Статья Optimal aligning of the sums of GNSS navigation signals (1141994), страница 2
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If we amplifythese signals separately, we should carry out band-pass filtering of each one before their integration for emission. Due tothe closeness of carrier frequencies, such a filtration leads toinadmissible distortions in emitted signals. These distortionsare removed by means of common signal generation from twoindependent signals followed by signal amplification in onepower amplifier. This option provides the necessary commonsignal alignment.AltBOC modulation is used in the Galileo system for emission of two independent signals over the range E5a-E5b ondifferent carrier frequencies.
The authors who recommended1.5Three-component signals of the L1 GPS band and E1-L1-E2Galileo band, which apply the interplex modulation meet theoptimality condition. Actually, as discussed in the article by E.Robeyrol et alia, in the GPS power ratio,=0 dB: 0.5dB: -3dB, i.e., the C/A signal with maximum power =0.5dB isthe unique one on its quadrature. This particular case of interplex modulation, suggested by P.
Dafesh et alia, received its ownname CASM (coherent adaptive subcarrier modulation). InGalileo, for alignment of the CBOC-signal (composite binaryoffset carrier) — which is the original version of the BOC signal— at power ratio=1:2:1, the second component PRSsignal has been singled out into a special quadrature, and thedata and pilot components of the open access signal combineon a common quadrature.58 InsideGNSS 1B0.5000.511.5LFIGURE 2Dependence LCA from B and L for interplex modulationj a nu a ry/ febru a ry 2012www.insidegnss.comAltBOC modulation describe it in the papers by L. Lestarquit etalia and G.
W. Hein et alia and listed in Additional References)as a particular method, which leads to amplitude stability ofthe leveling signal. The main principle of AltBOC modulationis not described and remains unclear.The foregoing review demonstrates the unsatisfactory statusof sum alignment theory of navigation signals in GNSS. Thevarious alignment methods do not have a common theoreticalbasis and have been developed by the designers based on anintuitive approach. Alignment principles serving as the basisof AltBOC modulation remain unclear.Synthesis of Alignment Methods Based onLCA Minimum CriterionIn the general case, the composite signal, which should bealigned, can be noted aswhere ψΣ (t) is the phase of the composite signal; S i(t),is the i t h component of the composite signalis the power of the ithcomponent, θi(t)=±1; ψi(t) is the vector angle in the complexplane with the values θi(t) = ±1 along it; and М is the numberof components in the composite signal, SΣ(t).Let us introduce the aligned composite signal, Sal(t), likethis:where Se(t) is the leveling signal which is defined by amplitudeC and phase φ(t).
Taking into account (13) and (14), LCA canbe shown thus:whereMinimum LCA along the amplitude C with fixed valueφal(t) can be found by evaluating the following equation:Hence, we obtain:Substituting (18) into (15) yields:In the general case,, and equality can bereached only when the value of x equals 0.
Hence, taking (16)into account, the minimum (19) according to φal(t) can bereached in this way:and in this case,From (20) and ( 21) it follows that the optimal alignmentmethod should keep constant the phase of the composite signal,SΣ(t) (13), at every moment and align the signal’s amplitude tothat value which is equal to the relation of the average powerof the composite signal to the average value of its amplitude.From the basic propertyand, it follows that in the general case, amplitudeof the optimally aligned composite signal is more than the average amplitude Copt of the composite signal, SΣ(t).Clearly, then, only the relative correlation between amplitudes SΣ(t) and Sal(t) is important and not the absolute value ofthe amplitude С of the aligned signal.
For this reason, we willtake on a value Copt = 1. With this proviso, we have proved avery simple, but not quite expected result: generation of theoptimally aligned composite signal is carried out by means ofa simple and well known procedure of tight restriction of thecomposite signal:where, for the complex value,Note that the aforementioned method of optimum alignment does not define the aligned composite signal Sal(t) (12)on time intervals, where SΣ(t) = 0.
The exit from this uncertainsituation can be inferred from physical reasoning. On timewww.insidegnss.com j a nu a ry/ febru a ry 2012InsideGNSS59working papersintervals where SΣ(t) = 0, outputs of all correlator multipliersof the receiver are equal to zero. Hence, on these intervals weshould configure an aligned composite signal that also wouldalso give zero contributions to correlator outputs of the receiver.For this purpose one can obviously use the aligned compositesignal, which is taken on the opposite values with equal amplitudes for equal time.We would like to note that if the earlier input minimumcriterion of LCA is added to the requirement of equality of correlator outputs of the navigation receiver, then the method ofoptimum alignment changes considerably. For example, witha numerical search method, the optimal alignment of a threecomponent sum of signals is determined to be {ψi} = {0,0,π/2}based on the criterion of minimum coefficient of losses andequivalence of correlator outputs of navigation receivers, whichleads to phase values of the composite signal {φi} = {0,π/2}.
Insuch a case, we reach minimum LCA which equals 0.25. Thisexactly corresponds to alignment with the interplex modulation method wherein the vectors’ phases of the composite signal, SΣ(t), are changed.Effect of Optimally Aligned CompositeSignal on Receiver CorrelatorsLet us calculate the average value from the product(correlation integral)where (20) and (21) are taken into account. Therefore, we seethat the optimal leveling signal, Se(t), is orthogonal to the composite signal, SΣ(t).Now let us calculate the sum of correlator outputs of navigation receiver,under the effect of the aligned composite signal in the input,Sal(t):From (25) we can see that value of correlator outputs fromnavigation receivers under the effect of the composite signalSal(t) at the input equals the sum of outputs of the same correlators under the effect of the misaligned composite signal SΣ(t)at the same input.
This condition is only exactly correct for thesum of the inputs and in the general case is not correct for theoutput of each separate correlator.Next we will consider the so-called symmetric signals withthe optimal alignment that keeps constant not only the sum ofcorrelator outputs but also the outputs of each correlator.Symmetrical Sums of Binary CompositeSignalsLet us specify the sum(13) of binary composite signals as a symmetrical one, assuming that the value set {xk} of this sum on a complex plane andfractions of time when it is contained in each x k value is symmetrical relative to the directions assumed with each compositesignal entering into the Si(t),sum.From the symmetric property of the composite signal, SΣ(t),relative to each composite signal, Si(t),, we can deducethe orthogonality of the optimal leveling signal, Se(t), with relation to each of the composite signals. Actually, the operationof hard constraint, i.e., the basis of the algorithm of optimalalignment, generates the symmetrical leveling signal from thesymmetrical composite signal, SΣ(t), and this means that theleveling signal, Se(t)= Sal(t)–SΣ(t), will be symmetrical.Hence, it follows thatwhere Q is a time independent constant.
Taking into accountthis fact, the equality (24) for symmetrical signals can berewritten as:The equality (27) can be executed in only one case, when Q= 0. This means that the optimal leveling signal, Se(t), is uncorrelated with each of the component signals for the symmetricalcomposite signals in the sum:Equality (28) for the symmetrical sums of signals can berewritten in this way:Taking into account the orthogonality of the componentsignals, Si(t),, it follows that60 InsideGNSS j a nu a ry/ febru a ry 2012www.insidegnss.comi.e., at the outputs of correlators in the case of the influence ofan optimally aligned symmetrical sum, we receive the samevalue as in the case of influence for a non-aligned sum, SΣ(t).This property of the symmetrically aligned sums of signals generally is not incident to arbitrary asymmetrically aligned sumswith signal power rescheduling at the outputs of correlatorscorresponding to components of the sum.In a later section, we consider the method of construction ofthe symmetric sums of signals (symmetrization method) fromany asymmetrical sums.The symmetrical sums of composite binary signals withoptimal alignment represent the signals with phase modulation.
Therefore, for convenience later on, we will refer to theseas multicomponent signals with phase modulation (MSPM).FIGURE 3Vector diagram of three-component symmetrical MSPMFIGURE 4Distribution of values of three-component symmetrical MSPMFIGURE 5Vector diagrams of four-component symmetrical MSPMExamples of Symmetrical Sums of ComplexBinary SignalsLet us consider the following example of a three-componentsymmetrical MSPM:The vector diagram of this signal is shown in Figure 3.The distribution of values of a three-component MSPM isshown in Figure 4. Comparing Figures 3 and 4 we can see thesymmetry of distribution of the sum which consists of threesignals, Si(t),relative to the value of each signal enteringinto the sum, SΣ(t).The values of the aligned signal are shown in Figure 4 withasterisks located on the circle of radius two.
In six cases out ofeight, these values coincide with the initial composite signal,SΣ(t).As shown in Figure 4, six values of the composite signallie on the radius circle 2 symmetrically with regard to each ofthree directions,defined by the component signals. The portion of time р whenthe composite signal takes on each of these values (i.e., theprobability value), is equal to pi = 1/8,.
Two more values in the distribution generated by values θ1 = θ2 = θ3 = ±1,with a relative part of time pi = 1/8, i = 7, 8 are equal to x7 = x8= 0. Using these values, we can find the average value of theamplitudeand average power of the composite signalHence, from (16) and (17),www.insidegnss.com Provided that SΣ(t) = 0, (x7 = x8 = 0), as noted above, thealigned signal can have any phase if its summary contribution(integral) equals zero for the time frame when SΣ(t) = 0.
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