Subcarrier shaping for BOC modulated GNSS signals
 Pratibha B Anantharamu^{1}Email author,
 Daniele Borio^{1} and
 Gérard Lachapelle^{1}
https://doi.org/10.1186/168761802011133
© Anantharamu et al; licensee Springer. 2011
Received: 31 October 2010
Accepted: 12 December 2011
Published: 12 December 2011
Abstract
One of the main challenges in Binary Offset Carrier (BOC) tracking is the presence of multiple peaks in the signal autocorrelation function. Thus, several tracking algorithms, including BumpJump, Double Estimator, Autocorrelation SidePeak Cancellation Technique and prefiltering have been developed to fully exploit the advantages brought by BOC signals and mitigate the problem of secondary peak lock. In this paper, the advantages of prefiltering techniques are explored. Prefiltering techniques based on the concepts of ZeroForcing and Minimum Mean Square Error equalization are proposed. The BOC subcarrier is modeled as a filter that introduces secondary peaks in the autocorrelation function. This filtering effect can be equalized leading to unambiguous tracking and allowing autocorrelation shaping. Monte Carlo simulations and real data analysis are used to characterize the proposed algorithms.
Keywords
1 Introduction
Recent developments in the Galileo program have introduced a variety of new modulation schemes including the Binary Offset Carrier (BOC) [1] that has several advantages over traditional Binary Phase Shift Keying (BPSK) signals. BOC signals have increased resilience against multipath and provide improved tracking performance. However, they are characterized by autocorrelation functions (ACF) with multiple peaks that may lead to false code lock. This has led to the design of various BOC tracking algorithms such as BumpJump (BJ) [2], Autocorrelation SidePeak Cancellation Technique (ASPeCT) [3] and its extensions [4], Double Estimator (DE) [5], Side Band Processing (SBP) [6] and prefiltering [7].
In BJ, the BOC autocorrelation function is continuously monitored using additional correlators. A control logic detects and corrects false peak locks exploiting these additional correlators. In ASPeCT and its extensions, i.e., Sidelobes Cancellation Methods (SCM) [4], the BOC signal is correlated with its local replica and a modified local code. Thus, two correlation functions are computed: the first one is the ambiguous BOC autocorrelation, whereas the second only contains secondary peaks. An unambiguous cost function is determined as a linear combination of the two correlations. The DE technique maps the BOC ambiguous correlation over an unambiguous bidimensional function [5]. The subcarrier and the PseudoRandom Number (PRN) code, the two components of a BOC signal, are tracked independently and an additional tracking loop for the subcarrier is required. In SBP, the spectrum of BOC signals is split into side band components through modulation and filtering. Each side band component leads to unambiguous correlation functions. Noncoherent processing can be used for combining the results of the different processing branches [6]. The techniques mentioned above are characterized by different performance and different computational requirements. In this paper, prefiltering techniques are considered for their generality and applicability to different contexts, such as unambiguous tracking and multipath mitigation. Prefiltering techniques [7] are based on the fact that the spectrum of a signal can be modified by filtering. BOC signals are filtered in order to reproduce BPSKlike spectra and autocorrelations.
In this paper, a new class of prefiltering techniques is derived from a convolutional representation of the transmitted signal. More specifically, the useful BOCmodulated signal is represented as the convolution of a PseudoRandom Sequence (PRS) and a subcarrier. The subcarrier is interpreted as the equivalent impulse response of a selective communication channel that needs to be equalized. From this principle, filters analogous to the ZeroForcing (ZF) and Minimum Mean Square Error (MMSE) equalizers [8] are derived. The proposed prefiltering techniques shape the BOC ACF for unambiguous tracking and are herein called ZF Shaping (ZFS) and MMSE Shaping (MMSES). These techniques can be considered an extension of algorithms proposed in the communication context such as the mismatch filter (MMF) [9] and the 'CLEAN' algorithm [10]. The MMF operates on the temporal input data to obtain a desired sequence, whereas the 'CLEAN' algorithm works in the frequency domain to obtain a desired spectrum. In these techniques, a different signal structure was considered and the spectrum of the received signal was shaped for Inter Symbol Interference (ISI) cancellation. The problem of secondary autocorrelation peaks was not considered. In [7], several prefiltering techniques were proposed. The filter design was however based on the combination of PRS and subcarrier. This was causing severe noise amplification making the algorithms impractical for moderate to low signaltonoise ratio conditions. In this paper, the noise amplification problem is mitigated using an innovative filter design based on the subcarrier alone. The feasibility of the proposed algorithms is shown using live Global Navigation Satellite System (GNSS) data.
The filters for subcarrier shaping are initially designed in the frequency domain. This approach requires a high processing load, and thus, a more computationally efficient time domain implementation is subsequently derived. A modified tracking loop architecture is also proposed to independently track code and carrier phase. Subcarrier equalization performed for autocorrelation shaping is only required for unambiguous code tracking. Thus, the modified tracking architecture operates Phase Lock Loop (PLL) and Delay Lock Loop (DLL) independently. The filtered signal is exploited for generating the correlator outputs used for driving the DLL, whereas the unfiltered samples are exploited by the PLL. This further mitigates the noise amplification problem, since the PLL is unaffected by the filtering performed by the subcarrier shaping algorithms.
Subcarrier shaping algorithms are thoroughly analyzed and figures of merit such as tracking jitter, tracking threshold, Mean Time to Lose Lock (MTLL), tracking error convergence analysis and multipath error envelope (MEE) are introduced and adopted for performance evaluation. Although several unambiguous BOC tracking algorithms are present in the literature, only BJ and DE have been used as comparison terms. The BJ has been chosen because it has been one of the first algorithms proposed for BOC tracking. In addition to this, its low computational requirements make it attractive for low complexity receivers. The DE technique has been selected for its close approximation to a matched filter and its improved performance in the absence of multipath. A comprehensive characterization of unambiguous BOC tracking algorithms is out of the scope of this paper. Additional material on the performance of BOC tracking techniques can be found in [4] and [11]. A comparison between standard prefiltering techniques and ZFS is provided in [12] showing the superiority of the latter algorithm.
Real data from the second Galileo experimental satellite, GIOVEB, have been used for extensively testing the proposed algorithms. Different CarrierpowertoNoisedensity ratios (C/N_{0}) have been obtained using a variable gain attenuator. Signals from the GIOVEB satellite have been progressively degraded simulating weak signal conditions.
From the tests and analysis, it is observed that MMSES provides a tracking sensitivity close to that provided by DE technique. When using real data, ZFS provides satisfactory results only for moderate to high C/N_{0}. This is due to the inherent noise amplification that can only be partially compensated for. On the other hand, MMSES is able to track weaker signals for a given bandwidth, leading to a performance close to that of the DE. Subcarrier shaping provides satisfactory tracking performance maintaining the flexibility of prefiltering techniques with the possibility of autocorrelation shaping. The slightly increased noise variance of the delay estimates is compensated by the flexibility of the algorithm that results in enhanced multipath mitigation capabilities. This work is an extension of the conference paper [12] that only considered the ZFS. The innovative contributions of the paper are the design of the MMSES algorithm and the novel implementation of prefiltering techniques in time domain. In addition to this, separate carrier and code tracking is introduced to further mitigate the noise amplification problem. A thorough characterization of prefiltering techniques is also provided.
The remainder of this paper is organized as follows: Section 2 introduces two different signal representations that are used as basis for the derivation of subcarrier shaping algorithms. The basic principles of prefiltering, BJ and DE are also briefly reviewed. Section 3 details subcarrier shaping techniques, their time domain implementation and the modified tracking structure suggested for reducing the noise amplification problem. Section 4 provides a brief theoretical and computational analysis of the proposed prefiltering techniques. Experimental setup, simulation and live data results are detailed in Section 5. Finally, some conclusions are drawn in Section 6.
2 Signal and system model
where

A is the received signal amplitude;

d(·) is the navigation message;

c(·) is the ranging sequence used for spreading the transmitted data; c(·) is usually made of several components and two different representations are discussed in the following;

τ_{0} models the delay introduced by the communication channel whereas θ_{0}(t) is used to model the phase variations due to the relative dynamics between receiver and satellite;

η(t) is a Gaussian random process whose spectral characteristics depend on the filtering and downconversion strategies applied at the frontend level.
In (1), the presence of a single useful signal is assumed. Although several signals from different satellites enter the antenna, a GNSS receiver is able to independently process each received signal, thus justifying model (1).
where s_{ b } (·) is the subcarrier of duration T_{ c } . Equation (2) can be interpreted in different ways leading to different signal representations.
2.1 Convolutional representation
2.2 Multiplicative representation
2.3 The correlation process
where R(Δτ) is the correlation function between the incoming and locally generated signal. The shape of R(Δτ) is essentially determined by the signal subcarrier. For a BPSK signal, R(Δτ) is characterized by a single peaked triangular function. But when a BOC is used, R(Δτ) is characterized by several secondary peaks that can lead to false code locks.
with the objective to make the filtered subcarrier, s_{ h } (t) = s_{ b } (t) * h(t), have a correlation function without sidepeaks. The third BOC tracking technique considered is the BJ [2] based on postcorrelation techniques. These techniques do not directly operate on the signal but on the correlation function and they require additional correlators that are used for monitoring the code lock condition.
3 Subcarrier shaping
In communications, the effect of a frequency selective transmission channel is usually compensated by the adoption of equalization techniques. In the considered research, the effect of subcarrier is interpreted as a selective communication channel that distorts the useful signal. Thus, a similar equalization approach can be adopted for mitigating the impact of the subcarrier. The convolutional representation of BOC signals is used here as basis to derive subcarrier equalizers to shape the BOC ACF.
3.1 MMSES
where

G_{ D }(f) is the desired signal spectrum. Its inverse Fourier transform is the desired correlation function;

G_{ x }(f) is the Fourier transform of the correlation between incoming and local signals. G_{ x }(f) and G_{ D }(f) have been normalized in order to have unit integral;

G_{ L }(f) is the spectrum of the local code;

N_{0} is the power spectral density (PSD) of η(t), the input noise is assumed to be white within the receiver bandwidth;

λ is a constant factor used to weight the noise impact;

B is the receiver frontend bandwidth;
In (12), G_{ x } (f ) can contain zeros that would make H(f ) diverge to infinity. This is avoided by clipping the amplitude of H(f) to certain limits, thus removing the singularities in G_{ x } (f ).
In the following, λ will be set to 1 and N_{0} is adapted according to the input C/N_{0} and scaling applied to the signal power density, G_{ x } (f). Comparison of ZFS and existing prefiltering techniques [7] have been performed in [12] and the analysis proved that ZFS is able to successfully compensate for secondary autocorrelation peaks, whereas standard approaches are unable to mitigate secondary peak locks for moderate to low C/N_{0} values. Since standard prefiltering techniques [7] are outperformed by ZFS, they would not be further considered in the reminder of this paper. The interested reader is referred to the findings presented in [12].
3.2 Time domain implementation
where ${\mathcal{F}}^{1}\left\{\cdot \right\}$ is the Inverse Fourier transform.
is an equivalent code accounting for the filtering performed by H(f).
3.3 Delay and phase independent tracking
4 Algorithm characterization
4.1 Theoretical analysis
It is noted that the numerator and denominator in (18) are the signal and noise terms of the cost function (9). The MMSES tries to find a compromise between making G_{ x } (f) H (f) as close as possible to the desired spectrum, G_{ D } (f ), and reducing the noise term at the denominator of (18).
In (20), Δτ denotes the additional delay used for computing a specific correlator. Δτ = 0 for the Prompt correlator and Δτ = ±d_{ s } /2 for Early and Late correlators. d_{ s } is the EarlyLate correlator spacing. In (21), Δτ is used to denote the delay difference between two correlators. Early and Late are separated by a delay equal to d_{ s } , whereas the Prompt correlator is characterized by a delay difference equal to d_{ s } /2 with respect to the other correlators.
The results listed above can be used for computing the tracking jitter. The tracking jitter is one of the most used metrics for determining the quality of estimates produced by tracking loops. More specifically, the tracking jitter quantifies the residual amount of noise present in the final loop estimate, in this case the code delay [17]. The tracking jitter is directly proportional to the standard deviation of the tracking error defined as the difference between true and estimated tracking parameters. A large tracking jitter indicates poor quality measurements and a large uncertainty in the estimated parameters.
where D (Δτ) defines the discriminator inputoutput function.
Theoretical tracking jitter for different discriminator types
Discriminator (D)  Tracking jitter (σ_{ j }) 

Coherent Re{E  L}  $\sqrt{\frac{{B}_{\text{eq}}{T}_{i}{\sigma}_{n}^{2}(1{R}_{n}({d}_{s}))}{2\dot{R}{({d}_{s}/2)}^{2}}}$ 
Quasicoherent dotproduct Re{(E  L)P*}  $\sqrt{\frac{{B}_{\text{eq}}{T}_{i}{\sigma}_{n}^{2}(1{R}_{n}({d}_{s}))}{2\dot{R}{({d}_{s}/2)}^{2}}(1+{\sigma}_{n}^{2})}$ 
Noncoherent early minus late power E^{2}  L^{2}  $\sqrt{\frac{{B}_{\text{eq}}{T}_{i}{\sigma}_{n}^{2}(1{R}_{n}({d}_{s}))}{2\dot{R}{({d}_{s}/2)}^{2}}\left[1+{\sigma}_{n}^{2}\frac{(1+{R}_{n}({d}_{s}))}{2{R}_{n}^{2}({d}_{s}/2)}\right]}$ 
4.2 Computational analysis
Computational complexity of prefiltering, BJ and DE
Algorithm  Number of complex correlators  Notes 

BJ  5 bipolar/binary  The local code is a bipolar sequence and code multiplication can be effectively implemented using sign changes Additional logic/circuitry is required for the generation of the local subcarrier replica. The number of multiplications is double since local code and subcarrier are wipedoff separately 
DE  5 bipolar/binary  
MMSES (time domain implementation and independent phase tracking)  3real 1bipolar/binary (for independent phase tracking)  The filtered local code is stored in memory and multibit multiplications are required for the code wipeoff 
Average processing time per second of data for different unambiguous boc tracking techniques
Algorithm  Average processing time (per second of data) 

BJ  6.8 s 
DE  7.6 s 
MMSES (independent phase tracking)  Frequency domain: 10.8 s Time domain: 7.3 s 
Parameters of the real data used for the computational analysis
Parameter  Value 

Sampling frequency  12.5 MHz 
Intermediate frequency  3.42 MHz 
Data duration Sampling  5 min 8 bit real samples 
5 Simulation and real data analysis
In this section, ZFS and MMSES are analyzed and compared against the DE [5] and BJ [2] techniques for BOCs(1,1) modulated signals in terms of tracking jitter, tracking threshold, MTLL, code error convergence and MEE for different EarlyminusLate chip spacing and discriminator types. The analysis is based on the semianalytic technique described in [18].
Simulation parameters considered for semianalytic analysis of BOC tracking techniques
Parameter  Value 

Coherent integration time  4 ms 
Frontend bandwidth  4 MHz 
Code bandwidth  0.5 Hz 
Code filter order  1 
Simulation runs  10,000 
Signal type  BOCs(1,1) 
5.1 Simulation results
5.1.1 Tracking jitter
In this section, the tracking jitter for different BOC tracking techniques have been provided. Different chip spacings, d_{ s } = 0.2, 0.3 and 0.4 chips, have been considered along with noncoherent, quasicoherent and coherent discriminators [13]. The noncoherent discriminator is analyzed in detail, whereas only sample results are shown for the other two cases.
5.1.2 Tracking threshold
5.1.3 Mean time to lose lock
5.1.4 Convergence analysis
where τ_{acq} is the code delay error from acquisition and M is the number of simulation runs used for averaging the tracking error, ${\tau}_{e}^{i}\left[k\right]$. Here i denotes the simulation run index and k denotes the time index already used for indexing the correlator outputs in (7).
In Figure 15a, an initial acquisition error of 0.5 chips is considered to evaluate the tracking error convergence. This delay error corresponds to a secondary peak of the BOC autocorrelation function. When the DLL is initialized on a secondary peak, both MMSES and DE converge to a zero delay error, whereas BJ is characterized by a steadystate error of about 0.15 chips. This phenomenon is better investigated in Figure 15b and 15c where different error trajectories for the initial 4 s are shown for MMSES and BJ, respectively. These trajectories show the evolution of the delay error as a function of time and for different simulation runs. In the MMSES case, all the trajectories tend to reach a zero steady state error whereas the BJ code error is characterized by two different behaviors. In some cases, the BJ decision logic correctly detects the false peak lock and the code delay error is corrected accordingly. In other cases, however, tracking is too noisy and the algorithm is unable to recover the false peak lock as seen in Figure 15c. The curves in Figure 15a summarize the average behaviors of the three considered algorithms determining the average tracking error defined in (28). Only MMSES and DE are able to provide a completely unambiguous BOC tracking. While all the three techniques behave similarly for high C/N_{0} ratios, BJ technique has higher probability to lose lock and track secondary peaks for low C/N_{0} s.
5.1.5 Multipath error envelope
5.2 Real data analysis
As explained by [23], the C/N_{0} estimator is often used as a delay lock indicator. More specifically, the C/N_{0} is estimated from the average postcorrelation power, i.e. the C/N_{0} is directly proportional to the correlation value that is in turn an indicator of the delay error. If a large delay error is committed then the correlation value and the C/N_{0} are significantly reduced. Loss of lock on the delay is thus reflected in randomly varying C/N_{0} estimates. In Figure 18, loss of lock is declared on the basis of the true signal parameters. More specifically, the experiment has been conducted using two frontends collecting synchronized signals. From the first unattenuated signal, reference parameters, i.e., Doppler frequency and code delay, were determined. When the parameters estimated from the second frontend started differing from the reference ones, loss of lock was declared. MMSES loses lock for a C/N_{0} of approximatively 2 dBHz lower compared with BJ. The C/N_{0} of the MMSES was determined using the unfiltered Prompt correlator used for carrier tracking. These findings are in agreement with the simulation results obtained in Section 5.1. It shall be noted that MMSES achieves performance similar to the DE. The ZFS performs poorly with respect to the other techniques.
6 Conclusions
In this paper, a new class of prefiltering techniques for shaping the autocorrelation function of GNSS signals has been proposed. The developed techniques substantially mitigate the noise amplification problem affecting previous prefiltering algorithms extending their applicability to moderate to low C/N_{0} values. The proposed algorithms are based on a convolutional representation of GNSS signals that allows one to apply the concepts of ZF and MMSE equalization to the signal subcarrier. The proposed algorithms retain all the flexibility of standard prefiltering techniques and can be used for unambiguous BOC tracking and autocorrelation shaping for multipath mitigation. From the performed analysis, simulations and real data testing, it emerges that this flexibility can be achieved with a negligible performance reduction with respect to the Double Estimator whose applicability is limited to unambiguous BOC tracking.
Declarations
Authors’ Affiliations
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