Optimal aligning of the sums of GNSS navigation signals (797943), страница 4
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The preferred relationship should be the firstphase distribution, {ψi} = {0, π/4, π/2, 3π/4}, as it is symmetrical and the loss coefficient, η1=0.1464, corresponding to thisarrangement is insignificantly less than the absolute minimumη2=0.1432.Synthesis of AltBOC SignalAltBOC modulation was developed for the transmission of twoindependent pairs of orthogonal binary signalswhere θ1(t) = θ11(t) + jθ12(t) and θ2(t) = θ21(t) + jθ22(t) are complexbinary signals with two quadratures, θ11(t), θ12(t), θ21(t), θ22(t)taking the value ±1, emitted on the different, but nearby carrier frequencies ω1, ω2, (ω1 < ω2), through the common antenna.Given that θ1(t), θ2(t) are binary, their phases take the values,(2k + 1) .
π/4, k = .Let us consider the optimal LCA minimum AltBOC-likesignal as a generalization of the optimal four-componentMSPM signal considered in previous sections with η=0.1464.It is not difficult to ascertain that this coincides withwithin the substitution θ11 = θ1, θ12 = θ3, θ21 = θ2, and θ22 = θ4.If we representwhere a clear connection ki with θi1, θi2, is defined by Table 1,θi11–11–1θi211–1–1ki0132TABLE 1Optimal Phases for Multi-Component SignalSums Using Minimum LCA CriterionBy the method of numerical search and also using numericalsorting of all phases ψi, we found the optimal value of the phases for three- and four-component sums of the signals providingthe minimum value of LCA in the course of optimal alignment64 InsideGNSS Connection ki with θi1, θi2SΣ(t) can be presented asFor generation of an AltBOC-like signal, the componentsθ1(t) and θ2(t) should be shifted on the frequencies ω1 and ω2,i.e., the signal becomes:j a nu a ry/ febru a ry 2012www.insidegnss.comTotal phase does not influence amplitude distribution,therefore it is naturally accepted as equal to δ+(t) = (ω2 + ω1) .t/2.
In this connection, δ2,1(t) = (ω2 + ω1) . t/2 + δ–(t).Strict restriction of a summary signal leads to an expressionfor the aligned signal depending on discrete parameters θ1 andθ2 (through k– and k+) and the step number (discrete time, ).FIGURE 10Stepped approximation of phase δ–(t)where cr(x) = sign(cos(x)).The AltBOC signal presentation given by the EuropeanGNSS Open Service Signal in Space Interface Control Document (OS SIS ICD) generalizes the case of nonzero frequencyby means of the following expression:where δi(t) are approximations of the linearly varying phase,ωi(t), of frequency shift, which we will soon choose. Removingthe average geometrical of summands from the square brackets, we now have:From (42) it follows that the phase set number (one out ofeight) is determined by the expressionThe restriction on the set of frequencies f1 and f2 arises fromthe obvious requirement of the integer number of steps, ri, onthe length τi of the symbols of code sequences θ1 and θ2where k+ = k1 + k2, k- = k2 – k1, δ+(t) = (δ2(t) + δ1(t))/2, δ–(t) = δ2(t)– δ1(t))/2.
The value of signal amplitude is equal toIntegration of conditions (41) and (44) sets the restrictionon the choice of frequency differenceTaking into account that |cos(x)| has the period π, the summandunder a cosine can be considered modulo π and wecan suppose that k– = mod(k2, – k1,4), that is takes values 0…3.For equally probable values, θij(t), i, j =, the distribution ofk1 and k2, is obviously uniform in. It is not difficult to makesure that modulo k– are also equiprobable. Summation with anarbitrary constant maintains a probability distribution that isequiprobable.This implies that, if δ–(t) is divisible by π/4, then the probability distribution of the modulo 4 cosine argument does notchange.
Distribution of |SΣ(t)| at that point remains the sameas well, i.e., the optimal value LCA = 0.1464. This is why k– hasfour equally probable values 0, 1, 2, 3. In other words, we shouldaccept the step approximation of phase δ–(t) with step valuesequal to π/4 (Figure 10):Having referred to a particular case of AltBOC signal realized in a signal of frequency band E5 for Galileo, where f2 + –f1= 15f b, τ1 = τ2 = 15f b = 1/10f b, we determine that f2 – f1 = 30f b, f2+ f1 = 0, h = 1/4(f2 – f1) = 1/20f b. The condition in (45) is obviously fulfilled, and r1 = r2 = 12.Expression (42) defines the phase value state.
Comparisonof k values for all θ1(t), θ2(t), and to Table 6 in the Galileo OSSIS ICD —republished here as Table 2 — demonstrates theirfull coincidence. This shows that the E5 Galileo signal can beconsidered as a particular case of the aligned four-componentsignal.For the prospective signals L3 and L5 in the GLONASSsystem, we propose frequencies that are equal to 1175 f b and1150 f b (f b = 1.023 MHz), respectively, and we also apply twocomponent signals with symbol duration of ranging code τ =1/10f b. The application of the AltBOC signal with its symmetrical subcarriers is assumed to generate the carrier on f0=1162.5f b frequency. Such a value is inconvenient for the frequencysynthesizer and gives rise to increasing phase noise within thatsystem element.where the step height is defined with difference frequency f– =f2 – f1 – (ω2 – ω1)/2πwww.insidegnss.com j a nu a ry/ febru a ry 2012InsideGNSS65working papersInput QuadrupleseE5a-1-1-1-1-1-1-1-1-111111111eE5b-1-1-1-1-11111-1-1-1-11111eE5a-Q-1-111-1-111-1-111-1-111eE5b-Q-11-11-11-11-11-1111-11t' = t' modolo Ts,E5iTst'0[0, Ts,E5 /8[k according to sE5(t) = exp(jkπ/r)54436312657278811[Ts,E5 /8, 2 Ts,E5 /8[54832312657674812[2 Ts,E5 /8, 3 Ts,E5 /8[14872312653634853[3 Ts,E5 /8, 4 Ts,E5 /8[18872316253634454[4 Ts,E5 /8, 5 Ts,E5 /8[18872756213634455[5 Ts,E5 /8, 6 Ts,E5 /8[18476756213238456[6 Ts,E5/8, 7 Ts,E5/8[58436756217278417[7 Ts,E5 /8, Ts,E5[5443675261727881TABLE 2.Look-up table for AltBOC phase states (Table 6 from Galileo OS SIS ICD)The value f 0 =1160 f b, where f1=–10f b and f2=15f b is moreacceptable.
For such a signal, f2 – f1 = 25f b, τ1 = τ2 = 1/10f b, andf2 + f1 = 5f b. The condition (45) is carried out at r1 – r2 = 10.The concept of GLONASS system development providesfor signal integration which has BOC(1,1) modulation on thefrequency range L1 GPS 1540 f b =1575.42 MHz.It is reasonable to integrate the aforementioned signal withthe BOC(5, 2.5) signal in the L1 range of the GLONASS system,which can be considered as a two-component code signal onfrequency 1565 f b with τ2 =10/f b.Let us suppose a signal on the 1540 f b frequency has twocomponents (pilot and data) with τ1 = 1/4f b and equipotentsignal with L1 GLONASS signal on 1565 f b frequency. Expression (45) will then have the following result: 25f b=4f br1/4,25f b=10f br2/4 and meet the requirements if r1=25 and r2=10.
Ifwe choose f0 = 1550 f b, then f1=-10 f b, f2=15 f b, and f+=5f b.We should remind the reader that all considered variants ofsignal integration have the loss coefficient on alignment (LCA)η =0.1464, the lowest possible coefficient for the sum comprisedof four signals with equal power.restriction of the power at the correlators’ outputs for signalspertaining to one quadrature is far less than for the signalspertaining to the other quadratures, i.e., the third signal alwaystakes up a quadrature with the first and second signals, whichgives a lower response at the output of the corresponding correlator.
Clearly however, if we increase α we can achieve anypower ratios at the correlators’ outputs, for example they canbe equal.Definition of the corresponding value of α is reduced to theanalysis of the correlators’ outputs q1, q2, q3 under the action ofa strictly limited signal with single amplitude:Setting the outputs equal to each other, q1=q2=q3, gives usthe equation forAlignment of Three-Component SignalAlignment of the three-component signal is necessary for boththe L1 GPS signal and the E6 and E1-L1-E2 Galileo signals. Letus specifically consider the opportunities of alignment for thethree-component signal, which differ from the symmetrizationmethod.
These are based on application of signal type: SΣ(t) =θi(t) + jθ2(t) + αθ3(t)ejδ(t), where the δ(t) equiprobably accepts 0and π/2 values. Contrary to interplex modulation, the complexvalued vector of signal αθ3(t)ejδ(t) interchangeably takes quadratures with signals θ1(t) or θ2(t). Therefore, the optimal alignmentkeeps the signal powers equal, θ1(t) and θ2(t), in the outputs ofthe respective correlators.The correlator’s output for the third signal depends on α.As already mentioned, for equal signals, α1 = α2 = α3, because66 InsideGNSS At this value, αe outputs of all three correlators are equipotent.For definition of the corresponding LCA, we must find thevalue of the amplitudes and of their average value, the powerand make use of the general formula (22).
In thisregard we obtain:Substituting αe ≈ 1.104 gives the value η=0.136, which isslightly worse than η=0.1273, which is achieved by symme-j a nu a ry/ febru a ry 2012www.insidegnss.comtrization, but essentially better than η=0.25, as in interplexmodulation.A function selection δ(t) remains to be concretized. Twopossible selections are obvious. The first one appears to be aconvenient alternative (or expansion) of the AltBOC-like signalfor generation of the double frequency three-component signal.For this purpose, we should choosewhich is equivalent to the phase step approximation +2π f tof the third signal with frequency shift f = ±1/4h. As a resultof such a selection for δ(t) we receive the integration of threebinary phase signals (BPSK), with one of these signals shiftedrelative to the two others on frequency, f = ±1/4h.
From theuser’s point of view, these three signals will have equal power.The second selection of δ(t) can be used if the frequencyshift of the third signal is unacceptable. In this casewhere sr(x)=sign(sin(x)) and δ(t) takes the value 0 or π/2 inalternating fashion. In this case, the third signal becomes thequadrature phase signal (QPSK), but the first and the secondones remain the usual BPSK signals.ConclusionThe theory of optimal alignment of the GNSS navigationsignals sum is developing. A review of applicable alignmentmethods is presented. Alignment methods for the synthesis ofsummarized signals on the LCA minimum criterion is developed.
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