On Generalized Signal Waveforms for Satellite Navigation (797942), страница 35
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To simplify thus ouranalysis, the three main cases are exposed next. These correspond basically to the threeCBOC implementations that were considered for the Galileo E1 OS:•••The use of a CBOC(6,1,1/11) where both the data and pilot channels have a BOC(6,1)component in anti-phase. In this case, the power of the BOC(6,1) part is 1/11 of thechannel total power. This is the implementation finally selected for Galileo E1 OS.The use of a CBOC(6,1,2/11) where only one of the channels, the pilot or the data, hasa BOC(6,1) component with alternating sign, while the other channel is a pureBOC(1,1).
In this case, the power of the BOC(6,1) part is 2/11 of the pilot channeltotal power. Moreover, it is important to note that the alternation of BOC(6,1) isnecessary to get rid of the cross spectral terms that would result otherwise. This willbe shown in the following lines.CBOC(6,1,1/11) is used on both the data and pilot channels, being BOC(6,1) in bothchannels with alternating sign. In this case, the data and pilot channels have aBOC(6,1) power of 1/11 of the channel total power.Recalling now the general CBCS definition from (4.121) and substituting BCS by BOC(6,1)we can define the general CBOC model as follows:⎡ cD (t )⎤[cosθ1 sBOC (1,1) (t ) ± cosθ 2 sBOC(6,1) (t )] + ⎥⎢⎢ 2⎥cP (t )⎢[cosθ1 sBOC(1,1) (t ) ± cosθ 2 sBOC (6,1) (t )] + ⎥⎥s (t ) = A1 ⎢+2⎢⎥⎢+ j s (t ) ⎛⎜ sin θ1 + sin θ 2 ⎞⎟ + s (t )⎥PRSIM⎢⎣⎥⎦2⎝⎠⎛ sin θ1 − sin θ 2 ⎞sIM (t ) = − j cD (t )cP (t ) sPRS (t ) ⎜⎟2⎝⎠(4.161)(4.162)150GNSS Signal Structureand looking only at the pilot channel for the three cases described above, we have:CBOC(6,1, ρ , '+ ' ) (t ) = c P (t ) { k1 s BOC (1,1) (t ) + k 2 s BOC (6,1) (t )}CBOC(6,1, ρ , '−' ) (t ) = c P (t ) { k1 s BOC(1,1) (t ) − k 2 s BOC (6,1) (t )}(4.163)(4.164)where ρ indicates the amount of power on the high frequency BOC(6,1) component:k2ρ= 2 2 2(4.165)k1 + k 2The last model to introduce is the alternating model, which is shown to be:CBOC(6,1, ρ , '+ / −' ) (t ) = c P (t ) { k1 s BOC(1,1) (t ) + k 2 s BOC(6,1) (t )}(4.166)for even chips, andCBOC(6,1, ρ , '+ / −' ) (t ) = c P (t ) {k1 s BOC(1,1) (t ) − k 2 s BOC(6,1) (t )}(4.167)for odd chips.
The autocorrelation function of the three models is shown next:ℜ CBOC( 6,1, ρ ,'− ') (τ ) = k12 ℜ BOC(1,1) (τ ) + k 22 ℜ BOC(6,1) (τ ) − 2 k1 k 2 ℜ BOC(1,1) / BOC(6,1) (τ ) (4.168)ℜ CBOC( 6,1, ρ ,'+ ') (τ ) = k12 ℜ BOC(1,1) (τ ) + k 22 ℜ BOC(6,1) (τ ) + 2 k1 k 2 ℜ BOC(1,1) / BOC(6,1) (τ ) (4.169)ℜ CBOC( 6,1, ρ ,'+ / − ') (τ ) = k12 ℜ BOC(1,1) (τ ) + k 22 ℜ BOC(6,1) (τ )(4.170)As we can see, the two first auto-correlations present an additional cross-term due to theexistence of cross-correlation between BOC(1,1) and BOC(6,1).
As we mentioned above, thiscross-term is not desirable and must be eliminated to generate a spectrum according to theMBOC definition. This is possible if data and pilot are in anti-phase (each with a differentsign) or if the BOC(6,1) component alternates its sign. Nonetheless, we will keep thesesignals for reference in the following figures to show the performance. We show next theautocorrelation functions of the different analyzed solutions compared with the TMBOCsolution for a receiver bandwidth of 24 MHz:Figure 4.51. CBOC and TMBOC Autocorrelation Function151GNSS Signal StructureAs we can recognize in the figure above, the shape of the auto-correlation function highlydepends on the amount of power on BOC(6,1). Indeed, the higher the value of ρ, the moreripples the auto-correlation will present and the better the potential performance will be.Furthermore, it is interesting to see that the sharpest slope around the origin is shown byTMBOC and the CBOC('−') version while TMBOC(6,1,2/11) and CBOC(6,1, 2 / 11, '+ / −')perform similar.
MBOC is analyzed next regarding the following three criteria:•••False tracking pointsCode tracking noiseMultipath-induced code tracking error4.7.5.1False Tracking PointsIf we take a closer look at the auto-correlation function of Figure 4.51 above, it is clear torecognize the existence of secondary peaks that could lead to stable false locks. Indeed, themore ripples the function presents and the more accentuated the undulations are, the higherthe probability that we lock on a non desired but stable tracking point.
Fortunately, not all thesolutions are equally susceptible to suffering from this effect, since this depends actually onhow the auto-correlation function looks like. In fact, while the selected CBOC(1/11) forGalileo E1 OS is not likely to lead to stable false lock points as shown in[J.-A. Avila-Rodriguez et al., 2006c], the existence of false lock points would be nearlyunavoidable for the CBOC(6,1, 2 / 11, '+ / −') implementation, implying thus a highercomplexity to detect the right peak. This is in fact the prize for allocating more BOC(6,1)power on the channel.
Finally, it is important to note that no matter which of theimplementations we look at, since BOC(1,1) is the dominant signal in all of them, a false lockdetector is still necessary in order to make sure that the receiver is tracking the signal basedon the correct autocorrelation main peak, and not the secondary of BOC(1,1).4.7.5.2Thermal Noise-Induced Code Tracking ErrorIn order to understand how the code tracking noise behaves for the different implementationsof MBOC discussed above, we present in the next figure the Cramér Rao Lower Bound.Figure 4.52. BOC(1,1), CBOC and TMBOC Cramér Rao Lower Bound with 1 Hz LoopBandwidth, 1/12 Chip E-L Spacing, 4 ms Integration and 12 MHz One-Sided Filter152GNSS Signal StructureAs we can see, the best performance corresponds to the alternating CBOC( '+ / −') versionwith 2/11 of power on BOC(6,1) or to its equivalent TMBOC version, since these can takeadvantage of the high frequency components of the signal the best. Moreover, among thedifferent CBOC solutions with the 1/11 of average power that results from considering dataand pilot together, the worst performance is shown to be that of the in-phase CBOC(1 / 11, '+')solution while the anti-phase CBOC(1 / 11, '−') is the best.
This was to expect since itsautocorrelation function’s main peak has the steepest slope. We can also see this if wecompare both phase and anti-phase versions in the time domain.Figure 4.53. CBOC data chip with BOC(1,1) and BOC(6,1) in-phaseEqually, the pilot channel would present the following shape for a chip.Figure 4.54.
CBOC pilot chip with BOC(1,1) and BOC(6,1) in anti-phaseFrom the previous figures it is easy to recognize that the pilot channel will have morecomponents at higher frequencies since at 0.5 chips the amplitude variation is higher.153GNSS Signal StructureFurthermore, the alternating CBOC solution with 1/11 of power on BOC(6,1) performs as theaverage of the phase and anti-phase solutions shown above. To avoid any confusion, it isimportant to underline that the solutions studied above correspond to MBOC implementationsthat were object of study during the design of Galileo E1 OS and GPS L1C. The finallyselected CBOC implementation of Galileo has BOC(6,1) on both data and pilot, being thesein anti-phase with respect to each other.
According to the previous notation, this means thatthe pilot channel of Galileo E1 OS will be CBOC(1 / 11, '−') and the data channelCBOC(1 / 11, '+') .4.7.5.3Multipath Induced Tracking ErrorThe multipath performance of a signal highly depends on the shape of the auto-correlationfunction, as we saw in chapter 4.1.1. Indeed, this was one of the aspects that were moreseriously taken into consideration during the optimization of the Galileo Open Service andGPS Civil signals in E1/L1.
We show in the next figures the multipath performance ofdifferent MBOC implementations by means of the multipath envelopes and the multipathrunning average error as done in [G.W. Hein et al., 2006a] and [G.W. Hein et al., 2006b].Note that TMBOC(6,1,1/11) and CBOC(6,1,1 11, '+ / −' ) perform the same and therefore areoverlapped in the next figures.Figure 4.55.
MBOC Multipath Envelopes with an E-L Spacing of 0.1 chips and aDouble-Sided Filter of 24 MHzFigure 4.56. MBOC Multipath Running Average Error with an E-L Spacing of 0.1chips and a Double-Sided Filter of 24 MHz154GNSS Signal StructureAs we can see, for an early-late spacing of 0.1 chips and a one-sided front-end filter of12 MHz, the best performance is exhibited by CBOC(6,1, 2 / 11, '+ / −') , followed byCBOC(6,1,1 / 11, '−') , CBOC(6,1,1 / 11, '+ / −') and CBOC(6,1,1 / 11, '+') .
Moreover, it is clear tosee that all the solutions clearly outperform BOC(1,1). In addition, it is important to note thatCBOC(6,1,1 / 11, '+ / −') and TMBOC(6,1,1/11) perform the same in average when both dataand pilot are processed in the receiver.4.7.5.4Conclusions on Optimal CBOC TrackingAs we have seen in previous lines, the best CBOC implementation of MBOC in terms ofperformance is the anti-phase CBOC(6,1,1 / 11, '−') solution.
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