Frequency Tracking for High Dynamic GPS Signals Based on Vector Frequency Lock Loop

Li Hui 702 laboratory Electronic Engineering Institute

Hefei 230037, China li.hui83@yahoo.com.cn

Yang Jingshu, Cui Chen 702 laboratory Electronic Engineering Institute

Hefei 230037, China

Abstract—A method of estimating high dynamic GPS signal parameters based on Vector Frequency Lock Loop(VFLL) is put forward. Instead of using the discriminator, the VFLL focus on the nonlinear estimation. The navigation filter is realized utilize Unscented Kalman Filter(UKF).The VFLL method is based on the fact that comparing with the rate of change of the Doppler frequency caused by the motions of the satellite and the receiver, the acceleration of the receiver is the dominant factor. Instead of tracking the Doppler frequency of individual channel using the traditional loop, the VFLL tracking the velocity, acceleration .etc states of the GPS receiver from the measurement value of all channels. In vector mode, the navigation processor controls the carrier numerically controls oscillators (NCOs) based on the most recent velocity solution of the receiver. A major advantage of this method is that the velocity estimation from all the channels can be more accurate than Doppler frequency estimation from individual channel.

Keywords-vector frequency lock loop(VFLL); high dynamic; GPS; frequency tracking; vector tracking;

I. INTRODUCTION Accurate frequency estimation and tracking are of

fundamental concern in the design of GPS receivers observing signals that exhibit high dynamics[1]. Different estimation techniques were proposed to tracking the frequency of the high dynamic GPS signals, for example, approximate maximum likelihood, Extend Kalman filter, cross-product automatic frequency control loop[1], and FEKF[2]. Doppler frequency of individual channel is tracked directly in those methods. So there is no common information among different channels is used, because the signals from each satellite are processed independently.

The vector tracking method combining the two functions, receiver tracking and navigation algorithm into one combined integrated tracking system[3]. Unlike traditional GPS receivers use tracking loops to track the GPS signals, all the satellite signals are tracked by navigation processor in the vector tracking method. Vector-based methods have several potential advantages over the traditional tracking methods. First, noise is reduced in all of the tracking channels making them less likely to enter the nonlinear region and fall below threshold. Second, the vector tracking can operate with momentary blockage of one or more satellites. Last, the

vector tracking can be better optimized than the conventional sequential DLL-Kalman filter combination which is constrained to be the product of two stable closed-loop tracking filters in series[3]. In [3], the Vector Delay Lock Loop (VDLL) tracking method is described and a more detailed discussion of how to implement vector tracking is given in [4]. The VDLL is used for estimating the power of weak signals in [5]. The ability of vector-based methods to operate during brief signal outages and rapidly reacquire blocked signals is explored in [6]. A performance analysis of the vector tracking algorithms based on the numerical method is given in [7]. Papers on the frequency estimation for high dynamic trajectories based on the vector tracking method are very few.

The VFLL method proposed by this paper is based on the fact that the frequency of the received carrier varies with time according to the dynamic of the receiver and satellites being tracked, but comparing with the rate of change of the Doppler frequency caused by the motions of the satellite and the receiver, the acceleration of the receiver is the dominant factor. Now let us consider the motion of the satellite and the receiver. The maximum rate of change of the speed of the satellite is 0.178 m/s2, the corresponding rate of change of the Doppler frequency is 0.936 Hz/s, this value is also very small[8]. If the user has an acceleration of 1 g (gravitational acceleration with a value of 9.8 m/s2) toward a satellite, the corresponding rate of change of the Doppler frequency is about 51.5 Hz/s. For a high-performance aircraft, the acceleration can achieve several g values, such as 7 g. The corresponding rate of change of the Doppler frequency will be close to 360 Hz/s. The rate of change of the Doppler frequency caused by the satellite motion is rather low; therefore, it does not affect the update rate of the tracking program significantly. The operation and performance of a receiver tracking loop greatly depends on the acceleration of the receiver.

If the velocity, acceleration .etc of the receiver were tracked from all the satellite signals, one would expect that the estimation will be more accurate than tracking the Doppler frequency of the individual channel. This is possible because, given the receiver’s position, velocity, clock bias, and clock drift, the Doppler frequency in each channel can be very accurately predicted. A major advantage of this method is that after once a good position, velocity and time

2010 WASE International Conference on Information Engineering

978-0-7695-4080-1/10 $26.00 © 2010 IEEE

DOI 10.1109/ICIE.2010.314

98

(PVT) solution is available, all Doppler frequency in each channel are continuously tracked in a navigation processor, so the VFLL can tracking the high dynamic GPS signals under weaker signal conditions than the traditional tracking loops .Different with the method given in [3~7], the nonlinear estimator is used to replace the discriminator to extract the Doppler frequency error. The nonlinear estimator can perform better than discriminator in environment with high dynamics and low CNR.

II. VFLL ALGORITHMS FOR HIGH DYNAMIC Figure 1 illustrates a block diagram of the VFLL

architecture typical of one of the digital receiver channels where the digitized received IF signal is applied to the input. For simplification, only the functions associated with the carrier tracking are illustrated, and the receiver is assumed to be already in the vector tracking mode.

Referring to Figure 1, first the digital IF is stripped of the carrier (plus carrier Doppler) by the local replica carrier (plus carrier Doppler) signals to produce in-phase (I) and quadraphase (Q) sampled data. Then the in-phase (I) and quadraphase (Q) sampled data as the input of the preprocessor, which includes three functions: code correlators, integration and dump accumulators, and the cross-product and dot-product operation. In the code correlators, the I and Q signals are correlated with early, prompt, and late replica codes (plus code Doppler) synthesized by the code generator, a 2-bit shift register, and the code NCO. The integration and dump accumulators reduce the sample frequency from MHz to hundreds Hz. The cross-product and dot-product operation use the output signals of the prompt correlators, which produces two measurement value to the navigation filter.

Figure 1. VFLL architecture

The navigation processor can be realized utilize Extend Kalman Filter(EKF) or Unscented Kalman Filter(UKF). UKF is used in this paper, because of the high nonlinear relationship between state vector and measurement vector. Using the navigation processor states and the broadcast navigation data(ephemeris) to control the carrier NCO’s, and then using the correlator outputs to correct the navigation processor’s states, constitutes a Vector Frequency Lock Loop (VFLL).

A. State Vector and Process Equation Because of the high dynamics, the velocity, acceleration

and jerk of the receiver were chosen as the component of the state vector x of the navigation processor,

[ , , ]T=x V A J (1) In (1), the velocity, acceleration and jerk states are all in

the Earth-centered Earth-fixed (ECEF) frame. Each of the three vectors is composed of an x, y and z component. Additional states such as the receiver clock drift state can also be appended to the state vector, but we omit these here for simplicity. The system dynamics of the VFLL are

( 1) ( ) ( )k k k+x = Fx + Gw (2) where

23 3 3 3 3 3

3 3 3 3 3 3

3 3 3 3 3 3

0.5T TT

× × ×

× × ×

× × ×

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

I I IF 0 I I

0 0 I (3)

23 3

3 3

3 3

0.5TT

×

×

×

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

IG I

I (4)

In the formulation above, T is the update time of the navigation processor. Vector ( )kw is the process noise,

( ) ( ) ( ) ( )T

x y zk w k w k w k⎡ ⎤= ⎣ ⎦w (5)

In this model, the white process noise ( )xw k , ( )yw k and

( )zw k are the jerk increment during the k sampling period in the x ,y and z component respectively, and they are assumed to be independent zero-mean white Gaussian sequences. The statistics for the process noise vector ( )kw are

3 1{ ( )}E k ×=w 0 (6) 2 2 2{ } { , , }Tx y zE diag σ σ σ=ww (7)

The values of 2xσ , 2

yσ and 2zσ should be chosen carefully

depend on the dynamic of the receiver.

B. Measurement Vector and Measurement Equation For the ith channel, two in-phase prompt outputs and two

quadrature prompt output are generated by the correlators. The first is formed by integrating the signal over the first half of the navigation processor update time at k, denote as ( ,1)iI k and ( ,1)iQ k . The second is generated from integrating the signal over the second half of the navigation processor update time at k, denote as ( , 2)iI k and ( , 2)iQ k . In this paper, instead of using the discriminator to extract the frequency error between the update time, the cross-product and dot-product operation was used. The cross-product and dot-product operation output for ith channel at the update time k are [2]

99

, ( ) ( , 2) ( , 2) ( ,1) ( , 2)i I i i i iz k I k Q k I k Q k= − (8)

, ( ) ( , 2) ( ,1) ( ,1) ( , 2)i Q i i i iz k I k I k Q k Q k= + (9) Assume there are N satellites in the sky, so the

measurement vector is T

1 2( ) [ ( ), ( ),......, ( )]Nk k k k=z z z z (10) where

, ,( ) [ ( ), ( )]i i I i Qk z k z k=z (11)

Denote ( )tn as the measure noise vector, each component of which has quasi-Gaussian probability density functions. The statistics for the measure noise

2 1{ ( )} NE t ×=n 0 (12)

2 2 2 2 2 2,1 ,1 ,2 ,2 , ,

{ ( ) ( )}{ , , , ,......, , }

T

n n n n n N n N

E t tdiag σ σ σ σ σ σ=

=R n n

(13)

For the ith channel[2] 2 2 4, 2( )n i i iσ σ σ= + (14)

It should be note that because the receiver’s velocity was estimated directly, not the velocity error like [8]. Denote the Doppler frequency error as ( )kδf ,

1 2( ) [ ( ) ( ) ... ( )]TNk f k f k f kδ δ δ=δf (15)

The relation between the Doppler frequency error and the navigation processor’s states as follow

( ) kk γ≈ Δδf H x (16)

Where, kΔx is state error between the state kx and the state

one step prediction / 1ˆ k k −x of the navigation filter,

/ 1ˆk k k k −Δ = −x x x (17) And γ is a constant,

1 / 5.255 Hz/(m/s)Lf cγ = ≈ (18) The matrix H is the augmented geometry matrix,

1 ......TT T T

j N⎡ ⎤= ⎣ ⎦H h h h (19) where,

1 6[ ]i xi yi zi xh h h= h 0 (20)

The terms xih , yih and zih in (11) are the components of the line-of-sight unit vector from the user’s estimated position to the satellite. Because of the large distances between the receiver and the satellites, the linearization in (16) is a very good approximation of the nonlinear pseudorange-rate equation.

C. Navigation Filter Realization Now the detail of the realization of the navigation filter is

given as follow. The augmented state vector and sigma-point vector is

given by [ , ]a T T T=x x n (21)

[( ) , ( ) ]a x T n T=χ χ χ (22)

Assuming / 1ˆ k k −x and / 1k k −P are known， the mean and covariance matrices of the augmented state vector are

/ 1 / 1 1 2ˆ ˆ[ , ]a T Tk k k k x N− −=x x 0 (23)

/ 1 9 2/ 1

2 9 2 2

k k x Nak k

Nx Nx N

−−

⎡ ⎤= ⎢ ⎥⎣ ⎦

P 0P

0 R (24)

(1) Calculate Sigma-point

/ 1 / 1 / 1 / 1 / 1ˆ ˆ ˆ[ , , ]a a a a a ak k k k k k k k k k kς ς− − − − −= + −χ x x P x P (25)

(2) Instantiate each of the prediction points through the measure function

/ 1 ( , )x nk kϑ − = h χ χ (26)

2

, / 10

ˆL

mk i i k k

iω ϑ −

==∑z (27)

2 2

, / 1 , / 10 0

ˆ ˆ[( )( ) ]

ˆ ˆ( )( )

k k

Tk k k k

L Lc Tij i k k k i k k k

i j

E

ω ϑ ϑ− −= =

= − −

= − −∑∑

z zP z z z z

z z (28)

(3) Measurement-update equations

/ / 1ˆ ˆ ˆ( )k k k k k k k−= + −x x K z z (29)

/ / 1 ( )k k

Tk k k k k k−= − z zP P K P K (30)

1( )k k k kk

−= x z z zK P P (31) (4) Time-update equations

1/ /ˆ ˆk k k k+ =x Fx (32)

1/ /T

k k k k k+ = +P FP F Q (33)

The paremeters ς , cijω and m

iω are weight values, for more detail on how to choose them, see reference [10].

Now the state prediction 1/ˆ k k+x will be used to control the carrier NCO.

III. VECTOR TRACKING PERFORMANCE ANALYSIS AND SIMULATION

The VFLL tracking architecture should have advantages in performance over the scalar tracking approach in principle. Since all the signals are tracked in unison, one would expect the signal tracking errors in all channels to be reduced. The performance of the VFLL is inherently scenario dependent, since the discriminator errors contain information in the line-of-sight of the satellites. Under steady-state conditions, if the geometric matrix is given, filter performance may be estimated from the steady-state covariance matrix, which is the solution of an algebraic matrix Riccati equation. However, for the nine-dimensional case under consideration, the expressions become unwieldy and contribute little to our understanding of the problem. Therefore, the performance of the VFLL will be assesses by means of numerical simulations.

The trajectory chosen for simulation is follow by [1], which consists of positive and negative going jerk pulses of

100

0.5s, duration and magnitude of 100g/s, separated by 2s of constant acceleration, the initial conditions for acceleration were chosen for symmetric 25g excursions.

In the scenario, there are eight satellites are tracked by the VFLL, and also assumed the dynamic are along the line of sight from the receiver to the 8th satellite in the simulation. So, all the satellites except 8th only “see” the partly component of dynamics that is along the line of sight between the receiver and the satellite assigned to the corresponding channel. Assumed the direction of the receiver’s velocity is [0.66299, -0.266802, -0.69946], and the project vector of the dynamic on the eight satellites is [0.15931, 0.76162, 0.81986, 0.64384, 0.11619, -0.54365, -0.74596, 1.0000] .

Intermediate Frequency (IF) of the data is 1.5 MHz and the sampling frequency is 9.75 MHz, the navigation update time is 4ms. The simulation results will be given in the next section.

IV. SIMULATION RESULTS The tracking performance of the VFLL is tested on the

basis the estimated root mean square (RMS) frequency estimation error.

0 1 2 3 4 5 6 7 8-20

-15

-10

-5

0

5

10

15

20

25

Time, s

velo

city

err

or,

m/s

Q=100g2

Figure 2. VFLL Velocity error at 24dB-Hz

23.5 24 24.5 25 25.5 265

5.5

6

6.5

7

7.5

8

8.5

CNR, (dB-Hz)

RM

S v

elo

city

Err

or,

m/s

Q=100g2

Figure 3. Tracking Performance of VFLL as a function of CNR

The tracking result of velocity error in one simulation at 24dB-Hz is shown in figure 2. The estimated RMS frequency estimation error is displayed in figure 3, again as a functions of CNR. The VFLL operates with 6.4m/s rms errors at 25 dB-Hz., converted to Doppler frequency is

33.5Hz. It is noted that different sampling method used in this paper and reference [2], we use the IF sampling method. In [2], the baseband sampling schemes is used, so there is no loss due to digital accumulator.

V. CONCLUSION A method of estimating high dynamic GPS signal

parameters based on Vector Frequency Lock Loop (VFLL) is proposed. Instead of tracking the Doppler frequency of individual channel using the traditional loop, the VFLL tracking the velocity, acceleration .etc states of the GPS receiver from the measure value of all the channels. In vector mode, the navigation processor controls the carrier numerically controls oscillators (NCOs) based on the most recent velocity solution of the receiver. The major advantage of this method is that the velocity estimation from all the channels can be more accurate than Doppler frequency estimation from individual channel. In this paper the VFLL architecture is given, and the system dynamics equation and measure equation of the navigation processor are discussed in detail. Instead of using the discriminator, the VFLL focus on the nonlinear estimation. The navigation filter is realized utilize UKF. The performance of the VFLL is tested using simulation, because it is inherently scenario dependent. Form the simulation results, it can be seen that estimation performance is improved compared with FEKF method.

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