On Generalized Signal Waveforms for Satellite Navigation (797942), страница 36
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Nevertheless, as demonstrated in(4.169), this solution presents the great inconvenient of showing a non desired cross-term thatcan only be eliminated if the other channel adopts the counterpart CBOC(6,1,1 / 11, '+') . This isindeed the case in the finally selected CBOC implementation of Galileo E1 OS. Then, thetotal performance of this CBOC implementation, seen as data and pilot together, is equivalentto that of the alternating CBOC(6,1,1 / 11, '+ / −') solution.Another studied option was to use directly the CBOC(6,1,1 / 11, '+ / −') solution on both thedata and pilot channels, in which case both channels would have the same trackingperformance.
This implementation would perfectly fulfil the MBOC spectrum too but thedrawback was be the extra complexity required for the implementation of the alternation.Equally, the option of using a CBOC(6,1, 2 / 11, '+ / −') on one channel and a pure BOC(1,1) onthe other channel would permit to have one channel with excellent tracking performance,while the other channel would just use a pure BOC(1,1) modulation. This signal would be ofinterest for some types of receivers with a philosophy behind very much in line with howTMBOC is conceived.All the figures that we have shown above rely on the assumption that a CBOC replica signalhas to be locally generated by the receiver. Indeed, CBOC is a linear combination of two subcarriers and has thus more than two levels, so that 2 bits are needed to encode the signal.While this should not be a great challenge as some works have already shown[P.G.
Mattos, 2007], we will study in the next chapter a suboptimal tracking of CBOC usinglocal replicas encoded with only 1 bit, as shown in [J.-A. Avila-Rodriguez et al., 2006c] and[O. Julien et al., 2006]. One could of course think of separating the correlation of theincoming CBOC signal with a pure BOC(1,1) replica and, on the other side, with a pureBOC(6,1) replica. Then a simple linear combination of the outputs of both correlators wouldbe sufficient to obtain the same result as if we had correlated directly with the CBOC replicaof two bits. However, this processing requires twice as many correlators as the traditionalCBOC tracking. In the next lines we present a solution that avoids to have to pay this prize[O.
Julien et al., 2006] and [J.-A. Avila-Rodriguez et al., 2006c].155GNSS Signal Structure4.7.6Suboptimal Tracking of CBOC: TMBOC likeapproachAs shown in [J.-A. Avila-Rodriguez et al., 2006c], the idea behind the 1-bit receivertechnique is to correlate the incoming CBOC signal with a locally generated signal obtained bytime-multiplexing a BOC(1,1) sub-carrier and a BOC(6,1) sub-carrier. We can model thisreplica as follows:⎧ ck (t ) sBOC (1,1) (t ), if t ∈ S1TM 61 (t ) = ⎨(4.171)⎩ck (t ) sBOC(6,1) (t ), if t ∈ S 2where S1 is the union of the segments of time when a BOC(1,1) sub-carrier is used.
Equally,S2, the complement of S1 in the time domain, is the union of the segments of time when aBOC(6,1) sub-carrier is generated. In addition, we write TM61 in order to distinguish thistechnique from the typical CBOC replica approach studied above. The advantage of thisapproach is that by using such a replica we only need to encode with 1 bit reducing thus thecomplexity of the receiver. Moreover, time-multiplexing reduces the number of correlators.Let us define α to designate the percentage of time that the BOC(6,1) sub-carrier is used inone code length, and β = 1 − α to represent the amount of time reserved for the BOC(1,1)local sub-carrier part.
Furthermore, we will assume that the sign of the BOC(6,1) local subcarrier in the local replica is taken according to the sign of the BOC(6,1) used in the CBOCsignal. This means that for the case the CBOC channel is in-phase, we have a positive signand thus the percentage of BOC(6,1) time is given by α + = 1 − β + . Equally, for the anti-phaseCBOC component the percentage of BOC(6,1) time is given by α − = 1 − β − .
It must be notedthat α + and α − can be different in general. As a result, the cross-correlation that results fromusing the TM61 replica with the different CBOC implementations of previous chapter is givenby the following expressions:()ℜ CBOC (´−´) / TM (α − ) (τ ) = β − k1 ℜ BOC (1,1) (τ ) + α − k 2 ℜ BOC (6,1) (τ ) − β − k1 + α − k 2 ℜ BOC(1,1) / BOC(6,1) (τ )61ℜ CBOC (´+´) / TM(4.172)(α + ) (τ ) = β k1 ℜ BOC(1,1) (τ ) + α k 2 ℜ BOC(6,1) (τ ) + β k1 + α k 2 ℜ BOC(1,1) / BOC(6,1) (τ )+61(++)+(4.173)1ℜCBOC(´+ / −´) / TM61 (α ) (τ ) =β − + β + k1 ℜBOC(1,1) (τ ) + α − + α + k2 ℜBOC(6,1) (τ ) + α − − α + (k2 − k1 )2[()()()](4.174)As we can see in the expressions above, the correlation functions of BOC(1,1) and BOC(6,1)are also weighted as was also the case for the CBOC replica, with the difference that theweights are in this case controlled by the factors α and β and the cross-correlation betweenTM61 and CBOC induces additional correlation losses as shown in [O.
Julien et al., 2006] and[J.-A. Avila-Rodriguez et al., 2006c].156GNSS Signal StructureIt is also interesting to see in [O. Julien et al., 2006] and [J.-A. Avila-Rodriguez et al., 2006c]that if the Dot Product (DP) discriminator is employed, different TM61 replicas could be usedfor the prompt correlator and for the early and late correlators. Indeed, for the particular caseof the DP discriminator, the prompt correlator only affects the tracking noise squaring lossesand is thus recommendable to choose a local prompt replica that minimizes the correlationloss with the incoming CBOC signal. As shown in [J.-A.
Avila-Rodriguez et al., 2006c], thiscan be achieved by selecting α = 0 or what is equivalent, by using a pure BOC(1,1) subcarrier as proposed in [J.-A. Avila-Rodriguez et al., 2006c].On the other hand, the early and late correlator outputs determine the gain of the discriminatoras well as the noise correlation between the early and late correlators’ output. Next tableshows the tracking noise degradation that results from the use of the TM61 technique for thedifferent studied CBOC cases and we compare it with the optimal TMBOC tracking for thesame BOC(6,1) power [O.
Julien et al., 2006].Table 4.5. TM61(α) Tracking Noise Degradation with respect to TMBOC in Terms ofEquivalent C/N0 for Different CBOC ConfigurationsValue of α for Early andLate TM61(α) LocalReplicasTM61(α) Tracking Noise Degradation w.r.t. TMBOC inTerms of Equivalent C/N0 [dB]CBOC(6,1,1/11,’x’) vsTMBOC(6,1,1/11)CBOC(6,1,2/11,’x’) vsTMBOC(6,1,2/11)+-+/-+/-0422.950.25.12.93.64.20.45.12.83.43.30.64.92.63.32.60.84.62.33.22.114.31.931.6As we can see, for CBOC(1/11) the most interesting values are either a high or a low value ofα, since for those cases the equivalent C/N0 degradation of the tracking noise is the lowestwith a value of 1.9 dB for the CBOC(6,1,1 / 11, '−') , of 3 dB for the CBOC(6,1,1 / 11, '+ / −') andof 4 dB for the CBOC(6,1,1 / 11, '+ ') .
It is important to note that α = 0 implies to use a pureBOC(1,1) replica and α = 1 a pure BOC(6,1) replica.In the same manner, the best option for CBOC(6,1, 2 / 11, '+ / −') and TMBOC(6,1,2/11) wouldbe to take a value of α as high as possible. This means, a pure BOC(6,1) replica for the E-Lcorrelators would be the ideal choice in this case. As a conclusion, in order to have a commonarchitecture for CBOC(6,1,1/11), CBOC(6,1,2/11) and TMBOC(6,1,2/11), the best would beto select a TM61 tracking technique that would only use pure local replicas: BOC(1,1) for theprompt correlator and BOC(6,1) for the E-L correlators. Moreover, no time-multiplexing157GNSS Signal Structurewould be needed any more, simplifying thus the complexity of the receiver.
As one canimagine, if this approach is valid for the different MBOC implementations discussed inprevious pages, it is also valid for the particular CBOC and TMBOC solutions adopted byGalileo and GPS.Another aspect that we have not touched yet is the multipath rejection of the differentdiscussed TM61 options. As shown in [O.
Julien et al., 2006] for both the CBOC(1/11) and theCBOC(2/11) cases, a value of α around 0.5 would lead to the best results. Nevertheless,values of α close to 1 would also deliver performances close to the optimum. In addition,using α = 0 is shown to be suboptimal in terms of multipath rejection since the performanceis then that of BOC(1,1) [J.-A. Avila-Rodriguez et al., 2006c].As a conclusion, it seems that a sub-optimum implementation of a CBOC-TMBOC receiverbased on a 1 bit architecture could use a pure local BOC(1,1) replica for the prompt correlatorand a pure local BOC(6,1) replica for the early and late correlators.
Moreover, we have seenthat this scheme provides with relatively low degradation in terms of code tracking noisecompared to the optimal TMBOC tracking.The preceding derivations pursued to show that processing CBOC with a1 bit receiver ispossible already today at the cost of some degradation. However, 2-bit receivers are alreadyreality as shown in [P.G. Mattos, 2007] and an optimal processing with 2 bits would be thuspreferred. This would additionally imply some superiority of CBOC with respect to theTMBOC implementation since no blanking would be needed any more.
Indeed, blanking ofthe BOC(6,1) pulses to avoid correlation losses is equivalent to reducing the equivalent codelength by a factor 29/33 of the total length (10230). It must be noted though that this supposesa minimum additional complexity.4.7.7MBOC Tracking SensitivityAs shown in [O. Julien, 2005], we can distinguish two types of tracking sensitivity:••Code Tracking Sensitivity (DLL)Carrier Tracking Sensitivity (PLL)We describe the MBOC properties regarding Code Tracking sensitivity in detail.4.7.7.1Code Tracking SensitivityFollowing the definition from [O.
Julien, 2005], the tracking sensitivity can be defined as theminimum pre-correlation Signal to Noise Ratio (SNR) that is necessary to correctly track adesired signal. To correctly track, the post-correlation SNR should be as high as possible,what can be achieved by different means. Whatever the followed approach is, the mainobjective is always to increase the correlation gain, which is the ratio between the postcorrelation SNR and pre-correlation SNR.158GNSS Signal StructureAccording to [O. Julien, 2005], the post-correlation SNRpost is shown to beSNRpost~2 PTI R 2 (ετ )=~N 0 R (0)(4.175)where• ε τ is the code delay~• R is the correlation of the filtered incoming signal with the local replica• N 0 is the noise power density••P is the power of the desired signaland TI is the coherent integration time.On the other hand, the pre-correlation SNR adopts the following form:~PR (0)SNRpre =N0βr(4.176)where β r is the pre-correlation bandwidth.Therefore the correlation gain can be expressed as [R.
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