Advanced global navigation satellite system receiver design (797918), страница 10

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 10)

The CramérRao lower bound is commonly used to describe the theoretical performance limit of atime of arrival estimator. Defining a code loop bandwidth, BL , carrier power C andnoise density N0, the Cramér-Rao lower bound states the variance of the code trackingerror can be written as follows [Betz and Kolodziejski 2000].BL2Σ CR =(2π ) 2CNO4–8βR / 2−∫β fR2S ( f )df/2β R is the receiver front-end bandwidth, S(f) is the normalised power spectral densityof the GNSS signal and the square root of the integral is known as the r.m.s. or Gaborbandwidth.Therefore, substituting from Equation 3–7 the code tracking jitter of a PSK signalnormalised to its chip width TC, can be written asσ CR2Σ=  CR TCBL4–9β /2(2π ) 2RC3TC ∫ f 2 sin c 2 (πfTC )dfNO−βR / 2BL=2 β R TC≈2 =BLC2bNOC[1 − sin c(πβ RTC )]NOfor b > 1where b = β R × TC is the normalised receiver front-end bandwidth.The theory given by Cramér-Rao is only a lower bound and strictly applies only in thelimit of an infinite signal to noise.

Therefore, a more robust analysis is required for58PSK and BOC signalstrue approximation of timing jitter across various noise levels [Betz and Kolodziejski2000].Precise understanding of the potential timing accuracy of GNSS systems can beachieved using a simple equivalent filter model of the GNSS receiver. Timingrecovery in a GNSS navigation system can be reduced down to the problem oflocating a pulse in the presence of additive noise.

Here we show how this approachcan be used to produce models of timing accuracy equivalent to those derived usingcomplex loop analysis.Dr MS Hodgart suggested simplifying the problem by assuming carrier demodulationand considering a single channel base-band model. The simplified base-band PSKreceived signal can be written as follows.xPSK (t ) = A ×The ‘(t − τ ) + n(t )4–10’symbol represents a pulse equivalent to one chip of the PRN code sequencewith unknown time delay of τ.

n(t ) is additive noise assumed to be white andGaussian.We model the receiver timing recovery by an ideally rectangular matched filter and adelay line subtractor as shown in Figure 4-9. The output of the delay line subtractorcan be written as follows.z (t ) = A × VΛ (t − τ ) + w(t )4–11The ‘VΛ’ symbol is equivalent to the coherent (assuming perfect carrier demodulation)discriminator created by the subtraction of early and late signals in a GNSS receiver(see Figure 4-5). The time delay TD is equivalent to the spacing in time between theearly and late signals, limited to 0 < TD ≤ TC .59PSK and BOC signalsx(t ) = A×+(t − τ ) + n(t )y(t ) = A× Λ(t −τ ) + v(t )Filterh(t)++z(t ) = A× VΛ (t −τ ) + w(t )–DelayTDn(t)Figure 4-9, Timing recovery model of PSK GNSS receiverThe linear zero crossing of the discriminator characteristic represents the correcttiming of the received pulse.

Additive noise bounded within the tracking range of thediscriminator (±TD/2) converts directly to a timing error.Figure 4-10 shows representations of the function realised by the timing recoverymodel.τATCxPSK (t ) = A ×(t − τ ) + n(t )Ay (t ) = A × Λ (t − τ ) + v(t )Az (t ) = A × Λ (t − τ ) + w(t )VTDFigure 4-10, Representations of the PSK timing model functions60PSK and BOC signalsA noise sample w, can be converted to a timing error asε=where dzdt=wdzdt4–122× Ais the slope of the discriminator characteristic at the zeroTCcrossing.

The mean square timing jitter is then2Σ = ε2=w24–13(dz dt )2The mean square noise output from the filter isw2 =η[h(t ) − h(t − T )] dt2∫2D=η ×4–14TD2TCwhere η is the one sided white noise density. Now substituting into Equation 4–13we have the mean square jitter on a single pulseΣ2 =1 TD×4 A24–15ηIn practical systems the noise is reduced by averaging a great number of pulses overintegration time T.

We define an averaging loop with loop bandwidth BL = 1 (2 × T ) ,input carrier power C and input noise density N0. The mean square timing jitter forPSK normalised to the chip width can then be written asσ PSK2Σ=  PSK TC2B × ∆ PSK = L2 × C N04–1661PSK and BOC signalswhere ∆ PSK = TD TC is the normalised early-late discriminator spacing width. Thisequation corresponds to the standard equation for timing jitter for coherent earlyminus late discriminators given in [Parkinson and Spilker 1996] and [Ries et al 2002].In [Betz and Kolodziejski 2000] extensive analysis of GPS code tracking accuracy,which has been widely accepted by the scientific community is given covering anydegree of band-limiting of the signal. PSK receivers are categorised into three distinctgroups, those who are limited by the receiver’s early-late spacing (spacing limited),those who are limited by the receiver’s front-end bandwidth, (bandwidth limited) andthose who are in transition between (transition).

Figure 4-11 shows the three groupsas a function of normalised receiver front-end bandwidth and early-late spacing.Figure 4-11, PSK receiver groupsThe analysis of coherent early-late discriminator produced the following timing jitterequations for the three groups.62PSK and BOC signalsfor π ≤ ∆ × bBL × ∆2× C N04–17(spacing limited)1BLb 1   for 1 < ∆ × b < π× +∆ −  (transition )b  2 × C N 0  b π − 1 for ∆ × b ≤ 1BL1×(bandwidth limited)2 × C N0 b2σ PSK =Equation 4–17 is equivalent to the timing analysis given here for the spacing limitedgroup which is true for ∆ × b ≥ π .

Note also that for bandwidth limited receiversEquation 4–17 corresponds to the timing jitter predicted by the Cramér-Rao lowerbound in Equation 4–9. This indicates that there is no benefit to reducing the earlylate spacing beyond the reciprocal of the front-end bandwidth.4.4Theoretical timing measurement of BOC modulated signalsThe theoretical timing accuracy of a BOC GNSS system can be evaluated in preciselythe same manner as shown for PSK system in the previous section. Again theproblem is reduced to the optimal location of a pulse in the presence of additive noise.We assume that the receiver is maintaining lock on the central peak of the BOCcorrelation, which corresponds to the correct timing location.

The mean squaretiming jitter can then be evaluated by comparison of the slope of the discriminatorwith the r.m.s. noise.TThe slope of the sine sub-carrier BOC discriminator is 2 × A 2 − STCTthe cosine sub-carrier BOC discriminator is 2 × A 2 + STC and the slope of as shown in Figure 4-12.63PSK and BOC signals1.5Discriminator error (chips)1ΛBF ( RN , 0.0)ΛBC( RN , 0.0)SinSlope( RN)0.40.200.20.4CosSlope( RN)true sine errortrue cosine errorsine approximationcos approximation1− 1.5− 0.5RN0.5Code error (sub-chips)Figure 4-12, Sine and cosine BOC(fC, fC) discriminators with slope approximationsAppendix H shows the derivation of timing jitter for sine and cosine BOC modulatedsignals. Modelling the BOC timing recovery as a delay line subtractor we find themean square timing jitter for sine BOC as follows.2Σ BOCs =2 × BL1×× TD × TSC N 0 4 2 − TS TC 4–18TS is the sub-chip width and TD is a delay equivalent to the early-late discriminatorspacing limited to 0 < TD ≤ TS for sine BOC or 0 < TD ≤TSfor cosine BOC.2The equivalent expression for cosine BOC is2Σ BOCc =2 × BL1×× TD × TSC N 0 4 2 + TS TC 4–1964PSK and BOC signalsTable 4-1 shows the resulting timing jitter formulations with various sub-carrier tocode ratios for sine and cosine BOC.Table 4-1, Timing jitter of BOC signalsSignalTiming jitterSineRelative timingCosinebenefit of cosinesub-carrierBOC(fC, fC)2 × BL 1× × TD × TSC N0 62 × BL 1× × TD × TSC N 0 101.66BOC(1.5×fC, fC)2 × BL 3× × TD × TSC N 0 202 × BL 3× × TD × TSC N 0 281.40BOC(2×fC, fC)2 × BL 1× × TD × TSC N0 72 × BL 1× × TD × TSC N0 91.29As shown in Table 4-1 there is only a small benefit in terms of timing jitter (2.2dBmaximum) to cosine BOC and only for low ratios of sub-carrier to code rate.

Forhigh ratios Equation 4–18 and Equation 4–19 are asymptotic to the following result.2Σ BOC =BL× TD × TS4 × C N04–20It is common in the literature to normalise the mean square jitter to the chip width,normalising Equation 4–20 givesσ BOC2Σ=  BOC TC2BLT ×T =× D 2S4 × C N0TCBL ∆ BOC=4 fS2CN0 fC4–21where ∆ BOC = TD TC is the normalised early-late discriminator spacing, fS is the subcarrier frequency and fC is the chipping rate. This equation corresponds to theequations derived for timing jitter for coherent early minus late BOC discriminatorsgiven in [Ries et al 2002].

However, we disagree with the authors about theimplications of this result. Comparing Equation 4–21 to the standard PSK timing65PSK and BOC signalsjitter (Equation 4–16) the authors claim that BOC modulation reduces the meansquare timing jitter by a factor of 4 f S f C when compared to PSK. This supposedadvantage depends on a normalisation to the chip width, which does not provide a fairbasis for comparison. For example, this implies comparing PSK-R(1) with BOCsignals such as BOC(1,1), BOC(2,1) and BOC(6,1). Clearly these signals have vastlydifferent bandwidth requirements and this is not a like-for-like comparison. Thetiming jitter must be evaluated in absolute terms to provide a fair comparison betweenBOC and PSK in an available bandwidth.

The PSK and BOC timing jitter equationsin absolute terms are as follows.BL× TD × TC2CN0BLBL=× TD × TC =× TD × TS4 fS4C2CN0N0 fC2Σ PSK =Σ BOC24–22The processing rate of a GNSS signal provides a fair basis of comparison. PSK andBOC systems have the same processing rate if the chip width TC, of the PSK signal isset equal to BOC sub-chip width TS.

Then from Equation 4–22 we can see that thetheoretical timing advantage of BOC modulation reduces to a factor of 2 for equalearly-late separation. If we wish to compare normalised quantities, we mustnormalise BOC signals with respect to the sub-chip width as follows.2σ * BOC2ΣBLT=  BOC  =× D4 × C N 0 TS TS B × ∆ BOC= L4×CN04–23Comparing Equation 4–23 with the equivalent normalised PSK jitter given inEquation 4–16 results in a factor of improvement for BOC modulated signals.

BOCmodulation therefore provides a small timing advantage (3dB maximum) overequivalent PSK systems on this basis of comparison.66PSK and BOC signals4.5PSK and BOC multipath analysisThe simplest and most common way [Irsigler et al 2004] of evaluating the timinglocation error induced by the presence of multipath is to consider the effect of a singleinterfering multipath signal with various relative time delays.

Характеристики

Список файлов учебной работы