1.8)
T
2
and the phase spectrum arg
F
(ω)
are depicted
The magnitude spectrum
|
F
(ω)
|
in Figure 1.2. Notice that arg
F
(ω)
of
f
2
π
n
(
t
)
is linear for
ω
=
with jumps
T
, because of the change of sign of sin
ω
2
at these points.
From (1.4) follows that the ACF
r
f
(τ )
T
2
π
n
equal to
π
for
ω
=
for a rectangular pulse has a triangular
waveform; see Figure 1.3,
T
1
−
|
τ
T
,
for
|
τ
|≤
T
,
r
f
(τ )
=
(1.9)
0
,
otherwise.
Search WWH ::
Custom Search