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

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Hindawi Publishing Corporation
International Journal of Navigation and Observation
Volume 2008, Article ID 651437, 8 pages
doi:10.1155/2008/651437
ResearchArticle
Joint L-/C-Band Code and Carrier Phase Linear
Combinations for Galileo
Patrick Henkel1and Christoph G¨ unther1,2
1Institute of Communications and Navigation, Technische Universit¨ at M¨ unchen (TUM), Theresienstraße 90,
80333 M¨ unchen, Germany
2German Aerospace Center (DLR), Institute of Communications and Navigation (IKN), M¨ unchner Straße 20,
82234 Weßling/Oberpfaffenhofen, Germany
Correspondence should be addressed to Patrick Henkel, patrick.henkel@tum.de
Received 1 August 2007; Revised 27 November 2007; Accepted 15 January 2008
Recommended by Gerard Lachapelle
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 (5010–5030MHz) 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 3.215meters and a noise level of 3.92centimeters 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-bandcodecombination.TheprobabilityofwrongfixingisreducedbyseveralordersofmagnitudewhenC-bandmeasurements
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.
Copyright © 2008 P. Henkel and C. G¨ unther. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
1.INTRODUCTION
The integer ambiguity resolution of carrier-phase measure-
ments has been simplified by the consideration of lin-
ear 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].Theweightingcoefficientsofthree-frequencyphasecom-
binations are designed either to eliminate the ionosphere at
the price of a rather small wavelength or to reduce the iono-
sphere 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
(14.65m) amplifies the ionospheric error by a factor 350.
Collins gives an overview of reduced ionosphere L1-L2
combinations with wavelengths up to 86.2cm (+1,−1
widelane) in [4].
The authors have extended this work to three-frequency
(3F) Galileo combinations (E1-E5a-E5b) in [5]. A 3F wide-
lane combination with a wavelength of 3.256m and an iono-
spheric suppression of 16.4dB was found. Furthermore, a 3F
narrowlane combination with a wavelength of 5.43cm could
reduce the ionospheric error by as much as 36.7dB. Sets
of linear carrier-phase combinations that are robust against
residualbiaseswerestudiedin[6].Theintegerambiguitiesof
thelinearcombinationscanbeestimatedbytheleast-squares
ambiguity decorrelation adjustment (LAMBDA) algorithm
developed by Teunissen [7]. The method includes an inte-
ger transformation which can also be used to determine op-
timum 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 (20MHz)
and their low associated Cramer-Rao bound of 5cm [9]. The
C-band phase measurements are particularly interesting due

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2International Journal of Navigation and Observation
to their small wavelength and their thus reduced phase noise.
The properties of code-carrier linear combinations are opti-
mized by including both L- and C-band measurements. The
costfunctionisdefinedastheratioofhalfthewavelengthand
the noise level of the ionosphere-free code-carrier combina-
tion. It is called combination discrimination and it is a mea-
sure of the radius of the decision regions expressed in units
given by the standard deviation of the noise. The L-Band sig-
nals of Galileo are defined in the Galileo-ICD [10]. The C-
band signals are foreseen in a band between 5010MHz to
5030MHz [11]. The signal propagation and tracking char-
acteristics in the C-band have been analyzed by Irsigler et al.
[12]. The larger frequencies result in an additional free space
loss of 10dB that has to be compensated by a larger transmit
power.
The paper is organized as follows: the next section in-
troduces 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 3, code-carrier linear combinations are com-
puted in a way that include both L- and C-band measure-
ments. An ionosphere-free code-only combination is deter-
mined 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 3.215m and a noise standard deviation of 3.9cm 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 Section 4: 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 8.9dB lower noise level as compared to the
equivalent L-band combination.
2.CODE-CARRIER LINEAR COMBINATIONS
Linearcombinationsofcarrier-phasemeasurementsarecon-
structed to increase the wavelength (widelane), suppress the
ionosphericerror,andtosimplifytheintegerambiguityreso-
lution. The properties of the linear combinations can be im-
proved by including weighted code measurements into the
pure phase combinations. Figure 1 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
α+β +γ +a +b +c
!= 1.
(1)
Moreover, the superposition of ambiguities should be an in-
teger multiple of a common wavelength λ, that is
αλ1N1+βλ2N2+γλ3N3
!= λN,(2)
λ1φ1
λ2φ2
λ3φ3
ρ1
ρ2
ρ3
αβγ
abc
λφ
Figure 1: Linear combination of carrier-phase and code measure-
ments.
which can be split into three sufficient conditions
i =αλ1
λ
∈ Z,
j =βλ2
λ
∈ Z,
k =γλ3
λ
∈ Z,(3)
with Z denoting the space of integers. These integer con-
straints are rewritten to obtain the weighting coefficients
α =iλ
λ1,
β =jλ
λ2,
γ =kλ
λ3.
(4)
Mixed code-carrier combinations weight the phase part
by τ 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
λ =
τ
i/λ1+ j/λ2+k/λ3.
(5)
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
τ > iq13+ jq23+k > 0 with qmn=λn
λm.
(6)
The linear combination scales the ionospheric error by
AI= α+βq2
12+γq2
13−a −bq2
12− cq2
13.
(7)
The thermal noise of the elementary carrier phase measure-
mentsisassumedGaussianwiththestandarddeviationgiven
by Kaplan and Hegarty [13]
σφi=λi
2π
?
BL
C/N0
?
1+
1
2T ·C/N0
?
,(8)
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

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P. Henkel and C. G¨ unther3
Table 1: Cramer-Rao bound for Galileo signals.
Modulation
BOC(1,1)
BPSK(10)
BPSK(10)
BOC(15,10)
Bandwidth (MHz)
4
24
24
51
CRB (cm)
20
5
5
1
E1
E5a
E5b
E5
written as
Nm=
?
(α2+β2q2
12+γ2q2
13) ·σ2
φ0+a2σ2
ρ1+b2σ2
ρ2+c2σ2
ρ3
(9)
with σ2
surements. Table 1 shows the Cramer-Rao bound (CRB) for
some Galileo signals as derived by Hein et al. [9]. A DLL
bandwidth of 1Hz has been assumed. The 4MHz receiver
bandwidthforE1hasbeenchosentoavoidsidelobetracking.
For E1, E5a, E5b, E6 phase measurements, the wave-
length scaling of σφican be neglected due to the close vicinity
of the frequency bands. However, it plays a major role when
C-Band measurements are included.
Figure 2showsthebenefitofthecodecontributiontothe
i = 1 (E1), j = −10 (E5b), k = 9 (E5a) linear combination: a
slight increase in noise level results in a considerable reduc-
tion 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
ρ1,...,σ2
ρ3being the noise variance of the code mea-
b = −c =α +βq2
12+γq2
12− q2
13
q2
13
.
(10)
E1 code measurements have not been taken into account due
to the increased noise level but might be included with a low
weight.
Thecombinationdiscrimination—measuredbytheratio
of half the wavelength and the noise level λ/(2Nm)—is pro-
posed 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 op-
timization with respect to this parameter. Note that the leg-
end refers to the elementary wavelengths which have to be
scaled by τ.
Thecomputationoftheoptimum τ takesagainonlyE5a/
E5b code measurements into account as the benefit of the
E1 code measurement is negligible (a = 0). The notation is
simplified by
λ =?λ · τ,
AI= ? κ ·τ −bq2
Nm=
?
with?λ, ? κ, and ? η implicitly given by (5), (7), and (9). The
12−cq2
13,
?
σ2
φ·(α2+β2q2
12+γ2q2
13) +σ2
ρ·(b2+c2)
=
σ2
φ· ? η2τ2+σ2
ρ·(b2+c2),
(11)
0.175
0.1755
0.176
0.1765
0.177
0.1775
0.178
Noise level of 3F code-carrier linear combination (m)
σφ= 1mm (E1, E5a, E5b), σρ= 5cm (E5a, E5b)
Convergence to
ionosphere-free
code-carrier combination
−50
−40
−30
−20
−10
Ionospheric reduction (dB)
Pure phase combination
[1,−10,9], λ = 3.256m
Increase of code
weighting coefficients
AI> 0
AI< 0
Figure 2: Adaptive code contribution to linear combinations:
tradeoff between noise level and ionospheric reduction.
2
4
6
8
10
12
14
16
18
Discriminator of ambiguity resolution λ/2Nm
0
Geometric weight τ = α +β +γ of carrier phase combination
9.768m
4.884m
3.256m
2.442m
1.953m
1.628m
1.395m
12345
1.221m
1.085m
0.976m
0.888m
0.814m
0.751m
0.697m
Figure 3: Optimal weighting of the phase combination part of
ionosphere-free code-carrier combinations with i = 1, k = −j − 1,
and j ∈ {−12,...,1}.
E5a/E5b code weights are determined from the ionosphere-
free and geometry-preserving conditions as
b = 1 −c − τ,
? κ + q2
?
c =
12
q2
13− q2
??
12
?
w1
·τ +
−q2
13− q2
??
12
q2
?
12
?
w2
= w1·τ +w2.
(12)

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4 International Journal of Navigation and Observation
Table 2: Properties and weighting coefficients of ionosphere-free
E1-E5b-E5a code-carrier combinations.
i
1 −12
1 −11
1 −10
1 −9
1 −8
1 −7
1 −6
1 −5
1 −4
1 −3
1 −2
1 −1
1
jk
11
10
9
8
7
6
5
4
3
2
1
0
−1
bc
?λ (m)
4.884
3.256
2.442
1.954
1.628
1.396
1.221
1.085
0.977
0.888
0.814
0.751
λ (m)
3.217
3.216
3.214
3.213
3.212
3.211
3.210
3.209
3.208
3.207
3.206
3.205
3.204
Nm(m)
0.28
0.25
0.23
0.20
0.18
0.16
0.14
0.12
0.11
0.10
0.10
0.11
0.12
R
5.81
6.38
7.05
7.86
8.84
10.02
11.42
12.99
14.54
15.65
15.85
15.04
13.59
0.327
0.166
0.006
−0.154
−0.314
−0.474
−0.634
−0.793
−0.953
−1.112
−1.272
−1.431
−1.590
0.344
0.175
0.007
−0.162
−0.330
−0.499
−0.667
−0.835
−1.003
−1.171
−1.339
−1.506
−1.674
9.768
0
Thecombinationdiscriminationbecomesfrom(5),(11),
and (12):
R(τ)
=λ(τ)/2
Nm(τ)
=
?λ ·τ
2
?
σ2
φ· ? η2τ2+σ2
ρ·??w1τ+w2
?2+?1 −τ −w1τ −w2
?2?.
(13)
Setting the derivative to zero yields the optimum weighting
τopt=
1 −2w2+2w2
1+w1−w2−2w1w2,
2
(14)
which is independent of both σρand σφ. Table 2 contains the
weighting coefficients and characteristics of the code-carrier
combinations shown in Figure 3.
Figure 4 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 de-
pends on the level of ionospheric reduction: a linear in-
crease can be observed near the pure phase combination,
whiletheincreasebecomesnegligible fortheionosphere-free
combination. Thus, the combination discrimination of the
ionosphere-free code-carrier combination is increased by al-
most the same factor as τ is risen.
3.C-BAND AIDED CODE-CARRIER
LINEAR COMBINATIONS
The 20MHz 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 10dB occurs
whichhastobecompensatedbyalargertransmitpower.The
0
−25
0.5
1
1.5
2
2.5
3
Noise level of 3F code-carrier linear combination (m)
σφ= 1mm (E1, E5a, E5b), σρ= 5cm (E5a, E5b)
Negligible noise
amplification due
to increase in τ
−20
−15
Ionospheric reduction (dB)
−10
−50510
Pure phase combination
[0,1,−1], λ = 9.768m
Linear increase of
noise level with τ
τ = 1
τ = 5
τ = 10
Figure 4: Benefit of adaptive code and phase weighting for linear
combinations with λ = τ 9.768m.
ionosphericdelayisapproximately10timeslowerthaninthe
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.Reducednoiseionosphere-free
code-onlycombinations
The design of three frequency code-only combinations that
preserve geometry and eliminate ionospheric errors is char-
acterized by one degree of freedom used for noise minimiza-
tion. The weighting coefficients are derived from the geome-
try preserving and ionosphere-free constraints in (1), (7) as
a = 1 −b −c,
b = −
1
q2
12− 1
????
v1
+
?
?
−q2
13−1
q2
??
ρ1+b2σ2
12−1
v2
?
?
·c = v1+v2· c.
(15)
Minimization of N2
m= a2σ2
ρ2+c2σ2
ρ3yields
c =(1 −v1+v2−v1v2) ·σ2
(1+2v2+v2
ρ1−v1v2·σ2
ρ1+v2
ρ2
2) · σ2
2· σ2
ρ2+σ2
ρ3
.
(16)
Ionosphere-free code-only combinations with more than
three frequencies are obtained by a multidimensional deriva-
tive. Table 3 shows that the pure L-band E1-E5b-E5a combi-
nation is characterized by a noise level of 44.41cm. If the E5
signal is received with full bandwidth, the CRB is reduced to
1cmbutthenumber of degreesof freedom isreducedbyone
so that the noise level of the E1-E5 combinations are lightly

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P. Henkel and C. G¨ unther5
Table 3: Ionosphere-free code-only combinations with minimum
noise σρ(E1) =σρ(C1) = ··· = σρ(C4) =20cm and σρ(E5a,E5b) =
5cm.
E1E5bE5aC1 C2C3 C4
Nm
(cm)
44.41
19.14
14.21
11.80
10.31
2.090 1.500
0.387 0.255
0.213 0.128
0.147 0.079
0.112 0.054
−2.590
−0.506
−0.292
−0.211
−0.168
00
0
0
0
0
0
0
0
0
0.863
0.476
0.328
0.251
0.476
0.328
0.251
0.329
0.2510.251
Table 4: Ionosphere-free code-only combinations with minimum
noise σρ(E1) = σρ(C1) = ··· = σρ(C4) = 20cm and σρ(E5) =
1cm.
E1E5C1 C2 C3C4
Nm
(cm)
46.78
19.31
14.27
11.84
10.34
2.338
0.398
0.217
0.150
0.114
−1.338
−0.278
−0.179
−0.142
−0.122
00
0
0
0
0
0
0
0
0
0.879
0.481
0.331
0.252
0.481
0.331
0.252
0.331
0.2520.252
Table 5: Ionosphere-free code-carrier widelane combinations with
σρ(E1) = σρ(C1) = ··· = σρ(C4) = 20cm and σρ(E5) = 1cm.
i
E1 E5
1
1
1
1
1
1
1
0
jklm
C3
nabλNm
(cm)
3.92
4.84
4.84
5.12
4.84
5.12
5.43
C1C2C4E1E5(m)
3.21
3.39
3.39
3.59
3.39
3.59
3.81
−1
−1
−1
−1
−1
−1
−1
0
0
0
0
1 −1
1
1
0
0
1 −1
1
0
1 −1 −4.7e−3
0 −4.7e−3
0 −1 −5.0e−3
00 −4.7e−3
0 −1
00 −1
001
0
−4.4e−3
−3.11
−3.28
−3.28
−3.46
−3.28
−3.46
−3.67
−6.0e−2 20.70 15.39
0
−5.0e−3
−5.3e−3
−8.5e−50 −1
increased (Table 4). These combinations will play a role in
conjunction with code-carrier combinations.
The C-band is split into 4 bands of 5MHz 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.JointL-/C-bandwidelanecombinations
Code-carrier linear combinations can also include both L-
band and C-band measurements. Therefore, (1)–(7) are ex-
tended to include the additional measurements. The weight-
ing 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,...}.
Table 5 contains ionosphere-free joint L-/C-band code-
carrier widelane combinations (λ > maxiλi). The E1-E5
10−1
100
101
102
Wavelength of linear combination (m)
10−2
10−1
100
Noise level of linear combination (m)
Joint L-/C-band combination
Pure L-band combination
Combination of L-band code and C-band phase measurements
Figure 5: Comparison of joint L-/C-Band linear combinations for
σρ(E1) = 20cm, σρ(E5) = 1cm and σφ,i = λi/λ1· σφ0with σφ0=
1mm.
pure L-band combination benefits from a noise level of only
3.92cm which simplifies the resolution of the 3.215m inte-
ger ambiguities. In contrast to the code-only combinations,
the use of the full-bandwidth E5 signal is advantageous com-
pared 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 dis-
crimination equals 67.25 but the noise level is also increased
to 15.39cm.
Figure 5 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.
3.3.JointL-/C-bandnarrowlanecombinations
There exists a large variety of joint code-carrier narrowlane
combinations where C-band measurements help to reduce
the noise substantially. Figure 6 shows the tradeoff be-
tween 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).
3.4. Reliabilityofambiguityresolution
The integer ambiguity resolution is based on the linear com-
bination of four different variable types: double-difference
measurements for eliminating clock errors and satellite/re-
ceiver biases; multifrequency combinations for suppressing
the ionosphere; code and carrier phase measurements for