On Generalized Signal Waveforms for Satellite Navigation (797942), страница 33
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This can be shown by meansof the Root Mean Square bandwidth in the next figure. For completeness also MBOC isdepicted. This modulation (baseline of the Galileo Open Service and GPS Civil signals inE1/L1) will be described in detail in the next chapter.Figure 4.47. Root Mean Square Bandwidth (RMS) of studied OS candidate signals.pMBOC refers to the pilot signal and the percentage to the amount of pilot power140GNSS Signal StructureAs we know, the Root Mean Square (RMS) bandwidth of a spreading symbol is defined by:BWRMS (β r ) =∫βr2−βrf 2 G ( f ) df(4.148)2where G ( f ) is normalized for unit power over the signal bandwidth being used, and β r isthe double-sided receiver pre-correlation bandwidth.
The RMS bandwidth can also be seen asanother way of interpreting the Cramér Rao lower bound or as the Gabor bandwidth of asignal. According to this, the higher the RMS bandwidth, the better the signal will be.If we observe now the results of Figure 4.47 above, we can clearly see that unlike CBCS*, thepotential RMS bandwidth of MBOC does not saturate for higher bandwidths. CBCS* is thephase alternating version of CBCS that we describe in the following lines.
Furthermore, notonly is MBOC by far better than BOC(1,1), but it presents also a performance comparable tothat of BOC(2,2) and even superior for some implementations. It is interesting to note that forbandwidths higher than about 14 MHz the RMS bandwidth of CBCS* does not grow anymore due to the necessary filtering, while that of MBOC does.As a conclusion, the CBCS signal candidate presented in [G.W. Hein et al., 2005] clearlyoutperformed the baseline BOC(1,1) but presented some inherent limitations that rose somedoubts.
Especially the two drawbacks explained above were reason of concern since theydemanded modification from the receiver manufacturers to get rid of these potential biases.The solution to all those problems would not take much time to come: the name was MBOCand this time, not only Galileo was eager to adopt it, but also GPS for its modernized GPS.4.7MBOC modulation definition and analysisNearly twenty months after the EU and the US signed the Agreement on the Promotion,Provision and use of Galileo and GPS Satellite-Based Navigation Systems and RelatedApplications an optimized signal waveform named MBOC (Multiplexed Binary OffsetCarrier modulation) was proposed by a common group of experts of the EU and US for GPSL1C and Galileo E1 OS [G.W.
Hein et al., 2006a], [G.W. Hein et al., 2006b] and[J.-A. Avila-Rodriguez et al., 2006d].Except for the fact that the CBCS definition requires Interplex to multiplex all the signals, theMBOC modulation can be seen a particular case of the CBCS solution where the BCSsequence adopts the known sine-phased BOC-like form. In this sense, MBOC(6,1,1/11) couldalso be expressed as CBCS([1,-1,1,-1,1,-1,1,-1,1,-1,1,-1],1,1/11) if the requirement on theInterplex Multiplexing were abandoned. The main objective of the common GPS and Galileosignal design activity was that the PSD of the proposed solution would be identical for GPS141GNSS Signal StructureL1C and Galileo E1 OS when the pilot and data components are computed together.
Thisassures a high interoperability between both signals. This normalized (unit power) powerspectral density, specified without the effect of bandlimiting filters and payloadimperfections, is given byGMBOC( 6,1,1 / 11) ( f ) =101GBOC(1,1) ( f ) + GBOC( 6,1) ( f )1111(4.149)where the high BOC frequency component, that is BOC(6,1), is shown to be:⎛ πfsin ⎜⎜⎝ fc= fc(πf )22GBOCsin (6,1)⎡ ⎛ πf ⎞ ⎛ πf ⎞ ⎤⎞⎟⎟⎟⎟ ⎥⎢ sin ⎜⎜ ⎟⎟ sin ⎜⎜⎠ tan 2 ⎛⎜ πf ⎞⎟ = f ⎢ ⎝ f c ⎠ ⎝ 12 f c ⎠ ⎥c⎜ 12 f ⎟⎢⎛ πf ⎞ ⎥c ⎠⎝⎟⎟ ⎥⎢ πf cos⎜⎜⎝ 12 f c ⎠ ⎦⎥⎣⎢2(4.150)with fc=1.023 MHz. Equally, the low BOC frequency component, namely BOC(1,1) will be:⎛ πfsin ⎜⎜⎝ fc= fc(πf )22GBOCsin (1,1)⎡ ⎛ πf ⎞ ⎛ πf ⎞ ⎤⎞⎟⎟⎟⎟ ⎥⎢ sin ⎜⎜ ⎟⎟ sin ⎜⎜⎠ tan 2 ⎛⎜ πf ⎞⎟ = f ⎢ ⎝ f c ⎠ ⎝ 2 f c ⎠ ⎥c⎜2f ⎟⎢⎛ πf ⎞ ⎥⎝ c⎠⎟⎟ ⎥⎢ πf cos⎜⎜⎢⎣⎝ 2 f c ⎠ ⎥⎦2(4.151)and thus:GMBOC( 6,1,1 / 11) ( f ) =101GBOC(1,1) ( f ) + GBOC( 6,1) ( f ) =111122⎡ ⎛ πf ⎞ ⎛ πf ⎞ ⎤⎡ ⎛ πf ⎞ ⎛ πf ⎞ ⎤⎟⎥⎟⎟ ⎥⎢ sin ⎜ ⎟ sin ⎜⎢ sin ⎜⎜ ⎟⎟ sin ⎜⎜10 ⎢ ⎝ f c ⎠ ⎝ 2 f c ⎠ ⎥1 ⎢ ⎜⎝ f c ⎟⎠ ⎜⎝ 12 f c ⎟⎠ ⎥fc==+ fc11 ⎢⎛ πf ⎞ ⎥⎛ πf ⎞ ⎥ 11 ⎢⎟⎟ ⎥⎟⎟ ⎥⎢ πf cos⎜⎜⎢ πf cos⎜⎜⎢⎣⎢⎣⎝ 12 f c ⎠ ⎥⎦⎝ 2 f c ⎠ ⎥⎦⎛ πf ⎞ ⎡⎛ πf ⎞⎛ πf ⎞⎤fc⎟⎟ + tan 2 ⎜⎜⎟⎟⎥sin 2 ⎜⎜ ⎟⎟ ⎢10 tan 2 ⎜⎜=2 211π f⎝ fc ⎠ ⎣⎝ 2 fc ⎠⎝ 12 f c ⎠⎦(4.152)Additional conclusions can be drawn from analyzing the spectral shape of MBOC.
Indeed, asshown in [J.-A. Avila-Rodriguez et al., 2006b], an interesting interpretation of (4.141) and(4.148) is that an ideal power spectral density regarding the tracking performance should beinversely proportional to the square of the frequency, according to:1(4.153)G (f )∝ 2fIt must be noted that such a spectrum would have nevertheless other non desirable propertieswith regards to its implementation.
However, if we look now at the envelope of the wellknown power spectral densities of BPSK(1) and BOC(1,1) we can clearly recognize that theirenvelopes interestingly decay with 1 f 2 , as Figure 4.48 shows next. Moreover, MBOCseems to follow pretty well this desirable pattern too. In fact, this was one of the figures in themind of all those people involved in the optimization of the Galileo OS in E1.142GNSS Signal StructureFigure 4.48. Decay of the envelopes of the power spectral densities of BOC(1,1) – in blue– and BPSK(1) – in black. As it can be clearly seen, the selected MBOC signal for GPSL1C and Galileo E1 OS – in red – follows a similar patternWe show in the next figure all the Galileo and GPS signals in the E1/L1-band.Figure 4.49. GPS and Galileo Spectra in E1/L14.7.1Implementing MBOCOnce we have defined the power spectral density of MBOC, it is the right moment to talkabout the implementation.
Indeed, different time representations result in the same powerspectral density and the Agreement between the EU and the US on MBOC left this freedom toboth parties so that each could implement its own solution according to its own conception.Two solutions have been realized to implement MBOC:143GNSS Signal Structure••CBOC: The Composite BOC is the solution adopted by Galileo for the Open Servicein E1/L1. It is an Interplex multiplexing where the sub-carriers BOC(1,1) andBOC(6,1) are added in anti-phase on each channel.TMBOC: The Time-Multiplexed BOC is the solution adopted by GPS for L1C.
It is abinary signal where BOC(1,1) and BOC(6,1) are time-multiplexed according to a preestablished pattern that was optimized to improve the correlation properties of thesignal when the effect of the PRN code is taken into account.We describe the two possible implementations in detail in the next chapters. Before that, it isinteresting to mention that between CBCS and the final MBOC(6,1) there was an intermediatesolution that was object of interest for a short period of time. This was the so calledMBOC(4,1) that was shortly described in chapter 3.6.4.One final but important comment is related to the power allocated on the high frequencycomponent of MBOC, namely BOC(6,1). Indeed, the 1/11 of power refers to the power atgeneration, without accounting for the effect of the satellite’s filter and other imperfections.This is so because as we know, MBOC admits different implementations, being one of themTMBOC.
If we would define the exact power split at user level, the power at generationwould be different depending on the final implementation. As we have seen, TMBOCaccomplishes the required power percentage by time-multiplexing BOC(6,1) and BOC(1,1) sothat 1/11 of the time the satellite transmits BOC(6,1) and the rest of the time BOC(1,1). Sincethe GPS L1C codes have a length of 10,230 chips as we saw in chapter 2.3.2.1, thispercentage incorporates the factorial decomposition of 10,230. It is trivial to show that10,230 = 2 x 3 x 5 x 11 x 31 and indeed 11 was found to be the optimum number to divide thetransmission periods of the multiplexing signals of GPS L1C. Indeed, 1/11 of the time waslong enough to considerably improve the performance with respect to BOC(1,1) but not solong to concentrate too much power on the high frequencies and overlap the M-Code and PRSto non acceptable levels.
More details on the exact location of the BOC(6,1) chips are given inchapter 4.7.4 of this chapter.Finally, it is important to mention that in the case of CBOC, the generation of power presentsno limitations since this is achieved by correspondingly modulating the amplitude asdescribed in (4.121).4.7.2On MBOC and Antisymmetric sequencesBefore we describe the performance of MBOC regarding the characteristics and details of itsdifferent implementations, let us first make some final comments on the MBOC spectrum andthe ideas behind.As we have seen some lines above, the Galileo MBOC implementation (CBOC) is the resultof an additive and subtractive mixture of two separate spreading symbols.
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