Phase Errors Simulation Analysis for GNSS Antenna in Multipath Environment

High-precision GNSS application requires the exact phase center calibration of antenna. Various methods are published to determine the locations of the phase center. In the outfield, when the phase errors that arose by multipath exceed the phase center variations (PCV) tolerance, the calibration values may be not useful. The objective of this paper is thus to evaluate the phase errors that arose by multipath signals. An improved model of antenna receiving signal is presented. The model consists of three main components: (1) an antenna model created by combination of right hand circular polarization (RHCP) and left hand circular polarization (LHCP), (2) a multipath signals model including amplitude, phase, and polarization, and (3) a ground reflection model applying to circular polarization signals. Based on the model, two kinds of novel up-to-down (U/D) ratios are presented. The performance of the model is assessed against the impact of up-to-down ratio of antenna on phase errors.


Introduction
Positioning in GNSS is based on the range measurements from the satellite antenna to the phase center of receiver antenna.The measurements are performed by utilizing phase delay of a pseudorandom noise (PRN) code or/and phase of the signal carrier.The satellite signals coming from different directions experience different delays of the PRN code and carrier phases.This can be interpreted as the effect of the variation of phase center [1].
For high-precision GNSS applications, the biases introduced by the phase center into the code phase and carrier phase measurements should be known and compensated.The tolerable residual measurements errors are at cm level for code and mm level for carrier measurements [2].
In order to mitigate the impact of phase center variations on high-precision GNSS, a careful determination of antenna phase center offsets and variations is required.Various methods for determination of the phase center have been proposed.Conventional phase center calibration methods include field calibration technique [3][4][5] and anechoic chamber calibration technique [6][7][8].
The field calibration is implemented in outfield using direct satellite signals.This requires suitable GNSS testing site and the same satellites signals for each measurement, so it has to be carried out for many days [9].What is more, it is generally difficult to characterize the multipath using field data because the exact sources of the errors cannot be easily isolated.For anechoic chamber calibration technique, the calibration is implemented in anechoic chamber where the multipath signals are absent.When the antenna is used in outfield, the calibration values may be not useful because of the phase errors that arose by multipath.Multipath signals are the exact sources of the errors of phase center calibration.
In [10], a mathematical model of the corrupted signals composed of the direct line-of-sight signal and multipath signals of antenna is proposed.It allows evaluating the maximal phase measurement errors connected to each kind of parasite signal.In [11], carrier phase multipath parameters are identified and their influences on measurements are investigated through a theoretical analysis.A multipath simulation model is developed and described wherein the multipath parameters can be varied and their influences can be observed.For these models mentioned above, it 2 International Journal of Antennas and Propagation considers that the antenna is modeled as point source, and the characteristics of polarization and radiation are ignored, while the receiving signal phase changes because of varying polarization and radiation.
A method of how to analyze the effect of multipath on geoquality GPS receivers based on amplitude pattern of a given antenna is described in [12].In [13], to characterize the basic influence of multipath, a simple multiray signal model has been used.Based on the model, multipath phase rates for different environments were computed.Reference [14] studies the characteristics of reflected signal based on the / ratio.In [15], a quantitative evaluation of multipath rejection capabilities of a GNSS antenna is introduced, and the technique is focused on the antenna pattern.For these models mentioned above, the multipath signal is assumed as pure RHCP or pure LHCP, while, in practical terms, the reflected signal includes not only RHCP part, but also LHCP part.
In this paper, we present a novel model of antenna receiving signal including direct and multipath signals; the model is further improved by adding antenna model and the characteristic of RHCP and LHCP.Based on the model, two kinds of / ratios are presented.We also analyze the impact on phase errors that arose by multipath for different / ratios.As an example, we use a choke antenna to assess the performance of the simulation results.We found that the multipath signal affects the phase evidently.

Model of the Corrupted Signal
2.1.Model of the Receiving Antenna.In general, the antenna polarization includes RHCP, LHCP, and linear polarization (LP).The LP can be converted to the linear combinations of RHCP and LHCP, so the universal antenna model can be represented as follows [16]: (, )        (,) ] , where |  (, )| and   (, ) are the amplitude and phase of the LHCP output in (, ) direction; |  (, )| and   (, ) are the amplitude and phase of the RHCP output in (, ) direction;  is the elevation angle;  is the azimuth angle.In this paper, the horizontal plane is at  = 0 ∘ .

Model of Line-of-Sight
Signal and Multipath Signal.The line-of-sight (LOS) signal can be written by the following formula: with the following new parameters:  is the signal amplitude;  is the radian frequency;  0 is the initial signal phase;   is the polarization of the LOS signal.
The   can be expressed by where   ,  − are the phase accumulated that arose by signal polarization, which is the typical character of circular polarization wave [17].
When  = 45 ∘ : LP.  = 90 ∘ : LHCP. = 0 ∘ : RHCP, the satellite signal is RHCP in GNSS; in this paper, the LOS signal is RHCP.Consider the following: Multipath signal is introduced by reflection from the ground around the antenna.Compared to the LOS signal, the amplitude, phase, and polarization of the multipath signal change.Assuming that the reflected signal originates from a single point located on a ground, the multipath signal can be expressed by where  is the attenuation factor which varies between 0 and 1,   is the polarization of the multipath signal, and   is the phase change due to the reflection.

Ground Reflection
Model.For a linear system, the Fresnel reflection factors are given by the following relationship [18]: where  ⊥ is the Fresnel reflection factor for vertical polarization,  U is the Fresnel reflection factor for horizontal polarization,  is the incident angle,  0 is the free space dielectric,  1 is the relative dielectric constant, and  is the conductivity.So the reflected wave can be expressed by where   ,   are the complex phasor horizontal and vertical components of the incident signal and    ,    are the complex phasor horizontal and vertical components of the reflected wave.
The electric signal can be expressed using the sum of RHCP and LHCP components: where   ,   are the complex phasor LHCP and RHCP components of the electric field.Convert (8) to following formula [19]: The reflected signal can be written by the following formula: where    ,    are the complex phasor horizontal and vertical components of the reflected wave.Convert (10) to the combinations of RHCP and LHCP; the reflected wave can be expressed by where    ,    are the complex phasor LHCP and RHCP components of the reflected wave.
Adding the accumulated phase that arose by circular polarization to (11), For a circular polarization system, the Fresnel reflection factor, , can be written by the following formula: [20].There are two important multipath sources: signal reflected by a building and off the ground.In high-precision GNSS applications, the buildings are beneath the antenna; the signal reflected off the ground is the key multipath sources.In (5),   = 4ℎ sin /, where  is the incident angle and ℎ is the height of the antenna above the reflected plane.

Phase Errors Arose by Multipath
The received signal is given as the sum of the LOS and multipath signals: where So the phase errors that arose by multipath, Δ, can be obtained by The typically high-precision GNSS antenna has stable phase center.In this paper, we assume the phase center variation of antenna is zero and the  = 0 ∘ ; the conclusion is the same with other  values.

Errors Analysis in
As ( 17) predicts, we will see that the two major factors affecting the phase errors are the relative amplitude between the up RHCP and down RHCP and the relative amplitude between the up RHCP and down LHCP.Two formulas for / ratio are defined as By rearranging (17), we can get the following relationship: In outfield environment, the wet ground and dry ground are two typical reflected materials, and the approximate values for dielectric constant and conductivity are given in Table 1 [21].
In our simulation, we choose the frequency GPS L1  = 1575.42MHz ( = 2), and the height of the antenna ℎ = 2 m.As shown in Figure 1, we note that as / 1 and / 2 ratios increase, the phase errors decrease.For different elevations, the phase errors change differently against / 1 and / 2 .When / 1 is higher than 10 dB, the max phase error is below 2 mm (−2 mm < phase error < 2 mm) at elevation = 80 ∘ and 70 ∘ (see Figures 1(a) and 1(b)).For elevation = 40 ∘ and 30 ∘ , the phase errors are always below 2 mm when / 1 > 1 dB; this is because the reflected wave is almost RHCP which is the same with the LOS wave.At elevation = 60 ∘ , 50 ∘ , 20 ∘ , and 10 ∘ , only when the / 1 and / 2 ratios are higher than a particular value, the phase errors are below 2 mm, while the values are higher at 20 ∘ and 10 ∘ than those at 60 ∘ and 50 ∘ . 2 depicts the phase errors that arose by wet ground reflected multipath changing against / 1 and / 2 for several values of elevation (10 ∘ -80 ∘ ).

Wet Ground. Figure
As shown in Figure 2, we note that the conclusion is similar to that at dry ground, while the reflected wave by wet ground is stronger than dry ground.In order to reduce the phase errors that arose by reflected multipath, higher / 1 and / 2 are demanded than dry ground.

Summary.
When the phase center variations tolerance is 2 mm (typically), in order to mitigate the impact of phase center variations on high-precision GNSS by using calibration values in outfield environment where typical reflected materials are the wet ground or dry grounds, the different / 1 and / 2 ratios are demanded which can be obtained from Figures 1 and 2, as in Table 2.

Examples
In order to assess the performance of the simulation results, a choke antenna is analyzed.Figure 3 shows the RH and LHCP amplitude pattern of the antenna.Figure 4 shows the / 1 and / 2 of the antenna; as presented in Figure 4, / 2 of the antenna is higher than 18 dB above horizontal, while / 1 is less than 12 dB at lowelevation angles (elevation < 10 ∘ ). Figure 5 shows the effect of the multipath on the carrier phase against elevation angle for choke antenna.We note that the phase error is 5.28 mm for wet ground reflected and 6.01 mm for wet ground reflected at elevation = 10 ∘ , which is according to Figures 1(g) and 2(g).At high-elevation angles (elevation > 15 ∘ ), the phase errors are all below 2 mm.
When the phase center variations tolerance is 2 mm, in order to mitigate the impact of phase center variations on high-precision GNSS application by using calibration values, the elevation cutoff angle will be set at 15 ∘ or higher.further improved by adding antenna model and the characteristic of RHCP and LHCP.We have applied the new antenna and multipath model to illustrate the phase errors that arose by multipath against / 1 and / 2 ; it was shown that the multipath signal affects the phase evidently; whichever of / 1 and / 2 ratio is below 15 dB, the phase errors are  higher than 2 mm that exceeds the phase center variations tolerance.In high-precision GNSS application, before using phase center calibration values, the following three steps are necessary:

International Journal of Antennas and Propagation
Dry ground Wet ground  In order to expand the coverage of antenna applied in high-precision GNSS application, we must heighten the / 1 and / 2 ratio at low-elevation angle.

Figure 4 :
Figure 4: The / ratios in dB for choke antenna.

Figure 5 :
Figure 5: The effect of the multipath on the carrier phase against elevation angle for choke antenna.

Table 1 :
Characteristics of dry and wet ground.

Table 2 :
The different / 1 and / 2 ratios demand for phase errors below 2 mm (approximate).