Advanced global navigation satellite system receiver design (797918), страница 19

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Firstly addressing the outercarrier loop, we are considering a coherent DE BOC receiver, therefore implyingphase tracking with a PLL. The carrier phase error is commonly determined using aCostas decision-directed discriminator, which can be applied to the DE BOC receiveras follows.()()()eφ = wQII × sgn (wIII ) ≈ A × sin φ − φˆ × sqc φ − φˆ × trc τˆ* − τ × Λ (τˆ − τ )6-9sqc( ) is a square wave cosine function. Incorporating the sign correction by hardlimiting the wIII correlation in the discriminator allows the PLL to function with noknowledge of the navigational data state. It can be seen that eφ tends to zero as()sin φ − φˆ tends to zero, which occurs at integer n multiples of π radians asφˆ → φ + nπ . The most common Costas PLL discriminators for the DE BOC receiverare shown in Table 6-1.126BOC tracking with double estimation receiverTable 6-1, Costas PLL discriminators for the DE BOC receiverDiscriminatorDecision-correctedError signaleφ = wQII × sgn (wIII )()()()≈ A × sin φ − φˆ × sqc φ − φˆ × trc τˆ* − τ × Λ (τˆ − τ )Dot product≈NormalisedSlope proportionaleφ = wQII × wIII2[ ()]A× sin 2 × φ − φˆ × trc 2 τˆ* − τ × Λ2 (τˆ − τ )2eφ = wQII / wIII(()(≈ φ − φˆto A)≈ tan φ − φˆeφ = arctan (wQII / wIII )2 quadrant arctangentDependencySlope proportionalto A2Slope independentof amplitudeSlope independentof amplitude)The sub-carrier phase error for the coherent DE BOC receiver can also determinedusing a decision-directed discriminator as follows.()()()eτ * = wIQI × sgn (wIII ) ≈ A × cos φ − φˆ × Trs τˆ* − τ × sqc τˆ* − τ × Λ (τˆ − τ )(6-10)It can be seen that eτ * tends to zero as Trs τˆ* − τ tends to zero, which occurs atinteger n multiples of sub-chips as τˆ* → τ + n × TS .

Alternative coherent SLLdiscriminators for the DE BOC receiver are shown in Table 6-2.Table 6-2, SLL discriminators for the DE BOC receiverDiscriminatorDecisioncorrectedError signaleτ * = wIQI × sgn (wIII )()(Dot product()eτ * = wIQI × wIII(Normalised)≈ A × cos φ − φˆ × Trs τˆ* − τ × sqc τˆ* − τ × Λ (τˆ − τ ))()()≈ A2 × cos 2 φ − φˆ × Trs τˆ* − τ × trc τˆ* − τ × Λ2 (τˆ − τ )eτ * = wIQI wIII≈(())Trs τˆ* − τ= Trt τˆ* − τ*trc τˆ − τ()DependencySlopeproportionalto ASlopeproportional2to ASlopeindependent ofamplitudeThe code phase error for the coherent DE BOC receiver can also determined using adecision-directed discriminator as follows.127BOC tracking with double estimation receiver()()eτ = wIIQ × sgn (wIII ) ≈ A × cos φ − φˆ × trc τˆ* − τ × VΛ (τˆ − τ )6-11It can be seen that eτ tends to zero as VΛ(τˆ − τ ) tends to zero, which occurs asτˆ → τ .

Alternative coherent DLL discriminators for the DE BOC receiver are shownin Table 6-3.Table 6-3, Coherent DLL discriminators for the DE BOC receiverDiscriminatorDecisioncorrectedError signaleτ = wIIQ × sgn (wIII )()()≈ A × cos φ − φˆ × trc τˆ* − τ × VΛ (τˆ − τ )eτ = wIIQ × wIIIDot product()()≈ A2 × cos 2 φ − φˆ × trc 2 τˆ* − τ × VΛ (τˆ − τ ) × Λ (τˆ − τ )22eτ = wIIE − wIILPowerDependencySlopeproportionalto ASlopeproportionalto ASlopeproportional TT≈ A2 × cos 2 φ − φˆ × trc 2 τˆ* − τ ×  Λ2 τˆ − DC − τ  − Λ2 τˆ + DC − τ  22 ()()to AFollowing common practice we update the carrier phase estimate with a second orderloop as follows.fφ ← fφ + eφ6-12φˆ ← φˆ + k1 × fφ + k 2 × eφfφ is the integrated phase error and k1 and k2 are loop gains which can be adjusted bythe designer.

The SLL and DLL loop are updated with independent first order loopsfor the sub-carrier and code delay estimates respectively with an appropriate carrieraiding term as follows.τˆ = τˆ + kφ × fφ + kτ × eτ*26-13*τˆ = τˆ + kφ * × fφ + kτ * × eτ *1282BOC tracking with double estimation receiverkφ and kφ * are constants calculated to provide the necessary open loop correction ofDoppler shift appropriately scaled down to the code rate and sub-carrier raterespectively. kτ and kτ * are a loop gains which can be adjusted by the designer.Assuming the loops are in lock the data estimate, d̂ for a coherent DE BOC receiveris determined as follows.dˆ = sgn (wIII )6-14An example acquisition of the DE BOC receiver is shown in Figure 6-6 for aBOC(2×fC, fC) signal, derived from Mathcad simulation with equal DLL and SLLloop bandwidths, BDLL = BSLL = 1 Hz and noise, C/N0 = 30 dB-Hz.

The initial delayoffset is set at 2.5 sub-chips and initial phase error of π/4.3DLL error (sub-chips)SLL error (sub-chips)PLL error (rads)32tT k−tRC ktT k−tRS kφ C−φR k 10100200300400500− 0.50k500Loop iterationsFigure 6-6, Example acquisition of the DE BOC receiver (BDLL = BSLL = 1 Hz , C/N0 = 30 dB-Hz)Under loop operations the DLL delay estimate provides unambiguous tracking withtiming jitter equivalent to that of the underlying PSK modulation (see Equation 4–16).The SLL delay estimate τˆ* delivers the full tracking accuracy of the BOC modulationgiven by Equation 4–20, it is however ambiguous, locking to integer sub-chip values.129BOC tracking with double estimation receiverOnce the loops have settled the ambiguity of the SLL estimate can be resolvedthrough the noisier but unambiguous DLL estimate as follows. τˆ* − τˆ +* × TSˆˆτ = τ − roundT S 6-15τˆ + is the corrected delay estimate.

Figure 6-7 shows an example acquisition with thecorrected delay estimate.3SLL error (sub-chips)DLL error (sub-chips)Corrected delay estimate2.5842.52tT k− tRS k1.5tT k− tRC ktT k− τB k10.50100200300400500− 0.150.50k500Loop iterationsFigure 6-7, Corrected delay estimate of the DE BOC receiver(BDLL = BSLL = 1 Hz , C/N0 = 30 dB-Hz)6.2The incoherent BOC double estimatorIncoherent systems are commonly used to deliver robust acquisition and tracking ofGNSS signals, particularly in weak signal environments. An incoherent system lockson to the frequency of the incoming carrier and not the phase.

This is achieved bymaintaining a constant or near constant carrier phase difference across the correlationinterval. The concept of double estimation, generating sub-carrier and codewaveforms with independent timing estimates can still be applied. A general130BOC tracking with double estimation receiverschematic of the DE BOC receiver is shown in Figure 6-8. Two additionalcorrelations are required to enable all three loops to operate with no carrier phasedependence.∫••∫∫^uBOC(t)•cos(ω0t + φ (t))^s(t − τ *)•^•sin(ω0t + φ (t))Carrier~•wIIQ~^a(t − τ )wQIICode∫•∫∫wQQIPROCESSINGwQIQeτeτ∗eφSIGNAL^sub-Carrier•wIQIa(t − τ )••^s(t − τ *)wIIIFigure 6-8, Incoherent DE BOC receiverAfter mixing and integration over time T the additional two correlations required foran incoherent DE BOC receiver can be written as follows.T() (wQIQ1= ∫ u BOC (t ) × sin ω0t + φˆ × s t − τˆ* × a~ (t − τˆ )dtT0wQQI) ( )( ) ( )≈ A × sin (φ − φˆ )× Trs (τˆ − τ )× Λ (τˆ − τ ) × d)6-16(≈ A × sin φ − φˆ × trc τˆ* − τ × VΛ (τˆ − τ ) × dT1= ∫ u BOC (t ) × sin ω0t + φˆ × ~s t − τˆ* × a (t − τˆ )dtT0*The rate of change of carrier phase can be determined by comparing wIII and wQII withthe correlation results from the next epoch wIII′ and wQII′.

The frequency error can bedetermined using the cross-product discriminator as follows.131BOC tracking with double estimation receiver()()′ × wIII − w′III × wQII ≈ A2 × sin ∆φˆ − ∆φ × trc 2 τˆ* − τ × Λ2 (τˆ − τ ) × d × d ′eω = wQII6-17Where the phase difference and estimated phase difference between epochs is∆φ = φ ′ − φ6-18∆φˆ = φˆ′ − φˆIt can be seen that eω tends to zero with ∆φˆ − ∆φ .

Incoherent SLL tracking can beachieved by expanding the decision directed discriminator as follows.eτ * = wIQI × sgn (wIII ) + wQQI × sgn (wQII )( ( )())(6-19)()≈ A × cos φ − φˆ + sin φ − φˆ × Trs τˆ − τ × sqc τˆ − τ × Λ (τˆ − τ )*(*)It can be seen that eτ * tends to zero as Trs τˆ* − τ tends to zero, which occurs atinteger n multiples of sub-chips as τˆ* → τ + n × TS . This is achieved while allowingfor an arbitrary offset between φ and φˆ . Incoherent DLL tracking can be achievedby expanding the decision directed discriminator as follows.eτ = wIIQ × sgn (wIII ) × wQIQ × sgn (wQII )( ( )())(6-20)≈ A × cos φ − φˆ + sin φ − φˆ × trc τˆ* − τ × VΛ (τˆ − τ )It can be seen that eτ tends to zero as VΛ(τˆ − τ ) tends to zero, which occurs asτˆ → τ .For updating the loops we assume second-order FLL tracking and first order SLL andDLL tracking with appropriate carrier Doppler aiding as follows.132BOC tracking with double estimation receivereφ ← eφ + eω6-21fφ ← fφ + eφφˆ ← φˆ + k1 × fφ + k2 × eφFLL updateτˆ* ← τˆ* + kφ * × fφ + kτ * × eτ *τˆ ← τˆ + kφ × fφ + kτ × eτSLL updateDLL updateeφ is the estimated phase error derived by the integrating frequency error eω , k1 andk2 are loop gains which can be adjusted by the designer.

Figure 6-9 shows theconvergence of all three of the DE loops with an initial frequency error of 1 radian percode epoch and a two sub-chip error on the SLL estimate. The DLL estimate is thenused to correct the SLL estimate in exactly the same way as the coherent DE BOCreceiver (Equation 6-15).2.52.4912φR k− φR k− 1tT k− tRS k1.51tT k− tRC k0.50−350100150200250− 1.933×100.51k250Carrier phase change (radians/epoch)SLL error (sub-chips)DLL error (sub-chips)Figure 6-9, Example incoherent acquisition6.3∆φ = 1 radThe DE AltBOC receiverThe extension of DE principle to Alternate BOC (AltBOC) signals provided by DrMS Hodgart requires little change to the theory of the DE BOC receiver.

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