GPS Output Signal Processing considering both Correlated/White Measurement Noise for Optimal Navigation Filtering
 Author: Kim DoMyung, Suk Jinyoung
 Organization: Kim DoMyung; Suk Jinyoung
 Publish: International Journal Aeronautical and Space Sciences Volume 13, Issue4, p499~506, 30 Dec 2012

ABSTRACT
In this paper, a dynamic modeling for the velocity and position information of a single frequency standalone GPS(Global Positioning System) receiver is described. In static condition, the position error dynamic model is identified as a first/second order transfer function, and the velocity error model is identified as a bandlimited Gaussian white noise via nonparametric method of a PSD(Power Spectrum Density) estimation in continuous time domain. A Kalman filter is proposed considering both correlated/white measurements noise based on identified GPS error model. The performance of the proposed Kalman filtering method is verified via numerical simulation.

KEYWORD
GPS Error Modeling , Power Spectral Density , Correlated Measurement Noise , Kalman filter.

1. Introduction
GPS has been used as main navigation system for a variety of aerospace applications. Kalman filter estimation is widely applied to GPS positioning as well as the integration of GPS with inertial navigation system. A conventional Kalman filter is formulated with assumption that both the process and measurement noise are Gaussian white. There are some GPS measurement error modeling studies under SA(Selective Availability) condition[14]. Where GPS position measurement error can be modeled as a time correlated low order GaussMarkov process both continuous and discrete time domain using various parameter identification approaches. Because it is difficult to determine the parameters of GaussMarkov process in a realfield problem, it is important to estimate of the amount of experimental data recoding time and ensemble average needed for a given required estimation accuracy.
Previous works for kalman filtering methods that consider timecorrelated measurement error are categorized in two main approaches. First approach is state augmentation method where the measurement error is augmented into state vector. However, the augmented sate space equation has the perfect measurement with singular measurement covariance matrix that may cause system to become illconditioned[57]. The second approach uses measurement differencing method to make a new measurement equation corrupted by white noise without timecorrelated part[69]. Therefore the conventional Kalman filter can be applied to this state and new measurement equation for optimal state estimation. However, In case of multiple measurements, the measurement differencing approach is limited by each measurement should be simultaneously timecorrelated.
This paper describes GPS error modeling method for standalone single frequency GPS. The relationship between required data recoding time and ensemble average number to satisfy the desired PSD(Power Spectrum Density) estimation error is investigated. It is also shown that the physical parameter of a first or second order dynamic system is related with required data recoding time and ensemble average number. A Kalman filtering method is proposed that consider both correlated/white measurements noise based on characteristics of a standalone GPS position and velocity measurements. The performance of the proposed Kalman filtering method is verified via numerical simulation.
2. GPS Error Modeling
2.1 Linear System and PSD
The relationship between PSD and transfer function of single inputoutput linear time invariant system is described as follows[10].
where
G(w) is a transfer function of the system,S_{uu}(w) is the PSD of input, andS_{xx}(w) is the PSD of output. The time average of the periodogram of a random processx(t) can be expressed as follows[7].where
x_{T}(t) is a time sample of the random processx(t) at timet ,is Fourier transform of the
x_{T}(t) . IfS_{uu}(w) is unit power white noise andS_{xx}(w) is the error signal of a GPS, using equation (1) and (2), we can find transfer function of the GPS error model with experimental data ofS_{xx}(w) .2.2 Normalized PSD Estimation Error
The normalized bias error of PSD estimation can be approximated via Taylor series expansion of expectation of the PDF(Probability Density Function) of a random process
x(t) [11].where
W is the resolution bandwidth. Eq. (3) can be rearranged asThe normalized random error of a PSD estimation can be represented using identity that the two degree of freedom chisquare variables have average
n and variance 2n .where
n_{d} is ensemble average number. The required data recoding time for given normalized random error is2.3 Transfer Function Parameter and PSD
An arbitrary transfer function and PSD of the firstorder/secondorder Gauss Markov process can be represented as follows.
First order:
Second order:
To investigate the relationship between PSD estimation error and resolution bandwidth, we define
and conservative approach should be applied to determine the frequency for calculation of in Eq. (4).
As shown in the Fig. 1. (a)~(b),
SF of the firstorder GaussMarkov reaches maximum value at the frequencyand the secondorder GaussMarkov process has an extreme value of SF depending on the damping ratio. In case of a secondorder GaussMarkov process with lightly damped system(damping ratio range is C<0.3), has a maximum value at the resonance frequency
In case of normally damped system, that has a damping ratio is 0.3
In case of over damped system, that has a damping ratio is C>0.707, SF has a maximum value at the frequency
ω_{m} = 2π/t. . Therefore, W should be evaluated at the appropriate frequency for given PSD estimation error bound based on damping ratio of the secondorder linear system.2.4 Estimation of the PSD and Transfer Function
In static condition, the position error dynamic model is identified as a first/second order transfer function, and the velocity error model is identified as a bandlimited Gaussian white noise via nonparametric method of a PSD estimation in continuous time domain. Based on quick identification of the position error model of the second order transfer function and its damping ratio
w_{m} using one hour recoding data analysis, we obtainedn_{d} =25 andT_{r} =45 minutes that satisfy PSD estimation error boundaryand
Fig. 2. (a) shows effectiveness of ensemble average of the PSD in frequency domain. In this case, 25 data set of
position error PSD are averaged. Fig. 2. (b) shows fitting of the averaged PSD estimation using nonlinear least square regression method. Fig. 2. (c) bode diagram of the identified transfer function. Fig. 2. (d) shows comparison of the real experimental data and linear simulation of the identified transfer function. In fact, the GPS position error model can be approximated as a first order transfer function within low frequencies with
20dB/dec rolloff ratio of the magnitude as shown in Fig. 2. (c).The Identified GPS position error model in NEU coordinate system is summarized in the Table 1.
As shown in the Fig. 3, the velocity measurement error PSD has relatively low rolloff ratio and flat magnitude within Nyquist frequency. Therefore it is reasonable to assume that velocity measurement noise is band limited white.
A Kalman filter can be applied to the statespace equation transformed from the error transfer function. The first order transfer function can be obtained as Eq.(11). σ and δ of the first order transfer function corresponding to each axis on NEU coordinate system is summarized in Table 2.
Eq.(11) can be discretized for the sampling time △
t yielding3. Kalman Filtering Methodology
The Kalman filtering method is applied for one measurement noise is first order GaussMarkov process and the other measurement noise is white. Consider the discrete state and measurement equation with both colored measurement noise and white noise.
where
w, d, c denotes states, measurements, state of timetime correlated measurement error, process noise, white noise measurement and white noise for shaping timecorrelated measurement error, respectively. These white noises are mutually uncorrelated. State variables can be classified as the three groups:(i)x_{c} , which are disturbed by the timecorrelated measurement noise, (ii)x_{n} , which are disturbed by white measurement noise, and (iii)x_{s} , which can be estimated. Each state variables have rowsn_{c} ,n_{n} andn_{s} . It is also assumed that the timecorrelated measurement noise can be represented by the first order transfer function havingn_{c} state variables. Eq.(19) shows the size of each variable vectors with appropriate subsystems.State and measurement equations can be rewritten using Eq.(19) as
After augmenting timecorrelated measurement error state into state vector, state and measurement equations can be rearranged as follows.
where,
As can be seen in Eq.(24), timecorrelated measurement for augmented system model does not contain the white noise. A numerical difference method is used in this paper to avoid the 'perfect measurement condition',which is often the cause of numerical instability. In order to do this, a new measurement vector can be defined as the linear difference of the measurement
z betweent(k1) andt(k) as follows.Combination of the augmented state equation and the new measurement equation yields
Using Eqs(28)(29) Eq.(30) is obtained.
Eq.(27) can be rearranged as,
Eqs.(30)(31) yields,
Using Eq.(26) and Eq(32), expression on
u_{c}(k) can be given byTime delay of the update on state estimation can be avoided when the measurement can be rewritten using the inverse of the state transition matrix of the augmented state equations:
Now, new parameters
N andU are introduced to combine Eqs.(35)(36),Eq.(22) and Eq.(37) can be simplified as,
where the new measurement noise is white, and
w_{a}(k1) ,w_{c}(k1), d(k) are not timecorrelated. However, the process noise and measurement noise is correlated because both state equation and measurement equation have the termw_{a}(k1) . Measurement noise covariance matrixF and processmeasurement cross covariance matrixS can be defined as,In general, Kalman filter cannot be applied for the case process noise and measurement noise is correlated. Therefore, a generalized Kalman filter is used in this paper in order to obtain optimal estimation that considers both timecorrelated measurement noise and white noise. State equation and measurement equation with updates for a generalized Kalman filter are shown in Eq.(44)(53)[12].
Time Update
Measurement Update
4. Numerical Simulation
A simple kinematic CWNJM(Continuous White Noise Jerk Model) is considered for numerical simulation. The states are position, velocity and acceleration. It is assumed that the position measurement noise is firstorder Gauss Markov process and velocity measurement noise is white as mentioned in chapter II. Discretized linear dynamic model of CWNJM as follows.
where,
State equation for augmented system can be expressed as,
Discretized version of measurement equation can be given by the following equations
Now the state equation and measurement equation are reconstructed so that the generalized Kalman filter can be applied. Parameters for process noise and white noise are set as
q =0.001^{2},D =0.01^{2},V =0.03^{2}.？ =0.999925 for Northing position error model from the discretization of the GPS position error model with △t =0.05 in Eq.(12). Numerical simulation is performed using Eqs.(44)(53).Fig. 4. (a) and (b) shows the proposed filter effectively estimates both position corrupted by timecorrelated measurement error and velocity corrupted by white noise. Also, the acceleration estimation is appropriate as shown in Fig. 4 (c). In this example, the process noise covariance is set relatively small value to show the estimate with respect to noise magnitude during small time window. It should be noted that the initial states are assumed to be known. Actually, this proposed Kalman filter decorrelates the time correlated measurement error of the position using velocity information which is not corrupted by time correlated noise, but by white noise. Moreover, using the velocity information and simple kinematics, we can get acceleration estimation without additional measurements.
5. Conclusions
In this paper, a dynamic modeling method for the velocity and position information of a single frequency standalone GPS receiver is described. The relationship between ensemble averages, required data recoding time and PSD estimation error that satisfy the given error bound is described. Also, analysis on transfer function parameters of a first and a second order linear system with respect to PSD error is described. A Kalman filter is proposed that consider both correlated/white measurements noise based on identified GPS error model. The proposed Kalman filter is derived from the fusion of the state augmentation approach and measurement differencing approach. The performance of the proposed Kalman filter is verified via numerical simulation. Using this filter, the time correlated position error of the GPS measurement is effectively decorrelated via its own GPS velocity information without any additional sensors. In near future, the proposed Kalman filtering method will be formulated for more general cases and applied to the precise navigation of moving vehicles.

[Fig. 1.] Transfer Function Parameter and Required Recoding Time: (a) First Order System SR with respect to Time Constant, (b) First Order System Required Recoding Time with respect to Bias Error, (c) Second Order System SR with respect to Damping Ratio, (d) Second Order System Required Recoding Time with respect to Bias Error.

[Fig. 2.] Identification of the Easting Position Error Transfer Function: (a) Ensemble Average, (b) Nonlinear Least Square Fitting with respect to PSD, (c) Bode Diagram of the Identified Transfer Function, (d) Comparison Real Data and Linear Simulation of ID

[Table 1.] Summary of the GPS Position Error Model

[Table 2.] Summary of the Model Parameters of 1st Order Transfer Function

[Fig. 3.] Ensemble Average of 25 Data Set of Velocity Error PSD.

[Fig. 4.] Simulation Results: (a) Position Estimation, (b) Velocity Estimation, (c) Acceleration Estimation, (d) Kalman Gain.