CN103616664B - A kind of Passive cross-localization method and system without ginseng Multilayer networks - Google Patents

Abstract

The invention provides a kind of based on the Passive cross-localization method and system without ginseng Multilayer networks, described method comprises: first, carry out cross bearing to the angle measurement of the multiple batches of collection of multiple sensor, wherein, same batch angle measurement data carry out cross bearing calculating; Then, nothing ginseng Multilayer networks is carried out in the calculated value set direct cross bearing obtained, and obtains the estimated coordinates point that statistical probability is the highest; Finally, in the threshold range of setting, coordinate points near maximum probability density Estimation point is done sums and on average obtains final positioning result.The present invention solves measured target when departing from passive sensor baseline perpendicular bisector, the problem that positioning error worsens; The positioning precision in effective region can be improved when only increasing a small amount of passive sensor, reducing the performance requirement to data processor; Avoid the mapping relations using passive sensor angle error and target radial distance, improve the stability of the model of location Calculation.

Description

Passive cross positioning method and system for non-parameter probability density estimation

Technical Field

The invention belongs to the field of positioning, and particularly relates to a passive cross positioning system and method for non-parameter probability density estimation.

Background

The passive cross positioning method mainly solves the position information of a positioning target based on a triangle rule, taking double-station cross positioning as an example: known passive sensor A₁(x₁,y₁) And A₂(x₂,y₂) The real coordinate of the target radiation source p to be direction-measured is (x, y), wherein A₁And A₂The direction angles to the target p are respectively:

α₁＝a₁+₁(1)

α₂＝a₂+₂(2)

wherein, a₁And a₂Is a passive sensor A₁、A₂For the true azimuth angle of p,₁and₂are respectively A₁、A₂The schematic diagram of the angle measurement error and the position of the angle measurement error is shown in fig. 1. p is the real position of the radiation source, q is the resolving coordinate position of the cross positioning, the quadrilateral ABCD determined by the direction-measuring angle and the angle-measuring error range is the possible area of the resolving result, and the bigger the angle-measuring error is, the bigger the area of the quadrilateral ABCD is.

Using passive sensors A₁、A₂The coordinate approximation q of the target p can be solved by the triangle rule, and the calculation formula is given by the formula (3)

$\left[\begin{array}{c}x\\ y\end{array}\right]={\left[\begin{array}{cc}\tan {\mathrm{\α}}_1& -1\\ \tan {\mathrm{\α}}_2& -1\end{array}\right]}^{-1}\left[\begin{array}{c}\tan {\mathrm{\α}}_1{x}_1-y_1\\ \tan {\mathrm{\α}}_2{x}_2-y_2\end{array}\right]---\left(3\right)$

In addition, as the target gets farther from the sensor's position or the radiation source gets farther from the baseline A₁OA₂The larger the area of the area ABCD, the larger the error in the position estimation.

In order to improve the positioning accuracy, a least square algorithm combining reasonable station distribution of sensors is adopted in the current general engineering. The least square method essentially expands the number of the base lines at the expense of the number of the sensors, and has complex system and high use cost. In the research, a passive cross-location algorithm using parameter probability density estimation is also available, but the parameter probability density estimation needs to have prior knowledge of a probability density function model, and in the actual situation, the true probability density model cannot be accurately predicted.

The defects of the prior art and the reasons for the defects are as follows: the positioning accuracy on the perpendicular bisector of the baseline of the passive sensor is known to be relatively highest in all areas, and the positioning error will be worsened when a target deviates from the perpendicular bisector of the baseline of the sensor, so that a least square positioning method which is often adopted in engineering and combines reasonable station arrangement of the sensors is adopted. When a passive cross-location algorithm with parameter probability density estimation is adopted for location, the mapping relation between factors such as target distance and a passive sensor perception data probability density function model has uncertainty, and the difficulty in probability density estimation calculation is brought.

Disclosure of Invention

It is therefore an object of the present invention to overcome the above problems by providing a passive cross-positioning system and method for non-parametric probability density estimation.

In order to achieve the above object, the present invention provides a passive cross-location system and method for estimating non-parameter probability density, which performs cross-location on angle measurement values acquired by multiple sensors in multiple batches (performing cross-location calculation on angle measurement data in the same batch); carrying out non-parameter probability density estimation on a calculated value set obtained by cross positioning to obtain an estimated coordinate point with the highest statistical probability; and finally, in a set threshold range, carrying out arithmetic mean on coordinate points near the highest probability density estimation point to obtain a final result. The method specifically comprises the following steps:

step 101) adopting n sensors to carry out direction finding on a target, and adopting pairwise crossing of angle measuring error ranges of two measuring sensors for a single target to obtainMeasuring and sampling for T times to obtainThe set of angle pairs is denoted V, where the subset of angle pairs for the ith and jth sensors is:

$v_{\mathrm{ij}}=\left[\begin{array}{cc}{\mathrm{\α}}_{1i}& {\mathrm{\α}}_{1j}\\ {\mathrm{\α}}_{2i}& {\mathrm{\α}}_{2j}\\ .& .\\ .& .\\ .& .\\ {\mathrm{\α}}_{\mathrm{Ti}}& {\mathrm{\α}}_{\mathrm{Tj}}\end{array}\right],$ and i is not equal to j;

step 102) carrying out double-station cross positioning on the set V based on angle measurement, and further obtaining a positioning point sample set p, wherein elements obtained by the ith sensor and the jth sensor in the positioning point sample set p are as follows, i is not equal to j:

$p_{\mathrm{ij}}=\left[\begin{array}{cc}{x}_{1i}& x_{1j}\\ x_{2i}& x_{2j}\\ .& .\\ .& .\\ .& .\\ x_{\mathrm{Ti}}& x_{\mathrm{Tj}}\end{array}\right];$

step 103) carrying out kernel probability density estimation according to the positioning point sample set p, and further obtaining the highest probability density p_maxThe locating point HP;

step 104) setting an extraction threshold value, and extracting the condition | p satisfied by the extraction threshold value_maxAll k anchor points, denoted as M, for p | <; wherein, the standard deviation of the distance from the positioning sample to the positioning point HP is used;

step 105) calculating the arithmetic mean value of the effective positioning point set by adopting the following formula according to M to obtain a final positioning result

$\hat{p}=\underset{i=1}{\overset{k}{\mathrm{\&Sigma;}}}M.$

Optionally, performing two-station cross positioning on V based on angle measurement according to the following formula:

$\left[\begin{array}{c}{x}_{\mathrm{tij}}\\ y_{\mathrm{tij}}\end{array}\right]={\left[\begin{array}{cc}\tan {\mathrm{\&alpha;}}_{\mathrm{ti}}& -1\\ \tan {\mathrm{\&alpha;}}_{\mathrm{tj}}& -1\end{array}\right]}^{-1}\left[\begin{array}{c}\tan {\mathrm{\&alpha;}}_{\mathrm{ti}}{x}_i-y_i\\ \tan {\mathrm{\&alpha;}}_{\mathrm{tj}}{x}_j-y_j\end{array}\right]$

wherein (x)_tij,y_tij) The coordinate values of the positioning points obtained by the sensor i and the sensor j during the t-th sampling are (x)_i,y_i) Is the coordinate value of sensor i, (x)_j,y_j) As a coordinate value of sensor j, α_tiAngle of measurement value sampled for the t-th measurement of sensor i, α_tjThe angle measurement value sampled for the t-th measurement of sensor j.

The step 103) further comprises:

step 103-1), selecting a kernel function K (-) and calculating the optimal estimation bandwidth h of kernel probability density estimation;

step 103-2) calculating probability density estimation of x coordinate and probability density estimation of y coordinate of positioning point according to kernel function and optimal estimation bandwidth hThe calculation formulas are the same, and the calculation formulas are as follows:

$\hat{f}\left(x\right)=\frac{1}{N}\underset{m=1}{\overset{N}{\mathrm{\&Sigma;}}}\frac{1}{h}K\left(\frac{x-x_m}{h}\right)$

step 103-3) obtaining the probability density estimation of the x coordinate of the positioning pointProbability density estimation of the maximum and y-coordinate ofIs recorded as:and will be dottedAs having the highest probability density p_maxThe anchor point HP of.

The kernel function may be a gaussian function.

In addition, the invention also provides a passive cross-positioning system for non-parametric probability density estimation, which comprises:

the positioning data acquisition module is used for acquiring a cross positioning result of passive sensors aiming at a certain target, and the number of the passive sensors is more than two;

a processing module to: and performing non-parameter probability density estimation on the positioning result by using a kernel probability density estimation algorithm, then estimating according to the non-parameter probability density to obtain a positioning point with the highest probability density, then extracting other positioning points within a certain set threshold range by taking the positioning point as the center, and finally performing arithmetic mean calculation on the extracted positioning points to obtain a final positioning estimation point.

Optionally, the processing module further includes:

the locating point obtaining submodule of the highest probability density is used for carrying out parameter-free probability density estimation on a locating result by using a kernel probability density estimation algorithm and then obtaining the locating point of the highest probability density according to the parameter-free probability density estimation;

the extraction submodule is used for extracting other positioning points within a certain set threshold range by taking the positioning points as centers; and

and the positioning result calculation output module is used for performing arithmetic mean calculation on the extracted positioning points to obtain the final positioning estimation points. Compared with the prior art, the invention has the technical advantages that:

the invention combines the non-parameter probability density estimation algorithm with cross positioning, and achieves the following 3 purposes:

1, solving the problem of location error deterioration when a measured target deviates from a neutral line of a passive sensor base line;

2, the positioning precision in the effective area can be improved under the condition of only adding a small number of passive sensors, and the performance requirement on the data processor is reduced;

3, the mapping relation between the angle measurement error of the passive sensor and the radial distance of the target is avoided, and the stability of the model of the positioning calculation is improved.

Drawings

FIG. 1 is a schematic cross-positioning of the prior art;

FIG. 2 is a schematic diagram of a three-station positioning of the present invention;

FIGS. 3(a) and 3(b) are schematic diagrams of the union/intersection region of the multi-sensor detection results provided by the present invention;

fig. 4 is a flow chart of a positioning algorithm provided by the present invention.

Detailed Description

The method of the present invention is described in detail below with reference to the accompanying drawings and examples.

The method is based on a non-parameter probability density estimation algorithm, and carries out post-processing on the cross positioning result of the passive sensor to estimate the optimal positioning result in the statistical sense. The non-parameter estimation of the probability density function has a certain relation with the histogram, and in order to obtain continuous probability density function estimation, a kernel probability density estimation method can be adopted.

Kernel probability density estimation

Let sample data x ═ x₁,x₂,…,x_nThe probability density function of is f (x), its kernel probability density estimateThe expression of (a) is:

${\hat{f}}_h\left(x\right)=\frac{1}{\mathrm{nh}}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}K\left(\frac{x-x_i}{h}\right)---\left(4\right)$

wherein n is the number of samples, k (x) is a kernel function, h is an estimated bandwidth, and the quality of kernel probability density estimation depends on the selection of the kernel function and the estimated bandwidth h.

In addition to satisfying the continuity condition, k (x) is generally selected to satisfy the following assumptions:

(1) k (x) is symmetrical, i.e. K (x) K (-x)

(2) ${\&Integral;}_{-\&infin;}^{+\&infin;}K\left(x\right)\mathrm{dx}=1$

(3) For j ═ 1,2, …, k-1 have ${\&Integral;}_{-\&infin;}^{+\&infin;}{x}^{j}K\left(x\right)\mathrm{dx}=0$ While ${\&Integral;}_{-\&infin;}^{+\&infin;}{x}^{k}K\left(x\right)\mathrm{dx}\&NotEqual;0.$

The four most commonly used k (x) functions are given below:

Uniform： $\frac{1}{2}I\left(\right|t|\&le;1)---\left(5\right)$

Epanechikov： $\frac{3}{4}(1-t^2)I\left(\right|t|<1)---\left(6\right)$

Quartic： $\frac{15}{16}(1-t^2)I\left(\right|t|<1)---\left(7\right)$

Gaussian： $\frac{1}{\sqrt{2\mathrm{\&pi;}}}{e}^{-\frac{1}{2}{t}^2}---\left(8\right)$

the kernel density estimation bandwidth h is an important parameter of the estimation model, because the integrated mean square error mise (h) is usually used as a criterion for judging whether the probability density estimation is good or bad. And is provided with

$\begin{array}{c}\mathrm{MISE}\left(h\right)={\&Integral;}_{-\&infin;}^{+\&infin;}E\left\{{[\hat{f}\left(x\right)-f\left(x\right)]}^2\right\}\mathrm{dx}\\ ={\&Integral;}_{-\&infin;}^{+\&infin;}E\{\hat{f}\left(x\right)-E[\hat{f}\left(x\right)]+E[\hat{f}\left(x\right)]-f\left(x\right){\}}^2\mathrm{dx}\\ ={\&Integral;}_{-\&infin;}^{+\&infin;}E{\{\hat{f}\left(x\right)-E[\hat{f}\left(x\right)\left]\right\}}^2\mathrm{dx}+\\ {\&Integral;}_{-\&infin;}^{+\&infin;}E{\left\{E\right[\hat{f}\left(x\right)]-f\left(x\right)\}}^2\mathrm{dx}\\ ={\&Integral;}_{-\&infin;}^{+\&infin;}\mathrm{Var}\hat{f}\left(x\right)\mathrm{dx}+{\&Integral;}_{-\&infin;}^{+\&infin;}{\mathrm{bias}}^2\hat{f}\left(x\right)\mathrm{dx}\\ =\mathrm{IV}\left(h\right)+\mathrm{IB}\left(h\right)\end{array}---\left(9\right)$

Wherein IV (h) is the integral variance, and IB (h) is the integral deviation. Assume sample { x₁,x₂,…,x_nIndependent equal distribution, for a second-order kernel function, h high-order terms which are obtained by respectively performing taylor series expansion on the integral variance and the integral deviation in the formula (9) and omitted can obtain the following approximate integral mean square error:

$\mathrm{AMISE}\left(h\right)={0.25h}^4{\left({\&Integral;}_{-\&infin;}^{+\&infin;}{t}^{2}K\left(t\right)\mathrm{dt}\right)}^2*{\&Integral;}_{-\&infin;}^{+\&infin;}{f}^{\&prime;\&prime;}{\left(x\right)}^2\mathrm{dt}+\frac{1}{\mathrm{Nh}}{\&Integral;}_{-\&infin;}^{+\&infin;}{K}^2\left(t\right)\mathrm{dt}---\left(10\right)$

in equation (10), f "(x) is the second derivative of f (x). When N → ∞ and h ═ h (N)) → 0 cause nh (N)) → ∞, amise (h) and mie (h) are asymptotically the same.

The minimum value of AMISE (h) is solved by a derivative method, so that the optimal bandwidth h can be obtained_rotCalculation formula [11 ]]Comprises the following steps:

$h_{\mathrm{opt}}={\left\{{\&Integral;}_{-\&infin;}^{+\&infin;}{t}^{2}K\left(t\right)\mathrm{dt}\right\}}^{-2/5}\left\{{\&Integral;}_{-\&infin;}^{+\&infin;}{K}^2\left(t\right)\mathrm{dt}\right\}{1/5}^{}*\{{\&Integral;}_{-\&infin;}^{+\&infin;}{f}^{\&prime;\&prime;2}\left(x\right)\mathrm{dx}{\}}^{-1/5}*N^{-1/5}---\left(11\right)$

the following two functional are now defined:

$R\left(f\right)={\&Integral;}_{-\&infin;}^{+\&infin;}{f}^2\left(x\right)\mathrm{dx}---\left(12\right)$

${\mathrm{\&sigma;}}_K^2={\&Integral;}_{-\&infin;}^{+\&infin;}{x}^{2}K\left(x\right)\mathrm{dx}---\left(13\right)$

then, equation (11) can be abbreviated as:

$h_{\mathrm{opt}}={\left[\frac{R\left(K\right)}{{\mathrm{\&sigma;}}_K^{4}R\left(f^{\&prime;\&prime;}\right)}\right]}^{1/5}{N}^{-1/5}---\left(14\right)$

when the standard normal distribution is used as the estimated gaussian kernel function, the estimation of the ideal bandwidth can be obtained:

$h_{\mathrm{opt}1}=1.06\widehat{\mathrm{\&sigma;}}{N}^{-1/5}---\left(15\right)$

wherein,to utilize { x₁,x₂,…,x_nSample calculation of the resulting estimate of σ.

If the interquartile range is utilizedInstead of the formerAs a measure of the sample distribution, equation (14) is then written as:

$h_{\mathrm{opt}2}=0.97\widehat{\mathrm{\&lambda;}}{N}^{-1/5}---\left(16\right)$

h_opt1and h_opt2For normally distributed probability density functions and unimodal probability density functions, a more satisfactory probability density estimate can be obtained, but for bimodal or multimodal probability density functions, the bandwidth estimate h_opt1And h_opt2Are so large that the final probability density function estimate is too smooth and may mask some useful structural information, such as bimodality and multimodality. One trade-off is to make a bandwidth estimation using the following equation:

$h_{\mathrm{opt}}=0.9\min (\widehat{\mathrm{\&sigma;}},\widehat{\mathrm{\&lambda;}}/1.34)N^{-1/5}---\left(17\right)$

bandwidth h obtained by equation (17)_optIt can still be applied to unimodal probability density function estimation and the estimation of multimodal probability density function is optimized.

Multi-sensor cross-positioning algorithm based on kernel probability density estimation

As can be seen from fig. 1, when the passive two-station positioning is used for cross positioning, the results are randomly distributed in the area ABCD, the passive sensors are added at other positions to form a layout of multi-sensor direction-finding positioning, different positioning result distribution areas are provided, and the passive sensor a is added in the three-station positioning, for example₃Fig. 2 shows a schematic diagram of 3 passive sensor cross-positioning.

All positioning calculation results of pairwise crossing positioning of the 3 passive sensors are randomly distributed in a region shown in fig. 3(a), and the region is called as a union region; the possible area where the positioning results are simultaneously obtained when every two stations perform cross positioning is shown in fig. 3(b), and this area is referred to as the intersection area.

According to the statistical probability, (A)₁,A₂)、(A₁,A₃) And (A)₂,A₃) The probability density of the positioning points appearing in the intersection area is larger than that outside the intersection area, and the approximate probability density p of all the positioning points is calculated by using kernel probability density estimation, wherein the point with the highest probability density has the probability density p_maxObtaining all probability densities satisfying | p by a set threshold_maxAnd positioning estimation points within the range of p < p >, and carrying out average calculation on the coordinates of the estimation points to obtain a final positioning result.

The algorithm implementation flow adopted by the method is shown in fig. 4, and specifically comprises the following steps:

step 101) direction finding of the sensor 1 and the sensor 2 … … on the targets by the sensor n, and the single targets are crossed pairwise to obtainEach angle measurement pair is marked as V;

102) performing double-station cross positioning on the result in the V by using the formula (3) to obtain a positioning point sample set;

step 103) carrying out nuclear probability density estimation on the positioning point sample set obtained in the step 102) to calculate the positioning point sample set with the highest probability density p_maxThe locating point HP;

step 104) setting a region extraction threshold;

step 105) extract all the satisfied conditions | p_maxAnd (4) taking the cross positioning point with the value of p < as an effective positioning point set, and performing arithmetic mean calculation on the effective positioning point set to obtain a final positioning result.

In a word, the method uses a kernel probability density estimation algorithm to carry out non-parameter probability density estimation on the positioning data; and after the positioning point with the highest probability density is obtained, taking the positioning point as the center, extracting other positioning points in a set threshold range, and performing arithmetic mean calculation to obtain the accurate positioning estimation point with statistical significance.

Compared with the least square cross positioning algorithm, the method has low requirement on the priori knowledge of the sensor and does not need to know the angle measurement precision of the sensor; a probability density estimation model of a target positioning error is not assumed, and the cross positioning data of the passive sensor is directly processed, so that the probability density estimation is closer to reality; when the target deviates from a certain two-station baseline perpendicular bisector, the positioning error does not seriously deteriorate with the deviation of the target.

Finally, it should be noted that the above embodiments are only used for illustrating the technical solutions of the present invention and are not limited. Although the present invention has been described in detail with reference to the embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted without departing from the spirit and scope of the invention as defined in the appended claims.

Claims (4)

1. A passive cross-localization method based on non-parametric probability density estimation, the method comprising:

firstly, carrying out cross positioning on angle measurement values acquired by a plurality of sensors in a plurality of batches, wherein cross positioning calculation is carried out on angle measurement data in the same batch;

then, carrying out non-parameter probability density estimation on a calculated value set obtained by cross positioning to obtain an estimated coordinate point with the highest statistical probability;

finally, in a set threshold range, carrying out arithmetic mean on coordinate points near the highest probability density estimation point to obtain a final positioning result

Wherein, the method specifically comprises the following steps:

step 101) adopting n sensors to carry out direction finding on a target, and adopting pairwise crossing of angle measuring error ranges of two measuring sensors for a single target to obtainMeasuring and sampling for T times to obtainThe set of angle pairs is denoted V, where the subset of angle pairs for the ith and jth sensors is:

$v_{ij}=\left[\begin{array}{cc}{\mathrm{\&alpha;}}_{1i}& {\mathrm{\&alpha;}}_{1j}\\ {\mathrm{\&alpha;}}_{2i}& {\mathrm{\&alpha;}}_{2j}\\ \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;\\ {\mathrm{\&alpha;}}_{Ti}& {\mathrm{\&alpha;}}_{Tj}\end{array}\right],$ and i is not equal to j;

step 102) carrying out double-station cross positioning on the set V based on angle measurement, and further obtaining a positioning point sample set p, wherein elements obtained by the ith sensor and the jth sensor in the positioning point sample set p are as follows, i is not equal to j:

$p_{ij}=\left[\begin{array}{cc}{x}_{1i}& y_{1j}\\ x_{2i}& y_{2j}\\ \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;\\ \&CenterDot;& \&CenterDot;\\ x_{Ti}& y_{Tj}\end{array}\right];$

step 103) carrying out kernel probability density estimation according to the positioning point sample set p, and further obtaining the highest probability density p_maxThe locating point HP;

step 104) setting an extraction threshold value, and extracting the condition | p satisfied by the extraction threshold value_maxAll k anchor points, denoted as M, for p | <; wherein, the standard deviation of the distance from the positioning sample to the positioning point HP is used;

step 105) calculating the arithmetic mean value of the effective positioning point set by adopting the following formula according to M to obtain a final positioning result

$\hat{p}=\frac{1}{k}\underset{i=1}{\overset{k}{\&Sigma;}}M.$

2. The passive cross-positioning method based on the non-parametric probability density estimation as claimed in claim 1, wherein the set V is subjected to the two-station cross-positioning based on the angle measurement according to the following formula:

$\left[\begin{array}{c}{x}_{tij}\\ y_{tij}\end{array}\right]={\left[\begin{array}{cc}{\mathrm{tan\&alpha;}}_{ti}& -1\\ {\mathrm{tan\&alpha;}}_{tj}& -1\end{array}\right]}^{-1}\left[\begin{array}{c}tan{\mathrm{\&alpha;}}_{ti}{x}_i-y_i\\ {\mathrm{tan\&alpha;}}_{tj}{x}_j-y_j\end{array}\right]$

wherein (x)_tij,y_tij) The coordinate values of the positioning points obtained by the sensor i and the sensor j during the t-th sampling are (x)_i,y_i) Is the coordinate value of sensor i, (x)_j,y_j) As a coordinate value of sensor j, α_tiAngle of measurement value sampled for the t-th measurement of sensor i, α_tjThe angle measurement value sampled for the t-th measurement of sensor j.

3. The passive cross-localization method based on the non-parametric probability density estimation of claim 1, wherein the step 103) further comprises:

step 103-1), selecting a kernel function K (-) and calculating the optimal estimation bandwidth h of kernel probability density estimation;

step 103-2) calculating probability density estimation of x coordinate of positioning point according to kernel function and optimal estimation bandwidth hAnd probability density estimation of y coordinateThe calculation formulas are the same, and the calculation formulas are as follows:

$\hat{f}\left(x\right)=\frac{1}{N}\underset{m=1}{\overset{N}{\&Sigma;}}\frac{1}{h}K\left(\frac{x-x_m}{h}\right)$

step 103-3) obtaining the probability density estimation of the x coordinate of the positioning pointProbability density estimation of the maximum and y-coordinate ofIs recorded as:and will be dottedAs having the highest probability density p_maxThe anchor point HP of.

4. The passive cross-localization method based on the non-parametric probability density estimation of claim 3, characterized in that the kernel function is a Gaussian function.