Advanced global navigation satellite system receiver design (797918), страница 16
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This is achieved through the correlation of two additional offsetreplica codes called very early (VE) and very late (VL). These replicas are separatedfrom the prompt (P) replica by a sub-chip, ±TS.The algorithm is achieved by using three counters, each associated with the VE, P orVL samples. The amplitudes of VE, P and VL in-phase samples are compared at theend of each integrate-and-dump period. If the VE sample is larger then its counter isincremented and the VL counter decremented, if VL is larger the opposite occurs.When P is the larger both VE and VL counters are decremented.
A jump to a newpeak occurs only if the VE or VL counters reach a specified threshold before theprompt counter. The counters are never decremented below zero and are reset when athreshold is reached. An example of a false-lock condition is shown in Figure 5-24for a BOC(2×fC, fC) with the appropriate gates required to implement the BJalgorithm. The narrowly spaced early-late gates provide fine tracking on the signalpreserving the BOC discriminator with no sensitivity loss.105Receiver theoryV L gate1V E gate1/2TSE arly gateLate gate3/4Prom pt gateTCFigure 5-24, BOC(2×fC, fC) false-lock example with BJ gatesThe BJ algorithm can only correct one peak at a time, shifting the tracking point by asub-chip each jump and resetting all counters.
Conventional search processes acquirethe signal with an accuracy of ±TC/2, therefore a number of corrections (jumps) maybe necessary before a valid lock is established. Assuming search accuracy of ±TC/2,the maximum number of jumps required is equal to the ratio of sub-carrier to coderatio f S f C . Therefore, a major drawback to the BJ algorithm can be the time takento reach the correct timing location either from acquisition or from a slip in tracking.The BJ algorithm is effectively ‘blind’ to the number of sub-chips required to find thevalid timing location, it must correct one at a time in sub-chip steps. In contrast,techniques providing a discriminator similar to that of a PSK signal, such as the SSBtechnique and the smooth MGD discriminators can make corrections across the wholediscriminator characteristic in a single step.The acquisition or slip correction time of the BJ algorithm is not only dependant onthe discriminator curve and loop setting time but also depends on the time takendetermine a false lock.
The receiver’s VE or VL counter must pass a predeterminedthreshold in order to determine a false lock state. This threshold must be setsufficiently high so that the level of noise on the VE and VL correlations will notcause a false lock to be declared when in fact the receiver is tracking the correct106Receiver theorytiming location.
Therefore, the threshold must be designed for the most severe noiseenvironment the receiver will operate in, with some margin. The relative amplitudebetween the main BOC correlation peak and its adjacent peaks decreases with thesub-carrier to code ratio. Hence the threshold for detecting a false lock conditionincreases for high rate BOC signal in turn increasing the acquisition and correctiontime.In [Fine and Wilson 1999] a figure of C/N0 = 24 dB-Hz is used to represent in aminimum signal to noise density, which is reasonable for weak signal applications atapproximately 15 dB less than the representative minimum GPS signal to noisedensity (38.9dB-Hz in [Kaplan and Hegarty 2006]). Considering a 20ms integrationperiod this equates to a minimum signal to noise of 10 dB per correlation.
Anuncorrelated noise threshold level can then be calculated by assuming uncorrelatedGaussian noise on the VE and VL correlations. Figure 5-25 shows the worse case ofmany trials assuming an uncorrelated 10 dB signal to noise on each correlation for aBOC(2×fC, fC). A threshold of 8 is identified with corresponds to the analysis andthreshold choice in [Fine and Wilson 1999]. See Appendix E for details of the BJthreshold calculation and algorithm simulation in Mathcad.10Count values8ve kvl k58−101200400600k80010001000Noise samplesFigure 5-25, Worse case VE and VL count values for BOC(2×fC, fC) with uncorrelated signal tonoise of 10 dB per correlationHowever, as identified by Dr Hodgart, the noise samples are in fact stronglycorrelated which actually improves the performance of the BJ algorithm.
ForBOC(2×fC, fC) the correlation coefficient between noise separated by TS and therefore107Receiver theorybetween VE and P and between VL and P is ρ = −0.75 . The correlation coefficientof noise between the VE and VL samples is ρ = −0.5 . The resulting worse case noiseinduced count values for correlated noise are shown in Figure 5-26, which finds arequired threshold of 5.6Count values5ve k 4vl k25−1020040016008001000k1000Noise samplesFigure 5-26, Worse case VE and VL count values for BOC(2×fC, fC) with correlated noisesamples, signal to noise of 10 dB per correlationClearly if the correlation or integration time is reduced from 20 ms the threshold mustbe increased accordingly because the signal to noise per correlation will be reduced.An example acquisition of a BOC(2, 1) signal is shown in Figure 5-27, withBL = 1 Hz and noise, C/N0 = 30 dB-Hz.
The BJ threshold level has been calculatedfor a minimum carrier to noise density of C/N0 = 24 dB-Hz. It can be seen that theacquisition time is not dependant only on the loop settling time but on the time takento reach the required threshold level and declare an invalid tracking state.43.199Timing error (subchips)32tT k− tRC k10100200300400500− 0.11410k500Loop iterationsFigure 5-27, Acquisition example of BJ algorithm for BOC(2, 1) signal (BL = 1 Hz , C/N0 = 30 dBHz, TD = TS)108Receiver theoryThe average time to make a correction can be formulated counting the number oftimes the threshold level is reached while in a false-lock condition.
Starting at anoffset of 1 sub-chip and applying the BJ counters over 2000 correlations withouttiming correction, Mathcad simulation finds the number of thresholds reached is 99with an equivalent C/N0 = 24 dB-Hz. This gives an average of 20.2 correlations percorrection. This results in a correction time of 303 ms for a sub-chip correction (forintegration time, T = 15 ms).Running multiple acquisitions across different initial time offsets provides acomparison of the acquisition time of the BJ algorithm with the equivalent acquisitiontimes using the SSB technique. Table 5-6 shows the acquisition times of the BJalgorithm and SSB technique, assuming a BOC(2, 1) signal, a carrier to noise densityof C/N0 = 24 dB-Hz, a loop bandwidth of BL = 1 Hz and averaging across 20acquisitions at each time step.
For the BOC(2, 1) signal the BJ algorithm showsapproximately equivalent performance to the SSB technique across all initial offsets.Table 5-6, Acquisition times of the BJ and SSB for BOC(2, 1), BL = 1 Hz, C/N0 = 24 dB-HzInitial chip offset1/41/23/4BJ acquisition time (ms)4136551639Jumps required123Correction time (ms)303606909SSB acquisition time (ms)6118891216BJ / SSB0.6760.7371.348The BJ poor performance at large initial offsets (>1/2 chip) can be explained byexamining the discriminator curve at different false locking points. It is assumed atthe discriminator will settle at integer sub-chip offsets, however this is not true. Thereis an offset between discriminator’s zero crossings and the integer sub-chip, whichincreases with each integer sub-chip due to the different gradients of the discriminatoreither side of each peak.
The discriminator zero crossings of a BOC(2×fC, fC) signalare shown in Figure 5-28.109Receiver theory0.150.15ΛBF ( R , 0.0)0.040.020.100.02Discriminator error (chips)Discriminator error (chips)0.10.04ΛBF( R , 0.0)1.121.11.081.061.040.10.1− 0.15− 0.15− 0.05R0.05− 1.135− 1.035RCode error (sub-chips)Code error (sub-chips)a)b)0.150.15ΛBF ( R , 0.0)2.162.142.122.12.080.1− 0.150.1Discriminator error (chips)Discriminator error (chips)0.1ΛBF( R , 0.0)3.283.263.243.223.20.1− 0.15− 2.175R− 2.075− 3.3− 3.2RCode error (sub-chips)Code error (sub-chips)c)d)Figure 5-28, Zero crossings of a BOC(2×fC, fC) signalAt integer multiples of sub-chip offsets the difference in amplitude between the Pcorrelation and VE or VL correlations used for BJ correction is assumed constant atTS TC .
However, shifting the false-lock location away from the integer sub-chipreduces the relative amplitude difference between the peaks required for the BJcomparison. The zero-crossing locations and respective P, VE and VL correlationsfor early false-lock conditions of a BOC(2×fC, fC) signal are shown in Table 5-7. Theamplitude difference between the P and VE correlation decreases with the sub-carrieroffset, which in turn increases the time taken to detect a false-lock condition under thesame noise conditions.
The effect band limiting by the receiver front-end filter willround of the peaks and exacerbate this problem by further reducing the relative peakamplitude.110Receiver theoryTable 5-7, BJ parameters for BOC(2×fC, fC) under false-lock conditionsEarly false-lockpointsZero crossinglocation (TS)P correlationamplitudeVE correlationamplitudeVL correlationamplitudeComparisonamplitudeVE − P1231.0852.1253.25-0.6440.406-0.1860.851-0.5940.3120.436-0.219-0.0020.2070.1880.126The BJ BOC(2×fC, fC) receiver threshold is designed for a peak amplitude differenceof A/4. The reduction of this comparison amplitude results in the potential for thereceiver to make a wrong decision while in a false lock condition. For example, at the3rd false lock point the BOC(2,1) receiver occasionally decides that valid tracking isoccurring and remains in its current position.
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