On Generalized Signal Waveforms for Satellite Navigation (797942), страница 61
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Pratt and J.I.R. Owen, 2005] and [P.A. Dafesh et al., 2006]. If we take a closer look atthe equations, we can recognize the desired in-phase and quadrature components of s(t)multiplied by cos(m) while the additional non-desired components added by the sub-carriermodulation appear multiplying sin(m). We have assumed from the very beginning that thedata and code spreading codes are binary, and thus a useful multiplex option is shown to bethe one that results from further assuming the following values for the partitioning functions:α d (t ) = d I (t )β c (t ) = c I (t )(7.88)These partitioning functions make use of the binary characteristics of the codes and data toseparate them.
In fact, substituting in (7.86), we obtain the following identities:d IS (t ) = d S (t )d QS (t ) = d I (t ) d Q (t ) d S (t )(7.89)cIS (t ) = cS (t )cQS (t ) = cI (t )cQ (t )cS (t )Moreover, assuming that the codes cI (t ) and cQ (t ) are optimized to be orthogonal with eachother, as it is the case, and that the I and Q data signals are also independent and thus ideallyorthogonal, we can rename the terms cQS (t ) and d QS (t ) as c S ,IM (t ) and d S ,IM (t )correspondingly, where the term IM denotes the Inter-Modulation signal. Finally, applying thechanges described above, our multiplexed signal s(t) can be expressed as follows:[]⎧ 2 PI cos(m ) d I (t ) c I (t ) − 2 PQ sin (m ) d S , IM (t ) c S ,IM (t )sign[sin (2πf s t )] cos(2πf c t )⎪s (t ) = ⎨⎪⎩− 2 PQ cos(m ) d Q (t ) cQ (t ) − 2 PI sin (m ) d S (t ) c S (t )sign[sin (2πf s t )] sin (2πf c t )[](7.90)276Signal Multiplex Techniques for GNSSIt is interesting to note from the expression above that the original quadrature productcomponents are reduced by a factor cos(m).
We recall that these quadrature productcomponents could correspond to two already existing orthogonal signals as could be the C/ACode and the P(Y) Code in GPS (CASM implementation) before the M-Code was introduced.But also Galileo with Interplex follows a similar pattern with the OS signals in-phase and thePRS in quadrature. In addition, we can see a new sub-carrier component with a BOCmodulation and data d S (t ) and code cS (t ) . Finally, we can equally recognize a fourthcomponent with the codes and data signals c S ,IM (t ) and d S ,IM (t ) . Since this last signal doesnot transmit any useful information in the general case as it is modulated by the crosscorrelation of two codes that are ideally orthogonal and two data streams that in an ideal caseare also independent, this term is further called inter-modulation component.
Nonetheless, itmust be noted that if an appropriate structure were found for the 3 spreading codes cI (t ) ,cQ (t ) and cS (t ) , this could be used as a fourth channel to transmit an additional signal.In summary, CASM and Interplex offer a QPSK signal and an additional BOC, all of themforming a constant envelope multiplexed signal. The phase constellation diagram is given bythe following 8 points:⎛⎛ PQ ⎞ ⎞⎟⎟θ (m ) = ±⎜ π ± m ± arctan⎜(7.91)⎜⎜ PI ⎟ ⎟⎝⎠⎠⎝and the power of the different channels is thus:Table 7.4. Power Distribution of the CASM and Interplex multiplexingI ChannelPI cos2(m)Q ChannelPQ cos2(m)S ChannelPI sin2(m)IM ChannelPQ sin2(m)We can further analyze the equations if we recall again the equation that we derived above:[]⎧ 2 PI cos(m ) d I (t ) cI (t ) − 2 PQ sin (m ) d QS (t )cQS (t )sign[sin (2πf s t )] cos(2πf c t )⎪s (t ) = ⎨⎪⎩− 2 PQ cos(m ) d Q (t ) cQ (t ) − 2 PI sin (m ) d IS (t ) cIS (t )sign[sin (2πf s t )] sin (2πf ct )whered IS (t ) = d I (t ) d S (t )α d (t )[d QS (t ) = d Q (t ) d S (t )α d (t )c IS (t ) = c I (t ) c S (t ) β c (t )](7.92)(7.93)cQS (t ) = cQ (t ) c S (t ) β c (t )We can further project the amplitudes of each of the signals on the I and Q axes, such that:2 PI = 2 P cos(β )2 PQ = 2 P sin (β )(7.94)277Signal Multiplex Techniques for GNSSwhere P is the total power of the signal and β an additional variable to link the I and Qpowers, PI and PQ respectively.
Furthermore, we rename the variables β and m as follows:β2 = β −⎧sin (β ) = − cos(β 2 )⇒⎨2⎩ cos(β ) = sin (β 2 )π(7.95)β3 = mand redefine the signals as follows:s1 (t ) = d Q (t ) cQ (t )s 2 (t ) = d I (t ) c I (t )s 3 (t ) = d QS (t ) cQS (t ) = d Q (t ) d S (t )α d (t ) cQ (t ) c S (t ) β c (t )We can show that then:yielding thus,(7.96)s2 (t ) s3 (t ) = d S (t )α d (t )cS (t ) β c (t )(7.97)d IS (t ) cIS (t ) = d I (t ) d S (t )α d (t )cI (t ) cS (t ) β c (t )(7.98)which can be further simplified to:d IS (t ) cIS (t ) = [d I (t ) cI (t )][d S (t )α d (t ) cS (t ) β c (t )] = s1 (t ) s2 (t ) s3 (t )(7.99)If we introduce these expressions in (7.92), we have then:[]⎧⎪ 2 P sin (β 2 ) cos(β 3 ) s2 (t ) + 2 P cos(β 2 )sin (β 3 ) s3 (t ) cos(2 π f ct ) +s (t ) = ⎨⎪⎩+ 2 P cos(β 2 )cos(β 3 ) s1 (t ) − 2 P sin (β 2 )sin (β 3 ) s1 (t ) s2 (t ) s3 (t ) sin (2πf c t )[](7.100)which coincides with the form derived by [E.
Rebeyrol et al., 2005]. In fact, it is not difficultto show that after some math this expression can be further simplified to:π⎞⎛s (t ) = 2 P cos⎜ 2 π f c t − s1 (t ) + β 2 s1 (t ) s2 (t ) + β 3 s1 (t ) s3 (t )⎟2⎝⎠(7.101)where the factor multiplying the signal s1(t), namely β1, is equal to − π 2 so that this signal isin quadrature with the other two we want to modulate, namely s2(t) and s3(t).
Moreover, thenormalized powers of the different signals can be expressed as follows:P1 (β 2 , β 3 ) = P cos 2 (β 2 )cos 2 (β 3 )P2 (β 2 , β 3 ) = P sin 2 (β 2 )cos 2 (β 3 )P3 (β 2 , β 3 ) = P cos 2 (β 2 )sin 2 (β 3 )(7.102)P4 (β 2 , β 3 ) = P sin 2 (β 2 )sin 2 (β 3 )According to this, the modulated signal could be generated following the scheme shown in[G.L. Cangiani and J.A. Rajan, 2002]:278Signal Multiplex Techniques for GNSSFigure 7.6. Interplex schematic generationThe generation scheme presented above is conceptually useful to derive the theoreticalexpressions of this chapter but presents a series of limitations in a real implementation asidentified by [G.
L. Cangiani et al., 2002].While common practice in the design of the architecture dictates generating the entirecomposite signal at baseband first to further up-convert it later to the desired carrierfrequency, for the frequencies of interest in GNSS (microwave systems) this is a problem. Infact, the baseband frequency is too low to avoid harmonic and intermodulation interferencewith the desired output during the up-conversion. Moreover, the time jitter of theDigital-to-Analog Converters (DAC) adds phase noise onto the desired output signal and theup-conversion process requires band-pass filters at each mixing stage that destroy the originalconstant envelope of the signal.A solution to this problem is the implementation proposed by [G.
L. Cangiani et al., 2002]where the Interplex signal at the desired carrier frequency is generated as depicted in thefollowing figure.Figure 7.7. Alternative Interplex scheme proposed by [G. L. Cangiani et al., 2002]As an example, the variation of the signal power of s1 as a function of the two interplexmodulation angles β 2 and β 3 is shown in the following figure:279Signal Multiplex Techniques for GNSSFigure 7.8. Variation of the P1 power of the signal s1 as a function of β2 and β 3As we can see, the optimum combination of β 2 and β 3 depends on how much power wewant to place on each of the desired signals.
In fact, the principle is to minimize the amount ofpower of the IM channel since this does not bring any information.It must be noted that the equations derived above do not make use of the data and codepartitioning functions. We express now the equivalent form for baseband as follows:⎡⎤P Ps BB (t ) = P2 s 2 (t ) + P3 s 3 (t ) + j ⎢ P1 s1 (t ) − 2 3 s1 (t ) s 2 (t ) s3 (t )⎥P1⎢⎣⎥⎦(7.103)As we can recognize, the equations derived above are based on the assumption of binary dataand spreading codes, and a square-wave sub-carrier signal (BOC).
Accordingly, the resultsare not valid for the most general case but can be easily generalized to other types of signalwaveforms at the extent of some extra terms in the final expressions. Indeed, as we will shownext, adapting this multiplexing to include the CBOC implementation of the MBOC signal istrivial. Furthermore, the derived expressions are only valid for infinite bandwidth, being thusthe real power split among all the signals slightly different after filtering.If we further normalize (7.103) to have unit power, the previous expression adopts thefollowing form:⎡⎤P PP2 s 2 (t ) + P3 s3 (t ) + j ⎢ P1 s1 (t ) − 2 3 s1 (t ) s 2 (t ) s3 (t )⎥P1⎢⎣⎥⎦(7.104)s BB (t ) =P2 P3P1 + P2 + P3 +P1which can be further simplified as follows:s BB (t ) =[]P1 P2 s 2 (t ) + P1 P3 s3 (t ) + j P1 s1 (t ) − P2 P3 s1 (t ) s 2 (t ) s3 (t )P12 + P1 P2 + P1 P3 + P2 P3(7.105)280Signal Multiplex Techniques for GNSSFor simplicity we refer next all the powers to P3 .
In fact, to define uniquely the multiplexwith three signals, two power ratios are necessary if the powers are referred to a third signaland the total power sums to unity. According to this, the three-signal interplex is defined by:I=P1 P2 s 2 (t ) + P1 s3 (t )=P1 P2 s 2 (t ) + P1 s3 (t )(1 + P1 )(P1 + P2 )P s (t ) − P2 s1 (t ) s 2 (t ) s3 (t ) P1 s1 (t ) − P2 s1 (t ) s 2 (t ) s3 (t )Q= 1 1=(1 + P1 )(P1 + P2 )P12 + P1 P2 + P1 + P2P12 + P1 P2 + P1 + P2being the efficiency of the interplex modulation as follows:P 2 + P1 + P1P2P (1 + P1 + P2 )= 1η Interplex = 1(1 + P1 )(P1 + P2 ) (1 + P1 )(P1 + P2 )(7.106)(7.107)This is the percentage of power with respect to the total transmitted power that is employedfor the useful signal.Finally, it is important to mention that additional signals can be incorporated into the CASMand Interplex schemes because the desirable constant envelope characteristic remainsunchanged independent from the shape of the sub-carrier modulation.
The drawback is theextra complexity that is needed to separate spectrally the additional signals, as pointed out in[A.R. Pratt and J.I.R. Owen, 2005] and developed in [T. Fan et al, 2005]. A particularrealization with non-binary sub-carriers is discussed in the next chapter, where a sinewavesub-carrier is employed instead of the usual square-wave version.7.7.3Single Sinewave Sub-carrier CASMThe Single Sinewave Sub-carrier version was originally proposed as a constant-envelopeMulti-ModeSpread-SpectrumSub-carrierModulation(MMSSS)forGPS[P.A. Dafesh et al., 1999a]. This sub-carrier is shown to adopt the following form:ϕ s (t ) = sin (2πf st )(7.108)If we introduce now the sine-wave sub-carrier in equations (7.75) and (7.76) the multiplexedsignal is shown to approximate to the following [P.A.
Dafesh et al., 1999a]:s (t ) ≈⎡ 2 PI J 0 (m )d I (t )c I (t ) −⎤⎥ cos(2πf c t ) −≈⎢⎢⎣− 2 PQ 2 J 1 (m )d Q (t )cQ (t )d s (t )c s (t )α d (t )β c (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 I (t )c I (t )d s (t )c s (t )α d (t )β c (t )sin (2πf s t )⎥⎦(7.109)+ 2 PI 2 J 2 (m )d I (t )c I (t ) cos(4πf s t ) cos(2πf c t ) −− 2 PQ 2 J 2 (m )d Q (t )cQ (t ) cos(4πf s t )sin (2πf c t )281Signal Multiplex Techniques for GNSSwhere Jn(m) is the Bessel function of order n and m the modulation index of the signal. As wecan recognize, the previous expression is only approximate as an infinite number of carrierharmonics (J0, J2, J4,…) and sub-carrier harmonics (J1, J3, J5, …) would be required to definethe multiplexed signal completely.
However, only the first carrier and sub-carrier harmonicsneed to be considered if m ≤ π 2 . This will be assumed in the following lines. Furthermore,for this range of modulation index the sinewave CASM approach presents a high efficiency ofapproximately 95 % as shown in [P.A.
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