Double estimator. A new receiver principle for tracking BOC signals (797924), страница 3
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In the followingmathematics, we maintain the convention that every waveformis analog and not quantized in amplitude and time (as it willbe in practice).In the correlator stage, a carrier digital controlled oscillator (DCO) synchronizes to the clock signal c(t) and generatesreference signals at the IF ω0 with trial phase represented asin Equation 6.www.insidegnss.com These demodulated signals are derived by successive useof multipliers. They are then integrated to various correlations.
The integrators run over a fixed time T, which can bethe same as the code period TP or an integer multiple of thiscode period.sp ring 2008InsideGNSS31double estimator∫∫∫∫∫∫∆Schematic structure of a triple-loop double-estimator (a coherent early late processing or CELP implementation) showing signal flow (red),reference and clock signals (blue), feedback and processed error signals (green), and final signal processing (gold).FIGURE 9The outputs of the integrators can be described as a set of sixcorrelations wIII[k], wIEI[k], wILI[k], wIIE[k], wIIL[k], and wQII[k] foran ever-increasing correlation count k = 1, 2, 3.
. . . The output ofeach of the integrators is sampled by the processing stage at theend of each fixed time and then the integrators are reset to zero.(Note: Count “k” is provided here for explanatory purposes andneed not be specifically recorded in any algorithm.)The values of the k’th correlations depend on the differencebetween the k’th trial phase = [k] and the true phase , thedifference between the k’th trial subcarrier delay = [k] andthe true delay , and the difference between the k’th trial codedelay = [k] and the true code delay . The I subcarrier P gateand code P gate correlation is given precisely in Equation 9 andapproximately as in Equation 10.In these equations trc( ) is a continuous triangular cosine ofperiodicity 2TS, and Λ( ), as always, is the familiar correlationfunction of a PSK-modulated signal having the same code rateas the received signal. The acceptability of these approxima32 InsideGNSS tions can be appreciated by referring back to Figure 5, wherethe cross-section view of the two-dimensional function χ( ) inthe dimension of the trial code delay is identical to PSK correlation function Λ( ).
Correspondingly, Figure 6 shows that χ() in the dimension of the trial subcarrier delay is sufficientlysimilar to a trc( ) function.The other correlations are likewise sufficiently well approximated (with an implicit dependence on k) as in Equations 11.The difference between appropriate early and late correlations creates discriminator functions according to Equation 12and Equation 13.sp ring 2008www.insidegnss.comIn these equations trs( ) is a triangular periodic discriminator function that goes through zero for = + nTS , and VΛ( ) isa discriminator function derived from early and late differencesof the Λ function and goes through zero uniquely for [k] = .The four correlations wIII[k], wQII[k], wIQI[k] and wIIQ[k] contain the information necessary to drive the three loops in abasic CELP.As shown here, in the processing stage, there is a limiter toestimate the sign of the I subcarrier P gate and code P gate correlation (which may be either positive or negative).
Expressedmathematically, this reads as in Equation 14In this implementation these error signals eϕ[k], e [k] andeτ[k] are generated from the correlations in order to steer thetrial phase , trial subcarrier delay , and trial code delay ,respectively, toward the true phase ϕ, true delay τ plus or minussome integer multiple of TS, and absolute true delay τ, respectively.Every completed correlation period T the errors may thenbe computed, notated above as an event by a unit incrementin count k. In Equations 15 the count record [k] is deliberatelyomitted here, because in the actual algorithm this count doesnot need to be recorded.The loop filters process the errors in order to incrementor decrement the trial phase , subcarrier trial delay , andcode trial delay appropriately.
These actions can be expressediteratively as in Equation 16In this CELP implementation the PLL is second order andcontrolled by two gain constants a1 and a2; the SLL is first order,controlled by a gain constant a ; and the DLL is also first order,controlled by a gain constant aτ. With increasing count and inthe realistic presence of noise these errors go to zero on average,i.e. eϕ[k] →0, eτ*[k] → 0 and eτ[k] → 0.Although the SLL and the DLL each require the other tohave converged, the values to their two emerging estimateswww.insidegnss.com are independent of each other, the first being derived from thetiming of the subcarrier component and the second from thetiming of the code component.In a final stage, the two estimates can then be linked,because the difference between them, after rounding, shouldbe an integer multiple of the sub-chip width TS, for both loopsbeing locked (converged).
Therefore, on every correlation thetwo independent estimates are combined into a single estimateaccording to Equation 4, where the ambiguity in the higheraccuracy is automatically corrected by the unambiguouslower accuracy . Provided the time jitter is not excessive, sothat constraint of Equation 5 is observed, this calculation willautomatically find the needed integer correction.GeneralizationsMore sophisticated strategies and developments have beendeveloped, staying within the basic concept, such as the following:1. Calculation Equation 4 can be performed implicitly.
Thereneed only be the two estimates, where the best = + estimate is automatically “booted” up or down by integer multiples of TS, from within the SLL, by continuous comparisonwith the rounded difference from the code estimate.2. At the cost of requiring a total of eight rather than fourcorrelations, an “incoherent DLL+SLL” may be realized inwhich a frequency-locked loop FLL replaces the PLL.3. The wide variety of different discriminator designs knownto the standard two-loop PSK correlation receivers may beadopted — including non-coherent early-late processing.4. The standard technique known as “carrier aiding” may beincorporated.5. The receiver concept readily generalizes to deal with theEuropean AltBOC concept.6.
Adaptive gate width in the DLL may be adopted to implement faster loop acquisition in low C/N0 conditions.Evaluation and Comparison with VE-VL ReceiverThe inherent property of BOC is that an estimate of delay fromthe subcarrier component alone may be “out” by an arbitraryinteger number of sub-chips. This is our SLL estimate. However,this ambiguity is of no consequence for us because we also havethe independent DLL estimate. And the key difference with ourmethod, compared to bump-jumping, is that once the loops arein lock then evaluation of the necessary integer correction tothe SLL estimate is “hard directed” and instantaneous.Our method requires no doubtful dependence on the monitoring of the amplitudes of secondary and main peaks.
In anycase, these different amplitudes no longer exist. Further, theexistence of two independent estimates offers a variety of safetychecks, because of the independence of the two estimates. Inparticular, what was otherwise shaping up to be a major problem with BOC — group delay distortion — can be expected tobe readily and automatically “calibrated out” from longer termaveraging and comparison of the two estimates.sp ring 2008InsideGNSS33double estimator3DLL error (sub-chips)SLL error (sub-chips)PLL error (rads)SLL error (sub-chips)DLL error (sub-chips)Corrected delay estimate2.522Timing ErrorTiming Error311.510.5000100200300400Loop iterationsAcquisition of two (DLL and SLL) estimates and the carrierphase (PLL)500-0.5An example computer simulation of thedouble estimating concept is shown inFigure 10 for a BOC(2f, f ) signal.
TheMATHCAD simulation assumes equalDLL and SLL bandwidths (1 Hz) andC/N0 = 30 dBHz. The initial delay offsetwas set at 2.5 sub-chips and an initialphase error to the carrier component ofπ/4. The numerical count (horizontalaxis) is of completed correlations.Under loop operation the DLL estimate (red) is seen to provide unambiguous tracking. The timing jitter is thesame as one would get for an equivalent receiver tracking PSK.
disturbance.The SLL delay estimate (blue) deliversthe lower timing jitter associated withBOC; however, this estimate is ambiguous, locking to the nearest subcarriercorrelating peak, which in this casehappened to be 2 sub-chips away fromthe truth.Figure 11 provides an example ofacquisition and tracking, now showinga corrected estimate (in green) accordingto Equation 4. The figure shows distinctly that this corrected value has the lowerjitter of the SLL estimate and the unambiguous location of the PLL estimate.A potential problem for VE-VLbump-jumping with BOC is groupdelay distortion from whatever cause .As is well known from the physics, theeffect of such a distortion on the standard correlation function ( ) is a shiftof the alternating peaks of the subcarrier component relative to the overall Λenvelope. The inevitable effect for the34 InsideGNSS 100200300400500Loop iterationsFIGURE 10Simulations0FIGURE 11Acquisition of two estimates and corrected estimateVE-VL principle is to reduce the amplitude margin between secondary peaksand the main peak, and increase the riskof false lock.The condition of group delay distortion should not however be a problem fora double estimating receiver.
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