Issue link: https://insidegnss.epubxp.com/i/1027343

38 Inside GNSS S E P T E M B E R / O C T O B E R 2 0 1 8 www.insidegnss.com Not requiring the LOS to be present is one of the main advantages of our approach because, especially in urban areas, LOS signals are not always received. If the LOS signal is absent, then the path delay relative to the shortest received NLOS path can be computed as where is the path delay of signal b relative to signal a, and NLOS1 is the shortest NLOS signal and NLOS2 is the second-shortest NLOS signal. ird, we perform a comparison between the predicted signal parameters and the received signal parameters. e latter are not known directly, but are completely contained in a GNSS receiver's correlator outputs. If the predicted signal parameters agree with the correlator outputs, it suggests that the candidate position used to generate the predicted param- eters was accurate. In contrast, if the agreement between the predicted and received signal parameters is poor, it suggests the candidate position was inaccurate. is final point suggests that instead of a single candidate position, several candidate positions should be considered and the position whose predicted signal parameters best match the correlator outputs should be selected as the final position estimate. is is precisely how the algorithm works — we use a grid of candidate points as the input and select the "best" amongst these as the final position estimate (how we obtain a grid of points is discussed later). Filling-In Some Details Although a full mathematical description of the algorithm is beyond the scope of this article, some key details are described in this section to better explain what is happening. e Additional Reading section at the end of the article pro- vides resources containing more of the mathematical details. One of the more important aspects of the algorithm is the development of mathematical models of the correlator outputs that relate to the predicted signal parameters. Based on our empirical observations, we only considered one-, two- and three-path models for the correlator outputs, but addi- tional paths could be included if necessary. For illustrative purposes, we only show the two-path model: where is the power of the correlator output; sub- scripts denote the shortest, second-shortest, etc. signal paths; A is the signal amplitude; τ 1 is the code phase of the shortest path; бτ is the path delay as defined above; R(τ) is the auto- correlation function of the signal's ranging code; and бϕ is the relative carrier phase. Although we model the correlator power, similar models could be derived for in-phase (I) and quadrature-phase (Q) signals. e parameterization above intentionally includes the path delay which, as discussed above, can be obtained from ray tracing and a 3DBM. More specifically, it allows the predicted path delay values to be used as input to the signal model. is leads to the final step, which is the comparison of the predicted signal parameters to the received signal param- eters. is is accomplished using a least-squares fit of the cor- relator outputs to the selected signal model. e state vector used for the two-path case is e solution is computed using the predicted path delays as a priori information, meaning the least-squares estimator is primarily estimating all of the other (nuisance) parameters. e astute reader will notice the absence of an explicit clock term — this implicitly contained in the code phase for short- est received signal (τ 1 ). As mentioned above, if the candidate position is accu- rate, the predicted path delays will also be accurate and the estimator should be able to reliably estimate the remaining signal parameters. e corresponding least-squares residuals should be small. In contrast, an inaccurate candidate position will lead to larger residuals. is works because of all of the parameters in the state vector, the one most sensitive to the input candidate position, is the path delay. In light of all this, the root-sum-squares (RSS) of the residuals is the metric used to assess the goodness of fit of the predicted signal parameters to the correlator outputs (smaller is better). More specifically, the process described above is performed separately for each satellite and the RSS residuals across satellites is used as the final metric. Over-parameterization of the signal model must be accounted for when computing the residuals for each satel- lite. is can happen when the predicted number of paths is larger than the number of paths actually received. In this case, the "extra" degrees of freedom in the model will artificially reduce the RSS of the residuals. When an over- parameterization is detected, the RSS residuals are set to be large to effectively de-weight the corresponding candidate position — this makes sense as the predicted signal param- eters are effectively wrong, most likely because of an inac- curate candidate position. Before showing results, it is worth noting the receiver's correlator taps should span a sufficiently wide range of code phase values and use a sufficiently tight spacing so as to capture enough of the received signal's "shape". In our work, we used 61 correlator taps equally spaced across ±1 chips. Results Data was collected in downtown Calgary, Canada on two separate days over a total of about 50 minutes. Figure 1 shows the trajectories of the two data sets in purple along with building outlines colored by building height. e tallest building exceeds 200 meters and there are parts of the trajec- GNSS SOLUTIONS

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