Elsevier

Signal Processing

Volume 183, June 2021, 107992
Signal Processing

A probabilistic model for on-line estimation of the GNSS carrier-to-noise ratio

https://doi.org/10.1016/j.sigpro.2021.107992Get rights and content

Highlights

  • A probabilistic model for 1-bit quantization front-end coherent detector.

  • High rate on-line estimator of the GNSS carrier-to-noise ratio.

  • Takes into account the cross-correlation with the other satellites.

  • Performs coherently with the widely known NovAtel receiver and with higher rate.

  • Able to cope with the abrupt changes in observations.

Abstract

This article is dedicated to the estimation of the GNSS signal carrier-to-noise ratio using the in-phase component of the signals as observations. In a GNSS receiver, it is the statistic of the correlation provided by the code tracking loop that is used to estimate the carrier-to-noise ratio. In fact, carrier-to-noise estimation is used to monitor the performance of GNSS receivers and the quality of the received signals. In this article, we aim at high rate carrier-to-noise estimation, namely the code repetition rate (e.g. 1ms for GPS C/A), in order to maximize the time resolution of carrier-to-noise observations. We show that in a 1-bit quantization receiver, the in-phase component of the signal can provide a direct observation of the signal amplitude, and therefore of the carrier-to-noise ratio. However, the model that links the 1ms rate observations of the in-phase component with the signal amplitude is non-linear. The non-linear expression that links the maximum value of the in-phase correlation component to the signal amplitude is derived. In order to estimate the time varying amplitudes of the signals, we propose an Extended Kalman Filter to reverse the non-linear expression with the noisy observations of correlation provided by the tracking loop. The proposed model and filter inversion method are assessed on synthetic and real data, while investigating the effect of the cross-correlation contribution of the visible satellites on the estimations. We show using real data that, for a 1-bit quantization receiver, the proposed estimator can achieve the same accuracy as a widely known commercial GNSS receiver with a much higher data rate. We also show that the proposed approach can cope with abrupt changes in the observations compared to a classical C/N0 estimate.

Introduction

Telecommunication systems use the power of the received signal, normalized by the noise power, as an indicator of the quality of reception. The Signal-to-Noise Ratio (SNR) is called Carrier-to-Noise Ratio (C/N0) when the noise power is defined for a unit of bandwidth. Measurements of C/N0 are often used in global navigation satellite system (GNSS) applications to monitor the receiver processing and its response to noisy environments [1], [2], determine whether the code and carrier tracking loops are in lock, and to detect the signal-to-noise environment in order to predict the performance of the receiver [3]. High rate variations in the C/N0 are associated with multipath disturbances [4] and low rate variations with indoor or in forest positioning [5]. In this regard, C/N0 observations have been used for remote sensing applications by the GNSS-Reflectometry (GNSS-R) community. In this context, the C/N0 of the GNSS signals received on Earth directly from the GNSS satellites as well as after reflection on the Earth surface are compared to retrieve some geophysical parameters of the reflective surface such as its height, shape or soil moisture [6], [7], [8], [9].

The carrier-to-noise ratio is used to observe the amplitude of a GNSS signal. However, this amplitude can’t be directly estimated because the GNSS antenna perceives a combination of all the GNSS signals from the satellites in view, which results in mixing the received signals. In this context, the inversion of the antenna measurements to retrieve the C/N0 of the signals from each satellite in view is an ill posed problem. However, each GNSS satellite signal can be differentiated with its Code Division Multiple Access (CDMA) code [10]. The C/N0 of the received signals can, therefore, be derived from the demultiplexing and demodulation processes [11]. In fact, each received signal can be expressed as a sum of a sine function and a cosine function (i.e. 90 degrees out of phase). The sine part is denoted as the in-phase component I and the cosine part as the quadrature component Q of the signal. The stages of demodulation and demultiplexing are realized respectively with a Phase Lock Loop (PLL) and a Delay Lock Loop (DLL) providing the in-phase component I and quadrature component Q measurements for each satellite. It can be shown that I are noisy observations of the signal amplitude and Q are observations of the noise [12]. In the classical approach, the statistics of these two components are used to estimate the SNR, which is proportional to the signal amplitude. Finally C/N0 is derived by the product of the SNR with the noise equivalent bandwidth of the receiver RF (Radio Frequency) front end.

In practice, the difficulty associated to the estimation of the C/N0 is the derivation of the statistical parameters of the two components in quadrature. This estimation process assumes that the noise is stationary and the signal amplitude is constant. This assumption is less restrictive for high rate estimations. However, the accuracy of the estimation depends on the duration of the observation. In this context, accuracy and estimation rate are two ambivalent parameters.

Several estimates of C/N0 have been proposed for GNSS applications in order to maximize the accuracy with minimal implementation complexity. The most widely used estimate is the Standard Estimate (SE) also referred to as Narrow-to-Wideband power ratio estimate [12]. This estimate uses the accumulated I and Q samples from the prompt P correlator [13]. Another way to measure C/N0 using the accumulated in-phase and quadrature components from the prompt correlator is the Correlator Comparison method. Unfortunately, these estimators perform poorly with weak signal and high interference environment [13]. Norman C. Beaulieu has also introduced an intuitively motivated algorithm to measure C/N0 using the accumulated I and Q samples from the prompt P correlator [14]. This method achieves high accuracy with minimum complexity, but requires a relatively high integration time to acheive the desired accuracy. The moment-based estimator is the most accurate SNR estimate and is shown to achieve the Cramer-Rao Lower Bound but this estimate has a high complexity cost [15]. It also needs a long integration time in order to achieve a good accuracy. Other data-aided estimates presuppose a knowledge of the transmitted data symbols which is a common assumption in GNSS applications. In this regard, the Maximum Likelihood Estimate (MLE) outperforms other estimators under the assumption of known data bits, because it provides a good compromise in term of complexity and accuracy [16]. Several versions of this estimate have been proposed with [16] or without [17] perfect phase synchronization and for modern GNSS signals that have pilot and data channels [18].

In a classical GNSS receiver, 1-bit or 2-bit quantization are used. However, in some military applications, 8 bits are necessary to accurately process the Power Spectral Density (PSD) of the signal in order to prevent GNSS spoofing. We developed for this work, a bit grabber to record the data with 1-bit or 2-bit quantization in collaboration with a GNSS specialized firm called Syntony. The DLL, PLL and FLL estimate the parameters of the signal in our self-built 1-bit GNSS software receiver. In fact, signal digitalization is done off-line but the software receiver processes the data on-line. The on-line loops correct the phase delay error, the code delay error and the error in Doppler during the tracking process. As a result, accurate estimates of the GNSS signal parameters are obtained. The tracking error increases when the SNR of the signal decreases, but this error is low and not significant given the noise power on I. Eventually, both errors (tracking and noise errors) are included in the value of the C/N0. We show in a preceding work [19], that we can indeed reach centimeter precise position estimates every 1ms. In that case, observations of phase and Doppler were used to process the code delay of the GNSS signal obtained every 1ms. In fact, increasing the quantization bits will improve the estimate accuracy of I (the in-phase component of the signal). However, the observations will be dependent on the automatic gain control (AGC). Moreover, the noise power on the observation of I is too high to observe this improvement.

In the context of this work, a probabilistic model of the front end architecture of a coherent detector is proposed. Coherent detectors are used in several applications like GNSS, telecommunication systems, radar systems, and sensors (Dual-Phase Lock-In Amplifier). Concerning the GNSS applications, the minimum length of integration Tb (1ms code period for GPS C/A) is known and is an objective in our development. Furthermore, several parameters render the model complex and show the interest of the proposed statistical approach. In our model, we indeed take into account the cross correlation between the different satellites. We also take into account the receiver-satellite velocity in the carrier frequency. The model in the context of this application shows different complexities that can be included in the probabilistic model of a coherent detector.

In this article, we show that for 1-bit quantization in a digital receiver, the digitized signals are independent of the automatic gain control, and thus the mean value of the in-phase component I, provides a direct observation of the signal amplitude and therefore of C/N0. We estimate the carrier-to-noise ratio at high rate (1 ms rate for GPS C/A) using 1 ms rate observations of I. For example, high rate C/N0 estimation is essential in multipath, and dynamic GNSS-R applications, where the C/N0 estimation rate determines the system’s ability to cope with the rapid displacement of the GNSS receiver and defines the rate at which the environment can be analyzed. However, the model that links the signal amplitude to the 1ms rate observations of I is non-linear and we derive its expression. For linear models, the Kalman filter, being a recursive filter, produces state estimates that are optimal in the minimum mean-squared error (MMSE) sense [20]. However, since the measurement equations are non-linear, we propose an on-line estimate of the amplitudes of the signals based on an Extended Kalman Filter (EKF) that uses measurements of I as observations. The linearization of the measurement equation that links I to the signal amplitude is then derived.

Based on the foregoing, this article is organized as follows: Section II presents the GNSS front end processing and the one bit front end model. In the third section, an on-line estimate of the signal amplitude in the form of an EKF is introduced, while interpreting the non-linear expression that links the observations of I to the amplitudes of the signals. The linearization of the measurement equation is derived in the fourth section. In the fifth section, the proposed methodology is assessed using both synthetic and real data. Finally, conclusions are provided in the sixth section.

Section snippets

GNSS front end processing

The purpose of the RF front end is to provide digital signal samples to the signal processing block. The signals sr(t) sensed by the GNSS antenna are amplified using a Low Noise Amplifier (LNA) due to the fact that the signals are immersed in noise. The LNA is characterized by its gain and noise figure [21]. Then, the amplitudes of the signals are regulated using an automatic gain control (AGC). After that, the received signals are down converted with a local oscillator frequency fLO to an

Kalman estimate of the signal amplitude

In our approach, we develop an on-line estimate of the amplitudes of GNSS signals. For this purpose, we construct a state estimate that uses the in-phase components of correlation as observations. The observations Iv,k, are processed by the receiver at instant k every period of code Tb. The state model is a classical second order state equation used for data smoothing where the second state is the rate of change of the first state.

For a satellite l in the set V of visible satellites, Al,k/lV

Statistical front end model

The probability for the random variable iv,k to take the value +1 is a function of the values of the local code and the received signal. These two signals sampled and quantified on one bit take the values +1 or -1. This probability is expressed as:P(iv,i=1)=P(cv,i=1)P(si=1/cv,i=1)+P(cv,i=1)P(si=1/cv,i=1) where iv,k the product of si with cv,i is equal to one in both cases. Let us construct the following model approximation of the sampled signal of satellite v after digitization:s^i=lVAlCAl(

Assessment on synthetic data

The aim of this experimentation is to assess the proposed amplitudes estimator on GPS C/A signals and study the effect of the correlation contribution on the carrier-to-noise estimation. In order to be independent of the AGC Gain, we compare the carrier-to-noise ratio in dB-Hz using the below expressions derived from the classical definition of the SNR in Eq. (8):C/N0k=20log(AvfsTb2)+10log(BW)C/N0c=20log(mean(Iv,k=1:1000)std(Iv,k=1:1000))+10log(BW)

Where C/N0k is the carrier-to-noise equation

Conclusion

In this article, we proposed an on-line estimate of the amplitudes of GNSS signals in the form of an Extended Kalman Filter that uses the 1ms rate of the in-phase components of the signals as observations. In order to be independent of the automatic gain control, 1-bit quantization digital receiver was used. The estimated amplitudes of the signals provide direct observations of C/N0, therefore the carrier-to-noise ratio is estimated. This estimator can provide robust amplitude values and,

CRediT authorship contribution statement

Hamza Issa: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing - original draft, Writing - review & editing, Visualization. Georges Stienne: Conceptualization, Validation, Formal analysis, Data curation, Resources, Supervision, Writing - review & editing. Serge Reboul: Conceptualization, Methodology, Software, Validation, Formal analysis, Data curation, Investigation, Resources, Supervision, Writing - review & editing, Project administration, Funding

Declaration of Competing Interest

The authors declare no conflict of interest regarding the article entitled “A probabilistic model for on-line estimation of the GNSS carrier-to-noise ratio”.

Acknowledgment

The authors would like to acknowledge the Université du Littoral Côte d’Opale (ULCO) and the National Council for Scientific Research of Lebanon (CNRS-L) for granting a joint doctoral fellowship to Mr. Hamza Issa.

The authors would also like to thank the CPER MARCO project (Recherche marine et littorale en CÔpale, des milieux aux ressources, aux usages et á la qualité des produits de la mer) for their financial support.

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