Fast phase and Doppler frequency shift estimation method based on prior information

Abstract

The invention provides a fast phase and Doppler frequency shift estimation method based on prior information, and belongs to the technical field of aerospace navigation, and the method comprises the steps: building two types of prior estimation models according to whether a spacecraft state error is considered or not; a general Bayesian risk function is established according to a Bayesian model and a loss function, and different objective functions including maximum likelihood estimation and parameter prior information are obtained by selecting the loss function. For a non-convex problem of an objective function, a fast non-convex optimization method with interval constraint is provided, a parameter interval range is determined based on prior information, so that the objective function can be regarded as a convex function in a constraint interval, an F-BFGS-B algorithm is provided based on a BFGS algorithm framework, and the parameter range is limited. According to the method, the proper search step size can be quickly found, and the estimation precision and the real-time performance can be considered at the same time.

Description

Technical Field

[0001] The invention belongs to the technical field of aerospace navigation, and in particular relates to a fast phase and Doppler frequency shift estimation method based on prior information. Background Art

[0002] As humanity's exploration of space deepens, autonomous spacecraft navigation is becoming increasingly important. Traditional astronomical navigation systems have been used in model missions, but they are susceptible to the complex space environment, which can degrade navigation performance. X-ray pulsar-based navigation (XNAV) is a very promising autonomous navigation method that can be applied throughout space and is not susceptible to interference from the space environment. X-ray pulsars are rotating neutron stars with a highly stable period, emitting X-ray signals with a stable period.

[0003] In X-ray pulsar navigation, the onboard detector detects the arrival times of a series of X-ray photons. By processing the photon arrival times, the pulse phase can be estimated, and the pulse delay can be obtained. After obtaining the pulse delay, the status information of the spacecraft can be obtained. Therefore, the estimation of the pulse phase is crucial. Pulse phase estimation refers to the phase at the initial or end time of an observation cycle, where the estimated phase at the initial time is the basic phase estimation, also called the initial phase estimation. Unless otherwise specified, the phase estimation mentioned in the present invention also refers to the initial phase estimation. The scenarios for estimating the pulse phase can be divided into two categories, namely zero acceleration scenarios and non-zero acceleration scenarios.

[0004] In zero-acceleration scenarios, phase or Doppler shift can be estimated through methods such as epoch folding and maximum likelihood estimation. However, these methods suffer from slow computational speed and accuracy below the Cramer-Rao lower bound. Traditional methods fail in non-zero-acceleration scenarios. While solutions exist, such as phase tracking and estimation methods based on on-orbit phase models, they each have limitations. For example, the phase tracking method reduces the amount of photon data due to the sub-observation period, thus reducing estimation accuracy. Estimation methods based on on-orbit phase models use two-dimensional search methods, making it difficult to achieve both real-time and high-precision performance. In summary, no method currently offers a solution that can simultaneously achieve both real-time and high-precision phase and Doppler shift estimation. Summary of the Invention

[0005] To address the inability of traditional methods to simultaneously achieve both real-time and accuracy in phase and Doppler shift estimation, this paper proposes a fast phase and Doppler shift estimation method based on prior information. Two phase models are established, depending on whether spacecraft state errors are considered. A Bayesian risk function is constructed based on a Bayesian model and a loss function. Different objective functions are derived by selecting the loss function, including maximum likelihood estimation and parameter prior information. To address the non-convex objective function, a fast non-convex optimization method with interval constraints is proposed. The parameter interval is determined based on prior information, making the objective function convex within the constrained interval. The F-BFGS-B (fast BFGS-B with interval constraints) algorithm, based on the BFGS (Breudin-Fletcher-Goldforb-Chanteau) algorithm framework, restricts the parameter range. This method can quickly find an appropriate search step size, balancing both estimation accuracy and real-time performance.

[0006] In order to achieve the above object, the present invention adopts the following technical solutions:

[0007] A fast phase and Doppler shift estimation method based on prior information includes the following steps:

[0008] Step 1: Establish a time conversion model to convert the photon arrival time at the spacecraft to the photon arrival time at the center of mass of the solar system;

[0009] Step 2: Combine the time conversion model to establish a pulse phase model that does not consider the spacecraft state error;

[0010] Step 3: Combining the time conversion model with the pulse phase model that does not consider the spacecraft state error, a pulse phase model that includes errors is established based on the error transfer characteristics of the orbital dynamics equation while considering the spacecraft state error.

[0011] Step 4: Combining the time conversion model and the pulse phase model containing errors, and according to the superposition property of the normal distribution, establish a joint probability density function of the initial pulse phase and Doppler frequency shift;

[0012] Step 5: Combined with the joint probability density function, an objective function of phase and Doppler shift with prior information is established based on the Bayesian model;

[0013] Step 6: Estimate the initial phase and Doppler shift according to the objective function using the fast Bruydan-Fletcher-Goldforb-Chanton algorithm with interval constraints.

[0014] The beneficial effects of the present invention compared with the prior art are:

[0015] (1) The phase model proposed in this invention can select any reference time, ensuring the accuracy of pulsar ephemeris parameters. It is more rigorous and universal than the traditional phase model.

[0016] (2) This invention establishes a universal Bayesian risk function based on the Bayesian model and loss function, taking both “return” and “risk” into account. Different objective functions can be obtained by selecting different loss functions, which is more universal than traditional methods.

[0017] (3) This paper proposes a fast BFGS algorithm with interval constraints, which can effectively solve the problems of slow and one-dimensional step size and no interval constraints in traditional BFGS search. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 The figure is a flow chart of the method for fast phase and Doppler shift estimation based on prior information of the present invention.

[0019] Figure 2 The simulation results of the method proposed in this invention are shown in Figure 2; (a) shows the change of phase estimation error with observation time, (b) shows the change of Doppler frequency shift estimation error with observation time, and (c) shows the change of GPU time with observation time. DETAILED DESCRIPTION

[0020] In order to make the objectives, technical solutions, and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are intended only to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other. The fast phase and Doppler shift estimation method based on prior information of the present invention is a new method for estimating phase and Doppler shift, which can solve the problems of rapidity and real-time performance of phase and Doppler shift estimation.

[0021] Specifically, the present invention first considers the time delay term of the relativistic effect and establishes a solar system barycenter coordinate transformation model for the photon arrival time. Secondly, a pulse phase model is obtained without considering the spacecraft state error. Then, a pulse phase model containing errors and Doppler shift are established based on the error transfer characteristics of the orbital dynamics equation while considering the spacecraft state error. Based on the superposition property of the normal distribution, a joint probability density function of the initial pulse phase and Doppler shift is established. Based on the Bayesian model, an objective function with prior information on phase and Doppler shift is established. Finally, a fast non-convex optimization algorithm with interval constraints is proposed, and the constraint interval is obtained based on the properties of the normal distribution. Within the constraint interval, the objective function is transformed from a non-convex function to a convex function.

[0022] The present invention is described in detail below with reference to specific embodiments.

[0023] like Figure 1 As shown, the fast phase and Doppler shift estimation method based on prior information in an embodiment of the present invention includes the following steps:

[0024] Step 1: Establish a time conversion model to convert the photon arrival time at the spacecraft into the photon arrival time at the center of mass of the solar system (i.e. Figure 1 The photon arrival time conversion model is established, including:

[0025] Taking the Mars probe as an example, the coordinate system is the Solar System Barycenter (SSB) coordinate system. During the observation period, the spacecraft detected The arrival time of a pulse photon (i.e. the arrival time of the photon at the spacecraft) is recorded as ,in, represents the time when the i-th photon in the j-th observation period arrives at the spacecraft, and the subscript sc represents the spacecraft;

[0026] (1)

[0027] Where, the superscript T represents the matrix transpose, Indicates that the spacecraft is The position vector at that moment is relative to the SSB, n is the direction vector of the pulsar observed from the SSB, and c is the speed of light. represents the photon time conversion model, and HOT represents the time delay term considering the relativistic effect.

[0028] Step 2: Establish a pulse phase model that does not consider spacecraft state errors, including:

[0029] If we assume that the reference time The pulse phase at , then the arrival time of photons at the spacecraft is , its pulse phase can be expressed as:

[0030] (2)

[0031] in, express The pulse phase at time and Represents the pulsar in The spin frequency and its first derivative at the moment . Therefore, Initial phase of the pulse It can be expressed as:

[0032] (3)

[0033] in, is the time when the initial photon arrives at the spacecraft in the jth observation period, A time transition model representing the initial moment.

[0034] Subtracting formula (2) from formula (3) yields the following phase model formula:

[0035] (4)

[0036] Step 3: Considering the spacecraft state error, establish a pulse phase model containing the error according to the error transfer characteristics of the orbital dynamics equation (i.e. Figure 1 The establishment of a pulse phase model taking into account spacecraft state errors) includes:

[0037] According to formula (1), the phase model formula (4) is determined by the position of the spacecraft. In actual navigation scenarios, the true position of the spacecraft is unknown. Therefore, in order to consider the impact of the spacecraft state estimation error on the phase model formula (4), the present invention defines and They are The true position and true speed of the spacecraft at this moment, and They are The estimated position and estimated velocity of the spacecraft at time .

[0038] The mathematical relationship between the true and estimated values of the spacecraft state is expressed as:

[0039] (5)

[0040] in, and Respectively represent the position estimation error and velocity estimation error of the spacecraft at any time. Therefore, according to the error transfer characteristics of the state equation, the phase model formula (4) is converted to and the initial estimated position Performing Taylor expansion yields:

[0041] (6)

[0042] in, Represents the estimated error of the spacecraft's position at the initial moment. and They represent the partial derivatives of the time conversion model of formula (1) with respect to the spacecraft position at the initial moment and the i-th moment respectively. Indicates the phase deviation caused by time variation.

[0043] Similarly, Formula (3) is used at the initial estimated position Performing Taylor expansion yields:

[0044] (7)

[0045] In formula (7), let represents the estimated value of the initial phase, Represents the estimation error of the initial phase, that is:

[0046] (8)

[0047] Therefore, substituting formula (8) into formula (6) can obtain the phase model that takes into account the prior information of the spacecraft state and the state estimation error:

[0048] (9)

[0049] Step 4: Combining the time conversion model and the pulse phase model including the error, according to the superposition property of the normal distribution, a joint probability density function of the initial pulse phase and Doppler shift is established, including:

[0050] According to the properties of the orbital dynamics model, the spacecraft position error is related to the initial position error and initial velocity error, which can be expressed as:

[0051] (10)

[0052] in, and is the state transition matrix, which can be expressed as:

[0053] (11)

[0054] in, and is a constant matrix. m is the order of the state matrix expansion, which is taken as 1 in the present invention. represents the 3D identity matrix. So substitute formula (10) and formula (11) into formula (9), and consider , so the on-orbit phase model considering the prior state information of the spacecraft can be expressed as:

[0055] (12)

[0056] in, represents the Doppler shift estimation error, which is expressed as:

[0057] (13)

[0058] The position estimation error and velocity estimation error of the spacecraft at any time caused by the use of Kalman filtering and is zero-mean Gaussian white noise, so according to the superposition property of normal distribution, the initial phase estimation error and Doppler shift estimation error is also zero-mean Gaussian white noise, then according to formula (8) and formula (13), and It also obeys the normal distribution.

[0059] exist Initial phase of the pulse The mean and variance of are:

[0060] (14)

[0061] in, and Respectively The mean and variance of It represents the variance of the position estimation error, E represents the expectation, and Var represents the variance.

[0062] Doppler shift estimation error The mean and variance of are:

[0063] (15)

[0064] in, and Respectively The mean and variance of represents the variance of the velocity estimation error, and They represent when m is 1. The constant matrix and The constant matrix of the pulsar source frequency It can be expressed as:

[0065] (16)

[0066] If the hyperparameter , then the joint probability density function Expressed as:

[0067] (17)

[0068] in, represents the mean vector, exp() represents the exponential function, Denotes the determinant of the covariance matrix, the covariance matrix Expressed as:

[0069] (18)

[0070] in, express and The correlation coefficient between them is expressed as:

[0071] (19)

[0072] Among them, Cov means covariance.

[0073] Step 5: Establish an objective function of phase and Doppler shift with prior information based on the Bayesian model, including:

[0074] Step 4 establishes the joint probability density function of the unknown parameters, so the objective function can be established based on the Bayesian estimation framework using the prior information of the location parameters.

[0075] Joint probability density of photon arrival times for:

[0076] (20)

[0077] in, , its prior probability density function is shown in formula (17). Indicates that given hyperparameters Under the conditions of The joint probability density of , N is the number of observed pulse photons, and They are the start observation time and the end observation time, Represents the photon arrival rate (photon flux rate) function, defined at any time t, the integral term represents the cumulative arrival rate during the entire observation period, the product term Represents the product of the photon arrival rates at each observation moment.

[0078] So according to Bayes' formula, the hyperparameter The posterior probability distribution of is:

[0079] (twenty one)

[0080] in, Representation parameters The posterior probability distribution of is the marginal likelihood, which is the value of all possible hyperparameters The integral of the likelihood function with respect to the value of .

[0081] Introducing loss function , then the risk function estimated by Bayes is for:

[0082] (twenty two)

[0083] in, represents the parameter space, for Bayesian estimate of .

[0084] So minimizing the Bayesian risk function is to minimize the hyperparameters Bayesian estimate of ,Right now:

[0085] (twenty three)

[0086] in, Indicates that the loss function reaches the minimum value The value of Represents the optimal estimate of the hyperparameter. Different loss functions will result in different Bayesian estimation models. This paper takes the 0-1 loss function as the loss function. , then the risk function estimated by Bayes is for:

[0087] (twenty four)

[0088] in, is the Kronecker function.

[0089] Then the posterior probability distribution of minimizing the risk function is transformed into maximizing the parameter:

[0090] (25)

[0091] in, Indicates that the loss function reaches its maximum value The value of hyperparameter The posterior probability distribution takes the natural logarithm and ignores the probability that is independent of the parameters, and the objective function based on Bayesian estimation is obtained. :

[0092] (26)

[0093] right and The estimation is converted into a maximum Bayesian estimation model, namely:

[0094] (27)

[0095] in, This means that the objective function is equivalent to , represents the range of Doppler frequency shift estimation error, Indicates that the objective function is maximized and The value of .

[0096] Step 6: Estimate the initial phase and Doppler shift using the fast Broydan-Fletcher-Goldforb-Chanton (BFGS) algorithm with interval constraints, including:

[0097] Step 6.1 Determine the constraint interval:

[0098] According to formula (8) and formula (13) and the explanation below formula (13), it has been proved that the parameters to be estimated obey the normal distribution under prior information:

[0099] (28)

[0100] in, Represents a normal distribution.

[0101] According to the properties of normal distribution, and In the constraint interval and The probability of being within is as high as 99.74%. Therefore, the present invention can be considered and The true values of and Therefore, the objective function solution problem defined by formula (27) is equivalent to the objective function solution problem with interval bounding constraints:

[0102] (29)

[0103] Among them, the lower bound of the initial phase , the upper bound of the initial phase Lower bound of Doppler shift estimation error , the upper bound of the Doppler shift estimation error .

[0104] Step 6.2 Design the F-BFGS-B algorithm:

[0105] Formula (29) defines the problem of solving a convex objective function under interval bounding constraints. Theoretically, it can be solved using convex optimization methods. However, the interval bounding constraints are based on the prior variance, which cannot always constrain non-convex functions to convex functions. For example, when the prior variance is very large, the parameter interval is also very large, making it difficult to ensure that the objective function is completely convex within a large interval. Therefore, to ensure that the present invention is applicable to various situations, F-BFGS-B (a fast Bruydan-Fletcher-Goldforb-Chanteau algorithm with interval constraints) is proposed based on BFGS.

[0106] The BFGS algorithm is a quasi-Newton optimization method that does not require the calculation of the second-order Hessian matrix and has a faster convergence speed. It is currently a widely used quasi-Newton method.

[0107] Assumptions is the Hessian matrix of the kth iteration, Is its approximate matrix, then consider the approximate matrix The rank 2 correction formula is:

[0108] (30)

[0109] in, , are the coefficients used to adjust the approximation matrix, which are calculated based on the gradient of the current iteration and the gradient of the previous iteration. represents the set of real numbers, Is a multidimensional vector related to the current iteration. The rank 2 corrected approximate matrix Substituting the quasi-Newton equation into the solution, we can get the approximate matrix of BFGS :

[0110] (31)

[0111] in, represents the difference between the hyperparameters at the kth iteration and the k-1th iteration, Represents the difference in the objective function gradient between two adjacent iterations, represents the objective function, represents the hyperparameter at k iterations, Represents a gradient operation.

[0112] like is the inverse matrix of the approximate Hessian matrix, then according to the Shermann-Morrison-Woodbury formula, the inverse matrix of the approximate Hessian matrix is for:

[0113] (32)

[0114] The present invention combines the first-order moment estimation and the second-order moment estimation of the search direction to adaptively adjust the step size, and automatically adjusts the step size for each parameter to be optimized.

[0115] First moment estimate in the descending direction and second-order moment estimates The update methods are:

[0116] (33)

[0117] (34)

[0118] in, and are the exponential decay rates of the first-order moment estimate and the second-order moment estimate, respectively, which are usually greater than 0.9. Indicates the search direction for the kth iteration. and Accumulates information about historical search directions.

[0119] Secondly, the k-th search direction is exponentially decayed to obtain the estimated search direction :

[0120] (35)

[0121] Using Nesterov momentum acceleration and bias correction, we can get the attenuated first-order moment estimate. and second-order moment estimates :

[0122] (36)

[0123] (37)

[0124] Get the adjusted step length for:

[0125] (38)

[0126] in, is the initial descent step size, It is a very small value that prevents the denominator from being zero.

[0127] Updated k-th search direction for:

[0128] (39)

[0129] Compared with the linear search step that requires iterative solution, by introducing Nesterov momentum acceleration and exponential decay to the search direction, the search step is directly solved, which improves the overall convergence speed of the algorithm.

[0130] This paper uses the same method as the Limited-memory BFGS with Boundary (L-BFGS-B) algorithm, namely the projection method:

[0131] (40)

[0132] Among them, the lower bound of the hyperparameter , the upper bound of hyperparameters .

[0133] The termination condition of the iterative optimization algorithm is also a key issue. The objective function gradient of the present invention is too large to use the gradient as the termination condition. Therefore, the present invention uses the change of parameters as the termination condition, namely:

[0134] (41)

[0135] in, represents the calculation of the vector's two norm, is the preset threshold.

[0136] Preferably, the embodiment of the present invention further includes:

[0137] Step 7: After the calculation is completed, the optimal initial phase and Doppler frequency shift are output.

[0138] The parameters of the present invention are set as follows: the initial value is the lower bound of the constraint interval, is 0.99, is 0.999, for , initial descent step length for , the maximum number of iterations is 10. When the changes in phase and Doppler shift are less than and The iteration ends early when . The performance of the method is measured by the root mean square error and GPU time consumption. The simulation results are as follows Figure 2 As shown, Figure 2 (a) is the variation of phase estimation error with observation time, Figure 2 (b) is the variation of Doppler frequency shift estimation error with observation time, Figure 2 (c) shows the change of GPU time consumption with observation time. Figure 2It can be seen that the phase and Doppler shift estimation errors of the present invention are significantly smaller than those of the traditional method, and the calculation speed of the proposed method is very fast.

[0139] Any content not described in detail in this specification belongs to the prior art known to those skilled in the art. It will be readily understood by those skilled in the art that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention shall be included within the scope of protection of the present invention.

Claims

1. A fast phase and Doppler shift estimation method based on prior information, characterized in that: The steps include: Step 1: Establish a time conversion model to convert the photon arrival time at the spacecraft to the photon arrival time at the center of mass of the solar system; Step 2: Combine the time conversion model to establish a pulse phase model that does not consider the spacecraft state error; Step 3: Combining the time conversion model with the pulse phase model that does not consider the spacecraft state error, a pulse phase model that includes errors is established based on the error transfer characteristics of the orbital dynamics equation while considering the spacecraft state error. Step 4: Combining the time conversion model and the pulse phase model containing errors, and according to the superposition property of the normal distribution, establish a joint probability density function of the initial pulse phase and Doppler frequency shift; Step 5: Combined with the joint probability density function, the objective function of phase and Doppler shift with prior information is established based on the Bayesian model; Step 6: Estimate the initial phase and Doppler shift according to the objective function using the fast Bruydan-Fletcher-Goldforb-Chanton algorithm with interval constraints.

2. The method for fast phase and Doppler shift estimation based on prior information according to claim 1, wherein: The step 1 comprises: For the solar system barycenter coordinate system, calculate the Within the observation period The time it takes for a photon to reach the spacecraft is recorded as In order to estimate the pulse phase and Doppler shift information, the time conversion model is used Will Transformed to the coordinates of the solar system's center of mass, recorded as .

3. The method for fast phase and Doppler shift estimation based on prior information according to claim 2, wherein: The step 2 includes: Assume that the reference time is , the initial phase of the observation period is , for any time at SSB , pulse phase Expressed as: （4） in, represents the initial time at the spacecraft during the jth observation period; n is the direction vector from the SSB to the observed pulsar, represents the time transition model at the initial moment, represents the time transition model at time i, and Represents the pulsar in The zeroth and first derivatives of the spin frequency at time .

4. The method for fast phase and Doppler shift estimation based on prior information according to claim 3, wherein: The step 3 includes: Considering the spacecraft state error, let , ,in and denote the estimated position and estimated velocity of the spacecraft, respectively, and represent the position estimation error and velocity estimation error of the spacecraft respectively. According to the error transmission characteristics of the spacecraft state equation, the phase model considering the spacecraft state prior information and state estimation error is obtained by Taylor formula expansion: （9） in, represents the estimated value of the initial phase, represents the estimation error of the initial phase, represents the phase deviation due to time variation, and They represent the partial derivatives of the time conversion model with respect to the spacecraft position at the initial moment and the i-th moment respectively.

5. The method for fast phase and Doppler shift estimation based on prior information according to claim 4, characterized in that: The step 4 comprises: According to the properties of the orbital dynamics model, the position error of the spacecraft at any time is related to the initial position error and initial velocity error, which can be expressed as ,in, and is the state transfer matrix, which is expanded by Taylor’s formula; The phase model considering the prior state information of the spacecraft is expressed as: （12） in, represents the Doppler shift estimation error, which is expressed as: （13） Initial phase estimation error and Doppler frequency estimation error It is also zero-mean Gaussian white noise.

6. The method for fast phase and Doppler shift estimation based on prior information according to claim 5, characterized in that: According to the superposition property of normal distribution, the initial phase The mean and variance of and ,in, Represents the variance of the position estimation error and the Doppler shift estimation error The mean and variance are 0 and ,in, and is a constant matrix related to the state transition matrix, represents the variance of the velocity estimation error, represents the natural frequency of the pulsar; If the hyperparameter , then the joint probability density function Expressed as: （17） in, represents the mean vector, exp represents the exponential function, Represents the covariance matrix The determinant of .

7. The method for fast phase and Doppler shift estimation based on prior information according to claim 6, characterized in that: The step 5 comprises: The joint probability density of photon arrival times is ,in, The mathematical relationship between the true value and the estimated value of the prior distribution of the spacecraft state is determined; According to the Bayesian formula, the hyperparameters are calculated The posterior probability distribution of .

8. The method for fast phase and Doppler shift estimation based on prior information according to claim 7, characterized in that: Take the 0-1 function as the loss function and set the parameter The posterior probability distribution takes the natural logarithm and ignores the probability that is independent of the parameters, and the objective function based on Bayesian estimation is obtained. ; Use the and The estimate of is converted into the maximization objective function.

9. The method for fast phase and Doppler shift estimation based on prior information according to claim 8, characterized in that: The step 6 comprises: According to the properties of normal distribution, and In the constraint interval and Within the constraint interval, the objective function is transformed from a non-convex function to a convex function; based on the traditional BFGS algorithm, the first-order moment estimation and second-order moment estimation of the search direction are combined to automatically adjust the search step size for each parameter to be optimized.

10. The method for fast phase and Doppler shift estimation based on prior information according to claim 9, characterized in that: Search step size for the kth iteration and search direction for: ； in, and are the exponential decay rates of the first-order moment estimate and the second-order moment estimate, respectively. and They represent the first-order moment estimation and second-order moment estimation after exponential decay of Nesterov momentum, Represents the estimated descent gradient after exponential decay, is the initial descent step size, is a parameter that prevents the denominator from being zero.

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