Global Positioning System Reference
In-Depth Information
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the sequence. The P(Y)-code is derived from two twelve-stage shift registers,
X
1
(t)
and
X
2
(t)
, having 15,345,000 and 15,345,037 stages (chips), respectively. Both reg-
isters continuously recycle. The modulo-2 sum of both registers has the length of
15,345,000 times 15,345,037 chips. At the chipping rate of 10.23 MHz it takes 266.4
days to complete the whole P(Y)-code cycle. It takes 1.5 s for the
X
1
register to go
though one cycle. The
X
1
cycles (epochs) are known as the Z count.
The bandwidth terminology is often used in connection with pseudorandom noise
modulation. Let
T
denote the duration of the chip (rectangular pulse), then the band-
width is inverse proportional to
T
. Therefore, shorter chips (pulses) require greater
bandwidth and vice versa. If we subject the rectangular pulse function to a Fourier
transform we obtain the well-known sinc (sine-cardinal) function
sin
(
2
1
f
c
π∆
f/f
c
)
S(
∆
f, f
c
)
=
(3.86)
[80
π∆
f/f
c
The symbol
f
is the difference with respect to the carrier frequency L1 or L2.
The code frequency 10.23 MHz or 1.023 MHz, respectively, is denoted by
f
c
. The
factor 1
/f
c
serves as a normalizing (unit area) scalar. Figure 3.13 shows the power
spectral density (3.86) for the P(Y)- and C/A-codes. This symmetric function is zero
at multiples of the code rate
f
c
. The first lobe stretches over the bandwidth, covering
the range of
∆
Lin
—
1.6
——
Nor
PgE
f
c
with respect to the center frequency. The spectral portion signal
beyond one bandwidth is filtered out at the satellite and is not transmitted.
Power ratios in electronics and in connection with signals and antennas are ex-
pressed in terms of decibels (dB) on a logarithmic scale. Of course, sound levels are
typically also given in units of decibels. One decibel is just detectable by the human
ear and a power of 100 watts is perceived to be twice as loud as 10 watts. The latter
relationship justifies the preference of using the logarithmic scale in addition to the
ability to express very large or very small ratios with a few digits. The power ratio in
terms of decibel units is defined as
±
[80
P
2
P
1
g
[dB]
=
10 log
10
(3.87)
A
bsolute power can be expressed with respect to a unit power
P
1
. For example, the
un
its dBW or dBm imply
P
1
=
1Wor
P
1
=
1 mW, respectively. Frequently the
relation
V
2
V
1
g
[dB]
=
20 log
10
(3.88)
is seen. In (3.88) the symbols
V
1
and
V
2
denote voltages. Both decibel expressions are
related by the fact that the square of the voltage divided by resistance equals power.
The power of the received GPS signals on the ground is lower than the background
noise (thermal noise). The specifications call for a minimum power at the user on the
earth of
−
160 dBW for the C/A-code,
−
163 dBW for the P(Y)-code on L1, and
−
166 dBW for the P(Y)-code on L2. To track the signal, the receiver correlates the
