## 1 Introduction  
In 2016, Google made accessible the GNSS raw measurements of smartphones and tablets running Android 7.0. As reported by the European GNSS Agency [1], the access to GNSS raw measurements has allowed researchers to conduct several studies for increasing the performance of commercial GNSS mounted on smartphones and tablets, mainly through post-processing techniques of measurement. [PERSON] et al. [2] showed the potential applications offered by using raw smartphone measurements in fields where precision positioning is not required. In fact, the authors described a number of applications, including commercial product detection, search and rescue for public protection, intelligent transportation systems, healthcare, environmental monitoring, etc. [PERSON] and [PERSON] [3] stated that the possibility to develop new algorithms for navigation and the possibility to perform post processing from raw data coming from GNSS mobile devices orient the research on these topics. In fact, in the last 4 years, about 1900 papers have been published in this field; on the one hand, research has addressed issues related to the development of algorithms to improve positioning (static or kinematic) and, on the other hand, it has attempted to correct the initial ambiguities of the carrier phase in order to improve the precision and accuracy of positioning. An example of algorithms to improve positioning is provided by [PERSON] et al. [4], who applied a time difference filter in GNSS kinematic positioning. The latter algorithm is divided into several steps. In the first step, the GNSS measurement error is estimated through the signal-to-noise-ratio (SNR) values. Subsequently, a single-point positioning (SPP) algorithm, a single-point velocity determination (SPV) algorithm and a time-differentiated phase measurement technique (TDCP) are applied to accurately estimate the GNSS system velocity. Finally, the receiver position is determined through the use of the Kalman filter. [PERSON] et al. [5] and [PERSON] et al. [6] tried to improve the algorithm by proposing a method for pedestrian positioning. The results in kinematic positioning show an RMS value of 3 m in horizontal positioning and about 5 m in vertical positioning; the authors claim that these values are those provided by commercial Android devices capable of acquiring continuous, smooth and reliable solutions. [PERSON] et al. [5] elaborated the Smart-RTK algorithm dedicated to kinematic positioning by applying a Doppler-Smoothed-Code (DSC) filter to reduce the noise of the code measurements derived by the smart device; furthermore, the weights of the observations were determined with a stochastic SNR model. To estimate the kinematic state, the Kalman filter was applied, in which a constant acceleration model (CA model) was introduced. In horizontal positioning, the RMSE reached values between 0.3 and 0.6 m in stationary conditions, 0.4 and 0.7 m in pedestrian conditions and 0.85 m in walking conditions. [PERSON] et al. [7] developed a temporal differential filter algorithm that exploits the dual frequency of the GNSS receiver mounted on the Xiaomi Mi 8. Through this algorithm, the authors obtained, in an urban environment in kinematic mode, errors (RMSE) of 1.22 m horizontally and 1.94 m vertically; in more hostile environments, such as Urban Canyons, the RMS was 1.61 m for horizontal positioning and 2.16 m for vertical positioning, while in static conditions, they obtained an RMS error of 1 m in horizontal positioning and 1.5 m in vertical positioning. In the context of research related to the correction of initial ambiguities in order to improve the accuracy and precision of positioning, an interesting study is that of [PERSON] et al. [8], who demonstrated that with intelligent devices such as the Nexus 9, it is possible to achieve positioning of decimetre accuracy with 15 min static--rapid surveys and a baseline of about 8 km, without fixing the ambiguity of the carrier phase observations. [PERSON] et al.