On Generalized Signal Waveforms for Satellite Navigation (797942), страница 49
Текст из файла (страница 49)
Thus we could easily estimatethe SSCs between two C/A codes by just adding a constant offset of approximately 11.25 dBto the smooth spectrum SSC since this is the difference between the constant value SSC(smooth spectrum, random codes, no data) and the peak of the repetitive codes (random or219Spectral Separation Coefficients with data and non ideal codesselected data sequences). Keep in mind that the average of the SSC values for C/A codes overall possible Doppler values leads to the smooth SSC. Moreover, using the previous figures,one does not require to know the exact data sequence or satellite interfering since the resultsare an average of all cases. As one can imagine, this can simplify computations enormouslywhat is of great interest for example with interference computations.Furthermore, if we work with the spectrum normalized to 30.69 MHz and integrate it within anarrow bandwidth of 4.092 MHz, only 95.05% of the power falls within the consideredbandwidth, and a correction of 0.2203 dB would be necessary in the PSD.
To compare theresults, this corresponds approximately to the difference of 0.4429 dB that we can observebetween the ideal SSC for 30.69 MHz and that with 4.092 MHz. For narrower bandwidthslike 2.046 MHz a larger correction of 0.8862 dB should be applied following the reasoningabove. This value is very close to that obtained from the simulations. As a conclusion, thenarrower the bandwidth, the higher the power spectral density and the higher will be the SSCas we can observe in Table 6.3.Additionally, we can see from the table above that when a wide bandwidth is considered,already 5 bits, thus 100 ms, seem to be sufficient in average to estimate correctly the PSD ofthe signal with data.
In other words, 5 bits seem to be enough to consider the ideal spectrumas a good approximation. However, for narrower bandwidths more bits have to be consideredto correctly estimate the SSC.This is important because many analyses on interference are highly dependant on the SSCsbetween the interfering and the desired signals. In fact, if a brute force approach werefollowed and all the possible SSCs for all possible combinations of satellites, data andreceiver implementations had to be computed, the computational load would be unaffordable.As a conclusion, if we compare the results that come from assuming ideal PSD with randomdata and those obtained from the simulations for a relatively low number of chips, we can seethat the difference in dB is always smaller than 1 dB.As we mentioned above and the previous figure for the combination of 5 data bits shows indetail, a close look into the fine structure reveals that the spectra are only slightly different.Indeed, the amplitude of the main lobes of the data sinc are nearly equal in both ideal andsimulated power spectral densities, being the main difference the amplitude of the secondarylobes of the data sinc.
It is also interesting to note that if the number of considered bits isincreased from 5 to 10, the shape of the estimated PSD does not significantly change and thedifference still remains. Furthermore, the previous figures were calculated for a bandwidth of30.69 MHz but similar can be obtained for the case of 2.046 MHz. The results are the sameexcept for the fact that the ideal spectrum and the averaged spectrum are higher by 0.55 dBdue to the normalization of the power in a narrower bandwidth.220Spectral Separation Coefficients with data and non ideal codesIt is also interesting to compare the estimated PSDs that result from integrating for very shortperiods with the ideal PSD with ideal data.
Indeed, as next figure shows, if only 1 data bit isconsidered, although the peaks in steps of 1 kHz can already be localized, the side lobescannot be distinguished yet. Moreover, the amplitude of the peaks is something different tothat of the ideal PSD, what explains the difference in the SSCs.Figure 6.9. Power Spectral Density of GPS C/A Code SVN 1 with data and differentnumber of bits consideredThe figures that have been shown above correspond to the average of all the possiblespectrum combinations of 5 bits of data. As we can clearly see, the spectrum is already verysmooth. However, for this short coherent integration, if a specific combination of bits isconsidered, the resulting spectrum is in the general case relatively spiky.Figure 6.10.
PSD of GPS C/A code SVN 1 that results from taking the 5 bits combination[1 1 1 1 -1] at low frequenciesWe conclude that although the average of 25 bit combinations results in a pretty smoothestimation of the PSD, when we analyze a specific data sequence, the spectrum will present inthe general case a pretty abrupt shape.221Spectral Separation Coefficients with data and non ideal codesOnce we have graphically shown the results for the C/A Code, we repeat next the analysis forother signals of interest and for different bandwidths.
We begin with BOC(1,1). The resultsare summarized in Table 6.4.Figure 6.11. Averaged PSD of GPS C/A code and BOC(1,1) with SVN 1 that results fromaveraging all possible combinations of 5 bitsFor the case of BOC(1,1), 5 bits of integration correspond to a different total integration timethan for the GPS C/A Code. Indeed, while for GPS C/A code 5 bits correspond to 100 ms, forthe Galileo E1 OS these correspond to only 20 ms.
For a correct comparison, the estimatedPower Spectral Densities were normalized to have unity power in the corresponding timeframe of the simulation.Table 6.4. BOC(1,1) SSC [dB-Hz] computed between the averaged spectra of SVN 1 andSVN 2 for a data rate of 250 sps. A spectral resolution of 5 Hz was assumedSignalBOC(1,1)Double-Sided Bandwidth [MHz]30.6924.55204.0922.046SSC with ideal primary randomcodes and no data-64.6920-64.6477-63.5526-61.8483SSC with ideal primary randomcodes and ideal random data-64.6920-64.6477-63.5526-61.8483SSC with SVN 1-SVN 2 primarycodes and ideal data-64.8997-64.8863-63.7495-61.41791 bit-64.9102-64.9446-63.8386-61.50512 bits-64.9721-67.9954-63.7993-64.33903 bits-64.9886-64.8525-64.3102-61.98444 bits-65.1495-64.7102-63.9080-61.64505 bits-64.9000-64.7954-63.7208-61.510410 bits-64.9253-64.9313-63.7977-61.4719It is important to note that as we also did with BPSK(1) in Table 6.3 and unlike in the tablesof chapter 5, the SSCs are integrated and normalized in the same bandwidth.222Spectral Separation Coefficients with data and non ideal codesAn interesting conclusion that results from comparing Table 6.3 and Table 6.4 is that, whilethe ideal PSDs of the GPS C/A Code and BOC(1,1) in the case of ideal codes and no datahave only a difference of approximately 3 dB, the PSD of the GPS C/A Code grows morethan that of BOC(1,1) when non random codes are considered and data is modulated on top.As a result, much higher SSCs are obtained for the GPS C/A Code as the simulations of thetables above clearly show.
Indeed, for Galileo E1 OS it seems that the smooth spectrumapproach is a good approximation while for the C/A Code this is not the case. This is due tothe fact that the data rate of the C/A Code is significantly lower than that of BOC(1,1).Furthermore, we can also recognize that BOC(1,1) presents similar SSCs to those of idealspectrum when ideal random codes are considered no matter whether the data is present ornot.
On the contrary, we saw in Table 6.4 for BPSK(1) that the SSC with ideal random codesand no data and the SSC for ideal random codes with data can differ by even 11 dB.Another interesting result from the comparison between the structure of BPSK(1) andBOC(1,1) with data comes from analyzing the fine structure. As an example, the lowfrequency region of Figure 6.11 is shown next.Figure 6.12. Comparison between the averaged (all combinations of 5 bits) PSD of GPSC/A code and BOC(1,1) with SVN 1 at low frequencies. For completeness also thesmooth BPSK(1) and BOC(1,1) spectra are depictedAs we can recognize, the low frequency region until 10 kHz shows that while the GPS C/ACode has sincs of width 50 Hz every 1 kHz, Galileo presents sincs of 250 Hz every 250 Hz aspredicted in theory.
This is shown more in detail in the next figure. If we take a closer lookinto the fine structure of the ideal PSD of Galileo E1 OS for SVN 1 with data, we can observethe interesting effect that even though the sincs of 250 Hz are located in steps of 250 Hz, ingeneral the local maxima are not necessarily situated at multiples of the data rate.223Spectral Separation Coefficients with data and non ideal codesFigure 6.13.
Averaged (5 bits) PSD of BOC(1,1) with SVN 1 at low frequenciesAs an example of this, there is a local maximum at 125 Hz which is caused by the particularspectral characteristics of the Galileo E1 OS code number 1. For this particular sequence, thesecond harmonic is higher than the first, so that the first secondary left lobe of the sinc at500 Hz (located at 125 Hz) summed up with the tails of other sincs around is higher than themain lobe of the sinc at 250 Hz. Such effects can be observed for Galileo E1OS spectra butnot so easily for GPS C/A code because the Galileo E1 OS codes are first longer and secondlyposses a higher data rate.In fact, the spectrum of the Galileo E1 OS code number 1 takes the following values for thefirst harmonics:Table 6.5.
Galileo E1 OS SVN 1 first harmonicsHarmonic-11Amplitude [x10 ]0123400.0130.0520.0300.419where we can see that the second harmonic at 500 Hz is nearly five times higher(approximately 7 dB higher) than the first at 250 Hz. Furthermore, we know from theory thata sinc of 250 Hz has a main lobe that is 13.4648 dB higher than the first secondary lobe,17.9018 dB higher than the second secondary lobe and 20.8244 dB higher than the thirdsecondary lobe. Considering the code structure and the properties of the sinc together, one canderive in theory all relative amplitudes.Finally, we present in the next lines similar SSC analyses for the case of MBOC.224Spectral Separation Coefficients with data and non ideal codesTable 6.6.
Data MBOC(6,1,1/11) SSC [dB-Hz] between the averaged spectra of SVN 1and SVN 2 for a data rate of 250 sps. A spectral resolution of 5 Hz was assumedSignalMBOC(6,1,1/11) data( '+' )Double-Sided Bandwidth [MHz]24.552012.2760SSC with ideal primary random codes and no data-65.3070-64.8149SSC with ideal primary random codes and ideal data-65.3070-64.8149SSC with SVN 1-SVN 2 primary codes and ideal data-65.3214-65.18601 bit-65.3794-65.24402 bits-65.3695-65.23433 bits-65.2749-65.17834 bits-65.1700-65.73625 bits-65.3133-65.293410 bits-65.3721-65.2328As we can recognize in the table above, only the in-phase channel (data '+ ' ) was shown.However, simulations were run for the pilot channel too.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
