Carrier-smoothing of code pseudoranges: Difference between revisions

Latest revision as of 11:25, 23 February 2012

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

[math]\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)} }[/math]

The algorithm is initialised with:

[math]\displaystyle{ \widehat{R}(s;1)= R(s;1) }[/math]

where, [math]\displaystyle{ n=k }[/math] when [math]\displaystyle{ k\lt N }[/math] and [math]\displaystyle{ n=N }[/math] when [math]\displaystyle{ 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]\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)} }[/math]

where the mean bias ^{[footnotes 1]} [math]\displaystyle{ \lt R - \Phi \gt }[/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

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

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

[footnotes 1]

@@ Line 1: / Line 1: @@
 {{Article Infobox2
 |Category=Fundamentals
+|Authors=J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
+|Level=Advanced
+|YearOfPublication=2011
 |Title={{PAGENAME}}
-|Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.
-|Level=Medium
-|YearOfPublication=2011
-|Logo=gAGE
 }}
 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:
@@ Line 13: / Line 11: @@
 ::<math>
-\begin{array}{l}
+\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 ]
-\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 ]\\[0.3cm]
+\qquad \mbox{(1)}
-\mbox{The algorithm is initialised with:}\\
+</math>
+: The algorithm is initialised with:
+::<math>
 \widehat{R}(s;1)= R(s;1)
-\end{array} \qquad \mbox{(1)}
 </math>
-where, <math>n=k</math> when <math>k<N</math> and <math>n=N</math> when <math>k \geq N</math>.\\[0.2cm]
+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&=\Phi_k +  \left < R - \Phi \right >_k=\\[0.3cm]
+\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< R - \Phi \right >_{k-1} + \frac{1}{n} \left (  R_k - \Phi_k \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} \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]
-&=  \frac{1}{n} R_k+ \frac{n-1}{n} \left [\widehat{R}_{k-1}+ \left( \Phi_k-\Phi_{k-1} \right) \right ]\\
+&= \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.
-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> \left < R - \Phi \right ></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:
-* [[Single Frequency Measurements Smoothing]]
+* [[Code-Carrier Divergence Effect]]
@@ Line 47: / Line 48: @@
 [[Category:Fundamentals]]
+[[Category:GNSS Measurements and Data Preprocessing]]

Carrier-smoothing of code pseudoranges: Difference between revisions

Latest revision as of 11:25, 23 February 2012

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