With full autonomy and uninterrupted advantages, X-ray pulsar-based navigation(XNAV) provides spacecraft plentiful navigation information, such as position, velocity and time, and has become one of the prevailing in autonomous navigation^{[1-3]}. In the XNAV system, spacecraft receives photons emitted from pulsars and records the photon time of arrivals(TOAs) which is the fundamental measurement of the navigation system. After the pulse time of arrival(TOA) is estimated from a set of observed TOAs, the navigation measurement equation can be established for an autonomous navigation system^{[4-6]}. The accuracy of estimated TOA information mainly determines the positioning accuracy of XNAV system. Acquiring high precise TOA in short observation time is the key to XNAV. Nowadays, TOA estimation algorithms in the time domain are comparatively mature and can be classified into two categories, including the Maximum Likelihood Estimation(MLE) method using TOAs^{[7]} and the phase estimation methods based on epoch folding^{[8]}. The time-domain TOA estimation methods demand a stringent signal period. For epoch-folded methods, the quality of folded profile is affected by the period accuracy. For the MLE method, in order to reduce the enormous computation, the TOA is estimated using the given period. Thus an exact period is necessary to guarantee a good accuracy of the estimated TOA. Nevertheless, the period of pulsar signals observed by spacecraft is shifting due to the doppler effect caused by spacecraft motion. Besides, the unpredicted glitches of pulsars also have an impact on the period of pulsar signals^{[9]}. Therefore, spacecraft is expected to have a certain ability to estimate and correct the given period of pulsar signals according to the received TOAs onboard.
Many researchers have studied the period estimation methods of pulsar signals. Aiming at the radio signals measured by radio astronomy observatories, researchers proposed fast Fourier transform algorithm^{[10]}, cyclo-period search method based on maximum correlation^{[11]}, quick search method of pulsar period using Lomb algorithm^{[12]} and so forth. In frequency domain methods of pulsar period estimation, sampling frequency should be selected carefully to reduce the spectrum leakage and fence effect. What's more, enormous computation is needed to estimate an accurate period. Aiming at pulse time of arrival array in X-ray band, Zhang^{[13]} compared the rebuilt profile set generated by a fast folding algorithm with the standard profile, and use the cross-correlation method to estimate pulsar period. While the performance of Zhang′s method is affected by the sampling rate and amplitudes.
In this paper, a direct pulsar period estimation algorithm is studied to reduce the complexity of the traditional methods. We firstly discuss the influence of period errors on the estimated TOA, and then present mathematical models of the proposed period correction algorithm. The accuracy and computational complexity of the proposed period estimation algorithm are also analyzed by comparing with the estimation method. Finally, the feasibility and the performance of the presented algorithm are illustrated by experiments based on physical and numerical simulation data.
1 Period estimation method using TOA information 1.1 The proposed method using TOA information from epoch-folding TOA estimation algorithmsAll the time tags during an observation time are folded into a single given pulse period to derive the empirical pulsar profile in pulse delay estimation algorithms based on epoch folding. TOA is estimated by comparing the difference between the empirical and the standard pulsar profiles. It can be seen clearly that pulsar period errors have an influence on the process of epoch folding and the result of the estimated TOA.
Assuming a given pulsar period is P, the difference between the given pulsar period and the true pulsar period is dP, the length of the observation time is T_{obs} which consists of N_{p} pulsar periods. If the given pulsar period equals to the true one exactly, that is dP=0, each cycle is time aligned and the folded profile has a sharp peak without any broadening. While if the given pulsar period differs from the true one (dP≠0), the phase difference between two adjacent cycle profiles can be given by
${\rm{d}}\varphi = \frac{{{\rm{d}}P}}{{P + {\rm{d}}P}}$ | (1) |
It should be pointed out that empirical profiles and dφ can hardly be obtained in a single cycle due to the low energy flow density of pulsars. Additionally, dP is very small, usually in the order of nanosecond. Thus, in order to estimate TOA and correct periods, we divide a long observation time into many pieces which contain considerable and reasonable cycles. Subsequently, the TOA of each piece is calculated by folding the correspondent piece into a profile. The phase difference between two pieces within time T can be described as
${\rm{d}}\varphi \left( T \right) = \frac{{{\rm{d}}P}}{{P + {\rm{d}}P}}\cdot\frac{T}{{P + {\rm{d}}P}}$ | (2) |
where, dφ(T)=φ(t+T)－φ(t), φ(t) is the pulse phase(TOA) at time t corresponding to a piece TOAs of the whole observation time. Next, the calculated pulse phases are plotted on the φ-t graph, and the slope k_{φ-t} can be estimated by the least square method. The equation of k_{φ-t} and period errors is
${k_{\varphi - t}} = \frac{{{\rm{d}}P}}{{{{\left( {P + {\rm{d}}P} \right)}^2}}}$ | (3) |
where dP is the positive real root of the above quadratic equation. On account of dP≤P, the denominator on the right side of Eq.(3) can be approximated to P^{2} and the approximate solution can be calculated easily.
1.2 The proposed method using TOA information from a maximum likelihood estimatorEmploying the probability density function associated with the detected TOAs, a maximum-likelihood estimator can be formulated to estimate the initial phase φ_{0} and the unknown frequency f. The unknowns can be found by solving the following^{[14]}
$\left( {{{\hat \varphi }_0},\hat f} \right) = {\rm{arg}}\;{\rm{max}}\sum\limits_{i = 1}^N {{\rm{ln}}\left[ {\beta + \alpha h\left( {{{\hat \varphi }_0} + \left( {{t_i} - {t_0}} \right)\hat f} \right)} \right]} $ | (4) |
where β, α are the known effective background rate and source arrival rate respectively. h(·) is the standard periodic pulsar profile and usually defined on the phase interval φ∈[0, 1] (cycle). Furthermore, the function h(φ) is non-negative and normalized according to
Using the given pulsar period to calculate the frequency in Eq.(4), the two-dimensional parameter estimation problem degenerates into a one-dimensional phase estimation problem. Period errors influence the result of estimated phases with time continuing, showing as follows
$\begin{array}{l} {{\hat \varphi }_0} = {\rm{arg}}\;{\rm{max}}\sum\limits_{i = 1}^N {{\rm{ln}}\left[ {\beta + \alpha h\left( {{{\hat \varphi }_0} + \left( {{t_i} - {t_0}} \right)\left( {{f_{{\rm{ref}}}} + {\rm{d}}f} \right)} \right)} \right]} = \\ \quad {\rm{arg}}\;{\rm{max}}\sum\limits_{i = 1}^N {{\rm{ln}}\left[ {\beta + \alpha h\left( {\left[ {{{\hat \varphi }_0} + \left( {{t_i} - {t_0}} \right){\rm{d}}f} \right] + \left( {{t_i} - {t_0}} \right){f_{{\rm{ref}}}}} \right)} \right]} \end{array}$ | (5) |
where, f_{ref} is the reference frequency corresponding to the given pulsar period P, df is the error of the f_{ref} relative to the true frequency of the pulsar signal. Comparing Eq.(5) to Eq.(4), it is clear to see that the df has a linear impact on the estimated phase with time t increasing
$\varphi \left( t \right) = {{\hat \varphi }_0} + \left( {t - {t_0}} \right){\rm{d}}f$ | (6) |
Similarly, by plotting the φ-t graph and calculating the slope k_{φ-t}, where k_{φ-t}=df. The period error can be expressed as
${\rm{d}}P = P - \frac{1}{{{P^{ - 1}} + {k_{\varphi - t}}}}$ | (7) |
The proposed period estimation method can be implemented as follows: firstly, a set of photon arrival time series is segmented, illustrating in Fig. 1, where, T_{p} is the time length of each segment, T_{m} is the time interval of the neighboring segment, and n is the total number of all segments. Next, the TOA of each segment is estimated in time domain by using the given pulsar period P. Finally, the period is corrected by calculating the slope of the φ-t graph and Eq. (3) or (7).
From the above process, the accuracy of the proposed method is affected by TOA estimation methods, parameters T_{p} and T_{m}. The parameter T_{p} reflects the signal accumulative time of the process of the TOA estimating and has a direct influence on the accuracy of TOA information according to Ref.[14]. There is an inversely proportional function of the accumulative time T_{p} and the theoretical standard deviation δφ(T_{p}) based on different TOA estimators, including a Nonlinear Least-Squares (NLS) estimator, a Cross Correlation (CC) estimator, and a Maximum Likelihood (ML) estimator.
We now take a CC estimator as an example to discuss the precision of the proposed method in theory. According to Eq.(2), it follows that if the parameter T_{m} is too small, there will be dφ(T_{m}) < δφ(T_{p}). In that case, dφ(T_{m}) is submerged in δφ(T_{p}) and the phase difference between two neighboring segments cannot reflect the phase bias caused by period errors. Hence we have the following restriction: parameters T_{m} and T_{p} should meet dφ(T_{m}) > δφ(t_{p}) to make sure dφ(T_{m}) can be detected. While if we use many segments to estimate phases and calculate period errors, the above restriction can be loosened by
${\rm{d}}\varphi \left( {{T_m}} \right) \cdot n > \delta \varphi \left( {{T_p}} \right)$ | (8) |
where n is the number of segments. With the help of Eq.(2), we can show Eq.(8) clearly. The left side of Eq.(8) can be expressed as
${\rm{d}}\varphi \left( {{T_m}} \right) \cdot n = \frac{{{\rm{d}}P}}{{P + {\rm{d}}P}} \cdot \frac{{{T_m}}}{{P + {\rm{d}}P}} \cdot n = {\rm{d}}\varphi \left( {{T_m} \cdot n} \right)$ | (9) |
Eq.(9) shows that increasing the number of segments is equal to enlarge the parameter T_{m} in some sense. It can also be explained by the simulating results of Fig. 2. Where three or four points correspond to the same TOA value(y axis), looking like 'footsteps'. These 'footsteps' are caused by dφ(T_{m}) < δφ(T_{p}). While the number of segments is large enough and the parameters are satisfied to Eq.(8), the slope k_{φ-t} still can be obtained.
Combining Eq.(8) and (2), the resolution of the estimated period in theory δP is defined as
$\delta P \approx \frac{{\delta \varphi \left( {{T_p}} \right) \cdot {P^2}}}{{{T_{{\rm{obs}}}}}}$ | (10) |
Next, we turn to study the precision of the proposed method in the aspect of the numerical solution method. Let N_{b} be the number of folding 'bins' in CC and NLS estimators. If we expect to distinguish the phase bias among two neighboring segments from epoch-folded estimation methods, there should be dφ(T_{m})>N_{b}^{－1}. Actually, the above restriction is too strong and can be loosened by
${\rm{d}}\varphi \left( {{T_m}} \right) \cdot n > 1/{N_b}$ | (11) |
The Eq.(11) illustrates that the phase bias between the last and the first segment of an observation time T_{obs} should be greater than the estimated TOA resolution. Combining Eqs.(11) and (2), we have another theoretical resolution of the estimated period δP,
$\delta P \approx \frac{{{P^2}}}{{{T_{{\rm{obs}}}}}} \cdot \frac{1}{{{N_b}}}$ | (12) |
The same conclusion can also be given when a ML estimator is utilized to estimate TOA. The phase resolution of a ML estimator is determined by the search step noted as 'S_{step}'. Therefore substituting 'S_{step}'' for '1/N_{b}' in Eq.(12), we have a δP based on a ML phase estimator
$\delta P \approx \frac{{{P^2}}}{{{T_{{\rm{obs}}}}}} \cdot {S_{{\rm{step}}}}$ | (13) |
By Eqs.(10), (12) and (13), the theoretical resolution of the estimated period δP of the proposed method can be presented as
$\delta P = {\rm{max}}\left( {\frac{{\delta \varphi \left( {{T_p}} \right) \cdot {P^2}}}{{{T_{{\rm{obs}}}}}},\frac{{{P^2}}}{{{T_{{\rm{obs}}}}}} \cdot \frac{1}{{{N_b}}}\quad {\rm{or}}\quad \frac{{{P^2}}}{{{T_{{\rm{obs}}}}}} \cdot {S_{{\rm{step}}}}} \right)$ | (14) |
From Eq.(14), the estimated period δP is determined by the max of dφ(T_{m}) and 1/N_{b} (or S_{step}). If we let 1/N_{b} be 0.001 and dφ(T_{m}) < 0.001(cycle), the δP of a millisecond pulsar during 1000s observation time will be in the order of nanosecond. For most TOA estimation methods, the accuracy of the estimated TOA is better than 0.001(cycle). Thus the proposed method can have a good period estimation with less demand on the accuracy of TOA information from the above evaluation. It also should be pointed out that the period error is closer to the theoretical resolution of the estimated period, the accuracy of the proposed method is worse. This is because during an observation time T_{obs}, the phase bias caused by period errors is limited and the effective number of estimated phases corresponding to different segments becomes small.
Noting that the time derivative of pulsar barycentric rotation frequency is not taken into account in our method. While in the actual situations, the periods of pulsar signals decrease with the increasing time. For example, the time derivative of Crab pulsar frequency is about 1×10^{－10} Hz/s and the frequency will have a 1×10^{－7} Hz shift during 1 000 s observation time. In that case, if the period error dp is over 1ns, the changed frequency caused by period error will be －9×10^{－7} Hz and the period error can be obtained by our method. Otherwise, our method will not work well because the changed frequency caused by the period error is submerged in the frequency shift. Fortunately, for most pulsars, the time derivative of pulsar frequency is in the order of 1×10^{－13} Hz/s or smaller^{[15]}. Hence, the time derivative of pulsar frequency has a very small impact on the proposed method.
2.2 Discussion about selecting parameters T_{p}, T_{m} and nNow considering the influence of the parameters T_{p} and n on the estimated k_{φ-t}, the estimated error variance of k_{φ-t} from the least square method can be given^{[16]}
${\rm{var}}\left( {{k_{\varphi - t}}} \right) = \frac{{{\sigma ^2}}}{{\sum\limits_{i = 1}^n {{{\left( {{x_i} - E\left( {{x_i}} \right)} \right)}^2}} }}$ | (15) |
where, φ_{i}~N(b+k_{φ-t}x_{i}, σ^{2}), σ=δφ(T_{p}) and x_{i}=i·(T_{m}). The relationship between parameters T_{p}, T_{m}, n, and T_{obs} can be approximated by
${T_p} + n \cdot {T_m} \approx {T_{{\rm{obs}}}}$ | (16) |
The inversely proportional function of δφ(T_{p}) and T_{p} can be expressed as^{[14]}
$\delta \varphi \left( {{T_p}} \right) = \frac{1}{{{T_p}}} \cdot \frac{{\int_0^1 {\lambda \left( \varphi \right){{\dot \lambda }^2}\left( \varphi \right){\rm{d}}\varphi } }}{{{{\left( {\int_0^1 {{{\dot \lambda }^2}\left( \varphi \right){\rm{d}}\varphi } } \right)}^2}}} = \frac{1}{{{T_p}}} \cdot A$ | (17) |
where
$\begin{array}{l} {\rm{var}}\left( {{k_{\varphi - t}}} \right) = \frac{{{{\left[ {\delta \varphi \left( {{T_p}} \right)} \right]}^2}}}{{\sum\limits_{i = 1}^n {\left( {i{T_m} - \frac{{\sum\nolimits_{i = 1}^n {i{T_m}} }}{{{n^2}}}} \right)} }} = \\ \frac{{12{{\left( {\frac{A}{{{T_p}}}} \right)}^2}}}{{T_m^2\left( {n - 1} \right)n\left( {n + 1} \right)}} \approx \frac{{12{A^2}}}{{n{{\left[ {{T_p}\left( {{T_{{\rm{obs}}}} - {T_p}} \right)} \right]}^2}}} \end{array}$ | (18) |
In order to acquire an accurate k_{φ-t}, we choose the parameters T_{p}, T_{m} and n to make the Eq.(18) be as small as possible. Usually, if period errors and the chosen T_{p} are so large that the folded profile has a gross distortion, the peak of the folded profile gradually degenerates, which has an adverse impact on TOA estimation. Therefore, we may try different parameter combinations before finding a proper one sometimes. Some proper parameter combinations are given in the following experiments.
2.3 Computational complexity analysisThe computational complexity of the proposed period estimation is mainly determined by the TOA estimators. The computational costs for epoch folding, CC estimator, NLS estimator and ML estimator have been studied by Emadzadeh Amir Abbas and Speyer Jason in Ref.[14]. Here we list their results in Table 1, as well as the computational cost of the Least Square method.
In Table 1, N_{g} is the number of search grids in the interval (0, 1) cycle and M is the total photon number. For the proposed method, n times TOA calculation are needed to obtain the TOA information. From Eq.(3), we can see two multiplications and one division are needed for computing the slope k_{φ-t}. From Eq.(7), one addition, one subtraction and two divisions are needed.
Now we construct the cost function of the traditional χ^{2} period estimation method to make a comparison. In χ^{2} estimation method, different test periods are utilized to fold a profile and the index function
For ease of comparison and notations, let N_{g} be equal to N_{b}. Substitute [T_{p}/P] and T_{p}(α+β) for N_{p} and M respectively, the computation cost results are shown in Table 2.
Where n_{x} is the number of χ^{2} estimation method times for searching the accurate period, and n_{x} is determined by the searching interval and searching step. Usually, n_{x} is in the order of hundred or thousand. n is relatively small and could be about one hundred. Thus, the computational cost of the proposed method using CC or NLS estimator is less than the computational cost of χ^{2} estimation method. Since the amount of calculations for the proposed method using ML estimator significantly increases as the number of photons T_{p} (α+β) becomes longer. Hence, when we using a ML estimator to calculate TOA information for the proposed method T_{p} is set to be not very large, which is consist with the conclusion in Section 2.2. And compared to the χ^{2} estimation method case, it is hard to say which one need more calculations. Because the amount of calculation for the proposed method using ML estimators and χ^{2} estimation method is affected by different parameters.
3 Experiments and performanceIn this section, the physical simulation data of the ground X-ray source and MCP detectors and the numerical simulation data based on Crab pulsar parameters are utilized to verify the feasibility and performance of the proposed method.
3.1 Experiment oneExperiments using the physical simulation data can reflect the actual stimulating situation and help to improve the physical simulation system. The data from the ground X-ray source and MCP detectors is given in column, showing photon time of arrivals recorded by MCP detectors. The file also provides the period of pulsar signals used to experiments, with 33.4 ms. As experimenters illustrate that the X-ray source and MCP detectors don't move during tests, the estimated phase of each segment picked up from the entire TOAs will fluctuate around a constant if the given pulsar period is exact. Otherwise, estimated phases change with time increasing and the slope k_{φ-t} of the φ-t graph is not equal to zero.
In accordance with the above discuss, we first divide TOAs into segments as indicated in Fig. 1. Then the given pulsar period P=33.4 ms is adopted to estimate TOA of each segment based on a CC estimator and a ML estimator. In order to acquire a suitable TOA precision, let T_{p} and T_{m} be 200 s and 7 s respectively. The N_{b} of the CC estimator and S_{step} of the ML estimator are set to be 1 024 and 0.001 respectively. T_{obs}, the total length of the TOAs is 997 s. The analysis results see in Fig. 2. The amount operations for the proposed method using CC and ML estimators are 8.23×10^{8} and 4.26×10^{10} respectively. If we use the above parameters to search period by χ^{2} estimation method, 3.06×10^{7}n_{x} calculations are needed. It is clear to see that only when n_{x} is equal to 27, the computational cost of χ^{2} estimation method is same with the proposed method based on CC estimator. However, it is quite difficult for n_{x} to be so small in most cases. And there is the same computational cost with the proposed method based on ML estimator when n_{x} equals to 1 395.
By Fig. 2, the slope k_{φ-t} is a non-zero constant, which means the given pulsar period has period errors. Slopes for the CC estimator and the ML estimator are calculated using Least Square method, with －3.3959×10^{－5} s^{－1} and －3.3855×10^{－5} s^{－1}, respectively. Correspondent period errors according to Eqs.(3) and (7) are －37.883 6 ns and －37.766 8 ns. As we do not have the exact pulsar period from the physical simulation system, we fold all the TOAs data into a profile using the given period and the calibrated period for comparison. It is apparent from Fig. 3 that folded profiles using the corrected period have a higher peak than the folded profile using the given period. Hence the corrected period is regarded as a more accurate one in common sense. It is also noticed that a constant phase difference exists between the CC estimator and the ML estimator in Fig. 2. We guess this is because of different data processing procedures. Here we only care about the slopes of the graph and the phase difference doesn′t impact the slope estimation. So we do not analyze this phenomenon in detail.
Similarly, analysis with another group of physical simulation data in low signal-to-radio (SNR) is executed. The total length of this group is 2 042 s. Let T_{p} and T_{m} be 200 s and 7 s respectively. Slopes of φ-t graphs in Fig. 4 are －1.5856×10^{－5} s^{－1} and －1.5889×10^{－5} s^{－1} corresponding to the CC estimator and the ML estimator, respectively. And period errors from Eq. (3) and (7) are －17.688 0 ns(CC estimator) and －17.725 4 ns(ML estimator) respectively. Fig. 5 also illustrates the comparison of folded profiles using different pulsar periods. From this group analysis, it can be concluded that the proposed period estimation method can calculate period errors even though the observation has a low SNR.
The reason why there are period errors in physical simulation experiments is mainly because of the system noise when X-ray source generates pulsar signals and clock errors when MCP detectors record the arrival time of photons. The period error is not expected in the physical simulation system. Since it has a negative influence on testing TOA estimation algorithms and other experiments based on the generated data. Therefore, it is necessary to correct the period error to guarantee a reliable and accurate period for further use.
3.2 Experiment twoIn this experiment, the RXTE observation data^{[18]} is utilized to simulate photon Time Of Arrivals(TOAs) in order to investigate the accuracy of the proposed method. By analyzing the data package FS_b36a20f-b36acab of Crab pulsar observed by RXTE, we have the ratio of the effective background rate and source arrival rate, about 9:1. And the event of TOAs can be demonstrated by a non-homogeneous Poisson process. By setting the exact pulsar period be 33.4 ms, a two-thousand-second TOAs observation is generated, with the effective background and source arrival rates at 450 photons/s and 50 photons/s respectively.
The experiment is carried out as follows: let t_{p}=50 s and T_{m}=19 s. Firstly, the TOA of each segment is calculated over 100 Monte Carlo trails. The Root Mean Squared error(RMS) of TOA by CC and ML estimators are evaluated as 9.817×10^{－4} cycle and 8.652×10^{－4} cycle respectively. Secondly, the above RMS is used to calculate the theoretical resolution of the estimated period by Eq.(9), with the results of 0.55 ns (CC estimator) and 0.48 ns (ML estimator). While taking into account the N_{b} and S_{step} that are chosen to be 1 024 and 0.001 in these two estimators, theoretical resolutions of the estimated period corresponding to CC and ML estimators should be 0.55 ns and 0.56 ns according to Eq. (14), respectively. Thirdly, different period errors, including 0.8 ns, 1 ns, 10 ns, 0.1 μs, and 1 μs, are added to the exact period(33.4 ms) to re-estimate the TOA of all segments. In order to reduce the computation, we only use CC estimator to estimate TOA. Finally, slopes of φ-t graphs are computed using Least Square method, and the corrected periods are calculated according to Equation (3).
Fig. 6 shows different φ-t graphs of various period errors. Considering the integer part of the estimated phase, we redefine φ on the phase interval φ∈[0, +∞)(cycle). With dP increasing, the linear relationship becomes apparent and the absolute value of the slope k_{φ-t} increases gradually. In Fig. 6(a) and (b), as dP is small and δφ(T_{p}) is relatively large, the linear function between φ and t is not obvious, which agrees with the analysis in Section 2. Fig. 7 compares the folded profiles using the different periods. Table 3 also shows the statics of period errors over 100 Monte Carlo trials. The RMS of different periods displayed in Table 3 is in the same order with about 0.35 ns and does not affect by the value of period errors. The results of this experiment show that though the period has a big error, making the folded profile a serious distortion, the proposed period estimation method is able to calculate periods in a quick and direct manner.
It should be commented that an iteration algorithm can be developed to acquire a more precise estimated period in the further studies since there are some approximations in the data process. The iteration algorithm can be designed as follows: the said method is first to execute to calculate a new period, then using the new period we do the same procedure again based on the said method. When the difference between the new period and the last one reaches the threshold that we set, the iteration algorithm stops.
4 ConclusionIn this paper, we demonstrate a pulsar period estimation method based on TOA information. The proposed method established the formula between period errors and estimated phases of each segment from a set of TOAs. The main contribution of the proposed method is to provide a direct method to estimate periods in the time domain with a new criterion for the precision of pulsar period. It is totally different from other period estimation methods in frequency domain or methods that search the perfect period based on maximum peak principle. The proposed method can estimate period in the order of nanosecond from a hundred seconds of observation time. Using the simulating data with a photon arrival rate 500 photons/s and total observation time 2 000 s, the proposed method has a theoretical resolution of the estimated period at 0.6ns, estimating the period at 0.8 ns successfully. The proposed pulsar period estimation method can help to improve the ground physical simulation system and be applied in engineering.
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