Edit Article: Carrier-smoothing of code pseudoranges

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The noisy (but unambiguous) code pseudorange measurements can be smoothed with the precise (but ambiguous) carrier phase measurements. A simple algorithm (the Hatch filter) is given as follows: Let's <math>R(s;n)</math> and <math>\Phi(s;n)</math> the code and carrier measurement of a given satellite <math>s</math> at the time <math>n</math>. Thence, the smoothed code <math>\hat{R}(s;n)</math> can be computed as: ::<math> \widehat{R}(s;k)= \frac{1}{n} R(s;k)+ \frac{n-1}{n} \left [\widehat{R}(s;k-1)+ \left( \Phi(s;k)-\Phi(s;k-1) \right) \right ] \qquad \mbox{(1)} </math> : The algorithm is initialised with: ::<math> \widehat{R}(s;1)= R(s;1) </math> where, <math>n=k</math> when <math>k<N</math> and <math>n=N</math> when <math>k \geq N.</math> This algorithm must be initialised every time that a carrier phase cycle-slip occurs. The previous algorithm can be interpreted as a real-time alignment of the carrier phase to the code measurement. That is: ::<math> \begin{array}{ll} \widehat{R}(k)&= \frac{1}{n} R(k)+ \frac{n-1}{n} \left [ \widehat{R}(k-1)+ \left( \Phi(k) - \Phi(k-1) \right) \right ]=\\[0.3cm] &= \Phi(k) +\frac{n-1}{n} \left (\widehat{R}(k-1)-\Phi(k-1)\right )+ \frac{1}{n} \left( R(k)-\Phi(k) \right)=\\[0.3cm] & =\Phi(k) + \frac{n-1}{n} <R - \Phi>_{(k-1)} + \frac{1}{n} \left ( R(k) - \Phi(k) \right )=\\[0.3cm] &= \Phi(k) + < R - \Phi >_{(k)} \end{array} \qquad \mbox{(2)} </math> where the mean bias <ref group="footnotes">The mean value of a set of measurements <math>\{x_1,\cdots,x_n\}</math> can be computed recursively as: <math><x>_k= \frac{1}{k} x_k+ \frac{k-1}{k}<x>_{k-1}</math>. The equation (2) is a variant of previous expression, that provides an estimate of a moving average over a <math>N</math> samples window. Notice that, when <math>k \geq N</math>, the weighting factors <math>\frac{1}{N}</math> and <math>\frac{N-1}{N}</math> are used instead of <math>\frac{1}{k}</math> and <math>\frac{k-1}{k}</math>.</ref> <math> < R - \Phi > </math> between the code and carrier phase is estimated in real time and used to align the carrier phase with the code. For more information, please go to the article: * [[Code-Carrier Divergence Effect]] ==Notes== <references group="footnotes"/> [[Category:Fundamentals]] [[Category:GNSS Measurements and Data Preprocessing]]

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