On Generalized Signal Waveforms for Satellite Navigation (797942), страница 31
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Summing now inparallel the terms of the 10 diagonals of the matrix definition of BCS, we have:⎡⎛ 2πf ⎞⎛ 2πf ⎞⎛ 2πf ⎞ ⎤⎟⎟ + 6 cos⎜⎜ 2⎟⎟ − 5 cos⎜⎜ 3⎟⎟ + ⎥⎢− 7 cos⎜⎜⎝ 10 f c ⎠⎝ 10 f c ⎠⎝ 10 f c ⎠ ⎥⎢⎢⎛ 2πf ⎞⎛ 2πf ⎞⎛ 2πf ⎞ ⎥BCS([1, −1,1, −1,1, −1,1, −1,1,1], f c )⎢⎥⎜⎟⎜⎟( f ) = 10 + 2 + 4 cos⎜ 4GMod⎟ − 3 cos⎜ 5 10 f ⎟ + 2 cos⎜⎜ 6 10 f ⎟⎟ − ⎥ (4.133)10f⎢c ⎠c ⎠c ⎠⎝⎝⎝⎢⎥⎛ 2πf ⎞⎛ 2πf ⎞⎢⎥⎢− cos⎜⎜ 7 10 f ⎟⎟ + cos⎜⎜ 9 10 f ⎟⎟⎥c ⎠c ⎠⎝⎝⎣⎦For our particular CBCS case, we have thus:GCBCS ( f ) = (1 − ρ )GBOC (1,1) + ρ GBCS ( f )(4.134)being ρ the percentage of power on the BCS component, as we saw above. According to this,we can express the power spectral density of the CBCS([s],1,%) modulation as follows:GCBCS ( f ) = (1 − ρ )GBOC(1,1) + ρ GBCS ( f ) == (1 − ρ )GBPSK (1) GMod ([ +1, −1]) + ρ GBPSK (10 ) ( f )GMod ([+1, −1, +1, −1, +1, −1, +1, −1, +1, +1]) ( f )(4.135)As derived in Appendix B the power spectral density of BOC(1,1) is shown to be:132GNSS Signal Structure⎡ ⎛ πf ⎞ ⎛ πf ⎞ ⎤⎟⎥⎢ sin ⎜⎜ ⎟⎟ sin ⎜⎜f c ⎠ ⎝ 2 f s ⎟⎠ ⎥⎝⎢GBOCsin (1,1) = f c⎢⎛ πf ⎞ ⎥⎟⎟ ⎥⎢ π f cos⎜⎜⎝ 2 f s ⎠ ⎦⎥⎣⎢2(4.136)f c = f s =1.023 MHzNevertheless, since we are interested in finding a simplified expression for the CBCS PowerSpectral Density, it is convenient to express BOC(1,1) as a BCS sequence with a vector oflength 10.
Indeed, BOC(1,1) can also be expressed as BCS([+1,+1,+1,+1,+1,-1,-1,-1,-1,-1],1).Another way of describing CBCS in the frequency domain is to realize that the BCS signalcan be expressed as the sum of a BOC(5,1) signal and an MCS(0,0,0,0,0,0,0,0,0,2) signal inthe time domain. Thus, we have to calculate first the Fourier Transform of both, sum them upand calculate the modulus according to equation (4.8):As we know, the spectrum of BOC(5,1) is shown to be:⎛ πf ⎞ ⎛ πf ⎞⎟sin ⎜⎜ ⎟⎟ sin ⎜⎜f c ⎠ ⎝ 10 f c ⎟⎠⎝S BOC sin (5,1) ( f ) = j⎛ πf ⎞⎟⎟π f cos⎜⎜⎝ 10 f c ⎠(4.137)f c =1.023 MHzEqually, the spectrum of a pulse signal of duration (Tc/10) with the pulse centred on the lastsubchip can be expressed as follows:⎛ πf ⎞⎟sin ⎜⎜10 f c ⎟⎠ ⎡ ⎛ πf ⎞⎛ πf ⎞⎤⎝⎟⎟ − j sin ⎜⎜ 9⎟⎟⎥(4.138)S Pulse ( f ) = 2⎢cos⎜⎜ 910f10fπ fc ⎠c ⎠⎦⎝⎣ ⎝f c =1.023 MHzHence the PSDs for the data and pilot signals yields:[G ( f ) = f [ 1− ρ S]( f ) }]GD ( f ) = f c 1 − ρ S BOC(1,1) ( f ) + ρ {S BOC(5,1) ( f ) + S pulse ( f ) }P4.6.4cBOC (1,1)( f )−ρ {S BOC(5,1) ( f ) + S pulse22(4.139)(4.140)CBCS Positioning PerformanceOne of the performance figures used to select the CBCS signal was the multipath error.
Inorder to realistically estimate the multipath that the candidate signals could present in realenvironments, the methodology presented in [G.W. Hein and J.-A. Avila-Rodriguez, 2005]and [M. Irsigler et al., 2005] was followed. Furthermore, in order to reduce the computationsand given the enormous number of potential signals to assess, a simplified model wasemployed.
Nevertheless, [M. Irsigler et al., 2005] and [M. Irsigler, 2008] have shown thatmore simplified models also give satisfactory results in the same direction at the expense ofgeneralizing the assumptions and simplifying scenarios.133GNSS Signal StructureAnother important aspect to note with regards to the CBCS modulation is that the data andpilot channels are in anti-phase and present thus different correlation functions as shown inAppendix J. The direct consequence of that is that the data and pilot channels present differentperformance depending on whether the signal has the BOC(1,1) and BCS in phase or in antiphase. As we will show in the next pages, the anti-phase signal has got the sharpest ACF andthus better performance.
For this reason, this was assigned to the pilot channel.In the following lines we show the performance of CBCS in terms of multipath using themultipath error envelopes. As we know, its computation relies on the assumptions that the lineof sight is always visible, that only one multipath signal is present and that the multipathsignal experiences a fixed amplitude attenuation (e.g. coefficient of reflection α = 0.5 in oursimulations) with respect to the direct signal. In addition, a static environment is commonlyassumed.More sophisticate models to quantify the differences between the multipath performance ofdifferent signals for a given receiver architecture or different receiver implementations havebeen analyzed in [M.
Irsigler, 2008].In the next figures we compare the performance of the CBCS(20) signal with that of othersolutions that were also considered in the past. As we can see, the same CBCS with a slightlower power was also object of the analysis. The reason to reduce the power on the BCScomponent to 15.6 % was to improve the coexistence of the signal with the rest of signals inthe E1/L1 band. Of course, this reduction of power on the BCS part implied a slightdeterioration of the performance. Note also that the A and B channels perform the same andthus the dotted and continues curves overlap each other.Figure 4.44.
Multipath error envelopes for different Galileo signal candidates. A precorrelation bandwidth of 12 MHz and a chip spacing of δ = 0.1 chips have been assumed134GNSS Signal StructureFigure 4.45. Running average multipath errors for different Galileo signal candidates. Apre-correlation bandwidth of 12 MHz and a chip spacing of δ = 0.1 chips were assumedAs analyses have shown, CBCS presented a considerable improvement of more than 25 %with respect to the baseline for a bandwidth of 24 MHz.
Furthermore, the improvement wasof even 40 % in terms of multipath with 12 MHz as previous figure shows. This wasespecially relevant in urban and suburban environments. Moreover, for 12 MHz CBCSperformed even better than BOC(2,2). This is a direct consequence of the fact that whileBOC(1,1) needs a larger bandwidth of 24 MHz to exploit to the limit the possibilities of themodulation, the CBCS modulation needs a lower bandwidth to make an optimum use of thesignal in the sense that its Gabor bandwidth for the same receiver bandwidth is higher. Acomplete analysis of positioning accuracy using the concept of the User Equivalent RangeError (UERE) was presented by [J.-A. Avila-Rodriguez et al., 2005b] where the superiority ofCBCS in different baseline scenarios was demonstrated.Finally, to have a complete insight into the performance of CBCS, the Cramér Rao LowerBound (CRLB) is shown in the following figure.Figure 4.46.
Cramér Rao Lower Bound of E1 OS signals. It must be noted that OSArefers to the data channel while OSB refers to the pilot channel135GNSS Signal StructureThe Cramér Rao Lower Bound is defined in (4.141) and is the lower bound of themean-squared error for any estimate of a non-random parameter as shown by[H. Cramér, 1946] and [J.-A. Avila-Rodriguez et al., 2006b]. The Cramér-Rao lower bounddefines the ultimate accuracy of any estimation and shows the minimum code pseudorangevariance we would have with the best possible receiver implementation. Indeed, this bound isa different way of expressing the Gabor bandwidth which sets the physical limit of a signalfor a given bandwidth.
This last one is also known in the literature as the Root Mean Square(RMS) bandwidth. As shown in Appendix K, the Cramér Rao Lower Bound is defined as:BLBLCRLB = −=(4.141)CC2 ∞2(2π ) ∫−∞ f Gs ( f ) dfR ′ss′ (0)N0N0where BL refers to the loop bandwidth of the code tracking loop and R′ss′ (0) and G s ( f ) arerespectively the autocorrelation and power spectral density of the signal.The Cramér-Rao lower bound is usually employed to assess the performance of the positionestimation based on the delay between the transmitting satellite and the receiver.
However,since the function that maps the signal delay to the physical location is not necessarilycontinuous and differentiable everywhere due to the usual oscillations and discontinuities ofthe received signal, the Cramér Rao lower bound cannot be applied in all cases. In fact,differentiability is a requirement for the Cramér Rao lower bound to be properly used. Asshown in [H. Koorapaty, 2004], it may be feasible to create a continuous approximation ofthis mapping function although this is discontinuous in reality. However, in order to beaccurate, the function would need to have large local variations and the Cramér-Rao boundwould then be too inaccurate.
In the same manner, if a smooth function without largevariations were assumed, the bound would also be inaccurate. In particular, such an approachwould be too pessimistic in its performance estimates.As an alternative, the Barankin bound was proposed in [E. Barankin, 1949].
As shown in[H. Koorapaty, 2004], the Barankin bound does not require the mapping function to becontinuous and differentiable being hence better suited for some problems. The Barankinbound is computed by selecting a set of test points [x1 , x2 ,..., xN ] in the area of analysis anddefining the following function:L(r , xi , x ) =P (r xi )(4.142)P(r x )where P(r x ) denotes the conditional probability density function at the set of chosen testpoints with i ∈ {1,2,..., N } . According to this, the Barankin matrix is defined as follows:Bi , j ( x ) = ∫ L(r , xi , x ) L(r , x j , x ) P(r x ) dr = E{∫ L(r, x , x ) L(r, x , x ) }ij(4.143)with (i, j ) ∈ {1,2,..., N }× {1,2,..., N }. Furthermore, the Barankin bound on the covarianceCov[x̂(r )] for the parameter x based on the measurements r is given by:136GNSS Signal StructureCov[xˆ (r )] ≥ ΛB −1 ( x )ΛTwhere(4.144)Λ = [{x1 − x}, {x 2 − x},..., {x N − x}]T(4.145)As we can recognize, the computation of the Barankin bound only requires knowledge of theconditional probability density function P(r x ) at the set of chosen test points and noassumption on the differentiability is required.
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