# Joint L-/C-Band Code and Carrier Phase Linear Combinations for Galileo

Authors: Patrick Henkel; Christoph Günther
Journal: International Journal of Navigation and Observation
 (2008)

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## Abstract

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.


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## Body

## 1. Introduction

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. [1],
as well as the cascade integer resolution (CIR) developed by Jung et al. [2]. The weighting coefficients of three-frequency phase
combinations are designed either to eliminate the ionosphere at the price of a
rather small wavelength or to reduce the ionosphere only by a certain amount
with the advantage of a larger wavelength.The systematic search of all possible GPS L1-L2
widelane combinations has been performed by Cocard and Geiger [3]. The L1-L2 linear
combination of maximum wavelength (14.65 m) amplifies
the ionospheric error by a factor 350. Collins gives an overview of reduced ionosphere
L1-L2 combinations with wavelengths up to 86.2 cm (+1,−1
widelane) in [4].The authors have extended this work to three-frequency
(3F) Galileo combinations (E1-E5a-E5b) in [5]. A 3F widelane combination with a wavelength of 3.256 m and an
ionospheric suppression of 16.4 dB was found.
Furthermore, a 3F narrowlane combination with a wavelength of 5.43 cm could reduce
the ionospheric error by as much as 36.7 dB. Sets of
linear carrier-phase combinations that are robust against residual biases were
studied in [6]. The
integer ambiguities of the linear combinations can be estimated by the
least-squares ambiguity decorrelation adjustment (LAMBDA) algorithm developed
by Teunissen [7]. The
method includes an integer transformation which can also be used to determine
optimum sets of linear combinations [8].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
(20 MHz) and their
low associated Cramer-Rao bound of 5 cm [9]. The C-band phase
measurements are particularly interesting due to their small wavelength and
their thus reduced phase noise. The properties of code-carrier linear
combinations are optimized by including both L- and C-band measurements. The
cost function is defined as the ratio of half the wavelength and the noise
level of the ionosphere-free code-carrier combination. It is called combination
discrimination and it is a measure of the radius of
the decision regions expressed in units given by the standard deviation of the
noise.  The L-Band signals of Galileo are
defined in the Galileo-ICD [10]. The C-band signals are foreseen in a band between 5010 MHz to 5030 MHz [11]. The signal propagation and
tracking characteristics in the C-band have been analyzed by Irsigler et al. [12]. The larger
frequencies result in an additional free space loss of 10 dB that has to
be compensated by a larger transmit power.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 Section3, code-carrier linear combinations are computed
in a way that include both L- and C-band
measurements. An ionosphere-free code-only combination is determined that benefits
from a 4.5 times lower noise level than a pure L-band combination. Furthermore,
a pure L-band ionosphere-free code-carrier combination with a wavelength of
3.215 m and a noise standard deviation of 3.9 cm is found. The combined use of
the two reduces the probability of wrong fixing of the latter solution by 9
orders of magnitude with respect to a pure L-band solution.The use of C-band measurements for ionosphere-free
carrier smoothing is discussed in Section4: an ionosphere-free code-carrier
combination of arbitrary wavelength is smoothed by a pure phase combination.
The low noise level of C-band measurements provides a linear combination that
benefits from an 8.9 dB lower noise
level as compared to the equivalent L-band combination.


## 2. Code-Carrier Linear Combinations

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. Figure1 shows a three frequency (3F) linear
combination where the phase measurements are weighted by α, β, γ, and the code
measurements are scaled by a, b, c. The weighting coefficients are generally restricted
by a few conditions: first, the geometry should be preserved, that
is
(1)α+β+γ+a+b+c=!1.Moreover, the superposition of
ambiguities should be an integer multiple of a common wavelength λ, that is
(2)αλ1N1+βλ2N2+γλ3N3=!λN,which can be split into three
sufficient conditions(3)i=αλ1λ∈ℤ,j=βλ2λ∈ℤ,k=γλ3λ∈ℤ,with ℤ denoting the
space of integers. These integer constraints are rewritten to obtain the
weighting coefficients(4)α=iλλ1,β=jλλ2,γ=kλλ3.Figure 1
Linear combination of carrier-phase and code measurements.Mixed code-carrier combinations weight the phase part
byτ and the code
part by 1−τ. The border cases are pure phase (τ=1) and pure code
(τ=0) combinations.
The parameter τ has a
significant impact on the properties of the linear combination, and it is
optimized later in this section. Replacing the weighting coefficients in τ=α+β+γ by (4) yields
the wavelength of the code-carrier combination(5)λ=τi/λ1+j/λ2+k/λ3.The generalized widelane
criterion is given for λ1<λ2<λ3 by λ>λ3. Equivalently, it can be expressed as a function of i, j, and k as(6)τ>iq13+jq23+k>0withqmn=λnλm.The linear combination scales
the ionospheric error by(7)AI=α+βq122+γq132−a−bq122−cq132.The thermal noise of the
elementary carrier phase measurements is assumed Gaussian with the standard
deviation given by Kaplan and Hegarty [13](8)σϕi=λi2πBLC/N0[1+12T⋅C/N0],where BL denotes the
loop bandwidth, C/N0 the
carrier-to-noise ratio, and T the
predetection integration time. The overall noise contribution of the linear
combination is written as(9)Nm=(α2+β2q122+γ2q132)⋅σϕ02+a2σρ12+b2σρ22+c2σρ32with σρ12,…,σρ32 being the noise
variance of the code measurements. Table 1 shows the Cramer-Rao bound (CRB) for
some Galileo signals as derived by Hein et al. [9]. A DLL bandwidth of 1 Hz has been
assumed. The 4 MHz receiver bandwidth for E1 has been chosen to avoid sidelobe
tracking.Table 1
Cramer-Rao bound for Galileo signals.
ModulationBandwidth (MHz)CRB (cm)E1BOC(1,1)420E5aBPSK(10)245E5bBPSK(10)245E5BOC(15,10)511For E1, E5a, E5b, E6 phase measurements, the
wavelength scaling ofσϕi can be
neglected due to the close vicinity of the frequency bands. However, it plays a
major role when C-Band measurements are included.Figure2 shows the benefit of the code contribution to
the i=1 (E1), j=−10 (E5b), k=9 (E5a) linear
combination: a slight increase in noise level results in a considerable
reduction of the ionospheric error. The phase weighting has been fixed to τ=1 so that α, β, γ, and λ are uniquely
determined. The E5b and E5a code weights are adapted continuously and the
ionosphere is eliminated in the border case(10)b=−c=α+βq122+γq132q122−q132.E1 code measurements have not
been taken into account due to the increased noise level but might be included
with a low weight.Figure 2
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 levelλ/(2Nm)—is proposed
as a cost function to select linear combinations due to its independence of the
geometry. It is shown for multiple ionosphere-free code-carrier combinations in
Figure 3. The strong dependency on the phase weighting τ suggests an
optimization with respect to this parameter. Note that the legend refers to the
elementary wavelengths which have to be scaled by τ.Figure 3
Optimal weighting of the phase combination part of ionosphere-free code-carrier
combinations withi=1, k=−j−1, and j∈{−12,…,1}.The computation of the optimumτ takes again
only E5a/ E5b code measurements into account as the benefit of the E1 code
measurement is negligible (a=0). The notation
is simplified by(11)λ=λ˜⋅τ,AI=κ˜⋅τ−bq122−cq132,Nm=σϕ2⋅(α2+β2q122+γ2q132)+σρ2⋅(b2+c2)=σϕ2⋅η˜2τ2+σρ2⋅(b2+c2),with λ˜, κ˜, and η˜ implicitly
given by (5), (7), and (9). The E5a/E5b code weights are determined from the
ionosphere-free and geometry-preserving conditions as(12)b=1−c−τ,c=κ˜+q122q132−q122︸w1⋅τ+−q122q132−q122︸w2=w1⋅τ+w2.The combination discrimination becomes from (5), (11),
and (12):
(13)R(τ)=λ(τ)/2Nm(τ)=λ˜⋅τ2σϕ2⋅η˜2τ2+σρ2⋅((w1τ+w2)2+(1−τ−w1τ−w2)2).Setting the derivative to zero
yields the optimum weighting(14)τopt=1−2w2+2w221+w1−w2−2w1w2,which is independent of both σρ and σϕ. Table 2 contains the weighting coefficients and
characteristics of the code-carrier combinations shown in Figure 3.Table 2
Properties and weighting coefficients of ionosphere-free E1-E5b-E5a code-carrier combinations.
ijkbcλ˜ (m)λ (m)Nm (m)R1−12110.3270.3449.7683.2170.285.811−11100.1660.1754.8843.2160.256.381−1090.0060.0073.2563.2140.237.051−98−0.154−0.1622.4423.2130.207.861−87−0.314−0.3301.9543.2120.188.841−76−0.474−0.4991.6283.2110.1610.021−65−0.634−0.6671.3963.2100.1411.421−54−0.793−0.8351.2213.2090.1212.991−43−0.953−1.0031.0853.2080.1114.541−32−1.112−1.1710.9773.2070.1015.651−21−1.272−1.3390.8883.2060.1015.851−10−1.431−1.5060.8143.2050.1115.0410−1−1.590−1.6740.7513.2040.1213.59Figure4 shows the benefit of adaptive code and phase
weighting for the code-carrier linear combination with i=0 (E1), j=1 (E5b), and k=−1 (E5a).
Obviously, the wavelength increases linearly with τ and the
ionosphere can be eliminated with any τ. The noise amplification depends on the level of
ionospheric reduction: a linear increase can be observed near the pure phase
combination, while the increase becomes negligible for the ionosphere-free combination.
Thus, the combination discrimination of the ionosphere-free code-carrier
combination is increased by almost the same factor as τ is risen.Figure 4
Benefit of adaptive code and phase weighting for linear combinations withλ=τ9.768m.


## 3. C-Band Aided Code-Carrier Linear Combinations

The20 MHz wide C-Band
(5010⋯5030MHz={489.736⋯491.691}⋅10.23MHz) has been
reserved for Galileo. The higher frequency range has a multitude of advantages
and drawbacks: an additional free space loss of 10 dB occurs which
has to be compensated by a larger transmit power. The ionospheric delay is
approximately 10 times lower
than in the E1 band. The small wavelength of 5.9691cm⋯5.9839cm complicates
direct ambiguity resolution but results in an approximately 3.2 times lower
standard deviation of phase noise. Moreover, the C-Band offers additional
degrees of freedom for the design of linear combinations.


### 3.1. Reduced Noise Ionosphere-Free Code-Only Combinations

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 (1), (7) as(15)a=1−b−c,b=−1q122−1︸v1+(−q132−1q122−1)︸v2⋅c=v1+v2⋅c.Minimization of Nm2=a2σρ12+b2σρ22+c2σρ32 yields(16)c=(1−v1+v2−v1v2)⋅σρ12−v1v2⋅σρ22(1+2v2+v22)⋅σρ12+v22⋅σρ22+σρ32.Ionosphere-free code-only
combinations with more than three frequencies are obtained by a
multidimensional derivative. Table 3 shows that the pure L-band E1-E5b-E5a
combination is characterized by a noise level of 44.41cm. If the E5 signal is received with full bandwidth,
the CRB is reduced to 1 cm but the
number of degrees of freedom is reduced by one so that the noise level of the
E1-E5 combinations are lightly increased (Table 4). These combinations will
play a role in conjunction with code-carrier combinations.Table 3
Ionosphere-free
code-only combinations with minimum noiseσρ(E1)=σρ(C1)=⋯=σρ(C4)=20cm and σρ(E5a,E5b)=5cm.
E1E5bE5aC1C2C3C4Nm(cm)2.0901.500−2.590000044.410.3870.255−0.5060.86300019.140.2130.128−0.2920.4760.4760014.210.1470.079−0.2110.3280.3280.329011.800.1120.054−0.1680.2510.2510.2510.25110.31Table 4
Ionosphere-free
code-only combinations with minimum noiseσρ(E1)=σρ(C1)=⋯=σρ(C4)=20cm and σρ(E5)=1cm.
E1E5C1C2C3C4Nm(cm)2.338−1.338000046.780.398−0.2780.87900019.310.217−0.1790.4810.4810014.270.150−0.1420.3310.3310.331011.840.114−0.1220.2520.2520.2520.25210.34The C-band is split into4 bands of 5 MHz bandwidth
centered at {490,490.5,491,491.5}⋅10.23MHz and allows a
significant reduction of the noise level. Note that the noise of any
contributing elementary combination is reduced by a weighting coefficient
smaller than one.


### 3.2. Joint L-/C-Band Widelane Combinations

Code-carrier linear combinations can also include both
L-band and C-band measurements. Therefore, (1)–(7) are extended to include the
additional measurements. The weighting coefficients {α,β,γ,…} and {a,b} are computed
such that the discriminator output of (13) is maximized for a given set of
integer coefficients {i,j,k,…}.Table5 contains ionosphere-free joint L-/C-band
code-carrier widelane combinations (λ>maxiλi). The E1-E5 pure
L-band combination benefits from a noise level of only 3.92cm which
simplifies the resolution of the 3.215m integer
ambiguities. In contrast to the code-only combinations, the use of the
full-bandwidth E5 signal is advantageous compared to separate E5a and E5b
measurements. The C-band offers no benefit for these wavelengths. In the last
row of Table 5, a linear combination with a pure L-band code and pure C-band
phase part is described. The combination discrimination equals 67.25 but the noise
level is also increased to 15.39cm.Table 5
Ionosphere-free
code-carrier widelane combinations withσρ(E1)=σρ(C1)=⋯=σρ(C4)=20cm and σρ(E5)=1cm.
ijklmnabλNmE1E5C1C2C3C4E1E5(m)(cm)1−10000−4.4e−3−3.113.213.921−1001−1−4.7e−3−3.283.394.841−101−10−4.7e−3−3.283.394.841−1010−1−5.0e−3−3.463.595.121−11−100−4.7e−3−3.283.394.841−110−10−5.0e−3−3.463.595.121−1100−1−5.3e−3−3.673.815.4300−1001−8.5e−5−6.0e−220.7015.39Figure5 shows the tradeoff between wavelength and
noise level for joint L-/C-band ionosphere-free linear code-carrier
combinations with {i,j}∈[−5,+5] and {k,l,m,n}∈[−2,+2]. The E1-E5 combination is of special interest but the
maximum combination discrimination is obtained for a joint L-/C-band
combination.Figure 5
Comparison of joint L-/C-Band linear combinations forσρ(E1)=20cm, σρ(E5)=1cm and σϕ,i=λi/λ1⋅σϕ0 with σϕ0=1mm.


### 3.3. Joint L-/C-Band Narrowlane Combinations

There exists a large variety of joint code-carrier
narrowlane combinations where C-band measurements help to reduce the noise
substantially. Figure6 shows the tradeoff between wavelength and noise level
for {i,j}∈[−5,+5] and {k,l,m,n}∈[−2,+2]. For λ=5.7cm, the consideration of C-band measurements reduces the
noise level by a factor of 5 compared to a
pure L-band combination (Table 6).Table 6
Ionosphere-free
code-carrier narrowlane combinations withσρ(E1)=σρ(C1)=⋯=σρ(C4)=20cm and σρ(E5)=1cm.
iE110−15jE5−101−3kC10010lC20000mC30000nC41100aE1−2.04e−67.60e−51.58e−42.55e−4bE5−1.43e−35.31e−20.1100.178λ(cm)5.555.655.765.73Nm(mm)0.510.611.222.50R54.446.323.611.46Figure 6
Comparison of joint L-/C-Band linear combinations forσρ(E1)=20cm, σρ(E5)=1cm and σϕ,i=λi/λ1⋅σϕ0 with σϕ0=1mm.


### 3.4. Reliability of Ambiguity Resolution

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. Theλ=3.215m, Nm=3.92cm combination of
Table 5 and one further combination of Table 4. The double difference (DD)
ionosphere-free combinations are modeled as(17)[yρϕyρ]=[GG]δx+[λ⋅10]N+ε=Xβ+ε,with(18)X=[λ⋅1G0G],β=[Nδx]and the DD geometry matrix G, the baseline δx and the integer
ambiguities N. The double-differenced troposphere is assumed to be
negligible or known a priori (e.g., from an accurate continued fraction model).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,(19)ε∼𝒩(0,Σ)withΣ=ΣLC⊗ΣDD,where ⊗ denotes the
Kronecker product. ΣLC models the
linear combination induced correlation and ΣDD includes the
correlation due to double difference measurements from Ns visible
satellites. The standard deviation of the most critical ambiguity estimate can
be written as(20)σmax=maxi={1⋯Ns−1}Σβ^(i,i),Σβ^=(XTΣ−1X)−1,and the probability of wrong
fixing follows as(21)Pwc=1−∫−0.5+0.512πσmax2e−x2/2σmax2dx.In the following analysis, the
location of the reference station is at 48.1507∘ N, 11.5690∘ E with a
baseline length of 10 km.Figure7 shows the benefit of C-band measurements for
integer ambiguity fixing. If the E1-E5 pure L-band combination is used as
second combination in (17), the failure rate varies between 0.01 and 0.07 due to its poor
noise characteristics. The use of two additional C-band measurements reduces
the maximum probability of wrong fixing to 10−5. For three C-band frequencies, the failure rate is at
most 10−11 which
corresponds to a gain of 9 to 17 orders of
magnitude compared to the pure L-band combination.Figure 7
Reliability ofλ=3.215m integer
ambiguity resolution: impact of C-band measurements on the probability of wrong
fixing of the most critical ambiguity.The reliability of ambiguity resolution can be further
improved by using the LAMBDA method of Teunissen [7]. The float ambiguity
estimates are decorrelated by an integer transformation ZT and the
ambiguity covariance matrix is written as(22)ΣN^′=ZTΣN^Z=LDLT, with the decomposition
into a lower triangular matrix L and a diagonal matrix D. The probability of wrong fixing of the sequential
bootstrapping estimator is given by Teunissen [14] as(23)Pw=1−∏i=1Ns−1∫−0.5+0.512πσc2(i)e−x2/2σc2(i)dx,with σc(i)=D(i,i). It represents a lower bound for the success rate of
the integer least-square estimator and is depicted in Figure 8. Obviously, the
use of joint L-/C-band linear combinations reduces the probability of wrong
fixing by several orders of magnitude compared to pure L-band combinations.Figure 8
Reliability ofλ=3.215m integer
ambiguity resolution: impact of C-band measurements on the probability of wrong
fixing based on sequential fixing with the integer decorrelation
transformation.


### 3.5. Accuracy of Baseline Estimation

After integer ambiguity fixing, the baseline is
re-estimated from (17). The covariance matrix of the baseline estimate in local
coordinates is given by(24)Σδx^=RL(GTΣ−1G)−1RLTwith the rotation matrix RL. Figure 9 shows the achievable horizontal and
vertical accuracies for the two optimized joint L-/C-band combinations.Figure 9
Standard deviation of baseline estimation using theλ=3.215mE1-E5
ionosphere-free code-carrier combination and the E1-E5-C1⋯C4 ionosphere-free
code-only combination.The pure L-band combinations in the first row of
Tables4 and 5 have been again selected as reference scenario. It can be
observed that the use of joint L-/C-band linear combinations enables a slight
improvement in position estimates compared to the significant benefit for
ambiguity resolution.


## 3.1. Reduced Noise Ionosphere-Free Code-Only Combinations

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 (1), (7) as(15)a=1−b−c,b=−1q122−1︸v1+(−q132−1q122−1)︸v2⋅c=v1+v2⋅c.Minimization of Nm2=a2σρ12+b2σρ22+c2σρ32 yields(16)c=(1−v1+v2−v1v2)⋅σρ12−v1v2⋅σρ22(1+2v2+v22)⋅σρ12+v22⋅σρ22+σρ32.Ionosphere-free code-only
combinations with more than three frequencies are obtained by a
multidimensional derivative. Table 3 shows that the pure L-band E1-E5b-E5a
combination is characterized by a noise level of 44.41cm. If the E5 signal is received with full bandwidth,
the CRB is reduced to 1 cm but the
number of degrees of freedom is reduced by one so that the noise level of the
E1-E5 combinations are lightly increased (Table 4). These combinations will
play a role in conjunction with code-carrier combinations.Table 3
Ionosphere-free
code-only combinations with minimum noiseσρ(E1)=σρ(C1)=⋯=σρ(C4)=20cm and σρ(E5a,E5b)=5cm.
E1E5bE5aC1C2C3C4Nm(cm)2.0901.500−2.590000044.410.3870.255−0.5060.86300019.140.2130.128−0.2920.4760.4760014.210.1470.079−0.2110.3280.3280.329011.800.1120.054−0.1680.2510.2510.2510.25110.31Table 4
Ionosphere-free
code-only combinations with minimum noiseσρ(E1)=σρ(C1)=⋯=σρ(C4)=20cm and σρ(E5)=1cm.
E1E5C1C2C3C4Nm(cm)2.338−1.338000046.780.398−0.2780.87900019.310.217−0.1790.4810.4810014.270.150−0.1420.3310.3310.331011.840.114−0.1220.2520.2520.2520.25210.34The C-band is split into4 bands of 5 MHz bandwidth
centered at {490,490.5,491,491.5}⋅10.23MHz and allows a
significant reduction of the noise level. Note that the noise of any
contributing elementary combination is reduced by a weighting coefficient
smaller than one.


## 3.2. Joint L-/C-Band Widelane Combinations

Code-carrier linear combinations can also include both
L-band and C-band measurements. Therefore, (1)–(7) are extended to include the
additional measurements. The weighting coefficients {α,β,γ,…} and {a,b} are computed
such that the discriminator output of (13) is maximized for a given set of
integer coefficients {i,j,k,…}.Table5 contains ionosphere-free joint L-/C-band
code-carrier widelane combinations (λ>maxiλi). The E1-E5 pure
L-band combination benefits from a noise level of only 3.92cm which
simplifies the resolution of the 3.215m integer
ambiguities. In contrast to the code-only combinations, the use of the
full-bandwidth E5 signal is advantageous compared to separate E5a and E5b
measurements. The C-band offers no benefit for these wavelengths. In the last
row of Table 5, a linear combination with a pure L-band code and pure C-band
phase part is described. The combination discrimination equals 67.25 but the noise
level is also increased to 15.39cm.Table 5
Ionosphere-free
code-carrier widelane combinations withσρ(E1)=σρ(C1)=⋯=σρ(C4)=20cm and σρ(E5)=1cm.
ijklmnabλNmE1E5C1C2C3C4E1E5(m)(cm)1−10000−4.4e−3−3.113.213.921−1001−1−4.7e−3−3.283.394.841−101−10−4.7e−3−3.283.394.841−1010−1−5.0e−3−3.463.595.121−11−100−4.7e−3−3.283.394.841−110−10−5.0e−3−3.463.595.121−1100−1−5.3e−3−3.673.815.4300−1001−8.5e−5−6.0e−220.7015.39Figure5 shows the tradeoff between wavelength and
noise level for joint L-/C-band ionosphere-free linear code-carrier
combinations with {i,j}∈[−5,+5] and {k,l,m,n}∈[−2,+2]. The E1-E5 combination is of special interest but the
maximum combination discrimination is obtained for a joint L-/C-band
combination.Figure 5
Comparison of joint L-/C-Band linear combinations forσρ(E1)=20cm, σρ(E5)=1cm and σϕ,i=λi/λ1⋅σϕ0 with σϕ0=1mm.


## 3.3. Joint L-/C-Band Narrowlane Combinations

There exists a large variety of joint code-carrier
narrowlane combinations where C-band measurements help to reduce the noise
substantially. Figure6 shows the tradeoff between wavelength and noise level
for {i,j}∈[−5,+5] and {k,l,m,n}∈[−2,+2]. For λ=5.7cm, the consideration of C-band measurements reduces the
noise level by a factor of 5 compared to a
pure L-band combination (Table 6).Table 6
Ionosphere-free
code-carrier narrowlane combinations withσρ(E1)=σρ(C1)=⋯=σρ(C4)=20cm and σρ(E5)=1cm.
iE110−15jE5−101−3kC10010lC20000mC30000nC41100aE1−2.04e−67.60e−51.58e−42.55e−4bE5−1.43e−35.31e−20.1100.178λ(cm)5.555.655.765.73Nm(mm)0.510.611.222.50R54.446.323.611.46Figure 6
Comparison of joint L-/C-Band linear combinations forσρ(E1)=20cm, σρ(E5)=1cm and σϕ,i=λi/λ1⋅σϕ0 with σϕ0=1mm.


## 3.4. Reliability of Ambiguity Resolution

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. Theλ=3.215m, Nm=3.92cm combination of
Table 5 and one further combination of Table 4. The double difference (DD)
ionosphere-free combinations are modeled as(17)[yρϕyρ]=[GG]δx+[λ⋅10]N+ε=Xβ+ε,with(18)X=[λ⋅1G0G],β=[Nδx]and the DD geometry matrix G, the baseline δx and the integer
ambiguities N. The double-differenced troposphere is assumed to be
negligible or known a priori (e.g., from an accurate continued fraction model).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,(19)ε∼𝒩(0,Σ)withΣ=ΣLC⊗ΣDD,where ⊗ denotes the
Kronecker product. ΣLC models the
linear combination induced correlation and ΣDD includes the
correlation due to double difference measurements from Ns visible
satellites. The standard deviation of the most critical ambiguity estimate can
be written as(20)σmax=maxi={1⋯Ns−1}Σβ^(i,i),Σβ^=(XTΣ−1X)−1,and the probability of wrong
fixing follows as(21)Pwc=1−∫−0.5+0.512πσmax2e−x2/2σmax2dx.In the following analysis, the
location of the reference station is at 48.1507∘ N, 11.5690∘ E with a
baseline length of 10 km.Figure7 shows the benefit of C-band measurements for
integer ambiguity fixing. If the E1-E5 pure L-band combination is used as
second combination in (17), the failure rate varies between 0.01 and 0.07 due to its poor
noise characteristics. The use of two additional C-band measurements reduces
the maximum probability of wrong fixing to 10−5. For three C-band frequencies, the failure rate is at
most 10−11 which
corresponds to a gain of 9 to 17 orders of
magnitude compared to the pure L-band combination.Figure 7
Reliability ofλ=3.215m integer
ambiguity resolution: impact of C-band measurements on the probability of wrong
fixing of the most critical ambiguity.The reliability of ambiguity resolution can be further
improved by using the LAMBDA method of Teunissen [7]. The float ambiguity
estimates are decorrelated by an integer transformation ZT and the
ambiguity covariance matrix is written as(22)ΣN^′=ZTΣN^Z=LDLT, with the decomposition
into a lower triangular matrix L and a diagonal matrix D. The probability of wrong fixing of the sequential
bootstrapping estimator is given by Teunissen [14] as(23)Pw=1−∏i=1Ns−1∫−0.5+0.512πσc2(i)e−x2/2σc2(i)dx,with σc(i)=D(i,i). It represents a lower bound for the success rate of
the integer least-square estimator and is depicted in Figure 8. Obviously, the
use of joint L-/C-band linear combinations reduces the probability of wrong
fixing by several orders of magnitude compared to pure L-band combinations.Figure 8
Reliability ofλ=3.215m integer
ambiguity resolution: impact of C-band measurements on the probability of wrong
fixing based on sequential fixing with the integer decorrelation
transformation.


## 3.5. Accuracy of Baseline Estimation

After integer ambiguity fixing, the baseline is
re-estimated from (17). The covariance matrix of the baseline estimate in local
coordinates is given by(24)Σδx^=RL(GTΣ−1G)−1RLTwith the rotation matrix RL. Figure 9 shows the achievable horizontal and
vertical accuracies for the two optimized joint L-/C-band combinations.Figure 9
Standard deviation of baseline estimation using theλ=3.215mE1-E5
ionosphere-free code-carrier combination and the E1-E5-C1⋯C4 ionosphere-free
code-only combination.The pure L-band combinations in the first row of
Tables4 and 5 have been again selected as reference scenario. It can be
observed that the use of joint L-/C-band linear combinations enables a slight
improvement in position estimates compared to the significant benefit for
ambiguity resolution.


## 4. Joint L-/C-Band Carrier Smoothed Carrier

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 (Table5). Both noise and
multipath of the code-carrier combinations can be reduced by the smoothing
filter of Hatch [15]
which is shown in Figure 10. The upper input can be an ionosphere-free
code-carrier combination of arbitrary wavelength. The lower input is a pure
ionosphere-free phase combination that is determined by three conditions: the
first ensures that the geometry is preserved, the second eliminates the ionosphere,
and the third minimizes the noise, that is,
(25)α+β+γ=!1,α+βq122+γq132=!0,minα,β,γNm2=minα,β,γ(σϕ,02⋅(α2+β2q122+γ2q132)).Figure 10
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 outputλAϕ¯A due to
different signs in the addition to λAϕA (Figure 10).Table7 shows an ionosphere-free E1-E5a-E5b phase
combination that increases the noise level by a factor 2.64. However, the low noise level of C-band measurements
suggests the use of the second combination with f3=491⋅10.23MHz. In this case, the noise level is not only reduced by
smoothing but also by the coefficients of the pure phase combination.Table 7
Weighting coefficients and properties of ionosphere-free carrier smoothed carrier phase combinations.
f1f2f3αβγNmE1E5bE5a2.324−0.559−0.7642.64⋅σϕ0E1E5bC−0.008−0.0561.0640.34⋅σϕ0The variance of
the smoothed combination is given by(26)σA¯2=E{(ε¯A(t)−ε¯B(t)+εB(t))2},with the low-pass filtered noise
(e.g., Konno et al. [16])(27)ε¯A(t)=1τs⋅∑n=0∞(1−1τs)nεA(t−n),and the smoothing time τs. Setting (27) into (26), and using the definition of
a geometric series yields
(28)σA¯2=σB2+12τs−1⋅(σA2+σB2−2σAB2)+2τs⋅(σAB2−σB2).For long smoothing times, only
the low noise of the joint L/C pure carrier-phase combination λBϕB remains (Table 7).


## 5. Conclusions

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 of3.215 m with a noise
level of 3.92 cm.The combination of L- and C-band measurements reduces
the noise level of ionosphere-free code-only combinations by a factor4.5 compared to
pure L-band combinations. This increases the reliability of an ambiguity
resolution option for the IFL combination by 9 orders of magnitude.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.



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*Source: 651437-2008-04-09.xml*