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.

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 Section 3, 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 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 8.9 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. 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 integer multiple of a common wavelength 𝜆, that is 𝛼𝜆1𝑁1+𝛽𝜆2𝑁2+𝛾𝜆3𝑁3!=𝜆𝑁,(2)which can be split into three sufficient conditions𝑖=𝛼𝜆1𝜆,𝑗=𝛽𝜆2𝜆,𝑘=𝛾𝜆3𝜆,(3)with denoting the space of integers. These integer constraints are rewritten to obtain the weighting coefficients𝛼=𝑖𝜆𝜆1,𝛽=𝑗𝜆𝜆2,𝛾=𝑘𝜆𝜆3.(4)

Mixed code-carrier combinations weight the phase part by 𝜏 and the code part by 1𝜏. 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 combination𝜆=𝜏𝑖/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 as𝜏>𝑖𝑞13+𝑗𝑞23+𝑘>0with𝑞𝑚𝑛=𝜆𝑛𝜆𝑚.(6)The linear combination scales the ionospheric error by𝐴𝐼=𝛼+𝛽𝑞212+𝛾𝑞213𝑎𝑏𝑞212𝑐𝑞213.(7)The thermal noise of the elementary carrier phase measurements is assumed Gaussian with the standard deviation given by Kaplan and Hegarty [13]𝜎𝜙𝑖=𝜆𝑖2𝜋𝐵𝐿𝐶/𝑁01+12𝑇𝐶/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 written as𝑁𝑚=(𝛼2+𝛽2𝑞212+𝛾2𝑞213)𝜎2𝜙0+𝑎2𝜎2𝜌1+𝑏2𝜎2𝜌2+𝑐2𝜎2𝜌3(9)with 𝜎2𝜌1,,𝜎2𝜌3 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.

For E1, E5a, E5b, E6 phase measurements, the wavelength scaling of 𝜎𝜙𝑖 can be neglected due to the close vicinity 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 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 case𝑏=𝑐=𝛼+𝛽𝑞212+𝛾𝑞213𝑞212𝑞213.(10)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/(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.

The computation of the optimum 𝜏 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 by𝜆=̃𝜆𝜏,𝐴𝐼=̃𝜅𝜏𝑏𝑞212𝑐𝑞213,𝑁𝑚=𝜎2𝜙(𝛼2+𝛽2𝑞212+𝛾2𝑞213)+𝜎2𝜌(𝑏2+𝑐2)=𝜎2𝜙̃𝜂2𝜏2+𝜎2𝜌(𝑏2+𝑐2),(11)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 as𝑏=1𝑐𝜏,𝑐=̃𝜅+𝑞212𝑞213𝑞212𝑤1𝜏+𝑞212𝑞213𝑞212𝑤2=𝑤1𝜏+𝑤2.(12)

The combination discrimination becomes from (5), (11), and (12): 𝑅(𝜏)=𝜆(𝜏)/2𝑁𝑚(𝜏)=̃𝜆𝜏2𝜎2𝜙̃𝜂2𝜏2+𝜎2𝜌𝑤1𝜏+𝑤22+1𝜏𝑤1𝜏𝑤22.(13)Setting the derivative to zero yields the optimum weighting𝜏opt=12𝑤2+2𝑤221+𝑤1𝑤22𝑤1𝑤2,(14)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.

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 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.

3. C-Band Aided Code-Carrier Linear Combinations

The 20 MHz wide C-Band (50105030MHz={489.736491.691}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 5.9691cm5.9839cm 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) as𝑎=1𝑏𝑐,𝑏=1𝑞2121𝑣1+𝑞2131𝑞2121𝑣2𝑐=𝑣1+𝑣2𝑐.(15)Minimization of 𝑁2𝑚=𝑎2𝜎2𝜌1+𝑏2𝜎2𝜌2+𝑐2𝜎2𝜌3 yields𝑐=(1𝑣1+𝑣2𝑣1𝑣2)𝜎2𝜌1𝑣1𝑣2𝜎2𝜌2(1+2𝑣2+𝑣22)𝜎2𝜌1+𝑣22𝜎2𝜌2+𝜎2𝜌3.(16)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.

The C-band is split into 𝑁𝑚 bands of  MHz bandwidth centered at 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 and are computed such that the discriminator output of (13) is maximized for a given set of integer coefficients .

Table 5 contains ionosphere-free joint L-/C-band code-carrier widelane combinations 4 The E1-E5 pure L-band combination benefits from a noise level of only 5 which simplifies the resolution of the {490,490.5,491,491.5}10.23MHz 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 {𝛼,𝛽,𝛾,} but the noise level is also increased to {𝑎,𝑏}.

Figure 5 shows the tradeoff between wavelength and noise level for joint L-/C-band ionosphere-free linear code-carrier combinations with and . The E1-E5 combination is of special interest but the maximum combination discrimination is obtained for a joint L-/C-band combination.

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. Figure 6 shows the tradeoff between wavelength and noise level for and . For , the consideration of C-band measurements reduces the noise level by a factor of {𝑖,𝑗}[5,+5] compared to a pure L-band combination (Table 6).

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. The , 𝑏 combination of Table 5 and one further combination of Table 4. The double difference (DD) ionosphere-free combinations are modeled aswithand the DD geometry matrix , the baseline 𝜆 and the integer ambiguities 𝑁𝑚. 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,𝑅where 𝜎𝜌(E1)=20cm denotes the Kronecker product. 𝜎𝜌(E5)=1cm models the linear combination induced correlation and 𝜎𝜙,𝑖=𝜆𝑖/𝜆1𝜎𝜙0 includes the correlation due to double difference measurements from 𝜎𝜙0=1mm. visible satellites. The standard deviation of the most critical ambiguity estimate can be written as𝜆=3.215mand the probability of wrong fixing follows as𝑁𝑚=3.92cmIn the following analysis, the location of the reference station is at 𝑦𝜌𝜙𝑦𝜌=𝐺𝐺𝛿𝑥+𝜆10𝑁+𝜀=𝑋𝛽+𝜀,(17) N, 𝑋=𝜆1𝐺0𝐺,𝛽=𝑁𝛿𝑥(18) E with a baseline length of 𝐺 km.

Figure 7 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 𝛿𝑥 and 𝑁 due to its poor noise characteristics. The use of two additional C-band measurements reduces the maximum probability of wrong fixing to 𝜀𝒩(0,Σ)withΣ=ΣLCΣDD,(19). For three C-band frequencies, the failure rate is at most which corresponds to a gain of ΣLC to ΣDD orders of magnitude compared to the pure L-band combination.

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 𝜎max=max𝑖=1𝑁𝑠1Σ̂𝛽(𝑖,𝑖),Σ̂𝛽=𝑋𝑇Σ1𝑋1,(20) and the ambiguity covariance matrix is written as𝑃𝑐𝑤=1+0.50.512𝜋𝜎2max𝑒𝑥2/2𝜎2max𝑑𝑥.(21) with the decomposition into a lower triangular matrix 48.1507 and a diagonal matrix 11.5690. The probability of wrong fixing of the sequential bootstrapping estimator is given by Teunissen [14] as10with 0.01. 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.

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 by105with the rotation matrix 1011. Figure 9 shows the achievable horizontal and vertical accuracies for the two optimized joint L-/C-band combinations.

The pure L-band combinations in the first row of Tables 4 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 (Table 5). 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, 𝐷

Note that the superposition of ambiguities of the pure phase combination is not necessarily an integer number of a common wavelength. The respective ambiguities are not affected by the low pass filter and do not occur in the smoothed output 𝑃𝑤=1𝑁𝑠1𝑖=1+0.50.512𝜋𝜎2𝑐(𝑖)𝑒𝑥2/2𝜎2𝑐(𝑖)𝑑𝑥,(23) due to different signs in the addition to 𝜎𝑐(𝑖)=𝐷(𝑖,𝑖) (Figure 10).

Table 7 shows an ionosphere-free E1-E5a-E5b phase combination that increases the noise level by a factor 𝜆=3.215m. However, the low noise level of C-band measurements suggests the use of the second combination with Σ𝛿̂𝑥=𝑅𝐿𝐺𝑇Σ1𝐺1𝑅𝑇𝐿(24). In this case, the noise level is not only reduced by smoothing but also by the coefficients of the pure phase combination.

The variance of the smoothed combination is given by𝛽with the low-pass filtered noise (e.g., Konno et al. [16])𝛾and the smoothing time 𝑁𝑚. Setting (27) into (26), and using the definition of a geometric series yields E5bFor long smoothing times, only the low noise of the joint L/C pure carrier-phase combination E5a 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 of  m with a noise level of 2.64𝜎𝜙0 cm.

The combination of L- and C-band measurements reduces the noise level of ionosphere-free code-only combinations by a factor E5b 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.