Performance of Multiplexed Binary Offset Carrier Modulations for Modernized GNSS Systems
By E. Simona Lohan, Mohammad Z. H. Bhuiyan, and Heikki Hurskainen
A candidate for modernized GNSS civil signals in the L1/E1 band was BOC(1,1), a binary-offset-carrier signal with a “split spectrum” that has negligible impact on the existing GPS signals. However, a signal with better acquisition capabilities and improved multipath performance (while still compatible with the existing GPS signals) is a multiplexed BOC modulation, MBOC(6,1,1/11). The MBOC spectrum can be achieved by following one of several different signal-construction paths with some resulting differences in how a receiver tracks the signal and its associated performance.
IN GEOFFREY CHAUCER’S 1391 ESSAY, A Treatise on the Astrolabe (one of the earliest known instruction manuals in English), he says (with modern spelling) “Right as diverse paths lead the folk the right way to Rome.” He was talking about the use of English rather than Latin or another language to convey the same information. And we now commonly use the shortened version of this expression — all roads lead to Rome — to express the sentiment that a particular problem can be solved in different ways.
So it was with the decision by the United States and Europe to use a common, interoperable signal for the new GPS III civil service and the Galileo Open Service on the L1/E1 frequency of 1575.42 MHz. The road to “Rome” was tedious, long, and a little bumpy at times. A number of studies and a lot of rhetoric centered on how to make the new signal compatible with the legacy GPS L1 signals, the C/A-code and the P(Y)-code, as well as the modernized GPS military signal on L1, the M-code.
A similar compatibility issue had been solved when the M-code was added to the legacy GPS signals, starting with the Block IIR-M satellites. The M-code is a binary-offset-carrier (BOC) signal — a split spectrum signal — that places most of its power near the edges of the allocated GPS frequency bands, thereby having negligible impact on the legacy signals. The M-code modulation, designated BOC(10.23,5.115) and commonly abbreviated BOC(10,5), uses a subcarrier frequency of 10.23 MHz and a spreading code rate of 5.115 megachips per second to achieve the desired spectral separation. This design provides military users with an improved signal with little impact on civil users.
Similar approaches were initially proposed for the new GPS L1C and Galileo E1/L1 OS signals with a BOC(1,1) modulation initially agreed on. However, further studies showed that a signal with better acquisition capabilities and improved multipath performance (while still compatible with the existing GPS signals) was a multiplexed BOC modulation, MBOC(6,1,1/11), formed by multiplexing a wideband signal, BOC(6,1), with a narrow-band signal, BOC(1,1), in such a way that 1/11th of the power is allocated, on average, to the high frequency component. Such a signal has the added benefit that one can choose whether to make use of just the low-frequency component in, say, a simple “mass market” receiver or also use the high-frequency component for more demanding applications.
It turns out that the agreed-upon MBOC spectrum can be achieved by following one of several different signal-construction paths with some resulting differences in how a receiver tracks the signal and its associated performance. In this month’s column, we take a look at some of the options.
In July 2007, the United States and Europe announced agreement on the use of the multiplexed binary offset carrier (MBOC) modulation as a common baseline for Galileo Open Service signals in the E1 band and GPS L1C signals in the L1 band. According to the most recent Galileo Signal-In-Space Interface Control Document (SIS-ICD; see Further Reading), the MBOC power spectral density (PSD) has been fixed to
where GBOC(m,n)(f) is the normalized PSD of a BOC(m,n)-modulated pseudorandom noise (PRN) code with sine phasing. The indices m and n are related to the sub-carrier frequency, fsc, and the chip frequency, fc, via m = fsc/fref and n = fc/fref, respectively; fref = 1.023 MHz is the reference C/A-code frequency, and NB = 2fsc/fc = 2m/n is the BOC modulation index.
The MBOC PSD is obtained by taking the data and pilot channels together. The data and pilot channels can use, independently, one of the following modulations: composite binary offset carrier (CBOC) or time-multiplexed binary offset carrier (TMBOC) modulations. CBOC and TMBOC, in turn, have several variants. Since the data and pilot channels are typically processed independently, it is important to understand the differences between various CBOC and TMBOC modulations and this is the primary goal of this article. There are several possible ways to achieve a PSD as given in Equation (1) and they are based on combining the data and pilot channels in the Galileo and modernized GPS systems. The main modulation types for pilot or data channels that can be used in order to achieve (when combined) the MBOC PSD can be summarized as follows:
1. The CBOC method: CBOC is formed via a weighted sum or difference of BOC(1,1)- and BOC(6,1)-modulated code symbols (where the BOC(1,1) part is passed through a delay block in order to match the rate of the BOC(6,1) part) as defined in Equation (2):
where sBOC(1,1),h is the up-sampled BOC(1,1)-modulated code (that is, the code provided at the same rate as the sBOC(6,1) signal), sBOC(6,1) is the BOC(6,1)-modulated code, and w1 and w2 are amplitude weighting factors, chosen in such a way to match (as closely as possible, when both data and pilot channels are considered) the PSD of Equation (1), with w12 + w22 = 1. When the two right-hand terms are added in Equation (2), CBOC(+) is formed; when subtracted, CBOC(–) is formed. A third alternative for CBOC implementation is to use the CBOC(+/–) approach, where the odd-numbered chips are CBOC(+)-modulated and the even chips are CBOC(–)-modulated. The current Galileo SIS-ICD uses a CBOC(+) variant (also called CBOC in-phase) for the E1-B data channel and a CBOC(–) variant (also called CBOC anti-phase) for the E1-C data-less (or pilot) channel.
2. The time-multiplexed BOC (TMBOC) method: the whole signal is divided into blocks of N code symbols with M (<N) code symbols sine-BOC(1,1)-modulated, while N-M code symbols are sine-BOC(6,1)-modulated. The typical shorthand notation for this variety of TMBOC would be TMBOC(6,1,(N-M)/N), referring to the sine-BOC(6,1) component of the signal. This time-domain division may be applied for both pilot and data channels, individually. The choice of the N and M parameter values depends on the desired power percentage of the pilot channel with respect to the data channel. We have shown in earlier work (see Further Reading) that, from the point of view of the MBOC autocorrelation function, TMBOC and CBOC(+) implementations are equivalent, as long as the weights are related to the N and M values using w1 = √(M/N) and w2 = √((N-M)/N). Various TMBOC implementations exist according to the values chosen for N and M and according to whether the BOC(1,1) code symbols are in phase or out of phase with the BOC(6,1) code symbols. For example, for a 50-percent/50-percent power split between the pilot and data channels using in-phase code symbols, M = 9 and N = 11 (that is, TMBOC(6,1,2/11) is used), while for a 75-percent/25-percent power split between the pilot and data channels (again, using in-phase code symbols), M = 29 and N = 33 (that is, TMBOC(6,1,4/33) is used).
A major difference between CBOC and TMBOC signals is that CBOC signals have four different levels (as a weighted sum or difference of two sub-carriers), while TMBOC signals have only two levels. The impact of these differences in the tracking stage of a receiver has been analyzed, for example, by a team of researchers led by Olivier Julien (see Further Reading). They showed that an optimal CBOC receiver should generate a local replica that also has four levels, resulting in a replica encoded on more than just one bit. This complicates the CBOC receiver architecture, compared to TMBOC 1-bit receiver architectures. In terms of performance, a CBOC(–) receiver proved to have the same delay-tracking variance performance as a TMBOC(6,1,4/33) receiver and both slightly outperform a TMBOC(6,1,1/11) receiver. And considering multipath error performance, a TMBOC(6,1,4/33) receiver was shown to give the best performance, followed very closely by a CBOC(–) receiver. Our research extends this earlier study.
Examples of CBOC and TMBOC waveforms are shown in Figure 1. Here, w1 = (10/11) and the TMBOC waveform has every first chip BOC(6,1)-modulated (inside blocks of 11 chips). In the figure, only the first five modulated chips are shown for clarity.Our article addresses the following issues: First, we analyze the spectral differences between various CBOC and TMBOC modulations in terms of their effect on receiver performance. Secondly, we look at the navigation data error probability, the tracking error variance in the presence of noise, and the robustness of the signal in the presence of multipath and bandwidth limitations of MBOC variants, by taking into account the spectral differences between the different variants. Thirdly, we justify the choice of CBOC(+) for data channels and CBOC(–) for pilot channels in the Galileo SIS-ICD in terms of these receiver performance criteria.
Spectral Differences of CBOC/TMBOC Modulations
The spectral differences refer to the differences in the PSD of various waveforms. We recall that the PSD is the Fourier transform of the CBOC/TMBOC autocorrelation function. CBOC/TMBOC signals are formed from the convolution of PRN code waveforms, CBOC/TMBOC modulation waveforms, and navigation data (when present). If the same PRN code is used for the BOC(1,1) and BOC(6,1) modulations, some cross-correlation terms appear in the autocorrelation function, which will also appear in the frequency spectrum. Indeed, following the model, after straightforward derivations, we obtain the generic CBOC/TMBOC PSD as:
where HBOC(1,1),h(f) and HBOC(6,1)(f) are the following Fourier transforms of the modulation waveforms:
Above, TB = TC/12 is the BOC(6,1) sub-interval and sinc(x) = sin(x)/x. The formula given in Equation (3) covers all CBOC/TMBOC cases: k = +1 for CBOC(+) and TMBOC, k = –1 for CBOC(–), and k = 0 for CBOC(+/–), respectively. Equation (3) characterizes either the pilot channel’s PSD or the data channel’s PSD. In order to achieve the PSD of Equation (1), data and pilot channels should be combined. For example, if k = 0, any combination of data and pilot channels is possible in order to attain the PSD. If k ≠ 0, then the data channel should use in-phase combining (k = +1) and the pilot channel should use anti-phase combining (k = –1) or vice versa.
Now, if we take as a reference the PSD of CBOC(+/–) (which, incidentally, is also the PSD of Equation (1)), the spectral differences between the other CBOC/TMBOC modulations and CBOC(+/–) are quantized by the following equation:
Examples of spectral difference between CBOC(+/–) and each of the following modulations: CBOC(–), CBOC(+), and TMBOC(6,1,(N-M)/N) and each of the following modulations: CBOC(–), CBOC(+), and TMBOC(6,1,(N-M)/N), respectively, are shown in Figure 2. Clearly, these differences are very small.
Impact on System Performance
As mentioned before, pilot and data channels typically use different CBOC/TMBOC modulations, in order to achieve an overall PSD as described by Equation (1). Now, based on the derivations we have presented so far, the following questions can be addressed: Which are the most suitable modulations (among the four discussed here; namely, CBOC(+), CBOC(–), CBOC(+/–), and TMBOC) to be used for a pilot channel and for a data channel, respectively; and how will the differences in the PSDs affect the error probability of the decoded signal and the tracking performance of each channel?
Uncoded Error Probability and Fractional Out-of-Band Energy. Data and pilot channels are usually processed independently and then combined (for example, non-coherently) in order to perform the line-of-sight (LOS) signal delay estimation and the navigation data detection. Since different CBOC or TMBOC modulations can be used for the data and pilot channels, one question to be addressed here is what is the most suitable modulation type. Additionally, the carrier-to-noise-density ratio (C/N0) deterioration when another modulation type is employed is also important. These two issues are addressed in this section.
One important spectral parameter that allows us to answer the question about error probability in the decoded data is the so-called fractional out-of-band energy (FOBE), which tells us about the fraction of the signal power remaining outside a certain double-sided bandwidth, Bw. FOBE is related to the power containment factor, used by some authors, via (1 – FOBE(Bw)). Clearly, FOBE depends on the signal modulation type. The higher FOBE is, the greater the deterioration of the signal energy we have after the receiver bandwidth limiting filters, and thus the higher error probability of the decoded signal we have. From the data-channel point of view, correctly decoding the navigation data is very important and therefore, low FOBE is the most important characteris
tic when choosing the modulation type. The bit error probability in decoding a binary signal, such as a BOC or MBOC signal, can be computed by taking into account the signal energy deterioration due to filtering. Using the basic formula for computing the bit error probability in decoding a 2-level signal (in the cases of BOC or TMBOC modulation) or a 4-level signal (in the case of CBOC modulation), we can compare the performance of various TMBOC and CBOC modulations in terms of error probability of the decoded data bits, as shown in Figure 3. Clearly, the error probability criterion is more important for a data channel than for a pilot channel. Sine-BOC(1,1) and BOC(6,1) modulations are included in the comparison of Figure 3 as benchmarks. A double-sided bandwidth of 24.552 MHz was considered here, following the choice in the Galileo SIS-ICD.
As seen in Figure 3, in terms of the error probability of the decoded signal, BOC(1,1) modulation gives the best results, followed closely by TMBOC(6,1,4/33). In order to achieve an error probability of 10-2, the CNR differences shown in Table 1 are needed for the different modulation types. From Table 1, it can be seen that, among CBOC modulations, the CBOC(+) modulation is the best option from the point of view of decoding the data, and, therefore, it makes it a suitable option for data channels, as chosen in the Galileo SIS-ICD. We remark that the huge CNR gap for BOC(6,1) at Bw = 8 MHz is due to the fact that the power containment of a BOC(6,1) signal is very poor at such a low bandwidth.
Gabor Bandwidth and Tracking Error Variance. Another important spectral parameter of interest in this analysis is the root-mean-square (RMS) or Gabor bandwidth. A larger RMS or Gabor bandwidth permits a higher accuracy against thermal noise and the tracking accuracy is approximately inversely proportional to the RMS bandwidth. The code-tracking error variance is an important parameter when trying to achieve accurate location estimates. Indeed, a Cramér-Rao lower bound (CRLB) on the tracking error variance has been derived by other researchers. Following the derivation for CRLB on the tracking error variance, we can also compare the performance of various CBOC and TMBOC modulations, as presented in Figure 4. Clearly, this criterion is more important for a pilot channel than for a data channel. A double-sided receiver bandwidth of 24.552 MHz was considered here.
In terms of the tracking error variance bound, which linearly decreases with the CNR (on a dB scale), the CNR differences between various modulations are shown in TablE 2 for a 4-Hz tracking-loop bandwidth. Clearly, from Table 2, CBOC modulations are better in terms of tracking error variance than TMBOC modulation, and, among the CBOC variants, CBOC(–) has the best performance. This justifies the fact that the Galileo SIS-ICD has chosen the CBOC(–) as the best option for pilot channels. We can also see in Table 2 that the bandwidth limitation has an important effect on the tracking error bounds, as expected. At low receiver bandwidth (such as 8 MHz), the differences between various modulations are rather small, while at high or infinite bandwidths, BOC(6,1) modulation is by far the best option, followed by CBOC(–) with a 1.69 dB gap in CNR (that is, CBOC(–) requires an additional 1.69 dB in order to achieve the same tracking error performance as BOC(6,1)).
Multipath Error Envelope. The typical procedure for evaluating the performance of a multipath-mitigation technique is via the multipath error envelope (MEE). The MEE curves are obtained for two extreme phase variations of a multipath signal with respect to the LOS component while varying the multipath (that is, second path) delays from 0 to 1.2 chips at maximum, since the multipath errors become less significant after that. The upper multipath error envelope can be obtained when the paths are in-phase (that is, 0° phase difference) and the lower multipath error envelope when the paths are out-of-phase (that is, 180° phase difference). In MEE analysis, several simplifying assumptions are usually made in order to distinguish the performance degradation caused by the multipath only. Such assumptions include zero additive white Gaussian noise, ideal infinite-length PRN codes, zero residual Doppler shift, and zero initial code-delay error.
The MEE curves are generated here for different variants of MBOC implementation. The multipath performance of these MBOC variants with a BOC(1,1)-modulated reference receiver is also presented. In the MEE generation, the second path amplitude was fixed at 3 dB lower than the LOS component. The MEE curves were generated for a 24.552 MHz double-sided bandwidth. The narrow early-minus-late (nEML) correlator with an early-late correlator spacing of 0.0833 chips was used here as a tool for evaluating the performance of the different MBOC variants in the presence of multipath. The nEML is based on the idea of narrowing the spacing between the early and late correlator pair, where the choice of correlator spacing depends on the receiver’s available front-end bandwidth along with the associated sampling frequency.
MEE curves are shown for all of the examined MBOC variants in Figure 5. It can be observed from the figure that CBOC(–) has the best multipath mitigation performance followed by the TMBOC(6,1,4/33) and CBOC(+) variants. A similar conclusion can be drawn when a BOC(1,1) reference receiver is used instead of the respective MBOC reference receiver. However, from Figure 5, it is obvious that there is a moderate performance degradation when a BOC(1,1) reference receiver is used instead of the respective MBOC version, as expected intuitively.
Simulation Results in Multipath Fading Channel
Simulations have been carried out in closely spaced multipath scenarios for different MBOC variants with a finite front-end bandwidth. The simulation profile is summarized in Table 3. A Rayleigh fading channel model is used in the simulation, where the number of channel paths is fixed to two. The successive path separation is random between 0.02 and 0.35 chips. The channel paths are assumed to obey a decaying power delay profile (PDP).
The received signal duration is 0.8 seconds for each particular C/N0 level. The tracking errors are computed after each NcNnc-milliseconds interval (in this case, NcNnc = 20 milliseconds). In the final statistics, the first 600 milliseconds are ignored in order to remove the initial error bias that may come from the delay difference between the received signal and the locally generated reference code. Therefore, for the above configuration, the left-over tracking errors after 600 milliseconds are mostly due to the effect of multipath only. We ran the simulations for 1,000 statistical points, for each C/N0 b> level. The RMS error (RMSE) of the delay estimates can be plotted in meters, by using the relationship RMSEm = RMSEchips•c•Tc, where c is the speed of light, Tc is the chip duration, and RMSEchips is the RMSE in chips. An RMSE versus C/N0 plot for the given multipath channel profile is shown in Figure 6.
As seen in the figure, the CBOC(–) reference receiver has the best multipath mitigation performance under a good
C/N0 (that is, 40 dB-Hz and higher) followed by the other two MBOC variants (CBOC(+) and TMBOC(6,1,4/33)), which exhibit almost similar performance. A similar conclusion can be drawn for the BOC(1,1) reference receiver, where the CBOC(–)-modulated transmitted signal with BOC(1,1) reference receiver showed the best multipath mitigation performance among all three of the studied MBOC variants. In Figure 6, we observe that the small performance deterioration caused by use of a BOC(1,1) reference receiver is visible only under good C/N0 conditions (that is, 40 dB-Hz and higher).
This article discusses the spectral differences between CBOC and TMBOC modulations and their impact on system performance. The exact frequency-domain form of the PSD for CBOC and TMBOC waveforms has been shown and the impact on tracking error variance bounds and on the error probability of the demodulated signal has been discussed. In addition, the multipath mitigation performances of different MBOC variants were presented in terms of RMSE and multipath error envelopes. It was shown that the CBOC(–) variant is the best variant in terms of multipath mitigation and tracking error variance, while TMBOC behaves better than CBOC in terms of error probability of the demodulated data. We also showed that the spectral differences and the differences between CBOC and TMBOC variants in terms of the two considered performance criteria are rather small, especially when the receiver bandwidth is not very high, and, therefore, several variants of MBOC can indeed be used for design and research purposes.
The research leading to the results presented in this article received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 227890 (the Galileo-Ready Advanced Mass Market Receiver–GRAMMAR–project). This research work has also been supported by the Academy of Finland and by the Tampere Doctoral Programme in Information Science and Engineering. Particular thanks are also addressed to Stephan Sand from the German Aerospace Center (DLR), Institute of Communications and Navigation, for his useful comments.
Elena Simona Lohan has been an adjunct professor in the Department of Communications Engineering at Tampere University of Technology (TUT) in Hervanta, Finland, since 2007. She obtained her Ph.D. degree in wireless communications from TUT. She also graduated with an M.Sc. in electrical engineering from “Politehnica” University of Bucharest, and with a diplôme d’études approfondies in econometrics from Ecole Polytechnique, Paris. Lohan is currently leading the research activities in signal processing for wireless communications in the Department of Communications Engineering at TUT.
Mohammad Zahidul H. Bhuiyan is a researcher in the Department of Communications Engineering at TUT. His main research areas are multipath mitigation and software receiver design for satellite-based positioning applications.
Heikki Hurskainen received an M.Sc. degree in electrical engineering and a doctoral degree in computing and electrical engineering from TUT in 2005 and 2009, respectively. Currently, Hurskainen is a senior research scientist in TUT’s Department of Computer Systems where he works on satellite navigation research projects.
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