Technical Article • May/June 2016
InterSignal Correction Sensitivity AnalysisApertureDependent Delays Induced by Antenna Anisotropy in Modernized GPS DualFrequency NavigationCostconscious civilian GPS receiver designers are faced with decisions as to whether to invest in L1/L2 frontends or L1/L5 front–ends, as well as whether they want to expend the extra bandwidth to use narrow chip tap spacings. If a dualfrequency correction is made with one signal from GPS and the other from a different GNSS system, the designer must consider the signal attributes that will best serve their customers. GPS specification documents ISGPS200H and ISGPS705 provide the new modernized dualfrequency correction algorithm that uses intersignal corrections to align all new GPS signals with the dual L1 and L2 PY ionospherefree reference. This article provides the physics, derivations, and error budgets needed to go beyond the equations as presented in those specification documents. It will provide insights into how L1 and L2 and L1 and L5 dualfrequency alignment actually works when satellite antenna anisotropy is present, and, in particular, will provide receiver designers the additional insights needed for their specific application markets.
Share via: Slashdot Technorati Twitter Facebook Modernized GPS satellites give civil users the ability to achieve dual L1/L2 PY accuracy using dual L1CA/L2C ionospherefree measurements and, with IIF satellites, dual L1/L5 signals. Because broadcast GPS ephemeris data is based on an ionospherefree pseudorange calculated from dual L1PY/L2PY measurements and the civil signals are not all perfectly aligned to it, new broadcast parameters and a new modernized dualfrequency algorithm are needed in order to align new signals with the dual L1/L2 PY signal. New intersignal correction (ISC) broadcast parameters and the modernized dualfrequency algorithm were published in 2004 in the unclassified interface specification documents ISGPS200D and ISGPS705. (Note: Originally, ISGPS200 was an interface control document, ICDGPS200, but changed to be an interface specification ISGPS200 for rev D and beyond.) There are more L1/L2 IIRM and IIF satellites now broadcasting civil navigation (CNAV) messages along with intersignal corrections (MSG 30), but there will be more L1/L5 satellites in the long term when Galileo and other GNSS systems become options. The parameters used in this new dualfrequency algorithm are based on two assumptions: 1) the antenna delays are approximately constant over the main beam, and 2) any electrical changes are slowly varying such that they can be treated as constants during data set upload intervals. However, published research (see references 3, 5, and 6 in the Additional Resources section at the end of this article) has shown delay variation across the main beam due to space vehicle (SV) antenna anisotropy — that is, directionally dependent differences in physical properties of the antenna (defined in ISGPS200F) — and thus needs to be considered for future navigation performance. This article will examine the ISC constancy approximations and examine two techniques for measuring ISC values that will be broadcasted and explain their pros and cons. The first technique measures the ISCs using code and carrier phase information. The second technique rearranges the modernized dualfrequency correction algorithm in terms of the new parameterization of intraband differentials (IBDs). Further, this article will provide methods to convert between ISCs and IBDs. With these explanations, receiver designers will understand a) how the measurement technique used enables constant ICS values to achieve accurate dual L1CA/L2C point positioning consistent with European differential code biases on the 0.1–2 nanosecond level [reference 3.5], b) the physics behind the new ICD term of SV antenna anisotropy, and c) understand some of the tradeoffs such as L1/L2 vs. L1/L5 alignment errors based on ISC accuracy in the new modernized dual frequency correction algorithm. After an introductory overview, the article is organized into the following sections: defining and modeling pseudorange alignment errors; ISCs, ionofree delay centers (IFDCs), and their impact on navigation; alignment and measurement algorithms; measurement data; and conclusions and recommendations. All technical appendices can be found in the Additional Information section at the end of this article.
Overview Due to variations within the SV equipment, each signal has a unique delay τ_{Lix}, where i=1,2,5 for L1, L2, and L5, and x = the corresponding code on that carrier. (Note: each different signal or code, x, such as CA, P, and M on L1, L2C, P, M on L2, L5I&L5Q on L5, and so on, is a PRN spreading code, and we will refer to each of these as either signals or codes. On a GPS satellite, all signals/codes are assigned the same PRN number. Each satellite is assigned an SVN number and the PRN to SVN mapping can change. The sem file format supports identification of the PRN, SVN, and block type, as documented online here. Technically, signals are the carrier with modulated code, but within the literature, signal and code have both been used interchangeably.) ISCs (ISGPS200D, circa 2004) are defined in Equation (4) as delay differentials between all signals relative to L1PY. Each SV has its own unique set of ISCs. For those familiar with the earlier versions of ISGPS200, the legacy scaled group delay parameter T_{GD} can be expressed in terms of ISC_{L2PY} as shown in Equation (5). The original frequency scaling factor γ is now γ_{ij} as shown in Equation (6) where i and j refer to the two frequencies used in the delay differential. Equation (3) reduces to equation (1) by inserting equations (4 through 6) into (3) and for Lix using i=1 x=PY for L1PY and for Ljz using j=2 and z=PY for L2PY. We will use the symbol “&” to represent the ionospherefree operator defined by equation (1). For the purposes of this article, we will focus on L1 & L2 or L1 & L5 pairs because L2 & L5 combinations are not numerically stable. Equation (3) doesn’t preclude ISC_{Lix} parameters that vary with SV boresight angle θ (SV antenna anisotropy). However, the current modernized navigation messages support only constant ISC values. As we will show, antenna anisotropy can challenge this assumption. In addition, ground monitoring stations have difficulty measuring nonconstant ISC values unless they can see the full antenna pattern. We have developed an alternative formulation of Equation (3) that solves these difficulties, expressed as equations (78) on the right side of Figure 1. We begin by factoring equation (3) into a new form by moving T_{GD} into the numerator, setting i=1 and j=2 or 5 for L1 & L2 or L1 & L5, and grouping the T_{GD} term with ISC_{Ljz}. The terms in the square brackets become the IBDs (intraband delays). IBDs as a function of ISCs is defined by equation (8) as: Examining equations (8), (8a), and (8b), for L1 and L2 we notice that the IBDs for j=1,2 are intraband differences between any code on the j^{th} L band and the PY code found within that Lband. Within a single carrier, the receive antenna, ionosphere, and the lineofsight (LOS) delays are equal and will cancel out, making IBDs easier to measure. More importantly, the L1 and L2 IBDs are typically constant across the SV antenna aperture’s main beam when all civil signals are measured with delay lock loops (DLLs) using ICD 200F–specified tap spacings (Appendix B). Even if a ground site making the IBD measurements can only see a portion of an SV’s antenna, it can accurately measure the mainbeam IBDs. When j=5, complications arise. Although we don’t get a physical L5 IBD (8c), it behaves as if there is an effective L5PY reference (8d). The L5 IBD, equation (8c), is free of ionospheric contamination because the (1 − γ_{15}) / 1 − γ_{12}) scaling converts the L1L2 iono error into an L1L5 iono error. However, the L5 IBDs will vary with boresight angle. Equation (8) has one other significant property: by using T_{GD}, all IBD values can be converted into ISC constants compatible with the current CNAV messages on L2 and L5 and broadcast ephemeris assumptions. This allows single frequency users to align their one Lband measurement to the dual L1PY&L2PY ionosphere free pseudorange. The rest of this article, along with its online technical appendices, will rigorously derive and demonstrate the foregoing assertions.
Defining and Modeling Pseudorange Alignment Errors Physical Model. Figure 2 shows the SV, ionosphere, and troposphere that add delays to the measured pseudorange. Because troposphere, SV clock, and receiver antenna delays errors are typically removed by lookup tables or broadcast algorithms, our measurement model, equation (9), simplifies to: Equation (9) contains the desired line of sight pseudorange ρ_{LOS}, the signal specific SV equipment delays τ_{Lix}(θ) which are functions of each L band and each signal, the ionosphere’s total electron content 40.3*TEC/f_{2Li} contribution, and receiver noise n_{Lix}. Equation (9) is used to assess how SVunique equipment delays τ_{Lix}(θ) affect both single and dualfrequency navigation. The speed of light c in equation (9) converts delay in seconds into meters. (Note: We will sometimes show a delay quantity being an explicit function of boresight angle. Other times, it is not shown in order to shorten the equations. In general, all of the delay parameters are a function of SV boresight and SV azimuth angle. In general, there is circular symmetry; so, the azimuth dependence is usually negligible. However, it is prudent to continually check this assumption.) ISGPS200 Model. Paragraph 3.3.1.7 of ISGPS200 defines SV equipment delay as the “delay between the signal radiated output of a specific SV (measured at the antenna phase center) and the output of that SV’s onboard frequency source.” Paragraph 30.3.3.3.1.1.1 introduces SV signal–dependent equipment delays in terms of intersignal corrections, ISC_{Lix}, defined as differential delays relative to the L1PY delay: τ_{L1PY} − τ_{Lix}. Figure 3 is a schematic representation of SV equipment delay as the accumulation of delays from the code generators, modulators, transmitters, tri/quadraplexors, and the SV antenna emanation point. (ISGPS200 refers to this point as the phase center.) If there is SV antenna anisotropy, the SV equipment delay is denoted as a function of boresight angle θ, τ_{Lix}(θ). Red, blue, and green are used to color code L1, L2, and L5 frequency bands. Black lines represent the ISCs for each differential of τ_{L1PY} − τ_{Lix}. Emanation Point Using Phase and Delay Measurements. GPS signals are spreadspectrum codes modulated on to an Lband carrier. When traversing the ionosphere or an electrical network, the group delay of the code/envelope can be different than the delay of the carrier. Code delay is a shift in the signal envelope (i.e., group delay) and is measured in seconds. Phase delay is a shift in the carrier and is measured in units of radians. (Reference 1.5 in Additional Resources provides links to animations of phase and group delay independence.) Pseudorange is affected by code delay. Integrated carrier phase (called delta range in the literature) is affected by phase delay. When the SV equipment delays code and carrier by different amounts, the code and carrier emanation points can be different. The antenna phase center is defined at the origin of the radiusofcurvature best fit to the farfield lines of phase and is the emanation point for the carrier. (In practice, the best fit to the radius of curvature lines of constant phase or delay is not a point but a region. The radius of the region should be much smaller in size than the smallest error of concern in the PPS performance standard. In the current precise positioning system or PPS standard, the miscellaneous SV error terms and/or group delay stability in the signalinspace error budget is 0.6 meter or 0.2 nanosecond, 95 percent; so, an emanation point with an uncertainty radius of 0.2 nanosecond would be a reasonable good emanation tolerance size to start with.) The notion of defining a best fit radiusofcurvature is also needed for lines of farfield constant group delay. When a receiver combines code and carrier ranging information, the individual emanation points of the pseudorange and carrier matter. This leads us to define an antenna delay center (DC) which locates the emanation point of the pseudorange. Although no application required the concept of an antenna delay center at the time the 1979 IEEE antenna standard was drafted, the concept has since been acknowledged (for example, see Additional Resources references 1.2 and 6.1). We will use the technically more precise “delay center” when referring to what ISGPS200 calls “phase center” as applied to codeonly navigation. Appendix B (available online) summarizes the metrics used to measure “group delay.” Not only do we need the concept of delay centers for individual signals, but we also need them for any ionospherefree (IF) pair that we want to use for navigation, hence, ionofree delay center (IFDC). Aligning both singlefrequency pseudoranges as well as ionospherefree pseudoranges to the original L1PY and L2PY ionospherefree pseudorange was the purpose of the introduction of ISCs into ISGPS200. Mathematically, ISCs allow alignment errors to be expressed in terms of differentials. In practice, we will show that a) IBDs are easily measured differential delays, b) pairs of IBDs physically represent the alignment errors between the blended emanation point of any new pair of signals relative to the blended emanation point of L2PY with L1PY, using the weighting coefficients in Equation (1).
ISCs, Delay Variations, IBDs, IFDCs, and Their Effects on Navigation Understanding Measured ISCs, Delay Variations, and IBDs. Figure 4 shows sample ISC data measured with a 10foot parabolic dish antenna. This data used differential code, carrier, and an external TGD calibration to calculate the absolute L1PY minus L2PY delay (Additional Resources reference 3.2, and Appendix G, online). In Figure 4, we plot the rising satellite boresight angle, –θ, and the setting satellite boresight angle, +θ. The small gap in θ is due to the fact the satellite did not pass directly overhead. The L1 ISC (ISC_{L1CA} = τ_{L1PY} − τ_{L1CA}) is almost constant until the combination of SV beam edge effects (~12.5 degrees and beyond) and observation site multipath at dish elevation angles below five degrees contaminates the measurement. Note that the L2 ISCs (ISC_{L2C} = τ_{L1PY} − τ_{L2C} and ISC_{L2PY} = τ_{L1PY} − τ_{L2PY}) are not constant across the beam. This will lead to this article’s position that IBDs are more accurately represented as constants. Using widelane code and carrier techniques and the external T_{GD} calibration (Appendix G), Figure 5 shows how each of the individual τ(θ)_{Lix} varies with boresight angle θ. In all figures, each Lband frequency is depicted in a different color: L1 in red, L2 in blue, L5 in green (if present). When ionospherefree pseudoranges created from L1 & L2 measurements are discussed, purple will be used. When ionospherefree pseudoranges are formed from L1 & L5, cyan will be used. Figure 5 shows that each Lband has a distinct shape and that signals within the same Lband generally behave the same way. Based on this observation, one would expect that: 1. IBDs will be relatively constant, because all signals within the same Lband behave the same way. 2. Differences of any Lband code relative to L1PY will have variations when the code being differenced is not from the L1 band. Thus, only ISC_{L1x} = τ(θ)_{L1PY} – τ(θ)_{L1x} will be constant. ISCs in the other bands can have boresight angle–dependent variations. Figure 6 shows that the IBDs are relatively constant across the main beam. As noted earlier, the results get noisier at large boresight angles due to SV beam rolloff and low elevation multipath. Within the same Lband, atmospheric effects, and lineofsight pseudorange cancels. Thus, neither code and carrier techniques nor the ionospherefree pseudorange are needed. Calibrated receivers can directly measure the IBDs. Systems that use a tracking antenna don’t have to calibrate the antenna portion of the electrical equipment delays because the antenna’s orientation to the satellite does not change. Understanding IonosphereFree Delay Center (IFDC). In this section, we use the physical model Equation (9) in conjunction with the ionospherefree pseudorange Equation (1) to quantitatively measure the emanation shape and location of the various “virtual” ionospherefree pseudoranges. After defining the reference for L1PY & L2PY as IFDC_{L1PYL2PY}, we discuss the alignment of new ionospherefree pairs to the reference IFDC_{L1PYL2PY} using intraband delays. The generalization of the original ionospherefree pseudorange equation (1) for any code pair Lix,Ljz results in: This equation is designed to eliminate 1/f^{2} ionosphere errors and retain common mode terms such as the desired lineofsight pseudorange ρ_{LOS}. However, any non1/f^{2} terms, as well as noise, are amplified. The details for how equation (10) works is discussed in Appendix C. Inserting the full measurement model Equation (9) into Equation (10), the following is obtained: When L1PY and L2PY are used, the bias error is defined as the L1PY&L2PY (Y code) ionosphere free delay center, where IFDC_{L1PYL2PY} is: In our previous publication (A. Tetewsky et alia, Additional Resources reference 3.1), we discussed how the master Kalman filter absorbs this term into the clock correction polynomial’s a_{f0} coefficient. Effectively, the Equation (12) term becomes the virtual reference location for the SV. It is useful to define the IFDC for any pair of codes as: Our goal is to find constants to align any IFDC_{LixLjz} profile with the reference IFDC_{L1PYL2PY}. In order to intuitively understand how the alignments work, we will next discuss measured IFDC data. In Figure 7 the ionospherefree carrierbased pseudorange is used to remove the LOS term from the ionospherefree delay codebased pseudorange in order to show the boresightdependent variations in the IFDCs. Notice that the two curves have very similar shapes. This is because each is a blend of an L1 and L2 signal. Only the shapes of the curves and the separation between them are important. In Figure 8, the difference between the two IFDCs in Figure 7 is plotted. The purpose of Figure 8 is to demonstrate the feasibility of using a constant to align L1CA&L2L2C with the L1PY&L2PY IFDC. The difference is shown to be constant to within 0.3 nanosecond peak to peak (1 nanosecond ~1 foot or ~30 centimeters). Residual errors remain due to small SV antenna anisotropy azimuth variations, and they are amplified by the frequency scaling factor γ. Navigation errors as a function of IFDCs and IFDC alignments are discussed next. Navigation Errors due to Delay Center Alignment Errors. The broadcast ephemeris is calculated by the master Kalman filter using the monitor station measurements of the L1PY&L2PY ionospherefree pseudorange. To a GPS receiver, the satellite position is the emanation point of this ionospherefree pseudorange. However, due to SV anisotropy, the dual L1PY&L2PY emanation point is not constant with boresight angle. Instead, as shown in figure 7, it follows a profile defined by the linear combination of the L2PY and L1PY profiles using the weighting factors of equation 1). But the master Kalman filter approximates it as a constant. This is one contributor to the signalinspace (SIS) user range error (URE). For any other singlefrequency or dualfrequency ionospherefree pair, the difference (alignment error) between its group delay center location and the IFDC_{L1PYL2PY} results in a second SIS URE error. For a future modernized performance standard, this suggests breaking the delay center variations and alignment errors into: 1) the absolute error present in the reference IFDC_{L1PY&L2PY} varying with boresight angle (i.e., SV anisotropy), and 2) an alignment error between the signals being used and the IFDC_{L1PY&L2PY}. The alignment error has two subcategories, dualfrequency and singlefrequency alignment errors. Dual L1PY&L2PY IFDC Variations. A 10foot parabolic dish can be used effectively to measure the variations in the IFDC_{L1PYL2PY}. In general, it is not possible to see the entire SV antenna pattern for every satellite. Thus the average value of the IFDC_{L1PYL2PY}, used in the broadcast ephemeris, cannot be easily obtained from a single ground location. To address this, the NASA Jet Propulsion Laboratory (JPL) used an orbiting semicodeless Pcode receiver to measure the IFDC_{L1PYL2PY} profiles. They did this for boresight angles from 0 to 15 degrees for all SV satellites in orbit during the 2006 to 2009 time period. The results of the JPL campaign are included in Appendix I. The IIRM satellites have some of the largest variations, on the order of 0.5 to 0.25 meter of variation across the main beam. Note because each GPS satellite has two redundant L band electrical systems, referred to as the A and B side, the JPL measurements of the IFDCs would have to be recalculated if an A/B equipment swap or other SV configuration change occurs. In Appendix H, the JPL profiles were inserted into a geometric dilution of precision (GDOP) program. This allowed user range error (URE) histograms to be measured. Although the focus of this article is to discuss alignment errors when using constant broadcast ISC parameters, appendices HI are included to allow for a more complete understanding of the SV antenna anisotropy components that contribute to a full navigation error budget.
Alignment and Measurement Results As the L1 and L2 IBDs are always referenced to the PY code on the ith L band, it is shown in Appendix E that the L1 and L2 IBDs can be calculated from the following pseudorange measurements: Because these IBD terms are more constant then the ISCs, and because using a 10foot parabolic dish antenna with 30decibel gain eliminates receiver antenna anisotropy and reduces measurement error, they can be readily measured with a projected error budget 0.12 nanosecond (95 percent) from any location that can see even a small portion of the SV antenna pattern (from Appendix E). Aligning new dual L1&L5 frequency pairs to Dual L1PY&L2PY. From Appendix D, with IBD_{L1x} = ISC_{L1x} and IBD_{L5z} = ISC_{L5z} − (1 − γ_{15} )T_{GD}, the alignment term for dual L1&L5 is: and the T_{GD} term contains 1 − γ_{12}, as shown in Appendix E, the L5 IBD can be measured using a scaled L1L2 interband pseudorange differenced with the L1L5 measurements to eliminate ionosphere errors, thus: Although ionosphere errors are eliminated, because the L1&L2 and L1&L5 SV antenna anisotropy contributions are different, the estimated error budget for the L5 IBD measurement and alignment is larger, thus the L5 IBD 95% uncertainty is estimated to be 0.55 nsec. Aligning SingleFrequency Measurements to Dual L1PY&L2PY. As shown in Additional Resources reference 3.1, and Appendix D, to align any singlefrequency i^{th} Liband signal x measurement with a delay center of c • τ_{Lix}(θ) to the IFDC_{L1PY&L2PY} reference center, the singlefrequency alignment term (SFAT_{Lix}) needed is: Equation (16) appears in ISGPS200 for aligning L1CA only (section 30.3.3.3.1.1.1), but it is expressed in seconds instead of being scaled into meters by the speed of light. Equation (16) is important for several reasons. First, it can be used to derive all the forms of the modernized and original ionospherefree equation. Second, it summarizes the mathematical linkage between the average value the Kalman filter uses for the IFDC_{L1PYL2PY} and the JPL supplied T_{GD} value. Because T_{GD} contains both an SVunique L1PYL2PY delay and a composite clock term, JPL updates each SV’s T_{GD} value as new satellites are added into the constellation. Currently, the current Kalman filter and JPL’s measurement of T_{GD} are constrained to be independent of SV boresight angle. Therefore, all of the SV’s L1PY&L2PY anisotropy errors are due to working with T_{GD} as a constant. The IBDs and ISCs can be interchanged only if T_{GD} is consistently applied, i.e., when JPL issues a new set of T_{GD}s, the new T_{GD}s must be used to convert between measured IBDs and the broadcast ISCs.
Measured L1&L2 and L1&L5 Data Figure 9 shows the boresight angle–dependent delay center variations for all pairs of pseudoranges. The L1CA&L2C and L1PY&L2PY have nearly the same shape. Using the same absolute tap spacings would have resulted in even better correlation. The L1CA&L5I and L1CA&L5Q have the same shapes but differ from the L1/L2 ionospherefree pseudoranges. All measurements are multipath limited at elevation angles less than 8 degrees (SV antenna boresight angles greater than 12 degrees). (Note: With a 10foot dish, the beam is roughly 5 degrees wide; so, when this 10foot antenna is at 5degree elevation angle, its beam is hitting the ground.) In Figure 10, IFDCs are plotted relative to IFDC_{L1PYL2PY}. Here pseudorange differences can be used so that the results do not have any carrier phase ambiguities. It can be seen that the alignment for L1CA&L2C is within 50 millimeters or 0.05 meters or about 0.168 nanosecond peaktopeak of L1PY&L2PY. Thus, constants can be reasonably used. However, the L1CA&L5I and L1CA&L5Q alignments have large deviations, pp errors of 300 millimeters or 0.3 meters or 1.0 nsec. In these cases, the constant DFAT assumption is in question.
Conclusions and Recommendations We also showed how measured IBDs can be combined with T_{GD} to compute ISC values that are consistent with the constant L1PY & L2PY IFDC approximations used to create the broadcast ephemeris parameters. We demonstrated how these computed ISCs can be used to align any dual pair or single signal profile to the L1&L2 PY ionospherefree pair profile. A proposed methodology for assessing performance was then presented. Appendices E, H, and I (available online) summarize preliminary error budgets and simulation results. We recommend extending the 1979 IEEE Antenna Standard by defining antenna group delay centers, defining the maximum likelihood delay estimator as an alternative metric to phase slope when the phase slope is not constant over the band of interest, and adding the concept of multifrequency blended delay centers. The phrase “delay center” is more appropriate than “phase center” in ISGPS200 and for the antenna phase center data published by the National GeoIntelligence Agency (NGA). When characterizing an SV antenna on future GPS satellites, we recommend both individual Lband and blended Lband chamber measurements. Both characterizations will be needed to maximize the performance improvement potentially offered by the modernized GPS signals.
Acknowledgments Please note that the views in this paper represent those of the authors, and are not necessarily supported by the foregoing listed people or their organizations.
Additional Resources Additional InformationAuthor Profiles
Gary Okerson
Jeff Ross
Avram Tetewsky, Arnold Soltz, Jan Anszperger, Stephen R. Smith Jr. Copyright © 2017 Gibbons Media & Research LLC, all rights reserved. 
