## 1 Introduction  
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 \(f_{d}\), the code delay \(\tau\), and the phase of the carrier, \(\phi\), of the signal transmitted by each visible satellite. The task of the acquisition system is to detect the visible satellites and to provide, for each detected satellite, a coarse estimate \(\left\langle\hat{f}_{d}^{a},\hat{\tau}^{a}\right\rangle\) of \(f_{d}\) and \(\tau\). This parameter vector is then passed to the tracking systems, whose task is to refine this estimate. The refinement of Doppler frequency estimate is generally performed by a classic phase lock loop (PLL), which requires an initial estimate much more accurate than the one provided by the acquisition system. Therefore it is necessary to improve the accuracy of the estimate \(\hat{f}_{d}^{a}\) to an acceptable level before starting the operations of the phase tracking loop. A system typically adopted by a GNSS receiver to reach this goal is a frequency lock loop (FLL), which is generally integrated within a PLL. The first refinement is done by a robust FLL operating at wideband, then the loop bandwidth is gradually reduced and finally the system switches to a PLL scheme [1,2]. Other methods [3,4] refine the frequency estimate by exploiting the phase difference between two successive periods of data. An interpolation method is introduced in Reference [5] to estimate the true value of the Doppler frequency, but it is based on an empirical approximation. In most of the previous methods, usually the estimates of \(f_{d}\) and \(\tau\) are picked in a search plane only considering the peak cell without any usage of the other cells. In the fields of communications, audio, medical, instrumentation, and others [6], the problem of estimating the frequency of a tone contaminated with noise is tackled for example by [PERSON] [7,8], [PERSON] [9,10], and [PERSON] [6,11], by exploiting the idea of refining the final frequency estimate using the peak sample and two neighbors of the discrete Fourier components. At the same time there are other methods, studied in Reference [12,13], which utilize the phase information. These methods cannot be directly applied to the acquisition of a GNSS signal, because of the very low signal-to-noise ratio and the different signal model, but they can inspire us to do some innovation in GNSS frequency estimation. 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 [14] using statistical principles, then Reference [15] presented a new approach of the CAF derivation. Furthermore in this paper the approximation in CAF main lobe is analyzed in details, based on this approximation, and a new family of methods for refining the estimate of the Doppler frequency is proposed, which exploits the cells close to the peak in the search plane. Compared with the traditional methods, these methods significantly improve the accuracy without increasing the computation complexity or using additional received data. A preliminary version of this work was presented in Reference [16]. With respect to that previous work, here we extend and complete all the mathematical derivations, extend the performance analysis with appropriate comparisons, derive and discuss the Cramer-Rao lower bound (CRLB) for the frequency estimator showing that the proposed approach is close to the CRLB (quasi-ML approach), and include the theoretical analysis of other non-AWGN nuisances. 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.