On Generalized Signal Waveforms for Satellite Navigation (797942), страница 34
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Moreover, we have144GNSS Signal Structurealready noted that some BCS sequences can potentially cause tracking bias in receiversadapted to receive only one of the spreading symbols. This was indeed one of the maindisadvantages of CBCS. The solution to this problem was given by MBOC and the keyconcept is antisymmetric sequences. These sequences are explained in detail in Appendix Kbut in the next lines we anticipate some concepts already.
Since BOC(1,1) is antisymmetric,receiver biases can be avoided by choosing the correct properties for the partner spreadingsymbol sequence.As we have repeatedly mentioned in this chapter, compatibility with BOC(1,1) receivers wasa major driver in the design of the optimized Galileo signal. As a result of this, havingBOC(1,1) a partner spreading symbol sequence with zero cross-correlation became one of themost important drivers in the design of the Galileo OS in E1.After CBCS was proposed, a new BCS solution that could avoid all the drawbacks describedin chapter 4.6.7 was the objective of the works of US and EU.
We can summarize theproperties that this desired BCS sequence should present:•••Anti-symmetryBalance (zero-sum) for the sequenceZero crosscorrelation with the partner sequenceAs shown in [A.R. Pratt et al., 2006] a comprehensive search for binary sequences wasconducted with some or all of these properties. These are listed below for a variety ofsequence lengths n, all divisible by 2, from 2 to 12. Moreover, we show only distinctsequences so that the tables contain {xi} but not the time reversed versions {-xi}, {xn-1-i} or{-xn-1-i}. Since the sequences are antisymmetric, they may be considered to be constructedfrom a base sequence {ŷi} of length n as shown in the following equation:X = {xi } = { yˆ i ,− yˆ n −1−i }(4.154)As shown in Appendix K, all antisymmetric sequences of even length n are balanced.
Thenext tables show the cross-correlation with a BOC(1,1) partner sequence, under theassumption that the spreading symbol durations are common, that is the duration of an n = 2sequence is identical with that of an n = 12 sequence, for example.Moreover, the cross-correlation between x0 = BOC(1,1) and the corresponding BCS sequencex k is shown to be:n −1n −1i =0i =0Σ( z k ) = ∑ z k ,i = ∑ x0, i xk , i = n Rxc0 , xk (0) = n mod 4(4.155)where n mod m represents the modulo operation defined as the remainder of the division ofn by m . Next pages summarizes the results:145GNSS Signal StructureTable 4.2.
Tables of Distinct Spreading Symbol Sequences for n=2 to 12n=2x0Rxc0 , xk (0)1,-11.0n=4Rxc0 , xk (0)x01,1,-1,-11.0x11,-1,1,-10.0n=6Rxc0 , xk (0)x01,1,1,-1,-1,-11.0x11,1,-1,1,-1,-11/3x21,-1,1,-1,1,-11/3n=8Rxc0 , xk (0)x01,1,1,1,-1,-1,-1,-11.0x11,1,1,-1,1,-1,-1,-10.5x21,1,-1,1,-1,1,-1,-10.5x31,1,-1,-1,1,1,-1,-10x41,-1,1,-1,1,-1,1,-10x51,-1,-1,1,-1,1,1,-10n = 10Rxc0 , xk (0)x01,1,1,1,1,-1,-1,-1,-1,-11.0x11,1,1,1,-1,1,-1,-1,-1,-13/5x21,1,1,-1,1,-1,1,-1,-1,-13/5x31,1,1,-1,-1,1,1,-1,-1,-11/5x41,1,-1,1,1,-1,-1,1,-1,-13/5x51,1,-1,1,-1,1,-1,1,-1,-11/5x61,1,-1,-1,1,-1,1,1,-1,-11/5x71,-1,1,1,-1,1,-1,-1,1,-11/5x81,-1,1-,1,1,-1,1,-1,1,-11/5x91,-1,-1,-1,1,-1,1,1,1,-1-1/5146GNSS Signal Structuren = 12Rxc0 , xk (0)x01,1,1,1,1,1,-1,-1,-1,-1,-1,-11.0x11,1,1,1,1,-1,1,-1,-1,-1,-1,-14/6x21,1,1,1,-1,1,-1,1,-1,-1,-1,-14/6x31,1,1,1,-1,-1,1,1,-1,-1,-1,-12/6x41,1,1,-1,1,1,-1,-1,1,-1,-1,-14/6x51,1,1,-1,1,-1,1,-1,1,-1,-1,-12/6x61,1,1,-1,-1,1,-1,1,1,-1,-1.-12/6x71,1,1,-1-,1,-1,1,1,1,-1,-1,-10x81,1,-1,1,1-,1,1,-1,-1,1,-1,-12/6x91,1,-1,1,-1,1,-1,1,-1,1,-1,-12/6x 101,1,-1,1,-1,-1,1,1,-1,1,-1,-10x 111,1,-1,-1,1,1,-1,-1,1,1,-1,-12/6x 121,1,-1,-1,1,-1,1,-1,1,1,-1,-10x 131,1,-1,-1,-1,1,-1,1,1,1,-1,-10x 141,-1,1,1,1,1,-1,-1,-1,-1,1,-14/6x 151,-1,1,1,1,-1,1,-1,-1,-1,1,-12/6x 161,-1,1,1,-1,1,-1,1,-1,-1,1,-12/6x 171,-1,1,-1,1,-1,1,-1,1,-1,1,-10x 181,-1,1,-1,-1,1,-1,1,1,-1,1,-10x 191,-1,-1,-1,-1,1,-1,1,1,1,1,-1-2/6From the tables, it can be seen that only tables for 0 = n mod 4 , that is n = 4, 8,12 , have anyentries with zero crosscorrelation with BOC(1,1).
For n = 4 , there is only 1 sequencecorresponding to a BOC(2,1) spreading symbol modulation. For n = 8 , there are 3permissible sequences, x3, x4, x5. Of these, x4 corresponds to the BOC(4,1) modulation that wementioned some lines above. Finally for n = 12 , there are 6 possible sequences, x7, x10, x12,x13, x17, x18 where x17 corresponds to BOC(6,1), one of the solutions and indeed the best interms of performance.4.7.3CBOC ImplementationThe CBOC implementation is a particular case of the CBCS modulation that we studied inchapter 4.6. As we saw there, the CBOC modulation can be expressed as follows⎡ cD (t )(cos θ1 sBOC(1,1) (t ) + cosθ 2 sBOC(6,1) (t )) + ⎤⎥⎢⎢ 2⎥⎢ cP (t )(4.156)(cos θ1 sBOC(1,1) (t ) − cos θ 2 sBOC(6,1) (t )) + ⎥⎥s (t ) = A1 ⎢+2⎢⎥⎢+ j s (t ) ⎛⎜ sin θ1 + sin θ 2 ⎞⎟ + s (t )⎥PRSIM⎢⎣⎥⎦2⎝⎠147GNSS Signal Structure⎛ sin θ1 − sin θ 2 ⎞sIM (t ) = − j cD (t ) cP (t ) sPRS (t ) ⎜⎟2⎝⎠(4.157)where we can recognize that in this particular case the BCS component is BOC(6,1) with apercentage of power of 1/11.
Furthermore the high frequency signal is placed on both the dataand pilot channels and equal power for both channels is assumed. In addition, the total OSpower is equal to that of the PRS and the modulation parameters that result of solving theequations system (J.35) of Appendix J are the following:Table 4.3. Interplex parameters of the CBOC(6,1,1/11) modulation4.7.4Percentage ρ1/11θ1θ20.1314692798 πA11.04099840822680.4064655161 πTMBOC ImplementationIn a TMBOC spreading time series [G.W. Hein et al., 2006a], different BOC spreadingsymbols are used for different chip values, in either a deterministic or periodic pattern. Toproduce an MBOC(6,1,1/11) spectrum, the used spreading symbols are BOC(1,1) chips,denoted as g BOC(1,1) (t ) , and BOC(6,1) chips, denoted as g BOC(6,1) (t ) , whereand⎧sign[sin (2 π t / Tc )] 0 ≤ t ≤ Tcg BOC (1,1) (t ) = ⎨0elsewhere⎩(4.158)⎧sign[sin (12 π t / Tc )] 0 ≤ t ≤ Tcg BOC(6,1) (t ) = ⎨0elsewhere⎩(4.159)Since the pilot and data components of a signal can be formed using different spreading timeseries, and the total signal power can be divided differently between the pilot and datacomponents, many different TMBOC-based implementations are possible.The selected TMBOC implementation is a signal with 75 % power on the pilot componentand 25 % power on the data component, such that all the BOC(1,1) spreading symbols areused for the data, since data demodulation does not benefit from the higher frequencycontributions of BOC(6,1).
On the other hand, the pilot component time series comprises29/33 BOC(1,1) spreading symbols and 4/33 BOC(6,1) spreading symbols. This design placesall of the higher frequency contributions in the pilot component, providing the greatestpossible benefit to signal tracking when only the pilot channel is used to this purpose.294GP ( f ) =GBOC(1,1) ( f ) + GBOC(6,1) ( f )3333(4.160)GD ( f ) = GBOC(1,1) ( f )GMBOC(6,1,1 / 11) ( f ) =31101GP ( f ) + GD ( f ) = GBOC(1,1) ( f ) + GBOC(6,1) ( f )441111148GNSS Signal StructureFigure 4.50 next shows the BOC(6,1) spreading symbols in locations 1, 5, 7, and 30 of each33 spreading symbol locations.
This pattern will be repeated 310 times since the spreadingcode is 10230 chips long.Figure 4.50. TMBOC time representation with all BOC(6,1) Spreading symbols in the75% Pilot Power ComponentSeveral considerations affect the choice of the specific locations of the BOC(6,1) spreadingsymbols.
For example, if the BOC(6,1) chips were placed in both the pilot and datacomponents, the receiver implementation would be more simple than if these symbols wereplaced in the same locations in both. In addition, the proper placement of the BOC(6,1)symbols leads to an improvement of the spreading codes’ autocorrelation and crosscorrelationproperties of approximately 1 dB, compared to those that can be observed when onlyBOC(1,1) spreading symbols are used. The good results obtained for L1C using the BOC(6,1)locations and the performance of the spreading codes for L1C [G.W. Hein et al., 2006a]confirm the optimality of the described positions of BOC(6,1).Finally, we show in the next table the correlation losses that result from correlating aBOC(1,1) replica with different implementations of MBOC, included the finally selected forGPS, discussed above.
The definition of correlation losses was provided in (4.126):Table 4.4. Correlation losses of different MBOC implementationsPower inpilot channel50%75%SignalCBOC(6,1,1/11)TMBOC(6,1,1/11)CBOC(6,1,1/11)TMBOC(6,1,1/11)SystemGalileoGPSGalileoGPSPilot losses1/11 (0.4 dB)2/11 (0.9 dB)1/11 (0.4 dB)4/33 (0.6 dB)Data losses1/11 (0.4 dB)01/11 (0.4 dB)0After having analyzed the two possible implementations of MBOC, namely CBOC andTMBOC, we can conclude that there are no significant differences of performance betweenboth as this depends actually on the final user configuration.
Indeed, how good or bad placingall the high frequency BOC(6,1) component on both data and pilot or only on the pilot channelis depends on the particular application. In addition, from the point of view of the correlationcharacteristics, we have seen that TMBOC provides an additional improvement of 1 dB interms of reduced correlation if the BOC(6,1) component is placed at proper locations, whatfuther improves the noise input.149GNSS Signal Structure4.7.5Optimal Tracking of CBOCAs one can imagine, until the final Galileo implementation of MBOC was selected, differentCBOC implementations were under consideration. In the next lines we summarize the mainresults of the analyses carried out to optimally track CBOC each of the proposed solutions.Following the results from [O. Julien et al., 2006] we will show in this chapter the trackingperformance of the CBOC modulation when only one channel of the Galileo E1 OS (data orpilot) is used.
As we have seen above, there are many ways of implementing MBOC and evenif we would constrain our analysis to CBOC, it can be shown that the variety of solutions isstill broad. Indeed, the high frequency BOC(6,1) component could go in principle on bothdata and pilot channels, only on the pilot, only on the data, the power splits could change, andthey all would still fulfil the MBOC definition that we gave above.
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