A tolerance analysis method for array power pattern with position error

Abstract

The present invention belongs to the technical field of antenna arrays, and specifically relates to a method for tolerance analysis of array power patterns with position errors. The present invention proposes a technique based on Taylor expansion, which is used for tolerance analysis of array power patterns with array element position errors, and derives the upper and lower limits of the array power pattern affected by the position error in an analytical form. Since the proposed technique only involves the first-order partial derivative of the position variable and the position disturbance, it has the advantages of easy implementation and unconditional judgment. After example verification, compared with interval arithmetic, the power pattern boundary calculated by Taylor expansion has higher reliability and robustness, and at the same time effectively improves the prediction performance of the pattern performance parameter boundary. The relevant results can provide a certain theoretical reference for the practice of antenna arrays in engineering.

Description

Technical Field

[0001] The present invention belongs to the field of antenna array technology, and in particular relates to a method for analyzing the tolerance of an array power pattern with position errors. Background Art

[0002] Antenna arrays are widely used due to their flexibility and reliability in wireless data transmission. However, due to factors such as low manufacturing and installation precision, component aging and environmental volatility, antenna array systems will produce frequency offsets, excitation weight (amplitude, phase) errors and array position errors. Among them, position errors are common in the engineering implementation of large-scale antenna arrays. Since the actual array element position deviates from the theoretical design value, this changes the expected radiation characteristics of the antenna array to a certain extent. Generally speaking, the greater the error, the more serious the distortion of the array's power pattern. Therefore, in order to avoid using complex and time-consuming steps to calibrate the position of the actual working array, it is necessary to analyze the impact of the error before the array system is operated, so that antenna engineers can design a robust array.

[0003] In order to evaluate the impact of errors on the radiation performance of arrays, three representative methods have been proposed: numerical simulation methods, methods based on statistical theory, and methods based on interval arithmetic. Numerical simulation methods require a large number of simulation experiments for all possible error combinations. When the array scale becomes larger, its computational complexity will increase dramatically and become unfeasible. Methods based on statistical theory often assume that the error parameters of the array have a probability distribution of composite settings. The limitation of this method is that it is very difficult to collect a large number of test samples to fit the accurate distribution when the array working environment changes. Therefore, it cannot be directly used for engineering design. The method based on interval arithmetic (IA) represents the error parameters to be analyzed as known bounded intervals, and obtains the upper and lower limits (performance intervals) of the power pattern distortion through limited interval operations. This type of research is also often called tolerance analysis. Interval arithmetic has great advantages over the first two methods and has been successfully applied to analyze the impact of array excitation amplitude error, reflector antenna structure deformation, excitation phase error, joint amplitude-phase error, material property error, etc. on the antenna array power pattern, and provides key conclusions about the power pattern's beam directivity, half-power beamwidth, sidelobe level and other indicators. However, due to the "wrapping effect" of interval arithmetic itself - the problem has multiple dependencies on interval variables, this will cause the power pattern limit calculated based on IA to be higher than the actual limit.

[0004] In view of the shortcomings of the existing methods, while avoiding probabilistic assumptions about the position error parameters, the array position can be established as an interval model, which is in line with the actual situation of representing the uncertainty of the array element position in engineering. In order to improve the high reliability of the prediction of the array power pattern boundary under the influence of position error, it is necessary to improve the boundary overestimation problem caused by the interval operation process, so as to provide engineers with an accurate reference for antenna performance evaluation. Summary of the invention

[0005] In order to solve the problem that traditional interval arithmetic methods are prone to overestimation of boundaries in antenna array tolerance analysis, a method based on Taylor expansion is proposed for array power pattern tolerance analysis with array element position errors, and the upper and lower limits of the array power pattern affected by the position error are derived in an analytical form. Since the proposed method only involves the first-order partial derivative of the position variable and the position perturbation, it has the advantages of easy implementation and unconditional judgment. After example verification, compared with interval arithmetic, the power pattern boundary calculated by Taylor expansion has higher reliability and robustness, and effectively improves the prediction performance of the pattern performance parameter boundary. The relevant results can provide a certain theoretical reference for the practice of antenna arrays in engineering.

[0006] The solution of the present invention is: first, an array position error model is established based on interval theory, and a tolerance performance evaluation model of the array power pattern under the influence of position error is established. Secondly, the approximate theory in mathematics, Taylor expansion, is effectively used to express the power pattern function as a function expanded at the position theory setting value. This technology gives an approximate expression of the power pattern in the form of a polynomial. Finally, the value of the position variable when the error has the greatest impact on the power pattern is analyzed, and the lower and upper bounds of the power pattern are given in analytical form.

[0007] The specific steps of the present invention are:

[0008] Step 1: Determine the array element position interval affected by the error;

[0009] Step 2: Determine the array configuration: number of elements, element spacing, operating frequency, and arrangement, and obtain the far-field power pattern function of the array based on these parameters.

[0010] Step 3: In the feasible domain of the position interval parameters given in step 1, establish a tolerance analysis model of the power pattern at each observation angle.

[0011] Step 4: Perform a first-order Taylor expansion on the power pattern function with position error at the theoretical value of each array element position.

[0012] Step 5: Calculate the terms in the Taylor expansion of step 4, including the first-order partial derivatives in the expansion.

[0013] Step 6: Analyze the value of the position interval variable when the error has the greatest impact on the directional pattern;

[0014] Step 7: According to the values ​​of the interval variables in step 6, calculate the upper and lower bounds of the power pattern function at each angle under the influence of the position error.

[0015] In step 1: the actual array position is represented as a vector d = [d1, d2, ..., d N ] T , dDue to the deviation from the design theoretical value, the interval vector d consisting of N interval parameters is designed I , they satisfy:

[0016] d∈d I =[d L ,d U ]=[d c -△d,d c +△d]

[0017] in(·) I represents the interval, and are the lower and upper boundaries of the position interval respectively. is the theoretical setting value of each element position in the array, △d=[△d1,△d2,...,△d N ] T Indicates the maximum displacement of the position under the influence of error, △d = (d U -d L ) / 2.

[0018] In step 2, based on the array information input in step 1, the array far-field power pattern P(θ) is obtained by taking the square of the array factor modulo:

[0019]

[0020] in and AF ζ (θ) represents the real and imaginary parts of the array factor AF(θ), θ n (θ) is the phase of the nth array element, Θ n (θ) = k0d n (sinθ-sinθ0), a n is the excitation amplitude of the nth array element, θ0 is the beam pointing of the array, k0=2π / λ is the spatial wave number, λ is the wavelength, and θ is the observation angle of the array's directional pattern.

[0021] In step 3, the tolerance analysis model established is

[0022]

[0023] st.d∈d I

[0024] In step 4, the first-order Taylor expansion of the power pattern is calculated by

[0025]

[0026] in is the value of the directional pattern function at the theoretical position, is a vector composed of the first-order partial derivatives of each array element position, o(2) is the remainder and can be ignored.

[0027] In step 5, the theoretical setting value of each array element position is substituted into the Taylor expansion of step 4 to calculate P c (θ) and the first-order partial derivative vector The first-order partial derivative is calculated as follows:

[0028]

[0029] In step 6, according to the linear function relationship in the first-order Taylor expansion given in step 5, the endpoints of the position interval {-△d n ,△d n ,n=1,...,N} calculate the value of the maximum impact of the error on the directional pattern, that is, when Pick when Pick

[0030] Step 7: Based on the values ​​obtained in step 6, calculate the upper and lower bounds of the power pattern function affected by the position error at each angle using the following formula:

[0031]

[0032]

[0033] The principle and improvement effect of the present invention are summarized as follows:

[0034] 1) The technology proposed in the present invention uses Taylor expansion technology to reasonably approximate the directional pattern function, and directly converts the change in the array directional pattern function caused by the position error into the partial derivative and position error interval of each array element position, avoiding the continuous generation of new interval variables for indirect solution in traditional interval operations, which is consistent with the position error analysis in actual engineering.

[0035] 2) All the position interval parameters generated in the present invention appear only once after Taylor expansion, do not involve square operations, do not have the judgment of multiple conditions, and only take values ​​on the boundaries (endpoints) of the position interval variables, which effectively improves the overestimation of the power pattern performance limit in traditional interval operations to a certain extent.

[0036] 3) The present invention is easy to implement, does not rely on a large number of cumbersome Monte Carlo simulation experiments, and has a small computational burden. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 The tolerance analysis results of the method of the present invention and the traditional interval arithmetic method are compared.

[0038] Figure 2 The upper boundary difference of the directional pattern calculated by the method of the present invention and the traditional interval arithmetic method at different observation angles is

[0039] Figure 3 : It is a boundary comparison of the array power pattern parameter indicators obtained by the method of the present invention and the traditional interval arithmetic method under different proportional errors, where sub-figure (a) is the result of the sidelobe level (SLL); sub-figure (b) is the result of the half-power beamwidth (BW). DETAILED DESCRIPTION

[0040] The following simulation experiments will further explain and illustrate the present invention in conjunction with the accompanying drawings. The relevant examples are exemplary and are only used to explain the present invention, and cannot be understood as limiting the present invention.

[0041] Step 1: Determine the theoretical position design value vector of the array Determine the maximum deviation from the theoretical value due to position error △d = [△d1, △d2, ..., △d N ] T , where △d can be obtained according to the actual situation of engineering design. For example, when installing the antenna array, each array element will produce a position error around the theoretical position ratio γ, that is,

[0042] Step 2: Determine the antenna array model to be analyzed: number of array elements, array element spacing, operating frequency, and arrangement method. Based on these parameters, set the sampling rate of the observation angle to 0.1°, generate the array power pattern function at each observation angle, and obtain the far-field power pattern function of the array.

[0043]

[0044] Step 3: Calculate the first-order Taylor expansion of the power pattern function and the theoretical power pattern value and the first-order partial derivative vector at the theoretical position

[0045] Step 4: Calculation and

[0046] Step 5: Determine whether P under all observation angles has been calculated L (θ) and P U (θ), if yes, exit; if no, repeat steps 2 to 4.

[0047] In this embodiment, 5000 random Monte Carlo simulation experiments were run simultaneously for verification. The upper and lower limits of the power pattern obtained by this method include the coverage of all Monte Carlo simulation experiments, which verifies the effectiveness and practicality of the present invention.

[0048] 1. Experimental conditions

[0049] MATLAB2020 software was used as the simulation platform. The antenna array was selected as a uniform linear array. N = 10 array elements were evenly arranged along the x-axis. The array operating frequency was P-band 606Mhz. The theoretical value of the array element spacing was 0.5λ, where λ was the array's operating wavelength. Each array element had a tolerance around the array element spacing ratio γ = {1%, 3%, 5%}. The present invention does not consider the influence of the excitation amplitude and excitation phase of a single array element on the radiation pattern. It is considered that the array elements are all equal-amplitude omnidirectional, and the beam pointing is in the normal direction, where the observation angle of the array radiation pattern is the elevation direction θ∈[-90°, 90°].

[0050] 2 Simulation Results

[0051] Figure 1 It is clearly shown that the 5000 Monte Carlo simulations are perturbed around the theoretical design pattern, but the envelope of the coverage range is surrounded by the boundary calculated by the present invention, which verifies the effectiveness of the present invention. It can be seen that when the position tolerance is 3%, the boundary predicted by the method of the present invention is closer to the range of the pattern covered by the actual 5000 Monte Carlo simulation experiments, and the boundary calculation is more reliable.

[0052] Figure 2 The boundary difference ΔP between the upper boundary calculated by the IA method and the upper boundary calculated by the present invention at different angles U ,according to Figure 2 , △P U are all negative, which indicates that the upper bounds generated by IA are higher than those of the method proposed in this paper, and as the distance from the main lobe area increases, △P U The downward trend is more obvious. Therefore, for the prediction of the upper limit, the result of the method of the present invention is more reliable. For tolerance γ = 1%, 3%, 5%, △P UThe average values ​​are -1.0657dB, -2.3111dB, and -3.187dB, respectively. These values ​​indicate that the method of the present invention effectively improves the performance of the traditional IA method, especially when the tolerance is large.

[0053] Depend on Figure 3 It can be seen that as the position tolerance γ increases, SLL and BW produce higher upper bounds and lower lower bounds. I With BW I , the interval calculated by the method of the present invention is more accurate than the result based on IA. When the position tolerance increases, the boundary overestimation phenomenon of IA becomes more serious. The above will illustrate that the method of the present invention is more robust in performing array tolerance analysis with a larger degree of error.

Claims

1. A method for tolerance analysis of array power patterns with position errors is used for a linear antenna array consisting of N array elements, and the actual array position vector is defined as d = [d1, d2, ..., d N ] T , characterized in that, The following steps are involved: Step 1: Determine the array element position interval affected by the error, specifically: Define an interval vector d consisting of N interval parameters I : d∈d I =[d L ,d U ]=[d c -△d,d c +△d] in(·) I represents the interval, and are the lower and upper boundaries of the position interval, respectively. is the theoretical setting value of the position, △d=[△d1,△d2,...,△d N ] T Indicates the maximum displacement of the position under the influence of error, △d = (d U -d L ) / 2; Step 2: Determine the array configuration, including the number of array elements, array element spacing, operating frequency, and arrangement, so as to obtain the far-field power pattern function P(θ) of the array: in and AF ζ (θ) represents the real and imaginary parts of the array factor AF(θ), θ n (θ) is the phase of the nth array element, Θ n (θ) = k0d n (sinθ-sinθ0), a n is the excitation amplitude of the nth array element, θ0 is the beam pointing of the array, k0=2π / λ is the spatial wave number, λ is the wavelength, and θ is the observation angle of the array; Step 3: In the feasible domain of the position interval parameters given in step 1, establish a tolerance analysis model of the power pattern at each observation angle: st.d∈d I Step 4: Perform a first-order Taylor expansion on the power pattern function with position error at the theoretical value of each array element position: in is the value of the directional pattern function at the theoretical position, is a vector composed of the first-order partial derivatives of each array element position, o(2) is the remainder; Step 5: Calculate the terms in the Taylor expansion of step 4, where the first-order partial derivative is calculated by the following formula: Step 6: Analyze the value of the position variable when the error has the greatest impact on the directional pattern. Specifically, according to the linear function relationship in the first-order Taylor expansion given in step 5, take the endpoint of the position interval {-△d n ,△d n ,n=1,...,N} calculate the value of the maximum impact of the error on the directional pattern, that is, when Pick when Pick Step 7: According to the value of the interval variable in step 6, the upper bound P of the power pattern function under the influence of position error is obtained. L (θ) and the lower bound P U (θ): The upper and lower bounds of the power pattern function at each angle under the influence of position error are obtained to complete the tolerance analysis.

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