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

Joint L-/C-Band Code and Carrier Phase Linear Combinations for Galileo Hindawi Publishing Corporation International Journal of Navigation and Observation Volume 2008, Article ID 651437, 8 pages doi:10.1155/2008/651437 Research Article Joint L-/C-Band Code and Carrier Phase Linear Combinations for Galileo 1 1, 2 Patrick Henkel and Christoph Gun ¨ ther Institute of Communications and Navigation, Technische Universitat ¨ Munche ¨ n (TUM), Theresienstraße 90, 80333 Munche ¨ n, Germany German Aerospace Center (DLR), Institute of Communications and Navigation (IKN), Munchne ¨ r 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 (5010–5030 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. Copyright © 2008 P. Henkel and C. Gunther ¨ . 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 authors have extended this work to three-frequency (3F) Galileo combinations (E1-E5a-E5b) in [5]. A 3F wide- The integer ambiguity resolution of carrier-phase measure- lane combination with a wavelength of 3.256 m and an iono- ments has been simplified by the consideration of lin- spheric suppression of 16.4 dBwas found. Furthermore, a3F ear combinations of measurements at multiple frequencies. narrowlane combination with a wavelength of 5.43 cm could Early methods were the three-carrier ambiguity resolution reduce the ionospheric error by as much as 36.7dB. Sets (TCAR) method introduced by Forssell et al. [1], as well as of linear carrier-phase combinations that are robust against the cascade integer resolution (CIR) developed by Jung et al. residual biases were studied in [6]. The integer ambiguities of [2].The weighting coefficients of three-frequency phase com- the linear combinations can be estimated by the least-squares binations are designed either to eliminate the ionosphere at ambiguity decorrelation adjustment (LAMBDA) algorithm the price of a rather small wavelength or to reduce the iono- developed by Teunissen [7]. The method includes an inte- sphere only by a certain amount with the advantage of a ger transformation which can also be used to determine op- larger wavelength. timum sets of linear combinations [8]. The systematic search of all possible GPS L1-L2 widelane In this paper, the authors used code- and carrier-phase combinations has been performed by Cocard and Geiger measurements in the linear combinations for obtaining [3]. The L1-L2 linear combination of maximum wavelength ionospheric elimination, large wavelengths, and a low noise (14.65 m) amplifies the ionospheric error by a factor 350. level at the same time. The E5a and E5b code measurements Collins gives an overview of reduced ionosphere L1-L2 are of special interest due to their large bandwidth (20 MHz) combinations with wavelengths up to 86.2 cm (+1,−1 and their low associated Cramer-Rao bound of 5 cm [9]. The widelane) in [4]. C-band phase measurements are particularly interesting due 2 International Journal of Navigation and Observation λ φ λ φ λ φ ρ ρ ρ to their small wavelength and their thus reduced phase noise. 1 1 2 2 3 3 1 2 3 The properties of code-carrier linear combinations are opti- mized by including both L- and C-band measurements. The αβ γ ab c cost function is defined as the ratio of half the wavelength and 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- λφ band signals are foreseen in a band between 5010 MHz to 5030 MHz [11]. The signal propagation and tracking char- Figure 1: Linear combination of carrier-phase and code measure- acteristics in the C-band have been analyzed by Irsigler et al. ments. [12]. The larger frequencies result in an additional free space loss of 10 dB that has to be compensated by a larger transmit power. which can be split into three sufficient conditions The paper is organized as follows: the next section in- troduces the design of code-carrier linear combinations. The βλ γλ αλ 1 2 3 underlying trade-off between a low noise level and strong i = ∈ Z, j = ∈ Z, k = ∈ Z,(3) λ λ λ ionospheric reduction turns out to be controlled by the weighting coefficients of E5a/E5b code measurements. with Z denoting the space of integers. These integer con- In Section 3, code-carrier linear combinations are com- straints are rewritten to obtain the weighting coefficients puted in a way that include both L- and C-band measure- ments. An ionosphere-free code-only combination is deter- iλ jλ kλ mined that benefits from a 4.5 times lower noise level than α = , β = , γ = . (4) λ λ λ a pure L-band combination. Furthermore, a pure L-band 1 2 3 ionosphere-free code-carrier combination with a wavelength Mixedcode-carrier combinationsweightthe phasepart of 3.215 m and a noise standard deviation of 3.9 cm is found. by τ and the code part by 1 − τ . The border cases are pure The combined use of the two reduces the probability of phase (τ = 1) and pure code (τ = 0) combinations. The wrong fixing of the latter solution by 9 orders of magnitude parameter τ has a significant impact on the properties of the with respect to a pure L-band solution. linear combination, and it is optimized later in this section. The use of C-band measurements for ionosphere-free Replacing the weighting coefficients in τ = α + β + γ by (4) carrier smoothing is discussed in Section 4:anionosphere- yields the wavelength of the code-carrier combination 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 λ = . (5) i/λ + j/λ + k/λ benefits from an 8.9 dB lower noise level as compared to the 1 2 3 equivalent L-band combination. The generalized widelane criterion is given for λ <λ <λ 1 2 3 by λ>λ . Equivalently, it can be expressed as a function of i, 2. CODE-CARRIER LINEAR COMBINATIONS j,and k as Linear combinations of carrier-phase measurements are con- structed to increase the wavelength (widelane), suppress the τ> iq + jq + k>0with q = . (6) 13 23 mn ionospheric error, and to simplify the integer ambiguity reso- m lution. The properties of the linear combinations can be im- The linear combination scales the ionospheric error by proved by including weighted code measurements into the pure phase combinations. Figure 1 shows a three frequency 2 2 2 2 (3F) linear combination where the phase measurements are A = α + βq + γq − a − bq − cq . (7) I 12 13 12 13 weighted by α, β, γ, and the code measurements are scaled by a, b, c. The weighting coefficients are generally restricted by a The thermal noise of the elementary carrier phase measure- few conditions: first, the geometry should be preserved, that ments is assumed Gaussian with the standard deviation given is by Kaplan and Hegarty [13] α + β + γ + a + b + c = 1. (1) λ B 1 i L σ = 1+ ,(8) 2π C/N 2T · C/N 0 0 Moreover, the superposition of ambiguities should be an in- teger multiple of a common wavelength λ, that is where B denotes the loop bandwidth, C/N the carrier- L 0 to-noise ratio, and T the predetection integration time. αλ N + βλ N + γλ N = λN,(2) The overall noise contribution of the linear combination is 1 1 2 2 3 3 P. Henkel and C. Gunther ¨ 3 Table 1: Cramer-Rao bound for Galileo signals. 0.178 Increase of code Modulation Bandwidth (MHz) CRB (cm) weighting coefficients 0.1775 E1 BOC(1,1) 4 20 E5a BPSK(10) 24 5 Convergence to 0.177 E5b BPSK(10) 24 5 ionosphere-free code-carrier combination A > 0 E5 BOC(15,10) 51 1 0.1765 A < 0 written as 0.176 2 2 2 2 2 2 2 2 2 2 2 2 N = (α + β q + γ q ) · σ + a σ + b σ + c σ 12 13 φ ρ ρ ρ 0 1 2 3 0.1755 (9) Pure phase combination [1,−10, 9], λ = 3.256 m 2 2 with σ ,... , σ being the noise variance of the code mea- ρ1 ρ3 0.175 −50 −40 −30 −20 −10 surements. Table 1 shows the Cramer-Rao bound (CRB) for Ionospheric reduction (dB) some Galileo signals as derived by Hein et al. [9]. A DLL bandwidth of 1 Hz has been assumed. The 4 MHz receiver Figure 2: Adaptive code contribution to linear combinations: bandwidth for E1 has been chosen to avoid sidelobe tracking. tradeoff between noise level and ionospheric reduction. For E1, E5a, E5b, E6 phase measurements, the wave- length scaling of σ can be neglected due to the close vicinity φ 18 of the frequency bands. However, it plays a major role when C-Band measurements are included. Figure 2 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 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 2 2 α + βq + γq 12 13 b =−c = . (10) 2 2 q − q 12 13 E1 code measurements have not been taken into account due to the increased noise level but might be included with a low weight. The combination discrimination—measured by the ratio of half the wavelength and the noise level λ/(2N )—is pro- 01 2 3 4 5 posed as a cost function to select linear combinations due to Geometric weight τ = α + β + γ of carrier phase combination its independence of the geometry. It is shown for multiple 9.768 m 1.221 m ionosphere-free code-carrier combinations in Figure 3.The 4.884 m 1.085 m strong dependency on the phase weighting τ suggests an op- 3.256 m 0.976 m timization with respect to this parameter. Note that the leg- 2.442 m 0.888 m end refers to the elementary wavelengths which have to be 1.953 m 0.814 m scaled by τ . 1.628 m 0.751 m The computation of the optimum τ takes again only E5a/ 1.395 m 0.697 m E5b code measurements into account as the benefit of the Figure 3: Optimal weighting of the phase combination part of E1 code measurement is negligible (a = 0). The notation is ionosphere-free code-carrier combinations with i = 1, k =−j − 1, simplified by and j ∈{−12,... ,1}. λ = λ · τ , E5a/E5b code weights are determined from the ionosphere- 2 2 A = κ· τ − bq − cq , I 12 13 free and geometry-preserving conditions as (11) 2 2 2 2 2 2 2 2 2 N = σ · (α + β q + γ q )+ σ · (b + c ) m 12 13 b = 1 − c − τ , φ ρ 2 2 2 2 2 2 2 κ + q −q = σ · η τ + σ · (b + c ), 12 12 (12) φ ρ c = · τ + = w · τ + w . 1 2 2 2 2 2 q − q q − q 13 12 13 12 with λ, κ,and η implicitly given by (5), (7), and (9). The w w 1 2 Noise level of 3F code-carrier linear combination (m) Discriminator of ambiguity resolution λ/2N m σ = 1 mm (E1, E5a, E5b), σ = 5 cm (E5a, E5b) φ ρ 4 International Journal of Navigation and Observation Table 2: Properties and weighting coefficients of ionosphere-free 3 E1-E5b-E5a code-carrier combinations. 2.5 ij k b c λ (m) λ (m) N (m) R 1 −12 11 0.327 0.344 9.768 3.217 0.28 5.81 1 −11 10 0.166 0.175 4.884 3.216 0.25 6.38 2 1 −10 9 0.006 0.007 3.256 3.214 0.23 7.05 1 −98 −0.154 −0.162 2.442 3.213 0.20 7.86 1.5 1 −87 −0.314 −0.330 1.954 3.212 0.18 8.84 Linear increase of Negligible noise 1 −76 −0.474 −0.499 1.628 3.211 0.16 10.02 noise level with τ amplification due 1 −65 −0.634 −0.667 1.396 3.210 0.14 11.42 to increase in τ 1 −54 −0.793 −0.835 1.221 3.209 0.12 12.99 0.5 Pure phase combination 1 −43 −0.953 −1.003 1.085 3.208 0.11 14.54 [0, 1,−1], λ = 9.768 m 1 −32 −1.112 −1.171 0.977 3.207 0.10 15.65 1 −21 −1.272 −1.339 0.888 3.206 0.10 15.85 −25 −20 −15 −10 −50 5 10 Ionospheric reduction (dB) 1 −10 −1.431 −1.506 0.814 3.205 0.11 15.04 10 −1 −1.590 −1.674 0.751 3.204 0.12 13.59 τ = 1 τ = 5 τ = 10 The combination discrimination becomes from (5), (11), Figure 4: Benefit of adaptive code and phase weighting for linear and (12): combinations with λ = τ 9.768 m. R(τ ) λ(τ )/2 ionospheric delay is approximately 10 times lower than in the N (τ ) E1 band. The small wavelength of 5.9691 cm··· 5.9839 cm complicates direct ambiguity resolution but results in an λ · τ = . approximately 3.2 times lower standard deviation of phase 2 2 2 2 2 noise. Moreover, the C-Band offers additional degrees of 2 σ · η τ + σ · w τ +w + 1 − τ − w τ − w 1 2 1 2 φ ρ freedom for the design of linear combinations. (13) Setting the derivative to zero yields the optimum weighting 3.1. Reduced noise ionosphere-free code-only combinations 1 − 2w +2w τ = , (14) opt 1+ w − w − 2w w The design of three frequency code-only combinations that 1 2 1 2 preserve geometry and eliminate ionospheric errors is char- which is independent of both σ and σ . Table 2 contains the ρ φ acterized by one degree of freedom used for noise minimiza- weighting coefficients and characteristics of the code-carrier tion. The weighting coefficients are derived from the geome- combinations shown in Figure 3. try preserving and ionosphere-free constraints in (1), (7)as Figure 4 shows the benefit of adaptive code and phase weighting for the code-carrier linear combination with i = 0 a = 1 − b − c, (E1), j = 1(E5b),and k =−1 (E5a). Obviously, the wavelength increases linearly with τ and the ionosphere 1 q − 1 (15) b =− + − · c = v + v · c. 1 2 can be eliminated with any τ . The noise amplification de- 2 2 q − 1 q − 1 12 12 pends on the level of ionospheric reduction: a linear in- v v 1 2 crease can be observed near the pure phase combination, while the increase becomes negligible for the ionosphere-free 2 2 2 2 2 2 2 Minimization of N = a σ + b σ + c σ yields m ρ ρ ρ 1 2 3 combination. Thus, the combination discrimination of the ionosphere-free code-carrier combination is increased by al- 2 2 (1 − v + v − v v ) · σ − v v · σ 1 2 1 2 1 2 ρ ρ 1 2 most the same factor as τ is risen. c = . (16) 2 2 2 2 2 (1 + 2v + v ) · σ + v · σ + σ 2 ρ 2 ρ ρ 1 2 3 3. C-BAND AIDED CODE-CARRIER Ionosphere-free code-only combinations with more than LINEAR COMBINATIONS three frequencies are obtained by a multidimensional deriva- The 20 MHz wide C-Band (5010··· 5030 MHz ={489.736 tive. Table 3 shows that the pure L-band E1-E5b-E5a combi- ··· 491.691}· 10.23 MHz) has been reserved for Galileo. nation is characterized by a noise level of 44.41 cm. If the E5 The higher frequency range has a multitude of advantages signal is received with full bandwidth, the CRB is reduced to and drawbacks: an additional free space loss of 10 dB occurs 1 cm but the number of degrees of freedom is reduced by one which has to be compensated by a larger transmit power. The so that the noise level of the E1-E5 combinations are lightly Noise level of 3F code-carrier linear combination (m) σ = 1 mm (E1, E5a, E5b), σ = 5 cm (E5a, E5b) φ ρ P. Henkel and C. Gunther ¨ 5 Table 3: Ionosphere-free code-only combinations with minimum 10 noise σ (E1) =σ (C1) = ··· = σ (C4) = 20 cm and σ (E5a, E5b) = ρ ρ ρ ρ 5cm. E1 E5b E5a C1 C2 C3 C4 (cm) 2.090 1.500 −2.590 0000 44.41 0.387 0.255 −0.506 0.863 0 0 0 19.14 0.213 0.128 −0.292 0.476 0.476 0 0 14.21 0.147 0.079 −0.211 0.328 0.328 0.329 0 11.80 0.112 0.054 −0.168 0.251 0.251 0.251 0.251 10.31 Table 4: Ionosphere-free code-only combinations with minimum noise σ (E1) = σ (C1) = ··· = σ (C4) = 20 cm and σ (E5) = −1 ρ ρ ρ ρ −2 −1 0 1cm. 10 10 10 Noise level of linear combination (m) E1 E5 C1 C2 C3 C4 (cm) Joint L-/C-band combination Pure L-band combination 2.338 −1.338 0000 46.78 Combination of L-band code and C-band phase measurements 0.398 −0.278 0.879 0 0 0 19.31 0.217 −0.179 0.481 0.481 0 0 14.27 Figure 5: Comparison of joint L-/C-Band linear combinations for σ (E1) = 20 cm, σ (E5) = 1cm and σ = λ /λ · σ with σ = 0.150 −0.142 0.331 0.331 0.331 0 11.84 ρ ρ φ,i i 1 φ0 φ0 1mm. 0.114 −0.122 0.252 0.252 0.252 0.252 10.34 Table 5: Ionosphere-free code-carrier widelane combinations with pure L-band combination benefits from a noise level of only σ (E1) = σ (C1) = ··· = σ (C4) = 20 cm and σ (E5) = 1cm. ρ ρ ρ ρ 3.92 cm which simplifies the resolution of the 3.215 m inte- ger ambiguities. In contrast to the code-only combinations, ij k l m n a b λ N the use of the full-bandwidth E5 signal is advantageous com- E1 E5 C1 C2 C3 C4 E1 E5 (m) (cm) pared to separate E5a and E5b measurements. The C-band 1 −10000 −4.4e−3 − 3.11 3.21 3.92 offers no benefit for these wavelengths. In the last row of 1 −1001 − 1 − 4.7e−3 − 3.28 3.39 4.84 Table 5, a linear combination with a pure L-band code and pure C-band phase part is described. The combination dis- 1 −101 −10 − 4.7e−3 − 3.28 3.39 4.84 crimination equals 67.25 but the noise level is also increased 1 −1010 − 1 − 5.0e−3 − 3.46 3.59 5.12 to 15.39 cm. 1 −11 −100 − 4.7e−3 − 3.28 3.39 4.84 Figure 5 shows the tradeoff between wavelength and 1 −110 −10 −5.0e−3 − 3.46 3.59 5.12 noise level for joint L-/C-band ionosphere-free linear code- 1 −1100 − 1 −5.3e−3 − 3.67 3.81 5.43 carrier combinations with {i, j}∈ [−5, +5] and {k, l, m, 00 −1001 −8.5e−5 − 6.0e−2 20.70 15.39 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. increased (Table 4). These combinations will play a role in conjunction with code-carrier combinations. 3.3. Joint L-/C-band narrowlane combinations The C-band is split into 4 bands of 5 MHz bandwidth centered at {490, 490.5, 491, 491.5}· 10.23 MHz and allows a There exists a large variety of joint code-carrier narrowlane significant reduction of the noise level. Note that the noise combinations where C-band measurements help to reduce of any contributing elementary combination is reduced by a the noise substantially. Figure 6 shows the tradeoff be- weighting coefficient smaller than one. tween wavelength and noise level for {i, j}∈ [−5, +5] and {k, l, m, n}∈ [−2, +2]. For λ = 5.7 cm, the consideration of C-band measurements reduces the noise level by a factor of 3.2. Joint L-/C-band widelane combinations 5 compared to a pure L-band combination (Table 6). Code-carrier linear combinations can also include both L- band and C-band measurements. Therefore, (1)–(7)are ex- 3.4. Reliability of ambiguity resolution tended to include the additional measurements. The weight- ing coefficients {α, β, γ,...} and {a, b} are computed such The integer ambiguity resolution is based on the linear com- that the discriminator output of (13) is maximized for a bination of four different variable types: double-difference given set of integer coefficients {i, j , k,...}. measurements for eliminating clock errors and satellite/re- Table 5 contains ionosphere-free joint L-/C-band code- ceiver biases; multifrequency combinations for suppressing carrier widelane combinations (λ> max λ ). The E1-E5 the ionosphere; code and carrier phase measurements for i i Wavelength of linear combination (m) 6 International Journal of Navigation and Observation −1 −5 −10 −15 −2 −20 0 5 10 15 20 −4 −3 −2 10 10 10 Time (h) Noise level of linear combination (m) E1/E5 λ = 3.215 m + E1/E5 code-only combination Joint L-/C-band combination E1/E5 λ = 3.215 m + E1/E5/C1 code-only combination Pure L-band combination E1/E5 λ = 3.215 m + E1/E5/C1/C2 code-only combination Combination of L-band code and C-band phase measurements E1/E5 λ = 3.215 m + E1/E5/C1/C2/C3 code-only combination Figure 7: Reliability of λ = 3.215 m integer ambiguity resolution: Figure 6: Comparison of joint L-/C-Band linear combinations for impact of C-band measurements on the probability of wrong fixing σ (E1) = 20 cm, σ (E5) = 1cm and σ = λ /λ · σ with σ = ρ ρ φ,i i 1 φ0 φ0 of the most critical ambiguity. 1mm. Table 6: Ionosphere-free code-carrier narrowlane combinations and the DD geometry matrix G, the baseline δx and the in- with σ (E1) = σ (C1) = ··· = σ (C4) = 20 cm and σ (E5) = 1cm. ρ ρ ρ ρ teger ambiguities N . The double-differenced troposphere is assumedtobenegligibleorknown apriori(e.g.,fromanac- i E1 1 0 −15 curate continued fraction model). j E5 −10 1 − 3 Note that the troposphere has the same impact on all k C1 0 0 1 0 geometry-preserving combinations and does not affect the l C2 0 0 0 0 optimization of the mixed code-carrier combinations. The m C3 0 0 0 0 noise vector is Gaussian distributed, that is, n C4 1 1 0 0 ε∼N (0, Σ)with Σ = Σ ⊗ Σ , (19) LC DD a E1 − 2.04e−6 7.60e−5 1.58e−4 2.55e−4 b E5 − 1.43e−3 5.31e−2 0.110 0.178 where ⊗ denotes the Kronecker product. Σ models the lin- LC λ (cm) 5.55 5.65 5.76 5.73 ear combination induced correlation and Σ includes the DD N (mm) 0.51 0.61 1.22 2.50 correlation due to double difference measurements from N R 54.4 46.3 23.6 11.46 visible satellites. The standard deviation of the most critical ambiguity estimate can be written as −1 T −1 reducing the noise level; and finally, L-/C-band combinations σ = max Σ (i, i), Σ = X Σ X , (20) max β β i={1···N −1} for noise and discrimination characteristics. Two joint L-/C-band code-carrier ionosphere-free com- and the probability of wrong fixing follows as binations are chosen for real-time (single epoch) ambiguity resolution. The λ = 3.215 m, N = 3.92 cm combination of +0.5 1 2 2 c −x /2σ max P = 1 − e dx. (21) Table 5 and one further combination of Table 4. The double −0.5 2πσ max difference (DD) ionosphere-free combinations are modeled as 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. y G λ · 1 ρφ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ = δx + N + ε = Xβ + ε, (17) Figure 7 shows the benefit of C-band measurements for y G 0 integer ambiguity fixing. If the E1-E5 pure L-band combina- tion is used as second combination in (17), the failure rate with varies between 0.01 and 0.07 due to its poor noise charac- ⎡ ⎤ ⎡ ⎤ teristics. The use of two additional C-band measurements λ · 1 G N −5 reduces the maximum probability of wrong fixing to 10 . ⎣ ⎦ ⎣ ⎦ X = , β = (18) −11 0 G δx For three C-band frequencies, the failure rate is at most 10 Wavelength of linear combination (m) Probability of wrong fixing of most critical ambiguity P. Henkel and C. Gunther ¨ 7 which corresponds to a gain of 9 to 17 orders of magnitude compared to the pure L-band combination. The reliability of ambiguity resolution can be further −5 improved by using the LAMBDA method of Teunissen [7]. The float ambiguity estimates are decorrelated by an integer transformation Z and the ambiguity covariance matrix is −10 written as T T Σ = Z Σ Z = LDL , (22) N N −15 with the decomposition into a lower triangular matrix L and a diagonal matrix D. The probability of wrong fixing of the −20 sequential bootstrapping estimator is given by Teunissen [14] 0 5 10 15 20 as Time (h) N −1 +0.5 1 E1/E5 λ = 3.215 m + E1/E5 code-only combination 2 2 −x /2σ (i) P = 1 − e dx, (23) E1/E5 λ = 3.215 m + E1/E5/C1 code-only combination −0.5 2πσ (i) i=1 E1/E5 λ = 3.215 m + E1/E5/C1/C2 code-only combination with σ (i) = D(i, i). It represents a lower bound for the c Figure 8: Reliability of λ = 3.215 m integer ambiguity resolution: success rate of the integer least-square estimator and is de- impact of C-band measurements on the probability of wrong fixing based on sequential fixing with the integer decorrelation transfor- picted in Figure 8. Obviously, the use of joint L-/C-band lin- mation. ear combinations reduces the probability of wrong fixing by severalordersofmagnitudecomparedtopureL-bandcom- 0.16 binations. 0.14 3.5. Accuracy of baseline estimation 0.12 After integer ambiguity fixing, the baseline is re-estimated from (17). The covariance matrix of the baseline estimate in 0.1 local coordinates is given by −1 T −1 T Σ = R G Σ G R (24) 0.08 δx L with the rotation matrix R . Figure 9 shows the achievable 0.06 horizontal and vertical accuracies for the two optimized joint L-/C-band combinations. 0.04 The pure L-band combinations in the first row of Tables 4 and 5 have been again selected as reference scenario. It can 0.02 0 5 10 15 20 be observed that the use of joint L-/C-band linear combi- Time (h) nations enables a slight improvement in position estimates compared to the significant benefit for ambiguity resolution. Horizontal comp. (joint L/C) Vertical comp. (joint L/C) Horizontal comp. (pure L) 4. JOINT L-/C-BAND CARRIER SMOOTHED CARRIER Vertical comp. (pure L) Ionosphere-free code-carrier linear combinations are charac- Figure 9: Standard deviation of baseline estimation using the λ = terized by a noise level that is one to two orders of magnitude 3.215 m E1-E5 ionosphere-free code-carrier combination and the larger than of the underlying carrier-phase measurements E1-E5-C1··· C4 ionosphere-free code-only combination. (Table 5). Both noise and multipath of the code-carrier com- binations can be reduced by the smoothing filter of Hatch [15] which is shown in Figure 10. The upper input can be an Note that the superposition of ambiguities of the pure ionosphere-free code-carrier combination of arbitrary wave- phase combination is not necessarily an integer number of a length. The lower input is a pure ionosphere-free phase com- commonwavelength. Therespectiveambiguities arenot af- bination that is determined by three conditions: the first en- fected by the low pass filter and do not occur in the smoothed sures that the geometry is preserved, the second eliminates output λ φ due to different signs in the addition to λ φ A A A the ionosphere, and the third minimizes the noise, that is, (Figure 10). Table 7 shows an ionosphere-free E1-E5a-E5b phase α + β + γ = 1, combination that increases the noise level by a factor 2.64. 2 2 However, the low noise level of C-band measurements sug- α + βq + γq = 0, (25) 12 13 gests the use of the second combination with f = 491 · 2 2 2 2 2 2 2 minN = min σ · α + β q + γ q . m φ,0 12 13 10.23 MHz. In this case, the noise level is not only reduced α,β,γ α,β,γ Probability of wrong fixing of all ambiguities Standard deviation of baseline estimate (m) using integer decorrelation transformations 8 International Journal of Navigation and Observation LP filter REFERENCES χ χ λ φ λ φ [1] B. Forssell, M. Martin-Neira, and R. A. Harris, “Carrier phase A A A − + ambiguity resolution in GNSS-2,” in Proceedings of the 10th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GPS ’97), vol. 2, pp. 1727–1736, λ φ B B Kansas City, Mo, USA, September 1997. [2] J. Jung, P. Enge, and B. Pervan, “Optimization of cascade in- Figure 10: Ionosphere-free carrier smoothed code-carrier combi- teger resolution with three civil frequencies,” in Proceedings of nations. the 13th International Technical Meeting of the Satellite Divi- sion of the Institute of Navigation (ION GPS ’00), Salt Lake City, Utah, USA, September 2000. Table 7: Weighting coefficients and properties of ionosphere-free carrier smoothed carrier phase combinations. [3] M. Cocard and A. Geiger, “Systematic search for all possi- ble Widelanes,” in Proceedings of the 6th International Geode- f f f αβ γ N 1 2 3 m tic Symposium on Satellite Positioning, Columbus, Ohio, USA, E1 E5b E5a 2.324 − 0.559 − 0.764 2.64 · σ March 1992. [4] P. Collins, “An overview of gps inter-frequency carrier phase E1 E5b C − 0.008 − 0.056 1.064 0.34 · σ φ0 combinations,” Technical Memorandum, Geodetic Survey Division, University of New Brunswick, Ottawa, Ontario, Canada, 1999. by smoothing but also by the coefficients of the pure phase [5] P. Henkel and C. Gunther ¨ , “Three frequency linear combina- combination. tions for Galileo,” in Proceedings of the 4th Workshop on Posi- The variance of the smoothed combination is given by tioning, Navigation and Communication (WPNC ’07), pp. 239– 245, Hannover, Germany, March 2007. σ = E ε (t) − ε (t)+ ε (t) , (26) A B B [6] P. Henkel and C. Gunther, “Sets of robust full-rank linear com- with the low-pass filtered noise (e.g., Konno et al. [16]) binations for wide-area differential ambiguity fixing,” in Pro- ceedings of the 2nd ESA Workshop on GNSS Signals and Signal 1 1 Processing, Noordwijk, The Netherlands, April 2007. ε (t) = · 1 − ε (t − n), (27) A A [7] P. Teunissen, “Least-squares estimation of the integer ambi- τ τ s s n=0 guities,” Invited lecture, Section IV, Theory and Methodology, and the smoothing time τ . Setting (27) into (26), and using IAG General Meeting, Beijing, China, 1993. the definition of a geometric series yields [8] P. J. G. Teunissen, “On the GPS widelane and its decorrelating property,” Journal of Geodesy, vol. 71, no. 9, pp. 577–587, 1997. 1 2 2 2 2 2 2 2 2 [9] G. Hein, J. Godet, J. Issler, et al., “Status of galileo frequency σ = σ + · σ + σ − 2σ + · σ − σ . B A B AB AB B 2τ − 1 τ s s and signal design,” in Proceedings of the 15th International (28) Technical Meeting of the Satellite Division of the Institute of Navigation (ION GPS ’02), pp. 266–277, Portland, Ore, USA, For long smoothing times, only the low noise of the joint L/C September 2002. pure carrier-phase combination λ φ remains (Table 7). B B [10] “Galileo Open Service Signal-in-Space ICD,” http://www .galileoju.com. 5. CONCLUSIONS [11] European Radiocommunications Office, “The European Ta- ble of Frequency Allocations and Utilisations in the Frequency In this paper, new joint L-/C-band linear combinations that Range 9 kHz to 1000 GHz,” ERC Report 25, p. 132, 2007. include both code- and carrier-phase measurements have [12] M. Irsigler, G. Hein, B. Eissfeller, et al., “Aspects of C-band been determined. The weighting coefficients are selected satellite navigation: signal propagation and satellite signal such that the ratio between wavelength and noise level is tracking,” in Proceedings of the European Navigation Confer- maximized. An ionosphere-free L-band combination (IFL) ence (ENC GNSS ’02), Kopenhagen, Denmark, May 2002. [13] E. Kaplan and C. Hegarty, Understanding GPS: Principles and could be found at a wavelength of 3.215 m with a noise level Applications, Artech House, London, UK, 2nd edition, 2006. of 3.92 cm. [14] P. J. G. Teunissen, “Success probability of integer GPS ambigu- The combination of L- and C-band measurements re- ity rounding and bootstrapping,” JournalofGeodesy, vol. 72, duces the noise level of ionosphere-free code-only combina- no. 10, pp. 606–612, 1998. tions by a factor 4.5 compared to pure L-band combinations. [15] R. R. Hatch, “A new three-frequency, geometry-free, technique This increases the reliability of an ambiguity resolution op- for ambiguity resolution,” in Proceedings of the 19th Interna- tion for the IFL combination by 9 orders of magnitude. tional Technical Meeting of the Satellite Division of the Insti- The residual variance of the noise can be further re- tute of Navigation (ION GNSS ’06), vol. 1, pp. 309–316, Fort duced by smoothing. An L-/C-band carrier combination can Worth, Tex, USA, September 2006. smooth the noise with a residual variance below the L-band [16] H. Konno, S. Pullen, J. Rife, and P. Enge, “Evaluation of phase noise variance. The smoothed solution can either be two types of dual-frequency differential GPS techniques un- used directly or can be used to resolve the narrowlane ambi- der anomalous ionosphere conditions,” in Proceedings of guities. The variance is basically the same in both cases. The the National Technical Meeting of the Institute of Navigation resolved ambiguities, however, provide instantaneous inde- (NTM ’06), vol. 2, pp. 735–747, Monterey, Calif, USA, January pendent solutions. 2006. 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Joint L-/C-Band Code and Carrier Phase Linear Combinations for Galileo

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Copyright © 2008 Patrick Henkel and Christoph Günther. 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.
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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 Research Article Joint L-/C-Band Code and Carrier Phase Linear Combinations for Galileo 1 1, 2 Patrick Henkel and Christoph Gun ¨ ther Institute of Communications and Navigation, Technische Universitat ¨ Munche ¨ n (TUM), Theresienstraße 90, 80333 Munche ¨ n, Germany German Aerospace Center (DLR), Institute of Communications and Navigation (IKN), Munchne ¨ r 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 (5010–5030 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. Copyright © 2008 P. Henkel and C. Gunther ¨ . 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 authors have extended this work to three-frequency (3F) Galileo combinations (E1-E5a-E5b) in [5]. A 3F wide- The integer ambiguity resolution of carrier-phase measure- lane combination with a wavelength of 3.256 m and an iono- ments has been simplified by the consideration of lin- spheric suppression of 16.4 dBwas found. Furthermore, a3F ear combinations of measurements at multiple frequencies. narrowlane combination with a wavelength of 5.43 cm could Early methods were the three-carrier ambiguity resolution reduce the ionospheric error by as much as 36.7dB. Sets (TCAR) method introduced by Forssell et al. [1], as well as of linear carrier-phase combinations that are robust against the cascade integer resolution (CIR) developed by Jung et al. residual biases were studied in [6]. The integer ambiguities of [2].The weighting coefficients of three-frequency phase com- the linear combinations can be estimated by the least-squares binations are designed either to eliminate the ionosphere at ambiguity decorrelation adjustment (LAMBDA) algorithm the price of a rather small wavelength or to reduce the iono- developed by Teunissen [7]. The method includes an inte- sphere only by a certain amount with the advantage of a ger transformation which can also be used to determine op- larger wavelength. timum sets of linear combinations [8]. The systematic search of all possible GPS L1-L2 widelane In this paper, the authors used code- and carrier-phase combinations has been performed by Cocard and Geiger measurements in the linear combinations for obtaining [3]. The L1-L2 linear combination of maximum wavelength ionospheric elimination, large wavelengths, and a low noise (14.65 m) amplifies the ionospheric error by a factor 350. level at the same time. The E5a and E5b code measurements Collins gives an overview of reduced ionosphere L1-L2 are of special interest due to their large bandwidth (20 MHz) combinations with wavelengths up to 86.2 cm (+1,−1 and their low associated Cramer-Rao bound of 5 cm [9]. The widelane) in [4]. C-band phase measurements are particularly interesting due 2 International Journal of Navigation and Observation λ φ λ φ λ φ ρ ρ ρ to their small wavelength and their thus reduced phase noise. 1 1 2 2 3 3 1 2 3 The properties of code-carrier linear combinations are opti- mized by including both L- and C-band measurements. The αβ γ ab c cost function is defined as the ratio of half the wavelength and 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- λφ band signals are foreseen in a band between 5010 MHz to 5030 MHz [11]. The signal propagation and tracking char- Figure 1: Linear combination of carrier-phase and code measure- acteristics in the C-band have been analyzed by Irsigler et al. ments. [12]. The larger frequencies result in an additional free space loss of 10 dB that has to be compensated by a larger transmit power. which can be split into three sufficient conditions The paper is organized as follows: the next section in- troduces the design of code-carrier linear combinations. The βλ γλ αλ 1 2 3 underlying trade-off between a low noise level and strong i = ∈ Z, j = ∈ Z, k = ∈ Z,(3) λ λ λ ionospheric reduction turns out to be controlled by the weighting coefficients of E5a/E5b code measurements. with Z denoting the space of integers. These integer con- In Section 3, code-carrier linear combinations are com- straints are rewritten to obtain the weighting coefficients puted in a way that include both L- and C-band measure- ments. An ionosphere-free code-only combination is deter- iλ jλ kλ mined that benefits from a 4.5 times lower noise level than α = , β = , γ = . (4) λ λ λ a pure L-band combination. Furthermore, a pure L-band 1 2 3 ionosphere-free code-carrier combination with a wavelength Mixedcode-carrier combinationsweightthe phasepart of 3.215 m and a noise standard deviation of 3.9 cm is found. by τ and the code part by 1 − τ . The border cases are pure The combined use of the two reduces the probability of phase (τ = 1) and pure code (τ = 0) combinations. The wrong fixing of the latter solution by 9 orders of magnitude parameter τ has a significant impact on the properties of the with respect to a pure L-band solution. linear combination, and it is optimized later in this section. The use of C-band measurements for ionosphere-free Replacing the weighting coefficients in τ = α + β + γ by (4) carrier smoothing is discussed in Section 4:anionosphere- yields the wavelength of the code-carrier combination 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 λ = . (5) i/λ + j/λ + k/λ benefits from an 8.9 dB lower noise level as compared to the 1 2 3 equivalent L-band combination. The generalized widelane criterion is given for λ <λ <λ 1 2 3 by λ>λ . Equivalently, it can be expressed as a function of i, 2. CODE-CARRIER LINEAR COMBINATIONS j,and k as Linear combinations of carrier-phase measurements are con- structed to increase the wavelength (widelane), suppress the τ> iq + jq + k>0with q = . (6) 13 23 mn ionospheric error, and to simplify the integer ambiguity reso- m lution. The properties of the linear combinations can be im- The linear combination scales the ionospheric error by proved by including weighted code measurements into the pure phase combinations. Figure 1 shows a three frequency 2 2 2 2 (3F) linear combination where the phase measurements are A = α + βq + γq − a − bq − cq . (7) I 12 13 12 13 weighted by α, β, γ, and the code measurements are scaled by a, b, c. The weighting coefficients are generally restricted by a The thermal noise of the elementary carrier phase measure- few conditions: first, the geometry should be preserved, that ments is assumed Gaussian with the standard deviation given is by Kaplan and Hegarty [13] α + β + γ + a + b + c = 1. (1) λ B 1 i L σ = 1+ ,(8) 2π C/N 2T · C/N 0 0 Moreover, the superposition of ambiguities should be an in- teger multiple of a common wavelength λ, that is where B denotes the loop bandwidth, C/N the carrier- L 0 to-noise ratio, and T the predetection integration time. αλ N + βλ N + γλ N = λN,(2) The overall noise contribution of the linear combination is 1 1 2 2 3 3 P. Henkel and C. Gunther ¨ 3 Table 1: Cramer-Rao bound for Galileo signals. 0.178 Increase of code Modulation Bandwidth (MHz) CRB (cm) weighting coefficients 0.1775 E1 BOC(1,1) 4 20 E5a BPSK(10) 24 5 Convergence to 0.177 E5b BPSK(10) 24 5 ionosphere-free code-carrier combination A > 0 E5 BOC(15,10) 51 1 0.1765 A < 0 written as 0.176 2 2 2 2 2 2 2 2 2 2 2 2 N = (α + β q + γ q ) · σ + a σ + b σ + c σ 12 13 φ ρ ρ ρ 0 1 2 3 0.1755 (9) Pure phase combination [1,−10, 9], λ = 3.256 m 2 2 with σ ,... , σ being the noise variance of the code mea- ρ1 ρ3 0.175 −50 −40 −30 −20 −10 surements. Table 1 shows the Cramer-Rao bound (CRB) for Ionospheric reduction (dB) some Galileo signals as derived by Hein et al. [9]. A DLL bandwidth of 1 Hz has been assumed. The 4 MHz receiver Figure 2: Adaptive code contribution to linear combinations: bandwidth for E1 has been chosen to avoid sidelobe tracking. tradeoff between noise level and ionospheric reduction. For E1, E5a, E5b, E6 phase measurements, the wave- length scaling of σ can be neglected due to the close vicinity φ 18 of the frequency bands. However, it plays a major role when C-Band measurements are included. Figure 2 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 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 2 2 α + βq + γq 12 13 b =−c = . (10) 2 2 q − q 12 13 E1 code measurements have not been taken into account due to the increased noise level but might be included with a low weight. The combination discrimination—measured by the ratio of half the wavelength and the noise level λ/(2N )—is pro- 01 2 3 4 5 posed as a cost function to select linear combinations due to Geometric weight τ = α + β + γ of carrier phase combination its independence of the geometry. It is shown for multiple 9.768 m 1.221 m ionosphere-free code-carrier combinations in Figure 3.The 4.884 m 1.085 m strong dependency on the phase weighting τ suggests an op- 3.256 m 0.976 m timization with respect to this parameter. Note that the leg- 2.442 m 0.888 m end refers to the elementary wavelengths which have to be 1.953 m 0.814 m scaled by τ . 1.628 m 0.751 m The computation of the optimum τ takes again only E5a/ 1.395 m 0.697 m E5b code measurements into account as the benefit of the Figure 3: Optimal weighting of the phase combination part of E1 code measurement is negligible (a = 0). The notation is ionosphere-free code-carrier combinations with i = 1, k =−j − 1, simplified by and j ∈{−12,... ,1}. λ = λ · τ , E5a/E5b code weights are determined from the ionosphere- 2 2 A = κ· τ − bq − cq , I 12 13 free and geometry-preserving conditions as (11) 2 2 2 2 2 2 2 2 2 N = σ · (α + β q + γ q )+ σ · (b + c ) m 12 13 b = 1 − c − τ , φ ρ 2 2 2 2 2 2 2 κ + q −q = σ · η τ + σ · (b + c ), 12 12 (12) φ ρ c = · τ + = w · τ + w . 1 2 2 2 2 2 q − q q − q 13 12 13 12 with λ, κ,and η implicitly given by (5), (7), and (9). The w w 1 2 Noise level of 3F code-carrier linear combination (m) Discriminator of ambiguity resolution λ/2N m σ = 1 mm (E1, E5a, E5b), σ = 5 cm (E5a, E5b) φ ρ 4 International Journal of Navigation and Observation Table 2: Properties and weighting coefficients of ionosphere-free 3 E1-E5b-E5a code-carrier combinations. 2.5 ij k b c λ (m) λ (m) N (m) R 1 −12 11 0.327 0.344 9.768 3.217 0.28 5.81 1 −11 10 0.166 0.175 4.884 3.216 0.25 6.38 2 1 −10 9 0.006 0.007 3.256 3.214 0.23 7.05 1 −98 −0.154 −0.162 2.442 3.213 0.20 7.86 1.5 1 −87 −0.314 −0.330 1.954 3.212 0.18 8.84 Linear increase of Negligible noise 1 −76 −0.474 −0.499 1.628 3.211 0.16 10.02 noise level with τ amplification due 1 −65 −0.634 −0.667 1.396 3.210 0.14 11.42 to increase in τ 1 −54 −0.793 −0.835 1.221 3.209 0.12 12.99 0.5 Pure phase combination 1 −43 −0.953 −1.003 1.085 3.208 0.11 14.54 [0, 1,−1], λ = 9.768 m 1 −32 −1.112 −1.171 0.977 3.207 0.10 15.65 1 −21 −1.272 −1.339 0.888 3.206 0.10 15.85 −25 −20 −15 −10 −50 5 10 Ionospheric reduction (dB) 1 −10 −1.431 −1.506 0.814 3.205 0.11 15.04 10 −1 −1.590 −1.674 0.751 3.204 0.12 13.59 τ = 1 τ = 5 τ = 10 The combination discrimination becomes from (5), (11), Figure 4: Benefit of adaptive code and phase weighting for linear and (12): combinations with λ = τ 9.768 m. R(τ ) λ(τ )/2 ionospheric delay is approximately 10 times lower than in the N (τ ) E1 band. The small wavelength of 5.9691 cm··· 5.9839 cm complicates direct ambiguity resolution but results in an λ · τ = . approximately 3.2 times lower standard deviation of phase 2 2 2 2 2 noise. Moreover, the C-Band offers additional degrees of 2 σ · η τ + σ · w τ +w + 1 − τ − w τ − w 1 2 1 2 φ ρ freedom for the design of linear combinations. (13) Setting the derivative to zero yields the optimum weighting 3.1. Reduced noise ionosphere-free code-only combinations 1 − 2w +2w τ = , (14) opt 1+ w − w − 2w w The design of three frequency code-only combinations that 1 2 1 2 preserve geometry and eliminate ionospheric errors is char- which is independent of both σ and σ . Table 2 contains the ρ φ acterized by one degree of freedom used for noise minimiza- weighting coefficients and characteristics of the code-carrier tion. The weighting coefficients are derived from the geome- combinations shown in Figure 3. try preserving and ionosphere-free constraints in (1), (7)as Figure 4 shows the benefit of adaptive code and phase weighting for the code-carrier linear combination with i = 0 a = 1 − b − c, (E1), j = 1(E5b),and k =−1 (E5a). Obviously, the wavelength increases linearly with τ and the ionosphere 1 q − 1 (15) b =− + − · c = v + v · c. 1 2 can be eliminated with any τ . The noise amplification de- 2 2 q − 1 q − 1 12 12 pends on the level of ionospheric reduction: a linear in- v v 1 2 crease can be observed near the pure phase combination, while the increase becomes negligible for the ionosphere-free 2 2 2 2 2 2 2 Minimization of N = a σ + b σ + c σ yields m ρ ρ ρ 1 2 3 combination. Thus, the combination discrimination of the ionosphere-free code-carrier combination is increased by al- 2 2 (1 − v + v − v v ) · σ − v v · σ 1 2 1 2 1 2 ρ ρ 1 2 most the same factor as τ is risen. c = . (16) 2 2 2 2 2 (1 + 2v + v ) · σ + v · σ + σ 2 ρ 2 ρ ρ 1 2 3 3. C-BAND AIDED CODE-CARRIER Ionosphere-free code-only combinations with more than LINEAR COMBINATIONS three frequencies are obtained by a multidimensional deriva- The 20 MHz wide C-Band (5010··· 5030 MHz ={489.736 tive. Table 3 shows that the pure L-band E1-E5b-E5a combi- ··· 491.691}· 10.23 MHz) has been reserved for Galileo. nation is characterized by a noise level of 44.41 cm. If the E5 The higher frequency range has a multitude of advantages signal is received with full bandwidth, the CRB is reduced to and drawbacks: an additional free space loss of 10 dB occurs 1 cm but the number of degrees of freedom is reduced by one which has to be compensated by a larger transmit power. The so that the noise level of the E1-E5 combinations are lightly Noise level of 3F code-carrier linear combination (m) σ = 1 mm (E1, E5a, E5b), σ = 5 cm (E5a, E5b) φ ρ P. Henkel and C. Gunther ¨ 5 Table 3: Ionosphere-free code-only combinations with minimum 10 noise σ (E1) =σ (C1) = ··· = σ (C4) = 20 cm and σ (E5a, E5b) = ρ ρ ρ ρ 5cm. E1 E5b E5a C1 C2 C3 C4 (cm) 2.090 1.500 −2.590 0000 44.41 0.387 0.255 −0.506 0.863 0 0 0 19.14 0.213 0.128 −0.292 0.476 0.476 0 0 14.21 0.147 0.079 −0.211 0.328 0.328 0.329 0 11.80 0.112 0.054 −0.168 0.251 0.251 0.251 0.251 10.31 Table 4: Ionosphere-free code-only combinations with minimum noise σ (E1) = σ (C1) = ··· = σ (C4) = 20 cm and σ (E5) = −1 ρ ρ ρ ρ −2 −1 0 1cm. 10 10 10 Noise level of linear combination (m) E1 E5 C1 C2 C3 C4 (cm) Joint L-/C-band combination Pure L-band combination 2.338 −1.338 0000 46.78 Combination of L-band code and C-band phase measurements 0.398 −0.278 0.879 0 0 0 19.31 0.217 −0.179 0.481 0.481 0 0 14.27 Figure 5: Comparison of joint L-/C-Band linear combinations for σ (E1) = 20 cm, σ (E5) = 1cm and σ = λ /λ · σ with σ = 0.150 −0.142 0.331 0.331 0.331 0 11.84 ρ ρ φ,i i 1 φ0 φ0 1mm. 0.114 −0.122 0.252 0.252 0.252 0.252 10.34 Table 5: Ionosphere-free code-carrier widelane combinations with pure L-band combination benefits from a noise level of only σ (E1) = σ (C1) = ··· = σ (C4) = 20 cm and σ (E5) = 1cm. ρ ρ ρ ρ 3.92 cm which simplifies the resolution of the 3.215 m inte- ger ambiguities. In contrast to the code-only combinations, ij k l m n a b λ N the use of the full-bandwidth E5 signal is advantageous com- E1 E5 C1 C2 C3 C4 E1 E5 (m) (cm) pared to separate E5a and E5b measurements. The C-band 1 −10000 −4.4e−3 − 3.11 3.21 3.92 offers no benefit for these wavelengths. In the last row of 1 −1001 − 1 − 4.7e−3 − 3.28 3.39 4.84 Table 5, a linear combination with a pure L-band code and pure C-band phase part is described. The combination dis- 1 −101 −10 − 4.7e−3 − 3.28 3.39 4.84 crimination equals 67.25 but the noise level is also increased 1 −1010 − 1 − 5.0e−3 − 3.46 3.59 5.12 to 15.39 cm. 1 −11 −100 − 4.7e−3 − 3.28 3.39 4.84 Figure 5 shows the tradeoff between wavelength and 1 −110 −10 −5.0e−3 − 3.46 3.59 5.12 noise level for joint L-/C-band ionosphere-free linear code- 1 −1100 − 1 −5.3e−3 − 3.67 3.81 5.43 carrier combinations with {i, j}∈ [−5, +5] and {k, l, m, 00 −1001 −8.5e−5 − 6.0e−2 20.70 15.39 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. increased (Table 4). These combinations will play a role in conjunction with code-carrier combinations. 3.3. Joint L-/C-band narrowlane combinations The C-band is split into 4 bands of 5 MHz bandwidth centered at {490, 490.5, 491, 491.5}· 10.23 MHz and allows a There exists a large variety of joint code-carrier narrowlane significant reduction of the noise level. Note that the noise combinations where C-band measurements help to reduce of any contributing elementary combination is reduced by a the noise substantially. Figure 6 shows the tradeoff be- weighting coefficient smaller than one. tween wavelength and noise level for {i, j}∈ [−5, +5] and {k, l, m, n}∈ [−2, +2]. For λ = 5.7 cm, the consideration of C-band measurements reduces the noise level by a factor of 3.2. Joint L-/C-band widelane combinations 5 compared to a pure L-band combination (Table 6). Code-carrier linear combinations can also include both L- band and C-band measurements. Therefore, (1)–(7)are ex- 3.4. Reliability of ambiguity resolution tended to include the additional measurements. The weight- ing coefficients {α, β, γ,...} and {a, b} are computed such The integer ambiguity resolution is based on the linear com- that the discriminator output of (13) is maximized for a bination of four different variable types: double-difference given set of integer coefficients {i, j , k,...}. measurements for eliminating clock errors and satellite/re- Table 5 contains ionosphere-free joint L-/C-band code- ceiver biases; multifrequency combinations for suppressing carrier widelane combinations (λ> max λ ). The E1-E5 the ionosphere; code and carrier phase measurements for i i Wavelength of linear combination (m) 6 International Journal of Navigation and Observation −1 −5 −10 −15 −2 −20 0 5 10 15 20 −4 −3 −2 10 10 10 Time (h) Noise level of linear combination (m) E1/E5 λ = 3.215 m + E1/E5 code-only combination Joint L-/C-band combination E1/E5 λ = 3.215 m + E1/E5/C1 code-only combination Pure L-band combination E1/E5 λ = 3.215 m + E1/E5/C1/C2 code-only combination Combination of L-band code and C-band phase measurements E1/E5 λ = 3.215 m + E1/E5/C1/C2/C3 code-only combination Figure 7: Reliability of λ = 3.215 m integer ambiguity resolution: Figure 6: Comparison of joint L-/C-Band linear combinations for impact of C-band measurements on the probability of wrong fixing σ (E1) = 20 cm, σ (E5) = 1cm and σ = λ /λ · σ with σ = ρ ρ φ,i i 1 φ0 φ0 of the most critical ambiguity. 1mm. Table 6: Ionosphere-free code-carrier narrowlane combinations and the DD geometry matrix G, the baseline δx and the in- with σ (E1) = σ (C1) = ··· = σ (C4) = 20 cm and σ (E5) = 1cm. ρ ρ ρ ρ teger ambiguities N . The double-differenced troposphere is assumedtobenegligibleorknown apriori(e.g.,fromanac- i E1 1 0 −15 curate continued fraction model). j E5 −10 1 − 3 Note that the troposphere has the same impact on all k C1 0 0 1 0 geometry-preserving combinations and does not affect the l C2 0 0 0 0 optimization of the mixed code-carrier combinations. The m C3 0 0 0 0 noise vector is Gaussian distributed, that is, n C4 1 1 0 0 ε∼N (0, Σ)with Σ = Σ ⊗ Σ , (19) LC DD a E1 − 2.04e−6 7.60e−5 1.58e−4 2.55e−4 b E5 − 1.43e−3 5.31e−2 0.110 0.178 where ⊗ denotes the Kronecker product. Σ models the lin- LC λ (cm) 5.55 5.65 5.76 5.73 ear combination induced correlation and Σ includes the DD N (mm) 0.51 0.61 1.22 2.50 correlation due to double difference measurements from N R 54.4 46.3 23.6 11.46 visible satellites. The standard deviation of the most critical ambiguity estimate can be written as −1 T −1 reducing the noise level; and finally, L-/C-band combinations σ = max Σ (i, i), Σ = X Σ X , (20) max β β i={1···N −1} for noise and discrimination characteristics. Two joint L-/C-band code-carrier ionosphere-free com- and the probability of wrong fixing follows as binations are chosen for real-time (single epoch) ambiguity resolution. The λ = 3.215 m, N = 3.92 cm combination of +0.5 1 2 2 c −x /2σ max P = 1 − e dx. (21) Table 5 and one further combination of Table 4. The double −0.5 2πσ max difference (DD) ionosphere-free combinations are modeled as 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. y G λ · 1 ρφ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ = δx + N + ε = Xβ + ε, (17) Figure 7 shows the benefit of C-band measurements for y G 0 integer ambiguity fixing. If the E1-E5 pure L-band combina- tion is used as second combination in (17), the failure rate with varies between 0.01 and 0.07 due to its poor noise charac- ⎡ ⎤ ⎡ ⎤ teristics. The use of two additional C-band measurements λ · 1 G N −5 reduces the maximum probability of wrong fixing to 10 . ⎣ ⎦ ⎣ ⎦ X = , β = (18) −11 0 G δx For three C-band frequencies, the failure rate is at most 10 Wavelength of linear combination (m) Probability of wrong fixing of most critical ambiguity P. Henkel and C. Gunther ¨ 7 which corresponds to a gain of 9 to 17 orders of magnitude compared to the pure L-band combination. The reliability of ambiguity resolution can be further −5 improved by using the LAMBDA method of Teunissen [7]. The float ambiguity estimates are decorrelated by an integer transformation Z and the ambiguity covariance matrix is −10 written as T T Σ = Z Σ Z = LDL , (22) N N −15 with the decomposition into a lower triangular matrix L and a diagonal matrix D. The probability of wrong fixing of the −20 sequential bootstrapping estimator is given by Teunissen [14] 0 5 10 15 20 as Time (h) N −1 +0.5 1 E1/E5 λ = 3.215 m + E1/E5 code-only combination 2 2 −x /2σ (i) P = 1 − e dx, (23) E1/E5 λ = 3.215 m + E1/E5/C1 code-only combination −0.5 2πσ (i) i=1 E1/E5 λ = 3.215 m + E1/E5/C1/C2 code-only combination with σ (i) = D(i, i). It represents a lower bound for the c Figure 8: Reliability of λ = 3.215 m integer ambiguity resolution: success rate of the integer least-square estimator and is de- impact of C-band measurements on the probability of wrong fixing based on sequential fixing with the integer decorrelation transfor- picted in Figure 8. Obviously, the use of joint L-/C-band lin- mation. ear combinations reduces the probability of wrong fixing by severalordersofmagnitudecomparedtopureL-bandcom- 0.16 binations. 0.14 3.5. Accuracy of baseline estimation 0.12 After integer ambiguity fixing, the baseline is re-estimated from (17). The covariance matrix of the baseline estimate in 0.1 local coordinates is given by −1 T −1 T Σ = R G Σ G R (24) 0.08 δx L with the rotation matrix R . Figure 9 shows the achievable 0.06 horizontal and vertical accuracies for the two optimized joint L-/C-band combinations. 0.04 The pure L-band combinations in the first row of Tables 4 and 5 have been again selected as reference scenario. It can 0.02 0 5 10 15 20 be observed that the use of joint L-/C-band linear combi- Time (h) nations enables a slight improvement in position estimates compared to the significant benefit for ambiguity resolution. Horizontal comp. (joint L/C) Vertical comp. (joint L/C) Horizontal comp. (pure L) 4. JOINT L-/C-BAND CARRIER SMOOTHED CARRIER Vertical comp. (pure L) Ionosphere-free code-carrier linear combinations are charac- Figure 9: Standard deviation of baseline estimation using the λ = terized by a noise level that is one to two orders of magnitude 3.215 m E1-E5 ionosphere-free code-carrier combination and the larger than of the underlying carrier-phase measurements E1-E5-C1··· C4 ionosphere-free code-only combination. (Table 5). Both noise and multipath of the code-carrier com- binations can be reduced by the smoothing filter of Hatch [15] which is shown in Figure 10. The upper input can be an Note that the superposition of ambiguities of the pure ionosphere-free code-carrier combination of arbitrary wave- phase combination is not necessarily an integer number of a length. The lower input is a pure ionosphere-free phase com- commonwavelength. Therespectiveambiguities arenot af- bination that is determined by three conditions: the first en- fected by the low pass filter and do not occur in the smoothed sures that the geometry is preserved, the second eliminates output λ φ due to different signs in the addition to λ φ A A A the ionosphere, and the third minimizes the noise, that is, (Figure 10). Table 7 shows an ionosphere-free E1-E5a-E5b phase α + β + γ = 1, combination that increases the noise level by a factor 2.64. 2 2 However, the low noise level of C-band measurements sug- α + βq + γq = 0, (25) 12 13 gests the use of the second combination with f = 491 · 2 2 2 2 2 2 2 minN = min σ · α + β q + γ q . m φ,0 12 13 10.23 MHz. In this case, the noise level is not only reduced α,β,γ α,β,γ Probability of wrong fixing of all ambiguities Standard deviation of baseline estimate (m) using integer decorrelation transformations 8 International Journal of Navigation and Observation LP filter REFERENCES χ χ λ φ λ φ [1] B. Forssell, M. Martin-Neira, and R. A. Harris, “Carrier phase A A A − + ambiguity resolution in GNSS-2,” in Proceedings of the 10th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GPS ’97), vol. 2, pp. 1727–1736, λ φ B B Kansas City, Mo, USA, September 1997. [2] J. Jung, P. Enge, and B. 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The smoothed solution can either be two types of dual-frequency differential GPS techniques un- used directly or can be used to resolve the narrowlane ambi- der anomalous ionosphere conditions,” in Proceedings of guities. The variance is basically the same in both cases. The the National Technical Meeting of the Institute of Navigation resolved ambiguities, however, provide instantaneous inde- (NTM ’06), vol. 2, pp. 735–747, Monterey, Calif, USA, January pendent solutions. 2006. 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