On Generalized Signal Waveforms for Satellite Navigation (797942), страница 40
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Wallner et al., 2005]. Minimum: 0.187 dB,mean 0.247 dB and maximum 0.310 dB173GNSS Signal StructureThe mathematical framework of the methodology described in Appendix M can be easilyexpanded to other bands as for example E5-L5 or to other SBAS systems as done in[S. Wallner et al., 2005] where the results with smooth spectra and real codes were compared.One final comment on the figures above is that the minimum power levels were used for thesimulations resulting thus in significantly higher values of interference than those we willobserve in a typical scenario. Nonetheless, it must be kept in mind that the purpose of aninterference methodology is to assess compatibility in all the cases, and therefore looking atthe worst cases is thus of major interest.4.8Other Modulation Schemes4.8.1AltBOC ModulationThe Alternative BOC modulation (AltBOC) is conceptually very similar to the BOCmodulation but with an important difference, since contrary to BOC, AltBOC provides highspectral isolation between the two upper main lobes and the two lower main lobes(considering the I and Q phases separately).
This is accomplished by using different codes foreach main lobe. We can clearly see this if we remember how single band processing works[J. W. Betz, 1999]. Indeed, any BOC signal could be correlated with a BPSK replica havingas chip rate the sub-carrier frequency of the original BOC signal. Of course the prize is theloss of power, but processing the upper or lower main lobe would make no difference sinceboth are modulated with the same PRN code. On the other hand, if we would do the samewith the AltBOC signal, we could still receive each main lobe separately since different codeswould be needed.
This is very interesting because AltBOC allows thus keeping the BOCimplementation simple while permitting to differentiate the lobes [E. Rebeyrol et al., 2005].Similar to the BOC modulation, for simplicity the AltBOC modulation is generally referred toas AltBOC(fs, fc) with f s = m ⋅ 1.023 and f c = n ⋅ 1.023 so that commonly one only saysAltBOC(m, n) for simplicity.The Alternative BOC modulation uses a complex sub-carrier so that the spectrum is not splitup, as is the case of BOC, but simply shifted to higher or lower frequencies.
We alreadyanalyzed in chapter 4.3.2 the problematic of using different conventions to represent the BOCsignals, and indeed the same conclusions are also valid here for the AltBOC modulation. Inorder to avoid this, we will make use of the definition presented in [L. Ries et al., 2003] wherethe AltBOC signal is defined as the product of a PRN code sequence with a complex subcarrier.
This convention covers even and odd ratios with no necessary modification.The AltBOC signal can be formed by two (only data signals) or four codes (data and pilot). Ifwe have only two codes, the signal is composed of only data and can be expressed as follows:sAltBOC (t ) = cL cs (t ) + cU cs (t )*(4.204)174GNSS Signal Structurewhere cs(t) is the complex sub-carrier, complex sum of the rectangular cosine and sine-phasedrectangular waveform.
As we can see, this is the binary version of the complex exponentialfunction and can be defined as followscs (t ) = sign[cos(2πf st )] + j sign[sin (2πf st )] = cr (t ) + j sr (t )(4.205)where cL and cU are the lower and upper codes respectively. As we can recognize from(4.204), what we are basically doing by multiplying the lower code and the upper code by thecomplex sub-carrier and its conjugate is approximately to shift the lower code to –fs and theupper code to +fs. In fact, this would be the case if we would multiply with the exponentialfunction:(4.206)e j 2πfst = cos(2πf s t ) + j sin (2πf s t )Although not exactly the same, the binary complex function is indeed a good approximation.Furthermore, AltBOC can be seen as a particular case of MCS with complex chip waveform.The following figure shows the spectrum (in units of Watts) of the complex sub-carrier and itsconjugate.
As we can see, most of the power is concentrated at the coefficients +1 and -1 andonly the first 10 negative and positive coefficients are depicted in the figure (a repetitioninterval of 20 samples was assumed for the simulation). It is important to note that +1 and -1corresponds in general to +fs and – fs after normalizing by the repetition period of theexponential function. Moreover, we can recognize that the spectrum, indeed the square of the2Fourier coefficients c k , is normalized to integrate to 2 W of power.Figure 4.70. Spectrum c k2[W] of the complex sub-carrierIf we add now the pilot channel to the definition, the general expression of the AltBOCmodulation will adopt the following form:()()sAltBOC (t ) = cLD + j cLP cs (t ) + cUD + j cUP cs (t )*(4.207)where cLD is the data lower code, cLP the pilot lower code, cUD the data upper code and cUP thepilot upper code.175GNSS Signal StructureThe signal defined above corresponds to the general case of the AltBOC modulation.
Theonly problem is that by introducing a complex sub-carrier and complex codes for the data andpilot channels, the composite signal looses the constant envelope that the original BOCmodulation possessed. As we have seen when we analyzed the MBOC modulation, having aconstant envelope is a must since otherwise the distortion caused by the High PowerAmplifier (HPA) in the satellite would not be tolerable.In order to solve this problem, a constant envelope modified version of the AltBOCmodulation was presented in [J. Godet, 2001]. The idea behind is to bring the phase points ofthe constellation back to the circle so that the amplitude of the envelope remains constant.This is achieved by introducing a new signal called Inter-Modulation (IM) product whosenew terms do not contain any usable information.
The modified constant envelope AltBOC[M. Soellner et al., 2003] and [E. D. Kaplan and C. Hegarty, 2006], yields:⎧ D⎪ cL +⎪⎪ D⎪ cU +⎪sAltBOC (t ) = ⎨⎪ cD +⎪ L⎪⎪ D⎪ cU +⎩((((⎡⎛ T ⎞⎤j cLP ⎢ scd (t ) − j scd ⎜ t − s ⎟⎥ +4 ⎠⎦⎝⎣⎡⎛ T ⎞⎤j cUP ⎢ scd (t ) + j scd ⎜ t − s ⎟⎥ +4 ⎠⎦⎝⎣⎡⎛ T ⎞⎤j cLP ⎢ sc p (t ) − j sc p ⎜ t − s ⎟⎥ +4 ⎠⎦⎝⎣)))(4.208))⎡⎛ T ⎞⎤j cUP ⎢ sc p (t ) + j sc p ⎜ t − s ⎟⎥4 ⎠⎦⎝⎣where Ts is the period of the sub-carrier. Moreover,cLD = cUP cUD cLPcLP = cUD cUP cLDcUD = cLD cUP cLPcUP = cUD cLD cLP(4.209)and the following data and pilot sub-carriers⎧ 2⎡ ⎛⎡ ⎛2π ⎞⎤ 1π ⎞⎤sign ⎢cos⎜ 2 π f s t + ⎟⎥scd (t ) = ⎨ sign ⎢cos⎜ 2 π f st − ⎟⎥ + sign[cos(2 π f s t )] +4 ⎠⎦ 244 ⎠⎦⎣ ⎝⎣ ⎝⎩ 4(4.210)⎧⎡ ⎛⎡ ⎛22π ⎞⎤ 1π ⎞⎤sign ⎢cos⎜ 2πf st − ⎟⎥ + sign[cos(2 π f s t )] −sign ⎢cos⎜ 2 π f s t + ⎟⎥sc p (t ) = ⎨−4 ⎠⎦ 244 ⎠⎦⎣ ⎝⎣ ⎝⎩ 4(4.211)As we can see, while in our original conception the complex sub-carrier was composed of acosine-phased rectangular signal for the real part and a sine-phased rectangular signal for thecomplex part, now both the real and complex part are a mixture of both sine and cosinedelayed and early rectangular waveforms.176GNSS Signal StructureFigure 4.71.
Shapes of data and pilot sub-carriersAnother interesting observation is that if we take a look at the phase plots of the constantenvelope AltBOC signal, we can clearly recognize that it is a classical 8-PSK modulationwith a non-constant allocation of the 8 phase-states.As shown in Appendix I, the power spectral density for the modified even AltBOCmodulation with constant envelope is shown to be:⎛ πf ⎞sin 2 ⎜⎜ ⎟⎟4fΦ even ,c⎝ f c ⎠ ⎡cos 2 ⎛⎜ πf ⎞⎟ − cos⎛⎜ πf ⎞⎟ − 2 cos⎛⎜ πf ⎞⎟ cos⎛⎜ πf ⎞⎟ + 2⎤GAltBOC( f ) = 2 c2⎢⎜2f ⎟⎜2f ⎟⎜2f ⎟ ⎜4f ⎟ ⎥π f⎛ πf ⎞ ⎣⎝ s⎠⎝ s⎠⎝ s⎠ ⎝ s⎠ ⎦⎟⎟cos 2 ⎜⎜⎝ 2 fs ⎠(4.212)while for the odd case, we have:⎛ πf ⎞cos 2 ⎜⎜ ⎟⎟4fΦ odd , c⎝ f c ⎠ ⎡cos 2 ⎛⎜ πf ⎞⎟ − cos⎛⎜ πf ⎞⎟ − 2 cos⎛⎜ πf ⎞⎟ cos⎛⎜ πf ⎞⎟ + 2⎤GAltBOC( f ) = 2 c2⎢⎜2f ⎟⎜2f ⎟⎜2f ⎟ ⎜4f ⎟ ⎥π f2 ⎛ πf ⎞ ⎣s ⎠s ⎠⎝⎝⎝ s⎠ ⎝ s⎠ ⎦⎟⎟cos ⎜⎜f2⎝ s⎠(4.213)Finally, it is important to underline that the AltBOC modulation in its most general form doesnot have a constant envelope as shown in Appendix I.
As shown in chapter 7.7, due to theneed to have a constant envelope, slight changes were made in the multiplexing scheme andthe result was the modified AltBOC modulation that we have shown in the previous lines.AltBOC can not only be understood as a signal waveform but also as a multiplexing schemeas those that we will see in chapter 7.
The same comment is indeed also valid for the BOCmodulation that we studied in previous chapters.177GNSS Signal Structure4.8.2Square-Root Raised Cosine Signals (SRRC)One of the main drawbacks of all the signal waveforms studied so far is that although theycan very well control the power emissions within the bandwidth of interest, they sendrelatively high amounts of power out of this one. A practical way of reducing the side-lobesof the spectrum of the navigation signals could be to use a Raised Cosine Filter (RCF) sincethis has a limited bandwidth.
The Raised Cosine Filter is a particular case of Nyquist filterand is defined in the frequency domain as followsfor | f |< 2W0 − W⎧1⎪⎪⎡ π | f | +W − 2W0 ⎤H RC ( f ) = ⎨cos 2 ⎢⎥ for 2W0 − W <| f |< WW − W0⎣4⎦⎪for | f |> W⎩⎪0(4.214)where W − W0 is defined as the excess bandwidth and indicates how much the spectrum ofthe Raised Cosine spills over a given bandwidth W0 . As we know, Nyquist pulses (filters) arepulses that result in no Inter Symbol Interference (ISI) at the sampling time. The Nyquistpulse-shaping criterion or Nyquist condition for zero ISI is fulfilled ifx(kTc ) = δ (k )where⎧1 k = 0⎩0 k ≠ 0δ (k ) = ⎨(4.215)(4.216)Indeed, this is a necessary and sufficient condition which can also be expressed as follows:⎛m⎞X ⎜⎜ f + ⎟⎟ = TcTc ⎠m = −∞⎝∞∑(4.217)where X ( f ) is the Fourier transform of a generic signal x(t) and Tc the time period of thepulse.
The Raised Cosine filter that we described some lines above has an equivalentrepresentation in the time domain. This is shown to be:hRC (t ) = 2W0 sinc(2W0t )cos[2π (W − W0 )t ]1 − [4(W − W0 )t ]2(4.218)Another way of expressing the excess bandwidth is by means of the roll-off factor, which isdefined as follows:W − W0α=(4.219)W0The roll-off factor indicates how much power the Raised Cosine emits above a givenbandwidth W0.
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