1. INTRODUCTION
A Vector Delay Locked Loop (DLL) technique which utilizes the navigation result and controls all Numerically Controlled Oscillator (NCOs) in parallel was firstly proposed by Spilker (Parkinson & Spilker 1996). And he observed that jamming immunity of a Vector DLL is superior to that of a Scalar DLL, and a GPS receiver with a Vector DLL can maintain operation although some of satellites are not being seen temporarily and it is more robust for dynamic stress than that with a Scalar DLL (Spilker 1995).
On the back of a research of Spilker, there are many researches for a Vector DLL until these days. Some of those are researches for GPS/INS integrated systems (Gustafson et al. 2000, Abbott & Lillo 2003, Landis et al. 2006, Petovello & Lachapelle 2006) and others are researches for weak signals environments (Pany et al. 2005, Benson 2007, Lashley & Bevly 2007). Most of above researches, however, did not analyze the performance of a Vector DLL with their structure, but give us experimental results. For example, Petovello just showed a position result, Pany showed a position result and the number of observations although a position result is not a related index with the performance of a tracking loop. Lashley showed the code phase error which is a related index with tracking performance, but there is no analysis of relation between the structure and the code phase error.
This paper, therefore, proposes the structure of the GPS receiver with the Vector DLL and theoretically analyzes the tracking and the positioning accuracy based on the performance index. And the tracking and the positioning accuracy of the receiver with the Vector DLL are confirmed by simulation results.
2. PERFORMANCE ANALYSIS OF A SCALAR DLL BASED GPS RECEIVER
2.1 Structure of a Scalar DLL Based GPS Receiver
A functional block diagram of a Scalar DLL based GPS receiver which consists of the signal tracking block and the navigation block is shown in Fig. 1. As shown in Fig. 1, the DLL is only illustrated in the signal tracking block and it consists of the bank of correlators, a code discriminator and a code loop filter including a code NCO. In the navigation block, the pseudoranges are calculated by using the code phase measurement of each DLL, and the range residuals are calculated for updating the position.
Fig. 1. Structure of a Scalar DLL based GPS receiver.
2.2 Tracking Accuracy of a Scalar DLL
The accuracy of a Scalar DLL is analyzed for the structure shown at Fig. 1. Firstly, the correlator's outputs which mean the correlation power represented in Watt are modeled as Eqs. (1-4) (Van Dierendonck et al. 1992): (1)
\(I_{E,l}(nT)=\sqrt{2C/N_0T}R \left ( \varepsilon_l(nT)-\frac{d}{2} \right ) \cos{(\phi_l(nT) )}+w_{IE}(nT)\) (1)
\(Q_{E,l}(nT)=\sqrt{2C/N_0T}R \left(\varepsilon_l(nT)-\frac{d}{2} \right)\sin{\left(\phi_l(nT)\right)}+w_{QE}(nT)\) (2)
\(I_{L,l}(nT)=\sqrt{2C/N_0T}R\left(\varepsilon_l(nT)+\frac{d}{2}\right)\cos{\left(\phi_l(nT)\right)}+w_{IL}(nT)\) (3)
\(Q_{L,l}(nT)=\sqrt{2C/N_0T}R\left(\varepsilon_l(nT)+\frac{d}{2}\right)\sin{\left(\phi_l(nT)\right)}+w_{QL}(nT)\) (4)
where \(C/N_0\) is the carrier power to noise density ratio in the linear scale, \(T\) is the Pre-detection Integration Time (PIT), \(R(\cdot)\) is the auto-correlation function of the GPS L1 C/A code, \(\varepsilon_l(nT)\) is the true code phase difference between the input and replica, \(d\) is the chip space between early to late, \(\phi_l(nT)\) is the carrier phase difference between the input and replica, and \(W_{IE}, W_{QE}, W_{IL}, W_{QL} \) are the noises whose means are zero, and variances are one.
In case of using the early minus late power discriminator, the variance of the output of \(l\)-th discriminator is given by (Van Dierendonck et al. 1992)
\(Var[\widetilde{ε}_l(n)]|_{ε_l{(n)=0}}=4d(2-d)[2+(2-d)C/N_0T],\) (5)
and output of \(l\)-th discriminator for the first order feedback loop is given by (Van Dierendonck et al. 1992)
\(σl2≡Var[\widetilde{ε}_l(n)]|_{ε_l{(n)=0,closed}}={B_nd \over 2 C/N_0} \left [1+{2\over(2-d)C/N_0T} \right ],\) (6)
where \(B_n\) is the loop noise bandwidth. From Eq. (6), it can be found that the higher \(C/N_0\) and lower \(B_n\) give better tracking performance.
The standard deviation of the code tracking error by the dynamics of the satellite and the receiver is given by (Ward et al. 2006)
\(R_e=\frac{{d^nR}/{d^nt}}{\omega_0^n},\) (7)
where \(R\) is the range between a satellite and a receiver, \(n\) is the loop filter order, and \(\omega_0\) is the loop filter natural radian frequency.
The total code-tracking error for the third-order loop including the dynamic stress error \((R_e)\) can be represented as (Ward et al. 2006)
\(\sigma_{SDLL}=\sigma_l+\frac{1}{3}R_e=\sqrt{\frac{B_nd}{2C/N_0}\left[1+\frac{2}{(2-d)C/N_0T}\right]}+\frac{1}{3}\frac{d^3R/dt^3}{\omega_0^3},\) (8)
where \(d^3R/dt^3\) is the Line-of-sight jerk dynamics. The reason why the thermal noise term is represented for the first-order loop in (6) is that the thermal noise error does not much depend on the loop order and the loop order is mainly sensitive to the same order of dynamics.
By using Eq. (8), the code tracking errors according to loop noise bandwidth and \(C/N_0\) are given in Fig. 2.
Fig. 2. Code Tracking Error according to Loop Noise Bandwidth.
3. PERFORMANCE ANALYSIS OF A VECTOR DLL BASED GPS RECEIVER
3.1 Structure of a Vector DLL Based GPS Receiver
A functional block diagram of a Vector DLL based GPS receiver is shown in Fig 3, and it has both of a Scalar DLL and a Vector DLL. Because the Vector DLL requires the initial positioning results based on the Scalar DLL to control the code NCO. Therefore, the receiver starts with a Scalar DLL, and then the receiver switches the tracking loop to a Vector DLL after finding receiver's position.
Fig. 3. Structure of a Vector DLL based GPS receiver.
Comparing the Fig. 3 with Fig. 1, it can be seen that the 'Pseudorange Calculation Block' has been modified and the 'Phase Error Calculation Block' is added. In the 'Pseudorange Calculation Block', firstly, the pseudorange of \(l\)-th satellite can be calculated by
\(\rho_l(mT_P)=\rho_l(mT_P-T_P)+\Delta\rho_l(mT_P)=\rho_l(mT_P-T_P)+CT_C\cdot\sum\limits_{k=n-T_P/T+1}^{n}{{\widetilde{\varepsilon}}_l(kT)}, \) (9)
where \(T_P\) is the interval of positioning, \(C\) is the speed of light, \(T_C\) is the chip duration of the C/A code and \({\widetilde{\varepsilon}}_l(kT)\) is the output of \(l\)-th discriminator at \(kT\). In the 'Phase Error Calculation Block', secondly, the code phase error is calculated as
\(\left[\begin{matrix}{\widetilde{\varepsilon}}_1(mT_P)\\{\widetilde{\varepsilon}}_2(mT_P)\\\vdots\\{\widetilde{\varepsilon}}_N(mT_P)\\ \end{matrix}\right]=\left (\frac{1}{CT_C} \right)\cdot H\cdot\Delta\vec{x}(mT_P),\) (10)
where \(N\) is the number of tracking signals, \(H\) is the line of sight vector.
3.2 Tracking Accuracy of a Vector DLL
In a vector DLL, the tracking error can be given by
\(\boldsymbol{\sigma}_{VDLL}^2=\left(\frac{1}{CT_C}\right)^2H\textrm{cov}{(}d\vec{x})H^T,\) (11)
because the code phase error is calculated by the position increments as shown in Fig. 3. And the position error in Eq. (11) can by calculated by (Ward et al. 2006)
\(\textrm{cov}(d\vec{x})=(H^TH)^{-1}H^T \textrm{cov}(d \boldsymbol{\rho})H(H^TH)^{-1}. \) (12)
From Eq. (9), the pseudorange errors of \(N\) satellites can be represented as
\(\textrm{cov}(d\boldsymbol{\rho})=\left(\begin{matrix} \textrm{var} (\Delta_{\rho_1})&0&\cdots&0\\0& \textrm{var} ( \Delta_{\rho_2})&0&0\\0&0&\ddots&0\\0&0&\cdots& \textrm{var} (\Delta_{\rho_N})\\\end{matrix}\right),\) (13)
and the variance of the delta pseudorange \((\Delta_{\rho_l})\) can be calculated as
\(\textrm{var} (\Delta\rho_l)=\left(\frac{C}{T_C}\right)^2\cdot \textrm{var}{\left(\sum\limits_{k=n-T_P/T+1}^{n}{{\widetilde{\varepsilon}}_l(kT)}\right)}\ =\left(\frac{T_P}{T}\right)\left(CT_C\right)^2 \textrm{var}{\left({\widetilde{\varepsilon}}_l(kT)\right)}=\left(\frac{T_P}{T}\right)\left(CT_C\right)^2 \boldsymbol{\sigma}_{SDLL}^2,\) (14)
by using the statistical properties of the variance operation. Finally, the variance of a Vector DLL is given as
\(\boldsymbol{\sigma}_{VDLL}^2=\left(\frac{T_P}{T}\right)H(H^TH)^{-1}H^T \boldsymbol{\sigma}_{SDLL}^2H(H^TH)^{-1}H^T,\mathrm{\ \ \ }T_P\geq T.\) (15)
Through this result, it can be found that the tracking accuracy of a Vector DLL depends on PIT and interval of positioning and satellite's geometry. In case that \(T_P = T\), it can be expected that the tracking error of the Vector DLL is smaller than that of the Scalar DLL because the components of \(H\) is naturally smaller than 1.
4. SIMULATION RESULTS
4.1 Simulation Environment
The goal of the simulation was to validate the theoretical analysis and to compare the accuracy of the Vector DLL with those of Scalar DLL. In a point of the accuracy, the tracking and positioning accuracy were compared for static case.
An environment for the simulation is shown in Fig. 4. As shown in Fig. 4, the GPS RF signal has been generated by the Spirent simulator and IF data have been collected for 120 seconds by the RF front-end of the Nordnav receiver. And then they are processed in a software GPS receiver with the tracking loop of a Scalar DLL and a Vector DLL and the positioning algorithm of Least Squares. Finally, the processed results of the receiver are presented by the MATLAB.
Fig. 4. Simulation Environment.
The satellites allocation for this simulation is given in Fig. 5. As shown in Fig. 5, there are 9 satellites and PDOP is 1.64. And the receiver design parameters are also given in Table 1.
Fig. 5. Sky plot for the visible satellites.
Table 1. Design parameters of the receiver.
| Parameters | Value |
| C/N0 Early to Late chip space DLL Loop noise bandwidth DLL Order FLL Loop noise bandwidth FLL order Pre-detection integration time |
45 [dB-Hz] 1 [chip] 1 [Hz] 3rd 5 [Hz] 2nd 1 [ms] |
4.2 Simulation Results
As shown in Fig. 6, the tracking accuracy is compared by using Eqs. (8) and (15), and then the horizontal positioning error when the \(C/N_0\) is 45 dB-Hz and \(T_P = T\) is compared as shown in Fig. 7. From these results in Fig. 6, it can be check that the tracking error of the Vector DLL is smaller than that of the Scalar DLL for all \(C/N_0\) when a receiver is static in which the jerk is 0.03 g/s by the motion of the satellite. From Fig. 7, it can be also checked that the positioning error of the GPS receiver with the Scalar DLL and Vector DLL are given as 0.95 m (CEP) and 0.69 m (CEP), respectively. By these results, it can be concluded that the tracking and positioning performance of the receiver with the Vector DLL are improved about 30% than those of the receiver with the Scalar DLL when the receiver is static and PDOP is 1.64.
Fig. 6. Comparison of the positioning accuracy in static case.
Fig. 7. Comparison of the positioning accuracy in static case.
Through this paper, eventually, it is confirmed that the performance of the GPS receiver with the Vector DLL is superior to the receiver with the Scalar DLL for specific environments, and performance expectation can be roughly done for a given environment.
5. CONCLUSIONS
In this paper, performance of the GPS receiver with a Vector DLL has been analyzed in a point of the tracking accuracy, and it also compared with that of the receiver with the Scalar DLL in a point of the positioning accuracy for the static case. From the results, it is confirmed that the tracking and positioning accuracy of the Vector DLL has been improved about 30% when a receiver is static and \(T_P = T\) while \(C/N_0\) is 45 dB-Hz and the PDOP is 1.64. Based on these observations, eventually, the performance expectation can be roughly done for a given environment.
ACKNOWLEDGMENTS
This work was supported by a grant from Korea Research Council of Fundamental Science & Technology funded by Ministry of Education, Science and Technology in 2012 [Project: A Study on Satellite based Position Tracking Technology for Calamity Prevention and Public Safety Improvement].
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