Advanced global navigation satellite system receiver design (797918), страница 9
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Also, theoretical equations for timing jitterare given for both PSK and BOC systems, and are used in subsequent chapters toevaluate receiver performances. The effect of BOC modulation on the signalspectrum, receiver timing jitter and multipath performance are considered andcompared to equivalent PSK modulations.Binary Offset Carrier (BOC) modulation is ideally a square wave sub-carriermodulation of a PRN sequence. Its effect is that the signal is amplitude modulated,with its power spectrum shifted away from the carrier frequency in upper and lowersidebands with a null at the centre frequency. The null is the primary benefit of BOCmodulation, allowing frequency re-use along with current PSK signals.
In addition,BOC modulation claims greater resistance to short-range multipath and a smalladvantage in code tracking accuracy. However, these advantages come at a price,namely the difficulties in acquiring and tracking signals with a sub-carriermodulation.4.1Signal spectra and bandwidthIn order to perform a fair comparison between BOC and PSK systems it is necessaryto identify a number of different definitions of bandwidth. The Nyquist bandwidthembodies the minimum spectral bandwidth into which a PSK transmission can becompressed. For PSK transmissions this bandwidth is identical to the chipping rateand so we define a processing rate, which for PSK transmissions is also equal to thechipping rate.The effect of BOC modulation is the creation of upper and lower sidebands of thesignal, which effectively doubles the Nyquist bandwidth to twice the chip rate.
Wedefine the processing rate for a BOC signal as the sub-chip rate which is the same astwice the sub-carrier frequency. Both the Nyquist bandwidth and processing rate areinherent to a given signal and are independent of the signal's spectral shape. The50PSK and BOC signalsfront-end bandwidth of a GNSS receiver must be equal to or greater than both theNyquist bandwidth and the processing rate. The different definitions of bandwidthare depicted in Figure 4-1.
These notions of bandwidth are used in Section 4.4 andSection 4.5 of this chapter to provide a fair basis of comparison between PSK andBOC signals.NyquistbandwidthP(f)ffCNyquist bandwidth = 2 fCProcessing rate = 2 fSf C - fSH(f)fC + fSfFront-end bandwidthffCFigure 4-1, PSK and BOC bandwidth definitionsThe phasing of the BOC sub-carrier to the code has an effect on the spectral shape ofthe signal.
The most noticeable effect is on the secondary lobes of the BOCtransmission. Cosine sub-carrier phasing has the effect of pushing more spectralenergy into the secondary lobes further away from the centre frequency compared tosine sub-carrier phasing. The choice of sub-carrier phasing is dependent on numberand location of the signals within a given bandwidth. BOC signals with high-ratesub-carriers are generally selected to have cosine phasing to reduce spectral overlapand hence interference with signals transmitted within the BOC null. For example, inthe L1 band the Galileo BOC(15, 2.5) signal has cosine phasing to reduce the51PSK and BOC signalsinterference with the GPS BOC(10, 5) M code signal. The effect of sub-carrierphasing on the signals spectral shape is shown in Figure 4-2 with a BOC(10, 5)modulation.70− 70.697Power spectral density (dBW/Hz)7580BOCs ( fP)85BOCc( fP)9095− 1001002015105− 20051015fP2020Frequency offset from carrier (MHz)BOC(10,5) sine sub-carrierBOC(10,5) cosine sub-carrierFigure 4-2, Power spectral density of a BOC(10, 5) signal with sine and cosine sub-carriersA common measure of the amount of signal degradation incurred from interferencebetween two signals in a receiver channel is given by their spectral separationcoefficient.
Envisioning the receiver as a matched filter process (see Figure 4-3), thespectral separation coefficient defines the signal degradation due to the interferenceon the desired ‘matched’ signal.whitenoiseN0v(t)Σ2H(f)i(t)interferenceiT(t)GI(f)BTBRGIT(f)PIFigure 4-3, Matched filter equivalent to GNSS receiver52PSK and BOC signalsA spectral separation coefficient between two signals in the notation of [Pratt andOwen 2003] and generalising to include effect of bandwidth limiting BR in thereceiver and bandwidth limiting BT in an interference transmitter is as follows.+∞4–1κ IS = ∫ Φ I ( f ) × Φ S ( f )df−∞ΦI(f) is a normalised power spectral density of the interfering signal and ΦS(f) is thenormalised power spectral density of the wanted signal whereΦS ( f ) =GSBR / 2(f )f < BR / 24–2∫ GS ( f )df− BR / 2=0f ≥ BR / 2andΦI ( f ) =GI+ BT / 2(f )f < BT / 24–3∫ GI ( f )df− BT / 2=0f ≥ BT / 2One can also define a ‘separation coefficient’ of a signal with itself+∞κ SS =∫ ΦS ( f )24–4df−∞In [Pratt and Owen 2003] it is shown that the separation coefficient can be partitionedinto partial spectral separation coefficients as follows.κ IS =∫ Φ ( f ) × Φ ( f )df + ∫ Φ ( f ) × Φ ( f )dfIf ∈ f2SIS4–5f ∉ f253PSK and BOC signalsf2 is a partitioning frequency.
Partial coefficients allow the contribution to the signaldegradation to be quantified for different parts of the interference source. Therefore,the contribution to signal degradation of the main lobes of the interfering signal canbe separated from the contribution by the side-lobes. The partial coefficientfrequency range is adjusted by setting f2. Thus, spectral separation coefficientsprovide a method for selecting the optimum combination of GNSS signals in a givenbandwidth and enable evaluation of sub-carrier phasing choices.BOC modulation therefore can provide a more efficient use of an allotted bandwidthand the BOC Subcarrier phasing can be used to minimise interference between signalsand systems.4.2Correlation functionsA BOC signal is created through modulation of a PRN sequence, a (t ) by a squarewave sub-carrier, s (t ) represented as follows.b(t ) = a (t ) × s (t )4–6The periodic autocorrelation of a BOC modulated signal can be written as1(τ ) =NTCNTC∫ b(t )b(t + τ )dt4–70where N is the PRN code length in chips and TC is the chip period.The autocorrelation envelopes of BOC signals produce sets of triangle functions withpositive and negative peaks.
The autocorrelations of a BOC(10, 5) signal with nofiltering and a front-end bandwidth of 24MHz respectively are shown in Figure 4-4(derived from a Matlab Simulink simulation).54PSK and BOC signalsAutocorrelation of BOC(10,5) bandlimited to 24MHzAutocorrelation of BOC(10,5) unfiltered110.80.750.6Normalised correlation amplitudeNormalised Correlation Amplitude0.50.250-0.25-0.50.20-0.2-0.4-0.6-0.8-0.75-1-10.4-1-1-0.75-0.5-0.2500.25Time Delay (Chips)0.50.75-0.8-0.61a)-0.4-0.200.2Time delay (chips)0.40.60.81b)Figure 4-4, Autocorrelations of BOC(10, 5) signals: a) unfiltered b) 24MHz bandwidthConventional GNSS receivers subtract the difference of correlations by replica signalsadvanced (‘early’) and retarded (‘late’) in time to form a code discriminator curve(Figure 4-5). This discriminator gives the receiver code loop an error signal, requiredto steer the loop toward the correct correlation between the incoming signal andlocally generated (‘prompt’) replica.
The PSK discriminator has only one stable codeNormalised discriminator error signalloop state, one ‘zero crossing’.1Early - Late spacing = 1 Chip0.50-0.5a)-1-1.5-1-0.500.5Time Delay (Chips)11.5b)Figure 4-5, a) Autocorrelations of time delayed early, prompt and late PSK signals b) PSKdiscriminator curveThe multiple correlation peaks of BOC modulated signals result in several stable codeloop states where only the central peak represents the correct correlation between theincoming signal and locally generated replica (Figure 4-6). This introduces anambiguity in the receiver’s code tracking loop, which can potentially lock to an55PSK and BOC signalsincorrect peak causing a large ranging error.
This effect has been referred to as ‘BOCtracking ambiguity’ or ‘false-lock points’ [Bello and Fante 2005]. The ambiguity isexacerbated by band-limiting the signal, which increases the ratio of the largestincorrect peak to the correct peak, resulting in a greater probability of false tracking.2221ΛBF ( R , 0.0)64202461Discriminator error (chips)Discriminator error (chips)21ΛBF ( R , 0.0)10505101−2−222−6RCode error (sub-chips)a)6− 12R12Code error (sub-chips)b)Figure 4-6, BOC discriminator curves a) BOC(2×fC, fC) b) BOC(6×fC, fC)There are numerous techniques proposed in the literature for unambiguous tracking ofBOC signals, each with varying degrees of ambiguity resolution and each with uniqueimplications on receiver performance and complexity.
The techniques proposed inthe literature are detailed in Section 5.4. A novel BOC tracking technique developedduring this research provides the best current solution to this problem and is describedin detail in Chapter 6.The phasing of the BOC sub-carrier also has an effect on correlation function andtherefore, the code tracking accuracy of the receiver. Providing the receiver front-endbandwidth is large enough to support it, the correlation function of a BOC signal withcosine sub-carrier becomes markedly sharper than the equivalent sine sub-carriercorrelation.
Figure 4-7 shows BOC correlation functions with sine and cosine subcarrier phasing.56PSK and BOC signals1.5sinecosine0.5ΛBF ( R , 0.0)ΛBC( R , 0.0)210sinecosine1.51120.5Correlaiton amplitude (chips)Correlaiton amplitude (chips)1.51ΛBF ( R , 0.0)ΛBC( R , 0.0)42024−1−111−2R2−4RCode error (sub-chips)4Code error (sub-chips)a)b)Figure 4-7, BOC correlation functions with different sub-carrier phasinga) BOC(fC, fC) b) BOC(2×fC, fC)The sharpening of the correlation has a follow-on effect on the discriminator curve asshown in Figure 4-8 with no filtering. For a given correlation signal to noise ther.m.s.
timing jitter is inversely proportional to the slope of the discriminator curve atthe zero crossing. Therefore, cosine BOC modulation can provide better noiseperformance than sine BOC, but only for low ratios of sub-carrier to code ratio. Thisis bought at the expense of widening the effective bandwidth of the signal (see Figure4-2), which is shown in [Ries et al 2003] to increase correlation losses for narrowbandreceivers, effectively cancelling any benefit. For wideband receivers the benefit ofcosine BOC tends to zero with increasing sub-carrier to code ratios. Expressions forsine and cosine BOC timing accuracy are given and the relative benefits of eachquantified in Section 4.4 of this chapter.sinecosineDiscriminator error (chips)11ΛBF ( R , 0.0)ΛBC ( R , 0)32101sinecosine1.523Discriminator error (chips)1.5ΛBF ( R , 0.0)ΛBC ( R , 0)642102461− 1.5− 1.5−3R3−6Code error (sub-chips)a)R6Code error (sub-chips)b)Figure 4-8, BOC discriminator curves with different sub-carrier phasinga) BOC(fC, fC) b) BOC(2×fC, fC)57PSK and BOC signals4.3Theoretical timing measurement of PSK modulated signalsA number of different approaches can be adopted to estimate the precision with whicha GNSS receiver can calculate the time of arrival of the GNSS signals.
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