^{1}

^{*}

^{2}

^{1}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (

It is known that signal acquisition in Global Navigation Satellite System (GNSS) field provides a rough maximum-likelihood (ML) estimate based on a peak search in a two-dimensional grid. In this paper, the theoretical mathematical expression of the cross-ambiguity function (CAF) is exploited to analyze the grid and improve the accuracy of the frequency estimate. Based on the simple equation derived from this mathematical expression of the CAF, a family of novel algorithms is proposed to refine the Doppler frequency estimate with respect to that provided by a conventional acquisition method. In an ideal scenario where there is no noise and other nuisances, the frequency estimation error can be theoretically reduced to zero. On the other hand, in the presence of noise, the new algorithm almost reaches the Cramer-Rao Lower Bound (CRLB) which is derived as benchmark. For comparison, a least-square (LS) method is proposed. It is shown that the proposed solution achieves the same performance of LS, but requires a dramatically reduced computational burden. An averaging method is proposed to mitigate the influence of noise, especially when signal-to-noise ratio (SNR) is low. Finally, the influence of the grid resolution in the search space is analyzed in both time and frequency domains.

The main purpose of the acquisition and tracking systems of a Global Navigation Satellite System (GNSS) receiver is to provide an estimate of the Doppler frequency _{d}_{d}

In most of the previous methods, usually the estimates of _{d}

In this paper, the peak and neighbor points of the cross-ambiguity function (CAF) in the frequency domain are used to derive a simple formula that greatly improves the accuracy of the frequency estimate provided by the acquisition system. The CAF was initially derived in Reference [

A preliminary version of this work was presented in Reference [

This paper is organized as follows. In Section 2 the signal model is presented and the approximate mathematical expression of the CAF in the main lobe is obtained. In Section 3 a new family of algorithms is derived and proved to work perfectly in the absence of noise. In Section 4 the performance of the proposed algorithms is investigated in the presence of additive noise; both the CRLB and a least-square (LS) solution are derived as benchmark, and the comparison shows that the new algorithms can approach very closely the CRLB. Besides that, a simple averaging approach based on non-coherent sums is proposed to improve the accuracy of the algorithms in low SNR conditions. Furthermore, in Section 5, the effects of other nuisances, uncorrelated with the additive noise, are analyzed, and some countermeasures are proposed. Finally, in Section 6 the conclusion is drawn.

The basic scheme of the acquisition method proposed in this paper is illustrated in

The acquisition system for GNSS application is based on the maximum-likelihood estimation theory, which can be briefly described as follows [

The incoming sampled signal can be denoted as a vector
_{1}_{2}_{3} ⋯ _{K}^{2}

The ML estimate of the parameter vector a can be found by maximizing the likelihood function, which depends on the probability density function (PDF), that is
_{a} that contains all the possible values of the unknown vector a, that is to say, _{a}.

If the energy of the test signal _{ML}_{ML}_{a}.

In GNSS field, without considering the influence of noise, the received signal, after down-conversion and sampling, can be written as [_{υ}_{m}_{m}_{s}_{m}_{m}_{s}_{m}_{s}_{m}_{m}_{s}_{m}_{b}_{s}_{m}_{b}_{s}_{m}_{m}_{m}_{s}_{m}_{IF}_{d,m}_{m}_{s}_{s}

From _{d}_{d}

With respect to the parameter data bit, in the implementation, a non-coherent acquisition scheme is used to solve the problem, so here we assume that there is no data-transition in the accumulation period.

Considering the parameter carrier phase, its influence can be removed by involving two components in the acquisition process that are in-phase component (I) and 90° phase-shifted quad-phase (Q) component [

_{d}_{a} will be discussed in Section 3.

There are mainly three acquisition schemes [_{sp}

In order to refine the Doppler frequency estimate, a system typically used is the Frequency Lock Loop (FLL) mentioned in Reference [

In this paper, we develop new methods to refine the Doppler frequency estimate, based only on the search grid already evaluated by the acquisition; that is, we do not have to compute new correlations, but we only use the neighbor cells of the CAF peak, already available in the search grid.

The CAF [

Following the approach presented in Reference [_{d}_{m,i}_{d}_{m}_{s}_{m}_{m}_{d}^{−jφm}, the subscript _{d}_{p}_{d}_{m},_{i}_{Td}_{m,i}

In _{k(m,i)}_{m}_{ch}_{ch}_{0(m}_{=}_{i)}_{k(m,i)}_{0} is the peak component in _{0}(_{-2}_{-1}_{1}_{2} …) affect the shape of the main lobe of _{-2}_{-1}_{1}_{2} …) are far smaller than α_{0}, the mathematical expression of the CAF in the main lobe (subscript “ml”) can be written as
_{0} = α_{0(}_{i,i}_{)} = R(Δ_{m} −

The validity of this approximation can be improved in two ways:

By enlarging the integration time _{d}_{d}_{d}

By decreasing the values of the adjacent coefficients (_{−2}_{−1}_{1}_{2} …). This can be obtained by improving the accuracy of code delay estimate, so as to work close to the maximum of

Based on the CAF expression in

The signal acquisition process is basically a two dimensional search in a grid plane (commonly referred to as _{p}_{d}_{dmax}_{dmax}_{d}_{sp}_{sp}_{d}_{s}_{τ}_{d}/τ_{sp}_{f}_{dmax}/f_{sp}_{τ}_{f}_{d}

The purpose of a traditional acquisition system is to find the coordinates of the peak cell of the grid plane when the satellite we want to detect is visible. To improve the accuracy of the estimates, the steps _{sp}_{sp}_{sp}_{s}

An example of search space is shown in _{d}_{d}_{ml}. As indicated in _{d}_{s}_{sp}_{s}

The cells A, B and C are characterized by the triplete and (_{A},f̅_{A}S̅_{a}),(_{B},f̅_{B}S̅_{B}_{C},f̅_{C}S̅_{C}_{a}=_{B}τ̅_{C}_{X}_{X}_{dX}_{ml}_{d}

In Reference [_{d}_{d}

First of all we observe that, when the points are located in the main lobe (like points A, B and C in _{s}_{d}_{A}_{s}_{d}_{A}_{d}_{sp}_{S}_{A,B,C}_{A}S̅_{A}_{B}S̅_{B}_{C}S̅_{C}_{d}_{A}_{B}_{C}_{d}_{sp}_{S}

More in general, if we choose the Doppler frequency step as

When

When

No matter what value _{d}−_{1} and _{1} is the trial value of the Doppler frequency in the first cell of the set (

For simplicity, hereafter we refer to solution

Based on References [_{d}_{ml} in the complex form given in

At the same time, similarly to _{k+1}=_{k}_{sp}_{1} is the Doppler frequency value in cell 1 marked in

For simplicity, hereafter we refer to solution

The formulas in the previous sections show that, in the absence of noise and other nuisances, the proposed equations are able to evaluate the true Doppler frequency, hence reducing the estimation error range from the traditional (−_{sp}_{sp}_{IF}_{s}

The accumulation time of the acquisition stage is set to the minimum period (4 ms), assuming that no data transition occurs in this period, the Doppler frequency step is
_{d}_{sp}_{sp}

Therefore, we can conclude that in the absence of noise, in experiments, the new methods eliminate the error in the evaluation of the Doppler frequency due to the discretization of the search space, just using three cells selected in the main lobe of the frequency-domain CAF The only constraint is that the Doppler frequency step has to be as given in

The practical situation in which the noise is unavoidable is discussed in the next section.

In real scenarios, we have to consider the influence of noise, which can be modeled as an Additive White Gaussian Noise (AWGN), as commonly done in the literature. Then the received signal model becomes
_{s}_{N}_{0}/2 of the analogue noise by the well-known formula
_{s}_{s}_{d}_{I}_{q}_{I}_{q}

Hereafter, we refer to _{n}_{n}_{n}-n̅_{n}-n̅

In the following two subsections, we discuss two terms of comparison worth to be considered for the frequency estimators proposed so far, firstly a least squares solution and secondly the CRLB on the variance of the estimator. Performance comparisons obtained in simulation are presented in Section 4.3.

Another approach to exploit the CAF points

Since we know that the main lobe of the CAF is a sinc function, it is possible to use the LS method in which the fitting curve is the sinc function _{τ})). This leads to the equations
_{n}_{n}_{d}

Usually, a statistical estimator is characterized by its bias (mean error), variance (mean square error), and the threshold SNR (signal-to-noise ratio) [

Without considering the data-transition problem and ignoring the influence of the code delay estimation error (which will be discussed in Section 5.2), the received samples _{d}_{IF}_{d}_{s}_{s}T_{d}

To test the algorithms R_{n}_{n}

The parameters used in the experiments are _{IF}_{s}_{sp}_{d}_{rmse}

In the first group of experiments, we set _{d}_{n}_{n}_{n}_{n}

In the second group of experiments, we changed the integration time to _{d}_{n}

Comparing _{d}_{d}

Since noise is dominant in the acquisition process, in order to increase the robustness of the proposed approach in the presence of noise, the performance of a simple averaging method, based on the idea of non-coherent sums, is assessed here. The main steps of the method represented in

Find the initial code delay using a first period of data, then pick out the column (marked in blue color in

Use the parameter 〈_{d}

Calculate the average of the

Pick out the top

In the following simulation, we set CNR = 43 dB-Hz,

The performance of the algorithms presented so far depends not only on the additive noise but also on other nuisances, whose impact is analyzed in this section. In particular we observe that the methods R_{n}_{n}_{C}_{A}_{B}_{C}_{A}_{B}_{p}

Since the search space is discretized, in general even in the absence of noise _{p}_{d}_{p}_{q}_{p}_{d}_{q}_{p}

In this section we study the influence of the quantization error in the frequency domain, which can be re-elaborated as
_{p}_{sp}_{f}_{p}

From _{p}_{p}_{C}_{n}_{n}_{p}

So when Δ_{p}_{p}_{C}

To implement this method, a threshold control has to be added, as drawn in

The peak column selected in the search space (marked in _{q}_{p}_{q}_{q}_{n}_{n}

The influence of such a code delay error can be quantified by modifying the expression of the signal-to-noise ratio _{q}

From _{q}_{q}_{q}

In this paper a new family of algorithms is proposed for the fine estimation of the Doppler frequency based on an approximate analytical expression of the CAF The proposed methods have been analyzed in both ideal (_{n}_{n}

Brief structure of new Doppler frequency refinement process.

The value of _{k}

The combination of sinc functions.

Two-dimensional search space.

The plot of the column (

The cells chosen in the generalized method.

Cumulative distribution of the frequency errors (comparison between the conventional and methods R-3 and C-3).

Results of the first group of experiments, with _{d}

Results of second group of experiments, with _{d}

Averaging method.

The distribution of the Doppler frequency estimates.

Influence of the quantization error on the RMSE as a function Δ_{p}

The strategy used in improved algorithm R_{n}