NEAR-FAR RESISTANT PSEUDOLITE RANGING USING THE EXTENDED KALMAN FILTER

Pseudolites have been proposed for augmentation/replacement of the GPS system in radiolocation applications. However, a terrestrial pseudolite system suffers from the near-far effect due to received power disparities. Conventional code tracking loops as employed in GPS receivers are unable to suppress near-far interference. Here, a multiuser code tracking algorithm is presented based on the extended Kalman filter (EKF.) The EKF jointly tracks the delays and amplitudes of multiple received pseudolite waveforms. A modified EKF based on an approximate Bayesian estimator (BEKF) is also developed, which can in principle both acquire and track code delays, as well as detect loss-of-lock. Representative simulation results for the BEKF are presented for code tracking with 2 and 5 users. KEY WORDS GPS, pseudolites, extended Kalman filter, code tracking, code acquisition. INTRODUCTION There is great interest in using pseudolites for augmentation and/or replacement of GPS SVs for ranging, in applications where severe fading and interference are present. However, the near-far problem is a major obstacle to high-accuracy pseudolite ranging, and has typically been solved using time-division access protocols [FAC98]. Unfortunately, time-division or contention strategies may become 1 This work was supported in part by a grant from the International Foundation for Telemetering. impractical when high-accuracy ranging requires the use of many pseudolites, or when platforms are highly mobile. Multiuser detection (MUD) is an alternative strategy for minimizing near-far interference (see [Ver98] for an overview of MUD,) which has been extensively investigated in the context of spread-spectrum communications. However, most MUD algorithms assume prior knowledge of channel parameters such as code delay and multipath. In pseudolite navigation, the data rate is very low (50 Hz for GPS signals,) and the primary problem is code delay estimation rather than data detection. Multiuser channel estimation (MCE) is the counterpart to MUD which includes the code delay and channel estimation problems. However, most previous work in MCE has focussed on communications rather than radiolocation problems. Representative MCE approaches include the MUSIC algorithm [PSM99] and approximate maximum-likelihood (ML) techniques [BeA98]. The extended Kalman filter has also been applied to code tracking [LiR97], [IlM94] but again, with the focus on communications rather than radiolocation applications. Here, an approximate Bayesian estimator for the code delay and signal amplitudes is developed for MCE. The overall BEKF algorithm employs an extended Kalman filter to update conditional delay/amplitude estimates, along with a suboptimal acquisition hypothesis merging strategy. PSEUDOLITE SIGNAL MODEL The following received signal model corresponds to a digital receiver, where the incoming waveform is sampled at a rate 1/Ts. r(n) = ck (n)ak (n)sk (n,τ k (n)) + (1 − ck(n))v k(n) + n(n) k=1 K ∑ k =1 K ∑ . (1) In (1), received Nyquist samples r(nTs) have been grouped into vectors of length Ns defined by r(n) = [r(nNsTs ),r((nNs +1)Ts ),K,r(((n +1)Ns −1)Ts )] T . The vectors r(n) thus correspond to a superposition of pseudolite waveforms sk(n,τk(n)) from K users. The variables ck(n) = 0,1 are binary and equal “1” if user k has been acquired, and zero otherwise. Thus, if a user is not acquired, it is modeled by additive white Gaussian noise vk(n) as an approximation to the pseudo-random ranging waveform. The vector n(n) represents thermal noise in the receiver, and is likewise white Gaussian, with covariance matrix σnI. The actual signal vectors are defined by sk (n,τ k (n)) = [sk(nNsTs − τ k (n)), sk ((nNs +1)Ts − τ k (n)),K,sk (((n + 1)Ns −1)Ts −τ k(n))] T , where τk(n) is the unknown delay of user k. The transmitted signal sk(t) is determined by the pseudolite transmitter design. For example, sk(t) could be a GPS waveform, or an alternative pseudo-random binary sequence. The continuous time signal is sk (t) = ck ,ng(t − nTc − mNTc ) n= 0 N −1 ∑ m = −∞ ∞ ∑ , (2) where g(t) is the chip pulse shape, bandlimited to the Nyquist frequency 1/(2Ts) Hz., and ck,n is the binary pseudorandom sequence for user k with period N. For convenience in the BEKF derivation, the vector model (1) is rewritten as follows r(n) = S(n,τ (n))A(n)c(n) + V(n) 1 − c(n) [ ] + n(n) (3) where S(n,τ ) = s1(n,τ 1),s2 (n,τ 2 ),K,sK (n,τ k) [ ], V(n) = v1(n), v2 (n), ...,vK (n) [ ] A(n) = diag a1(n),a2(n),K,aK (n) { } c(n) = [c1(n),c2 (n),K,cK (n)] T and 1 is the all-ones vector. Note that there are 2 possible vectors c(n), corresponding to the acquisition hypotheses. DERIVATION OF THE BAYESIAN EKF (BEKF) The acquisition/code tracking problem is to estimate the vector of unknown delays τ and amplitudes A(n). Thus, define the state vector x(n) = [a1 (n),τ1 (n),a2 (n),τ 2 (n),K, aK (n),τ K (n)] T . The process x(n) is assumed to evolve according to x(n + 1) = Fx(n) + w(n +1) , (4) where F is typically a diagonal matrix, and w(n) is a white Gaussian noise process with covariance matrix Q. The posterior density of x(n) is the most complete estimator in the Bayesian framework, and is denoted by p(x(n)|r), where r = {r(n),r(n-1),…,r(0)} is the cumulative measurement history. If the linearization error in the EKF is ignored, Kalman filter theory yields [AnM79] p(x(n) | r ,c ) = N x(n); ˆ x c(n | n),Pc (n | n) ( ), (5) where xc(n|n) and Pc(n|n) are the (extended) Kalman filter measurement update and error covariance matrix, conditioned on the cumulative acquisition hypotheses c={c(n),c(n-1),…,c(0)}. The density N(x;m,P) is circular Gaussian with mean vector m and covariance matrix P. The unconditional posterior density of x(n) is then obtained by the sum p(x(n) | r n ) = p(x(n) | r ,ci n i =1 2( n+1)K ∑ )p(ci n | r ) . (6) Unfortunately, the estimator (6) has complexity which grows exponentially with both the number of users and with time, and is clearly impractical. Specifically, (6) describes a tree with separate EKFs evolving along the branches, and with weighting probabilities given in terms of innovations likelihoods. In order to develop a practical code tracking and acquisition algorithm, an approximate Bayesian estimation strategy is used, in which the predicted density p(x(n)|r) is approximated as unimodal circular Gaussian. This approach was suggested by the probabilistic data association filter (PDAF) widely used in multitarget tracking [BaF88]. The sequence of approximations is then (a) p(x(n) | r ) ≈ N x(n); ˆ x (n | n −1),P(n | n −1) ( ) (b) p(x(n) | rn ,c i(n)) ≈ N x(n); ˆ x i(n | n),Pi(n | n) ( ) (c) p(x(n + 1) | r) ≈ N x(n +1); ˆ x (n +1 | n),P(n +1 | n) ( ), (d) ˆ x (n | n) = ˆ xi(n | n) i =1 2K ∑ p(c i(n) | r ), ˆ x (n +1| n) = Fˆ x (n | n) (e) P(n +1 | n) = E Fˆ x (n | n) − x(n +1) [ ][ ] H | r n { } (7) The key approximations are (b), in which a single EKF estimate is computed for each of 2 acquisition vectors c(n), and (d), in which the one-step prediction is formed by the Bayesian combining of the conditional EKF estimates. The overall BEKF algorithm is summarized in Table 1 for the code delay tracking problem. The Jacobian matrix is key to the EKF derivation, and is defined by H i(n) = ∂S(n,τ )Aci (n) ∂x x= ˆ x(n |n −1) (8) where ∂/∂x is a row gradient operator. The actual derivatives are computed following the EKF derivation in [IlM94]. ACQUISITION STRATEGY The EKF will diverge, or fail to acquire, if the initial error between the code delay estimate and actual delay is greater than half a chip. In a practical system, a separate acquisition algorithm, based on a method such as MUSIC [PMS99] or ML [BeA98] will be required. However, the BEKF has some ability to perform acquisition using the following method. Note that p(ci(n)|r) is the probability of the ith acquisition hypothesis. Let αk denote the event that user k is acquired, that is, that its code delay estimate is within half a chip of the true delay. The probability of αk is then given by p(α k | r n ) = p(c i(n) | r n ) i : c i (n) ( )k =1 { } ∑ (9) The acquisition strategy is as follows. Let Th represent an acquisition threshold. If p(αk|r) is above Th, then τ(n|n) is updated according to the BEKF equations in Table 1. However, if p(αk|r) falls below the threshold, an increment τinc, given by some small fraction of a chip, is added to the estimate τ(n|n). Thus, the code delay estimate is forced to increment through the entire uncertainty region until acquisition is detected. SIMULATION RESULTS The BEKF algorithm was simulated for a radiolocation system with K = 5 and K = 2 users. The binary sequences ck,n were length N = 31 Gold codes. A data-free preamble is assumed for radiolocation, so that ak(n) is constant, with magnitude equal to the square root of the relative received power over the simulation run. Figure 1 shows the performance of the BEKF in tracking mode, where all code delay estimates are initialized to within half a chip of the true delays. In this case, with K = 5 users, the nominal SNR, defined by |a1(n)|/σn, is set to 10 dB for user 1. The relative power levels for users 2 through 5, defined by |ak(n)|/|a1(n)|, were set to 10, 5, 1 and 10. It is seen that all five user delays are successfully tracked starting with half a chip or less of initial error. Figure 2 shows the evolution of the acquisition probabilities p(αk(n)|r) for all five users. The probabilities rapidly converge to unity, showing that the acquisition condition is successfully detected. Figure 3 shows successful acquisition of K = 2 users, followed by tracking, using the threshold test on p(αk(n)|r). The initial code delay estimates were set to τk(0|0) = 0 chips. The actual delays were τ = 2.1 and τ = 10.2 chips, respectively. The SNR for user 1 was 10 dB, and user 2 was 10 dB above user 1, corresponding to a strong near-far condition. CONCLUSIONS A new algorithm (Bayesian EKF) was developed for code tracking and acquisition in a radiolocation syst