On Generalized Signal Waveforms for Satellite Navigation (797942), страница 59
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According to(7.67), the fraction of time values will be tMaj , tc 2 , tc3 = {2 5 ,1 5 , 2 5} . This means that out of{}the 100 chips of a bit, 40 of them will directly correspond to the highest gain code c3 . In thesame manner other 20 will be devoted to the medium gain code c 2 and the final 40 chips aredetermined following the true majority vote of the chips of the three codes as described inchapters 7.4.4 and 7.4.5.As shown by [G. L. Cangiani et al., 2002], the interest of this approach is that the combininglosses are distributed uniformly over the three original signals in such a way that all thesignals will suffer the same percentage loss of power at the end. The efficiency of the threecode multiplex is shown to adopt the following form:η MV =G1 + G2 + G3(G2 + G3)2(7.68)which can also be expressed as follows taking as power reference the signal with the lowestgain, namely c1G G1+ 2 + 3G1 G1η MV =(7.69)2⎛ G2⎞G3 ⎟⎜⎜ G + G ⎟11 ⎠⎝Next figure depicts the efficiency of the majority vote as a function of the power ratiosG2 G1 and G3 G1 :Figure 7.5.
Majority Vote Efficiency for three codes c1 , c2 and c3269Signal Multiplex Techniques for GNSSAs we can recognize in the previous figure, when the power of one of the codes is muchlarger than that of either of the other two, the multiplex efficiency approaches 100 %.However, in the case that one code is much smaller that the other two, the efficiencydeteriorates to approximately 50 %.
If the three codes are of similar power levels, theefficiency is then close to 75 %. Although the previous values are only valid for the threecode multiplexing case, similar behaviours are found when the number of codes to multiplexincreases. Indeed, majority voting presents the great inconvenience that in presence of a fewcodes monopolizing the total power, the efficiency reduces considerably.In the previous three-code case, the solution for the time fraction of each individual code ormajority voted code was unique. Indeed, the number of target parameters was two (since thethree power ratios sum to unity there are only two free ratios) and the number of free variableswas also two (since the three time fractions also sum to unity, only two are free).
Howeverthis is not the case always. Indeed, when the number of signals to multiplex increases, so doesalso increase the number of possible solutions of the fractions of time that deliver the targetedpower ratios. As an example, in the five-code case, the Generalized Majority Vote (GMV)signal could consist of the following signals [G. L. Cangiani et al., 2002]:• One five majority-vote code⎛ 5⎞• ⎜⎜ ⎟⎟ three-way code combinations⎝ 3⎠••Four solo chips. It must be noted that the weakest code is not used as shown above⎛5⎞⎜⎜ ⎟⎟ four-way code combinations where one of the codes is weighted twice.⎝ 4⎠If we consider all the potential components of the GMV signal described above, there are atotal of 20 elements to consider.
Nevertheless, since the fraction times must sum to unity, thereal number of free variables is actually 19. This number is however by far greater than thecommanded powers (four in the case of five multiplexed signals) so that at the end the mostefficient multiplex can only be discovered by a search technique of all potential combinations.Last but not the least, it is important to mention that the cyclostationary solutions depicted inprevious lines also fall in the general mathematical description given by (7.24). In fact,recalling the general equation of GMV, the majority voted code will adopt the following form⎞⎛ NcMaj (k ) = sign ⎜ ∑ λi (k )ci (k )⎟⎠⎝ i =1(7.70)taking on the sub-majority vote factors λi (k ) values of 0 or 1 in this case.
It is important tomention here that while in the approach of previous lines the weighting was achieved byaveraging over the time, in chapter 7.4.5 this effect was mainly reached by selecting a properweighting factor.270Signal Multiplex Techniques for GNSS7.5Hard LimitingAs we have already mentioned in previous chapters, during the design of the GPS M-Codedifferent multiplexing options were considered to achieve a constant envelope[P.A. Dafesh et al., 2006]. Together with the majority voting that we saw above, hard-limitingof the C/A Code, P(Y) Code and M-Code signals was another considered option.The hard-limiting approach basically forces the non-constant envelope of the sum of C/ACode and M-Code on one phase and P(Y) on the orthogonal phase (assuming we want toplace the M-Code on the same phase as the C/A Code) to be constant by limiting the variationof the amplitude to its minimum value.
The problem though is that the C/A Code and M-Codesuffer from significant power losses and distortions. Indeed, the total efficiency of the hardlimiting modulation is roughly of 83 % as shown in [P.A. Dafesh et al., 2006], whatcorresponds to approximately 0.8 dB of overall combining losses. Moreover, as alsodescribed in this paper, the power split between the different signals is not easy to achieve.7.6Quadrature Product Sub-carrier ModulationThe quadrature product sub-carrier modulation (QPSM) enables the transmission of aquadrature multiplexed carrier modulation with one or more sub-carrier signals in the sameconstant envelope waveform as shown in [P.A.
Dafesh, 1999] and [P.A. Dafesh et al., 2006].Moreover, in its generalized form, QPSM is capable of applying sub-carrier modulation toalready existing systems with Quadrature Phase Shift Keying (QPSK) or Minimum ShiftKeying (MSK). In fact, one of its main advantages is that it can easily introduce newadditional spread signals with excellent spectral isolation to those already in the band, usingthe same transmitter power amplifier. This is accomplished using multiple rate product codesthat cause minimal interference to the existing ones. Finally, the power control and energydistribution between the carrier and sub-carrier signals can be accomplished selecting thedesired modulation index.In the next lines we show how the QPSM modulation can be represented mathematically.However, let us begin first with a simplified model.
Let us assume a pair of quadraturecomponents I0 and Q0 onto a carrier signal as follows,s (t ) = I 0 (t ) cos(2πf c t ) − Q0 (t )sin (2πf c t )(7.71)where fc denotes the carrier frequency. As we can clearly recognize, the magnitude is constantand can be expressed as follows [S. Butman and U.
Timor, 1972]:wheres (t ) = A0 (t )cos [2 π f c t + φ (t )](7.72)A0 (t ) = I 02 (t ) + Q02 (t )(7.73)271Signal Multiplex Techniques for GNSSand⎛ Q0 (t ) ⎞⎟⎟⎝ I 0 (t ) ⎠φ (t ) = arctan ⎜⎜(7.74)From the equations above we can recognize that the resulting composite signal has a constantmagnitude A0 and does not depend on time as long as I 02 (t ) and Q02 (t ) do not vary, what isalways satisfied if I0 and Q0 are binary sequences.Indeed this is the basic idea of the QPSM modulation.
If we generalize now this principle, wecan further add new signals modulating the phase part according to the following approach[S. Butman and U. Timor, 1972]:withs (t ) = I 0 (t )cos[2π f c t + φs (t )] − Q0 (t )sin[2π f c t + φs (t )](7.75)φ s (t ) = ∑ m j s j (t )ϕ j (t )(7.76)jwhere• mj is the modulation index of the j-th signal.
It determines the power allocation of eachcomponent,• s j (t ) is the j-th signal to modulate. It can be expressed as the product of the respective•data message and the PRN code.and ϕ j (t ) is the periodic sub-carrier, which may be any regular signal as for examplesine, square or triangular, for example.It is trivial to see that the envelope also remains constant in this modulated carrier signal.Moreover, the sub-carrier signal can be made up of many components being the limit only thephase noise that appears when the states of the constellation come too close to each other, aswe will see later.
Furthermore, the model is not only valid for binary sequences. In fact, theperiodic sub-carrier could adopt any form in principle as long as it is regular. This is veryimportant, because as we saw in chapter 4.6.1, the CBCS and CBOC signals are not binary.The conventional sub-carrier modulation presents so-called cross-product inter-modulationcomponents which can be considered as signal losses, resulting thus in a loss of efficiency.The conventional constant envelope Sub-carrier Modulation is used today on the SpaceGround Link Subsystem (SGLS) and other terrestrial and space systems as shown by[P.A.
Dafesh et al., 1999a], [Philco-Ford Corp., 1968], [J.K. Holmes, 1982] and[M. M. Shihabbi et al., 1994]. A generalization of the Sub-carrier modulation that has gainedin interest over the past years is the Coherent Adaptive Sub-Carrier, which is presented in thefollowing chapter.Finally, it is important to note that the spectral separation of the different signals to modulatedepends on the sub-carrier signal in particular allowing thus for spectral control as desired.272Signal Multiplex Techniques for GNSS7.7Coherent Adaptive Sub-Carrier Modulation(CASM) and Interplex7.7.1Origins of CASM and InterplexThe CASM Modulation is very similar to Interplex [S. Butman and U. Timor, 1972].
Itreceives also the name of Modified Tri-code Hexaphase modulation since it can be seen as aconstant envelope modification of the Tri-code Hexaphase modulation described in chapter7.3.1. CASM was first proposed in [M. Ananda et al., 1993] and later patented by[P.A. Dafesh, 2002] while Interplex was patented by [G.L. Cangiani, 2005].In spite of CASM and Interplex being mathematically very similar, an important distinctionmust be made regarding the implementation.
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