GPS Digital Tracking Loops Design for High Dynamic Launching Vehicles

Transcription

1 26 IEEE Ninth International Symposium on Spread Spectrum Techniques and Applications GPS Digital Tracking Loops Design for High Dynamic Launching Vehicles Pedro A. Roncagliolo, Cristian E. De Blasis and Carlos H. Muravchik Laboratorio de Electrónica Industrial, Control e Instrumentación (LEICI), Dto. Electrotecnia, Facultad Ingeniería, UNLP, CC 9, 9, La Plata, Argentina. Abstract This paper describes the design of digital tracking loops for GPS receivers in a high dynamics environment, without external aiding. We adopted the loop structure of a frequencylocked loop (FLL)-assisted phase-locked loop (PLL) and design it to track accelerations steps, as those occurring in launching vehicles. We used a completely digital model of the loop where the FLL and PLL parts are jointly designed, as opposed to the classical discretized analog model with separately designed FLL and PLL. The new approach does not increase the computational burden. We performed simulations and real RF signal experiments of a fixed-point implementation of the loop, showing that reliable tracking of steps up to 4g can be achieved. I. INTRODUCTION A Global Positioning System (GPS) receiver estimates user s navigation parameters (i.e. position, velocity, etc) measuring its distance to the satellites of the GPS constellation. This process requires precise synchronization with the GPS signals, which is typically accomplished by means of tracking loops. Since they are direct sequence spread spectrum (DS- SS) signals, code and carrier tracking loops are employed. The data modulation is BPSK at 5bps, and then the correlations utilized to de-spread the signal have to be performed within a data bit interval of 2ms. Transmitter and receiver relative dynamics causes Doppler effect in the carrier and code frequency of the received signal thus requiring tracking capability to the loops. The typical tradeoff in tracking loop design is bandwidth versus dynamic performance: the effects of noise increase with increasing loop bandwidth, while dynamic tracking errors increase with decreasing loop bandwidth []. Nowadays, the loop structure known as FLL-assisted-PLL [2] is very often adopted for GPS receivers. Rather than using a single loop, it consists of a PLL and a FLL in a coupled mode, to reduce locking times and avoid false locks. The advantages of adding the FLL to track spread spectrum signals in dynamic environments were already studied in [3]. Notably, designers continue relying on discretized analog prototypes, and separately designing each loop, even for some high dynamic applications [4]. The method of discretizing the well-known results of analog PLL is satisfactory for loop bandwidths less than a tenth of the sampling rate of the discrete loops [3]. This sampling rate is in turn limited by the correlation time, which is typically ms (half of a This work was supported by the Comisión Nacional de Actividades Espaciales (CONAE). Additional funding was provided by ANPCyT, CON- ICET, CIC-PBA and UNLP. s: agustinr@ing.unlp.edu.ar, cristiandeblasis@hotmail.com, carlosm@ing.unlp.edu.ar data bit). Receivers for high dynamic vehicles often require larger loop bandwidths, leaving impractical this condition. This is because it implies short integration times that preclude correlators to collect sufficient energy to de-spread the signal. To overcome this limitation other receiver structures have been proposed in [], claiming tracking capability up to 5g of acceleration, in contrast with the 5g regularly assigned to tracking loops. However, the required computational burden is large since several simultaneous correlation calculations and FFT computations are needed. In the case of launching vehicles, the critical moments are the engine turn-on and turn-off that produce large changes of acceleration. This dynamics can be modelled by acceleration steps, i.e. with unbounded derivative, the jerk, differently from the tests used in [4]. The fact that the analog loops optimized to respond to this kind of excitation loose their optimality when discretized [5] seems to be less well known. In this paper we show a careful design of the digital loops that can expand their tracking ability to acceleration steps up to 4g. Our focus will be on carrier loops, because the carrier shares the same dynamics than the code. Then, the estimation of the carrier can be used to aid the estimation of the code, and it suffices using a first order code loop [6]. In the next section the digital model of FLL-assisted-PLL described in [7] is reviewed, emphasizing the way the loops influence each other and its effect on the design. In section 3 the optimum digital loop filter design proposed in [8] is applied leading to a computationally inexpensive filter structure. Only a single operation, or pole, is added to the classical third order PLL, assisted by a second order FLL. In section 4, we validate our design via simulations and real RF signal tests performed on a GPS receiver with loops implemented in fix-point arithmetic. Finally, the conclusions are presented in section 5. II. DIGITAL LOOPS MODEL The in-phase and in-quadrature correlation of the received signal with the locally generated replicas are inputs to the tracking loops. Leaving aside the noise for simplicity, they can be expressed for the n-th integration interval as [9] I n = AD n TR(Δτ n )sinc(δf n ) cos(πδf n +Δθ n ) () Q n = AD n TR(Δτ n )sinc(δf n )sin(πδf n +Δθ n ) (2) where Δτ n = τ n ˆτ n is the code delay estimation error, Δf n = f n ˆf n is the frequency estimation error, Δθ n = /6/$2. 26 IEEE 4

2 f φ[n] ˆφ[n] z [ ] f π 2 e f [n] z [ ] π p 3 p 2 e p [n] p z z z z Fig.. Block diagram of the FLL-assisted-PLL structure. θ n ˆθ n is the phase estimation error, T is the integrations time, A is the received signal amplitude, R( ) is the code correlation function, and sinc(x) =sin(πx)/(πx). This expression assumes that the data bit D n and the received signal frequency f n remain constant during the integration time. This assumption is equivalent to approximate the received signal phase by a piecewise linear function, with initial value θ n.it is also assumed that the local replica in the n-th integration interval has initial phase ˆθ n and constant frequency ˆf n. In tracking conditions (i.e. after the acquisition process has been completed [6]), estimation errors are small and then the functions sinc( ) and R( ) can be approximated by. In this case () and (2) reduce to I n = AD n T cos(δφ[n]) (3) Q n = AD n T sin(δφ[n]) (4) where Δφ[n] = φ[n] ˆφ[n], with φ[n] = πf n + θ n and ˆφ[n] =πf ˆ n + θ ˆ n as defined in [7]. Sequences φ[n] and ˆφ[n] can be interpreted as the average phase in the n-th integration interval of the received and locally generated signal (assuming the piecewise linear approximation mentioned before). These sequences allow to model the carrier tracking loop as a purely digital single-input single-output (SISO) system. The phase estimation error is obtained from the correlations using one of several possible discriminators [6]. In this work we chose e p [n] =tan (Q[n]/I[n]) = [Δφ[n]] π (5) because it is optimal (maximum likelihood estimator), it is not amplitude dependent, and the computational burden of calculating tan ( ) can be avoided with a short lookup table, since in practice I and Q are necessarily quantized to a few bits. The notation [ ] π indicates that the value Δφ[n] is kept within the interval ( π 2, π 2 ] by adding or subtracting π as many times as is needed. A four-quadrant tan ( ) is not appropriate because the discriminator becomes sensitive to the BPSK data modulation. As it was explained in [7] the frequency discriminator for the FLL can be obtained by simple difference of two consecutive outputs of the phase discriminator, but keeping the result in the ( π 2, π 2 ] interval, that is e f [n] =[e p [n] e p [n ]] π (6) Fig. shows a block diagram of the FLL-assisted-PLL structure presented in [2], where the loop is close to equilibrium. The three lower branches (with gains p, p 2 and p 3 ) form the PLL loop filter, and the two upper branches (with gains f and f 2 ) form the FLL one. The two delays included in the loop deserve some explanation. One of them is due to the time spent in the I and Q calculation. The other appears because the estimated values used in the present correlations have to be known before the calculations begin. That is, the value ˆφ[n] is obtained with the loop filter output of the (n )- th integration interval, which in turn is calculated with the estimation errors of ˆφ[n 2]. In locked condition Δφ[n] is small enough to consider that e p [n] =Δφ[n] and e f [n] =Δφ[n] Δφ[n ], justifying a linear analysis of the loop. Moreover, it is possible to consider the complete loop as an equivalent PLL with filter coefficients p 3, p 2 + f 2 and p + f, instead of p 3, p 2 and p. In this way the influence of the FLL can be inserted into the model of the PLL at a design stage. This eliminates the restriction of using a narrow bandwidth FLL not to significantly perturb the PLL behavior, as is done in [2] and [4]. The use of a wide bandwidth FLL allows the loop to have two modes of operation: Phase Locked as it was described before, and Frequency Locked when the dynamics unlocked the PLL but the FLL kept the frequency error within the linear range of its discriminator. In this mode the loop response is governed by the FLL (the filter coefficients f and f 2 ) and the phase error input acts like a zero-mean perturbation, as explained in [7]. As soon as the dynamics lets the loop reduce its frequency error close to zero, the phase lock can be restored. III. LOOP FILTER DESIGN The method to design the digital loop filters is based in an optimization process that poses the mentioned tradeoff in a quadratic functional which is minimized for a particular 42

3 z [ ] π e f [n] f f 2 φ[n] ˆφ[n] z [ ] π e p [n] p 3 z +Cz ( z ) 2 z Fig. 2. Block diagram of the designed tracking loop. dynamic input. The details of this fully digital method have been presented in [8]. As we said before, for the launching vehicles we considered, the dynamic input was an acceleration step, which becomes a quadratic ramp in terms of phase and a linear one, in terms of frequency. For these inputs the optimal loop filters for our digital FLL-assisted PLL are the classical second and third order filters plus an extra pole, leading to a third order FLL and a fourth order PLL. The aim of the filters design was to produce loop operation in Phase Locked mode with steps up to g and in Frequency Locked mode up to 2g of acceleration. These requirements were too demanding using the commonly adopted integration time of ms, and then it was lowered to 5ms (at the cost of almost doubling the processor load). It has to be noticed that an integer number of correlations have to fit a bit interval of 2ms. As a rule of thumb based on keeping a reasonable distance from the pull-out values of the loop, we assigned up to half of the linear range of the phase discriminator (an eighth of cycle) to the peak of the error transient, as it will be shown in the next section. The desired filter transfer function becomes F (z) = A Bz + Cz 2 ( z ) 3 ( + Cz (7) ) where A =.673, B =.5 and C =.5. Then, in the structure of Fig. this implies p + f = C =.5, p 2 + f 2 = B 2C =.5, and p 3 = A B +C =.23, plus a block that implements the extra pole in z = C. The resulting PLL equivalent noise bandwidth is B N =75.6Hz. Since the FLL design does not affect the previous results, it was designed wider than strictly necessary in order to facilitate the posterior implementation. The selected transfer function is D Ez F (z) = ( z ) 2 ( + Ez (8) ) where D =.6 and E =.5, resulting that the extra pole needed for the FLL and the PLL is the same. Then, f = E =.5and f 2 = D E =., that imply p =and p 2. With these simplifications the complete loop design reduces to the diagram showed in Fig. 2. This FLL loop can track steps up to 4g with transient error peaks smaller than 25Hz, half of the linear range of the frequency discriminator, with an equivalent noise bandwidth of B N =6.3Hz. Estimated Frequency [Hz] 2 2 Fig Tracked frequency during a g test. IV. RF TESTS MEASUREMENTS After extensive simulations the previously designed loop was implemented in a System Developer Kit (SDK) for GPS receivers from SiRF []. Due to the real time nature of this task all calculations of the loop have to be done in a fraction of 5ms. They were programmed in fixed-point arithmetic, using some scaling and approximating coefficient values by powers of 2. Details of these implementations were given in []. To verify the tracking capability of the loop with real signals and without relying in expensive equipment like a GPS signal simulator, two methods were used. The first of them consisted in producing software perturbations to the loop, equivalent to the desired driving dynamic. Results obtained with this method will not be shown here due to space limitations, but can be found in []. The second method was to use a RF signal generator to produce a frequency modulated carrier at MHz (the L GPS frequency). The signal was not spread with the code of a particular satellite and then the code generators at the GPS receiver had to be turned off during the test. This is not a limitation since the focus is on the carrier loops. A triangular waveform was used as modulation to simulate steps in acceleration. The frequency 43

4 Phase Error [cycles] Estimated Frequency [Hz] Fig. 4. Phase error during a g step. Fig. 5. Tracked frequency during a 2 g test. deviation was selected according to the magnitude of the step (an instantaneous frequency deviation of Δf corresponds to a λ.δf instantaneous velocity, where λ is the L wavelength). The selected carrier power was 3dBm. Taking into account the noise of the 5Ω output resistance of the generator gives a C/N =43dB/Hz, which is a typical value for GPS receivers. A test of g acceleration steps is shown in Fig. 3; depicting the tracked frequency de-trended by a linear fit that accounts for the local clock drift. The amplitude of the triangular waveform was increased gradually up to the desired value to avoid large frequency steps. The measured transient response at the output of the phase discriminator in one of the steps is shown in Fig. 4. The simulated response of the modelled loop (as in Fig. 2) is also displayed in the same Fig. It can be seen that the loop is properly characterized and that the phase locked condition has been kept, with the peak of the error transient occupying half of the linear range of the phase discriminator. The same experiment was repeated for acceleration steps of 2g. The estimated frequency is shown in Fig. 5, the measured and the simulated phase error for a particular step are depicted in Fig. 6 and the frequency error at the same epoch is illustrated in Fig. 7. In this case the PLL looses the lock condition for a moment however, the loop can still track the dynamics because of the FLL. The vertical axis of the last figure shows the complete output range of the frequency discriminator to emphasize that the amplitude of the transient is significantly small. As a last experiment steps of 4g were generated. The tracked frequency and phase and frequency error transients are presented is Figs. 8, 9 and, respectively. Again the loop operates for a moment in frequency locked mode, using only half of the frequency discriminator range as it was anticipated, returning after a short time to the frequency and phase locked mode. Phase Error [cycles] Frequency Error [Hz] Fig. 6. Phase error during a 2 g step Fig. 7. Frequency error during a 2 g step. 44

5 Estimated Frequency [Hz] V. CONCLUSION A design of digital tracking loops for GPS receivers for high dynamic environments without external aiding has been presented. The known structure of FLL-assisted PLL [2], but with the fully digital conception in [7], led to a carrier loop that operates in phase locked condition normally, and in frequency locked condition if the dynamic becomes too severe. The effect of coupling the FLL to the PLL is considered at the design stage allowing a fine control of effective loop bandwidths. The loop filters are designed to be optimum for acceleration steps, the kind of dynamic expected in launching vehicles. The designed loops were implemented in a GPS receiver using fixed point arithmetic and tested with frequency modulated RF signals. Simulations and experimental results confirm that a careful design of the digital loops can expand their tracking capacity up to acceleration steps of 4g. Phase Error [cycles] Frequency error [Hz] Fig. 8. Tracked frequency during a 4 g test Fig. 9. Phase error during a 4 g step. REFERENCES [] W. J. Hurd, J.I. Statman, and V. A. Vilnrotter, High dynamic gps receiver using maximun likelihood estimation and frequency tracking, IEEE Transactions on Aerospace and Electronic Systems, vol. 23, no. 4, pp , July 968. [2] P. W. Ward, Performance comparisons between fll, pll and a novel fll-assisted-pll carrier tracking loop under rf interference conditions, in Proceedings of The th International Technical Meeting of The Satellite Division of The Institute of Navigation, ION GPS-98, Nashville, Tennessee, September 998. [3] Ch. R. Cahn, D. K. Leimer, Ch. L. Marsh, F. J. Huntowski and G. D. Larue, Software implementation of a pn spread sectrum receiver to accomodate dynamics, IEEE Transactions on Communications, vol. 25, no. 8, pp , August 977. [4] S. B. Son, I. K. Kim, S. H. Oh, S. H. Kim and Y. B. Kim, Commercial gps receiver design for high dymanic launching vehicles, in Proceedings of The 24 International Symposium on GNSS/GPS, Sydney, Australia, December 24. [5] S. C. Gupta, On digital phase-locked loops, IEEE Transactions on Communication Technology, vol. 6, no. 2, pp , April 968. [6] E. D. Kaplan, Understanding GPS: Principles and applications. Boston: Artech House, 996. [7] J. A. Areta, P. A. Roncagliolo and C. H. Muravchik, Sincronización digital de señales de espectro expandido con perturbaciones de alta dinámica, in Proceedings of X Reunión de trabajo en Procesamiento de la Información y Control, San Nicolás, Buenos Aires, Argentina, September 23, pp [8] P. A. Roncagliolo, C. E. De Blasis and C. H. Muravchik, Seguimiento de señales de espectro expandido: Filtros de lazo óptimos, in Proceedings of XI Reunión de trabajo en Procesamiento de la Información y Control, Río Cuarto, Córdoba, Argentina, October 25, pp [9] B. W. Parkinson and J. J. Spilker (eds.), Global Positioning System: Theory and Applications. Washington: American Institute of Aeronautics and Astronautics (AIAA), 996. [] SiRF technology, [] C. E. De Blasis, Seguimiento de señales de GPS, 25, Electronic Eng. graduation project, UNLP, Argentina Fig.. Frequency error during a 4 g step. 45