Dual binary phase-shift keying tracking method for Galileo E 5 AltBOC ( 15 , 10 ) signal and its thermal noise performance

Coherent wideband processing for the Galileo E5 AltBOC signal encounters a great challenge because of its modulation complexity, multi-peaked auto correlation function (ACF) and large bandwidth. In this study, a new tracking method called ‘dual binary phase-shift keying (BPSK) tracking’ (DBT), which is derived from its reception models and double estimator technique (DET) methodology, is specially designed for the Galileo E5 signal. While this method can achieve the full potential of the Galileo E5 signal in ranging performance, it features a robust wideband processing technique, backward compatibility with conventional BPSK signal tracking, and easy implementation in hardware. More specifically, the DBT method makes use of the coherence of the lower and upper bands of the Galileo E5 signal and decouples sub-carrier phase and carrier phase through coherently combining correlator outputs of the two bands, and then implements independent tracking for code, sub-carrier and carrier based on the DET methodology. Furthermore, thermal noise performances of this new method are given and verified by processing both simulated and real Galileo E5 signals.


Introduction
With the ever-increasing demand of navigation users for positioning service improvements and the drive of some countries or organisations for their own independent navigation systems, Global Navigation Satellite Systems (GNSS) are currently undergoing a historical updating phase.GPS and Glonass, traditional mainstay navigation systems, are continuously being modernised, whereas Galileo, an emerging navigation system, is still under its construction phase, and Beidou, another emerging navigation system, is also entering its next phase of development.New navigation signals based on the well-known binary offset carrier (BOC) modulation or its variants are adopted for the next generation of GNSS with the aim of providing a better performance in terms of accuracy, availability, and robustness while maintaining compatibility and interoperability with the legacy navigation signals within the limited frequency resources for satellite navigation.Among these new signals, Galileo E5 signal, which is implemented by the state-of-the-art alternative BOC (AltBOC) modulation scheme, is regarded as one of the most advanced and promising signals because of its potential for unprecedented accuracy (0.02 m in open sky scenarios) [1,2].The potential applications of the Galileo E5 signal are being carried out in fields of precise position determination [3] and accurate time and frequency transfer [4,5].Meanwhile, the processing technique for the signal, which can harvest the full potential of unprecedented accuracy while avoid the problems caused by its complex modulation and large bandwidth, is still gaining interest in the GNSS signal processing field.
The Galileo E5 signal was initially recommended to be viewed as two separate quadrature phase-shift keying (QPSK) signals with two different carrier frequencies (E5a: 1176.45MHz and E5b: 1207.14MHz) and to be processed independently by user receivers via the conventional QPSK/ binary phase-shift keying (BPSK) way in the Galileo signal in space interface control document (SIS ICD) [6].In fact, processing the Galileo E5 signal as a whole, rather than two separate QPSK signals, was not at first envisioned by its inventors and the idea of full AltBOC signal processing arrived later when theoretical studies showed an unprecedented accuracy of the wideband processing [1], which is particularly attractive to high-end users, such as surveying and timing users.However, the Galileo E5 signal, an AltBOC multiplexed constant envelop signal, modulates two different groups of data-plus-pilot pseudo-random noise (PRN) codes on its lower and upper bands, respectively, through two complex side-band sub-carriers, and its overall bandwidth reaches over 50 MHz, much larger than traditional signals.All these factors combined together make it a challenging task for receiver designers to implement wideband processing for this signal.
To the best knowledge of the authors of this paper, based on different processing principles of the sub-carrier, the existing wideband processing methods for the Galileo E5 signal can basically be divided into two categories.The first category is the combined sub-carrier and code processing, and the second category is the separated sub-carrier and code processing.In the combined processing methods, sub-carrier and code are treated as a whole and driven by a same NCO (numerically controlled oscillator), while in the separated processing methods sub-carrier is treated independently and tracked by an additional loop.
The first category mainly includes: (i) auto correlation function (ACF) based or approximate ACF-based methods [7][8][9][10]; (ii) sub-carrier processing (SP) methods [1,11]; (iii) side-peak cancellation (SC) methods based on the pseudo correlation function [12,13].The ACF-based method employs the well-known delay lock loop (DLL) technique to track code and sub-carrier as a whole, where prompt, early and late replica of the baseband signal (code multiplied by sub-carrier) are generated and correlated with the incoming signal.Its main problem arises from the multi-peaked ACF of the AltBOC signal and the potential for the code/sub-carrier tracking loop closing in a false lock onto a secondary rather than the primary peak [12,14,15].The SP method tracks sub-carrier phase instead of code/ sub-carrier delay by the commonly used phase lock loop (PLL), where in-phase and quadrature binary sub-carrier replica multiplied by the synchronous PRN code are generated locally to correlate with the incoming signal.The false lock problem still exists in this method, as the sub-carrier discriminator has multiple zero crossing points where PLL can keep stable tracking [14].On the other hand, the SC method generates new local signal whose chip waveforms are different from those of the incoming signal and non-coherently combines correlator outputs to remove the side peaks from the correlation function while maintaining the sharpness of its main peak [16].The main problem of the SC method lies in the fact that it sacrifices its thermal noise performance and dynamic tracking range for eliminating the side peaks because of the mismatch between local and incoming signals and non-coherent combination.
The second category is represented by the method proposed in [15], which originates from the innovative double estimator technique (DET) [17][18][19].DET and its improved version were first proposed to realise unambiguous tracking for BOC signals.Besides standard code and carrier loops, DET method adds an independent loop for sub-carrier tracking where early, prompt and late sub-carrier replica are generated and DLL discriminators are employed.In other words, sub-carrier is treated as a 'code' in the DET method since it is a square wave in the BOC signals.Two-dimensional (2D) correlation function of code and sub-carrier is derived to replace the multi-peaked 1D ACF.When code and sub-carrier are decoupled, the correlation function is a triangle function on the code dimension while it is a periodic triangle function on the sub-carrier dimension, and thus unambiguous tracking for code can be realised.Although sub-carrier tracking may lock on any peak of the periodic triangle function, this ambiguity can be distinguished using the synchronisation relationship between code and sub-carrier.However, the DET method cannot be directly applied to the AltBOC signal because of the modulation complexity, but it provides a new processing idea.The method deriving from DET in [15] generates sine signals to correlate with the sub-carrier component of the incoming signal and employs an additional PLL for sub-carrier tracking.It should be noted that a perfect carrier alignment is assumed in this DET-based method and 2D correlation function of code and sub-carrier is still taken into account.Nevertheless, sub-carrier and carrier are coupled together when sub-carrier is treated as an independent 'carrier', and their decoupling is conditional, which implies a possible false lock on one of the side bands, as will be illustrated in this paper.
From the above discussion, it can be seen that different methods have different forms of false lock, and therefore the notation 'ambiguity problem' is introduced in this context and defined as a collective name of all kinds of false lock mentioned above.Admittedly, the ambiguity problem cannot be completely eliminated by coherent wideband processing techniques.Therefore, the key to the problem is to avoid or reduce the probability of a false lock especially at the time of loop closure, and to implement a sensitive false lock indicator to detect this possible false condition.Another common problem confronted by these methods is multi-level sub-carrier generation.Multi-level sub-carriers are required to be generated to correlate with the incoming signal in all the methods except the SP method, making them inconvenient for hardware implementation.However, the SP method would actually degrade the thermal noise performance by replacing the multi-level sub-carriers by simpler binary sub-carriers.Besides, the existence of secondary codes on top of the primary PRN codes additionally complicates the tracking and requires techniques for wiping off the secondary codes before further processing the signal [20].Both ACF-based and SP methods neglect the effects of the secondary codes before combining the correlator outputs and lack corresponding techniques to wipe off the secondary codes.Similarly, the DET-based method in [15] does not take into account the secondary codes either when generating combined codes (sum and delta codes).More importantly, sum and delta codes (pure primary codes) frequently interchange between I and Q branches because of the existence of the secondary codes, consequently, the DET-based method in [15] where only sum code is generated locally and used to track the incoming signal, would not simply lead to a 3 dB power loss, but would be unable to close its sub-carrier and carrier tracking loops in phase lock state before wiping off the secondary codes.In addition, sum or delta of two bi-level codes produces the three-level combined codes with values 1, −1 and 0, and they also add some additional complexity for implementation in hardware.It should also be noted that all methods almost never mention data demodulation function which is equally important to a navigation receiver, and their correlator architectures make it difficult to recover the data component either in the lower or upper band.For instance, an additional block called side band translator [21] is required for recovering the navigation data from the Galileo E5 signal.
Considering the great potential of the Galileo E5 signal, and given the drawbacks of current processing techniques, this paper aims to design a new processing method for the Galileo E5 signal, which should stick to a set of principles, including: Regarding the first point, as the ability to achieve an unparalleled ranging performance is always considered to be of key value of the Galileo E5 signal, the methods that degrade the inherent performance through ambiguousness would not be options in this paper.Regarding the second point, it is actually not necessary to eliminate the ambiguity problem, but requires an algorithm which can help to avoid it or reduce its occurrence to a sufficiently low level and detect if a false lock occurs.Regarding the last two points, since the Galileo E5 signal can be processed as two separate QPSK/BPSK signals independently and conventional BPSK correlator is particularly suitable for hardware implementation, a tracking method, which permits avoidance of multi-level sub-carrier and combined code generation and hence maintains compatibility with BPSK signal tracking, would be attractive.Such a method would not simply allow flexible switch between single-band and wideband tracking modes, but also provide an approach to address the tough ambiguity problem and wipe off the secondary codes.
Based on the principles described above, we design a tracking method called 'dual BPSK tracking' (DBT) in this paper based on the reception models and DET methodology.The core idea of the DET method is to treat code, sub-carrier and carrier independently and add an additional loop to track the sub-carrier, while its implementations can be diversified, and should not be confined to the proposed architecture in [17][18][19].The DBT method takes advantage of the conventional BPSK correlator structure to implement the new DET methodology especially for the Galileo AltBOC signal, and utilises the coherence of the lower and upper bands of the signal.Specifically, the DBT method coherently combines the BPSK correlator outputs of the lower and upper bands to decouple the sub-carrier and carrier, and then implements independent tracking for each component of the signal.The remainder of this paper is organised as follows: in Section 2, the Galileo E5 AltBOC signal property is described and its two equivalent reception models are given and analysed.In Section 3, the DBT method is described in detail, including correlator architecture, tracking algorithm and the solution to avoid the ambiguity problem.In Section 4, thermal noise performances are given for the new method.In Section 5, the thermal noise performances are verified by processing simulated and real Galileo E5 signals in hardware.Finally, summary and conclusions are drawn in Section 6.

Galileo E5 AltBOC signal property and its reception models
It is well-known that the Galileo E5 signal incorporates the innovative data-plus-pilot signal structure, and locates two pairs of data-plus-pilot components at its lower and upper bands, respectively.The wideband Galileo E5 signal is generated with the AltBOC modulation of code chip rate f code = 10.23 MHz and sub-carrier frequency f s = 15.345MHz according to [6] (1), where e E5a−I (t) and e E5b−I (t) are data components which carry two different navigation services, e E5a−Q (t) and e E5b−Q (t) are data-less pilot components which allow a net gain of 3 dB in tracking threshold with a pure PLL [22].These four components are termed 'single signals'.It should be pointed out that there are secondary codes with different length modulated on the primary PRN codes of the four components with the aim of shaping the power spectra and shortening the synchronisation time.e E5a−I (t), e E5a−Q (t), e E5b−I (t) and e E5b−Q (t) are termed 'product signals', and they are required to realise a constant envelop modulation at the transmitting end and hence to make the high power amplifier work at saturation with high efficiency, while they are useless at the receiving end.sc E5−S (t) and sc E5−P (t) denote the four-level sub-carriers designed for the single and product signals, respectively, as shown in Fig. 1, and T s = 1/f s denotes the sub-carrier period.
A full AltBOC signal could be replicated locally with external data-aiding to track the pilot and data components as a whole in achieving its best performance, but instead the pilot component is normally used for tracking while the data component is used for data demodulation in a data-plus-pilot signal structure.Therefore, pilot tracking plus data demodulation strategy is adopted in this context.For these reasons, neglecting the useless product signals and ignoring the data components for the moment, the desired baseband signal at the transmitting end can be expressed as follows In addition to the data-plus-pilot signal structure, another notable feature of the Galileo E5 signal is that it adopts complex single side band sub-carriers instead of real double side band sub-carriers as common BOC signals do, which enables it to carry different services at the lower and upper bands.According to [1], the lower band sub-carrier has its harmonics at −f s , 7f s and −9f s and the conjugate upper band sub-carrier has its harmonics at f s , −7f s and 9f s .Considering the transmission bandwidth of the Galileo E5 signal is below 90 MHz, the AltBOC inventors claim that ground users receive only the fundamental harmonics of the single side band sub-carriers [1].Therefore, a frontend bandwidth of 51.15 MHz is suitable for the Galileo E5 signal tracking, which is consistent with the reception bandwidth recommended in the Galileo SIS ICD [6].Larger bandwidth might be useful for code tracking while the final pseudo-range accuracy is largely determined by the sub-carrier tracking, so increasing the bandwidth will not add any processing gain.As a result, the desired baseband signal can be further expressed as follows For simplicity, the amplitude term is dropped.The desired transmitted radio frequency (RF) signal can be written as follows where f c represents carrier frequency.By substituting (3) into (4), the desired RF signal can be expanded as follows In this paper, the Galileo E5 signal is assumed to be received only in the presence of thermal noise at the receiver frontend which can be modelled as stationary additive white Gaussian noise.As a result, the received signal can be expressed as follows where t is the signal propagation delay, ñ(t) is the thermal noise and the delayed signal is as (7), where j = −2πf s t and θ = −2πf c t represent the sub-carrier phase and the carrier phase, respectively.Through trigonometric transformation, the reception model ( 7) can be equivalently converted into the following Note that the effects of Doppler shift and down-conversion at the receiving end are neglected here for simplicity, and they do not affect the following derivations and final conclusions.
The two equivalent reception models ( 7) and ( 8) provide two different perspectives to view this signal.From the reception model ( 7), the received signal can be viewed as an Alt-LOC (linear offset carrier) signal [5] or a sum of two LOC signals with cosine and sine phasing which are modulated by two combined PRN codes, respectively.The model indicates three separate components of the received signal: code, sub-carrier and carrier, and suggests that the DET method can be applied in processing this signal.From the reception model ( 8), the received signal can also be viewed as a sum of two BPSK signals centred at ( f c − f s ) and ( f c + f s ), respectively, which not only forms the basis of independent processing for the lower and upper bands recommended in the Galileo SIS ICD [6], but also provides some insights of wideband processing for the Galileo E5 signal as a whole.The model shows the coherence of the lower and upper bands, and implies that the sub-carrier phase j and the carrier phase θ can be recovered from these two BPSK signals.
In this paper, the specifically defined baseband signals e E5a−Q (t) and e E5b−Q (t) are replaced by two general baseband signals s a (t) and s b (t) to extract a general reception, as shown below As will be explained in this paper, the two baseband signals modulated on the lower and upper bands are not required to be orthogonal and they can even be equal.For instance, this model also applies in high-order BOC signals, where two baseband signals modulated on the lower and upper bands are identical.
3 Dual BPSK tracking method

Correlator architecture
From the reception model ( 7), it can be seen that directly using the DET method to track the Galileo E5 signal would require generating multi-level sub-carriers and combined codes, and they would complicate correlator architecture design.However, this complication is not actually necessary.It should be noted that the essence of the DET method is to track code delay, sub-carrier phase and carrier phase independently by using separate tracking loops.It should also be also noted from the reception model ( 8) that the two BPSK signals located at the lower and upper bands are coherently related and their carrier phases are linear combinations of the sub-carrier phase and carrier phase of the AltBOC signal.Therefore, the estimates of the sub-carrier phase and the carrier phase can be derived from these two coherently related BPSK signals.
Based on the above analysis, we implement two groups of conventional BPSK correlator to correlate with the incoming signal, and expect that code delay, sub-carrier phase and carrier phase can be completely separated by some kind of operation without performance penalty.Fig. 2 depicts the new correlator architecture for the Galileo E5 signal proposed in this paper.The left and right sides are the two groups of conventional BPSK correlator used to correlate the lower and upper bands of the incoming signal, whose outputs are then fed into the middle processing unit.The processing unit is responsible for combining, discriminating and filtering, and filtered discriminator outputs are used to correct NCO frequencies of code, sub-carrier and carrier, respectively, where carrier aiding of code or sub-carrier loop might be employed in order to get more accurate measurements.Closed loops for code, sub-carrier and carrier tracking are finally formed by feeding their corrected NCO frequencies back to the correlators.It should be noted that the input signals of the left and right sides are the same IF (immediate frequency) signals.
The local signals generated on the left side are used to correlate with the lower band of the incoming signal.As stated previously, the lower band can be separately viewed as a standard BPSK signal, therefore the locally generated signals are defined as the six possible combinations of the in-phase (I) and quadra-phase (Q) local carrier replicas and early (E), prompt (P) and late (L) local code replicas, and they can be written as follows Similarly, the six locally generated signals on the right side can be written as follows where t represents the previous estimate of code delay, fc and fs represent the previous estimates of the carrier and sub-carrier frequencies, respectively, û and ŵ represent the previous estimates of the carrier and sub-carrier phases and D represents the correlator spacing between the early and late correlators.
From Fig. 2, it can be seen that the DBT correlator architecture have decided advantages over other existing correlator architectures [1,[7][8][9][10][11], including: (i) neither multi-level sub-carriers nor combined codes are required to be generated, as sub-carrier and carrier signals are generated in an indirect way where carrier signals of the lower and upper bands whose centre frequencies are equivalent to ( f c − f s ) and ( f c + f s ) are generated instead, and combined code generation is replaced by combining correlations after wiping off the secondary codes; (ii) the correlator architecture is backward compatible with the conventional BPSK correlator and hence allows separate BPSK tracking (SBT) for the lower and upper bands at the same time without any hardware modifications.Therefore, the ambiguity problem can be avoided through smooth transition from SBT mode to DBT mode and the secondary codes can be wiped off through SBT processing of the lower and upper bands; and (iii) data demodulation can be easily implemented by adding a correlator to correlate the data component in either lower or upper band.For simplicity, correlators for data demodulation are not shown in Fig. 2.

Combination and decoupling
After correlator architecture design, we need to estimate code delay, sub-carrier phase and carrier phase.However, the sub-carrier and carrier phases are still tightly coupled together in the carrier phases of the lower and upper bands.A straightforward method to obtain their respective estimates is to discriminate first and then decouple: obtain the carrier phase estimates of the lower and upper bands separately by discriminating first and then convert them to the desired sub-carrier and carrier phases according to their relationship as shown in (9).However, it will lead to the so-called 'squaring loss' as the carrier phase discriminator is a non-linear function, which departs from our design principles.Therefore, it is expected that the sub-carrier and carrier phases can be decoupled first by combining the correlator outputs in a coherent manner and then be estimated by respective discriminators.
According to the local reference signals generated in the correlator architecture, the six correlator outputs for the lower band can be expressed as follows Similarly, the other six correlator outputs for the upper band can be expressed as follows where R a (t The coherent combinations of the correlator outputs are implemented with the aim of decoupling the sub-carrier phase and the carrier phase and hence obtaining their respective estimates.Assume that both the lower and upper bands are locked, viz., both residual frequency errors (Δf c + Δf s ) and (Δf c − Δf s ) approach zero, and that the ACFs R a (t) and R b (t) are identical to the standard triangle function R(t), then their coherent combinations are as follows in (14).
As can be observed from (14), code delay, sub-carrier phase and carrier phase are completely separated, and especially the coupling relationship between sub-carrier and carrier, as shown in the uncombined correlations in ( 12) and ( 13), has been dissolved through combining coherent correlations of the lower and upper bands.Therefore, independent discriminating can be carried out for the three components.The first two results in (14) are used to feed into the coherent code discriminator, while the next two are used to feed into the sub-carrier and carrier discriminators, respectively, and the last in-phase, prompt correlation is used to normalise the four correlations above to remove the amplitude sensitivity.

Discriminators and filters
After decoupling, we need to design independent discriminators for code, sub-carrier and carrier.Both coherent and non-coherent code discriminators can be employed here.The coherent code discriminator is defined as early minus late envelop, as shown below It can provide superior performance when both sub-carrier and carrier loops are phase locked.While the non-coherent code discriminator is defined as the early minus late power.As the power does not change during the processing, so the non-coherent discriminator can be directly written as (16).
It is generally employed because of its insensitivity to the phase lock of the other two loops in comparison to the coherent code discriminator.
As the pilot signal is data-less, a pure PLL discriminator can be implemented in tracking both the sub-carrier and carrier to gain the so-called 3 dB net gain, and discriminators can be written for the sub-carrier and carrier, as follows (the optimal four-quadrant arctangent is employed here for both the sub-carrier and carrier discriminators) As the outputs of the discriminators are noisy, and therefore discriminators are normally followed by loop filters to reduce noise in order to generate more accurate estimates of code delay, sub-carrier phase and carrier phase.The filtered outputs of discriminators, combined with external aiding signals if any, are then used to drive their respective NCOs.In this paper, two separate second-order filters are selected for code and sub-carrier loops, while a third-order filter is selected for carrier loop, and carrier aiding of code and  sub-carrier loops are implemented as the carrier loop is more accurate.Although the loop filter order has great influence on tracking performance, this paper is not going to discuss different filter configurations for simplicity.

Potential false lock and its solution
As stated previously, the decoupling between sub-carrier and carrier is not always true but conditional on other factors when the sub-carrier is treated as a 'carrier'.Let us step back to the assumption made at the beginning that both the lower and upper bands are locked.Note that they might be incorrect, especially when coarse frequencies and phases are transferred from acquisition to pull-in process, say the lower or the upper band is unlocked, then the above derivations will not hold.Without loss of generality, suppose that the upper band is unlocked here, viz., Df c + Df s ≫ 0, then the correlation combinations turn into the following Clearly, the sub-carrier and carrier remain tightly coupled together.If the discriminators mentioned above were still employed, their outputs would be equal and opposite, as shown below Thus, the sub-carrier and carrier tracking loops would interact with each other and make a receiver close in a false lock onto either the lower or the upper band.As discussed earlier, the solution to the ambiguity problem is not to eliminate it but to avoid it or reduce its occurrence.As we know, it is possible for DBT method to close in a false lock on either lower or upper band if sub-carrier and carrier are not well-decoupled at the time of loop closure.Therefore, DBT method is designed to close its tracking loops with the unambiguous SBT mode, and then DBT receiver will systematically transition into its normal mode when carrier frequencies and phases of both the upper and lower bands are locked and secondary codes are wiped off as well.As SBT method can provide sufficiently accurate frequencies and phases for the next DBT phase, thus this mechanism permits smooth transition from acquisition to the final DBT phase without falling into a false lock trap.
Another issue worthy of discussion is the accurate and timely detection of a false lock when it does happen.If dynamic stress or other factors cause DBT to fall into a false lock or even lose lock, the receiver should have the ability to detect it using appropriate lock detectors and transition back to SBT mode, and then the above mentioned DBT closure process is repeated.On the one hand, the fact that DBT method keeps stable tracking requires that both the lower and upper bands are phase locked and sub-carrier and carrier are well-decoupled, otherwise, it either falls into a false lock condition or even loses lock.On the other hand, if a single band (lower or upper) is in phase lock state, then its corresponding IP will be maximum and QP will be minimum, and their lock detectors are defined as follows [23] These lock detectors are also termed as PLL lock values.According to the above analysis, the phase lock detectors of the lower and upper bands, or their filtered versions, can be used together to detect the false lock condition.The DBT method checks the phase lock detectors before combining the correlator outputs to decide if a false lock occurs.Until now, a robust tracking method, which is backward compatible with the conventional BPSK signal tracking as well as easy to be implemented in hardware, has not been achieved for the Galileo E5 signal processing.

Thermal noise performances
After providing the overview of the DBT method, we present in this section its thermal noise performances in order to analyse if the full potential of the Galileo E5 signal is achieved.Assuming no multipath and jamming or interference in the received signal, thermal noise is generally treated as the dominant error sources of signal tracking.Note that, compared with the conventional BPSK signal tracking, the DBT method designed for the Galileo E5 signal here, introduces a sub-carrier phase in its reception model and correspondingly employs an independent loop to track it.Based on its pass-band reception model, a complex baseband reception model for the Galileo E5 pilot signal can be obtained by the following where n(t) represents the equivalent complex thermal noise, and the rest are the same as the symbols defined previously.Note that s a (t) and s b (t) are orthogonal mutually frequency-wise not code-wise, as shown below where t 1 and t 2 are arbitrary delays of the two signals.This implies that the orthogonal relation still holds even if they are identical in the BOC case.This orthogonal property is crucial to the derivations in the Appendix.The code tracking errors induced by the thermal noise can be obtained, respectively, by ( 26) and ( 27) for the coherent and non-coherent cases using the theory proposed in [24,25] and the orthogonal property mentioned above where C/N 0 represents the single-band carrier-to-noise power ratio, G( f ) represents the power spectra density of either the lower or the upper band, which is normalised to unit power over infinite bandwidth, β r is the equivalent double-side front-end bandwidth and B nDLL denotes the equivalent noise bandwidth for code tracking loop, and the part of (27) in the bracket represents the squaring loss term in the non-coherent case.
Since the pilot tracking plus data demodulation strategy is adopted here and pure PLL discriminator is superior to the Costas PLL discriminator in terms of thermal noise performance and tracking threshold, and thus only pure PLL discriminator is taken into account in this paper.The tracking errors induced by the thermal noise are given by ( 28) and (29) for sub-carrier and carrier, respectively.The detailed derivations are provided in the Appendix where B nPLLS and B nPLLC are the equivalent noise bandwidths for sub-carrier and carrier tracking loops, respectively.Note that the DBT processing has a 3 dB gain in terms of thermal noise errors of code and carrier tracking over the SBT processing of either lower or upper band alone, since the correlations of the upper and lower bands are coherently combined with the DBT method, leading to an equivalent 3 dB improvement in signal-to-noise ratio.More importantly, this new tracking method provides a new measurement: sub-carrier phase measurement, which is intermediate between code and carrier phase measurements in terms of accuracy.As can be seen in the next section, code tracking errors are on the order of decimetres, sub-carrier tracking error is on the order of centimetres, and the carrier tracking error is on the order of millimetres.Therefore, a natural idea is to combine the sub-carrier phase measurement, which is high-precision but ambiguous, with the code measurement, which is just the opposite, to form a high-precision and unambiguous measurement given the sub-carrier wavelength of 19.53 m, as shown below [17] where tc represents the combined measurement, t and ŵ are the code and sub-carrier phase measurements in metre, respectively, and l represents the sub-carrier wavelength.A higher-precision combination among code, sub-carrier phase and carrier phase measurements, where the sub-carrier phase measurement serves as a bridge over the code and carrier phase measurements, is even possible given the carrier wavelength of 25 cm.

Simulation and real test results
In  The next step is to verify its noise performances in both simulated and real signal scenarios.In the simulated signal scenario, the Galileo E5 signals of two different satellites (Galileo PRN 11 and PRN 12) are transmitted synchronously by the signal simulator and processed independently by two DBT channels in the verification receiver.Therefore, the differences between measurement outputs of the two channels can be viewed as the thermal noise errors for code, sub-carrier and carrier tracking correspondingly.By adjusting the transmitting power, the variances of the thermal noise errors can be obtained at different C/N 0 .Figs. 5a-c depict the results for code, sub-carrier and carrier tracking errors, respectively.For the convenience of comparison, the theoretically derived results are also presented in Fig. 5.It can be seen that the simulation results are consistent with the theoretically derived results especially at high C/N 0 region and non-coherent DLL is more robust than coherent DLL at low C/N 0 region because of its insensitivity of the phase lock of the other two loops.
In the real signal scenario, it is not as convenient to get the thermal noise errors as in the simulated signal scenario, therefore we take the differences between code and sub-carrier phase measurements with carrier phase measurement to illustrate the thermal noise performances of code and sub-carrier tracking, as carrier phase measurement is much more accurate than code and sub-carrier phase  6 depicts the differences between code and sub-carrier phase measurements with carrier phase measurement, respectively.The thermal noise errors are still not obvious, as the differences are dominated by ionospheric delays.To highlight the thermal noise errors, we differentiate the differences with respect to time, given the fact that ionospheric delays typically change very slowly with time.As shown Fig. 7, the thermal noise errors of code and sub-carrier tracking are basically in accord with the simulated and theoretically derived results.

Summary and conclusion
A new processing method called DBT, is proposed and implemented for the Galileo E5 signal in this paper, and its thermal noise performances are subsequently derived.It can take full advantage of the Galileo E5 signal ranging potential and avoid problems caused by its modulation complexity and large bandwidth at the same time.The DBT correlator architecture is compatible and interoperable with the conventional BPSK correlator, making it convenient to be implemented in hardware.More importantly, this compatibility and interoperability enable user receivers to avoid the aforementioned problems.Simulations and real tests verify and validate the method and its performances.
Besides conventional code and carrier phase measurements, a new measurement extracted from sub-carrier tracking loop can be acquired in this method, which is much more accurate than the unambiguous code measurement.Since sub-carrier wavelength is much larger than code measurement error, it is possible to combine code and sub-carrier phase measurements to generate an unambiguous measurement with accuracy consistent with sub-carrier measurement.Further research can be carried out on the multipath and interference mitigation techniques for the DBT method.

Appendix
Sub-carrier and carrier tracking error analysis Assume that code and carrier are both perfectly aligned, and then the local complex reference signals can be written as follows s la (t) = e j(u−w s k ) s a (t − t) (31) where w s k is the previous sub-carrier phase estimate acquired from the smoothing loop filter.The expressions of the prompt correlator outputs of the lower and upper bands can be written as follows

Fig. 1
Fig. 1 Four-level sub-carriers designed for the single and product signals of the Galileo E5 AltBOC(15,10) signal ) and R b (t) are ACFs of the lower and upper bands, respectively, and Dt = t − t, Df s = f s − fs , Df c = f c − fc , Du = u − û and Dw = w − ŵ represent residual errors for code delay, sub-carrier frequency, carrier frequency, sub-carrier phase and carrier phase, respectively, and T represents the pre-detection integration time.Besides, α = π (Δf c − Δf s )T and β = π(Δf c + Δf s )T are introduced here to simplify the above expressions.

Fig. 2
Fig. 2 Correlator architecture of dual BPSK tracking (coherent combination of two standard BPSK correlators) this section, a new receiver with a frontend bandwidth of 51.15 MHz, intermediate frequency of 210 MHz and sampling frequency of 120 MHz, is specially designed for verifying this new tracking method.The pre-detection integration time is set to 10 ms, and the noise bandwidth parameters of code, sub-carrier and carrier tracking loops are set to 0.4, 0.4 and 10 Hz, respectively, and carrier aiding of code and sub-carrier loops are employed.Two scenarios are used, one is simulated signal transmitted by a signal generator specially designed for this verification, and the other is a real signal transmitted by Galileo satellites in orbit.The signal generator is implemented by field programmable gate array, digital-to-analog converter, and Agilent E4438C vector signal generator.The real Galileo signal test was done at the GNSS Lab of Tsinghua University from 2014-04-27 8:52 to 2014-04-27 15:44 (UTC), and the antenna was mounted on the roof of the Weiqing Building, Tsinghua University, Beijing.During the test, the Galileo PRN 12 satellite was visible (satellite rise and set) and tracked by the receiver.The first step is to verify the robustness of the proposed tracking method.Real signal scenario would be adequate to illustrate this issue.Fig.3depicts the PLL lock values of both lower and upper bands during the transition phase.As shown in Fig.3, the PLL lock values have slight fluctuations when transitioning from SBT mode into DBT mode, while the transition process is quite smooth on the whole, and the ambiguity problem is effectively avoided, as the frequencies and phases which are transferred from SBT to DBT are adequately accurate.The other aspect about robustness is the ability to keep stable tracking at different C/N 0 .Fig.4depicts the PLL lock values and corresponding C/N 0 during satellite rise and set.It can be seen that C/N 0 decreases gradually with the satellite going down while

Fig. 3
Fig. 3 PLL lock values of the lower and upper bands during the transition phase

Fig. 4
Fig. 4 PLL lock values of the lower and upper bands and corresponding single-band carrier-to-noise ratio during satellite rise and set

Fig. 6 Fig. 7
Fig. 6 Differences between code and sub-carrier phase measurements with carrier phase measurement

b r / 2 −b r / 2 G 2 = C T b r / 2 −b r / 2 Gb r / 2 −b r / 2 Gb r / 2 −b r / 2 G
correlation P a (w, w s k ) is composed of signal part S a and noise part N a , and they are defined by (35) and (36), respectivelyS a W 1 T kT (k−1)Ts * la (t) e j(u−w) s a (t − t) + e j(u+w) s b (t − t) into (35) and using the orthogonal property (25), we can obtain the signal part, as follows (f ) df e j(w−w s k )(37)    According to the assumption made on the thermal noise, we can obtain the mean and variance of the noise part, as well as its real and image parts and their relationship, as shown belowE(N a ) = E Re(N a ) = E Im(N a ) = 0 (38) var Re(N a ) = var Im(N a ) = var(N a )/2 (39)Using the definition (36), the variance of the noise part can be expanded as followsvar(N a ) 1 )s(t 2 )R n (t 1 − t 2 ) dt 1 dt 2(40)By substituting (31) into (40) and using the orthogonal property (25) again, we can further obtain the variance of the noise part by the followingvar(N a ) = 1 T 2 kT (k−1)T kT (k−1)T s * a (t 1 − t)s a (t 2 − t) × R n (t 1 − t 2 ) dt 1 dt (f )G n (f ) df(41)Using the same way, we can obtain similar results for the signal part S b and noise part N b of the upper band correlation P b (w, w s k ) as followsS b = C (f ) df e j(w−w s k ) (42) E(N b ) = E Re(N b ) = E Im(N b ) = 0 (43) var Re(N b ) = var Im(N b ) = var(N b )/2 (44) var(N b ) = C T (f )G n (f ) df(45)From the orthogonal property(25), we can find that the noise parts of the lower and upper bands N a and N b are mutually independentE(N a N b ) = 0 (46)According to (14), (33) and (34), the combination correlations can be expressed as follows QP b − QP a = Im(P b ) − Im(P a ) = Im(S b + N b ) − Im(S a + N a ) (47) IP a + IP b = Re(P b ) + Re(P a ) = Re(S b + N b ) + Re(S a + N a ) (48)