CN104217083A - Reflector antenna face plate modeling method based on multi-scale fractal function - Google Patents

Abstract

The invention discloses a reflector antenna face plate modeling method based on a multi-scale fractal function. The reflector antenna face plate modeling method is characterized by comprising the following steps: manufacturing a template; measuring the roughness of the face plate of the template; seeking a scale range with fractal characteristics on the template, and determining the scale-free region of the template; determining the scale, the boundary frequency, the fractal dimension D value and the characteristic length G value of the fractal function used in template modeling, as well as a mathematical model used for simulating the fractal function; checking the accuracy of the mathematical model. The reflector antenna face plate modeling method provided by the invention has the benefits as follows: main parameters of the fractal function are determined according to data actually measured on the surface of the antenna, and the accuracy of the mathematical model is ensured; during a modeling process, the levels of the fractal function are determined according to the power frequency spectrum of discrete date actually measured, the lowest frequency, the highest frequency and the boundary frequency of the multi-scale fractal function are determined according to the working frequency of the antenna, the high efficiency during the modeling process is ensured, and a foundation for improving the accuracy and the efficiency of analysis to the modeling of the reflector antenna is paved.

Description

A kind of reflecting plane aerial panel modeling method based on Multi-scale Fractal function

Technical field

The present invention relates to a kind of reflecting plane aerial panel modeling method, be specifically related to a kind of reflecting plane aerial panel modeling method based on Multi-scale Fractal function, belong to antenna technical field.

Background technology

Reflector antenna is typical electrical and mechanical comprehensive electronics product, along with it is to high band, high-gain, high reliability and light-weighted future development, interaction between its displacement structure field and electromagnetic field with influence each other more and more obvious, because two mutual relationships are unclear and cause the electrical property of antenna to improve being subject to severely restricts.Therefore, the mutual relationship between furtheing investigate two, the electrical and mechanical comprehensive analysis realizing reflector antenna is very necessary.The reflecting surface of reflector antenna is the primary structure of antenna, is again the boundary condition that Electromagnetic Fields of Antenna is propagated.The electrical property how structural behaviour of reflecting surface affects antenna becomes the key issue that reflector antenna electrical and mechanical comprehensive is analyzed.

In the traditional analysis of the existing professional software of application, come simulated reflections face mainly with periodic function or random function, there is analog function too simple, computation process too simplifies, the problem that result of calculation is too coarse.

For the problems referred to above existing in the simulation of reflector antenna reflecting surface, in current scientific paper and patent, the main disposal route adopted is that life cycle function or random function come simulated reflections face, and then obtain the axial displacement of reflecting surface, among the calculating impact of this axial displacement being imported to the electrical property of reflecting surface.The advantage of these class methods is that used function mathematic(al) structure is simply clear, is convenient to subsequent calculations, also can meet basic electrical property computational accuracy requirement for bass reflex surface antenna; Its shortcoming is that this class function does not meet true engineering non-periodic, nonrandom feature; Periodic function and random function institute operation parameter are not " intrinsic parameters ", relevant with surveying instrument, sample length; This type of functional simulation is used not consider the impact of yardstick.Different frequency of operation, the surface error yardstick affecting its electrical property is also different.

Summary of the invention

For solving the deficiencies in the prior art, the object of the present invention is to provide and a kind ofly effectively can improve the precision of reflector antenna computer-aided analysis and the reflecting plane aerial panel modeling method based on Multi-scale Fractal function of efficiency.

In order to realize above-mentioned target, the present invention adopts following technical scheme:

Based on a reflecting plane aerial panel modeling method for Multi-scale Fractal function, it is characterized in that, comprise the following steps:

(1) according to the material properties of reflector antenna, the experiment model used of same material properties is made;

(2) measure the roughness of previous experiments model used, obtain the discrete measured data of roughness;

(3) find range scale model with fractal property, determine the dimensionless interzone of model, and dimensionless interzone is checked;

(4) determine pattern modeling use the yardstick of fractal function;

(5) according to sample length, employing Nyquist frequency algorithm determination edge frequency;

(6) Fractal dimensions value and characteristic length G value is determined;

(7) according to the Fractal dimensions value determined and characteristic length G value, the mathematical model z (x) of simulated reflections faceplate panels is determined:

For single fractal function:

$z\left(x\right)=G^{(D-1)}\underset{n=n_1}{\overset{n_c}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D)n}}$ Formula (1)

In formula, γ is the mould of surface profile spatial frequency, is constant value, and value is γ=1.5;

G is the scale parameter of single fractal function;

X is the lateral coordinates of surface profile;

N is sampling number, n ₁for sampled point initial value, n _cfor sampled point stop value;

For two fractal function:

$z\left(x\right)={G_1}^{(D_1-1)}\underset{n=n_1}{\overset{n_{c1}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_1)n}}+{G_2}^{(D_2-1)}\underset{n=n_2}{\overset{n_{c2}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_2)n}}$ Formula (2)

In formula, γ is the mould of surface profile spatial frequency, is constant value, and value is γ=1.5;

G ₁for the scale parameter of first paragraph fractal function in two fractal function;

G ₂for the scale parameter of second segment fractal function in two fractal function;

X is the lateral coordinates of surface profile;

N is sampling number, n ₁for sampled point initial value, the n of first paragraph fractal function _c1for the sampled point stop value of first paragraph fractal function, n ₂for sampled point initial value, the n of second segment fractal function _c2for the sampled point stop value of second segment fractal function;

For other fractal function: corresponding mathematical model z (x) can be obtained by that analogy;

(8) according to error precision index request, test to the accuracy of mathematical model z (x), if coincidence loss precision index, then prove that this modeling method is enough accurate, modeling terminates; If do not meet error precision index, then repeat step (2) to step (7), till meeting accuracy requirement, modeling terminates.

The aforesaid reflecting plane aerial panel modeling method based on Multi-scale Fractal function, is characterized in that, in step (2), the roughness measuring previous experiments model used is carried out as follows:

(2a) select the metal probe contact pilotage of profile measurer according to model roughness value, the radius of aforementioned contact pilotage is less than the half of model roughness rms;

(2b) sample length A and the evaluation length L of model is determined;

(2c) according to x to y to equidistant measuring route Uniform Scanning model, obtain the roughness discrete data of model.

The aforesaid reflecting plane aerial panel modeling method based on Multi-scale Fractal function, it is characterized in that, in step (3), find range scale model with fractal property, the dimensionless interzone determining model and dimensionless interzone checked and carry out as follows:

(3a) the field of definition L of operational evaluation length L exemplarily range scale, L ∈ (1, n), n is scale sum, on L draw estimate M (δ) with the double logarithmic curve of measurement scale δ, double logarithmic curve indicates measurement point, measurement point be labeled as (U _i, V _i), pre-determine two end points of the good curve of linear relationship as non-scaling section scope (n according to the linear relationship of measurement point distribution ₁, n ₂) approximate value;

(3b) according to three lines fitting, at (n ₁± △ n, n ₂± △ n) interior use formula (3) digital simulation difference Ω:

$\mathrm{\Ω}(3,n_1,n_2)=\underset{i=1}{\overset{a}{\mathrm{\Σ}}}\underset{i=1}{\overset{b}{\mathrm{\Σ}}}{(V_i-a-b{U}_i)}^2$ Formula (3)

In formula, a, b are the parameter of straight-line equation,

Obtain matching difference Ω minimum time, the broken line end points n in the middle of two end points ₁and n ₂the interval formed is dimensionless interzone;

(3c) when dimensionless interzone scope on model exceedes 2/3 of model field of definition L total length, proceed to step (4), use the inventive method to continue modeling.

The aforesaid reflecting plane aerial panel modeling method based on Multi-scale Fractal function, is characterized in that, in step (4), determine pattern modeling use the yardstick of fractal function to carry out as follows:

(4a) Fourier transform is carried out to the roughness discrete data of the model obtained in step (2), obtain frequency data;

(4b) logarithm change is carried out to aforementioned frequencies data, obtain the power spectrum chart of specimen surface profile;

(4c) by aforementioned power spectrogram obvious fluctuating hop count determination pattern modeling use the yardstick of fractal function: if obvious fluctuating hop count is one section in power spectrum chart, use single fractal function modeling, if obvious fluctuating hop count is two sections in power spectrum chart, use two fractal function modeling, by that analogy.

The aforesaid reflecting plane aerial panel modeling method based on Multi-scale Fractal function, is characterized in that, in step (5), determines that edge frequency is carried out as follows:

(5a) the low-limit frequency f of inverse as power spectrum of sample length is used _l;

(5b) use the inverse of sample interval as the highest frequency f of power spectrum chart _h;

(5c) from low-limit frequency f _lstart, along the direction that frequency increases, the power spectrum chart that step (4b) obtains carries out Least square analysis, on abscissa axis, divides one section every 0.5 calibration, calculate average gradient k ₀, frequency when average gradient is undergone mutation is edge frequency f ₀.

The aforesaid reflecting plane aerial panel modeling method based on Multi-scale Fractal function, is characterized in that, in step (6), determines that Fractal dimensions value and characteristic length G value are carried out as follows:

For single fractal function:

(6a) power spectrum chart obtained in step (4b) carries out Least square analysis, calculates the average gradient k of logS (ω)-ω curve, obtain fractal dimension D:

D=(5-k)/2 formula (4)

(6b) curve between the low-limit frequency of rated output spectrogram and highest frequency exists _yintercept G on axle, obtains characteristic length G:

G=b formula (5)

For two fractal function:

(6c) between the low-limit frequency of the power spectrum chart obtained in step (4b) respectively and the first edge frequency, between the first edge frequency and highest frequency, carry out Least square analysis, then calculate the average gradient k of corresponding logS (ω)-ω curve respectively ₁, k ₂, obtain fractal dimension D ₁, D ₂:

D ₁=(5-k ₁)/2 formula (4 ')

D ₂=(5-k ₂)/2 formula (4 ")

(6d) the intercept b of curve in y-axis between the low-limit frequency of rated output spectrogram and the first edge frequency is distinguished ₁, the intercept b of curve in y-axis between the first edge frequency and highest frequency ₂, obtain characteristic length G ₁, G ₂:

G ₁=b ₁formula (5 ')

G ₂=b ₂formula (5 ")

For other fractal function, by that analogy.

The aforesaid reflecting plane aerial panel modeling method based on Multi-scale Fractal function, is characterized in that, in step (8), the accuracy of inspection mathematical model z (x) is carried out according to following steps:

(8a) the mathematical model z (x) determined according to step (7) calculates theoretical root-mean-square error R _q(1) value:

$R_q\left(1\right)={\∫}_{{\mathrm{\ω}}_1}^{{\mathrm{\ω}}_2}S\left(\mathrm{\ω}\right)\mathrm{d\ω}=\frac{G^{(D-1)}}{{\mathrm{\ω}}^{(2-D)}{\left[2\ln {\mathrm{\γ}}^{(4-2D)}\right]}^{1/2}}$ Formula (6)

Wherein, ω ₁for sample length, ω ₂for Measurement Resolution, D is fractal dimension, and G is characteristic length;

(8b) the discrete measured data of roughness obtained according to step (2) calculates model root-mean-square error R _q(2) value:

$R_q\left(2\right)=\sqrt{\frac{1}{m}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}Z{\left(x_i\right)}^2}$ Formula (7)

(8c) theoretical root-mean-square error R is contrasted _qand root-mean-square error R (1) _q(2), if root-mean-square error R _q(2) compared to theoretical root-mean-square error R _q(1) within the scope of accuracy requirement, then illustrate that this modeling method is enough accurate, modeling terminates, and mathematical model z (x) is wanted model; If do not meet error precision index, then repeat step (2) to step (7), till meeting accuracy requirement.

Usefulness of the present invention is: according to the major parameter of the measured data determination fractal function of antenna surface, ensure that the accuracy of mathematical model; According to the level of the power spectrum figure determination fractal function of actual measurement discrete data in modeling process, according to the low-limit frequency of the frequency of operation determination Multi-scale Fractal function of antenna, highest frequency and boundary frequency, ensure that the high-level efficiency of modeling process, for the precision and efficiency improving reflector antenna modeling analysis is laid a good foundation.

Accompanying drawing explanation

Fig. 1 is the general flow chart of modeling method of the present invention;

Fig. 2 is the sub-process figure of the present invention according to reflector antenna machining experiment model;

Fig. 3 is that in the present invention, specimen surface data carry out the sub-process figure tested;

Fig. 4 is the sub-process figure in the present invention, specimen surface data being carried out to subsequent treatment;

Fig. 5 is the one dimension mirror surface mathematical model schematic diagram that the present invention sets up.

Embodiment

Below in conjunction with the drawings and specific embodiments, concrete introduction is done to the present invention.

Reflecting plane aerial panel modeling method based on Multi-scale Fractal function of the present invention, its Integral Thought is:

First the roughness of contourgraph to reflector antenna surface is used to survey, then the power spectrum chart that Fourier transform obtains surface is carried out to measured data, according to the yardstick of power spectrum chart determination fractal function, again according to the low-limit frequency of the frequency of operation determination Multi-scale Fractal function of antenna, highest frequency and boundary frequency, finally realize the accurate modeling of reflector antenna.

With reference to Fig. 1, the reflecting plane aerial panel modeling method concrete steps based on Multi-scale Fractal function of the present invention are as follows:

Step one: according to reflector antenna, obtains testing model used

With reference to Fig. 2, concrete process is as follows:

1a, select corresponding alloy raw material, alloy raw material founding is become slab, heating of plate blank is pressed into light sheet, light sheet is put into the panel blank that required size made by mould, counter plate blank carries out the roughing such as trimming, polishing, semi-finishing, finishing three process obtain panel, on panel processing mounting holes, spray paint, obtain reflection surface panel finished product.

1b, to process according to the processing process of reflection surface panel with the sheet metal of studied reflector antenna same material attribute, obtain the experiment model consistent with studied reflecting surface structure performance, in addition, flat smooth is answered on the surface of this model.

Step 2: measure the roughness of experiment model, obtains discrete measured data

2a, select the metal probe contact pilotage of profile measurer according to roughness value, the radius of contact pilotage should be less than the half of model roughness rms, finally determines that the radius of contact pilotage is 0.25 μm.

2b, determine that the sample length A of model is 1mm, sample interval is 100 μm, evenly gets 100 points, namely evaluating length L is 5 sample lengths, L=5mm.

2c, according to x to y to equidistant measuring route Uniform Scanning model, obtain the roughness discrete data of model, with reference to Fig. 3, detailed process is as follows:

1. adjust position and the state of profile measurer, comprise the location of workbench, the focusing of measuring head and elements of a fix initial point and coordinate direction, and initialization is carried out to motor and the various parameter of sensor.

2. lay workpiece on worktable, tested part is rectangular parallelepiped part, and it is long is 60mm, and wide is 20mm, and height is 35mm, and material is 45 ^#steel, job operation is milling.

3. check the working condition of each parts, connect with the mains and start to measure, measure its end face, measurement range is (1mm × 1mm).

4. measuring route press x to y to equidistantly sampling is to obtain ordered data point range, spacing is 100 μm.Measurement data leaves in user's specified file.

5. measure after terminating, measuring head and workbench stop mobile, obtain discrete data.

6. carry out subsequent treatment according to the flow process of Fig. 4 to discrete data, obtain final measurement data, final measurement data is in table 1.

The final measurement data of table 1 antenna surface

X	0	100	200	300	400	500	600	700	800	900	1000
												0	5.0121	5.2055	4.9897	4.9901	5.0172	5.0223	5.0156	4.9992	4.9932	4.9944	5.0165
100	5.0153	5.0023	4.9976	4.9963	5.0054	5.0094	4.9977	4.9954	4.9984	5.0031	5.0164
												200	5.0090	5.1022	5.0935	4.9994	4.9965	5.0125	5.0038	5.0025	5.0125	4.9995	4.9953
300	4.9924	4.9982	4.9961	4.9975	5.2152	5.2752	5.0129	5.0065	5.0256	4.9955	4.9948
												400	5.0356	5.0151	5.0095	4.0027	4.9963	4.9986	4.9961	5.0051	5.0056	5.0036	5.0052
500	4.9957	4.9961	4.9926	5.0089	5.0025	5.0031	5.0025	5.0053	4.9953	5.0121	5.0012
												600	5.0055	5.0204	4.9959	4.9922	5.0021	5.0023	5.0046	5.0114	5.0014	5.0054	4.9990
700	5.1326	5.1126	5.0121	5.0061	4.9981	4.9965	5.0507	5.0026	5.0025	4.9995	4.9981
												800	4.9991	4.9950	5.2425	5.0023	5.1343	5.0086	5.0361	5.5057	5.0021	5.5036	4.9992
900	4.9915	4.9903	5.0063	5.0084	5.0090	5.0012	5.0109	5.0784	4.9953	4.9944	4.9985
												1000	5.0024	5.0836	5.0827	5.0614	5.0912	4.9946	4.9974	4.9946	5.0044	4.9952	5.1906

Step 3: find range scale model with fractal property, determine the dimensionless interzone of model

The field of definition L of 3a, operational evaluation length L exemplarily range scale, L ∈ (1, n), n is scale sum, on L draw estimate M (δ) with the double logarithmic curve of measurement scale δ, double logarithmic curve indicates measurement point, measurement point be labeled as (U _i, V _i), pre-determine two end points of the good curve of linear relationship as non-scaling section scope (n according to the linear relationship of measurement point distribution ₁, n ₂) approximate value.

3b, foundation three lines fitting, at (n ₁± △ n, n ₂± △ n) interior use formula (3) digital simulation difference Ω:

$\mathrm{\Ω}(3,n_1,n_2)=\underset{i=1}{\overset{a}{\mathrm{\Σ}}}\underset{i=1}{\overset{b}{\mathrm{\Σ}}}{(V_i-a-b{U}_i)}^2$ Formula (3)

In formula, a, b are the parameter of straight-line equation.

Obtain matching difference Ω minimum time, the broken line end points n in the middle of two end points ₁and n ₂the interval formed is dimensionless interzone.

3c, when dimensionless interzone scope on model exceedes 2/3 of model field of definition L total length, proceed to step 4, use the inventive method to proceed modeling; Otherwise, other modeling methods should be attempted.

Step 4: determine pattern modeling use the yardstick of fractal function

4a, Fourier transform is carried out to the roughness discrete data of the model that step 2 obtains, obtain frequency data.

4b, logarithm change is carried out to frequency data, obtain the power spectrum chart of specimen surface profile.

4c, by power spectrum chart obvious fluctuating hop count determination pattern modeling use the yardstick of fractal function:

If obvious fluctuating hop count is one section in power spectrum chart, then use single fractal function modeling;

If obvious fluctuating hop count is two sections in power spectrum chart, then use two fractal function modeling;

Other fractal function by that analogy.

In the present embodiment, because fluctuating hop count obvious in power spectrum chart is two sections, therefore use two fractal function modeling.

Step 5: according to sample length, adopt Nyquist frequency algorithm determination edge frequency

5a, use the inverse of sample length as the low-limit frequency f of power spectrum _l.

5b, use the inverse of sample interval as the highest frequency f of power spectrum chart _h.

5c, from low-limit frequency f _lstart, along the direction that frequency increases, the power spectrum chart that step 4b obtains carries out Least square analysis, on abscissa axis, divides one section every 0.5 calibration, calculate average gradient k ₀, frequency when average gradient is undergone mutation is edge frequency f ₀.

Step 6: determine Fractal dimensions value and characteristic length G value

For single fractal function:

6a, the power spectrum chart that obtains in step 4b carry out Least square analysis, calculate the average gradient k of logS (w)-w curve, obtain fractal dimension D:

D=(5-k)/2 formula (4).

The intercept b of curve in y-axis between the low-limit frequency of 6b, rated output spectrogram and highest frequency, obtains characteristic length G:

G=b formula (5).

For two fractal function:

The fractal dimension of two fractal function comprises D ₁and D ₂, characteristic length comprises G ₁and G ₂, all need to calculate respectively, concrete:

Carry out Least square analysis between the low-limit frequency of the power spectrum chart obtained in step 4b and the first edge frequency, then calculate the average gradient k of logS (w)-w curve ₁, obtain fractal dimension D ₁:

D ₁=(5-k ₁)/2 formula (4 ').

Carry out Least square analysis between first edge frequency of the power spectrum chart obtained in step 4b and highest frequency, then calculate the average gradient k of logS (w)-w curve ₂, obtain fractal dimension D ₂:

D ₂=(5-k ₂)/2 formula (4 ").

The intercept b of curve in y-axis between the low-limit frequency of rated output spectrogram and the first edge frequency ₁, obtain characteristic length G ₁:

G ₁=b ₁formula (5 ').

The intercept b of curve between first edge frequency of rated output spectrogram and highest frequency in y-axis ₂, obtain characteristic length G ₂:

G ₂=b ₂formula (5 ").

For other fractal function such as three fractal function, four fractal function, Fractal dimensions value and the characteristic length G value of modeling can be determined by that analogy according to method presented hereinbefore.

In addition, characteristic length G value can also be obtained by the height distribution variance of the height variance and W-M function that compare real surface profile, concrete:

First, the height distribution variance of W-M function is asked.

Then, the height variance of real profile is asked.

Finally, the value of characteristic length G can be obtained than height distribution variance and height variance.

Step 7: determine the mathematical model simulating fractal function

According to the Fractal dimensions value determined in step 6 and characteristic length G value, determine the mathematical model z (x) simulating fractal function:

For single fractal function:

$z\left(x\right)=G^{(D-1)}\underset{n=n_1}{\overset{n_c}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D)n}}$ Formula (1)

In formula, γ is the mould of surface profile spatial frequency, is constant value, and value is γ=1.5;

G is the scale parameter of single fractal function;

X is the lateral coordinates of surface profile;

N is sampling number, n ₁for sampled point initial value, n _cfor sampled point stop value.

For two fractal function:

$z\left(x\right)={G_1}^{(D_1-1)}\underset{n=n_1}{\overset{n_{c1}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_1)n}}+{G_2}^{(D_2-1)}\underset{n=n_2}{\overset{n_{c2}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_2)n}}$ Formula (2)

In formula, γ is the mould of surface profile spatial frequency, is constant value, and value is γ=1.5;

G ₁for the scale parameter of first paragraph fractal function in two fractal function;

G ₂for the scale parameter of second segment fractal function in two fractal function;

X is the lateral coordinates of surface profile;

N is sampling number, n ₁for sampled point initial value, the n of first paragraph fractal function _c1for the sampled point stop value of first paragraph fractal function, n ₂for sampled point initial value, the n of second segment fractal function _c2for the sampled point stop value of second segment fractal function.

For other fractal function: corresponding mathematical model z (x) can be obtained by that analogy.

Such as, for three fractal function:

$z\left(x\right)={G_1}^{(D_1-1)}\underset{n=n_1}{\overset{n_{c1}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_1)n}}+{G_2}^{(D_2-1)}\underset{n=n_2}{\overset{n_{c2}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_2)n}}+{G_3}^{(D_3-1)}\underset{{n=n}_3}{\overset{n_{c3}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_3)n}}$ Formula (8)

In formula, γ is the mould of surface profile spatial frequency, is constant value, and value is γ=1.5;

G ₁it is the scale parameter of first paragraph fractal function in three fractal function;

G ₂it is the scale parameter of second segment fractal function in three fractal function;

G ₃it is the scale parameter of the 3rd section of fractal function in three fractal function;

X is the lateral coordinates of surface profile;

N is sampling number, n ₁for sampled point initial value, the n of first paragraph fractal function _c1for the sampled point stop value of first paragraph fractal function, n ₂for sampled point initial value, the n of second segment fractal function _c2for the sampled point stop value of second segment fractal function, n ₃be sampled point initial value, the n of the 3rd section of fractal function _c3it is the sampled point stop value of the 3rd section of fractal function.

Again such as, for four fractal function:

$z\left(x\right)={G_1}^{(D_1-1)}\underset{n=n_1}{\overset{n_{c1}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_1)n}}+{G_2}^{(D_2-1)}\underset{n=n_2}{\overset{n_{c2}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_2)n}}+{G_3}^{(D_3-1)}\underset{{n=n}_3}{\overset{n_{c3}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_3)n}}+{G_4}^{(D_4-1)}\underset{n=n_4}{\overset{n_{c4}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_4)n}}$

Formula (9)

In formula, γ is the mould of surface profile spatial frequency, is constant value, and value is γ=1.5;

G ₁it is the scale parameter of first paragraph fractal function in four fractal function;

G ₂it is the scale parameter of second segment fractal function in four fractal function;

G ₃it is the scale parameter of the 3rd section of fractal function in four fractal function;

G ₄it is the scale parameter of the 4th section of fractal function in four fractal function;

X is the lateral coordinates of surface profile;

N is sampling number, n ₁for sampled point initial value, the n of first paragraph fractal function _c1for the sampled point stop value of first paragraph fractal function, n ₂for sampled point initial value, the n of second segment fractal function _c2for the sampled point stop value of second segment fractal function, n ₃be sampled point initial value, the n of the 3rd section of fractal function _c3be the sampled point stop value of the 3rd section of fractal function, n ₄be sampled point initial value, the n of the 4th section of fractal function _c4it is the sampled point stop value of the 4th section of fractal function.

In the present embodiment, because fluctuating hop count obvious in power spectrum chart is two sections, two fractal function modeling need be used, therefore the mathematical model z (x) of correspondence is:

$z\left(x\right)={G_1}^{(D_1-1)}\underset{n=n_1}{\overset{n_{c1}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_1)n}}+{G_2}^{(D_2-1)}\underset{n=n_2}{\overset{n_{c2}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_2)n}}$ Formula (2).

Step 8: the accuracy of the mathematical model in step 7 is tested

8a, the mathematical model z (x) determined according to step 7 calculate theoretical root-mean-square error R _q(1) value:

$R_q\left(1\right)={\∫}_{{\mathrm{\ω}}_1}^{{\mathrm{\ω}}_2}S\left(\mathrm{\ω}\right)\mathrm{d\ω}=\frac{G^{(D-1)}}{{\mathrm{\ω}}^{(2-D)}{\left[2\ln {\mathrm{\γ}}^{(4-2D)}\right]}^{1/2}}$ Formula (6)

Wherein, ω ₂for Measurement Resolution, ω ₁for sample length, D is fractal dimension, and G is characteristic length.

8b, the roughness discrete measured data calculating model root-mean-square error R obtained according to step 2 _q(2) value:

$R_q\left(2\right)=\sqrt{\frac{1}{m}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}Z{\left(x_i\right)}^2}$ Formula (7).

8c, contrast theoretical root-mean-square error R _qand root-mean-square error R (1) _q(2), comparing result display: root-mean-square error R _q(2) compared to theoretical root-mean-square error R _q(1) within the scope of accuracy requirement, this illustrates that modeling method of the present invention is enough accurate, and modeling terminates, and mathematical model z (x) is wanted model.

If root-mean-square error R _q(2) compared to theoretical root-mean-square error R _q(1) not within the scope of accuracy requirement, then need repetition step 2 to step 7, till meeting accuracy requirement.

The one dimension mirror surface mathematical model of method establishment of the present invention is adopted to see Fig. 5.

Simulation result shows: the present embodiment uses two fractal function to carry out the reflecting surface of simulated reflections surface antenna, consider the uncontinuity that Practical Project finished surface has, non-differentiability, self affine and multiple dimensioned property everywhere, use the intrinsic parameter fractal dimension not relying on surveying instrument resolution as basic assessment parameters, the precision of reflector antenna reflecting surface mathematical simulation can be significantly improved.

Modeling method of the present invention can not only be used for the analytical work of reflector antenna, and can also be used for the dull and stereotyped crack array antenna of microwave frequency band and the modeling work of wave filter, has good application value.

It should be noted that, above-described embodiment does not limit the present invention in any form, the technical scheme that the mode that all employings are equal to replacement or equivalent transformation obtains, and all drops in protection scope of the present invention.

Claims (7)

1., based on a reflecting plane aerial panel modeling method for Multi-scale Fractal function, it is characterized in that, comprise the following steps:

(1) according to the material properties of reflector antenna, the experiment model used of same material properties is made;

(2) measure the roughness of previous experiments model used, obtain the discrete measured data of roughness;

(3) find range scale model with fractal property, determine the dimensionless interzone of model, and dimensionless interzone is checked;

(4) determine pattern modeling use the yardstick of fractal function;

(5) according to sample length, employing Nyquist frequency algorithm determination edge frequency;

(6) Fractal dimensions value and characteristic length G value is determined;

(7) according to the Fractal dimensions value determined and characteristic length G value, the mathematical model z (x) of simulated reflections faceplate panels is determined:

For single fractal function:

$z\left(x\right)=G^{(D-1)}\underset{n=n_1}{\overset{n_c}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D)n}}$ Formula (1)

In formula, γ is the mould of surface profile spatial frequency, is constant value, and value is γ=1.5;

G is the scale parameter of single fractal function;

X is the lateral coordinates of surface profile;

N is sampling number, n ₁for sampled point initial value, n _cfor sampled point stop value;

For two fractal function:

$z\left(x\right)={G_1}^{(D_1-1)}\underset{n=n_1}{\overset{n_{c1}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_1)n}}+{G_2}^{(D_2-1)}\underset{n=n_2}{\overset{n_{c2}}{\mathrm{\Σ}}}\frac{\cos 2\mathrm{\π}{\mathrm{\γ}}^{n}x}{{\mathrm{\γ}}^{(2-D_2)n}}$ Formula (2)

In formula, γ is the mould of surface profile spatial frequency, is constant value, and value is γ=1.5;

G ₁for the scale parameter of first paragraph fractal function in two fractal function;

G ₂for the scale parameter of second segment fractal function in two fractal function;

X is the lateral coordinates of surface profile;

N is sampling number, n ₁for sampled point initial value, the n of first paragraph fractal function _c1for the sampled point stop value of first paragraph fractal function, n ₂for sampled point initial value, the n of second segment fractal function _c2for the sampled point stop value of second segment fractal function;

For other fractal function: corresponding mathematical model z (x) can be obtained by that analogy;

(8) according to error precision index request, test to the accuracy of mathematical model z (x), if coincidence loss precision index, then prove that this modeling method is enough accurate, modeling terminates; If do not meet error precision index, then repeat step (2) to step (7), till meeting accuracy requirement, modeling terminates.

2. the reflecting plane aerial panel modeling method based on Multi-scale Fractal function according to claim 1, is characterized in that, in step (2), the roughness measuring previous experiments model used is carried out as follows:

(2a) select the metal probe contact pilotage of profile measurer according to model roughness value, the radius of aforementioned contact pilotage is less than the half of model roughness rms;

(2b) sample length A and the evaluation length L of model is determined;

(2c) according to x to y to equidistant measuring route Uniform Scanning model, obtain the roughness discrete data of model.

3. the reflecting plane aerial panel modeling method based on Multi-scale Fractal function according to claim 1, it is characterized in that, in step (3), find range scale model with fractal property, the dimensionless interzone determining model and dimensionless interzone checked and carry out as follows:

(3a) the field of definition L of operational evaluation length L exemplarily range scale, L ∈ (1, n), n is scale sum, on L draw estimate M (δ) with the double logarithmic curve of measurement scale δ, double logarithmic curve indicates measurement point, measurement point be labeled as (U _i, V _i), pre-determine two end points of the good curve of linear relationship as non-scaling section scope (n according to the linear relationship of measurement point distribution ₁, n ₂) approximate value;

(3b) according to three lines fitting, at (n ₁± △ n, n ₂± △ n) interior use formula (3) digital simulation difference Ω:

$\mathrm{\Ω}(3,n_1,n_2)=\underset{i=1}{\overset{a}{\mathrm{\Σ}}}\underset{i=1}{\overset{b}{\mathrm{\Σ}}}{(V_i-a-b{U}_i)}^2$ Formula (3)

In formula, a, b are the parameter of straight-line equation,

Obtain matching difference Ω minimum time, the broken line end points n in the middle of two end points ₁and n ₂the interval formed is dimensionless interzone;

(3c) when dimensionless interzone scope on model exceedes 2/3 of model field of definition L total length, proceed to step (4), use the inventive method to continue modeling.

4. the reflecting plane aerial panel modeling method based on Multi-scale Fractal function according to claim 1, is characterized in that, in step (4), determine pattern modeling use the yardstick of fractal function to carry out as follows:

(4a) Fourier transform is carried out to the roughness discrete data of the model obtained in step (2), obtain frequency data;

(4b) logarithm change is carried out to aforementioned frequencies data, obtain the power spectrum chart of specimen surface profile;

(4c) by aforementioned power spectrogram obvious fluctuating hop count determination pattern modeling use the yardstick of fractal function: if obvious fluctuating hop count is one section in power spectrum chart, use single fractal function modeling, if obvious fluctuating hop count is two sections in power spectrum chart, use two fractal function modeling, by that analogy.

5. the reflecting plane aerial panel modeling method based on Multi-scale Fractal function according to claim 1, is characterized in that, in step (5), determines that edge frequency is carried out as follows:

(5a) the low-limit frequency f of inverse as power spectrum of sample length is used _l;

(5b) use the inverse of sample interval as the highest frequency f of power spectrum chart _h;

(5c) from low-limit frequency f _lstart, along the direction that frequency increases, the power spectrum chart that step (4b) obtains carries out Least square analysis, on abscissa axis, divides one section every 0.5 calibration, calculate average gradient k ₀, frequency when average gradient is undergone mutation is edge frequency f ₀.

6. the reflecting plane aerial panel modeling method based on Multi-scale Fractal function according to claim 1, is characterized in that, in step (6), determines that Fractal dimensions value and characteristic length G value are carried out as follows:

For single fractal function:

(6a) power spectrum chart obtained in step (4b) carries out Least square analysis, calculates the average gradient k of logS (ω)-ω curve, obtain fractal dimension D:

D=(5-k)/2 formula (4)

(6b) the intercept G of curve in y-axis between the low-limit frequency of rated output spectrogram and highest frequency, obtains characteristic length G:

G=b formula (5)

For two fractal function:

(6c) between the low-limit frequency of the power spectrum chart obtained in step (4b) respectively and the first edge frequency, between the first edge frequency and highest frequency, carry out Least square analysis, then calculate the average gradient k of corresponding logS (ω)-ω curve respectively ₁, k ₂, obtain fractal dimension D ₁, D ₂:

D ₁=(5-k ₁)/2 formula (4 ')

D ₂=(5-k ₂)/2 formula (4 ")

(6d) the intercept b of curve in y-axis between the low-limit frequency of rated output spectrogram and the first edge frequency is distinguished ₁, the intercept b of curve in y-axis between the first edge frequency and highest frequency ₂, obtain characteristic length G ₁, G ₂:

G ₁=b ₁formula (5 ')

G ₂=b ₂formula (5 ")

For other fractal function, by that analogy.

7. the reflecting plane aerial panel modeling method based on Multi-scale Fractal function according to claim 1, is characterized in that, in step (8), the accuracy of inspection mathematical model z (x) is carried out according to following steps:

(8a) the mathematical model z (x) determined according to step (7) calculates theoretical root-mean-square error R _q(1) value:

$R_q\left(1\right)={\∫}_{{\mathrm{\ω}}_1}^{{\mathrm{\ω}}_2}S\left(\mathrm{\ω}\right)\mathrm{d\ω}=\frac{G^{(D-1)}}{{\mathrm{\ω}}^{(2-D)}{\left[2\ln {\mathrm{\γ}}^{(4-2D)}\right]}^{1/2}}$ Formula (6)

Wherein, ω ₁for sample length, ω ₂for Measurement Resolution, D is fractal dimension, and G is characteristic length;

(8b) the discrete measured data of roughness obtained according to step (2) calculates model root-mean-square error R _q(2) value:

$R_q\left(2\right)=\sqrt{\frac{1}{m}\underset{i=1}{\overset{m}{\mathrm{\Σ}}}Z{\left(x_i\right)}^2}$ Formula (7)

(8c) theoretical root-mean-square error R is contrasted _qand root-mean-square error R (1) _q(2), if root-mean-square error R _q(2) compared to theoretical root-mean-square error R _q(1) within the scope of accuracy requirement, then illustrate that this modeling method is enough accurate, modeling terminates, and mathematical model z (x) is wanted model; If do not meet error precision index, then repeat step (2) to step (7), till meeting accuracy requirement.