Fast near-maximum likelihood phase estimation of X-ray pulsars

doi:10.1016/j.sigpro.2012.07.002

Signal Processing

Volume 93, Issue 1, January 2013, Pages 326-331

https://doi.org/10.1016/j.sigpro.2012.07.002 Get rights and content

Abstract

This letter addresses the problem of X-ray pulsar radiation phase estimation, encountered in research works concerning autonomous deep space navigation systems. Autonomous navigation systems represent an intriguing solution to be employed when Earth-assisted navigation is not viable for long range missions. In such applications, X-ray pulsars, as well as other celestial objects, may be employed as peculiar beacons to allow the spacecrafts to adjust their own route. State of the art techniques for estimation of X-ray pulsar radiation phase involve maximization of generally non-convex objective functions, thus resulting in computationally onerous procedures. Here, we show how the problem of pulsar phase estimation can be recast as a cyclic shift parameter estimation problem under multinomial distributed observations, whose maximum likelihood solution can be implemented by means of a fast, Discrete Fourier Transform based procedure. Numerical results show how the herein described fast, near maximum likelihood, estimator favorably compares with selected state of the art estimators, while presenting a significantly reduced computational complexity.

Introduction

Pulsars are highly magnetized rotating neutron stars emitting a periodic profile of electromagnetic radiation. Among other properties, a pulsar's emission profile is characterized by extremely strong stability and regularity to such an extent that its periodicity accuracy is comparable with the one of an atomic clock [1], [2], [3].

Such properties make pulsars good candidate in a large field of applications, including autonomous navigation systems [4]. Deep space navigation is becoming a relevant research topic, due to the increasing interest in space missions aimed at monitoring and exploring universe outside the solar system. Space navigation in close to Earth regions can be suitably assisted by Global Positioning System (GPS) satellites [5], [6]. Nevertheless, for deep space missions as well as all those application situations where the GPS signal becomes unavailable, different approaches have to be envisaged.

Autonomous navigation solutions have been proposed so as to relax the issues relating with spacecraft-to-Earth communications, and let the crafts self-determine their own position relying on the observation of celestial objects [7]. In such frameworks, stars and other celestial beacons play the role of GPS satellites and drive the self-positioning procedure. Since radiations from selected sky objects must exhibit a strong periodicity and stability, pulsars represent good candidate celestial beacons. Moreover, photon detection can be accomplished with small size and cheap detectors [4].

Recent literature has investigated X-ray pulsars based navigation: a useful review can be found in [8]. Most of these works identify the correct synchronization with the pulsar periodic radiation profile as the fundamental task in pulsar-based navigation.

In [9], a maximum likelihood (ML) phase estimator is described; the estimator accuracy approaches the Cramér–Rao lower bound (CRLB) for increasing observation times, but the straightforward exploitation of the ML criterion ends up with a direct search procedure at an high computational cost. In [10], [11], the authors describe two estimators, namely a nonlinear least squares (NLS) estimator based on the estimation of the pulsar profile via the so-called epoch folding (EF) procedure, and a ML pulsar phase estimator based on the statistical modeling of photons time of arrivals (TOAs). Both estimators are still based on the optimization of suitable non-convex objective functions, and hence, to avoid suboptimal undesired local stationary points, they result in direct search based procedures. It must be also remarked that the grid discretization employed in the direct search maximization procedure directly descends from a tradeoff between computational complexity and estimation accuracy.

In [12], the authors describe a low signal to noise ratio (SNR) approximation ML pulsar phase estimator. The low SNR approximation allows to reduce the computational complexity, but causes the estimator performance to degrade for increased SNR values.

In this letter, we recast the problem of pulsar radiation phase estimation in terms of a more general cyclic shift parameter estimation problem under multinomial distributed observations. The authors of this paper have recently proved in [14] that cyclic shift parameter ML estimation is attainable by means of a fast, discrete Fourier transform (DFT) based, estimation procedure. In this respect, we will first show that the estimated pulsar profile obtained by means of EF procedure can be well approximated in terms of multinomial distributed random variables. Then, following the guidelines indicated in [14], we describe how to form an approximate, computationally efficient, ML pulsar radiation phase estimator. We will also provide simulation results that show how the herein presented near-ML pulsar phase estimator favorably compares with estimators described in [10], [11], [12], while presenting a significantly reduced computational complexity.

Section snippets

Problem formulation and related works

Absolute and relative positioning in the deep space can be performed stemming from the observations of the cyclic emissions from celestial objects such as X-ray pulsars. In the case of absolute positioning, in principle, the unambiguous phase should be known. That is, it should be necessary to determine which specific pulse is arriving at the detector from a specific star, as the knowledge of the ambiguous phase only allows to determine a set of candidate positions along the line connecting the

Fast maximum likelihood X-ray pulsar phase estimation

Here, we show that the estimation procedure described in Appendix A is applicable to estimate the pulsar phase from the observed samples ${\hat{λ}}_{i}$ , $i = 0, \dots, N_{b} - 1$ , of the rate function estimated through the EF procedure.

First of all, given the periodic nature of the pulsar radiation, we can recast the pulsar phase estimation as a cyclic shift parameter estimation problem. So, in order to apply the fast ML estimation procedure described in Appendix A, we just need to show that the random variables ${\hat{λ}}_{i}$

Numerical results

Here, we assess the performance of the herein described FAML pulsar phase estimation method by discussing numerical simulation results. We have generated the photons TOAs resorting to the algorithm in [11], referring to two different X-ray pulsars, namely the B1821-24 pulsar and the “Crab” pulsar, whose profiles are reported in Fig. 1. The “Crab” pulsar is characterized by $λ_{s} = 15 ph / s$ , while the B1821-24 pulsar, for which $λ_{s} = 0.0058 ph / s$ , exhibits a considerably weaker emission. Estimated rate

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Fast communicationFast near-maximum likelihood phase estimation of X-ray pulsars

Abstract

Introduction

Section snippets

Problem formulation and related works

Fast maximum likelihood X-ray pulsar phase estimation

Numerical results

Chinese Astronomy and Astrophysics

Accretion-powered X-ray pulsars

Astronomical Society of Japan

Spacecraft navigation using X-ray pulsars

Journal of Guidance, Control, and Dynamics

A statistic for describing pulsar and clock stabilities

Astronomy and Astrophysics

A GPS receiver for high-altitude satellite navigation

IEEE Journal of Selected Topics in Signal Processing

Next generation GPS-aided space navigation systems

IEEE Aerospace and Electronic Systems Magazine

Navigation in Space by X-ray Pulsars

Fast communication
Fast near-maximum likelihood phase estimation of X-ray pulsars