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# Carrier-smoothing of code pseudoranges

Fundamentals
Title Carrier-smoothing of code pseudoranges
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Year of Publication 2011

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 $\displaystyle{ R(s;n) }$ and $\displaystyle{ \Phi(s;n) }$ the code and carrier measurement of a given satellite $\displaystyle{ s }$ at the time $\displaystyle{ n }$. Thence, the smoothed code $\displaystyle{ \hat{R}(s;n) }$ can be computed as:

$\displaystyle{ \widehat{R}(s;k)= \frac{1}{n} R(s;k)+ \frac{n-1}{n} \left [\widehat{R}(s;k-1)+ \left( \Phi(s;k)-\Phi(s;k-1) \right) \right ] \qquad \mbox{(1)} }$
The algorithm is initialised with:
$\displaystyle{ \widehat{R}(s;1)= R(s;1) }$

where, $\displaystyle{ n=k }$ when $\displaystyle{ k\lt N }$ and $\displaystyle{ n=N }$ when $\displaystyle{ k \geq N. }$

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:

$\displaystyle{ \begin{array}{ll} \widehat{R}(k)&= \frac{1}{n} R(k)+ \frac{n-1}{n} \left [ \widehat{R}(k-1)+ \left( \Phi(k) - \Phi(k-1) \right) \right ]=\\[0.3cm] &= \Phi(k) +\frac{n-1}{n} \left (\widehat{R}(k-1)-\Phi(k-1)\right )+ \frac{1}{n} \left( R(k)-\Phi(k) \right)=\\[0.3cm] & =\Phi(k) + \frac{n-1}{n} \lt R - \Phi\gt _{(k-1)} + \frac{1}{n} \left ( R(k) - \Phi(k) \right )=\\[0.3cm] &= \Phi(k) + \lt R - \Phi \gt _{(k)} \end{array} \qquad \mbox{(2)} }$

where the mean bias [footnotes 1] $\displaystyle{ \lt R - \Phi \gt }$ between the code and carrier phase is estimated in real time and used to align the carrier phase with the code.

1. ^ The mean value of a set of measurements $\displaystyle{ \{x_1,\cdots,x_n\} }$ can be computed recursively as: $\displaystyle{ \lt x\gt _k= \frac{1}{k} x_k+ \frac{k-1}{k}\lt x\gt _{k-1} }$. The equation (2) is a variant of previous expression, that provides an estimate of a moving average over a $\displaystyle{ N }$ samples window. Notice that, when $\displaystyle{ k \geq N }$, the weighting factors $\displaystyle{ \frac{1}{N} }$ and $\displaystyle{ \frac{N-1}{N} }$ are used instead of $\displaystyle{ \frac{1}{k} }$ and $\displaystyle{ \frac{k-1}{k} }$.