The upcoming Galileo global navigation satellite system has a design problem with the cosine-phased BOCc(15, 2.5) modulation of its Public Regulated Service (PRS A-code). This signal needs far more bandwidth than the available 40.92 MHz. The present signal and system specification cannot be expected to deliver design performance under practical operational conditions (noise, receiver phase distortion, and multipath). There would not have been this problem with sine-phased BOC(15, 2.5).
The Galileo global navigation satellite system has been more than a decade in the design and planning. The first two IOV satellites have been launched and at time of writing (January 2012) are being tested. The next two IOV satellites and FOC satellites are under construction but at present there is nothing in space to allow for a full check of their navigational capability. As is well known at least four satellites have to be up and running to attempt a full navigational solution. It is at that point this paper anticipates that some inherent problems will be confirmed because of an inherent design problem (not to say fault).
It may seem surprising that after such a concerted effort and expert input that there could be any such a possibility. But if a system does not work properly, then it does not work. If the predictions of this paper turn out to be wrong, then there is of course no cause for concern. But if the predictions are right, then the reasons are fully explained here—as is the necessary correction.
The difficulty lies with the flag-ship PRS A-code (public regulated service): the wide-bandwidth signal transmitting on E1 channel (centred on 1575.42 MHz). It is predicted here that navigation receivers trying to track this signal and compute their location are likely to fail in any but the most ideal conditions. Low-strength signal (drop-outs), multipath, and all the contingent practical imperfections associated with real-world reception will make tracking and computations of the navigation solution unreliable. The same problem but less severe in its effect can be expected in reception of the other PRS -A code signal transmitting in the E6 channel centred on 1278.75 MHz.
There are hints that responsible engineers already recognise the problem to come but are understandably reluctant to say so explicitly—see, for example, recent conference papers hosted by ESTEC [
The Galileo Signal Task Force (GSTF) appointed by the European Commission chose for its PRS A-code signal a BOC modulation [
The problem is this: in order to work properly,
The fact that a change of phase will alter the bandwidth requirements of an operational system by a ratio of 2 : 1 will seem as counter-intuitive to many technically minded readers as it must have been to the members of the GSTF (who designed the system). But BOC modulation makes for a complicated signal, and navigational requirements have a number of subtle aspects. It is understandable how the point seemingly was not appreciated in the overall signal design. But it is perhaps significant that Betz’s original specification for BOC only ever envisaged sine phasing and that the US-upgraded GPS adopts this phase only.
In this paper this key finding is given a relatively simple nonmathematical explanation and then demonstrated in a modification of the standard computer-generated multipath test. A new test criterion is offered and explained in order to justify the conclusion and completes the paper.
Ever since John Betz proposed BOC [
In a standard correlating receiver the problem is exhibited as a multipeaked correlation with a need to track only on its primary peak. The coarse estimate is implied in the timing of the peak
It is here that Galileo’s BOCcos(15, 2.5) gets into difficulties. Cosine phasing generates a significant
The risk can be assessed in a number of ways—in conditions of low input C/N0, phase distortion in the channel, and above all
The same difficulty is associated with the BOCc(10, 5) signal—but less acutely—see later comments.
The bandwidth allocated to the E1/L1 channel for the test satellite GIOVE-B was 40.92 = 40 × 1.023 MHz—as confirmed by actual tests at Chilbolton [
A navigational receiver may of course fail to work in all manner of ways. For avoidance of doubt
The allocated channel bandwidth is sufficient for sine-phased BOC(15, 2.5), that is, BOCsin(15, 2.5) to work satisfactorily. The reason is simply that sine-phased BOC has no significant outer spectrum. Simulations run by the author predict that receiver operation with this phasing is not significantly degraded even if the signal is filtered to a bandwidth as low as 35 × 1.023 MHz. This modulation of sine-phased BOC(15,2.5) was of course the earlier choice by the GSTF before they finally decided on “cosine phasing.”
The other PRS A-code signal BOCcos(10, 5) centred on 1278.75 MHz has the same problem with cosine phasing but it is not so badly degraded. The specification for the E6 channel is again for a bandwidth of 40.92 MHz. But this modulation with its lower value of sub-carrier frequency and a higher code rate—the ratio is now 2 : 1—has a greater separation of ambiguous fine estimates and is more capable of delivering a coarse estimate of sufficient accuracy.
Focussing our attention on the BOC(15, 2.5) signal the crucial point is to explain how a change from sine to cosine phasing—from the earlier sine option to the cosine option and no other change to the specification—should double the required bandwidth.
Figure
Simplified GNSS channel from transmitter to receiver transmitting two possible phases of sub-carrier to code for BOC(15, 2.5). Example code (magenta); cosine option (red); sine option (blue). Code rate
Ultra-wide TX filter (black)
The BOC specification in general requires an integer number of half cycles of sub-carrier per chip. For BOC(15, 2.5) there is a 6 : 1 ratio of sub-carrier frequency to code rate and therefore 12 half cycles or alternating “subchips” of sub-carrier assigned to each chip.
Before transmission the signal is bandpass filtered (assigned here bandwidth
There is a very considerable difference in the detailed signal structure between sine phasing and cosine phasing. The difference is visualised in a postulated example sequence showing part or whole of four chips in sequence with two alternations of polarity.
The transitions in the sub-carrier coincide exactly with pseudorandom transitions in the code sequence. The schematic example shows the previous choice by the GSTF for the PRS signal which was BOCsin(15, 2.5).
The transitions in the sub-carrier are offset by a quarter cycle—half a subchip width—relative to the code transitions. The schematic shows the final choice by the GSTF for the PRS signal which is BOCcos(15, 2.5).
So what difference does the choice of phase make? A great deal. One can see this immediately from closer inspection of the waveforms. From one code chip to the next there is a 50% chance of a polarity switch. It occurs when the succeeding code chip pseudorandomly changes phase. On this change of phase the cosine phasing creates a full cycle at
One can see the effect in the frequency domain. Figure
A linear-phase filter of any desired bandwidth can be assigned to a theoretical transmitter. The difference between full and dotted curves shows the effect of filtering. The black curve in Figure
One can test for the presence of the double-frequency pulse on the filtered cosine-phased BOC by computing the real-time signal at point [P3] of the system. Figure
Two chips opposite polarity time domain response after RX filtering at [P3]. Ultra-wide bandwidth
But Galileo is not going to deliver such a wide bandwidth. The official bandwidth for the E1 channel is exactly half at 40 × 1.023 = 40.92 MHz. The Chilbolton tests on GIOVE-B measured the transmitted power spectrum of the BOCc(15, 2.5) signal. The 1st, 2nd, and 3rd outer side lobes are clearly discernible. There is no indication of any higher frequency sidelobes (which would have appeared above the noise floor). So these must have been filtered out. The GIOVE-B spectrum is presumed here to be representative of what is planned for the IOV series (and the FOC to follow).
In Figure
TX filter as GIOVE-B (black)
Time domain response at [P3]; sequential chips opposite polarity. Linear phase filters. TX filter as GIOVE-B. Matching filter in receiver.
These computer-generated tests from Figures Cosine-phased BOC has a significantly higher frequency content than sine-phased BOC because it pseudorandomly generates a double-frequency pulse (d.f.p). The practically available band-limited channel does not propagate this d.f.p.
The key question is if failure to propagate this pulse should be important to the correct operation of a BOC receiver. The answer is that it is very important. The reason is not so much in the power but in the information that is now being lost. A navigation system is unlike a communication system in that the essential information is contained in the signal
But ranging information is also contained in the code transitions. These are what the heritage GPS depends on, and these transitions continue to provide the basis of the coarse estimate for BOC. In the case of cosine BOC double-frequency pulse exists whenever there is a change of sign on a code transition. The effect of limited bandwidth which loses that pulse will be shown to degrade that coarse estimate. This degradation is what affects the two-level operation of a BOC receiver.
Recall that the aim of one GNSS receiver (channel) is to estimate the range to a transmitting satellite. A receiver does this by estimating the relative time of arrival of that signal and indirectly computing the propagation delay of that transmission.
It is well known that BOC supports a large number of receiver implementations including one in which the sub-carrier is ignored and only the code component is tracked.
A BOC receiver in any configuration must deliver at least a low-level coarse estimate by tracking just the code component in the transmitted signal. In effect the receiver “sees” only the transitions in the code component of the signal and treats it like the ordinary BPSK transmission of the heritage GPS system. In order however to realise the potential accuracy of the BOC signal, a maximum likelihood receiver must generate a high-level fine estimate by tracking the sub-carrier component. But there are multiple solutions to this estimate—only one of which is correct. The multiple solutions are spaced at half a wavelength of the sub-carrier. The receiver must have therefore that other solution—the coarse estimate—in order to resolve this ambiguity. The coarse estimate—derived from the code transitions—must be available at all times assuming that the fine estimate is always liable to cycle slip from a valid to an invalid solution.
The coarse estimate must not however be too coarse. It needs to be sufficiently accurate to be able to distinguish reliably between the right solution from the adjacent wrong solutions by that fine estimate. The most accurate BOC receiver has to deliver
A simplified model of a channel feeding a maximum likelihood BOC system—transmitter, channel, and receiver, is shown in Figure
System modelling (base band) maximum likelihood (ML) receiver of BOC/PSK signal + multipath.
The upconversion and downconversion process is simplified to a base-band representation. Carrier recovery is not the issue. The unfiltered chip input at [P1] in the transmitter is equivalently lowpass filtered at [P2]. The signal propagates over space after a delay
The filtered chip at point [P3] is multiplied by an unfiltered replica which embodies a trial delay
It is to be understood that the receiver has to generate a vector of various trial values of
The presence of sub-carrier and the ambiguity is revealed in these multiple peaks (include negative). The separation of inner peaks (count both negative and positive) is half the period of the sub-carrier. Only tracking the centre peak gives the correct answer. A coarse estimate is not explicitly computed but is implicitly located in the Λ-shaped
In the
In an alternative
In yet another concept of
See Figure
Schematic of correlation shapes in three basic types of optimal BOC receiver.
However it is derived that the coarse estimate will generally have a greater error than the fine estimate.
The problem with band-limited cosine-phased BOC is that the relative error in the coarse estimate is much more likely to exceed this limit under representative worst case conditions.
We test this assertion on the model receiver implementing the BJ “bump-jumping” concept. However exactly the same general conclusions follow for the other two possible implementations. DET may give a better performance on initial acquisition and pull-in but cannot overcome this tracking problem. Early tests in simulations did not appreciate the fundamental difficulties described here [
It is immediately clear that if the effect of noise and distortion is to drive adjacent secondary peaks to an equal or greater amplitude than the primary peak, then eventual failure follows. This is more likely to happen with a
The critical condition is depicted in Figure
Limit of BJ operation: max error
If the relative error should exceed the limit then the receiver shifts to a semipermanent condition of
The graph of Figure
Rectified correlation. Linear phase TX filter
We can define an amplitude margin as between the primary peak and the worst case adjacent secondary peak. In a channel of infinite bandwidth, the correlations are much the same and the amplitude margin is essentially the same.
In the move from BOCsin(15, 2.5) to BOCcos(15, 2.5) assuming very wide-band theoretical filters with
For the same value
A practical system introduces phase distortion as well—in both transmitter and receiver unless deliberately compensated. The effect is modelled here by making the input filter to the receiver a standard resonator type filter with a sharp cut-off outside the passband. The same linear phase filter is assumed in the transmitter (dotted black). Figure
Phase distortion. Rectified correlation. Linear phase TX filter
The theoretical test is to maintain the same proportional shape to the receiver filter while varying its bandwidth. With phase distortion not only is the peak of the envelope flattened but the peak of the envelope function Λ() is displaced relative to the peaks in the actual correlation. The resulting effect on the correlation is shown in Figure
Phase distortion. Rectified correlation. Linear phase TX filter
The combination of these two effects on the cosine option is seen to identify a critical bandwidth
In principle phase compensation can be implemented in the receiver—as for the transmitter. But effective control and implementation of appropriate compensation in the receiver is a complex matter and may need to be adaptive. Automatically the receiver is not robust.
The point is the following the band-limited cosine option is far more sensitive to phase distortion than the sine option, without specific compensation a receiver of BOCc(15,2.5) will not work.
The transmitters for the Galileo satellites are constrained to a rigorous specification on linearity. Necessarily there must be less control over the receiver design.
Multipath vector adds one or more extraneous signals to the main path. Adaptive compensation against multi-path is a complication which it is assumed here that the BOC receiver will not have implemented. Accordingly it is crucial to test what is the immediate effect of multipath on an unadapted receiver.
A standard test—following the GSTF (Galileo Signal Task Force) [
These bias errors generally are different: multipath therefore forces a relative error or relative bias between the coarse and the fine estimates. If this relative error exceeds the ±
We introduce therefore a new kind of test of BOC performance—one which may not have previously been identified in the literature. Instead of just testing for the absolute shift of the correlation peak—as in the usual test—we now test for the
Figure
Absolute multi-path error in main peak (fine estimate). Rectified correlation. Linear phase TX filter
The effect of multi-path on the
Worst case rectified correlation distorted by multi-path. Linear phase TX filter
A computer search now implements the proposed new test: of the relative error as between correct peak of the correlation function and peak of its envelope over a range of delay times
Relative multi-path error between fine and coarse range estimate. Linear phase TX filter
On this limit adjacent peaks (include both signs) of the actual correlation are the same amplitude and it is not possible to tell true from false. Over this limit and the receiver must slip into a condition of false lock and not recover. There are multiple opportunities for cycle slip—a condition which phase distortion, electrical noise, and other system errors can only exacerbate—to bring this about.
The deviations of the peak of the envelope for the sine option are clearly much less and stay well within limits.
The extent of multi-path error for the
Relative multipath error between fine and coarse range estimate. Linear phase TX filter
To make the point that the problem is due to an insufficient bandwidth, we run an identical test assuming now that (somehow) the very much wider bandwidth could be allocated to the receiver
Relative multi-path error between fine and coarse range estimate. Ultra-wide TX bandwidth linear phase. Red: cosine option blue: sine option. Matching filter in receiver is
Repeating the multi-path test for BOCcos(10, 5) shows that the cosine option is again worse off although it is not so drastically affected. Again there is no significant difference in the effect of multi-path on the error to the
The cosine option is however degraded by a brick wall filter. The assumed receiver brick wall filter now has
Computer tests show that there are significant ranges of multi-path delay for which the relative error breaches the limit. For the sine-phased option it stays well within limits.
The conclusion to be drawn from this paper is that the PRS A code signal for E1 as presently specified is not going to work as expected and specified. What undermines this BOCcos(15, 2.5) signal are factors no one of which is wrong individually but in combination will lead to indifferent operational performance if not outright failure. These factors are (i) high ratio of sub-carrier frequency to code rate, (ii) relatively high sub-carrier frequency, (iii) relatively low assigned bandwidth in the transmitter and receiver, and (iv) cosine phasing. Individually these conditions may be acceptable but not all in combination.
It is readily confirmed that the error due to multi-path in making the fine estimate is about the same for either cosine or sine phasing.
But BOC needs other estimates for a receiver to do its work properly—the coarse estimate derived from transitions in the code component. It must have this in order to resolve the ambiguities in the fine estimate. This is where the cosine option comes unstuck. It has been shown that the accurate determination of these transitions in the code depends on the propagation of double-frequency pulse (d.f.p) needing that much wider bandwidth.
The inability to transmit and receive the critical high-frequency content of the cosine phased signal degrades the two-level BOC receiver for the reasons stated and explained. This fact has been demonstrated here most simply by introducing multi-path into a test calculation according to a standard test model. Deeper analysis will show a corresponding increased vulnerability to electrical noise in critical conditions of low C/N0 and high loop bandwidth.
The source of the very great difference in bandwidth dependence is located in every chip transition when the pseudo-random code sequence changes phase. For the cosine option a bi-polar pulse is created which is double the sub-carrier frequency. For BOCc(15, 2.5) a.k.a BOCcos(15, 2.5) the frequency of this pulse is 30 × 1.023 = 30.69 MHz. The requirement of this d.f.p has been shown to push up the inherent bandwidth—when doubled-up to allow for upper and lower sidebands—from 40.92 ideally to 81.84 MHz. Galileo is not going to provide this higher value of bandwidth. The available bandwidth will not be able to propagate this key signal element and deliver the much-vaunted accuracy of this new global navigation satellite system using PRS.
All information presented here comes from public sources. The deductions and conclusions are entirely the responsibility of the author. None of the views presented in this paper should be attributed or taken to represent the views individually or collectively of Surrey Space Centre, University of Surrey, or any associated company, or their representatives. The work was done without financial support. The author is grateful for useful comments from the anonymous referee which have been incorporated this paper.