GNSS Doppler Positioning (An Overview)

Transcription

1 GNSS Doppler Positioning (An Overview) Mojtaba Bahrami Geomatics CEGE Dept. University College London A paper prepared for the GNSS SIG Technical Reading Group Friday, 29-Aug-2008

2 To be completed... Abstract

3 Contents Nomenclature iv 1 GNSS Doppler Positioning: An Overview Introduction Receiver Doppler Shift Recovery Pseudorange and Delta Range GPS Delta Range Observables Deriving the Doppler Shift Equation The Doppler Shift Equation The Clock Drift Error The Ionospheric Delay Change Rate Mathematical Formulation for Doppler Observables General Least-Square Solution The Gauss-Newton Method Deriving the Partial Derivatives Numerical Results References 12 ii

4 List of Figures 1.1 Frequency shift for a moving transmitter; it illustrates the squeezing of the signal when the transmitter is at motion with speed v towards observer B to the left-hand side of the figure Radial velocity of the satellite related to the receiver GPS Signal Search Area covers Doppler shift, f D, and code phase, τ,.[image Source: Global Locate ] iii

5 Nomenclature Subscripts u subscript user Other Symbols c The speed of light c in vacuum = m/s Dot product, scalar product f R f T λ Euclidian norm, 2-norm, length/magnitude of a vector Received frequency Transmitted frequency The signal wavelength where λ = c/f p s p u p s p u Normalised user to satellite line-of-sight (LOS) vector p s p u θ v prop v radial v s v u The satellite position vector The user/receiver position vector Angle between satellite velocity and satellite-to-user vector The propagation speed of a wave The radial velocity of the satellite related to the receiver The satellite/transmitter velocity vector The user/receiver velocity vector iv

6 Chapter 1 GNSS Doppler Positioning: An Overview 1.1 Introduction As a fundamental principle of the propagation of waves (more specifically, electromagnetic radio waves) the Doppler Effect has been exploited for space-based ranging, positioning and radionavigation since the launch of Sputnik I at the beginning of the Space Age in 1957 [Guier & Weiffenbach (1960)]. The Doppler effect is a change in the apparent frequency of the received signal caused by the relative motion of the emitter and receiver (see Fig. 1.1 page 2). In the context of GNSS radionavigation, the frequency experienced by the receiver can be modelled as [ Misra & Enge (2005) p.16]: f R = (1 + v radial v prop ) f T if moving toward (1 v radial v prop ) f T if moving away (1.1) where 1 f T and f R are the transmitted and received frequencies, respectively; v prop is the propagation speed of the waves which equals the speed of light c in vacuum in this context; v radial is the relative radial velocity between the signal emitter (i.e. the satellite) and the receiver in the line-of-sight (LOS) direction 1 The notation is explained in the nomenclature section :-) 1

7 1.1 Introduction and given by: v radial = ( v s v u ) p s p u p s p u = v s v u cos θ (1.2) where v s is the satellite/transmitter velocity vector and the v u is the user/receiver velocity vector and hence v s v u is the magnitude of relative velocity of the satellite with respect to the receiver; p s p u p s p u is the normalised user to satellite line-of-sight (LOS) vector where p s is the satellite position vector and p u is the user position vector; and θ is the projection angle of the satellite-receiver relative velocity vector ( v s v u ) to the LOS vector (see Fig. 1.2 page 3). The satellite-receiver relative radial velocity v radial can also be thought of as the range rate. Figure 1.1: Frequency shift for a moving transmitter; it illustrates the squeezing of the signal when the transmitter is at motion with speed v towards observer B to the left-hand side of the figure. Eequation 1.1 is a valid approximation when v s v u c. There is a more general Doppler effect equation under the special relativity theory [Zhang et al. (2006)] which can take into account special relativity for high velocity sources: 1+v ( radial 1 ( ) f v s v u 2 /c 2 T ) if moving toward f R = 1 v ( radial 1 ( ) f v s v u 2 /c 2 T ) if moving away (1.3) By rewriting equation 1.1 the apparent change of the frequency of the received signal called the Doppler shift f R is: f R (f R f T ) = ± v radial c f T = ± V radial λ ft (1.4) 2

8 1.1 Introduction where λ ft is the nominal/transmitted frequency wavelength. Note that due to the convention used in this equation, v radial is positive if the receiver and the transmitter approach each other and it is negative if the transmitter-to-receiver range is increasing. Figure 1.2: Radial velocity of the satellite related to the receiver. Equation 1.4 represents the instantaneous frequency shift/doppler. From equation 1.4 the radial (i.e. LOS) velocity of the satellite related to the receiver or the range rate can be derived as [Braasch & van Dierendonck (1999)]: ± v radial c( f R f T ) λ ft f R (1.5) To estimate the radial velocity (sometimes referred to as the delta-range), a receiver may form an average of the radial velocity by simply integrating equation 1.5 over a very short period of time (e.g. 0.1 seconds) and then dividing it 3

9 1.2 Receiver Doppler Shift Recovery by the duration of the integration interval [Braasch & van Dierendonck (1999)]. Equation 1.5 tells us that if we have a means to measure the Doppler shift, f R, then we can get the radial velocity to a satellite, and by forming a system of equations one should be able to estimate the user velocity vector assuming that we can model the satellite velocities. Moreover, if we can model the receiver velocity vector by any external means then a system of equations should yield the receiver position solution. [The author believes that the Doppler observables in the RINEX observation data file format should be a measure of the instantaneous range rate in Hz, and if the time interval for the above mentioned integration is selected small enough, the average of the range rate is the same as the instantaneous frequency shift. Hence this average of the range rate over a very short period of time in Hz is what the Doppler observables in the RINEX observation data file are. Of course this is subject to debate and verification as the author couldn t yet find any references to confirm or reject this assertion. Finally, it is worth mentioning that the phase of the incoming signal are not measured or incorporated to any of the equations we derived so far.] 1.2 Receiver Doppler Shift Recovery In writing this section of the paper [Parkinson & Spilker (1996), Misra & Enge (2005) and Kaplan & Hegarty (2005)] works were used extensively. It is mentioned here because these works had a great influence on the author and simple references through the section are an insufficient acknowledgement of their importance. At a transmitting satellite, a GPS L1 C/A signal is typically modelled as: s L1 (t) = 2P C1xmit D(t)x(t) cos(2πf L1 t + θ xmit ) (1.6) After a transmit time of about 70 milliseconds the GPS signal arrives at the receiver antenna. There the signal is modelled as: 4

10 1.2 Receiver Doppler Shift Recovery s L1 (t) = 2P C1rcvd D(t τ)x(t τ) cos(2π(f L1 + f D )t + θ rcvd ) + ε(t) (1.7) where the subscript xmit refers to transmitted and rcvd refers to received. τ is the C/A code delay/shift/code phase (with respect to the generated replica code in the receiver), f D is the Doppler shift, and ε(t) is the noise. This GPS civil signal has amplitude 2P C1 and is modulated with the spread spectrum C/A code, x(t), and the navigation data, D(t). The radio frequency carrier of the civil signal is cos(2πf L1 t + θ xmit ) where the phase, θ, is waveform phase shift either earlier or later along the time axis and has a value of 0 to 360 degrees or 0 to 2π radians and f L1 is the GPS L1 centre frequency at MHz. Broadly speaking, the receiver job is to estimate τ which contains the basic range and time information required for position and time, f D which contains the pseudorange rate information used to compute the user velocity and clock frequency, and θ for precise carrier phase positioning. However, before the signal acquisition stage, the receiver s front-end firstly filters the received signal to remove the interfering signals in neighbouring frequency bands. Secondly, it amplifies the power of the signal by approximately while reducing the carrier frequency of MHz to a more manageable intermediate frequency (IF) through a process known as frequency down conversion. Finally, it passes through an A/D (Analog-to-Digital) converter to perform sampling and digitisation. The GPS signal from one satellite is now ready for the signal acquisition stage and if the sampling rate were high enough, for the sake of clarity it can be modelled as a continuous function of time (although it is a discrete time signal at this stage): s(t) = CD(t τ)x(t τ) cos(2π(f IF + f D )t + δθ) + ε(t) (1.8) The signal acquisition stage is essentially a 2-dimensional search (see Fig. 1.3) for accurate estimation of (τ, f D ) performed by a code replica generator, a set of correlators and a numerically controlled oscillator (NCO). Depending on the type of the receiver, it also may include estimation of the carrier phase offset, δθ. Typically, before this search can be carried out the signal undergoes two more processes, namely, carrier wipeoff and code wipeoff. The NCO generate two refer- 5

11 1.2 Receiver Doppler Shift Recovery Figure 1.3: GPS Signal Search Area covers Doppler shift, f D, and code phase, τ,.[image Source: Global Locate ] 6

12 1.2 Receiver Doppler Shift Recovery ence signals: Inphase (cosine signal) and Quadrature (sine signal). For simplicity we write the mathematical model of the Inphase reference signal: s ref (t) = 2 cos(2π(f IF + ˆf D )t + ˆθ) (1.9) This Inphase reference signal is then multiplied by the Inphase received signal given by equation 1.8 and the product goes through a low-pass filter. The output of the filter from the Inphase is: s(t) = CD(t τ)x(t τ) cos(2π(f D ˆf D )t + (δθ ˆθ)) (1.10) As you can see from equation 1.10, the signal is no longer modulated by the carrier frequency or any intermediate frequency. This is the carrier wipeoff process and the signal is only modulated by the (f D ˆf D ) or the difference between true Doppler, f D, and the receiver s (more strictly speaking the NCO s) best estimate of Doppler, ˆfD. Moreover, the phase of this resulting signal is the difference between the input phase, δθ, and the receiver s best estimate of the phase, ˆθ. Parallel to the Inphase processing there would be a Quadrature processing so the receiver can overcome the amplitude fading problem when the Doppler shift is equal to zero. This also enables the receiver to distinguish between a positive Doppler and a negative Doppler when the phase offset is equal to zero. As a result of this carrier wipeoff (i.e. I/Q processing) the receiver knows the Doppler error or f D which at the start of the acquisition could be as large as ±6000Hz: f D = (f D ˆf D ) (1.11) The next step is the code wipeoff performed by correlators to get some estimate of the error in the code delay/shift τ which at the start of the acquisition could be as large as 1023 chips: τ = (τ ˆτ) (1.12) By having an approximate knowledge of the Doppler error and the code delay/shift, ( f D, τ), the receiver can perform the signal acquisition search over 7

13 1.2 Receiver Doppler Shift Recovery the ( f D, τ) two dimensional space (see Fig. 1.3) where τ of 1023 chips is along one axis, and f D of -6KHz to +6KHz is along other axis. The search over the code shift error, τ, usually is broken to half a chip steps because the main auto-correlation peak is only one chip wide along the τ axis, hence 2046 steps required to cover the length of the C/A code. The search over the Doppler error, f D, usually is broken into bins of approximately 500Hz wide, hence 24 steps over the Doppler shift. The search dwells at each possible grid value of ( f D, τ) long enough to determine whether a high auto-correlation pick is present or not. In the worst case scenario, we may need 2046x24 = 49, 104 grid tests to find the true auto-correlation pick. The result of this signal acquisition stage is a rough estimates of the code phase/shift, τ, and the Doppler frequency, f D. From this stage the signal tracking process begins which normally consists of two feedback loops: the Delay Lock Loop (DLL) and Phase Lock feedback Loop (PLL) (or maybe the Frequency Lock Loop (FLL)). Their job is to refine the initial estimate of code shift/phase and the Doppler frequency and track changes into the future. The description and details of these feedback control loops (i.e. DLL and PLL) is out of the scope of this paper. It can be concluded that either the receiver outputs the measured instantaneous frequency shift or the range rate in Hz as Doppler observables, or it is possible that the receiver forms an average of the range rate by accumulating the number of Doppler cycles over a short period of time (e.g. less than a second), then using equation 1.5, scaling the accumulated value by the wavelength of the nominal frequency and dividing by the duration of the integration interval. This is known as a delta-range measurement and it is possible that the receiver outputs that as Doppler observables. The Integrated Doppler or accumulated delta range (i.e., change of range) is formed by the receiver if equation 1.5 is integrated (i.e. the Doppler count is kept running continuously). As the integral of velocity is displacement or change of range relative to the start of the integration, the integrated Doppler represents the change of range of a satellite to a receiver over the observation period. If the receiver uses PLL feedback control and keep track of changes in the incoming signal phase, after the initial Doppler frequency estimation the phase change (which has a 0 counter crossing) is compared with the measured value to get the precise value of Doppler cycles, and accumulates those 8

14 1.3 Pseudorange and Delta Range cycles to the integrated Doppler along the way. At the end of a given integration interval, a whole number of Doppler cycles will have been counted, and possibly phase as a fraction of a cycle would remain. If this fractional phase is also measured and added to the integrated Doppler then the resulting observable is, of course, the carrier phase measurement [Braasch & van Dierendonck (1999) and Kaplan & Hegarty (2005)]. Based on this account, one can say that the Doppler frequency shift (i.e. Doppler observable) is a by-product of the carrierphase measurements. However, [Xu (2003)] Page 38 mentions that the Doppler frequency shift is an independent observable and a measure of the instantaneous range rate. 1.3 Pseudorange and Delta Range In this section we try to show that the Doppler shift is the delta range or the time derivative of the pseudorange i.e. where dρ λ ft d t f R (f R f T ) = ± v radial c f T = ± v radial λ ft = ± dρ λ ft d t (1.13) is the time derivative of the pseudorange in units of Hz. This will help us relate pseudorange observables to Doppler observables and derive the physical (mathematical) model for Doppler observables GPS Delta Range Observables GPS is Time-Of-Arrival (TOA) radionavigation system. Based on the PRN timestamp (C/A code) on the GPS L1 signal a GPS receiver is able to measure the signal propagation time and hence the range to the satellite. This is called the pseudorange observable ρ and has the following mathematical model: ρ (t) = c τ + c [δt u (t) δt s (t τ)] + Ion + T rop + Mp + ɛ ρ(t) = p s p u +c [δt u (t) δt s (t τ)] + Ion + T rop + Mp + ɛ ρ(t) (1.14) where τ is the signal propagation time from the satellite to the receiver in seconds and t is the reception time; p s p u is the true range from the satellite to 9

15 1.4 Deriving the Doppler Shift Equation the receiver; δt u and δt s (t τ) are the receiver clock offset at reception time t and the satellite clock offset at transmit time (t τ), respectively. Ion and T rop reflect signal propagation delay through the ionosphere and the troposphere, respectively; Mp is the multipath effect on the pseudorange observable and ɛ ρ(t) is other biases and noises on pseudorange observable that are not modelled. The magnitude of the multipath effect/error differ significantly for the pseudorange observables and the carrier phase observables. Typical multipath error in carrier phase measurements varies from 1 to 5 centimeters while the corresponding errors in pseudorange measurements are typically two order of magnitude bigger (1-5 meters) and in highly reflective environment when only reflected signals are received this can reach up to hundred meters Misra & Enge (2005). As mentioned in the previous section the delta-range or Doppler shift is a by-product of the carrier-phase measurements hence the Doppler observable should suffer from typical multipath error as in carrier phase measurements. Assuming a perfect condition with little multipath we bundle the Mp and the ɛ ρ(t) together. In the following we try to derive the time derivative of the pseudorange ρ (t) : ρ (t) = dρ (t) d t = d ( p s p dt u +c [δt u (t) δt s (t τ)] + Ion + T rop + ɛ ) ρ(t) conclusion... = ( v s v u ) p s p u p + c [δṫ s p u u (t) δṫ s (t τ)] + Ion + T rop + ɛ ρ(t) (1.15) 1.4 Deriving the Doppler Shift Equation using Dot Product... 10

16 1.5 Numerical Results The Doppler Shift Equation The Clock Drift Error The Ionospheric Delay Change Rate Mathematical Formulation for Doppler Observables General Least-Square Solution The Gauss-Newton Method Deriving the Partial Derivatives 1.5 Numerical Results 11

17 References Braasch, M. & van Dierendonck, A. (1999). Gps receiver architectures and measurements. Proceedings of the IEEE, 87, , 4, 9 Guier, W. & Weiffenbach, G. (1960). A satellite doppler navigation system. Proceedings of the IRE, 48, Kaplan, E.D. & Hegarty, C.J., eds. (2005). Understanding GPS: Principles and Applications. Artech House, second edition edn. 4, 9 Misra, P. & Enge, P. (2005). Global Positioning Sytem: Signals, Measurements and Performance. Ganga-Jamuna Press, Lincoln, Massachusetts, 2nd edn. 1, 4, 10 Parkinson, B.W. & Spilker, J.J., eds. (1996). Global Positioning System: Theory and Applications Volume I, vol. I. American Institute of Aeronautics and Astronautics. 4 Xu, G. (2003). GPS Theory, Algorithms and Applications. Springer. 9 Zhang, J., Zhang, K., Grenfell, R. & Deakin, R. (2006). Short note: On the relativistic doppler effect for precise velocity determination using GPS. Journal of Geodesy, 80,