A probabilistic model for on-line estimation of the GNSS carrier-to-noise ratio
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 () when the noise power is defined for a unit of bandwidth. Measurements of 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 are associated with multipath disturbances [4] and low rate variations with indoor or in forest positioning [5]. In this regard, observations have been used for remote sensing applications by the GNSS-Reflectometry (GNSS-R) community. In this context, the 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 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 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 and the cosine part as the quadrature component 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 and quadrature component measurements for each satellite. It can be shown that are noisy observations of the signal amplitude and 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 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 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 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 and samples from the prompt correlator [13]. Another way to measure 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 using the accumulated and samples from the prompt 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 . Eventually, both errors (tracking and noise errors) are included in the value of the . 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 (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 provides a direct observation of the signal amplitude and therefore of . We estimate the carrier-to-noise ratio at high rate (1 ms rate for GPS C/A) using 1 ms rate observations of . For example, high rate estimation is essential in multipath, and dynamic GNSS-R applications, where the 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 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 as observations. The linearization of the measurement equation that links 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 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 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 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 are processed by the receiver at instant every period of code . 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 in the set of visible satellites,
Statistical front end model
The probability for the random variable 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: where the product of with is equal to one in both cases. Let us construct the following model approximation of the sampled signal of satellite after digitization:
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):
Where 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 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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