On Generalized Signal Waveforms for Satellite Navigation (797942), страница 62
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Dafesh et al., 1999a].We can further develop the previous expression if we realize the same transformation as in(7.93). In fact, after some math (7.109) is shown to simplify to [P.A. Dafesh, 1999]:⎡ 2 PI J 0 (m ) d I (t )c I (t ) −⎤⎥ cos(2πf c t ) −s (t ) ≈ ⎢⎢⎣− 2 PQ 2 J 1 (m )d IS (t )cQS (t )sin (2πf s t )⎥⎦⎤⎡ 2 PQ J 0 (m )d Q (t )cQ (t ) +⎥ sin (2πf c t )⎢−⎢+ 2 PI 2 J 1 (m ) d QS (t )c IS (t )sin (2πf s t )⎥⎦⎣(7.110)It is interesting to note that by appropriately selecting the sub-carrier and code partitioningfunctions we may employ cIS (t ) and cQS (t ) as coherently military acquisition and trackingsignals as suggested by [P.A. Dafesh et al., 1999a]. Moreover, if different partitioningfunctions are selected, the I/Q phasing of these military signals can be reversed.7.7.4Generalization to any number of Sub-carriersIn the previous chapter, general expressions were derived for CASM and Interplex with onlythree ranging signals.
However, they can be easily extended to any number n of signals in themost general case. Furthermore, the sub-carriers do not necessarily have to be square-wavebut could also be sinewaves as shown by [P.A. Dafesh et al., 1999a]. In fact, if we recall thegeneral expression:s (t ) = 2 PT cos[2πf c t + φ s (t )](7.111)withnφ s (t ) = ∑ mi d i (t )α di (t )β ci (t )s j (t )ϕ i (t )(7.112)i =1we can see that the number of sub-carriers n that can be multiplexed is in principle unlimited.As we can recognize from the previous expression,•••••mi indicates the modulation index of the i-th sub-carrier,d i (t ) is the data sequence of the i-th multiplexed sub-carrier,α di (t ) is the data partition function of the i-th multiplexed sub-carrier,β ci (t ) is the code partition function of the i-th multiplexed sub-carrier andϕi (t ) is the i-th sub-carrier.282Signal Multiplex Techniques for GNSSAfter appropriate selection of the data and code partition functions, (7.112) can be furthermodified and expressed as follows for the case of square-wave sub-carriers:nφ s (t ) = m1 s1 (t ) + ∑ mi s1 (t ) si (t )(7.113)withs i (t ) = ci (t ) d i (t )ϕ i (t )(7.114)i =2beingϕi (t ) the square-wave sub-carrier,•d i (t ) is the data message of the i-th signal to multiplex, and•ci (t ) is the spreading code of the i-th multiplexed signal.•As we can recognize, this is the same notation that we followed in (7.101) for the particularcase of only three signals to multiplex.Once we have described CASM and Interplex, it is the right moment to talk a little bit moreon the Galileo multiplex needs.
Indeed, sometimes we are not free to choose how we want oursystem to be and in the case of Galileo there were clear requirements and constraints on howthe signals should interact with each other. As we will see, this determines already to a highdegree the multiplex scheme to choose.To conclude, it is important to mention that the names Interplex and CASM are ambiguouslyused to define a similar idea. Nonetheless, we can find slight differentiations in addition to theimplementation aspects we have mentioned. In [E. Rebeyrol et al., 2006], for example, we cansee CASM defined as a three components Interplex modulation with a particular and optimalchoice of the modulation indexes.
However, nothing is said about what signals are in phase orin quadrature. In fact, although [P.A. Dafesh et al., 1999] and [P.A. Dafesh et al., 2000]proposed to have the C/A Code and M-Code in phase with the P(Y) Code in quadrature, otherworks have explored alternative configurations [G.H. Wang et al., 2004]. The same applies toGalileo as we see next.7.7.5Galileo Multiplex NeedsAs shown in [A.R. Pratt and J.I.R.
Owen, 2005], Galileo has to use an additive multiplexingtechnique that produces a constant envelope by means of an inter-modulation signal. We haveshown that this is possible with the Interplex or CASM techniques provided that themodulating signals remain binary. Moreover, as we saw in chapter 2.4.1, the Galileo systemaims at having the following signals on E1, for example:••••E1 OS data signal,E1 OS pilot signal,E1 Public Regulated Service (PRS), andan Inter-Modulation (IM) signal.283Signal Multiplex Techniques for GNSSInterplex fulfils these requirements and it is in fact the multiplex baseline for the Galileosignals transmitted on E1 as shown in [G.W. Hein et al., 2002] and [G.H.
Wang et al., 2004].The Interplex modulation consists of one In-phase and one quadrature signal. In the particularcase of Galileo:••The In-phase signal is the linear sum of the two Open Signal components, OSD andOSP where D stands for data and P for pilot.The quadrature signal carries the Public Regulated Service (PRS) signal and anadditional signal, named inter-modulation product, whose role is to provide themodulation with a constant envelope.Moreover, according to [Galileo SIS ICD, 2008], the power split of the OSD, OSP and PRSmust be 25 %, 25 % and 50 % respectively.As we have seen in chapter 4.6.1, the mathematical expression of the interplex modulation forthe old BOC(1,1) baseline of 2004, it is shown to be :⎡ c (t ) + c P (t )⎤⎛ 1 + sin θ 0 ⎞cos θ 0 s BOC(1,1) (t ) + j ⎜s(t ) = A0 ⎢ D⎟ s PRS (t ) + s IM (t )⎥22⎠⎝⎣⎦⎛ sin θ 0 − 1 ⎞s IM (t ) = − j c D (t ) c P (t ) ⎜⎟ s PRS (t )2⎠⎝(7.115)(7.116)where,••A0 is the amplitude of the modulation envelope,θ 0 is the angle of four of the six phase states of the 6-PSK modulation.
It corresponds•to half the angular distance between 2 states across the real axis,s BOC(1,1) (t ) represents the BOC(1,1) modulation,•s PRS (t ) is the PRS BOCcos(15,2.5) modulation,•sIM (t) is the Inter-Modulation product, used to keep the amplitude constant, and•c D (t ) and c P (t ) are the codes for the E1 OS data and pilot channels respectively.The baseband equations above can also be expressed as follows for the particular case of theGalileo signals on E1:(7.117)s E1 (t ) = α s E1OS (t ) − s E1OS (t ) cos(2πf E1t ) − β s E1PRS (t ) − γ s E1IM (t ) sin (2πf E1t )[beingD]P[]E1(t )cDE1OS (t ) sDE1OS (t )sE1OS (t ) = d OSDsE1OS (t ) = cPE1OS (t ) sPE1OS (t )PE1E1E1(t )cPRS(t ) sPRS(t )sE1PRS (t ) = d PRS(7.118)sE1IM (t ) = sE1OS (t ) sE1OS (t ) sE1PRS (t )DP284Signal Multiplex Techniques for GNSSHere the coefficients α, β and γ play the same role as the phase angles θ 0 and the amplitudeA0 in the equations of previous chapters.
Moreover, D stands for data, P for pilot, d(t) is thedata signal, c(t) is the PRN code and s(t) is the modulated signal. As it can be shown, theresulting modulation is a 6-PSK or Hexaphase modulation with constant envelope, alsoknown as Modified Hexaphase for this reason. Next figure shows the phase plot:Figure 7.9. Modified Hexaphase modulation with BOC(1,1)where the angle θ 0 is chosen so as to provide the appropriate power ratio as described in theAppendix J.
In our case, in order to have the power ratios given in [Galileo SIS ICD, 2008],the value must be of:θ 0 = 0.1082 π = 0.3399 radians(7.119)To have a better understanding on the location of the phase states, we show next the differentprobabilities of the constellation phase points by means of the following truth table. It must benoted that the amplitudes were not corrected to account for the loss of efficiency that resultsfrom the inter-modulation signal IM.Table 7.5. Phase states of the Interplex modulation as a function of code and data inputs{E1 OSD, E1 OSP, PRS}E1 OSD+E1 OSPPhase stateInter-Modulation+1,+1,+2+21-0.5+1,-1,+202+0.5-1,-1,+2-23-0.5-1,+1,+202+0.5+1,+1,-2+26+0.5+1,-1,-205-0.5-1,-1,-2-24+0.5-1,+1,-205-0.5We can graphically see this also as follows:285Signal Multiplex Techniques for GNSSFigure 7.10.
Modified Hexaphase modulationAs we can see from the figure and table above, states 2 and 5 occur each with a probability of25 %. As a result, the OS channel transmits no signal 50 % of the time. Moreover, binarycodes were assumed with values {+1,-1} of equal probability. In addition, the PRS has 3 dBmore power than the OS signals in consonance with [Galileo SIS ICD, 2008]. As we can see,the mission of the Inter-Modulation signal IM is to bring the phase points back to the circle asdepicted by the arrows of the figure above.The final power distribution inside the modulation takes thus the following values for thebaseline of 2004 (see chapter 3.5 for more details):Table 7.6. Power distribution of Interplex with OS and PRSBOC(1,1)RelativePowerOSAOSBPRSIM2/92/94/91/9-6.53 dB -6.53 dB -3.53 dB -9.54 dBAs we can observe, the IM term has a power level of -9.54 dB with respect to the totaltransmitted power and 6 dB below the PRS.
It is important to note that in a real applicationthe values derived above should be further compensated to account for the different losses ofevery signal through the satellite filter.Moreover, the amplitude of the envelope, A0, must be modified to compensate the loss ofefficiency of the modulation due to the presence of the Inter-Modulation product. In thepresent case, A0 is set to 9 8 = 1.0607 .
After applying the amplitude compensation, thepower distribution adopts the following form:286Signal Multiplex Techniques for GNSSTable 7.7. Compensated power distribution to match the baseline valuesBOC(1,1)OSAOSBPRSIM1/41/41/21/8-6 dB-6 dB-3 dB-9.04 dBRelative PowerFinally, it is interesting to note that the power split between the OS and PRS channels can beeasily modified playing with the parameters A0 and θ 0 .
In terms of phase states, the effectwould be a movement of the phase angle θ 0 within the circle of the constellation.7.7.6Power Spectral Density of CASM and InterplexAlthough CASM and Interplex correspond to two different implementations, the simplifiedmathematical description is very similar, being only the Inter-Modulation components slightlydifferent. As we have seen in the lines above, the baseband expression can be expressed as:(7.120)sBB (t ) = P2 s2 (t ) + P3 s3 (t ) + j P1 s1 (t ) − PIM s1 (t ) s2 (t ) s3 (t )wherePPPIM = 2 3(7.121)P1[]Moreover, as shown in [E.
Rebeyrol et al., 2006],{sBB (t ) = Re sBB (t )e j 2πf c t}(7.122)such that:1ℜ s (τ ) = ℜ s BB (τ )cos (2πf cτ )2(7.123)As shown in Appendix I, the autocorrelation of the baseband signal adopts the form:ℜ s BB (τ ) = E { sBB (t ) sBB (t − τ )}(7.124)If we further assume ideal codes, the expression for the autocorrelation function will be:ℜ s BB (τ ) = P1 ℜ s1 (τ ) + P2 ℜ s 2 (τ ) + P3 ℜ s 3 (τ ) + PIM ℜ s IM (τ )(7.125)In addition, since the power spectral density is the Fourier Transform of the autocorrelationfunction, we have:{}Gs BB ( f ) = FT ℜ s BB (τ ) = P1 Gs1 ( f ) + P2 Gs 2 ( f ) + P3 Gs3 ( f ) + PIM Gs IM ( f )Equally, for the whole signal including carrier frequency, we would have:11GInterplex ( f ) = Gs BB ( f − f c ) + Gs BB ( f + f c )44(7.126)(7.127)287Signal Multiplex Techniques for GNSS7.7.7CASM Modulation in GPSAs we saw in chapter 7.7, if we apply CASM to all the GPS signals except for GPS L1C, thePower Spectral Density is shown to adopt the following form:( f ) = PC/A GC/A ( f ) + PP (Y ) GP (Y ) ( f ) + PM − Code GM − Code ( f ) + PIM GIM ( f )GsGPSBB(7.128)where the power spectral densities of the C/A Code, P(Y) Code and M-Code were alreadyshown in chapter 4.3.2.
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