MODIFIED ZIV-ZAKAI BOUND FOR TIME-OF-ARRIVAL ESTIMATION OF GNSS SIGNAL-IN-SPACE Andrea Emmanuele, Marco Luise Dipartimento di Ingegneria dell’Informazione University of Pisa, Pisa, Italy andrea.emmanuele; marco.luise@iet.unipi.it Francesca Zanier, Massimo Crisci European Space Agency (ESA) ESTEC/TEC-ETN, Noordwijk, NL francesca.zanier@esa.int ABSTRACT Signal Time-Of-Arrival (TOA) estimation accuracy is fun- damental to the functioning of Global Navigation Satellite Systems (GNSSs). This work investigates a variant of the Ziv-Zakai bound (ZZB) named modified ZZB (MZZB) as a theoretical performance limit in TOA estimation, to over- come the heavy computational effort caused by the presence of nuisance parameters (carrier amplitude/phase, channel coefficients). (M)ZZB is adopted to analyze the theoretical performance of signal delay estimators in the different phases of acquisition and tracking, and numerical results are shown for the main GNSS standard signal formats: BPSK and (fil- tered) Binary Offset Carriers (BOC) modulations in Additive White Gaussian Noise (AWGN) channel. Index Terms— Modified Ziv-Zakai bound, Time-of- Arrival (TOA) estimation, acquisition, tracking, Binary Off- set Carrier (BOC). 1. INTRODUCTION “One-way signal Time-Of-Arrival (TOA)” estimation rep- resents the basis of all current Global Navigation Satellite Systems (GNSSs). The accuracy of user position is directly related to the (pseudo-)ranges estimation performed by the receiver via TOA estimation. Commonly the well known Cram´ er-Rao bound (CRB) is adopted as mean square error (MSE) theoretical benchmark for unbiased estimators, for its ease of calculation. Unfortunately, it requires sufficiently smooth signal waveform and possibly a differentiable pa- rameter probability density function (pdf). In some cases of practical interest, both these conditions are not satisfied, especially as far as the standard GNSS Signal-In-Space are concerned. GPS, GLONASS, Galileo, and other GNSSs adopt Binary Phase Shift Keying (BPSK) and Binary Offset Carriers (BOC) modulations [1] with (theoretically) rectan- gular pulses, so that the CRB is not applicable if they are not filtered. Other bounds can be found in literature, which prove to be tighter than the CRB, but cannot in general be easily cast into a simple closed form expression. One of these is the Ziv- Zakai bound (ZZB) [2], [3] that stems out of detection theory and also considers possible parameter a priori information. The ZZB shows no constraints of parameter pdf, signal shape or SNR value resulting a very interesting MSE benchmark for any signal format. Unfortunately, as well as other bounds, computing the ZZB in the presence of nuisance parameters is very hard. In this contribution a modified version of the bound is adopted, i.e. the modified ZZB (MZZB) [4],[5], [6], whose computation in the presence of nuisance parameters is much simpler. Besides, whenever the size of the nuisance vector gets large the gap between the two versions shows to be negligible [4], [5]. We use the MZZB here to evaluate the performance of TOA estimation during signal acquisition and tracking for standard GNSS SIS (BPSK, BOC). In particular, assuming the proper a priori information, we can evaluate the minimum C/N 0 threshold that is needed to acquire or track the signal delay with an MSE lower than a fixed value. 2. MODIFIED ZZB FOR TOA ESTIMATION The problem considered here is TOA estimation for position- ing systems for a generic signal in Additive White Gaussian Noise (AWGN). The model of the received signal is r (t)= s (t - τ,γ )+w(t), where s (t, γ ) is the transmitted signal, τ is the signal delay with a uniform distribution in [0,T x ] (differ- ent distributions could be considered as well), γ is an array of “stray” (nuisance) parameters, and w(t) is a white Gaussian process with Power Spectral Density (PSD) equal to N 0 /2. In [2],[3],[4] and [5] the ZZB and its modified version are de- fined step by step for this scenario. The final expression of the modified ZZB runs as follows: MZZ B( τ )  1 T x  Tx 0 Δ  Tx- Δ 0 Q ⎛ ⎝  E γ {d 2 (Δ,h|γ )} 2N 0 ⎞ ⎠ dhdΔ (1) where E γ {·} indicates statistical expectation over all possible values of γ , T x is the maximum uncertainty on the delay and d is the euclidean distance between the two (equiprobable) delayed signals s(t -h|γ ) and s(t -h -Δ|γ ), conditioned to the particular γ . When the estimation time T 0 is large, the 3041 978-1-4673-0046-9/12/$26.00 ©2012 IEEE ICASSP 2012