Global Positioning System Reference
In-Depth Information
mum length.
A linear feedback shift register is a simple digital circuit that consists of
n
bits of memory and some feedback logic [2], all clocked at a certain rate. Every
clock cycle, the
n
th bit value is output from the device, the logical value in bit 1 is
moved to bit 2, the value in bit 2 to bit 3, and so on. Finally, a linear function is
applied to the prior values of bits 1 to
n
to create a new input value into bit 1 of the
device. With an
n
-bit linear feedback shift register, the longest length sequence that
can be produced before the output repeats is
N
1. A linear feedback shift regis-
ter that produces a sequence of this length is referred to as maximum length. During
each period, the
n
bits within the register pass through all 2
n
possible states, except
the all-zeros state, since all zeros would result in a constant output value of 0.
Because the number of negative values (1s) is always one larger than the number of
positive values (0s) in a maximum-length sequence, the autocorrelation function of
the spreading waveform
PN
(
t
) outside of the correlation interval is
=
2
n
−
A
2
/
N
. Recall
that the correlation was 0 (uncorrelated) in this interval for the DSSS signal with
random code in the previous example. The autocorrelation function for a maxi-
mum-length PRN sequence is the infinite series of triangular functions with period
NT
c
(seconds) shown in Figure 4.4(a). The negative correlation amplitude (
−
A
2
/
N
)is
−
shown in Figure 4.4(a), when the time shift,
τ
, is greater than
±
T
c
or multiples of
±
1) and represents a zero-frequency term in the series. Expressing the equa-
tion for the periodic autocorrelation function mathematically [9] requires the use of
the unit impulse function shifted in time by discrete (
m
) increments of the PRN
sequence period
NT
c
:
T
c
(
N
±
mNT
c
). Simply stated, this notation (also called a Dirac
delta function) represents a unit impulse with a discrete phase shift of
mNT
c
sec-
onds. Using this notation, the autocorrelation function can be expressed as the sum
of the zero-frequency term and an infinite series of the triangle function,
R
(
δ
(
τ +
),
defined by (4.7). The infinite series of the triangle function is obtained by the convo-
lution (denoted by
τ
⊗
)of
R
(
τ
) with an infinite series of the phase shifted unit impulse
functions as follows:
2
−
A
N
N
N
+
1
∞
∑
()
()
(
)
R
τ
=
+
R
τ
⊗
δ τ
+
mNT
(4.9)
PN
c
m
=−∞
The power spectrum of the DSSS signal generated from a maximum-length
PRN sequence is derived from the Fourier transform of (4.9) and is the line spec-
trum shown in Figure 4.4(b). The unit impulse function is also required to express
this in equation form as follows:
2
A
N
∞
∑
m
N
π
m
NT
2
π
()
()
(
)
2
Sf
=
δ
f
+
N
+
1
sinc
δπ
2
f
+
(4.10)
PN
2
m
=−∞
≠
0
c
where
m
3, …
Observe in Figure 4.4(b) that the envelope of the line spectrum is the same as the
continuous power spectrum obtained for the random PRN code, except for
the small zero-frequency term in the line spectrum and the scale factor
T
c
. As the
period,
N
(chips), of the maximum-length sequence increases, then the line spacing,
=±
1,
±
2,
±
Search WWH ::
Custom Search