4.3 Legacy GPS Signals
4.3.3 Autocorrelation Functions and Power Spectral Densities
The autocorrelation characteristics of the GPS PRN codes are fundamental to the signal demodulation process. The power spectral densities of the GPS PRN codes determine the channel bandwidths required to transmit and receive the spread spec- trum signals
As would be expected, the GPS PRN codes have periodic correlation triangles and a line spectrum that closely resemble the characteristics of maximum-length shift register PN sequences, but with several subtle differences. This is because the GPS PRN codes arenotshift register sequences of maximum length. For example, for the C/A code 10-bit shift register, there are only 30 usable maximum-length sequences, and among these available maximum-length sequences, the cross- correlation properties between different codes are not as good as that desired for GPS. Another problem is that the autocorrelation function of maximum-length sequences has sidelobes when the integration time is one (or a few) code periods.
(This can be a problem to a lesser extent with the C/A codes as well.) In a GPS receiver, the integration and dump time associated with the correlation of its replica C/A code with the incoming SV C/A code (similar to autocorrelation) is typically 1 to 5 ms (i.e., 1 to 5 C/A code periods). Except for a highly specialized mode of oper- ation called data wipeoff, the integration and dump time never exceeds the 50-Hz data period of 20 ms. During search modes, these short integration and dump peri- ods for the maximum-length sequences increase the probability of high sidelobes leading to the receiver locking onto a wrong correlation peak (a sidelobe). For these reasons, the Gold codes described earlier were selected for the C/A codes.
Using the exclusive-or of two maximum length shift registers, G1 and G2 (with a programmable delay), there are 2n– 1 possible delays. Therefore, there are 1,023 possible Gold codes for the GPS C/A code generator architecture (plus two addi- tional maximum-length sequences if the G1 and G2 sequences were used independ- ently). However, there are only 45 Gold code combinations for the architecture of the C/A code generator defined in [10], using two taps on the G2 register to form the delay. The 32 Gold codes with the best properties were selected for the GPS space segment. (There were only four more unique two-tap combinations selected for the pseudolites since two of these codes are redundant.) Extensions of the GPS C/A code for such applications as the WAAS, wherein augmentation C/A code signals are transmitted from geostationary satellites, required a careful analysis of their proper- ties and their effect on the space segment codes before their implementation. (Refer to Chapter 8 for details on the WAAS C/A code generation.)
Neglecting the navigation data, the autocorrelation function of the GPS C/A code signal is:
( ) ( ) ( )
RG T G t G t d
CA
i i
t
τ = +τ τ
=
∫
1
1023 0
1023
, (4.22)
where:
Gi(t)=C/A code Gold code sequence as a function of time,t, for SVi TCA=C/A code chipping period (977.5 ns)
τ=phase of the time shift in the autocorrelation function
The C/A code autocorrelation function is a series of correlation triangles with a period of 1,023 C/A code chips or 1 ms, as shown in Figure 4.13(a). As observed in Figure 4.13(a), the autocorrelation function of the GPS C/A (Gold) codes has the same period and the same shape in the correlation interval as that of a maximum-
A2 R ( )Gτ
1
−1023
977.5 10× −9 1 10× −3 τ(seconds)
A2
S ( )Gω
Envelope = A sinc2 2 ω997.5 10× −9 2
ω(radians) 2π
10 log S ( )G2 ω PS
Line number dB
0
−20
−30
−40
−50
1 2 3 4 5 6 7 8 9
977.5 10× −9 0
( )
( )
(c) (b) (a)
Figure 4.13 (a) The autocorrelation function, (b) spectrum, and (c) power ratio of a typical C/A code.
length sequence (see Section 4.2.3). There are small fluctuations in the intervals between the correlation intervals rather than the uniform minimum correlation level of 1/1,023 for the maximum-length sequence using a 10-bit feedback shift reg- ister [14]. This is because the C/A code correlation process cannot be synchronously clocked, as was assumed for the maximum-length sequence. These small fluctua- tions in the autocorrelation function of the C/A codes result in the deviation of the line spectrum from the sinc2(x) envelope, as shown in Figure 4.13(b). Recall that the power line spectrum of the maximum-length sequences matched the sinc2(x) enve- lope exactly, except for the zero-frequency term. However, the line spectrum spac- ing of 1,000 Hz is the same for both the C/A code and the 10-bit maximum-length sequence code. Figure 4.13(c) illustrates that the ratio of the power in each C/A line to the total power in the spectrum plotted in decibels can fluctuate significantly (nearly 8 dB) with respect to the−30 dB levels that would be obtained if every line contained the same power. Every C/A code has a fewstronglines [i.e., lines above the sinc2(x) envelope], which render them more vulnerable to a continuous wave (CW) RF interference at this line frequency than their maximum length sequence counterpart. For example, the correlation process between a CW line and a PRN code ordinarily spreads the CW line, but the mixing process at some strong C/A code line results in the RF interference line being minimally suppressed. As a result, the CW energy “leaks” through the correlation process at this strong line frequency.
The presence of the navigation data mitigates this leakage to a certain extent. (The effects of RF interference will be discussed further in Chapter 6.)
Keeping in mind that the GPS C/A codes have these limitations, it is often conve- nient and approximately correct to illustrate their autocorrelation functions as fol- lowing ideal maximum-length sequences, as shown in Figure 4.14. Note that there are other typical simplifications in this figure. The -axis is represented in C/A code chips instead of seconds and the peak amplitude of the correlation function has been normalized to unity (corresponding to the PRN sequence amplitude being±1).
The autocorrelation function of the GPS P(Y) code is:
( ) ( ) ( )
RP T P t P t d
CP
i i
t
τ = τ τ
× +
=
∫
×1
61871 1012 0
6 1871 1012
.
.
(4.23)
where:
Pi(t) = P(Y) code PRN sequence as a function of time,t, for SVi TCP=P(Y) code chipping period (97.8 ns)
=phase of the time shift in the autocorrelation function
The P(Y) code is also not a maximum-length sequence code, but because its period is so long and its chipping rate is so fast, its autocorrelation characteristics are essentially ideal. The P(Y) code was designed to have a one-week period made up of 403,200 periods of its 1.5-second X1 epochs, called Z-counts. Figure 4.15 depicts a normalized autocorrelation function for P(Y) code (amplitude A = ±1) with the phase shift axis,τ, shown in units of P(Y) code chips. The autocorrelation function for P(Y) code has similar characteristics to the C/A code, but with signifi-
cant differences in values. Table 4.7 compares these characteristics. From Table 4.7, it can be observed that P(Y) code can be considered essentially uncorrelated with itself (typically−127.9 dB) for all intervals outside the correlation interval, whereas, the C/A code is adequately uncorrelated with itself (typically−30.1 dB) outside its correlation interval. However, the C/A codes can be as poorly uncorrelated with themselves as−21.1 dB outside the correlation interval—fortunately this occurs only a small percentage of the time.
When the GPS codes are combined with the 50-Hz navigation message data, there is essentially an imperceptible effect on the resulting autocorrelation functions
1
τ(chips)
−1 0 1 1,022 1,024
1,023
−1 / 1,023
R ( )Gτ
Figure 4.14 Normalized and simplified autocorrelation function of a typical C/A code with in chips.
Table 4.7 Comparisons Between C/A Code and P(Y) Code Autocorrelation C/A Code P(Y) Code Maximum autocorrelation amplitude 1 1 Typical autocorrelation amplitude
outside the correlation interval − 1
1 023, −
× 1 61871 10. 12 Typical autocorrelation in decibels with
respect to maximum correlation −30.1 −127.9
Autocorrelation period 1 ms 1 week
Autocorrelation interval (chips) 2 2
Autocorrelation time interval (ns) 1,955.0 195.5 Autocorrelation range interval (m) 586.1 58.6 Rc=chipping rate (Mchip/s) 1.023 10.23
Tc=chipping period (ns) 977.5 97.8
Range of one chip (m) 293.0 29.3
τ(chips)
−1 0 1 p−1 p+1
p
−1 / p
R ( )Pτ
p=6.1871×1012chips 1
Figure 4.15 Normalized and simplified autocorrelation function of a typical P(Y) code withτin chips.
and the power spectrum. When these are modulated onto the L-band carrier, there is a translation to L-band of the power spectrum from the baseband frequencies that have been described so far. Assuming that the PRN waveform is BPSK modulated onto the carrier and that the carrier frequency and the code are not coherent, the resulting power spectrum is given by [9]:
( ) [ ( ) ( ) ]
SL ω = 1 P Sc PN ω ω+ c +P Sc PN ω ω− c
2 (4.24)
where:
Pc=unmodulated carrier power ωc=carrier frequency (radians)
SPN(ωc)=power spectrum of the PRN code(s) (plus data) at baseband
As can be observed from (4.24), the baseband spectra are shifted up to the car- rier frequency (and down to the negative carrier frequency). In the following GPS L-band power spectrum illustrations, only the (upper) single-sided frequency is con- sidered. The GPS signals were synthesized by a GPS signal generator and measured by a Hewlett-Packard spectrum analyzer.
Figure 4.16 is a plot of the power spectrum of the GPS P(Y) code and C/A code (plus 50-Hz data) BPSK modulated onto the L1 carrier. The spectrum analyzer per- formed the plot using a 300-kHz resolution bandwidth, so it is impossible to observe the line spectrum characteristics of either code. Therefore, the power spec- trums appear to be continuous. The center frequency is at the L1 carrier, 1,575.42
hp 5 dB/
Ref−55.0 dBm Atten 10 dB
Center 1575.42 MHz
Res BW 300 kHz VBW 3 Hz
Span 50.00 MHz SWP 100 sec
MKR 1575.42 MHz
−61.85 dBm
Marker 1575.42 MHz
61.85 dBm
−
Figure 4.16 Power spectrum of L1 P(Y) code and C/A code from a GPS signal generator.
MHz. The combined power spectra of C/A code and P(Y) codes are centered at the L1 carrier frequency. The first nulls of the C/A code power spectrum are at±1.023 MHz from the center frequency and the first nulls of the P(Y) code power spectrum are at±10.23 MHz from the center frequency.
Figure 4.17 is a plot of the power spectrum of the GPS P(Y) code (plus 50-Hz data) BPSK modulated onto the L2 carrier. The plot is virtually identical to Figure 4.15, except the center frequency is at the L2 carrier, 1,227.60 MHz, and the C/A code modulation is removed. The first null of the P(Y) code is at±10.23 MHz.
Figure 4.18 is a plot of the power spectrum of the GPS C/A code (plus 50-Hz data) BPSK modulated onto the L1 carrier with the P(Y) code turned off. The fre- quency scale has been adjusted to be narrower than Figure 4.16 by a factor of ten in order to inspect the C/A code power spectrum more closely. The resolution band- width of the spectrum analyzer has been reduced to 3 kHz so that the line spectrum of the C/A code is just beginning to be visible in the plot. The strong lines of the C/A code [those above the nominal sinc2(x) envelope] are also somewhat observable. It would be impossible to observe the line spectrum of the P(Y) code with a spectrum analyzer because the resolution bandwidth corresponding to its extremely fine line spacing would be unreasonably narrow.