On Generalized Signal Waveforms for Satellite Navigation (797942), страница 64
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Indeed, as we saw in chapter 4.6.1, the OS channel is in this case transmittingsignal all the time and not only 50 % of the time as was the case with the BOC(1,1) signal.The equation above can be further expressed as follows:[]E1OSE1⎧α d OS(t )cDE1OS (t ) − cPE1OS (t ) sBOC(1,1) (t ) cos(2 π f E1 t ) +⎪⎪E1OSE1(t )cDE1OS (t ) + cPE1OS (t ) sBOCsE1 (t ) = ⎨+ α μ d OS(6,1) (t ) cos(2 π f E1 t ) −⎪E1E1⎪⎩− β sPRS (t ) − γ sIM (t ) sin (2 π f E1 t )[[]](7.140)where the emission of BOC(1,1) and the BOC(6,1) signal are shown separately. As it is trivialto recognize, the two emissions are time disjoint and satisfy the requirement to be orthogonal.Another important observation is that now we have 8 phase points instead of only 6.
From aquick inspection of the phase diagram one might find analogies with the CASM and Interplexschemes that we saw above. Nonetheless, the CASM and Interplex cannot be applied directlysince we do not have binary signals any more.Finally, it is important to mention that the more signals are multiplexed in the general scheme,the more phase constellation points are needed to achieve the constant envelope. This raisessome concerns on the complexity of the signal generator and the identification of the phasepoints as shown in [A.R. Pratt and J.I.R. Owen, 2005].
Indeed, after filtering at the receiversome phase states might be difficult to distinguish if they are too close to each other.For more details on the modified interplex modulation, refer to Appendix J where all theanalytical expressions for the general CBCS modulation and CBOC are derived.293Signal Multiplex Techniques for GNSS7.7.10Interesting Aspects of the Modified InterplexIn all the modulations studied so far, the most typical case was to multiplex binary signals.Nevertheless, there are cases of interest that cannot be described by the original version of theInterplex modulation unless slight modifications are made.
In fact, CBOC has data and pilotin anti-phase. This results in an additive/subtractive combination of BOC(1,1) and BOC(6,1)that is not binary any more. Nonetheless, a careful look into the equations shows that for thesecases, the inter-modulation still remains binary and thus, except for the amplitude, it can bepredicted with the Interplex equations as shown in [G.W. Hein et al., 2005].As we saw in chapter 4.7.3, CBOC requires to form the sum and difference of the data andpilot channels. The conditions for the BOC(1,1) and BOC(6,1) can be stated as follows:E1(t ) cDE1OS (t ) − cPE1OS (t ) ≠ 0 BOC(1,1)d OSE1(t ) cDE1OS (t ) + cPE1OS (t ) ≠ 0 BOC(6,1)d OS(7.141)where the BOC(1,1) and BOC(6,1) channels are time disjoint and could be thus separatelydecoded.
The equation above is subject to further interesting interpretations. In fact, since twotime multiplexed channels are created, one could use the difference channel to carryadditional signals or services as proposed in [A.R. Pratt and J.I.R. Owen, 2005]. If we assumethat 20 % of the total signal power is on the BCS channel and 80 % on the BOC(1,1) channel,this makes a difference of 6 dB in power.
For the bit error rate not to be affected with thesepower levels, the reduced power of the difference channel BOC(6,1) could be compensated bya reduction of the data rate from 250 symbols per second to 50 as shown next:[][]s E1 (t ) = α s E1OS (t ) − s E1OS (t ) cos(2 π f E1 t ) − β s E1PRS (t ) − γ s E1IM (t ) sin (2 π f E1 t )DP(7.142)being[]E1OSE1OSE1E1(t ) cDE1OS (t ) sBOCsE1OS (t ) = d OS(1,1) (t ) + μ d Diff (t ) sBOC (6,1) (t )D[]E1OSE1OSE1sE1OS (t ) = cPE1OS (t ) sBOC(1,1) (t ) − μ d Diff (t ) sBOC (6,1) (t )P(7.143)E1E1E1(t ) cPRS(t ) sPRS(t )sE1PRS (t ) = d PRSwhich is very similar to the expressions derived in previous chapters, but with the slightE1(t ) accompaniesdifference that in this case an additional signal with information d DiffBOC(6,1).
This additional signal has no effect on the time-multiplexing as we can see, sincethe phase inversion of this data does only cause a change of the sign of the BOC(6,1) signal atevery data symbol transition of the difference channel. Finally, it must be noted that ascommented in [A.R.
Pratt and J.I.R. Owen, 2005], the presence of a data signal on thedifference channel, namely BOC(6,1), definitely has an effect on the global signalcharacteristics equalizing the average spectra, the multipath sensitivity envelope and the DLLtracking characteristics. A method for modulating data for the BOC(6,1) signal and todispread this data message have been proposed in [A.R. Pratt and J.I.R. Owen, 2005].294Signal Multiplex Techniques for GNSS7.8Intervoting (Interplex + Majority Voting)In chapter 7.4 we saw that majority voting presents interesting characteristics with respect tocurrent known multiplexing techniques, being majority voting an extremely efficienttechnique when the number of signals to multiplex increases.
On the other hand, it can beshown that while the number of signals to multiplex is not higher than three, interplex isoptimum in the sense of presenting minimum multiplex losses, especially when the powerdistribution of the different signals varies considerably. For more than three signals, however,majority voting is shown to outperform interplex.When both multiplex techniques are compared, one comes to the natural question of whetherit would not be possible to have the best of both techniques in one single multiplexing. Theanswer to this is that it is in fact possible.
Such a modulation receives the name of Intervoting(Interplex + Majority Voting). We dedicate this chapter to study its mathematical properties.7.8.1Origins of Intervoting[G. L. Cangiani et al., 2002] have proved that the interplex modulation can be furtherexploited and generalized to the intervote multiplex, where elementary majority votingtechniques are combined with the interplex modulation. Next figure shows schematically howthe intervoting modulation works.Figure 7.14. Scheme of the intervote multiplexAs we can recognize, an intervote modulator includes a majority voting logic unit and aninterplex modulator.
The majority voting logic receives as input a number of signal codeswith the commanded power ratios and delivers as output three signals that result frommajority voting the inputs. In a particular implementation, two of the original input signalsremain unchanged and are directly output, being the third output signal the majority vote ofthe rest of input signals. The majority vote signal and the other two uncombined signals arethen fed into the interplex modulator as signals s1, s2 and s3 to form the in-phase andquadrature components of the final composite signal.295Signal Multiplex Techniques for GNSS7.8.2Theory on IntervotingThe intervoting multiplex takes advantage of the positive aspects of both interplex andmajority voting while it gives a solution to the disadvantages that both present separately. Infact, while majority voting degrades its performance when the power difference between thedifferent signals is considerable (this is the so-called small signal suppression problem that wesaw in chapter 7.4.5), interplex reduces its efficiency as the number of signals increases.
Thecombination of both techniques, is capable of elegantly coping with the drawbacks of the twomultiplexing schemes, providing an efficient modulation for any number of signals and anyarbitrary power distribution.The particular intervoting multiplex that we analyze in this chapter combines five differentsignals, being three of them previously multiplexed using Majority Voting.
It is important tomention that intervote could in principle be constructed with an interplex scheme that couldaccommodate more than three signals. Moreover, the total number of signals to multiplexcould be in principle also higher. Without loss of generality, we will concentrate however inthe following on this particular case given its simplicity and optimum combination ofperformance and flexibility.As we can recognize from Figure 7.14, three of the signals are aggregated using majorityvoting and the resulting majority voted signal is then further multiplexed with the other twouncombined codes using interplex. This solution is especially optimum if the commandedpower distribution could change during operation or would adopt any arbitrary set of valuesnot necessarily fixed at the beginning.
It is important to note that for a particular powerdistribution, a particular five signal interplex solution could possibly outperform intervote.Nevertheless, intervote will result in general in an improved efficiency for the majority ofpower distributions, bringing an additional flexibility in the design.Let us assume five binary signals {c1 , c 2 , c3 , c 4 , c5 } with target commanded powerdistribution {g1 , g 2 , g 3 , g 4 , g 5 }, being the gains defined in non-decreasing order such thatg n +1 ≥ g n . As graphically depicted in Figure 7.14, each time there is a change in the targetgains of the different signals, the majority voting logic of the intervote multiplexer has todetermine which of the five signals is to be mapped [G. L.
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