On Generalized Signal Waveforms for Satellite Navigation (797942), страница 48
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Thus, the analytical way of computingthe PSD would be to obtain first the spectrum of the sequence of 1 millisecond and substituteevery spike by a sinc of 50 Hz. In fact, the code is 1 millisecond of duration and repeats 20times. Moreover, since the GPS C/A primary code has a length of 1 ms, the spectral lines willpresent a separation with each other of 1 kHz.Figure 6.1 next shows the PSD that would result from modulating our original signal withdata on top. For our analysis a one-sided bandwidth of 15.345 MHz was assumed and thus asampling frequency of 30.69 MHz was employed for the simulations. As a result, every chipis sampled with 30 values. No filtering is employed in the simulation and for exemplificationonly the spectrum of SVN 1 is shown.
Moreover, the smooth spectrum of the C/A Code withideal random codes is also shown in red for comparison.Figure 6.1. Power spectral density of GPS C/A code SVN 1 with data for a bandwidth of30.69 MHz versus ideal PSD of a BPSK(1) modulated with an ideal random codeIf we look now into the fine structure of the spectrum, we can recognize the data sincsseparated by 1 kHz with respect to each other that we can predict from theory.Figure 6.2. Power spectral density of GPS C/A code SVN 1 with data at low frequencies.A transmission bandwidth of 30.69 MHz was assumed214Spectral Separation Coefficients with data and non ideal codesNow that we have derived the analytical expression for the power spectral density of a signalwaveform modulated with ideal random data and non-ideal codes, we analyze the differencewith respect to the ideal case when the signal waveform is modulated with ideal codes.As we have seen, if data is supposed to be ideally random, the spectrum of the modulatedsignal with data results from the convolution of both power spectral densities, what isequivalent to substituting every spike by a sinc of the form:⎛ f ⎞1(6.2)sinc⎜⎜ ⎟⎟fd⎝ fd ⎠where fd refers to the data rate.
As a result, the spikes of the code are spread over a widerbandwidth, increasing the amplitude as Figure 6.1 has shown. For the case of GPS C/A codewith a data rate of 50 sps, this increase in the amplitude with respect to the ideal smoothspectrum is of 10 log10 (50) ≈ 17 dB.If we plot now the product of the ideal PSDs of SVN 1 and SVN 2 with data and 0 Hz offsetand compare it with the product of the ideal PSDs without data, we can recognize that in theideal smooth case the product lies around 34 dB below. This is due to the fact that each PSDis 17 dB below with respect to the case with data as we saw above. Indeed, this is the reasonthat the SSCs, result of the integrate of both spectra with data, are around 12 dB higher thanfor the ideal case that we calculated in chapter 5 (-61.8597 dB-Hz).Figure 6.3.
Product of C/A Code Spectra modulated with codes SVN 1 and SVN 2 withand without dataIn the next table we present for comparison the SSCs of different GNSS signals when there isdata and when an ideal spectrum is assumed.Table 6.1. Spectral Separation Coefficients with and without dataSSC [dB-Hz]Ideal SSCwithout dataIdeal SSC withrandom dataTX BandwidthRx BandwidthBPSK(1)-61.8008-50.546030.6930.69BOC(1,1)-64.6920-64.692030.6930.69MBOC(6,1,1/11)-65.3070-65.307024.552024.5520215Spectral Separation Coefficients with data and non ideal codesOnce we have studied the problematic of calculating spectra when data or non-ideal codes areaccounted, it is time to make a classification of the different types of spectra depending on thefine structure that they present. Indeed, we distinguish the following three classes according to[F. Soualle and T. Burger, 2002]:•••Type 1: Signals with smooth power spectral density (non periodical codes)Type 2: Signals with periodic codes and no data modulation (pilot signals)Type 3: Signals with periodic codes and data modulation.As we will show next, the SSCs differ in value depending on the type of signals that areconsidered and the data rate.
Moreover, it can be shown that if one of the signals is of type 1(non periodical code) and the other one of type 2 or 3 the value of the spectral separationcoefficient is similar to that of considering both signals of type 1.We show an example now. The following figure depicts the product of the Power SpectralDensities of the M-Code and BOC(1,1) assuming ideal random codes and ideal random datafor the M-Code (thus smooth spectrum) while for the BOC(1,1) we have smooth spectrum inone case (type 1) and the effect of ideal random data in the other case (type 3).Figure 6.4.
Product of BOC(1,1) Code Spectrum modulated with SVN 1 and M-Codesmooth spectrum. In blue the effect of the data on BOC(1,1) is considered while in redideal smooth fine structure is assumedWe can see the difference between both curves in detail in the next figure:Figure 6.5. Comparison of the product of PSDs between BOC(1,1) and M-Code whendata modulation is present or not216Spectral Separation Coefficients with data and non ideal codesIf we measure now the resulting SSCs between the M-Code and BOC(1,1) when no data isconsidered and when data is considered, we obtain the following results:Table 6.2.
Spectral Separation Coefficients between BOC(1,1) and M-Code with andwithout dataSSC [dB-Hz]Data-82.6566Yes-83.1241NoIf we sum up now the difference of Figure 6.5 and divide by the interference coefficient withdata and without data, we obtain a value of approximately -9.9113 dB and -9.4437 dBrespectively, confirming thus that the interference coefficient or SSC is about of the sameorder of magnitude as when we consider two non-periodical PSDs (SSC = -83.1241 dB-Hz),and when only the non periodic properties of BOC(1,1) are taken into account(SSC = -82.6566 dB-Hz).
While this is true for these types of signals, others like the C/ACode require special attention since the fine structure is considerably affected by the codestructure.6.2Computation of non-ideal Spectral SeparationCoefficientsOnce we have derived the ideal PSD that a signal with a non-ideal pseudorandom code butcompletely random data will have, it is important to revisit the assumptions that we havemade. Indeed, assuming that the data is completely random is equivalent to saying that ourreceiver integrates over an infinite period of time, what is of course impossible.This means that the integration time for the computation of the SSC plays an outstanding rolein the estimation of the power spectral density.
Indeed, even though our signal were ideallymodulated with random data, since our observation window is limited, the partial integrationof the data will result in a non-ideal partial correlation different to the ideal Dirac delta. Thisis deeply connected with the ideas gathered in [G.W. Hein et al., 2006c] where the concept ofrandom codes was discussed concluding that time-limited codes and ideal random noise arecontradictory per se.In order to see the effect that the number of bits taken into account in the integration plays inthe composition of the spectrum at receiver level, simulations were run for differentintegration times, averaging the resulting spectra over all possible bit combinations to derivegeneral consequences from the results of the computations.
Indeed, for n bits, 2n possible bitcombinations are possible, of which only a few have to be considered due to existingsymmetries. Furthermore, (6.1) was employed to compute the SSC, not accounting thus forthe filtering effect of the time-windowing. Additionally, it was assumed that the DLL behaves217Spectral Separation Coefficients with data and non ideal codesas a filter and thus the power spectral density that is formed could be seen as the average ofpreceding bit combinations. Following these ideas, non ideal SSCs have been computed inthis chapter and compared with the SSCs that would result from using the ideal power spectraldensity of chapter 4. This process was repeated for different bandwidths and for each of themthe results are shown next. As we can see, for every considered receiver bandwidth, averagingover all data combinations results in mean SSCs that are virtually equal to those obtainedusing ideal PSDs.
However, if we take a closer look into the fine structure (see Figure 6.6) wecan recognize that the spectra are slightly different.Figure 6.6. Averaged PSD of GPS C/A code SVN 1 that results from averaging allpossible code sequences with 5 data bitsFor the particular case of the C/A Code, we can recognize the 50 Hz sinc with already 100milliseconds of coherent integration. However, the amplitude of the side lobes is lower in thecase of the spectra estimated from simulations. Nevertheless, the amplitude of the main lobesis approximately equal in both ideal and simulated power spectral densities, and since this partof the spectrum is the main contributor to the SSC, the difference is considered negligible.Figure 6.7.
Comparison between the ideal PSD of BPSK(1) with random data at 50 spsand the averaged PSD that results from taking all combinations of 5 data bits218Spectral Separation Coefficients with data and non ideal codesMoreover, if we take a look now at the fine structure that results from averaging the spectra ofall possible combinations of five bits and compare it with the ideal spectrum with ideal data,we obtain the following figure:Figure 6.8.
Low frequencies comparison between the ideal PSD of BPSK(1) with data at50 sps and the averaged PSD of all 5 data bit combinationsTable 6.3. BPSK(1) SSC [dB-Hz] computed between the averaged spectra of SVN 1 andSVN 2 for a data rate of 50 sps. A spectral resolution of 5 Hz was assumedSignalBPSK(1)Double-Sided Bandwidth [MHz]30.694.0922.046Self SSC with ideal primary random codes and no data-61.8008-61.4155-60.9845SSC with ideal primary random codes and ideal data-50.5460-50.1607-49.7297SSC with SVN 1-SVN 2 primary codes and ideal data-50.3370-49.8988-49.46491 bit-51.1024-50.9993-50.64902 bits-50.9997-51.1065-50.72823 bits-50.3370-50.2056-49.82734 bits-50.8283-50.6059-50.22755 bits-50.3370-50.2056-49.827310 bits-50.3370-50.2056-49.8273It must be noted that the SSCs are normalized and integrated to the same bandwidth. Thatmeans that for the 30.69 MHz column, the normalization of the PSD is done to 30.69 MHzand the SSC computation integrates equally in 30.69 MHz.The interest of these results lies in the fact that no matter what the actual code selected for thesimulation is, the values will not vary significantly in reality.
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