CN106096208A - The Forecasting Methodology of reflector antenna power radiation pattern excursion - Google Patents

Abstract

The invention belongs to the Forecasting Methodology of reflector antenna power radiation pattern excursion, it comprises the following steps: (1) carries out stress and strain model to the reflecting surface of antenna；(2) bore face random phase error is calculated(3) reflector antenna far-field pattern E (θ, φ) under panel random errors affect is obtained；(4) expression formula of reflector antenna power radiation pattern P (θ, φ) under the influence of surface error is given；(5) average of the power radiation pattern P (θ, φ) that step (4) obtains is givenExpression formula；(6) reflector antenna power radiation pattern variance D (P (θ, φ)) under the influence of surface error is sought；(7) excursion of reflector antenna power radiation pattern variance D (P (θ, φ)) is given.

Description

The Forecasting Methodology of reflector antenna power radiation pattern excursion

Technical field

The invention belongs to antenna technical field, be specifically related to the prediction side of reflector antenna power radiation pattern excursion Method.

Background technology

Reflector antenna is a kind of typical electronics, and its frame for movement is not only carrier and the guarantor that electrical property realizes Barrier, and govern the realization of electrical property.Along with reflector antenna answering to directions such as high band, high-gain, high pointing accuracies With, its structure precision is had higher requirement.But the error source of reflector antenna is from many, in order to reach Structure precision requirement, needs according to different error sources antenna electric performance to be affected relation, carry out error source tolerance index point Joining, wherein reflection surface panel error is one of its main error source.

Reflection surface panel can introduce foozle and alignment error in processing and manufacturing in assembling link, and this kind of error has Probability density characteristics, belongs to random error.The research affected antenna electric performance about reflection surface panel random error, the earliest may be used To trace back to Ruze formula, which give the functional relationship between panel root-mean-square error and reflector antenna gain loss.And then, Yahya Rahmat-Samii have studied panel random error affects relation to reflector antenna average power pattern, it is thus achieved that Panel precision is on antenna gain and the impact of minor level meansigma methods.Wang Wei have studied positions of panels error to reflector antenna Power radiation pattern Influencing Mechanism, it is thus achieved that the change of the reflector antenna power radiation pattern that the translation of piecemeal panel and inclination cause. But, Yahya Rahmat-Samii only accounts for the impact on antenna power pattern average of the panel random error.Wang Wei examines Consider is the translation of piecemeal panel and the systematic error tilted, and does not have randomness.Therefore, both approaches all cannot assess panel The situation of change of reflector antenna power direction under random error.Because under the influence of panel random error, reflector antenna Gain and minor level should fluctuate up and down in meansigma methods, the average merely with power radiation pattern weighs panel random error Impact on reflector antenna electrical property is incomplete.

Summary of the invention

Goal of the invention: the problem that the present invention is directed to the existence of above-mentioned prior art makes improvement, i.e. the invention discloses panel The Forecasting Methodology of reflector antenna power radiation pattern excursion under the influence of error.The method is based on existing reflector antenna face The relational expression that average power pattern is affected by plate random error, derives according to probabilistic method and obtains panel random error With the functional relationship of reflector antenna power radiation pattern variance, use Chebyshev inequality, give panel random error shadow Ring lower power radiation pattern excursion.The present invention can be used for instructing rationally determining of reflecting plane aerial panel tolerance.

Technical scheme: the Forecasting Methodology of reflector antenna power radiation pattern excursion, comprises the following steps:

(1) in the case of known paraboloidal-reflector antenna structural parameters and electromagnetic parameter, the reflecting surface of antenna is carried out Stress and strain model, is radially divided into N ring, more each ring is divided into K_n(n=1,2 ... N) block grid；

(2) the panel random error of each piece of grid supposing step (1) is identical, and equal in this net region Error at heart point, the axial random error of n-th i-th piece of grid element center position of ring is expressed as ε_n,i, the bore face that it causes is random Phase errorFor

Wherein:

λ is operation wavelength；

ξ_nIt it is the subtended angle of the n-th ring center position；

(3) application aperture field method is by the phase place random error of step (2)Introduce reflector antenna far-field pattern table Reaching in formula, obtain reflector antenna far-field pattern E (θ, φ) under panel random errors affect, its expression formula is,

Wherein:

$E_{n,i}=\underset{A(n,i)}{\∫\∫}{E}_0({\mathrm{\ρ}}^{\′},{\mathrm{\φ}}^{\′})\exp \[{\mathrm{jk\ρ}}^{\′}sin\mathrm{\θ}cos(\mathrm{\φ}-{\mathrm{\φ}}^{\′})\]{\mathrm{\ρ}}^{\′}{\mathrm{d\ρ}}^{\′}{\mathrm{d\φ}}^{\′}$

(θ, φ) is far field direction of observation；

E_n,iFor the contribution to global radiation field of the i-th piece of panel of reflecting surface the n-th ring；

(n i) is the i-th piece of panel projection of reflecting surface the n-th ring area on bore face to A；

E₀(ρ ', φ ') is the field distribution function in bore face；

ρ ' is the distance of any point in feed focus to bore face；

φ ' is the corner of x-axis in bore face to feed Yu grid element center position line；

K is wave number；

J is imaginary unit；

(4) according to the expression formula of reflector antenna far-field pattern E (θ, φ) in step (3), under the influence of surface error The expression formula of reflector antenna power radiation pattern P (θ, φ) is:

Wherein:

E^*(θ, φ) is the conjugation of field strength pattern E (θ, φ)；

E_n,iFor the contribution to global radiation field of the i-th piece of panel of reflecting surface the n-th ring；

E_m,lFor the contribution to global radiation field of the reflecting surface m ring the-th block panel；

For E_m,lConjugation；

Random phase error for i-th piece of face plate center position of reflecting surface the n-th ring；

Random phase error for reflecting surface m ring the-th block face plate center position；

J is imaginary unit；

(5) assume that in each grid, panel is separate by manufacturing, install introduced random error, and each ring It is zero that panel random error obeys average, and it is zero that the panel random error of each ring obeys average, and variance isGauss divide Cloth, the average of the power radiation pattern P (θ, φ) that step (4) obtainsExpression formula be:

Wherein:

For the contribution to global radiation field of reflecting surface the n-th anchor ring plate；

E_n,iFor the contribution to global radiation field of the i-th piece of panel of reflecting surface the n-th ring；

For the contribution to global radiation Field conjugate of the reflecting surface m anchor ring plate；

E_m,lFor the contribution to global radiation field of the reflecting surface m ring the-th block panel；

For E_m,lConjugation；

J is imaginary unit；

Random phase error for i-th piece of face plate center position of reflecting surface the n-th ring；

Random phase error for reflecting surface m ring the-th block face plate center position；

σ_nIt it is the standard deviation of phase error under the n-th anchor ring plate random errors affect；

(6) according to the computing formula of variance, the reflector antenna power radiation pattern variance D (P under the influence of surface error is sought (θ, φ)), its expression formula is

Wherein:

E_nFor the contribution to global radiation field of reflecting surface the n-th anchor ring plate；

For the contribution to global radiation Field conjugate of the reflecting surface m anchor ring plate；

E_pFor the contribution to global radiation field of the reflecting surface pth anchor ring plate；

For the contribution to global radiation Field conjugate of the reflecting surface q anchor ring plate；

J is imaginary unit；

Random phase error for reflecting surface the n-th anchor ring plate center；

Random phase error for reflecting surface m anchor ring plate center；

Random phase error for reflecting surface pth anchor ring plate center；

Random phase error for reflecting surface q anchor ring plate center；

σ_nIt it is the standard deviation of phase error under the n-th anchor ring plate random errors affect；

σ_mIt it is the standard deviation of phase error under m anchor ring plate random errors affect；

σ_pFor the standard deviation of phase error under pth anchor ring plate random errors affect；

σ_qIt it is the standard deviation of phase error under q anchor ring plate random errors affect；

(7) according to Chebyshev inequality, the change of reflector antenna power radiation pattern variance D (P (θ, φ)) in step (6) Change scope is

Further, the standard deviation σ of phase error under step (5) n-th anchor ring plate random errors affect_nDetermined by following formula

Wherein:

λ is operation wavelength；

ε_n(rms)It it is the root-mean-square value of the n-th anchor ring plate random error；

ξ_nIt it is the subtended angle of the n-th ring center position.

Further, step (7) comprises the following steps:

(71) under panel random errors affect, the average of power radiation pattern P (θ, φ) isPanel random error shadow The variance ringing lower power radiation pattern P (θ, φ) is D (P (θ, φ)), according to Chebyshev inequality, takes positive count ε, power side Exist to figure P (θ, φ) spanInterior probability is

Wherein:

Prob{} is to meet the probability of condition in braces；

(72) takeObtain

$\mathrm{Pr}ob\left\{\right|P(\mathrm{\&theta;},\mathrm{\&phi;})-\stackrel{\&OverBar;}{P(\mathrm{\&theta;},\mathrm{\&phi;})}|<<3\sqrt{D\left(P\right)}\}\&GreaterEqual;88.89\%$

This formula shows that under panel random errors affect, power radiation pattern P (θ, φ) power radiation pattern P (θ, φ) is in intervalProbability be 88.89%.

Beneficial effect: the Forecasting Methodology of reflector antenna power radiation pattern excursion disclosed by the invention has and following has Benefit effect:

1, under the influence of technical scheme disclosed by the invention gives surface error, reflector antenna power radiation pattern variance becomes The Forecasting Methodology of change scope, being described more fully hereinafter surface error affects relation to reflector antenna electrical property；

2, antenna under the influence of the manufacturing and fixing error of the measurable reflecting plane aerial panel of technical scheme disclosed by the invention The actual change scope of power radiation pattern；

3, technical scheme disclosed by the invention combines the mean value computation formula of existing reflector antenna power radiation pattern, permissible The most reasonably provide the tolerance index of reflecting plane aerial panel.

Accompanying drawing explanation

Fig. 1 is the flow chart of the Forecasting Methodology of reflector antenna power radiation pattern excursion disclosed by the invention；

Fig. 2 is the stress and strain model figure in reflecting surface bore face；

Fig. 3 is reflecting surface geometric error schematic diagram；

Fig. 4 is reflector antenna average power pattern under different panels root-mean-square error；

Fig. 5 is the excursion figure (ε of reflector antenna power radiation pattern under the influence of panel root-mean-square error_rms=λ/40)；

Fig. 6 is the actual change areal map (ε of reflector antenna power radiation pattern under the influence of panel root-mean-square error_rms=λ/ 30)；

Fig. 7 is the comparison diagram (ε of power radiation pattern formula of variance in Monte-carlo Simulation Method and the present invention_rms=λ 40).

Detailed description of the invention:

Below the detailed description of the invention of the present invention is described in detail.

As it is shown in figure 1, the Forecasting Methodology of reflector antenna power radiation pattern excursion, comprise the following steps:

(1) in the case of known paraboloidal-reflector antenna structural parameters and electromagnetic parameter, the reflecting surface of antenna is carried out Stress and strain model, is radially divided into N ring, more each ring is divided into K_n(n=1,2 ... N) block grid, as in figure 2 it is shown, Wherein K₁、K₂、…、K_NFor any positive integer；

(2) the panel random error of each piece of grid supposing step (1) is identical, and equal in this net region Error at heart point, the axial random error of n-th i-th piece of grid element center position of ring is expressed as ε_n,i, then according to Fig. 3 Reflecting surface error geometrical relationship, the bore face random phase error that it causesFor

Wherein:

λ is operation wavelength；

ξ_nIt it is the subtended angle of the n-th ring center position；

(3) application aperture field method is by the phase place random error of step (2)Introduce reflector antenna far-field pattern table Reaching in formula, obtain reflector antenna far-field pattern E (θ, φ) under panel random errors affect, its expression formula is,

Wherein:

$E_{n,i}=\underset{A(n,i)}{\&Integral;\&Integral;}{E}_0({\mathrm{\&rho;}}^{\&prime;},{\mathrm{\&phi;}}^{\&prime;})\exp \&lsqb;{\mathrm{jk\&rho;}}^{\&prime;}sin\mathrm{\&theta;}cos(\mathrm{\&phi;}-{\mathrm{\&phi;}}^{\&prime;})\&rsqb;{\mathrm{\&rho;}}^{\&prime;}{\mathrm{d\&rho;}}^{\&prime;}{\mathrm{d\&phi;}}^{\&prime;}$

(θ, φ) is far field direction of observation；

E_n,iFor the contribution to global radiation field of the i-th piece of panel of reflecting surface the n-th ring；

(n i) is the i-th piece of panel projection of reflecting surface the n-th ring area on bore face to A；

E₀(ρ ', φ ') is the field distribution function in bore face；

ρ ' is the distance of any point in feed focus to bore face；

φ ' is the corner of x-axis in bore face to feed Yu grid element center position line；

K is wave number；

J is imaginary unit；

(4) according to the expression formula of reflector antenna far-field pattern E (θ, φ) in step (3), under the influence of surface error The expression formula of reflector antenna power radiation pattern P (θ, φ) is:

Wherein:

E^*(θ, φ) is the conjugation of field strength pattern E (θ, φ)；

E_n,iFor the contribution to global radiation field of the i-th piece of panel of reflecting surface the n-th ring；

E_m,lFor the contribution to global radiation field of the reflecting surface m ring the-th block panel；

For E_m,lConjugation；

Random phase error for i-th piece of face plate center position of reflecting surface the n-th ring；

Random phase error for reflecting surface m ring the-th block face plate center position；

J is imaginary unit；

(5) assume that in each grid, panel is separate by manufacturing, install introduced random error, and each ring It is zero that panel random error obeys average, and variance isGauss distribution, the power radiation pattern P (θ, φ) that step (4) obtains AverageExpression formula be:

Wherein:

For the contribution to global radiation field of reflecting surface the n-th anchor ring plate；

E_n,iFor the contribution to global radiation field of the i-th piece of panel of reflecting surface the n-th ring；

For the contribution to global radiation Field conjugate of the reflecting surface m anchor ring plate；

E_m,lFor the contribution to global radiation field of the reflecting surface m ring the-th block panel；

For E_m,lConjugation；

J is imaginary unit；

Random phase error for i-th piece of face plate center position of reflecting surface the n-th ring；

Random phase error for reflecting surface m ring the-th block face plate center position；

σ_nIt it is the standard deviation of phase error under the n-th anchor ring plate random errors affect；

(6) according to the computing formula of variance, the reflector antenna power radiation pattern variance D (P under the influence of surface error is sought (θ, φ)), its expression formula is

Wherein:

E_nFor the contribution to global radiation field of reflecting surface the n-th anchor ring plate；

For the contribution to global radiation Field conjugate of the reflecting surface m anchor ring plate；

E_pFor the contribution to global radiation field of the reflecting surface pth anchor ring plate；

For the contribution to global radiation Field conjugate of the reflecting surface q anchor ring plate；

J is imaginary unit；

Random phase error for reflecting surface the n-th anchor ring plate center；

Random phase error for reflecting surface m anchor ring plate center；

Random phase error for reflecting surface pth anchor ring plate center；

Random phase error for reflecting surface q anchor ring plate center；

σ_nIt it is the standard deviation of phase error under the n-th anchor ring plate random errors affect；

σ_mIt it is the standard deviation of phase error under m anchor ring plate random errors affect；

σ_pFor the standard deviation of phase error under pth anchor ring plate random errors affect；

σ_qIt it is the standard deviation of phase error under q anchor ring plate random errors affect；

(7) according to Chebyshev inequality, the change of reflector antenna power radiation pattern variance D (P (θ, φ)) in step (6) Change scope is

Further, the standard deviation σ of phase error under step (5) n-th anchor ring plate random errors affect_nDetermined by following formula

Wherein:

λ is operation wavelength；

ε_n(rms)It it is the root-mean-square value of the n-th anchor ring plate random error；

ξ_nIt it is the subtended angle of the n-th ring center position.

Further, step (7) comprises the following steps:

(71) under panel random errors affect, the average of power radiation pattern P (θ, φ) isPanel random error shadow The variance ringing lower power radiation pattern P (θ, φ) is D (P (θ, φ)), according to Chebyshev inequality, takes positive count ε, power side Exist to figure P (θ, φ) spanInterior probability is

Wherein:

Prob{} is to meet the probability of condition in braces；

(72) takeObtain

$\mathrm{Pr}ob\left\{\right|P(\mathrm{\&theta;},\mathrm{\&phi;})-\stackrel{\&OverBar;}{P(\mathrm{\&theta;},\mathrm{\&phi;})}|<<3\sqrt{D\left(P\right)}\}\&GreaterEqual;88.89\%$

This formula shows that under panel random errors affect, power radiation pattern P (θ, φ) power radiation pattern P (θ, φ) is in intervalProbability be 88.89%.

Advantages of the present invention can be further illustrated by emulation experiment:

1. determine structural parameters and the electromagnetic parameter of reflector antenna, and grid division

1a) reflecting surface structure parameter: a diameter of D=2a=100 λ of antenna aperture, focal length is F=70 λ, and wherein a is mouth The radius in footpath, λ is operation wavelength；

1b) reflecting surface electromagnetic parameter: aperture field edge taper ET=-20dB, Aperture field distribution function is E (ρ ')=B+ (1-B)(1-ρ^′2/a²)^p, p=2, B=10^ET/20=0.1, ρ ' are the distance of any point in feed focus to bore face；

1c) reflecting surface being carried out stress and strain model, be divided into 50 rings the most at equal intervals, each ring is angularly divided into 50 pieces Grid.

2, emulation content and result

2a) analyze the average of reflector antenna power radiation pattern corresponding to different panels root-mean-square error.Fig. 4 is face to face Plate root-mean-square error is respectivelyTime average power pattern.Simulation results show mean power side To the correctness of figure formula, finding that panel root-mean-square error is the biggest, the meansigma methods of antenna gain loss is the biggest, minor level simultaneously Meansigma methods raise the most obvious.

2b) for panel root-mean-square error it isOperating mode carry out variance analysis, give the change model of power radiation pattern Enclose, as shown in Figure 5.It can be seen that in main lobe region, the excursion of power is always under ideal value, say, that face Under plate random errors affect, the gain of reflector antenna always reduces；In the first secondary lobe region, the excursion of power is resonable Thinking that value fluctuates, namely under panel random errors affect, the first minor level of reflector antenna might not simply rise High.Right existing method only considers the average of power radiation pattern, have ignored the fluctuation of power radiation pattern under panel random errors affect Characteristic.

2c) Fig. 6 analyzes panel root-mean-square error and isTime power radiation pattern variance, give power radiation pattern Excursion.It can be seen that the first minor level of average power pattern is not raised relative to ideal value a lot.If Only consider average, it will be considered that secondary lobe is not the most affected by this error, and consider the actual range display secondary lobe electricity that variance obtains Putting down and raise nearly 2dB relative to ideal value, this impact cannot be ignored.The most in actual applications, should not only focus on The average of electrical property, and variance to be combined considers the impact of reflection surface panel tolerance.

2d) Fig. 7 utilize Monte-carlo Simulation Method and the present invention to oppose panel root-mean-square error isOperating mode carry out point Analysis and contrast.The simulation results show correctness of the inventive method.

Above embodiments of the present invention are elaborated.But the present invention is not limited to above-mentioned embodiment, In the ken that art those of ordinary skill is possessed, it is also possible to do on the premise of without departing from present inventive concept Go out various change.

Claims (3)

1. the Forecasting Methodology of reflector antenna power radiation pattern excursion, it is characterised in that comprise the following steps:

(1) in the case of known paraboloidal-reflector antenna structural parameters and electromagnetic parameter, the reflecting surface of antenna is carried out grid Divide, be radially divided into N ring, more each ring be divided into K_nBlock grid；

(2) the panel random error of each piece of grid of supposition step (1) is identical, and equal to central point in this net region The error at place, the axial random error of n-th i-th piece of grid element center position of ring is expressed as ε_n,i, bore face random phase that it causes ErrorFor

Wherein:

λ is operation wavelength；

ξ_nIt it is the subtended angle of the n-th ring center position；

(3) application aperture field method is by the phase place random error of step (2)Introduce reflector antenna far-field pattern expression formula In, obtaining reflector antenna far-field pattern E (θ, φ) under panel random errors affect, its expression formula is,

Wherein:

$E_{n,i}=\underset{A(n,i)}{\&Integral;\&Integral;}{E}_0({\mathrm{\&rho;}}^{\&prime;},{\mathrm{\&phi;}}^{\&prime;})\exp \&lsqb;{\mathrm{jk\&rho;}}^{\&prime;}sin\mathrm{\&theta;}cos(\mathrm{\&phi;}-{\mathrm{\&phi;}}^{\&prime;})\&rsqb;{\mathrm{\&rho;}}^{\&prime;}{\mathrm{d\&rho;}}^{\&prime;}{\mathrm{d\&phi;}}^{\&prime;}$

(θ, φ) is far field direction of observation；

E_n,iFor the contribution to global radiation field of the i-th piece of panel of reflecting surface the n-th ring；

(n i) is the i-th piece of panel projection of reflecting surface the n-th ring area on bore face to A；

E₀(ρ ', φ ') is the field distribution function in bore face；

ρ ' is the distance of any point in feed focus to bore face；

φ ' is the corner of x-axis in bore face to feed Yu grid element center position line；

K is wave number；

J is imaginary unit；

(4) according to the expression formula of reflector antenna far-field pattern E (θ, φ), the reflection under the influence of surface error in step (3) The expression formula of surface antenna power radiation pattern P (θ, φ) is:

Wherein:

E^*(θ, φ) is the conjugation of field strength pattern E (θ, φ)；

E_n,iFor the contribution to global radiation field of the i-th piece of panel of reflecting surface the n-th ring；

E_m,lFor the contribution to global radiation field of the reflecting surface m ring the-th block panel；

For E_m,lConjugation；

Random phase error for i-th piece of face plate center position of reflecting surface the n-th ring；

Random phase error for reflecting surface m ring the-th block face plate center position；

J is imaginary unit；

(5) assume that in each grid, panel is separate by manufacturing, install introduced random error, and the panel of each ring It is zero that random error obeys average, and it is zero that the panel random error of each ring obeys average, and variance isGauss distribution, The average of the power radiation pattern P (θ, φ) that step (4) obtainsExpression formula be:

Wherein:

For the contribution to global radiation field of reflecting surface the n-th anchor ring plate；

E_n,iFor the contribution to global radiation field of the i-th piece of panel of reflecting surface the n-th ring；

For the contribution to global radiation Field conjugate of the reflecting surface m anchor ring plate；

E_m,lFor the contribution to global radiation field of the reflecting surface m ring the-th block panel；

For E_m,lConjugation；

J is imaginary unit；

Random phase error for i-th piece of face plate center position of reflecting surface the n-th ring；

Random phase error for reflecting surface m ring the-th block face plate center position；

σ_nIt it is the standard deviation of phase error under the n-th anchor ring plate random errors affect；

(6) according to the computing formula of variance, ask reflector antenna power radiation pattern variance D under the influence of surface error (P (θ, φ)), its expression formula is

Wherein:

E_nFor the contribution to global radiation field of reflecting surface the n-th anchor ring plate；

For the contribution to global radiation Field conjugate of the reflecting surface m anchor ring plate；

E_pFor the contribution to global radiation field of the reflecting surface pth anchor ring plate；

For the contribution to global radiation Field conjugate of the reflecting surface q anchor ring plate；

J is imaginary unit；

Random phase error for reflecting surface the n-th anchor ring plate center；

Random phase error for reflecting surface m anchor ring plate center；

Random phase error for reflecting surface pth anchor ring plate center；

Random phase error for reflecting surface q anchor ring plate center；

σ_nIt it is the standard deviation of phase error under the n-th anchor ring plate random errors affect；

σ_mIt it is the standard deviation of phase error under m anchor ring plate random errors affect；

σ_pFor the standard deviation of phase error under pth anchor ring plate random errors affect；

σ_qIt it is the standard deviation of phase error under q anchor ring plate random errors affect；

(7) according to Chebyshev inequality, the change model of reflector antenna power radiation pattern variance D (P (θ, φ)) in step (6) Enclose for

The Forecasting Methodology of reflector antenna power radiation pattern excursion the most according to claim 1, it is characterised in that step Suddenly the standard deviation σ of phase error under (5) n-th anchor ring plate random errors affect_nDetermined by following formula

Wherein:

λ is operation wavelength；

ε_n(rms)It it is the root-mean-square value of the n-th anchor ring plate random error；

ξ_nIt it is the subtended angle of the n-th ring center position.

The Forecasting Methodology of reflector antenna power radiation pattern excursion the most according to claim 1, it is characterised in that step Suddenly (7) comprise the following steps:

(71) under panel random errors affect, the average of power radiation pattern P (θ, φ) isUnder panel random errors affect The variance of power radiation pattern P (θ, φ) is D (P (θ, φ)), according to Chebyshev inequality, takes positive count ε, power radiation pattern P (θ, φ) span existsInterior probability is

Wherein:

Prob{} is to meet the probability of condition in braces；

(72) takeObtain

$\mathrm{Pr}ob\left\{\right|P(\mathrm{\&theta;},\mathrm{\&phi;})-\stackrel{\&OverBar;}{P(\mathrm{\&theta;},\mathrm{\&phi;})}|<<3\sqrt{D\left(P\right)}\}\&GreaterEqual;88.89\%$

This formula shows that under panel random errors affect, power radiation pattern P (θ, φ) power radiation pattern P (θ, φ) is in intervalProbability be 88.89%.