GPS/Galileo Multipath Detection and Mitigation Using Closed-Form Solutions

We propose an e ﬃ cient method for the detection of Line of Sight (cid:3) LOS (cid:4) and Multipath (cid:3) MP (cid:4) signals in global navigation satellite systems (cid:3) GNSSs (cid:4) which is based on the use of virtual MP mitigation (cid:3) VMM (cid:4) technique. By using the proposed method, the MP signals’ delay and coe ﬃ cient amplitudes can be e ﬃ ciently estimated. According to the computer simulation results, it is obvious that our proposed method is a solution for obtaining high performance in the estimation and mitigation of MP signals and thus it results in a high accuracy in GNSS positioning. Copyright


Introduction
In GNSS systems such as Global Positioning System GPS and future Galileo the positioning accuracy is seriously degraded in the presence of MP propagation 1 .In effect, MP signals provoke tracking error in the Delay Locked Loop DLL 1, 2 .Therefore, it is necessary to eliminate MP errors in the DLL discriminator and track the LOS signals transmitted from satellites.Here, various techniques are proposed to mitigate the MP effect 3-10 .In 5 , an analysis of feedback as well as feedforward code tracking algorithms has been done.Then, the peak tracking methods have been proposed as a combination of both feedback and feedforward structures that utilize the inherent advantages of both structures.Yet, at low signal-to-noise ratio SNR , narrow correlator NC 6 is still the best choice among the considered algorithms.The MP Estimating Delay Locked Loop MEDLL 10 is one of the analyzed algorithms.It estimates the parameters of LOS and MP signals such as delays, amplitudes, and phases.The former MEDLL proves to have the best performance in MP environments 10, 11 .However, MEDLL requires a lot of hardware resources.Recently, VMM technique for determining the position of the LOS has been introduced 7 .This technique is shown to give superior performance on the detection of the peak location

Correlation Function (CF)
In the absence of MP signals, the CF of C/A-GPS and BOC 1, 1 -Galileo codes can be approached by the following expressions, respectively: with a 0 : LOS signal coefficient amplitude, T c : code chip spacing, τ: phase shift of the pseudo noise PN code. Figure 1 shows the form of these CF's in the absence of the MP signals.

MP Signals Model
In the presence of MP signals the baseband signal model is defined as follows 7 : ii If the delay of the MP is less than one T c code PN chip lengths, the internally generated receiver signal partially correlates with it.
iii The MP signals can be stronger or weaker than the LOS signal.
In all the figures of the paper, the delays are normalized with respect to the LOS.In effect, "0" represents τ 0 .

Detection of a Single MP Signal
In presence of a single MP component, the normalized input signal with respect to τ f is defined as follows 7 : with: a 1 : MP signal coefficient amplitude, a 0 : LOS signal coefficient amplitude, τ 1 τ 1r − τ f : fraction of the delay of MP signal.

Case of C/A-GPS Signal (1) Presence of a Weak MP Signal
The relationship between the amplitudes of LOS and MP signals is given as a 0 > a 1 .
In presence of a LOS and single weak MP signals, the receiver tries to correlate with these two components.The resulting CF is distorted as shown in Figure 2. Analytically, the LOS and MP signals may be treated separately.Thus, one may consider the CF associated with LOS LOSCF and the CF associated with MP signal MCF .At any point, these two functions can be vector summed to yield the CF associated with the composite signal CCF .To estimate each component of the received signal, we should estimate its amplitude coefficient and its delay.The mathematical expression of the MCF can be obtained from 2.1 as follows:

4.2
The concept of the VMM, that is, the plot of CBD VLOSCF , is illustrated in Figure 3.
The VLOSCF has peak amplitude equal to the maximum value of the CCF and has a width of two code chips on the horizontal axis.
The mathematical expression of VLOSCF can be derived from 2.1 as follows:

4.3
The mutual maximum of the VLOSCF and CCF is given as follows:

Mathematical Problems in Engineering
The mathematical expressions of the line segments PA and PB are given by the following expressions, respectively: In these two equations, the slopes of R PA and R PB are defined as follows:

4.7
Since a 0 > a 1 , then Inequality 4.8 guarantees that the resulting CF, obtained after subtracting VLOSCF from the CCF, is aligned with MCF as illustrated in Figure 3, that is, CB D P E F dashed line .The resulting CF is named "the virtual MP CF" VMCF .As shown in Figure 3, the VMCF is negative in the point B ; this implies that the maximum of the VMCF is aligned with the MCF.
The peak location of the VMCF corresponds to the MP delay τ 1 .The amplitude a V M characterized by the point P can be derived from 4.3 and 4.6 as follows:

4.9
The mathematical expressions of the line segments P'D' and P'E' are given in the following expressions, respectively:

4.10
To estimate the delay τ 1 and the coefficient amplitude a 1 of the MP signal, we use the following proposed equations:

4.11
After estimating both delay τ 1 and amplitude coefficient a 1 we can subtract MCF corresponding to the MP signal from CCF.The resulting CF corresponds to only LOSCF.By using the following proposed equations, we can estimate τ 0 and a 0 (

2) Presence of a Strong MP
In this case, the relationship between the amplitudes of the LOS and MP signals is given as In presence of a strong MP signal, the CCF is not aligned with LOSCF but with the MCF as shown in Figure 4.This is true as long as inequality 4.8 is not satisfied.Since the maximum of CCF is aligned with MCF, we apply a virtual CF aligned with MCF "virtual MP CF VMCF ." As shown in Figure 5, the resulting CF "virtual LOS CF VLOSCF ," that is, A AP B C D dashed line is aligned with the LOSCF.
The mathematical expression of VMCF can be derived from 2.1 as follows: elsewhere.

4.13
The amplitude a VLOS characterized by the point P' can be derived from 4.5 and 4.13 and is given in the expression below

4.14
From Figure 5 we can verify the nonsatisfaction of inequality 4.8 .In this case, to estimate the delays τ 0 and τ 1 and the coefficient amplitudes a 0 and a 1 , we use the following proposed equations: 4.15

Case of BOC(1, 1)-Galileo Signal (1) Presence of a Weak MP
The relationship between the amplitudes of the LOS and MP signals is given as a 0 > a 1 .
Similarly, the mathematical expression of MCF and VLOSCF can be obtained from 2.2 , respectively, as follows: 0, elsewhere.

4.16
The CFs of the received signal and the concept of the VMM are illustrated in Figures 6 and 7, respectively.The maximum of the VLOSCF and CCF is given as 4.17 The obtained VMCF A1A2A3A4A5A6 in Figure 7 after subtraction of the VLOSCF is characterized by the coefficient amplitude a V M and the delay τ 1 .To estimate the delays τ 0 and τ 1 and the coefficient amplitudes a 0 and a 1 , we use the following proposed equations:

2) Presence of a Strong MP
With the same discussion and as the case of C/A-GPS signals, the resulting CCF and the concept of the VMM are illustrated in Figures 8 and 9, respectively.The delays τ 0 and τ 1 and the coefficient amplitudes a 0 and a 1 can be estimated by the following proposed equations, respectively: 4.19

Detection of Two MP Signals
In presence of two MP and a LOS signals, the received signal can be expressed by the following equation 7 : with: a 2 : coefficient amplitude of second MP signal MP2 , τ 2 : delay of MP2, φ 2 : phase shift due to the MP2.The relationships of the amplitudes and the delays is: 4.21

Case of C/A-GPS Signal
Here, the mathematical expression MCF2 can be obtained from 2.1 as follows: 0, elsewhere.

4.22
Figures 10 and 11 illustrate the CF's LOSCF, MCF1, MCF2, and CCF and the concept of VMM, that is, the plot of CBD, respectively.
In the presence of two MP and LOS signals, the peak position of the CCF can be located on the peak of LOSCF or MCF1 or MCF2.In this paper, we discuss only the second case.Another discussion can be done for the first and the third cases.a z represents the maximum value of the CCF and the VMCF and is given as By subtracting the VMCF from the CCF, we get the VLOSCF which is aligned on the maximum of the LOSCF, as illustrated in Figure 11.The values of the delays τ 0 , τ 1 , and τ 2 can be estimated using the following proposed equations, respectively:

4.24
By solving the proposed equations' system 4.25 and 4.26 , we can estimate the amplitude of LOS a 0 and the amplitude of the MP2 a 2

4.26
Finally, by the subtraction of the two estimated CFs LOSCF and MCF2, we can get the amplitude a 1 of MP1 signal as follows:

Case of BOC(1, 1)-Galileo Signal
With a similar discussion, Figures 12 and 13 illustrate the CFs LOSCF, MPCF1, MPCF2, and CCF and the concept of VMM, that is, the plot of CBD, respectively.

Mathematical Problems in Engineering
The values of the delays τ 0 , τ 1 , and τ 2 can be estimated using the following proposed equations, respectively:

4.28
We can estimate the amplitudes of LOS a 0 and of MP2 signal a 2 by solving the proposed equations' system 4.29 and 4.30 .

4.30
Finally, by subtracting the two CFs VLOSCF and MCF2 , we can get the amplitude a 1 of MP1 signal as follows:

4.31
Note that the practical receiver implementation of all these closed form solutions requires that the complete correlation function of the received signal CCF to be measured in order to detect the shape and the distortions caused by MP.The CCF can be sampled via using a bank of correlators 10 see the appendix .

Simulation Results
The impact of MP on code tracking accuracy is often represented as an error envelope which represents the maximum error resulting from one single MP with a certain phase delay and amplitude.The same method of analysis applied to the MP-induced error will be used for the discriminators considered herein.It is worth noting that computing the MP-induced code tracking error envelope CTEE is equivalent to finding the point, where the discriminator output crosses the origin because it represents the point where the DLL will lock.It is obvious that both the finite-bandwidth filter and the correlator spacing have an influence on the envelope.So, a large correlator spacing will result in a greater susceptibility of the tracking loop with respect to MP. Usually, a narrow finite-bandwidth filter will tend to increase the MP-induced error envelope.Thus, the GNSS positioning accuracy requires a rigorous choice of these two parameters.
In order to demonstrate that our RCMPM method performs better than a single NC, two schemes have been simulated: an NC with 0.1 chip spacing and our RCMPM method.For these simulated schemes, we consider a LOS and single MP signals and a band-limited CF of 20 MHz.The errors are computed versus MP signal that has amplitude of 0.5 and a delay that varies from 0 to 1.5 chips with respect to the LOS.The MP error envelopes are calculated at the maximum points when the MP signal is at 0 • in phase or 180 • out of phase with respect to the LOS.The result is given in Figures 14 and 15, respectively.As illustrated in Figure 14 that represents the error envelope for NC, the BOC 1, 1 -Galileo offers better resistance to the long delay MP than the C/A-GPS; however, they have exactly the same envelope for short delay MP.Since the error envelope of the NC for the Galileo receivers is inferior to that of the GPS, it suffices to compare the error envelope of our RCMPM method with that of Galileo receiver.As illustrated in Figure 15, it is clear that RCMPM method performs better than the NC in terms of the error envelope.Whatever the relative delay of the MP is, the offset error of our proposed method is always less than the NC error.In effect, our method shows the best overall MP performance, which is only sensitive for Mathematical Problems in Engineering   medium MP delays.The code error envelopes decrease to zero for long and short MP delays.Before ending up, we can say that there are great differences between the MP performance obtained by the NC and MP performance obtained by our RCMPM method, because the error envelope of RCMPM is greatly smaller than the NC.In other words, the bias due to the MP is reduced by 40 to 95 percent than that of the NC.Also, the band of variation of the error is completely reduced.This shows that our method has better MP rejection.

Conclusion
An efficient method for the detection and mitigation of MP signals is proposed in this paper.This method is derived from the VMM technique 7 .The latter proves limited due to the finite-bandwidth filter in the receiver that creates an offset between the peak location of the virtual LOS and the actual LOS, leading to a code error tracking in positioning the receiver.
In our proposed method we have used the concept of the VMM mitigation technique not only for the LOS delay estimation but for estimating amplitudes and delays of both LOS and MP signals as well.The estimated MP has then been subtracted from the composite signal in order to mitigate the MP effect and efficiently estimate the LOS delay.The simulation results have shown that our proposed method performs better than a single NC.Hence, the error and its band of variation are completely reduced vis-à-vis to what we observed for NC scheme.

Figure 16 :
Figure 16: Multiple correlator sampling of the CF.

Figure 17 :
Figure 17: Block diagram of the bank of correlators.
Some important characteristics of the MP signals are summarised as follows: i The MP signals arrive after the LOS signal because it must travel a longer propagation path.
Figure 1: Correlation functions in the absence of MP signals for both C/A-GPS and BOC1, 1 -Galileo.with τ τ f τ 0 , τ; τ rk : delay of LOS or MP signal τ 0 : fraction of the time delay τ. τ f : portion of time delay corresponding to the integer number of chips which is determined by signal acquisition processes in the receiver , A t ; a k : LOS or MP signal amplitudes, φ k : phase shift due to the MP signals, N: number of MP signals, n t : noise.p t : PN code.