On Generalized Signal Waveforms for Satellite Navigation (797942), страница 37
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Watson, 2005]~2SNRpost 2TI β r R(ετ )=G=~2SNRpreR (0)(4.177)which can be further simplified toG = 2TI β r(4.178)as also shown in [O. Julien, 2005].4.7.7.2Effect of longer integrations on code tracking sensitivityIf we take a closer look into the expressions above, we can clearly see that the moststraightforward way of improving the tracking sensitivity is to increase the coherentintegration TI as much as possible. Unfortunately, this is not always possible due to thepresence of data bits or secondary codes.
Additionally, even though it were possible tointegrate for long periods of time in the absence of data, other major problems coming fromthe code and phase delay variation during the integration would appear.The best known solution to overcome this problem is the use of standard non-coherentsummations according to the following expressionMMk =1k =1Yst = ∑ Z k = ∑I k2 + Qk2(4.179)where Ik and Qk correspond to the in-phase and quadrature non-coherent correlation inputsobtained over a coherent integration time TI. Moreover M is the number of values used for thenon-coherent integration and Yst , as defined in (4.179), can be used to apply the NeumannPearson lemma, as this lemma allows one to obtain a powerful test in the case of two simplealternative hypotheses H 1 and H 2 .159GNSS Signal StructureBy doing so, further correlation gain can be reached but due to the squaring in the expressionabove the process is subject to the so-called squaring losses that reduce the total gain[M.M.
Chansarkar and L. Garin, 2000] and [G.D. MacGougan, 2003]. The squaring lossdepends on the SNR before the non-coherent integration is realized and is higher the lowerthe SNR. Thus, long coherent integration is desirable before the non-coherent correlation isapplied, in order to reach a good SNR before accumulating.
Indeed, by non-coherentintegrating we increase the power of our desired signal but since the noise is not eliminated aswith the coherent integration, the gain in power is lower than the increase of noise.According to this scheme, depending on whether the desired signal is or is not present in thesearching bin, the variable Zk will present a Ricean or Rayleigh distribution correspondingly.Indeed, it can be shown that if the desired signal is present, a Ricean distribution holds, whichhas the following probability density function [J.-A. Avila-Rodriguez et al., 2006b]:z 2 + A2⎛ Az ⎞I0 ⎜ 2 ⎟σ⎝σ ⎠where z is the test variable, A is the signal amplitude, σ 2 the noise power and I 0 (pdf (z ) =z2e−2σ 2(4.180)) the zero-order modified Bessel function of the first kind.
According to this, if the output SNR isdefined as:E(z ) − E( z ) s = 0SNR z =(4.181)σzs =0whereE(z ) = σπ2e−z24σ 2⎡⎛z2 ⎞ ⎛ z2 ⎞ z2 ⎛ z2 ⎞ ⎤⎟ I ⎜ 2 ⎟⎟ +I ⎜ 2 ⎟⎟ ⎥⎢ ⎜⎜1 +2 ⎟ 0⎜2 1⎜⎝ 4σ ⎠ ⎦⎣ ⎝ 2σ ⎠ ⎝ 4σ ⎠ 2σ(4.182)the squaring loss can be obtained according to [G. Lachapelle, 2004] as follows:⎛ A⎞S L (dB) = 20 log10 ⎜ ⎟ − 20 log10 (SNR z )⎝z⎠(4.183)as shown graphically in the following figure:Figure 4.57.
Squaring Loss as a function of the SNR after coherent integration (beforethe non-coherent accumulation)160GNSS Signal StructureReading the squaring losses from the figure above, the total processing gain with respect tothe pre-correlation SNR can be easily calculated [J.-A. Avila-Rodriguez et al., 2006b]:G = 10 log10 (β r TI ) + GNC = 10 log10 (β r TI ) + 10 log10 (M ) − S L(4.184)where• G is the total processing gain (dB) with respect to the pre-correlation SNR,• GNC is the non-coherent signal gain from the non-coherent integration alone,•β r is the pre-detection bandwidth,•••TI is the total coherent integration time,M is the number of non-coherent integrations,and SL are the squaring losses that we defined above.The main drawback from the standard non-coherent integration comes from the fact that thenoise is squared.
Alternative expressions have been studied in the literature to sort out thisproblem as explained in [J.-A. Avila-Rodriguez et al., 2005c] and [G. Lachapelle, 2004]. Oneof those is the non-coherent differential correlation dc, also known as dot-product correlation,which is based on multiplying consecutive samples. One of its multiple expressions is:K /2Ydc = ∑ (I 2i I 2i −1 + Q2iQ2i −1 ) = I 2 I1 + Q2Q1 + I 4 I 3 + Q4Q3 + ...
+ I K I K −1 + QK QK −1i =1(4.185)where Ii and Qi denote again the output of the coherent integration process. Since now theinput i is multiplied with the input i-1, better results are expected in the ideal case, given thatthe noise is uncorrelated in the time with itself. This is in fact what the simulations show.However, this algorithm presents a main drawback due to its high sensitivity to Doppler.In the previous lines we have briefly discussed the most straightforward way of increasingtracking sensitivity by increasing the total integration time. Additionally, there exist otherways of increasing the post-correlation SNR such as increasing the signal power at thesatellite.
Unfortunately, this would have extremely negative effects on interference withalready existing terrestrial systems.4.7.7.3Signal structure and DLL code tracking errorFortunately, there is another way of increasing the per se tracking sensitivity of a receiver,which is based on the signal structure of the desired signal. As we know, any DLLconfiguration is usually based on the combination of early and late correlators, so that thenoise correlation of each output is also important to the resulting combined noise.
Here playsthe signal structure an outstanding role since the DLL tracking sensitivity will be affected bythe selection of the signal waveform. In the next lines, we will show the theoretical trackingperformance of MBOC and we will compare it with that of BOC(1,1) and the C/A Code.161GNSS Signal StructureIf perfect normalization is assumed and the loop bandwidth is negligible compared with thebandwidth of the discriminator noise, [J.K. Holmes, 2000] and [O. Julien, 2005] have shownthat the DLL estimated code delay tracking error variance with Gaussian noise yields:2σ disc,ε τ , t⎛ 1⎞2 BL ⎜1 − BLTI ⎟ S N disc (0)2⎝⎠=2K disc(4.186)where• disc refers to the type of discriminator,• S N disc is the discriminator noise PSD,••BL is the loop bandwidth,TI is the integration time, and•K disc is the loop gain associated to the discriminator, with K disc =dDdiscdετ, whereετ =0Ddisc is the discriminator function.Additionally, since the noise power spectral density that results from multiplying theincoming signal with the local replica is very wide band, we can approximate the expressionabove by the following [O.
Julien, 2005]:⎛ 1⎞2 BL ⎜1 − BLTI ⎟ TI σ D2 disc2⎝ 2⎠(4.187)σ disc,ε τ , t =2K discwhere σ D2 disc represents the discriminator output standard deviation without normalizing. Thisexpression is very close to that of the PLL, with the difference that the effect of thenormalization (through K disc ) has been introduced here. This means in other words, that theDLL tracking error is directly dependent upon the discriminator resistance to noise, and thuson the signal structure.If perfect normalization is assumed again, no frequency uncertainty is considered, a front-endfilter with ideal unity gain and receiver bandwidth β r and a code delay error remaining small,the DLL tracking error variance produced by use of an EMLP discriminator is shown to be[O.
Julien, 2005] and [J.W. Betz and K.R. Kolodziejski, 2000]:ββr⎤⎡⎛ 1⎞ 2r222⎥⎢BL ⎜1 − BL TI ⎟ ∫ β r G ( f )sin (π f δ ) dfβ r G ( f ) cos (π f δ ) df∫−−⎥⎢2⎝⎠ 2221+σ EMLP=⎢22 ⎥βrβr⎤⎤ ⎥C ⎡C ⎡⎢22 G ( f ) cos(π f δ ) dfT2π⎢2π ∫ − β r f G ( f )sin (π f δ ) df ⎥⎢⎥ ⎥⎢ N I∫ − βrN0 ⎣022⎦⎣⎦ ⎦⎣(4.188)where β r is the receiver bandwidth and δ the correlator spacing of the receiver.162GNSS Signal StructureFigure 4.58. Pseudorange Code Measurement Accuracy as a function of thediscriminator spacingIn the figure above, derived according to [J.W. Betz and K.R.
Kolodziejski, 2000], a receiverbandwidth β r of 24 MHz, a loop bandwidth BL of 1 Hz, a C/N0 of 45 dB-Hz and a coherentintegration time TI of 4 ms were used for BOC(1,1) and MBOC(6,1,11). For BPSK(1), 20 mscoherent integration were assumed. It is important to note that other alternative models havebeen derived to describe the behaviour of the code tracking noise for spacing values close tozero as shown in [T.
Pany et al., 2002]. In addition, similar figures could also be obtained fordifferent receiver discriminators. One final comment on the figure above is that MBOC hasgot less dangerous regions than CBCS* regarding the code spacing what was also animportant advantage in favour of MBOC.Equally, as derived in [O. Julien, 2005], the DLL tracking error variance using a Dot Product(DP) discriminator presents the following expression:β⎤⎡⎞ 2r⎛ 1B L ⎜1 − B L TI ⎟ ∫ β r G ( f )sin 2 (π f δ ) df ⎢⎥1⎠ −2⎝ 22⎥⎢1+σ DP =(4.189)βr2⎥⎢βrC⎡⎤CTI ∫ 2β r G ( f ) df ⎥⎢⎢2π ∫ − 2β r f G ( f )sin (π f δ ) df ⎥−N0N0 ⎣2⎦⎥⎣⎢2⎦which simplifies for the case that an infinite receiver bandwidth is assumed, yielding:⎤⎛ 1⎞ ⎡BL ⎜1 − BLTI ⎟ δ ⎢⎥222⎠ ⎢1 +⎥(4.190)= ⎝σ EMLPCC ⎥⎢(2α2 − δ α ) TI⎢⎣N0N 0 ⎥⎦2σ DP⎛ 1⎞BL ⎜1 − BLTI ⎟δ2⎠= ⎝C2αN0⎡⎤⎢1 ⎥⎢1 +⎥C ⎥⎢TI⎢⎣N 0 ⎥⎦(4.191)where α is the slope of the autocorrelation function around the main peak.
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