^{1}

^{1, 2}

^{1}

^{2}

Linear code combinations have been considered for suppressing the ionospheric error. In the L-band, this leads to an increased noise floor. In a combined L- and C-band (5010–5030 MHz) approach, the ionosphere can be eliminated and the noise floor reduced at the same time. Furthermore, combinations that involve both code- and carrier-phase measurements are considered. A new L-band code-carrier combination with a wavelength of 3.215 meters and a noise level of 3.92 centimeters is found. The double difference integer ambiguities of this combination can be resolved by extending the system of equations with an ionosphere-free L-/C-band code combination. The probability of wrong fixing is reduced by several orders of magnitude when C-band measurements are included. Carrier smoothing can be used to further reduce the residual variance of the solution. The standard deviation is reduced by a factor 7.7 if C-band measurements are taken into account. These initial findings suggest that the combined use of L- and C-band measurements, as well as the combined code and phase processing are an attractive option for precise positioning.

The integer ambiguity resolution
of carrier-phase measurements has been simplified by the consideration of
linear combinations of measurements at multiple frequencies. Early methods were
the three-carrier ambiguity resolution (TCAR) method introduced by Forssell et al. [

The systematic search of all possible GPS L1-L2
widelane combinations has been performed by Cocard and Geiger [

The authors have extended this work to three-frequency
(3F) Galileo combinations (E1-E5a-E5b) in [

In this paper, the authors used code- and
carrier-phase measurements in the linear combinations for obtaining ionospheric
elimination, large wavelengths, and a low noise level at the same time. The E5a
and E5b code measurements are of special interest due to their large bandwidth
(

The paper is organized as follows: the next section introduces the design of code-carrier linear combinations. The underlying trade-off between a low noise level and strong ionospheric reduction turns out to be controlled by the weighting coefficients of E5a/E5b code measurements.

In Section

The use of C-band measurements for ionosphere-free
carrier smoothing is discussed in Section

Linear combinations of carrier-phase measurements are
constructed to increase the wavelength (widelane), suppress the ionospheric
error, and to simplify the integer ambiguity resolution. The properties of the
linear combinations can be improved by including weighted code measurements
into the pure phase combinations. Figure

Linear combination of carrier-phase and code measurements.

Mixed code-carrier combinations weight the phase part
by

Cramer-Rao bound for Galileo signals.

Modulation | Bandwidth (MHz) | CRB (cm) | |
---|---|---|---|

E1 | BOC(1,1) | 4 | 20 |

E5a | BPSK(10) | 24 | 5 |

E5b | BPSK(10) | 24 | 5 |

E5 | BOC(15,10) | 51 | 1 |

For E1, E5a, E5b, E6 phase measurements, the
wavelength scaling of

Figure

Adaptive code contribution to linear combinations: tradeoff between noise level and ionospheric reduction.

The combination discrimination—measured by the ratio
of half the wavelength and the noise level

Optimal weighting of the phase combination part of ionosphere-free code-carrier
combinations with

The computation of the optimum

The combination discrimination becomes from (

Properties and weighting coefficients of ionosphere-free E1-E5b-E5a code-carrier combinations.

1 | 11 | 0.327 | 0.344 | 9.768 | 3.217 | 0.28 | 5.81 | |

1 | 10 | 0.166 | 0.175 | 4.884 | 3.216 | 0.25 | 6.38 | |

1 | 9 | 0.006 | 0.007 | 3.256 | 3.214 | 0.23 | 7.05 | |

1 | 8 | 2.442 | 3.213 | 0.20 | 7.86 | |||

1 | 7 | 1.954 | 3.212 | 0.18 | 8.84 | |||

1 | 6 | 1.628 | 3.211 | 0.16 | 10.02 | |||

1 | 5 | 1.396 | 3.210 | 0.14 | 11.42 | |||

1 | 4 | 1.221 | 3.209 | 0.12 | 12.99 | |||

1 | 3 | 1.085 | 3.208 | 0.11 | 14.54 | |||

1 | 2 | 0.977 | 3.207 | 0.10 | 15.65 | |||

1 | 1 | 0.888 | 3.206 | 0.10 | 15.85 | |||

1 | 0 | 0.814 | 3.205 | 0.11 | 15.04 | |||

1 | 0 | 0.751 | 3.204 | 0.12 | 13.59 |

Figure

Benefit of adaptive code and phase weighting for linear combinations with

The

The design of three frequency code-only combinations
that preserve geometry and eliminate ionospheric errors is characterized by one
degree of freedom used for noise minimization. The weighting coefficients are
derived from the geometry preserving and ionosphere-free constraints in (

Ionosphere-free
code-only combinations with minimum noise

E1 | E5b | E5a | C2 | C3 | C4 | ||
---|---|---|---|---|---|---|---|

(cm) | |||||||

2.090 | 1.500 | 0 | 0 | 0 | 0 | 44.41 | |

0.387 | 0.255 | 0.863 | 0 | 0 | 0 | 19.14 | |

0.213 | 0.128 | 0.476 | 0.476 | 0 | 0 | 14.21 | |

0.147 | 0.079 | 0.328 | 0.328 | 0.329 | 0 | 11.80 | |

0.112 | 0.054 | 0.251 | 0.251 | 0.251 | 0.251 | 10.31 |

Ionosphere-free
code-only combinations with minimum noise

C1 | C2 | C3 | C4 | |||
---|---|---|---|---|---|---|

(cm) | ||||||

2.338 | 0 | 0 | 0 | 0 | 46.78 | |

0.398 | 0.879 | 0 | 0 | 0 | 19.31 | |

0.217 | 0.481 | 0.481 | 0 | 0 | 14.27 | |

0.150 | 0.331 | 0.331 | 0.331 | 0 | 11.84 | |

0.114 | 0.252 | 0.252 | 0.252 | 0.252 | 10.34 |

The C-band is split into

Code-carrier linear combinations can also include both
L-band and C-band measurements. Therefore, (

Table

Ionosphere-free
code-carrier widelane combinations with

C1 | C2 | C3 | C4 | (m) | (cm) | ||||
---|---|---|---|---|---|---|---|---|---|

1 | 0 | 0 | 0 | 0 | 3.21 | 3.92 | |||

1 | 0 | 0 | 1 | −4.7e | 3.39 | 4.84 | |||

1 | 0 | 1 | 0 | −4.7e | 3.39 | 4.84 | |||

1 | 0 | 1 | 0 | −5.0e | 3.59 | 5.12 | |||

1 | 1 | 0 | 0 |
−4.7e | 3.39 | 4.84 | |||

1 | 1 | 0 | 0 |
−5.0e | 3.59 | 5.12 | |||

1 | 1 | 0 | 0 |
−5.3e | 3.81 | 5.43 | |||

0 | 0 | 0 | 0 | 1 | −6.0e−2 | 20.70 | 15.39 |

Figure

Comparison of joint L-/C-Band linear combinations for

There exists a large variety of joint code-carrier
narrowlane combinations where C-band measurements help to reduce the noise
substantially. Figure

Ionosphere-free
code-carrier narrowlane combinations with

1 | 0 | 5 | |||
---|---|---|---|---|---|

E5 | 0 | 1 | |||

C1 | 0 | 0 | 1 | 0 | |

C2 | 0 | 0 | 0 | 0 | |

C3 | 0 | 0 | 0 | 0 | |

C4 | 1 | 1 | 0 | 0 | |

E1 | 7.60e | 1.58e | 2.55e | ||

E5 | 5.31e | 0.110 | 0.178 | ||

(cm) | 5.55 | 5.65 | 5.76 | 5.73 | |

(mm) | 0.51 | 0.61 | 1.22 | 2.50 | |

54.4 | 46.3 | 23.6 | 11.46 |

Comparison of joint L-/C-Band linear combinations for

The integer ambiguity resolution is based on the linear combination of four different variable types: double-difference measurements for eliminating clock errors and satellite/receiver biases; multifrequency combinations for suppressing the ionosphere; code and carrier phase measurements for reducing the noise level; and finally, L-/C-band combinations for noise and discrimination characteristics.

Two joint L-/C-band code-carrier ionosphere-free
combinations are chosen for real-time (single epoch) ambiguity resolution. The

Note that the troposphere has the same impact on all
geometry-preserving combinations and does not affect the optimization of the
mixed code-carrier combinations. The noise vector is Gaussian distributed, that
is,

Figure

Reliability of

The reliability of ambiguity resolution can be further
improved by using the LAMBDA method of Teunissen [

Reliability of

After integer ambiguity fixing, the baseline is
re-estimated from (

Standard deviation of baseline estimation using the

The pure L-band combinations in the first row of
Tables

Ionosphere-free code-carrier linear combinations are
characterized by a noise level that is one to two orders of magnitude larger
than of the underlying carrier-phase measurements (Table

Ionosphere-free carrier smoothed code-carrier combinations.

Note that the superposition of ambiguities of the pure
phase combination is not necessarily an integer number of a common wavelength.
The respective ambiguities are not affected by the low pass filter and do not
occur in the smoothed output

Table

Weighting coefficients and properties of ionosphere-free carrier smoothed carrier phase combinations.

E1 | 2.324 | −0.764 | ||||

E1 | 1.064 |

The variance of
the smoothed combination is given by

In this paper, new joint L-/C-band linear combinations
that include both code- and carrier-phase measurements have been determined.
The weighting coefficients are selected such that the ratio between wavelength
and noise level is maximized. An ionosphere-free L-band combination (IFL) could
be found at a wavelength of

The combination of L- and C-band measurements reduces
the noise level of ionosphere-free code-only combinations by a factor

The residual variance of the noise can be further reduced by smoothing. An L-/C-band carrier combination can smooth the noise with a residual variance below the L-band phase noise variance. The smoothed solution can either be used directly or can be used to resolve the narrowlane ambiguities. The variance is basically the same in both cases. The resolved ambiguities, however, provide instantaneous independent solutions.