Method and system for determining mathematical expectation of weight of atomic clock

Abstract

The invention provides a method and system for determining mathematical expectation of atomic clock weight, and relates to the technical field of signal processing. The method is applied to a scheme of establishing the time reference without passing through a free paper surface; the method comprises the following steps: acquiring an observation clock difference between a reference time reference and an atomic clock; calculating an estimated value of a noise diffusion coefficient based on the observation clock difference; the weight of the atomic clock is determined by adopting an ALGOS new weight algorithm, and the mathematical expectation of the weight statistics corresponding to each atomic clock is further determined. The mathematical expectation of the weight of each atomic clock can be effectively determined; and performing mutual verification on the test performance and the theoretical conclusion of the weight of each atomic clock.

Description

Technical Field

[0001] The present invention belongs to the technical field of signal processing, and in particular relates to a method and system for determining the mathematical expectation of atomic clock weights in the process of establishing a time reference. Background Art

[0002] Both the timekeeping laboratory and the Global Navigation Satellite System (GNSS) need to establish a time reference. The international time reference is the Coordinated Universal Time (UTC) established and maintained by the Bureau International desPoids et Mesures (BIPM). The time reference of the timekeeping laboratory is denoted as UTC(k), where k is the laboratory code. The time reference of the GNSS is denoted as GNSST (GNSSTime).

[0003] The core algorithms for establishing a time base include: a time scale algorithm (including a weight algorithm and a prediction algorithm) and a steering algorithm. The time base can be established using the traditional method or the new method proposed in the literature [1]. Taking the establishment of the GNSS time base GNSST (GNSS Time) as an example, the traditional method steps are as follows: Step 1, using all the atomic clocks in the integrated clock group, use the time scale algorithm to establish a free paper time (temps atomique, TA); Step 2, trace the free paper time to a certain UTC (k), that is, use the UTC (k) to steer the free paper time to obtain GNSST. The new method steps are as follows: design a prediction algorithm and a steering algorithm for each clock, so that each clock can independently obtain a driven clock, and then perform a weighted average of these driven clocks.

[0004] References:

[0005] [1] Wu Yiwei, et al. Method and performance analysis of establishing time reference without free paper surface[J]. Acta Geodaetica et Cartographica Sinica, 2021, 50(3): 343–355.

[0006] [2]Yiwei Wu. Determination of Mathematical Expectations of TwoWeighting Statistics for Ensemble Time Scales[J]. Metrologia, 2023, 60:065009–1–065009–19.

[0007] [3] Wu Yiwei, et al. Atomic clock model and frequency stability analysis method [J]. Journal of Wuhan University (Information Science Edition), 2019, 44(8): 1226–1232.

[0008] Reference [1] pointed out through analysis that the inherent properties of the traditional method will cause the method itself to have shortcomings. The main reason is that the performance of a single clock and the performance of free paper time affect each other, and other shortcomings caused by this cannot be improved by optimizing the design algorithm. The new method avoids the shortcomings caused by the mutual influence between the single clock and the free paper time in the traditional method, and can also optimize the design of the prediction algorithm and the steering algorithm according to the characteristics of each clock.

[0009] Although the method proposed in document [1] does not establish a free paper time, it establishes a paper time that is synchronized with an external reference, and its expression form is similar to TA. In the present invention, it is called "pseudo-free paper time" or "pseudo-TA", denoted as "PseTA".

[0010] In the literature [1], the weight algorithm directly adopts the weight algorithm in the ALGOS algorithm adopted by the International Bureau of Weights and Measures (BIPM). The ALGOS algorithm is similar to other time scale algorithms and also includes a prediction algorithm and a weight algorithm. Its prediction algorithm has gone through two stages. The original ALGOS prediction algorithm uses a first-order polynomial prediction. The new ALGOS prediction algorithm uses a second-order polynomial prediction. Its weight algorithm has gone through two stages. In the original ALGOS weight algorithm and the new weight algorithm, the weights are inversely proportional to two statistics. In the original weight algorithm used before 2014, the statistic for each clock is the variance of its multiple frequency differences. In the new weight algorithm used since 2014, the statistic for each clock is the weighted average of its multiple squared prediction errors. The conclusions drawn based on the experimental phenomena are as follows: 1) In the original ALGOS weight algorithm, the weight of the cesium clock is significantly greater than that of the hydrogen clock, and the full weight is basically the cesium clock. In the new weight algorithm, the weight of the hydrogen clock is significantly greater than that of the cesium clock, and the full weight is basically the hydrogen clock. 2) After the maximum weight limit in the new weight algorithm is increased, the weight of the hydrogen clock is further improved. 3) Whether the original prediction algorithm or the new prediction algorithm is used, there is basically no effect on the weight distribution of the hydrogen clock and the cesium clock. 4) When the new weight algorithm and the new prediction algorithm are used at the same time, the frequency stability of the free atom clock is optimal.

[0011] Prior to the publication of reference [2], the causes of these experimental phenomena had not been rigorously explained theoretically. Reference [2] believes that these two statistics are the key to analyzing weight performance. Since the weight is inversely proportional to these two statistics, the statistical properties of the statistics (mathematical expectation and variance) completely determine the statistical properties of the weight. The main work of reference [2] is to derive the analytical expressions of the mathematical expectations of these two statistics under ideal conditions, and to theoretically reveal the reasons for the different weight distributions of hydrogen clocks and cesium clocks in the original ALGOS weight algorithm and the new weight algorithm. Based on these analytical expressions, explanations can be given for various experimental phenomena in the ALGOS algorithm.

[0012] The analytical expressions of the mathematical expectations of the two statistics derived in reference [2] are generally applicable, and can therefore be applied to the new method in reference [1] to determine the theoretical performance of the weights of the atomic clocks when the new method in reference [1] is used to establish a time base. For a clock group composed of atomic clocks of different models, the method in reference [3] can be used to obtain the estimated values ​​of their parameters, and substitute these parameter estimates and the time intervals for estimating or predicting the parameters (frequency difference, frequency drift, etc.) into the analytical expressions given in reference [2] to calculate the theoretical values ​​of the mathematical expectations of the two statistics, thereby calculating the theoretical values ​​of the mathematical expectations of the weights, which provides a theoretical basis for the subsequent theoretical analysis of the performance of the time base, the selection and configuration of the clock group, etc.

[0013] Furthermore, the analytical expressions of the mathematical expectations of the two statistics derived in reference [2] contain several key parameters, namely the diffusion coefficient of noise. Reference [3] proposed a method to iteratively estimate the noise diffusion coefficient based on historical data, combined with the Kalman filter (KF) estimation results and the Allan deviation map. The experimental results show that this method can effectively obtain the estimated value of the noise diffusion coefficient. Therefore, in practical applications, it is first necessary to obtain the observed clock error of the "reference time base-atomic clock". The reference time base is denoted as Ref, and the clock face reading of the i-th atomic clock is denoted as hi. When the frequency stability of Ref is much better than hi, it can be considered that the noise in the observed clock error "Ref-hi" is mainly the atomic clock noise and observation noise, and the influence of Ref is very small. Then, the method of reference [3] can be used to obtain the estimated value of the noise diffusion coefficient of the observed clock error "Ref-hi". Summary of the invention

[0014] Based on the above technical status, the present invention proposes a scheme for determining the mathematical expectation of atomic clock weights. The purpose of the present invention is to provide a scheme for determining the mathematical expectation of atomic clock weights and their theoretical performance under specific circumstances and experimentally verifying the new method for establishing a time reference without free atoms described in document [1], so as to solve the technical problems existing in the prior art. The specific circumstances refer to: the atomic clock signal is continuous, the atomic clock has no abnormal jump, the expressions of the atomic clock's deterministic terms and the statistical characteristics of the random terms have not changed, etc.

[0015] The first aspect of the present invention provides a method for determining the mathematical expectation of atomic clock weights, the method being applied under the condition of establishing a time reference without using a free paper surface, the method comprising:

[0016] Step S1, obtaining the observed clock error of the reference time standard-atomic clock;

[0017] Step S2, calculating an estimated value of the noise diffusion coefficient based on the observed clock difference;

[0018] Step S3: Use the ALGOS new weight algorithm to determine the atomic clock weights, and further determine the mathematical expectation of the weight statistics corresponding to each atomic clock.

[0019] According to the method of the first aspect, in step S1, the observed clock error is expressed as:

[0020]

[0021] Among them, x(t k ) represents the time difference, z(t k ) represents x(t k ) of the observed clock error, ε(t k ) represents the observation noise, which is described as the time derivative of the Wiener process W0(t), σ represents the intensity of the observation noise, x0 and y0 represent the initial values ​​of the time state and frequency state, d represents the frequency drift, W1(t) and W2(t) represent two independent Wiener processes, σ1 and σ2 represent the diffusion coefficients, represents the frequency white noise, represents frequency random walk noise, both of which are atomic clock noise.

[0022] According to the method of the first aspect, in step S2, an estimated value of the noise diffusion coefficient is obtained by fitting according to the power spectrum density or Allan variance of the observed clock error:

[0023] The mathematical expectation of the Allan variance of the observed clock error includes the Allan variance excluding frequency drift and the Allan variance including frequency drift The mathematical expectations of the two are expressed as:

[0024]

[0025] Among them, τ is the smoothing time, E[.] means to find the mathematical expectation; the terms containing (σ1,σ2,σ0,d) in the observed clock difference all contribute to the Allan variance, and the estimated values ​​of (σ1,σ2,σ0,) are obtained by calculation.

[0026] According to the method of the first aspect, in step S3, when the weight of the atomic clock is determined using the ALGOS new weight algorithm, the weight statistic of each atomic clock is the weighted average of the square prediction errors of its multiple frequency differences; the observation time intervals when estimating the frequency difference and the frequency drift are respectively recorded as T i,1 and T i,2 , where T i,1 The time interval is equal to that when calculating the pseudo-free paper; the mathematical expectation of the weight statistic is expressed as:

[0027]

[0028] in, quad ST2 i,k represents the weight statistic when the ALGOS new weight algorithm is used, the subscript i represents the i-th clock, and the subscript k represents the k-th time interval;

[0029] When establishing pseudo-free paper time, an optimized T is selected for each atomic clock. i,1 , then T i,1 The time interval between each calculation of pseudo-free paper is not equal:

[0030]

[0031] Among them, t p Represents the prediction time, which is equal to the time interval between each calculation of the pseudo-free paper.

[0032] According to the method of the first aspect, the method further includes: step S4, determining the theoretical performance and experimental performance of the weight and pseudo-free paper time frequency stability of each atomic clock to verify the mathematical expectation of the weight statistics of the atomic clock; wherein:

[0033] In obtaining the E( quad ST2 i,k ) value, set the upper limit of the weight A / N, A is an empirical value, and calculate the weight of each atomic clock by using the weight inversely proportional to the statistic to obtain the theoretical value of the mathematical expectation of the weight:

[0034]

[0035] Among them, ω i,k represents weight;

[0036] Obtain the observed clock errors of N groups of reference time standards - atomic clocks, and calculate the corresponding pseudo-free paper time; and compare and verify with the theoretical value.

[0037] A second aspect of the present invention provides a system for determining the mathematical expectation of atomic clock weights, wherein the system is applied under the condition of establishing a time reference without using a free paper surface, and the system comprises a processing unit, wherein the processing unit is configured to perform the following steps:

[0038] Obtain the observed clock error of the reference time standard - atomic clock;

[0039] Calculating an estimated value of a noise diffusion coefficient based on the observed clock error;

[0040] The ALGOS new weight algorithm is used to determine the atomic clock weights, and further determine the mathematical expectation of the weight statistics corresponding to each atomic clock.

[0041] According to the system of the second aspect, the observed clock error is expressed as:

[0042]

[0043] Among them, x(t k ) represents the time difference, z(t k ) represents x(t k ) of the observed clock error, ε(t k ) represents the observation noise, which is described as the time derivative of the Wiener process W0(t), σ represents the intensity of the observation noise, x0 and y0 represent the initial values ​​of the time state and frequency state, d represents the frequency drift, W1(t) and W2(t) represent two independent Wiener processes, σ1 and σ2 represent the diffusion coefficients, σ1W1(t k ) represents frequency white noise, represents frequency random walk noise, both of which are atomic clock noise.

[0044] According to the system of the second aspect, the processing unit is specifically configured to obtain an estimated value of the noise diffusion coefficient by fitting according to the power spectrum density or Allan variance of the observed clock error:

[0045] The mathematical expectation of the Allan variance of the observed clock error includes the Allan variance excluding frequency drift and the Allan variance including frequency drift The mathematical expectations of the two are expressed as:

[0046]

[0047] Among them, τ is the smoothing time, E[.] means to find the mathematical expectation; the terms containing (σ1,σ2,σ0,d) in the observed clock difference all contribute to the Allan variance, and the estimated values ​​of (σ1,σ2,σ0,) are obtained by calculation.

[0048] According to the system of the second aspect, the processing unit is specifically configured to, when the ALGOS new weight algorithm is used to determine the weight of the atomic clock, the weight statistic of each atomic clock is the weighted average of the square prediction errors of its multiple frequency differences; the observation time intervals when estimating the frequency difference and the frequency drift are respectively recorded as T i,1 and T i,2 , where T i,1 The time interval is equal to that when calculating the pseudo-free paper; the mathematical expectation of the weight statistic is expressed as:

[0049]

[0050] in, quad ST2 i,k represents the weight statistic when the ALGOS new weight algorithm is used, the subscript i represents the i-th clock, and the subscript k represents the k-th time interval;

[0051] When establishing pseudo-free paper time, an optimized T is selected for each atomic clock. i,1 , then T i,1 The time interval between each calculation of pseudo-free paper is not equal:

[0052]

[0053] Among them, t p Represents the prediction time, which is equal to the time interval between each calculation of the pseudo-free paper.

[0054] According to the system of the second aspect, the processing unit is further configured to perform: determining the theoretical performance and experimental performance of the frequency stability of each atomic clock weight and pseudo-free paper time to verify the mathematical expectation of the weight statistics of the atomic clock; wherein:

[0055] In obtaining the E( quad ST2 i,k ) value, set the upper limit of the weight A / N, A is an empirical value, and calculate the weight of each atomic clock by using the weight inversely proportional to the statistic to obtain the theoretical value of the mathematical expectation of the weight:

[0056]

[0057] Among them, ω i,k represents weight;

[0058] Obtain the observed clock errors of N groups of reference time standards - atomic clocks, and calculate the corresponding pseudo-free paper time; and compare and verify with the theoretical value.

[0059] The third aspect of the present invention discloses an electronic device, which includes a memory and a processor, wherein the memory stores a computer program, and when the processor executes the computer program, the steps in the method for determining the mathematical expectation of atomic clock weights described in the first aspect of the present disclosure are implemented.

[0060] The fourth aspect of the present invention discloses a computer-readable storage medium having a computer program stored thereon, and when the computer program is executed by a processor, the steps in the method for determining the mathematical expectation of atomic clock weights described in the first aspect of the present disclosure are implemented.

[0061] In summary, the technical solution of the present invention can be applied to the method of establishing a time reference without free atoms as described in document [1] to determine the theoretical performance of atomic clock weights under specific circumstances. The present invention can effectively determine the mathematical expectation of each atomic clock weight; and mutually verify the experimental performance and theoretical conclusions of each atomic clock weight. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] In order to more clearly illustrate the specific implementation methods of the present invention or the technical solutions in the prior art, the drawings required for use in the specific implementation methods or the description of the prior art will be briefly introduced below. Obviously, the drawings in the following description are some implementation methods of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0063] Figure 1 A flowchart of a method for determining the mathematical expectation of atomic clock weights;

[0064] Figure 2 is the theoretical curve of Allan deviation;

[0065] Figure 3 is the observed clock error curve of Ref-hi;

[0066] Figure 4 is the “Ref-PseTA” curve;

[0067] Figure 5 is the weight curve;

[0068] Figure 6 The present invention is a structural diagram of an electronic device. DETAILED DESCRIPTION

[0069] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0070] The first aspect of the present invention provides a method for determining the mathematical expectation of atomic clock weights, the method being applied under the condition of establishing a time reference without using a free paper surface, the method comprising:

[0071] Step S1, obtaining the observed clock error of the reference time standard-atomic clock;

[0072] Step S2, calculating an estimated value of the noise diffusion coefficient based on the observed clock difference;

[0073] Step S3: Use the ALGOS new weight algorithm to determine the atomic clock weights, and further determine the mathematical expectation of the weight statistics corresponding to each atomic clock.

[0074] According to the method of the first aspect, in step S1, the observed clock error is expressed as:

[0075]

[0076] Among them, x(t k ) represents the time difference, z(t k ) represents x(t k ) of the observed clock error, ε(t k ) represents the observation noise, which is described as the time derivative of the Wiener process W0(t), σ represents the intensity of the observation noise, x0 and y0 represent the initial values ​​of the time state and frequency state, d represents the frequency drift, W1(t) and W2(t) represent two independent Wiener processes, σ1 and σ2 represent the diffusion coefficients, σ1W1(t k ) represents frequency white noise, represents frequency random walk noise, both of which are atomic clock noise.

[0077] According to the method of the first aspect, in step S2, an estimated value of the noise diffusion coefficient is obtained by fitting according to the power spectrum density or Allan variance of the observed clock error:

[0078] The mathematical expectation of the Allan variance of the observed clock error includes the Allan variance excluding frequency drift and the Allan variance including frequency drift The mathematical expectations of the two are expressed as:

[0079]

[0080] Among them, τ is the smoothing time, E[.] means to find the mathematical expectation; the terms containing (σ1,σ2,σ0,d) in the observed clock difference all contribute to the Allan variance, and the estimated values ​​of (σ1,σ2,σ0,) are obtained by calculation.

[0081] According to the method of the first aspect, in step S3, when the weight of the atomic clock is determined using the ALGOS new weight algorithm, the weight statistic of each atomic clock is the weighted average of the square prediction errors of its multiple frequency differences; the observation time intervals when estimating the frequency difference and the frequency drift are respectively recorded as T i,1 and T i,2 , where T i,1 The time interval is equal to that when calculating the pseudo-free paper; the mathematical expectation of the weight statistic is expressed as:

[0082]

[0083] in, quad ST2 i,k represents the weight statistic when the ALGOS new weight algorithm is used, the subscript i represents the i-th clock, and the subscript k represents the k-th time interval;

[0084] When establishing pseudo-free paper time, an optimized T is selected for each atomic clock. i,1 , then T i,1 The time interval between each calculation of the pseudo-free paper is not equal:

[0085]

[0086] Among them, t p Represents the prediction time, which is equal to the time interval between each calculation of the pseudo-free paper.

[0087] According to the method of the first aspect, the method further includes: step S4, determining the theoretical performance and experimental performance of the weight and pseudo-free paper time frequency stability of each atomic clock to verify the mathematical expectation of the weight statistics of the atomic clock; wherein:

[0088] In obtaining the E( quad ST2 i,k ) value, set the upper limit of the weight A / N, A is an empirical value, and calculate the weight of each atomic clock by using the weight inversely proportional to the statistic to obtain the theoretical value of the mathematical expectation of the weight:

[0089]

[0090] Among them, ω i,k represents weight;

[0091] Obtain the observed clock errors of N groups of reference time standards - atomic clocks, and calculate the corresponding pseudo-free paper time; and compare and verify with the theoretical value.

[0092] First embodiment

[0093] A method for determining the theoretical performance of atomic clock weights in a time base established using the new method described in [1].

[0094] This method is applied to the new method of establishing a time reference without free atoms described in reference [1], and is based on references [2] and [3]. First, obtain the observed clock error of "Ref-hi"; for measured data, obtain the estimated value of the noise diffusion coefficient of each atomic clock according to the method of reference [3]; for simulation data, the value of the noise diffusion coefficient of each atomic clock is known to the user. Secondly, set a reasonable time interval for estimating or predicting each parameter (frequency difference, frequency drift, etc.), substitute the estimated values ​​of these noise diffusion coefficients and the set time interval into the analytical expression given in reference [2], and calculate the mathematical expectation value of the corresponding statistics of each atomic clock. Thirdly, on this basis, set the upper limit of the weight and determine the theoretical value of the mathematical expectation of the weight of each atomic clock. Finally, obtain the experimental performance of the weight of each atomic clock to verify the theoretical conclusion.

[0095] S.1.1 Obtain the observed clock error of the “reference time standard – atomic clock”, i.e. “Ref-hi”.

[0096] For measured data, it can be directly extracted from the timekeeping system, or downloaded from the International Bureau of Weights and Measures, IGS, IGMAS, etc. In addition, the observed clock error can also be simulated by the user.

[0097] Whether it is measured or simulated, the observed clock error can be expressed as:

[0098]

[0099] Among them, x(t k ) represents the time difference, z(t k ) represents x(t k ) observation, i.e., the observed clock error; ε(t k ) is the observation noise, which is generally considered to be white phase modulation noise (WPM), and can be written as the time derivative of a Wiener process W0(t), σ is used to indicate the intensity of the observation noise; x0 and y0 represent the initial values ​​of the time state and frequency state, d represents the frequency drift, W1(t) and W2(t) represent two independent Wiener processes, σ1 and σ2 represent the diffusion coefficients; σ1W1(t k) represents white frequency modulation noise (WFM), It stands for walk random modulation noise (RWFM), which are all atomic clock noises.

[0100] In fact, most typical hydrogen clocks and cesium clocks can be represented by the model shown in formula (1). Formula (1) shows that the observed clock error can be expressed as a quadratic polynomial superimposed with three types of noise: WFM, RWFM, and WPM, where WFM and RWFM are atomic clock noises and WPM is observation noise.

[0101] S.1.2 Obtaining an estimate of the noise diffusion coefficient

[0102] For the measured observational clock error, The estimated value of can be fitted based on the power spectral density or Allan variance of the observed clock difference.

[0103] The mathematical expectation of the Allan variance of the observed clock error is expressed as: The Allan variance excluding frequency drift (denoted as ) and the Allan variance including frequency drift (denoted as ) is expressed as:

[0104]

[0105] Among them, τ is the smoothing time, and E[.] represents the mathematical expectation.

[0106] From equations (2) and (3), we can see that the terms (σ1, σ2, σ0, d) in the observed clock error all contribute to the Allan variance. The present invention adopts the method of reference [3] to obtain the estimated value of (σ1, σ2, σ0, d).

[0107] For the observed clock errors generated by simulation, the (σ1,σ2,σ0,) values ​​are selected by the user and are known to the user.

[0108] S.1.3 Determine the theoretical value of the mathematical expectation of the statistics corresponding to each atomic clock

[0109] In the literature [1], the ALGOS new weight algorithm is directly used to determine the weights. In this case, the statistic for each clock is the weighted average of the squared prediction errors of its multiple frequency differences. The user needs to select the observation time interval for estimating the frequency difference and frequency drift, which are denoted as T and i,1 and T i,2 , where T i,1 The time interval is equal to that when calculating PseTA. Substituting these values ​​into the conclusion of reference [2], we obtain the theoretical value of the mathematical expectation of the statistic, expressed as:

[0110]

[0111] in, quad ST2 i,k represents the statistic when the ALGOS new weight algorithm is used, and the prediction model is a quadratic polynomial model; subscript i represents the i-th clock, subscript k represents the k-th time interval; T i,1 is the observation time interval for obtaining the frequency error estimate, T i,2 is the observation time interval for obtaining the frequency drift estimate.

[0112] It should be noted that in the above formula, T1 is equal to the time interval for calculating PseTA each time, which is exactly the same as the ALGOS algorithm. However, in fact, the literature [1] points out that when establishing PseTA, the optimal value T can be optimized for each specific situation. i,1 At this time, T i,1 The time interval between each calculation of PseTA may not be equal. According to the literature [2], we get:

[0113]

[0114] Among them, t p Represents the prediction time, which is equal to the time interval between each calculation of PseTA.

[0115] S.1.4 Determine the theoretical performance of the individual atomic clock weights and PseTA frequency stability.

[0116] After obtaining the E( quad ST2 i,k ) value, set the upper limit of the weight A / N, where A is selected by the user, for example, it can be 1.8, 2, 2.5, 4, etc., and then the weight is inversely proportional to the statistic, so as to calculate the weight of each atomic clock and obtain the theoretical value of the mathematical expectation of the weight, that is:

[0117]

[0118] Among them, ω i,k represents the weight, the subscript i represents the i-th clock, and the subscript k represents the k-th time interval.

[0119] S.1.5 Experimental performance of obtaining the weights of each atomic clock and the PseTA frequency stability

[0120] Obtain N groups of "Ref-hi" observation clock errors. Calculate PseTA according to the method in reference [1] and express it in the form of "Ref-PseTA". Among them, the prediction algorithm and weight algorithm are exactly the same as the new prediction algorithm and new weight algorithm of ALGOS. The specific calculation steps can be found in reference [2]. When calculating the prediction value, the user needs to select the values ​​of parameters T1 and T2 in advance. When calculating the weight, the user needs to set the value A / N of the upper limit of the weight. Finally, obtain the test results, including the test results of the weight of each atomic clock, obtain "Ref-PseTA", and draw the curve of "Ref-PseTA". Compare the test results of the weight of each atomic clock with the theoretical conclusions to verify the theoretical conclusions.

[0121] Second embodiment

[0122] The simulation generates 30 groups of "Ref-hi" observation clock errors. A total of 30 atomic clocks, including 10 imported hydrogen clocks, 10 domestic hydrogen clocks, and 10 cesium clocks, are selected to form a clock group. Ref can be a reference time standard established by multiple optical clocks and fountain clocks, and its frequency stability is much better than that of a single atomic clock. Therefore, the noise in the "Ref-hi" observation clock error is mainly atomic clock noise and observation noise. The data length of each group of observation clock errors is 4800 data points, and the time interval between data points is 3600s, that is, the data length is 200d. The simulation parameters are shown in Table 1. This parameter is the same as the parameter used in the literature [1]. Since (x0, y0) has no effect on the weight and PseTA frequency stability, x0=y0=0 is simply selected. The observation noise variance of all observation clock errors is set to This is equivalent to an observation noise standard deviation of 10 ps. In practical applications, a variety of preprocessing methods can be used to filter out observation noise. Here, the KF algorithm in the literature [3] is used to filter out observation noise, which basically eliminates the impact of observation noise on statistics, weights, and PseTA.

[0123] Table 1 Parameters of different atomic clocks

[0124]

[0125] Set the PseTA calculation time interval to t p =1d, that is, PseTA is calculated every 1d. According to the literature [1], the optimal observation time interval for different atomic clocks is calculated as follows: T i,1 =7.5d, domestic hydrogen clock T i,1 =0.1d, cesium clock T i,1 =100.76d. Taking all factors into consideration, the imported hydrogen clock T i,1 =7.5d (consistent with the optimal observation time interval), domestic hydrogen clock T i,1=10800s (integer multiple of the minimum time interval of data points, close to the optimal observation time interval), cesium clock T i,1 = 10d (as close as possible to the optimal observation time interval, but not too long, and not more than T i,2 ). Reference [2] proves that T i,2 The longer the time interval is, the smaller the prediction uncertainty is. However, in practical applications, it is not recommended to use T because the observation characteristics of atomic clocks may change. i,2 The selection is too long. Here, set T i,2 =10d.

[0126] First, determine the atomic clock E(ω i,k ) and theoretical performance of PseTA frequency stability.

[0127] Put (σ i,1 ,σ i,2 ,T1,T2) values ​​into formula (4) to calculate the E( quad ST2 i,k ) value. Select the upper limit of the weight as 2 / N, and calculate the mathematical expectation of the weight according to formula (6), that is, E(ω i,k ) values. Table 2 lists the E( quad ST2 i,k ) value, E(ω of each atomic clock when the highest weight is not restricted and the highest weight is restricted i,k )value.

[0128] Table 2 E( quad ST2 i,k ) and E(ω i,k )value

[0129]

[0130]

[0131] Imported hydrogen clock, domestic hydrogen clock, cesium clock, PseTA (no limit on the highest weight), PseTA (limit on the highest weight) (σ i,1 ,σ i,2 ,d i ) values ​​are substituted into equations (2) and (3) to calculate their Allan deviations (i.e., the square root of the Allan variance). Figure 2 The theoretical curves of their Allan deviations were drawn.

[0132] Secondly, obtain the atomic clock ω i,k The test performance.

[0133] Figure 3The "Ref-hi" observation clock difference curves of an imported hydrogen clock, a domestic hydrogen clock, and a cesium clock generated by simulation are drawn, where the black curve represents the imported hydrogen clock, the green curve represents the domestic hydrogen clock, and the red curve represents the cesium clock.

[0134] 32 days of data are accumulated. Starting from 0:00 on the 33rd day, the new method described in the literature [1] is used to establish the time benchmark PseTA, and the calculation time interval is selected as 1d, that is, PseTA is calculated every 1d. The length of PseTA is 168 days, with a total of 168 calculation time intervals. Among them, the weights and prediction values ​​are obtained using the ALGOS new weight algorithm and new prediction algorithm, that is, the calculation method of the weights and prediction values ​​is exactly the same as that in the literature [2]. Finally, the "Ref-PseTA" is calculated. Figure 4 A "Ref-PseTA" curve was plotted.

[0135] Figure 5 The weights of a certain imported hydrogen clock, a certain domestic hydrogen clock, and a certain cesium clock are plotted. At the same time, the weight averages of all atomic clocks of the same type (i.e., 10 imported hydrogen clocks, 10 domestic hydrogen clocks, and 10 cesium clocks) are calculated, and the results are as follows: imported hydrogen clock: 0.0667; domestic hydrogen clock: 0.0313; cesium clock: 0.0021. This is basically consistent with the theoretical conclusions shown in Table 2, indicating that the experimental results verify the theoretical conclusions.

[0136] A second aspect of the present invention provides a system for determining the mathematical expectation of atomic clock weights, wherein the system is applied under the condition of establishing a time reference without using a free paper surface, and the system comprises a processing unit, wherein the processing unit is configured to perform the following steps:

[0137] Obtain the observed clock error of the reference time standard - the atomic clock;

[0138] Calculating an estimated value of a noise diffusion coefficient based on the observed clock error;

[0139] The ALGOS new weight algorithm is used to determine the atomic clock weights, and further determine the mathematical expectation of the weight statistics corresponding to each atomic clock.

[0140] According to the system of the second aspect, the observed clock error is expressed as:

[0141]

[0142] Among them, x(t k ) represents the time difference, z(t k ) represents x(t k ) of the observed clock error, ε(t k) represents the observation noise, which is described as the time derivative of the Wiener process W0(t), σ represents the intensity of the observation noise, x0 and y0 represent the initial values ​​of the time state and frequency state, d represents the frequency drift, W1(t) and W2(t) represent two independent Wiener processes, σ1 and σ2 represent the diffusion coefficients, σ1W1(t k ) represents frequency white noise, represents frequency random walk noise, both of which are atomic clock noise.

[0143] According to the system of the second aspect, the processing unit is specifically configured to obtain an estimated value of the noise diffusion coefficient by fitting according to the power spectrum density or Allan variance of the observed clock error:

[0144] The mathematical expectation of the Allan variance of the observed clock error includes the Allan variance excluding frequency drift and the Allan variance including frequency drift The mathematical expectations of the two are expressed as:

[0145]

[0146] Among them, τ is the smoothing time, E[.] means to find the mathematical expectation; the terms containing (σ1,σ2,σ0,d) in the observed clock difference all contribute to the Allan variance, and the estimated values ​​of (σ1,σ2,σ0,) are obtained by calculation.

[0147] According to the system of the second aspect, the processing unit is specifically configured to, when the ALGOS new weight algorithm is used to determine the weight of the atomic clock, the weight statistic of each atomic clock is the weighted average of the square prediction errors of its multiple frequency differences; the observation time intervals when estimating the frequency difference and the frequency drift are respectively recorded as T i,1 and T i,2 , where T i,1 The time interval is equal to that when calculating the pseudo-free paper; the mathematical expectation of the weight statistic is expressed as:

[0148]

[0149] in, quad ST2 i,k represents the weight statistic when the ALGOS new weight algorithm is used, the subscript i represents the i-th clock, and the subscript k represents the k-th time interval;

[0150] When establishing pseudo-free paper time, an optimized T is selected for each atomic clock. i,1 , then T i,1 The time interval between each calculation of the pseudo-free paper is not equal:

[0151]

[0152] Among them, t p Represents the prediction time, which is equal to the time interval between each calculation of the pseudo-free paper.

[0153] According to the system of the second aspect, the processing unit is further configured to perform: determining the theoretical performance and experimental performance of the frequency stability of each atomic clock weight and pseudo-free paper time to verify the mathematical expectation of the weight statistics of the atomic clock; wherein:

[0154] In obtaining the E( quad ST2 i,k ) value, set the upper limit of the weight A / N, A is an empirical value, and calculate the weight of each atomic clock by using the weight inversely proportional to the statistic to obtain the theoretical value of the mathematical expectation of the weight:

[0155]

[0156] Among them, ω i,k represents weight;

[0157] Obtain the observed clock errors of N groups of reference time standards - atomic clocks, and calculate the corresponding pseudo-free paper time; and compare and verify with the theoretical value.

[0158] Third embodiment

[0159] A system for determining the theoretical performance of atomic clock weights in a time base established using the new method described in [1].

[0160] The system includes: an actual observation acquisition module, configured to obtain all "Ref-hi" observation clock errors; a parameter acquisition module, configured to, for measured data, obtain an estimated value of the noise diffusion coefficient of each atomic clock according to the method of document [3]; for simulation data, directly obtain the noise diffusion coefficient value set by the user; a theoretical performance determination module, configured to estimate the parameters of each atomic clock based on the actual observation information provided by the actual observation acquisition module, set a reasonable time interval for estimating or predicting each parameter (frequency difference, frequency drift, etc.), substitute it into the analytical expression given in document [2], calculate the mathematical expectation value of the corresponding statistic of each atomic clock, set the upper limit of the weight, and determine the mathematical expectation value of the weight of each atomic clock; an experimental verification module, configured to generate PseTA according to the method of document [1], obtain the experimental performance of the weight of each atomic clock, and display it together with the theoretical conclusion for mutual verification.

[0161] The actual observation acquisition module is configured to obtain all the "Ref-hi" observation clock errors in the clock group; it can simulate and generate the "Ref-hi" observation clock errors that meet formula (1), and it can also read the measured observation clock errors downloaded from the International Bureau of Weights and Measures, IGS, IGMAS, etc.; whether it is simulated data or measured data, this module requires the user to manually input the sampling interval of the data point, the calculation time interval of PseTA (that is, how often PseTA is calculated), and the total length of the data point.

[0162] The parameter acquisition module is configured to obtain the estimated diffusion coefficient of each atomic clock noise and observation noise according to the method of reference [3] for the measured data, that is, (σ i,1 ,σ i,2 ,σ i,0 ) is an estimate of; for the simulated data, (σ i,1 ,σ i,2 ,σ i,0 ) is a known quantity set by the user and can be obtained directly.

[0163] The theoretical performance determination module is configured to determine the actual observation information provided by the actual observation acquisition module according to the (σ i,1 ,σ i,2 ) parameters, set reasonable (T1, T2), substitute into the analytical expression given in reference [2], calculate the mathematical expectation value of the corresponding statistic of each atomic clock, set the upper limit of the weight, and determine the theoretical value of the mathematical expectation of the weight of each atomic clock.

[0164] The experimental verification module is configured to generate PseTA according to the method in reference [1], where the estimation and prediction models as well as the weight algorithm are completely consistent with the new prediction algorithm and new weight algorithm of ALGOS, and a "Ref-PseTA" curve is drawn to obtain the experimental performance of each atomic clock weight and display it together with the theoretical conclusion as a mutual verification.

[0165] The third aspect of the present invention discloses an electronic device, which includes a memory and a processor, wherein the memory stores a computer program, and when the processor executes the computer program, the steps in the method for determining the mathematical expectation of atomic clock weights described in the first aspect of the present disclosure are implemented.

[0166] Figure 6 is a structural diagram of an electronic device according to an embodiment of the present invention, such as Figure 6As shown, the electronic device includes a processor, a memory, a communication interface, a display screen and an input device connected via a system bus. Among them, the processor of the electronic device is used to provide computing and control capabilities. The memory of the electronic device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system and a computer program. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The communication interface of the electronic device is used to communicate with an external terminal in a wired or wireless manner, and the wireless manner can be achieved through WIFI, an operator network, near field communication (NFC) or other technologies. The display screen of the electronic device can be a liquid crystal display screen or an electronic ink display screen, and the input device of the electronic device can be a touch layer covered on the display screen, or a button, a trackball or a touch pad set on the housing of the electronic device, or an external keyboard, touch pad or mouse, etc.

[0167] Those skilled in the art will understand that Figure 6 The structure shown in the figure is only a structural diagram of the part related to the technical solution of the present disclosure, and does not constitute a limitation on the electronic device to which the technical solution of the present application is applied. The specific electronic device may include more or fewer components than those shown in the figure, or combine certain components, or have a different component arrangement.

[0168] The fourth aspect of the present invention discloses a computer-readable storage medium having a computer program stored thereon, and when the computer program is executed by a processor, the steps in the method for determining the mathematical expectation of atomic clock weights described in the first aspect of the present disclosure are implemented.

[0169] In summary, the technical solution of the present invention can be applied to the method of establishing a time reference without using free atoms as described in document [1], to determine the theoretical performance of the atomic clock weight under ideal conditions.

[0170] Please note that the technical features of the above embodiments can be combined arbitrarily. In order to make the description concise, all possible combinations of the technical features in the above embodiments are not described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification. The above-mentioned embodiments only express several implementation methods of the present application, and their descriptions are relatively specific and detailed, but they cannot be understood as limiting the scope of the invention patent. It should be pointed out that for ordinary technicians in this field, without departing from the concept of the present application, several variations and improvements can be made, which all belong to the scope of protection of the present application. Therefore, the scope of protection of the patent in this application shall be based on the attached claims.

Claims

1. A method for determining the mathematical expectation of atomic clock weights, characterized in that: The method is applied under the condition of establishing a time reference without passing through a free paper surface, and the method comprises: Step S1, obtaining the observed clock error of the reference time standard-atomic clock; Step S2, calculating an estimated value of the noise diffusion coefficient based on the observed clock difference; Step S3: Use the ALGOS new weight algorithm to determine the atomic clock weights, and further determine the mathematical expectation of the weight statistics corresponding to each atomic clock.

2. A method for determining the mathematical expectation of atomic clock weights according to claim 1, characterized in that: In step S1, the observed clock error is expressed as: Among them, x(t k ) represents the time difference, z(t k ) represents x(t k ) of the observed clock error, ε(t k ) represents the observation noise, which is described as the time derivative of the Wiener process W0(t), σ represents the intensity of the observation noise, x0 and y0 represent the initial values ​​of the time state and frequency state, d represents the frequency drift, W1(t) and W2(t) represent two independent Wiener processes, σ1 and σ2 represent the diffusion coefficients, σ1W1(t k ) represents frequency white noise, represents frequency random walk noise, both of which are atomic clock noise.

3. A method for determining the mathematical expectation of atomic clock weights according to claim 2, characterized in that: In step S2, an estimated value of the noise diffusion coefficient is obtained by fitting according to the power spectrum density or Allan variance of the observed clock error: The mathematical expectation of the Allan variance of the observed clock error includes the Allan variance excluding frequency drift and the Allan variance including frequency drift The mathematical expectations of the two are expressed as: Among them, τ is the smoothing time, E[.] means to find the mathematical expectation; the terms containing (σ1,σ2,σ0,d) in the observed clock difference all contribute to the Allan variance, and the estimated values ​​of (σ1,σ2,σ0,) are obtained by calculation.

4. A method for determining the mathematical expectation of atomic clock weights according to claim 3, characterized in that: In step S3, when the ALGOS new weight algorithm is used to determine the weight of the atomic clock, the weight statistic of each atomic clock is the weighted average of the squared prediction errors of its multiple frequency differences; the observation time intervals for estimating the frequency difference and frequency drift are respectively denoted as T i,1 and T i,2 , where T i,1 The time interval is equal to that when calculating the pseudo-free paper; the mathematical expectation of the weight statistic is expressed as: in, quad ST2 i,k represents the weight statistic when the ALGOS new weight algorithm is used, the subscript i represents the i-th clock, and the subscript k represents the k-th time interval; When establishing pseudo-free paper time, an optimized T is selected for each atomic clock. i,1 , then T i,1 The time interval between each calculation of the pseudo-free paper is not equal: Among them, t p Represents the prediction time, which is equal to the time interval between each calculation of the pseudo-free paper.

5. A method for determining the mathematical expectation of atomic clock weights according to claim 4, characterized in that: The method further comprises: step S4, determining the theoretical performance and experimental performance of the weights and pseudo-free time frequency stability of each atomic clock to verify the mathematical expectation of the weight statistics of the atomic clock; wherein: In obtaining the E( quad ST2 i,k ) value, set the upper limit of the weight A / N, A is an empirical value, and calculate the weight of each atomic clock by using the weight inversely proportional to the statistic to obtain the theoretical value of the mathematical expectation of the weight: Among them, ω i,k represents weight; Obtain the observed clock errors of N groups of reference time standards - atomic clocks, and calculate the corresponding pseudo-free paper time; and compare and verify with the theoretical value.

6. A system for determining the mathematical expectation of atomic clock weights, characterized in that The system is applied under the condition of establishing a time reference without passing through a free paper surface, and the system comprises a processing unit, and the processing unit is configured to perform the following steps: Obtain the observed clock error of the reference time standard - atomic clock; Calculating an estimated value of a noise diffusion coefficient based on the observed clock error; The ALGOS new weight algorithm is used to determine the atomic clock weights, and further determine the mathematical expectation of the weight statistics corresponding to each atomic clock.

7. A system for determining the mathematical expectation of atomic clock weights according to claim 6, characterized in that: The observed clock error is expressed as: Among them, x(t k ) represents the time difference, z(t k ) represents x(t k ) of the observed clock error, ε(t k ) represents the observation noise, which is described as the time derivative of the Wiener process W0(t), σ represents the intensity of the observation noise, x0 and y0 represent the initial values ​​of the time state and frequency state, d represents the frequency drift, W1(t) and W2(t) represent two independent Wiener processes, σ1 and σ2 represent the diffusion coefficients, σ1W1(t k ) represents frequency white noise, represents frequency random walk noise, both of which are atomic clock noise.

8. A system for determining the mathematical expectation of atomic clock weights according to claim 7, characterized in that: The processing unit is specifically configured to obtain an estimated value of the noise diffusion coefficient by fitting according to the power spectrum density or Allan variance of the observed clock error: The mathematical expectation of the Allan variance of the observed clock error includes the Allan variance excluding frequency drift and the Allan variance including frequency drift The mathematical expectations of the two are expressed as: Among them, τ is the smoothing time, E[.] means to find the mathematical expectation; the terms containing (σ1,σ2,σ0,d) in the observed clock difference all contribute to the Allan variance, and the estimated values ​​of (σ1,σ2,σ0,) are obtained by calculation.

9. A system for determining the mathematical expectation of atomic clock weights according to claim 8, characterized in that: The processing unit is specifically configured to, when the ALGOS new weight algorithm is used to determine the weight of the atomic clock, the weight statistic of each atomic clock is the weighted average of the square prediction errors of its multiple frequency differences; the observation time intervals when estimating the frequency difference and the frequency drift are respectively recorded as T i,1 and T i,2 , where T i,1 The time interval is equal to that when calculating the pseudo-free paper; the mathematical expectation of the weight statistic is expressed as: in, quad ST2 i,k represents the weight statistic when the ALGOS new weight algorithm is used, the subscript i represents the i-th clock, and the subscript k represents the k-th time interval; When establishing pseudo-free paper time, an optimized T is selected for each atomic clock. i,1 , then T i,1 The time interval between each calculation of the pseudo-free paper is not equal: Among them, t p Represents the prediction time, which is equal to the time interval between each calculation of the pseudo-free paper.

10. A system for determining the mathematical expectation of atomic clock weights according to claim 9, characterized in that: The processing unit is further configured to perform: determining theoretical performance and experimental performance of the weights and pseudo-free paper time frequency stability of each atomic clock to verify the mathematical expectation of the weight statistics of the atomic clock; wherein: In obtaining the E( quad ST2 i,k ) value, set the upper limit of the weight A / N, A is an empirical value, and calculate the weight of each atomic clock by using the weight inversely proportional to the statistic to obtain the theoretical value of the mathematical expectation of the weight: Among them, ω i,k represents weight; Obtain the observed clock errors of N groups of reference time standards - atomic clocks, and calculate the corresponding pseudo-free paper time; and compare and verify with the theoretical value.

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