Amir A. Emadzadeh

On Modeling and Pulse Phase Estimation of X-Ray Pulsars

Mathematical models are developed to characterize the X-ray pulsar signals, and the pulse phase estimation problem is addressed. The Cramér-Rao lower bound for estimation of the pulse phase is presented. Depending on employing the photon counts or direct use of the measured photon time of arrivals, two different estimation strategies are proposed and analyzed. In the first approach, utilizing the epoch folding procedure, the observed pulsar rate function on the detector is retrieved, and the pulse phase is estimated through a nonlinear least-squares fit of the empirical rate function to the known pulsar rate function. It is shown that this estimator is consistent, but not asymptotically efficient. In the second strategy, a maximum likelihood (ML) estimation problem is formulated using the probability density function of the photon time of arrivals. It is shown that the ML estimator is asymptotically efficient. Computational complexity of the proposed estimators is investigated as well. The analytical results are verified numerically via computer simulations.

Relative Navigation Between Two Spacecraft Using X-ray Pulsars

This paper suggests utilizing X-ray pulsars for relative navigation between two spacecraft in deep space. Mathematical models describing X-ray pulsar signals are presented. The pulse delay estimation problem is formulated, and the Cramér-Rao lower bound (CRLB) for estimation of the pulse delay is given. Two different pulse delay estimators are introduced, and their asymptotic performance is studied. Numerical complexity of each delay estimator, and the effect of absolute velocity errors on its performance is investigated. Using the pulsar measurements, a recursive algorithm is proposed for relative navigation between two spacecraft. The spacecraft acceleration data are provided by the inertial measurement units (IMUs). The pulse delay estimates are used as measurements, and based on models of the spacecraft and IMU dynamics, a Kalman filter is employed to obtain the 3-D relative position and velocity. Furthermore, it is shown that the relative accelerometer biases as well as the differential time between clocks can be estimated. Numerical simulations are also performed to assess the proposed navigation algorithm.

X-Ray Pulsar-Based Relative Navigation using Epoch Folding

How the relative position between two spacecraft can be estimated utilizing signals emitted from X-ray pulsars is explained. The mathematical models of X-ray pulsar signals are developed, and the pulse delay estimation problem is formulated. The Cramér-Rao lower bound (CRLB) for any unbiased estimator of the pulse delay is presented. To retrieve the pulsar photon intensity function, the epoch folding procedure is characterized. Based on epoch folding, two different pulse delay estimators are introduced, and their performance against the CRLB is studied. One is obtained by solving a least squares problem, and the other uses the cross correlation function between the empirical rate function and the true one. The effect of absolute velocity errors on position estimation is also studied. Numerical simulations are performed to verify the theoretical results.

Asymptotically Ecient Estimation of Pulse Time Delay For X-ray Pulsar Based Relative Navigation

The problem of relative position estimation between two spacecraft based on X-ray pulsar measurements is addressed. A mathematical model is developed for the pulsar’s intensity as observed at each spacecraft. The time of arrival (TOA) of photons is modeled as a Poisson point process, and the relevant probability density functions are provided. A solution to the relative navigation problem is suggested based on estimation of the pulse delay between the detected signals. The problem of pulse time delay estimation is formulated and the Cram er-Rao lower bound (CRLB) is presented. Based on probability distribution of photon time of arrivals, a maximum likelihood estimation (MLE) problem is formulated and the pulse delay estimator is proposed. It is proven that the proposed estimator is asymptotically ecient. The analytical results are veried numerically employing computer simulations.

Consistent estimation of pulse delay for X-ray pulsar based relative navigation

The relative navigation problem between two spacecraft in deep space is formulated based on employing X-ray pulsar signals. The proposed approach is to lock the detectors on the same pulsar and time-tag the detected X-ray photons. Then using the epoch folding procedure the observed pulsar rate functions on each spacecraft are retrieved and the time delay is estimated through a nonlinear least-squares (NLS) fitting of the empirical rate function to the known pulsar rate function. The relative distance between the space vehicles is proportional to the time delay. The pulsar signals are modeled mathematically and the epoch folding noise is characterized analytically. The mean and variance of the pulse delay estimator are calculated. The Cramér-Rao lower bound (CRLB) is also presented and the estimators performance is compared against it. The analytical results are verified numerically by computer simulations.

Online time delay estimation of pulsar signals for relative navigation using adaptive filters

Relative navigation of spacecrafts may be accomplished by observing X-ray sources and indirectly determining the spacecraftspsila relative position. In this approach, two spacecrafts lock on a known pulsar which irradiates X-ray waveforms that reach them with a differential time delay that is proportional to the distance between the spacecrafts. By observing different pulsar sources geometrically distributed over the galactic disc, it is possible to determine the spacecraftspsila relative inertial position. Our goal is to estimate their relative position by Time Delay Estimation (TDE) between the detected signals. Although there are several off-line TDE methods, like the basic cross-correlation (BCC) and the generalized cross-correlation (GCC) techniques, in this work we formulate TDE as a channel estimation problem and apply adaptive filtering techniques to estimate the time delay online. There are certain benefits in using adaptive filters, especially when the underlying parameters like signalspsila statistics are unknown or change over time. We study different adaptive algorithms and show how they are able to efficiently deliver accurate delay estimates at reduced computational complexity and in real time.

A new relative navigation system based on X-ray pulsar measurements

The relative position estimation problem between two spacecraft, based on utilizing signals emitted from X-ray pulsars, is introduced. The pulse delay estimation problem is formulated, and the Cramér-Rao Lower Bound (CRLB) for any unbiased estimator of the pulse delay is presented as well. Two different estimation strategies are proposed, and an asymptotically efficient estimator is chosen, which is based on the maximum-likelihood criterion. The navigation system is equipped with inertial measurement units (IMUs). The time delay estimates are used as measurements, and based on the models of spacecraft and IMU dynamics, a Kalman filter is employed to obtain the three-dimensional relative position estimate. Numerical simulations are performed to verify the theoretical results.

Optimal control for a scalar one-step linear system with additive Cauchy noise

An optimal control scheme is developed for scalar discrete linear dynamic systems driven by Cauchy distributed process and measurement noises. Since the Cauchy density has infinite variance, a cost function is defined for which the unconditional expectation with respect to the Cauchy densities produces a cost criterion that exists. After showing that this cost criterion allows a dynamic programming solution for the multistage problem, an optimal controller is determined for one step time update. Characteristics of the optimal controller is compared with the linear exponential Gaussian (LEG) controller. The dramatic performance difference between the Cauchy and the LEG controllers is studied. Furthermore, through different numerical examples, some interesting properties of the Cauchy controller are examined.

Pulse Delay Estimation Using Epoch Folding

In this chapter we offer our first approach for estimation of the pulse delay. The proposed approach is to retrieve the photon intensity functions on each detector via epoch folding to obtain the empirical rate functions, i.e., \(\breve{{\lambda }}_{k}(t)\). Then, the empirical intensities will be used for estimation of the initial phase on each detector.

Pulse Delay Estimation

As explained in Chap. 3, the navigation system measurement is obtained through estimation of the time delay between the received signals. The delay estimation problem plays the most important role in the navigation system. In this chapter, we study this problem in more detail.