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.
 
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