Technical Article • May/June 2012
Demystifying GLONASS InterFrequency Carrier Phase BiasesDigital versus AnalogThis article provides new insights into the origin of GLONASS interfrequency carrier phase biases in GNSS receivers. The authors identify the origin of wellknown decimeterlevel linear biases affecting GLONASS carrier phase measurements as a function of how measurements are generated in the digital signal processing (DSP) section of a GNSS receiver. Contrary to the common assumption that analoginduced biases are dominant, these DSPinduced biases are, by far, the major cause of GLONASS interfrequency biases and can be compensated for. Share via: Slashdot Technorati Twitter Facebook GLONASS currently uses a frequency division multiple access (FDMA) technique to distinguish the signals coming from different satellites in the Russian GNSS constellation. The GLONASS L1 and L2 bands are divided into 14 subbands, and each satellite transmits in one of these. The subbands are identified by frequency numbers k, from 7 to 6. The GLONASS L1 and L2 carrier frequencies, in hertz, at a frequency number k are defined by: Use of the FDMA technique has long been known to cause significant interfrequency biases in carrier phase measurements of GLONASS satellites. As already postulated in early GLONASS developments, these biases can be well modeled as a linear function of the frequency number k, and are very similar on L1 and L2 when expressed in units of length. (See, for example, the article by A. Povalyaev listed in the Additional Resources section near the end of this article.) As reflected by the articles by L. Wanninger cited in Additional Resources, these biases have also been shown to tend to be the same for all receivers of a given brand, but significantly different across brands. The fact that the biases depend on the GLONASS frequency number and are not the same between brands significantly complicates the realtime kinematic (RTK) ambiguity resolution process in heterogeneous base/rover combinations. Although the general properties of the GLONASS interfrequency carrier phase biases (linearity with respect to k, homogeneity within a given brand, and equality between the L1 and L2 bands) are well known and documented, the origin of these biases in the receiver signal processing chain remains largely unexplained. The widely accepted hypothesis is that the biases originate in the analog hardware and thus are difficult to tackle without specialized laboratory equipment. This article provides new insights into this question. Our analysis demonstrates that the major cause of interfrequency carrier phase biases is not to be found in the analog RF part of the receiver, but rather in the way the measurements are generated in the digital part of the receiver. This discovery opens new perspectives and new hope for the calibration of the biases between receivers. Our discussion here will show that the biases can be compensated to millimeterlevel in an absolute sense. InterFrequency Carrier Phase Biases: A Definition To focus on the interfrequency phase biases, we shall purposely ignore in our formulas all other error sources such as atmospheric delays, multipath, or tracking noise. We also ignore interfrequency code biases, which shall not be discussed in this article. Under this idealized assumption, the code and carrier phase measurement generated in a GNSS receiver differs only by an integer number of wavelengths: Equation 2 Where φ^{k}_{Li} is the phase measurement, in units of cycles, for the frequency band Li (i=1 or 2) for a GLONASS satellite transmitting in a frequency channel k. C^{k} is the code measurement, N^{k}_{Li} is an integer phase ambiguity, and λ^{k}_{Li} is the carrier wavelength, defined as λ^{k}_{Li} = c/ƒ^{k}_{Li} with c being the speed of light and ƒ^{k}_{Li }defined by Equation (1). In this article, we will use the symbol “φ” for phase measurements expressed in cycles and “Ф” for phase measurements expressed in meters. To convert φ to Ф, it is sufficient to multiply it by the carrier wavelength λ^{k}_{Li}. Equation (2) represents an ideal nonbiased case. If biases affect carrier phase measurements, Equation (2) must be rewritten as: Equation 3 where Δφ^{k}_{Li} is the carrier phase bias term, in cycles. This term is dependent on the frequency number, hence the superscript k. The GLONASS interfrequency carrier phase bias is commonly defined as the difference of the bias at frequency number k with respect to the bias at frequency number 0. In this article, we will denote the interfrequency carrier phase bias as Δφ^{k,0}_{Li} when expressed in units of cycles, and ΔФ^{k,0}_{Li} when expressed in units of length: Equation 4 and Equation 5 The approximation in Equation (5) is accurate to a submillimeter level and hence is valid in all practical cases. Figure 1 shows the L1 and L2 GLONASS interfrequency carrier phase bias ΔФ^{k,0}_{Li} and ΔФ^{k,0}_{L2} as a function of the frequency number k for a line of a dualfrequency, multiGNSS receivers that we used in our research discussed here. The biases shown in Figure 1 are computed from the results in the 2012 article by L. Wanninger (Additional Resources) and, as stated by that author, are defined relative to a set of receivers taken as reference. Figure 1 illustrates that the interfrequency biases are linear functions of the frequency number, and, when expressed in units of length, are equal for L1 and L2. According to Wanninger’s research, GLONASS interfrequency phase biases for a given brand can be characterized by a single parameter: the slope of their linear dependence upon the frequency number expressed in centimeters per frequency number. In the case of the dualfrequency, multiGNSS receiver that we used, the biases of which are shown in Figure 1, the slope is 4.9 centimeters per frequency number and is the same on L1 and L2. The value of the slope for other receiver manufacturers can be found in L. Wanninger (2012). In the next section it will be shown that interfrequency carrier phase biases generated in GNSS receivers consist of two components: biases caused by analog radiofrequency hardware and biases caused by the digital signal processing (DSP): Equation 6 Although analog hardware is commonly assumed to be a main source of biases, we will show that this is not the case: in reality, the digital signal processing is by far the dominant source of biases. InterFrequency Biases Generated by Analog Filters The phase response of an analog filter characterizes the phase shift introduced by the filter as a function of the carrier frequency. The phase response for a particular receiver can be computed a priori if the filter design is known, or it can be accurately measured in an absolute sense using specialized laboratory equipment such as a network analyzer. As an example, Figure 2 shows the phase response across the GLONASS L2 band for the L2 analog filter of the dualfrequency, multiGNSS receiver, the bias of which is shown in Figure 1. In this figure, the effect of the frequencyindependent delay introduced by the filter has been removed. From this figure, we can see that the phase shift variation caused by that RF filter is very small (submillimeter level) and cannot account for the decimeterlevel biases shown in Figure 1. More generally, the observed properties of the interfrequency carrier phase biases clearly do not correspond to what would be caused by analog filters because of the following:
InterFrequency Biases Generated in the DSP Chain It is not commonly known, however, that the fundamental assumption that code and carrier phase measurements share the same clock bias is generally incorrect. There exist at least two mechanisms by which the measurement generation algorithm in the receiver’s digital signal processing (DSP) can induce a difference in code and carrier clock bias. First, DSP techniques commonly adjust code measurements by some constant offset, for instance, to compensate for group delay effects in the reception chain, in order to align the time at which the pulsepersecond (PPS) strobe is generated. This adjustment is done in the receiver firmware by adding a constant term c · δt_{PPS} to all raw code measurements. This adjustment, being constant for all satellites, is seen as a code clock bias by the positioning algorithm. If it is applied to code measurement only, it obviously introduces a difference between code and phase clock biases. The second cause of codephase bias is found in the correlation process that takes place in the digital hardware. Signal tracking involves maximizing the correlation between the incoming signal and local signal replicas generated by code and carrier generators implemented in the receiver’s digital circuits. This process is illustrated in Figure 3. A delay δt_{C} exists from the code generator to the correlator, and another delay δt_{φ} from the carrier generator to the correlator. These delays are fixed for a given receiver architecture and do not vary with temperature. Typically, they are multiples of the sampling interval used by a particular receiver design. Depending on the chip architecture, the delays δt_{C} and δt_{φ} are not necessarily equal. This is important, because any difference between these delays is directly reflected in a bias between the code and carrier phase measurements. With the delays δt_{PPS}, δt_{C}, and δt_{φ}, Equation (2) does not hold any more and must be rewritten as follows: Equation 7 The third term in the righthand side of Equation (7) is the DSPinduced carrier phase bias defined in Equation (3): Equation 8 where δt_{CP} = δt_{C}  δt_{φ}  δt_{PPS} is the aggregate codephase bias (CPB) induced by the digital processing, in units of time. Using the values of ƒ^{k}_{Li} defined in the equations in (1), we can rewrite the DSPinduced L1 and L2 phase biases as follows: Equation 9 In these formulas, δt_{CP} must be expressed in seconds. The interfrequency biases, as defined in Equation (4), now read: Equation 10 In units of length, the biases become [per Equation (5)]: Equation 11 This last result shows that the DSPinduced interfrequency biases, when expressed in meters, are linear functions of k and are equal on L1 and L2. These are exactly the properties we observe, and which cannot be explained by analog hardware biases. The slope of the linear interfrequency biases as given by (11) is proportional to the DSPinduced codephase bias δt_{CP}. The value of δt_{CP} depends upon the receiver brand and typically ranges from zero to a few hundreds of nanoseconds. The resulting interfrequency carrier phase biases, as computed from (11), may amount to a few centimeters per frequency number. In the case of the dualfrequency, multiGNSS receiver used in our research, the bias of which was shown in Figure 1, the DSPinduced code phase bias is known to us: its value is δt_{CP}=475 nanoseconds. Equation (11) shows that this causes an interfrequency carrier phase bias of 475 · 10^{9} · 105264 · k = 0.05 meter per frequency number on L1 and L2. This closely matches the bias of 4.9 centimeters per frequency number reported for that receiver by L. Wanninger in his 2012 journal article. Compensation of DSPInduced Code Phase Biases The term δt_{PPS} is a firmware parameter that can directly be retrieved from the source code of the DSP software. The codephase correlator delay, δt_{C}  δt_{φ}, can be retrieved from the architecture of the baseband digital chip. Typically, but not necessarily, δt_{C}  δt_{φ} is constrained to a multiple of the sampling interval. GNSS receiver manufacturers know the parameters applicable to their own design. If the codephase bias is not zero for their receivers, they can decide to apply formula (10) or (11) to correct their carrier phase measurements. This concept was presented and discussed during the International GNSS Service (IGS) Workshop on GNSS Biases held in January 2012. One of the recommendations agreed upon at the end of the meeting was for manufacturers to confirm the effectiveness and check the feasibility of such compensation. Having receivers applying the correction by default prior to outputting their carrier phase measurements is not necessarily recommended, as some RTK rover engines rely on a hardcoded table of carrier biases per manufacturer (see, for example, Table 2 in the 2012 article by L. Wanninger). Changing the biases, even if it is to remove them, would introduce a backward incompatibility. Instead, a proposal is circulating to apply the correction only to the new RTCM “multiple signal messages” (MSM) (This is discussed in the article by F. Takac et alia cited in Additional Resources). As the MSM messages are new, they are free of backwardcompatibility constraints. Making the MSM messages free of DSPinduced codephase biases would greatly facilitate the fixing of GLONASS ambiguities in heterogeneous networks. However, some RTCM members expressed concerns that correcting only the MSM messages could introduce an undesirable difference between MSM and legacy RTCM or RINEX. At the time of writing, evaluation of the concept and interoperability testing is ongoing at most highend receiver manufacturers. One could argue that DSPinduced codephase biases can be cancelled by correcting the code measurement instead of correcting the carrier phase measurement. This is indeed true: Adding a constant correction term, c · δt_{CP}, to the code measurements is another way to eliminate codephase biases. However, this is not the preferred approach, because modifying the code measurement has an undesirable effect on the alignment of the PPS timing. We must note that compensating for the DSPinduced biases does not imply that all interfrequency carrier phase biases are removed. Analog hardware–related biases do remain, but these are at the millimeterlevel and do not prevent integer ambiguity resolution in GLONASS RTK algorithms. Conclusion Two causes of large linear interfrequency phase biases have been identified: biases caused by code measurement adjustment in the receiver firmware, and biases caused by differential delays between the signals from the code and carrier generators in the receiver’s digital chip. These DSPinduced biases are, by far, the major cause of GLONASS interfrequency carrier phase biases, contrary to the common assumption that analoginduced biases dominate. DSPinduced biases are not dependent on temperature, they do not vary from unit to unit, and they are stable in time. They can be directly derived from the receiver firmware and digital chip architecture and, hence, can be compensated for in an absolute sense. This means that no tedious empirical interreceiver calibration is required, and the interoperability of GLONASS receivers can be ensured through relatively simple measures taken by each receiver manufacturer individually. Acknowledgment Additional Resources ManufacturersThe dualfrequency, multiGNSS receiver used in the research on which this article is based is the PolaRx3 from Septentrio nv, Leuven, Belgium.Copyright © 2017 Gibbons Media & Research LLC, all rights reserved. 
