On Generalized Signal Waveforms for Satellite Navigation (797942), страница 38
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The above derivedexpressions are in consonance with the results obtained in [L. Ries et al., 2003] and163GNSS Signal Structure[O. Julien, 2005] and confirm the fact that depending on the slope of the autocorrelationfunction of the signal waveform around the main peak, the tracking performance andachievable sensitivity will be better or worse, as we expected. Indeed, the steeper theautocorrelation function around the main peak is, the better the performance of the signal willbe in terms of standard error. This result underlines the conclusions that we drew in chapter4.1.1 on how an optimized signal should be designed.Additionally, we can see from the expressions above that if an infinite front-end filterbandwidth is assumed, the squaring losses for the Dot Product (DP) discriminator do notdepend on the signal structure any more, while for case of the Early Minus Late Power(EMLP) discriminator they do depend, being these larger the steeper the autocorrelationfunction around the main peak.If we take a look now at Figure 4.59 next, we can recognize that while the GPS C/A codepresents a slope value of α = 1 , for BOC(1,1) the slope around the main peak is of α = 3 ,resulting thus in an improvement of the tracking error standard deviation of 2.4 dBapproximately (without accounting for the squaring effects).Figure 4.59.
Autocorrelation Function of BPSK(1), BOC(1,1) and MBOC(6,1,1/11)The improvement is still more spectacular if we take a look at the MBOC autocorrelationfunction, where the average slope around the main peak, considering data and pilot together,is shown to be 53/11. This represents an improvement of the tracking performance ofapproximately 1.03 dB with respect to BOC(1,1) and of 3.41 dB with respect to BPSK(1).The slope of the main peak can be easily computed from (4.149).In fact, the slope of the autocorrelation function of BOC(6,1) takes a value of 23 around themain peak for infinite bandwidth. Equally, for BOC(1,1) the slope has a value 3 and theresulting slope of MBOC around the main peak will be 23 ⋅ 1 11 + 3 ⋅ 10 11 = 53 11 .
In general,the slope of the main peak of a BOC(x,1) is shown to be 4 x − 1 for the case of infinitebandwidth. It is important to note that the MBOC ACF shown above (CBOC implementation)corresponds to the mean ACF that results from averaging the data and pilot channel. In the164GNSS Signal Structurecase that both data and pilot would alternate the phase of the BOC(6,1) signal every chip, asimilar result could be obtained. For the TMBOC case, it would represent the case whenBOC(6,1) is on both data and pilot with the same power on both. This is equivalent to sayingthat the cross-correlation between BOC(1,1) and BOC(6,1) does not show up in the figuresabove since they average to zero.We can see this more clearly if we recall the equations derived in chapter 4.6 for the CBOCimplementation of MBOC.
Indeed,R CBOC ('+ / − ') (τ ) = k12 RBOC (1,1) (τ ) + k 22 RBOC ( 6,1) (τ )(4.192)so that the cross-term between the BOC(1,1) and BOC(6,1) does not appear in the expression,unlike for the CBOC( '+ ') and CBOC( '−') cases.If we take a look now into the performance of the data and pilot channels separately, we cansee that for the case of GPS L1C, the TMBOC implementation of MBOC puts all the highfrequency power on the pilot channel for tracking purposes. This results in a steeper slope ofthe autocorrelation function around the main peak.
Recalling the GPS power split betweendata and pilot of 25/75 in L1C, the slope will have a value of 23 ⋅ 4 33 + 3 ⋅ 29 33 = 179 33 ,bringing thus an improvement for the pilot channel of approximately 3.67 dB with respect toBPSK(1) and of 1.28 dB with respect to BOC(1,1).
On the other hand, the data channel willpresent a performance equivalent to that of BOC(1,1).Equivalently, its counterpart Galileo will have a performance 3.41 dB better than BPSK(1)and will be 1.03 dB better than BOC(1,1) since BOC(6,1) is placed on both data an pilot witha power split 50/50. This can also be seen if we take a look at the ACFs of the data and pilotchannels separately.Finally, it is important to note that although MBOC(6,1,1/11) and BOC(1,1) present improvedperformance with respect to BPSK(1), the tracking region is smaller if we track the wholeMBOC. In fact, the linear region around the main peak will be six times narrower.4.7.7.4Signal structure and DLL sensitivityNow that we have calculated the code tracking error for MBOC and the rest of Open signalsin E1, the next step is to obtain the necessary C/N0 to ensure a correct tracking.
This will giveus an idea of the potential DLL sensitivity of the different analyzed signal structures.Following [P. Ward, 1996] and [O. Julien, 2005] to study the PLL sensitivity, the rule ofthumb we will use is to have a 3-sigma of the errors within the linear tracking region, which isin theory ±δ2. This can also be expressed as follows:3σ ετ ,TH ≤δ2(4.193)165GNSS Signal StructureIt must be noted that actually not all the errors are included in the expression above, and thusthe multipath-induced tracking error must be studied separately. The reason for this is that themultipath error does not imply a tracking error in the sense that it would push the trackingloop away from its stability point as explained in detail in [O. Julien, 2005].Recalling now the equations for the tracking error of (4.190) and (4.191) above, we can seethat (4.190) can be simplified for the EMLP as follows:⎡⎤⎢A1 ⎥2⎥(4.194)= 1 ⎢1 +σ EMLPC ⎢C⎥A2N 0 ⎢⎣N 0 ⎥⎦whereβ⎛ 1⎞ rBL ⎜1 − BL TI ⎟ ∫ 2β r G ( f )sin 2 (π f δ ) df⎝ 2⎠ −2A1 =(4.195)2βr⎡⎤⎢2π ∫− 2β r f G ( f )sin (π f δ ) df ⎥2⎣⎦and2βr⎤⎡2TI ⎢2π ∫ β r G ( f )cos(π f δ )df ⎥−⎦(4.196)A2 = ⎣ β r 222∫ β r G( f )cos (π f δ )df−2Equally for the Dot Product Discriminator, (4.191) can also be expressed as:⎛⎞⎜⎟A121 ⎜⎟σ DP =1+C ⎜C⎟A3 ⎟⎜N0 ⎝N0 ⎠withA3 = TI ∫βr2β− r2G ( f ) df(4.197)(4.198)According to this, if we assume as mentioned above that 3σ ετ ,TH ≤ δ 2 for the case of theEMLP discriminator, the EMLP tracking threshold results to be:2⎛⎞⎜1 + 1 + δ⎟⎜⎟AA91 2 ⎠C⎝= 18 A1N 0 TH ,DPδ2(4.199)and for the DP we have equally:CN02⎛⎜1 + 1 + δ⎜9 A1 A3= 18 A1 ⎝2TH , DPδ⎞⎟⎟⎠(4.200)Using the expressions above, the tracking thresholds are computed in the next figures as afunction of the coherent integration time and the DLL loop bandwidth.
For the case of infinite166GNSS Signal Structurebandwidth the expressions for the standard delay error simplify considerably adopting A1, A2and A3 the following values:⎛ 1⎞BL ⎜1 − BLTI ⎟δ2⎠A1 = ⎝(4.201)2α(2 − δα )TIA2 =(4.202)2A3 = TI(4.203)For comparison, a spacing of 0.1 chips (left) and a spacing of 0.2 chips (right) will be used.Figure 4.60.
DLL Tracking Threshold of BPSK(1) for the DP DiscriminatorFigure 4.61. DLL Tracking Threshold of BOC(1,1) for the DP DiscriminatorFigure 4.62. DLL Tracking Threshold of MBOC(6,1,1/11) for the DP167GNSS Signal StructureAs we can recognize, using a chip spacing of 0.1 or 0.2 chips does not really make a bigdifference.If we take a close look at the results shown in the figures above, we can see that MBOCpresents the lowest tracking threshold and thus the best sensitivity, followed by BOC(1,1) andBPSK(1).
It is also interesting to see that the results of the simulations show an even higherimprovement of the C/N0 sensitivity of MBOC with respect to BOC(1,1) and BPSK(1) thanthat predicted in chapter 4.7.7.3. Indeed, while we saw there that MBOC was expected tohave a tracking threshold 1 dB lower than that of BOC(1,1) and 3.4 dB better with respect toBPSK(1), the results of the figures above account for higher improvements.
This could be dueto filtering effects in the loop bandwidth and coherent integration that are not reflected in theanalytical expressions for infinite bandwidth and that would positively favour MBOC againstthe other studied options.Another important comment is that for the simulation of MBOC, a slope of 53/11 wasassumed around the main peak. This is actually the average of data and pilot as we saw above.If we concentrate on the pilot performance of GPS L1C, the slope will be steeper since it hasall the BOC(6,1) contribution, and even better results are expected.In order to be able to compare the three signals more efficiently, the next figures show againthe tracking thresholds for specific DLL configurations and chip spacing values.Figure 4.63.
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