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Essential elements of the Personal Tracker signal

The purpose of this chapter is to explain the elements of the GPS radio signal that will allow the receiver to solve for user position and receiver clock error. The GPS radio signal has a complex signal structure. Fortunately we do not need to address all of it, only the parts we need. By investigating a simplified model of the SV transmitter, we can focus on the essential elements of the GPS signal that address the specific information we will need to solve for user position and receiver clock error. Also in this chapter, a method is presented that allows the receiver to properly “set” the replica clock Dials above the 1 ms Dial using the Rearview Mirror GPS Navigation data stream. More detailed analytical expressions can be written for the power spectrum of the GPS signal. Regardless of their complexity, the reader is cautioned that complexity does not translate into reality. Issues of carrier leakage at the modulator and correlators, phase changes through media or signal processing elements, etc., all conspire to make an exact mathematical statement of the actual signal very difficult.

In this blog, the most applicable and necessary algorithms for static and kinematic as well as dynamic GPS data processing are outlined. Least squares adjustment is the most basic adjustment method. It starts by establishing observation equations and forming normal equations; then it solves the unknowns. It is a suitable method for static Best GPS Tracker data processing. The sequential application of least squares adjustment by accumulating the sequential normal equations makes applications of least squares adjustment more effective. Normal equations can be formed epoch-wise and then accumulated. This method can be used not only for solving the problem at the end, but also for obtaining epoch-wise solutions. It is suitable for static GPS data processing. The equivalent sequential least squares adjustment, which can be read from different publications, is also derived. This is an epoch-wise solving method and therefore is generally not suitable for static GPS data processing.The differences increase with time and are generally not negligible. Therefore by using this method, the numerical process has to be carefully examined to avoid the accumulation of numerical errors.

The conditional least squares adjustment is needed if there are some constraints that have to be taken into account. The commonly used least squares ambiguity search criterion is derived from this principle. The typical application of this method in Personal GPS data processing is taking into account the known distance of multiple kinematic antennas. The sequential application of conditional least squares adjustment is discussed because of practical needs. The problem may be solved first without conditions, and then the conditions may be applied afterward. The constraints such as the known distances of multiple antennas fixed on an aircraft have to be considered for every epoch. Block-wise least squares adjustment is discussed for separating the unknowns into two groups. For example, one group is time dependent parameters such as kinematic coordinates, and the other is the group of time independent parameters such as ambiguities. The sequential application of block wise least squares adjustment makes it possible to give up some unknowns and keep the information related to the common unknowns during the processing process.

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