1. Introduction
X-ray pulsar-based navigation (XPNAV) is a developing spacecraft autonomous navigation technique [
1]. After being first introduced in 1981 [
2], the past 40 years have witnessed a significant growth in XPNAV, including the pulse phase estimation as well as the navigation algorithm. Moreover, some flight experiments were performed such as SEXTANT (Station Explorer for X-ray Timing and Navigation Technology) [
3] by the United States and the TG-2 (Tiangong-2) spacelab [
4] and the Insight-HXMT (Insight-Hard X-ray Modulation Telescope) Satellite [
5] of China.
Since the pulsar signal is extremely weak, an orbiting spacecraft can only record a series of photon TOAs (times of arrival) rather than a continuous pulsed signal [
1]. Assuming the spacecraft is stationary or performs a uniform linear motion towards the pulsar, there are two types of method to estimate the pulse phase: the epoch-folding method [
6] and the maximum likelihood estimator (MLE) method [
7]. The epoch-folding method estimates pulse TOA by comparing the template with the empirical profile recovered by photon TOAs. Many methods based on epoch folding have been proposed in recent years. A wavelet-bispectrum algorithm is proposed in reference [
8] to reduce the central processing unit (CPU) time cost of pulse phase estimation. In this method, wavelet transform is used to decompose the empirical profile and obtain low-frequency components of the empirical profile. Then, the low-frequency components are used to estimate the pulse phase [
8]. In 2020, reference [
9] proposed a genetic algorithm (GA)-optimized EMD (empirical mode decomposition)-CS (compressed sensing) with high accuracy and small computational load. In this method, the template is decomposed by the EMD (empirical mode decomposition) to obtain IMFs (intrinsic mode functions), and the IMFs selected by GA constitute the optimized measurement matrix. Then the pulse phase is estimated using the optimized measurement matrix [
9]. In order to solve the sensitivity for noise, reference [
10] proposed a variable step size iteration method which estimates pulse phase using a one-dimensional slice of template and empirical profile. The epoch-folding method is simple to implement, but needs lots of photons for the recovery of empirical profile and has an estimation accuracy less than MLE. However, the CPU time cost of MLE increases sharply as photon amount increases [
7].
For orbiting spacecraft, the phase estimation problem become more complicated in that the orbit motion of a spacecraft introduces an unknown Doppler frequency into the pulse signal. To cope with this, Golshan et al. proposed a phase-tracking algorithm that divides the whole observation period into several short intervals [
11]. During each interval, the spacecraft is approximated to perform a linear uniform motion, and in this case, the Doppler frequency is assumed to be constant and is tracked by a digital phase-locked loop (DPLL) [
11]. In 2013, Huang et al. modified the DPLL to be a two-order Kalman filter and improve the performance of phase tracking [
12]. In 2015, the phase tracking method was verified by the lab equipment of the Neutron Star Internal Composition Detector (NICER) and SEXTANT teams [
13]. The phase tracking method works well for young pulsars such as PSR B0531+21(Crab) but fails when applied to faint millisecond pulsars such as PSR B1821-24 and PSR B1937+21. (The flux of Crab pulsar is 3667 photons/m
2/s, but that of PSR B1937+21 is 0.161 photons/m
2/s [
14].) This is because the threshold for faint pulsars to obtain a reliable result is over 100 s, during which the assumption that a spacecraft performing a uniform linear motion is usually violated when the spacecraft is orbiting in a low Earth orbit. To overcome the drawbacks of the phase-tracking method, Wang and Zheng proposed a phase estimation method with the aid of orbital dynamics of spacecraft [
15,
16]. This method derives a linearized pulse phase propagation model, which modifies the pulse phase propagation model as a term of the position of the spacecraft. In addition, this method estimates the parameters of the linearized model to eliminate the impact of Doppler frequency. Compared with the phase-tracking algorithm, the phase estimation method with the aid of orbital dynamics of spacecraft does not divide the observation into intervals and thus is feasible for millisecond pulsar. Wang and Zheng recommend estimating the parameters by the direct search method [
17]. Unfortunately, when using the direct search method, the CPU time cost is still high when the photon amount is large. In addition, Xue proposed estimating the Doppler frequency and pulse phase by a maximum likelihood estimator simultaneously [
18], and this method also incurs high CPU time cost. Also, the photon background is an important factor which will impact on the performance of pulse phase estimation. Reference [
19] shows that the photon background constantly changes because of orbital motion of the spacecraft. Reference [
20] argues the effect of orbital motion on photon background could be reduced by data screening and the photon background could be assumed as a constant after data screening. Thus, in the existing methods, the photon background is assumed as a constant in the observation period.
For an on-orbit pulse phase estimation problem, there is a contradiction between estimation accuracy and CPU time cost. In order to estimate an accurate pulse phase, large amount of photons is needed, which leads to a high CPU time cost [
21]. In reference [
3], the error of position of XNAV is less than 10 km. In order to achieve this goal, the accuracy of phase estimation for Crab pulsar and PSR B1821-24 should be about 1 × 10
−3 cycle. The linearized method could provide an accurate pulse phase estimation result with a high CPU time cost. According to the simulation results, the linearized method could obtain the pulse phase with enough accuracy when observation period is 1000 s. However, based on the computation environment of this paper, the CPU time cost of the linearized method is about 8000 s. This means that we can obtain the pulse phase of the first observation period after 8 observation periods, which is unacceptable for a navigation system. Thus, the CPU time of the pulse phase estimation method must shorter than the observation period. In addition, the position of spacecraft should be estimated using the estimated pulse phase. However, the CPU time cost by the pulse phase estimation might hinder the real-time navigation process. This is because the spacecraft keeps orbiting during the CPU time and the final estimated pulse phase reflects the position of the spacecraft before the CPU time. Thus, the CPU time cost by the pulse phase estimation should be as short as possible. To cope with this problem, we aimed to balance the pulse phase estimation accuracy and CPU time cost, which demands a highly efficient computational method with a decent estimation performance.
We cast the pulse phase estimation problem into a multi-extremal optimization problem. The cross-entropy (CE) algorithm is a classical method to solve a multi-extremal optimization problem, but has a heavy computational burden [
22,
23,
24]. The Adam (adaptive moment estimation) algorithm is of high computational efficiency, but could only obtain a local optimum [
25,
26]. To combine the advantages of the two methods, this paper proposes a CE-Adam algorithm which features a low CPU time cost and global optimum. We compare the performance of the CE-Adam algorithm with the CE algorithm, DE (differential evolution) algorithm [
27] and PSO (particle swarm optimization) [
28] and find that the CE-Adam algorithm could provide accurate estimation results of pulse phase and significantly reduce the CPU time cost. In addition, we consider that the proposed algorithm could also be used to detect period signal in photon sequences over larger observational periods, which might indicate new physical phenomena.
The organization of this paper is as follows.
Section 2 describes the principle of pulse phase estimation.
Section 3 briefly reviews the CE algorithm and Adam algorithm.
Section 4 shows the proposed CE-Adam algorithm. Simulation data and the real data are processed in
Section 5 to verify the proposed algorithm. Finally, the summary of the conclusions is given in
Section 6.
2. Principle of Pulse Phase Estimation
The arrival event of X-ray photons
follows the non-homogeneous Poisson process. Within the interval
, the probability of
k photons arrival is [
29]:
where
satisfies
.
is the rate function and can be given by:
The parameters
,
represent average signal and total background count rates in units of counts per second, respectively.
is the normalized profile function of pulsar,
represents the pulsar signal phase at the detector.
can be expressed as [
14]:
where
is the phase evolution at the reference observatory (we usually choose the Solar System Barycenter (SSB) as the reference observatory), and
represents the light propagation time from the detector to the reference observatory.
For an orbiting spacecraft,
can be expressed as [
14]:
where
is the spacecraft position vector,
is the pulsar direction vector,
is the predicted position vector of spacecraft,
,
can be linearized as
,
q is the initial phase at time
, and
is the frequency of the pulsar signal.
According to Equation (4),
can be expressed as:
The joint probability density function of the pulsar photon arrival event is:
We define Equation (5) as the likelihood function. The natural logarithm of the likelihood function is:
When the observation period is long enough, the second term in Equation (6) is a constant [
29]. Thus, Equation (6) can be transformed into:
Parameters
and
can be estimated by solving the following optimization problem [
14]:
The CRLB (Cramer–Rao lower bound) for estimation for
q is given by [
21]:
where
is the observation period and
is a constant for each pulsar. It can be seen that the estimation accuracy of
q is increased with observation period. As a result, large amounts of photons are needed for an accurate estimation of pulse phase. Equation (9) shows CPU time cost enhancing with the amount of photons. Thus, there is a contradiction between estimation accuracy and CPU time cost, and a computationally efficient algorithm with a decent estimation accuracy is needed.
As shown in
Figure 1, the objective function shown as (8) has many extrema. Then, the gradient-descent technique could only arrive at a local optimum. Thus, a global optimization algorithm is needed.